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2015arXiv150204965H | https://arxiv.org/pdf/1502.04965.pdf | <document>
<section_header_level_1><location><page_1><loc_18><loc_81><loc_82><loc_84></location>ASYMPTOTIC PROPERTIES OF LINEAR FIELD EQUATIONS IN ANTI-DE SITTER SPACE</section_header_level_1>
<text><location><page_1><loc_18><loc_77><loc_82><loc_78></location>GUSTAV HOLZEGEL, JONATHAN LUK, JACQUES SMULEVICI, AND CLAUDE WARNICK</text>
<text><location><page_1><loc_21><loc_57><loc_79><loc_74></location>Abstract. We study the global dynamics of the wave equation, Maxwell's equation and the linearized Bianchi equations on a fixed anti-de Sitter (AdS) background. Provided dissipative boundary conditions are imposed on the dynamical fields we prove uniform boundedness of the natural energy as well as both degenerate (near the AdS boundary) and non-degenerate integrated decay estimates. Remarkably, the non-degenerate estimates 'lose a derivative'. We relate this loss to a trapping phenomenon near the AdS boundary, which itself originates from the properties of (approximately) gliding rays near the boundary. Using the Gaussian beam approximation we prove that non-degenerate energy decay without loss of derivatives does not hold. As a consequence of the nondegenerate integrated decay estimates, we also obtain pointwise-in-time decay estimates for the energy. Our paper provides the key estimates for a proof of the non-linear stability of the anti-de Sitter spacetime under dissipative boundary conditions. Finally, we contrast our results with the case of reflecting boundary conditions.</text>
<section_header_level_1><location><page_1><loc_43><loc_51><loc_57><loc_52></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_17><loc_48><loc_74><loc_50></location>The non-linear Einstein vacuum equations with cosmological constant Λ,</text>
<section_header_level_1><location><page_1><loc_15><loc_46><loc_55><loc_47></location>Ric [ g ] = Λ g , (1)</section_header_level_1>
<text><location><page_1><loc_15><loc_33><loc_85><loc_45></location>constitute a complicated coupled quasi-linear hyperbolic system of partial differential equations for a Lorentzian metric g . The past few decades have seen fundamental progress in understanding the global dynamics of solutions to (1). In particular, a satisfactory answer - asymptotic stability - has been given for the dynamics of (1) with Λ = 0 near Minkowski space [1]; the dynamics of (1) with Λ > 0 near de Sitter space [2, 3], the maximally symmetric solution of (1) with Λ > 0; as well as for the (Λ > 0) Kerr-dS black hole spacetimes [4]. Today, the dynamics near black hole solutions of (1) for Λ = 0 is a subject of intense investigation [5, 6], with the current state-of-the-art being the results in [7] establishing the</text>
<text><location><page_1><loc_17><loc_30><loc_34><loc_31></location>Date : September 5, 2019.</text>
<section_header_level_1><location><page_1><loc_17><loc_28><loc_36><loc_29></location>[email protected]</section_header_level_1>
<text><location><page_1><loc_17><loc_27><loc_78><loc_28></location>Dept. of Mathematics, South Kensington Campus, Imperial College London, SW7 2AZ, UK.</text>
<text><location><page_1><loc_17><loc_25><loc_32><loc_26></location>[email protected]</text>
<text><location><page_1><loc_17><loc_24><loc_83><loc_25></location>Dept. of Mathematics, Stanford University, 450 Serra Mall, Building 380, Stanford, CA 94305-2125.</text>
<text><location><page_1><loc_17><loc_22><loc_42><loc_23></location>[email protected]</text>
<text><location><page_1><loc_17><loc_21><loc_75><loc_22></location>Laboratoire de Math´ematiques, Universit´e Paris-Sud 11, bˆat. 425, 91405 Orsay, France.</text>
<text><location><page_1><loc_17><loc_19><loc_38><loc_20></location>[email protected]</text>
<text><location><page_1><loc_17><loc_18><loc_80><loc_19></location>DPMMS and DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK.</text>
<text><location><page_2><loc_15><loc_85><loc_85><loc_88></location>linear stability of Schwarzschild, and [8] establishing nonlinear stability for Schwarzschild within a restricted symmetry class.</text>
<text><location><page_2><loc_15><loc_72><loc_85><loc_84></location>In contrast to the above, the global dynamics of (1) with Λ < 0 near anti-de Sitter space (AdS), the maximally symmetric solution of the vacuum Einstein equations with Λ < 0, is mostly unknown. Part of the problem is that in the case of Λ < 0, the PDE problem associated with (1) takes the form of an initial boundary value problem.Therefore, even to construct local in time solutions, one needs to understand what appropriate (well-posed, geometric) boundary conditions are. It also suggests that the global behaviour of solutions starting initially close to the anti-de Sitter geometry may depend crucially on the choice of these boundary conditions [9, 10].</text>
<text><location><page_2><loc_15><loc_62><loc_85><loc_71></location>Using his conformal field equations, Friedrich [10] constructed local in time solutions to (1) with Λ < 0. (See also [11] for a recent proof for Dirichlet conditions using harmonic gauge.) While in general it is quite intricate to isolate the geometric content inherent in the boundary conditions imposed (partly due to the large gauge freedom present in the problem), there is nevertheless a 'conformal' piece of boundary data that does admit a physical interpretation. This goes back to the Bianchi equations,</text>
<formula><location><page_2><loc_15><loc_59><loc_56><loc_61></location>[ ∇ g ] a W abcd = 0 , (2)</formula>
<text><location><page_2><loc_15><loc_39><loc_85><loc_58></location>satisfied by the Weyl-tensor of a metric g satisfying (1). As is well known [1], the equations (2) can be used to estimate the curvature components of a dynamical metric g . The boundary data required for well-posed evolution of (2) will generally imply a condition on the energy-flux of curvature through the timelike boundary. Two 'extreme' cases seem particularly natural and interesting: The case when this flux vanishes, corresponding to reflecting (Dirichlet or Neumann) conditions, and the case when this flux is 'as large as possible', corresponding to 'optimally dissipative' conditions. While any such boundary data for (2) will have to be complemented with other data (essentially the choice of a boundary defining function and various gauge choices) to estimate the full spacetime metric [9], it is nevertheless reasonable, in view of the strong non-linearities appearing in the Einstein equations, to conjecture the following loose statement for the global dynamics of perturbations of AdS under the above 'extreme' cases of boundary conditions:</text>
<text><location><page_2><loc_15><loc_35><loc_85><loc_38></location>Conjecture 1. Anti-de Sitter spacetime is non-linearly unstable for reflecting and asymptotically stable for optimally dissipative boundary conditions.</text>
<text><location><page_2><loc_15><loc_23><loc_85><loc_34></location>The instability part of Conjecture 1 was first made in [12, 13] in connection with work on five-dimensional gravitational solitons. See also [14]. By now there exist many refined versions of this part of the conjecture as well as strong heuristic and numerical support in its favour [15, 16, 17]. In a series of papers [18, 19, 20], Moschidis has considered the stability problem for the AdS spacetime within spherical symmetry, in the presence of null dust or massless Vlasov matter, culminating in a proof of the instability of AdS for the Einstein-Massless Vlasov system in [21].</text>
<text><location><page_2><loc_15><loc_18><loc_85><loc_22></location>The present paper is the first of a series of papers establishing the stability part of Conjecture 1. Here we contribute the first fundamental ingredient, namely robust decay estimates for the associated linear problem. Our interest in the case of dissipative boundary</text>
<text><location><page_3><loc_15><loc_75><loc_85><loc_88></location>conditions in part goes back to the original work of Friedrich in [10], which establishes that local well posedness does not single out a preferred boundary condition at null infinity. It is then reasonable to ask how questions of global existence depend on the choice of boundary condition. We also observe that to date the only solutions to the vacuum Einstein equations (regardless of boundary conditions) which are future-complete are necessarily stationary. See for example [22] for general constructions of such spacetimes. A positive resolution of the stability part of Conjecture 1 would necessarily imply the existence of truly dynamical, future-complete, solutions to (1).</text>
<unordered_list>
<list_item><location><page_3><loc_15><loc_62><loc_85><loc_74></location>1.1. Linear field equations on AdS. An important prerequisite for any non-linear stability result is that the associated linear problem is robustly controlled [23]. In the present (dissipative) context, this means that the mechanisms and obstructions for the decay of linear waves in the fixed AdS geometry should be understood and decay estimates with constants depending on the initial data available. We accomplish this by giving a complete description of the decay properties of three fundamental field equations of mathematical physics:</list_item>
<list_item><location><page_3><loc_17><loc_60><loc_78><loc_62></location>(W) The conformal wave equation for a scalar function on the AdS manifold 1 ,</list_item>
</unordered_list>
<formula><location><page_3><loc_15><loc_58><loc_57><loc_59></location>glyph[square] g AdS u +2 u = 0 . (3)</formula>
<unordered_list>
<list_item><location><page_3><loc_17><loc_56><loc_68><loc_57></location>(M) Maxwell's equations for a two-form F on the AdS manifold,</list_item>
</unordered_list>
<formula><location><page_3><loc_15><loc_53><loc_62><loc_55></location>dF = 0 and d glyph[star] g AdS F = 0 . (4)</formula>
<unordered_list>
<list_item><location><page_3><loc_17><loc_49><loc_85><loc_52></location>(B) The Bianchi equations for a Weyl field W (see Definition 1 below) on the AdS manifold</list_item>
</unordered_list>
<formula><location><page_3><loc_15><loc_47><loc_58><loc_49></location>[ ∇ g AdS ] a W abcd = 0 . (5)</formula>
<text><location><page_3><loc_15><loc_43><loc_85><loc_46></location>Note that since AdS is conformally flat, equation (5) is precisely the linearization of the full non-linear Bianchi equations (2) with respect to the anti-de Sitter metric.</text>
<text><location><page_3><loc_15><loc_33><loc_85><loc_42></location>The models (W), (M), and (B) will be accompanied by dissipative boundary conditions. In general, to even state the latter, one requires a choice of boundary defining function for AdS and a choice of timelike vectorfield (or, alternatively, the choice of an outgoing nullvector) at each point of the AdS boundary. For us, it is easiest to state these conditions in coordinates which suggest a canonical choice for both these vectors. We write the AdS metric in spherical polar coordinates on R 4 , where it takes the simple familiar form</text>
<formula><location><page_3><loc_15><loc_30><loc_69><loc_32></location>g AdS = -( 1 + r 2 ) dt 2 + ( 1 + r 2 ) -1 dr 2 + r 2 d Ω 2 S 2 , (6)</formula>
<text><location><page_3><loc_15><loc_26><loc_85><loc_29></location>with the asymptotic boundary corresponding to the timelike hypersurface ' r = ∞ '. 2 Note that 1 /r is a boundary defining function and that</text>
<formula><location><page_3><loc_30><loc_22><loc_70><loc_25></location>e 0 = 1 √ 1 + r 2 ∂ t , e r = √ 1 + r 2 ∂ r , e A = /e A</formula>
<text><location><page_4><loc_15><loc_85><loc_85><loc_88></location>with /e A ( A = 1 , 2) an orthonormal frame on the sphere of radius r defines an orthonormal frame for AdS. Finally, the vector ∂ t singles out a preferred timelike direction.</text>
<text><location><page_4><loc_15><loc_82><loc_85><loc_84></location>In the case of the wave equation, the optimally dissipative boundary condition can then be stated as</text>
<formula><location><page_4><loc_15><loc_78><loc_65><loc_81></location>(7) ∂ ( ru ) ∂t + r 2 ∂ ( ru ) ∂r → 0 , as r →∞ .</formula>
<text><location><page_4><loc_15><loc_76><loc_73><loc_77></location>Note that ∂ t + ( 1 + r 2 ) ∂ r is an outgoing null-vector. See also Section 4.1.</text>
<text><location><page_4><loc_15><loc_71><loc_85><loc_75></location>In the case of Maxwell's equations, we define the electric and magnetic field E i = F ( e 0 , e i ) and H i = glyph[star] g AdS F ( e 0 , e i ) respectively. Here glyph[star] g AdS is the Hodge dual with respect to the AdS metric. The dissipative boundary condition then takes the form</text>
<formula><location><page_4><loc_15><loc_68><loc_65><loc_70></location>(8) r 2 ( E A + glyph[epsilon1] A B H B ) → 0 , as r →∞ ,</formula>
<text><location><page_4><loc_15><loc_64><loc_85><loc_67></location>which means that the Poynting vector points outwards, allowing energy to leave the spacetime.</text>
<text><location><page_4><loc_15><loc_58><loc_85><loc_64></location>Finally, in the case of the Bianchi equations, we define the electric and magnetic part of the Weyl tensor E AB = W ( e 0 , e A , e 0 , e B ) and H AB = glyph[star] g AdS W ( e 0 , e A , e 0 , e B ) respectively. Introducing the trace-free part of E AB as ˆ E AB = E AB -1 2 δ AB E C C and similarly for H AB , the dissipative boundary conditions can then be expressed as:</text>
<formula><location><page_4><loc_15><loc_55><loc_67><loc_57></location>(9) r 3 ( ˆ E AB + glyph[epsilon1] ( A C ˆ H B ) C ) → 0 , as r →∞ .</formula>
<text><location><page_4><loc_15><loc_47><loc_85><loc_53></location>Interpreting the Bianchi equations as a linearisation of the full vacuum Einstein equations, we can understand these boundary conditions in terms of the metric perturbations as implying a relation between the Cotton-York tensor of the conformal metric on I and the 'stress-energy tensor' of the boundary 3 .</text>
<text><location><page_4><loc_15><loc_39><loc_85><loc_47></location>That the above boundary conditions are indeed correct, naturally dissipative, boundary conditions leading to a well-posed boundary initial value problem will be a result of the energy identity. While this is almost immediate in the case of the wave equation and Maxwell's equations, we will spend a considerable amount of time on the 'derivation' of (9) in the Bianchi case, see Section 4.3.</text>
<text><location><page_4><loc_17><loc_37><loc_78><loc_39></location>Note that we could consider a more general Klein-Gordon equation than (W):</text>
<formula><location><page_4><loc_15><loc_35><loc_57><loc_36></location>glyph[square] g AdS u + au = 0 . (10)</formula>
<text><location><page_4><loc_49><loc_24><loc_49><loc_26></location>glyph[negationslash]</text>
<text><location><page_4><loc_15><loc_21><loc_85><loc_34></location>Here the boundary behaviour of u is more complicated [24], but a well posedness theory with dissipative boundary conditions is still available [25] for a in a certain range. We restrict attention to the case a = 2 for two reasons. Firstly, for this choice of a the equation has much in common with the systems (M), (B), owing to their shared conformal invariance. This is the correct 'toy model' for the Einstein equations. Secondly, the problem appears to be considerably more challenging for a = 2. In particular, it does not appear that the methods of this paper can be directly applied, even once allowance is made using the formalism of [24] for the more complicated boundary behaviour of solutions.</text>
<unordered_list>
<list_item><location><page_5><loc_15><loc_86><loc_59><loc_88></location>1.2. The main theorems. We now turn to the results.</list_item>
</unordered_list>
<section_header_level_1><location><page_5><loc_15><loc_84><loc_50><loc_85></location>Theorem 1.1. Let one of the following hold</section_header_level_1>
<unordered_list>
<list_item><location><page_5><loc_17><loc_80><loc_85><loc_83></location>(W) Ψ is a scalar function and a smooth solution of (3) subject to dissipative boundary condition (7). We associate with Ψ the energy density</list_item>
</unordered_list>
<formula><location><page_5><loc_25><loc_75><loc_75><loc_79></location>ε [Ψ] := √ 1 + r 2 ( ( ∂ t Ψ) 2 +Ψ 2 1 + r 2 + [ ∂ r ( √ 1 + r 2 Ψ )] 2 + ∣ ∣ / ∇ Ψ ∣ ∣ 2 )</formula>
<text><location><page_5><loc_17><loc_71><loc_85><loc_73></location>(M) Ψ is a Maxwell-two-form and a smooth solution of (4) subject to dissipative boundary conditions (8). We associate with Ψ the energy density</text>
<formula><location><page_5><loc_38><loc_68><loc_62><loc_69></location>ε [Ψ] = √ 1 + r 2 ( | E | 2 + | H | 2 )</formula>
<text><location><page_5><loc_21><loc_65><loc_77><loc_66></location>where E and H denote the electric and magnetic part of Ψ respectively.</text>
<unordered_list>
<list_item><location><page_5><loc_17><loc_62><loc_85><loc_65></location>(B) Ψ is a Weyl-field and a smooth solution of (5) subject to dissipative boundary conditions (9). We associate with Ψ the energy density</list_item>
</unordered_list>
<formula><location><page_5><loc_38><loc_59><loc_62><loc_61></location>ε [Ψ] = ( 1 + r 2 ) 3 2 ( | E | 2 + | H | 2 )</formula>
<text><location><page_5><loc_21><loc_56><loc_85><loc_57></location>where E and H denote the electric and magnetic part of the Weyl-field respectively.</text>
<text><location><page_5><loc_15><loc_53><loc_44><loc_54></location>Then we have the following estimates</text>
<unordered_list>
<list_item><location><page_5><loc_18><loc_51><loc_62><loc_52></location>(1) Uniform Boundedness: For any 0 < T < ∞ we have</list_item>
</unordered_list>
<formula><location><page_5><loc_33><loc_46><loc_67><loc_49></location>∫ Σ T ε [Ψ] √ 1 + r 2 r 2 drdω glyph[lessorsimilar] ∫ Σ 0 ε [Ψ] √ 1 + r 2 r 2 drdω ,</formula>
<text><location><page_5><loc_21><loc_44><loc_58><loc_45></location>where the implicit constant is independent of T .</text>
<unordered_list>
<list_item><location><page_5><loc_18><loc_42><loc_73><loc_43></location>(2) Degenerate (near infinity) integrated decay without derivative loss:</list_item>
</unordered_list>
<formula><location><page_5><loc_31><loc_37><loc_69><loc_41></location>∫ ∞ 0 dt ∫ Σ t ε [Ψ] 1 + r 2 r 2 drdω glyph[lessorsimilar] ∫ Σ 0 ε [Ψ] √ 1 + r 2 r 2 drdω .</formula>
<unordered_list>
<list_item><location><page_5><loc_18><loc_35><loc_74><loc_36></location>(3) Non-degenerate (near infinity) integrated decay with derivative loss:</list_item>
</unordered_list>
<formula><location><page_5><loc_28><loc_30><loc_72><loc_34></location>∫ ∞ 0 dt ∫ Σ t ε [Ψ] √ 1 + r 2 r 2 drdω glyph[lessorsimilar] ∫ Σ 0 ε [Ψ] + ε [ ∂ t Ψ] √ 1 + r 2 r 2 drdω .</formula>
<text><location><page_5><loc_15><loc_26><loc_85><loc_29></location>Remark 1. Similar statements hold for higher order energies by commuting with ∂ t and doing elliptic estimates. As this is standard we omit the details.</text>
<text><location><page_5><loc_15><loc_22><loc_85><loc_25></location>Corollary 1.2 (Uniform decay) . Under the assumptions of the previous theorem the following uniform-in-time decay estimate holds for any integer n ≥ 1</text>
<formula><location><page_5><loc_22><loc_17><loc_78><loc_21></location>∫ Σ t ε [Ψ] √ 1 + r 2 r 2 drdω glyph[lessorsimilar] 1 (1 + t ) n ∫ Σ 0 ε [Ψ] + ε [ ∂ t Ψ] + . . . + ε [ ∂ n t Ψ] √ 1 + r 2 r 2 drdω .</formula>
<text><location><page_6><loc_15><loc_80><loc_85><loc_88></location>What is remarkable about the above theorem is that the derivative loss occurring in (3) allows one to achieve integrated decay of the energy without loss in the asymptotic weight r . While it is likely that more refined methods can reduce the loss of a full derivative in (3), we shall however establish that some loss is necessary and in fact reflects a fundamental property of the hyberbolic equations on AdS: the presence of trapping at infinity.</text>
<text><location><page_6><loc_15><loc_74><loc_85><loc_78></location>Theorem 1.3. With the assumptions of Theorem 1.1, the term ε [ ∂ t Ψ] on the right hand side of estimate (3) of Theorem 1.1 is necessary: The estimate fails (for general solutions) if it is dropped.</text>
<text><location><page_6><loc_15><loc_65><loc_85><loc_73></location>We will prove Theorem 1.3 only for the case of the wave equation (W), see (45) and Corollary 5.8. The proof is based on the Gaussian beam approximation for the wave equation. 4 In particular, we construct a solution of the conformal wave equation in AdS which contradicts estimate (3) of Theorem 1.1 without the ε [ ∂ t Ψ]-term on the right hand side. Similar constructions can be given for the Maxwell and the Bianchi case. 5</text>
<unordered_list>
<list_item><location><page_6><loc_15><loc_57><loc_85><loc_63></location>1.3. Overview of the proof of Theorem 1.1 and main difficulties. The proof of Theorem 1.1 is a straightforward application of the vectorfield method once certain difficulties have been overcome. Let us begin by discussing the proof in case of the wave- (W) and Maxwell's equation (M) as it is conceptually easier.</list_item>
</unordered_list>
<text><location><page_6><loc_15><loc_47><loc_85><loc_56></location>In the case of (W) and (M), statement (1) follows immediately from integration of the divergence identity ∇ a ( T ab ( ∂ t ) b ) = 0 with T being the energy momentum tensor of the scalar or Maxwell field respectively. In addition, in view of the dissipative condition, this estimate gives control over certain derivatives of u (certain components of the Maxwell field, namely E A and H A ) integrated along the boundary.</text>
<text><location><page_6><loc_15><loc_31><loc_85><loc_47></location>The statement (2) of the main theorem is then obtained by constructing a vectorfield X (see (17)) which is almost conformally Killing near infinity. The key observations are that firstly, the right hand side of the associated divergence identity ∇ a ( T ab X b ) = X π · T controls all derivatives of u in the wave equation case (components of F in the Maxwell case). Secondly, when integrating this divergence identity, the terms appearing on the boundary at infinity come either with good signs (angular derivatives in the case of the wave equation, E r and H r components in the Maxwell case) or are components already under control from the previous T ab ( ∂ t ) b -estimate. The integrated decay estimate thus obtained comes with a natural degeneration in the r -weight, as manifest in the estimate (2) of Theorem 1.1.</text>
<text><location><page_6><loc_15><loc_23><loc_85><loc_31></location>To remove this degeneration we first note that in view of the fact that the vectorfield ∂ t is Killing, the estimates (1) and (2) also hold for the ∂ t -commuted equations. In the case of the wave equation, controlling ∂ t ∂ t ψ in L 2 on spacelike slices implies an estimate for all spatial derivatives of ψ through an elliptic estimate. The crucial point here is that weighted estimates are required. Similarly, one can write Maxwell's equation as a three-dimensional</text>
<text><location><page_7><loc_15><loc_85><loc_85><loc_88></location>div-curl-system with the time derivatives of E and H on the right hand side. Again the crucial point is that weighted elliptic estimates are needed to prove the desired results.</text>
<text><location><page_7><loc_15><loc_80><loc_85><loc_85></location>Once all (spatial) derivatives are controlled in a weighted L 2 sense on spacelike slices one can invoke Hardy inequalities to improve the weight in the lower order terms and remove the degeneration in the estimate (2).</text>
<text><location><page_7><loc_15><loc_60><loc_85><loc_80></location>For the case of the spin 2 equations (B), the proof follows a similar structure. However, the divergence identity for the Bel-Robinson tensor (the analogue of the energy momentum tensor in cases (W) and (M)) alone will not generate the estimate (1). In fact, the term appearing on the boundary after integration does not have a sign unless one imposes an additional boundary condition! On the other hand, one can show (by proving energy estimates for a reduced system of equations, see Section 4.3.1) that the boundary conditions (9) already uniquely determine the solution. 6 The resolution is that in the case of the Bianchi equations one needs to prove (1) and (2) at the same time: Contracting the BelRobinson tensor with a suitable combination of the vectorfields ∂ t and X ensures that the boundary term on null-infinity does have a favorable sign and so does the spacetime-term in the interior. Once (1) and (2) are established, (3) follows from doing elliptic estimates for the reduced system of Bianchi equations similar to the Maxwell case (M).</text>
<unordered_list>
<list_item><location><page_7><loc_15><loc_51><loc_85><loc_59></location>1.4. Remarks on Theorem 1.1. The estimates of Theorem 1.1 remain true for a class of C k perturbations of AdS which preserve the general properties of the deformation tensor of the timelike vectorfield √ 3 ∂ t + X , cf. the proof of Proposition 5.6. This includes perturbations which may be dynamical. This fact is of course key for the non-linear problem [27].</list_item>
</unordered_list>
<text><location><page_7><loc_15><loc_31><loc_85><loc_51></location>The estimates of Theorem 1.1 are stable towards perturbations of the optimally dissipative boundary conditions. In the cases of (W) and (M) one can in fact establish these estimates for any (however small) uniform dissipation at the boundary, cf. Section 6. Whether this is possible also in case of (B) is an open problem and (if true) will require a refinement of our techniques. In Section 6.2 we discuss the case of Dirichlet conditions for (B) which fix the conformal class of the induced metric at infinity to linear order. The Dirichlet boundary conditions may be thought of as a limit of dissipative conditions in which the dissipation vanishes. We outline a proof of boundedness for solutions of the Bianchi equations in this setting. We also briefly discuss the relation of our work to the Teukolsky formalism in Section 6.3, and argue that the proposed boundary conditions of [28] may not lead to a well posed dynamical problem. We state a set of boundary conditions for the Teukolsky formulation that does lead to a well posed dynamical problem.</text>
<text><location><page_7><loc_15><loc_25><loc_85><loc_31></location>Finally, one may wonder whether and how the derivative loss in Theorem 1.1 manifests itself in the non-linear stability problem. In ongoing work [27], we shall see that the degeneration is sufficiently weak in the sense that the degenerate estimate (2) will be sufficient to deal with the non-linear error-terms.</text>
<unordered_list>
<list_item><location><page_7><loc_15><loc_21><loc_85><loc_24></location>1.5. Main ideas for the proof of Theorem 1.3. In order to better understand both the geometry of AdS and the derivative loss occurring in (3) of Theorem 1.1, it is useful</list_item>
</unordered_list>
<text><location><page_8><loc_15><loc_84><loc_85><loc_88></location>to invoke the conformal properties of AdS and the conformal wave equation (3). Setting r = tan ψ with ψ ∈ ( 0 , π 2 ) one finds from (3)</text>
<formula><location><page_8><loc_15><loc_80><loc_71><loc_83></location>g AdS = 1 cos 2 ψ ( -dt 2 + dψ 2 +sin 2 ψd Ω 2 S 2 ) =: 1 cos 2 ψ g E , (11)</formula>
<text><location><page_8><loc_15><loc_69><loc_85><loc_79></location>and hence that AdS is conformal to a part of the Einstein cylinder, namely R × S 3 h ⊂ R × S 3 , where S 3 h denotes a hemisphere of S 3 with ψ = π 2 being its equatorial boundary. The wave equation (3) is called conformal or conformally invariant because if u is a solution of (3), then v := Ω u is a solution of a wave equation with respect to the conformally transformed metric g = Ω 2 g AdS . This suggests that understanding the dynamics of (3) for g AdS is essentially equivalent to that of understanding solutions of</text>
<formula><location><page_8><loc_15><loc_66><loc_55><loc_67></location>glyph[square] g E v -v = 0 (12)</formula>
<text><location><page_8><loc_15><loc_61><loc_85><loc_64></location>on one hemisphere of the Einstein cylinder. This latter is a finite problem, which we will refer to as Problem 1 below.</text>
<unordered_list>
<list_item><location><page_8><loc_18><loc_54><loc_85><loc_60></location>(1) Problem 1: The wave equation (12) on R t × S 3 h (with the natural product metric of the Einstein cylinder) where S 3 h is the (say northern) hemisphere of the 3-sphere S 3 with boundary at ψ = π 2 , where (say optimally) dissipative boundary conditions are imposed. We contrast this problem with</list_item>
<list_item><location><page_8><loc_18><loc_51><loc_85><loc_54></location>(2) Problem 2: the wave equation (3) on R t × B 3 (with the flat metric) where B 3 is the unit ball with boundary S 2 where dissipative boundary conditions are imposed.</list_item>
</unordered_list>
<text><location><page_8><loc_15><loc_43><loc_85><loc_50></location>For Problem 2 it is well-known that exponential decay of energy holds without loss of derivatives, see [29, 30, 31]. (It is an entertaining exercise to prove this in a more robust fashion using the methods of this paper.) For Problem 1, however, there will be a derivative loss present in any decay estimate, as seen in Theorem 1.1. 7</text>
<text><location><page_8><loc_15><loc_34><loc_85><loc_43></location>This phenomenon can be explained in the geometric optics approximation for the wave equation. Recall that in this picture, the optimally dissipative boundary condition says that the energy of a ray is fully absorbed if it hits the boundary orthogonally. For rays which graze the boundary, the fraction of the energy that is absorbed upon reflection depends on the glancing angle: the shallower the incident angle, the less energy is lost in the reflection.</text>
<text><location><page_8><loc_15><loc_27><loc_85><loc_33></location>Now let us fix a (large) time interval [0 , T ] for both Problem 1 and Problem 2. To construct a solution which decays very slowly, we would like to identify rays which a) hit the boundary as little as possible and b) if they do hit the boundary, they should do this at a very shallow angle (grazing rays).</text>
<text><location><page_8><loc_15><loc_24><loc_85><loc_27></location>What happens in Problem 2, is that the shallower one chooses the angle of the ray, the more reflections will take place in [0 , T ]. This is easily seen by looking at the projection of</text>
<text><location><page_9><loc_15><loc_86><loc_50><loc_88></location>the null rays to the surface t = 0 (see figure).</text>
<figure>
<location><page_9><loc_24><loc_76><loc_37><loc_86></location>
</figure>
<figure>
<location><page_9><loc_63><loc_76><loc_76><loc_86></location>
</figure>
<figure>
<location><page_9><loc_43><loc_76><loc_56><loc_86></location>
</figure>
<text><location><page_9><loc_15><loc_65><loc_85><loc_75></location>In sharp contrast, for Problem 1, the time until a null geodesic will meet the boundary again after reflection does not depend on the incident angle ! This goes back to the fact that all geodesics emanating from point on the three sphere refocus at the antipodal point. As a consequence, in Problem 1 we can keep the number of reflections in [0 , T ] fixed while choosing the incident angle as small as we like. This observation is at the heart of the Gaussian beam approximation invoked to prove Theorem 1.3.</text>
<unordered_list>
<list_item><location><page_9><loc_15><loc_46><loc_85><loc_64></location>1.6. Structure of the paper. We conclude this introduction providing the structure of the paper. In Section 2 we define the coordinate systems, frames and basic vectorfields which we are going to employ. Section 3 introduces the field equations of spin 0, 1 and 2 fields, together with their energy momentum tensors. The well-posedness under dissipative boundary conditions for each of these equations is discussed in Section 4 with particular emphasis on the importance of the reduced system in the spin 2 case. Section 5 is at the heart of the paper proving the global results of Theorem 1.1, first for the spin 0 (Section 5.1), then the spin 1 (Section 5.2) and finally the spin 2 case (Section 5.3). Corollary 1.2 is proven in Section 5.4 and Theorem 1.3 in Section 5.5. We conclude the paper outlining generalizations of our result. Some elementary computations have been relegated to the appendix.</list_item>
<list_item><location><page_9><loc_15><loc_38><loc_85><loc_45></location>1.7. Acknowledgements. G.H. is grateful for the support offered by a grant of the European Research Council. G.H., J.S. and C.M.W. are grateful to the Newton Institute for support during the programme 'Mathematics and Physics of the Holographic Principle'. J.L. thanks the support of the NSF Postdoctoral Fellowship DMS-1204493.</list_item>
</unordered_list>
<section_header_level_1><location><page_9><loc_43><loc_35><loc_57><loc_36></location>2. Preliminaries</section_header_level_1>
<unordered_list>
<list_item><location><page_9><loc_15><loc_31><loc_85><loc_34></location>2.1. Coordinates, Frames and Volume forms. We will consider the so-called 'global' anti-de Sitter space-time, defined on R 4 , with metric (in polar coordinates)</list_item>
</unordered_list>
<formula><location><page_9><loc_15><loc_27><loc_66><loc_30></location>(13) g AdS = -( 1 + r 2 ) dt 2 + dr 2 1 + r 2 + r 2 d Ω 2 S 2</formula>
<text><location><page_9><loc_15><loc_22><loc_85><loc_26></location>with d Ω 2 S 2 the standard round metric on the unit sphere. As long as there is no risk of confusion, we will also denote the metric by g . It will be convenient to introduce an orthonormal basis { e a } :</text>
<formula><location><page_9><loc_15><loc_17><loc_69><loc_21></location>(14) e 0 = √ 1 + r 2 dt, e r = dr √ 1 + r 2 , e A = /e A ,</formula>
<text><location><page_10><loc_15><loc_83><loc_85><loc_88></location>for A = 1 , 2, where /e A are an orthonormal basis for the round sphere 8 of radius r . Throughout this paper, we will use capital Latin letters for indices on the sphere while small Latin letters are reserved as spacetime indices. The dual basis of vector fields is denoted { e a } :</text>
<formula><location><page_10><loc_15><loc_79><loc_70><loc_82></location>(15) e 0 = 1 √ 1 + r 2 ∂ ∂t , e r = √ 1 + r 2 ∂ ∂r , e A = /e A ,</formula>
<text><location><page_10><loc_15><loc_74><loc_85><loc_78></location>We introduce the surfaces Σ T = { t = T } and ˜ Σ [ T 1 ,T 2 ] R = { r = R,T 1 ≤ t ≤ T 2 } , which have respective unit normals:</text>
<formula><location><page_10><loc_42><loc_72><loc_58><loc_74></location>n := e 0 , m := e r .</formula>
<text><location><page_10><loc_15><loc_70><loc_48><loc_71></location>The surface measures on these surfaces are</text>
<formula><location><page_10><loc_30><loc_66><loc_70><loc_70></location>dS Σ T = r 2 √ 1 + r 2 drdω, dS Σ R = r 2 √ 1 + r 2 dtdω</formula>
<text><location><page_10><loc_15><loc_62><loc_85><loc_65></location>with dω the volume form of the round unit sphere. We denote by S [ T 1 ,T 2 ] := { T 1 ≤ t ≤ T 2 } the spacetime slab between Σ T 1 and Σ T 2 . Finally, the spacetime volume form is</text>
<formula><location><page_10><loc_44><loc_60><loc_56><loc_61></location>dη = r 2 dtdrdω.</formula>
<text><location><page_10><loc_15><loc_55><loc_85><loc_58></location>2.2. Vectorfields. The global anti-de Sitter spacetime enjoys the property of being static. The Killing field</text>
<formula><location><page_10><loc_15><loc_51><loc_58><loc_54></location>(16) T := ∂ ∂t = √ 1 + r 2 e 0</formula>
<text><location><page_10><loc_15><loc_47><loc_75><loc_50></location>is everywhere timelike, and orthogonal to the surfaces of constant t . Besides the Killing field T , we will exploit the properties of the vectorfield</text>
<formula><location><page_10><loc_15><loc_43><loc_59><loc_46></location>(17) X := r √ 1 + r 2 ∂ ∂r = re r</formula>
<text><location><page_10><loc_15><loc_39><loc_85><loc_42></location>While the vectorfield X is not Killing, it is almost conformally Killing near infinity. More importantly, it generates terms with a definite sign. To see this, note that</text>
<formula><location><page_10><loc_15><loc_35><loc_65><loc_38></location>(18) X π := 1 2 L X g = ( e 0 ) 2 √ 1 + r 2 + g √ 1 + r 2 .</formula>
<text><location><page_10><loc_15><loc_31><loc_85><loc_34></location>In particular, contracting X π with a symmetric traceless tensor T will only see the first term and this contraction will in fact have a sign if T satisfies the dominant energy condition.</text>
<text><location><page_10><loc_15><loc_26><loc_85><loc_29></location>2.3. Differential operators. Throughout this paper, we will use ∇ to denote the LeviCivita connection of g . Define also glyph[square] g as the standard Laplace-Beltrami operator:</text>
<formula><location><page_10><loc_44><loc_22><loc_56><loc_24></location>glyph[square] g u := ∇ a ∇ a u.</formula>
<text><location><page_11><loc_15><loc_83><loc_85><loc_88></location>2.4. The divergence theorem. We denote by Div the spacetime divergence associated to the metric g for a vector field. In coordinates, it takes the following form: If K = K 0 e 0 + K r e r + K A e A , then</text>
<formula><location><page_11><loc_27><loc_79><loc_73><loc_82></location>Div K = 1 √ 2 ∂K + 1 2 ∂ ( r 2 √ 1 + r 2 K r ) + / ∇ A K A .</formula>
<formula><location><page_11><loc_15><loc_79><loc_52><loc_82></location>(19) 1 + r 0 ∂t r ∂r</formula>
<text><location><page_11><loc_15><loc_76><loc_71><loc_78></location>It will be useful to record the following form of the divergence theorem:</text>
<text><location><page_11><loc_15><loc_72><loc_85><loc_75></location>Lemma 2.1 (Divergence theorem) . Suppose that K is a suitably regular vector field defined on R 4 . Then we have</text>
<formula><location><page_11><loc_30><loc_63><loc_70><loc_71></location>0 = ∫ Σ T 2 K a n a dS Σ T 2 -∫ Σ T 1 K a n a dS Σ T 1 -lim r →∞ ∫ ˜ Σ [ T 1 ,T 2 ] r K a m a dS ˜ Σ r + ∫ S [ T 1 ,T 2 ] Div Kdη.</formula>
<text><location><page_11><loc_40><loc_60><loc_60><loc_61></location>3. The field equations</text>
<text><location><page_11><loc_15><loc_57><loc_75><loc_59></location>3.1. Spin 0: The wave equation. The conformal wave equation on AdS is</text>
<formula><location><page_11><loc_15><loc_55><loc_57><loc_56></location>(20) glyph[square] g AdS u +2 u = 0 .</formula>
<text><location><page_11><loc_15><loc_49><loc_85><loc_53></location>Akey object to study the dynamics of (20) is the twisted, or renormalised energy-momentum tensor. See [32, 24] for further details on the following construction. We first define the twisted covariant derivative 9 by:</text>
<formula><location><page_11><loc_37><loc_45><loc_63><loc_48></location>˜ ∇ a ( · ) := 1 √ 1 + r 2 ∇ a ( · √ 1 + r 2 ) .</formula>
<text><location><page_11><loc_15><loc_42><loc_71><loc_44></location>With this derivative we construct the twisted energy momentum tensor</text>
<formula><location><page_11><loc_15><loc_38><loc_68><loc_41></location>(21) T ab [ u ] = ˜ ∇ a u ˜ ∇ b u -1 2 g ab ( ˜ ∇ c u ˜ ∇ c u + u 2 1 + r 2 ) ,</formula>
<text><location><page_11><loc_15><loc_35><loc_26><loc_37></location>which satisfies</text>
<formula><location><page_11><loc_15><loc_33><loc_65><loc_34></location>(22) ∇ a T a b [ u ] = ( glyph[square] g u +2 u ) ˜ ∇ b u -T c c ˜ ∇ b 1 .</formula>
<text><location><page_11><loc_15><loc_30><loc_57><loc_31></location>If K is any vector field then we can define the current</text>
<formula><location><page_11><loc_15><loc_27><loc_58><loc_29></location>(23) K J a [ u ] = T a b [ u ] K b ,</formula>
<text><location><page_11><loc_15><loc_21><loc_85><loc_26></location>which is a compatible current in the sense of Christodoulou [33]. Moreover, if K is Killing and K ( r ) = 0, then K a ˜ ∇ a 1 = 0 and we can see that K J [ u ] is a conserved current when u solves the conformal wave equation (20).</text>
<text><location><page_12><loc_15><loc_85><loc_85><loc_88></location>3.2. Spin 1: Maxwell's equations. The Maxwell equations in vacuum are given by the following first order differential equations for a 2-form F on R 4 :</text>
<formula><location><page_12><loc_40><loc_82><loc_60><loc_84></location>dF = 0 , d ( glyph[star] g F ) = 0 .</formula>
<text><location><page_12><loc_15><loc_80><loc_46><loc_81></location>We decompose the Maxwell 2 -form as:</text>
<formula><location><page_12><loc_15><loc_78><loc_73><loc_79></location>(24) F = E r e 0 ∧ e r + E A e 0 ∧ e A + H r e 1 ∧ e 2 + H A glyph[epsilon1] A B e B ∧ e r ,</formula>
<text><location><page_12><loc_15><loc_75><loc_75><loc_77></location>where glyph[epsilon1] AB is the volume form on the sphere 10 . The dual Maxwell 2 -form is:</text>
<formula><location><page_12><loc_26><loc_73><loc_74><loc_75></location>glyph[star] g F = H r e 0 ∧ e r + H A e 0 ∧ e A -E r e 1 ∧ e 2 -E A glyph[epsilon1] A B e B ∧ e r .</formula>
<text><location><page_12><loc_15><loc_64><loc_85><loc_72></location>We will use the notation E i with i = r, 1 , 2, and similarly for H . Since F is a smooth 2 -form, and the basis vectors e µ are bounded (but not continuous) at the origin, we deduce that the functions E i , H i are bounded, but not necessarily continuous, at r = 0. With respect to this decomposition, Maxwell's vacuum equations (3.2) split into six evolution equations:</text>
<formula><location><page_12><loc_15><loc_60><loc_53><loc_64></location>r √ 1 + r 2 ∂ t E r = -rglyph[epsilon1] AB / ∇ A H B , (25)</formula>
<formula><location><page_12><loc_15><loc_57><loc_70><loc_60></location>r √ 1 + r 2 ∂ t E A = glyph[epsilon1] A B [ ∂ r ( r √ 1 + r 2 H B ) -r / ∇ B H r ] , (26)</formula>
<formula><location><page_12><loc_15><loc_54><loc_51><loc_57></location>r √ 1 + r 2 ∂ t H r = rglyph[epsilon1] AB / ∇ A E B , (27)</formula>
<formula><location><page_12><loc_15><loc_50><loc_71><loc_54></location>r √ 1 + r 2 ∂ t H A = -glyph[epsilon1] A B [ ∂ r ( r √ 1 + r 2 E B ) -r / ∇ B E r ] , (28)</formula>
<text><location><page_12><loc_15><loc_48><loc_31><loc_50></location>and two constraints:</text>
<formula><location><page_12><loc_15><loc_45><loc_63><loc_48></location>0 = 1 + r 2 ∂ r ( r 2 E r ) + r / ∇ A E A , (29)</formula>
<formula><location><page_12><loc_40><loc_44><loc_44><loc_49></location>√ r</formula>
<formula><location><page_12><loc_15><loc_42><loc_63><loc_44></location>0 = 1 + r 2 ∂ r ( r 2 H r ) + r / ∇ A H A . (30)</formula>
<formula><location><page_12><loc_40><loc_41><loc_44><loc_45></location>√ r</formula>
<text><location><page_12><loc_15><loc_34><loc_85><loc_40></location>Here r / ∇ is the covariant derivative on the unit sphere, which commutes with ∇ ∂ r (note that our conventions are that /e A are orthonormal vector fields on the sphere of radius r ). The evolution equations (25-28) form a symmetric hyperbolic system. If the evolution equations hold, and assuming sufficient differentiability, it is straightforward to verify that</text>
<formula><location><page_12><loc_44><loc_30><loc_56><loc_33></location>∂ E ∂t = ∂ H ∂t = 0 ,</formula>
<text><location><page_12><loc_15><loc_28><loc_20><loc_29></location>where</text>
<formula><location><page_12><loc_36><loc_25><loc_64><loc_28></location>E := √ 1 + r 2 ∂ r ( r 2 E r ) + r / ∇ A E A</formula>
<formula><location><page_12><loc_35><loc_21><loc_65><loc_24></location>H := 1 + r 2 ∂ r ( r 2 H r ) + r / ∇ A H A</formula>
<formula><location><page_12><loc_43><loc_20><loc_65><loc_26></location>r , √ r .</formula>
<text><location><page_13><loc_19><loc_90><loc_85><loc_90></location>ASYMPTOTIC PROPERTIES OF LINEAR FIELD EQUATIONS IN ANTI-DE SITTER SPACE 13</text>
<text><location><page_13><loc_15><loc_85><loc_85><loc_88></location>Thus if the constraint equations hold on the initial data surface, then they hold for all times.</text>
<text><location><page_13><loc_15><loc_81><loc_85><loc_83></location>3.2.1. The energy momentum tensor. The analogue of the energy momentum tensor (21) for the wave equation is the symmetric tensor</text>
<formula><location><page_13><loc_15><loc_77><loc_62><loc_80></location>(31) T [ F ] ab = F ac F b c -1 4 g ab F cd F cd ,</formula>
<text><location><page_13><loc_15><loc_75><loc_26><loc_76></location>which satisfies</text>
<formula><location><page_13><loc_34><loc_73><loc_66><loc_75></location>∇ a T [ F ] ab = F b c ∇ d F dc +( glyph[star]F ) b c ∇ d ( glyph[star]F ) dc</formula>
<text><location><page_13><loc_15><loc_70><loc_85><loc_73></location>so that if F satisfies Maxwell's equations, T [ F ] is divergence free and traceless. We define in the obvious fashion the current</text>
<formula><location><page_13><loc_15><loc_67><loc_58><loc_69></location>(32) K J a [ F ] = T a b [ F ] K b</formula>
<text><location><page_13><loc_15><loc_65><loc_32><loc_67></location>for any vector field K .</text>
<text><location><page_13><loc_15><loc_58><loc_85><loc_64></location>3.3. Spin 2: The Bianchi equations. The equations for a spin 2 field, also called the Bianchi equations, can be expressed as first order differential equations for a Weyl tensor, which is an arbitrary 4 -tensor which satisfies the same symmetry properties as the Weyl curvature tensor. More precisely:</text>
<text><location><page_13><loc_15><loc_55><loc_69><loc_57></location>Definition 1. We say a 4 -tensor W is a Weyl tensor if it satisfies:</text>
<unordered_list>
<list_item><location><page_13><loc_16><loc_53><loc_40><loc_55></location>i) W abcd = -W bacd = -W abdc .</list_item>
<list_item><location><page_13><loc_16><loc_52><loc_40><loc_53></location>ii) W abcd + W acdb + W abdc = 0 .</list_item>
<list_item><location><page_13><loc_15><loc_50><loc_30><loc_51></location>iii) W abcd = W cdab .</list_item>
<list_item><location><page_13><loc_16><loc_48><loc_27><loc_50></location>iv) W a bad = 0 .</list_item>
</unordered_list>
<text><location><page_13><loc_15><loc_47><loc_52><loc_48></location>The dual of a Weyl tensor is defined by (cf. [1] )</text>
<formula><location><page_13><loc_35><loc_43><loc_65><loc_46></location>glyph[star] W abcd = 1 2 glyph[epsilon1] abef W ef cd = 1 2 W ab ef glyph[epsilon1] efcd .</formula>
<text><location><page_13><loc_17><loc_41><loc_37><loc_42></location>The Bianchi equations are</text>
<formula><location><page_13><loc_15><loc_38><loc_56><loc_40></location>(33) ∇ [ a W bc ] de = 0 ,</formula>
<text><location><page_13><loc_15><loc_36><loc_51><loc_37></location>which are equivalent to either of the equations</text>
<formula><location><page_13><loc_15><loc_34><loc_56><loc_35></location>∇ a W abcd = 0 , (34)</formula>
<formula><location><page_13><loc_15><loc_32><loc_56><loc_33></location>∇ aglyph[star] W abcd = 0 , (35)</formula>
<text><location><page_13><loc_15><loc_29><loc_59><loc_31></location>and for us studying (34) will be particularly convenient.</text>
<text><location><page_13><loc_15><loc_23><loc_85><loc_29></location>As in the case of the Maxwell equations, it is convenient to decompose the Weyl tensor based on the 3 + 1 splitting of space and time. In the Maxwell case, the field strength tensor F decomposes as a pair of vectors tangent to Σ t . In the spin 2 case, the Weyl tensor W decomposes as pair of symmetric tensors tangent to Σ t . We define:</text>
<formula><location><page_13><loc_15><loc_20><loc_62><loc_22></location>E ab := W 0 a 0 b = W cadb ( e 0 ) c ( e 0 ) d , (36)</formula>
<formula><location><page_13><loc_15><loc_18><loc_63><loc_20></location>H ab := glyph[star] W 0 a 0 b = glyph[star] W cadb ( e 0 ) c ( e 0 ) d . (37)</formula>
<text><location><page_14><loc_15><loc_83><loc_85><loc_88></location>The symmetries indeed ensure that E,H are symmetric and tangent to Σ t . Moreover, both E and H are necessarily trace-free. In fact, one can reconstruct the whole tensor W from the symmetric trace-free fields E , H , see [1, § 7.2].</text>
<text><location><page_14><loc_15><loc_77><loc_85><loc_83></location>We will further decompose E,H along the orthonormal frame defined in (14). To write the equations of motion in terms of E , H we consider the equations 0 = ∇ a W arr 0 , 0 = ∇ a W a ( Ar )0 and 0 = ∇ a W a ( AB )0 , from which we respectively find the evolution equations for E :</text>
<formula><location><page_14><loc_20><loc_73><loc_51><loc_76></location>r √ 2 ∂E rr ∂t = -rglyph[epsilon1] AB / ∇ A H Br E rr )</formula>
<formula><location><page_14><loc_15><loc_64><loc_83><loc_75></location>1 + r , (Evol r √ 1 + r 2 ∂E Ar ∂t = 1 2 glyph[epsilon1] A B [ ∂ r ( r (1 + r 2 ) H Br ) √ 1 + r 2 -r / ∇ B H rr ] -r 2 glyph[epsilon1] BC / ∇ B H CA , (Evol E Ar ) r √ 1 + r 2 ∂E AB ∂t = glyph[epsilon1] ( A C [ ∂ r ( r (1 + r 2 ) H B ) C ) √ 1 + r 2 -r / ∇ | C | H B ) r ] . (Evol E AB )</formula>
<text><location><page_14><loc_15><loc_60><loc_85><loc_63></location>From the equations 0 = ∇ a W a 0 A 0 and 0 = ∇ a W a 0 r 0 respectively we find the constraint equations:</text>
<formula><location><page_14><loc_34><loc_58><loc_36><loc_60></location>√</formula>
<formula><location><page_14><loc_15><loc_52><loc_67><loc_59></location>1 + r 2 r 2 ∂ ∂r ( r 3 E rr ) + r / ∇ B E Br =: E r = 0 (Con E r ) √ 1 + r 2 r 2 ∂ ∂r ( r 3 E Ar ) + r / ∇ B E AB =: E A = 0 (Con E A )</formula>
<text><location><page_14><loc_15><loc_48><loc_85><loc_51></location>By considering the equivalent equations for glyph[star] W , we find that the evolution equations for H can be obtained from these by the substitution ( E,H ) → ( H, -E ):</text>
<formula><location><page_14><loc_15><loc_35><loc_84><loc_47></location>r √ 1 + r 2 ∂H rr ∂t = rglyph[epsilon1] AB / ∇ A E Br , (Evol H rr ) r √ 1 + r 2 ∂H Ar ∂t = -1 2 glyph[epsilon1] A B [ ∂ r ( r (1 + r 2 ) E Br ) √ 1 + r 2 -r / ∇ B E rr ] + r 2 glyph[epsilon1] BC / ∇ B E CA , (Evol H Ar ) r √ 1 + r 2 ∂H AB ∂t = -glyph[epsilon1] ( A C [ ∂ r ( r (1 + r 2 ) E B ) C ) √ 1 + r 2 -r / ∇ | C | E B ) r ] , (Evol H AB )</formula>
<text><location><page_14><loc_15><loc_33><loc_46><loc_34></location>and the constraint equations for H are:</text>
<formula><location><page_14><loc_34><loc_31><loc_35><loc_33></location>√</formula>
<formula><location><page_14><loc_15><loc_24><loc_67><loc_31></location>1 + r 2 r 2 ∂ ∂r ( r 3 H rr ) + r / ∇ B H Br =: H r = 0 , (Con H r ) √ 1 + r 2 r 2 ∂ ∂r ( r 3 H Ar ) + r / ∇ B H AB =: H A = 0 (Con H B )</formula>
<formula><location><page_14><loc_67><loc_25><loc_67><loc_27></location>.</formula>
<text><location><page_14><loc_15><loc_21><loc_85><loc_23></location>3.3.1. The Bel-Robinson tensor. The spin 2 analogue of the energy momentum tensor for the Maxwell field is the Bel-Robinson tensor [1, § 7.1.1]. This is defined to be</text>
<formula><location><page_14><loc_15><loc_17><loc_64><loc_20></location>(38) Q abcd := W aecf W e f b d + glyph[star] W aecf glyph[star] W e f b d</formula>
<text><location><page_15><loc_19><loc_90><loc_81><loc_90></location>ASYMPTOTIC PROPERTIES OF LINEAR FIELD EQUATIONS IN ANTI-DE SITTER SPACE</text>
<text><location><page_15><loc_15><loc_85><loc_85><loc_88></location>It is symmetric, trace-free on all pairs of indices and if W satisfies the Bianchi equations then</text>
<formula><location><page_15><loc_15><loc_82><loc_55><loc_84></location>(39) ∇ a Q abcd = 0 .</formula>
<text><location><page_15><loc_15><loc_78><loc_85><loc_81></location>We shall require the following quantitative version of the dominant energy condition for the Bel-Robinson tensor</text>
<text><location><page_15><loc_15><loc_74><loc_85><loc_77></location>Lemma 3.1. Suppose that t 1 , t 2 ∈ T p M are future-directed timelike unit vectors, and that -g ( t 1 , t 2 ) ≤ B for some B ≥ 1 . Then there exists a constant C > 0 such that</text>
<formula><location><page_15><loc_28><loc_70><loc_72><loc_73></location>1 CB 4 Q ( t 1 , t 1 , t 1 , t 1 ) ≤ Q ( t i , t j , t k , t l ) ≤ CB 4 Q ( t 1 , t 1 , t 1 , t 1 )</formula>
<text><location><page_15><loc_15><loc_67><loc_85><loc_70></location>holds for any i, j, k, l ∈ { 1 , 2 } . Moreover, Q ( t 1 , t 1 , t 1 , t 1 ) ≥ 0 , with equality at a point p if and only if W vanishes at p .</text>
<text><location><page_15><loc_15><loc_61><loc_85><loc_66></location>Proof. We follow the proof of [23, Prop. 4.2]. We can pick a pair of null vectors e ' 3 , e ' 4 with g ( e ' 3 , e ' 4 ) = -1 2 such that t 1 = e ' 3 + e ' 4 and t 2 = be ' 3 + b -1 e ' 4 for some b ≥ 1. The condition -g ( t 1 , t 2 ) ≤ B implies</text>
<formula><location><page_15><loc_46><loc_58><loc_55><loc_61></location>1 B ≤ b ≤ B.</formula>
<text><location><page_15><loc_15><loc_56><loc_40><loc_57></location>As a consequence, we can write</text>
<formula><location><page_15><loc_33><loc_51><loc_67><loc_55></location>Q ( t i , t j , t k , t l ) = 4 ∑ a,b,c,d =3 q abcd ijkl Q ( e ' a , e ' b , e ' c , e ' d )</formula>
<text><location><page_15><loc_15><loc_49><loc_20><loc_50></location>where</text>
<formula><location><page_15><loc_43><loc_46><loc_57><loc_49></location>c B 4 ≤ q abcd ijkl ≤ c ' B 4</formula>
<text><location><page_15><loc_15><loc_39><loc_85><loc_46></location>for some combinatorial constants c, c ' which are independent of B . By [23, Prop. 4.2], all of the quantities Q ( e ' a , e ' b , e ' c , e ' d ) are non-negative, and so we have established the first part. For the second part, we note that, again following [23, Prop. 4.2], the quantity Q ( t 1 , t 1 , t 1 , t 1 ) controls all components of W . glyph[square]</text>
<section_header_level_1><location><page_15><loc_42><loc_36><loc_58><loc_37></location>4. Well posedness</section_header_level_1>
<section_header_level_1><location><page_15><loc_15><loc_34><loc_48><loc_35></location>4.1. Dissipative boundary conditions.</section_header_level_1>
<text><location><page_15><loc_15><loc_29><loc_85><loc_32></location>4.1.1. Wave equation. For the wave equation, an analysis of solutions near to the conformal boundary leads us to expect that u has the following behaviour:</text>
<formula><location><page_15><loc_29><loc_25><loc_71><loc_28></location>u = u + ( t, x A ) r + u -( t, x A ) r 2 + O ( 1 r 3 ) as r →∞ .</formula>
<text><location><page_15><loc_15><loc_21><loc_85><loc_24></location>The boundary condition we shall impose for (20) on anti-de Sitter space can be simply written in the form</text>
<formula><location><page_15><loc_15><loc_17><loc_65><loc_20></location>(40) ∂ ( ru ) ∂t + r 2 ∂ ( ru ) ∂r → 0 , as r →∞ ,</formula>
<text><location><page_16><loc_15><loc_86><loc_32><loc_88></location>which is equivalent to</text>
<formula><location><page_16><loc_41><loc_83><loc_60><loc_86></location>∂u + ∂t -u -= 0 , on I .</formula>
<text><location><page_16><loc_15><loc_79><loc_85><loc_82></location>4.1.2. Maxwell's equations. In the Maxwell case, by (optimally) dissipative boundary conditions we understand the condition</text>
<formula><location><page_16><loc_15><loc_76><loc_65><loc_78></location>(41) r 2 ( E A + glyph[epsilon1] A B H B ) → 0 , as r →∞ .</formula>
<text><location><page_16><loc_15><loc_67><loc_85><loc_75></location>The above boundary conditions ensure that the asymptotic Poynting vector is outward directed, i.e. energy is leaving the spacetime. It effectively means that I behaves like an imperfect conductor with a surface resistance, for which the electric currents induced by the electromagnetic field dissipate energy as heat. These boundary conditions are an example of Leontovic boundary conditions [34, § 87].</text>
<text><location><page_16><loc_15><loc_62><loc_85><loc_65></location>4.1.3. Bianchi equations. In the case of the Bianchi equation, by (optimally) dissipative boundary conditions we understand the condition</text>
<formula><location><page_16><loc_33><loc_59><loc_67><loc_61></location>r 3 ( ˆ E AB + glyph[epsilon1] ( A C ˆ H B ) C ) → 0 , as r →∞ .</formula>
<text><location><page_16><loc_15><loc_57><loc_32><loc_58></location>which is equivalent to:</text>
<formula><location><page_16><loc_15><loc_52><loc_73><loc_56></location>(42) r 3 ( E AB -1 2 δ AB E C C + glyph[epsilon1] ( A C H B ) C ) → 0 , as r →∞ .</formula>
<text><location><page_16><loc_15><loc_43><loc_85><loc_51></location>Notice that there are only two boundary conditions imposed (since the relevant objects appearing in the boundary term are the trace-free parts of E AB , H AB ). We could choose more general dissipative boundary conditions. For simplicity we shall just consider those above, which ought in some sense to represent 'optimal dissipation' at the boundary, but see § 6 for generalisations.</text>
<unordered_list>
<list_item><location><page_16><loc_15><loc_30><loc_85><loc_42></location>4.2. The well-posedness statements. We now state a general well-posedness statement for each of our three models with dissipative boundary conditions. As is well-known, the key to prove these theorems is the existence of a suitable energy estimate under the boundary conditions imposed. In the Wave- and Maxwell case such an estimate is immediate. In the Bianchi case, however, there is a subtlety (discussed and resolved already in [10]). We will dedicate Section 4.3 to derive a local energy estimate showing that the condition (42) indeed leads to a well-posed problem.</list_item>
</unordered_list>
<text><location><page_16><loc_15><loc_26><loc_85><loc_29></location>4.2.1. Spin 0: The wave equation. The following result can be established either directly, or by making use of the conformal invariance of (20):</text>
<text><location><page_16><loc_15><loc_20><loc_85><loc_24></location>Theorem 4.1 (Well posedness for the conformal wave equation) . Fix T > 0 . Given smooth functions u 0 , u 1 : Σ 0 → R satisfying suitable asymptotic conditions, there exists a unique smooth u , such that:</text>
<unordered_list>
<list_item><location><page_16><loc_16><loc_18><loc_59><loc_19></location>i) u solves the conformal wave equation (20) in S [0 ,T ] .</list_item>
</unordered_list>
<section_header_level_1><location><page_17><loc_16><loc_86><loc_36><loc_88></location>ii) We have the estimate:</section_header_level_1>
<formula><location><page_17><loc_30><loc_82><loc_70><loc_86></location>sup S [0 ,T ] ∑ k,l,m ≤ K ∣ ∣ ∣ ( r 2 ∂ r ) k ( ∂ t ) l ( r / ∇ ) m ( ru ) ∣ ∣ ∣ ≤ C T,u 0 ,u 1 ,K ,</formula>
<text><location><page_17><loc_18><loc_78><loc_85><loc_81></location>for k, l, m = 0 , 1 , . . . and a constant C T,u 0 ,u 1 ,K depending on K,T and the initial data. This in particular implies a particular asymptotic behaviour for the fields.</text>
<unordered_list>
<list_item><location><page_17><loc_15><loc_77><loc_40><loc_78></location>iii) The initial conditions hold:</list_item>
</unordered_list>
<formula><location><page_17><loc_39><loc_74><loc_61><loc_76></location>u | t =0 = u 0 , u t | t =0 = u 1</formula>
<unordered_list>
<list_item><location><page_17><loc_16><loc_72><loc_51><loc_74></location>iv) Dissipative boundary conditions (40) hold.</list_item>
</unordered_list>
<text><location><page_17><loc_15><loc_68><loc_85><loc_71></location>4.2.2. Spin 1: Maxwell's equations. Working either directly with (25-30), or else by making use of the conformal invariance of Maxwell's equations, it can be shown that:</text>
<text><location><page_17><loc_15><loc_62><loc_85><loc_67></location>Theorem 4.2 (Well posedness for Maxwell's equations) . Fix T > 0 . Given smooth vector fields E 0 i , H 0 i satisfying the constraint equations, together with suitable asymptotic conditions, there exists a unique set of smooth vector fields: E i ( t ) , H i ( t ) , such that:</text>
<unordered_list>
<list_item><location><page_17><loc_16><loc_59><loc_85><loc_62></location>i) E i ( t ) , H i ( t ) solve (25-30) in S [0 ,T ] and the corresponding Maxwell tensor F is a smooth 2 -form on S [0 ,T ] .</list_item>
<list_item><location><page_17><loc_16><loc_57><loc_36><loc_58></location>ii) We have the estimate:</list_item>
</unordered_list>
<formula><location><page_17><loc_15><loc_52><loc_85><loc_57></location>sup S [0 ,T ] ∑ k,l,m k ' ,l ' ,m ' ≤ K ∣ ∣ ∣ ( r 2 ∂ r ) k ( ∂ t ) l ( r / ∇ ) m ( r 2 E i ) ∣ ∣ ∣ + ∣ ∣ ∣ ( r 2 ∂ r ) k ' ( ∂ t ) l ' ( r / ∇ ) m ' ( r 2 H i ) ∣ ∣ ∣ ≤ C T,E 0 ,H 0 ,K ,</formula>
<text><location><page_17><loc_18><loc_48><loc_85><loc_51></location>for any K ≥ 0 , where C T,E 0 ,H 0 ,K depends on K,T and the initial data. This in particular implies a particular asymptotic behaviour for the fields.</text>
<unordered_list>
<list_item><location><page_17><loc_15><loc_46><loc_40><loc_48></location>iii) The initial conditions hold:</list_item>
</unordered_list>
<formula><location><page_17><loc_39><loc_44><loc_61><loc_46></location>E i (0) = E 0 i , H i (0) = H 0 i</formula>
<unordered_list>
<list_item><location><page_17><loc_16><loc_42><loc_51><loc_43></location>iv) Dissipative boundary conditions (41) hold.</list_item>
</unordered_list>
<text><location><page_17><loc_15><loc_31><loc_85><loc_41></location>The asymptotic conditions on the initial data are corner conditions that come from ensuring that the initial data are compatible with the boundary conditions. It is certainly possible to construct initial data satisfying the constraints and the corner conditions to any order. We could work at finite regularity, and our results will in fact be valid with much weaker assumptions on the solutions, but for convenience it is simpler to assume the solutions are smooth.</text>
<unordered_list>
<list_item><location><page_17><loc_15><loc_29><loc_80><loc_30></location>4.2.3. Spin2 : Bianchi equations. In the case of the Bianchi equation we can prove:</list_item>
</unordered_list>
<text><location><page_17><loc_15><loc_21><loc_85><loc_27></location>Theorem 4.3 (Well posedness for the Bianchi system) . Fix T > 0 . Given smooth traceless symmetric 2 -tensors E 0 ab , H 0 ab on Σ 0 satisfying the constraint equations, together with suitable asymptotic conditions as r →∞ , there exists a unique set of traceless symmetric 2 -tensors: E ab ( t ) , H ab ( t ) such that:</text>
<unordered_list>
<list_item><location><page_17><loc_16><loc_18><loc_85><loc_21></location>i) E ab ( t ) , H ab ( t ) are traceless and the corresponding Weyl tensor W is a smooth 4 -tensor satisfying the Bianchi equations on S [0 ,T ] .</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_18><loc_16><loc_86><loc_46><loc_88></location>ii) We have the asymptotic behaviour:</list_item>
</unordered_list>
<formula><location><page_18><loc_15><loc_81><loc_85><loc_86></location>sup S [0 ,T ] ∑ k,l,m k ' ,l ' ,m ' ≤ K ∣ ∣ ∣ ( r 2 ∂ r ) k ( ∂ t ) l ( r / ∇ ) m ( r 3 E ab ) ∣ ∣ ∣ + ∣ ∣ ∣ ( r 2 ∂ r ) k ' ( ∂ t ) l ' ( r / ∇ ) m ' ( r 3 H ab ) ∣ ∣ ∣ ≤ C T,E 0 ,H 0 ,K</formula>
<text><location><page_18><loc_18><loc_79><loc_39><loc_80></location>as r →∞ , for any K ≥ 0 .</text>
<unordered_list>
<list_item><location><page_18><loc_15><loc_77><loc_40><loc_78></location>iii) The initial conditions hold:</list_item>
</unordered_list>
<formula><location><page_18><loc_38><loc_74><loc_62><loc_76></location>E ab (0) = E 0 ab , H ab (0) = H 0 ab</formula>
<unordered_list>
<list_item><location><page_18><loc_16><loc_72><loc_51><loc_74></location>iv) Dissipative boundary conditions (42) hold.</list_item>
</unordered_list>
<text><location><page_18><loc_15><loc_68><loc_85><loc_71></location>We will spend the remainder of this section to derive the key-energy estimate that is behind the proof of Theorem 4.3. 11</text>
<unordered_list>
<list_item><location><page_18><loc_15><loc_59><loc_85><loc_67></location>4.3. The modified system of Bianchi equations. In order to establish a well posedness theorem for the initial-boundary value problem associated to the spin 2 equations, the natural thing to do is to consider the evolution equations (Evol) as a symmetric hyperbolic 12 system. Having established existence and uniqueness for this system (Step 1), one can then attempt to show that the constraints (Con) are propagated from the initial data (Step 2).</list_item>
</unordered_list>
<text><location><page_18><loc_15><loc_54><loc_85><loc_59></location>If one attempts this strategy with the equations in the form given in Section 3.3, one finds that Step 1 causes no problems: one can easily derive an energy estimate for the system (Evol). In fact, it is a matter of simple calculation to check that taking:</text>
<formula><location><page_18><loc_18><loc_51><loc_82><loc_53></location>r (1 + r 2 ) 3 2 [ E rr × (Evol E rr ) + 2 E Ar × (Evol E Ar ) + E AB × (Evol E AB ) + E ↔ H ]</formula>
<text><location><page_18><loc_15><loc_49><loc_60><loc_51></location>and integrating over Σ t with measure drdω we arrive at 13</text>
<formula><location><page_18><loc_26><loc_45><loc_74><loc_48></location>d dt 1 2 ∫ ( | E rr | 2 +2 | E Ar | 2 + | E AB | 2 + E ↔ H ) r 2 (1 + r 2 ) drdω</formula>
<formula><location><page_18><loc_15><loc_41><loc_71><loc_46></location>Σ t = lim r →∞ ∫ ˜ Σ r ∩ Σ t ( 1 2 glyph[epsilon1] AB E Ar H Br + glyph[epsilon1] AB E AC H B C ) r 6 dω . (43)</formula>
<text><location><page_18><loc_15><loc_31><loc_85><loc_40></location>Once one has an energy estimate of this kind, establishing the existence of solutions to the evolution equations under the assumption that the boundary term has a good sign is essentially straightforward. Looking at the boundary term we obtain, this suggests that boundary conditions should be imposed on both the Ar and the AB components of E (or alternatively H ) at infinity. We shouldn't be so hasty, however. Before declaring victory, we must return to look at the constraints (Step 2).</text>
<text><location><page_19><loc_17><loc_86><loc_77><loc_88></location>Firstly, it is simple to verify that if (Evol) hold with sufficient regularity then</text>
<formula><location><page_19><loc_42><loc_82><loc_58><loc_85></location>0 = ∂E a a ∂t = ∂H a a ∂t ,</formula>
<text><location><page_19><loc_15><loc_76><loc_85><loc_81></location>so that the trace constraints on E and H are propagated by the evolution equations. Next, we turn to the differential constraints. We find that if (Evol) and the trace constraints hold, then the functions E a , H a defined in Section 3.3 satisfy the system of equations:</text>
<formula><location><page_19><loc_15><loc_72><loc_53><loc_75></location>r √ 1 + r 2 ∂ E r ∂t = -r 2 glyph[epsilon1] AB / ∇ A H B , (Evol E r )</formula>
<formula><location><page_19><loc_15><loc_67><loc_80><loc_72></location>r √ 1 + r 2 ∂ E A ∂t = glyph[epsilon1] A B 2 ∂ r ( r 2 √ 1 + r 2 H B ) √ 1 + r 2 -r / ∇ B H r -H B √ 1 + r 2 , (Evol E A )</formula>
<formula><location><page_19><loc_15><loc_62><loc_51><loc_66></location>r √ 1 + r 2 ∂ H r ∂t = r 2 glyph[epsilon1] AB / ∇ A E B , (Evol H r )</formula>
<formula><location><page_19><loc_15><loc_58><loc_80><loc_63></location>r √ 1 + r 2 ∂ H A ∂t = -glyph[epsilon1] A B 2 ∂ r ( r 2 √ 1 + r 2 E B ) √ 1 + r 2 -r / ∇ B E r -E B √ 1 + r 2 . (Evol H A )</formula>
<text><location><page_19><loc_15><loc_53><loc_85><loc_56></location>Now, things appear to be working in our favour. This system is symmetric hyperbolic and we can check that by taking:</text>
<formula><location><page_19><loc_18><loc_50><loc_82><loc_51></location>r 2 (1 + r 2 ) [ E r × (Evol E r ) + E A × (Evol E A ) + H r × (Evol H r ) + H A × (Evol H A )]</formula>
<text><location><page_19><loc_15><loc_47><loc_63><loc_48></location>and integrating over Σ t with the measure drdω we can derive:</text>
<formula><location><page_19><loc_22><loc_39><loc_78><loc_46></location>d dt 1 2 ∫ Σ t [ | E r | 2 + | E A | 2 + | H r | 2 + | H A | 2 ] r 3 √ 1 + r 2 drdω = lim r →∞ ∫ ˜ Σ r ∩ Σ t ( 1 2 glyph[epsilon1] AB E A H B ) r 6 dω -∫ Σ t [ glyph[epsilon1] AB E A H B ] r 2 √ 1 + r 2 drdω .</formula>
<text><location><page_19><loc_15><loc_29><loc_85><loc_37></location>Here we see the problem. We have no reason a priori to expect that the boundary term on ˜ Σ r ∩ Σ t vanishes. If it did, we could infer by Gronwall's lemma that the constraints are propagated. We conclude that the form of the propagation equations of Section 3.3 does not in general propagate the constraints at the boundary if boundary conditions are imposed on both E AB and E Ar (or H AB and H Ar ).</text>
<text><location><page_19><loc_15><loc_18><loc_85><loc_27></location>4.3.1. The modified equations. In order to resolve this issue, we have to modify the propagation equations before attempting to solve them as a symmetric hyperbolic system. In the previous calculation, the problematic boundary terms arise due to the radial derivatives appearing on the right hand side of (Evol E Ar ), (Evol H Ar ). We can remove these radial derivatives at the expense of introducing angular derivatives by using the constraint equations (Con E r ). It is also convenient to eliminate E rr and H rr from our equations using</text>
<text><location><page_20><loc_15><loc_86><loc_84><loc_88></location>the trace constraints. Doing this, we arrive at the modified set of propagation equations:</text>
<formula><location><page_20><loc_20><loc_81><loc_62><loc_84></location>r √ 2 ∂E Ar ∂t = -rglyph[epsilon1] BC / ∇ B H CA -glyph[epsilon1] A B H Br √ 2 E Ar )</formula>
<formula><location><page_20><loc_20><loc_69><loc_82><loc_72></location>r √ 1 + r 2 ∂H AB ∂t = -glyph[epsilon1] ( A C [ 1 √ 1 + r 2 ∂ ∂r ( r (1 + r 2 ) E B ) C ) -r / ∇ | C | E B ) r ] . H AB )</formula>
<formula><location><page_20><loc_15><loc_70><loc_81><loc_83></location>1 + r 1 + r , (Evol' r √ 1 + r 2 ∂E AB ∂t = glyph[epsilon1] ( A C [ 1 √ 1 + r 2 ∂ ∂r ( r (1 + r 2 ) H B ) C ) -r / ∇ | C | H B ) r ] . (Evol' E AB ) r √ 1 + r 2 ∂H Ar ∂t = rglyph[epsilon1] BC / ∇ B E CA + glyph[epsilon1] A B E Br √ 1 + r 2 , (Evol' H Ar ) (Evol'</formula>
<text><location><page_20><loc_15><loc_65><loc_60><loc_67></location>This again forms a symmetric hyperbolic system. Taking</text>
<formula><location><page_20><loc_26><loc_61><loc_74><loc_63></location>r (1 + r 2 ) 3 2 [ E Ar × (Evol E Ar ) + E AB × (Evol E AB ) + E ↔ H ]</formula>
<text><location><page_20><loc_15><loc_57><loc_59><loc_58></location>and integrating over Σ t with measure drdω we arrive at</text>
<formula><location><page_20><loc_15><loc_47><loc_79><loc_55></location>d dt 1 2 ∫ Σ t ( | E Ar | 2 + | E AB | 2 + | H Ar | 2 + | H AB | 2 ) r 2 (1 + r 2 ) drdω = lim r →∞ ∫ ˜ Σ r ∩ Σ t glyph[epsilon1] AB E AC H B C r 6 dω +2 ∫ Σ t glyph[epsilon1] AB E Ar H Br r (1 + r 2 ) drdω , (44)</formula>
<text><location><page_20><loc_15><loc_39><loc_85><loc_45></location>which is certainly sufficient to establish a well posedness result for the equations (Evol'), provided we choose boundary conditions such that the term on I has a sign. Notice that this will involve imposing conditions only on the E AB (or H AB ) components, so this formulation of the propagation equations is clearly different to the previous one! 14</text>
<text><location><page_20><loc_15><loc_31><loc_85><loc_39></location>Notice also that, unlike the estimate (43) for the (Evol) equations, we now have a bulk term in the energy estimate (44). For well-posedness this is no significant obstacle, but it will make establishing global decay estimates more difficult. In particular, it is no longer immediate that solutions with Dirichlet boundary conditions remain uniformly bounded globally.</text>
<text><location><page_20><loc_15><loc_27><loc_85><loc_30></location>Now let us consider the propagation of the constraints. We of course have to interpret the E rr term in E r as ( -E A A ) and similarly for H rr . The evolution equations for the</text>
<text><location><page_21><loc_15><loc_86><loc_40><loc_88></location>constraints take a simpler form:</text>
<text><location><page_21><loc_15><loc_83><loc_20><loc_84></location>(Evol'</text>
<text><location><page_21><loc_15><loc_79><loc_20><loc_81></location>(Evol'</text>
<text><location><page_21><loc_15><loc_76><loc_20><loc_77></location>(Evol'</text>
<text><location><page_21><loc_15><loc_72><loc_20><loc_73></location>(Evol'</text>
<formula><location><page_21><loc_20><loc_82><loc_61><loc_85></location>r √ 1 + r 2 ∂ E r ∂t = -rglyph[epsilon1] AB / ∇ A H B , E r )</formula>
<formula><location><page_21><loc_20><loc_79><loc_72><loc_81></location>, E A )</formula>
<formula><location><page_21><loc_37><loc_78><loc_71><loc_82></location>r √ 1 + r 2 ∂ E A ∂t = -glyph[epsilon1] A B 2 [ r / ∇ B H r + H B √ 1 + r 2 ]</formula>
<formula><location><page_21><loc_20><loc_75><loc_59><loc_78></location>r ∂ H , H r )</formula>
<formula><location><page_21><loc_37><loc_74><loc_58><loc_77></location>√ 1 + r 2 r ∂t = rglyph[epsilon1] AB / ∇ A E B</formula>
<formula><location><page_21><loc_20><loc_71><loc_70><loc_74></location>r √ 1 + r 2 ∂ H A ∂t = glyph[epsilon1] A B 2 [ r / ∇ B E r + E B √ 1 + r 2 ] . H A )</formula>
<text><location><page_21><loc_15><loc_68><loc_64><loc_70></location>Now, once again this is a symmetric hyperbolic system. Taking</text>
<text><location><page_21><loc_15><loc_63><loc_84><loc_67></location>r 2 (1 + r 2 ) [ E r × (Evol' E r ) + 2 E A × (Evol' E A ) + H r × (Evol' H r ) + 2 H A × (Evol' H A )] and integrating over Σ t with the measure drdω we can derive:</text>
<formula><location><page_21><loc_27><loc_55><loc_73><loc_62></location>d dt 1 2 ∫ Σ t [ | E r | 2 +2 | E A | 2 + | H r | 2 +2 | H A | 2 ] r 3 √ 1 + r 2 drdω = -2 ∫ Σ t [ glyph[epsilon1] AB E A H B ] r 2 √ 1 + r 2 drdω</formula>
<text><location><page_21><loc_15><loc_51><loc_85><loc_54></location>Immediately, with Gronwall's Lemma, we deduce that the constraints, if initially satisfied, will be satisfied for all time.</text>
<text><location><page_21><loc_15><loc_46><loc_85><loc_51></location>To our knowledge, identifying the above modified system as the correct formulation to prove well-posedness goes back to Friedrich's work [10]. In particular, Theorem 4.3 above could be inferred from this paper.</text>
<section_header_level_1><location><page_21><loc_36><loc_42><loc_64><loc_44></location>5. Proof of the main theorems</section_header_level_1>
<text><location><page_21><loc_15><loc_38><loc_85><loc_41></location>5.1. Proof of Theorem 1.1 for Spin 0. The Killing field T immediately gives us a boundedness estimate:</text>
<text><location><page_21><loc_15><loc_34><loc_85><loc_37></location>Proposition 5.1 (Boundedness of energy) . Let u be a solution of (20) subject to dissipative boundary conditions (40) as in Theorem 4.1. Define the energy to be:</text>
<formula><location><page_21><loc_15><loc_29><loc_75><loc_33></location>(45) E t [ u ] := 1 2 ∫ Σ t ( ( ∂ t u ) 2 + u 2 1 + r 2 + ( 1 + r 2 ) ( ˜ ∂ r u ) 2 + ∣ ∣ / ∇ u ∣ ∣ 2 ) r 2 drdω.</formula>
<text><location><page_21><loc_15><loc_26><loc_39><loc_28></location>Then we have for any T 1 < T 2 :</text>
<formula><location><page_21><loc_28><loc_22><loc_72><loc_25></location>E T 2 [ u ] + 1 2 ∫ ˜ Σ [ T 1 ,T 2 ] ∞ ( r 2 ( ∂ t u ) 2 + r 6 ( ˜ ∂ r u ) 2 ) dωdt = E T 1 [ u ] .</formula>
<text><location><page_21><loc_15><loc_20><loc_31><loc_21></location>Proof. We have that</text>
<formula><location><page_21><loc_45><loc_18><loc_55><loc_19></location>Div ( T J ) = 0 .</formula>
<text><location><page_22><loc_15><loc_85><loc_85><loc_88></location>Integrating this over S [ T 1 ,T 2 ] we pick up terms from Σ T 1 , Σ T 2 and ˜ Σ [ T 1 ,T 2 ] ∞ . A straightforward calculation shows</text>
<formula><location><page_22><loc_41><loc_80><loc_59><loc_84></location>∫ Σ t T J a n a dS Σ t = E t [ u ]</formula>
<text><location><page_22><loc_15><loc_78><loc_25><loc_79></location>We also find</text>
<formula><location><page_22><loc_26><loc_65><loc_74><loc_76></location>∫ ˜ Σ [ T 1 ,T 2 ] r T J a m a dS ˜ Σ r = ∫ S 2 r 2 (1 + r 2 ) ( ∂ t u ) ( ˜ ∂ r u ) dωdt = -1 2 ∫ S 2 [ ( r 2 ( ∂ t u ) 2 + r 2 (1 + r 2 ) 2 ( ˜ ∂ r u ) 2 ) -{ ∂ t ( ru ) + r (1 + r 2 )( ˜ ∂ r u ) } 2 ] dωdt.</formula>
<text><location><page_22><loc_15><loc_61><loc_76><loc_62></location>As r →∞ , the term in braces vanishes by the boundary condition and we find</text>
<formula><location><page_22><loc_24><loc_56><loc_76><loc_59></location>lim r →∞ ∫ ˜ Σ [ T 1 ,T 2 ] r T J a m a dS ˜ Σ r = -1 2 ∫ ˜ Σ [ T 1 ,T 2 ] ∞ ( r 2 ( ∂ t u ) 2 + r 6 ( ˜ ∂ r u ) 2 ) dωdt.</formula>
<text><location><page_22><loc_15><loc_53><loc_42><loc_54></location>Applying Lemma 2.1, we are done.</text>
<text><location><page_22><loc_84><loc_53><loc_85><loc_54></location>glyph[square]</text>
<text><location><page_22><loc_17><loc_49><loc_79><loc_50></location>We next show an integrated decay estimate with a loss in the weight at infinity:</text>
<text><location><page_22><loc_15><loc_44><loc_85><loc_47></location>Proposition 5.2 (Integrate decay estimate with loss) . Let T 2 > T 1 . Suppose u is a solution of (20) subject to (40) as in Theorem 4.1. Then the estimate</text>
<formula><location><page_22><loc_20><loc_34><loc_80><loc_43></location>∫ S [ T 1 ,T 2 ] ( ( ∂ t u ) 2 + u 2 1 + r 2 + ( 1 + r 2 ) ( ˜ ∂ r u ) 2 + ∣ ∣ / ∇ u ∣ ∣ 2 ) r 2 √ 1 + r 2 drdωdt + ∫ ˜ Σ [ T 1 ,T 2 ] ∞ ( r 2 ( ∂ t u ) 2 + ∣ ∣ r 2 / ∇ u ∣ ∣ 2 + r 6 ( ˜ ∂ r u ) 2 ) dωdt ≤ CE T 1 [ u ]</formula>
<text><location><page_22><loc_15><loc_32><loc_60><loc_33></location>holds for some constant C > 0 , independent of T 1 and T 2 .</text>
<text><location><page_22><loc_15><loc_27><loc_85><loc_30></location>Proof. We integrate a current constructed from the (renormalized) energy momentum tensor (21) and a radial vector field. The current is</text>
<formula><location><page_22><loc_33><loc_22><loc_67><loc_26></location>J a X,w 1 ,w 2 = X J a + w 1 √ 1 + r 2 u ˜ ∇ a u + w 2 u 2 X a ,</formula>
<text><location><page_22><loc_15><loc_18><loc_85><loc_21></location>where X is the radial vector field defined in (17). The proof of the theorem is a straightforward corollary of the following two Lemmas.</text>
<section_header_level_1><location><page_23><loc_15><loc_86><loc_33><loc_88></location>Lemma 5.1. We have</section_header_level_1>
<formula><location><page_23><loc_25><loc_74><loc_75><loc_86></location>Div J X,w 1 ,w 2 = ( T J · e 0 ) -( X b ˜ ∇ b 1 -√ 1 + r 2 + w 1 √ 1 + r 2 ) T c c + ( ∇ a ( w 1 √ 1 + r 2 ) +2 w 2 X a ) u ˜ ∇ a u + ( (1 + r 2 ) Div ( w 2 1 + r 2 X ) -w 1 (1 + r 2 ) 3 2 ) u 2 .</formula>
<text><location><page_23><loc_15><loc_71><loc_65><loc_73></location>In particular, if we take w 1 = 1 , w 2 = 1 2 (1 + r 2 ) -1 then we have</text>
<formula><location><page_23><loc_35><loc_67><loc_65><loc_71></location>Div J X,w 1 ,w 2 = ( T J · e 0 ) + u 2 2(1 + r 2 ) 3 2 .</formula>
<text><location><page_23><loc_15><loc_65><loc_57><loc_66></location>Proof. The vector field X has the deformation tensor:</text>
<formula><location><page_23><loc_39><loc_60><loc_61><loc_64></location>X π = ( e 0 ) 2 √ 1 + r 2 + g √ 1 + r 2 ,</formula>
<text><location><page_23><loc_15><loc_58><loc_20><loc_59></location>so that</text>
<formula><location><page_23><loc_32><loc_56><loc_68><loc_58></location>Div X J = ( T J · e 0 ) -( X b ˜ ∇ b 1 -√ 1 + r 2 ) T c c .</formula>
<text><location><page_23><loc_15><loc_54><loc_43><loc_55></location>We also require the observation that</text>
<formula><location><page_23><loc_31><loc_49><loc_69><loc_53></location>√ 1 + r 2 ∇ a ( ˜ ∇ a u √ 1 + r 2 ) = ( glyph[square] g u +2 u ) + u 1 + r 2 .</formula>
<text><location><page_23><loc_15><loc_47><loc_61><loc_48></location>Thus when u solves the conformal wave equation, we have:</text>
<formula><location><page_23><loc_21><loc_42><loc_79><loc_46></location>Div ( w 1 √ 1 + r 2 u ˜ ∇ a u ) = -w 1 √ 1 + r 2 T c c + ∇ a ( w 1 √ 1 + r 2 ) u ˜ ∇ a u -u 2 w 1 (1 + r 2 ) 3 2 .</formula>
<text><location><page_23><loc_15><loc_40><loc_28><loc_41></location>Finally, we have</text>
<formula><location><page_23><loc_27><loc_36><loc_73><loc_39></location>Div ( w 2 u 2 X ) = 2 w 2 uX a ˜ ∇ a u +(1 + r 2 )Div ( w 2 1 + r 2 X ) u 2 .</formula>
<text><location><page_23><loc_15><loc_30><loc_85><loc_35></location>Combining these identities we have the first part of the result. We can arrange that the term proportional to T c c vanishes by taking w 1 = 1. The term proportional to u ˜ ∇ a u vanishes if w 2 = 1 2 (1 + r 2 ) -1 , and the final part of the result follows from a brief calculation. glyph[square]</text>
<text><location><page_23><loc_15><loc_28><loc_85><loc_29></location>Lemma 5.2. With w 1 , w 2 chosen as in the second part of the previous Lemma, we have:</text>
<formula><location><page_23><loc_32><loc_23><loc_68><loc_27></location>∣ ∣ ∣ ∣ ∣ ∫ Σ T 2 J a X,w 1 ,w 2 n a dS Σ T 2 ∣ ∣ ∣ ∣ ∣ ≤ CE T 2 [ u ] ≤ CE T 1 [ u ]</formula>
<text><location><page_23><loc_15><loc_21><loc_18><loc_22></location>and</text>
<formula><location><page_23><loc_24><loc_17><loc_76><loc_21></location>lim r →∞ ∫ ˜ Σ [ T 1 ,T 2 ] r J a X,w 1 ,w 2 m a dS ˜ Σ r + 1 2 ∫ ˜ Σ [ T 1 ,T 2 ] ∞ ∣ ∣ r 2 / ∇ u ∣ ∣ 2 dωdt ≤ CE T 1 [ u ] .</formula>
<text><location><page_24><loc_15><loc_86><loc_30><loc_88></location>Proof. We calculate</text>
<formula><location><page_24><loc_25><loc_82><loc_75><loc_86></location>∫ Σ t J a X,w 1 ,w 2 n a dS Σ t = -∫ Σ t ∂ t u (1 + r 2 ) 3 2 ( r (1 + r 2 ) ˜ ∂ r u + u ) r 2 drdω</formula>
<text><location><page_24><loc_15><loc_78><loc_85><loc_81></location>which, after applying Cauchy-Schwarz, can certainly be controlled by the energy E t [ u ], which in turn is controlled by E 0 [ u ] using Theorem 5.1.</text>
<text><location><page_24><loc_17><loc_77><loc_49><loc_78></location>For the other surface terms, we calculate</text>
<formula><location><page_24><loc_23><loc_72><loc_76><loc_76></location>∫ ˜ Σ [ T 1 ,T 2 ] r J a X,w 1 ,w 2 m a dS ˜ Σ r = 1 2 ∫ ˜ Σ [ T 1 ,T 2 ] r [ r 2 ( ∂ t u ) 2 1 + r 2 + r 2 (1 + r 2 )( ˜ ∂ r u ) 2 +</formula>
<formula><location><page_24><loc_48><loc_69><loc_76><loc_71></location>+2 ru ( ˜ ∂ r u ) -∣ ∣ r / ∇ u ∣ ∣ 2 ] r √ 1 + r 2 dtdω</formula>
<text><location><page_24><loc_15><loc_65><loc_20><loc_67></location>so that</text>
<formula><location><page_24><loc_19><loc_61><loc_81><loc_65></location>lim r →∞ ∫ ˜ Σ [ T 1 ,T 2 ] r J a X,w 1 ,w 2 m a dS ˜ Σ r = 1 2 ∫ ˜ Σ [ T 1 ,T 2 ] ∞ ( r 2 ( ∂ t u ) 2 -∣ ∣ r 2 / ∇ u ∣ ∣ 2 + r 6 ( ˜ ∂ r u ) 2 ) dωdt</formula>
<text><location><page_24><loc_15><loc_58><loc_85><loc_61></location>and the result follows since we already control the time and radial derivatives of u on the boundary by Theorem 5.1. glyph[square]</text>
<text><location><page_24><loc_15><loc_54><loc_85><loc_57></location>This concludes the proof of Proposition 5.2, and establishes the claimed degenerate integrated decay without derivative loss result. glyph[square]</text>
<text><location><page_24><loc_15><loc_50><loc_85><loc_53></location>We next improve the radial weight of the spacetime term in the integrated decay estimate, at the expense of losing a derivative. 15</text>
<text><location><page_24><loc_15><loc_46><loc_85><loc_49></location>Proposition 5.3 (Higher order estimates) . Let u be a solution of (20) subject to (40) as in Theorem 4.1. Then the estimate</text>
<formula><location><page_24><loc_20><loc_37><loc_81><loc_45></location>∫ S [ T 1 ,T 2 ] (1 + r 2 ) 3 2 [ [ ˜ ∂ r ( r 2 ˜ ∂ r u )] 2 + r 2 1 + r 2 ( ∣ ∣ ∣ ˜ ∂ r [ r / ∇ u ] ∣ ∣ ∣ 2 + ∣ ∣ ∣ ˜ ∂ r [ ∂ t u ] ∣ ∣ ∣ 2 )] drdωdt + ∫ S [ T 1 ,T 2 ] r 4 √ 1 + r 2 | / ∇ 2 u | 2 drdωdt ≤ C ( E T 1 [ u ] + E T 1 [ u t ])</formula>
<text><location><page_24><loc_15><loc_35><loc_60><loc_36></location>holds for some constant C > 0 , independent of T 1 and T 2 .</text>
<text><location><page_24><loc_15><loc_32><loc_68><loc_34></location>Proof. Let us define, for a solution of the conformal wave equation:</text>
<formula><location><page_24><loc_26><loc_28><loc_74><loc_33></location>Lu := 1 1 + r 2 ( u tt + u ) = √ 1 + r 2 r 2 ∂ r ( r 2 ∂ r ( u √ 1 + r 2 )) + / ∆ u.</formula>
<text><location><page_24><loc_15><loc_26><loc_63><loc_28></location>By commuting with T and applying Proposition 5.2, we have</text>
<formula><location><page_24><loc_15><loc_22><loc_71><loc_26></location>(46) ∫ S [ T 1 ,T 2 ] { [ Lu ] 2 r 2 √ 1 + r 2 } dη ≤ C ( E T 1 [ u ] + E T 1 [ u t ]) .</formula>
<text><location><page_25><loc_15><loc_86><loc_44><loc_88></location>We can expand the integrand to give</text>
<formula><location><page_25><loc_15><loc_77><loc_80><loc_86></location>[ Lu ] 2 r 2 √ 1 + r 2 = r 2 √ 1 + r 2 {[ √ 1 + r 2 r 2 ∂ r ( r 2 ∂ r ( u √ 1 + r 2 )) ] 2 + [ / ∆ u ] 2 +2 [ √ 1 + r 2 r 2 ∂ r ( r 2 ∂ r ( u √ 1 + r 2 )) ] [ / ∇ A / ∇ A u ] } (47)</formula>
<text><location><page_25><loc_15><loc_70><loc_85><loc_75></location>We clearly have two terms with a good sign and a cross term. To deal with the cross term, we integrate by parts twice, so that we obtain a term (with a good sign) that looks like ∣ ∣ ∂ r / ∇ u ∣ ∣ 2 and some lower order terms. More explicitly, we have</text>
<text><location><page_25><loc_15><loc_67><loc_52><loc_69></location>Lemma 5.3. Let K be the vector field given by</text>
<formula><location><page_25><loc_17><loc_63><loc_83><loc_67></location>K = ∂ r ( r 2 ∂ r ( u √ 1 + r 2 )) ( / ∇ A u ) e A -( r / ∇ A u ) ˜ ∂ r ( r / ∇ A u ) e r -r 2(1 + r 2 ) ∣ ∣ r / ∇ u ∣ ∣ 2 e r .</formula>
<text><location><page_25><loc_15><loc_61><loc_26><loc_62></location>Then we have</text>
<formula><location><page_25><loc_19><loc_56><loc_81><loc_60></location>∂ r ( r 2 ∂ r ( u √ 1 + r 2 ))[ / ∇ A / ∇ A u ] = √ 1 + r 2 ∣ ∣ ∣ ˜ ∂ r ( r / ∇ u ) ∣ ∣ ∣ 2 + 3 ∣ ∣ r / ∇ u ∣ ∣ 2 2(1 + r 2 ) 3 2 + Div K.</formula>
<text><location><page_25><loc_15><loc_53><loc_35><loc_55></location>Proof. See Appendix 7.1.</text>
<text><location><page_25><loc_84><loc_54><loc_85><loc_55></location>glyph[square]</text>
<text><location><page_25><loc_15><loc_49><loc_85><loc_52></location>To prove Proposition 5.3, we simply insert (47) into (46) and handle the cross-term using Lemma 5.3. The boundary terms coming from Div K are</text>
<formula><location><page_25><loc_33><loc_44><loc_67><loc_48></location>∫ S [ T 1 ,T 2 ] Div Kdη = -1 2 ∫ ˜ Σ [ T 1 ,T 2 ] ∞ ∣ ∣ r 2 / ∇ u ∣ ∣ 2 dωdt</formula>
<text><location><page_25><loc_15><loc_42><loc_56><loc_43></location>which we control by the estimate of Proposition 5.2.</text>
<text><location><page_25><loc_15><loc_38><loc_85><loc_41></location>This in particular controls the term [ / ∆ u ] 2 . Since S 2 has constant positive Gauss curvature, we have the following elliptic estimates (see for instance Corollary 2.2.2.1 in [1]):</text>
<formula><location><page_25><loc_40><loc_34><loc_60><loc_37></location>∫ S 2 | / ∇ 2 u | 2 ≤ C ∫ S 2 [ / ∆ u ] 2</formula>
<text><location><page_25><loc_15><loc_31><loc_56><loc_33></location>from which we obtain the desired bounds for | / ∇ 2 u | 2 .</text>
<text><location><page_25><loc_15><loc_27><loc_85><loc_31></location>Finally, the ( ˜ ∂ r ∂ t u ) 2 term appearing in Proposition 5.3 is directly controlled by the T -commuted version of the estimates in Proposition 5.2.</text>
<text><location><page_25><loc_84><loc_26><loc_85><loc_27></location>glyph[square]</text>
<text><location><page_25><loc_15><loc_18><loc_85><loc_24></location>We finally improve the weight in the spacetime term of Proposition 5.2 making use of the fact that by Proposition 5.3 we now control radial derivatives of ∂ t u, / ∇ u and ˜ ∂ r u which lead to improved zeroth order terms through a Hardy inequality. This is a standard result, but for convenience we include here a proof.</text>
<text><location><page_26><loc_48><loc_86><loc_48><loc_88></location>glyph[negationslash]</text>
<text><location><page_26><loc_15><loc_85><loc_85><loc_88></location>Lemma 5.4 (Hardy's inequality) . Fix a = 0 . Suppose that f : [1 , ∞ ) → R is smooth, f (1) = 0 and | f ( r ) | 2 r -a → 0 as r →∞ . Then we have the estimate</text>
<formula><location><page_26><loc_34><loc_80><loc_66><loc_84></location>∫ ∞ 1 | f | 2 r -1 -a dr ≤ 4 a 2 ∫ ∞ 1 | ∂ r f | 2 r 1 -a dr,</formula>
<text><location><page_26><loc_15><loc_78><loc_44><loc_79></location>provided the right hand side is finite.</text>
<text><location><page_26><loc_15><loc_75><loc_28><loc_76></location>Proof. We write</text>
<formula><location><page_26><loc_28><loc_63><loc_71><loc_74></location>∫ ∞ 1 | f | 2 r -1 -a dr = ∫ ∞ 1 f 2 d dr ( -r -a a ) dr = -1 a [ | f | 2 r -a ] ∞ 1 + 2 a ∫ ∞ 1 f∂ r fr -a dr = 2 a ∫ ∞ 1 f∂ r fr -a dr</formula>
<text><location><page_26><loc_15><loc_59><loc_85><loc_62></location>Here we have used f (1) = 0 and the fact that lim r →∞ r -a | f ( r ) | 2 = 0 to discard the boundary terms. Now applying Cauchy-Schwarz, we deduce</text>
<formula><location><page_26><loc_25><loc_54><loc_75><loc_58></location>∫ ∞ 1 | f | 2 r -1 -a dr ≤ ( 4 a 2 ∫ ∞ 1 | f | 2 r -1 -a dr ∫ ∞ 1 | ∂ r f | 2 r 1 -a dr ) 1 2 ,</formula>
<text><location><page_26><loc_15><loc_51><loc_35><loc_53></location>whence the result follows.</text>
<formula><location><page_26><loc_84><loc_52><loc_85><loc_53></location>glyph[square]</formula>
<text><location><page_26><loc_17><loc_48><loc_41><loc_50></location>From Lemma 5.4 we establish:</text>
<text><location><page_26><loc_15><loc_46><loc_85><loc_47></location>Theorem 5.1 (Full integrated decay) . Let u be a smooth function such that | ru | is bounded.</text>
<text><location><page_26><loc_15><loc_44><loc_29><loc_45></location>Then the estimate</text>
<formula><location><page_26><loc_17><loc_29><loc_83><loc_43></location>∫ S [ T 1 ,T 2 ] ( ( ∂ t u ) 2 + u 2 1 + r 2 + ( 1 + r 2 ) ( ˜ ∂ r u ) 2 + ∣ ∣ / ∇ u ∣ ∣ 2 ) r 2 drdωdt ≤ C [ ∫ S [ T 1 ,T 2 ] (1 + r 2 ) 3 2 [ [ ˜ ∂ r ( r 2 ˜ ∂ r u )] 2 + r 2 1 + r 2 ( ∣ ∣ ∣ ˜ ∂ r [ r / ∇ u ] ∣ ∣ ∣ 2 + ∣ ∣ ∣ ˜ ∂ r [ ∂ t u ] ∣ ∣ ∣ 2 )] drdωdt + ∫ S [ T 1 ,T 2 ] ( ( ∂ t u ) 2 + u 2 1 + r 2 + ( 1 + r 2 ) ( ˜ ∂ r u ) 2 + ∣ ∣ / ∇ u ∣ ∣ 2 ) r 2 √ 1 + r 2 drdωdt ] .</formula>
<text><location><page_26><loc_15><loc_24><loc_85><loc_28></location>holds for some constant C > 0 independent of T 1 and T 2 . If, moreover, u solves (20) subject to (40), then the right-hand side may be bounded by C ' ( E T 1 [ u ] + E T 1 [ ∂ t u ]) for some C ' > 0 independent of T 1 and T 2 .</text>
<text><location><page_26><loc_15><loc_18><loc_85><loc_22></location>Proof. By introducing a cut-off we can quickly reduce to showing that the estimate holds for u supported either on r ≤ 2 or on r ≥ 1. For u supported on r ≤ 2, the estimate follows immediately, since the first order terms on the right hand side are comparable to those on</text>
<text><location><page_27><loc_15><loc_85><loc_85><loc_88></location>the left hand side on any finite region. For u supported on r ≥ 1, we first apply Lemma 5.4 with f = r 2 √ 1 + r 2 ˜ ∂ r u and a = 1 to deduce</text>
<formula><location><page_27><loc_21><loc_71><loc_79><loc_83></location>∫ S [ T 1 ,T 2 ] ∣ ∣ ∣ ˜ ∂ r u ∣ ∣ ∣ 2 r 2 (1 + r 2 ) drdωdt ≤ 4 ∫ S [ T 1 ,T 2 ] [ ∂ r ( r 2 √ 1 + r 2 ˜ ∂ r u ) ] 2 drdωdt ≤ 4 ∫ S [ T 1 ,T 2 ] (1 + r 2 ) [ ˜ ∂ r ( r 2 ˜ ∂ r u )] 2 drdωdt ≤ 4 ∫ S [ T 1 ,T 2 ] (1 + r 2 ) 3 2 [ ˜ ∂ r ( r 2 ˜ ∂ r u )] 2 drdωdt.</formula>
<text><location><page_27><loc_15><loc_68><loc_71><loc_70></location>Similarly, applying Lemma 5.4 to f = √ 1 + r 2 ∂ t u with a = 1 we deduce</text>
<formula><location><page_27><loc_24><loc_54><loc_76><loc_66></location>∫ S [ T 1 ,T 2 ] | ∂ t u | 2 r 2 1 + r 2 drdωdt ≤ ∫ S [ T 1 ,T 2 ] | ∂ t u | 2 r -2 (1 + r 2 ) drdω ≤ 4 ∫ S [ T 1 ,T 2 ] ∣ ∣ ∣ ˜ ∂ r [ ∂ t u ] ∣ ∣ ∣ 2 (1 + r 2 ) drdωdt ≤ 8 ∫ S [ T 1 ,T 2 ] ∣ ∣ ∣ ˜ ∂ r [ ∂ t u ] ∣ ∣ ∣ 2 r 2 √ 1 + r 2 drdωdt,</formula>
<text><location><page_27><loc_15><loc_48><loc_85><loc_53></location>where we've used that u is supported on r ≥ 1. A similar calculation gives the estimate for / ∇ u . Finally, making use of Propositions 5.2, 5.3, we see that if u satisfies the equation, then we can bound the right hand side in terms of E T 1 [ u ] + E T 1 [ ∂ t u ]. glyph[square]</text>
<text><location><page_27><loc_15><loc_43><loc_85><loc_45></location>Combining Proposition 5.3 with Theorem 5.1 we have established the claimed nondegenerate integrated decay with derivative loss result for the wave equation.</text>
<text><location><page_27><loc_15><loc_33><loc_85><loc_39></location>5.2. Proof of Theorem 1.1 for Spin 1. The proof of the theorem for the Maxwell field follows a similar pattern to that of the conformal scalar field. There is a simplification owing to the fact that the energy-momentum tensor is trace-free, and the elliptic estimate takes a slightly different form.</text>
<text><location><page_27><loc_15><loc_27><loc_85><loc_31></location>Proposition 5.4 (Boundedness of energy) . Let T 2 > T 1 . Suppose that F is a solution of Maxwell's equations (25-30), subject to the dissipative boundary conditions (41) as in Theorem 4.2. Then we have:</text>
<formula><location><page_27><loc_27><loc_17><loc_76><loc_25></location>∫ Σ T 2 ( | E | 2 + | H | 2 ) r 2 drdω + ∫ ˜ Σ [ T 1 ,T 2 ] ∞ ( ∣ ∣ r 2 E A ∣ ∣ 2 + ∣ ∣ r 2 H A ∣ ∣ 2 ) dtdω = ∫ Σ T 1 ( | E | 2 + | H | 2 ) r 2 drdω.</formula>
<text><location><page_28><loc_15><loc_85><loc_85><loc_88></location>Proof. We apply Lemma 2.1 to the vector field J T [ F ] a = T [ F ] ab T b . Consider the term on I . We have</text>
<formula><location><page_28><loc_24><loc_68><loc_76><loc_83></location>lim r →∞ ∫ ˜ Σ [ T 1 ,T 2 ] r T J a m a dS ˜ Σ r = lim r →∞ ∫ ˜ Σ [ T 1 ,T 2 ] r r 2 (1 + r 2 ) glyph[epsilon1] AB E A H B dtdω = lim r →∞ ∫ ˜ Σ [ T 1 ,T 2 ] r r 4 glyph[epsilon1] AB ( -glyph[epsilon1] A C H C ) H B dtdω = -lim r →∞ ∫ ˜ Σ [ T 1 ,T 2 ] r ∣ ∣ r 2 H A ∣ ∣ 2 dtdω = -1 2 ∫ ˜ Σ [ T 1 ,T 2 ] ∞ ( ∣ ∣ r 2 E A ∣ ∣ 2 + ∣ ∣ r 2 H A ∣ ∣ 2 ) dtdω.</formula>
<text><location><page_28><loc_15><loc_64><loc_85><loc_67></location>Here we have used the dissipative boundary conditions (41). Since ∇ a T J a = 0, there is no bulk term, and a simple calculation gives</text>
<formula><location><page_28><loc_32><loc_59><loc_68><loc_62></location>∫ Σ t T J a n a dS Σ T 2 = 1 2 ∫ Σ t ( | E | 2 + | H | 2 ) r 2 drdω,</formula>
<text><location><page_28><loc_15><loc_56><loc_36><loc_57></location>which completes the proof.</text>
<text><location><page_28><loc_84><loc_56><loc_85><loc_57></location>glyph[square]</text>
<text><location><page_28><loc_17><loc_52><loc_79><loc_53></location>We next show an integrated decay estimate with a loss in the weight at infinity:</text>
<text><location><page_28><loc_15><loc_45><loc_85><loc_50></location>Proposition 5.5 (Integrated decay estimate with loss) . Let T 2 > T 1 . Suppose that F is a solution of Maxwell's equations (25-30), subject to the dissipative boundary conditions (41) as in Theorem 4.2. Then we have:</text>
<formula><location><page_28><loc_23><loc_36><loc_80><loc_44></location>∫ S [ T 1 ,T 2 ] ( | E | 2 + | H | 2 ) r 2 √ 1 + r 2 drdωdt + ∫ ˜ Σ [ T 1 ,T 2 ] ∞ ( ∣ ∣ r 2 E ∣ ∣ 2 + ∣ ∣ r 2 H ∣ ∣ 2 ) dtdω ≤ 3 ∫ Σ T 1 ( | E | 2 + | H | 2 ) r 2 drdω.</formula>
<text><location><page_28><loc_15><loc_29><loc_85><loc_34></location>Proof. We apply the divergence theorem to integrate the current X J a = T a b X b over S [ T 1 ,T 2 ] . Recalling the expression (18) for X π , and noting the fact that the energy-momentum tensor is traceless, we have</text>
<formula><location><page_28><loc_37><loc_18><loc_63><loc_28></location>Div X J = X π ab T ab = T 00 √ 1 + r 2 + T a a √ 1 + r 2 = | E | 2 + | H | 2 √ 1 + r 2 .</formula>
<text><location><page_29><loc_15><loc_86><loc_64><loc_88></location>Now consider the flux through a spacelike surface Σ t . We have</text>
<formula><location><page_29><loc_32><loc_78><loc_68><loc_85></location>∫ Σ t X J a n a dS Σ t = ∫ Σ t T 0 r r 3 √ 1 + r 2 drdω = ∫ Σ t glyph[epsilon1] AB E A H B r 3 √ 1 + r 2 drdω</formula>
<text><location><page_29><loc_15><loc_75><loc_53><loc_77></location>so that if t ∈ [ T 1 , T 2 ] we have by Proposition 5.4:</text>
<formula><location><page_29><loc_19><loc_70><loc_81><loc_74></location>∣ ∣ ∣ ∣ ∫ Σ t X J a n a dS Σ t ∣ ∣ ∣ ∣ ≤ 1 2 ∫ Σ t ( | E | 2 + | H | 2 ) r 2 drdω ≤ 1 2 ∫ Σ T 1 ( | E | 2 + | H | 2 ) r 2 drdω.</formula>
<text><location><page_29><loc_15><loc_68><loc_66><loc_70></location>Next consider the flux through a surface of constant r . We have:</text>
<formula><location><page_29><loc_19><loc_60><loc_81><loc_67></location>∫ ˜ Σ [ T 1 ,T 2 ] r X J a m a dS ˜ Σ r = ∫ ˜ Σ [ T 1 ,T 2 ] r T rr r 3 √ 1 + r 2 dtdω = ∫ ˜ Σ [ T 1 ,T 2 ] r ( -| E r | 2 -| H r | 2 + | E A | 2 + | H A | 2 ) r 3 √ 1 + r 2 dtdω</formula>
<text><location><page_29><loc_15><loc_58><loc_20><loc_59></location>so that</text>
<formula><location><page_29><loc_26><loc_49><loc_74><loc_57></location>lim r →∞ ∫ ˜ Σ [ T 1 ,T 2 ] r X J a m a dS ˜ Σ r ≤ -∫ ˜ Σ [ T 1 ,T 2 ] ∞ ( ∣ ∣ r 2 E ∣ ∣ 2 + ∣ ∣ r 2 H ∣ ∣ 2 ) dtdω +2 ∫ Σ T 1 ( | E | 2 + | H | 2 ) r 2 drdω,</formula>
<text><location><page_29><loc_15><loc_44><loc_85><loc_48></location>where we use Proposition 5.4 to control the angular components of E,H at infinity. Integrating Div X J over S [ T 1 ,T 2 ] , applying the divergence theorem and using the estimates above for the fluxes completes the proof of the Proposition. glyph[square]</text>
<text><location><page_29><loc_15><loc_38><loc_85><loc_42></location>We next seek to improve the weight at infinity at the expense of a derivative loss. The first stage in doing this is an elliptic estimate. First note that by commuting the equations with the Killing field T we have:</text>
<formula><location><page_29><loc_15><loc_29><loc_83><loc_37></location>∫ S [ T 1 ,T 2 ] ( | ∂ t E | 2 + | ∂ t H | 2 ) r 2 √ 1 + r 2 drdωdt + ∫ ˜ Σ [ T 1 ,T 2 ] ∞ ( ∣ ∣ r 2 ∂ t E ∣ ∣ 2 + ∣ ∣ r 2 ∂ t H ∣ ∣ 2 ) dtdω (48) ≤ 3 ∫ Σ T 1 ( | ∂ t E | 2 + | ∂ t H | 2 ) r 2 drdω. (49)</formula>
<text><location><page_29><loc_15><loc_24><loc_85><loc_28></location>We can use the evolution equations to control the right hand side in terms of spatial derivatives in the slice Σ T 1 if we choose. We will require the following Lemma:</text>
<text><location><page_29><loc_15><loc_22><loc_45><loc_23></location>Lemma 5.5. Let K be the vector field</text>
<formula><location><page_29><loc_22><loc_17><loc_78><loc_21></location>K := 2 r √ 1 + r 2 H r [ ( r / ∇ A H A ) e r -∂ ∂r ( r √ 1 + r 2 H A ) e A ] -r 3 1 + r 2 H 2 r e r .</formula>
<text><location><page_30><loc_15><loc_86><loc_68><loc_88></location>Then if H satisfies the constraint equation (30) we have the identity</text>
<formula><location><page_30><loc_18><loc_82><loc_46><loc_85></location>-2 √ 1 + r 2 ∂ r ( r √ 1 + r 2 H A ) r / ∇ A H r</formula>
<formula><location><page_30><loc_41><loc_78><loc_82><loc_83></location>= 2 √ 1 + r 2 r 2 ∣ ∣ ∣ ∣ ∂ ∂r ( r 2 H r ) ∣ ∣ ∣ ∣ 2 + r 2 (1 + r 2 ) 3 2 | H r | 2 + Div K</formula>
<text><location><page_30><loc_15><loc_75><loc_35><loc_76></location>Proof. See Appendix 7.2.</text>
<text><location><page_30><loc_17><loc_72><loc_49><loc_74></location>Now consider the Maxwell equation (26):</text>
<formula><location><page_30><loc_30><loc_68><loc_70><loc_71></location>r √ 1 + r 2 ∂ t E A = glyph[epsilon1] A B [ ∂ r ( r √ 1 + r 2 H B ) -r / ∇ B H r ] ,</formula>
<text><location><page_30><loc_15><loc_65><loc_57><loc_67></location>squaring this and multiplying by (1 + r 2 ) -1 2 , we have</text>
<formula><location><page_30><loc_15><loc_56><loc_74><loc_64></location>r 2 | ∂ t E A | 2 (1 + r 2 ) 3 2 = 1 √ 1 + r 2 [ ∣ ∣ r / ∇ A H r ∣ ∣ 2 + ∣ ∣ ∣ ∣ ∂ ∂r ( r √ 1 + r 2 H A ) ∣ ∣ ∣ ∣ 2 ] -2 √ 1 + r 2 ∂ r ( r √ 1 + r 2 H A ) r / ∇ A H r . (50)</formula>
<text><location><page_30><loc_17><loc_54><loc_41><loc_55></location>From here we readily conclude:</text>
<text><location><page_30><loc_15><loc_48><loc_85><loc_53></location>Theorem 5.2 (Higher order estimates) . Suppose that F is a solution of Maxwell's equations, as in Theorem 4.2. Then there exists a constant C > 0 , independent of T 1 and T 2 such that we have</text>
<formula><location><page_30><loc_24><loc_34><loc_76><loc_47></location>∫ S [ T 1 ,T 2 ] { 1 √ 1 + r 2 [ ∣ ∣ r / ∇ A H r ∣ ∣ 2 + ∣ ∣ ∣ ∣ ∂ ∂r ( r √ 1 + r 2 H A ) ∣ ∣ ∣ ∣ 2 + ∣ ∣ r / ∇ A H B ∣ ∣ 2 ] +2 √ 1 + r 2 r 2 ∣ ∣ ∣ ∣ ∂ ∂r ( r 2 H r ) ∣ ∣ ∣ ∣ 2 + H ↔ E } r 2 dtdrdω ≤ C ∫ Σ T 1 ( | E | 2 + | H | 2 + | ∂ t E | 2 + | ∂ t H | 2 ) r 2 drdω.</formula>
<text><location><page_30><loc_15><loc_24><loc_85><loc_33></location>Proof. We integrate the identity (50) over S [ T 1 ,T 2 ] . We control the left hand side by the time commuted integrated decay estimate (48) since r 2 (1 + r 2 ) -1 ≤ 1. The right hand side, after making use of Lemma 5.5 and applying the divergence theorem will give us good derivative terms, a good zero'th order term which we can ignore and a surface term. The surface term at infinity gives 16 a term proportional to ∣ ∣ r 2 H r ∣ ∣ 2 , integrated over the cylinder, which we control with the estimate in Theorem 5.5. This immediately gives the</text>
<formula><location><page_30><loc_84><loc_76><loc_85><loc_76></location>glyph[square]</formula>
<text><location><page_31><loc_15><loc_85><loc_85><loc_88></location>result for all of the terms except the term ∣ ∣ r / ∇ A H B ∣ ∣ 2 , which we obtain from a standard elliptic estimate on the sphere (see for instance Proposition 2.2.1 in [1]) :</text>
<formula><location><page_31><loc_32><loc_80><loc_68><loc_83></location>∫ S 2 ∣ ∣ / ∇ A H B ∣ ∣ 2 ≤ C ∫ S 2 | glyph[epsilon1] AB / ∇ A H B | 2 + | / ∇ A H A | 2 ,</formula>
<text><location><page_31><loc_15><loc_74><loc_85><loc_79></location>after noticing that we already control glyph[epsilon1] AB / ∇ A H B and / ∇ A H A with a suitable weight from (25) and (30). Finally, we note that the estimate can be derived in an identical manner for E . glyph[square]</text>
<text><location><page_31><loc_15><loc_69><loc_85><loc_72></location>Theorem 5.3 (Full integrated decay) . Suppose that F is a solution of Maxwell's equations, as in Theorem 4.2. Then we have</text>
<formula><location><page_31><loc_18><loc_60><loc_86><loc_68></location>∫ S [ T 1 ,T 2 ] [ | E r | 2 + | H r | 2 + | E A | 2 + | H A | 2 ] r 2 dtdrdω ≤ C ∫ Σ T ( | E r | 2 + | E A | 2 + | H r | 2 + | H A | 2 + ∣ ∣ ∣ ˙ E r ∣ ∣ ∣ 2 + ∣ ∣ ∣ ˙ E A ∣ ∣ ∣ 2 + ∣ ∣ ∣ ˙ H r ∣ ∣ ∣ 2 + ∣ ∣ ∣ ˙ H A ∣ ∣ ∣ 2 ) r 2 drdω</formula>
<formula><location><page_31><loc_28><loc_60><loc_29><loc_60></location>1</formula>
<text><location><page_31><loc_15><loc_57><loc_48><loc_58></location>for some C > 0 independent of T 1 and T 2 .</text>
<text><location><page_31><loc_15><loc_51><loc_85><loc_55></location>Proof. We combine the result of Theorem 5.2 with the Hardy estimates of Lemma 5.4 with a = 1 to improve the weights near infinity in the integrated decay estimates, making use of a cut-off in much the same way as for the spin 0 problem. glyph[square]</text>
<text><location><page_31><loc_15><loc_45><loc_85><loc_48></location>5.3. Proof of Theorem 1.1 for Spin 2. As a useful piece of notation we first introduce the trace-free part of E AB by</text>
<formula><location><page_31><loc_32><loc_41><loc_68><loc_44></location>ˆ E AB = E AB -1 2 δ AB E C C = E AB + 1 2 δ AB E rr ,</formula>
<text><location><page_31><loc_15><loc_36><loc_85><loc_39></location>and similarly for ˆ H AB . The boundary conditions of Theorem 4.3 may then be conveniently expressed as</text>
<formula><location><page_31><loc_33><loc_33><loc_67><loc_35></location>r 3 ( ˆ E AB + glyph[epsilon1] ( A C ˆ H B ) C ) → 0 , as r →∞ ,</formula>
<text><location><page_31><loc_15><loc_28><loc_85><loc_32></location>and we also have | E AB | 2 = | ˆ E AB | 2 + 1 2 | E ¯ r ¯ r | 2 . We will first prove the boundedness and the degenerate integrated decay statement of the main theorem. 17</text>
<text><location><page_31><loc_15><loc_24><loc_85><loc_27></location>Proposition 5.6 (Boundedness of energy and integrated decay estimate with loss) . Let T 2 > T 1 . Suppose E ab and H ab are solutions to the spin 2 equations subject to (42) as in</text>
<text><location><page_32><loc_15><loc_86><loc_84><loc_88></location>Theorem 4.3, there exists a constant C > 0 , independent of T 1 and T 2 such that we have</text>
<formula><location><page_32><loc_15><loc_82><loc_85><loc_85></location>∫ S [ T ,T ] { E ab E ab + H ab H ab } r 2 √ 1 + r 2 dtdrdω + ∫ Σ T 2 { E ab E ab + H ab H ab } r 2 (1 + r 2 ) drdω</formula>
<formula><location><page_32><loc_23><loc_74><loc_56><loc_81></location>+ ˜ Σ [ T 1 ,T 2 ] ∞ { E ab E ab + H ab H ab } r 6 dtdω ≤ C ∫ Σ T 1 { E ab E ab + H ab H ab } r 2 (1 + r 2 )</formula>
<formula><location><page_32><loc_15><loc_75><loc_61><loc_82></location>1 2 ∫ (51) drdω.</formula>
<text><location><page_32><loc_15><loc_71><loc_49><loc_73></location>Proof. Let us now introduce the vector field</text>
<formula><location><page_32><loc_44><loc_67><loc_56><loc_70></location>Y = T + 1 √ 3 X.</formula>
<text><location><page_32><loc_15><loc_65><loc_48><loc_66></location>We certainly have that Y is timelike, since</text>
<formula><location><page_32><loc_33><loc_61><loc_67><loc_64></location>g ( Y, Y ) = -(1 + r 2 ) + 1 3 r 2 = -1 -2 3 r 2 < 0 .</formula>
<text><location><page_32><loc_15><loc_58><loc_29><loc_60></location>Moreover, we have</text>
<formula><location><page_32><loc_37><loc_54><loc_63><loc_58></location>Y π = 1 √ 3 ( ( e 0 ) 2 √ 1 + r 2 + g √ 1 + r 2 )</formula>
<text><location><page_32><loc_15><loc_52><loc_54><loc_54></location>The current that we shall integrate over S [ T 1 ,T 2 ] is</text>
<formula><location><page_32><loc_43><loc_49><loc_57><loc_51></location>J a = Q a bcd Y b Y c Y d .</formula>
<text><location><page_32><loc_15><loc_39><loc_85><loc_48></location>Clearly, we have by Lemma 3.1 that the flux of J through a spacelike surface with respect to the future directed normal will be positive. Moreover, since Q is trace-free and divergence free, Div J will also be positive. To establish a combined energy and integrated decay estimate, we simply have to verify that the surface term on I has a definite sign (and check the weights appearing in the various integrals). We shall require some components of Q , which are summarised in the following Lemma:</text>
<text><location><page_32><loc_15><loc_34><loc_85><loc_37></location>Lemma 5.6 (Components of Q abcd ) . With respect to the orthonormal basis in which we work, we have</text>
<formula><location><page_32><loc_25><loc_23><loc_75><loc_33></location>Q 0000 = | E AB | 2 + | H AB | 2 +2 | E Ar | 2 +2 | H Ar | 2 + | E rr | 2 + | H rr | 2 Q 000 r = 2 ( E AC glyph[epsilon1] AB H B C + E Ar glyph[epsilon1] AB H Br ) Q 00 rr = | E AB | 2 + | H AB | 2 -| E rr | 2 -| H rr | 2 Q 0 rrr = 2 ( E AC glyph[epsilon1] AB H B C -E Ar glyph[epsilon1] AB H Br ) 2</formula>
<formula><location><page_32><loc_25><loc_22><loc_74><loc_24></location>Q rrrr = | E AB | 2 + | H AB | 2 -2 | E Ar | 2 -2 | H Ar | 2 + | E rr | 2 + | H rr |</formula>
<text><location><page_32><loc_15><loc_18><loc_85><loc_21></location>With our definition of the current J above, we apply the divergence theorem, Lemma 2.1. We now verify that all the terms have a definite sign.</text>
<section_header_level_1><location><page_33><loc_15><loc_86><loc_44><loc_88></location>a) Fluxes through Σ t We compute</section_header_level_1>
<formula><location><page_33><loc_32><loc_82><loc_68><loc_86></location>∫ Σ t J a n a dS Σ t = ∫ Σ t Q ( e 0 , Y, Y, Y ) r 2 √ 1 + r 2 drdω.</formula>
<text><location><page_33><loc_17><loc_79><loc_54><loc_81></location>Now, defining ˆ Y = (1 + 2 3 r 2 ) -1 2 Y , we have that</text>
<formula><location><page_33><loc_38><loc_74><loc_62><loc_79></location>-g ( e 0 , ˆ Y ) = √ 1 + r 2 1 + 2 3 r 2 ≤ √ 3 2 .</formula>
<text><location><page_33><loc_17><loc_72><loc_41><loc_74></location>so by Lemma 3.1 we deduce 18 :</text>
<formula><location><page_33><loc_41><loc_70><loc_59><loc_71></location>Q ( e 0 , ˆ Y , ˆ Y , ˆ Y ) ∼ Q 0000 ,</formula>
<text><location><page_33><loc_17><loc_67><loc_23><loc_69></location>so that</text>
<text><location><page_33><loc_17><loc_63><loc_25><loc_65></location>and hence</text>
<formula><location><page_33><loc_37><loc_65><loc_63><loc_67></location>Q ( e 0 , Y, Y, Y ) ∼ ( 1 + r 2 ) 3 2 Q 0000 ,</formula>
<formula><location><page_33><loc_30><loc_60><loc_70><loc_64></location>∫ Σ t J a n a dS Σ t ∼ ∫ Σ t ( | E ab | 2 + | H ab | 2 ) r 2 (1 + r 2 ) dr.</formula>
<section_header_level_1><location><page_33><loc_15><loc_58><loc_34><loc_60></location>b) Bulk term We have</section_header_level_1>
<formula><location><page_33><loc_29><loc_54><loc_71><loc_58></location>Div J = 3 Q abcd ( Y π ) ab Y c Y d = √ 3 √ 1 + r 2 Q ( e 0 , e 0 , Y, Y ) .</formula>
<text><location><page_33><loc_17><loc_52><loc_61><loc_53></location>Again applying Lemma 3.1 to ˆ Y and rescaling, we have</text>
<formula><location><page_33><loc_41><loc_49><loc_59><loc_51></location>Div J ∼ ( 1 + r 2 ) 1 2 Q 0000 .</formula>
<text><location><page_33><loc_17><loc_47><loc_33><loc_48></location>As a result, we have</text>
<formula><location><page_33><loc_26><loc_42><loc_74><loc_46></location>∫ S [ T 1 ,T 2 ] Div Jdη ∼ ∫ S [ T 1 ,T 2 ] ( | E ab | 2 + | H ab | 2 ) r 2 √ 1 + r 2 dtdrdω.</formula>
<unordered_list>
<list_item><location><page_33><loc_15><loc_40><loc_83><loc_41></location>c) Boundary term at I Finally, we consider the flux through surfaces ˜ Σ r . We have</list_item>
</unordered_list>
<formula><location><page_33><loc_30><loc_36><loc_70><loc_39></location>∫ ˜ Σ r J a m a dS ˜ Σ r = ∫ ˜ Σ r Q ( e r , Y, Y, Y ) r 2 √ 1 + r 2 dtdω,</formula>
<text><location><page_33><loc_17><loc_34><loc_64><loc_35></location>Now, inserting our expression for Y and expanding, we have</text>
<text><location><page_33><loc_21><loc_32><loc_22><loc_33></location>Q</text>
<text><location><page_33><loc_22><loc_32><loc_23><loc_33></location>(</text>
<text><location><page_33><loc_23><loc_32><loc_24><loc_33></location>e</text>
<text><location><page_33><loc_24><loc_31><loc_25><loc_32></location>r</text>
<text><location><page_33><loc_25><loc_32><loc_30><loc_33></location>, Y, Y, Y</text>
<text><location><page_33><loc_31><loc_32><loc_31><loc_33></location>)</text>
<formula><location><page_33><loc_20><loc_27><loc_80><loc_31></location>= ( 1 + r 2 ) 3 2 ( Q r 000 + √ 3 r √ 1 + r 2 Q rr 00 + r 2 1 + r 2 Q rrr 0 + 1 3 √ 3 r 3 (1 + r 2 ) 3 2 Q rrrr )</formula>
<text><location><page_33><loc_17><loc_25><loc_23><loc_26></location>so that</text>
<formula><location><page_33><loc_18><loc_21><loc_82><loc_24></location>lim r →∞ Q ( e r , Y, Y, Y ) r 2 √ 1 + r 2 = lim r →∞ r 6 ( Q r 000 + √ 3 Q rr 00 + Q rrr 0 + 1 3 √ 3 Q rrrr ) .</formula>
<text><location><page_34><loc_17><loc_86><loc_77><loc_88></location>Let us consider the first and third terms on the right hand side 19 . We have:</text>
<formula><location><page_34><loc_16><loc_84><loc_40><loc_86></location>Q r 000 + Q rrr 0 = 4 E AC glyph[epsilon1] AB H B C</formula>
<formula><location><page_34><loc_27><loc_78><loc_84><loc_83></location>= 2 ( E AB -1 2 δ AB E C C + glyph[epsilon1] ( A C H B ) C )( ˆ E AB -1 2 δ AB E D D + glyph[epsilon1] ( A D H B ) D ) -2 | E AB | 2 -2 | H AB | 2</formula>
<text><location><page_34><loc_17><loc_75><loc_23><loc_77></location>so that</text>
<formula><location><page_34><loc_28><loc_73><loc_72><loc_75></location>lim r →∞ r 6 ( Q r 000 + Q rrr 0 ) = -2 lim r →∞ r 6 ( | E AB | 2 + | H AB | 2 )</formula>
<text><location><page_34><loc_17><loc_69><loc_85><loc_72></location>where we make use of the boundary condition. Taking this together with the expressions for Q rr 00 , Q rrrr we have:</text>
<formula><location><page_34><loc_15><loc_60><loc_91><loc_68></location>lim r →∞ Q ( e r , Y, Y, Y ) r 2 √ 1 + r 2 = -lim r →∞ r 6 [ ( 2 -10 3 √ 3 ) ( | E AB | 2 + | H AB | 2 ) + 8 3 √ 3 ( | E rr | 2 + | H rr | 2 + 2 3 √ 3 ( | E Ar | 2 + | H Ar | 2 ) ] ,</formula>
<text><location><page_34><loc_17><loc_57><loc_23><loc_58></location>so that</text>
<formula><location><page_34><loc_27><loc_54><loc_73><loc_57></location>lim r →∞ Q ( e r , Y, Y, Y ) r 2 √ 1 + r 2 ∼ -lim r →∞ r 6 ( | E ab | 2 + | H ab | 2 )</formula>
<text><location><page_34><loc_17><loc_52><loc_20><loc_54></location>and</text>
<formula><location><page_34><loc_26><loc_49><loc_74><loc_52></location>∫ ˜ Σ [ T 1 ,T 2 ] ∞ J a m a dS ˜ Σ r ∼ -∫ ˜ Σ [ T 1 ,T 2 ] ∞ lim r →∞ r 6 ( | E ab | 2 + | H ab | 2 ) dtdω.</formula>
<text><location><page_34><loc_15><loc_47><loc_54><loc_48></location>Taking all of this together, we arrive at the result.</text>
<text><location><page_34><loc_84><loc_47><loc_85><loc_48></location>glyph[square]</text>
<text><location><page_34><loc_15><loc_33><loc_85><loc_46></location>As we did in the Maxwell case, we shall now use the structure of the equations to allow us to establish (weighted) integrated decay estimates for all derivatives of the fields E,H . To control time derivatives we can simply commute with the Killing field T and apply Proposition 5.6. To control spatial derivatives we replace the time derivatives by the equations of motion and integrate the resulting cross terms by parts making use also of the constraints equations. The remarkable fact is that in the process we only see spacetime terms with good signs and lower order surface terms that we already control by the estimate before commutation.</text>
<text><location><page_34><loc_15><loc_30><loc_85><loc_33></location>We will note the following useful result, which allows the cross term to be integrated by parts:</text>
<section_header_level_1><location><page_34><loc_15><loc_27><loc_45><loc_28></location>Lemma 5.7. Let K be the vector field</section_header_level_1>
<formula><location><page_34><loc_15><loc_23><loc_83><loc_26></location>(52) K := ( 2 r 3 / ∇ C H BC H B r -r 4 √ 1 + r 2 | H Br | 2 ) e r -2 r 2 H Br √ 1 + r 2 ∂ ∂r [ r (1 + r 2 ) H BC ] e C .</formula>
<formula><location><page_34><loc_91><loc_66><loc_92><loc_67></location>)</formula>
<text><location><page_35><loc_15><loc_86><loc_69><loc_88></location>If H satisfies the constraint equation (Con H B ), we have the identity</text>
<formula><location><page_35><loc_22><loc_82><loc_78><loc_86></location>Div K = -2 r 2 √ 1 + r 2 / ∇ C H Br ∂ ∂r [ r (1 + r 2 ) H B C ] -2 1 + r 2 r 3 ∣ ∣ ∂ r ( r 3 H Br )∣ ∣ 2 .</formula>
<text><location><page_35><loc_15><loc_80><loc_35><loc_81></location>Proof. See Appendix 7.3.</text>
<text><location><page_35><loc_84><loc_80><loc_85><loc_81></location>glyph[square]</text>
<text><location><page_35><loc_15><loc_74><loc_85><loc_79></location>Proposition 5.7 (Higher order estimates) . Let T 2 > T 1 . If E ab , H ab solve the spin 2 equations subject to the dissipative boundary condition (42) as in Theorem 4.3, then we have</text>
<formula><location><page_35><loc_19><loc_61><loc_82><loc_74></location>∫ S [ T 1 ,T 2 ] { r 3 1 + r 2 ∣ ∣ ∂ r ( r (1 + r 2 ) H BC ) ∣ ∣ 2 + 1 + r 2 r ∣ ∣ ∂ r ( r 3 H Br )∣ ∣ 2 +( H ↔ E ) } drdtdω + ∫ S [ T 1 ,T 2 ] { r 5 | / ∇ B H Cr | 2 + r 5 | / ∇ A H BC | 2 +( H ↔ E ) } drdtdω ≤ C ∫ Σ T { E ab E ab + H ab H ab + ˙ E ab ˙ E ab + ˙ H ab ˙ H ab } r 2 (1 + r 2 ) drdω,</formula>
<formula><location><page_35><loc_25><loc_61><loc_26><loc_61></location>1</formula>
<text><location><page_35><loc_15><loc_59><loc_48><loc_60></location>for some C > 0 independent of T 1 and T 2 .</text>
<text><location><page_35><loc_15><loc_56><loc_40><loc_58></location>Proof. Recall now (Evol' E AB ):</text>
<formula><location><page_35><loc_24><loc_52><loc_76><loc_56></location>r √ 1 + r 2 ∂E AB ∂t = glyph[epsilon1] ( A C [ 1 √ 1 + r 2 ∂ ∂r ( r (1 + r 2 ) H B ) C ) -r / ∇ | C | H B ) r ]</formula>
<text><location><page_35><loc_15><loc_50><loc_22><loc_52></location>Consider</text>
<formula><location><page_35><loc_31><loc_31><loc_69><loc_50></location>glyph[epsilon1] AB glyph[epsilon1] A C [ 1 √ 1 + r 2 ∂ ∂r ( r (1 + r 2 ) H BC ) -r / ∇ C H Br ] = 1 √ 1 + r 2 ∂ ∂r ( r (1 + r 2 ) H B B ) -r / ∇ B H Br = -1 √ 1 + r 2 ∂ ∂r ( r 3 (1 + r 2 ) r 2 H rr ) -r / ∇ B H Br = -√ 1 + r 2 r 2 ∂ r ( r 3 H rr ) -r / ∇ B H Br + 2 H rr √ 1 + r 2 = 2 H rr √ 1 + r 2 .</formula>
<text><location><page_35><loc_15><loc_27><loc_85><loc_30></location>Now, since for any 2 -tensor Z on S 2 we have Z [ AB ] = 1 2 glyph[epsilon1] AB ( glyph[epsilon1] CD Z CD ), we deduce that if the constraints hold then (Evol' E AB ) may be re-written:</text>
<formula><location><page_35><loc_19><loc_23><loc_81><loc_26></location>r √ 1 + r 2 ∂E AB ∂t = glyph[epsilon1] A C [ 1 √ 1 + r 2 ∂ ∂r ( r (1 + r 2 ) H BC ) -r / ∇ C H Br ] -glyph[epsilon1] AB √ 1 + r 2 H rr ,</formula>
<text><location><page_35><loc_15><loc_21><loc_29><loc_22></location>whence we deduce</text>
<formula><location><page_35><loc_17><loc_17><loc_83><loc_21></location>X AB := 1 √ 1 + r 2 ∂ ∂r ( r (1 + r 2 ) H BC ) -r / ∇ C H Br = glyph[epsilon1] A C r √ 1 + r 2 ∂E AB ∂t + δ BC √ 1 + r 2 H rr .</formula>
<text><location><page_36><loc_15><loc_85><loc_85><loc_88></location>Now, using the second equality and applying the estimates in Proposition 5.6 for H and the commuted quantity ˙ E , we can verify that</text>
<formula><location><page_36><loc_21><loc_72><loc_79><loc_84></location>∫ S [ T 1 ,T 2 ] X AB X AB r 3 dtdrdω ≤ C ∫ S [ T 1 ,T 2 ] { ∣ ∣ ∣ ˙ E AB ∣ ∣ ∣ 2 + | H rr | 2 } r 2 √ 1 + r 2 dtdrdω ≤ C ∫ Σ T 1 { E ab E ab + H ab H ab + ˙ E ab ˙ E ab + ˙ H ab ˙ H ab } r 2 (1 + r 2 ) drdω.</formula>
<text><location><page_36><loc_15><loc_70><loc_33><loc_71></location>We also have, however,</text>
<formula><location><page_36><loc_24><loc_47><loc_76><loc_69></location>∫ S [ T 1 ,T 2 ] X AB X AB r 3 dtdrdω = ∫ S [ T 1 ,T 2 ] { r 1 + r 2 ∣ ∣ ∂ r ( r (1 + r 2 ) H BC ) ∣ ∣ 2 + r 3 ∣ ∣ / ∇ B H Cr ∣ ∣ 2 -2 r 2 √ 1 + r 2 / ∇ C H Br ∂ ∂r [ r (1 + r 2 ) H B C ] } dη = ∫ S [ T 1 ,T 2 ] { r 1 + r 2 ∣ ∣ ∂ r ( r (1 + r 2 ) H BC ) ∣ ∣ 2 + r 3 ∣ ∣ / ∇ B H Cr ∣ ∣ 2 +2 1 + r 2 r 3 ∣ ∣ ∂ r ( r 3 H Br )∣ ∣ 2 } dη -∫ ˜ Σ [ T 1 ,T 2 ] ∞ ∣ ∣ r 3 H Br ∣ ∣ 2 dtdω,</formula>
<text><location><page_36><loc_15><loc_43><loc_85><loc_46></location>where in the last step, we have used the result of Lemma 5.7 to replace the cross term with a good derivative term and a surface term.</text>
<text><location><page_36><loc_15><loc_40><loc_85><loc_43></location>It remains to control the term | / ∇ A H BC | 2 . Notice that by (Evol' E Ar ) and (Con H B ), we have</text>
<formula><location><page_36><loc_16><loc_35><loc_84><loc_39></location>r 2 | glyph[epsilon1] BC / ∇ B H CA | 2 + r 2 | / ∇ B H AB | 2 ≤ C ( r 2 1 + r 2 | ˙ E Ar | 2 + 1 + r 2 r 4 ∣ ∣ ∣ ∣ ∂ ∂r ( r 3 H Ar ) ∣ ∣ ∣ ∣ 2 + | H Ar | 2 1 + r 2 ) .</formula>
<text><location><page_36><loc_17><loc_33><loc_36><loc_34></location>On the other hand, since</text>
<formula><location><page_36><loc_35><loc_30><loc_65><loc_32></location>glyph[epsilon1] A D glyph[epsilon1] BC / ∇ B H CA = / ∇ A H AD -/ ∇ D H A A ,</formula>
<text><location><page_36><loc_15><loc_28><loc_78><loc_29></location>we can apply the standard elliptic estimate (see for instance Lemma 2.2.2 in [1])</text>
<formula><location><page_36><loc_24><loc_23><loc_76><loc_27></location>∫ S 2 | / ∇ A H BC | 2 ≤ C ∫ S 2 ( | glyph[epsilon1] BC / ∇ B H CA | 2 + | / ∇ B H AB | 2 + | / ∇ A H B B | 2 )</formula>
<text><location><page_36><loc_15><loc_21><loc_49><loc_22></location>to obtain the desired bounds for | / ∇ A H BC | 2 .</text>
<text><location><page_36><loc_15><loc_18><loc_85><loc_21></location>This gives all the desired estimates for the derivatives of H . As in the Maxwell case, similar bounds for the derivatives of E can be derived in an identical manner. glyph[square]</text>
<text><location><page_37><loc_15><loc_85><loc_85><loc_88></location>Finally, much as in the Maxwell case, we apply the Hardy inequalities to establish integrated decay of the non-degenerate energy with the loss of a derivative:</text>
<text><location><page_37><loc_15><loc_79><loc_85><loc_83></location>Theorem 5.4 (Full integrated decay) . Suppose that W is Weyl tensor, satisfying the Bianchi equations with dissipative boundary conditions, as in Theorem 4.3. Then there exists a constant C > 0 , independent of T such that we have</text>
<formula><location><page_37><loc_19><loc_74><loc_49><loc_78></location>∫ S [ T ,T ] { E ab E ab + H ab H ab } r 2 (1 + r 2 )</formula>
<formula><location><page_37><loc_29><loc_70><loc_76><loc_74></location>≤ C ∫ Σ T 1 { E ab E ab + H ab H ab + ˙ E ab ˙ E ab + ˙ H ab ˙ H ab } r 2 (1 + r 2 )</formula>
<formula><location><page_37><loc_22><loc_71><loc_81><loc_77></location>1 2 dtdrdω drdω.</formula>
<text><location><page_37><loc_15><loc_61><loc_85><loc_69></location>Proof. Using the result of Proposition 5.6, 5.7 with the Hardy estimates of Lemma 5.4 to improve the weights near infinity in the integrated decay estimates, making use of a cut-off in much the same way as for the spin 0 and spin 1 problems, we obtain the desired estimates for | E AB | , | H AB | , | E A ¯ r | and | H A ¯ r | . Finally, the bounds for | E ¯ r ¯ r | and | H ¯ r ¯ r | are obtained trivially using the trace-free condition for E and H . glyph[square]</text>
<unordered_list>
<list_item><location><page_37><loc_15><loc_50><loc_85><loc_59></location>5.4. Proof of Corollary 1.2 (uniform decay). In this subsection, we show a uniform decay rate for the solutions to the confomal wave, Maxwell and Bianchi equations, hence proving Corollary 1.2. We will in fact prove the uniform decay estimates for all of the equations at once by showing that this is a consequence of the bounds that we have obtained previously. The result below is a combination of relatively standard ideas (for example, see [35] and Prop 3.1 (a) of [36]), but for completeness we include a direct proof.</list_item>
</unordered_list>
<text><location><page_37><loc_15><loc_44><loc_85><loc_48></location>Lemma 5.8. Let Ψ be a solution of either the conformal wave, Maxwell or Bianchi equations. Suppose that we are given some positive quantity E [Ψ]( t ) depending smoothly on Ψ and its derivatives at some time t which satisfies:</text>
<text><location><page_37><loc_15><loc_42><loc_52><loc_43></location>1. E [Ψ]( t ) is a non-increasing C 1 function of t ,</text>
<unordered_list>
<list_item><location><page_37><loc_15><loc_40><loc_72><loc_41></location>2. For every 0 ≤ T 1 ≤ T 2 , E [Ψ]( t ) satisfies the integrated decay estimate:</list_item>
</unordered_list>
<formula><location><page_37><loc_33><loc_35><loc_67><loc_39></location>∫ T 2 T 1 E [Ψ]( t ) dt ≤ C {E [Ψ]( T 1 ) + E [ ∂ t Ψ]( T 1 ) } ,</formula>
<text><location><page_37><loc_17><loc_33><loc_50><loc_34></location>for some C > 0 independent of T 1 and T 2 .</text>
<text><location><page_37><loc_15><loc_31><loc_36><loc_32></location>Then we have the estimate</text>
<formula><location><page_37><loc_36><loc_26><loc_64><loc_30></location>E [Ψ]( t ) ≤ C n (1 + t ) n n ∑ k =0 E [ ( ∂ t ) k Ψ ] (0)</formula>
<text><location><page_37><loc_15><loc_23><loc_58><loc_24></location>for some constants C n > 0 depending only on n and C .</text>
<text><location><page_37><loc_15><loc_21><loc_28><loc_22></location>Proof. Let us set</text>
<formula><location><page_37><loc_41><loc_18><loc_59><loc_20></location>E ( k ) ( t ) = E [ ( ∂ t ) k Ψ ] ( t ) .</formula>
<text><location><page_38><loc_15><loc_86><loc_25><loc_88></location>We calculate</text>
<formula><location><page_38><loc_27><loc_74><loc_73><loc_86></location>(1 + t -T 1 ) E (0) ( t ) = E (0) ( t ) + ∫ t T 1 d ds ( ( s -T 1 ) E (0) ( s ) ) ds = E (0) ( t ) + ∫ t T 1 E (0) ( s ) + ( s -T 1 ) ˙ E (0) ( s ) ds ≤ E (0) ( t ) + ∫ t T 1 E (0) ( s ) ds,</formula>
<text><location><page_38><loc_15><loc_70><loc_85><loc_74></location>where we have used the monotonicity of E (0) to obtain the last inequality. Now, it follows from the assumptions of the Lemma that</text>
<formula><location><page_38><loc_30><loc_66><loc_70><loc_70></location>E (0) ( t ) + ∫ t T 1 E (0) ( s ) ds ≤ C 1 ( E (0) ( T 1 ) + E (1) ( T 1 ) ) ,</formula>
<text><location><page_38><loc_15><loc_64><loc_64><loc_65></location>which together with the preceding estimate immediately imply</text>
<formula><location><page_38><loc_15><loc_60><loc_67><loc_63></location>(53) E (0) ( t ) ≤ C 1 1 + t -T 1 ( E (0) ( T 1 ) + E (1) ( T 1 ) ) .</formula>
<text><location><page_38><loc_15><loc_58><loc_84><loc_60></location>Taking T 1 = 0, this in particular implies the conclusion of the Lemma in the case n = 1.</text>
<text><location><page_38><loc_15><loc_53><loc_85><loc_58></location>To proceed, we induct on n . The n = 1 case has just been established. Suppose now that the statement holds for some n . Noticing that the equation commutes with ∂ t , we use the induction hypothesis for both Ψ and ∂ t Ψ to obtain</text>
<formula><location><page_38><loc_35><loc_48><loc_65><loc_52></location>E (0) ( t ) + E (1) ( t ) ≤ C n (1 + t ) n n +1 ∑ k =0 E ( k ) (0) .</formula>
<text><location><page_38><loc_15><loc_46><loc_48><loc_48></location>Now, we apply (53) with T 1 = t 2 to deduce</text>
<formula><location><page_38><loc_34><loc_32><loc_63><loc_45></location>E (0) ( t ) ≤ C 1 1 + t 2 ( E (0) ( t 2 ) + E (1) ( t 2 ≤ C 1 C n (1 + t 2 ) n +1 n +1 ∑ k =0 E ( k ) (0) ≤ C 1 C n 2 n (1 + t ) n +1 n +1 ∑ E ( k ) (0) ,</formula>
<text><location><page_38><loc_15><loc_30><loc_35><loc_31></location>whence the result follows.</text>
<formula><location><page_38><loc_50><loc_30><loc_85><loc_45></location>)) k =0 glyph[square]</formula>
<text><location><page_38><loc_15><loc_26><loc_85><loc_28></location>Proof of Corollary 1.2. For the conformal wave and Maxwell equations, Corollary 1.2 follows immediately by applying Lemma 5.8 to the quantity</text>
<formula><location><page_38><loc_38><loc_21><loc_62><loc_25></location>E [Ψ]( t ) = ∫ Σ T ε [Ψ] √ 1 + r 2 r 2 drdω</formula>
<text><location><page_38><loc_15><loc_18><loc_85><loc_21></location>which is monotone decreasing and satisfies an integrated decay statement with loss of one derivative. For the Bianchi equations, this quantity is not monotone decreasing (merely</text>
<text><location><page_39><loc_15><loc_85><loc_85><loc_88></location>bounded by a constant times its initial value). We can circumvent this by applying Lemma 5.8 to the quantity:</text>
<formula><location><page_39><loc_34><loc_80><loc_66><loc_84></location>E [ W ]( t ) = ∫ Σ t Q ( e 0 , Y, Y, Y ) r 2 √ 1 + r 2 drdω,</formula>
<text><location><page_39><loc_15><loc_77><loc_85><loc_80></location>which is monotone decreasing and satisfies an integrated decay statement with loss of one derivative. Noting that</text>
<formula><location><page_39><loc_38><loc_73><loc_62><loc_77></location>E [ W ]( t ) ∼ ∫ Σ T ε [ W ] √ 1 + r 2 r 2 drdω,</formula>
<text><location><page_39><loc_15><loc_72><loc_25><loc_73></location>we are done.</text>
<text><location><page_39><loc_84><loc_72><loc_85><loc_73></location>glyph[square]</text>
<text><location><page_39><loc_15><loc_61><loc_85><loc_70></location>5.5. Proof of Theorem 1.3: Gaussian beams. It is noteworthy that in the first instance, for all of the integrated decay estimates we obtained above the r -weight near infinity is weaker than that for the energy estimate. In particular, in order to show a uniform-intime decay estimate, we needed to lose a derivative. In this section, we show that without any loss, there cannot be any uniform decay statements for the conformal wave equation. Moreover, an integrated decay estimate with no degeneration in the r -weight does not hold.</text>
<text><location><page_39><loc_15><loc_52><loc_85><loc_60></location>In order to show this, we will construct approximate solutions to the conformally coupled wave equation for a time interval [0 , T ] with an arbitrarily small loss in energy. We will in fact first construct a Gaussian beam solution on the Einstein cylinder and make use of the fact that (one half of) the Einstein cylinder is conformally equivalent to the AdS spacetime to obtain an approximate solution to the conformally coupled wave equation on AdS.</text>
<text><location><page_39><loc_15><loc_44><loc_85><loc_52></location>In the following, we will first study the null geodesics on the Einstein cylinder. We then construct Gaussian beam approximate solutions to the wave equation on the Einstein cylinder. Such construction is standard and in particular we follow closely Sbierski's geometric approach [26] (see also [37, 38]). After that we return to the AdS case and build solutions that have an arbitrarily small loss in energy.</text>
<text><location><page_39><loc_15><loc_40><loc_85><loc_43></location>5.5.1. Geodesics in the Einstein cylinder. We consider the spacetime ( M E , g E ), where M E is diffeomorphic to R × S 3 and the metric g E is given by</text>
<formula><location><page_39><loc_15><loc_38><loc_67><loc_39></location>(54) g E = -dt 2 + dψ 2 +sin 2 ψ ( dθ 2 +sin 2 θdφ 2 ) .</formula>
<text><location><page_39><loc_15><loc_34><loc_85><loc_37></location>We will slightly abuse notation and denote the subsets of M E with notations similar to that for the AdS spacetime. More precisely, we will take</text>
<formula><location><page_39><loc_40><loc_31><loc_60><loc_33></location>Σ T := { ( t, ψ, θ, φ ) : t = T } ,</formula>
<formula><location><page_39><loc_36><loc_29><loc_64><loc_30></location>S [ T 1 ,T 2 ] := { ( t, ψ, θ, φ ) : T 1 ≤ t ≤ T 2 } .</formula>
<text><location><page_39><loc_15><loc_25><loc_85><loc_28></location>Take null geodesics γ : ( -∞ , ∞ ) →M E in the equatorial plane { θ = π 2 } . In coordinates, we express γ as</text>
<formula><location><page_39><loc_37><loc_22><loc_63><loc_25></location>( t, ψ, θ, φ ) = ( T ( s ) , Ψ( s ) , π 2 , Φ( s )) .</formula>
<text><location><page_39><loc_15><loc_20><loc_62><loc_22></location>Since ∂ ∂t and ∂ ∂φ are Killing vector fields, E and L defined as</text>
<formula><location><page_39><loc_41><loc_18><loc_59><loc_19></location>˙ T = E, sin 2 Ψ ˙ Φ = L</formula>
<text><location><page_40><loc_15><loc_83><loc_85><loc_88></location>are both conserved quantities. Here, and below, we use the convention that ˙ denotes a derivative in s . We require from now on that 0 ≤ | L | < E . The geodesic equation therefore reduces to the ODE</text>
<formula><location><page_40><loc_15><loc_79><loc_57><loc_83></location>(55) ˙ Ψ = √ E 2 -L 2 sin 2 Ψ</formula>
<formula><location><page_40><loc_57><loc_80><loc_58><loc_81></location>.</formula>
<text><location><page_40><loc_15><loc_76><loc_85><loc_79></location>Solving (55) with the condition that Ψ achieves its minimum at s = 0 and T (0) = Φ(0) = 0, we have</text>
<formula><location><page_40><loc_35><loc_70><loc_65><loc_75></location>T ( s ) = Es, sin Ψ( s ) = √ ( E 2 -L 2 ) sin 2 ( Es ) + L 2 E ,</formula>
<text><location><page_40><loc_15><loc_68><loc_18><loc_70></location>and</text>
<formula><location><page_40><loc_34><loc_65><loc_66><loc_69></location>Φ( s ) = ∫ s 0 LE 2 ( E 2 -L 2 ) sin 2 ( Es ' ) + L 2 ds ' .</formula>
<text><location><page_40><loc_15><loc_58><loc_85><loc_65></location>In order to invert the sine function to recover Ψ, we will use the convention that for Es ∈ [2 kπ -π 2 , 2 kπ + π 2 ), and k ∈ Z , we require Ψ( s ) ∈ [0 , π 2 ]; while for Es ∈ [(2 k +1) π -π 2 , (2 k +1) π + π 2 ), and k ∈ Z , we require Ψ( s ) ∈ [ π 2 , π ]. Notice that this choice of the inverse of the sine function gives rise to a smooth null geodesic.</text>
<text><location><page_40><loc_17><loc_57><loc_49><loc_58></location>Moreover, direct computations show that</text>
<formula><location><page_40><loc_17><loc_52><loc_83><loc_57></location>˙ T ( s ) = E, ˙ Ψ( s ) = E √ E 2 -L 2 sin( Es ) √ ( E 2 -L 2 ) sin 2 ( Es ) + L 2 , ˙ Φ( s ) = LE 2 ( E 2 -L 2 ) sin 2 ( Es ) + L 2 .</formula>
<text><location><page_40><loc_15><loc_42><loc_85><loc_51></location>5.5.2. Constructing the Gaussian beam. Given a null geodesic γ on ( M E , g E ) as above, we follow the construction in Sbierski [26] to obtain an approximate solution to the wave equation on ( M E , g E ) which is localised near γ and has energy close to that of γ . We first define the phase function ϕ and its first and second partial derivatives on γ and then construct the function ϕ in a neighbourhood of γ . We also define the amplitude a on γ . More precisely, on γ , we require the following conditions:</text>
<unordered_list>
<list_item><location><page_40><loc_18><loc_40><loc_30><loc_41></location>(1) ϕ ( γ ( s )) = 0</list_item>
<list_item><location><page_40><loc_18><loc_38><loc_34><loc_39></location>(2) dϕ ( γ ( s )) = ˙ γ ( s ) glyph[flat]</list_item>
<list_item><location><page_40><loc_18><loc_36><loc_79><loc_38></location>(3) The matrix M µν := ∂ µ ∂ ν ϕ ( γ ( s )) is a symmetric matrix satisfying the ODE</list_item>
</unordered_list>
<formula><location><page_40><loc_36><loc_32><loc_65><loc_36></location>d ds M = -A -BM -MB T -MCM,</formula>
<text><location><page_40><loc_21><loc_31><loc_50><loc_32></location>where A , B , C are matrices given by</text>
<formula><location><page_40><loc_40><loc_23><loc_60><loc_30></location>A κρ = 1 2 ( ∂ κ ∂ ρ g µν ) ∂ µ ϕ∂ ν ϕ, B κρ = ∂ κ g ρµ ∂ µ ϕ, C κρ = g κρ ,</formula>
<text><location><page_40><loc_21><loc_21><loc_47><loc_22></location>and obeying the initial conditions</text>
<unordered_list>
<list_item><location><page_40><loc_22><loc_19><loc_40><loc_21></location>(a) M (0) is symmetric;</list_item>
<list_item><location><page_40><loc_21><loc_18><loc_42><loc_19></location>(b) M (0) µν ˙ γ ν = ( ˙ ∂ µ ϕ )(0);</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_41><loc_22><loc_85><loc_85><loc_88></location>(c) glyph[Ifractur] ( M (0) µν ) dx µ | γ (0) ⊗ dx ν | γ (0) is positive definite on a three dimensional subspace of T γ (0) M that is transversal to ˙ γ .</list_item>
<list_item><location><page_41><loc_18><loc_83><loc_49><loc_84></location>(4) a ( γ (0)) = 0 and a satisfies the ODE</list_item>
</unordered_list>
<text><location><page_41><loc_27><loc_83><loc_27><loc_84></location>glyph[negationslash]</text>
<formula><location><page_41><loc_41><loc_81><loc_59><loc_82></location>2grad ϕ ( a ) + glyph[square] ϕ · a = 0</formula>
<text><location><page_41><loc_21><loc_78><loc_27><loc_79></location>along γ .</text>
<text><location><page_41><loc_15><loc_69><loc_85><loc_77></location>The results in [26] ensure that ϕ | γ , ∂ µ ϕ | γ , ∂ µ ∂ ν ϕ | γ , a | γ can be constructed satisfying these conditions. We then let ϕ to be a smooth extension of ϕ away from γ compatible with these derivatives. Likewise a N is defined to be an extension of a | γ as constructed above. Moreover, we require a N to be compactly supported in a (small) tubular neighborhood N of the null geodesic γ .</text>
<text><location><page_41><loc_17><loc_68><loc_53><loc_69></location>We define the energy for a function on M E by</text>
<formula><location><page_41><loc_22><loc_63><loc_78><loc_67></location>ˆ E t ( w ) := 1 2 ∫ Σ t ( ( ∂ t w ) 2 +( ∂ ψ w ) 2 + ( ∂ θ w ) 2 sin 2 ψ + ( ∂ φ w ) 2 sin 2 ψ sin 2 θ ) sin 2 ψdψdω.</formula>
<text><location><page_41><loc_15><loc_60><loc_85><loc_62></location>Here, as elsewhere, dω = sin θdθdφ is the volume form of the round unit sphere. We also associate the geodesic γ with a conserved energy</text>
<formula><location><page_41><loc_43><loc_57><loc_57><loc_58></location>E ( γ ) = -g E ( ˙ γ, ∂ t ) .</formula>
<text><location><page_41><loc_15><loc_55><loc_80><loc_56></location>Notice that this agrees with the convention E = ˙ T used in the previous subsection.</text>
<text><location><page_41><loc_15><loc_51><loc_85><loc_54></location>The main result 20 of Sbierski regarding the approximate solution constructed above is the following theorem 21 (see Theorems 2.1 and 2.36 in [26]):</text>
<text><location><page_41><loc_15><loc_49><loc_73><loc_50></location>Theorem 5.5. Given a geodesic γ parametrized by E and L as above, let</text>
<formula><location><page_41><loc_43><loc_46><loc_57><loc_48></location>w E,L,λ, N = a N e iλϕ ,</formula>
<text><location><page_41><loc_15><loc_44><loc_78><loc_45></location>where a N and ϕ are defined as above. Then w λ, N obeys the following conditions:</text>
<unordered_list>
<list_item><location><page_41><loc_18><loc_41><loc_44><loc_43></location>(1) ‖ glyph[square] w E,L,λ, N ‖ L 2 ( S [0 ,T ] ) ≤ C ( T ) ;</list_item>
<list_item><location><page_41><loc_18><loc_39><loc_45><loc_41></location>(2) ˆ E 0 ( w E,L,λ, N ) →∞ as λ →∞ ;</list_item>
<list_item><location><page_41><loc_18><loc_38><loc_65><loc_39></location>(3) w E,L,λ, N is supported in N , a tubular neighborhood of γ ;</list_item>
<list_item><location><page_41><loc_18><loc_36><loc_66><loc_38></location>(4) Fix µ > 0 and normalize the initial energy of w E,L,λ, N by</list_item>
</unordered_list>
<formula><location><page_41><loc_36><loc_32><loc_64><loc_36></location>˜ w E,L,λ, N := w E,L,λ, N √ ˆ E 0 ( w E,L,λ, N ) · E ( γ ) .</formula>
<formula><location><page_41><loc_27><loc_25><loc_73><loc_26></location>-C ≤ g ( ∂ t , ∂ t ) ≤ c < 0 , |∇ ∂ t ( ∂ t , ∂ t ) | + |∇ ∂ t ( ∂ t , e i ) | + |∇ ∂ t ( e i , e j ) | ≤ C,</formula>
<text><location><page_41><loc_15><loc_23><loc_83><loc_25></location>where e i is an orthonormal frame on the S 3 slice. These estimates are obviously satisfied in our setting.</text>
<text><location><page_41><loc_15><loc_18><loc_85><loc_23></location>21 Regarding point (4) in the theorem below, the original work of Sbierski gives a more general characterization of the energy of Gaussian beams in terms of the the energy of geodesics on general Lorentzian manifold. Since in our special setting, the energy of a geodesic is conserved, we will not record the most general result but will refer the readers to [26] for details.</text>
<text><location><page_42><loc_21><loc_85><loc_85><loc_88></location>Then for N a sufficiently small neighborhood of γ and λ sufficiently large, the following bound holds:</text>
<formula><location><page_42><loc_37><loc_81><loc_63><loc_84></location>sup t ∈ [0 ,T ] ∣ ∣ ∣ ˆ E t ( ˜ w E,L,λ, N ) -E ( γ ) ∣ ∣ ∣ < µ.</formula>
<text><location><page_42><loc_15><loc_77><loc_85><loc_80></location>We also need another fact regarding the second derivatives of ϕ which is a consequence of the construction in [26] (see (2.14) in [26]):</text>
<text><location><page_42><loc_15><loc_73><loc_85><loc_76></location>Lemma 5.9. glyph[Ifractur] ( ϕ | γ ) = glyph[Ifractur] ( ∇ ϕ | γ ) = 0 . Moreover, glyph[Ifractur] ( ∇∇ ϕ | γ ) is positive definite on a 3 -dimensional subspace transversal to ˙ γ .</text>
<text><location><page_42><loc_15><loc_67><loc_85><loc_72></location>The fact that a N is independent of λ together with Lemma 5.9 imply that the Gaussian beam approximate solution constructed above has bounded L 2 norm independent of λ . We record this bound in the following lemma:</text>
<text><location><page_42><loc_15><loc_64><loc_76><loc_66></location>Lemma 5.10. Let w E,L,λ, N be as in Theorem 5.5. The following bound holds:</text>
<formula><location><page_42><loc_39><loc_62><loc_61><loc_64></location>‖ w E,L,λ, N ‖ L 2 ( S [0 ,T ] ) ≤ C ( T ) .</formula>
<text><location><page_42><loc_15><loc_56><loc_85><loc_61></location>5.5.3. The conformal transformation. Once we have constructed the Gaussian beam for the wave equation on the Einstein cylinder, it is rather straightforward to construct the necessary sequence of functions using the conformal invariance of the operator</text>
<formula><location><page_42><loc_43><loc_53><loc_57><loc_56></location>L = glyph[square] g -1 6 R ( g ) ,</formula>
<text><location><page_42><loc_15><loc_51><loc_54><loc_52></location>where R ( g ) is the scalar curvature of the metric g .</text>
<text><location><page_42><loc_15><loc_46><loc_85><loc_51></location>We first set up some notations. We will be considering M E restricted to ψ ≤ π 2 as a manifold with boundary diffeomorphic to R × S 3 h . The interior of this manifold will also be identified with M AdS via identifying the coordinate functions ( t, θ, φ ), as well as</text>
<formula><location><page_42><loc_46><loc_44><loc_54><loc_45></location>tan ψ = r.</formula>
<text><location><page_42><loc_15><loc_40><loc_85><loc_43></location>It is easy to see that g E and g AdS are conformal. More precisely, g E as before can be written as</text>
<formula><location><page_42><loc_33><loc_38><loc_67><loc_39></location>g E = -dt 2 + dψ 2 +sin 2 ψ ( dθ 2 +sin 2 θdφ 2 );</formula>
<text><location><page_42><loc_15><loc_36><loc_71><loc_37></location>while in the ( t, ψ, θ, φ ) coordinate system, g AdS takes the following form</text>
<formula><location><page_42><loc_29><loc_32><loc_71><loc_35></location>g AdS = 1 cos 2 ψ ( -dt 2 + dψ 2 +sin 2 ψ ( dθ 2 +sin 2 θdφ 2 ) ) .</formula>
<text><location><page_42><loc_15><loc_27><loc_85><loc_31></location>We now proceed to the construction of the approximate solution to the conformally coupled wave equation on AdS using the conformal invariance of L . More precisely, we have</text>
<text><location><page_42><loc_15><loc_24><loc_76><loc_26></location>Lemma 5.11. Let w be a function on M E restricted to ψ ≤ π 2 . Then we have</text>
<formula><location><page_42><loc_30><loc_20><loc_70><loc_23></location>1 (1 + r 2 ) 3 2 ( glyph[square] g E w -w ) = glyph[square] g AdS w √ 1 + r 2 +2 w √ 1 + r 2 .</formula>
<text><location><page_42><loc_15><loc_18><loc_58><loc_19></location>Proof. This can be verified by an explicit computation.</text>
<text><location><page_42><loc_84><loc_18><loc_85><loc_19></location>glyph[square]</text>
<figure>
<location><page_43><loc_21><loc_70><loc_80><loc_87></location>
<caption>Figure 1. An typical null geodesic γ shown in black projected to a surface of constant θ in the optical geometry of the Einstein cylinder (left). The corresponding curve for the anti-de Sitter spacetime, after reflections at I (right).</caption>
</figure>
<text><location><page_43><loc_15><loc_31><loc_85><loc_54></location>We are now ready to construct the approximate solution. To heuristically explain the construction, consider Figure 1. On the left we sketch the curve γ along which our approximate solution is concentrated in the Einstein cylinder. To allow this to be plotted, we project onto a surface of constant t , θ which carries a spherical geometry and can be visualised via its embedding in R 3 . We actually wish to construct an approximate solution on anti-de Sitter, which is conformal to one half of the Einstein cylinder. To do so, we restrict our attention to one hemisphere and arrange that whenever the curve γ strikes the equator of the Einstein cylinder, it is reflected. Each time this occurs, we shall arrange that the Gaussian beam is attenuated by a factor which depends on the angle of incidence (which in turn depends on E and L ). The more shallow the reflection, the weaker the attenuation. Crucially, the time, t , between reflections is independent of the angle that we choose. As a result, by taking the angle of incidence to be sufficiently small, we can arrange that an arbitrarily small fraction of the initial energy is lost at the boundary over any given time interval.</text>
<text><location><page_43><loc_15><loc_18><loc_85><loc_31></location>To be more precise, let us fix T ≥ 0. We will construct an approximate solution for t ∈ [0 , T ]. Take N ∈ N be the smallest integer such that T < (4 N +3) π 2 . We then construct an approximate solution for t ∈ ( -π 2 , (4 N +3) π 2 ) starting from the function w E,L,λ, N constructed previously. This will correspond to a Gaussian beam which strikes the boundary ∼ 2 N times between t = 0 and t = T . Notice that the geodesic γ has the property that it lies in the hemisphere { ψ < π 2 } for t ∈ (2 kπ -π 2 , 2 kπ + π 2 ) (for k ∈ Z ) and that it lies in the other hemisphere, i.e., { ψ > π 2 } , for t ∈ ((2 k +1) π -π 2 , (2 k +1) π + π 2 ) (for k ∈ Z ). Therefore, we will assume without loss of generality that the neighborhood N has been taken sufficiently</text>
<figure>
<location><page_44><loc_44><loc_60><loc_58><loc_88></location>
<caption>Figure 2. The cutoff functions.</caption>
</figure>
<text><location><page_44><loc_15><loc_50><loc_85><loc_54></location>small such that it lies entirely in { ψ < π 2 } for t ∈ (2 kπ -π 4 , 2 kπ + π 4 ) (and entirely in { ψ > π 2 } for t ∈ ((2 k +1) π -π 4 , (2 k +1) π + π 4 )), where k ∈ Z .</text>
<text><location><page_44><loc_17><loc_49><loc_71><loc_51></location>Now for ( t, ψ, θ, φ ) in M E restricted to ψ ≤ π 2 , we define for t ∈ [0 , T ]</text>
<text><location><page_44><loc_15><loc_47><loc_18><loc_48></location>(56)</text>
<formula><location><page_44><loc_17><loc_37><loc_83><loc_46></location>u E,L,λ ( t, ψ, θ, φ ) = N ∑ k =0 R 2 k χ t ∈ (2 kπ -3 π 4 , 2 kπ + 3 π 4 ] w E,L,λ, N ( t, ψ, θ, φ ) | { ψ ≤ π 2 } √ 1 + r 2 + N ∑ k =0 R 2 k +1 χ t ∈ ((2 k +1) π -3 π 4 , (2 k +1) π + 3 π 4 ] w E,L,λ, N ( t, π -ψ, θ, φ ) | { ψ ≤ π 2 } √ 1 + r 2 ,</formula>
<text><location><page_44><loc_43><loc_36><loc_45><loc_37></location>√</text>
<text><location><page_44><loc_15><loc_29><loc_85><loc_36></location>where R is taken to be R = -E -E 2 -L 2 E + √ E 2 -L 2 and χ is the indicator function. Notice in particular that when the time cutoff function is 0 in the first term (resp. in the second term), the support of w E,L,λ, N is entirely in { ψ > π 2 } (resp. { ψ < π 2 } ). We also depict this in Figure 2, where we denote</text>
<formula><location><page_44><loc_22><loc_24><loc_78><loc_28></location>χ 1 ( t ) = N ∑ k =0 χ t ∈ (2 kπ -3 π 4 , 2 kπ + 3 π 4 ] , χ 2 ( t ) = N ∑ k =0 χ t ∈ ((2 k +1) π -3 π 4 , (2 k +1) π + 3 π 4 ] .</formula>
<text><location><page_44><loc_15><loc_17><loc_85><loc_23></location>The definition above is such that in the time interval t ∈ ( -3 π 4 , 3 π 4 ], we take w E,L,λ, N constructed previously, restrict it to ψ ≤ π 2 and rescale it by 1 √ 1+ r 2 . Then on the time interval t ∈ ( π -3 π 4 , π + 3 π 4 ], we take the part of w E,L,λ, N that is supported in ψ ≥ π 2 ,</text>
<text><location><page_45><loc_15><loc_81><loc_85><loc_88></location>reflect it across the ψ = π 2 hypersurface, rescale by a factor 1 √ 1+ r 2 and then multiply by the factor R . As we will see later, the factor R is chosen so that the boundary conditions are approximately satisfied. We then continue this successively, taking parts of the solutions in ψ ≤ π 2 and ψ ≥ π 2 , reflecting when appropriate, and multiplying by factors of R 's.</text>
<text><location><page_45><loc_15><loc_79><loc_80><loc_80></location>Lemma 5.12. The function u E,L,λ defined as in (56) has the following properties:</text>
<formula><location><page_45><loc_15><loc_77><loc_52><loc_78></location>E 0 [ u E,L,λ ] →∞ as λ →∞ ; (57)</formula>
<formula><location><page_45><loc_15><loc_72><loc_72><loc_76></location>∫ S [0 ,T ] ( glyph[square] g AdS u E,L,λ +2 u E,L,λ ) 2 r 2 (1 + r 2 ) dr dω dt ≤ C ( T ); (58)</formula>
<formula><location><page_45><loc_15><loc_68><loc_67><loc_72></location>∫ ¯ Σ [0 ,T ] ∞ ( ∂ t ( ru E,L,λ ) + r 2 ∂ r ( ru E,L,λ )) 2 dω dt ≤ C ( T ) . (59)</formula>
<text><location><page_45><loc_15><loc_58><loc_85><loc_68></location>Proof. First note that ˆ E 0 [ w E,L,λ ] ≤ CE 0 [ u E,L,λ ], so the first claim follows from Theorem 5.5. Next recall that we have assumed that the neighbourhood N has been taken sufficiently small so that for each of the summands, the indicator function χ is constant on the support of w E,L,λ, N when restricted to a hemisphere. It therefore suffices to consider only the contributions from w E,L,λ, N (as no derivatives fall on the indicator functions). We will write</text>
<formula><location><page_45><loc_39><loc_56><loc_61><loc_57></location>u E,L,λ = u E,L,λ, 1 + u E,L,λ, 2 ,</formula>
<text><location><page_45><loc_15><loc_53><loc_85><loc_56></location>where u E,L,λ, 1 is the first sum in (56) and u E,L,λ, 2 is the second sum in (56). For u E,L,λ, 1 , we apply Lemma 5.11 to get</text>
<formula><location><page_45><loc_26><loc_38><loc_74><loc_52></location>∫ S [0 ,T ] ( glyph[square] g AdS u E,L,λ, 1 +2 u E,L,λ, 1 ) 2 r 2 (1 + r 2 ) dr dω dt ≤ ∫ S [0 ,T ] 1 (1 + r 2 ) 3 ( glyph[square] g E w E,L,λ, N -w E,L,λ, N ) 2 r 2 (1 + r 2 ) dr dω dt = ∫ S [0 ,T ] ( glyph[square] g E w E,L,λ, N -w E,L,λ, N ) 2 sin 2 ψdψdωdt ≤ C ( T ) .</formula>
<text><location><page_45><loc_15><loc_33><loc_85><loc_38></location>In the last line, we have used the estimates in Theorem 5.5 and Lemma 5.10. By a straightforward identification of the two hemispheres in M E , we can prove a similar bound for u E,L,λ, 2 :</text>
<formula><location><page_45><loc_27><loc_25><loc_73><loc_32></location>∫ S [0 ,T ] ( glyph[square] g AdS u E,L,λ, 2 +2 u E,L,λ, 2 ) 2 r 2 (1 + r 2 ) dr dω dt ≤ ∫ S [0 ,T ] ( glyph[square] g E w E,L,λ, N -w E,L,λ, N ) 2 sin 2 ψdψdωdt ≤ C ( T ) .</formula>
<text><location><page_45><loc_15><loc_23><loc_50><loc_24></location>This concludes the proof of the second claim.</text>
<text><location><page_45><loc_15><loc_18><loc_85><loc_22></location>We now show that the boundary terms are appropriately bounded in L 2 . By construction, there are only contributions to the boundary terms in a neighborhood of 1 π ( t -π 2 ) ∈ N . Since there are at most O ( N ) = O ( T ) such contributions, it suffices to show that one of</text>
<text><location><page_46><loc_15><loc_85><loc_85><loc_88></location>them is bounded. We will look at the boundary contribution near t = π 2 , which takes the form</text>
<formula><location><page_46><loc_15><loc_71><loc_72><loc_84></location>lim r →∞ ( r∂ t u E,L,λ + r 2 ∂ r ( ru E,L,λ ))( t, r, θ, φ ) = lim ψ → π 2 -( ∂ t w E,L,λ, N + ∂ ψ w E,L,λ, N )( t, ψ, θ, φ ) + lim ψ → π 2 -R ( ∂ t w E,L,λ, N + ∂ ψ w E,L,λ, N )( t, π -ψ, θ, φ ) = iλa N (( ∂ t ϕ + ∂ ψ ϕ ) + R ( ∂ t ϕ -∂ ψ ϕ )) e iλϕ ( t,ψ = π 2 ,θ,φ ) +(( ∂ t a N + ∂ ψ a N ) + R ( ∂ t a N -∂ ψ a N )) e iλϕ ( t,ψ = π 2 ,θ,φ ) . (60)</formula>
<text><location><page_46><loc_15><loc_64><loc_85><loc_70></location>The latter term is clearly bounded pointwise independent of λ . The first term has a factor of λ and we will show that it is nevertheless bounded in L 2 since by the choice of R , (( ∂ t ϕ + ∂ ψ ϕ ) + R ( ∂ t ϕ -∂ ψ ϕ )) vanishes on γ . More precisely, by points (1), (2) in Section 5.5.2 and Lemma 5.9, we have</text>
<formula><location><page_46><loc_16><loc_59><loc_84><loc_63></location>glyph[Ifractur] ϕ ( t, ψ = π 2 , θ, φ ) ≥ α (( t -π 2 ) 2 +( θ -π 2 ) 2 +( φ -∫ Eπ 2 0 LE 2 ( E 2 -L 2 ) sin 2 ( Es ' ) + L 2 ds ' ) 2 )</formula>
<text><location><page_46><loc_15><loc_57><loc_45><loc_58></location>for some α > 0. We further claim that</text>
<formula><location><page_46><loc_21><loc_53><loc_79><loc_56></location>(( ∂ t ϕ + ∂ ψ ϕ ) + R ( ∂ t ϕ -∂ ψ ϕ )) | ( t = π 2 , ψ = π 2 , θ = π 2 , φ = ∫ Eπ 2 0 LE 2 ( E 2 -L 2 ) sin 2 ( Es ' )+ L 2 ) = 0 .</formula>
<text><location><page_46><loc_15><loc_51><loc_85><loc_52></location>This follows from the fact that dϕ = ˙ γ glyph[flat] on γ and the choice of R . More precisely, we have</text>
<formula><location><page_46><loc_22><loc_43><loc_78><loc_50></location>(( ∂ t ϕ + ∂ ψ ϕ ) + R ( ∂ t ϕ -∂ ψ ϕ )) | ( t = π 2 , ψ = π 2 , θ = π 2 , φ = ∫ Eπ 2 0 LE 2 ( E 2 -L 2 ) sin 2 ( Es ' )+ L 2 ) =( -E + √ E 2 -L 2 + R ( -E -√ E 2 -L 2 )) =0 .</formula>
<text><location><page_46><loc_15><loc_40><loc_27><loc_42></location>The desired L 2</text>
<text><location><page_46><loc_27><loc_40><loc_85><loc_41></location>bound on the boundary then follows from Lemma 5.13. glyph[square]</text>
<text><location><page_46><loc_15><loc_35><loc_85><loc_39></location>Lemma 5.13. Suppose that f is a function defined on { ψ = π 2 } which vanishes to order 0 at ( t = π 2 , θ = π 2 , φ = ∫ Eπ 2 0 LE 2 ( E 2 -L 2 ) sin 2 ( Es ' )+ L 2 ) . Then</text>
<formula><location><page_46><loc_39><loc_31><loc_61><loc_34></location>∫ dω dt | fe iλϕ | 2 ≤ C f,ϕ λ -5 2 .</formula>
<text><location><page_46><loc_15><loc_28><loc_51><loc_30></location>Proof. In order to simplify notation, we define</text>
<formula><location><page_46><loc_34><loc_24><loc_66><loc_27></location>φ 0 := ∫ Eπ 2 0 LE 2 ( E 2 -L 2 ) sin 2 ( Es ' ) + L 2 ds ' .</formula>
<text><location><page_46><loc_15><loc_21><loc_60><loc_23></location>The statement that f vanishes to order 0 is equivalent to</text>
<formula><location><page_46><loc_33><loc_17><loc_67><loc_21></location>| f | ≤ C ( ( t -π 2 ) 2 +( θ -π 2 ) 2 +( φ -φ 0 ) 2 ) 1 2 .</formula>
<text><location><page_47><loc_15><loc_86><loc_19><loc_88></location>Thus</text>
<formula><location><page_47><loc_15><loc_74><loc_86><loc_85></location>∫ dω dt | fe iλϕ | 2 = ∫ dω dt | f | 2 e -2 λ glyph[Ifractur] ϕ ≤ C ∫ π 0 sin θdθ ∫ 2 π 0 dφ ∫ π 0 dt ( ( t -π 2 ) 2 +( θ -π 2 ) 2 +( φ -φ 0 ) 2 ) e -2 λα (( t -π 2 ) 2 +( θ -π 2 ) 2 +( φ -φ 0 ) 2 ) ≤ C ∫ R 3 | x | 2 e -2 αλ | x | 2 dx ≤ Cλ -5 2 ,</formula>
<text><location><page_47><loc_15><loc_72><loc_61><loc_73></location>where in the last line we simply scale λ out of the integral.</text>
<text><location><page_47><loc_84><loc_72><loc_85><loc_73></location>glyph[square]</text>
<text><location><page_47><loc_15><loc_62><loc_85><loc_69></location>5.5.4. Building a true solution. Given the approximate solution constructed above, we are now ready to build a true solution to the homogeneous conformally coupled wave equation with dissipative boundary condition. To this end, we need a strengthening of Theorem 5.1 which includes inhomogeneous terms:</text>
<text><location><page_47><loc_15><loc_59><loc_85><loc_61></location>Theorem 5.6. Let u be a solution of the inhomogeneous conformally coupled wave equation</text>
<formula><location><page_47><loc_15><loc_56><loc_61><loc_58></location>(61) glyph[square] g AdS u +2 u = f in AdS</formula>
<text><location><page_47><loc_15><loc_53><loc_50><loc_54></location>with finite (renormalized) energy initial data</text>
<formula><location><page_47><loc_45><loc_50><loc_55><loc_51></location>E T 1 ( u ) < ∞</formula>
<text><location><page_47><loc_15><loc_47><loc_63><loc_48></location>and subject to inhomogeneous dissipative boundary conditions</text>
<formula><location><page_47><loc_35><loc_42><loc_65><loc_45></location>∂ ( ru ) ∂t + r 2 ∂ ( ru ) ∂r → g, as r →∞ .</formula>
<text><location><page_47><loc_15><loc_39><loc_42><loc_41></location>Then we have for any T 1 < t < T 2 :</text>
<formula><location><page_47><loc_22><loc_34><loc_78><loc_38></location>E t [ u ] ≤ C T 1 ,T 2 ( E T 1 [ u ] + ∫ ˜ Σ [ T 1 ,T 2 ] ∞ g 2 dωdt + ∫ S [ T 1 ,T 2 ] f 2 r 2 (1 + r 2 ) drdωdt ) .</formula>
<text><location><page_47><loc_15><loc_31><loc_37><loc_32></location>Proof. We have by (22) that</text>
<formula><location><page_47><loc_35><loc_28><loc_65><loc_29></location>Div ( T J ) = ( glyph[square] AdS u +2 u ) ∂ t u = f∂ t u.</formula>
<text><location><page_47><loc_15><loc_22><loc_85><loc_26></location>Integrating this over S [ T 1 ,T 2 ] and applying the divergence theorem we pick up terms on the left hand side from Σ T 1 , Σ T 2 and ˜ Σ [ T 1 ,T 2 ] ∞ . A straightforward calculation shows</text>
<formula><location><page_47><loc_41><loc_17><loc_59><loc_21></location>∫ Σ t T J a n a dS Σ t = E t [ u ]</formula>
<text><location><page_48><loc_15><loc_86><loc_25><loc_88></location>We also find</text>
<formula><location><page_48><loc_26><loc_74><loc_74><loc_86></location>∫ ˜ Σ [ T 1 ,T 2 ] r T J a m a dS ˜ Σ r = ∫ S 2 r 2 (1 + r 2 ) ( ∂ t u ) ( ˜ ∂ r u ) dωdt = -1 2 ∫ S 2 [ ( r 2 ( ∂ t u ) 2 + r 2 (1 + r 2 ) 2 ( ˜ ∂ r u ) 2 ) -{ ∂ t ( ru ) + r (1 + r 2 )( ˜ ∂ r u ) } 2 ] dωdt.</formula>
<text><location><page_48><loc_15><loc_71><loc_67><loc_72></location>As r →∞ , the term in braces may be replaced with g , so we have</text>
<formula><location><page_48><loc_22><loc_67><loc_78><loc_70></location>lim r →∞ ∫ ˜ Σ [ T 1 ,T 2 ] r T J a m a dS ˜ Σ r = -1 2 ∫ ˜ Σ [ T 1 ,T 2 ] ∞ ( r 2 ( ∂ t u ) 2 + r 6 ( ˜ ∂ r u ) 2 -g 2 ) dωdt.</formula>
<text><location><page_48><loc_15><loc_65><loc_39><loc_66></location>Applying Lemma 2.1, we have:</text>
<formula><location><page_48><loc_18><loc_60><loc_82><loc_64></location>E t [ u ] -E T 1 [ u ] = 1 2 ∫ ˜ Σ [ T 1 ,t ] ∞ ( -r 2 ( ∂ t u ) 2 -r 6 ( ˜ ∂ r u ) 2 + g 2 ) dωdt + ∫ S [ T 1 ,t ] f∂ t ur 2 drdtdω</formula>
<text><location><page_48><loc_15><loc_58><loc_20><loc_59></location>so that</text>
<formula><location><page_48><loc_15><loc_51><loc_18><loc_52></location>(62)</formula>
<formula><location><page_48><loc_19><loc_49><loc_81><loc_57></location>sup t ∈ [ T 1 ,T 2 ] E t [ u ] ≤ E T 1 [ u ] + 1 2 ∫ ˜ Σ [ T 1 ,T 2 ] ∞ g 2 dωdt + 1 4( T 2 -T 1 ) ∫ S [ T 1 ,T 2 ] ( ∂ t u ) 2 1 + r 2 r 2 drdtdω +( T 2 -T 1 ) ∫ S [ T 1 ,T 2 ] f 2 r 2 (1 + r 2 ) drdωdt.</formula>
<text><location><page_48><loc_15><loc_47><loc_18><loc_48></location>Now</text>
<formula><location><page_48><loc_18><loc_43><loc_82><loc_47></location>1 4( T 2 -T 1 ) ∫ S [ T 1 ,T 2 ] ( ∂ t u ) 2 1 + r 2 r 2 drdtdω ≤ 1 4 sup t ∈ [ T 1 ,T 2 ] ∫ Σ t ( ∂ t u ) 2 1 + r 2 r 2 drdω ≤ 1 2 sup t ∈ [ T 1 ,T 2 ] E t [ u ] .</formula>
<text><location><page_48><loc_15><loc_39><loc_85><loc_42></location>Applying this estimate to (62) and absorbing the energy term on the left hand side, we are done. glyph[square]</text>
<text><location><page_48><loc_15><loc_31><loc_85><loc_37></location>After taking λ to be large, we now construct true solutions of the wave equation with dissipative boundary conditions which only have a small loss of energy. To do this, we define ˜ u E,L,λ to be u E,Lλ (defined by (56) in the previous subsection) multiplied by a constant factor in such a way that</text>
<formula><location><page_48><loc_44><loc_29><loc_56><loc_30></location>E 0 (˜ u E,L,λ ) = 1 .</formula>
<text><location><page_48><loc_15><loc_26><loc_67><loc_28></location>Let U E,L,λ solve the homogeneous initial-boundary value problem:</text>
<formula><location><page_48><loc_40><loc_24><loc_60><loc_25></location>glyph[square] g AdS U E,L,λ +2 U E,L,λ = 0</formula>
<text><location><page_48><loc_15><loc_21><loc_48><loc_23></location>subject to dissipative boundary conditions</text>
<formula><location><page_48><loc_31><loc_17><loc_69><loc_20></location>∂ ( rU E,L,λ ) ∂t + r 2 ∂ ( rU E,L,λ ) ∂r → 0 , as r →∞ ,</formula>
<text><location><page_49><loc_15><loc_86><loc_32><loc_88></location>and initial conditions</text>
<formula><location><page_49><loc_29><loc_82><loc_71><loc_86></location>U E,L,λ | t =0 = ˜ u E,L,λ | t =0 , ∂U E,L,λ ∂t ∣ ∣ ∣ ∣ t =0 = ∂ ˜ u E,L,λ ∂t ∣ ∣ ∣ ∣ t =0 .</formula>
<text><location><page_49><loc_15><loc_81><loc_41><loc_82></location>We obtain the following theorem:</text>
<text><location><page_49><loc_15><loc_75><loc_85><loc_79></location>Theorem 5.7. Fix E > 0 and 0 ≤ | L | < E . Fix also T, glyph[epsilon1] > 0 . There exists a solution U E,L,λ of the homogeneous conformally coupled wave equation with dissipative boundary conditions such that U E,L,λ has energy 1 at time 0 , and</text>
<formula><location><page_49><loc_32><loc_70><loc_68><loc_74></location>inf t ∈ [0 ,T ] E t ( U E,L,λ ) ≥ ( E -√ E 2 -L 2 E + √ E 2 -L 2 ) CT -glyph[epsilon1],</formula>
<text><location><page_49><loc_15><loc_68><loc_46><loc_69></location>where C > 0 is some universal constant.</text>
<text><location><page_49><loc_15><loc_63><loc_85><loc_66></location>Proof. Consider U E,L,λ as defined above, which is a solution to the homogeneous conformal wave equation. By Lemma 5.12, as λ →∞ , we have</text>
<formula><location><page_49><loc_29><loc_59><loc_71><loc_63></location>∫ S [0 ,T ] ( glyph[square] g AdS ˜ u E,L,λ +2˜ u E,L,λ ) 2 r 2 (1 + r 2 ) dr dω dt → 0</formula>
<text><location><page_49><loc_15><loc_57><loc_18><loc_58></location>and</text>
<formula><location><page_49><loc_32><loc_54><loc_68><loc_57></location>∫ ¯ Σ [0 ,T ] ∞ ( ∂ t ( r ˜ u E,L,λ ) + r 2 ∂ r ( r ˜ u E,L,λ )) 2 dω dt → 0 .</formula>
<text><location><page_49><loc_15><loc_50><loc_85><loc_54></location>Therefore, after taking λ to be sufficiently large and applying Theorem 5.6 to U E,L,λ -˜ u E,L,λ , we can assume</text>
<formula><location><page_49><loc_39><loc_48><loc_61><loc_50></location>sup t ∈ [0 ,T ] E t [ U E,L,λ -˜ u E,L,λ ] < glyph[epsilon1].</formula>
<text><location><page_49><loc_15><loc_46><loc_60><loc_47></location>On the other hand, by the construction of u E,L,λ , we have</text>
<formula><location><page_49><loc_24><loc_39><loc_85><loc_45></location>inf t ∈ [0 ,T ] E t [˜ u E,L,λ ] ≥ ( E -√ E 2 -L 2 E + √ E 2 -L 2 ) 2 N +1 ≥ ( E -√ E 2 -L 2 E + √ E 2 -L 2 ) CT glyph[square]</formula>
<text><location><page_49><loc_15><loc_38><loc_56><loc_40></location>for some C > 0. The results follow straightforwardly.</text>
<text><location><page_49><loc_15><loc_33><loc_85><loc_37></location>In particular, by taking | L | sufficiently close to E and glyph[epsilon1] sufficiently small, this implies that on the time interval [0 , T ], the loss of energy can be arbitrarily small. This also implies that any uniform integrated decay estimates without loss do not hold:</text>
<text><location><page_49><loc_15><loc_30><loc_61><loc_31></location>Corollary 5.8. There exists no constant C > 0 , such that</text>
<formula><location><page_49><loc_42><loc_26><loc_58><loc_30></location>∫ ∞ 0 E t [ u ] dt ≤ CE 0 [ u ]</formula>
<text><location><page_49><loc_15><loc_23><loc_85><loc_26></location>holds for every solution u to the conformal wave equation with finite initial energy subject to dissipative boundary conditions.</text>
<text><location><page_49><loc_15><loc_19><loc_85><loc_22></location>Similarly, there exists no continuous positive function f : R + → R + , such that f ( t ) → 0 as t →∞ and</text>
<formula><location><page_49><loc_43><loc_18><loc_57><loc_19></location>E t [ u ] ≤ f ( t ) E 0 [ u ] ,</formula>
<text><location><page_50><loc_15><loc_85><loc_85><loc_88></location>holds for every solution u to the conformal wave equation with finite initial energy subject to dissipative boundary conditions.</text>
<text><location><page_50><loc_15><loc_72><loc_85><loc_84></location>Remark 2. We are grateful to an anonymous referee, who points out that our construction above can in fact be adapted to establish the stronger statement that the degeneracy in r for the integrated decay rate established above is in fact optimal. That is to say for any δ > 0 , there can exist no constant C such that Proposition 5.2 holds with the weight r 2 √ 1+ r 2 replaced by r 2+ δ √ 1+ r 2 . This in particular suggests that the trapping phenomenon present here is different to the normally hyperbolic trapping observed, for example, at the photon sphere of the Schwarzschild black hole.</text>
<section_header_level_1><location><page_50><loc_41><loc_69><loc_59><loc_70></location>6. Generalizations</section_header_level_1>
<section_header_level_1><location><page_50><loc_15><loc_66><loc_48><loc_68></location>6.1. Alternative boundary conditions.</section_header_level_1>
<text><location><page_50><loc_15><loc_64><loc_85><loc_65></location>6.1.1. Conformal Wave. We assumed that our solution u satisfied the boundary condition</text>
<formula><location><page_50><loc_35><loc_60><loc_65><loc_63></location>∂ ( ru ) ∂t + r 2 ∂ ( ru ) ∂r → 0 , as r →∞ .</formula>
<text><location><page_50><loc_15><loc_58><loc_51><loc_60></location>We can also consider the boundary conditions</text>
<formula><location><page_50><loc_33><loc_55><loc_67><loc_58></location>∂ ( ru ) ∂t + β ( ω ) r 2 ∂ ( ru ) ∂r → 0 , as r →∞ .</formula>
<text><location><page_50><loc_15><loc_53><loc_79><loc_54></location>where β : S 2 → R is a smooth function satisfying the uniform positivity condition:</text>
<formula><location><page_50><loc_46><loc_51><loc_54><loc_52></location>β ( ω ) ≥ κ 2</formula>
<text><location><page_50><loc_15><loc_42><loc_85><loc_50></location>for some κ > 0 for all ω ∈ S 2 . This can be treated exactly as above, with the same results although the constants in the various estimates will now depend on β . One could also imagine adding some small multiples of tangential derivatives of u to the boundary condition. This can also be handled by the methods above, but this will require combining the energy and integrated decay estimates.</text>
<text><location><page_50><loc_15><loc_39><loc_62><loc_41></location>6.1.2. Maxwell. The boundary conditions that we assumed,</text>
<formula><location><page_50><loc_35><loc_37><loc_65><loc_39></location>r 2 ( E A + glyph[epsilon1] A B H B ) → 0 , as r →∞ ,</formula>
<text><location><page_50><loc_15><loc_33><loc_85><loc_36></location>can also be generalised without materially affecting the results. In particular, we could choose as boundary conditions</text>
<formula><location><page_50><loc_32><loc_31><loc_68><loc_33></location>r 2 ( E A + β AB ( ω ) glyph[epsilon1] BC H C ) → 0 , as r →∞</formula>
<text><location><page_50><loc_15><loc_27><loc_85><loc_30></location>where the symmetric matrix valued function β : S 2 → M (2 × 2) satisfies a uniform positivity bound:</text>
<formula><location><page_50><loc_15><loc_25><loc_59><loc_26></location>(63) β AB ( ω ) ξ A ξ B ≥ κ 2 | ξ | 2 ,</formula>
<text><location><page_50><loc_15><loc_18><loc_85><loc_24></location>for some κ > 0 and for all ξ ∈ R 2 , ω ∈ S 2 . In particular, our results hold for any Leontovic boundary condition [34, § 87] satisfying (63). Again, one could also permit other components of E , H to appear in the boundary condition with small coefficients and this can be handled by combining the energy and integrated decay estimates.</text>
<text><location><page_51><loc_15><loc_86><loc_77><loc_88></location>6.1.3. Bianchi. Recall that we are considering boundary conditions of the form:</text>
<formula><location><page_51><loc_15><loc_84><loc_67><loc_86></location>(64) r 3 ( ˆ E AB + glyph[epsilon1] ( A C ˆ H B ) C ) → 0 , as r →∞ .</formula>
<text><location><page_51><loc_15><loc_79><loc_85><loc_82></location>To generalise these, let us introduce a 4 -tensor on S 2 , β ABCD ( ω ), symmetric on its first and last pairs of indices and also under interchange of the first and last pair of indices:</text>
<formula><location><page_51><loc_34><loc_77><loc_65><loc_78></location>β ABCD = β ( AB ) CD = β AB ( CD ) = β CDAB</formula>
<text><location><page_51><loc_15><loc_71><loc_85><loc_76></location>we also require that β is trace free on its first (or last) indices, i.e. β A ABC = 0. In other words, β represents a symmetric bilinear form on the space of symmetric trace-free tensors at each point of S 2 . We can consider boundary conditions</text>
<formula><location><page_51><loc_15><loc_68><loc_70><loc_70></location>(65) r 3 ( β AB CD ˆ E CD + glyph[epsilon1] ( A C ˆ H B ) C ) → 0 , as r →∞ .</formula>
<text><location><page_51><loc_15><loc_62><loc_85><loc_67></location>Provided that β ABCD is uniformly close to the canonical inner product on symmetric tracefree tensors, our methods will apply. More concretely, there is a δ > 0, which could be explicitly calculated, such that if</text>
<formula><location><page_51><loc_32><loc_60><loc_68><loc_62></location>(1 + δ ) | Ξ | 2 ≥ β ABCD ( ω )Ξ AB Ξ CD ≥ (1 -δ ) | Ξ | 2</formula>
<text><location><page_51><loc_15><loc_56><loc_85><loc_59></location>holds for any symmetric traceless Ξ and ω ∈ S 2 , then our results hold for the boundary conditions (65).</text>
<text><location><page_51><loc_15><loc_51><loc_85><loc_56></location>Of course, this does not imply that boundary conditions which don't satisfy this inequality lead to growth, simply that our approach breaks down when the boundary conditions are too far from the 'optimally dissipative' ones (64).</text>
<text><location><page_51><loc_15><loc_39><loc_85><loc_50></location>6.2. The Dirichlet problem for (B). Let us briefly discuss the Dirichlet problem for the Weyl tensor and its relation to the Dirichlet problem from the point of view of metric perturbations. In the latter, one wishes to fix, to linear order, the conformal class of the metric at infinity. Fixing the conformal class to be that of the unperturbed anti-de Sitter spacetime is equivalent to requiring that the Cotton-York tensor of the perturbed boundary metric vanishes. One may verify that the Cotton-York tensor of the boundary metric vanishes if and only if</text>
<formula><location><page_51><loc_31><loc_35><loc_69><loc_38></location>∣ ∣ ∣ r 3 ˆ H AB ∣ ∣ ∣ + ∣ ∣ r 3 E Ar ∣ ∣ + ∣ ∣ r 3 H rr ∣ ∣ → 0 as r →∞ ,</formula>
<text><location><page_51><loc_15><loc_29><loc_85><loc_34></location>as can be established by considering the metric in Fefferman-Graham coordinates. Let us illustrate in what way fixing the conformal class of the metric on the boundary is a more restrictive condition than fixing Dirichlet-conditions on the Weyl tensor, ∣ ∣ ∣ r 3 ˆ H AB ∣ ∣ ∣ → 0.</text>
<text><location><page_51><loc_15><loc_18><loc_85><loc_29></location>It is possible to extract from the Bianchi equations a symmetric hyperbolic system on I involving only r 3 E Ar , r 3 H rr , where r 3 ˆ H AB appears as a source term. Using this system it is possible to show that if r 3 E Ar and r 3 H rr vanish at infinity for the initial data then this condition propagates. Moreover, it is easy to see how to construct a large class initial data satisfying this vanishing condition at infinity, as well as a large class not satisfying it illustrating that fixing the conformal class of the metric on the boundary is a more restrictive condition than purely fixing Dirichlet-conditions on the Weyl tensor.</text>
<text><location><page_52><loc_15><loc_78><loc_85><loc_88></location>In conclusion, for initial data satisfying the vanishing condition, we may return to the estimate (43) and establish directly that solutions of the Bianchi system representing a linearised gravitational perturbation fixing the conformal class of the boundary metric are bounded. This is in accordance with the results of [39], in which it is shown that metric perturbations obeying the linearised Einstein equations can be decomposed into components which separately obey wave equations admitting a conserved energy.</text>
<text><location><page_52><loc_15><loc_66><loc_85><loc_77></location>6.3. The relation to the Teukolsky equations. We finally contrast our result with an alternative approach to study the spin 2 equations on AdS, which has a large tradition in the asymptotically flat context and relies on certain curvature components satisfying decoupled wave equations. As we will see below, however, in the AdS context this approach merely obscures the geometric nature of the problem and does not provide any obvious simplification as the resulting decoupled equations couple via the boundary conditions (and moreover do not admit a conserved energy).</text>
<text><location><page_52><loc_15><loc_62><loc_85><loc_65></location>To decouple the spin 2 equations, we take the standard θ, φ coordinates for the spherical directions 22 , and choose as basis e 1 = r -1 ∂ θ , e 2 = ( r sin θ ) -1 ∂ φ . We then write:</text>
<formula><location><page_52><loc_31><loc_60><loc_69><loc_61></location>Ψ ± = E 11 -E 22 ∓ 2 H 12 ∓ i ( H 11 -H 22 ± 2 E 12 ) .</formula>
<text><location><page_52><loc_15><loc_57><loc_52><loc_59></location>These quantities obey the Teukolsky equations:</text>
<formula><location><page_52><loc_15><loc_46><loc_78><loc_56></location>0 = -r 2 1 + r 2 ∂ 2 t Ψ ± ± 4 r 1 + r 2 ∂ t Ψ ± + 1 + r 2 r 3 ∂ r ( r 4 1 + r 2 ∂ r [ r (1 + r 2 )Ψ ± ] ) + 1 sin θ ∂ θ ( sin θ∂ θ Ψ ± ) + 1 sin 2 θ ∂ 2 φ Ψ ± -4 i cos θ sin 2 θ ∂ φ Ψ ± (66) -( 4 sin 2 θ -2 ) Ψ ± .</formula>
<text><location><page_52><loc_15><loc_38><loc_85><loc_45></location>Once Ψ ± have been found, the other components of W can be recovered by solving an elliptic system coupled to the symmetric hyperbolic system in the boundary discussed in Section 6.2. It might appear that one can simply study the decoupled equations for Ψ ± separately. Unfortunately, in general, the correct boundary conditions couple the equations.</text>
<text><location><page_52><loc_15><loc_33><loc_85><loc_38></location>Let us see what the appropriate boundary conditions to impose on Ψ ± are in order to fix the conformal class of the boundary metric. Clearly the vanishing of ˆ H AB at I is equivalent to the condition</text>
<formula><location><page_52><loc_15><loc_31><loc_64><loc_33></location>(67) ∣ ∣ r 3 (Ψ + -Ψ -) ∣ ∣ → 0 as r →∞ .</formula>
<text><location><page_52><loc_15><loc_25><loc_85><loc_30></location>This however only gives us one condition for two equations. For the other condition we use the fact that in the context of the Dirichlet problem (fixing the conformal class discussed in Section 6.2) we know that r 3 E Ar , r 3 H rr vanish on the boundary. 23 Inserting this into</text>
<text><location><page_53><loc_15><loc_86><loc_47><loc_88></location>the Bianchi equations we can derive that</text>
<formula><location><page_53><loc_36><loc_82><loc_64><loc_85></location>∣ ∣ ∣ ∣ r 2 ∂ ∂r ( r 3 ˆ E AB ) ∣ ∣ ∣ ∣ → 0 as r →∞ .</formula>
<text><location><page_53><loc_15><loc_79><loc_65><loc_80></location>This gives us a Neumann condition for the Teukolsky equations:</text>
<formula><location><page_53><loc_15><loc_75><loc_66><loc_78></location>(68) ∣ ∣ ∣ ∣ r 2 ∂ ∂r [ r 3 (Ψ + +Ψ -) ] ∣ ∣ ∣ ∣ → 0 as r →∞ .</formula>
<text><location><page_53><loc_15><loc_61><loc_85><loc_73></location>We should of course not be surprised that the two Teukolsky equations couple at the boundary. The two scalar functions Ψ ± represent the outgoing and ingoing radiative degrees of freedom. Since the Dirichlet boundary conditions are reflecting, one should of course expect that the two components couple at the boundary. The pair of equations (66) with the boundary conditions (67), (68) forms a well posed system of wave equations, as may be seen for instance with the methods of [24]. In any case, there seems to be no advantage in studying this coupled system of wave equations over the first order techniques of this paper.</text>
<text><location><page_53><loc_15><loc_38><loc_85><loc_60></location>We remark that the Teukolsky approach was recently used in [28] to consider perturbations of the Kerr-AdS family of metrics (which includes anti-de Sitter for a = m = 0). In this paper, the Teukolsky equation is separated and a boundary condition (preserving the conformal class of the metric at infinity) is proposed for the radial part of each mode of Ψ ± separately . This appears to contradict our discussion above. When one examines equations (3.9-12) of [28] one sees spectral parameters appearing up to fourth order 24 . Returning to a physical space picture, these will appear as fourth order operators on the boundary. Accordingly, it is far from clear whether these boundary conditions can be meaningfully interpreted as giving boundary conditions for a dynamical evolution problem. Indeed, our heuristic argument for the coupling strongly suggests that the conditions of [28] understood as boundary conditions for a dynamical problem cannot give rise to a well posed evolution. That is not to say that these boundary conditions are not suitable for calculating (quasi)normal modes: any such mode will certainly obey these conditions, providing a useful trick to simplify such computations.</text>
<section_header_level_1><location><page_53><loc_45><loc_33><loc_55><loc_34></location>7. Appendix</section_header_level_1>
<section_header_level_1><location><page_53><loc_15><loc_30><loc_37><loc_32></location>7.1. Proof of Lemma 5.3.</section_header_level_1>
<text><location><page_53><loc_15><loc_27><loc_62><loc_28></location>Proof. Let us first consider the first two terms of K . We set</text>
<formula><location><page_53><loc_25><loc_24><loc_75><loc_26></location>K 1 := ∂ r ( r 2 ∂ r ( u √ 1 + r 2 )) ( / ∇ A u ) e A -( r / ∇ A u ) ˜ ∂ r ( r / ∇ A u ) e r .</formula>
<text><location><page_54><loc_15><loc_86><loc_77><loc_88></location>Calculating with the expression for the divergence of a vector field (19), we find</text>
<formula><location><page_54><loc_17><loc_62><loc_83><loc_85></location>Div K 1 = / ∇ A [ ∂ r ( r 2 ∂ r ( u √ 1 + r 2 )) ( / ∇ A u ) ] -1 r 2 ∂ ∂r [ r 2 √ 1 + r 2 ( r / ∇ A u ) ˜ ∂ r ( r / ∇ A u )] = ∂ r ( r 2 ∂ r ( u √ 1 + r 2 ))[ / ∇ A / ∇ A u ] -√ 1 + r 2 ∣ ∣ ∣ ˜ ∂ r ( r / ∇ u ) ∣ ∣ ∣ 2 + / ∇ A [ ∂ r ( r 2 ∂ r ( u √ 1 + r 2 ))] ( / ∇ A u ) -√ 1 + r 2 r 2 ( r / ∇ A u ) ∂ ∂r [ r 2 ˜ ∂ r ( r / ∇ A u )] = ∂ r ( r 2 ∂ r ( u √ 1 + r 2 ))[ / ∇ A / ∇ A u ] -√ 1 + r 2 ∣ ∣ ∣ ˜ ∂ r ( r / ∇ u ) ∣ ∣ ∣ 2 + [ ∂ r ( r 2 ∂ r ( r / ∇ A u √ 1 + r 2 ))] ( / ∇ A u ) r -√ 1 + r 2 r 2 ( r / ∇ A u ) ∂ ∂r [ r 2 ˜ ∂ r ( r / ∇ A u )] = ∂ r ( r 2 ∂ r ( u √ 1 + r 2 ))[ / ∇ A / ∇ A u ] -√ 1 + r 2 ∣ ∣ ∣ ˜ ∂ r ( r / ∇ u ) ∣ ∣ ∣ 2 + r √ 1 + r 2 ( r / ∇ A u ) ˜ ∂ r ( r / ∇ A u ) .</formula>
<text><location><page_54><loc_15><loc_59><loc_26><loc_61></location>Finally, taking</text>
<formula><location><page_54><loc_39><loc_56><loc_61><loc_59></location>K 2 := -r 2(1 + r 2 ) ∣ ∣ r / ∇ u ∣ ∣ 2 e r ,</formula>
<text><location><page_54><loc_15><loc_54><loc_24><loc_55></location>we calculate</text>
<formula><location><page_54><loc_21><loc_41><loc_79><loc_53></location>Div K 2 = -1 r 2 ∂ ∂r [ r 2 √ 1 + r 2 r 2(1 + r 2 ) ∣ ∣ r / ∇ u ∣ ∣ 2 ] = -r √ 1 + r 2 ( r / ∇ A u ) ˜ ∂ r ( r / ∇ A u ) -1 + r 2 2 r 2 ∣ ∣ r / ∇ u ∣ ∣ 2 ∂ ∂r ( r 3 (1 + r 2 ) 3 2 ) = -r √ 1 + r 2 ( r / ∇ A u ) ˜ ∂ r ( r / ∇ A u ) -3 ∣ ∣ r / ∇ u ∣ ∣ 2 2(1 + r 2 ) 3 2 .</formula>
<text><location><page_54><loc_15><loc_38><loc_58><loc_39></location>Adding these two contributions, we arrive at the result.</text>
<section_header_level_1><location><page_54><loc_15><loc_35><loc_51><loc_36></location>7.2. Proof of Lemma 5.5. Let us introduce</section_header_level_1>
<formula><location><page_54><loc_34><loc_31><loc_66><loc_34></location>α := -2 √ 1 + r 2 ∂ r ( r √ 1 + r 2 H A ) r / ∇ A H r .</formula>
<text><location><page_54><loc_15><loc_27><loc_85><loc_30></location>From the expression (19) for the divergence of a vector field, we can quickly establish that, owing to cancellation between the mixed partial derivatives, we have</text>
<formula><location><page_54><loc_22><loc_18><loc_78><loc_26></location>Div K = 2 √ 1 + r 2 ∂ ∂r ( r 2 H r √ 1 + r 2 ) / ∇ B H B -2 ∂ ∂r ( r √ 1 + r 2 H B ) r / ∇ B H r √ 1 + r 2 -1 r 2 ∂ ∂r ( r 5 √ 1 + r 2 H 2 r )</formula>
<text><location><page_54><loc_84><loc_38><loc_85><loc_39></location>glyph[square]</text>
<text><location><page_55><loc_15><loc_86><loc_20><loc_88></location>so that</text>
<formula><location><page_55><loc_22><loc_65><loc_78><loc_85></location>α = Div K -2 √ 1 + r 2 ∂ ∂r ( r 2 H r √ 1 + r 2 ) / ∇ B H B + 1 r 2 ∂ ∂r ( r 5 √ 1 + r 2 H 2 r ) = Div K +2 1 + r 2 r 2 ∂ ∂r ( r 2 H r √ 1 + r 2 ) ∂ ∂r ( r 2 H r ) + 1 r 2 ∂ ∂r ( r 5 √ 1 + r 2 H 2 r ) = Div K +2 √ 1 + r 2 r 2 ∣ ∣ ∣ ∣ ∂ ∂r ( r 2 H r ) ∣ ∣ ∣ ∣ 2 -2 r √ 1 + r 2 H r ∂ ∂r ( r 2 H r ) + 2 r √ 1 + r 2 H r ∂ ∂r ( r 2 H r ) + r 2 (1 + r 2 ) 3 2 | H r | 2 = 2 √ 1 + r 2 r 2 ∣ ∣ ∣ ∣ ∂ ∂r ( r 2 H r ) ∣ ∣ ∣ ∣ 2 + r 2 (1 + r 2 ) 3 2 | H r | 2 +Div K.</formula>
<text><location><page_55><loc_15><loc_60><loc_85><loc_63></location>Here we have used the constraint equation to pass from the first to the second line. This completes the proof.</text>
<text><location><page_55><loc_15><loc_55><loc_85><loc_57></location>7.3. Proof of Lemma 5.7. This is a straightforward calculation with the formula for the divergence of a vector field (19). We find</text>
<formula><location><page_55><loc_18><loc_22><loc_82><loc_53></location>Div K = 1 r 2 ∂ ∂r ( 2 r 5 √ 1 + r 2 / ∇ C H BC H B r -∣ ∣ r 3 H Br ∣ ∣ 2 ) -/ ∇ C ( 2 r 2 H Br √ 1 + r 2 ∂ ∂r [ r (1 + r 2 ) H B C ] ) = 2 rH Br √ 1 + r 2 ∂ ∂r [ r 2 (1 + r 2 ) / ∇ C H B C ] +2(1 + r 2 ) / ∇ C H BC ∂ ∂r [ r 3 √ 1 + r 2 H B r ] -2 rH Br √ 1 + r 2 ∂ ∂r [ r 2 (1 + r 2 ) / ∇ C H B C ] -2 r 2 √ 1 + r 2 / ∇ C H Br ∂ ∂r [ r (1 + r 2 ) H B C ] -2 rH Br ∂ ∂r ( r 3 H B r ) = -2(1 + r 2 ) 3 2 r 3 ∂ ∂r ( r 3 H Br ) [ 1 √ 1 + r 2 ∂ ∂r ( r 3 H B r ) -r 4 (1 + r 2 ) 3 2 H B r ] -2 r 2 √ 1 + r 2 / ∇ C H Br ∂ ∂r [ r (1 + r 2 ) H B C ] -2 rH Br ∂ ∂r ( r 3 H B r ) = -2 r 2 √ 1 + r 2 / ∇ C H Br ∂ ∂r [ r (1 + r 2 ) H B C ] -2 1 + r 2 r 3 ∣ ∣ ∂ r ( r 3 H Br )∣ ∣ 2 .</formula>
<text><location><page_55><loc_15><loc_18><loc_85><loc_21></location>Here, we have used the constraint equation in passing from the second equality to the third by replacing / ∇ C H BC with a term involving ∂ r ( r 3 H Br ).</text>
<section_header_level_1><location><page_56><loc_45><loc_87><loc_55><loc_87></location>References</section_header_level_1>
<unordered_list>
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<list_item><location><page_56><loc_16><loc_77><loc_81><loc_80></location>[3] H. Friedrich, 'Existence and structure of past asymptotically simple solutions of Einstein's field equations with positive cosmological constant,' J. Geom. Phys. 3 (1986) 101-117.</list_item>
<list_item><location><page_56><loc_16><loc_75><loc_83><loc_77></location>[4] P. Hintz and A. Vasy, 'The global non-linear stability of the Kerr-de Sitter family of black holes,' Acta Mathematica, 220:1-206, 2018. arXiv:1606.04014 [math.DG].</list_item>
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</document> | [] |
2022ApJ...925..160F | https://arxiv.org/pdf/2109.12807.pdf | <document>
<section_header_level_1><location><page_1><loc_17><loc_85><loc_83><loc_86></location>Consistency of Planck Data With Power-Law Primordial Scalar Power Spectrum</section_header_level_1>
<text><location><page_1><loc_29><loc_83><loc_70><loc_84></location>Marzieh Farhang 1 and Muhammad Sadegh Esmaeilian 1</text>
<text><location><page_1><loc_26><loc_81><loc_73><loc_82></location>1 Department of Physics, Shahid Beheshti University, 1983969411, Tehran, Iran</text>
<section_header_level_1><location><page_1><loc_45><loc_78><loc_55><loc_79></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_57><loc_86><loc_77></location>In this work we explore the possibility of variations in the primordial scalar power spectrum around the power-law shape, as predicted by single-field slow-roll inflationary scenarios. We search for the trace of these fluctuations in a semi-blind, model-independent way in the observations of the Cosmic Microwave Background (CMB) sky. In particular we use two sets of perturbation patterns, specific patterns with typical features such as oscillations, bumps and transitions, as well as perturbation modes, constructed from the eigenanalysis of the forecasted or measured covariance of perturbation parameters. These modes, in principle, span the parameter space of all possible perturbations to the primordial spectrum, and when rank-ordered, the ones with the highest detectability would suffice to explore the constrainable features around the power-law spectrum in a data-driven (and not theoretically-biased) manner. With Planck measurements of CMB anisotropies, the amplitudes of all perturbation patterns considered in this work are found to be consistent with zero. This finding confirms, in the absence of theoretical biases, the consistency of the Planck data with the assumption of power-law inflationary pattern for the primordial spectrum.</text>
<section_header_level_1><location><page_1><loc_20><loc_54><loc_36><loc_55></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_39><loc_48><loc_53></location>The inflationary paradigm is the most widely accepted scenario to seed fluctuations in the temperature and polarization of the Cosmic Microwave Background (CMB) and the large scale distribution of matter in the Universe. The predictions of the simplest class of inflationary models, i.e., the single-field slow-roll inflation, for the primordial perturbations are Gaussian scalar and tensor fluctuations, described by power-law spectra (Starobinski ˇ i 1979; Linde 1982),</text>
<formula><location><page_1><loc_11><loc_35><loc_45><loc_37></location>P s ( k ) = A s ( k/k p , s ) 1 -n s , P t ( k ) = A t ( k/k p , t ) n t</formula>
<text><location><page_1><loc_8><loc_17><loc_48><loc_34></location>with A s , t and n s , t standing for the amplitudes and tilts of scalar (s) and tensor (t) perturbations and k p is the corresponding pivot scale. These predictions for the primordial scalar power spectrum are in great agreement with CMB observations with ln(10 10 A s ) = 3 . 044 ± 0 . 014 and n s = 0 . 9649 ± 0 . 0042 at k p , s = 0 . 05Mpc -1 , whereas the %95 upper bound on the amplitude of tensor power spectrum, parametrized by the tensor-to-scalar ratio, is r ≤ 0 . 10 at k p , t = 0 . 002Mpc -1 (Planck Collaboration et al. 2020a,b). In this work our focus is on the primordial scalar power spectrum, abbreviated as PSPS.</text>
<text><location><page_1><loc_8><loc_9><loc_48><loc_17></location>Despite the great achievement, there could still be small deviations around these predictions. Various scenarios of early Universe make clear predictions for the specific patterns of the PSPS. For instance see Danielsson (2002); Martin & Brandenberger (2003); Bozza et al.</text>
<text><location><page_1><loc_52><loc_36><loc_92><loc_55></location>(2003); Chen (2011); Jackson & Shiu (2013); Flauger et al. (2017) for models of global oscillation and Adams et al. (2001); Chen et al. (2007); Ach'ucarro et al. (2011); Miranda et al. (2012); Bartolo et al. (2013) for localized oscillatory features. The parameters of the various models of early Universe have been constrained by different cosmological data (e.g., Meerburg et al. 2014; Beutler et al. 2019; Planck Collaboration et al. 2020b). Forecasts have also been made on the detectability of their imprints with future surveys (e.g., Huang et al. 2012; Chen et al. 2016; Ballardini et al. 2016; Xu et al. 2016; Beutler et al. 2019; Li et al. 2021).</text>
<text><location><page_1><loc_52><loc_9><loc_92><loc_36></location>A parallel and complementary approach to this theoretically motivated path would be a model-independent analysis. In this semi-blind approach, one relaxes general degrees of freedom in the parameter space of all perturbations to the power-law PSPS, as many as numerically feasible (and required), and allow for the data to find and construct the perturbation patterns that are most tightly constrainable. This non-parametric search for deviation around power-law spectrum in Planck data was investigated in Planck Collaboration et al. (2020b). Also see Zhao et al. (2009); Ishida & de Souza (2011); Farhang et al. (2012); Hall et al. (2013); Sapone et al. (2014); Regan & Munshi (2015); Feng & Li (2016); Huang & Wang (2017); Taylor et al. (2018); Farhang & Vafaei Sadr (2019); Sharma et al. (2020) for examples of the application of this method in different contexts in cosmology. In particular Esmaeilian et al. (2021) con-</text>
<text><location><page_2><loc_8><loc_89><loc_48><loc_92></location>struct the perturbation eigenmodes to the PSPS for future CMB-S4 like and large scale surveys.</text>
<text><location><page_2><loc_8><loc_71><loc_48><loc_88></location>Our goal here is to investigate the consistency of Planck observations with the power-law PSPS in an enhanced parameter space with different sets of degrees of freedom to cover different sorts of fluctuations. The main parameter set would be perturbation eigenmodes constructed for Planck data. The major results of this work are presented in Figure 2, showing data-driven trajectories of possible perturbations to the PSPS. As is evident from the figure, no significant deviation is detected and the PSPS is found to be consistent with the power-law spectrum.</text>
<text><location><page_2><loc_8><loc_60><loc_48><loc_71></location>The organization of the paper is as follows. In Section 2 the dataset and simulations are briefly introduced. We then discuss the analysis details and introduce the patterns of perturbations explored in this work in Section 3. The results are presented in Section 4. Section 5 closes the paper with our final words and a discussion of the results.</text>
<section_header_level_1><location><page_2><loc_17><loc_58><loc_39><loc_59></location>2. SIMULATIONS AND DATA</section_header_level_1>
<text><location><page_2><loc_8><loc_45><loc_48><loc_57></location>In this work we use Planck observations of fluctuations in CMB temperature and polarization (Planck Collaboration et al. 2020c) to probe the physics of the early Universe through its impact on the CMB power spectra. In parts we also use simulations of the CMB power spectra, generated by the publicly available Boltzmann code CAMB 1 and Planck noise 2 for comparison with results from real data, as will be discussed in Section 4.</text>
<section_header_level_1><location><page_2><loc_23><loc_42><loc_33><loc_44></location>3. ANALYSIS</section_header_level_1>
<text><location><page_2><loc_8><loc_36><loc_48><loc_42></location>We assume the underlying PSPS of the Universe is close to the slow-roll power-law inflationary spectrum, P 0 ( k ), with possible small deviations ∆ P ( k ) around it,</text>
<formula><location><page_2><loc_13><loc_34><loc_48><loc_36></location>P ( k ) = P 0 ( k ) + ∆ P ( k ) = P 0 ( k )[1 + δ P ( k )] . (1)</formula>
<text><location><page_2><loc_8><loc_17><loc_48><loc_34></location>We follow the search for deviations in two different paths: search for specific, yet quite general patterns (Section 3.1) and semi-blind search (Section 3.2). The goal is then to measure the amplitudes and other free parameters of these patterns, as will be discussed below. For parameter estimation we sample the parameter space using the Cosmological Monte Carlo code, CosmoMC 3 (Lewis & Bridle 2002). This space consists of parameters characterizing perturbations to PSPS, along with standard cosmological parameters and the experimental nuisance parameters. In the following k min</text>
<text><location><page_2><loc_52><loc_87><loc_92><loc_92></location>and k max are the minimum and maximum wavenumbers considered in the analysis and we take ( k min , k max ) ∼ (0 . 004 , 1) h/ Mpc.</text>
<text><location><page_2><loc_52><loc_70><loc_92><loc_87></location>The search for features around the primordial powerlaw spectrum was also done in Planck Collaboration et al. (2020b). The main models used there for the reconstruction of perturbations were cubic-splines, a sum of several top hats, a penalized likelihood method and several specific patterns. Our method differs from their work in the reconstruction scheme based on the eigenanalysis of the covariance matrix of perturbations. We also search for different specific features in the spectrum except for the oscillatory pattern, which we keep for completeness.</text>
<section_header_level_1><location><page_2><loc_65><loc_67><loc_80><loc_68></location>3.1. Specific patterns</section_header_level_1>
<text><location><page_2><loc_52><loc_57><loc_92><loc_66></location>Here we introduce the three patterns of perturbations used in this work as generic possible forms of deviations around the power-law inflationary scalar power spectrum. The left panel in Figure 1 shows these patterns (up) and the response in the CMB temperature power spectrum (bottom) to small changes in their amplitudes.</text>
<text><location><page_2><loc_52><loc_52><loc_92><loc_55></location>Single Gaussian bump , referred to as Gbump, representing a local transient feature in k -space,</text>
<formula><location><page_2><loc_59><loc_49><loc_92><loc_51></location>δ P ( k ) = A exp[ -(ln k -ln k c ) 2 / 2 σ 2 ] (2)</formula>
<text><location><page_2><loc_52><loc_45><loc_92><loc_48></location>where A , k c and σ are the amplitude, center and width of the bump.</text>
<text><location><page_2><loc_52><loc_42><loc_83><loc_43></location>Transition in k -space, modeled by a tanh,</text>
<formula><location><page_2><loc_61><loc_39><loc_92><loc_41></location>δ P ( k ) = A tanh[ α (ln k -ln k tr )] (3)</formula>
<text><location><page_2><loc_52><loc_35><loc_92><loc_38></location>where A , k tr and α are the transition amplitude, wavenumber and width inverse.</text>
<text><location><page_2><loc_52><loc_30><loc_92><loc_33></location>Oscillations to model non-local patterns, extended in k -space,</text>
<formula><location><page_2><loc_65><loc_28><loc_92><loc_30></location>δ P ( k ) = A sin(2 πny ) (4)</formula>
<text><location><page_2><loc_52><loc_23><loc_92><loc_28></location>where A and n (positive integer) are the amplitude and frequency of the oscillation and y = (ln k -ln k min ) / ∆ln k and ∆ln k = ln( k max /k min ).</text>
<section_header_level_1><location><page_2><loc_63><loc_21><loc_81><loc_22></location>3.2. Semi-blind patterns</section_header_level_1>
<text><location><page_2><loc_52><loc_9><loc_92><loc_20></location>Alongside the search for the several specific patterns in the primordial spectrum, we also investigate the possibility of unknown deviations from the power-law PSPS, not properly expressible by the above patterns. Any general deviations can in principle be expanded using a complete set of base functions that span the full k -range of interest (see, e.g., Farhang et al. 2012, for</text>
<text><location><page_3><loc_23><loc_77><loc_24><loc_78></location>k</text>
<figure>
<location><page_3><loc_9><loc_77><loc_36><loc_92></location>
</figure>
<figure>
<location><page_3><loc_9><loc_61><loc_36><loc_76></location>
</figure>
<text><location><page_3><loc_24><loc_61><loc_24><loc_61></location>/lscript</text>
<figure>
<location><page_3><loc_37><loc_77><loc_64><loc_92></location>
</figure>
<figure>
<location><page_3><loc_65><loc_77><loc_91><loc_92></location>
</figure>
<figure>
<location><page_3><loc_37><loc_61><loc_64><loc_76></location>
</figure>
<figure>
<location><page_3><loc_64><loc_61><loc_92><loc_76></location>
<caption>Figure 1. Top: Deviations from the power-law inflationary prediction in the form of three specific patterns: an oscillation or OSC, a transition or TR, and a Gbump (left). Three Fisher-based eigenmodes or FEMs (middle) and three data-driven eigenmodes or dEMs (right). Bottom: Response in the CMB temperature power spectrum due to small changes in the amplitude of the above perturbations.</caption>
</figure>
<text><location><page_3><loc_8><loc_49><loc_48><loc_51></location>more details). We use three sets with different motivations as discussed below.</text>
<text><location><page_3><loc_8><loc_16><loc_48><loc_47></location>Gaussian bumps (Gbumps) As the first approximation to the expansion base functions, we use N Gbumps (Eq. 2). The bump centres are logarithmically spaced in the k -range, and their widths are taken to be σ = δ ln k/ 3 with δ ln k = (ln k max -ln k min ) /N . We refer to this case as the multi-Gaussian expansion. The Gbumps can be considered as naive, however tame, approximations to Dirac delta functions. We treat these Gbumps, in the limit of very large n , as the base functions of the N dimensional parameter space of all possible deviations to the scalar primordial power spectrum. Nevertheless, they have their numerical limitations. For instance, to cover all points in the k -space, the widths need to be chosen so that there is non-vanishing overlap between neighbouring Gbumps . This overlap, on the other hand, destroys the orthogonality of the bumps and can result in the correlation of the final eigenmodes constructed from the Gbumps. One could in principle correct for this error. However, as will be discussed in Section 5, this high precision is not required here.</text>
<text><location><page_3><loc_8><loc_9><loc_48><loc_14></location>Data-driven Eigenmodes of the covariance matrix (dEMs) The large number of Gbumps, and the correlations between their amplitudes, in particular for</text>
<text><location><page_3><loc_52><loc_38><loc_92><loc_51></location>neighbours, are expected to lead to relatively high uncertainties in the measurements. These large errors would render possible deviations hard to detect. We therefore go one step further and construct linear combinations of Gbumps with vanishing linear correlations. We call these combinations eigenmodes, or EMs, and rank order them based on their estimated uncertainties. We then keep only the EMs with the lowest errors and the rest are discarded.</text>
<text><location><page_3><loc_52><loc_27><loc_92><loc_37></location>For the EM construction, we first generate the correlation matrix, C , of the amplitudes of the N Gbumps through post processing the CosmoMC results. The goal is to linearly transform the expansion basis, here the n Gbumps, so that the representation of C be diagonal in the new basis. These new basis functions are constructed from the eigenvectors of C ,</text>
<formula><location><page_3><loc_59><loc_21><loc_92><loc_25></location>E i ( k ) = N ∑ α =1 X iα g α ( k ) , k = 1 , ..., N (5)</formula>
<text><location><page_3><loc_52><loc_9><loc_92><loc_20></location>where X is the matrix whose columns are the eigenvectors of C , g α ( k ) refers to the Gbumps and E i ( k ) is any of the N perturbation eigenmodes. The diagonality of the covariance matrix in the new basis and the (closeto)orthogonality of the Gbumps guarantee the linear uncorrelation of the eigenmodes. Any function representing possible deviations in the power spectrum can</text>
<text><location><page_3><loc_79><loc_77><loc_80><loc_78></location>k</text>
<text><location><page_4><loc_8><loc_79><loc_48><loc_92></location>be expanded in terms of this new basis. The expansion coefficients yield a measure of the contribution of each mode to the perturbation. Moreover, the eigenvalues of C represent the (squares of the) estimated errors of the eigenmodes as measured by the data in hand. The modes and the C glyph[lscript] response to changes in the PSPS in the form of these modes are shown in the right panel of Figure 1.</text>
<text><location><page_4><loc_8><loc_68><loc_48><loc_77></location>Fisher-based Eigenmodes (FEMs) For comparison, we also use eigenmodes of perturbations to PSPS constructed from the Fisher matrix. In our Fisher matrix F ( glyph[vector]q ), the parameter set glyph[vector]q are the amplitudes of the Gbumps and the simulations for Planck power spectrum represent the data d ,</text>
<formula><location><page_4><loc_19><loc_64><loc_48><loc_67></location>[ F ( glyph[vector]q )] αβ = -〈 ∂ 2 ln P f ∂q α ∂q β 〉 (6)</formula>
<text><location><page_4><loc_8><loc_41><loc_48><loc_63></location>where P f ≡ P f ( glyph[vector]q | d ) is the Bayesian posterior distribution of the parameter set glyph[vector]q for the dataset d and 〈 ... 〉 is the ensemble average. In the limit of Gaussian distribution for the parameters, one has F -1 = C and therefore the two matrices share eigenvectors. The uncorrelated modes of perturbations can thus be constructed from Fisher eigenvectors. The details of this approach is described in Esmaeilian et al. (2021). The only important difference is that the eigenmodes here are marginalized over the standard cosmological parameters, while the eigenmodes in Esmaeilian et al. (2021) were constructed with fixed standard parameters. See Farhang et al. (2012) for a detailed description on marginalized mode construction.</text>
<text><location><page_4><loc_8><loc_20><loc_48><loc_40></location>The modes from this approach are expected to differ (although not hugely) from the dEMs in several ways. The covariance matrix in the latter case was marginalized on numerous nuisance parameters (as well as standard parameters) while in the former the marginalization was only performed on the standard parameters. Moreover, the dEM covariance matrix was based on the sampling of the parameter space while for the FEMs the assumption of the Gaussianity of the C was assumed. The Gaussianity assumption of the covariance matrix should be treated with care as the likelihood surface for the many correlated, poorly constrainable amplitudes of the Gbumps is probably far from a perfect Gaussian.</text>
<section_header_level_1><location><page_4><loc_23><loc_18><loc_33><loc_19></location>4. RESULTS</section_header_level_1>
<text><location><page_4><loc_8><loc_9><loc_48><loc_17></location>We use Planck data to measure the free parameters of the perturbation patterns, introduced in Sections 3.1 and 3.2. The results are presented in Table 1. As is evident from the table, the measured amplitudes for the three specific patterns are consistent with zero. For the</text>
<table>
<location><page_4><loc_58><loc_87><loc_90><loc_91></location>
<caption>Table 1. The best-fit measurement of the amplitudes of perturbations to the PSPS in the form of a single Gaussian bump (Gbump), a transition (TR) and an oscillatory pattern (OSC). The estimated 1 σ uncertainty is marginalized over all other parameters included in the analysis, including the bump width and position in the Gbump, the width inverse and wavelength of the transition in the TR case and the frequency in the OSC scenario.</caption>
</table>
<table>
<location><page_4><loc_52><loc_67><loc_92><loc_72></location>
<caption>Table 2. The best-fit measurement of the amplitude of perturbations to the PSPS in the form of Fisher-based eigenmodes (FEMs) and data-driven eigenmodes (dEMs).</caption>
</table>
<text><location><page_4><loc_52><loc_30><loc_92><loc_58></location>blind search we use 60 Gbumps with their 60 amplitudes as the free parameters, along with the standard cosmological and observational nuisance parameters. All the amplitudes are found consistent with zero, as expected, since we do not see any hint (Table 1) for a deviating bump in the PSPS in the single Gbump scenario with varying k c . From these 60 Gbumps, we construct the eigenmodes dEMs, i.e., the E i ( k )'s in Eq. 5. The measured amplitudes of these eigenmodes are presented in Table 2, and are compared to the measurements of the Fisher-based eigenmodes. The analysis is performed with the first three modes in both cases, and no deviation from the power-law PSPS is observed. The result of the analysis with four eigenmodes was also null. It is interesting to note the huge gap between the estimated uncertainties of the amplitudes for the two sets of eigenmodes (Table 2) and the specific patterns of perturbations (Table 1).</text>
<text><location><page_4><loc_52><loc_17><loc_92><loc_30></location>Figure 2 illustrates the 1 σ trajectories in the k -space of perturbations to the PSPS spanned by the various values taken by the parameters in each scenario. The reconstructed perturbations in all scenarios are consistent with power-law PSPS. It should be noted that these results are not motivated by theoretical models and are intended to be unbiased by predictions of early Universe scenarios.</text>
<section_header_level_1><location><page_4><loc_66><loc_14><loc_78><loc_16></location>5. DISCUSSION</section_header_level_1>
<text><location><page_4><loc_52><loc_9><loc_92><loc_14></location>In this work we investigated, in a very general sense, the consistency of CMB data, as observed by Planck , with the predictions of the slow-roll single field infla-</text>
<text><location><page_5><loc_11><loc_84><loc_13><loc_84></location>)</text>
<text><location><page_5><loc_11><loc_83><loc_13><loc_84></location>k</text>
<text><location><page_5><loc_11><loc_83><loc_13><loc_83></location>(</text>
<text><location><page_5><loc_12><loc_83><loc_13><loc_83></location>p</text>
<text><location><page_5><loc_11><loc_82><loc_13><loc_83></location>δ</text>
<text><location><page_5><loc_13><loc_80><loc_14><loc_81></location>-</text>
<text><location><page_5><loc_13><loc_77><loc_14><loc_78></location>-</text>
<text><location><page_5><loc_13><loc_75><loc_14><loc_76></location>-</text>
<text><location><page_5><loc_14><loc_90><loc_14><loc_90></location>0</text>
<text><location><page_5><loc_14><loc_90><loc_14><loc_90></location>.</text>
<text><location><page_5><loc_14><loc_90><loc_15><loc_90></location>3</text>
<text><location><page_5><loc_14><loc_87><loc_14><loc_88></location>0</text>
<text><location><page_5><loc_14><loc_87><loc_14><loc_88></location>.</text>
<text><location><page_5><loc_14><loc_87><loc_15><loc_88></location>2</text>
<text><location><page_5><loc_14><loc_85><loc_14><loc_86></location>0</text>
<text><location><page_5><loc_14><loc_85><loc_14><loc_86></location>.</text>
<text><location><page_5><loc_14><loc_85><loc_15><loc_86></location>1</text>
<text><location><page_5><loc_14><loc_83><loc_14><loc_83></location>0</text>
<text><location><page_5><loc_14><loc_83><loc_14><loc_83></location>.</text>
<text><location><page_5><loc_14><loc_83><loc_15><loc_83></location>0</text>
<text><location><page_5><loc_14><loc_80><loc_14><loc_81></location>0</text>
<text><location><page_5><loc_14><loc_80><loc_14><loc_81></location>.</text>
<text><location><page_5><loc_14><loc_80><loc_15><loc_81></location>1</text>
<text><location><page_5><loc_14><loc_78><loc_14><loc_78></location>0</text>
<text><location><page_5><loc_14><loc_78><loc_14><loc_78></location>.</text>
<text><location><page_5><loc_14><loc_78><loc_15><loc_78></location>2</text>
<text><location><page_5><loc_14><loc_75><loc_14><loc_76></location>0</text>
<text><location><page_5><loc_14><loc_75><loc_14><loc_76></location>.</text>
<text><location><page_5><loc_14><loc_75><loc_15><loc_76></location>3</text>
<text><location><page_5><loc_18><loc_77><loc_20><loc_78></location>TR</text>
<text><location><page_5><loc_18><loc_76><loc_20><loc_77></location>OSC</text>
<text><location><page_5><loc_18><loc_75><loc_21><loc_76></location>Gbump</text>
<text><location><page_5><loc_18><loc_74><loc_19><loc_74></location>10</text>
<text><location><page_5><loc_19><loc_74><loc_20><loc_75></location>-</text>
<text><location><page_5><loc_20><loc_74><loc_20><loc_75></location>2</text>
<text><location><page_5><loc_25><loc_74><loc_26><loc_74></location>10</text>
<text><location><page_5><loc_26><loc_74><loc_27><loc_75></location>-</text>
<text><location><page_5><loc_27><loc_74><loc_27><loc_75></location>1</text>
<text><location><page_5><loc_33><loc_74><loc_34><loc_74></location>10</text>
<text><location><page_5><loc_34><loc_74><loc_34><loc_75></location>0</text>
<text><location><page_5><loc_40><loc_74><loc_41><loc_74></location>10</text>
<text><location><page_5><loc_41><loc_74><loc_41><loc_75></location>1</text>
<text><location><page_5><loc_30><loc_73><loc_30><loc_74></location>k</text>
<figure>
<location><page_5><loc_11><loc_53><loc_45><loc_72></location>
<caption>Figure 2. The reconstructed perturbations to the scalar power spectrum and the trajectories spanning the 1 σ region around them, for the Gbump, oscillatory pattern and transition, labeled respectively as Gbump, OSC and TR (top), and the first three FEMs and dEMs (bottom).</caption>
</figure>
<text><location><page_5><loc_8><loc_10><loc_48><loc_41></location>tionary models. Specifically, we explored whether there are hints for deviations from the power-law spectrum of primordial scalar fluctuations. We first searched for certain, yet general, features in the spectrum, such as a Gaussian bump (with varying position and width), a transition (with varying transition wavelength and width) and an oscillatory pattern (with varying frequency). In parallel we also constructed orthogonal basis functions for the parameter space of all perturbations to the PSPS (up to a certain resolution, determined by the number of used basis functions) and rank-ordered them based on their errors. This mode construction was done both with sampling the parameter space of perturbations and the likelihood surface exploration, and from Fisher matrix analysis. We then searched for deviations in terms of these eigenmodes, focusing mainly on the first few. We found no hints for deviations from the power-law spectrum in any of the above approaches, and the reconstructed PSPS was found to be fully consistent with the scale invariant scenario. Planck Collab-</text>
<text><location><page_5><loc_52><loc_87><loc_92><loc_92></location>oration et al. (2020b) also found null results in their nonparametric search for features in the primordial power spectrum.</text>
<text><location><page_5><loc_52><loc_53><loc_92><loc_87></location>There are two points in order here. First, our method was intended to be as blind as possible to any particular theoretical model of the early Universe. The Gaussian bumps, the transitionary patterns, and the oscillations, all with varying parameters, were used as rather typical features characterizing general functions and were not driven by theoretical biases. The eigenmode analysis was data-driven in the sense that, by construction, most detectable features would show up as the first few modes and their amplitudes would be measured in the next steps of the analysis. Therefore the null results indicate that data, by themselves, do not imply any fluctuations around the scale invariant spectrum. However, they do not rule out the possibility of detection of fluctuations with some certain patterns (different and not constructible form the ones explored here) predicted by given theoretical models. Nevertheless it should be noted that these possible detections are highly model dependent and theoretically biased. This strong prior imposed by theory requires in turn strong theoretical justification for the preference for the certain model over the zoo of many other models of the early Universe.</text>
<text><location><page_5><loc_52><loc_15><loc_92><loc_52></location>The second point is the relevance of the physics of the very early (inflationary) Universe in relaxing the Hubble tension, as reported in the disagreement between the local Universe measurements of the Hubble constant ( H 0 = 67 . 36 ± 0 . 54 km / s / Mpc) (Riess et al. 2021) and the inferred value from CMB data ( H 0 = 73 . 3 ± 0 . 8 km / s / Mpc) (Planck Collaboration et al. 2020a). For a thorough unbiased analysis, the relevant degrees of freedom for the model of the early Universe should be opened along with the standard cosmological parameters, including H 0 . To appropriately address the tension, the data of the local Universe should not be included in the analysis so that it can later be compared with the final (CMB-based) inference. The hope is that the extended parameter space, including the new degrees of freedom, may change the likelihood surface in a way that the local high H 0 value would lie in this extended parameter region allowed by CMB data. However, this is not what we find in our semi-blind search. In particular, we find H 0 = 67 . 4 ± 0 . 6 with the first three dEMs, implying that in the absence of theoretical priors, CMB data do not prefer a high H 0 value even in an extended parameter space encompassing new degrees of freedom in the early Universe.</text>
<section_header_level_1><location><page_5><loc_62><loc_10><loc_82><loc_11></location>6. ACKNOWLEDGEMENT</section_header_level_1>
<text><location><page_6><loc_8><loc_89><loc_48><loc_92></location>Part of the numerical computations of this work was carried out on the computing cluster of the Canadian In-</text>
<text><location><page_6><loc_52><loc_89><loc_92><loc_92></location>stitute for Theoretical Astrophysics (CITA), University of Toronto.</text>
<section_header_level_1><location><page_6><loc_44><loc_85><loc_56><loc_86></location>REFERENCES</section_header_level_1>
<table>
<location><page_6><loc_8><loc_17><loc_48><loc_84></location>
</table>
<table>
<location><page_6><loc_52><loc_18><loc_92><loc_84></location>
</table>
</document> | [{"title": "ABSTRACT", "content": "In this work we explore the possibility of variations in the primordial scalar power spectrum around the power-law shape, as predicted by single-field slow-roll inflationary scenarios. We search for the trace of these fluctuations in a semi-blind, model-independent way in the observations of the Cosmic Microwave Background (CMB) sky. In particular we use two sets of perturbation patterns, specific patterns with typical features such as oscillations, bumps and transitions, as well as perturbation modes, constructed from the eigenanalysis of the forecasted or measured covariance of perturbation parameters. These modes, in principle, span the parameter space of all possible perturbations to the primordial spectrum, and when rank-ordered, the ones with the highest detectability would suffice to explore the constrainable features around the power-law spectrum in a data-driven (and not theoretically-biased) manner. With Planck measurements of CMB anisotropies, the amplitudes of all perturbation patterns considered in this work are found to be consistent with zero. This finding confirms, in the absence of theoretical biases, the consistency of the Planck data with the assumption of power-law inflationary pattern for the primordial spectrum.", "pages": [1]}, {"title": "Consistency of Planck Data With Power-Law Primordial Scalar Power Spectrum", "content": "Marzieh Farhang 1 and Muhammad Sadegh Esmaeilian 1 1 Department of Physics, Shahid Beheshti University, 1983969411, Tehran, Iran", "pages": [1]}, {"title": "1. INTRODUCTION", "content": "The inflationary paradigm is the most widely accepted scenario to seed fluctuations in the temperature and polarization of the Cosmic Microwave Background (CMB) and the large scale distribution of matter in the Universe. The predictions of the simplest class of inflationary models, i.e., the single-field slow-roll inflation, for the primordial perturbations are Gaussian scalar and tensor fluctuations, described by power-law spectra (Starobinski \u02c7 i 1979; Linde 1982), with A s , t and n s , t standing for the amplitudes and tilts of scalar (s) and tensor (t) perturbations and k p is the corresponding pivot scale. These predictions for the primordial scalar power spectrum are in great agreement with CMB observations with ln(10 10 A s ) = 3 . 044 \u00b1 0 . 014 and n s = 0 . 9649 \u00b1 0 . 0042 at k p , s = 0 . 05Mpc -1 , whereas the %95 upper bound on the amplitude of tensor power spectrum, parametrized by the tensor-to-scalar ratio, is r \u2264 0 . 10 at k p , t = 0 . 002Mpc -1 (Planck Collaboration et al. 2020a,b). In this work our focus is on the primordial scalar power spectrum, abbreviated as PSPS. Despite the great achievement, there could still be small deviations around these predictions. Various scenarios of early Universe make clear predictions for the specific patterns of the PSPS. For instance see Danielsson (2002); Martin & Brandenberger (2003); Bozza et al. (2003); Chen (2011); Jackson & Shiu (2013); Flauger et al. (2017) for models of global oscillation and Adams et al. (2001); Chen et al. (2007); Ach'ucarro et al. (2011); Miranda et al. (2012); Bartolo et al. (2013) for localized oscillatory features. The parameters of the various models of early Universe have been constrained by different cosmological data (e.g., Meerburg et al. 2014; Beutler et al. 2019; Planck Collaboration et al. 2020b). Forecasts have also been made on the detectability of their imprints with future surveys (e.g., Huang et al. 2012; Chen et al. 2016; Ballardini et al. 2016; Xu et al. 2016; Beutler et al. 2019; Li et al. 2021). A parallel and complementary approach to this theoretically motivated path would be a model-independent analysis. In this semi-blind approach, one relaxes general degrees of freedom in the parameter space of all perturbations to the power-law PSPS, as many as numerically feasible (and required), and allow for the data to find and construct the perturbation patterns that are most tightly constrainable. This non-parametric search for deviation around power-law spectrum in Planck data was investigated in Planck Collaboration et al. (2020b). Also see Zhao et al. (2009); Ishida & de Souza (2011); Farhang et al. (2012); Hall et al. (2013); Sapone et al. (2014); Regan & Munshi (2015); Feng & Li (2016); Huang & Wang (2017); Taylor et al. (2018); Farhang & Vafaei Sadr (2019); Sharma et al. (2020) for examples of the application of this method in different contexts in cosmology. In particular Esmaeilian et al. (2021) con- struct the perturbation eigenmodes to the PSPS for future CMB-S4 like and large scale surveys. Our goal here is to investigate the consistency of Planck observations with the power-law PSPS in an enhanced parameter space with different sets of degrees of freedom to cover different sorts of fluctuations. The main parameter set would be perturbation eigenmodes constructed for Planck data. The major results of this work are presented in Figure 2, showing data-driven trajectories of possible perturbations to the PSPS. As is evident from the figure, no significant deviation is detected and the PSPS is found to be consistent with the power-law spectrum. The organization of the paper is as follows. In Section 2 the dataset and simulations are briefly introduced. We then discuss the analysis details and introduce the patterns of perturbations explored in this work in Section 3. The results are presented in Section 4. Section 5 closes the paper with our final words and a discussion of the results.", "pages": [1, 2]}, {"title": "2. SIMULATIONS AND DATA", "content": "In this work we use Planck observations of fluctuations in CMB temperature and polarization (Planck Collaboration et al. 2020c) to probe the physics of the early Universe through its impact on the CMB power spectra. In parts we also use simulations of the CMB power spectra, generated by the publicly available Boltzmann code CAMB 1 and Planck noise 2 for comparison with results from real data, as will be discussed in Section 4.", "pages": [2]}, {"title": "3. ANALYSIS", "content": "We assume the underlying PSPS of the Universe is close to the slow-roll power-law inflationary spectrum, P 0 ( k ), with possible small deviations \u2206 P ( k ) around it, We follow the search for deviations in two different paths: search for specific, yet quite general patterns (Section 3.1) and semi-blind search (Section 3.2). The goal is then to measure the amplitudes and other free parameters of these patterns, as will be discussed below. For parameter estimation we sample the parameter space using the Cosmological Monte Carlo code, CosmoMC 3 (Lewis & Bridle 2002). This space consists of parameters characterizing perturbations to PSPS, along with standard cosmological parameters and the experimental nuisance parameters. In the following k min and k max are the minimum and maximum wavenumbers considered in the analysis and we take ( k min , k max ) \u223c (0 . 004 , 1) h/ Mpc. The search for features around the primordial powerlaw spectrum was also done in Planck Collaboration et al. (2020b). The main models used there for the reconstruction of perturbations were cubic-splines, a sum of several top hats, a penalized likelihood method and several specific patterns. Our method differs from their work in the reconstruction scheme based on the eigenanalysis of the covariance matrix of perturbations. We also search for different specific features in the spectrum except for the oscillatory pattern, which we keep for completeness.", "pages": [2]}, {"title": "3.1. Specific patterns", "content": "Here we introduce the three patterns of perturbations used in this work as generic possible forms of deviations around the power-law inflationary scalar power spectrum. The left panel in Figure 1 shows these patterns (up) and the response in the CMB temperature power spectrum (bottom) to small changes in their amplitudes. Single Gaussian bump , referred to as Gbump, representing a local transient feature in k -space, where A , k c and \u03c3 are the amplitude, center and width of the bump. Transition in k -space, modeled by a tanh, where A , k tr and \u03b1 are the transition amplitude, wavenumber and width inverse. Oscillations to model non-local patterns, extended in k -space, where A and n (positive integer) are the amplitude and frequency of the oscillation and y = (ln k -ln k min ) / \u2206ln k and \u2206ln k = ln( k max /k min ).", "pages": [2]}, {"title": "3.2. Semi-blind patterns", "content": "Alongside the search for the several specific patterns in the primordial spectrum, we also investigate the possibility of unknown deviations from the power-law PSPS, not properly expressible by the above patterns. Any general deviations can in principle be expanded using a complete set of base functions that span the full k -range of interest (see, e.g., Farhang et al. 2012, for k /lscript more details). We use three sets with different motivations as discussed below. Gaussian bumps (Gbumps) As the first approximation to the expansion base functions, we use N Gbumps (Eq. 2). The bump centres are logarithmically spaced in the k -range, and their widths are taken to be \u03c3 = \u03b4 ln k/ 3 with \u03b4 ln k = (ln k max -ln k min ) /N . We refer to this case as the multi-Gaussian expansion. The Gbumps can be considered as naive, however tame, approximations to Dirac delta functions. We treat these Gbumps, in the limit of very large n , as the base functions of the N dimensional parameter space of all possible deviations to the scalar primordial power spectrum. Nevertheless, they have their numerical limitations. For instance, to cover all points in the k -space, the widths need to be chosen so that there is non-vanishing overlap between neighbouring Gbumps . This overlap, on the other hand, destroys the orthogonality of the bumps and can result in the correlation of the final eigenmodes constructed from the Gbumps. One could in principle correct for this error. However, as will be discussed in Section 5, this high precision is not required here. Data-driven Eigenmodes of the covariance matrix (dEMs) The large number of Gbumps, and the correlations between their amplitudes, in particular for neighbours, are expected to lead to relatively high uncertainties in the measurements. These large errors would render possible deviations hard to detect. We therefore go one step further and construct linear combinations of Gbumps with vanishing linear correlations. We call these combinations eigenmodes, or EMs, and rank order them based on their estimated uncertainties. We then keep only the EMs with the lowest errors and the rest are discarded. For the EM construction, we first generate the correlation matrix, C , of the amplitudes of the N Gbumps through post processing the CosmoMC results. The goal is to linearly transform the expansion basis, here the n Gbumps, so that the representation of C be diagonal in the new basis. These new basis functions are constructed from the eigenvectors of C , where X is the matrix whose columns are the eigenvectors of C , g \u03b1 ( k ) refers to the Gbumps and E i ( k ) is any of the N perturbation eigenmodes. The diagonality of the covariance matrix in the new basis and the (closeto)orthogonality of the Gbumps guarantee the linear uncorrelation of the eigenmodes. Any function representing possible deviations in the power spectrum can k be expanded in terms of this new basis. The expansion coefficients yield a measure of the contribution of each mode to the perturbation. Moreover, the eigenvalues of C represent the (squares of the) estimated errors of the eigenmodes as measured by the data in hand. The modes and the C glyph[lscript] response to changes in the PSPS in the form of these modes are shown in the right panel of Figure 1. Fisher-based Eigenmodes (FEMs) For comparison, we also use eigenmodes of perturbations to PSPS constructed from the Fisher matrix. In our Fisher matrix F ( glyph[vector]q ), the parameter set glyph[vector]q are the amplitudes of the Gbumps and the simulations for Planck power spectrum represent the data d , where P f \u2261 P f ( glyph[vector]q | d ) is the Bayesian posterior distribution of the parameter set glyph[vector]q for the dataset d and \u3008 ... \u3009 is the ensemble average. In the limit of Gaussian distribution for the parameters, one has F -1 = C and therefore the two matrices share eigenvectors. The uncorrelated modes of perturbations can thus be constructed from Fisher eigenvectors. The details of this approach is described in Esmaeilian et al. (2021). The only important difference is that the eigenmodes here are marginalized over the standard cosmological parameters, while the eigenmodes in Esmaeilian et al. (2021) were constructed with fixed standard parameters. See Farhang et al. (2012) for a detailed description on marginalized mode construction. The modes from this approach are expected to differ (although not hugely) from the dEMs in several ways. The covariance matrix in the latter case was marginalized on numerous nuisance parameters (as well as standard parameters) while in the former the marginalization was only performed on the standard parameters. Moreover, the dEM covariance matrix was based on the sampling of the parameter space while for the FEMs the assumption of the Gaussianity of the C was assumed. The Gaussianity assumption of the covariance matrix should be treated with care as the likelihood surface for the many correlated, poorly constrainable amplitudes of the Gbumps is probably far from a perfect Gaussian.", "pages": [2, 3, 4]}, {"title": "4. RESULTS", "content": "We use Planck data to measure the free parameters of the perturbation patterns, introduced in Sections 3.1 and 3.2. The results are presented in Table 1. As is evident from the table, the measured amplitudes for the three specific patterns are consistent with zero. For the blind search we use 60 Gbumps with their 60 amplitudes as the free parameters, along with the standard cosmological and observational nuisance parameters. All the amplitudes are found consistent with zero, as expected, since we do not see any hint (Table 1) for a deviating bump in the PSPS in the single Gbump scenario with varying k c . From these 60 Gbumps, we construct the eigenmodes dEMs, i.e., the E i ( k )'s in Eq. 5. The measured amplitudes of these eigenmodes are presented in Table 2, and are compared to the measurements of the Fisher-based eigenmodes. The analysis is performed with the first three modes in both cases, and no deviation from the power-law PSPS is observed. The result of the analysis with four eigenmodes was also null. It is interesting to note the huge gap between the estimated uncertainties of the amplitudes for the two sets of eigenmodes (Table 2) and the specific patterns of perturbations (Table 1). Figure 2 illustrates the 1 \u03c3 trajectories in the k -space of perturbations to the PSPS spanned by the various values taken by the parameters in each scenario. The reconstructed perturbations in all scenarios are consistent with power-law PSPS. It should be noted that these results are not motivated by theoretical models and are intended to be unbiased by predictions of early Universe scenarios.", "pages": [4]}, {"title": "5. DISCUSSION", "content": "In this work we investigated, in a very general sense, the consistency of CMB data, as observed by Planck , with the predictions of the slow-roll single field infla- ) k ( p \u03b4 - - - 0 . 3 0 . 2 0 . 1 0 . 0 0 . 1 0 . 2 0 . 3 TR OSC Gbump 10 - 2 10 - 1 10 0 10 1 k tionary models. Specifically, we explored whether there are hints for deviations from the power-law spectrum of primordial scalar fluctuations. We first searched for certain, yet general, features in the spectrum, such as a Gaussian bump (with varying position and width), a transition (with varying transition wavelength and width) and an oscillatory pattern (with varying frequency). In parallel we also constructed orthogonal basis functions for the parameter space of all perturbations to the PSPS (up to a certain resolution, determined by the number of used basis functions) and rank-ordered them based on their errors. This mode construction was done both with sampling the parameter space of perturbations and the likelihood surface exploration, and from Fisher matrix analysis. We then searched for deviations in terms of these eigenmodes, focusing mainly on the first few. We found no hints for deviations from the power-law spectrum in any of the above approaches, and the reconstructed PSPS was found to be fully consistent with the scale invariant scenario. Planck Collab- oration et al. (2020b) also found null results in their nonparametric search for features in the primordial power spectrum. There are two points in order here. First, our method was intended to be as blind as possible to any particular theoretical model of the early Universe. The Gaussian bumps, the transitionary patterns, and the oscillations, all with varying parameters, were used as rather typical features characterizing general functions and were not driven by theoretical biases. The eigenmode analysis was data-driven in the sense that, by construction, most detectable features would show up as the first few modes and their amplitudes would be measured in the next steps of the analysis. Therefore the null results indicate that data, by themselves, do not imply any fluctuations around the scale invariant spectrum. However, they do not rule out the possibility of detection of fluctuations with some certain patterns (different and not constructible form the ones explored here) predicted by given theoretical models. Nevertheless it should be noted that these possible detections are highly model dependent and theoretically biased. This strong prior imposed by theory requires in turn strong theoretical justification for the preference for the certain model over the zoo of many other models of the early Universe. The second point is the relevance of the physics of the very early (inflationary) Universe in relaxing the Hubble tension, as reported in the disagreement between the local Universe measurements of the Hubble constant ( H 0 = 67 . 36 \u00b1 0 . 54 km / s / Mpc) (Riess et al. 2021) and the inferred value from CMB data ( H 0 = 73 . 3 \u00b1 0 . 8 km / s / Mpc) (Planck Collaboration et al. 2020a). For a thorough unbiased analysis, the relevant degrees of freedom for the model of the early Universe should be opened along with the standard cosmological parameters, including H 0 . To appropriately address the tension, the data of the local Universe should not be included in the analysis so that it can later be compared with the final (CMB-based) inference. The hope is that the extended parameter space, including the new degrees of freedom, may change the likelihood surface in a way that the local high H 0 value would lie in this extended parameter region allowed by CMB data. However, this is not what we find in our semi-blind search. In particular, we find H 0 = 67 . 4 \u00b1 0 . 6 with the first three dEMs, implying that in the absence of theoretical priors, CMB data do not prefer a high H 0 value even in an extended parameter space encompassing new degrees of freedom in the early Universe.", "pages": [4, 5]}, {"title": "6. ACKNOWLEDGEMENT", "content": "Part of the numerical computations of this work was carried out on the computing cluster of the Canadian In- stitute for Theoretical Astrophysics (CITA), University of Toronto.", "pages": [6]}] |
2024arXiv241205373M | https://arxiv.org/pdf/2412.05373.pdf | <document>
<section_header_level_1><location><page_1><loc_6><loc_85><loc_94><loc_87></location>Sun-related variability in the light curves of compact radio sources</section_header_level_1>
<section_header_level_1><location><page_1><loc_28><loc_82><loc_72><loc_83></location>A new view on extreme scattering events</section_header_level_1>
<text><location><page_1><loc_35><loc_79><loc_65><loc_81></location>N. Marchili 1 , G. Witzel 2 , and M. F. Aller 3</text>
<unordered_list>
<list_item><location><page_1><loc_10><loc_75><loc_61><loc_77></location>1 Istituto di Radioastronomia - INAF, Via Piero Gobetti 101, 40129, Bologna (Italy) e-mail: [email protected]</list_item>
<list_item><location><page_1><loc_10><loc_72><loc_65><loc_75></location>2 Max-Planck Institut fuer Radioastronomie, Auf del Hügel 69, 53121, Bonn, Deutschland e-mail: [email protected]</list_item>
<list_item><location><page_1><loc_10><loc_70><loc_73><loc_72></location>3 Department of Astronomy, University of Michigan, 323 West Hall, Ann Arbor, MI 48109-1107, USA e-mail: [email protected]</list_item>
</unordered_list>
<text><location><page_1><loc_10><loc_68><loc_22><loc_69></location>Received ; accepted</text>
<section_header_level_1><location><page_1><loc_46><loc_65><loc_54><loc_66></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_10><loc_59><loc_90><loc_64></location>Context. Compact radio sources can show remarkable flux density variations at GHz frequencies on a wide range of timescales. The origin of the variability is a mix of source-intrinsic mechanisms and propagation e ff ects, the latter being generally identified with scattering from the interstellar medium. Some of the most extreme episodes of variability, however, show characteristics that are not consistent with any of the explanations commonly proposed.</text>
<text><location><page_1><loc_10><loc_54><loc_90><loc_59></location>Aims. An in-depth analysis of variability at radio frequencies has been carried out on light curves from the impressive database of the US Navy's extragalactic source monitoring program at the Green Bank Interferometer (GBI) - a long-term project mainly aimed at the investigation of extreme scattering events - complemented by UMRAO light curves for selected sources. The purpose of the present work is to identify events of flux density variations that appear to correlate with the position of the Sun.</text>
<text><location><page_1><loc_10><loc_51><loc_90><loc_54></location>Methods. The 2 GHz and 8 GHz light curves observed in the framework of the GBI monitoring program have been inspected in a search for one-year periodic patterns in the data. Variations on timescales below one year have been isolated through a de-trending algorithm and analysed, looking for possible correlations with the Sun's position relative to the sources.</text>
<text><location><page_1><loc_10><loc_46><loc_90><loc_51></location>Results. Objects at ecliptic latitude below ∼ 20 · show one-year periodic drops in flux densities, centred close to the time of minimum solar elongation; both interplanetary scintillation and instrumental e ff ects may contribute to these events. However, in some cases the drops extend to much larger angular distances, a ff ecting sources at high ecliptic latitudes, and causing variability on timescales of months. Three di ff erent kinds of such events have been identified in the data. Their exact nature is not yet known.</text>
<text><location><page_1><loc_10><loc_35><loc_90><loc_46></location>Conclusions. In the present study we show that the variability of compact radio sources is heavily influenced by e ff ects that correlate with solar angular distance; this unexpected contribution significantly alters the sources' variability characteristics estimated at GHz frequencies. In particular, we found that many extreme scattering events previously identified in the GBI monitoring program are actually the consequence of Sun-related e ff ects; others occur simultaneously in several objects, which excludes interstellar scattering as their possible cause. These discoveries have a severe impact on our understanding of extreme scattering events. Furthermore, Sunrelated variability, given its amplitude and timescale, can significantly alter results of variability studies, which are very powerful tools for the investigation of active galactic nuclei. Without a thorough comprehension of the mechanisms that cause these variations, the estimation of some essential information about the emitting regions, such as their size and all the derived quantities, might be seriously compromised.</text>
<text><location><page_1><loc_10><loc_33><loc_20><loc_34></location>Key words. - -</text>
<section_header_level_1><location><page_1><loc_6><loc_29><loc_18><loc_30></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_6><loc_20><loc_49><loc_28></location>Variability studies provide us with a powerful tool to investigate the nature of the emission from extragalactic compact radio sources. For these objects, variability can appear over a wide range of timescales, going from hours (intrahour / intraday variability, hereafter IDV; see Witzel et al. 1986, Heeschen et al. 1987) to many years (see e.g. Ulrich et al. 1997).</text>
<text><location><page_1><loc_6><loc_10><loc_49><loc_20></location>It is generally assumed that the origin of the observed flux density variations in the radio is either intrinsic to the sources or caused by interstellar scintillation (ISS; see, e.g., Wagner & Witzel 1995; Rickett et al. 2006), a propagation e ff ect due to the presence of a screen of interstellar medium (ISM) along the line of sight between the source and the observer. The importance of scattering in the ISM on centimetre band data was demonstrated, among others, by Pushkarev & Kovalev (2015) and Koryukova</text>
<text><location><page_1><loc_51><loc_10><loc_94><loc_30></location>et al. (2022). In Marchili et al. (2011) it was shown that, on IDV timescales, a further contribution to the variability is provided by the Sun, either through local propagation e ff ects (i.e. interplanetary scintillation, IPS), or, indirectly, through instrumental e ff ects rising when a source's angular distance to the Sun, that is its solar elongation, is low. The analysis of 5 GHz light curves from the Urumqi Observatory and the 4.9 GHz ones from the MASIV survey (Lovell et al. 2003) at the Very Large Array (VLA) revealed a significant increase of the intraday variability (up to a factor 2) of the analysed sources as their solar elongation decreases. The aim of the present study is to assess how the Sun can a ff ect the variability curves of compact radio sources on longer timescales, from a few days up to one year. This goal was pursued by searching for variability features that cannot be explained in terms of source-intrinsic processes or through ISS, such as solar-elongation-dependent flux density variations</text>
<text><location><page_2><loc_6><loc_76><loc_49><loc_93></location>or correlated variability among many objects. Given the variety of manifestations of the e ff ects we found, the results of our analysis have been split in two publications; the present one (Paper I) deals with sharp flux density variations similar to extreme scattering events (ESEs, Fiedler et al. 1987a). The second publication (Witzel et al., in prep.; from now on, Paper II) focuses on smooth one-year periodic variations in radio light curves. At this stage of the work, it is not known whether and how the two phenomena are related. To reach our goal, we inspected one of the best available databases for the study of compact objects' variability at GHz frequencies, namely the US Navy's extragalactic source monitoring program (henceforth, NESMP; see Fiedler et al. 1987b, Waltman et al. 1991, and Lazio et al. 2001b).</text>
<text><location><page_2><loc_6><loc_59><loc_49><loc_76></location>The NESMP was a project running at the Green Bank Interferometer (GBI) between 1979 and 1996, whose primary aim was to search for ESEs. These are a class of dramatic variations in the flux density of compact radio sources, characterised by a substantial decrease bracketed by less pronounced increases. Extreme scsttering events have a typical timescale between a few weeks and several months; their origin is generally attributed to the scattering of radio waves in the interstellar medium. These events are rare: Fiedler et al. (1994) estimated that, taking into account a cumulative observing time of about 600 source-years for the 330 objects they monitored, the timespan covered by unusual variability associated to ESEs amounts to 4.8 y, corresponding to about 0.8% of the total time.</text>
<text><location><page_2><loc_6><loc_33><loc_49><loc_59></location>Athorough analysis of Sun-related variability (from now on, SRV) is crucial on several levels. Given the importance of variability studies to constrain basic properties of emitting sources (e.g. emitting regions' sizes, and, from these, brightness temperatures and Doppler factors; Hovatta et al. 2009, Liodakis et al. 2017), and to understand the origin and the evolution of flares (see, e.g., Marscher et al. 1992, Ulrich et al. 1997, Marscher 2016), it is necessary to identify and quantify all non-intrinsic sources of variability in the light curves; the contribution of SRV to blazar variability at GHz frequencies seems to be quite significant. The nature of SRV is a puzzling and fascinating topic: the evidence we collected rules out an instrumental origin at least for some of the SRV manifestations. Propagation e ff ects seem to be the most obvious candidate to explain them, but, in the light of the known properties of IPS or of the Earth's upper atmospheric layers, these e ff ects would be expected to be generally negligible; this implies that our picture of local propagation e ff ects must be missing some important pieces. Last but not least, SRV has a strong impact on our understanding of ESEs, given the fact that their signature in the light curves appears to be the same.</text>
<text><location><page_2><loc_6><loc_22><loc_49><loc_33></location>After a general description of the database and the procedures used for the data analysis (Sect. 2), we will present the four di ff erent kinds of SRV manifestations we identified in the data (Sec 3) and we will illustrate our classification of the sources (Sect. 4). Some hypotheses about the origin of the variability will be discussed in Sect. 5, while Sect. 6 will be dedicated to a revision of the identified ESEs in Lazio et al. (2001b), in the light of the new evidence we collected. The main conclusions of the present work will be presented in Sect. 7.</text>
<section_header_level_1><location><page_2><loc_6><loc_18><loc_25><loc_19></location>2. Data and procedures</section_header_level_1>
<text><location><page_2><loc_6><loc_10><loc_49><loc_17></location>The NESMP observations were performed at two radio frequencies, of approximately 2.5 and 8.2 GHz. Flux density measurements were collected for 149 objects, with a typical cadence of one observation every two days. Information about the GBI can be found in Hogg et al. (1969) and Coe (1973). The procedures for the data calibration and the results of the data analysis have</text>
<text><location><page_2><loc_51><loc_88><loc_94><loc_93></location>been thoroughly discussed in Fiedler et al. (1987b), Waltman et al. (1991), and Lazio et al. (2001b). Here below a short summary of the main aspects regarding data acquisition and calibration is presented.</text>
<text><location><page_2><loc_51><loc_71><loc_94><loc_88></location>Observations were carried out on a 2.4 km baseline, initially at frequencies of 2.7 GHz (S band) and 8.1 GHz (X band). In September 1989 the installation of cryogenic receivers brought a change of the observed frequencies, which became 2.25 GHz and 8.3 GHz respectively. Four sources were used for the calibration of raw data; 1328 + 307 was used to fix the individual flux densities of 0237-233, 1245-197, 1328 + 254; the flux density measurements of these three sources were combined to form a hybrid calibrator for all the other targets. A deeper look into this procedure will be given in Appendix A. According to Fiedler et al. (1987b), the uncertainties of GBI's flux density measurements before the installation of the cryogenic receivers could be expressed as</text>
<formula><location><page_2><loc_51><loc_67><loc_94><loc_69></location>σ s 2 = (0 . 037 Jy) 2 + (0 . 014 S s) 2 (1)</formula>
<formula><location><page_2><loc_51><loc_63><loc_94><loc_64></location>σ x 2 = (0 . 057 Jy) 2 + (0 . 049 S x) 2 (2)</formula>
<text><location><page_2><loc_51><loc_59><loc_94><loc_62></location>while after completion of the operation (August 1989), they could be expressed (see Lazio et al. 2001b) as</text>
<formula><location><page_2><loc_51><loc_56><loc_94><loc_57></location>σ s 2 = (0 . 0037 Jy) 2 + (0 . 015 S s) 2 (3)</formula>
<formula><location><page_2><loc_51><loc_51><loc_94><loc_53></location>σ x 2 = (0 . 0057 Jy) 2 + (0 . 05 S x) 2 (4)</formula>
<text><location><page_2><loc_51><loc_45><loc_94><loc_50></location>where S s and S x indicate the measured flux densities in S and X band respectively. A further contribution to the uncertainties, of the order of 10 and 20% (for S and X band respectively), could be explained by atmospheric and hardware e ff ects.</text>
<text><location><page_2><loc_51><loc_36><loc_94><loc_45></location>Some basic information about the analysed sources are reported in Table 3, such as name (Col. 1), ecliptic latitude (Col. 2), time of minimum solar elongation (Col. 3), average flux density and standard deviations at 2 and 8 GHz (Col. 4-7). The Xray binary system 1909 + 048 has been excluded from the present analysis, as its source-intrinsic variability component is so strong to be absolutely dominating over any other kind of variability.</text>
<text><location><page_2><loc_51><loc_27><loc_94><loc_36></location>For a deeper study of SRV episodes in the blazars OJ 287 and 0528 + 134, we made use of 8 GHz and 14.5 GHz data from the University of Michigan Radio Astronomical Observatory (UMRAO; for more details see, for example, Aller et al. 1985, Aller et al. 1999), which allowed for both an extension of the analysis to higher frequencies and an essential consistency check between the 8 GHz data from the two facilities.</text>
<section_header_level_1><location><page_2><loc_51><loc_23><loc_76><loc_25></location>2.1. Algorithms for the data analysis</section_header_level_1>
<text><location><page_2><loc_51><loc_17><loc_94><loc_22></location>Two main procedures have been used for the analysis of the data: a de-trending procedure to remove the long-term variability and highlight the fast variations; a data-stacking procedure for identifying yearly patterns in the data.</text>
<text><location><page_2><loc_51><loc_10><loc_94><loc_17></location>The de-trending procedure (see Villata et al. 2002) consists of several steps: firstly, the flux densities of a light curve are averaged on time intervals of selected length; the resulting average fluxes are fitted by a spline curve, which is then interpolated with the same sampling of the original light curve; the re-sampled spline curve (i.e. the long-term trend) is subtracted</text>
<figure>
<location><page_3><loc_9><loc_75><loc_46><loc_94></location>
<caption>Fig. 1. The average amplitude (expressed in percentage of the mean flux density) of the systematic dips at 2 GHz is plotted, for each source, versus β . Particularly remarkable is the existence of sources showing SRV at latitudes higher than 20 · .</caption>
</figure>
<text><location><page_3><loc_6><loc_63><loc_49><loc_67></location>from the original flux densities, leaving only the variations on timescales smaller than the time intervals used for the initial data averaging.</text>
<text><location><page_3><loc_6><loc_55><loc_49><loc_63></location>The data-stacking procedure makes use of the de-trended data: these are divided into intervals of one year, which then are stacked on top of each other; the resulting variability curve as a function of day-of-the-year is finally averaged on bins of four days 1 , providing us with a mean yearly pattern of the fast variability.</text>
<section_header_level_1><location><page_3><loc_6><loc_51><loc_26><loc_52></location>3. SRV phenomenology</section_header_level_1>
<text><location><page_3><loc_6><loc_45><loc_49><loc_50></location>The light curve of some NESMP sources show one-year periodic dips at the time of minimum solar elongation, which were explained by Fiedler et al. (1987b) as due to the e ff ect of the Sun passing through a sidelobe of the primary antenna beam.</text>
<text><location><page_3><loc_6><loc_28><loc_49><loc_45></location>In order to identify and characterise the periodic variations in the data, we applied the data-stacking procedure (with a detrending interpolation timescale of ten months) to all sources in the NESMP catalogue. Along with the fast variability (with timescale from days to a few weeks) at very low solar elongation mentioned above, this analysis revealed three more kinds of systematics: slow (timescale of months) one-year periodic variability, correlated with solar elongation, mainly a ff ecting sources at low ( < 20 · ) ecliptic latitude; slow six-month periodic variability correlated with solar elongation; six-month periodic variability apparently not correlated with solar elongation. In all four cases (which will be separately discussed in the following), the flux density variations have shapes consistent with an ESE.</text>
<text><location><page_3><loc_6><loc_13><loc_49><loc_28></location>For sources showing flux density variations that resemble an ESE, centred at the time of minimum solar elongation, we fitted a gaussian profile - centred at minimum solar elongation too - to the stacked 2 GHz data. For each source, the amplitude and uncertainty of the fit have been used to quantify the dips' characteristics: these values are reported in Table 3, Col. 8. The dips' amplitudes are plotted in Fig. 1, divided by the average flux of the sources for a proper comparison. A clear dependence on ecliptic latitude, β , can be seen up to β ∼ 20 · ; at higher latitudes, instead, the correlation disappears, which is a clear indication of a di ff erent mechanism producing the variability.</text>
<figure>
<location><page_3><loc_54><loc_76><loc_91><loc_94></location>
<caption>Fig. 2. Seasonal flux density variations of 1253-055 at 2 and 8 GHz, calculated by de-trending the light curves on bins of ten months. The de-trended data are plotted versus day of the year (DoY); black and blue squares show the best fitting gaussian models. At 2 GHz, the e ff ect of the Sun starts around Doy 265 and ends around DoY 298, indicating an influence spreading to solar elongations up to ∼ 16 · . For the 8 GHz data the influence is limited to ∼ 8 · .</caption>
</figure>
<text><location><page_3><loc_51><loc_57><loc_94><loc_64></location>It is important to stress that the vast majority of dips identified in the annual pattern of the data through the data-stacking procedure occurs either at minimum or at maximum solar elongation, which shows that the appearance of systematic dips in the light curves correlates with the position of the Sun with respect to the sources.</text>
<section_header_level_1><location><page_3><loc_51><loc_53><loc_85><loc_54></location>3.1. Fast SRV at low ecliptic latitude (type I SRV)</section_header_level_1>
<text><location><page_3><loc_51><loc_34><loc_94><loc_52></location>The mean yearly pattern of 1253-055 shown in Fig. 2 nicely illustrates the typical e ff ect of the Sun on sources at low ecliptic latitude at di ff erent frequencies. Systematic dropouts at 2 and at 8 GHz are comparable in amplitude, with flux density variations up to ∼ 30%, but not in width: at 2 GHz data are a ff ected up to a solar elongation of 15-20 · , while at 8 GHz the solar influence does not go further than 8 · . The characteristics of this kind of SRV (which henceforth, for simplicity, will be denoted as type I SRV) are compatible with the explanation provided by Fiedler et al. (1987b) in terms of an instrumental e ff ect, due to the Sun passing through a sidelobe of the primary antenna beam. Afurther contribution to the variability may also come from IPS, whose e ff ect is expected to become important at very low solar elongation.</text>
<section_header_level_1><location><page_3><loc_51><loc_30><loc_92><loc_32></location>3.2. Annual SRV up to high solar elongation (type IIa SRV)</section_header_level_1>
<text><location><page_3><loc_51><loc_24><loc_94><loc_29></location>The annual pattern of some NESMP sources shows a systematic flux density decrease, correlated with solar elongation, extending over a very large interval of time. An example of this kind of variability is provided by the light curves of OJ 287.</text>
<text><location><page_3><loc_51><loc_12><loc_94><loc_24></location>OJ 287 is among the best studied blazars; its fame grew considerably with the discovery of a long-term ( ∼ 12 y) periodic pattern in the optical band light curve (Sillanpää et al. 1988). It has an ecliptic latitude of 2.6 · , which makes it a natural candidate to show type I SRV. We analysed the OJ 287 data from both the NESMP (2 and 8 GHz) and the UMRAO (8 and 14.5 GHz) monitoring campaigns, to check the consistency among flux density measurements with di ff erent telescopes, and to extend the investigation of slow SRV to higher frequencies.</text>
<text><location><page_3><loc_51><loc_10><loc_94><loc_12></location>The data-stacking procedure reveals the existence of type I SRV at 2 GHz, as expected. Surprisingly, at 8 and 14.5 GHz</text>
<figure>
<location><page_4><loc_9><loc_74><loc_46><loc_93></location>
<caption>Fig. 3. The 1983 type IIa SRV event in 0742 + 103 and OJ 287 (left and right panel, respectively). The event is stronger at 8 (red dots) than at 2 GHz (black dots), and even stronger at 14.5 GHz (magenta dots, from the UMRAO archive). Note the excellent agreement between GBI and UMRAO 8 GHz data (green dots). Turquoise lines mark the times of minimum solar elongation for the two sources, while orange lines indicate the beginning and end of the events.</caption>
</figure>
<text><location><page_4><loc_6><loc_54><loc_49><loc_62></location>the dip is much larger than at 2 GHz, as it extends from about 0 . 45 to 0 . 70 -0 . 75 y, which correspond to a solar elongation range up to 50 -55 · . The dips do not recur systematically every year, but they occur frequently: an example of two consecutive dips, in 1983 and 1984, is shown in Fig. 3, right panel. The two dips, which are almost identical in amplitude and duration, are stronger at 8 and 14.5 GHz than at 2 GHz.</text>
<text><location><page_4><loc_6><loc_33><loc_49><loc_53></location>Particularly important is the simultaneity of the flux density variations in the di ff erent bands. Generally, the variability of OJ 287 shows spectral evolution: a correlation analysis of the light curves through locally normalised discrete correlation function (henceforth NDCF, see Lehar et al. 1992; Edelson & Krolik 1988) reveals that the 14 GHz variations occur, on average, about 0.08 y earlier than at 8 GHz, and about 0.25 y earlier than at 2 GHz. At minimum solar elongation, however, the time delays between these frequencies drops to zero. Also remarkable, in Fig. 3, is the occurrence, in 1983, of a flux density drop in 0742 + 103 ( β = -10.9 · ), similar in amplitude and duration to the one in OJ 287, and similarly stronger at high rather than at low frequencies (see Fig. 3, left panel). Given the small angular distance between the sources, it is very likely that these dips have the same origin. A weaker dip, partially hidden by the noise, is visible in the 0742 + 103 light curve in 1984 too, at 8 GHz.</text>
<text><location><page_4><loc_6><loc_19><loc_49><loc_32></location>More sources show evidence of the same broad systematic dips at minimum solar elongation found in OJ 287 (which henceforth we will address as type IIa SRV); they are indicated in Table 3, Col. 9 and 10, for the 2 and the 8 GHz light curves separately. Several of them are very close to OJ 287 in right ascension and declination. Type IIa SRV at 2 GHz is often associated with the same kind of variability at 8 GHz, although exceptions are not rare: in 2059 + 034, for example, two strong ESE-like variability episodes at minimum solar elongation can be found at 2 GHz (at 1990.13 and 1992.10), but not at 8 GHz.</text>
<section_header_level_1><location><page_4><loc_6><loc_15><loc_47><loc_17></location>3.3. Semi-annual SRV up to high solar elongation (type IIb SRV)</section_header_level_1>
<text><location><page_4><loc_6><loc_10><loc_49><loc_13></location>A rather peculiar manifestation of solar-elongation-dependent variability is the existence of six-month periodic dips, similar to ESEs, in the light curves of some NESMP sources (indicated</text>
<figure>
<location><page_4><loc_54><loc_75><loc_91><loc_93></location>
<caption>Fig. 4. The 2 GHz light curve of 0537-158. Between 1990 and 1993 the variability of the source appears as a sequence of six-month separated ESEs, centred at the time of minimum and maximum solar elongation of the source (indicated, respectively, with green and orange arrows). In the upper right box, as a comparison, a zoom-in on the 2 GHz 1981 event of 0954 + 658, which can be considered as the archetype of ESEs.</caption>
</figure>
<text><location><page_4><loc_51><loc_59><loc_94><loc_64></location>in Table 3, Col. 9 and 10); we will refer to it as type IIb SRV. The most striking example is given by the 2 GHz light curve of 0537-158 ( β = -39 · ), which, between 1990 and 1994, appears as a regular sequence of ESEs with six-month cadence (see Fig. 4).</text>
<text><location><page_4><loc_51><loc_48><loc_94><loc_59></location>The average duration of the events is of the order of months. Type IIb SRV generally a ff ects both 2 and 8 GHz light curves, although, similarly to the case of type IIa SRV, the way the two frequencies are a ff ected varies from source to source: 1830 + 285, for example, doesn't show any variability resembling an ESE at 8 GHz. Also, the e ff ect is not equally strong at minimum and at maximum solar elongation: generally the features at minimum are stronger than at maximum solar elongation, but in 0528 + 134 and 0954 + 658 the opposite occurs.</text>
<text><location><page_4><loc_51><loc_10><loc_94><loc_47></location>The case of 0528 + 134 deserves particular attention. In Lazio et al. (2001b), the source is reported as showing three ESEs, in 1991.0, 1993.5, and 1993.9, which makes it the most a ff ected object by ESEs in the whole monitoring program. All the reported events are consistent with the semi-annual cadence of type IIb SRV (the source reaches minimum solar elongation at 0.45 y). A comparison between the data collected within the NESMP with the ones from the UMRAO database (see Fig. 5) shows that the flux density measurements from the two facilities are in excellent agreement. Also in this case, it is important to stress the achromaticity of the flux density variations: in general, the variability of 0528 + 134 shows an overall trend of spectral evolution, with a delay, calculated via NDCF, of ∼ 0 . 17 y between 14.5 and 8 GHz data, and much larger between 8 and 2 GHz. Most of the variability observed on timescales of one year or shorter, however, is achromatic (the 14.5-8 GHz cross-correlation time delay of de-trended data is an impressive 0 . 00 ± 0 . 05 y). Along with the three ESEs identified by Lazio et al. (2001b), indicated with brown lines, three more achromatic dips can be seen in the data (turquoise lines), namely in 1993.0, 1995.1, and 1996.1. The last two are particularly strong at 14.5 GHz (orange dots). Another event, visible only at 2 GHz, occurs in 1988.9 (see the small box within the figure); this is a rare example of a complex ESE shape detected at this frequency. The semi-annual nature of the variability, correlated with solar elongation, is very clear, which confirms its attribution to SRV. The agreement between NESMP and UMRAOdata, and the importance of the e ff ect at 14.5 GHz, confirm the non-instrumental origin of type IIb SRV and its achromaticity within the range of investigated wavelengths.</text>
<figure>
<location><page_5><loc_9><loc_75><loc_46><loc_94></location>
<caption>Fig. 5. GBI and UMRAO light curves of 0528 + 134. Brown lines indicate the ESEs identified by Lazio et al. (2001b), while turquoise ones show three more simultaneous multi-frequency dips recognisable in the data. In the small box the 1988.9 event, a rare example of complex ESE shape observable at 2 GHz.</caption>
</figure>
<section_header_level_1><location><page_5><loc_6><loc_64><loc_42><loc_65></location>3.4. Time-dependent semi-annual variability (TDV)</section_header_level_1>
<text><location><page_5><loc_6><loc_53><loc_49><loc_63></location>So far, all the systematic variability we discussed showed a strict correlation to solar elongation, as the flux density minima occurred either at minimum or at maximum solar elongation. The NESMP data show a further interesting kind of ESE-like variability that is not directly ascribable to the relative position of the Sun to a source; it seems instead to be related to a specific time of the year. This kind of variability will be henceforth addressed as Time-Dependent Variability (TDV).</text>
<text><location><page_5><loc_6><loc_41><loc_49><loc_53></location>In Tables 5, 6, and 7 of Lazio et al. (2001b), which present lists of possible ESEs from the GBI monitoring program, eight sources are reported as showing identified or potential ESEs approximately at the same time , in 1993.5 (i.e. July 1993). A thorough check of the complete NESMP database reveals that a flux density drop compatible with the shape of an ESE is visible in many other sources. The light curves of some of them are plotted in the bottom panel of Fig. 6, after applying the de-trending procedure with time intervals of 50 days.</text>
<text><location><page_5><loc_6><loc_10><loc_49><loc_41></location>This event is both extraordinary and puzzling for several reasons. Among the a ff ected sources there is 0528 + 134, which has already been discussed as a ff ected by type IIb SRV, with sixmonth periodic ESE-like variability close to the time of minimum and maximum solar elongation, occurring at 0.45 and 0.95 y. The 1993.5 event occurs close to the minimum in solar elongation, therefore it is consistent with the solar-elongationrelated semi-annual trail of events. For many other sources (e.g., 1225 + 368, 1404 + 286, and 1438 + 385), however, the time of the dip does not come close either to a minimum or to a maximum of solar elongation, which demonstrates that the 1993.5 event is truly time-dependent, rather than solar elongation-dependent. Furthermore, the 1993.5 ESE of 0528 + 134 has been detected by Pohl et al. (1995) in data from the 100-m E ff elsberg telescope and the 30-m Pico Veleta telescope. This poses some questions: correlated variations simultaneously detected in many objects could point to an instrumental or a calibration problem, but how to explain the excellent consistency with the data from different telescopes? Also the interpretation of the variability of 0528 + 134 is problematic: is it solar-elongation dependent or time-dependent? This point will be addressed in Sect. 5, where we will discuss a possible link between the di ff erent manifestations of SRV and TDV. The hypothesis that TDV is due to a calibration problem is assessed in Appendix A.</text>
<figure>
<location><page_5><loc_54><loc_75><loc_91><loc_94></location>
<caption>Fig. 6. Upper panel: the 0528 + 134 ESE reported by Pohl et al. (1996) using data from the 100m-E ff elsberg telescope and the 30m Pico Veleta one. Bottom panel: some examples of the ESE-like feature in the GBI light curves at the very same time. The light curves are de-trended and shifted in flux for an easier comparison.</caption>
</figure>
<text><location><page_5><loc_51><loc_32><loc_94><loc_66></location>We analysed the combined variability characteristics of all NESMP sources as a function of time to check whether more episodes of correlated variability, similar to the July 1993 event, can be found in the data. All the 2 GHz light curves have been detrended, and then normalised, imposing the same standard deviation to all light curves, to ensure that the fast variability of all sources has a comparable weight in the calculation of the combined variability curve; finally, the data have been stacked. The resulting data-points have been averaged on intervals of 0.01 y to generate an average combined variability curve. This turned out to be characterised by a smooth one-year periodic pattern, which results from the combination of the annual oscillations in the light curves, discussed in Paper II; after modelling and removing this systematic modulation, we obtained the combined variability curve in Fig. 7. Episodes of time-correlated variability among many sources should result in clear variability features in the combined curve. The 1993.5 ESE is indeed recognisable as a short and sharp drop of the average flux (magenta arrow in the plot); six more events (orange arrows), all shaped approximately as an ESE, can be identified in the data, generally less intense but longer than the 1993.5 event. The temporal sequence of the events is the following: 1991.0, 1992.1, 1993.0, 1993.5, 1994.5, 1995.1, and 1995.9. The 1993.5 event appears to be an extreme manifestation of a long trail of similar phenomena with a cadence of ∼ 6 months, approximately falling at the end and in the middle of the year.</text>
<text><location><page_5><loc_51><loc_20><loc_94><loc_32></location>The semi-annual pattern is detectable, but without the same regularity, also in the 8 GHz data, which display ESE-shaped variability in 1990.5, 1993.0, 1994.5, and 1995.4. Before the year 1990 the periodic pattern, both at 2 and at 8 GHz, is less evident, as the variability in the combined variability curve is considerably stronger; most likely, this is due to the low number of sources monitored before the GBI upgrade (40 objects, less than a third of the later sample), which makes it harder to isolate correlated variability from the one that is intrinsic to the sources.</text>
<section_header_level_1><location><page_5><loc_51><loc_17><loc_68><loc_18></location>4. SRV classification</section_header_level_1>
<text><location><page_5><loc_51><loc_10><loc_94><loc_16></location>Through the analysis reported above we identified four distinct kinds of ESE-like variations in the data, which di ff er in duration, dependence (on solar elongation or time), and recurrence (which is the discriminating factor between type IIa and type IIb SRV). Their main properties are summarised in Table 1; they all have</text>
<figure>
<location><page_6><loc_9><loc_75><loc_46><loc_93></location>
<caption>Fig. 7. The combined 2 GHz variability curve of all the NESMP data: for each source, the flux densities have been de-trended, and normalised according their standard deviation σ ; afterwards, the data of all the sources have been stacked into a single light curve, which has been averaged over time bins of 0.01 y. A one-year periodic ripple, resulting from the combination of the annual oscillations in the light curves, has been removed from the stacked data. The regular sequence of dips, with average separation of 0.5 y, is indicated with orange arrows, while a magenta arrow is reserved for the 1993.5 ESE.</caption>
</figure>
<table>
<location><page_6><loc_7><loc_50><loc_48><loc_57></location>
<caption>Table 1. Main characteristics of the di ff erent types of systematic variability found in the data.. The acronym SE in the dependency (Dep.) column indicates solar elongation.</caption>
</table>
<text><location><page_6><loc_6><loc_43><loc_49><loc_47></location>in common the shape (resembling an ESE) and the achromaticity, although the importance of the e ff ect at di ff erent frequencies changes not only according to the variability type, but, often, also from source to source.</text>
<text><location><page_6><loc_6><loc_32><loc_49><loc_42></location>All sources of the NESMP have been classified in Col. 9 and 10 of Table 3 according to the kind of SRV found through a visual inspection of the 2 GHz and 8 GHz light curves, respectively. The reader should be warned that, while for some objects the classification is straightforward, for others it cannot be given without uncertainty. This is mainly because of the superposition of di ff erent e ff ects, and, for a few sources, because of the limited amount of available data.</text>
<text><location><page_6><loc_6><loc_16><loc_49><loc_32></location>Sources showing episodic variability (e.g. 0333 + 321) have been classified as a ff ected by SRV. The presence of timedependent variability, indicated in Col. 11, should be considered as purely indicative: TDV occurs in many sources in 1993.5; we classified a source as a ff ected by TDV if at least one more event at 0.0 or 0.5 y could be detected. There is an obvious ambiguity between TDV and type IIb SRV in sources whose minimum / maximum solar elongation occurs around 0.0 or 0.5 y. The duration of the events can often remove the ambiguity (type IIb SRV is characterised by longer timescales); sources for which the two e ff ects cannot be unambiguously identified are indicated as being a ff ected by both.</text>
<section_header_level_1><location><page_6><loc_6><loc_13><loc_34><loc_14></location>4.1. Automated classification of sources</section_header_level_1>
<text><location><page_6><loc_6><loc_10><loc_49><loc_12></location>Given some unavoidable degree of subjectivity in the classification of the sources according to a visual inspection, we devel-</text>
<text><location><page_6><loc_51><loc_88><loc_94><loc_93></location>oped a simple method for their automatic classification. The latter is not meant to replace the former, which is deeper and more accurate, but it provides an important tool to assess the reliability of its results.</text>
<text><location><page_6><loc_51><loc_81><loc_94><loc_88></location>It is based on the usage of NDCF, for the identification and localisation of dips in the light curves; the number of dips whose timing is consistent with SRV / TDV is compared with the null hypothesis that dips can randomly occur at any time of the year, independently of solar elongation.</text>
<text><location><page_6><loc_51><loc_41><loc_94><loc_81></location>More specifically, we created a generic dip model as a 0.2y V-shaped flux density drop, flanked by 0.1y-wide segments of constant flux. We cross-correlated the original light curve (i.e. without de-trending) of each source with the generic dip model. A high peak in the NDCF indicates the presence (and the location) of a feature in the light curve similar to the dip model. Peaks are considered significant if they are higher than a given threshold. For most sources, a degree of correlation of 0.4 produced a reasonable amount of events; when less than 2 or more 10 events were detected, the analysis has been repeated with a correlation threshold of 0.3 or 0.5, respectively. The choice of a variable threshold originates from the need to overcome two problems: a high level of noise in the data proportionally reduces the level of correlation between the light curve and the dip model, causing a decrease of detected events which can be compensated by a decrease of the threshold; a high level of source-intrinsic variability, on the other hand, causes spurious detections of dips, which requires a higher threshold to limit the number of false alarms. The detection of few or many events in a light curve with threshold 0.4 is a rather reliable indicator of the presence of either of the problems described above. By allowing the algorithm to overcome them through a variation of the threshold we can keep the procedure as automatic as possible. An example of how the algorithm works is provided in Fig. 8 for the type IIb SRV source 1830 + 285. In the lower panel, the peaks of the NDCF above the threshold correspond to detections of dips in the light curve (upper panel). It is worth to note the presence of negative dips in the NDCF, which correspond to rapid outbursts in the light curves. Since the algorithm we developed is very sensitive to sharp flux density variations, it can reveal, with opposite signs, both dips and flares.</text>
<text><location><page_6><loc_51><loc_37><loc_94><loc_41></location>The full list of identified events, for all sources, is reported in Appendix B; those that are undoubtably caused by TDV are shown in boldface.</text>
<text><location><page_6><loc_51><loc_15><loc_94><loc_37></location>For type I SRV, events are expected to occur exactly at the time of minimum solar elongation. For type IIa and IIb, SRV dips are broader, and sometimes their time of occurrence can be determined less precisely; therefore, NDCF peaks occurring within 0.05y from the solar elongation minimum (or maximum) are still consistent with type IIa (or type IIb) SRV. To take into account these di ff erences in the expected location of the dips, for each source we calculated four indicators: I. number of events occurring exactly at solar elongation minimum; II. number of events occurring within 0.05y from solar elongation minimum; III. number of events occurring exactly at solar elongation minimum or maximum; IV. number of events occurring within 0.05y from solar elongation minimum or maximum. Since the NDCF was calculated with a step of 0.05y, a chance occurrence of an event at solar elongation minimum for indicator I is 0.05; for indicator II it is 3x0.05 = 0.15; for indicator III 2x0.05 = 0.1; for indicator IV 6x0.05 = 0.3.</text>
<text><location><page_6><loc_51><loc_10><loc_94><loc_15></location>Through a binomial probability distribution, we can calculate the probability that the number of successes (i.e. the events occurring at minimum / maximum solar elongation) is consistent with the null hypothesis that they happen at random time. If the</text>
<figure>
<location><page_7><loc_7><loc_71><loc_48><loc_94></location>
<caption>Fig. 8. An example of the automatic detection of ESE-like dips in the light curves: the lower panel shows the NDCF output for the light curve of the type IIb-SRV source 1830 + 285 (plotted in the upper panel). Events above the threshold of 0.4 (red horizontal line) are regarded as detections. Blue and brown vertical lines indicate, respectively, the time of minimum and maximum solar elongation of the source.</caption>
</figure>
<text><location><page_7><loc_6><loc_55><loc_49><loc_60></location>probability is lower than 0.05 the source is classified as a ff ected by SRV. If the lowest probability is returned for indicators I or II, the source is classified as type I or IIa SRV, otherwise as type IIb SRV.</text>
<text><location><page_7><loc_6><loc_40><loc_49><loc_54></location>Subsequently, the same procedure has been applied three more times to the data: a first time after removing data-points obtained when the source's solar elongation is lower than 16 · ; if the resulting light curve is still a ff ected by SRV, it is classified as a type IIa, otherwise as a type I. A second time, after redefining successful detections as events occurring around day-of-the-year 0 (0.0y) or 182.5 (0.5y), in order to assess TDV. Lastly, after redefining successful detections as events occurring 0.25y before minimum / maximum solar elongation: as there is no reason to expect dips to systematically appear at this time, this test provides an estimate of the false positives returned by the analysis.</text>
<section_header_level_1><location><page_7><loc_6><loc_37><loc_16><loc_38></location>4.1.1. Results</section_header_level_1>
<text><location><page_7><loc_6><loc_18><loc_49><loc_36></location>Despite its simplicity, the algorithm is e ff ective. The original classification of sources is confirmed for 80% of the sample (120 out of 148 sources): 83 out of 90 sources are confirmed as showing no SRV; 19 out of 26 sources are confirmed as type I SRV; 5 out of 7 as type IIa; 8 out of 12 as type IIb (see Col. 13 and 15 of Table 3 for the automatic SRV and TDV classification; the probability of a random occurrence of the events is given in Col. 12 and 14). Among the uncertain cases, 4 out of 8 type IIa? and 1 out of 5 type IIb? are classified as type IIa and type IIb respectively. In total, with the automatic classification of sources we obtain 95 no-SRV sources, 19 type Ia SRV, 20 type IIa SRV, and 14 type IIb SRV (the classes of uncertain cases disappear). The group of 95 no-SRV sources includes also 11 objects for which too few data are available for a proper classification.</text>
<text><location><page_7><loc_6><loc_10><loc_49><loc_17></location>The search for sources highly a ff ected by TDV returned 9 objects, 4 of which are also showing type IIb SRV. Quite remarkable is the result provided by the algorithm when successes are defined as the events occurring 0.25y before minimum / maximum solar elongation: this way, only one object (i.e. 2234 + 282) out of 148 is identified as a ff ected by periodic dips,</text>
<text><location><page_7><loc_51><loc_91><loc_94><loc_93></location>which shows that the number of false positive detections to be expected in the entire sample is very low.</text>
<text><location><page_7><loc_51><loc_71><loc_94><loc_90></location>The list of events found through automatic detection can also be used to identify the most important TDV events. We divided the timespan of GBI observations (1979-1996) in bins of 0.05y width; after flagging dips that are consistent with SRV (because of their time of occurrence and the classification of the source in which they were detected), we counted the number of dips in each bin. The bins with the highest number of counts are the more likely to be a ff ect by TDV. By setting a threshold of 8 counts, we found the following sequence of TDV events: 1991.00, 1991.55, 1992.50, 1992.95-1993.00, 1993.50-.55. Note that before 1988 relatively few sources were monitored, so it is not possible to identify TDV events through this method. The sequence above does not coincide with the one obtained via the inspection of Fig. 7; the semi-annual cadence of TDV, however, is confirmed.</text>
<text><location><page_7><loc_51><loc_57><loc_94><loc_71></location>There seems to be no obvious similarity between the sources showing simultaneous dips in the light curves. We identified 10 events around 1991.00y that, because of their shape and duration, appear to be certainly due to TDV; they a ff ect the light curves of the following objects: 0316 + 413, 0337 + 319, 0400 + 258, 0723 + 679, 1123 + 264, 1150 + 812, 1538 + 149, 2007 + 776, 2032 + 107, 2105 + 420. The sources are widely spread both in right ascension and in declination, which seems to exclude both a common calibration scheme or a possible issue related to the pointing towards a specific region of the sky.</text>
<section_header_level_1><location><page_7><loc_51><loc_53><loc_60><loc_54></location>4.1.2. Caveat</section_header_level_1>
<text><location><page_7><loc_51><loc_37><loc_94><loc_52></location>A complete automation of the procedure is hampered by the ambiguity between TDV and SRV for sources reaching minimum / maximum solar elongation around 0.0 / 0.5y. If the algorithm detects an event around these times, it is necessary to proceed with a visual inspection to properly classify it, although in some cases the ambiguity cannot be resolved. If the shape of the event is equal to the one seen in the combined light curve (see Fig. 7) its classification as TDV is straightforward; if the shape / duration of the event is clearly di ff erent, the classification as SRV is obvious too. Sometimes, instead, the event's shape suggests a probable superposition of the two e ff ects, and therefore a classification as TDV + SRV.</text>
<text><location><page_7><loc_51><loc_10><loc_94><loc_37></location>The most complicated case, in this respect, is the one of 0954 + 658. Out of the 7 events found by the automated detection algorithm, 4 occur either around 0.00 or 0.50y, while only one (the archetype of the ESEs) is at maximum solar elongation; this would suggest TDV as the origin of the variability, rather than SRV. The duration of the events, however, is not consistent with the features in the 2 GHz combined variability curve. A further complication comes from the very high degree of correlation between the fast variability of the source (obtained through a de-trending interpolation timescale of ten months) and the ones of the angularly-nearby sources 0633 + 734, 0723 + 679, and 0836 + 710, with time delays that are consistent with the different solar elongation patterns followed by these objects. The correlation between the variability of di ff erent sources would be consistent with TDV, but the time delay between these variations implies a solar-elongation-related e ff ect. Finally, the archetype event of 1981.1 is not the only clear episode of sharp flux density drops at maximum solar elongation: even if not detected by the automatic algorithm, three more events can be seen (in 1989.10, 1991.10, and 1992.10), which explain the dips found in the annual pattern of the source (visible both at 2 and 8 GHz, see Fig.</text>
<text><location><page_8><loc_6><loc_91><loc_49><loc_93></location>C.1) and the visual classification of its variability as type IIb SRV.</text>
<text><location><page_8><loc_6><loc_83><loc_49><loc_90></location>Given the facts illustrated above, the most probable explanation for the fast variability of 0954 + 658 is that both TDV and SRV contribute to it. The similarities with the light curves of 0633 + 734, 0723 + 679, and 0836 + 710 suggest a common origin of the variability. Note that the complexity of such cases cannot be resolved through a fully automatic analysis of the data.</text>
<text><location><page_8><loc_6><loc_74><loc_49><loc_82></location>The algorithm has been applied also to the 8 GHz data. The results, however, in this case are much less reliable. The variety of shapes and widths of the dips caused by SRV / TDV makes it di ffi cult to create a single archetype that could e ffi ciently detect the majority of the events. Only 16 sources are automatically classified as a ff ected by type II SRV, and they often do not coincide with the 23 objects identified through visual inspection.</text>
<section_header_level_1><location><page_8><loc_6><loc_70><loc_17><loc_71></location>5. Discussion</section_header_level_1>
<text><location><page_8><loc_6><loc_56><loc_49><loc_69></location>Among SRV manifestations, only type I events can be satisfactorily explained so far (see Sect. 3.1). In Sect. 3.2, 3.3, and 3.4 we identified and described three more types of systematic variability in the data, namely type IIa SRV, IIb SRV, and TDV; there are ambiguities and exceptions in the general properties we outlined for each variability type, which do not facilitate the formulation of a robust hypothesis concerning the origin and the possible relationships among these e ff ects. It seems reasonable to assume that they are not independent of each other, and that there is a common ground to which the events can be attributed.</text>
<text><location><page_8><loc_6><loc_32><loc_49><loc_55></location>Some hints concerning the nature of the variability may come from the distribution in the sky of the a ff ected sources; however, the indications resulting from the visual inspection and the automatic classification of the sources are not very consistent among each other. The distribution obtained with the first method (see Fig. 9) suggests that type IIa SRV mainly a ff ects objects at low ecliptic latitude within the right ascension range from 04h to 09h (corresponding to solar elongation minima between 0.39 and 0.59 y). This range largely overlaps with the one comprising most of type IIb sources (from 05h to 10h, corresponding to solar elongation minima between 0.45 and 0.63 y); type IIa and type IIb could therefore be manifestations of the same phenomenon, essentially di ff erentiated by the ecliptic latitude of the sources (lower for the former, higher for the latter). A second block of type IIb sources can be found in a narrow right ascension stripe about 12h apart from the first one, around right ascension 18h; it corresponds to solar elongation minima between 0.89 and 0.02 y.</text>
<text><location><page_8><loc_6><loc_22><loc_49><loc_32></location>The automatic detection algorithm returns a distribution of sources that is consistent with the one of the visual inspection for type IIb SRV. Type IIa SRV objects, instead, are more widely spread in the sky than what the visual inspection suggests. The two classification methods agree though on the facts that this kind of variability mostly concerns objects at low ecliptic latitude, and that there is a large cluster of them around R.A.: 8h, Dec: + 20 · .</text>
<text><location><page_8><loc_6><loc_18><loc_49><loc_21></location>Some hypothesis as to the origin of the variability are shortly discussed below; particular attention is paid to assess a possible role of ISS.</text>
<section_header_level_1><location><page_8><loc_6><loc_14><loc_30><loc_15></location>5.1. SRV as a sequence of ESEs</section_header_level_1>
<text><location><page_8><loc_6><loc_10><loc_49><loc_13></location>Since ESEs are generally attributed to ISS, it is important to assess whether a series of events that look like ESEs can be reasonably explained in terms of ISS too. Concerning TDV the an-</text>
<text><location><page_8><loc_51><loc_71><loc_94><loc_93></location>swer is certainly no, as the simultaneity of the events in di ff erent sources excludes the intervention of a localised screen as required in ESEs. Concerning type II SRV, however, the reply is less obvious. In principle, a localised screen moving very slowly could cause repeated events occurring approximately at the same time of the year, every time that the line of sight (LOS) to the source crosses the screen again. The fact that, as shown in Fig. 9 for the results of the visual inspection, sources a ff ected by SRV are mainly concentrated near R.A. 6 and 18h would agree with this picture, as these are the conditions for which the ISM velocity transverse to the LOS is minimum, so the Earth's orbital velocity is dominant. The search for events repeating regularly with semi-annual cadence could facilitate the detection of ESEs that occur around the time of minimum / maximum solar elongation of a source, because at larger angular distances the cadence would generally be more asymmetric (e.g. there could be 4 months and then 8 months intervals between consecutive events).</text>
<text><location><page_8><loc_51><loc_68><loc_94><loc_71></location>There are, however, a number of critical arguments against this hypothesis:</text>
<unordered_list>
<list_item><location><page_8><loc_52><loc_61><loc_94><loc_67></location>-ISS fails to explain both TDV and type IIa SRV. If the dips at minimum solar elongation would be caused by yearly crossings of the same screen, dips would be found at maximum solar elongation too, as the Earth would align along the same LOS, therefore type IIa SRV would not exist.</list_item>
<list_item><location><page_8><loc_52><loc_47><loc_94><loc_61></location>-The procedure for the automatic classification of type IIb SRV sources is the only one to be biased because of the requirement of a semi-annual cadence of the events. The automatic detection of events, the visual and automatic classification of type IIa SRV sources, as well as the visual classification of type IIb SRV objects are unbiased. They all agree upon the fact that dips are not randomly occurring across the year, but they are much more common at the time of minimum and maximum solar elongation, which cannot be explained without a direct correlation between SRV and the position of the Sun.</list_item>
<list_item><location><page_8><loc_52><loc_39><loc_94><loc_47></location>-Events in the light curves of SRV sources often do not occur regularly every 6 or 12 months, but there can be large gaps between them. For example, in 0528 + 134, SRV events are found in 1990.50, 1991.00, 1992.40, and 1993.95. This is unexplainable, under the assumption that an almost motionless screen is causing the variability.</list_item>
<list_item><location><page_8><loc_52><loc_14><loc_94><loc_39></location>-The case of 0954 + 658 and the angularly-nearby sources 0633 + 734, 0723 + 679, and 0836 + 710, which show correlated variability with time delays consistent with the time difference among the solar elongation minima, shows that SRV events can a ff ect multiple sources in the same region of the sky. Another interesting example is provided by 0922 + 005, and the angularly-nearby source 0837 + 035 (see Fig. 11). The former is a strong type II SRV source; the variability of the latter is dominated by random noise, and, as a consequence, is not identified as a ff ected by SRV. Nevertheless, the fast variability of the two objects is correlated: an NDCF analysis of their de-trended light curves shows regular peaks of correlation with 0.5y cadence, and a time delay of 0.04y, in excellent agreement with the di ff erence in their solar elongation minima. A further example is provided by the quasi simultaneous events in OJ287 and 0742 + 103 (see Fig. 3). They all suggest that SRV is not caused by a very localised screen at large distance from the observer, but by a nearby, angularly-wide screen related to the position of the Sun.</list_item>
</unordered_list>
<text><location><page_8><loc_51><loc_10><loc_94><loc_13></location>A possible source of bias in favour of events occurring more often around the times of minimum / maximum solar elongation could be the higher transversal speed of the Earth with respect to</text>
<text><location><page_9><loc_6><loc_74><loc_49><loc_93></location>the LOS, which implies that a larger region in the sky is swept around these times. This argument, again, cannot explain TDV, neither type IIa SRV, because, if the transversal speed would be the cause of SRV, the events would have the same chance to occur at minimum and at maximum solar elongation. Type IIb SRV, instead, would be compatible with an e ff ect of the transversal speed; but this would become more important as the ecliptic latitude of the source tends to zero. Instead type IIb SRV concerns mostly sources at high ecliptic latitude where the e ff ect would be marginal, while type IIa SRV seems stronger for sources at low ecliptic latitude. Finally, an important contribution of the transversal speed to SRV would necessarily lead to an equally important decrease of the duration of the events at the times of minimum / maximum solar elongation; the data, however, show absolutely no sign of it.</text>
<section_header_level_1><location><page_9><loc_6><loc_70><loc_23><loc_71></location>5.2. Further hypotheses</section_header_level_1>
<text><location><page_9><loc_6><loc_54><loc_49><loc_69></location>An instrumental origin of the variability can be ruled out for different reasons: all the main examples of the di ff erent kinds of variability (OJ287 for type IIa, 0528 + 134 for type IIb and TDV) are supported by observations from di ff erent facilities, whose flux density measurements are in excellent agreement with each other; the variability also a ff ects night time observations (for type IIb SRV in particular), which excludes sunlight reflection as a possible cause; it is also worth mentioning that, di ff erently from the instrumental type I SRV, the centre and the duration of type II and TDV events in the light curves changes from year to year, which means that the variability is not as systematic as one would expect from an instrumental e ff ect.</text>
<text><location><page_9><loc_6><loc_40><loc_49><loc_54></location>Assuming that the source distribution returned by the visual inspection of light curves is correct, type II SRV mostly a ff ects sources whose solar elongation minima and maxima occur around 0.0 and 0.5 y; these are also the critical times for the appearance of TDV: time could therefore be the link between the di ff erent kinds of ESE-shaped events we detected in the data. Several astronomically-relevant e ff ects or events occur around 0.0 and 0.5 y, namely meteorological e ff ects, perihelion / aphelion, the Earth's intersection of the Sun's equatorial plane, and solstices. We separately discuss them, briefly, in the context of systematic flux density variability, here below:</text>
<unordered_list>
<list_item><location><page_9><loc_7><loc_32><loc_49><loc_38></location>- meteorological e ff ects , such as storms, or cold / heat waves, are local (so they are not expected to a ff ect in the same way di ff erent instruments), non-periodic, and cannot account for the solar-elongation dependence of SRV. Their involvement in SRV can safely be excluded.</list_item>
<list_item><location><page_9><loc_7><loc_24><loc_49><loc_32></location>-The occurrence of perihelion and aphelion seems unlikely to explain ESE-shaped variability, essentially because of the smoothness and the extent of the Sun's distance variation; also, it would be hard to understand how the Earth reaching the minimum and the maximum distance from the Sun could lead to the same consequences, as required by type IIb SRV.</list_item>
<list_item><location><page_9><loc_7><loc_14><loc_49><loc_24></location>-The Earth's intersection of the Sun's equatorial plane raises interest because of its 6-month periodic cadence; however, the angle between the ecliptic and the Sun's equatorial plane is small ( ∼ 7 · ) and the variation smooth. In order to produce the observed ESE-like features, a sharp change of the environmental characteristics at the crossing of the equatorial plane would be needed, which though does not seem to be supported by external evidence.</list_item>
<list_item><location><page_9><loc_7><loc_10><loc_49><loc_13></location>-An interpretation of the variability as a consequence of the solstices seems the most promising, although it does not come without problems. On the one hand, it is reasonable to</list_item>
</unordered_list>
<figure>
<location><page_9><loc_52><loc_78><loc_93><loc_93></location>
<caption>Fig. 9. Visual inspection of sources: the distribution in the sky of type I (orange dots), type IIa (green dots), type IIb (blue dots), and una ff ected sources (grey dots). The size of the dots is proportional to the ratio between the 2 GHz dropout at minimum solar elongation and the average 2 GHz flux density (Col. 8 and 4, respectively, in Table 3). The magenta line shows the ecliptic, while the red one shows the galactic plane.</caption>
</figure>
<figure>
<location><page_9><loc_51><loc_53><loc_94><loc_69></location>
<caption>Fig. 10. Same as Fig. 9, but for the automatic classification of sources. The size of the dots is proportional to the exponential of -(10p), where p is the false alarm probability returned by the algorithm (Col. 12 in Table 3).</caption>
</figure>
<text><location><page_9><loc_53><loc_36><loc_94><loc_45></location>consider that the variation of the Sun's declination during the year could have an impact on flux density measurements, for instance by a ff ecting the propagation of radio waves through the atmosphere. On the other hand, there is no obvious reason why a source should be similarly a ff ected when the Sun is at maximum and at minimum declination, as for type IIb SRV.</text>
<text><location><page_9><loc_51><loc_27><loc_94><loc_34></location>The mechanism that could be responsible for the variability is equally hard to identify. It is important to preliminarily underline that the timescales and the amplitudes of the variations detected in the light curves are not consistent with the theoretical values from scattering in a nearby medium (see Narayan 1992 for an introduction to the physics of scintillation).</text>
<text><location><page_9><loc_51><loc_16><loc_94><loc_26></location>The plausible existence of a ring that crosses the whole sky along the right ascension lines around 06h and 18h, comprising most of the type II SRV sources, if confirmed, would not be consistent with IPS because of its wide extension. Since type II SRV a ff ects the sources months before / after the time of minimum elongation, when the Sun's declination is far from its highest northern / southern extension, atmospheric scattering seems unlikely too.</text>
<text><location><page_9><loc_51><loc_10><loc_94><loc_16></location>The limited range of declination of type II sources could suggest a relationship with the Earth's North-South direction, implying a link to the Earth's geometry and properties. The fact that, within the 06h-18h right ascension ring, some sources are a ff ected for a large part of the year, would indicate a geometri-</text>
<figure>
<location><page_10><loc_9><loc_73><loc_46><loc_94></location>
<caption>Fig. 11. Upper panel: the light curves of the strong type IIb source 0922 + 005 (black dots) and of 0837 + 035 (green dots). The NDCF between the de-trended light curves is shown in the lower panel; it reveals a 0.5y periodic modulation (magenta line) with a time delay that is consistent with the di ff erence in the solar elongation minima of the objects.</caption>
</figure>
<text><location><page_10><loc_6><loc_27><loc_49><loc_63></location>cal e ff ect as the most plausible. The plane on which the ring lies corresponds to the one defined by the equatorial and the ecliptic latitudes axes. At a level of conjecture, one could wonder whether the complex shape of the magnetosphere could play a role in the observed variability, acting somehow as a lens. The magnetosphere stretches along the direction of the flow of solar wind, in the ecliptic plane; it is compressed towards the Sun, and very elongated in the opposite direction (magnetotail); its shape varies during the year, and its structure along the line of sight to a source depends on the source's position with respect to the Sun. The inner structure of the magnetosphere is also heavily influenced, because of the magnetic field, by the celestial poles. This implies the existence, in the morphology of the magnetosphere, of a privileged direction (the 6h-to-18h right ascension direction), towards the tilt of the Earth axis, which at 0.0 and 0.5 y, approximately, aligns with the flow of the solar wind; such alignment implies a symmetric configuration of the magnetosphere with respect to Earth's axis that could possibly trigger both SRV and TDV. When the e ff ect of the Earth axis modifies more strongly the inner structure of the magnetosphere, TDV would become dominant; when the e ff ect is weaker, only sources with solar elongation minima around 0.0 and 0.5 y would be affected, and we would see type IIb SRV. Although this scenario may be geometrically plausible, it has to face the fact that, to cause significant variability in the data, a lensing e ff ect would require a change in refraction index that is orders of magnitude larger than what is expected at the level of all the atmospheric layers.</text>
<section_header_level_1><location><page_10><loc_6><loc_23><loc_31><loc_25></location>6. Revision of identified ESEs</section_header_level_1>
<text><location><page_10><loc_6><loc_14><loc_49><loc_22></location>The analysis of the NESMP data by Lazio et al. (2001b) led to the detection of 24 ESEs (see their Tables 5 and 6). We have already shown that some of them are caused by SRV. In Table 2 we summarise our new classification of these events in the light of the SRV and TDV characteristic previously illustrated. A short explanation of the reasons behind this classification is reported in Appendix C.</text>
<text><location><page_10><loc_6><loc_10><loc_49><loc_13></location>The number of ESEs that cannot be explained through SRV is very low. Even the archetype of ESEs, the 1981.1 event in 0954 + 658, seems more likely to be caused by the Sun than by an</text>
<table>
<location><page_10><loc_56><loc_53><loc_89><loc_88></location>
<caption>Table 2. Summary of the analysis of ES-like events reported by Lazio et al. 2001b. Col. 1 and 2 report the name of the source and the time of occurrence of the event. The plausible origin of the variability is reported in Col. 3.</caption>
</table>
<text><location><page_10><loc_51><loc_42><loc_94><loc_50></location>interstellar screen. This means that the actual frequency of ESEs in the GBI data is significantly lower than previously claimed. Furthermore, this result demonstrates the di ffi culty to identify ESEs uniquely from the examination of light curves; for a robust detection, it would be necessary to use independent diagnostic tools confirming the interstellar medium as the origin of the variability.</text>
<section_header_level_1><location><page_10><loc_51><loc_37><loc_63><loc_38></location>7. Conclusions</section_header_level_1>
<text><location><page_10><loc_51><loc_10><loc_94><loc_35></location>We reported about a variety of events, shaped as ESEs, detected in the radio light curves of compact extragalactic objects, from GBI, UMRAO, Pico Veleta, and E ff elsberg observations. We distinguished between four di ff erent kinds of variations, three of which are centred at the time of minimum solar elongation, while the fourth depends on time. Type I SRV has an instrumental origin, with a possible further contribution from IPS; it causes dips in the light curves when sources are at solar elongation below 15-20 · at 2 GHz ( < ∼ 10 · at 8 GHz), with average flux density decreases of about 30 percent. Di ff erently from type I, type II SRV can have a strong impact on the light curves even at high solar elongation. We found hints, which though need confirmation, that it mainly a ff ects sources in right ascension ranges around 06h and 18h. The origin of the variability is unknown, but the detection of the e ff ects in data from di ff erent facilities rules out an instrumental problem. TDV is a semi-annual e ff ect a ff ecting the light curves of a large number of sources around 0.0 and 0.5 y; its nature, as for type II SRV, is yet unknown. The evidence we collected rules out both ISS and instrumental / calibration problems from the possible sources of the variability. By exclusion,</text>
<text><location><page_11><loc_6><loc_91><loc_49><loc_93></location>the most viable option seems to be a propagation e ff ect in a local screen (the IPM or the atmosphere).</text>
<text><location><page_11><loc_6><loc_76><loc_49><loc_90></location>After analysing the general properties of the four kinds of variability, we assessed di ff erent hypotheses about their origin; all hypotheses are weakened by critical arguments that cannot be overcome without assuming that the actual properties of local screens substantially di ff er from the ones proposed by current models. It cannot be ruled out that the variability is caused by a superposition of di ff erent e ff ects, which could explain the di ffi -culty to describe it in terms of a single self-consistent model. However, the convergence of many SRV and TDV manifestations around the time of the solstices, if confirmed, would strongly support the idea of a common origin.</text>
<text><location><page_11><loc_6><loc_48><loc_49><loc_76></location>From the picture sketched above, it is evident that the nature of the variability is far from being identified. Much clearer is the situation concerning the consequences of SRV / TDV, especially in relation to ESEs studies. Many of the ESEs identified by Lazio et al. (2001b) during the NESMP (which is the most important program dedicated to the detection of these events) are certainly due to SRV; only a few are not compatible with the characteristics of a solar influence. Even the impressive events in the light curves of 1741-038 (see Lazio et al. 2001a), 0954 + 658 (Fiedler et al. 1987a; Fiedler et al. 1987b; Walker & Wardle 1998), and 0528 + 134 (Pohl et al. 1996) appear to be extreme cases of semiannual events caused by the Sun (type IIb SRV). In the light of the discoveries here reported, also the ESE detected in the data of PKS 1939-315 on June 2014 (Bannister et al. 2016) would be naturally interpreted as due to SRV or TDV, considering that, because of its coordinates, the source would be expected to show type IIb SRV, and that the minimum of the symmetric event falls at the beginning of July 2014 (i.e. the critical time 0.5 y), when the source reaches its maximum solar elongation. Our study does not rule out the existence of ESEs, but it shows that this kind of event, which already Fiedler et al. (1994) considered rare, is even rarer than previously thought.</text>
<text><location><page_11><loc_6><loc_14><loc_49><loc_47></location>Concerning the study of the radio properties of compact sources, the existence of SRV implies that the variability characteristics of some objects may be heavily a ff ected by the influence of the Sun. For sources such as OJ 287, it may be important to set a much larger Sun constraint than the mechanical one allowed by the telescope (which for E ff elsberg, e.g., is 2 · , see Komossa et al. 2023). Particularly worrisome is the recognition of the high degree of correlation between the fast variability of 0954 + 658 and of some angularly-nearby sources, whose solar elongation is always high. Periodicity in light curves with periods close to one year or 6 months should be regarded as highly suspicious, especially if they appear in sources whose solar elongation peaks around the critical times 0.0 and 0.5 y; in this respect, it would be interesting to check whether the ∼ 180-day periodicity found in OVRO data for the NLSy1 J0849 + 5108 ( β : 32 · ; see Zhang & Wang 2021), whose coordinates are compatible with type IIb SRV, shows dips around 0.1 and 0.6 y, consistently with its time of maximum / minimum solar elongation. Additional studies will be needed to assess whether SRV may also play a role in the recently discovered symmetric achromatic variability events (SAV; see Vedantham et al. 2017), found in the blazar 1413 + 135; however, the position of the source (far from the location of most type IIa and IIb SRV sources), the duration of the events (of the order of one year or more), and the occurrence of the detected SAV events (see Peirson et al. 2022) all seem to indicate that the variability is not caused by the Sun.</text>
<text><location><page_11><loc_6><loc_10><loc_49><loc_13></location>A thorough investigation of the causes and the manifestations of SRV is made even more urgent by the consideration that several of the most variable compact radio sources of extragalac-</text>
<text><location><page_11><loc_51><loc_87><loc_94><loc_93></location>tic origin (e.g. OJ 287 and 0235 + 164) fall at low ecliptic latitudes, and that Sgr A*, among the most interesting and studied objects in astronomy, because of its coordinates and an ecliptic latitude of only -5 . 6 · , should be particularly sensitive to type II SRV.</text>
<text><location><page_12><loc_6><loc_10><loc_15><loc_93></location>T able 3. The main v ariability characteristics of NESMP sources are reported as follo ws: name of the source (Col. 1), ecliptic latitude (Col. 2), time of the year in which solar elong ation reaches the minimum (Col. 3), a v erage S 2GHz (Col. 4), S 2GHz standard de viation (Col. 5), a v erage S 8GHz (Col. 6), S 8GHz standard de viation (Col. 7), 2 GHz dropout at minimum solar elong ation (Col. 8), SR V classification at 2 GHz (Col. 9), SR V classification at 8 GHz (Col. 10), e vidence of TD V (Col. 11). The results of the automatic classification of sources according to their 2 GHz light curv es are summarised in Col. 12-15; the y report the probability that the number of SIV or TD V e v ents in the light curv es is consistent with an homogeneous distrib ution of dips across the year (Col 12 and 14, respecti v ely), and the consequent classification of the sources (Col. 13 and 15). The "N A" indication is used when no meaningful estimate could be achie v ed because of lack of data.</text>
<figure>
<location><page_12><loc_16><loc_16><loc_93><loc_87></location>
</figure>
<table>
<location><page_12><loc_16><loc_16><loc_93><loc_87></location>
</table>
<table>
<location><page_13><loc_9><loc_16><loc_93><loc_87></location>
</table>
<figure>
<location><page_14><loc_9><loc_16><loc_93><loc_87></location>
</figure>
<table>
<location><page_14><loc_9><loc_16><loc_93><loc_87></location>
</table>
<table>
<location><page_15><loc_9><loc_16><loc_64><loc_87></location>
</table>
<text><location><page_16><loc_6><loc_82><loc_49><loc_93></location>Acknowledgements. We thank the anonymous referees for the useful comments and the thorough discussion of the manuscript. We thank Tim Sprenger and Laura Spitler for their suggestions, which helped to clarify and highlight important points of the article. This research is based on data of the Green Bank Interferometer (GBI), which was a facility of the National Science Foundation operated by the National Radio Astronomy Observatory under contract with the US Naval Observatory and the Naval Research Laboratory during these observations. The UMRAO observations included in this analysis were obtained as part of programs funded by a series of grants from the NSF. Additional funding for the operation of UMRAO was provided by the University of Michigan.</text>
<section_header_level_1><location><page_16><loc_6><loc_78><loc_16><loc_79></location>References</section_header_level_1>
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<list_item><location><page_16><loc_6><loc_10><loc_49><loc_77></location>Aller, H. D., Aller, M. F., Latimer, G. E., & Hodge, P. E. 1985, ApJS, 59, 513 Aller, M. F., Aller, H. D., Hughes, P. A., & Latimer, G. E. 1999, ApJ, 512, 601 Bannister, K. W., Stevens, J., Tuntsov, A. V., et al. 2016, Science, 351, 354 Coe, J. R. 1973, IEEE Proceedings, 61, 1335 Edelson, R. A. & Krolik, J. H. 1988, ApJ, 333, 646 Fiedler, R., Dennison, B., Johnston, K. J., Waltman, E. B., & Simon, R. S. 1994, ApJ, 430, 581 Fiedler, R. L., Dennison, B., Johnston, K. J., & Hewish, A. 1987a, Nature, 326, 675 Fiedler, R. L., Waltman, E. B., Spencer, J. H., et al. 1987b, ApJS, 65, 319 Heeschen, D. S., Krichbaum, T. P., Schalinski, C. J., & Witzel, A. 1987, AJ, 94, 1493 Hogg, D. E., MacDonald, G. H., Conway, R. G., & Wade, C. M. 1969, AJ, 74, 1206 Hovatta, T., Valtaoja, E., Tornikoski, M., & Lähteenmäki, A. 2009, A&A, 494, 527 Komossa, S., Kraus, A., Grupe, D., et al. 2023, ApJ, 944, 177 Koryukova, T. A., Pushkarev, A. B., Plavin, A. V., & Kovalev, Y. Y. 2022, MNRAS, 515, 1736 Lazio, T. J. W., Gaume, R. A., Claussen, M. J., et al. 2001a, ApJ, 546, 267 Lazio, T. J. W., Waltman, E. B., Ghigo, F. D., et al. 2001b, ApJS, 136, 265 Lehar, J., Hewitt, J. N., Roberts, D. H., & Burke, B. F. 1992, ApJ, 384, 453 Liodakis, I., Marchili, N., Angelakis, E., et al. 2017, MNRAS, 466, 4625 Lovell, J. E. J., Jauncey, D. L., Bignall, H. E., et al. 2003, AJ, 126, 1699 Marchili, N., Krichbaum, T. P., Liu, X., et al. 2011, A&A, 530, A129 Marscher, A. 2016, Galaxies, 4, 37 Marscher, A. P., Gear, W. K., & Travis, J. P. 1992, in Variability of Blazars, ed. E. Valtaoja & M. Valtonen, 85 Narayan, R. 1992, Philosophical Transactions of the Royal Society of London Series A, 341, 151 Peirson, A. L., Liodakis, I., Readhead, A. C. S., et al. 2022, ApJ, 927, 24 Pohl, M., Reich, W., Krichbaum, T. P., et al. 1995, A&A, 303, 383 Pohl, M., Reich, W., Schlickeiser, R., Reich, P., & Ungerechts, H. 1996, A&AS, 120, 529 Pushkarev, A. B. & Kovalev, Y. Y. 2015, MNRAS, 452, 4274 Rickett, B. J., Lazio, T. J. W., & Ghigo, F. D. 2006, ApJS, 165, 439 Sillanpää, A., Haarala, S., Valtonen, M. J., Sundelius, B., & Byrd, G. G. 1988, ApJ, 325, 628 Ulrich, M., Maraschi, L., & Urry, C. M. 1997, ARA&A, 35, 445</list_item>
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<text><location><page_16><loc_6><loc_7><loc_24><loc_8></location>Article number, page 16 of 20</text>
<text><location><page_16><loc_51><loc_78><loc_78><loc_79></location>Zhang, P. & Wang, Z. 2021, ApJ, 914, 1</text>
<section_header_level_1><location><page_17><loc_6><loc_90><loc_45><loc_93></location>Appendix A: Investigation of TDV as a possible calibration problem</section_header_level_1>
<text><location><page_17><loc_6><loc_83><loc_49><loc_88></location>TDV is characterised by the occurrence of simultaneous events, resembling ESEs, in di ff erent sources; correlated flux density variations in di ff erent sources may suggest the existence of an instrumental or a calibration problem.</text>
<text><location><page_17><loc_6><loc_70><loc_49><loc_81></location>Given the periodic recurrence of TDV, the hypothesis of an instrumental problem seems implausible. The detection of ESEshaped variability in 0528 + 134 with three di ff erent telescopes, applying di ff erent calibration schemes, during the 1993.5 event seems to rule out a calibration problem as well. However, in consideration of the ambiguity of the 0528 + 134 variability in 1993.5 (is it TDV or SRV?), it seems useful to analyse in some detail the variability characteristics of the calibrators, to asses the possibility of a calibration problem.</text>
<text><location><page_17><loc_6><loc_50><loc_49><loc_68></location>All three sources used to create a hybrid calibrator show a smooth 1-year periodic oscillation correlated with solar elongation. The minima of the oscillation occur at 0.3 y for 0237233, at 0.78 y for 1245-197 and at 0.76 y for 1328 + 254; they are therefore strongly out-of-phase, almost in phase opposition. A combination of smooth oscillations, however, cannot cause ESEshaped variability episodes. Through the de-trending procedure, we isolated the fast variability component in the calibrators, to compare it with the combined 2 GHz variability curve shown in Fig. 7. It should be noted that 1245-197 has an ecliptic latitude of -13.6 · , and it is a ff ected by type I SRV; the ESE-like variations at minimum solar elongation introduce in some sources mild 1-year periodic flares of short duration. However, this does not cause the semi-annual dips at 0.0 and 0.5 y.</text>
<text><location><page_17><loc_6><loc_36><loc_49><loc_49></location>The 2 GHz combined variability curve and the calibrators de-trended light curves are shown in Fig. A.1, after averaging them on 0.02 y bins to reduce the random fluctuations. Given their usage in the calibration procedure, discrepancies between the calibrators' light curves tend to cancel out, which implies a significant anti-correlation between their variations; 1245-197 (brown dots) and 1328 + 254 (green dots) generally show a better agreement among each other than with 0237-233 (black dots), because their angular distance is much smaller, and they are observed approximately at the same time of the day.</text>
<text><location><page_17><loc_6><loc_10><loc_49><loc_34></location>If TDV were caused by a calibrator, its light curve would show a systematic discrepancy with respect to the other two during the ESE-shaped events. If the e ff ect were instead caused by di ff erent observational conditions for 0237-233 compared to the two angularly-nearby sources 1245-197 and 1328 + 254, we should see a periodic deviation between the former and the latter ones. In general, strong disagreements between the calibrators do not coincide with dips in the data; in Fig. A.1, only two events look correlated with a significant discrepancy between the calibrators: the 1993.0 and, to a lesser extent, the 1995.1 one. In both cases, however, the agreement is good between 0237-233 and 1245-197, while the flux density of 1328 + 254 is considerably lower. A dip in a calibrator would introduce a peak, not a dip, in the calibrated light curves. This implies that either two distant calibrators are a ff ected by chance by a similar problem for a few months (which seems unrealistic) or the observed drop of flux density must a ff ect 1328 + 254 as well as several of the calibrated sources. In conclusion, the calibrators do not seem to cause the problem, but they might be a ff ected by it.</text>
<figure>
<location><page_17><loc_54><loc_75><loc_91><loc_93></location>
<caption>Fig. A.1. Comparison between the de-trended calibrators' light curves and the combined 2 GHz variability curve plotted in Fig. 7. Data are averaged on 0.02 y bins; the combined variability curve is arbitrarily shifted for an easier comparison. Blue lines indicate the start and the end of the semi-annual ESE-like episodes in the combined variability curve.</caption>
</figure>
<section_header_level_1><location><page_17><loc_51><loc_62><loc_90><loc_64></location>Appendix B: List of all events detected through automated analysis</section_header_level_1>
<text><location><page_17><loc_51><loc_59><loc_94><loc_60></location>0003 + 380 : 1993.77, 1993.22, 1992.22, 1991.62, 1990.62,</text>
<text><location><page_17><loc_51><loc_10><loc_94><loc_59></location>1990.32 0003-066 : 1992.92, 1992.22, 1991.87, 1990.22, 1989.22 0016 + 731 : 1993.22, 1992.62, 1991.77, 1991.32, 1990.47, 1989.72, 1988.92 0019 + 058 : 1993.24, 1992.29, 1991.24 0035 + 121 : 1993.26, 1992.26, 1991.26, 1990.26, 1989.26 0035 + 413 : 1992.34, 1991.44, 1990.89, 1989.74 0055 + 300 : 1993.19, 1991.19, 1990.99, 1989.74 0056-001 : 1993.25, 1992.25, 1991.25, 1990.25, 1989.25 0113-118 : 1991.25, 1989.25, 1988.95 0123 + 257 : 1988.94 0130-171 : 1989.49, 1988.94 0133 + 476 : 1992.33, 1990.78, 1990.48, 1990.23, 1987.73, 1986.73 0134 + 329 : 1992.36, 1991.26, 1991.01, 1988.91 0147 + 187 : 1993.46, 1992.31 0201 + 113 : 1993.31, 1992.31, 1991.56, 1991.31, 1990.31, 1989.31 0202 + 319 : 1993.48 , 1992.33, 1989.48, 1988.38, 1987.23, 1985.33, 1984.28 0212 + 735 : 1994.15, 1993.35, 1993.00 , 1992.35, 1992.05 0224 + 671 : 1995.84, 1992.19, 1991.19, 1990.29, 1988.89, 1988.59, 1987.99, 1983.79, 1979.94 0235 + 164 : 1992.34, 1991.34, 1986.69, 1985.34, 1984.09, 1982.89, 1982.04 0237-233 : 1994.95, 1989.25, 1984.40, 1983.40, 1982.90 0256 + 075 : 1993.49 , 1992.89, 1992.74, 1992.49 , 1991.84 0300 + 470 : 1993.53, 1992.68, 1992.33, 1991.48, 1990.73, 1990.23, 1989.88, 1989.48, 1988.93, 1987.93 0316 + 413 : 1991.03 , 1990.38, 1988.78 0319 + 121 : 1993.51 , 1991.36, 1990.36, 1989.31 0333 + 321 : 1993.74, 1993.39, 1991.84, 1987.94, 1987.44, 1986.34 0336-019 : 1995.36, 1994.46, 1992.91, 1992.56, 1992.06, 1991.51, 1983.16, 1981.11 0337 + 319 : 1993.44, 1992.94 , 1991.39, 1990.99 0355 + 508 : 1984.01, 1983.36, 1981.66, 1980.31, 1979.61 0400 + 258 : 1993.50 , 1992.90, 1992.45, 1991.95, 1991.45,</text>
<text><location><page_18><loc_6><loc_92><loc_18><loc_93></location>1991.00 , 1990.55</text>
<text><location><page_18><loc_6><loc_91><loc_25><loc_92></location>0403-132 : 1992.92, 1991.47</text>
<text><location><page_18><loc_6><loc_89><loc_49><loc_90></location>0420-014 : 1992.20, 1989.25, 1987.30, 1986.65, 1985.35,</text>
<text><location><page_18><loc_6><loc_88><loc_12><loc_89></location>1984.40</text>
<text><location><page_18><loc_6><loc_87><loc_13><loc_88></location>0440-003</text>
<text><location><page_18><loc_13><loc_87><loc_13><loc_88></location>:</text>
<text><location><page_18><loc_14><loc_87><loc_20><loc_88></location>1994.21,</text>
<text><location><page_18><loc_22><loc_87><loc_27><loc_88></location>1993.96,</text>
<text><location><page_18><loc_29><loc_87><loc_34><loc_88></location>1993.51</text>
<text><location><page_18><loc_34><loc_87><loc_35><loc_88></location>,</text>
<text><location><page_18><loc_36><loc_87><loc_42><loc_88></location>1993.26,</text>
<text><location><page_18><loc_43><loc_87><loc_49><loc_88></location>1990.46,</text>
<text><location><page_18><loc_6><loc_85><loc_18><loc_86></location>1989.86, 1989.41</text>
<text><location><page_18><loc_6><loc_84><loc_49><loc_85></location>0444 + 634 : 1994.44, 1993.99, 1993.29, 1992.09, 1991.59,</text>
<text><location><page_18><loc_6><loc_83><loc_30><loc_84></location>1990.99, 1990.64, 1989.34, 1989.09</text>
<text><location><page_18><loc_6><loc_82><loc_10><loc_83></location>0454</text>
<text><location><page_18><loc_10><loc_82><loc_11><loc_83></location>+</text>
<text><location><page_18><loc_11><loc_82><loc_13><loc_83></location>844</text>
<text><location><page_18><loc_13><loc_82><loc_32><loc_83></location>: 1993.52, 1993.27, 1990.22</text>
<text><location><page_18><loc_6><loc_80><loc_26><loc_81></location>0500 + 019 : 1992.88, 1992.23</text>
<text><location><page_18><loc_6><loc_79><loc_49><loc_80></location>0528 + 134 : 1993.95, 1993.50 , 1993.05, 1992.40, 1991.00,</text>
<text><location><page_18><loc_6><loc_78><loc_18><loc_79></location>1990.50, 1988.95</text>
<text><location><page_18><loc_6><loc_76><loc_32><loc_77></location>0532 + 826 : 1991.72, 1989.97, 1989.07</text>
<text><location><page_18><loc_6><loc_75><loc_49><loc_76></location>0537-158 : 1993.50, 1993.35, 1993.00, 1992.45, 1991.95,</text>
<text><location><page_18><loc_6><loc_74><loc_30><loc_75></location>1991.45, 1990.95, 1990.50, 1989.30</text>
<text><location><page_18><loc_6><loc_72><loc_10><loc_73></location>0538</text>
<text><location><page_18><loc_10><loc_72><loc_11><loc_73></location>+</text>
<text><location><page_18><loc_11><loc_72><loc_13><loc_73></location>498</text>
<text><location><page_18><loc_13><loc_72><loc_26><loc_73></location>: 1991.01, 1990.46</text>
<text><location><page_18><loc_6><loc_70><loc_49><loc_72></location>0552 + 398 : 1993.52, 1992.17, 1991.47, 1990.62, 1988.77, 1987.47, 1986.22, 1985.82, 1984.42</text>
<text><location><page_18><loc_6><loc_67><loc_49><loc_70></location>0555-132 : 1993.92, 1993.47, 1992.67, 1992.27, 1990.02, 1989.42</text>
<text><location><page_18><loc_6><loc_65><loc_49><loc_67></location>0615 + 820 : 1993.98, 1993.53, 1992.83, 1992.08, 1991.58, 1990.98</text>
<text><location><page_18><loc_6><loc_63><loc_25><loc_64></location>0624-058 : 1992.60, 1990.45</text>
<text><location><page_18><loc_6><loc_62><loc_38><loc_63></location>0633 + 734 : 1993.48 , 1992.98, 1990.48, 1990.28</text>
<text><location><page_18><loc_6><loc_61><loc_45><loc_62></location>0650 + 371 : 1992.55, 1991.65, 1991.00, 1990.45, 1989.05</text>
<text><location><page_18><loc_6><loc_58><loc_49><loc_60></location>0653-033 : 1994.12, 1993.47, 1992.32, 1991.72, 1990.52, 1989.07, 1988.82</text>
<text><location><page_18><loc_6><loc_57><loc_10><loc_58></location>0716</text>
<text><location><page_18><loc_10><loc_57><loc_11><loc_58></location>+</text>
<text><location><page_18><loc_11><loc_57><loc_13><loc_58></location>714</text>
<text><location><page_18><loc_13><loc_57><loc_38><loc_58></location>: 1991.40, 1990.00, 1988.75, 1988.15</text>
<text><location><page_18><loc_6><loc_55><loc_38><loc_57></location>0723 + 679 : 1993.55, 1993.00, 1991.00 , 1990.45</text>
<text><location><page_18><loc_6><loc_54><loc_25><loc_55></location>0723-008 : 1993.14, 1989.74</text>
<text><location><page_18><loc_6><loc_53><loc_45><loc_54></location>0742 + 103 : 1992.95, 1992.55, 1990.55, 1986.10, 1982.20</text>
<text><location><page_18><loc_6><loc_52><loc_32><loc_53></location>0743 + 259 : 1993.34, 1990.54, 1989.54</text>
<text><location><page_18><loc_6><loc_49><loc_49><loc_51></location>0759 + 183 : 1993.76, 1993.51, 1992.51, 1991.86, 1990.01, 1989.56</text>
<text><location><page_18><loc_6><loc_48><loc_32><loc_49></location>0804 + 499 : 1994.54, 1991.64, 1990.59</text>
<text><location><page_18><loc_6><loc_46><loc_31><loc_47></location>0818-128 : 1992.59, 1990.04, 1989.44</text>
<text><location><page_18><loc_6><loc_45><loc_38><loc_46></location>0827 + 243 : 1994.52, 1993.52, 1992.52, 1991.52</text>
<text><location><page_18><loc_6><loc_44><loc_10><loc_45></location>0836</text>
<text><location><page_18><loc_10><loc_44><loc_11><loc_45></location>+</text>
<text><location><page_18><loc_11><loc_44><loc_13><loc_45></location>710</text>
<text><location><page_18><loc_13><loc_44><loc_14><loc_45></location>:</text>
<text><location><page_18><loc_15><loc_44><loc_21><loc_45></location>1995.93,</text>
<text><location><page_18><loc_22><loc_44><loc_28><loc_45></location>1995.38,</text>
<text><location><page_18><loc_29><loc_44><loc_35><loc_45></location>1994.93,</text>
<text><location><page_18><loc_36><loc_44><loc_42><loc_45></location>1994.53,</text>
<text><location><page_18><loc_43><loc_44><loc_49><loc_45></location>1993.38,</text>
<text><location><page_18><loc_6><loc_42><loc_37><loc_44></location>1992.98, 1991.48, 1991.03, 1990.48, 1989.58</text>
<text><location><page_18><loc_6><loc_41><loc_10><loc_42></location>0837</text>
<text><location><page_18><loc_10><loc_41><loc_11><loc_42></location>+</text>
<text><location><page_18><loc_11><loc_41><loc_13><loc_42></location>035</text>
<text><location><page_18><loc_13><loc_41><loc_32><loc_42></location>: 1993.49, 1992.94, 1992.39</text>
<text><location><page_18><loc_6><loc_40><loc_49><loc_41></location>0851 + 202 : 1991.54, 1989.59, 1986.34, 1985.14, 1983.59,</text>
<text><location><page_18><loc_6><loc_39><loc_12><loc_40></location>1979.54</text>
<text><location><page_18><loc_6><loc_37><loc_38><loc_38></location>0859-140 : 1993.67, 1992.52, 1991.47, 1990.07</text>
<text><location><page_18><loc_6><loc_36><loc_49><loc_37></location>0922 + 005 : 1993.58, 1993.08, 1992.48, 1991.68, 1991.13,</text>
<text><location><page_18><loc_6><loc_35><loc_12><loc_36></location>1990.18</text>
<text><location><page_18><loc_6><loc_32><loc_49><loc_34></location>0923 + 392 : 1992.99, 1991.99 , 1991.59, 1988.69, 1987.59, 1986.04, 1984.44</text>
<text><location><page_18><loc_6><loc_31><loc_26><loc_32></location>0938 + 119 : 1995.48, 1992.63</text>
<text><location><page_18><loc_6><loc_29><loc_32><loc_31></location>0945 + 408 : 1993.50 , 1991.50, 1991.10</text>
<text><location><page_18><loc_6><loc_28><loc_26><loc_29></location>0952 + 179 : 1993.48 , 1991.58</text>
<text><location><page_18><loc_6><loc_26><loc_49><loc_28></location>0954 + 658 : 1994.52, 1992.97, 1990.47, 1989.47, 1986.67, 1981.12, 1980.37</text>
<text><location><page_18><loc_6><loc_24><loc_10><loc_25></location>1020</text>
<text><location><page_18><loc_10><loc_24><loc_11><loc_25></location>+</text>
<text><location><page_18><loc_11><loc_24><loc_13><loc_25></location>400</text>
<text><location><page_18><loc_13><loc_24><loc_14><loc_25></location>:</text>
<text><location><page_18><loc_14><loc_24><loc_20><loc_25></location>1993.52</text>
<text><location><page_18><loc_20><loc_24><loc_26><loc_25></location>, 1990.62</text>
<text><location><page_18><loc_6><loc_23><loc_38><loc_24></location>1022 + 194 : 1993.75, 1993.50 , 1992.90, 1991.65</text>
<text><location><page_18><loc_6><loc_22><loc_31><loc_23></location>1036-154 : 1993.54 , 1992.64, 1991.14</text>
<text><location><page_18><loc_6><loc_20><loc_20><loc_21></location>1038 + 528 : 1992.01</text>
<text><location><page_18><loc_6><loc_19><loc_20><loc_20></location>1055 + 018 : 1991.74</text>
<text><location><page_18><loc_6><loc_18><loc_38><loc_19></location>1100 + 772 : 1993.50, 1993.20, 1992.95, 1991.00</text>
<text><location><page_18><loc_6><loc_16><loc_26><loc_18></location>1116 + 128 : 1993.49 , 1990.69</text>
<text><location><page_18><loc_6><loc_15><loc_32><loc_16></location>1123 + 264 : 1993.53 , 1992.58, 1991.03</text>
<text><location><page_18><loc_6><loc_14><loc_25><loc_15></location>1127-145 : 1993.72, 1991.07</text>
<text><location><page_18><loc_6><loc_13><loc_10><loc_14></location>1128</text>
<text><location><page_18><loc_10><loc_13><loc_11><loc_14></location>+</text>
<text><location><page_18><loc_11><loc_13><loc_13><loc_14></location>385</text>
<text><location><page_18><loc_13><loc_13><loc_20><loc_14></location>: 1990.31</text>
<text><location><page_18><loc_6><loc_11><loc_31><loc_12></location>1145-071 : 1993.52 , 1991.62, 1989.72</text>
<text><location><page_18><loc_6><loc_10><loc_26><loc_11></location>1150 + 812 : 1993.53, 1991.03</text>
<text><location><page_18><loc_51><loc_92><loc_71><loc_93></location>1155 + 251 : 1993.50, 1993.00</text>
<text><location><page_18><loc_51><loc_91><loc_76><loc_92></location>1200-051 : 1993.54 , 1992.04, 1991.74</text>
<text><location><page_18><loc_51><loc_89><loc_77><loc_90></location>1225 + 368 : 1993.50, 1992.95 , 1991.55</text>
<text><location><page_18><loc_51><loc_88><loc_89><loc_89></location>1243-072 : 1992.57, 1991.82, 1991.02, 1990.62, 1990.32</text>
<text><location><page_18><loc_51><loc_87><loc_76><loc_88></location>1253-055 : 1995.62, 1993.07, 1992.77</text>
<text><location><page_18><loc_51><loc_85><loc_76><loc_86></location>1302-102 : 1991.68, 1990.63, 1990.38</text>
<text><location><page_18><loc_51><loc_84><loc_64><loc_85></location>1308 + 326 : 1993.98</text>
<text><location><page_18><loc_51><loc_83><loc_83><loc_84></location>1328 + 254 : 1995.06 , 1992.96, 1991.56, 1981.11</text>
<text><location><page_18><loc_51><loc_82><loc_77><loc_83></location>1328 + 307 : 1995.05 , 1985.60, 1979.75</text>
<text><location><page_18><loc_51><loc_80><loc_77><loc_81></location>1354 + 195 : 1993.73, 1990.83, 1989.58</text>
<text><location><page_18><loc_51><loc_79><loc_71><loc_80></location>1404 + 286 : 1994.23, 1993.53</text>
<text><location><page_18><loc_51><loc_78><loc_71><loc_79></location>1409 + 524 : 1991.78, 1990.78</text>
<text><location><page_18><loc_51><loc_76><loc_83><loc_77></location>1413 + 135 : 1993.50 , 1992.85, 1991.95, 1990.85</text>
<text><location><page_18><loc_51><loc_75><loc_94><loc_76></location>1430-155 : 1993.79, 1992.89, 1991.89, 1990.84, 1990.04 ,</text>
<text><location><page_18><loc_51><loc_74><loc_63><loc_75></location>1989.14, 1988.84</text>
<text><location><page_18><loc_51><loc_72><loc_71><loc_73></location>1438 + 385 : 1994.23, 1993.53</text>
<text><location><page_18><loc_51><loc_71><loc_64><loc_72></location>1449-012 : 1992.84</text>
<text><location><page_18><loc_51><loc_70><loc_71><loc_71></location>1455 + 247 : 1993.52 , 1992.92</text>
<text><location><page_18><loc_51><loc_67><loc_94><loc_70></location>1502 + 106 : 1994.07, 1991.24, 1990.19, 1989.79, 1987.49, 1986.19, 1985.69, 1984.84, 1980.29</text>
<text><location><page_18><loc_51><loc_66><loc_77><loc_67></location>1511 + 238 : 1993.48 , 1992.93, 1991.93</text>
<text><location><page_18><loc_51><loc_63><loc_94><loc_66></location>1514 + 197 : 1994.99, 1992.94 , 1992.19, 1991.44, 1990.09, 1989.44, 1989.09, 1988.79</text>
<text><location><page_18><loc_51><loc_62><loc_77><loc_63></location>1538 + 149 : 1991.31, 1990.96 , 1990.16</text>
<text><location><page_18><loc_51><loc_61><loc_94><loc_62></location>1555 + 001 : 1993.94, 1993.59, 1993.39, 1989.89, 1988.24,</text>
<text><location><page_18><loc_51><loc_59><loc_56><loc_60></location>1987.14</text>
<text><location><page_18><loc_51><loc_58><loc_94><loc_59></location>1611 + 343 : 1992.97, 1991.17, 1989.92, 1987.92, 1985.67,</text>
<text><location><page_18><loc_51><loc_57><loc_69><loc_58></location>1985.42, 1984.42, 1983.72</text>
<text><location><page_18><loc_51><loc_54><loc_94><loc_57></location>1614 + 051 : 1993.85, 1993.60, 1993.30, 1992.80, 1992.30, 1992.00, 1991.00, 1990.55, 1989.80</text>
<text><location><page_18><loc_51><loc_53><loc_71><loc_54></location>1624 + 416 : 1992.92, 1989.47</text>
<text><location><page_18><loc_51><loc_50><loc_94><loc_53></location>1635-035 : 1993.77, 1993.22, 1992.92, 1992.67, 1991.42, 1990.67, 1989.32, 1989.02, 1988.77</text>
<text><location><page_18><loc_51><loc_48><loc_94><loc_50></location>1641 + 399 : 1993.39, 1992.94, 1991.99, 1987.84, 1986.74, 1986.04, 1985.59, 1979.59</text>
<text><location><page_18><loc_51><loc_45><loc_94><loc_47></location>1655 + 077 : 1993.73, 1993.43, 1992.78, 1992.18, 1991.83, 1991.53, 1989.08</text>
<text><location><page_18><loc_51><loc_44><loc_83><loc_45></location>1656 + 477 : 1992.94, 1992.49, 1991.94, 1991.34</text>
<text><location><page_18><loc_51><loc_42><loc_57><loc_44></location>1741-038</text>
<text><location><page_18><loc_57><loc_42><loc_58><loc_44></location>:</text>
<text><location><page_18><loc_59><loc_42><loc_65><loc_44></location>1993.36,</text>
<text><location><page_18><loc_66><loc_42><loc_72><loc_44></location>1992.91,</text>
<text><location><page_18><loc_74><loc_42><loc_79><loc_44></location>1992.41,</text>
<text><location><page_18><loc_81><loc_42><loc_87><loc_44></location>1990.21,</text>
<text><location><page_18><loc_88><loc_42><loc_94><loc_44></location>1989.41,</text>
<text><location><page_18><loc_51><loc_41><loc_81><loc_42></location>1988.41, 1987.96, 1987.36, 1986.46, 1984.91</text>
<text><location><page_18><loc_51><loc_40><loc_54><loc_41></location>1749</text>
<text><location><page_18><loc_54><loc_40><loc_55><loc_41></location>+</text>
<text><location><page_18><loc_55><loc_40><loc_58><loc_41></location>096</text>
<text><location><page_18><loc_58><loc_40><loc_58><loc_41></location>:</text>
<text><location><page_18><loc_60><loc_40><loc_65><loc_41></location>1993.82,</text>
<text><location><page_18><loc_67><loc_40><loc_73><loc_41></location>1991.57,</text>
<text><location><page_18><loc_74><loc_40><loc_80><loc_41></location>1989.27,</text>
<text><location><page_18><loc_81><loc_40><loc_87><loc_41></location>1987.97,</text>
<text><location><page_18><loc_88><loc_40><loc_94><loc_41></location>1987.47,</text>
<text><location><page_18><loc_51><loc_39><loc_75><loc_40></location>1987.02, 1986.17, 1985.17, 1983.82</text>
<text><location><page_18><loc_51><loc_37><loc_89><loc_38></location>1749 + 701 : 1993.36, 1992.96, 1989.96, 1989.51, 1988.91</text>
<text><location><page_18><loc_51><loc_35><loc_94><loc_37></location>1756 + 237 : 1993.97, 1993.52, 1992.62, 1991.52, 1991.27, 1990.52, 1990.02</text>
<text><location><page_18><loc_51><loc_33><loc_83><loc_34></location>1803 + 784 : 1993.87, 1992.92, 1991.72, 1990.77</text>
<text><location><page_18><loc_51><loc_32><loc_71><loc_33></location>1807 + 698 : 1992.99, 1985.64</text>
<text><location><page_18><loc_51><loc_29><loc_94><loc_32></location>1821 + 107 : 1993.89, 1993.54, 1991.94, 1991.39, 1988.39, 1985.29, 1984.69, 1984.24</text>
<text><location><page_18><loc_51><loc_28><loc_77><loc_29></location>1823 + 568 : 1993.88, 1992.08, 1990.58</text>
<text><location><page_18><loc_51><loc_27><loc_64><loc_28></location>1828 + 487 : 1991.02</text>
<text><location><page_18><loc_51><loc_24><loc_94><loc_27></location>1830 + 285 : 1993.90, 1993.50, 1992.95, 1992.45, 1991.95, 1991.45, 1990.80, 1990.45, 1989.50, 1988.50</text>
<text><location><page_18><loc_51><loc_23><loc_83><loc_24></location>1928 + 738 : 1993.87, 1993.62, 1993.37, 1992.97</text>
<text><location><page_18><loc_51><loc_22><loc_77><loc_23></location>1943 + 228 : 1993.87, 1993.47, 1993.12</text>
<text><location><page_18><loc_51><loc_20><loc_89><loc_21></location>1947 + 079 : 1993.96, 1992.71, 1991.66, 1988.91, 1988.56</text>
<text><location><page_18><loc_51><loc_19><loc_83><loc_20></location>2007 + 776 : 1993.35, 1991.00 , 1990.10, 1989.45</text>
<text><location><page_18><loc_51><loc_18><loc_64><loc_19></location>2008-068 : 1988.94</text>
<text><location><page_18><loc_51><loc_16><loc_94><loc_18></location>2032 + 107 : 1993.85, 1992.95, 1992.40, 1991.90, 1991.45,</text>
<text><location><page_18><loc_51><loc_15><loc_63><loc_16></location>1991.00 , 1989.15</text>
<text><location><page_18><loc_51><loc_14><loc_71><loc_15></location>2037 + 511 : 1991.92, 1991.52</text>
<text><location><page_18><loc_51><loc_13><loc_71><loc_14></location>2047 + 098 : 1992.95, 1992.10</text>
<text><location><page_18><loc_51><loc_10><loc_94><loc_12></location>2059 + 034 : 1993.51, 1992.96 , 1992.06, 1991.16, 1990.11, 1988.91</text>
<text><location><page_18><loc_6><loc_7><loc_24><loc_8></location>Article number, page 18 of 20</text>
<text><location><page_19><loc_6><loc_68><loc_49><loc_93></location>2105 + 420 : 1992.46, 1992.06, 1991.56, 1991.01 2113 + 293 : 1993.09, 1992.49, 1991.54, 1989.94, 1989.14 2121 + 053 : 1993.87, 1992.62, 1991.37, 1990.02, 1989.32 2134 + 004 : 1993.08, 1991.18, 1987.88, 1986.63 2155-152 : 1993.63, 1992.28, 1991.83, 1991.18, 1990.13 2200 + 420 : 1992.25, 1990.55, 1988.90, 1986.70, 1985.20, 1984.50, 1981.70 2209 + 081 : 1993.81, 1993.01, 1991.11, 1990.16 2214 + 350 : 1992.90, 1991.45, 1989.75, 1989.45, 1988.60 2234 + 282 : 1993.20, 1989.40, 1987.55, 1985.90, 1984.95, 1983.40 2251 + 158 : 1992.99 , 1989.44, 1985.59, 1984.89, 1983.09, 1980.34 2251 + 244 : 1993.50 , 1993.15, 1992.50, 1992.25, 1991.55 2307 + 107 : 1993.70, 1989.40 2319 + 272 : 1992.88, 1992.48, 1992.18, 1990.53, 1989.28 2344 + 092 : 1993.42, 1992.17, 1991.22, 1990.52, 1990.17 2352 + 495 : 1993.68, 1993.28, 1991.23, 1990.33, 1985.38, 1984.68</text>
<section_header_level_1><location><page_19><loc_6><loc_62><loc_45><loc_65></location>Appendix C: Extended discussion of individual ESEs</section_header_level_1>
<text><location><page_19><loc_6><loc_57><loc_49><loc_61></location>Here below we report a short analysis of the individual ESEs identified in Lazio et al. (2001b), in the light of the SRV / TDV features discussed in the present article.</text>
<unordered_list>
<list_item><location><page_19><loc_7><loc_54><loc_49><loc_56></location>-The event in the light curves of 0201 + 113 and the two detected for 0952 + 179 are due to type I SRV.</list_item>
<list_item><location><page_19><loc_7><loc_50><loc_49><loc_54></location>-Three events (for 0133 + 476, 0300 + 470, and 1749 + 096) happened during GBI transition (in 1988.1-1988.2 y); it is plausible that their origin is instrumental.</list_item>
<list_item><location><page_19><loc_7><loc_40><loc_49><loc_50></location>-The 1990.2 event in the light curves of 0133 + 476 ( β : 34.8 · , minimum elongation at 0.33 y) is not easy to identify: the 2 GHz data are characterised by very fast variations whose shape resembles ESEs; much slower and less frequent variations are visible in the 8 GHz data. It is not clear whether the variations at the two frequencies are correlated at the time of the event. Since there is no clear sign of SRV, this has been classified as an ESE.</list_item>
<list_item><location><page_19><loc_7><loc_34><loc_49><loc_40></location>-0202 + 319 ( β : 18.3 · , minimum elongation at 0.33 y) shows ESE-like variability in 1989.5; other episodes of similar variations can be seen at 1990.5, 1993.0, and 1993.5, suggesting TDV as the most likely explanation.</list_item>
<list_item><location><page_19><loc_7><loc_27><loc_49><loc_34></location>-The 1988.3 event in 0300 + 470 ( β : 28.7 · , minimum elongation at 0.38 y) at 2 GHz has a complex shape that cannot be immediately associated with an ESE, while at 8 GHz a dip is hardly detectable. Assuming it can be classified as an event, it is unlikely to be due to SRV, as it ends before the minimum solar elongation is reached.</list_item>
<list_item><location><page_19><loc_7><loc_19><loc_49><loc_27></location>-The 1986.37 event of 0333 + 321 ( β : 12.6 · , minimum elongation at 0.39 y) is certainly caused by the Sun. The source is a ff ected by type II SRV, with clear dips, both at 2 and 8 GHz, in 1984.4, 1992.4, and 1993.4. A second ESE has been identified at 1987.95; since there is no evidence that the source is a ff ected by TDV, we classified it as an ESE.</list_item>
<list_item><location><page_19><loc_7><loc_16><loc_49><loc_19></location>-The three events in 0528 + 134 have been extensively discussed in Sect. 3.3; they are due to type IIb SRV.</list_item>
<list_item><location><page_19><loc_7><loc_10><loc_49><loc_16></location>-The case of 0954 + 658 ( β : 48.8 · , minimum elongation at 0.57 y) needs to be carefully considered, as its 1981.1 event represents the archetype of all ESEs. The source is a ff ected by type IIb SRV. After removing the data covering the 1981.1 event, the data at both frequencies have been de-trended</list_item>
</unordered_list>
<text><location><page_19><loc_53><loc_75><loc_94><loc_93></location>(with 0.3 y interpolation timescale) and stacked, looking for yearly patterns; at 2GHz, the stacked data (black dots in Fig. C.1, left panel; the flux has been multiplied by a factor 9 for a better comparison with the event) show a semi-annual pattern, which at the beginning of the year (blue arrow) has a clear ESE-like shape. The 1981.1 event (emphasised in the de-trended data, green dots) is much stronger than the semiannual event in the stacked data, and it lasts longer, but they are both centred at the time of maximum solar elongation. Even more suggestive is the behaviour of the 8 GHz data (Fig. C.1, right panel): here the ESE-like pattern at the beginning of the year is stronger, and has the same duration as the 1981.1 event. This strongly favours an interpretation of the event in terms of SRV.</text>
<unordered_list>
<list_item><location><page_19><loc_52><loc_72><loc_94><loc_75></location>-The ∼ 1993.5 event in 1438 + 385 ( β : 50.4 · , minimum elongation at 0.78 y) is due to TDV.</list_item>
<list_item><location><page_19><loc_52><loc_64><loc_94><loc_72></location>-The yearly pattern of 1502 + 106 ( β : 26.7 · , minimum elongation at 0.84 y) has a complex shape that suggests the possible influence of both type IIb SRV and TDV. The 1979.5 event is likely due to TDV; the 1986 event seems more like a sequence of two or more variability episodes, it is therefore hard to classify.</list_item>
<list_item><location><page_19><loc_52><loc_56><loc_94><loc_64></location>-In the light curves of 1611 + 343 ( β : 54.1 · , minimum elongation at 0.87 y) one event was detected in 1985.4, approximately at the time of maximum solar elongation. The source, however, shows one single episode of SRV in 1993.9, and it does not appear to be strongly influenced by the Sun. It is likely that the ESE is real.</list_item>
<list_item><location><page_19><loc_52><loc_39><loc_94><loc_56></location>-The impressive event in 1992.5 2 in the light curves of 1741038 ( β : 19.5 · , minimum elongation at 0.96 y) is due to SRV. This conclusion is supported by the determination of the average yearly pattern of the variability: the original light curve (see Fig. C.2, left panel, black dots) has been de-trended (red dots) and then stacked (left panel, green dots). The average of the stacked data has been calculated both including all data (violet dots) and excluding the data covering the 1992.5 event (brown dots; the data have been shifted in flux for more clarity). In both cases, two clear minima are visible, and occur very close (i.e. at 1985.42 and 1985.92) to the time of minimum and maximum solar elongation, in agreement with the characteristics of type IIb SRV.</list_item>
<list_item><location><page_19><loc_52><loc_29><loc_94><loc_39></location>-The 1993.7 event of 1756 + 237 ( β : 47.2 · , minimum elongation at 0.97 y) is hard to classify. The source shows clear type IIb SRV, with regular dips around 0.0 and 0.5 y, which though are not compatible with the time of the discussed event. On the other hand, the yearly pattern in the 2 GHz light curve of 1756 + 237 is more complex than for other sources, showing also regular dips at 0.7 y. The event has been conservatively classified as an ESE.</list_item>
<list_item><location><page_19><loc_52><loc_21><loc_94><loc_29></location>-The event detected for 1821 + 107 ( β : 34.0 · , minimum elongation at 0.99 y) in 1984.25 cannot be explained in terms of SRV. It is worth noting that it may be part of a yearly sequence, as three additional, less prominent, dips can be found in 1985.28, 1986.24, 1987.28. However, given the high variability on short timescales, this may also happen by chance.</list_item>
<list_item><location><page_19><loc_52><loc_15><loc_94><loc_21></location>-For 2251 + 244 ( β : 29.1 · , minimum elongation at 0.20 y) an event is reported at 1988.9 3 . The simultaneity with similar flux density drops in other sources demonstrates that it is due to TDV.</list_item>
</unordered_list>
<figure>
<location><page_20><loc_9><loc_77><loc_46><loc_93></location>
<caption>Fig. C.1. The de-trended 2 and 8 GHz light curves of 0954 + 658 (left and right panel respectively, green dots) are here plotted together with the average yearly pattern obtained after removing the data-points covering the 1981.1 event, stacking the remaining data and averaging them (black dots). The stacked data are multiplied by a factor 9 for a better comparison between the datasets. The peak-to-dip di ff erence in the 1981.1 event is about 20 times greater than the spread in the noise level of the source. Blue and orange arrows indicate the time of maximum and minimum solar elongation of the source.</caption>
</figure>
<figure>
<location><page_20><loc_9><loc_46><loc_46><loc_64></location>
<caption>Fig. C.2. Left panel: in black, the 2 GHz light curve of 1741-038; the detrended data are shown in red. Right panel: the de-trended data, stacked on one-year intervals, are shown as green dots; averaged data are shown as violet dots, while stacked data not including the 1992.5 event are shown in brown, after shifting them in flux density for an easier comparison. At 2 GHz, the peak-to-dip di ff erence in the 1992.4 event is about 40 times greater than the spread in the noise level of the source. As in the previous figure, blue and orange arrows indicate the time of maximum and minimum solar elongation of the source.</caption>
</figure>
<unordered_list>
<list_item><location><page_20><loc_7><loc_19><loc_49><loc_32></location>-2352 + 495 ( β : 45.0 · , minimum elongation at 0.28 y) is affected by type IIa SRV, with dips around the time of minimum solar elongation in 1990.3, 1991.3, 1993.3 and 1994.3; Fiedler et al. (1994) reports an event with rather complex shape starting in 1984.5 and ending in 1985.5, while Lazio et al. (2001b) indicates as an ESE the interval between 1984.5 and 1984.8. The flux density variation could be accounted for as a sequence of two events centred in 1984.78 and 1985.3, therefore at maximum and minimum solar elongation, but no final conclusion can be drawn.</list_item>
</unordered_list>
</document> | [{"title": "ABSTRACT", "content": "Context. Compact radio sources can show remarkable flux density variations at GHz frequencies on a wide range of timescales. The origin of the variability is a mix of source-intrinsic mechanisms and propagation e ff ects, the latter being generally identified with scattering from the interstellar medium. Some of the most extreme episodes of variability, however, show characteristics that are not consistent with any of the explanations commonly proposed. Aims. An in-depth analysis of variability at radio frequencies has been carried out on light curves from the impressive database of the US Navy's extragalactic source monitoring program at the Green Bank Interferometer (GBI) - a long-term project mainly aimed at the investigation of extreme scattering events - complemented by UMRAO light curves for selected sources. The purpose of the present work is to identify events of flux density variations that appear to correlate with the position of the Sun. Methods. The 2 GHz and 8 GHz light curves observed in the framework of the GBI monitoring program have been inspected in a search for one-year periodic patterns in the data. Variations on timescales below one year have been isolated through a de-trending algorithm and analysed, looking for possible correlations with the Sun's position relative to the sources. Results. Objects at ecliptic latitude below \u223c 20 \u00b7 show one-year periodic drops in flux densities, centred close to the time of minimum solar elongation; both interplanetary scintillation and instrumental e ff ects may contribute to these events. However, in some cases the drops extend to much larger angular distances, a ff ecting sources at high ecliptic latitudes, and causing variability on timescales of months. Three di ff erent kinds of such events have been identified in the data. Their exact nature is not yet known. Conclusions. In the present study we show that the variability of compact radio sources is heavily influenced by e ff ects that correlate with solar angular distance; this unexpected contribution significantly alters the sources' variability characteristics estimated at GHz frequencies. In particular, we found that many extreme scattering events previously identified in the GBI monitoring program are actually the consequence of Sun-related e ff ects; others occur simultaneously in several objects, which excludes interstellar scattering as their possible cause. These discoveries have a severe impact on our understanding of extreme scattering events. Furthermore, Sunrelated variability, given its amplitude and timescale, can significantly alter results of variability studies, which are very powerful tools for the investigation of active galactic nuclei. Without a thorough comprehension of the mechanisms that cause these variations, the estimation of some essential information about the emitting regions, such as their size and all the derived quantities, might be seriously compromised. Key words. - -", "pages": [1]}, {"title": "A new view on extreme scattering events", "content": "N. Marchili 1 , G. Witzel 2 , and M. F. Aller 3 Received ; accepted", "pages": [1]}, {"title": "1. Introduction", "content": "Variability studies provide us with a powerful tool to investigate the nature of the emission from extragalactic compact radio sources. For these objects, variability can appear over a wide range of timescales, going from hours (intrahour / intraday variability, hereafter IDV; see Witzel et al. 1986, Heeschen et al. 1987) to many years (see e.g. Ulrich et al. 1997). It is generally assumed that the origin of the observed flux density variations in the radio is either intrinsic to the sources or caused by interstellar scintillation (ISS; see, e.g., Wagner & Witzel 1995; Rickett et al. 2006), a propagation e ff ect due to the presence of a screen of interstellar medium (ISM) along the line of sight between the source and the observer. The importance of scattering in the ISM on centimetre band data was demonstrated, among others, by Pushkarev & Kovalev (2015) and Koryukova et al. (2022). In Marchili et al. (2011) it was shown that, on IDV timescales, a further contribution to the variability is provided by the Sun, either through local propagation e ff ects (i.e. interplanetary scintillation, IPS), or, indirectly, through instrumental e ff ects rising when a source's angular distance to the Sun, that is its solar elongation, is low. The analysis of 5 GHz light curves from the Urumqi Observatory and the 4.9 GHz ones from the MASIV survey (Lovell et al. 2003) at the Very Large Array (VLA) revealed a significant increase of the intraday variability (up to a factor 2) of the analysed sources as their solar elongation decreases. The aim of the present study is to assess how the Sun can a ff ect the variability curves of compact radio sources on longer timescales, from a few days up to one year. This goal was pursued by searching for variability features that cannot be explained in terms of source-intrinsic processes or through ISS, such as solar-elongation-dependent flux density variations or correlated variability among many objects. Given the variety of manifestations of the e ff ects we found, the results of our analysis have been split in two publications; the present one (Paper I) deals with sharp flux density variations similar to extreme scattering events (ESEs, Fiedler et al. 1987a). The second publication (Witzel et al., in prep.; from now on, Paper II) focuses on smooth one-year periodic variations in radio light curves. At this stage of the work, it is not known whether and how the two phenomena are related. To reach our goal, we inspected one of the best available databases for the study of compact objects' variability at GHz frequencies, namely the US Navy's extragalactic source monitoring program (henceforth, NESMP; see Fiedler et al. 1987b, Waltman et al. 1991, and Lazio et al. 2001b). The NESMP was a project running at the Green Bank Interferometer (GBI) between 1979 and 1996, whose primary aim was to search for ESEs. These are a class of dramatic variations in the flux density of compact radio sources, characterised by a substantial decrease bracketed by less pronounced increases. Extreme scsttering events have a typical timescale between a few weeks and several months; their origin is generally attributed to the scattering of radio waves in the interstellar medium. These events are rare: Fiedler et al. (1994) estimated that, taking into account a cumulative observing time of about 600 source-years for the 330 objects they monitored, the timespan covered by unusual variability associated to ESEs amounts to 4.8 y, corresponding to about 0.8% of the total time. Athorough analysis of Sun-related variability (from now on, SRV) is crucial on several levels. Given the importance of variability studies to constrain basic properties of emitting sources (e.g. emitting regions' sizes, and, from these, brightness temperatures and Doppler factors; Hovatta et al. 2009, Liodakis et al. 2017), and to understand the origin and the evolution of flares (see, e.g., Marscher et al. 1992, Ulrich et al. 1997, Marscher 2016), it is necessary to identify and quantify all non-intrinsic sources of variability in the light curves; the contribution of SRV to blazar variability at GHz frequencies seems to be quite significant. The nature of SRV is a puzzling and fascinating topic: the evidence we collected rules out an instrumental origin at least for some of the SRV manifestations. Propagation e ff ects seem to be the most obvious candidate to explain them, but, in the light of the known properties of IPS or of the Earth's upper atmospheric layers, these e ff ects would be expected to be generally negligible; this implies that our picture of local propagation e ff ects must be missing some important pieces. Last but not least, SRV has a strong impact on our understanding of ESEs, given the fact that their signature in the light curves appears to be the same. After a general description of the database and the procedures used for the data analysis (Sect. 2), we will present the four di ff erent kinds of SRV manifestations we identified in the data (Sec 3) and we will illustrate our classification of the sources (Sect. 4). Some hypotheses about the origin of the variability will be discussed in Sect. 5, while Sect. 6 will be dedicated to a revision of the identified ESEs in Lazio et al. (2001b), in the light of the new evidence we collected. The main conclusions of the present work will be presented in Sect. 7.", "pages": [1, 2]}, {"title": "2. Data and procedures", "content": "The NESMP observations were performed at two radio frequencies, of approximately 2.5 and 8.2 GHz. Flux density measurements were collected for 149 objects, with a typical cadence of one observation every two days. Information about the GBI can be found in Hogg et al. (1969) and Coe (1973). The procedures for the data calibration and the results of the data analysis have been thoroughly discussed in Fiedler et al. (1987b), Waltman et al. (1991), and Lazio et al. (2001b). Here below a short summary of the main aspects regarding data acquisition and calibration is presented. Observations were carried out on a 2.4 km baseline, initially at frequencies of 2.7 GHz (S band) and 8.1 GHz (X band). In September 1989 the installation of cryogenic receivers brought a change of the observed frequencies, which became 2.25 GHz and 8.3 GHz respectively. Four sources were used for the calibration of raw data; 1328 + 307 was used to fix the individual flux densities of 0237-233, 1245-197, 1328 + 254; the flux density measurements of these three sources were combined to form a hybrid calibrator for all the other targets. A deeper look into this procedure will be given in Appendix A. According to Fiedler et al. (1987b), the uncertainties of GBI's flux density measurements before the installation of the cryogenic receivers could be expressed as while after completion of the operation (August 1989), they could be expressed (see Lazio et al. 2001b) as where S s and S x indicate the measured flux densities in S and X band respectively. A further contribution to the uncertainties, of the order of 10 and 20% (for S and X band respectively), could be explained by atmospheric and hardware e ff ects. Some basic information about the analysed sources are reported in Table 3, such as name (Col. 1), ecliptic latitude (Col. 2), time of minimum solar elongation (Col. 3), average flux density and standard deviations at 2 and 8 GHz (Col. 4-7). The Xray binary system 1909 + 048 has been excluded from the present analysis, as its source-intrinsic variability component is so strong to be absolutely dominating over any other kind of variability. For a deeper study of SRV episodes in the blazars OJ 287 and 0528 + 134, we made use of 8 GHz and 14.5 GHz data from the University of Michigan Radio Astronomical Observatory (UMRAO; for more details see, for example, Aller et al. 1985, Aller et al. 1999), which allowed for both an extension of the analysis to higher frequencies and an essential consistency check between the 8 GHz data from the two facilities.", "pages": [2]}, {"title": "2.1. Algorithms for the data analysis", "content": "Two main procedures have been used for the analysis of the data: a de-trending procedure to remove the long-term variability and highlight the fast variations; a data-stacking procedure for identifying yearly patterns in the data. The de-trending procedure (see Villata et al. 2002) consists of several steps: firstly, the flux densities of a light curve are averaged on time intervals of selected length; the resulting average fluxes are fitted by a spline curve, which is then interpolated with the same sampling of the original light curve; the re-sampled spline curve (i.e. the long-term trend) is subtracted from the original flux densities, leaving only the variations on timescales smaller than the time intervals used for the initial data averaging. The data-stacking procedure makes use of the de-trended data: these are divided into intervals of one year, which then are stacked on top of each other; the resulting variability curve as a function of day-of-the-year is finally averaged on bins of four days 1 , providing us with a mean yearly pattern of the fast variability.", "pages": [2, 3]}, {"title": "3. SRV phenomenology", "content": "The light curve of some NESMP sources show one-year periodic dips at the time of minimum solar elongation, which were explained by Fiedler et al. (1987b) as due to the e ff ect of the Sun passing through a sidelobe of the primary antenna beam. In order to identify and characterise the periodic variations in the data, we applied the data-stacking procedure (with a detrending interpolation timescale of ten months) to all sources in the NESMP catalogue. Along with the fast variability (with timescale from days to a few weeks) at very low solar elongation mentioned above, this analysis revealed three more kinds of systematics: slow (timescale of months) one-year periodic variability, correlated with solar elongation, mainly a ff ecting sources at low ( < 20 \u00b7 ) ecliptic latitude; slow six-month periodic variability correlated with solar elongation; six-month periodic variability apparently not correlated with solar elongation. In all four cases (which will be separately discussed in the following), the flux density variations have shapes consistent with an ESE. For sources showing flux density variations that resemble an ESE, centred at the time of minimum solar elongation, we fitted a gaussian profile - centred at minimum solar elongation too - to the stacked 2 GHz data. For each source, the amplitude and uncertainty of the fit have been used to quantify the dips' characteristics: these values are reported in Table 3, Col. 8. The dips' amplitudes are plotted in Fig. 1, divided by the average flux of the sources for a proper comparison. A clear dependence on ecliptic latitude, \u03b2 , can be seen up to \u03b2 \u223c 20 \u00b7 ; at higher latitudes, instead, the correlation disappears, which is a clear indication of a di ff erent mechanism producing the variability. It is important to stress that the vast majority of dips identified in the annual pattern of the data through the data-stacking procedure occurs either at minimum or at maximum solar elongation, which shows that the appearance of systematic dips in the light curves correlates with the position of the Sun with respect to the sources.", "pages": [3]}, {"title": "3.1. Fast SRV at low ecliptic latitude (type I SRV)", "content": "The mean yearly pattern of 1253-055 shown in Fig. 2 nicely illustrates the typical e ff ect of the Sun on sources at low ecliptic latitude at di ff erent frequencies. Systematic dropouts at 2 and at 8 GHz are comparable in amplitude, with flux density variations up to \u223c 30%, but not in width: at 2 GHz data are a ff ected up to a solar elongation of 15-20 \u00b7 , while at 8 GHz the solar influence does not go further than 8 \u00b7 . The characteristics of this kind of SRV (which henceforth, for simplicity, will be denoted as type I SRV) are compatible with the explanation provided by Fiedler et al. (1987b) in terms of an instrumental e ff ect, due to the Sun passing through a sidelobe of the primary antenna beam. Afurther contribution to the variability may also come from IPS, whose e ff ect is expected to become important at very low solar elongation.", "pages": [3]}, {"title": "3.2. Annual SRV up to high solar elongation (type IIa SRV)", "content": "The annual pattern of some NESMP sources shows a systematic flux density decrease, correlated with solar elongation, extending over a very large interval of time. An example of this kind of variability is provided by the light curves of OJ 287. OJ 287 is among the best studied blazars; its fame grew considerably with the discovery of a long-term ( \u223c 12 y) periodic pattern in the optical band light curve (Sillanp\u00e4\u00e4 et al. 1988). It has an ecliptic latitude of 2.6 \u00b7 , which makes it a natural candidate to show type I SRV. We analysed the OJ 287 data from both the NESMP (2 and 8 GHz) and the UMRAO (8 and 14.5 GHz) monitoring campaigns, to check the consistency among flux density measurements with di ff erent telescopes, and to extend the investigation of slow SRV to higher frequencies. The data-stacking procedure reveals the existence of type I SRV at 2 GHz, as expected. Surprisingly, at 8 and 14.5 GHz the dip is much larger than at 2 GHz, as it extends from about 0 . 45 to 0 . 70 -0 . 75 y, which correspond to a solar elongation range up to 50 -55 \u00b7 . The dips do not recur systematically every year, but they occur frequently: an example of two consecutive dips, in 1983 and 1984, is shown in Fig. 3, right panel. The two dips, which are almost identical in amplitude and duration, are stronger at 8 and 14.5 GHz than at 2 GHz. Particularly important is the simultaneity of the flux density variations in the di ff erent bands. Generally, the variability of OJ 287 shows spectral evolution: a correlation analysis of the light curves through locally normalised discrete correlation function (henceforth NDCF, see Lehar et al. 1992; Edelson & Krolik 1988) reveals that the 14 GHz variations occur, on average, about 0.08 y earlier than at 8 GHz, and about 0.25 y earlier than at 2 GHz. At minimum solar elongation, however, the time delays between these frequencies drops to zero. Also remarkable, in Fig. 3, is the occurrence, in 1983, of a flux density drop in 0742 + 103 ( \u03b2 = -10.9 \u00b7 ), similar in amplitude and duration to the one in OJ 287, and similarly stronger at high rather than at low frequencies (see Fig. 3, left panel). Given the small angular distance between the sources, it is very likely that these dips have the same origin. A weaker dip, partially hidden by the noise, is visible in the 0742 + 103 light curve in 1984 too, at 8 GHz. More sources show evidence of the same broad systematic dips at minimum solar elongation found in OJ 287 (which henceforth we will address as type IIa SRV); they are indicated in Table 3, Col. 9 and 10, for the 2 and the 8 GHz light curves separately. Several of them are very close to OJ 287 in right ascension and declination. Type IIa SRV at 2 GHz is often associated with the same kind of variability at 8 GHz, although exceptions are not rare: in 2059 + 034, for example, two strong ESE-like variability episodes at minimum solar elongation can be found at 2 GHz (at 1990.13 and 1992.10), but not at 8 GHz.", "pages": [3, 4]}, {"title": "3.3. Semi-annual SRV up to high solar elongation (type IIb SRV)", "content": "A rather peculiar manifestation of solar-elongation-dependent variability is the existence of six-month periodic dips, similar to ESEs, in the light curves of some NESMP sources (indicated in Table 3, Col. 9 and 10); we will refer to it as type IIb SRV. The most striking example is given by the 2 GHz light curve of 0537-158 ( \u03b2 = -39 \u00b7 ), which, between 1990 and 1994, appears as a regular sequence of ESEs with six-month cadence (see Fig. 4). The average duration of the events is of the order of months. Type IIb SRV generally a ff ects both 2 and 8 GHz light curves, although, similarly to the case of type IIa SRV, the way the two frequencies are a ff ected varies from source to source: 1830 + 285, for example, doesn't show any variability resembling an ESE at 8 GHz. Also, the e ff ect is not equally strong at minimum and at maximum solar elongation: generally the features at minimum are stronger than at maximum solar elongation, but in 0528 + 134 and 0954 + 658 the opposite occurs. The case of 0528 + 134 deserves particular attention. In Lazio et al. (2001b), the source is reported as showing three ESEs, in 1991.0, 1993.5, and 1993.9, which makes it the most a ff ected object by ESEs in the whole monitoring program. All the reported events are consistent with the semi-annual cadence of type IIb SRV (the source reaches minimum solar elongation at 0.45 y). A comparison between the data collected within the NESMP with the ones from the UMRAO database (see Fig. 5) shows that the flux density measurements from the two facilities are in excellent agreement. Also in this case, it is important to stress the achromaticity of the flux density variations: in general, the variability of 0528 + 134 shows an overall trend of spectral evolution, with a delay, calculated via NDCF, of \u223c 0 . 17 y between 14.5 and 8 GHz data, and much larger between 8 and 2 GHz. Most of the variability observed on timescales of one year or shorter, however, is achromatic (the 14.5-8 GHz cross-correlation time delay of de-trended data is an impressive 0 . 00 \u00b1 0 . 05 y). Along with the three ESEs identified by Lazio et al. (2001b), indicated with brown lines, three more achromatic dips can be seen in the data (turquoise lines), namely in 1993.0, 1995.1, and 1996.1. The last two are particularly strong at 14.5 GHz (orange dots). Another event, visible only at 2 GHz, occurs in 1988.9 (see the small box within the figure); this is a rare example of a complex ESE shape detected at this frequency. The semi-annual nature of the variability, correlated with solar elongation, is very clear, which confirms its attribution to SRV. The agreement between NESMP and UMRAOdata, and the importance of the e ff ect at 14.5 GHz, confirm the non-instrumental origin of type IIb SRV and its achromaticity within the range of investigated wavelengths.", "pages": [4]}, {"title": "3.4. Time-dependent semi-annual variability (TDV)", "content": "So far, all the systematic variability we discussed showed a strict correlation to solar elongation, as the flux density minima occurred either at minimum or at maximum solar elongation. The NESMP data show a further interesting kind of ESE-like variability that is not directly ascribable to the relative position of the Sun to a source; it seems instead to be related to a specific time of the year. This kind of variability will be henceforth addressed as Time-Dependent Variability (TDV). In Tables 5, 6, and 7 of Lazio et al. (2001b), which present lists of possible ESEs from the GBI monitoring program, eight sources are reported as showing identified or potential ESEs approximately at the same time , in 1993.5 (i.e. July 1993). A thorough check of the complete NESMP database reveals that a flux density drop compatible with the shape of an ESE is visible in many other sources. The light curves of some of them are plotted in the bottom panel of Fig. 6, after applying the de-trending procedure with time intervals of 50 days. This event is both extraordinary and puzzling for several reasons. Among the a ff ected sources there is 0528 + 134, which has already been discussed as a ff ected by type IIb SRV, with sixmonth periodic ESE-like variability close to the time of minimum and maximum solar elongation, occurring at 0.45 and 0.95 y. The 1993.5 event occurs close to the minimum in solar elongation, therefore it is consistent with the solar-elongationrelated semi-annual trail of events. For many other sources (e.g., 1225 + 368, 1404 + 286, and 1438 + 385), however, the time of the dip does not come close either to a minimum or to a maximum of solar elongation, which demonstrates that the 1993.5 event is truly time-dependent, rather than solar elongation-dependent. Furthermore, the 1993.5 ESE of 0528 + 134 has been detected by Pohl et al. (1995) in data from the 100-m E ff elsberg telescope and the 30-m Pico Veleta telescope. This poses some questions: correlated variations simultaneously detected in many objects could point to an instrumental or a calibration problem, but how to explain the excellent consistency with the data from different telescopes? Also the interpretation of the variability of 0528 + 134 is problematic: is it solar-elongation dependent or time-dependent? This point will be addressed in Sect. 5, where we will discuss a possible link between the di ff erent manifestations of SRV and TDV. The hypothesis that TDV is due to a calibration problem is assessed in Appendix A. We analysed the combined variability characteristics of all NESMP sources as a function of time to check whether more episodes of correlated variability, similar to the July 1993 event, can be found in the data. All the 2 GHz light curves have been detrended, and then normalised, imposing the same standard deviation to all light curves, to ensure that the fast variability of all sources has a comparable weight in the calculation of the combined variability curve; finally, the data have been stacked. The resulting data-points have been averaged on intervals of 0.01 y to generate an average combined variability curve. This turned out to be characterised by a smooth one-year periodic pattern, which results from the combination of the annual oscillations in the light curves, discussed in Paper II; after modelling and removing this systematic modulation, we obtained the combined variability curve in Fig. 7. Episodes of time-correlated variability among many sources should result in clear variability features in the combined curve. The 1993.5 ESE is indeed recognisable as a short and sharp drop of the average flux (magenta arrow in the plot); six more events (orange arrows), all shaped approximately as an ESE, can be identified in the data, generally less intense but longer than the 1993.5 event. The temporal sequence of the events is the following: 1991.0, 1992.1, 1993.0, 1993.5, 1994.5, 1995.1, and 1995.9. The 1993.5 event appears to be an extreme manifestation of a long trail of similar phenomena with a cadence of \u223c 6 months, approximately falling at the end and in the middle of the year. The semi-annual pattern is detectable, but without the same regularity, also in the 8 GHz data, which display ESE-shaped variability in 1990.5, 1993.0, 1994.5, and 1995.4. Before the year 1990 the periodic pattern, both at 2 and at 8 GHz, is less evident, as the variability in the combined variability curve is considerably stronger; most likely, this is due to the low number of sources monitored before the GBI upgrade (40 objects, less than a third of the later sample), which makes it harder to isolate correlated variability from the one that is intrinsic to the sources.", "pages": [5]}, {"title": "4. SRV classification", "content": "Through the analysis reported above we identified four distinct kinds of ESE-like variations in the data, which di ff er in duration, dependence (on solar elongation or time), and recurrence (which is the discriminating factor between type IIa and type IIb SRV). Their main properties are summarised in Table 1; they all have in common the shape (resembling an ESE) and the achromaticity, although the importance of the e ff ect at di ff erent frequencies changes not only according to the variability type, but, often, also from source to source. All sources of the NESMP have been classified in Col. 9 and 10 of Table 3 according to the kind of SRV found through a visual inspection of the 2 GHz and 8 GHz light curves, respectively. The reader should be warned that, while for some objects the classification is straightforward, for others it cannot be given without uncertainty. This is mainly because of the superposition of di ff erent e ff ects, and, for a few sources, because of the limited amount of available data. Sources showing episodic variability (e.g. 0333 + 321) have been classified as a ff ected by SRV. The presence of timedependent variability, indicated in Col. 11, should be considered as purely indicative: TDV occurs in many sources in 1993.5; we classified a source as a ff ected by TDV if at least one more event at 0.0 or 0.5 y could be detected. There is an obvious ambiguity between TDV and type IIb SRV in sources whose minimum / maximum solar elongation occurs around 0.0 or 0.5 y. The duration of the events can often remove the ambiguity (type IIb SRV is characterised by longer timescales); sources for which the two e ff ects cannot be unambiguously identified are indicated as being a ff ected by both.", "pages": [5, 6]}, {"title": "4.1. Automated classification of sources", "content": "Given some unavoidable degree of subjectivity in the classification of the sources according to a visual inspection, we devel- oped a simple method for their automatic classification. The latter is not meant to replace the former, which is deeper and more accurate, but it provides an important tool to assess the reliability of its results. It is based on the usage of NDCF, for the identification and localisation of dips in the light curves; the number of dips whose timing is consistent with SRV / TDV is compared with the null hypothesis that dips can randomly occur at any time of the year, independently of solar elongation. More specifically, we created a generic dip model as a 0.2y V-shaped flux density drop, flanked by 0.1y-wide segments of constant flux. We cross-correlated the original light curve (i.e. without de-trending) of each source with the generic dip model. A high peak in the NDCF indicates the presence (and the location) of a feature in the light curve similar to the dip model. Peaks are considered significant if they are higher than a given threshold. For most sources, a degree of correlation of 0.4 produced a reasonable amount of events; when less than 2 or more 10 events were detected, the analysis has been repeated with a correlation threshold of 0.3 or 0.5, respectively. The choice of a variable threshold originates from the need to overcome two problems: a high level of noise in the data proportionally reduces the level of correlation between the light curve and the dip model, causing a decrease of detected events which can be compensated by a decrease of the threshold; a high level of source-intrinsic variability, on the other hand, causes spurious detections of dips, which requires a higher threshold to limit the number of false alarms. The detection of few or many events in a light curve with threshold 0.4 is a rather reliable indicator of the presence of either of the problems described above. By allowing the algorithm to overcome them through a variation of the threshold we can keep the procedure as automatic as possible. An example of how the algorithm works is provided in Fig. 8 for the type IIb SRV source 1830 + 285. In the lower panel, the peaks of the NDCF above the threshold correspond to detections of dips in the light curve (upper panel). It is worth to note the presence of negative dips in the NDCF, which correspond to rapid outbursts in the light curves. Since the algorithm we developed is very sensitive to sharp flux density variations, it can reveal, with opposite signs, both dips and flares. The full list of identified events, for all sources, is reported in Appendix B; those that are undoubtably caused by TDV are shown in boldface. For type I SRV, events are expected to occur exactly at the time of minimum solar elongation. For type IIa and IIb, SRV dips are broader, and sometimes their time of occurrence can be determined less precisely; therefore, NDCF peaks occurring within 0.05y from the solar elongation minimum (or maximum) are still consistent with type IIa (or type IIb) SRV. To take into account these di ff erences in the expected location of the dips, for each source we calculated four indicators: I. number of events occurring exactly at solar elongation minimum; II. number of events occurring within 0.05y from solar elongation minimum; III. number of events occurring exactly at solar elongation minimum or maximum; IV. number of events occurring within 0.05y from solar elongation minimum or maximum. Since the NDCF was calculated with a step of 0.05y, a chance occurrence of an event at solar elongation minimum for indicator I is 0.05; for indicator II it is 3x0.05 = 0.15; for indicator III 2x0.05 = 0.1; for indicator IV 6x0.05 = 0.3. Through a binomial probability distribution, we can calculate the probability that the number of successes (i.e. the events occurring at minimum / maximum solar elongation) is consistent with the null hypothesis that they happen at random time. If the probability is lower than 0.05 the source is classified as a ff ected by SRV. If the lowest probability is returned for indicators I or II, the source is classified as type I or IIa SRV, otherwise as type IIb SRV. Subsequently, the same procedure has been applied three more times to the data: a first time after removing data-points obtained when the source's solar elongation is lower than 16 \u00b7 ; if the resulting light curve is still a ff ected by SRV, it is classified as a type IIa, otherwise as a type I. A second time, after redefining successful detections as events occurring around day-of-the-year 0 (0.0y) or 182.5 (0.5y), in order to assess TDV. Lastly, after redefining successful detections as events occurring 0.25y before minimum / maximum solar elongation: as there is no reason to expect dips to systematically appear at this time, this test provides an estimate of the false positives returned by the analysis.", "pages": [6, 7]}, {"title": "4.1.1. Results", "content": "Despite its simplicity, the algorithm is e ff ective. The original classification of sources is confirmed for 80% of the sample (120 out of 148 sources): 83 out of 90 sources are confirmed as showing no SRV; 19 out of 26 sources are confirmed as type I SRV; 5 out of 7 as type IIa; 8 out of 12 as type IIb (see Col. 13 and 15 of Table 3 for the automatic SRV and TDV classification; the probability of a random occurrence of the events is given in Col. 12 and 14). Among the uncertain cases, 4 out of 8 type IIa? and 1 out of 5 type IIb? are classified as type IIa and type IIb respectively. In total, with the automatic classification of sources we obtain 95 no-SRV sources, 19 type Ia SRV, 20 type IIa SRV, and 14 type IIb SRV (the classes of uncertain cases disappear). The group of 95 no-SRV sources includes also 11 objects for which too few data are available for a proper classification. The search for sources highly a ff ected by TDV returned 9 objects, 4 of which are also showing type IIb SRV. Quite remarkable is the result provided by the algorithm when successes are defined as the events occurring 0.25y before minimum / maximum solar elongation: this way, only one object (i.e. 2234 + 282) out of 148 is identified as a ff ected by periodic dips, which shows that the number of false positive detections to be expected in the entire sample is very low. The list of events found through automatic detection can also be used to identify the most important TDV events. We divided the timespan of GBI observations (1979-1996) in bins of 0.05y width; after flagging dips that are consistent with SRV (because of their time of occurrence and the classification of the source in which they were detected), we counted the number of dips in each bin. The bins with the highest number of counts are the more likely to be a ff ect by TDV. By setting a threshold of 8 counts, we found the following sequence of TDV events: 1991.00, 1991.55, 1992.50, 1992.95-1993.00, 1993.50-.55. Note that before 1988 relatively few sources were monitored, so it is not possible to identify TDV events through this method. The sequence above does not coincide with the one obtained via the inspection of Fig. 7; the semi-annual cadence of TDV, however, is confirmed. There seems to be no obvious similarity between the sources showing simultaneous dips in the light curves. We identified 10 events around 1991.00y that, because of their shape and duration, appear to be certainly due to TDV; they a ff ect the light curves of the following objects: 0316 + 413, 0337 + 319, 0400 + 258, 0723 + 679, 1123 + 264, 1150 + 812, 1538 + 149, 2007 + 776, 2032 + 107, 2105 + 420. The sources are widely spread both in right ascension and in declination, which seems to exclude both a common calibration scheme or a possible issue related to the pointing towards a specific region of the sky.", "pages": [7]}, {"title": "4.1.2. Caveat", "content": "A complete automation of the procedure is hampered by the ambiguity between TDV and SRV for sources reaching minimum / maximum solar elongation around 0.0 / 0.5y. If the algorithm detects an event around these times, it is necessary to proceed with a visual inspection to properly classify it, although in some cases the ambiguity cannot be resolved. If the shape of the event is equal to the one seen in the combined light curve (see Fig. 7) its classification as TDV is straightforward; if the shape / duration of the event is clearly di ff erent, the classification as SRV is obvious too. Sometimes, instead, the event's shape suggests a probable superposition of the two e ff ects, and therefore a classification as TDV + SRV. The most complicated case, in this respect, is the one of 0954 + 658. Out of the 7 events found by the automated detection algorithm, 4 occur either around 0.00 or 0.50y, while only one (the archetype of the ESEs) is at maximum solar elongation; this would suggest TDV as the origin of the variability, rather than SRV. The duration of the events, however, is not consistent with the features in the 2 GHz combined variability curve. A further complication comes from the very high degree of correlation between the fast variability of the source (obtained through a de-trending interpolation timescale of ten months) and the ones of the angularly-nearby sources 0633 + 734, 0723 + 679, and 0836 + 710, with time delays that are consistent with the different solar elongation patterns followed by these objects. The correlation between the variability of di ff erent sources would be consistent with TDV, but the time delay between these variations implies a solar-elongation-related e ff ect. Finally, the archetype event of 1981.1 is not the only clear episode of sharp flux density drops at maximum solar elongation: even if not detected by the automatic algorithm, three more events can be seen (in 1989.10, 1991.10, and 1992.10), which explain the dips found in the annual pattern of the source (visible both at 2 and 8 GHz, see Fig. C.1) and the visual classification of its variability as type IIb SRV. Given the facts illustrated above, the most probable explanation for the fast variability of 0954 + 658 is that both TDV and SRV contribute to it. The similarities with the light curves of 0633 + 734, 0723 + 679, and 0836 + 710 suggest a common origin of the variability. Note that the complexity of such cases cannot be resolved through a fully automatic analysis of the data. The algorithm has been applied also to the 8 GHz data. The results, however, in this case are much less reliable. The variety of shapes and widths of the dips caused by SRV / TDV makes it di ffi cult to create a single archetype that could e ffi ciently detect the majority of the events. Only 16 sources are automatically classified as a ff ected by type II SRV, and they often do not coincide with the 23 objects identified through visual inspection.", "pages": [7, 8]}, {"title": "5. Discussion", "content": "Among SRV manifestations, only type I events can be satisfactorily explained so far (see Sect. 3.1). In Sect. 3.2, 3.3, and 3.4 we identified and described three more types of systematic variability in the data, namely type IIa SRV, IIb SRV, and TDV; there are ambiguities and exceptions in the general properties we outlined for each variability type, which do not facilitate the formulation of a robust hypothesis concerning the origin and the possible relationships among these e ff ects. It seems reasonable to assume that they are not independent of each other, and that there is a common ground to which the events can be attributed. Some hints concerning the nature of the variability may come from the distribution in the sky of the a ff ected sources; however, the indications resulting from the visual inspection and the automatic classification of the sources are not very consistent among each other. The distribution obtained with the first method (see Fig. 9) suggests that type IIa SRV mainly a ff ects objects at low ecliptic latitude within the right ascension range from 04h to 09h (corresponding to solar elongation minima between 0.39 and 0.59 y). This range largely overlaps with the one comprising most of type IIb sources (from 05h to 10h, corresponding to solar elongation minima between 0.45 and 0.63 y); type IIa and type IIb could therefore be manifestations of the same phenomenon, essentially di ff erentiated by the ecliptic latitude of the sources (lower for the former, higher for the latter). A second block of type IIb sources can be found in a narrow right ascension stripe about 12h apart from the first one, around right ascension 18h; it corresponds to solar elongation minima between 0.89 and 0.02 y. The automatic detection algorithm returns a distribution of sources that is consistent with the one of the visual inspection for type IIb SRV. Type IIa SRV objects, instead, are more widely spread in the sky than what the visual inspection suggests. The two classification methods agree though on the facts that this kind of variability mostly concerns objects at low ecliptic latitude, and that there is a large cluster of them around R.A.: 8h, Dec: + 20 \u00b7 . Some hypothesis as to the origin of the variability are shortly discussed below; particular attention is paid to assess a possible role of ISS.", "pages": [8]}, {"title": "5.1. SRV as a sequence of ESEs", "content": "Since ESEs are generally attributed to ISS, it is important to assess whether a series of events that look like ESEs can be reasonably explained in terms of ISS too. Concerning TDV the an- swer is certainly no, as the simultaneity of the events in di ff erent sources excludes the intervention of a localised screen as required in ESEs. Concerning type II SRV, however, the reply is less obvious. In principle, a localised screen moving very slowly could cause repeated events occurring approximately at the same time of the year, every time that the line of sight (LOS) to the source crosses the screen again. The fact that, as shown in Fig. 9 for the results of the visual inspection, sources a ff ected by SRV are mainly concentrated near R.A. 6 and 18h would agree with this picture, as these are the conditions for which the ISM velocity transverse to the LOS is minimum, so the Earth's orbital velocity is dominant. The search for events repeating regularly with semi-annual cadence could facilitate the detection of ESEs that occur around the time of minimum / maximum solar elongation of a source, because at larger angular distances the cadence would generally be more asymmetric (e.g. there could be 4 months and then 8 months intervals between consecutive events). There are, however, a number of critical arguments against this hypothesis: A possible source of bias in favour of events occurring more often around the times of minimum / maximum solar elongation could be the higher transversal speed of the Earth with respect to the LOS, which implies that a larger region in the sky is swept around these times. This argument, again, cannot explain TDV, neither type IIa SRV, because, if the transversal speed would be the cause of SRV, the events would have the same chance to occur at minimum and at maximum solar elongation. Type IIb SRV, instead, would be compatible with an e ff ect of the transversal speed; but this would become more important as the ecliptic latitude of the source tends to zero. Instead type IIb SRV concerns mostly sources at high ecliptic latitude where the e ff ect would be marginal, while type IIa SRV seems stronger for sources at low ecliptic latitude. Finally, an important contribution of the transversal speed to SRV would necessarily lead to an equally important decrease of the duration of the events at the times of minimum / maximum solar elongation; the data, however, show absolutely no sign of it.", "pages": [8, 9]}, {"title": "5.2. Further hypotheses", "content": "An instrumental origin of the variability can be ruled out for different reasons: all the main examples of the di ff erent kinds of variability (OJ287 for type IIa, 0528 + 134 for type IIb and TDV) are supported by observations from di ff erent facilities, whose flux density measurements are in excellent agreement with each other; the variability also a ff ects night time observations (for type IIb SRV in particular), which excludes sunlight reflection as a possible cause; it is also worth mentioning that, di ff erently from the instrumental type I SRV, the centre and the duration of type II and TDV events in the light curves changes from year to year, which means that the variability is not as systematic as one would expect from an instrumental e ff ect. Assuming that the source distribution returned by the visual inspection of light curves is correct, type II SRV mostly a ff ects sources whose solar elongation minima and maxima occur around 0.0 and 0.5 y; these are also the critical times for the appearance of TDV: time could therefore be the link between the di ff erent kinds of ESE-shaped events we detected in the data. Several astronomically-relevant e ff ects or events occur around 0.0 and 0.5 y, namely meteorological e ff ects, perihelion / aphelion, the Earth's intersection of the Sun's equatorial plane, and solstices. We separately discuss them, briefly, in the context of systematic flux density variability, here below: consider that the variation of the Sun's declination during the year could have an impact on flux density measurements, for instance by a ff ecting the propagation of radio waves through the atmosphere. On the other hand, there is no obvious reason why a source should be similarly a ff ected when the Sun is at maximum and at minimum declination, as for type IIb SRV. The mechanism that could be responsible for the variability is equally hard to identify. It is important to preliminarily underline that the timescales and the amplitudes of the variations detected in the light curves are not consistent with the theoretical values from scattering in a nearby medium (see Narayan 1992 for an introduction to the physics of scintillation). The plausible existence of a ring that crosses the whole sky along the right ascension lines around 06h and 18h, comprising most of the type II SRV sources, if confirmed, would not be consistent with IPS because of its wide extension. Since type II SRV a ff ects the sources months before / after the time of minimum elongation, when the Sun's declination is far from its highest northern / southern extension, atmospheric scattering seems unlikely too. The limited range of declination of type II sources could suggest a relationship with the Earth's North-South direction, implying a link to the Earth's geometry and properties. The fact that, within the 06h-18h right ascension ring, some sources are a ff ected for a large part of the year, would indicate a geometri- cal e ff ect as the most plausible. The plane on which the ring lies corresponds to the one defined by the equatorial and the ecliptic latitudes axes. At a level of conjecture, one could wonder whether the complex shape of the magnetosphere could play a role in the observed variability, acting somehow as a lens. The magnetosphere stretches along the direction of the flow of solar wind, in the ecliptic plane; it is compressed towards the Sun, and very elongated in the opposite direction (magnetotail); its shape varies during the year, and its structure along the line of sight to a source depends on the source's position with respect to the Sun. The inner structure of the magnetosphere is also heavily influenced, because of the magnetic field, by the celestial poles. This implies the existence, in the morphology of the magnetosphere, of a privileged direction (the 6h-to-18h right ascension direction), towards the tilt of the Earth axis, which at 0.0 and 0.5 y, approximately, aligns with the flow of the solar wind; such alignment implies a symmetric configuration of the magnetosphere with respect to Earth's axis that could possibly trigger both SRV and TDV. When the e ff ect of the Earth axis modifies more strongly the inner structure of the magnetosphere, TDV would become dominant; when the e ff ect is weaker, only sources with solar elongation minima around 0.0 and 0.5 y would be affected, and we would see type IIb SRV. Although this scenario may be geometrically plausible, it has to face the fact that, to cause significant variability in the data, a lensing e ff ect would require a change in refraction index that is orders of magnitude larger than what is expected at the level of all the atmospheric layers.", "pages": [9, 10]}, {"title": "6. Revision of identified ESEs", "content": "The analysis of the NESMP data by Lazio et al. (2001b) led to the detection of 24 ESEs (see their Tables 5 and 6). We have already shown that some of them are caused by SRV. In Table 2 we summarise our new classification of these events in the light of the SRV and TDV characteristic previously illustrated. A short explanation of the reasons behind this classification is reported in Appendix C. The number of ESEs that cannot be explained through SRV is very low. Even the archetype of ESEs, the 1981.1 event in 0954 + 658, seems more likely to be caused by the Sun than by an interstellar screen. This means that the actual frequency of ESEs in the GBI data is significantly lower than previously claimed. Furthermore, this result demonstrates the di ffi culty to identify ESEs uniquely from the examination of light curves; for a robust detection, it would be necessary to use independent diagnostic tools confirming the interstellar medium as the origin of the variability.", "pages": [10]}, {"title": "7. Conclusions", "content": "We reported about a variety of events, shaped as ESEs, detected in the radio light curves of compact extragalactic objects, from GBI, UMRAO, Pico Veleta, and E ff elsberg observations. We distinguished between four di ff erent kinds of variations, three of which are centred at the time of minimum solar elongation, while the fourth depends on time. Type I SRV has an instrumental origin, with a possible further contribution from IPS; it causes dips in the light curves when sources are at solar elongation below 15-20 \u00b7 at 2 GHz ( < \u223c 10 \u00b7 at 8 GHz), with average flux density decreases of about 30 percent. Di ff erently from type I, type II SRV can have a strong impact on the light curves even at high solar elongation. We found hints, which though need confirmation, that it mainly a ff ects sources in right ascension ranges around 06h and 18h. The origin of the variability is unknown, but the detection of the e ff ects in data from di ff erent facilities rules out an instrumental problem. TDV is a semi-annual e ff ect a ff ecting the light curves of a large number of sources around 0.0 and 0.5 y; its nature, as for type II SRV, is yet unknown. The evidence we collected rules out both ISS and instrumental / calibration problems from the possible sources of the variability. By exclusion, the most viable option seems to be a propagation e ff ect in a local screen (the IPM or the atmosphere). After analysing the general properties of the four kinds of variability, we assessed di ff erent hypotheses about their origin; all hypotheses are weakened by critical arguments that cannot be overcome without assuming that the actual properties of local screens substantially di ff er from the ones proposed by current models. It cannot be ruled out that the variability is caused by a superposition of di ff erent e ff ects, which could explain the di ffi -culty to describe it in terms of a single self-consistent model. However, the convergence of many SRV and TDV manifestations around the time of the solstices, if confirmed, would strongly support the idea of a common origin. From the picture sketched above, it is evident that the nature of the variability is far from being identified. Much clearer is the situation concerning the consequences of SRV / TDV, especially in relation to ESEs studies. Many of the ESEs identified by Lazio et al. (2001b) during the NESMP (which is the most important program dedicated to the detection of these events) are certainly due to SRV; only a few are not compatible with the characteristics of a solar influence. Even the impressive events in the light curves of 1741-038 (see Lazio et al. 2001a), 0954 + 658 (Fiedler et al. 1987a; Fiedler et al. 1987b; Walker & Wardle 1998), and 0528 + 134 (Pohl et al. 1996) appear to be extreme cases of semiannual events caused by the Sun (type IIb SRV). In the light of the discoveries here reported, also the ESE detected in the data of PKS 1939-315 on June 2014 (Bannister et al. 2016) would be naturally interpreted as due to SRV or TDV, considering that, because of its coordinates, the source would be expected to show type IIb SRV, and that the minimum of the symmetric event falls at the beginning of July 2014 (i.e. the critical time 0.5 y), when the source reaches its maximum solar elongation. Our study does not rule out the existence of ESEs, but it shows that this kind of event, which already Fiedler et al. (1994) considered rare, is even rarer than previously thought. Concerning the study of the radio properties of compact sources, the existence of SRV implies that the variability characteristics of some objects may be heavily a ff ected by the influence of the Sun. For sources such as OJ 287, it may be important to set a much larger Sun constraint than the mechanical one allowed by the telescope (which for E ff elsberg, e.g., is 2 \u00b7 , see Komossa et al. 2023). Particularly worrisome is the recognition of the high degree of correlation between the fast variability of 0954 + 658 and of some angularly-nearby sources, whose solar elongation is always high. Periodicity in light curves with periods close to one year or 6 months should be regarded as highly suspicious, especially if they appear in sources whose solar elongation peaks around the critical times 0.0 and 0.5 y; in this respect, it would be interesting to check whether the \u223c 180-day periodicity found in OVRO data for the NLSy1 J0849 + 5108 ( \u03b2 : 32 \u00b7 ; see Zhang & Wang 2021), whose coordinates are compatible with type IIb SRV, shows dips around 0.1 and 0.6 y, consistently with its time of maximum / minimum solar elongation. Additional studies will be needed to assess whether SRV may also play a role in the recently discovered symmetric achromatic variability events (SAV; see Vedantham et al. 2017), found in the blazar 1413 + 135; however, the position of the source (far from the location of most type IIa and IIb SRV sources), the duration of the events (of the order of one year or more), and the occurrence of the detected SAV events (see Peirson et al. 2022) all seem to indicate that the variability is not caused by the Sun. A thorough investigation of the causes and the manifestations of SRV is made even more urgent by the consideration that several of the most variable compact radio sources of extragalac- tic origin (e.g. OJ 287 and 0235 + 164) fall at low ecliptic latitudes, and that Sgr A*, among the most interesting and studied objects in astronomy, because of its coordinates and an ecliptic latitude of only -5 . 6 \u00b7 , should be particularly sensitive to type II SRV. T able 3. The main v ariability characteristics of NESMP sources are reported as follo ws: name of the source (Col. 1), ecliptic latitude (Col. 2), time of the year in which solar elong ation reaches the minimum (Col. 3), a v erage S 2GHz (Col. 4), S 2GHz standard de viation (Col. 5), a v erage S 8GHz (Col. 6), S 8GHz standard de viation (Col. 7), 2 GHz dropout at minimum solar elong ation (Col. 8), SR V classification at 2 GHz (Col. 9), SR V classification at 8 GHz (Col. 10), e vidence of TD V (Col. 11). The results of the automatic classification of sources according to their 2 GHz light curv es are summarised in Col. 12-15; the y report the probability that the number of SIV or TD V e v ents in the light curv es is consistent with an homogeneous distrib ution of dips across the year (Col 12 and 14, respecti v ely), and the consequent classification of the sources (Col. 13 and 15). The \"N A\" indication is used when no meaningful estimate could be achie v ed because of lack of data. Acknowledgements. We thank the anonymous referees for the useful comments and the thorough discussion of the manuscript. We thank Tim Sprenger and Laura Spitler for their suggestions, which helped to clarify and highlight important points of the article. This research is based on data of the Green Bank Interferometer (GBI), which was a facility of the National Science Foundation operated by the National Radio Astronomy Observatory under contract with the US Naval Observatory and the Naval Research Laboratory during these observations. The UMRAO observations included in this analysis were obtained as part of programs funded by a series of grants from the NSF. Additional funding for the operation of UMRAO was provided by the University of Michigan.", "pages": [10, 11, 12, 16]}, {"title": "References", "content": "Article number, page 16 of 20 Zhang, P. & Wang, Z. 2021, ApJ, 914, 1", "pages": [16]}, {"title": "Appendix A: Investigation of TDV as a possible calibration problem", "content": "TDV is characterised by the occurrence of simultaneous events, resembling ESEs, in di ff erent sources; correlated flux density variations in di ff erent sources may suggest the existence of an instrumental or a calibration problem. Given the periodic recurrence of TDV, the hypothesis of an instrumental problem seems implausible. The detection of ESEshaped variability in 0528 + 134 with three di ff erent telescopes, applying di ff erent calibration schemes, during the 1993.5 event seems to rule out a calibration problem as well. However, in consideration of the ambiguity of the 0528 + 134 variability in 1993.5 (is it TDV or SRV?), it seems useful to analyse in some detail the variability characteristics of the calibrators, to asses the possibility of a calibration problem. All three sources used to create a hybrid calibrator show a smooth 1-year periodic oscillation correlated with solar elongation. The minima of the oscillation occur at 0.3 y for 0237233, at 0.78 y for 1245-197 and at 0.76 y for 1328 + 254; they are therefore strongly out-of-phase, almost in phase opposition. A combination of smooth oscillations, however, cannot cause ESEshaped variability episodes. Through the de-trending procedure, we isolated the fast variability component in the calibrators, to compare it with the combined 2 GHz variability curve shown in Fig. 7. It should be noted that 1245-197 has an ecliptic latitude of -13.6 \u00b7 , and it is a ff ected by type I SRV; the ESE-like variations at minimum solar elongation introduce in some sources mild 1-year periodic flares of short duration. However, this does not cause the semi-annual dips at 0.0 and 0.5 y. The 2 GHz combined variability curve and the calibrators de-trended light curves are shown in Fig. A.1, after averaging them on 0.02 y bins to reduce the random fluctuations. Given their usage in the calibration procedure, discrepancies between the calibrators' light curves tend to cancel out, which implies a significant anti-correlation between their variations; 1245-197 (brown dots) and 1328 + 254 (green dots) generally show a better agreement among each other than with 0237-233 (black dots), because their angular distance is much smaller, and they are observed approximately at the same time of the day. If TDV were caused by a calibrator, its light curve would show a systematic discrepancy with respect to the other two during the ESE-shaped events. If the e ff ect were instead caused by di ff erent observational conditions for 0237-233 compared to the two angularly-nearby sources 1245-197 and 1328 + 254, we should see a periodic deviation between the former and the latter ones. In general, strong disagreements between the calibrators do not coincide with dips in the data; in Fig. A.1, only two events look correlated with a significant discrepancy between the calibrators: the 1993.0 and, to a lesser extent, the 1995.1 one. In both cases, however, the agreement is good between 0237-233 and 1245-197, while the flux density of 1328 + 254 is considerably lower. A dip in a calibrator would introduce a peak, not a dip, in the calibrated light curves. This implies that either two distant calibrators are a ff ected by chance by a similar problem for a few months (which seems unrealistic) or the observed drop of flux density must a ff ect 1328 + 254 as well as several of the calibrated sources. In conclusion, the calibrators do not seem to cause the problem, but they might be a ff ected by it.", "pages": [17]}, {"title": "Appendix B: List of all events detected through automated analysis", "content": "0003 + 380 : 1993.77, 1993.22, 1992.22, 1991.62, 1990.62, 1990.32 0003-066 : 1992.92, 1992.22, 1991.87, 1990.22, 1989.22 0016 + 731 : 1993.22, 1992.62, 1991.77, 1991.32, 1990.47, 1989.72, 1988.92 0019 + 058 : 1993.24, 1992.29, 1991.24 0035 + 121 : 1993.26, 1992.26, 1991.26, 1990.26, 1989.26 0035 + 413 : 1992.34, 1991.44, 1990.89, 1989.74 0055 + 300 : 1993.19, 1991.19, 1990.99, 1989.74 0056-001 : 1993.25, 1992.25, 1991.25, 1990.25, 1989.25 0113-118 : 1991.25, 1989.25, 1988.95 0123 + 257 : 1988.94 0130-171 : 1989.49, 1988.94 0133 + 476 : 1992.33, 1990.78, 1990.48, 1990.23, 1987.73, 1986.73 0134 + 329 : 1992.36, 1991.26, 1991.01, 1988.91 0147 + 187 : 1993.46, 1992.31 0201 + 113 : 1993.31, 1992.31, 1991.56, 1991.31, 1990.31, 1989.31 0202 + 319 : 1993.48 , 1992.33, 1989.48, 1988.38, 1987.23, 1985.33, 1984.28 0212 + 735 : 1994.15, 1993.35, 1993.00 , 1992.35, 1992.05 0224 + 671 : 1995.84, 1992.19, 1991.19, 1990.29, 1988.89, 1988.59, 1987.99, 1983.79, 1979.94 0235 + 164 : 1992.34, 1991.34, 1986.69, 1985.34, 1984.09, 1982.89, 1982.04 0237-233 : 1994.95, 1989.25, 1984.40, 1983.40, 1982.90 0256 + 075 : 1993.49 , 1992.89, 1992.74, 1992.49 , 1991.84 0300 + 470 : 1993.53, 1992.68, 1992.33, 1991.48, 1990.73, 1990.23, 1989.88, 1989.48, 1988.93, 1987.93 0316 + 413 : 1991.03 , 1990.38, 1988.78 0319 + 121 : 1993.51 , 1991.36, 1990.36, 1989.31 0333 + 321 : 1993.74, 1993.39, 1991.84, 1987.94, 1987.44, 1986.34 0336-019 : 1995.36, 1994.46, 1992.91, 1992.56, 1992.06, 1991.51, 1983.16, 1981.11 0337 + 319 : 1993.44, 1992.94 , 1991.39, 1990.99 0355 + 508 : 1984.01, 1983.36, 1981.66, 1980.31, 1979.61 0400 + 258 : 1993.50 , 1992.90, 1992.45, 1991.95, 1991.45, 1991.00 , 1990.55 0403-132 : 1992.92, 1991.47 0420-014 : 1992.20, 1989.25, 1987.30, 1986.65, 1985.35, 1984.40 0440-003 : 1994.21, 1993.96, 1993.51 , 1993.26, 1990.46, 1989.86, 1989.41 0444 + 634 : 1994.44, 1993.99, 1993.29, 1992.09, 1991.59, 1990.99, 1990.64, 1989.34, 1989.09 0454 + 844 : 1993.52, 1993.27, 1990.22 0500 + 019 : 1992.88, 1992.23 0528 + 134 : 1993.95, 1993.50 , 1993.05, 1992.40, 1991.00, 1990.50, 1988.95 0532 + 826 : 1991.72, 1989.97, 1989.07 0537-158 : 1993.50, 1993.35, 1993.00, 1992.45, 1991.95, 1991.45, 1990.95, 1990.50, 1989.30 0538 + 498 : 1991.01, 1990.46 0552 + 398 : 1993.52, 1992.17, 1991.47, 1990.62, 1988.77, 1987.47, 1986.22, 1985.82, 1984.42 0555-132 : 1993.92, 1993.47, 1992.67, 1992.27, 1990.02, 1989.42 0615 + 820 : 1993.98, 1993.53, 1992.83, 1992.08, 1991.58, 1990.98 0624-058 : 1992.60, 1990.45 0633 + 734 : 1993.48 , 1992.98, 1990.48, 1990.28 0650 + 371 : 1992.55, 1991.65, 1991.00, 1990.45, 1989.05 0653-033 : 1994.12, 1993.47, 1992.32, 1991.72, 1990.52, 1989.07, 1988.82 0716 + 714 : 1991.40, 1990.00, 1988.75, 1988.15 0723 + 679 : 1993.55, 1993.00, 1991.00 , 1990.45 0723-008 : 1993.14, 1989.74 0742 + 103 : 1992.95, 1992.55, 1990.55, 1986.10, 1982.20 0743 + 259 : 1993.34, 1990.54, 1989.54 0759 + 183 : 1993.76, 1993.51, 1992.51, 1991.86, 1990.01, 1989.56 0804 + 499 : 1994.54, 1991.64, 1990.59 0818-128 : 1992.59, 1990.04, 1989.44 0827 + 243 : 1994.52, 1993.52, 1992.52, 1991.52 0836 + 710 : 1995.93, 1995.38, 1994.93, 1994.53, 1993.38, 1992.98, 1991.48, 1991.03, 1990.48, 1989.58 0837 + 035 : 1993.49, 1992.94, 1992.39 0851 + 202 : 1991.54, 1989.59, 1986.34, 1985.14, 1983.59, 1979.54 0859-140 : 1993.67, 1992.52, 1991.47, 1990.07 0922 + 005 : 1993.58, 1993.08, 1992.48, 1991.68, 1991.13, 1990.18 0923 + 392 : 1992.99, 1991.99 , 1991.59, 1988.69, 1987.59, 1986.04, 1984.44 0938 + 119 : 1995.48, 1992.63 0945 + 408 : 1993.50 , 1991.50, 1991.10 0952 + 179 : 1993.48 , 1991.58 0954 + 658 : 1994.52, 1992.97, 1990.47, 1989.47, 1986.67, 1981.12, 1980.37 1020 + 400 : 1993.52 , 1990.62 1022 + 194 : 1993.75, 1993.50 , 1992.90, 1991.65 1036-154 : 1993.54 , 1992.64, 1991.14 1038 + 528 : 1992.01 1055 + 018 : 1991.74 1100 + 772 : 1993.50, 1993.20, 1992.95, 1991.00 1116 + 128 : 1993.49 , 1990.69 1123 + 264 : 1993.53 , 1992.58, 1991.03 1127-145 : 1993.72, 1991.07 1128 + 385 : 1990.31 1145-071 : 1993.52 , 1991.62, 1989.72 1150 + 812 : 1993.53, 1991.03 1155 + 251 : 1993.50, 1993.00 1200-051 : 1993.54 , 1992.04, 1991.74 1225 + 368 : 1993.50, 1992.95 , 1991.55 1243-072 : 1992.57, 1991.82, 1991.02, 1990.62, 1990.32 1253-055 : 1995.62, 1993.07, 1992.77 1302-102 : 1991.68, 1990.63, 1990.38 1308 + 326 : 1993.98 1328 + 254 : 1995.06 , 1992.96, 1991.56, 1981.11 1328 + 307 : 1995.05 , 1985.60, 1979.75 1354 + 195 : 1993.73, 1990.83, 1989.58 1404 + 286 : 1994.23, 1993.53 1409 + 524 : 1991.78, 1990.78 1413 + 135 : 1993.50 , 1992.85, 1991.95, 1990.85 1430-155 : 1993.79, 1992.89, 1991.89, 1990.84, 1990.04 , 1989.14, 1988.84 1438 + 385 : 1994.23, 1993.53 1449-012 : 1992.84 1455 + 247 : 1993.52 , 1992.92 1502 + 106 : 1994.07, 1991.24, 1990.19, 1989.79, 1987.49, 1986.19, 1985.69, 1984.84, 1980.29 1511 + 238 : 1993.48 , 1992.93, 1991.93 1514 + 197 : 1994.99, 1992.94 , 1992.19, 1991.44, 1990.09, 1989.44, 1989.09, 1988.79 1538 + 149 : 1991.31, 1990.96 , 1990.16 1555 + 001 : 1993.94, 1993.59, 1993.39, 1989.89, 1988.24, 1987.14 1611 + 343 : 1992.97, 1991.17, 1989.92, 1987.92, 1985.67, 1985.42, 1984.42, 1983.72 1614 + 051 : 1993.85, 1993.60, 1993.30, 1992.80, 1992.30, 1992.00, 1991.00, 1990.55, 1989.80 1624 + 416 : 1992.92, 1989.47 1635-035 : 1993.77, 1993.22, 1992.92, 1992.67, 1991.42, 1990.67, 1989.32, 1989.02, 1988.77 1641 + 399 : 1993.39, 1992.94, 1991.99, 1987.84, 1986.74, 1986.04, 1985.59, 1979.59 1655 + 077 : 1993.73, 1993.43, 1992.78, 1992.18, 1991.83, 1991.53, 1989.08 1656 + 477 : 1992.94, 1992.49, 1991.94, 1991.34 1741-038 : 1993.36, 1992.91, 1992.41, 1990.21, 1989.41, 1988.41, 1987.96, 1987.36, 1986.46, 1984.91 1749 + 096 : 1993.82, 1991.57, 1989.27, 1987.97, 1987.47, 1987.02, 1986.17, 1985.17, 1983.82 1749 + 701 : 1993.36, 1992.96, 1989.96, 1989.51, 1988.91 1756 + 237 : 1993.97, 1993.52, 1992.62, 1991.52, 1991.27, 1990.52, 1990.02 1803 + 784 : 1993.87, 1992.92, 1991.72, 1990.77 1807 + 698 : 1992.99, 1985.64 1821 + 107 : 1993.89, 1993.54, 1991.94, 1991.39, 1988.39, 1985.29, 1984.69, 1984.24 1823 + 568 : 1993.88, 1992.08, 1990.58 1828 + 487 : 1991.02 1830 + 285 : 1993.90, 1993.50, 1992.95, 1992.45, 1991.95, 1991.45, 1990.80, 1990.45, 1989.50, 1988.50 1928 + 738 : 1993.87, 1993.62, 1993.37, 1992.97 1943 + 228 : 1993.87, 1993.47, 1993.12 1947 + 079 : 1993.96, 1992.71, 1991.66, 1988.91, 1988.56 2007 + 776 : 1993.35, 1991.00 , 1990.10, 1989.45 2008-068 : 1988.94 2032 + 107 : 1993.85, 1992.95, 1992.40, 1991.90, 1991.45, 1991.00 , 1989.15 2037 + 511 : 1991.92, 1991.52 2047 + 098 : 1992.95, 1992.10 2059 + 034 : 1993.51, 1992.96 , 1992.06, 1991.16, 1990.11, 1988.91 Article number, page 18 of 20 2105 + 420 : 1992.46, 1992.06, 1991.56, 1991.01 2113 + 293 : 1993.09, 1992.49, 1991.54, 1989.94, 1989.14 2121 + 053 : 1993.87, 1992.62, 1991.37, 1990.02, 1989.32 2134 + 004 : 1993.08, 1991.18, 1987.88, 1986.63 2155-152 : 1993.63, 1992.28, 1991.83, 1991.18, 1990.13 2200 + 420 : 1992.25, 1990.55, 1988.90, 1986.70, 1985.20, 1984.50, 1981.70 2209 + 081 : 1993.81, 1993.01, 1991.11, 1990.16 2214 + 350 : 1992.90, 1991.45, 1989.75, 1989.45, 1988.60 2234 + 282 : 1993.20, 1989.40, 1987.55, 1985.90, 1984.95, 1983.40 2251 + 158 : 1992.99 , 1989.44, 1985.59, 1984.89, 1983.09, 1980.34 2251 + 244 : 1993.50 , 1993.15, 1992.50, 1992.25, 1991.55 2307 + 107 : 1993.70, 1989.40 2319 + 272 : 1992.88, 1992.48, 1992.18, 1990.53, 1989.28 2344 + 092 : 1993.42, 1992.17, 1991.22, 1990.52, 1990.17 2352 + 495 : 1993.68, 1993.28, 1991.23, 1990.33, 1985.38, 1984.68", "pages": [17, 18, 19]}, {"title": "Appendix C: Extended discussion of individual ESEs", "content": "Here below we report a short analysis of the individual ESEs identified in Lazio et al. (2001b), in the light of the SRV / TDV features discussed in the present article. (with 0.3 y interpolation timescale) and stacked, looking for yearly patterns; at 2GHz, the stacked data (black dots in Fig. C.1, left panel; the flux has been multiplied by a factor 9 for a better comparison with the event) show a semi-annual pattern, which at the beginning of the year (blue arrow) has a clear ESE-like shape. The 1981.1 event (emphasised in the de-trended data, green dots) is much stronger than the semiannual event in the stacked data, and it lasts longer, but they are both centred at the time of maximum solar elongation. Even more suggestive is the behaviour of the 8 GHz data (Fig. C.1, right panel): here the ESE-like pattern at the beginning of the year is stronger, and has the same duration as the 1981.1 event. This strongly favours an interpretation of the event in terms of SRV.", "pages": [19]}] |
2020PhRvD.101h4035K | https://arxiv.org/pdf/2004.06617.pdf | <document>
<section_header_level_1><location><page_1><loc_26><loc_92><loc_74><loc_93></location>Influence of dark matter on black hole scalar hair</section_header_level_1>
<text><location><page_1><loc_31><loc_85><loc_70><loc_90></location>Bartlomiej Kiczek ∗ and Marek Rogatko † Institute of Physics, Maria Curie-Sklodowska University, 20-031 Lublin, pl. Marii Curie-Sklodowskiej 1, Poland (Dated: April 15, 2020)</text>
<text><location><page_1><loc_18><loc_76><loc_83><loc_84></location>Searches for dark matter sector field imprints on the astrophysical phenomena are one of the most active branches of the current researches. Using numerical methods we elaborate the influence of dark matter on the emergence of black hole hair and formation of boson stars. We explore thermodynamics of different states of the system in Einstein-Maxwell-scalar dark matter theory with box boundary conditions. Finally we find that the presence of dark sector within the system diminishes a chance of formation of scalar hair around a black hole.</text>
<section_header_level_1><location><page_1><loc_20><loc_73><loc_37><loc_74></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_32><loc_49><loc_70></location>The astrophysical evidence of the illusive ingredient of our Universe, dark matter , is overwhelming and authorizes the galaxy rotation curves, gravitational lensing, a thread-like structure (cosmic web) on which ordinary matter accumulates [1, 2]. On the contrary, the absence of the evidence of the most popular particle candidates for baryonic dark matter stipulates the necessity of diversifying experimental efforts [3]. Black holes and ultracompact horizonless objects being the ideal laboratories for dark matter studies, may help us to answer the tantalizing question of how dark matter sector leaves its imprint in the physics of these objects. However, it happens that Schwarzschild black hole has a negative specific heat and it cannot be in equilibrium with thermal radiation. To overcome this difficulty the idea of enclosing the black hole within a box was proposed [4, 5]. Einstein-Maxwell systems with box boundary conditions were elaborated in [6], where it was established that the phase structure of the models were similar to AdS gravity. Inclusion of the additional scalar field to the theory in question, envisages the correspondence of phase transitions in gravity in a box with s-wave holographic superconductor [7]-[9]. The thermodynamical studies of Einstein-Maxwell scalar systems in the asymptotically flat spacetime with reflecting boundary conditions were conducted in [10]. A certain range of parameters allows to obtain stable black hole solution, giving a way to circumvent no-hair theorem.</text>
<text><location><page_1><loc_9><loc_22><loc_49><loc_31></location>The next compact objects studied in our paper, from the point of view of the influence of dark matter on their physics, are boson stars. Boson stars being a selfgravitating solution of massive scalar field with a potential coupled to gauge fields and gravity [11] are widely studied in literature [12]-[16], for a quite long period of time.</text>
<text><location><page_1><loc_9><loc_14><loc_49><loc_21></location>The purpose of our paper is to examine thermodynamical properties and stability of the black holes and horizonless objects-boson stars in Einstein-Maxwell-scalar system influenced by dark matter sector and envisage the role of the dark matter in the elaborated problems.</text>
<text><location><page_1><loc_52><loc_67><loc_92><loc_74></location>The organization of the paper is as follows. In Sec. II we describe the basic features of the hidden sector model and derived the basic equations needed in what follows. Sec. III is devoted to the description of the obtained numerical results. In Sec. IV we concluded our researches.</text>
<section_header_level_1><location><page_1><loc_67><loc_62><loc_77><loc_63></location>II. MODEL</section_header_level_1>
<text><location><page_1><loc_52><loc_55><loc_92><loc_60></location>We consider the spacetime manifold with time-like boundary ∂ M , which will be referred as a box. The action for Einstein-Maxwell scalar dark matter gravity is provided by</text>
<formula><location><page_1><loc_53><loc_47><loc_92><loc_53></location>S = ∫ M d 4 x √ -g ( R -1 4 F µν F µν -α 4 B µν F µν (1) -1 4 B µν B µν -| D Ψ | 2 -m 2 | Ψ | 2 ) -∫ ∂ M d 3 x √ -γ K ,</formula>
<text><location><page_1><loc_52><loc_8><loc_92><loc_46></location>where F µν is a Maxwell field strength tensor, B µν is a strength tensor of a hidden sector vector boson. The complex scalar field Ψ = ψe iθ , where θ denotes the phase, is coupled only to the ordinary electromagnetic field by the covariant derivative D µ = ∇ µ -iqA µ . The theoretical justifications of the model in question originate from M/string theories, where such mixing portals coupling Maxwell and auxiliary gauge fields can be encountered [17]. The hidden sectors states are charged under their own groups and interact with the visible sector via gravitational interactions. The realistic string compactifications establish the range of values for α between 10 -2 to 10 -10 [18]-[21]. It seems that astrophysical observations of gamma rays of energy 511 keV [22], positron excess in galaxies [23], and muon anomalous magnetic moment [24], argue for the aforementioned idea of coupling Maxwell field with dark matter sector. Recent experiments aimed at gamma rays emissions from dwarf galaxies [25], dilaton-like coupling to photons caused by ultra-light dark matter [26], oscillations of the fine structure constant [27], revisions of the constraints on dark photon 1987A supernova emission [28], measurements of excitation of electrons in CCD-like detector [29], as well as, the examinations in e + e -Earth colliders [30], give us some hints for the correctness of the proposed model. They and the future planned ballon d'essai will amelio-</text>
<text><location><page_2><loc_9><loc_90><loc_49><loc_93></location>constraints on the hidden sector particles, especially for dark photons .</text>
<text><location><page_2><loc_9><loc_84><loc_49><loc_90></location>The second integral denotes the Gibbons-Hawking boundary term of our box with γ metric on the threedimensional hypersurface ( r = r b ), with the extrinsic curvature K .</text>
<text><location><page_2><loc_9><loc_81><loc_49><loc_84></location>Varying the action (1) we get the equations of motion of the forms</text>
<formula><location><page_2><loc_14><loc_75><loc_49><loc_79></location>( ∇ µ -iqA µ )( ∇ µ -iqA µ ) Ψ -m 2 Ψ = 0 , (2) ˜ α ∇ µ F µν = j ν , (3)</formula>
<text><location><page_2><loc_9><loc_70><loc_49><loc_73></location>where ˜ α = 1 -α 2 4 and the current j ν is provided by the relation</text>
<formula><location><page_2><loc_11><loc_67><loc_49><loc_69></location>j ν = iq [ Ψ † ( ∇ ν -iqA ν ) Ψ -Ψ ( ∇ ν + iqA ν ) Ψ † ] . (4)</formula>
<text><location><page_2><loc_9><loc_60><loc_49><loc_64></location>In what follows we use a time independent spherically symmetric line element, with the metric coefficients being functions of r -coordinate</text>
<formula><location><page_2><loc_10><loc_55><loc_49><loc_58></location>ds 2 = -g ( r ) h ( r ) dt 2 + dr 2 g ( r ) + r 2 ( dθ 2 +sin θ 2 dφ 2 ) , (5)</formula>
<text><location><page_2><loc_9><loc_51><loc_49><loc_53></location>and the adequate components of the fields in the theory will constitute radial functions of the forms</text>
<formula><location><page_2><loc_13><loc_47><loc_49><loc_49></location>A µ dx µ = φ ( r ) dt, B µ dx µ = χ ( r ) dt, Ψ = Ψ( r ) . (6)</formula>
<text><location><page_2><loc_9><loc_35><loc_49><loc_45></location>In general the scalar field can have harmonic time dependence which can be absorbed by a redefinition of the gauge field function. Having this in mind it can be seen that the r -component of the equations of motion for the gauge and scalar fields leads the conclusion that Ψ( r ) = ψ ( r ). By virtue of this, the following equations of motion are provided:</text>
<formula><location><page_2><loc_19><loc_31><loc_49><loc_33></location>R µν -1 2 g µν R = T µν , (7)</formula>
<formula><location><page_2><loc_16><loc_29><loc_49><loc_30></location>∇ µ ∇ µ ψ -q 2 A µ A µ ψ -m 2 ψ = 0 , (8)</formula>
<formula><location><page_2><loc_16><loc_26><loc_49><loc_29></location>∇ µ F µν + α 2 ∇ µ B µν -2 q 2 A ν ψ 2 = 0 , (9)</formula>
<formula><location><page_2><loc_16><loc_23><loc_49><loc_26></location>∇ µ B µν + α 2 ∇ µ F µν = 0 . (10)</formula>
<text><location><page_2><loc_9><loc_18><loc_49><loc_21></location>As in the case of the equation (3), the last two equations can be rewritten as</text>
<formula><location><page_2><loc_20><loc_15><loc_49><loc_16></location>˜ α ∇ µ F µν -2 q 2 A ν ψ 2 = 0 , (11)</formula>
<formula><location><page_2><loc_20><loc_12><loc_49><loc_15></location>∇ µ B µν + α ˜ α q 2 A ν ψ 2 = 0 . (12)</formula>
<text><location><page_2><loc_10><loc_9><loc_49><loc_10></location>Consequently, the explicit forms of the equations of</text>
<text><location><page_2><loc_52><loc_92><loc_61><loc_93></location>motion yield</text>
<formula><location><page_2><loc_55><loc_88><loc_92><loc_91></location>h ' -rhψ ' 2 -q 2 rφ 2 ψ 2 g 2 = 0 , (13)</formula>
<formula><location><page_2><loc_55><loc_81><loc_92><loc_87></location>g ' + g ( 1 r + 1 2 rψ ' 2 ) + q 2 rφ 2 ψ 2 2 gh -1 r + r 2 h ( φ ' 2 + αχ ' φ ' + χ ' 2 + m 2 hψ 2 ) = 0 , (14)</formula>
<formula><location><page_2><loc_55><loc_78><loc_92><loc_81></location>φ '' + ( 2 r -h ' 2 h ) φ ' -2 q 2 φψ 2 ˜ αg = 0 , (15)</formula>
<formula><location><page_2><loc_54><loc_74><loc_92><loc_78></location>ψ '' + ( 2 r + h ' 2 h + g ' g ) ψ ' + ( q 2 φ 2 gh -m 2 ) ψ g = 0 , (16)</formula>
<formula><location><page_2><loc_55><loc_71><loc_92><loc_74></location>χ '' + ( 2 r -h ' 2 h ) χ ' + αq 2 χψ 2 ˜ αg = 0 . (17)</formula>
<text><location><page_2><loc_52><loc_63><loc_92><loc_70></location>To solve the equations of the theory in question one has to provide adequate boundary conditions. Namely we can pick either a horizonless or a black hole solution. In case of a black hole we expand the underlying functions in a Taylor series around the horizon of radius r h</text>
<formula><location><page_2><loc_56><loc_60><loc_92><loc_62></location>ψ = ψ 0 + ψ 1 ( r -r h ) + ψ 2 ( r -r h ) 2 + O ( r 3 ) , (18)</formula>
<formula><location><page_2><loc_56><loc_58><loc_92><loc_60></location>φ = φ 1 ( r -r h ) + φ 2 ( r -r h ) 2 + O ( r 3 ) , (19)</formula>
<formula><location><page_2><loc_56><loc_57><loc_92><loc_58></location>g = g 1 ( r -r h ) + g 2 ( r -r h ) 2 + O ( r 3 ) , (20)</formula>
<formula><location><page_2><loc_56><loc_55><loc_92><loc_56></location>h = 1 + h 1 ( r -r h ) + O ( r 2 ) , (21)</formula>
<formula><location><page_2><loc_56><loc_53><loc_92><loc_54></location>χ = χ 1 ( r -r h ) + χ 2 ( r -r h ) 2 + O ( r 3 ) . (22)</formula>
<text><location><page_2><loc_52><loc_39><loc_92><loc_52></location>We set g 0 = 0, due to occurrence of the black hole event horizon. For the regularity of the U (1)-gauge fields on the event horizon, one also puts φ 0 and χ 0 equal to zero (in order to keep the terms with division by g ( r h ) in equations of motion finite). By implementing the expansions (18)-(22) into the equations of motion, we find out that { r h , ψ 0 , φ 1 , χ 1 , α } comprise free parameters of the theory in question, while the remaining ones can be expressed by them.</text>
<text><location><page_2><loc_52><loc_26><loc_92><loc_39></location>As far as the boson star scenario is concerned, we perform a similar expansion. However since the configuration in question is horizonless, the expansion accomplishes around the origin of the reference frame. At r = 0 we require that the derivatives of all the functions are set equal to zero, which ensures that there is no kink at this point. At r = r b , we establish the Dirichlet boundary condition for the scalar field ψ ( r b ) = 0 (the reflecting mirror-like boundary conditions).</text>
<text><location><page_2><loc_52><loc_23><loc_92><loc_26></location>Asymptotic analysis of matter fields, at the box boundary, enables us to write</text>
<formula><location><page_2><loc_60><loc_21><loc_92><loc_22></location>ψ ∼ ψ (0) + ψ (1) ( r b -r ) + O ( r 2 ) , (23)</formula>
<formula><location><page_2><loc_61><loc_19><loc_92><loc_20></location>φ ∼ φ (0) + φ (1) ( r b -r ) + O ( r 2 ) , (24)</formula>
<formula><location><page_2><loc_61><loc_17><loc_92><loc_18></location>χ ∼ χ (0) + χ (1) ( r b -r ) + O ( r 2 ) . (25)</formula>
<text><location><page_2><loc_52><loc_9><loc_92><loc_16></location>As was proposed in Refs. [10, 16], because of the fact that the scalar field satisfies the reflecting mirrorlike boundary conditions ψ ( r b ) = 0, one can fix ψ (0) = 0 and the other parameter ψ (1) can be used for the phase transition description. This approach to the problem in</text>
<text><location><page_3><loc_9><loc_90><loc_49><loc_93></location>question is widely exploited in holographic studies of superconductors and superfluids.</text>
<text><location><page_3><loc_9><loc_67><loc_49><loc_90></location>For the gauge fields one has that φ (0) = µ and χ (0) = µ d as chemical potentials for visible and hidden sector fields, treating the system as a grand canonical ensemble. In order to conduct the thermodynamical analysis we calculate the free energy of the system, to see which phase is thermodynamically preferable, for a fixed temperature. In the case of a hairless solution we take into account the classical formula F = E -TS -µQ -µ d Q d , where E is Brown-York quasilocal energy [4, 5]. Nevertheless this approach may cause problems in hairy solution analysis, being insufficient to capture the mass of the scalar field. Therefore we treat this problem evaluating the on-shell action in Euclidean signature F = TS cl , which enables to take into considerations the non-trivial profile of scalar field constituting the solution of the underlying system of differential equations.</text>
<text><location><page_3><loc_9><loc_30><loc_49><loc_67></location>Solution of the equations (13)-(17) with ψ = 0 can be achieved analytically, giving the Reissner-Nordstrom (RN) dark matter black object [31]. To proceed further and accomplish the complete numerical analysis of the underlying equations, we implement the shooting method, integrating the aforementioned relations from r h to r b , using the fourth order Runge-Kutta method. From the set of free parameters we fix the scalar magnitude on the event horizon ψ 0 and pick r h , φ 1 and χ 1 to be shooting parameters. Moreover we impose values on both chemical potentials, that serve as constrains in our shooting procedure for φ 1 and χ 1 . We set a domain of shooting parameters from the series expansion of the solutions near the horizon, then by using the iterative bisections one finds a solution that meets constrains, with a desired tolerance. Therefore parameters { µ d , α } , which are controlling respectively amount of dark charge and the coupling strength remain free, thus they can be varied to see their impact on the system in question. For the convenience let us refer to the parameter ψ (1) , as a condensation , which serves as a handy analogy to holographic theory. As mentioned above, in our numerical scheme we treat µ d as an input parameter in our code, however one might not be interested in expressing these relations in a language of chemical potentials. Therefore one might compute the total dark charge of the system</text>
<formula><location><page_3><loc_16><loc_25><loc_49><loc_29></location>Q d = lim r → r b 1 4 π ∫ S 2 B µν t µ n ν √ -gd 2 θ, (26)</formula>
<text><location><page_3><loc_9><loc_20><loc_49><loc_24></location>where t µ is a unit time-like vector and n µ is a normal vector to the boundary. In the similar manner we compute electrical charge, for F µν .</text>
<section_header_level_1><location><page_3><loc_23><loc_16><loc_35><loc_17></location>III. RESULTS</section_header_level_1>
<text><location><page_3><loc_9><loc_9><loc_49><loc_14></location>We commence with the hairy black hole solution (HBH), i.e., a system with an event horizon and nontrivial scalar field profile. The parameter space of HBH can be illustrated on a plane of chemical potential and</text>
<text><location><page_3><loc_52><loc_80><loc_92><loc_93></location>Hawking temperature ( µ -T ) as a triangular shape. That region is bound between boson star phase from the lefthand side and generalized Reissner-Nordstrom (RN) solution from the right-hand side. A schematic phase diagram has been presented in the Fig. 1, where both mentioned lines are marked. Moreover the influence of the dark sector on phase boundaries is visualized by arrows, showing the trend of the flow by increasing the hidden sector chemical potential.</text>
<figure>
<location><page_3><loc_52><loc_52><loc_93><loc_79></location>
<caption>FIG. 1. A scheme of phase diagram of the described system. Blue-yellow line indicates the border between boson star and hairy black hole parameter space, while the red line depicts hairy BH - generalized RN BH phase boundary. The arrows on the scheme show us the flow of phase boundaries driven by the chemical potential of dark matter . Lines have been split and labeled from A to C, with a point D being the center of rotation of left-hand side boundary.</caption>
</figure>
<text><location><page_3><loc_52><loc_9><loc_92><loc_37></location>The hairy configuration can be achieved for a specific value of the chemical potential. Below the value µ RN scalar can not condensate and we get RNdark matter black hole. On the other hand, for the value greater than the critical one, µ c , the system becomes unstable. By stable hairy solution we mean a constrained solution of the equations of motion (13)-(17) that fulfils the boundary conditions with desired tolerance and its free energy is lower than the free energy of RN and BS, making it the ground state of the system. We can define µ c as the chemical potential for which the phase transition driven by temperature is no longer of second order and the condensate collapses. In the range between µ RN and µ c , we contend a typical second order phase transition, depicted in Fig. 2. In the vicinity of critical temperature the condensation can be described by a function ψ (1) ∼ ( T c -T ) 1 / 2 . It should also be noted that establishing a HBH solution requires relatively large value of the scalar charge. In our calculation we used q = 100 and a small mass of m = 10 -6 .</text>
<figure>
<location><page_4><loc_9><loc_71><loc_50><loc_94></location>
<caption>FIG. 2. Condensation ψ (1) as a function of temperature, for the different values of µ d and α = 10 -3 . For µ = 0 . 1 a typical second order phase transition takes place, the dark matter presence influences the transition point and the condensation.</caption>
</figure>
<figure>
<location><page_4><loc_9><loc_39><loc_51><loc_62></location>
<caption>FIG. 3. Double valued profiles of the condensation as functions of Hawking temperature caused by increasing amount of dark matter in the system with α = 10 -3 and µ = 0 . 14. While the first transition for µ d = 0 . 08 might still be considered as a regular the another strictly not - the value of condensation becomes double valued for some range of temperatures. Moreover these solutions obey the boundary conditions but their free energy is larger than both BS's and RN's, therefore they cannot be considered as thermodynamically preferred.</caption>
</figure>
<text><location><page_4><loc_9><loc_9><loc_49><loc_23></location>Let us now discuss physical mechanisms behind the phase boundaries flow from the Fig. 1. When we cross the line of the critical chemical potential value µ c , one encounters the exotic phase , where for one value of temperature we have two values of the condensation parameter ψ (1) . Moreover by evaluating its free energy we can find it is so high that the hairy state is no longer stable - our numerical method finds constrained solutions, but due to free energy leap, they are not thermodynamically preferred. The exotic phase effect occurs in case</text>
<figure>
<location><page_4><loc_52><loc_70><loc_94><loc_93></location>
<caption>FIG. 4. Relative change of the critical temperature of hairy black hole - generalized RN black hole as a function of a ratio of chemical potentials. The temperature ratio has been normalized to the critical temperature of dark matter free solution, where µ d = 0.</caption>
</figure>
<text><location><page_4><loc_52><loc_41><loc_92><loc_59></location>when scalar mass is close to or equal zero, for a mass away from this limit we do not obtain that phase. Instead we have a sharp crossing, from stable solutions below µ c to the situation when the equations of motion do not provide solutions with condensed scalar above the µ c threshold at all. It is worth mentioning that a similar condensation-temperature profile has been shown in the so-called vector p-wave holographic superfluids [32], but a first order phase transition was hidden behind it. However, it was revealed that for the real value of the vector field, the model in question gave us the same description as holographic s-wave model with dark matter sector [33]-[34].</text>
<text><location><page_4><loc_52><loc_12><loc_92><loc_40></location>Dark matter gauge field plays an interesting role in this transition, as it accelerates the appearance, let us say, the exotic phase. For a larger value of dark sector chemical potential, ψ (1) becomes double valued for the lower chemical potential, which is depicted in the Fig. 3, where µ d has gradually increasing value. Moreover when system enters the exotic phase its free energy rises repeatedly and exceeds the free energy of RN black hole, so the hairy phase is no longer a preferred option. In this way, that effect restricts the range of chemical potential where the second order phase transition may occur, µ c becomes a descending function of the dark charge (see curve (A) in the Fig. 1). However it cannot be increased without a limit. For every value of the electric charge there exists a certain limit of dark charge , below which a formation of scalar hair is possible. Above it, such condensation cannot take place and no stable solutions are found. This phenomenon adds up both gravitational influence of the charge on the metric and non-gravitational coupling between both gauge fields.</text>
<text><location><page_4><loc_52><loc_9><loc_92><loc_11></location>Now let us draw our attention to the HBH-RN BH (B) border. An interesting effect that hidden sector exerts on</text>
<figure>
<location><page_5><loc_9><loc_71><loc_50><loc_94></location>
<caption>FIG. 5. Quasi-rotation of the phase transition boundary between boson star and hairy black hole caused by dark sector charge with α -coupling equal to 10 -4 . It can be observed that the µ threshold for hairy BH solution is significantly lowered and some parameter space of boson star is taken for the advantage of hairy BH for lower values of the chemical potential.</caption>
</figure>
<text><location><page_5><loc_9><loc_50><loc_49><loc_58></location>the hairy black hole system is the shifting of the critical temperature of the phase transition. The larger growth of dark matter charge (and also µ d ) we observe, the lower value of the transition temperature one achieves. Such effect has been depicted in the Fig. 4, where the critical temperature ratio described by the relation</text>
<formula><location><page_5><loc_21><loc_46><loc_37><loc_49></location>δT c = T c ( µ d ) -T c (0) T c (0) ,</formula>
<text><location><page_5><loc_9><loc_27><loc_49><loc_44></location>is shown as a function of the chemical potential of hidden sector normalized to the visible sector chemical potential. One can notice that the shift of the critical temperature is proportional to the square of µ d . Obviously it can not decrease as low as one wishes and a certain limit exist which has been discussed in analytic solution of dark matter charged RN-like black hole [31]. The descent of the critical temperature becomes steeper for larger value of the chemical potential of the visible sector . It can be explained by the non-gravitational interaction between fields via kinetic mixing term, which plays the significant role when both fields are sufficiently strong.</text>
<text><location><page_5><loc_9><loc_9><loc_49><loc_27></location>To proceed further, we shed some light on the influence of dark matter sector on the black hole-boson star phase transition in the stable area of small values of the chemical potential (line (C) in Fig. 1). This process is depicted in Fig. 5, which asserts a phase diagram at the boundary between hairy black hole and boson star. While the boson star is a horizonless object its Hawking temperature remains undefined. However it is possible to calculate its characteristic - condensation and free energy as a function of chemical potential. Then to obtain the phase boundary curve, we start in the hairy black hole regime, then one moves towards lower Hawking temperatures and study the value of the free energy of the hairy black hole</text>
<text><location><page_5><loc_52><loc_73><loc_92><loc_93></location>on the way. When it exceeds the free energy of a boson star, for the corresponding value of the chemical potential, the transition point is found. Both phases of the system are influenced by the hidden sector , nonetheless the free energy of a boson star is affected much less than black hole's. Dark sector causes a significant drop of free energy of a hairy black hole. It means that the stability of a hairy solution is preserved for lower temperatures given the presence of the dark matter in the system. Such effect causes that hairy black hole solution is thermodynamically preferable for the lower Hawking temperatures and limits the emergence of boson star. The presence of α -coupling constant slightly diminishes the space of parameters for which boson star can emerge.</text>
<text><location><page_5><loc_52><loc_50><loc_92><loc_73></location>At last it is sensible to mention some points that seem to be dark sector resistant. One of them appears on the phase boundary, labeled with (D) on the phase diagram scheme in Fig. 1. This point or rather its neighborhood does not seem to be susceptible on the dark charge presence in the system. For a particular numerical example, like in the Fig. 5, it is placed around µ ≈ 0 . 1160942 and T ≈ 0 . 2929537. Another one can be noticed in the condensation-Hawking temperature dependence presented in Fig. 2. All the curves certainly cross each other in one point, located at ψ (1) ≈ 0 . 345 and T ≈ 0 . 3325. This interesting phenomenon shows that while the dark sector may modify the phase structure of the system and has an imprint on its critical quantities, there exists a specific configuration of the system, that remains completely untouched.</text>
<text><location><page_5><loc_52><loc_37><loc_92><loc_50></location>The points in question constitute the so-called isosbestic ones [35], where the curves dependent on temperature T and parametrized by values of dark matter chemical potential, intersect. They illustrate the influence of temperature on condensation ψ (1) and chemical potential of visible sector . At this point we may perform a short analysis, which would reveal the leading order term of the dark sector influence. We take a following expansion of the condensate</text>
<formula><location><page_5><loc_54><loc_34><loc_92><loc_36></location>ψ (1) ( T, µ d ) = ψ (1) ( T, 0) + µ 2 d ψ (1) 1 ( T ) + O ( µ 3 d ) . (27)</formula>
<text><location><page_5><loc_52><loc_32><loc_86><loc_33></location>Second order term takes the approximated form</text>
<formula><location><page_5><loc_58><loc_27><loc_92><loc_30></location>ψ (1) 1 ( T ) = ψ (1) ( T, µ d 1 ) -ψ (1) ( T, µ d 2 ) µ 2 d 1 -µ 2 d 2 , (28)</formula>
<text><location><page_5><loc_52><loc_17><loc_92><loc_26></location>where in a certain example of curves from Fig. 2 we took µ d 1 = 0 . 12 and µ d 2 = 0 . 08. The zero of this function refers to the isosbestic point, where the contribution of µ d is by definition none. By calculating above function with help the leading order of the influence of the dark sector may be subtracted from the main function</text>
<formula><location><page_5><loc_58><loc_14><loc_92><loc_16></location>˜ ψ (1) ( T, µ d ) = ψ (1) ( T, µ d ) -µ 2 d ψ (1) 1 ( T ) . (29)</formula>
<text><location><page_5><loc_52><loc_9><loc_92><loc_13></location>In the similar manner we can expand and analyze the chemical potential as a function of Hawking temperature, parametrized by µ d from boson star - hairy black hole</text>
<figure>
<location><page_6><loc_9><loc_48><loc_50><loc_93></location>
<caption>FIG. 6. Panel a) presents condensation profile after the transformation performed in equation (29). The plot refers to the same data as in Fig. 2, however it can be seen that the separation between curves is significantly smaller. In case of panel b), which refers to Fig. 5, all curves appear to overlap with the dark matter free solution. The aforementioned transformation had removed the leading term of the dark sector influence.</caption>
</figure>
<text><location><page_6><loc_9><loc_33><loc_20><loc_34></location>phase boundary</text>
<formula><location><page_6><loc_15><loc_30><loc_49><loc_31></location>µ ( T, µ d ) = µ ( T, 0) + µ 2 d µ 1 ( T ) + O ( µ 3 d ) . (30)</formula>
<text><location><page_6><loc_9><loc_17><loc_49><loc_29></location>We define µ 1 ( T ) analogically to (28) with µ d 1 = 0 . 2 and µ d 2 = 0 . 08 and perform the same transformation for µ ( T, µ d ) curve as for ψ ( T, µ d ) in (29). The effect of these transformations is depicted in Fig. 6, where all curves tend to be much closer to each other than before. Obviously the total effect of µ d is not ruled out completely, since it is much more complex than in the considered expansion.</text>
<section_header_level_1><location><page_6><loc_21><loc_13><loc_37><loc_14></location>IV. CONCLUSION</section_header_level_1>
<text><location><page_6><loc_9><loc_9><loc_49><loc_11></location>In our paper, based on Einstein-Maxwell scalar dark matter theory, where the hidden sector is mimicked by</text>
<text><location><page_6><loc_52><loc_83><loc_92><loc_93></location>the auxiliary U (1)-gauge field coupled to the ordinary Maxwell one by the kinetic mixing term with a coupling constant α , we elaborate two scenarios of emergence of a hairy black hole or a boson star. The main motivation standing behind our research was to shed some light on the influence of dark matter sector on the physics and thermodynamics of these systems.</text>
<text><location><page_6><loc_52><loc_50><loc_92><loc_83></location>The obtained results reveal that the coupling between visible and hidden sectors plays a complex role in the behavior of scalar hair. The parameter space ( µ -T ), where these solutions constitute a thermodynamically favorable phase, is being narrowed on two boundaries and extended to another one. The dark sector 's presence strongly reduces the value of critical chemical potential, above which the hairy solution becomes unstable. Moreover the critical temperature of HBH-RN-like solution is shifted towards lower value of Hawking temperature. However the boundary between HBH and boson stars is shifted towards the latter. The presence of the dark sector lowers the free energy of HBH system, which broadens the parameter space available for the emergence of the object in question by a noticeable extent, i.e., leaving boson star as an adverse phase in the low µ regime. It appears that the free energy of boson stars in the considered configuration reacts faintly to the presence of U (1)-gauge dark matter field. While the response of the system is visible it is much smaller in magnitude than of the condensate around a black hole. However we suppose that interesting results may be achieved for more robust model of a scalar field, e.g., containing self interacting terms.</text>
<text><location><page_6><loc_52><loc_40><loc_92><loc_49></location>In the view of presented results it seems that the hairy solutions are not only battled by no-hair theorems originating from the theory of black holes, but also by a factor that is commonly present in our universe - the dark matter . Even if such formation would be possible despite different obstacles a significant abundance of dark matter may prevent hairy solutions from emerging.</text>
<text><location><page_6><loc_52><loc_23><loc_92><loc_39></location>To visualize the impact of dark sector, we compute the area of HBH parameter space between both phase boundaries. One can consider simple integration ∫ ( µ BS ( T ) -µ RN ( T )) dT of the curves from the Fig. 5, which reveals that the dark sector with µ d = 0 . 14 takes away approximately 27% of the hairy black hole's parameter space, compared to dark matter free solution. It is indeed a significant difference, because even if such formation would be possible despite different obstacles a significant abundance of dark sector may prevent hairy solutions from emerging.</text>
<text><location><page_6><loc_52><loc_12><loc_92><loc_23></location>The curves ψ (1) ( T ) and µ ( T ), parametrized by the values of dark matter chemical potential reveal the isosbestic points , where they all intersect. One has the specific configurations of the considered system which is unaffected by the influence of hidden sector . At the points in question we perform analysis revealing that the leading order influence of dark matter on the condensation ψ (1) and chemical potential of ordinary matter is quadratic in µ d .</text>
<text><location><page_6><loc_52><loc_9><loc_92><loc_11></location>As a concluding remark, we present promising future research directions. We have elaborated the simple box-</text>
<text><location><page_7><loc_9><loc_86><loc_49><loc_93></location>boundary models of a hairy black hole and a boson star (the so-called small boson star). The tantalizing question can be asked about the different boson star configurations with additional fields and potentials. Further investigations in this direction will be published elsewhere.</text>
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<section_header_level_1><location><page_7><loc_62><loc_92><loc_82><loc_93></location>ACKNOWLEDGMENTS</section_header_level_1>
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<list_item><location><page_7><loc_52><loc_32><loc_92><loc_35></location>[33] M. Rogatko and K.I. Wysokinski, P-wave holographic superconductor/insulator phase transitions affected by dark matter sector , JHEP 03 (2016) 215.</list_item>
<list_item><location><page_7><loc_52><loc_28><loc_92><loc_31></location>[34] M. Rogatko and K.I. Wysokinski, Condensate flow in holographic models in the presence of dark matter , JHEP 10 (2016) 152.</list_item>
<list_item><location><page_7><loc_52><loc_24><loc_92><loc_27></location>[35] D. Vollhardt, Characteristic crossing points in specific heat curves of correlated systems , Phys. Rev. Lett. 78 ( 1997) 1307,</list_item>
<list_item><location><page_7><loc_55><loc_18><loc_92><loc_23></location>S. Uchida, T. Ido, H. Takagi, T. Arima, Y. Tokura, and S. Tajima, Optical spectra of La 2 -x Sr x CuO 4 : Effect of carrier doping on the electronic structure of the CuO 2 plane , Phys. Rev. B 43 (1991) 7942,</list_item>
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</document> | [] |
2018arXiv181110303G | https://arxiv.org/pdf/1811.10303.pdf | <document>
<section_header_level_1><location><page_1><loc_12><loc_84><loc_88><loc_88></location>Gravitational radiation and the evolution of gravitational collapse in cylindrical symmetry</section_header_level_1>
<text><location><page_1><loc_34><loc_81><loc_75><loc_82></location>Alfonso Garc´ıa-Parrado glyph[sharp] ∗ and Filipe C. Mena glyph[flat]glyph[natural] †</text>
<text><location><page_1><loc_13><loc_77><loc_96><loc_78></location>glyph[sharp] Faculty of Mathematics and Physics, Charles University in Prague, V Holeˇsoviˇck'ach 2, 180 00 Praha 8, Czech Republic.</text>
<text><location><page_1><loc_29><loc_76><loc_80><loc_77></location>glyph[flat] Centro de Matem'atica, Universidade do Minho, 4710-057 Braga, Portugal</text>
<text><location><page_1><loc_14><loc_74><loc_95><loc_75></location>glyph[natural] Dep. Matem'atica, Instituto Superior T'ecnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal</text>
<text><location><page_1><loc_42><loc_71><loc_58><loc_72></location>November 27, 2018</text>
<section_header_level_1><location><page_1><loc_47><loc_66><loc_54><loc_67></location>Abstract</section_header_level_1>
<text><location><page_1><loc_16><loc_54><loc_84><loc_65></location>Using the Sparling form and a geometric construction adapted to spacetimes with a 2-dimensional isometry group, we analyse a quasi-local measure of gravitational energy. We then study the gravitational radiation through spacetime junctions in cylindrically symmetric models of gravitational collapse to singularities. The models result from the matching of collapsing dust fluids interiors with gravitational wave exteriors, given by the Einstein-Rosen type solutions. For a given choice of a frame adapted to the symmetry of the matching hypersurface, we are able to compute the total gravitational energy radiated during the collapse and state whether the gravitational radiation is incoming or outgoing, in each case. This also enables us to distinguish whether a gravitational collapse is being enhanced by the gravitational radiation.</text>
<text><location><page_1><loc_16><loc_51><loc_61><loc_52></location>Keywords: Quasi-local energy; Gravitational waves; Sparling form</text>
<section_header_level_1><location><page_1><loc_12><loc_47><loc_31><loc_48></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_12><loc_34><loc_88><loc_45></location>The theorem about the positivity of the global gravitational field at spatial infinity is a cornerstone of General Relativity (GR) and there are already different proofs available of this important result. In practical applications, it would be desirable to have a quasi-local notion of energy which could be applied to finite objects. A great deal of effort has been put towards this goal [19], after the earlier proposals for the quasi-local mass by Hawking [12] and Penrose [16]. An interesting proposal was made by Brown and York [3] using a Hamiltonian formulation of General Relativity, and their definition depends on a gauge choice along three-dimensional spacelike hypersurfaces. Other authors have put forward definitions of mass which are based on more geometric constructions (see [22, 23] and references therein).</text>
<text><location><page_1><loc_12><loc_26><loc_88><loc_34></location>Associated to this problem is the problem of finding an appropriate Hamiltonian in each setting. In fact, the Hamiltonian for every diffeormorphism invariant theory depends on boundary conditions and is, therefore, non-unique. In General Relativity, for example, both the Komar Hamiltonian and the ADM Hamiltonian have been widely used (see e.g. [4]). The latter, in turn, is a particular case of the Von Freud super-potential [10] and a problem related to the search of superpotentials for GR is the question of the existence of the Lanzcos potential for the Weyl tensor (see e.g. [6]).</text>
<text><location><page_1><loc_12><loc_15><loc_88><loc_26></location>An alternative approach to the definition of quasi-local gravitational energy quantities is given by the use of gravitational pseudo-tensors . In [9], Frauendiener explains how to treat pseudo-tensors in the geometric framework of the principal bundle defined by the frame bundle of a 4-dimensional Lorentzian manifold. Using this approach, it is shown that the energy-momentum pseudo-tensors of Einstein and Landau-Lifschitz can be recovered from pull-backs to the spacetime manifold under appropriate sections of the so-called Sparling 3-form , which is defined in the bundle of linear frames. Szabados [18], generalizes these results and gives explicit formulae for gravitational pseudo-tensors in rigid basis or anholonomic frames. In particular, he shows that the pull-backs defined by coordinate sections of the contravariant</text>
<text><location><page_2><loc_12><loc_89><loc_88><loc_92></location>and dual forms of the Sparling's form, defined on the bundle of linear frames over the spacetime, are the Bergman and the Landau-Lifshitz pseudo-tensors, respectively.</text>
<text><location><page_2><loc_12><loc_70><loc_88><loc_89></location>In most of the applications we are aware of, the quasi-local quantities are evaluated for an isolated system at infinity, while in the present work we are interested in their local values at certain physically interesting hypersurfaces such as spacetime junctions. Even though quasi-local gravitational energy quantities are well-defined, they are frame dependent and, therefore, whenever they are used it is necessary to give a justification about the choice of a particular frame. In the case of an isolated system, it is assumed that it is asymptotically flat and this means that one can introduce a group of asymptotic symmetries which, roughly speaking, correspond to the symmetries of the Poincar'e group defined in flat spacetime or to a generalization called the BMS group . These asymptotic symmetries induce a set of privileged frames in the spacetime, in the sense that they correspond asymptotically to the generators of the asymptotic symmetry group and one can then take a frame of that set to compute a quasi-local gravitational energy quantity. Within this approach, it is possible to recover the standard ADM and Bondi notions of momentum for an isolated system using definitions of quasi-local quantities based on pseudo-tensors [11]. An approach based on the study of the asymptotic conditions as boundary conditions, defining when an isolated system is radiating, is also possible [21].</text>
<text><location><page_2><loc_12><loc_58><loc_88><loc_69></location>We would like to carry over the formalism described in the paragraph above to the case of a system composed by an (inhomogeneous) interior matched to a radiating exterior through a common hypersurface. The objective here is to give measures of the gravitational energy interchanged at the matching hypersurface, which plays now a similar role to the spacelike or null infinity of the case of an isolated system. In the case of matching hypersurfaces with enough symmetry, there is a natural geometrical way to define frames adapted to the symmetries of the hypersurface, and which will play the role of the frames corresponding to the asymptotic symmetries used to compute the quasi-local gravitational energy at infinity in an isolated system.</text>
<text><location><page_2><loc_12><loc_43><loc_88><loc_58></location>Here, to formulate this problem in a mathematically precise way, we use a frame bundle formalism and define equivalence classes between frames. We then apply our formalism to spacetimes admitting a G 2 isometry group and, in particular, calculate the radiated gravitational energy through the transitivity hypersurfaces of such isometry groups. A novel aspect of our work is the introduction of the notion of gravitational collapse enhanced by the gravitational radiation , i.e. by the gravitational energy-momentum flux, defined using the Sparling form. The meaning of this notion is that during the collapse, gravitational radiation is produced and the gravitational energy can either be radiated outside of the collapsing body or be radiated inwards, in which case it enhances the gravitational collapse. Although our presentation is restricted to 4 dimensional spacetimes, most of it can be easily generalised to n dimensions and can be applied e.g. to the brane world scenario where 4-dimensional branes are embedded, as hypersurfaces, on a 5-dimensional bulk.</text>
<text><location><page_2><loc_12><loc_24><loc_88><loc_42></location>Motivated by the recent detection of gravitational waves by LIGO [1], an interesting application of our formalism would be to models of objects with a finite radius in astrophysics, emitting gravitational radiation into a vacuum exterior. Due to Birkhoff's theorem, it is not possible to have such vacuum exteriors in spherical symmetry. The next simplest symmetry assumption is cylindrical symmetry. So, we will consider models of gravitational collapse in cylindrical symmetry. Such models result from gluing an interior metric representing matter collapsing gravitationally to an exterior metric containing gravitational waves. An interesting aspect of these models is that the matter density is non-zero at the matching boundary, resulting in discontinuities in the Einstein field equations. Even though a realistic system undergoing gravitational collapse is not cylindrically symmetric, some non-trivial effects already present in the real system can be found in the cylindrically symmetric model. For example, the degrees of freedom of the (pure) gravitational energy in the exterior may mix with the gravitational energy sourced by the matter at the boundary. These non-trivial effects may be captured by local, gauge-dependent, measures of gravitational energy.</text>
<text><location><page_2><loc_12><loc_14><loc_88><loc_24></location>The plan of the paper is as follows: in section 2, we revise some properties of the Sparling 3-form and we compute its transformation under a change of frame. In section 3, we review how to measure the gravitational energy radiated through embedded hypersurfaces of codimension 1 using the Sparling form. In section 4, we provide a geometric set-up for the definition of a frame based on the existence of a submanifold representing the surface of a radiative body. In section 5, we apply our formalism to cylindrically symmetric collapsing dust spacetimes containing gravitational radiation in the exterior which corresponds to the Einstein-Rosen solution.</text>
<text><location><page_2><loc_12><loc_11><loc_88><loc_14></location>Some algebraic computations of this paper have been done with xAct [15]. We use Latin indices a, b, c, .. = 1 , 2 , 3 , 4, Greek indices α, β, .. = 1 , 2 , 3, capitals A,B,.. = 1 , 2 and units such that 8 πG = c = 1.</text>
<section_header_level_1><location><page_3><loc_12><loc_90><loc_61><loc_92></location>2 Sparling 3-form and Sparling identity</section_header_level_1>
<text><location><page_3><loc_12><loc_84><loc_88><loc_89></location>In this section and in the following section 3, we review some known notions and properties about the Sparling form and the sparling identity that are needed for our work. Even though most of the material is known, we present the equations in accordance to the notation that shall be followed in sections 4 and 5.</text>
<text><location><page_3><loc_12><loc_79><loc_88><loc_83></location>Let ( M , g ab ) be a 4-dimensional spacetime with signature ( -, + , + , +) and consider the bundle of frames L ( M ) over M . Let B ≡ { glyph[vector] e 1 , glyph[vector] e 2 , glyph[vector] e 3 , glyph[vector] e 4 } be a chosen frame (smooth section of L ( M )) and denote by B ∗ ≡ { θ 1 , θ 2 , θ 3 , θ 4 } its dual co-basis. Define the super-potential</text>
<formula><location><page_3><loc_39><loc_75><loc_88><loc_78></location>L ( B,glyph[vector] e a ) ≡ 1 2 η abcd θ d ∧ Γ bc ( B ) , (1)</formula>
<text><location><page_3><loc_12><loc_69><loc_88><loc_74></location>where Γ( B ) b c is the connection 1-form on the frame B and η abcd are the components of the volume element of the metric g ab in the frame B . Note the dependency of the super-potential on both the frame B and the frame elements glyph[vector] e a . 1 Under the choice of the frame B , the super-potential is a 2-form in Λ 2 ( M ) and it fulfills the Sparling identity [9, 18, 5]</text>
<formula><location><page_3><loc_38><loc_66><loc_88><loc_67></location>d ( L ( B,glyph[vector] e a )) = E ( glyph[vector] e a ) + S ( B,glyph[vector] e a ) . (2)</formula>
<text><location><page_3><loc_12><loc_62><loc_88><loc_65></location>where the quantity E ( glyph[vector] e a ) ∈ Λ 3 ( M ) is the so-called Einstein 3-form , which is given in terms of the curvature 2-form R ab by</text>
<formula><location><page_3><loc_40><loc_59><loc_88><loc_62></location>E ( glyph[vector] e a ) = -1 2 η abcd θ b ∧ R cd , (3)</formula>
<text><location><page_3><loc_12><loc_58><loc_29><loc_59></location>and it has the property</text>
<formula><location><page_3><loc_37><loc_56><loc_88><loc_58></location>G ( glyph[vector] e a ) ≡ ∗ E ( glyph[vector] e a ) = G ab θ b ∈ Λ 1 ( M ) , (4)</formula>
<text><location><page_3><loc_12><loc_54><loc_59><loc_55></location>with ∗ representing the Hodge dual and G ab the Einstein tensor.</text>
<text><location><page_3><loc_12><loc_50><loc_88><loc_54></location>As E ( glyph[vector] e a ) and ∗ E ( glyph[vector] e a ) are tensorial differential forms, we do not write their dependency on B , such dependency being only kept for pseudo-tensorial forms . This is for instance the case of the 3-form S ( B,glyph[vector] e a ) that it is given by</text>
<formula><location><page_3><loc_28><loc_46><loc_88><loc_49></location>S ( B,glyph[vector] e a ) ≡ 1 2 ( η bcde θ b ∧ Γ( B ) c a -η abce θ b ∧ Γ( B ) c d ) ∧ Γ( B ) de . (5)</formula>
<text><location><page_3><loc_12><loc_41><loc_88><loc_45></location>We will follow the convention of [9, 18] and call this form the Sparling 3-form . For us, the important point about the Sparling identity is that it can be interpreted as a conservation law in the following way: By defining the 1-form</text>
<formula><location><page_3><loc_42><loc_40><loc_88><loc_41></location>J ( B,glyph[vector] e a ) ≡ ∗ dL ( B,glyph[vector] e a ) , (6)</formula>
<text><location><page_3><loc_12><loc_37><loc_51><loc_39></location>and computing the co-differential of J ( B,glyph[vector] e a ), we get</text>
<formula><location><page_3><loc_30><loc_35><loc_88><loc_36></location>δJ ( B,glyph[vector] e a ) = -∗ d ∗ J ( B,glyph[vector] e a ) = -∗ d ∗ ∗ dL ( B,glyph[vector] e a ) = 0 . (7)</formula>
<text><location><page_3><loc_12><loc_29><loc_88><loc_33></location>From this equation, we deduce that J ( B,glyph[vector] e a ) always yields a conserved current, independently of the choice of the frame B or the frame vectors glyph[vector] e a . According to the above considerations, the current J ( B,glyph[vector] e a ) can be split into two parts as</text>
<formula><location><page_3><loc_39><loc_27><loc_88><loc_28></location>J ( B,glyph[vector] e a ) = G ( glyph[vector] e a ) + ∗ S ( B,glyph[vector] e a ) . (8)</formula>
<text><location><page_3><loc_12><loc_20><loc_88><loc_25></location>The first part is a tensorial 1-form and, via the Einstein equations, can be interpreted as the flux of energy momentum due to the matter. The second part is non-tensorial and can be regarded as the flux of gravitational energy-momentum with respect to the frame B . If glyph[vector] u is a frame element of B representing an observer, then the 1-form</text>
<formula><location><page_3><loc_43><loc_19><loc_88><loc_20></location>j ( B, glyph[vector] u ) ≡ ∗ S ( B, glyph[vector] u ) (9)</formula>
<text><location><page_4><loc_12><loc_85><loc_88><loc_92></location>shall be called the gravitational energy-momentum flux 1-form for the observer glyph[vector] u with respect to the frame B or just the gravitational flux 1-form, if no confusion arises. It can be interpreted as the gravitational energy-momentum flux, which in this context is also called gravitational radiation, measured by the given observer in the frame B . In our context, we need to be able to decide if gravitational radiation is leaving or entering a radiating source. To that end, we put forward the following definition.</text>
<section_header_level_1><location><page_4><loc_12><loc_83><loc_47><loc_84></location>Definition 1. We shall take the scalar quantity</section_header_level_1>
<formula><location><page_4><loc_46><loc_80><loc_88><loc_81></location>j ( B, glyph[vector] u ) 〈 glyph[vector] e a 〉 (10)</formula>
<text><location><page_4><loc_12><loc_75><loc_88><loc_79></location>as the component of the gravitational radiation flux along the direction defined by the vector field ( glyph[vector] e a ) . If this scalar quantity is positive (resp. negative), then we say that the gravitational radiation is outgoing (resp. incoming) with respect to ( glyph[vector] e a ) .</text>
<text><location><page_4><loc_12><loc_58><loc_88><loc_73></location>The fact that j ( B, glyph[vector] u ) is frame and observer dependent does not render it useless when computing the radiated gravitational energy in certain cases, as we illustrate through explicit examples in section 5. In fact, in [9], it was proved that the scalar quantity (10) has a correspondence with the components of the energy-momentum complex given by Einstein in 1916 [7]. Furthermore, it has been shown that, under an appropriate choice of frame B , it is possible to use this complex to recover the well-known notions of ADM and Bondi momentum at infinity for an isolated system (see [11]). Thus, following a similar approach, we will compute, in section 5, the gravitational energy radiated through a matching hypersurface separating two regions (exterior and interior) under the assumption that the hypersurface has certain symmetry properties. We will present explicit examples of systems undergoing gravitational collapse and show how the gravitational flux 1-form enables us to give a criterion stating whether gravitational radiation is being emitted or absorbed during the collapse.</text>
<text><location><page_4><loc_12><loc_54><loc_88><loc_58></location>It is obvious that a similar reasoning as the above can be performed if we take the Einstein form instead of the Sparling form. In this fashion, one can define the matter current with respect to the observer glyph[vector] u in the standard way</text>
<formula><location><page_4><loc_42><loc_52><loc_88><loc_54></location>J ( glyph[vector] u ) ≡ G ( glyph[vector] u ) ∈ Λ 1 ( M ) . (11)</formula>
<text><location><page_4><loc_12><loc_47><loc_88><loc_51></location>This current is tensorial and, as it is well-known, it represents the matter energy-momentum flux. From previous considerations, it is clear that the sum of both currents is always conserved, a property that can be expressed by the condition</text>
<formula><location><page_4><loc_42><loc_46><loc_88><loc_47></location>δ ( J ( glyph[vector] u ) + j ( B, glyph[vector] u )) = 0 . (12)</formula>
<text><location><page_4><loc_12><loc_38><loc_88><loc_45></location>Interestingly, even though j ( B, glyph[vector] u ) is frame dependent, the conservation property (12) holds for any frame B . A particular case occurs when each of the currents are independently conserved. As we will show, this is the situation for all the examples of section 5 modelling gravitational collapse. The implication of this is that, during the gravitational collapse, no fraction of the collapsing matter is transformed into gravitational radiation.</text>
<section_header_level_1><location><page_4><loc_12><loc_32><loc_88><loc_36></location>3 Frame equivalence and radiated gravitational energy through an embedded hypersurface</section_header_level_1>
<text><location><page_4><loc_12><loc_28><loc_88><loc_31></location>Suppose now that we introduce another frame B ' and its associate co-frame B '∗ , related to B and B ∗ through the formulae</text>
<formula><location><page_4><loc_24><loc_25><loc_88><loc_27></location>glyph[vector] e ' a = γ ( B,B ' ) b a glyph[vector] e b , θ ' a = γ ( B ' , B ) a b θ b , γ ( B,B ' ) b a γ ( B ' , B ) c b = δ c a , (13)</formula>
<text><location><page_4><loc_12><loc_20><loc_88><loc_24></location>where the matrices γ ( B,B ' ), γ ( B ' , B ) encode the transition between the frames B and B ' . We regard these matrices as having entries which are scalars on M (0-forms). Under these circumstances, it is well-known that the connection 1-form transforms as</text>
<formula><location><page_4><loc_26><loc_17><loc_88><loc_19></location>Γ( B ' ) a b = γ ( B ' , B ) a q Γ( B ) q p γ ( B,B ' ) p b + γ ( B ' , B ) a p d ( γ ( B,B ' ) p b ) . (14)</formula>
<text><location><page_4><loc_12><loc_13><loc_88><loc_16></location>We can use this expression to find the transformation law for the Sparling 3-form. A straightforward but tedious computation reveals:</text>
<text><location><page_5><loc_12><loc_91><loc_74><loc_92></location>Lemma 1. Under the change of frames B to B ' , the Sparling 3-form transforms as:</text>
<formula><location><page_5><loc_17><loc_80><loc_88><loc_90></location>S ( B ' , glyph[vector] e ' s ) = -1 2 γ ( B,B ' ) r a η abcd dγ ( B,B ' ) s b ∧ dγ ( B,B ' ) r c ∧ θ d -1 2 γ ( B ' , B ) c h γ ( B,B ' ) s a γ ( B,B ' ) r b η abdm dγ ( B,B ' ) r c ∧ dγ ( B,B ' ) h d ∧ θ m -1 2 η a bcd Γ( B ) b a ∧ dγ ( B,B ' ) s c ∧ θ d -1 2 γ ( B ' , B ) b r γ ( B,B ' ) s a η a c dm Γ( B ) b c ∧ dγ ( B,B ' ) r d ∧ θ m -1 2 γ ( B,B ' ) s a γ ( B,B ' ) r b η bcdm Γ( B ) c a ∧ dγ ( B,B ' ) r d ∧ θ m + 1 2 γ ( B,B ' ) s a γ ( B,B ' ) r b η abcd Γ( B ) c m ∧ dγ ( B,B ' ) r m ∧ θ d + γ ( B,B ' ) s a S ( B,glyph[vector] e a ) . (15)</formula>
<text><location><page_5><loc_12><loc_76><loc_88><loc_79></location>From the expression above, it is clear that the relation between S ( B ' , glyph[vector] e ' s ) and S ( B,glyph[vector] e a ) is linear if dγ ( B,B ' ) s a = 0 and, in that case, S transforms like a tensor. We shall explore this fact next.</text>
<text><location><page_5><loc_12><loc_74><loc_88><loc_76></location>Let H ⊂ M be a co-dimension 1 smooth embedded hypersurface and consider two frames B , B ' fulfilling the condition</text>
<formula><location><page_5><loc_42><loc_72><loc_88><loc_74></location>d ( φ ∗ ( γ b a ( B ' , B ))) = 0 , (16)</formula>
<text><location><page_5><loc_12><loc_70><loc_57><loc_71></location>where φ : H → M is a smooth embedding. Since one has that</text>
<formula><location><page_5><loc_39><loc_68><loc_88><loc_69></location>φ ∗ ( γ b a ( B ' , B )) = γ b a ( B ' , B ) | H (17)</formula>
<text><location><page_5><loc_12><loc_65><loc_58><loc_67></location>we deduce that the scalars γ b a ( B ' , B ) | H can only be constants.</text>
<text><location><page_5><loc_12><loc_63><loc_84><loc_64></location>Proposition 1. For H ⊂ M and φ : H → M as described in the previous paragraph, the relation</text>
<formula><location><page_5><loc_38><loc_60><loc_88><loc_62></location>B ∼ B ' ⇐⇒ d ( φ ∗ ( γ b a ( B ' , B ))) = 0 (18)</formula>
<text><location><page_5><loc_12><loc_58><loc_83><loc_59></location>is an equivalence relation in X ( L ( M )) ( ≡ the set of smooth sections in the frame bundle L ( M ) ).</text>
<text><location><page_5><loc_12><loc_55><loc_88><loc_57></location>Proof. To ease the notation, we suppress indices in the matrix γ b a ( B ' , B ) and write it simply as γ ( B ' , B ). We show that the properties characterizing an equivalence relation are fulfilled:</text>
<unordered_list>
<list_item><location><page_5><loc_15><loc_52><loc_48><loc_53></location>· Reflexive: As γ ( B,B ) = I trivially B ∼ B .</list_item>
<list_item><location><page_5><loc_15><loc_50><loc_61><loc_51></location>· Symmetric: If B ∼ B ' then d ( φ ∗ γ ( B ' , B )) = 0. One has then</list_item>
</unordered_list>
<formula><location><page_5><loc_25><loc_48><loc_88><loc_49></location>0 = d ( φ ∗ ( I )) = d ( φ ∗ ( γ ( B ' , B ) φ ∗ ( γ ( B,B ' ))) = φ ∗ ( γ ( B ' , B )) d ( φ ∗ ( γ ( B,B ' )) . (19)</formula>
<text><location><page_5><loc_16><loc_45><loc_85><loc_46></location>Since φ ∗ ( γ ( B ' , B )) is invertible, the last equation entails d ( φ ∗ ( γ ( B,B ' )) = 0 and thus B ' ∼ B .</text>
<unordered_list>
<list_item><location><page_5><loc_15><loc_43><loc_46><loc_44></location>· Transitive: if B ∼ B ' and B ' ∼ B '' then</list_item>
</unordered_list>
<formula><location><page_5><loc_35><loc_40><loc_73><loc_42></location>d ( φ ∗ γ ( B '' , B )) = d ( φ ∗ ( γ ( B '' , B ' )) φ ∗ ( γ ( B ' , B )) ) = 0 .</formula>
<text><location><page_5><loc_26><loc_38><loc_27><loc_39></location>''</text>
<formula><location><page_5><loc_16><loc_38><loc_28><loc_39></location>Hence B ∼ B .</formula>
<text><location><page_5><loc_12><loc_36><loc_79><loc_37></location>Proposition 2. If there are two frames B , B ' on M such that B | H = B ' | H , then B ∼ B ' .</text>
<text><location><page_5><loc_12><loc_33><loc_78><loc_35></location>Proof. If B | H = B ' | H , then eq. (17) holds with γ b a ( B ' , B ) | H = δ a b and therefore B ∼ B ' .</text>
<text><location><page_5><loc_12><loc_30><loc_49><loc_32></location>Now, let B , B ' be frames such that B ∼ B ' . Then</text>
<formula><location><page_5><loc_34><loc_28><loc_88><loc_29></location>φ ∗ ( d ( γ b a ( B ' , B ))) = d ( φ ∗ ( γ b a ( B ' , B ))) = 0 . (20)</formula>
<text><location><page_5><loc_12><loc_26><loc_43><loc_27></location>Using this information in (15), we deduce</text>
<formula><location><page_5><loc_35><loc_21><loc_88><loc_24></location>∫ H S ( B ' , glyph[vector] e ' a ) = γ ( B ' , B ) b a | H ∫ H S ( B,glyph[vector] e b ) , (21)</formula>
<text><location><page_5><loc_12><loc_18><loc_88><loc_21></location>where we used the fact that γ ( B ' , B ) b a | H are constants as B ∼ B ' . Bearing in mind this property, let us make the following definition</text>
<text><location><page_5><loc_12><loc_14><loc_88><loc_17></location>Definition 2. Let H ⊂ M be an embedded co-dimension 1 smooth hypersurface. For a given frame B and frame element glyph[vector] e a ∈ B , the gravitational flux through H , with respect to B and glyph[vector] e a is defined as</text>
<formula><location><page_5><loc_40><loc_10><loc_88><loc_13></location>P ( B, H , glyph[vector] e a ) ≡ ∫ H S ( B,glyph[vector] e a ) . (22)</formula>
<text><location><page_6><loc_15><loc_91><loc_76><loc_92></location>If B and B ' are frames such that B ∼ B ' , then eq. (21) can be rendered in the form</text>
<formula><location><page_6><loc_29><loc_88><loc_88><loc_90></location>P ( B, H , glyph[vector] e a ) = γ ( B ' , B ) b a | H P ( B ' , H , glyph[vector] e ' b ) , for B ∼ B ' . (23)</formula>
<text><location><page_6><loc_12><loc_83><loc_88><loc_87></location>This means that if we have a privileged frame B in our problem, we can compute the gravitational flux according to (22) and consider the scalars { P ( B, H , glyph[vector] e a ) } as the components of the gravitational 4-momentum of the hypersurface H . We shall represent it as P ( B, H ).</text>
<text><location><page_6><loc_12><loc_80><loc_88><loc_83></location>An important particular case occurs in vacuum, when S ( B,glyph[vector] e a ) = dL ( B,glyph[vector] e a ). In this case, using Stokes theorem, we can write (22) as follows</text>
<formula><location><page_6><loc_34><loc_76><loc_88><loc_79></location>P ( B, H , glyph[vector] e a ) ≡ ∫ H dL ( B,glyph[vector] e a ) = ∫ ∂ H L ( B,glyph[vector] e a ) . (24)</formula>
<text><location><page_6><loc_12><loc_72><loc_88><loc_76></location>In an asymptotically flat isolated system we can choose H in such a way that one of the connected components of ∂ H surrounds the source. Denoting by Σ r such a connected component, we can study the existence of the limit</text>
<formula><location><page_6><loc_43><loc_69><loc_88><loc_72></location>lim r →∞ ∫ Σ r L ( B,glyph[vector] e a ) , (25)</formula>
<text><location><page_6><loc_12><loc_66><loc_88><loc_68></location>for a hypersurface Σ r that approaches infinity . This analysis has been already performed in a number of situations (see [11]):</text>
<unordered_list>
<list_item><location><page_6><loc_14><loc_61><loc_88><loc_65></location>1. The surface Σ r approaches spatial infinity . In this case, if we choose an asymptotically Cartesian frame B , one can show that the limit (25) corresponds to the ADM 4-momentum of the isolated source.</list_item>
<list_item><location><page_6><loc_14><loc_56><loc_88><loc_60></location>2. The surface Σ r approaches null infinity . In this case, if we choose a frame B as the normalized translation defined by the asymptotic BMS group, then the limit (25) corresponds to the Bondi 4-momentum of the isolated source.</list_item>
</unordered_list>
<text><location><page_6><loc_12><loc_48><loc_88><loc_55></location>In any case, we note that the choice of a frame B with suitable properties at infinity, yields integral values of quasi-local quantities that have a valid physical meaning. In the next section, we shall follow this approach and show how the choice of a frame B adapted to the symmetries of our physical system, at the matching hypersurface, enables us to analyse the problem of the gravitational radiation through the matching boundary.</text>
<text><location><page_6><loc_12><loc_38><loc_88><loc_47></location>The most important point to stress here is that the matching hypersurface, in our case, plays a role similar to null infinity or spacelike infinity in the case of an isolated system. As discused above, the choice of a frame B adapted to the symmetries of null or spacelike infinity (the asymptotic symmetries ) provides, respectively, the Bondi or ADM momentum. Therefore, a frame B adapted to the symmetries of the matching hypersurface, should give us the quantities corresponding to the flux of gravitational energymomentum through the matching hypersurface. The detailed analysis of this assertion is presented in the next section.</text>
<section_header_level_1><location><page_6><loc_12><loc_32><loc_88><loc_35></location>4 Gravitational flux through a hypersurface in a spacetime with a G 2 isometry group</section_header_level_1>
<text><location><page_6><loc_12><loc_28><loc_88><loc_30></location>Assume that M admits a maximal isometry group G 2 generated by the Killing vectors glyph[vector] ξ 1 , glyph[vector] ξ 2 . Let H ⊂ M be a co-dimension 1 hypersurface. Choose a frame of the form</text>
<formula><location><page_6><loc_43><loc_25><loc_88><loc_27></location>B = { glyph[vector] n , glyph[vector] ξ 1 , glyph[vector] ξ 2 , glyph[vector] u } , (26)</formula>
<text><location><page_6><loc_12><loc_16><loc_88><loc_24></location>where glyph[vector] n is a vector field on M such that glyph[vector] n | H is unit and normal to the hypersurface and glyph[vector] u is an unit timelike vector field representing an observer. This frame is fully adapted to the geometry of the matching hypersurface H because its elements are either fixed at H (as happens for an observer glyph[vector] u and the unit normal glyph[vector] n ) or are symmetries of the matching hypersurfaces (as happens for the Killing vectors glyph[vector] ξ 1 , glyph[vector] ξ 2 ). There is a residual freedom in the choice of the frame B , coming from the fact that the vector field glyph[vector] n is defined up to a sign on H and that</text>
<formula><location><page_6><loc_24><loc_13><loc_88><loc_15></location>glyph[vector] ξ ' 1 = α 11 glyph[vector] ξ 1 + α 12 glyph[vector] ξ 2 , glyph[vector] ξ ' 2 = α 21 glyph[vector] ξ 1 + α 22 glyph[vector] ξ 2 , α 11 , α 12 , α 21 , α 22 ∈ R , (27)</formula>
<text><location><page_6><loc_12><loc_11><loc_49><loc_12></location>are Killing vectors too. However, it is easy to get:</text>
<text><location><page_7><loc_12><loc_89><loc_88><loc_92></location>Proposition 3. Any other frame B ' = { glyph[vector] n , glyph[vector] ξ ' 1 , glyph[vector] ξ ' 2 , glyph[vector] u } with glyph[vector] ξ ' 1 , glyph[vector] ξ ' 2 related to glyph[vector] ξ 1 , glyph[vector] ξ 2 by (27) fulfills the property B ' ∼ B .</text>
<text><location><page_7><loc_12><loc_82><loc_88><loc_88></location>From the previous proposition we deduce that, in a spacetime admitting a G 2 isometry group, one can define unambiguously the gravitational 4-momentum of a hypersurface H as P ( B, H ), where B is a frame of the form (26). Since, in this case, the only freedom left to construct the frame B is given by the choice of the observer glyph[vector] u , we shall use the shorthand notation</text>
<formula><location><page_7><loc_43><loc_80><loc_88><loc_82></location>P ( glyph[vector] u , H ) ≡ P ( B, H ) . (28)</formula>
<text><location><page_7><loc_12><loc_78><loc_67><loc_80></location>Next, we are going to compute P ( glyph[vector] u , H ) in a number of practical examples.</text>
<section_header_level_1><location><page_7><loc_12><loc_74><loc_83><loc_76></location>5 Gravitational flux in cylindrically symmetric spacetimes</section_header_level_1>
<text><location><page_7><loc_12><loc_66><loc_88><loc_73></location>In this section, we consider the problem of computing the gravitational flux through hypersurfaces in cylindrically symmetric spacetimes. The hypersurfaces, here, result from the matching of an exterior vacuum spacetime containing gravitational waves and a dust fluid interior spacetime which is collapsing to form a singularity. To simplify the presentation, we consider exact solutions to the Einstein equations with diagonal metrics only.</text>
<text><location><page_7><loc_12><loc_51><loc_88><loc_66></location>We start by briefly recalling the matching conditions between two spacetimes ( M ± , g ± ) across timelike hypersurfaces H ± (see more details in [14]). The matching between two spacetimes requires an identification of their boundaries, i.e. a pair of embeddings Φ ± : H -→ M ± with Φ ± ( H ) = H ± , where H is an abstract copy of one of the boundaries. Let ξ α be a coordinate system on H , where α, β = 1 , 2 , 3 are indices on the hypersurface. Given a vector basis { ∂/∂ξ α } of the tangent space T H , the push-forwards d Φ ± provide a correspondence between the vectors of this basis and sets of linearly independent vectors tangent to H ± , given in appropriate coordinates by e ± a α = ∂ ξ α Φ ± a . Here, we follow the convention of previous sections that small Latin letters a, b = 1 , 2 , 3 , 4 represent spacetime indices. There are also (up to orientation) vectors n a ± normal to the boundaries. The first and second fundamental forms on H are given by q ± = Φ ± glyph[star] ( g ± ) and K ± = Φ ± glyph[star] ( ∇ ± n ± ), where Φ ± glyph[star] denotes the pull-back corresponding to the maps Φ ± . In components, we may write</text>
<formula><location><page_7><loc_33><loc_48><loc_88><loc_50></location>q ± αβ ≡ e ± a α e ± b β g ab | H ± , K ± αβ = -n ± a e ± b α ∇ ± b e ± a β . (29)</formula>
<text><location><page_7><loc_12><loc_45><loc_88><loc_48></location>The matching conditions between two spacetimes through H require the equality of the first and second fundamental form on H , i.e.</text>
<formula><location><page_7><loc_41><loc_43><loc_88><loc_45></location>q + αβ = q -αβ , K + αβ = K -αβ . (30)</formula>
<text><location><page_7><loc_12><loc_42><loc_72><loc_43></location>In the examples below, the spacetimes are matched across cylinders of symmetry.</text>
<section_header_level_1><location><page_7><loc_12><loc_39><loc_45><loc_40></location>5.1 Gravitational wave exteriors</section_header_level_1>
<text><location><page_7><loc_12><loc_35><loc_88><loc_38></location>The most general diagonal metric for cylindrically symmetric vacuum spacetimes (with an Abelian G 2 acting on 2-dimensional spacelike hypersurfaces S 2 ) can be written as [13]</text>
<formula><location><page_7><loc_25><loc_33><loc_88><loc_34></location>g + = e 2( γ -ψ ) ( -dT ⊗ dT + dρ ⊗ dρ ) + R 2 e -2 ψ d ˜ ϕ ⊗ d ˜ ϕ + e 2 ψ d ˜ z ⊗ d ˜ z, (31)</formula>
<text><location><page_7><loc_12><loc_25><loc_88><loc_32></location>where ψ, γ, R are functions of the coordinates ρ, T satisfying the Einstein equations (76)-(80). This metric is written in coordinates adapted to the Killing vectors ∂ ˜ ϕ and ∂ ˜ z . From the physical point of view, the metric models cylindrical gravitational waves with one polarization state and, for the choice R ( T, ρ ) = ρ , corresponds to the metric found by Einstein and Rosen [8]. So, we say that (31) are metrics of Einstein-Rosen type .</text>
<text><location><page_7><loc_12><loc_17><loc_88><loc_25></location>Dust matter sources to such spacetimes were studied in [20, 2], where the matching between collapsing interior spacetimes and Einstein-Rosen type of exteriors was shown to exist along timelike hypersurfaces. It would be therefore interesting to study the gravitational radiation through those hypersurfaces and to find out whether it enhances the gravitational collapse or not. It has been shown [20] that, in those cases, the matching hypersurfaces are ruled by geodesics and, from the point of view of the exterior, can be parametrized as</text>
<formula><location><page_7><loc_38><loc_15><loc_88><loc_17></location>H + : { t = λ, ρ = r 0 , ˜ ϕ = φ, ˜ z = ζ } , (32)</formula>
<text><location><page_7><loc_12><loc_11><loc_88><loc_15></location>where r 0 is constant and λ is a parameter along the geodesics. So, H + is generated by glyph[vector] e + 1 = ∂ t | H + , glyph[vector] e + 2 = ∂ ˜ ϕ | H + and glyph[vector] e + 3 = ∂ ˜ z | H + , which together with glyph[vector] n + = ∂ r | H + form an orthogonal basis for the spacetime at H + .</text>
<section_header_level_1><location><page_8><loc_12><loc_91><loc_49><loc_92></location>5.2 Spatially homogeneous interiors</section_header_level_1>
<text><location><page_8><loc_12><loc_87><loc_88><loc_90></location>As interiors to (31), we first consider locally rotationally symmetric and spatially homogeneous dust solutions given by the Bianchi class I of metrics [13], in cylindrical coordinates, as</text>
<formula><location><page_8><loc_29><loc_85><loc_88><loc_86></location>g -= -dt ⊗ dt + a ( t ) 2 dz ⊗ dz + b ( t ) 2 ( dr ⊗ dr + r 2 dϕ ⊗ dϕ ) (33)</formula>
<text><location><page_8><loc_12><loc_82><loc_15><loc_83></location>with</text>
<formula><location><page_8><loc_34><loc_81><loc_88><loc_82></location>a ( t ) = ( α -t ) -1 / 3 ( β -t ) , b ( t ) = ( α -t ) 2 / 3 (34)</formula>
<text><location><page_8><loc_12><loc_77><loc_88><loc_80></location>for 0 ≤ t < min { α, β } and r ≤ r 0 . If α = β , the metric reduces to the Friedman-Lemaˆıtre-RobertsonWalker (FLRW) class of spatially homogeneous and isotropic metrics.</text>
<text><location><page_8><loc_12><loc_73><loc_88><loc_77></location>The interior metrics (33) always contain cylinders of symmetry which are trapped near the singularity (see [20]). The marginally-trapped cylinders trace out a 3-surface in the interior which eventually arrives at the boundary of the matter located at r = r 0 .</text>
<text><location><page_8><loc_15><loc_71><loc_76><loc_73></location>From the point of view of the interior, the matching hypersurface is parametrized as</text>
<formula><location><page_8><loc_37><loc_69><loc_63><loc_70></location>H -: { T = λ, r = r 0 , ϕ = φ, z = ζ } ,</formula>
<text><location><page_8><loc_12><loc_65><loc_88><loc_68></location>so that H -is generated by glyph[vector] e -1 = ∂ T | H -, glyph[vector] e -2 = ∂ ϕ | H -and glyph[vector] e -3 = ∂ z | H -, which together with glyph[vector] n -= ∂ ρ | H -form an orthogonal basis for the spacetime at H -.</text>
<text><location><page_8><loc_15><loc_64><loc_68><loc_65></location>Then, the matching conditions (30) between (31) and (33) across H give</text>
<formula><location><page_8><loc_26><loc_61><loc_88><loc_62></location>e ψ H = a ( t ) , R H = a ( t ) b ( t ) r, R ρ H = a ( t ) , γ H = ψ, γ ρ H = ψ ρ = 0 , (35)</formula>
<text><location><page_8><loc_12><loc_49><loc_88><loc_59></location>where H = denotes evaluation at H . In order to solve the matching problem, the strategy followed in [20] was as follows: For a suitable spacetime interior to (31) and for the conditions (35) at the boundary, one obtains an explicit solution for R ( ρ, T ) which satisfies the wave equation (77) and the constraints (79) and (80), at H . In turn, the remaining Einstein equations (76) and (78) can be seen as providing γ ,ρρ and ψ ,ρρ on H . Since we know data for the exterior metric and its normal derivatives at the boundary, it then follows [20] that a unique ψ exists on a neighbourhood D of H . Since γ H = ψ and γ ,ρ H = ψ ,ρ , once we have ψ , we use a similar argument in (76) to get a unique γ in D .</text>
<section_header_level_1><location><page_8><loc_12><loc_46><loc_28><loc_47></location>5.2.1 FLRW case</section_header_level_1>
<text><location><page_8><loc_12><loc_41><loc_88><loc_45></location>If the interior is given by a FLRW metric, then a ( t ) = b ( t ) = ( α -t ) 2 / 3 , for t ∈ ( -∞ , α ). From the exterior, the matching surface H is the cylinder ρ = ρ 0 , T < α , terminating in the singularity at T = α . At H , we have</text>
<formula><location><page_8><loc_37><loc_39><loc_88><loc_41></location>R H = r 0 ( α -T ) 4 / 3 , R ρ H = ( α -T ) 2 3 , (36)</formula>
<text><location><page_8><loc_12><loc_36><loc_88><loc_38></location>so that, in particular, R and R ρ on H are positive for T < α , vanishing only at T = α . We may obtain R explicitly as</text>
<formula><location><page_8><loc_13><loc_32><loc_88><loc_35></location>R = r 0 2 ( α + ρ 0 -T -ρ ) 4 3 -3 10 ( α + ρ 0 -T -ρ ) 5 3 + r 0 2 ( α -ρ 0 -T + ρ ) 4 3 + 3 10 ( α -ρ 0 -T + ρ ) 5 3 (37)</formula>
<text><location><page_8><loc_12><loc_30><loc_27><loc_31></location>and we may also get</text>
<formula><location><page_8><loc_39><loc_27><loc_88><loc_30></location>ψ H = 2 3 ln ( α -T ) , ψ ρ H = 0 . (38)</formula>
<text><location><page_8><loc_12><loc_24><loc_88><loc_27></location>We now wish to calculate (28) at H and, in order to do that, we need to choose an adequate frame in the exterior. Since ψ = γ on H , then the holonomic frame B = { ∂ ˜ ϕ , ∂ ˜ z , ∂ ρ , ∂ T } is such that</text>
<formula><location><page_8><loc_31><loc_20><loc_88><loc_23></location>{ ∂ T , ∂ ρ , ∂ ˜ z , ∂ ˜ ϕ }| H = { e γ -ψ ∂ ∂T , e γ -ψ ∂ ∂ρ , glyph[vector] ξ 1 , glyph[vector] ξ 2 }∣ ∣ ∣ ∣ H , (39)</formula>
<text><location><page_8><loc_12><loc_15><loc_88><loc_19></location>where the last frame has the properties of (26). Proposition 2 then implies that we can carry out our computations in the holonomic frame with the choice glyph[vector] u = exp( γ -ψ ) ∂ ∂T as the vector field representing the observer. In this case, a computation reveals that the Sparling 3-form at H is given by</text>
<formula><location><page_8><loc_19><loc_11><loc_88><loc_14></location>S + T | H = -2 dT ∧ d ˜ z ∧ d ˜ φ 3( α -T ) 1 3 , S + ρ | H = 4 r 0 dT ∧ d ˜ z ∧ d ˜ φ 9( α -T ) 2 3 , S + ˜ z | H = 0 , S + ˜ φ | H = 0 . (40)</formula>
<text><location><page_9><loc_12><loc_89><loc_88><loc_92></location>If we define now H + ⊂ H by the conditions ˜ z 0 ≤ ˜ z ≤ ˜ z 0 + h , T 0 ≤ T ≤ T 1 with ˜ z 0 , T 0 , T 1 and h constants, then one has</text>
<formula><location><page_9><loc_16><loc_85><loc_88><loc_88></location>∫ H + S + T = 2 πh ( ( α -T 1 ) 2 3 -( α -T 0 ) 2 3 ) , ∫ H + S + ρ = 8 πr 0 h 3 ( ( α -T 0 ) 1 3 -( α -T 1 ) 1 3 ) . (41)</formula>
<text><location><page_9><loc_12><loc_82><loc_88><loc_84></location>From this, we can compute the four real numbers which define the 4-momentum P ( glyph[vector] u , H + ) (see eq. (22)). Its components are</text>
<formula><location><page_9><loc_19><loc_78><loc_88><loc_80></location>P ( glyph[vector] u , H + ) = ( 2 πh ( ( α -T 1 ) 2 3 -( α -T 0 ) 2 3 ) , 8 πr 0 h 3 ( ( α -T 0 ) 1 3 -( α -T 1 ) 1 3 ) , 0 , 0 ) . (42)</formula>
<text><location><page_9><loc_12><loc_72><loc_88><loc_76></location>Also, we can compute the gravitational flux 1-form at the matching hypersurface according to definition 1 and check whether the gravitational radiation is incoming or outgoing with respect to the radial direction ∂/∂ρ . The result is</text>
<formula><location><page_9><loc_39><loc_69><loc_88><loc_72></location>j ( B, glyph[vector] u ) 〈 ∂ ∂ρ 〉 = 2 3 r 0 ( α -T ) 5 3 . (43)</formula>
<text><location><page_9><loc_12><loc_63><loc_88><loc_68></location>It is interesting to note that the components of P ( glyph[vector] u , H + ) have a constant sign, namely, P T < 0 and P ρ > 0. Comparing with the sign of (43), we deduce that the gravitational energy-momentum flux is always outgoing. Even though one can also observe from (40) that S + T | H , S + ρ | H diverge as T → α , the flux integrals remain finite. This last conclusion is in line with the conclusions of [20] using Weyl scalars.</text>
<text><location><page_9><loc_12><loc_60><loc_88><loc_62></location>As a consistency check, we can also carry out a similar computation for the interior, since the matching conditions entail H + = H -= H and P ( glyph[vector] u , H + ) = P ( glyph[vector] u , H -). To that end, we use the frame given by</text>
<formula><location><page_9><loc_41><loc_56><loc_88><loc_59></location>B = { ∂ t , 1 b ( t ) ∂ r , ∂ z , ∂ ϕ } , (44)</formula>
<text><location><page_9><loc_12><loc_53><loc_79><loc_55></location>because this is the frame which matches at H with the frame used for the exterior, namely</text>
<formula><location><page_9><loc_32><loc_49><loc_88><loc_52></location>B = { ∂ T , ∂ ρ , ∂ ˜ z , ∂ ˜ ϕ }| H = { ∂ t , 1 b ( t ) ∂ r , ∂ z , ∂ ϕ }∣ ∣ ∣ ∣ H . (45)</formula>
<text><location><page_9><loc_12><loc_47><loc_57><loc_48></location>Under this frame choice, the Sparling 3-form in the interior is</text>
<formula><location><page_9><loc_25><loc_37><loc_88><loc_46></location>S -t = -2 3( α -t ) 1 / 3 dt ∧ dz ∧ dφ + 4 r 3( α -t ) 2 / 3 θ 1 ∧ dz ∧ dφ, S -r = 4 r 9( α -t ) 2 / 3 dt ∧ dz ∧ dφ -4 3( α -t ) 1 / 3 θ 1 ∧ dz ∧ dφ, S -z = 4 r 3( α -t ) 2 / 3 dt ∧ θ 1 ∧ dφ , S -φ = 4 r 3( α -t ) 2 / 3 dt ∧ θ 1 ∧ dz , (46)</formula>
<text><location><page_9><loc_12><loc_34><loc_16><loc_35></location>where</text>
<formula><location><page_9><loc_44><loc_33><loc_88><loc_34></location>θ 1 = ( α -t ) 2 / 3 dr. (47)</formula>
<text><location><page_9><loc_12><loc_31><loc_56><loc_32></location>The gravitational flux 1-form in the interior is then given by</text>
<formula><location><page_9><loc_30><loc_27><loc_88><loc_29></location>j ( B, ∂ ∂t ) = ∗ g -( S -t ) = 1 3 r ( α -t ) 2 (2( α -t ) dr -4 rdt ) , (48)</formula>
<text><location><page_9><loc_12><loc_24><loc_26><loc_25></location>from which we get</text>
<formula><location><page_9><loc_35><loc_21><loc_88><loc_24></location>j ( B, ∂ ∂t )〈 1 ( α -t ) 2 3 ∂ ∂r 〉 = 2 3 r ( α -t ) 5 3 , (49)</formula>
<text><location><page_9><loc_12><loc_18><loc_88><loc_20></location>which agrees with (43) at the matching hypersurface. If we assume now that H -⊂ H is defined by the conditions z 0 ≤ z ≤ z 0 + h , T 0 ≤ t ≤ T 1 with z 0 , T 0 , T 1 and h constants, then one has</text>
<formula><location><page_9><loc_18><loc_10><loc_88><loc_16></location>∫ H -S -t = 2 πh ( ( α -T 1 ) 2 3 -( α -T 0 ) 2 3 ) , ∫ H -S -r = 8 πr 0 h 3 ( ( α -T 0 ) 1 3 -( α -T 1 ) 1 3 ) , ∫ H -S -z = 0 , ∫ H -S -φ = 0 . (50)</formula>
<text><location><page_10><loc_12><loc_89><loc_88><loc_92></location>Again, we compute the four real numbers which define the 4-momentum P ( glyph[vector] u , H -) (see eq. (22)) The result is</text>
<formula><location><page_10><loc_19><loc_86><loc_88><loc_88></location>P ( glyph[vector] u , H -) = ( 2 πh ( ( α -T 1 ) 2 3 -( α -T 0 ) 2 3 ) , 8 πr 0 h 3 ( ( α -T 0 ) 1 3 -( α -T 1 ) 1 3 ) , 0 , 0 ) . (51)</formula>
<text><location><page_10><loc_12><loc_82><loc_86><loc_84></location>Therefore, we can check explicitly that eqs. (42) and (51) are consistent with P ( glyph[vector] u , H -) = P ( glyph[vector] u , H + ). Since the interior is non-vacuum, we can also compute the Einstein 3-form with the result</text>
<formula><location><page_10><loc_33><loc_78><loc_88><loc_80></location>E t = -4 3 rdr ∧ dz ∧ dφ , E r = E z = E φ = 0 . (52)</formula>
<text><location><page_10><loc_12><loc_74><loc_88><loc_77></location>From (52), we easily deduce that dE t = 0 which implies that dS t = 0, so the matter energy-momentum current J ( ∂/∂t ) and j ( B,∂/∂t ) are independently conserved currents.</text>
<section_header_level_1><location><page_10><loc_12><loc_71><loc_30><loc_72></location>5.2.2 Bianchi I case</section_header_level_1>
<text><location><page_10><loc_12><loc_69><loc_52><loc_70></location>In this case, with equations (31)-(35), we may find [20]</text>
<formula><location><page_10><loc_19><loc_57><loc_82><loc_68></location>R = r 0 2 ( α + ρ 0 -T -ρ ) 1 3 ( β + ρ 0 -T -ρ ) -3 4 ( α + ρ 0 -T -ρ ) 2 3 ( β + ρ 0 -T -ρ ) + 9 20 ( α + ρ 0 -T -ρ ) 5 3 + r 0 2 ( α -ρ 0 -T + ρ ) 1 3 ( β -ρ 0 -T + ρ ) + 3 4 ( α -ρ 0 -T + ρ ) 2 3 ( β -ρ 0 -T + ρ ) -9 20 ( α -ρ 0 -T + ρ ) 5 3 ,</formula>
<text><location><page_10><loc_12><loc_55><loc_19><loc_56></location>as well as</text>
<formula><location><page_10><loc_38><loc_52><loc_88><loc_55></location>ψ H = γ H = ln ( β -T ) -1 3 ln ( α -T ) . (53)</formula>
<text><location><page_10><loc_12><loc_50><loc_34><loc_51></location>Using this information, we get</text>
<formula><location><page_10><loc_15><loc_46><loc_88><loc_49></location>S + T ∣ ∣ H = ( β -T ) -3( α -T ) 3( α -T ) 4 / 3 dT ∧ d ˜ z ∧ d ˜ φ , S + ρ ∣ ∣ H = 2 r 0 3( α -T ) -( β -T ) 9( α -T ) 5 / 3 dT ∧ d ˜ z ∧ d ˜ φ , (54)</formula>
<text><location><page_10><loc_12><loc_43><loc_88><loc_45></location>the remaining components being zero. We define H + ⊂ H in a fashion similar as in subsection 5.2.1 and perform the following computations</text>
<formula><location><page_10><loc_28><loc_38><loc_88><loc_41></location>∫ H + S + T = 2 πh [ β -T 1 ( α -T 1 ) 1 3 -β -T 0 ( α -T 0 ) 1 3 ] , (55)</formula>
<formula><location><page_10><loc_28><loc_35><loc_88><loc_38></location>∫ H + S + ρ = 2 3 πhr 0 [ 3( α -T 0 ) + ( β -T 0 ) ( α -T 0 ) 2 / 3 -3( α -T 1 ) + ( β -T 1 ) ( α -T 1 ) 2 / 3 ] . (56)</formula>
<text><location><page_10><loc_12><loc_33><loc_38><loc_34></location>Hence, we deduce that in this case:</text>
<formula><location><page_10><loc_13><loc_27><loc_88><loc_31></location>P ( glyph[vector] u , H + ) = 2 πh ( β -T 1 ( α -T 1 ) 1 3 -β -T 0 ( α -T 0 ) 1 3 , r 0 3 [ 3( α -T 0 ) + ( β -T 0 ) ( α -T 0 ) 2 / 3 -3( α -T 1 ) + ( β -T 1 ) ( α -T 1 ) 2 / 3 ] , 0 , 0 ) . (57)</formula>
<text><location><page_10><loc_12><loc_26><loc_65><loc_27></location>Using the gravitational flux 1-form at the matching hypersurface, we get</text>
<formula><location><page_10><loc_30><loc_22><loc_88><loc_25></location>j ( B, glyph[vector] u ) 〈 ∂ ∂ρ 〉 = ∗ g + ( S + T ) 〈 ∂ ∂ρ 〉 = ( β -T ) -3( α -T ) 3 r 0 ( α -T ) 5 3 ( T -β ) . (58)</formula>
<text><location><page_10><loc_12><loc_16><loc_88><loc_20></location>One can carry out a similar computation from the interior, as we did in subsection 5.2.1, and use again the matching conditions to find that P ( glyph[vector] u , H + ) = P ( glyph[vector] u , H -). Also, the case α = β reduces to the FLRW case of the previous section.</text>
<text><location><page_10><loc_12><loc_11><loc_88><loc_16></location>Differently from the FLRW case, here one has to distinguish two cases: (i) 0 < α < β , where the singularity is string-like along the axis of symmetry and S T , S ρ diverge as T → α and (ii) 0 < β < α , where the spacetime singularity is pancake-like and S T , S ρ are finite when T → β . In the first case, the covector P ( glyph[vector] u , H + ) has components such that P T > 0 and P ρ < 0, whereas in the second case P T < 0 and</text>
<text><location><page_11><loc_12><loc_84><loc_88><loc_92></location>P ρ < 0. Eq. (58) tells us that the gravitational flux with respect to the radial direction is incoming in the first case and outgoing in the second case. In the first case, P T is ever increasing as the gravitational collapse evolves towards the singularity and, therefore, P T represents a net gain of gravitational energy whereas, in the second case P T , measures a net loss of gravitational energy. From this, we deduce that, in the first case, the gravitational collapse is enhanced by incoming gravitational radiation, whereas in the second case gravitational radiation is outgoing as the gravitational collapse progresses.</text>
<text><location><page_11><loc_12><loc_79><loc_88><loc_83></location>For the Bianchi I interior, a computation shows that (52) is also valid. Therefore, dE t = 0, which implies that dS t = 0, so again the matter energy-momentum current J ( ∂/∂t ) and j ( B,∂/∂t ) are independently conserved currents.</text>
<section_header_level_1><location><page_11><loc_12><loc_76><loc_50><loc_77></location>5.3 Spatially inhomogeneous interior</section_header_level_1>
<text><location><page_11><loc_12><loc_72><loc_88><loc_75></location>The following metric corresponds to an inhomogeneous solution of the Einstein equations of the Szekeres family with a dust source and a regular axis [17]:</text>
<formula><location><page_11><loc_27><loc_68><loc_88><loc_71></location>g -= -dt ⊗ dt + dr ⊗ dr + ( 1 -t 2 + r 2 α 2 ) 2 dz ⊗ dz + r 2 dϕ ⊗ dϕ, (59)</formula>
<text><location><page_11><loc_12><loc_63><loc_88><loc_67></location>where α ∈ R \ { 0 } . As the previous two cases, one can also show that this metric is an interior to Einstein-Rosen type metrics [2] and it is, thus, an interesting case to study by comparison with the spatially homogeneous cases.</text>
<text><location><page_11><loc_15><loc_61><loc_64><loc_62></location>In this case, the equality of the first fundamental forms on H gives</text>
<formula><location><page_11><loc_24><loc_57><loc_88><loc_60></location>γ H = ψ, R H = r 0 ( 1 -t 2 + r 2 0 α 2 ) , ψ H = ln ( 1 -t 2 + r 2 0 α 2 ) , γ ,T H = ψ ,T , (60)</formula>
<text><location><page_11><loc_12><loc_55><loc_54><loc_56></location>while the equality of the second fundamental forms gives</text>
<formula><location><page_11><loc_26><loc_50><loc_88><loc_54></location>γ ,ρ H = ψ ,ρ , ψ ,ρ H = -2 r 0 α 2 ( 1 -t 2 + r 2 0 α 2 ) -1 , R ,ρ H = 1 -t 2 +3 r 2 0 α 2 . (61)</formula>
<text><location><page_11><loc_12><loc_48><loc_63><loc_49></location>The solution to (77), which satisfies the matching conditions at H , is</text>
<formula><location><page_11><loc_18><loc_41><loc_88><loc_47></location>R ( T, ρ ) = r 0 ( 1 -r 2 0 α 2 ) + ( 1 -3 r 2 0 α 2 ) ( ρ -ρ 0 ) -1 6 α 2 ( ( T + ρ -ρ 0 ) 3 -( T -ρ + ρ 0 ) 3 ) -r 0 2 α 2 ( ( T -ρ + ρ 0 ) 2 +( T + ρ -ρ 0 ) 2 ) . (62)</formula>
<text><location><page_11><loc_12><loc_38><loc_55><loc_40></location>From the matching conditions ψ H = γ , we deduce that on H</text>
<formula><location><page_11><loc_31><loc_34><loc_88><loc_37></location>{ ∂ ρ , ∂ ˜ ϕ , ∂ ˜ z , ∂ T }| H = { e γ -ψ ∂ ∂ρ , glyph[vector] ξ 1 , glyph[vector] ξ 2 , e γ -ψ ∂ ∂T }∣ ∣ ∣ ∣ H , (63)</formula>
<text><location><page_11><loc_12><loc_29><loc_88><loc_33></location>where the last frame has the properties of (26). As we did in subsections 5.2.1 and 5.2.2, we appeal to Proposition 2 to carry out our computations in the holonomic frame { ∂ T , ∂ ρ , ∂ ˜ ϕ , ∂ ˜ z } , in order to compute (5) on H . In this case, we actually carry out the computation in all of the exterior with the result:</text>
<formula><location><page_11><loc_24><loc_25><loc_88><loc_27></location>S + T = -2 T α 2 dT ∧ d ˜ z ∧ d ˜ ϕ , S + ρ = -2 r 0 α 2 dT ∧ d ˜ z ∧ d ˜ ϕ , S + ˜ z = S + ˜ ϕ = 0 . (64)</formula>
<text><location><page_11><loc_12><loc_22><loc_71><loc_24></location>Next, we define H + ⊂ H and compute P ( glyph[vector] u , H + ) as we did before. The result is</text>
<formula><location><page_11><loc_30><loc_18><loc_88><loc_21></location>P ( glyph[vector] u , H + ) = ( 4 πh ( T 2 0 -T 2 1 ) α 2 , 4 πhr 0 α 2 ( T 0 -T 1 ) , 0 , 0 ) . (65)</formula>
<text><location><page_11><loc_12><loc_16><loc_41><loc_17></location>At the matching hypersurface, we have</text>
<formula><location><page_11><loc_20><loc_12><loc_88><loc_15></location>j ( B, glyph[vector] u ) = ∗ g + ( S T ) = -2 T r 0 ( r 2 0 -α 2 + T 2 ) dρ , j ( B, glyph[vector] u ) 〈 ∂ ∂ρ 〉 = -2 T r 0 ( r 2 0 -α 2 + T 2 ) , (66)</formula>
<text><location><page_12><loc_12><loc_88><loc_88><loc_92></location>from which we deduce that the gravitational flux is outgoing through the matching hypersurface. This indicates that the gravitational collapse is not enhanced by the gravitational radiation at the matching hypersurface.</text>
<text><location><page_12><loc_12><loc_84><loc_88><loc_88></location>In this particular case, we also supply an analysis of the collapsing process in the interior. To that end, we start by repeating the computation in the interior and find out whether the result agrees with what we have just found. In this case, the matching conditions entail</text>
<formula><location><page_12><loc_36><loc_81><loc_88><loc_83></location>{ ∂ T , ∂ ρ , ∂ ˜ z , ∂ ˜ ϕ }| H = { ∂ t , -∂ r , ∂ z , ∂ ϕ }| H . (67)</formula>
<text><location><page_12><loc_12><loc_78><loc_88><loc_80></location>Comparing with (63), we deduce that we can choose the frame { ∂ t , -∂ r , ∂ z , ∂ ϕ } to carry out the computations in the interior. The result of these computations is</text>
<formula><location><page_12><loc_25><loc_74><loc_88><loc_77></location>S -t = -2 t α 2 dt ∧ dz ∧ dϕ , S -r = -2 r 0 α 2 dt ∧ dz ∧ dϕ , S -z = S -ϕ = 0 . (68)</formula>
<text><location><page_12><loc_12><loc_73><loc_27><loc_74></location>So, this result yields</text>
<formula><location><page_12><loc_41><loc_71><loc_59><loc_73></location>P ( glyph[vector] u , H + ) = P ( glyph[vector] u , H -) ,</formula>
<text><location><page_12><loc_12><loc_69><loc_21><loc_70></location>as expected.</text>
<text><location><page_12><loc_15><loc_68><loc_62><loc_69></location>The gravitational flux 1-form in the interior is explicitly given by</text>
<formula><location><page_12><loc_35><loc_64><loc_88><loc_67></location>j -( B, glyph[vector] u ) = ∗ g -( S t ) = 2( rdt + tdr ) r ( -α 2 + r 2 + t 2 ) , (69)</formula>
<text><location><page_12><loc_12><loc_55><loc_88><loc_63></location>To check whether the gravitational collapse is enhanced or not in the interior we choose to compute the gravitational radiation flux in the apparent horizon and, to that end, we analyse the trapped surface formation in the interior. In particular, we look at null 2-surfaces generated by the null vectors glyph[vector] k ( ± ) = √ 2 2 ( ∂ t ± ∂ r ). We then take the vectors glyph[vector]e 1 = ∂ φ and glyph[vector]e 2 = ∂ z , generators of the 2-cylinders and calculate the expansions θ ( ± ) AB = -k ( ± ) a e b A ∇ b e a B , where A,B = 1 , 2, whose trace is denoted by θ ( ± ) . The condition θ ( ± ) = 0, for the existence of a marginally trapped cylinder , is equivalent to</text>
<formula><location><page_12><loc_41><loc_52><loc_88><loc_54></location>T : t 2 ± 2 tr +3 r 2 = α 2 , (70)</formula>
<text><location><page_12><loc_12><loc_48><loc_88><loc_52></location>for t 2 + r 2 < α 2 . This means that, for a given α , there exists a positive t ( ± ) 0 = ∓ r + √ α 2 -2 r 2 , for r < α/ √ 2 , such that θ ( ± ) = 0 and cylinders to the future of t (+) 0 are trapped.</text>
<text><location><page_12><loc_12><loc_40><loc_88><loc_48></location>To carry out a more detailed analysis, we pick the branch with a positive sign and introduce a new variable x through the definition x ≡ t + r . In terms of this new variable, condition (70) adopts the form x 2 +2 r 2 = α 2 , from which we conclude that in the ( x, r ) plane, condition (70) represents arcs of ellipses as shown in figure 1. We can now eliminate the variable t from (68). Next, using the map x = √ α 2 -2 r 2 , we compute the pullback of the Sparling 3-form to the marginally trapped cylinders which yields (here and in the following, we choose the positive sign in (70)):</text>
<formula><location><page_12><loc_20><loc_34><loc_88><loc_40></location>S -t = 2 ( α 2 -4 r 2 ) α 2 √ α 2 -2 r 2 dr ∧ dz ∧ dϕ , S -r = 2 ( α 2 -2 r ( √ α 2 -2 r 2 +2 r )) α 2 √ α 2 -2 r 2 dr ∧ dz ∧ dϕ , (71) S -z = S -ϕ = 0 . (72)</formula>
<text><location><page_12><loc_12><loc_30><loc_88><loc_33></location>From these expressions, we can compute the gravitational flux 1-form on the marginally trapped cylinders using the standard definition, getting</text>
<formula><location><page_12><loc_31><loc_27><loc_88><loc_29></location>j -( B, glyph[vector] u ) = dr r 2 (2 r 2 -α 2 ) ( α 2 -2 r (2 r + √ α 2 -2 r 2 ) ) . (73)</formula>
<text><location><page_12><loc_12><loc_23><loc_88><loc_26></location>We can also compute the gravitational energy-momentum flux through these cylinders and we conclude that</text>
<formula><location><page_12><loc_26><loc_20><loc_88><loc_23></location>∫ T S -t = 4 πhr 0 √ α 2 -2 r 2 0 α 2 , ∫ T S -r = 4 πhr 0 ( √ α 2 -2 r 2 0 -r 0 ) α 2 . (74)</formula>
<text><location><page_12><loc_12><loc_15><loc_88><loc_20></location>Since r 0 < α/ √ 2 we deduce, from the first integral, that the observer will measure a net positive emitted gravitational energy. However, looking at equation (73), we deduce that the gravitational flux with respect to ∂/∂r is incoming for 0 < r < r ∗ and outgoing if r ∗ < r < α/ √ 3, where r ∗ is given by</text>
<formula><location><page_12><loc_44><loc_11><loc_57><loc_14></location>r ∗ ≡ α 2 √ 1 -1 √ 3 .</formula>
<figure>
<location><page_13><loc_30><loc_69><loc_75><loc_87></location>
<caption>Figure 1: The left panel shows a diagram of the spacetime structure of (59), as obtained in [17]. The vertical lines represent the geodesics along which the dust moves between singularities. The blue dashed lines represent the marginally trapped cylinders which are arcs of elipses. On the right panel, a cut of the spacetime is depicted for r < α/ √ 2. This figure is adapted from [2].</caption>
</figure>
<text><location><page_13><loc_12><loc_48><loc_88><loc_55></location>Therefore, when the singularity is approached the radiation is incoming but when one gets towards the matching hypersurface the radiation is outgoing, consistently with the analysis carried out in the exterior. This means that the gravitational collapse will be enhanced by the gravitational radiation when the singularity is approached ( r < r ∗ ) but not when r > r ∗ (in particular, not in the matching hypersurface between the interior and the exterior).</text>
<text><location><page_13><loc_15><loc_46><loc_64><loc_47></location>To finish, we compute the Einstein 3-form in the interior obtaining</text>
<formula><location><page_13><loc_33><loc_42><loc_88><loc_45></location>E t = -4 α 2 rdr ∧ dz ∧ dφ , E r = E z = E φ = 0 . (75)</formula>
<text><location><page_13><loc_12><loc_39><loc_88><loc_41></location>From this result, we deduce that dE t = 0 which entails dS t = 0, so in this case the matter energymomentum current J -( ∂/∂t ) and j -( B,∂/∂t ) are also independently conserved currents in the interior.</text>
<section_header_level_1><location><page_13><loc_12><loc_35><loc_30><loc_36></location>6 Conclusions</section_header_level_1>
<text><location><page_13><loc_12><loc_30><loc_88><loc_33></location>We have considered a quasi-local measure of gravitational energy, using the Sparling 3-form and a geometric construction adapted to spacetimes with a 2-dimensional isometry group. We have then studied the gravitational energy-momentum flux in models of gravitational collapse with cylindrical symmetry.</text>
<text><location><page_13><loc_12><loc_18><loc_88><loc_29></location>Taking advantage of the existence of the Killing vectors, we defined a frame adapted to our problem to compute the gravitational radiation at the boundary and at the interior of the collapsing body. The interiors we have analysed contain a dust fluid and are FLRW, Bianchi I and Szekeres solutions, whereas the exterior is always a vacuum Einstein-Rosen type solution containing gravitational waves. Our method shows that in the collapse modelled with the FLRW and inhomogeneous interiors, the gravitational radiation is always outgoing at the matching boundary of the collapsing body whereas, in the case of the Bianchi I , the gravitational radiation can be either outgoing or incoming at the boundary during the collapse.</text>
<text><location><page_13><loc_12><loc_11><loc_88><loc_18></location>We find that, in a model whose collapsing interior is a Bianchi I spatially homogeneous spacetime with a string like singularity, the collapse is being enhanced by the gravitational radiation coming from the exterior. In the other cases analysed, the gravitational radiation is outgoing from the matching hypersurface during the collapse process. Note that these considerations pertain only to the analysis at the matching hypersurface.</text>
<text><location><page_14><loc_12><loc_85><loc_88><loc_92></location>In the case of the inhomogeneous interior, we also carried out the analysis at the apparent horizon and found that, there, the gravitational collapse is enhanced towards its final phase, whereas it is not so at earlier stages. In this case, since the collapse is not enhanced at the matching hypersurface, we conclude that a gravitational energy-momentum flux is originated in the interior enhancing the collapse during its late evolution.</text>
<text><location><page_14><loc_12><loc_81><loc_88><loc_85></location>Our results show how a quasi-local measure of gravitational energy can be constructed in a geometrical way using the Sparling form, and can be applied to the problem of gravitational collapse of a fluid body having an exterior with gravitational waves.</text>
<section_header_level_1><location><page_14><loc_12><loc_77><loc_33><loc_78></location>Acknowledgments</section_header_level_1>
<text><location><page_14><loc_12><loc_66><loc_88><loc_75></location>We thank an anonymous referee for constructive criticisms on an earlier version of this paper. We thank FCT Projects Est-OE/MAT/UI0013/2014, PTDC/MAT-ANA/1275/2014 and CMAT, Univ. Minho, through FEDER Funds COMPETE. We also thank the Erwin Schrodinger International Institute for Mathematical Physics, ESI, where part of this work has been done. FCM thanks FCT for grant SFRH/BSAB/130242/2017. AGP thanks the financial support from Grant 14-37086G of the Czech Science Foundation and the partial support from the projects IT956-16 ('Eusko Jaurlaritza', Spain), FIS2014-57956-P ('Ministerio de Econom'ıa y Competitividad', Spain).</text>
<section_header_level_1><location><page_14><loc_12><loc_62><loc_28><loc_63></location>7 Appendix</section_header_level_1>
<text><location><page_14><loc_12><loc_59><loc_60><loc_60></location>The EFEs for the cylindrically symmetric vacuum metric (31) are</text>
<formula><location><page_14><loc_36><loc_57><loc_88><loc_58></location>0 = γ ,TT -γ ,ρρ -ψ 2 ,ρ + ψ 2 ,T (76)</formula>
<formula><location><page_14><loc_36><loc_55><loc_88><loc_56></location>0 = R ,TT -R ,ρρ (77)</formula>
<formula><location><page_14><loc_36><loc_52><loc_88><loc_54></location>0 = ψ ,TT + R ,T R ψ ,T -ψ ,ρρ -R ,ρ R ψ ,ρ (78)</formula>
<text><location><page_14><loc_12><loc_50><loc_43><loc_51></location>together with the two constraint equations</text>
<formula><location><page_14><loc_22><loc_45><loc_88><loc_48></location>γ ,ρ = 1 R 2 ,ρ -R 2 ,T ( RR ,ρ ( ψ 2 ,T + ψ 2 ,ρ ) -2 RR ,T ψ ,T ψ ,ρ + R ,ρ R ,ρρ -R ,T R ,Tρ ) (79)</formula>
<formula><location><page_14><loc_22><loc_42><loc_88><loc_45></location>γ ,T = -1 R 2 ,T -R 2 ,ρ ( RR ,T ( ψ 2 ,T + ψ 2 ,ρ ) -2 RR ,ρ ψ ,T ψ ,ρ + R ,T R ,ρρ -R ,ρ R ,Tρ ) , (80)</formula>
<text><location><page_14><loc_12><loc_40><loc_42><loc_41></location>where the commas denote differentiation.</text>
<section_header_level_1><location><page_14><loc_12><loc_36><loc_25><loc_37></location>References</section_header_level_1>
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</unordered_list>
</document> | [{"title": "Gravitational radiation and the evolution of gravitational collapse in cylindrical symmetry", "content": "Alfonso Garc\u00b4\u0131a-Parrado glyph[sharp] \u2217 and Filipe C. Mena glyph[flat]glyph[natural] \u2020 glyph[sharp] Faculty of Mathematics and Physics, Charles University in Prague, V Hole\u02c7sovi\u02c7ck'ach 2, 180 00 Praha 8, Czech Republic. glyph[flat] Centro de Matem'atica, Universidade do Minho, 4710-057 Braga, Portugal glyph[natural] Dep. Matem'atica, Instituto Superior T'ecnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal November 27, 2018", "pages": [1]}, {"title": "Abstract", "content": "Using the Sparling form and a geometric construction adapted to spacetimes with a 2-dimensional isometry group, we analyse a quasi-local measure of gravitational energy. We then study the gravitational radiation through spacetime junctions in cylindrically symmetric models of gravitational collapse to singularities. The models result from the matching of collapsing dust fluids interiors with gravitational wave exteriors, given by the Einstein-Rosen type solutions. For a given choice of a frame adapted to the symmetry of the matching hypersurface, we are able to compute the total gravitational energy radiated during the collapse and state whether the gravitational radiation is incoming or outgoing, in each case. This also enables us to distinguish whether a gravitational collapse is being enhanced by the gravitational radiation. Keywords: Quasi-local energy; Gravitational waves; Sparling form", "pages": [1]}, {"title": "1 Introduction", "content": "The theorem about the positivity of the global gravitational field at spatial infinity is a cornerstone of General Relativity (GR) and there are already different proofs available of this important result. In practical applications, it would be desirable to have a quasi-local notion of energy which could be applied to finite objects. A great deal of effort has been put towards this goal [19], after the earlier proposals for the quasi-local mass by Hawking [12] and Penrose [16]. An interesting proposal was made by Brown and York [3] using a Hamiltonian formulation of General Relativity, and their definition depends on a gauge choice along three-dimensional spacelike hypersurfaces. Other authors have put forward definitions of mass which are based on more geometric constructions (see [22, 23] and references therein). Associated to this problem is the problem of finding an appropriate Hamiltonian in each setting. In fact, the Hamiltonian for every diffeormorphism invariant theory depends on boundary conditions and is, therefore, non-unique. In General Relativity, for example, both the Komar Hamiltonian and the ADM Hamiltonian have been widely used (see e.g. [4]). The latter, in turn, is a particular case of the Von Freud super-potential [10] and a problem related to the search of superpotentials for GR is the question of the existence of the Lanzcos potential for the Weyl tensor (see e.g. [6]). An alternative approach to the definition of quasi-local gravitational energy quantities is given by the use of gravitational pseudo-tensors . In [9], Frauendiener explains how to treat pseudo-tensors in the geometric framework of the principal bundle defined by the frame bundle of a 4-dimensional Lorentzian manifold. Using this approach, it is shown that the energy-momentum pseudo-tensors of Einstein and Landau-Lifschitz can be recovered from pull-backs to the spacetime manifold under appropriate sections of the so-called Sparling 3-form , which is defined in the bundle of linear frames. Szabados [18], generalizes these results and gives explicit formulae for gravitational pseudo-tensors in rigid basis or anholonomic frames. In particular, he shows that the pull-backs defined by coordinate sections of the contravariant and dual forms of the Sparling's form, defined on the bundle of linear frames over the spacetime, are the Bergman and the Landau-Lifshitz pseudo-tensors, respectively. In most of the applications we are aware of, the quasi-local quantities are evaluated for an isolated system at infinity, while in the present work we are interested in their local values at certain physically interesting hypersurfaces such as spacetime junctions. Even though quasi-local gravitational energy quantities are well-defined, they are frame dependent and, therefore, whenever they are used it is necessary to give a justification about the choice of a particular frame. In the case of an isolated system, it is assumed that it is asymptotically flat and this means that one can introduce a group of asymptotic symmetries which, roughly speaking, correspond to the symmetries of the Poincar'e group defined in flat spacetime or to a generalization called the BMS group . These asymptotic symmetries induce a set of privileged frames in the spacetime, in the sense that they correspond asymptotically to the generators of the asymptotic symmetry group and one can then take a frame of that set to compute a quasi-local gravitational energy quantity. Within this approach, it is possible to recover the standard ADM and Bondi notions of momentum for an isolated system using definitions of quasi-local quantities based on pseudo-tensors [11]. An approach based on the study of the asymptotic conditions as boundary conditions, defining when an isolated system is radiating, is also possible [21]. We would like to carry over the formalism described in the paragraph above to the case of a system composed by an (inhomogeneous) interior matched to a radiating exterior through a common hypersurface. The objective here is to give measures of the gravitational energy interchanged at the matching hypersurface, which plays now a similar role to the spacelike or null infinity of the case of an isolated system. In the case of matching hypersurfaces with enough symmetry, there is a natural geometrical way to define frames adapted to the symmetries of the hypersurface, and which will play the role of the frames corresponding to the asymptotic symmetries used to compute the quasi-local gravitational energy at infinity in an isolated system. Here, to formulate this problem in a mathematically precise way, we use a frame bundle formalism and define equivalence classes between frames. We then apply our formalism to spacetimes admitting a G 2 isometry group and, in particular, calculate the radiated gravitational energy through the transitivity hypersurfaces of such isometry groups. A novel aspect of our work is the introduction of the notion of gravitational collapse enhanced by the gravitational radiation , i.e. by the gravitational energy-momentum flux, defined using the Sparling form. The meaning of this notion is that during the collapse, gravitational radiation is produced and the gravitational energy can either be radiated outside of the collapsing body or be radiated inwards, in which case it enhances the gravitational collapse. Although our presentation is restricted to 4 dimensional spacetimes, most of it can be easily generalised to n dimensions and can be applied e.g. to the brane world scenario where 4-dimensional branes are embedded, as hypersurfaces, on a 5-dimensional bulk. Motivated by the recent detection of gravitational waves by LIGO [1], an interesting application of our formalism would be to models of objects with a finite radius in astrophysics, emitting gravitational radiation into a vacuum exterior. Due to Birkhoff's theorem, it is not possible to have such vacuum exteriors in spherical symmetry. The next simplest symmetry assumption is cylindrical symmetry. So, we will consider models of gravitational collapse in cylindrical symmetry. Such models result from gluing an interior metric representing matter collapsing gravitationally to an exterior metric containing gravitational waves. An interesting aspect of these models is that the matter density is non-zero at the matching boundary, resulting in discontinuities in the Einstein field equations. Even though a realistic system undergoing gravitational collapse is not cylindrically symmetric, some non-trivial effects already present in the real system can be found in the cylindrically symmetric model. For example, the degrees of freedom of the (pure) gravitational energy in the exterior may mix with the gravitational energy sourced by the matter at the boundary. These non-trivial effects may be captured by local, gauge-dependent, measures of gravitational energy. The plan of the paper is as follows: in section 2, we revise some properties of the Sparling 3-form and we compute its transformation under a change of frame. In section 3, we review how to measure the gravitational energy radiated through embedded hypersurfaces of codimension 1 using the Sparling form. In section 4, we provide a geometric set-up for the definition of a frame based on the existence of a submanifold representing the surface of a radiative body. In section 5, we apply our formalism to cylindrically symmetric collapsing dust spacetimes containing gravitational radiation in the exterior which corresponds to the Einstein-Rosen solution. Some algebraic computations of this paper have been done with xAct [15]. We use Latin indices a, b, c, .. = 1 , 2 , 3 , 4, Greek indices \u03b1, \u03b2, .. = 1 , 2 , 3, capitals A,B,.. = 1 , 2 and units such that 8 \u03c0G = c = 1.", "pages": [1, 2]}, {"title": "2 Sparling 3-form and Sparling identity", "content": "In this section and in the following section 3, we review some known notions and properties about the Sparling form and the sparling identity that are needed for our work. Even though most of the material is known, we present the equations in accordance to the notation that shall be followed in sections 4 and 5. Let ( M , g ab ) be a 4-dimensional spacetime with signature ( -, + , + , +) and consider the bundle of frames L ( M ) over M . Let B \u2261 { glyph[vector] e 1 , glyph[vector] e 2 , glyph[vector] e 3 , glyph[vector] e 4 } be a chosen frame (smooth section of L ( M )) and denote by B \u2217 \u2261 { \u03b8 1 , \u03b8 2 , \u03b8 3 , \u03b8 4 } its dual co-basis. Define the super-potential where \u0393( B ) b c is the connection 1-form on the frame B and \u03b7 abcd are the components of the volume element of the metric g ab in the frame B . Note the dependency of the super-potential on both the frame B and the frame elements glyph[vector] e a . 1 Under the choice of the frame B , the super-potential is a 2-form in \u039b 2 ( M ) and it fulfills the Sparling identity [9, 18, 5] where the quantity E ( glyph[vector] e a ) \u2208 \u039b 3 ( M ) is the so-called Einstein 3-form , which is given in terms of the curvature 2-form R ab by and it has the property with \u2217 representing the Hodge dual and G ab the Einstein tensor. As E ( glyph[vector] e a ) and \u2217 E ( glyph[vector] e a ) are tensorial differential forms, we do not write their dependency on B , such dependency being only kept for pseudo-tensorial forms . This is for instance the case of the 3-form S ( B,glyph[vector] e a ) that it is given by We will follow the convention of [9, 18] and call this form the Sparling 3-form . For us, the important point about the Sparling identity is that it can be interpreted as a conservation law in the following way: By defining the 1-form and computing the co-differential of J ( B,glyph[vector] e a ), we get From this equation, we deduce that J ( B,glyph[vector] e a ) always yields a conserved current, independently of the choice of the frame B or the frame vectors glyph[vector] e a . According to the above considerations, the current J ( B,glyph[vector] e a ) can be split into two parts as The first part is a tensorial 1-form and, via the Einstein equations, can be interpreted as the flux of energy momentum due to the matter. The second part is non-tensorial and can be regarded as the flux of gravitational energy-momentum with respect to the frame B . If glyph[vector] u is a frame element of B representing an observer, then the 1-form shall be called the gravitational energy-momentum flux 1-form for the observer glyph[vector] u with respect to the frame B or just the gravitational flux 1-form, if no confusion arises. It can be interpreted as the gravitational energy-momentum flux, which in this context is also called gravitational radiation, measured by the given observer in the frame B . In our context, we need to be able to decide if gravitational radiation is leaving or entering a radiating source. To that end, we put forward the following definition.", "pages": [3, 4]}, {"title": "Definition 1. We shall take the scalar quantity", "content": "as the component of the gravitational radiation flux along the direction defined by the vector field ( glyph[vector] e a ) . If this scalar quantity is positive (resp. negative), then we say that the gravitational radiation is outgoing (resp. incoming) with respect to ( glyph[vector] e a ) . The fact that j ( B, glyph[vector] u ) is frame and observer dependent does not render it useless when computing the radiated gravitational energy in certain cases, as we illustrate through explicit examples in section 5. In fact, in [9], it was proved that the scalar quantity (10) has a correspondence with the components of the energy-momentum complex given by Einstein in 1916 [7]. Furthermore, it has been shown that, under an appropriate choice of frame B , it is possible to use this complex to recover the well-known notions of ADM and Bondi momentum at infinity for an isolated system (see [11]). Thus, following a similar approach, we will compute, in section 5, the gravitational energy radiated through a matching hypersurface separating two regions (exterior and interior) under the assumption that the hypersurface has certain symmetry properties. We will present explicit examples of systems undergoing gravitational collapse and show how the gravitational flux 1-form enables us to give a criterion stating whether gravitational radiation is being emitted or absorbed during the collapse. It is obvious that a similar reasoning as the above can be performed if we take the Einstein form instead of the Sparling form. In this fashion, one can define the matter current with respect to the observer glyph[vector] u in the standard way This current is tensorial and, as it is well-known, it represents the matter energy-momentum flux. From previous considerations, it is clear that the sum of both currents is always conserved, a property that can be expressed by the condition Interestingly, even though j ( B, glyph[vector] u ) is frame dependent, the conservation property (12) holds for any frame B . A particular case occurs when each of the currents are independently conserved. As we will show, this is the situation for all the examples of section 5 modelling gravitational collapse. The implication of this is that, during the gravitational collapse, no fraction of the collapsing matter is transformed into gravitational radiation.", "pages": [4]}, {"title": "3 Frame equivalence and radiated gravitational energy through an embedded hypersurface", "content": "Suppose now that we introduce another frame B ' and its associate co-frame B '\u2217 , related to B and B \u2217 through the formulae where the matrices \u03b3 ( B,B ' ), \u03b3 ( B ' , B ) encode the transition between the frames B and B ' . We regard these matrices as having entries which are scalars on M (0-forms). Under these circumstances, it is well-known that the connection 1-form transforms as We can use this expression to find the transformation law for the Sparling 3-form. A straightforward but tedious computation reveals: Lemma 1. Under the change of frames B to B ' , the Sparling 3-form transforms as: From the expression above, it is clear that the relation between S ( B ' , glyph[vector] e ' s ) and S ( B,glyph[vector] e a ) is linear if d\u03b3 ( B,B ' ) s a = 0 and, in that case, S transforms like a tensor. We shall explore this fact next. Let H \u2282 M be a co-dimension 1 smooth embedded hypersurface and consider two frames B , B ' fulfilling the condition where \u03c6 : H \u2192 M is a smooth embedding. Since one has that we deduce that the scalars \u03b3 b a ( B ' , B ) | H can only be constants. Proposition 1. For H \u2282 M and \u03c6 : H \u2192 M as described in the previous paragraph, the relation is an equivalence relation in X ( L ( M )) ( \u2261 the set of smooth sections in the frame bundle L ( M ) ). Proof. To ease the notation, we suppress indices in the matrix \u03b3 b a ( B ' , B ) and write it simply as \u03b3 ( B ' , B ). We show that the properties characterizing an equivalence relation are fulfilled: Since \u03c6 \u2217 ( \u03b3 ( B ' , B )) is invertible, the last equation entails d ( \u03c6 \u2217 ( \u03b3 ( B,B ' )) = 0 and thus B ' \u223c B . '' Proposition 2. If there are two frames B , B ' on M such that B | H = B ' | H , then B \u223c B ' . Proof. If B | H = B ' | H , then eq. (17) holds with \u03b3 b a ( B ' , B ) | H = \u03b4 a b and therefore B \u223c B ' . Now, let B , B ' be frames such that B \u223c B ' . Then Using this information in (15), we deduce where we used the fact that \u03b3 ( B ' , B ) b a | H are constants as B \u223c B ' . Bearing in mind this property, let us make the following definition Definition 2. Let H \u2282 M be an embedded co-dimension 1 smooth hypersurface. For a given frame B and frame element glyph[vector] e a \u2208 B , the gravitational flux through H , with respect to B and glyph[vector] e a is defined as If B and B ' are frames such that B \u223c B ' , then eq. (21) can be rendered in the form This means that if we have a privileged frame B in our problem, we can compute the gravitational flux according to (22) and consider the scalars { P ( B, H , glyph[vector] e a ) } as the components of the gravitational 4-momentum of the hypersurface H . We shall represent it as P ( B, H ). An important particular case occurs in vacuum, when S ( B,glyph[vector] e a ) = dL ( B,glyph[vector] e a ). In this case, using Stokes theorem, we can write (22) as follows In an asymptotically flat isolated system we can choose H in such a way that one of the connected components of \u2202 H surrounds the source. Denoting by \u03a3 r such a connected component, we can study the existence of the limit for a hypersurface \u03a3 r that approaches infinity . This analysis has been already performed in a number of situations (see [11]): In any case, we note that the choice of a frame B with suitable properties at infinity, yields integral values of quasi-local quantities that have a valid physical meaning. In the next section, we shall follow this approach and show how the choice of a frame B adapted to the symmetries of our physical system, at the matching hypersurface, enables us to analyse the problem of the gravitational radiation through the matching boundary. The most important point to stress here is that the matching hypersurface, in our case, plays a role similar to null infinity or spacelike infinity in the case of an isolated system. As discused above, the choice of a frame B adapted to the symmetries of null or spacelike infinity (the asymptotic symmetries ) provides, respectively, the Bondi or ADM momentum. Therefore, a frame B adapted to the symmetries of the matching hypersurface, should give us the quantities corresponding to the flux of gravitational energymomentum through the matching hypersurface. The detailed analysis of this assertion is presented in the next section.", "pages": [4, 5, 6]}, {"title": "4 Gravitational flux through a hypersurface in a spacetime with a G 2 isometry group", "content": "Assume that M admits a maximal isometry group G 2 generated by the Killing vectors glyph[vector] \u03be 1 , glyph[vector] \u03be 2 . Let H \u2282 M be a co-dimension 1 hypersurface. Choose a frame of the form where glyph[vector] n is a vector field on M such that glyph[vector] n | H is unit and normal to the hypersurface and glyph[vector] u is an unit timelike vector field representing an observer. This frame is fully adapted to the geometry of the matching hypersurface H because its elements are either fixed at H (as happens for an observer glyph[vector] u and the unit normal glyph[vector] n ) or are symmetries of the matching hypersurfaces (as happens for the Killing vectors glyph[vector] \u03be 1 , glyph[vector] \u03be 2 ). There is a residual freedom in the choice of the frame B , coming from the fact that the vector field glyph[vector] n is defined up to a sign on H and that are Killing vectors too. However, it is easy to get: Proposition 3. Any other frame B ' = { glyph[vector] n , glyph[vector] \u03be ' 1 , glyph[vector] \u03be ' 2 , glyph[vector] u } with glyph[vector] \u03be ' 1 , glyph[vector] \u03be ' 2 related to glyph[vector] \u03be 1 , glyph[vector] \u03be 2 by (27) fulfills the property B ' \u223c B . From the previous proposition we deduce that, in a spacetime admitting a G 2 isometry group, one can define unambiguously the gravitational 4-momentum of a hypersurface H as P ( B, H ), where B is a frame of the form (26). Since, in this case, the only freedom left to construct the frame B is given by the choice of the observer glyph[vector] u , we shall use the shorthand notation Next, we are going to compute P ( glyph[vector] u , H ) in a number of practical examples.", "pages": [6, 7]}, {"title": "5 Gravitational flux in cylindrically symmetric spacetimes", "content": "In this section, we consider the problem of computing the gravitational flux through hypersurfaces in cylindrically symmetric spacetimes. The hypersurfaces, here, result from the matching of an exterior vacuum spacetime containing gravitational waves and a dust fluid interior spacetime which is collapsing to form a singularity. To simplify the presentation, we consider exact solutions to the Einstein equations with diagonal metrics only. We start by briefly recalling the matching conditions between two spacetimes ( M \u00b1 , g \u00b1 ) across timelike hypersurfaces H \u00b1 (see more details in [14]). The matching between two spacetimes requires an identification of their boundaries, i.e. a pair of embeddings \u03a6 \u00b1 : H -\u2192 M \u00b1 with \u03a6 \u00b1 ( H ) = H \u00b1 , where H is an abstract copy of one of the boundaries. Let \u03be \u03b1 be a coordinate system on H , where \u03b1, \u03b2 = 1 , 2 , 3 are indices on the hypersurface. Given a vector basis { \u2202/\u2202\u03be \u03b1 } of the tangent space T H , the push-forwards d \u03a6 \u00b1 provide a correspondence between the vectors of this basis and sets of linearly independent vectors tangent to H \u00b1 , given in appropriate coordinates by e \u00b1 a \u03b1 = \u2202 \u03be \u03b1 \u03a6 \u00b1 a . Here, we follow the convention of previous sections that small Latin letters a, b = 1 , 2 , 3 , 4 represent spacetime indices. There are also (up to orientation) vectors n a \u00b1 normal to the boundaries. The first and second fundamental forms on H are given by q \u00b1 = \u03a6 \u00b1 glyph[star] ( g \u00b1 ) and K \u00b1 = \u03a6 \u00b1 glyph[star] ( \u2207 \u00b1 n \u00b1 ), where \u03a6 \u00b1 glyph[star] denotes the pull-back corresponding to the maps \u03a6 \u00b1 . In components, we may write The matching conditions between two spacetimes through H require the equality of the first and second fundamental form on H , i.e. In the examples below, the spacetimes are matched across cylinders of symmetry.", "pages": [7]}, {"title": "5.1 Gravitational wave exteriors", "content": "The most general diagonal metric for cylindrically symmetric vacuum spacetimes (with an Abelian G 2 acting on 2-dimensional spacelike hypersurfaces S 2 ) can be written as [13] where \u03c8, \u03b3, R are functions of the coordinates \u03c1, T satisfying the Einstein equations (76)-(80). This metric is written in coordinates adapted to the Killing vectors \u2202 \u02dc \u03d5 and \u2202 \u02dc z . From the physical point of view, the metric models cylindrical gravitational waves with one polarization state and, for the choice R ( T, \u03c1 ) = \u03c1 , corresponds to the metric found by Einstein and Rosen [8]. So, we say that (31) are metrics of Einstein-Rosen type . Dust matter sources to such spacetimes were studied in [20, 2], where the matching between collapsing interior spacetimes and Einstein-Rosen type of exteriors was shown to exist along timelike hypersurfaces. It would be therefore interesting to study the gravitational radiation through those hypersurfaces and to find out whether it enhances the gravitational collapse or not. It has been shown [20] that, in those cases, the matching hypersurfaces are ruled by geodesics and, from the point of view of the exterior, can be parametrized as where r 0 is constant and \u03bb is a parameter along the geodesics. So, H + is generated by glyph[vector] e + 1 = \u2202 t | H + , glyph[vector] e + 2 = \u2202 \u02dc \u03d5 | H + and glyph[vector] e + 3 = \u2202 \u02dc z | H + , which together with glyph[vector] n + = \u2202 r | H + form an orthogonal basis for the spacetime at H + .", "pages": [7]}, {"title": "5.2 Spatially homogeneous interiors", "content": "As interiors to (31), we first consider locally rotationally symmetric and spatially homogeneous dust solutions given by the Bianchi class I of metrics [13], in cylindrical coordinates, as with for 0 \u2264 t < min { \u03b1, \u03b2 } and r \u2264 r 0 . If \u03b1 = \u03b2 , the metric reduces to the Friedman-Lema\u02c6\u0131tre-RobertsonWalker (FLRW) class of spatially homogeneous and isotropic metrics. The interior metrics (33) always contain cylinders of symmetry which are trapped near the singularity (see [20]). The marginally-trapped cylinders trace out a 3-surface in the interior which eventually arrives at the boundary of the matter located at r = r 0 . From the point of view of the interior, the matching hypersurface is parametrized as so that H -is generated by glyph[vector] e -1 = \u2202 T | H -, glyph[vector] e -2 = \u2202 \u03d5 | H -and glyph[vector] e -3 = \u2202 z | H -, which together with glyph[vector] n -= \u2202 \u03c1 | H -form an orthogonal basis for the spacetime at H -. Then, the matching conditions (30) between (31) and (33) across H give where H = denotes evaluation at H . In order to solve the matching problem, the strategy followed in [20] was as follows: For a suitable spacetime interior to (31) and for the conditions (35) at the boundary, one obtains an explicit solution for R ( \u03c1, T ) which satisfies the wave equation (77) and the constraints (79) and (80), at H . In turn, the remaining Einstein equations (76) and (78) can be seen as providing \u03b3 ,\u03c1\u03c1 and \u03c8 ,\u03c1\u03c1 on H . Since we know data for the exterior metric and its normal derivatives at the boundary, it then follows [20] that a unique \u03c8 exists on a neighbourhood D of H . Since \u03b3 H = \u03c8 and \u03b3 ,\u03c1 H = \u03c8 ,\u03c1 , once we have \u03c8 , we use a similar argument in (76) to get a unique \u03b3 in D .", "pages": [8]}, {"title": "5.2.1 FLRW case", "content": "If the interior is given by a FLRW metric, then a ( t ) = b ( t ) = ( \u03b1 -t ) 2 / 3 , for t \u2208 ( -\u221e , \u03b1 ). From the exterior, the matching surface H is the cylinder \u03c1 = \u03c1 0 , T < \u03b1 , terminating in the singularity at T = \u03b1 . At H , we have so that, in particular, R and R \u03c1 on H are positive for T < \u03b1 , vanishing only at T = \u03b1 . We may obtain R explicitly as and we may also get We now wish to calculate (28) at H and, in order to do that, we need to choose an adequate frame in the exterior. Since \u03c8 = \u03b3 on H , then the holonomic frame B = { \u2202 \u02dc \u03d5 , \u2202 \u02dc z , \u2202 \u03c1 , \u2202 T } is such that where the last frame has the properties of (26). Proposition 2 then implies that we can carry out our computations in the holonomic frame with the choice glyph[vector] u = exp( \u03b3 -\u03c8 ) \u2202 \u2202T as the vector field representing the observer. In this case, a computation reveals that the Sparling 3-form at H is given by If we define now H + \u2282 H by the conditions \u02dc z 0 \u2264 \u02dc z \u2264 \u02dc z 0 + h , T 0 \u2264 T \u2264 T 1 with \u02dc z 0 , T 0 , T 1 and h constants, then one has From this, we can compute the four real numbers which define the 4-momentum P ( glyph[vector] u , H + ) (see eq. (22)). Its components are Also, we can compute the gravitational flux 1-form at the matching hypersurface according to definition 1 and check whether the gravitational radiation is incoming or outgoing with respect to the radial direction \u2202/\u2202\u03c1 . The result is It is interesting to note that the components of P ( glyph[vector] u , H + ) have a constant sign, namely, P T < 0 and P \u03c1 > 0. Comparing with the sign of (43), we deduce that the gravitational energy-momentum flux is always outgoing. Even though one can also observe from (40) that S + T | H , S + \u03c1 | H diverge as T \u2192 \u03b1 , the flux integrals remain finite. This last conclusion is in line with the conclusions of [20] using Weyl scalars. As a consistency check, we can also carry out a similar computation for the interior, since the matching conditions entail H + = H -= H and P ( glyph[vector] u , H + ) = P ( glyph[vector] u , H -). To that end, we use the frame given by because this is the frame which matches at H with the frame used for the exterior, namely Under this frame choice, the Sparling 3-form in the interior is where The gravitational flux 1-form in the interior is then given by from which we get which agrees with (43) at the matching hypersurface. If we assume now that H -\u2282 H is defined by the conditions z 0 \u2264 z \u2264 z 0 + h , T 0 \u2264 t \u2264 T 1 with z 0 , T 0 , T 1 and h constants, then one has Again, we compute the four real numbers which define the 4-momentum P ( glyph[vector] u , H -) (see eq. (22)) The result is Therefore, we can check explicitly that eqs. (42) and (51) are consistent with P ( glyph[vector] u , H -) = P ( glyph[vector] u , H + ). Since the interior is non-vacuum, we can also compute the Einstein 3-form with the result From (52), we easily deduce that dE t = 0 which implies that dS t = 0, so the matter energy-momentum current J ( \u2202/\u2202t ) and j ( B,\u2202/\u2202t ) are independently conserved currents.", "pages": [8, 9, 10]}, {"title": "5.2.2 Bianchi I case", "content": "In this case, with equations (31)-(35), we may find [20] as well as Using this information, we get the remaining components being zero. We define H + \u2282 H in a fashion similar as in subsection 5.2.1 and perform the following computations Hence, we deduce that in this case: Using the gravitational flux 1-form at the matching hypersurface, we get One can carry out a similar computation from the interior, as we did in subsection 5.2.1, and use again the matching conditions to find that P ( glyph[vector] u , H + ) = P ( glyph[vector] u , H -). Also, the case \u03b1 = \u03b2 reduces to the FLRW case of the previous section. Differently from the FLRW case, here one has to distinguish two cases: (i) 0 < \u03b1 < \u03b2 , where the singularity is string-like along the axis of symmetry and S T , S \u03c1 diverge as T \u2192 \u03b1 and (ii) 0 < \u03b2 < \u03b1 , where the spacetime singularity is pancake-like and S T , S \u03c1 are finite when T \u2192 \u03b2 . In the first case, the covector P ( glyph[vector] u , H + ) has components such that P T > 0 and P \u03c1 < 0, whereas in the second case P T < 0 and P \u03c1 < 0. Eq. (58) tells us that the gravitational flux with respect to the radial direction is incoming in the first case and outgoing in the second case. In the first case, P T is ever increasing as the gravitational collapse evolves towards the singularity and, therefore, P T represents a net gain of gravitational energy whereas, in the second case P T , measures a net loss of gravitational energy. From this, we deduce that, in the first case, the gravitational collapse is enhanced by incoming gravitational radiation, whereas in the second case gravitational radiation is outgoing as the gravitational collapse progresses. For the Bianchi I interior, a computation shows that (52) is also valid. Therefore, dE t = 0, which implies that dS t = 0, so again the matter energy-momentum current J ( \u2202/\u2202t ) and j ( B,\u2202/\u2202t ) are independently conserved currents.", "pages": [10, 11]}, {"title": "5.3 Spatially inhomogeneous interior", "content": "The following metric corresponds to an inhomogeneous solution of the Einstein equations of the Szekeres family with a dust source and a regular axis [17]: where \u03b1 \u2208 R \\ { 0 } . As the previous two cases, one can also show that this metric is an interior to Einstein-Rosen type metrics [2] and it is, thus, an interesting case to study by comparison with the spatially homogeneous cases. In this case, the equality of the first fundamental forms on H gives while the equality of the second fundamental forms gives The solution to (77), which satisfies the matching conditions at H , is From the matching conditions \u03c8 H = \u03b3 , we deduce that on H where the last frame has the properties of (26). As we did in subsections 5.2.1 and 5.2.2, we appeal to Proposition 2 to carry out our computations in the holonomic frame { \u2202 T , \u2202 \u03c1 , \u2202 \u02dc \u03d5 , \u2202 \u02dc z } , in order to compute (5) on H . In this case, we actually carry out the computation in all of the exterior with the result: Next, we define H + \u2282 H and compute P ( glyph[vector] u , H + ) as we did before. The result is At the matching hypersurface, we have from which we deduce that the gravitational flux is outgoing through the matching hypersurface. This indicates that the gravitational collapse is not enhanced by the gravitational radiation at the matching hypersurface. In this particular case, we also supply an analysis of the collapsing process in the interior. To that end, we start by repeating the computation in the interior and find out whether the result agrees with what we have just found. In this case, the matching conditions entail Comparing with (63), we deduce that we can choose the frame { \u2202 t , -\u2202 r , \u2202 z , \u2202 \u03d5 } to carry out the computations in the interior. The result of these computations is So, this result yields as expected. The gravitational flux 1-form in the interior is explicitly given by To check whether the gravitational collapse is enhanced or not in the interior we choose to compute the gravitational radiation flux in the apparent horizon and, to that end, we analyse the trapped surface formation in the interior. In particular, we look at null 2-surfaces generated by the null vectors glyph[vector] k ( \u00b1 ) = \u221a 2 2 ( \u2202 t \u00b1 \u2202 r ). We then take the vectors glyph[vector]e 1 = \u2202 \u03c6 and glyph[vector]e 2 = \u2202 z , generators of the 2-cylinders and calculate the expansions \u03b8 ( \u00b1 ) AB = -k ( \u00b1 ) a e b A \u2207 b e a B , where A,B = 1 , 2, whose trace is denoted by \u03b8 ( \u00b1 ) . The condition \u03b8 ( \u00b1 ) = 0, for the existence of a marginally trapped cylinder , is equivalent to for t 2 + r 2 < \u03b1 2 . This means that, for a given \u03b1 , there exists a positive t ( \u00b1 ) 0 = \u2213 r + \u221a \u03b1 2 -2 r 2 , for r < \u03b1/ \u221a 2 , such that \u03b8 ( \u00b1 ) = 0 and cylinders to the future of t (+) 0 are trapped. To carry out a more detailed analysis, we pick the branch with a positive sign and introduce a new variable x through the definition x \u2261 t + r . In terms of this new variable, condition (70) adopts the form x 2 +2 r 2 = \u03b1 2 , from which we conclude that in the ( x, r ) plane, condition (70) represents arcs of ellipses as shown in figure 1. We can now eliminate the variable t from (68). Next, using the map x = \u221a \u03b1 2 -2 r 2 , we compute the pullback of the Sparling 3-form to the marginally trapped cylinders which yields (here and in the following, we choose the positive sign in (70)): From these expressions, we can compute the gravitational flux 1-form on the marginally trapped cylinders using the standard definition, getting We can also compute the gravitational energy-momentum flux through these cylinders and we conclude that Since r 0 < \u03b1/ \u221a 2 we deduce, from the first integral, that the observer will measure a net positive emitted gravitational energy. However, looking at equation (73), we deduce that the gravitational flux with respect to \u2202/\u2202r is incoming for 0 < r < r \u2217 and outgoing if r \u2217 < r < \u03b1/ \u221a 3, where r \u2217 is given by Therefore, when the singularity is approached the radiation is incoming but when one gets towards the matching hypersurface the radiation is outgoing, consistently with the analysis carried out in the exterior. This means that the gravitational collapse will be enhanced by the gravitational radiation when the singularity is approached ( r < r \u2217 ) but not when r > r \u2217 (in particular, not in the matching hypersurface between the interior and the exterior). To finish, we compute the Einstein 3-form in the interior obtaining From this result, we deduce that dE t = 0 which entails dS t = 0, so in this case the matter energymomentum current J -( \u2202/\u2202t ) and j -( B,\u2202/\u2202t ) are also independently conserved currents in the interior.", "pages": [11, 12, 13]}, {"title": "6 Conclusions", "content": "We have considered a quasi-local measure of gravitational energy, using the Sparling 3-form and a geometric construction adapted to spacetimes with a 2-dimensional isometry group. We have then studied the gravitational energy-momentum flux in models of gravitational collapse with cylindrical symmetry. Taking advantage of the existence of the Killing vectors, we defined a frame adapted to our problem to compute the gravitational radiation at the boundary and at the interior of the collapsing body. The interiors we have analysed contain a dust fluid and are FLRW, Bianchi I and Szekeres solutions, whereas the exterior is always a vacuum Einstein-Rosen type solution containing gravitational waves. Our method shows that in the collapse modelled with the FLRW and inhomogeneous interiors, the gravitational radiation is always outgoing at the matching boundary of the collapsing body whereas, in the case of the Bianchi I , the gravitational radiation can be either outgoing or incoming at the boundary during the collapse. We find that, in a model whose collapsing interior is a Bianchi I spatially homogeneous spacetime with a string like singularity, the collapse is being enhanced by the gravitational radiation coming from the exterior. In the other cases analysed, the gravitational radiation is outgoing from the matching hypersurface during the collapse process. Note that these considerations pertain only to the analysis at the matching hypersurface. In the case of the inhomogeneous interior, we also carried out the analysis at the apparent horizon and found that, there, the gravitational collapse is enhanced towards its final phase, whereas it is not so at earlier stages. In this case, since the collapse is not enhanced at the matching hypersurface, we conclude that a gravitational energy-momentum flux is originated in the interior enhancing the collapse during its late evolution. Our results show how a quasi-local measure of gravitational energy can be constructed in a geometrical way using the Sparling form, and can be applied to the problem of gravitational collapse of a fluid body having an exterior with gravitational waves.", "pages": [13, 14]}, {"title": "Acknowledgments", "content": "We thank an anonymous referee for constructive criticisms on an earlier version of this paper. We thank FCT Projects Est-OE/MAT/UI0013/2014, PTDC/MAT-ANA/1275/2014 and CMAT, Univ. Minho, through FEDER Funds COMPETE. We also thank the Erwin Schrodinger International Institute for Mathematical Physics, ESI, where part of this work has been done. FCM thanks FCT for grant SFRH/BSAB/130242/2017. AGP thanks the financial support from Grant 14-37086G of the Czech Science Foundation and the partial support from the projects IT956-16 ('Eusko Jaurlaritza', Spain), FIS2014-57956-P ('Ministerio de Econom'\u0131a y Competitividad', Spain).", "pages": [14]}, {"title": "7 Appendix", "content": "The EFEs for the cylindrically symmetric vacuum metric (31) are together with the two constraint equations where the commas denote differentiation.", "pages": [14]}] |
2014SCPMA..57..387Z | https://arxiv.org/pdf/1306.1289.pdf | <document>
<section_header_level_1><location><page_1><loc_24><loc_92><loc_77><loc_93></location>Revisiting the interacting model of new agegraphic dark energy</section_header_level_1>
<text><location><page_1><loc_32><loc_89><loc_68><loc_90></location>Jing-Fei Zhang, 1 Li-Ang Zhao, 1 and Xin Zhang ∗ 1, 2, †</text>
<text><location><page_1><loc_29><loc_88><loc_29><loc_88></location>1</text>
<text><location><page_1><loc_27><loc_86><loc_73><loc_88></location>College of Sciences, Northeastern University, Shenyang 110004, China 2 Center for High Energy Physics, Peking University, Beijing 100080, China</text>
<text><location><page_1><loc_18><loc_70><loc_83><loc_85></location>In this paper, a new version of the interacting model of new agegraphic dark energy (INADE) is proposed and analyzed in detail. The interaction between dark energy and dark matter is reconsidered. The interaction term Q = bH 0 ρ α de ρ 1 -α dm is adopted, which abandons the Hubble expansion rate H and involves both ρ de and ρ dm. Moreover, the new initial condition for the agegraphic dark energy is used, which solves the problem of accommodating baryon matter and radiation in the model. The solution of the model can be given using an iterative algorithm. A concrete example for the calculation of the model is given. Furthermore, the model is constrained by using the combined Planck data (Planck + BAO + SNIa + H 0) and the combined WMAP-9 data (WMAP + BAO + SNIa + H 0). Three typical cases are considered: (A) Q = bH 0 ρ de, (B) Q = bH 0 √ ρ de ρ dm, and (C) Q = bH 0 ρ dm, which correspond to α = 1, 1 / 2, and 0, respectively. The departures of the models from the Λ CDMmodel are measured by the ∆ BIC and ∆ AIC values. It is shown that the INADE model is better than the NADE model in the fit, and the INADE(A) model is the best in fitting data among the three cases.</text>
<text><location><page_1><loc_18><loc_67><loc_59><loc_69></location>PACS numbers: 95.36. + x, 98.80.Es, 98.80.-k Keywords: agegraphic dark energy, interacting dark energy model, Planck data</text>
<text><location><page_1><loc_9><loc_55><loc_49><loc_64></location>Cosmological observations continue to indicate that the universe is currently experiencing an accelerated expansion [1-3]. This cosmic acceleration is commonly believed to be caused by 'dark energy,' something producing gravitational repulsion. However, the nature of dark energy, hitherto, is still unknown.</text>
<text><location><page_1><loc_9><loc_23><loc_49><loc_55></location>The simplest candidate for dark energy is Einstein's cosmological constant, Λ , which is equivalent to the vacuum energy density in the universe and produces negative pressure with w = -1 (here, w is the equation of state parameter, defined by w = p /ρ ). However, the cosmological constant is theoretically challenged: its observationally required value is 10 120 times smaller than its theoretical expectation. So, cosmologists are in need of new theoretical insights. Alternatively, dark energy might be due to some unknown scalar field, usually dubbed quintessence, that could supply the requisite negative pressure to accelerate the cosmic expansion. Scalar field is, nevertheless, only one option. Many dynamical dark energy models from various theoretical perspectives have been proposed; for reviews, see, e.g., Refs. [4]. In particular, there is an attractive idea that links the vacuum energy density with the holographic principle of quantum gravity. This class of models is called 'holographic dark energy,' in which the UV problem of dark energy is converted to an IR problem and the dark energy density can be expressed as ρ de ∝ L -2 where L is the IR length-scale cuto ff of the theory. As a consequence of the e ff ective quantum field theory, the vacuum energy density in this theory is not a constant, but dynamically evolutionary.</text>
<text><location><page_1><loc_9><loc_15><loc_49><loc_22></location>The original version of the holographic dark energy chooses the event horizon size of the universe as the IR cuto ff [5]; for extensive studies of this model, see, e.g., Refs. [6, 7]. Subsequently, other versions were proposed; for example, the so called 'agegraphic dark energy' model (here, we refer to the</text>
<text><location><page_1><loc_52><loc_58><loc_92><loc_64></location>new agegraphic dark energy model, abbreviated as NADE) chooses the conformal time (age) of the universe as the IR cuto ff [8]. Thus, in this model, the dark energy density is of the form,</text>
<formula><location><page_1><loc_66><loc_56><loc_92><loc_57></location>ρ de = 3 n 2 M 2 Pl η -2 , (1)</formula>
<text><location><page_1><loc_52><loc_52><loc_92><loc_55></location>where n is a numerical parameter, M Pl is the reduced Planck mass, and η is the conformal age of the universe,</text>
<formula><location><page_1><loc_64><loc_48><loc_92><loc_51></location>η ≡ ∫ t 0 dt a = ∫ a 0 da Ha 2 . (2)</formula>
<text><location><page_1><loc_52><loc_34><loc_92><loc_47></location>Actually, this model can also be derived from the uncertainty relation of quantum mechanics together with the gravitational e ff ect in general relativity [9]. The most attractive merit of the NADE model is that it has the same number of parameters as the Λ CDM (the cosmological constant plus cold dark matter) model, less than other dynamical dark energy models. The model has also been proven to fit the data well [10]. See also, e.g., Refs. [11, 12] for various studies of the NADE model.</text>
<text><location><page_1><loc_52><loc_20><loc_92><loc_34></location>It seems necessary to consider the important possibility that there is some direct interaction between dark energy and dark matter. Though there is no convincing observational evidence for this coupling, such a hypothesis has inspired considerable theoretical interests. A large number of interacting dark energy models have been investigated. The interacting model of new agegraphic dark energy (INADE) was proposed and studied in detail in Refs. [13, 14]. If dark energy interacts with cold dark matter, the continuity equations for them are written as</text>
<formula><location><page_1><loc_63><loc_16><loc_92><loc_18></location>˙ ρ de + 3 H (1 + w ) ρ de = -Q , (3)</formula>
<formula><location><page_1><loc_67><loc_15><loc_92><loc_16></location>˙ ρ dm + 3 H ρ dm = Q , (4)</formula>
<text><location><page_1><loc_52><loc_8><loc_92><loc_14></location>where w is the equation of state parameter of dark energy, and Q phenomenologically describes the interaction between dark energy and dark matter. Since we have no a fundamental theory to determine the form of Q , its form needs to be assumed</text>
<text><location><page_2><loc_9><loc_76><loc_49><loc_93></location>phenomenologically. The most common choice is Q ∝ H ρ , where ρ denotes the density of dark energy or dark matter (or the sum of the two). Such a scenario is mathematically simple, but it is di ffi cult to see how this form can emerge from a physical description of dark sector interaction. It is expected that the interaction is determined by the local properties of the dark sectors, i.e., ρ de and ρ dm, but it is hard to understand why the interaction term must be proportional to the Hubble expansion rate H . A more natural hypothesis is that the Hubble parameter is abandoned and thus Q is only proportional to the dark sector density, namely, Q ∝ ρ de or Q ∝ ρ dm. Such a scenario has also been studied widely; see, e.g., Refs. [15].</text>
<text><location><page_2><loc_9><loc_53><loc_49><loc_76></location>In Ref. [14] the INADE model was investigated in detail, but there are some issues that should be re-scrutinized under the current situation: (i) In Ref. [14] the interaction term is assumed to be of the form Q ∝ H ρ ; but now, it seems necessary to adopt the more natural form in which the Hubble parameter H is abandoned. (ii) In Ref. [14] the model can only accommodatetwo components, dark energy and dark matter, but cannot involve baryon matter, radiation and spatial curvature. This is caused by the old initial condition used. In Ref. [12], however, a new initial condition (with a numerical algorithm) of the model was proposed, which can solve this problem. (iii) Recently, the Planck Collaboration publicly released the cosmic microwave background (CMB) temperature and lensing data [3], so it is also necessary to constrain the model by using the new data. Thus, under the current situation, we revisit the INADE model in this paper.</text>
<text><location><page_2><loc_9><loc_47><loc_49><loc_52></location>In this work, we will consider a more general form for the interaction term. Following Ref. [16] in which the interacting model of holographic dark energy was discussed in detail, we will take the form</text>
<formula><location><page_2><loc_25><loc_43><loc_49><loc_46></location>Q ∝ ρ α de ρ β dm , (5)</formula>
<text><location><page_2><loc_9><loc_29><loc_49><loc_43></location>which includes the forms Q ∝ ρ de and Q ∝ ρ dm as special cases, describing the decay process of dark energy or dark matter. Moreover, this form can also describe the more complicated cases of interaction, such as scattering, in which one may expect the existence of both ρ de and ρ dm. Therefore, Eq. (5) is, undoubtedly, a more natural and physically plausible form in describing the interaction between dark energy and dark matter. In the following, for simplicity, we confine our discussion in the class with the condition α + β = 1, so the interaction term can be explicitly expressed as</text>
<formula><location><page_2><loc_23><loc_26><loc_49><loc_27></location>Q = bH 0 ρ α de ρ 1 -α dm , (6)</formula>
<text><location><page_2><loc_9><loc_20><loc_49><loc_24></location>where b is the coupling constant; when b > 0, the energy flow is from dark energy to dark matter, and when b < 0 the energy flow is from dark matter to dark energy.</text>
<text><location><page_2><loc_9><loc_9><loc_49><loc_20></location>In what follows we will discuss the INADE model with the interaction form (6). We will use the new initial condition proposed in Ref. [12]. After the numerical solution of the model is given, we will further constrain the parameter space of the model by using the latest observational data, including the CMB data, the baryon acoustic oscillation (BAO) data, the type Ia supernova (SNIa) data, and the Hubble constant data. In particular, for the CMB data, we will use both the Planck</text>
<text><location><page_2><loc_52><loc_90><loc_92><loc_93></location>data [3] and the 9-year Wilkinson Microwave Anisotropy Probe (WMAP-9) data [2], respectively, for a comparison.</text>
<text><location><page_2><loc_52><loc_85><loc_92><loc_90></location>Defining f de ≡ ρ de /ρ de0 and f dm ≡ ρ dm /ρ dm0 (the subscript '0' in this paper denotes the present value of the corresponding quantity), we can rewrite Eqs. (3) and (4) into the following forms,</text>
<formula><location><page_2><loc_60><loc_77><loc_92><loc_83></location>d f de( x ) dx + 3(1 + w de , e ff ) f de( x ) = 0 , (7) d f dm( x ) dx + 3(1 + w dm , e ff ) f dm( x ) = 0 , (8)</formula>
<text><location><page_2><loc_52><loc_75><loc_64><loc_76></location>where x = ln a , and</text>
<formula><location><page_2><loc_58><loc_71><loc_92><loc_74></location>w de , e ff ( x ) = -1 + 2 3 ne x E ( x ) √ Ω de0 f de( x ) , (9)</formula>
<formula><location><page_2><loc_58><loc_68><loc_92><loc_71></location>w dm , e ff ( x ) = -br α 3 E ( x ) f de( x ) α f dm( x ) -α , (10)</formula>
<text><location><page_2><loc_52><loc_63><loc_92><loc_67></location>with r ≡ ρ de0 /ρ dm0 and E ( x ) ≡ H ( x ) / H 0. Note that in this paper we consider a flat universe, so the Friedmann equation 3 M 2 Pl H 2 = ρ de + ρ dm + ρ b + ρ r can be recast as</text>
<formula><location><page_2><loc_54><loc_58><loc_92><loc_61></location>E ( x ) = √ Ω de0 f de( x ) + Ω dm0 f dm( x ) + Ω b0 e -3 x + Ω r0 e -4 x . (11)</formula>
<text><location><page_2><loc_52><loc_49><loc_92><loc_58></location>The solutions to the system of di ff erential equations (7) and (8), f de( x ) and f dm( x ), completely describe the cosmological evolution of the INADE model. Hence, next, the main task is to find out the solutions to Eqs. (7) and (8). In order to solve the di ff erential equations, we first need to give the initial conditions for them.</text>
<text><location><page_2><loc_52><loc_29><loc_92><loc_49></location>As shown in Ref. [8], the parameters Ω m0 and n are not independent of each other. Once n is given, Ω m0 can be derived, and vice versa. If one takes both Ω m0 and n as free parameters, the NADE model will become problematic; see Fig. 2 (a) in Ref. [14] and the corresponding discussions. So the initial conditions in this model should be taken at the early times; usually, as a convention, the initial conditions are taken at z ini = 2000, in the matter-dominated epoch. Since in the matter-dominated epoch the contribution of dark energy to the cosmological evolution is negligible, it is expected that the impact of the interaction on dark energy in the early times is also ignorable. So, it is suitable to follow Ref. [12] to take the initial condition for the agegraphic dark energy in this model as</text>
<formula><location><page_2><loc_55><loc_24><loc_92><loc_28></location>f de( x ini) = n 2 Ω 2 m0 4 Ω de0 ( √ Ω m0 e x ini + Ω r0 -√ Ω r0) -2 , (12)</formula>
<text><location><page_2><loc_52><loc_15><loc_92><loc_24></location>where Ω m0 = Ω dm0 +Ω b0. In fact, the impact of interaction on dark matter in the early times is also fairly small, so the deviation from the scaling law a -3 for dark matter is tiny. Nevertheless, we still use a small quantity δ to parameterize this tiny deviation; so the initial condition for dark matter is taken as</text>
<formula><location><page_2><loc_65><loc_13><loc_92><loc_14></location>f dm( x ini) = e ( -3 + δ ) x ini . (13)</formula>
<text><location><page_2><loc_52><loc_9><loc_92><loc_11></location>We shall show that δ is also a derived parameter, and its value can be determined using an iteration calculation. Throughout</text>
<text><location><page_3><loc_9><loc_87><loc_49><loc_93></location>the calculation, we fix Ω r0 = 2 . 469 × 10 -5 h -2 (1 + N e ff ), where N e ff = 3 . 046 is the standard value of the e ff ective number of the neutrino species, and h is the Hubble constant H 0 in units of 100 km s -1 Mpc -1 .</text>
<text><location><page_3><loc_9><loc_78><loc_49><loc_87></location>In this model, the free parameters are: n , b , Ω b0, and h . The independent parameters δ and Ω m0 (or Ω dm0) can be derived using an iterative algorithm. In our calculation, we employ the Newton iteration method. The conditions of convergence we set are: δ ( l + 1) -δ ( l ) δ ( l ) < 10 -5 and Ω ( l + 1) m0 -Ω ( l ) m0 Ω ( l ) m0 < 10 -5 , where the index l denotes the iteration times in the numerical calculation.</text>
<text><location><page_3><loc_9><loc_35><loc_49><loc_77></location>To show the solution of the model, we give a concrete example. In this example, we take α = 1 in Eq. (6), i.e., Q = bH 0 ρ de. Furthermore, we fix h = 0 . 7 and Ω b0 = 0 . 05, in order to explicitly show the impacts of the parameters n and b on the model. The solutions f de( z ) and f dm( z ) are shown in Fig. 1. In the left panel, we fix b = 0 . 1, and take n = 2 . 0, 2.5 and 30., respectively; one can see that the parameter n impacts both evolutions of dark matter and dark energy evidently. In the right panel, we fix n = 2 . 5, and take b = 0 . 1, 0.3 and 0.5, respectively; one can see that the coupling strength b impacts on the evolution of dark matter more evidently than the evolution of dark energy. The parameters Ω m0 and δ can be derived through the iterative calculation. For instance, for the case n = 2 . 5 and b = 0 . 1, we obtain Ω m0 = 0 . 340 and δ = 0 . 010. Actually, the calculations for all the cases show that δ is around O (10 -2 ), verifying the previous statement for the initial condition of dark matter that the early deviation from the scaling law a -3 for dark matter is indeed tiny. Obviously, once the solutions to Eqs. (7) and (8), f de( z ) and f dm( z ), are given, the other quantities of interest, such as Ω de( z ), Ω dm( z ), w ( z ), H ( z ), Q ( z ), and so on, can be directly calculated. Thus, so far, we have proposed the revised version of the INADE model and given the solution of this model. Since the issues such as the alleviation of the cosmic coincidence problem are the common characteristics of interacting dark energy models, we do not discuss this class of issues in this paper. Next, we will test this model with the latest observational data and explore the parameter space of the model in the fit to data.</text>
<text><location><page_3><loc_9><loc_29><loc_49><loc_35></location>In the fit, we only focus on several typical cases of the model. The cases we consider include: (A) Q = bH 0 ρ de, (B) Q = bH 0 √ ρ de ρ dm, and (C) Q = bH 0 ρ dm, which correspond to α = 1, 1 / 2, and 0, respectively.</text>
<text><location><page_3><loc_9><loc_9><loc_49><loc_29></location>The data we use include the CMB data, BAO data, SNIa data, and Hubble constant data. For CMB, we use both the Planck data [3] and the WMAP-9 data [2], for a comparison. In this work, we do not consider the cosmological perturbations in the calculation. This avoids the extra assumptions on the sound speed of dark energy perturbation, the momentum transport between dark energy and dark matter, and so forth. In fact, the result will not be a ff ected evidently when the cosmological perturbations are involved. Since the perturbations are ignored, we can use the CMB distance prior data in the fit. The results of the CMB distance priors ( l A, R , ω b) for Planck and WMAP-9 have been given in Ref. [17], so we will use these data in our fit analysis. For BAO, we use the SDSSDR7, SDSS-DR9, 6dFGS, and WiggleZ data; the prescription</text>
<text><location><page_3><loc_52><loc_86><loc_92><loc_93></location>of the use of these data has been given in Ref. [2]. For SNIa, we use the Union2.1 data [18]. For the Hubble constant measurement, we use the HST result [19], H 0 = 73 . 8 ± 2 . 4 km s -1 Mpc -1 . The Markov Chain Monte Carlo (MCMC) method is employed in our data fit analysis.</text>
<text><location><page_3><loc_52><loc_26><loc_92><loc_85></location>The fit results are presented in Table I. Since the Λ CDM model fits the data very well, in this work we also make a comparison with the Λ CDM model. Actually, the Λ CDM model has been adopted as a fiducial model in dark energy cosmology. In addition, a comparison with the NADE model without interaction is also made. The numbers of parameters in the Λ CDM model and the NADE model are equal, but they are less than that in the INADE model. We denote the number of parameters in a model as k . For the Λ CDM model and the NADE model, k = 3; For the INADE model (with α fixed), k = 4. In order to fairly compare the models with di ff erent numbers of parameters, we employ the information criteria (IC), such as the Bayesian information criterion (BIC) and the Akaike information criterion (AIC), as the assessment tools. They are defined as BIC = χ 2 min + k ln N and AIC = χ 2 min + 2 k , where N is the number of the data used in the fit. Statistically, a model with few parameters and with a better fit to the data has lower IC values. Thus, the models can be ranked according to their IC values. Note that the parameter h is included in the number of degrees of freedom and in k as a parameter in each model, since it appears in the fits to the data of CMB, BAO, and H 0, and cannot be marginalized in the fits. Also, in this work, the number of data N is fixed, so the BIC and the AIC produce the same order of the models. The Λ CDM model is proven to be the best in fitting data (i.e., with lowest values of BIC and AIC; see also Refs. [20, 21]), so in Table I the ∆ BIC and ∆ AIC values are measured with respect to the Λ CDMmodel. In this table we list the free parameters for the models, and present their best fit values with 1-2 σ errors. For both the combined Planck data and the combined WMAP-9 data, the Λ CDM model performs best, and the NADE model performs worst. Among the three cases of the INADE model, the case (A) is the best in fitting data, though the di ff erence between them is little. The parameter Ω m0 can be derived in the INADE model. For example, for the combined Planck data, the best-fit values of Ω m0 are 0.317, 0.318 and 0.319, for the cases (A), (B) and (C), respectively. Also, we find that the coupling constant b in the INADE models is always positive, indicating that the energy transport is from dark energy to dark matter. This is helpful in alleviating the cosmic coincidence problem.</text>
<text><location><page_3><loc_52><loc_9><loc_92><loc_26></location>The parameter spaces of the INADE models are also explored. Since the INADE(A) model (with Q = bH 0 ρ de) is the best in fitting data among the three interacting models, we only show the parameter space of this model in this paper. In fact, the three cases are similar, so the two-dimensional contours in the parameter planes for the INADE(B) and INADE(C) models are not reported; for a full report on these results, see Ref. [22]. In Fig. 2 we show the two-dimensional marginalized contours (68% and 95% CLs) in the parameter planes for the INADE(A) model. The left panel is the case for the combined Planck data, and the right panel is for the combined WMAP-9 data. The degeneracies between the pa-</text>
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<caption>FIG. 1: The solutions of Eqs. (7) and (8), f de( z ) and f dm( z ), in the INADE model with Q = bH 0 ρ de. In this case, we fix h = 0 . 7 and Ω b0 = 0 . 05. In the left panel, we fix b = 0 . 1 and vary n ; in the right panel, we fix n = 2 . 5 and vary b .</caption>
</figure>
<table>
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<caption>TABLEI: The fit results of the Λ CDM, NADE, and INADE models. For the INADE model, cases with (A) Q = bH 0 ρ de, (B) Q = bH 0 √ ρ de ρ dm, and (C) Q = bH 0 ρ dm are considered. Since the numbers of parameters are di ff erent for di ff erent models, the information criteria BIC and AIC are used in the model comparison. Here, k denotes the number of parameters in the models. The order of the three cases of the INADE model is arranged according to the fit results. In the fit, we use the data combination CMB + BAO + SNIa + H 0, where for CMB we use the Planck data and WMAP-9 data, respectively, for a comparison. The best-fit values with 1-2 σ errors for the parameters in the models are presented.</caption>
</table>
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<caption>FIG. 2: The two-dimensional marginalized constraints (68% and 95% CLs) on the INADE(A) model with Q = bH 0 ρ de, from the CMB + BAO + SNIa + H 0 data. The CMB data used in the two panels are di ff erent: in the left panel, the Planck data are used; in the right panel, the WMAP-9 data are used.</caption>
</figure>
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<text><location><page_5><loc_9><loc_85><loc_49><loc_93></location>rameters can be explicitly seen from this figure. We can see that, for the Planck case, the parameter space is more tight, but the degeneracies are more evident, especially for the n -h plane. Since Ω m0 is derived from n , this implies that h degenerates strongly with Ω m0, which is consistent with the case of the Λ CDM model [3, 23].</text>
<text><location><page_5><loc_9><loc_60><loc_49><loc_84></location>In summary, a revised version of the INADE model is proposed and analyzed in this paper. In this version, the interaction between dark energy and dark matter is reconsidered. The interaction term Q = bH 0 ρ α de ρ 1 -α dm is adopted, which abandons the Hubble expansion rate H and involves both ρ de and ρ dm. Moreover, in this version the new initial condition for the agegraphic dark energy is used, which solves the problem of accommodating baryon matter and radiation in the model. The solution of the model can be given using an iterative algorithm. We give a concrete example for the calculation of the model. Furthermore, we constrain the model by using the combined Planck data (Planck + BAO + SNIa + H 0) and the combined WMAP-9 data (WMAP + BAO + SNIa + H 0). We focus on the three typical cases: (A) Q = bH 0 ρ de, (B) Q = bH 0 √ ρ de ρ dm, and (C) Q = bH 0 ρ dm, which correspond to α = 1, 1 / 2, and 0, respectively. The departures of the models from the Λ CDM model are measured by the ∆ BIC and</text>
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<text><location><page_5><loc_52><loc_77><loc_92><loc_93></location>∆ AIC values. We show that the INADE model is better than the NADE model in the fit, and the INADE(A) model is the best in fitting data among the three cases. As an example, we show the two-dimensional marginalized contours (68% and 95% CLs) in the parameter planes for the INADE(A) model. It is indicated that, for the Planck case, the parameter space is more tight, but the degeneracies between the parameters are more evident, especially for the n -h plane. Owing to the fact that Ω m0 is derived from n , this implies that h degenerates strongly with Ω m0, which is consistent with the case of the Λ CDM model.</text>
<section_header_level_1><location><page_5><loc_66><loc_71><loc_78><loc_72></location>Acknowledgments</section_header_level_1>
<text><location><page_5><loc_52><loc_60><loc_92><loc_68></location>This work was supported by the National Natural Science Foundation of China (Grants No. 10975032 and No. 11175042), the National Ministry of Education of China (Grants No. NCET-09-0276, No. N110405011 and No. N120505003), and the Provincial Department of Education of Liaoning (Grant No. L2012087).</text>
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</document> | [{"title": "Revisiting the interacting model of new agegraphic dark energy", "content": "Jing-Fei Zhang, 1 Li-Ang Zhao, 1 and Xin Zhang \u2217 1, 2, \u2020 1 College of Sciences, Northeastern University, Shenyang 110004, China 2 Center for High Energy Physics, Peking University, Beijing 100080, China In this paper, a new version of the interacting model of new agegraphic dark energy (INADE) is proposed and analyzed in detail. The interaction between dark energy and dark matter is reconsidered. The interaction term Q = bH 0 \u03c1 \u03b1 de \u03c1 1 -\u03b1 dm is adopted, which abandons the Hubble expansion rate H and involves both \u03c1 de and \u03c1 dm. Moreover, the new initial condition for the agegraphic dark energy is used, which solves the problem of accommodating baryon matter and radiation in the model. The solution of the model can be given using an iterative algorithm. A concrete example for the calculation of the model is given. Furthermore, the model is constrained by using the combined Planck data (Planck + BAO + SNIa + H 0) and the combined WMAP-9 data (WMAP + BAO + SNIa + H 0). Three typical cases are considered: (A) Q = bH 0 \u03c1 de, (B) Q = bH 0 \u221a \u03c1 de \u03c1 dm, and (C) Q = bH 0 \u03c1 dm, which correspond to \u03b1 = 1, 1 / 2, and 0, respectively. The departures of the models from the \u039b CDMmodel are measured by the \u2206 BIC and \u2206 AIC values. It is shown that the INADE model is better than the NADE model in the fit, and the INADE(A) model is the best in fitting data among the three cases. PACS numbers: 95.36. + x, 98.80.Es, 98.80.-k Keywords: agegraphic dark energy, interacting dark energy model, Planck data Cosmological observations continue to indicate that the universe is currently experiencing an accelerated expansion [1-3]. This cosmic acceleration is commonly believed to be caused by 'dark energy,' something producing gravitational repulsion. However, the nature of dark energy, hitherto, is still unknown. The simplest candidate for dark energy is Einstein's cosmological constant, \u039b , which is equivalent to the vacuum energy density in the universe and produces negative pressure with w = -1 (here, w is the equation of state parameter, defined by w = p /\u03c1 ). However, the cosmological constant is theoretically challenged: its observationally required value is 10 120 times smaller than its theoretical expectation. So, cosmologists are in need of new theoretical insights. Alternatively, dark energy might be due to some unknown scalar field, usually dubbed quintessence, that could supply the requisite negative pressure to accelerate the cosmic expansion. Scalar field is, nevertheless, only one option. Many dynamical dark energy models from various theoretical perspectives have been proposed; for reviews, see, e.g., Refs. [4]. In particular, there is an attractive idea that links the vacuum energy density with the holographic principle of quantum gravity. This class of models is called 'holographic dark energy,' in which the UV problem of dark energy is converted to an IR problem and the dark energy density can be expressed as \u03c1 de \u221d L -2 where L is the IR length-scale cuto ff of the theory. As a consequence of the e ff ective quantum field theory, the vacuum energy density in this theory is not a constant, but dynamically evolutionary. The original version of the holographic dark energy chooses the event horizon size of the universe as the IR cuto ff [5]; for extensive studies of this model, see, e.g., Refs. [6, 7]. Subsequently, other versions were proposed; for example, the so called 'agegraphic dark energy' model (here, we refer to the new agegraphic dark energy model, abbreviated as NADE) chooses the conformal time (age) of the universe as the IR cuto ff [8]. Thus, in this model, the dark energy density is of the form, where n is a numerical parameter, M Pl is the reduced Planck mass, and \u03b7 is the conformal age of the universe, Actually, this model can also be derived from the uncertainty relation of quantum mechanics together with the gravitational e ff ect in general relativity [9]. The most attractive merit of the NADE model is that it has the same number of parameters as the \u039b CDM (the cosmological constant plus cold dark matter) model, less than other dynamical dark energy models. The model has also been proven to fit the data well [10]. See also, e.g., Refs. [11, 12] for various studies of the NADE model. It seems necessary to consider the important possibility that there is some direct interaction between dark energy and dark matter. Though there is no convincing observational evidence for this coupling, such a hypothesis has inspired considerable theoretical interests. A large number of interacting dark energy models have been investigated. The interacting model of new agegraphic dark energy (INADE) was proposed and studied in detail in Refs. [13, 14]. If dark energy interacts with cold dark matter, the continuity equations for them are written as where w is the equation of state parameter of dark energy, and Q phenomenologically describes the interaction between dark energy and dark matter. Since we have no a fundamental theory to determine the form of Q , its form needs to be assumed phenomenologically. The most common choice is Q \u221d H \u03c1 , where \u03c1 denotes the density of dark energy or dark matter (or the sum of the two). Such a scenario is mathematically simple, but it is di ffi cult to see how this form can emerge from a physical description of dark sector interaction. It is expected that the interaction is determined by the local properties of the dark sectors, i.e., \u03c1 de and \u03c1 dm, but it is hard to understand why the interaction term must be proportional to the Hubble expansion rate H . A more natural hypothesis is that the Hubble parameter is abandoned and thus Q is only proportional to the dark sector density, namely, Q \u221d \u03c1 de or Q \u221d \u03c1 dm. Such a scenario has also been studied widely; see, e.g., Refs. [15]. In Ref. [14] the INADE model was investigated in detail, but there are some issues that should be re-scrutinized under the current situation: (i) In Ref. [14] the interaction term is assumed to be of the form Q \u221d H \u03c1 ; but now, it seems necessary to adopt the more natural form in which the Hubble parameter H is abandoned. (ii) In Ref. [14] the model can only accommodatetwo components, dark energy and dark matter, but cannot involve baryon matter, radiation and spatial curvature. This is caused by the old initial condition used. In Ref. [12], however, a new initial condition (with a numerical algorithm) of the model was proposed, which can solve this problem. (iii) Recently, the Planck Collaboration publicly released the cosmic microwave background (CMB) temperature and lensing data [3], so it is also necessary to constrain the model by using the new data. Thus, under the current situation, we revisit the INADE model in this paper. In this work, we will consider a more general form for the interaction term. Following Ref. [16] in which the interacting model of holographic dark energy was discussed in detail, we will take the form which includes the forms Q \u221d \u03c1 de and Q \u221d \u03c1 dm as special cases, describing the decay process of dark energy or dark matter. Moreover, this form can also describe the more complicated cases of interaction, such as scattering, in which one may expect the existence of both \u03c1 de and \u03c1 dm. Therefore, Eq. (5) is, undoubtedly, a more natural and physically plausible form in describing the interaction between dark energy and dark matter. In the following, for simplicity, we confine our discussion in the class with the condition \u03b1 + \u03b2 = 1, so the interaction term can be explicitly expressed as where b is the coupling constant; when b > 0, the energy flow is from dark energy to dark matter, and when b < 0 the energy flow is from dark matter to dark energy. In what follows we will discuss the INADE model with the interaction form (6). We will use the new initial condition proposed in Ref. [12]. After the numerical solution of the model is given, we will further constrain the parameter space of the model by using the latest observational data, including the CMB data, the baryon acoustic oscillation (BAO) data, the type Ia supernova (SNIa) data, and the Hubble constant data. In particular, for the CMB data, we will use both the Planck data [3] and the 9-year Wilkinson Microwave Anisotropy Probe (WMAP-9) data [2], respectively, for a comparison. Defining f de \u2261 \u03c1 de /\u03c1 de0 and f dm \u2261 \u03c1 dm /\u03c1 dm0 (the subscript '0' in this paper denotes the present value of the corresponding quantity), we can rewrite Eqs. (3) and (4) into the following forms, where x = ln a , and with r \u2261 \u03c1 de0 /\u03c1 dm0 and E ( x ) \u2261 H ( x ) / H 0. Note that in this paper we consider a flat universe, so the Friedmann equation 3 M 2 Pl H 2 = \u03c1 de + \u03c1 dm + \u03c1 b + \u03c1 r can be recast as The solutions to the system of di ff erential equations (7) and (8), f de( x ) and f dm( x ), completely describe the cosmological evolution of the INADE model. Hence, next, the main task is to find out the solutions to Eqs. (7) and (8). In order to solve the di ff erential equations, we first need to give the initial conditions for them. As shown in Ref. [8], the parameters \u2126 m0 and n are not independent of each other. Once n is given, \u2126 m0 can be derived, and vice versa. If one takes both \u2126 m0 and n as free parameters, the NADE model will become problematic; see Fig. 2 (a) in Ref. [14] and the corresponding discussions. So the initial conditions in this model should be taken at the early times; usually, as a convention, the initial conditions are taken at z ini = 2000, in the matter-dominated epoch. Since in the matter-dominated epoch the contribution of dark energy to the cosmological evolution is negligible, it is expected that the impact of the interaction on dark energy in the early times is also ignorable. So, it is suitable to follow Ref. [12] to take the initial condition for the agegraphic dark energy in this model as where \u2126 m0 = \u2126 dm0 +\u2126 b0. In fact, the impact of interaction on dark matter in the early times is also fairly small, so the deviation from the scaling law a -3 for dark matter is tiny. Nevertheless, we still use a small quantity \u03b4 to parameterize this tiny deviation; so the initial condition for dark matter is taken as We shall show that \u03b4 is also a derived parameter, and its value can be determined using an iteration calculation. Throughout the calculation, we fix \u2126 r0 = 2 . 469 \u00d7 10 -5 h -2 (1 + N e ff ), where N e ff = 3 . 046 is the standard value of the e ff ective number of the neutrino species, and h is the Hubble constant H 0 in units of 100 km s -1 Mpc -1 . In this model, the free parameters are: n , b , \u2126 b0, and h . The independent parameters \u03b4 and \u2126 m0 (or \u2126 dm0) can be derived using an iterative algorithm. In our calculation, we employ the Newton iteration method. The conditions of convergence we set are: \u03b4 ( l + 1) -\u03b4 ( l ) \u03b4 ( l ) < 10 -5 and \u2126 ( l + 1) m0 -\u2126 ( l ) m0 \u2126 ( l ) m0 < 10 -5 , where the index l denotes the iteration times in the numerical calculation. To show the solution of the model, we give a concrete example. In this example, we take \u03b1 = 1 in Eq. (6), i.e., Q = bH 0 \u03c1 de. Furthermore, we fix h = 0 . 7 and \u2126 b0 = 0 . 05, in order to explicitly show the impacts of the parameters n and b on the model. The solutions f de( z ) and f dm( z ) are shown in Fig. 1. In the left panel, we fix b = 0 . 1, and take n = 2 . 0, 2.5 and 30., respectively; one can see that the parameter n impacts both evolutions of dark matter and dark energy evidently. In the right panel, we fix n = 2 . 5, and take b = 0 . 1, 0.3 and 0.5, respectively; one can see that the coupling strength b impacts on the evolution of dark matter more evidently than the evolution of dark energy. The parameters \u2126 m0 and \u03b4 can be derived through the iterative calculation. For instance, for the case n = 2 . 5 and b = 0 . 1, we obtain \u2126 m0 = 0 . 340 and \u03b4 = 0 . 010. Actually, the calculations for all the cases show that \u03b4 is around O (10 -2 ), verifying the previous statement for the initial condition of dark matter that the early deviation from the scaling law a -3 for dark matter is indeed tiny. Obviously, once the solutions to Eqs. (7) and (8), f de( z ) and f dm( z ), are given, the other quantities of interest, such as \u2126 de( z ), \u2126 dm( z ), w ( z ), H ( z ), Q ( z ), and so on, can be directly calculated. Thus, so far, we have proposed the revised version of the INADE model and given the solution of this model. Since the issues such as the alleviation of the cosmic coincidence problem are the common characteristics of interacting dark energy models, we do not discuss this class of issues in this paper. Next, we will test this model with the latest observational data and explore the parameter space of the model in the fit to data. In the fit, we only focus on several typical cases of the model. The cases we consider include: (A) Q = bH 0 \u03c1 de, (B) Q = bH 0 \u221a \u03c1 de \u03c1 dm, and (C) Q = bH 0 \u03c1 dm, which correspond to \u03b1 = 1, 1 / 2, and 0, respectively. The data we use include the CMB data, BAO data, SNIa data, and Hubble constant data. For CMB, we use both the Planck data [3] and the WMAP-9 data [2], for a comparison. In this work, we do not consider the cosmological perturbations in the calculation. This avoids the extra assumptions on the sound speed of dark energy perturbation, the momentum transport between dark energy and dark matter, and so forth. In fact, the result will not be a ff ected evidently when the cosmological perturbations are involved. Since the perturbations are ignored, we can use the CMB distance prior data in the fit. The results of the CMB distance priors ( l A, R , \u03c9 b) for Planck and WMAP-9 have been given in Ref. [17], so we will use these data in our fit analysis. For BAO, we use the SDSSDR7, SDSS-DR9, 6dFGS, and WiggleZ data; the prescription of the use of these data has been given in Ref. [2]. For SNIa, we use the Union2.1 data [18]. For the Hubble constant measurement, we use the HST result [19], H 0 = 73 . 8 \u00b1 2 . 4 km s -1 Mpc -1 . The Markov Chain Monte Carlo (MCMC) method is employed in our data fit analysis. The fit results are presented in Table I. Since the \u039b CDM model fits the data very well, in this work we also make a comparison with the \u039b CDM model. Actually, the \u039b CDM model has been adopted as a fiducial model in dark energy cosmology. In addition, a comparison with the NADE model without interaction is also made. The numbers of parameters in the \u039b CDM model and the NADE model are equal, but they are less than that in the INADE model. We denote the number of parameters in a model as k . For the \u039b CDM model and the NADE model, k = 3; For the INADE model (with \u03b1 fixed), k = 4. In order to fairly compare the models with di ff erent numbers of parameters, we employ the information criteria (IC), such as the Bayesian information criterion (BIC) and the Akaike information criterion (AIC), as the assessment tools. They are defined as BIC = \u03c7 2 min + k ln N and AIC = \u03c7 2 min + 2 k , where N is the number of the data used in the fit. Statistically, a model with few parameters and with a better fit to the data has lower IC values. Thus, the models can be ranked according to their IC values. Note that the parameter h is included in the number of degrees of freedom and in k as a parameter in each model, since it appears in the fits to the data of CMB, BAO, and H 0, and cannot be marginalized in the fits. Also, in this work, the number of data N is fixed, so the BIC and the AIC produce the same order of the models. The \u039b CDM model is proven to be the best in fitting data (i.e., with lowest values of BIC and AIC; see also Refs. [20, 21]), so in Table I the \u2206 BIC and \u2206 AIC values are measured with respect to the \u039b CDMmodel. In this table we list the free parameters for the models, and present their best fit values with 1-2 \u03c3 errors. For both the combined Planck data and the combined WMAP-9 data, the \u039b CDM model performs best, and the NADE model performs worst. Among the three cases of the INADE model, the case (A) is the best in fitting data, though the di ff erence between them is little. The parameter \u2126 m0 can be derived in the INADE model. For example, for the combined Planck data, the best-fit values of \u2126 m0 are 0.317, 0.318 and 0.319, for the cases (A), (B) and (C), respectively. Also, we find that the coupling constant b in the INADE models is always positive, indicating that the energy transport is from dark energy to dark matter. This is helpful in alleviating the cosmic coincidence problem. The parameter spaces of the INADE models are also explored. Since the INADE(A) model (with Q = bH 0 \u03c1 de) is the best in fitting data among the three interacting models, we only show the parameter space of this model in this paper. In fact, the three cases are similar, so the two-dimensional contours in the parameter planes for the INADE(B) and INADE(C) models are not reported; for a full report on these results, see Ref. [22]. In Fig. 2 we show the two-dimensional marginalized contours (68% and 95% CLs) in the parameter planes for the INADE(A) model. The left panel is the case for the combined Planck data, and the right panel is for the combined WMAP-9 data. The degeneracies between the pa- /s32 /s32 /s32 /s32 - - - - - - - - /s32 /s32 rameters can be explicitly seen from this figure. We can see that, for the Planck case, the parameter space is more tight, but the degeneracies are more evident, especially for the n -h plane. Since \u2126 m0 is derived from n , this implies that h degenerates strongly with \u2126 m0, which is consistent with the case of the \u039b CDM model [3, 23]. In summary, a revised version of the INADE model is proposed and analyzed in this paper. In this version, the interaction between dark energy and dark matter is reconsidered. The interaction term Q = bH 0 \u03c1 \u03b1 de \u03c1 1 -\u03b1 dm is adopted, which abandons the Hubble expansion rate H and involves both \u03c1 de and \u03c1 dm. Moreover, in this version the new initial condition for the agegraphic dark energy is used, which solves the problem of accommodating baryon matter and radiation in the model. The solution of the model can be given using an iterative algorithm. We give a concrete example for the calculation of the model. Furthermore, we constrain the model by using the combined Planck data (Planck + BAO + SNIa + H 0) and the combined WMAP-9 data (WMAP + BAO + SNIa + H 0). We focus on the three typical cases: (A) Q = bH 0 \u03c1 de, (B) Q = bH 0 \u221a \u03c1 de \u03c1 dm, and (C) Q = bH 0 \u03c1 dm, which correspond to \u03b1 = 1, 1 / 2, and 0, respectively. The departures of the models from the \u039b CDM model are measured by the \u2206 BIC and \u2206 AIC values. We show that the INADE model is better than the NADE model in the fit, and the INADE(A) model is the best in fitting data among the three cases. As an example, we show the two-dimensional marginalized contours (68% and 95% CLs) in the parameter planes for the INADE(A) model. It is indicated that, for the Planck case, the parameter space is more tight, but the degeneracies between the parameters are more evident, especially for the n -h plane. Owing to the fact that \u2126 m0 is derived from n , this implies that h degenerates strongly with \u2126 m0, which is consistent with the case of the \u039b CDM model.", "pages": [1, 2, 3, 4, 5]}, {"title": "Acknowledgments", "content": "This work was supported by the National Natural Science Foundation of China (Grants No. 10975032 and No. 11175042), the National Ministry of Education of China (Grants No. NCET-09-0276, No. N110405011 and No. N120505003), and the Provincial Department of Education of Liaoning (Grant No. L2012087). Phys Lett B, 2007, 648: 1-7; Zhang X. Dynamical vacuum energy, holographic quintom, and the reconstruction of scalarfield dark energy. Phys Rev D, 2006, 74: 103505; Zhang J, Zhang X, Liu H. Holographic dark energy in a cyclic universe. Eur Phys J C, 2007, 52: 693-699; Setare M R, Zhang J, Zhang X. Statefinder diagnosis in a non-flat universe and the holographic model of dark energy. J Cosmol Astrpart Phys, 2007, 03: 007; Zhang X. Heal the world: Avoiding the cosmic doomsday in the holographic dark energy model. Phys Lett B, 2010, 683: 81-87; Zhang J F, Li Y Y, Liu Y, et al. Holographic \u039b (t)CDM model in a non-flat universe. Eur Phys J C, 2012, 72: 2077. acting dark energy. Phys Rev D, 2009, 79: 063518; CalderaCabral G, Maartens R, Schaefer B M. The growth of structure in interacting dark energy models. J Cosmol Astropart Phys, 2009, 07: 027; Koyama K, Maartens R, Song Y S. Velocities as a probe of dark sector interactions. J Cosmol Astropart Phys, 2009, 10: 017; Majerotto E, Valiviita J, Maartens R. Adiabatic initial conditions for perturbations in interacting dark energy models. Mon Not R Astron Soc, 2010, 402: 2344-2354; Valiviita J, Maartens R, Majerotto E. Observational constraints on an interacting dark energy model. Mon Not R Astron Soc, 2010, 402: 2355-2368; Clemson T, Koyama K, Zhao G B, et al. Interacting dark energy: Constraints and degeneracies. Phys Rev D, 2012, 85: 043007.", "pages": [5, 6]}] |
2013MNRAS.430.1486R | https://arxiv.org/pdf/1302.5722.pdf | <document>
<section_header_level_1><location><page_1><loc_7><loc_80><loc_89><loc_84></location>Peaks and dips in Gaussian random fields: a new algorithm for the shear eigenvalues, and the excursion set theory</section_header_level_1>
<section_header_level_1><location><page_1><loc_7><loc_75><loc_26><loc_77></location>Graziano Rossi 1 , 2 , 3 /star</section_header_level_1>
<unordered_list>
<list_item><location><page_1><loc_7><loc_72><loc_64><loc_74></location>1 Korea Institute for Advanced Study, Hoegiro 85, Dongdaemun-Gu, Seoul 130 -722 , Korea</list_item>
<list_item><location><page_1><loc_7><loc_71><loc_50><loc_72></location>2 CEA, Centre de Saclay, Irfu/SPP, F-91191 Gif-sur-Yvette, France</list_item>
<list_item><location><page_1><loc_7><loc_70><loc_76><loc_71></location>3 Paris Center for Cosmological Physics (PCCP) and Laboratoire APC, Universit'e Paris 7, 75205 Paris, France</list_item>
</unordered_list>
<text><location><page_1><loc_7><loc_66><loc_60><loc_67></location>Accepted 2012 November 27. Received 2012 November 7; in original form 2012 June 29</text>
<section_header_level_1><location><page_1><loc_28><loc_61><loc_38><loc_62></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_28><loc_25><loc_89><loc_59></location>We present a new algorithm to sample the constrained eigenvalues of the initial shear field associated with Gaussian statistics, called the 'peak/dip excursion-setbased' algorithm, at positions which correspond to peaks or dips of the correlated density field. The computational procedure is based on a new formula which extends Doroshkevich's unconditional distribution for the eigenvalues of the linear tidal field, to account for the fact that halos and voids may correspond to maxima or minima of the density field. The ability to differentiate between random positions and special points in space around which halos or voids may form (i.e. peaks/dips), encoded in the new formula and reflected in the algorithm, naturally leads to a straightforward implementation of an excursion set model for peaks and dips in Gaussian random fields - one of the key advantages of this sampling procedure. In addition, it offers novel insights into the statistical description of the cosmic web. As a first physical application, we show how the standard distributions of shear ellipticity and prolateness in triaxial models of structure formation are modified by the constraint. In particular, we provide a new expression for the conditional distribution of shape parameters given the density peak constraint, which generalizes some previous literature work. The formula has important implications for the modeling of non-spherical dark matter halo shapes, in relation to their initial shape distribution. We also test and confirm our theoretical predictions for the individual distributions of eigenvalues subjected to the extremum constraint, along with other directly related conditional probabilities. Finally, we indicate how the proposed sampling procedure naturally integrates into the standard excursion set model, potentially solving some of its well-known problems, and into the ellipsoidal collapse framework. Several other ongoing applications and extensions, towards the development of algorithms for the morphology and topology of the cosmic web, are discussed at the end.</text>
<text><location><page_1><loc_28><loc_21><loc_89><loc_24></location>Key words: methods: analytical - methods: statistical - galaxies: formation - galaxies: halos - cosmology: theory - large-scale structure of Universe.</text>
<section_header_level_1><location><page_1><loc_7><loc_15><loc_24><loc_17></location>1 INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_7><loc_5><loc_89><loc_14></location>The large-scale spatial organization of matter in clusters, filaments, sheets and voids, known as the cosmic web (Peebles 1980; Bardeen et al. 1986; Bond et al. 1991), is the manifestation of the anisotropic nature of gravitational collapse. This typical filamentary pattern has been confirmed by observations, for example with the 2dFGRS, SDSS and 2MASS redshift surveys of the nearby Universe (Colless et al. 2003; Tegmark et al. 2004; Huchra et al. 2005), and is routinely seen in large scale N -body numerical simulations - see for example the New Horizon Runs (Kim et al. 2011) or the recent Millennium-XXL (Angulo et al. 2012). Basic characteristics of the cosmic web are the spatial arrangement of galaxies and mass into elongated filaments, sheet-like walls and dense compact clusters, the existence of large near-empty void regions, and the hierarchical nature of this</text>
<section_header_level_1><location><page_2><loc_7><loc_89><loc_23><loc_90></location>2 Graziano Rossi</section_header_level_1>
<text><location><page_2><loc_7><loc_80><loc_89><loc_87></location>mass distribution (Arag'on-Calvo et al. 2007, 2010a,b; Zhang et al. 2009). In particular, as pointed out by Bond, Kofman & Pogosyan (1996), 'embryonic' cosmic web is already present in the primordial density field. These key properties, along with the alignment of shape and angular momentum of objects (i.e. Argyres et al. 1986; Catelan et al. 2001; Lee & Springel 2010), are mainly due to the effects of the tidal field, associated with the gravitational potential - while the Hessian of the density field (i.e. the inertia tensor) plays a secondary role in determining the characteristic pattern of the cosmic web.</text>
<text><location><page_2><loc_7><loc_58><loc_89><loc_80></location>Pioneering works devoted to the key role of the initial tidal field in shaping large-scale structures trace back to Doroshkevich & Zeldovich (1964), Doroshkevich (1970), Zeldovich (1970), Sunyaev & Zeldovich (1972), Icke (1973), and Doroshkevich & Shandarin (1978); their studies have contributed to reach a solid understanding of the formation and evolution of structures. Other classical works in this direction (i.e. Peebles 1980; White 1984; Bardeen et al. 1986; Kaiser 1986; Bertschinger 1987; Dubinski 1992; Bond & Myers 1996; Bond, Kofman & Pogosyan 1996; van de Weygaert & Bertschinger 1996) have considerably improved the statistical description of the cosmic web from first principles. Their impact is broader, as there is a correspondence between structures in the evolved density field and local properties of the linear tidal shear; this allows one to estimate the morphology of the cosmic web (Bond & Myers 1996; Rossi, Sheth & Tormen 2011), and is crucial in understanding the nonlinear evolution of cosmic structures (Springel et al. 2005; Shandarin et al. 2006; Pogosyan et al. 2009), the statistical properties of voids (Lee & Park 2006; Platen, van de Weygaert & Jones 2008; Lavaux & Wandelt 2010), the alignment of shape and angular momentum of halos (West 1989; Catelan et al. 2001; Faltenbacher et al. 2009; Lee & Springel 2010), and for characterizing the geometry and topology of the cosmic web (Gott et al. 1986, 1989; Park & Gott 1991; Park et al. 1992, 2005; van de Weygaert & Bond 2008; Forero-Romero et al. 2009; Aragon-Calvo et al. 2010a,b; Choi et al. 2010; Shandarin et al. 2010; van de Weygaert et al. 2011; Cautun et al. 2012; Hidding et al. 2012). In addition, the eigenvalues of the mass tidal tensor can be used to classify the large-scale environment (Shen et al. 2006; Hahn et al. 2007; Zhang et al. 2009) - hence their fundamental importance.</text>
<text><location><page_2><loc_7><loc_43><loc_89><loc_58></location>In the context of the initial shear field, Doroshkevich (1970) provided a key contribution by deriving the joint probability distribution of an ordered set of eigenvalues in the tidal field matrix corresponding to a Gaussian potential, given the variance of the density field. Throughout the paper, we will refer to it as the unconditional probability distribution of shear eigenvalues. Because the initial shear field associated with Gaussian statistics plays a major role in the formation of large scale structures, considerable analytic work has been based on Doroshkevich's unconditional formulas since their appearance, but those relations neglect the fact that halos (voids) may correspond to maxima (minima) of the underlying density field. The local extrema of such field (i.e. peaks/dips) are plausible sites for the formation of nonlinear structures (Bardeen et al. 1986), and some numerical studies have indeed reported a good correspondence between peaks in the initial conditions and halos at late times (see in particular Ludlow & Porciani 2011). Hence, their statistical properties can be used to predict the abundances and clustering of objects of various types, and in studies of triaxial formation of large-scale structures (Bardeen et al. 1986; Bond & Myers 1996).</text>
<text><location><page_2><loc_7><loc_25><loc_89><loc_43></location>Motivated by these reasons, recently Rossi (2012) has provided a set of analytic expressions which extend the work of Doroshkevich (1970) and Bardeen et al. (1986) - and are akin in philosophy to that of van de Weygaert & Bertschinger (1996). These new relations incorporate the peak/dip constraint into the statistical description of the initial shear field, and are able to differentiate between random positions and peak/dips in the correlated density field. In essence, they allow one to express the joint probability distribution of an ordered set of eigenvalues in the initial shear field, given the fact that positions are peaks or dips of the density - and not just random spatial locations. These relations are obtained by requiring the density Hessian (i.e. the matrix of the second derivatives of the density field, associated with the curvature) to be positive/negative definite, which is the case in the vicinity of minima/maxima of the density. The correlation strength between the gravitational and density fields is quantified via a reduced parameter γ , which plays a major role in peaks theory (i.e. γ is the same as in Bardeen et al. 1986 for Gaussian smoothing filters - see Appendix A). Doroshkevich's (1970) unconditional formulas are then naturally recovered in the absence of correlation, when γ = 0. From these new conditional joint probabilities, it is possible to derive the individual distributions of eigenvalues subjected to the extremum constraint, along with some other related conditional probabilities; their expressions were also provided in Rossi (2012), extending the work of Lee & Shandarin (1998).</text>
<text><location><page_2><loc_7><loc_11><loc_89><loc_25></location>The primary goal of this paper is show how the main conditional formulas presented in Rossi (2012) and reported here (Equations 1 and 11) lead to a new algorithm - called the 'peak/dip excursion-set-based' algorithm - to sample the constrained eigenvalues of the initial shear field associated with Gaussian statistics, at positions which correspond to peaks or dips of the correlated density field. While it is clearly possible to sample the constrained eigenvalues of the tidal field directly from the conditional probability distribution function, as done for example by Lavaux & Wandelt (2010) in the context of cosmic voids, the theoretical work carried out by Rossi (2012) allows for a much simpler procedure, part of which was previously thought not to be achievable analytically (see again the Appendix B in Lavaux & Wandelt 2010). Besides providing novel theoretical insights, the main strength of the algorithm resides in its natural inclusion into the standard excursion set framework, allowing for a generalization. As we will discuss in Section 6, this technique potentially solves a long-standing problem of the standard excursion set theory; moreover, it is well-suited for the ellipsoidal collapse framework.</text>
<text><location><page_2><loc_7><loc_1><loc_89><loc_11></location>The other goal of the paper is to present a first physical application of the new algorithm, related to the morphology of the cosmic web. In particular, we provide a new expression for the conditional distribution of shape parameters (i.e. ellipticity and prolateness) in the presence of the density peak constraint, which generalizes some previous literature work and combines the formalism of Bardeen et al. (1986) - based on the density field - with that of Bond & Myers (1996) - based on the shear field. The formula has important implications for the modeling of non-spherical dark matter halos and their evolved halo shapes, in relation to the initial shape distribution. In addition, we also test and confirm our theoretical predictions for the individual distributions of eigenvalues subjected to the extremum constraint, along with other directly related conditional</text>
<text><location><page_3><loc_7><loc_79><loc_89><loc_87></location>probabilities. Finally, we illustrate how this algorithm can be readily merged into the excursion set framework (Peacock & Heavens 1990; Bond et al. 1991), and in particular to obtain an excursion set model for peaks and dips in Gaussian random fields. The key point is the ability to differentiate between random positions and peaks/dips, which is contained in Equations (1) and (11) and encapsulated in the algorithm. While we leave this latter part to a dedicated forthcoming publication, we anticipate the main ideas here. We also discuss several other ongoing applications and extensions, towards the development of algorithms for classifying cosmic web structures.</text>
<text><location><page_3><loc_7><loc_62><loc_89><loc_79></location>The layout of the paper is organized as follows. Section 2 provides a short review of the key equations derived in Rossi (2012), which constitute the theoretical framework for the new algorithm described here; the main notation adopted is summarized in Appendix A, for convenience. Moving from this mathematical background, Section 3 presents the new 'peak/dip excursion-set-based' algorithm (some insights derived from this part are left in Appendix B), while Section 4 tests its performance against various analytic distributions of eigenvalues and related conditional probabilities, subjected to the extremum constraint. Section 5 shows a physical application of the computational procedure, towards the morphology of the cosmic web. A new expression for the conditional distribution of ellipticity and prolateness in the presence of the density peak constraint is also given; in particular, the description of Bardeen et al. (1986) is combined with that of Bond & Myers (1996). Section 6 illustrates how this new algorithm can be readily used to implement an excursion set model for peaks and dips in Gaussian random fields, and makes the connection with some previous literature. Finally, Section 7 discusses several ongoing and future promising applications, which will be presented in forthcoming publications, and in particular the use of this algorithm for triaxial models of collapse and in relation to the morphology and topology of the cosmic web.</text>
<section_header_level_1><location><page_3><loc_7><loc_58><loc_83><loc_59></location>2 JOINT CONDITIONAL DISTRIBUTION OF EIGENVALUES IN THE PEAK/DIP PICTURE</section_header_level_1>
<text><location><page_3><loc_7><loc_40><loc_89><loc_57></location>We begin by reexamining two main results derived in Rossi (2012), which constitute the key for developing a new algorithm to sample the constrained eigenvalues of the initial shear field. This part may be also regarded as a compact review of the main formulas for the constrained shear eigenvalues, which can be used directly for several applications related to the cosmic web. The notation adopted here is the same as the one introduced by Rossi (2012), with a few minor changes to make the connection with previous literature more explicit. It is summarized in Appendix A; a reader not familiar with the notation may want to start from the appendix first. In particular, in what follows we do not adopt 'reduced' variables, so that the various dependencies on σ values (i.e. σ T ≡ σ 0 for the shear, and σ H ≡ σ 2 for the density Hessian) are now shown explicitly. However, for the sake of clarity, we omit to indicate the understood dependence of these two global parameters in the left-hand side of all the formulas. Note that we prefer to write σ T and σ H , rather than σ 0 and σ 2 , for their more intuitive meaning (i.e. the label T indicates that a quantity is connected to the shear field, while the label H points to the Hessian of the density field). As explained in Appendix A, we also introduce the 6-dimensional vectors T = ( T 11 , T 22 , T 33 , T 12 , T 13 , T 23 ) and H = ( H 11 , H 22 , H 33 , H 12 , H 13 , H 23 ), derived from the components of their corresponding shear and density Hessian tensors.</text>
<text><location><page_3><loc_7><loc_38><loc_89><loc_40></location>Rossi (2012) obtained the following expression for the probability of observing a tidal field T for the gravitational potential, given a curvature H for the density field:</text>
<formula><location><page_3><loc_26><loc_33><loc_89><loc_37></location>p ( T | H , γ ) = 15 3 16 √ 5 π 3 1 σ 6 T (1 -γ 2 ) 3 exp [ -3 2 σ 2 T (1 -γ 2 ) (2 K 2 1 -5 K 2 ) ] (1)</formula>
<text><location><page_3><loc_7><loc_32><loc_11><loc_33></location>where</text>
<formula><location><page_3><loc_7><loc_30><loc_89><loc_31></location>K 1 = ( T 11 ηH 11 ) + ( T 22 ηH 22 ) + ( T 33 ηH 33 ) = k 1 ηh 1 (2)</formula>
<text><location><page_3><loc_7><loc_28><loc_8><loc_29></location>K</text>
<text><location><page_3><loc_8><loc_28><loc_9><loc_29></location>2</text>
<formula><location><page_3><loc_8><loc_25><loc_89><loc_26></location>τ = T 11 H 11 + T 22 H 22 + T 33 H 33 +2 T 12 H 12 +2 T 13 H 13 +2 T 23 H 23 (4)</formula>
<formula><location><page_3><loc_10><loc_25><loc_89><loc_29></location>= ( T 11 -ηH 11 )( T 22 -ηH 22 ) + ( T 11 -ηH 11 )( T 33 -ηH 33 ) + ( T 22 -ηH 22 )( T 33 -ηH 33 ) -( T 12 -ηH 12 ) 2 -( T 13 -ηH 13 ) 2 -( T 23 -ηH 23 ) 2 = k 2 + η 2 h 2 -ηh 1 k 1 + ητ (3)</formula>
<text><location><page_3><loc_7><loc_23><loc_23><loc_24></location>k 1 = T 11 + T 22 + T 33</text>
<formula><location><page_3><loc_87><loc_23><loc_89><loc_24></location>(5)</formula>
<unordered_list>
<list_item><location><page_3><loc_7><loc_20><loc_42><loc_22></location>k 2 = T 11 T 22 + T 11 T 33 + T 22 T 33 -T 2 12 -T 2 13 -T 2 23</list_item>
</unordered_list>
<text><location><page_3><loc_7><loc_19><loc_24><loc_20></location>h 1 = H 11 + H 22 + H 33</text>
<text><location><page_3><loc_87><loc_19><loc_89><loc_20></location>(7)</text>
<formula><location><page_3><loc_7><loc_17><loc_89><loc_19></location>h 2 = H 11 H 22 + H 11 H 33 + H 22 H 33 -H 2 12 -H 2 13 -H 2 23 (8)</formula>
<text><location><page_3><loc_8><loc_16><loc_19><loc_17></location>η = γσ T /σ H .</text>
<text><location><page_3><loc_87><loc_16><loc_89><loc_17></location>(9)</text>
<text><location><page_3><loc_7><loc_12><loc_89><loc_15></location>The corresponding unconditional marginal distributions p ( T ) and p ( H ) are multidimensional Gaussians, expressed using Doroshkevich's formulas as</text>
<formula><location><page_3><loc_19><loc_8><loc_89><loc_12></location>p ( T ) = 15 3 16 √ 5 π 3 1 σ 6 T exp [ -3 2 σ 2 T (2k 2 1 -5k 2 ) ] , p ( H ) = 15 3 16 √ 5 π 3 1 σ 6 H exp [ -3 2 σ 2 H (2h 2 1 -5h 2 ) ] . (10)</formula>
<text><location><page_3><loc_7><loc_1><loc_89><loc_8></location>Equation (1) generalizes Doroshkevich's formulas (10) to include the fact that halos/voids may correspond to maxima/minima of the density field. Note in particular that p ( T | H , γ ) is a multivariate Gaussian distribution with mean b = η H and covariance matrix (1 -γ 2 ) σ 2 T A / 15, with A given in Appendix A. Clearly, one can also consider the reverse distribution p ( H | T , γ ), the expression of which is given in Rossi (2012). This is useful in order to make the connection and extend some results derived in Bardeen et al. (1986), the subject of a forthcoming publication.</text>
<formula><location><page_3><loc_87><loc_21><loc_89><loc_22></location>(6)</formula>
<text><location><page_3><loc_16><loc_29><loc_47><loc_31></location>----</text>
<section_header_level_1><location><page_4><loc_7><loc_89><loc_23><loc_90></location>4 Graziano Rossi</section_header_level_1>
<text><location><page_4><loc_7><loc_84><loc_89><loc_87></location>It is also possible to express (1) in terms of the constrained eigenvalues of T | H , indicated with ζ i ( i = 1 , 2 , 3) and ordered as ζ 1 /greaterorequalslant ζ 2 /greaterorequalslant ζ 3 . The result is:</text>
<formula><location><page_4><loc_17><loc_79><loc_89><loc_83></location>p ( ζ 1 , ζ 2 , ζ 3 | γ ) = 15 3 8 √ 5 π 1 σ 6 T (1 -γ 2 ) 3 exp [ -3 2 σ 2 T (1 -γ 2 ) (2 K 2 1 -5 K 2 ) ] ( ζ 1 -ζ 2 )( ζ 1 -ζ 3 )( ζ 2 -ζ 3 ) (11)</formula>
<text><location><page_4><loc_7><loc_78><loc_43><loc_79></location>where in terms of constrained eigenvalues we now have:</text>
<formula><location><page_4><loc_7><loc_74><loc_89><loc_76></location>K 1 = ζ 1 + ζ 2 + ζ 3 = k 1 -ηh 1 (12)</formula>
<formula><location><page_4><loc_8><loc_72><loc_89><loc_73></location>τ = λ 1 ξ 1 + λ 2 ξ 2 + λ 3 ξ 3 (14)</formula>
<formula><location><page_4><loc_7><loc_73><loc_89><loc_75></location>K 2 = ζ 1 ζ 2 + ζ 1 ζ 3 + ζ 2 ζ 3 = k 2 + η 2 h 2 -ηh 1 k 1 + γτ (13)</formula>
<formula><location><page_4><loc_7><loc_70><loc_89><loc_71></location>k 1 = λ 1 + λ 2 + λ 3 (15)</formula>
<formula><location><page_4><loc_7><loc_68><loc_89><loc_69></location>k 2 = λ 1 λ 2 + λ 1 λ 3 + λ 2 λ 3 (16)</formula>
<formula><location><page_4><loc_7><loc_66><loc_89><loc_68></location>h 1 = ξ 1 + ξ 2 + ξ 3 (17)</formula>
<formula><location><page_4><loc_7><loc_65><loc_89><loc_66></location>h 2 = ξ 1 ξ 2 + ξ 1 ξ 3 + ξ 2 ξ 3 (18)</formula>
<formula><location><page_4><loc_8><loc_62><loc_89><loc_64></location>ζ i = λ i -ηξ i . (19)</formula>
<text><location><page_4><loc_7><loc_60><loc_81><loc_62></location>The partial distributions p ( λ 1 , λ 2 , λ 3 ) and p ( ξ 1 , ξ 2 , ξ 3 ) are expressed by Doroshkevich's unconditional formulas as</text>
<formula><location><page_4><loc_23><loc_56><loc_89><loc_59></location>p ( λ 1 , λ 2 , λ 3 ) = 15 3 8 √ 5 π 1 σ 6 T exp [ -3 2 σ 2 T (2k 2 1 -5k 2 ) ] ( λ 1 -λ 2 )( λ 1 -λ 3 )( λ 2 -λ 3 ) (20)</formula>
<formula><location><page_4><loc_24><loc_50><loc_89><loc_53></location>p ( ξ 1 , ξ 2 , ξ 3 ) = 15 3 8 √ 5 π 1 σ 6 H exp [ -3 2 σ 2 H (2h 2 1 -5h 2 ) ] ( ξ 1 -ξ 2 )( ξ 1 -ξ 3 )( ξ 2 -ξ 3 ) , (21)</formula>
<text><location><page_4><loc_7><loc_44><loc_89><loc_49></location>where λ 1 , λ 2 , λ 3 are the eigenvalues of the shear tensor, while ξ 1 , ξ 2 , ξ 3 are those of the density Hessian. Similarly, one can easily obtain formulas for the reverse probability functions. Note also that k 1 = λ 1 + λ 2 + λ 3 is simply the overdensity δ T associated to the shear field, while ξ 1 + ξ 2 + ξ 3 = h 1 ≡ δ H . Later on, we will make use of the peak height ν = δ T /σ T and of the peak curvature x = δ H /σ H .</text>
<text><location><page_4><loc_7><loc_40><loc_89><loc_43></location>Equations (1) and (11) allow one to develop a new algorithm to sample the constrained eigenvalues of the initial shear field, presented in the next section. It will be then straightforward to use this algorithm in order to implement an excursion set model for peaks and dips in Gaussian random fields.</text>
<section_header_level_1><location><page_4><loc_7><loc_31><loc_56><loc_33></location>3 THE PEAK/DIP EXCURSION-SET-BASED ALGORITHM</section_header_level_1>
<text><location><page_4><loc_7><loc_20><loc_89><loc_30></location>While the constrained eigenvalues of the initial shear field can be sampled directly from their probability distribution function (i.e. Equations 1 or 11), the new theoretical formalism developed by Rossi (2012) leads to a much simpler algorithm, which is particularly interesting because well-suited for the ellipsoidal collapse model (Section 5), and naturally integrable in the excursion set framework (Section 6). In addition, the algebra leading to the new computational procedure allows one to explain analytically how the halo (void) shape distributions are altered by the inclusion of the peak (dip) constraint, and offers a variety of applications and insights on the statistical description of the cosmic web (see Section 7 and Appendix B). In what follows, we present the mathematical aspects of the 'peak/dip excursion-set-based' algorithm, and describe in detail the novel computational procedure.</text>
<text><location><page_4><loc_7><loc_17><loc_89><loc_19></location>In particular, we are mainly interested in the distribution p ( T | H > 0 , γ ), the joint probability of observing a tidal field T for the gravitational potential with a positive density curvature H > 0 (i.e. at density peak locations). Clearly,</text>
<formula><location><page_4><loc_28><loc_11><loc_89><loc_15></location>p ( T | H > 0 , γ ) = p ( T , H > 0 | γ ) p ( H > 0) = ∫ H > 0 p ( H ) p ( T | H , γ ) d H ∫ H > 0 p ( H ) d H , (22)</formula>
<text><location><page_4><loc_7><loc_4><loc_89><loc_11></location>where p ( T | H , γ ) and p ( H ) are given by Equations (1) and (10), and the integrals are 6-dimensional. A similar expression can be written in terms of the constrained distributions of eigenvalues, using Equation (11) instead. In principle, one should then compute the previous integral to obtain p ( T | H > 0 , γ ). However, it is easier to sample p ( T | H , γ ) and impose the condition H > 0 directly, so that we are effectively computing p ( T | H > 0 , γ ). Given the nature of p ( T | H , γ ) and p ( H ) expressed by (1) and (10), this can be readily achieved - as the following algebra will show.</text>
<text><location><page_4><loc_7><loc_1><loc_89><loc_4></location>In fact, because the elements of the density Hessian are drawn from a multivariate Gaussian distribution with zero mean and covariance matrix σ 2 H A / 15, where A is given by Equation (A8), one can simply obtain them by generating six independent</text>
<text><location><page_5><loc_7><loc_86><loc_80><loc_87></location>zero-mean unit-variance Gaussian random variates y i ( i = 1 , 6), and then determine the various components as:</text>
<formula><location><page_5><loc_7><loc_68><loc_89><loc_85></location>H 11 = σ H 3 ( -y 1 + 2 √ 5 y 2 ) H 22 = σ H 3 ( -y 1 -1 √ 5 y 2 -3 √ 15 y 3 ) H 33 = σ H 3 ( -y 1 -1 √ 5 y 2 + 3 √ 15 y 3 ) H 12 = H 21 = σ H √ 15 y 4 H 13 = H 31 = σ H √ 15 y 5 H 23 = H 32 = σ H √ 15 y 6 . (23)</formula>
<text><location><page_5><loc_7><loc_63><loc_89><loc_67></location>Similarly, the elements of the shear tensor are also drawn from a multivariate Gaussian distribution with mean zero and covariance matrix σ 2 T A / 15. Hence, if z i ( i = 1 , 6) are other six independent zero-mean unit-variance Gaussian random variates, the components of the shear tensor are given by:</text>
<formula><location><page_5><loc_7><loc_46><loc_89><loc_63></location>T 11 = σ T 3 ( -z 1 + 2 √ 5 z 2 ) T 22 = σ T 3 ( -z 1 -1 √ 5 z 2 -3 √ 15 z 3 ) T 33 = σ T 3 ( -z 1 -1 √ 5 z 2 + 3 √ 15 z 3 ) T 12 = T 21 = σ T √ 15 z 4 T 13 = T 31 = σ T √ 15 z 5 T 23 = T 32 = σ T √ 15 z 6 . (24)</formula>
<text><location><page_5><loc_7><loc_40><loc_89><loc_45></location>Next, consider the 6-dimensional vector T | H , made from the components of the conditional shear field. Its elements are drawn from a multivariate Gaussian distribution p ( T | H , γ ) with mean b = η H and covariance matrix (1 -γ 2 ) σ 2 T A / 15, where the elements of the density Hessian are expressed by (23). Therefore, this implies that the components of T | H are distributed according to:</text>
<formula><location><page_5><loc_7><loc_19><loc_89><loc_39></location>T 11 | H 11 = ηH 11 + σ T √ 1 -γ 2 3 ( -l 1 + 2 √ 5 l 2 ) ≡ σ T 3 ( -m 1 + 2 √ 5 m 2 ) T 22 | H 22 = ηH 22 + σ T √ 1 -γ 2 3 ( -l 1 -1 √ 5 l 2 -3 √ 15 l 3 ) ≡ σ T 3 ( -m 1 -1 √ 5 m 2 -3 √ 15 m 3 ) T 33 | H 33 = ηH 33 + σ T √ 1 -γ 2 3 ( -l 1 -1 √ 5 l 2 + 3 √ 15 l 3 ) ≡ σ T 3 ( -m 1 -1 √ 5 m 2 + 3 √ 15 m 3 ) T 12 | H 12 = T 21 | H 21 = ηH 12 + σ T √ 1 -γ 2 √ 15 l 4 ≡ σ T √ 15 m 4 T 13 | H 13 = T 31 | H 31 = ηH 13 + σ T √ 1 -γ 2 √ 15 l 5 ≡ σ T √ 15 m 5 T 23 | H 23 = T 32 | H 32 = ηH 23 + σ T √ 1 -γ 2 √ 15 l 6 ≡ σ T √ 15 m 6 . (25)</formula>
<text><location><page_5><loc_7><loc_13><loc_89><loc_18></location>In the previous expression, the various l i ( i = 1 , 6) are other six independent zero-mean unit-variance Gaussian variates, while the m i ( i = 1 , 6) are six independent Gaussian distributed variates with shifted mean γy i and reduced variance (1 -γ 2 ), i.e. m i = γy i + √ 1 -γ 2 l i .</text>
<text><location><page_5><loc_7><loc_10><loc_89><loc_14></location>Equations (23), (24) and (25) suggest a new excursion-set-based algorithm, in order to obtain the constrained eigenvalues of the matrix having components T α | H α (see Appendix A for the definition of the index α ) - supplemented by the condition of a positive curvature H for the density field (or negative curvature, for voids). The procedure can be summarized as follows:</text>
<unordered_list>
<list_item><location><page_5><loc_7><loc_7><loc_89><loc_9></location>(i) Draw six independent zero-mean unit-variance Gaussian distributed variates y i ( i = 1 , 6), and determine the components H α of the density Hessian matrix via Equation (23). Compute the value of σ H using (A6).</list_item>
<list_item><location><page_5><loc_7><loc_0><loc_89><loc_6></location>(ii) Calculate the eigenvalues ξ 1 , ξ 2 , ξ 3 of the previous Hessian matrix, and check if they are all positive (negative). If so, proceed to the next step, otherwise try again. This will guarantee the Hessian to be positive (negative) definite (i.e. the Hessian is a real-symmetric matrix), which is the condition for maxima (minima) of the field. Note that this step is clearly not required if we want to sample only p ( T | H , γ ).</list_item>
</unordered_list>
<section_header_level_1><location><page_6><loc_7><loc_89><loc_23><loc_90></location>6 Graziano Rossi</section_header_level_1>
<text><location><page_6><loc_7><loc_81><loc_89><loc_87></location>(iii) Draw other six independent Gaussian distributed variates l i ( i = 1 , 6), with mean zero and variance one, and determine the T α | H α components via Equation (25), using the previously accepted values H α from (23) - while σ T is determined via (A6). Since we require the density Hessian to be positive definite, this means that we are effectively sampling the conditional probability p ( T | H > 0 , γ ) - or p ( T | H < 0 , γ ) for a negative definite Hessian.</text>
<unordered_list>
<list_item><location><page_6><loc_8><loc_79><loc_75><loc_81></location>(iv) Calculate and store the constrained eigenvalues ζ i of the matrix having components T α | H α > 0.</list_item>
</unordered_list>
<text><location><page_6><loc_7><loc_74><loc_89><loc_79></location>We leave in Appendix B some more insights on the main conditional formula (1), which readily follow from the previous Equations (23), (24) and (25). In the next sections, we will show a few applications of the new algorithm - with particular emphasis on the conditional distributions of shape parameters in triaxial models of structure formation (i.e. ellipticity and prolateness). Additional applications will be presented in forthcoming publications.</text>
<section_header_level_1><location><page_6><loc_7><loc_68><loc_74><loc_69></location>4 CONDITIONAL DISTRIBUTIONS AND PROBABILITIES: NUMERICAL TESTS</section_header_level_1>
<text><location><page_6><loc_7><loc_60><loc_89><loc_67></location>The algorithm illustrated in the previous section allows one to test and confirm several formulas derived by Rossi (2012), regarding conditional distributions and probabilities subjected to the extremum constraint. We present here results of the comparison (i.e. theory versus numerical), where for simplicity we set σ T = σ H ≡ 1; this corresponds to adopt 'reduced variables', as done in Rossi (2012). Note that we show explicitly the dependencies on σ values in the following expressions, although we set them to unity in the mock tests.</text>
<section_header_level_1><location><page_6><loc_7><loc_55><loc_38><loc_56></location>4.1 Individual conditional distributions</section_header_level_1>
<text><location><page_6><loc_7><loc_51><loc_89><loc_54></location>The individual conditional distributions of shear eigenvalues, for a given correlation strength γ with the density field (which encapsulate the peak/dip constraint), are given by (Rossi 2012):</text>
<formula><location><page_6><loc_7><loc_43><loc_89><loc_51></location>p ( ζ 1 | γ ) = √ 5 12 πσ T [ 20 (1 -γ 2 ) ζ 1 σ T exp ( -9 2(1 -γ 2 ) ζ 2 1 σ 2 T ) -√ 2 π (1 -γ 2 ) 3 / 2 exp ( -5 2(1 -γ 2 ) ζ 2 1 σ 2 T ) × (26) erfc ( -√ 2 √ 1 -γ 2 ζ 1 σ T )[ (1 -γ 2 ) -20 ζ 2 1 σ 2 T ] + 3 √ 3 π √ 1 -γ 2 exp ( -15 4(1 -γ 2 ) ζ 2 1 σ 2 T ) erfc ( -√ 3 2 √ 1 -γ 2 ζ 1 σ T )]</formula>
<formula><location><page_6><loc_7><loc_37><loc_89><loc_43></location>p ( ζ 2 | γ ) = √ 15 2 √ π 1 σ T √ 1 -γ 2 exp [ -15 4(1 -γ 2 ) ζ 2 2 σ 2 T ] (27)</formula>
<formula><location><page_6><loc_7><loc_29><loc_89><loc_37></location>p ( ζ 3 | γ ) = -√ 5 12 πσ T [ 20 (1 -γ 2 ) ζ 3 σ T exp ( -9 2(1 -γ 2 ) ζ 2 3 σ 2 T ) + √ 2 π (1 -γ 2 ) 3 / 2 exp ( -5 2(1 -γ 2 ) ζ 2 3 σ 2 T ) × (28) erfc ( √ 2 √ 1 -γ 2 ζ 3 σ T )[ (1 -γ 2 ) -20 ζ 2 3 σ 2 T ] -3 √ 3 π √ 1 -γ 2 exp ( -15 4(1 -γ 2 ) ζ 2 3 σ 2 T ) erfc ( √ 3 2 √ 1 -γ 2 ζ 3 σ T )] .</formula>
<text><location><page_6><loc_7><loc_23><loc_89><loc_29></location>The previous formulas show explicitly that the conditional distributions of shear eigenvalues are Doroshkevich-like expressions, with shifted mean and reduced variance. Different panels in Figure 1 display plots of these distributions, contrasted with results from the algorithm presented in Section 3, where 500,000 mock realizations are considered. In particular, note the symmetry between p ( ζ 1 | γ ) and p ( ζ 3 | γ ). We expect the mean values and variances of those distributions to be:</text>
<formula><location><page_6><loc_7><loc_20><loc_89><loc_23></location>〈 ζ 1 | γ 〉 = 3 √ 10 π σ T √ 1 -γ 2 , σ 2 ζ 1 | γ = 13 π -27 30 π σ 2 T (1 -γ 2 ) (29)</formula>
<formula><location><page_6><loc_7><loc_17><loc_89><loc_20></location>〈 ζ 2 | γ 〉 = 0 , σ 2 ζ 2 | γ = 2 15 σ 2 T (1 -γ 2 ) (30)</formula>
<formula><location><page_6><loc_7><loc_14><loc_89><loc_17></location>〈 ζ 3 | γ 〉 ≡ -〈 ζ 1 | γ 〉 = -3 √ 10 π σ T √ 1 -γ 2 , σ 2 ζ 3 | γ ≡ σ 2 ζ 1 | γ = 13 π -27 30 π σ 2 T (1 -γ 2 ) . (31)</formula>
<text><location><page_6><loc_7><loc_9><loc_89><loc_13></location>In the various panels, both theoretical and numerical expectations are reported and found to be in excellent agreement. For example, when γ = 0 . 50, 〈 ζ 1 | γ 〉 num = 0 . 4644 while the expected theoretical value is 〈 ζ 1 | γ 〉 th = 0 . 4635, and for the variances σ 2 , num ζ 1 | γ = 0 . 1102 where the theoretical expectation is σ 2 , th ζ 1 | γ = 0 . 1101.</text>
<text><location><page_6><loc_7><loc_3><loc_89><loc_9></location>Obviously, to see explicitly the effect of the peak constraint, one needs to make cuts in ξ i from the partial conditional distributions λ i | ξ i , since ζ i = λ i -ηξ i . In particular, expressions for p ( λ i | H > 0 , γ ) can be derived from (26-28) via substitution of variables and integration over H > 0. For example, it is straightforward to obtain an analytic formula for p ( λ 3 | H > 0 , γ ). In Section 5, we will return on these issues in more detail, in connection with the conditional ellipticity and prolateness for dark matter halos.</text>
<text><location><page_6><loc_10><loc_0><loc_89><loc_2></location>Another important distribution is p (∆ T | H | ζ 3 > 0 , γ ), with ∆ T | H ≡ K 1 = ζ 1 + ζ 2 + ζ 3 being the sum of the constrained</text>
<figure>
<location><page_7><loc_7><loc_27><loc_88><loc_86></location>
<caption>Figure 1. Individual conditional probabilities p ( ζ 1 | γ ) [top], p ( ζ 2 | γ ) [middle], and p ( ζ 3 | γ ) [bottom] of the initial shear field in the peak/dip picture. Solid curves are Equations (26), (27), (28), while histograms are obtained from 500,000 mock realizations via the algorithm described in Section 3. Each panel displays the numerical and theoretical expectations for the mean values and variances of these distributions; their agreement is excellent.</caption>
</figure>
<text><location><page_7><loc_7><loc_16><loc_87><loc_18></location>eigenvalues, namely the probability distribution of ∆ T | H confined in the regions with ζ 3 > 0 (i.e. all positive eigenvalues):</text>
<formula><location><page_7><loc_7><loc_8><loc_89><loc_16></location>p (∆ T | H | ζ 3 > 0 , γ ) = -75 √ 5 8 πσ 2 T ∆ T | H (1 -γ 2 ) exp ( -9 8 ∆ 2 T | H σ 2 T (1 -γ 2 ) ) + (32) + 25 4 √ 2 π 1 σ T √ 1 -γ 2 exp ( -∆ 2 T | H 2 σ 2 T (1 -γ 2 ) )[ erf ( √ 10∆ T | H 4 σ T √ 1 -γ 2 ) +erf ( √ 10∆ T | H 2 σ T √ 1 -γ 2 )] .</formula>
<text><location><page_7><loc_7><loc_7><loc_39><loc_8></location>This distribution is expected to have mean value</text>
<formula><location><page_7><loc_25><loc_3><loc_89><loc_7></location>〈 ∆ T | H | ζ 3 > 0 , γ 〉 = 25 √ 10 144 √ π (3 √ 6 -2) σ T √ 1 -γ 2 /similarequal 1 . 66 σ T √ 1 -γ 2 . (33)</formula>
<text><location><page_7><loc_7><loc_0><loc_89><loc_2></location>It is also easy to see that the maximum of p (∆ T | H | ζ 3 > 0 , γ ) is reached when ∆ T | H /similarequal 1 . 5 σ T 1 -γ 2 , and with a more</text>
<figure>
<location><page_8><loc_9><loc_68><loc_88><loc_86></location>
<caption>Figure 2. Conditional probability distribution p (∆ T | H | ζ 3 > 0 , γ ). Solid lines are obtained from Equation (32), for different values of the correlation parameter γ , while histograms are drawn from 500,000 realizations via the algorithm described in Section 3. Note the excellent agreement between numerical results and theoretical predictions for the corresponding mean and rms values, reported in the various panels. In addition, solid vertical lines show the expected maximum value of each distribution, while dotted lines represent their corresponding mean value.</caption>
</figure>
<text><location><page_8><loc_7><loc_55><loc_58><loc_56></location>elaborate calculation its variance can be estimated analytically. The result is:</text>
<formula><location><page_8><loc_13><loc_51><loc_89><loc_55></location>σ 2 ∆ T | H | ζ 3 > 0 ,γ = 25 4 π σ 2 T (1 -γ 2 ) [ arctan( √ 5 / 2) + arctan( √ 5) -11 √ 5 54 -125 2592 (3 √ 6 -2) 2 ] /similarequal 0 . 31 σ 2 T (1 -γ 2 ) . (34)</formula>
<text><location><page_8><loc_7><loc_42><loc_89><loc_51></location>Figure 2 confirms the previous relations, by contrasting Equations (32), (33) and (34) with numerical results from 500,000 realizations from the algorithm presented in Section 3. We find good agreement, and recover correctly the expected mean values and variances of the distributions. For example, when γ = 0 . 25, we find 〈 ∆ T | H | ζ 3 > 0 , γ 〉 num = 1 . 6090 while the expected theoretical value is 〈 ∆ T | H | ζ 3 > 0 , γ 〉 th = 1 . 6040, and for the rms values we find σ num ∆ T | H | ζ 3 > 0 ,γ = 0 . 5314 where the expected theoretical value is σ th ∆ T | H | ζ 3 > 0 ,γ = 0 . 5399. In the absence of correlations between the potential and density fields (i.e. when γ = 0), all the previous expressions reduce consistently to the unconditional limit of Lee & Shandarin (1998).</text>
<section_header_level_1><location><page_8><loc_7><loc_38><loc_32><loc_39></location>4.2 Distribution of peak heights</section_header_level_1>
<text><location><page_8><loc_7><loc_22><loc_89><loc_37></location>An important quantity which plays a major role in peaks theory (Bardeen et al. 1986) is the distribution of peak heights, namely p ( ν | H > 0 , γ ), where ν = δ T /σ T and the overdensity δ T is the trace of the shear tensor - defined in Section 2. One can easily obtain a simplified analytic expression for this distribution as follows. First, normalize Equation (12) by σ T , defining N = ∆ T | H /σ T and x = δ H /σ H , so that N = ν -γx . In this framework, x is the peak curvature of Bardeen et al. (1986) if Gaussian filters are used for computing the moments of the smoothed power spectrum (Equation A6). Since p ( N | γ ) is simply a Gaussian with mean zero and variance (1 -γ 2 ) - see Rossi (2012) - and because of Equation (12), then clearly p ( ν | x, γ ) is also Gaussian, with mean γx and reduced variance (1 -γ 2 ). Note in fact that p ( ν, x | γ ) is a bivariate Gaussian, because both ν and x are normally distributed random variates. This implies that 〈 ν | x, γ 〉 = γx , and clearly 〈 ν | H > 0 , γ 〉 = γ 〈 x | ξ 3 > 0 〉 ≡ γ 〈 ν | λ 3 > 0 〉 (Bardeen et al. 1986) - where 〈 ν | λ 3 > 0 〉 = 1 . 6566 according to Equation (21) of Lee & Shandarin (1998). Starting from a bivariate Gaussian with the previous expected mean value, it is then direct to derive a fairly good approximation for the distribution of peak heights, namely</text>
<text><location><page_8><loc_7><loc_16><loc_11><loc_18></location>where</text>
<formula><location><page_8><loc_28><loc_17><loc_89><loc_21></location>p ( ν | H > 0 , γ ) = 1 √ 2 π exp [ ( ν -χ ) 2 2 ][ 1 + erf ( γ ( ν -χ ) √ 2(1 -γ 2 ) )] (35)</formula>
<formula><location><page_8><loc_35><loc_13><loc_89><loc_17></location>χ = √ 2 √ π [ 25 √ 5 144 (3 √ 6 -2) -1 ] γ /similarequal 0 . 86 γ. (36)</formula>
<text><location><page_8><loc_7><loc_7><loc_89><loc_12></location>In essence, p ( ν | H > 0 , γ ) is a Gaussian with shifted mean modulated by the role of the peak curvature, and it consistently reduces to a zero-mean unit-variance Gaussian distribution when γ = 0 - as expected. Additionally, the previous distribution can be shown to be equivalent to p ( ν )[ ∫ ∞ 0 p ( ξ 3 | γ, ν )d ξ 3 /p ( ξ 3 > 0)], where the term in square brackets quantifies the effect of the peak constrain - and so the scale at which the difference between peaks and random positions would appear.</text>
<text><location><page_8><loc_7><loc_1><loc_89><loc_6></location>Figure 3 shows plots of p ( ν | H > 0 , γ ) for different values of γ . In the various panels, vertical lines are the expected mean values; we find good agreement between numerical estimates and theoretical expectations. For example, when γ = 0 . 25, 〈 ν | H > 0 , γ 〉 num = 0 . 4146 while 〈 ν | H > 0 , γ 〉 th = 0 . 4142. Solid curves in the figure are drawn from Equation (35), an approximation which is particularly good for lower values of the correlation strength.</text>
<figure>
<location><page_9><loc_17><loc_40><loc_76><loc_85></location>
<caption>Figure 3. Distribution of peak heights, p ( ν | H > 0 , γ ), for different values of the correlation parameter γ - as specified in the various panels. The numerical values for the mean of the distribution as a function of γ are indicated in each panel, and also marked with vertical lines; they are in good agreement with theoretical expectations. Solid curves are drawn from Equation (35).</caption>
</figure>
<section_header_level_1><location><page_9><loc_7><loc_31><loc_52><loc_32></location>5 CONDITIONAL ELLIPTICITY AND PROLATENESS</section_header_level_1>
<text><location><page_9><loc_7><loc_22><loc_89><loc_30></location>The theoretical framework outlined in Section 2, along with the algebra leading to the new computational procedure presented in Section 3, provides a natural way to include the peak/dip constraint into the initial shape distributions of halos and voids. This is a key aspect for implementing an excursion set algorithm for peaks and dips in Gaussian random fields (which will be discussed in Section 6), and has direct applications in the context of triaxial models of structure formation. In particular, our description combines the formalism of Bardeen et al. (1986) - based on the density field - with that of Bond & Myers (1996) - focused on the shear field.</text>
<text><location><page_9><loc_7><loc_15><loc_89><loc_22></location>In what follows, first we introduce some useful definitions regarding shape parameters (ellipticity and prolateness); we then provide a new expression for the joint conditional distribution of halo shapes given the density peak constraint, derived from Equation (11), which generalizes some previous literature work. The formula has important implications for the modeling of non-spherical dark matter halos, in relation to their initial shape distribution. Finally, we briefly discuss how the shear ellipticity and prolateness (and so the initial halo shapes) are modified by the inclusion of the density peak condition.</text>
<text><location><page_9><loc_7><loc_8><loc_89><loc_15></location>Our analysis is based on the constrained eigenvalues of the initial shear field (Equations 11-19), and we use the new algorithm to support our description. While here we only consider the case for dark matter halos, it is straightforward to deal with voids. In a forthcoming companion publication, we will present a more detailed study on the morphology of both halos and voids in the peak/dip picture, where we also investigate the modifications induced by primordial non-Gaussianity on their shapes (see also Section 7).</text>
<section_header_level_1><location><page_9><loc_7><loc_5><loc_39><loc_6></location>5.1 Shape parameters: general definitions</section_header_level_1>
<text><location><page_9><loc_7><loc_1><loc_89><loc_4></location>In triaxial models of halo formation, such as the ellipsoidal collapse (Icke 1973; White & Silk 1979; Barrow & Silk 198; Kuhlman et al. 1996; Bond & Myers 1996), it is customary to characterize the shape of a region by its ellipticity and prolateness. In</text>
<section_header_level_1><location><page_10><loc_7><loc_89><loc_24><loc_90></location>10 Graziano Rossi</section_header_level_1>
<text><location><page_10><loc_7><loc_83><loc_89><loc_87></location>particular, in the 'peak-patch' approach of Bond & Myers (1996), the shape parameters are associated with the eigenvalues of the shear field ( λ i in our notation). Considering the mapping ( λ 1 , λ 2 , λ 3 → e T , p T , δ T ), where the subscript T refers to the tidal tensor, we can write:</text>
<formula><location><page_10><loc_18><loc_79><loc_89><loc_82></location>δ T ≡ k 1 = λ 1 + λ 2 + λ 3 , e T = λ 1 -λ 3 2 δ T , p T = λ 1 + λ 3 -2 λ 2 2 δ T ≡ e T -λ 2 -λ 3 δ T , (37)</formula>
<text><location><page_10><loc_7><loc_75><loc_89><loc_78></location>where e T and p T are the 'unconditional' shear ellipticity and prolateness. In particular, if δ T > 0 then e T > 0 and therefore -e T /lessorequalslant p T /lessorequalslant e T .</text>
<text><location><page_10><loc_7><loc_72><loc_89><loc_76></location>In the 'peaks theory' of Bardeen et al. (1986), instead, ellipticity ( e H ) and prolateness ( p H ) are associated with the eigenvalues of the density field ( ξ i in our notation). Hence, given the mapping ( ξ 1 , ξ 2 , ξ 3 → e H , p H , δ H ) where now the subscript H refers to the Hessian of the density field, one has:</text>
<formula><location><page_10><loc_19><loc_68><loc_89><loc_71></location>δ H ≡ h 1 = ξ 1 + ξ 2 + ξ 3 , e H = ξ 1 -ξ 3 2 δ H , p H = ξ 1 + ξ 3 -2 ξ 2 2 δ H ≡ e H -ξ 2 -ξ 3 δ H . (38)</formula>
<text><location><page_10><loc_7><loc_65><loc_51><loc_67></location>As in the previous case, if δ H > 0 then e H > 0 and -e H /lessorequalslant p H /lessorequalslant e H .</text>
<text><location><page_10><loc_10><loc_64><loc_77><loc_66></location>It is therefore natural to consider the 'conditional' mapping ( ζ 1 , ζ 2 , ζ 3 → E T | H , P T | H , ∆ T | H ) and define:</text>
<formula><location><page_10><loc_16><loc_61><loc_89><loc_63></location>∆ T | H ≡ K 1 = ζ 1 + ζ 2 + ζ 3 , E T | H = ζ 1 -ζ 3 2∆ T | H , P T | H = ζ 1 + ζ 3 -2 ζ 2 2∆ T | H ≡ E T | H -ζ 2 -ζ 3 ∆ T | H , (39)</formula>
<text><location><page_10><loc_7><loc_56><loc_89><loc_60></location>where ζ i are the constrained shear eigenvalues - given by Equation (19). In what follows, we will refer to E T | H and P T | H as to the conditional shear ellipticity and prolateness, respectively; we will also relate these quantities to the unconditional expressions previously introduced, involving ( e T , p T ) and ( e H , p H ).</text>
<section_header_level_1><location><page_10><loc_7><loc_51><loc_78><loc_52></location>5.2 Joint conditional distribution of shear ellipticity and prolateness in the peak/dip picture</section_header_level_1>
<text><location><page_10><loc_7><loc_46><loc_89><loc_50></location>Combining (37) with Doroshkevich's formula (20), it is straightforward to derive the 'unconditional' distribution of e T and p T given δ T , g ( e T , p T | δ T ), related to the gravitational potential. In particular,</text>
<formula><location><page_10><loc_37><loc_44><loc_89><loc_46></location>g ( δ T , e T , p T ) = p ( δ T ) g ( e T , p T | δ T ) (40)</formula>
<text><location><page_10><loc_7><loc_42><loc_54><loc_44></location>where p ( δ T ) is simply a Gaussian with zero mean and variance σ 2 T , and</text>
<formula><location><page_10><loc_25><loc_38><loc_89><loc_41></location>g ( e T , p T | δ T ) = 15 3 3 √ 10 π ( δ T σ T ) 5 e T ( e 2 T -p 2 T ) exp [ -5 2 ( δ T σ T ) 2 (3 e 2 T + p 2 T ) ] . (41)</formula>
<text><location><page_10><loc_7><loc_35><loc_89><loc_37></location>Equation (40) implies that the unconditional joint distribution of shear ellipticity and prolateness is independent of that of the overdensity δ T . Similarly, inserting (38) in Doroshkevich's formula (21), one obtains</text>
<formula><location><page_10><loc_37><loc_31><loc_89><loc_33></location>g ( δ H , e H , p H ) = p ( δ H ) g ( e H , p H | δ H ) (42)</formula>
<text><location><page_10><loc_7><loc_30><loc_83><loc_31></location>for the quantities related to the density Hessian, where p ( δ H ) is a Gaussian with zero mean and variance σ 2 H , while</text>
<formula><location><page_10><loc_24><loc_25><loc_89><loc_29></location>g ( e H , p H | δ H ) = 15 3 3 √ 10 π ( δ H σ H ) 5 e H ( e 2 H -p 2 H ) exp [ -5 2 ( δ H σ H ) 2 (3 e 2 H + p 2 H ) ] . (43)</formula>
<text><location><page_10><loc_7><loc_22><loc_89><loc_25></location>Along the same lines, combining (39) with (11) and using the definitions introduced in the previous section, it is direct to obtain</text>
<formula><location><page_10><loc_29><loc_19><loc_89><loc_21></location>G (∆ T | H , E T | H , P T | H | γ ) = p (∆ T | H | γ ) G ( E T | H , P T | H | ∆ T | H , γ ) (44)</formula>
<text><location><page_10><loc_7><loc_16><loc_89><loc_19></location>where p (∆ T | H | γ ) is a Gaussian distribution with mean zero and variance σ 2 T (1 -γ 2 ) - i.e. Equation (57) in Rossi (2012) - and</text>
<formula><location><page_10><loc_9><loc_12><loc_89><loc_16></location>G ( E T | H , P T | H | ∆ T | H , γ ) = 15 3 3 √ 10 π ( ∆ T | H σ T ) 5 1 (1 -γ 2 ) 5 / 2 E T | H ( E 2 T | H -P 2 T | H ) exp [ 5 2(1 -γ 2 ) ( ∆ T | H σ T ) 2 (3 E 2 T | H + P 2 T | H ) ] . (45)</formula>
<text><location><page_10><loc_7><loc_5><loc_89><loc_12></location>The previous expression (45) is the joint conditional distribution of shear ellipticity and prolateness, given ∆ T | H and a correlation strength γ with the density field. Clearly, G ( E T | H , P T | H | ∆ T | H , γ ) is also independent of the distribution of (∆ T | H | γ ). Equation (45) generalizes the standard joint distribution of halo shape parameters to include the peak constraint, and is another main result of this paper. Note that, in the absence of correlation (i.e. when γ = 0), E T | H → e T , P T | H → p T , ∆ T | H → δ T , so that (45) reduces consistently to the 'unconditional' Doroshkevich's limit (41).</text>
<text><location><page_10><loc_7><loc_1><loc_89><loc_5></location>From this joint conditional probability function, it is possible to derive the partial conditional distributions for the shear ellipticity and prolateness given the peak constrain (which we will discuss next), and their expressions at density peak locations, when the condition H > 0 is satisfied.</text>
<figure>
<location><page_11><loc_7><loc_68><loc_88><loc_86></location>
</figure>
<figure>
<location><page_11><loc_7><loc_47><loc_88><loc_66></location>
<caption>Figure 4. [Top] Distribution of ( ζ 1 -ζ 3 | γ ), for different values of the correlation strength γ - as indicated in the panels. Its shape is directly related to the shear conditional ellipticity in the peak/dip picture - see Equation (46). [Bottom] Distribution of ( ζ 2 -ζ 3 | γ ), for different values of the correlation strength γ . This distribution is instead related to the shear conditional prolateness in the peak/dip picture - see Equation (47). The numerical values for the mean and variance of the distributions, as a function of γ , are also indicated in the figures. In both cases, solid curves are obtained via numerical integrations starting from the joint conditional distribution of eigenvalues (11). Histograms are drawn from 500,000 realizations, using the algorithm described in Section 3.</caption>
</figure>
<section_header_level_1><location><page_11><loc_7><loc_34><loc_62><loc_35></location>5.3 Initial distributions of triaxial halo shapes at density peak locations</section_header_level_1>
<text><location><page_11><loc_7><loc_29><loc_89><loc_33></location>We can readily express the conditional distributions of shear ellipticity and prolateness in the peak/dip picture in terms of the unconditional quantities (37) and (38). This is done simply by recalling that the constrained shear eigenvalues are given by ζ i = λ i -ηξ i , according to Equation (19).</text>
<text><location><page_11><loc_10><loc_28><loc_27><loc_29></location>It is then direct to obtain:</text>
<formula><location><page_11><loc_26><loc_25><loc_89><loc_27></location>E T | H ∆ T | H = ( ζ 1 -ζ 3 ) 2 ≡ ( λ 1 -λ 3 ) 2 -η ( ξ 1 -ξ 3 ) 2 = δ T e T -ηδ H e H . (46)</formula>
<text><location><page_11><loc_7><loc_13><loc_89><loc_24></location>The previous relation implies that, for a given ∆ T | H , the conditional shear ellipticity will have its mean value shifted by the presence of the peak constraint; the entity of the shift has to be ascribed to the additional factor ηδ H e H , which is given by the density ellipticity (essentially, the latter term quantifies the role of the peak curvature). The top panel of Figure 4 shows the distribution of ( ζ 1 -ζ 3 | γ ) for different values of the correlation strength γ , namely the combination of constrained shear eigenvalues which controls the conditional ellipticity - according to Equation (46). Histograms are drawn from 500,000 realizations using the algorithm described in Section 3, while solid curves are obtained via numerical integration - starting from the joint conditional distribution of eigenvalues (11). In the various panels, we also provide the numerical values for the mean and variance of the distribution, as a function of γ .</text>
<text><location><page_11><loc_10><loc_12><loc_16><loc_13></location>Similarly,</text>
<formula><location><page_11><loc_11><loc_8><loc_89><loc_11></location>P T | H ∆ T | H = ( ζ 1 + ζ 3 -2 ζ 2 ) 2 = ( λ 1 + λ 3 -2 λ 2 ) 2 -η ( ξ 1 + ξ 3 -2 ξ 2 ) 2 = δ T p T -ηδ H p H ≡ E T | H ∆ T | H -( ζ 2 -ζ 3 ) . (47)</formula>
<text><location><page_11><loc_7><loc_3><loc_89><loc_8></location>This expression shows that the conditional shear prolateness can be obtained from the conditional shear ellipticity (46) and the combination ( ζ 2 -ζ 3 ) of the constrained shear eigenvalues. The bottom panel of Figure 4 displays the distribution of ( ζ 2 -ζ 3 | γ ), for different values of the correlation strength γ . Again, solid curves are derived from a numerical integration of Equation (11).</text>
<text><location><page_11><loc_10><loc_0><loc_89><loc_2></location>Of particular interest are also the distributions g ( e T | H > 0 , γ ) and g ( p T | H > 0 , γ ), which will provide the initial triaxial</text>
<figure>
<location><page_12><loc_7><loc_47><loc_89><loc_86></location>
<caption>Figure 5. Distributions p ( ζ 1 -ζ 3 | ζ 3 > 0 , γ ) [top] and p ( ζ 2 -ζ 3 | ζ 3 > 0 , γ ) [bottom] for different values of the correlation strength γ . In both figures, histograms are drawn from 500,000 realizations using the algorithm described in Section 3, while solid curves are obtained via numerical integrations of the joint conditional distribution of eigenvalues (11). In the various panels, we also provide the numerical values for the mean and variance of the distributions, as a function of γ . In particular, their mean values at γ = 0 allow one to quantify the shift in the mean of the shear ellipticity and prolateness caused by the peak constraint.</caption>
</figure>
<text><location><page_12><loc_7><loc_18><loc_89><loc_37></location>shapes of dark matter halos at peak locations. They are related to the previous expressions, and can be readily obtained within the outlined formalism; clearly, they will reduce to the standard (or unconditional) shear ellipticity and prolateness when γ = 0, in the absence of correlation. We will present a dedicated study focused on the morphology of halos and voids in the peak/dip picture, where we characterize in detail those distributions, especially in relation to the ellipsoidal collapse model. We anticipate here that several analytic and insightful results on their shapes can be derived, starting from (45) and using (11), (46) and (47). For example, according to (46), when H > 0 (which is equivalent to impose the condition ξ 3 > 0 on an ordered set of density Hessian eigenvalues), then the quantity 〈 ξ 1 -ξ 3 | ξ 3 > 0 〉 will essentially provide by how much the mean value of the shear ellipticity has been shifted by the peak constrain. For the prolateness the situation is slightly more complicated, but according to (47) the shift can be derived by the additional knowledge of 〈 ξ 2 -ξ 3 | ξ 3 > 0 〉 . To this end, Figure 5 shows the distributions of ( ζ 1 -ζ 3 | ζ 3 > 0 , γ ) and ( ζ 2 -ζ 3 | ζ 3 > 0 , γ ), for different values of the correlation strength γ (note that the quantities just discussed are their mean values when γ = 0). As in the previous plot, histograms are drawn from 500,000 realizations using the algorithm described in Section 3, while solid curves are obtained via numerical integrations starting from the joint conditional distribution of eigenvalues (11). In the various panels, we also provide the numerical values for the mean and variance of the distribution, as a function of γ .</text>
<text><location><page_12><loc_7><loc_11><loc_89><loc_17></location>As mentioned before, we will return in more detail on these distributions in a companion publication, to quantify the impact of the peak constrain on their shapes. We will also make the connection with the work of Bardeen et al. (1986) more explicit - see their Appendix C. It will be also interesting to include the role of the peculiar gravity field in our description, along the lines of van de Weygaert & Bertschinger (1996), as well as to extend the work of Desjacques (2008) on the joint statistics of the shear tensor and on the dynamical aspect of the environmental dependence, within this formalism.</text>
<section_header_level_1><location><page_12><loc_7><loc_6><loc_83><loc_7></location>6 EXCURSION SET APPROACH FOR PEAKS AND DIPS IN GAUSSIAN RANDOM FIELDS</section_header_level_1>
<text><location><page_12><loc_7><loc_1><loc_89><loc_5></location>The ability to distinguish between random positions and peaks/dips, contained in Equations (1) or (11) and achieved by the algorithm presented in Section 3, is the key to implement an excursion set model for peaks and dips in Gaussian random fields. Indeed, the primary motivation (and one of the main strengths) of the proposed sampling procedure resides in its</text>
<text><location><page_13><loc_7><loc_70><loc_89><loc_87></location>direct inclusion into the excursion set framework (Epstein 1983; Peacock & Heavens 1990; Bond et al. 1991; Lacey & Cole 1993). In essence, because in our formalism the peak overdensity is simply the trace of the conditional shear tensor (recall the definitions in Section 2), and since in triaxial models of collapse the initial shape parameters are just combinations of the shear eigenvalues (Bond & Myers 1996), our prescription provides a direct way to generate the distribution of initial overdensities under the conditions that they are peaks/dips (i.e. the distribution of peak heights, when H > 0) - along with the corresponding conditional distribution of initial shapes (see Section 5). In this respect, it is then straightforward to include this part in standard excursion set algorithms, as those used for example in Chiueh & Lee (2001), Sheth & Tormen (2002) or Sandvik et al. (2007) to compute the mass function, or in Rossi, Sheth & Tormen (2011) to describe halo shapes; the 'peak/dip excursion-set-based' algorithm is also useful for computing the halo bias assuming triaxial models of structure formation (i.e. ellipsoidal collapse). The only main conceptual difference is the pre-selection of peak/dip locations, instead of random positions in the field. The mass scale of the peak will then be fixed by finding the proper σ T which satisfies the combination of ( δ T , e T , p T | H > 0 , γ ), assuming some structure formation models - as for example the ellipsoidal collapse.</text>
<text><location><page_13><loc_7><loc_55><loc_89><loc_70></location>This is particularly useful because the excursion set theory is a powerful tool for understanding various aspects of the full dynamical complexity of halo formation. Perturbations are assumed to evolve stochastically with the smoothing scale, and the problem of computing the probability of halo formation is mapped into the classical first-passage time problem in the presence of a barrier. A very elegant reformulation of this theory has been recently proposed by Maggiore & Riotto (2010a,b,c), who made several technical and conceptual improvements (i.e. no ad hoc absorbing barrier boundary conditions, account for nonmarkovianity induced by filtering, unambiguous mass association to a smoothed scale, etc.) by deriving the original excursion set theory from a path integral formulation - following a microscopical approach. These authors also noted that the failure of the standard excursion set approach may be related to the inadequacy of the oversimplified physical model adopted for halo formation (either spherical or ellipsoidal), and propose to treat the critical threshold for collapse as a stochastic variable which better captures some of the dynamical complexity of the halo formation phenomenon. Even so, they find that the non-markovian contributions do not alleviate the discrepancy between excursion set predictions and N -body simulations.</text>
<text><location><page_13><loc_7><loc_44><loc_89><loc_55></location>In addition to the problems pointed out by Maggiore & Riotto (2010a,b,c), there is also the fact that - in its standard formulation - the excursion set approach is unable to differentiate between peaks/dips and random locations in space - i.e. all points are treated equally. However, since local extrema are plausible sites for the formation of nonlinear structures and there is a good correspondence between peaks in the initial conditions and halos at late times, it may be important to differentiate between those special positions in space. The algorithm proposed here goes in this direction, since it allows one to pre-select those special points in space around which halos form (peaks), and not just random locations - and permits to associate their corresponding initial shape distribution: in essence, the computational procedure selects a special subset, among all the possible random walks considered in the standard excursion set procedure.</text>
<text><location><page_13><loc_7><loc_25><loc_89><loc_44></location>Therefore, the 'peak/dip excursion-based' algorithm can be used to study the mass function of halos and their triaxial shapes at peak/dip positions, and also the halo bias. In fact, our prescription allows one to generate the initial distributions of overdensity, ellipticity and prolateness (i.e. shape parameters) at a scale set by the variance σ T , with the constraint that δ T is a peak (i.e. the condition H > 0 on the Hessian). One can then just evolve this initial conditional shape distribution by solving a dynamical equation of collapse, and study the final shape distribution (as done for example in Rossi, Sheth & Tormen 2011) but now at peak/dip locations. Alternatively, one can adopt a 'crossing threshold', since in the excursion set approach an halo is formed when the smoothed density perturbation reaches a critical value for the first time, and the problem is reduced to a first-passage problem in the presence of a barrier (i.e. if the overdensity exceeds a critical value, the random walk stops at this scale; if not, the walk continues to smaller scales). Moreover, our numerical technique can be easily integrated in Montecarlo realizations of the trajectories obtained from a Langevin equation with colored noise (i.e. Bond et al. 1991; Robertson et al. 2009) at peak/dip locations, and even for situations where the walks are correlated - in presence of non-markovian effects, along the lines of De Simone et al. (2011a,b). In this respect, our technique is general, since any kind of filter function can be readily implemented. All these lines of research are ongoing efforts, and will be presented in several forthcoming publications.</text>
<section_header_level_1><location><page_13><loc_7><loc_19><loc_22><loc_20></location>7 CONCLUSIONS</section_header_level_1>
<text><location><page_13><loc_7><loc_1><loc_89><loc_18></location>From the joint conditional probability distribution of an ordered set of shear eigenvalues in the peak/dip picture (Rossi 2012; Section 2, Equations 1 and 11), we derived a new algorithm to sample the constrained eigenvalues of the initial shear field associated with Gaussian statistics at peak/dips positions in the correlated density field. The algorithm, described in Section 3, was then used to test and confirm several formulas presented in Rossi (2012) regarding conditional distributions and probabilities, subjected to the extremum constraint (Section 4). We found excellent agreement between numerical results and theoretical predictions (Figures 1 and 2). In addition, we also showed how the standard distributions of shear ellipticity and prolateness in triaxial models of structure formation are modified by the constraint (Section 5; Figures 4 and 5), and provided a new expression for the conditional distribution of shape parameters given the density peak requirement (Equation 45), which generalizes some previous literature work. The formula has important implications for the modeling of non-spherical dark matter halo shapes, in relation to their initial shape distribution, and is directly applicable to the ellipsoidal collapse model (Icke 1973; White & Silk 1979; Barrow & Silk 1981; Kuhlman et al. 1996; Bond & Myers 1996). In particular, our novel description is able to combine consistently, for the first time, the formalism of Bardeen et al. (1986) - based on the density</text>
<text><location><page_14><loc_7><loc_84><loc_89><loc_87></location>field - with that of Bond & Myers (1996) - based on the shear field. Along the way, we also discussed the distribution of peak heights (see Figure 3), which plays a major role in peaks theory (Bardeen et al. 1986).</text>
<text><location><page_14><loc_7><loc_71><loc_89><loc_84></location>While the primary motivation of this paper was to illustrate the new 'peak/dip excursion-set-based' algorithm, and to show a few applications focused on the morphology of the cosmic web (following up, and complementing with some more insights, the theoretical work presented in Rossi 2012), the other goal was to describe how the new sampling procedure naturally integrates into the standard excursion set framework (Epstein 1983; Peacock & Heavens 1990; Bond et al. 1991; Lacey & Cole 1993) - potentially solving some of its well-known problems. In particular, in Section 6 we argued that the ability to distinguish between random positions and peaks/dips, encoded in the algorithm and in Equations (1) and (11) derived from first principles, is indeed the key to implement a generalized excursion set model for peaks and dips in Gaussian random fields. This is the actual strength of the proposed computational procedure, since part of the failure of the original excursion set theory may be attributed to its inability to differentiate between random positions and special points (peaks) in space around which halos may form.</text>
<text><location><page_14><loc_7><loc_65><loc_89><loc_70></location>To this end, our simple prescription can be used to study the halo mass function, halo/void shapes and bias at peak/dip density locations, in conjunction with triaxial models of structure formation. All these research lines are ongoing efforts, subjects of several forthcoming publications. The essential part is the characterization of the distribution of peak heights, and of the initial shape distribution at peak/dip locations (Equations 1, 11, and 45).</text>
<text><location><page_14><loc_7><loc_46><loc_89><loc_65></location>The algorithm presented in this paper offers also a much broader spectrum of applications. This is because, as pointed out by Rossi (2012), the fact that the eigenvalues of the Hessian matrix can be used to discriminate between different types of structures in a particle distribution is fundamental to a number of structure-finding algorithms, shape-finders algorithms, and structure reconstruction on the basis of tessellations. For example, it can be used for studying the dynamics and morphology of cosmic voids - see for example van de Weygaert & Platen (2011), Bos et al. (2012), Pan et al. (2012), and the Monge-Amp'ereKantorovitch reconstruction procedure by Lavaux & Wandelt (2010) - and in several observationally-oriented applications, in particular for developing algorithms to find and classify structures in the cosmic web or in relation to its skeleton (Sahni et al. 1998; Schaap & van de Weygaert 2000; Novikov et al. 2006; Hahn et al. 2007; Romano-Diaz & van de Weygaert 2007; Forero-Romero et al. 2009; Zhang et al. 2009; Lee & Springel 2010; Arag'on-Calvo et al. 2010a,b; Platen et al. 2011; Cautun et al. 2012; Hidding et al. 2012). Another application is related to the work of Bond, Strauss & Cen (2010), who presented an algorithm that uses the eigenvectors of the Hessian matrix of the smoothed galaxy distribution to identify individual filamentary structures. In addition, since galaxy clusters are related to primordial density peaks, and there is a correspondence between structures in the evolved density field and local properties of the linear tidal shear, our theoretical framework provides a direct way to relate initial conditions and observables from galaxy clusters.</text>
<text><location><page_14><loc_7><loc_30><loc_89><loc_45></location>Other intriguing connections involve topological studies of the cosmic web, the genus statistics and Minkowski functionals (Gott et al. 1986, 1989; Park et al. 1991, 2005; Matsubara 2010), and the possibility to address open questions regarding the origin of angular momentum and halo spin within this framework; this is because the dependence of the spin alignment on the morphology of the large-scale mass distribution is due to the difference in shape of the tidal fields in different environments, and most of the halo properties depend significantly on environment, and in particular on the tidal field - i.e. the environmental dependence of halo assembly time and unbound substructure fraction has its origin from the tidal field (Wang et al. 2011). It will be also interesting to explore how the new formalism proposed here can be used to study halo spin, shape and the orbital angular momentum of subhaloes relative to the LSS, in the context of the eigenvectors of the velocity shear tensor (see the recent study by Libeskind et al. 2013). In addition, the more complex question of the local expected density field alignment/orientation distribution as a function of the local field value (Bond 1987; Lee & Pen 2002; Porciani et al. 2002; Lee, Hahn & Porciani 2009; Lee 2011) can be addressed within this framework, and is the subject of future studies.</text>
<text><location><page_14><loc_7><loc_14><loc_89><loc_30></location>On the theoretical side, we note that our algorithm is restricted to one scale (i.e. peaks and dips in the density field, as in Bardeen et al. 1986), but the extension to a multiscale peak-patch approach along the lines of Bond & Myers (1996) is doable and subject of ongoing work. As argued in Rossi (2012), this will allow to account for the role of the peculiar gravity field itself, an important aspect not considered in our formalism but discussed in detail in van de Weygaert & Bertschinger (1996). Including all these effects in our framework and translating them into a more elaborated algorithm is ongoing effort, and will allow us to make the connection with the multiscale analysis of the Hessian matrix of the density field by van de Weygaert & Bertschinger (1996) and Arag'on-Calvo et al. (2007; 2010a,b). It will also allow us to incorporate the distortion effect of the peculiar gravity field in our initial distribution of halo/void shapes (Section 5). The natural extension of the peak/dip picture for the initial shear to non-Gaussian fields is also ongoing effort, along with some other broader applications in the context of the excursion set model - for example in relation to the hot and cold spots in the Cosmic Microwave Background, including the effects of f NL -type non-Gaussianity on their shapes (i.e. Rossi, Chingangbam & Park 2011) - which will be presented in forthcoming publications.</text>
<section_header_level_1><location><page_14><loc_7><loc_7><loc_26><loc_9></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_14><loc_7><loc_1><loc_89><loc_6></location>The final stage of this work was completed during the 'APCTP-IEU Focus Program on Cosmology and Fundamental Physics III' (June 11-22, 2012) at Postech, in Pohang, Korea. I would like to thank the organizers of the workshop, and in particular Changrim Ahn. Also, many thanks to Changbom Park for a careful reading of the manuscript, and for many interesting discussions, suggestions and encouragement.</text>
<section_header_level_1><location><page_15><loc_7><loc_86><loc_19><loc_87></location>REFERENCES</section_header_level_1>
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<section_header_level_1><location><page_16><loc_7><loc_38><loc_39><loc_39></location>APPENDIX A: ESSENTIAL NOTATION</section_header_level_1>
<text><location><page_16><loc_7><loc_26><loc_89><loc_37></location>We summarize here the basic notation adopted in the paper, which is essentially the same as the one introduced by Rossi (2012) - with a few minor changes to make the connection with previous literature more explicit. In particular, let Ψ denote the displacement field, Φ the potential of the displacement field (i.e. the gravitational field), F the source of the displacement field (i.e. the density field). Both F and Φ are Gaussian random fields, the latter specified by the matter power spectrum P ( k ), with k denoting the wave number and W ( k ) the smoothing kernel. Use T ij for the shear tensor (its eigenvalues are λ 1 , λ 2 , λ 3 ), H ij for the Hessian matrix of the density F (with eigenvalues ξ 1 , ξ 2 , ξ 3 ), J ij for the Jacobian of the displacement field, where i, j = 1 , 2 , 3. Indicate with q the Lagrangian coordinate, with x the Eulerian coordinate, where x ( q ) = q +Ψ( q ). Clearly,</text>
<formula><location><page_16><loc_15><loc_22><loc_89><loc_25></location>J ij ( q ) = ∂x i ∂q j = δ ij + T ij , T ij = ∂ 2 Φ ∂q i ∂q j , H ij = ∂ 2 F ∂q i ∂q j , F ( q ) = 3 ∑ i=1 ∂ Ψ i ∂q i ≡ 3 ∑ i=1 ∂ 2 Φ ∂q 2 i . (A1)</formula>
<text><location><page_16><loc_7><loc_20><loc_55><loc_21></location>The correlations between the shear and density Hessian are expressed by:</text>
<formula><location><page_16><loc_36><loc_16><loc_89><loc_19></location>〈 T ij T kl 〉 = σ 2 T 15 ( δ ij δ kl + δ ik δ jl + δ il δ jk ) (A2)</formula>
<formula><location><page_16><loc_36><loc_11><loc_89><loc_14></location>〈 H ij H kl 〉 = σ 2 H 15 ( δ ij δ kl + δ ik δ jl + δ il δ jk ) (A3)</formula>
<text><location><page_16><loc_36><loc_6><loc_36><loc_8></location>〈</text>
<text><location><page_16><loc_36><loc_7><loc_37><loc_8></location>T</text>
<text><location><page_16><loc_37><loc_7><loc_38><loc_8></location>ij</text>
<text><location><page_16><loc_38><loc_7><loc_39><loc_8></location>H</text>
<text><location><page_16><loc_39><loc_7><loc_40><loc_8></location>kl</text>
<text><location><page_16><loc_40><loc_6><loc_41><loc_8></location>〉</text>
<text><location><page_16><loc_41><loc_7><loc_42><loc_8></location>=</text>
<text><location><page_16><loc_43><loc_8><loc_44><loc_9></location>Γ</text>
<text><location><page_16><loc_44><loc_8><loc_46><loc_9></location>TH</text>
<text><location><page_16><loc_44><loc_7><loc_45><loc_8></location>15</text>
<text><location><page_16><loc_46><loc_7><loc_46><loc_8></location>(</text>
<text><location><page_16><loc_46><loc_7><loc_47><loc_8></location>δ</text>
<text><location><page_16><loc_47><loc_7><loc_48><loc_8></location>ij</text>
<text><location><page_16><loc_48><loc_7><loc_49><loc_8></location>δ</text>
<text><location><page_16><loc_49><loc_7><loc_50><loc_8></location>kl</text>
<text><location><page_16><loc_50><loc_7><loc_51><loc_8></location>+</text>
<text><location><page_16><loc_51><loc_7><loc_52><loc_8></location>δ</text>
<text><location><page_16><loc_52><loc_7><loc_53><loc_8></location>ik</text>
<text><location><page_16><loc_53><loc_7><loc_54><loc_8></location>δ</text>
<text><location><page_16><loc_54><loc_7><loc_55><loc_8></location>jl</text>
<text><location><page_16><loc_55><loc_7><loc_56><loc_8></location>+</text>
<text><location><page_16><loc_56><loc_7><loc_57><loc_8></location>δ</text>
<text><location><page_16><loc_57><loc_7><loc_58><loc_8></location>il</text>
<text><location><page_16><loc_58><loc_7><loc_59><loc_8></location>δ</text>
<text><location><page_16><loc_59><loc_7><loc_60><loc_8></location>jk</text>
<text><location><page_16><loc_60><loc_7><loc_60><loc_8></location>)</text>
<text><location><page_16><loc_86><loc_7><loc_89><loc_8></location>(A4)</text>
<text><location><page_16><loc_7><loc_4><loc_62><loc_6></location>where σ 2 T = S 2 ≡ σ 2 0 , σ 2 H = S 6 ≡ σ 2 2 , Γ TH = S 4 ≡ σ 2 1 , δ ij is the Kronecker delta, and</text>
<formula><location><page_16><loc_37><loc_1><loc_89><loc_4></location>S n = 1 2 π 2 ∫ ∞ 0 k n P ( k ) W 2 ( k )d k (A5)</formula>
<formula><location><page_17><loc_32><loc_82><loc_89><loc_85></location>σ 2 j = 1 2 π 2 ∫ ∞ 0 k 2(j+1) P ( k ) W 2 ( k ) d k ≡ S 2(j+1) . (A6)</formula>
<text><location><page_17><loc_7><loc_78><loc_89><loc_82></location>In the main text we prefer to use σ T and σ H , rather than the more familiar σ 0 and σ 2 , for their intuitive meaning. In particular, the subscript T always indicates that a quantity is related to the shear field, while the subscript H denotes a quantity linked to the Hessian of the density field.</text>
<text><location><page_17><loc_7><loc_70><loc_89><loc_78></location>The shear and density Hessian T ij and H ij are real symmetric tensors, so they are specified by 6 components. Whenever necessary, we label those components with the symbols α or β to indicate the various couples, where α, β = (1 , 1) , (2 , 2) , (3 , 3) , (1 , 2) , (1 , 3) , (2 , 3). It is also useful to introduce the vectors T and H , derived from the components of their corresponding tensors, i.e. T = ( T 11 , T 22 , T 33 , T 12 , T 13 , T 23 ) and H = ( H 11 , H 22 , H 33 , H 12 , H 13 , H 23 ). The constrained eigenvalues of the matrix having components T α | H α will be indicated with ζ 1 , ζ 2 , ζ 3 . As shown in Rossi (2012), the covariance matrix of the joint probability distribution of T and H is simply</text>
<formula><location><page_17><loc_32><loc_66><loc_89><loc_69></location>V = ( 〈 T α T α 〉 〈 T α H β 〉 〈 H β T α 〉 〈 H β H β 〉 ) = 1 15 ( σ 2 T A Γ TH A Γ TH A σ 2 H A ) (A7)</formula>
<text><location><page_17><loc_7><loc_64><loc_11><loc_66></location>where</text>
<formula><location><page_17><loc_37><loc_59><loc_89><loc_64></location>A = ( B /circledivide /circledivide I ) , B = 3 1 1 1 3 1 1 1 3 (A8)</formula>
<text><location><page_17><loc_7><loc_56><loc_89><loc_59></location>with I a (3 × 3) identity matrix and /circledivide a (3 × 3) null matrix. Finally, an important 'spectral parameter' often used here is the 'reduced' correlation:</text>
<formula><location><page_17><loc_40><loc_53><loc_89><loc_56></location>γ = Γ TH /σ T σ H = σ 2 1 σ 0 σ 2 , (A9)</formula>
<text><location><page_17><loc_7><loc_46><loc_89><loc_52></location>which plays a crucial role in peaks theory (i.e. Bardeen et al. 1986). If Gaussian filters are used in (A6), then our γ is the same as the one introduced in Bardeen et al. (1986) - specified by their Equation (4.6a). Note that in the main text we also define η = γσ T /σ H ; if one adopts reduced variables (i.e. T α and H α normalized by their corresponding rms values σ T and σ H ), clearly η ≡ γ .</text>
<section_header_level_1><location><page_17><loc_7><loc_43><loc_65><loc_44></location>APPENDIX B: INVARIANTS FROM THE CONDITIONAL FORMULAS</section_header_level_1>
<text><location><page_17><loc_7><loc_29><loc_89><loc_42></location>The algebra presented in Section 3 allows one to gain more insights into the joint conditional distribution of eigenvalues in the peak/dip picture (Equation 1). For simplicity, in what follows we consider 'reduced' variables, so that the various components of T and H are now normalized by their corresponding rms values ( σ T and σ H , respectively). With some abuse of notation, we omit the tilde symbol (used instead in Rossi 2012) to distinguish between normalized and unnormalized quantities. It is then possible to characterize and study the properties of the first few elementary symmetric functions of degree n for the density Hessian, the shear tensor, and the conditional shear tensor - along the lines of Weyl (1948), Doroshkevich (1970), Sheth & Tormen (2002) and Desjacques (2008). In particular, it is direct to note that, using Equation (23) and considering six independent Gaussian random variates y i ( i = 1 , 6) represented by the six-dimensional vector y , one obtains that the first two classical invariants</text>
<formula><location><page_17><loc_7><loc_26><loc_89><loc_28></location>h 1 = H 11 + H 22 + H 33 = -y 1 (B1)</formula>
<formula><location><page_17><loc_15><loc_24><loc_21><loc_26></location>-5</formula>
<formula><location><page_17><loc_7><loc_25><loc_89><loc_27></location>h 2 3 = h 2 1 3 h 2 = 1 ( y 2 2 + y 2 3 + y 2 4 + y 2 5 + y 2 6 ) (B2)</formula>
<text><location><page_17><loc_7><loc_23><loc_33><loc_24></location>are independent. This fact implies that</text>
<formula><location><page_17><loc_32><loc_19><loc_89><loc_22></location>p ( H ) = e -h 2 1 / 2 √ 2 π 15 3 8 √ 10 π 5 / 2 e -5 h 2 3 / 2 ≡ p ( h 1 ) p ( h 3 ) , (B3)</formula>
<text><location><page_17><loc_7><loc_14><loc_89><loc_18></location>hence p ( H ) is the product of two independent distributions, where in particular p ( h 1 ) is a one-dimensional Gaussian with mean zero and unity variance. Note also that p ( H )d H = p ( y )d y , where p ( y ) ≡ ∏ 6 i=1 g i is simply the product of six independent one-dimensional zero mean unit variance Gaussians g i .</text>
<text><location><page_17><loc_10><loc_13><loc_43><loc_14></location>Similarly, for the shear tensor T , one obtains that</text>
<formula><location><page_17><loc_7><loc_10><loc_89><loc_12></location>k 1 = T 11 + T 22 + T 33 = -z 1 (B4)</formula>
<formula><location><page_17><loc_7><loc_8><loc_89><loc_10></location>k 2 3 = k 2 1 3 k 2 = ( z 2 2 + z 2 3 + z 2 4 + z 2 5 + z 2 6 ) (B5)</formula>
<formula><location><page_17><loc_15><loc_8><loc_21><loc_10></location>-1 5</formula>
<text><location><page_17><loc_7><loc_5><loc_89><loc_7></location>are also independent, with z i ( i = 1 , 6) other six Gaussian random variates represented by the six-dimensional vector z . Therefore,</text>
<formula><location><page_17><loc_33><loc_1><loc_89><loc_4></location>p ( T ) = e -k 2 1 / 2 √ 2 π 15 3 8 √ 10 π 5 / 2 e -5 k 2 3 / 2 ≡ p ( k 1 ) p ( k 3 ) (B6)</formula>
<section_header_level_1><location><page_18><loc_7><loc_89><loc_24><loc_90></location>18 Graziano Rossi</section_header_level_1>
<text><location><page_18><loc_7><loc_84><loc_89><loc_87></location>with p ( k 1 ) a one-dimensional Gaussian distribution. Note again that p ( T )d T = p ( z )d z , where p ( z ) ≡ ∏ 6 i=1 g i is the product of six independent one-dimensional zero mean unit variance Gaussians g i .</text>
<text><location><page_18><loc_7><loc_82><loc_89><loc_84></location>Following the previous logic, one would naturally expect that the quantities K 1 and K 2 3 = K 2 1 -3 K 2 , defined in the main text (see Equation 2), should also be independent. Indeed, it is direct to obtain that (Ravi Sheth, private communication):</text>
<formula><location><page_18><loc_7><loc_72><loc_89><loc_81></location>K 1 = = -( m 1 -γy 1 ) = -√ 1 -γ 2 l 1 (B7) K 2 3 = K 2 1 -3 K 2 = 1 5 [( m 2 -γy 2 ) 2 +( m 3 -γy 3 ) 2 +( m 4 -γy 4 ) 2 +( m 5 -γy 5 ) 2 +( m 6 -γy 6 ) 2 ] = (1 -γ 2 ) 5 ( l 2 2 + l 2 3 + l 2 4 + l 2 5 + l 2 6 ) , (B8)</formula>
<text><location><page_18><loc_7><loc_66><loc_89><loc_71></location>where l i ( i = 1 , 6) are other six independent Gaussian distributed variates with mean zero and unity variance, while m i ( i = 1 , 6) are six Gaussian distributed variates with shifted mean γy i and reduced variance (1 -γ 2 ), i.e. m i = γy i + √ 1 -γ 2 l i . Hence, the joint conditional distribution of eigenvalues in the peak/dip picture (Equation 1) can be written as the product of two independent distributions as follows:</text>
<formula><location><page_18><loc_24><loc_60><loc_89><loc_65></location>p ( T | H , γ ) = e -K 2 1 / [2(1 -γ 2 )] √ 2 π (1 -γ 2 ) 15 3 8 √ 10 π 5 / 2 e -5 K 2 3 / [2(1 -γ 2 )] (1 -γ 2 ) 5 / 2 ≡ p ( K 1 | γ ) p ( K 3 | γ ) . (B9)</formula>
<text><location><page_18><loc_7><loc_55><loc_89><loc_61></location>This latter expression clearly shows that the distribution of constrained K 1 ≡ ∆ T | H is independent of the distribution of the constrained angular momentum K 2 3 , where in particular p ( K 1 | γ ) is a Gaussian with zero mean and variance given by (1 -γ 2 ), while p ( K 3 ) is a chi-square distribution with five degrees of freedom. Once again, note that p ( T | H , γ )d( T | H ) = p ( m | y , γ )d( m | y ), where now</text>
<formula><location><page_18><loc_34><loc_50><loc_89><loc_55></location>p ( m | y , γ ) ≡ 6 ∏ i=1 e ( m i -γy i ) 2 / 2(1 -γ 2 ) √ 2 π (1 -γ 2 ) ≡ 6 ∏ i=1 t i ; (B10)</formula>
<text><location><page_18><loc_7><loc_47><loc_89><loc_50></location>namely, p ( m | y , γ ) is now the product of six independent one-dimensional Gaussians t i with shifted mean γy i and reduced variance (1 -γ 2 ) - represented by the six-dimensional vector m | y .</text>
</document> | [] |
2018arXiv180102063Z | https://arxiv.org/pdf/1801.02063.pdf | <document>
<section_header_level_1><location><page_1><loc_13><loc_92><loc_88><loc_93></location>Scalar field vs hydrodynamic models in the homogeneous isotropic cosmology</section_header_level_1>
<text><location><page_1><loc_36><loc_89><loc_65><loc_90></location>V. I. Zhdanov 1, ∗ and S. S. Dylda 2, †</text>
<text><location><page_1><loc_25><loc_86><loc_75><loc_88></location>1 Astronomical Observatory, Taras Shevchenko National University of Kyiv 2 Physical Faculty, Taras Shevchenko National University of Kyiv</text>
<text><location><page_1><loc_18><loc_70><loc_83><loc_85></location>We study relations between hydrodynamical (H) and scalar field (SF) models of the dark energy in the homogeneous isotropic Universe. The focus is on SF described by the Lagrangian with the canonical kinetic term within spatially flat cosmology. We analyze requirements that guarantee the same cosmological history for the SF and H models at least for special solutions. The differential equation for the SF potential is obtained that ensures such equivalence of the SF and H-models. However, if the 'equivalent' SF potential is found for given equation of state (EOS) of the H-model, this does not mean that all solutions of this SF-model have corresponding H-model analogs. In this view we derived a condition that guarantees an 'approximate equivalence', when there is a small difference between energy-momentum tensors of the models. The 'equivalent' SF potentials and corresponding SF solutions for linear EOS are found in an explicit form; we also present examples with more complicated EOSs.</text>
<text><location><page_1><loc_18><loc_68><loc_33><loc_69></location>PACS numbers: 98.80.Cq</text>
<section_header_level_1><location><page_1><loc_20><loc_64><loc_37><loc_65></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_38><loc_49><loc_62></location>Observations show [1] that about 70% of the average mass density in the Universe owes to the dark energy (DE) which drives the acceleration of the cosmological expansion. It is widely assumed that some form of DE or its constituents that dominated in the very early Universe must have a dynamical nature owing to an action of unknown physical fields and/or due to modifications of the General Relativity [2, 3]. Theories with scalar fields (SF) occupy an important sector of this area (see, e.g., [2-6]). Though observational data restrict some of the SF models [4], there is still a considerable uncertainty in their choice, not to mention the revision of the underlying gravitational theory [3, 5]. The abundance of various cosmological models draws attention to unifying schemes and interrelations between competing dark energy candidates [5-7] that can be used to explain observational data.</text>
<text><location><page_1><loc_9><loc_16><loc_49><loc_37></location>To this end, the phenomenological hydrodynamic approach is often used [5, 6, 8-10]. It is well known that the matter in the spatially homogeneous and isotropic Universe can be described by means of the energymomentum tensor of an ideal fluid. Under certain conditions SF models allow for a hydrodynamic (H) description with some 'equivalent' equation of state (EOS) beyond homogeneity as well [9]. The H analogs of the SF models typically involve such phenomenological parameters as the EOS parameter, effective sound speed, adiabatic sound speed, which can be limited in view of available astronomical data [9, 11]. The transition from simplest hydrodynamical EOSs to SFs and vice versa deals with rather unusual models, whereas in the spirit of Occam's razor, it would be desirable to restrict the choice</text>
<text><location><page_1><loc_52><loc_45><loc_92><loc_65></location>of the SF Lagrangian to canonical one, which is more familiar from the point of view of particle physics. This is possible within the approach of papers [5, 6, 8, 12], which treat the equivalence problems by direct comparing solutions for the cosmological scale factor and the energy density in the homogeneous isotropic Universe. As distinct from these papers, we propose a differential equation for the SF potential in closed form guaranteeing some equivalence of H and SF models. We note, however, that in any approach, the H-SF correspondence is not universal; this is well known though not always clearly stated. A typical situation is that two different models mimic each other for some area of the original data, but they have different solutions outside this area.</text>
<text><location><page_1><loc_52><loc_29><loc_92><loc_45></location>We study the H-SF equivalence on the basis of equality of the corresponding energy-momentum tensors (Section II). This problem becomes more complicated if we impose some additional conditions either on the form of the EOS, or on the SF Lagrangian. We focus on the relationship between the H-model and the SF-model with the canonical kinetic term and a self-interaction potential for the real SF. We call this 'restricted 1 equivalence', in contrast to the case, when no such restrictions are imposed. As a result, the equivalence considerations deal with some restrictions on the initial data (Section III).</text>
<text><location><page_1><loc_52><loc_16><loc_92><loc_29></location>Then we consider a case, when the relations that guarantee some kind of equivalence of SF and H models are valid approximately. In this case the equations of H and SF models can lead to different energy-momentum tensors (and correspondingly different evolution equations), and the question is when this difference remains small, provided that it is small at the initial moment. We derived conditions for such approximate equivalence (Section IV).</text>
<text><location><page_1><loc_52><loc_13><loc_92><loc_16></location>The results are applied to the linear EOS, including situation near the phantom line (subsection VI A), to some</text>
<text><location><page_2><loc_9><loc_86><loc_49><loc_93></location>non-linear EOS known from papers [5, 12] (subsection VIB), and to a simple two-parametric EOS (subsection VIC). Here we present examples showing when one can speak about an equivalence between the H and SF models.</text>
<section_header_level_1><location><page_2><loc_14><loc_82><loc_43><loc_83></location>II. GENERAL CONSIDERATIONS</section_header_level_1>
<text><location><page_2><loc_9><loc_76><loc_49><loc_80></location>General Lagrangian L ( X,ϕ ) for the real SF ϕ with X = 1 2 ϕ, µ ϕ ,µ and space-time metric g µν yields the energy-momentum tensor 2</text>
<formula><location><page_2><loc_10><loc_70><loc_49><loc_74></location>T ( sf ) µν = 2 √ -g ∂ ∂g µν [ √ -gL ] = ∂L ∂X ϕ ,µ ϕ ,ν -g µν L, (1)</formula>
<text><location><page_2><loc_9><loc_67><loc_49><loc_70></location>that can be equated to the energy-momentum tensor of an ideal fluid</text>
<formula><location><page_2><loc_21><loc_64><loc_49><loc_66></location>T ( h ) µν = hu µ u ν -pg µν , (2)</formula>
<text><location><page_2><loc_9><loc_56><loc_49><loc_63></location>where h = e + p is the specific enthalpy, p is the pressure, e is the invariant energy density and u µ is the four-velocity of the fluid. It is assumed that some EOS is known that relates the pressure to the other parameters of the problem: p = P ( e, ϕ ).</text>
<text><location><page_2><loc_10><loc_54><loc_27><loc_56></location>We have T ( sf ) µν = T ( h ) µν if</text>
<formula><location><page_2><loc_14><loc_52><loc_49><loc_53></location>p = L ( X,ϕ ) , h = 2 X∂L/∂X, X > 0 . (3)</formula>
<text><location><page_2><loc_9><loc_43><loc_49><loc_50></location>At the points where X changes its sign (i.e., ϕ ,µ is not timelike), the hydrodynamical interpretation is no longer valid. The relations (3) yield an ordinary differential equation with respect to L ( X,ϕ ), where ϕ is involved as a parameter:</text>
<formula><location><page_2><loc_19><loc_40><loc_49><loc_42></location>E ( L, ϕ ) = 2 X∂L/∂X -L. (4)</formula>
<text><location><page_2><loc_9><loc_29><loc_49><loc_39></location>The solution L of (4) exists in case of rather a general EOS; this solution contains an arbitrary function of ϕ . Additional constraints that ensure equality of (1) and (2) for all values of ϕ and its derivatives are outlined in Appendix A. These constraints are fulfilled identically in case of an homogeneous isotropic Universe to be discussed further.</text>
<text><location><page_2><loc_9><loc_19><loc_49><loc_29></location>However, under additional restrictions on the functions L ( X,ϕ ) and/or E ( p, ϕ ) in (4) the solution of this equation for all X,ϕ may not exist. For example, if we want to define the EOS parametrically from (3), then a general Lagrangian L cannot yield the barotropic EOS, because in this case the right-hand sides of (3) may depend on two independent variables X and ϕ .</text>
<text><location><page_2><loc_9><loc_15><loc_49><loc_19></location>In this view we shall require that equations (3,4) be satisfied not for arbitrary hydrodynamical and/or SF variables, but only for certain cosmological solutions in the</text>
<text><location><page_2><loc_52><loc_89><loc_92><loc_93></location>isotropic homogeneous Universe. We wonder, is it possible to compare H-model with the SF-model, if the SF Lagrangian has the canonical form</text>
<formula><location><page_2><loc_66><loc_86><loc_92><loc_88></location>L = X -V ( ϕ ) . (5)</formula>
<text><location><page_2><loc_52><loc_84><loc_71><loc_85></location>Then equations (3,4) yield</text>
<formula><location><page_2><loc_58><loc_81><loc_92><loc_83></location>e = X + V ( ϕ ) , p ( e, ϕ ) = X -V ( ϕ ) . (6)</formula>
<text><location><page_2><loc_52><loc_66><loc_92><loc_80></location>If we demanded that (6) be fulfilled for all variables ( e, ϕ ) or ( X,ϕ ), we would have very special EOS p = e -2 V ( ϕ ), whereas in case of another equations of state the relations (6) cannot be identities. However, we deal with the unique Universe, so in fact we do not need that the 'equivalence' conditions be satisfied for all possible values of the variables that enter EOS and/or Lagrangian. If we compare different cosmological models, then the main question is when they predict the same observational data, when they mimic each other etc.</text>
<text><location><page_2><loc_52><loc_57><loc_92><loc_66></location>We say that there is an equivalence of H and SF models, if both predict the same Hubble diagram and, consequently, the same Hubble parameter H ( z ) as a function of the redshift z . In this case we have the same H ( t ) as a function of the cosmological time t , yielding the same 3 dependence of the cosmological scale factor a ( t ).</text>
<text><location><page_2><loc_52><loc_46><loc_92><loc_57></location>Within the homogeneous isotropic cosmology, a solution a ( t ) , ϕ ( t ) of the SF-model 4 yields e ( t ) , p ( t ) as a parametric representation of EOS due to (6) and vice versa; this approach is often considered (see, e.g. [5, 6]). The main difference of the present paper is that we are looking for a direct criterion on the SF potential V ( ϕ ), which yields the same cosmological history as in the case of H-model with the prescribed EOS.</text>
<section_header_level_1><location><page_2><loc_56><loc_42><loc_88><loc_43></location>III. STATEMENT OF THE PROBLEM</section_header_level_1>
<text><location><page_2><loc_52><loc_37><loc_92><loc_40></location>We consider the spatially flat cosmology described by the Friedmann-Lemaitre-Robertson-Walker metric</text>
<formula><location><page_2><loc_54><loc_32><loc_92><loc_36></location>ds 2 = g µν dx µ dx ν = dt 2 -a 2 ( t ) [ dχ 2 + χ 2 dO 2 ] . (7)</formula>
<text><location><page_2><loc_52><loc_25><loc_92><loc_33></location>It should be noted that the supposition of spatial flatness agrees with observations [1] and is perfectly explained in the framework of widely known ideas of inflation in the early Universe [4]. In case of the Universe filled with an ideal fluid we have the Friedmann equations (spatially flat case)</text>
<formula><location><page_2><loc_64><loc_21><loc_92><loc_24></location>d 2 a dt 2 = -4 π 3 a ( e +3 p ) , (8)</formula>
<formula><location><page_2><loc_62><loc_16><loc_92><loc_19></location>H 2 = 8 π 3 e, H = a -1 da/dt (9)</formula>
<text><location><page_3><loc_9><loc_92><loc_46><loc_93></location>( G = c = 1) . One more (hydrodynamical) equation</text>
<formula><location><page_3><loc_22><loc_89><loc_49><loc_91></location>˙ e +3 H ( e + p ) = 0 (10)</formula>
<text><location><page_3><loc_9><loc_82><loc_49><loc_88></location>also follows from (8,9); on the other hand, (8) follows from (9,10). Further we use (9,10) as the independent equations taking in mind that they must be supplemented by an equation of state.</text>
<text><location><page_3><loc_9><loc_78><loc_49><loc_82></location>In case of the isotropic homogeneous Universe filled with uniform scalar field ϕ = ϕ ( t ); X = ˙ ϕ 2 / 2 the evolution equations corresponding to (5) are</text>
<formula><location><page_3><loc_19><loc_72><loc_49><loc_77></location>d 2 a dt 2 = -8 π 3 a ( ˙ ϕ 2 -V ( ϕ ) ) , (11)</formula>
<formula><location><page_3><loc_25><loc_69><loc_49><loc_72></location>H 2 = 8 π 3 e f , (12)</formula>
<text><location><page_3><loc_9><loc_65><loc_49><loc_68></location>where e f ≡ ˙ ϕ 2 / 2 + V ( ϕ ) is the field energy density, and the field equation is</text>
<formula><location><page_3><loc_21><loc_62><loc_49><loc_64></location>¨ ϕ +3 H ˙ ϕ + V ' ( ϕ ) = 0 . (13)</formula>
<text><location><page_3><loc_9><loc_57><loc_49><loc_61></location>Analogously, these equations are not independent and we use further (12, 13) as the evolution equations of the SF-model with the initial conditions</text>
<formula><location><page_3><loc_20><loc_54><loc_49><loc_56></location>˙ ϕ ( t 0 ) = ˙ ϕ 0 , ϕ ( t 0 ) = ϕ 0 . (14)</formula>
<text><location><page_3><loc_9><loc_50><loc_49><loc_53></location>If (9,10) and (12,13) are fulfilled with the same H ( t ), then</text>
<formula><location><page_3><loc_21><loc_47><loc_49><loc_49></location>e = e f ≡ ˙ ϕ 2 / 2 + V ( ϕ ) , (15)</formula>
<text><location><page_3><loc_9><loc_45><loc_47><loc_46></location>and substituting this into (10) and using (13), we get</text>
<formula><location><page_3><loc_25><loc_42><loc_49><loc_44></location>˙ ϕ 2 = e + p. (16)</formula>
<text><location><page_3><loc_9><loc_30><loc_49><loc_41></location>Analogously, on account of (15, 16) equations (12,13) yield (9,10). Conversely, (9,10) and (13,15) yield (16). The relations (15,16) are necessary for the equivalence of H and SF models. These conclusions do not depend on either p = P ( e ) or p = P ( e, ϕ ). However, the statement of the initial value problem of H-model and its comparison with SF-models looks somewhat different in case of (i) one-parametric and (ii) two-parametric EOS.</text>
<text><location><page_3><loc_9><loc_27><loc_49><loc_30></location>(i) Barotropic EOS: p = P ( e ). The H-model is defined by equations (9,10) with the initial condition</text>
<formula><location><page_3><loc_25><loc_24><loc_49><loc_25></location>e ( t 0 ) = e 0 (17)</formula>
<text><location><page_3><loc_9><loc_9><loc_49><loc_23></location>(ii) Two-parametric EOS . We shall see below that the requirement of equivalence of H and SF models imposes severe limitations on cosmological solutions. In order to generalize the discussion and verify that the limitations are not due to the one-parametric form (i), we consider EOS that contains two parameters. Following [10] we suppose that p = P ( e, φ ); this generalization can be used to construct phenomenological models of the dark energy. Obviously, this demands that dynamical equation for the additional variable φ must be also involved in the</text>
<text><location><page_3><loc_52><loc_76><loc_92><loc_93></location>H-model. Now we wonder, is it possible to describe the solutions of this model with the help of some scalar field ϕ alone, without using the hydrodynamic variables? The very first step in this direction and the most economic way within our 'restricted' approach is to suppose that φ = ϕ obeys the same equation (13). Therefore, we assume that the equations of the H-model include (9,10,13) with corresponding initial conditions (14,17). Our formal aim is to find criteria for existence of the H-model solution e ( t ) and the SF-model solution ϕ ( t ) with the same H ( t ), such that e f ( t ) ≡ e ( t ).</text>
<text><location><page_3><loc_52><loc_64><loc_92><loc_77></location>Obviously, considering (ii) of the H-model with twoparametric EOS and its comparison to the SF-model differs from considering (i), in particular, because we have different dimensions of the space of initial data. However, the mathematics we deal with below is formally the same and the equivalence criterion (26) derived below is applicable both to (i) and (ii). So further we work with (ii), having in mind the reservation concerning the difference of (i) and (ii).</text>
<text><location><page_3><loc_52><loc_61><loc_92><loc_64></location>We assume h ( e, ϕ ) to be a continuously differentiable function of e, ϕ . Further for brevity we denote</text>
<formula><location><page_3><loc_60><loc_58><loc_84><loc_60></location>G ( x, y ) ≡ x 2 -h ( x 2 / 2 + V ( y ) , y ) .</formula>
<text><location><page_3><loc_52><loc_55><loc_92><loc_58></location>Using this function, in view of the relations (15,16), we have</text>
<formula><location><page_3><loc_66><loc_53><loc_92><loc_54></location>G ( ˙ ϕ ( t ) , ϕ ( t )) = 0 . (18)</formula>
<text><location><page_3><loc_52><loc_43><loc_92><loc_51></location>As we have seen, this condition along with (15) ensures that both H-model and SF-model lead to the same Hubble diagram (at least for specially chosen initial data). In this sense we speak about 'restricted 5 equivalence' of H and SF-models. The condition (18) must be fulfilled for initial data (17) as well:</text>
<formula><location><page_3><loc_65><loc_40><loc_92><loc_42></location>G ( ˙ ϕ ( t 0 ) , ϕ ( t 0 )) = 0 . (19)</formula>
<text><location><page_3><loc_52><loc_36><loc_92><loc_39></location>Also, we shall consider deviations from equation (18); in this case we consider the function</text>
<formula><location><page_3><loc_65><loc_34><loc_92><loc_36></location>g ( t ) = G ( ˙ ϕ ( t ) , ϕ ( t )) . (20)</formula>
<text><location><page_3><loc_52><loc_26><loc_92><loc_33></location>It should be noted that for fixed V ( ϕ ), h ( e, ϕ ) it is generally impossible to satisfy (19) with ∀ ˙ ϕ 0 , ϕ 0 ; this relation singles out a particular solution to equation (13). Therefore most of solutions of the SF-model cannot be H-model solutions.</text>
<text><location><page_3><loc_52><loc_18><loc_92><loc_26></location>After the comments about the initial data we proceed to conditions for the potential, which must be fulfilled to ensure (18). We suppose that the function h ( e, ϕ ) is known. The problem we are interested in can be formulated as follows.</text>
<text><location><page_3><loc_52><loc_14><loc_92><loc_18></location>A. Let ϕ ( t ) be a solution of (12,13). What are sufficient conditions for V ( ϕ ) so as to ensure g ( t ) ≡ 0 at least for special initial data (14) satisfying (19)?</text>
<text><location><page_4><loc_9><loc_88><loc_49><loc_93></location>After finding potential V ( ϕ ) that solves the problem (A) for special initial data satisfying (19), it is natural to ask about another solutions of the SF-model with the same potential, which do not satisfy (19).</text>
<text><location><page_4><loc_9><loc_85><loc_49><loc_87></location>B. Let for some V ( ϕ ) there are solutions ϕ ( t ) , ¯ ϕ ( t ) of equations (12,13), and</text>
<formula><location><page_4><loc_15><loc_82><loc_49><loc_84></location>G ( ˙ ¯ ϕ ( t ) , ¯ ϕ ( t )) = 0 , G ( ˙ ϕ ( t ) , ϕ ( t )) = 0 . (21)</formula>
<text><location><page_4><loc_39><loc_81><loc_39><loc_83></location>/negationslash</text>
<text><location><page_4><loc_9><loc_80><loc_35><loc_81></location>What we can say about g ( t ) in (20)?</text>
<text><location><page_4><loc_9><loc_75><loc_49><loc_79></location>If ϕ ( t 0 ) satisfies (19) approximately, will this approximation work for t > t 0 ? If yes, we can say that we have an 'approximate equivalence' of H and SF-models.</text>
<section_header_level_1><location><page_4><loc_11><loc_71><loc_46><loc_72></location>IV. COMPARISON OF S AND H MODELS</section_header_level_1>
<text><location><page_4><loc_9><loc_65><loc_49><loc_69></location>The equation (20) can be solved with respect to ˙ ϕ 2 . With this aim we introduce function Θ( V, g, ϕ ), which is defined as a solution of the equation</text>
<formula><location><page_4><loc_20><loc_63><loc_49><loc_64></location>Θ = g + h (Θ / 2 + V, ϕ ) . (22)</formula>
<text><location><page_4><loc_9><loc_59><loc_49><loc_61></location>Uniqueness of solution of (22) can be easily established if, for ε = const > 0 (arbitrarily small),</text>
<formula><location><page_4><loc_25><loc_55><loc_49><loc_58></location>∂h ∂e ≤ 2 -ε. (23)</formula>
<text><location><page_4><loc_9><loc_38><loc_49><loc_54></location>The uniqueness follows from consideration of ζ ( ϑ ) = ϑ -h ( ϑ/ 2 + V, ϕ ), which is monotonically increasing function of ϑ . Then ζ ( ϑ ) takes the value ζ ( ϑ ) = g only once, therefore we have a unique solution ϑ = Θ( V, g, ϕ ) of (22). A sufficient condition of existence is h ( V, ϕ ) ≥ -g , because in this case ζ (0) ≤ g and ζ ( ϑ ) →∞ as ϑ →∞ due to (23); so in virtue of continuity of ζ ( ϑ ) there exists the solution ϑ of (22). In case of g = 0 this sufficient condition is simply the requirement for the positive specific enthalpy. Note that (23) means ∂P/∂e ≤ 1 -ε < 1, which avoids superluminal speed of sound.</text>
<text><location><page_4><loc_9><loc_35><loc_49><loc_38></location>We also introduce E ( V, g, ϕ ) = Θ( V, g, ϕ ) / 2 + V that satisfies the equation</text>
<formula><location><page_4><loc_20><loc_31><loc_49><loc_34></location>E -1 2 h ( E,ϕ ) = 1 2 g + V. (24)</formula>
<text><location><page_4><loc_9><loc_28><loc_49><loc_30></location>Further we consider solutions of (24) such that e = E ( V, g, ϕ ) > 0 , ˙ ϕ 2 = Θ( V, g, ϕ ) > 0.</text>
<text><location><page_4><loc_9><loc_25><loc_49><loc_27></location>After differentiation of (20) and in view of (12,13) we get</text>
<formula><location><page_4><loc_12><loc_20><loc_49><loc_24></location>˙ g = -˙ ϕ [ ˙ ϕ √ 24 πe ( 2 -∂h ∂e ) + ∂h ∂ϕ +2 dV dϕ ] , (25)</formula>
<text><location><page_4><loc_9><loc_16><loc_48><loc_19></location>where h ≡ h ( e, ϕ ) and we denote e = 1 2 ˙ ϕ 2 + V ( ϕ ) > 0. If we require g ≡ 0, then, for ˙ ϕ = 0, we have</text>
<text><location><page_4><loc_33><loc_16><loc_33><loc_18></location>/negationslash</text>
<formula><location><page_4><loc_13><loc_11><loc_49><loc_15></location>dV dϕ + 1 2 ∂h ∂ϕ + S ( 2 -∂h ∂e ) √ 6 πE 0 Θ 0 = 0 , (26)</formula>
<text><location><page_4><loc_9><loc_9><loc_49><loc_11></location>where h = h ( e, ϕ ), e = E 0 ( V, ϕ ) ≡ E ( V, 0 , ϕ ), Θ 0 ( V, ϕ ) ≡ Θ( V, 0 , ϕ ), S = sign( ˙ ϕ ), and we used (22,24). Note that</text>
<text><location><page_4><loc_52><loc_90><loc_92><loc_93></location>equation (26) is a formal consequence of (25) only for those ϕ that belong to the range of solutions ϕ ( t ) of (13).</text>
<text><location><page_4><loc_52><loc_83><loc_92><loc_90></location>The differential equation (26) for the potential V ( ϕ ) is a sufficient condition to have (18) provided that g ( t 0 ) = 0. Thus, the problem (A) of equivalence is reduced to the equation for potential V ( ϕ ) in closed form, which, however, is different for different signs of ˙ ϕ .</text>
<text><location><page_4><loc_52><loc_80><loc_92><loc_83></location>In case of a barotropic EOS h = h ( e ) equation (26) is simplified to the form</text>
<formula><location><page_4><loc_55><loc_74><loc_92><loc_79></location>dV dϕ = -S ( 2 -∂h ∂e ) √ 6 πE 0 ( V, ϕ )Θ 0 ( V, ϕ ) . (27)</formula>
<text><location><page_4><loc_52><loc_71><loc_92><loc_74></location>In virtue of (24) we have for V = E 0 -h ( E 0 ) / 2. Substitution to (26) yields more compact equation</text>
<formula><location><page_4><loc_63><loc_66><loc_92><loc_70></location>dE 0 dϕ = -S √ 24 π E 0 h ( E 0 ) . (28)</formula>
<text><location><page_4><loc_52><loc_53><loc_92><loc_66></location>For given EOS, equations (27,28) allow to find V ( ϕ ) such that certain modes of cosmological evolution H ( t ) , e ( t ) = e f ( t ) can be obtained by means of either H-model or SFmodel. This, however, does not apply to all possible solutions to this SF-model, in particular, when ˙ ϕ changes its sign. From (27) it follows that the potential V ( ϕ ) must be a monotonically increasing function provided that we consider an interval where ˙ ϕ < 0. This includes, e.g., the case of the slow-roll modes of the chaotic inflation.</text>
<section_header_level_1><location><page_4><loc_54><loc_48><loc_89><loc_49></location>V. INITIAL DATA NOT SATISFYING (19)</section_header_level_1>
<text><location><page_4><loc_52><loc_42><loc_92><loc_46></location>Now we proceed to (B). Let V = V ( ϕ ) satisfies (26) and ¯ ϕ ( t ) , ϕ ( t ) satisfy (21). We shall consider an interval of t , where S = sign[ ˙ ϕ ( t )] = sign[ ˙ ¯ ϕ ( t )] is constant.</text>
<text><location><page_4><loc_52><loc_38><loc_92><loc_42></location>Using (20,22,24), we substitute expressions for e, ˙ ϕ into equation (25) to have a first order ordinary differential equation with respect to g ( t ):</text>
<formula><location><page_4><loc_54><loc_32><loc_92><loc_36></location>˙ g = -˙ ϕ { S K ( ϕ, g ) [ 2 -∂h ∂e ] + ∂h ∂ϕ +2 dV dϕ } , (29)</formula>
<text><location><page_4><loc_52><loc_28><loc_92><loc_32></location>where in the r.h.s. K ( ϕ, g ) = √ 48 πe ( e -V ), h = h ( e, ϕ ), e = E ( V, g, ϕ ), V = V ( ϕ ).</text>
<text><location><page_4><loc_53><loc_27><loc_60><loc_28></location>Denoting</text>
<formula><location><page_4><loc_59><loc_22><loc_92><loc_26></location>D ( ϕ, g ) = S K ( g, ϕ ) [ 2 -∂h ∂e ] + ∂h ∂ϕ , (30)</formula>
<text><location><page_4><loc_52><loc_18><loc_92><loc_21></location>where h = h ( e, ϕ ), e = E ( V ( ϕ ) , g, ϕ ), in virtue of (26), which is true for any ϕ , we have</text>
<formula><location><page_4><loc_66><loc_14><loc_78><loc_17></location>2 dV dϕ = -D ( ϕ, 0)</formula>
<text><location><page_4><loc_52><loc_11><loc_57><loc_13></location>yielding</text>
<formula><location><page_4><loc_66><loc_8><loc_92><loc_10></location>˙ g = -g ˙ ϕR ( ϕ, g ) (31)</formula>
<text><location><page_5><loc_9><loc_90><loc_49><loc_93></location>where R ( ϕ, g ) = g -1 { D ( ϕ, g ) -D ( ϕ, 0) } is a regular function. From (31) we get</text>
<formula><location><page_5><loc_13><loc_85><loc_45><loc_89></location>g ( t ) = g ( t 0 ) exp -t ∫ t 0 ds ˙ ϕ ( s ) R [ ϕ ( s ) , g ( s )] .</formula>
<text><location><page_5><loc_9><loc_80><loc_49><loc_86></location> The behavior of g ( t ) depends on the monotonicity sign of D ( ϕ, g ) as a function of g , which defines the sign of R ( ϕ, g ). If</text>
<formula><location><page_5><loc_24><loc_77><loc_49><loc_79></location>S R ( ϕ, g ) > 0 , (32)</formula>
<text><location><page_5><loc_9><loc_70><loc_49><loc_76></location>then | g ( t ) | ≤ | g (0) | for t > 0 and we arrive at the approximate equivalence for a sufficiently small initial g (0). Moreover, if ˙ ϕ ( s ) R ( ϕ, g ) ≥ β > 0 , β = const, then we have g ( t ) → 0 for t →∞ exponentially.</text>
<text><location><page_5><loc_9><loc_68><loc_49><loc_71></location>One can estimate the sign of (32) under supposition of differentiability of (30). Equations (22,24) yield</text>
<formula><location><page_5><loc_20><loc_63><loc_49><loc_67></location>∂E ∂g = ∂ Θ ∂g = [ 2 -∂h ∂e ] -1 . (33)</formula>
<text><location><page_5><loc_9><loc_62><loc_42><loc_63></location>The monotonicity condition (32) transforms to</text>
<formula><location><page_5><loc_13><loc_57><loc_45><loc_61></location>S ∂D ∂g = √ 24 π √ E Θ { Θ 2 + E -E Θ 2 -∂h/∂e ∂ 2 h ∂e 2 } +</formula>
<formula><location><page_5><loc_21><loc_52><loc_49><loc_56></location>+ S 2 -∂h/∂e ∂ 2 h ∂e∂ϕ > 0 , (34)</formula>
<text><location><page_5><loc_9><loc_42><loc_49><loc_52></location>where h = h ( e, ϕ ), e = E ( V ( ϕ ) , g, ϕ ), Θ = Θ( V ( ϕ ) , g, ϕ ). Since R ( ϕ, 0) = ∂D/∂g for g = 0, if this inequality is fulfilled for g = 0, then we have the 'approximate equivalence', i.e. at least for sufficiently small g (0) we have | g ( t ) | ≤ | g (0) | for t > 0 and in this sense we say that ϕ ( t ) well approximates ¯ ϕ ( t ) on interval where the signs of ˙ ¯ ϕ ( t ) and ˙ ϕ ( t ) are equal.</text>
<section_header_level_1><location><page_5><loc_22><loc_38><loc_36><loc_39></location>VI. EXAMPLES</section_header_level_1>
<section_header_level_1><location><page_5><loc_18><loc_35><loc_40><loc_36></location>A. Linear equation of state</section_header_level_1>
<text><location><page_5><loc_9><loc_28><loc_49><loc_33></location>Now we shall consider an example with a concrete equation of state. The simplest one is the linear barotropic EOS:</text>
<formula><location><page_5><loc_11><loc_25><loc_49><loc_27></location>h ( e ) = ξ ( e -e 0 ) + h 0 = ξe -η, η = ξe 0 -h 0 . (35)</formula>
<text><location><page_5><loc_10><loc_24><loc_35><loc_25></location>Solutions of equations (22, 24) are</text>
<formula><location><page_5><loc_14><loc_19><loc_43><loc_23></location>Θ 0 ( V ) = 2( ξV -η ) 2 -ξ , E 0 ( V ) = 2 V -η 2 -ξ ;</formula>
<text><location><page_5><loc_32><loc_17><loc_32><loc_19></location>/negationslash</text>
<text><location><page_5><loc_9><loc_15><loc_49><loc_19></location>they are uniquely defined for ξ = 2, so we assume this condition instead of (23). Equation (27) takes on the form</text>
<formula><location><page_5><loc_10><loc_9><loc_49><loc_14></location>dV dϕ = -S 1 S √ √ √ √ 24 π ξ [ ( V -2 + ξ 4 ξ η ) 2 -( 2 -ξ 4 ξ η ) 2 ] , (36)</formula>
<text><location><page_5><loc_52><loc_91><loc_64><loc_93></location>S 1 = sign(2 -ξ ).</text>
<text><location><page_5><loc_52><loc_88><loc_92><loc_92></location>For ξ > 0 the solution of (36) that obeys inequalities Θ 0 ≥ 0 , E 0 ≥ 0 is</text>
<formula><location><page_5><loc_53><loc_85><loc_92><loc_88></location>V ( ϕ ) = (2 + ξ ) η 4 ξ + (2 -ξ ) 4 | η | ξ cosh[2 α ( ϕ -ψ 0 )] , (37)</formula>
<text><location><page_5><loc_52><loc_82><loc_75><loc_84></location>α = √ 6 πξ , under condition that</text>
<formula><location><page_5><loc_64><loc_78><loc_92><loc_80></location>sign[ ˙ ϕ ( ϕ -ψ 0 )] = -1 , (38)</formula>
<text><location><page_5><loc_52><loc_72><loc_92><loc_78></location>ψ 0 is an integration constant. The other options that do not yield positive Θ 0 and E 0 have been discarded. For 0 < ξ < 2 the potential (37) has minimum at ϕ = ψ 0 ; for ξ > 2 the potential is unbounded from below.</text>
<text><location><page_5><loc_52><loc_65><loc_92><loc_72></location>The particular solution ¯ ϕ ( t ) of the SF-problem (12,13) with the initial data satisfying (19) can be found from the first order differential equation (18); it generates the solution of the H-problem e ( t ) = ¯ e f ( t ) ≡ ˙ ¯ ϕ 2 / 2 + V ( ¯ ϕ ).</text>
<text><location><page_5><loc_52><loc_63><loc_92><loc_66></location>Consider, e.g., the case of η > 0; here (18) on account of (38) leads to the equation</text>
<formula><location><page_5><loc_62><loc_60><loc_81><loc_63></location>˙ ¯ ϕ = -√ η sinh [ α ( ¯ ϕ -ψ 0 )] ,</formula>
<text><location><page_5><loc_52><loc_58><loc_67><loc_59></location>yielding two solutions</text>
<formula><location><page_5><loc_55><loc_54><loc_92><loc_57></location>¯ ϕ ( t ) = ψ 0 ± 1 α arsinh { sinh[ α √ η ( t -t 1 )] } -1 , (39)</formula>
<text><location><page_5><loc_52><loc_52><loc_62><loc_53></location>t > t 1 = const .</text>
<text><location><page_5><loc_53><loc_50><loc_65><loc_51></location>Correspondingly</text>
<formula><location><page_5><loc_57><loc_45><loc_86><loc_49></location>e ( t ) = ¯ e f ( t ) = η ξ { coth[ √ 6 πξη ( t -t 1 )] } 2</formula>
<text><location><page_5><loc_52><loc_43><loc_78><loc_45></location>is the solution of (9,10), ξ > 0 , η > 0.</text>
<text><location><page_5><loc_52><loc_29><loc_92><loc_43></location>For any S 1 (39) represents the monotonically decreasing/increasing function that never reaches ϕ = ψ 0 and ¯ e f ( t ) never reaches the value e = η/ξ . The other solutions of (12,13) with the same V ( ϕ ) but not satisfying (19) at t = t 0 , do not fulfill (9,10) with the same h ( e ) (35) with e ( t ) = e f ( t ). For example, for ξ < 2 the solutions of (13) that oscillate near the minimum of the potential cannot be described by the H-model (35): this would contradict to (38) after passing either the turning point ˙ ϕ = 0 or the point ϕ = ψ 0 .</text>
<text><location><page_5><loc_54><loc_19><loc_54><loc_21></location>/negationslash</text>
<text><location><page_5><loc_52><loc_14><loc_92><loc_29></location>There is some freedom in the choice of the solution of (36), which can be used, if we study a correspondence not between models with fixed h ( e ) and/or V ( ϕ ), but between families of potentials and equations of state. Suppose that for initial data (14) we have ˙ ϕ 2 0 = η sinh 2 [ α ( ϕ 0 -ψ 0 )], i.e. (19) is not valid. However, by transforming parameters ξ, η of the EOS (35) or ψ 0 of the potential, one can find some new values of these parameters to satisfy (19) and to find the other special solution of SF-model that corresponds to H-model.</text>
<text><location><page_5><loc_52><loc_9><loc_92><loc_14></location>The condition (34) is fulfilled at least for closest to (39) solutions because Θ 0 / 2 + E 0 > 0. Therefore ¯ ϕ ( t ) well approximates such solutions on whole intervals where sign[ ˙ ϕ ( t )] = sign[ ˙ ¯ ϕ ( t )]. Though in case of (37) is is easy</text>
<text><location><page_6><loc_9><loc_86><loc_49><loc_93></location>to study the qualitative behavior of solutions of (12,13); it is easy to see that ϕ ( t ) → ψ 0 and, in view of continuous dependence of solution on ant finite interval upon the initial data, the deviation of ϕ ( t ) from ¯ ϕ ( t ) will be small for all t > t 0 , provided that it is small at t = t 0 .</text>
<text><location><page_6><loc_9><loc_79><loc_49><loc_86></location>Now we proceed to the case ξ < 0, η < 0. This example 6 is unlikely to be of cosmological significance, but it illustrates problems that can arise when in the course of evolution there are points with zero energy density. From (36) we obtain the periodic potential</text>
<formula><location><page_6><loc_17><loc_74><loc_49><loc_77></location>V ( ϕ ) = η (2 + ξ ) 4 ξ + η (2 -ξ ) 4 ξ cos Φ , (40)</formula>
<text><location><page_6><loc_9><loc_66><loc_49><loc_73></location>where Φ = 2 α ( ϕ -ψ 0 ), α = √ 6 π | ξ | ; the additional condition for (36) to be fulfilled is ˙ ϕ sin Φ > 0. On account of this condition and restricting ourselves to the range Φ ∈ (0 , π ), using (18) we have the solution ¯ ϕ ( t ) of (13):</text>
<formula><location><page_6><loc_11><loc_62><loc_49><loc_66></location>¯ ϕ ( t ) = ψ 0 + 1 α arccos { tanh[ α √ | η | ( t 1 -t )] } , (41)</formula>
<text><location><page_6><loc_9><loc_56><loc_49><loc_62></location>for t < t 1 = const . This relation describes the SF evolution from ¯ ϕ = ψ 0 to ¯ ϕ = ψ 0 + π/ (2 α ). There is no analytic continuation 7 of this solution for t > t 1 . Correspondingly,</text>
<formula><location><page_6><loc_12><loc_50><loc_49><loc_55></location>e ( t ) = ¯ e f ( t ) = ∣ ∣ ∣ η ξ ∣ ∣ ∣ { tanh[ √ 6 π | ξη | ( t -t 1 )] } 2 (42)</formula>
<text><location><page_6><loc_9><loc_46><loc_49><loc_53></location>∣ ∣ is the solution of hydrodynamical equations (9,10) for t < t 1 , where h ( e ) is given by (35). There is a trivial extension of (42) for t > t 1 .</text>
<text><location><page_6><loc_9><loc_39><loc_49><loc_46></location>At last, consider the important case ξ < 0 , η = 0, yielding the famous 'Big Rip' hydrodynamical solution [15]. In this case RHS of (36) is not real, there is no nontrivial solution for the potential and there is no canonical SF counterpart.</text>
<section_header_level_1><location><page_6><loc_16><loc_35><loc_41><loc_36></location>B. Example of a nonlinear EOS</section_header_level_1>
<text><location><page_6><loc_10><loc_32><loc_36><loc_33></location>Consider the barotropic EOS [5, 12]</text>
<formula><location><page_6><loc_21><loc_28><loc_49><loc_30></location>h ( e ) = ξe [1 -( e/e 0 ) µ ] , (43)</formula>
<text><location><page_6><loc_9><loc_25><loc_49><loc_27></location>where µ > 0. We are looking for possible solutions of (28).</text>
<text><location><page_6><loc_9><loc_16><loc_49><loc_24></location>For 0 < ξ < 2 the conditions for existence, uniqueness and positivity of Θ 0 , E 0 of (26) can be verified directly using the solution below. Equation (28) yields the solution E 0 ( ϕ ) = e 0 (cosh Φ) -2 /µ ≤ e 0 , where Φ = µ √ 6 πξ ( ϕ -ψ 0 ), ψ 0 is an integration constant, under</text>
<text><location><page_6><loc_52><loc_90><loc_92><loc_93></location>condition that sign [ ˙ ϕ ( ϕ -ψ 0 )] = 1. Then we have the potential (cf. [12])</text>
<formula><location><page_6><loc_63><loc_86><loc_80><loc_89></location>V ( ϕ ) = E 0 -1 2 h ( E 0 ) =</formula>
<formula><location><page_6><loc_56><loc_80><loc_92><loc_84></location>= e 0 2 (cosh Φ) 2(1+1 /µ ) [ ξ +(2 -ξ ) cosh 2 Φ ] . (44)</formula>
<text><location><page_6><loc_52><loc_74><loc_92><loc_80></location>The SF-model with this potential has the particular solution ¯ ϕ ( t ) satisfying (18); correspondingly e f ( t ) satisfies the equations (9, 10) of the H-model. Equation (18) takes on the form</text>
<formula><location><page_6><loc_57><loc_69><loc_87><loc_74></location>˙ ¯ ϕ = √ ξe 0 sinh ¯ Φ (cosh ¯ Φ) 1+1 /µ , ¯ Φ = µ √ 6 πξ ( ¯ ϕ -ψ 0 ) .</formula>
<text><location><page_6><loc_52><loc_62><loc_92><loc_69></location>For ϕ > ψ 0 this means that ¯ ϕ ( t ) descents down the potential hill to the right of ψ 0 ; it grows logarithmically and ¯ e f ( t ) → 0 for t → ∞ . For t → -∞ we have ¯ ϕ ( t ) → ψ 0 and ¯ e f ( t ) → e 0 .</text>
<text><location><page_6><loc_53><loc_62><loc_78><loc_63></location>The condition (34) for g = 0 yields</text>
<formula><location><page_6><loc_54><loc_56><loc_90><loc_61></location>Θ 0 2 + E 0 + ξµ ( µ +1)Θ 0 2 -ξ + ξ ( µ +1)( E 0 /e 0 ) µ ( E 0 e 0 ) µ > 0 .</formula>
<text><location><page_6><loc_52><loc_52><loc_92><loc_56></location>This is always fulfilled for 0 < ξ < 2 thus guaranteeing that ϕ ( t ) ≈ ¯ ϕ ( t ) on appropriate intervals in case of small deviation of the initial data.</text>
<text><location><page_6><loc_52><loc_47><loc_92><loc_52></location>Note that the SF-model with ξ > 0 does not admit divergent solutions like the 'Big Rip' [15] of the hydrodynamical counterpart.</text>
<text><location><page_6><loc_52><loc_40><loc_92><loc_47></location>The case ξ < 0 is possible for e > e 0 ; equation (28) yields E 0 ( ϕ ) = e 0 (cos Φ) -2 /µ ≥ e 0 for | Φ | < π/ 2, Φ = µ √ 6 π | ξ | ( ϕ -ψ 0 ). We have a potential pit with infinite walls (with a periodic continuation):</text>
<formula><location><page_6><loc_54><loc_36><loc_92><loc_40></location>V ( ϕ ) = e 0 2 (cos Φ) 2(1+1 /µ ) [ ξ +(2 -ξ ) cos 2 Φ ] , (45)</formula>
<text><location><page_6><loc_52><loc_33><loc_92><loc_36></location>and the SF solutions with this potential lead to the same evolution of H ( t ) as the hydrodynamical ones if</text>
<formula><location><page_6><loc_64><loc_29><loc_80><loc_31></location>sign [ ˙ ϕ ( ϕ -ψ 0 )] = -1 .</formula>
<text><location><page_6><loc_52><loc_26><loc_92><loc_29></location>In this view, equation (18) for the corresponding particular solution of (13) yields</text>
<formula><location><page_6><loc_65><loc_21><loc_79><loc_25></location>˙ ¯ ϕ = -√ | ξ | e 0 sin ¯ Φ (cos ¯ Φ) 1+1 /µ .</formula>
<text><location><page_6><loc_52><loc_17><loc_92><loc_20></location>The scalar field slides off the wall and tends to the potential minimum with energy e = e 0 , ϕ → ψ 0 for t →∞ .</text>
<section_header_level_1><location><page_6><loc_62><loc_13><loc_82><loc_14></location>C. Two-parametric EOS</section_header_level_1>
<text><location><page_6><loc_52><loc_8><loc_92><loc_11></location>To illustrate how equation (26) works in case of twoparametric EOS, we consider h ( e, ϕ ) = ξe -U ( ϕ ), ξ =</text>
<text><location><page_7><loc_9><loc_89><loc_49><loc_93></location>const , which is obtained as a generalization of (35) by changing η → U ( ϕ ). We assume in this subsection 0 < ξ < 2.</text>
<text><location><page_7><loc_10><loc_87><loc_48><loc_89></location>After substitution ˜ V = V -U/ 2 equation (26) yields</text>
<formula><location><page_7><loc_15><loc_82><loc_49><loc_86></location>d ˜ V dϕ = -S √ 24 π ˜ V [ ( ξ ˜ V -2 -ξ 2 U ( ϕ ) ] , (46)</formula>
<text><location><page_7><loc_9><loc_80><loc_22><loc_81></location>where S = sign( ˙ ϕ ).</text>
<text><location><page_7><loc_9><loc_75><loc_49><loc_80></location>This equation can be used either to derive V ( ϕ ) for given U ( ϕ ) or, vice versa, to find EOS on condition that V ( ϕ ) is given:</text>
<formula><location><page_7><loc_15><loc_68><loc_42><loc_74></location>U ( ϕ ) = 2 2 -ξ ξ ˜ V -1 24 π ˜ V ( d ˜ V dϕ ) 2 </formula>
<text><location><page_7><loc_9><loc_62><loc_49><loc_68></location>By considering various ˜ V one can generate examples with subsequent verification of equation (46) and inequalities Θ 0 > 0 , E 0 > 0. We give two such examples dealing with simple elementary functions.</text>
<text><location><page_7><loc_10><loc_61><loc_43><loc_62></location>(i) For ˜ V ( ϕ ) = A 2 ϕ 2 , A = const > 0, we have</text>
<formula><location><page_7><loc_9><loc_56><loc_49><loc_60></location>U ( ϕ ) = 2 A 2 2 -ξ ( ξϕ 2 -1 6 π ) , V ( ϕ ) = 2 A 2 2 -ξ ( ϕ 2 -1 12 π ) .</formula>
<text><location><page_7><loc_9><loc_51><loc_49><loc_55></location>Equation (46) is valid if ˙ ϕ ( t ) ϕ ( t ) < 0. The non-trivial particular solution satisfying both the equations of H and SF models exists for t < t 1 = const</text>
<formula><location><page_7><loc_13><loc_44><loc_45><loc_50></location>ϕ ( t ) = ± A ( t 1 -t ) √ 3 π (2 -ξ ) , e ( t ) = 2 A 4 ( t -t 1 ) 2 3 π (2 -ξ ) 2 .</formula>
<text><location><page_7><loc_9><loc_42><loc_49><loc_45></location>(ii) The choice ˜ V ( ϕ ) = A 2 exp( αϕ ), where α, A > 0 are constants, generates</text>
<formula><location><page_7><loc_10><loc_37><loc_51><loc_41></location>U ( ϕ ) = 2 A 2 e αϕ 2 -ξ ( ξ -α 2 24 π ) , V ( ϕ ) = A 2 e αϕ 2 -ξ ( 2 -α 2 24 π )</formula>
<text><location><page_7><loc_9><loc_34><loc_49><loc_36></location>Equation (46) is valid if α ˙ ϕ ( t ) < 0. The particular solution of (13) is</text>
<formula><location><page_7><loc_13><loc_28><loc_45><loc_32></location>ϕ ( t ) = 2 α ln [ 4 √ 3 π (2 -ξ ) Aα 2 ( t -t 1 ) ] , t > t 1 = const.</formula>
<text><location><page_7><loc_9><loc_24><loc_49><loc_27></location>The corresponding solution of hydrodynamical equation (10) is e ( t ) = e f ( t ) = 96 πα -4 ( t -t 1 ) -2 .</text>
<section_header_level_1><location><page_7><loc_21><loc_21><loc_37><loc_22></location>VII. DISCUSSION</section_header_level_1>
<text><location><page_7><loc_9><loc_9><loc_49><loc_19></location>It is clear that the hydrodynamic description of DE is an oversimplification of the real cosmological situation in comparison with field-theoretic models. A consistent description of hydrodynamic phenomena assumes the local thermodynamical equilibrium. It is unclear how this assumption works as regards DE in the early Universe and in the modern era. Nevertheless, this does not prevent us</text>
<text><location><page_7><loc_51><loc_39><loc_52><loc_40></location>.</text>
<text><location><page_7><loc_52><loc_85><loc_92><loc_93></location>from using the hydrodynamical model on a formal level by equating the scalar field energy momentum tensor to the hydrodynamical one. On the other hand, some solutions of hydrodynamic models that are widely used in cosmological considerations, can be interpreted in therms of the SF-models.</text>
<text><location><page_7><loc_52><loc_66><loc_92><loc_84></location>In this paper we found conditions for the SF-model that make this possible in case of the homogeneous isotropic spatially flat cosmology, under additional restriction on the form of the SF Lagrangian to be a canonical one. This is a very restrictive requirement; it leads lead to the differential equation for the potential V ( ϕ ), which is effective on intervals with the constant S = sign( ˙ ϕ ( t )). Moreover, the space of solutions of the SF-model is much wider than that of the barotropic Hmodel. In any case, the global equivalence between H and SF models for all modes of cosmological evolution is impossible. This is clearly seen in the examples of Section VI.</text>
<text><location><page_7><loc_52><loc_50><loc_92><loc_66></location>This, however, does not prohibit using the H-SF analogy to study some special regimes. The hydrodynamical solutions with EOS (35) yield the SF solutions for the potential (37), when SF rolls down the potential well or descents down the potential hill (Section VI). But the Hmodel cannot describe the SF oscillations near minimum of the potential, though this regime being important for particle creation at the post-inflationary stage of the cosmological evolution [4]. On the other hand, some singular solutions like the 'Big Rip' [15] that may take place for certain EOSs are ruled out in case of the SF counterparts.</text>
<text><location><page_7><loc_52><loc_37><loc_92><loc_50></location>The restriction on the initial data reduces possibilities to use the hydrodynamical representation of the 'restricted' SF model. This trouble is mitigated by the possibility to investigate close solutions. We derived conditions that ensure certain closeness of the SF-model energy-momentum tensors to that of the H model. In this sense the fiducial solution, which satisfies equations of both H and SF models, well approximates nearby solutions and describes their qualitative properties.</text>
<section_header_level_1><location><page_7><loc_62><loc_33><loc_82><loc_34></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_7><loc_52><loc_25><loc_92><loc_31></location>This work has been supported in part by the Department of target training of Taras Shevchenko National University of Kyiv under National Academy of Sciences of Ukraine (project 6Φ).</text>
<section_header_level_1><location><page_7><loc_55><loc_20><loc_88><loc_22></location>Appendix A: SF-H correspondence without restrictions</section_header_level_1>
<text><location><page_7><loc_52><loc_15><loc_92><loc_18></location>The hydrodynamical and scalar field approaches are equivalent, if</text>
<formula><location><page_7><loc_67><loc_12><loc_92><loc_14></location>T ( h ) µν = T ( sf ) µν ; (A1)</formula>
<text><location><page_7><loc_52><loc_9><loc_92><loc_11></location>This equivalence can be used to find special solutions of hydrodynamics equations by means of the SF equations</text>
<text><location><page_8><loc_9><loc_88><loc_49><loc_93></location>[13]. However, (A1) presupposes that the perfect fluid flow involved is a relativistic analog of the classical potential flow [13, 14]. Indeed, besides (3), equations (A1) yield</text>
<formula><location><page_8><loc_13><loc_82><loc_49><loc_86></location>u µ F = ϕ ,µ , X > 0 , F = √ e + p ∂L/∂X , (A2)</formula>
<text><location><page_8><loc_9><loc_69><loc_49><loc_82></location>where X > 0 is a solution of equation (4) for given EOS. This can be easily shown by considering (A1) in a locally Lorentz frame (where at some point x 0 we have g µν ( x 0 ) = η µν , ∂ α g µν ( x 0 ) = 0), which is also an instantaneous proper frame for u µ ( x 0 ) = (1 , 0 , 0 , 0). The inequality X > 0 must be fulfilled because u µ is timelike; therefore, the H-model cannot be equivalent to the SF model in case of a stationary SF. On account of (1,2) and A2) we get</text>
<formula><location><page_8><loc_18><loc_66><loc_49><loc_67></location>∂ ν [ F ( e, ϕ ) u µ ] = ∂ µ [ F ( e, ϕ ) u ν ] . (A3)</formula>
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<text><location><page_8><loc_55><loc_72><loc_55><loc_74></location>/negationslash</text>
<text><location><page_8><loc_52><loc_69><loc_92><loc_93></location>Usually for a given EOS e = E ( p, ϕ ), the differential equation (4) has a solution L that transforms (4) into an identity. On the other hand, for given L ( X,ϕ ), equations (3) represent the EOS parametrically, the domain of E as a function of p depending upon the range of L . In this sense we can speak about some equivalence of H and SF models, provided that conditions (A3) are fulfilled; this latter depends on the initial conditions of the hydrodynamical problem. Obviously, these conditions may be not satisfied if we deal with an arbitrary hydrodynamic flow, i.e. there is no full 'equivalence' between H and SF models. However, the conditions (A3) are always fulfilled in case of a homogeneous cosmology, where the gradient ∂ µ ϕ = 0 is time-like and all the functions involved depend on the time variable only. In a more general case, the relativistic ideal fluid flows satisfying (A3) may be considered as an analogue of classical irrotational flows.</text>
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</document> | [{"title": "Scalar field vs hydrodynamic models in the homogeneous isotropic cosmology", "content": "V. I. Zhdanov 1, \u2217 and S. S. Dylda 2, \u2020 1 Astronomical Observatory, Taras Shevchenko National University of Kyiv 2 Physical Faculty, Taras Shevchenko National University of Kyiv We study relations between hydrodynamical (H) and scalar field (SF) models of the dark energy in the homogeneous isotropic Universe. The focus is on SF described by the Lagrangian with the canonical kinetic term within spatially flat cosmology. We analyze requirements that guarantee the same cosmological history for the SF and H models at least for special solutions. The differential equation for the SF potential is obtained that ensures such equivalence of the SF and H-models. However, if the 'equivalent' SF potential is found for given equation of state (EOS) of the H-model, this does not mean that all solutions of this SF-model have corresponding H-model analogs. In this view we derived a condition that guarantees an 'approximate equivalence', when there is a small difference between energy-momentum tensors of the models. The 'equivalent' SF potentials and corresponding SF solutions for linear EOS are found in an explicit form; we also present examples with more complicated EOSs. PACS numbers: 98.80.Cq", "pages": [1]}, {"title": "I. INTRODUCTION", "content": "Observations show [1] that about 70% of the average mass density in the Universe owes to the dark energy (DE) which drives the acceleration of the cosmological expansion. It is widely assumed that some form of DE or its constituents that dominated in the very early Universe must have a dynamical nature owing to an action of unknown physical fields and/or due to modifications of the General Relativity [2, 3]. Theories with scalar fields (SF) occupy an important sector of this area (see, e.g., [2-6]). Though observational data restrict some of the SF models [4], there is still a considerable uncertainty in their choice, not to mention the revision of the underlying gravitational theory [3, 5]. The abundance of various cosmological models draws attention to unifying schemes and interrelations between competing dark energy candidates [5-7] that can be used to explain observational data. To this end, the phenomenological hydrodynamic approach is often used [5, 6, 8-10]. It is well known that the matter in the spatially homogeneous and isotropic Universe can be described by means of the energymomentum tensor of an ideal fluid. Under certain conditions SF models allow for a hydrodynamic (H) description with some 'equivalent' equation of state (EOS) beyond homogeneity as well [9]. The H analogs of the SF models typically involve such phenomenological parameters as the EOS parameter, effective sound speed, adiabatic sound speed, which can be limited in view of available astronomical data [9, 11]. The transition from simplest hydrodynamical EOSs to SFs and vice versa deals with rather unusual models, whereas in the spirit of Occam's razor, it would be desirable to restrict the choice of the SF Lagrangian to canonical one, which is more familiar from the point of view of particle physics. This is possible within the approach of papers [5, 6, 8, 12], which treat the equivalence problems by direct comparing solutions for the cosmological scale factor and the energy density in the homogeneous isotropic Universe. As distinct from these papers, we propose a differential equation for the SF potential in closed form guaranteeing some equivalence of H and SF models. We note, however, that in any approach, the H-SF correspondence is not universal; this is well known though not always clearly stated. A typical situation is that two different models mimic each other for some area of the original data, but they have different solutions outside this area. We study the H-SF equivalence on the basis of equality of the corresponding energy-momentum tensors (Section II). This problem becomes more complicated if we impose some additional conditions either on the form of the EOS, or on the SF Lagrangian. We focus on the relationship between the H-model and the SF-model with the canonical kinetic term and a self-interaction potential for the real SF. We call this 'restricted 1 equivalence', in contrast to the case, when no such restrictions are imposed. As a result, the equivalence considerations deal with some restrictions on the initial data (Section III). Then we consider a case, when the relations that guarantee some kind of equivalence of SF and H models are valid approximately. In this case the equations of H and SF models can lead to different energy-momentum tensors (and correspondingly different evolution equations), and the question is when this difference remains small, provided that it is small at the initial moment. We derived conditions for such approximate equivalence (Section IV). The results are applied to the linear EOS, including situation near the phantom line (subsection VI A), to some non-linear EOS known from papers [5, 12] (subsection VIB), and to a simple two-parametric EOS (subsection VIC). Here we present examples showing when one can speak about an equivalence between the H and SF models.", "pages": [1, 2]}, {"title": "II. GENERAL CONSIDERATIONS", "content": "General Lagrangian L ( X,\u03d5 ) for the real SF \u03d5 with X = 1 2 \u03d5, \u00b5 \u03d5 ,\u00b5 and space-time metric g \u00b5\u03bd yields the energy-momentum tensor 2 that can be equated to the energy-momentum tensor of an ideal fluid where h = e + p is the specific enthalpy, p is the pressure, e is the invariant energy density and u \u00b5 is the four-velocity of the fluid. It is assumed that some EOS is known that relates the pressure to the other parameters of the problem: p = P ( e, \u03d5 ). We have T ( sf ) \u00b5\u03bd = T ( h ) \u00b5\u03bd if At the points where X changes its sign (i.e., \u03d5 ,\u00b5 is not timelike), the hydrodynamical interpretation is no longer valid. The relations (3) yield an ordinary differential equation with respect to L ( X,\u03d5 ), where \u03d5 is involved as a parameter: The solution L of (4) exists in case of rather a general EOS; this solution contains an arbitrary function of \u03d5 . Additional constraints that ensure equality of (1) and (2) for all values of \u03d5 and its derivatives are outlined in Appendix A. These constraints are fulfilled identically in case of an homogeneous isotropic Universe to be discussed further. However, under additional restrictions on the functions L ( X,\u03d5 ) and/or E ( p, \u03d5 ) in (4) the solution of this equation for all X,\u03d5 may not exist. For example, if we want to define the EOS parametrically from (3), then a general Lagrangian L cannot yield the barotropic EOS, because in this case the right-hand sides of (3) may depend on two independent variables X and \u03d5 . In this view we shall require that equations (3,4) be satisfied not for arbitrary hydrodynamical and/or SF variables, but only for certain cosmological solutions in the isotropic homogeneous Universe. We wonder, is it possible to compare H-model with the SF-model, if the SF Lagrangian has the canonical form Then equations (3,4) yield If we demanded that (6) be fulfilled for all variables ( e, \u03d5 ) or ( X,\u03d5 ), we would have very special EOS p = e -2 V ( \u03d5 ), whereas in case of another equations of state the relations (6) cannot be identities. However, we deal with the unique Universe, so in fact we do not need that the 'equivalence' conditions be satisfied for all possible values of the variables that enter EOS and/or Lagrangian. If we compare different cosmological models, then the main question is when they predict the same observational data, when they mimic each other etc. We say that there is an equivalence of H and SF models, if both predict the same Hubble diagram and, consequently, the same Hubble parameter H ( z ) as a function of the redshift z . In this case we have the same H ( t ) as a function of the cosmological time t , yielding the same 3 dependence of the cosmological scale factor a ( t ). Within the homogeneous isotropic cosmology, a solution a ( t ) , \u03d5 ( t ) of the SF-model 4 yields e ( t ) , p ( t ) as a parametric representation of EOS due to (6) and vice versa; this approach is often considered (see, e.g. [5, 6]). The main difference of the present paper is that we are looking for a direct criterion on the SF potential V ( \u03d5 ), which yields the same cosmological history as in the case of H-model with the prescribed EOS.", "pages": [2]}, {"title": "III. STATEMENT OF THE PROBLEM", "content": "We consider the spatially flat cosmology described by the Friedmann-Lemaitre-Robertson-Walker metric It should be noted that the supposition of spatial flatness agrees with observations [1] and is perfectly explained in the framework of widely known ideas of inflation in the early Universe [4]. In case of the Universe filled with an ideal fluid we have the Friedmann equations (spatially flat case) ( G = c = 1) . One more (hydrodynamical) equation also follows from (8,9); on the other hand, (8) follows from (9,10). Further we use (9,10) as the independent equations taking in mind that they must be supplemented by an equation of state. In case of the isotropic homogeneous Universe filled with uniform scalar field \u03d5 = \u03d5 ( t ); X = \u02d9 \u03d5 2 / 2 the evolution equations corresponding to (5) are where e f \u2261 \u02d9 \u03d5 2 / 2 + V ( \u03d5 ) is the field energy density, and the field equation is Analogously, these equations are not independent and we use further (12, 13) as the evolution equations of the SF-model with the initial conditions If (9,10) and (12,13) are fulfilled with the same H ( t ), then and substituting this into (10) and using (13), we get Analogously, on account of (15, 16) equations (12,13) yield (9,10). Conversely, (9,10) and (13,15) yield (16). The relations (15,16) are necessary for the equivalence of H and SF models. These conclusions do not depend on either p = P ( e ) or p = P ( e, \u03d5 ). However, the statement of the initial value problem of H-model and its comparison with SF-models looks somewhat different in case of (i) one-parametric and (ii) two-parametric EOS. (i) Barotropic EOS: p = P ( e ). The H-model is defined by equations (9,10) with the initial condition (ii) Two-parametric EOS . We shall see below that the requirement of equivalence of H and SF models imposes severe limitations on cosmological solutions. In order to generalize the discussion and verify that the limitations are not due to the one-parametric form (i), we consider EOS that contains two parameters. Following [10] we suppose that p = P ( e, \u03c6 ); this generalization can be used to construct phenomenological models of the dark energy. Obviously, this demands that dynamical equation for the additional variable \u03c6 must be also involved in the H-model. Now we wonder, is it possible to describe the solutions of this model with the help of some scalar field \u03d5 alone, without using the hydrodynamic variables? The very first step in this direction and the most economic way within our 'restricted' approach is to suppose that \u03c6 = \u03d5 obeys the same equation (13). Therefore, we assume that the equations of the H-model include (9,10,13) with corresponding initial conditions (14,17). Our formal aim is to find criteria for existence of the H-model solution e ( t ) and the SF-model solution \u03d5 ( t ) with the same H ( t ), such that e f ( t ) \u2261 e ( t ). Obviously, considering (ii) of the H-model with twoparametric EOS and its comparison to the SF-model differs from considering (i), in particular, because we have different dimensions of the space of initial data. However, the mathematics we deal with below is formally the same and the equivalence criterion (26) derived below is applicable both to (i) and (ii). So further we work with (ii), having in mind the reservation concerning the difference of (i) and (ii). We assume h ( e, \u03d5 ) to be a continuously differentiable function of e, \u03d5 . Further for brevity we denote Using this function, in view of the relations (15,16), we have As we have seen, this condition along with (15) ensures that both H-model and SF-model lead to the same Hubble diagram (at least for specially chosen initial data). In this sense we speak about 'restricted 5 equivalence' of H and SF-models. The condition (18) must be fulfilled for initial data (17) as well: Also, we shall consider deviations from equation (18); in this case we consider the function It should be noted that for fixed V ( \u03d5 ), h ( e, \u03d5 ) it is generally impossible to satisfy (19) with \u2200 \u02d9 \u03d5 0 , \u03d5 0 ; this relation singles out a particular solution to equation (13). Therefore most of solutions of the SF-model cannot be H-model solutions. After the comments about the initial data we proceed to conditions for the potential, which must be fulfilled to ensure (18). We suppose that the function h ( e, \u03d5 ) is known. The problem we are interested in can be formulated as follows. A. Let \u03d5 ( t ) be a solution of (12,13). What are sufficient conditions for V ( \u03d5 ) so as to ensure g ( t ) \u2261 0 at least for special initial data (14) satisfying (19)? After finding potential V ( \u03d5 ) that solves the problem (A) for special initial data satisfying (19), it is natural to ask about another solutions of the SF-model with the same potential, which do not satisfy (19). B. Let for some V ( \u03d5 ) there are solutions \u03d5 ( t ) , \u00af \u03d5 ( t ) of equations (12,13), and /negationslash What we can say about g ( t ) in (20)? If \u03d5 ( t 0 ) satisfies (19) approximately, will this approximation work for t > t 0 ? If yes, we can say that we have an 'approximate equivalence' of H and SF-models.", "pages": [2, 3, 4]}, {"title": "IV. COMPARISON OF S AND H MODELS", "content": "The equation (20) can be solved with respect to \u02d9 \u03d5 2 . With this aim we introduce function \u0398( V, g, \u03d5 ), which is defined as a solution of the equation Uniqueness of solution of (22) can be easily established if, for \u03b5 = const > 0 (arbitrarily small), The uniqueness follows from consideration of \u03b6 ( \u03d1 ) = \u03d1 -h ( \u03d1/ 2 + V, \u03d5 ), which is monotonically increasing function of \u03d1 . Then \u03b6 ( \u03d1 ) takes the value \u03b6 ( \u03d1 ) = g only once, therefore we have a unique solution \u03d1 = \u0398( V, g, \u03d5 ) of (22). A sufficient condition of existence is h ( V, \u03d5 ) \u2265 -g , because in this case \u03b6 (0) \u2264 g and \u03b6 ( \u03d1 ) \u2192\u221e as \u03d1 \u2192\u221e due to (23); so in virtue of continuity of \u03b6 ( \u03d1 ) there exists the solution \u03d1 of (22). In case of g = 0 this sufficient condition is simply the requirement for the positive specific enthalpy. Note that (23) means \u2202P/\u2202e \u2264 1 -\u03b5 < 1, which avoids superluminal speed of sound. We also introduce E ( V, g, \u03d5 ) = \u0398( V, g, \u03d5 ) / 2 + V that satisfies the equation Further we consider solutions of (24) such that e = E ( V, g, \u03d5 ) > 0 , \u02d9 \u03d5 2 = \u0398( V, g, \u03d5 ) > 0. After differentiation of (20) and in view of (12,13) we get where h \u2261 h ( e, \u03d5 ) and we denote e = 1 2 \u02d9 \u03d5 2 + V ( \u03d5 ) > 0. If we require g \u2261 0, then, for \u02d9 \u03d5 = 0, we have /negationslash where h = h ( e, \u03d5 ), e = E 0 ( V, \u03d5 ) \u2261 E ( V, 0 , \u03d5 ), \u0398 0 ( V, \u03d5 ) \u2261 \u0398( V, 0 , \u03d5 ), S = sign( \u02d9 \u03d5 ), and we used (22,24). Note that equation (26) is a formal consequence of (25) only for those \u03d5 that belong to the range of solutions \u03d5 ( t ) of (13). The differential equation (26) for the potential V ( \u03d5 ) is a sufficient condition to have (18) provided that g ( t 0 ) = 0. Thus, the problem (A) of equivalence is reduced to the equation for potential V ( \u03d5 ) in closed form, which, however, is different for different signs of \u02d9 \u03d5 . In case of a barotropic EOS h = h ( e ) equation (26) is simplified to the form In virtue of (24) we have for V = E 0 -h ( E 0 ) / 2. Substitution to (26) yields more compact equation For given EOS, equations (27,28) allow to find V ( \u03d5 ) such that certain modes of cosmological evolution H ( t ) , e ( t ) = e f ( t ) can be obtained by means of either H-model or SFmodel. This, however, does not apply to all possible solutions to this SF-model, in particular, when \u02d9 \u03d5 changes its sign. From (27) it follows that the potential V ( \u03d5 ) must be a monotonically increasing function provided that we consider an interval where \u02d9 \u03d5 < 0. This includes, e.g., the case of the slow-roll modes of the chaotic inflation.", "pages": [4]}, {"title": "V. INITIAL DATA NOT SATISFYING (19)", "content": "Now we proceed to (B). Let V = V ( \u03d5 ) satisfies (26) and \u00af \u03d5 ( t ) , \u03d5 ( t ) satisfy (21). We shall consider an interval of t , where S = sign[ \u02d9 \u03d5 ( t )] = sign[ \u02d9 \u00af \u03d5 ( t )] is constant. Using (20,22,24), we substitute expressions for e, \u02d9 \u03d5 into equation (25) to have a first order ordinary differential equation with respect to g ( t ): where in the r.h.s. K ( \u03d5, g ) = \u221a 48 \u03c0e ( e -V ), h = h ( e, \u03d5 ), e = E ( V, g, \u03d5 ), V = V ( \u03d5 ). Denoting where h = h ( e, \u03d5 ), e = E ( V ( \u03d5 ) , g, \u03d5 ), in virtue of (26), which is true for any \u03d5 , we have yielding where R ( \u03d5, g ) = g -1 { D ( \u03d5, g ) -D ( \u03d5, 0) } is a regular function. From (31) we get \uf8f3 \uf8fe The behavior of g ( t ) depends on the monotonicity sign of D ( \u03d5, g ) as a function of g , which defines the sign of R ( \u03d5, g ). If then | g ( t ) | \u2264 | g (0) | for t > 0 and we arrive at the approximate equivalence for a sufficiently small initial g (0). Moreover, if \u02d9 \u03d5 ( s ) R ( \u03d5, g ) \u2265 \u03b2 > 0 , \u03b2 = const, then we have g ( t ) \u2192 0 for t \u2192\u221e exponentially. One can estimate the sign of (32) under supposition of differentiability of (30). Equations (22,24) yield The monotonicity condition (32) transforms to where h = h ( e, \u03d5 ), e = E ( V ( \u03d5 ) , g, \u03d5 ), \u0398 = \u0398( V ( \u03d5 ) , g, \u03d5 ). Since R ( \u03d5, 0) = \u2202D/\u2202g for g = 0, if this inequality is fulfilled for g = 0, then we have the 'approximate equivalence', i.e. at least for sufficiently small g (0) we have | g ( t ) | \u2264 | g (0) | for t > 0 and in this sense we say that \u03d5 ( t ) well approximates \u00af \u03d5 ( t ) on interval where the signs of \u02d9 \u00af \u03d5 ( t ) and \u02d9 \u03d5 ( t ) are equal.", "pages": [4, 5]}, {"title": "A. Linear equation of state", "content": "Now we shall consider an example with a concrete equation of state. The simplest one is the linear barotropic EOS: Solutions of equations (22, 24) are /negationslash they are uniquely defined for \u03be = 2, so we assume this condition instead of (23). Equation (27) takes on the form S 1 = sign(2 -\u03be ). For \u03be > 0 the solution of (36) that obeys inequalities \u0398 0 \u2265 0 , E 0 \u2265 0 is \u03b1 = \u221a 6 \u03c0\u03be , under condition that \u03c8 0 is an integration constant. The other options that do not yield positive \u0398 0 and E 0 have been discarded. For 0 < \u03be < 2 the potential (37) has minimum at \u03d5 = \u03c8 0 ; for \u03be > 2 the potential is unbounded from below. The particular solution \u00af \u03d5 ( t ) of the SF-problem (12,13) with the initial data satisfying (19) can be found from the first order differential equation (18); it generates the solution of the H-problem e ( t ) = \u00af e f ( t ) \u2261 \u02d9 \u00af \u03d5 2 / 2 + V ( \u00af \u03d5 ). Consider, e.g., the case of \u03b7 > 0; here (18) on account of (38) leads to the equation yielding two solutions t > t 1 = const . Correspondingly is the solution of (9,10), \u03be > 0 , \u03b7 > 0. For any S 1 (39) represents the monotonically decreasing/increasing function that never reaches \u03d5 = \u03c8 0 and \u00af e f ( t ) never reaches the value e = \u03b7/\u03be . The other solutions of (12,13) with the same V ( \u03d5 ) but not satisfying (19) at t = t 0 , do not fulfill (9,10) with the same h ( e ) (35) with e ( t ) = e f ( t ). For example, for \u03be < 2 the solutions of (13) that oscillate near the minimum of the potential cannot be described by the H-model (35): this would contradict to (38) after passing either the turning point \u02d9 \u03d5 = 0 or the point \u03d5 = \u03c8 0 . /negationslash There is some freedom in the choice of the solution of (36), which can be used, if we study a correspondence not between models with fixed h ( e ) and/or V ( \u03d5 ), but between families of potentials and equations of state. Suppose that for initial data (14) we have \u02d9 \u03d5 2 0 = \u03b7 sinh 2 [ \u03b1 ( \u03d5 0 -\u03c8 0 )], i.e. (19) is not valid. However, by transforming parameters \u03be, \u03b7 of the EOS (35) or \u03c8 0 of the potential, one can find some new values of these parameters to satisfy (19) and to find the other special solution of SF-model that corresponds to H-model. The condition (34) is fulfilled at least for closest to (39) solutions because \u0398 0 / 2 + E 0 > 0. Therefore \u00af \u03d5 ( t ) well approximates such solutions on whole intervals where sign[ \u02d9 \u03d5 ( t )] = sign[ \u02d9 \u00af \u03d5 ( t )]. Though in case of (37) is is easy to study the qualitative behavior of solutions of (12,13); it is easy to see that \u03d5 ( t ) \u2192 \u03c8 0 and, in view of continuous dependence of solution on ant finite interval upon the initial data, the deviation of \u03d5 ( t ) from \u00af \u03d5 ( t ) will be small for all t > t 0 , provided that it is small at t = t 0 . Now we proceed to the case \u03be < 0, \u03b7 < 0. This example 6 is unlikely to be of cosmological significance, but it illustrates problems that can arise when in the course of evolution there are points with zero energy density. From (36) we obtain the periodic potential where \u03a6 = 2 \u03b1 ( \u03d5 -\u03c8 0 ), \u03b1 = \u221a 6 \u03c0 | \u03be | ; the additional condition for (36) to be fulfilled is \u02d9 \u03d5 sin \u03a6 > 0. On account of this condition and restricting ourselves to the range \u03a6 \u2208 (0 , \u03c0 ), using (18) we have the solution \u00af \u03d5 ( t ) of (13): for t < t 1 = const . This relation describes the SF evolution from \u00af \u03d5 = \u03c8 0 to \u00af \u03d5 = \u03c8 0 + \u03c0/ (2 \u03b1 ). There is no analytic continuation 7 of this solution for t > t 1 . Correspondingly, \u2223 \u2223 is the solution of hydrodynamical equations (9,10) for t < t 1 , where h ( e ) is given by (35). There is a trivial extension of (42) for t > t 1 . At last, consider the important case \u03be < 0 , \u03b7 = 0, yielding the famous 'Big Rip' hydrodynamical solution [15]. In this case RHS of (36) is not real, there is no nontrivial solution for the potential and there is no canonical SF counterpart.", "pages": [5, 6]}, {"title": "B. Example of a nonlinear EOS", "content": "Consider the barotropic EOS [5, 12] where \u00b5 > 0. We are looking for possible solutions of (28). For 0 < \u03be < 2 the conditions for existence, uniqueness and positivity of \u0398 0 , E 0 of (26) can be verified directly using the solution below. Equation (28) yields the solution E 0 ( \u03d5 ) = e 0 (cosh \u03a6) -2 /\u00b5 \u2264 e 0 , where \u03a6 = \u00b5 \u221a 6 \u03c0\u03be ( \u03d5 -\u03c8 0 ), \u03c8 0 is an integration constant, under condition that sign [ \u02d9 \u03d5 ( \u03d5 -\u03c8 0 )] = 1. Then we have the potential (cf. [12]) The SF-model with this potential has the particular solution \u00af \u03d5 ( t ) satisfying (18); correspondingly e f ( t ) satisfies the equations (9, 10) of the H-model. Equation (18) takes on the form For \u03d5 > \u03c8 0 this means that \u00af \u03d5 ( t ) descents down the potential hill to the right of \u03c8 0 ; it grows logarithmically and \u00af e f ( t ) \u2192 0 for t \u2192 \u221e . For t \u2192 -\u221e we have \u00af \u03d5 ( t ) \u2192 \u03c8 0 and \u00af e f ( t ) \u2192 e 0 . The condition (34) for g = 0 yields This is always fulfilled for 0 < \u03be < 2 thus guaranteeing that \u03d5 ( t ) \u2248 \u00af \u03d5 ( t ) on appropriate intervals in case of small deviation of the initial data. Note that the SF-model with \u03be > 0 does not admit divergent solutions like the 'Big Rip' [15] of the hydrodynamical counterpart. The case \u03be < 0 is possible for e > e 0 ; equation (28) yields E 0 ( \u03d5 ) = e 0 (cos \u03a6) -2 /\u00b5 \u2265 e 0 for | \u03a6 | < \u03c0/ 2, \u03a6 = \u00b5 \u221a 6 \u03c0 | \u03be | ( \u03d5 -\u03c8 0 ). We have a potential pit with infinite walls (with a periodic continuation): and the SF solutions with this potential lead to the same evolution of H ( t ) as the hydrodynamical ones if In this view, equation (18) for the corresponding particular solution of (13) yields The scalar field slides off the wall and tends to the potential minimum with energy e = e 0 , \u03d5 \u2192 \u03c8 0 for t \u2192\u221e .", "pages": [6]}, {"title": "C. Two-parametric EOS", "content": "To illustrate how equation (26) works in case of twoparametric EOS, we consider h ( e, \u03d5 ) = \u03bee -U ( \u03d5 ), \u03be = const , which is obtained as a generalization of (35) by changing \u03b7 \u2192 U ( \u03d5 ). We assume in this subsection 0 < \u03be < 2. After substitution \u02dc V = V -U/ 2 equation (26) yields where S = sign( \u02d9 \u03d5 ). This equation can be used either to derive V ( \u03d5 ) for given U ( \u03d5 ) or, vice versa, to find EOS on condition that V ( \u03d5 ) is given: By considering various \u02dc V one can generate examples with subsequent verification of equation (46) and inequalities \u0398 0 > 0 , E 0 > 0. We give two such examples dealing with simple elementary functions. (i) For \u02dc V ( \u03d5 ) = A 2 \u03d5 2 , A = const > 0, we have Equation (46) is valid if \u02d9 \u03d5 ( t ) \u03d5 ( t ) < 0. The non-trivial particular solution satisfying both the equations of H and SF models exists for t < t 1 = const (ii) The choice \u02dc V ( \u03d5 ) = A 2 exp( \u03b1\u03d5 ), where \u03b1, A > 0 are constants, generates Equation (46) is valid if \u03b1 \u02d9 \u03d5 ( t ) < 0. The particular solution of (13) is The corresponding solution of hydrodynamical equation (10) is e ( t ) = e f ( t ) = 96 \u03c0\u03b1 -4 ( t -t 1 ) -2 .", "pages": [6, 7]}, {"title": "VII. DISCUSSION", "content": "It is clear that the hydrodynamic description of DE is an oversimplification of the real cosmological situation in comparison with field-theoretic models. A consistent description of hydrodynamic phenomena assumes the local thermodynamical equilibrium. It is unclear how this assumption works as regards DE in the early Universe and in the modern era. Nevertheless, this does not prevent us . from using the hydrodynamical model on a formal level by equating the scalar field energy momentum tensor to the hydrodynamical one. On the other hand, some solutions of hydrodynamic models that are widely used in cosmological considerations, can be interpreted in therms of the SF-models. In this paper we found conditions for the SF-model that make this possible in case of the homogeneous isotropic spatially flat cosmology, under additional restriction on the form of the SF Lagrangian to be a canonical one. This is a very restrictive requirement; it leads lead to the differential equation for the potential V ( \u03d5 ), which is effective on intervals with the constant S = sign( \u02d9 \u03d5 ( t )). Moreover, the space of solutions of the SF-model is much wider than that of the barotropic Hmodel. In any case, the global equivalence between H and SF models for all modes of cosmological evolution is impossible. This is clearly seen in the examples of Section VI. This, however, does not prohibit using the H-SF analogy to study some special regimes. The hydrodynamical solutions with EOS (35) yield the SF solutions for the potential (37), when SF rolls down the potential well or descents down the potential hill (Section VI). But the Hmodel cannot describe the SF oscillations near minimum of the potential, though this regime being important for particle creation at the post-inflationary stage of the cosmological evolution [4]. On the other hand, some singular solutions like the 'Big Rip' [15] that may take place for certain EOSs are ruled out in case of the SF counterparts. The restriction on the initial data reduces possibilities to use the hydrodynamical representation of the 'restricted' SF model. This trouble is mitigated by the possibility to investigate close solutions. We derived conditions that ensure certain closeness of the SF-model energy-momentum tensors to that of the H model. In this sense the fiducial solution, which satisfies equations of both H and SF models, well approximates nearby solutions and describes their qualitative properties.", "pages": [7]}, {"title": "ACKNOWLEDGMENTS", "content": "This work has been supported in part by the Department of target training of Taras Shevchenko National University of Kyiv under National Academy of Sciences of Ukraine (project 6\u03a6).", "pages": [7]}, {"title": "Appendix A: SF-H correspondence without restrictions", "content": "The hydrodynamical and scalar field approaches are equivalent, if This equivalence can be used to find special solutions of hydrodynamics equations by means of the SF equations [13]. However, (A1) presupposes that the perfect fluid flow involved is a relativistic analog of the classical potential flow [13, 14]. Indeed, besides (3), equations (A1) yield where X > 0 is a solution of equation (4) for given EOS. This can be easily shown by considering (A1) in a locally Lorentz frame (where at some point x 0 we have g \u00b5\u03bd ( x 0 ) = \u03b7 \u00b5\u03bd , \u2202 \u03b1 g \u00b5\u03bd ( x 0 ) = 0), which is also an instantaneous proper frame for u \u00b5 ( x 0 ) = (1 , 0 , 0 , 0). The inequality X > 0 must be fulfilled because u \u00b5 is timelike; therefore, the H-model cannot be equivalent to the SF model in case of a stationary SF. On account of (1,2) and A2) we get /negationslash Usually for a given EOS e = E ( p, \u03d5 ), the differential equation (4) has a solution L that transforms (4) into an identity. On the other hand, for given L ( X,\u03d5 ), equations (3) represent the EOS parametrically, the domain of E as a function of p depending upon the range of L . In this sense we can speak about some equivalence of H and SF models, provided that conditions (A3) are fulfilled; this latter depends on the initial conditions of the hydrodynamical problem. Obviously, these conditions may be not satisfied if we deal with an arbitrary hydrodynamic flow, i.e. there is no full 'equivalence' between H and SF models. However, the conditions (A3) are always fulfilled in case of a homogeneous cosmology, where the gradient \u2202 \u00b5 \u03d5 = 0 is time-like and all the functions involved depend on the time variable only. In a more general case, the relativistic ideal fluid flows satisfying (A3) may be considered as an analogue of classical irrotational flows. (2009) [arXiv:0811.2797]; S. Nojiri, S. D. Odintsov, Phys. Rev., D 72 , 023003 (2005) [hep-th/0505215]; S. Nojiri, S. D. Odintsov, Phys. Lett., B 639 , 144 (2006) [hep-th/0606025]; S. Capozziello, V. F. Cardone, E. Elizalde, S. Nojiri, S. D. Odintsov, Phys. Rev., D 73 , 043512 (2006); L. Xu, Y. Chang, arXiv:1310.1532; M. Li, X.-D. Li, Y.-Zh. Ma, X. Zhang, Zh. Zhang, arXiv:1305.5302; P.-H. Chavanis, Eur. Phys. J. Plus, 129 38 (2014) [arXiv:1208.0797].", "pages": [7, 8]}] |
2019arXiv191207963K | https://arxiv.org/pdf/1912.07963.pdf | <document>
<section_header_level_1><location><page_1><loc_20><loc_89><loc_80><loc_91></location>Quantum Fluctuations of a Self-interacting Inflaton</section_header_level_1>
<text><location><page_1><loc_35><loc_85><loc_64><loc_87></location>G. Karakaya ∗ and V. K. Onemli †</text>
<text><location><page_1><loc_16><loc_82><loc_84><loc_84></location>Department of Physics, Istanbul Technical University, Maslak, Istanbul 34469, Turkey</text>
<text><location><page_1><loc_17><loc_46><loc_82><loc_80></location>We present a method to analytically compute the quantum corrected two-point correlation function of a scalar field in leading order at each loop in a homogeneous, isotropic and spatially flat spacetime where the expansion rate is time dependent and express the quantum corrected power spectrum ∆ 2 ( k ) as a time derivative of the coincident correlation function evaluated at time t k of the first horizon crossing of a mode with comoving wave number k . To facilitate the method, we consider the simplest version of inflation driven by a massive, minimally coupled inflaton endowing a quartic self-interaction-with positive or negative self-coupling. We compute the quantum corrected two-point correlation function, power spectrum, spectral index n ( k ) and the running of the spectral index α ( k ) for the inflaton fluctuations at one-loop order. Numerical estimates of the n ( k ) and α ( k ) and the cosmological measurements are in agreement, within reasonable ranges of values for the physical parameters in the model.</text>
<text><location><page_1><loc_17><loc_42><loc_43><loc_43></location>PACS numbers: 98.80.Cq, 04.62.+v</text>
<section_header_level_1><location><page_1><loc_40><loc_33><loc_60><loc_34></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_12><loc_15><loc_88><loc_30></location>A method to compute the quantum corrected two-point correlation function of a selfinteracting spectator scalar field, in leading order at each loop, during de Sitter inflationwhere the expansion rate H is constant-was introduced in Refs. [1, 2]. The computation incorporated Starobinsky's stochastic approach [3, 4] and the techniques of quantum field theory. The quantum corrected power spectrum was attained [2] as a time derivative of the coincidence limit of the computed correlation function evaluated at the time t k of first</text>
<text><location><page_2><loc_12><loc_87><loc_88><loc_91></location>horizon crossing. The method was employed to study a massless [1] and massive [2] minimally coupled scalars with a quartic self-interaction.</text>
<text><location><page_2><loc_12><loc_60><loc_88><loc_85></location>In this paper, we generalize the method to a more general Friedman-Robinson-Walker spacetime where the expansion rate H is time dependent. A massive, minimally coupled, selfinteracting scalar (inflaton) field with potential V ( ϕ )= m 2 2 ϕ 2 ± λ 4! ϕ 4 , where 0 <λ glyph[lessmuch] 1, drives the inflation in the model we consider, and hence, the expansion rate is time dependent. The plus sign in the quartic self-interaction implies a slightly steeper feature in the potential whereas the minus sign implies a slightly flattened feature. The alternative sign choices in the potential yield only a trivial sign difference between the results of any O ( λ ) computation. Note that a spectator scalar with this potential exhibits [5-10] enhanced quantum effectsin the massless limit-during de Sitter inflation. The effects are reduced [2] as the mass increases.</text>
<text><location><page_2><loc_12><loc_29><loc_88><loc_59></location>Physical origin of the quantum fluctuations is the Heisenberg's uncertainty principle which manifests itself as the particle-antiparticle pair production and annihilation out of and into the vacuum. The observed scalar perturbations on the cosmic microwave background (CMB) anisotropy are believed to be the amplified imprints of quantum fluctuations of an inflaton which seeded the large scale structure in the universe. The issue of fluctuations in inflationary spacetimes and their observable implications has been of more than passing interest [11] in cosmology. One naturally inquires, if there are correlations between the theoretical predictions for primordial inflaton fluctuations and the measurements of CMB temperature anisotropy, galaxy power spectra and gravitational lensing survey correlation functions. Indeed, all measurements agree that the number of the observed entity of a particular size in the corresponding distribution, in each case of the inquiries, increases as the size of the entity increases.</text>
<text><location><page_2><loc_12><loc_13><loc_88><loc_27></location>During inflation the physical size grows more rapidly than the horizon size. Therefore, for each scale of cosmological interest with comoving wave number k there exists a certain comoving time t = t k , as pointed out earlier, at which the scale exits the horizon. The Fourier mode k of the fluctuation field is, therefore, a subhorizon [ultraviolet (UV)] mode when t<t k but becomes a superhorizon [infrared (IR)] mode when t>t k . Many subhorizon fluctuation modes are shifted continuously to the superhorizon scales during inflation.</text>
<text><location><page_2><loc_12><loc_7><loc_88><loc_11></location>As soon as a particular mode exits the horizon, the crest of the mode loses causal contact with the trough and therefore cannot propagate. Hence, the super-horizon mode 'freezes</text>
<text><location><page_3><loc_12><loc_66><loc_88><loc_91></location>in' after the horizon exit. It does not behave like a wave at all. After the end of inflationary phase, during the phases of radiation domination and matter domination, the horizon size grows more rapidly than the scale factor a ( t )-hence, than the physical size. Thus, as the physical horizon size successively reaches the physical wavelengths of the super-horizon modes, the comoving scales that exited the horizon during inflation start reentering the horizon with the same amplitudes as they were when they exited-except for a small factor which depends on the ratio of pressure to energy density in the universe when they exit and reenter-and the crests and troughs of the corresponding modes regain causal contact and propagate. The modes that exit the horizon latest reenter earliest and play the key role in seeding the large scale structure formation in the early universe.</text>
<text><location><page_3><loc_12><loc_26><loc_88><loc_64></location>Starobinsky realized [3, 4] that the UV modes are irrelevant for the late time behavior of fields in the scalar potential models and amputated them. The free field expansion of a field contains arbitrarily large wavenumbers (UV modes) and therefore the expectation values of coincident products of the fields can arbor UV divergences. All of these features are absent in the Starobinsky's realization of the free field which is constructed by taking the IR limit of the mode function and retaining only the IR modes at a particular time. Using the IR truncated field in computations is an approximation which gets only the UV-finite, secular parts of the full result. In scalar potential models, Starobinsky's formalism gives exactly the same leading terms that the Schwinger-Keldish (in-in) formalism gives at each perturbative order. For example, the coincidence limit of two-point correlation function of a massless minimally coupled scalar with a quartic self-interaction in de Sitter spacetime, computed applying Schwinger-Keldish formalism [7, 8] and applying Starobinsky's formalism [8] yield exactly the same leading terms at one- and two-loop order. Other examples showing the exact agreement include: scalars interacting with fermions [12, 13], scalar quantum electrodynamics [14, 15] and scalars with derivative interactions [16-18].</text>
<text><location><page_3><loc_12><loc_7><loc_88><loc_25></location>Several aspects of IR dynamics in scalar potential models have been studied with various approaches that include, employing stochastic spectral expansion [19], extending the stochastic formalism [20], applying complementary and principal series analysis [21, 22], computing effective actions [23-26] and potentials [27], implementing Fokker-Planck equation and δN formalism [28, 29], employing 1 /N expansion [30], adopting reduced density matrix method [31] and applying renormalization group analysis [32]. We follow Starobinsky's approach [3, 4] and use the standard techniques of quantum field theory, in this paper.</text>
<text><location><page_4><loc_12><loc_60><loc_88><loc_91></location>The two-point correlation function of the inflaton fluctuation field we compute is a measure of the relationship between the random fluctuations at two events ( t, glyph[vector]x ) and ( t ' , glyph[vector]x ' ). Thus, it provides information about how independent the fluctuation δϕ ( t, glyph[vector]x ) is from the fluctuation δϕ ( t ' , glyph[vector]x ' ). The spatial two-point correlation function, therefore, measures the emergence probability of a fluctuation δϕ ( t, glyph[vector]x ) at a separation ∆ x = ‖ glyph[vector]x -glyph[vector]x ' ‖ from the fluctuation δϕ ( t, glyph[vector]x ' ), at a given time t . The fluctuation spectrum contains contributions from all ranges of fluctuations of different sizes (modes). The Fourier transform of the spatial two-point correlation function can be used to define the time dependent power spectrum ∆ 2 ( t, k ), which quantifies information about the probability of how many random quantum fluctuation of each size ought to contribute to the spectrum at a given time t . The usual, time-independent power spectrum [33] of the fluctuations lim t glyph[greatermuch] t k ∆ 2 δϕ ( t, k ) ≡ ∆ 2 δϕ ( k ) can be obtained [2] from the coincident correlation function of the IR truncated fluctuation field.</text>
<text><location><page_4><loc_12><loc_29><loc_88><loc_59></location>The ∆ 2 δϕ ( k ) of the primordial inflaton fluctuations can be measured from the power spectrum of the CMB which encodes size distribution of the temperature fluctuations, i.e., how many hot and cold spots there are of each angular size in the CMB. Hence, the measured CMB data is the ultimate arbiter of the inflationary models. The models, for example, generically predict a slightly negative (red) tilt in the power spectrum, i.e., a deviation between -0 . 08 and -0 . 02 from that of the Harrison-Zeldovich scale invariant (constant amplitude) spectrum with zero tilt generated during a de Sitter inflation. (Red-tilt implies that the amplitudes of inflaton fluctuations grow toward the larger scales.) The exact value of the tilt depends on the details of the particular inflationary scenario and can be used to discriminate between the models. We compute the quantum corrected tilt and its running in our model up to O ( λ 2 ). Numerical estimates obtained from the results are in accordance with the observations.</text>
<text><location><page_4><loc_12><loc_7><loc_88><loc_27></location>The remainder of this paper is organized as follows. In Sec. II we present the model, obtain the field equations, solve the expansion rate, scale factor, slow roll parameter and the background field in terms of the initial values. In Sec. III we obtain the mode function for the free fluctuation field, give the free field expansion and express the full field in terms of the free field and the Green's function. We compute the tree-order and one-loop two point correlators for the fluctuations of the IR truncated inflaton in our model, up to O ( λ 2 ), in Sec. IV. The coincidence limit of the quantum corrected two-point correlation function is given in Sec. V. In Sec. VI, using the coincident correlation function, we compute the</text>
<text><location><page_5><loc_12><loc_81><loc_88><loc_91></location>quantum corrected power spectrum, spectral index and running of the spectral index for the inflaton fluctuations; estimate the tilt and its running numerically and compare them with the observational values. The conclusions are summarized in Sec. VII. The Appendixes comprise the computational details of the tree-order and one-loop correlators.</text>
<section_header_level_1><location><page_5><loc_41><loc_76><loc_58><loc_77></location>II. THE MODEL</section_header_level_1>
<text><location><page_5><loc_12><loc_69><loc_88><loc_73></location>We consider a massive, minimally coupled self-interacting scalar field driven inflation. The renormalized Lagrangian density of the model</text>
<formula><location><page_5><loc_31><loc_64><loc_88><loc_68></location>L = [ R 16 πG -1+ δZ 2 ∂ µ ϕ∂ ν ϕg µν -V ( ϕ ) ] √ -g , (1)</formula>
<text><location><page_5><loc_12><loc_53><loc_88><loc_62></location>where the R and g denote the Ricci scalar and the determinant of the metric g µν , respectively. [We adopt the convention where a Greek index µ =0 , 1 , 2 , . . . , ( D -1), hence the components of four-position vector x are x µ =( x 0 , glyph[vector]x ) with comoving time x 0 ≡ t and ∂ µ =( ∂ 0 , glyph[vector] ∇ ).] The δZ stand for the field strength renormalization counterterm. The renormalized potential</text>
<formula><location><page_5><loc_35><loc_48><loc_88><loc_52></location>V ( ϕ )= 1+ δZ 2 m 2 ϕ 2 + V pert ( ϕ ) , (2)</formula>
<text><location><page_5><loc_12><loc_45><loc_87><loc_47></location>where m is the renormalized mass. The part of the potential that we treat perturbatively,</text>
<formula><location><page_5><loc_36><loc_41><loc_88><loc_44></location>V pert ( ϕ ) ≡ δm 2 2 ϕ 2 ± λ + δλ 4! ϕ 4 , (3)</formula>
<text><location><page_5><loc_12><loc_27><loc_88><loc_39></location>where λ is the coupling strength and δm 2 and δλ stand for the mass renormalization and the coupling strength renormalization counterterms, respectively. Counterterms are just constants and their numerical coefficients are identical in any geometry. Thus, as in flat spacetime and de Sitter geometry, the orders of λ of the counterterms are δZ ∼ O ( λ 2 ), δm 2 ∼O ( λ ) and δλ ∼O ( λ 2 ).</text>
<text><location><page_5><loc_12><loc_19><loc_88><loc_26></location>The metric tensor g µν is chosen as that of a D -dimensional spatially flat FriedmannRobertson-Walker spacetime with no metric perturbations. Therefore, the invariant line element</text>
<formula><location><page_5><loc_33><loc_16><loc_88><loc_18></location>ds 2 = g µν dx µ dx ν = -dt 2 + a 2 ( t ) dglyph[vector]x · dglyph[vector]x . (4)</formula>
<text><location><page_5><loc_12><loc_11><loc_88><loc_15></location>Note that, just as the tensor power spectrum is suppressed by a factor of the slow-roll parameter</text>
<formula><location><page_5><loc_40><loc_7><loc_88><loc_11></location>glyph[epsilon1] ( t ) ≡-˙ H ( t ) H 2 ( t ) glyph[lessmuch] 1 , (5)</formula>
<text><location><page_6><loc_12><loc_87><loc_88><loc_91></location>where an overdot denotes derivative with respect to comoving time t and H ( t ) ≡ ˙ a/a stands for the Hubble parameter, the metric perturbations are suppressed by √ glyph[epsilon1] .</text>
<text><location><page_6><loc_14><loc_84><loc_62><loc_85></location>The inflaton in this background obeys the field equation,</text>
<formula><location><page_6><loc_25><loc_78><loc_88><loc_82></location>¨ ϕ ( x )+( D -1) H ( t ) ˙ ϕ ( x ) -[ ∇ 2 a 2 -m 2 ] ϕ ( x )= -∂V pert ( ϕ ) ∂ϕ ( x ) 1+ δZ , (6)</formula>
<text><location><page_6><loc_12><loc_75><loc_53><loc_77></location>where the derivative of the perturbative potential</text>
<formula><location><page_6><loc_36><loc_70><loc_88><loc_73></location>∂V pert ( ϕ ) ∂ϕ = δm 2 ϕ ± 1 6 ( λ + δλ ) ϕ 3 . (7)</formula>
<text><location><page_6><loc_12><loc_64><loc_88><loc_68></location>We consider the inflaton field ϕ ( x ) as a sum of an unperturbed (averaged) background field ¯ ϕ ( t ) that depends only on time and a small perturbation δϕ ( x ) on this background field,</text>
<formula><location><page_6><loc_41><loc_60><loc_88><loc_61></location>ϕ ( x )= ¯ ϕ ( t )+ δϕ ( x ) . (8)</formula>
<text><location><page_6><loc_12><loc_53><loc_88><loc_57></location>Thus Eq. (6) yields the following equations for the homogenous part ¯ ϕ ( t ) and for the fluctuations δϕ ( t, glyph[vector]x ) around it,</text>
<formula><location><page_6><loc_21><loc_48><loc_88><loc_52></location>¨ ¯ ϕ ( t )+( D -1) H ( t ) ˙ ¯ ϕ ( t )+ m 2 ¯ ϕ ( t )= -1 1+ δZ { δm 2 ¯ ϕ ( t ) ± λ + δλ 6 ¯ ϕ 3 ( t ) } , (9)</formula>
<formula><location><page_6><loc_24><loc_43><loc_88><loc_47></location>δ ¨ ϕ ( x )+( D -1) H ( t ) δ ˙ ϕ ( x ) -[ ∇ 2 a 2 -m 2 ] δϕ ( x )= -∂V pert ( δϕ ) ∂δϕ ( x ) 1+ δZ , (10)</formula>
<text><location><page_6><loc_12><loc_40><loc_17><loc_42></location>where</text>
<formula><location><page_6><loc_17><loc_35><loc_88><loc_38></location>∂V pert ( δϕ ) ∂δϕ ( x ) ≡ δm 2 δϕ ( x ) ± λ + δλ 6 [ 3¯ ϕ 2 ( t ) δϕ ( x )+3¯ ϕ ( t ) ( δϕ ( x )) 2 +( δϕ ( x )) 3 ] . (11)</formula>
<text><location><page_6><loc_12><loc_29><loc_88><loc_33></location>The scale factor a ( t ) and the background field ¯ ϕ ( t ) are related by two non-trivial Einstein's field equations-corresponding to purely temporal and purely spatial components-as</text>
<formula><location><page_6><loc_32><loc_24><loc_88><loc_28></location>( D -2) D -1 2 H 2 =8 πG [ ˙ ¯ ϕ 2 ( t ) 2 + V ( ¯ ϕ ) ] , (12)</formula>
<formula><location><page_6><loc_27><loc_20><loc_88><loc_24></location>-( D -2) D -1 2 H 2 [ 1 -2 glyph[epsilon1] ( t ) D -1 ] =8 πG [ ˙ ¯ ϕ 2 ( t ) 2 -V ( ¯ ϕ ) ] . (13)</formula>
<text><location><page_6><loc_12><loc_17><loc_85><loc_18></location>Equations (12) and (13) can be inverted to eliminate the background inflaton field ¯ ϕ ( t ),</text>
<formula><location><page_6><loc_38><loc_12><loc_88><loc_15></location>˙ ¯ ϕ 2 ( t )= ( D -2) H 2 ( t ) 8 πG glyph[epsilon1] ( t ) , (14)</formula>
<formula><location><page_6><loc_35><loc_8><loc_88><loc_11></location>V ( ¯ ϕ )= ( D -2) H 2 ( t ) 16 πG [ D -1 -glyph[epsilon1] ( t ) ] . (15)</formula>
<text><location><page_7><loc_12><loc_89><loc_55><loc_91></location>Using Eqs. (2) and (3) in Eq. (15), at O ( λ ), we find</text>
<formula><location><page_7><loc_27><loc_84><loc_88><loc_88></location>H 2 = 8 πG ( D -2)( D -1 -glyph[epsilon1] ) [ ( m 2 + δm 2 ) ¯ ϕ 2 ± λ 12 ¯ ϕ 4 + O ( λ 2 ) ] . (16)</formula>
<text><location><page_7><loc_12><loc_74><loc_88><loc_83></location>The mass counterterm δm 2 , which is of O ( λ ), is multiplied by ¯ ϕ 2 . Hence, it is suppressed by a factor of ¯ ϕ 2 compared to the quartic self-interaction term which is also of O ( λ ). Because 4 glyph[lessorsimilar] ¯ ϕ glyph[lessorsimilar] 10 6 in Planckian units, the former cannot contribute as significant as the latter. Therefore, we have</text>
<formula><location><page_7><loc_27><loc_68><loc_88><loc_72></location>H = √ 8 πG ( D -2)( D -1) m 2 ¯ ϕ m [ 1 ± λ 24 ( ¯ ϕ m ) 2 + O ( glyph[epsilon1] )+ O ( λ 2 ) ] . (17)</formula>
<text><location><page_7><loc_12><loc_62><loc_88><loc_66></location>Recall that the quartic self-interaction term λ 4! ¯ ϕ 4 is treated perturbatively next to the the mass term m 2 2 ¯ ϕ 2 . Hence, 1 glyph[greatermuch] λ 12 ¯ ϕ 2 m 2 in Eq. (17).</text>
<text><location><page_7><loc_12><loc_54><loc_88><loc_61></location>To get the expansion rate H from Eq. (17), as a function of time and initial values H i 0 and glyph[epsilon1] i 0 , we first need to solve field equation (9) for ¯ ϕ ( t ). In the slow roll regime where | ¨ ¯ ϕ | glyph[lessmuch] H | ˙ ¯ ϕ | , and at O ( λ ), Eq. (9) implies</text>
<formula><location><page_7><loc_39><loc_49><loc_88><loc_53></location>( D -1) H ˙ ¯ ϕ + m 2 ¯ ϕ glyph[similarequal] ∓ λ 6 ¯ ϕ 3 . (18)</formula>
<text><location><page_7><loc_12><loc_44><loc_88><loc_48></location>We neglected the other O ( λ ) termδm 2 ¯ ϕ -suppressed by a factor of ¯ ϕ 2 on the right side of Eq. (9), as we did in obtaining Eq. (17). Employing Eq. (17) in Eq. (18) yields</text>
<formula><location><page_7><loc_29><loc_38><loc_88><loc_42></location>˙ ¯ ϕ glyph[similarequal] -m √ 8 πG √ D -2 D -1 [ 1 ± λ 8 ( ¯ ϕ m ) 2 + O ( glyph[epsilon1] )+ O ( λ 2 ) ] . (19)</formula>
<text><location><page_7><loc_12><loc_36><loc_81><loc_37></location>The solution of Eq. (19), for the plus sign choice in the O ( λ )-term, can be given as</text>
<formula><location><page_7><loc_25><loc_30><loc_88><loc_34></location>¯ ϕ ( t ) glyph[similarequal] √ 8 λ m tan ( -√ λ 8 D -2 ( D -1)8 πG t +arctan ( √ λ 8 ¯ ϕ i m ) ) . (20)</formula>
<text><location><page_7><loc_12><loc_24><loc_88><loc_28></location>For the minus sign choice in Eq. (19), taking λ →-λ and analytically continuing the tangent function in Eq. (20), yields the background field solution as</text>
<formula><location><page_7><loc_24><loc_19><loc_88><loc_23></location>¯ ϕ ( t ) glyph[similarequal] √ 8 λ m tanh ( -√ λ 8 D -2 ( D -1)8 πG t +arctanh ( √ λ 8 ¯ ϕ i m ) ) . (21)</formula>
<text><location><page_7><loc_12><loc_13><loc_88><loc_17></location>Using the addition formulae for the tan( x ) and tanh( x ) functions in Eqs. (20) and (21) respectively and then expanding the results in series, up to O ( λ 2 ), leads to</text>
<formula><location><page_7><loc_25><loc_7><loc_88><loc_11></location>¯ ϕ ( t ) glyph[similarequal] ¯ ϕ i { 1 -√ D -2 ( D -1)8 πG [ m ¯ ϕ i ± λ 8 ¯ ϕ i m ] t ± λ 8 D -2 ( D -1)8 πG t 2 } , (22)</formula>
<text><location><page_8><loc_12><loc_34><loc_13><loc_35></location>as</text>
<text><location><page_8><loc_12><loc_12><loc_13><loc_13></location>as</text>
<text><location><page_8><loc_12><loc_84><loc_88><loc_91></location>where ¯ ϕ i is the initial value of ¯ ϕ ( t ) at t = t i =0. Note that ¯ ϕ i = ¯ ϕ i 0 where the subscript 0 denotes the non-interacting ( λ → 0) result. Note also that, in the noninteracting regime, one recovers</text>
<formula><location><page_8><loc_35><loc_80><loc_88><loc_84></location>lim λ → 0 ¯ ϕ ( t ) glyph[similarequal] -m √ D -2 ( D -1)8 πG t +¯ ϕ i , (23)</formula>
<text><location><page_8><loc_12><loc_75><loc_88><loc_79></location>the well known solution in the simplest version of the chaotic inflation driven by a mass term.</text>
<text><location><page_8><loc_12><loc_69><loc_88><loc_73></location>The fluctuation field δϕ ( x ) in Eq. (8), on the other hand, is obtained by solving Eq. (10) as</text>
<formula><location><page_8><loc_26><loc_65><loc_88><loc_70></location>δϕ ( x )= δϕ 0 ( x ) -∫ t 0 dt ' a D -1 ( t ' ) ∫ d D -1 x ' G ret ( x ; x ' ) ∂V pert ( δϕ ) ∂δϕ ( x ' ) 1+ δZ , (24)</formula>
<text><location><page_8><loc_12><loc_52><loc_88><loc_64></location>where ∂V pert ( δϕ ) ∂δϕ is given in Eq. (11) and the retarded Green's function is computed in Eq. (95). We use solution (24) to compute the quantum corrected two-point correlation function in Sec. IV. In the rest of this section, throughout Secs. II A-C, we use homogeneous background solution (22) to express the expansion rate H ( t ), scale factor a ( t ) and slow-roll parameter glyph[epsilon1] ( t ) in terms of the initial values H i 0 and glyph[epsilon1] i 0 .</text>
<section_header_level_1><location><page_8><loc_21><loc_47><loc_79><loc_48></location>A. The expansion rate H ( t ) in terms of initial values H i 0 and glyph[epsilon1] i 0</section_header_level_1>
<text><location><page_8><loc_14><loc_42><loc_68><loc_44></location>Employing Eq. (22) in Eq. (17) we obtain the Hubble parameter</text>
<formula><location><page_8><loc_37><loc_38><loc_88><loc_40></location>H ( t )= H 0 ( t ) ± λH λ ( t )+ O ( λ 2 ) , (25)</formula>
<formula><location><page_8><loc_32><loc_30><loc_88><loc_34></location>H ( t )= H i + ˙ H i t + 1 2! H i t 2 + 1 3! ... H i t 3 + O ( λ 2 ) . (26)</formula>
<text><location><page_8><loc_12><loc_25><loc_88><loc_30></location>The initial values of the Hubble parameter H i and its time derivatives ˙ H i , H i and ... H i in Eq. (26) can be written in terms of the tree-order initial values</text>
<formula><location><page_8><loc_39><loc_20><loc_88><loc_24></location>H i 0 = √ 8 πG ( D -2)( D -1) m ¯ ϕ i , (27)</formula>
<formula><location><page_8><loc_45><loc_15><loc_88><loc_19></location>˙ H i 0 = -m 2 D -1 , (28)</formula>
<formula><location><page_8><loc_30><loc_7><loc_88><loc_10></location>H i = H i 0 [ 1 ± λ 4! ( ¯ ϕ i m ) 2 ] + O ( λ 2 ) ≡ H i 0 ± λH iλ + O ( λ 2 ) , (29)</formula>
<formula><location><page_9><loc_29><loc_87><loc_88><loc_91></location>˙ H i = ˙ H i 0 [ 1 ± λ 4 ( ¯ ϕ i m ) 2 ] + O ( λ 2 ) ≡ ˙ H i 0 ± λ ˙ H iλ + O ( λ 2 ) , (30)</formula>
<formula><location><page_9><loc_29><loc_83><loc_88><loc_87></location>H i = ± H i 0 λ 2 D -2 ( D -1)8 πG + O ( λ 2 ) ≡ H i 0 ± λ H iλ + O ( λ 2 ) , (31)</formula>
<formula><location><page_9><loc_29><loc_79><loc_88><loc_83></location>... H i = ± ˙ H i 0 λ 4 D -2 ( D -1)8 πG + O ( λ 2 ) ≡ ... H i 0 ± λ ... H iλ + O ( λ 2 ) . (32)</formula>
<text><location><page_9><loc_12><loc_74><loc_88><loc_78></location>The ratio ¯ ϕ i /m in Eqs. (29) and (30) can be expressed in terms of the initial values of the Hubble parameter H i 0 and the slow roll parameter</text>
<formula><location><page_9><loc_39><loc_68><loc_88><loc_73></location>glyph[epsilon1] i 0 = -˙ H i 0 H 2 i 0 = m 2 ( D -1) H 2 i 0 , (33)</formula>
<text><location><page_9><loc_12><loc_66><loc_58><loc_67></location>where we used Eq. (28). Equations (27) and (33) yield</text>
<formula><location><page_9><loc_37><loc_61><loc_88><loc_64></location>¯ ϕ i = ¯ ϕ i 0 = √ ξglyph[epsilon1] -1 i 0 m = √ D -2 8 πGglyph[epsilon1] i 0 . (34)</formula>
<text><location><page_9><loc_12><loc_58><loc_45><loc_59></location>Here, we define a dimensionless number</text>
<formula><location><page_9><loc_42><loc_53><loc_88><loc_56></location>ξ ≡ D -2 ( D -1)8 πGH 2 i 0 . (35)</formula>
<text><location><page_9><loc_12><loc_47><loc_88><loc_51></location>Employing Eq. (34) in Eqs. (29) and (30) yields the ratios of H iλ and H i 0 and of ˙ H iλ and ˙ H i 0 , in terms ξ and glyph[epsilon1] i 0 , as</text>
<formula><location><page_9><loc_41><loc_43><loc_88><loc_47></location>H iλ H i 0 = 1 6 ˙ H iλ ˙ H i 0 = 1 4! ξglyph[epsilon1] -2 i 0 . (36)</formula>
<text><location><page_9><loc_74><loc_41><loc_74><loc_42></location>glyph[negationslash]</text>
<text><location><page_9><loc_85><loc_41><loc_85><loc_42></location>glyph[negationslash]</text>
<text><location><page_9><loc_12><loc_38><loc_88><loc_43></location>Equations (31) and (32) imply that the initial values H i 0 =0, ... H i 0 =0, H iλ =0 and ... H iλ =0. In fact, the ratios</text>
<formula><location><page_9><loc_27><loc_33><loc_88><loc_37></location>˙ H iλ H iλ = -6 glyph[epsilon1] i 0 H i 0 , H iλ H iλ =12 glyph[epsilon1] 2 i 0 H 2 i 0 and ... H iλ H iλ = -6 glyph[epsilon1] 3 i 0 H 3 i 0 . (37)</formula>
<text><location><page_9><loc_12><loc_30><loc_50><loc_31></location>Equations (25)-(26) and (36)-(37) imply that</text>
<formula><location><page_9><loc_33><loc_25><loc_88><loc_29></location>H 0 ( t )= H i 0 [ 1+ ˙ H i 0 H i 0 t ] = H i 0 [ 1 -glyph[epsilon1] i 0 H i 0 t ] , (38)</formula>
<formula><location><page_9><loc_12><loc_20><loc_88><loc_25></location>H λ ( t )= H iλ [ 1+ ˙ H iλ H iλ t + 1 2 H iλ H iλ t 2 + 1 3! ... H iλ H iλ t 3 ] = H i 0 ξ 4! glyph[epsilon1] -2 i 0 [ 1 -6 glyph[epsilon1] i 0 H i 0 t +6( glyph[epsilon1] i 0 H i 0 t ) 2 -( glyph[epsilon1] i 0 H i 0 t ) 3 ] . (39)</formula>
<text><location><page_9><loc_12><loc_18><loc_72><loc_19></location>Thus, combining Eqs. (38)-(39) in Eq. (25) yields the expansion rate as</text>
<formula><location><page_9><loc_19><loc_13><loc_88><loc_16></location>H ( t )= H i 0 [ 1 -glyph[epsilon1] i 0 H i 0 t ± λξ 4! [ glyph[epsilon1] -2 i 0 -6 glyph[epsilon1] -1 i 0 H i 0 t +6( H i 0 t ) 2 -glyph[epsilon1] i 0 ( H i 0 t ) 3 ] + O ( λ 2 ) ] . (40)</formula>
<text><location><page_9><loc_12><loc_7><loc_88><loc_11></location>Among the O ( λ ) terms, the first two ones dominate as long as 0 <glyph[epsilon1] i 0 < 1 2( H i 0 t ) [ √ 5 / 3 -1]. This means that for a range of 0 <glyph[epsilon1] i 0 < 0 . 0025 they remain as leading terms during the</text>
<text><location><page_10><loc_12><loc_71><loc_88><loc_91></location>first 60 e-foldings. For a larger glyph[epsilon1] i 0 the third term becomes of order the sum of the first two terms during the last epochs of the inflation. The forth term (the cubic one), however, is suppressed during inflation for an glyph[epsilon1] i 0 < 6 / ( H i 0 t ). Thus for glyph[epsilon1] i 0 glyph[lessmuch] 0 . 1-satisfied in the slow-roll regime-the forth term may be neglected during the first 60 e-foldings. Thus, to express time t in terms of the scale factor a , which provides the essential change of variable to evaluate the integrals of the one-loop correlator in Sec. IV B, we neglect the fourth term as a zeroth-order approximation but we do include it in our first iteration . Thus, integrating Eq. (40) we find the cubic polynomial</text>
<formula><location><page_10><loc_21><loc_66><loc_88><loc_69></location>( H i 0 t ) 3 ∓ 6 λξ glyph[epsilon1] i 0 [ 1 ± λξ 4 glyph[epsilon1] -2 i 0 ] ( H i 0 t ) 2 ± 12 λξ [ 1 ± λξ 4! glyph[epsilon1] -2 i 0 ] ( H i 0 t ) ∓ 12 λξ ln( a ) ∼ = 0 , (41)</formula>
<text><location><page_10><loc_12><loc_63><loc_77><loc_64></location>which has three distinct real roots; see Appendix A. We define a new variable</text>
<formula><location><page_10><loc_42><loc_59><loc_88><loc_61></location>q ≡ √ 1 -2 glyph[epsilon1] i 0 ln( a ) , (42)</formula>
<text><location><page_10><loc_12><loc_52><loc_88><loc_56></location>and express the relevant root of polynomial (41) in terms of q in Eq. (A5) which implies that the comoving time</text>
<formula><location><page_10><loc_34><loc_47><loc_88><loc_51></location>t ∼ = glyph[epsilon1] -1 i 0 1 -q H i 0 [ 1 ± λξ 4! glyph[epsilon1] -2 i 0 [ 1 -2 q ] + O ( λ 2 ) ] . (43)</formula>
<text><location><page_10><loc_12><loc_44><loc_88><loc_46></location>Using Eq. (43) in Eq. (40)-without neglecting the fourth term-yields the expansion rate as</text>
<formula><location><page_10><loc_33><loc_39><loc_88><loc_43></location>H ( t )= H i 0 [ q ± λ 4! ξglyph[epsilon1] -2 i 0 [ q 3 + q 2 -1 ] + O ( λ 2 ) ] . (44)</formula>
<text><location><page_10><loc_12><loc_36><loc_31><loc_38></location>Hence, Eq. (44) implies</text>
<text><location><page_10><loc_12><loc_30><loc_17><loc_32></location>where</text>
<text><location><page_10><loc_12><loc_24><loc_20><loc_26></location>Therefore,</text>
<formula><location><page_10><loc_43><loc_22><loc_88><loc_23></location>˙ H 0 ( t )= H i 0 ˙ q 0 ( t ) . (47)</formula>
<text><location><page_10><loc_12><loc_18><loc_51><loc_20></location>Note that the time derivative of Eq. (42) yields</text>
<formula><location><page_10><loc_43><loc_13><loc_88><loc_17></location>˙ q ( t )= -glyph[epsilon1] i 0 H ( t ) q ( t ) , (48)</formula>
<text><location><page_10><loc_12><loc_10><loc_23><loc_12></location>which implies</text>
<formula><location><page_10><loc_43><loc_34><loc_88><loc_35></location>H 0 ( t )= H i 0 q 0 ( t ) , (45)</formula>
<formula><location><page_10><loc_42><loc_28><loc_88><loc_30></location>q 0 = √ 1 -2 glyph[epsilon1] i 0 ln( a 0 ) . (46)</formula>
<formula><location><page_10><loc_35><loc_7><loc_88><loc_10></location>˙ q 0 ( t )= -glyph[epsilon1] i 0 H 0 ( t ) q 0 ( t ) = -glyph[epsilon1] i 0 H i 0 =const . . (49)</formula>
<text><location><page_11><loc_12><loc_36><loc_15><loc_38></location>and</text>
<text><location><page_11><loc_12><loc_89><loc_39><loc_91></location>Using Eq. (49) in Eq. (47) yields</text>
<formula><location><page_11><loc_37><loc_85><loc_88><loc_87></location>˙ H 0 ( t )= -glyph[epsilon1] i 0 H 2 i 0 = ˙ H i 0 =const . , (50)</formula>
<text><location><page_11><loc_12><loc_76><loc_88><loc_82></location>in agreement with Eq. (33). Expansion rate H ( t ), by definition, is the time derivative of ln( a ( t )). Therefore, in the next section, the scale factor a ( t ) is obtained integrating the H ( t ) and then exponentiating the result.</text>
<section_header_level_1><location><page_11><loc_23><loc_70><loc_77><loc_71></location>B. The scale factor a ( t ) in terms of initial values H i 0 and glyph[epsilon1] i 0</section_header_level_1>
<text><location><page_11><loc_14><loc_66><loc_38><loc_67></location>The logarithm of scale factor</text>
<formula><location><page_11><loc_34><loc_61><loc_88><loc_63></location>ln( a ( t ))=ln( a 0 ( t )) ± λ ln( a ( t )) λ + O ( λ 2 ) , (51)</formula>
<text><location><page_11><loc_12><loc_55><loc_88><loc_59></location>is obtained by integrating expansion rate (44) via making change of variable (43) which implies</text>
<formula><location><page_11><loc_33><loc_51><loc_88><loc_55></location>dt = -dq H i 0 glyph[epsilon1] -1 i 0 [ 1 ± λ 8 ξglyph[epsilon1] -2 i 0 [ 1 -4 3 q ] + O ( λ 2 ) ] . (52)</formula>
<text><location><page_11><loc_12><loc_49><loc_38><loc_50></location>The result of the integral yields</text>
<formula><location><page_11><loc_43><loc_45><loc_88><loc_49></location>ln( a 0 ( t ))= glyph[epsilon1] -1 i 0 E 0 2 , (53)</formula>
<text><location><page_11><loc_12><loc_43><loc_25><loc_44></location>where we define</text>
<formula><location><page_11><loc_37><loc_40><loc_88><loc_42></location>E 0 ( t ) ≡ 1 -q 2 0 ( t )=2 glyph[epsilon1] i 0 ln( a 0 ( t )) , (54)</formula>
<formula><location><page_11><loc_39><loc_33><loc_88><loc_37></location>ln( a ( t )) λ = -ξ 4! glyph[epsilon1] -3 i 0 [1 -q 0 ] 4 4 . (55)</formula>
<text><location><page_11><loc_12><loc_30><loc_75><loc_32></location>Exponentiating Eq. (51), using Eqs. (53) and (55), gives the scale factor as</text>
<formula><location><page_11><loc_16><loc_25><loc_88><loc_29></location>a ( t )= a 0 ( t ) exp ( ∓ λ 4! ξ 4 glyph[epsilon1] -3 i 0 [1 -q 0 ] 4 + O ( λ 2 ) ) = a 0 ( t ) [ 1 ∓ λ 4! ξ 4 glyph[epsilon1] -3 i 0 [1 -q 0 ] 4 + O ( λ 2 ) ] , (56)</formula>
<text><location><page_11><loc_12><loc_22><loc_17><loc_23></location>where</text>
<formula><location><page_11><loc_30><loc_18><loc_88><loc_22></location>a 0 ( t )=exp ( H i 0 t 0 [ 1 -glyph[epsilon1] i 0 2 H i 0 t 0 ]) = exp ( glyph[epsilon1] -1 i 0 E 0 2 ) . (57)</formula>
<text><location><page_11><loc_12><loc_16><loc_80><loc_17></location>To get Eq. (57), we used Eq. (38) in the first equality and Eq. (53) in the second.</text>
<text><location><page_11><loc_14><loc_13><loc_58><loc_15></location>Scale factor (56) reaches its maximum value at time</text>
<formula><location><page_11><loc_43><loc_9><loc_88><loc_11></location>t m ∼ =( glyph[epsilon1] i 0 H i 0 ) -1 , (58)</formula>
<text><location><page_12><loc_12><loc_87><loc_88><loc_91></location>where ˙ a =0. Inflation, on the other hand, continues as long as a > 0. This implies that inflation ends at</text>
<formula><location><page_12><loc_35><loc_84><loc_88><loc_87></location>t e ∼ = t m [1 -√ glyph[epsilon1] i 0 ] = t m -( √ glyph[epsilon1] i 0 H i 0 ) -1 . (59)</formula>
<text><location><page_12><loc_12><loc_73><loc_88><loc_82></location>The e-folding condition a ( t e ) glyph[greaterorsimilar] e 60 and the fact that q > 0 constrain the glyph[epsilon1] i 0 . They imply that glyph[epsilon1] i 0 < 0 . 0083. There is, however, a more stringent constraint on the glyph[epsilon1] i 0 as we shall obtain in Sec. VI. For the spectral index not to be singular, a slow-roll parameter with an initial value glyph[epsilon1] i 0 < 0 . 0041 is required. Hence, m< 0 . 111 H i 0 in D =4 dimensions.</text>
<text><location><page_12><loc_14><loc_70><loc_51><loc_71></location>Note that Eqs. (51), (53) and (55) also yield</text>
<formula><location><page_12><loc_29><loc_65><loc_88><loc_68></location>q 2 =1 -2 glyph[epsilon1] i 0 ln( a )= q 2 0 [ 1 ± λ 4! ξ 2 glyph[epsilon1] -2 i 0 q -2 0 [1 -q 0 ] 4 + O ( λ 2 ) ] . (60)</formula>
<text><location><page_12><loc_12><loc_62><loc_52><loc_63></location>Hence, using Eq. (60), Eq. (44) can be recast as</text>
<formula><location><page_12><loc_28><loc_56><loc_88><loc_60></location>H ( t )= H i 0 [ q 0 ± λ 4! ξglyph[epsilon1] -2 i 0 [ q 3 0 + q 2 0 -1+ q -1 0 [1 -q 0 ] 4 4 ] + O ( λ 2 ) ] , (61)</formula>
<text><location><page_12><loc_12><loc_52><loc_56><loc_54></location>which implies that Hubble parameter (25) at O ( λ ) is</text>
<formula><location><page_12><loc_33><loc_47><loc_88><loc_51></location>H λ ( t )= H i 0 ξ 4! glyph[epsilon1] -2 i 0 [ q 3 0 + q 2 0 -1+ q -1 0 [1 -q 0 ] 4 4 ] . (62)</formula>
<text><location><page_12><loc_12><loc_41><loc_88><loc_45></location>In the next section, we similarly obtain the slow roll parameter, which characterizes the flatness of the potential that derives inflation.</text>
<section_header_level_1><location><page_12><loc_19><loc_36><loc_81><loc_37></location>C. The slow-roll parameter glyph[epsilon1] ( t ) in terms of initial values H i 0 and glyph[epsilon1] i 0</section_header_level_1>
<text><location><page_12><loc_12><loc_29><loc_88><loc_33></location>Slow-roll parameter (5) can also be expressed in terms of the initial values glyph[epsilon1] i 0 and H i 0 . Note that change of variable (42) and Eq. (48) imply</text>
<formula><location><page_12><loc_42><loc_23><loc_88><loc_27></location>d dt = -glyph[epsilon1] i 0 q -1 H d dq . (63)</formula>
<text><location><page_12><loc_12><loc_20><loc_48><loc_22></location>Hence, using Eq. (63) in Eq. (44) we obtain</text>
<formula><location><page_12><loc_16><loc_15><loc_88><loc_19></location>glyph[epsilon1] = -˙ H H 2 = glyph[epsilon1] i 0 qH dH dq = glyph[epsilon1] i 0 [ q -2 ± λ 4! ξglyph[epsilon1] -2 i 0 q -3 [ 2 q 3 + q 2 +1 ] ] + O ( λ 2 ) ≡ glyph[epsilon1] 0 ± λglyph[epsilon1] λ + O ( λ 2 ) . (64)</formula>
<text><location><page_12><loc_12><loc_12><loc_20><loc_13></location>Therefore,</text>
<formula><location><page_12><loc_31><loc_8><loc_88><loc_12></location>glyph[epsilon1] 0 = -˙ H 0 H 2 0 = glyph[epsilon1] i 0 q -2 0 = m 2 q -2 0 ( D -1) H 2 i 0 = m 2 ( D -1) H 2 0 , (65)</formula>
<text><location><page_13><loc_12><loc_89><loc_44><loc_91></location>where we used Eqs. (33) and (45), and</text>
<formula><location><page_13><loc_27><loc_84><loc_88><loc_88></location>glyph[epsilon1] λ = -˙ H λ H 2 0 -2 glyph[epsilon1] 0 H λ H 0 = ξ 4! glyph[epsilon1] -1 i 0 q -3 0 [ 2 q 3 0 + q 2 0 +1 -q -1 0 [1 -q 0 ] 4 2 ] , (66)</formula>
<text><location><page_13><loc_12><loc_78><loc_88><loc_82></location>where we used Eqs. (60) and (64). This completes the discussion of geometry in the interacting theory. Next, we obtain the solution of the background field ¯ ϕ ( t ) up to O ( λ 2 ).</text>
<section_header_level_1><location><page_13><loc_16><loc_72><loc_83><loc_74></location>D. The background inflaton field ¯ ϕ ( t ) in terms of initial values H i 0 and glyph[epsilon1] i 0</section_header_level_1>
<text><location><page_13><loc_12><loc_65><loc_88><loc_69></location>The background inflaton field ¯ ϕ ( t ) is obtained in Eq. (22). Employing Eqs. (34) and (35) in Eq. (22) yields</text>
<formula><location><page_13><loc_25><loc_60><loc_88><loc_64></location>¯ ϕ ( t ) glyph[similarequal] ¯ ϕ i 0 [ 1 -glyph[epsilon1] i 0 H i 0 t ∓ λ 8 ξ [ glyph[epsilon1] -1 i 0 ( H i 0 t ) -( H i 0 t ) 2 ] + O ( λ 2 ) ] . (67)</formula>
<text><location><page_13><loc_12><loc_57><loc_38><loc_59></location>Using Eq. (43) in Eq. (67) gives</text>
<formula><location><page_13><loc_16><loc_51><loc_88><loc_55></location>¯ ϕ ( t ) glyph[similarequal] ¯ ϕ i 0 [ q ∓ λ 4! ξglyph[epsilon1] -2 i 0 [ 1 -q 2 ] + O ( λ 2 ) ] = ¯ ϕ i 0 [ q 0 ∓ λ 4! ξglyph[epsilon1] -2 i 0 [ E 0 -q -1 0 [1 -q 0 ] 4 4 ] + O ( λ 2 ) ] . (68)</formula>
<text><location><page_13><loc_12><loc_48><loc_65><loc_49></location>Recall that the initial value ¯ ϕ i 0 is given terms of glyph[epsilon1] i 0 in Eq. (34).</text>
<text><location><page_13><loc_12><loc_43><loc_88><loc_47></location>Results we obtained throughout Secs. II A-D are needed to study the fluctuations of the self-interacting inflaton to which the remainder of this paper is devoted.</text>
<section_header_level_1><location><page_13><loc_33><loc_37><loc_66><loc_38></location>III. INFLATON FLUCTUATIONS</section_header_level_1>
<text><location><page_13><loc_12><loc_11><loc_88><loc_34></location>Quantum nature of the universe imply production of virtual pair of particles out of vacuum and causes fluctuations in the field. These quantum fluctuations are enhanced during inflation. As a particular fluctuation mode exits the horizon, it becomes acausal and 'freezes-in.' After the end of inflation as the horizon extends and includes these modes they become causal and provide the origin of large scale structure in the universe. To compute the effects of a quartic self-interaction on the correlations (Secs. IV-V) and power spectrum (Sec. VI) of inflaton fluctuations, we first obtain the mode expansion for the fluctuation field of the inflaton at tree-order in Sec. III A and then use it to get the O ( λ ) correction to the fluctuation field at one-loop order in Sec. III B.</text>
<section_header_level_1><location><page_14><loc_28><loc_89><loc_71><loc_91></location>A. Fluctuations of the tree-order inflaton field</section_header_level_1>
<text><location><page_14><loc_12><loc_77><loc_88><loc_86></location>Fluctuation field δϕ ( x ) of the full inflaton satisfies Eq. (10) whose solution is given in Eq. (24) in terms of the fluctuation field δϕ 0 ( x ) of the tree-order inflaton and the retarded Green's function G ret ( x ; x ' ). The mode expansion for the fluctuation field of the tree-order inflaton can be given as</text>
<formula><location><page_14><loc_27><loc_72><loc_88><loc_76></location>δϕ 0 ( x )= ∫ d D -1 k (2 π ) D -1 { δ Φ 0 ( x, glyph[vector] k ) A ( glyph[vector] k )+ δ Φ ∗ 0 ( x, glyph[vector] k ) A ∗ ( glyph[vector] k ) } , (69)</formula>
<text><location><page_14><loc_12><loc_69><loc_52><loc_71></location>where the mode function of the fluctuation field</text>
<formula><location><page_14><loc_40><loc_65><loc_88><loc_67></location>δ Φ 0 ( x, glyph[vector] k ) ≡ u 0 ( t, k ) e i glyph[vector] k · glyph[vector]x , (70)</formula>
<text><location><page_14><loc_12><loc_56><loc_88><loc_63></location>is the spatial plane wave solution of the linearized effective field equation, evaluated at the full solution of the background effective field equation. The tree-order amplitude function u 0 ( t, k ) of the plane waves that are superposed in Eq. (69), therefore, obeys</text>
<formula><location><page_14><loc_28><loc_51><loc_88><loc_55></location>u 0 ( t, k )+( D -1) H ( t ) ˙ u 0 ( t, k )+ [ k 2 a 2 + m 2 ] u 0 ( t, k )=0 . (71)</formula>
<text><location><page_14><loc_12><loc_40><loc_88><loc_50></location>The mode coordinates A ( glyph[vector] k ) and A ∗ ( glyph[vector] k ) in Eq. (69) are functions of the spatial Fourier transforms of the δϕ 0 ( x ) and its first time derivative evaluated on the initial value surface. Canonical quantization promotes the mode coordinates to the annihilation and creation operators ˆ A ( glyph[vector] k ) and ˆ A † ( glyph[vector] k ) satisfying the usual commutation relations</text>
<formula><location><page_14><loc_16><loc_36><loc_88><loc_38></location>[ ˆ A ( glyph[vector] k ) , ˆ A ( glyph[vector] k ' ) ] =0 , [ ˆ A † ( glyph[vector] k ) , ˆ A † ( glyph[vector] k ' ) ] =0 , [ ˆ A ( glyph[vector] k ) , ˆ A † ( glyph[vector] k ' ) ] =(2 π ) D -1 δ D -1 ( glyph[vector] k -glyph[vector] k ' ) , (72)</formula>
<text><location><page_14><loc_12><loc_32><loc_78><loc_34></location>and imposes the Wronskian normalization condition on the amplitude function,</text>
<formula><location><page_14><loc_33><loc_27><loc_88><loc_31></location>u 0 ( t, k ) ˙ u ∗ 0 ( t, k ) -u ∗ 0 ( t, k ) ˙ u 0 ( t, k )= i a D -1 . (73)</formula>
<text><location><page_14><loc_12><loc_22><loc_88><loc_26></location>Rescaling the amplitude function as a D -1 2 ( t ) u 0 ( t, k ) ≡ v 0 ( t, k ) reduces differential equation (71) to the form of the harmonic oscillator equation with a time dependent frequency</text>
<formula><location><page_14><loc_24><loc_17><loc_88><loc_20></location>v 0 ( t, k )+ [ k 2 a 2 ( t ) + m 2 -D -1 2 ˙ H ( t ) -( D -1 2 H ( t ) ) 2 ] v 0 ( t, k )=0 . (74)</formula>
<text><location><page_14><loc_12><loc_11><loc_88><loc_15></location>To solve Eq. (74), following the approach by Finelli, Marozzi, Vacca, and Venturi given in Ref. [34], we make the ansatz</text>
<formula><location><page_14><loc_43><loc_7><loc_88><loc_10></location>ζ = k , µ = µ ( H ) , ν = ν ( H ) , and χ = χ ( H ) . (75)</formula>
<formula><location><page_14><loc_16><loc_6><loc_49><loc_9></location>v 0 ( t, k ) ≡ ζ µ Z ν ( χζ ) with Ha</formula>
<text><location><page_15><loc_12><loc_89><loc_71><loc_91></location>and express the time derivative as derivatives with respect to ζ and H ,</text>
<formula><location><page_15><loc_36><loc_84><loc_88><loc_88></location>∂ ∂t = -H [ (1 -glyph[epsilon1] ) ζ ∂ ∂ζ + glyph[epsilon1]H ∂ ∂H ] . (76)</formula>
<text><location><page_15><loc_12><loc_81><loc_79><loc_83></location>Employing Eq. (76) in Eq. (74), neglecting the terms of O ( H ) and O ( glyph[epsilon1] 2 ), yields</text>
<formula><location><page_15><loc_35><loc_76><loc_88><loc_80></location>ζ 2 ∂ 2 Z ν ∂ζ 2 + ζ ∂Z ν ∂ζ + ( χ 2 ζ 2 -ν 2 ) Z ν =0 , (77)</formula>
<text><location><page_15><loc_12><loc_74><loc_25><loc_75></location>with parameters</text>
<text><location><page_15><loc_12><loc_68><loc_15><loc_69></location>and</text>
<formula><location><page_15><loc_31><loc_64><loc_88><loc_68></location>ν 2 = ( D -1 2 ) 2 -m 2 H 2 + ( D -2)( D -1) 2 glyph[epsilon1] + O ( glyph[epsilon1] 2 ) . (79)</formula>
<text><location><page_15><loc_12><loc_62><loc_39><loc_63></location>Using Eqs. (33) and (44) implies</text>
<formula><location><page_15><loc_40><loc_57><loc_88><loc_60></location>m 2 H 2 =( D -1) glyph[epsilon1] + O ( λglyph[epsilon1] λ ) , (80)</formula>
<text><location><page_15><loc_12><loc_54><loc_20><loc_55></location>and hence</text>
<formula><location><page_15><loc_35><loc_50><loc_88><loc_54></location>ν 2 = ( D -1 2 ) 2 [ 1+2 D -4 D -1 glyph[epsilon1] + O ( glyph[epsilon1] 2 ) ] . (81)</formula>
<text><location><page_15><loc_12><loc_45><loc_88><loc_49></location>The solution of Bessel's equation (77) is given in terms of Hankel's function of the first kind and its complex conjugate</text>
<formula><location><page_15><loc_35><loc_41><loc_88><loc_43></location>Z ν ( χζ )= C 1 H (1) ν ( χζ )+ C 2 H (2) ν ( χζ ) . (82)</formula>
<text><location><page_15><loc_12><loc_38><loc_50><loc_39></location>Thus, the solution of Eq. (71) can be given as</text>
<formula><location><page_15><loc_15><loc_33><loc_88><loc_36></location>u 0 ( t, k )= a -D -1 2 ζ µ Z ν ( χζ )= a -D -1 2 ( k Ha ) -glyph[epsilon1] 2 [ C 1 H (1) ν ( (1+ glyph[epsilon1] ) k Ha ) + C 2 H (2) ν ( (1+ glyph[epsilon1] ) k Ha )] . (83)</formula>
<text><location><page_15><loc_12><loc_30><loc_67><loc_31></location>Wronskian normalization (73) constrains the constants C 1 and C 2 ,</text>
<formula><location><page_15><loc_32><loc_25><loc_88><loc_28></location>| C 1 | 2 -| C 2 | 2 = π 4 H (1+ glyph[epsilon1] ) ( k Ha ) glyph[epsilon1] [ 1+ O ( glyph[epsilon1] 2 ) ] . (84)</formula>
<text><location><page_15><loc_12><loc_20><loc_88><loc_24></location>A solution which corresponds to Bunch-Davies mode function in de Sitter spacetime can be chosen, up to O ( glyph[epsilon1] 2 ), as</text>
<formula><location><page_15><loc_31><loc_14><loc_88><loc_18></location>C 1 = i [ π 4 H (1+ glyph[epsilon1] ) ( k Ha ) glyph[epsilon1] ] 1 2 and C 2 =0 . (85)</formula>
<text><location><page_15><loc_12><loc_11><loc_86><loc_13></location>Inserting Eq. (85) into Eq. (83) yields the amplitude of mode function (70) up to O ( glyph[epsilon1] 2 ),</text>
<formula><location><page_15><loc_32><loc_6><loc_88><loc_10></location>u 0 ( t, k )= ia -D -1 2 [ π 4 H (1+ glyph[epsilon1] ) ] 1 2 H (1) ν ( (1+ glyph[epsilon1] ) k Ha ) . (86)</formula>
<formula><location><page_15><loc_36><loc_70><loc_88><loc_73></location>µ = -glyph[epsilon1] 2 + O ( glyph[epsilon1] 2 ) , χ =1+ glyph[epsilon1] + O ( glyph[epsilon1] 2 ) , (78)</formula>
<text><location><page_16><loc_12><loc_89><loc_67><loc_91></location>Recall that, at comoving time t = t k , the mode with wave number</text>
<formula><location><page_16><loc_43><loc_84><loc_88><loc_88></location>k ≡ H ( t k ) a ( t k ) 1+ glyph[epsilon1] ( t k ) , (87)</formula>
<text><location><page_16><loc_12><loc_78><loc_88><loc_82></location>exits the horizon, becomes a superhorizon (IR) mode and 'freezes in.' At a given time t , we consider superhorizon modes</text>
<formula><location><page_16><loc_46><loc_75><loc_88><loc_78></location>k< Ha 1+ glyph[epsilon1] , (88)</formula>
<text><location><page_16><loc_12><loc_67><loc_88><loc_74></location>for which the argument of the Hankel function (1+ glyph[epsilon1] ) k Ha < 1. Thus, using the power series expansion of the Hankel function we obtain the IR limit of the amplitude function in leading order</text>
<formula><location><page_16><loc_22><loc_63><loc_88><loc_67></location>u 0 ( t, k ) -→ 2 ν -1 Γ( ν ) √ πa δ 2 ( H 1+ glyph[epsilon1] ) ν -1 2 k -ν [ 1+ O ( ( (1+ glyph[epsilon1] ) k 2 Ha ) 2 )] [ 1+ O ( glyph[epsilon1] 2 ) ] , (89)</formula>
<text><location><page_16><loc_12><loc_58><loc_88><loc_62></location>where we define the parameter δ ≡ D -1 -2 ν . Employing Eq. (89) in Eq. (69) we obtain the IR truncated fluctuation field of the tree-order inflaton as</text>
<formula><location><page_16><loc_16><loc_53><loc_88><loc_57></location>δ ¯ ϕ 0 ( t, glyph[vector]x )= 2 ν -1 Γ( ν ) √ π a δ 2 ( H 1+ glyph[epsilon1] ) ν -1 2 ∫ d D -1 k (2 π ) D -1 Θ ( Ha 1+ glyph[epsilon1] -k ) k ν [ e i glyph[vector] k · glyph[vector]x ˆ A ( glyph[vector] k )+ e -i glyph[vector] k · glyph[vector]x ˆ A † ( glyph[vector] k ) ] . (90)</formula>
<text><location><page_16><loc_12><loc_46><loc_88><loc_51></location>The ratio H ( t ) a ( t ) 1+ glyph[epsilon1] ( t ) in the argument of the Θ-function is monotonically increasing function of time until</text>
<formula><location><page_16><loc_41><loc_44><loc_88><loc_46></location>t f ∼ = t m [1 -3 1 / 4 √ glyph[epsilon1] i 0 ] , (91)</formula>
<text><location><page_16><loc_12><loc_35><loc_88><loc_42></location>which is slightly less than t e defined in Eq. (59). Hence, we consider inflation during 0 <t<t f in the model we consider. Mode expansion (90) is used to obtain the full fluctuation field in the next section.</text>
<section_header_level_1><location><page_16><loc_32><loc_29><loc_68><loc_31></location>B. Fluctuations of the full inflaton field</section_header_level_1>
<text><location><page_16><loc_12><loc_12><loc_88><loc_26></location>Fluctuations δϕ ( x ) of a self-interacting inflaton field is given in Eq. (24) in terms of the fluctuations δϕ 0 ( x ) of the tree-order field and the retarded Green's function G ( x ; x ' ). The δϕ 0 ( x ) is given via Eqs. (69), (70) and (86). [The IR limit δ ¯ ϕ 0 ( x ) of the free field is obtained in Eqs. (89) and (90).] To get the full fluctuations δϕ ( x ) and its IR limit, the last ingredient we need is the Green's function which can be given in terms of the commutator function of the δϕ 0 ( x ) as</text>
<formula><location><page_16><loc_34><loc_9><loc_88><loc_11></location>G ( x ; x ' )= i Θ( t -t ' ) [ δϕ 0 ( x ) , δϕ 0 ( x ' )] . (92)</formula>
<text><location><page_17><loc_12><loc_89><loc_33><loc_91></location>The commutator function</text>
<formula><location><page_17><loc_19><loc_85><loc_88><loc_89></location>[ δϕ 0 ( x ) , δϕ 0 ( x ' )] = ∫ d D -1 k (2 π ) D -1 { u ( t, k ) u ∗ ( t ' , k ) -u ∗ ( t, k ) u ( t ' , k ) } e i glyph[vector] k · ( glyph[vector]x -glyph[vector]x ' ) , (93)</formula>
<text><location><page_17><loc_12><loc_83><loc_46><loc_84></location>where the terms inside the curly brackets</text>
<formula><location><page_17><loc_13><loc_74><loc_88><loc_82></location>u ( t, k ) u ∗ ( t ' , k ) -u ∗ ( t, k ) u ( t ' , k )= i π 2 [ aa ' ] -D -1 2 [ (1+ glyph[epsilon1] )(1+ glyph[epsilon1] ' ) HH ' ] 1 2 × { [ cot( νπ ) -cot( ν ' π ) ] J ν ( z ) J ν ' ( z ' )+csc( ν ' π ) J ν ( z ) J -ν ' ( z ' ) -csc( νπ ) J -ν ( z ) J -ν ' ( z ' ) } . (94)</formula>
<text><location><page_17><loc_12><loc_65><loc_88><loc_72></location>Here, we define z ≡ χζ = (1+ glyph[epsilon1] ) k Ha and z ' ≡ χ ' ζ ' = (1+ glyph[epsilon1] ' ) k H ' a ' . During slow roll inflation ν = D -1 2 + O ( glyph[epsilon1] )= ν ' in D -dimensions. [In D =4, on the other hand, we have ν = 3 2 + O ( glyph[epsilon1] 2 ) = ν ' , see Eq. (81).] Therefore, the IR limit of Green's function (92), in leading order, is</text>
<formula><location><page_17><loc_16><loc_60><loc_88><loc_64></location>G ( x ; x ' ) → Θ( t -t ' ) D -1 [ H 1+ glyph[epsilon1] ] D 2 -1 [ H ' 1+ glyph[epsilon1] ' ] D 2 -1 { [ 1+ glyph[epsilon1] ' H ' a ' ] D -1 -[ 1+ glyph[epsilon1] Ha ] D -1 } δ D -1 ( glyph[vector]x -glyph[vector]x ' ) . (95)</formula>
<text><location><page_17><loc_12><loc_46><loc_88><loc_58></location>The fluctuation field of the full inflaton, in the IR limit, can be obtained mingling Eqs. (11), (24) and (95): When the source term in Eq. (10) weighted by the Green's function is integrated throughout the spacetime from the initial time t i = 0 to a late time t yields the δϕ . In leading order, therefore, the latter term in the curly brackets in Eq. (95) can be neglected next to the former which dominates throughout the range of integration. Hence,</text>
<formula><location><page_17><loc_22><loc_41><loc_88><loc_45></location>δϕ ( x ) glyph[similarequal] δϕ 0 ( x ) -1 D -1 [ H 1+ glyph[epsilon1] ] D 2 -1 ∫ t 0 dt ' Θ( t -t ' ) [ 1+ glyph[epsilon1] ' H ' ] D 2 ∂V pert ( δϕ ) ∂δϕ ( t ' , glyph[vector]x ) 1+ δZ . (96)</formula>
<text><location><page_17><loc_12><loc_17><loc_88><loc_40></location>In the above integrand, the counterterms in 1 1+ δZ ∂V pert ( δϕ ) ∂δϕ can be neglected. This can be seen by comparing the orders of λ of the counterterms and the powers of the fields ¯ ϕ ( t ) and δϕ ( x ) involved in various terms in Eq. (11). The terms proportional to the δλ and δZ can be neglected because they are of O ( λ 2 ). [ δm 2 ∼O ( λ ).] Because ¯ ϕ glyph[greatermuch]| δϕ | , the term quadratic in background field is the dominant term which yields the leading contribution. Recall that 4 glyph[lessorsimilar] ¯ ϕ< 10 6 in Planckian units. Thus, the fact that | δϕ | glyph[lessmuch] ¯ ϕ does not necessarily imply that | δϕ | is small by itself. Hence, we keep the term quadratic in δϕ -that is also linear in the background field-and the term cubic in δϕ , but neglect the term linear in δϕ next to the dominant term. Therefore, the fluctuation field of the full inflaton in our model is</text>
<formula><location><page_17><loc_19><loc_8><loc_88><loc_16></location>δϕ ( x ) glyph[similarequal] δϕ 0 ( x ) ∓ λ 6( D -1) [ H 0 1+ glyph[epsilon1] 0 ] D 2 -1 ∫ t 0 dt ' Θ( t -t ' ) [ 1+ glyph[epsilon1] ' 0 H ' 0 ] D 2 { 3¯ ϕ 2 0 ( t ' ) δϕ 0 ( t ' , glyph[vector]x ) +3¯ ϕ 0 ( t ' ) [ δϕ 0 ( t ' , glyph[vector]x ) ] 2 + [ δϕ 0 ( t ' , glyph[vector]x ) ] 3 } + O ( λ 2 ) . (97)</formula>
<text><location><page_18><loc_12><loc_87><loc_88><loc_91></location>We employ Eq. (97) to compute quantum corrected two-point correlation function of the inflaton fluctuations at tree- and one-loop order in the next section.</text>
<section_header_level_1><location><page_18><loc_14><loc_81><loc_85><loc_82></location>IV. TWO-POINT CORRELATION FUNCTION OF THE FLUCTUATIONS</section_header_level_1>
<text><location><page_18><loc_12><loc_74><loc_88><loc_78></location>The two-point correlation function of the IR truncated fluctuations of the full field for two distinct events</text>
<formula><location><page_18><loc_17><loc_70><loc_88><loc_72></location>〈 Ω | δ ¯ ϕ ( x ) δ ¯ ϕ ( x ' ) | Ω 〉 = 〈 Ω | δ ¯ ϕ ( x ) δ ¯ ϕ ( x ' ) | Ω 〉 tree + 〈 Ω | δ ¯ ϕ ( x ) δ ¯ ϕ ( x ' ) | Ω 〉 1 -loop + O ( λ 2 ) , (98)</formula>
<text><location><page_18><loc_26><loc_66><loc_26><loc_68></location>glyph[negationslash]</text>
<text><location><page_18><loc_12><loc_64><loc_88><loc_68></location>with t ' ≤ t and glyph[vector]x ' = glyph[vector]x , can be obtained for the inflaton with a quartic self-interaction using Eq. (97). It yields, at tree-order</text>
<formula><location><page_18><loc_31><loc_60><loc_88><loc_62></location>〈 Ω | δ ¯ ϕ ( x ) δ ¯ ϕ ( x ' ) | Ω 〉 tree = 〈 Ω | δ ¯ ϕ 0 ( x ) δ ¯ ϕ 0 ( x ' ) | Ω 〉 , (99)</formula>
<text><location><page_18><loc_12><loc_56><loc_30><loc_57></location>and at one-loop order</text>
<formula><location><page_18><loc_14><loc_42><loc_88><loc_54></location>〈 Ω | δ ¯ ϕ ( x ) δ ¯ ϕ ( x ' ) | Ω 〉 1 -loop = ∓ λ 6( D -1) { [ H ' 0 1+ glyph[epsilon1] ' 0 ] D 2 -1 ∫ t ' 0 d ˜ t [ 1+˜ glyph[epsilon1] 0 ˜ H 0 ] D 2 〈 Ω | δ ¯ ϕ 0 ( x ) [ 3¯ ϕ 2 0 ( ˜ t ) δ ¯ ϕ 0 ( ˜ t, glyph[vector]x ' )+[ δ ¯ ϕ 0 ( ˜ t, glyph[vector]x ' )] 3 ] | Ω 〉 + [ H 0 1+ glyph[epsilon1] 0 ] D 2 -1 ∫ t 0 dt '' [ 1+ glyph[epsilon1] '' 0 H '' 0 ] D 2 〈 Ω | [ 3¯ ϕ 2 0 ( t '' ) δ ¯ ϕ 0 ( t '' , glyph[vector]x )+[ δ ¯ ϕ 0 ( t '' , glyph[vector]x )] 3 ] δ ¯ ϕ 0 ( x ' ) | Ω 〉 } , (100)</formula>
<text><location><page_18><loc_12><loc_39><loc_43><loc_40></location>which are computed in Secs. IV A-C.</text>
<section_header_level_1><location><page_18><loc_38><loc_34><loc_61><loc_35></location>A. Tree-order correlator</section_header_level_1>
<text><location><page_18><loc_12><loc_26><loc_88><loc_30></location>The tree-order two-point correlation function of the IR truncated inflaton fluctuations is obtained using Eqs. (72) and (90) as</text>
<formula><location><page_18><loc_12><loc_18><loc_88><loc_24></location>〈 Ω | δ ¯ ϕ ( x ) δ ¯ ϕ ( x ' ) | Ω 〉 tree = 〈 Ω | δ ¯ ϕ 0 ( x ) δ ¯ ϕ 0 ( x ' ) | Ω 〉 = 1 4 π ( H 1+ glyph[epsilon1] ) ν -1 2 ( H ' 1+ glyph[epsilon1] ' ) ν ' -1 2 2 ν Γ( ν ) a δ 2 2 ν ' Γ( ν ' ) a ' δ ' 2 ∫ d D -1 k (2 π ) D -1 Θ ( Ha 1+ glyph[epsilon1] -k ) Θ ( H ' a ' 1+ glyph[epsilon1] ' -k ) e i glyph[vector] k · ( glyph[vector]x -glyph[vector]x ' ) k ν + ν ' . (101)</formula>
<text><location><page_18><loc_12><loc_12><loc_88><loc_17></location>For 0 ≤ t ' ≤ t , we have H ' a ' 1+ glyph[epsilon1] ' ≤ Ha 1+ glyph[epsilon1] . Thus, defining ∆ x ≡‖ ∆ glyph[vector]x ‖ = ‖ glyph[vector]x -glyph[vector]x ' ‖ and θ as the angle between the vectors glyph[vector] k and ∆ glyph[vector]x , the integral on the right side of Eq. (101) yields</text>
<formula><location><page_18><loc_15><loc_7><loc_88><loc_11></location>∫ d Ω D -1 (2 π ) D -1 ∫ ∞ 0 dk Θ ( H ' a ' 1+ glyph[epsilon1] ' -k ) e ik ∆ x cos ( θ ) k 1 -δ + δ ' 2 = Γ ( D 2 ) π D 2 Γ( D -1) (∆ x 2 ) -δ + δ ' 4 ∫ α ' α i dy sin( y ) y 2 -δ + δ ' 2 , (102)</formula>
<text><location><page_19><loc_12><loc_89><loc_87><loc_91></location>where we performed angular integrations, made a change of variable y ≡ k ∆ x and defined</text>
<formula><location><page_19><loc_45><loc_85><loc_88><loc_88></location>α ≡ Ha ∆ x 1+ glyph[epsilon1] , (103)</formula>
<text><location><page_19><loc_12><loc_80><loc_88><loc_84></location>which implies α ' = H ' a ' ∆ x 1+ glyph[epsilon1] ' and α i = H i ∆ x 1+ glyph[epsilon1] i . The remaining integral can be evaluated in terms of the exponential integral function, defined in Eq. (B1), as</text>
<formula><location><page_19><loc_26><loc_74><loc_88><loc_78></location>∫ dy sin( y ) y 2 -δ + δ ' 2 = -i 2 y δ + δ ' 2 -1 [ E 2 -δ + δ ' 2 ( iy ) -E 2 -δ + δ ' 2 ( -iy ) ] . (104)</formula>
<text><location><page_19><loc_12><loc_69><loc_88><loc_73></location>Note that the result is real for a purely real parameter y . Combining Eqs. (102)-(104) in Eq. (101) we find the tree-order correlator as</text>
<formula><location><page_19><loc_13><loc_60><loc_88><loc_68></location>〈 Ω | δ ¯ ϕ 0 ( x ) δ ¯ ϕ 0 ( x ' ) | Ω 〉 = -i Γ ( D 2 ) 8 π D 2 +1 Γ( D -1) ( H 1+ glyph[epsilon1] ) D 2 -1 ( H ' 1+ glyph[epsilon1] ' ) D 2 -1 2 ν Γ( ν ) α δ 2 2 ν ' Γ( ν ' ) α ' δ ' 2 × { α ' δ + δ ' 2 -1 [ E 2 -δ + δ ' 2 ( iα ' ) -E 2 -δ + δ ' 2 ( -iα ' ) ] -α δ + δ ' 2 -1 i [ E 2 -δ + δ ' 2 ( iα i ) -E 2 -δ + δ ' 2 ( -iα i ) ] } . (105)</formula>
<text><location><page_19><loc_12><loc_51><loc_88><loc_58></location>One can also express the tree-order correlator in terms of the incomplete gamma function Γ( β, z ), defined in Eq. (B2). The exponential integral function is related to the incomplete gamma function,</text>
<formula><location><page_19><loc_40><loc_49><loc_88><loc_50></location>E β ( z )= z β -1 Γ(1 -β, z ) . (106)</formula>
<text><location><page_19><loc_12><loc_45><loc_53><loc_47></location>Therefore, tree-order correlator (105) is recast as</text>
<formula><location><page_19><loc_12><loc_36><loc_88><loc_44></location>〈 Ω | δ ¯ ϕ 0 ( x ) δ ¯ ϕ 0 ( x ' ) | Ω 〉 = Γ ( D 2 ) 8 π D 2 +1 Γ( D -1) ( H 1+ glyph[epsilon1] ) D 2 -1 ( H ' 1+ glyph[epsilon1] ' ) D 2 -1 2 ν Γ( ν ) α δ 2 2 ν ' Γ( ν ' ) α ' δ ' 2 ( -1) -δ + δ ' 4 × { Γ ( δ + δ ' 2 -1 , iα ' ) -Γ ( δ + δ ' 2 -1 , iα i ) +( -1) -δ + δ ' 2 [ Γ ( δ + δ ' 2 -1 , -iα ' ) -Γ ( δ + δ ' 2 -1 , -iα i )] } . (107)</formula>
<text><location><page_19><loc_12><loc_30><loc_88><loc_34></location>This form is useful in obtaining a series representation of the correlator using expansion (B4) of the incomplete gamma function.</text>
<text><location><page_19><loc_12><loc_22><loc_88><loc_29></location>A simpler form of tree-order correlator (105) or (107) is obtained recalling the fact that Eq. (81) implies ν =( D -1) / 2+ O ( glyph[epsilon1] ) in D -dimensions and ν =3 / 2+ O ( glyph[epsilon1] 2 ) for D =4. The tree-order correlator for ν =( D -1) / 2 is given in Eq. (C1). Its D → 4 limit, up to O ( glyph[epsilon1] 2 ), is</text>
<formula><location><page_19><loc_18><loc_17><loc_88><loc_21></location>〈 Ω | δ ¯ ϕ 0 ( x ) δ ¯ ϕ 0 ( x ' ) | Ω 〉 = HH ' 4 π 2 (1+ glyph[epsilon1] )(1+ glyph[epsilon1] ' ) [ ci( α ' ) -sin( α ' ) α ' -ci( α i )+ sin( α i ) α i ] , (108)</formula>
<text><location><page_19><loc_12><loc_12><loc_88><loc_16></location>where the ci( z ) is the cosine integral function defined in Eq. (B5). Using identity (B6) in Eq. (108) a power series expansion of the tree-order correlator is obtained</text>
<formula><location><page_19><loc_14><loc_7><loc_88><loc_11></location>〈 Ω | δ ¯ ϕ 0 ( x ) δ ¯ ϕ 0 ( x ' ) | Ω 〉 = HH ' 4 π 2 (1+ glyph[epsilon1] )(1+ glyph[epsilon1] ' ) [ ln ( H ' a ' H i ) -ln ( 1+ glyph[epsilon1] ' 1+ glyph[epsilon1] i ) + ∞ ∑ n =1 ( -1) n [ α ' 2 n -α 2 n i ] 2 n (2 n +1)! ] . (109)</formula>
<text><location><page_20><loc_12><loc_81><loc_88><loc_91></location>The tree-order correlator is evaluated at the full solution of the background effective field equation, as in the case of tree-order mode function (86). The Hubble parameter, scale factor and slow-roll parameter in Eqs. (108) and (109) have O ( λ ) corrections which can be given in terms of q 0 [Eq. (46)] and the initial values H i 0 and glyph[epsilon1] i 0 . The ratio</text>
<formula><location><page_20><loc_33><loc_76><loc_88><loc_80></location>H 1+ glyph[epsilon1] = H 0 1+ glyph[epsilon1] 0 [ 1 ± λ [ H λ H 0 -glyph[epsilon1] λ 1+ glyph[epsilon1] 0 ] + O ( λ 2 ) ] , (110)</formula>
<text><location><page_20><loc_12><loc_70><loc_88><loc_74></location>where H 0 , H λ , glyph[epsilon1] 0 and glyph[epsilon1] λ are given in Eqs. (45), (62), (65) and (66), respectively. They yield the various terms in Eq. (110) as</text>
<formula><location><page_20><loc_27><loc_59><loc_88><loc_69></location>H 0 1+ glyph[epsilon1] 0 = H i 0 q 3 0 q 2 0 + glyph[epsilon1] i 0 , H λ H 0 = ξ 4! glyph[epsilon1] -2 i 0 [ q 2 0 + q 0 -q -1 0 + q -2 0 [1 -q 0 ] 4 4 ] , glyph[epsilon1] λ 1+ glyph[epsilon1] 0 = ξ 4! glyph[epsilon1] -1 i 0 q 2 0 + glyph[epsilon1] i 0 [ 2 q 2 0 + q 0 + q -1 0 -q -2 0 [1 -q 0 ] 4 2 ] . (111)</formula>
<text><location><page_20><loc_12><loc_53><loc_88><loc_58></location>Their initial values are obtained by recalling that q i 0 =1. Note also that H ' 1+ glyph[epsilon1] ' in Eqs. (108)(109) can be inferred from Eqs. (110)-(111).</text>
<text><location><page_20><loc_12><loc_48><loc_88><loc_52></location>The logarithm of the scale factor is given in Eq. (51). Therefore, the first logarithmic term in Eq. (109),</text>
<formula><location><page_20><loc_30><loc_43><loc_88><loc_46></location>ln ( H ' a ' H i ) =ln ( H ' 0 a ' 0 H i 0 ) ± λ [ H ' λ H ' 0 -H iλ H i 0 +ln( a ' ) λ ] + O ( λ 2 ) (112)</formula>
<formula><location><page_20><loc_20><loc_38><loc_88><loc_42></location>= ln( q ' 0 )+ glyph[epsilon1] -1 i 0 E ' 0 2 ± λ 4! ξglyph[epsilon1] -2 i 0 [ q ' 2 0 + q ' 0 -1 -q '-1 0 -[ glyph[epsilon1] -1 i 0 -q '-2 0 ] [1 -q ' 0 ] 4 4 ] + O ( λ 2 ) , (113)</formula>
<text><location><page_20><loc_12><loc_35><loc_50><loc_36></location>and the second logarithmic term in Eq. (109),</text>
<formula><location><page_20><loc_31><loc_29><loc_88><loc_33></location>ln ( 1+ glyph[epsilon1] ' 1+ glyph[epsilon1] i ) =ln ( 1+ glyph[epsilon1] ' 0 1+ glyph[epsilon1] i 0 ) ± λ [ glyph[epsilon1] ' λ 1+ glyph[epsilon1] ' 0 -glyph[epsilon1] iλ 1+ glyph[epsilon1] i 0 ] + O ( λ 2 ) (114)</formula>
<formula><location><page_20><loc_15><loc_25><loc_88><loc_29></location>=ln ( q ' 2 0 + glyph[epsilon1] i 0 q ' 2 0 (1+ glyph[epsilon1] i 0 ) ) ± λ 4! ξglyph[epsilon1] -1 i 0 [ 1 q ' 2 0 + glyph[epsilon1] i 0 [ 2 q ' 2 0 + q ' 0 + q '-1 0 -q '-2 0 [1 -q ' 0 ] 4 2 ] -4 1+ glyph[epsilon1] i 0 ] + O ( λ 2 ) . (115)</formula>
<text><location><page_20><loc_12><loc_21><loc_74><loc_23></location>The α ' [Eq. (103)] in Eqs. (108)-(109), on the other hand, can be given as</text>
<formula><location><page_20><loc_41><loc_17><loc_88><loc_19></location>α ' = α ' 0 ± λα ' λ + O ( λ 2 ) , (116)</formula>
<text><location><page_20><loc_12><loc_13><loc_17><loc_14></location>where</text>
<formula><location><page_20><loc_32><loc_7><loc_88><loc_11></location>α ' 0 = a ' 0 H ' 0 ∆ x 1+ glyph[epsilon1] ' 0 = q ' 3 0 H i 0 ∆ x q ' 2 0 + glyph[epsilon1] i 0 exp ( glyph[epsilon1] -1 i 0 E ' 0 2 ) , (117)</formula>
<text><location><page_21><loc_12><loc_89><loc_15><loc_91></location>and</text>
<formula><location><page_21><loc_15><loc_79><loc_88><loc_88></location>α ' λ = α ' 0 [ H ' λ H ' 0 -glyph[epsilon1] ' λ 1+ glyph[epsilon1] ' 0 +ln( a ' ) λ ] = α ' 0 ξ 4! glyph[epsilon1] -2 i 0 [ q ' 2 0 + q ' 0 -q '-1 0 -[ glyph[epsilon1] -1 i 0 -q '-2 0 ] [1 -q ' 0 ] 4 4 -glyph[epsilon1] i 0 q ' 2 0 + glyph[epsilon1] i 0 [ 2 q ' 2 0 + q ' 0 + q '-1 0 -q '-2 0 [1 -q ' 0 ] 4 2 ] ] . (118)</formula>
<text><location><page_21><loc_12><loc_70><loc_88><loc_78></location>The powers of α ' in Eq. (109), to wit, α ' 2 n = α ' 2 n 0 [ 1 ± 2 nλ α ' λ α ' 0 ] + O ( λ 2 ) and its initial value α ' 2 n i can be inferred from Eqs. (117) and (118). Hence, tree-order correlator (109) evaluated at the full solution of the background effective field equation can be given as</text>
<formula><location><page_21><loc_25><loc_63><loc_88><loc_68></location>〈 Ω | δ ¯ ϕ 0 ( x ) δ ¯ ϕ 0 ( x ' ) | Ω 〉 ≡〈 Ω | δ ¯ ϕ 0 ( x ) δ ¯ ϕ 0 ( x ' ) | Ω 〉 0 ± λ 〈 Ω | δ ¯ ϕ 0 ( x ) δ ¯ ϕ 0 ( x ' ) | Ω 〉 λ + O ( λ 2 ) . (119)</formula>
<text><location><page_21><loc_12><loc_59><loc_55><loc_61></location>The tree-order correlator in the noninteracting limit</text>
<formula><location><page_21><loc_13><loc_53><loc_88><loc_58></location>〈 Ω | δ ¯ ϕ 0 ( x ) δ ¯ ϕ 0 ( x ' ) | Ω 〉 0 = H 0 H ' 0 4 π 2 (1+ glyph[epsilon1] 0 )(1+ glyph[epsilon1] ' 0 ) [ ln ( H ' 0 a ' 0 H i 0 ) -ln ( 1+ glyph[epsilon1] ' 0 1+ glyph[epsilon1] i 0 ) + ∞ ∑ n =1 ( -1) n [ α ' 2 n 0 -α 2 n i 0 ] 2 n (2 n +1)! ] . (120)</formula>
<text><location><page_21><loc_12><loc_50><loc_53><loc_52></location>Employing Eqs. (45) and (65) in Eq. (120) yields</text>
<formula><location><page_21><loc_33><loc_45><loc_88><loc_49></location>〈 Ω | δ ¯ ϕ 0 ( x ) δ ¯ ϕ 0 ( x ' ) | Ω 〉 0 = H 2 i 0 4 π 2 f 00 ( t, t ' , ∆ x ) , (121)</formula>
<text><location><page_21><loc_12><loc_42><loc_17><loc_44></location>where</text>
<formula><location><page_21><loc_19><loc_37><loc_88><loc_41></location>f 00 ≡ ( q 0 q ' 0 ) 3 ( q 2 0 + glyph[epsilon1] i 0 )( q ' 2 0 + glyph[epsilon1] i 0 ) [ glyph[epsilon1] -1 i 0 E ' 0 2 -ln ( q ' 2 0 + glyph[epsilon1] i 0 q ' 3 0 (1+ glyph[epsilon1] i 0 ) ) + ∞ ∑ n =1 ( -1) n [ α ' 2 n 0 -α 2 n i 0 ] 2 n (2 n +1)! ] . (122)</formula>
<text><location><page_21><loc_12><loc_31><loc_88><loc_35></location>The α ' 0 , given in Eq. (117), implies α i 0 = H i 0 ∆ x 1+ glyph[epsilon1] i 0 . Choosing the physical distance a ( x ' )∆ x a constant fraction K of Hubble length so that the comoving separation</text>
<formula><location><page_21><loc_42><loc_26><loc_88><loc_29></location>∆ x = K H 0 ( t ' ) a 0 ( t ' ) , (123)</formula>
<text><location><page_21><loc_12><loc_7><loc_88><loc_24></location>in Eq. (122) yields f 00 ( t, t ' , K ). The plots of the function f 00 ( t, t ' , K ) versus a ( t ' ), for K =1 / 2 and four different initial values of the slow-roll parameter glyph[epsilon1] i 0 = m 2 3 H 2 i 0 , are given in Fig. 1. They show that correlation (121), for a given glyph[epsilon1] i 0 , grows and asymptotes to an almost constant value at late times during inflation. The growth is sensitive to the value of glyph[epsilon1] i 0 . Even a slight increase in the glyph[epsilon1] i 0 causes a noticeable suppression in the growth. This is because the production rate and lifetime of virtual inflatons-the origin of fluctuations-are suppressed as the mass increases.</text>
<figure>
<location><page_22><loc_23><loc_68><loc_76><loc_91></location>
<caption>FIG. 1: Plots of the function f 00 ( t, t ' , K ), defined in Eq. (122) with ∆ x = K/H 0 ( t ' ) a 0 ( t ' ), versus a 0 ( t ' ) for different values of glyph[epsilon1] i 0 = m 2 / 3 H 2 i 0 . The scale factor a 0 ( t ) and the fraction K are chosen to be e 50 and 1 / 2, respectively. a 0 ( t ' ) runs from 1 to a 0 ( t ). The solid, large-dashed, dashed and dot-dashed plots are for glyph[epsilon1] i 0 =0 . 0025 , 0 . 00275 , 0 . 003 and 0 . 0035, respectively.</caption>
</figure>
<text><location><page_22><loc_12><loc_48><loc_88><loc_52></location>As pointed out earlier, evaluation of the tree-order correlator at the full solution of the background effective field equation yields an O ( λ ) correction,</text>
<formula><location><page_22><loc_12><loc_38><loc_88><loc_47></location>〈 Ω | δ ¯ ϕ 0 ( x ) δ ¯ ϕ 0 ( x ' ) | Ω 〉 λ = [ H λ H 0 -glyph[epsilon1] λ 1+ glyph[epsilon1] 0 + H ' λ H ' 0 -glyph[epsilon1] ' λ 1+ glyph[epsilon1] ' 0 ] 〈 Ω | δ ¯ ϕ 0 ( x ) δ ¯ ϕ 0 ( x ' ) | Ω 〉 0 + H 0 H ' 0 4 π 2 (1+ glyph[epsilon1] 0 )(1+ glyph[epsilon1] ' 0 ) { [ H ' λ H ' 0 -glyph[epsilon1] ' λ 1+ glyph[epsilon1] ' 0 +ln( a ' ) λ ] ∞ ∑ n =0 ( -1) n α ' 2 n 0 (2 n +1)! -[ H iλ H i 0 -glyph[epsilon1] iλ 1+ glyph[epsilon1] i 0 ] ∞ ∑ n =0 ( -1) n α 2 n i 0 (2 n +1)! } . (124)</formula>
<text><location><page_22><loc_12><loc_35><loc_45><loc_36></location>Employing Eq. (111) in Eq. (124) yields</text>
<formula><location><page_22><loc_31><loc_30><loc_88><loc_33></location>〈 Ω | δ ¯ ϕ 0 ( x ) δ ¯ ϕ 0 ( x ' ) | Ω 〉 λ = H 2 i 0 4 π 2 ξ 4! f 0 λ ( t, t ' , ∆ x ) , (125)</formula>
<text><location><page_22><loc_12><loc_24><loc_88><loc_28></location>where ξ/ 4! ∼ 700 in a GUT scale inflation for which H i 0 ∼ 10 -3 M Pl and the dimensionless function</text>
<formula><location><page_22><loc_12><loc_8><loc_88><loc_23></location>f 0 λ = glyph[epsilon1] -2 i 0 {{[ q 2 0 + q 0 -q -1 0 + q -2 0 [1 -q 0 ] 4 4 -glyph[epsilon1] i 0 q 2 0 + glyph[epsilon1] i 0 ( 2 q 2 0 + q 0 + q -1 0 -q -2 0 [1 -q 0 ] 4 2 ) ] + [ q 0 → q ' 0 ]} f 00 + ( q 0 q ' 0 ) 3 ( q 2 0 + glyph[epsilon1] i 0 )( q ' 2 0 + glyph[epsilon1] i 0 ) {[ q ' 2 0 + q ' 0 -q '-1 0 -[ glyph[epsilon1] -1 i 0 -q '-2 0 ] [1 -q ' 0 ] 4 4 -glyph[epsilon1] i 0 q ' 2 0 + glyph[epsilon1] i 0 [ 2 q ' 2 0 + q ' 0 + q '-1 0 -q '-2 0 [1 -q ' 0 ] 4 2 ] ] × ∞ ∑ n =0 ( -1) n α ' 2 n 0 (2 n +1)! -[ 1 -4 glyph[epsilon1] i 0 1+ glyph[epsilon1] i 0 ] ∞ ∑ n =0 ( -1) n α 2 n i 0 (2 n +1)! }} . (126)</formula>
<figure>
<location><page_23><loc_23><loc_68><loc_76><loc_91></location>
<caption>FIG. 2: Plots of the function f 0 λ ( t, t ' , K ), defined in Eq. (126). The ranges of parameters a 0 ( t ), a 0 ( t ' ), K and glyph[epsilon1] i 0 are chosen same as in Fig. 1.</caption>
</figure>
<text><location><page_23><loc_12><loc_43><loc_88><loc_58></location>For comoving separation (123) and K =1 / 2, plots of the function f 0 λ ( t, t ' , K ) versus a ( t ' ) are given in Fig. 2 considering the values of glyph[epsilon1] i 0 chosen in Fig 1. The plots show that O ( λ ) correction (125), for a given glyph[epsilon1] i 0 , grows at early times and asymptotes to a constant at late times during inflation. The asymptotic value is substantially suppressed as glyph[epsilon1] i 0 increases even slightly. In Secs. IV A 1-2 we compute the temporal and coincident tree-order correlators, respectively.</text>
<section_header_level_1><location><page_23><loc_36><loc_38><loc_63><loc_39></location>1. Temporal tree-order correlator</section_header_level_1>
<text><location><page_23><loc_14><loc_33><loc_86><loc_35></location>The temporal tree-order correlator is the equal space ( glyph[vector]x ' → glyph[vector]x ) limit of correlator (107),</text>
<formula><location><page_23><loc_18><loc_24><loc_88><loc_29></location>Γ ( D 2 ) 2 π D 2 +1 Γ( D -1) ( H 1+ glyph[epsilon1] ) D 2 -1 ( H ' 1+ glyph[epsilon1] ' ) D 2 -1 2 ν Γ( ν ) ( Ha ) δ 2 2 ν ' Γ( ν ' ) ( H ' a ' ' ) δ ' 2 [ ( H ' a ' 1+ glyph[epsilon1] ' ) δ + δ ' 2 -( H i 1+ glyph[epsilon1] i ) δ + δ ' 2 δ + δ ' ] . (127)</formula>
<formula><location><page_23><loc_16><loc_24><loc_59><loc_31></location>〈 Ω | δ ¯ ϕ 0 ( t, glyph[vector]x ) δ ¯ ϕ 0 ( t ' , glyph[vector]x ) | Ω 〉 = 1+ glyph[epsilon1] 1+ glyph[epsilon1]</formula>
<text><location><page_23><loc_12><loc_21><loc_87><loc_23></location>Recall that in D → 4 limit we have ν = ν ' → 3 / 2+ O ( glyph[epsilon1] 2 ) and δ = δ ' → 0+ O ( glyph[epsilon1] 2 ). At this order,</text>
<formula><location><page_23><loc_21><loc_16><loc_88><loc_20></location>〈 Ω | δ ¯ ϕ 0 ( t, glyph[vector]x ) δ ¯ ϕ 0 ( t ' , glyph[vector]x ) | Ω 〉→ HH ' 4 π 2 (1+ glyph[epsilon1] )(1+ glyph[epsilon1] ' ) [ ln ( H ' a ' H i ) -ln ( 1+ glyph[epsilon1] ' 1+ glyph[epsilon1] i )] . (128)</formula>
<text><location><page_23><loc_12><loc_14><loc_35><loc_15></location>Expanding the correlator as</text>
<formula><location><page_23><loc_22><loc_7><loc_88><loc_12></location>〈 Ω | δ ¯ ϕ 0 ( t, glyph[vector]x ) δ ¯ ϕ 0 ( t ' , glyph[vector]x ) | Ω 〉 ≡〈 Ω | δ ¯ ϕ 0 ( t, glyph[vector]x ) δ ¯ ϕ 0 ( t ' , glyph[vector]x ) | Ω 〉 0 ± λ 〈 Ω | δ ¯ ϕ 0 ( t, glyph[vector]x ) δ ¯ ϕ 0 ( t ' , glyph[vector]x ) | Ω 〉 λ + O ( λ 2 ) . (129)</formula>
<text><location><page_24><loc_12><loc_89><loc_18><loc_91></location>implies</text>
<formula><location><page_24><loc_22><loc_80><loc_88><loc_88></location>〈 Ω | δ ¯ ϕ 0 ( t, glyph[vector]x ) δ ¯ ϕ 0 ( t ' , glyph[vector]x ) | Ω 〉 0 = H 0 H ' 0 4 π 2 (1+ glyph[epsilon1] 0 )(1+ glyph[epsilon1] ' 0 ) [ ln ( H ' 0 a ' 0 H i 0 ) -ln ( 1+ glyph[epsilon1] ' 0 1+ glyph[epsilon1] i 0 )] = H 2 i 0 4 π 2 ( q 0 q ' 0 ) 3 ( q 2 0 + glyph[epsilon1] i 0 )( q ' 2 0 + glyph[epsilon1] i 0 ) [ glyph[epsilon1] -1 i 0 E ' 0 2 -ln ( q ' 2 0 + glyph[epsilon1] i 0 q ' 3 0 (1+ glyph[epsilon1] i 0 ) ) ] , (130)</formula>
<text><location><page_24><loc_12><loc_77><loc_15><loc_78></location>and</text>
<formula><location><page_24><loc_12><loc_57><loc_88><loc_75></location>〈 Ω | δ ¯ ϕ 0 ( t, glyph[vector]x ) δ ¯ ϕ 0 ( t ' , glyph[vector]x ) | Ω 〉 λ = [ H λ H 0 -glyph[epsilon1] λ 1+ glyph[epsilon1] 0 + H ' λ H ' 0 -glyph[epsilon1] ' λ 1+ glyph[epsilon1] ' 0 ] 〈 Ω | δ ¯ ϕ 0 ( t, glyph[vector]x ) δ ¯ ϕ 0 ( t ' , glyph[vector]x ) | Ω 〉 0 + H 0 H ' 0 4 π 2 (1+ glyph[epsilon1] 0 )(1+ glyph[epsilon1] ' 0 ) { [ H ' λ H ' 0 -glyph[epsilon1] ' λ 1+ glyph[epsilon1] ' 0 +ln( a ' ) λ ] -[ H iλ H i 0 -glyph[epsilon1] iλ 1+ glyph[epsilon1] i 0 ] } = ξ 4! glyph[epsilon1] -2 i 0 {{[ q 2 0 + q 0 -q -1 0 + q -2 0 [1 -q 0 ] 4 4 -glyph[epsilon1] i 0 q 2 0 + glyph[epsilon1] i 0 ( 2 q 2 0 + q 0 + q -1 0 -q -2 0 [1 -q 0 ] 4 2 ) ] + [ q 0 → q ' 0 ]} 〈 Ω | δ ¯ ϕ 0 ( t,glyph[vector]x ) δ ¯ ϕ 0 ( t ' ,glyph[vector]x ) | Ω 〉 0 + H 2 i 0 4 π 2 ( q 0 q ' 0 ) 3 ( q 2 0 + glyph[epsilon1] i 0 )( q ' 2 0 + glyph[epsilon1] i 0 ) × {[ q ' 2 0 + q ' 0 -q '-1 0 -[ glyph[epsilon1] -1 i 0 -q '-2 0 ] [1 -q ' 0 ] 4 4 -glyph[epsilon1] i 0 q ' 2 0 + glyph[epsilon1] i 0 [ 2 q ' 2 0 + q ' 0 + q '-1 0 -q '-2 0 [1 -q ' 0 ] 4 2 ] ] -[ 1 -4 glyph[epsilon1] i 0 1+ glyph[epsilon1] i 0 ] }} . (131)</formula>
<text><location><page_24><loc_12><loc_48><loc_88><loc_55></location>In Sec. IV A 2 we take the equal time limits of correlators (127) and (128) to get the coincident tree-order correlator respectively in D , and in four dimensions which is used to compute the one-loop correlator in Sec. IV B.</text>
<section_header_level_1><location><page_24><loc_36><loc_43><loc_64><loc_44></location>2. Coincident tree-order correlator</section_header_level_1>
<text><location><page_24><loc_14><loc_38><loc_59><loc_40></location>Equal time t ' → t limit of tree-order correlator (127) is</text>
<formula><location><page_24><loc_24><loc_32><loc_88><loc_36></location>〈 Ω | δ ¯ ϕ 2 0 ( x ) | Ω 〉 = Γ ( D 2 ) 4 π D 2 +1 Γ( D -1) ( H 1+ glyph[epsilon1] ) D -2 2 2 ν Γ 2 ( ν ) δ [ 1 -( α i α ) δ ] . (132)</formula>
<text><location><page_24><loc_12><loc_29><loc_76><loc_31></location>In D =4, the equal time limit of Eq. (128) yields the coincident correlator as</text>
<formula><location><page_24><loc_29><loc_24><loc_88><loc_28></location>〈 Ω | δ ¯ ϕ 2 0 ( x ) | Ω 〉→ H 2 4 π 2 (1+ glyph[epsilon1] ) 2 [ ln ( Ha H i ) -ln ( 1+ glyph[epsilon1] 1+ glyph[epsilon1] i )] . (133)</formula>
<text><location><page_24><loc_12><loc_21><loc_63><loc_22></location>Perturbative expansion of the parameters H , a and glyph[epsilon1] implies</text>
<formula><location><page_24><loc_27><loc_17><loc_88><loc_18></location>〈 Ω | δ ¯ ϕ 2 0 ( x ) | Ω 〉≡〈 Ω | δ ¯ ϕ 2 0 ( x ) | Ω 〉 0 ± λ 〈 Ω | δ ¯ ϕ 2 0 ( x ) | Ω 〉 λ + O ( λ 2 ) . (134)</formula>
<text><location><page_24><loc_12><loc_13><loc_75><loc_14></location>The O ( λ 0 ) term, i.e., the coincident correlator in the noninteracting theory</text>
<formula><location><page_24><loc_12><loc_7><loc_88><loc_11></location>〈 Ω | δ ¯ ϕ 2 0 ( x ) | Ω 〉 0 = H 2 0 4 π 2 (1+ glyph[epsilon1] 0 ) 2 [ ln ( H 0 a 0 H i 0 ) -ln ( 1+ glyph[epsilon1] 0 1+ glyph[epsilon1] i 0 ) ] = H 2 i 0 4 π 2 q 6 0 [ q 2 0 + glyph[epsilon1] i 0 ] 2 [ glyph[epsilon1] -1 i 0 E 0 2 -ln ( q 2 0 + glyph[epsilon1] i 0 q 3 0 (1+ glyph[epsilon1] i 0 ) ) ] , (135)</formula>
<text><location><page_25><loc_12><loc_89><loc_29><loc_91></location>where the logarithm</text>
<formula><location><page_25><loc_33><loc_84><loc_88><loc_88></location>ln ( q 2 0 + glyph[epsilon1] i 0 q 3 0 (1+ glyph[epsilon1] i 0 ) ) = -ln( q 0 )+ glyph[epsilon1] i 0 E 0 q -2 0 + O ( glyph[epsilon1] 2 i 0 ) . (136)</formula>
<text><location><page_25><loc_12><loc_77><loc_88><loc_83></location>The O ( λ ) correction in Eq. (134) induced as a reaction to the change in the expansion rate-hence in the scale factor and the slow-roll parameter-due to the self-interactions of the inflaton is</text>
<formula><location><page_25><loc_12><loc_62><loc_89><loc_76></location>〈 Ω | δ ¯ ϕ 2 0 ( x ) | Ω 〉 λ = [ 2 〈 Ω | δ ¯ ϕ 2 0 ( x ) | Ω 〉 0 + H 2 0 4 π 2 (1+ glyph[epsilon1] 0 ) 2 ][ H λ H 0 -glyph[epsilon1] λ 1+ glyph[epsilon1] 0 ] + H 2 0 4 π 2 (1+ glyph[epsilon1] 0 ) 2 [ ln( a ) λ -H iλ H i 0 + glyph[epsilon1] iλ 1+ glyph[epsilon1] i 0 ] = H 2 i 0 4 π 2 q 6 0 ( q 2 0 + glyph[epsilon1] i 0 ) 2 ξ 4! glyph[epsilon1] -2 i 0 { [ glyph[epsilon1] -1 i 0 E 0 +1 -2 ln ( q 2 0 + glyph[epsilon1] i 0 q 3 0 (1+ glyph[epsilon1] i 0 ) ) ][ q 2 0 + q 0 -q -1 0 + q -2 0 [1 -q 0 ] 4 4 -glyph[epsilon1] i 0 q 2 0 + glyph[epsilon1] i 0 [ 2 q 2 0 + q 0 + q -1 0 -q -2 0 [1 -q 0 ] 4 2 ] ] -glyph[epsilon1] -1 i 0 [1 -q 0 ] 4 4 -1+ 4 glyph[epsilon1] i 0 1+ glyph[epsilon1] i 0 } . (137)</formula>
<text><location><page_25><loc_12><loc_59><loc_75><loc_61></location>Expanding the terms inside the curly brackets in powers of glyph[epsilon1] i 0 and E 0 yields</text>
<formula><location><page_25><loc_12><loc_31><loc_88><loc_58></location>〈 Ω | δ ¯ ϕ 2 0 ( x ) | Ω 〉 λ = -H 2 i 0 576 π 4 π GH 2 i 0 q 4 0 [ q 2 0 + glyph[epsilon1] i 0 ] 2 { glyph[epsilon1] -3 i 0 [ 1 -q 0 -7 2 E 0 [ 1 -5 7 q 0 ] + 25 8 E 2 0 [ 1 -4 25 q 0 ] -3 4 E 3 0 ] -glyph[epsilon1] -2 i 0 q -2 0 [ 1 -q 0 + 3 2 [ 1 -2 3 q 0 ] ln( q 2 0 ) -3 2 E 0 [ 1+ 4 3 q 0 + 7 3 [ 1 -2 7 q 0 ] ln( q 2 0 ) ] + 17 8 E 2 0 [ 1+ 12 17 q 0 + 21 17 ln( q 2 0 ) ] -11 8 E 3 0 [ 1+ 5 11 ln( q 2 0 ) ] ] -glyph[epsilon1] -1 i 0 q -4 0 3 [ 1 -q 0 + 1 -3 q 0 3 ln( q 2 0 ) -11 3 E 0 [ 1 -19 22 q 0 + 1 11 [ 1 -9 2 q 0 ] ln( q 2 0 ) ] + 49 12 E 2 0 × [ 1 -20 49 q 0 -3 49 [ 1+2 q 0 ] ln( q 2 0 ) ] -23 12 E 3 0 [ 1 -3 23 ln( q 2 0 ) ] + 5 12 E 4 0 ] + q -6 0 3 [ 1 -q 0 + 1 -3 q 0 3 ln( q 2 0 ) -11 3 E 0 × [ 1 -7 22 q 0 + 1 11 [ 1 -9 2 q 0 ] ln( q 2 0 ) ] + 71 12 E 2 0 [ 1+ 12 71 q 0 -3 71 [ 1+2 q 0 ] ln( q 2 0 ) ] -67 12 E 3 0 [ 1+ 8 67 q 0 -3 67 ln( q 2 0 ) ] + 59 24 E 4 0 -5 24 E 5 0 ] + O ( glyph[epsilon1] i 0 ) } . (138)</formula>
<text><location><page_25><loc_12><loc_25><loc_88><loc_29></location>Self-interactions of the inflaton field yield quantum corrections to the two-point correlation function. The one-loop correction at O ( λ ) is computed in the next section.</text>
<section_header_level_1><location><page_25><loc_39><loc_19><loc_61><loc_21></location>B. One-loop correlator</section_header_level_1>
<text><location><page_25><loc_12><loc_7><loc_88><loc_16></location>Computation of the one-loop contribution (100) to two-point correlation (98) involves evaluations of four VEVs. Two of the VEVs which are quadratic both in the background and fluctuation fields are evaluated in Sec. IV B 1. The remaining VEVs which are quartic in the fluctuation field are evaluated in Sec. IV B 2.</text>
<text><location><page_26><loc_12><loc_42><loc_17><loc_44></location>where</text>
<formula><location><page_26><loc_17><loc_32><loc_88><loc_41></location>A ( q ' 0 , ∆ x ) ≡ ∞ ∑ n =1 ( -1) n ( H i 0 ∆ x ) 2 n 2 n (2 n +1)! 2 e n glyph[epsilon1] i 0 { glyph[epsilon1] -2 i 0 [ q ' 2 n +2 0 E -n ( nq ' 2 0 glyph[epsilon1] i 0 ) -E -n ( n glyph[epsilon1] i 0 )] -(2 n -1) [ glyph[epsilon1] -1 i 0 [ q ' 2 n 0 E 1 -n ( nq ' 2 0 glyph[epsilon1] i 0 ) -E 1 -n ( n glyph[epsilon1] i 0 )] -n [ q ' 2 n -2 0 E 2 -n ( nq ' 2 0 glyph[epsilon1] i 0 ) -E 2 -n ( n glyph[epsilon1] i 0 )] ] } . (141)</formula>
<text><location><page_26><loc_12><loc_27><loc_88><loc_31></location>The second integral in Eq. (139) is obtained mingling Eq. (140) and Eqs. (D10)-(D13). The result is</text>
<formula><location><page_26><loc_12><loc_7><loc_88><loc_26></location>H 0 1+ glyph[epsilon1] 0 ∫ t 0 0 dt '' 0 [ 1+ glyph[epsilon1] '' 0 H '' 0 ] 2 ¯ ϕ 2 0 ( t '' 0 ) 〈 Ω | δ ¯ ϕ 0 ( t '' 0 , glyph[vector]x ) δ ¯ ϕ 0 ( t ' 0 , glyph[vector]x ' ) | Ω 〉 = 1 32 π 3 G q 3 0 q ' 3 0 ( q 2 0 + glyph[epsilon1] i 0 )( q ' 2 0 + glyph[epsilon1] i 0 ) { glyph[epsilon1] -3 i 0 E ' 2 0 4 -glyph[epsilon1] -2 i 0 [ 1+ q ' 2 0 ] ln( q ' 0 ) -glyph[epsilon1] -1 i 0 [ ln 2 ( q ' 0 ) -E ' 0 ] + [ glyph[epsilon1] -2 i 0 E ' 0 -glyph[epsilon1] -1 i 0 ln( q ' 2 0 ) ] × [ ln( α i 0 ) -2+ γ ] -[ glyph[epsilon1] -2 i 0 E 0 -glyph[epsilon1] -1 i 0 ln( q ' 2 0 ) ][ ci( α i 0 ) -sin( α i 0 ) α i 0 ] -[ glyph[epsilon1] -2 i 0 [ q 2 0 -q ' 2 0 ] + glyph[epsilon1] -1 i 0 [ ln( q 2 0 ) -ln( q ' 2 0 ) ] ] × [ ci( α ' 0 ) -sin( α ' 0 ) α ' 0 ] -ln( q ' 2 0 )+ q ' 2 0 2 -q '-2 0 2 + A ( q ' 0 , ∆ x ) 2 + O ( glyph[epsilon1] i 0 ) } . (142)</formula>
<section_header_level_1><location><page_26><loc_25><loc_89><loc_75><loc_91></location>1. The VEVs quadratic in the background and fluctuation fields</section_header_level_1>
<text><location><page_26><loc_12><loc_80><loc_88><loc_86></location>One-loop correlator (100) has two VEVs that are quadratic in the background and fluctuation fields-in addition to the two VEVs that are quartic in the fluctuation field. In D =4 dimensions, they are</text>
<formula><location><page_26><loc_21><loc_69><loc_88><loc_78></location>V ¯ ϕ 2 0 δ ¯ ϕ 2 0 ( q 0 , q ' 0 , ∆ x ) ≡∓ λ 6 { H ' 0 1+ glyph[epsilon1] ' 0 〈 Ω | δ ¯ ϕ 0 ( x ) ∫ t ' 0 0 d ˜ t 0 [ 1+˜ glyph[epsilon1] 0 ˜ H 0 ] 2 ¯ ϕ 2 0 ( ˜ t 0 ) δ ¯ ϕ 0 ( ˜ t 0 , glyph[vector]x ' ) | Ω 〉 0 + H 0 1+ glyph[epsilon1] 0 〈 Ω | ∫ t 0 0 dt '' 0 [ 1+ glyph[epsilon1] '' 0 H '' 0 ] 2 ¯ ϕ 2 0 ( t '' 0 ) δ ¯ ϕ 0 ( t '' 0 , glyph[vector]x ) δ ¯ ϕ 0 ( x ' ) | Ω 〉 0 } . (139)</formula>
<text><location><page_26><loc_12><loc_64><loc_88><loc_68></location>Evaluations of the integrals in Eq. (139) are outlined in Appendix D 1. Combining Eqs. (D3), (D4), (D7) and (D8) in Eq. (D1) yields the first integral in Eq. (139) as</text>
<formula><location><page_26><loc_19><loc_50><loc_20><loc_52></location>-</formula>
<formula><location><page_26><loc_20><loc_45><loc_88><loc_63></location>H ' 0 1+ glyph[epsilon1] ' 0 ∫ t ' 0 0 d ˜ t 0 [ 1+˜ glyph[epsilon1] 0 ˜ H 0 ] 2 ¯ ϕ 2 0 ( ˜ t 0 ) 〈 Ω | δ ¯ ϕ 0 ( t 0 , glyph[vector]x ) δ ¯ ϕ 0 ( ˜ t 0 , glyph[vector]x ' ) | Ω 〉 = 1 32 π 3 G q 3 0 q ' 3 0 ( q 2 0 + glyph[epsilon1] i 0 )( q ' 2 0 + glyph[epsilon1] i 0 ) { glyph[epsilon1] -3 i 0 E ' 2 0 4 -glyph[epsilon1] -2 i 0 [ [ 1+ q ' 2 0 ] +(2 -γ ) E ' 0 ] ln( q ' 0 ) glyph[epsilon1] -1 i 0 [ ln 2 ( q ' 0 ) -(2 -γ ) ln( q ' 2 0 ) -1+ q ' 2 0 ] -ln( q ' 2 0 ) -q '-2 0 2 + q ' 2 0 2 -[ glyph[epsilon1] -2 i 0 E ' 0 -glyph[epsilon1] -1 i 0 ln( q ' 2 0 ) ] × [ ci( α i ) -sin( α i ) α i -ln( α i ) ] + A ( q ' 0 , ∆ x ) 2 + O ( glyph[epsilon1] i 0 ) } , (140)</formula>
<figure>
<location><page_27><loc_23><loc_68><loc_76><loc_91></location>
<caption>FIG. 3: Plots of the function f 1 λ ¯ ϕ 2 0 δ ¯ ϕ 2 0 ( t, t ' , K ), defined in Eq. (144). The ranges of parameters a 0 ( t ), a 0 ( t ' ), K and glyph[epsilon1] i 0 are chosen same as in Fig. 1.</caption>
</figure>
<text><location><page_27><loc_12><loc_54><loc_88><loc_58></location>The sum of the VEVs that are quadratic in the background and fluctuation fields at O ( λ ) is obtained combining Eqs. (140) and (142) in Eq. (139),</text>
<formula><location><page_27><loc_35><loc_48><loc_88><loc_52></location>V ¯ ϕ 2 0 δ ¯ ϕ 2 0 ≡∓ λ H 2 i 0 192 π 4 π GH 2 i 0 f 1 λ ¯ ϕ 2 0 δ ¯ ϕ 2 0 ( t, t ' , ∆ x ) , (143)</formula>
<text><location><page_27><loc_12><loc_45><loc_17><loc_47></location>where</text>
<formula><location><page_27><loc_12><loc_30><loc_88><loc_44></location>f 1 λ ¯ ϕ 2 0 δ ¯ ϕ 2 0 = q 3 0 q ' 3 0 ( q 2 0 + glyph[epsilon1] i 0 )( q ' 2 0 + glyph[epsilon1] i 0 ) { glyph[epsilon1] -3 i 0 E ' 2 0 2 -glyph[epsilon1] -2 i 0 [ 1+ q ' 2 0 ] ln( q ' 2 0 ) -glyph[epsilon1] -1 i 0 2 [ ln 2 ( q ' 0 ) -E ' 0 ] + [ glyph[epsilon1] -2 i 0 E ' 0 -glyph[epsilon1] -1 i 0 ln( q ' 2 0 ) ] × 2 [ ln( α i 0 ) -2+ γ ] -[ glyph[epsilon1] -2 i 0 [ 2 -q 2 0 -q ' 2 0 ] -glyph[epsilon1] -1 i 0 [ ln( q 2 0 )+ln( q ' 2 0 ) ] ] [ ci( α i 0 ) -sin( α i 0 ) α i 0 ] -[ glyph[epsilon1] -2 i 0 [ q 2 0 -q ' 2 0 ] + glyph[epsilon1] -1 i 0 [ ln( q 2 0 ) -ln( q ' 2 0 ) ] ] [ ci( α ' 0 ) -sin( α ' 0 ) α ' 0 ] -2 ln( q ' 2 0 ) -q '-2 0 + q ' 2 0 + A ( q ' 0 , ∆ x )+ O ( glyph[epsilon1] i 0 ) } . (144)</formula>
<text><location><page_27><loc_12><loc_13><loc_88><loc_28></location>For comoving separation (123) and K =1 / 2, plots of the function f 1 λ ¯ ϕ 2 0 δ ¯ ϕ 2 0 ( t, t ' , K ) versus a 0 ( t ' ) are given in Fig. 3. The values of glyph[epsilon1] i 0 are chosen as in Figs. 1 and 2. The plots imply that, for a given glyph[epsilon1] i 0 , the one loop O ( λ ) correction V ¯ ϕ 2 0 δ ¯ ϕ 2 0 grows-positively or negatively, depending on the sign choice in the potential-and asymptotes to a constant at late times during inflation. The growth is suppressed as the glyph[epsilon1] i 0 increases, though not as sensitively as in O ( λ ) correction (125) depicted in Fig. 2.</text>
<text><location><page_27><loc_12><loc_8><loc_88><loc_12></location>To complete the computation of one-loop correlator (100), we need to evaluate the remaining VEVs that are quartic in the fluctuation field. That is the task of the next section.</text>
<text><location><page_28><loc_12><loc_82><loc_88><loc_86></location>The VEVs that are quartic in the fluctuation field in one-loop correlator (100), for D =4, are</text>
<formula><location><page_28><loc_22><loc_72><loc_88><loc_81></location>V δ ¯ ϕ 4 0 ( q 0 , q ' 0 , ∆ x ) ≡∓ λ 18 { H ' 0 1+ glyph[epsilon1] ' 0 〈 Ω | δ ¯ ϕ 0 ( x ) ∫ t ' 0 0 d ˜ t 0 [ 1+˜ glyph[epsilon1] 0 ˜ H 0 ] 2 δ ¯ ϕ 3 0 ( ˜ t 0 , glyph[vector]x ' ) | Ω 〉 + H 0 1+ glyph[epsilon1] 0 〈 Ω | ∫ t 0 0 dt '' 0 [ 1+ glyph[epsilon1] '' 0 H '' 0 ] 2 δ ¯ ϕ 3 0 ( t '' 0 , glyph[vector]x ) δ ¯ ϕ 0 ( x ' ) | Ω 〉 } . (145)</formula>
<text><location><page_28><loc_12><loc_67><loc_88><loc_70></location>Evaluation of the integrals in Eq. (145) are outlined in Appendix D 2. Combining Eqs. (D15)(D26) in Eq. (D14) gives the first integral in Eq. (145) as</text>
<formula><location><page_28><loc_12><loc_42><loc_89><loc_65></location>H ' 0 1+ glyph[epsilon1] ' 0 ∫ t ' 0 0 d ˜ t 0 [ 1+˜ glyph[epsilon1] 0 ˜ H 0 ] 2 〈 Ω | δ ¯ ϕ 0 ( t 0 , glyph[vector]x ) δ ¯ ϕ 3 0 ( ˜ t 0 , glyph[vector]x ' ) | Ω 〉 = 3 32 π 4 H 2 i 0 q 3 0 q ' 3 0 ( q 2 0 + glyph[epsilon1] i 0 )( q ' 2 0 + glyph[epsilon1] i 0 ) { glyph[epsilon1] -3 i 0 E ' 3 0 12 + glyph[epsilon1] -2 i 0 E ' 2 0 4 [ ln( q ' 2 0 ) -(1 -γ ) ] + glyph[epsilon1] -1 i 0 [ E ' 2 0 2 + E ' 0 [ ln 2 ( q ' 0 )+ γ ln( q ' 0 )+1 ] + q '-2 0 -q ' 2 0 4 + [ 1+ q ' 2 0 2 ] ln( q ' 2 0 ) ] -1 3 + 1 -q '-4 0 8 -2 [ 1 -q '-2 0 ][ 1 -q ' 2 0 2 ] -E ' 2 0 4 + E ' 0 [ 1+ q '-2 0 2 ][ ln( q ' 2 0 )+ γ ] + γ [ ln 2 ( q ' 0 )+ln( q ' 3 0 ) ] + ln 3 ( eq ' 2 0 ) 12 + ln 2 ( eq ' 2 0 ) 4 -[ glyph[epsilon1] -2 i 0 E ' 2 0 4 + glyph[epsilon1] -1 i 0 E ' 0 ln( q ' 0 )+ E ' 0 [ 1+ q '-2 0 2 ] +ln 2 ( q ' 0 )+ln( q ' 3 0 ) ] × [ ci( α i 0 ) -sin( α i 0 ) α i 0 -ln( α i 0 ) ] + B ( q ' 0 , ∆ x ) 2 + O ( glyph[epsilon1] i 0 ) } , (146)</formula>
<text><location><page_28><loc_12><loc_39><loc_25><loc_41></location>where we define</text>
<formula><location><page_28><loc_12><loc_7><loc_88><loc_39></location>B ( q ' 0 , ∆ x ) ≡ ∞ ∑ n =1 ( -1) n ( H i 0 ∆ x ) 2 n 2 n (2 n +1)! e n glyph[epsilon1] i 0 { glyph[epsilon1] -2 i 0 [ E -n -1 ( n glyph[epsilon1] i 0 ) -q ' 2 n +4 0 E -n -1 ( nq ' 2 0 glyph[epsilon1] i 0 ) -{ E -n ( n glyph[epsilon1] i 0 ) -q ' 2 n +2 0 E -n ( nq ' 2 0 glyph[epsilon1] i 0 ) }] -glyph[epsilon1] -1 i 0 [ (2 n +1) { E -n ( n glyph[epsilon1] i 0 ) -q ' 2 n +2 0 E -n ( nq ' 2 0 glyph[epsilon1] i 0 ) -[ E 1 -n ( n glyph[epsilon1] i 0 ) -q ' 2 n 0 E 1 -n ( nq ' 2 0 glyph[epsilon1] i 0 ) ]} -q ' 2 n +2 0 E -n ( nq ' 2 0 glyph[epsilon1] i 0 ) ln( q ' 2 0 )+ 1 ( n +1) 2 { 2 F 2 ( n +1 ,n +1; n +2 ,n +2; -n glyph[epsilon1] i 0 ) -q ' 2 n +2 0 2 F 2 ( n +1 ,n +1; n +2 ,n +2; -nq ' 2 0 glyph[epsilon1] i 0 ) }] -2 [ E -n ( n glyph[epsilon1] i 0 ) -q ' 2 n +2 0 E -n ( nq ' 2 0 glyph[epsilon1] i 0 ) -{ E 1 -n ( n glyph[epsilon1] i 0 ) -q ' 2 n 0 E 1 -n ( nq ' 2 0 glyph[epsilon1] i 0 ) }] +(2 n +1) [ ( n +1) { E 1 -n ( n glyph[epsilon1] i 0 ) -q ' 2 n 0 E 1 -n ( nq ' 2 0 glyph[epsilon1] i 0 ) -{ E 2 -n ( n glyph[epsilon1] i 0 ) -q ' 2 n -2 0 E 2 -n ( nq ' 2 0 glyph[epsilon1] i 0 ) }} -q ' 2 n 0 E 1 -n ( nq ' 2 0 glyph[epsilon1] i 0 ) ln( q ' 2 0 )+ 1 n 2 { 2 F 2 ( n, n ; n +1 , n +1; -n glyph[epsilon1] i 0 ) -q ' 2 n 0 2 F 2 ( n, n ; n +1 , n +1; -nq ' 2 0 glyph[epsilon1] i 0 ) }]} . (147)</formula>
<text><location><page_29><loc_12><loc_89><loc_82><loc_91></location>Combining Eqs. (D28)-(D30) in Eq. (D27) yields the second integral in Eq. (145) as</text>
<formula><location><page_29><loc_13><loc_56><loc_89><loc_88></location>H 0 1+ glyph[epsilon1] 0 ∫ t 0 0 dt '' 0 [ 1+ glyph[epsilon1] '' 0 H '' 0 ] 2 〈 Ω | δ ¯ ϕ 3 0 ( t '' 0 , glyph[vector]x ) δ ¯ ϕ 0 ( t ' 0 , glyph[vector]x ' ) | Ω 〉 = 3 32 π 4 H 2 i 0 q 3 0 q ' 3 0 ( q 2 0 + glyph[epsilon1] i 0 )( q ' 2 0 + glyph[epsilon1] i 0 ) { glyph[epsilon1] -3 i 0 E ' 3 0 12 + glyph[epsilon1] -2 i 0 E ' 2 0 4 [ ln( q ' 2 0 ) -(1 -γ ) ] + glyph[epsilon1] -1 i 0 [ E ' 2 0 2 + E ' 0 [ ln 2 ( q ' 0 )+ γ ln( q ' 0 )+1 ] -q ' 2 0 -q '-2 0 4 + [ 1+ q ' 2 0 2 ] ln( q ' 2 0 ) ] -1 3 + 1 -q '-4 0 8 -2 [ 1 -q '-2 0 ][ 1 -q ' 2 0 2 ] -E ' 2 0 4 + E ' 0 [ 1+ q '-2 0 2 ][ ln( q ' 2 0 )+ γ ] + γ [ ln 2 ( q ' 0 )+ln( q ' 3 0 ) ] + ln 3 ( eq ' 2 0 ) 12 + ln 2 ( eq ' 2 0 ) 4 -{ glyph[epsilon1] -2 i 0 E 2 0 4 + glyph[epsilon1] -1 i 0 E 0 ln( q 0 )+ E 0 [ 1+ q -2 0 2 ] +ln 2 ( q 0 )+ln( q 3 0 ) } × [ ci( α i 0 ) -sin( α i 0 ) α i 0 ] + { glyph[epsilon1] -2 i 0 E ' 2 0 4 + glyph[epsilon1] -1 i 0 E ' 0 ln( q ' 0 )+ E ' 0 [ 1+ q '-2 0 2 ] +ln 2 ( q ' 0 )+ln( q ' 3 0 ) } ln( α i 0 ) + { glyph[epsilon1] -2 i 0 4 [ E 2 0 -E ' 2 0 ] + glyph[epsilon1] -1 i 0 [ E 0 ln( q 0 ) -E ' 0 ln( q ' 0 ) ] + E 0 [ 1+ q -2 0 2 ] -E ' 0 [ 1+ q '-2 0 2 ] +ln 2 ( q 0 ) -ln 2 ( q ' 0 ) +ln( q 3 0 ) -ln( q ' 3 0 ) } [ ci( α ' 0 ) -sin( α ' 0 ) α ' 0 ] + B ( q ' 0 , ∆ x ) 2 + O ( glyph[epsilon1] i 0 ) } . (148)</formula>
<text><location><page_29><loc_12><loc_51><loc_88><loc_55></location>Adding up Eqs. (146) and (148) yields the contribution due to the VEVs that are quartic in the fluctuation field given in Eq. (145) as</text>
<formula><location><page_29><loc_37><loc_45><loc_88><loc_49></location>V δ ¯ ϕ 4 0 ≡∓ λ H 2 i 0 192 π 4 f 1 λδ ¯ ϕ 4 0 ( t, t ' , ∆ x ) , (149)</formula>
<text><location><page_29><loc_12><loc_42><loc_17><loc_44></location>where</text>
<formula><location><page_29><loc_12><loc_13><loc_88><loc_41></location>f 1 λδ ¯ ϕ 4 0 = q 3 0 q ' 3 0 ( q 2 0 + glyph[epsilon1] i 0 )( q ' 2 0 + glyph[epsilon1] i 0 ) { glyph[epsilon1] -3 i 0 E ' 3 0 6 + glyph[epsilon1] -2 i 0 E ' 2 0 [ ln( q ' 0 ) -1 -γ 2 ] + glyph[epsilon1] -1 i 0 [ E ' 2 0 +2 E ' 0 [ ln 2 ( q ' 0 )+ γ ln( q ' 0 )+1 ] + q '-2 0 2 -q ' 2 0 2 + [ 2+ q ' 2 0 ] ln( q ' 2 0 ) ] -2 3 + 1 -q '-4 0 4 -4 [ 1 -q '-2 0 ] -2 E ' 0 -E ' 2 0 2 + [ 1+ q '-2 0 -2 q ' 2 0 ][ ln( q ' 2 0 )+ γ ] +2 γ [ ln 2 ( q ' 0 )+ln( q ' 3 0 ) ] + ln 3 ( eq ' 2 0 ) 6 + ln 2 ( eq ' 2 0 ) 2 + { glyph[epsilon1] -2 i 0 E ' 2 0 4 + glyph[epsilon1] -1 i 0 E ' 0 ln( q ' 0 ) -1 -q '-2 0 2 + E ' 0 +ln 2 ( q ' 0 ) +ln( q ' 3 0 ) } ln( α 2 i 0 ) -{ glyph[epsilon1] -2 i 0 4 [ E 2 0 + E ' 2 0 ] + glyph[epsilon1] -1 i 0 [ E 0 ln( q 0 )+ E ' 0 ln( q ' 0 ) ] + q -2 0 2 + q '-2 0 2 -[ q 2 0 + q ' 2 0 ] +1+ln 2 ( q 0 ) +ln 2 ( q ' 0 )+ln( q 3 0 )+ln( q ' 3 0 ) } [ ci( α i 0 ) -sin( α i 0 ) α i 0 ] + { glyph[epsilon1] -2 i 0 4 [ E 2 0 -E ' 2 0 ] + glyph[epsilon1] -1 i 0 [ E 0 ln( q 0 ) -E ' 0 ln( q ' 0 ) ] + q -2 0 2 -q '-2 0 2 -[ q 2 0 -q ' 2 0 ] +ln 2 ( q 0 ) -ln 2 ( q ' 0 )+ln( q 3 0 ) -ln( q ' 3 0 ) } [ ci( α ' 0 ) -sin( α ' 0 ) α ' 0 ] + B ( q ' 0 , ∆ x )+ O ( glyph[epsilon1] i 0 ) } . (150)</formula>
<text><location><page_29><loc_12><loc_7><loc_88><loc_11></location>Plots of the function f 1 λδ ¯ ϕ 4 0 ( t, t ' , K ) versus a 0 ( t ' ), for comoving separation (123), K =1 / 2 and the four values of glyph[epsilon1] i 0 that are used in the preceding figures, are given in Fig. 4. They imply</text>
<figure>
<location><page_30><loc_23><loc_68><loc_76><loc_91></location>
<caption>FIG. 4: Plots of the function f 1 λδ ¯ ϕ 4 0 ( t, t ' , K ), defined in Eq. (150). The ranges of parameters a 0 ( t ), a 0 ( t ' ), K and glyph[epsilon1] i 0 are chosen same as in Fig. 1.</caption>
</figure>
<text><location><page_30><loc_12><loc_51><loc_88><loc_58></location>that, for a chosen glyph[epsilon1] i 0 , the one loop O ( λ ) correction V δ ¯ ϕ 4 0 grows-positively or negatively, depending on the sign choice in the potential-and asymptotes to an almost constant value at late times during inflation. The growth is only marginally suppressed as the glyph[epsilon1] i 0 increases.</text>
<section_header_level_1><location><page_30><loc_44><loc_46><loc_55><loc_47></location>3. The result</section_header_level_1>
<text><location><page_30><loc_12><loc_39><loc_88><loc_43></location>The one-loop correlator at O ( λ ) is obtained mingling contributions (143) and (149) in Eq (100), setting D =4, as</text>
<formula><location><page_30><loc_28><loc_34><loc_88><loc_38></location>〈 Ω | δ ¯ ϕ ( t, glyph[vector]x ) δ ¯ ϕ ( t ' , glyph[vector]x ' ) | Ω 〉 1 -loop glyph[similarequal] ∓ λ H 2 i 0 192 π 4 f 1 λ ( t, t ' , ∆ x ) , (151)</formula>
<text><location><page_30><loc_12><loc_31><loc_17><loc_33></location>where</text>
<formula><location><page_30><loc_12><loc_7><loc_88><loc_31></location>f 1 λ = q 3 0 q ' 3 0 ( q 2 0 + glyph[epsilon1] i 0 )( q ' 2 0 + glyph[epsilon1] i 0 ) { π GH 2 i 0 { glyph[epsilon1] -3 i 0 E ' 2 0 2 -glyph[epsilon1] -2 i 0 [ 1+ q ' 2 0 ] ln( q ' 2 0 ) -glyph[epsilon1] -1 i 0 2 [ ln 2 ( q ' 0 ) -E ' 0 ] + [ glyph[epsilon1] -2 i 0 E ' 0 -glyph[epsilon1] -1 i 0 ln( q ' 2 0 ) ] × [ ln( α 2 i 0 ) -4+2 γ ] -[ glyph[epsilon1] -2 i 0 [ 2 -q 2 0 -q ' 2 0 ] -glyph[epsilon1] -1 i 0 [ ln( q 2 0 )+ln( q ' 2 0 ) ] ] [ ci( α i 0 ) -sin( α i 0 ) α i 0 ] -[ glyph[epsilon1] -2 i 0 [ q 2 0 -q ' 2 0 ] + glyph[epsilon1] -1 i 0 [ ln( q 2 0 ) -ln( q ' 2 0 ) ] ] [ ci( α ' 0 ) -sin( α ' 0 ) α ' 0 ] -2 ln( q ' 2 0 ) -q '-2 0 + q ' 2 0 + A ( q ' 0 , ∆ x ) } + glyph[epsilon1] -3 i 0 E ' 3 0 6 + glyph[epsilon1] -2 i 0 E ' 2 0 [ ln( q ' 0 ) -1 -γ 2 ] + glyph[epsilon1] -1 i 0 [ E ' 2 0 +2 E ' 0 [ ln 2 ( q ' 0 )+ γ ln( q ' 0 )+1 ] + q '-2 0 -q ' 2 0 2 + [ 2+ q ' 2 0 ] ln( q ' 2 0 ) ] -2 3 + 1 -q '-4 0 4 -4 [ 1 -q '-2 0 ] -2 E ' 0 -E ' 2 0 2 + [ 1+ q '-2 0 -2 q ' 2 0 ][ ln( q ' 2 0 )+ γ ] +2 γ [ ln 2 ( q ' 0 )+ln( q ' 3 0 ) ] + ln 3 ( eq ' 2 0 ) 6 + ln 2 ( eq ' 2 0 ) 2 + { glyph[epsilon1] -2 i 0 E ' 2 0 4</formula>
<formula><location><page_31><loc_12><loc_72><loc_88><loc_91></location>+ glyph[epsilon1] -1 i 0 E ' 0 ln( q ' 0 ) -1 -q '-2 0 2 + E ' 0 +ln 2 ( q ' 0 )+ln( q ' 3 0 ) } ln( α 2 i 0 ) -{ glyph[epsilon1] -2 i 0 4 [ E 2 0 + E ' 2 0 ] + glyph[epsilon1] -1 i 0 [ E 0 ln( q 0 )+ E ' 0 ln( q ' 0 ) ] + q -2 0 + q '-2 0 2 -[ q 2 0 + q ' 2 0 ] +1+ln 2 ( q 0 )+ln 2 ( q ' 0 )+ln( q 3 0 )+ln( q ' 3 0 ) } [ ci( α i 0 ) -sin( α i 0 ) α i 0 ] + { glyph[epsilon1] -2 i 0 4 [ E 2 0 -E ' 2 0 ] + glyph[epsilon1] -1 i 0 [ E 0 ln( q 0 ) -E ' 0 ln( q ' 0 ) ] + q -2 0 -q '-2 0 2 -[ q 2 0 -q ' 2 0 ] +ln 2 ( q 0 ) -ln 2 ( q ' 0 )+ln( q 3 0 ) -ln( q ' 3 0 ) } × [ ci( α ' 0 ) -sin( α ' 0 ) α ' 0 ] + B ( q ' 0 , ∆ x )+ O ( glyph[epsilon1] i 0 ) } . (152)</formula>
<text><location><page_31><loc_12><loc_69><loc_66><loc_70></location>In the next section, we obtain the coincident one-loop correlator.</text>
<section_header_level_1><location><page_31><loc_36><loc_63><loc_63><loc_64></location>4. Coincident one-loop correlator</section_header_level_1>
<text><location><page_31><loc_12><loc_56><loc_88><loc_60></location>Using the equal spacetime limit of Eq. (152) in Eq. (151) yields the coincident one-loop correlator at O ( λ ) as</text>
<formula><location><page_31><loc_13><loc_36><loc_88><loc_55></location>〈 Ω | δ ¯ ϕ 2 ( t, glyph[vector]x ) | Ω 〉 1 -loop = ∓ λ H 2 i 0 192 π 4 q 6 0 [ q 2 0 + glyph[epsilon1] i 0 ] 2 { π GH 2 i 0 [ glyph[epsilon1] -3 i 0 E 2 0 2 -glyph[epsilon1] -2 i 0 2 [ ln( q 2 0 )+ E 0 [ 1 -ln( q 2 0 ) 2 ] ] + glyph[epsilon1] -1 i 0 2 [ ln( q 2 0 ) -ln 2 ( q 2 0 ) 4 + E 0 ] -2 ln( q 2 0 ) -q -2 0 [ 2 E 0 -E 2 0 ] + O ( glyph[epsilon1] i 0 ) ] + glyph[epsilon1] -3 i 0 E 3 0 6 + glyph[epsilon1] -2 i 0 E 2 0 ln( q 2 0 ) 2 + glyph[epsilon1] -1 i 0 3 [ ln( q 2 0 )+ E 0 q -2 0 [ 1+ ln 2 ( q 2 0 ) 6 -E 0 2 [ 1+ ln 2 ( q 2 0 ) 3 ] -E 2 0 3 ] ] + 9 2 q -2 0 ln( q 2 0 )+ 3 2 ln 2 ( q 2 0 )+ ln 3 ( q 2 0 ) 6 + 9 2 E 0 q -4 0 [ 1 -ln( q 2 0 ) 3 -7 6 E 0 [ 1+ 2 21 ln( q 2 0 ) ] + 2 9 E 2 0 [ 1+2ln( q 2 0 ) ] -E 3 0 9 ] + O ( glyph[epsilon1] i 0 ) } + O ( λ 2 ) . (153)</formula>
<text><location><page_31><loc_12><loc_25><loc_88><loc_35></location>Adding up the tree-order coincident correlator at O ( λ 0 ) and O ( λ ), and the one-loop coincident correlator at O ( λ ) we obtain the coincident correlator up to O ( λ 2 ). The result is given in the next section and is used to compute the quantum corrected power spectrum up to O ( λ 2 ) in Sec. VI.</text>
<section_header_level_1><location><page_31><loc_14><loc_20><loc_86><loc_21></location>V. COINCIDENT CORRELATION FUNCTION OF THE FLUCTUATIONS</section_header_level_1>
<text><location><page_31><loc_12><loc_13><loc_88><loc_17></location>Coincident correlation function of the inflaton fluctuations is obtained by adding up Eqs. (135), (138) and (153) as</text>
<formula><location><page_31><loc_23><loc_7><loc_79><loc_11></location>〈 Ω | δ ¯ ϕ 2 ( t, glyph[vector]x ) | Ω 〉 stoch = H 2 i 0 4 π 2 q 4 0 [ q 2 0 + glyph[epsilon1] i 0 ] 2 { [ 1 -E 0 ] { ln( a 0 ) -ln ( q 2 0 + glyph[epsilon1] i 0 q 3 0 (1+ glyph[epsilon1] i 0 ) ) }</formula>
<formula><location><page_32><loc_12><loc_47><loc_88><loc_91></location>∓ λ 144 π 2 { π GH 2 i 0 { glyph[epsilon1] -3 i 0 [ 1 -q 0 -7 2 E 0 [ 1 -5 7 q 0 ] + 37 8 E 2 0 [ 1 -4 37 q 0 ] -9 4 E 3 0 ] -glyph[epsilon1] -2 i 0 q -2 0 [ 1 -q 0 + 15 2 [ 1 -2 15 q 0 ] × ln( q 2 0 )+ 9 2 E 0 [ 1 -4 9 q 0 -37 9 [ 1 -2 37 q 0 ] ln( q 2 0 ) ] -79 8 E 2 0 [ 1 -12 79 q 0 -117 79 ln( q 2 0 ) ] + 37 8 E 3 0 [ 1 -29 37 ln( q 2 0 ) ] ] + glyph[epsilon1] -1 i 0 q -4 0 3 [ 1 -q 0 -5 3 [ 1+ 3 5 q 0 ] ln( q 2 0 )+ ln 2 ( q 2 0 ) 2 -17 3 E 0 [ 1 -19 34 q 0 -[ 1+ 9 34 q 0 ] ln( q 2 0 )+ 9 34 ln 2 ( q 2 0 ) ] + 121 12 E 2 0 [ 1 -20 121 q 0 -75 121 [ 1+ 2 25 q 0 ] ln( q 2 0 )+ 18 121 ln 2 ( q 2 0 ) ] -95 12 E 3 0 [ 1 -27 95 ln( q 2 0 )+ 6 95 ln 2 ( q 2 0 ) ] + 29 12 E 4 0 ] + q -6 0 3 [ 1 -q 0 -5 3 [ 1+ 3 5 q 0 ] ln( q 2 0 ) -17 3 E 0 [ 1 -7 34 q 0 -23 17 [ 1+ 9 46 q 0 ] ln( q 2 0 ) ] + 155 12 E 2 0 [ 1+ 12 155 q 0 -147 155 × [ 1+ 2 49 q 0 ] ln( q 2 0 ) ] -175 12 E 3 0 [ 1+ 8 175 q 0 -99 175 ln( q 2 0 ) ] + 179 24 E 4 0 [ 1 -48 179 ln( q 2 0 ) ] -29 24 E 5 0 ] + O ( glyph[epsilon1] i 0 ) } + [ 1 -E 0 ] [ glyph[epsilon1] -3 i 0 E 3 0 2 + glyph[epsilon1] -2 i 0 3 2 E 2 0 ln( q 2 0 ) ] + glyph[epsilon1] -1 i 0 9 [ ln( q 2 0 )+ E 0 [ 1 -ln( q 2 0 )+ ln 2 ( q 2 0 ) 6 ] -E 2 0 2 [ 1+ ln 2 ( q 2 0 ) 3 ] -E 3 0 3 ] + 27 2 ln( q 2 0 )+ 9 2 [ 1 -E 0 ] [ ln 2 ( q 2 0 )+ ln 3 ( q 2 0 ) 9 ] + 27 2 E 0 q -2 0 [ 1 -ln( q 2 0 ) 3 -7 6 E 0 [ 1+ 2 21 ln( q 2 0 ) ] + 2 9 E 2 0 [ 1+2ln( q 2 0 ) ] -E 3 0 9 ] + O ( glyph[epsilon1] i 0 ) } + O ( λ 2 ) . (154)</formula>
<text><location><page_32><loc_12><loc_33><loc_88><loc_45></location>Correlation function and the power spectrum are related by the Wiener-Khinchin theorem which states that the spatial correlation between simultaneous values of the field measured at two spatial points and power spectrum form a Fourier transform pair of each other. In the next section, we use coincident correlator (154) to compute the quantum corrected power spectrum of inflaton fluctuations up to O ( λ 2 ).</text>
<section_header_level_1><location><page_32><loc_24><loc_28><loc_75><loc_29></location>VI. POWER SPECTRUM OF THE FLUCTUATIONS</section_header_level_1>
<text><location><page_32><loc_12><loc_21><loc_88><loc_25></location>The spatial two-point correlation function for an untruncated free fluctuation field is obtained using free field mode expansion (69) as</text>
<formula><location><page_32><loc_27><loc_16><loc_88><loc_19></location>〈 Ω | δϕ 0 ( t, glyph[vector]x ) δϕ 0 ( t, glyph[vector]x ' ) | Ω 〉 = ∫ d D -1 k (2 π ) D -1 | u 0 ( t, k ) | 2 e i glyph[vector] k · ( glyph[vector]x -glyph[vector]x ' ) . (155)</formula>
<text><location><page_32><loc_12><loc_10><loc_88><loc_14></location>The norm square of amplitude function | u 0 ( t, k ) | 2 is a measure of power for each mode k of the fluctuation field at time t . It is, however, convenient to evaluate the integral in Eq. (155)</text>
<text><location><page_33><loc_12><loc_89><loc_48><loc_91></location>and define a measure of power ∆ 2 δϕ 0 ( t, k ) as</text>
<formula><location><page_33><loc_28><loc_84><loc_88><loc_88></location>〈 Ω | δϕ 0 ( t, glyph[vector]x ) δϕ 0 ( t, glyph[vector]x ' ) | Ω 〉 = ∫ ∞ 0 dk k sin( k ∆ x ) k ∆ x ∆ 2 δϕ 0 ( t, k ) , (156)</formula>
<text><location><page_33><loc_12><loc_81><loc_56><loc_82></location>where the time dependent tree-order power spectrum</text>
<formula><location><page_33><loc_36><loc_75><loc_88><loc_79></location>∆ 2 δϕ 0 ( t, k )= k D -1 π D 2 Γ ( D 2 ) Γ( D -1) | u 0 ( t, k ) | 2 . (157)</formula>
<text><location><page_33><loc_12><loc_60><loc_88><loc_74></location>The quantum corrected fluctuation field δϕ ( t, glyph[vector]x ) [Eq. (97)] is a full field hence, unlike the free field, it does not have a mode expansion. The quantum corrected power spectrum ∆ 2 δϕ ( t, k ) can be defined [35] in two different ways: (i) using the quantum corrected two point correlation function via Eq. (156) or (ii) using the quantum corrected mode function in Eq. (157). The two definitions yield slightly different results. We use definition (i) to compute quantum corrected power spectrum in this paper.</text>
<text><location><page_33><loc_14><loc_57><loc_49><loc_58></location>In the coincidence limit, Eq. (156) implies</text>
<formula><location><page_33><loc_36><loc_52><loc_88><loc_56></location>〈 Ω | δϕ 2 ( t, glyph[vector]x ) | Ω 〉≡ ∫ ∞ 0 dk k ∆ 2 δϕ ( t, k ) . (158)</formula>
<text><location><page_33><loc_12><loc_46><loc_88><loc_50></location>The stochastic contribution to coincident correlator, on the other hand, ought to involve the usual [33] time-independent power spectrum</text>
<formula><location><page_33><loc_40><loc_41><loc_88><loc_44></location>∆ 2 δϕ ( k ) ≡ lim t glyph[greatermuch] t k ∆ 2 δϕ ( t, k ) , (159)</formula>
<text><location><page_33><loc_12><loc_35><loc_88><loc_39></location>where t k is the time of first horizon crossing defined by Eq. (87). Thus, for the IR truncated full fluctuation field</text>
<formula><location><page_33><loc_25><loc_29><loc_88><loc_34></location>〈 Ω | δ ¯ ϕ 2 ( t, glyph[vector]x ) | Ω 〉 stoch ≡ ∫ ∞ 0 dk k Θ ( Ha 1+ glyph[epsilon1] -k ) ∆ 2 δ ¯ ϕ ( k )= ∫ k = Ha 1+ glyph[epsilon1] k i = H i 1+ glyph[epsilon1] i dk k ∆ 2 δ ¯ ϕ ( k ) . (160)</formula>
<text><location><page_33><loc_12><loc_27><loc_51><loc_28></location>Taking the time derivative of both sides yields</text>
<formula><location><page_33><loc_34><loc_21><loc_88><loc_25></location>d dt 〈 Ω | δ ¯ ϕ 2 ( t, glyph[vector]x ) | Ω 〉 stoch = ˙ k k ∆ 2 δ ¯ ϕ ( k ) ∣ ∣ ∣ k = Ha 1+ glyph[epsilon1] , (161)</formula>
<text><location><page_33><loc_12><loc_18><loc_17><loc_20></location>where</text>
<formula><location><page_33><loc_39><loc_14><loc_88><loc_19></location>˙ k k ∣ ∣ ∣ k = Ha 1+ glyph[epsilon1] = [ H [1 -glyph[epsilon1] ] -˙ glyph[epsilon1] 1+ glyph[epsilon1] ] . (162)</formula>
<text><location><page_33><loc_12><loc_12><loc_69><loc_14></location>Thus, the power for the horizon sized mode k at time t (i.e. t = t k ) is</text>
<formula><location><page_33><loc_29><loc_7><loc_88><loc_11></location>∆ 2 δ ¯ ϕ ( k = Ha 1+ glyph[epsilon1] ) = [ H [1 -glyph[epsilon1] ] -˙ glyph[epsilon1] 1+ glyph[epsilon1] ] -1 d dt 〈 Ω | δ ¯ ϕ 2 ( t, glyph[vector]x ) | Ω 〉 stoch . (163)</formula>
<text><location><page_34><loc_12><loc_24><loc_15><loc_25></location>and</text>
<formula><location><page_34><loc_39><loc_20><loc_88><loc_24></location>a 0 ( t k )= k H 0 ( t k ) [1+ glyph[epsilon1] 0 ( t k )] , (167)</formula>
<text><location><page_34><loc_12><loc_18><loc_42><loc_19></location>one obtains the power for any mode</text>
<formula><location><page_34><loc_42><loc_12><loc_88><loc_16></location>k H i 0 = a 0 ( t k ) q 3 0 ( t k ) q 2 0 ( t k )+ glyph[epsilon1] i 0 , (168)</formula>
<text><location><page_34><loc_12><loc_7><loc_88><loc_11></location>which is a monotonically increasing function of a 0 . Note that Eqs. (167) and (168) reduce [2] to k H i 0 = a 0 ( t k ) in de Sitter limit.</text>
<text><location><page_34><loc_12><loc_81><loc_88><loc_91></location>One can, however, look at the power at any time t and that time corresponds to a mode k through the relation a = k 1+ glyph[epsilon1] H . So by considering Eq. (163) at different times one can access the power for different k values. Employing Eq. (63), converting the derivative with respect to q to the derivative with respect to q 0 via the chain rule</text>
<formula><location><page_34><loc_33><loc_76><loc_88><loc_80></location>d dq =[ dq dq 0 ] -1 d dq 0 =[ dq dq 0 ] -1 [ ∂ ∂q 0 -2 q 0 ∂ ∂ E 0 ] , (164)</formula>
<text><location><page_34><loc_12><loc_73><loc_62><loc_75></location>and using Eq. (154) in Eq. (163), in D =4 dimensions, yield</text>
<formula><location><page_34><loc_11><loc_34><loc_88><loc_72></location>∆ 2 δ ¯ ϕ ( k = H ( t ) a ( t ) 1+ glyph[epsilon1] ( t ) ) = -[ 1 -glyph[epsilon1] ( t ) -˙ glyph[epsilon1] ( t ) H ( t ) [1+ glyph[epsilon1] ( t )] ] -1 glyph[epsilon1] i 0 q d dq 〈 Ω | δ ¯ ϕ 2 ( t, glyph[vector]x ) | Ω 〉 stoch = H 2 i 0 4 π 2 { q 6 0 [ q 2 0 + glyph[epsilon1] i 0 ] 2 [ 1+ q 2 0 +3 glyph[epsilon1] i 0 q 4 0 -3 glyph[epsilon1] 2 i 0 ln ( q 2 0 + glyph[epsilon1] i 0 a 0 q 3 0 (1+ glyph[epsilon1] i 0 ) ) 2 glyph[epsilon1] i 0 ] ∓ λ 144 π 2 { π GH 2 i 0 [ glyph[epsilon1] -2 i 0 q -1 0 4 [ 1 -5 4 q 0 -E 0 ( 1 -7 2 q 0 ) -E 2 0 8 ( 1+25 q 0 ) ] + glyph[epsilon1] -1 i 0 q -4 0 4 [ 1+ q 0 2 +5 ( 1+ q 0 20 ) ln( q 2 0 ) -E 0 ( 1+ q 0 + 27 2 [ 1+ 5 108 q 0 ] ln( q 2 0 ) ) -9 4 E 2 0 ( 1 -5 18 q 0 -16 3 [ 1+ q 0 32 ] ln( q 2 0 ) ) + 35 16 E 3 0 ( 1 -8 5 ln( q 2 0 ) ) ] -q -6 0 5 [ 1 -4 q 0 -3 5 ( 1 -2 3 q 0 ) ln( q 2 0 )+ 3 5 ln 2 ( q 2 0 )+ 2 5 E 0 ( 1+16 q 0 + 13 2 [ 1 -7 26 q 0 ] ln( q 2 0 ) -9 2 ln 2 ( q 2 0 ) ) + 21 10 E 2 0 ( 1 -17 21 q 0 + 23 14 [ 1 -2 23 q 0 ] ln( q 2 0 )+ 6 7 ln 2 ( q 2 0 ) ) -91 10 E 3 0 ( 1+ 6 91 q 0 -29 182 ln( q 2 0 )+ 6 91 ln 2 ( q 2 0 ) ) + 28 5 E 4 0 ] + O ( glyph[epsilon1] i 0 ) ] + glyph[epsilon1] -2 i 0 3 [ E 2 0 -4 3 E 3 0 ] + glyph[epsilon1] -1 i 0 6 q -4 0 [ E 0 ln( q 2 0 ) -E 2 0 ( 1+ 7 2 ln( q 2 0 ) ) + 11 6 E 3 0 ( 1+ 24 11 ln( q 2 0 ) ) -5 6 E 4 0 ( 1+ 9 5 ln( q 2 0 ) ) ] -18 q -6 0 [ ln( q 2 0 ) -ln 2 ( q 2 0 ) 6 + E 0 ( 1 -7 3 ln( q 2 0 )+ 5 6 ln 2 ( q 2 0 ) ) -17 6 E 2 0 ( 1 -7 17 ln( q 2 0 )+ 9 17 ln 2 ( q 2 0 ) ) + 5 3 E 3 0 ( 1+ 2 5 ln( q 2 0 )+ 7 10 ln 2 ( q 2 0 ) ) + 7 6 E 4 0 ( 1 -3 7 ln( q 2 0 ) -2 7 ln 2 ( q 2 0 ) ) -E 5 0 ] + O ( glyph[epsilon1] i 0 ) } . (165)</formula>
<text><location><page_34><loc_12><loc_30><loc_30><loc_31></location>Through the relations</text>
<formula><location><page_34><loc_34><loc_27><loc_88><loc_29></location>q 2 0 ( t k )=1 -E 0 ( t k )=1 -2 glyph[epsilon1] i 0 ln ( a 0 ( t k ) ) , (166)</formula>
<text><location><page_35><loc_14><loc_89><loc_37><loc_91></location>In the noninteracting limit,</text>
<formula><location><page_35><loc_41><loc_86><loc_88><loc_89></location>∆ 2 δ ¯ ϕ 0 ( k )= H 2 i 0 4 π 2 P 0 ( k ) , (169)</formula>
<text><location><page_35><loc_12><loc_80><loc_88><loc_85></location>where the dimensionless function P 0 ( k ), whose exact form can be read off from Eq. (165), is expanded in glyph[epsilon1] i 0 as</text>
<formula><location><page_35><loc_30><loc_76><loc_88><loc_79></location>P 0 ( k )=2 q 2 0 -1 -glyph[epsilon1] i 0 [ 1+ q -2 0 +ln( q 2 0 ) ] + O ( glyph[epsilon1] 2 i 0 ) . (170)</formula>
<text><location><page_35><loc_12><loc_64><loc_88><loc_74></location>The P 0 ( k ) is positive definite during inflation and decreases monotonically as a 0 , i.e., k/H i 0 increases. Thus, as the comoving wave number k decreases, i.e., as the scale increases, the power ∆ 2 δ ¯ ϕ 0 increases. The amplitude of fluctuations grow toward the larger scales and the spectrum is said to be red-tilted. The O ( λ ) correction to the power spectrum</text>
<formula><location><page_35><loc_38><loc_59><loc_60><loc_63></location>∆ 2 δ ¯ ϕ λ ( k )= ∓ λ H 2 i 0 576 π 4 P λ ( k ) ,</formula>
<text><location><page_35><loc_12><loc_48><loc_88><loc_57></location>where the dimensionless function P λ ( k ), which can be read off from Eq. (165), is also positive definite and decreases monotonically as a 0 or k/H i 0 increases. As the scale increases, so does the P λ . Thus, for the + ( -) sign choice in potential (3), the O ( λ ) correction ∆ 2 δ ¯ ϕ λ ( k ) reduces (increases) the power ∆ 2 δ ¯ ϕ 0 ( k ) in the noninteracting limit.</text>
<text><location><page_35><loc_12><loc_43><loc_88><loc_47></location>The tilt is quantified by the spectral index. Taylor expanding the ln(∆ 2 ( k )) around the ln( k P ), k P being a pivotal wavenumber, yields</text>
<formula><location><page_35><loc_23><loc_38><loc_88><loc_41></location>ln(∆ 2 ( k ))=ln(∆ 2 ( k p ))+ n ( k P ) ln ( k k P ) + 1 2! α ( k P ) ln 2 ( k k P ) + · · · , (171)</formula>
<text><location><page_35><loc_12><loc_35><loc_39><loc_36></location>where the expansion coefficients,</text>
<formula><location><page_35><loc_39><loc_29><loc_88><loc_33></location>n ( k P ) ≡ d ln(∆ 2 ( k )) d ln( k ) ∣ ∣ ∣ k P , (172)</formula>
<text><location><page_35><loc_12><loc_26><loc_15><loc_28></location>and</text>
<formula><location><page_35><loc_35><loc_23><loc_88><loc_26></location>α ( k P ) ≡ d 2 ln(∆ 2 ( k )) d (ln( k )) 2 ∣ ∣ ∣ k P = dn ( k ) d ln( k ) ∣ ∣ ∣ k P . (173)</formula>
<text><location><page_35><loc_12><loc_12><loc_88><loc_22></location>Conventionally, the scale independent Harrison-Zeldovich spectrum is defined, for inflaton fluctuations, so that the corresponding expansion coefficient n δ ¯ ϕ , the spectral index, equals 1 rather then more logical 0, as in the case of graviton fluctuations. Therefore, the tilt for the inflaton fluctuations, is defined as deviation of n δ ¯ ϕ from 1,</text>
<formula><location><page_35><loc_34><loc_7><loc_88><loc_11></location>n δ ¯ ϕ ( k = Ha 1+ glyph[epsilon1] ) -1 ≡ d ln ( ∆ 2 δ ¯ ϕ ( k = Ha 1+ glyph[epsilon1] )) d ln( k ) , (174)</formula>
<text><location><page_36><loc_12><loc_87><loc_88><loc_91></location>which measures the variation of the power spectrum with scale. The logarithmic derivative can be written as</text>
<formula><location><page_36><loc_29><loc_82><loc_88><loc_86></location>d d ln( k ) = k d dk = k dt dk d dt = -[ 1 -glyph[epsilon1] -˙ glyph[epsilon1] H [1+ glyph[epsilon1] ] ] -1 glyph[epsilon1] i 0 q d dq . (175)</formula>
<text><location><page_36><loc_12><loc_79><loc_70><loc_80></location>where we used Eqs. (63) and (162) in the last equality. Hence the tilt</text>
<formula><location><page_36><loc_11><loc_46><loc_88><loc_78></location>n δ ¯ ϕ ( k = H ( t ) a ( t ) 1+ glyph[epsilon1] ( t ) ) -1= -[ 1 -glyph[epsilon1] ( t ) -˙ glyph[epsilon1] ( t ) H ( t ) [1+ glyph[epsilon1] ( t )] ] -1 glyph[epsilon1] i 0 q d dq ln ( ∆ 2 δ ¯ ϕ ( k = H ( t ) a ( t ) 1+ glyph[epsilon1] ( t ) ) ) = -4 glyph[epsilon1] i 0 N { q 10 0 +3 glyph[epsilon1] i 0 q 6 0 [ 1 -5 6 E 0 ] -6 glyph[epsilon1] 2 i 0 q 4 0 [ 1 -5 4 E 0 ] -18 glyph[epsilon1] 3 i 0 q 2 0 [ 1 -19 12 E 0 ] +9 glyph[epsilon1] 4 i 0 [ 1+ E 0 2 ] +27 glyph[epsilon1] 5 i 0 -M } ∓ λ 36 π 2 q -3 0 [1 -2 E 0 ] 2 { π GH 2 i 0 5 [ glyph[epsilon1] -1 i 0 [ 1 -13 5 E 0 ( 1+ 8 13 q 0 ) + 91 40 E 2 0 ( 1+ 12 7 q 0 ) -13 20 E 3 0 ( 1+ 46 13 q 0 ) ] -3 q -3 0 1 -2 E 0 × [ 1 -9 5 q 0 -8 15 ( 1+ 19 16 q 0 ) ln( q 2 0 ) -25 6 E 0 [ 1 -363 250 q 0 -88 125 ( 1+ 261 352 q 0 ) ln( q 2 0 ) ] + 49 8 E 2 0 [ 1 -296 245 q 0 -664 735 × ( 1+ 41 83 q 0 ) ln( q 2 0 ) ] -119 40 E 3 0 [ 1 -67 51 q 0 -520 357 ( 1+ 177 520 q 0 ) ln( q 2 0 ) ] -2 3 E 4 0 [ 1+ 49 40 q 0 + 17 10 ( 1+ q 0 4 ) ln( q 2 0 )+ 7 10 E 5 0 × ( 1 -2 21 ln( q 2 0 ) )] ] + O ( glyph[epsilon1] i 0 ) ] + glyph[epsilon1] -1 i 0 3 q 0 [ E 0 -4 E 2 0 + 17 3 E 3 0 -8 3 E 4 0 ] + 3 q -3 0 1 -2 E 0 [ [ 1 -7 E 0 ] ln( q 2 0 )+3 E 2 0 [ 1+ 22 3 ln( q 2 0 ) ] -32 3 E 3 0 [ 1+ 29 8 ln( q 2 0 ) ] + 41 3 E 4 0 [ 1+ 117 41 ln( q 2 0 ) ] -22 3 E 5 0 [ 1+ 63 22 ln( q 2 0 ) ] + 4 3 E 6 0 [ 1+ 7 2 ln( q 2 0 ) ] ] + O ( glyph[epsilon1] i 0 ) } , (176)</formula>
<text><location><page_36><loc_12><loc_43><loc_17><loc_44></location>where</text>
<formula><location><page_36><loc_27><loc_38><loc_88><loc_42></location>N ( q 0 , glyph[epsilon1] i 0 )= [ q 4 0 -3 glyph[epsilon1] 2 i 0 ] 3 { 1+ q 2 0 +3 glyph[epsilon1] i 0 q 4 0 -3 glyph[epsilon1] 2 i 0 ln ( q 2 0 + glyph[epsilon1] i 0 a 0 q 3 0 (1+ glyph[epsilon1] i 0 ) ) 2 glyph[epsilon1] i 0 } , (177)</formula>
<formula><location><page_36><loc_27><loc_33><loc_88><loc_37></location>M ( q 0 , glyph[epsilon1] i 0 )= glyph[epsilon1] 2 i 0 [ q 6 0 +3 glyph[epsilon1] i 0 q 4 0 +21 glyph[epsilon1] 2 i 0 q 2 0 +27 glyph[epsilon1] 3 i 0 ] ln ( q 2 0 + glyph[epsilon1] i 0 q 3 0 (1+ glyph[epsilon1] i 0 ) ) . (178)</formula>
<text><location><page_36><loc_12><loc_30><loc_84><loc_32></location>Recall that q 0 ( t k ) and k are related via Eq. (168). In the noninteracting limit, the tilt</text>
<formula><location><page_36><loc_34><loc_25><loc_88><loc_29></location>n δ ¯ ϕ 0 ( k ) -1 ≡ N 0 ( k )= -4 glyph[epsilon1] i 0 1 -2 E 0 + O ( glyph[epsilon1] 2 i 0 ) , (179)</formula>
<text><location><page_36><loc_12><loc_23><loc_77><loc_24></location>whose exact form is given in Eq. (176), is nonsingular for E 0 < 1 / 2 which yields</text>
<formula><location><page_36><loc_44><loc_18><loc_88><loc_21></location>glyph[epsilon1] i 0 < 1 4 ln( a 0 ) . (180)</formula>
<text><location><page_36><loc_12><loc_7><loc_88><loc_16></location>During an inflation which lasts 60 e-foldings, for example, Eq. (180) implies glyph[epsilon1] i 0 < 0 . 0041. The dimensionless function N 0 ( k ), for an glyph[epsilon1] i 0 satisfying Eq. (180), is negative definite. Thus, the tilt is red. The N 0 ( k ) decreases monotonically (becomes more negative) as a 0 , i.e., k/H i 0 , increases.</text>
<text><location><page_37><loc_14><loc_89><loc_49><loc_91></location>The O ( λ ) correction to the spectral index</text>
<formula><location><page_37><loc_40><loc_85><loc_58><loc_88></location>n ¯ ϕ λ ( k ) ≡∓ λ 36 π 2 N λ ( k ) ,</formula>
<text><location><page_37><loc_12><loc_74><loc_88><loc_83></location>where the dimensionless function N λ ( k ), which can be read off from Eq. (176), is nonsingular for E 0 < 1 / 2. Thus, the dimensionless function N λ ( k ), for an glyph[epsilon1] i 0 satisfying Eq. (180), is positive definite and increases monotonically as a 0 , hence k/H i 0 , increases. Thus, for the + ( -) sign choice in potential (3), the O ( λ ) correction n ¯ ϕ λ ( k ) enhances (reduces) the red tilt.</text>
<text><location><page_37><loc_12><loc_63><loc_88><loc_73></location>The measured tilt [36] in primordial power spectrum of the inflaton fluctuations implied by the Planck TT+lowP+BAO data, is -0 . 032 ± 0 . 0045 at 68% confidence level. For a 0 ( t k ) = e 50 , choosing glyph[epsilon1] i 0 =0 . 00305, which corresponds to the physical wave number k phys = k/a 0 ( t k )=0 . 83 H i 0 [Eq. (168)], for example, the tilt we get from Eq. (176) is</text>
<formula><location><page_37><loc_32><loc_58><loc_67><loc_62></location>n δ ¯ ϕ -1= -0 . 032 ∓ λ ( 12 . 996 π GH 2 i 0 +1 . 866 ) .</formula>
<text><location><page_37><loc_12><loc_51><loc_88><loc_57></location>The measured tilt [36] implied by the Planck TT,TE,EE+lowP data, on the other hand, is -0 . 0348 ± 0 . 0047 at 68% confidence level. For a 0 ( t k ) = e 50 and glyph[epsilon1] i 0 =0 . 00315, the tilt that Eq. (176) yields is</text>
<formula><location><page_37><loc_32><loc_47><loc_68><loc_51></location>n δ ¯ ϕ -1= -0 . 0348 ∓ λ ( 13 . 657 π GH 2 i 0 +1 . 993 ) ,</formula>
<text><location><page_37><loc_12><loc_45><loc_84><loc_46></location>in agreement with observation within the range provided by the Planck Collaboration.</text>
<text><location><page_37><loc_12><loc_40><loc_88><loc_44></location>Another physical quantity that can be computed and compared with measurements is the running [Eq. (173)] of the spectral index,</text>
<formula><location><page_37><loc_37><loc_35><loc_88><loc_38></location>α δ ¯ ϕ ( k = Ha 1+ glyph[epsilon1] ) ≡ dn δ ¯ ϕ ( k = Ha 1+ glyph[epsilon1] ) d ln( k ) . (181)</formula>
<text><location><page_37><loc_12><loc_32><loc_82><loc_33></location>Employing Eq. (175) in Eq. (181) we find the running in our model, up to O ( λ 2 ), as</text>
<formula><location><page_37><loc_11><loc_7><loc_88><loc_31></location>α δ ¯ ϕ ( k = H ( t ) a ( t ) 1+ glyph[epsilon1] ( t ) ) = -[ 1 -glyph[epsilon1] ( t ) -˙ glyph[epsilon1] ( t ) H ( t ) [1+ glyph[epsilon1] ( t )] ] -1 glyph[epsilon1] i 0 q d dq n δ ¯ ϕ ( k = H ( t ) a ( t ) 1+ glyph[epsilon1] ( t ) ) = -4 glyph[epsilon1] 2 i 0 [ q 2 0 +3 glyph[epsilon1] i 0 ] 2 { 4 N 2 [ q 4 0 -3 glyph[epsilon1] 2 i 0 ] 2 [ q 8 0 + 13 2 glyph[epsilon1] i 0 q 6 0 + 15 2 glyph[epsilon1] 2 i 0 q 4 0 -15 2 glyph[epsilon1] 3 i 0 q 2 0 -27 2 glyph[epsilon1] 4 i 0 ] 2 + glyph[epsilon1] i 0 q 2 0 [ q 2 0 + glyph[epsilon1] i 0 ] q 4 0 -3 glyph[epsilon1] 2 i 0 × [ 1 N [ q 10 0 +3 glyph[epsilon1] i 0 q 8 0 +66 glyph[epsilon1] 2 i 0 q 6 0 +198 glyph[epsilon1] 3 i 0 q 4 0 +153 glyph[epsilon1] 4 i 0 q 2 0 +27 glyph[epsilon1] 5 i 0 ] -2 [ q 4 0 -3 glyph[epsilon1] 2 i 0 ] 3 [ q 10 0 +6 glyph[epsilon1] i 0 q 8 0 +54 glyph[epsilon1] 2 i 0 q 6 0 +180 glyph[epsilon1] 3 i 0 q 4 0 +189 glyph[epsilon1] 4 i 0 q 2 0 +54 glyph[epsilon1] 5 i 0 ] ]} ∓ λ 6 π 2 1 [1 -2 E 0 ] 3 { π GH 2 i 0 q -5 0 37 6 [ 1 -24 37 q 0 -255 74 E 0 [ 1 -32 51 q 0 ] + 1305 296 E 2 0 [ 1 -736 1305 q 0 ] -183 74 E 3 0 [ 1 -80 183 q 0 ] + E 4 0 2 [ 1 -8 37 q 0 ] + O ( glyph[epsilon1] i 0 ) ] +1 -4 E 0 +8 E 2 0 -16 3 E 3 0 + O ( glyph[epsilon1] i 0 ) } . (182)</formula>
<text><location><page_38><loc_12><loc_87><loc_88><loc_91></location>In the noninteracting limit, the running α δ ¯ ϕ 0 ( k ) ≡ A 0 ( k ), whose exact form is given in Eq. (182), can be expanded in powers of glyph[epsilon1] i 0 as</text>
<formula><location><page_38><loc_38><loc_81><loc_88><loc_85></location>A 0 ( k )= -16 glyph[epsilon1] 2 i 0 (2 q 2 0 -1) 2 + O ( glyph[epsilon1] 3 i 0 ) . (183)</formula>
<text><location><page_38><loc_12><loc_76><loc_88><loc_80></location>The dimensionless function A 0 ( k ) is negative definite and decreases monotonically as a 0 , i.e., k/H i 0 , increases. The O ( λ ) correction to the running</text>
<formula><location><page_38><loc_40><loc_71><loc_58><loc_74></location>α δ ¯ ϕ λ ( k ) ≡∓ λ 6 π 2 A λ ( k ) ,</formula>
<text><location><page_38><loc_12><loc_60><loc_88><loc_69></location>where the dimensionless function A λ ( k ), which can be read off from Eq. (176), is positive definite and increases monotonically as a 0 , hence k/H i 0 , increases. Thus, for the + ( -) sign choice in potential (3), the O ( λ ) correction α δ ¯ ϕ λ ( k ) enhances (reduces) the α δ ¯ ϕ 0 ( k ), the negative running in the noninteracting limit.</text>
<text><location><page_38><loc_12><loc_49><loc_88><loc_58></location>Observations [36] also constrain the running of the spectral index. The α δ ¯ ϕ implied by the Planck TT+lowP+BAO data, is -0 . 0125 ± 0 . 0091 at 68% confidence level. For a 0 ( t k ) = e 50 , choosing glyph[epsilon1] i 0 = 0 . 00305, as we did in the estimation of the tilt, the running we get from Eq. (182) is</text>
<formula><location><page_38><loc_34><loc_45><loc_88><loc_49></location>α δ ¯ ϕ = -0 . 001 ∓ λ ( 0 . 585 π GH 2 i 0 +0 . 106 ) . (184)</formula>
<text><location><page_38><loc_12><loc_38><loc_88><loc_44></location>The α δ ¯ ϕ implied by the Planck TT,TE,EE+lowP data, on the other hand, is -0 . 0085 ± 0 . 0076 at 68% confidence level. For a 0 ( t k ) = e 50 , choosing glyph[epsilon1] i 0 =0 . 00315, as we did in the estimation of the tilt, the running that Eq. (182) yields is</text>
<formula><location><page_38><loc_33><loc_32><loc_88><loc_36></location>α δ ¯ ϕ = -0 . 0012 ∓ λ ( 0 . 685 π GH 2 i 0 +0 . 122 ) , (185)</formula>
<text><location><page_38><loc_12><loc_29><loc_84><loc_31></location>in agreement with observation within the range provided by the Planck Collaboration.</text>
<section_header_level_1><location><page_38><loc_39><loc_24><loc_60><loc_25></location>VII. CONCLUSIONS</section_header_level_1>
<text><location><page_38><loc_12><loc_9><loc_88><loc_21></location>We extended our method [1, 2] to compute two-point correlation function of infrared truncated fields, the usual power spectrum ∆ 2 ( k ) ≡ lim t glyph[greatermuch] t k ∆ 2 ( t, k ), spectral index n ( k ) and running of the spectral index α ( k ) with quantum corrections to more general spacetimes where the expansion rate H is not constant. We applied the method to study the quantum fluctuations of a self-interacting inflaton in the simplest model of inflationary spacetime.</text>
<text><location><page_39><loc_12><loc_68><loc_88><loc_91></location>In Sec. II, we introduced the background metric of a spatially flat Friedman-RobinsonWalker spacetime and presented the Lagrangian of the model where a minimally coupled massive scalar with a quartic self-interaction ± λ 4! ϕ 4 , which we treat perturbatively, drives the inflation. We considered the inflaton field as a sum of the averaged background field ¯ ϕ ( x ), which we treat classically, plus the fluctuation field δϕ ( x ) which we treat quantum mechanically. We solved the Einstein's equations and the equations of motion for the inflaton to obtain the expansion rate H ( t ), scale factor a ( t ), slow-roll parameter glyph[epsilon1] ( t ) and the background field ¯ ϕ ( t ), with O ( λ ) corrections, in terms of the initial values of the expansion rate H i 0 and the slow-roll parameter glyph[epsilon1] i 0 .</text>
<text><location><page_39><loc_12><loc_60><loc_88><loc_67></location>Section III is devoted to the fluctuations of the inflaton field. The mode expansion for the fluctuation field δϕ 0 ( x ) of the free inflaton is given in Sec. III A. The fluctuation field δϕ ( x ) of the full inflaton is expressed in terms of the free inflaton fluctuations in Sec. III B.</text>
<text><location><page_39><loc_12><loc_29><loc_88><loc_59></location>Following the Starobinsky's approach, in Sec. IV, we computed the quantum corrected two point correlation function for the fluctuations of the infrared truncated inflaton field in leading order. The correlation function is evaluated at the full solution of the effective field equation. Therefore the tree-order and one-loop correlators respectively get O ( λ ) and O ( λ 2 ) corrections, due to the backreaction of the interactions on the spacetime geometry. We obtained the tree-order correlator with the O ( λ ) correction in Sec. IV A and the one-loop correlator at O ( λ ) in Sec. IV B. The tree-order correlator in the noninteracting limit grows in time and asymptotes to a constant at late times for a physical distance chosen as a constant fraction of the Hubble length. The growth is reduced as the glyph[epsilon1] i 0 = m 2 3 H 2 i 0 increases. For the + ( -) sign choice in the interaction potential, the O ( λ ) correction at tree-order enhances (reduces) the growth whereas the O ( λ ) correction at one-loop order reduces (enhances) the growth.</text>
<text><location><page_39><loc_12><loc_7><loc_88><loc_27></location>In Sec. V, we obtained the coincidence limit of the quantum corrected two-point correlation function. In Sec. VI, we introduced a method to compute the power spectrum ∆ 2 δ ¯ ϕ ( k ) [Eq. (165)] of the inflaton fluctuations as a time derivative of the coincident correlator. In the noninteracting limit, the power spectrum ∆ 2 δ ¯ ϕ 0 ( k ) is positive definite but decreases as k increases. The O ( λ ) correction to the power spectrum ∆ 2 δ ¯ ϕ λ ( k ), in the interacting theory, reduces (increases) the power for the + ( -) sign choice in the interaction potential. The power spectrum is red tilted. In the non-interacting limit, the tilt 1 -n δ ¯ ϕ 0 ( k ) is negative definite and decreases as k increases. The O ( λ ) correction to the tilt n δ ¯ ϕ λ ( k ), in the inter-</text>
<text><location><page_40><loc_12><loc_81><loc_88><loc_91></location>reduces) the red-tilt for the + ( -) sign choice in the interaction potential. The spectral index [Eq. (176)] and the running of the spectral index [Eq. (182)] are in accordance with the observations within reasonable ranges of values for the glyph[epsilon1] i 0 and the number of e-foldings.</text>
<section_header_level_1><location><page_40><loc_42><loc_76><loc_58><loc_77></location>Acknowledgments</section_header_level_1>
<text><location><page_40><loc_25><loc_71><loc_74><loc_73></location>We thank Richard P. Woodard for stimulating discussions.</text>
<section_header_level_1><location><page_40><loc_22><loc_65><loc_78><loc_66></location>Appendix A: The comoving time t in terms of q ( t ) , H i 0 and glyph[epsilon1] i 0</section_header_level_1>
<text><location><page_40><loc_12><loc_58><loc_88><loc_62></location>In this Appendix, we express the comoving time t in the interacting theory in terms of q ( t ) defined in Eq. (42) and the initial values H i 0 and glyph[epsilon1] i 0 .</text>
<text><location><page_40><loc_12><loc_52><loc_88><loc_56></location>We integrate Eq. (40), neglecting the term linear in glyph[epsilon1] i 0 among the O ( λ )-terms as a zeroth order approximation in our iteration, and obtain</text>
<formula><location><page_40><loc_41><loc_48><loc_88><loc_50></location>z 3 + Az 2 + Bz + C ∼ =0 , (A1)</formula>
<text><location><page_40><loc_12><loc_44><loc_25><loc_46></location>where we define</text>
<formula><location><page_40><loc_18><loc_39><loc_88><loc_43></location>z ≡ H i 0 t, glyph[rho1] ≡± λξ 12 glyph[epsilon1] -2 i 0 , A ≡-glyph[epsilon1] -1 i 0 2 glyph[rho1] [ 1+3 glyph[rho1] ] , B ≡ glyph[epsilon1] -2 i 0 glyph[rho1] [ 1+ glyph[rho1] 2 ] , C ≡-glyph[epsilon1] -3 i 0 2 glyph[rho1] [ 1 -q 2 ] . (A2)</formula>
<text><location><page_40><loc_12><loc_36><loc_15><loc_38></location>Let</text>
<formula><location><page_40><loc_22><loc_31><loc_88><loc_35></location>Q ≡ 3 B -A 2 9 , R ≡ 9 AB -27 C -2 A 3 54 , S ≡ 3 √ R + √ D, T ≡ 3 √ R -√ D, (A3)</formula>
<text><location><page_40><loc_12><loc_28><loc_74><loc_30></location>where the discriminant D ≡ Q 3 + R 2 . The roots of the cubic polynomial are</text>
<formula><location><page_40><loc_17><loc_23><loc_88><loc_28></location>z 1 = S + T -A 3 , z 2 = -S + T 2 -A 3 + i √ 3 2 ( S -T ) , z 3 = -S + T 2 -A 3 -i √ 3 2 ( S -T ) . (A4)</formula>
<text><location><page_40><loc_12><loc_15><loc_88><loc_22></location>If the discriminant D > 0, one root is real and two are complex conjugate of each other. If D =0, all roots are real and at least two are equal. If D < 0, as is the case for cubic polynomial (A1), all roots are real and unequal. The physically relevant root,</text>
<formula><location><page_40><loc_33><loc_10><loc_88><loc_13></location>z 2 = H i 0 t = glyph[epsilon1] -1 i 0 [ 1 -q ][ 1+ glyph[rho1] 2 [ 1 -2 q ] + O ( λ 2 ) ] , (A5)</formula>
<text><location><page_40><loc_12><loc_7><loc_48><loc_8></location>yields the comoving time given in Eq. (43).</text>
<section_header_level_1><location><page_41><loc_36><loc_89><loc_64><loc_91></location>Appendix B: Special functions</section_header_level_1>
<text><location><page_41><loc_14><loc_85><loc_80><loc_86></location>In this Appendix, we define various special functions we use in the manuscript.</text>
<section_header_level_1><location><page_41><loc_32><loc_79><loc_67><loc_81></location>1. Exponential integral function E β ( z )</section_header_level_1>
<text><location><page_41><loc_14><loc_75><loc_42><loc_76></location>The exponential integral function</text>
<formula><location><page_41><loc_40><loc_70><loc_88><loc_74></location>E β ( z ) ≡ ∫ ∞ 1 dt t -β e -tz , (B1)</formula>
<text><location><page_41><loc_12><loc_68><loc_37><loc_69></location>where β and z are c -numbers.</text>
<section_header_level_1><location><page_41><loc_32><loc_62><loc_67><loc_63></location>2. Incomplete gamma function Γ( β, z )</section_header_level_1>
<text><location><page_41><loc_14><loc_58><loc_41><loc_59></location>The incomplete gamma function</text>
<formula><location><page_41><loc_40><loc_53><loc_88><loc_57></location>Γ( β, z ) ≡ ∫ ∞ z dt t β -1 e -t , (B2)</formula>
<text><location><page_41><loc_12><loc_50><loc_38><loc_52></location>satisfies the recurrence relation</text>
<formula><location><page_41><loc_38><loc_47><loc_88><loc_48></location>Γ( β +1 , z )= β Γ( β, z )+ z β e -z . (B3)</formula>
<text><location><page_41><loc_12><loc_40><loc_88><loc_44></location>Incomplete gamma function can be expressed in terms of the ordinary gamma function plus an alternating power series as</text>
<formula><location><page_41><loc_37><loc_35><loc_88><loc_39></location>Γ( β, z )=Γ( β ) -∞ ∑ n =0 ( -1) n z β + n n !( β + n ) . (B4)</formula>
<text><location><page_41><loc_12><loc_32><loc_88><loc_34></location>The right side of Eq. (B4) is replaced by its limiting value if β is a negative integer or zero.</text>
<section_header_level_1><location><page_41><loc_35><loc_27><loc_65><loc_28></location>3. Cosine integral function ci ( z )</section_header_level_1>
<text><location><page_41><loc_14><loc_22><loc_38><loc_24></location>The cosine integral function</text>
<formula><location><page_41><loc_29><loc_17><loc_88><loc_21></location>ci( z ) ≡-∫ ∞ z dt cos( t ) t = γ +ln( z )+ ∫ z 0 dt cos( t ) -1 t , (B5)</formula>
<text><location><page_41><loc_12><loc_15><loc_86><loc_16></location>where γ ≈ 0 . 577 is the Euler-Mascheroni number. The following identity involving ci( z ),</text>
<formula><location><page_41><loc_32><loc_9><loc_88><loc_14></location>ci( z ) -sin( z ) z = γ -1+ln( z )+ ∞ ∑ n =1 ( -1) n z 2 n 2 n (2 n +1)! , (B6)</formula>
<text><location><page_41><loc_12><loc_7><loc_19><loc_8></location>is useful.</text>
<section_header_level_1><location><page_42><loc_23><loc_89><loc_76><loc_91></location>4. Generalized hypergeometric function 2 F 2 ( α 1 , α 2 ; β 1 , β 2 ; z )</section_header_level_1>
<text><location><page_42><loc_14><loc_85><loc_45><loc_86></location>Generalized hypergeometric function,</text>
<formula><location><page_42><loc_23><loc_79><loc_88><loc_84></location>2 F 2 ( α 1 , α 2 ; β 1 , β 2 ; z )= Γ( β 1 ) Γ( β 2 ) Γ( α 1 ) Γ( α 2 ) ∞ ∑ n =0 Γ( α 1 + n ) Γ( α 2 + n ) Γ( β 1 + n ) Γ( β 2 + n ) ( z n n ! ) , (B7)</formula>
<text><location><page_42><loc_12><loc_77><loc_40><loc_78></location>where α i , β j and z are c -numbers.</text>
<section_header_level_1><location><page_42><loc_27><loc_71><loc_73><loc_72></location>Appendix C: Tree-order correlator in D-dimensions</section_header_level_1>
<text><location><page_42><loc_12><loc_64><loc_88><loc_68></location>Three-order correlator in an arbitrary spacetime dimensions is given in Eqs. (105) and (107). Using Eq. (81), which implies ν =( D -1) / 2+ O ( glyph[epsilon1] ), in these equations we find</text>
<formula><location><page_42><loc_13><loc_57><loc_88><loc_63></location>〈 Ω | δ ¯ ϕ 0 ( t, glyph[vector]x ) δ ¯ ϕ 0 ( t ' , glyph[vector]x ' ) | Ω 〉glyph[similarequal] Γ( D -1) 2 D -1 π D 2 Γ( D 2 ) [ HH ' (1+ glyph[epsilon1] )(1+ glyph[epsilon1] ' ) ] D 2 -1 [ ci( α ' ) -sin( α ' ) α ' -ci( α i )+ sin( α i ) α i ] , (C1)</formula>
<text><location><page_42><loc_12><loc_54><loc_84><loc_56></location>Power series expansions of the functions inside the second square brackets in Eq. (C1),</text>
<formula><location><page_42><loc_15><loc_49><loc_88><loc_53></location>ci( α ' ) -sin( α ' ) α ' -ci( α i )+ sin( α i ) α i = ln ( H ' a ' H i ) -ln ( 1+ glyph[epsilon1] ' 1+ glyph[epsilon1] i ) + ∞ ∑ n =1 ( -1) n [ α ' 2 n -α 2 n i ] 2 n (2 n +1)! , (C2)</formula>
<text><location><page_42><loc_12><loc_44><loc_88><loc_47></location>provides a useful representation for the tree-order correlator. Using equal space and equal spacetime limits of Eq. (C2) in Eq. (C1), respectively, yields</text>
<formula><location><page_42><loc_15><loc_38><loc_88><loc_42></location>〈 Ω | δ ¯ ϕ 0 ( t, glyph[vector]x ) δ ¯ ϕ 0 ( t ' , glyph[vector]x ) | Ω 〉glyph[similarequal] Γ( D -1) 2 D -1 π D 2 Γ( D 2 ) [ HH ' (1+ glyph[epsilon1] )(1+ glyph[epsilon1] ' ) ] D 2 -1 [ ln ( H ' a ' H i ) -ln ( 1+ glyph[epsilon1] ' 1+ glyph[epsilon1] i )] , (C3)</formula>
<text><location><page_42><loc_12><loc_35><loc_15><loc_37></location>and</text>
<formula><location><page_42><loc_22><loc_30><loc_88><loc_34></location>〈 Ω | δ ¯ ϕ 2 0 ( t, glyph[vector]x ) | Ω 〉glyph[similarequal] Γ( D -1) 2 D -1 π D 2 Γ( D 2 ) ( H 1+ glyph[epsilon1] ) D -2 [ ln ( Ha H i ) -ln ( 1+ glyph[epsilon1] 1+ glyph[epsilon1] i )] . (C4)</formula>
<text><location><page_42><loc_12><loc_27><loc_67><loc_29></location>The D → 4 limits of Eqs. (C1), (C3) and (C4) are used in Sec. IV.</text>
<section_header_level_1><location><page_42><loc_28><loc_22><loc_72><loc_23></location>Appendix D: Integrals of the one-loop correlator</section_header_level_1>
<text><location><page_42><loc_12><loc_7><loc_88><loc_19></location>One-loop correlator is presented in Sec. IV B. Integrals of the one-loop correlator that involve VEVs that are quadratic in the background and fluctuation fields are computed in Sec. IV B 1. Integrals that involve VEVs that are quartic in the fluctuation field, on the other hand, are computed in Sec. IV B 2. In Secs. D 1 and D 2 of this appendix we, respectively, outline the details of those integrals.</text>
<section_header_level_1><location><page_43><loc_38><loc_89><loc_62><loc_91></location>1. Integrals of Sec. IV B 1</section_header_level_1>
<text><location><page_43><loc_14><loc_85><loc_62><loc_86></location>The first integral that we need to evaluate in Eq. (139) is</text>
<formula><location><page_43><loc_16><loc_75><loc_88><loc_84></location>H ' 0 1+ glyph[epsilon1] ' 0 ∫ t ' 0 0 d ˜ t 0 [ 1+˜ glyph[epsilon1] 0 ˜ H 0 ] 2 ¯ ϕ 2 0 ( ˜ t 0 ) 〈 Ω | δ ¯ ϕ 0 ( t 0 , glyph[vector]x ) δ ¯ ϕ 0 ( ˜ t 0 , glyph[vector]x ' ) | Ω 〉 0 = 1 4 π 2 H 0 1+ glyph[epsilon1] 0 H ' 0 1+ glyph[epsilon1] ' 0 ∫ t ' 0 0 d ˜ t 0 1+˜ glyph[epsilon1] 0 ˜ H 0 ¯ ϕ 2 0 ( ˜ t 0 ) [ ln ( ˜ H 0 ˜ a 0 H i 0 ) -ln ( 1+˜ glyph[epsilon1] 0 1+ glyph[epsilon1] i 0 ) + ∞ ∑ ( -1) n [ ˜ α 2 n 0 -α 2 n i 0 ] 2 n (2 n +1)! ] , (D1)</formula>
<formula><location><page_43><loc_63><loc_75><loc_66><loc_76></location>n =1</formula>
<text><location><page_43><loc_12><loc_69><loc_88><loc_73></location>where we used the fact that ¯ ϕ 0 ( t ) is just a c -number as far as the perturbations are concerned and Eq. (120). To evaluate the integral we make change of variable (43) which implies</text>
<formula><location><page_43><loc_43><loc_64><loc_88><loc_68></location>d ˜ t 0 = -glyph[epsilon1] -1 i 0 H i 0 d ˜ q 0 . (D2)</formula>
<text><location><page_43><loc_12><loc_61><loc_86><loc_63></location>The first integral in Eq. (D1) is evaluated employing Eqs. (68), (111), (113) and (D2) as</text>
<formula><location><page_43><loc_18><loc_51><loc_88><loc_60></location>∫ t ' 0 0 d ˜ t 0 1+˜ glyph[epsilon1] 0 ˜ H 0 ¯ ϕ 2 0 ( ˜ t 0 ) ln ( ˜ H 0 ˜ a 0 H i 0 ) = glyph[epsilon1] -2 i 0 4 πGH 2 i 0 ∫ 1 q ' 0 d ˜ q 0 ˜ q 2 0 + glyph[epsilon1] i 0 ˜ q 0 [ ln(˜ q 0 )+ glyph[epsilon1] -1 i 0 2 [ 1 -˜ q 2 0 ] ] = 1 8 πGH 2 i 0 { glyph[epsilon1] -3 i 0 E ' 2 0 4 -glyph[epsilon1] -2 i 0 [ [ 1+ q ' 2 0 ] ln( q ' 0 )+ E ' 0 ] -glyph[epsilon1] -1 i 0 ln 2 ( q ' 0 ) } . (D3)</formula>
<text><location><page_43><loc_12><loc_48><loc_88><loc_49></location>The second integral in Eq. (D1) is evaluated employing Eqs. (68), (111), (115) and (D2) as</text>
<formula><location><page_43><loc_21><loc_38><loc_88><loc_47></location>∫ t ' 0 0 d ˜ t 0 1+˜ glyph[epsilon1] 0 ˜ H 0 ¯ ϕ 2 0 ( ˜ t 0 ) ln ( 1+˜ glyph[epsilon1] 0 1+ glyph[epsilon1] i 0 ) = glyph[epsilon1] -2 i 0 4 πGH 2 i 0 ∫ 1 q ' 0 d ˜ q 0 ˜ q 2 0 + glyph[epsilon1] i 0 ˜ q 0 ln ( ˜ q 2 0 + glyph[epsilon1] i 0 ˜ q 2 0 (1+ glyph[epsilon1] i 0 ) ) = -1 8 πGH 2 i 0 { glyph[epsilon1] -1 i 0 [ ln( q ' 2 0 )+ E ' 0 ] -ln( q ' 2 0 ) -E ' 0 2 + 1 -q '-2 0 2 + O ( glyph[epsilon1] i 0 ) } . (D4)</formula>
<text><location><page_43><loc_12><loc_35><loc_39><loc_36></location>The third integral in Eq. (D1) is</text>
<formula><location><page_43><loc_17><loc_29><loc_88><loc_33></location>∫ t ' 0 0 d ˜ t 0 1+˜ glyph[epsilon1] 0 ˜ H 0 ¯ ϕ 2 0 ( ˜ t 0 ) ∞ ∑ n =1 ( -1) n ˜ α 2 n 0 2 n (2 n +1)! = ∞ ∑ n =1 ( -1) n (∆ x ) 2 n 2 n (2 n +1)! ∫ t ' 0 0 d ˜ t 0 1+˜ glyph[epsilon1] 0 ˜ H 0 ¯ ϕ 2 0 ( ˜ t 0 ) [ ˜ H 0 ˜ a 0 1+˜ glyph[epsilon1] 0 ] 2 n . (D5)</formula>
<text><location><page_43><loc_12><loc_26><loc_84><loc_27></location>Employing Eqs. (68), (111), (117) and (D2) in the above integral brings it to the form</text>
<formula><location><page_43><loc_19><loc_20><loc_88><loc_25></location>∫ t ' 0 0 d ˜ t 0 1+˜ glyph[epsilon1] 0 ˜ H 0 ¯ ϕ 2 0 ( ˜ t 0 ) [ ˜ H 0 ˜ a 0 1+˜ glyph[epsilon1] 0 ] 2 n = glyph[epsilon1] -2 i 0 4 πGH 2 i 0 H 2 n i 0 e n glyph[epsilon1] i 0 ∫ 1 q ' 0 d ˜ q 0 ˜ q 2 0 [ ˜ q 3 0 ˜ q 2 0 + glyph[epsilon1] i 0 ] 2 n -1 e -n glyph[epsilon1] i 0 ˜ q 2 0 . (D6)</formula>
<text><location><page_43><loc_12><loc_18><loc_88><loc_19></location>Evaluating the final form of the integral in Eq. (D6) and using the result in Eq. (D5) yields</text>
<formula><location><page_43><loc_12><loc_7><loc_88><loc_16></location>∫ t ' 0 0 d ˜ t 0 1+˜ glyph[epsilon1] 0 ˜ H 0 ¯ ϕ 2 0 ( ˜ t 0 ) ∞ ∑ n =1 ( -1) n ˜ α 2 n 0 2 n (2 n +1)! = 1 8 πGH 2 i 0 ∞ ∑ n =1 ( -1) n ( H i 0 ∆ x ) 2 n 2 n (2 n +1)! e n glyph[epsilon1] i 0 { glyph[epsilon1] -2 i 0 [ q ' 2 n +2 0 E -n ( nq ' 2 0 glyph[epsilon1] i 0 ) -E -n ( n glyph[epsilon1] i 0 )] -(2 n -1) [ glyph[epsilon1] -1 i 0 [ q ' 2 n 0 E 1 -n ( nq ' 2 0 glyph[epsilon1] i 0 ) -E 1 -n ( n glyph[epsilon1] i 0 )] -n [ q ' 2 n -2 0 E 2 -n ( nq ' 2 0 glyph[epsilon1] i 0 ) -E 2 -n ( n glyph[epsilon1] i 0 )] ]} . (D7)</formula>
<text><location><page_44><loc_12><loc_89><loc_38><loc_91></location>The fourth integral in Eq. (D1)</text>
<formula><location><page_44><loc_22><loc_79><loc_88><loc_88></location>∫ t ' 0 0 d ˜ t 0 1+˜ glyph[epsilon1] 0 ˜ H 0 ¯ ϕ 2 0 ( ˜ t 0 ) ∞ ∑ n =1 ( -1) n α 2 n i 0 2 n (2 n +1)! = glyph[epsilon1] -2 i 0 4 πGH 2 i 0 ∞ ∑ n =1 ( -1) n α 2 n i 0 2 n (2 n +1)! ∫ 1 q ' 0 d ˜ q 0 ˜ q 2 0 + glyph[epsilon1] i 0 ˜ q 0 = 1 8 πGH 2 i 0 [ glyph[epsilon1] -2 i 0 E ' 0 -glyph[epsilon1] -1 i 0 ln( q ' 2 0 ) ][ 1 -γ +ci( α i 0 ) -sin( α i 0 ) α i 0 -ln( α i 0 ) ] . (D8)</formula>
<text><location><page_44><loc_12><loc_74><loc_88><loc_78></location>Thus, the first integral in Eq. (139) is obtained using Eqs. (D3), (D4), (D7) and (D8) in Eq. (D1). The result is given in Eq. (140).</text>
<text><location><page_44><loc_14><loc_72><loc_65><loc_73></location>The second integral that we need to compute in Eq. (139) is</text>
<formula><location><page_44><loc_28><loc_66><loc_88><loc_70></location>H 0 1+ glyph[epsilon1] 0 ∫ t 0 0 dt '' 0 [ 1+ glyph[epsilon1] '' 0 H '' 0 ] 2 ¯ ϕ 2 0 ( t '' 0 ) 〈 Ω | δ ¯ ϕ 0 ( t '' 0 , glyph[vector]x ) δ ¯ ϕ 0 ( t ' 0 , glyph[vector]x ' ) | Ω 〉 . (D9)</formula>
<text><location><page_44><loc_12><loc_59><loc_88><loc_65></location>To evaluate the remaining VEV in the integrand we need to break up the integral into two as ∫ t 0 0 dt '' 0 = ∫ t ' 0 0 dt '' 0 + ∫ t 0 t ' 0 dt '' 0 . In the first integral on the right side t '' 0 ≤ t ' 0 , whereas t ' 0 ≤ t '' 0 in the second. The first part</text>
<formula><location><page_44><loc_27><loc_53><loc_88><loc_57></location>H 0 1+ glyph[epsilon1] 0 ∫ t ' 0 0 dt '' 0 [ 1+ glyph[epsilon1] '' 0 H '' 0 ] 2 ¯ ϕ 2 0 ( t '' 0 ) 〈 Ω | δ ¯ ϕ 0 ( t '' 0 , glyph[vector]x ) δ ¯ ϕ 0 ( t ' 0 , glyph[vector]x ' ) | Ω 〉 , (D10)</formula>
<text><location><page_44><loc_12><loc_51><loc_87><loc_52></location>yields exactly the same result obtained in Eq. (140). The second part, on the other hand,</text>
<formula><location><page_44><loc_14><loc_40><loc_88><loc_49></location>H 0 1+ glyph[epsilon1] 0 ∫ t 0 t ' 0 dt '' 0 [ 1+ glyph[epsilon1] '' 0 H '' 0 ] 2 ¯ ϕ 2 0 ( t '' 0 ) 〈 Ω | δ ¯ ϕ 0 ( t '' 0 , glyph[vector]x ) δ ¯ ϕ 0 ( t ' 0 , glyph[vector]x ' ) | Ω 〉 = 1 4 π 2 H 0 1+ glyph[epsilon1] 0 H ' 0 1+ glyph[epsilon1] ' 0 [ ln ( H ' 0 a ' 0 H i 0 ) -ln ( 1+ glyph[epsilon1] ' 0 1+ glyph[epsilon1] i 0 ) + ∞ ∑ n =1 ( -1) n [ α ' 0 2 n -α 2 n i 0 ] 2 n (2 n +1)! ] ∫ t 0 t ' 0 dt '' 0 1+ glyph[epsilon1] '' 0 H '' 0 ¯ ϕ 2 0 ( t '' 0 ) , (D11)</formula>
<text><location><page_44><loc_12><loc_38><loc_27><loc_39></location>where the integral</text>
<formula><location><page_44><loc_27><loc_28><loc_88><loc_37></location>∫ t 0 t ' 0 dt '' 0 1+ glyph[epsilon1] '' 0 H '' 0 ¯ ϕ 2 0 ( t '' 0 )= glyph[epsilon1] -2 i 0 4 πGH 2 i 0 ∫ q ' 0 q 0 dq '' 0 q '' 2 0 + glyph[epsilon1] i 0 q '' 0 = -1 8 πGH 2 i 0 { glyph[epsilon1] -2 i 0 [ q 2 0 -q ' 2 0 ] + glyph[epsilon1] -1 i 0 [ ln( q 2 0 ) -ln( q ' 2 0 ) ] } . (D12)</formula>
<text><location><page_44><loc_12><loc_22><loc_88><loc_26></location>Hence, employing integral (D12)-summing up the infinite series multiplying it-in Eq. (D11) yields</text>
<formula><location><page_44><loc_23><loc_7><loc_88><loc_21></location>H 0 1+ glyph[epsilon1] 0 ∫ t 0 t ' 0 dt '' 0 [ 1+ glyph[epsilon1] '' 0 H '' 0 ] 2 ¯ ϕ 2 0 ( t '' 0 ) 〈 Ω | δ ¯ ϕ 0 ( t '' 0 , glyph[vector]x ) δ ¯ ϕ 0 ( t ' 0 , glyph[vector]x ' ) | Ω 〉 = 1 32 π 3 G q 3 0 q ' 3 0 ( q 2 0 + glyph[epsilon1] i 0 )( q ' 2 0 + glyph[epsilon1] i 0 ) { glyph[epsilon1] -2 i 0 [ q 2 0 -q ' 2 0 ] + glyph[epsilon1] -1 i 0 [ ln( q 2 0 ) -ln( q ' 2 0 ) ] } × [ ci( α i 0 ) -sin( α i 0 ) α i 0 -[ ci( α ' 0 ) -sin( α ' 0 ) α ' 0 ] ] . (D13)</formula>
<text><location><page_45><loc_12><loc_87><loc_88><loc_91></location>The second integral in Eq. (139) is, therefore, obtained combining Eq. (140) and Eqs. (D10)(D13). The result is given in Eq. (142).</text>
<text><location><page_45><loc_12><loc_81><loc_88><loc_85></location>In the next section, we outline the evaluation of the integrals involving VEVs that are quartic in the fluctuation field in one-loop correlator (100).</text>
<section_header_level_1><location><page_45><loc_38><loc_76><loc_62><loc_77></location>2. Integrals of Sec. IV B 2</section_header_level_1>
<text><location><page_45><loc_14><loc_71><loc_39><loc_73></location>The first integral in Eq. (145)</text>
<formula><location><page_45><loc_31><loc_66><loc_69><loc_70></location>H ' 0 1+ glyph[epsilon1] ' ∫ t ' 0 d ˜ t 0 [ 1+˜ glyph[epsilon1] 0 ˜ ] 2 〈 Ω | δ ¯ ϕ 0 ( t 0 , glyph[vector]x ) δ ¯ ϕ 3 0 ( ˜ t 0 , glyph[vector]x ' ) | Ω</formula>
<formula><location><page_45><loc_13><loc_54><loc_88><loc_61></location>3 16 π 4 H 0 1+ glyph[epsilon1] 0 H ' 0 1+ glyph[epsilon1] ' 0 ∫ t ' 0 0 d ˜ t 0 ˜ H 0 1+˜ glyph[epsilon1] 0 { [ ln ( ˜ H 0 ˜ a 0 H i 0 ) -ln ( 1+˜ glyph[epsilon1] 0 1+ glyph[epsilon1] i 0 )] 2 + ∞ ∑ n =1 ( -1) n [ ˜ α 2 n 0 -α 2 n i 0 ] 2 n (2 n +1)! [ ln ( ˜ H 0 ˜ a 0 H i 0 ) -ln ( 1+˜ glyph[epsilon1] 0 1+ glyph[epsilon1] i 0 )] } (D14)</formula>
<formula><location><page_45><loc_11><loc_58><loc_78><loc_68></location>0 0 H 0 〉 = H ' 0 1+ glyph[epsilon1] ' 0 ∫ t ' 0 0 d ˜ t 0 [ 1+˜ glyph[epsilon1] 0 ˜ H 0 ] 2 3 · 1 〈 Ω | δ ¯ ϕ 0 ( t 0 , glyph[vector]x ) δ ¯ ϕ 0 ( ˜ t 0 , glyph[vector]x ' ) | Ω 〉〈 Ω | δ ¯ ϕ 2 0 ( ˜ t 0 , glyph[vector]x ' ) | Ω 〉 =</formula>
<text><location><page_45><loc_12><loc_48><loc_88><loc_52></location>where we used correlator (120) and its coincident limit (133) at O ( λ 0 ). The integral of the square bracketed terms can be written, by expanding the integrand, as</text>
<formula><location><page_45><loc_11><loc_42><loc_88><loc_46></location>∫ t ' 0 0 d ˜ t 0 ˜ H 0 1+˜ glyph[epsilon1] 0 [ ln ( ˜ H 0 ˜ a 0 H i 0 ) -ln ( 1+˜ glyph[epsilon1] 0 1+ glyph[epsilon1] i 0 )] 2 = ∫ t ' 0 0 d ˜ t 0 ˜ H 0 1+˜ glyph[epsilon1] 0 [ ln 2 ( ˜ H 0 ˜ a 0 H i 0 ) -2 ln ( ˜ H 0 ˜ a 0 H i 0 ) ln ( 1+˜ glyph[epsilon1] 0 1+ glyph[epsilon1] i 0 ) +ln 2 ( 1+˜ glyph[epsilon1] 0 1+ glyph[epsilon1] i 0 )] . (D15)</formula>
<text><location><page_45><loc_12><loc_39><loc_51><loc_41></location>The first and second integrals in Eq. (D15) are</text>
<formula><location><page_45><loc_15><loc_29><loc_88><loc_38></location>∫ t ' 0 0 d ˜ t 0 ˜ H 0 1+˜ glyph[epsilon1] 0 ln 2 ( ˜ H 0 ˜ a 0 H i 0 ) = glyph[epsilon1] -1 i 0 ∫ 1 q ' 0 d ˜ q 0 ˜ q 3 0 ˜ q 2 0 + glyph[epsilon1] i 0 [ ln(˜ q 0 )+ glyph[epsilon1] -1 i 0 2 [1 -˜ q 2 0 ] ] 2 = glyph[epsilon1] -3 i 0 E ' 3 0 24 + glyph[epsilon1] -2 i 0 E ' 2 0 4 ln( q ' 0 ) -glyph[epsilon1] -1 i 0 2 [ q ' 2 0 4 -q '-2 0 4 -E ' 0 ln 2 ( q ' 0 ) -ln( q ' 0 ) ] + ln 3 ( eq ' 2 0 ) 24 + q '-2 0 2 ln( eq ' 0 ) -q '-4 0 16 -23 48 + O ( glyph[epsilon1] i 0 ) , (D16)</formula>
<text><location><page_45><loc_12><loc_26><loc_15><loc_27></location>and</text>
<formula><location><page_45><loc_16><loc_16><loc_88><loc_25></location>∫ t ' 0 0 d ˜ t 0 ˜ H 0 1+˜ glyph[epsilon1] 0 ln ( ˜ H 0 ˜ a 0 H i 0 ) ln ( 1+˜ glyph[epsilon1] 0 1+ glyph[epsilon1] i 0 ) = glyph[epsilon1] -1 i 0 ∫ 1 q ' 0 d ˜ q 0 ˜ q 3 0 ˜ q 2 0 + glyph[epsilon1] i 0 [ ln(˜ q 0 )+ glyph[epsilon1] -1 i 0 2 [1 -˜ q 2 0 ] ] ln ( ˜ q 2 0 + glyph[epsilon1] i 0 ˜ q 2 0 (1+ glyph[epsilon1] i 0 ) ) = -glyph[epsilon1] -1 i 0 2 [ E ' 2 0 4 + E ' 0 2 +ln( q ' 0 ) ] + E ' 2 0 16 + 3 4 [ 1 -q '-2 0 2 ] -ln 2 ( q ' 0 ) 2 + [ q ' 2 0 2 -5 4 ] ln( q ' 0 )+ O ( glyph[epsilon1] i 0 ) , (D17)</formula>
<text><location><page_45><loc_12><loc_13><loc_76><loc_15></location>respectively. The third integral in Eq. (D15), on the other hand, is of O ( glyph[epsilon1] i 0 )</text>
<formula><location><page_45><loc_12><loc_7><loc_88><loc_12></location>∫ t ' 0 0 d ˜ t 0 ˜ H 0 1+˜ glyph[epsilon1] 0 ln 2 ( 1+˜ glyph[epsilon1] 0 1+ glyph[epsilon1] i 0 ) = glyph[epsilon1] -1 i 0 ∫ 1 q ' 0 d ˜ q 0 ˜ q 3 0 ˜ q 2 0 + glyph[epsilon1] i 0 ln 2 ( ˜ q 2 0 + glyph[epsilon1] i 0 ˜ q 2 0 (1+ glyph[epsilon1] i 0 ) ) = glyph[epsilon1] i 0 [ q '-2 0 2 -q ' 2 0 2 +ln( q ' 2 0 ) ] + O ( glyph[epsilon1] 2 i 0 ) , (D18)</formula>
<text><location><page_46><loc_12><loc_87><loc_88><loc_91></location>hence we neglect it. The remaining three terms in Eq. (D14) involve series expansions of which the first is</text>
<formula><location><page_46><loc_32><loc_81><loc_88><loc_86></location>∞ ∑ n =1 ( -1) n (∆ x ) 2 n 2 n (2 n +1)! ∫ t ' 0 0 d ˜ t 0 ˜ H 0 1+˜ glyph[epsilon1] 0 [ ˜ H 0 ˜ a 0 1+˜ glyph[epsilon1] 0 ] 2 n ln ( ˜ H 0 ˜ a 0 H i 0 ) , (D19)</formula>
<text><location><page_46><loc_12><loc_79><loc_27><loc_80></location>where the integral</text>
<formula><location><page_46><loc_13><loc_48><loc_88><loc_77></location>∫ t ' 0 0 d ˜ t 0 ˜ H 0 1+˜ glyph[epsilon1] 0 [ ˜ H 0 ˜ a 0 1+˜ glyph[epsilon1] 0 ] 2 n ln ( ˜ H 0 ˜ a 0 H i 0 ) = glyph[epsilon1] -1 i 0 H 2 n i 0 e n glyph[epsilon1] i 0 ∫ 1 q ' 0 d ˜ q 0 [ ˜ q 3 0 ˜ q 2 0 + glyph[epsilon1] i 0 ] 2 n +1 e -n glyph[epsilon1] i 0 ˜ q 2 0 [ ln(˜ q 0 )+ glyph[epsilon1] -1 i 0 2 [ 1 -˜ q 2 0 ] ] = H 2 n i 0 4 e n glyph[epsilon1] i 0 { glyph[epsilon1] -2 i 0 [ E -n -1 ( n glyph[epsilon1] i 0 ) -q ' 2 n +4 0 E -n -1 ( nq ' 2 0 glyph[epsilon1] i 0 ) -{ E -n ( n glyph[epsilon1] i 0 ) -q ' 2 n +2 0 E -n ( nq ' 2 0 glyph[epsilon1] i 0 ) } ] -glyph[epsilon1] -1 i 0 [ (2 n +1) { E -n ( n glyph[epsilon1] i 0 ) -q ' 2 n +2 0 E -n ( nq ' 2 0 glyph[epsilon1] i 0 ) -{ E 1 -n ( n glyph[epsilon1] i 0 ) -q ' 2 n 0 E 1 -n ( nq ' 2 0 glyph[epsilon1] i 0 ) }} + 1 ( n +1) 2 { 2 F 2 ( n +1 , n +1; n +2 , n +2; -n glyph[epsilon1] i 0 ) -q ' 2 n +2 0 2 F 2 ( n +1 , n +1; n +2 , n +2; -nq ' 2 0 glyph[epsilon1] i 0 ) } ] +(2 n +1) [ ( n +1) { E 1 -n ( n glyph[epsilon1] i 0 ) -q ' 2 n 0 E 1 -n ( nq ' 2 0 glyph[epsilon1] i 0 ) -{ E 2 -n ( n glyph[epsilon1] i 0 ) -q ' 2 n -2 0 E 2 -n ( nq ' 2 0 glyph[epsilon1] i 0 ) }} + 1 n 2 { 2 F 2 ( n, n ; n +1 , n +1; -n glyph[epsilon1] i 0 ) -q ' 2 n 0 2 F 2 ( n, n ; n +1 , n +1; -nq ' 2 0 glyph[epsilon1] i 0 ) } ] + O ( glyph[epsilon1] i 0 ) } . (D20)</formula>
<text><location><page_46><loc_12><loc_43><loc_88><loc_47></location>We define the generalized hypergeometric function 2 F 2 in Eq. (B7). The second remaining term which involves a series expansion in Eq. (D14) is</text>
<formula><location><page_46><loc_30><loc_37><loc_88><loc_41></location>∞ ∑ n =1 ( -1) n (∆ x ) 2 n 2 n (2 n +1)! ∫ t ' 0 0 d ˜ t 0 ˜ H 0 1+˜ glyph[epsilon1] 0 [ ˜ H 0 ˜ a 0 1+˜ glyph[epsilon1] 0 ] 2 n ln ( 1+˜ glyph[epsilon1] 0 1+ glyph[epsilon1] i 0 ) , (D21)</formula>
<text><location><page_46><loc_12><loc_34><loc_40><loc_36></location>where the integral in Eq. (D21) is</text>
<formula><location><page_46><loc_15><loc_24><loc_88><loc_33></location>∫ t ' 0 0 d ˜ t 0 ˜ H 0 1+˜ glyph[epsilon1] 0 [ ˜ H 0 ˜ a 0 1+˜ glyph[epsilon1] 0 ] 2 n ln ( 1+˜ glyph[epsilon1] 0 1+ glyph[epsilon1] i 0 ) = glyph[epsilon1] -1 i 0 H 2 n i 0 e n glyph[epsilon1] i 0 ∫ 1 q ' 0 d ˜ q 0 [ ˜ q 3 0 ˜ q 2 0 + glyph[epsilon1] i 0 ] 2 n +1 e -n glyph[epsilon1] i 0 ˜ q 2 0 ln ( ˜ q 2 0 + glyph[epsilon1] i 0 ˜ q 2 0 (1+ glyph[epsilon1] i 0 ) ) = -H 2 n i 0 2 e n glyph[epsilon1] i 0 { E 1 -n ( n glyph[epsilon1] i 0 ) -q ' 2 n 0 E 1 -n ( nq ' 2 0 glyph[epsilon1] i 0 ) -[ E -n ( n glyph[epsilon1] i 0 ) -q ' 2 n +2 0 E -n ( nq ' 2 0 glyph[epsilon1] i 0 ) ] + O ( glyph[epsilon1] i 0 ) } . (D22)</formula>
<text><location><page_46><loc_12><loc_21><loc_44><loc_22></location>Final term we evaluate in Eq. (D14) is</text>
<formula><location><page_46><loc_24><loc_15><loc_88><loc_20></location>∞ ∑ n =1 ( -1) n (∆ x ) 2 n 2 n (2 n +1)! [ H i 0 1+ glyph[epsilon1] i 0 ] 2 n ∫ t ' 0 0 d ˜ t 0 ˜ H 0 1+˜ glyph[epsilon1] 0 [ ln ( ˜ H 0 ˜ a 0 H i 0 ) -ln ( 1+˜ glyph[epsilon1] 0 1+ glyph[epsilon1] i 0 )] , (D23)</formula>
<text><location><page_46><loc_12><loc_12><loc_35><loc_14></location>where the sum in Eq. (D23)</text>
<formula><location><page_46><loc_24><loc_7><loc_88><loc_11></location>∞ ∑ n =1 ( -1) n (∆ x ) 2 n 2 n (2 n +1)! [ H i 0 1+ glyph[epsilon1] i 0 ] 2 n =1 -γ +ci( α i 0 ) -sin( α i 0 ) α i 0 -ln( α i 0 ) , (D24)</formula>
<text><location><page_47><loc_12><loc_89><loc_36><loc_91></location>and the integral in Eq. (D23)</text>
<formula><location><page_47><loc_12><loc_79><loc_88><loc_88></location>∫ t ' 0 0 d ˜ t 0 ˜ H 0 1+˜ glyph[epsilon1] 0 [ ln ( ˜ H 0 ˜ a 0 H i 0 ) -ln ( 1+˜ glyph[epsilon1] 0 1+ glyph[epsilon1] i 0 )] = glyph[epsilon1] -1 i 0 ∫ 1 q ' 0 d ˜ q 0 ˜ q 3 0 ˜ q 2 0 + glyph[epsilon1] i 0 [ ln(˜ q 0 )+ glyph[epsilon1] -1 i 0 2 [1 -˜ q 2 0 ] -ln ( ˜ q 2 0 + glyph[epsilon1] i 0 ˜ q 2 0 (1+ glyph[epsilon1] i 0 ) ) ] (D25) = glyph[epsilon1] -2 i 0 E ' 2 0 8 + glyph[epsilon1] -1 i 0 E ' 0 2 ln( q ' 0 )+ E ' 0 2 [ 1+ q '-2 0 2 ] + ln 2 ( q ' 0 ) 2 + ln( q ' 3 0 ) 2 + O ( glyph[epsilon1] i 0 ) . (D26)</formula>
<text><location><page_47><loc_12><loc_74><loc_88><loc_78></location>Thus, the first integral in Eq. (145) is obtained using Eqs. (D15)-(D26) in Eq. (D14). The result is given in Eq. (146).</text>
<text><location><page_47><loc_14><loc_71><loc_52><loc_72></location>To evaluate the second integral in Eq. (145),</text>
<formula><location><page_47><loc_29><loc_65><loc_88><loc_69></location>H 0 1+ glyph[epsilon1] 0 ∫ t 0 0 dt '' 0 [ 1+ glyph[epsilon1] '' 0 H '' 0 ] 2 〈 Ω | δ ¯ ϕ 3 0 ( t '' 0 , glyph[vector]x ) δ ¯ ϕ 0 ( t ' 0 , glyph[vector]x ' ) | Ω 〉 , (D27)</formula>
<text><location><page_47><loc_12><loc_59><loc_88><loc_64></location>we break up the integral into two as ∫ t 0 dt '' = ∫ t ' 0 dt '' + ∫ t t ' dt '' . In the first integral t '' ≤ t ' , whereas t ' ≤ t '' in the second. The first part of integral (D27)</text>
<formula><location><page_47><loc_30><loc_54><loc_88><loc_58></location>H 0 1+ glyph[epsilon1] 0 ∫ t ' 0 0 dt '' 0 [ 1+ glyph[epsilon1] '' 0 H '' 0 ] 2 〈 Ω | δ ¯ ϕ 3 0 ( t '' 0 , glyph[vector]x ) δ ¯ ϕ 0 ( t ' 0 , glyph[vector]x ' ) | Ω 〉 , (D28)</formula>
<text><location><page_47><loc_12><loc_48><loc_88><loc_52></location>yields-following the same steps through Eqs. (D15)-(D26)-exactly the same result obtained in Eq. (146). The remaining part of integral (D27) is</text>
<formula><location><page_47><loc_20><loc_28><loc_88><loc_47></location>H 0 1+ glyph[epsilon1] 0 ∫ t 0 t ' 0 dt '' 0 [ 1+ glyph[epsilon1] '' 0 H '' 0 ] 2 〈 Ω | δ ¯ ϕ 3 0 ( t '' 0 , glyph[vector]x ) δ ¯ ϕ 0 ( t ' 0 , glyph[vector]x ' ) | Ω 〉 = H 0 1+ glyph[epsilon1] 0 ∫ t 0 t ' 0 dt '' 0 [ 1+ glyph[epsilon1] '' 0 H '' 0 ] 2 3 · 1 〈 Ω | δ ¯ ϕ 0 ( t '' 0 , glyph[vector]x ) δ ¯ ϕ 0 ( t ' 0 , glyph[vector]x ' ) | Ω 〉〈 Ω | δ ¯ ϕ 2 0 ( t '' 0 , glyph[vector]x ) | Ω 〉 = 3 16 π 4 H 0 1+ glyph[epsilon1] 0 H ' 0 1+ glyph[epsilon1] ' 0 { ln ( H ' 0 a ' 0 H i 0 ) -ln ( 1+ glyph[epsilon1] ' 0 1+ glyph[epsilon1] i 0 ) + ∞ ∑ n =1 ( -1) n [ α ' 2 n 0 -α 2 n i 0 ] 2 n (2 n +1)! } × ∫ t 0 t ' 0 dt '' 0 H '' 0 1+ glyph[epsilon1] '' 0 [ ln ( H '' 0 a '' 0 H i 0 ) -ln ( 1+ glyph[epsilon1] '' 0 1+ glyph[epsilon1] i 0 )] . (D29)</formula>
<text><location><page_47><loc_12><loc_20><loc_88><loc_27></location>The terms inside the curly brackets in Eq. (D29) can be written in terms of simple analytic functions using Eq. (C2). The integral in Eq. (D29) is the same as integral (D25)-except for the limits of integration. It is evaluated as</text>
<formula><location><page_47><loc_20><loc_7><loc_88><loc_19></location>∫ t 0 t ' 0 dt '' 0 H '' 0 1+ glyph[epsilon1] '' 0 [ ln ( H '' 0 a '' 0 H i 0 ) -ln ( 1+ glyph[epsilon1] '' 0 1+ glyph[epsilon1] i 0 )] = glyph[epsilon1] -2 i 0 8 [ E 2 0 -E ' 2 0 ] -glyph[epsilon1] -1 i 0 2 [ q 2 0 ln( q 0 ) -q ' 2 0 ln( q ' 0 ) -[ ln( q 0 ) -ln( q ' 0 )] ] + 1 2 [ q -2 0 -q '-2 0 2 -[ q 2 0 -q ' 2 0 ] +ln 2 ( q 0 ) -ln 2 ( q ' 0 )+ln( q 3 0 ) -ln( q ' 3 0 ) ] + O ( glyph[epsilon1] i 0 ) . (D30)</formula>
<text><location><page_48><loc_12><loc_87><loc_88><loc_91></location>The second integral in Eq. (145) is, therefore, obtained combining Eqs. (D28)-(D30) in Eq. (D27). The result is given in Eq. (148).</text>
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</document> | [] |
2023SPIE12424E..0BR | https://arxiv.org/pdf/2302.06393.pdf | <document>
<section_header_level_1><location><page_1><loc_11><loc_86><loc_89><loc_90></location>Astrophotonics: photonic integrated circuits for astronomical instrumentation</section_header_level_1>
<text><location><page_1><loc_14><loc_81><loc_86><loc_84></location>Martin M. Roth a,b , Kalaga Madhav a , Andreas Stoll a,b , Daniel Bodenmuller a,b , Aline Dinkelaker a , Aashia Rahman a , Eloy Hernandez a , Alan Gunther a , and Stella Vjesnica a</text>
<text><location><page_1><loc_12><loc_76><loc_88><loc_80></location>a Leibniz-Institute for Astrophysics Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany</text>
<text><location><page_1><loc_11><loc_73><loc_89><loc_76></location>b Universitat Potsdam, Institut fur Physik und Astronomie, Haus 28, Karl-Liebknecht-Straße 24/25, 14476 Potsdam, Germany</text>
<section_header_level_1><location><page_1><loc_44><loc_68><loc_56><loc_69></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_10><loc_48><loc_90><loc_67></location>Photonic Integrated Circuits (PIC) are best known for their important role in the telecommunication sector, e.g. high speed communication devices in data centers. However, PIC also hold the promise for innovation in sectors like life science, medicine, sensing, automotive etc. The past two decades have seen efforts of utilizing PIC to enhance the performance of instrumentation for astronomical telescopes, perhaps the most spectacular example being the integrated optics beam combiner for the interferometer GRAVITY at the ESO Very Large Telescope. This instrument has enabled observations of the supermassive black hole in the center of the Milky Way at unprecedented angular resolution, eventually leading to the Nobel Price for Physics in 2020. Several groups worldwide are actively engaged in the emerging field of astrophotonics research, amongst them the innoFSPEC Center in Potsdam, Germany. We present results for a number of applications developed at innoFSPEC, notably PIC for integrated photonic spectrographs on the basis of arrayed waveguide gratings and the PAWS demonstrator (Potsdam Arrayed Waveguide Spectrograph), PIC-based ring resonators in astronomical frequency combs for precision wavelength calibration, discrete beam combiners (DBC) for large astronomical interferometers, as well as aperiodic fiber Bragg gratings for complex astronomical filters and their possible derivatives in PIC.</text>
<text><location><page_1><loc_10><loc_45><loc_88><loc_47></location>Keywords: Ground-based telescopes, spectroscopy, long-baseline interferometry, photonic integrated circuit</text>
<section_header_level_1><location><page_1><loc_40><loc_42><loc_60><loc_43></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_10><loc_15><loc_90><loc_41></location>Since the first use of lenses for a telescope by Galileo Galilei [1] back in 1610, astronomical instrumentation has been key for new discoveries, and shaping our picture of the universe. In the context of photonics, it is interesting to note that early on, from the first availability of optical fibers, astronomers have discovered their virtue for spectroscopy: not only do fibers provide a higher level of flexibility than free space optics in coupling spectrographs at practically arbitrary locations to the telescope focal plane [2, 3], but also an elegant way to accomplish multi object spectroscopy (MOS) [4], i.e. the opportunity to measure spectra of N objects at the same time, rather than sequentially. Given the necessary total exposure times for faint objects, that often extend to hours rather than only seconds or minutes, the need to create databases of hundreds of thousands, or even millions, of objects has made fiber coupled MOS an industry, enabling e.g. fundamental studies of the large scale structure of the universe to validate numerical simulations in cosmology. For example, the Sloan Digital Sky Survey (SDSS) [5], arguably one of the highest scientific impact ground based observing programs ever, has employed fiber-coupled MOS to provide hundreds of thousands of spectra for stars and galaxies in a series of data releases, spanning a period of more than 20 years. High resolution echelle spectrographs with thermal and pressure control have achieved unprecedented precision in measuring Doppler shifts of stars for exoplanet detection, thanks to a flexure-free stationary setup that is coupled to the telescope by means of a fiber link, e.g. [6], [7], [8]. Also, fiber bundles have enabled spectroscopy over an extended two-dimensional field-of-view, e.g. [9], [10], with powerful astronomical applications, e.g. the study of the evolution of galaxies [11], [12].</text>
<text><location><page_2><loc_10><loc_83><loc_90><loc_91></location>Beyond the application of optical fibers, the last two decades have seen more sophisticated applications of waveguides for astronomical instrumentation, such as to coin the term 'Astrophotonics' [13]. The emerging field of Astrophotonics is sketched in [14], and more comprehensively described in a recent review [15]. Here we summarize results achieved at the innovation center innoFSPEC Potsdam [16], [17] [18] with a focus on photonic integrated circuits.</text>
<section_header_level_1><location><page_2><loc_11><loc_78><loc_89><loc_81></location>2. INTEGRATED PHOTONIC SPECTROGRAPHS ON THE BASIS OF ARRAYED WAVEGUIDE GRATINGS</section_header_level_1>
<text><location><page_2><loc_10><loc_64><loc_90><loc_77></location>Fiber-coupled miniaturized spectrometers based on classical diffraction gratings have been available as commercial products for some time, but these devices are generally not suitable for demanding applications in astronomical instrumentation. However, the demand for dense wavelength domain multiplexing (DWDM) in telecommunications and the development of arrayed waveguide gratings (AWG) has inspired the transfer of this technologies to other fields, such as spectroscopy in medicine, life sciences, and chemistry, e.g. [19]. First experiments into this direction for astronomy [20] were reported by [21] and [22], and a vision for future development proposed by [23]. The early experiments were able to demonstrate a proof of principle by using commercially available AWGs, however falling short of delivering a performance needed for a competitive focal plane instrument at an observatory.</text>
<text><location><page_2><loc_10><loc_58><loc_90><loc_63></location>The innoFSPEC innovation center has engaged in research for integrated photonic spectrographs that are optimized for astronomy. For the time being, the focus has remained on a wavelength region in the near infrared (NIR), namely the astronomical H-band (1500 . . . 1800 nm), that is close enough to telecommunication wavelengths to facilitate manufacture and characterization.</text>
<text><location><page_2><loc_10><loc_45><loc_90><loc_57></location>The motivation was driven by the consideration that mass and volume are major cost drivers for space missions, hence miniaturizing an H-band spectrograph would be a worthwhile objective. Fig. 1 illustrates the potential savings by showing the prominent example of the NIRspec spectrograph on board of the James Webb Space Telescope (JWST). The instrument has a size of 1900 mm × 1400 mm × 700 mm, and a mass of 196 kg. For reasons of thermal and mechanical stability, the mirror mounts and the optical bench base plate are manufactured out of silicon carbide ceramic and already amount to a mass of 100 kg, i.e. half of the total mass of the instrument. It is then quite obvious that shrinking appreciable parts of the free space optical system to a PIC would immediately result in reduced size, mass, and potentially improved thermal and mechanical stability.</text>
<figure>
<location><page_2><loc_11><loc_20><loc_57><loc_43></location>
</figure>
<figure>
<location><page_2><loc_59><loc_20><loc_89><loc_44></location>
<caption>Figure 1. Miniaturizing a free-space optical system spectrograph to a chip. Left: NIRspec instrument on board of JWST (credit: ESA/Astrium). Right: AWG chip manufactured for innoFSPEC.</caption>
</figure>
<section_header_level_1><location><page_3><loc_10><loc_89><loc_45><loc_91></location>2.1 First experiments and exploration</section_header_level_1>
<text><location><page_3><loc_10><loc_68><loc_90><loc_88></location>The very first steps to design and manufacture AWGs for spectroscopy were conducted in collaboration with the Leibniz Institute for High Performance Microelectronics (IHP) [24,25]. Initially, a conventional AWG, however without output receiver waveguides, was designed on a silicon nitride/SiO2/Si (Si3N4-SiO2-Si) platform for its relatively high refractive index, which, for a given channel spacing, was expected to allow a more compact device than Silica-on-Silicon. CMOS compatibility and the presence of a high non-linear optical coefficient was considered to be an added advantage for the manufacture of a densely integrated photonic chip, that could potentially also include a ring resonator based frequency comb for on-chip wavelength calibration. An analytical calculation, based on Gaussian beam approximation, was used to determine the optimal flat plane position where the non-uniformity in 1/e electric field widths would be minimal, to become the diced output plane for further coupling to a cross-dispersion optics and an imaging camera. The AWG was designed to resolve 48 spectral channels with 0.4 nm (50GHz) resolution and adjacent channel cross-talk level within a 0.2nm window (ITUgrid) -28dB. The calculated insertion loss non-uniformity was close to 3dB. The footprint of the high index contrast (∆n=23%) area was 12 × 8.5 mm 2 . The modelled mean spectral resolving power at the flat image-plane was computed to R = 7600.</text>
<figure>
<location><page_3><loc_42><loc_50><loc_58><loc_66></location>
<caption>Figure 2. First experiments with AWGs on Si3N4 platform.</caption>
</figure>
<text><location><page_3><loc_10><loc_23><loc_90><loc_45></location>The not entirely satisfactory experience with the silicon nitride platform led to a re-design using Silicaon-Silicon (SoS) technology, to tailor specific performance parameters of interest to a resolving power of R = λ/ ∆ λ = 60,000 [26]. The chip was designed to resolve up to 646 spectral lines per spectral order, with a wavelength spacing of 25 pm, at a central wavelength of 1630 nm. Fabricated test waveguides were stress engineered in order to compensate the inherent birefringence of SoS waveguides. The birefringence values of fabricated test structures were quantified, to be on the order of 10 -6 , the theoretical value required to avoid the formation of ghost-images, through inscription of Bragg-gratings on straight waveguides and subsequent measurement of Bragg-reflection spectra. An interferometer system was integrated on the same chip in order to allow for the characterization of phase errors of the waveguide array. Moreover, promising results of first fabricated key photonics components to form other complex integrated photonic circuits, such as astro-interferometers, using silicon nitride-on-insulator (SNOI) technology were also presented. The fabricated PICs included multimode interference based devices (power splitter/combiners, optical cross/bar-switches), directional-couplers with varying power ratios, Mach-Zehnder interferometers, and the AWG. First results of annealed, low-hydrogen SNOI based devices were promising and comparable to SOI and commercial devices, with device excess-loss less than 2 dB and under 1 dB/cm waveguide-loss at NIR-wavelengths.</text>
<text><location><page_3><loc_10><loc_12><loc_90><loc_22></location>Prompted by the latter step, the design of a folded-architecture AWG for the astronomical H-band with a theoretical maximum resolving power R = 60,000 at 1630 nm was investigated [27]. The geometry of the device was optimized for a compact structure with a footprint of 55 mm × 39.3 mm on SiO2 platform. To evaluate the fabrication challenges of such high-resolution AWGs, the effects of random perturbations of the effective refractive index (RI) distribution in the free propagation region (FPR), and small variations of the array waveguide optical lengths were numerically investigated. The results of the study have shown a dramatic degradation of the point spread function (PSF) for a random effective RI distribution with variance values above 10 -4 for both the FPR</text>
<text><location><page_4><loc_10><loc_88><loc_90><loc_91></location>and the waveguide array. Based on these results, requirements on the fabrication technology for high-resolution AWG-based spectrographs were derived.</text>
<text><location><page_4><loc_10><loc_74><loc_90><loc_87></location>Furthermore, a numerical and experimental study of the impact of phase errors on the performance of large, high-resolution arrayed waveguide gratings (AWG) for applications in astronomy was conducted [28]. Using a scalar diffraction model, the transmission spectrum of an AWG under random variations of the optical waveguide lengths was studied. Phase error correction was simulated by numerically trimming the lengths of the optical waveguides to the nearest integer multiple of the central wavelength. The optical length error distribution of a custom-fabricated silica AWG was measured using frequency-domain interferometry and Monte-Carlo fitting of interferogram intensities. As a conclusion, the estimate for the phase-error limited size of a waveguide array manufactured using state-of-the-art technology was presented. It was also shown that post-processing can, in principle, eliminate phase errors as a performance limiting factor of AWGs for astronomical spectroscopy.</text>
<section_header_level_1><location><page_4><loc_10><loc_71><loc_42><loc_72></location>2.2 Fabricated AWGs Generation I</section_header_level_1>
<text><location><page_4><loc_10><loc_48><loc_90><loc_70></location>Based on the previous studies, the fabrication of a first generation of custom-developed AWG on a silica platform for spectroscopic applications in near-infrared astronomy was attempted [29]. This work was targeting a comprehensive description of the design, numerical simulation and characterization of several different types of devices, aiming at spectral resolving powers of 15,000-60,000 in the astronomical H-band. The AWGs were fabricated in a foundry run by Enablence Technologies Inc. using an atmospheric pressure chemical vapor deposition process (APCVD). The spectral characteristics of the fabricated devices were investigaged in terms of insertion loss and estimated spectral resolving power and compared with the numerical simulations of the design. The different AWG types had been designed with resolving powers of up to 18,900 from the output channel 3-dB transmission bandwidth. Based on a first characterization results, two of the best performing AWGs were selected for further processing by removal of the output waveguide array and polishing the output facet to optical quality. The diced output is needed for integration as the primary diffractive element in a cross-dispersed spectrograph. For the selected devices, the imaging properties were measured with regards to spectral resolution in direct imaging mode, geometry-related defocus aberration, and polarization sensitivity of the spectral image. This work identified phase error control, birefringence control, and aberration suppression as the major issues to be solved towards high performance integrated photonics spectrographs.</text>
<figure>
<location><page_4><loc_22><loc_21><loc_45><loc_43></location>
</figure>
<figure>
<location><page_4><loc_51><loc_22><loc_76><loc_43></location>
<caption>Figure 3. AWG Gen I fabricated on SoS platform.</caption>
</figure>
<section_header_level_1><location><page_5><loc_10><loc_89><loc_43><loc_91></location>2.3 Fabricated AWGs Generation II</section_header_level_1>
<text><location><page_5><loc_10><loc_68><loc_90><loc_88></location>With lessons learned from the design, manufacture, and characterization of Gen I AWGs, an improved variant was developed using the three-stigmatic-point method with the goal to solve the problem of degraded image quality along the flat output facet of the devices as a result of longitudinal aberration [30]. This work presented theoretical and experimental results on the design, simulation and characterization of optimized Gen II silicaon-silicon AWGs. Several mid-to-high resolution field-flattened AWG designs were derived, targeting resolving powers of 11,000 - 35,000 in the astronomical H-band, by iterative computation of differential coefficients of the optical path function. Numerical simulations were used to study the imaging properties of the designs in a wide wavelength range between 1500 nm and 1680 nm. The design-specific degradation of spectral resolving power at far-off-centre wavelengths was discussed, and possible solutions suggested. Fig. 3 shows the lithographic mask that was used to manufacture a variety of different sizes and types of AWGs, based on the optimized design. The photographs show three designs of different geometry. The wafers were again fabricated by Enablence using APCVD like for Gen I. Seven selected devices were characterized in the lab, with performance parameters listed in the table shown in Fig. 3. The results show that it is possible to achieve resolving powers of around R = 30.000, and insertion losses as low as 1.45 dB.</text>
<figure>
<location><page_5><loc_13><loc_54><loc_57><loc_64></location>
</figure>
<table>
<location><page_5><loc_14><loc_36><loc_56><loc_48></location>
</table>
<figure>
<location><page_5><loc_59><loc_35><loc_93><loc_65></location>
<caption>Figure 4. AWG Gen II fabricated on SoS platform. Right: lithographic mask for the 6 inch wafer containing different types of AWG, fabricated by Enablence. Upper left: photograph of three types of AWG with different geometries. Lower left: table with performance parameters as obtained from lab characterization.</caption>
</figure>
<text><location><page_5><loc_10><loc_22><loc_90><loc_26></location>Beyond the initial exploratory work and Gen I and II designs, we mention for completeness without detailed discussion further work on echelle gratings [31], and the peculiar design for an AWG that is embedded as a helical structure within an optical fiber, rather than on a planar chip [32].</text>
<section_header_level_1><location><page_6><loc_10><loc_89><loc_63><loc_91></location>2.4 PAWS, the Potsdam Arrayed Waveguide Spectrograph</section_header_level_1>
<text><location><page_6><loc_10><loc_80><loc_90><loc_88></location>A crucial step to enable progress for any new technology in astronomical instrumenation is the need to validate the performance with an on-sky experiment that can be used to demonstrate technology readiness levels TRL 6 and TRL 7. To this end, innoFSPEC has designed and built an instrument, the Potsdam Arrayed Waveguide Spectrograph (PAWS) [33, 34]. Fig. 5 illustrates the major components, including the Gen II AWG chip as described in Section 2.3, that is mounted in subsystem A. Commissioning at the telescope and on-sky testing is planned at Calar Alto Observatory in southern Spain.</text>
<figure>
<location><page_6><loc_13><loc_58><loc_45><loc_77></location>
</figure>
<figure>
<location><page_6><loc_48><loc_58><loc_86><loc_77></location>
<caption>Figure 5. AWG demonstrator instrument PAWS. Left: CAD layout, showing AWG mount A with microscopy objective, connecting to the cryostat (rectangular box). Inside of the cryostat, the free space optical system for cross-dispersion B is cooled down to a temperature of 140 K. The detector C, cooled to 80 K, is an MCT NIR array H2RG from Teledyne. Cooling is accomplished through the cryocooler D. Right: open cryostat, showing gold-plated radiation shield that reduces radiative thermal losses to a minimum.</caption>
</figure>
<section_header_level_1><location><page_6><loc_12><loc_43><loc_88><loc_46></location>3. PIC RING RESONATORS IN ASTRONOMICAL FREQUENCY COMBS FOR PRECISION WAVELENGTH CALIBRATION</section_header_level_1>
<text><location><page_6><loc_10><loc_30><loc_90><loc_42></location>Optical frequency combs have been instrumental for high precision and accuracy in wavelength calibration for astronomical high resolution spectrographs, most notably achieving Doppler velocity measurements of stars that are accurate to about 1 m/s. Such accuracies are necessary for the spectroscopic detection of exo-planets. However, even spectrographs with medium or even low resolution would benefit from properties of frequency combs that deliver equidistant emission lines of comparable intensity - properties that are not at all warranted by classical spectral line lamps. However, the cost of commercially available laser frequency combs is prohibitively high. innoFSPEC has made attempts to find solutions that would address affordable general purpose frequency combs, however not necessarily at high spectral resolution, or the highest possible accuracy.</text>
<text><location><page_6><loc_10><loc_27><loc_90><loc_29></location>Early on, two different platforms (and approaches) were numerically and experimentally investigated, targeting medium and low resolution spectrographs at astronomical facilities [35].</text>
<text><location><page_6><loc_10><loc_11><loc_90><loc_26></location>In the first approach, a frequency comb was generated by propagating two lasers through three nonlinear stages. The first two stages serve for the generation of low-noise ultra-short pulses, while the final stage is a lowdispersion highly-nonlinear fibre where the pulses undergo strong spectral broadening. The wavelength of one of the lasers can be tuned, allowing the comb line spacing being continuously varied during the calibration procedure. The input power, the dispersion, the nonlinear coefficient, and fibre lengths in the nonlinear stages were defined and optimized by solving the generalized nonlinear Schrodinger equation. Experimentally, the 250 GHz linespacing frequency comb was generated by using two narrow linewidth lasers that were adiabatically compressed, firstly, in a standard fibre and, secondly, in a double-clad Er/Yb doped fibre. The spectral broadening finally took place in a highly nonlinear fibre resulting in an astro-comb with 250 calibration lines (covering a bandwidth of 500 nm) with good spectral equalization.</text>
<text><location><page_7><loc_10><loc_83><loc_90><loc_91></location>The second approach aimed to generate optical frequency combs in dispersion-optimized silicon nitride ring resonators. A technique for lowering and flattening the chromatic dispersion in silicon nitride waveguides with silica cladding was proposed and demonstrated. By minimizing the waveguide dispersion over a broader wavelength range two goals were targeted: enhancing the phase matching for non-linear interactions, and broadening the wavelength coverage for the generated frequency combs.</text>
<text><location><page_7><loc_10><loc_78><loc_90><loc_82></location>For this purpose, instead of one cladding layer, the design incorporated two layers with appropriate thicknesses. It was possible to demonstrate nearly zero dispersion (with +/- 4 ps/nm-km variation) over the spectral region from 1.4 to 2.3 µ m.</text>
<text><location><page_7><loc_10><loc_66><loc_90><loc_77></location>The first approach was demonstrated with an experiment using a fiber-fed copy of a spectrograph module from the Multi Unit Spectroscopic Explorer (MUSE) [36]. The frequency comb was generated by propagating two free-running lasers at 1554.3 and 1558.9 nm through two dispersion optimized nonlinear fibers [37]. The generated comb was centered at 1590 nm and comprised more than one hundred lines with an optical-signal-tonoise ratio larger than 30 dB. A nonlinear crystal was used to frequency double the whole comb spectrum, which was efficiently converted into the 800 nm spectral band. A series of tests demonstrated not only that the comb delivered significantly more emission lines than a comparison Neon spectral line lamp, but also that equidistancy of the comb lines was confirmed with an absolute accuracy of 0.4 pm.</text>
<text><location><page_7><loc_10><loc_62><loc_90><loc_65></location>Furthermore, on-sky tests were conducted using the Potsdam Multi-Aperture Spectrograph (PMAS) [9] at the 3.5 m Calar Alto Telescope [38].</text>
<figure>
<location><page_7><loc_27><loc_42><loc_73><loc_59></location>
<caption>Figure 6. Frequency comb based on micro ring resonator.</caption>
</figure>
<text><location><page_7><loc_10><loc_29><loc_90><loc_36></location>The second approach was subsequently studied, and first results reported for frequency combs in a silicon nitride micro-ring resonator, achieving an ultra-stable repetition frequency of 28.55 GHz [39], Fig. 6. The combs were generated by means of an amplitude modulated pump laser at 1568.8 nm and compared to numerical calculation based on a modified Lugiato-Lefever-Equation. The comb spectrum at a power level of -40 dB with respect to the pump line was found to span a wavelength range of 70 nm in this first experiment.</text>
<text><location><page_7><loc_10><loc_12><loc_90><loc_28></location>Further research resulted in a break-through towards practical applications, in particular a viable route towards developing a turn-key system for reliable operation at an observatory [40]. Nonlinear Kerr microresonators have enabled the understanding of dissipative solitons and their application to optical frequency comb generation. However, the conversion efficiency of the pump power into a soliton frequency comb typically remains below a few percent. A solution for the problem of low efficiency consists in the implementation of a hybrid Mach-Zehnder ring resonator geometry, that is realized as a micro-ring resonator embedded in an additional cavity with twice the optical path length of the ring, see Fig. 7. The resulting interferometric back coupling enables an unprecedented control of the pump depletion: pump-to-frequency comb conversion efficiencies of up to 55% of the input pump power have been experimentally demonstrated. The robustness of the novel on-chip geometry was verified through manufacture and testing a large variety of dissipative Kerr soliton combs. Microresonators with feedback enable new regimes of coherent soliton comb generation with applications in astronomy,</text>
<figure>
<location><page_8><loc_32><loc_78><loc_69><loc_90></location>
<caption>Figure 7. Micro-ring resonator with interferometric feedback loop, controlled by a micro heater, after [40].</caption>
</figure>
<text><location><page_8><loc_10><loc_68><loc_90><loc_71></location>spectroscopy and telecommunications. The concept is presently being implemented in a prototype instrument, the Potsdam Frequency Comb (POCO).</text>
<table>
<location><page_8><loc_27><loc_43><loc_72><loc_66></location>
<caption>Figure 8. Parameters of the Potsdam Frequency Comb POCO, and a future version for use at optical/blue wavelengths.</caption>
</table>
<section_header_level_1><location><page_8><loc_15><loc_33><loc_85><loc_36></location>4. DISCRETE BEAM COMBINERS (DBC) FOR LARGE ASTRONOMICAL INTERFEROMETERS</section_header_level_1>
<text><location><page_8><loc_10><loc_11><loc_90><loc_32></location>Arguably, the first PICs ever appearing for astronomical instrumentation, were used in NIR interferometers, such as the European Southern Observatory's Very Large Telescope Interferometer (VLTI) in Chile. An aerial view of the VLTI platform is shown in Fig. 9, where the channels and delay lines that connect the four Unit Telescopes and auxiliary telescopes are highlighted in white color. The location where the various beams are combined to create the interferometric fringes is indicated with the symbol of a white star. The beam combiner laboratory in its classical configuration harbors a plethora of free space optical elements like mirrors and lenses that obviously must be controlled thermally and mechanically to maintain their positions during the course of an observation to a fraction of a wavelength. If the optical path length varies in the different trains of the setup, the visibility of the fringes will be compromised. Fig. 10 is an illustration of this situation, suggesting that a tiny integrated optics beam combiner would solve a multitude of issues, concerning complexity, alignment, and stability. After initial demonstration of feasibility [41], integrated optics beam combiners have advanced to 2nd generation VLTI instrumentation [42], most prominently for the instrument GRAVITY [43]. Amongst other targets, GRAVITY has been used to observe the massive black hole in the center of the Milky Way. Perhaps one of the most striking results is the measurement of gravitational redshift of the star S2 in the orbit around the Galactic center black</text>
<text><location><page_9><loc_10><loc_80><loc_90><loc_91></location>hole [44], but there are many other measurements that make the instrument an outstanding facility. There is no doubt that the GRAVITY integrated optics beam combiner has become an essential element of the success story of this instrument. However, the technology of the device is limited in so far as it is fabricated as a planar chip whose geometry constrains the liberty of routing waveguides for four or more input beams. innoFSPEC has engaged in developing Discrete Beam Combiners (DBC) that depart from the planar geometry and allow for three-dimensional routing of waveguides in glass plates thanks to inscription with femto-second lasers, also known as Ultra-fast Laser Inscription (ULI) [45,46]. An example of a DBC is shown in the upper right of Fig. 10.</text>
<figure>
<location><page_9><loc_22><loc_51><loc_78><loc_78></location>
<caption>Figure 9. Very Large Telescope Interferometer (VLTI). Credit: ESO.</caption>
</figure>
<figure>
<location><page_9><loc_22><loc_20><loc_78><loc_44></location>
<caption>Figure 10. VLTI free space optics beam combiner. Credit: ESO. Insert: DBC glass plate for comparison, fabricated for innoFSPEC, and tested at WHT.</caption>
</figure>
<text><location><page_9><loc_10><loc_11><loc_90><loc_14></location>A four-beam DBC has been developed [47] for the astronomical H-band and tested on-sky at the 4.2m William Herschel Telescope, La Palma (WHT) [48]. As opposed to the example discussed above for GRAVITY, where</text>
<text><location><page_10><loc_10><loc_58><loc_90><loc_91></location>light is combined that comes from different telescopes, the experiment consists of a four-input pupil remapper, followed by a DBC and a 23-output reformatter. It was devised to test aperture masking - a technique that has originally been developed for single 8-10m class telescopes with the goal of diffraction limited imaging, delivering superior angular resolution in comparison to speckle interferometry, or adaptive optics. The DBC has been manufactured by Politecnico di Milano in an alumino-borosilicate substrate using ULI. A cavity-dumped Yb:KYW laser operating at 1030 nm with 300 fs pulses and a repetition rate of 1 MHz was employed for the process. In the experiment at WHT, the device was operated with a deformable mirror and a microlens array, whose purpose was to select sub-pupils of the telescope pupil, as shown in the upper left of Fig. 11, labelled 1, 2, 3, 4. The corresponding beams were injected into the DBC in plane (a) as highlighted by the colored symbols 1, . . . ,4. The corresponding waveguides were routed in three dimensions to accurately maintain the designed optical path lenghts before entering the beam combiner section proper, that is arranged in a zig-zag pattern of 23 evanescently coupling waveguides as shown in section (b). Four of these waveguides are actively fed from the input section, whereas nearest and second nearest neigboring waveguides accomplish the interference, which is read out in the reformatted linear array as shown for section (c). The intensity pattern on the output is then sensing the phases, to be recorded conveniently with a direct imaging camera. The on-sky experiment has measured visibility amplitudes and closure phases obtained on the bright stars Vega and Altair. While the coherence function could be indeed reconstructed, the results showed significant dispersion from the expected values due to the only limited signal-to-noise ratio that could be obtained, considering the limited light collecting area at a 4m class telescope. Nonetheless, the experiment has been an important first validation step towards future application for long-baseline interferometry. While this work was conducted in the H-band, further work with collaborators in the UK, Germany, and in the U.S.A. have led to similar on-sky tests with DBCs in the K-band, using the CHARA interferometer on Mt. Wilson Observatory [49-51].</text>
<figure>
<location><page_10><loc_21><loc_32><loc_78><loc_55></location>
<caption>Figure 11. Four-beam discrete beam combiner for on-sky test at William Herschel Telescope, La Palma.</caption>
</figure>
<section_header_level_1><location><page_10><loc_12><loc_22><loc_88><loc_25></location>5. APERIODIC FIBER BRAGG GRATINGS FOR COMPLEX ASTRONOMICAL FILTERS</section_header_level_1>
<text><location><page_10><loc_10><loc_13><loc_90><loc_21></location>While conclusive results are as yet not available, we mention for completeness first activities at innoFSPEC to experiment with ULI inscription in photonic chips for the purpose of creating aperiodic fiber Bragg gratings (FBG) that have been demonstrated previously to enable OH suppression on the basis of single-mode fibers [52],[53-58]. The goal of OH suppression is to filter out extremely bright atmospheric OH emission lines that limit severely the ability of large ground-based telescopes to record NIR spectra of faint galaxies. While FBG have been reported to be feasible with ULI inscription in optical fibers [59], it would be interesting to demonstrate</text>
<text><location><page_11><loc_10><loc_88><loc_90><loc_91></location>the same in photonic chips. A competing technology is currently under development using phase mask inscription with ultraviolet laser light [60].</text>
<section_header_level_1><location><page_11><loc_41><loc_84><loc_59><loc_86></location>6. CONCLUSIONS</section_header_level_1>
<text><location><page_11><loc_10><loc_73><loc_90><loc_83></location>Photonic integrated circuits are presenting interesting capabilities for instrumentation in astronomy. They are already established key components for long-baseline interferometry in the NIR. We have demonstrated custommade PICs for integrated photonic spectrographs and frequency combs, now proceeding to prototypes, hence TRL6. DBC have been developed and demonstrated for the astronomical H-band, with progress being made towards longer wavelengths in the K-band. Silica-on-Silicon, silcon nitride, and ULI inscribed glass substrates are the technology platforms that have been employed successfully. Astrophotonics can be considered a firmly established sub-discipline with interesting prospects of further growth and development.</text>
<section_header_level_1><location><page_11><loc_38><loc_70><loc_62><loc_71></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_11><loc_10><loc_65><loc_92><loc_69></location>Weacknowledge support from BMBF grants 03Z22AN11 'Astrophotonics', 03Z22A511 'astrOOptics', 03Z22AB1A 'NIR-DETECT', and 03Z22AI1 'Strategic Investment' through the BMBF program 'Unternehmen Region', and DFG grant 326946494, 'NAIR', at the Zentrum fur Innovationskompetenz innoFSPEC.</text>
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</document> | [] |
2020MNRAS.492.5391S | https://arxiv.org/pdf/2001.07048.pdf | <document>
<section_header_level_1><location><page_1><loc_6><loc_93><loc_88><loc_95></location>On the variation of solar coronal rotation using SDO/AIA observations</section_header_level_1>
<text><location><page_1><loc_6><loc_91><loc_53><loc_92></location>Jaidev Sharma 1 *, Brajesh Kumar 2 , Anil K Malik 1 and Hari Om Vats 3</text>
<unordered_list>
<list_item><location><page_1><loc_6><loc_88><loc_50><loc_90></location>1 Department of Physics, C.C.S. University, Meerut, 250001, India.</list_item>
<list_item><location><page_1><loc_6><loc_86><loc_84><loc_88></location>2 Udaipur Solar Observatory, Physical Research Laboratory, Dewali, Badi Road, Udaipur 313004 Rajasthan, India.</list_item>
<list_item><location><page_1><loc_6><loc_84><loc_54><loc_86></location>3 Space Education and Research Foundation, Ahmedabad 380054, India.</list_item>
</unordered_list>
<section_header_level_1><location><page_1><loc_14><loc_80><loc_24><loc_82></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_52><loc_95><loc_80></location>We report on the variability of rotation periods of solar coronal layers with respect to temperature (or, height). For this purpose, we have used the observations from Atmospheric Imaging Assembly (AIA) telescope on board Solar Dynamics Observatory (SDO) space mission of National Aeronautics and Space Administration (NASA). The images used are at the wavelengths 94 Å, 131 Å, 171 Å, 193 Å, 211 Å, and 335 Å during the period from 2012 to 2018. Analysis of solar full disk images obtained at these wavelengths by AIA is carried out using flux modulation method. Seventeen rectangular strips/bins at equal interval of 10 degrees (extending from 80 0 S to 80 0 N) are selected to extract a time series of extreme ultraviolet (EUV) intensity variations to obtain autocorrelation coefficient. The peak of Gaussian fit to first secondary maxima in the autocorrelogram gives synodic rotation period. Our analysis shows the differential rotation with respect to latitude as well as temperature (or, height). In the present study, we find that the sidereal rotation periods of different coronal layers decrease with increasing temperature (or, height).Average sidereal rotation period at the lowest temperature (~ 600000 Kelvin) corresponding to AIA-171 Å which originates from the upper transition region/quiet corona is 27.03 days. The sidereal rotation period decreases with temperature (or, height) to 25.47 days at the higher temperature (~ 10 million Kelvin) corresponding to the flaring regions of solar corona as seen in AIA-131 Å observations.</text>
<text><location><page_1><loc_14><loc_50><loc_54><loc_51></location>Key words: Sun: corona, Sun: UV rotation, Sun: rotation</text>
<section_header_level_1><location><page_1><loc_6><loc_45><loc_21><loc_47></location>1 INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_6><loc_9><loc_50><loc_45></location>The rotational profile of the solar corona has created a great interest for the scientific community due to its latitudinal as well as altitudinal variations and thereby illustrating a migration from rigid to differential nature. This phenomenon is not well understood because of less prominent features in the spatial and temporal extent of the corona as compared to the solar photosphere. During the last decade, results show that coronal rotation period varies as low to high values from equator towards the poles. Tracer tracking method is one of the oldest method in which features on solar full disc (SFD) image like sunspots, granules, faculae and coronal bright points (CBPs) etc. are tracked to obtain solar rotation period (Newton and Nunn 1951 ; Howard et al. 1984 ; Balthasar et al. 1986) . The various reports, viz., Howard ( 1991, 1996 ), Shivraman et al. ( 1993 ) showed that all the features such as sunspots (SSNs), plages, filaments, faculae, coronal bright points (CBPs), supergranules, coronal holes, giant cells, etc. on the solar full disc (SFD) images also rotate as that of the Sun. Its rotational</text>
<text><location><page_1><loc_52><loc_41><loc_95><loc_47></location>profile with respect to latitude as well as altitude/height can be obtained by tracing the passage of aforementioned features across the SFD.</text>
<text><location><page_1><loc_52><loc_9><loc_95><loc_41></location>Various studies on the latitude dependent rotational profiles of solar corona by using soft X-Ray (SXR) data (Timothy et al. 1975 ; Kozuka et al. 1994 ) have reported the rigid and differential nature; however, the picture is not so clear until now. For example, study of Weberand Sturrock ( 2002 ) using the data obtained from the Yohkoh/SXT shows that coronal rotation has more rigid profile in comparison to photosphere and chromosphere. Kariyappa ( 2008 ) studied coronal rotation rate employing SFD images from the soft X-ray telescope (SXT) onboard the Yohkoh and the Hinode solar space missions. They showed that the coronal rotation has differential nature during the solar magnetic cycle as that of its neighbouring lower atmospheric layers; however it is almost independent of the phases of the solar magnetic cycle. Chandra and Vats ( 2009 ) performed time series analysis on the latitude blocks of SFD images taken by Nobeyama Radioheliograph</text>
<text><location><page_2><loc_6><loc_40><loc_50><loc_95></location>(NoRH) telescope at 17 GHz during the period 1990-2001. They obtained differential rotation period with respect to latitude extending from 60 0 S to 60 0 N, which are correlated in the phase with respect to the annual sunspot numbers and showed that the differential gradient is in antiphase with the annual sunspot numbers. Their reports show that equatorial rotation rates obtained from aforementioned analysis are in good agreement with rotation of photospheric sunspot regions estimated by Balthasar et al. ( 1986 ), Howard et al. ( 1984 ), Pulkkinen and Tuominen ( 1998 ), with chromospheric level reported by Brajsa et al. ( 2004 ) and Karachik et al. ( 2006 ). Chandra and Vats ( 2010 ) further reported that the equatorial rotation period follows a systematic trend as that of sunspot numbers and rotation period depends on the phases of solar activity cycle in case of SFD images between 80 # S to 80 # N, obtained from the soft X-ray telescope (SXT) on board the Yohkoh space mission. There are two major extensive works (Altrock et al. 2003 , Vats et al. 2001 ) for the estimation of coronal rotation and its variations, one is the optical observations at Fe X and Fe XIV lines almost over three decades and other is the estimation of radio observations at 2.8 GHz as well as other radio frequencies. Altrock et al. ( 2003 ) used the observations of Fe X (averaged over 18 years) and Fe XIV (averaged over 26 years) at a radius of 1.15 Rʘ and reported that equatorial coronal rotation period is in good agreement with the rotation of photosphere but at high latitudes, these rotation data differ.</text>
<text><location><page_2><loc_6><loc_9><loc_50><loc_39></location>Vats et al . ( 2001 ) suggested that the localized emission originates from a subpart of the solar disc. There is a possibility that feeble type III burst-like activities occur in the same region of the solar atmosphere. They used model of Aschwanden and Benz ( 1995 ) to find the height of radio emission in the solar corona and reported that radio emissions at 11 radio frequencies from 275 to 2800 MHz were originated from the mean height range of 6×10 4 km to 15×10 4 km above the photosphere. Their results suggest that the sidereal rotation period at 2800 MHz, emitted from lower corona at about 6×10 4 km above the photosphere, is 24.1 days. They also reported that at the lower frequencies, sidereal rotation period decreases with height. Recently, space-based images at higher resolution taken by EUV telescopes comprise of well-defined features on corona and hence these could be very useful in precise measurement of</text>
<text><location><page_2><loc_52><loc_60><loc_95><loc_95></location>solar rotation period. In this paper, we report on the investigation of latitudinal as well as temperature (or, height) dependent differential rotation profile of different solar coronal layers during the period from 2012 to 2018 using Flux Modulation Method on solar full disk images (SFD) from Atmospheric Imaging Assembly (AIA) telescope on board Solar Dynamics Observatory (SDO) space mission at 94 Å, 131 Å, 171 Å, 193 Å, 211 Å and 335 Å. The organization of the manuscript is as follows. In section II, we discuss the details of the data utilized for this investigation as well as the technique used to extract the EUV flux at different latitudes at equal interval of 10 deg from the solar full disc (SFD) images. In section III, we discuss methods used to calculate the rotation period at different latitudes. Section IV contains the result and discussion about the latitudinal as well as altitudinal variation of rotation periods and trend in rotation periods at different confidence levels. Conclusions are given in the last section.</text>
<section_header_level_1><location><page_2><loc_52><loc_58><loc_85><loc_59></location>2 OBSERVATIONS AND METHODOLOGY</section_header_level_1>
<text><location><page_2><loc_52><loc_15><loc_95><loc_57></location>The Solar Dynamics Observatory (SDO), a National Aeronautics and Space Administration (NASA) solar space mission, has three main instruments namely Atmospheric Imaging Assembly (AIA), The Helioseismic and Magnetic Imager (HMI) and the Extreme Ultraviolet Variability Experiment (EVE) that are designed to obtain images of Sun's atmosphere (photosphere, chromosphere, transition region and corona) aimed to study the dynamical activities taking place in the solar environment . The AIA instrument contains filters with 10 different wavelength bands to identify key aspects of solar activities. The spatial resolution of the instrument is ~ 0.6' per pixel that is approximately twice as much as Hinode/XRT resolution (i.e. ~ 1.032' per pixel) and four times of SOHO/EIT resolution (i.e. ~2.629' per pixel) (Lemen et al. 2012 ). In our study, we used preliminary observations from AIA instrument at aforementioned wavelengths. The available SFD images in the digital format having size 512×512 is utilized from the webaccessible database at a cadence of one image per day (at almost fixed time), from January 2012 to December 2018, with very less data gaps. However, this negligible gap of flux/intensity is filled by interpolation process.</text>
<text><location><page_2><loc_52><loc_9><loc_95><loc_14></location>Here, in this analysis, on SFD images total 17 latitudinal rectangular strips/bins/blocks for ± 80 0 on both hemispheres at an interval of 10 0 latitude are chosen. Each rectangular block</text>
<text><location><page_3><loc_6><loc_40><loc_50><loc_95></location>contains the width only two pixels and length covers total pixels on SFD at each latitude. All the data sets are categorized into one-year intervals from January 2012 to December 2018. A time series of daily values of EUV flux/intensity variations of multiple layers is generated from mean of available pixels values in particular bin. We obtain autocorrelation coefficients using standard subroutines of Interactive Data Analysis (IDL) software to find out rotation period. Most of the time series of EUV flux at different latitudes contains the information of coronal features that dominate in the investigation of rotation period at that latitude. The coronal structure indicates that large-scale emitters can remain on SFD for several rotation periods (Fisher et al. 1984) . Due to long lived features/emitters on SFD, it is reasonable to utilize the autocorrelation analysis to obtain the precious rotation rates. (Hansen et al. 1969 ; Sime et al. 1989 ; Vats et al. 1998 and Weber et al. 1999 ) have demonstrated that autocorrelation analysis is a very precious statistical process to find any prominent emitters present in the observations and thereby determine the coronal rotation. During our analysis, we observed that at some latitude bins, the peaks of autocorrelogram are not too smooth to give rotation period that may be because short-lived features behave like a noise. This could also be due to less periodicity in features, interference of noise due to crosstalk and also due to poor availability of features. In order to obtain high accuracy in synodic rotation period, Gaussian fitted peak value of secondary maxima (first peak of autocorrelogram) is used.</text>
<section_header_level_1><location><page_3><loc_6><loc_38><loc_31><loc_39></location>The Gaussian function is defined as</section_header_level_1>
<formula><location><page_3><loc_7><loc_31><loc_48><loc_37></location>(1) 2 2 0 ) ( 2 0 2 w x x e w A y y --+ = p</formula>
<text><location><page_3><loc_6><loc_24><loc_50><loc_30></location>where = center of the maxima, = 2 times the standard deviation of the Gaussian fit, A = area under the curve and = baseline offset. 0 x w 0 y</text>
<text><location><page_3><loc_6><loc_15><loc_50><loc_22></location>Recently, parametric and nonparametric methods have been applied to ascertain trends in time series observations. However, scientific community migrated towards the application of nonparametric tests because it is more precious</text>
<text><location><page_3><loc_52><loc_83><loc_95><loc_95></location>for non-normally distributed and censored data, including missing values. These methods are a little bit influenced by the presence of outliers in the data. The MK test (Mann, 1945 ; Kendall, 1975 ) is one of the popular methods to identify the trend in time series data. The MK trend test was first carried out by computing an S statistic as follows:</text>
<text><location><page_3><loc_52><loc_76><loc_95><loc_81></location>Suppose 𝑥 & , 𝑥 ' , ………… , 𝑥 * represents n observations where 𝑥 + represents the data points at time j, then the Mann-Kendall statistics (S) is given by</text>
<formula><location><page_3><loc_52><loc_71><loc_95><loc_73></location>𝑆 = ∑ ∑ 𝑠𝑔𝑛(𝑥 + * +345& *6& 43& -𝑥 4 ) (2)</formula>
<formula><location><page_3><loc_52><loc_64><loc_95><loc_68></location>sgn( 𝜃) = : 1 𝑖𝑓 𝜃 > 0 0 𝑖𝑓 𝜃 = 0 1 𝑖𝑓 𝜃 < 0 (3)</formula>
<text><location><page_3><loc_52><loc_56><loc_95><loc_63></location>Above conditions are utilized by assuming that the sample data are independently and identically distributed. Here, S statistic defined by Eq. (2) has mean and variance as follows (Kendall, 1975) .</text>
<formula><location><page_3><loc_52><loc_54><loc_95><loc_55></location>E(S) = 0 (4)</formula>
<formula><location><page_3><loc_52><loc_51><loc_95><loc_53></location>𝑉𝑎𝑟 (𝑆) = C CD [ 𝑛(𝑛 - 1)(2𝑛 + 5) - ∑ 𝑡 I J I3& K𝑡 I -1L(2 𝑡 I +5) ] (5)</formula>
<text><location><page_3><loc_52><loc_45><loc_95><loc_51></location>Where n is sample size, g is the number of tied groups (group having the same value) and 𝑡 I is the number of data points in the 𝑝 NO group.</text>
<text><location><page_3><loc_52><loc_43><loc_80><loc_44></location>The original MK test can be computed as</text>
<formula><location><page_3><loc_52><loc_36><loc_95><loc_42></location>Z= ⎩ ⎨ ⎧ (S6&) TUVW (S) 𝑆 > 0 0 𝑆 = 0 (S5&) TUVW (S) 𝑆 < 0 (6)</formula>
<text><location><page_3><loc_52><loc_20><loc_95><loc_36></location>If - 𝑍 &6Y '/ ≤ Z ≤ 𝑍 &6Y '/ is satisfied then at significance level of 𝛼, the null hypothesis is accepted with no trend. Failing which, the null hypothesis is rejected while alternative hypothesis is accepted at the same significance level. Now the probability density function f(z) has to be defined to ascertain the significant trend (increasing or decreasing). If a normal distribution has mean 0 and standard deviation 1 than the probability density function is defined as follows:</text>
<formula><location><page_3><loc_52><loc_16><loc_94><loc_19></location>𝑓(𝑧) = & √' _ 𝑒 a b b (7)</formula>
<figure>
<location><page_4><loc_7><loc_58><loc_47><loc_97></location>
</figure>
<figure>
<location><page_4><loc_50><loc_57><loc_89><loc_97></location>
<caption>Figure 1: Above panels are typical examples of the time series of average EUV flux and corresponding autocorrelograms at 30 0 South at each wavelength for the year 2017.</caption>
</figure>
<text><location><page_4><loc_6><loc_37><loc_50><loc_50></location>If Z is negative having the condition that computed probability has larger value in comparison to the level of significance α then the trend is said to be decreasing. On the other hand the trend is said to be increasing if Z is positive under the condition that computed probability has larger value as compared to the level of significance α. The condition of no trend at significant level α is satisfied if α has larger value than computed probability.</text>
<section_header_level_1><location><page_4><loc_6><loc_35><loc_21><loc_36></location>3 DATA ANALYSIS</section_header_level_1>
<text><location><page_4><loc_6><loc_8><loc_50><loc_34></location>Autocorrelation upto a lag of 150 days is calculated and plotted in the form of autocorrelogram for the time-series of each latitude bins. Figure 1 shows average EUV flux and typical autocorrelograms for a data set of the year 2017 at 30 0 South at each wavelength. Autocorrelation analysis has been performed by using above mentioned time series flux. Almost all the autocorrelograms obtained by time series flux have smooth nature that causes a fair amount of accuracy in the estimation of rotation period. The autocorrelograms in Figure 1 show several smooth periodic peaks with consecutive decreasing amplitudes. The amplitudes of oscillatory features decrease with increase in lag perhaps due to change in temporal solar rotation period. Here in each case, the first peak of autocorrelogram is fitted by the</text>
<text><location><page_4><loc_53><loc_45><loc_97><loc_50></location>Gaussian function to estimate the synodic rotation period (location of peak at horizontal axis in days) because it seems to be more clear, smoother and higher than other peaks.</text>
<text><location><page_4><loc_53><loc_8><loc_97><loc_44></location>The error in determining synodic rotation period for farthest position of the peak is much greater than for the closest peak (Chandra et.al. 2010 ). Thus, in order to obtain the rotational nature at highest accuracy, it appears essential to select the first peak. However, this possible accuracy will vary with latitude bins. Hence, we take the horizontal value (lag in days) of secondary maxima corresponding to first peak of each autocorrelogram to estimate the synodic rotation period. Further to reduce the standard errors and to enhance accuracy, Gaussian fitting is carried out. Standard errors in the fitting of Gaussian function are significantly low which results in high accuracy in the measurements. We have not found sufficient latitudinal coverage of rotation periods in some years (AIA-171 in 2015 and AIA-335 in 2014 & 2018) owing to the less periodicity in data due to uniformity or randomness of EUV emitters or inclusion of noise at corresponding latitude. The data of AIA335 in year 2016 is not updated at the webpage of SDO so this year is also excluded from the analysis. For the sake of</text>
<text><location><page_5><loc_6><loc_86><loc_50><loc_98></location>comparison of all findings, the average EUV flux at higher latitudes (>60 0 ), in both the hemispheres, have insufficient possibility of containing the rotational periodicity and so is excluded from the analysis. The conversion of synodic rotation period into sidereal rotation period is given by the following equation.</text>
<formula><location><page_5><loc_6><loc_83><loc_48><loc_86></location>𝑻𝒔𝒊𝒅𝒆𝒓𝒆𝒂𝒍 = 𝟑𝟔𝟓.𝟐𝟔∗𝑻 𝒔𝒚𝒏𝒐𝒅𝒊𝒄 𝟑𝟔𝟓.𝟐𝟔5𝑻 𝒔𝒚𝒏𝒐𝒅𝒊𝒄 (8)</formula>
<figure>
<location><page_5><loc_7><loc_65><loc_49><loc_81></location>
<caption>Figure 2: The plot shows the Gaussian fit to the 1 st secondary maxima of the autocorrelogram of EUV flux at 211 Å, and therefore optimum position of peak needs to be selected.</caption>
</figure>
<text><location><page_5><loc_6><loc_48><loc_50><loc_55></location>In previous studies, most of the measurements of solar rotations are traditionally fitted with a polynomial expressing the latitudinal dependence with rotation rate (Howard and Harvey 1970 ; Schroter et al. 1985 ) and given as</text>
<formula><location><page_5><loc_6><loc_45><loc_48><loc_47></location>𝝎 𝒓𝒐𝒕 (𝒃) = 𝑪 𝟎 +𝑪 𝟏 𝑺𝒊𝒏 𝟐 𝒃 + 𝑪 𝟐 𝑺𝒊𝒏 𝟒 𝒃 (9)</formula>
<text><location><page_5><loc_6><loc_13><loc_51><loc_45></location>here 𝜔 W~N (𝑏) is solar rotation rate, b is latitude, 𝐶 # represents equatorial rotation rate and 𝐶 & 𝑎𝑛𝑑 𝐶 ' represent differential gradients with 𝐶 & for lower latitudes and 𝐶 ' for higher latitudes. In Equation (9), parameters 𝐶 & 𝑎𝑛𝑑 𝐶 ' are hold opposite correlation (termed as cross talk between the coefficients) that causes problems to compare with other results (Snodgrass 1984 ; Snodgrass and Ulrich 1990 ). Many techniques to remove this crosstalk have been reported in the literature. For example, Howard et al, 1984 ; Pulkkinen and Tuominen 1998 ; Brajsa et al. 2002a ; Sudar et al. ( 2014 ) put 𝐶 ' =0, Scherrer et al, (1980) put 𝐶 & = 𝐶 ' , and Ulrich et al. ( 1988 ) put 𝐶 ' = 1.0216295 𝐶 & . However, no justification has been given in favor of these arguments. Vats et al. ( 2011) reported North-South asymmetry in rotation rate on both the hemispheres. To achieve better accuracy, it is reasonable to fit asymmetric equation to estimate the rotational coefficients. Thus, we fit our data with</text>
<text><location><page_5><loc_53><loc_94><loc_97><loc_98></location>asymmetric expression given in Equation (10) and observe that our data strongly supports the asymmetric rotational profile.</text>
<formula><location><page_5><loc_53><loc_90><loc_95><loc_94></location>𝜔 W~N (𝑏) = 𝐶 # +𝐶 & 𝑆𝑖𝑛𝑏+𝐶 ' 𝑆𝑖𝑛 ' 𝑏 + 𝐶 GLYPH<135> 𝑆𝑖𝑛 GLYPH<135> 𝑏 + 𝐶 GLYPH<136> 𝑆𝑖𝑛 GLYPH<136> 𝑏 (10)</formula>
<text><location><page_5><loc_53><loc_74><loc_97><loc_90></location>Here, 𝐶 # is equatorial rotation rate, 𝐶 & and 𝐶 GLYPH<135> determine asymmetry on both the hemispheres and 𝐶 ' and 𝐶 GLYPH<136> represent the gradients at middle and upper latitudes. The analysis of our data for the period of January 2012 to December 2018 shows that by setting the coefficient 𝐶 GLYPH<136> =0, fitting process has less root mean square (RMSE) in comparison of other setting ( 𝐶 ' = 𝐶 GLYPH<136> ) for removal of crosstalk. The Sidereal rotation period (in days) corresponding to rotation rate (deg/day) is given as</text>
<formula><location><page_5><loc_53><loc_71><loc_96><loc_73></location>𝑇(𝑏) = GLYPH<135>GLYPH<138>#˚ GLYPH<140> GLYPH<141>GLYPH<142>GLYPH<143> (GLYPH<144> ) (11)</formula>
<text><location><page_5><loc_53><loc_69><loc_71><loc_70></location>together with b as latitude.</text>
<text><location><page_5><loc_53><loc_50><loc_97><loc_68></location>In Figure 2 , autocorrelation coefficient and Gaussian fit of secondary maxima at first peak is plotted for the year 2014 at 30 0 South. This plot shows that autocorrelation coefficient follows the Gaussian function with most acceptable value of fitting parameter R 2 (Pearson's coefficient). Almost similar trend is observed for all latitudes and all the duration considered in this work. The RMSE in fitting process of equation (10) has small values that indicate the fitting is reasonably good with higher accuracy in results.</text>
<section_header_level_1><location><page_5><loc_53><loc_47><loc_77><loc_49></location>4 RESULTS AND DISCUSSION</section_header_level_1>
<text><location><page_5><loc_53><loc_13><loc_97><loc_47></location>The solar rotation profile with respect to height in the solar atmosphere is not clearly understood as yet. Since solar rotation has implications on the solar dynamo processes, it is quite necessary to understand this important phenomenon on the Sun. As of now, it is well known that the solar interior rotates faster as compared to photosphere (Howe 2009 ). Further; rotation periods of individual latitudes with respect to depth in the solar interior follow a complex pattern. Vats et al. ( 1999 ) used radio flux for solar cycles 21 and 22 at 2.8 GHz and determined that the solar corona has higher rotation rate in comparison to lower regions of the Sun. Again Vats et al. ( 2001 ) used solar radio flux from the Cracow Astronomical Observatory at 275, 405, 670, 810, 925, 1080, 1215, 1350, 1620, 1755 and at 2800 MHz from the Algonquin Radio Observatory in Canada (From duration 1June 1997 to 31 July 1999) and reported that coronal rotation periods have downward trend with respect to increasing height in the coronal layers.</text>
<figure>
<location><page_6><loc_7><loc_53><loc_94><loc_96></location>
<caption>Figure 3: Contour plots showing yearly profile of sidereal rotation period in days (shown in color code) from 2012 to 2017 with respect to latitude in degree (vertical direction) as well as temperature in Kelvin in horizontal direction.</caption>
</figure>
<section_header_level_1><location><page_6><loc_6><loc_43><loc_50><loc_46></location>4.1 LATITUDINAL AS WELL AS ALTITUDINAL PROFILE IN ROTATION</section_header_level_1>
<text><location><page_6><loc_6><loc_31><loc_50><loc_42></location>We show in Figure 3 and 4 , the latitude as well as temperature (or, height) dependent profiles of yearly sidereal rotation period for the duration from 2012 to 2018. In these contour plots, vertical direction shows latitudinal rotational profile for each year whereas along the horizontal direction sidereal rotation period with respect to temperature (or, height) is shown.</text>
<text><location><page_6><loc_6><loc_20><loc_50><loc_30></location>As expected, these plots show lower sidereal rotation period at equatorial region and these gradually increase towards the poles. Rotation period of individual latitudes (spanning -60 deg to +60 deg) with respect to temperature (or, height) follows a complex pattern as that of solar interior.</text>
<section_header_level_1><location><page_6><loc_6><loc_18><loc_34><loc_20></location>4.2 TREND IN ROTATION PERIOD</section_header_level_1>
<text><location><page_6><loc_6><loc_8><loc_50><loc_18></location>In this analysis, Mann-Kendall test is applied to estimate the significance level in the trend in sidereal rotation period with respect to increasing temperature (or, height). Here, we report yearly trend, overall trend and band wise trend at different confidence levels. Table 1 provides the important parameters of</text>
<text><location><page_6><loc_52><loc_33><loc_95><loc_46></location>this test like S , Z, corresponding probability, and trends. These statistical parameters have been discussed in detail in the section 2 of the paper. Present analysis concludes that in most of the years sidereal rotation period decreases with increasing temperature (or, height) at different confidence levels (c.f., Table 2). However, this trend is statistically significant in 2015 and 2017.</text>
<figure>
<location><page_6><loc_53><loc_12><loc_98><loc_30></location>
<caption>Figure 4 : Similar as in figure 3 but for the year 2018.</caption>
</figure>
<figure>
<location><page_7><loc_7><loc_58><loc_94><loc_93></location>
</figure>
<figure>
<location><page_7><loc_6><loc_39><loc_49><loc_58></location>
<caption>Figure 5 : Panel (a) shows a typical diagram of downward trend in 2017 whereas panel (b) represents overall trend (yearly averaged) in sidereal rotation period (days) with respect to temperature (kelvin). Panels (c), (d) and (e) represent the trend in the southern hemispheric band (average of -60 to -40 latitude), equatorial band (average of -10 to +10 latitude) and northern hemispheric band (average of 40 to 60 latitude) respectively. In above panels linear regression is carried out to estimate the probable trend. These results show downward trend in sidereal rotation period with increasing temperature (or, height) at different confidence levels (c.f., Table 1 ).</caption>
</figure>
<text><location><page_7><loc_6><loc_8><loc_50><loc_15></location>The gradients of yearly trends have different values with minimum in 2018 and maximum in 2012 having random systematic variation. Overall trend of sidereal rotation period has interesting downward trend with increasing temperature</text>
<text><location><page_7><loc_52><loc_38><loc_95><loc_58></location>(or, height) at statistically very high significance level (97.6%) with variation from 27.03 to 25.47 days. The outcomes in our study and Vats et al. ( 2001 ) have similar downward trend in sidereal rotation periods with increasing temperature (or, height). The variation in sidereal rotation periods of different coronal layers found in our analysis is 1.16 days more than Vats et al. ( 2001 ) which shows that altitudinal rotational profile of coronal layers is more differential than Vats et al. ( 2001 ).This could be due to more coverage of height in present analysis as compared to Vats et al.( 2001 ).</text>
<text><location><page_7><loc_52><loc_15><loc_95><loc_37></location>Southern region (-60,-50,-40), equatorial region (-10, 0, 10) and northern region (40, 50, 60) also follow the downward trend in rotation period with increasing temperature (or, height) at considerably good confidence levels (99.7, 86 and 99.2%, respectively) with increasing gradients from South to North directions. Here, variation in rotation periods in this range of temperatures is 4.5%, 2.23% and 1.4%, respectively, that means from Southern region towards Northern region, altitudinal profile of rotation period is consecutively becoming less differential. The reason for this observed phenomenon remains an enigma.</text>
<text><location><page_7><loc_52><loc_9><loc_95><loc_15></location>Altogether, our analysis confirms a clear downward trend in rotation period with increasing temperature (or, height) of different coronal layers.</text>
<table>
<location><page_8><loc_5><loc_71><loc_49><loc_93></location>
<caption>Table 1: Results of Mann-kendall test</caption>
</table>
<text><location><page_8><loc_4><loc_56><loc_13><loc_57></location>AIA Channels</text>
<table>
<location><page_8><loc_7><loc_42><loc_50><loc_57></location>
<caption>Table 2 : Summary of the regions of the solar corona sampled by the various AIA observations at different wavelengths and the corresponding temperatures of those regions (Lemen et al. 2012 ).</caption>
</table>
<section_header_level_1><location><page_8><loc_6><loc_39><loc_20><loc_40></location>5 CONCLUSIONS</section_header_level_1>
<text><location><page_8><loc_6><loc_8><loc_49><loc_38></location>Our results show that latitude dependent coronal rotation profiles with respect to increasing temperature (or, height) have no systematic variation as observed in the case of solar interior. The reason for this complex pattern is still an enigma. However, average rotation of latitudinal bands (-60,50,-40), (-10, 0, 10) and (40, 50, 60), follow systematic downward trend in rotation period corresponding to increasing temperature at considerably good confidence levels. The yearly trend in this analysis follows significant downward trend in the years 2015 and 2017, however the remaining years follow this trend at relatively low confidence levels. This could be due to the complexities in the rotational features on coronal layers, noise contents and errors in the tools applied. As far as overall trend in rotation period is concerned, there is a gross downward trend in the solar</text>
<text><location><page_8><loc_52><loc_55><loc_94><loc_95></location>corona with respect to increasing temperature (or, height) . The present data set has six AIA observing channels, so MK test is applied to six data points which is giving reasonably consistent indication of trend. However, in statistical analysis, more number of data points is always recommended for better accuracy. Our findings of rotation rate variation as a function of temperature is reasonably supported by Livingston et al. ( 1969 ) who compared rotation in chromosphere and photosphere. They stated that the chromosphere rotates 8% faster than the photosphere. Thus, rotation rate increases (or period decreases) with increasing temperature from photosphere to chromosphere. Similarly, Howe et al. ( 2009 ) reported a considerable decreasing trend of average rotation rate from interior of the Sun to outward to the photosphere. It is to be noted that in this case the temperature decreases from the solar interior to upward in the photosphere. The physical mechanism which is causing this variation is largely unknown. However, it appears that the rotation of the solar interior and its atmosphere are linked to show similar variation with temperature.</text>
<section_header_level_1><location><page_8><loc_52><loc_52><loc_72><loc_54></location>ACKNOWLEDGEMENTS</section_header_level_1>
<text><location><page_8><loc_52><loc_28><loc_95><loc_52></location>The authors wish to acknowledge the use of data (AIA-94, AIA-131, AIA-171, AIA-193, AIA-211 and AIA-335) from AIA onboard SDO for the period 2012-2018. These are acquired from the webpage of Solar Dynamics Observatory (SDO), a mission of National Aeronautics and Space Administration (NASA). The research at Udaipur Solar Observatory (USO), Physical Research laboratory, Udaipur is supported by Department of Space, Government of India. We also acknowledge the various supports for this research work provided by Department of Physics, Chaudhary Charan Singh University, Meerut, India. We thank the referee for useful comments and suggestions to improve our manuscript.</text>
<section_header_level_1><location><page_8><loc_52><loc_24><loc_62><loc_25></location>REFRENCES</section_header_level_1>
<text><location><page_8><loc_52><loc_22><loc_78><loc_23></location>Altrock R.C., 2003, Sol. Phys, 213, 23</text>
<text><location><page_8><loc_52><loc_9><loc_94><loc_21></location>Aschwanden, M. J., & Benz, A. O. 1995, ApJ, 438, 997 Balthasar H., Vazquez M. &Woehl H., 1986, A & A, 155, 87 Brajsa R., Wohl H., Vrsnak B., 2002a, Sol. Phys., 206, 229 Brajsa R., Wohl H., Vrsnak B., 2004, A & A, 414, 707 Chandra, S. Vats H.O. &Iyer K.N., 2009, MNRAS, 400, 34 Chandra S., Vats H.O. &Iyer K.N., 2010, MNRAS, 407, 1108</text>
<table>
<location><page_9><loc_6><loc_11><loc_50><loc_95></location>
</table>
<text><location><page_9><loc_52><loc_90><loc_95><loc_95></location>Weber M.A. & Sturrock P.A., 2002, COSPAR Colloquia Series, 13, 347 Weber M.A., Acton L.W., Alexander D., Kubo S. & Hara H.,</text>
<text><location><page_9><loc_52><loc_88><loc_70><loc_89></location>1999, Sol. Phys., 189, 271</text>
</document> | [] |
2021IJMPA..3650161K | https://arxiv.org/pdf/2005.06228.pdf | <document>
<section_header_level_1><location><page_1><loc_14><loc_68><loc_77><loc_75></location>Gauge-Field-Induced Torsion and Cosmic Inflation</section_header_level_1>
<section_header_level_1><location><page_1><loc_14><loc_57><loc_42><loc_59></location>A. Kasem a and S. Khalil b</section_header_level_1>
<text><location><page_1><loc_14><loc_54><loc_88><loc_56></location>a,b Center for Fundamental Physics, Zewail City of Science and Technology, 6 October City, Giza 12578, Egypt.</text>
<text><location><page_1><loc_16><loc_52><loc_61><loc_53></location>E-mail: [email protected], [email protected]</text>
<text><location><page_1><loc_14><loc_34><loc_88><loc_50></location>Abstract. Inflation in the framework of Einstein-Cartan theory is revisited. Einstein-Cartan theory is a natural extension of the General Relativity, with non-vanishing torsion. The connection on Riemann-Cartan spacetime is only compatible with the cosmological principal for a particular form of torsion. We show this form to also be compatible with gauge invariance principle for a non-Abelian and Abelian gauge fields under a certain deviced minimal coupling procedure. We adopt an Abelian gauge field in the form of 'cosmic triad'. The dynamical field equations are obtained and shown to sustain cosmic inflation with a large number of e-folds. We emphasize that at the end of inflation, torsion vanishes and the theory of Einstein-Cartan reduces to the General Relativity with the usual FRW geometry.</text>
<section_header_level_1><location><page_2><loc_14><loc_86><loc_23><loc_87></location>Contents</section_header_level_1>
<table>
<location><page_2><loc_14><loc_55><loc_88><loc_84></location>
</table>
<section_header_level_1><location><page_2><loc_14><loc_52><loc_30><loc_53></location>1 Introduction</section_header_level_1>
<text><location><page_2><loc_14><loc_37><loc_88><loc_50></location>From the outset of the general theory of relativity (GR), it has been playing a major role as a framework for our thinking about the universe. The theory has witnessed many success stories throughout the past century that have been crowned by what is known as the standard model of cosmology or Λ CDM model [1-4]. The model describes a homogeneous, isotropic, and spatially-(almost)-flat universe started at a very dense state 'big bang', undergone a period of exponential expansion and has been expanding in an accelerated manner ever since. Despite what the name suggests, the model is plagued with many theoretical issues. It is only standard as a best fit model to the observational evidence we have.</text>
<text><location><page_2><loc_14><loc_30><loc_88><loc_36></location>Problems with Λ CDM model are de facto inherited from GR which is still struggling to find a foothold in the general scheme of particle physics. Success of GR in describing large scale physics should not tempt us to overlook other alternatives. Having said that, it is more likely the alternative being an extension to GR so one is guaranteed not to stray from observational tests.</text>
<text><location><page_2><loc_14><loc_23><loc_88><loc_29></location>We argue that a good, well-motivated candidate would be Einstein-Cartan theory. The theory differs by admitted a torsion tensor in space, an object conspicuous by its absence in GR. Once one starts by building a local gauge theory of Poincar'e group, Einstein-Cartan falls into place as the other forces in particle physics with no a priori reason for having a vanishing torsion.</text>
<text><location><page_2><loc_14><loc_14><loc_88><loc_22></location>Torsion can be a rather de trop object. The antisymmetic part of the connection explicitly violates gauge invariance when introducing a gauge field into curved background. In the cosmological context, several semi-classical models have been regrading torsion as a spinning fluid [5-10]. In some cases they required an unreasonable amount of fine tuning or an extremely large amount of spinning particles in the early universe. But they were mainly not successful because torsion violated</text>
<text><location><page_3><loc_14><loc_87><loc_88><loc_90></location>homogeneity and isotropy of space and inhomogeneous cosmology has not yet borne fruit on the observational level.</text>
<text><location><page_3><loc_14><loc_72><loc_88><loc_86></location>In this paper, we aim to introduce a scenario of torsion sourced by a gauge field. We show that a minimal setup motivated by respecting FRW symmetries and gauge invariance sustains a period of exponential expansion. We start in section 2 by looking at the development of Einstein-Cartan theory as gauge theory of gravity. In section 3, we see that only a certain form of torsion is compatible with FRW symmetries and it can be sourced by a gauge field provided that a new minimal coupling prescription is adopted. In section 4, we setup the model and compute the field equations. In section 5, we show that the field equations can sustain a period of early cosmic inflation with a large number of e-folds. Our conclusions and final remarks are given in section 6.</text>
<text><location><page_3><loc_14><loc_65><loc_88><loc_72></location>Appendixes A and B provide calculations for the form of torsion compatible with FRW symmetries and gauge invariance of a nonabelian gauge field. In appendixes C and D we provide the explicit form of the energy momentum, Ricci, and Einstein tensors that we used to compute the field equations.</text>
<section_header_level_1><location><page_3><loc_14><loc_61><loc_57><loc_62></location>2 Development of Einstein-Cartan Theory</section_header_level_1>
<text><location><page_3><loc_14><loc_47><loc_88><loc_59></location>The concept of an antisymmetric affine connection was first introduced by ' Elie Cartan in 1922, later development of a gravitational theory allowing torsion did not ensue much until the sixties. This in part was due to the consensus on Einstein GR which passed all macrophysical experimental tests. Later development of particle physics made it clear that a direct measurement of gravitational effects between elementary particles is beyond our technological capabilities. However, it gave us a general scheme to govern all forces of nature. Gravity can only fit into the general scheme of things if it can be written as a gauge theory.</text>
<text><location><page_3><loc_14><loc_35><loc_88><loc_47></location>If one thinks of a global symmetry for gravity, the group of Lorentz transformations is the obvious candidate. Promoting this symmetry into local one was an endeavor first started by Utiyama [11], and later developed by Kibble [12] and Sciama [13, 14]. The theory is better known as EinsteinCartan-Sciama-Kibblle or U 4 for short. U 4 is indeed the local gauge theory of the Poincar'e group. 1 It features a connection with an antisymmetric part known as torsion; S i jk ≡ Γ i [ jk ] = 1 2 (Γ i jk -Γ i kj ). With the help of the metric postulate, one can write the connection in terms of the metric and torsion as follows</text>
<formula><location><page_3><loc_44><loc_33><loc_88><loc_34></location>Γ i jk = ˜ Γ i jk + K i jk . (2.1)</formula>
<text><location><page_3><loc_14><loc_21><loc_88><loc_32></location>Where ˜ Γ i jk being the usual Levi-Civita Connection and the contortion tensor K i jk is defined as K i jk ≡ S i jk + S i k j -S i jk . As the case with GR, one can still construct an invariant scalar quantity using the Riemann curvature tensor so the simplest Lagrangian would be L = √ -g ( 1 2 R (Γ) + L M ). By a straightforward but rather lengthy calculations, one finds the field equations by varying the action with respect to the independent variables (10 represented by the metric and 24 represented by the torsion) resulting in the field equations;</text>
<formula><location><page_3><loc_46><loc_18><loc_88><loc_20></location>G µν = T µν (2.2)</formula>
<formula><location><page_3><loc_46><loc_16><loc_88><loc_18></location>T αβγ = S αβγ , (2.3)</formula>
<text><location><page_4><loc_14><loc_87><loc_88><loc_90></location>where T µν 'Energy-momentum tensor', T αβγ 'modified torsion tensor', and S αβγ 'spin energy potential' are defined as follows;</text>
<formula><location><page_4><loc_42><loc_84><loc_88><loc_87></location>T µ ν ≡ q ,ν ∂ L M ∂q ,µ -δ µ ν L M , (2.4)</formula>
<formula><location><page_4><loc_39><loc_81><loc_88><loc_83></location>T γ αβ ≡ S γ αβ + δ γ α S λ βλ -δ γ β S λ αλ , (2.5)</formula>
<formula><location><page_4><loc_39><loc_77><loc_88><loc_80></location>S αβγ ≡ 1 2 √ -g ( δ L M δS αβγ -δ L M δS αγβ ) . (2.6)</formula>
<text><location><page_4><loc_14><loc_66><loc_88><loc_76></location>Unless one acquiesces to setting the torsion to zero, it should be regarded as fundamental as the metric itself. It is important to note that GR cannot be written as a gauge theory [17]. One can reach the field equations of GR with the local theory of translations with torsion as a gauge field strength. However, this formulation is only equivalent at the level of field equations as TEGR action involves a torsion scalar rather than a curvature scalar and the theory is formulated with a Weitzenbock connection rather than a Levi-Civita connection.</text>
<text><location><page_4><loc_14><loc_54><loc_88><loc_66></location>Few remarks on the field equations of U 4 are in order. First we note from the second field equation (2.3) is an algebraic equation so it can be fed into Eq.(2.2) so one recovers Einstein field equations with an 'effective energy momentum tensor'. Secondly, the canonical Energy-momentum (2.4) is not symmetric for a spinor field, i.e. for Dirac field it is totally antisymmetric T µ ν can = i 2 ¯ Ψ γ µ ∂ ν Ψ -i 2 Ψ γ µ ∂ ν ¯ Ψ. 2 This may give us some account as of why torsion is linked to spin in terminology. A bosonic field does not feel rotation in space so it cannot source a torsion tensor. The type of possible matter fields feeling torsion is a topic we address in the next section.</text>
<section_header_level_1><location><page_4><loc_14><loc_50><loc_34><loc_51></location>3 Source of torsion</section_header_level_1>
<text><location><page_4><loc_14><loc_40><loc_88><loc_48></location>Early cosmological treatments of U 4 adopted a semiclassical model of a spinning dust known as Weyssenhoff fluid [18]. It features a singularity-free solution [19-22], and has been studied in the context of inflation [5, 6, 10]. A number of these models have underplayed the incompatibility of this fluid with the symmetries of FRW universe [23, 24]. To get more insight we can look at the torsion covariantly split into three irreducible parts [25];</text>
<formula><location><page_4><loc_40><loc_36><loc_88><loc_38></location>S c ab = T S c ab + A S c ab + V S c ab , (3.1)</formula>
<text><location><page_4><loc_14><loc_32><loc_88><loc_35></location>where V S c ab , A S c ab , and T S c ab are vector, axial vector and tensorial components; respectively. They are defined as follows</text>
<formula><location><page_4><loc_40><loc_28><loc_88><loc_31></location>V S c ab ≡ 1 3 ( S a δ c b -S b δ c a ) , (3.2)</formula>
<formula><location><page_4><loc_40><loc_26><loc_88><loc_27></location>A S c ab ≡ g cd S [ abd ] , (3.3)</formula>
<formula><location><page_4><loc_40><loc_24><loc_88><loc_25></location>T S c ab ≡ S c ab -A S c ab -V S c ab , (3.4)</formula>
<text><location><page_4><loc_14><loc_17><loc_88><loc_22></location>where the one-indexed S a is defined as S a ≡ S b ab . We wish to only keep the parts in S c ab that make it form-invariant on a spatially maxiamally-symmetric space. To this end we require the vanishing of the Lie derivative with respect to the 6 Killing vectors of that symmetry; L ξ S a ab = 0. In Appendix</text>
<text><location><page_5><loc_14><loc_86><loc_88><loc_90></location>A, we present a succinct proof following the work of [23]. The result is a vanishing tensorial part T S c ab = 0 and a particular form for the vector and axialvector parts, namely</text>
<formula><location><page_5><loc_41><loc_82><loc_88><loc_85></location>S 1 01 = S 2 02 = S 3 03 = F ( t ) , S 123 = S [123] = f ( t ) , (3.5)</formula>
<text><location><page_5><loc_14><loc_67><loc_88><loc_81></location>where F ( t ) and f ( t ) are arbitrary functions of time. It is evident that this form is incompatible with the spinning fluid description which has a non-vanishing traceless part. The cosmological implications of torsion in the form of (3.5) have been investigated in various works [26-30]. As this type of models undermine a spinning fluid description, it gives up one of its significant results; a universe free of cosmic singularity. Moreover, the functions F ( t ), f ( t ) are introduced in an ad hoc manner with no available physical interpretation to its source. Therefore, one would be reluctant to take up this pursuit any further, however we show that the vector torsion part can be a grist to the mill of cosmology if it couples to a gauge field.</text>
<text><location><page_5><loc_14><loc_60><loc_88><loc_66></location>The possibility of sourcing torsion with a gauge field is often dismissed as torsion is known to violate gauge invariance, i.e. if one tries the natural generalization of the field strength of gauge field (Abelian or non-Abelian) to Einstein-Cartan manifold through replacing the partial derivative by a space-time derivative: ∂ µ A ν → ∂ µ A ν -Γ λ µν A λ , one finds</text>
<formula><location><page_5><loc_34><loc_55><loc_88><loc_59></location>F i µν ≡ ∇ µ A i ν -∇ ν A i µ + gC i jk A j µ A k ν = ∂ µ A i ν -∂ ν A i µ -S λ µν A i λ + gC i jk A j µ A k ν . (3.6)</formula>
<text><location><page_5><loc_14><loc_49><loc_88><loc_54></location>Thus, gauge invariance is found to be manifestly violated due to the antisymmetric part of the connection. One way to avoid this violation is adopting the flat-space expression for F µν . However, this defeats our purpose in introducing this field in the first place.</text>
<text><location><page_5><loc_14><loc_38><loc_88><loc_48></location>A more satisfactory approach have been introduced by Hojman, et al. [31], where a new minimal coupling procedure has been devised to allow for a nonvanishing torsion in the definition of an Abelian gauge field strength. Then, it was shorty generalized for a non-abelian field as well [32]. In this procedure, one defines the gauge transformation to allow a set of space-time functions b α µ into the the definition of a covariant derivative such that; D µ = ∇ µ -igb α µ A α . This affects the transformation of the gauge field in the following way</text>
<formula><location><page_5><loc_38><loc_36><loc_88><loc_37></location>A i µ ' = A i µ -g -1 c ν µ θ i ; ν + C i jk θ j A k µ , (3.7)</formula>
<text><location><page_5><loc_14><loc_31><loc_88><loc_34></location>where c ν µ b α ν ≡ δ α µ . Now the following modified field strength can be made invariant (see Appendix B for detailed calculations):</text>
<formula><location><page_5><loc_33><loc_29><loc_88><loc_30></location>F i µν ≡ ∂ µ A i ν -∂ ν A i µ -S β µν A i β + ge -f C i jk A j µ A k ν , (3.8)</formula>
<text><location><page_5><loc_14><loc_14><loc_88><loc_27></location>under the constraints of c ν µ = e f ( X ) δ ν µ and S σ µν ≡ δ σ ν f ,µ -δ σ µ f ,ν , which coincides with the expression of vector torsion Eq.(3.2). It also conforms with the FRW-compatible torsion when one assumes f as a function of time f ( X ) ≡ f ( t ). One can think of this procedure as giving a nonconstant gauge coupling g ( t ) = ge -f ( t ) . So in the limit of vanishing torsion, the usual minimal coupling procedure is restored with c ν µ = δ ν µ and g ( t ) ≡ g . Our discussion extends to three Abelian fields A i µ which can be thought of as a special case of SU (2) nonabelian gauge theory with vanishing structure constants C i jk → 0. For the purpose of constructing a minimal setup, we stick with the Abelian version in the following analysis.</text>
<section_header_level_1><location><page_6><loc_14><loc_88><loc_42><loc_90></location>4 Model with cosmic triad</section_header_level_1>
<text><location><page_6><loc_14><loc_82><loc_88><loc_87></location>Having established an FRW-compatible form of torsion, we would like to couple it with gauge gauge fields respecting the same symmetries. We adopt an ansatz of three equal and mutually-orthogonal vectors known as 'cosmic triad' was proposed by [33];</text>
<formula><location><page_6><loc_44><loc_80><loc_88><loc_81></location>A a i = φ ( t ) · a ( t ) δ a i , (4.1)</formula>
<text><location><page_6><loc_14><loc_74><loc_88><loc_78></location>where a ( t ) is the FRW scale factor. The cosmological implication of this choice has been investigated in many different contexts [34-42]. Here we take the simplest form with three identical copies of an Abeilan field coupled with torsion as per the prescription of the previous section;</text>
<formula><location><page_6><loc_41><loc_71><loc_88><loc_73></location>F a 0 i = ( a ˙ φ + ˙ aφ ) δ a i -φa ˙ fδ a i (4.2)</formula>
<text><location><page_6><loc_14><loc_64><loc_88><loc_71></location>The standard Lagrangian L = √ -g [ 1 2 ( R -2Λ) -1 4 F a µν F µν a + L M ] is constructed assuming FRW background and the aformentioned choice for torsion and fields. We assume a flat space, a vanishing cosmological constant and a universe dominated by the triad field. Also we relabel the torsion function ˙ f → χ ( t ). Now the above Lagrangian simplifies into;</text>
<formula><location><page_6><loc_35><loc_60><loc_88><loc_63></location>L = 3 a ˙ a 2 -12 χ 2 a 3 + 3 a 2 (˙ aφ + a ˙ φ -φaχ ) 2 (4.3)</formula>
<text><location><page_6><loc_14><loc_55><loc_88><loc_60></location>We calculate the field equations (2.2, 2.3), using the energy-momentum tensor and the Einstein tensor as given in appendix D, and C. A straightforward but rather onerous calculations gives the following Freidman-like equations and an extra equation for the torsion field:</text>
<formula><location><page_6><loc_34><loc_51><loc_88><loc_54></location>( ˙ a a ) 2 = 1 3 ρ A -4 χ ˙ a a -4 χ 2 , (4.4)</formula>
<formula><location><page_6><loc_36><loc_49><loc_88><loc_51></location>a a = -1 6 ( ρ A +3 P A ) -2 χ ˙ a a -2 ˙ χ, (4.5)</formula>
<formula><location><page_6><loc_36><loc_47><loc_88><loc_48></location>¨ φ = -(3 H + χ ) ˙ φ +(2 χ 2 + ˙ χ -2 H 2 -˙ H ) φ, (4.6)</formula>
<formula><location><page_6><loc_37><loc_42><loc_88><loc_45></location>χ = 1 2 a 3 δ L M δχ ⇒ χ = 3 φ 2 3 φ 2 -2 ( ˙ φ φ + H ) . (4.7)</formula>
<text><location><page_6><loc_14><loc_45><loc_18><loc_46></location>with</text>
<text><location><page_6><loc_14><loc_36><loc_88><loc_41></location>As we expect, the torsion equation (4.7) is algebraic, so one can use it to substitute for the nondynamical field. By introducing the Hubble parameter H ( t ) ≡ ˙ a a and explicitly substitute for ρ A and P A , defined in Appendix C, one finds that the above equations can be written as</text>
<formula><location><page_6><loc_24><loc_33><loc_61><loc_36></location>H 2 = 2 Hφ + ˙ φ (2 -3 φ 2 ) 2 ( (1 -18 φ 2 ) ˙ φ +(13 -36 φ 2 ) Hφ ) ,</formula>
<formula><location><page_6><loc_25><loc_25><loc_65><loc_27></location>¨ φ = H (2 -3 φ 2 ) 2 ( 3(2 -5 φ 2 ) ˙ φ +4(1 -3 φ 2 ) Hφ ) -φ ˙ H.</formula>
<formula><location><page_6><loc_20><loc_25><loc_88><loc_35></location>(4.8) ˙ H + H 2 = 2 (2 -3 φ 2 ) 2 ( (5 + 9 φ 2 ) ˙ φ 2 -9( H 2 + ˙ H ) φ 4 +(5 H 2 +6 ˙ H ) φ 2 -9( H ˙ φ + ¨ φ ) φ 3 + 2(8 H ˙ φ +3 ¨ φ ) φ ) , (4.9) (4.10)</formula>
<section_header_level_1><location><page_6><loc_14><loc_22><loc_38><loc_23></location>5 Inflationary Scenario</section_header_level_1>
<text><location><page_6><loc_14><loc_17><loc_88><loc_20></location>To understand the cosmic evolution let us start with Eq.(4.8), which is a quadratic equation of the Hubble parameter, one can solve to get H ( φ, ˙ φ )</text>
<formula><location><page_6><loc_37><loc_13><loc_88><loc_17></location>H = ± 2 √ 2 + (14 -3(18 φ ± √ 2) φ ) φ 4 -38 φ 2 +81 φ 4 ˙ φ. (5.1)</formula>
<figure>
<location><page_7><loc_35><loc_74><loc_69><loc_89></location>
<caption>Figure 1 : The inflationary parameter glyph[epsilon1] as function of the field φ</caption>
</figure>
<text><location><page_7><loc_14><loc_66><loc_77><loc_67></location>Substituting this solution as well as ¨ φ from Eq.(4.10) into Eq.(4.9), one finds ˙ H ( φ, ˙ φ )</text>
<formula><location><page_7><loc_15><loc_62><loc_88><loc_65></location>˙ H = ∓ 2 ˙ φ 2 (4 -38 φ 2 +81 φ 4 ) 2 ( ∓ 16+ ( 32 √ 2+( ± 412+3( -52 √ 2+3( ∓ 254+9(2 √ 2 ± 45 φ ) φ ) φ ) φ ) φ ) φ ) (5.2)</formula>
<text><location><page_7><loc_14><loc_55><loc_88><loc_60></location>Now we are set to find the inflationary parameter glyph[epsilon1] ≡ ˙ H H 2 , which we use to identify the inflationary period a > 0 as a a = H 2 (1 -glyph[epsilon1] ), so exponential growth is going in the region of glyph[epsilon1] < 1. From the above expressions of H and ˙ H , one obtains the following relation of glyph[epsilon1] in terms of φ scalar field.</text>
<formula><location><page_7><loc_39><loc_51><loc_88><loc_55></location>glyph[epsilon1] ≡ -˙ H H 2 = -8 ∓ 24 √ 2 φ -90 φ 2 ( √ 2 ± 6 φ 2 ) 2 . (5.3)</formula>
<text><location><page_7><loc_14><loc_36><loc_88><loc_50></location>We stress that we only use this parameter to define the inflationary period with no requirement on epsilon as to being positive or infinitesimal. For reasons that should be obvious shortly, we find only the second solution can sustain a period of inflation large enough to generate a desirable number of e-fold so we stick to it in the following analysis. In Fig. 1, we plot inflationary parameter glyph[epsilon1] versus the scalar field φ . As can be seen from this figure, the inflationary period ( glyph[epsilon1] < 1) is going on as long as the field is less than a certain value φ < 1 9 ( √ 2+ √ 17) ≈ 0 . 62. It is clear that for a large field inflation, i.e. φ > 1, inflation starts when φ is rolling down to vales less than 0 . 62, while for a small field, i.e. φ starts from values close to zero and rolls up, the inflation ends when glyph[epsilon1] approaches one at φ glyph[similarequal] 0 . 62.</text>
<text><location><page_7><loc_14><loc_32><loc_88><loc_35></location>Now, we consider the number of e-folds N of inflation that are needed to solve the horizon problem. In general N is defined as N ≡ ∫ t f t i Hdt . In our model, one can show that it is given by</text>
<formula><location><page_7><loc_18><loc_27><loc_88><loc_31></location>N = 1 111 ( (2 √ 37 -37) log( √ 2 + √ 74 -18 φ ) -(2 √ 37 + 37) log( -√ 2 + √ 74 + 18 φ ) ) ∣ ∣ ∣ ∣ ∣ φ f φ i (5.4)</formula>
<text><location><page_7><loc_14><loc_17><loc_88><loc_26></location>In Fig. 2, we show the integral of e-folds which has a notable pole at φ ∗ = 1 18 ( √ 2 + √ 74) ≈ 0 . 556, and the required value N glyph[similarequal] 60 can be obtained when the field φ approach this point. In this case, we have a new scenario of inflation, where the field φ is rolling down from a relatively large value with an initial velocity, an inflationary period starts at φ i (the point for glyph[epsilon1] = 1) and as the field approaches the pole at φ ∗ the number e-folds blows up, as shown in Fig. 2, and inflation abruptly ends.</text>
<text><location><page_7><loc_14><loc_14><loc_88><loc_17></location>The abrupt end of the inflation can be understood by considering the evolution of the field φ , by substituting Eq.(5.1) and Eq.(5.2) into Eq.(4.10), one gets the following equation for the evolution of</text>
<figure>
<location><page_8><loc_16><loc_74><loc_51><loc_90></location>
</figure>
<figure>
<location><page_8><loc_54><loc_74><loc_89><loc_90></location>
<caption>Figure 2 : The integral N and the drag function d versus the field φ</caption>
</figure>
<text><location><page_8><loc_14><loc_66><loc_27><loc_67></location>the scalar field φ</text>
<formula><location><page_8><loc_36><loc_63><loc_88><loc_66></location>¨ φ = ± -6 √ 2 + 2( ∓ 18 + ( √ 2 ± 72 φ ) φ ) φ 4 -38 φ 2 +81 φ 4 ˙ φ 2 . (5.5)</formula>
<text><location><page_8><loc_14><loc_53><loc_88><loc_62></location>Thus, as the field rolls down, it suffers a non-constant drag of d ( φ ) = 6 √ 2 -2(18+( √ 2 -72 φ ) φ ) φ 4 -38 φ 2 +81 φ 4 . As one notices from Fig. 2, the drag approaches an infinite value at the point φ ∗ creating a virtual infinite barrier at this point forcing the field to stop. As the field stops, i.e. ˙ φ ≈ 0, inflation ends as H ∼ ˙ φ ≈ 0. How close the field gets to φ ∗ determines how much N-folds accumulated and depends on the initial conditions for the field and its velocity.</text>
<text><location><page_8><loc_14><loc_46><loc_88><loc_53></location>It worth noting that by end of inflation, torsion will vanish (as ˙ φ = 0), and thus the theory of Einstein-Cartan reduces to the GR with the usual FRW geometry. Therefore, in this scenario the torsion begins with large values at very early times of the universe that give rise inflation. Then, it diminishes at the end of inflation, in conformity with our late-time universe.</text>
<section_header_level_1><location><page_8><loc_14><loc_42><loc_29><loc_44></location>6 Conclusions</section_header_level_1>
<text><location><page_8><loc_14><loc_25><loc_88><loc_40></location>In this paper we have analyzed a novel scenario for cosmic inflation induced by a gauge field in the framework of Einstein-Cartan theory. We show that the vector torsion can play a crucial role in the early cosmology if it couples to a gauge field. We show a specific form of the torsion, in terms of one scalar function, complies with both the cosmological principal (spatial homogeneity and isotropy) and gauge invariance principle. We consider a simple example of three equal and mutually orthogonal abelian gauge fields, in the form of 'cosmic triad'. We calculate the corresponding field equations and show that torsion is determined, through an algebraic equation, in terms of the scalar field of the cosmic triad. By solving the dynamical field equations, we found that a period of inflation can be sustained.</text>
<text><location><page_8><loc_14><loc_16><loc_88><loc_24></location>The field evolution corresponds to large field (chaotic) type of inflation as the large number of e-folds accumulated near the end of inflation depends on the initial conditions. At the end of inflation, the field has a constant background value while torsion vanishes and the theory reduces to the General Relativity with the usual FRW geometry. Finally, we would like to point out prospects of studying this model in light of cosmological perturbation theory and possible reheating mechanisms.</text>
<section_header_level_1><location><page_9><loc_14><loc_88><loc_32><loc_90></location>Acknowledgments</section_header_level_1>
<text><location><page_9><loc_14><loc_84><loc_88><loc_87></location>The authors would like to thank A. Awad, A. El-Zant, A. Golovnev, E. Lashin, and G. Nashed for the useful discussions.</text>
<section_header_level_1><location><page_9><loc_14><loc_80><loc_26><loc_82></location>Appendix</section_header_level_1>
<section_header_level_1><location><page_9><loc_14><loc_76><loc_82><loc_77></location>A Spatially Maximally form-invariant 3-rank antisymmetric tensor</section_header_level_1>
<text><location><page_9><loc_14><loc_71><loc_88><loc_74></location>We assume that we have a maximally symmetric subspace. It is always possible to find a coordinate transformation for which the metric of the whole space take the following form [43]</text>
<formula><location><page_9><loc_31><loc_68><loc_88><loc_69></location>ds 2 = g µν dx µ dx ν = g ab ( V ) dv a dv b + f ( V )˜ g ij ( U ) du i du j (A.1)</formula>
<text><location><page_9><loc_14><loc_62><loc_88><loc_66></location>Where ˜ g ij ( U ) being the metric of an M dimensional maximally symmetric space. The indices a, b ; · · · label the invariant coordinates V and run over N -M while the indices i, j, · · · label the infinitesimally transformed coordinates U and run over the M labels.</text>
<text><location><page_9><loc_14><loc_58><loc_88><loc_61></location>A tensor T µν ··· ( X ) is called maximally form-invariant if it has a vanishing Lie derivative with respect to the Killing vectors ξ λ ( X );</text>
<formula><location><page_9><loc_30><loc_54><loc_88><loc_57></location>L ξ T µν ··· ≡ ξ λ ∂ ∂x λ T µν ··· + ∂ξ ρ ∂x µ T ρν ··· + ∂ξ σ ∂x ν T µσ ··· + · · · = 0 (A.2)</formula>
<text><location><page_9><loc_18><loc_52><loc_82><loc_53></location>We apply this condition on an antisymmetric tensor S µ νρ = -S µ ρν on the space of A.1;</text>
<formula><location><page_9><loc_47><loc_47><loc_88><loc_49></location>L ξ S i jk = 0 (A.3)</formula>
<formula><location><page_9><loc_47><loc_45><loc_88><loc_46></location>L ξ S i ja = 0 (A.4)</formula>
<formula><location><page_9><loc_47><loc_43><loc_88><loc_44></location>L ξ S i ab = 0 (A.5)</formula>
<formula><location><page_9><loc_47><loc_41><loc_88><loc_42></location>L ξ S a bc = 0 (A.6)</formula>
<formula><location><page_9><loc_47><loc_38><loc_88><loc_40></location>L ξ S a bi = 0 (A.7)</formula>
<formula><location><page_9><loc_47><loc_36><loc_88><loc_38></location>L ξ S a ij = 0 (A.8)</formula>
<text><location><page_9><loc_84><loc_34><loc_88><loc_35></location>(A.9)</text>
<text><location><page_9><loc_18><loc_31><loc_37><loc_32></location>The condition A.3 leads to</text>
<formula><location><page_9><loc_31><loc_28><loc_88><loc_29></location>S l jk δ m i + S l i k δ m j + S l ij δ m k = S m jk δ l i + S m i k δ l j + S m ij δ l k (A.10)</formula>
<text><location><page_9><loc_18><loc_25><loc_40><loc_26></location>Contracting m and i , one gets</text>
<formula><location><page_9><loc_35><loc_21><loc_88><loc_23></location>mS l jk + S l j k + S l kj = S l jk + S i i k δ l j + S i ij δ l k (A.11)</formula>
<text><location><page_9><loc_14><loc_17><loc_88><loc_20></location>Further contraction for l and j gives T i ik = 0, substituting this back in the previous equation gives;</text>
<formula><location><page_9><loc_40><loc_13><loc_88><loc_15></location>( m -1) S l jk + S l j k + S l kj = 0 (A.12)</formula>
<text><location><page_10><loc_14><loc_87><loc_88><loc_90></location>The solution for which is simply in the form of Levi-Civita tensor times an arbitrary function of time</text>
<formula><location><page_10><loc_42><loc_82><loc_88><loc_85></location>S ijk = { F ( t ) glyph[epsilon1] ijk m = 3 0 m = 3 (A.13)</formula>
<text><location><page_10><loc_57><loc_82><loc_57><loc_83></location>glyph[negationslash]</text>
<text><location><page_10><loc_18><loc_80><loc_37><loc_81></location>The condition A.4 implies</text>
<formula><location><page_10><loc_31><loc_76><loc_88><loc_78></location>S α j 0 δ β i + S α i 0 δ β j + S α ij δ β 0 = S β j 0 δ α i + S β i 0 δ α j + S β ij δ α 0 (A.14)</formula>
<text><location><page_10><loc_18><loc_74><loc_54><loc_75></location>Solution is zero unless α = j and β = i for which;</text>
<formula><location><page_10><loc_42><loc_70><loc_88><loc_72></location>S i i 0 = S j j 0 = f ( t ) ∀ i, j (A.15)</formula>
<text><location><page_10><loc_18><loc_68><loc_82><loc_69></location>For A.5 and A.6, its trivial to show that T i 00 = T 0 00 = 0. The fifth condition A.7 gives;</text>
<formula><location><page_10><loc_18><loc_63><loc_88><loc_65></location>S α 0 i δ β 0 + S α 0 i δ β 0 + S α 00 δ β i = S β 0 i δ α 0 + S β 0 i δ α 0 + S β 00 δ α i ⇒ S k 00 δ j i = S j 00 δ k i ⇒ S 00 i = 0 (A.16)</formula>
<text><location><page_10><loc_18><loc_61><loc_42><loc_62></location>And the last condition A.8 gives;</text>
<formula><location><page_10><loc_17><loc_56><loc_88><loc_58></location>S α ij δ β 0 + S α 0 j δ β i + S α 0 i δ β j = S β ij δ α 0 + S β 0 j δ α i + S β 0 i δ α j ⇒ S k 0 ( j δ l i ) = S l 0 ( j δ k i ) ⇒ S 0 ij = 0 (A.17)</formula>
<text><location><page_10><loc_14><loc_52><loc_88><loc_55></location>And this proves Eq.(3.5). The enthusiastic reader is encouraged to look at the discussion of maximally symmetric spaces in [43, 44].</text>
<section_header_level_1><location><page_10><loc_14><loc_49><loc_32><loc_50></location>B Field Strength</section_header_level_1>
<text><location><page_10><loc_14><loc_46><loc_61><loc_47></location>The commutation of two covariant derivatives can be found as;</text>
<formula><location><page_10><loc_17><loc_37><loc_88><loc_43></location>[ D µ , D ν ] ψ = [ ∇ µ , ∇ ν ] ψ -ig ∇ µ ( b α ν A α ) ψ + ig ∇ ν ( b α µ A α ) ψ -igb α ν A α ψ ; µ -igb α µ A α ψ ; ν -g 2 b α µ b β ν A α A β ψ + igb α µ A α ψ ; ν + igb α ν A α ψ ; µ + g 2 b α ν b β µ A α A β ψ = -ig ( ∂ µ ( b α ν A i α ) -∂ ν ( b α µ A i α ) -S β µν b α β A i α + gC i jk b α µ b β ν A j α A k β ) ψ -S α µν ∂ α ψ (B.1)</formula>
<text><location><page_10><loc_14><loc_32><loc_88><loc_35></location>We try to construct an invariant quantity with minimal modification to 3.6; this can be achieved by taking b ν µ = e -f ( X ) δ ν µ</text>
<formula><location><page_10><loc_33><loc_29><loc_88><loc_30></location>F µν = ∂ µ A i ν -∂ ν A i µ -S σ µν A i σ + ge -f C i jk A j µ A k ν (B.2)</formula>
<text><location><page_10><loc_14><loc_25><loc_88><loc_28></location>Now we seek out a condition on the torsion to guarantee the invariance of this quantity (more precisely the invariance of F i µν F iµν )</text>
<formula><location><page_10><loc_16><loc_16><loc_88><loc_22></location>δF i µν = -g -1 ( c β ν ,µ θ i ,β + c β ν θ i ,βµ -c β µ ,ν θ i ,β -c β µ θ i ,βν ) + C i jk ( A k ν,µ θ j + A k ν θ j ,µ -A k µ,ν θ j -A k µ θ j ,ν ) -S β µν ( -g -1 c γ β θ i ,γ + C i jk A k β θ j ) + ge -f C i jk ( -g -1 A j µ c γ ν θ k ,γ + C k lm A j µ A m ν θ l -g -1 c γ µ A k ν θ j ,γ + C j lm A m µ A k ν θ l ) (B.3)</formula>
<formula><location><page_10><loc_20><loc_14><loc_88><loc_16></location>= C i jk θ j F k µν + g -1 ( c λ µ ,ν -c λ ν ,µ + S α µν c λ α ) θ i ,λ (B.4)</formula>
<text><location><page_11><loc_18><loc_88><loc_73><loc_90></location>For the coefficient of θ i ,λ to vanish, the torsion must conform with the form;</text>
<formula><location><page_11><loc_35><loc_85><loc_88><loc_86></location>S σ µν = b σ λ ( c λ ν ,µ -c λ µ ,ν ) = δ σ ν χ ,µ -δ σ µ χ ,ν (B.5)</formula>
<text><location><page_11><loc_14><loc_82><loc_54><loc_83></location>which is the form of a vector torsion as defined in 3.2.</text>
<section_header_level_1><location><page_11><loc_14><loc_78><loc_44><loc_79></location>C Energy-momentum tensor</section_header_level_1>
<text><location><page_11><loc_14><loc_75><loc_68><loc_76></location>We need to find the energy momentum tensor for Λ = -1 4 F a µν F µν a , where</text>
<formula><location><page_11><loc_24><loc_70><loc_26><loc_71></location>T</formula>
<formula><location><page_11><loc_26><loc_67><loc_78><loc_72></location>µ ν = q ,ν ∂ Λ ∂q ,µ -δ µ ν Λ = -F µρ a F a νρ + 1 δ µ ν F λρ a F a λρ = -F µρ a F a νρ + δ µ ν ( -3 2 (˙ aφ + a ˙ φ -aφχ ) 2 )</formula>
<formula><location><page_11><loc_38><loc_66><loc_63><loc_67></location>4 2 a</formula>
<text><location><page_11><loc_45><loc_64><loc_45><loc_65></location>glyph[negationslash]</text>
<text><location><page_11><loc_18><loc_64><loc_70><loc_65></location>It is now obvious that T µ ν = 0 for µ = ν . For µ = ν one explicitly find;</text>
<formula><location><page_11><loc_39><loc_60><loc_88><loc_61></location>T µ ν = diag ( ρ A , -P A , -P A , -P A ) (C.1)</formula>
<formula><location><page_11><loc_43><loc_52><loc_88><loc_55></location>ρ A ≡ 3 2 ( ˙ φ + Hφ -φχ ) 2 (C.2)</formula>
<formula><location><page_11><loc_42><loc_49><loc_88><loc_52></location>P A ≡ 1 2 ( ˙ φ + Hφ -φχ ) 2 (C.3)</formula>
<text><location><page_11><loc_14><loc_45><loc_88><loc_46></location>It is due now to point that the usual conservation law of energy-momentum is modified, c.f. [27]. To</text>
<text><location><page_11><loc_14><loc_43><loc_70><loc_48></location>Which is in the form of a perfect fluid namely a radiation field with P = 1 3 ρ . see that we substitute Eq.(4.4) into Eq.(4.5), and differentiate Eq.(4.4);</text>
<formula><location><page_11><loc_37><loc_39><loc_88><loc_41></location>˙ H = -1 2 ( ρ A + P A ) + 2 χH -2 ˙ χ +4 χ 2 (C.4)</formula>
<formula><location><page_11><loc_38><loc_34><loc_88><loc_37></location>2 H ˙ H = 1 3 ˙ ρ A -4 ˙ Hχ -4 H ˙ χ -8 χ ˙ χ (C.5)</formula>
<text><location><page_11><loc_14><loc_32><loc_62><loc_33></location>multiplying through Eq.(C.4) by 2 H and equating it to Eq.(C.5);</text>
<formula><location><page_11><loc_26><loc_25><loc_88><loc_29></location>1 3 ˙ ρ A -4 ˙ Hχ -4 H ˙ χ -8 χ ˙ χ = 2 H ( -1 2 ( ρ A + P A ) + 2 χH -2 ˙ χ +4 χ 2 ) ⇒ ˙ ρ A = 12 χ ( ˙ H + H 2 ) + 24 χ ˙ χ -3 H ( ρ A + P A ) + 24 Hχ 2 (C.6)</formula>
<text><location><page_11><loc_14><loc_20><loc_88><loc_23></location>Substituting for ˙ H + H 2 from Eq.(4.5), we get the continuity equation for a perfect fluid in the presence of a vector torsion;</text>
<formula><location><page_11><loc_37><loc_16><loc_88><loc_17></location>˙ ρ A = -3 H ( ρ A + P A ) -2 χ ( ρ A +3 P A ) (C.7)</formula>
<text><location><page_11><loc_18><loc_14><loc_78><loc_15></location>Which reduces to the usual continuity equation in the limit of a vanishing torsion.</text>
<text><location><page_11><loc_18><loc_58><loc_23><loc_59></location>Where</text>
<section_header_level_1><location><page_12><loc_14><loc_88><loc_44><loc_90></location>D Ricci and Einstein tensors</section_header_level_1>
<text><location><page_12><loc_14><loc_86><loc_80><loc_87></location>Explicit calculation for the Ricci tensor produces the following non-vanishing components</text>
<formula><location><page_12><loc_30><loc_73><loc_72><loc_83></location>R 00 = -3 [ a a +2˙ χ +3 χ ˙ a a ] R 11 = 1 1 -Kr ( a a +2˙ χa 2 +2˙ a 2 +10 χa ˙ a +8˙ χa 2 +2 K ) R 22 = r 2 ( a a +2˙ χa 2 +2˙ a 2 +10 χa ˙ a +8˙ χa 2 +2 K ) R 33 = r 2 sin 2 ( θ )( a a +2˙ χa 2 +2˙ a 2 +10 χa ˙ a +8˙ χa 2 +2 K )</formula>
<text><location><page_12><loc_14><loc_70><loc_30><loc_71></location>And the Ricci scalar;</text>
<formula><location><page_12><loc_35><loc_66><loc_88><loc_69></location>R = 6 a 2 ( a a + ˙ a 2 +6 χa ˙ a +2 a 2 (2 χ 2 + ˙ χ ) + K ) (D.1)</formula>
<text><location><page_12><loc_18><loc_64><loc_57><loc_65></location>The nonvanishing components of Einstein tensors are;</text>
<formula><location><page_12><loc_30><loc_52><loc_72><loc_61></location>G 00 = 3 [( ˙ a a ) 2 +4 χ a ˙ a a 2 +4 χ 2 + K a 2 ] G 11 = -1 1 -Kr 2 (2 a a +8 χa ˙ a + ˙ a 2 +4 χ 2 a 2 +4˙ χa 2 + K ) G 22 = -r 2 (2 a a +8 χa ˙ a + ˙ a 2 +4 χ 2 a 2 +4˙ χa 2 + K ) G 11 = -r 2 sin 2 ( θ )(2 a a +8 χa ˙ a + ˙ a 2 +4 χ 2 a 2 +4˙ χa 2 + K )</formula>
<section_header_level_1><location><page_13><loc_14><loc_88><loc_25><loc_90></location>References</section_header_level_1>
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</document> | [{"title": "A. Kasem a and S. Khalil b", "content": "a,b Center for Fundamental Physics, Zewail City of Science and Technology, 6 October City, Giza 12578, Egypt. E-mail: [email protected], [email protected] Abstract. Inflation in the framework of Einstein-Cartan theory is revisited. Einstein-Cartan theory is a natural extension of the General Relativity, with non-vanishing torsion. The connection on Riemann-Cartan spacetime is only compatible with the cosmological principal for a particular form of torsion. We show this form to also be compatible with gauge invariance principle for a non-Abelian and Abelian gauge fields under a certain deviced minimal coupling procedure. We adopt an Abelian gauge field in the form of 'cosmic triad'. The dynamical field equations are obtained and shown to sustain cosmic inflation with a large number of e-folds. We emphasize that at the end of inflation, torsion vanishes and the theory of Einstein-Cartan reduces to the General Relativity with the usual FRW geometry.", "pages": [1]}, {"title": "1 Introduction", "content": "From the outset of the general theory of relativity (GR), it has been playing a major role as a framework for our thinking about the universe. The theory has witnessed many success stories throughout the past century that have been crowned by what is known as the standard model of cosmology or \u039b CDM model [1-4]. The model describes a homogeneous, isotropic, and spatially-(almost)-flat universe started at a very dense state 'big bang', undergone a period of exponential expansion and has been expanding in an accelerated manner ever since. Despite what the name suggests, the model is plagued with many theoretical issues. It is only standard as a best fit model to the observational evidence we have. Problems with \u039b CDM model are de facto inherited from GR which is still struggling to find a foothold in the general scheme of particle physics. Success of GR in describing large scale physics should not tempt us to overlook other alternatives. Having said that, it is more likely the alternative being an extension to GR so one is guaranteed not to stray from observational tests. We argue that a good, well-motivated candidate would be Einstein-Cartan theory. The theory differs by admitted a torsion tensor in space, an object conspicuous by its absence in GR. Once one starts by building a local gauge theory of Poincar'e group, Einstein-Cartan falls into place as the other forces in particle physics with no a priori reason for having a vanishing torsion. Torsion can be a rather de trop object. The antisymmetic part of the connection explicitly violates gauge invariance when introducing a gauge field into curved background. In the cosmological context, several semi-classical models have been regrading torsion as a spinning fluid [5-10]. In some cases they required an unreasonable amount of fine tuning or an extremely large amount of spinning particles in the early universe. But they were mainly not successful because torsion violated homogeneity and isotropy of space and inhomogeneous cosmology has not yet borne fruit on the observational level. In this paper, we aim to introduce a scenario of torsion sourced by a gauge field. We show that a minimal setup motivated by respecting FRW symmetries and gauge invariance sustains a period of exponential expansion. We start in section 2 by looking at the development of Einstein-Cartan theory as gauge theory of gravity. In section 3, we see that only a certain form of torsion is compatible with FRW symmetries and it can be sourced by a gauge field provided that a new minimal coupling prescription is adopted. In section 4, we setup the model and compute the field equations. In section 5, we show that the field equations can sustain a period of early cosmic inflation with a large number of e-folds. Our conclusions and final remarks are given in section 6. Appendixes A and B provide calculations for the form of torsion compatible with FRW symmetries and gauge invariance of a nonabelian gauge field. In appendixes C and D we provide the explicit form of the energy momentum, Ricci, and Einstein tensors that we used to compute the field equations.", "pages": [2, 3]}, {"title": "2 Development of Einstein-Cartan Theory", "content": "The concept of an antisymmetric affine connection was first introduced by ' Elie Cartan in 1922, later development of a gravitational theory allowing torsion did not ensue much until the sixties. This in part was due to the consensus on Einstein GR which passed all macrophysical experimental tests. Later development of particle physics made it clear that a direct measurement of gravitational effects between elementary particles is beyond our technological capabilities. However, it gave us a general scheme to govern all forces of nature. Gravity can only fit into the general scheme of things if it can be written as a gauge theory. If one thinks of a global symmetry for gravity, the group of Lorentz transformations is the obvious candidate. Promoting this symmetry into local one was an endeavor first started by Utiyama [11], and later developed by Kibble [12] and Sciama [13, 14]. The theory is better known as EinsteinCartan-Sciama-Kibblle or U 4 for short. U 4 is indeed the local gauge theory of the Poincar'e group. 1 It features a connection with an antisymmetric part known as torsion; S i jk \u2261 \u0393 i [ jk ] = 1 2 (\u0393 i jk -\u0393 i kj ). With the help of the metric postulate, one can write the connection in terms of the metric and torsion as follows Where \u02dc \u0393 i jk being the usual Levi-Civita Connection and the contortion tensor K i jk is defined as K i jk \u2261 S i jk + S i k j -S i jk . As the case with GR, one can still construct an invariant scalar quantity using the Riemann curvature tensor so the simplest Lagrangian would be L = \u221a -g ( 1 2 R (\u0393) + L M ). By a straightforward but rather lengthy calculations, one finds the field equations by varying the action with respect to the independent variables (10 represented by the metric and 24 represented by the torsion) resulting in the field equations; where T \u00b5\u03bd 'Energy-momentum tensor', T \u03b1\u03b2\u03b3 'modified torsion tensor', and S \u03b1\u03b2\u03b3 'spin energy potential' are defined as follows; Unless one acquiesces to setting the torsion to zero, it should be regarded as fundamental as the metric itself. It is important to note that GR cannot be written as a gauge theory [17]. One can reach the field equations of GR with the local theory of translations with torsion as a gauge field strength. However, this formulation is only equivalent at the level of field equations as TEGR action involves a torsion scalar rather than a curvature scalar and the theory is formulated with a Weitzenbock connection rather than a Levi-Civita connection. Few remarks on the field equations of U 4 are in order. First we note from the second field equation (2.3) is an algebraic equation so it can be fed into Eq.(2.2) so one recovers Einstein field equations with an 'effective energy momentum tensor'. Secondly, the canonical Energy-momentum (2.4) is not symmetric for a spinor field, i.e. for Dirac field it is totally antisymmetric T \u00b5 \u03bd can = i 2 \u00af \u03a8 \u03b3 \u00b5 \u2202 \u03bd \u03a8 -i 2 \u03a8 \u03b3 \u00b5 \u2202 \u03bd \u00af \u03a8. 2 This may give us some account as of why torsion is linked to spin in terminology. A bosonic field does not feel rotation in space so it cannot source a torsion tensor. The type of possible matter fields feeling torsion is a topic we address in the next section.", "pages": [3, 4]}, {"title": "3 Source of torsion", "content": "Early cosmological treatments of U 4 adopted a semiclassical model of a spinning dust known as Weyssenhoff fluid [18]. It features a singularity-free solution [19-22], and has been studied in the context of inflation [5, 6, 10]. A number of these models have underplayed the incompatibility of this fluid with the symmetries of FRW universe [23, 24]. To get more insight we can look at the torsion covariantly split into three irreducible parts [25]; where V S c ab , A S c ab , and T S c ab are vector, axial vector and tensorial components; respectively. They are defined as follows where the one-indexed S a is defined as S a \u2261 S b ab . We wish to only keep the parts in S c ab that make it form-invariant on a spatially maxiamally-symmetric space. To this end we require the vanishing of the Lie derivative with respect to the 6 Killing vectors of that symmetry; L \u03be S a ab = 0. In Appendix A, we present a succinct proof following the work of [23]. The result is a vanishing tensorial part T S c ab = 0 and a particular form for the vector and axialvector parts, namely where F ( t ) and f ( t ) are arbitrary functions of time. It is evident that this form is incompatible with the spinning fluid description which has a non-vanishing traceless part. The cosmological implications of torsion in the form of (3.5) have been investigated in various works [26-30]. As this type of models undermine a spinning fluid description, it gives up one of its significant results; a universe free of cosmic singularity. Moreover, the functions F ( t ), f ( t ) are introduced in an ad hoc manner with no available physical interpretation to its source. Therefore, one would be reluctant to take up this pursuit any further, however we show that the vector torsion part can be a grist to the mill of cosmology if it couples to a gauge field. The possibility of sourcing torsion with a gauge field is often dismissed as torsion is known to violate gauge invariance, i.e. if one tries the natural generalization of the field strength of gauge field (Abelian or non-Abelian) to Einstein-Cartan manifold through replacing the partial derivative by a space-time derivative: \u2202 \u00b5 A \u03bd \u2192 \u2202 \u00b5 A \u03bd -\u0393 \u03bb \u00b5\u03bd A \u03bb , one finds Thus, gauge invariance is found to be manifestly violated due to the antisymmetric part of the connection. One way to avoid this violation is adopting the flat-space expression for F \u00b5\u03bd . However, this defeats our purpose in introducing this field in the first place. A more satisfactory approach have been introduced by Hojman, et al. [31], where a new minimal coupling procedure has been devised to allow for a nonvanishing torsion in the definition of an Abelian gauge field strength. Then, it was shorty generalized for a non-abelian field as well [32]. In this procedure, one defines the gauge transformation to allow a set of space-time functions b \u03b1 \u00b5 into the the definition of a covariant derivative such that; D \u00b5 = \u2207 \u00b5 -igb \u03b1 \u00b5 A \u03b1 . This affects the transformation of the gauge field in the following way where c \u03bd \u00b5 b \u03b1 \u03bd \u2261 \u03b4 \u03b1 \u00b5 . Now the following modified field strength can be made invariant (see Appendix B for detailed calculations): under the constraints of c \u03bd \u00b5 = e f ( X ) \u03b4 \u03bd \u00b5 and S \u03c3 \u00b5\u03bd \u2261 \u03b4 \u03c3 \u03bd f ,\u00b5 -\u03b4 \u03c3 \u00b5 f ,\u03bd , which coincides with the expression of vector torsion Eq.(3.2). It also conforms with the FRW-compatible torsion when one assumes f as a function of time f ( X ) \u2261 f ( t ). One can think of this procedure as giving a nonconstant gauge coupling g ( t ) = ge -f ( t ) . So in the limit of vanishing torsion, the usual minimal coupling procedure is restored with c \u03bd \u00b5 = \u03b4 \u03bd \u00b5 and g ( t ) \u2261 g . Our discussion extends to three Abelian fields A i \u00b5 which can be thought of as a special case of SU (2) nonabelian gauge theory with vanishing structure constants C i jk \u2192 0. For the purpose of constructing a minimal setup, we stick with the Abelian version in the following analysis.", "pages": [4, 5]}, {"title": "4 Model with cosmic triad", "content": "Having established an FRW-compatible form of torsion, we would like to couple it with gauge gauge fields respecting the same symmetries. We adopt an ansatz of three equal and mutually-orthogonal vectors known as 'cosmic triad' was proposed by [33]; where a ( t ) is the FRW scale factor. The cosmological implication of this choice has been investigated in many different contexts [34-42]. Here we take the simplest form with three identical copies of an Abeilan field coupled with torsion as per the prescription of the previous section; The standard Lagrangian L = \u221a -g [ 1 2 ( R -2\u039b) -1 4 F a \u00b5\u03bd F \u00b5\u03bd a + L M ] is constructed assuming FRW background and the aformentioned choice for torsion and fields. We assume a flat space, a vanishing cosmological constant and a universe dominated by the triad field. Also we relabel the torsion function \u02d9 f \u2192 \u03c7 ( t ). Now the above Lagrangian simplifies into; We calculate the field equations (2.2, 2.3), using the energy-momentum tensor and the Einstein tensor as given in appendix D, and C. A straightforward but rather onerous calculations gives the following Freidman-like equations and an extra equation for the torsion field: with As we expect, the torsion equation (4.7) is algebraic, so one can use it to substitute for the nondynamical field. By introducing the Hubble parameter H ( t ) \u2261 \u02d9 a a and explicitly substitute for \u03c1 A and P A , defined in Appendix C, one finds that the above equations can be written as", "pages": [6]}, {"title": "5 Inflationary Scenario", "content": "To understand the cosmic evolution let us start with Eq.(4.8), which is a quadratic equation of the Hubble parameter, one can solve to get H ( \u03c6, \u02d9 \u03c6 ) Substituting this solution as well as \u00a8 \u03c6 from Eq.(4.10) into Eq.(4.9), one finds \u02d9 H ( \u03c6, \u02d9 \u03c6 ) Now we are set to find the inflationary parameter glyph[epsilon1] \u2261 \u02d9 H H 2 , which we use to identify the inflationary period a > 0 as a a = H 2 (1 -glyph[epsilon1] ), so exponential growth is going in the region of glyph[epsilon1] < 1. From the above expressions of H and \u02d9 H , one obtains the following relation of glyph[epsilon1] in terms of \u03c6 scalar field. We stress that we only use this parameter to define the inflationary period with no requirement on epsilon as to being positive or infinitesimal. For reasons that should be obvious shortly, we find only the second solution can sustain a period of inflation large enough to generate a desirable number of e-fold so we stick to it in the following analysis. In Fig. 1, we plot inflationary parameter glyph[epsilon1] versus the scalar field \u03c6 . As can be seen from this figure, the inflationary period ( glyph[epsilon1] < 1) is going on as long as the field is less than a certain value \u03c6 < 1 9 ( \u221a 2+ \u221a 17) \u2248 0 . 62. It is clear that for a large field inflation, i.e. \u03c6 > 1, inflation starts when \u03c6 is rolling down to vales less than 0 . 62, while for a small field, i.e. \u03c6 starts from values close to zero and rolls up, the inflation ends when glyph[epsilon1] approaches one at \u03c6 glyph[similarequal] 0 . 62. Now, we consider the number of e-folds N of inflation that are needed to solve the horizon problem. In general N is defined as N \u2261 \u222b t f t i Hdt . In our model, one can show that it is given by In Fig. 2, we show the integral of e-folds which has a notable pole at \u03c6 \u2217 = 1 18 ( \u221a 2 + \u221a 74) \u2248 0 . 556, and the required value N glyph[similarequal] 60 can be obtained when the field \u03c6 approach this point. In this case, we have a new scenario of inflation, where the field \u03c6 is rolling down from a relatively large value with an initial velocity, an inflationary period starts at \u03c6 i (the point for glyph[epsilon1] = 1) and as the field approaches the pole at \u03c6 \u2217 the number e-folds blows up, as shown in Fig. 2, and inflation abruptly ends. The abrupt end of the inflation can be understood by considering the evolution of the field \u03c6 , by substituting Eq.(5.1) and Eq.(5.2) into Eq.(4.10), one gets the following equation for the evolution of the scalar field \u03c6 Thus, as the field rolls down, it suffers a non-constant drag of d ( \u03c6 ) = 6 \u221a 2 -2(18+( \u221a 2 -72 \u03c6 ) \u03c6 ) \u03c6 4 -38 \u03c6 2 +81 \u03c6 4 . As one notices from Fig. 2, the drag approaches an infinite value at the point \u03c6 \u2217 creating a virtual infinite barrier at this point forcing the field to stop. As the field stops, i.e. \u02d9 \u03c6 \u2248 0, inflation ends as H \u223c \u02d9 \u03c6 \u2248 0. How close the field gets to \u03c6 \u2217 determines how much N-folds accumulated and depends on the initial conditions for the field and its velocity. It worth noting that by end of inflation, torsion will vanish (as \u02d9 \u03c6 = 0), and thus the theory of Einstein-Cartan reduces to the GR with the usual FRW geometry. Therefore, in this scenario the torsion begins with large values at very early times of the universe that give rise inflation. Then, it diminishes at the end of inflation, in conformity with our late-time universe.", "pages": [6, 7, 8]}, {"title": "6 Conclusions", "content": "In this paper we have analyzed a novel scenario for cosmic inflation induced by a gauge field in the framework of Einstein-Cartan theory. We show that the vector torsion can play a crucial role in the early cosmology if it couples to a gauge field. We show a specific form of the torsion, in terms of one scalar function, complies with both the cosmological principal (spatial homogeneity and isotropy) and gauge invariance principle. We consider a simple example of three equal and mutually orthogonal abelian gauge fields, in the form of 'cosmic triad'. We calculate the corresponding field equations and show that torsion is determined, through an algebraic equation, in terms of the scalar field of the cosmic triad. By solving the dynamical field equations, we found that a period of inflation can be sustained. The field evolution corresponds to large field (chaotic) type of inflation as the large number of e-folds accumulated near the end of inflation depends on the initial conditions. At the end of inflation, the field has a constant background value while torsion vanishes and the theory reduces to the General Relativity with the usual FRW geometry. Finally, we would like to point out prospects of studying this model in light of cosmological perturbation theory and possible reheating mechanisms.", "pages": [8]}, {"title": "Acknowledgments", "content": "The authors would like to thank A. Awad, A. El-Zant, A. Golovnev, E. Lashin, and G. Nashed for the useful discussions.", "pages": [9]}, {"title": "A Spatially Maximally form-invariant 3-rank antisymmetric tensor", "content": "We assume that we have a maximally symmetric subspace. It is always possible to find a coordinate transformation for which the metric of the whole space take the following form [43] Where \u02dc g ij ( U ) being the metric of an M dimensional maximally symmetric space. The indices a, b ; \u00b7 \u00b7 \u00b7 label the invariant coordinates V and run over N -M while the indices i, j, \u00b7 \u00b7 \u00b7 label the infinitesimally transformed coordinates U and run over the M labels. A tensor T \u00b5\u03bd \u00b7\u00b7\u00b7 ( X ) is called maximally form-invariant if it has a vanishing Lie derivative with respect to the Killing vectors \u03be \u03bb ( X ); We apply this condition on an antisymmetric tensor S \u00b5 \u03bd\u03c1 = -S \u00b5 \u03c1\u03bd on the space of A.1; (A.9) The condition A.3 leads to Contracting m and i , one gets Further contraction for l and j gives T i ik = 0, substituting this back in the previous equation gives; The solution for which is simply in the form of Levi-Civita tensor times an arbitrary function of time glyph[negationslash] The condition A.4 implies Solution is zero unless \u03b1 = j and \u03b2 = i for which; For A.5 and A.6, its trivial to show that T i 00 = T 0 00 = 0. The fifth condition A.7 gives; And the last condition A.8 gives; And this proves Eq.(3.5). The enthusiastic reader is encouraged to look at the discussion of maximally symmetric spaces in [43, 44].", "pages": [9, 10]}, {"title": "B Field Strength", "content": "The commutation of two covariant derivatives can be found as; We try to construct an invariant quantity with minimal modification to 3.6; this can be achieved by taking b \u03bd \u00b5 = e -f ( X ) \u03b4 \u03bd \u00b5 Now we seek out a condition on the torsion to guarantee the invariance of this quantity (more precisely the invariance of F i \u00b5\u03bd F i\u00b5\u03bd ) For the coefficient of \u03b8 i ,\u03bb to vanish, the torsion must conform with the form; which is the form of a vector torsion as defined in 3.2.", "pages": [10, 11]}, {"title": "C Energy-momentum tensor", "content": "We need to find the energy momentum tensor for \u039b = -1 4 F a \u00b5\u03bd F \u00b5\u03bd a , where glyph[negationslash] It is now obvious that T \u00b5 \u03bd = 0 for \u00b5 = \u03bd . For \u00b5 = \u03bd one explicitly find; It is due now to point that the usual conservation law of energy-momentum is modified, c.f. [27]. To Which is in the form of a perfect fluid namely a radiation field with P = 1 3 \u03c1 . see that we substitute Eq.(4.4) into Eq.(4.5), and differentiate Eq.(4.4); multiplying through Eq.(C.4) by 2 H and equating it to Eq.(C.5); Substituting for \u02d9 H + H 2 from Eq.(4.5), we get the continuity equation for a perfect fluid in the presence of a vector torsion; Which reduces to the usual continuity equation in the limit of a vanishing torsion. Where", "pages": [11]}, {"title": "D Ricci and Einstein tensors", "content": "Explicit calculation for the Ricci tensor produces the following non-vanishing components And the Ricci scalar; The nonvanishing components of Einstein tensors are;", "pages": [12]}] |
2014MNRAS.439.3798F | https://arxiv.org/pdf/1401.5803.pdf | <document>
<section_header_level_1><location><page_1><loc_7><loc_83><loc_87><loc_88></location>H 2 Suppression with Shocking Inflows: Testing a Pathway for Supermassive Black Hole Formation</section_header_level_1>
<text><location><page_1><loc_7><loc_78><loc_75><loc_80></location>Ricardo Fernandez 1 , Greg L. Bryan 1 , Zoltan Haiman 1 , and Miao Li 1</text>
<text><location><page_1><loc_7><loc_77><loc_70><loc_78></location>1 Department of Astronomy, Columbia University, 550 West 120th Street, New York, NY 10027, USA</text>
<section_header_level_1><location><page_1><loc_28><loc_70><loc_38><loc_71></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_28><loc_44><loc_89><loc_70></location>The presence of quasars at redshifts z > 6 indicates the existence of supermassive black holes (SMBHs) as massive as a few times 10 9 M glyph[circledot] , challenging models for SMBH formation. One pathway is through the direct collapse of gas in T vir glyph[greaterorsimilar] 10 4 K halos; however, this requires the suppression of H 2 cooling to prevent fragmentation. In this paper, we examine a proposed new mechanism for this suppression which relies on cold-mode accretion flows leading to shocks at high densities ( n > 10 4 cm -3 ) and temperatures ( T > 10 4 K). In such gas, H 2 is efficiently collisionally dissociated. We use high-resolution numerical simulations to test this idea, demonstrating that such halos typically have lower temperature progenitors, in which cooling is efficient. Those halos do show filamentary flows; however, the gas shocks at or near the virial radius (at low densities), thus preventing the proposed collisional mechanism from operating. We do find that, if we artificially suppress H 2 formation with a high UV background, so as to allow gas in the halo center to enter the high-temperature, high-density 'zone of no return', it will remain there even if the UV flux is turned off, collapsing to high density at high temperature. Due to computational limitations, we simulated only three halos. However, we demonstrate, using Monte Carlo calculations of 10 6 halo merger histories, that a few rare halos could assemble rapidly enough to avoid efficient H 2 cooling in all of their progenitor halos, provided that the UV background exceeds J 21 ∼ few at redshifts as high as z ∼ 20.</text>
<text><location><page_1><loc_28><loc_41><loc_81><loc_43></location>Key words: black hole physics - methods:numerical - cosmology:theory</text>
<section_header_level_1><location><page_1><loc_7><loc_35><loc_24><loc_36></location>1 INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_50><loc_34><loc_89><loc_36></location>jevi'c, Couch & Bromm 2009; Tanaka & Haiman 2009; Jeon et al. 2012; Tanaka, Perna & Haiman 2012).</text>
<text><location><page_1><loc_7><loc_20><loc_46><loc_33></location>Dynamical evidence indicates that most nearby galaxies harbor a central supermassive black hole (e.g., Ferrarese & Ford 2005), including our own Milky Way, which hosts a central SMBH with mass ∼ 10 6 M glyph[circledot] (Ghez et al. 2005). Furthermore, the discovery of quasars at redshifts greater than 6 signals the existence of SMBHs as massive as a few times 10 9 M glyph[circledot] at an epoch when the Universe was less than a billion years old (e.g., Fan 2006; Mortlock et al. 2011). Such massive and early SMBHs pose a challenge to current models of their formation.</text>
<text><location><page_1><loc_7><loc_5><loc_46><loc_18></location>One possible formation scenario is the growth of a remnant black hole (BH) seed, generated from a population III star ( ∼ 100 M glyph[circledot] ), by mergers and gas accretion (e.g., Haiman & Loeb 2001; Volonteri, Haardt & Madau 2003; Li et al. 2007). However, this formation scenario poses certain difficulties. The time to assemble a 10 9 M glyph[circledot] SMBH by standard Eddington accretion is comparable to the age of the universe at z ∼ 6 and it is unlikely that the seed BH will have continual accretion due to negative feedback and merger-induced gravitational recoils (Alvarez, Wise & Abel 2009; Milosavl-</text>
<text><location><page_1><loc_50><loc_5><loc_89><loc_34></location>An alternative pathway is the direct collapse of metalfree primordial gas with virial temperature glyph[greaterorsimilar] 10 4 K into a BH seed of mass 10 4 -10 6 M glyph[circledot] (Oh & Haiman 2002; Bromm & Loeb 2003; Regan & Haehnelt 2009; Shang, Bryan & Haiman 2010). Such a large seed requires many fewer Salpeter times to grow to quasar size and so bypasses many of the difficulties of Eddington growth of stellar BH seeds. The exact mechanism by which the collapse occurs is not entirely clear (e.g., Bromm & Loeb 2003; Begelman, Rossi & Armitage 2008; Begelman 2010); however, a vital condition for this scenario is that the collapsing gas avoids fragmentation into stars. A natural way to avoid fragmentation is to have a long Jeans length due to a high gas temperature. The temperature of the gas depends on the interplay of atomic and molecular cooling. In the absence of H 2 , a halo with T vir glyph[greaterorsimilar] 10 4 K cools to ∼ 8000 K by atomic hydrogen. Including H 2 , the halo cools further to ∼ 200 K. The corresponding Jeans mass at characteristic central densities M J ≈ 10 6 M glyph[circledot] ( T/ 10 4 K ) 3 / 2 is 10 3 M glyph[circledot] for the latter, suggesting the formation of a Pop III star, and 10 6 M glyph[circledot] for the former, suggesting direct collapse into a massive BH. There-</text>
<text><location><page_2><loc_7><loc_88><loc_46><loc_90></location>a necessary condition for direct collapse is to prevent cooling by H 2 .</text>
<text><location><page_2><loc_7><loc_72><loc_46><loc_88></location>The suppression of H 2 can be accomplished by a strong far ultraviolet (UV) radiation flux in the Lyman-Werner (LW) bands, J 21 glyph[greaterorsimilar] 10 2 -10 3 (in units of 10 -21 erg s -1 cm -2 Hz -1 sr -1 ) (Omukai 2001; Bromm & Loeb 2003; Shang, Bryan & Haiman 2010). This photo-dissociates the molecular hydrogen. However, only a small subset glyph[lessorsimilar] 10 -6 of all atomic cooling halos are estimated to be exposed to such levels, due to the presence of close luminous neighbor (Dijkstra et al. 2008; Agarwal et al. 2012). This makes it difficult to explain the production of the observed quasar population at z > 6. Note that the threshold only increases in the presence of a cosmic-ray/X-ray flux (Inayoshi & Omukai 2011).</text>
<text><location><page_2><loc_7><loc_31><loc_46><loc_71></location>Inayoshi & Omukai (2012) have proposed an alternative mechanism for the suppression of H 2 cooling that does not depend on having a high UV flux. In this scenario, cold accretion flows penetrate to the center of the halo, colliding with each other and shocking to produce hot and dense gas. The post-shock layer cools efficiently due to atomic hydrogen cooling and contracts isobarically until the gas reaches ∼ 8000 K. If the shocked gas at this high temperature is already at a high enough number density n glyph[greaterorsimilar] 10 4 cm -3 , then H 2 rotational-vibrational levels reach local thermodynamic equilibrium, and collisional dissociation can destroy the molecular hydrogen. Crucially, once the gas is shocked to this high-temperature, high-density regime, it will no longer be able to cool via H 2 , even in the absence of any LW radiation. Throughout this paper, we will thus refer to this regime (defined more precisely below) as the 'zone of no return'. Inayoshi & Omukai (2012) have argued that this mechanism may be able to produce a massive BH seed without strong radiative feedback; however, their numerical experiments focused on one-zone models and so questions still remain about the applicability of this pathway in cosmological simulations. In particular, only fluid elements that have a sufficiently high temperature and density have their fragmentation suppressed, and it is not clear if: (1) the gas which ends up in halos that with such large virial temperatures is not first processed through lower mass halos in which fragmentation and star formation can occur 1 , and (2) if a sufficient amount of gas enters the 'zone of no return' for this mechanism to be important in realistic halos.</text>
<text><location><page_2><loc_7><loc_22><loc_46><loc_31></location>To address these questions, we explore the possibility of H 2 suppression via cold accretion shocks by conducting numerical simulations. This paper is organized in the following manner. In Section 2 we describe the ingredients and initial setup of the code. In Section 3 we describe the results of our numerical simulations followed by a discussion in Section 4. Finally, in Section 5 we summarize our conclusions.</text>
<section_header_level_1><location><page_2><loc_7><loc_17><loc_29><loc_18></location>2 NUMERICAL METHOD</section_header_level_1>
<text><location><page_2><loc_7><loc_10><loc_46><loc_16></location>The simulations were performed with the publicly available Eulerian adaptive mesh refinement (AMR) Enzo code. (Bryan 1999; Norman & Bryan 1999; O'Shea et al. 2004; Bryan et al. 2013). The code implements an N-body particle mesh technique (Efstathiou et al. 1985; Hockney &</text>
<table>
<location><page_2><loc_51><loc_84><loc_87><loc_91></location>
<caption>Table 1. Virial quantities for the three halos selected for resimulation from the low resolution run. The maximum level of refinement was set to 4 and radiative cooling was turned off.</caption>
</table>
<text><location><page_2><loc_50><loc_59><loc_89><loc_75></location>Eastwood 1988) to follow the dynamics of the dark matter particles and an Eulerian AMR method (Berger & Colella 1989) for the gas. In addition, Enzo provides modules which compute the radiative cooling of the gas as well as solve the chemical reaction network of a primordial mixture of H and He. Our simulations use the H 2 cooling function of Galli & Palla (1998) and solve the non-equilibrium evolution of the following nine species: H, H + , He, He + , He ++ , H -, H 2 , H + 2 , and e -(Abel et al. 1997; Abel, Bryan & Norman 2000). Density-dependent collisional dissociation (Martin, Schwarz & Mandy 1996) is important - we include this with a rate as described in Shang, Bryan & Haiman (2010).</text>
<text><location><page_2><loc_50><loc_28><loc_89><loc_58></location>We use a set of zoom simulations to focus on a number of halos selected from a 1 h -1 Mpc comoving box, with a root grid resolution of 128 3 , using standard Λ cold dark matter model parameters: Ω Λ , 0 = 0 . 721, Ω m, 0 = 0 . 233, Ω b = 0 . 233, σ 8 = 0 . 817, n s = 0 . 96 and h = 0 . 701 (Komatsu et al. 2009). Initially we performed a low resolution run with the maximum refinement level set to 4 and radiative cooling turned off to inhibit the gas from collapsing to high densities. We evolved this simulation from z = 99 to z = 10. Then we applied the HOP halo finder (Eisenstein & Hut 1998) to the resulting data files at various redshifts to identify halos with masses corresponding to virial temperatures glyph[greaterorsimilar] 10 4 K. Throughout this paper, we adopt the relation T vir = 0 . 75 × 1800( M/ 10 6 M glyph[circledot] ) 2 / 3 (1 + z ) / 21 K between halo mass and virial temperature. This is consistent with the commonly adopted version for neutral primordial gas with mean molecular weight µ = 1 . 2 (Bryan & Norman 1998), except that we reduced the normalization by a factor of 0.75. We have found that this correction agrees better with our simulations - that is, it yields T vir = 10 4 K for halos at the redshift and mass when they begin to cool efficiently via atomic H.</text>
<text><location><page_2><loc_50><loc_5><loc_89><loc_28></location>Three halos where selected at random (see Table 1) to be re-run at high-resolution. We regenerated the initial conditions for the volumes, adding three nested grids that enclosed the Lagrangian volume of each halo. Since each additional grid doubles the spatial resolution, this resulted in an innermost grid with an effective resolution of 1024 3 and a dark matter particle mass of ∼ 85 M glyph[circledot] . Radiative cooling and multi-species were turned on to self-consistency follow the build-up of molecular hydrogen. During the course of the simulation each cell was adaptively refined using the following three criteria: baryon mass, dark matter mass and Jeans length. For the first two criteria, refinement is added when the baryon or dark matter mass exceeds four times the mass of the initial most refined cell, corresponding to mass resolutions of 68 and 340 M glyph[circledot] for the baryons and dark matter, respectively. The third criterion enforces the Truelove et al. (1997) condition which states that at least four cells</text>
<text><location><page_3><loc_86><loc_80><loc_87><loc_80></location>/circledot</text>
<figure>
<location><page_3><loc_9><loc_64><loc_86><loc_88></location>
<caption>Figure 1. Phase plots of number density and temperature for each of the J 21 = 10 simulations (from left to right: halos A, B and C). The color indicates how much mass is at each point in the phase-diagram. In all three simulations, no cells were shocked heated to the 'zone of no return', shown by the dashed line and defined as in Inayoshi & Omukai (2012).</caption>
</figure>
<text><location><page_3><loc_7><loc_44><loc_46><loc_56></location>should resolve the Jeans length to avoid artificial fragmentation. In our simulations the Jeans length was resolved by sixteen cells to be sure that we adequately followed the collapse. Based on these criteria, the simulations were allowed to refine to a maximum level of 18, which corresponds to a comoving scale of 0.0298 h -1 pc. The dark matter distribution was smoothed at refinement level 13 (about 0.065 proper pc at z = 20) to suppress numerical effects from the discreteness of DM particles.</text>
<text><location><page_3><loc_7><loc_17><loc_46><loc_43></location>We carried out two sets of runs for each of the halos, which differed only in the background LW flux that we adopted. In the first set of runs, we used J 21 = 10, where J 21 is the specific intensity in the Lyman-Werner bands (11.213.6 eV) in units of 10 -21 erg cm -2 sr -1 Hz -1 . This corresponds to a typical (but slightly high) value in the late pre-ionization period (Dijkstra et al. 2008) and is well below that required to suppress H 2 radiatively (e.g., Shang, Bryan & Haiman 2010; Wolcott-Green, Haiman & Bryan 2011). We use these simulations to determine if the H 2 suppression mechanism suggested by Inayoshi & Omukai (2012) can be responsible for halting fragmentation in these halos. We carry out these runs until they collapse to high densities and then examine the resulting gas distribution. As we will show, these halos do form an abundant supply of H 2 , and so the Inayoshi & Omukai (2012) mechanism by itself does not appear to be sufficient to allow direct collapse. In fact, cooling and collapse set in well before the virial temperature reaches 10 4 K.</text>
<text><location><page_3><loc_7><loc_5><loc_46><loc_17></location>In a second set of simulations, we adopt a much higher value of the LW background, in particular we take J 21 = 10 5 , which is well above the critical flux required to suppress H 2 formation and cooling. We evolve these simulations until their virial temperatures are above 10 4 K, which allows us to (artificially) run the halo until it has a virial temperature sufficiently high for the Inayoshi & Omukai (2012) mechanism to operate. We then turn the LW background down to J 21 = 10 to see if and how the purely collisional H 2 suppres-</text>
<table>
<location><page_3><loc_50><loc_49><loc_87><loc_56></location>
<caption>Table 2. Virial quantities of our three halos, as defined at the indicated collapse redshifts z col for the J 21 = 10 runs.</caption>
</table>
<text><location><page_3><loc_50><loc_39><loc_89><loc_43></location>sion acts. These runs are intended as a academic exercise to see if gas that is in or near the 'zone of no return' will indeed stay there.</text>
<section_header_level_1><location><page_3><loc_50><loc_35><loc_60><loc_36></location>3 RESULTS</section_header_level_1>
<text><location><page_3><loc_50><loc_29><loc_89><loc_34></location>In the following sections, we present the results of our numerical simulations, first focusing on runs with low background flux, and then follow with high-UV background simulations.</text>
<section_header_level_1><location><page_3><loc_50><loc_25><loc_64><loc_26></location>3.1 J 21 = 10 Runs</section_header_level_1>
<text><location><page_3><loc_50><loc_5><loc_89><loc_24></location>The addition of radiative cooling causes the halos to cool and collapse before the redshifts at which they were identified in the low-resolution run. This occurs because the halos build up in a hierarchical fashion and so have progenitor halos with virial temperatures that allow them to cool via H 2 cooling. We stop the runs when they reach the maximum refinement level allowed in the simulation, as at this point they will start to form stars. The collapse redshifts, virial masses and virial temperatures of the halos at the point at which they collapse are shown in Table 2. Only the third halo reaches a virial temperature close to 10 4 K. In this context, we define collapse as the point when we would need additional refinement in order to continue evolving the halo, which corresponds to a density of approximately 10 8 cm -3 .</text>
<figure>
<location><page_4><loc_16><loc_37><loc_80><loc_90></location>
<caption>Figure 2. Slices of the y-z plane for density, temperature, entropy, Mach number, -∇· glyph[vector] V and H 2 fraction for halo C (with T vir = 8 . 1 × 10 3 K) with a low LW background. The field of view of each plot is 5 kpc (physical) and the virial radius is 0.64 kpc.</caption>
</figure>
<text><location><page_4><loc_7><loc_13><loc_46><loc_29></location>To investigate if the density-dependent H 2 suppression mechanism is operating, we analyze the simulation results at the point at which they collapse. In Figure 1 we plot the temperature and density distribution of all cells within a radius of a few times the virial radius for each simulation run. Additionally, we overlay the 'zone of no return' in this figure, using the approximate curves for the boundary of the zone from Inayoshi & Omukai (2012). From the plots we see directly that no cells have been shocked-heated into this 'no return zone'. This indicates that the H 2 suppression mechanism is not operating, at least for these halos at this point.</text>
<text><location><page_4><loc_7><loc_5><loc_46><loc_11></location>Inayoshi & Omukai (2012) argued that cold, filamentary flows would penetrate to the center of these halos, shock heating the gas at high densities. To better understand why this is not happening, in Figure 2 we plot slices through the center of halo C (with T vir = 8 . 1 × 10 3 K), showing the den-</text>
<text><location><page_4><loc_50><loc_13><loc_89><loc_29></location>ity, temperature, entropy, negative of the divergence of the velocity field, and H 2 fraction. From the density and temperature slices, we clearly see the cold filamentary structure feeding the halo. However, it is evident that no cold flows (T ∼ 10 2 -10 3 K) penetrate unperturbed into the center. Instead, the gas quickly heats to the average temperature around the virial radius. This is consistent with the phase plots which show continual heating of the cells at number densities approach ∼ 1 cm -3 . Most of the cells at this number density have roughly T ∼ 10 4 K. At higher densities glyph[greaterorsimilar] 1 cm -3 , H 2 cooling becomes dominant, leaving the gas roughly at ∼ 700 K.</text>
<text><location><page_4><loc_50><loc_5><loc_89><loc_11></location>One way to identify shocks is to look for sources of entropy production (although shocks are not the only source of entropy). Figure 2 shows that the largest entropy production happens around the virial radius. This is largely due to the spherical accretion of the surrounding cold gas. This</text>
<text><location><page_5><loc_7><loc_70><loc_46><loc_90></location>is more apparent in the Mach number slice, where the flow outside the virial radius is supersonic and the flow interior to the virial radius is mostly transonic. The transition from supersonic to subsonic flow happens abruptly. We also see the cold inflows are not immune to this shock, and gas in the filaments begins to exhibit higher entropy at the virial radius (although the entropy is considerably lower than most of the gas at the virial temperature). The entropy generation is at the expense of its kinetic energy as evident in the Mach number slice where the inflow also becomes transonic as it passes the virial radius. The negative divergence of the velocity shown in Figure 2 reinforces our previous statements. Shocks produce a large value of -∇· glyph[vector] V . It is evident there is a strong shock at the virial radius and the filamentary structure is disrupted as it passes through this radius.</text>
<text><location><page_5><loc_7><loc_63><loc_46><loc_70></location>This shows clearly that the cold, filamentary flows in these halos shock at or around the virial radius, where their densities are low - too low for the H 2 suppression mechanism suggested in Inayoshi & Omukai (2012) to operate. Instead, the shocked filaments can efficiently form H 2 and cool.</text>
<text><location><page_5><loc_7><loc_33><loc_46><loc_63></location>We argue that this conclusion is largely consistent with (comparable) previous work. For example, Wise & Abel (2007) carried out similar simulations, finding that although filamentary gas can, in some cases, penetrate through the virial radius without shocking, it does not get past one-third of the virial radius (as stated in the abstract of that paper). Moreover, this occurs only in the absence of H 2 cooling (i.e. in their H+He only runs A6 and B6), which is consistent with our own findings. In any case, this penetration is not nearly far enough - for example, Fig. 2 of that paper demonstrates that the densities do not get to 10 4 cm -3 until about 0.005 r vir . The SPH simulations of Greif et al. (2008) also examine the inflow of filaments in halos of similar size. They appear to find that cold filaments may not shock-heat as the flow in; however two points are relevant here. The first is that later work (Nelson et al. 2013) has shown that SPH simulations incorrectly predict substantial amount of coldaccretion compared to Eulerian or moving-mesh codes. The second point is that even in the simulations of Greif et al. (2008), gas does not get close to the H 2 zone of no return (e.g. their Fig. 7). Therefore, we conclude that our key results are in agreement with previous work.</text>
<text><location><page_5><loc_7><loc_24><loc_46><loc_32></location>We have seen that - for the three halos simulated the halos collapse and fragment without any gas entering the 'zone of no return'. They do this in part because the 10 4 K halos do not collapse monolithically - instead, the hierarchical formation naturally involves lower temperature halos which can cool efficiently via H 2 .</text>
<text><location><page_5><loc_7><loc_15><loc_46><loc_24></location>One possible objection is that we have simply not simulated enough halos, and that a small fraction of halos might collapse quickly enough for the suppression mechanism to be efficient. We examine this possibility in more detail in Section 4; however, first (in the next section), we look at what would happen if we could somehow suppress cooling in the lower mass halos until highT vir halos build up.</text>
<text><location><page_5><loc_7><loc_5><loc_46><loc_14></location>Before examining that issue, we make a brief remark about resolution - a concern with any result from a numerical simulation. Although we have not explicitly carried out a resolution study as part of this work, this point has been examined regularly in the past. For example, Machacek, Bryan & Abel (2001) explicitly ran multiple simulations (also with the Enzo code) to examine how mass resolution impacted</text>
<text><location><page_5><loc_89><loc_77><loc_90><loc_78></location>/circledot</text>
<figure>
<location><page_5><loc_52><loc_60><loc_89><loc_87></location>
<caption>Figure 3. Phase plot of the number density and temperature (with color showing the gas mass distribution) for a simulation of halo A with a large UV background ( J 21 = 10 5 ). The high flux suppresses cooling and we see a significant number of cells in the 'zone of no return'.</caption>
</figure>
<text><location><page_5><loc_50><loc_15><loc_89><loc_48></location>the cooling of primordial halos like the ones simulated here. They used dark matter particles masses of 306 and 38 M glyph[circledot] , finding only minor changes in the statistics of cooling halos, implying that, with a dark matter particle mass of 85 M glyph[circledot] , our results are robust to mass resolution. A related but slightly different concern is the resolution of the Jeans length - recent work has suggested that the Jeans length may, in some circumstances, need to be resolved by more than the 16 cells adopted here. For example, Turk et al. (2012) find that the magnetic properties require ratios greater than 32, with possible changes to the temperature profile as the resolution is changed. However, the temperature changes are not found in Greif, Springel & Bromm (2013), and moreover, the impact is only significant at higher densities than explored here, n glyph[greaterorsimilar] 10 12 cm -3 . At lower densities, the profiles are well converged. Therefore, we conclude that this issue is unlikely to impact the results found here. We can also compare our profiles to previous work. For example, O'Shea & Norman (2008) carried out simulations of similar halos with similar LW backgrounds - the temperature profiles they find (e.g. their Fig. 9) are very similar to ours. As that paper demonstrates, the relatively high temperatures ( ∼ 700 K) we find in the H 2 cooling region are an indirect consequence of the relatively high LW background used here.</text>
<section_header_level_1><location><page_5><loc_50><loc_11><loc_64><loc_12></location>3.2 J 21 = 10 5 Run</section_header_level_1>
<text><location><page_5><loc_50><loc_5><loc_89><loc_10></location>To investigate if the collisional H 2 suppression mechanism can keep a halo from fragmenting if we could place its central region in or near the 'zone of no return', we re-simulate halo A using a much higher value of the LW-background to</text>
<figure>
<location><page_6><loc_16><loc_37><loc_80><loc_90></location>
<caption>Figure 4. Slices of the y-z plane for density, temperature, Entropy, Mach number, -∇· glyph[vector] V and H 2 fraction for Halo A with a high-UV background ( J 21 = 10 5 ). The field of view of each plot is 3 kpc and the virial radius is 0.53 kpc.</caption>
</figure>
<text><location><page_6><loc_7><loc_15><loc_46><loc_29></location>artificially suppress H 2 cooling. We adopt J 21 = 10 5 - with such a large (and unrealistic) background, most of the H 2 will be dissociated, as shown by Shang, Bryan & Haiman (2010) and Wolcott-Green, Haiman & Bryan (2011), where the critical flux was determined to be J 21 = 10 3 . We picked the halo with the smallest virial temperature at collapse in the previous runs ( T vir = 2 . 86 × 10 3 K) and reran it with the large J 21 . The halo collapses at a later redshift, z = 16 . 74, with a larger virial temperature T vir = 0 . 97 × 10 4 K, mass 2 . 48 × 10 7 M glyph[circledot] , and radius 534 pc.</text>
<text><location><page_6><loc_7><loc_5><loc_46><loc_13></location>In Figure 3, we show the number density and temperature phase plot for this run. We see the cells for n glyph[greaterorsimilar] 1 cm -3 have now increased in temperature. The large J 21 has dissociated most of the H 2 , leaving atomic cooling as the dominant mechanism and keeping the gas at T ∼ 8000 K. For n glyph[lessorsimilar] 1 cm -3 the same structure is apparent as the pre-</text>
<text><location><page_6><loc_50><loc_26><loc_89><loc_29></location>ous simulations, where cells are heated to T ∼ 10 4 K at the virial radius.</text>
<text><location><page_6><loc_50><loc_5><loc_89><loc_22></location>Figure 4 shows slices with the same quantities as in the previous section, but now for the large J 21 run. Although the general structure is quite similar, there are now some signs of cold flows penetrating past the virial radius. This is more apparent in the Mach number slice where we see high velocity flows passing the virial radius where the majority of the fluid is transonic. It can also be seen in the temperature distribution, where low-temperature gas flows into the halo before heating. However, we note that the 'zone of no return' occurs at densities above ∼ 10 4 cm -3 , which corresponds to the very central region (colored red) of the density slice, and so apparently, the cold flows are not penetrating this zone.</text>
<text><location><page_7><loc_81><loc_65><loc_83><loc_66></location>/circledot</text>
<figure>
<location><page_7><loc_14><loc_35><loc_82><loc_85></location>
<caption>Figure 5. Phase plots showing the distribution of number density and temperature in halo A at a variety of redshifts, as indicated. The simulation ran with J 21 = 10 5 until the maximum density reached the edge of the 'zone of no return' (as shown in the top left panel). Thereafter the gas is evolved with a reduced J 21 = 10. Gas quickly forms H 2 and cools to a temperature well below the zone.</caption>
</figure>
<section_header_level_1><location><page_7><loc_7><loc_23><loc_34><loc_24></location>3.2.1 Stability of the Zone of no Return</section_header_level_1>
<text><location><page_7><loc_7><loc_5><loc_46><loc_21></location>Next, we investigate if gas in the 'zone of no return' will indeed stay there without any external flux. We do this by turning down the high-UV background flux in the previous simulation. We run two identical simulations as in the last section, but now reduce J 21 at different evolutionary points. The two points are distinguished by whether or not gas cells have reached the critical H 2 number density for local thermodynamic equilibrium, n cr ∼ 10 4 cm -3 . Thus, in our first simulation, we allow the gas to evolve with a large J 21 until the maximum density just reaches n cr , and at this point the UV background is reduced to J 21 = 10 and the gas is allowed to evolve. Similarly, our second simulation is iden-</text>
<text><location><page_7><loc_50><loc_20><loc_89><loc_24></location>cal with the exception that the reduction is applied when a small fraction of cells have entered into the 'zone of no return'.</text>
<text><location><page_7><loc_50><loc_6><loc_89><loc_20></location>Figure 5 shows the phase plot evolution for the first simulation (the instant of flux reduction corresponds to the upper-left panel, at z = 16 . 831). As the simulation evolves with the reduced J 21 , H 2 begins to form and cool efficiently. The cooling occurs first at densities n ∼ 10 3 cm -3 , leaving the higher density regions still hot ( T ∼ 10 4 K). As the evolution progresses, this gas eventually cools and by the final time shown, when the halo collapses to the highest refinement level we allow, the phase distribution looks very much like the low-J runs.</text>
<text><location><page_7><loc_53><loc_5><loc_89><loc_6></location>It is interesting to analyze this evolution in more detail.</text>
<figure>
<location><page_8><loc_8><loc_63><loc_47><loc_90></location>
<caption>Figure 6. The left set of four panels shows temperature slices through the center of halo A corresponding to the same simulation and output times as in Figure 5, while the right set of four panels shows number density slices for the same halo at the same sets of times. After the background flux is reduced, the lower-density region outside the core cools first, driving an outflow which evaporates the center.</caption>
</figure>
<figure>
<location><page_8><loc_49><loc_63><loc_87><loc_90></location>
</figure>
<text><location><page_8><loc_49><loc_77><loc_50><loc_78></location>z</text>
<text><location><page_8><loc_7><loc_41><loc_46><loc_54></location>In Figure 6, we show temperature (left) and density (right) slices that go through the center of this halo for the same times presented in Figure 5. We see that the cooling process creates a cold envelope surrounding the hot core. This core is over-pressured, which generates an outflow, driving clumps of dense material out of the central region, as evident in the density slices of Figure 6. The outflow reduces the density (and temperature) of the gas in the core, which is then effectively evaporated. The final remnant is a dense shell surrounding a core that has cooled on average to T ∼ 700 K.</text>
<text><location><page_8><loc_7><loc_21><loc_46><loc_40></location>In the second simulation of this section, we allow the run with J 21 = 10 5 to continue a bit further, to z = 16 . 754, when a significant amount of gas has entered the 'zone of no return' (see the top left panel of Figure 7) before turning the UV background down to J 21 = 10. The resulting evolution of the phase diagram is shown in Figure 7. In this case the evolution is quite different: although some gas does cool, the dense gas evolves more quickly, and by the time we stop the simulation (when it reaches our highest refinement level at z = 16 . 749), the amount of mass inside the 'zone of no return' has not changed by more than a few percent. There is some cooling - in particular at n ∼ 10 3 cm -3 and n ∼ 10 6 cm -3 ; however, this does not significantly affect the evolution of the clump.</text>
<text><location><page_8><loc_7><loc_17><loc_46><loc_21></location>The resulting baryonic mass in the 'zone of no return' is ∼ 10 5 M glyph[circledot] , roughly 10% of the halo mass within the virial radius.</text>
<section_header_level_1><location><page_8><loc_7><loc_13><loc_20><loc_14></location>4 DISCUSSION</section_header_level_1>
<text><location><page_8><loc_7><loc_5><loc_46><loc_11></location>Using a set of numerical simulations, we have investigated the viability of the mechanism for collisional suppression of H 2 suggested by Inayoshi & Omukai (2012). We found that, for the three halos examined, H 2 -mediated cooling set in before the conditions required for suppression could be</text>
<text><location><page_8><loc_50><loc_50><loc_89><loc_54></location>established - in particular, before a T vir glyph[greaterorsimilar] 10 4 K halo forms, which would be a pre-requisite for the generation of shocks strong enough to put gas into the 'zone of no return'.</text>
<text><location><page_8><loc_50><loc_36><loc_89><loc_50></location>However, we did confirm (using a set of simulations that employed a high UV background to temporary suppress cooling) some aspects of the mechanism: if the gas in the core of an atomic cooling halo can avoid H 2 cooling until it enters the 'zone of no return', then subsequent H 2 cooling can be naturally averted. The gas would then remain at ∼ 10 4 K, possibly leading to conditions suitable for SMBH formation, as envisioned in the many previous works mentioned in the Introduction (see, e.g. Haiman 2013 for a recent review).</text>
<text><location><page_8><loc_50><loc_23><loc_89><loc_35></location>Unfortunately, H2 cooling at any time in the past history of the same gas would likely invalidate this possibility, if this H2 cooling allowed the gas to reach temperatures of a few 100 K and to evolve to high density at such low temperatures. This cold and dense gas, at the center of the progenitor minihalo, would form one, or possibly a few, PopIII stars. The subsequent evolution of this progenitor into an atomic cooling halo is uncertain, however, there are many obstacles to rapid SMBH formation.</text>
<text><location><page_8><loc_50><loc_5><loc_89><loc_22></location>First, even a handful of SNe (or a single pair instability SNe) can enrich the entire atomic cooling halo to a metallicity of Z glyph[greaterorsimilar] 10 -3 Z glyph[circledot] (e.g., Bromm & Yoshida 2011, see page 396) the critical value above which direct SMBH formation is replaced by fragmentation (Inayoshi & Omukai 2012). It is possible that none of these Pop III stars produce any metals (such as non-rotating metal-free stars between 40-140 M glyph[circledot] ; Heger et al. 2003). Such massive Pop III stars would leave behind stellar-mass BHs, which, in principle, could grow rapidly into SMBHs, provided that they are surrounded by very dense gas, and accrete sufficiently rapidly to trap their own radiation (e.g., Volonteri & Rees 2005). However, the parent Pop III stars of these seed BHs will</text>
<text><location><page_9><loc_81><loc_65><loc_83><loc_66></location>/circledot</text>
<figure>
<location><page_9><loc_14><loc_35><loc_82><loc_85></location>
<caption>Figure 7. Phase plots showing the distribution of number density and temperature in halo A at a variety of redshifts, as indicated. The simulation ran with J 21 = 10 5 until the maximum density reached well into the 'zone of no return' (top left panel). Thereafter the gas is evolved with a reduced J 21 = 10. Although some gas at moderate densities begins to cool, the gas in the 'zone of no return' collapses more quickly than the outer region can cool.</caption>
</figure>
<text><location><page_9><loc_7><loc_9><loc_46><loc_23></location>create a large ionized bubble and therefore the seed BHs will likely begin their life in a low-density medium. As they later begin to accrete, their radiation likely self-limits the time-averaged accretion rate to a fraction of the Eddington rate (Alvarez, Wise & Abel 2009; Park & Ricotti 2012; Milosavljevi'c, Couch & Bromm 2009). Although the above scenarios are worth investigating further in detail, we will hereafter assume that these obstacles prevent rapid SMBH formation in any halo that experienced H 2 cooling at any time in its history.</text>
<section_header_level_1><location><page_9><loc_50><loc_22><loc_89><loc_23></location>4.1 Avoiding H 2 Cooling by Rapid Halo Assembly</section_header_level_1>
<text><location><page_9><loc_50><loc_7><loc_89><loc_20></location>However, since we only simulated three halos a natural question is whether H 2 cooling may be avoided in a few, highly atypical, atomic cooling halos, even if the background flux J LW was much lower. One possibility is that - in rare cases the progenitor halos could experience unusually rapid mergers, continuously shock-heating the gas on a time-scale that always remains shorter than the H 2 -cooling time. Here we evaluate the likelihood of this scenario, using Monte Carlo realizations of the merger histories of 10 6 atomic cooling halos.</text>
<text><location><page_9><loc_53><loc_5><loc_89><loc_6></location>In particular, we start by creating dark matter halo</text>
<text><location><page_10><loc_7><loc_73><loc_46><loc_90></location>merger trees, using the Monte Carlo algorithm in Zhang, Fakhouri & Ma (2008). This paper presents three different numerical algorithms, which are based on the conditional halo mass functions in the ellipsoidal collapse model (Sheth & Tormen 2002; Zhang, Ma & Fakhouri 2008). We adopted their 'method B', which was found in subsequent work to provide the best match to the statistics of merger trees in N-body simulations (Zhang 2012, private communication). This method represents an improvement over previous merger-tree methods based on spherical collapse - in particular, the ellipsoidal collapse models tend to predict a larger number of more massive progenitors, or a 'flatter' assembly history (Tanaka, Li & Haiman 2013).</text>
<text><location><page_10><loc_7><loc_66><loc_46><loc_73></location>We have created 10 6 merger trees of a DM halo with T vir , 0 = 10 4 K ( M 0 = 6 . 3 × 10 7 M glyph[circledot] ) at redshift z 0 = 10, extending back to redshift z = 20. We then follow the mass of the most massive progenitor back in time, and at each redshift z , we compute the H 2 -cooling time</text>
<formula><location><page_10><loc_7><loc_62><loc_46><loc_65></location>t H2 = 1 . 5 n g k B T Λ n H n H 2 (1)</formula>
<text><location><page_10><loc_7><loc_58><loc_46><loc_62></location>in this progenitor, using the procedures and rates in Machacek, Bryan & Abel (2001). Specifically, we compute the maximum density</text>
<formula><location><page_10><loc_7><loc_55><loc_46><loc_57></location>n max = 187Ω b h 2 ( T vir / 1000 K ) 1 . 5 cm -3 (2)</formula>
<text><location><page_10><loc_7><loc_31><loc_46><loc_54></location>that could be reached by the gas in the nucleus of this progenitor, in the absence of H 2 -cooling (i.e. by adiabatic compression). We then specify a constant background flux J 21 , and compute the equilibrium H 2 fraction and the corresponding t H2 at this density, given J 21 and ρ max . Finally, we require that the longer of the dynamical time t dyn = √ 3 π/ 16 Gρ max and t H2 remains longer than the cosmic time ∆ t ( z ) = t ( z 0 ) -t ( z ) elapsed between the z and z 0 . This last requirement ensures that the most massive progenitor of our atomic cooling halo at redshift z > z 0 cannot cool via H 2 and evolve dynamically, before it is incorporated by mergers into the M 0 = 6 . 3 × 10 7 M glyph[circledot] atomic cooling halo at redshift z 0 = 10. Note that this requirement must be satisfied at all redshifts z > z 0 - in other words, if our atomic cooling halo had a progenitor, at any redshift z , that was massive enough to cool and collapse, the 'SMBH formation by direct collapse' scenario is no longer feasible.</text>
<text><location><page_10><loc_7><loc_24><loc_46><loc_31></location>In practice, as we run our merger trees backward, we simply look, at each redshift step, whether the most massive progenitor violates the criterion ∆ t ( z ) < max[ t H2 , t dyn ]. If it does, we record the redshift z max where this happened, and we discard the rest of this merger tree.</text>
<text><location><page_10><loc_7><loc_5><loc_46><loc_24></location>In Figure 8, we show the probability distribution of the 'terminal lookback redshift' z max we have found among the 10 6 merger trees, for four different values of a constant J LW = 1 , 3 , 10, and 30. For a LW background J LW = 1 (shown in the figure as the solid black histogram), we found that all of the 10 6 merger trees contain an H 2 -cooling progenitor by z < 20. However, for the higher background fluxes of J LW = 3 (dot-dashed green), J LW = 10 (dotted red), or J LW = 30 (dashed blue), we have found approximately 2, 200, and 4000 cases, respectively, where none of the progenitors, out to z = 20, could cool via H 2 - this is shown by the pile-up of the probability distribution in the last bin at ∆ z max = 10. For the atomic cooling halos with these particular merging histories, we expect that H 2 cool-</text>
<figure>
<location><page_10><loc_51><loc_64><loc_87><loc_91></location>
<caption>Figure 8. The figure is derived from 10 6 Monte Carlo realizations of the merger tree of an atomic cooling halo with T vir , 0 = 10 4 K ( M 0 = 6 . 3 × 10 7 M glyph[circledot] ) at redshift z 0 = 10. It shows the probability distribution of the terminal lookback redshift ∆ z max , at which the atomic cooling halo had a progenitor, which was massive enough to cool via H 2 and collapse. The 'SMBH formation by direct collapse' scenario is feasible only if none of the progenitors can cool and collapse prior to z 0 = 10. For a LW background J LW = 1 (shown by the solid black histogram), all of the 10 6 merger trees contain an H 2 -cooling progenitor. However, for higher background fluxes of J LW = 3 (in dot-dashed green), J LW = 10 (in dotted red), or J LW = 30 (in dashed blue), we find approximately 2, 200, and 4000 cases, respectively, where none of the progenitors, out to z = 20, could cool via H 2 - this is shown by the pile-up of the probability distribution in the last bin at ∆ z max = 10.</caption>
</figure>
<text><location><page_10><loc_50><loc_24><loc_89><loc_39></location>g is avoided entirely until the atomic cooling halo forms at z 0 = 10. These rare halos are good candidates where the subsequent ionizing shocks can prevent the gas temperature from falling below ≈ 10 4 K, allowing the gas to collapse directly to a SMBH. This scenario reduces the required LW flux by a factor of ∼ 300, from J 21 ≈ 10 3 (Shang, Bryan & Haiman 2010; Wolcott-Green, Haiman & Bryan 2011) to J 21 ≈ 3, but it does require that the latter LW background is in place at a redshift as high as z ∼ 20 (in order for a fraction 2 × 10 -6 of the atomic cooling halos at z 0 = 10 to be such candidates).</text>
<section_header_level_1><location><page_10><loc_50><loc_19><loc_62><loc_20></location>5 SUMMARY</section_header_level_1>
<text><location><page_10><loc_50><loc_5><loc_89><loc_18></location>One way to explain the presence of SMBHs at z > 6 with masses greater than 10 9 M glyph[circledot] is through the formation of a massive ( ∼ 10 5 M glyph[circledot] ) BH seed from gas in T vir glyph[greaterorsimilar] 10 4 Khalos, either via direct collapse to a BH or through the formation of a supermassive star or a quasi-star (Begelman, Rossi & Armitage 2008; Hosokawa et al. 2013). As noted in the Introduction, one difficulty with these modes is the need to prevent fragmentation of the cooling halo into lower mass stars, and, in particular, to prevent cooling due to H 2 . Inayoshi & Omukai (2012) have suggested that this can be</text>
<text><location><page_11><loc_7><loc_76><loc_46><loc_90></location>accomplished through the action of cold flows, which result in gas shocking to high densities ( n > 10 4 cm -3 ) and temperatures ( T > 10 4 K). This shocked gas cools by Ly α emission to about 8000 K; however, if the density is high enough, enhanced H 2 collisional dissociation suppresses the gas from cooling further. This was demonstrated using onezone models calculated by Inayoshi & Omukai (2012), who identified a 'zone of no return'. This mechanism is appealing as it does not require a high LW background to destroy the H 2 ; however, it is unclear if this mechanism operates in nature.</text>
<text><location><page_11><loc_7><loc_60><loc_46><loc_75></location>To test this idea, we carried out cosmological hydrodynamic simulations with the adaptive mesh refinement code Enzo. We first identified three halos from a low resolution simulation which all had T vir glyph[greaterorsimilar] 10 4 K at redshifts ranging from 12 to 17. We then re-simulated these halos at highresolution with a relatively modest UV flux in the LymanWerner band, J 21 = 10. We found that in all three cases, cooling from H 2 was able to efficiently lower the gas temperature below that of the 'zone of no return', indicating that, at least for these three halos, the mechanism was not operating.</text>
<text><location><page_11><loc_7><loc_41><loc_46><loc_60></location>To determine why this occurred, we examined the structure of the simulated halos in detail, and found that, while cold flows do occur, they generally shock at or near the virial radius and do not penetrate into the halo center where the densities are high. We note that although these small halos have low virial temperatures and so might be naively classified as 'cold-mode' halos (Birnboim & Dekel 2003; Kereˇs et al. 2005; Dekel & Birnboim 2006), they actually have low cooling rates because of the inefficiency of H 2 cooling. Therefore, the characteristic cooling times of these halos is longer (or comparable) to their dynamical times, making them more akin to hot-mode halos. This is consistent with the clear virial shocks that we see in these halos (e.g., Figure 2).</text>
<text><location><page_11><loc_7><loc_17><loc_46><loc_40></location>To determine if the suppression mechanism could function if we artificially suppressed H 2 cooling, we reran one of the simulations with a high LW background ( J 21 = 10 5 ), well above the critical flux required to suppress H 2 cooling radiatively (Shang, Bryan & Haiman 2010; Wolcott-Green, Haiman & Bryan 2011). We showed that, in this case, the gas did indeed enter the 'zone of no return'. To see if this situation was stable, we ran two variations of this simulation, in which we turned the flux off either just before, or just after the gas in the halo center entered the 'zone of no return'. In the first case, cooling eventually won out, with the region just outside the core cooling first and driving an evaporative wind which led to cooling by H 2 throughout the halo. However, in the second case, the gas inside the 'zone of no return' stayed there, and eventually collapsed to high densities while remaining at T ∼ 8000 K, despite the lack of any LW flux.</text>
<text><location><page_11><loc_7><loc_5><loc_46><loc_17></location>This demonstrates that the mechanism would work if a halo could collapse quickly enough, but that typical T vir glyph[greaterorsimilar] 10 4 K halos have progenitors which can cool efficiently. To investigate if any halos can collapse quickly enough to escape this fate, we ran Monte Carlo merger tree calculations of 10 6 halos with an ellipsoidal collapse model. We used a simple analytic prescription to determine if any halos could assemble quickly enough to prevent cooling and collapse prior to building up to a T vir = 10 4 K halo. We</text>
<text><location><page_11><loc_50><loc_83><loc_89><loc_90></location>found that for a Lyman-Werner background of J 21 = 1, none of the 10 6 merger tree histories collapsed quickly enough. For larger LW backgrounds, a small fraction (up to ∼ 10 -3 for J 21 as high as 30) of the halos did form quickly enough to evade cooling. We note that at higher redshift, J 21 would certainly be expected to be lower.</text>
<text><location><page_11><loc_50><loc_59><loc_89><loc_82></location>In summary, we confirm the essential physics proposed by Inayoshi & Omukai (2012), but we find that the supersonic filamentary flows are unlikely to shock gas into the 'zone of no return', as they had originally envisioned. Rather, we propose here a modification of their scenario. In the core of a few rare halos, which assembled unusually rapidly, the gas may have been kept continuously shockheated throughout their history, and eventually shocked into the 'zone of no return'. This scenario reduces the required LW flux by a factor of ∼ 300, from J 21 ≈ 10 3 (Shang, Bryan & Haiman 2010; Wolcott-Green, Haiman & Bryan 2011) to J 21 glyph[greaterorsimilar] 1, but it does require that the latter LW background is in place at a redshift as high as z ∼ 20. Future, selfconsistent global models of the build-up of the LW background, together with the assembly of a large number of atomic-cooling halos, is required to assess the viability of this scenario.</text>
<section_header_level_1><location><page_11><loc_50><loc_54><loc_69><loc_55></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_11><loc_50><loc_41><loc_89><loc_53></location>We thank Jun Zhang for useful discussions about ellipsoidal collapse-based merger trees, and for sharing unpublished results on comparisons between Monte Carlo methods and Nbody simulations. ZH acknowledges financial support from NASA through grant NNX11AE05G. GB acknowledge financial support from NSF grants AST-0908390 and AST1008134, and NASA grant NNX12AH41G, as well as computational resources from NSF XSEDE, and Columbia University's Hotfoot cluster.</text>
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</document> | [{"title": "ABSTRACT", "content": "The presence of quasars at redshifts z > 6 indicates the existence of supermassive black holes (SMBHs) as massive as a few times 10 9 M glyph[circledot] , challenging models for SMBH formation. One pathway is through the direct collapse of gas in T vir glyph[greaterorsimilar] 10 4 K halos; however, this requires the suppression of H 2 cooling to prevent fragmentation. In this paper, we examine a proposed new mechanism for this suppression which relies on cold-mode accretion flows leading to shocks at high densities ( n > 10 4 cm -3 ) and temperatures ( T > 10 4 K). In such gas, H 2 is efficiently collisionally dissociated. We use high-resolution numerical simulations to test this idea, demonstrating that such halos typically have lower temperature progenitors, in which cooling is efficient. Those halos do show filamentary flows; however, the gas shocks at or near the virial radius (at low densities), thus preventing the proposed collisional mechanism from operating. We do find that, if we artificially suppress H 2 formation with a high UV background, so as to allow gas in the halo center to enter the high-temperature, high-density 'zone of no return', it will remain there even if the UV flux is turned off, collapsing to high density at high temperature. Due to computational limitations, we simulated only three halos. However, we demonstrate, using Monte Carlo calculations of 10 6 halo merger histories, that a few rare halos could assemble rapidly enough to avoid efficient H 2 cooling in all of their progenitor halos, provided that the UV background exceeds J 21 \u223c few at redshifts as high as z \u223c 20. Key words: black hole physics - methods:numerical - cosmology:theory", "pages": [1]}, {"title": "H 2 Suppression with Shocking Inflows: Testing a Pathway for Supermassive Black Hole Formation", "content": "Ricardo Fernandez 1 , Greg L. Bryan 1 , Zoltan Haiman 1 , and Miao Li 1 1 Department of Astronomy, Columbia University, 550 West 120th Street, New York, NY 10027, USA", "pages": [1]}, {"title": "1 INTRODUCTION", "content": "jevi'c, Couch & Bromm 2009; Tanaka & Haiman 2009; Jeon et al. 2012; Tanaka, Perna & Haiman 2012). Dynamical evidence indicates that most nearby galaxies harbor a central supermassive black hole (e.g., Ferrarese & Ford 2005), including our own Milky Way, which hosts a central SMBH with mass \u223c 10 6 M glyph[circledot] (Ghez et al. 2005). Furthermore, the discovery of quasars at redshifts greater than 6 signals the existence of SMBHs as massive as a few times 10 9 M glyph[circledot] at an epoch when the Universe was less than a billion years old (e.g., Fan 2006; Mortlock et al. 2011). Such massive and early SMBHs pose a challenge to current models of their formation. One possible formation scenario is the growth of a remnant black hole (BH) seed, generated from a population III star ( \u223c 100 M glyph[circledot] ), by mergers and gas accretion (e.g., Haiman & Loeb 2001; Volonteri, Haardt & Madau 2003; Li et al. 2007). However, this formation scenario poses certain difficulties. The time to assemble a 10 9 M glyph[circledot] SMBH by standard Eddington accretion is comparable to the age of the universe at z \u223c 6 and it is unlikely that the seed BH will have continual accretion due to negative feedback and merger-induced gravitational recoils (Alvarez, Wise & Abel 2009; Milosavl- An alternative pathway is the direct collapse of metalfree primordial gas with virial temperature glyph[greaterorsimilar] 10 4 K into a BH seed of mass 10 4 -10 6 M glyph[circledot] (Oh & Haiman 2002; Bromm & Loeb 2003; Regan & Haehnelt 2009; Shang, Bryan & Haiman 2010). Such a large seed requires many fewer Salpeter times to grow to quasar size and so bypasses many of the difficulties of Eddington growth of stellar BH seeds. The exact mechanism by which the collapse occurs is not entirely clear (e.g., Bromm & Loeb 2003; Begelman, Rossi & Armitage 2008; Begelman 2010); however, a vital condition for this scenario is that the collapsing gas avoids fragmentation into stars. A natural way to avoid fragmentation is to have a long Jeans length due to a high gas temperature. The temperature of the gas depends on the interplay of atomic and molecular cooling. In the absence of H 2 , a halo with T vir glyph[greaterorsimilar] 10 4 K cools to \u223c 8000 K by atomic hydrogen. Including H 2 , the halo cools further to \u223c 200 K. The corresponding Jeans mass at characteristic central densities M J \u2248 10 6 M glyph[circledot] ( T/ 10 4 K ) 3 / 2 is 10 3 M glyph[circledot] for the latter, suggesting the formation of a Pop III star, and 10 6 M glyph[circledot] for the former, suggesting direct collapse into a massive BH. There- a necessary condition for direct collapse is to prevent cooling by H 2 . The suppression of H 2 can be accomplished by a strong far ultraviolet (UV) radiation flux in the Lyman-Werner (LW) bands, J 21 glyph[greaterorsimilar] 10 2 -10 3 (in units of 10 -21 erg s -1 cm -2 Hz -1 sr -1 ) (Omukai 2001; Bromm & Loeb 2003; Shang, Bryan & Haiman 2010). This photo-dissociates the molecular hydrogen. However, only a small subset glyph[lessorsimilar] 10 -6 of all atomic cooling halos are estimated to be exposed to such levels, due to the presence of close luminous neighbor (Dijkstra et al. 2008; Agarwal et al. 2012). This makes it difficult to explain the production of the observed quasar population at z > 6. Note that the threshold only increases in the presence of a cosmic-ray/X-ray flux (Inayoshi & Omukai 2011). Inayoshi & Omukai (2012) have proposed an alternative mechanism for the suppression of H 2 cooling that does not depend on having a high UV flux. In this scenario, cold accretion flows penetrate to the center of the halo, colliding with each other and shocking to produce hot and dense gas. The post-shock layer cools efficiently due to atomic hydrogen cooling and contracts isobarically until the gas reaches \u223c 8000 K. If the shocked gas at this high temperature is already at a high enough number density n glyph[greaterorsimilar] 10 4 cm -3 , then H 2 rotational-vibrational levels reach local thermodynamic equilibrium, and collisional dissociation can destroy the molecular hydrogen. Crucially, once the gas is shocked to this high-temperature, high-density regime, it will no longer be able to cool via H 2 , even in the absence of any LW radiation. Throughout this paper, we will thus refer to this regime (defined more precisely below) as the 'zone of no return'. Inayoshi & Omukai (2012) have argued that this mechanism may be able to produce a massive BH seed without strong radiative feedback; however, their numerical experiments focused on one-zone models and so questions still remain about the applicability of this pathway in cosmological simulations. In particular, only fluid elements that have a sufficiently high temperature and density have their fragmentation suppressed, and it is not clear if: (1) the gas which ends up in halos that with such large virial temperatures is not first processed through lower mass halos in which fragmentation and star formation can occur 1 , and (2) if a sufficient amount of gas enters the 'zone of no return' for this mechanism to be important in realistic halos. To address these questions, we explore the possibility of H 2 suppression via cold accretion shocks by conducting numerical simulations. This paper is organized in the following manner. In Section 2 we describe the ingredients and initial setup of the code. In Section 3 we describe the results of our numerical simulations followed by a discussion in Section 4. Finally, in Section 5 we summarize our conclusions.", "pages": [1, 2]}, {"title": "2 NUMERICAL METHOD", "content": "The simulations were performed with the publicly available Eulerian adaptive mesh refinement (AMR) Enzo code. (Bryan 1999; Norman & Bryan 1999; O'Shea et al. 2004; Bryan et al. 2013). The code implements an N-body particle mesh technique (Efstathiou et al. 1985; Hockney & Eastwood 1988) to follow the dynamics of the dark matter particles and an Eulerian AMR method (Berger & Colella 1989) for the gas. In addition, Enzo provides modules which compute the radiative cooling of the gas as well as solve the chemical reaction network of a primordial mixture of H and He. Our simulations use the H 2 cooling function of Galli & Palla (1998) and solve the non-equilibrium evolution of the following nine species: H, H + , He, He + , He ++ , H -, H 2 , H + 2 , and e -(Abel et al. 1997; Abel, Bryan & Norman 2000). Density-dependent collisional dissociation (Martin, Schwarz & Mandy 1996) is important - we include this with a rate as described in Shang, Bryan & Haiman (2010). We use a set of zoom simulations to focus on a number of halos selected from a 1 h -1 Mpc comoving box, with a root grid resolution of 128 3 , using standard \u039b cold dark matter model parameters: \u2126 \u039b , 0 = 0 . 721, \u2126 m, 0 = 0 . 233, \u2126 b = 0 . 233, \u03c3 8 = 0 . 817, n s = 0 . 96 and h = 0 . 701 (Komatsu et al. 2009). Initially we performed a low resolution run with the maximum refinement level set to 4 and radiative cooling turned off to inhibit the gas from collapsing to high densities. We evolved this simulation from z = 99 to z = 10. Then we applied the HOP halo finder (Eisenstein & Hut 1998) to the resulting data files at various redshifts to identify halos with masses corresponding to virial temperatures glyph[greaterorsimilar] 10 4 K. Throughout this paper, we adopt the relation T vir = 0 . 75 \u00d7 1800( M/ 10 6 M glyph[circledot] ) 2 / 3 (1 + z ) / 21 K between halo mass and virial temperature. This is consistent with the commonly adopted version for neutral primordial gas with mean molecular weight \u00b5 = 1 . 2 (Bryan & Norman 1998), except that we reduced the normalization by a factor of 0.75. We have found that this correction agrees better with our simulations - that is, it yields T vir = 10 4 K for halos at the redshift and mass when they begin to cool efficiently via atomic H. Three halos where selected at random (see Table 1) to be re-run at high-resolution. We regenerated the initial conditions for the volumes, adding three nested grids that enclosed the Lagrangian volume of each halo. Since each additional grid doubles the spatial resolution, this resulted in an innermost grid with an effective resolution of 1024 3 and a dark matter particle mass of \u223c 85 M glyph[circledot] . Radiative cooling and multi-species were turned on to self-consistency follow the build-up of molecular hydrogen. During the course of the simulation each cell was adaptively refined using the following three criteria: baryon mass, dark matter mass and Jeans length. For the first two criteria, refinement is added when the baryon or dark matter mass exceeds four times the mass of the initial most refined cell, corresponding to mass resolutions of 68 and 340 M glyph[circledot] for the baryons and dark matter, respectively. The third criterion enforces the Truelove et al. (1997) condition which states that at least four cells /circledot should resolve the Jeans length to avoid artificial fragmentation. In our simulations the Jeans length was resolved by sixteen cells to be sure that we adequately followed the collapse. Based on these criteria, the simulations were allowed to refine to a maximum level of 18, which corresponds to a comoving scale of 0.0298 h -1 pc. The dark matter distribution was smoothed at refinement level 13 (about 0.065 proper pc at z = 20) to suppress numerical effects from the discreteness of DM particles. We carried out two sets of runs for each of the halos, which differed only in the background LW flux that we adopted. In the first set of runs, we used J 21 = 10, where J 21 is the specific intensity in the Lyman-Werner bands (11.213.6 eV) in units of 10 -21 erg cm -2 sr -1 Hz -1 . This corresponds to a typical (but slightly high) value in the late pre-ionization period (Dijkstra et al. 2008) and is well below that required to suppress H 2 radiatively (e.g., Shang, Bryan & Haiman 2010; Wolcott-Green, Haiman & Bryan 2011). We use these simulations to determine if the H 2 suppression mechanism suggested by Inayoshi & Omukai (2012) can be responsible for halting fragmentation in these halos. We carry out these runs until they collapse to high densities and then examine the resulting gas distribution. As we will show, these halos do form an abundant supply of H 2 , and so the Inayoshi & Omukai (2012) mechanism by itself does not appear to be sufficient to allow direct collapse. In fact, cooling and collapse set in well before the virial temperature reaches 10 4 K. In a second set of simulations, we adopt a much higher value of the LW background, in particular we take J 21 = 10 5 , which is well above the critical flux required to suppress H 2 formation and cooling. We evolve these simulations until their virial temperatures are above 10 4 K, which allows us to (artificially) run the halo until it has a virial temperature sufficiently high for the Inayoshi & Omukai (2012) mechanism to operate. We then turn the LW background down to J 21 = 10 to see if and how the purely collisional H 2 suppres- sion acts. These runs are intended as a academic exercise to see if gas that is in or near the 'zone of no return' will indeed stay there.", "pages": [2, 3]}, {"title": "3 RESULTS", "content": "In the following sections, we present the results of our numerical simulations, first focusing on runs with low background flux, and then follow with high-UV background simulations.", "pages": [3]}, {"title": "3.1 J 21 = 10 Runs", "content": "The addition of radiative cooling causes the halos to cool and collapse before the redshifts at which they were identified in the low-resolution run. This occurs because the halos build up in a hierarchical fashion and so have progenitor halos with virial temperatures that allow them to cool via H 2 cooling. We stop the runs when they reach the maximum refinement level allowed in the simulation, as at this point they will start to form stars. The collapse redshifts, virial masses and virial temperatures of the halos at the point at which they collapse are shown in Table 2. Only the third halo reaches a virial temperature close to 10 4 K. In this context, we define collapse as the point when we would need additional refinement in order to continue evolving the halo, which corresponds to a density of approximately 10 8 cm -3 . To investigate if the density-dependent H 2 suppression mechanism is operating, we analyze the simulation results at the point at which they collapse. In Figure 1 we plot the temperature and density distribution of all cells within a radius of a few times the virial radius for each simulation run. Additionally, we overlay the 'zone of no return' in this figure, using the approximate curves for the boundary of the zone from Inayoshi & Omukai (2012). From the plots we see directly that no cells have been shocked-heated into this 'no return zone'. This indicates that the H 2 suppression mechanism is not operating, at least for these halos at this point. Inayoshi & Omukai (2012) argued that cold, filamentary flows would penetrate to the center of these halos, shock heating the gas at high densities. To better understand why this is not happening, in Figure 2 we plot slices through the center of halo C (with T vir = 8 . 1 \u00d7 10 3 K), showing the den- ity, temperature, entropy, negative of the divergence of the velocity field, and H 2 fraction. From the density and temperature slices, we clearly see the cold filamentary structure feeding the halo. However, it is evident that no cold flows (T \u223c 10 2 -10 3 K) penetrate unperturbed into the center. Instead, the gas quickly heats to the average temperature around the virial radius. This is consistent with the phase plots which show continual heating of the cells at number densities approach \u223c 1 cm -3 . Most of the cells at this number density have roughly T \u223c 10 4 K. At higher densities glyph[greaterorsimilar] 1 cm -3 , H 2 cooling becomes dominant, leaving the gas roughly at \u223c 700 K. One way to identify shocks is to look for sources of entropy production (although shocks are not the only source of entropy). Figure 2 shows that the largest entropy production happens around the virial radius. This is largely due to the spherical accretion of the surrounding cold gas. This is more apparent in the Mach number slice, where the flow outside the virial radius is supersonic and the flow interior to the virial radius is mostly transonic. The transition from supersonic to subsonic flow happens abruptly. We also see the cold inflows are not immune to this shock, and gas in the filaments begins to exhibit higher entropy at the virial radius (although the entropy is considerably lower than most of the gas at the virial temperature). The entropy generation is at the expense of its kinetic energy as evident in the Mach number slice where the inflow also becomes transonic as it passes the virial radius. The negative divergence of the velocity shown in Figure 2 reinforces our previous statements. Shocks produce a large value of -\u2207\u00b7 glyph[vector] V . It is evident there is a strong shock at the virial radius and the filamentary structure is disrupted as it passes through this radius. This shows clearly that the cold, filamentary flows in these halos shock at or around the virial radius, where their densities are low - too low for the H 2 suppression mechanism suggested in Inayoshi & Omukai (2012) to operate. Instead, the shocked filaments can efficiently form H 2 and cool. We argue that this conclusion is largely consistent with (comparable) previous work. For example, Wise & Abel (2007) carried out similar simulations, finding that although filamentary gas can, in some cases, penetrate through the virial radius without shocking, it does not get past one-third of the virial radius (as stated in the abstract of that paper). Moreover, this occurs only in the absence of H 2 cooling (i.e. in their H+He only runs A6 and B6), which is consistent with our own findings. In any case, this penetration is not nearly far enough - for example, Fig. 2 of that paper demonstrates that the densities do not get to 10 4 cm -3 until about 0.005 r vir . The SPH simulations of Greif et al. (2008) also examine the inflow of filaments in halos of similar size. They appear to find that cold filaments may not shock-heat as the flow in; however two points are relevant here. The first is that later work (Nelson et al. 2013) has shown that SPH simulations incorrectly predict substantial amount of coldaccretion compared to Eulerian or moving-mesh codes. The second point is that even in the simulations of Greif et al. (2008), gas does not get close to the H 2 zone of no return (e.g. their Fig. 7). Therefore, we conclude that our key results are in agreement with previous work. We have seen that - for the three halos simulated the halos collapse and fragment without any gas entering the 'zone of no return'. They do this in part because the 10 4 K halos do not collapse monolithically - instead, the hierarchical formation naturally involves lower temperature halos which can cool efficiently via H 2 . One possible objection is that we have simply not simulated enough halos, and that a small fraction of halos might collapse quickly enough for the suppression mechanism to be efficient. We examine this possibility in more detail in Section 4; however, first (in the next section), we look at what would happen if we could somehow suppress cooling in the lower mass halos until highT vir halos build up. Before examining that issue, we make a brief remark about resolution - a concern with any result from a numerical simulation. Although we have not explicitly carried out a resolution study as part of this work, this point has been examined regularly in the past. For example, Machacek, Bryan & Abel (2001) explicitly ran multiple simulations (also with the Enzo code) to examine how mass resolution impacted /circledot the cooling of primordial halos like the ones simulated here. They used dark matter particles masses of 306 and 38 M glyph[circledot] , finding only minor changes in the statistics of cooling halos, implying that, with a dark matter particle mass of 85 M glyph[circledot] , our results are robust to mass resolution. A related but slightly different concern is the resolution of the Jeans length - recent work has suggested that the Jeans length may, in some circumstances, need to be resolved by more than the 16 cells adopted here. For example, Turk et al. (2012) find that the magnetic properties require ratios greater than 32, with possible changes to the temperature profile as the resolution is changed. However, the temperature changes are not found in Greif, Springel & Bromm (2013), and moreover, the impact is only significant at higher densities than explored here, n glyph[greaterorsimilar] 10 12 cm -3 . At lower densities, the profiles are well converged. Therefore, we conclude that this issue is unlikely to impact the results found here. We can also compare our profiles to previous work. For example, O'Shea & Norman (2008) carried out simulations of similar halos with similar LW backgrounds - the temperature profiles they find (e.g. their Fig. 9) are very similar to ours. As that paper demonstrates, the relatively high temperatures ( \u223c 700 K) we find in the H 2 cooling region are an indirect consequence of the relatively high LW background used here.", "pages": [3, 4, 5]}, {"title": "3.2 J 21 = 10 5 Run", "content": "To investigate if the collisional H 2 suppression mechanism can keep a halo from fragmenting if we could place its central region in or near the 'zone of no return', we re-simulate halo A using a much higher value of the LW-background to artificially suppress H 2 cooling. We adopt J 21 = 10 5 - with such a large (and unrealistic) background, most of the H 2 will be dissociated, as shown by Shang, Bryan & Haiman (2010) and Wolcott-Green, Haiman & Bryan (2011), where the critical flux was determined to be J 21 = 10 3 . We picked the halo with the smallest virial temperature at collapse in the previous runs ( T vir = 2 . 86 \u00d7 10 3 K) and reran it with the large J 21 . The halo collapses at a later redshift, z = 16 . 74, with a larger virial temperature T vir = 0 . 97 \u00d7 10 4 K, mass 2 . 48 \u00d7 10 7 M glyph[circledot] , and radius 534 pc. In Figure 3, we show the number density and temperature phase plot for this run. We see the cells for n glyph[greaterorsimilar] 1 cm -3 have now increased in temperature. The large J 21 has dissociated most of the H 2 , leaving atomic cooling as the dominant mechanism and keeping the gas at T \u223c 8000 K. For n glyph[lessorsimilar] 1 cm -3 the same structure is apparent as the pre- ous simulations, where cells are heated to T \u223c 10 4 K at the virial radius. Figure 4 shows slices with the same quantities as in the previous section, but now for the large J 21 run. Although the general structure is quite similar, there are now some signs of cold flows penetrating past the virial radius. This is more apparent in the Mach number slice where we see high velocity flows passing the virial radius where the majority of the fluid is transonic. It can also be seen in the temperature distribution, where low-temperature gas flows into the halo before heating. However, we note that the 'zone of no return' occurs at densities above \u223c 10 4 cm -3 , which corresponds to the very central region (colored red) of the density slice, and so apparently, the cold flows are not penetrating this zone. /circledot", "pages": [5, 6, 7]}, {"title": "3.2.1 Stability of the Zone of no Return", "content": "Next, we investigate if gas in the 'zone of no return' will indeed stay there without any external flux. We do this by turning down the high-UV background flux in the previous simulation. We run two identical simulations as in the last section, but now reduce J 21 at different evolutionary points. The two points are distinguished by whether or not gas cells have reached the critical H 2 number density for local thermodynamic equilibrium, n cr \u223c 10 4 cm -3 . Thus, in our first simulation, we allow the gas to evolve with a large J 21 until the maximum density just reaches n cr , and at this point the UV background is reduced to J 21 = 10 and the gas is allowed to evolve. Similarly, our second simulation is iden- cal with the exception that the reduction is applied when a small fraction of cells have entered into the 'zone of no return'. Figure 5 shows the phase plot evolution for the first simulation (the instant of flux reduction corresponds to the upper-left panel, at z = 16 . 831). As the simulation evolves with the reduced J 21 , H 2 begins to form and cool efficiently. The cooling occurs first at densities n \u223c 10 3 cm -3 , leaving the higher density regions still hot ( T \u223c 10 4 K). As the evolution progresses, this gas eventually cools and by the final time shown, when the halo collapses to the highest refinement level we allow, the phase distribution looks very much like the low-J runs. It is interesting to analyze this evolution in more detail. z In Figure 6, we show temperature (left) and density (right) slices that go through the center of this halo for the same times presented in Figure 5. We see that the cooling process creates a cold envelope surrounding the hot core. This core is over-pressured, which generates an outflow, driving clumps of dense material out of the central region, as evident in the density slices of Figure 6. The outflow reduces the density (and temperature) of the gas in the core, which is then effectively evaporated. The final remnant is a dense shell surrounding a core that has cooled on average to T \u223c 700 K. In the second simulation of this section, we allow the run with J 21 = 10 5 to continue a bit further, to z = 16 . 754, when a significant amount of gas has entered the 'zone of no return' (see the top left panel of Figure 7) before turning the UV background down to J 21 = 10. The resulting evolution of the phase diagram is shown in Figure 7. In this case the evolution is quite different: although some gas does cool, the dense gas evolves more quickly, and by the time we stop the simulation (when it reaches our highest refinement level at z = 16 . 749), the amount of mass inside the 'zone of no return' has not changed by more than a few percent. There is some cooling - in particular at n \u223c 10 3 cm -3 and n \u223c 10 6 cm -3 ; however, this does not significantly affect the evolution of the clump. The resulting baryonic mass in the 'zone of no return' is \u223c 10 5 M glyph[circledot] , roughly 10% of the halo mass within the virial radius.", "pages": [7, 8]}, {"title": "4 DISCUSSION", "content": "Using a set of numerical simulations, we have investigated the viability of the mechanism for collisional suppression of H 2 suggested by Inayoshi & Omukai (2012). We found that, for the three halos examined, H 2 -mediated cooling set in before the conditions required for suppression could be established - in particular, before a T vir glyph[greaterorsimilar] 10 4 K halo forms, which would be a pre-requisite for the generation of shocks strong enough to put gas into the 'zone of no return'. However, we did confirm (using a set of simulations that employed a high UV background to temporary suppress cooling) some aspects of the mechanism: if the gas in the core of an atomic cooling halo can avoid H 2 cooling until it enters the 'zone of no return', then subsequent H 2 cooling can be naturally averted. The gas would then remain at \u223c 10 4 K, possibly leading to conditions suitable for SMBH formation, as envisioned in the many previous works mentioned in the Introduction (see, e.g. Haiman 2013 for a recent review). Unfortunately, H2 cooling at any time in the past history of the same gas would likely invalidate this possibility, if this H2 cooling allowed the gas to reach temperatures of a few 100 K and to evolve to high density at such low temperatures. This cold and dense gas, at the center of the progenitor minihalo, would form one, or possibly a few, PopIII stars. The subsequent evolution of this progenitor into an atomic cooling halo is uncertain, however, there are many obstacles to rapid SMBH formation. First, even a handful of SNe (or a single pair instability SNe) can enrich the entire atomic cooling halo to a metallicity of Z glyph[greaterorsimilar] 10 -3 Z glyph[circledot] (e.g., Bromm & Yoshida 2011, see page 396) the critical value above which direct SMBH formation is replaced by fragmentation (Inayoshi & Omukai 2012). It is possible that none of these Pop III stars produce any metals (such as non-rotating metal-free stars between 40-140 M glyph[circledot] ; Heger et al. 2003). Such massive Pop III stars would leave behind stellar-mass BHs, which, in principle, could grow rapidly into SMBHs, provided that they are surrounded by very dense gas, and accrete sufficiently rapidly to trap their own radiation (e.g., Volonteri & Rees 2005). However, the parent Pop III stars of these seed BHs will /circledot create a large ionized bubble and therefore the seed BHs will likely begin their life in a low-density medium. As they later begin to accrete, their radiation likely self-limits the time-averaged accretion rate to a fraction of the Eddington rate (Alvarez, Wise & Abel 2009; Park & Ricotti 2012; Milosavljevi'c, Couch & Bromm 2009). Although the above scenarios are worth investigating further in detail, we will hereafter assume that these obstacles prevent rapid SMBH formation in any halo that experienced H 2 cooling at any time in its history.", "pages": [8, 9]}, {"title": "4.1 Avoiding H 2 Cooling by Rapid Halo Assembly", "content": "However, since we only simulated three halos a natural question is whether H 2 cooling may be avoided in a few, highly atypical, atomic cooling halos, even if the background flux J LW was much lower. One possibility is that - in rare cases the progenitor halos could experience unusually rapid mergers, continuously shock-heating the gas on a time-scale that always remains shorter than the H 2 -cooling time. Here we evaluate the likelihood of this scenario, using Monte Carlo realizations of the merger histories of 10 6 atomic cooling halos. In particular, we start by creating dark matter halo merger trees, using the Monte Carlo algorithm in Zhang, Fakhouri & Ma (2008). This paper presents three different numerical algorithms, which are based on the conditional halo mass functions in the ellipsoidal collapse model (Sheth & Tormen 2002; Zhang, Ma & Fakhouri 2008). We adopted their 'method B', which was found in subsequent work to provide the best match to the statistics of merger trees in N-body simulations (Zhang 2012, private communication). This method represents an improvement over previous merger-tree methods based on spherical collapse - in particular, the ellipsoidal collapse models tend to predict a larger number of more massive progenitors, or a 'flatter' assembly history (Tanaka, Li & Haiman 2013). We have created 10 6 merger trees of a DM halo with T vir , 0 = 10 4 K ( M 0 = 6 . 3 \u00d7 10 7 M glyph[circledot] ) at redshift z 0 = 10, extending back to redshift z = 20. We then follow the mass of the most massive progenitor back in time, and at each redshift z , we compute the H 2 -cooling time in this progenitor, using the procedures and rates in Machacek, Bryan & Abel (2001). Specifically, we compute the maximum density that could be reached by the gas in the nucleus of this progenitor, in the absence of H 2 -cooling (i.e. by adiabatic compression). We then specify a constant background flux J 21 , and compute the equilibrium H 2 fraction and the corresponding t H2 at this density, given J 21 and \u03c1 max . Finally, we require that the longer of the dynamical time t dyn = \u221a 3 \u03c0/ 16 G\u03c1 max and t H2 remains longer than the cosmic time \u2206 t ( z ) = t ( z 0 ) -t ( z ) elapsed between the z and z 0 . This last requirement ensures that the most massive progenitor of our atomic cooling halo at redshift z > z 0 cannot cool via H 2 and evolve dynamically, before it is incorporated by mergers into the M 0 = 6 . 3 \u00d7 10 7 M glyph[circledot] atomic cooling halo at redshift z 0 = 10. Note that this requirement must be satisfied at all redshifts z > z 0 - in other words, if our atomic cooling halo had a progenitor, at any redshift z , that was massive enough to cool and collapse, the 'SMBH formation by direct collapse' scenario is no longer feasible. In practice, as we run our merger trees backward, we simply look, at each redshift step, whether the most massive progenitor violates the criterion \u2206 t ( z ) < max[ t H2 , t dyn ]. If it does, we record the redshift z max where this happened, and we discard the rest of this merger tree. In Figure 8, we show the probability distribution of the 'terminal lookback redshift' z max we have found among the 10 6 merger trees, for four different values of a constant J LW = 1 , 3 , 10, and 30. For a LW background J LW = 1 (shown in the figure as the solid black histogram), we found that all of the 10 6 merger trees contain an H 2 -cooling progenitor by z < 20. However, for the higher background fluxes of J LW = 3 (dot-dashed green), J LW = 10 (dotted red), or J LW = 30 (dashed blue), we have found approximately 2, 200, and 4000 cases, respectively, where none of the progenitors, out to z = 20, could cool via H 2 - this is shown by the pile-up of the probability distribution in the last bin at \u2206 z max = 10. For the atomic cooling halos with these particular merging histories, we expect that H 2 cool- g is avoided entirely until the atomic cooling halo forms at z 0 = 10. These rare halos are good candidates where the subsequent ionizing shocks can prevent the gas temperature from falling below \u2248 10 4 K, allowing the gas to collapse directly to a SMBH. This scenario reduces the required LW flux by a factor of \u223c 300, from J 21 \u2248 10 3 (Shang, Bryan & Haiman 2010; Wolcott-Green, Haiman & Bryan 2011) to J 21 \u2248 3, but it does require that the latter LW background is in place at a redshift as high as z \u223c 20 (in order for a fraction 2 \u00d7 10 -6 of the atomic cooling halos at z 0 = 10 to be such candidates).", "pages": [9, 10]}, {"title": "5 SUMMARY", "content": "One way to explain the presence of SMBHs at z > 6 with masses greater than 10 9 M glyph[circledot] is through the formation of a massive ( \u223c 10 5 M glyph[circledot] ) BH seed from gas in T vir glyph[greaterorsimilar] 10 4 Khalos, either via direct collapse to a BH or through the formation of a supermassive star or a quasi-star (Begelman, Rossi & Armitage 2008; Hosokawa et al. 2013). As noted in the Introduction, one difficulty with these modes is the need to prevent fragmentation of the cooling halo into lower mass stars, and, in particular, to prevent cooling due to H 2 . Inayoshi & Omukai (2012) have suggested that this can be accomplished through the action of cold flows, which result in gas shocking to high densities ( n > 10 4 cm -3 ) and temperatures ( T > 10 4 K). This shocked gas cools by Ly \u03b1 emission to about 8000 K; however, if the density is high enough, enhanced H 2 collisional dissociation suppresses the gas from cooling further. This was demonstrated using onezone models calculated by Inayoshi & Omukai (2012), who identified a 'zone of no return'. This mechanism is appealing as it does not require a high LW background to destroy the H 2 ; however, it is unclear if this mechanism operates in nature. To test this idea, we carried out cosmological hydrodynamic simulations with the adaptive mesh refinement code Enzo. We first identified three halos from a low resolution simulation which all had T vir glyph[greaterorsimilar] 10 4 K at redshifts ranging from 12 to 17. We then re-simulated these halos at highresolution with a relatively modest UV flux in the LymanWerner band, J 21 = 10. We found that in all three cases, cooling from H 2 was able to efficiently lower the gas temperature below that of the 'zone of no return', indicating that, at least for these three halos, the mechanism was not operating. To determine why this occurred, we examined the structure of the simulated halos in detail, and found that, while cold flows do occur, they generally shock at or near the virial radius and do not penetrate into the halo center where the densities are high. We note that although these small halos have low virial temperatures and so might be naively classified as 'cold-mode' halos (Birnboim & Dekel 2003; Kere\u02c7s et al. 2005; Dekel & Birnboim 2006), they actually have low cooling rates because of the inefficiency of H 2 cooling. Therefore, the characteristic cooling times of these halos is longer (or comparable) to their dynamical times, making them more akin to hot-mode halos. This is consistent with the clear virial shocks that we see in these halos (e.g., Figure 2). To determine if the suppression mechanism could function if we artificially suppressed H 2 cooling, we reran one of the simulations with a high LW background ( J 21 = 10 5 ), well above the critical flux required to suppress H 2 cooling radiatively (Shang, Bryan & Haiman 2010; Wolcott-Green, Haiman & Bryan 2011). We showed that, in this case, the gas did indeed enter the 'zone of no return'. To see if this situation was stable, we ran two variations of this simulation, in which we turned the flux off either just before, or just after the gas in the halo center entered the 'zone of no return'. In the first case, cooling eventually won out, with the region just outside the core cooling first and driving an evaporative wind which led to cooling by H 2 throughout the halo. However, in the second case, the gas inside the 'zone of no return' stayed there, and eventually collapsed to high densities while remaining at T \u223c 8000 K, despite the lack of any LW flux. This demonstrates that the mechanism would work if a halo could collapse quickly enough, but that typical T vir glyph[greaterorsimilar] 10 4 K halos have progenitors which can cool efficiently. To investigate if any halos can collapse quickly enough to escape this fate, we ran Monte Carlo merger tree calculations of 10 6 halos with an ellipsoidal collapse model. We used a simple analytic prescription to determine if any halos could assemble quickly enough to prevent cooling and collapse prior to building up to a T vir = 10 4 K halo. We found that for a Lyman-Werner background of J 21 = 1, none of the 10 6 merger tree histories collapsed quickly enough. For larger LW backgrounds, a small fraction (up to \u223c 10 -3 for J 21 as high as 30) of the halos did form quickly enough to evade cooling. We note that at higher redshift, J 21 would certainly be expected to be lower. In summary, we confirm the essential physics proposed by Inayoshi & Omukai (2012), but we find that the supersonic filamentary flows are unlikely to shock gas into the 'zone of no return', as they had originally envisioned. Rather, we propose here a modification of their scenario. In the core of a few rare halos, which assembled unusually rapidly, the gas may have been kept continuously shockheated throughout their history, and eventually shocked into the 'zone of no return'. This scenario reduces the required LW flux by a factor of \u223c 300, from J 21 \u2248 10 3 (Shang, Bryan & Haiman 2010; Wolcott-Green, Haiman & Bryan 2011) to J 21 glyph[greaterorsimilar] 1, but it does require that the latter LW background is in place at a redshift as high as z \u223c 20. Future, selfconsistent global models of the build-up of the LW background, together with the assembly of a large number of atomic-cooling halos, is required to assess the viability of this scenario.", "pages": [10, 11]}, {"title": "ACKNOWLEDGMENTS", "content": "We thank Jun Zhang for useful discussions about ellipsoidal collapse-based merger trees, and for sharing unpublished results on comparisons between Monte Carlo methods and Nbody simulations. ZH acknowledges financial support from NASA through grant NNX11AE05G. GB acknowledge financial support from NSF grants AST-0908390 and AST1008134, and NASA grant NNX12AH41G, as well as computational resources from NSF XSEDE, and Columbia University's Hotfoot cluster.", "pages": [11]}, {"title": "REFERENCES", "content": "Abel T., Anninos P., Zhang Y., Norman M. L., 1997, New Astronomy, 2, 181 Abel T., Bryan G. L., Norman M. L., 2000, ApJ, 540, 39 Alvarez M. A., Wise J. H., Abel T., 2009, ApJL, 701, L133 Begelman M. C., 2010, MNRAS, 402, 673 Begelman M. C., Rossi E. M., Armitage P. J., 2008, MN- RAS, 387, 1649 Berger M. J., Colella P., 1989, Journal of Computational Physics, 82, 64 Birnboim Y., Dekel A., 2003, MNRAS, 345, 349 Bromm V., Loeb A., 2003, ApJ, 596, 34 Bromm V., Yoshida N., 2011, ARA&A, 49, 373 Bryan G., 1999, Comp. Sci. Eng., 1, 46 Bryan G. L., Norman M. L., 1998, ApJ, 495, 80 Bryan G. L. et al., 2013, ArXiv e-prints Dekel A., Birnboim Y., 2006, MNRAS, 368, 2 Dijkstra M., Haiman Z., Mesinger A., Wyithe J. S. B., 2008, MNRAS, 391, 1961 Efstathiou G., Davis M., White S. D. M., Frenk C. S., 1985, ApJS, 57, 241 Wise J. H., Abel T., 2007, ApJ, 665, 899 Wolcott-Green J., Haiman Z., Bryan G. L., 2011, MNRAS, 418, 838 Zhang J., Fakhouri O., Ma C.-P., 2008, MNRAS, 389, 1521 Zhang J., Ma C.-P., Fakhouri O., 2008, MNRAS, 387, L13", "pages": [11, 12]}] |
2022FrASS...999319A | https://arxiv.org/pdf/2207.12894.pdf | <document>
<section_header_level_1><location><page_1><loc_14><loc_83><loc_86><loc_86></location>Interface Region Imaging Spectrograph (IRIS) Observations of the Fractal Dimension in the Solar Atmosphere</section_header_level_1>
<text><location><page_1><loc_42><loc_80><loc_58><loc_81></location>Markus J. Aschwanden</text>
<text><location><page_1><loc_14><loc_76><loc_86><loc_79></location>Lockheed Martin, Solar and Astrophysics Laboratory (LMSAL), Advanced Technology Center (ATC), A021S, Bldg.252, 3251 Hanover St., Palo Alto, CA 94304, USA; e-mail: [email protected]</text>
<text><location><page_1><loc_49><loc_73><loc_51><loc_75></location>and</text>
<text><location><page_1><loc_42><loc_71><loc_58><loc_72></location>Nived Vilangot Nhalil</text>
<text><location><page_1><loc_22><loc_68><loc_78><loc_69></location>Armagh Observatory and Planetarium, College Hill, Armagh BT61 9DG, UK</text>
<section_header_level_1><location><page_1><loc_45><loc_65><loc_55><loc_66></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_16><loc_38><loc_84><loc_62></location>While previous work explored the fractality and self-organized criticality (SOC) of flares and nanoflares in wavelengths emitted in the solar corona (such as in hard X-rays, soft Xrays, and EUV wavelenghts), we focus here on impulsive phenomena in the photosphere and transition region, as observed with the Interface Region Imaging Spectrograph (IRIS) in the temperature range of T e ≈ 10 4 -10 6 K. We find the following fractal dimensions (in increasing order): D A = 1 . 21 ± 0 . 07 for photospheric granulation, D A = 1 . 29 ± 0 . 15 for plages in the transition region, D A = 1 . 54 ± 0 . 16 for sunspots in the transition region, D A = 1 . 59 ± 0 . 08 for magnetograms in active regions, D A = 1 . 56 ± 0 . 08 for EUV nanoflares, D A = 1 . 76 ± 0 . 14 for large solar flares, and up to D A = 1 . 89 ± 0 . 05 for the largest X-class flares. We interpret low values of the fractal dimension (1 . 0 < ∼ D A < ∼ 1 . 5) in terms of sparse curvi-linear flow patterns, while high values of the fractal dimension (1 . 5 < ∼ D A < ∼ 2 . 0) indicate near space-filling transport processes, such as chromospheric evaporation. Phenomena in the solar transition region appear to be consistent with SOC models, based on their size distributions of fractal areas A and (radiative) energies E , which show power law slopes of α obs A = 2 . 51 ± 0 . 21 (with α theo A = 2 . 33 predicted), and α obs E = 2 . 03 ± 0 . 18 (with α theo E = 1 . 80 predicted).</text>
<text><location><page_1><loc_16><loc_33><loc_84><loc_36></location>Subject headings: methods: statistical - fractal dimension - Sun: transition region - solar granulation - solar photosphere -</text>
<section_header_level_1><location><page_1><loc_40><loc_28><loc_60><loc_29></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_12><loc_10><loc_88><loc_26></location>There are at least three different approaches that quantify the statistics of nonlinear processes with the concept of self-organized criticality (SOC) and fractality: (i) microscopic models, (ii) macroscopic models, and (iii) observations of power laws and scaling laws. The microscopic SOC models consist of numerically simulated avalanches that evolve via next-neighbor interactions in a lattice grid (Bak et al. 1987, 1988), also called cellular automatons , which have been quantized up to numerical limits of ≈ 10 6 -10 9 cells per avalanche process. The macroscopic models describe the nonlinear evolution of (avalanching) instabilities with analytical (geometric and energetic) quantities, which predict physical scaling laws and power law-like occurrence frequency size distributions. The third category of SOC approaches includes observations with fitting of power law-like distribution functions and waiting time distributions, which provide powerful tests of theoretical SOC models. A total of over 1500 SOC-related publications appeared at the time of writing.</text>
<text><location><page_2><loc_12><loc_82><loc_88><loc_86></location>For brevity, we mention a few textbooks only (Bak 1996; Aschwanden 2011; Pruessner 2012), and a recent collection of astrophysical SOC reviews, presented in the special volume Space Science Reviews Vol. 198 (Watkins et al. 2016; Aschwanden et al. 2016; Sharma et al. 2016; McAteer et al. 2016).</text>
<text><location><page_2><loc_12><loc_59><loc_88><loc_80></location>In this paper we focus on SOC modeling of impulsive events detected in the solar atmosphere, as observed with the Interface Region Imaging Spectrograph (IRIS) (De Pontieu et al. 2014), while solar flare events observed in hard X-rays, soft X-rays, and Extreme-Ultraviolet (EUV) have been compared in recent studies (Aschwanden 2022a, 2022b). Large solar flares observed in hard and soft X-rays show typically electron temperatures of T e ≈ 5 -35 K, while coronal nanoflares observed in EUV have moderate temperatures of T e ≈ 1 -2 MK. Hence it is interesting to investigate transition region events, which are observed in a different temperature regime ( T e ≈ 10 4 -10 6 K) than coronal phenomena. In a previous study with the same IRIS data, it was found that plages and sunspots have different power law indices for the areas of events, being smaller ( α A < ∼ 2) for sunspot-dominated active regions, and larger for plage regions, α > ∼ 2, (Nhalil et al. 2020). If both coronal and transition region brightenings exhibit the same SOC behavior and are produced by the same physical mechanism, one would expect the same fractal dimension and power law slope of the occurrence frequency size distribution, which is an important test of the coronal heating problem.</text>
<text><location><page_2><loc_12><loc_55><loc_88><loc_58></location>The structure of this paper consists of an observational Section 2, a theoretical modeling Section 3, a discussion Section 4, and a conclusion Section 5.</text>
<section_header_level_1><location><page_2><loc_41><loc_50><loc_59><loc_51></location>2. OBSERVATIONS</section_header_level_1>
<text><location><page_2><loc_12><loc_30><loc_88><loc_48></location>This is a follow-on study of previous work, 'The power-law energy distributions of small-scale impulsive events on the active Sun: Results from IRIS' (Nhalil et al. 2020). We call these small-scale impulsive events simply 'events', which possibly could be related to 'nanoflares' or 'brightenings'. In the previous study, 12 IRIS datasets were investigated with an automated pattern recognition algorithm, yielding statistics of three parameters, namely the event area A (in units of pixels), the event (radiative) energy E (in units of erg), and event durations or lifetimes T (in units in seconds). IRIS has pixels with a size of 0 . 17 '' ≈ 0 . 123 Mm, which have been rebinned to L pixel = 0 . 33 '' ≈ 0 . 247 Mm. The pixel size of areas thus corresponds to A pixel = L 2 pixel = 0 . 247 2 Mm 2 = 0.06076 Mm 2 . The range of event areas covers A = 4 -677 pixels, which amounts to length scales of L = √ A = (2 -26) pixels, or L = (2 -26) ∗ 0 . 247 Mm ≈ (0.5-6.4) Mm = (500-6400) km. The date of observations, the field-of-view (FOV), the cadence, and the NOAA active region numbers are given in Table 1 of Nhalil et al. (2020), for each of the 12 IRIS datasets.</text>
<text><location><page_2><loc_12><loc_19><loc_88><loc_29></location>The automated pattern recognition code was run with different threshold levels of 3, 5, and 7 σ for the detection of events, from which we use the 3σ level here. We use Slitjaw images (SJI) of the 1400 ˚ A channel of IRIS, which are dominated by the Si IV 1394 ˚ A and 1403 ˚ A resonance lines, formed in the transition region. Nhalil et al. (2020) compared also images from the SJI 1330 ˚ A channel, which is dominated by the C II 1335 ˚ A and 1336 ˚ A lines, originating in the upper chromosphere and transition region at formation temperatures of T e ≈ 3 × 10 4 K and T e ≈ 8 × 10 4 K (Rathore and Carlsson 2015; Rathore et al. 2015).</text>
<section_header_level_1><location><page_3><loc_40><loc_85><loc_60><loc_86></location>2.1. Size Distributions</section_header_level_1>
<text><location><page_3><loc_12><loc_72><loc_88><loc_83></location>Our first measurement is the fitting of a power law distribution function N ( A ) ∝ A -α A of the event (or nanoflare) areas A , separately for each of the 12 IRIS datasets, as shown in Fig. 1. The area of the event is a combination of all the spatially connected 3σ pixels throughout its lifetime. The lowest bin was discarded when a visible deviation from a power law was apparent in the histogram. The number of events amounts to 23,633 for all 12 datasets together, varying from 65 to 4725 events per IRIS dataset (Table 1). The power law slope fits vary from the lowest value α A = 2 . 14 (dataset 1) to the highest value α A = 2 . 83 (dataset 4), having a mean and standard deviation of (Fig. 2, top panel).</text>
<formula><location><page_3><loc_43><loc_69><loc_88><loc_70></location>a obs A = 2 . 51 ± 0 . 21 . (1)</formula>
<text><location><page_3><loc_12><loc_64><loc_88><loc_67></location>The area size distributions are shown superimposed for the 12 IRIS datasets (Fig. 2, top panel), which illustrates almost identical power law slopes in different IRIS datasets.</text>
<text><location><page_3><loc_12><loc_60><loc_88><loc_63></location>Fitting the energy size distributions, N ( E ) ∝ E -α E , yields the following mean for all 12 IRIS datasets (Fig. 2, middle panel),</text>
<formula><location><page_3><loc_43><loc_58><loc_88><loc_60></location>a obs E = 2 . 03 ± 0 . 18 . (2)</formula>
<text><location><page_3><loc_12><loc_54><loc_88><loc_57></location>Fitting the duration size distributions, N ( T ) ∝ T -α T , yields the following mean for all 12 IRIS datasets (Fig. 2, bottom panel),</text>
<formula><location><page_3><loc_43><loc_52><loc_88><loc_53></location>a obs T = 2 . 65 ± 0 . 39 . (3)</formula>
<text><location><page_3><loc_12><loc_50><loc_69><loc_51></location>We will interpret these power law slopes in terms of SOC models in Section 4.6.</text>
<section_header_level_1><location><page_3><loc_28><loc_44><loc_72><loc_46></location>2.2. Fractal Dimension with Box-Counting Method</section_header_level_1>
<text><location><page_3><loc_12><loc_33><loc_88><loc_43></location>The next parameter that we are interested in is the fractal dimension. A standard method to determine the fractal dimension D A of an image is the box-counting method, which is defined by the asymptotic ( L ↦→ 0) ratio of the fractal area A to the the length scale L , i.e., D A = log( A ) / log( L ), also called Hausdorff (fractal) dimension. We calculate the so-defined Hausdorff dimension D A for each of the 12 IRIS datasets (column D A 2 in Table 1), which reveals a very narrow spread of values for the fractal dimension, with a mean and stadard deviation of < ∼ 2%,</text>
<formula><location><page_3><loc_43><loc_31><loc_88><loc_33></location>D obs A 2 = 1 . 58 ± 0 . 03 . (4)</formula>
<text><location><page_3><loc_12><loc_20><loc_88><loc_29></location>The fractal nature of the 12 IRIS datasets is rendered in Figs. (3) and (4), where the white areas correspond to zones with enhanced emission, and black areas correspond to the background with weak emission. The successive reduction of spatial resolution is shown in Fig. 4 for N bin = 128 , 64 , 32 , 16 pixels, which all converge to the same fractal dimension of D A = 1 . 33. An example of a theoretical fractal pattern with a close ressemblance to the observed transition region patterns of dataset 8 is shown in Fig. 5, which is called the 'golden dragon fractal' and has a Hausdorff dimension of D A = 1 . 61803.</text>
<section_header_level_1><location><page_3><loc_14><loc_15><loc_86><loc_16></location>2.3. Comparison of Photospheric, Transition Region, and Coronal Fractal Dimensions</section_header_level_1>
<text><location><page_3><loc_12><loc_10><loc_88><loc_13></location>In Table 3 we compile fractal dimensions obtained from photospheric and transition region fractal features, which may be different from coronal and flare-like size distributions. The fractal dimension has</text>
<text><location><page_4><loc_12><loc_82><loc_88><loc_86></location>been measured in white-light wavelengths with the perimeter-area method, containing dominantly granules and supergranulation features (Roudier and Muller 1986; Hirzberger et al. 1997; Bovelet and Wiehr 2001; Paniveni et al. 2010), which exhibit a mean value of (Table 3),</text>
<formula><location><page_4><loc_42><loc_79><loc_88><loc_81></location>D gran A = 1 . 21 ± 0 . 07 , (5)</formula>
<text><location><page_4><loc_12><loc_72><loc_88><loc_78></location>We have to be aware that white-light emission originates in the solar photosphere, which has a lower altitude than any transition region or coronal feature. The relatively low value obtained for granulation features thus indicates that the granulation features seen in optical wavelengths are almost curvi-linear (with little area-filling geometries), which is expected for sparse photospheric mass flows along curvi-linear flow lines.</text>
<text><location><page_4><loc_12><loc_66><loc_88><loc_70></location>A second feature we consider are plages, measured in magnetograms with the linear-area (LA) method (Balke et al. 1993), and in transition region IRIS 1400 ˚ A data (Nhalil et al. 2020), which have formation temperatures of ≈ 10 3 . 7 -10 5 . 2 K in the lower transition region, exhibiting a mean value of (Table 3),</text>
<formula><location><page_4><loc_42><loc_63><loc_88><loc_65></location>D plage A = 1 . 26 ± 0 . 16 . (6)</formula>
<text><location><page_4><loc_12><loc_58><loc_88><loc_62></location>This set of IRIS measurements exhibit a relatively low value for the fractal dimension, similar to the photospheric granulation features. Based on this low fractal dimension, photospheric flows appear to be organized along curvi-linear features, rather than solid-area geometries.</text>
<text><location><page_4><loc_12><loc_54><loc_88><loc_56></location>A third feature that we investigate are sunspots (Nhalil et al. 2020, and this work), which reveal higher values of fractal dimension, namely (Table 3),</text>
<formula><location><page_4><loc_41><loc_51><loc_88><loc_53></location>D sunspot A = 1 . 54 ± 0 . 16 . (7)</formula>
<text><location><page_4><loc_12><loc_45><loc_88><loc_50></location>Apparently, sunspots organize fractal features into space-filling geometries, where fragmentation into smaller and smaller fractal features is suppressed, because of the strong magnetic fields that control the penumbral flows of sunspots.</text>
<text><location><page_4><loc_12><loc_38><loc_88><loc_44></location>A fourth feature is an active region, observed in photospheric magnetograms and analyzed with the linear-area method (Lawrence 1991; Lawrence and Schrijver 1993; Meunier 1999, 2004; Janssen et al. 2003; Ioshpa et al. 2008), or with the box-counting method (McAteer et al. 2005). The mean value of fractal dimensions measured in active regions is found to be (Table 3),</text>
<formula><location><page_4><loc_43><loc_35><loc_88><loc_37></location>D AR A = 1 . 59 ± 0 . 20 . (8)</formula>
<text><location><page_4><loc_12><loc_31><loc_88><loc_34></location>Apparently, active regions organize magnetic features into space-filling, area-like geometries, similar to sunspot features.</text>
<text><location><page_4><loc_12><loc_24><loc_88><loc_30></location>A fifth phenomenon is a nanoflare event, which has been related to the SOC interpretation since Lu and Hamilton (1991). Nanoflares have been observed in EUV 171 ˚ A and 195 ˚ A with the TRACE instrument, as well as in soft X-rays using the Yohkoh/SXT (Solar X-Ray Telescope) (Aschwanden and Parnell 2002), which show a mean value of (see Table 3),</text>
<formula><location><page_4><loc_42><loc_21><loc_88><loc_23></location>D nano A = 1 . 56 ± 0 . 08 . (9)</formula>
<text><location><page_4><loc_12><loc_17><loc_88><loc_20></location>Nanoflares have been observed in the Quiet Sun and appear to have a similar fractal dimension as impulsive brightenings in active regions, as measured in magnetograms.</text>
<text><location><page_4><loc_12><loc_11><loc_88><loc_16></location>For completeness we list also the fractal dimension measured in large solar flares, for M-class flares, X-class flares, and the Bastille Day flare (Aschwanden and Aschwanden 2008a), which all together exhibit a mean value of (Table 3),</text>
<formula><location><page_4><loc_42><loc_9><loc_88><loc_11></location>D flare A = 1 . 76 ± 0 . 14 . (10)</formula>
<text><location><page_5><loc_12><loc_82><loc_88><loc_86></location>This is the largest mean value of any measured fractal dimension, which indicates that the flare process fills the flare area almost completely, due to the superposition of many coronal postflare loops that become filled as a consequence of the chromospheric evaporation process.</text>
<text><location><page_5><loc_12><loc_71><loc_88><loc_80></location>Thus, we can distinguish at least three groups with significantly different fractal properties in photospheric, transition region, and coronal data. A first group has curvi-linear features in the granulation and in plage features, which have a relatively low fractal dimension D gran A ≈ D plage A ≈ 1 . 2 -1 . 3. There is a second group of sunspot, active region, and nanoflare phenomena, wich exhibit an intermediate range of fractal dimensions of D sunspot A ≈ D AR A ≈ D nano A ≈ 1 . 5 -1 . 6. And there is a third group of large flares (M- and X-class), which have a fractal dimensions of D flare A ≈ 1 . 6 -1 . 9.</text>
<section_header_level_1><location><page_5><loc_35><loc_66><loc_65><loc_67></location>3. THEORETICAL MODELING</section_header_level_1>
<section_header_level_1><location><page_5><loc_34><loc_63><loc_66><loc_64></location>3.1. The Hausdorff Fractal Dimension</section_header_level_1>
<text><location><page_5><loc_12><loc_58><loc_88><loc_61></location>The definition of the fractal dimension D A for 2-D areas A is also called the Hausdorff dimension D A (Mandelbrot 1977),</text>
<formula><location><page_5><loc_46><loc_56><loc_88><loc_58></location>A = L D A , (11)</formula>
<text><location><page_5><loc_12><loc_54><loc_21><loc_55></location>or explicitly,</text>
<formula><location><page_5><loc_45><loc_51><loc_88><loc_54></location>D A = log( A ) log( L ) , (12)</formula>
<text><location><page_5><loc_12><loc_42><loc_88><loc_50></location>where the area A is the sum of all image pixels I ( x, y ) ≥ I 0 above a background threshold I 0 , and L is the length scale of a fractal area. A structure is fractal, when the ratio D A is approximately constant versus different length scales and converges to a constant for the smallest length scales L ↦→ 0. The method described here is also called the box-counting method, because the pixels are counted for the area A and the length scale L .</text>
<text><location><page_5><loc_15><loc_40><loc_67><loc_41></location>In analogy, a fractal dimension can also be defined for the 3-D volume V ,</text>
<formula><location><page_5><loc_46><loc_37><loc_88><loc_38></location>V = L D V , (13)</formula>
<text><location><page_5><loc_12><loc_34><loc_20><loc_35></location>or explicitly</text>
<formula><location><page_5><loc_45><loc_31><loc_88><loc_34></location>D V = log( V ) log( L ) , (14)</formula>
<text><location><page_5><loc_12><loc_27><loc_88><loc_30></location>The valid range for area fractal dimensions is 1 ≤ D A ≤ 2 and 2 ≤ D V ≤ 3, where D = 1 , 2 , 3 are the Euclidean dimensions.</text>
<text><location><page_5><loc_12><loc_23><loc_88><loc_26></location>We can estimate the numerical values of the fractal dimensions D A and D V from the means of the minimum and maximum values in each Euclidean domain,</text>
<formula><location><page_5><loc_32><loc_19><loc_88><loc_21></location>D A = ( D A,min + D A,max ) 2 = (1 + 2) 2 = 3 2 = 1 . 50 , (15)</formula>
<text><location><page_5><loc_12><loc_16><loc_26><loc_17></location>and correspondingly,</text>
<formula><location><page_5><loc_32><loc_13><loc_88><loc_16></location>D V = ( D V,min + D V,max ) 2 = (2 + 3) 2 = 5 2 = 2 . 50 . (16)</formula>
<text><location><page_5><loc_12><loc_10><loc_88><loc_13></location>The 2-D fractal dimension D A is the easiest accessible SOC parameter, while the 3-D fractal dimension D V requires information of fractal structures along the line-of-sight, either using a geometric or tomographic</text>
<text><location><page_6><loc_12><loc_83><loc_88><loc_86></location>model, or modeling of optically-thin plasma (in the case of an astrophysical object observed in soft X-ray or EUV wavelengths).</text>
<text><location><page_6><loc_12><loc_77><loc_88><loc_82></location>We find that the theoretical prediction of D A = (3 / 2) = 1 . 50 (Eq. 15) for the fractal area parameter A is approximately consistent with the observed values obtained with the box-counting method, D obs A 2 = 1 . 58 ± 0 . 03 (Table 1).</text>
<section_header_level_1><location><page_6><loc_32><loc_72><loc_68><loc_73></location>3.2. The SOC-Inferred Fractal Dimension</section_header_level_1>
<text><location><page_6><loc_12><loc_68><loc_88><loc_70></location>The size distribution N ( L ) of length scales L , also called the scale-free probability conjecture is (Aschwanden 2012),</text>
<formula><location><page_6><loc_43><loc_66><loc_88><loc_67></location>N ( L ) dL ∝ L -d dL , (17)</formula>
<text><location><page_6><loc_12><loc_59><loc_88><loc_65></location>where d is the Euclidean space dimension, generally set to d = 3 for most real-world data. Note, that this occurrence frequency distribution function is simply a power law, which results from the reciprocal relationship of the number of events N ( L ) and the length scale L . Since the fractal dimension D A for event areas A is defined as (Eq. 11),</text>
<formula><location><page_6><loc_46><loc_57><loc_88><loc_59></location>A = L D A , (18)</formula>
<text><location><page_6><loc_12><loc_55><loc_38><loc_56></location>we obtain the inverse function L ( A ) ,</text>
<text><location><page_6><loc_12><loc_51><loc_25><loc_52></location>and the derivative,</text>
<formula><location><page_6><loc_42><loc_48><loc_88><loc_51></location>( dL dA ) = A (1 /D A -1 ) , (20)</formula>
<text><location><page_6><loc_12><loc_45><loc_88><loc_48></location>so that we obtain the area distribution N ( A ) by substituting of L (Eq. 19) and dL/dA (Eq. 20) into N ( L ) (Eq. 17),</text>
<formula><location><page_6><loc_24><loc_42><loc_88><loc_45></location>N ( A ) dA = N [ L ( A )] ( dL dA ) dA = [ L ( A )] -d A (1 /D A -1) dA = A ( -α A ) dA , (21)</formula>
<text><location><page_6><loc_12><loc_40><loc_38><loc_41></location>which yields the power law index α A ,</text>
<formula><location><page_6><loc_39><loc_36><loc_88><loc_39></location>α A = 1 + ( d -1) D A = 1 + 2 D A . (22)</formula>
<text><location><page_6><loc_12><loc_34><loc_83><loc_35></location>Vice versa we can then obtain the SOC fractal dimension D A from an observed power law slope α A ,</text>
<formula><location><page_6><loc_44><loc_31><loc_88><loc_34></location>D A = 2 ( α A -1) . (23)</formula>
<text><location><page_6><loc_12><loc_24><loc_88><loc_30></location>This is an alternative method (Eq. 23) to calculate the fractal area dimension, in contrast to the box-counting method (Eq. 12), which we call the SOC-inferred fractal dimension, because it uses the size distribution of areas that are defined in SOC models. The so calculated fractal dimension D A 1 exhibits a mean and standard deviation of (Table 1),</text>
<formula><location><page_6><loc_38><loc_21><loc_88><loc_23></location>D A 1 = 2 ( α A 1 -1) = 1 . 35 ± 0 . 19 , (24)</formula>
<text><location><page_6><loc_12><loc_19><loc_44><loc_20></location>as tabulated for each IRIS dataset in Table 1.</text>
<text><location><page_6><loc_12><loc_10><loc_88><loc_18></location>However, there is a significant difference between the two methods, which is apparent in terms of a much smaller spread of values ( ≈ 2%) for the box-counting method, compared with the much wider spread of values ( ≈ 14%) for the power law fit method. Obviously, the power law fit method is more sensitive to the spatial variation of individual fractal features than the box-counting method, while the latter method averages the fractal features, so that the mean value of the fractal dimension is more robust.</text>
<formula><location><page_6><loc_45><loc_53><loc_88><loc_55></location>L = A (1 /D A ) , (19)</formula>
<section_header_level_1><location><page_7><loc_42><loc_85><loc_58><loc_86></location>4. DISCUSSION</section_header_level_1>
<section_header_level_1><location><page_7><loc_28><loc_82><loc_72><loc_83></location>4.1. Basic Fractal Dimension Measurement Methods</section_header_level_1>
<text><location><page_7><loc_12><loc_72><loc_88><loc_80></location>A fractal geometry is a ratio that provides a statistical index of complexity, and changes as a function of a length scale that is used as a yardstick to measure it (Mandelbrot 1977). Well-known geometries are the Euclidean dimensions of curves or lines L , being one-dimensional structures ( d = 1), areas A ∝ L 2 , being two-dimensional structures ( d = 2), and volumes V = L 3 , being three-dimensional structures ( d = 3). All other values between 1 and 3 are non-integer dimensions and are called fractal dimensions.</text>
<text><location><page_7><loc_12><loc_50><loc_88><loc_71></location>Basic methods to measure fractal dimensions include the linear-area (LA) method, the perimeter-area (PA) method, and the box-counting (BC) method. The LA method calculates the ratio of a fractal area A to a space-filled encompassing rectangular area with size L 2 . Similarly, the PA method yields a ratio of the encompassing curve length or perimeter length ( P = πr in the case of a circular boundary). The boxcounting method uses a cartesian (2-D or 3-D) lattice grid [ x, y ] and counts all pixels above some threshold or background, and takes the ratio to the total counts of all pixels inside the encompassing coordinate grid. These three methods appear to be very simple, but are not unique. The resulting fractal dimensions may depend on the assumed level of background subtraction. The encompassing perimeter depends on the definition of the perimeter (square, circle, polygon, etc.). Multiple different geometric patterns may cause a variation of the fractal dimension across an image or data cube. Temporal variability can modulate the fractal dimension as a function of time. Theoretical values of fractal dimensions converge by definition to a unique value (e.g., D A = 1 . 61803 for the golden dragon fractal, Fig. 5), while observed data almost always exhibit some spatial inhomogeneity that gives rise to a spread of fractal dimension values across an image.</text>
<section_header_level_1><location><page_7><loc_36><loc_45><loc_64><loc_46></location>4.2. Granulation in Photosphere</section_header_level_1>
<text><location><page_7><loc_12><loc_12><loc_88><loc_43></location>A compilation of fractal dimensions measured in photospheric white-light wavelengths (or in magnetograms) is given in Table 3. The solar granulation has a typical spatial scale of L = 1000 km, or a perimeter of P = πL ≈ 3000 km. Roudier and Muller (1986) measured the areas A and perimeters P of 315 granules and found a power law relation P ∝ A D/ 2 , with D = 1 . 25 for small granules (with perimeters of P ≈ 500 -4500 km) and D = 2 . 15 for large granules (with P = 4500 -15 , 000 km). The smaller granules were interpreted in terms of turbulent origin, because the predicted fractal dimension of an isobaric atmosphere with isotropic and homogeneous turbulence is D = 4 / 3 ≈ 1 . 33 (Mandelbrot 1977). Similar values ( D A = 1 . 30 and D A = 1 . 16) were found by Hirzberger et al. (1997). Bovelet and Wiehr (2001) tested different pattern recognition algorithms (Fourier-based recognition technique FBR and multiple-level tracking MLT) and found that the value of the fractal dimension strongly depends on the measurement method. The MLT method yielded a fractal dimension of D A = 1 . 09, independent of the spatial resolution, the heliocentric angle, and the definition in terms of temperature or velocity. Paniveni et al. (2005) found a fractal dimension of D A ≈ 1 . 25 and concluded, by relating it to the variations of kinetic energy, temperature, and pressure, that the supergranular network is close to being isobaric and possibly of turbulent origin. Paniveni et al. (2010) investigated supergranular cells and found a fractal dimension of D A = 1 . 12 for active region cells, and D A = 1 . 25 for quiet region cells, a difference that they attributed to the inhibiting effect of the stronger magnetic field in active regions. Averaging all fractal dimensions related to granular datasets we obtain a mean value of D A = 1 . 21 ± 0 . 07, which is closer to a curvi-linear topology ( D A > ∼ 1 . 0) than to an area-filled geometry ( D A < ∼ 2 . 0).</text>
<text><location><page_8><loc_12><loc_75><loc_88><loc_86></location>The physical understanding of solar (or stellar) granulation has been advanced by numerical magnetoconvection models and N-body dynamic simulations, which predict the evolution of small-scale (granules) into large-scale features (meso or supergranulation), organized by surface flows that sweep up small-scale structures and form clusters of recurrent and stable granular features (Hathaway et al. 2000; Berrilli et al. 2005; Rieutord et al. 2008, 2010). The fractal structure of the solar granulation is obviously a selforganizing pattern that is created by a combination of subphotospheric magneto-convection and surface flows, which are turbulence-type phenomena.</text>
<section_header_level_1><location><page_8><loc_40><loc_70><loc_60><loc_71></location>4.3. Transition Region</section_header_level_1>
<text><location><page_8><loc_12><loc_55><loc_88><loc_68></location>Measurements of the fractal dimension in the transition region have been accomplished with IRIS 1400 ˚ A observations of plages and sunspot regions (Nhalil et al. 2020; and this work, see Table 3). Fractal dimensions of transition region features were evaluated with a box-counting method here, yielding a range of D A ≈ 1 . 26 ± 0 . 13 for the 8 datasets of plages in the transition region listed in Table 3, and D A ≈ 1 . 54 ± 0 . 16 for the 4 datasets with sunspots listed in Table 3), following the same identification as used in the previous work (Nhalil et al. 2020). The structures observed in the 1400 ˚ A channel of IRIS are dominated by the Si IV 1394 ˚ A and 1403 ˚ A resonance lines, which are formed in the transition region temperature range of T = 10 4 . 5 -10 6 K, sandwiched between the cooler chromosphere and the hotter corona.</text>
<text><location><page_8><loc_12><loc_29><loc_88><loc_54></location>One prominent feature in the transition region is the phenomenon of 'moss' , which appeas as a bright, dynamic pattern with dark inclusions, on spatial scales of L ≈ 1 -3 Mm, which has been interpreted as the upper transition region above active region plage and below relatively hot loops (De Pontieu et al. 1999). Besides transition region features, measurements in chromospheric (Quiet-Sun) network structures in the temperature range of T = 10 4 . 5 -10 6 K yield fractal dimensions of D A = 1 . 30 -1 . 70 (Gallagher et al. 1998). Furthermore, a value of D A ≈ 1 . 4 was found for so-called Ellerman bombs (Georgoulis et al. 2002), which are short-lived brightenings seen in the wings of the H α line from the low chromosphere. In addition, a range of D A ≈ 1 . 25 -1 . 45 was measured from a large survey of 9342 active region magnetograms (McAteer et al. 2005). Measurements of SOHO/CDS in EUV lines in the temperature range of T e ≈ 10 4 . 5 -10 6 revealed a distinct temperature dependence: fractal dimensions of D A ≈ 1 . 5 -1 . 6 were identified in He I, He II, OIII, OIV, OV, Ne VI lines at log( T e ) ≈ 5 . 8, then a peak with D A ≈ 1 . 6 -1 . 7 at log( T e ) ≈ 5 . 9, and a drop of D A ≈ 1 . 3 -1 . 35 at log( T e ) ≈ 6 . 0 (see Fig. 11 in Gallagher et al. 1998). The temperature dependence of the fractal dimension can be interpreted in terms of sparse heating that produces curvi-linear flow patterns with a low fractal dimensions of D A < ∼ 1 . 5, while strong heating produces volume-filling by chromospheric evaporation with high fractal dimensions D A > ∼ 1 . 5.</text>
<text><location><page_8><loc_12><loc_24><loc_88><loc_28></location>In recent work it was found that the concept of mono-fractals has to be generalized to multi-fractals to quantify the spatial structure of solar magnetograms more accurately (Lawrence et al. 1993, 1996; Cadavid et al. 1994; McAteer et al. 2005; Conlon et al. 2008).</text>
<section_header_level_1><location><page_8><loc_28><loc_18><loc_72><loc_20></location>4.4. Photospheric Magnetic Field in Active Regions</section_header_level_1>
<text><location><page_8><loc_12><loc_10><loc_88><loc_17></location>A number of studies investigated the fractal dimension of the photospheric magnetic field, as observed in magnetograms in the Fe I (6302 ˚ A, 5250 ˚ A) or Ni I (6768 ˚ A) lines. Meunier (1999) evaluated the fractal dimension with the perimeter-area method and found D A = 1 . 48 for supergranular structures to D A = 1 . 68 for the largest structures, while the linear size-area method yielded D A = 1 . 78 and D A = 1 . 94, respectively.</text>
<text><location><page_9><loc_12><loc_78><loc_88><loc_86></location>In addition, a solar cycle dependence was found by Meunier (2004), with the fractal dimension varying from D A = 1 . 09 ± 0 . 11 (minimum) to D A = 1 . 73 ± 0 . 01 for weak-field regions ( B m < 900 G), and D A = 1 . 53 ± 0 . 06 (minimum) to D A = 1 . 80 ± 0 . 01 for strong-field regions ( B m > 900 G), respectively. A fractal dimension of D A = 1 . 41 ± 0 . 05 was found by Janssen et al. (2003), but the value varies as a function of the center-to-limb angle and is different for a speckle-reconstructed image that eliminates seeing and noise.</text>
<text><location><page_9><loc_12><loc_67><loc_88><loc_77></location>A completely different approach to measure the fractal dimension D was pursued in terms of a 2-D diffusion process, finding fractal diffusion with dimensions in the range of D ≈ 1 . 3 -1 . 8 (Lawrence 1991) or D = 1 . 56 ± 0 . 08 (Lawrence and Schrijver 1993) by measuring the dependence of the mean square displacement of magnetic elements as a function of time. Similar results were found by Balke et al. (1993). The results exclude Euclidean 2-D diffusion but are consistent with percolation theory for diffusion of clusters at a density below the percolation threshold (Lawrence and Schrijver 1993; Balke et al. 1993).</text>
<section_header_level_1><location><page_9><loc_41><loc_62><loc_59><loc_63></location>4.5. Coronal Flares</section_header_level_1>
<text><location><page_9><loc_12><loc_48><loc_88><loc_60></location>Although this study is focused on the fractal geometry of transition region features observed with IRIS, we compare these results also with coronal values. The fractal dimension of coronal events has been measured for 10 X-class flares, 10 M-class flares, and the Bastille-Day flare (Aschwanden and Aschwanden 2008a, 2008b). Interestingly, these datasets exhibit relatively large values of the fractal dimension, with a mean and standard deviation of D A = 1 . 76 ± 0 . 14. They show a trend that the largest flares, especially X-class flares, exhibit the highest values of D A < ∼ 1 . 8 -1 . 9 (Table 3). If we attribute flare events to the magnetic reconnection process, the observations imply that the flare plasma fills up the flare volume with a high space-filling factor, which is consistent with the chromospheric evaporation process..</text>
<text><location><page_9><loc_12><loc_40><loc_88><loc_46></location>Phenomena of smaller magnitude than large flares include microflares, nanoflares, coronal EUV brightenings, etc. Such small-scale variability events are found to have a mean fractal dimension of D A = 1 . 56 ± 0 . 08 (Table 3), which is compatible with those found in M-class flares, but clearly has a lower fractal dimension than large flares, i.e., D A = 1 . 76 ± 0 . 14 (Table 3).</text>
<section_header_level_1><location><page_9><loc_34><loc_35><loc_66><loc_36></location>4.6. Self-Organization and Criticality</section_header_level_1>
<text><location><page_9><loc_12><loc_14><loc_88><loc_33></location>The generation of magnetic structures that bubble up from the solar convection zone to the solar surface by buoyancy, observed as emerging flux phenomena in form of active regions, sunspots, and pores, can be statistically described as a random process, a self-organization (SO) process, self-organized criticality (SOC), percolation, or a diffusion process. Random processes produce incoherent structures, in contrast to the coherent magnetic flux concentrations observed in sunspots. A self-organization (SO) process needs a driving force and a counter-acting feedback mechanism that produces ordered structures (such as the convective granulation cells; Aschwanden et al. 2018). A SOC process exhibits power law size distributions of avalanche sizes and durations. The finding of a fractal dimension of a power law size distribution in magnetic features alone is not a sufficient condition to prove or rule out any of these processes. Nevertheless, the fractal dimension yields a scaling law between areas ( A ∝ L D 2 ) or volumes ( V ∝ L D 3 ), and length scales L that quantifies scale-free (fractal) processes in form of power laws and can straightforwardly be incorporated in SOC-like models.</text>
<text><location><page_9><loc_15><loc_11><loc_88><loc_12></location>If we compare the standard SOC parameters measured in observations (Fig. 2) with the theoretically</text>
<text><location><page_10><loc_12><loc_70><loc_88><loc_86></location>expected values from the standard SOC model (Table 2), we find that the power law slopes for event areas A agree well ( a obs A = 2 . 51 ± 0 . 21) versus a theo A = 2 . 33 (Fig. 2), while the power law slopes for the radiated energy E agree within' the stated uncertainties, ( a obs E = 2 . 03 ± 0 . 18) versus a theo E = 1 . 80 (Fig. 2), but the power law slopes for the time duration T disagree ( a obs T = 2 . 65 ± 0 . 39) versus a theo T = 2 . 00 (Fig. 2). The latter disagreement is possibly caused by the restriction of a constant minimum event lifetime (either 60 s or 110 s) that was assumed in the previous work (Nhalil et al. 2020). The interpretation of these results implies that transition region brightenings have a similar statistics as the SOC model, at least for active regions, nanoflares, and large flares, with a typical fractal dimension of D A ≈ 1 . 5 -1 . 6, but are significantly lower for photospheric granulation and transition region plages ( D A ≈ 1 . 2 -1 . 3), which implies the dominance of sparse quasi-linear flow structures in the photosphere and transition region.</text>
<section_header_level_1><location><page_10><loc_41><loc_65><loc_59><loc_66></location>5. CONCLUSIONS</section_header_level_1>
<text><location><page_10><loc_12><loc_52><loc_88><loc_63></location>Our aim is to obtain an improved undestanding of fractal dimensions and size distributions observed in the solar photosphere and transition region, which complement previous measurements of coronal phenomena, from nanoflares to the largest solar flares. Building on the previous study 'Power-law energy distributions of small-scale impulsive events on the active Sun: Results from IRIS' , we are using the same IRIS 1400 ˚ A data, extracted with an automated pattern recognition code during 12 time episodes observed in plage and sunspot regions. We obtain a total of 23,633 events, quantified in terms of event areas A , radiative energies E , and event durations T . We obtain the following results:</text>
<unordered_list>
<list_item><location><page_10><loc_14><loc_42><loc_88><loc_50></location>1. Fractal dimensions, measured in solar images at various wavelengths and spatial resolutions, cover a range of D A = 1 -2, where linear of curvi-linear features, as produced by surface flows and magnetoconvection, characterize the lower limit D A > ∼ 1, while area-filling structures, as they occur as a consequence of the chromospheric evaporation process, characterize the upper limit D A < ∼ 2. The mean value D A ≈ 1 . 5 appears to be a good approximation for SOC models.</list_item>
<list_item><location><page_10><loc_14><loc_35><loc_88><loc_41></location>2. We calculate a power law fit to the size distribution N ( A ) ∝ A -α A of event areas A , and find a mean value of a A = 2 . 51 ± 0 . 21 that agrees well with the value a A = 2 . 33 expected from the theoretical SOC model. Consequently, brightenings in plages of the transition region are consistent with the generic SOC model.</list_item>
<list_item><location><page_10><loc_14><loc_27><loc_88><loc_33></location>3. Based on the power law slope α A we derive the fractal dimension D A = 2 / ( α A -1), which yields a mean observed value of D A = 1 . 35 ± 0 . 19 and approximately matches the theoretial mean value of D A = 1 . 5. Alternatively, we obtain with the standard box-counting method an observed value of D A = 1 . 58 ± 0 . 03.</list_item>
<list_item><location><page_10><loc_14><loc_21><loc_88><loc_26></location>4. Synthesizing the measurements of the fractal dimension from photospheric, transition region, and coronal data we arrive at seven groups that yield the following means and standard deviations of their fractal dimension:</list_item>
</unordered_list>
<text><location><page_10><loc_17><loc_20><loc_35><loc_21></location>photospheric granulation:</text>
<text><location><page_10><loc_39><loc_20><loc_47><loc_21></location>1.21 ± 0.07</text>
<text><location><page_10><loc_17><loc_18><loc_34><loc_19></location>transition region plages:</text>
<text><location><page_10><loc_39><loc_18><loc_47><loc_19></location>1.29 ± 0.15</text>
<text><location><page_10><loc_17><loc_16><loc_36><loc_17></location>transition region sunspots:</text>
<text><location><page_10><loc_39><loc_16><loc_47><loc_17></location>1.54 ± 0.16</text>
<text><location><page_10><loc_17><loc_15><loc_37><loc_16></location>active region magnetograms:</text>
<text><location><page_10><loc_39><loc_15><loc_47><loc_16></location>1.59 ± 0.08</text>
<text><location><page_10><loc_17><loc_13><loc_29><loc_14></location>EUV nanoflares:</text>
<text><location><page_10><loc_39><loc_13><loc_47><loc_14></location>1.56 ± 0.08</text>
<text><location><page_10><loc_17><loc_11><loc_29><loc_12></location>large solar flares:</text>
<text><location><page_10><loc_39><loc_11><loc_47><loc_13></location>1.76 ± 0.14</text>
<text><location><page_10><loc_17><loc_10><loc_37><loc_11></location>Bastille-Day X5.7-class flare:</text>
<text><location><page_10><loc_39><loc_10><loc_47><loc_11></location>1.89 ± 0.05</text>
<unordered_list>
<list_item><location><page_11><loc_14><loc_78><loc_88><loc_86></location>5. From these seven groups we can discriminate 3 groups with significantly different fractal dimensions, which implies different physical mechanisms: Low values of the fractal dimension ( D A ≈ 1 . 2 -1 . 3) indicate granulation or transition region plage features, intermediate values of the fractal dimension ( D A ≈ 1 . 5 -1 . 6) indicate sunspots, active region, or nanoflare features, and ( D A ≈ 1 . 6 -1 . 9) indicate large flares.</list_item>
</unordered_list>
<text><location><page_11><loc_12><loc_63><loc_88><loc_76></location>The analysis presented here demonstrates that we can distinguish between (i) physical processes with sparse curvi-linear flows, as they occur in granulation, meso-granulation, and super-granulation. and (ii) physical processes with near space-filling flows, as they occur in the chromospheric evaporation process in solar flares. IRIS data can therefore be used to diagnose the strength of mass flows in the transition region. Moreover, reliable measurements of the fractal dimension yields realistic plasma filling factors that are important in the estimate of radiative energies and hot plasma emission measures. Future work on fractal dimensions in multi-wavlength datasets from IRIS and AIA/SDO may clarify the dynamics of coronal heating events.</text>
<text><location><page_11><loc_12><loc_52><loc_88><loc_60></location>Acknowledgements: We acknowledge constructive and stimulating discussions (in alphabetical order) with Sandra Chapman, Paul Charbonneau, Henrik Jeldtoft Jensen, Adam Kowalski, Alexander Milovanov, Leonty Miroshnichenko, Jens Juul Rasmussen, Karel Schrijver, Vadim Uritsky, Loukas Vlahos, and Nick Watkins. This work was partially supported by NASA contract NNX11A099G 'Self-organized criticality in solar physics' and NASA contract NNG04EA00C of the SDO/AIA instrument to LMSAL.</text>
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<table>
<location><page_15><loc_26><loc_51><loc_74><loc_81></location>
<caption>Table 1: Fractal Dimension obtained from power law fits and from the box counting method for 12 IRIS datasets.Table 2: Parameters of the standard SOC Model, with fractal dimensions D x and power law slopes α x of size distributions).</caption>
</table>
<table>
<location><page_15><loc_21><loc_13><loc_79><loc_40></location>
</table>
<table>
<location><page_16><loc_12><loc_7><loc_95><loc_82></location>
<caption>Table 3: The fractal dimensions of granules, plages, sunspots, active regions, nanoflares, and large flares.</caption>
</table>
<figure>
<location><page_17><loc_19><loc_19><loc_80><loc_82></location>
<caption>Fig. 1.- Size distributions of flare areas A for 12 datasets observed with IRIS SJI 1400 ˚ A in different active regions.</caption>
</figure>
<figure>
<location><page_18><loc_21><loc_18><loc_74><loc_83></location>
<caption>Fig. 2.- Power law fits to the size distributions of three SOC parameters: the event area A (top panel), the radiative energy E (middle panel), and the time duration (bottom panel). Individual fits to each of the 12 IRIS datasets are indicated with thin line style, while the fit to all events combined is indicated with thick line style, and the power law slopes are given in each panel.</caption>
</figure>
<text><location><page_19><loc_16><loc_73><loc_17><loc_80></location>y-axis (pixels)</text>
<text><location><page_19><loc_16><loc_57><loc_17><loc_64></location>y-axis (pixels)</text>
<text><location><page_19><loc_16><loc_40><loc_17><loc_47></location>y-axis (pixels)</text>
<text><location><page_19><loc_18><loc_82><loc_20><loc_83></location>120</text>
<text><location><page_19><loc_18><loc_80><loc_20><loc_81></location>100</text>
<text><location><page_19><loc_18><loc_78><loc_20><loc_79></location>80</text>
<text><location><page_19><loc_18><loc_76><loc_20><loc_77></location>60</text>
<text><location><page_19><loc_18><loc_74><loc_20><loc_75></location>40</text>
<text><location><page_19><loc_18><loc_71><loc_20><loc_72></location>20</text>
<text><location><page_19><loc_19><loc_70><loc_20><loc_71></location>0</text>
<text><location><page_19><loc_20><loc_68><loc_21><loc_69></location>0</text>
<text><location><page_19><loc_18><loc_66><loc_20><loc_67></location>120</text>
<text><location><page_19><loc_18><loc_63><loc_20><loc_64></location>100</text>
<text><location><page_19><loc_18><loc_61><loc_20><loc_62></location>80</text>
<text><location><page_19><loc_18><loc_59><loc_20><loc_60></location>60</text>
<text><location><page_19><loc_18><loc_57><loc_20><loc_58></location>40</text>
<text><location><page_19><loc_18><loc_55><loc_20><loc_56></location>20</text>
<text><location><page_19><loc_19><loc_53><loc_20><loc_54></location>0</text>
<text><location><page_19><loc_20><loc_52><loc_21><loc_53></location>0</text>
<text><location><page_19><loc_22><loc_52><loc_24><loc_53></location>20</text>
<text><location><page_19><loc_25><loc_52><loc_27><loc_53></location>40</text>
<text><location><page_19><loc_28><loc_52><loc_30><loc_53></location>60</text>
<text><location><page_19><loc_31><loc_52><loc_38><loc_53></location>80 100 120</text>
<text><location><page_19><loc_18><loc_49><loc_20><loc_50></location>120</text>
<text><location><page_19><loc_18><loc_47><loc_20><loc_48></location>100</text>
<text><location><page_19><loc_18><loc_45><loc_20><loc_46></location>80</text>
<text><location><page_19><loc_18><loc_42><loc_20><loc_44></location>60</text>
<text><location><page_19><loc_18><loc_40><loc_20><loc_41></location>40</text>
<text><location><page_19><loc_18><loc_38><loc_20><loc_39></location>20</text>
<text><location><page_19><loc_19><loc_36><loc_20><loc_37></location>0</text>
<text><location><page_19><loc_20><loc_35><loc_21><loc_36></location>0</text>
<text><location><page_19><loc_22><loc_35><loc_24><loc_36></location>20</text>
<text><location><page_19><loc_25><loc_35><loc_27><loc_36></location>40</text>
<text><location><page_19><loc_28><loc_35><loc_30><loc_36></location>60</text>
<text><location><page_19><loc_31><loc_35><loc_38><loc_36></location>80 100 120</text>
<figure>
<location><page_19><loc_15><loc_17><loc_39><loc_34></location>
<caption>Fig. 3.- Intensity maps of 12 different active regions, observed with IRIS SJI 1400 ˚ A .</caption>
</figure>
<text><location><page_19><loc_40><loc_82><loc_42><loc_83></location>120</text>
<text><location><page_19><loc_40><loc_80><loc_42><loc_81></location>100</text>
<text><location><page_19><loc_40><loc_78><loc_42><loc_79></location>80</text>
<text><location><page_19><loc_40><loc_76><loc_42><loc_77></location>60</text>
<text><location><page_19><loc_40><loc_74><loc_42><loc_75></location>40</text>
<text><location><page_19><loc_40><loc_71><loc_42><loc_72></location>20</text>
<text><location><page_19><loc_41><loc_70><loc_42><loc_71></location>0</text>
<text><location><page_19><loc_42><loc_68><loc_43><loc_69></location>0</text>
<text><location><page_19><loc_44><loc_68><loc_46><loc_69></location>20</text>
<text><location><page_19><loc_47><loc_68><loc_49><loc_69></location>40</text>
<text><location><page_19><loc_50><loc_68><loc_52><loc_69></location>60</text>
<text><location><page_19><loc_53><loc_68><loc_60><loc_69></location>80 100 120</text>
<text><location><page_19><loc_40><loc_66><loc_42><loc_67></location>120</text>
<text><location><page_19><loc_40><loc_63><loc_42><loc_64></location>100</text>
<text><location><page_19><loc_40><loc_61><loc_42><loc_62></location>80</text>
<text><location><page_19><loc_40><loc_59><loc_42><loc_60></location>60</text>
<text><location><page_19><loc_40><loc_57><loc_42><loc_58></location>40</text>
<text><location><page_19><loc_40><loc_55><loc_42><loc_56></location>20</text>
<text><location><page_19><loc_41><loc_53><loc_42><loc_54></location>0</text>
<text><location><page_19><loc_42><loc_52><loc_43><loc_53></location>0</text>
<text><location><page_19><loc_44><loc_52><loc_46><loc_53></location>20</text>
<text><location><page_19><loc_47><loc_52><loc_49><loc_53></location>40</text>
<text><location><page_19><loc_50><loc_52><loc_52><loc_53></location>60</text>
<text><location><page_19><loc_53><loc_52><loc_60><loc_53></location>80 100 120</text>
<text><location><page_19><loc_40><loc_49><loc_42><loc_50></location>120</text>
<text><location><page_19><loc_40><loc_47><loc_42><loc_48></location>100</text>
<text><location><page_19><loc_40><loc_45><loc_42><loc_46></location>80</text>
<text><location><page_19><loc_40><loc_42><loc_42><loc_44></location>60</text>
<text><location><page_19><loc_40><loc_40><loc_42><loc_41></location>40</text>
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<caption>Fig. 4.- The IRIS dataset 8 is shown with different spatial resolutions of 128, 64, 32, and 16 bins, which demonstrates the scale-free definition of the Hausdorff dimension D H = 1 . 33.</caption>
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<caption>Fig. 5.- This numerically calculated fractal pattern is called a golden dragon and has a Hausdorff dimension of D A = 1 . 61803. ( https : //en.wikipedia.org/wiki/List of fractals by Hausdorff dimension ). Note the similarity with dataset 8 in Fig. 4</caption>
</figure>
</document> | [{"title": "ABSTRACT", "content": "While previous work explored the fractality and self-organized criticality (SOC) of flares and nanoflares in wavelengths emitted in the solar corona (such as in hard X-rays, soft Xrays, and EUV wavelenghts), we focus here on impulsive phenomena in the photosphere and transition region, as observed with the Interface Region Imaging Spectrograph (IRIS) in the temperature range of T e \u2248 10 4 -10 6 K. We find the following fractal dimensions (in increasing order): D A = 1 . 21 \u00b1 0 . 07 for photospheric granulation, D A = 1 . 29 \u00b1 0 . 15 for plages in the transition region, D A = 1 . 54 \u00b1 0 . 16 for sunspots in the transition region, D A = 1 . 59 \u00b1 0 . 08 for magnetograms in active regions, D A = 1 . 56 \u00b1 0 . 08 for EUV nanoflares, D A = 1 . 76 \u00b1 0 . 14 for large solar flares, and up to D A = 1 . 89 \u00b1 0 . 05 for the largest X-class flares. We interpret low values of the fractal dimension (1 . 0 < \u223c D A < \u223c 1 . 5) in terms of sparse curvi-linear flow patterns, while high values of the fractal dimension (1 . 5 < \u223c D A < \u223c 2 . 0) indicate near space-filling transport processes, such as chromospheric evaporation. Phenomena in the solar transition region appear to be consistent with SOC models, based on their size distributions of fractal areas A and (radiative) energies E , which show power law slopes of \u03b1 obs A = 2 . 51 \u00b1 0 . 21 (with \u03b1 theo A = 2 . 33 predicted), and \u03b1 obs E = 2 . 03 \u00b1 0 . 18 (with \u03b1 theo E = 1 . 80 predicted). Subject headings: methods: statistical - fractal dimension - Sun: transition region - solar granulation - solar photosphere -", "pages": [1]}, {"title": "Interface Region Imaging Spectrograph (IRIS) Observations of the Fractal Dimension in the Solar Atmosphere", "content": "Markus J. Aschwanden Lockheed Martin, Solar and Astrophysics Laboratory (LMSAL), Advanced Technology Center (ATC), A021S, Bldg.252, 3251 Hanover St., Palo Alto, CA 94304, USA; e-mail: [email protected] and Nived Vilangot Nhalil Armagh Observatory and Planetarium, College Hill, Armagh BT61 9DG, UK", "pages": [1]}, {"title": "1. INTRODUCTION", "content": "There are at least three different approaches that quantify the statistics of nonlinear processes with the concept of self-organized criticality (SOC) and fractality: (i) microscopic models, (ii) macroscopic models, and (iii) observations of power laws and scaling laws. The microscopic SOC models consist of numerically simulated avalanches that evolve via next-neighbor interactions in a lattice grid (Bak et al. 1987, 1988), also called cellular automatons , which have been quantized up to numerical limits of \u2248 10 6 -10 9 cells per avalanche process. The macroscopic models describe the nonlinear evolution of (avalanching) instabilities with analytical (geometric and energetic) quantities, which predict physical scaling laws and power law-like occurrence frequency size distributions. The third category of SOC approaches includes observations with fitting of power law-like distribution functions and waiting time distributions, which provide powerful tests of theoretical SOC models. A total of over 1500 SOC-related publications appeared at the time of writing. For brevity, we mention a few textbooks only (Bak 1996; Aschwanden 2011; Pruessner 2012), and a recent collection of astrophysical SOC reviews, presented in the special volume Space Science Reviews Vol. 198 (Watkins et al. 2016; Aschwanden et al. 2016; Sharma et al. 2016; McAteer et al. 2016). In this paper we focus on SOC modeling of impulsive events detected in the solar atmosphere, as observed with the Interface Region Imaging Spectrograph (IRIS) (De Pontieu et al. 2014), while solar flare events observed in hard X-rays, soft X-rays, and Extreme-Ultraviolet (EUV) have been compared in recent studies (Aschwanden 2022a, 2022b). Large solar flares observed in hard and soft X-rays show typically electron temperatures of T e \u2248 5 -35 K, while coronal nanoflares observed in EUV have moderate temperatures of T e \u2248 1 -2 MK. Hence it is interesting to investigate transition region events, which are observed in a different temperature regime ( T e \u2248 10 4 -10 6 K) than coronal phenomena. In a previous study with the same IRIS data, it was found that plages and sunspots have different power law indices for the areas of events, being smaller ( \u03b1 A < \u223c 2) for sunspot-dominated active regions, and larger for plage regions, \u03b1 > \u223c 2, (Nhalil et al. 2020). If both coronal and transition region brightenings exhibit the same SOC behavior and are produced by the same physical mechanism, one would expect the same fractal dimension and power law slope of the occurrence frequency size distribution, which is an important test of the coronal heating problem. The structure of this paper consists of an observational Section 2, a theoretical modeling Section 3, a discussion Section 4, and a conclusion Section 5.", "pages": [1, 2]}, {"title": "2. OBSERVATIONS", "content": "This is a follow-on study of previous work, 'The power-law energy distributions of small-scale impulsive events on the active Sun: Results from IRIS' (Nhalil et al. 2020). We call these small-scale impulsive events simply 'events', which possibly could be related to 'nanoflares' or 'brightenings'. In the previous study, 12 IRIS datasets were investigated with an automated pattern recognition algorithm, yielding statistics of three parameters, namely the event area A (in units of pixels), the event (radiative) energy E (in units of erg), and event durations or lifetimes T (in units in seconds). IRIS has pixels with a size of 0 . 17 '' \u2248 0 . 123 Mm, which have been rebinned to L pixel = 0 . 33 '' \u2248 0 . 247 Mm. The pixel size of areas thus corresponds to A pixel = L 2 pixel = 0 . 247 2 Mm 2 = 0.06076 Mm 2 . The range of event areas covers A = 4 -677 pixels, which amounts to length scales of L = \u221a A = (2 -26) pixels, or L = (2 -26) \u2217 0 . 247 Mm \u2248 (0.5-6.4) Mm = (500-6400) km. The date of observations, the field-of-view (FOV), the cadence, and the NOAA active region numbers are given in Table 1 of Nhalil et al. (2020), for each of the 12 IRIS datasets. The automated pattern recognition code was run with different threshold levels of 3, 5, and 7 \u03c3 for the detection of events, from which we use the 3\u03c3 level here. We use Slitjaw images (SJI) of the 1400 \u02da A channel of IRIS, which are dominated by the Si IV 1394 \u02da A and 1403 \u02da A resonance lines, formed in the transition region. Nhalil et al. (2020) compared also images from the SJI 1330 \u02da A channel, which is dominated by the C II 1335 \u02da A and 1336 \u02da A lines, originating in the upper chromosphere and transition region at formation temperatures of T e \u2248 3 \u00d7 10 4 K and T e \u2248 8 \u00d7 10 4 K (Rathore and Carlsson 2015; Rathore et al. 2015).", "pages": [2]}, {"title": "2.1. Size Distributions", "content": "Our first measurement is the fitting of a power law distribution function N ( A ) \u221d A -\u03b1 A of the event (or nanoflare) areas A , separately for each of the 12 IRIS datasets, as shown in Fig. 1. The area of the event is a combination of all the spatially connected 3\u03c3 pixels throughout its lifetime. The lowest bin was discarded when a visible deviation from a power law was apparent in the histogram. The number of events amounts to 23,633 for all 12 datasets together, varying from 65 to 4725 events per IRIS dataset (Table 1). The power law slope fits vary from the lowest value \u03b1 A = 2 . 14 (dataset 1) to the highest value \u03b1 A = 2 . 83 (dataset 4), having a mean and standard deviation of (Fig. 2, top panel). The area size distributions are shown superimposed for the 12 IRIS datasets (Fig. 2, top panel), which illustrates almost identical power law slopes in different IRIS datasets. Fitting the energy size distributions, N ( E ) \u221d E -\u03b1 E , yields the following mean for all 12 IRIS datasets (Fig. 2, middle panel), Fitting the duration size distributions, N ( T ) \u221d T -\u03b1 T , yields the following mean for all 12 IRIS datasets (Fig. 2, bottom panel), We will interpret these power law slopes in terms of SOC models in Section 4.6.", "pages": [3]}, {"title": "2.2. Fractal Dimension with Box-Counting Method", "content": "The next parameter that we are interested in is the fractal dimension. A standard method to determine the fractal dimension D A of an image is the box-counting method, which is defined by the asymptotic ( L \u21a6\u2192 0) ratio of the fractal area A to the the length scale L , i.e., D A = log( A ) / log( L ), also called Hausdorff (fractal) dimension. We calculate the so-defined Hausdorff dimension D A for each of the 12 IRIS datasets (column D A 2 in Table 1), which reveals a very narrow spread of values for the fractal dimension, with a mean and stadard deviation of < \u223c 2%, The fractal nature of the 12 IRIS datasets is rendered in Figs. (3) and (4), where the white areas correspond to zones with enhanced emission, and black areas correspond to the background with weak emission. The successive reduction of spatial resolution is shown in Fig. 4 for N bin = 128 , 64 , 32 , 16 pixels, which all converge to the same fractal dimension of D A = 1 . 33. An example of a theoretical fractal pattern with a close ressemblance to the observed transition region patterns of dataset 8 is shown in Fig. 5, which is called the 'golden dragon fractal' and has a Hausdorff dimension of D A = 1 . 61803.", "pages": [3]}, {"title": "2.3. Comparison of Photospheric, Transition Region, and Coronal Fractal Dimensions", "content": "In Table 3 we compile fractal dimensions obtained from photospheric and transition region fractal features, which may be different from coronal and flare-like size distributions. The fractal dimension has been measured in white-light wavelengths with the perimeter-area method, containing dominantly granules and supergranulation features (Roudier and Muller 1986; Hirzberger et al. 1997; Bovelet and Wiehr 2001; Paniveni et al. 2010), which exhibit a mean value of (Table 3), We have to be aware that white-light emission originates in the solar photosphere, which has a lower altitude than any transition region or coronal feature. The relatively low value obtained for granulation features thus indicates that the granulation features seen in optical wavelengths are almost curvi-linear (with little area-filling geometries), which is expected for sparse photospheric mass flows along curvi-linear flow lines. A second feature we consider are plages, measured in magnetograms with the linear-area (LA) method (Balke et al. 1993), and in transition region IRIS 1400 \u02da A data (Nhalil et al. 2020), which have formation temperatures of \u2248 10 3 . 7 -10 5 . 2 K in the lower transition region, exhibiting a mean value of (Table 3), This set of IRIS measurements exhibit a relatively low value for the fractal dimension, similar to the photospheric granulation features. Based on this low fractal dimension, photospheric flows appear to be organized along curvi-linear features, rather than solid-area geometries. A third feature that we investigate are sunspots (Nhalil et al. 2020, and this work), which reveal higher values of fractal dimension, namely (Table 3), Apparently, sunspots organize fractal features into space-filling geometries, where fragmentation into smaller and smaller fractal features is suppressed, because of the strong magnetic fields that control the penumbral flows of sunspots. A fourth feature is an active region, observed in photospheric magnetograms and analyzed with the linear-area method (Lawrence 1991; Lawrence and Schrijver 1993; Meunier 1999, 2004; Janssen et al. 2003; Ioshpa et al. 2008), or with the box-counting method (McAteer et al. 2005). The mean value of fractal dimensions measured in active regions is found to be (Table 3), Apparently, active regions organize magnetic features into space-filling, area-like geometries, similar to sunspot features. A fifth phenomenon is a nanoflare event, which has been related to the SOC interpretation since Lu and Hamilton (1991). Nanoflares have been observed in EUV 171 \u02da A and 195 \u02da A with the TRACE instrument, as well as in soft X-rays using the Yohkoh/SXT (Solar X-Ray Telescope) (Aschwanden and Parnell 2002), which show a mean value of (see Table 3), Nanoflares have been observed in the Quiet Sun and appear to have a similar fractal dimension as impulsive brightenings in active regions, as measured in magnetograms. For completeness we list also the fractal dimension measured in large solar flares, for M-class flares, X-class flares, and the Bastille Day flare (Aschwanden and Aschwanden 2008a), which all together exhibit a mean value of (Table 3), This is the largest mean value of any measured fractal dimension, which indicates that the flare process fills the flare area almost completely, due to the superposition of many coronal postflare loops that become filled as a consequence of the chromospheric evaporation process. Thus, we can distinguish at least three groups with significantly different fractal properties in photospheric, transition region, and coronal data. A first group has curvi-linear features in the granulation and in plage features, which have a relatively low fractal dimension D gran A \u2248 D plage A \u2248 1 . 2 -1 . 3. There is a second group of sunspot, active region, and nanoflare phenomena, wich exhibit an intermediate range of fractal dimensions of D sunspot A \u2248 D AR A \u2248 D nano A \u2248 1 . 5 -1 . 6. And there is a third group of large flares (M- and X-class), which have a fractal dimensions of D flare A \u2248 1 . 6 -1 . 9.", "pages": [3, 4, 5]}, {"title": "3.1. The Hausdorff Fractal Dimension", "content": "The definition of the fractal dimension D A for 2-D areas A is also called the Hausdorff dimension D A (Mandelbrot 1977), or explicitly, where the area A is the sum of all image pixels I ( x, y ) \u2265 I 0 above a background threshold I 0 , and L is the length scale of a fractal area. A structure is fractal, when the ratio D A is approximately constant versus different length scales and converges to a constant for the smallest length scales L \u21a6\u2192 0. The method described here is also called the box-counting method, because the pixels are counted for the area A and the length scale L . In analogy, a fractal dimension can also be defined for the 3-D volume V , or explicitly The valid range for area fractal dimensions is 1 \u2264 D A \u2264 2 and 2 \u2264 D V \u2264 3, where D = 1 , 2 , 3 are the Euclidean dimensions. We can estimate the numerical values of the fractal dimensions D A and D V from the means of the minimum and maximum values in each Euclidean domain, and correspondingly, The 2-D fractal dimension D A is the easiest accessible SOC parameter, while the 3-D fractal dimension D V requires information of fractal structures along the line-of-sight, either using a geometric or tomographic model, or modeling of optically-thin plasma (in the case of an astrophysical object observed in soft X-ray or EUV wavelengths). We find that the theoretical prediction of D A = (3 / 2) = 1 . 50 (Eq. 15) for the fractal area parameter A is approximately consistent with the observed values obtained with the box-counting method, D obs A 2 = 1 . 58 \u00b1 0 . 03 (Table 1).", "pages": [5, 6]}, {"title": "3.2. The SOC-Inferred Fractal Dimension", "content": "The size distribution N ( L ) of length scales L , also called the scale-free probability conjecture is (Aschwanden 2012), where d is the Euclidean space dimension, generally set to d = 3 for most real-world data. Note, that this occurrence frequency distribution function is simply a power law, which results from the reciprocal relationship of the number of events N ( L ) and the length scale L . Since the fractal dimension D A for event areas A is defined as (Eq. 11), we obtain the inverse function L ( A ) , and the derivative, so that we obtain the area distribution N ( A ) by substituting of L (Eq. 19) and dL/dA (Eq. 20) into N ( L ) (Eq. 17), which yields the power law index \u03b1 A , Vice versa we can then obtain the SOC fractal dimension D A from an observed power law slope \u03b1 A , This is an alternative method (Eq. 23) to calculate the fractal area dimension, in contrast to the box-counting method (Eq. 12), which we call the SOC-inferred fractal dimension, because it uses the size distribution of areas that are defined in SOC models. The so calculated fractal dimension D A 1 exhibits a mean and standard deviation of (Table 1), as tabulated for each IRIS dataset in Table 1. However, there is a significant difference between the two methods, which is apparent in terms of a much smaller spread of values ( \u2248 2%) for the box-counting method, compared with the much wider spread of values ( \u2248 14%) for the power law fit method. Obviously, the power law fit method is more sensitive to the spatial variation of individual fractal features than the box-counting method, while the latter method averages the fractal features, so that the mean value of the fractal dimension is more robust.", "pages": [6]}, {"title": "4.1. Basic Fractal Dimension Measurement Methods", "content": "A fractal geometry is a ratio that provides a statistical index of complexity, and changes as a function of a length scale that is used as a yardstick to measure it (Mandelbrot 1977). Well-known geometries are the Euclidean dimensions of curves or lines L , being one-dimensional structures ( d = 1), areas A \u221d L 2 , being two-dimensional structures ( d = 2), and volumes V = L 3 , being three-dimensional structures ( d = 3). All other values between 1 and 3 are non-integer dimensions and are called fractal dimensions. Basic methods to measure fractal dimensions include the linear-area (LA) method, the perimeter-area (PA) method, and the box-counting (BC) method. The LA method calculates the ratio of a fractal area A to a space-filled encompassing rectangular area with size L 2 . Similarly, the PA method yields a ratio of the encompassing curve length or perimeter length ( P = \u03c0r in the case of a circular boundary). The boxcounting method uses a cartesian (2-D or 3-D) lattice grid [ x, y ] and counts all pixels above some threshold or background, and takes the ratio to the total counts of all pixels inside the encompassing coordinate grid. These three methods appear to be very simple, but are not unique. The resulting fractal dimensions may depend on the assumed level of background subtraction. The encompassing perimeter depends on the definition of the perimeter (square, circle, polygon, etc.). Multiple different geometric patterns may cause a variation of the fractal dimension across an image or data cube. Temporal variability can modulate the fractal dimension as a function of time. Theoretical values of fractal dimensions converge by definition to a unique value (e.g., D A = 1 . 61803 for the golden dragon fractal, Fig. 5), while observed data almost always exhibit some spatial inhomogeneity that gives rise to a spread of fractal dimension values across an image.", "pages": [7]}, {"title": "4.2. Granulation in Photosphere", "content": "A compilation of fractal dimensions measured in photospheric white-light wavelengths (or in magnetograms) is given in Table 3. The solar granulation has a typical spatial scale of L = 1000 km, or a perimeter of P = \u03c0L \u2248 3000 km. Roudier and Muller (1986) measured the areas A and perimeters P of 315 granules and found a power law relation P \u221d A D/ 2 , with D = 1 . 25 for small granules (with perimeters of P \u2248 500 -4500 km) and D = 2 . 15 for large granules (with P = 4500 -15 , 000 km). The smaller granules were interpreted in terms of turbulent origin, because the predicted fractal dimension of an isobaric atmosphere with isotropic and homogeneous turbulence is D = 4 / 3 \u2248 1 . 33 (Mandelbrot 1977). Similar values ( D A = 1 . 30 and D A = 1 . 16) were found by Hirzberger et al. (1997). Bovelet and Wiehr (2001) tested different pattern recognition algorithms (Fourier-based recognition technique FBR and multiple-level tracking MLT) and found that the value of the fractal dimension strongly depends on the measurement method. The MLT method yielded a fractal dimension of D A = 1 . 09, independent of the spatial resolution, the heliocentric angle, and the definition in terms of temperature or velocity. Paniveni et al. (2005) found a fractal dimension of D A \u2248 1 . 25 and concluded, by relating it to the variations of kinetic energy, temperature, and pressure, that the supergranular network is close to being isobaric and possibly of turbulent origin. Paniveni et al. (2010) investigated supergranular cells and found a fractal dimension of D A = 1 . 12 for active region cells, and D A = 1 . 25 for quiet region cells, a difference that they attributed to the inhibiting effect of the stronger magnetic field in active regions. Averaging all fractal dimensions related to granular datasets we obtain a mean value of D A = 1 . 21 \u00b1 0 . 07, which is closer to a curvi-linear topology ( D A > \u223c 1 . 0) than to an area-filled geometry ( D A < \u223c 2 . 0). The physical understanding of solar (or stellar) granulation has been advanced by numerical magnetoconvection models and N-body dynamic simulations, which predict the evolution of small-scale (granules) into large-scale features (meso or supergranulation), organized by surface flows that sweep up small-scale structures and form clusters of recurrent and stable granular features (Hathaway et al. 2000; Berrilli et al. 2005; Rieutord et al. 2008, 2010). The fractal structure of the solar granulation is obviously a selforganizing pattern that is created by a combination of subphotospheric magneto-convection and surface flows, which are turbulence-type phenomena.", "pages": [7, 8]}, {"title": "4.3. Transition Region", "content": "Measurements of the fractal dimension in the transition region have been accomplished with IRIS 1400 \u02da A observations of plages and sunspot regions (Nhalil et al. 2020; and this work, see Table 3). Fractal dimensions of transition region features were evaluated with a box-counting method here, yielding a range of D A \u2248 1 . 26 \u00b1 0 . 13 for the 8 datasets of plages in the transition region listed in Table 3, and D A \u2248 1 . 54 \u00b1 0 . 16 for the 4 datasets with sunspots listed in Table 3), following the same identification as used in the previous work (Nhalil et al. 2020). The structures observed in the 1400 \u02da A channel of IRIS are dominated by the Si IV 1394 \u02da A and 1403 \u02da A resonance lines, which are formed in the transition region temperature range of T = 10 4 . 5 -10 6 K, sandwiched between the cooler chromosphere and the hotter corona. One prominent feature in the transition region is the phenomenon of 'moss' , which appeas as a bright, dynamic pattern with dark inclusions, on spatial scales of L \u2248 1 -3 Mm, which has been interpreted as the upper transition region above active region plage and below relatively hot loops (De Pontieu et al. 1999). Besides transition region features, measurements in chromospheric (Quiet-Sun) network structures in the temperature range of T = 10 4 . 5 -10 6 K yield fractal dimensions of D A = 1 . 30 -1 . 70 (Gallagher et al. 1998). Furthermore, a value of D A \u2248 1 . 4 was found for so-called Ellerman bombs (Georgoulis et al. 2002), which are short-lived brightenings seen in the wings of the H \u03b1 line from the low chromosphere. In addition, a range of D A \u2248 1 . 25 -1 . 45 was measured from a large survey of 9342 active region magnetograms (McAteer et al. 2005). Measurements of SOHO/CDS in EUV lines in the temperature range of T e \u2248 10 4 . 5 -10 6 revealed a distinct temperature dependence: fractal dimensions of D A \u2248 1 . 5 -1 . 6 were identified in He I, He II, OIII, OIV, OV, Ne VI lines at log( T e ) \u2248 5 . 8, then a peak with D A \u2248 1 . 6 -1 . 7 at log( T e ) \u2248 5 . 9, and a drop of D A \u2248 1 . 3 -1 . 35 at log( T e ) \u2248 6 . 0 (see Fig. 11 in Gallagher et al. 1998). The temperature dependence of the fractal dimension can be interpreted in terms of sparse heating that produces curvi-linear flow patterns with a low fractal dimensions of D A < \u223c 1 . 5, while strong heating produces volume-filling by chromospheric evaporation with high fractal dimensions D A > \u223c 1 . 5. In recent work it was found that the concept of mono-fractals has to be generalized to multi-fractals to quantify the spatial structure of solar magnetograms more accurately (Lawrence et al. 1993, 1996; Cadavid et al. 1994; McAteer et al. 2005; Conlon et al. 2008).", "pages": [8]}, {"title": "4.4. Photospheric Magnetic Field in Active Regions", "content": "A number of studies investigated the fractal dimension of the photospheric magnetic field, as observed in magnetograms in the Fe I (6302 \u02da A, 5250 \u02da A) or Ni I (6768 \u02da A) lines. Meunier (1999) evaluated the fractal dimension with the perimeter-area method and found D A = 1 . 48 for supergranular structures to D A = 1 . 68 for the largest structures, while the linear size-area method yielded D A = 1 . 78 and D A = 1 . 94, respectively. In addition, a solar cycle dependence was found by Meunier (2004), with the fractal dimension varying from D A = 1 . 09 \u00b1 0 . 11 (minimum) to D A = 1 . 73 \u00b1 0 . 01 for weak-field regions ( B m < 900 G), and D A = 1 . 53 \u00b1 0 . 06 (minimum) to D A = 1 . 80 \u00b1 0 . 01 for strong-field regions ( B m > 900 G), respectively. A fractal dimension of D A = 1 . 41 \u00b1 0 . 05 was found by Janssen et al. (2003), but the value varies as a function of the center-to-limb angle and is different for a speckle-reconstructed image that eliminates seeing and noise. A completely different approach to measure the fractal dimension D was pursued in terms of a 2-D diffusion process, finding fractal diffusion with dimensions in the range of D \u2248 1 . 3 -1 . 8 (Lawrence 1991) or D = 1 . 56 \u00b1 0 . 08 (Lawrence and Schrijver 1993) by measuring the dependence of the mean square displacement of magnetic elements as a function of time. Similar results were found by Balke et al. (1993). The results exclude Euclidean 2-D diffusion but are consistent with percolation theory for diffusion of clusters at a density below the percolation threshold (Lawrence and Schrijver 1993; Balke et al. 1993).", "pages": [8, 9]}, {"title": "4.5. Coronal Flares", "content": "Although this study is focused on the fractal geometry of transition region features observed with IRIS, we compare these results also with coronal values. The fractal dimension of coronal events has been measured for 10 X-class flares, 10 M-class flares, and the Bastille-Day flare (Aschwanden and Aschwanden 2008a, 2008b). Interestingly, these datasets exhibit relatively large values of the fractal dimension, with a mean and standard deviation of D A = 1 . 76 \u00b1 0 . 14. They show a trend that the largest flares, especially X-class flares, exhibit the highest values of D A < \u223c 1 . 8 -1 . 9 (Table 3). If we attribute flare events to the magnetic reconnection process, the observations imply that the flare plasma fills up the flare volume with a high space-filling factor, which is consistent with the chromospheric evaporation process.. Phenomena of smaller magnitude than large flares include microflares, nanoflares, coronal EUV brightenings, etc. Such small-scale variability events are found to have a mean fractal dimension of D A = 1 . 56 \u00b1 0 . 08 (Table 3), which is compatible with those found in M-class flares, but clearly has a lower fractal dimension than large flares, i.e., D A = 1 . 76 \u00b1 0 . 14 (Table 3).", "pages": [9]}, {"title": "4.6. Self-Organization and Criticality", "content": "The generation of magnetic structures that bubble up from the solar convection zone to the solar surface by buoyancy, observed as emerging flux phenomena in form of active regions, sunspots, and pores, can be statistically described as a random process, a self-organization (SO) process, self-organized criticality (SOC), percolation, or a diffusion process. Random processes produce incoherent structures, in contrast to the coherent magnetic flux concentrations observed in sunspots. A self-organization (SO) process needs a driving force and a counter-acting feedback mechanism that produces ordered structures (such as the convective granulation cells; Aschwanden et al. 2018). A SOC process exhibits power law size distributions of avalanche sizes and durations. The finding of a fractal dimension of a power law size distribution in magnetic features alone is not a sufficient condition to prove or rule out any of these processes. Nevertheless, the fractal dimension yields a scaling law between areas ( A \u221d L D 2 ) or volumes ( V \u221d L D 3 ), and length scales L that quantifies scale-free (fractal) processes in form of power laws and can straightforwardly be incorporated in SOC-like models. If we compare the standard SOC parameters measured in observations (Fig. 2) with the theoretically expected values from the standard SOC model (Table 2), we find that the power law slopes for event areas A agree well ( a obs A = 2 . 51 \u00b1 0 . 21) versus a theo A = 2 . 33 (Fig. 2), while the power law slopes for the radiated energy E agree within' the stated uncertainties, ( a obs E = 2 . 03 \u00b1 0 . 18) versus a theo E = 1 . 80 (Fig. 2), but the power law slopes for the time duration T disagree ( a obs T = 2 . 65 \u00b1 0 . 39) versus a theo T = 2 . 00 (Fig. 2). The latter disagreement is possibly caused by the restriction of a constant minimum event lifetime (either 60 s or 110 s) that was assumed in the previous work (Nhalil et al. 2020). The interpretation of these results implies that transition region brightenings have a similar statistics as the SOC model, at least for active regions, nanoflares, and large flares, with a typical fractal dimension of D A \u2248 1 . 5 -1 . 6, but are significantly lower for photospheric granulation and transition region plages ( D A \u2248 1 . 2 -1 . 3), which implies the dominance of sparse quasi-linear flow structures in the photosphere and transition region.", "pages": [9, 10]}, {"title": "5. CONCLUSIONS", "content": "Our aim is to obtain an improved undestanding of fractal dimensions and size distributions observed in the solar photosphere and transition region, which complement previous measurements of coronal phenomena, from nanoflares to the largest solar flares. Building on the previous study 'Power-law energy distributions of small-scale impulsive events on the active Sun: Results from IRIS' , we are using the same IRIS 1400 \u02da A data, extracted with an automated pattern recognition code during 12 time episodes observed in plage and sunspot regions. We obtain a total of 23,633 events, quantified in terms of event areas A , radiative energies E , and event durations T . We obtain the following results: photospheric granulation: 1.21 \u00b1 0.07 transition region plages: 1.29 \u00b1 0.15 transition region sunspots: 1.54 \u00b1 0.16 active region magnetograms: 1.59 \u00b1 0.08 EUV nanoflares: 1.56 \u00b1 0.08 large solar flares: 1.76 \u00b1 0.14 Bastille-Day X5.7-class flare: 1.89 \u00b1 0.05 The analysis presented here demonstrates that we can distinguish between (i) physical processes with sparse curvi-linear flows, as they occur in granulation, meso-granulation, and super-granulation. and (ii) physical processes with near space-filling flows, as they occur in the chromospheric evaporation process in solar flares. IRIS data can therefore be used to diagnose the strength of mass flows in the transition region. Moreover, reliable measurements of the fractal dimension yields realistic plasma filling factors that are important in the estimate of radiative energies and hot plasma emission measures. Future work on fractal dimensions in multi-wavlength datasets from IRIS and AIA/SDO may clarify the dynamics of coronal heating events. Acknowledgements: We acknowledge constructive and stimulating discussions (in alphabetical order) with Sandra Chapman, Paul Charbonneau, Henrik Jeldtoft Jensen, Adam Kowalski, Alexander Milovanov, Leonty Miroshnichenko, Jens Juul Rasmussen, Karel Schrijver, Vadim Uritsky, Loukas Vlahos, and Nick Watkins. This work was partially supported by NASA contract NNX11A099G 'Self-organized criticality in solar physics' and NASA contract NNG04EA00C of the SDO/AIA instrument to LMSAL.", "pages": [10, 11]}, {"title": "References", "content": "Aschwanden, M.J. and Parnell, C.E. 2002, Nanoflare statistics from first principles: fractal geometry and temperature synthesis , Astrophys. J. 572, 1048 Aschwanden, M.J. and Aschwanden P.D. 2008b, Solar flare geometries: II. The volume fractal dimension , Astrophys. J. 574, 544 Aschwanden, M.J. 2011, Self-Organized Criticality in Astrophysics. The Statistics of Nonlinear Processes in the Universe , ISBN 978-3-642-15000-5, Springer-Praxis: New York, 416p. Aschwanden, M.J. 2012, A statistical fractal-diffusive avalanche model of a slowly-driven self-organized criticality system , A&A 539, A2, (15 p) Aschwanden,M.J., Crosby,N., Dimitropoulou,M., Georgoulis,M.K., Hergarten,S., McAteer,J., Milovanov,A., Mineshige,S., Morales,L., Nishizuka,N., Pruessner,G., Sanchez,R., Sharma,S., Strugarek,A., and Uritsky, V. 2016, 25 Years of Self-Organized Criticality: Solar and Astrophysics Space Science Reviews 198, 47-166. Aschwanden, M.J., Scholkmann, F., Bethune, W., Schmutz, W., Abramenko,W., Cheung,M.C.M., Mueller,D., Benz,A.O., Chernov,G., Kritsuk,A.G., Scargle,J.D., Melatos,A., Wagoner,R.V., Trimble,V., Green,W. 2018, Order out of randomness: Self-organization processes in astrophysics , Space Science Reviews 214:55 Conlon, P.A., Gallagher, P.T., McAteer, R.T.J., Ireland, J., Young, C.A., Kestener, P., Hewett, R.J., and Maguire, K. 2008, Multifractal properties of evolving active regions Solar Phys. 248 , 297. De Pontieu, B., Berger, T.E., Schrijver, C.J., and Title, A.M. 1999, Dynamics of transition region 'moss' at high time resolution . Sol.Phys. 190, 419 De Pontieu, B. et al. 2014, Sol.Phys. 289, 2733 Gallagher, P.T., Phillips, K.J.H., Harra-Murnion, L.K., et al. 1998, Properties of the Quiet Sun EUV network , A&A 335, 733 Georgoulis, M.K., Rust, D.M., Bernasconi, P.N., and Schmieder, B. 2002, Statistics, morphology, and energetics of Ellerman bombs , Astrophys. J. 575 , 506. Hathaway, D.H., Beck, J.G., Bogart, R.S., et al. 2000, The photospheric convection spectrum SoPh 193, 299 Lawrence, J.K. 1991, Diffusion of magnetic flux elements on a fractal geometry , Solar Phys. 135 , 249. Lawrence, J.K. and Schrijver, C.J. 1993, Anomalous diffusion of magnetic elements across the solar surface , ApJ 411 , 402. Lu, E.T. and Hamilton, R.J. 1991, Avalanches and the distribution of solar flares , ApJ 380, L89 Mandelbrot, B.B. 1977, The Fractal Geometry of Nature . W.H.Freeman and Company: New York McAteer, R.T.J., Gallagher, P.T., and Ireland, J. 2005, Statistics of Active Region Complexity: A LargeScale Fractal Dimension Survey , ApJ 631, 628 McAteer,R.T.J., Aschwanden,M.J., Dimitropoulou,M., Georgoulis,M.K., Pruessner, G., Morales, L., Ireland, J., and Abramenko,V. 2016, 25 Years of Self-Organized Criticality: Numerical Detection Methods , SSRv 198 217-266. Paniveni, U., Krishan, V., Singh, J., Srikanth, R. 2010, Activity dependence of solar supergranular fractal dimension , MNRAS 402 (1), 424. Pruessner, G. 2012, Self-Organised Criticality. Theory, Models and Characterisation , Cambridge University Press: Cambridge. Rieutord, M., Meunier, N., Roudier, T., et al. 2008, Solar supergranulation revealed by granule tracking , A&A 479, L17 Rathore, B. and Carlsson, M. 2015, ApJ 811, 80. Rathore, B., Carlsson, M., Leenaarts, J., De Pontieu B. 2015, ApJ 811, 81. Rieutord, M., Roudier, T., Rincon, F., 2010, On the power spectrum of solar surface flows , A&A 512, A4 Roudier, T. and Muller, R. 1986, Structure of solar granulation , Solar Phys. 107 , 11. Sharma,A.S., Aschwanden,M.J., Crosby,N.B., Klimas,A.J., Milovanov,A.V., Morales,L., Sanchez,R., and Uritsky,V. 2016, 25 Years of Self-Organized Criticality: Space and Laboratory Plamsas , SSRv 198, 167-216. Uritsky, V.M., Paczuski, M., Davila, J.M., and Jones, S.I. 2007, Coexistence of self-organized criticality and intermittent turbulence in the solar corona , Phys. Rev. Lett. 99(2), id. 025001 Uritsky, V.M., Davila, J.M., Ofman, L., and Coyner, A.J. 2013, Stochastic coupling of solar photosphere and corona , ApJ 769, 62 Watkins,N.W., Pruessner, G., Chapman, S.C., Crosby, N.B., and Jensen, H.J. 25 Years of Self-organized Criticality: Concepts and Controversies , 2016, SSRv 198, 3-44. y-axis (pixels) y-axis (pixels) y-axis (pixels) 120 100 80 60 40 20 0 0 120 100 80 60 40 20 0 0 20 40 60 80 100 120 120 100 80 60 40 20 0 0 20 40 60 80 100 120 120 100 80 60 40 20 0 0 20 40 60 80 100 120 120 100 80 60 40 20 0 0 20 40 60 80 100 120 120 100 80 60 40 20 0 0 20 40 60 80 100 120 120 100 80 60 40 20 0 0 20 40 60 80 100 120 x-axis (pixels), Dataset=11 120 100 80 60 40 20 0 0 20 40 60 80 100 120 120 100 80 60 40 20 0 0 20 40 60 80 100 120 120 100 80 60 40 20 0 0 20 40 60 80 100 120 120 100 80 60 40 20 0 0 20 40 60 80 100 120 x-axis (pixels), Dataset=12 20 40 60 80 100 120", "pages": [12, 13, 14, 19]}] |
2016arXiv160408285N | https://arxiv.org/pdf/1604.08285.pdf | <document>
<figure>
<location><page_1><loc_14><loc_93><loc_29><loc_96></location>
</figure>
<section_header_level_1><location><page_1><loc_14><loc_78><loc_82><loc_88></location>Estimate of the radius responsible for quasinormal modes in the extreme Kerr limit and asymptotic behavior of the Sasaki-Nakamura transformation</section_header_level_1>
<text><location><page_1><loc_14><loc_75><loc_82><loc_76></location>Hiroyuki Nakano 1 , Norichika Sago 2 , Takahiro Tanaka 1 , 3 , and Takashi Nakamura 1</text>
<unordered_list>
<list_item><location><page_1><loc_14><loc_72><loc_62><loc_74></location>1 Department of Physics, Kyoto University, Kyoto 606-8502, Japan</list_item>
<list_item><location><page_1><loc_14><loc_71><loc_68><loc_72></location>2 Faculty of Arts and Science, Kyushu University, Fukuoka 819-0395, Japan</list_item>
<list_item><location><page_1><loc_14><loc_69><loc_75><loc_71></location>3 Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan</list_item>
</unordered_list>
<text><location><page_1><loc_24><loc_66><loc_85><loc_68></location>. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .</text>
<text><location><page_1><loc_24><loc_50><loc_85><loc_66></location>The Sasaki-Nakamura transformation gives a short-ranged potential and a convergent source term for the master equation of perturbations in the Kerr space-time. In this paper, we study the asymptotic behavior of the transformation, and present a new relaxed necessary and sufficient condition for the transformation to obtain the shortranged potential in the assumption that the transformation converges in the far distance. Also, we discuss the peak location of the potential which is responsible for quasinormal mode frequencies in tWKB analysis. Finally, in the extreme Kerr limit, a/M → 1, where M and a denote the mass and spin parameter of a Kerr black hole, respectively, we find the peak location of the potential, r p /M /lessorsimilar 1 + 1 . 8 (1 -a/M ) 1 / 2 by using the new transformation. The uncertainty of the location is as large as that expected from the equivalence principle.</text>
<text><location><page_1><loc_23><loc_49><loc_85><loc_50></location>. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .</text>
<text><location><page_1><loc_24><loc_48><loc_32><loc_49></location>Subject Index</text>
<text><location><page_1><loc_35><loc_48><loc_48><loc_49></location>E31, E02, E01, E38</text>
<section_header_level_1><location><page_1><loc_14><loc_40><loc_29><loc_42></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_14><loc_31><loc_85><loc_40></location>The source of GW150914, a gravitational wave (GW) event observed by advanced LIGO on September 14, 2015 [1], is considered as a merging binary black hole (BBH) and the black hole (BH) masses are estimated as 36 +5 -4 M /circledot and 29 +4 -4 M /circledot , respectively. According to Ref. [2], the BH masses are well predicted by the recent population synthesis results of Population III BBHs [3-5] (see also Ref. [6]).</text>
<text><location><page_1><loc_14><loc_17><loc_85><loc_30></location>The mass and (non-dimensional) spin of the remnant BH were estimated as 62 +4 -4 M /circledot and 0 . 67 +0 . 05 -0 . 07 [7] by using the model derived in Refs. [8, 9], respectively. But, the signal from the ringdown phase of GWs, described by quasinormal modes (QNMs) of a BH, was too weak to test general relativity (see Refs. [10, 11], and also Refs. [12, 13]), and only a consistency check for the least-damped QNM has been done in Ref. [14]. However, since the expected event rate is high [3-5], and the sensitivities of GW observations will improve, there will be a good chance to have an event with much higher signal-to-noise ratio.</text>
<text><location><page_1><loc_14><loc_14><loc_85><loc_17></location>To consider QNMs, we use the BH perturbation approach. The Kerr metric [15] in the Boyer-Lindquist coordinates is written as</text>
<formula><location><page_1><loc_28><loc_8><loc_71><loc_12></location>ds 2 = -( 1 -2 Mr Σ ) dt 2 -4 Mar sin 2 θ Σ dtdφ + Σ ∆ dr 2</formula>
<formula><location><page_2><loc_36><loc_88><loc_86><loc_92></location>+Σ dθ 2 + ( r 2 + a 2 + 2 Ma 2 r Σ sin 2 θ ) sin 2 θdφ 2 , (1)</formula>
<text><location><page_2><loc_15><loc_82><loc_86><loc_87></location>where Σ = r 2 + a 2 cos 2 θ , ∆ = r 2 -2 Mr + a 2 , and M and a denote the mass and the spin parameter of a Kerr BH, respectively. The Kerr space-time is the background to calculate BH perturbations.</text>
<text><location><page_2><loc_15><loc_78><loc_86><loc_81></location>Perturbations are discussed by using the Teukolsky formalism. The radial Teukolsky equation [16] for gravitational perturbations in the Kerr space-time is formally written as</text>
<formula><location><page_2><loc_39><loc_74><loc_86><loc_77></location>∆ 2 d dr 1 ∆ dR dr -V R = -T , (2)</formula>
<text><location><page_2><loc_15><loc_72><loc_56><loc_73></location>where T is the source and the potential V is given by</text>
<formula><location><page_2><loc_36><loc_67><loc_86><loc_71></location>V = -K 2 ∆ -2 iK ∆ ' ∆ +4 iK ' + λ, (3)</formula>
<text><location><page_2><loc_15><loc_65><loc_18><loc_67></location>with</text>
<formula><location><page_2><loc_39><loc_60><loc_86><loc_64></location>K = ( r 2 + a 2 ) ω -am. (4)</formula>
<text><location><page_2><loc_15><loc_56><loc_86><loc_61></location>The constants m and λ in the Teukolsky equation label the spin-weighted spheroidal harmonics Z aω /lscriptm ( θ, φ ). λ is the separation constant which depends on m and aω . A prime denotes the derivative with respect to r .</text>
<text><location><page_2><loc_15><loc_41><loc_86><loc_55></location>There are various modifications of the original Teukolsky equation proposed to improve the behavior of the potential V and the source term T . For example, in Ref. [17] (and related references therein), Chandrasekhar and Detweiler developed various transformations in the 1970s. In Refs. [18, 19], we used the Detweiler potential given in Ref. [20] to study QNM frequencies in WKB analysis [21, 22]. Sasaki and Nakamura [23-25] considered a transformation to remove the divergence in the source term and to obtain the short-ranged potential. This Sasaki-Nakamura transformation has been generalized for various spins in Ref. [26].</text>
<text><location><page_2><loc_16><loc_39><loc_64><loc_40></location>In the WKB analysis, the QNM frequencies are calculated by</text>
<formula><location><page_2><loc_29><loc_31><loc_86><loc_38></location>( ω r + iω i ) 2 = V ( r ∗ 0 ) -i ( n + 1 2 ) √ -2 d 2 V dr ∗ 2 ∣ ∣ ∣ r ∗ = r ∗ 0 , (5)</formula>
<text><location><page_2><loc_15><loc_25><loc_86><loc_35></location>∣ with n = 0 , 1 , 2 , · · · . Here, ω r and ω i are the real and imaginary parts of the frequency, respectively, r ∗ 0 denotes the location where the derivative of the potential d V /dr ∗ = 0 in the tortoise coordinate r ∗ defined by dr ∗ /dr = ( r 2 + a 2 ) / ∆, and we focus only on the n = 0 mode in this paper. It is noted that r ∗ 0 is complex-valued in general.</text>
<text><location><page_2><loc_15><loc_12><loc_86><loc_25></location>In Refs. [18, 19, 27], we have used the potential V with the substitution of accurate numerical results of the complex QNM frequencies [28] 1 obtained by the Leaver's method [29] to determine the above r ∗ 0 of the potential. In practice, the peak location r ∗ p of |V| , a real-valued radius, is also used because we have seen a good agreement between the real part of r ∗ 0 and r ∗ p [18]. Then, we have compared the QNM frequency calculated by Eq. (5) with r ∗ 0 (or r ∗ p ) in the WKB method with that from the numerical result. The difference provides an error estimation used to establish the physical picture that the QNM brings information around the</text>
<text><location><page_3><loc_14><loc_89><loc_85><loc_92></location>peak radius. Here, we implicitly assume that the peak location is relevant to the generation of the QNM if the estimated error is small.</text>
<text><location><page_3><loc_14><loc_79><loc_85><loc_88></location>The analysis of the peak location of the potential in the extreme Kerr limit has been discussed based on a single form of the potential V in Ref. [27]. Here, we also evaluate the uncertainty in the analysis of the peak location that originates from the fact that the GWs cannot be localized due to the equivalence principle, by comparing various forms of the potential.</text>
<text><location><page_3><loc_14><loc_68><loc_85><loc_79></location>This paper is organized as follows. In Sect. 2, we briefly review the Sasaki-Nakamura transformation [23-25] and present and discuss a new transformation introduced in Ref. [27]. In Sect. 3, the peak location of the potential is calculated in the extreme Kerr limit. The peak location is related to the mass and spin of the Kerr BH with expected uncertainties. Section 4 is devoted to discussions. In Appendix A we give a brief summary of WKB analysis for the QNMs. We use the geometric unit system, where G = c = 1 in this paper.</text>
<section_header_level_1><location><page_3><loc_14><loc_64><loc_63><loc_65></location>2. The Sasaki-Nakamura equation and its modification</section_header_level_1>
<text><location><page_3><loc_14><loc_52><loc_85><loc_63></location>The Teukolsky equation in Eq. (2) has undesired features. One is that the source term T diverges as ∝ r 7 / 2 when we consider a test particle falling into a Kerr BH as the source. Also, the potential V in Eq. (3) is a long-ranged one. To remove these undesired features, Sasaki and Nakamura [23-25] considered a change of variable and potential. Since we deal with QNMs in this paper, we focus on the homogeneous version of the Sasaki-Nakamura formalism in the beginning.</text>
<text><location><page_3><loc_14><loc_48><loc_85><loc_52></location>Using two functions α ( r ) and β ( r ) unspecified for the moment, we introduce various variables as</text>
<formula><location><page_3><loc_31><loc_43><loc_85><loc_48></location>X = √ r 2 + a 2 ∆ ( αR + β ∆ R ' ) , (6)</formula>
<formula><location><page_3><loc_32><loc_39><loc_85><loc_43></location>γ = α ( α + β ' ∆ ) -β ∆ ( α ' + β ∆ 2 V ) , (7)</formula>
<formula><location><page_3><loc_31><loc_36><loc_85><loc_39></location>F = ∆ r 2 + a 2 γ ' γ , (8)</formula>
<formula><location><page_3><loc_31><loc_31><loc_85><loc_35></location>U 0 = V + ∆ 2 β [( 2 α + β ' ∆ ) ' -γ ' γ ( α + β ' ∆ )] , (9)</formula>
<formula><location><page_3><loc_31><loc_27><loc_85><loc_31></location>G = -∆ ' r 2 + a 2 + r ∆ ( r 2 + a 2 ) 2 , (10)</formula>
<formula><location><page_3><loc_31><loc_23><loc_85><loc_27></location>U = ∆ U 0 ( r 2 + a 2 ) 2 + G 2 + dG dr ∗ -∆ G r 2 + a 2 γ ' γ . (11)</formula>
<text><location><page_3><loc_14><loc_21><loc_79><loc_22></location>Then, we have a new wave equation for X derived from the Teukolsky equation as</text>
<formula><location><page_3><loc_40><loc_16><loc_85><loc_19></location>d 2 X dr ∗ 2 -F dX dr ∗ -UX = 0 . (12)</formula>
<text><location><page_3><loc_14><loc_14><loc_32><loc_15></location>We specify α and β by</text>
<formula><location><page_3><loc_43><loc_9><loc_85><loc_12></location>α = A -iK ∆ B, (13)</formula>
<formula><location><page_3><loc_43><loc_7><loc_85><loc_9></location>β = ∆ B, (14)</formula>
<text><location><page_4><loc_15><loc_90><loc_19><loc_92></location>where</text>
<text><location><page_4><loc_15><loc_82><loc_18><loc_83></location>with</text>
<formula><location><page_4><loc_40><loc_87><loc_86><loc_89></location>A = 3 iK ' + λ +∆ P , (15)</formula>
<formula><location><page_4><loc_40><loc_84><loc_86><loc_86></location>B = -2 iK +∆ ' +∆ Q, (16)</formula>
<formula><location><page_4><loc_37><loc_76><loc_86><loc_81></location>P = r 2 + a 2 gh (( g r 2 + a 2 ) ' h ) ' , (17)</formula>
<text><location><page_4><loc_15><loc_70><loc_84><loc_73></location>where g and h are free functions. Using a new variable Y defined by X = √ γ Y , we have</text>
<text><location><page_4><loc_15><loc_64><loc_19><loc_65></location>where</text>
<formula><location><page_4><loc_37><loc_72><loc_86><loc_77></location>Q = ( r 2 + a 2 ) 2 g 2 h ( g 2 h ( r 2 + a 2 ) 2 ) ' , (18)</formula>
<formula><location><page_4><loc_39><loc_64><loc_86><loc_69></location>d 2 Y dr ∗ 2 + ( ω 2 -V SN ) Y = 0 , (19)</formula>
<formula><location><page_4><loc_29><loc_58><loc_86><loc_63></location>V SN = ω 2 + U -[ 1 2 d dr ∗ ( 1 γ dγ dr ∗ ) -1 4 γ 2 ( dγ dr ∗ ) 2 ] . (20)</formula>
<text><location><page_4><loc_15><loc_52><loc_86><loc_57></location>In the above Sasaki-Nakamura transformation, there are two free functions, g and h . The restrictions that guarantee a short-ranged potential V SN and a convergent source term have been given by</text>
<formula><location><page_4><loc_39><loc_49><loc_86><loc_51></location>g = const . , h = const . , (21)</formula>
<text><location><page_4><loc_15><loc_45><loc_29><loc_48></location>for r ∗ →-∞ , and</text>
<formula><location><page_4><loc_31><loc_43><loc_86><loc_45></location>g = const . + O ( r -2 ) , h = const . + O ( r -2 ) , (22)</formula>
<text><location><page_4><loc_15><loc_39><loc_39><loc_42></location>for r ∗ → + ∞ . In Refs. [23, 24],</text>
<formula><location><page_4><loc_42><loc_36><loc_86><loc_39></location>h = 1 , g = r 2 + a 2 r 2 , (23)</formula>
<text><location><page_4><loc_15><loc_33><loc_61><loc_35></location>are adopted to satisfy the conditions in Eqs. (21) and (22).</text>
<text><location><page_4><loc_16><loc_31><loc_64><loc_33></location>In Ref. [27], however, we have introduced a new g defined by</text>
<formula><location><page_4><loc_39><loc_27><loc_86><loc_30></location>g = r ( r -a ) ( r + a ) 2 (with h = 1) , (24)</formula>
<text><location><page_4><loc_15><loc_11><loc_86><loc_26></location>which turned out to be suitable to discuss the QNM frequencies in the WKB approximation. The new form of the potential derived from this new g has been plotted in Fig. 1 of Ref. [27] up to a/M = q = 0 . 99999. From the standpoint that we calculate the peak of the potential as the location where the QNM GWs are emitted in the WKB analysis, while we cannot apply this discussion to the original Sasaki-Nakamura or the Detweiler potential (used in Refs. [18, 19]), the new form of the potential with Eq. (24) allows us to discuss the extreme Kerr limit. The choice given in Eq. (24) shares the same feature as the original one (Eq. (23)), in the sense that the Regge-Wheeler potential [30] is recovered for a = 0.</text>
<text><location><page_4><loc_15><loc_7><loc_86><loc_11></location>Although g given in Eq. (24) does not satisfy the condition (22) for r ∗ → + ∞ but behaves as const . + O ( r -1 ), we have obtained a short-ranged potential, which motivates us to revisit</text>
<text><location><page_5><loc_14><loc_87><loc_85><loc_92></location>the asymptotic conditions on g (and h ). To investigate the asymptotic behavior of the potential V SN for r ∗ → + ∞ , we assume that the two free functions are expanded as</text>
<formula><location><page_5><loc_30><loc_84><loc_85><loc_87></location>g = g [0] + g [1] r + g [2] r 2 , h = h [0] + h [1] r + h [2] r 2 , (25)</formula>
<text><location><page_5><loc_69><loc_80><loc_69><loc_82></location>/negationslash</text>
<text><location><page_5><loc_79><loc_80><loc_79><loc_82></location>/negationslash</text>
<text><location><page_5><loc_14><loc_79><loc_85><loc_83></location>where g [ n ] and h [ n ] ( n = 0, 1, 2) are r -independent coefficients, and g [0] = 0 and h [0] = 0. For simplicity, we set M = 1 in the following.</text>
<text><location><page_5><loc_14><loc_72><loc_85><loc_79></location>When h [1] /r does not vanish, γ given in Eq. (7) has O ( r 1 ) terms which become O ( r -1 ) for F in Eq. (8). Then, we have O ( r -1 ) terms in the potential, which indicates that the potential is long-ranged. On the other hand, the term g [1] /r derives O ( r 0 ) in γ defined by Eq. (7), and does not contribute to any O ( r -1 ) term in the potential.</text>
<text><location><page_5><loc_16><loc_70><loc_60><loc_71></location>More precisely, if we choose h [1] = 0 in Eq. (25), we find</text>
<formula><location><page_5><loc_28><loc_65><loc_85><loc_68></location>P = 6 r 2 + 3 G r 3 + O ( r -4 ) , Q = -4 r -G r 2 + O ( r -3 ) , (26)</formula>
<text><location><page_5><loc_14><loc_51><loc_85><loc_64></location>for r → + ∞ , where G = 2 g [1] /g [0] . Although the above asymptotic behavior of P and Q is different from that presented in Eq. (A.4) in Ref. [24] (cf. P = 6 /r 2 +( r -4 ) and Q = -4 /r + O ( r -3 ) in Ref. [24]), A and B in Eqs. (15) and (16) have the same asymptotic behavior as given in Eq. (A.5) of Ref. [24] and γ = const . + O ( r -1 ). This fact guarantees V SN to be short-ranged. Namely, V SN = O ( r -2 ) is achieved under the less restrictive condition, h [1] = 0. It is worth noting that the asymptotic behavior given in Eq. (26) does not depend on the choice of g [0] , g [1] , g [2] , h [0] , or h [2] .</text>
<text><location><page_5><loc_14><loc_47><loc_85><loc_51></location>As a summary, we conclude that the sufficient condition for r ∗ → + ∞ can be relaxed from Eq. (22) to</text>
<formula><location><page_5><loc_31><loc_44><loc_85><loc_46></location>g = const . + O ( r -1 ) , h = const . + O ( r -2 ) . (27)</formula>
<text><location><page_5><loc_14><loc_33><loc_85><loc_42></location>Under the assumption that the two free functions have the forms of Eq. (25) at r ∗ →∞ , we find that h [1] = 0 is also the necessary condition. Although we do not discuss here the inhomogeneous version of the Sasaki-Nakamura formalism, i.e., the source term, in detail, it is easily found that the transformation under the above conditions (27) leads to a wellbehaved source (see, e.g., the dependence of g in Eqs. (2.26), (2.27), and (2.29) of Ref. [24]).</text>
<section_header_level_1><location><page_5><loc_14><loc_29><loc_35><loc_31></location>3. Extreme Kerr limit</section_header_level_1>
<text><location><page_5><loc_14><loc_23><loc_85><loc_28></location>In the previous work [27] for the analysis of the fundamental ( n = 0) QNM with ( /lscript = 2 , m = 2) in the extreme Kerr case, q = a/M → 1, we have derived a fitting curve of the peak location in the Boyer-Lindquist coordinates as</text>
<formula><location><page_5><loc_36><loc_19><loc_85><loc_22></location>r fit M = 1 + 1 . 4803( -ln q ) 0 . 503113 , (28)</formula>
<text><location><page_5><loc_14><loc_13><loc_85><loc_18></location>for the absolute value of the potential | V SN | obtained by using the new g presented in Eq. (24) (called V NNT in Ref. [27]). In the WKB approximation, this peak location is an important output obtained from the observation of the QNM GWs.</text>
<text><location><page_5><loc_16><loc_11><loc_59><loc_12></location>Here, we note that the event horizon radius is given by</text>
<formula><location><page_5><loc_33><loc_5><loc_50><loc_10></location>r + M = 1+ √ 1 -q 2</formula>
<text><location><page_6><loc_15><loc_58><loc_18><loc_60></location>with</text>
<formula><location><page_6><loc_28><loc_54><loc_86><loc_57></location>-2 A 22 = 0 . 545652 + (6 . 02497 + 1 . 38591 i )( -ln q ) 1 / 2 , (32)</formula>
<text><location><page_6><loc_15><loc_50><loc_86><loc_53></location>which is a constant defined in Eq. (25) of Ref. [33]. Also, we have used the approximation for the ( n = 0) QNM frequency with ( /lscript = 2 , m = 2) in the extreme Kerr limit [34],</text>
<formula><location><page_6><loc_38><loc_46><loc_86><loc_49></location>Mω ext = Mq r + -i 4 r + -M r + . (33)</formula>
<text><location><page_6><loc_15><loc_36><loc_86><loc_44></location>Then, defining /epsilon1 by q = 1 -/epsilon1 2 , and expanding the potential V NNT with respect to /epsilon1 , we derive the location r 0 of dV NNT /dr ∗ = 0 instead of finding the peak location r p of | V NNT | . It is noted that the expression given in Eq. (33) can be considered as the exact frequency derived by Leaver's method, since we have discussed the extreme Kerr limit /epsilon1 → 0.</text>
<formula><location><page_6><loc_35><loc_29><loc_86><loc_33></location>r 0 M = 1 + (1 . 44905 -0 . 020157 i ) /epsilon1 , (34)</formula>
<text><location><page_6><loc_15><loc_32><loc_86><loc_37></location>In Appendix A of Ref. [18], we have found a good agreement between the peak location of | V SN | and the real part of the location of dV SN /dr ∗ = 0. The result was obtained as</text>
<text><location><page_6><loc_15><loc_19><loc_86><loc_28></location>where the appearance of the O ( /epsilon1 1 ) term is consistent with the expression for r fit given in Eq. (28) because ( -ln q ) 1 / 2 = (1 -q ) 1 / 2 + O ((1 -q ) 3 / 2 ). Although it is consistent that both expressions, r fit and r 0 have a correction of O ( /epsilon1 1 ), a different choice of g from Eq. (24) makes a difference in the coefficient of O ( /epsilon1 1 ) in the estimation of r p (and r 0 ). In this section we study how robust the above estimation of the peak location is.</text>
<text><location><page_6><loc_16><loc_17><loc_46><loc_19></location>We expand the event horizon radius as</text>
<formula><location><page_6><loc_40><loc_13><loc_86><loc_17></location>r + M = 1 + √ 2 /epsilon1 + O ( /epsilon1 3 ) , (35)</formula>
<text><location><page_6><loc_15><loc_11><loc_59><loc_12></location>and the Boyer-Lindquist radial coordinate around r + as</text>
<formula><location><page_6><loc_42><loc_7><loc_54><loc_10></location>r M = r + M + ξ/epsilon1</formula>
<formula><location><page_6><loc_37><loc_89><loc_86><loc_93></location>= 1+ √ 2(1 -q ) 1 / 2 + O ((1 -q ) 3 / 2 ) , (29)</formula>
<text><location><page_6><loc_15><loc_87><loc_52><loc_88></location>and the inner light ring radius [31] is written as</text>
<formula><location><page_6><loc_35><loc_78><loc_86><loc_86></location>r lr M = 2+2cos [ 2 3 cos -1 ( -q ) ] = 1+ 2 √ 2 √ 3 (1 -q ) 1 / 2 + O (1 -q ) . (30)</formula>
<text><location><page_6><loc_15><loc_67><loc_86><loc_77></location>The latter radius is evaluated in the equatorial ( θ = π/ 2) plane. Although there are various studies on the relation between the QNMs and the orbital frequency of the light ring orbit (see a useful lecture note [32]), the peak location of the potential r fit which derives the QNM frequencies, is much closer to the horizon radius, r + /M ≈ 1 + 1 . 414(1 -q ) 1 / 2 than the inner light ring radius, r lr /M ≈ 1 + 1 . 633(1 -q ) 1 / 2 .</text>
<text><location><page_6><loc_15><loc_64><loc_86><loc_68></location>In Ref. [27], to check the validity of r fit , we have evaluated the peak location (denoted by r p in the Boyer-Lindquist coordinates) semi-analytically by using a fitting formula for</text>
<formula><location><page_6><loc_38><loc_60><loc_86><loc_63></location>λ = s A /lscriptm -2 amω + a 2 ω 2 , (31)</formula>
<formula><location><page_7><loc_45><loc_90><loc_85><loc_93></location>= 1+ √ 2 /epsilon1 + ξ/epsilon1 , (36)</formula>
<text><location><page_7><loc_14><loc_85><loc_85><loc_89></location>introducing a rescaled radial coordinate ξ whose origin corresponds to the event horizon. The tortoise coordinate is expressed as</text>
<formula><location><page_7><loc_35><loc_80><loc_85><loc_85></location>r ∗ M = -√ 2 2 /epsilon1 ln ( 1 + 2 √ 2 ξ ) + O ( /epsilon1 0 ) . (37)</formula>
<text><location><page_7><loc_14><loc_76><loc_85><loc_81></location>In the following analysis, we investigate the peak location r p /M = 1 + √ 2 /epsilon1 + ξ p /epsilon1 , keeping only the leading order with respect to /epsilon1 for ξ p . The QNM frequency in Eq. (33) is written as</text>
<formula><location><page_7><loc_31><loc_71><loc_85><loc_75></location>Mω ext = 1 + ( -√ 2 -i 4 √ 2 ) /epsilon1 + ( 1 + i 2 ) /epsilon1 2 , (38)</formula>
<text><location><page_7><loc_14><loc_69><loc_24><loc_71></location>up to O ( /epsilon1 2 ).</text>
<text><location><page_7><loc_16><loc_67><loc_56><loc_69></location>The function g in Eq. (24) is expanded for /epsilon1 /lessmuch 1 as</text>
<formula><location><page_7><loc_37><loc_62><loc_85><loc_67></location>g = ( 1 4 √ 2 + 1 4 ξ ) /epsilon1 + O ( /epsilon1 2 ) . (39)</formula>
<text><location><page_7><loc_14><loc_57><loc_85><loc_62></location>In this expansion, the terms of O ( ξ 2 ) appear only at O ( /epsilon1 2 ). We focus on the leading-order modification of O ( /epsilon1 1 ), and consider a function linear in ξ . Such a function is parametrized by two real parameters µ and ν as</text>
<formula><location><page_7><loc_37><loc_51><loc_85><loc_57></location>g = /epsilon1 + √ 2 2 ( 1 2 + µ + iν ) ξ/epsilon1 . (40)</formula>
<text><location><page_7><loc_14><loc_44><loc_85><loc_51></location>The function in Eq. (39) is recovered when µ = 1 / 2 and ν = 0, except for the overall normalization of g which does not contribute to the potential because of the dependence of g in P and Q given by Eqs. (17) and (18), respectively. In the series expansion with respect to /epsilon1 , the potential in the Sasaki-Nakamura equation (see Eq. (20)) is formally written as</text>
<formula><location><page_7><loc_33><loc_39><loc_85><loc_44></location>V = 1 + ( -2 √ 2 -√ 2 2 i ) /epsilon1 + v (2) µ,ν ( ξ ) /epsilon1 2 , (41)</formula>
<text><location><page_7><loc_14><loc_33><loc_85><loc_39></location>where we do not explicitly present the huge expression of v (2) µ,ν ( ξ ). It is noted that any O ( /epsilon1 2 ) term in g of Eq. (40) does not contribute to the potential in the second order with respect to /epsilon1 .</text>
<text><location><page_7><loc_16><loc_31><loc_70><loc_32></location>Here, we define the error in the estimation of the QNM frequencies as</text>
<formula><location><page_7><loc_34><loc_15><loc_85><loc_30></location>Err = √ (Err r ) 2 +(Err i ) 2 ; Err r = ∣ ∣ ∣ ∣ Re( ω WKB ) -(1 -√ 2 /epsilon1 ) Re( ω ext ) -(1 -√ 2 /epsilon1 ) -1 ∣ ∣ ∣ ∣ ≈ ∣ ∣ ∣ ∣ Re( ω WKB ) -(1 -√ 2 /epsilon1 ) /epsilon1 2 -1 ∣ ∣ ∣ ∣ , Err i ≈ ∣ ∣ ∣ Im( ω WKB ) -√ 2 /epsilon1/ 4 -1 ∣ ∣ ∣ , (42)</formula>
<text><location><page_7><loc_14><loc_6><loc_85><loc_18></location>∣ ∣ where we have calculated Re( ω ext ) by using Eq. (38), and used the leading order of Im( ω ext ) obtained from Eq. (38) for Err i . Since the real part of the error, Err r is always tiny in the case of small /epsilon1 if we use | Re( ω WKB ) / Re( ω ext ) -1 | or | (Re( ω WKB ) -1) / (Re( ω ext ) -1) -1 | , we have adopted the above estimator to normalize the error of the real part. Note that this estimator is independent of /epsilon1 in the limit /epsilon1 → 0.</text>
<figure>
<location><page_8><loc_15><loc_64><loc_85><loc_92></location>
<caption>Fig. 1 Scatter plots for the error estimation given by Eq. (42) with respect to ξ p , obtained by varying the parameters µ and ν in Eq. (40). The left panel shows the case with fixed µ = 1 / 2, and the black dot denotes µ = 1 / 2 and ν = 0. In the right panel, µ is also variable. We do not find any point with a small error for ξ p > 0 . 4. The empty region around 0 < ξ p < 0 . 1 and 1 . 5 < Log 10 (Err) < 2 . 0 will be filled by the points if we use a much finer grid for the parameters µ and ν .</caption>
</figure>
<figure>
<location><page_8><loc_15><loc_21><loc_85><loc_48></location>
<caption>Fig. 2 Scatter plots for Err r (left) and Err i (right) given by Eq. (42) with respect to ξ p , obtained by varying the parameters µ and ν in Eq. (40).</caption>
</figure>
<text><location><page_8><loc_15><loc_7><loc_86><loc_13></location>Varying the parameters µ and ν , we obtain Fig. 1, which shows the error in the estimation of the QNM frequencies calculated by Eq. (42) with respect to ξ p Figure 2 shows Err r (the left panel) and Err i (the right panel), respectively. Err is dominated by Err r for large ξ p .</text>
<text><location><page_9><loc_14><loc_83><loc_85><loc_92></location>We find from Fig. 1 that the region where Err given in Eq. (42) is small spreads widely. Although the minimum error of Err ≈ 1 . 2% is obtained for µ ≈ -0 . 21 and ν ≈ 0 . 25 in the analysis, there are many other combinations of µ and ν for which Err remains small, and the region with small Err extends to a range of 0 < ξ p < 0 . 4. Therefore, we should consider that the peak of the potential is located in 0 < ξ p < 0 . 4.</text>
<text><location><page_9><loc_14><loc_70><loc_85><loc_82></location>In Refs. [18, 19, 27], we have used the WKB analysis to claim how deeply we can actually inspect the region close to the event horizon of a BH by observing the QNM GWs. Since the GWs cannot be localized and the QNMs are determined not only by the potential at the peak radius but also by the curvature of the potential, what we can claim is that the QNM frequency is determined by the information 'around' the peak of the potential. Therefore, it is necessary to properly take into account this fact in the interpretation of the estimated radius obtained in Refs. [18, 19, 27].</text>
<text><location><page_9><loc_14><loc_60><loc_85><loc_69></location>The uncertainty in the peak location can be discussed in the following manner. Here, we use r ∗ 0 instead of r ∗ p because r ∗ 0 is derived easily in the analytical calculation. Expanding Eq. (41) with respect to ∆ r ∗ = r ∗ -r ∗ 0 around r ∗ 0 , and using the QNM frequency in the WKBapproximation of Eq. (5), we have the radial wavenumber [which corresponds to W 1 / 2 in Eq. (A2)], as</text>
<formula><location><page_9><loc_23><loc_48><loc_85><loc_59></location>k ( r ∗ 0 +∆ r ∗ ) = √ ω 2 QNM -V = √ √ √ √ -i 2 √ -2 d 2 V dr ∗ 2 ∣ ∣ ∣ ∣ r ∗ = r ∗ 0 -1 2 d 2 V dr ∗ 2 ∣ ∣ ∣ ∣ r ∗ = r ∗ 0 (∆ r ∗ ) 2 + · · · , (43)</formula>
<text><location><page_9><loc_14><loc_42><loc_85><loc_50></location>where ( · · · ) denotes the terms of higher order in the WKB approximation or of O ((∆ r ∗ ) 3 ). We note that d 2 V/dr ∗ 2 | r ∗ = r ∗ 0 = O ( /epsilon1 4 ) because d/dr ∗ = O ( /epsilon1 ) × d/dξ and the ξ dependence of V is as given in Eq. (41). If we expect that the uncertainty of the peak location is given by the inverse of the wavenumber, it may be estimated by the ∆ r ∗ that solves</text>
<formula><location><page_9><loc_39><loc_39><loc_85><loc_41></location>k ( r ∗ 0 +∆ r ∗ )∆ r ∗ = O (1) . (44)</formula>
<text><location><page_9><loc_14><loc_36><loc_46><loc_37></location>Combining Eqs. (43) and (44), we derive</text>
<formula><location><page_9><loc_42><loc_31><loc_85><loc_35></location>| ∆ r ∗ | M = O ( /epsilon1 -1 ) , (45)</formula>
<text><location><page_9><loc_14><loc_29><loc_52><loc_30></location>which is translated into the uncertainty in ξ 0 as</text>
<formula><location><page_9><loc_43><loc_25><loc_85><loc_27></location>| ∆ ξ 0 | = O (1) , (46)</formula>
<text><location><page_9><loc_14><loc_21><loc_85><loc_24></location>by using Eq. (37). This estimate is consistent with the extension of the region where Err given in Eq. (42) is small in Fig. 1.</text>
<text><location><page_9><loc_14><loc_7><loc_85><loc_20></location>In our previous work [27], we used only one potential which corresponds to µ = 1 / 2 and ν = 0 in Eq. (40). The peak location was inside the light ring radius as shown in Eqs. (28) and (34), and we concluded that the QNM GWs were emitted 'around' the peak location. However, the meaning of the word 'around' was not clear, and Fig. 1 gives a clear explanation of it based on the error estimation of the WKB frequencies compared with the exact QNM frequencies. Using various potentials with the parameters, µ and ν , even if we change the threshold of the error estimator (42) from a few % to 10%, the extension of the</text>
<text><location><page_10><loc_15><loc_89><loc_86><loc_92></location>region does not change much from ξ p ∼ 0 . 4. Therefore, we conclude that the estimated peak location is restricted to</text>
<formula><location><page_10><loc_39><loc_85><loc_86><loc_88></location>r p M /lessorsimilar 1 + 1 . 8(1 -q ) 1 / 2 . (47)</formula>
<text><location><page_10><loc_15><loc_78><loc_86><loc_84></location>The above result confirms that we can see the space-time sufficiently inside the ergoregion ( r ergo = 2 M for the equatorial radius of the ergosurface) and around the inner light ring r lr /M ≈ 1 + 1 . 633(1 -q ) 1 / 2 .</text>
<section_header_level_1><location><page_10><loc_15><loc_76><loc_28><loc_77></location>4. Discussions</section_header_level_1>
<text><location><page_10><loc_15><loc_64><loc_86><loc_75></location>In the modification of the Sasaki-Nakamura equation, we have found that the necessary and sufficient condition for the fast fall-off at r ∗ → + ∞ can be relaxed to the one given by Eq. (27), if we assume that the free functions g and h can be expanded in a power series of 1 /r . When we use g in Eq. (24) which satisfies Eq. (27), the potential is suitable for the WKB analysis of QNMs (see Ref. [27]), but the general expression of the potential is much more complicated than that for g in Eq. (23).</text>
<text><location><page_10><loc_15><loc_60><loc_86><loc_63></location>One way to obtain a simple potential will be to keep γ constant in the Sasaki-Nakamura transformation. γ in Eq. (7) is rewritten as</text>
<formula><location><page_10><loc_23><loc_55><loc_86><loc_59></location>γ = A 2 ( 1 -( λ +3 iK ' ) ∆ ( B A ) 2 -i (2 K + i ∆ ' ) ∆ ( B A ) + ( B A ) ' ) . (48)</formula>
<text><location><page_10><loc_15><loc_47><loc_86><loc_54></location>Thus, it is not difficult to find A and B so that γ is constant because we may choose B/A which leads 1 /A 2 for the expression in the parenthesis of the above equation. The difficult part arises from the condition for A and B that gives a short-ranged potential. To derive such A and B is one of our future studies.</text>
<text><location><page_10><loc_15><loc_41><loc_86><loc_47></location>In the study of QNMs in the WKB method, we have evaluated the uncertainty of the peak location of the potential in the extreme Kerr limit. This uncertainty is expressed as Eq. (47), and is consistent with that expected from the equivalence principle.</text>
<text><location><page_10><loc_48><loc_37><loc_48><loc_39></location>/negationslash</text>
<text><location><page_10><loc_15><loc_32><loc_86><loc_41></location>Here, we should note that the imaginary parts of the QNM frequencies become zero in the extreme Kerr limit, and many overtones ( n = 0) accumulate at one frequency (see, e.g., Fig. 3 in Ref. [35] and a recent work [36]). Therefore, observing QNM GWs in the near-extremal Kerr case will be very different from the other case, and further studies are required to extract the information from extreme Kerr BHs.</text>
<text><location><page_10><loc_15><loc_17><loc_86><loc_32></location>Finally, thanks to the recent GW observation, GW150914, we have entered the next stage of using GWs to extract new physics. To test the strong gravitational field around BHs, the QNMs are simple and useful, and the QNM GWs are the target not only for the second-generation GW detectors such as Advanced LIGO (aLIGO) [37], Advanced Virgo (AdV) [38], and KAGRA [39, 40], but also for space-based GW detectors such as eLISA [41] and DECIGO [42]. The enhancement of the signal-to-noise ratio by the third-generation detectors such as the Einstein Telescope (ET) [43] will significantly improve the precision of the test of general relativity.</text>
<section_header_level_1><location><page_10><loc_15><loc_13><loc_30><loc_15></location>Acknowledgments</section_header_level_1>
<text><location><page_10><loc_15><loc_7><loc_86><loc_13></location>This work was supported by MEXT Grant-in-Aid for Scientific Research on Innovative Areas, 'New Developments in Astrophysics Through Multi-Messenger Observations of Gravitational Wave Sources,' Nos. 24103001 and 24103006 (HN, TT, TN), JSPS Grant-in-Aid</text>
<text><location><page_11><loc_14><loc_87><loc_85><loc_92></location>for Scientific Research (C), No. 16K05347 (HN), JSPS Grant-in-Aid for Young Scientists (B), No. 25800154 (NS), and Grant-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan No. 15H02087 (TT, TN).</text>
<section_header_level_1><location><page_11><loc_14><loc_83><loc_59><loc_84></location>A. Quasinormal mode in the WKB approximation</section_header_level_1>
<text><location><page_11><loc_14><loc_81><loc_42><loc_82></location>We schematically write Eq. (19) as</text>
<formula><location><page_11><loc_40><loc_76><loc_85><loc_80></location>d 2 ψ dr ∗ 2 + W ( r ∗ ) ψ = 0 , (A1)</formula>
<text><location><page_11><loc_14><loc_70><loc_85><loc_76></location>where W = ω 2 -V SN . In Ref. [22], the QNM frequencies are discussed as a 'second-order turning point' problem in the WKB approximation (see also Ref. [44]). We prepare two WKB solutions,</text>
<formula><location><page_11><loc_29><loc_61><loc_85><loc_69></location>ψ WKB 1 ≈ [ W ( r ∗ )] -1 / 4 exp ( ± i ∫ r ∗ r ∗ 2 [ W ( x )] 1 / 2 dx ) , ψ WKB 2 ≈ [ W ( r ∗ )] -1 / 4 exp ( ± i ∫ r ∗ 1 r ∗ [ W ( x )] 1 / 2 dx ) , (A2)</formula>
<text><location><page_11><loc_14><loc_54><loc_85><loc_61></location>where r ∗ 1 and r ∗ 2 are the turning points, and also parabolic cylinder functions for r ∗ 1 < r ∗ < r ∗ 2 (see Eq. (5) of Ref. [22]). The QNM frequencies are derived in the matching condition for the outgoing (from the peak location of the potential) solutions of ψ WKB 1 and ψ WKB 2 . This means that we choose the signs in Eq. (A2) appropriately.</text>
<text><location><page_11><loc_16><loc_52><loc_57><loc_54></location>As a usual picture, the peak location is calculated by</text>
<formula><location><page_11><loc_43><loc_48><loc_85><loc_51></location>dW ( r ∗ ) dr ∗ = 0 , (A3)</formula>
<text><location><page_11><loc_14><loc_44><loc_85><loc_47></location>the solution is denoted by r ∗ 0 , and r ∗ 1 < r ∗ 0 < r ∗ 2 in the case of a real potential. We extend this to a complex potential. Therefore, r ∗ 0 is in the complex plane.</text>
<section_header_level_1><location><page_11><loc_14><loc_41><loc_24><loc_42></location>References</section_header_level_1>
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</document> | [{"title": "Estimate of the radius responsible for quasinormal modes in the extreme Kerr limit and asymptotic behavior of the Sasaki-Nakamura transformation", "content": "Hiroyuki Nakano 1 , Norichika Sago 2 , Takahiro Tanaka 1 , 3 , and Takashi Nakamura 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Sasaki-Nakamura transformation gives a short-ranged potential and a convergent source term for the master equation of perturbations in the Kerr space-time. In this paper, we study the asymptotic behavior of the transformation, and present a new relaxed necessary and sufficient condition for the transformation to obtain the shortranged potential in the assumption that the transformation converges in the far distance. Also, we discuss the peak location of the potential which is responsible for quasinormal mode frequencies in tWKB analysis. Finally, in the extreme Kerr limit, a/M \u2192 1, where M and a denote the mass and spin parameter of a Kerr black hole, respectively, we find the peak location of the potential, r p /M /lessorsimilar 1 + 1 . 8 (1 -a/M ) 1 / 2 by using the new transformation. The uncertainty of the location is as large as that expected from the equivalence principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subject Index E31, E02, E01, E38", "pages": [1]}, {"title": "1. Introduction", "content": "The source of GW150914, a gravitational wave (GW) event observed by advanced LIGO on September 14, 2015 [1], is considered as a merging binary black hole (BBH) and the black hole (BH) masses are estimated as 36 +5 -4 M /circledot and 29 +4 -4 M /circledot , respectively. According to Ref. [2], the BH masses are well predicted by the recent population synthesis results of Population III BBHs [3-5] (see also Ref. [6]). The mass and (non-dimensional) spin of the remnant BH were estimated as 62 +4 -4 M /circledot and 0 . 67 +0 . 05 -0 . 07 [7] by using the model derived in Refs. [8, 9], respectively. But, the signal from the ringdown phase of GWs, described by quasinormal modes (QNMs) of a BH, was too weak to test general relativity (see Refs. [10, 11], and also Refs. [12, 13]), and only a consistency check for the least-damped QNM has been done in Ref. [14]. However, since the expected event rate is high [3-5], and the sensitivities of GW observations will improve, there will be a good chance to have an event with much higher signal-to-noise ratio. To consider QNMs, we use the BH perturbation approach. The Kerr metric [15] in the Boyer-Lindquist coordinates is written as where \u03a3 = r 2 + a 2 cos 2 \u03b8 , \u2206 = r 2 -2 Mr + a 2 , and M and a denote the mass and the spin parameter of a Kerr BH, respectively. The Kerr space-time is the background to calculate BH perturbations. Perturbations are discussed by using the Teukolsky formalism. The radial Teukolsky equation [16] for gravitational perturbations in the Kerr space-time is formally written as where T is the source and the potential V is given by with The constants m and \u03bb in the Teukolsky equation label the spin-weighted spheroidal harmonics Z a\u03c9 /lscriptm ( \u03b8, \u03c6 ). \u03bb is the separation constant which depends on m and a\u03c9 . A prime denotes the derivative with respect to r . There are various modifications of the original Teukolsky equation proposed to improve the behavior of the potential V and the source term T . For example, in Ref. [17] (and related references therein), Chandrasekhar and Detweiler developed various transformations in the 1970s. In Refs. [18, 19], we used the Detweiler potential given in Ref. [20] to study QNM frequencies in WKB analysis [21, 22]. Sasaki and Nakamura [23-25] considered a transformation to remove the divergence in the source term and to obtain the short-ranged potential. This Sasaki-Nakamura transformation has been generalized for various spins in Ref. [26]. In the WKB analysis, the QNM frequencies are calculated by \u2223 with n = 0 , 1 , 2 , \u00b7 \u00b7 \u00b7 . Here, \u03c9 r and \u03c9 i are the real and imaginary parts of the frequency, respectively, r \u2217 0 denotes the location where the derivative of the potential d V /dr \u2217 = 0 in the tortoise coordinate r \u2217 defined by dr \u2217 /dr = ( r 2 + a 2 ) / \u2206, and we focus only on the n = 0 mode in this paper. It is noted that r \u2217 0 is complex-valued in general. In Refs. [18, 19, 27], we have used the potential V with the substitution of accurate numerical results of the complex QNM frequencies [28] 1 obtained by the Leaver's method [29] to determine the above r \u2217 0 of the potential. In practice, the peak location r \u2217 p of |V| , a real-valued radius, is also used because we have seen a good agreement between the real part of r \u2217 0 and r \u2217 p [18]. Then, we have compared the QNM frequency calculated by Eq. (5) with r \u2217 0 (or r \u2217 p ) in the WKB method with that from the numerical result. The difference provides an error estimation used to establish the physical picture that the QNM brings information around the peak radius. Here, we implicitly assume that the peak location is relevant to the generation of the QNM if the estimated error is small. The analysis of the peak location of the potential in the extreme Kerr limit has been discussed based on a single form of the potential V in Ref. [27]. Here, we also evaluate the uncertainty in the analysis of the peak location that originates from the fact that the GWs cannot be localized due to the equivalence principle, by comparing various forms of the potential. This paper is organized as follows. In Sect. 2, we briefly review the Sasaki-Nakamura transformation [23-25] and present and discuss a new transformation introduced in Ref. [27]. In Sect. 3, the peak location of the potential is calculated in the extreme Kerr limit. The peak location is related to the mass and spin of the Kerr BH with expected uncertainties. Section 4 is devoted to discussions. In Appendix A we give a brief summary of WKB analysis for the QNMs. We use the geometric unit system, where G = c = 1 in this paper.", "pages": [1, 2, 3]}, {"title": "2. The Sasaki-Nakamura equation and its modification", "content": "The Teukolsky equation in Eq. (2) has undesired features. One is that the source term T diverges as \u221d r 7 / 2 when we consider a test particle falling into a Kerr BH as the source. Also, the potential V in Eq. (3) is a long-ranged one. To remove these undesired features, Sasaki and Nakamura [23-25] considered a change of variable and potential. Since we deal with QNMs in this paper, we focus on the homogeneous version of the Sasaki-Nakamura formalism in the beginning. Using two functions \u03b1 ( r ) and \u03b2 ( r ) unspecified for the moment, we introduce various variables as Then, we have a new wave equation for X derived from the Teukolsky equation as We specify \u03b1 and \u03b2 by where with where g and h are free functions. Using a new variable Y defined by X = \u221a \u03b3 Y , we have where In the above Sasaki-Nakamura transformation, there are two free functions, g and h . The restrictions that guarantee a short-ranged potential V SN and a convergent source term have been given by for r \u2217 \u2192-\u221e , and for r \u2217 \u2192 + \u221e . In Refs. [23, 24], are adopted to satisfy the conditions in Eqs. (21) and (22). In Ref. [27], however, we have introduced a new g defined by which turned out to be suitable to discuss the QNM frequencies in the WKB approximation. The new form of the potential derived from this new g has been plotted in Fig. 1 of Ref. [27] up to a/M = q = 0 . 99999. From the standpoint that we calculate the peak of the potential as the location where the QNM GWs are emitted in the WKB analysis, while we cannot apply this discussion to the original Sasaki-Nakamura or the Detweiler potential (used in Refs. [18, 19]), the new form of the potential with Eq. (24) allows us to discuss the extreme Kerr limit. The choice given in Eq. (24) shares the same feature as the original one (Eq. (23)), in the sense that the Regge-Wheeler potential [30] is recovered for a = 0. Although g given in Eq. (24) does not satisfy the condition (22) for r \u2217 \u2192 + \u221e but behaves as const . + O ( r -1 ), we have obtained a short-ranged potential, which motivates us to revisit the asymptotic conditions on g (and h ). To investigate the asymptotic behavior of the potential V SN for r \u2217 \u2192 + \u221e , we assume that the two free functions are expanded as /negationslash /negationslash where g [ n ] and h [ n ] ( n = 0, 1, 2) are r -independent coefficients, and g [0] = 0 and h [0] = 0. For simplicity, we set M = 1 in the following. When h [1] /r does not vanish, \u03b3 given in Eq. (7) has O ( r 1 ) terms which become O ( r -1 ) for F in Eq. (8). Then, we have O ( r -1 ) terms in the potential, which indicates that the potential is long-ranged. On the other hand, the term g [1] /r derives O ( r 0 ) in \u03b3 defined by Eq. (7), and does not contribute to any O ( r -1 ) term in the potential. More precisely, if we choose h [1] = 0 in Eq. (25), we find for r \u2192 + \u221e , where G = 2 g [1] /g [0] . Although the above asymptotic behavior of P and Q is different from that presented in Eq. (A.4) in Ref. [24] (cf. P = 6 /r 2 +( r -4 ) and Q = -4 /r + O ( r -3 ) in Ref. [24]), A and B in Eqs. (15) and (16) have the same asymptotic behavior as given in Eq. (A.5) of Ref. [24] and \u03b3 = const . + O ( r -1 ). This fact guarantees V SN to be short-ranged. Namely, V SN = O ( r -2 ) is achieved under the less restrictive condition, h [1] = 0. It is worth noting that the asymptotic behavior given in Eq. (26) does not depend on the choice of g [0] , g [1] , g [2] , h [0] , or h [2] . As a summary, we conclude that the sufficient condition for r \u2217 \u2192 + \u221e can be relaxed from Eq. (22) to Under the assumption that the two free functions have the forms of Eq. (25) at r \u2217 \u2192\u221e , we find that h [1] = 0 is also the necessary condition. Although we do not discuss here the inhomogeneous version of the Sasaki-Nakamura formalism, i.e., the source term, in detail, it is easily found that the transformation under the above conditions (27) leads to a wellbehaved source (see, e.g., the dependence of g in Eqs. (2.26), (2.27), and (2.29) of Ref. [24]).", "pages": [3, 4, 5]}, {"title": "3. Extreme Kerr limit", "content": "In the previous work [27] for the analysis of the fundamental ( n = 0) QNM with ( /lscript = 2 , m = 2) in the extreme Kerr case, q = a/M \u2192 1, we have derived a fitting curve of the peak location in the Boyer-Lindquist coordinates as for the absolute value of the potential | V SN | obtained by using the new g presented in Eq. (24) (called V NNT in Ref. [27]). In the WKB approximation, this peak location is an important output obtained from the observation of the QNM GWs. Here, we note that the event horizon radius is given by with which is a constant defined in Eq. (25) of Ref. [33]. Also, we have used the approximation for the ( n = 0) QNM frequency with ( /lscript = 2 , m = 2) in the extreme Kerr limit [34], Then, defining /epsilon1 by q = 1 -/epsilon1 2 , and expanding the potential V NNT with respect to /epsilon1 , we derive the location r 0 of dV NNT /dr \u2217 = 0 instead of finding the peak location r p of | V NNT | . It is noted that the expression given in Eq. (33) can be considered as the exact frequency derived by Leaver's method, since we have discussed the extreme Kerr limit /epsilon1 \u2192 0. In Appendix A of Ref. [18], we have found a good agreement between the peak location of | V SN | and the real part of the location of dV SN /dr \u2217 = 0. The result was obtained as where the appearance of the O ( /epsilon1 1 ) term is consistent with the expression for r fit given in Eq. (28) because ( -ln q ) 1 / 2 = (1 -q ) 1 / 2 + O ((1 -q ) 3 / 2 ). Although it is consistent that both expressions, r fit and r 0 have a correction of O ( /epsilon1 1 ), a different choice of g from Eq. (24) makes a difference in the coefficient of O ( /epsilon1 1 ) in the estimation of r p (and r 0 ). In this section we study how robust the above estimation of the peak location is. We expand the event horizon radius as and the Boyer-Lindquist radial coordinate around r + as and the inner light ring radius [31] is written as The latter radius is evaluated in the equatorial ( \u03b8 = \u03c0/ 2) plane. Although there are various studies on the relation between the QNMs and the orbital frequency of the light ring orbit (see a useful lecture note [32]), the peak location of the potential r fit which derives the QNM frequencies, is much closer to the horizon radius, r + /M \u2248 1 + 1 . 414(1 -q ) 1 / 2 than the inner light ring radius, r lr /M \u2248 1 + 1 . 633(1 -q ) 1 / 2 . In Ref. [27], to check the validity of r fit , we have evaluated the peak location (denoted by r p in the Boyer-Lindquist coordinates) semi-analytically by using a fitting formula for introducing a rescaled radial coordinate \u03be whose origin corresponds to the event horizon. The tortoise coordinate is expressed as In the following analysis, we investigate the peak location r p /M = 1 + \u221a 2 /epsilon1 + \u03be p /epsilon1 , keeping only the leading order with respect to /epsilon1 for \u03be p . The QNM frequency in Eq. (33) is written as up to O ( /epsilon1 2 ). The function g in Eq. (24) is expanded for /epsilon1 /lessmuch 1 as In this expansion, the terms of O ( \u03be 2 ) appear only at O ( /epsilon1 2 ). We focus on the leading-order modification of O ( /epsilon1 1 ), and consider a function linear in \u03be . Such a function is parametrized by two real parameters \u00b5 and \u03bd as The function in Eq. (39) is recovered when \u00b5 = 1 / 2 and \u03bd = 0, except for the overall normalization of g which does not contribute to the potential because of the dependence of g in P and Q given by Eqs. (17) and (18), respectively. In the series expansion with respect to /epsilon1 , the potential in the Sasaki-Nakamura equation (see Eq. (20)) is formally written as where we do not explicitly present the huge expression of v (2) \u00b5,\u03bd ( \u03be ). It is noted that any O ( /epsilon1 2 ) term in g of Eq. (40) does not contribute to the potential in the second order with respect to /epsilon1 . Here, we define the error in the estimation of the QNM frequencies as \u2223 \u2223 where we have calculated Re( \u03c9 ext ) by using Eq. (38), and used the leading order of Im( \u03c9 ext ) obtained from Eq. (38) for Err i . Since the real part of the error, Err r is always tiny in the case of small /epsilon1 if we use | Re( \u03c9 WKB ) / Re( \u03c9 ext ) -1 | or | (Re( \u03c9 WKB ) -1) / (Re( \u03c9 ext ) -1) -1 | , we have adopted the above estimator to normalize the error of the real part. Note that this estimator is independent of /epsilon1 in the limit /epsilon1 \u2192 0. Varying the parameters \u00b5 and \u03bd , we obtain Fig. 1, which shows the error in the estimation of the QNM frequencies calculated by Eq. (42) with respect to \u03be p Figure 2 shows Err r (the left panel) and Err i (the right panel), respectively. Err is dominated by Err r for large \u03be p . We find from Fig. 1 that the region where Err given in Eq. (42) is small spreads widely. Although the minimum error of Err \u2248 1 . 2% is obtained for \u00b5 \u2248 -0 . 21 and \u03bd \u2248 0 . 25 in the analysis, there are many other combinations of \u00b5 and \u03bd for which Err remains small, and the region with small Err extends to a range of 0 < \u03be p < 0 . 4. Therefore, we should consider that the peak of the potential is located in 0 < \u03be p < 0 . 4. In Refs. [18, 19, 27], we have used the WKB analysis to claim how deeply we can actually inspect the region close to the event horizon of a BH by observing the QNM GWs. Since the GWs cannot be localized and the QNMs are determined not only by the potential at the peak radius but also by the curvature of the potential, what we can claim is that the QNM frequency is determined by the information 'around' the peak of the potential. Therefore, it is necessary to properly take into account this fact in the interpretation of the estimated radius obtained in Refs. [18, 19, 27]. The uncertainty in the peak location can be discussed in the following manner. Here, we use r \u2217 0 instead of r \u2217 p because r \u2217 0 is derived easily in the analytical calculation. Expanding Eq. (41) with respect to \u2206 r \u2217 = r \u2217 -r \u2217 0 around r \u2217 0 , and using the QNM frequency in the WKBapproximation of Eq. (5), we have the radial wavenumber [which corresponds to W 1 / 2 in Eq. (A2)], as where ( \u00b7 \u00b7 \u00b7 ) denotes the terms of higher order in the WKB approximation or of O ((\u2206 r \u2217 ) 3 ). We note that d 2 V/dr \u2217 2 | r \u2217 = r \u2217 0 = O ( /epsilon1 4 ) because d/dr \u2217 = O ( /epsilon1 ) \u00d7 d/d\u03be and the \u03be dependence of V is as given in Eq. (41). If we expect that the uncertainty of the peak location is given by the inverse of the wavenumber, it may be estimated by the \u2206 r \u2217 that solves Combining Eqs. (43) and (44), we derive which is translated into the uncertainty in \u03be 0 as by using Eq. (37). This estimate is consistent with the extension of the region where Err given in Eq. (42) is small in Fig. 1. In our previous work [27], we used only one potential which corresponds to \u00b5 = 1 / 2 and \u03bd = 0 in Eq. (40). The peak location was inside the light ring radius as shown in Eqs. (28) and (34), and we concluded that the QNM GWs were emitted 'around' the peak location. However, the meaning of the word 'around' was not clear, and Fig. 1 gives a clear explanation of it based on the error estimation of the WKB frequencies compared with the exact QNM frequencies. Using various potentials with the parameters, \u00b5 and \u03bd , even if we change the threshold of the error estimator (42) from a few % to 10%, the extension of the region does not change much from \u03be p \u223c 0 . 4. Therefore, we conclude that the estimated peak location is restricted to The above result confirms that we can see the space-time sufficiently inside the ergoregion ( r ergo = 2 M for the equatorial radius of the ergosurface) and around the inner light ring r lr /M \u2248 1 + 1 . 633(1 -q ) 1 / 2 .", "pages": [5, 6, 7, 8, 9, 10]}, {"title": "4. Discussions", "content": "In the modification of the Sasaki-Nakamura equation, we have found that the necessary and sufficient condition for the fast fall-off at r \u2217 \u2192 + \u221e can be relaxed to the one given by Eq. (27), if we assume that the free functions g and h can be expanded in a power series of 1 /r . When we use g in Eq. (24) which satisfies Eq. (27), the potential is suitable for the WKB analysis of QNMs (see Ref. [27]), but the general expression of the potential is much more complicated than that for g in Eq. (23). One way to obtain a simple potential will be to keep \u03b3 constant in the Sasaki-Nakamura transformation. \u03b3 in Eq. (7) is rewritten as Thus, it is not difficult to find A and B so that \u03b3 is constant because we may choose B/A which leads 1 /A 2 for the expression in the parenthesis of the above equation. The difficult part arises from the condition for A and B that gives a short-ranged potential. To derive such A and B is one of our future studies. In the study of QNMs in the WKB method, we have evaluated the uncertainty of the peak location of the potential in the extreme Kerr limit. This uncertainty is expressed as Eq. (47), and is consistent with that expected from the equivalence principle. /negationslash Here, we should note that the imaginary parts of the QNM frequencies become zero in the extreme Kerr limit, and many overtones ( n = 0) accumulate at one frequency (see, e.g., Fig. 3 in Ref. [35] and a recent work [36]). Therefore, observing QNM GWs in the near-extremal Kerr case will be very different from the other case, and further studies are required to extract the information from extreme Kerr BHs. Finally, thanks to the recent GW observation, GW150914, we have entered the next stage of using GWs to extract new physics. To test the strong gravitational field around BHs, the QNMs are simple and useful, and the QNM GWs are the target not only for the second-generation GW detectors such as Advanced LIGO (aLIGO) [37], Advanced Virgo (AdV) [38], and KAGRA [39, 40], but also for space-based GW detectors such as eLISA [41] and DECIGO [42]. The enhancement of the signal-to-noise ratio by the third-generation detectors such as the Einstein Telescope (ET) [43] will significantly improve the precision of the test of general relativity.", "pages": [10]}, {"title": "Acknowledgments", "content": "This work was supported by MEXT Grant-in-Aid for Scientific Research on Innovative Areas, 'New Developments in Astrophysics Through Multi-Messenger Observations of Gravitational Wave Sources,' Nos. 24103001 and 24103006 (HN, TT, TN), JSPS Grant-in-Aid for Scientific Research (C), No. 16K05347 (HN), JSPS Grant-in-Aid for Young Scientists (B), No. 25800154 (NS), and Grant-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan No. 15H02087 (TT, TN).", "pages": [10, 11]}, {"title": "A. Quasinormal mode in the WKB approximation", "content": "We schematically write Eq. (19) as where W = \u03c9 2 -V SN . In Ref. [22], the QNM frequencies are discussed as a 'second-order turning point' problem in the WKB approximation (see also Ref. [44]). We prepare two WKB solutions, where r \u2217 1 and r \u2217 2 are the turning points, and also parabolic cylinder functions for r \u2217 1 < r \u2217 < r \u2217 2 (see Eq. (5) of Ref. [22]). The QNM frequencies are derived in the matching condition for the outgoing (from the peak location of the potential) solutions of \u03c8 WKB 1 and \u03c8 WKB 2 . This means that we choose the signs in Eq. (A2) appropriately. As a usual picture, the peak location is calculated by the solution is denoted by r \u2217 0 , and r \u2217 1 < r \u2217 0 < r \u2217 2 in the case of a real potential. We extend this to a complex potential. Therefore, r \u2217 0 is in the complex plane.", "pages": [11]}] |
2020JCAP...10..008D | https://arxiv.org/pdf/2007.13653.pdf | <document>
<section_header_level_1><location><page_1><loc_21><loc_92><loc_79><loc_93></location>Primordial black holes in a dimensionally oxidizing Universe</section_header_level_1>
<text><location><page_1><loc_62><loc_89><loc_62><loc_90></location>∗</text>
<text><location><page_1><loc_26><loc_86><loc_75><loc_90></location>Konstantinos F. Dialektopoulos Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University, Yangzhou 225009, China</text>
<text><location><page_1><loc_45><loc_84><loc_66><loc_85></location>† ‡</text>
<text><location><page_1><loc_29><loc_77><loc_72><loc_84></location>Piero Nicolini and Athanasios G. Tzikas Frankfurt Institute for Advanced Studies (FIAS), Ruth-Moufang-Str. 1, 60438 Frankfurt am Main, Germany and Institut fur Theoretische Physik, Goethe Universitat Frankfurt, Max-von-Laue-Str.1, 60438 Frankfurt am Main, Germany (Dated: September 1, 2020)</text>
<text><location><page_1><loc_18><loc_55><loc_83><loc_76></location>The spontaneous creation of primordial black holes in a violently expanding Universe is a well studied phenomenon. Based on quantum gravity arguments, it has been conjectured that the early Universe might have undergone a lower dimensional phase before relaxing to the current (3 + 1) dimensional state. In this article we combine the above phenomena: we calculate the pair creation rates of black holes nucleated in an expanding Universe, by assuming a dimensional evolution, we term 'oxidation', from (1 + 1) to (2 + 1) and finally to (3 + 1) dimensions. Our investigation is based on the no boundary proposal that allows for the construction of the required gravitational instantons. If, on the one hand, the existence of a dilaton non-minimally coupled to the metric is necessary for black holes to exist in the (1 + 1) phase, it becomes, on the other hand, trivial in (2 + 1) dimensions. Nevertheless, the dilaton might survive the oxidation and be incorporated in a modified theory of gravity in (3+1) dimensions: by assuming that our Universe, in its current state, originates from a lower-dimensional oxidation, one might be led to consider the pair creation rate in a sub-class of the Horndeski action. Our findings for this case show that, for specific values of the Galileon coupling to the metric, the rate can be unsuppressed. This would imply the possibility of compelling parameter bounds for non-Einstein theories of gravity by using the spontaneous black hole creation.</text>
<section_header_level_1><location><page_1><loc_20><loc_51><loc_37><loc_52></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_29><loc_49><loc_48></location>Gravity is by far the less understood of the fundamental interactions governing our Universe. This is not only due to its nonlinear nature and the presence of curvature singularities, but also to the difficulty of determining basic parameters. For instance, the value of the coupling constant G N is known with an accuracy that has barely improved since the time of Cavendish [1-3]. Furthermore, gravity resists a direct quantization due to its nonrenormalizable character and is nested in most of the ongoing issues that prevent a satisfactory understanding of the Universe. Such issues are, for instance, the transplanckian problem, the hierarchy problem, the nature of dark components, the thermodynamic description of event horizons and their informational content [4].</text>
<text><location><page_1><loc_9><loc_17><loc_49><loc_28></location>As a viable solution to address some of the above issues, it has been postulated that the actual number of spacetime dimensions might differ from four. This led to the formulation of higher dimensional scenarios, most notably the large extradimension scenario [5-8], the warp geometry models [9-13], and the universal extradimensions [14]. Alternatively, a radically opposite idea has been put forward by 't Hooft [15]. In the extreme high en-</text>
<text><location><page_1><loc_52><loc_28><loc_92><loc_52></location>ergy regime, the spacetime might undergo a dimensional reduction. This feature would allow for an improvement or even a solution of the nonrenormalizability problem of gravity. For these reasons, there have been an array of investigations aiming to show the existence of the 'dimensional flow' of a spacetime manifold [16-29]. Rather than a smooth differential manifold, the spacetime at the shortest scale would behave like a fractal due to its fluctuating quantum character. Fractals have a continuous dimension that differs from the topological dimension of the ambient space. In the case of gravity, the dimensional flow refers to the dependence of the fractal dimension on the energy at which the spacetime is probed. Although the spacetime topological dimension remains unaltered, i.e. four, some phenomena might behave as dimensionally reduced systems, being subject to a loss of local resolution at short scales.</text>
<text><location><page_1><loc_52><loc_17><loc_92><loc_26></location>Along this line of reasoning, the repercussions of the dimensional flow have recently been considered by studying the quantum decay of de Sitter space within a two dimensional dilaton gravity formulation [30]. Although classically stable, de Sitter space is quantum mechanically unstable, resulting in the nucleation of black hole pairs 1 . At this point, we recall that primordial black</text>
<text><location><page_2><loc_9><loc_79><loc_49><loc_93></location>holes (PBHs) are theoretical objects with masses ranging from Planck relics ( ∼ 10 -8 kg) up to thousands of solar masses. PBHs are unlikely to form from the gravitational collapse of a star today, since low-mass black holes can form only if matter is compressed to enormously high densities by very large external pressures [35] and [3638]. Such conditions of high temperatures and pressures can be found in the early stages of a violent Universe and it is believed that PBHs may have been produced plentiful back then.</text>
<text><location><page_2><loc_9><loc_57><loc_49><loc_78></location>According to the instanton formalism for Euclidean quantum gravity proposed by Mann & Ross [39], Bousso & Hawking [40, 41], black holes are copiously produced in (3 + 1) dimensions provided the cosmological constant has Planckian values [42]. Therefore, these nucleated black holes can be considered 'primordial', relative to those stemming from alternative formation mechanisms [43, 44]. As an additional feature, the de Sitter space quantum instability may lead to the resolution of one of aforementioned ongoing issues in physics, since stable relics of primordially produced black holes have been considered as a cold dark matter component [45, 46]. This possibility is of primary interest as long as attempts of direct detection of dark matter candidates at the LHC have not found experimental corroboration [47].</text>
<text><location><page_2><loc_9><loc_27><loc_49><loc_56></location>There is, however, a potential problem within such a scenario. Black holes produced prior inflation would be washed out with no significant effects on the current Universe [48]. Conversely, the production of primordial black holes in a dimensionally reduced Universe can occur without requiring Planckian values for the cosmological constant [30]. This is a clear advantage that descends from the fact that in (1 + 1) dimensions the gravitational coupling is dimensionless and no actual gravitational scale is associated to it. Interestingly such (1 + 1) dimensional black holes have a peculiar thermodynamics. They radiate with a power proportional to the their mass squared, P ∼ M 2 , making the tiny ones actually stable while large ones would quickly evaporate off. Such a scenario suggests the existence of a new kind of black hole remnants, called 'dimensional remnants', that might generate detectable electroweak bursts [49]. In other words, remnants of dilaton gravity black holes might still be observable today, provided the effective (1+1) dimensional nucleation survives the inflation.</text>
<text><location><page_2><loc_9><loc_18><loc_49><loc_26></location>Given this background, one might be led to investigate the expansion of the early Universe, soon after the Planck era, as a process of dimensional increase, namely from an initial (1 + 1) dimensional phase to a (3 + 1) dimensional one. We term such a dimensional evolution 'dimensional oxidation', being the opposite mechanism of the reduc-</text>
<text><location><page_2><loc_52><loc_92><loc_92><loc_93></location>tion, as can be seen in Fig. 1. As the Universe oxidizes</text>
<figure>
<location><page_2><loc_53><loc_73><loc_92><loc_88></location>
<caption>Figure 1: Dimensional oxidation of the Universe. Initially, i.e., during the Planck era, the spacetime behaves as a (1 +1) dimensional manifold; then below some transition temperature T 1 → 2 , e.g., T 1 → 2 ∼ T Hag ∼ 10 30 K glyph[lessmuch] T P ∼ 1 . 42 × 10 32 K (Hagedorn string transition [50]), it becomes (2 + 1); finally, below a critical temperature T 2 → 3 it takes its current conventional form.</caption>
</figure>
<text><location><page_2><loc_52><loc_38><loc_92><loc_59></location>from (1+1) to (2+1) dimensions at a temperature T 1 → 2 , and finally to the known (3 + 1) dimensional spacetime below a temperature T 2 → 3 , the existence of black holes may not be supported by pure general relativity arguments. In (1 + 1) dimensions, gravity is necessarily described by invoking the presence of the dilaton, a scalar field ψ coupled to geometry representing an extra gravitational degree of freedom alongside the graviton. The inclusion of the dilaton is crucial due to the triviality of the Einstein tensor in (1 + 1) dimensions. It should be emphasized that the dilaton action describing the (1+1) phase can been derived from the D → 2 limit of the D -dimensional Einstein-Hilbert action [51]. This suggests a high degree of correspondence between the full (3 + 1) dimensional Universe and its reduced counterpart.</text>
<text><location><page_2><loc_52><loc_16><loc_92><loc_37></location>As the Universe oxidizes from (1+1) to (2+1) dimensions, black holes would not form in an Einstein gravity Universe, unless in the presence of an anti-de Sitter (AdS) background. For this reason, the (2 + 1) dimensional Universe may be thought as a stage of a non-analytic phase transition from (1+1) to (3+1) dimensional black holes [30]. But what if the (2 + 1) dimensional phase is not necessarily described by Einstein's gravity? What if the dilaton and/or additional coupling survived and remained coupled to geometry when passing from the (1 + 1) to the (2 + 1) and, finally to the (3 + 1) phase? Will this have an impact on PBH production? It is the purpose of this work to give an insight to those questions by providing a possible scenario of PBH nucleation inside an early oxidizing Universe.</text>
<text><location><page_2><loc_52><loc_9><loc_92><loc_16></location>The structure of the paper is as follows: in Sec. II we review the (1 + 1) dimensional phase of the Universe which is governed by Liouville gravity, i.e., a linear coupling of the dilaton to gravity. We calculate the gravitational instantons both for the black hole and for the</text>
<text><location><page_3><loc_9><loc_66><loc_49><loc_93></location>background (de Sitter), using the no-boundary proposal and we compute the pair production rate. In Sec. III we assume a transition to the (2+1) dimensional phase. We study a variety of models of dilaton gravity as well as the inclusion of a quadratic coupling to find black hole solutions. We then calculate the associated instantons and finally the rate. In Sec. IV, we consider the (3 + 1) dimensional case and modifications of the Einstein-Hilbert action that might have been inherited, at least in part, from the dimensionally reduced phases. As a result, we choose to work within a theory described by a part of the Horndeski action, presenting some non-trivial and regular solutions for the dilaton around a Schwarzschild-de Sitter black hole. We also calculate the pair production rate of PBHs through their instantons. Last but not least, in Sec. V we compare the different expressions of the rates in all three phases of the oxidizing Universe. In Sec. VI we draw our conclusions. Throughout the paper we work in natural units where c = glyph[planckover2pi1] = k B = 1.</text>
<section_header_level_1><location><page_3><loc_13><loc_62><loc_45><loc_63></location>II. THE (1+1) DIMENSIONAL PHASE</section_header_level_1>
<text><location><page_3><loc_9><loc_47><loc_49><loc_60></location>As a start we briefly review the results found in [30] for the (1 + 1) dimensional Universe. Customarily, a faithful description of a spacetime in (1 + 1) dimensions is obtained by using a dilatonic gravitational action, since the dilaton represents a physical connection between lower and higher dimensional spacetimes. In particular, Mann & Ross showed that, in the limit D → 2, the D -dimensional Einstein-Hilbert action in natural units reads [51]:</text>
<formula><location><page_3><loc_10><loc_42><loc_49><loc_46></location>S (2 D ) = 1 16 πG 2 ∫ d 2 x √ -g [ ψR + 1 2 ( ∇ ψ ) 2 -2Λ 2 ] , (1)</formula>
<text><location><page_3><loc_9><loc_21><loc_49><loc_41></location>where R is the Ricci scalar, g is the determinant of the metric tensor, ψ is the aforementioned dilaton field and Λ 2 , G 2 are the cosmological constant and the Newton's constant in (1 + 1) dimensions. The presence of the dilaton guarantees that the above action is a good starting point to study the dimensional oxidation. Interestingly, while the limit D → 2 has been carefully studied, the opposite one, 2 → D , has never been proved to be unique. This means that there can be more theories in higher dimensions that have the same (1 + 1) dimensional limit. The Einstein-Hilbert action, from this perspective, would turn to be the leading (3 + 1) dimensional term in the action emerging from the process of oxidation. This fact will be instrumental for our discussion.</text>
<text><location><page_3><loc_9><loc_18><loc_49><loc_21></location>Variation of the action (1) with respect to the metric and the dilaton, respectively, gives</text>
<formula><location><page_3><loc_11><loc_13><loc_49><loc_17></location>Λ 2 g µν + 1 2 ∂ µ ψ∂ ν ψ -1 4 g µν ( ∇ ψ ) 2 + + g µν ✷ ψ -∇ µ ∇ ν ψ = 0 , (2)</formula>
<formula><location><page_3><loc_26><loc_9><loc_49><loc_10></location>R = ✷ ψ. (3)</formula>
<text><location><page_3><loc_52><loc_90><loc_92><loc_93></location>By combining the trace of (2) with (3), we end up with the Liouville equation in vacuum</text>
<formula><location><page_3><loc_67><loc_88><loc_92><loc_89></location>R +2Λ 2 = 0 , (4)</formula>
<text><location><page_3><loc_52><loc_81><loc_92><loc_86></location>that is considered the best analogue of Einstein gravity in (1 + 1) dimensions. Assuming a linearly-symmetric solution around the origin ( x → | x | ) and solving (4) for a line element of the form</text>
<formula><location><page_3><loc_62><loc_77><loc_92><loc_80></location>d s 2 = -f ( | x | ) dt 2 + d x 2 f ( | x | ) , (5)</formula>
<text><location><page_3><loc_52><loc_74><loc_68><loc_75></location>we get the solution [52]</text>
<formula><location><page_3><loc_62><loc_72><loc_92><loc_73></location>f ( | x | ) = C +2 M | x | +Λ 2 x 2 , (6)</formula>
<text><location><page_3><loc_52><loc_57><loc_92><loc_70></location>where C and M are integration constants. Eq. (6) represents a (1+1) dimensional Schwarzschild-(anti-)de Sitter black hole with mass M for negative C . In such a case, the integration constant can be normalized, C = -1, for sake of clarity. The sign convention for Λ 2 = ±| Λ 2 | is de Sitter ( -) and anti-de Sitter (+). Also the black hole mass scales as M ∼ x -1 , making indistinguishable the concept of a particle and a black hole in (1 + 1) dimensions [23].</text>
<text><location><page_3><loc_52><loc_47><loc_92><loc_57></location>The decay of de Sitter space can be studied by considering gravitational instantons [53], namely saddle points of the Euclidean gravitational action (see [39, 41] for more details). The related pair creation rate for black holes can be obtained in the context of the no boundary proposal [54], which states that the wavefunction Ψ of a quantum Universe can be approximated by</text>
<formula><location><page_3><loc_68><loc_45><loc_92><loc_46></location>Ψ ≈ e -I , (7)</formula>
<text><location><page_3><loc_52><loc_32><loc_92><loc_43></location>where I is the instanton-action. The square of this wavefunction provides the probability density to nucleate a specific Universe. Such a probability has to be normalized by the probability of decaying de Sitter space. Therefore, the black hole pair creation rate is defined as the probability ratio of two Universes: the probability of a Universe with a black hole P bh , over the probability of the empty background P bg [42]:</text>
<formula><location><page_3><loc_65><loc_27><loc_92><loc_31></location>Γ = P bh P bg = e -2 I bh e -2 I bg . (8)</formula>
<text><location><page_3><loc_52><loc_19><loc_92><loc_26></location>In other words, the above 'rate' can be thought as a relative probability of two Universes and indicates how much suppressed or favored is the black hole formation. From the relations above, one can obtain the actual probabilities as a function of the rate:</text>
<formula><location><page_3><loc_61><loc_15><loc_92><loc_18></location>P bg = | Ψ bg | 2 | Ψ bg | 2 + | Ψ bh | 2 = 1 1 + Γ (9)</formula>
<text><location><page_3><loc_52><loc_13><loc_54><loc_14></location>and</text>
<formula><location><page_3><loc_60><loc_8><loc_92><loc_12></location>P bh = | Ψ bh | 2 | Ψ bg | 2 + | Ψ bh | 2 = Γ 1 + Γ . (10)</formula>
<text><location><page_4><loc_9><loc_86><loc_49><loc_93></location>For the (1 + 1) dimensional Schwarzschild-de Sitter black hole we get two different cases; a lukewarm black hole for M > √ | Λ 2 | [55] and a Nariai black hole [56] with the event and the cosmological horizon coalescing in a degenerate horizon ( x h = x c = 1 / √ | Λ 2 | = 1 /M ).</text>
<text><location><page_4><loc_9><loc_83><loc_49><loc_86></location>The instanton-action can be derived from the Wickrotated version of (1), namely</text>
<formula><location><page_4><loc_15><loc_79><loc_49><loc_82></location>I (2 D ) = -| Λ 2 | 8 πG 2 ∫ d 2 x √ g ( 1 2 ψ +1 ) . (11)</formula>
<text><location><page_4><loc_9><loc_72><loc_49><loc_77></location>To get the above compact form of the action, one has to perform an integration by parts of the instanton kinetic term and use the equations of motion of Liouville gravity. The dilaton resulting from (2) and (3) has the form of</text>
<formula><location><page_4><loc_18><loc_67><loc_49><loc_70></location>ψ ( t, x ) = ψ 0 + 4 π β t -ln f ( | x | ) , (12)</formula>
<text><location><page_4><loc_9><loc_58><loc_49><loc_66></location>where ψ 0 is an integration constant and β is the periodicity of the Euclidean time given by the inverse of the Hawking temperature. The t -dependence of the dilaton is necessary for its regularization on the horizons, a feature that can be verified by making a coordinate transformation to the Eddington-Finkelstein coordinate system.</text>
<text><location><page_4><loc_9><loc_41><loc_49><loc_57></location>After choosing the suitable boundary ψ 0 for the dilaton, we find that the instanton of the de Sitter background depends on integration constants only. Therefore it can be set to zero, I (2 D ) dS = 0. This means that de Sitter background will not affect the production rate. In contrast to standard general relativity in (3+1) dimensions, the quantum instability of de Sitter space occurs even if the cosmological constant does not attain Planckian values. This is the result of the absence of a fundamental energy scale in the coupling constant, G 2 , of (1 + 1) dimensional gravity.</text>
<text><location><page_4><loc_9><loc_37><loc_49><loc_41></location>After calculating the instantons for the black hole cases, one can finally get the production rates. For the lukewarm black hole the production rate reads:</text>
<formula><location><page_4><loc_10><loc_31><loc_49><loc_35></location>Γ (1+1) lw = ( 1 M 2 / | Λ 2 | -1 ) 1 / 2 G 2 with M > √ | Λ 2 | . (13)</formula>
<text><location><page_4><loc_9><loc_28><loc_49><loc_31></location>In marked contrast to the (3 + 1) case [40], this rate is not exponentially suppressed. Specifically, one finds:</text>
<unordered_list>
<list_item><location><page_4><loc_11><loc_25><loc_45><loc_27></location>· for M ≈ √ | Λ 2 | , the rate diverges (Γ lw glyph[greatermuch] 1);</list_item>
<list_item><location><page_4><loc_11><loc_20><loc_49><loc_24></location>· for √ | Λ 2 | < M < √ 2 | Λ 2 | , the rate Γ lw exceeds unity, corresponding to a highly unstable de Sitter space;</list_item>
<list_item><location><page_4><loc_11><loc_16><loc_49><loc_18></location>· for M = √ 2 | Λ 2 | , Γ lw = 1, meaning that the two Universes have equal probabilities.</list_item>
</unordered_list>
<text><location><page_4><loc_9><loc_9><loc_49><loc_14></location>Only for M glyph[greatermuch] √ | Λ 2 | , the rate is highly suppressed (Γ lw glyph[lessmuch] 1) . This corresponds to the case in which the black hole mass is larger than the energy stored in the cosmological term. As a result the decay is not possible.</text>
<text><location><page_4><loc_52><loc_83><loc_92><loc_93></location>For the Nariai black hole the rate has to depend on an additional mass scale µ 0 , which is arbitrary and not related to M or to the Planck mass M P . This occurs because, in contrast to the lukewarm case, the Nariai rate cannot depend on the ratio M 2 / | Λ 2 | , being M = √ | Λ 2 | the condition for the existence of the instanton. As a result one finds:</text>
<formula><location><page_4><loc_63><loc_78><loc_92><loc_82></location>Γ (1+1) N = ( µ 2 0 | Λ 2 | ) 1 / 2 G 2 . (14)</formula>
<text><location><page_4><loc_52><loc_73><loc_92><loc_77></location>Being µ 0 not set a priori , Nariai PBHs can be prolifically produced for any value of Λ 2 even in the sub-Planckian mass regime.</text>
<section_header_level_1><location><page_4><loc_55><loc_68><loc_88><loc_69></location>III. THE (2+1) DIMENSIONAL PHASE</section_header_level_1>
<text><location><page_4><loc_52><loc_61><loc_92><loc_66></location>As we already mentioned, in (1 + 1) dimensions there is no possibility for gravity to be described solely by the Einstein-Hilbert action and that is why we need the introduction of the dilaton.</text>
<text><location><page_4><loc_52><loc_43><loc_92><loc_60></location>Works on gravity in (2 + 1) dimensions date back to more than forty years ago [57, 58], but the topic became of major interest only later [59-61]. It turns out that, general relativity in (2 + 1) dimensions has no propagating degrees of freedom, or in other words, there are no gravitons in its quantum description. This happens because of the vanishing of the Weyl tensor and so the remaining curvature tensor is described by the Ricci tensor and scalar. In fact, the spacetime will be locally flat, de Sitter or anti-de Sitter in vacuum, depending on the value of the cosmological constant. That is why it is sometimes called (2 + 1) topological gravity.</text>
<text><location><page_4><loc_52><loc_32><loc_92><loc_43></location>For many years it was thought that, the vanishing of the (2 + 1) Riemann tensor in the absence of a cosmological term would lead to non-existence of black holes. However, Ba˜nados, Teitelboim and Zanelli found the socalled BTZ black hole [62] in the presence of a negative cosmological constant, which resembles the properties of the Schwarzschild and Kerr black holes in (3 + 1) dimensions.</text>
<text><location><page_4><loc_52><loc_24><loc_92><loc_31></location>Along this line of reasoning, the derivation of (2 + 1) black hole solutions embedded in a de Sitter space have required the inclusion of higher derivative terms in the gravity action. Let us consider, for example, the action of Bergshoeff-Hohm-Troncoso (BHT) massive gravity [63]</text>
<formula><location><page_4><loc_53><loc_20><loc_92><loc_23></location>S (3 D ) = 1 16 πG 3 ∫ d 3 x √ -g ( R -2Λ 3 -1 m 2 K ) , (15)</formula>
<text><location><page_4><loc_52><loc_12><loc_92><loc_18></location>where K = R µν R µν -3 8 R 2 and m is a mass parameter. This theory predicts massive gravitons with two spin states of helicity ± 2. By varying the action (15) with respect to the metric we get</text>
<formula><location><page_4><loc_61><loc_8><loc_92><loc_11></location>G µν +Λ 3 g µν -1 2 m 2 K µν = 0 , (16)</formula>
<text><location><page_5><loc_9><loc_92><loc_12><loc_93></location>with</text>
<formula><location><page_5><loc_11><loc_85><loc_49><loc_91></location>K µν =2 glyph[square] R µν -1 2 ∇ µ ∇ ν R -1 2 g µν glyph[square] R -8 R µλ R λ ν + + 9 2 RR µν + g µν ( 3 R κλ R κλ -13 8 R 2 ) . (17)</formula>
<text><location><page_5><loc_9><loc_76><loc_49><loc_83></location>It has been proven [64, 65] that for the special case of Λ 3 = m 2 , the theory admits a unique spherically symmetric solution. Since we are interested in a Schwarzschild de Sitter like solution, we consider a static and spherically symmetric line element of the form</text>
<formula><location><page_5><loc_15><loc_74><loc_49><loc_75></location>ds 2 = -f ( r ) dt 2 + f ( r ) -1 dr 2 + r 2 dφ 2 . (18)</formula>
<text><location><page_5><loc_9><loc_71><loc_24><loc_72></location>with metric potential</text>
<formula><location><page_5><loc_19><loc_68><loc_49><loc_70></location>f ( r ) = -G 3 µ + br -Λ 3 r 2 , (19)</formula>
<text><location><page_5><loc_16><loc_63><loc_16><loc_64></location>glyph[negationslash]</text>
<text><location><page_5><loc_9><loc_45><loc_49><loc_67></location>where µ is a black hole mass parameter and b is an integration constant scaling as b ∼ r -1 , that is non-vanishing for 1 /m 2 = 0. One sees that black holes exist not only for Λ 3 < 0 , but also for Λ 3 > 0 where we can get two different horizons, i.e., one event horizon r + and one cosmological horizon r c . If b = 0, there is no contribution from the higher derivative term K , then for Λ 3 < 0 the solutions becomes the known BTZ black hole [66]. From this perspective we can already anticipate that the (2+1) phase can be either a non-analytic phase of the oxidation without black holes or a smooth transient phase admitting event horizons. To better understand how this phase connects with the preceding (1 + 1) dimensional era, let us see what is the role of the dilaton in (2+1) dimensional gravity, without considering higher derivative terms.</text>
<text><location><page_5><loc_10><loc_44><loc_34><loc_45></location>We start from the general action</text>
<formula><location><page_5><loc_9><loc_38><loc_49><loc_43></location>S (3 D ) = 1 16 πG 3 ∫ d 3 x √ -g ( h ( ψ ) R -ω ( ∇ ψ ) 2 -2 V ( ψ ) ) (20)</formula>
<text><location><page_5><loc_9><loc_27><loc_49><loc_38></location>where h ( ψ ) is a function of ψ , determining whether the coupling of the dilaton will be minimal ( h ( ψ ) = 1) or not, V ( ψ ) is an arbitrary potential and ω is a coupling constant. If the potential is not dynamical, it is straightforwardly associated with the cosmological constant in (2 + 1) dimensions Λ 3 . By varying the action (20) with respect to the metric and the scalar field respectively, we get</text>
<formula><location><page_5><loc_10><loc_20><loc_49><loc_25></location>h ( ψ ) G µν + V ( ψ ) g µν = ω ( ∇ µ ψ ∇ ν ψ -1 2 g µν ( ∇ ψ ) 2 ) --g µν ✷ h + ∇ µ ∇ ν h, (21)</formula>
<formula><location><page_5><loc_14><loc_17><loc_49><loc_20></location>2 ω ✷ ψ + dh dψ R =2 dV dψ . (22)</formula>
<text><location><page_5><loc_9><loc_9><loc_49><loc_16></location>For several different theories, with a variety of coupling functions h ( ψ ) and potentials V ( ψ ), one can see that black holes form in the presence of an anti-de Sitter background only (see Tab. I). This would confirm the fact that, a violation of the dominant energy condition</text>
<text><location><page_5><loc_50><loc_40><loc_50><loc_42></location>,</text>
<text><location><page_5><loc_52><loc_90><loc_92><loc_93></location>is required for the existence of de Sitter black holes in (2 + 1) dimensions [67].</text>
<text><location><page_5><loc_52><loc_83><loc_92><loc_90></location>Other examples known in the literature confirm such a scenario. Minimally or non-minimally coupled scalar fields have been considered in (2 + 1) dimensions, but existence of black holes always necessitates AdS asymptotics [68-75].</text>
<formula><location><page_5><loc_55><loc_77><loc_89><loc_81></location>h ( ψ ) ω V ( ψ ) ψ , ψ 2 1 / 2 Λ 3 , Λ 3 + V 0 ψ 2 , Λ 3 + V 0 ψ 4 , Λ 3 + V 0 ψ 6 1 -ψ 2 / 8 1 / 2 Λ 3 , Λ 3 + V 0 ψ 2 , Λ 3 + V 0 ψ 4 , Λ 3 + V 0 ψ 6</formula>
<paragraph><location><page_5><loc_52><loc_73><loc_92><loc_75></location>Table I: Theories under consideration in (2+1) dimensions, all requiring negative cosmological term for black hole existence.</paragraph>
<text><location><page_5><loc_52><loc_53><loc_92><loc_70></location>Given this background we can briefly summarize what one can learn from the proposed analysis up to now. In (1 + 1) dimensions, gravity is necessarily mediated through the metric and the scalar field. In (3+1) dimensions, there are many reasons (see e.g. [76-78]) to believe that general relativity is not the final theory of gravity to successfully describe the cosmological dynamics. During the intermediate phase, the (2+1) the scalar field trivializes and one has to introduce higher derivative terms to find black holes with de Sitter asymptotics. This means that the theory governing the gravitational interactions in this phase would be of the form</text>
<formula><location><page_5><loc_53><loc_46><loc_92><loc_52></location>S (3 D ) = 1 16 πG 3 ∫ d 3 x √ -g ( R -2Λ 3 -1 2 ( ∇ ψ ) 2 -1 m 2 K ) , (23)</formula>
<text><location><page_5><loc_52><loc_32><loc_92><loc_44></location>where the dilaton couples minimally to the metric, and its solution is constant. In other words, the above action is actually equivalent to (15). Such a possibility would suggest that the (3 + 1) dimensional Universe could have inherited correcting terms to the gravity action from the preceding dimensionally reduced phases, i.e., the dilaton as well as higher derivative terms. This fact will be instrumental to calculate the black hole production rate in (3 + 1) dimensions - see the following section.</text>
<text><location><page_5><loc_52><loc_29><loc_92><loc_31></location>After this premise, we will proceed by calculating the pair creation for the theory (15).</text>
<section_header_level_1><location><page_5><loc_55><loc_24><loc_88><loc_25></location>Pair Creation Rate in BHT massive gravity</section_header_level_1>
<text><location><page_5><loc_52><loc_16><loc_92><loc_22></location>Along the lines of what found in [64], we start by calculating the pair creation rate of Nariai black holes in BHT massive gravity (15). Taking the condition f ( r ) = 0 along with Λ 3 > 0 from (19), we find that</text>
<formula><location><page_5><loc_60><loc_12><loc_92><loc_15></location>r + = 1 2Λ 3 ( b -√ b 2 -4 µG 3 Λ 3 ) , (24)</formula>
<formula><location><page_5><loc_60><loc_9><loc_92><loc_12></location>r c = 1 2Λ 3 ( b + √ b 2 -4 µG 3 Λ 3 ) . (25)</formula>
<text><location><page_6><loc_9><loc_88><loc_49><loc_93></location>At the special case of µ = b 2 4 G 3 Λ 3 , the two horizons coincide at ρ = r + = r c = b/ 2Λ 3 . Therefore, a black hole exists for the mass range 0 < µ < b 2 4 G 3 Λ 3 .</text>
<text><location><page_6><loc_9><loc_80><loc_49><loc_88></location>Remarkably, the two horizons have always a common Hawking temperature T + = T c = 1 4 π √ b 2 -4 µG 3 Λ 3 and in the Nariai limit, r + = r c , this temperature vanishes. After Wick-rotating ( τ = it ), we can build a regular instanton, whose line element can be expressed with respect to the two horizons. Taking into account that</text>
<formula><location><page_6><loc_13><loc_75><loc_49><loc_78></location>µ = Λ 3 r + r c G 3 and b = Λ 3 ( r + + r c ) , (26)</formula>
<text><location><page_6><loc_9><loc_72><loc_49><loc_74></location>the form of f ( r ) is given by f ( r ) = Λ 3 ( r -r + )( r c -r ) and the black hole instanton reads</text>
<formula><location><page_6><loc_9><loc_66><loc_49><loc_70></location>d s 2 = Λ 3 ( r -r + )( r c -r )d τ 2 + d r 2 Λ 3 ( r -r + )( r c -r ) + r 2 d φ 2 . (27)</formula>
<text><location><page_6><loc_9><loc_63><loc_49><loc_66></location>For the Nariai case we can apply a transformation of the form [56]</text>
<formula><location><page_6><loc_19><loc_59><loc_49><loc_62></location>τ = ξ glyph[epsilon1] Λ 3 , r = ρ -glyph[epsilon1] cos χ, (28)</formula>
<text><location><page_6><loc_9><loc_48><loc_49><loc_58></location>where ξ and χ are periodic variables with periods 2 π and π respectively. The length glyph[epsilon1] is an arbitrarily small scale ( glyph[epsilon1] glyph[lessmuch] G 3 ) with glyph[epsilon1] → 0 in the Nariai limit. That means that the event horizon lies at r + = ρ -glyph[epsilon1] and the cosmological one at r c = ρ + glyph[epsilon1] . Then the metric potential takes the approximate form of f ( r ) ≈ Λ 3 glyph[epsilon1] 2 sin 2 χ and the Nariai instanton reads</text>
<formula><location><page_6><loc_13><loc_43><loc_49><loc_47></location>d s 2 = 1 Λ 3 ( d χ 2 +sin 2 χ d ξ 2 + b 2 4Λ 3 d φ 2 ) . (29)</formula>
<text><location><page_6><loc_9><loc_37><loc_49><loc_42></location>Now we can estimate the pair creation rate by calculating the two instanton-actions from the Wick-rotated version of the BHT-action (15). The evaluation of the instantonaction vanishes for the above two black hole solutions</text>
<formula><location><page_6><loc_22><loc_34><loc_49><loc_35></location>I (3 D ) bh = I (3 D ) Nariai = 0 , (30)</formula>
<text><location><page_6><loc_9><loc_30><loc_49><loc_32></location>while the instanton action of the respective de Sitter background is</text>
<formula><location><page_6><loc_22><loc_26><loc_49><loc_29></location>I (3 D ) bg = -π 2 G 3 √ Λ 3 (31)</formula>
<text><location><page_6><loc_9><loc_23><loc_38><loc_24></location>and so the pair creation rate (8) becomes</text>
<formula><location><page_6><loc_22><loc_20><loc_49><loc_22></location>Γ (BHT) = e -π G 3 √ Λ 3 . (32)</formula>
<text><location><page_6><loc_9><loc_9><loc_49><loc_18></location>Effectively, this means that we can retrieve an expanding de Sitter Universe whose spontaneous black hole formation is suppressed relative to the empty de Sitter background. We also note that, despite both instantons (30) vanish, the above rate does implicitly depends on the black hole parameters through the cosmological constant Λ 3 .</text>
<section_header_level_1><location><page_6><loc_55><loc_92><loc_88><loc_93></location>IV. THE (3+1) DIMENSIONAL PHASE</section_header_level_1>
<text><location><page_6><loc_52><loc_70><loc_92><loc_90></location>For almost two decades, there have been many attempts [76-78] to find a better description for gravity than general relativity, since it appears that Einstein's theory is plagued with many shortcomings both in the short and in large scale regime. Scalar-tensor theories have been among the first proposals alternatives to general relativity. Already in the early sixties Brans and Dicke conjectured the presence of a scalar field non minimally coupled to gravity. However, the most general scalar-tensor theory in four dimensions, with a single scalar field, leading to second order field equations was firstly proposed by G. W. Horndeski [79], in 1974, and later rediscovered as the decoupling limit of the five dimensional DGP massive gravity [80, 81].</text>
<text><location><page_6><loc_52><loc_51><loc_92><loc_70></location>It is well known though that, once a black hole reaches a stationary state, it is characterized only by its mass, its charges (i.e. associated with long range gauge fields) and its angular momentum. This is the so-called no-hair theorem, that has also been extended in the Brans-Dicke [82, 83] and other scalar-tensor theories [84]. In these papers it has been shown that there is no non-trivial and regular solution for the scalar field around a black hole, or shortly, black holes have no scalar hair. Conversely, if we allow higher order derivative couplings in the action (while the equations are still of second order), there are non-trivial and regular solutions for the scalar field around a black hole.</text>
<text><location><page_6><loc_52><loc_38><loc_92><loc_51></location>Armed with the results from previous sections, we are ready to exploit the dimensional oxidation to select a suitable action containing both a scalar field and higher order derivative terms. This is why in the present section we consider a theory that is part of the Horndeski action. Such a set up would allow for the study of black hole production that can potentially depart from the previous Bousso Hawking result about the known Schwarzschild de Sitter spacetime [40].</text>
<section_header_level_1><location><page_6><loc_62><loc_34><loc_81><loc_35></location>A. Galileon black holes</section_header_level_1>
<text><location><page_6><loc_52><loc_22><loc_92><loc_32></location>Among the theories described by the Horndeski action we are considering those that are shift symmetric under the transformation ψ → ψ +constant. The advantage of this choice is that such theories are customarily taken in account in cosmological contexts since (as we will show) they offer self-accelerating solutions at late-times. Specifically, the chosen action reads</text>
<formula><location><page_6><loc_53><loc_16><loc_92><loc_21></location>S (4 D ) = 1 16 πG N ∫ d 4 x √ -g [ R -ω ( ∇ ψ ) 2 + + BG µν ∇ µ ψ ∇ ν ψ -2Λ ] , (33)</formula>
<text><location><page_6><loc_52><loc_9><loc_92><loc_14></location>where G N is the known Newton's constant in fourdimensions, Λ is the cosmological constant and ω, B are coupling constants [85]. Apart from the obvious ω > 0 , in order to have no ghost instabilities for the scalar field,</text>
<text><location><page_7><loc_9><loc_77><loc_49><loc_93></location>we also have to consider B < 0 to get de Sitter-like solutions, as we will see later on. As anticipated from above, the presence of the third term in the action, i.e., the 'John' term [86], which denotes the non-trivial interaction of the geometry with the scalar field, will help us overcome the known no-hair theorems [83, 84] and will give non-trivial solutions for the dilaton in a static and spherically symmetric black hole geometry. The John term contains the higher derivative terms we assumed to be inherited from the (2 + 1) dimensional phase, even if they exhibit a different functional dependence.</text>
<text><location><page_7><loc_9><loc_60><loc_49><loc_77></location>Before we cut to the chase, let us pause to motivate a bit more the chosen model, i.e. (33). First of all, as already mentioned, the theory is part of the Horndeski action and as such, even though it contains higher order coupling terms, it does not propagate new degrees of freedom (two of the metric plus one of the scalar field). Furthermore, this very coupling was proposed as a remedy to Higgs inflation that suffered from dangerous quantum corrections, and is known as new Higgs infation [87]. Last but not least, the same higher order term appears when one takes the D → 4 limit of the Gauss-Bonnet gravity [88].</text>
<text><location><page_7><loc_9><loc_38><loc_49><loc_60></location>The last few months there has been an increased interest in the so-called Einstein-Gauss-Bonnet theories of gravity in (3 + 1) dimensions, see [89-93] and references therein. It is reasonable thus to ask what if the Universe oxidized from (1 + 1) to (2 + 1) and finally to (3 + 1) dimensions as the limit of the Gauss-Bonnet gravity in arbitrary D dimensions. The problem however arises from the fact that in D < 4 the Gauss-Bonnet term identically vanishes. On top of that, in (2+1) dimensions, the Riemann tensor vanishes as well, thus surviving only terms like K in (15). There has been some studies in lower dimensional Gauss-Bonnet gravity [92, 93], where they find BTZ like black holes, however, all of them have anti-de Sitter asymptotics. Proceeding now with the variation of the action (33) with respect to the metric we get</text>
<formula><location><page_7><loc_10><loc_28><loc_49><loc_37></location>G µν +Λ g µν = ω ( ∇ µ ψ ∇ ν ψ -1 2 g µν ( ∇ ψ ) 2 ) --B 2 ( G µν ( ∇ ψ ) 2 +2 P µανβ ∇ α ψ ∇ β ψ + + g µα δ αρσ νγδ ∇ γ ∇ ρ ψ ∇ δ ∇ σ ψ ) , (34)</formula>
<text><location><page_7><loc_9><loc_25><loc_43><loc_26></location>where P αβµν is the dual of the Riemann tensor,</text>
<formula><location><page_7><loc_18><loc_21><loc_49><loc_24></location>P αβµν = -1 4 glyph[epsilon1] αβρσ R ρσγδ glyph[epsilon1] µνγδ . (35)</formula>
<text><location><page_7><loc_9><loc_11><loc_49><loc_20></location>One can easily notice that for ω = 0 = B , we can recover Einstein equations with a cosmological constant. The related black hole solutions is the Schwarzschild-de Sitter geometry. Moreover, if we vary the action with respect to the dilaton, we get the equation of motion for it, that can be written as:</text>
<formula><location><page_7><loc_10><loc_9><loc_49><loc_10></location>∇ µ J µ = 0 , with J µ = ( ωg µν -BG µν ) ∇ ν ψ. (36)</formula>
<text><location><page_7><loc_52><loc_63><loc_55><loc_65></location>with</text>
<formula><location><page_7><loc_56><loc_60><loc_92><loc_63></location>Λ eff = ω/ | B | and φ ' ( r ) = ± q f √ 1 -f . (40)</formula>
<text><location><page_7><loc_52><loc_45><loc_92><loc_58></location>The mass parameter µ can be written as µ = aG N M where M is the mass of the black hole and the positive constant a can be specified from the corresponding Newtonian limit. The parameter q has dimensions of an energy, [ E ], and its value is specified by q 2 ω = Λ -Λ eff . From this relation we see that Λ ≥ Λ eff must be satisfied for (39) to be real (we remind that ω > 0 as the standard kinetic term). Note that Eq. (38) can be also expressed as f ( r ) = 1 -2 µ r -1 3 ( Λ 1+ q 2 | B | ) r 2 .</text>
<section_header_level_1><location><page_7><loc_61><loc_40><loc_83><loc_41></location>B. Gravitational instantons</section_header_level_1>
<text><location><page_7><loc_52><loc_34><loc_92><loc_38></location>We build again two instantons from the metric (38): one for the de Sitter background and one for the Nariai black hole.</text>
<text><location><page_7><loc_52><loc_28><loc_92><loc_34></location>For the de Sitter instanton ( µ = 0 in (38)), the metric potential is f dS ( r ) = 1 -Λ eff 3 r 2 . The cosmological horizon lies at r c = √ 3 / Λ eff with a temperature of T = 1 2 πr c = β -1 . The de Sitter instanton then reads</text>
<formula><location><page_7><loc_56><loc_25><loc_92><loc_26></location>d s 2 = f dS ( r )d τ 2 + f dS ( r ) -1 d r 2 + r 2 dΩ 2 . (41)</formula>
<text><location><page_7><loc_52><loc_19><loc_92><loc_23></location>After performing the transformation τ = r c ξ, r = r c cos χ , the above instanton can be cast into the regular form of</text>
<formula><location><page_7><loc_55><loc_15><loc_92><loc_18></location>d s 2 = 3 Λ eff ( d χ 2 +sin 2 χ d ξ 2 +cos 2 χ dΩ 2 ) . (42)</formula>
<text><location><page_7><loc_52><loc_86><loc_92><loc_93></location>A detailed discussion about the behavior of the scalar field itself, or of the induced Noether current at the horizon, can be found in [85]. Rather, we will directly proceed to the scalar solution around a self-tuning Schwarzschildde Sitter black hole, which is of interest in our paper.</text>
<text><location><page_7><loc_52><loc_83><loc_92><loc_86></location>We start by considering a static and spherically symmetric metric of the form 2</text>
<formula><location><page_7><loc_58><loc_80><loc_92><loc_82></location>d s 2 = -f ( r )d t 2 + f ( r ) -1 d r 2 + r 2 dΩ 2 , (37)</formula>
<text><location><page_7><loc_52><loc_72><loc_92><loc_79></location>with dΩ 2 = d θ 2 + sin 2 θ d φ 2 and a scalar field which has, apart from the r dependence from the metric, an additional time-dependence, i.e., ψ = ψ ( t, r ) . It has been shown in [85] that equations (34) and (36) are satisfied when</text>
<formula><location><page_7><loc_63><loc_68><loc_92><loc_71></location>f ( r ) = 1 -2 µ r -Λ eff 3 r 2 , (38)</formula>
<formula><location><page_7><loc_66><loc_66><loc_92><loc_67></location>ψ ( r ) = qt + φ ( r ) , (39)</formula>
<text><location><page_8><loc_9><loc_90><loc_49><loc_93></location>In addition, since we have Wick-rotated the time axis, the dilaton (39) becomes ψ = -iqτ + φ ( r ) .</text>
<text><location><page_8><loc_9><loc_85><loc_49><loc_90></location>For the Nariai black hole we have the relations 9 µ 2 Λ eff = 1 and ρ = 3 µ = 1 / √ Λ eff where ρ = r 2 = r 3 is the degenerate horizon. After a transformation for approximately degenerate black holes</text>
<formula><location><page_8><loc_10><loc_80><loc_49><loc_83></location>τ = ξ glyph[epsilon1] Λ eff , r = ρ -glyph[epsilon1] cos χ, r 2 = ρ -glyph[epsilon1] , r 3 = ρ + glyph[epsilon1] , (43)</formula>
<text><location><page_8><loc_9><loc_75><loc_49><loc_79></location>with glyph[epsilon1] being again a small length parameter, the metric potential (38) takes the approximate form of f N ( χ ) ≈ Λ eff glyph[epsilon1] 2 sin 2 χ and, in the limit glyph[epsilon1] → 0 , the instanton reads</text>
<formula><location><page_8><loc_15><loc_71><loc_49><loc_74></location>d s 2 = 1 Λ eff ( d χ 2 +sin 2 χ d ξ 2 +dΩ 2 ) . (44)</formula>
<text><location><page_8><loc_9><loc_69><loc_27><loc_70></location>Also the dilaton becomes</text>
<formula><location><page_8><loc_18><loc_65><loc_49><loc_68></location>ψ = ψ ( ξ, χ ) = -iqξ glyph[epsilon1] Λ eff + φ ( χ ) , (45)</formula>
<text><location><page_8><loc_9><loc_63><loc_12><loc_64></location>with</text>
<formula><location><page_8><loc_19><loc_59><loc_49><loc_62></location>∂φ ( χ ) ∂χ = ( q f N √ 1 -f N ) ∂r ∂χ . (46)</formula>
<text><location><page_8><loc_9><loc_52><loc_49><loc_57></location>Note that even though the dilaton, both in (39) and in (45), may look divergent on the horizon or when glyph[epsilon1] → 0, it is not. One can verify this by changing to the generalized Eddington-Finkelstein coordinates.</text>
<section_header_level_1><location><page_8><loc_20><loc_48><loc_37><loc_49></location>C. Pair creation rate</section_header_level_1>
<text><location><page_8><loc_9><loc_42><loc_49><loc_46></location>The evaluation of the Euclidean version of the action (33) for the de Sitter and the Nariai instanton will give respectively</text>
<formula><location><page_8><loc_18><loc_37><loc_49><loc_40></location>I (4 D ) dS = -3 π 2 G N Λ 2 eff [ 2Λ eff -Λ ] (47)</formula>
<text><location><page_8><loc_9><loc_35><loc_11><loc_36></location>and</text>
<formula><location><page_8><loc_18><loc_31><loc_49><loc_34></location>I (4 D ) N = -π G N Λ 2 eff [ 2Λ eff -Λ ] . (48)</formula>
<text><location><page_8><loc_9><loc_22><loc_49><loc_30></location>A note may be necessary here: even though the bare cosmological constant Λ does not appear in the spacetime metric, it appears explicitly in the solution of the scalar field, through q and also in the action (33); that is why it enters the instanton action. Substituting (47) and (48) in (8), we get a Nariai rate of the form</text>
<formula><location><page_8><loc_15><loc_14><loc_49><loc_21></location>Γ (3+1) N = exp [ -π G N Λ 2 eff ( 2Λ eff -Λ )] = exp [ -π G N Λ ( 1 -B 2 q 4 ) ] . (49)</formula>
<text><location><page_8><loc_9><loc_9><loc_49><loc_13></location>We recall that, in the | B | → 0 limit, there is no nontrivial solution for the scalar field for any ω . This means that black holes (as well as the de Sitter Universe) and</text>
<text><location><page_8><loc_52><loc_88><loc_92><loc_93></location>the above instantons are exactly the same as in general relativity. This confirms that black holes with non-trivial dilatonic configurations can exist only in the presence of the higher derivative coupling (John term).</text>
<text><location><page_8><loc_52><loc_80><loc_92><loc_87></location>We also remark that the rate (49) exceeds the conventional Bousso-Hawking (BH) rate [40]: for any nonvanishing q and B one has Γ (3+1) N > Γ BH = e -π/ Λ G N . In addition the dilaton coupling provides a richer phenomenology:</text>
<unordered_list>
<list_item><location><page_8><loc_54><loc_76><loc_92><loc_79></location>· For Λ > 2Λ eff (or | B | > 1 /q 2 ) the rate is unsuppressed signalling an overproduction of PBHs.</list_item>
<list_item><location><page_8><loc_54><loc_73><loc_92><loc_75></location>· For 2Λ eff = Λ (or | B | = 1 /q 2 ) the two Universes have equal probabilities to nucleate.</list_item>
<list_item><location><page_8><loc_54><loc_69><loc_92><loc_71></location>· For Λ eff < Λ < 2Λ eff (or | B | < 1 /q 2 ) the pair production is exponentially suppressed.</list_item>
</unordered_list>
<text><location><page_8><loc_52><loc_48><loc_92><loc_68></location>This result follows a natural continuation of the expansion of the Universe. In our model both the dilaton and the cosmological constant drive the inflationary era. At the end of inflation, there might be a time window where the value of Λ is relatively higher compared to 2Λ eff (Λ ≥ 2Λ eff ), allowing this way the prolific production of relatively large and cold black holes with respect to the Planckian ones. As the Universe continues to expand, the value of Λ decreases even further, reaching the range of values Λ eff < Λ < 2Λ eff at late times where the production stops. We remind here that Λ ≥ Λ eff always, in order for the dilaton to be real. This condition would also save the catastrophic instability of de Sitter space in the late Universe [94].</text>
<section_header_level_1><location><page_8><loc_57><loc_43><loc_86><loc_44></location>V. COMPARISON OF THE RATES</section_header_level_1>
<text><location><page_8><loc_52><loc_36><loc_92><loc_41></location>Now the scenario of the oxidation has been completed. Assuming that the theories describing gravitational interactions in each phase are those described in the previous section, the Nariai rates for each dimensional phase are</text>
<formula><location><page_8><loc_58><loc_32><loc_92><loc_35></location>Γ (1+1) N = ( µ 0 / √ | Λ 2 | ) 1 /G 2 , (50)</formula>
<formula><location><page_8><loc_58><loc_28><loc_92><loc_32></location>Γ (2+1) N = exp [ -π G 3 √ Λ 3 ] , (51)</formula>
<formula><location><page_8><loc_58><loc_25><loc_92><loc_28></location>Γ (3+1) N = exp [ -π G N Λ 2 eff ( 2Λ eff -Λ )] . (52)</formula>
<text><location><page_8><loc_52><loc_20><loc_92><loc_24></location>As a first comment one can see that the dimensionality of the gravitational coupling constant affects the functional dependence of rates in each of the above regimes.</text>
<text><location><page_8><loc_52><loc_11><loc_92><loc_20></location>More importantly, the above calculation offers a possible history of a lower dimensional Universe in relation to the spontaneous production of PBHs, as seen in Fig 2 and 3, where for ease of notation we introduced the de Sitter radius glyph[lscript] , to express the cosmological constants Λ 2 , Λ 3 and Λ as 1 /glyph[lscript] 2 .</text>
<text><location><page_8><loc_52><loc_9><loc_92><loc_11></location>Specifically, when the Universe was in its effective (1+1) dimensional era, PBHs would have been plentifully</text>
<figure>
<location><page_9><loc_10><loc_75><loc_48><loc_93></location>
<caption>Figure 2: The Nariai rates vs the logarithm of the de Sitter radius glyph[lscript] are plotted for | Λ 2 | = Λ 3 = Λ and for G 2 = G 3 = G N = µ 0 = 1. The blue dashed line stands for the conventional Bousso-Hawking rate.</caption>
</figure>
<figure>
<location><page_9><loc_9><loc_46><loc_48><loc_66></location>
<caption>Figure 3: The black hole probabilities vs the logarithm of the de Sitter radius glyph[lscript] are plotted for | Λ 2 | = Λ 3 = Λ and for G 2 = G 3 = G N = µ 0 = 1. The blue dashed line stands for the conventional Nariai black hole probability.</caption>
</figure>
<text><location><page_9><loc_9><loc_9><loc_49><loc_36></location>produced since the rate (50) is unsuppressed. Conversely, during the (2 + 1) dimensional phase, the rate (51) is exponentially suppressed but to a lesser extent than the conventional Bousso-Hawking result. For G 2 3 Λ 3 ∼ 1 the black hole nucleation is not negligible. Hence, a lower dimensional Universe could enhance the population of PBHs relative to standard (3 + 1) scenario, provided that the magnitude of the cosmological constants started with Planckian values during the Planck era and then decreased as the Universe expanded. Nevertheless, all the black holes formed before inflation would have been exponentially diluted leaving no observational traces. The only possibility for significant effects, is that the lower dimensional phases left an imprint of its pair production at short scales after inflation. Being the oxidizing dimension an effective quantity related to the local fractality of the spacetime, such an occurrence might be acceptable also in the case the topological dimension of the (ambient) spacetime is four. As a result, the scenario is compatible</text>
<text><location><page_9><loc_52><loc_92><loc_83><loc_93></location>with the standard paradigm of the inflation.</text>
<text><location><page_9><loc_52><loc_62><loc_92><loc_92></location>Regarding the (3+1) phase, one finds the most promising results since the presence of the parameter Λ eff allows to circumvent the issue of the production prior/after the inflation. The dilaton corrects the Bousso-Hawking rate with an unsuppressed part. The new Nariai rate leads to an unsuppressed production even at the final stages of inflation, as long as Λ > 2Λ eff . We must stress here that we expect for Λ eff to have much lower value than those displayed in Fig. 2, in order for the production to continue beyond the inflationary era. Inflation should have lasted at least for 60 e-foldings in order to solve the various cosmological problems (horizon, flatness and monopole problem), providing a lower post-inflationary value for the de Sitter radius of the order glyph[lscript] end glyph[greaterorsimilar] e 60 L P . In the plot we just give some arbitrary values for Λ eff to see the new corrected behavior of the (3 + 1) rate, even post-inflationary unsuppressed rates are clearly admissible. Conversely if the value of Λ enters the domain Λ eff < Λ < 2Λ eff before the Universe reaches the late time expansion, the rate becomes suppressed at late times with no observational consequences.</text>
<text><location><page_9><loc_52><loc_51><loc_92><loc_61></location>On the ground of this reasoning, one can find a concrete result stemming from our investigation. By using the current value of the cosmological constant and by invoking the Universe stability, i.e., no black hole production, the present-day value of Λ eff can be constrained by using the relation Λ eff < Λ < 2Λ eff . Being Λ glyph[similarequal] 2 . 888 × 10 -122 in Planck units [95], one finds:</text>
<formula><location><page_9><loc_58><loc_49><loc_92><loc_51></location>1 . 444 × 10 -122 < Λ eff < 2 . 888 × 10 -122 . (53)</formula>
<section_header_level_1><location><page_9><loc_62><loc_45><loc_81><loc_46></location>VI. FINAL REMARKS</section_header_level_1>
<text><location><page_9><loc_52><loc_27><loc_92><loc_43></location>In this paper, we assumed that the Universe underwent a dimensional oxidation before it reached its current form. Specifically, at very high temperatures the Universe had (1 + 1) dimensions, later on as it cooled down, it became (2+1) dimensional and finally, it took its known (3+1) dimensional form. Taking this for granted, for each of the above phases, we calculated the probability of Schwarzschild-de Sitter black holes to be nucleated inside an expanding de Sitter background. We did this by using the no boundary proposal to calculate the associated gravitational instantons.</text>
<text><location><page_9><loc_52><loc_9><loc_92><loc_27></location>In the (1 + 1) phase the existence of a dilaton is necessary. In (2 + 1) we saw that, in order for black holes to exist, the dilaton should be trivial and higher order derivative terms should appear in the action. In its current (3 + 1) phase, there is no necessity for the dilaton to exist, however if it does it solves many of the shortcomings that general relativity possesses. That is why we considered a specific case of the Horndeski action that contains a Galileon field which is symmetric under the shift transformation ψ → ψ + constant. In cosmology this model is very successful in describing the late-time acceleration of the Universe with a self-accelerating solution given by the scalar field.</text>
<text><location><page_10><loc_9><loc_74><loc_49><loc_93></location>We calculated the Nariai pair creation rate in each phase. Pre-inflationary production is not of interest because pre-inflationary nucleated black holes would have been washed away by inflation. If there is, however, an imprint of the lower dimensional phase after inflation, then the production of lower dimensional Planckian relics mayb still continue until today. There is an additional production from the (3 + 1) phase, depending on the values of the coupling parameters and specifically for Λ > 2Λ eff . Then the rate is unsuppressed for this range of values even at the final stages of inflation. However, the decrease of Λ should start satisfying the relation Λ eff < Λ < 2Λ eff as we approach present times.</text>
<text><location><page_10><loc_9><loc_69><loc_49><loc_74></location>Interestingly the proposed investigation offers a concrete result. The condition Λ eff < Λ < 2Λ eff can be applied to the current Universe to obtain compelling constraints for the John term of the Horndenski action.</text>
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<section_header_level_1><location><page_10><loc_65><loc_92><loc_79><loc_93></location>Acknowledgments</section_header_level_1>
<text><location><page_10><loc_52><loc_69><loc_92><loc_87></location>K.F.D would like to thank the Institute of Space Sciences and Astronomy of the University of Malta where part of this work was conducted. The work of P.N. has been partially supported by GNFM, the Italian National Group for Mathematical Physics. The work of A.G.T. has been supported by the GRADE Completion Scholarships, which are funded by the STIBET program of the German Academic Exchange Service (DAAD) and the Stiftung zur Forderung der internationalen wissenschaftlichen Beziehungen der Johann Wolfgang Goethe-Universitat. The authors acknowledge the networking support by the COST Actions GWverse CA16104 and CANTATA CA15117.</text>
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</document> | [{"title": "Primordial black holes in a dimensionally oxidizing Universe", "content": "\u2217 Konstantinos F. Dialektopoulos Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University, Yangzhou 225009, China \u2020 \u2021 Piero Nicolini and Athanasios G. Tzikas Frankfurt Institute for Advanced Studies (FIAS), Ruth-Moufang-Str. 1, 60438 Frankfurt am Main, Germany and Institut fur Theoretische Physik, Goethe Universitat Frankfurt, Max-von-Laue-Str.1, 60438 Frankfurt am Main, Germany (Dated: September 1, 2020) The spontaneous creation of primordial black holes in a violently expanding Universe is a well studied phenomenon. Based on quantum gravity arguments, it has been conjectured that the early Universe might have undergone a lower dimensional phase before relaxing to the current (3 + 1) dimensional state. In this article we combine the above phenomena: we calculate the pair creation rates of black holes nucleated in an expanding Universe, by assuming a dimensional evolution, we term 'oxidation', from (1 + 1) to (2 + 1) and finally to (3 + 1) dimensions. Our investigation is based on the no boundary proposal that allows for the construction of the required gravitational instantons. If, on the one hand, the existence of a dilaton non-minimally coupled to the metric is necessary for black holes to exist in the (1 + 1) phase, it becomes, on the other hand, trivial in (2 + 1) dimensions. Nevertheless, the dilaton might survive the oxidation and be incorporated in a modified theory of gravity in (3+1) dimensions: by assuming that our Universe, in its current state, originates from a lower-dimensional oxidation, one might be led to consider the pair creation rate in a sub-class of the Horndeski action. Our findings for this case show that, for specific values of the Galileon coupling to the metric, the rate can be unsuppressed. This would imply the possibility of compelling parameter bounds for non-Einstein theories of gravity by using the spontaneous black hole creation.", "pages": [1]}, {"title": "I. INTRODUCTION", "content": "Gravity is by far the less understood of the fundamental interactions governing our Universe. This is not only due to its nonlinear nature and the presence of curvature singularities, but also to the difficulty of determining basic parameters. For instance, the value of the coupling constant G N is known with an accuracy that has barely improved since the time of Cavendish [1-3]. Furthermore, gravity resists a direct quantization due to its nonrenormalizable character and is nested in most of the ongoing issues that prevent a satisfactory understanding of the Universe. Such issues are, for instance, the transplanckian problem, the hierarchy problem, the nature of dark components, the thermodynamic description of event horizons and their informational content [4]. As a viable solution to address some of the above issues, it has been postulated that the actual number of spacetime dimensions might differ from four. This led to the formulation of higher dimensional scenarios, most notably the large extradimension scenario [5-8], the warp geometry models [9-13], and the universal extradimensions [14]. Alternatively, a radically opposite idea has been put forward by 't Hooft [15]. In the extreme high en- ergy regime, the spacetime might undergo a dimensional reduction. This feature would allow for an improvement or even a solution of the nonrenormalizability problem of gravity. For these reasons, there have been an array of investigations aiming to show the existence of the 'dimensional flow' of a spacetime manifold [16-29]. Rather than a smooth differential manifold, the spacetime at the shortest scale would behave like a fractal due to its fluctuating quantum character. Fractals have a continuous dimension that differs from the topological dimension of the ambient space. In the case of gravity, the dimensional flow refers to the dependence of the fractal dimension on the energy at which the spacetime is probed. Although the spacetime topological dimension remains unaltered, i.e. four, some phenomena might behave as dimensionally reduced systems, being subject to a loss of local resolution at short scales. Along this line of reasoning, the repercussions of the dimensional flow have recently been considered by studying the quantum decay of de Sitter space within a two dimensional dilaton gravity formulation [30]. Although classically stable, de Sitter space is quantum mechanically unstable, resulting in the nucleation of black hole pairs 1 . At this point, we recall that primordial black holes (PBHs) are theoretical objects with masses ranging from Planck relics ( \u223c 10 -8 kg) up to thousands of solar masses. PBHs are unlikely to form from the gravitational collapse of a star today, since low-mass black holes can form only if matter is compressed to enormously high densities by very large external pressures [35] and [3638]. Such conditions of high temperatures and pressures can be found in the early stages of a violent Universe and it is believed that PBHs may have been produced plentiful back then. According to the instanton formalism for Euclidean quantum gravity proposed by Mann & Ross [39], Bousso & Hawking [40, 41], black holes are copiously produced in (3 + 1) dimensions provided the cosmological constant has Planckian values [42]. Therefore, these nucleated black holes can be considered 'primordial', relative to those stemming from alternative formation mechanisms [43, 44]. As an additional feature, the de Sitter space quantum instability may lead to the resolution of one of aforementioned ongoing issues in physics, since stable relics of primordially produced black holes have been considered as a cold dark matter component [45, 46]. This possibility is of primary interest as long as attempts of direct detection of dark matter candidates at the LHC have not found experimental corroboration [47]. There is, however, a potential problem within such a scenario. Black holes produced prior inflation would be washed out with no significant effects on the current Universe [48]. Conversely, the production of primordial black holes in a dimensionally reduced Universe can occur without requiring Planckian values for the cosmological constant [30]. This is a clear advantage that descends from the fact that in (1 + 1) dimensions the gravitational coupling is dimensionless and no actual gravitational scale is associated to it. Interestingly such (1 + 1) dimensional black holes have a peculiar thermodynamics. They radiate with a power proportional to the their mass squared, P \u223c M 2 , making the tiny ones actually stable while large ones would quickly evaporate off. Such a scenario suggests the existence of a new kind of black hole remnants, called 'dimensional remnants', that might generate detectable electroweak bursts [49]. In other words, remnants of dilaton gravity black holes might still be observable today, provided the effective (1+1) dimensional nucleation survives the inflation. Given this background, one might be led to investigate the expansion of the early Universe, soon after the Planck era, as a process of dimensional increase, namely from an initial (1 + 1) dimensional phase to a (3 + 1) dimensional one. We term such a dimensional evolution 'dimensional oxidation', being the opposite mechanism of the reduc- tion, as can be seen in Fig. 1. As the Universe oxidizes from (1+1) to (2+1) dimensions at a temperature T 1 \u2192 2 , and finally to the known (3 + 1) dimensional spacetime below a temperature T 2 \u2192 3 , the existence of black holes may not be supported by pure general relativity arguments. In (1 + 1) dimensions, gravity is necessarily described by invoking the presence of the dilaton, a scalar field \u03c8 coupled to geometry representing an extra gravitational degree of freedom alongside the graviton. The inclusion of the dilaton is crucial due to the triviality of the Einstein tensor in (1 + 1) dimensions. It should be emphasized that the dilaton action describing the (1+1) phase can been derived from the D \u2192 2 limit of the D -dimensional Einstein-Hilbert action [51]. This suggests a high degree of correspondence between the full (3 + 1) dimensional Universe and its reduced counterpart. As the Universe oxidizes from (1+1) to (2+1) dimensions, black holes would not form in an Einstein gravity Universe, unless in the presence of an anti-de Sitter (AdS) background. For this reason, the (2 + 1) dimensional Universe may be thought as a stage of a non-analytic phase transition from (1+1) to (3+1) dimensional black holes [30]. But what if the (2 + 1) dimensional phase is not necessarily described by Einstein's gravity? What if the dilaton and/or additional coupling survived and remained coupled to geometry when passing from the (1 + 1) to the (2 + 1) and, finally to the (3 + 1) phase? Will this have an impact on PBH production? It is the purpose of this work to give an insight to those questions by providing a possible scenario of PBH nucleation inside an early oxidizing Universe. The structure of the paper is as follows: in Sec. II we review the (1 + 1) dimensional phase of the Universe which is governed by Liouville gravity, i.e., a linear coupling of the dilaton to gravity. We calculate the gravitational instantons both for the black hole and for the background (de Sitter), using the no-boundary proposal and we compute the pair production rate. In Sec. III we assume a transition to the (2+1) dimensional phase. We study a variety of models of dilaton gravity as well as the inclusion of a quadratic coupling to find black hole solutions. We then calculate the associated instantons and finally the rate. In Sec. IV, we consider the (3 + 1) dimensional case and modifications of the Einstein-Hilbert action that might have been inherited, at least in part, from the dimensionally reduced phases. As a result, we choose to work within a theory described by a part of the Horndeski action, presenting some non-trivial and regular solutions for the dilaton around a Schwarzschild-de Sitter black hole. We also calculate the pair production rate of PBHs through their instantons. Last but not least, in Sec. V we compare the different expressions of the rates in all three phases of the oxidizing Universe. In Sec. VI we draw our conclusions. Throughout the paper we work in natural units where c = glyph[planckover2pi1] = k B = 1.", "pages": [1, 2, 3]}, {"title": "II. THE (1+1) DIMENSIONAL PHASE", "content": "As a start we briefly review the results found in [30] for the (1 + 1) dimensional Universe. Customarily, a faithful description of a spacetime in (1 + 1) dimensions is obtained by using a dilatonic gravitational action, since the dilaton represents a physical connection between lower and higher dimensional spacetimes. In particular, Mann & Ross showed that, in the limit D \u2192 2, the D -dimensional Einstein-Hilbert action in natural units reads [51]: where R is the Ricci scalar, g is the determinant of the metric tensor, \u03c8 is the aforementioned dilaton field and \u039b 2 , G 2 are the cosmological constant and the Newton's constant in (1 + 1) dimensions. The presence of the dilaton guarantees that the above action is a good starting point to study the dimensional oxidation. Interestingly, while the limit D \u2192 2 has been carefully studied, the opposite one, 2 \u2192 D , has never been proved to be unique. This means that there can be more theories in higher dimensions that have the same (1 + 1) dimensional limit. The Einstein-Hilbert action, from this perspective, would turn to be the leading (3 + 1) dimensional term in the action emerging from the process of oxidation. This fact will be instrumental for our discussion. Variation of the action (1) with respect to the metric and the dilaton, respectively, gives By combining the trace of (2) with (3), we end up with the Liouville equation in vacuum that is considered the best analogue of Einstein gravity in (1 + 1) dimensions. Assuming a linearly-symmetric solution around the origin ( x \u2192 | x | ) and solving (4) for a line element of the form we get the solution [52] where C and M are integration constants. Eq. (6) represents a (1+1) dimensional Schwarzschild-(anti-)de Sitter black hole with mass M for negative C . In such a case, the integration constant can be normalized, C = -1, for sake of clarity. The sign convention for \u039b 2 = \u00b1| \u039b 2 | is de Sitter ( -) and anti-de Sitter (+). Also the black hole mass scales as M \u223c x -1 , making indistinguishable the concept of a particle and a black hole in (1 + 1) dimensions [23]. The decay of de Sitter space can be studied by considering gravitational instantons [53], namely saddle points of the Euclidean gravitational action (see [39, 41] for more details). The related pair creation rate for black holes can be obtained in the context of the no boundary proposal [54], which states that the wavefunction \u03a8 of a quantum Universe can be approximated by where I is the instanton-action. The square of this wavefunction provides the probability density to nucleate a specific Universe. Such a probability has to be normalized by the probability of decaying de Sitter space. Therefore, the black hole pair creation rate is defined as the probability ratio of two Universes: the probability of a Universe with a black hole P bh , over the probability of the empty background P bg [42]: In other words, the above 'rate' can be thought as a relative probability of two Universes and indicates how much suppressed or favored is the black hole formation. From the relations above, one can obtain the actual probabilities as a function of the rate: and For the (1 + 1) dimensional Schwarzschild-de Sitter black hole we get two different cases; a lukewarm black hole for M > \u221a | \u039b 2 | [55] and a Nariai black hole [56] with the event and the cosmological horizon coalescing in a degenerate horizon ( x h = x c = 1 / \u221a | \u039b 2 | = 1 /M ). The instanton-action can be derived from the Wickrotated version of (1), namely To get the above compact form of the action, one has to perform an integration by parts of the instanton kinetic term and use the equations of motion of Liouville gravity. The dilaton resulting from (2) and (3) has the form of where \u03c8 0 is an integration constant and \u03b2 is the periodicity of the Euclidean time given by the inverse of the Hawking temperature. The t -dependence of the dilaton is necessary for its regularization on the horizons, a feature that can be verified by making a coordinate transformation to the Eddington-Finkelstein coordinate system. After choosing the suitable boundary \u03c8 0 for the dilaton, we find that the instanton of the de Sitter background depends on integration constants only. Therefore it can be set to zero, I (2 D ) dS = 0. This means that de Sitter background will not affect the production rate. In contrast to standard general relativity in (3+1) dimensions, the quantum instability of de Sitter space occurs even if the cosmological constant does not attain Planckian values. This is the result of the absence of a fundamental energy scale in the coupling constant, G 2 , of (1 + 1) dimensional gravity. After calculating the instantons for the black hole cases, one can finally get the production rates. For the lukewarm black hole the production rate reads: In marked contrast to the (3 + 1) case [40], this rate is not exponentially suppressed. Specifically, one finds: Only for M glyph[greatermuch] \u221a | \u039b 2 | , the rate is highly suppressed (\u0393 lw glyph[lessmuch] 1) . This corresponds to the case in which the black hole mass is larger than the energy stored in the cosmological term. As a result the decay is not possible. For the Nariai black hole the rate has to depend on an additional mass scale \u00b5 0 , which is arbitrary and not related to M or to the Planck mass M P . This occurs because, in contrast to the lukewarm case, the Nariai rate cannot depend on the ratio M 2 / | \u039b 2 | , being M = \u221a | \u039b 2 | the condition for the existence of the instanton. As a result one finds: Being \u00b5 0 not set a priori , Nariai PBHs can be prolifically produced for any value of \u039b 2 even in the sub-Planckian mass regime.", "pages": [3, 4]}, {"title": "III. THE (2+1) DIMENSIONAL PHASE", "content": "As we already mentioned, in (1 + 1) dimensions there is no possibility for gravity to be described solely by the Einstein-Hilbert action and that is why we need the introduction of the dilaton. Works on gravity in (2 + 1) dimensions date back to more than forty years ago [57, 58], but the topic became of major interest only later [59-61]. It turns out that, general relativity in (2 + 1) dimensions has no propagating degrees of freedom, or in other words, there are no gravitons in its quantum description. This happens because of the vanishing of the Weyl tensor and so the remaining curvature tensor is described by the Ricci tensor and scalar. In fact, the spacetime will be locally flat, de Sitter or anti-de Sitter in vacuum, depending on the value of the cosmological constant. That is why it is sometimes called (2 + 1) topological gravity. For many years it was thought that, the vanishing of the (2 + 1) Riemann tensor in the absence of a cosmological term would lead to non-existence of black holes. However, Ba\u02dcnados, Teitelboim and Zanelli found the socalled BTZ black hole [62] in the presence of a negative cosmological constant, which resembles the properties of the Schwarzschild and Kerr black holes in (3 + 1) dimensions. Along this line of reasoning, the derivation of (2 + 1) black hole solutions embedded in a de Sitter space have required the inclusion of higher derivative terms in the gravity action. Let us consider, for example, the action of Bergshoeff-Hohm-Troncoso (BHT) massive gravity [63] where K = R \u00b5\u03bd R \u00b5\u03bd -3 8 R 2 and m is a mass parameter. This theory predicts massive gravitons with two spin states of helicity \u00b1 2. By varying the action (15) with respect to the metric we get with It has been proven [64, 65] that for the special case of \u039b 3 = m 2 , the theory admits a unique spherically symmetric solution. Since we are interested in a Schwarzschild de Sitter like solution, we consider a static and spherically symmetric line element of the form with metric potential glyph[negationslash] where \u00b5 is a black hole mass parameter and b is an integration constant scaling as b \u223c r -1 , that is non-vanishing for 1 /m 2 = 0. One sees that black holes exist not only for \u039b 3 < 0 , but also for \u039b 3 > 0 where we can get two different horizons, i.e., one event horizon r + and one cosmological horizon r c . If b = 0, there is no contribution from the higher derivative term K , then for \u039b 3 < 0 the solutions becomes the known BTZ black hole [66]. From this perspective we can already anticipate that the (2+1) phase can be either a non-analytic phase of the oxidation without black holes or a smooth transient phase admitting event horizons. To better understand how this phase connects with the preceding (1 + 1) dimensional era, let us see what is the role of the dilaton in (2+1) dimensional gravity, without considering higher derivative terms. We start from the general action where h ( \u03c8 ) is a function of \u03c8 , determining whether the coupling of the dilaton will be minimal ( h ( \u03c8 ) = 1) or not, V ( \u03c8 ) is an arbitrary potential and \u03c9 is a coupling constant. If the potential is not dynamical, it is straightforwardly associated with the cosmological constant in (2 + 1) dimensions \u039b 3 . By varying the action (20) with respect to the metric and the scalar field respectively, we get For several different theories, with a variety of coupling functions h ( \u03c8 ) and potentials V ( \u03c8 ), one can see that black holes form in the presence of an anti-de Sitter background only (see Tab. I). This would confirm the fact that, a violation of the dominant energy condition , is required for the existence of de Sitter black holes in (2 + 1) dimensions [67]. Other examples known in the literature confirm such a scenario. Minimally or non-minimally coupled scalar fields have been considered in (2 + 1) dimensions, but existence of black holes always necessitates AdS asymptotics [68-75]. Given this background we can briefly summarize what one can learn from the proposed analysis up to now. In (1 + 1) dimensions, gravity is necessarily mediated through the metric and the scalar field. In (3+1) dimensions, there are many reasons (see e.g. [76-78]) to believe that general relativity is not the final theory of gravity to successfully describe the cosmological dynamics. During the intermediate phase, the (2+1) the scalar field trivializes and one has to introduce higher derivative terms to find black holes with de Sitter asymptotics. This means that the theory governing the gravitational interactions in this phase would be of the form where the dilaton couples minimally to the metric, and its solution is constant. In other words, the above action is actually equivalent to (15). Such a possibility would suggest that the (3 + 1) dimensional Universe could have inherited correcting terms to the gravity action from the preceding dimensionally reduced phases, i.e., the dilaton as well as higher derivative terms. This fact will be instrumental to calculate the black hole production rate in (3 + 1) dimensions - see the following section. After this premise, we will proceed by calculating the pair creation for the theory (15).", "pages": [4, 5]}, {"title": "Pair Creation Rate in BHT massive gravity", "content": "Along the lines of what found in [64], we start by calculating the pair creation rate of Nariai black holes in BHT massive gravity (15). Taking the condition f ( r ) = 0 along with \u039b 3 > 0 from (19), we find that At the special case of \u00b5 = b 2 4 G 3 \u039b 3 , the two horizons coincide at \u03c1 = r + = r c = b/ 2\u039b 3 . Therefore, a black hole exists for the mass range 0 < \u00b5 < b 2 4 G 3 \u039b 3 . Remarkably, the two horizons have always a common Hawking temperature T + = T c = 1 4 \u03c0 \u221a b 2 -4 \u00b5G 3 \u039b 3 and in the Nariai limit, r + = r c , this temperature vanishes. After Wick-rotating ( \u03c4 = it ), we can build a regular instanton, whose line element can be expressed with respect to the two horizons. Taking into account that the form of f ( r ) is given by f ( r ) = \u039b 3 ( r -r + )( r c -r ) and the black hole instanton reads For the Nariai case we can apply a transformation of the form [56] where \u03be and \u03c7 are periodic variables with periods 2 \u03c0 and \u03c0 respectively. The length glyph[epsilon1] is an arbitrarily small scale ( glyph[epsilon1] glyph[lessmuch] G 3 ) with glyph[epsilon1] \u2192 0 in the Nariai limit. That means that the event horizon lies at r + = \u03c1 -glyph[epsilon1] and the cosmological one at r c = \u03c1 + glyph[epsilon1] . Then the metric potential takes the approximate form of f ( r ) \u2248 \u039b 3 glyph[epsilon1] 2 sin 2 \u03c7 and the Nariai instanton reads Now we can estimate the pair creation rate by calculating the two instanton-actions from the Wick-rotated version of the BHT-action (15). The evaluation of the instantonaction vanishes for the above two black hole solutions while the instanton action of the respective de Sitter background is and so the pair creation rate (8) becomes Effectively, this means that we can retrieve an expanding de Sitter Universe whose spontaneous black hole formation is suppressed relative to the empty de Sitter background. We also note that, despite both instantons (30) vanish, the above rate does implicitly depends on the black hole parameters through the cosmological constant \u039b 3 .", "pages": [5, 6]}, {"title": "IV. THE (3+1) DIMENSIONAL PHASE", "content": "For almost two decades, there have been many attempts [76-78] to find a better description for gravity than general relativity, since it appears that Einstein's theory is plagued with many shortcomings both in the short and in large scale regime. Scalar-tensor theories have been among the first proposals alternatives to general relativity. Already in the early sixties Brans and Dicke conjectured the presence of a scalar field non minimally coupled to gravity. However, the most general scalar-tensor theory in four dimensions, with a single scalar field, leading to second order field equations was firstly proposed by G. W. Horndeski [79], in 1974, and later rediscovered as the decoupling limit of the five dimensional DGP massive gravity [80, 81]. It is well known though that, once a black hole reaches a stationary state, it is characterized only by its mass, its charges (i.e. associated with long range gauge fields) and its angular momentum. This is the so-called no-hair theorem, that has also been extended in the Brans-Dicke [82, 83] and other scalar-tensor theories [84]. In these papers it has been shown that there is no non-trivial and regular solution for the scalar field around a black hole, or shortly, black holes have no scalar hair. Conversely, if we allow higher order derivative couplings in the action (while the equations are still of second order), there are non-trivial and regular solutions for the scalar field around a black hole. Armed with the results from previous sections, we are ready to exploit the dimensional oxidation to select a suitable action containing both a scalar field and higher order derivative terms. This is why in the present section we consider a theory that is part of the Horndeski action. Such a set up would allow for the study of black hole production that can potentially depart from the previous Bousso Hawking result about the known Schwarzschild de Sitter spacetime [40].", "pages": [6]}, {"title": "A. Galileon black holes", "content": "Among the theories described by the Horndeski action we are considering those that are shift symmetric under the transformation \u03c8 \u2192 \u03c8 +constant. The advantage of this choice is that such theories are customarily taken in account in cosmological contexts since (as we will show) they offer self-accelerating solutions at late-times. Specifically, the chosen action reads where G N is the known Newton's constant in fourdimensions, \u039b is the cosmological constant and \u03c9, B are coupling constants [85]. Apart from the obvious \u03c9 > 0 , in order to have no ghost instabilities for the scalar field, we also have to consider B < 0 to get de Sitter-like solutions, as we will see later on. As anticipated from above, the presence of the third term in the action, i.e., the 'John' term [86], which denotes the non-trivial interaction of the geometry with the scalar field, will help us overcome the known no-hair theorems [83, 84] and will give non-trivial solutions for the dilaton in a static and spherically symmetric black hole geometry. The John term contains the higher derivative terms we assumed to be inherited from the (2 + 1) dimensional phase, even if they exhibit a different functional dependence. Before we cut to the chase, let us pause to motivate a bit more the chosen model, i.e. (33). First of all, as already mentioned, the theory is part of the Horndeski action and as such, even though it contains higher order coupling terms, it does not propagate new degrees of freedom (two of the metric plus one of the scalar field). Furthermore, this very coupling was proposed as a remedy to Higgs inflation that suffered from dangerous quantum corrections, and is known as new Higgs infation [87]. Last but not least, the same higher order term appears when one takes the D \u2192 4 limit of the Gauss-Bonnet gravity [88]. The last few months there has been an increased interest in the so-called Einstein-Gauss-Bonnet theories of gravity in (3 + 1) dimensions, see [89-93] and references therein. It is reasonable thus to ask what if the Universe oxidized from (1 + 1) to (2 + 1) and finally to (3 + 1) dimensions as the limit of the Gauss-Bonnet gravity in arbitrary D dimensions. The problem however arises from the fact that in D < 4 the Gauss-Bonnet term identically vanishes. On top of that, in (2+1) dimensions, the Riemann tensor vanishes as well, thus surviving only terms like K in (15). There has been some studies in lower dimensional Gauss-Bonnet gravity [92, 93], where they find BTZ like black holes, however, all of them have anti-de Sitter asymptotics. Proceeding now with the variation of the action (33) with respect to the metric we get where P \u03b1\u03b2\u00b5\u03bd is the dual of the Riemann tensor, One can easily notice that for \u03c9 = 0 = B , we can recover Einstein equations with a cosmological constant. The related black hole solutions is the Schwarzschild-de Sitter geometry. Moreover, if we vary the action with respect to the dilaton, we get the equation of motion for it, that can be written as: with The mass parameter \u00b5 can be written as \u00b5 = aG N M where M is the mass of the black hole and the positive constant a can be specified from the corresponding Newtonian limit. The parameter q has dimensions of an energy, [ E ], and its value is specified by q 2 \u03c9 = \u039b -\u039b eff . From this relation we see that \u039b \u2265 \u039b eff must be satisfied for (39) to be real (we remind that \u03c9 > 0 as the standard kinetic term). Note that Eq. (38) can be also expressed as f ( r ) = 1 -2 \u00b5 r -1 3 ( \u039b 1+ q 2 | B | ) r 2 .", "pages": [6, 7]}, {"title": "B. Gravitational instantons", "content": "We build again two instantons from the metric (38): one for the de Sitter background and one for the Nariai black hole. For the de Sitter instanton ( \u00b5 = 0 in (38)), the metric potential is f dS ( r ) = 1 -\u039b eff 3 r 2 . The cosmological horizon lies at r c = \u221a 3 / \u039b eff with a temperature of T = 1 2 \u03c0r c = \u03b2 -1 . The de Sitter instanton then reads After performing the transformation \u03c4 = r c \u03be, r = r c cos \u03c7 , the above instanton can be cast into the regular form of A detailed discussion about the behavior of the scalar field itself, or of the induced Noether current at the horizon, can be found in [85]. Rather, we will directly proceed to the scalar solution around a self-tuning Schwarzschildde Sitter black hole, which is of interest in our paper. We start by considering a static and spherically symmetric metric of the form 2 with d\u2126 2 = d \u03b8 2 + sin 2 \u03b8 d \u03c6 2 and a scalar field which has, apart from the r dependence from the metric, an additional time-dependence, i.e., \u03c8 = \u03c8 ( t, r ) . It has been shown in [85] that equations (34) and (36) are satisfied when In addition, since we have Wick-rotated the time axis, the dilaton (39) becomes \u03c8 = -iq\u03c4 + \u03c6 ( r ) . For the Nariai black hole we have the relations 9 \u00b5 2 \u039b eff = 1 and \u03c1 = 3 \u00b5 = 1 / \u221a \u039b eff where \u03c1 = r 2 = r 3 is the degenerate horizon. After a transformation for approximately degenerate black holes with glyph[epsilon1] being again a small length parameter, the metric potential (38) takes the approximate form of f N ( \u03c7 ) \u2248 \u039b eff glyph[epsilon1] 2 sin 2 \u03c7 and, in the limit glyph[epsilon1] \u2192 0 , the instanton reads Also the dilaton becomes with Note that even though the dilaton, both in (39) and in (45), may look divergent on the horizon or when glyph[epsilon1] \u2192 0, it is not. One can verify this by changing to the generalized Eddington-Finkelstein coordinates.", "pages": [7, 8]}, {"title": "C. Pair creation rate", "content": "The evaluation of the Euclidean version of the action (33) for the de Sitter and the Nariai instanton will give respectively and A note may be necessary here: even though the bare cosmological constant \u039b does not appear in the spacetime metric, it appears explicitly in the solution of the scalar field, through q and also in the action (33); that is why it enters the instanton action. Substituting (47) and (48) in (8), we get a Nariai rate of the form We recall that, in the | B | \u2192 0 limit, there is no nontrivial solution for the scalar field for any \u03c9 . This means that black holes (as well as the de Sitter Universe) and the above instantons are exactly the same as in general relativity. This confirms that black holes with non-trivial dilatonic configurations can exist only in the presence of the higher derivative coupling (John term). We also remark that the rate (49) exceeds the conventional Bousso-Hawking (BH) rate [40]: for any nonvanishing q and B one has \u0393 (3+1) N > \u0393 BH = e -\u03c0/ \u039b G N . In addition the dilaton coupling provides a richer phenomenology: This result follows a natural continuation of the expansion of the Universe. In our model both the dilaton and the cosmological constant drive the inflationary era. At the end of inflation, there might be a time window where the value of \u039b is relatively higher compared to 2\u039b eff (\u039b \u2265 2\u039b eff ), allowing this way the prolific production of relatively large and cold black holes with respect to the Planckian ones. As the Universe continues to expand, the value of \u039b decreases even further, reaching the range of values \u039b eff < \u039b < 2\u039b eff at late times where the production stops. We remind here that \u039b \u2265 \u039b eff always, in order for the dilaton to be real. This condition would also save the catastrophic instability of de Sitter space in the late Universe [94].", "pages": [8]}, {"title": "V. COMPARISON OF THE RATES", "content": "Now the scenario of the oxidation has been completed. Assuming that the theories describing gravitational interactions in each phase are those described in the previous section, the Nariai rates for each dimensional phase are As a first comment one can see that the dimensionality of the gravitational coupling constant affects the functional dependence of rates in each of the above regimes. More importantly, the above calculation offers a possible history of a lower dimensional Universe in relation to the spontaneous production of PBHs, as seen in Fig 2 and 3, where for ease of notation we introduced the de Sitter radius glyph[lscript] , to express the cosmological constants \u039b 2 , \u039b 3 and \u039b as 1 /glyph[lscript] 2 . Specifically, when the Universe was in its effective (1+1) dimensional era, PBHs would have been plentifully produced since the rate (50) is unsuppressed. Conversely, during the (2 + 1) dimensional phase, the rate (51) is exponentially suppressed but to a lesser extent than the conventional Bousso-Hawking result. For G 2 3 \u039b 3 \u223c 1 the black hole nucleation is not negligible. Hence, a lower dimensional Universe could enhance the population of PBHs relative to standard (3 + 1) scenario, provided that the magnitude of the cosmological constants started with Planckian values during the Planck era and then decreased as the Universe expanded. Nevertheless, all the black holes formed before inflation would have been exponentially diluted leaving no observational traces. The only possibility for significant effects, is that the lower dimensional phases left an imprint of its pair production at short scales after inflation. Being the oxidizing dimension an effective quantity related to the local fractality of the spacetime, such an occurrence might be acceptable also in the case the topological dimension of the (ambient) spacetime is four. As a result, the scenario is compatible with the standard paradigm of the inflation. Regarding the (3+1) phase, one finds the most promising results since the presence of the parameter \u039b eff allows to circumvent the issue of the production prior/after the inflation. The dilaton corrects the Bousso-Hawking rate with an unsuppressed part. The new Nariai rate leads to an unsuppressed production even at the final stages of inflation, as long as \u039b > 2\u039b eff . We must stress here that we expect for \u039b eff to have much lower value than those displayed in Fig. 2, in order for the production to continue beyond the inflationary era. Inflation should have lasted at least for 60 e-foldings in order to solve the various cosmological problems (horizon, flatness and monopole problem), providing a lower post-inflationary value for the de Sitter radius of the order glyph[lscript] end glyph[greaterorsimilar] e 60 L P . In the plot we just give some arbitrary values for \u039b eff to see the new corrected behavior of the (3 + 1) rate, even post-inflationary unsuppressed rates are clearly admissible. Conversely if the value of \u039b enters the domain \u039b eff < \u039b < 2\u039b eff before the Universe reaches the late time expansion, the rate becomes suppressed at late times with no observational consequences. On the ground of this reasoning, one can find a concrete result stemming from our investigation. By using the current value of the cosmological constant and by invoking the Universe stability, i.e., no black hole production, the present-day value of \u039b eff can be constrained by using the relation \u039b eff < \u039b < 2\u039b eff . Being \u039b glyph[similarequal] 2 . 888 \u00d7 10 -122 in Planck units [95], one finds:", "pages": [8, 9]}, {"title": "VI. FINAL REMARKS", "content": "In this paper, we assumed that the Universe underwent a dimensional oxidation before it reached its current form. Specifically, at very high temperatures the Universe had (1 + 1) dimensions, later on as it cooled down, it became (2+1) dimensional and finally, it took its known (3+1) dimensional form. Taking this for granted, for each of the above phases, we calculated the probability of Schwarzschild-de Sitter black holes to be nucleated inside an expanding de Sitter background. We did this by using the no boundary proposal to calculate the associated gravitational instantons. In the (1 + 1) phase the existence of a dilaton is necessary. In (2 + 1) we saw that, in order for black holes to exist, the dilaton should be trivial and higher order derivative terms should appear in the action. In its current (3 + 1) phase, there is no necessity for the dilaton to exist, however if it does it solves many of the shortcomings that general relativity possesses. That is why we considered a specific case of the Horndeski action that contains a Galileon field which is symmetric under the shift transformation \u03c8 \u2192 \u03c8 + constant. In cosmology this model is very successful in describing the late-time acceleration of the Universe with a self-accelerating solution given by the scalar field. We calculated the Nariai pair creation rate in each phase. Pre-inflationary production is not of interest because pre-inflationary nucleated black holes would have been washed away by inflation. If there is, however, an imprint of the lower dimensional phase after inflation, then the production of lower dimensional Planckian relics mayb still continue until today. There is an additional production from the (3 + 1) phase, depending on the values of the coupling parameters and specifically for \u039b > 2\u039b eff . Then the rate is unsuppressed for this range of values even at the final stages of inflation. However, the decrease of \u039b should start satisfying the relation \u039b eff < \u039b < 2\u039b eff as we approach present times. Interestingly the proposed investigation offers a concrete result. The condition \u039b eff < \u039b < 2\u039b eff can be applied to the current Universe to obtain compelling constraints for the John term of the Horndenski action.", "pages": [9, 10]}, {"title": "Acknowledgments", "content": "K.F.D would like to thank the Institute of Space Sciences and Astronomy of the University of Malta where part of this work was conducted. The work of P.N. has been partially supported by GNFM, the Italian National Group for Mathematical Physics. The work of A.G.T. has been supported by the GRADE Completion Scholarships, which are funded by the STIBET program of the German Academic Exchange Service (DAAD) and the Stiftung zur Forderung der internationalen wissenschaftlichen Beziehungen der Johann Wolfgang Goethe-Universitat. The authors acknowledge the networking support by the COST Actions GWverse CA16104 and CANTATA CA15117. Late-time Evolution,' Phys. Rept. 692 , 1 (2017).", "pages": [10, 12]}] |
2024arXiv241114705R | https://arxiv.org/pdf/2411.14705.pdf | <document>
<section_header_level_1><location><page_1><loc_12><loc_82><loc_89><loc_85></location>Starkiller : subtracting stars and other sources from IFU spectroscopic data through forward modeling</section_header_level_1>
<text><location><page_1><loc_11><loc_80><loc_88><loc_81></location>Ryan Ridden-Harper , 1 Michele T. Bannister , 1 Sophie E. Deam , 1, 2 and Thomas Nordlander 3, 4</text>
<figure>
<location><page_1><loc_85><loc_80><loc_88><loc_81></location>
</figure>
<text><location><page_1><loc_8><loc_76><loc_91><loc_78></location>1 School of Physical and Chemical Sciences - Te Kura Mat¯u, University of Canterbury, Private Bag 4800, Christchurch 8140, Aotearoa New Zealand</text>
<text><location><page_1><loc_10><loc_73><loc_90><loc_76></location>2 Space Science and Technology Centre, School of Earth and Planetary Sciences, Curtin University, Perth, Western Australia 6845, Australia</text>
<text><location><page_1><loc_18><loc_71><loc_81><loc_73></location>3 Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT 2611 4 ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia</text>
<text><location><page_1><loc_40><loc_68><loc_60><loc_69></location>(Received November 22, 2024)</text>
<text><location><page_1><loc_44><loc_65><loc_56><loc_67></location>Submitted to AJ</text>
<section_header_level_1><location><page_1><loc_45><loc_62><loc_55><loc_63></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_43><loc_86><loc_62></location>We present starkiller , an open-source Python package for forward-modeling flux retrieval from integral field unit spectrograph (IFU) datacubes. starkiller simultaneously provides stellar spectral classification, relative velocity, and line-of-sight extinction for all sources in a catalog, alongside a source-subtracted datacube. It performs synthetic difference imaging by simulating all catalog sources in the field of view, using the catalog for positions and fluxes to scale stellar models, independent of the datacube. This differencing method is particularly powerful for subtracting both point-sources and trailed or even streaked sources from extended astronomical objects. We demonstrate starkiller 's effectiveness in improving observations of extended sources in dense stellar fields for VLT/MUSE observations of comets, asteroids and nebulae. We also show that starkiller can treat satelliteimpacted VLT/MUSE observations. The package could be applied to tasks as varied as dust extinction in clusters and stellar variability; the stellar modeling using Gaia fluxes is provided as a standalone function. The techniques can be expanded to imagers and to other IFUs.</text>
<text><location><page_1><loc_14><loc_35><loc_86><loc_39></location>Keywords: Small Solar System bodies (1469), Interstellar objects (52), Artificial satellites (68), Extended radiation sources (504), Interdisciplinary astronomy (804), Open source software (1866)</text>
<section_header_level_1><location><page_1><loc_20><loc_32><loc_36><loc_33></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_17><loc_48><loc_31></location>Integral field unit spectrographs (IFUs) combine the strengths of imaging and spectroscopy: both spatial and spectral resolution for every spaxel in a given field of view. Coupled with adaptive optics, this makes them a mainstay of extended-source observational astronomy -serving a variety of communities including stellar evolution, star clusters, Galactic and extragalactic science. Multiple generations of IFUs have been built for facilities around the world, from WiFeS on the ANU 2.3 m at</text>
<text><location><page_1><loc_8><loc_12><loc_35><loc_14></location>Corresponding author: Ryan Ridden-Harper [email protected]</text>
<text><location><page_1><loc_52><loc_24><loc_92><loc_33></location>Siding Spring (Dopita et al. 2010), to MUSE on ESO's UT4 of the VLT (Bacon et al. 2010) and JWST's NIRSpec (Boker et al. 2022). Indeed, every proposed thirtymetre-class optical facility (the ELTs) includes an IFU as a first-light instrument. However, IFU data currently presents challenges for some science cases.</text>
<section_header_level_1><location><page_1><loc_55><loc_19><loc_89><loc_22></location>1.1. Case 1: a background extended target with foreground stellar fluxes e.g. a nebula</section_header_level_1>
<text><location><page_1><loc_52><loc_9><loc_92><loc_18></location>In the situation where an extended source such as a nebula or low-surface-brightness galaxy has a foreground and/or background stellar field, the data will be acquired with sidereal tracking, and all sources will have circular point-source point-spread functions (PSFs). The density of the stellar field can limit the</text>
<text><location><page_2><loc_8><loc_86><loc_48><loc_92></location>potential inference of the extended source's morphology and composition. The PampelMUSE package (Kamann et al. 2013) provides precise PSF fitting for photometry in dense stellar fields in IFU data.</text>
<section_header_level_1><location><page_2><loc_11><loc_82><loc_46><loc_84></location>1.2. Case 2: a foreground extended target with background trailed stellar fluxes e.g. a comet</section_header_level_1>
<text><location><page_2><loc_8><loc_43><loc_48><loc_81></location>In contrast to other spectroscopic modes, IFUs have seen comparatively little use by the Solar System smallbodies community, whose targets are frequently unresolved sources. A common use of IFUs in Solar System studies has been atmospheric characterisation of bright planets (e.g. instruments such as Gemini/NIFS and VLT/SINFONI (Simon et al. 2022)). However, IFUs are ideal for compositional studies of small Solar System bodies with activity creating extended comae - but like all Solar System targets, these worlds move across the sky. The requirement of non-sidereal telescope tracking to increase signal-to-noise streaks all background sources. For imaging, median stacking can adequately remove the streaked stellar signal, but this often limits any characterization of time-varying phenomena. Studies of active minor planets are historically kept to areas of sky with low stellar density, with observation programs paused when targets traverse the Galactic plane (e.g. the 2022-23 DART mission post-impact followup campaign; Moskovitz et al. (2024); Kareta et al. (2023)). For imaging in dense stellar fields, image subtraction is now a robust technique that aids both minor planet detection, and characterization such as lightcurve studies, but it is not yet used frequently for IFUs.</text>
<section_header_level_1><location><page_2><loc_9><loc_39><loc_47><loc_42></location>1.3. Case 3: a celestial target with foreground streaks e.g. satellite streaks or trailed asteroids</section_header_level_1>
<text><location><page_2><loc_8><loc_14><loc_48><loc_39></location>As the industrialization of near-Earth space increases, the astronomical communities also face the advent of streak-smeared data. The accelerating rate of satellite constellation emplacement into low-Earth orbit (LEO) means > 6400 have been launched, with > 5800 operationally in place as of April 2024 1 ; at least 20,000100,000, with potential for around a million LEO satellites, will be in place in the mid-2030s (Walker et al. 2020, 2021; Falle et al. 2023). The population will then need to be maintained at that level by ongoing launches. These satellites produce an industrially-caused environmental impact on astronomical observations: out-offocus streaks of reflected Solar flux if they traverse the field of view, obscuring science targets, during an exposure. While the probability of satellite streaks affecting the smaller fields of view of IFUs is lower than</text>
<text><location><page_2><loc_52><loc_71><loc_92><loc_92></location>that for the massively wide-field imager of the Vera C. Rubin Observatory, the probability of effects on astronomical imaging is already non-zero (Walker et al. 2020; Michaglyph[suppress]lowski et al. 2021; Walker et al. 2021). IFUs typically acquire longer integrations on target than e.g the 15s/30 s exposures of the LSST, so there is a different scope for adverse impact on IFUs. We demonstrate example MUSE data impacts in § 3.8. The environmental impacts will only continue to increase, particularly in the era of ELTs. By the expected European ELT first light, a steady-state LEO population kept by industry at some 30,000 satellites could exist; it is certainly on track to exceed 15,000.</text>
<text><location><page_2><loc_52><loc_64><loc_92><loc_71></location>For distant celestial targets acquired with sidereal tracking, near-field minor planets, particularly asteroids at geometries outside of quadrature, will also trail to varying degrees. The data on both types of astronomical object will be of interest to different communities.</text>
<section_header_level_1><location><page_2><loc_61><loc_60><loc_83><loc_61></location>1.4. The starkiller package</section_header_level_1>
<text><location><page_2><loc_52><loc_34><loc_92><loc_59></location>Here we present a new open-source forward-modeling approach to removing stellar flux and satellite streaks from IFU datacubes, regardless of whether they are trailed or round. We model the flux of the stars in the field that are identifiable in a source catalog, via stellar atmosphere models. As the flux can be streaked or circular, a trailed point-spread function (PSF) is constructed to fit the stellar PSF of each datacube. We apply location-appropriate dust extinction and find the best-match stellar spectra for each star. The streaks then form a simulated data cube, which is subtracted from the original (e.g. Fig. 1). This approach allows us to replicate traditional difference imaging to remove background stars without the need for a reference data cube. All magnitudes are assumed to be AB, and fluxes in terms of F λ (erg / s / cm 2 / ˚ A) unless otherwise stated.</text>
<text><location><page_2><loc_52><loc_20><loc_92><loc_34></location>Our approach is inherently generalisable between instruments. We use VLT/MUSE here as a case study, with two extended small Solar System bodies, a nebula ( § 3.7), and a satellite streak that pass over a blazar ( § 3.8) as examples. With suitable stellar models across the appropriate wavelength range, and a wellcharacterized instrumental PSF, starkiller can be extended to other IFUs ( § 4.5). Pull requests are welcome 2 .</text>
<section_header_level_1><location><page_2><loc_69><loc_17><loc_75><loc_18></location>2. DATA</section_header_level_1>
<text><location><page_2><loc_61><loc_15><loc_84><loc_16></location>2.1. Example IFU: VLT/MUSE</text>
<figure>
<location><page_3><loc_9><loc_70><loc_36><loc_91></location>
<caption>Figure 1. Left: Reduced MUSE datacube at 7000 ˚ A of 2I/Borisov at b = 26 · on 2020-03-19, with position-corrected Gaia DR3 stars shown as orange stars. Center: Scene constructed by starkiller on Gaia -listed stars in the field of view. Right: The scene-subtracted cube at 7000 ˚ A. Subtraction quality varies between sources, influenced primarily by the closeness of the model spectra, and crowding. Sources that are not present in the Gaia DR3 catalog are not subtracted, and are therefore present unaltered in the subtracted scene.</caption>
</figure>
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<location><page_3><loc_38><loc_70><loc_64><loc_92></location>
</figure>
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<location><page_3><loc_66><loc_70><loc_91><loc_90></location>
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<text><location><page_3><loc_79><loc_89><loc_81><loc_92></location>-</text>
<text><location><page_3><loc_8><loc_46><loc_48><loc_60></location>The ESO Very Large Telescope's Multi Unit Spectroscopic Explorer (VLT/MUSE; Bacon et al. 2010) is a panoramic integral field unit spectrograph covering 4000 ˚ A-9300 ˚ A, on the 8.2 m UT4 optical telescope at Paranal, Chile. In wide-field mode, the field of view (FOV) is 1 ' × 1 ' , ideal for imaging extended sources. The optionally adaptive-optics corrected light is split equally and fed to 24 individual spectrographs (integral field units) (Bacon et al. 2010).</text>
<text><location><page_3><loc_8><loc_35><loc_48><loc_46></location>We used contrasting MUSE datasets acquired with both sidereal and non-sidereal tracking for our primary development and testing of starkiller . For sidereal data, we use two example selections from the ESO Archive: the planetary nebula NGC 6563, and satelliteimpacted observations from 2021-22 of blazar WISEA J014132.24-542751.0 (J0141-5427).</text>
<text><location><page_3><loc_8><loc_17><loc_48><loc_35></location>We use two non-sidereal MUSE datasets, one from each MUSE mode, each of which present different dataanalysis challenges. The first is of the interstellar comet 2I/Borisov, observed on 16 epochs in 2019-2020 (Bannister et al. 2020; Deam et al. 2024). These data have increasingly dense stellar backgrounds, as 2I/Borisov moved from 49 · outside the Galactic plane to within the plane after it passed perihelion. The stars are streaked up to ∼ 20 '' at high galactic latitude, with the shortest ∼ 4 '' at low galactic latitude. 2I/Borisov has a compact coma, entirely contained within at most 46 '' diameter, and thus fully within the MUSE WFM FOV.</text>
<text><location><page_3><loc_8><loc_12><loc_48><loc_16></location>Our other non-sidereal dataset is from followup of the NASA DART mission's impact on 2022 September 26 of the near-Earth asteroid moon Dimorphos (Opitom</text>
<text><location><page_3><loc_52><loc_37><loc_92><loc_60></location>et al. 2023; Murphy et al. 2023). 3 The resulting debris formed extended and time-varying structures over the following month relative to the bright parent body in the system, Didymos. While Opitom et al. (2023); Murphy et al. (2023) acquired 11 epochs, only the last three have dense stellar backgrounds, when Didymos moved onto the Galactic plane. These data were acquired at very rapid motion rates tracked on Didymos due to its geocentric proximity of only 0.08-0.09 au, which produced longer stellar streaks than those in the 2I/Borisov data; most streaks are not completely enclosed in the FOV. The alignment of the FOV was offset on Didymos and rotated 90 · with respect to that of the 2I data, with the data acquired in MUSE's 8 '' × 8 '' narrow-field mode with AO.</text>
<section_header_level_1><location><page_3><loc_65><loc_34><loc_79><loc_36></location>2.2. Catalog: Gaia</section_header_level_1>
<text><location><page_3><loc_52><loc_21><loc_92><loc_34></location>To identify stars within each cube, we use the Gaia DR3 source catalog for star positions and brightness. Gaia provides an all-sky catalog of sources with a limiting magnitude of 22 and a saturation magnitude of ∼ 3 (Gaia Collaboration et al. 2016, 2023; Babusiaux et al. 2023). The precise positions of Gaia sources assists in sub-spaxel alignment of stars, while the broad G band magnitudes are ideal for scaling model spectra flux.</text>
<text><location><page_3><loc_52><loc_14><loc_92><loc_21></location>As starkiller assumes all magnitudes are in the AB system, we must apply a correction to the Gaia DR3 magnitudes, which are presented in the Vega system. We compute the correction by following the same procedure as Axelrod et al. (2023), comparing observed Gaia</text>
<text><location><page_3><loc_74><loc_90><loc_79><loc_92></location>MUSE</text>
<text><location><page_3><loc_81><loc_90><loc_85><loc_92></location>Scene</text>
<figure>
<location><page_4><loc_9><loc_71><loc_47><loc_92></location>
<caption>Figure 2. Distribution of differences between the Synthetic G band AB magnitudes ( G Syn ) and the observed G band Vega magnitudes ( G Obs ) for DA white dwarf calibrators. The offset between the two magnitudes is primarily due to the differences in magnitude systems. There is also a small calibration offset, which is described in Axelrod et al. (2023).</caption>
</figure>
<text><location><page_4><loc_29><loc_70><loc_30><loc_72></location>-</text>
<text><location><page_4><loc_8><loc_50><loc_48><loc_59></location>Gmagnitudes (Vega) to synthetic AB magnitudes in the Gfilter for 17 well-calibrated DA white dwarfs (Narayan et al. 2019). We take the median offset between the synthetic and observed magnitudes to be the correction factor to map Gaia G (Vega) to Gaia G (AB). Using the median in Fig. 2, the correction becomes:</text>
<formula><location><page_4><loc_20><loc_46><loc_48><loc_48></location>G AB = G V ega +0 . 118 (1)</formula>
<section_header_level_1><location><page_4><loc_17><loc_41><loc_39><loc_43></location>2.3. Stellar Atmosphere Models</section_header_level_1>
<text><location><page_4><loc_8><loc_24><loc_48><loc_41></location>High-quality stellar atmosphere models are essential for reliably representing stars within the data cubes. For accurate spectral matching, we need a diverse set of model spectra that span wide ranges in effective surface temperature T eff , surface gravities log(g), and metallicities. In this initial demonstration of starkiller we also prioritize ease of use, so we restrict ourselves to smaller spectral libraries that can be installed alongside the base code. In order to best represent the spectral types of any star, we provide multiple models and implementation pathways between these models.</text>
<text><location><page_4><loc_8><loc_16><loc_48><loc_23></location>Primarily, we use the Castelli & Kurucz (2003) stellar atmosphere models (CK models) as a basis for stellar spectra comparisons. We choose the STScI subsection of the total CK atlas to cover the key range of spectral types 4 . We also include the ESO stellar spectra li-</text>
<text><location><page_4><loc_52><loc_56><loc_92><loc_92></location>brary 5 which use the Pickles (1998) stellar spectra, supplemented with corrections from Ivanov et al. (2004). (While there is a MUSE-specific library, it only contains 35 stellar spectra and is restricted to the MUSE wavelengths, so we do not use it here). We include a partial grid of medium-resolution ( R = 20000) sampled fluxes from the MARCS grid (Gustafsson et al. 2008). We have selected spectra covering the most common types of latetype stars, with T eff = 3000-8000K and log g between -0 . 5 and +5 . 0, in a pattern that broadly follows the main sequence and red giant branch. The models have [Fe / H] = -0 . 5, 0.0 and +0 . 5, using solar-scaled abundances except that [ α/ Fe] = +0 . 2 when [Fe / H] < 0. For simplicity, a single value of the micro-turbulence value v mic = 2kms -1 was selected. Models of both planeparallel and spherical geometry (assuming stellar masses of one solar mass) are included where available. Finally, we include the PoWR grid of OB-type synthetic spectra (Hainich et al. 2019), called OB-i . Specifically, we include the solar-metallicity grid that covers much of the parameter space T eff = 15-56kK and log g = 2 . 04.4, broadly corresponding to the evolution of stars of roughly 7-60 M ⊙ .</text>
<section_header_level_1><location><page_4><loc_61><loc_52><loc_83><loc_53></location>3. ANALYSIS STRUCTURE</section_header_level_1>
<text><location><page_4><loc_52><loc_37><loc_92><loc_51></location>The process of analysis of starkiller is generalized to operate on any optical IFU data that has the same header and HDU format as VLT's MUSE, including WCS information. We have minimized the amount of additional inputs required, with a strong preference towards self-determination of key information from the input datacube. The following sections outline the key steps we use in determining the stars within the field of view (FOV), and modeling those sources.</text>
<section_header_level_1><location><page_4><loc_65><loc_33><loc_79><loc_34></location>3.1. Source catalog</section_header_level_1>
<text><location><page_4><loc_52><loc_21><loc_92><loc_32></location>By default, we use the Gaia DR3 catalog (Gaia Collaboration et al. 2023) as the source catalog. starkiller obtains the Gaia sources within a radius defined by the size of the IFU centered on the R.A. and Decl. provided in the input cube's header. We access the Gaia DR3 catalog I/355/gaiadr3 through Vizier via astroquery .</text>
<text><location><page_4><loc_52><loc_15><loc_92><loc_21></location>Alternatively, starkiller also accepts user input source catalogs. The input catalogs must contain columns specifying the source ID (id), R.A. (ra), and Dec. (dec) in degrees, magnitude ( x mag), and a filter</text>
<text><location><page_5><loc_8><loc_89><loc_48><loc_92></location>designation for the SVO filter service 6 ( x filt) where x is the desired filter shorthand.</text>
<text><location><page_5><loc_8><loc_81><loc_48><loc_88></location>If multiple filters are specified for a single source, then the model spectra will be reshaped to match the magnitudes in all filters; unless the key filter argument is defined, identifying the filter to which to normalise the flux.</text>
<section_header_level_1><location><page_5><loc_21><loc_76><loc_35><loc_77></location>3.2. WCS correction</section_header_level_1>
<text><location><page_5><loc_8><loc_42><loc_48><loc_75></location>As the purpose of starkiller is to simulate spectral data cubes to provide simulated differenced cubes, precise positions of sources are essential: while this may be straightforward for siderally tracked observations, for those tracked non-sidereally, both the spatial WCS solutions and source identification require additional refinement. For MUSE, the WCS solution produced by the MUSE data reduction pipeline (Bacon et al. 2016) is propagated from the target coordinates defined in the VLT's observing block. For sidereally tracked observations, the spatial WCS solutions are effective: only minor corrections on the order of a spaxel are generally required. In non-sidereal cases, the offset needed from the Bacon et al. (2016) WCS is determined by the on-sky motion rate of the target, relative to the ephemeris timestamp choice when the VLT begins observation setup, modulo the typical MUSE acquisition time of 10-15 minutes. For example, we find that the WCS solution can be offset by an arcminute in our 2I/Borisov data. However, elongated sources then present a challenge for conventional source identifiers.</text>
<text><location><page_5><loc_8><loc_18><loc_48><loc_42></location>To identify sources in the IFU, regardless of shape, we use clustering algorithms on an 'image' constructed from a median stack of the input cube in wavelength space. We create a Boolean image with a percentile cut, selecting for spaxels that are brighter than the 90th percentile. We then label sources in the Boolean image with scipy.ndimage.label . Sources are then downselected to retain the labeled objects with a total spaxel count between 0.01% and 10% of the total IFU spaxels 7 . This cut limits contamination from sources that partially fall within the FOV (lower limit) and the background spaxels (upper limit). The center points of the labeled sources are taken to be the average x and y spaxel positions of spaxels for each source. We also extract initial guesses for the stellar PSF, by estimating</text>
<text><location><page_5><loc_52><loc_89><loc_92><loc_92></location>the trailing length, trail angle, and the x and y spaxel extents.</text>
<text><location><page_5><loc_52><loc_71><loc_92><loc_88></location>With sources identified in the image, we fit basic offsets from the star catalog to the image. Within starkiller there are two methods to conduct an initial match, and a final method to refine matching sources with the catalog. The first and less reliable preliminary match method is to fit for for x , y , and rotational offsets, by minimizing the distance of the spaxel coordinates of the brightest stars in the catalog to the image sources. However, this method struggles for crowded fields, where the regions labeled as sources are composites of multiple sources and not representative of the PSF.</text>
<text><location><page_5><loc_52><loc_43><loc_92><loc_71></location>The second preliminary match method, which is robust to crowding, assumes that the position angle on the sky has low error, and matches the catalog through shifts. In this method we first iterate through x and y offsets to the raw catalog spaxel coordinates from -100 to 100 spaxels, in steps of 10 spaxels. For each shift, we generate a simulated image by convolving sources within the image bounds with the profile of the median labeled source, and subtract this from the labeled image, which is altered such that sources are represented by 1 and all other spaxels are NaN. The pair of x and y offsets that provide the smallest residual are then taken as the starting parameters, to minimize the residual between the two images with scipy.minimize . This method is the default method used in starkiller and provides close matches, even in crowded fields. For point sources this match is within 1 spaxel; the uncertainty does increase in crowded fields with high source elongation.</text>
<text><location><page_5><loc_52><loc_21><loc_92><loc_43></location>Following the initial catalog match, the WCS correction is then refined through PSF fitting. The creation of the PSF that is used in this routine is outlined in § 3.4. We take the x and y positions of the calibration sources found through PSF fitting, and compare those with the shifted catalog positions. Following a method similar to the first catalog matching method discussed, we minimize the distance between the shifted catalog positions and the PSF positions through x and y shifts as well as a rotation θ around the image center. This refined PSF shift matches observed sources with the source catalog positions to a sub-spaxel precision. For trailed sources, we consider the center points as defined by the PSF to be the observed position.</text>
<text><location><page_5><loc_52><loc_10><loc_92><loc_21></location>With these methods developed for starkiller , we are able to correct for any errors present in the spatial WCS solution due to challenging observational conditions, such as non-sidereal tracking. While effective, these methods may lose reliability in highly crowded fields or with very elongated sources. We discuss this further in § 4.4.</text>
<figure>
<location><page_6><loc_9><loc_64><loc_46><loc_91></location>
<caption>Figure 3. Comparison of the Gaia DR3 star position using the MUSE WCS (blue +), and the starkiller corrected positions (orange star). Through the starkiller reduction process sources are realigned assuming a linear offset, plus rotation. In all instances sources are aligned to be at the center of their corresponding streaks. The alignment method used by starkiller has proven to be robust to source elongation and crowding.</caption>
</figure>
<section_header_level_1><location><page_6><loc_20><loc_48><loc_36><loc_49></location>3.3. Isolating sources</section_header_level_1>
<text><location><page_6><loc_8><loc_36><loc_48><loc_47></location>Identifying isolated sources in non-sidereal data presents an interesting challenge as the sources may be streaked to any length, and aligned on any angle. As with the WCS correction, we adopt a 2 stage approach to identifying isolated sources, where we use the initial PSF approximations in stage 1, which we then refine with the preliminary PSF in stage 2.</text>
<text><location><page_6><loc_8><loc_16><loc_48><loc_36></location>In stage 1, we rotate the coordinates of the catalog sources and image according to the estimated trail angle such that the trails are vertical. A source is then considered to be isolated if it is more than 8 spaxels from a neighbor in the x direction (PSF minor axis), and separated by more than 1.2 times the total trail length in the y direction (PSF major axis). We also incorporate magnitude information into the isolation criterion. If nearby sources are at least 2 magnitudes fainter, it is assumed that their contribution is small and therefore ignored when calculating source distances. The isolated sources identified in this process are used to generate the first iteration of the PSF.</text>
<text><location><page_6><loc_8><loc_10><loc_48><loc_15></location>Stage 2 relies on a PSF being defined to determine a refined calibration source list. To identify isolated sources using the PSF, we check overlaps between Boolean masks, created by placing the PSF at each</text>
<text><location><page_6><loc_52><loc_78><loc_92><loc_92></location>source position where spaxels must contribute more than 1 × 10 -5 to the total PSF. If any two masks contain overlapping points, the sources are considered to be overlapping. By using the PSF information, we can reliably identify isolated sources, regardless of trail length or orientation. Sources we identify as 'isolated' through this process are then used as the final calibration sources, from which the final PSF and WCS correction are calculated.</text>
<section_header_level_1><location><page_6><loc_65><loc_73><loc_79><loc_74></location>3.4. PSF modeling</section_header_level_1>
<text><location><page_6><loc_52><loc_56><loc_92><loc_72></location>A precise PSF is key to creating accurate simulations of data cubes. We developed starkiller to model PSFs for static sources and elongated sources, generated from sidereal and non-sidereal tracking. Our PSF module is built from the PSF module within TRIPPy (Fraser et al. 2016), which downsamples a high resolution PSF to the image resolution. Alongside the Moffat profile PSF used in TRIPPy , we also incorporate a Gaussian profile PSF, and a data-generated PSF; we refer to the latter as the 'data PSF'.</text>
<text><location><page_6><loc_52><loc_39><loc_92><loc_56></location>As constructed in TRIPPy , a trailed PSF can be reliably modeled by a non-trailed point source PSF profile convolved with a line model. Fitting trailed PSFs therefore requires fitting the PSF profile parameters alongside the line model which incorporates trail length and angle, which we choose to be the angle measured counterclockwise from the x axis. In starkiller the nontrailed point source PSF profile can be either Moffat, or symmetric 2D Gaussian, and is defined by fitting to calibration sources that are considered 'isolated' and brighter than a user-defined calibration magnitude limit.</text>
<text><location><page_6><loc_52><loc_23><loc_92><loc_39></location>When constructing the model PSF, we simultaneously fit all parameters for the profile and line element, including small positional offsets. The models are generated at 10 times the spatial resolution and downsampled to the data resolution. While the PSF is dependent on wavelength, we find that these variations are small for the highly streaked stars in MUSE data. Therefore, to optimize the signal-to-noise of the calibration sources, starkiller fits a single PSF using the median stack of all wavelengths.</text>
<text><location><page_6><loc_52><loc_9><loc_92><loc_23></location>A further complication to the PSF fitting process is variability in the atmospheric seeing. For sidereal tracking, variations in seeing contribute evenly to the total PSF throughout the exposure. However, for nonsidereal tracking, the trailed PSF becomes a record of the seeing variability during the exposure, where each segment of the trail may be constructed by different seeing conditions to another section. A trailed PSF may therefore be quite different from a distribution that can</text>
<text><location><page_7><loc_8><loc_89><loc_48><loc_92></location>be readily modeled by a simple elongated Moffat, or Gaussian profile.</text>
<text><location><page_7><loc_8><loc_59><loc_48><loc_88></location>An example of highly trailed stars can be seen in Fig. 4. Stars A & B 8 (Fig. 4, top two panels) were observed alongside 2I/Borisov with MUSE on 2019-12-31 during a 300 s exposure. The high slew rate throughout these observations created elongated stellar PSFs that were ∼ 100 spaxels long. In this extreme case, variability can be clearly seen along the lengths of the stars it cannot be replicated in a simple elongated Gaussian (or Moffat). While more complex models could be constructed that vary across the length of the trailed PSF, in starkiller we instead create a data PSF: we normalize and average together the trails of all calibration stars in the data cube. While this simple approach can introduce noise structures into the PSF, such as seen in Fig. 4 (Data PSF), it reliably captures the seeing variability. In the major axis cross sections shown in the lower panel of Fig. 4, the Gaussian PSF profile fails to capture the variability that is largely shared between the stars, which is present in the data PSF.</text>
<text><location><page_7><loc_8><loc_35><loc_48><loc_59></location>In constructing the data PSF, starkiller must only use stars that are well contained within the data cube. Therefore, alongside the magnitude limit and isolation requirement, we introduce a 'containment' requirement. We test PSF containment by creating individual images for all catalog stars by convolving an image containing their spaxel position with the model PSF function. The implanted PSFs are then summed over the extent of the datacube to give a containment fraction. To ensure the data PSF is representative of the entire PSF, we only include sources with containment fractions > 95% in its construction. In cases where there are few suitable calibration stars, the data PSF may be biased to those stars, and therefore not a fair representation of all sources.</text>
<text><location><page_7><loc_8><loc_17><loc_48><loc_35></location>While spatial variability in the trailed PSF is clearest in the highly elongated sources, it is still present in sources with shorter trails. In general, we find that the data PSF provides a more accurate representation of trailed sources in MUSE datacubes. In Fig. 5, we use all three PSF methods to model 2 stars from the MUSE 2I/Borisov datacube observed on 2020-03-1 9 . While all methods greatly reduce the total counts, both the absolute residual and visual artifacts are lowest for the trailed data PSF fits at ∼ 5%, while the other two methods have ∼ 10% residuals. For sidereally tracked data</text>
<text><location><page_7><loc_52><loc_88><loc_92><loc_92></location>we find the data PSF outperforms the other models by ∼ 15% as seen in Fig. 16 in Appendix A.</text>
<text><location><page_7><loc_52><loc_76><loc_92><loc_88></location>Therefore, after constructing the data PSF, starkiller will check for differences between the model PSF (either Moffat or Gaussian) and the data PSF: if the difference is large, then starkiller will default to using the data PSF for spectral extraction and scene creation. If desired, this behavior can be disabled by setting the ' psf preference ' option of starkiller to ' model '.</text>
<text><location><page_7><loc_52><loc_71><loc_92><loc_76></location>The computed PSF is used by starkiller to extract observed spectra through PSF photometry, and ultimately to model sources in the scene.</text>
<section_header_level_1><location><page_7><loc_64><loc_68><loc_80><loc_69></location>3.5. Spectral matching</section_header_level_1>
<text><location><page_7><loc_52><loc_48><loc_92><loc_67></location>We identify the best-match spectra for a star through correlation comparisons with the model stellar spectra. To avoid model confusion from noise spikes, or external emission lines, we apply an iterative sigma clipping and smoothing procedure to observed spectra before calculating model correlations. In this procedure we cut points that have gradient absolute values greater than 10 σ above the median. By default the spetcra are clipped for 3 iterations before undergoing smoothing with a Savitzky-Golay filter, as implemented in scipy. These treated spectra are then correlated with model spectra.</text>
<text><location><page_7><loc_52><loc_31><loc_92><loc_48></location>Correlation allows us to make a morphological comparison of the similarities between two spectra, without considering flux scaling. For each IFU spectra, we calculate the Pearson correlation coefficient using scipy.stats.pearsonr for all available model spectra, downsampled to the input IFU's spectral resolution. This approach (rather than e.g. χ 2 ) emphasises relative shapes and avoids concerns of normalisation. Since we want the closest match, we take the corresponding model to the IFU spectra to be that which has the largest positive correlation p -value.</text>
<text><location><page_7><loc_52><loc_9><loc_92><loc_31></location>In starkiller we provide multiple pathways for spectral matching, using the range of model catalogs described in § 2.3. The model catalog used is determined by the spec catalog argument. By default starkiller uses 'ck', which checks against the CK models. If 'ck+' is specified then starkiller uses the temperature of the selected CK model to identify relevant high resolution spectra to compare against. If the CK model temperature is < 8000 K, we compare to a set of MARCS models within a ± 500 K temperature range of the input value. Similarly, if the CK temperature is > 15000 K, we check against all models in the OB-i model list (see Sect. 2.3). This approach provides the most comprehensive stellar spectral matching in starkiller . Alternately, other</text>
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<caption>Figure 4. Seeing variability of highly elongated stars in MUSE observations of 2I/Borisov on 2019-Dec-31. Stars A & B, as shown in the top two panels, are the calibration stars used by starkiller in this reduction, followed by their best-fitting Gaussian and data PSFs. The stars are highly elongated, with a fitted streak length of 98 spaxels. Over the course of this 600 s exposure, the star trails exhibit a non-uniform brightness profile along their major axis, which causes the stars to be poorly represented by a simple elongated Moffat or Gaussian profile. The bottom panel shows the normalized cross section along the major axis for: stars A & B (blue and green solid lines respectively); the best fitting Gaussian PSF (red dash-dot line); and the Data PSF (orange dashed line). The Data PSF is able to capture some of the subtle structure imposed on the PSF from seeing variability, which leads to better subtractions.</caption>
</figure>
<text><location><page_8><loc_8><loc_14><loc_48><loc_23></location>catalogs can be selected: setting spec catalog to 'ck' will restrict the spectral fitting to only the CK models, while 'eso' will use the library of stellar spectra listed by ESO. If the base spectral models included in starkiller are insufficient for the desired case, the spectral catalogs that are used can be readily altered.</text>
<text><location><page_8><loc_8><loc_9><loc_48><loc_14></location>As extinction from interstellar dust can significantly reshape spectra, we must incorporate extinction in the template matching. For every template spectrum we</text>
<text><location><page_8><loc_52><loc_73><loc_92><loc_92></location>create an extinction grid by applying the Fitzpatrick (1999) extinction model with R V = 3 . 1 over the range 0 ≤ E ( B -V ) ≤ 4 in steps of 0.01 using the extinction (Barbary 2016) and PySynphot (STScI Development Team 2013) packages. We then calculate the correlation of the extracted spectra with the grid of reddened models. The model spectrum with the highest correlation is then used to represent the source, and re-scaled to match the catalog magnitudes. Through simulated recovery tests we find that this method is robust to noise, and has minimal degeneracy between spectral type and extinction.</text>
<text><location><page_8><loc_52><loc_62><loc_92><loc_73></location>This spectral matching process provides the best approximation of the spectra from every star in the IFU. An example of one such fit is shown in Fig. 6. Our approach minimizes the input information from the IFU, thus reducing the likelihood that the stellar spectra are biased by the flux of other sources within the IFU, such as a foreground extended coma of a target comet.</text>
<section_header_level_1><location><page_8><loc_58><loc_57><loc_86><loc_59></location>3.5.1. Optional output: Velocity matching</section_header_level_1>
<text><location><page_8><loc_52><loc_44><loc_92><loc_56></location>An additional step to matching the fine detail of observed and model spectra is to apply corrections for any relative motion. While it is not used in the primary reduction procedure, starkiller has a routine to identify the most likely Doppler shift to the observed spectrum. We calculate the likely shift by fitting Gaussian models to prominent absorption lines in stellar spectra: H β , H α , Na D, and the Ca II triplet.</text>
<text><location><page_8><loc_52><loc_9><loc_92><loc_43></location>We fit the absorption lines with astropy.modeling independently with a single Gaussian plus an constant offset, with the exception of Na D which we fit with a double Gaussian model with a constant offset. The models are fit to a region of the spectrum ± 20 ˚ A of the rest frame wavelength for the line. For each line, we normalize the spectra by the median flux value of a range +20 to +40 ˚ A from the absorption line. The models are fit through astropy 's Levenberg-Marquardt algorithm and least squares statistic (LevMarLSQFitter) method, where the uncertainties are taken to be the square root of the diagonal elements covariance matrix. While we do not place bounds on the fit, we require the amplitude of the Gaussian models to be negative for it to be considered in the weighted average. The final velocity is taken to be the error-weighted average of the fit lines. An example of this method is shown in Fig. 7 for star Gaia 5856950561179332352 from the MUSE datacube of 2I/Borisov on 2020-03-19. This method will only be effective for stars where the selected spectral features are prominent, such as spectral types A to K. However, it is not currently fully implemented in starkiller , as the</text>
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<caption>Figure 5. Example of the 3 PSF models - Moffat, Gaussian, and Data (top row) - alongside two field stars in a 2I/Borisov datacube observed on 2020-03-19 (left column), and their respective residuals (center). While each PSF appears similar, the subtle differences become clear in the fit residuals. Neither the Moffat or Gaussian profiles provide a reliable representations of the stars in this datacube, or in most other streaked MUSE datacubes. However, the data PSF is a much closer match, lacking the poor subtraction patterns of the Moffat and Gaussian. We attribute the complex PSF that significantly varies along the streak to atmospheric scintillation throughout the 600 s exposure. starkiller will begin modeling the PSF as either Gaussian or Moffat; however, if the initial model is significantly different from the data PSF, the data PSF is then used exclusively in spectra extraction and scene generation.</caption>
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<text><location><page_9><loc_8><loc_39><loc_48><loc_42></location>primary science cases have not to date been concerned with narrow lines.</text>
<section_header_level_1><location><page_9><loc_21><loc_36><loc_35><loc_38></location>3.6. Flux correction</section_header_level_1>
<text><location><page_9><loc_8><loc_20><loc_48><loc_36></location>The final adjustment that we make to the model spectra is to correct for any global trends in differences between the observed and model spectra. These wavelength-dependent differences may arise from issues with the MUSE flux calibration, or from consistently poorly matched model spectra. For starkiller the origin of these global differences is inconsequential, as the primary goal is to replicate the IFU stellar spectra; for this, correcting for bulk trends as a function of wavelength is sufficient.</text>
<text><location><page_9><loc_8><loc_9><loc_48><loc_20></location>We generate a wavelength-dependent flux correction by averaging the flux ratios of all sources with correlation coefficients > 0 . 9. We further restrict the calibration sources to be brighter than the user-defined magnitude limit, to limit the influence of bad matches and noise on the flux correction. We then create a smoothed spline using a Savitzky-Golay filter using the scipy im-</text>
<text><location><page_9><loc_52><loc_29><loc_92><loc_42></location>plementation, with a window size of 625 ˚ A (501 pixels) and a polynomial order of 3. The window size and polynomial order were chosen to avoid being biased by narrow features such as absorption lines, while being able to capture broader features alongside the continuum offset. We perform a 3 sigma clip on the difference between the flux ratios and smoothed spline, and refit the spline to further limit the influence of narrow line features.</text>
<text><location><page_9><loc_52><loc_14><loc_92><loc_29></location>As seen in Fig. 8, a characteristic correction curve emerges when examining the median flux ratio of all calibration sources. Our flux correction method is able to correct for the overall flux offset, and for larger features, such as the broad wiggles occurring along the spectrum, and the rapid rise after 9000 ˚ A, while limiting bias from the poorly fit narrow lines. Dividing all model spectra by this flux correction brings them into closer alignment with the MUSE spectra. The model spectra shown in Figs. 6 & 9 have this flux correction applied.</text>
<text><location><page_9><loc_52><loc_10><loc_92><loc_13></location>While the method we present here is sufficient for starkiller to reliably match model spectra to the ob-</text>
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<caption>Figure 6. Example spectral model match to two stellar spectra extracted from the trailed MUSE data cube in Fig. 1. The MUSE spectra is shown by the solid blue line, and the best-fitting model's stellar spectra, which has been independently flux scaled by Gaia G-band photometry, is shown as an orange dashed line. The best-matching model is taken to be the model with the highest Pearsons-r correlation, from a grid of spectra subject to a range of extinction values. An additional flux correction discussed in § 3.6 is also applied. The residuals of the model fits are shown in the lower panels.</caption>
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<caption>Figure 7. Doppler fits for Gaia 5856950561179332352 in the MUSE datacube of 2I/Borisov on 2020-03-19. starkiller provides a rough relative velocity for stars with spectral types A-K by fitting Gaussian models to prominent absorption lines in stellar spectra: H β , H α , Na D, and the Ca II triplet. Each line in the Ca II triplet is modeled independently and labeled as a , b , and c in increasing wavelength. In this example, the error-weighted average velocity is -46 ± 2 km/s.</caption>
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<caption>Figure 8. Ratio of the best fitting stellar models through the ck+ method to the MUSE spectra for each of the calibration sources from the IFU. As each model spectrum is scaled by the catalog, in this case Gaia DR3 magnitudes, any differences between the models can be a result of poor template matching, or variations in flux calibration. Common trends that are present between the models (purple line) for the calibration sources are used to create a flux correction (orange dotted line). Every model spectrum is divided by the flux correction, to bring them into greater alignment with the observed spectra. As all variations occur close to 1, we can see that the flux calibration between Gaia and MUSE is consistent.</caption>
</figure>
<text><location><page_11><loc_8><loc_35><loc_48><loc_50></location>served MUSE spectra, it may not be reliable at distinguishing calibration errors from poor model fits. As the origin of this offset is unknown, the flux correction is only applied to the model spectra to make them a closer match to the IFU data, therefore the calibration of the input data is not altered. It is worth noting that in general the flux calibration of MUSE spectra and Gaia AB magnitude photometry are in agreement, with small deviations of ∼ 10% occurring around a ratio of 1.</text>
<section_header_level_1><location><page_11><loc_20><loc_33><loc_36><loc_35></location>3.7. Sidereal IFU data</section_header_level_1>
<text><location><page_11><loc_8><loc_25><loc_48><loc_32></location>While starkiller was developed for modeling nonsidereally-tracked IFU cubes, it is entirely capable of processing sidereally-tracked cubes. In these instances, the best-fitting trail length for the PSF profiles is ∼ 1, and the rotation angle becomes irrelevant.</text>
<text><location><page_11><loc_8><loc_9><loc_48><loc_25></location>One complication can arise from datacubes with large extended structures, such as bright nebulae and galaxies, where the PSF profiles fail to fit correctly due to the underlying structure. To limit the influence of this structure, the 'fuzzy' option can be set to true in starkiller . In the 'fuzzy' mode, a 'fuzzymask' will be created by applying scipy.ndimage.label to a Boolean image created by conditioning spaxel brightness in the datacube image based on the median spaxel brightness. If there are labels that occupy more than</text>
<text><location><page_11><loc_52><loc_82><loc_92><loc_92></location>40% of the spaxels, it is considered for masking. The background is taken to be the label with the lowest median counts; the other labels are included in the fuzzymask. Sources within the fuzzymask will not be used for PSF creation or flux correction, even if they meet all other requirements to be considered a calibration source.</text>
<text><location><page_11><loc_52><loc_62><loc_92><loc_82></location>As an example, we apply starkiller with the 'fuzzy' option enabled to a MUSE datacube of the planetary nebula NGC 6563, observed on 2018-08-22 without AO. An additional complication that we correct for in starkiller , can arise from the presence of narrow features in the extracted spectra. While the spikes seen in Fig 9 could result in low correlations with model spectra, the sigma clipping and smoothing algorithm that starkiller applies results in high correlations between the extracted spectra and models. The full subtraction of NGC 6563 with a data-PSF (Fig. 10) is successful in removing the majority of flux from sources that were identified by Gaia with on average residuals of < 10%.</text>
<section_header_level_1><location><page_11><loc_59><loc_60><loc_85><loc_61></location>3.8. Satellite detection and removal</section_header_level_1>
<text><location><page_11><loc_52><loc_44><loc_92><loc_59></location>By adapting the process that starkiller uses to subtract stars, we are also able to identify spectra of satellites crossing an IFU, and attempt to remove them. Unlike stars, the magnitudes, SEDs, and on-sky locations of satellites are poorly constrained, making a full forward-modeling approach unachievable - therefore, starkiller relies entirely on the IFU data. In this prototype case, we focus on satellite detection and removal by subtracting a satellite PSF that is scaled by a spectrum extracted by PSF photometry.</text>
<text><location><page_11><loc_52><loc_34><loc_92><loc_43></location>If the 'satellite' option in starkiller is set to 'True', it will search for satellite streaks, and if one or more is presents, fit and subtract the streaks. All functions used for this purpose are contained in the sat killer class. We use opencv to detect satellite streaks through the following process:</text>
<unordered_list>
<list_item><location><page_11><loc_55><loc_29><loc_92><loc_33></location>1. Calculate a detection threshold: the image median plus 15 × the standard deviation of the image.</list_item>
<list_item><location><page_11><loc_55><loc_25><loc_92><loc_29></location>2. The image is conditioned on the detection threshold and then dilated with a 9 × 9 kernel of ones using cv2.dilate .</list_item>
<list_item><location><page_11><loc_55><loc_21><loc_92><loc_24></location>3. The dilated image is passed through the cv2 Canny edge detection algorithm (Canny 1986).</list_item>
<list_item><location><page_11><loc_55><loc_9><loc_92><loc_20></location>4. The cv2 Hough Line Transform (Hough 1962; Duda & Hart 1972; Ballard 1981) is applied to the edge image with a vote threshold of 100, minimum line length of 100 spaxels, and maximum line gap of 50 spaxels. While these parameters are sufficient for MUSE, fine tuning may be required for other IFUs.</list_item>
</unordered_list>
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<caption>Figure 9. Example spectral model match to two stellar spectra extracted from the MUSE data cube in Fig. 10. The MUSE spectra is shown by the solid blue line, and the best-fitting model's stellar spectra which has been independently flux scaled by Gaia G-band photometry, is shown as an orange dashed line (the same as in Fig. 6). Both of these sources are from within the nebula. The spectral extraction and modeling is robust to narrow features from external emission, or instrument noise, resulting in high model correlations.</caption>
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<caption>Figure 10. starkiller applied to the datacube of planetary nebula NGC 6563 (no AO) with position-corrected Gaia DR3 stars shown as orange circles. (left) Mean stack of all wavelength for the reduced MUSE datacube. (centre) Scene constructed by starkiller on Gaia -listed stars in the field of view. (right) The median stack of all wavelengths for the scene-subtracted cube. The subtracted Gaia DR3 stars have residuals < 10%.</caption>
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<unordered_list>
<list_item><location><page_12><loc_10><loc_22><loc_48><loc_28></location>5. Lines are grouped by finding the average distances between all points in each line to every other line. If the average line distance is less than a maximum separation distance, they are grouped.</list_item>
<list_item><location><page_12><loc_10><loc_16><loc_48><loc_21></location>6. The final consolidated line is then used to calculate the streak center, length, and angle, which are key parameters used in PSF fitting.</list_item>
</unordered_list>
<text><location><page_12><loc_8><loc_9><loc_48><loc_15></location>If one or more satellite streaks are identified, starkiller then fits a model PSF to each satellite streak. Currently, starkiller assumes that each satellite streak can be modeled as an extremely streaked</text>
<text><location><page_12><loc_52><loc_11><loc_92><loc_28></location>source, and so uses the best-fitting model PSF parameters determined when earlier fitting the stars in the field. The satellite PSF is then constructed using the star PSF parameters and the line properties. Since satellite sky locations are frequently highly uncertain due to maneuvers and drag effects, satellite shape models and reflectance functions are infrequently made public, and satellite materials are almost always kept proprietary, we lack a comprehensive database of satellite spectra, with efforts ongoing (e.g. Battle et al. 2024). We therefore cannot forward-model the satellite spectrum as we</text>
<text><location><page_13><loc_8><loc_81><loc_48><loc_92></location>do with the stars. Instead, we simply fit a single flux value for each wavelength, through basic PSF photometry with the satellite PSF. While simple, this method limits the influence of astrophysical sources on the extracted satellite spectrum, as in most fields they will only occupy a small fraction of the satellite's trail length, and thus not be favored in the fit.</text>
<text><location><page_13><loc_8><loc_65><loc_48><loc_81></location>An example of starkiller applied to a satellite streak is shown in Fig. 11 for a MUSE observation centered on Blazar WISEA J014132.24-542751.0 (J01415427). In this worst-case scenario satellite strike, the unidentified satellite crosses directly over the science target. With starkiller , we are able to effectively model this satellite streak and subtract it from the MUSE datacube, potentially salvaging a 2960 s exposure. We discuss a comparison of the contaminated and clean spectra of J0141-5427 in § 4.2.</text>
<section_header_level_1><location><page_13><loc_9><loc_62><loc_47><loc_63></location>3.9. Simulated datacube construction and subtraction</section_header_level_1>
<text><location><page_13><loc_8><loc_47><loc_48><loc_61></location>With the PSF defined, and all catalog sources matched with model stellar spectra, which is independently flux calibrated, starkiller can generate a 'scene' of the input datacube. The simulated cube is generated at 10 times the spatial resolution of the input cube, to allow for fine positioning of sources. We also extend the x and y spatial dimensions of the scene by the trail length, to include sources that are partially contained in the observed datacube.</text>
<text><location><page_13><loc_8><loc_28><loc_48><loc_47></location>For every astrophysical source, we create a 'seed' image, by convolving the super-sampled PSF model with the position of the source in the image. These seed images are then multiplied with their respective stellar spectral models to create a simulated target. If satellites are detected in the image, they are added through the same process; however, the seed is multiplied by the PSF spectrum from the IFU. All sources are then combined to create the final starkiller scene, and finally, the scene is downsampled to match the input cube dimensions. Examples of the final scenes are shown in the middle panels of Figs. 1, 10 & 13.</text>
<text><location><page_13><loc_8><loc_22><loc_48><loc_28></location>Once the starkiller scene is generated, it is subtracted from the observed datacube. The resultant cube is saved as a FITS file, alongside diagnostic figures for the matched spectra.</text>
<section_header_level_1><location><page_13><loc_22><loc_19><loc_34><loc_20></location>4. DISCUSSION</section_header_level_1>
<text><location><page_13><loc_8><loc_9><loc_48><loc_18></location>With starkiller , we have developed a new method of analyzing crowded IFU data to primarily aid in the analysis of non-sidereally tracked extended sources. This technique allows us to improve crowded data such that exposures that would have otherwise been dropped can now be included in the data analysis. While PSF</text>
<text><location><page_13><loc_52><loc_78><loc_92><loc_92></location>extraction pipelines such as PampelMUSE can provide precise PSF subtractions and photometry for sidereally tracked data, starkiller is applicable to both sidereal and non-sidereal tracked data. Since both pipelines can be applied to sidereal data, we compare them in Appendix B. Furthermore, starkiller opens up the possibility to perform difference imaging of IFU data cubes with single exposures, while retrieving estimated stellar parameters.</text>
<section_header_level_1><location><page_13><loc_55><loc_74><loc_89><loc_75></location>4.1. starkiller capabilities and potential uses</section_header_level_1>
<text><location><page_13><loc_52><loc_41><loc_92><loc_73></location>The development case for starkiller was the analysis of MUSE datacubes for 2I/Borisov, which are presented in Deam et al. (2024). Many of these observations were within 10 degrees of the galactic plane, where star crowding heavily limited the data quality. Of the 51 exposures that were taken during the 2I/Borisov observing campaign, starkiller needed to be applied to 27. The reduction improved the data quality for dust/gas maps of 2I's coma in 23, and in the remaining 4, starkiller improved the data quality so significantly that the exposures no longer had to be rejected from analysis (Deam et al. 2024), recovering data for an unusual transient target. Similarly, applying starkiller provides improved detail in the debris trail from Dimorphos of Opitom et al. (2023); Murphy et al. (2023). This is demonstrated in Fig. 12 and Fig 13 respectively. Murphy et al. (2023) had to discard a significant portion of their debris trail datapoints, due to contamination from the stars crossing the tail. In theory, starkiller means the Solar System community can now use any IFU to look near the Galaxy.</text>
<text><location><page_13><loc_52><loc_31><loc_92><loc_40></location>For serendipitously observed Solar System objects, starkiller provides the capacity to return timeresolved spectra on streaks. For instance, the foreground of asteroids present in many long-duration sidereally tracked exposures can be extracted (using the sat killer mode).</text>
<text><location><page_13><loc_52><loc_8><loc_92><loc_31></location>Since starkiller identifies the best fitting extinction E(B -V) for each source, it may be possible to use it to calculate, or to independently check, the extinction present in star clusters. We present in Fig. 14 the distribution of extinction values obtained by starkiller for NGC 6563 (Fig. 10) as an example for this use case. The distribution of extinctions present in this field is highly bimodal, displaying low and high extinction populations with median E(B -V) of ∼ 0 . 5 mag, ∼ 1 . 2 mag, respectively. As the high extinction population is largely co-located with NGC 6563 this may suggest that they are being obscured by the nebula. While these are Gaia DR3 sources, most do not have distances to compare against. Furthermore, we note that the E(B -V)</text>
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<caption>Figure 11. starkiller applied to the datacube where a satellite strikes Blazar WISEA J014132.24-542751.0 (J0141-5427) on 2022-Jul-23. (Left) In a worst-case scenario satellite strike, the unknown satellite crosses directly over the central science target. With the satkiller extension to starkiller we are able to effectively model (Middle) and remove the satellite signal in all wavelengths (Right), such that the residuals are on the same scale as the detector imperfections. With starkiller we have potentially salvaged a 2960 s exposure from a satellite strike. The orange circle is a field star identified by the Gaia DR3 catalog.</caption>
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<text><location><page_14><loc_8><loc_40><loc_48><loc_59></location>value of the low extinction population is approximately double the S&F dust map for the region, which has E(B -V) = 0 . 2271 ± 0 . 0024 10 (Schlafly & Finkbeiner 2011). The large discrepancy in values for this case requires further investigation to ascertain the reliability of extinction values generated by starkiller . As with the flux calibration process, starkiller is only optimizing the correlations between individual models and the observed spectra. Therefore, if the input stellar atmosphere library is insufficient, starkiller may be using extinction as a tool to reshape poorly matched spectra to improve the correlation.</text>
<text><location><page_14><loc_8><loc_32><loc_48><loc_40></location>In starkiller 's stellar fitting, the point-source fluxes are quantified according to the expected Gaia catalog magnitudes. Where fully stellar PSFs remain as residuals, this can be used to identify variable sources, a common practice in conventional difference imaging.</text>
<section_header_level_1><location><page_14><loc_13><loc_30><loc_43><loc_31></location>4.2. Satellite impacts on spectroscopic data</section_header_level_1>
<text><location><page_14><loc_8><loc_15><loc_48><loc_29></location>The growing number of satellites in low-Earth orbit present fundamental challenges for all ground-based observing. As the number of satellites approach the projected numbers in the variety of orbital shells, their density will be sufficient such that even instruments with small FoVs will have frequent satellite streaks. With starkiller , we provide a tool to isolate and remove satellite streaks from IFU datacubes. Furthermore, we can directly analyze a satellite spectrum, and even test</text>
<text><location><page_14><loc_52><loc_53><loc_92><loc_59></location>for temporal variability. This initial approach can be expanded to become a robust method for addressing satellite contamination, and the generation of a library of satellite spectra.</text>
<text><location><page_14><loc_52><loc_21><loc_92><loc_52></location>Satellite streaks directly crossing science targets, or 'strikes' in IFU data provides us with a unique opportunity to examine how satellite strikes impact optical observations, which has not yet been shown for MUSE. Fig. 15 demonstrates that the satellite observed in the datacube of J0141-5427 has a spectrum that resembles a smooth continuum, with some prominent absorption lines visible in Fig. 15 (top left). Both this unidentified satellite and the blazar are SNR ∼ 100, generating a large effect on the blazar spectrum. This appears more severe than the expectations for spectroscopic impact of Hainaut & Moehler (2024). However, in comparison with a spectrum of J0141-5427 that was observed subsequently and is without a satellite passage, we find that starkiller 's satellite subtraction process successfully removes the broad satellite continuum, as seen with the orange line in Fig. 15 (top right). In this case, we believe that starkiller has effectively 'cleaned' the datacube of the satellite contaminant, allowing it to now be used for science.</text>
<text><location><page_14><loc_52><loc_10><loc_92><loc_21></location>With this well-isolated satellite spectrum, we can analyse its properties and the atmospheric effects. As the satellite is reflecting sunlight, we might expect the satellite spectrum to be similar to the Sun; this is a frequent assumption in studies of potential satellite impact on astronomical spectra (e.g. Bialek et al. 2023; Hainaut & Moehler 2024). As shown in Fig. 15 (lower left),</text>
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<caption>2020-Feb-02 Number of frames: 4</caption>
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<caption>Figure 12. Images of the dust emission (7080 ˚ A-7120 ˚ A) from 2I/Borisov (Deam et al. 2024) before and after the application of starkiller to MUSE datacubes. 2I was within 10 degrees of the Galactic plane on 2020-Feb-02 and 2020-Mar-19, and the FOV for each exposure contained between 80 and 136 stars with m GAIA -G < 21. More significant improvement occurred for the 2020-Mar-19 data: the number of usable exposures (without a star directly behind 2I) increased from 2 to 3 due to starkiller , allowing for improved median co-adding and the removal of more stars. The greater number of bright stars in the March observations allowed for more accurate modeling of the streaked PSFs and their subsequent subtractions. The upper images are displayed with a square root stretch over a 99.95 percentile interval, while the lower images show contours from a Gaussian convolved image of 1 standard deviation.</caption>
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</figure>
<figure>
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<caption>Figure 13. starkiller applied to a MUSE datacube of Didymos, observed on 2022-10-25, following the DART impact of 2022-09-27 (Opitom et al. 2023). Faint stellar sources crossing the tail are well-modeled and subtracted. Brighter sources have poor subtractions in the wings of the PSF; this limitation is discussed in Sec. 4.4.</caption>
</figure>
<figure>
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<caption>2020-Feb-02 Number of frames: 42020-Mar-19 Number of frames: 2</caption>
</figure>
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<caption>2020-Mar-19 Number of frames: 3</caption>
</figure>
<figure>
<location><page_16><loc_9><loc_67><loc_48><loc_92></location>
<caption>Figure 14. Best fitting E(B -V) values for the Gaia DR3 stars in NGC 6563. (left) Histogram of E(B -V) occurrence rates, (right) spatial distribution superimposed on the IFU median image. The bimodal distribution may indicate a population of low-extinction foreground stars, and a population of high extinction stars which are largely co-located with the nebula. While the validity of extinctions that starkiller applies to models is yet to be tested, it provides a way to independently and rapidly calculate extinction values for stars in a cluster.</caption>
</figure>
<figure>
<location><page_16><loc_49><loc_67><loc_91><loc_92></location>
</figure>
<text><location><page_16><loc_31><loc_67><loc_33><loc_69></location>-</text>
<text><location><page_16><loc_8><loc_52><loc_48><loc_58></location>we find that the satellite spectrum is indeed well represented by a Solar spectrum that has been highly processed by atmospheric extinction, with the parameters fit by pyExtinction (Buton & Copin 2014).</text>
<text><location><page_16><loc_8><loc_35><loc_48><loc_52></location>While the MUSE pipeline applies a correction for standard atmospheric extinction Weilbacher et al. (2020, Sec. 4.9), it is possible that the satellite has experienced higher levels of atmospheric extinction. For satellites, the relative configuration of the Sun, satellite, Earth, and observer become important. In many configurations it is possible for light from the Sun to pass through the Earth's atmosphere before reaching the satellite. This additional passage of light through the atmosphere would create the enhanced atmospheric extinction that we observe in the satellite spectrum.</text>
<section_header_level_1><location><page_16><loc_12><loc_32><loc_44><loc_33></location>4.3. Independent flux calibration verification</section_header_level_1>
<text><location><page_16><loc_8><loc_9><loc_48><loc_31></location>Through the forward-modeling approach of starkiller , we have developed a way to independently verify the flux calibration of an IFU datacube. As described in Sec. 3.5, we only use the observed spectra for shape comparison by correlating the observed spectra with model spectra. The best-fitting model spectra are then flux scaled according to catalog magnitudes ( Gaia DR3 in the default case); we then compare the flux of these scaled models to the observed spectra to generate a flux correction, as described in Sec. 3.6. While some fine features in the flux correction may result from poor model matching, overall trends and flux offsets are likely to be indicative of a flux calibration error between the datacube and the catalog.</text>
<section_header_level_1><location><page_16><loc_59><loc_57><loc_85><loc_58></location>4.4. Current starkiller limitations</section_header_level_1>
<text><location><page_16><loc_52><loc_26><loc_92><loc_56></location>While starkiller is robust to a wide range of challenging observational conditions, there are several limitations. Primary among these is the source density: if there are too few stars in the FOV or in the source catalog, then starkiller is unable to create corrections based on the population of sources. If there are few sources i.e. < 3 sources contained in the IFU, then the data PSF and flux correction will be biased by the available targets. Flux correction will only be applied if there are at least 3 calibration sources present in the IFU, or the force flux correction option is set. If the catalog is incomplete, then un-cataloged sources will not be subtracted: in the example of NGC 6563 (Fig. 10), the majority of sources that can be seen are not identified in the Gaia DR3 catalog. Additionally, the selected calibration sources may then face higher-than-expected crowding. Unexpected crowding of calibration sources can lead to poor PSF models, and therefore poor subtractions.</text>
<text><location><page_16><loc_52><loc_11><loc_92><loc_26></location>Conversely, while starkiller is fairly robust to crowding, overlapping sources are not well modeled. The current method for spectral extraction and source position fitting considers each source independently. For crowded sources, this approach leads to spectral contamination and poor positional fitting for crowded sources, particularly faint sources. An alternative approach such as iterative PSF fitting and subtraction of targets, or simultaneous fitting of grouped sources, could be more successful for overlapping sources. In cases of sidereal</text>
<text><location><page_16><loc_89><loc_81><loc_92><loc_82></location>-</text>
<figure>
<location><page_17><loc_9><loc_61><loc_91><loc_92></location>
<caption>Figure 15. Extracted spectra of the satellite strike on Blazar WISEA J014132.24-542751.0 (J0141-5427) on 2022-Jul-23 (see Fig. 11). Top left: Blazar with satellite contamination (blue), with the satellite spectrum extracted from its streak (red) by starkiller 's sat killer functionality. Top right: The residuals from subtracting another MUSE spectrum ('uncontaminated') of the blazar that was acquired immediately after the satellite passed. (blue) equivalent to the satellite spectrum (i.e. red line in top left panel); (orange) the formerly contaminated blazar data after starkiller was applied. Lower left: The satellite spectrum (red) compared with a solar spectrum (gray) and the best-fitting atmospheric extinction for a solar spectrum (black). Several prominent solar lines are visible in the satellite's spectrum, such as H α and the Na doublet, but others are not Solar or telluric. Lower right : Best-fitting atmospheric extinction that maximizes the correlation of an extincted solar spectrum to the satellite spectrum (orange), and the three dominant extinction components from aerosols (purple dot-dash), Rayleigh scattering (blue dashed), and ozone (blue dotted).</caption>
</figure>
<text><location><page_17><loc_8><loc_42><loc_48><loc_44></location>tracking of crowded sources, PampelMUSE will provide higher quality spectra and PSF subtractions.</text>
<text><location><page_17><loc_8><loc_30><loc_48><loc_41></location>The starkiller PSF construction method currently does not create wavelength-dependent PSFs. While this limitation is largely inconsequential for the low signalto-noise stars in non-sidereal observations, it may limit the reliability of starkiller in high signal-to-noise observations of bright stars in sidereal observations, a situation where PampelMUSE is better suited.</text>
<text><location><page_17><loc_8><loc_12><loc_48><loc_30></location>Another limitation of the current PSF implementation is how the PSF wings are treated, particularly for the data PSF. To avoid including detector artifacts into the data PSF, we set all spaxels with less than 10 -4 % contribution to the total PSF to 0. This sets a nominal PSF radius in MUSE data to be ∼ 8 spaxels. For faint sources, this PSF truncation has little effect, as the flux at the PSF wings is low; however, for bright sources, this leads to overly-subtracted centers, surrounded by un-subtracted wings. A possible solution to this would be to augment the data PSF with a Moffat or Gaussian model component fit to the PSF wings.</text>
<text><location><page_17><loc_52><loc_37><loc_92><loc_44></location>The quality of the starkiller subtraction for a target depends heavily on how well the spectral models fit. While we have equipped starkiller with a broad sample of stellar spectra, additional models may be required, or be more representative of specific stars.</text>
<text><location><page_17><loc_52><loc_24><loc_92><loc_36></location>Even with these limitations, starkiller is effective at removing more than 90% of the stellar flux in most cases. For non-sidereal targets, the combination of starkiller and averaging together multiple cubes observed at different sky locations overcomes much of the limitations. Some refinement is still required to optimize starkiller for sidereal-tracked observations with high signal-to-noise stars.</text>
<section_header_level_1><location><page_17><loc_57><loc_21><loc_87><loc_22></location>4.5. Extending starkiller to other IFUs</section_header_level_1>
<text><location><page_17><loc_52><loc_9><loc_92><loc_20></location>With a catalog in hand, starkiller can be extended to other IFUs; currently operational ones have smaller FoVs than MUSE. For instance, it would be immediately suitable for WiFes (38 '' × 25 '' ), GMTIFS (20.4 '' × 20.4 '' ) or Keck KCWI (20 '' × 33 '' ) data that has the same data format as MUSE. Planned ELT IFUs (e.g. HARMONI, WFOS) have similar-scale FOVs. Even for the smallest</text>
<text><location><page_18><loc_8><loc_86><loc_48><loc_92></location>IFUs, the capabilities of starkiller will be suited for treating satellite-affected data. The future VLT instrument BlueMUSE 11 is planned to have a FOV larger than 1 arcmin 2 , which would be highly suited to starkiller .</text>
<text><location><page_18><loc_8><loc_81><loc_48><loc_85></location>For JWST's NIRSpec or MIRI, we note that the default stellar models in starkiller are cut to the optical range; similarly, using an IR catalog would be necessary.</text>
<text><location><page_18><loc_8><loc_73><loc_48><loc_81></location>In all cases, for fast-moving non-sidereal objects, in the present version of starkiller , the exposure lengths of observations would need to be capped, so as to retain the star trail length within the IFU FOV (the containment requirement discussed in § 3.4).</text>
<section_header_level_1><location><page_18><loc_21><loc_70><loc_35><loc_71></location>5. CONCLUSION</section_header_level_1>
<text><location><page_18><loc_8><loc_57><loc_48><loc_69></location>In this paper we have presented the starkiller package, which creates synthetic difference images of single IFU data cubes using a forward-modeling approach. By utilizing independent photometric catalogs, and a suite of stellar atmosphere models, starkiller simultaneously provides stellar spectral classification, relative velocity, and line-of-sight extinction for all sources in a catalog, alongside a source-subtracted datacube.</text>
<text><location><page_18><loc_8><loc_38><loc_48><loc_57></location>We developed starkiller to be compatible with both sidereal and non-sidereally acquired observations. For streaked sources, we do not require input tracking, as starkiller is generalized to work with even extreme cases of elongated sources and crowded fields. In the most extreme case, it can model and subtract satellite streaks. We developed this method to clean highly elongated stars from VLT/MUSE observations of 2I/Borisov. As we were working to preserve the spectral features of a diffuse foreground object, we developed starkiller to rely heavily on catalogs of model stellar spectra and magnitudes.</text>
<text><location><page_18><loc_8><loc_33><loc_48><loc_38></location>IFUs provide exceptional capabilities for astronomical enquiry. We look forward to seeing what uses this package may find in the community.</text>
<text><location><page_18><loc_52><loc_79><loc_92><loc_90></location>R.R.H. is supported by the Royal Society of New Zealand, Te Ap¯arangi through a Marsden Fund Fast Start Grant and by the Rutherford Foundation Postdoctoral Fellowship. M.T.B. appreciates support by the Rutherford Discovery Fellowships from New Zealand Government funding, administered by the Royal Society Te Ap¯arangi.</text>
<text><location><page_18><loc_52><loc_61><loc_92><loc_79></location>Package development based on observations collected at the European Southern Observatory under ESO programmes 103.2033.001-003 and 105.2086.002 (2I/Borisov, PIs: M.T.B and Cyrielle Opitom), 110.23XL and 109.2361 (DART, with thanks to PI Cyrielle Opitom), 60.A-9100 (PI: MUSE Team), and 109.238W (PI: Fuyan Bian). We thank the ESO staff, particularly Danuta Dobrzycka and Lodovico Coccato of the MUSE SDP team, Marco Berton, Henri Boffin, Bin Yang, Diego Parraguez, Edmund Christian Herenz, Fuyan Bian, and Israel Blanchard, for their help in the acquisition or retrieval of these observations.</text>
<text><location><page_18><loc_52><loc_56><loc_92><loc_60></location>We thank the community of the IAU Centre for Protection of the Dark and Quiet Sky From Satellite Constellation Interference for their help during this work.</text>
<text><location><page_18><loc_52><loc_43><loc_92><loc_56></location>This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www. cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www. cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.</text>
<section_header_level_1><location><page_18><loc_54><loc_38><loc_72><loc_40></location>Facilities: VLT(MUSE)</section_header_level_1>
<text><location><page_18><loc_52><loc_22><loc_92><loc_37></location>Software: Starkiller (Ridden-Harper et al. 2024), astropy (Astropy Collaboration et al. 2013, 2018, 2022), astroquery (Ginsburg et al. 2019), scipy (Virtanen et al. 2020), photutils (Bradley 2023), pandas (Wes McKinney 2010; The pandas development Team 2024), numpy (Harris et al. 2020), matplotlib Hunter (2007), OpenCV (Bradski 2000), MUSE Python Data Analysis Framework (MPDAF; Bacon et al. 2016), TRIPPy (Fraser et al. 2016), pyExtinction (Buton & Copin 2014), PampelMUSE (Kamann et al. 2013)</text>
<figure>
<location><page_19><loc_9><loc_53><loc_91><loc_92></location>
<caption>Figure 16. Recreation of Fig. 5 for sidereally tracked point sources extracted from the sky observation ADP.2023-0927T15 07 36.629. As with the trailed sources, the data PSF outperforms both the Moffat and Gaussian PSF profiles.</caption>
</figure>
<section_header_level_1><location><page_19><loc_46><loc_44><loc_54><loc_45></location>APPENDIX</section_header_level_1>
<section_header_level_1><location><page_19><loc_34><loc_39><loc_66><loc_41></location>A. PSF MODELING OF POINT SOURCES</section_header_level_1>
<text><location><page_19><loc_8><loc_33><loc_92><loc_39></location>The functions starkiller uses to model trailed PSFs can be readily applied to point sources. If the optional variable trail is set to false, the trail length will be set to 1 spaxel, and starkiller will fit regular point sources. As seen in Fig. 16, the data PSF outperforms both the Gaussian and Moffat PSF with residuals ∼ 15% lower for the data PSF.</text>
<section_header_level_1><location><page_19><loc_18><loc_32><loc_82><loc_33></location>B. COMPARISON OF SIDEREAL SOURCE-SUBTRACTION WITH PAMPELMUSE</section_header_level_1>
<text><location><page_19><loc_8><loc_20><loc_92><loc_31></location>The PempelMUSE pipeline (Kamann et al. 2013) was developed to obtain high precision extraction of point sources from MUSE data. Some aspects of PampelMUSE are similar to starkiller , such as requiring a source catalog to identify sources; both pipelines can produce source-subtracted cubes. Unlike starkiller , PempelMUSE can only be applied to sidereally tracked data, and primarily uses a Moffat profile to model the PSF. PempelMUSE creates its source-subtracted data cube through subtracting the fit PSF profile for each source from every frame. Therefore, while the two pipelines were created for different use cases, we can draw comparisons between the two when looking at their outcomes on sidereally tracked data.</text>
<text><location><page_19><loc_8><loc_9><loc_92><loc_20></location>In Fig. 17 we compare the subtracted cubes produced by starkiller and PempelMUSE for NGC 6563 (the same nebula shown in § 3.7 and Fig. 10). While both model have subtraction artifacts, PempelMUSE systematically oversubtracts sources within the extent of the nebula. This over-subtraction will result in output spectra from PempelMUSE that have larger fluxes than were observed. In contrast, starkiller appears to under-subtract sources, which will be due to the flux scaling to match the catalog photometry and any intrinsic variability of sources. PampelMUSE outperforms starkiller for crowded sources. This test also highlights the poor subtractions from starkiller and PampelMUSE for bright sources, suggesting that the MUSE PSF profile differs between bright and faint sources.</text>
<figure>
<location><page_20><loc_9><loc_64><loc_44><loc_90></location>
</figure>
<section_header_level_1><location><page_20><loc_69><loc_90><loc_83><loc_91></location>PampelMUSE</section_header_level_1>
<figure>
<location><page_20><loc_58><loc_64><loc_91><loc_90></location>
<caption>Figure 17. Comparison of source-subtracted MUSE datacubes for NGC 6563 processed by starkiller (left) and PempelMUSE (right). The two pipelines take different approaches: starkiller uses spectral models scaled by external flux catalogs, and PempelMUSE directly subtracts sources through PSF fitting. In this example, PempelMUSE appears to systematically oversubtract sources, particularly those that overlap with the nebula.</caption>
</figure>
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</document> | [{"title": "ABSTRACT", "content": "We present starkiller , an open-source Python package for forward-modeling flux retrieval from integral field unit spectrograph (IFU) datacubes. starkiller simultaneously provides stellar spectral classification, relative velocity, and line-of-sight extinction for all sources in a catalog, alongside a source-subtracted datacube. It performs synthetic difference imaging by simulating all catalog sources in the field of view, using the catalog for positions and fluxes to scale stellar models, independent of the datacube. This differencing method is particularly powerful for subtracting both point-sources and trailed or even streaked sources from extended astronomical objects. We demonstrate starkiller 's effectiveness in improving observations of extended sources in dense stellar fields for VLT/MUSE observations of comets, asteroids and nebulae. We also show that starkiller can treat satelliteimpacted VLT/MUSE observations. The package could be applied to tasks as varied as dust extinction in clusters and stellar variability; the stellar modeling using Gaia fluxes is provided as a standalone function. The techniques can be expanded to imagers and to other IFUs. Keywords: Small Solar System bodies (1469), Interstellar objects (52), Artificial satellites (68), Extended radiation sources (504), Interdisciplinary astronomy (804), Open source software (1866)", "pages": [1]}, {"title": "Starkiller : subtracting stars and other sources from IFU spectroscopic data through forward modeling", "content": "Ryan Ridden-Harper , 1 Michele T. Bannister , 1 Sophie E. Deam , 1, 2 and Thomas Nordlander 3, 4 1 School of Physical and Chemical Sciences - Te Kura Mat\u00afu, University of Canterbury, Private Bag 4800, Christchurch 8140, Aotearoa New Zealand 2 Space Science and Technology Centre, School of Earth and Planetary Sciences, Curtin University, Perth, Western Australia 6845, Australia 3 Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT 2611 4 ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia (Received November 22, 2024) Submitted to AJ", "pages": [1]}, {"title": "1. INTRODUCTION", "content": "Integral field unit spectrographs (IFUs) combine the strengths of imaging and spectroscopy: both spatial and spectral resolution for every spaxel in a given field of view. Coupled with adaptive optics, this makes them a mainstay of extended-source observational astronomy -serving a variety of communities including stellar evolution, star clusters, Galactic and extragalactic science. Multiple generations of IFUs have been built for facilities around the world, from WiFeS on the ANU 2.3 m at Corresponding author: Ryan Ridden-Harper [email protected] Siding Spring (Dopita et al. 2010), to MUSE on ESO's UT4 of the VLT (Bacon et al. 2010) and JWST's NIRSpec (Boker et al. 2022). Indeed, every proposed thirtymetre-class optical facility (the ELTs) includes an IFU as a first-light instrument. However, IFU data currently presents challenges for some science cases.", "pages": [1]}, {"title": "1.1. Case 1: a background extended target with foreground stellar fluxes e.g. a nebula", "content": "In the situation where an extended source such as a nebula or low-surface-brightness galaxy has a foreground and/or background stellar field, the data will be acquired with sidereal tracking, and all sources will have circular point-source point-spread functions (PSFs). The density of the stellar field can limit the potential inference of the extended source's morphology and composition. The PampelMUSE package (Kamann et al. 2013) provides precise PSF fitting for photometry in dense stellar fields in IFU data.", "pages": [1, 2]}, {"title": "1.2. Case 2: a foreground extended target with background trailed stellar fluxes e.g. a comet", "content": "In contrast to other spectroscopic modes, IFUs have seen comparatively little use by the Solar System smallbodies community, whose targets are frequently unresolved sources. A common use of IFUs in Solar System studies has been atmospheric characterisation of bright planets (e.g. instruments such as Gemini/NIFS and VLT/SINFONI (Simon et al. 2022)). However, IFUs are ideal for compositional studies of small Solar System bodies with activity creating extended comae - but like all Solar System targets, these worlds move across the sky. The requirement of non-sidereal telescope tracking to increase signal-to-noise streaks all background sources. For imaging, median stacking can adequately remove the streaked stellar signal, but this often limits any characterization of time-varying phenomena. Studies of active minor planets are historically kept to areas of sky with low stellar density, with observation programs paused when targets traverse the Galactic plane (e.g. the 2022-23 DART mission post-impact followup campaign; Moskovitz et al. (2024); Kareta et al. (2023)). For imaging in dense stellar fields, image subtraction is now a robust technique that aids both minor planet detection, and characterization such as lightcurve studies, but it is not yet used frequently for IFUs.", "pages": [2]}, {"title": "1.3. Case 3: a celestial target with foreground streaks e.g. satellite streaks or trailed asteroids", "content": "As the industrialization of near-Earth space increases, the astronomical communities also face the advent of streak-smeared data. The accelerating rate of satellite constellation emplacement into low-Earth orbit (LEO) means > 6400 have been launched, with > 5800 operationally in place as of April 2024 1 ; at least 20,000100,000, with potential for around a million LEO satellites, will be in place in the mid-2030s (Walker et al. 2020, 2021; Falle et al. 2023). The population will then need to be maintained at that level by ongoing launches. These satellites produce an industrially-caused environmental impact on astronomical observations: out-offocus streaks of reflected Solar flux if they traverse the field of view, obscuring science targets, during an exposure. While the probability of satellite streaks affecting the smaller fields of view of IFUs is lower than that for the massively wide-field imager of the Vera C. Rubin Observatory, the probability of effects on astronomical imaging is already non-zero (Walker et al. 2020; Michaglyph[suppress]lowski et al. 2021; Walker et al. 2021). IFUs typically acquire longer integrations on target than e.g the 15s/30 s exposures of the LSST, so there is a different scope for adverse impact on IFUs. We demonstrate example MUSE data impacts in \u00a7 3.8. The environmental impacts will only continue to increase, particularly in the era of ELTs. By the expected European ELT first light, a steady-state LEO population kept by industry at some 30,000 satellites could exist; it is certainly on track to exceed 15,000. For distant celestial targets acquired with sidereal tracking, near-field minor planets, particularly asteroids at geometries outside of quadrature, will also trail to varying degrees. The data on both types of astronomical object will be of interest to different communities.", "pages": [2]}, {"title": "1.4. The starkiller package", "content": "Here we present a new open-source forward-modeling approach to removing stellar flux and satellite streaks from IFU datacubes, regardless of whether they are trailed or round. We model the flux of the stars in the field that are identifiable in a source catalog, via stellar atmosphere models. As the flux can be streaked or circular, a trailed point-spread function (PSF) is constructed to fit the stellar PSF of each datacube. We apply location-appropriate dust extinction and find the best-match stellar spectra for each star. The streaks then form a simulated data cube, which is subtracted from the original (e.g. Fig. 1). This approach allows us to replicate traditional difference imaging to remove background stars without the need for a reference data cube. All magnitudes are assumed to be AB, and fluxes in terms of F \u03bb (erg / s / cm 2 / \u02da A) unless otherwise stated. Our approach is inherently generalisable between instruments. We use VLT/MUSE here as a case study, with two extended small Solar System bodies, a nebula ( \u00a7 3.7), and a satellite streak that pass over a blazar ( \u00a7 3.8) as examples. With suitable stellar models across the appropriate wavelength range, and a wellcharacterized instrumental PSF, starkiller can be extended to other IFUs ( \u00a7 4.5). Pull requests are welcome 2 .", "pages": [2]}, {"title": "2. DATA", "content": "2.1. Example IFU: VLT/MUSE - The ESO Very Large Telescope's Multi Unit Spectroscopic Explorer (VLT/MUSE; Bacon et al. 2010) is a panoramic integral field unit spectrograph covering 4000 \u02da A-9300 \u02da A, on the 8.2 m UT4 optical telescope at Paranal, Chile. In wide-field mode, the field of view (FOV) is 1 ' \u00d7 1 ' , ideal for imaging extended sources. The optionally adaptive-optics corrected light is split equally and fed to 24 individual spectrographs (integral field units) (Bacon et al. 2010). We used contrasting MUSE datasets acquired with both sidereal and non-sidereal tracking for our primary development and testing of starkiller . For sidereal data, we use two example selections from the ESO Archive: the planetary nebula NGC 6563, and satelliteimpacted observations from 2021-22 of blazar WISEA J014132.24-542751.0 (J0141-5427). We use two non-sidereal MUSE datasets, one from each MUSE mode, each of which present different dataanalysis challenges. The first is of the interstellar comet 2I/Borisov, observed on 16 epochs in 2019-2020 (Bannister et al. 2020; Deam et al. 2024). These data have increasingly dense stellar backgrounds, as 2I/Borisov moved from 49 \u00b7 outside the Galactic plane to within the plane after it passed perihelion. The stars are streaked up to \u223c 20 '' at high galactic latitude, with the shortest \u223c 4 '' at low galactic latitude. 2I/Borisov has a compact coma, entirely contained within at most 46 '' diameter, and thus fully within the MUSE WFM FOV. Our other non-sidereal dataset is from followup of the NASA DART mission's impact on 2022 September 26 of the near-Earth asteroid moon Dimorphos (Opitom et al. 2023; Murphy et al. 2023). 3 The resulting debris formed extended and time-varying structures over the following month relative to the bright parent body in the system, Didymos. While Opitom et al. (2023); Murphy et al. (2023) acquired 11 epochs, only the last three have dense stellar backgrounds, when Didymos moved onto the Galactic plane. These data were acquired at very rapid motion rates tracked on Didymos due to its geocentric proximity of only 0.08-0.09 au, which produced longer stellar streaks than those in the 2I/Borisov data; most streaks are not completely enclosed in the FOV. The alignment of the FOV was offset on Didymos and rotated 90 \u00b7 with respect to that of the 2I data, with the data acquired in MUSE's 8 '' \u00d7 8 '' narrow-field mode with AO.", "pages": [2, 3]}, {"title": "2.2. Catalog: Gaia", "content": "To identify stars within each cube, we use the Gaia DR3 source catalog for star positions and brightness. Gaia provides an all-sky catalog of sources with a limiting magnitude of 22 and a saturation magnitude of \u223c 3 (Gaia Collaboration et al. 2016, 2023; Babusiaux et al. 2023). The precise positions of Gaia sources assists in sub-spaxel alignment of stars, while the broad G band magnitudes are ideal for scaling model spectra flux. As starkiller assumes all magnitudes are in the AB system, we must apply a correction to the Gaia DR3 magnitudes, which are presented in the Vega system. We compute the correction by following the same procedure as Axelrod et al. (2023), comparing observed Gaia MUSE Scene - Gmagnitudes (Vega) to synthetic AB magnitudes in the Gfilter for 17 well-calibrated DA white dwarfs (Narayan et al. 2019). We take the median offset between the synthetic and observed magnitudes to be the correction factor to map Gaia G (Vega) to Gaia G (AB). Using the median in Fig. 2, the correction becomes:", "pages": [3, 4]}, {"title": "2.3. Stellar Atmosphere Models", "content": "High-quality stellar atmosphere models are essential for reliably representing stars within the data cubes. For accurate spectral matching, we need a diverse set of model spectra that span wide ranges in effective surface temperature T eff , surface gravities log(g), and metallicities. In this initial demonstration of starkiller we also prioritize ease of use, so we restrict ourselves to smaller spectral libraries that can be installed alongside the base code. In order to best represent the spectral types of any star, we provide multiple models and implementation pathways between these models. Primarily, we use the Castelli & Kurucz (2003) stellar atmosphere models (CK models) as a basis for stellar spectra comparisons. We choose the STScI subsection of the total CK atlas to cover the key range of spectral types 4 . We also include the ESO stellar spectra li- brary 5 which use the Pickles (1998) stellar spectra, supplemented with corrections from Ivanov et al. (2004). (While there is a MUSE-specific library, it only contains 35 stellar spectra and is restricted to the MUSE wavelengths, so we do not use it here). We include a partial grid of medium-resolution ( R = 20000) sampled fluxes from the MARCS grid (Gustafsson et al. 2008). We have selected spectra covering the most common types of latetype stars, with T eff = 3000-8000K and log g between -0 . 5 and +5 . 0, in a pattern that broadly follows the main sequence and red giant branch. The models have [Fe / H] = -0 . 5, 0.0 and +0 . 5, using solar-scaled abundances except that [ \u03b1/ Fe] = +0 . 2 when [Fe / H] < 0. For simplicity, a single value of the micro-turbulence value v mic = 2kms -1 was selected. Models of both planeparallel and spherical geometry (assuming stellar masses of one solar mass) are included where available. Finally, we include the PoWR grid of OB-type synthetic spectra (Hainich et al. 2019), called OB-i . Specifically, we include the solar-metallicity grid that covers much of the parameter space T eff = 15-56kK and log g = 2 . 04.4, broadly corresponding to the evolution of stars of roughly 7-60 M \u2299 .", "pages": [4]}, {"title": "3. ANALYSIS STRUCTURE", "content": "The process of analysis of starkiller is generalized to operate on any optical IFU data that has the same header and HDU format as VLT's MUSE, including WCS information. We have minimized the amount of additional inputs required, with a strong preference towards self-determination of key information from the input datacube. The following sections outline the key steps we use in determining the stars within the field of view (FOV), and modeling those sources.", "pages": [4]}, {"title": "3.1. Source catalog", "content": "By default, we use the Gaia DR3 catalog (Gaia Collaboration et al. 2023) as the source catalog. starkiller obtains the Gaia sources within a radius defined by the size of the IFU centered on the R.A. and Decl. provided in the input cube's header. We access the Gaia DR3 catalog I/355/gaiadr3 through Vizier via astroquery . Alternatively, starkiller also accepts user input source catalogs. The input catalogs must contain columns specifying the source ID (id), R.A. (ra), and Dec. (dec) in degrees, magnitude ( x mag), and a filter designation for the SVO filter service 6 ( x filt) where x is the desired filter shorthand. If multiple filters are specified for a single source, then the model spectra will be reshaped to match the magnitudes in all filters; unless the key filter argument is defined, identifying the filter to which to normalise the flux.", "pages": [4, 5]}, {"title": "3.2. WCS correction", "content": "As the purpose of starkiller is to simulate spectral data cubes to provide simulated differenced cubes, precise positions of sources are essential: while this may be straightforward for siderally tracked observations, for those tracked non-sidereally, both the spatial WCS solutions and source identification require additional refinement. For MUSE, the WCS solution produced by the MUSE data reduction pipeline (Bacon et al. 2016) is propagated from the target coordinates defined in the VLT's observing block. For sidereally tracked observations, the spatial WCS solutions are effective: only minor corrections on the order of a spaxel are generally required. In non-sidereal cases, the offset needed from the Bacon et al. (2016) WCS is determined by the on-sky motion rate of the target, relative to the ephemeris timestamp choice when the VLT begins observation setup, modulo the typical MUSE acquisition time of 10-15 minutes. For example, we find that the WCS solution can be offset by an arcminute in our 2I/Borisov data. However, elongated sources then present a challenge for conventional source identifiers. To identify sources in the IFU, regardless of shape, we use clustering algorithms on an 'image' constructed from a median stack of the input cube in wavelength space. We create a Boolean image with a percentile cut, selecting for spaxels that are brighter than the 90th percentile. We then label sources in the Boolean image with scipy.ndimage.label . Sources are then downselected to retain the labeled objects with a total spaxel count between 0.01% and 10% of the total IFU spaxels 7 . This cut limits contamination from sources that partially fall within the FOV (lower limit) and the background spaxels (upper limit). The center points of the labeled sources are taken to be the average x and y spaxel positions of spaxels for each source. We also extract initial guesses for the stellar PSF, by estimating the trailing length, trail angle, and the x and y spaxel extents. With sources identified in the image, we fit basic offsets from the star catalog to the image. Within starkiller there are two methods to conduct an initial match, and a final method to refine matching sources with the catalog. The first and less reliable preliminary match method is to fit for for x , y , and rotational offsets, by minimizing the distance of the spaxel coordinates of the brightest stars in the catalog to the image sources. However, this method struggles for crowded fields, where the regions labeled as sources are composites of multiple sources and not representative of the PSF. The second preliminary match method, which is robust to crowding, assumes that the position angle on the sky has low error, and matches the catalog through shifts. In this method we first iterate through x and y offsets to the raw catalog spaxel coordinates from -100 to 100 spaxels, in steps of 10 spaxels. For each shift, we generate a simulated image by convolving sources within the image bounds with the profile of the median labeled source, and subtract this from the labeled image, which is altered such that sources are represented by 1 and all other spaxels are NaN. The pair of x and y offsets that provide the smallest residual are then taken as the starting parameters, to minimize the residual between the two images with scipy.minimize . This method is the default method used in starkiller and provides close matches, even in crowded fields. For point sources this match is within 1 spaxel; the uncertainty does increase in crowded fields with high source elongation. Following the initial catalog match, the WCS correction is then refined through PSF fitting. The creation of the PSF that is used in this routine is outlined in \u00a7 3.4. We take the x and y positions of the calibration sources found through PSF fitting, and compare those with the shifted catalog positions. Following a method similar to the first catalog matching method discussed, we minimize the distance between the shifted catalog positions and the PSF positions through x and y shifts as well as a rotation \u03b8 around the image center. This refined PSF shift matches observed sources with the source catalog positions to a sub-spaxel precision. For trailed sources, we consider the center points as defined by the PSF to be the observed position. With these methods developed for starkiller , we are able to correct for any errors present in the spatial WCS solution due to challenging observational conditions, such as non-sidereal tracking. While effective, these methods may lose reliability in highly crowded fields or with very elongated sources. We discuss this further in \u00a7 4.4.", "pages": [5]}, {"title": "3.3. Isolating sources", "content": "Identifying isolated sources in non-sidereal data presents an interesting challenge as the sources may be streaked to any length, and aligned on any angle. As with the WCS correction, we adopt a 2 stage approach to identifying isolated sources, where we use the initial PSF approximations in stage 1, which we then refine with the preliminary PSF in stage 2. In stage 1, we rotate the coordinates of the catalog sources and image according to the estimated trail angle such that the trails are vertical. A source is then considered to be isolated if it is more than 8 spaxels from a neighbor in the x direction (PSF minor axis), and separated by more than 1.2 times the total trail length in the y direction (PSF major axis). We also incorporate magnitude information into the isolation criterion. If nearby sources are at least 2 magnitudes fainter, it is assumed that their contribution is small and therefore ignored when calculating source distances. The isolated sources identified in this process are used to generate the first iteration of the PSF. Stage 2 relies on a PSF being defined to determine a refined calibration source list. To identify isolated sources using the PSF, we check overlaps between Boolean masks, created by placing the PSF at each source position where spaxels must contribute more than 1 \u00d7 10 -5 to the total PSF. If any two masks contain overlapping points, the sources are considered to be overlapping. By using the PSF information, we can reliably identify isolated sources, regardless of trail length or orientation. Sources we identify as 'isolated' through this process are then used as the final calibration sources, from which the final PSF and WCS correction are calculated.", "pages": [6]}, {"title": "3.4. PSF modeling", "content": "A precise PSF is key to creating accurate simulations of data cubes. We developed starkiller to model PSFs for static sources and elongated sources, generated from sidereal and non-sidereal tracking. Our PSF module is built from the PSF module within TRIPPy (Fraser et al. 2016), which downsamples a high resolution PSF to the image resolution. Alongside the Moffat profile PSF used in TRIPPy , we also incorporate a Gaussian profile PSF, and a data-generated PSF; we refer to the latter as the 'data PSF'. As constructed in TRIPPy , a trailed PSF can be reliably modeled by a non-trailed point source PSF profile convolved with a line model. Fitting trailed PSFs therefore requires fitting the PSF profile parameters alongside the line model which incorporates trail length and angle, which we choose to be the angle measured counterclockwise from the x axis. In starkiller the nontrailed point source PSF profile can be either Moffat, or symmetric 2D Gaussian, and is defined by fitting to calibration sources that are considered 'isolated' and brighter than a user-defined calibration magnitude limit. When constructing the model PSF, we simultaneously fit all parameters for the profile and line element, including small positional offsets. The models are generated at 10 times the spatial resolution and downsampled to the data resolution. While the PSF is dependent on wavelength, we find that these variations are small for the highly streaked stars in MUSE data. Therefore, to optimize the signal-to-noise of the calibration sources, starkiller fits a single PSF using the median stack of all wavelengths. A further complication to the PSF fitting process is variability in the atmospheric seeing. For sidereal tracking, variations in seeing contribute evenly to the total PSF throughout the exposure. However, for nonsidereal tracking, the trailed PSF becomes a record of the seeing variability during the exposure, where each segment of the trail may be constructed by different seeing conditions to another section. A trailed PSF may therefore be quite different from a distribution that can be readily modeled by a simple elongated Moffat, or Gaussian profile. An example of highly trailed stars can be seen in Fig. 4. Stars A & B 8 (Fig. 4, top two panels) were observed alongside 2I/Borisov with MUSE on 2019-12-31 during a 300 s exposure. The high slew rate throughout these observations created elongated stellar PSFs that were \u223c 100 spaxels long. In this extreme case, variability can be clearly seen along the lengths of the stars it cannot be replicated in a simple elongated Gaussian (or Moffat). While more complex models could be constructed that vary across the length of the trailed PSF, in starkiller we instead create a data PSF: we normalize and average together the trails of all calibration stars in the data cube. While this simple approach can introduce noise structures into the PSF, such as seen in Fig. 4 (Data PSF), it reliably captures the seeing variability. In the major axis cross sections shown in the lower panel of Fig. 4, the Gaussian PSF profile fails to capture the variability that is largely shared between the stars, which is present in the data PSF. In constructing the data PSF, starkiller must only use stars that are well contained within the data cube. Therefore, alongside the magnitude limit and isolation requirement, we introduce a 'containment' requirement. We test PSF containment by creating individual images for all catalog stars by convolving an image containing their spaxel position with the model PSF function. The implanted PSFs are then summed over the extent of the datacube to give a containment fraction. To ensure the data PSF is representative of the entire PSF, we only include sources with containment fractions > 95% in its construction. In cases where there are few suitable calibration stars, the data PSF may be biased to those stars, and therefore not a fair representation of all sources. While spatial variability in the trailed PSF is clearest in the highly elongated sources, it is still present in sources with shorter trails. In general, we find that the data PSF provides a more accurate representation of trailed sources in MUSE datacubes. In Fig. 5, we use all three PSF methods to model 2 stars from the MUSE 2I/Borisov datacube observed on 2020-03-1 9 . While all methods greatly reduce the total counts, both the absolute residual and visual artifacts are lowest for the trailed data PSF fits at \u223c 5%, while the other two methods have \u223c 10% residuals. For sidereally tracked data we find the data PSF outperforms the other models by \u223c 15% as seen in Fig. 16 in Appendix A. Therefore, after constructing the data PSF, starkiller will check for differences between the model PSF (either Moffat or Gaussian) and the data PSF: if the difference is large, then starkiller will default to using the data PSF for spectral extraction and scene creation. If desired, this behavior can be disabled by setting the ' psf preference ' option of starkiller to ' model '. The computed PSF is used by starkiller to extract observed spectra through PSF photometry, and ultimately to model sources in the scene.", "pages": [6, 7]}, {"title": "3.5. Spectral matching", "content": "We identify the best-match spectra for a star through correlation comparisons with the model stellar spectra. To avoid model confusion from noise spikes, or external emission lines, we apply an iterative sigma clipping and smoothing procedure to observed spectra before calculating model correlations. In this procedure we cut points that have gradient absolute values greater than 10 \u03c3 above the median. By default the spetcra are clipped for 3 iterations before undergoing smoothing with a Savitzky-Golay filter, as implemented in scipy. These treated spectra are then correlated with model spectra. Correlation allows us to make a morphological comparison of the similarities between two spectra, without considering flux scaling. For each IFU spectra, we calculate the Pearson correlation coefficient using scipy.stats.pearsonr for all available model spectra, downsampled to the input IFU's spectral resolution. This approach (rather than e.g. \u03c7 2 ) emphasises relative shapes and avoids concerns of normalisation. Since we want the closest match, we take the corresponding model to the IFU spectra to be that which has the largest positive correlation p -value. In starkiller we provide multiple pathways for spectral matching, using the range of model catalogs described in \u00a7 2.3. The model catalog used is determined by the spec catalog argument. By default starkiller uses 'ck', which checks against the CK models. If 'ck+' is specified then starkiller uses the temperature of the selected CK model to identify relevant high resolution spectra to compare against. If the CK model temperature is < 8000 K, we compare to a set of MARCS models within a \u00b1 500 K temperature range of the input value. Similarly, if the CK temperature is > 15000 K, we check against all models in the OB-i model list (see Sect. 2.3). This approach provides the most comprehensive stellar spectral matching in starkiller . Alternately, other catalogs can be selected: setting spec catalog to 'ck' will restrict the spectral fitting to only the CK models, while 'eso' will use the library of stellar spectra listed by ESO. If the base spectral models included in starkiller are insufficient for the desired case, the spectral catalogs that are used can be readily altered. As extinction from interstellar dust can significantly reshape spectra, we must incorporate extinction in the template matching. For every template spectrum we create an extinction grid by applying the Fitzpatrick (1999) extinction model with R V = 3 . 1 over the range 0 \u2264 E ( B -V ) \u2264 4 in steps of 0.01 using the extinction (Barbary 2016) and PySynphot (STScI Development Team 2013) packages. We then calculate the correlation of the extracted spectra with the grid of reddened models. The model spectrum with the highest correlation is then used to represent the source, and re-scaled to match the catalog magnitudes. Through simulated recovery tests we find that this method is robust to noise, and has minimal degeneracy between spectral type and extinction. This spectral matching process provides the best approximation of the spectra from every star in the IFU. An example of one such fit is shown in Fig. 6. Our approach minimizes the input information from the IFU, thus reducing the likelihood that the stellar spectra are biased by the flux of other sources within the IFU, such as a foreground extended coma of a target comet.", "pages": [7, 8]}, {"title": "3.5.1. Optional output: Velocity matching", "content": "An additional step to matching the fine detail of observed and model spectra is to apply corrections for any relative motion. While it is not used in the primary reduction procedure, starkiller has a routine to identify the most likely Doppler shift to the observed spectrum. We calculate the likely shift by fitting Gaussian models to prominent absorption lines in stellar spectra: H \u03b2 , H \u03b1 , Na D, and the Ca II triplet. We fit the absorption lines with astropy.modeling independently with a single Gaussian plus an constant offset, with the exception of Na D which we fit with a double Gaussian model with a constant offset. The models are fit to a region of the spectrum \u00b1 20 \u02da A of the rest frame wavelength for the line. For each line, we normalize the spectra by the median flux value of a range +20 to +40 \u02da A from the absorption line. The models are fit through astropy 's Levenberg-Marquardt algorithm and least squares statistic (LevMarLSQFitter) method, where the uncertainties are taken to be the square root of the diagonal elements covariance matrix. While we do not place bounds on the fit, we require the amplitude of the Gaussian models to be negative for it to be considered in the weighted average. The final velocity is taken to be the error-weighted average of the fit lines. An example of this method is shown in Fig. 7 for star Gaia 5856950561179332352 from the MUSE datacube of 2I/Borisov on 2020-03-19. This method will only be effective for stars where the selected spectral features are prominent, such as spectral types A to K. However, it is not currently fully implemented in starkiller , as the primary science cases have not to date been concerned with narrow lines.", "pages": [8, 9]}, {"title": "3.6. Flux correction", "content": "The final adjustment that we make to the model spectra is to correct for any global trends in differences between the observed and model spectra. These wavelength-dependent differences may arise from issues with the MUSE flux calibration, or from consistently poorly matched model spectra. For starkiller the origin of these global differences is inconsequential, as the primary goal is to replicate the IFU stellar spectra; for this, correcting for bulk trends as a function of wavelength is sufficient. We generate a wavelength-dependent flux correction by averaging the flux ratios of all sources with correlation coefficients > 0 . 9. We further restrict the calibration sources to be brighter than the user-defined magnitude limit, to limit the influence of bad matches and noise on the flux correction. We then create a smoothed spline using a Savitzky-Golay filter using the scipy im- plementation, with a window size of 625 \u02da A (501 pixels) and a polynomial order of 3. The window size and polynomial order were chosen to avoid being biased by narrow features such as absorption lines, while being able to capture broader features alongside the continuum offset. We perform a 3 sigma clip on the difference between the flux ratios and smoothed spline, and refit the spline to further limit the influence of narrow line features. As seen in Fig. 8, a characteristic correction curve emerges when examining the median flux ratio of all calibration sources. Our flux correction method is able to correct for the overall flux offset, and for larger features, such as the broad wiggles occurring along the spectrum, and the rapid rise after 9000 \u02da A, while limiting bias from the poorly fit narrow lines. Dividing all model spectra by this flux correction brings them into closer alignment with the MUSE spectra. The model spectra shown in Figs. 6 & 9 have this flux correction applied. While the method we present here is sufficient for starkiller to reliably match model spectra to the ob- served MUSE spectra, it may not be reliable at distinguishing calibration errors from poor model fits. As the origin of this offset is unknown, the flux correction is only applied to the model spectra to make them a closer match to the IFU data, therefore the calibration of the input data is not altered. It is worth noting that in general the flux calibration of MUSE spectra and Gaia AB magnitude photometry are in agreement, with small deviations of \u223c 10% occurring around a ratio of 1.", "pages": [9, 11]}, {"title": "3.7. Sidereal IFU data", "content": "While starkiller was developed for modeling nonsidereally-tracked IFU cubes, it is entirely capable of processing sidereally-tracked cubes. In these instances, the best-fitting trail length for the PSF profiles is \u223c 1, and the rotation angle becomes irrelevant. One complication can arise from datacubes with large extended structures, such as bright nebulae and galaxies, where the PSF profiles fail to fit correctly due to the underlying structure. To limit the influence of this structure, the 'fuzzy' option can be set to true in starkiller . In the 'fuzzy' mode, a 'fuzzymask' will be created by applying scipy.ndimage.label to a Boolean image created by conditioning spaxel brightness in the datacube image based on the median spaxel brightness. If there are labels that occupy more than 40% of the spaxels, it is considered for masking. The background is taken to be the label with the lowest median counts; the other labels are included in the fuzzymask. Sources within the fuzzymask will not be used for PSF creation or flux correction, even if they meet all other requirements to be considered a calibration source. As an example, we apply starkiller with the 'fuzzy' option enabled to a MUSE datacube of the planetary nebula NGC 6563, observed on 2018-08-22 without AO. An additional complication that we correct for in starkiller , can arise from the presence of narrow features in the extracted spectra. While the spikes seen in Fig 9 could result in low correlations with model spectra, the sigma clipping and smoothing algorithm that starkiller applies results in high correlations between the extracted spectra and models. The full subtraction of NGC 6563 with a data-PSF (Fig. 10) is successful in removing the majority of flux from sources that were identified by Gaia with on average residuals of < 10%.", "pages": [11]}, {"title": "3.8. Satellite detection and removal", "content": "By adapting the process that starkiller uses to subtract stars, we are also able to identify spectra of satellites crossing an IFU, and attempt to remove them. Unlike stars, the magnitudes, SEDs, and on-sky locations of satellites are poorly constrained, making a full forward-modeling approach unachievable - therefore, starkiller relies entirely on the IFU data. In this prototype case, we focus on satellite detection and removal by subtracting a satellite PSF that is scaled by a spectrum extracted by PSF photometry. If the 'satellite' option in starkiller is set to 'True', it will search for satellite streaks, and if one or more is presents, fit and subtract the streaks. All functions used for this purpose are contained in the sat killer class. We use opencv to detect satellite streaks through the following process: If one or more satellite streaks are identified, starkiller then fits a model PSF to each satellite streak. Currently, starkiller assumes that each satellite streak can be modeled as an extremely streaked source, and so uses the best-fitting model PSF parameters determined when earlier fitting the stars in the field. The satellite PSF is then constructed using the star PSF parameters and the line properties. Since satellite sky locations are frequently highly uncertain due to maneuvers and drag effects, satellite shape models and reflectance functions are infrequently made public, and satellite materials are almost always kept proprietary, we lack a comprehensive database of satellite spectra, with efforts ongoing (e.g. Battle et al. 2024). We therefore cannot forward-model the satellite spectrum as we do with the stars. Instead, we simply fit a single flux value for each wavelength, through basic PSF photometry with the satellite PSF. While simple, this method limits the influence of astrophysical sources on the extracted satellite spectrum, as in most fields they will only occupy a small fraction of the satellite's trail length, and thus not be favored in the fit. An example of starkiller applied to a satellite streak is shown in Fig. 11 for a MUSE observation centered on Blazar WISEA J014132.24-542751.0 (J01415427). In this worst-case scenario satellite strike, the unidentified satellite crosses directly over the science target. With starkiller , we are able to effectively model this satellite streak and subtract it from the MUSE datacube, potentially salvaging a 2960 s exposure. We discuss a comparison of the contaminated and clean spectra of J0141-5427 in \u00a7 4.2.", "pages": [11, 12, 13]}, {"title": "3.9. Simulated datacube construction and subtraction", "content": "With the PSF defined, and all catalog sources matched with model stellar spectra, which is independently flux calibrated, starkiller can generate a 'scene' of the input datacube. The simulated cube is generated at 10 times the spatial resolution of the input cube, to allow for fine positioning of sources. We also extend the x and y spatial dimensions of the scene by the trail length, to include sources that are partially contained in the observed datacube. For every astrophysical source, we create a 'seed' image, by convolving the super-sampled PSF model with the position of the source in the image. These seed images are then multiplied with their respective stellar spectral models to create a simulated target. If satellites are detected in the image, they are added through the same process; however, the seed is multiplied by the PSF spectrum from the IFU. All sources are then combined to create the final starkiller scene, and finally, the scene is downsampled to match the input cube dimensions. Examples of the final scenes are shown in the middle panels of Figs. 1, 10 & 13. Once the starkiller scene is generated, it is subtracted from the observed datacube. The resultant cube is saved as a FITS file, alongside diagnostic figures for the matched spectra.", "pages": [13]}, {"title": "4. DISCUSSION", "content": "With starkiller , we have developed a new method of analyzing crowded IFU data to primarily aid in the analysis of non-sidereally tracked extended sources. This technique allows us to improve crowded data such that exposures that would have otherwise been dropped can now be included in the data analysis. While PSF extraction pipelines such as PampelMUSE can provide precise PSF subtractions and photometry for sidereally tracked data, starkiller is applicable to both sidereal and non-sidereal tracked data. Since both pipelines can be applied to sidereal data, we compare them in Appendix B. Furthermore, starkiller opens up the possibility to perform difference imaging of IFU data cubes with single exposures, while retrieving estimated stellar parameters.", "pages": [13]}, {"title": "4.1. starkiller capabilities and potential uses", "content": "The development case for starkiller was the analysis of MUSE datacubes for 2I/Borisov, which are presented in Deam et al. (2024). Many of these observations were within 10 degrees of the galactic plane, where star crowding heavily limited the data quality. Of the 51 exposures that were taken during the 2I/Borisov observing campaign, starkiller needed to be applied to 27. The reduction improved the data quality for dust/gas maps of 2I's coma in 23, and in the remaining 4, starkiller improved the data quality so significantly that the exposures no longer had to be rejected from analysis (Deam et al. 2024), recovering data for an unusual transient target. Similarly, applying starkiller provides improved detail in the debris trail from Dimorphos of Opitom et al. (2023); Murphy et al. (2023). This is demonstrated in Fig. 12 and Fig 13 respectively. Murphy et al. (2023) had to discard a significant portion of their debris trail datapoints, due to contamination from the stars crossing the tail. In theory, starkiller means the Solar System community can now use any IFU to look near the Galaxy. For serendipitously observed Solar System objects, starkiller provides the capacity to return timeresolved spectra on streaks. For instance, the foreground of asteroids present in many long-duration sidereally tracked exposures can be extracted (using the sat killer mode). Since starkiller identifies the best fitting extinction E(B -V) for each source, it may be possible to use it to calculate, or to independently check, the extinction present in star clusters. We present in Fig. 14 the distribution of extinction values obtained by starkiller for NGC 6563 (Fig. 10) as an example for this use case. The distribution of extinctions present in this field is highly bimodal, displaying low and high extinction populations with median E(B -V) of \u223c 0 . 5 mag, \u223c 1 . 2 mag, respectively. As the high extinction population is largely co-located with NGC 6563 this may suggest that they are being obscured by the nebula. While these are Gaia DR3 sources, most do not have distances to compare against. Furthermore, we note that the E(B -V) value of the low extinction population is approximately double the S&F dust map for the region, which has E(B -V) = 0 . 2271 \u00b1 0 . 0024 10 (Schlafly & Finkbeiner 2011). The large discrepancy in values for this case requires further investigation to ascertain the reliability of extinction values generated by starkiller . As with the flux calibration process, starkiller is only optimizing the correlations between individual models and the observed spectra. Therefore, if the input stellar atmosphere library is insufficient, starkiller may be using extinction as a tool to reshape poorly matched spectra to improve the correlation. In starkiller 's stellar fitting, the point-source fluxes are quantified according to the expected Gaia catalog magnitudes. Where fully stellar PSFs remain as residuals, this can be used to identify variable sources, a common practice in conventional difference imaging.", "pages": [13, 14]}, {"title": "4.2. Satellite impacts on spectroscopic data", "content": "The growing number of satellites in low-Earth orbit present fundamental challenges for all ground-based observing. As the number of satellites approach the projected numbers in the variety of orbital shells, their density will be sufficient such that even instruments with small FoVs will have frequent satellite streaks. With starkiller , we provide a tool to isolate and remove satellite streaks from IFU datacubes. Furthermore, we can directly analyze a satellite spectrum, and even test for temporal variability. This initial approach can be expanded to become a robust method for addressing satellite contamination, and the generation of a library of satellite spectra. Satellite streaks directly crossing science targets, or 'strikes' in IFU data provides us with a unique opportunity to examine how satellite strikes impact optical observations, which has not yet been shown for MUSE. Fig. 15 demonstrates that the satellite observed in the datacube of J0141-5427 has a spectrum that resembles a smooth continuum, with some prominent absorption lines visible in Fig. 15 (top left). Both this unidentified satellite and the blazar are SNR \u223c 100, generating a large effect on the blazar spectrum. This appears more severe than the expectations for spectroscopic impact of Hainaut & Moehler (2024). However, in comparison with a spectrum of J0141-5427 that was observed subsequently and is without a satellite passage, we find that starkiller 's satellite subtraction process successfully removes the broad satellite continuum, as seen with the orange line in Fig. 15 (top right). In this case, we believe that starkiller has effectively 'cleaned' the datacube of the satellite contaminant, allowing it to now be used for science. With this well-isolated satellite spectrum, we can analyse its properties and the atmospheric effects. As the satellite is reflecting sunlight, we might expect the satellite spectrum to be similar to the Sun; this is a frequent assumption in studies of potential satellite impact on astronomical spectra (e.g. Bialek et al. 2023; Hainaut & Moehler 2024). As shown in Fig. 15 (lower left), - we find that the satellite spectrum is indeed well represented by a Solar spectrum that has been highly processed by atmospheric extinction, with the parameters fit by pyExtinction (Buton & Copin 2014). While the MUSE pipeline applies a correction for standard atmospheric extinction Weilbacher et al. (2020, Sec. 4.9), it is possible that the satellite has experienced higher levels of atmospheric extinction. For satellites, the relative configuration of the Sun, satellite, Earth, and observer become important. In many configurations it is possible for light from the Sun to pass through the Earth's atmosphere before reaching the satellite. This additional passage of light through the atmosphere would create the enhanced atmospheric extinction that we observe in the satellite spectrum.", "pages": [14, 16]}, {"title": "4.3. Independent flux calibration verification", "content": "Through the forward-modeling approach of starkiller , we have developed a way to independently verify the flux calibration of an IFU datacube. As described in Sec. 3.5, we only use the observed spectra for shape comparison by correlating the observed spectra with model spectra. The best-fitting model spectra are then flux scaled according to catalog magnitudes ( Gaia DR3 in the default case); we then compare the flux of these scaled models to the observed spectra to generate a flux correction, as described in Sec. 3.6. While some fine features in the flux correction may result from poor model matching, overall trends and flux offsets are likely to be indicative of a flux calibration error between the datacube and the catalog.", "pages": [16]}, {"title": "4.4. Current starkiller limitations", "content": "While starkiller is robust to a wide range of challenging observational conditions, there are several limitations. Primary among these is the source density: if there are too few stars in the FOV or in the source catalog, then starkiller is unable to create corrections based on the population of sources. If there are few sources i.e. < 3 sources contained in the IFU, then the data PSF and flux correction will be biased by the available targets. Flux correction will only be applied if there are at least 3 calibration sources present in the IFU, or the force flux correction option is set. If the catalog is incomplete, then un-cataloged sources will not be subtracted: in the example of NGC 6563 (Fig. 10), the majority of sources that can be seen are not identified in the Gaia DR3 catalog. Additionally, the selected calibration sources may then face higher-than-expected crowding. Unexpected crowding of calibration sources can lead to poor PSF models, and therefore poor subtractions. Conversely, while starkiller is fairly robust to crowding, overlapping sources are not well modeled. The current method for spectral extraction and source position fitting considers each source independently. For crowded sources, this approach leads to spectral contamination and poor positional fitting for crowded sources, particularly faint sources. An alternative approach such as iterative PSF fitting and subtraction of targets, or simultaneous fitting of grouped sources, could be more successful for overlapping sources. In cases of sidereal - tracking of crowded sources, PampelMUSE will provide higher quality spectra and PSF subtractions. The starkiller PSF construction method currently does not create wavelength-dependent PSFs. While this limitation is largely inconsequential for the low signalto-noise stars in non-sidereal observations, it may limit the reliability of starkiller in high signal-to-noise observations of bright stars in sidereal observations, a situation where PampelMUSE is better suited. Another limitation of the current PSF implementation is how the PSF wings are treated, particularly for the data PSF. To avoid including detector artifacts into the data PSF, we set all spaxels with less than 10 -4 % contribution to the total PSF to 0. This sets a nominal PSF radius in MUSE data to be \u223c 8 spaxels. For faint sources, this PSF truncation has little effect, as the flux at the PSF wings is low; however, for bright sources, this leads to overly-subtracted centers, surrounded by un-subtracted wings. A possible solution to this would be to augment the data PSF with a Moffat or Gaussian model component fit to the PSF wings. The quality of the starkiller subtraction for a target depends heavily on how well the spectral models fit. While we have equipped starkiller with a broad sample of stellar spectra, additional models may be required, or be more representative of specific stars. Even with these limitations, starkiller is effective at removing more than 90% of the stellar flux in most cases. For non-sidereal targets, the combination of starkiller and averaging together multiple cubes observed at different sky locations overcomes much of the limitations. Some refinement is still required to optimize starkiller for sidereal-tracked observations with high signal-to-noise stars.", "pages": [16, 17]}, {"title": "4.5. Extending starkiller to other IFUs", "content": "With a catalog in hand, starkiller can be extended to other IFUs; currently operational ones have smaller FoVs than MUSE. For instance, it would be immediately suitable for WiFes (38 '' \u00d7 25 '' ), GMTIFS (20.4 '' \u00d7 20.4 '' ) or Keck KCWI (20 '' \u00d7 33 '' ) data that has the same data format as MUSE. Planned ELT IFUs (e.g. HARMONI, WFOS) have similar-scale FOVs. Even for the smallest IFUs, the capabilities of starkiller will be suited for treating satellite-affected data. The future VLT instrument BlueMUSE 11 is planned to have a FOV larger than 1 arcmin 2 , which would be highly suited to starkiller . For JWST's NIRSpec or MIRI, we note that the default stellar models in starkiller are cut to the optical range; similarly, using an IR catalog would be necessary. In all cases, for fast-moving non-sidereal objects, in the present version of starkiller , the exposure lengths of observations would need to be capped, so as to retain the star trail length within the IFU FOV (the containment requirement discussed in \u00a7 3.4).", "pages": [17, 18]}, {"title": "5. CONCLUSION", "content": "In this paper we have presented the starkiller package, which creates synthetic difference images of single IFU data cubes using a forward-modeling approach. By utilizing independent photometric catalogs, and a suite of stellar atmosphere models, starkiller simultaneously provides stellar spectral classification, relative velocity, and line-of-sight extinction for all sources in a catalog, alongside a source-subtracted datacube. We developed starkiller to be compatible with both sidereal and non-sidereally acquired observations. For streaked sources, we do not require input tracking, as starkiller is generalized to work with even extreme cases of elongated sources and crowded fields. In the most extreme case, it can model and subtract satellite streaks. We developed this method to clean highly elongated stars from VLT/MUSE observations of 2I/Borisov. As we were working to preserve the spectral features of a diffuse foreground object, we developed starkiller to rely heavily on catalogs of model stellar spectra and magnitudes. IFUs provide exceptional capabilities for astronomical enquiry. We look forward to seeing what uses this package may find in the community. R.R.H. is supported by the Royal Society of New Zealand, Te Ap\u00afarangi through a Marsden Fund Fast Start Grant and by the Rutherford Foundation Postdoctoral Fellowship. M.T.B. appreciates support by the Rutherford Discovery Fellowships from New Zealand Government funding, administered by the Royal Society Te Ap\u00afarangi. Package development based on observations collected at the European Southern Observatory under ESO programmes 103.2033.001-003 and 105.2086.002 (2I/Borisov, PIs: M.T.B and Cyrielle Opitom), 110.23XL and 109.2361 (DART, with thanks to PI Cyrielle Opitom), 60.A-9100 (PI: MUSE Team), and 109.238W (PI: Fuyan Bian). We thank the ESO staff, particularly Danuta Dobrzycka and Lodovico Coccato of the MUSE SDP team, Marco Berton, Henri Boffin, Bin Yang, Diego Parraguez, Edmund Christian Herenz, Fuyan Bian, and Israel Blanchard, for their help in the acquisition or retrieval of these observations. We thank the community of the IAU Centre for Protection of the Dark and Quiet Sky From Satellite Constellation Interference for their help during this work. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www. cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www. cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.", "pages": [18]}, {"title": "Facilities: VLT(MUSE)", "content": "Software: Starkiller (Ridden-Harper et al. 2024), astropy (Astropy Collaboration et al. 2013, 2018, 2022), astroquery (Ginsburg et al. 2019), scipy (Virtanen et al. 2020), photutils (Bradley 2023), pandas (Wes McKinney 2010; The pandas development Team 2024), numpy (Harris et al. 2020), matplotlib Hunter (2007), OpenCV (Bradski 2000), MUSE Python Data Analysis Framework (MPDAF; Bacon et al. 2016), TRIPPy (Fraser et al. 2016), pyExtinction (Buton & Copin 2014), PampelMUSE (Kamann et al. 2013)", "pages": [18]}, {"title": "A. PSF MODELING OF POINT SOURCES", "content": "The functions starkiller uses to model trailed PSFs can be readily applied to point sources. If the optional variable trail is set to false, the trail length will be set to 1 spaxel, and starkiller will fit regular point sources. As seen in Fig. 16, the data PSF outperforms both the Gaussian and Moffat PSF with residuals \u223c 15% lower for the data PSF.", "pages": [19]}, {"title": "B. COMPARISON OF SIDEREAL SOURCE-SUBTRACTION WITH PAMPELMUSE", "content": "The PempelMUSE pipeline (Kamann et al. 2013) was developed to obtain high precision extraction of point sources from MUSE data. Some aspects of PampelMUSE are similar to starkiller , such as requiring a source catalog to identify sources; both pipelines can produce source-subtracted cubes. Unlike starkiller , PempelMUSE can only be applied to sidereally tracked data, and primarily uses a Moffat profile to model the PSF. PempelMUSE creates its source-subtracted data cube through subtracting the fit PSF profile for each source from every frame. Therefore, while the two pipelines were created for different use cases, we can draw comparisons between the two when looking at their outcomes on sidereally tracked data. In Fig. 17 we compare the subtracted cubes produced by starkiller and PempelMUSE for NGC 6563 (the same nebula shown in \u00a7 3.7 and Fig. 10). While both model have subtraction artifacts, PempelMUSE systematically oversubtracts sources within the extent of the nebula. This over-subtraction will result in output spectra from PempelMUSE that have larger fluxes than were observed. In contrast, starkiller appears to under-subtract sources, which will be due to the flux scaling to match the catalog photometry and any intrinsic variability of sources. PampelMUSE outperforms starkiller for crowded sources. This test also highlights the poor subtractions from starkiller and PampelMUSE for bright sources, suggesting that the MUSE PSF profile differs between bright and faint sources.", "pages": [19]}, {"title": "REFERENCES", "content": "Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068 Astropy Collaboration, Price-Whelan, A. M., Sip\"ocz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f Astropy Collaboration, Price-Whelan, A. M., Lim, P. L., et al. 2022, ApJ, 935, 167, doi: 10.3847/1538-4357/ac7c74 Axelrod, T., Saha, A., Matheson, T., et al. 2023, ApJ, 951, 78, doi: 10.3847/1538-4357/acd333 Babusiaux, C., Fabricius, C., Khanna, S., et al. 2023, A&A, 674, A32, doi: 10.1051/0004-6361/202243790 Bacon, R., Piqueras, L., Conseil, S., Richard, J., & Shepherd, M. 2016, MPDAF: MUSE Python Data Analysis Framework, Astrophysics Source Code Library, record ascl:1611.003. http://ascl.net/1611.003 Bacon, R., Accardo, M., Adjali, L., et al. 2010, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7735, The MUSE second-generation VLT instrument, 773508, doi: 10.1117/12.856027 Ballard, D. H. 1981, Pattern Recognition, 13, 111, doi: 10.1016/0031-3203(81)90009-1 Bannister, M. 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2014ApJ...786..153X | https://arxiv.org/pdf/1609.08636.pdf | <document>
<section_header_level_1><location><page_1><loc_11><loc_85><loc_89><loc_87></location>ASYMMETRIC ORBITAL DISTRIBUTION NEAR MEAN MOTION RESONANCE: APPLICATION TO PLANETS OBSERVED BY KEPLER AND RADIAL VELOCITIES</section_header_level_1>
<text><location><page_1><loc_45><loc_83><loc_55><loc_84></location>Ji-Wei Xie 1 , 2</text>
<unordered_list>
<list_item><location><page_1><loc_10><loc_80><loc_91><loc_83></location>1 Department of Astronomy & Key Laboratory of Modern Astronomy and Astrophysics in Ministry of Education, Nanjing University, Nanjing, 210093, China; [email protected] and</list_item>
<list_item><location><page_1><loc_12><loc_79><loc_89><loc_80></location>2 Department of Astronomy and Astrophysics, University of Toronto, Toronto, ON M5S 3H4, Canada; [email protected]</list_item>
</unordered_list>
<text><location><page_1><loc_41><loc_78><loc_59><loc_79></location>Draft version August 6, 2018</text>
<section_header_level_1><location><page_1><loc_45><loc_76><loc_55><loc_77></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_67><loc_86><loc_75></location>Many multiple planet systems have been found by the Kepler transit survey and various Radial Velocity (RV) surveys. Kepler planets show an asymmetric feature, namely there are small but significant deficits/excesses of planet pairs with orbital period spacing slightly narrow/wide of the exact resonance, particularly near the first order Mean Motion Resonance (MMR), such as 2:1 and 3:2 MMR. Similarly, if not exactly the same, an asymmetric feature (pileup wide of 2:1 MMR) is also seen in RV planets, but only for massive ones.</text>
<text><location><page_1><loc_14><loc_57><loc_86><loc_67></location>We analytically and numerically study planets' orbital evolutions near/in MMR. We find that their orbital period ratios could be asymmetrically distributed around the MMR center regardless of dissipation. In the case of no dissipation, Kepler planets' asymmetric orbital distribution could be partly reproduced for 3:2 MMR but not for 2:1 MMR, implying dissipation might be more important to the latter. The pileup of massive RV planets just wide of 2:1 MMR is found to be consistent with the scenario that planets formed separately then migrated toward MMR. The location of the pileup infers a K value of 1-100 on order of magnitude for massive planets, where K is the damping rate ratio between orbital eccentricity and semimajor axis during planet migration.</text>
<text><location><page_1><loc_14><loc_55><loc_67><loc_56></location>Subject headings: Planets and satellites: dynamical evolution and stability</text>
<section_header_level_1><location><page_1><loc_22><loc_51><loc_35><loc_52></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_32><loc_48><loc_51></location>The Kepler mission has discovered from its first 16 months data over 2300 planetary candidates (Borucki et al. 2011; Batalha et al. 2012). Over one third ( > 800) of these candidates are in multiple transiting candidate planetary systems, and one remarkable feature of them, as shown by Lissauer et al. (2011) and Fabrycky et al. (2012a), is that the vast majority of candidate pairs are neither in nor near low-order mean motion resonance (MMR hereafter, see also in Veras & Ford (2012)), however there are small but significant excesses/deficits of candidate pairs slightly wider/narrow of the exact resonance (or nominal resonance center), particularly near the first order MMR, such as 2:1 and 3:2 MMR.</text>
<text><location><page_1><loc_8><loc_16><loc_48><loc_32></location>Such an intriguing asymmetric period ratio distribution has stimulated a number of theorists recently, who developed different models to understand and interpret it. Lithwick & Wu (2012); Batygin & Morbidelli (2012); Delisle et al. (2012) consider that such an asymmetric period ratio distribution around MMR could be an outcome of resonant couples having underwent eccentricity damping during some dissipative evolutions, such as tidal dissipation (see also in Terquem & Papaloizou (2007)). On the other side, Rein (2012) attempts to interpret it as a result of the combination of stochastic and smooth planet migrations.</text>
<text><location><page_1><loc_8><loc_7><loc_48><loc_16></location>Beside and before the Kepler transit survey, many near MMRplanets had been found by various Radial Velocity (RV hereafter) surveys. As we will show below (section 3.2), similar, if not exactly the same, features of the period ratio distributions seen in Kepler planets, have been also shown in RV planets. One question is how all these features/clues in both the Kepler and RV samples could</text>
<text><location><page_1><loc_52><loc_49><loc_92><loc_53></location>be understood systematically in a common context. This paper is such an attempt and it is organized as the following.</text>
<text><location><page_1><loc_52><loc_31><loc_92><loc_49></location>We first analytically study the dynamics of planets near/in MMR in section 2.1, and confirm the analytical results with numerical simulations in section 2.2. We find that planets' orbital distribution could be asymmetric around the MMR center under certain conditions. We then discuss its implications to Kepler and RV planets in section 3. Finally, we summarize this paper in section 4. Some analytical derivations are also given in the appendix A and B as supplementary. We note that Petrovich et al. (2012) posted their paper to arxiv.org just a few days before submitting this paper, which, independently and in a different way, arrived at many of the results presented in this paper.</text>
<section_header_level_1><location><page_1><loc_54><loc_29><loc_90><loc_30></location>2. ASYMMETRIC ORBIT DISTRIBUTION NEAR MMR</section_header_level_1>
<text><location><page_1><loc_52><loc_23><loc_92><loc_28></location>We study the orbital evolutions of two planets (orbiting a central star) near/in first order MMR. As we will show below, the orbit distribution could be asymmetric near the MMR center under certain circumstances.</text>
<section_header_level_1><location><page_1><loc_65><loc_20><loc_79><loc_22></location>2.1. Analytic Study</section_header_level_1>
<text><location><page_1><loc_61><loc_19><loc_83><loc_20></location>2.1.1. No dissipation (analytical)</text>
<text><location><page_1><loc_52><loc_14><loc_92><loc_18></location>For simplicity, we assume both planets' orbits are coplanar. The total energy, or Hamiltonian, is (Murray & Dermott 1999)</text>
<formula><location><page_1><loc_57><loc_10><loc_92><loc_13></location>H = -GM /star m 1 2 a 1 -GM /star m 2 2 a 2 -Gm 1 m 2 a 2 R j , (1)</formula>
<text><location><page_1><loc_52><loc_7><loc_92><loc_9></location>where G is the gravity constant, M /star is the stellar mass, and following Lithwick et al. (2012), the disturbing func-</text>
<text><location><page_2><loc_8><loc_90><loc_34><loc_92></location>tion due to the j : j -1 resonance is</text>
<text><location><page_2><loc_8><loc_86><loc_12><loc_87></location>where</text>
<formula><location><page_2><loc_17><loc_88><loc_48><loc_90></location>R j = f 1 e 1 cos ( φ 1 ) + f 2 e 2 cos ( φ 2 ) , (2)</formula>
<formula><location><page_2><loc_18><loc_84><loc_48><loc_86></location>φ 1 = λ j -/pi1 1 , φ 2 = λ j -/pi1 2 , (3)</formula>
<text><location><page_2><loc_8><loc_83><loc_31><loc_84></location>are the two resonance angles for</text>
<formula><location><page_2><loc_21><loc_80><loc_48><loc_82></location>λ j = jλ 2 -( j -1) λ 1 . (4)</formula>
<text><location><page_2><loc_8><loc_70><loc_48><loc_80></location>Hereafter, we adopt the convention that properties with subscripts '1' and '2' belong to the inner and outer planets respectively. In the above, { m , a , e , λ , /pi1 } are the mass and standard orbital elements for planets. f 1 and f 2 are relevant Laplace coefficients, which are on order of unity and tabulated in Murray & Dermott (1999) and Lithwick et al. (2012).</text>
<text><location><page_2><loc_8><loc_66><loc_48><loc_70></location>Using the Lagrange's planetary equation (on the lowest order terms in e ), we derive the evolutions of planets' semi major axes and eccentricities,</text>
<formula><location><page_2><loc_11><loc_59><loc_48><loc_66></location>˙ a 1 = -2( j -1) Gm 2 n 1 a 1 a 2 ( f 1 e 1 sinφ 1 + f 2 e 2 sinφ 2 ) , ˙ a 2 =2 j Gm 1 n 2 a 2 2 ( f 1 e 1 sinφ 1 + f 2 e 2 sinφ 2 ) , (5)</formula>
<formula><location><page_2><loc_19><loc_51><loc_48><loc_58></location>˙ e 1 = Gm 2 n 1 a 2 1 a 2 e 1 ( f 1 e 1 sinφ 1 ) , ˙ e 2 = Gm 1 n 2 a 3 2 e 2 ( f 2 e 2 sinφ 2 ) , (6)</formula>
<text><location><page_2><loc_8><loc_48><loc_48><loc_50></location>where, n 1 and n 2 are the mean motion of the inner and outer planets respectively.</text>
<text><location><page_2><loc_8><loc_45><loc_48><loc_48></location>Using equation 6 to eliminate φ 1 and φ 2 , we can rewrite equation 5 as</text>
<formula><location><page_2><loc_17><loc_38><loc_48><loc_45></location>˙ a 1 a 1 = -2( j -1) ( e 1 ˙ e 1 + qρ 1 3 e 2 ˙ e 2 ) ˙ a 2 a 2 =2 j ( q -1 ρ -1 3 e 1 ˙ e 1 + e 2 ˙ e 2 ) (7)</formula>
<text><location><page_2><loc_8><loc_36><loc_25><loc_38></location>which integrate to give</text>
<formula><location><page_2><loc_15><loc_30><loc_48><loc_36></location>ln a 1 +( j -1) ( e 2 1 + qρ 1 3 e 2 2 ) = Const . ln a 2 -j ( q -1 ρ -1 3 e 2 1 + e 2 2 ) = Const . (8)</formula>
<text><location><page_2><loc_8><loc_28><loc_24><loc_30></location>where we have defined</text>
<formula><location><page_2><loc_17><loc_25><loc_48><loc_28></location>ρ = j/ ( j -1) and q = m 2 /m 1 (9)</formula>
<text><location><page_2><loc_8><loc_13><loc_48><loc_25></location>We note equations 7 or 8 are equivalent to the well known constants of motion in resonance (see appendix A). A worth noting implication of equation 7 or 8 is that if planet pairs initially formed with circular orbit near MMR, they will shift to a little bit larger orbital period ratio as their eccentricities are excited, inducing an asymmetric orbit distribution near MMR (see numerical confirmation in section 2.2). Using equation 7, this small shift extent in period ratio ( p 2 /p 1 ) can be estimated as</text>
<formula><location><page_2><loc_10><loc_6><loc_48><loc_13></location>d ( p 2 p 1 ) = 3 2 ( a 2 a 1 ) 3 / 2 ( d a 2 a 2 -d a 1 a 1 ) (10) = 3 2 j [( q -1 ρ 2 3 +1 ) d e 2 1 + ρ ( qρ -2 3 +1 ) d e 2 2 ]</formula>
<text><location><page_2><loc_52><loc_88><loc_92><loc_92></location>According to Murray & Dermott (1999) (see their Eqn. 8.209 and 8.210), the maximum eccentricity increase in e 1 and e 2 (or critical eccentricities) are</text>
<formula><location><page_2><loc_52><loc_78><loc_92><loc_87></location>e cr 1 = √ 6 ∣ ∣ ∣ ∣ 3 f 1 ( j -1) 4 3 j 2 3 + j 4 3 ( j -1) 2 3 /q ∣ ∣ ∣ ∣ -1 / 3 ( m 2 M /star ) 1 / 3 e cr 2 = √ 6 ∣ ∣ ∣ 3 f 2 ( j -1) 4 3 j 2 3 q + j 2 ∣ ∣ ∣ -1 / 3 ( m 1 M /star ) 1 / 3 . (11)</formula>
<text><location><page_2><loc_52><loc_75><loc_92><loc_80></location>∣ ∣ Setting d e 2 1 = e 2 cr 1 and d e 2 2 = e 2 cr 2 , then equation 10 will give an estimate of the largest asymmetric shift of period ratio.</text>
<section_header_level_1><location><page_2><loc_60><loc_72><loc_83><loc_73></location>2.1.2. With dissipation (analytical)</section_header_level_1>
<text><location><page_2><loc_52><loc_62><loc_92><loc_71></location>Dissipation processes (e.g., tidal evolution, disk migration) may play an import role during planet formation and evolution. Generally they cause changing on planets' orbital semi major axes and damping in eccentricities. To include these effects, we consider the following changing/damping terms (i.e., inverse of the damping timescales) of semi major axes and eccentricities,</text>
<formula><location><page_2><loc_60><loc_58><loc_92><loc_61></location>γ ak = -1 a k d a k d t , γ ek = -1 e k d e k d t , (12)</formula>
<text><location><page_2><loc_52><loc_52><loc_92><loc_57></location>where (hereafter) k = 1 , 2 for the inner and outer planets respectively. Note, γ ak could be negative, which indicates outward migration, and γ ek is generally positive, i.e., eccentricity is damped in dissipation process.</text>
<text><location><page_2><loc_52><loc_48><loc_92><loc_52></location>Following Lithwick et al. (2012) (see the appendix B for the derivation), the evolutions of the semi major axes of two planets (after adding above damping terms) are.</text>
<formula><location><page_2><loc_53><loc_39><loc_92><loc_47></location>˙ a 1 a 1 = -2 jqρ 2 / 3 1 ∆ 2 ( m 2 M /star ) 2 ( qρ 1 3 f 2 1 γ e1 + f 2 2 γ e2 ) -γ a1 , ˙ a 2 a 2 = 2 j 1 ∆ 2 ( m 1 M /star ) 2 ( qρ 1 3 f 2 1 γ e1 + f 2 2 γ e2 ) -γ a2 , (13)</formula>
<text><location><page_2><loc_52><loc_38><loc_56><loc_39></location>where</text>
<formula><location><page_2><loc_65><loc_34><loc_92><loc_37></location>∆ = j -1 j p 2 p 1 -1 (14)</formula>
<text><location><page_2><loc_52><loc_30><loc_92><loc_33></location>is the proximity to the nominal resonance center, and thus its evolution follows,</text>
<formula><location><page_2><loc_54><loc_23><loc_92><loc_30></location>˙ ∆= 3 2 ( ˙ a 2 a 2 -˙ a 1 a 1 ) = 3 j ∆ 2 ( m 1 M /star ) 2 ( 1 + qρ -2 3 ) × ( qρ 1 3 f 2 1 γ e1 + f 2 2 γ e2 ) + γ a 1 -γ a 2 (15)</formula>
<text><location><page_2><loc_52><loc_14><loc_92><loc_22></location>If γ a1 ≥ γ a2 , then ˙ ∆ will be always positive, namely the two planets will always keep divergent migration, i.e., their period ratio will always increase. This is the case if the planetary system undergoes tidal evolution (Terquem & Papaloizou 2007; Lithwick & Wu 2012; Batygin & Morbidelli 2012).</text>
<text><location><page_2><loc_52><loc_12><loc_92><loc_14></location>If γ a1 < γ a2 otherwise, then there is an stable equilibrium with</text>
<formula><location><page_2><loc_53><loc_5><loc_92><loc_11></location>∆ eq = m 1 M /star 3 ( 1 + qρ -2 3 )( qρ 1 3 f 2 1 γ e1 + f 2 2 γ e2 ) j ( γ a 2 -γ a 1 ) 1 / 2 , (16)</formula>
<text><location><page_3><loc_8><loc_88><loc_48><loc_92></location>wider/narrower than which, the two planets will undergo convergent/divergent migration, thus eventually they will be locked at ∆ = ∆ eq .</text>
<text><location><page_3><loc_8><loc_85><loc_48><loc_88></location>Interestingly, the above equation can be roughly written as</text>
<formula><location><page_3><loc_23><loc_82><loc_48><loc_85></location>∆ eq ∼ µK 1 / 2 , (17)</formula>
<text><location><page_3><loc_8><loc_73><loc_48><loc_82></location>where µ is the typical planet-star mass ratio of the system and K = γ e /γ a is the well known model parameter describing the ratio between the damping rate of orbital eccentricity and that of semimajor axis. From equation (17), we see that the theoretical parameter K is linked to an observable ∆ eq . We will discuss this more in section 3.2.</text>
<section_header_level_1><location><page_3><loc_21><loc_70><loc_36><loc_71></location>2.2. Numerical Study</section_header_level_1>
<text><location><page_3><loc_8><loc_58><loc_48><loc_70></location>For comparison against the above analytical results, we perform some 3-Body (1 star + 2 planets) simulations using the well-tested N-body integrator MERCURY (Chambers & Migliorini 1997). For all the simulations, the central star is set with a mass M /star = M /circledot , and all angular orbital elements, except for orbital inclinations, are initially randomly set. For most simulations, the semi major axis of the inner planet is set at 0.1 AU if not specified.</text>
<section_header_level_1><location><page_3><loc_17><loc_55><loc_39><loc_56></location>2.2.1. No dissipation (numerical)</section_header_level_1>
<text><location><page_3><loc_8><loc_48><loc_48><loc_55></location>From equations 7-10, we expect that planets' orbits have an asymmetric distribution near the MMR center. Here, we numerically show such an asymmetry and its dependence on the initial period ratio, orbital eccentricities, inclinations and planetary masses.</text>
<text><location><page_3><loc_8><loc_27><loc_48><loc_48></location>Figure 1 shows the orbital evolutions of two equal mass (10 M ⊕ ) planets initially with circular and coplanar orbits but different orbital ratios. Planets' semimajor axes and eccentricities follow periodical oscillations, and their period ratios increase with eccentricities as expected from equation 7. On average the planets spend more time on orbits wider than the initial ones, causing an asymmetric distribution in their period ratio. The asymmetry become weaker as the planet pair is further away from MMR. However, the most prominent asymmetric feature is not realized at the MMR center but at a little bit narrower than the center. The reason is that planets' eccentricities get most excited when they are at the separatrix which is at narrower than the nominal resonance center for the first order MMR (Murray & Dermott 1999).</text>
<text><location><page_3><loc_8><loc_12><loc_48><loc_27></location>Figure 2 shows how the asymmetry is affected by the initial orbital eccentricities. As expected from equation 7, if the eccentricity is initially larger, then it will have larger possibility (compared to the case of zero initial eccentricity) to decrease in the future, thus the period ratio will become more symmetric around the initial one. The critical eccentricity, greater than which the asymmetry will be very weak, could be estimated using equation 11, which is consistent with the numerical results and the results within the context of the restricted 3-body problem (Murray & Dermott 1999).</text>
<text><location><page_3><loc_8><loc_7><loc_48><loc_12></location>Figure 3 shows the effect of planetary mass on the asymmetry. As expected from equation 7-11, increasing mass leading to larger eccentricity excitation and thus larger period ratio shift extent. Roughly, systems with</text>
<text><location><page_3><loc_52><loc_89><loc_92><loc_92></location>similar total masses (regardless of mass ratio) have similar shift extents.</text>
<text><location><page_3><loc_52><loc_77><loc_92><loc_89></location>Figure 4 shows the role of relative inclination i 12 in the asymmetry. Generally the asymmetry becomes very weak for i 12 > 10 · . This is not surprised, as the above analytical studies are all based on an assumption of low i 12 . For large i 12 , more terms ( e.g., on the oder of ie ) should be considered in the disturbing function in equation 2, and in such cases, planets could be involved in second order of MMR, which is symmetric around nominal resonance center (Murray & Dermott 1999).</text>
<section_header_level_1><location><page_3><loc_60><loc_75><loc_84><loc_76></location>2.2.2. With dissipation (numerical)</section_header_level_1>
<text><location><page_3><loc_52><loc_59><loc_92><loc_74></location>The case of divergent migration (i.e., γ a 1 ≥ γ a 2 ) has been studied recently in detail recently by Terquem & Papaloizou (2007); Lithwick & Wu (2012); Batygin & Morbidelli (2012). Here we focus on the other case where γ a 1 < γ a 2 . For simplicity, we assume γ a 1 = γ e 1 = 0, and damping is only added on the outer planet with γ a 2 = 10 -8 d -1 and γ e 2 = Kγ a 2 . The two planets are started at 0.2 and 0.35 AU respectively with an initial orbital period ∼ 2 . 3. We study 7 different K values from 0 to 10000 and 3 different planetary mass sets. The results are plotted in figure 5.</text>
<text><location><page_3><loc_52><loc_46><loc_92><loc_59></location>The left 4 panels of figure 5 plot the results of one simulation with m 1 = m 2 = 100 M ⊕ and K = 100. The out planet moves inward and captures into 2:1 MMR with the inner planet at about t = 2 × 10 7 d. After that, the two planets still moving inward together but with resonance angles, eccentricities and period ratios reaching a relatively stable state. The orbital period ratio at the later state is asymmetric around the nominal MMR center, and the its mean value is roughly consistent with the analytical estimate from equation 16.</text>
<text><location><page_3><loc_52><loc_28><loc_92><loc_46></location>The right panel of figure 5 shows how ∆ eq depends on planetary mass and damping ratio K . Generally we see that ∆ eq is proportional to planetary mass and increases with K . Not surprised, the analytical predictions are consistent with the numerical simulations only for relative large ∆ eq and K (see Appendix B). For low K values ( K < 10), ∆ eq do not approach zero but become a positive constant which is proportional to planetary mass. Such a tiny constant ∆ eq may reflect the intrinsic asymmetry of the MMR. However, we note that here the constant ∆ eq is much smaller than the maximum asymmetry estimated by equations 10 and 11, which is reasonable because large eccentricity leads to weak asymmetry as seen in figure 2.</text>
<section_header_level_1><location><page_3><loc_66><loc_25><loc_78><loc_26></location>3. DISCUSSIONS</section_header_level_1>
<section_header_level_1><location><page_3><loc_60><loc_23><loc_84><loc_25></location>3.1. Application to Kepler Planets</section_header_level_1>
<text><location><page_3><loc_52><loc_7><loc_92><loc_23></location>The period ratio distribution of Kepler multiple planet candidate systems show an intriguing asymmetric feature near MMR, especially for 2:1 and 3:2 MMR, namely there are small deficits/excesses just a little bit narrow/wide of the nominal MMR center (Lissauer et al. 2011; Fabrycky et al. 2012a). To interpret such an asymmetric feature, Lithwick & Wu (2012); Batygin & Morbidelli (2012) consider that it could be a result of planets undergoing some dissipative evolution, such as tidal dissipation. In such a case, as discussed in section 2.1.2, γ a 1 > γ a 2 , thus the planet period will always increase. To quantitively explain the observed</text>
<text><location><page_4><loc_8><loc_67><loc_48><loc_92></location>asymmetric period ratio distribution, one needs to put a right amount of dissipation on them. In addition, as tidal effect is only efficient for short period planet, e.g., less 10 days, one needs to resort to other dissipations at larger orbital period where the observed asymmetry is still significant. Rein (2012) then considers if the observed period ratio is consistent with the scenario of planets migrating in disks. First, he considers smooth migration and finds that the excess or pileup of planet pairs is too large and too close to the MMR center. His result is expected from our analytical results in figure 5 and equation 16, which shows ∆ eq ∼ 10 -4 (2 order of magnitude lower than the observed one) if assuming a typical Kepler planet mass on order of 10 M ⊕ and K = 10. Nevertheless, he further shows that by including certain amount stochastic forces due to disk turbulence during migration, the large pileup at MMR center could be smeared out and a period ratio distribution similar to that of Kepler planets could be reproduced.</text>
<text><location><page_4><loc_8><loc_37><loc_48><loc_67></location>All the above attempts belong to the case with dissipation. As we have shown (section 2.1.1 and 2.2.1), the period ratio distribution is intrinsically asymmetric near the MMR center even if there is no dissipation. In order to see whether and how the intrinsic asymmetry can reproduce Kepler planets' period ratio distribution, we perform the following N-body simulations. Specifically, we draw 4000 planets pairs initially with a uniform period ratio distribution near MMR, Rayleigh eccentricity and inclination distributions, and uniformly random distribution for all the other angular orbital elements. We use the MERCURY integrator to simulate these 4000 systems individually on a timescale of 10 5 days and intensively output their period ratio very 200 days. The final period ratio distribution is calculated with these output period ratios of all 4000 systems. As Kepler multiple planet systems are believed to be highly coplanar within a few degree (Fabrycky et al. 2012a), we assume the mean inclination < i > = 2 . 5 · . For simplicity, we only study equal mass pairs, i.e, m 1 = m 2 because different mass ratios lead to similar results as long as their total masses are the same (Fig.3).</text>
<text><location><page_4><loc_8><loc_7><loc_48><loc_37></location>Figure 6 compares the observed period ratio distribution to those from above simulations with different planetary masses from 10 M ⊕ to 100 M ⊕ and mean eccentricities from < e > = 0 . 01 to < e > = 0 . 1. The simulated period ratio distributions have an asymmetric feature resembling the observation, i.e., a trough/pile up just a little bit narrow/wide of MMR center. As expected (Fig.2 and 3), the asymmetric feature become weaker with increasing eccentricity and more extended with increasing mass. In order to reproduce the observed period ratio distribution, it requires a mean eccentricity less than a few percents and planetary mass about 10-20 M ⊕ for 3:2 MMR and ∼ 100 M ⊕ for 2:1 MMR. The eccentricity requirement is consistent with recent eccentricity estimate with transit timing variation (Fabrycky et al. 2012ba; Wu & Lithwick 2012). As for the typical mass of Kepler planets, it is expected to be 4-9 M ⊕ given the typical radii of 2-3 R ⊕ and a mass radio distribution either based on fitting of the solar system, m = M ⊕ ( r/R ⊕ ) 2 . 06 (Lissauer et al. 2011), or transit timing variation, m = 3 M ⊕ ( r/R ⊕ ) (Wu & Lithwick 2012). Even considering a relatively large uncertain of mass measurements, say 100%, such an expected mass is</text>
<text><location><page_4><loc_52><loc_81><loc_92><loc_92></location>still too low to meet the requirement for 2:1 MMR, although it is comparable to the mass requirement for 3:2 MMR. Therefore, we conclude that the intrinsic MMR asymmetry (without any damping) could partially explain Kepler planets' asymmetric period ratio distribution near 3:2 MMR but not 2:1 MMR. For the latter, other mechanisms, e.g., dissipation, should play a more important role.</text>
<section_header_level_1><location><page_4><loc_61><loc_78><loc_83><loc_80></location>3.2. Application to RV Planets</section_header_level_1>
<text><location><page_4><loc_52><loc_55><loc_92><loc_78></location>At the time of writing this paper, there are 409 exoplanets detected with radial velocity (RV) method (exoplanet.org) and about 30% of them reside in multiple planet systems. These RV planets have a wide mass range featured with a bimodal distribution (Pepe et al. 2011) as shown in the left panel of figure7. The boundary is at about 0.2 M J ∼ 64 M ⊕ , which separate the light RV planets (with a media mass of ∼ 12 M ⊕ ) and the massive ones (with a media mass of ∼ 1 . 54 M J ). This bimodal distribution may indicate planets undergo different formations and evolutions for the light and massive groups (Mordasini et al. 2009). Interestingly, we find that these two groups may have different period ratio distributions. As shown in the right panels of figure 7, there is a strong pileup of planet pairs near 2:1 MMR in the massive planet group, which is not seen in the light group.</text>
<text><location><page_4><loc_52><loc_22><loc_92><loc_55></location>Those massive planets piled up near 2:1 MMR seems unlikely formed in situ within a small annulus, but they are more likely formed with larger distance in a disk then brought into 2:1 MMR through convergent migration. Interestingly, we note that the pile up is just a few percent (in period ratio) wide of the 2:1 MMR center, which is expected from our analytical and numerical prediction with planetary migration (e.g., Fig.5). Furthermore, from the location of the pileup (i.e., ∆ eq ), we can infer the damping ratio between eccentricity and semi major axis during planetary migration (i.e., K ) by using equation 16. The result of such an exercise is shown in figure 8. Here we considered two migration scenarios. In scenario 1, only the outer planet undergoes migration, i.e., γ e 2 = Kγ a 2 and γ e 1 = γ a 1 = 0. In scenario 2, the inner one migrates outward and the outer one migrates inward, i.e., γ e 2 = Kγ a 2 , γ e 1 = -Kγ a 1 and γ a 1 = -γ a 2 < 0. As can be seen from figure ?? , the K value is constrained in a relative wide range about 1-100 on order of magnitude. We note this K range is consistent with the hydrodynamical simulations by Kley et al. (2004) which predicts a K value of order of unity, and with dynamical modeling of the well-studied system GJ876 by Lee & Peale (2002) which prefers K = 10 -100.</text>
<section_header_level_1><location><page_4><loc_67><loc_21><loc_77><loc_22></location>4. SUMMARY</section_header_level_1>
<text><location><page_4><loc_52><loc_12><loc_92><loc_20></location>In this paper, we analytically and numerically study the dynamics of planet pairs near first order MMR. Focusing on the evolution of orbital period ratio, we find it could have an asymmetric distribution around the nominal MMR center regardless of whether dissipation is included or not.</text>
<text><location><page_4><loc_52><loc_7><loc_92><loc_12></location>Applying the asymmetric nature of MMR to the Kepler planets, we find that, without dissipation, Kepler planets' asymmetric period ratio distribution could be partly explained for the case of 3:2 MMR but not for 2:1</text>
<text><location><page_5><loc_8><loc_89><loc_48><loc_92></location>MMR, suggesting that dissipation or other mechanisms may play a more important role in 2:1 than in 3:2 MMR.</text>
<text><location><page_5><loc_8><loc_76><loc_48><loc_89></location>Beside the Kepler planets, similar asymmetric feature, i.e., planets piled up wide of MMR, is also seen in RV planets. Nevertheless, planets in multiple RV systems are bimodal distribution on mass, and the pileup is currently only seen in the higher mass group. The location of the pileup is consistent with the scenario that planetary migration toward MMR, and it infers that the ratio of damping rate between eccentricity and semimajor axis (i.e., K value) during planet migration is K = 1 -100 on order of magnitude for massive planets.</text>
<text><location><page_5><loc_52><loc_80><loc_92><loc_92></location>JWX thanks the referee for helpful comments and suggestions, Yanqin Wu and Hanno Rei for valuable discussions and the Kepler team for producing such an invaluable data set. JWX was supported by the National Natural Science Foundation of China (Nos. 10833001 and 10925313), PhD training grant of China (20090091110002), Fundamental Research Funds for the Central Universities (1112020102) and the Ontario government.</text>
<section_header_level_1><location><page_5><loc_45><loc_74><loc_55><loc_75></location>REFERENCES</section_header_level_1>
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<section_header_level_1><location><page_6><loc_46><loc_88><loc_54><loc_89></location>APPENDIX</section_header_level_1>
<section_header_level_1><location><page_6><loc_35><loc_86><loc_65><loc_87></location>A: TWO CONSTANTS OF MOTION IN MMR</section_header_level_1>
<text><location><page_6><loc_8><loc_81><loc_92><loc_85></location>Here we show that the equations 7 and/or 8 are equivalent to the well known constants of motion of MMR. For j:j-1 MMR there are two constants of motion in addition to the energy (see chapter 8.8 of Murray & Dermott (1999)), i.e.,</text>
<formula><location><page_6><loc_39><loc_77><loc_92><loc_80></location>Λ 1 +( j -1)(Γ 1 +Γ 2 ) = Const . Λ 2 -j (Γ 1 +Γ 2 ) = Const . (A1)</formula>
<text><location><page_6><loc_8><loc_74><loc_92><loc_77></location>where, Λ and Γ are the Poincar'e momenta (see chapter 2.10 of Murray & Dermott (1999)), and the subscript '1' and '2' denotes the inner and outer planets respectively. Changing the above equation to basic orbital elements, we have</text>
<formula><location><page_6><loc_22><loc_66><loc_92><loc_73></location>m 1 √ a 1 +( j -1) [ m 1 √ a 1 ( 1 -√ 1 -e 2 1 ) + m 2 √ a 2 ( 1 -√ 1 -e 2 2 )] =Const . m 2 √ a 2 -j [ m 1 √ a 1 ( 1 -√ 1 -e 2 1 ) + m 2 √ a 2 ( 1 -√ 1 -e 2 2 )] =Const . (A2)</formula>
<text><location><page_6><loc_8><loc_65><loc_36><loc_66></location>In the leading term of e , we then have</text>
<formula><location><page_6><loc_31><loc_57><loc_92><loc_64></location>m 1 √ a 1 +( j -1) ( m 1 √ a 1 1 2 e 2 1 + m 2 √ a 2 1 2 e 2 2 ) =Const . m 2 √ a 2 -j ( m 1 √ a 1 1 2 e 2 1 + m 2 √ a 2 1 2 e 2 2 ) =Const . (A3)</formula>
<text><location><page_6><loc_8><loc_55><loc_73><loc_56></location>Take the differential form of above equations and keep the leading term in e , we then have</text>
<formula><location><page_6><loc_32><loc_48><loc_92><loc_54></location>m 1 ˙ a 1 2 √ a 1 +( j -1) ( m 1 √ a 1 e 1 ˙ e 1 + m 2 √ a 2 e 2 ˙ e 2 ) = 0 m 2 ˙ a 2 2 √ a 2 -j ( m 1 √ a 1 e 1 ˙ e 1 + m 2 √ a 2 e 2 ˙ e 2 ) = 0 (A4)</formula>
<text><location><page_6><loc_8><loc_43><loc_92><loc_47></location>Using the approximation, a 2 /a 1 ∼ [ j/ ( j -1)] 2 / 3 , above equations can be rewritten as equation 7. Compared to the original formulas of the constants, the new formulas (Eqn. 7) solve a 1 and a 2 out (they are not coupled together as in Eqn.A1) and they are dimensionless and simpler.</text>
<section_header_level_1><location><page_6><loc_20><loc_40><loc_81><loc_41></location>B: EVOLUTION OF PLANETARY SEMI MAJOR AXES UNDER DISSIPATION NEAR MMR</section_header_level_1>
<text><location><page_6><loc_10><loc_38><loc_71><loc_40></location>Following Lithwick et al. (2012), it is convenient to introduce the compact eccentricity</text>
<formula><location><page_6><loc_45><loc_36><loc_92><loc_38></location>z k = e k e i/pi1 k , (B1)</formula>
<text><location><page_6><loc_8><loc_33><loc_92><loc_35></location>where /pi1 is the longitude of the periastron and k = 1 , 2 for the inner and outer planets respectively. In terms of which, the disturbing function can be expressed as</text>
<formula><location><page_6><loc_39><loc_29><loc_92><loc_32></location>R j = 1 2 ( f 1 z ∗ 1 + f 2 z ∗ 2 ) e iλ j + c.c. (B2)</formula>
<text><location><page_6><loc_8><loc_26><loc_92><loc_28></location>where the superscript ' * ' denotes the complex conjugate of the variable and ' c.c. ' denotes the complex conjugate of the proceeding term. Then the eccentricity equation (after adding the damping term) for the two planets is</text>
<formula><location><page_6><loc_37><loc_22><loc_92><loc_25></location>˙ z k = -1 √ GM /star 2 i m k √ a k ∂H ∂z ∗ k -γ ek z k , (B3)</formula>
<text><location><page_6><loc_8><loc_20><loc_18><loc_21></location>or specifically,</text>
<formula><location><page_6><loc_28><loc_16><loc_92><loc_19></location>˙ z 1 = iρ 1 3 f 1 n 2 m 2 M /star e iλ j -γ e1 z 1 , ˙ z 2 = iρ -1 f 2 n 1 m 1 M /star e iλ j -γ e2 z 2 . (B4)</formula>
<text><location><page_6><loc_8><loc_14><loc_36><loc_15></location>Adopting the following approximation</text>
<formula><location><page_6><loc_34><loc_11><loc_92><loc_13></location>λ j = jλ 2 -( j -1) λ 1 ∼ -j ∆ n 2 t, γ ek /lessmuch ∆ n 2 , (B5)</formula>
<text><location><page_6><loc_8><loc_10><loc_35><loc_11></location>then we can solve the eccentricities as</text>
<formula><location><page_6><loc_18><loc_6><loc_92><loc_9></location>z 1 = -ρ 1 / 3 f 1 j ∆ m 2 M /star e iλ j ( 1 -i γ e1 j ∆ n 2 ) + z free1 , z 2 = -f 2 j ∆ m 1 M /star e iλ j ( 1 -i γ e2 j ∆ n 2 ) + z free2 , (B6)</formula>
<text><location><page_7><loc_8><loc_90><loc_51><loc_92></location>where z free1 and z free2 are free solutions (free eccentricities).</text>
<text><location><page_7><loc_8><loc_88><loc_92><loc_90></location>In terms of the compact eccentricity, the evolution of semi major axes (Eqn.5) can be rewritten (after adding damping terms) as</text>
<formula><location><page_7><loc_14><loc_84><loc_92><loc_87></location>˙ a 1 a 1 = -( j -1) Gm 2 n 1 a 2 1 a 2 [ ( f 1 z ∗ 1 + f 2 z ∗ 2 ) ie iλ j + c.c. ] -γ a 1 , ˙ a 2 a 2 = jGm 1 n 2 a 3 2 [ ( f 1 z ∗ 1 + f 2 z ∗ 2 ) ie iλ j + c.c. ] -γ a 1 , (B7)</formula>
<text><location><page_7><loc_8><loc_82><loc_50><loc_83></location>which can be finally written as (with the help of Eqn.B6),</text>
<formula><location><page_7><loc_30><loc_73><loc_92><loc_81></location>˙ a 1 a 1 = -2 jqρ 2 / 3 1 ∆ 2 ( m 2 M /star ) 2 ( qρ 1 3 f 2 1 γ e1 + f 2 2 γ e2 ) -γ a1 + F 1 ˙ a 2 a 2 = 2 j 1 ∆ 2 ( m 1 M /star ) 2 ( qρ 1 3 f 2 1 γ e1 + f 2 2 γ e2 ) -γ a2 + F 2 , (B8)</formula>
<text><location><page_7><loc_8><loc_72><loc_57><loc_73></location>where F 1 and F 2 are the terms caused by the free eccentricities, i.e.,</text>
<formula><location><page_7><loc_25><loc_68><loc_92><loc_71></location>F 1 = -( j -1) Gm 2 n 1 a 2 1 a 2 ( Z ∗ free ie iλ j + c.c. ) , F 2 = jGm 1 n 2 a 3 2 ( Z ∗ free ie iλ j + c.c. ) , (B9)</formula>
<text><location><page_7><loc_8><loc_60><loc_92><loc_67></location>for Z free = f 1 z 1 + f 2 z 2 defined as the free eccentricities of the system. . In the case where it is not too close to MMR (modest ∆) and eccentricity damping is efficient (large γ e k ), Z free ∼ 0 and thus the two oscillation terms F 1 and F 2 can be ignored (i.e., Eqn.13 and 16). Otherwise, if it is very close to MMR (very small ∆) and the eccentricity damping is weak, then the system could get significant free eccentricities (probably by approaching the separatrix), thus F 1 and F 2 cannot be ignored and the equilibrium cannot be well estimated by using equation 16 (see also in Fig.5).</text>
<figure>
<location><page_8><loc_14><loc_59><loc_88><loc_92></location>
<caption>Fig. 1.Evolutions of semimajor axes (top, normalized to the initial value, a/a 0 ), and eccentricity (middle, e ) of two planets with masses m 1 = m 2 = 10 M ⊕ (red for the inner planet and green for the outer one) initial orbital ratio of p 2 /p 1 = 2 . 0 (top left 2 panels) and p 2 /p 1 = 1 . 994 (top right 2 panels), circular e 1 = e 2 = 0) and coplanar ( i 12 = 0) orbits. In the top 2 panels, the circles are numerical results and the dashed lines are analytical results based on equation 8. In the bottom two panels, the horizontal dot lines mark the critical eccentricities (Eqn.11). Performing above simulation 100 times with random initial angular orbital elements, we plot the average orbital ratio (sampled at uniformly-spaced time points) distributions in the bottom panel for the cases with initial p 2 /p 1 = 1 . 98 , 1 . 99 , 1 . 994 , 2 . 0 , 2 . 01 and 2 . 02. A dot vertical line is plot in each histogram to indicate the initial period ratio. The asymmetric orbital ratio distribution is most prominent at a little bit narrower than the MMR center (i.e., p 2 /p 1 = 1 . 994 here) and become weaker and weaker as it is further away from MMR. The arrow in the bottom panel marks the largest orbital ratio shift estimated from equations 10-11, which is roughly consistent with the numerical results (orange histogram).</caption>
</figure>
<figure>
<location><page_8><loc_9><loc_29><loc_90><loc_44></location>
<caption>Fig. 2.Similar to the bottom panel of figure 1 but here we compare the period ratio distributions of cases with different initial eccentricities (printed in each panel). As shown, the asymmetric feature diminishes as the eccentricity become comparable to or larger than the critical eccentricities. Here, e cr 1 = 0 . 035 , e cr 2 = 0 . 023 according to equations 11.</caption>
</figure>
<figure>
<location><page_9><loc_33><loc_68><loc_69><loc_92></location>
<caption>Fig. 3.Similar to Fig.2, but here we investigate the dependance of asymmetry on planets' masses. As expected from equations 10-11, the period ratio shift extent increases with planetary mass, and systems with the same total mass (regardless of the mass ratio) have a similar period ratio shift extent.</caption>
</figure>
<figure>
<location><page_9><loc_9><loc_21><loc_90><loc_42></location>
<caption>Fig. 4.Similar to Fig.2, but here we investigate the dependance of asymmetry on the initial relative orbital inclination ( i 12 ) of the two planets. As shown, the asymmetry become very weak if i 12 > 10 · .</caption>
</figure>
<figure>
<location><page_10><loc_10><loc_60><loc_47><loc_92></location>
</figure>
<figure>
<location><page_10><loc_49><loc_60><loc_90><loc_94></location>
<caption>Fig. 5.Numerical tests of the asymmetric feature with dissipation. The left four panels show the orbital evolutions (resonance angles, eccentricities, semi major axes and period ratio from top to bottom) of two planets in one simulation with m 1 = m 2 = 100 M ⊕ and γ e 2 = 100 , γ a 2 = 10 -6 d -1 . The outer planet moves inward and then capture in 2:1 MMR with the inner planets. As expected they will finally stay a little bit wider than the MMR center with the mean period ratio equal to 2.003 (red dot line) which is consistent with the estimate from equation 16 (orange dashed line in the inserted panel). The right panels compares the simulated ∆ eq (symbols) to the one predicted from equation 16 (lines) with different K values, total planetary masses (black, green and red for m 1 + m 2 = 20 , 200 , 2000 M ⊕ respectively) and mass ratios (triangle, solid line for q = 1 and squares and dashed lines for q = 0 . 25, respectively). As expected, the analytical results fit roughly well for relative large K ( > 10) and ∆ eq (see also in Appendix B).</caption>
</figure>
<figure>
<location><page_11><loc_10><loc_65><loc_91><loc_92></location>
</figure>
<figure>
<location><page_11><loc_10><loc_38><loc_91><loc_63></location>
<caption>Fig. 6.Orbital period ratio ( p 2 /p 1 ) distributions: comparison between simulations (colourized curves, normalized to the same peak as the observational histogram) to Kepler observations for planets near 2:1 resonance (top four rows) and those near 3:2 resonance (bottom four rows). For each panel above, we numerically integrate the orbital evolutions of a sample of 4000 planet pairs with an uniform distribution of initial p 2 /p 1 around the nominal resonance center, with equal mass (from bottom to top: 10 M ⊕ -green, 20 M ⊕ -red, 50 M ⊕ -blue, and 100 M ⊕ -purple), with a Rayleigh distribution of initial orbital eccentricity (from left to right: < e > = 0 . 01-solid, < e > = 0 . 05-dot, and < e > = 0 . 1-dashed).</caption>
</figure>
<figure>
<location><page_12><loc_15><loc_53><loc_48><loc_89></location>
</figure>
<figure>
<location><page_12><loc_51><loc_54><loc_84><loc_89></location>
<caption>Fig. 7.Mass distributions (left panel) and period ratio distributions (right two panels) of RV planet sample based on the current exoplanet data set from 'exoplanet.org'. As can be seen, the mass distribution seem bimodal, and it is most prominent for those planets in multiple systems (blue). The period ratio distribution of these RV multiple systems shows a significant pileup near 2:1 MMR for massive planets. The four vertical lines mark the locations of 3:2, 5:3, 2:1 and 3:1 MMR.</caption>
</figure>
<figure>
<location><page_13><loc_32><loc_61><loc_69><loc_90></location>
<caption>Fig. 8.K -K digram for those massive RV planets piled up near 2:1. The horizontal K is the damping ratio γ e /γ a solved from equation (16) by assuming only the outer planet was subject to orbital damping (scenario 1), i.e., γ e2 = Kγ a2 and γ e1 = γ a1 = 0, while the vertical K is the damping ratio solved by assuming both the planets were subject to orbital migration (scenario 2), i.e., γ e1 = -Kγ a1 , γ e2 = Kγ a2 , and γ e1 = γ e2 , following the two damping scenarios studied in Lee & Peale (2002). The error bars reflect the uncertainties of their orbital period measurements, except for HD 82943 for which no uncertainty is reported from exoplanet.org. There are another 2 pairs, HD 73526 b and c and 24 Sex b and c, are not plot here because their period uncertainties are too large that could lead to negative ∆ eq in equation (16). The K -K digram shows a damping ratio ( K value) of 1-100 on order of magnitude constrained by the pileup near 2:1 MMR observed in the massive RV sample (see bottom right panel of Fig.7).</caption>
</figure>
</document> | [{"title": "ABSTRACT", "content": "Many multiple planet systems have been found by the Kepler transit survey and various Radial Velocity (RV) surveys. Kepler planets show an asymmetric feature, namely there are small but significant deficits/excesses of planet pairs with orbital period spacing slightly narrow/wide of the exact resonance, particularly near the first order Mean Motion Resonance (MMR), such as 2:1 and 3:2 MMR. Similarly, if not exactly the same, an asymmetric feature (pileup wide of 2:1 MMR) is also seen in RV planets, but only for massive ones. We analytically and numerically study planets' orbital evolutions near/in MMR. We find that their orbital period ratios could be asymmetrically distributed around the MMR center regardless of dissipation. In the case of no dissipation, Kepler planets' asymmetric orbital distribution could be partly reproduced for 3:2 MMR but not for 2:1 MMR, implying dissipation might be more important to the latter. The pileup of massive RV planets just wide of 2:1 MMR is found to be consistent with the scenario that planets formed separately then migrated toward MMR. The location of the pileup infers a K value of 1-100 on order of magnitude for massive planets, where K is the damping rate ratio between orbital eccentricity and semimajor axis during planet migration. Subject headings: Planets and satellites: dynamical evolution and stability", "pages": [1]}, {"title": "ASYMMETRIC ORBITAL DISTRIBUTION NEAR MEAN MOTION RESONANCE: APPLICATION TO PLANETS OBSERVED BY KEPLER AND RADIAL VELOCITIES", "content": "Ji-Wei Xie 1 , 2 Draft version August 6, 2018", "pages": [1]}, {"title": "1. INTRODUCTION", "content": "The Kepler mission has discovered from its first 16 months data over 2300 planetary candidates (Borucki et al. 2011; Batalha et al. 2012). Over one third ( > 800) of these candidates are in multiple transiting candidate planetary systems, and one remarkable feature of them, as shown by Lissauer et al. (2011) and Fabrycky et al. (2012a), is that the vast majority of candidate pairs are neither in nor near low-order mean motion resonance (MMR hereafter, see also in Veras & Ford (2012)), however there are small but significant excesses/deficits of candidate pairs slightly wider/narrow of the exact resonance (or nominal resonance center), particularly near the first order MMR, such as 2:1 and 3:2 MMR. Such an intriguing asymmetric period ratio distribution has stimulated a number of theorists recently, who developed different models to understand and interpret it. Lithwick & Wu (2012); Batygin & Morbidelli (2012); Delisle et al. (2012) consider that such an asymmetric period ratio distribution around MMR could be an outcome of resonant couples having underwent eccentricity damping during some dissipative evolutions, such as tidal dissipation (see also in Terquem & Papaloizou (2007)). On the other side, Rein (2012) attempts to interpret it as a result of the combination of stochastic and smooth planet migrations. Beside and before the Kepler transit survey, many near MMRplanets had been found by various Radial Velocity (RV hereafter) surveys. As we will show below (section 3.2), similar, if not exactly the same, features of the period ratio distributions seen in Kepler planets, have been also shown in RV planets. One question is how all these features/clues in both the Kepler and RV samples could be understood systematically in a common context. This paper is such an attempt and it is organized as the following. We first analytically study the dynamics of planets near/in MMR in section 2.1, and confirm the analytical results with numerical simulations in section 2.2. We find that planets' orbital distribution could be asymmetric around the MMR center under certain conditions. We then discuss its implications to Kepler and RV planets in section 3. Finally, we summarize this paper in section 4. Some analytical derivations are also given in the appendix A and B as supplementary. We note that Petrovich et al. (2012) posted their paper to arxiv.org just a few days before submitting this paper, which, independently and in a different way, arrived at many of the results presented in this paper.", "pages": [1]}, {"title": "2. ASYMMETRIC ORBIT DISTRIBUTION NEAR MMR", "content": "We study the orbital evolutions of two planets (orbiting a central star) near/in first order MMR. As we will show below, the orbit distribution could be asymmetric near the MMR center under certain circumstances.", "pages": [1]}, {"title": "2.1. Analytic Study", "content": "2.1.1. No dissipation (analytical) For simplicity, we assume both planets' orbits are coplanar. The total energy, or Hamiltonian, is (Murray & Dermott 1999) where G is the gravity constant, M /star is the stellar mass, and following Lithwick et al. (2012), the disturbing func- tion due to the j : j -1 resonance is where are the two resonance angles for Hereafter, we adopt the convention that properties with subscripts '1' and '2' belong to the inner and outer planets respectively. In the above, { m , a , e , \u03bb , /pi1 } are the mass and standard orbital elements for planets. f 1 and f 2 are relevant Laplace coefficients, which are on order of unity and tabulated in Murray & Dermott (1999) and Lithwick et al. (2012). Using the Lagrange's planetary equation (on the lowest order terms in e ), we derive the evolutions of planets' semi major axes and eccentricities, where, n 1 and n 2 are the mean motion of the inner and outer planets respectively. Using equation 6 to eliminate \u03c6 1 and \u03c6 2 , we can rewrite equation 5 as which integrate to give where we have defined We note equations 7 or 8 are equivalent to the well known constants of motion in resonance (see appendix A). A worth noting implication of equation 7 or 8 is that if planet pairs initially formed with circular orbit near MMR, they will shift to a little bit larger orbital period ratio as their eccentricities are excited, inducing an asymmetric orbit distribution near MMR (see numerical confirmation in section 2.2). Using equation 7, this small shift extent in period ratio ( p 2 /p 1 ) can be estimated as According to Murray & Dermott (1999) (see their Eqn. 8.209 and 8.210), the maximum eccentricity increase in e 1 and e 2 (or critical eccentricities) are \u2223 \u2223 Setting d e 2 1 = e 2 cr 1 and d e 2 2 = e 2 cr 2 , then equation 10 will give an estimate of the largest asymmetric shift of period ratio.", "pages": [1, 2]}, {"title": "2.1.2. With dissipation (analytical)", "content": "Dissipation processes (e.g., tidal evolution, disk migration) may play an import role during planet formation and evolution. Generally they cause changing on planets' orbital semi major axes and damping in eccentricities. To include these effects, we consider the following changing/damping terms (i.e., inverse of the damping timescales) of semi major axes and eccentricities, where (hereafter) k = 1 , 2 for the inner and outer planets respectively. Note, \u03b3 ak could be negative, which indicates outward migration, and \u03b3 ek is generally positive, i.e., eccentricity is damped in dissipation process. Following Lithwick et al. (2012) (see the appendix B for the derivation), the evolutions of the semi major axes of two planets (after adding above damping terms) are. where is the proximity to the nominal resonance center, and thus its evolution follows, If \u03b3 a1 \u2265 \u03b3 a2 , then \u02d9 \u2206 will be always positive, namely the two planets will always keep divergent migration, i.e., their period ratio will always increase. This is the case if the planetary system undergoes tidal evolution (Terquem & Papaloizou 2007; Lithwick & Wu 2012; Batygin & Morbidelli 2012). If \u03b3 a1 < \u03b3 a2 otherwise, then there is an stable equilibrium with wider/narrower than which, the two planets will undergo convergent/divergent migration, thus eventually they will be locked at \u2206 = \u2206 eq . Interestingly, the above equation can be roughly written as where \u00b5 is the typical planet-star mass ratio of the system and K = \u03b3 e /\u03b3 a is the well known model parameter describing the ratio between the damping rate of orbital eccentricity and that of semimajor axis. From equation (17), we see that the theoretical parameter K is linked to an observable \u2206 eq . We will discuss this more in section 3.2.", "pages": [2, 3]}, {"title": "2.2. Numerical Study", "content": "For comparison against the above analytical results, we perform some 3-Body (1 star + 2 planets) simulations using the well-tested N-body integrator MERCURY (Chambers & Migliorini 1997). For all the simulations, the central star is set with a mass M /star = M /circledot , and all angular orbital elements, except for orbital inclinations, are initially randomly set. For most simulations, the semi major axis of the inner planet is set at 0.1 AU if not specified.", "pages": [3]}, {"title": "2.2.1. No dissipation (numerical)", "content": "From equations 7-10, we expect that planets' orbits have an asymmetric distribution near the MMR center. Here, we numerically show such an asymmetry and its dependence on the initial period ratio, orbital eccentricities, inclinations and planetary masses. Figure 1 shows the orbital evolutions of two equal mass (10 M \u2295 ) planets initially with circular and coplanar orbits but different orbital ratios. Planets' semimajor axes and eccentricities follow periodical oscillations, and their period ratios increase with eccentricities as expected from equation 7. On average the planets spend more time on orbits wider than the initial ones, causing an asymmetric distribution in their period ratio. The asymmetry become weaker as the planet pair is further away from MMR. However, the most prominent asymmetric feature is not realized at the MMR center but at a little bit narrower than the center. The reason is that planets' eccentricities get most excited when they are at the separatrix which is at narrower than the nominal resonance center for the first order MMR (Murray & Dermott 1999). Figure 2 shows how the asymmetry is affected by the initial orbital eccentricities. As expected from equation 7, if the eccentricity is initially larger, then it will have larger possibility (compared to the case of zero initial eccentricity) to decrease in the future, thus the period ratio will become more symmetric around the initial one. The critical eccentricity, greater than which the asymmetry will be very weak, could be estimated using equation 11, which is consistent with the numerical results and the results within the context of the restricted 3-body problem (Murray & Dermott 1999). Figure 3 shows the effect of planetary mass on the asymmetry. As expected from equation 7-11, increasing mass leading to larger eccentricity excitation and thus larger period ratio shift extent. Roughly, systems with similar total masses (regardless of mass ratio) have similar shift extents. Figure 4 shows the role of relative inclination i 12 in the asymmetry. Generally the asymmetry becomes very weak for i 12 > 10 \u00b7 . This is not surprised, as the above analytical studies are all based on an assumption of low i 12 . For large i 12 , more terms ( e.g., on the oder of ie ) should be considered in the disturbing function in equation 2, and in such cases, planets could be involved in second order of MMR, which is symmetric around nominal resonance center (Murray & Dermott 1999).", "pages": [3]}, {"title": "2.2.2. With dissipation (numerical)", "content": "The case of divergent migration (i.e., \u03b3 a 1 \u2265 \u03b3 a 2 ) has been studied recently in detail recently by Terquem & Papaloizou (2007); Lithwick & Wu (2012); Batygin & Morbidelli (2012). Here we focus on the other case where \u03b3 a 1 < \u03b3 a 2 . For simplicity, we assume \u03b3 a 1 = \u03b3 e 1 = 0, and damping is only added on the outer planet with \u03b3 a 2 = 10 -8 d -1 and \u03b3 e 2 = K\u03b3 a 2 . The two planets are started at 0.2 and 0.35 AU respectively with an initial orbital period \u223c 2 . 3. We study 7 different K values from 0 to 10000 and 3 different planetary mass sets. The results are plotted in figure 5. The left 4 panels of figure 5 plot the results of one simulation with m 1 = m 2 = 100 M \u2295 and K = 100. The out planet moves inward and captures into 2:1 MMR with the inner planet at about t = 2 \u00d7 10 7 d. After that, the two planets still moving inward together but with resonance angles, eccentricities and period ratios reaching a relatively stable state. The orbital period ratio at the later state is asymmetric around the nominal MMR center, and the its mean value is roughly consistent with the analytical estimate from equation 16. The right panel of figure 5 shows how \u2206 eq depends on planetary mass and damping ratio K . Generally we see that \u2206 eq is proportional to planetary mass and increases with K . Not surprised, the analytical predictions are consistent with the numerical simulations only for relative large \u2206 eq and K (see Appendix B). For low K values ( K < 10), \u2206 eq do not approach zero but become a positive constant which is proportional to planetary mass. Such a tiny constant \u2206 eq may reflect the intrinsic asymmetry of the MMR. However, we note that here the constant \u2206 eq is much smaller than the maximum asymmetry estimated by equations 10 and 11, which is reasonable because large eccentricity leads to weak asymmetry as seen in figure 2.", "pages": [3]}, {"title": "3.1. Application to Kepler Planets", "content": "The period ratio distribution of Kepler multiple planet candidate systems show an intriguing asymmetric feature near MMR, especially for 2:1 and 3:2 MMR, namely there are small deficits/excesses just a little bit narrow/wide of the nominal MMR center (Lissauer et al. 2011; Fabrycky et al. 2012a). To interpret such an asymmetric feature, Lithwick & Wu (2012); Batygin & Morbidelli (2012) consider that it could be a result of planets undergoing some dissipative evolution, such as tidal dissipation. In such a case, as discussed in section 2.1.2, \u03b3 a 1 > \u03b3 a 2 , thus the planet period will always increase. To quantitively explain the observed asymmetric period ratio distribution, one needs to put a right amount of dissipation on them. In addition, as tidal effect is only efficient for short period planet, e.g., less 10 days, one needs to resort to other dissipations at larger orbital period where the observed asymmetry is still significant. Rein (2012) then considers if the observed period ratio is consistent with the scenario of planets migrating in disks. First, he considers smooth migration and finds that the excess or pileup of planet pairs is too large and too close to the MMR center. His result is expected from our analytical results in figure 5 and equation 16, which shows \u2206 eq \u223c 10 -4 (2 order of magnitude lower than the observed one) if assuming a typical Kepler planet mass on order of 10 M \u2295 and K = 10. Nevertheless, he further shows that by including certain amount stochastic forces due to disk turbulence during migration, the large pileup at MMR center could be smeared out and a period ratio distribution similar to that of Kepler planets could be reproduced. All the above attempts belong to the case with dissipation. As we have shown (section 2.1.1 and 2.2.1), the period ratio distribution is intrinsically asymmetric near the MMR center even if there is no dissipation. In order to see whether and how the intrinsic asymmetry can reproduce Kepler planets' period ratio distribution, we perform the following N-body simulations. Specifically, we draw 4000 planets pairs initially with a uniform period ratio distribution near MMR, Rayleigh eccentricity and inclination distributions, and uniformly random distribution for all the other angular orbital elements. We use the MERCURY integrator to simulate these 4000 systems individually on a timescale of 10 5 days and intensively output their period ratio very 200 days. The final period ratio distribution is calculated with these output period ratios of all 4000 systems. As Kepler multiple planet systems are believed to be highly coplanar within a few degree (Fabrycky et al. 2012a), we assume the mean inclination < i > = 2 . 5 \u00b7 . For simplicity, we only study equal mass pairs, i.e, m 1 = m 2 because different mass ratios lead to similar results as long as their total masses are the same (Fig.3). Figure 6 compares the observed period ratio distribution to those from above simulations with different planetary masses from 10 M \u2295 to 100 M \u2295 and mean eccentricities from < e > = 0 . 01 to < e > = 0 . 1. The simulated period ratio distributions have an asymmetric feature resembling the observation, i.e., a trough/pile up just a little bit narrow/wide of MMR center. As expected (Fig.2 and 3), the asymmetric feature become weaker with increasing eccentricity and more extended with increasing mass. In order to reproduce the observed period ratio distribution, it requires a mean eccentricity less than a few percents and planetary mass about 10-20 M \u2295 for 3:2 MMR and \u223c 100 M \u2295 for 2:1 MMR. The eccentricity requirement is consistent with recent eccentricity estimate with transit timing variation (Fabrycky et al. 2012ba; Wu & Lithwick 2012). As for the typical mass of Kepler planets, it is expected to be 4-9 M \u2295 given the typical radii of 2-3 R \u2295 and a mass radio distribution either based on fitting of the solar system, m = M \u2295 ( r/R \u2295 ) 2 . 06 (Lissauer et al. 2011), or transit timing variation, m = 3 M \u2295 ( r/R \u2295 ) (Wu & Lithwick 2012). Even considering a relatively large uncertain of mass measurements, say 100%, such an expected mass is still too low to meet the requirement for 2:1 MMR, although it is comparable to the mass requirement for 3:2 MMR. Therefore, we conclude that the intrinsic MMR asymmetry (without any damping) could partially explain Kepler planets' asymmetric period ratio distribution near 3:2 MMR but not 2:1 MMR. For the latter, other mechanisms, e.g., dissipation, should play a more important role.", "pages": [3, 4]}, {"title": "3.2. Application to RV Planets", "content": "At the time of writing this paper, there are 409 exoplanets detected with radial velocity (RV) method (exoplanet.org) and about 30% of them reside in multiple planet systems. These RV planets have a wide mass range featured with a bimodal distribution (Pepe et al. 2011) as shown in the left panel of figure7. The boundary is at about 0.2 M J \u223c 64 M \u2295 , which separate the light RV planets (with a media mass of \u223c 12 M \u2295 ) and the massive ones (with a media mass of \u223c 1 . 54 M J ). This bimodal distribution may indicate planets undergo different formations and evolutions for the light and massive groups (Mordasini et al. 2009). Interestingly, we find that these two groups may have different period ratio distributions. As shown in the right panels of figure 7, there is a strong pileup of planet pairs near 2:1 MMR in the massive planet group, which is not seen in the light group. Those massive planets piled up near 2:1 MMR seems unlikely formed in situ within a small annulus, but they are more likely formed with larger distance in a disk then brought into 2:1 MMR through convergent migration. Interestingly, we note that the pile up is just a few percent (in period ratio) wide of the 2:1 MMR center, which is expected from our analytical and numerical prediction with planetary migration (e.g., Fig.5). Furthermore, from the location of the pileup (i.e., \u2206 eq ), we can infer the damping ratio between eccentricity and semi major axis during planetary migration (i.e., K ) by using equation 16. The result of such an exercise is shown in figure 8. Here we considered two migration scenarios. In scenario 1, only the outer planet undergoes migration, i.e., \u03b3 e 2 = K\u03b3 a 2 and \u03b3 e 1 = \u03b3 a 1 = 0. In scenario 2, the inner one migrates outward and the outer one migrates inward, i.e., \u03b3 e 2 = K\u03b3 a 2 , \u03b3 e 1 = -K\u03b3 a 1 and \u03b3 a 1 = -\u03b3 a 2 < 0. As can be seen from figure ?? , the K value is constrained in a relative wide range about 1-100 on order of magnitude. We note this K range is consistent with the hydrodynamical simulations by Kley et al. (2004) which predicts a K value of order of unity, and with dynamical modeling of the well-studied system GJ876 by Lee & Peale (2002) which prefers K = 10 -100.", "pages": [4]}, {"title": "4. SUMMARY", "content": "In this paper, we analytically and numerically study the dynamics of planet pairs near first order MMR. Focusing on the evolution of orbital period ratio, we find it could have an asymmetric distribution around the nominal MMR center regardless of whether dissipation is included or not. Applying the asymmetric nature of MMR to the Kepler planets, we find that, without dissipation, Kepler planets' asymmetric period ratio distribution could be partly explained for the case of 3:2 MMR but not for 2:1 MMR, suggesting that dissipation or other mechanisms may play a more important role in 2:1 than in 3:2 MMR. Beside the Kepler planets, similar asymmetric feature, i.e., planets piled up wide of MMR, is also seen in RV planets. Nevertheless, planets in multiple RV systems are bimodal distribution on mass, and the pileup is currently only seen in the higher mass group. The location of the pileup is consistent with the scenario that planetary migration toward MMR, and it infers that the ratio of damping rate between eccentricity and semimajor axis (i.e., K value) during planet migration is K = 1 -100 on order of magnitude for massive planets. JWX thanks the referee for helpful comments and suggestions, Yanqin Wu and Hanno Rei for valuable discussions and the Kepler team for producing such an invaluable data set. JWX was supported by the National Natural Science Foundation of China (Nos. 10833001 and 10925313), PhD training grant of China (20090091110002), Fundamental Research Funds for the Central Universities (1112020102) and the Ontario government.", "pages": [4, 5]}, {"title": "REFERENCES", "content": "arXiv:1211.5603 Batalha, N. M., Rowe, J. F., Bryson, S. T., et al. 2012, arXiv:1202.5852 Brouwer, D. 1963, AJ, 68, 152 arXiv:1207.3171 arXiv:1202.6328", "pages": [5]}, {"title": "A: TWO CONSTANTS OF MOTION IN MMR", "content": "Here we show that the equations 7 and/or 8 are equivalent to the well known constants of motion of MMR. For j:j-1 MMR there are two constants of motion in addition to the energy (see chapter 8.8 of Murray & Dermott (1999)), i.e., where, \u039b and \u0393 are the Poincar'e momenta (see chapter 2.10 of Murray & Dermott (1999)), and the subscript '1' and '2' denotes the inner and outer planets respectively. Changing the above equation to basic orbital elements, we have In the leading term of e , we then have Take the differential form of above equations and keep the leading term in e , we then have Using the approximation, a 2 /a 1 \u223c [ j/ ( j -1)] 2 / 3 , above equations can be rewritten as equation 7. Compared to the original formulas of the constants, the new formulas (Eqn. 7) solve a 1 and a 2 out (they are not coupled together as in Eqn.A1) and they are dimensionless and simpler.", "pages": [6]}, {"title": "B: EVOLUTION OF PLANETARY SEMI MAJOR AXES UNDER DISSIPATION NEAR MMR", "content": "Following Lithwick et al. (2012), it is convenient to introduce the compact eccentricity where /pi1 is the longitude of the periastron and k = 1 , 2 for the inner and outer planets respectively. In terms of which, the disturbing function can be expressed as where the superscript ' * ' denotes the complex conjugate of the variable and ' c.c. ' denotes the complex conjugate of the proceeding term. Then the eccentricity equation (after adding the damping term) for the two planets is or specifically, Adopting the following approximation then we can solve the eccentricities as where z free1 and z free2 are free solutions (free eccentricities). In terms of the compact eccentricity, the evolution of semi major axes (Eqn.5) can be rewritten (after adding damping terms) as which can be finally written as (with the help of Eqn.B6), where F 1 and F 2 are the terms caused by the free eccentricities, i.e., for Z free = f 1 z 1 + f 2 z 2 defined as the free eccentricities of the system. . In the case where it is not too close to MMR (modest \u2206) and eccentricity damping is efficient (large \u03b3 e k ), Z free \u223c 0 and thus the two oscillation terms F 1 and F 2 can be ignored (i.e., Eqn.13 and 16). Otherwise, if it is very close to MMR (very small \u2206) and the eccentricity damping is weak, then the system could get significant free eccentricities (probably by approaching the separatrix), thus F 1 and F 2 cannot be ignored and the equilibrium cannot be well estimated by using equation 16 (see also in Fig.5).", "pages": [6, 7]}] |
2023arXiv230713006M | https://arxiv.org/pdf/2307.13006.pdf | <document>
<section_header_level_1><location><page_1><loc_25><loc_92><loc_76><loc_93></location>The shadows of quantum gravity on Bell's inequality</section_header_level_1>
<text><location><page_1><loc_25><loc_89><loc_76><loc_90></location>Hooman Moradpour, 1 Shahram Jalalzadeh, 2, 3 and Hamid Tebyanian 4</text>
<text><location><page_1><loc_25><loc_87><loc_76><loc_88></location>1 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM),</text>
<text><location><page_1><loc_29><loc_86><loc_72><loc_87></location>University of Maragheh, P.O. Box 55136-553, Maragheh, Iran</text>
<text><location><page_1><loc_18><loc_82><loc_83><loc_86></location>2 Departamento de Fisica, Universidade Federal de Pernambuco, Recife, PE 50670-901, Brazil 3 Center for Theoretical Physics, Khazar University, 41 Mehseti Street, Baku, AZ1096, Azerbaijan 4 Department of Mathematics, University of York, Heslington, York, YO10 5DD, United Kingdom</text>
<text><location><page_1><loc_18><loc_69><loc_83><loc_81></location>This study delves into the validity of quantum mechanical operators in the context of quantum gravity, recognizing the potential need for their generalization. A primary objective is to investigate the repercussions of these generalizations on the inherent non-locality within quantum mechanics, as exemplified by Bell's inequality. Additionally, the study scrutinizes the consequences of introducing a non-zero minimal length into the established framework of Bell's inequality. The findings contribute significantly to our theoretical comprehension of the intricate interplay between quantum mechanics and gravity. Moreover, this research explores the impact of quantum gravity on Bell's inequality and its practical applications within quantum technologies, notably in the realms of device-independent protocols, quantum key distribution, and quantum randomness generation.</text>
<section_header_level_1><location><page_1><loc_20><loc_65><loc_37><loc_66></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_42><loc_49><loc_63></location>The quantum realm is governed by the Heisenberg uncertainty principle (HUP), which mandates that the Hamiltonian be written as the starting point, leading to the Schrodinger equation and, eventually, the eigenvalues and wave function of the quantum system under consideration. In Heisenberg's formulation of quantum mechanics (QM) in the Hilbert space, we encounter states rather than wave functions (although they are connected). In general, QM fails to produce satisfactory solutions for systems featuring the Newtonian gravitational potential in their Hamiltonian. Therefore, in conventional and widely accepted quantum mechanics, gravity is not accounted for in terms of its operators or corresponding Hilbert space (quantum states) carrying gravitational information.</text>
<text><location><page_1><loc_9><loc_8><loc_49><loc_41></location>The incompatibility of gravity and quantum mechanics is not limited to Newtonian gravity and persists even when general relativity is considered. On the other hand, the existence of gravity, even in a purely Newtonian regime, leads to a non-zero minimum (of the order of 10 -35 m (Planck length) [1]) for the uncertainty in position measurement [1-4]. Consistently, various scenarios of quantum gravity (QG), like String theory, also propose a non-zero minimal for the length measurement [3, 4]. The non-zero minimal length existence may affect the operators, and it leads to the generalization of HUP, called generalized uncertainty principle (GUP) [3, 4] that becomes significant at scales close to the Planck length and may even justify a modified gravity [5, 6]. This concept has profound implications for our understanding of space and time at the most fundamental level. It seems that minimal length is not merely a mathematical artifact of the theory but a physical reality that could have observable consequences. This is a crucial point, as it suggests that the effects of quantum gravity could be detected in experiments, a distinction subtle but essential, as it affects how we understand the physical implications of the GUP [7]. Furthermore, understanding the effects of</text>
<text><location><page_1><loc_52><loc_58><loc_92><loc_66></location>the GUP on various quantum mechanical phenomena is an important issue traced in diverse works like Refs. [8] where the GUP implications on i ) the behavior of various oscillators, ii ) the transformations of space and time, and iii ) the emergence of a cutoff in the energy spectrum of quantum systems have been investigated.</text>
<text><location><page_1><loc_52><loc_22><loc_92><loc_57></location>Operators and system states in QG may differ from those in QM. They are, in fact, functions of ordinary operators that appear in QM [4]. For instance, when considering the first order of the GUP parameter ( β ), we find that the momentum operator ˆ P can be expressed as ˆ p (1+ β ˆ p 2 ), where ˆ P and ˆ p represent momentum operators in QG and QM, respectively. In this representation, β is positive, the position operator remains unchanged [4], and GUP is written as ∆ˆ x ∆ ˆ P ≥ /planckover2pi1 2 [1 + β (∆ ˆ P ) 2 ]. Here, although β seems to be a positive parameter [3, 9, 10] related to a minimal length (of the order of the Planck length ( ≡ 10 -35 m)) as ∆ˆ x = /planckover2pi1 √ β , models including negative values for β have also been proposed [11]. Current experiments and theoretical ideas predict a large range for the upper bound of its value [4, 7, 12-14]. Therefore, it follows that gravity could impact our understanding of classical physics-based operator sets that have been established by QM [15, 16]. Consequently, it is possible to write ˆ O = ˆ o + β ˆ o p for some operators, where ˆ O and ˆ o are operators in QG and QM, respectively, and ˆ o p is the first-order correction obtained using perturbation theory [17]. It should also be noted that as the position operator does not change in the above mentioned representation [4], we have ˆ o p = 0 for this operator.</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_21></location>The discovery of quantum non-locality goes back to the famous thought experiment by Einstein, Podolsky and Rosen (EPR) designed to challenge the quantum mechanics [18]. It clearly shows the role of HUP in emerging quantum non-locality and thus, an advantage for QM versus classical mechanics [19, 20]. In order to establish a border between classical physics and quantum mechanical phenomena, J. S. Bell [21] introduces his inequality including the maximum possible correlation in clas-</text>
<table>
<location><page_2><loc_11><loc_87><loc_46><loc_94></location>
<caption>TABLE I: A comparison between QM and QG (up to the first order of β ). Here, | ψ 〉 and | ψ GUP 〉 denote the quantum states in QM and QG, respectively, and | ψ 〉 p is also calculable using the perturbation theory. It should be noted that for operators not affected by GUP (like the position operator in the above mentioned representation [4]), there is not any perturbation (ˆ o p = 0) meaning that their corresponding states remain unchanged.</caption>
</table>
<text><location><page_2><loc_9><loc_61><loc_49><loc_71></location>sical physics that respects the locality. In the presence of quantum non-locality, this inequality is violated [22], and the first experimental evidences of its violation (and thus, the existence of quantum non-locality) have been reported by Aspect et al. [23-25]. Interestingly enough, the quantum non-locality is also predicted in single particle systems [26, 27].</text>
<text><location><page_2><loc_9><loc_29><loc_49><loc_61></location>Motivated by the correlation between HUP and quantum non-locality (which is easily demonstrated in the square of Bell's inequality) [18-20], as well as the impact of GUP on operators, particularly angular momentum [28, 29], recent studies have revealed that minimal length can alter the square of Bell's operator [30]. Furthermore, GUP can affect the entanglement between energy and time, as evidenced by the results of a Franson experiment (which serves as a testing setup for time-energy entanglement) [14]. Table I clearly displays the generally expected modifications to operators and states resulting from minimal length. The term | ψ 〉 p indicates an increase in a quantum superposition, which is a probabilistic signal for entanglement enhancement [15, 16] and therefore, non-locality beyond quantum mechanics [31]. It is apparent that gravity impacts the information bound [17]. Indeed, studying the effects of gravity on quantum entanglement is a long-standing topic which will also establish ways to test the quantum aspects of gravity. In this regard, many efforts have been made based on the Newtonian gravity and its quantization and their effects on the quantum entanglement [32-37].</text>
<text><location><page_2><loc_9><loc_8><loc_49><loc_29></location>The inquiry into the influence of special and general relativity (SR and GR, respectively) on Bell's inequality (quantum non-locality) has been extensively studied over the years [38-42]. The existing research on the effects of SR on Bell's inequality can be classified into three general categories, depending on the method of applying Lorentz transformations: (i) the operators change while the states remain unchanged, (ii) only the states undergo the Lorentz transformation while the operators remain unaltered (the reverse of the previous one), and (iii) both the operators and states are affected by the Lorentz transformation [43-54]. In order to clarify the first two cases, consider a Lab frame, carrying a Bell state ( | φ 〉 ) and a Bell measurement apparatus ( B ), and a</text>
<text><location><page_2><loc_52><loc_60><loc_92><loc_93></location>moving frame (including a Bell measurement apparatus ( B ' )) so that they are connected to each other through the Lorentz transformation Λ. In this manner, the moving frame faces the Lorentz transformed Bell state | φ Λ 〉 , and whenever the Lab frame looks at the Bell measurement apparatus of the moving frame ( B ' ), its Lorentz transformed is seen ( B ' Λ ). Now, it is apparent that using the same directions for the Bell measurement, we find 〈 φ | B | φ 〉 /negationslash = 〈 φ Λ | B ' | φ Λ 〉 /negationslash = 〈 φ | B ' Λ | φ 〉 meaning that the maximum violation amount of Bell's inequality is reported by both observers at different measurement directions [43-47, 49-54]. In the third case, the moving observer is supposed to witness a Bell measurement done in the Lab frame. The moving frame sees | φ Λ 〉 and B Λ (the Lorentz transformed version of B ) leading to 〈 φ | B | φ 〉 = 〈 φ Λ | B Λ | φ Λ 〉 meaning that both the Lab observer and the moving viewer report the same amount for the Bell measurement, simultaneously [48]. Furthermore, certain implications of GR and non-inertial observers have also been addressed in Refs. [55-58]. Given the ongoing effort to bridge QG with QM [59], exploring the effects of QG on quantum non-locality is deemed inevitable and advantageous.</text>
<text><location><page_2><loc_52><loc_38><loc_92><loc_59></location>Bell's theorem suggests that certain experimental outcomes are constrained if the universe adheres to local realism. However, quantum entanglement, which seemingly allows distant particles to interact instantaneously, can breach these constraints [67]. This led to cryptographic solutions like quantum key distribution (QKD) [72] and quantum random number generation (QRNG) [65, 68]. However, classical noise can enter QKDs and QRNGs during implementation, which hackers can exploit to gain partial information. A device-independent (DI) method was developed to address this, ensuring security when a particular correlation is detected, irrespective of device noise. DI protocols often hinge on non-local game violations, like the CHSH inequality [61]. Section IV delves into the impacts of QG on these applications.</text>
<text><location><page_2><loc_52><loc_9><loc_92><loc_37></location>In this study, our primary goal is to explore the ramifications of QG on Bell's inequality, specifically by investigating the implications of minimal length (up to the first order of β ). To address this objective, we adopt a methodology analogous to the three scenarios previously examined concerning the effects of SR on quantum non-locality. To facilitate this exploration, we categorize the existing cases into three distinct groups, which we elaborate on in the following section. As the GUP effects become important at energy scales close to the Planck scale, the first case includes a quantum state produced at purely quantum mechanical situation (lowenergy) while the observer uses the Bell measurement apparatus prepared by employing the quantum aspects of gravity meaning that high-energy physics considerations have been employed to build the apparatus. Therefore, we face a high-energy affected observer (measurement), who tries to study the quantum non-locality stored in a low-energy state (a purely quantum mechanical state). The reversed situation is checked in the second case, and</text>
<text><location><page_3><loc_9><loc_80><loc_49><loc_93></location>the consequences of applying a quantum gravity-based Bell measurement, built by considering the effects of QG, on a state including the QG consideration are also investigated as the third case. The paper concludes by providing a comprehensive summary of our research findings, shedding light on the intricate interplay between quantum mechanics and gravity, elucidating the impact of QG on Bell's inequality, and exploring potential applications within various quantum-based systems.</text>
<section_header_level_1><location><page_3><loc_14><loc_74><loc_44><loc_76></location>II. BELL'S INEQUALITY AND THE IMPLICATIONS OF QG</section_header_level_1>
<text><location><page_3><loc_9><loc_62><loc_49><loc_71></location>In the framework of QM, assume two particles and four operators ˆ A, ˆ A ' , ˆ B, ˆ B ' with eigenvalues λ J ( J ∈ { ˆ A, ˆ A ' , ˆ B, ˆ B ' } ), while the first (second) two operators act on the first (second) particle. Now, operators ˆ j = ˆ J | λ J | ∈ { ˆ a, ˆ a ' , ˆ b, ˆ b ' } have eigenvalues ± 1, and Bell's inequality is defined as</text>
<formula><location><page_3><loc_16><loc_58><loc_49><loc_60></location>〈 ˆ B 〉 ≡ 〈 ˆ a ( ˆ b + ˆ b ' ) + ˆ a ' ( ˆ b -ˆ b ' ) 〉 ≤ 2 . (1)</formula>
<text><location><page_3><loc_9><loc_33><loc_49><loc_56></location>Taking into account the effects of QG (up to the first order), the operators are corrected as ˆ J GUP = ˆ J + β ˆ J p and ˆ j GUP = ˆ J + β ˆ J p | λ J GUP | where λ J GUP represents the eigenvalue of ˆ J GUP . Since QM should be recovered at the limit β → 0, one may expect λ J GUP /similarequal λ J + βλ J p . Moreover, as the βλ J p term is perturbative, it is reasonable to expect | β λ J p λ J | << 1 leading to | λ J + βλ J p | = | λ J | (1+ β λ J p λ J ). Applying modifications to the states, operators, or both in QG can result in three distinct situations. Similar studies conducted on the effects of SR on Bell's inequality have also revealed three cases [43-49, 54]. Therefore, it is necessary to consider the possibilities arising from these situations to understand the implications of quantum gravitational modifications. In the following paragraphs, we will examine these possibilities in depth.</text>
<section_header_level_1><location><page_3><loc_11><loc_26><loc_47><loc_28></location>1. Purely quantum mechanical entangled states in the presence of operators modified by QG</section_header_level_1>
<text><location><page_3><loc_9><loc_8><loc_49><loc_24></location>Firstly, let us contemplate the scenario in which an entangled state ( | ξ 〉 ) has been prepared away from the QG influences. This implies that the objective has been accomplished using purely quantum mechanical procedures. Furthermore, it is assumed that an observer utilizes Bell measurements that are constructed through the incorporation of operators containing the QG corrections ( ˆ j GUP ). In the framework of QM, the violation amount of inequality (1) depends on the directions of Bell's measurements. Here, we have ˆ j = ˆ j GUP + β ( λ J p λ J ˆ j GUP -ˆ J p | λ J | )</text>
<text><location><page_3><loc_52><loc_92><loc_73><loc_93></location>inserted into Eq. (1) to reach</text>
<formula><location><page_3><loc_51><loc_77><loc_92><loc_90></location>〈 ˆ B GUP 〉 ≡ (2) 〈 ˆ a GUP ( ˆ b GUP + ˆ b ' GUP ) +ˆ a ' GUP ( ˆ b GUP -ˆ b ' GUP )〉 ≤ 2 -〈 β ' a ˆ a GUP ( ˆ b GUP + ˆ b ' GUP ) + β ' a ' ˆ a ' GUP ( ˆ b GUP -ˆ b ' GUP )〉 -〈 ˆ a GUP ( β ' b ˆ b GUP + β ' b ' ˆ b ' GUP ) +ˆ a ' GUP ( β ' b ˆ b GUP -β ' b ' ˆ b ' GUP )〉 + β '' a 〈 ˆ A GUP ( ˆ b GUP + ˆ b ' GUP ) + ˆ A ' GUP ( ˆ b GUP -ˆ b ' GUP )〉 + β '' b 〈 ˆ a GUP ( ˆ B GUP + ˆ B ' GUP ) +ˆ a ' GUP ( ˆ B GUP -ˆ B ' GUP )〉 ,</formula>
<text><location><page_3><loc_76><loc_69><loc_76><loc_71></location>/negationslash</text>
<text><location><page_3><loc_52><loc_63><loc_92><loc_77></location>where β ' j = β λ J p λ J , β '' j = β | λ J | -1 and the last two expressions have been written using β '' a = β '' a ' and β '' b = β '' b ' . In this manner, it is clearly seen that although the state is unchanged, in general, 〈 ˆ B GUP 〉 = 〈 ˆ B 〉 as the operators are affected by quantum features of gravity [14, 29, 30]. In studying the effects of SR on Bell's inequality, whenever the states remain unchanged, and Lorentz transformations only affect Bell's operator, a similar situation is also obtained [43-49, 54].</text>
<section_header_level_1><location><page_3><loc_52><loc_57><loc_92><loc_60></location>2. Purely quantum mechanical measurements and quantum gravitational states</section_header_level_1>
<text><location><page_3><loc_52><loc_39><loc_92><loc_55></location>Now, let us consider the situation in which the Bell apparatus is built using purely quantum mechanical operators j , and the primary entangled state carries the Planck scale information, i.e., the quantum features of gravity. It means that the entangled state is made using the j GUP operators. A similar case in studies related to the effects of SR on Bell's inequality is the case where the Bell measurement does not go under the Lorentz transformation while the system state undergoes the Lorentz transformation [43-49, 54]. In this setup, we have | ξ GUP 〉 = | ξ 〉 + β | ξ 〉 p and thus</text>
<formula><location><page_3><loc_54><loc_33><loc_92><loc_38></location>〈 ξ GUP ∣ ∣ ˆ B ∣ ∣ ξ GUP 〉 ≡ 〈 ˆ B 〉 GUP = 〈 ˆ B 〉 +2 β 〈 ξ ∣ ∣ ˆ B ∣ ∣ ξ 〉 p ⇒ 〈 ˆ B 〉 GUP ≤ 2 ( 1 + β 〈 ξ ∣ ∣ ˆ B ∣ ∣ ξ 〉 p ) . (3)</formula>
<text><location><page_3><loc_52><loc_26><loc_92><loc_32></location>Correspondingly, if one considers a Bell measurement apparatus that yields 〈 ˆ B 〉 = 2 √ 2, then such an apparatus cannot lead 〈 ˆ B 〉 GUP to its maximum possible value whenever Lorentz symmetry is broken [60].</text>
<section_header_level_1><location><page_3><loc_54><loc_21><loc_90><loc_23></location>3. Bell's inequality in a purely quantum gravitational regime</section_header_level_1>
<text><location><page_3><loc_52><loc_9><loc_92><loc_19></location>In deriving Bell's inequality, it is a significant step to ensure that the operators' eigenvalues are only either ± 1, regardless of their origin, whether it be from QM or QG. If both the Bell measurement and the entangled state were prepared using the quantum gravitational operators, then it is evident that 〈 ξ GUP ∣ ∣ ˆ B GUP ∣ ∣ ξ GUP 〉 ≤ 2. This result indicates that, when considering the effects</text>
<text><location><page_4><loc_9><loc_83><loc_49><loc_93></location>of QG on both the state and the operators, Bell's inequality and the classical regime's limit (which is 2 in the inequality) remain unchanged compared to the previous setups. The same outcome is also achieved when it comes to the relationship between SR and Bell's inequality, provided that both the system state and Bell's measurement undergo a Lorentz transformation [48].</text>
<section_header_level_1><location><page_4><loc_23><loc_79><loc_35><loc_80></location>III. RESULTS</section_header_level_1>
<text><location><page_4><loc_9><loc_64><loc_49><loc_77></location>This section studies QG's implications on Bell's inequality, specifically within the contexts delineated earlier. The CHSH inequality, a specific form of Bell's inequality, provides a quantifiable limit on the correlations predicted by local hidden-variable theories [73]. A violation of the CHSH inequality underscores the inability of such approaches to account for the observed correlations in specific experiments with entangled quantum systems, as predicted by quantum mechanics [69].</text>
<text><location><page_4><loc_9><loc_58><loc_49><loc_64></location>Now, we define the scenario where there are two parties where an entangled pair is shared between them. The entangled state of two qubits can be represented by the Bell state:</text>
<formula><location><page_4><loc_20><loc_55><loc_49><loc_58></location>| ψ 〉 = 1 √ 2 ( | 00 〉 + | 11 〉 ) (4)</formula>
<text><location><page_4><loc_9><loc_42><loc_49><loc_54></location>Alice and Bob each measure their respective states. They can choose between two measurement settings: ˆ a, ˆ a ' for Alice and ˆ b, ˆ b ' for Bob. The measurement results can be either +1 or -1. The expected value of the CHSH game using the above quantum strategy and the Bell state is given in Eq. 1. Classically, the maximum value of 〈 ˆ B 〉 is 2. However, this value can reach 2 √ 2 with the quantum strategy, violating the CHSH inequality.</text>
<figure>
<location><page_4><loc_10><loc_23><loc_48><loc_40></location>
<caption>FIG. 1: The 2D plot of the CHSH inequality values as functions of detection angles θ 1 /π and θ 2 /π . Different colors indicate different 〈 ˆ B 〉 values, with a contour distinguishing the classical and quantum regions.</caption>
</figure>
<text><location><page_4><loc_9><loc_9><loc_49><loc_14></location>Fig. 1 illustrates that the CHSH inequality can be surpassed by judiciously selecting the appropriate detection angles, denoted as θ 1 and θ 2 . The color bar quantitatively represents the value of the inequality, highlighting</text>
<text><location><page_4><loc_52><loc_88><loc_92><loc_93></location>two distinct regions where the value exceeds the classical limit of 2. In Fig. 1, the simulation of Bell's inequality is conducted solely based on QM representations without incorporating QG impact.</text>
<text><location><page_4><loc_52><loc_48><loc_92><loc_87></location>Next, we consider the QG impact on Bell's inequality for various cases; better to say, we extend the well-known Bell inequality to account for the effects of QG. Equations 2 and 3 introduce new terms that are parameterized by β , a constant that quantifies the strength of quantum gravitational effects. These equations represent the modified Bell inequalities in the presence of QG. To explore the implications of these modifications, we plot, see Fig. 2, the degree of Bell inequality violation, denoted as 〈 ˆ B 〉 , as a function of θ for various angles β . Each sub-figure in Fig. 2 presents six curves representing simulations conducted on two different quantum computing platforms: IBM and Google. For IBM, the curves are colour-coded as blue, red, and green, corresponding to quantum mechanical predictions, first quantum gravitational corrections, and second quantum gravitational corrections, respectively. The Google platform uses cyan, pink, and grey to represent the same sequence of calculations. These simulations are repeated multiple times to account for noise, defects in quantum computer circuits, and errors in computation and simulation using the independent platforms of IBM and Google. The insets in each figure show that the results from both platforms are in agreement, confirming the reliability of the findings. The maximum violation observed in the presence of quantum gravity remains below 4, adhering to the theoretical limit set by the world box scenario [74].</text>
<text><location><page_4><loc_52><loc_37><loc_92><loc_48></location>The results notably indicate an escalating violation of the Bell inequality with the introduction of QG. As the parameter β increases, the violation surpasses the quantum mechanical limit of √ 8, signifying a more pronounced breach of the inequality. This implies that the presence of quantum gravitational effects could lead to a more pronounced violation of the Bell inequality than what is predicted by standard quantum mechanics.</text>
<section_header_level_1><location><page_4><loc_63><loc_32><loc_81><loc_33></location>IV. APPLICATIONS</section_header_level_1>
<text><location><page_4><loc_52><loc_12><loc_92><loc_30></location>QKD and QRNG represent two extensively researched and commercially implemented areas where the applications of quantum mechanics come to life. While quantum mechanics underpins the security of these systems, experimental imperfections can introduce vulnerabilities. To address this, DI protocols have been developed. These protocols harness the non-local correlations inherent in quantum entanglement. Importantly, they do not rely on an intricate understanding of the devices in use; their security is grounded solely in the observed violation of non-local correlations, such as the Bell inequalities. This approach offers a robust solution to the security challenges posed by device imperfections [62, 64].</text>
<text><location><page_4><loc_52><loc_9><loc_92><loc_11></location>In DI QKD, two distant parties share an entangled quantum state. They perform measurements on their</text>
<figure>
<location><page_5><loc_16><loc_61><loc_87><loc_91></location>
<caption>FIG. 2: Bell inequality values are plotted against the rotation angle θ , illustrating the effects of varying β values: 0.1, 0.2, 0.5, and 0.9. This plot comprehensively compares six curves, each representing simulations performed on two different quantum computing platforms: IBM and Google. For each platform, the curves are coloured distinctly blue, red, and green for IBM quantum computer simulations representing quantum mechanical predictions (QM), first quantum gravitational corrections (QG-1), and second quantum gravitational corrections (QG-2), respectively; similarly, cyan, pink, and grey represent the same sequence of calculations performed using a Google quantum simulator. The remarkable overlay of curves from the two platforms demonstrates consistent agreement, reinforcing the computational models' reliability. An inset within the figure provides a zoomed-in view to examine further the regions where the curves closely approach or reach the theoretical maximum violation. This feature is crucial for better comparing subtle differences between the curves and understanding the implications of each model. Notably, the maximum violation observed does not exceed the limit of 4, consistent with the boundaries set by the Boxworld theorem. This boundary is a crucial benchmark in general probabilistic theories. It indicates that while the quantum mechanical violations are significant, they do not exceed what is theoretically possible under models that assume no faster-than-light (superluminal) communication.</caption>
</figure>
<text><location><page_5><loc_9><loc_23><loc_49><loc_36></location>respective parts of the state, and due to the non-local nature of entanglement, the outcomes of these measurements are correlated in a way that disobeys classical explanation. These correlations serve as the foundation for key generation, with the security of the key guaranteed by the violation of Bell inequalities. Basically, any eavesdropper attempting to intercept or tamper with the quantum states would disrupt these correlations, making their presence detectable.</text>
<text><location><page_5><loc_9><loc_9><loc_49><loc_21></location>The security and randomness of DI QRNG do not depend on trusting the intrinsic workings of the devices. Traditional QRNGs require detailed models and assumptions about the device, but in DI QRNGs, as long as observed outcomes violate Bell inequalities, one can be assured of the randomness. With the rise of quantum computers, many cryptographic methods are at risk. Nevertheless, the unpredictability in DI QRNG is more than just computationally hard for quantum computers; it's</text>
<text><location><page_5><loc_52><loc_33><loc_92><loc_36></location>theoretically impossible to predict due to the inherent randomness of quantum processes [65, 68].</text>
<text><location><page_5><loc_52><loc_22><loc_92><loc_32></location>Incorporating the effects of QG in quantum information science and technology becomes an intellectual exercise and a practical necessity. Given the results in the previous section that QG effects can significantly enhance the violation of Bell inequalities, let us consider its implications for quantum information science and technology and its applications.</text>
<text><location><page_5><loc_52><loc_9><loc_92><loc_21></location>The security of QKD is guaranteed by the quantum mechanical violation of Bell inequalities; increasing the violation value of Bell's inequality makes QKD even more secure against attacks. This disturbance changes the quantum correlations between Alice's and Bob's measurements. In other words, if the eavesdropper is listening in, the observed violations of Bell's inequalities at Alice's and Bob's ends will reduce, moving closer to what would be expected classically. Thus, if you start</text>
<text><location><page_6><loc_9><loc_80><loc_49><loc_93></location>with a higher violation of Bell's inequalities (thanks to QG effects), you are raising the 'quantumness' of your initial state. The higher this initial level, the more sensitive your system becomes to any eavesdropping activities. A significant drop in the observed Bell inequality violation from this higher baseline would more quickly and definitively signal the presence of eavesdropping, thus enabling quicker and more reliable detection of any security breaches.</text>
<text><location><page_6><loc_9><loc_57><loc_49><loc_80></location>DI protocols prevent the need for trust in the hardware by utilizing Bell inequality violations the greater the violation, the higher the level of security. The introduction of QG effects adds an additional layer of robustness to DI protocols, fortifying them through quantum mechanical principles and integrating fundamental theories of nature. Similarly, for QRNGs, a heightened violation signifies a more quantum-coherent system, enhancing the quality of randomness, which comprises not merely an incremental advancement but a paradigmatic leap in the entropy of the generated random numbers. Consequently, this reduces the computational time required to achieve a given level of randomness and unpredictability, analogous to transitioning from conventional vehicular propulsion to advanced warp drives, all while adhering to the fundamental constraints of space-time.</text>
<text><location><page_6><loc_9><loc_46><loc_49><loc_57></location>More importantly, quantum gravity could offer richer quantum correlations in multipartite systems. Imagine a quantum network secured by quantum gravity effects each additional party would enhance not just the computational power but the security, generating what could be termed 'quantum gravity-secured entanglement.' Enabling a brand-new platform for multiparty quantum computations and secret sharing protocols.</text>
<text><location><page_6><loc_9><loc_38><loc_49><loc_45></location>In summary, enhanced violations of Bell inequalities render QKD virtually impregnable, elevate QRNGs to sources of high-entropy randomness, and establish DI protocols as the epitome of trust-free security mechanisms. Dismissing QG as a purely academic endeavor</text>
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<text><location><page_6><loc_52><loc_83><loc_92><loc_93></location>could overlook its potential as a critical element in safeguarding quantum data against even the most advanced computational threats. If quantum mechanics is considered the apex of security and efficiency, the advent of QG compels a reevaluation. It promises to redefine the boundaries of what is secure, efficient, and trustworthy in quantum technologies.</text>
<section_header_level_1><location><page_6><loc_64><loc_79><loc_79><loc_80></location>V. CONCLUSION</section_header_level_1>
<text><location><page_6><loc_52><loc_54><loc_92><loc_76></location>The study can be summarized by its two main components: i ) the origin of entangled states and ii ) Bell's measurement. Furthermore, the study has introduced the possibility of three outcomes depending on which cornerstone carries the quantum gravitational modifications. The first two scenarios suggest that if only one of the foundations stores the effects of QG, then a precise Bell measurement (depending on the value of β ) could detect the effects of QG. This is due to the differences between 〈 ˆ B 〉 , 〈 ˆ B GUP 〉 , and 〈 ˆ B 〉 GUP . In the third case, Bell's inequality remains invariant if we consider the quantum aspects of gravity on both the states and the operators. Moreover, the results demonstrate that the presence of QG enhances Bell's inequality violation, thereby offering avenues for improving the security and performance of DI QRNG and QKD protocols.</text>
<section_header_level_1><location><page_6><loc_65><loc_49><loc_79><loc_50></location>Acknowledgement</section_header_level_1>
<text><location><page_6><loc_52><loc_38><loc_92><loc_47></location>S.J. acknowledges financial support from the National Council for Scientific and Technological DevelopmentCNPq, Grant no. 308131/2022-3. H. T. acknowledge the Quantum Communications Hub of the UK Engineering and Physical Sciences Research Council (EPSRC) (Grant Nos. EP/M013472/1 and EP/T001011/1).</text>
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2013ApJ...766...38L | https://arxiv.org/pdf/1302.1588.pdf | <document>
<section_header_level_1><location><page_1><loc_17><loc_86><loc_84><loc_87></location>ON THE ASSEMBLY HISTORY OF STELLAR COMPONENTS IN MASSIVE GALAXIES</section_header_level_1>
<text><location><page_1><loc_38><loc_84><loc_63><loc_85></location>Jaehyun Lee and Sukyoung K. Yi</text>
<text><location><page_1><loc_11><loc_82><loc_90><loc_84></location>Department of Astronomy and Yonsei University Observatory, Yonsei University, Seoul 120-749, Republic of Korea; [email protected] Draft version August 20, 2018</text>
<section_header_level_1><location><page_1><loc_45><loc_79><loc_55><loc_80></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_59><loc_86><loc_79></location>Matsuoka & Kawara (2010) showed that the number density of the most massive galaxies (log M/M /circledot = 11 . 5 -12 . 0) increases faster than that of the next massive group (log M/M /circledot = 11 . 0 -11 . 5) during 0 < z < 1. This appears to be in contradiction to the apparent 'downsizing effect'. We attempt to understand the two observational findings in the context of the hierarchical merger paradigm using semi-analytic techniques. Our models closely reproduce the result of Matsuoka & Kawara (2010). Downsizing can also be understood as larger galaxies have, on average, smaller assembly ages but larger stellar ages. Our fiducial models further reveal details of the history of the stellar mass growth of massive galaxies. The most massive galaxies (log M/M /circledot = 11 . 5 -12 . 0 at z=0), which are mostly brightest cluster galaxies, obtain roughly 70% of their stellar components via merger accretion. The role of merger accretion monotonically declines with galaxy mass: 40% for log M/M /circledot = 11 . 0 -11 . 5 and 20% for log M/M /circledot = 10 . 5 -11 . 0 at z = 0. The specific accreted stellar mass rates via galaxy mergers decline very slowly during the whole redshift range, while specific star formation rates sharply decrease with time. In the case of the most massive galaxies, merger accretion becomes the most important channel for the stellar mass growth at z ∼ 2. On the other hand, in-situ star formation is always the dominant channel in L ∗ galaxies.</text>
<text><location><page_1><loc_14><loc_56><loc_86><loc_59></location>Subject headings: galaxies: evolution - galaxies: elliptical and lenticular, cD - galaxies: formation galaxies: stellar content</text>
<section_header_level_1><location><page_1><loc_21><loc_52><loc_36><loc_54></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_43><loc_48><loc_52></location>Dynamical realizations based on the concordance ΛCDM cosmology (e.g. Spergel et al. 2007) have been remarkably successful at reproducing large-scale structures in the universe (e.g. Springel et al. 2006). In the common perception of this paradigm, large galaxies grow hierarchically through numerous galaxy mergers that follow mergers between dark halos.</text>
<text><location><page_1><loc_8><loc_21><loc_48><loc_42></location>Supporting this view, galaxies with disturbed features have frequently been witnessed (e.g. Arp 1966; Schweizer & Seitzer 1988). With the advent of deep and wide field surveys, the arguments could be investigated in greater detail. Recent studies based on ultra-deep imaging data are particularly noteworthy. From the µ r = 28 mag arcsec -2 deep images, van Dokkum (2005) found that about 50% of red bulge-dominant galaxies in field environments show tidal debris. Merger galaxy fraction has been found to be almost as high in cluster environments (Sheen et al. 2012). Kaviraj et al. (2007) and Kaviraj (2010) claimed that residual star formation found in a large fraction ( ∼ 30%) of local massive earlytype galaxies is related to mergers or interactions. As observation techniques reach deeper and hidden nature of galaxies, 'peculiar' is no longer a synonym of 'rare'.</text>
<text><location><page_1><loc_8><loc_7><loc_48><loc_21></location>High redshift surveys allow us direct investigation of the role of galaxy mergers and interactions on galaxy evolution. Using the K-band Hubble diagram, Aragon-Salamanca et al. (1998) found that brightest cluster galaxies(BCGs) have increased their mass by a factor of two to four with no or negative luminosity evolution during 0 < z < 1. From the GOODS fields, Bundy et al. (2009) found that the pair fraction of massive (logM / M /circledot > 11 . 0) red spheroidal galaxies is higher than that of less massive systems (logM / M /circledot ∼</text>
<text><location><page_1><loc_52><loc_34><loc_92><loc_54></location>10 . 0). In addition, the pair fraction unaccompanied by star formation increases with time. They concluded that massive galaxies grow primarily through dry or minor mergers, at least at z /lessorsimilar 1. Almost simultaneously, from the UKIDSS and the SDSS II Supernova Survey, Matsuoka & Kawara (2010, hereafter MK10) found that, during z < 1 the number density of the most massive galaxies (log M/M /circledot = 11 . 5 -12 . 0) increased more rapidly than that of the next massive group (log M/M /circledot = 11 . 0 -11 . 5). They also showed that more massive systems have a lower blue galaxy fraction than less massive systems, and that the fraction decreases with time. All these observations seem to imply that, during z /lessorsimilar 1, massive galaxies are mainly brought up via mergers.</text>
<text><location><page_1><loc_52><loc_7><loc_92><loc_34></location>Not all observations naively support the hierarchical merger picture. The tight color-magnitude relation found among early-type galaxies is more simply, albeit not exclusively, explained by monolithic formation scenarios (e.g. Bower et al. 1992; Kodama & Arimoto 1997). Furthermore, the 'downsizing' effect, where larger galaxies appear to be older and thus suspected to have formed earlier, seems to be inconsistent with the new paradigm (Cowie et al. 1996; Glazebrook et al. 2004; Cimatti et al. 2004). On the face of it, the inconsistency seems a counter-evidence of the hierarchical picture; however, some studies have pointed out that downsizing can be understood by the hierarchical paradigm reasonably enough. Based on semi-analytic models, De Lucia et al. (2006) showed that the star formation rates of all the progenitors of more massive galaxies peak earlier and decrease faster than those of less massive galaxies. In fact, downsizing could be a natural result of the hierarchical clusterings of dark halos. Neistein et al. (2006) demonstrated that downsiz-</text>
<text><location><page_2><loc_8><loc_73><loc_48><loc_92></location>ing appears in the total (combined) mass evolution of all the progenitors of a dark matter halo. If the stellar mass of a galaxy correlates with the mass of its dark matter halo (Moster et al. 2010), downsizing may very well originate from the bottom-up assembly of dark matter halos. Using semi-analytic approaches, many studies have shown that the assembly and formation history of stellar components, especially in massive, red and luminous, or elliptical galaxies, can be different in the hierarchical Universe (Kauffmann 1996; Baugh et al. 1996; De Lucia et al. 2006; De Lucia & Blaizot 2007; Almeida et al. 2008). More direct hydrodynamic simulations performed for smaller volumes have yielded consistent results (Oser et al. 2010; Lackner et al. 2012).</text>
<text><location><page_2><loc_8><loc_64><loc_48><loc_73></location>In this study, we look further into the details of the assembly history of stellar components in massive galaxies. We use semi-analytic approaches because these are more effective for constructing models of a large volume of the universe. Motivated mainly by the empirical results of MK10, we investigate the factors that drive the growth of galaxy stellar mass as a function of time and mass.</text>
<section_header_level_1><location><page_2><loc_24><loc_61><loc_32><loc_62></location>2. MODEL</section_header_level_1>
<text><location><page_2><loc_8><loc_55><loc_48><loc_60></location>We have developed our own semi-analytic model for galaxy formation and evolution. In this section, we briefly introduce physical ingredients, with more focus on the prescriptions that play important roles in this study.</text>
<section_header_level_1><location><page_2><loc_11><loc_51><loc_46><loc_53></location>2.1. Dark Matter Halo Merger Trees and Galaxy Mergers</section_header_level_1>
<text><location><page_2><loc_8><loc_21><loc_48><loc_50></location>In the context of the two-step galaxy formation theory (White & Rees 1978), the first step involves the construction of dark halo merger trees. We ran dark matter N-body volume simulations using the GADGET-2 code (Springel 2005). The simulation was performed using the standard ΛCDM cosmology parameters derived from the WMAP 7-year observations, Ω m = 0 . 266, Ω Λ = 0 . 734, σ 8 = 0 . 801, Ω b = 0 . 0449, and H 0 = 71 km s -1 Mpc -1 (Jarosik et al. 2011). The periodic cube size of the simulation was 70 h -1 Mpc on the side with 512 3 collisionless particles, mass of a particle was 1.9 × 10 8 h -1 M /circledot , and mass resolution of a halo was ∼ 10 10 h -1 M /circledot . We identify halo structures in the simulation box, using a halo finding code developed by Tweed et al. (2009), which is based on the AdaptaHOP (Aubert et al. 2004). Halo merger trees were generated by backtracing the infall histories of subhalos with increasing redshift. We calculate galaxy merger timescales, using the positional information of subhalos extracted from our N-body simulation. In this study, we assume that the mass distribution of dark matter halos basically follows the NavarroFrank-White (NFW) profile (Navarro et al. 1996).</text>
<text><location><page_2><loc_8><loc_10><loc_48><loc_21></location>Subhalos can disappear before arriving at central regions if they are heavily embedded in their host halo density profiles or their mass decreases below the resolution limit of our numerical simulation. We take these numerical artifacts into consideration. Thus, once a halo enters into a more massive halo, we additionally calculate its merger timescale, t merge , using the following fitting formula suggested by Jiang et al. (2008):</text>
<formula><location><page_2><loc_14><loc_6><loc_39><loc_9></location>t merge (Gyr)= 0 . 94 /epsilon1 0 . 60 +0 . 60 2 C M host M sat</formula>
<formula><location><page_2><loc_68><loc_89><loc_92><loc_92></location>1 ln[1 + ( M host /M sat )] R vir V c , (1)</formula>
<text><location><page_2><loc_52><loc_69><loc_92><loc_88></location>where /epsilon1 is the circularity of the orbit of a satellite halo, C is a constant, approximately equal to 0.43, M host is the mass of a host halo, M sat is the mass of a satellite halo, R vir is the virial radius of a host halo, and V c is the circular velocity of a host halo at R vir . If a subhalo disappears within 0 . 1 R vir of its host halo, a galaxy in the subhalo is regarded as having merged with its central galaxy. On the other hand, if it disappears outside 0 . 1 R vir , we assume that the galaxy in the subhalo merges with its central galaxy at t merge given by Eq. 1 after the subhalo becomes a satellite of its host halo. In that case, we consider dynamical friction to analytically compute the positions and velocities of subhalos. We adopt the dynamical friction prescription introduced by Binney & Tremaine (2008):</text>
<formula><location><page_2><loc_56><loc_60><loc_92><loc_68></location>d /vectorv d t dynf = -GM sat ( t ) r 2 lnΛ ( V c v ) 2 { erf ( v V c ) -√ π 2 ( v V c ) exp [ -( v V c ) 2 ]} /vector e v , (2)</formula>
<text><location><page_2><loc_52><loc_49><loc_92><loc_59></location>where M sat is the mass of a subhalo, which is initially defined as the mass at a previous time step after which the subhalo cannot be resolved in the N-body simulation anymore, r is the distance between the subhalo and the center of its host halo, ln Λ is a Coulomb logarithm with Λ = 1 + M host /M sat adopted by Springel et al. (2001), V c is the circular velocity of the host halo at the virial radius, and v is the orbital velocity of the subhalo.</text>
<text><location><page_2><loc_52><loc_34><loc_92><loc_49></location>During the orbital motion of a satellite halo in its host halo, the dark matter of the satellite halo is stripped due to dynamical friction. If the satellite is resolved in an N-body simulation, dark matter of the satellite would be naturally stripped. On the other hand, if the halo is not resolved in the simulation but considered to orbit around its host halo, stripping due to dynamical friction should be computed analytically. We evaluate the amount of dark matter stripped by dynamical friction by adopting the concept of the sphere of influence ( r soi ) within which dark matter particles are bound to the satellite halos, using the following formula (Battin 1987):</text>
<formula><location><page_2><loc_52><loc_30><loc_96><loc_33></location>r soi ∼ r [ ( M sat , tot M host ( < r ) ) -0 . 4 (1 + 3 cos 2 θ ) 0 . 1 +0 . 4 cos θ ( 1 + 6 cos 2 θ 1 + 3 cos 2 θ )] -1 , (3)</formula>
<text><location><page_2><loc_52><loc_14><loc_92><loc_29></location>where r is the distance between the centers of the satellite and its host halos, M sat , tot is the total (baryon+dark matter) mass of the satellite halo, M host ( < r ) is the total mass of the central halo within r , and θ is the angle between the line connecting the particle to the center of the satellite halo and the line connecting the centers of the satellite and the host halos. During a time step, δt , we assume that a satellite halo loses δM sat = M sat ( r > r soi ) δt/t dyn , where t dyn is the dynamical timescale of the satellite halo, and M sat ( r > r soi ) is the mass of dark matter outside r soi of the satellite halo.</text>
<text><location><page_2><loc_52><loc_12><loc_92><loc_15></location>We assume that stellar components in satellite 1 galaxies merging with their hosts constitute the bulge com-</text>
<text><location><page_2><loc_52><loc_7><loc_92><loc_10></location>1 In this paper, galaxies that are not the central one in a halo are all 'satellite'. Only one galaxy is qualified as the central galaxy of a halo and all the rest, regardless of brightness, are satellites.</text>
<text><location><page_3><loc_8><loc_84><loc_48><loc_92></location>ponent of the host. If the mass ratio of baryonic mass ( m cold + m ∗ ) between merging galaxies, m secondary /m primary , is greater than 0.25, then it is assumed that all the stellar components of the host galaxy quickly become bulge components of the remnant, as well.</text>
<text><location><page_3><loc_8><loc_63><loc_48><loc_84></location>Empirical studies have shown that intra-cluster light originates from the extended diffuse stellar components of the brightest galaxies in groups or clusters (e.g. Feldmeier et al. 2002; Gonzalez et al. 2005; Zibetti et al. 2005). They suggested that diffuse stellar components are the stars scattered from satellite galaxies during tidal stripping or mergers into central galaxies. It has been suggested that 10-40% of stellar components in satellite galaxies turn into diffuse stellar components in each galaxy merger (Murante et al. 2004; Monaco et al. 2006). We adopt the value of 40% in this study because it resulted in the best reproduction of empirical data. The amount of stellar mass that a central galaxy acquires via merger is (1 -f scatter ) M ∗ , sat , where f scatter is the fraction of scattered stellar components and M ∗ , sat is the stellar mass of a satellite galaxy.</text>
<section_header_level_1><location><page_3><loc_11><loc_60><loc_46><loc_61></location>2.2. Gas Cooling, Star Formation, and Recycling</section_header_level_1>
<text><location><page_3><loc_8><loc_54><loc_48><loc_59></location>We assume that the baryonic fraction in accreted dark matter follows the global baryonic fraction, Ω b / Ω m . Baryons are accreted onto dark halos and shock-heated to become hot gas components.</text>
<text><location><page_3><loc_8><loc_48><loc_48><loc_54></location>Gas accretion onto a galactic disk plane via atomic cooling of hot gas is calculated based on the model proposed by White & Frenk (1991). The cooling timescale at distance r from the center of a halo is estimated by the following formula:</text>
<formula><location><page_3><loc_15><loc_44><loc_48><loc_47></location>t cool ( r ) = 3 2 ρ g ( r ) kT µ ( Z, T ) m P n e n i Λ( Z, T ) , (4)</formula>
<text><location><page_3><loc_8><loc_19><loc_48><loc_43></location>where ρ g is the gas density within radius r of a dark matter halo, T is the temperature of gas, assumed to be the virial temperature in the model, Z is metallicity, µ ( Z, T ) is the mean molecular weight of gas with Z and T , m P is the mass of a proton, n e is the number density of electrons, n i is the number density of ions, and Λ( Z, T ) is the cooling function with Z and T . The values of µ ( Z, T ), n e , n i , and Λ( Z, T ) are determined by referring to Sutherland & Dopita (1993). We do not include selfconsistent chemical evolution in our calculation. Instead, we take the metallicity of hot gas components in halos as a constant, 0 . 3 Z /circledot , which is comparable to the metallicity of observed clusters with various masses (Arnaud et al. 1992). Because the cooling function, Λ( Z, T ), is sensitive to metallicity, our results should not be taken too literally. However, relative analysis, a main tool of this study, is affected little by the details in the treatment of chemical evolution.</text>
<text><location><page_3><loc_8><loc_7><loc_48><loc_19></location>It is assumed that the gas density follows a singular isothermal profile truncated at R vir : ρ g ( r ) = m hot / (4 πR vir r 2 ). Substituting the formula for ρ g ( r ) in Eq. 4 and adopting the dynamical timescale of a halo, t dyn , as t cool , one can derive the cooling radius, r cool , within which hot gas can cool within t cool . For the case of r cool > R vir , the cooling rate is rather restrained by the free-fall rate than the cooling rate, so that ˙ m cool = m hot / (2 t cool ). In contrast, if r cool < R vir ,</text>
<text><location><page_3><loc_52><loc_90><loc_74><loc_92></location>˙ m cool = m hot r cool / (2 R vir t cool ).</text>
<text><location><page_3><loc_52><loc_83><loc_92><loc_90></location>In our model, stars can be formed through a quiescent mode, in which cold gas turns into disk stellar components via gas contraction on a disk, or a burst mode, which is induced by galaxy mergers. Star formation rate in the quiescent mode are delineated by a simple law proposed by Kauffmann et al. (1993) as follows:</text>
<formula><location><page_3><loc_66><loc_78><loc_92><loc_82></location>˙ m ∗ = α m cold t dyn , gal , (5)</formula>
<text><location><page_3><loc_52><loc_72><loc_92><loc_77></location>where α is the empirically-determined star formation efficiency, m cold is the amount of cold gas, and t dyn , gal is the dynamical timescale of cold gas disk assumed to be 0 . 1 t dyn .</text>
<text><location><page_3><loc_52><loc_55><loc_92><loc_72></location>Observations (e.g. Borne et al. 2000; Woods & Geller 2007) and hydrodynamic simulations of galaxy mergers (e.g. Cox et al. 2008, hereafter C08) have shown that galaxy mergers can give rise to rapid star formation. We follow the conventional treatment: stars formed in the quiescent mode belong to a galactic disk, while stars born in the burst mode become bulge components. We adopt the prescription for merger-induced starbursts described in Somerville et al. (2008, hereafter S08), which formulates the prescription based on C08. S08 defines burst efficiency, e burst , to parameterize the fraction of the cold gas reservoir involved in a merger induced starburst as follows:</text>
<formula><location><page_3><loc_64><loc_52><loc_92><loc_54></location>e burst = e burst , 0 µ γ burst , (6)</formula>
<text><location><page_3><loc_52><loc_45><loc_92><loc_51></location>where µ is the mass ratio between a host galaxy and its merger counterpart, γ burst is the bulge-to-total mass ratio(B/T) of the host galaxy, and e burst , 0 is the burst efficiency fitted by the following formula:</text>
<formula><location><page_3><loc_55><loc_41><loc_92><loc_44></location>e burst , 0 = 0 . 60[ V vir / (kms -1 )] 0 . 07 (1 + q EOS ) -0 . 17 (1 + f g ) 0 . 07 (1 + z ) 0 . 04 , (7)</formula>
<text><location><page_3><loc_52><loc_23><loc_92><loc_39></location>where V vir is virial velocity, q EOS is the effective equation of state of gas, f g ≡ m cold / ( m cold + m ∗ ) is the fraction of cold gas, and z is the redshift when the disks of progenitor galaxies are constructed. q EOS was suggested to parameterize the multiphase nature of ISM: q EOS = 0 indicates an isothermal state, and q EOS = 1 represents the fully pressurized multiphase ISM. In this study, we adopt q EOS = 1 at which gas is dynamically stable, so that starbursts are more suppressed than the case of q EOS = 0. The redshift dependency of e burst , 0 is very weak, and thus we assume (1 + z ) 0 . 04 ∼ 1. The burst timescale, τ burst , is also formulated by S08 as follows:</text>
<formula><location><page_3><loc_54><loc_19><loc_92><loc_22></location>τ burst = 191Gyr[ V vir / (kms -1 )] -1 . 88 (1 + q EOS ) 2 . 58 (1 + f g ) -0 . 74 (1 + z ) -0 . 16 . (8)</formula>
<text><location><page_3><loc_52><loc_8><loc_92><loc_17></location>The mass ratio, µ , is the ratio of the total mass of central regions ( m DM , core + m ∗ + m cold ) of a host galaxy to that of its merger counterpart. Following S08, we calculate the core mass of a dark matter halo, m DM , core = m DM ( r < 2 r s ), where r s ≡ R vir /c NFW . The concentration index of the Navarro-Frenk-White profile, c NFW , is derived based on the fitting function suggested by Macci'o et al. (2007).</text>
<text><location><page_3><loc_53><loc_7><loc_92><loc_8></location>The parameter γ burst is determined by the B/T of a</text>
<text><location><page_4><loc_8><loc_91><loc_24><loc_92></location>host galaxy as follows:</text>
<formula><location><page_4><loc_14><loc_86><loc_48><loc_90></location>γ burst = { 0 . 61 B/T ≤ 0.085 0 . 74 0.085 < B/T ≤ 0.25 1 . 02 0.25 < B/T (9)</formula>
<text><location><page_4><loc_8><loc_69><loc_48><loc_85></location>C08 showed that the burst efficiency of a host galaxy with a high B/T is lower than that of a galaxy with a lower B/T, because more massive bulges stabilize the galaxies and reduce the burst efficiency more effectively. Because C08 demonstrated that mergers with mass ratios below 1:10 are not associated with starbursts, we assume e burst = 0 if µ < 0 . 1. With the ingredients, the amount of stars born in burst modes is calculated as m burst = e burst m cold . We assume that m burst turns into stars for τ burst in a uniform rate, ˙ m burst = m burst /τ burst . While merger-induced starbursts occur in a galaxy, the quiescent mode also still goes on in our models.</text>
<text><location><page_4><loc_8><loc_57><loc_48><loc_69></location>In our model, the recycling of stellar mass loss is considered in great detail. We compute the mass loss of every single population at each epoch after a new stellar population is born. The mass loss of a single population is calculated as follows. We adopt the Scalo initial mass function (Scalo 1986). The lifetime of a star with mass M , τ M , is computed by a broken-power law (Ferreras & Silk 2000), which is obtained from the data of Tinsley (1980) and Schaller et al. (1992).</text>
<formula><location><page_4><loc_11><loc_52><loc_48><loc_56></location>τ M (Gyr) = { 9 . 694( M/M /circledot ) -2 . 762 M < 10 M /circledot 0 . 095( M/M /circledot ) -0 . 764 M ≥ 10 M /circledot (10)</formula>
<text><location><page_4><loc_8><loc_46><loc_48><loc_52></location>At the end of its lifetime (after τ M elapses), a star returns most of its mass into space, leaving small remnants such as a white dwarf, a neutron star or a black hole. The remnant mass of a star with mass M , ω M , is suggested by Ferreras & Silk (2000) as follows:</text>
<formula><location><page_4><loc_9><loc_40><loc_48><loc_45></location>ω M M /circledot = { 0 . 1( M/M /circledot ) + 0 . 45 M < 10 M /circledot 1 . 5 10 M /circledot ≤ M < 25 M /circledot 0 . 61( M/M /circledot ) -13 . 75 M ≥ 25 M /circledot . (11)</formula>
<text><location><page_4><loc_8><loc_36><loc_48><loc_40></location>In this study, we simply assume that half of the mass loss returns to cold gas components and the rest becomes hot gas.</text>
<section_header_level_1><location><page_4><loc_19><loc_34><loc_38><loc_35></location>2.3. Environmental Effect</section_header_level_1>
<text><location><page_4><loc_8><loc_7><loc_48><loc_33></location>The Chandra X-ray Observatory revealed that massive satellite galaxies in nearby clusters have hot gas components (Sun et al. 2007; Jeltema et al. 2007), in contradiction to expectations based on an instantaneous hot gas stripping scenario for satellite galaxies in cluster environments (e.g. Kauffmann et al. 1999; Somerville et al. 2008). The old assumption predicted a higher fraction of passive satellites in large halos than observed, known as the satellite overquenching problem (Kimm et al. 2009). Kimm et al. (2011) showed that a gradual, rather than instant, and more realistic stripping of the hot gas reservoir relieves the above-mentioned problem to some degree. Therefore, we implemented the gradual hot gas stripping of satellite galaxies in our model by considering tidal stripping (see Kimm et al. 2011) and ram pressure stripping that uses the prescriptions of McCarthy et al. (2008), which had been modified for semi-analytic models by Font et al. (2008). However, we are still missing a realistic prescription on cold gas stripping. Empirical evidence for cold gas stripping is clear (Vollmer et al.</text>
<text><location><page_4><loc_52><loc_87><loc_92><loc_92></location>2008; Chung et al. 2008a,b), and a theoretical study using a semi-analytic approach shows its effect in galaxy evolution (Tecce et al. 2010). Thus, it should be considered in our model in due course.</text>
<section_header_level_1><location><page_4><loc_64><loc_82><loc_81><loc_83></location>2.4. Feedback Processes</section_header_level_1>
<text><location><page_4><loc_52><loc_64><loc_92><loc_81></location>Feedback mechanisms have been introduced into galaxy formation theory to reconcile the discrepancy between galaxy and dark matter halo mass functions. Hydrogen atoms, neutralized at the recombination, may be reionized at later epochs ( z < 10) due to background high-energy photons. This mechanism may suppress the growth of small galaxies (Gnedin 2000; Somerville 2002; Benson et al. 2002a,b). Supernova feedback is thought to be effective at disturbing the growth of small galaxies by ejecting cold gas (White & Rees 1978; Dekel & Silk 1986), and AGN feedback is considered more effective in massive galaxies with a large black hole (Silk & Rees 1998; Schawinski et al. 2006, 2007).</text>
<text><location><page_4><loc_52><loc_51><loc_92><loc_64></location>We utilize the reionization prescription of Benson et al. (2002b). The prescription allows an inflow of baryons into a dark halo via accretion of dark matter when V vir > V reionization throughout the age of the Universe, where V reionization is the suppression velocity of reionization . If a halo has a lower value of V vir than the criterion, the inflow of baryons is allowed only at z > z reionization , where z reionization is the suppression redshift of reionization. In this study, we adopt V reionization = 30 km s -1 and z reionization = 8 following Benson et al. (2002b).</text>
<text><location><page_4><loc_52><loc_44><loc_92><loc_51></location>We follow the prescriptions of S08 for supernova feedback. These prescriptions take into account not only the amount of reheated gas but also the fraction of reheated gas blown away from halos. The reheating rate of cold gas due to supernova feedback is formulated as follows:</text>
<formula><location><page_4><loc_60><loc_39><loc_92><loc_43></location>˙ m rh = /epsilon1 SN 0 ( 200kms -1 V disk ) α rh ˙ m ∗ , (12)</formula>
<text><location><page_4><loc_52><loc_30><loc_92><loc_38></location>where /epsilon1 SN 0 and α rh are free parameters, V disk is the rotational velocity of a disk, and ˙ m ∗ is the star formation rate. S08 assumes that the rotational velocity of a disk is the same as the maximum rotational velocity of the DM halo. We calculate the fraction of the reheated gas that has enough kinetic energy to escape from the halo as</text>
<formula><location><page_4><loc_60><loc_25><loc_92><loc_29></location>f eject = [ 1 . 0 + ( V vir V eject ) α eject ] -1 (13)</formula>
<text><location><page_4><loc_52><loc_16><loc_92><loc_24></location>where α eject = 6 and V eject ∼ 100 -150kms -1 . In the case of a satellite halo, a fraction of reheated gas by supernova feedback, f eject m rh , is ejected from the satellite and added to the hot gas reservoir of its host halo. On the other hand, it is assumed that the ejected gas from main halos is diffused through inter-cluster medium.</text>
<text><location><page_4><loc_52><loc_7><loc_92><loc_16></location>We take the quasar-mode and radio-mode AGN feedback into account in our model. It has been suggested that the quasar mode is induced by an inflow of cold gas into the central super-massive black hole (SMBH) of the central galaxy during major mergers (Kauffmann & Haehnelt 2000). The increasing mass of the SMBH via the accretion of cold gas can be ex-</text>
<figure>
<location><page_5><loc_9><loc_70><loc_47><loc_92></location>
<caption>Fig. 1.The cosmic star formation history. The gray dots with error bars indicate the empirical cosmic star formation history (Panter et al. 2007). The solid line shows the cosmic star formation history derived from the fiducial model. The dotted line displays the contribution to the cosmic star formation history from merger-induced starbursts.</caption>
</figure>
<text><location><page_5><loc_8><loc_60><loc_15><loc_62></location>pressed as</text>
<formula><location><page_5><loc_16><loc_56><loc_48><loc_59></location>∆ m BH , Q = f ' BH m cold 1 + (280km s -1 /V vir ) 2 , (14)</formula>
<text><location><page_5><loc_8><loc_38><loc_48><loc_55></location>where f ' BH is the efficiency of gas accretion. In this study, we take the modified parameter proposed by Croton et al. (2006): f ' BH = f BH ( M sat /M host ), where f BH is the original form of the parameter introduced by Kauffmann & Haehnelt (2000), M host is the mass of a host galaxy, and M sat is the mass of the host galaxy's merger counterpart. In this study, we assume that the lifetime of the quasar mode, t QSO , is 0.2Gyr, as Martini & Weinberg (2001) and Martini & Schneider (2003) suggested t QSO < 0 . 3Gyr. Thus, we have ˙ m BH , Q = ∆ m BH , Q /t QSO . It is generally thought that the quasar mode is caused by a high accretion rate of cold gas, resulting in a rapid growth of an SMBH.</text>
<text><location><page_5><loc_8><loc_26><loc_48><loc_38></location>The radio-mode feedback releases low-Eddington-ratio energy through the accretion of hot gas distributed throughout the halo. Although the energy released from the radio-mode AGN is far less than that from the quasar mode, it is regarded that the radio mode supplies enough energy to the surrounding medium to interrupt gas cooling or to blow away (some of the) cold gas. We implement radio-mode feedback into our model following the prescription of Croton et al. (2006):</text>
<formula><location><page_5><loc_10><loc_21><loc_48><loc_25></location>˙ m BH , R = κ AGN ( m BH 10 8 M /circledot )( f hot 0 . 1 )( V vir 200kms -1 ) 3 (15)</formula>
<text><location><page_5><loc_8><loc_15><loc_48><loc_21></location>where κ AGN is a free parameter with units of M /circledot yr -1 , m BH is the black hole mass, f hot is the fraction of hot gas with respect to the total halo mass, and V vir is the virial velocity.</text>
<text><location><page_5><loc_8><loc_13><loc_48><loc_15></location>We assume that the amount of energy generated by the accretion of gas into the SMBH is given as follows:</text>
<formula><location><page_5><loc_14><loc_10><loc_48><loc_12></location>L BH = η ˙ m BH c 2 = η ( ˙ m BH , Q + ˙ m BH , R ) c 2 , (16)</formula>
<text><location><page_5><loc_8><loc_7><loc_48><loc_9></location>where η = 0 . 1 is the standard efficiency of the conversion of rest mass to radiation, and c is the speed of light. The</text>
<figure>
<location><page_5><loc_53><loc_63><loc_90><loc_91></location>
<caption>Fig. 2.The galaxy stellar mass functions in the local Universe derived by Panter et al. (2007) from SDSS DR3 (gray shade) and the fiducial model at z=0 (black solid line). The thickness of the empirical data indicates the error range.</caption>
</figure>
<text><location><page_5><loc_52><loc_56><loc_82><loc_57></location>reduced cooling rate of gas is computed by</text>
<formula><location><page_5><loc_63><loc_52><loc_92><loc_55></location>˙ m ' cool = ˙ m cool -L BH 0 . 5 V 2 vir (17)</formula>
<text><location><page_5><loc_52><loc_49><loc_85><loc_51></location>where the minimum of ˙ m ' cool is set to be zero.</text>
<section_header_level_1><location><page_5><loc_64><loc_45><loc_80><loc_46></location>2.5. Model calibrations</section_header_level_1>
<text><location><page_5><loc_52><loc_29><loc_92><loc_44></location>Our models are based on the conventional techniques and ingredients used in up-to-date semi-analytic models; hence, the output is not particularly noteworthy compared to other successful models. Our models roughly match the global star formation history, galaxy mass functions, black hole mass versus bulge mass relation, etc. Figure 1 and 2 display comparisons of the cosmic star formation history and the galaxy stellar mass functions in the local Universe from empirical data (Panter et al. 2007) and our fiducial model. While there still is a large room for improvement, we decide to focus on the mass growth histories of massive galaxies.</text>
<section_header_level_1><location><page_5><loc_52><loc_25><loc_91><loc_26></location>3. EVOLUTION OF GALAXY NUMBER DENSITY</section_header_level_1>
<text><location><page_5><loc_52><loc_7><loc_92><loc_24></location>MK10 presented a rapid growth of massive galaxies since z = 1, using the United Kingdom Infrared Telescope(UKIRT) Infrared Deep Sky Survey (UKIDSS) and the Sloan Digital Sky Survey (SDSS) II Supernova Survey. Figure 3 shows the number density evolution of the most massive (log M/M /circledot = 11 . 5 -12 . 0) and the next massive (log M/M /circledot = 11 . 0 -11 . 5) galaxies in the empirical data derived by MK10 and from our fiducial model at each redshift. The empirical data clearly show that the number of the most massive galaxies rapidly increases between z = 1 and 0, while the next massive group experiences a milder evolution. The reproduction of the data by our fiducial model looks reasonably good. We also</text>
<figure>
<location><page_6><loc_9><loc_70><loc_47><loc_92></location>
<caption>Fig. 3.Number density evolution of massive galaxies as a function of the age of the Universe. The red represents the evolution of the most massive galaxies (log M/M /circledot = 11 . 5 -12 . 0), the blue indicates that of the next massive group (log M/M /circledot = 11 . 0 -11 . 5), and the green indicates that of the third massive group (log M/M /circledot = 10 . 5 -11 . 0) at each redshift. The diamonds with error bars come from empirical data in MK10 and the squares with error bars are measurements taken from Cole et al. (2001). The solid lines present the predictions of the semi-analytic model.</caption>
</figure>
<text><location><page_6><loc_8><loc_49><loc_48><loc_58></location>present the number density evolution of the third massive group (log M/M /circledot = 10 . 5 -11 . 0), which makes our models 'superL ∗ galaxies'. We define 'relative number density growth rate', Γ, as the ratio of the speeds in number density evolution of the two most massive groups of galaxies as a function of observational limit in redshift, as follows:</text>
<formula><location><page_6><loc_18><loc_45><loc_48><loc_48></location>Γ = n most ( z = 0) /n most ( z ) n next ( z = 0) /n next ( z ) , (18)</formula>
<text><location><page_6><loc_8><loc_9><loc_48><loc_44></location>where n most ( z = 0) and n next ( z = 0) are the number densities of the most massive and next massive groups at z = 0, and n most ( z ) and n next ( z ) are the number densities of the two groups at a redshift, respectively. For example, MK10 compared the number density evolution of the two mass groups between redshift 0 and 1, in which case the observational limit is 1 and the relative number density growth rate becomes 3. We present their observations and our models in Figure 4. The MK10 data point should be compared with our model for the most and the next massive galaxy groups (solid line). The model that compares the next massive group of galaxies with L ∗ galaxies (dashed line) exhibits a similar but milder trend. Observational constraints are still weak but are at least roughly reproduced by the models. The relative growth rate is always greater than 1 in the models, which indicates that the number density of more massive galaxies undergoes a faster evolution than that of less massive groups in the superL ∗ range. The number evolution is most dramatic when a comparison is made against the most massive galaxies. This is caused by the fact that the mass bin for the 'most massive' galaxies has only influx from less massive galaxies, whereas the mass bin of less massive galaxies can have outflux to more massive galaxy bins as well as influx from even less massive galaxy bins.</text>
<text><location><page_6><loc_8><loc_7><loc_48><loc_9></location>Primarily motivated by MK10, we focus on the mass growth histories of superL ∗ galaxies. We divide the</text>
<figure>
<location><page_6><loc_54><loc_70><loc_91><loc_92></location>
<caption>Fig. 4.Relative galaxy number density growth rate, Γ, between z=0 and observational redshift limits. The solid line shows Γ of the most/next and the dashed line indicates that of the next/third in our model. The cross is derived from the empirical data in Figure 3.</caption>
</figure>
<text><location><page_6><loc_52><loc_55><loc_92><loc_63></location>model galaxies into three groups according to z = 0 mass: Rank 1: log M/M /circledot = 11 . 5 -12 . 0, Rank 2: log M/M /circledot = 11 . 0 -11 . 5, and Rank 3: log M/M /circledot = 10 . 5 -11 . 0, where Rank 3 roughly represents L ∗ galaxies. From our simulation volume, we found 49 galaxies in Rank 1, 472 galaxies in Rank 2, and 2,188 galaxies in Rank 3.</text>
<section_header_level_1><location><page_6><loc_55><loc_49><loc_89><loc_52></location>4. EVOLUTION OF MASSIVE GALAXIES IN MODELS</section_header_level_1>
<section_header_level_1><location><page_6><loc_57><loc_47><loc_87><loc_48></location>4.1. Evolution of Stellar Mass in Galaxies</section_header_level_1>
<text><location><page_6><loc_52><loc_23><loc_92><loc_47></location>In the hierarchical paradigm, a galaxy can have more than one progenitor. Progenitors of a galaxy can be divided into 'direct' and 'collateral' progenitors. A direct progenitor is the galaxy in the largest halo when a merger between halos takes place. While there can be numerous progenitors, there is only one direct progenitor at each epoch. Collateral progenitors are all the other galaxies that contribute to the final galaxy. In this concept, to build the evolutionary history of a galaxy, one should consider not only direct progenitors, but also merger counterparts or collateral progenitors. Figure 5 shows the average mass evolution of the direct progenitors (solid lines) and all (direct and collateral combined) the progenitors (dotted lines) of Rank 1, 2, and 3 galaxies. The mean stellar masses of the three groups at z = 0 are 4 . 97 × 10 11 M /circledot , 1 . 55 × 10 11 M /circledot , and 5 . 25 × 10 10 M /circledot , respectively. Most of the Rank 1 galaxies in our volume are brightest cluster galaxies.</text>
<text><location><page_6><loc_52><loc_8><loc_92><loc_23></location>As all the progenitors merge with each other, the dotted lines and the solid lines finally meet at z = 0. Stellar mass loss and the scattering of stellar components in satellite galaxies into diffuse stellar components, which takes place when mergers occur, lead to a gradual decrease in the total mass during the evolution. For example, one can see a slight decline in the total stellar mass (red dotted line at the top) after z ∼ 0 . 5. This effect is not clearly visible in Rank 2 and 3 galaxies in which star formation is more extended and mergers are less frequent than in Rank 1 galaxies.</text>
<text><location><page_6><loc_53><loc_7><loc_92><loc_8></location>It is useful to have a definition of the formation red-</text>
<figure>
<location><page_7><loc_10><loc_70><loc_47><loc_92></location>
<caption>Fig. 5.Average mass evolution of galaxies. The red, blue, and green represent three groups of galaxies: log M/M /circledot = 11 . 5 -12 . 0, log M/M /circledot = 11 . 0 -11 . 5, and log M/M /circledot = 10 . 5 -11 . 0 at z = 0, respectively. The solid lines indicate the mean mass evolution of direct progenitors and the dotted lines show the mean mass evolution of all (direct+collateral) progenitors. The black horizontal dashed lines denote half the mean galaxy mass at z = 0 of the groups. A 1 , A 2 , and A 3 , and the arrows indicate the epochs when the total stellar masses of all progenitors reach half of the final mass. D 1 , D 2 , and D 3 with arrows show the epochs at which the direct progenitors of the three groups acquire half of their final mass.</caption>
</figure>
<text><location><page_7><loc_8><loc_29><loc_48><loc_55></location>shift, z f . We define it as the redshift at which half of the stellar mass at z = 0 has been assembled. In the case of Rank 1 galaxies, half of the final mass is achieved at z f , D ∼ 0 . 7(denoted as D 1 in Figure 5) in direct progenitors and at z f , A ∼ 3 . 0(A 1 ) when all progenitors are combined. Our models suggest ( z f , D , z f , A ) = (0.9, 2.1) for Rank 2, and ( z f , D , z f , A ) = (1.1, 1.6) for Rank 3. Models exhibit a monotonic mass dependence of z f , D and z f , A in the sense that, with mass, z f , D decreases while z f , A increases. In other words, the mass of the direct progenitors of a more massive group grows more slowly, while its total mass of all the progenitors is assembled earlier than that of a less massive group. The evolutionary histories of direct progenitors are opposite to the pattern of cosmic downsizing. As Neistein et al. (2006) and Oser et al. (2010) pointed out, however, if the growth histories of collateral progenitors of galaxies are also considered, downsizing would be a natural outcome of the hierarchical concept of galaxy formation.</text>
<text><location><page_7><loc_8><loc_15><loc_48><loc_29></location>The difference in the growth history between the three groups can be understood in depth through Figure 6, which presents the evolutionary histories of star formation rates (SFRs). The mean SFRs contain the star formation histories of both direct and collateral progenitors. We show the best fitting log-normal function to the three SFR curves. The star formation rates of more massive galaxies peak earlier, as marked by S1, S2, and S3 in the figure, and decreases faster than those of less massive groups. The same features were noted in an earlier study by De Lucia et al. (2006).</text>
<text><location><page_7><loc_8><loc_7><loc_48><loc_15></location>Figure 6 also show the redshift at which half of the total cumulative star formation has occurred in the three groups of galaxies: (C1, C2, C3) = (2.0, 1.5, 1.4) in z . The general trend shown by these three values agrees with that of z f , A discussed above. In a sense, it is these values, rather than z f , A , that are closer to the general</text>
<figure>
<location><page_7><loc_53><loc_70><loc_91><loc_92></location>
<caption>Fig. 6.Mean star formation histories of all the progenitor galaxies of Rank 1(red), 2(blue), and 3(green). The gray dashed lines denote log-normal fitting. S 1 , S 2 , and S 3 with arrows indicate the epochs when SFRs peak. C 1 , C 2 , and C 3 with arrows show the epochs at which the cumulative stellar mass reach half of the total stellar mass born by z = 0.</caption>
</figure>
<text><location><page_7><loc_52><loc_46><loc_92><loc_62></location>definition of formation redshift. Again, the more massive galaxies are, the earlier they form their stars, consistent with downsizing. In conclusion, the observational finding of downsizing is a result of the hierarchical galaxy formation process, where more massive galaxies have larger stellar ages and smaller assembly ages (see also e.g. De Lucia & Blaizot 2007; Kaviraj et al. 2009). Both the larger stellar ages and the smaller assembly ages can be understood as a result of large-scale effect; that is, in a deeper potential well, progenitor galaxies and their stars form earlier, and many more galaxies participate in galaxy mergers for a long period of time.</text>
<section_header_level_1><location><page_7><loc_60><loc_43><loc_84><loc_44></location>4.2. Origin of Stellar Components</section_header_level_1>
<text><location><page_7><loc_52><loc_29><loc_92><loc_43></location>In this section, we investigate how stellar components are assembled into massive galaxies. Stellar components in a galaxy originate either from in-situ star formation or from 'merger accretion'. Two modes of star formation are considered: 'quiescent' mode and merger-induced 'starburst'. Stars in a galaxy can therefore have four different origins: (1) in-situ quiescent star formation, (2) in-situ starburst, (3) merger accretion of stars formed in quiescent mode, and (4) merger accretion of stars formed in burst mode.</text>
<text><location><page_7><loc_52><loc_23><loc_92><loc_29></location>Figure 7 shows the decomposition of the four channels as a function of time for the three different mass groups. The top and bottom rows show the absolute stellar mass evolution and relative mass fraction evolution, respectively.</text>
<text><location><page_7><loc_52><loc_7><loc_92><loc_23></location>Several remarkable features are visible, more easily in the bottom rows. First, in-situ quiescent star formation is an important channel for the stellar components in massive galaxies. Its fractional contribution is (30, 60, 80)% in Rank (1, 2, 3) galaxies, respectively. In-situ quiescent star formation takes place on a galactic disk, and thus, one may wonder why its contribution is so large in these massive, and probably bulge-dominant galaxies. This is because massive bulge-dominant galaxies at z = 0 have many latetype progenitors, and that is more pronounced in less massive galaxies (Rank 3) than in Rank 1 galaxies.</text>
<figure>
<location><page_8><loc_9><loc_57><loc_92><loc_92></location>
<caption>Fig. 7.Average mass evolution of stellar components in direct progenitors (upper) and the fraction of each component divided by the mean total stellar mass of direct progenitors at each redshift (bottom). The left, middle, and right panels show the mean evolutionary histories of the direct progenitors of Rank 1, 2 , and 3 galaxies, respectively. Each color code represents each stellar component as follows: (1) sky blue: merger accretion of stars formed in quiescent mode, (2) orange: in-situ quiescent star formation, (3) blue: in-situ starburst, and (4) red: merger accretion of stars formed in burst mode.</caption>
</figure>
<text><location><page_8><loc_8><loc_45><loc_48><loc_49></location>This is a reflection of the progenitor bias discussed earlier (c.f. Franx & van Dokkum 2001; Guo & White 2008; Parry et al. 2009; Kaviraj et al. 2009).</text>
<text><location><page_8><loc_8><loc_20><loc_48><loc_45></location>Second, the stars formed in burst mode are an extreme minority ( ∼ 2%) in these massive galaxies. This result is somewhat unexpected. In the hierarchical universe, larger galaxies are generally a product of numerous mergers between galaxies. Mergers, especially between similar-mass galaxies (major mergers), cause a starburst, and so a naive expectation is that larger galaxies would have a large fraction of stars formed in burst mode. However, it is not that simple. Starbursts are usually a result of major mergers, and major mergers are extremely rare once galaxies become massive, that is, at low redshifts. Major mergers between massive galaxies sometimes occur at low redshifts, but they are most often dry, between early-type galaxies without much cold gas (c.f. van Dokkum 2005; Bundy et al. 2007, 2009; Parry et al. 2009). Thus, it has been suggested that the fraction of stars formed in merger-driven burst mode may be less than a few per cent (e.g. Aragon-Salamanca et al. 1998; Rodighiero et al. 2011).</text>
<text><location><page_8><loc_8><loc_8><loc_48><loc_20></location>Third, merger accretion is the dominant channel in the most massive galaxies. Roughly 70% of the stellar components in the most massive group (bottom left panel) form outside direct progenitors and get accreted via mergers. This fraction is smaller for Rank 2 (40%) and Rank 3 (20%) galaxies, but still substantial. It is interesting to note that much of the mass difference between ranks is attributed to the difference in the amount of merger accretion. For example, roughly 80% of the</text>
<figure>
<location><page_8><loc_53><loc_28><loc_91><loc_49></location>
<caption>Fig. 8.Frequency of mergers (per Gyr per galaxy) with a mass ratio m 2 /m 1 /greaterorsimilar 0 . 1 experienced by direct progenitors. The solid, dashed, and dotted lines indicate the merger rates of Rank 1, 2, and 3 galaxies, respectively.</caption>
</figure>
<text><location><page_8><loc_52><loc_7><loc_92><loc_20></location>mass difference between Ranks 1 and 2 comes from the differences in merger accretion. The value is ∼ 60% between Ranks 2 and 3. Massive galaxies are so mainly because they have a substantial amount of in-situ quiescent star formation, but the most massive galaxies are so because they have acquired a large amount of mass through merger accretion. The result is supported by previous studies. Aragon-Salamanca et al. (1998) confirmed that BCGs have experienced no or negative evolution in luminosity during 0 < z < 1 from the K-band</text>
<figure>
<location><page_9><loc_9><loc_52><loc_47><loc_92></location>
<caption>Fig. 9.The specific star formation rates (upper) and the specific stellar accretion rates via mergers (bottom) of the direct progenitors in the three groups. See the text for their definitions. The solid, dashed, and dotted lines represent Rank 1, 2, and 3 galaxies.</caption>
</figure>
<text><location><page_9><loc_8><loc_8><loc_48><loc_45></location>Hubble diagram for a sample of BCGs while they have increased their mass by a factor of two to four, depending on cosmological parameters. Thus, it has been understood that merger and accretion may be the most plausible explanation for the evolution. Using a semianalytic model, De Lucia & Blaizot (2007) showed that most stellar components in model BCGs are formed in the very early age of the Universe (80 % at z ∼ 3) in small progenitors and accreted onto BCGs far later (50% after z ∼ 0 . 5) via mergers. Oser et al. (2010) presented similar results, using numerical simulations. About 80% of stellar components in simulated massive galaxies ( M ∗ > 2 . 4 × 10 11 M /circledot at z=0) are formed outside in the early age ( z > 3) and brought into the massive galaxies via mergers and accretion. The massive galaxies double their mass after z ∼ 1. They revealed that the fraction gets smaller in less massive galaxies. Parry et al. (2009) also found a similar trend in their investigation on two separate semi-analytic models. They demonstrated that the contribution of mergers to the bulge growth exceeds that of disk instability at M ∗ > 10 11 . 5 M /circledot . Considering the fact that such massive galaxies ( M ∗ > 10 11 M /circledot ) are likely early type (e.g. Bell et al. 2003), it implies that mergers play a more important role in the growth of massive galaxies. It should however be noted that disk instability which is more effective to the smaller late-type galaxy evolution may play a role in such progenitor galaxies of present-day massive early types.</text>
<text><location><page_9><loc_10><loc_7><loc_48><loc_8></location>Figure 8 shows the merger rate evolution for bary-</text>
<figure>
<location><page_9><loc_54><loc_41><loc_90><loc_92></location>
<caption>Fig. 10.Comparison of the mean specific star formation rates (solid lines) and the mean specific stellar accretion rates via mergers (dotted lines) of the direct progenitors of Rank 1(upper), 2 (middle), and 3 (bottom) galaxies.</caption>
</figure>
<text><location><page_9><loc_52><loc_14><loc_92><loc_35></location>onic mass ratios greater than or equal to 1:10. Although merger rates show stochastic effects, there is a clear decreasing tendency with time, whereas the star formation rates of the three groups drop more sharply, as shown in Figure 6. In general, more massive galaxies are likely to be involved in galaxy mergers more frequently, so that more massive galaxies have many more stellar components born outside and accreted via mergers, as discussed above. During the whole calculation, Rank 1 galaxies undergo about 9.0 mergers with a mass ratio greater than or equal to 1:10 while those in Ranks 2 and 3 experience 4.0 and 1.0 mergers, respectively. During 0 < z < 1, Rank 1, 2, and 3 galaxies experience 3.5, 1.8, or 0.6 mergers for the same mass ratio criterion. This explains the higher contribution of merger accretion in more massive galaxies, as illustrated in Figure 7.</text>
<section_header_level_1><location><page_9><loc_55><loc_10><loc_89><loc_13></location>4.3. Specific Star Formation Rates and Merger Accretion Rates</section_header_level_1>
<text><location><page_9><loc_52><loc_7><loc_92><loc_9></location>We found in the previous section that in-situ star formation and merger accretion were the two most signif-</text>
<text><location><page_10><loc_8><loc_83><loc_48><loc_92></location>icant channels for the stellar mass growth of massive galaxies. In this section, we scrutinize their time evolution in greater detail. Specific star formation rates (SSFRs) are normalized growth rates of star formation histories. Likewise, we hereby define the 'specific stellar accretion rate' (SSAR) to evaluate a normalized growth rate via mergers as follows:</text>
<formula><location><page_10><loc_22><loc_79><loc_48><loc_82></location>SSAR = ∆ M M ( t )∆ t , (19)</formula>
<text><location><page_10><loc_8><loc_76><loc_48><loc_78></location>where M ( t ) is the mass of a galaxy at an epoch, and</text>
<text><location><page_10><loc_8><loc_41><loc_48><loc_76></location>∆ M is an increment of mass by mergers during a time step ∆ t . Because we allow diffuse stellar components due to galaxy mergers, mass increment can be expressed as ∆ M = (1 -f scatter ) M ∗ , sat , as described in Section 2.1. Figure 9 shows the evolution of the SSFRs (upper) and of the SSARs (bottom) of direct progenitors. In general, more massive galaxies have lower SSFRs, as observations have shown (e.g. Salim et al. 2007; Schiminovich et al. 2007), while they have higher SSARs than less massive galaxies. Star formation rates are decreased by the depression of gas cooling rates due to an increase in the cooling timescale via the growth of halos, supernova feedback (White & Rees 1978; Dekel & Silk 1986; White & Frenk 1991) and/or AGN feedback (Silk & Rees 1998). Furthermore, the cold gas reservoir of a galaxy could be reduced by feedback processes. Besides, if a galaxy orbits around a more massive galaxy, it becomes redder as it loses its hot gas, which is a source of cold gas, and its cold gas reservoir by tidal and ram pressure stripping (Gunn & Gott 1972; Abadi et al. 1999; Quilis et al. 2000; Chung et al. 2007; Tonnesen & Bryan 2009; Yagi et al. 2010). On the other hand, accretion of stellar components via mergers is determined by gravitational interactions between host and subhalos alone; hence, the accretion rate could remain relatively steady as halos continue to fall into other more massive halos over time.</text>
<text><location><page_10><loc_8><loc_27><loc_48><loc_41></location>Figure 10 displays a comparison of the SSFRs and the SSARs of direct progenitors. As illustrated in Figure 7, quiescent star formation dominates the stellar mass growth history in L ∗ (Rank 3) galaxies (bottom panel). In Rank 2 galaxies, they are comparable to each other most of the time. However, in the most massive (Rank 1) galaxies, merger accretion takes over star formation as the most important channel of stellar mass growth around z ∼ 2 (top panel). Our result is qualitatively consistent with that of Oser et al. (2010).</text>
<section_header_level_1><location><page_10><loc_15><loc_25><loc_41><loc_26></location>5. SUMMARY AND DISCUSSION</section_header_level_1>
<text><location><page_10><loc_8><loc_19><loc_48><loc_24></location>We have investigated the assembly history of stellar components of massive (superL ∗ ) galaxies, using semianalytic approaches. Our major results can be summarized as follows.</text>
<unordered_list>
<list_item><location><page_10><loc_11><loc_12><loc_48><loc_17></location>· More massive galaxies grow in number faster than less massive galaxies, as a result of the hierarchical nature of galaxy clustering. This result is consistent with the recent observation of MK10.</list_item>
<list_item><location><page_10><loc_11><loc_7><loc_48><loc_11></location>· The conflict between the predictions from hierarchical models and downsizing is reconcilable. If we consider only direct progenitors of massive galaxies,</list_item>
</unordered_list>
<text><location><page_10><loc_56><loc_83><loc_92><loc_92></location>our models suggest 'upsizing' rather than downsizing; that is, the direct progenitors of more massive galaxies grow more slowly. However, if we consider all the progenitors, direct and collateral, the combined mass suggests downsizing. Our models suggest that more massive galaxies have older stellar ages but younger assembly ages.</text>
<unordered_list>
<list_item><location><page_10><loc_54><loc_72><loc_92><loc_82></location>· Merger-induced 'bursty' star formation is negligible compared to quiescent disk-mode star formation despite the fact that massive galaxies form through numerous mergers. This is because most of the gas-rich major mergers occur at high redshifts when galaxies are small, and recent major mergers tend to be rare and 'dry'.</list_item>
<list_item><location><page_10><loc_54><loc_59><loc_92><loc_71></location>· Merger accretion is a growingly more important channel of stellar mass growth in more massive galaxies. It accounts for 70% of the final stellar mass in the most massive galaxies in our sample (log M/M /circledot = 11 . 5 -12 . 0). It is merger accretion that causes much of the mass difference between massive galaxies. This implies that environments play a central role in the growth of massive galaxies.</list_item>
<list_item><location><page_10><loc_54><loc_52><loc_92><loc_58></location>· In the most massive galaxies, which are likely brightest cluster galaxies, merger accretion has remained the most important channel of stellar mass growth ever since z ∼ 2.</list_item>
</unordered_list>
<text><location><page_10><loc_52><loc_30><loc_92><loc_52></location>The origin of massive galaxies is a pivotal subject of cosmological paradigms and subsidiary galaxy formation theories. The simplest views based on some pieces of observations, such as downsizing, may favor simplistic formation scenarios, while dynamical models of the universe based on the current cosmology and other statistical aspects of observations, such as galaxy luminosity functions, indicate the other direction. This conflict is at its maximum when it comes to the formation of massive galaxies. The goal of this study is to reconcile the various perspectives and to have an accurate understanding on their formation. In this study, we showed that galaxy models in the hierarchical paradigm provide explanations to seemingly contradicting empirical constraints: the faster growth of the number of more massive galaxies (MK10) and downsizing (Cowie et al. 1996). This is encouraging.</text>
<text><location><page_10><loc_52><loc_15><loc_92><loc_30></location>Galaxy formation models, whether hydrodynamic or semi-analytic, are still incomplete in many aspects: it is often claimed that much of their incompleteness is caused by our limited knowledge of baryon physics. The success of understanding massive galaxy formation on the other hand seems to be more hinged upon our knowledge of large-scale clusterings, and thus dark matter physics. Massive galaxies achieve their grandeur through mergers; thus, only by a realistic consideration of large-scale clustering information is it possible to accurately reconstruct their formation history.</text>
<section_header_level_1><location><page_10><loc_63><loc_13><loc_81><loc_14></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_10><loc_52><loc_7><loc_92><loc_12></location>We thank Taysun Kimm and Sadegh Khochfar for their feedback in the early stage of our code development and Intae Jung for helping us run cosmological volume simulations. We thank the anonymous referee for a number</text>
<text><location><page_11><loc_8><loc_87><loc_48><loc_92></location>of comments and suggestions that improved the clarity of the paper. We acknowledge the support from the National Research Foundation of Korea through the Center for Galaxy Evolution Research (No. 2010-0027910),</text>
<text><location><page_11><loc_52><loc_91><loc_87><loc_92></location>Doyak grant (No. 20090078756), and DRC grant.</text>
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</document> | [] |
2019CQGra..36x5021L | https://arxiv.org/pdf/1911.05786.pdf | <document>
<section_header_level_1><location><page_1><loc_14><loc_86><loc_86><loc_91></location>Spectroscopic tests for short-range modifications of Newtonian and post-Newtonian potentials</section_header_level_1>
<text><location><page_1><loc_15><loc_71><loc_85><loc_83></location>A. S. Lemos 2 , 3 , G. C. Luna 1 , E. Maciel 3 , and F. Dahia 1 1 Department of Physics, Federal University of Para´ıba - Jo˜ao Pessoa -PB - Brazil 2 Dep. of Phys., State University of Para´ıba - Campina Grande - PB - Brazil. and 3 Department of Physics, Federal University of Campina Grande - Campina Grande -PB - Brazil</text>
<section_header_level_1><location><page_1><loc_45><loc_67><loc_54><loc_69></location>Abstract</section_header_level_1>
<text><location><page_1><loc_12><loc_37><loc_88><loc_66></location>There are theoretical frameworks, such as the large extra dimension models, which predict the strengthening of the gravitational field in short distances. Here we obtain new empiric constraints for deviations of standard gravity in the atomic length scale from analyses of recent and accurate data of hydrogen spectroscopy. The new bounds, extracted from 1 S -3 S transition, are compared with previous limits given by antiprotonic Helium spectroscopy. Independent constraints are also determined by investigating the effects of gravitational spin-orbit coupling on the atomic spectrum. We show that the analysis of the influence of that interaction, which is responsible for the spin precession phenomena, on the fine structure of the states can be employed as a test of a postNewtonian potential in the atomic domain. The constraints obtained here from 2P 1 / 2 -2 P 3 / 2 transition in hydrogen are tighter than previous bounds determined from measurements of the spin precession in an electron-nucleus scattering.</text>
<section_header_level_1><location><page_2><loc_12><loc_89><loc_32><loc_91></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_2><loc_12><loc_66><loc_88><loc_86></location>The existence of extra dimensions has been speculated on a modern scientific basis since the advent of the Kaluza-Klein theory in the early twentieth century. This theory was an attempt to unify gravity and electromagnetism, the interactions known at that time, within a single formalism that postulated the presence of an additional spatial dimension. To avoid empirical contradictions, it is assumed that the fifth dimension has the topology of a circle with a radius of the order of the Planck length (10 -35 m). With such a tiny radius, however, there were no prospects for experimentally testing the existence of hidden dimensions neither at that time nor in a foreseeable future.</text>
<text><location><page_2><loc_12><loc_53><loc_88><loc_65></location>At the end of the last century, the subject of extra dimensions has emerged with renewed interest due to the braneworld models, which proposes a new scenario, inspired from developments in string theory, in which matter and all the standard model fields are confined in a space with three spatial dimensions (the 3-brane), while the gravitational field can propagate in all directions [1-4].</text>
<text><location><page_2><loc_12><loc_43><loc_88><loc_52></location>Initially, this scenario drew a lot of attention because the predicted 'spreading' of gravity to other dimensions could be an explanation for the hierarchy problem, i.e., for the question of why the gravitational interaction is so weak compared to the other forces, at long distances [1-4].</text>
<text><location><page_2><loc_12><loc_27><loc_88><loc_41></location>Another very interesting feature brought by some braneworld models, such as the ADD model [1, 2], is the possibility that empirical signals of extra dimensions could be the object of experimental search at present days. Indeed, as gravity is the only field that has access to the extra dimensions, then the hypothesis that the compactification radius R could be much greater than the Planck length scale is phenomenological feasible, since gravity is being tested in a small length scale just recently.</text>
<text><location><page_2><loc_12><loc_8><loc_88><loc_26></location>It is well known that, if the supplementary space has compact topology, the gravitational field recovers its traditional four-dimensional behavior for distances r much greater than the compactification radius R . On the other hand, the force gets strengthened at short distances ( r < R ). Direct laboratory tests of the inverse-square law, based on modern versions of torsion balances, put the upper limit R < 44 µ m [5, 6] if there is only one extra dimension. Regarding more codimensions, the most stringent constraints come from highenergy particle collision data and analysis of some astrophysics processes [7-13]. In all cases,</text>
<text><location><page_3><loc_12><loc_89><loc_83><loc_91></location>however, experimental upper bounds for R are much weaker than the Planck length.</text>
<text><location><page_3><loc_12><loc_76><loc_88><loc_88></location>The amplification of gravity at short distances as predicted by higher-dimension theories has motivated many investigations about the behavior of the gravitational field in microscopic domains [14-21]. It is worthy of mention that, concerning the proton radius puzzle [23-27], there are conjectures on the possibility that higher-dimensional gravity could be an explanation for this issue [28].</text>
<text><location><page_3><loc_12><loc_60><loc_88><loc_75></location>A very common way of expressing modifications of gravity is through the so-called Yukawa parametrization, in which a Yukawa-like term is added to the Newtonian potential [6]. In this parametrization, the modified gravitational potential is written as ϕ = -GM/r (1 + α exp( -r/λ )), where the dimensionless parameter α measures the amplification of the interaction strength and λ determines the length scale where the modifications are significant.</text>
<text><location><page_3><loc_12><loc_44><loc_88><loc_59></location>The Yukawa parametrization is very useful because it can account for deviations of Newtonian gravity which may have different physics origins. As we have mentioned above, hidden dimensions is a possible cause, however, there are extensions of the standard model of particle physics that predict the existence of additional bosons that indirectly could interfere in the inverse square law of gravity in certain domains [29, 30]. Some F(R)-theories make similar predictions too [31].</text>
<text><location><page_3><loc_12><loc_23><loc_88><loc_43></location>Facing these theoretical possibilities, here we are interested in using recent spectroscopic data of the hydrogen in order to searching for modifications of the gravitational field on the atomic scale. One of the tightest constraints for the parameter α , derived from the atomic spectroscopy, is imposed by measurements of transition frequencies of the antiprotonic Helium [6, 32, 33], by exploring the gravitational interaction between the antiproton and the Helium nucleus. In section II, we shall see that new data of the 1 S -3 S transition in the hydrogen establishes empiric bounds on deviations of the Newtonian potential that are slightly stronger than those obtained from that exotic atom.</text>
<text><location><page_3><loc_12><loc_7><loc_88><loc_22></location>Besides spectroscopic constraints, other independent bounds for hypothetical short-range interactions around the Angstrom length scale can be inferred from experiments that examine different physics phenomena such as neutron scattering [34], for instance. Another interesting example is the MTV-G experiment [35, 36] that intends to test the existence of a strong gravitational field produced by atomic nuclei by measuring, with a Mott polarimeter, the spin precession of an electron in a scattering process with a heavy nucleus. Preliminary</text>
<text><location><page_4><loc_12><loc_87><loc_88><loc_91></location>results were obtained by treating the geodetic precession of the spin from a classical point of view [6, 35, 36].</text>
<text><location><page_4><loc_12><loc_52><loc_88><loc_85></location>Here, inspired by the MTV-G experiment, we intend to inspect gravitational effects on the spin precession of an electron which is found in an atomic bound state, adopting the quantum perspective. As it is known, in the Hamiltonian formalism, the geodetic spin precession is dictated by the gravitational spin-orbit coupling [37, 38]. In section III, considering the Dirac equation in the curved spacetime, we discuss the influence of the gravitational spinorbit coupling in the fine splitting between the states 2 P 1 / 2 and 2 P 3 / 2 . As we shall see, this analysis does not put empirical limits on deviations of the Newtonian potential only, but actually it allows us to investigate the influence of a Post-Newtonian potential associated with spatial components of the metric in atomic length scale. This new potential is related to a specific parameter of the PPN-formalism (parametrized Post-Newtonian formalism) [39], whose geometrical meaning is connected to the curvature of the sections t = const . The constraints obtained here are stronger than those derived from the MTV-G scattering experiment [6, 35].</text>
<section_header_level_1><location><page_4><loc_12><loc_47><loc_50><loc_48></location>II. SPECTROSCOPIC CONSTRAINTS</section_header_level_1>
<text><location><page_4><loc_12><loc_18><loc_88><loc_44></location>In a certain range of length close to the Angstrom scale, the strongest empirical constraint on deviations of the gravitational field, imposed by atomic spectroscopy, as far as we know, comes from the analysis of transitions in the antiprotonic Helium (¯ pHe + ) [6, 32, 33]. This exotic atom is formed in laboratory by replacing one of the two electrons of a natural Helium by an antiproton [40]. With this change, the Coulombian interaction between the particles and the nucleus is not directly altered, but the gravitational interaction between the antiproton, with mass m p , and the Helium's nucleus becomes almost two thousand times bigger than that between the nucleus and the electron, since m p /similarequal 1836 m , where m is the electron mass. Hence, this exotic atom seems to be a system with adequate features to investigate the behavior of gravity in the atomic domain.</text>
<text><location><page_4><loc_12><loc_8><loc_88><loc_17></location>At first sight, another positive characteristic of this system would be the possibility of probing the gravitational interaction in a range of length that could reach thousandths of the Angstrom, once the relative distance between the nucleus and the antiproton, which depends on the inverse of the antiproton mass, could be much smaller than traditional Bohr</text>
<text><location><page_5><loc_12><loc_79><loc_88><loc_91></location>radius ( a 0 /similarequal 0 , 5 ˚ A). However, regarding this point, there is a downside aspect. In fact, in states in which the antiproton is found very close to the nucleus, the matter-antimatter annihilation process abbreviates the lifetime of the antiprotonic Helium to few picoseconds [41], preventing, therefore, any possibility of studying the spectroscopy of this atom with present technology.</text>
<text><location><page_5><loc_12><loc_44><loc_88><loc_78></location>It happens that, in a fraction of the antiprotonic helium atoms that are synthesized in laboratory, ¯ p is found in Rydberg states with high principal quantum number n and high angular momentum l ∼ n -1 [41]. In states with n ∼ l ∼ 40, the average distance of the antiproton to the nucleus is approximately around a 0 [32] and the overlap of the wave function with the nucleus is drastically suppressed. As a consequence, the lifetime of the antiprotonic helium increases to the order of microseconds [41]. These Rydberg states, with this longer lifetime, are amenable to be investigated by laser spectroscopy and, indeed, the transition frequencies between these metastable states were measured with a relative precision of some parts in 10 -9 [41, 42]. Comparing the experimental data [41, 42] with the theoretical calculations [43, 44], based on the theory of Quantum Electrodynamics (QED), one verifies a precise agreement between them [6, 32, 33]. This result put some bounds on the values that the parameters α and λ could assume. For instance, at 1 σ confidence-level, α < 10 28 for λ ∼ 1 ˚ A [33].</text>
<text><location><page_5><loc_12><loc_7><loc_88><loc_43></location>Here, we investigate possible deviations of the gravitational Newtonian potential in the atomic scale by using new data from hydrogen spectroscopy. The intention is to compare the constraints for the Yukawa parameters determined by the hydrogen spectroscopy with that established by the antiprotonic Helium. Although the gravitational interaction between proton and electron in the hydrogen atom is much weaker than the antiproton-nucleus gravitational interaction in ¯ pHe + , the available spectroscopic data of the hydrogen are much more accurate. For instance, the experimental value of the 1 S -2 S transition frequency, f exp 1 S -2 S = 2466061413187035 Hz, was recently measured with an error of δ exp = 10 Hz, which corresponds to a relative precision of the order of 10 -14 [45]. If the theoretical value f th 1 S -2 S , predicted by QED, had an uncertainty δ th of the same magnitude order, then the agreement between f th 1 S -2 S and f exp 1 S -2 S would impose a much tighter constraint for the Yukawa parameters (see Figure 1). However the experimental value of the 1 S -2 S transition frequency is the most precise value of the data set that is employed to determine the values of certain fundamental spectroscopic constants, such as the Rydberg constant (see Table XVIII of Ref.</text>
<text><location><page_6><loc_12><loc_84><loc_88><loc_91></location>[46]). As the theoretical predictions depend on these constants, the comparison between the calculated value with the measured frequency of this specific transition should be viewed with caution [47].</text>
<text><location><page_6><loc_12><loc_63><loc_88><loc_83></location>Because of this, let us consider the 1 S -3 S transition. The relative precision achieved in the most recent measurement of the frequency of this transition, f exp 1 S -3 S , is of the order of 10 -12 [48]. The theoretical value calculated in Ref. [47] is of the same order. Although f exp 1 S -3 S is not so accurate as f exp 1 S -2 S , the advantage of using that frequency to test QED is the fact that the isolated value of 1 S -3 S transition frequency does not belong to the input data used in the least-squares adjustment of the values of the fundamental constants recommended by CODATA-2002[49], which were employed by Ref. [47] to calculate the frequency of that transition.</text>
<text><location><page_6><loc_14><loc_60><loc_69><loc_62></location>The theoretical [47] and experimental [48] values are respectively:</text>
<formula><location><page_6><loc_35><loc_56><loc_88><loc_58></location>f th 1 S -3 S = 2922743278671 . 6(1 . 4) kHz , (1)</formula>
<formula><location><page_6><loc_35><loc_53><loc_88><loc_55></location>f exp 1 S -3 S = 2922743278671 . 5(2 . 6) kHz . (2)</formula>
<text><location><page_6><loc_12><loc_33><loc_88><loc_50></location>The uncertainty is expressed in parenthesis. Admitting that the theoretical and experimental uncertainties are independent, then the combined error is δf = √ δ 2 th + δ 2 exp /similarequal 3 . 0 kHz. It is clear that the theoretical prediction f th 1 S -3 S agrees very well with the measured frequency f exp 1 S -3 S within the combined error δf . Therefore, any new hypothetical interaction, such as the modified gravitational interaction, should not introduce corrections for the transition frequency in an amount ∆ f greater than the error δf . The condition ∆ f < δf imposes certain empirical limits for the parameters α and λ .</text>
<text><location><page_6><loc_12><loc_9><loc_88><loc_32></location>The supposed correction, ∆ f , that the modified gravity provides for the 1 S -3 S transition can be calculated by using the perturbation method. The new gravitational interaction between the proton and electron is described, in the leading order, by the Hamiltonian H (0) G = m e ϕ , where ϕ = -Gm p /r (1 + α exp( -r/λ )) is the modified gravitational potential produced by the proton. This Hamiltonian H (0) G should be considered as a small term of the atomic Hamiltonian. According to the perturbation formalism, in the first order, this new interaction will decrease the energy of each state ψ by the amount 〈 H (0) G 〉 , the average value of H (0) G in the state ψ . The gravitational interaction will change the energy of the states 1 S and 3 S by different amounts, increasing the energy gap between these states. This implies</text>
<text><location><page_7><loc_12><loc_89><loc_88><loc_91></location>a correction of the transition frequency which, in the first approximation order, is given by:</text>
<formula><location><page_7><loc_38><loc_84><loc_88><loc_88></location>∆ f = 〈 H (0) G 〉 3 S -〈 H (0) G 〉 1 S h . (3)</formula>
<text><location><page_7><loc_12><loc_70><loc_88><loc_80></location>In Figure 1, we show the constraints on the Yukawa parameters imposed by the condition ∆ f < δf. As we can see, the new bounds are slightly stronger than those obtained from the spectroscopy of the atom ¯ pHe + . For λ = 1 ˚ A, for instance, the data demand that α < 1 . 7 × 10 27 at 1 σ confidence-level.</text>
<figure>
<location><page_7><loc_30><loc_50><loc_70><loc_70></location>
<caption>Figure 1. Constraints for deviations of Newtonian potential determined here from the 1 S -3 S transition in the hydrogen ( H 1 S -3 S line) are compared to other bounds imposed by spectroscopy of ddµ + , ¯ pHe + and HD + (data extracted from Ref. [32]). The H 1 S -2 S curve is just a reference line that shows how stringent the spectroscopy of hydrogen could be, considering the current relative empirical precision (10 -14 ) of the 1 S -2 S transition.</caption>
</figure>
<text><location><page_7><loc_12><loc_25><loc_88><loc_35></location>Actually, according to Figure 1, for λ < 0 . 6 ˚ A, the H 1 S -3 S constraints are tighter than several spectroscopic bounds, which also include empirical limits determined by the spectroscopy of ddµ + (an exotic molecule formed by two deuterons and one muon) and also from the HD + (the ionized molecule constituted by a hydrogen and a deuterium).</text>
<text><location><page_7><loc_12><loc_7><loc_88><loc_24></location>The frequency of the 1 S -3 S transition that we use here is one of the most accurate measurements in hydrogen spectroscopy, only surpassed by the 1 S -2 S transition [50]. Based on these two transitions, it is possible to infer the proton charge radius. The value found in Ref. [50] is in agreement with the CODATA-2014 recommended value and differs by 2 . 8 σ from the value extracted from the muonic hydrogen spectroscopy. Given this result, it seems interesting to resort to Rydberg states if we are aiming for a spectroscopic analysis that is less dependent on the proton size [51].</text>
<text><location><page_8><loc_12><loc_60><loc_88><loc_91></location>The effects of hidden dimensions in certain Rydberg states were studied in Ref. [52], by using a power-law parametrization for the modified gravitational potential. More recently, in Ref. [53] (we thank one of the referees to call our attention to this paper) Rydberg states were also considered with the purpose of constraining non-standard interactions by using Yukawa parametrization. As expected, the strongest restrictions are found in a length scale beyond the Bohr radius, since the studied states have a large principal quantum number. Admitting a relative precision of the order of 10 -12 in the energy levels, it was found that the strength of the new interaction should be lesser than 10 28 for λ > 10 -9 m (after converting data of figure 3 of Ref. [53] to units used here), which is very close to our result. They also considered the Rydberg states combined with data from other transitions. In this case, the constraint for α is almost of the order of 10 27 in the range 10 -10 -10 -7 m at a 95% confidence level, which is clearly compatible with our result.</text>
<text><location><page_8><loc_12><loc_42><loc_88><loc_59></location>So far, we have considered only spectroscopic constraints, since the main objective of this work is to use new data from hydrogen spectroscopy to put indenpendent constraints on deviations of standard gravity. However, it is also interesting to compare these constraints with empirical limits imposed by sources of different natures. Figure 2, in addition to the spectroscopic constraints, includes other empirical limits set by data with distinct origins such as particle colliders, Casimir effect, torsion balance and Lunar Laser Ranging experiment, among others.</text>
<figure>
<location><page_8><loc_30><loc_21><loc_70><loc_41></location>
<caption>Figure 2. In this figure, the spectroscopic constraints for α determined here (H 1 S -3 S line) are compared to empirical limits established by data from different origins (see Ref. [6] for collider data and Ref. [32] for all other data).</caption>
</figure>
<text><location><page_8><loc_12><loc_7><loc_88><loc_11></location>Almost all data in Figure 2 were extracted from Ref. [32], except for the collider data, which are obtained from Figure 1 of Ref. [6], and the line H 3 S -1 S which was determined</text>
<text><location><page_9><loc_12><loc_81><loc_88><loc_91></location>here. Each distinct bound stands out on different length scale. As we can see, in this more general context, the H 1 S -3 S constraint is surpassed by the collisor bound for short λ and by HD + limit for λ > 0 . 6 ˚ A. It is also important to remark that constraints from neutron scattering, claimed in Ref. [34], are stronger than spectroscopic limit.</text>
<section_header_level_1><location><page_9><loc_12><loc_76><loc_59><loc_77></location>III. GRAVITATIONAL SPIN-ORBIT COUPLING</section_header_level_1>
<text><location><page_9><loc_12><loc_63><loc_88><loc_73></location>In a curved space, the final direction of a vector that is parallel transported along a closed path may not coincide with its initial direction. Based on this fact, it is expected, in accordance with the theory of General Relativity, that when a particle moves in a gravitational field its spin will precess [54].</text>
<text><location><page_9><loc_12><loc_55><loc_88><loc_62></location>This effect in the classical regime was directly verified by some experiments such as GRAVITY PROBE B [55], which measured the orientation changes of a gyroscope's axis moving along a geodesic around the Earth.</text>
<text><location><page_9><loc_12><loc_40><loc_88><loc_54></location>In the microscopic domain, the precession of an electron's spin induced by the gravitational field of an atomic nucleus was preliminarily investigated in the so-called MTV-G experiment [35, 36]. This experiment tests the existence of a strong gravitational field in the nuclear domain by studying the influence of a modified gravity on the spin precession of an electron that is scattered by a nucleus. Following a classical description of the geodetic spin precession of the electron, some preliminary results were obtained [6, 35, 36].</text>
<text><location><page_9><loc_12><loc_32><loc_88><loc_38></location>However, as the spin of an elementary particle is a quantum quantity, it is more appropriate to treat the spin precession phenomena induced by a gravitational field in the quantum mechanics formalism.</text>
<text><location><page_9><loc_12><loc_16><loc_88><loc_31></location>For this purpose, let us consider the Dirac equation in curved spacetime. The coupling between fermions and the gravitational field is implemented through the tetrad fields e µ ˆ A , which consists of components of four orthonormal vector fields (each one is identified by the index ˆ A = 0 , 1 , 2 or 3) written in some coordinate system ( x µ ) (here, µ = 0 , 1 , 2 and 3). The tetrad fields satisfy the orthonormality condition g µν e µ ˆ A e v ˆ B = η ˆ A ˆ B , where g µν denotes the metric of the spacetime and η AB is the Minkowski metric.</text>
<text><location><page_9><loc_12><loc_8><loc_88><loc_15></location>With the help of the tetrad fields and the Dirac matrices defined on the Minkowski spacetime, γ ˆ A , we can construct the matrices γ µ ( x ) = e µ ˆ A ( x ) γ ˆ A , which satisfy the anticommutation algebra of the Dirac matrices adapted to the curved space: { γ µ ( x ) , γ ν ( x ) } =</text>
<text><location><page_10><loc_12><loc_89><loc_19><loc_91></location>2 g µν ( x ).</text>
<text><location><page_10><loc_12><loc_84><loc_88><loc_88></location>The Dirac equation that describes the state ψ of a particle of mass m in a gravitational field is given by</text>
<formula><location><page_10><loc_38><loc_81><loc_88><loc_83></location>[ iγ µ ( x ) ∇ µ -mc/ /planckover2pi1 ] ψ ( x ) = 0 , (4)</formula>
<text><location><page_10><loc_12><loc_75><loc_88><loc_79></location>where c is the speed of light, /planckover2pi1 is the reduced Planck constant and ∇ µ is the covariant derivative of the spinor ψ , which depends on the spinorial connection Γ µ ( x ) as follows:</text>
<formula><location><page_10><loc_37><loc_71><loc_88><loc_73></location>∇ µ ψ ( x ) = [ ∂ µ +Γ µ ( x )] ψ ( x ) . (5)</formula>
<text><location><page_10><loc_12><loc_62><loc_88><loc_69></location>Admitting the compatibility between the spinorial connection and the metric, it is possible to write Γ µ ( x ) in terms of the Levi-Civita covariant derivative of the tetrad fields, e v ˆ A ; µ ( x ), according to the expression:</text>
<formula><location><page_10><loc_35><loc_57><loc_88><loc_60></location>Γ µ ( x ) = -i 4 σ ˆ A ˆ B g αν e α ˆ A ( x ) e ν ˆ B ; µ ( x ) , (6)</formula>
<text><location><page_10><loc_12><loc_51><loc_88><loc_55></location>where σ ˆ A ˆ B = i 2 [ γ ˆ A , γ ˆ B ] is a representation of the Lorentz Lie Algebra in the spinor space, written in terms of the commutating operator [ , ].</text>
<text><location><page_10><loc_12><loc_37><loc_88><loc_49></location>From the Dirac equation, we can study the influence of the gravitational field produced by the proton on the behavior of an electron in the hydrogen atom. In the first approximation approach, it is reasonable to assume that the proton produces a static gravitational field with spherical symmetry. Under this condition, the spacetime metric can be put in the following form:</text>
<formula><location><page_10><loc_34><loc_35><loc_88><loc_36></location>ds 2 = -c 2 w 2 dt 2 + v 2 ( dx 2 + dy 2 + dz 2 ) , (7)</formula>
<text><location><page_10><loc_12><loc_29><loc_88><loc_33></location>in the isotropic coordinates. The functions w and v depend only on the coordinate r = ( x 2 + y 2 + z 2 ) 1 / 2 . Associated to this metric, the non-null tetrad fields components are:</text>
<formula><location><page_10><loc_44><loc_24><loc_88><loc_26></location>e 0 ˆ 0 ( x ) = w -1 , (8)</formula>
<formula><location><page_10><loc_44><loc_21><loc_88><loc_23></location>e i ˆ ( x ) = δ i j v -1 . (9)</formula>
<text><location><page_10><loc_12><loc_15><loc_88><loc_19></location>In the weak-field regime the function v and w can be expressed in terms of gravitational potentials as:</text>
<formula><location><page_10><loc_44><loc_11><loc_88><loc_12></location>w = 1 + ϕ/c 2 , (10)</formula>
<formula><location><page_10><loc_44><loc_8><loc_88><loc_9></location>v = 1 -˜ ϕ/c 2 . (11)</formula>
<text><location><page_11><loc_12><loc_79><loc_88><loc_91></location>According to the General Relativity theory, ϕ = ˜ ϕ . However, as we are investigating modifications of the gravitational field in the atomic domain, let us admit that ϕ and ˜ ϕ can be different functions, or more precisely, that the Yukawa parameter (˜ α ) associated to the potential ˜ ϕ is not necessarily equal to α (the parameter investigated in the previous section) and should be determined experimentally.</text>
<text><location><page_11><loc_12><loc_63><loc_88><loc_78></location>The potential ˜ ϕ is directly related to the curvature of the spatial section ( t = const. ) of the spacetime according to the geometric viewpoint. The possibility that ˜ ϕ is not necessarily equal to the Newtonian potential is also embodied in the parametrized post-Newtonian (PPN) formalism, which is a theoretical framework properly developed for the purpose of testing metric theories, such as the General Relativity and Brans-Dicke theory, in the weakfield limit [39].</text>
<text><location><page_11><loc_12><loc_52><loc_88><loc_62></location>The parameter ˜ α that we are investigating here can be put in correspondence with a certain PPN-parameter. Considering the Yukawa parametrization of the potential, we can see that ˜ ϕ = GM/r (1+˜ α ), in the limit r << λ . Thus, we can conclude that the combination (˜ α +1) plays the role of the parameter γ of the PPN-formalism [39].</text>
<text><location><page_11><loc_12><loc_36><loc_88><loc_51></location>In the astrophysics domain, empirical bounds for the PPN-parameter γ are extracted from time-delay and light deflection experiments, for example. Recent experiments were performed with the help of Cassini spacecraft and find that γ = 1 + (2 . 1 ± 2 . 3) × 10 -5 [56], by studying the behavior of radio waves under the influence of the gravitational field of the Sun. This constraint is valid in the length scale of the solar radius and is compatible with the value predicted by the theory of General Relativity ( γ = 1).</text>
<text><location><page_11><loc_12><loc_29><loc_88><loc_35></location>As we shall see later in this section, this Post-Newtonian parameter can be investigated in the atomic domain through the study of the influence of the gravitational spin-orbit coupling on the fine structure of the atom.</text>
<text><location><page_11><loc_12><loc_15><loc_88><loc_27></location>To show this, let us turn our attention to the Dirac equation again, now using the tetrad fields given above (Eqs. (8) and (9) ). The Dirac equation can be rewritten in the form i /planckover2pi1 ∂ψ ∂t = H G ψ , where H G is the operator that contains the gravitational sector of the atomic Hamiltonian. In a convenient representation, H G assumes the following form in the first order of the gravitational potentials [37, 38, 57]:</text>
<formula><location><page_11><loc_29><loc_9><loc_88><loc_14></location>H G = βmc 2 + βϕm + 1 2 { /vector α · /vector p, [1 + ( ϕ + ˜ ϕ ) /c 2 ] } , (12)</formula>
<text><location><page_11><loc_12><loc_7><loc_88><loc_9></location>where /vector p is the usual three-dimensional momentum operator in flat spacetime, α i = γ 0 γ i and</text>
<text><location><page_12><loc_12><loc_74><loc_88><loc_91></location>β = γ 0 . The first term of H G is related to the rest energy of the electron, the second term is associated with the usual potential energy of the proton-electron gravitational interaction ( H (0) G ), which was examined in the last section, and the third one gives rise to the kinetic term and also to relativistic and quantum corrections. In the form (12), it is important to stress that H G is Hermitian in the Hilbert space of square-integrable functions endowed with the usual flat inner product ( ∫ d 3 x ).</text>
<text><location><page_12><loc_12><loc_60><loc_88><loc_75></location>Admitting that the rest energy of the test particle is the leading term of the Hamiltonian, a semi-relativistic expansion of H G (12) can be obtained by following the Foldy-Wouthuysen procedure, which consists in the elimination of odd operators of H G , in each order of 1 /mc 2 , by a convenient sequence of unitary transformations of the Hamiltonian [58]. Among many terms that arise in the expansion, here we want to focus our attention on the Hamiltonian term associated with the gravitational spin-orbit coupling, which can be expressed as [37, 38]:</text>
<formula><location><page_12><loc_35><loc_55><loc_88><loc_59></location>H Gso = 1 mc 2 1 r ( 1 2 dϕ dr + d ˜ ϕ dr ) ( /vector S · /vector L ) , (13)</formula>
<text><location><page_12><loc_12><loc_49><loc_88><loc_53></location>where /vector L is the orbital angular momentum of the particle and /vector S is the spin operator, which can be written in terms of the Pauli matrices in the form /vector S = ( /planckover2pi1 / 2) /vectorσ .</text>
<text><location><page_12><loc_12><loc_38><loc_88><loc_48></location>As we have already mentioned, this term (13), in the classical regime, is responsible for the geodetic precession of the axis of a gyroscope in curved spacetime [37, 38, 54]. Considering /vector S as a classical angular momentum measured by a co-moving geodesic observer, it can be shown [37, 54] that the spin dynamics is governed by the equation:</text>
<formula><location><page_12><loc_37><loc_33><loc_88><loc_37></location>d /vector S dt = [( 1 2 + γ ) GM mc 2 r 3 /vector L ] × /vector S, (14)</formula>
<text><location><page_12><loc_12><loc_30><loc_41><loc_31></location>assuming that ϕ = ˜ ϕ/γ = -GM/r .</text>
<text><location><page_12><loc_12><loc_22><loc_88><loc_29></location>The analysis of the spin precession in the MTV-G experiment was based on the equation (14) taking γ = 1 [35, 36]. Therefore, without making any distinction between the Newtonian and the post-Newtonian potentials.</text>
<text><location><page_12><loc_12><loc_9><loc_88><loc_21></location>In the present work, we want to study the effect of the gravitational spin-orbit coupling on the energy levels of the hydrogen. In the atom, the electron is not a free-falling particle, but it is found in a bound state due to the electromagnetic interaction with the proton. This dominant interaction establishes a spin-orbit coupling too, described by a Hamiltonian that can be put in the form [58]:</text>
<formula><location><page_13><loc_38><loc_85><loc_88><loc_89></location>H Eso = -q 2 m 2 c 2 1 r dφ E dr ( /vector S · /vector L ) , (15)</formula>
<text><location><page_13><loc_12><loc_70><loc_88><loc_85></location>where -q is the electron charge and the function φ E is the electric potential produced by the proton in the curved space. In a spacetime with the metric (7), the electric field equations can be written in the same form of the Maxwell equations defined in a flat space endowed with a new electric permittivity given by ε = ε 0 v/w , where ε 0 is the permittivity of free space [38, 52]. Thus, if the electrostatic potential has spherical symmetry, it satisfies, in the first-order approximation, the following equation:</text>
<formula><location><page_13><loc_35><loc_64><loc_88><loc_68></location>dφ E dr = -q 4 πε 0 r 2 ( 1 + ϕ/c 2 + ˜ ϕ/c 2 ) . (16)</formula>
<text><location><page_13><loc_12><loc_51><loc_88><loc_63></location>As we can see, the spacetime curvature changes the electric potential. Through this correction of φ E , the gravitational field can indirectly influence the atomic spin-orbit coupling from Hamiltonian H Eso . However, as we show in appendix, this contribution is five magnitude orders (10 -5 ) lesser than the direct contribution coming from (13). So, for our purposes, the indirect gravitational contribution can be ignored hereafter.</text>
<text><location><page_13><loc_12><loc_40><loc_88><loc_50></location>It is well known that the electromagnetic spin-orbit coupling is responsible for a fine splitting of energy levels with the same angular momentum l , such as the 2 P 1 / 2 and 2 P 3 / 2 states, for example. The gravitational analogous (13), considered here as a weaker interaction in comparison to the Hamiltonian H Eso , will provide an additional shift in those states.</text>
<text><location><page_13><loc_12><loc_30><loc_88><loc_39></location>There are precise theoretical calculations of the energy of the hydrogen states, based on the QED theory. In Ref. [46], aiming to test the QED predictions, it was explicitly determined the transitions frequency between levels with n = 2 in hydrogen. Specifically, the frequency transition between 2 P 1 / 2 and 2 P 3 / 2 is (pp. 1540 of Ref. [46]):</text>
<formula><location><page_13><loc_35><loc_25><loc_88><loc_28></location>f th 2 P 1 / 2 -2 P 3 / 2 = 10969041 . 571(41) kHz . (17)</formula>
<text><location><page_13><loc_12><loc_14><loc_88><loc_23></location>This calculation is based on the recommended values for the fundamental constants which are extracted from the input data of CODATA-2010, but excluding the experimental values of 2 S 1 / 2 -2 P 1 / 2 and 2 P 3 / 2 -2 S 1 / 2 transition frequencies, which corresponds to the items A39, A40.1, and A40.2 in Table XVIII of CODATA-2010 [46].</text>
<text><location><page_13><loc_12><loc_8><loc_88><loc_13></location>On its turn, there are measurements of the centroid transition frequencies between 2 S 1 / 2 -2 P 1 / 2 [59] and 2 P 3 / 2 -2 S 1 / 2 [60]. Based on these experimental values, from which</text>
<text><location><page_14><loc_12><loc_87><loc_88><loc_91></location>the hyperfine structure of the states has already been excluded, we can determine the experimental frequency of the transition between 2 P 1 / 2 and 2 P 3 / 2 :</text>
<formula><location><page_14><loc_36><loc_82><loc_88><loc_84></location>f exp 2 P 1 / 2 -2 P 3 / 2 = 10969045(15) kHz . (18)</formula>
<text><location><page_14><loc_12><loc_73><loc_88><loc_80></location>Of course, the theoretical and experimental values coincide within the combined error δf = 15 kHz. So the contribution provided by the gravitational spin-orbit interaction, ∆ f so , cannot exceed this empirical error.</text>
<text><location><page_14><loc_12><loc_52><loc_88><loc_72></location>Now, in order to estimate ∆ f so , let us remember that, due to the spin-orbit interaction, the operators /vector L and /vector S no longer commute with the atomic Hamiltonian. On the other hand, the total angular momentum /vector J = /vector L + /vector S is a conserved quantity. Therefore, atomic stationary states are labeled by the eigenvalues of the total angular momentum. When the orbital angular momentum and the spin are, let us say, aligned, the total momentum is j = l + 1 / 2 and, in this case, H Gso will provide a positive energy shift for the state. On the other hand, for states with j = l -1 / 2, the spin-orbit interaction will give a negative contribution to the energy.</text>
<text><location><page_14><loc_12><loc_44><loc_88><loc_51></location>These effects increase the energy gap between those states. In the first approximation, the additional separation between states ( n, l, j = l + 1 / 2) and ( n, l, j = l -1 / 2) due to gravitational spin-orbit interaction leads to the following change in the frequency transition:</text>
<formula><location><page_14><loc_26><loc_39><loc_88><loc_43></location>∆ f so = ∆ E Gso h = 1 hmc 2 〈 1 r d dr ( ϕ/ 2 + ˜ ϕ ) 〉 n,l ( l +1 / 2) /planckover2pi1 2 , (19)</formula>
<text><location><page_14><loc_12><loc_31><loc_88><loc_37></location>where the average 〈〉 n,l is calculated with respect to the radial solution of the Schrodinger equation, R n,l ( r ), for states with a principal quantum number n and orbital angular momentum l .</text>
<text><location><page_14><loc_12><loc_17><loc_88><loc_29></location>Now let us discuss the implications of the condition ∆ f so < δf in the case of the 2 P 1 / 2 -2 P 3 / 2 transition, for which R 21 = ( 1 / √ 24 a 3 o ) ( r/a o ) e -r/ 2 a 0 . First, let us emphasize that in the previous section, we have tested modifications of the Newtonian potential, but not the potential ˜ ϕ , since, in the 1 S -3 S transition, the curvature of the spatial sections has no influence on the energy of the S -states in the leading order.</text>
<text><location><page_14><loc_12><loc_7><loc_88><loc_16></location>The expression (19) indicates that the effects of the gravitational spin-orbit coupling on the spectroscopy depend on a linear combination of the Newtonian potential and the potential ˜ ϕ . Therefore, the spectroscopic analysis will put empirical bounds on the mixed parameter ( α/ 2 + ˜ α ). However, taking into account the empirical bounds for the parameter</text>
<text><location><page_15><loc_12><loc_79><loc_88><loc_91></location>α previously established, we can verify that the contribution of the potential ϕ in the transition 2 P 1 / 2 -2 P 3 / 2 is smaller than the experimental error. Therefore, for practical purposes, we can neglect the potential ϕ in the expression (19). Thus, we can conclude that the analysis of the influence of the gravitational spin-orbit interaction on the fine structure of the states is actually a test of the Post-Newtonian potential ˜ ϕ in the atomic domain.</text>
<text><location><page_15><loc_12><loc_65><loc_88><loc_78></location>In Figure 3, we show the constraints for the mixed parameter ( α/ 2 + ˜ α ) in terms of λ . As we can see, for λ > 1 . 5 × 10 -3 ˚ A, the bounds determined by the 2 P 1 / 2 -2 P 3 / 2 transition are more stringent than the empirical limits put by MTV-G experiment. In particular, for λ = 1 ˚ A, we find ˜ α < 2 . 1 × 10 33 , which is stronger than the MTV-G constraint by four magnitude orders.</text>
<figure>
<location><page_15><loc_31><loc_44><loc_70><loc_64></location>
<caption>Figure 3. The dashed line is extracted from the analysis of the influence of the gravitational spin-orbit coupling on the separation between the states 2 P 1 / 2 and 2 P 3 / 2 of hydrogen. It sets an empirical constraint for the mixed parameter ( α/ 2 + ˜ α ) , where ˜ α is related to the γ -parameter in the PPN-formalism. For λ > 1 . 5 × 10 -3 ˚ A, the 2 P 1 / 2 -2 P 3 / 2 constraint is stronger than the empirical bounds extracted from the MTV-G scaterring experiment[36] .</caption>
</figure>
<text><location><page_15><loc_12><loc_11><loc_88><loc_28></location>The analysis based on the influence of the gravitational spin-orbit coupling on the 2 P 1 / 2 -2 P 3 / 2 transition provides a weaker limit compared to the bounds determined from the 1 S -3 S transition. However, we should have in mind that these two tests probe different physical quantities. In fact, the 1 S -3 S transition, as well as all tests described in the Figure 2, sets upper limits on deviation of the Newtonian potential, while the 2 P 1 / 2 -2 P 3 / 2 actually allows us to constrain the behavior of a post-Newtonian potential, which has a proper geometric meaning and it is not necessarily equal to the Newtonian potential in some metric theories.</text>
<section_header_level_1><location><page_16><loc_12><loc_89><loc_34><loc_91></location>IV. FINAL REMARKS</section_header_level_1>
<text><location><page_16><loc_12><loc_66><loc_88><loc_86></location>Considering accurate data of hydrogen spectroscopy (more specifically, recent measurement of the 1 S -3 S transition frequency [48]) we find constraints for short-range modifications of the Newtonian potential in the atomic domain. The bounds obtained here are tighter than several empirical limits imposed by the spectroscopy of some exotic atom such as the antiprotonic-Helium. Although the interaction between electron and proton is almost two thousand times weaker compared to antiproton-nucleus gravitational interaction, we have seen that the accuracy achieved in the hydrogen spectroscopy is high enough to determine an improved constraint for deviations of the Newtonian potential in range λ < 0 . 6 ˚ A.</text>
<text><location><page_16><loc_12><loc_37><loc_88><loc_65></location>In the present discussion, we have adopted the comprehensive Yukawa parametrization to express modifications of gravity. The reason is that several models, such as large extra dimension models [61] and some F(R)-theories [31], predict an exponentially decreasing correction of the Newtonian potential in a domain beyond a certain length scale. Although the Yukawa parametrization is useful, it has some limitations as any approximation scheme[22]. Indeed, as we are exploring atomic spectroscopy data, the best results are found when the supposed deviations occur around the Bohr radius ( a 0 ). However, if the modifications are significant in a length scale much lesser than a 0 , then the Yukawa parametrization will not be capable to appropriately capture their effects. In this case, each model should be considered separately and studied in detail. In Ref. [20], for instance, the power-law parametrization is adopted to study the ADD model in thick branes scenarios.</text>
<text><location><page_16><loc_12><loc_19><loc_88><loc_36></location>In this paper, we have also investigated non-standard gravitational effects on the fine separation between the 2 P 3 / 2 and 2 P 1 / 2 states. The energy difference between these states, which have the same principal quantum number and the same orbital angular momentum, is mainly determined by the spin-orbit coupling. This interaction, which is responsible for the spin precession phenomena, was explored by MTV-G experiment in order to investigate a possible strong gravitational field produced by the nucleus, by measuring the spin precession of an electron in a scattering process with a Mott polarimeter.</text>
<text><location><page_16><loc_12><loc_8><loc_88><loc_18></location>Inspired by the MTV-G experiment, we have discussed the effects of gravitational spinorbit coupling on the electron in a bound state. From the Dirac equation, we study the influence of that interaction on the fine structure of the 2 P -state of the hydrogen. We have shown that the empirical constraints determined by this analysis are numerically weaker</text>
<text><location><page_17><loc_12><loc_79><loc_88><loc_91></location>than those put by the 3 S -1 S transition, however, it is important to emphasize that, as the separation between 2 P 3 / 2 and 2 P 1 / 2 states has essentially a relativistic origin, then the analysis of the 2 P 3 / 2 -2 P 1 / 2 transition yields not a test of Newtonian potential, but, actually, it is a test of a post-Newtonian potential, which is related to the γ -parameter in the PPN-formalism.</text>
<text><location><page_17><loc_12><loc_71><loc_88><loc_78></location>We have seen that, for λ > 1 . 5 × 10 -3 ˚ A, the empirical limits on the post-Newtonian potential determined by the spectroscopic data are stronger than those extracted from the MTV-G scattering experiment.</text>
<text><location><page_17><loc_12><loc_60><loc_88><loc_70></location>At this point, we would like to mention that in a recent measurement of 2 S -4 P transition frequency, the splitting between the states 4 P 3 / 2 and 4 P 1 / 2 was determined with a precision of 4 . 3 kHz [62]. In principle, this more accurate measurement could be used to set a better constraint for the post-Newtonian potential.</text>
<text><location><page_17><loc_12><loc_50><loc_88><loc_59></location>Another perspective to test modifications of gravity in the atomic length scale is to consider the spectroscopy of heavier atoms or ions, especially that of the element He whose transitions are being measured with a relative precision of the order of 10 -12 [63], and, for this reason, are being used to probe new physics in the microscopic domain [64].</text>
<text><location><page_17><loc_12><loc_34><loc_88><loc_49></location>Finally, we would like to highlight that a supposed modified gravitational interaction would affect the isotope shift of atomic transitions. Therefore, taking advantage of the well-studied theoretical framework of isotope shift spectroscopy as well as the precision measurements of the correspondent frequencies [65], we could, from this complementary method, set new constraints for the strength of non-standard gravitational interactions in the atomic domain [65, 66].</text>
<section_header_level_1><location><page_17><loc_12><loc_28><loc_27><loc_30></location>V. APPENDIX</section_header_level_1>
<text><location><page_17><loc_12><loc_18><loc_88><loc_25></location>Here we compare the magnitude order of the indirect contribution of the gravitational field through the Hamiltonian H Eso (15) and the direct contribution given by H Gso to the fine structure of P -states, as discussed in section III.</text>
<text><location><page_17><loc_12><loc_13><loc_88><loc_17></location>Substituting (16) in (15), we may verify that the gravitational correction of H Eso is given by:</text>
<formula><location><page_17><loc_32><loc_9><loc_88><loc_13></location>H ( G ) Eso = q 2 8 πε 0 m 2 c 2 r 3 ( ϕ/c 2 + ˜ ϕ/c 2 ) ( /vector S · /vector L ) . (20)</formula>
<text><location><page_17><loc_12><loc_7><loc_88><loc_9></location>Now let us write the potential ϕ explicitly in terms of r in both Hamiltonians H Gso and</text>
<text><location><page_18><loc_12><loc_86><loc_88><loc_91></location>H ( G ) Eso . Considering just the Yukawa-term, since the standard part is negligible, we obtain the following expressions in magnitude order:</text>
<formula><location><page_18><loc_33><loc_77><loc_67><loc_85></location>H ( G ) Eso ˜ q 2 8 πε 0 m 2 c 4 αGM r 4 e -r/λ ( /vector S · /vector L ) , H Gso ∼ 1 mc 2 ( 1 + r λ ) αGM r 3 e -r/λ ( /vector S · /vector L ) .</formula>
<text><location><page_18><loc_12><loc_72><loc_88><loc_76></location>For the sake of simplicity, here we have assumed α ∼ ˜ α . Now taking the average of the Hamiltonians in the 2P-state, for instance, we obtain the following relation:</text>
<formula><location><page_18><loc_32><loc_67><loc_67><loc_71></location>〈 H ( G ) Eso 〉 ∼ ( q 2 / 4 πε 0 a 0 ) mc 2 ( λ λ + a 0 ) 〈 H Gso 〉 .</formula>
<text><location><page_18><loc_12><loc_59><loc_88><loc_66></location>Notice that the coefficient is proportional to the ratio between the energy of the hydrogen ground state and the rest energy of the electron, which is of the order of 10 -5 . It also depends on a certain relation between λ and a 0 , which is lesser than 1 for all value of λ .</text>
<text><location><page_18><loc_12><loc_55><loc_73><loc_56></location>Acknowledgement 1 G. C. Luna thanks CAPES for financial support.</text>
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2015MNRAS.446..330W | https://arxiv.org/pdf/1410.2256.pdf | <document>
<section_header_level_1><location><page_1><loc_9><loc_83><loc_85><loc_88></location>Galactic rotation curves, the baryon-to-dark-halo-mass relation and space -time scale invariance</section_header_level_1>
<section_header_level_1><location><page_1><loc_9><loc_78><loc_39><loc_80></location>Xufen Wu 1 , ∗ , Pavel Kroupa 2 , ∗</section_header_level_1>
<unordered_list>
<list_item><location><page_1><loc_10><loc_77><loc_80><loc_78></location>1 Deparment of Astronomy, University of Science and Technology of China, Jinzai Road 96, 230026, Hefei, China</list_item>
<list_item><location><page_1><loc_10><loc_76><loc_78><loc_77></location>2 Helmholtz-Institut fur Strahlen-und Kernphysik, Universitat Bonn, Nussallee 14-16, D-53115 Bonn, Germany</list_item>
<list_item><location><page_1><loc_10><loc_75><loc_45><loc_76></location>∗ Email: [email protected]; [email protected]</list_item>
</unordered_list>
<text><location><page_1><loc_9><loc_69><loc_19><loc_70></location>27 October 2014</text>
<section_header_level_1><location><page_1><loc_31><loc_65><loc_41><loc_66></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_31><loc_36><loc_92><loc_65></location>Low-acceleration space -time scale invariant dynamics (SID, Milgrom 2009a) predicts two fundamental correlations known from observational galactic dynamics: the baryonic Tully-Fisher relation (BTFR) and a correlation between the observed mass discrepancy and acceleration (MDA) in the low acceleration regime for disc galaxies. SID corresponds to the deep MOdified Newtonian Dynammics (MOND) limit. The MDA data emerging in cold/warm dark matter (C/WDM) cosmological simulations disagree significantly with the tight MDA correlation of the observed galaxies. Therefore, the most modern simulated disc galaxies, which are delicately selected to have a quiet merging history in a standard dark-matter-cosmological model, still do not represent the correct rotation curves. Also, the observed tight correlation contradicts the postulated stochastic formation of galaxies in low-mass DM haloes. Moreover, we find that SID predicts a baryonic to apparent virial halo (dark matter) mass relation which agrees well with the correlation deduced observationally assuming Newtonian dynamics to be valid, while the baryonic to halo mass relation predicted from CDM models does not. The distribution of the observed ratios of dark-matter halo masses to baryonic masses may be empirical evidence for the external field effect, which is predicted in SID as a consequence of the forces acting between two galaxies depending on the position and mass of a third galaxy. Applying the external field effect, we predict the masses of galaxies in the proximity of the dwarf galaxies in the Miller et al. sample. Classical non-relativistic gravitational dynamics is thus best described as being Milgromian, rather than Newtonian.</text>
<text><location><page_1><loc_31><loc_33><loc_92><loc_35></location>Key words: gravitation -galaxies: general -galaxies: stellar content -galaxies: kinematics and dynamics</text>
<section_header_level_1><location><page_1><loc_9><loc_27><loc_26><loc_28></location>1 INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_5><loc_49><loc_25></location>The currently widely accepted understanding of gravity is based entirely on the empirical law derived by Newton. Newton based his derivation on a number of observations, the central ones being the laws of planetary motion proposed by Kepler. Einstein (1916) revolutionized our concept of gravitation as being not a force but an effect due to space -time curvature. Einstein's field equation is in turn based on Newton's law derived on the Solar System scale: at the time of Einstein's proposal in 1916, galaxies had not been discovered to be what we know them to be nowadays. Thus, when applied to galaxies and cosmological scales, the Einsteinian/Newtonian law of gravity constitutes an extrapolation by many orders of magnitude in spatial and acceleration scale beyond the gravitational systems known in 1916. The observation that the rotation curves of galaxies</text>
<text><location><page_1><loc_52><loc_6><loc_92><loc_28></location>deviate from the expected Keplerian decline by being essentially flat at large radii, r (Rubin & Ford 1970; Bosma 1981; Rubin et al. 1982), i.e. that the circular velocities of galaxies obey v circ ≈ constant, is therefore a very major discovery. This behavior violates Newton's empirical law of gravity that predicts a Keplerian fall off of the circular velocity, v circ ∝ r -1 / 2 , in the outer regimes of galaxies. It is not entirely surprising that Einsteinian/Newtonian gravity breaks down at those scales. However, today the popular interpretation of this discrepancy is to assume Newtonian/Einsteinian gravitation to be valid and to postulate the existence of cold (C) or warm (W) dark matter (DM) particles which make up the mass-discrepancy when rotation curves are interpreted in terms of Einsteinian/Newtonian gravitation. However there exists no experimental evidence for the existence of additional (e.g., dark matter) particles</text>
<text><location><page_2><loc_9><loc_80><loc_49><loc_91></location>beyond those predicted or contained within the standard model of particle physics despite a highly significant effort for finding them (see e.g. the recent press release from the direct search for dark matter particles with the Large Underground Xenon dark matter detector, Akerib et al. 2013; Kroupa 2014). Therefore it is important to test alternative gravities, amongst which Milgromian dynamics 1 (Milgrom 1983c) is the most promising one.</text>
<text><location><page_2><loc_9><loc_67><loc_49><loc_79></location>On the scale of galaxies, a massdiscrepancy -acceleration (MDA) correlation has been predicted by Milgrom (1983c): there is an exact correlation between the mass discrepancy (i.e., the amount of unseen additional mass needed when interpreting the observed motions within the Newtonian dynamics framework) and the acceleration deduced from the orbits at all radii observed in galaxies. The mass discrepancy, M dyn ( <r ) M b ( <r ) ∝ [ v ( r ) v b ( r ) ] 2 , since it is well known that</text>
<formula><location><page_2><loc_10><loc_62><loc_49><loc_65></location>v ( r ) 2 ≡ g ( r ) r /similarequal GM dyn ( < r ) r , (1)</formula>
<formula><location><page_2><loc_9><loc_59><loc_49><loc_62></location>v b ( r ) 2 ≡ g N ( r ) r /similarequal GM b ( < r ) r , (2)</formula>
<text><location><page_2><loc_9><loc_50><loc_49><loc_58></location>where M dyn ( < r ) is the dynamical mass (i.e., the total Newtonian mass) within r , v ( r ) is the total observed or actual rotation speed at r , v b ( r ) is the rotation speed at r contributed only from the baryons assuming Newtonian dynamics (note that ' /similarequal ' in Eqs. 1-2 becomes an equality when the system is spherically symmetric).</text>
<text><location><page_2><loc_9><loc_30><loc_49><loc_50></location>This MDA correlation has been quantified empirically by Sanders (1990) and more recently by McGaugh (2004) for a sample of 74 disc galaxies. This correlation is extended to the Solar System scale by Famaey & McGaugh (2012, their fig. 4). Trippe (2013) fitted the MDA relation using McGaugh (2004) data in a massive graviton model, which is equivalent to MOND with a simple interpolating function. Scarpa (2006) tested this correlation for a sample of over 1000 pressure-supported systems from globular clusters to rich clusters of galaxies. More precisely, Scarpa (2006) studied the correlation between g and g N instead of the MDA correlation. However, these two correlations, though formally different, are fully equivalent. Tiret & Combes (2009) examined the MDA correlation for a sample of 43 galaxies including early- and late-type galaxies.</text>
<text><location><page_2><loc_9><loc_17><loc_49><loc_29></location>The prediction by Milgrom (1983c) of the MDA correlation and its subsequent empirical confirmation offer detailed tests of theories of galaxy formation and dynamics. One of the major questions addressed with this contribution is whether galaxies simulated in the W/CDM cosmological frameworks also reproduce the observed MDA correlation. This is a timely question to ask, because recently there have been clams that disc galaxies can form in the DM models and that these galaxies also resemble real galaxies.</text>
<text><location><page_2><loc_9><loc_12><loc_49><loc_17></location>Noteworthy is that the deep Milgromian limit (i.e., dynamics in the weak field regime where g /lessmuch a 0 ; here Milgrom's constant a 0 ≈ 1 . 2 × 10 -10 m/s 2 can be derived</text>
<text><location><page_2><loc_9><loc_5><loc_49><loc_10></location>1 Milgromian dynamics, often referred to as Modified Newtonian Dynamics (MOND), is briefly introduced in Appendix A, but here focus is on the deep-MOND or weak-acceleration limit, which is the regime of scale-invariant dynamics (SID, Sec. 2).</text>
<text><location><page_2><loc_52><loc_73><loc_92><loc_91></location>from space -time scale invariance and appears to be a constant of nature) can be described extremely well with lowacceleration space -time scale-invariant dynamics (hereafter SID, Sec. 2), as originally pointed out by Milgrom (2009a). With this contribution we revisit pure SID. It is shown, using simple arguments, how phantom (ie. unreal) dark matter haloes, the BTFR and the MDA correlation emerge naturally within SID for an observer who interprets observations in terms of Newtonian dynamics. Therewith this further affirms the Milgrom's and Bekenstein's extension of effective gravity beyond the 1916 version as being realistic, in contrast to the introduction of C/WDM particles which are speculated to exist outside the standard model of physics.</text>
<text><location><page_2><loc_52><loc_47><loc_92><loc_73></location>The Strong Equivalence Principle (SEP) is violated in SID because of external fields (Milgrom 1983c; Bekenstein & Milgrom 1984). This violation is a direct outcome of the truncation of the phantom dark matter haloes as can be demonstrated nicely in the pure SID regime. This means that in SID, the force acting between two galaxies depends on the position and mass of a third galaxy . The violation of SEP is one of the most fundamental differences between SID/MOND and W/CDM. The violation of the SEP means that SID/MOND cannot be obtained by taking the non-relativistic limit of Einstein's GR, which is known to be built on the basis of SEP. The internal dynamics of a system embedded in an external field depends on both the internal and the external gravitational fields (Milgrom 1983c; Bekenstein & Milgrom 1984). Therefore, it is important to study the effects of the external fields for real galaxies embedded in clusters of galaxies, and of satellite galaxies, to improve our constraints on the validity of SID and thus of Milgromian dynamics.</text>
<text><location><page_2><loc_52><loc_27><loc_92><loc_46></location>SID is discussed in Sec. 2. The MDA correlation is shown to directly and immediately emerge from SID. The connection between SID and MOND is discussed in Appendix A. We then visit in Sec. 3 the standard interpretation of the MDA correlation in terms of C/WDM and consult the most advanced astrophysical models of disc galaxy formation, which claim to explain disc galaxies readily. The baryonic-to-dark-matter-halo mass relation is studied in Sec. 4 for the SID prediction and for the C/WDM simulations, and a comparison to the observations is provided. The external field effect predicts a truncation of the phantom dark matter haloes in SID. This is studied in Sec. 4.3, where possible observational evidence for this important truncation is documented. We conclude with Sec. 5.</text>
<section_header_level_1><location><page_2><loc_52><loc_19><loc_84><loc_23></location>2 SPACE -TIME SCALE INVARIANT DYNAMICS AND THE MDA CORRELATION</section_header_level_1>
<text><location><page_2><loc_52><loc_5><loc_92><loc_18></location>In this section we revisit low-acceleration space -time scaleinvariant dynamics (SID) raised by Milgrom (2009a) concerning the deep-MOND limit. The reader is reminded that most of the universe is in the SID regime (Kroupa 2014). It is also shown that in the low acceleration regime the MDA correlation follows from SID and the BTFR also immediately follows from SID. We thereby stress again that is a remarkable fact that such a simple symmetry, discovered by Milgrom (2009a), leads to such profound and correct reproduction of the most important laws of observed galactic</text>
<text><location><page_3><loc_9><loc_81><loc_49><loc_91></location>dynamics. It also follows from SID using simple arguments that apparent (i.e. non-particle) 'dark matter haloes' arise simply and directly, as is shown in sec. 2.1 in Wu & Kroupa (2013) and this is applied to the external field effect in Sec. 4.2-4.3.1 to demonstrate that the effective gravitating masses of galaxies depend on the position and mass of neighboring galaxies.</text>
<text><location><page_3><loc_9><loc_76><loc_49><loc_81></location>Consider a space -time scale invariance of the equations of motion under the consideration of the transformation in Minkowsky space (Milgrom 2009a; see also Milgrom 2014b,a),</text>
<formula><location><page_3><loc_24><loc_71><loc_49><loc_73></location>( t, r ) → ( λt, λ r ) , (3)</formula>
<text><location><page_3><loc_9><loc_66><loc_49><loc_71></location>where t and r = ( x, y, z ) are time and Cartesian coordinates, respectively, and λ is a positive number. The Newtonian gravitational acceleration for a spherically symmetric system,</text>
<formula><location><page_3><loc_25><loc_61><loc_49><loc_64></location>g N = GM b r 2 , (4)</formula>
<text><location><page_3><loc_9><loc_38><loc_49><loc_61></location>then transforms as g N → λ -2 g N , whereas the kinematical acceleration, g ≡ d˙ x/ d t , scales as g → λ -1 g . Here M b ( < r ) is the enclosed baryonic mass within r . As a result, the Newtonian gravitational acceleration and the kinematical acceleration scale differently under Eq. 3. Linking purely gravitational interactions to symmetries such as defined in Eq. 3 suggests deeper physics and constitutes a motivation for viewing MOND as much more than a mere phenomenological description of galactic dynamics (Milgrom 2009c). In order to assure that both, the gravitational and the kinematical accelerations scale symmetrically under Eq. 3, that is, in order to maintain the invariant symmetry, the gravitational acceleration, g , has to scale proportionally to g 1 / 2 N . In order to obtain the correct dimension, a constant with the unit of acceleration, needs to be introduced. This constant is referred to as a 0 , such that</text>
<formula><location><page_3><loc_24><loc_35><loc_49><loc_36></location>g = ( a 0 g N ) 1 / 2 , (5)</formula>
<text><location><page_3><loc_9><loc_30><loc_49><loc_34></location>i.e. g 2 = a 0 g N . Thus g = ( GM b a 0 ) 1 / 2 /r , and the circular velocity, which follows from the centrifugal acceleration g = v 2 /r , is</text>
<formula><location><page_3><loc_20><loc_26><loc_49><loc_28></location>v = ( GM b a 0 ) 1 / 4 = constant , (6)</formula>
<text><location><page_3><loc_9><loc_5><loc_49><loc_25></location>which is exactly the BTFR (Milgrom 1983c; McGaugh et al. 2000; Milgrom 2009a; Famaey & McGaugh 2012; Milgrom 2014b). We refer to gravitational dynamics which thus conforms to low-acceleration scale-invariance (Eq. 3) as lowacceleration scale-invariant dynamics (SID). SID beautifully reproduces the deep MOND equations of motion. It is rather remarkable that such a simple principle as SID and discovered by Milgrom leads to one of the most important scaling relations which real galaxies are observed to obey. Note that Eq. 6 implies that each baryonic galaxy is surrounded by a logarithmic non-particle (and thus phantom) dark matter halo potential, which is however not a real halo as it is only evident if Newtonian dynamics is applied to the galaxy. If SID is true, then a Newtonian observer would thus deduce that each baryonic galaxy is surrounded by a phantom DM</text>
<text><location><page_3><loc_52><loc_88><loc_91><loc_91></location>halo the mass of which is proportional to radial distance (Eq. 27 below).</text>
<text><location><page_3><loc_52><loc_85><loc_91><loc_88></location>Under SID and due to Eq. 1 -2, for a spherical system, the mass discrepancy,</text>
<formula><location><page_3><loc_64><loc_80><loc_91><loc_83></location>( v v b ) 2 = g r · r g N = g g N , (7)</formula>
<text><location><page_3><loc_52><loc_73><loc_91><loc_79></location>becomes ( GM b a 0 ) 1 / 2 GM b /r = ( a 0 g N ) 1 / 2 . Thus SID immediately implies a simple relation between the mass discrepancy and the baryonic Newtonian acceleration under the invariance transformation,</text>
<formula><location><page_3><loc_65><loc_68><loc_91><loc_71></location>( v v b ) 2 = ( a 0 g N ) 1 / 2 . (8)</formula>
<text><location><page_3><loc_52><loc_51><loc_92><loc_67></location>This function (Eq. 8) is plotted in the upper panel of Fig 1, where it is compared with the observational data in the weak-field regime ( g N < 0 . 2 × 10 -10 m/s 2 ). The lower panel of this figure also shows the observationally deduced acceleration, g , in dependence of g N . A value for a 0 can be obtained by fitting Eq. 8 to the data points in Fig. 1 (cyan curve). A Levenberg-Marquardt fit to the observed MDA data is shown in the upper panel of Fig. 1. It follows that pure SID constitutes an excellent description of the observational data for a 0 = 1 . 24 ± 0 . 03 × 10 -10 m s -2 = 3 . 90 ± 0 . 01 pc Myr -2 (within 1 σ confidence level) for data points within the above mentioned weak field regime.</text>
<text><location><page_3><loc_52><loc_37><loc_92><loc_51></location>SID is broken near gravitating masses whenever g approaches a 0 from below such that gravitational dynamics becomes Newtonian. A connection between the Newtonian regime and the SID regime is required to study the kinematic acceleration in the transitionary regime where g /similarequal a 0 . To study the kinematics of galaxies in the full regime of acceleration, a transition function µ is introduced by Milgrom (1983c), yielding the full MOND description of galactic dynamics (Appendix A). The kinematic acceleration, g , is transformed from the Newtonian acceleration, g N , through</text>
<formula><location><page_3><loc_66><loc_33><loc_91><loc_35></location>g N = µ ( | g | /a 0 ) g , (9)</formula>
<text><location><page_3><loc_52><loc_25><loc_92><loc_33></location>with the help of the µ ( | g | /a 0 ) function. 2 Several forms of the µ ( | g | /a 0 ) function have been proposed by Milgrom (1983c, 1999) and Bekenstein (2004); Famaey & Binney (2005); Zhao (2008). We use two of the most popular interpolating functions: the 'simple' µ -function and the 'standard' µ -function,</text>
<formula><location><page_3><loc_52><loc_15><loc_91><loc_23></location>µ ( x ) = x 1 + x , 'simple ' µ, µ ( x ) = x √ 1 + x 2 , 'standard ' µ, (10) x ≡ | g | /a 0 .</formula>
<text><location><page_3><loc_52><loc_5><loc_92><loc_14></location>2 Note that this is, from the procedural point of view, equivalent to Planck (1901) introducing the constant /planckover2pi1 as an auxiliary constant ('Hilfsgroße' in German) to describe the transition between the low-energy Rayleigh-Jeans black body spectrum and the downturn towards high-energies observed for black body radiators. The quantisation of energy had not been realised to be the underlying physics until more than 25 years later.</text>
<figure>
<location><page_4><loc_8><loc_59><loc_51><loc_89></location>
<caption>Figure 1. Upper panel: The mass discrepancy ( v/v b ) 2 versus baryonic acceleration g N in the weak field ( g N < 0 . 2 a 0 /lessmuch a 0 ) regime. The black points are the observed mass-discrepancybaryonic-acceleration data by McGaugh (2004), and the cyan curve is the best fit of the MDA correlation for these data (see Eq. 8), where a 0 = 1 . 24 × 10 -10 m s -2 . The green curve is the MDA correlation with a 0 determined from observations of galactic rotation curves (see Sec. 2, Begeman et al. 1991). Lower panel: the kinematical acceleration versus Newtonian gravitational acceleration in the weak field regime. The observed data show a tight correlation between g and g N . The faint dotted line shows the g = g N relation, which is not followed by the observational data.</caption>
</figure>
<text><location><page_4><loc_9><loc_24><loc_49><loc_39></location>We note here that the complete Milgromian description of classical gravity has been shown to have a Lagrangian formulation (Bekenstein & Milgrom 1984) such that this theory is energy and angular momentum conserving. For a spherically symmetric, cylindrically symmetric or axi-symmetrically system, the solution of the Lagrangian formulation takes the simplified form, Eq. 9 (Bekenstein & Milgrom 1984). The standard form of the µ -function can be associated with quantum mechanical processes in the vacuum (Milgrom 1999, see also Appendix A in Kroupa et al. 2010).</text>
<text><location><page_4><loc_9><loc_20><loc_49><loc_24></location>Thus a relation between the mass discrepancy and the acceleration over the full classical regime is determined by the interpolating µ -function:</text>
<formula><location><page_4><loc_21><loc_14><loc_49><loc_18></location>( v v b ) 2 = g · r g N · r /similarequal 1 µ ( x ) . (11)</formula>
<text><location><page_4><loc_9><loc_5><loc_49><loc_14></location>This indicates that the formation of galaxies cannot be stochastic, since otherwise the observational MDA data would have a much wider spread. The MDA correlation conflicts with the requirement of ΛCDM cosmological simulations that galaxy formation in low-mass DM haloes must be stochastic (Boylan-Kolchin et al. 2011). In such a speculative stochastic galaxy formation model, the C/WDM halo</text>
<text><location><page_4><loc_52><loc_84><loc_91><loc_91></location>mass ceases to be correlated with its baryonic/luminous galaxy. A further discussion of the MDA correlation for ΛCDM-simulated galaxies can be found in Sec. 3. More quantitative studies of the different forms of the µ -function will also be presented in Sec. 3.</text>
<section_header_level_1><location><page_4><loc_52><loc_77><loc_88><loc_80></location>3 STANDARD COSMOLOGICAL MODELS AND THE MDA CORRELATION</section_header_level_1>
<text><location><page_4><loc_52><loc_56><loc_92><loc_76></location>Despite the observations indicating rather convincingly that gravitation in the classical regime is Milgromian (Sec. 2), it is more popular to describe galactic dynamics by postulating Newtonian gravity to be valid as a major extrapolation from the Solar System scale to the scale of galaxies plus the existence of dynamically significant C/WDM particles which are neither described by nor contained within the otherwise highly successful standard model of particle physics (Blumenthal et al. 1984; Davis et al. 1985). The resulting standard cosmological model, the ΛC/WDM model, has been subject to significant testing (Kroupa et al. 2010; Kroupa 2012, 2014). One of the major problems for the ΛCDM model is that the merging history of each major dark matter halo makes the formation of disc galaxies highly problematical and until now not convincingly successful.</text>
<text><location><page_4><loc_52><loc_24><loc_92><loc_56></location>In the past two decades, the formation of disc galaxies has been extensively investigated by means of standard ΛCDM cosmological simulations (Katz & Gunn 1991; Navarro & Benz 1991; Navarro & Steinmetz 1997; Weil et al. 1998; Abadi et al. 2003; Piontek & Steinmetz 2011; Hummels & Bryan 2012; Agertz et al. 2011; Guedes et al. 2011; Aumer et al. 2013; Marinacci et al. 2013). The rotation curve has attracted particular interest since it is one of the most important tools to examine the reality of the simulated disc galaxies. It had been shown that the simulated disc galaxies have unrealistic rotation curves with sharp peaks at their centres declining at larger radii (Navarro & Benz 1991; Navarro & Steinmetz 1997; Weil et al. 1998; Abadi et al. 2003; Piontek & Steinmetz 2011; Hummels & Bryan 2012), which disagrees with the observations (Rubin et al. 1985). This is a result of the simulated galaxies having too little angular momentum because of their merging history which is inherent in the standard model by virtue of dynamical friction between the dark matter haloes, and because the gaseous baryons lose orbital energy by being compressed and by dissipating kinetic energy as they fall into the deep potential well of a dark matter halo.</text>
<text><location><page_4><loc_52><loc_5><loc_92><loc_24></location>Recent studies claim that disc galaxies with 'realistic' shapes of rotation curves can form in cosmological simulations with gas by using a subgrid model which enhances the efficiency of stellar feedback (Governato et al. 2010; Agertz et al. 2011; Guedes et al. 2011; Stinson et al. 2013; Aumer et al. 2013; Marinacci et al. 2013). Note that the feedback processes of model disc galaxies are fitted artificially to 'agree with' the observations, and these feedback processes are neither natural results of the cosmological simulations nor predictions of dark matter. This also applies to the recent results from the Illustris project: the feedback applied there is unphysical and the BTFR comes out incorrectly (Vogelsberger et al. 2014; Kroupa 2014). Another criticism of the feedback processes can be found in the most re-</text>
<text><location><page_5><loc_9><loc_85><loc_49><loc_91></location>cent EAGLE project publication (Schaye et al. 2014). However, in spite of the artificiality of feedback processes, we would like to know: how realistic are these model disc galaxies?</text>
<text><location><page_5><loc_9><loc_73><loc_49><loc_85></location>If the simulated disc galaxies represent real galaxies, there should be a MDA correlation in these galaxies, and such relations have to agree with the empirical data. We here study the MDA correlation of simulated disc galaxies (Agertz et al. 2011; Guedes et al. 2011; Aumer et al. 2013; Marinacci et al. 2013), which are claimed to be more realistic than obtained in previous work and which are taken to demonstrate that the standard cosmological model can, after all, account for the observed galaxies.</text>
<text><location><page_5><loc_9><loc_47><loc_49><loc_73></location>Apart from the problems of rotation curves, there are other difficulties with the ΛCDM cosmology simulations on galactic scales, such as the cusp vs core problem (Springel et al. 2008) and the missing satellites problem (or more correctly, the satellite over-prediction problem) (Klypin et al. 1999; Moore et al. 1999) and many other failures to account for data (Kroupa et al. 2010; Kroupa 2012; Famaey & McGaugh 2012; Kroupa 2014). To overcome these difficulties, an alternative dark matter particle has been proposed, a thermal relic WDM particle with a keV mass scale (Col'ın et al. 2000; Bode et al. 2001; Gao & Theuns 2007; Schneider et al. 2012). The time scale for structure formation in the WDM cosmology is longer than that in the CDM cosmology, and the amount of substructures is decreased. The pioneering work on the formation of disc galaxies in WDM cosmological models is carried out by Herpich et al. (2013), according to which the disc galaxies have less centrally concentrated stellar profiles in improved agreement with real galaxies.</text>
<text><location><page_5><loc_9><loc_41><loc_49><loc_46></location>We here also study the MDA correlation of these galaxies. The circular velocities of galaxies contributed from baryons, v b , and from dark matter, v DM , are taken from the above simulations. The mass discrepancy is</text>
<formula><location><page_5><loc_20><loc_36><loc_49><loc_40></location>[ v ( r ) v b ( r ) ] 2 = v 2 b ( r ) + v 2 DM ( r ) v 2 b ( r ) . (12)</formula>
<text><location><page_5><loc_9><loc_34><loc_49><loc_36></location>The Newtonian acceleration (from baryons only) can be calculated from the circular velocity,</text>
<formula><location><page_5><loc_25><loc_31><loc_49><loc_33></location>g N ( r ) = v 2 b /r. (13)</formula>
<section_header_level_1><location><page_5><loc_9><loc_25><loc_42><loc_27></location>3.1 Disc galaxies from CDM cosmological simulations</section_header_level_1>
<text><location><page_5><loc_9><loc_5><loc_49><loc_24></location>In the dark-matter approach, there are two primary formation scenarios for disc galaxies. The first one was studied by Eggen et al. (1962); Samland & Gerhard (2003) and Sommer-Larsen et al. (2003, model S1), which is to form a disc galaxy in a growing dark matter halo by accreting gas (and also dark matter in Samland & Gerhard 2003) in the absence of mergers of dark haloes. This work was generally not accepted by the community because it lacked a 'realistic' merging history. The second scenario is to form disc galaxies through accreting a large number of satellite galaxies (so called minor mergers, e.g., Bullock & Johnston 2005; Moore et al. 2006). The latter scenario was more favoured since it is consistent with the hierarchical assembly of CDM haloes in cosmological simulations (Helmi 2008). However,</text>
<figure>
<location><page_5><loc_53><loc_52><loc_92><loc_89></location>
<caption>Figure 2. Upper panel: the mass discrepancy ( v/v b ) 2 versus baryonic acceleration, g N , from weak to strong fields. The black points are the observed MDA data by McGaugh (2004) for a sample of 74 disc galaxies, including both dwarf and Milky Wayscale spiral galaxies. The red and orange areas are the correlations predicted by Milgromian dynamics with different interpolating functions, red for simple µ and orange for standard µ (Eq. 10). The values of a 0 and the corresponding error are listed in the 6 th and 7 th columns of Table 1. The green curve shows the mass discrepancy-acceleration relation of simulated Milky Wayscale galaxies by Guedes et al. (2011). The coloured shadow areas are the mass discrepancy-baryonic gravitational acceleration data for simulated galaxies from cold dark matter cosmological simulations (upper panel Agertz et al. 2011; Marinacci et al. 2013; Aumer et al. 2013). Middle panel: the difference of the mass discrepancy between CDM models and observed galaxies, δ , which is defined in Eq. 14. For the simulated and observed galaxies g N is computed from Eq. 13. Note that Milgromian dynamics with the standard µ function (orange curve) is an excellent description of the data (black dots in upper panel), the differences of which to the Milgromian dynamics relation are plotted as orange dots. The red dots are the differences between the observed (black points) and the Milgromian dynamics curve with the simple µ function. The apparent larger scatter in the orange δ values at small g N is expected for a constant (small) dispersion of data around the Milgromian dynamics curve. Lower panel: the kinematic-luminous acceleration relation of the CDM models and the observed galaxies (symbols and colors as in the upper panel). The black line corresponds to g = g N .</caption>
</figure>
<text><location><page_6><loc_9><loc_65><loc_49><loc_91></location>it is difficult to produce disc-dominated and bulgeless galaxies from the simulations of cosmological mergers, while there is a large fraction ( ≈ 70%) of edge-on disc galaxies which are bulgeless or disc-dominated in observations (in a complete and homogeneous sample of 15127 edge-on disc galaxies in the SDSS data release 6, Kautsch 2009). HST photometry and Hobby-Eberly Telescope (HET) spectroscopy of giant Sc-Scd galaxies (Kormendy et al. 2010) also shows that more than 50% of a sample of 19 galaxies are bulgeless galaxies, which challenges the picture of galaxy formation by hierarchical merging. Baryonic feedback processes have been studied so as to save the second scenario (e.g., Okamoto et al. 2005; Piontek & Steinmetz 2011; Scannapieco et al. 2012). However it has been shown that the angular momentum problem of the simulated discdominated galaxies and bulgeless galaxies cannot be solved by adding feedback process and by increasing the numerical resolution of the simulations (e.g. D'Onghia & Burkert 2004; Piontek & Steinmetz 2011).</text>
<text><location><page_6><loc_9><loc_29><loc_49><loc_63></location>In a more recent study based on the Millennium-II simulations (Fakhouri et al. 2010) about 31% of the Galaxy-scale haloes have experienced a major merger since z = 1 (corresponding to a look-back time of about 7 -8 Gyr), and the fraction of major mergers rises to 69% since z = 3 (corresponding to a look-back time of about 11 -12 Gyr). Moreover, in cosmological simulations, over the last 10 Gyr for Galaxy-scale haloes with a mass of ≈ 10 12 M /circledot h -1 , 95% of them have undergone a minor merger by accreting a subhalo with mass > 5 × 10 10 M /circledot h -1 , and 70% of them have accreted a subhalo with mass > 10 11 M /circledot h -1 (Stewart et al. 2008). Therefore mergers are very common for Milky-Wayscale haloes in cosmological simulations. The simulations of major mergers (with equal mass galaxy pairs) show that the disc can be completely disrupted and that the remnants of such mergers become early-type galaxies (elliptical galaxies or bulge-dominated galaxies, Toomre 1977; Cox & Loeb 2008), and minor mergers (with a mass ratio 10 : 1) also lead to growth of the bulge and thickness of the disc (e.g., Walker et al. 1996; Naab & Burkert 2003; Younger et al. 2007; Kazantzidis et al. 2009). Thus, the very large fraction of observed bulgeless disc galaxies (70% in edge-on disc galaxies) is inconsistent with the high incidence ( > 70%) of significant mergers, a point also emphasized by Kormendy et al. (2010).</text>
<text><location><page_6><loc_9><loc_5><loc_49><loc_28></location>More recently, a series of new models using smoothed-particle hydrodynamics (SPH) simulations (Agertz et al. 2011; Guedes et al. 2011; Aumer et al. 2013; Marinacci et al. 2013) claimed that more realistic disc galaxies are formed in a quiet merger history scenario. In such a scenario no mergers with a mass ratio of the substructure and host galaxy larger than 1 : 10 are allowed at low redshift. These authors have thus, essentially, returned to the previously discussed models by Samland & Gerhard (2003) which, however, had been criticized as lacking cosmological realism by being void of mergers. Thus these galaxies are selected from the unlikely fraction of Milky-Way-scale haloes in standard cosmological simulations. It has been claimed that the rotation curves of these simulated galaxies have more reasonable shapes without sharp central peaks. To test these simulated galaxies more quantitatively and in more detail, we now study their MDA correlation, and then</text>
<text><location><page_6><loc_52><loc_88><loc_91><loc_91></location>compare the theoretical relation with that extracted from the observed galaxies.</text>
<text><location><page_6><loc_52><loc_50><loc_92><loc_88></location>The relation is shown in the upper panel of Fig. 2, where the green line, coloured shadows and areas represent the relations of disc galaxies in the aforementioned simulations, respectively. A correlation between the mass discrepancy at radius r, [ v ( r ) v b ( r ) ] 2 , and the baryonic Newtonian acceleration at the same radius, Eq. 13, does exist in the simulated galaxies. However, despite the existence of such a correlation, the simulated galaxies are not consistent with observations. The MDA correlation obtained from the simulated disc galaxies (Agertz et al. 2011; Guedes et al. 2011; Aumer et al. 2013; Marinacci et al. 2013) lies significantly above the empirical relation. Among the above simulations, there is one Milky Way like disc galaxy modeled by Guedes et al. (2011), in which the trend of the mass discrepancy-acceleration relation is different from that obtained from the observed data points: with decrease of the acceleration, the increase of mass discrepancy is slower in the Guedes et al. (2011) model galaxy. Note that the disc galaxies in Agertz et al. (2011); Marinacci et al. (2013) and Guedes et al. (2011) are obtained from re-simulations of haloes with a quiet merger history, i.e., from haloes without major mergers at low redshift. The disc galaxies in Aumer et al. (2013) are selected from haloes both with and without low-z major mergers. There is a much wider spread and a larger deviation from the observational data for these Aumer et al. (2013) galaxies compared to other samples, and this could be an effect of the low-z mergers.</text>
<text><location><page_6><loc_52><loc_40><loc_92><loc_49></location>Although there is an overlap of the MDA correlations from simulations and observations in the weak field regime where g N < 0 . 1 a 0 , the majority of MDA correlations predicted from the re-simulated disc galaxies are inconsistent with observations. More concretely, for a given enclosed baryonic mass M b , the mass discrepancy is always overpredicted.</text>
<text><location><page_6><loc_52><loc_29><loc_91><loc_40></location>Therefore, considering the detailed rotation curves of galaxies simulated in a CDM universe, the simulated galaxies do not agree with the observed ones. The host dark matter haloes have to bear very quiet merger histories. Since the vast majority of local galaxies (72% spiral galaxies and 15% S0 galaxies, overall 87%) are disc galaxies (Delgado-Serrano et al. 2010), this generates another severe issue that cannot be resolved within the dark matter models.</text>
<text><location><page_6><loc_52><loc_24><loc_92><loc_28></location>The middle panel of Fig. 2 shows the difference of the MDA correlation between the simulated [ v ( g N ) v b ( g N ) ] 2 CDM and</text>
<text><location><page_6><loc_52><loc_5><loc_92><loc_25></location>the observed galaxies [ v ( g N v b ( g N ) ] 2 obs , δ (Eq. 14 below). Here the values of g N for the observed galaxies are computed from Eq. 13. For most of the data points of the simulated galaxies, the mass discrepancy is always larger than for the observed galaxies. On the other hand, the observed data agree extremely well with Milgromian dynamics (i.e., with there being no cold or warm dark matter) with a standard µ -function (orange areas and symbols with error bars in Fig. 2) and with a simple µ -function (red areas and symbols with error bars in Fig. 2). The best fitting values of a 0 and the corresponding errors with different µ functions are computed with the Levenberg-Marquardt method, and are listed in the 6 th and 7 th columns of Table 1. With a simple µ -function (red areas and symbols in Fig. 2), the predictions of</text>
<text><location><page_7><loc_9><loc_87><loc_49><loc_91></location>mass discrepancy in Milgromian dynamics are slightly larger than the observations. Therefore, the standard µ -function is a better interpolating function for the MDA correlations.</text>
<text><location><page_7><loc_9><loc_60><loc_49><loc_86></location>We also notice that the dispersion of the relation for the simulated galaxies in the ΛCDM model is much wider than that of the observational data, especially in the weak acceleration regime. However, the correlation is tight for the observed galaxies. This is surprising because the observational points have measurement uncertainties which enlarge any spread. The wide spread of the theoretical relation comes about because the spatial distribution of dark matter does not tightly correlate with the baryonic distribution in different simulated galaxies. Such a wide spread is due to the multiplicity of free parameters of dark haloes: the parameters of the dark matter profiles are not unique for different modeled disc galaxies, the shapes of the dark haloes are triaxial, and there is a variation between the inclination angle for the principle axes of the dark haloes and the baryonic discs (as emphasized by Disney et al. 2008). Instead, the observational data demonstrate a very close one-to-one relationship between the baryons and their rotation about the centre of their galaxy.</text>
<text><location><page_7><loc_9><loc_31><loc_49><loc_60></location>The lower panel of Fig. 2 shows the kinematic-luminous acceleration relation of the simulated galaxies from ΛCDM cosmological simulations (the coloured areas and shadows) and of the observed galaxies (black points). The predictions from Milgromian dynamic with two µ -functions are plotted (red area for the simple µ -function and orange area for the standard µ -function) as well. The kinematic-luminous acceleration relation of the simulated galaxies lies above that of the observed galaxies, and there is a much wider spread of such a relation for the simulated galaxies. This is not surprising, since the kinematic-luminous acceleration relation is exactly the MDA correlation plotted differently and the MDA correlation already indicates the disagreement between the simulated and the observed galaxies. Due to the stochastic galaxy formation scenario in cosmological simulations (Boylan-Kolchin et al. 2011), for a given enclosed baryonic mass, there are various possibilities for the centrifugal acceleration of the simulated galaxies. Therefore a tight correlation is not expected from the model galaxies. For a comparison, Milgromian dynamics predicts tight relations (with different forms of the µ -functions) for the MDA.</text>
<text><location><page_7><loc_9><loc_15><loc_49><loc_31></location>In summary, there are two problems for galaxy formation in ΛCDM cosmology: (i) the fraction of CDM haloes with a sufficiently quiet merging history is far too small to account for the large fraction of galaxies that are bulgeless discs or disc-dominated galaxies. (ii) Even those delicately selected DM haloes that do have a quiet merging history fail to host disc galaxies which correspond to real observed galaxies. Only the slow growth of the baryonic disc with the DM halo without any mergers has yielded realistic-looking disc galaxies, as already shown by Samland & Gerhard (2003). This is, however, equivalent to a growing purely baryonic galaxy in Milgromian dynamics.</text>
<section_header_level_1><location><page_7><loc_9><loc_10><loc_43><loc_12></location>3.2 Disc galaxies from WDM cosmological simulations</section_header_level_1>
<text><location><page_7><loc_9><loc_5><loc_49><loc_9></location>Herpich et al. (2013) simulated three galaxies, and the circular velocities contributed from baryons and dark matter can be found for two galaxies from the three (fig. 3 of Herpich</text>
<figure>
<location><page_7><loc_53><loc_52><loc_92><loc_89></location>
<caption>Figure 3. Upper panel: The black points and coloured areas are defined as in Fig. 2. The cyan areas are the mass discrepancybaryonic gravitational acceleration relation for simulated galaxies from warm dark matter (Herpich et al. 2013) cosmological simulations. Middle panel: the difference of the mass discrepancy between WDM models and observed galaxies, δ (Eq. 14 below). Lower panel: the kinematic-luminous acceleration relation of the CDM models and the observed galaxies (symbols and colours as in the upper panel).</caption>
</figure>
<text><location><page_7><loc_52><loc_13><loc_92><loc_34></location>et al. 2013), g1536 and g5664, which are used here. For each galaxy, there are three sets of parameters, corresponding to different masses of the WDM particles. The MDA correlation for galaxies simulated in a WDM cosmology is presented in the upper panel of Fig. 3 with the cyan area. The difference of the MDA data between the WDM simulated galaxies [ v ( g N ) v b ( g N ) ] 2 WDM and the observations [ v ( g N ) v b ( g N ) ] 2 obs , δ , is given by Eq. 14 below and is shown in the middle panel. The kinematic-luminous-acceleration relation for the WDM simulated galaxies is shown in the lower panel of Fig. 3. The MDA correlation is not consistent with the observations. The relations obtained from the simulated galaxies lie above the observational relation. Therefore, the above MDA data rule out the two model galaxies obtained from WDM simulations as well.</text>
<text><location><page_7><loc_52><loc_5><loc_92><loc_13></location>In Fig. 3, a comparison of the observational MDA data predicted under the assumption of Milgromian dynamics with a standard µ -function (orange areas and symbols) and a simple µ -function (red areas and symbols) is shown. It confirms again that the dispersion of the MDA data predicted by WDM models is larger than for Milgromian dynamics</text>
<text><location><page_8><loc_9><loc_87><loc_49><loc_91></location>with a standard µ -function and a simple µ -function. This is a result of the spread of WDM halo properties for a given baryonic mass (compare with Disney et al. 2008).</text>
<section_header_level_1><location><page_8><loc_9><loc_83><loc_37><loc_84></location>3.3 The Wilcoxon signed-rank test</section_header_level_1>
<text><location><page_8><loc_9><loc_70><loc_49><loc_82></location>As mentioned above (Sec. 3.1 and Sec. 3.2), the majority of the MDA data obtained from the C/WDM simulated galaxies lie above the relation obtained from the observations. Here we apply a non-parametric statistical hypothesis test, the Wilcoxon signed-rank test (Wilcoxon 1945; Bhattacharyya & Johnson 1977), to formally study the difference of the data between the simulations and the observations. The procedure of the Wilcoxon signed-rank test applied here is as follows:</text>
<unordered_list>
<list_item><location><page_8><loc_9><loc_55><loc_49><loc_68></location>· 1. For the observed data (i.e., McGaugh 2004 data) there are m data points. For each data point the values of g N (defined in Eq. 13) and the ratio [ v ( g N ) v b ( g N ) ] 2 (data from McGaugh 2004) are known. To compare the observed data with the simulated galaxies, interpolate the upper envelope of the MDA data of the simulated galaxies at each observed data point g N,i , where i = 1 , ..., m . Compute the difference of the MDA correlation between the simulated and the observed galaxies at each g N,i , which is defined in Eq. 13,</list_item>
</unordered_list>
<formula><location><page_8><loc_16><loc_50><loc_49><loc_54></location>δ i = [ v ( g N,i ) v b ( g N,i ) ] 2 C / WDM -[ v ( g N,i ) v b ( g N,i ) ] 2 obs . (14)</formula>
<unordered_list>
<list_item><location><page_8><loc_9><loc_44><loc_49><loc_49></location>· 2. Exclude the simulated-observed data pairs with | δ i | = 0 . 0. 3 Order the remaining data pairs according to increasing | δ i | . The number of the remaining data pairs is n . The rank of the data pairs is denoted as R i .</list_item>
<list_item><location><page_8><loc_9><loc_41><loc_49><loc_44></location>· 3. The data pairs with δ i > 0 are selected, and the sum of their ranks are denoted as T + . Let</list_item>
</unordered_list>
<formula><location><page_8><loc_10><loc_37><loc_49><loc_39></location>E = n ( n +1) 4 , (15)</formula>
<formula><location><page_8><loc_9><loc_34><loc_49><loc_36></location>D = n ( n +1)( n +2) 24 . (16)</formula>
<text><location><page_8><loc_11><loc_28><loc_11><loc_30></location>/negationslash</text>
<text><location><page_8><loc_13><loc_28><loc_13><loc_30></location>/negationslash</text>
<text><location><page_8><loc_9><loc_29><loc_49><loc_33></location>There could be a group of ties within which the | δ | values of the elements are the same, i.e., | δ i | = | δ a | = | δ b | = ... ( i = a = b = ... ). Let</text>
<text><location><page_8><loc_47><loc_30><loc_47><loc_32></location>/negationslash</text>
<formula><location><page_8><loc_22><loc_23><loc_49><loc_27></location>Q = 1 48 l ∑ j =1 q j ( q 2 j -1) , (17)</formula>
<text><location><page_8><loc_9><loc_20><loc_49><loc_23></location>where l is the number of such ties, and q j is the number of elements in the j th tie.</text>
<unordered_list>
<list_item><location><page_8><loc_9><loc_16><loc_49><loc_20></location>· 4. Calculate the statistic Z for a large sample of data pairs (for more details of the statistical method, see Bhattacharyya & Johnson 1977, p. 519),</list_item>
</unordered_list>
<formula><location><page_8><loc_24><loc_10><loc_49><loc_14></location>Z = T + -E √ D -Q . (18)</formula>
<unordered_list>
<list_item><location><page_8><loc_52><loc_86><loc_91><loc_91></location>· 5. Interpolate the lower envelope of the MDA data of the simulated galaxies at each g N,i , and repeat the steps 1 -4.</list_item>
</unordered_list>
<text><location><page_8><loc_52><loc_62><loc_92><loc_86></location>To test how well the MDA correlation predicted by Milgromian dynamics matches the observations, steps 1 -4 are repeated for Milgromian dynamics with different µ functions. For each µ function, a Levenberg-Marquardt fit to the observed MDA data is applied to obtain the best fitting values of a 0 and the corresponding errors. The results are listed in Table 1. All of the CDM and WDM models, together with Milgromian dynamics with the simple µ -function, can be ruled out with a confidence of better than 99 . 99%. Only under Milgromian dynamics with the standard µ -function, Z = 0 . 71. Since the level of significance for exclusion and for a directional (1 -tailed) test is α = 0 . 05 for Z = 1 . 645, Milgromian dynamics with the standard µ -function constitutes a good description of the observed data. That is, the hypothesis that Milgromian dynamics/MOND does not agree with the MDA data can be ruled out with at most 1 -2 α = 90% confidence.</text>
<text><location><page_8><loc_52><loc_48><loc_92><loc_62></location>Finally, we test the MDA correlation in the pure SID regime (Sec. 2). Steps 1 -4 are repeated for the data pairs with weak accelerations, g N < 0 . 2 × 10 -10 ms -2 , and the best fitting value of a 0 = 1 . 24 × 10 -10 m s -2 (Tab. 1). Z = -0 . 38 is obtained. This is an extremely good agreement between SID and the observed data. The level of significance for a 1 -tailed standard normal critical value is -1 . 282 corresponding to α = 0 . 1. Thus the hypothesis that SID does not agree with the MDA data in the weak field regime can be rejected with at most 1 -2 α = 80% confidence.</text>
<section_header_level_1><location><page_8><loc_52><loc_42><loc_90><loc_44></location>4 THE DARK MATTER HALO TO STELLAR MASS RELATION</section_header_level_1>
<section_header_level_1><location><page_8><loc_52><loc_39><loc_77><loc_41></location>4.1 The masses of CDM haloes</section_header_level_1>
<text><location><page_8><loc_52><loc_19><loc_92><loc_38></location>Above we have seen that the most advanced models based on Einsteinian/Newtonian gravitation together with cold or warm dark matter (DM) are not able to reproduce the observed MDA correlation. Another related way to test for the existence of dark matter haloes is to study the dark-matter-halo-mass versus baryonic-mass correlation. Recently, Miller et al. (2014) constrained the stellar to halo mass relation for a sample of 41 dwarf galaxies within a redshift range of 0 < z < 1 (the positions, stellar and dark halo masses of the galaxies are listed in Tab. 2): the stellar masses, M s , are derived from the Fitting and Assessment of Synthetic Templates (Kriek et al. 2009) code for the photometric database (Newman et al. 2013), and the DM halo masses, M vir , are computed from</text>
<formula><location><page_8><loc_60><loc_15><loc_91><loc_17></location>log 10 ( M vir /h -1 ) = 3 log 10 ( G -1 V 200 ) (19)</formula>
<text><location><page_8><loc_52><loc_5><loc_92><loc_14></location>assuming the haloes are spherical. Here h = H 0 / 100 kms -1 Mpc -1 , V 200 is the rotation speed at the virial radius, r vir , within which the enclosed CDM halo mass has a mean overdensity of 200 times of the critical density of the Universe, ρ crit = 3 H 2 0 8 πG . The parameters of their cosmological model are Ω Λ = 0 . 7 , Ω m = 0 . 3 , H 0 = 70 kms -1 Mpc -1 . V 200 is converted according to</text>
<table>
<location><page_9><loc_24><loc_63><loc_77><loc_79></location>
<caption>Table 1. Wilcoxon signed-rank test. The size of the sample of non-zero-differences of simulated-observed data pairs is n = 730. The first column tabulates the data source from the simulated galaxies, the 2 nd and 4 th columns contain the number of data pairs with | δ | > 0 . 0 for the lower and upper envelopes of the simulated galaxies, respectively. The 3 rd and 5 th columns list, respectively, the statistic Z for the lower and upper envelopes of the simulated galaxies. The 6 th column lists the values of a 0 for different µ functions. For Guedes+'s model, there is only one simulated galaxy (no lower envelope). The results assuming SID and the transition regime are valid, with the simple and standard µ -functions, are listed in the 8 th and 9 th lines, respectively. The bottom line show the Wilcoxon signed-rank test for pure SID in the weak field regime (i.e., without the µ function), where g N < 0 . 2 × 10 -10 m s -2 . n -is the number of the data pairs with | δ | < 0 . 0.</caption>
</table>
<formula><location><page_9><loc_23><loc_56><loc_49><loc_57></location>V 2 . 2 /V 200 = 1 . 05 , (20)</formula>
<text><location><page_9><loc_9><loc_36><loc_49><loc_55></location>where V 2 . 2 is a direct measurement of the circular velocity at the radius r 2 . 2 which is 2.2 times the scale radius (for more details, see Sec. 3.3 and Sec.4 in Miller et al. 2014). Note that this is an empirical relation measured from weak lensing by Reyes et al. (2012) for galaxies within the mass range of [10 9 , 10 11 ]M /circledot , and the slope of velocity ratio to stellar masses is 0 . 53 ± 0 . 03. Miller et al. (2014) extrapolated this relation to low mass dwarf galaxies with masses of 10 7 M /circledot , thus the dispersion of V 200 of the haloes derived from this relation is largest for the low-mass galaxies (see figures 6-7 in Miller et al. 2014). Since the 'virial mass' in Miller et al. (2014) is not derived from dynamics, it does not exactly amount to the virial mass defined in Eq. 21 and Eq. 22 below.</text>
<text><location><page_9><loc_9><loc_19><loc_49><loc_36></location>Miller et al. (2014) compared their observations with CDM cosmological simulations by Behroozi et al. (2013) within a similar redshift range. They found that the stellar to halo mass relation predicted from the simulated dwarf galaxies is at odds with observations. For a given stellar mass, the simulations significantly over-predict the mass of dark matter for the dwarf galaxies. Although the dispersion of data for the dwarf galaxies is large, the trend is clear that the observed data are not consistent with the curves from the simulated galaxies. This problem is essentially the same as that of the MDA correlation at large galactic radii, i.e., at low acceleration (see Fig. 2-3).</text>
<text><location><page_9><loc_9><loc_10><loc_49><loc_19></location>Furthermore, Guo et al. (2010) proposed an abundance-matching stellar to halo mass relation for model galaxies from ΛCDM cosmological simulations. However, in the low stellar mass range [10 6 , 10 8 ]M /circledot , the theoretical halo masses are too large by a factor of 5 compared to those of dwarf galaxies in a large survey of SDSS central galaxies 4 (More et al. 2009). Ferrero et al.</text>
<text><location><page_9><loc_52><loc_57><loc_91><loc_59></location>(2012) improved the Guo et al. (2010) stellar to halo mass relation as follows:</text>
<formula><location><page_9><loc_53><loc_49><loc_91><loc_54></location>M s M vir = c [ 1 + ( M vir M 1 ) -2 ] κ [ ( M vir M 0 ) -α + ( M vir M 0 ) β ] -γ , (21)</formula>
<text><location><page_9><loc_52><loc_33><loc_92><loc_49></location>where c = 0 . 129, M 0 = 10 11 . 4 M /circledot , M 1 = 10 10 . 65 M /circledot , α = 0 . 926, β = 0 . 261 and γ = 2 . 440. κ is a free parameter. Larger values of κ represent shallower stellar to halo mass relations for low mass haloes, while κ = 0 returns to the abundancematching relation proposed by Guo et al. (2010). We note that the procedure to associate visible galaxies with their dark matter haloes which are derived from dark-matter-only simulations lacks the physics of galaxy formation entirely. Abundance-matching is merely the short-circuiting of a major problem of ΛCDM cosmology (the 'missing dwarf galaxy problem', or more truthfully 'the dwarf over-prediction problem').</text>
<text><location><page_9><loc_52><loc_24><loc_91><loc_32></location>We compare the halo to stellar masses predicted by Eq. 21 (cyan curves) for different values of κ to the empirical data of Miller et al. (2014) in Fig. 4, finding an inconsistency between the simulated and observed galaxies. For a given stellar mass, the halo mass is significantly over-predicted for Ferrero's relation for all different values of κ .</text>
<section_header_level_1><location><page_9><loc_52><loc_17><loc_89><loc_18></location>4.2 The mass of the phantom dark matter halo</section_header_level_1>
<text><location><page_9><loc_52><loc_5><loc_92><loc_16></location>Concerning Milgromian dynamics, a baryonic object is surrounded by an unreal non-particle (i.e., phantom) isothermal dark matter halo with constant circular velocity given by Eq. 6. This follows directly from pure-SID, i.e. even without considering the transition from pure-SID to the Newtonian regime, the description of which constitutes Milgromian dynamics (Appendix A) in the classical dynamical regime. The apparent phantom virial dark matter halo mass, M vir , can</text>
<table>
<location><page_10><loc_25><loc_31><loc_75><loc_85></location>
<caption>Table 2. The right ascensions and declinations (epoch J2000, the 2 nd and 3 rd columns), stellar masses (the 4 th column) and halo masses (the 5 th column) of dwarf galaxies in Miller et al. (2014). The strength of the external field (Eq. 30) is listed in the 6 th column for the labeled magenta dwarf galaxies in Fig. 4, assuming M b = M s . Here a 0 = 1 . 21 × 10 -10 m s -2 as in Tab. 1.</caption>
</table>
<text><location><page_10><loc_9><loc_25><loc_49><loc_28></location>be derived from Eq. 6 (for the derivation, see Sec. 2.1 in Wu & Kroupa 2013), which is 5</text>
<text><location><page_10><loc_9><loc_5><loc_49><loc_21></location>5 The virial mass in SID can be derived in the same way as in Miller et al. (2014), i.e., from a relation between V 200 and V 2 . 2 in SID. The derivation of such a 'virial mass' is presented in Appendix B. However, because the errors of the Tully-Fisher (TF) relation in the galaxies of Miller et al. (2014) are large, and the fit parameters for the TF relation are significantly different for different samples of galaxies (see table 2 in Miller et al. 2014), the so-obtained 'virial mass' is strongly sample dependent. Therefore a more universal virial mass of a galaxy in SID is adopted in Sec. 4.2. Essentially, we assume that the inner rotation velocity (see Sec. 4.1) is a measure of the flat part of the rotation velocity at larger radii, which in turn measures the mass of the dark matter halo.</text>
<formula><location><page_10><loc_52><loc_20><loc_91><loc_25></location>M vir = ( Ga 0 M b ) 3 / 4 p -1 / 2 G -3 / 2 , (22) p = 4 3 π × 200 ρ crit .</formula>
<text><location><page_10><loc_52><loc_10><loc_92><loc_18></location>Such phantom dark matter haloes are associated with a noninertial (i.e., unreal) mass by a Newtonian observer. That is, an observer interpreting the motion of a star around a galaxy in terms of Newtonian dynamics will deduce (wrongly) that the galaxy is immersed in a dark matter halo. However, the inertial mass of the galaxy is exactly its baryonic mass only.</text>
<text><location><page_10><loc_52><loc_5><loc_91><loc_10></location>Therefore, the phantom dark matter halo mass is sourced entirely by the baryonic mass of the galaxy (Eq. 22). The phantom halo to baryonic (stellar plus gas) mass ratio follows from Eq. 22,</text>
<figure>
<location><page_11><loc_26><loc_51><loc_76><loc_89></location>
<caption>Figure 4. The halo to baryonic (stellar plus gas) mass relation from observations by Miller et al. (2014, black circles with error bars, the size of the symbols represents the redshift, z ∈ (0 , 1), of the dwarf galaxies: larger symbols for higher redshifts and smaller symbols for lower redshifts) and from simulated galaxies by Ferrero et al. (2012, cyan curves) and Behroozi et al. (2013, red and green curves, corresponding to z = 1 . 0 and 0 . 1, respectively). The black lines are predictions from SID (Eq. 23): for isolated galaxies (solid lines, the upper is for assumming the mass of gaseous matter, M g , equals to the mass of stars, M s , in a galaxy and the lower solid line is for assumming the mass of gas M g = 0 in a galaxy), for galaxies located at the position of the LMC near a Milky-Way-like galaxy (dashed lines, the upper and lower dashed lines are defined the same as solid lines) and for galaxies embedded in a strong external field (Newtonian limit, dotted lines, the upper and lower dashed lines are defined the same as solid lines). The magenta symbols point out the dwarf galaxies whose halo-to-baryonic mass ratio lies beyond the 3 σ confidence level away from the prediction of SID, i.e., galaxies probably embedded in external fields. The magenta numbers show the numbers of the corresponding galaxies in Table 2.</caption>
</figure>
<formula><location><page_11><loc_17><loc_28><loc_49><loc_30></location>M vir M b = ( Ga 0 ) 3 / 4 M -1 / 4 b p -1 / 2 G -3 / 2 , (23)</formula>
<text><location><page_11><loc_9><loc_5><loc_49><loc_27></location>which is shown with two black solid lines in Fig. 4. Since the amount of gas is difficult to determine in the observations, the mass of gas is a parameter in SID/MOND which is not well constrained. Here a 0 = 1 . 21 × 10 -10 ms -2 , which is the value determined by Begeman et al. (1991), and agrees with the best-fit value of the MDA data in Sec. 3.1. The upper limit for the mass of gas, M g , is assumed to be the same as M s , i.e., M b = 2 M s . The lower limit for the mass of gas is M g = 0, i.e., M b = M s . The two limits correspond to the upper and lower solid lines in Fig. 4. These two assumptions on the mass of gas most probably straddle the real gas content of the dwarf galaxies which is undetermined for the sample of Miller et al. (2014). The existence of gas decreases the halo to stellar mass ratio in Milgromian dynamics. There are overlaps between the prediction of Milgromian dynamics and CDM cosmological simulations for the isolated gas-</text>
<text><location><page_11><loc_52><loc_16><loc_92><loc_32></location>ich galaxies ( M g = M s ). However, the prediction based on Milgromian dynamics agrees much better with the observations within the error range of the data. The data which deviate more than 3 σ from Eq. 23 are over-plotted using magenta symbols in Fig. 4. These data points are possibly the bad points of the sample, since the virial mass for the data points are obtained from the empirical relation of Eq. 20, and they are not exactly the virial mass derived from dark halo dynamics. Interestingly, SID predicts a truncated phantom halo which stays below the solid lines in the figure if its baryonic source system is exposed to an external field (see Sec. 4.3).</text>
<section_header_level_1><location><page_11><loc_52><loc_12><loc_80><loc_13></location>4.3 The external field effect in SID</section_header_level_1>
<text><location><page_11><loc_52><loc_8><loc_91><loc_11></location>4.3.1 The virial mass of a galaxy embedded in an external field</text>
<text><location><page_11><loc_52><loc_5><loc_91><loc_7></location>In SID/Milgromian dynamics there appears an interesting effect that leads to an observable prediction which does not</text>
<text><location><page_12><loc_9><loc_77><loc_49><loc_91></location>exist in the Newtonian plus dark matter model. It is relevant for a satellite object falling within the gravitational field of a host such that the host field does not vary significantly across the satellite. This external gravitational field effectively truncates the isothermal phantom dark matter halo, as deduced by an observer who interprets the observations within Newtonian dynamics. The Milgromian dynamics/MOND equation for a spherical, axisymmetric or cylindrical system embedded in an external field is (Milgrom 1983c; Sanders & McGaugh 2002)</text>
<formula><location><page_12><loc_16><loc_74><loc_49><loc_76></location>g N,i + g N,e = µ ( | ( g i + g e ) | /a 0 )( g i + g e ) , (24)</formula>
<text><location><page_12><loc_9><loc_63><loc_49><loc_74></location>where g N,i is the Newtonian acceleration from the baryonic matter of the internal system, g N,e is the Newtonian acceleration from the baryonic matter of the external gravitational source generating the background uniform field, and g i and g e are the internal and external gravitational accelerations. For an external field dominated system, g e = | g e | /greatermuch g i = | g i | , Eq. 24 is expanded around g e to lowest order as</text>
<formula><location><page_12><loc_9><loc_58><loc_49><loc_62></location>g N,e = µ ( | g e | /a 0 ) g e , g N,i = µ ( | g e | /a 0 ) g i , (25)</formula>
<text><location><page_12><loc_9><loc_48><loc_49><loc_58></location>with a dilation factor of ∆ 1 = (1 + d ln µ/d ln x ) x = g e /a 0 . ∆ 1 approaches 1 in the Newtonian limit and approaches 2 in the deep MOND limit. The value of ∆ 1 also depends on the direction relative to the external field (for more details see Milgrom 1983c; Bekenstein & Milgrom 1984; Zhao & Tian 2006). µ ( x ) = x in the low acceleration regime where SID is valid (i.e., in deep MOND limit). 6</text>
<text><location><page_12><loc_9><loc_39><loc_49><loc_48></location>A star on a circular orbit at a large distance from the satellite galaxy with mass M b orbits subject to the centrifugal acceleration g i = v 2 /r . In SID (Milgrom 2009c, 2014a) a Newtonian observer interprets this to be due to the centripetal acceleration from an isothermal (phantom) dark matter halo (PDMH) with mass within radius r of M PDMH ( < r ). Assuming spherical symmetry, it follows that</text>
<formula><location><page_12><loc_20><loc_34><loc_49><loc_37></location>GM PDMH ( < r ) r 2 = √ GM b a 0 r . (26)</formula>
<text><location><page_12><loc_9><loc_32><loc_43><loc_33></location>Thus, the mass of the phantom dark matter halo is</text>
<formula><location><page_12><loc_18><loc_29><loc_49><loc_30></location>M PDMH ( < r ) = ( M b a 0 ) 1 / 2 G -1 / 2 r. (27)</formula>
<text><location><page_12><loc_9><loc_25><loc_49><loc_28></location>An isolated galaxy has an infinitely extended PDMH with an unbounded phantom mass. An isolated galaxy within</text>
<text><location><page_12><loc_9><loc_5><loc_49><loc_22></location>6 Note that SID as such does NOT necessary imply the introduction of a 0 , nor the role of acceleration. One can obtain SID by introducing a length scale factor, r 0 , instead of the introduction of a 0 , into the Newtonian law of gravity. Thus g = GM/ ( rr 0 ) instead of the Newtonian spherical symmetric gravity, g N = GM/r 2 . One can also obtain SID by introducing a time constant, t 0 , such that the gravity becomes g = ( GM/t 0 ) 2 / 3 r -1 . Therefore SID is obtained from the above two examples, but the baryonic Tully-Fisher relations are different. M ∝ v 2 for the length scale factor in the former case and M ∝ v 3 for the time constant in the latter case, whereas M ∝ v 4 through the introduction of a 0 in SID/MOND. Therefore, MOND is based on two main axioms: the introduction of a 0 as the role of an acceleration constant and SID is the deep MOND limit.</text>
<text><location><page_12><loc_52><loc_76><loc_92><loc_91></location>a cosmological model has a virial PDMH mass given by Eq. 22 above which follows from equating the mean PDMH density within r vir to 200 times the critical density in the universe, yielding the maximum radius r vir of the isothermal PDMH. If the galaxy is immersed in a uniform external gravitational field corresponding to an acceleration g e = (0 , 0 , g e ) on a Cartesion grid, i.e., the external field is along the z -axis direction, then the centripetal acceleration from the PDMH, g i , equals g e = | g e | at the radius r eq , SID = v 2 /g e = √ GM b a 0 /g e . Using Eq. 27, the PDMH mass of such a galaxy is thus reduced in mass to the value</text>
<formula><location><page_12><loc_58><loc_72><loc_91><loc_73></location>M PDMH ( r eq , SID ) = ( M b a 0 ) 1 / 2 G -1 / 2 r eq , SID . (28)</formula>
<text><location><page_12><loc_52><loc_67><loc_91><loc_71></location>At radius r > r eq , SID the star accelerates mainly according to g e , while at r < r eq , SID it accelerates mainly according to internal g i .</text>
<text><location><page_12><loc_52><loc_55><loc_92><loc_67></location>Thus, a strict prediction following from SID and thus from Milgromian dynamics (i.e. MOND) is that a Newtonian observer will deduce galaxies to have a maximal (phantom) dark matter halo mass given by M PDMH = M vir (Eq. 22). Galaxies which are immersed in a uniform external field will appear to have reduced PDMH masses (Eq. 28). The two solid lines in Fig. 4 show the virial masses of isolated galaxies in SID. SID predicts that galaxies embedded in external fields stay below the solid lines in Fig. 4.</text>
<section_header_level_1><location><page_12><loc_52><loc_51><loc_87><loc_52></location>4.3.2 A distance-strength-of-external-field relation</section_header_level_1>
<text><location><page_12><loc_52><loc_47><loc_91><loc_50></location>At r = r eq , SID , the strength of the external field as obtained by a Newtonian observer is</text>
<formula><location><page_12><loc_64><loc_42><loc_91><loc_45></location>g e = GM PDMH ( r eq , SID ) r 2 eq , SID . (29)</formula>
<text><location><page_12><loc_52><loc_35><loc_91><loc_41></location>For a given dwarf galaxy, the strength of the external field as a function of the observationally determined baryonic mass and the observationally determined phantom dark matter halo mass can be derived from the combination of g i ( r eq , SID ) = g e , Eq. 6 and Eq. 29,</text>
<formula><location><page_12><loc_66><loc_30><loc_91><loc_33></location>g e = M b M PDMH a 0 , (30)</formula>
<text><location><page_12><loc_52><loc_21><loc_92><loc_29></location>where M PDMH is the unreal phantom dark matter halo mass of the dwarf galaxy observationally deduced by a Newtonian observer. This simple relation supplies a quick way to determine the strength of an external field which a dwarf galaxy is exposed to if the baryonic and halo (or dynamical) mass of the galaxy are known.</text>
<text><location><page_12><loc_52><loc_8><loc_92><loc_21></location>For galaxies whose phantom halo masses deviate from the SID prediction (Eq. 23) by more than the 3 σ confidence level (the magenta symbols and numbered labels in Fig. 4), the strength of their external fields (Eq. 30) are computed and listed in Table 2. The external fields are generally weak, from 0 . 01 a 0 to 0 . 03 a 0 . In SID/Milgromian dynamics all the data points are expected to stay on (for the field galaxies) and below (for the dwarf galaxies near another gravitational source) the solid lines in Fig. 4: upper solid line for gas-rich galaxies and lower solid line for gas-poor galaxies.</text>
<text><location><page_12><loc_52><loc_5><loc_92><loc_7></location>Eq. 30 is a relation between the baryonic mass, M b, host , of a nearby galaxy or cluster of galaxies generating the</text>
<text><location><page_13><loc_9><loc_85><loc_49><loc_91></location>required external field and the distance to the centre of the nearby gravitational source, d . For a given g e , for example, as calculated from Eq. 30, d = v 2 host /g e , where v host = ( GM b, host a 0 ) 1 / 4 , thus</text>
<formula><location><page_13><loc_22><loc_79><loc_49><loc_82></location>d = √ GM b, host a 0 /g e . (31)</formula>
<text><location><page_13><loc_9><loc_26><loc_49><loc_63></location>For a comparison, we consider the Large Magellanic Cloud (LMC, dashed lines in Fig. 4, the upper line is for gasrich galaxies, i.e., with the assumption of M b = 2 M s , M g = M s , and the lower line is for gas-poor galaxies, i.e. with the assumption M b = M s , M g = 0), which is located at a Galactocentric distance of 49 . 5 kpc (Kallivayalil et al. 2006). The LMC is embedded in the gravitational background field of the Milky Way (MW). To calculate the external field strength of the MW at the position of the LMC, the baryonic Milky Way model with an interpolating function of 'simple' µ form is used from Wu et al. (2008). For the MW as a host galaxy, g e ≈ 0 . 17 a 0 for the LMC can be obtained from this model. All the 41 galaxies in Fig. 4 are above the halo to stellar mass relation of the LMC, since they are either isolated galaxies or embedded in external fields (see the 6 th column of Tab. 2) in general one order of magnitude smaller than that of the LMC. As a result, the phantom dark matter haloes have larger truncation radii, and the enclosed halo masses within the truncation radii are larger. The dashed line in Fig. 5 shows the d ( M b, host ) relation for the external field of the LMC, and the circle represents the values of d and M b, host for the LMC in the Milky Way. For the dwarf galaxies of Miller et al. (2014) within an external field, i.e., the magenta symbols in Fig. 4, if the nearby galaxy is a MilkyWay like galaxy, the distances between the galaxy sourcing the external field and the dwarfs are about 200 -400 kpc (see Fig. 5).</text>
<text><location><page_13><loc_9><loc_62><loc_49><loc_79></location>Here v host is the circular velocity of the host-galaxy's phantom dark matter halo generated by the host galaxy's baryonic mass, M b, host . Fig. 5 shows the d ( M b, host ) relation for the dwarf galaxies of Miller et al. (2014) assuming they are embedded in external fields, i.e., for the magenta symbols in Fig. 4, and assuming that for the dwarfs M b = M s , M g = 0. If the nearby galaxy is also a dwarf galaxy, with a baryonic mass of 10 9 M /circledot , the distances to the dwarf galaxies are only 40 -90 kpc, while if the nearby external field source is a rich cluster of galaxies with a baryonic mass of 10 13 M /circledot , the distances between the cluster centre and the Miller et al. (2013) dwarf galaxies are 4000 -9000 kpc.</text>
<text><location><page_13><loc_9><loc_5><loc_49><loc_25></location>SID/Milgromian dynamics predicts that for dwarf galaxies, the halo to baryonic mass relation has to stay in the range between the upper solid line (isolated galaxies) and the lower dashed line (galaxies embedded in an external field from a nearby massive galaxy) in Fig. 4. Thus it is possible to falsify MOND by future observations of the stellar to halo mass relation of dwarf galaxies. For example, an isolated field dwarf galaxy must appear near the solid lines of Fig. 4 since MOND would be falsified otherwise. Also, a self-gravitation satellite star cluster or dwarf galaxy will, as a result of the external field, compress as it orbits away from its host due to the build-up of its phantom dark matter halo, and conversely, it will expand on its orbit towards its host (Wu & Kroupa 2013). The same satellite will be larger in size near a more massive host. This may be the explanation</text>
<figure>
<location><page_13><loc_53><loc_66><loc_93><loc_91></location>
<caption>Figure 5. The d ( M b, host ) relation (Eq. 31) for the dwarf galaxies of Miller et al. (2014) assuming they are embedded in an external field. Here d is the distance between a dwarf galaxy in the Miller et al. (2014) sample (labeled magenta in Fig. 4, from top to bottom are for models 36, 25, 31, 7, 18, 26, 35, 14 in Tab. 4) and the centre of the external field source (solid lines). M b, host is the baryonic mass of the external field souce, which could be a nearby galaxy or a cluster of galaxies. The dashed line is the d ( M b, host ) relation within an external field of 0 . 17 a 0 , which is the strength of the external field at the LMC due to the Milky Way. The circle is the ( d, M b, host ) position of the LMC.</caption>
</figure>
<text><location><page_13><loc_52><loc_46><loc_91><loc_48></location>why the Andromeda satellites have larger radii compared to the satellite galaxies of the Milky Way (Collins et al. 2011).</text>
<section_header_level_1><location><page_13><loc_52><loc_42><loc_68><loc_43></location>5 CONCLUSIONS</section_header_level_1>
<text><location><page_13><loc_52><loc_35><loc_91><loc_40></location>The observed centrifugal acceleration and the Newtonian acceleration for disc galaxies are extremely strongly correlated. In the weak field regime, where g N << a 0 , the correlation follows readily from SID (Sec. 2).</text>
<text><location><page_13><loc_52><loc_12><loc_92><loc_35></location>We showed that the mass discrepancy and the acceleration are correlated for disc galaxies simulated in CDM and WDM cosmological models. But, the correlation does not agree with the observations. This indicates that the best simulated disc galaxies, which are delicately chosen to have had a quiet merger history, even so still do not represent the correct rotation curves. In any case, the fraction of real galaxies without a bulge is far larger than the fraction of model galaxies in the dark matter framework which have no significant merger history, thus making the dark matter framework unlikely to work. The here reported analysis thus adds to the growing evidence that cold or warm dark matter does not exist. If so, then dynamical friction between galaxies on their dark matter halos would not be evident in the galaxy population as galaxies would merge rarely (Kroupa 2014). Recent additional evidence for a lack of mergers has been independently found by Lena et al. (2014).</text>
<text><location><page_13><loc_52><loc_5><loc_92><loc_11></location>Moreover, the dispersion of the MDA correlation in the weak acceleration regime that is obtained in the CDM and the WDM simulations is much wider than for the observational data. The tight correlation between the observed centrifugal acceleration and the Newtonian acceleration from</text>
<text><location><page_14><loc_9><loc_71><loc_49><loc_91></location>the observed baryon-only masses implies a small scatter of the baryonic Tully-Fisher relation (McGaugh 2004), which has been confirmed by the more recent observations of McGaugh (2012). On the other hand, the wide spread of the relation in CDM and WDM simulations indicates a large scatter of the baryonic Tully-Fisher relation, which is incompatible with the observations. The large scatter in the simulated ΛCDM and the WDM galaxies is a necessary feature of these models because the dark matter haloes have various shapes and masses for a given baryonic galaxy, as already noted by Disney et al. (2008). Instead, the Newtonian observer detects unphysical (i.e., phantom) dark matter halo masses which are due to Milgromian rather than Newtonian dynamics in the weak field regime.</text>
<text><location><page_14><loc_9><loc_60><loc_49><loc_71></location>The halo to baryonic mass relation is studied for cosmologically simulated galaxies and for SID. We find that SID (and thus MOND) predicts an apparent (phantom) virial halo to baryonic mass relation which agrees well with observations. The simulated galaxies from CDM models fail to reproduce the observed halo-to-stellar mass relation, thereby constituting another major problem of the particle dark matter models.</text>
<text><location><page_14><loc_9><loc_31><loc_49><loc_60></location>Finally, SID is studied for systems falling in an external field, assuming the field does not vary much across the system. This is the case when a star cluster or a satellite dwarf galaxy orbits within a much larger host field. The external field truncates the phantom dark matter halo radius, and therewith reduces the mass of the phantom dark matter halo. A d ( M b, host ) relation is considered for the dwarf galaxies compiled by Miller et al. (2014) assuming they are embedded in an external field. SID predicts that for these galaxies there must be nearby gravitational sources like a companion galaxy or a cluster of galaxies. For each deviating dwarf galaxy, the distance to the external field source and the baryonic mass of the external field source follows a simple relation (Eq. 31), shown in Fig. 5. Also satellite galaxies near more massive hosts will have larger radii on average as a result of the truncation of their phantom dark matter halo masses due to the external field. The hitherto unexplained size difference between Andromeda and Milky Way satellites may be thus perhaps resolved. This implies that the gravitational forces acting between two galaxies vary with the position and mass of a third galaxy.</text>
<text><location><page_14><loc_9><loc_22><loc_49><loc_31></location>In summary, real galaxies follow SID/Milgromian dynamics rather than Newtonian dynamics, and reality is thus properly described by Milgromian dynamics (Famaey & McGaugh 2012; Kroupa 2012). Additional tests of Milgromian dynamics on star-cluster scales and on cosmological scales are required to further ascertain its range of validity.</text>
<section_header_level_1><location><page_14><loc_9><loc_15><loc_32><loc_16></location>6 ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_14><loc_9><loc_5><loc_49><loc_14></location>We thank Mordehai Milgrom for his helpful and important comments. We thank Stacy McGaugh for sharing his observational data of the mass discrepancy and acceleration of disc galaxies. Part of this work is done in ArgelanderInstitut fur Astronomie der Universitat Bonn, and Xufen Wu gratefully acknowledges support through the Alexander von Humboldt Foundation.</text>
<section_header_level_1><location><page_14><loc_52><loc_88><loc_77><loc_91></location>APPENDIX A: MILGROMIAN DYNAMICS/MOND</section_header_level_1>
<text><location><page_14><loc_52><loc_77><loc_92><loc_87></location>In this section a brief introduction to Milgromian dynamics (i.e. to Modified Newtonian Dynamics, MOND), is provided. The reader is referred to Famaey & McGaugh (2012) for a deep and thorough review of this topic. Essentially, MOND is the full classical description of gravitational dynamics encompassing the low-acceleration pure-SID (Sec. 2) regime and the Newtonian regime.</text>
<text><location><page_14><loc_52><loc_38><loc_92><loc_77></location>In addition to the major problems found in Λ cold dark matter (ΛCDM) cosmological simulations based on Newtonian/Einsteinian gravity (e.g. the cusp, the satellite over-prediction and the disc-of satellites problems, see Pawlowski et al. 2014; Ibata et al. 2014; Kroupa 2014, 2012; Springel et al. 2008; Klypin et al. 1999; Moore et al. 1999), it is also difficult to explain some puzzling conspiracies of C/WDM and baryons in galaxies by means of the current best ΛC/WDM models (Sanders 2009; Kroupa et al. 2010; Famaey & McGaugh 2012). For instance, there is an empirical relation between observed galaxy baryonic mass, i.e. the luminosity, and rotation speed. This is the baryonic Tully-Fisher relation (hereafter BTFR, Tully & Fisher 1977; McGaugh et al. 2000; McGaugh 2012), which cannot be naturally obtained in ΛCDM models (McGaugh 2012). Apart from the BTFR, there are other newly observed coincidences which are difficult to be reproduced in the simulated galaxies in the dark-matter framework, such as the observationallydeduced apparent dark matter content in the tidally formed dwarf galaxies (Gentile et al. 2007) and the discovery of a universal scale for the surface density of both the baryons and dark matter halo at the core radius of effective dark matter in a galaxy (Gentile et al. 2009; Milgrom 2009b). Indeed, the standard model of cosmology is in poor agreement with data and the hypothesis that C/WDM particles play a significant role in the universe has been seriously challenged if not ruled out (Sanders 2009; Kroupa et al. 2010; Kroupa 2012; Famaey & McGaugh 2012; Kroupa 2014).</text>
<text><location><page_14><loc_52><loc_5><loc_92><loc_38></location>Milgromian dynamics was originally proposed (Milgrom 1983c,a,b) to account for gravitational dynamics in the classical regime without introducing DM. With MOND, Milgrom extends our understanding of effective gravitational dynamics beyond the gravitational-dynamical systems known in 1916. Bekenstein & Milgrom (1984) demonstrated that MOND conserves momentum, energy and angular momentum in a self-gravitating system. In MOND the dynamical acceleration, g = | g | = √ g N a 0 , takes the place of Newtonian acceleration, g N , in the weak field regime where g /lessmuch a 0 and which is the regime of pure-SID (Sec. 2); while dynamical acceleration approaches Newtonian acceleration in the strong field regime, i.e., g = g N when g /greatermuch a 0 . The constant acceleration, a 0 /similarequal 1 . 21 × 10 -10 ms -2 ≈ 3 . 8 pc/Myr 2 , is the critical value of acceleration which switches dynamics between Newtonian and Milgromian (e.g., Milgrom 1983c; Bekenstein & Milgrom 1984; Begeman et al. 1991; Kent 1987; Milgrom 1988; McGaugh 2011, 2012; Sanders & McGaugh 2002; Bekenstein 2006; Milgrom 2008). It is found to be in coincidence with various constants of cosmology, such as a 0 ≈ cH 0 / 2 π , a 0 ≈ c (Λ / 3) 1 / 2 / 2 π , where c is the speed of light in vacuum, H 0 is the local Hubble constant and Λ is the cosmological constant (Milgrom 1983c, 1989, 2009c, 2014b).</text>
<text><location><page_15><loc_9><loc_67><loc_49><loc_91></location>MOND has until now passed all tests over a wide range of scales in different types of galaxies and naturally accounts for the aforementioned observations (Milgrom 1983c; Bekenstein & Milgrom 1984; McGaugh 2011, 2012; Milgrom & Sanders 2003; Sanders & Noordermeer 2007; Gentile et al. 2007, 2009; McGaugh 2004; Milgrom 2009b), such as the BTFR and the apparent dark matter content in tidal dwarf galaxies. Moreover, MOND is very successful in explaining the vertical kinematics of stars in galactic discs in the absence of dark matter (Bienaym'e et al. 2009) and naturally accounts for the faster rotational speeds of polar rings (Lughausen et al. 2013). Furthermore, there are other new covariant theories equivalent to MOND at their non-relativistic limit (Bekenstein 2004; Zlosnik et al. 2007; Bruneton & Esposito-Far'ese 2007; Zhao 2007; Sanders 2005; Skordis 2008; Skordis & Zlosnik 2012; Halle et al. 2008; Milgrom 2009a).</text>
<section_header_level_1><location><page_15><loc_9><loc_60><loc_41><loc_64></location>APPENDIX B: VIRIAL MASSES CALCULALTED BASED ON MILLER'S EMPIRICAL RELATION</section_header_level_1>
<text><location><page_15><loc_9><loc_41><loc_49><loc_59></location>The virial masses of particle dark matter halo masses cannot be measured directly. However, because the rotation curves of late-type galaxies are about constant to large radii, the masses can be estimated by measuring the rotation speed within the outer regions of the luminous galaxy component, which corresponds to the inner halo region (Sec. 4.1). Considering that the virial masses of the haloes in Miller et al. (2014) are converted from the inner circular velocity, V 2 . 2 , of the galaxies, we now apply the same method to calculate the virial masses for the same galaxies in MOND. The total stellar masses, M s , and V 2 . 2 are known in Miller et al. (2014), and they follow a simple fitting function (Miller et al. 2011, 2014) 7 ,</text>
<formula><location><page_15><loc_16><loc_36><loc_49><loc_39></location>log 10 ( M s M /circledot ) = [ a + b log 10 ( V 2 . 2 kms -1 ) ] . (B1)</formula>
<text><location><page_15><loc_9><loc_31><loc_49><loc_36></location>Here a = 0 . 57 ± 0 . 48 in M s and b = 4 . 35 ± 0 . 62. From Eq. 6 we know that M b = V 4 200 Ga 0 . Thus for gas-poor galaxies, the relation is</text>
<formula><location><page_15><loc_18><loc_26><loc_49><loc_29></location>log 10 ( V 4 200 Ga 0 ) = a + b log 10 ( V 2 . 2 ); (B2)</formula>
<text><location><page_15><loc_9><loc_23><loc_48><loc_26></location>while for gas-rich galaxies, M s = M b / 2 = V 4 200 2 Ga 0 , the relation is</text>
<formula><location><page_15><loc_18><loc_17><loc_49><loc_21></location>log 10 ( V 4 200 2 Ga 0 ) = a + b log 10 ( V 2 . 2 ) . (B3)</formula>
<text><location><page_15><loc_9><loc_15><loc_49><loc_17></location>Combining with Eq. 19, the virial masses of the galaxies in the sample of Miller et al. (2014) are:</text>
<formula><location><page_15><loc_10><loc_9><loc_49><loc_13></location>log 10 ( M vir h ) = 3 4 [ a + b log 10 ( V 2 . 2 )+log 10 ( a 0 )] , (gas -poor);</formula>
<text><location><page_15><loc_9><loc_5><loc_49><loc_7></location>7 The last term on the right hand side of Eq. 5 in Miller et al. (2011) should not exist.</text>
<formula><location><page_15><loc_53><loc_87><loc_93><loc_91></location>log 10 ( M vir h ) = 3 4 [ a + b log 10 ( V 2 . 2 )+log 10 (2 a 0 )] , (gas -rich) . (B4)</formula>
<text><location><page_15><loc_52><loc_77><loc_91><loc_87></location>However, the fitting parameters for the Tully-Fisher relation introduced by Miller et al. (2011, 2014) are strongly sampledependent, and the errors for V 2 . 2 are large and up to 60% (see Table 1 in Miller et al. 2014). Hence the virial masses calculated with Eq. B4 are unreliable. Thus we do not calculate the virial masses of the galaxies using Eq. 22 and we use Eq. B4 instead.</text>
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2015MNRAS.447.2479S | https://arxiv.org/pdf/1412.2093.pdf | <document>
<section_header_level_1><location><page_1><loc_9><loc_88><loc_80><loc_90></location>A fast algorithm for estimating actions in triaxial potentials</section_header_level_1>
<section_header_level_1><location><page_1><loc_9><loc_83><loc_45><loc_85></location>Jason L. Sanders 1 , 2 glyph[star] &James Binney 1</section_header_level_1>
<text><location><page_1><loc_10><loc_81><loc_53><loc_83></location>1 Rudolf Peierls Centre for Theoretical Physics, Keble Road, Oxford, OX1 3NP, UK 2</text>
<text><location><page_1><loc_10><loc_80><loc_43><loc_81></location>Institute of Astronomy, Madingley Road, Cambridge, CB3 0HA</text>
<text><location><page_1><loc_9><loc_76><loc_16><loc_77></location>10 June 2022</text>
<section_header_level_1><location><page_1><loc_31><loc_72><loc_39><loc_73></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_31><loc_56><loc_91><loc_72></location>We present an approach to approximating rapidly the actions in a general triaxial potential. The method is an extension of the axisymmetric approach presented by Binney (2012a), and operates by assuming that the true potential is locally sufficiently close to some Stackel potential. The choice of Stackel potential and associated ellipsoidal coordinates is tailored to each individual input phase-space point. We investigate the accuracy of the method when computing actions in a triaxial Navarro-Frenk-White potential. The speed of the algorithm comes at the expense of large errors in the actions, particularly for the box orbits. However, we show that the method can be used to recover the observables of triaxial systems from given distribution functions to sufficient accuracy for the Jeans equations to be satisfied. Consequently, such models could be used to build models of external galaxies as well as triaxial components of our own Galaxy. When more accurate actions are required, this procedure can be combined with torus mapping to produce a fast convergent scheme for action estimation.</text>
<text><location><page_1><loc_31><loc_52><loc_91><loc_54></location>Key words: methods: numerical - Galaxy: kinematics and dynamics - galaxies: kinematics and dynamics</text>
<section_header_level_1><location><page_1><loc_9><loc_46><loc_23><loc_47></location>1 INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_30><loc_48><loc_45></location>The haloes that form in baryon-free cosmological simulations almost always have triaxial shapes (Jing & Suto 2002; Allgood et al. 2006; Vera-Ciro et al. 2011). When baryons are added to the simulations, many dark haloes become more spherical (Kazantzidis et al. 2004; Bailin et al. 2005; Valluri et al. 2010), but the most successful current models suggest our Galaxy's dark halo is triaxial (Law & Majewski 2010; Vera-Ciro & Helmi 2013). Moreover, there is considerable observational evidence that the so-called 'cored', slowly-rotating elliptical galaxies are generically triaxial (Cappellari et al. 2011). Hence dynamical models of triaxial stellar systems are of considerable astronomical interest.</text>
<text><location><page_1><loc_9><loc_12><loc_48><loc_30></location>The first triaxial models were made by violent relaxation of an N -body model (Aarseth & Binney 1978), and these models prompted Schwarzschild (1979) to develop the technique of orbit superposition so triaxial models with prescribed density profiles could be constructed. Schwarzschild's work gave significant insight into how triaxial systems work for the first time, and this insight was enhanced by de Zeeuw (1985), who showed that Stackel potentials provided analytic models of orbits in a very interesting class of triaxial systems. The most important subsequent development in the study of triaxial systems was the demonstration by Merritt & Valluri (1999) that when a triaxial system lacks a homogeneous core, as real galaxies do, box orbits tend to become centrophobic resonant box orbits.</text>
<text><location><page_1><loc_12><loc_11><loc_48><loc_12></location>Work on axisymmetric models in the context of our Galaxy</text>
<text><location><page_1><loc_52><loc_24><loc_91><loc_47></location>has increased awareness of the value in stellar dynamics of the intimately related concepts of the Jeans' theorem and action integrals. Ollongren (1962) established that the space of quasi-periodic orbits in galactic potentials is three-dimensional. Jeans' theorem tells us that any non-negative function f on this space provides an equilibrium stellar system. The key to gaining access to the observable properties of this tantalising array of stellar systems, is finding a practical coordinate system for orbit space. A coordinate system for orbit space comprises a set of three functions I i ( x , v ) that are constant along any orbit in the gravitational potential Φ( x ) of the equilibrium system. A major difficulty is that in the case of a selfconsistent system Φ( x ) has to be determined from f ( I ) by computing the model's density, and the latter can be computed only when Φ( x ) is known. Hence the computation of Φ( x ) has to be done iteratively, and expressions are needed for the I i that are valid in any reasonable potential Φ( x ) , not merely the potential of the equilibrium model.</text>
<text><location><page_1><loc_52><loc_7><loc_91><loc_23></location>If the system is axisymmetric, the energy E and component of angular momentum L z are integrals that are defined for any axisymmetric potential Φ( R,z ) , and equilibrium models of axisymmetric systems have been constructed from distribution functions (DFs) of the form f ( E,L z ) (Prendergast & Tomer 1970; Wilson 1975; Rowley 1988). However, these two-integral models are not generic, and they are much harder to construct than generic models when the DF is specified as a function f ( J ) of the actions (Binney 2014). Moreover, knowledge of the DF as a function of the actions is the key to Hamiltonian perturbation theory, and the ability to perturb models is crucial if we are to really understand how galaxies work, and evolve over time. Actions are also the key to</text>
<text><location><page_2><loc_9><loc_89><loc_48><loc_92></location>modelling stellar streams, which are themselves promising probes of our Galaxy's distribution of dark matter (Tremaine 1999; Helmi &White 1999; Eyre & Binney 2011; Sanders & Binney 2013).</text>
<text><location><page_2><loc_9><loc_79><loc_48><loc_88></location>Action integrals constitute uniquely advantageous coordinates for orbit space, which is often called action space because its natural Cartesian coordinates are the actions. Actions can be defined for any quasi-periodic orbit and, uniquely among isolating integrals, they can be complemented by canonically conjugate coordinates, the angle variables θ i . These have the convenient properties of (i) increasing linearly in time, so</text>
<formula><location><page_2><loc_9><loc_77><loc_24><loc_78></location>θ i ( t ) = θ i (0) + Ω i ( J ) t,</formula>
<text><location><page_2><loc_9><loc_72><loc_48><loc_76></location>and (ii) being periodic such that any ordinary phase-space coordinate such as x satisfies x ( θ , J ) = x ( θ +2 π m , J ) , where m is any triple of integers.</text>
<text><location><page_2><loc_9><loc_44><loc_48><loc_71></location>The discussion above amply motivates the quest for algorithms that yield angle-action coordinates ( θ , J ) given ordinary phase-space coordinates ( x , v ) , and vice versa. These algorithms are usefully divided into convergent and non-convergent algorithms. Convergent algorithms yield approximations to the desired quantity that can achieve any desired accuracy given sufficient computational resource, whereas non-convergent algorithms provide, more cheaply, an approximation of uncontrolled accuracy. Torus mapping (Kaasalainen & Binney 1994; McMillan & Binney 2008) is a convergent algorithm that yields x ( θ , J ) and v ( θ , J ) , while Sanders & Binney (2014) introduced a convergent algorithm for θ ( x , v ) and J ( x , v ) . Both algorithms work by constructing the generating function for the canonical mapping of some 'toy' analytic system of angle-action variables into the real phase space. Torus mapping has been demonstrated only in two dimensions, but both axisymmetric and two-dimensional static barred potentials have been successfully handled, and there is no evident obstacle to generalising to the three-dimensional case. Sanders & Binney (2014) treated the triaxial case, but the restriction to lower dimensions and axisymmetry is trivial.</text>
<text><location><page_2><loc_9><loc_18><loc_48><loc_44></location>These convergent algorithms are numerically costly, and, in the axisymmetric case, non-convergent algorithms have been used extensively, especially for extracting observables from a DF f ( J ) . These extractions require J to be evaluated at very many phasespace points, and speed is more important than accuracy. The adiabatic approximation (Binney 2010) has been extensively used in modelling the solar neighbourhood (e.g. Schonrich & Binney 2009) but its validity is restricted to orbits that keep close to the Galactic plane. Stackel fitting (Sanders 2012) has been successfully used to model stellar streams (Sanders 2014). This method estimates the actions as those in the best-fitting Stackel potential for the local region a give orbit probes. Sanders (2015, in prep.) shows that is is less cost-effective than the 'Stackel fudge' that was introduced by Binney (2012a). Binney (2012b) used the 'Stackel fudge' to model the solar neighbourhood and to explore the first family of self-consistent stellar systems with specified f ( J ) (Binney 2014). The Stackel fudge was recently used by Piffl et al. (2014) to place by far the strongest available constraints on the Galaxy's dark halo. In this paper we extend the Stackel fudge to triaxial systems.</text>
<text><location><page_2><loc_9><loc_7><loc_48><loc_17></location>We begin in Section 2 by showing how to find the actions in a triaxial Stackel potential. In Section 3 we extend the Stackel fudge to general triaxial potentials. In Section 4 we apply this algorithm to a series of orbits in a triaxial Navarro-Frenk-White (NFW) potential, and in Section 5 we construct the first triaxial stellar systems with specified DFs f ( J ) , and demonstrate that, notwithstanding the uncontrolled nature of the fudge as an approximation, the models satisfy the Jeans equations to good accuracy. In Section 6 we de-sc</text>
<text><location><page_2><loc_52><loc_90><loc_91><loc_92></location>be a new convergent algorithm for obtaining ( J , θ ) from ( x , v ) . Finally we conclude in Section 7.</text>
<section_header_level_1><location><page_2><loc_52><loc_85><loc_79><loc_86></location>2 TRIAXIAL ST ACKEL POTENTIALS</section_header_level_1>
<text><location><page_2><loc_52><loc_80><loc_91><loc_84></location>In this section, we show how actions can be found in a triaxial Stackel potential. The presentation here follows that given by de Zeeuw (1985).</text>
<section_header_level_1><location><page_2><loc_52><loc_76><loc_70><loc_77></location>2.1 Ellipsoidal coordinates</section_header_level_1>
<text><location><page_2><loc_52><loc_71><loc_91><loc_74></location>Triaxial Stackel potentials are expressed in terms of ellipsoidal coordinates ( λ, µ, ν ) . These coordinates are related to the Cartesian coordinates ( x, y, z ) as the three roots of the cubic in τ</text>
<formula><location><page_2><loc_52><loc_67><loc_91><loc_70></location>x 2 ( τ + α ) + y 2 ( τ + β ) + z 2 ( τ + γ ) = 1 , (1)</formula>
<text><location><page_2><loc_52><loc_50><loc_91><loc_66></location>where α , β and γ are constants defining the coordinate system. For the potential explored later, we choose to set x as the major axis, y as the intermediate axis and z as the minor axis, such that -γ glyph[lessorequalslant] ν glyph[lessorequalslant] -β glyph[lessorequalslant] µ glyph[lessorequalslant] -α glyph[lessorequalslant] λ . Surfaces of constant λ are ellipsoids, surfaces of constant µ are hyperboloids of one sheet (flared tubes of elliptical cross section that surround the x axis), and surfaces of constant ν are hyperboloids of two sheets that have their extremal point on the z axis. In the plane z = 0 , lines of constant λ are ellipses with foci at y = ± ∆ 1 ≡ ± √ β -α , whilst, in the plane x = 0 , lines of constant µ are ellipses with foci at z = ± ∆ 2 ≡ ± √ γ -β . The expressions for the Cartesian coordinates as a function of the ellipsoidal coordinates are</text>
<formula><location><page_2><loc_52><loc_40><loc_91><loc_48></location>x 2 = ( λ + α )( µ + α )( ν + α ) ( α -β )( α -γ ) , y 2 = ( λ + β )( µ + β )( ν + β ) ( β -α )( β -γ ) , z 2 = ( λ + γ )( µ + γ )( ν + γ ) ( γ -β )( γ -α ) . (2)</formula>
<text><location><page_2><loc_52><loc_33><loc_91><loc_39></location>Note that a Cartesian coordinate ( x, y, z ) gives a unique ( λ, µ, ν ) , whilst the point ( λ, µ, ν ) corresponds to eight points in ( x, y, z ) . Therefore, we will only consider potentials with this symmetry i.e. triaxial potentials with axes aligned with the Cartesian axes.</text>
<text><location><page_2><loc_52><loc_29><loc_91><loc_33></location>The generating function, S , to take us between Cartesian, ( x, y, z, p x , p y , p z ) , and ellipsoidal coordinates, ( λ, µ, ν, p λ , p µ , p ν ) , is</text>
<formula><location><page_2><loc_52><loc_25><loc_92><loc_28></location>S ( p x , p y , p z , λ, µ, ν ) = p x x ( λ, µ, ν )+ p y y ( λ, µ, ν )+ p z z ( λ, µ, ν ) . (3)</formula>
<text><location><page_2><loc_52><loc_22><loc_76><loc_24></location>Using p τ = ∂ S/ ∂ τ we find, for instance,</text>
<formula><location><page_2><loc_52><loc_10><loc_91><loc_21></location>p λ = p x 2 √ ( µ + α )( ν + α ) ( α -β )( α -γ )( λ + α ) + p y 2 √ ( µ + β )( ν + β ) ( β -α )( β -γ )( λ + β ) + p z 2 √ ( µ + γ )( ν + γ ) ( γ -α )( γ -β )( λ + γ ) . (4)</formula>
<text><location><page_2><loc_52><loc_7><loc_91><loc_9></location>There are similar equations for p µ and p ν . Inversion of these three equations gives us expressions for p x , p y and p z as functions of</text>
<text><location><page_3><loc_9><loc_90><loc_48><loc_92></location>p τ and τ . For a general triaxial potential, Φ , we can express the Hamiltonian, H , in terms of the ellipsoidal coordinates as</text>
<formula><location><page_3><loc_9><loc_84><loc_48><loc_89></location>H = 1 2 ( p 2 x + p 2 y + p 2 z ) + Φ( x, y, z ) , = 1 2 ( p 2 λ P 2 λ + p 2 µ P 2 µ + p 2 ν P 2 ν ) +Φ( λ, µ, ν ) . (5)</formula>
<text><location><page_3><loc_9><loc_82><loc_13><loc_83></location>where</text>
<formula><location><page_3><loc_9><loc_72><loc_48><loc_81></location>P 2 λ = ( λ -µ )( λ -ν ) 4( λ + α )( λ + β )( λ + γ ) , P 2 µ = ( µ -ν )( µ -λ ) 4( µ + α )( µ + β )( µ + γ ) , P 2 ν = ( ν -µ )( ν -λ ) 4( ν + α )( ν + β )( ν + γ ) . (6)</formula>
<section_header_level_1><location><page_3><loc_9><loc_68><loc_24><loc_69></location>2.2 Stackel potentials</section_header_level_1>
<text><location><page_3><loc_9><loc_66><loc_47><loc_67></location>The most general triaxial Stackel potential, Φ S , can be written as</text>
<formula><location><page_3><loc_9><loc_60><loc_50><loc_64></location>Φ S ( λ, µ, ν ) = f ( λ ) ( λ -µ )( ν -λ ) + f ( µ ) ( µ -ν )( λ -µ ) + f ( ν ) ( ν -λ )( µ -ν ) (7)</formula>
<text><location><page_3><loc_9><loc_47><loc_48><loc_59></location>Φ S is composed of three functions of one variable. Here we denote the three functions with the same letter, f , as their domains are distinct. Additionally, f ( τ ) must be differentiable everywhere and continuous at τ = -α and τ = -β for Φ S to be finite at λ = µ = -α and µ = ν = -β . With this form for the potential we can solve the Hamilton-Jacobi equation (de Zeeuw 1985). We write p τ = ∂ W/ ∂ τ and equate the Hamiltonian to the total energy, E , in equation (5). We then multiply through by ( λ -µ )( µ -ν )( ν -λ ) to find</text>
<formula><location><page_3><loc_9><loc_36><loc_47><loc_46></location>( ν -µ ) ( 2( λ + α )( λ + β )( λ + γ ) ( ∂ W ∂ λ ) 2 -f ( λ ) -λ 2 E ) +( λ -ν ) ( 2( µ + α )( µ + β )( µ + γ ) ( ∂ W ∂ µ ) 2 -f ( µ ) -µ 2 E ) +( µ -λ ) ( 2( ν + α )( ν + β )( ν + γ ) ( ∂ W ∂ ν ) 2 -f ( ν ) -ν 2 E ) = 0 .</formula>
<text><location><page_3><loc_47><loc_34><loc_48><loc_35></location>(8)</text>
<text><location><page_3><loc_9><loc_31><loc_38><loc_33></location>We make the Ansatz W = ∑ τ W τ ( τ ) and define</text>
<formula><location><page_3><loc_9><loc_28><loc_48><loc_31></location>U ( τ ) = 2( τ + α )( τ + β )( τ + γ ) ( ∂ W ∂ τ ) 2 -f ( τ ) -τ 2 E, (9)</formula>
<text><location><page_3><loc_9><loc_26><loc_38><loc_27></location>such that the Hamilton-Jacobi equation becomes</text>
<formula><location><page_3><loc_9><loc_24><loc_48><loc_25></location>( ν -µ ) U ( λ ) + ( λ -ν ) U ( µ ) + ( µ -λ ) U ( ν ) = 0 . (10)</formula>
<text><location><page_3><loc_9><loc_20><loc_48><loc_23></location>Taking the second derivative of this expression with respect to τ = { λ, µ, ν } we find that</text>
<formula><location><page_3><loc_9><loc_18><loc_48><loc_19></location>U ( τ ) = aτ -b, (11)</formula>
<text><location><page_3><loc_9><loc_14><loc_48><loc_17></location>where a and b are constants. Therefore, the equations for the momenta can be written as</text>
<formula><location><page_3><loc_9><loc_12><loc_48><loc_13></location>2( τ + α )( τ + β )( τ + γ ) p 2 τ = τ 2 E -τa + b + f ( τ ) . (12)</formula>
<text><location><page_3><loc_9><loc_7><loc_48><loc_10></location>For an initial phase-space point, ( x 0 , v 0 ) , we find τ 0 ( x 0 , v 0 ) and p τ 0 ( x 0 , v 0 ) using the coordinate transformations and can then find the integrals a and b by solving equation (12) (see de Zeeuw 1985,</text>
<text><location><page_3><loc_50><loc_63><loc_51><loc_64></location>.</text>
<table>
<location><page_3><loc_52><loc_68><loc_91><loc_86></location>
<caption>Table 1. Actions in a triaxial Stackel potential. We give the limits of the action integrals and the physical meaning of each of the actions for each of the four orbit classes. The numbers in brackets after the orbit class are the orbit classification numbers used in Section 4.2.</caption>
</table>
<text><location><page_3><loc_52><loc_62><loc_91><loc_66></location>for more details). These integrals are related to the classical integrals I 2 and I 3 in a simple way. As p τ is only a function of τ , the actions are then given by the 1D integrals</text>
<formula><location><page_3><loc_52><loc_58><loc_91><loc_61></location>J τ = 2 π ∫ τ + τ -d τ | p τ ( τ ) | . (13)</formula>
<text><location><page_3><loc_52><loc_48><loc_91><loc_57></location>where ( τ -, τ + ) are the roots of p τ ( τ ) = 0 , which we find by using Brent's method to find points where the right side of equation (12) vanishes. Note that for loop orbits we must divide the 'radial' action by two ( J λ for the short-axis loops and outer long-axis loops, J µ for the inner long-axis loops). In Table 1, we give the limits ( τ -, τ + ) of the action integrals and the physical meaning of each of the actions for each of the four orbit classes.</text>
<text><location><page_3><loc_52><loc_42><loc_91><loc_47></location>The approach to finding the actions presented here requires an explicit form for f . In the next section we will show how we can circumnavigate the need for this explicit form, which allows us to use the same equations for a general potential.</text>
<section_header_level_1><location><page_3><loc_52><loc_38><loc_78><loc_39></location>3 THE TRIAXIAL ST ACKEL FUDGE</section_header_level_1>
<text><location><page_3><loc_52><loc_30><loc_91><loc_36></location>We now show how we can use the insights from Stackel potentials to estimate actions in a more general potential. For a general triaxial potential, Φ , we can attempt to find the actions by assuming that the general potential is close to a Stackel potential. Given a general potential we define the quantities</text>
<formula><location><page_3><loc_52><loc_24><loc_91><loc_29></location>χ λ ( λ, µ, ν ) ≡ ( λ -µ )( ν -λ )Φ( λ, µ, ν ) , χ µ ( λ, µ, ν ) ≡ ( µ -ν )( λ -µ )Φ( λ, µ, ν ) , χ ν ( λ, µ, ν ) ≡ ( ν -λ )( µ -ν )Φ( λ, µ, ν ) . (14)</formula>
<text><location><page_3><loc_52><loc_19><loc_91><loc_23></location>where we have chosen a particular coordinate system, ( α, β, γ ) (see § 4.1). If Φ were a Stackel potential, these quantities would be given by, for instance,</text>
<formula><location><page_3><loc_52><loc_16><loc_91><loc_18></location>χ λ ( λ, µ, ν ) = f ( λ ) -λ f ( µ ) -f ( ν ) µ -ν + νf ( µ ) -µf ( ν ) µ -ν . (15)</formula>
<text><location><page_3><loc_52><loc_14><loc_80><loc_15></location>Therefore, for a general potential, we can write</text>
<formula><location><page_3><loc_52><loc_12><loc_91><loc_13></location>f ( τ ) ≈ χ τ ( λ, µ, ν ) + C τ τ + D τ , (16)</formula>
<text><location><page_3><loc_52><loc_7><loc_91><loc_10></location>where C τ and D τ are constants provided we always evaluate χ τ with two of the ellipsoidal coordinates fixed. For instance, we always evaluate χ λ at fixed µ and ν .</text>
<text><location><page_4><loc_9><loc_94><loc_10><loc_96></location>4</text>
<text><location><page_4><loc_13><loc_94><loc_33><loc_96></location>J. L. Sanders & J. Binney</text>
<text><location><page_4><loc_9><loc_90><loc_48><loc_92></location>When we substitute these expressions into equation (12) we find</text>
<formula><location><page_4><loc_9><loc_88><loc_48><loc_89></location>2( τ + α )( τ + β )( τ + γ ) p 2 τ = τ 2 E -τA τ + B τ + χ τ ( λ, µ, ν ) . (17)</formula>
<text><location><page_4><loc_9><loc_84><loc_48><loc_87></location>For each τ coordinate there are two new integrals of motion given by A τ = a -C τ and B τ = b + D τ .</text>
<text><location><page_4><loc_9><loc_79><loc_48><loc_84></location>Given an initial phase-space point, ( x 0 , v 0 ) , and a coordinate system, ( α, β, γ ) , we can calculate the ellipsoidal coordinates ( λ 0 , µ 0 , ν 0 , p λ 0 , p µ 0 , p ν 0 ) . Inserting this initial phase-space point into equation (17) gives us an expression for B τ as</text>
<formula><location><page_4><loc_9><loc_75><loc_48><loc_78></location>B τ = 2( τ 0 + α )( τ 0 + β )( τ 0 + γ ) p 2 τ 0 -τ 2 0 E + τ 0 A τ -χ τ ( λ 0 , µ 0 , ν 0 ) . (18)</formula>
<text><location><page_4><loc_9><loc_64><loc_48><loc_73></location>It remains to find an expression for A τ as a function of the initial phase-space point. To proceed we consider the derivative of the Hamiltonian with respect to τ . In a Stackel potential we can stay on the orbit while changing τ and p τ ( τ ) with all the other phase-space variables held constant. Therefore, in a Stackel potential ∂ H/ ∂ τ = 0 . Here we consider ∂ H/ ∂ λ and will give the results for µ and ν afterwards. Using equation (5) we write</text>
<formula><location><page_4><loc_9><loc_57><loc_48><loc_63></location>0 = ( ∂ H ∂ λ ) µ,ν = 1 2 ∂ ∂ λ [ p 2 λ P 2 λ ] + 1 2 p 2 µ ( µ -λ ) P 2 µ + 1 2 p 2 ν ( ν -λ ) P 2 ν + ∂ Φ ∂ λ . (19)</formula>
<text><location><page_4><loc_9><loc_55><loc_42><loc_57></location>To evaluate ∂ [ p 2 λ /P 2 λ ] / ∂ λ we use equation (18) to write</text>
<formula><location><page_4><loc_9><loc_49><loc_49><loc_54></location>2( λ + α )( λ + β )( λ + γ ) p 2 λ = 2( λ 0 + α )( λ 0 + β )( λ 0 + γ ) p 2 λ 0 +( λ 2 -λ 2 0 ) E -( λ -λ 0 ) A λ -χ λ ( λ, µ 0 , ν 0 ) + χ λ ( λ 0 , µ 0 , ν 0 ) , (20)</formula>
<text><location><page_4><loc_9><loc_47><loc_14><loc_48></location>such that</text>
<formula><location><page_4><loc_9><loc_43><loc_48><loc_46></location>p 2 λ P 2 λ = Q +( λ 2 -λ 2 0 ) E -( λ -λ 0 ) A λ ( λ -µ )( λ -ν ) -Φ( λ, µ 0 , ν 0 ) , (21)</formula>
<text><location><page_4><loc_9><loc_42><loc_13><loc_43></location>where</text>
<formula><location><page_4><loc_9><loc_39><loc_48><loc_41></location>Q = 2( λ 0 + α )( λ 0 + β )( λ 0 + γ ) p 2 λ 0 + χ λ ( λ 0 , µ 0 , ν 0 ) . (22)</formula>
<text><location><page_4><loc_9><loc_35><loc_48><loc_38></location>Upon substitution into equation (19) we note that the derivatives of Φ cancel. Therefore, evaluating ∂ H/ ∂ λ at the initial phase-space point we find</text>
<formula><location><page_4><loc_9><loc_27><loc_48><loc_34></location>A λ =2 λ 0 E -(2 λ 0 -µ 0 -ν 0 ) ( Φ( λ 0 , µ 0 , ν 0 ) + 1 2 p 2 λ 0 P 2 λ 0 ) -1 2 p 2 µ 0 ( λ 0 -ν 0 ) P 2 µ 0 -1 2 p 2 ν 0 ( λ 0 -µ 0 ) P 2 ν 0 . (23)</formula>
<text><location><page_4><loc_9><loc_26><loc_28><loc_27></location>This can be simplified further to</text>
<formula><location><page_4><loc_9><loc_22><loc_48><loc_25></location>A λ = ( µ 0 + ν 0 ) E + 1 2 p 2 µ 0 ( λ 0 -µ 0 ) P 2 µ 0 + 1 2 p 2 ν 0 ( λ 0 -ν 0 ) P 2 ν 0 . (24)</formula>
<text><location><page_4><loc_9><loc_17><loc_48><loc_21></location>Note that A λ is independent of λ 0 and p λ 0 (except implicitly in the energy, E ) as P τ 0 contains cancelling factors of ( λ 0 -τ 0 ) . Similarly</text>
<formula><location><page_4><loc_9><loc_10><loc_48><loc_16></location>A µ = ( λ 0 + ν 0 ) E + 1 2 p 2 λ 0 ( µ 0 -λ 0 ) P 2 λ 0 + 1 2 p 2 ν 0 ( µ 0 -ν 0 ) P 2 ν 0 , A ν = ( λ 0 + µ 0 ) E + 1 2 p 2 λ 0 ( ν 0 -λ 0 ) P 2 λ 0 + 1 2 p 2 µ 0 ( ν 0 -µ 0 ) P 2 µ 0 . (25)</formula>
<text><location><page_4><loc_9><loc_7><loc_48><loc_9></location>For a true Stackel potential , given an initial phase-space point we can find 6 integrals of motion, ( A λ , A µ , A ν , B λ , B µ , B ν ) from</text>
<text><location><page_4><loc_52><loc_82><loc_91><loc_92></location>equations (18), (24) and (25). Note that a general Stackel potential only admits three integrals of motion so the 6 derived integrals of motion are not independent. This procedure gives identical results to evaluating the integrals as in de Zeeuw (1985). Note that the expressions for these integrals do not explicitly involve the function f ( τ ) - they only involve the potential, Φ . With the integrals of motion calculated we are in a position to find p τ ( τ ) and hence the actions from equation (13).</text>
<text><location><page_4><loc_52><loc_71><loc_91><loc_81></location>For a general potential we may find six approximate integrals of motion using the same equations, and hence estimate the actions. In this case, although the potential may admit only three true integrals of motion, the 6 approximate integrals of motion are independent estimates of true integrals of motion. Again, as the expressions do not require f ( τ ) they can be evaluated for a general potential. In Appendix A we show how the angles and frequencies can be estimated using the same approach.</text>
<section_header_level_1><location><page_4><loc_52><loc_67><loc_74><loc_68></location>3.1 Relation to axisymmetric case</section_header_level_1>
<text><location><page_4><loc_52><loc_59><loc_91><loc_65></location>The above procedure extends the work of Binney (2012a). Binney (2012a) constructed the 'Stackel fudge' algorithm for estimating actions in a general axisymmetric potential Φ( R,z ) , where R and z are the usual cylindrical polar coordinates. We now relate the procedure to that of Binney (2012a) to develop further understanding.</text>
<text><location><page_4><loc_52><loc_56><loc_91><loc_58></location>Oblate axisymmetric Stackel potentials are associated with prolate elliptic coordinates ( λ, ν ) given by the roots for τ of</text>
<formula><location><page_4><loc_52><loc_52><loc_91><loc_55></location>R 2 τ + α + z 2 τ + γ = 1 , (26)</formula>
<text><location><page_4><loc_52><loc_49><loc_91><loc_51></location>where -γ glyph[lessorequalslant] ν glyph[lessorequalslant] -α glyph[lessorequalslant] λ . Binney (2012a) uses the coordinates ( u, v ) which are related to ( λ, ν ) via</text>
<formula><location><page_4><loc_52><loc_43><loc_91><loc_48></location>sinh 2 u = λ + α γ -α , cos 2 v = ν + γ γ -α , (27)</formula>
<formula><location><page_4><loc_52><loc_37><loc_91><loc_42></location>such that R = √ γ -α sinh u sin v, z = √ γ -α cosh u cos v. (28)</formula>
<text><location><page_4><loc_55><loc_35><loc_90><loc_36></location>An oblate axisymmetric Stackel potential can be written as</text>
<formula><location><page_4><loc_52><loc_31><loc_91><loc_34></location>Φ S ( λ, ν ) = -f ( λ ) -f ( ν ) λ -ν , (29)</formula>
<text><location><page_4><loc_52><loc_29><loc_90><loc_30></location>and the equations for the momenta are given by (de Zeeuw 1985)</text>
<formula><location><page_4><loc_52><loc_26><loc_91><loc_28></location>2( τ + α )( τ + γ ) p 2 τ = E ( τ + γ ) -( τ + γ τ + α ) I 2 -I 3 + f ( τ ) . (30)</formula>
<text><location><page_4><loc_52><loc_21><loc_91><loc_25></location>For axisymmetric potentials I 2 = 1 2 L 2 z , where L z is the z -component of the angular momentum. For a general oblate axisymmetric potential, Φ , we define</text>
<formula><location><page_4><loc_52><loc_17><loc_91><loc_20></location>χ λ ( λ, ν ) ≡ -( λ -ν )Φ , χ ν ( λ, ν ) ≡ -( ν -λ )Φ . (31)</formula>
<text><location><page_4><loc_52><loc_15><loc_90><loc_16></location>If Φ were a Stackel potential these quantities would be given by</text>
<formula><location><page_4><loc_52><loc_11><loc_91><loc_14></location>χ λ ( λ, ν ) = f ( λ ) -f ( ν ) , χ ν ( λ, ν ) = f ( ν ) -f ( λ ) . (32)</formula>
<text><location><page_4><loc_52><loc_9><loc_80><loc_10></location>Therefore, for a general potential, we can write,</text>
<formula><location><page_4><loc_52><loc_7><loc_91><loc_8></location>f ( τ ) ≈ χ τ ( λ, ν ) + D τ , (33)</formula>
<figure>
<location><page_5><loc_9><loc_77><loc_30><loc_92></location>
<caption>Figure 1. Equipotential contours for the triaxial NFW potential in the two planes z = 0 (left) and y = 0 (right). The central contour shows Φ /GM s m 0 = -0 . 0096 and the contours increase linearly by ∆(Φ /GM s m 0 ) = 0 . 0008 outwards.</caption>
</figure>
<text><location><page_5><loc_33><loc_78><loc_35><loc_79></location>-20</text>
<text><location><page_5><loc_36><loc_78><loc_38><loc_79></location>-10</text>
<text><location><page_5><loc_40><loc_78><loc_41><loc_79></location>0</text>
<text><location><page_5><loc_44><loc_78><loc_45><loc_79></location>10</text>
<text><location><page_5><loc_47><loc_78><loc_48><loc_79></location>20</text>
<text><location><page_5><loc_39><loc_77><loc_42><loc_78></location>x/kpc</text>
<text><location><page_5><loc_9><loc_65><loc_48><loc_67></location>where D τ are constants provided we evaluate χ λ at constant ν and vice versa. We can write the equations for the momenta as</text>
<formula><location><page_5><loc_9><loc_60><loc_48><loc_64></location>2( τ + α )( τ + γ ) p 2 τ = E ( τ + γ ) -( τ + γ τ + α ) I 2 -B τ + χ τ ( λ, ν ) , (34)</formula>
<text><location><page_5><loc_9><loc_48><loc_48><loc_59></location>where we have defined the integral of motion B τ = I 3 -D τ . B τ may be found given an initial phase-space point and we then integrate the equations for the momenta to find the actions. Note that in this case only two integrals of the motion, B τ , need to be found, as, in the axisymmetric case, we can find two exact integrals of motion, E and L z . This is the procedure followed in Binney (2012a) and, despite the differing conventions and presentation, this method gives identical results to that of Binney (2012a).</text>
<section_header_level_1><location><page_5><loc_9><loc_43><loc_16><loc_44></location>4 TESTS</section_header_level_1>
<text><location><page_5><loc_9><loc_40><loc_48><loc_42></location>For the purposes of testing the above algorithm, we use a triaxial NFW halo (Navarro et al. 1997; Jing & Suto 2002):</text>
<formula><location><page_5><loc_9><loc_33><loc_48><loc_39></location>Φ( x, y, z ) = Φ( m ) = -GM s m log ( 1 + m m 0 ) where m = √ x 2 + y 2 y 2 s + z 2 z 2 s . (35)</formula>
<text><location><page_5><loc_9><loc_20><loc_48><loc_32></location>We set y s = 0 . 95 , z s = 0 . 85 , m 0 = 10kpc and GM s = (1109 km s -1 ) 2 kpc . In Fig. 1 we show the equipotential contours in the z = 0 and y = 0 planes. It is perhaps more conventional to include the triaxiality in the density (e.g. Jing & Suto 2002), but, for simplicity, we have chosen to include triaxiality in the potential. For our choice of parameters this leads to negative densities along the z -axis for z glyph[greaterorsimilar] 130 kpc . This is well outside the region we will probe in our experiments so we are not concerned that our model is unphysical at large z .</text>
<section_header_level_1><location><page_5><loc_9><loc_16><loc_31><loc_17></location>4.1 Selection of coordinate system</section_header_level_1>
<text><location><page_5><loc_9><loc_7><loc_48><loc_15></location>The accuracy of the above routine for a general potential will depend upon our choice of coordinate system, ( α, β, γ ) . Note that the potential is fixed and this coordinate system acts only as a set of parameters in the algorithm to find the actions. We can freely set γ = -1 kpc 2 as the coordinate system only depends on ∆ 1 = √ β -α and ∆ 2 = √ γ -β . For each orbit we consider</text>
<text><location><page_5><loc_32><loc_90><loc_33><loc_91></location>20</text>
<text><location><page_5><loc_32><loc_87><loc_33><loc_89></location>10</text>
<text><location><page_5><loc_32><loc_85><loc_33><loc_86></location>0</text>
<text><location><page_5><loc_31><loc_82><loc_33><loc_83></location>-10</text>
<text><location><page_5><loc_31><loc_79><loc_33><loc_80></location>-20</text>
<text><location><page_5><loc_45><loc_91><loc_48><loc_92></location>y =0</text>
<text><location><page_5><loc_52><loc_90><loc_91><loc_92></location>we are in a position to choose different ∆ i . Here we consider how we can choose suitable ∆ i given an initial phase-space point.</text>
<text><location><page_5><loc_52><loc_79><loc_91><loc_90></location>In Sanders (2012) the mixed derivative ∂ λ ∂ ν [( λ -ν )Φ] was used to select an appropriate coordinate system in an axisymmetric potential. For the triaxial case we could construct a similar quantity: ∂ λ ∂ µ ∂ ν [( λ -µ )( µ -ν )( ν -λ )Φ] . However, this expression would involve third derivatives of the potential so is undesirable. Binney (2014) selected a coordinate system by fitting ellipses to shell orbits at each energy, E . We follow a similar procedure: we assume that the best choice of coordinate system is solely a function of E .</text>
<text><location><page_5><loc_52><loc_70><loc_91><loc_78></location>In a Stackel potential the short-axis closed loops are ellipses confined to the plane z = 0 with foci at y = ± ∆ 1 = ± √ α -β , whilst the long-axis closed loops are confined to the plane x = 0 with foci at z = ± ∆ 2 = ± √ γ -β . Additionally, for these closed loop orbits only one of the actions is non-zero ( J µ for the short-axis closed loop and J ν for the long-axis closed loop).</text>
<text><location><page_5><loc_52><loc_59><loc_91><loc_70></location>For a general potential we use these facts to select appropriate values for ∆ i using a two step procedure: given a value for E we find the two closed loop orbits - one around the short axis and one around the long axis, and with these closed orbits found we alter the position of the foci to optimise the action estimates from our algorithm. Note that the structure of the closed orbits is independent of any choice of the foci positions such that the two steps of the procedure are distinct.</text>
<text><location><page_5><loc_52><loc_46><loc_91><loc_59></location>First, to find the closed orbits with energy E , we select a point along the intermediate axis, y = y I , and launch an orbit with speed v = √ 2( E -Φ(0 , y I , 0)) in either the x (for the short-axis loop) or z direction (for the long-axis loop). The next time the orbit crosses the y -axis we note the y -intercept, y = y F and calculate | -y F -y I | . We repeat this procedure with a new y I until we have minimised | -y F -y I | using Brent's method. We only integrate half of the orbit and assume that the other half can be obtained by symmetry to avoid misidentifying fish-tail resonant orbits as closed loop orbits.</text>
<text><location><page_5><loc_52><loc_33><loc_91><loc_45></location>With the closed orbits with energy E in our potential found, we turn to estimating the location of the foci. Using the long-axis closed loop orbit integration we find an estimate of ∆ 2 by minimising the standard deviation of the J ν estimates from each time-step with respect to β using Brent's method. The action estimates are found using the algorithm outlined in Section 3. This procedure is not sensitive to the choice of α . Once we have found β we perform a similar procedure for the short-axis loop: vary α until we have minimised the standard deviation of J µ .</text>
<text><location><page_5><loc_52><loc_15><loc_91><loc_33></location>We perform the above procedure for a range of energies from E min = Φ(0 , y min , 0) to E max = Φ(0 , y max , 0) , tabulating the found values of α and β for interpolation. For the NFW potential, we adopt y min = 0 . 05 kpc and y max = 60kpc . In Figs. 2 and 3 we plot the standard deviation of the actions of the closed loop orbits against ∆ 2 and ∆ 1 for the constant energy surface with E = Φ(0 , m 0 , 0) = -(290 km s -1 ) 2 . In both cases there is a clear minimum in the standard deviation. In Fig. 2 we show the standard deviation in J ν as a function of ∆ 2 = √ γ -β using two different values for α . The results are indistinguishable. Provided we initially choose a sufficiently negative value of α that the optimal β satisfies β > α , we are free to first set ∆ 2 and then choose ∆ 1 .</text>
<section_header_level_1><location><page_5><loc_52><loc_9><loc_74><loc_10></location>4.1.1 Coordinate system procedure</section_header_level_1>
<text><location><page_5><loc_52><loc_7><loc_83><loc_8></location>For clarity, we now summarise the above procedure:</text>
<figure>
<location><page_6><loc_11><loc_63><loc_47><loc_92></location>
<caption>Figure 2. Standard deviation in J ν as a function of ∆ 2 for the closed longaxis loop orbit shown in the inset. The solid line shows the results if we set α = -80 kpc 2 whilst the red crosses show the results if we set α = -20 kpc 2 . The choice of ∆ 2 is insensitive to α . In the inset, the two red arrows show the initial position vector for the orbit and that position vector rotated by 90 degrees anticlockwise. The black squares show the chosen location of the foci z = ± ∆ 2 .</caption>
</figure>
<figure>
<location><page_6><loc_11><loc_21><loc_47><loc_50></location>
<caption>Figure 3. Standard deviation in J µ as a function of ∆ 1 for the closed shortaxis loop orbit shown in the inset. In the inset, the two red arrows show the initial position vector for the orbit and that position vector rotated by 90 degrees clockwise. The black squares show the chosen location of the foci y = ± ∆ 1 .</caption>
</figure>
<figure>
<location><page_6><loc_54><loc_71><loc_89><loc_92></location>
<caption>Figure 4. Closed-loop choice of ∆ 1 (solid black) and ∆ 2 (dashed red) as a function of energy, E , for the NFW potential described in Section 4. The range of energies covered corresponds to the energies of particles dropped from 0 . 5 kpc to 30 kpc along the intermediate axis. The vertical blue dotted line gives the energy of the surface explored in Section 4.2.</caption>
</figure>
<text><location><page_6><loc_52><loc_58><loc_91><loc_60></location>(i) Given a general potential, create a regularly-spaced grid in energy, E , between some minimum and maximum energy.</text>
<text><location><page_6><loc_52><loc_50><loc_91><loc_58></location>(ii) At each grid-point, E i , find the short-axis and long-axis closed loops by integrating orbits launched at (0 , y k , 0) with velocity √ 2( E i -Φ(0 , y k , 0)) in the direction of the long-axis or shortaxis respectively. The closed loops will cross the y -axis for the first time at (0 , -y k , 0) . We store the phase-space points ( x j , v j ) at each time sample t j .</text>
<text><location><page_6><loc_52><loc_43><loc_91><loc_49></location>(iii) Minimise the standard deviation of the J ν ( x j , v j ) from the long-axis closed loop orbit integration with respect to β to find ∆ 2 . (iv) Minimise the standard deviation of the J µ ( x j , v j ) from the short-axis closed loop orbit integration with respect to α to find ∆ 1 .</text>
<text><location><page_6><loc_52><loc_39><loc_91><loc_42></location>We call the ∆ 1 and ∆ 2 found using this procedure the closed-loop estimates.</text>
<section_header_level_1><location><page_6><loc_52><loc_35><loc_72><loc_36></location>4.1.2 Coordinate system results</section_header_level_1>
<text><location><page_6><loc_52><loc_25><loc_91><loc_34></location>In Fig. 4 we have plotted the closed-loop choice of ∆ 1 and ∆ 2 as a function of the energy. We see that for low energies (very centrally confined orbits) ∆ i tends to zero. Due to the cusp at the centre of the NFW potential, loop orbits exist right down to the centre of the potential. The foci must lie within these loop orbits so ∆ i must decrease as we go to lower energy. As we increase the energy ∆ i increases with ∆ 1 < ∆ 2 .</text>
<text><location><page_6><loc_52><loc_7><loc_91><loc_24></location>To check the closed-loop estimates we launch a series of orbits of constant energy E = Φ(0 , m 0 , 0) = -(290 km s -1 ) 2 at linearly-spaced intervals along the y -axis with velocity vectors in the ( x, z ) plane oriented at differing linearly-spaced angles, θ , to the x axis and integrate the orbits for approximately 10 . 3 Gyr storing phase-space points every 0 . 1 Gyr . Note again that the orbit integration is in the fixed NFW potential and so the structure of an orbit is independent of any choice of α and β . The choice of α and β only affects the recovery of the actions and we wish to find the optimal choice of α and β for each orbit i.e. the choice that makes the actions as constant in time as possible. Therefore, we minimise the sum of the variances of the actions with respect to α and β . The results of this procedure are shown in Fig. 5. We see that the major-</text>
<figure>
<location><page_7><loc_9><loc_73><loc_49><loc_93></location>
<caption>Figure 5. Choice of ∆ 1 and ∆ 2 which minimises the variation in the actions for a range of orbits confined to a constant energy surface. Each orbit was launched at y on the intermediate axis with angle θ from the long axis. The dashed black line gives the values chosen by only inspecting the closed loop orbits as specified in Section 4.1.</caption>
</figure>
<text><location><page_7><loc_9><loc_57><loc_48><loc_62></location>its yield optimal ∆ i similar to the closed-loop estimates. At the extremes of y ∆ i deviates from this choice. These are the box orbits and they seem to favour lower ∆ i . At fixed y the choice of ∆ i is not so sensitive to θ .</text>
<text><location><page_7><loc_9><loc_49><loc_48><loc_57></location>We could improve our choice of ∆ 1 and ∆ 2 by making the choice a function of an additional variable. For instance, we could make the choice a function of the total angular momentum, which is not an integral of motion. However, we will see that we cannot significantly improve the action recovery with a better choice of ∆ i .</text>
<section_header_level_1><location><page_7><loc_9><loc_43><loc_18><loc_44></location>4.2 Accuracy</section_header_level_1>
<text><location><page_7><loc_9><loc_11><loc_48><loc_42></location>We now briefly inspect the accuracy of the action recovery using the triaxial Stackel fudge. We take three orbits from the surface of constant energy explored in the previous section. The three orbits are a box orbit with y = 1 . 8234 kpc , θ = 0 . 6 rad (shown in Fig. 6), a short-axis loop orbit with y = 4 . 8234 kpc , θ = 0 . 4 rad (shown in Fig. 7), and a long-axis loop orbit with y = 3 . 8234 kpc , θ = 1 . 2 rad (shown in Fig. 8). The top row of each figure shows three projections of the orbit, while the three lower panels show the action estimates calculated at each point along the orbit using the closed-loop choice of ∆ i in blue, and in green those obtained with the choice of ∆ i that minimises the spread in the action estimates. Clearly no procedure for determining ∆ i will give superior performance to that obtained with the latter, which is expensive to compute because it requires orbit integration. The intersection of the black lines in the bottom panels of Figs. 6 to 8 show the 'true' actions calculated with the method of Sanders & Binney (2014). The distributions of coloured points from the Stackel Fudge scatter around the true actions, as one would hope. The extent of the green distributions, obtained with the computationally costly values of ∆ i , are at best a factor two smaller that the distributions of blue points, obtained with the cheap value of ∆ i . From this experiment we conclude that there is not a great deal to be gained by devising a better way to evaluate the ∆ i .</text>
<text><location><page_7><loc_9><loc_8><loc_48><loc_10></location>In Appendix A we show how well the angle coordinates are recovered for these orbits.</text>
<text><location><page_7><loc_12><loc_7><loc_48><loc_8></location>The actions of the box orbit are ( J λ , J µ , J ν ) =</text>
<text><location><page_7><loc_52><loc_83><loc_91><loc_93></location>(686 , 192 , 137) kpc km s -1 and our method yields errors of (∆ J λ , ∆ J µ , ∆ J ν ) = (56 , 39 , 22) kpc km s -1 so approximately 10 -20 per cent. If we adjust ∆ i to minimise the spread in the action estimates along the orbit, we find errors of (∆ J λ , ∆ J µ , ∆ J ν ) = (17 , 19 , 16) kpc km s -1 so approximately glyph[lessorsimilar] 10 per cent. We can achieve a factor of two improvement for J λ and J µ .</text>
<text><location><page_7><loc_52><loc_75><loc_91><loc_83></location>The actions of the short-axis loop orbit are ( J λ , J µ , J ν ) = (55 , 752 , 78) kpc km s -1 and our method yields errors of (∆ J λ , ∆ J µ , ∆ J ν ) = (2 , 3 , 1) kpc km s -1 so glyph[lessorsimilar] 4 per cent. If we adjust ∆ i to minimise the spread in the action estimates along the orbit, we find errors of (∆ J λ , ∆ J µ , ∆ J ν ) = (0 . 8 , 2 . 0 , 0 . 9) kpc km s -1 .</text>
<text><location><page_7><loc_52><loc_66><loc_91><loc_74></location>The actions of the long-axis loop orbit are ( J λ , J µ , J ν ) = (50 , 102 , 680) kpc km s -1 and our method yields errors of (∆ J λ , ∆ J µ , ∆ J ν ) = (4 , 5 , 6) kpc km s -1 so glyph[lessorsimilar] 8 per cent. If we adjust ∆ i to minimise the spread in the action estimates along the orbit, we yield errors of (∆ J λ , ∆ J µ , ∆ J ν ) = (2 . 0 , 2 . 5 , 4 . 2) kpc km s -1 .</text>
<text><location><page_7><loc_52><loc_51><loc_91><loc_66></location>For all the orbits shown in Fig. 5 (sampled from the constant energy surface E = Φ(0 , m 0 , 0) = -(290 km s -1 ) 2 ), we have plotted the logarithm of the fractional error in the actions in Fig. 9 (i.e. logarithm of the standard deviation of the action estimates around the orbit over the mean action estimate). We find the most accurate action recovery occurs for the orbits with the initial condition y ≈ m 0 / 2 , where we have mostly loop orbits. For these loop orbits, J µ and J ν are accurate to glyph[lessorsimilar] 1 per cent but the 'radial' action J λ is small for these orbits so the relative error can be large. For the box orbits at the extremes of y , the relative error increases to ∼ 10 per cent but can be as large as order one in J µ for low y .</text>
<text><location><page_7><loc_52><loc_25><loc_91><loc_51></location>In Fig. 10 we show the absolute errors in the actions as a function of action for the constant energy surface along with the orbit classification. These are again calculated as the standard deviation of action estimates around the orbit. Each phase-space point along the orbit is allocated a classification number based on the limits of τ found in the Stackel approximation (see Table 1): λ -= -α, µ -= -β and ν -= -γ correspond to a box orbit (classification number 0), µ -= β , µ + = -α to a short-axis loop orbit (1), λ -= -α , ν + = -β to an inner long-axis loop (2), and µ + = -α , ν + = -β to an outer long-axis loop (3). The orbit classification number is calculated as an average of these classifications along the orbit. With this scheme, orbits near the boundaries of the orbit classes that are chaotic or resonant are allocated noninteger orbit classification numbers. We see that the largest action errors occur at the interfaces between the orbit classes. In particular, ∆ J λ and ∆ J µ are largest along the box-short-axis-loop interface, whilst ∆ J ν is largest at the box-long-axis-loop interface. It is at these boundaries that the orbits pass close to the foci so clearly our choice of foci affects the action recovery for these orbits.</text>
<text><location><page_7><loc_52><loc_17><loc_91><loc_25></location>In general, we find that the action recovery for loop orbits is good, as these orbits probe a small radial range of the potential. For box orbits the recovery deteriorates as these orbits probe a larger central region of the potential. Additionally, we have seen that by altering ∆ i we can achieve up to a factor of two improvement in the accuracy of the actions for both the loop and box orbits.</text>
<section_header_level_1><location><page_7><loc_52><loc_13><loc_67><loc_14></location>4.3 Surfaces of section</section_header_level_1>
<text><location><page_7><loc_52><loc_7><loc_91><loc_12></location>For understanding the behaviour of dynamical systems, Poincar'e (1892) introduced the concept of a surface of section. These diagrams simplify the motion of a high-dimensional dynamical system. A regular orbit in an integrable triaxial potential permits three</text>
<figure>
<location><page_8><loc_22><loc_61><loc_78><loc_91></location>
<caption>Figure 6. Action estimates for example box orbit using the triaxial Stackel fudge: the top three panels show three projections of the orbit, and the bottom three panels show the action estimates for points along the orbit. The dark blue points show the action estimates calculated using our closed-loop estimate of ∆ i based on the energy, the light green points show the choice of ∆ i that minimises the spread in the action estimates, and the black lines show the 'true' actions found using the method presented by Sanders & Binney (2014). Note that the origin is not included in the plots. Between the top and bottom plots, we give the absolute and relative error in the actions.</caption>
</figure>
<figure>
<location><page_8><loc_23><loc_17><loc_78><loc_49></location>
<caption>Figure 7. Action estimates for example short-axis loop orbit using the triaxial Stackel fudge. See Fig. 6 for more information on each panel.</caption>
</figure>
<text><location><page_9><loc_34><loc_90><loc_36><loc_92></location>loop</text>
<text><location><page_9><loc_76><loc_61><loc_78><loc_62></location>120</text>
<figure>
<location><page_9><loc_22><loc_59><loc_78><loc_91></location>
<caption>Figure 8. Action estimates for example long-axis loop orbit using the triaxial Stackel fudge. See Fig. 6 for more information on each panel.</caption>
</figure>
<figure>
<location><page_9><loc_9><loc_30><loc_91><loc_52></location>
<caption>Figure 10. Absolute errors in the actions as functions of action in the constant energy surface E = Φ(0 , m 0 , 0) = -(290 km s -1 ) 2 for the NFW potential. The leftmost panel shows the constant energy surface coloured by orbit class: boxes in black, short-axis loops in blue, inner long-axis loops in green, and outer long-axis loops in red. Note the classification is a continuum as it is calculated from an average of classifications along an orbit. The second, third and fourth panels show the absolute error in the three actions, J λ , J µ and J ν respectively.</caption>
</figure>
<text><location><page_9><loc_9><loc_15><loc_48><loc_21></location>constants of the motion, thus confining the motion to a 3-torus. If we choose to only plot the series of points where the orbit passes through a 4-surface in phase-space, e.g. defined by y = 0 and z = 0 , the phase-space points will be confined to a line, or a consequent, which may be visualized clearly.</text>
<text><location><page_9><loc_9><loc_7><loc_48><loc_13></location>We can test the Stackel approach outlined here by seeing how well it reproduces the surfaces of section. To produce the true surface of section we must integrate the orbit in the true potential and find the phase-space points where the orbit crosses our chosen 4surface. Here we use 4-surfaces defined by one of the spatial axes.</text>
<text><location><page_9><loc_52><loc_8><loc_91><loc_21></location>The orbit will never pass through a given spatial axis in a finite time so we can only produce points arbitrarily close to the given axis. If we require the points where the orbit crosses the x -axis we integrate until the y and z steps have bracketed y = 0 and z = 0 . We then bisect the integration step N max times choosing the interval that brackets y = 0 and if the interval still brackets z = 0 after the N max bisections we store the final point. We integrate over our chosen step-size with a Dortmund-Prince 8th order adaptive integration scheme with a absolute accuracy of glyph[epsilon1] = 1 × 10 -10 . We choose a step-size of 0 . 005 kpc and set N max = 10 . This scheme</text>
<figure>
<location><page_10><loc_11><loc_80><loc_36><loc_93></location>
<caption>Fig. 11 shows that the Stackel approximation consequents for the short-axis loop lie close to the true consequent. Those of the long-axis loop are slightly worse. The box orbit seems problematic. For the phase-space points that lie close to the centre of the potential the consequents turn over in the centre as required. However they underestimate p x at a given x . The phase-space points which lie further out fail to turn over at low x . The Stackel tori for these orbits are near radial such that p x is maximum for x = 0 . However, we see from Fig. 6 that the orbit crosses through x = 0 , y = 0 at an angle such that p x is smaller than its maximum value. This behaviour is only captured for the initial phase-space points at low x .</caption>
</figure>
<text><location><page_10><loc_35><loc_87><loc_37><loc_87></location>|</text>
<text><location><page_10><loc_35><loc_86><loc_37><loc_86></location>|</text>
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<location><page_10><loc_38><loc_80><loc_63><loc_93></location>
</figure>
<text><location><page_10><loc_61><loc_87><loc_63><loc_87></location>|</text>
<text><location><page_10><loc_61><loc_86><loc_63><loc_86></location>|</text>
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<location><page_10><loc_65><loc_80><loc_89><loc_93></location>
</figure>
<text><location><page_10><loc_88><loc_87><loc_90><loc_87></location>|</text>
<text><location><page_10><loc_88><loc_85><loc_90><loc_86></location>|</text>
<figure>
<location><page_10><loc_10><loc_26><loc_47><loc_71></location>
<caption>Figure 11. Surfaces of section for the three test orbits in the triaxial NFW potential. In the left panel we show the box orbit, the central panel shows the short-axis loop orbit and the right panel shows the long-axis loop orbit. In each panel the solid black line gives the true curve of consequents found from orbit integration. The narrower coloured lines give the consequents from the Stackel approximation coloured by | x i | of the initial phase-space point where x i = x for the short-axis loop and box orbit, and x i = y for the long-axis loop orbit. The text above each plot gives the plane that defines the surface of section.Figure 9. Logarithm of the fractional error in the actions for a selection of orbits in the constant energy surface E = Φ(0 , m 0 , 0) = -(290 km s -1 ) 2 for the triaxial NFW potential. The x -axis shows the position along the intermediate axis at which the orbits were launched ( y ), and the colour-coding in the right panel shows the angle, θ , in the x -z plane at which the orbits were launched.</caption>
</figure>
<text><location><page_10><loc_9><loc_12><loc_48><loc_15></location>produces points that are glyph[lessorsimilar] 0 . 001 kpc away from the x axis for the box orbit considered below.</text>
<text><location><page_10><loc_9><loc_7><loc_48><loc_12></location>To produce the corresponding surface of section from the Stackel method we determine τ along our chosen spatial axis using equation (2) between the determined limits in τ , and use equation (17) to find the corresponding p τ . From p τ , we can use expres-s</text>
<text><location><page_10><loc_52><loc_62><loc_91><loc_71></location>such as equation (4) to calculate p x , p y and p z : if we wish to draw the consequent defined by y = 0 , z = 0 we have that µ = -β and ν = -γ such that x = √ λ + α , and p x = √ 4( λ + α ) p λ . If we wish to draw the consequent defined by x = 0 , z = 0 we have that for | y | > ∆ 1 , µ = -α and ν = -γ such that y = √ λ + β , and p y = √ 4( λ + β ) p λ , whilst for | y | < ∆ 1 , λ = -α and ν = -γ such that y = √ µ + β and p y = √ 4( µ + β ) p µ 1 .</text>
<section_header_level_1><location><page_10><loc_52><loc_40><loc_84><loc_41></location>5 A TRIAXIAL MODEL WITH SPECIFIED DF</section_header_level_1>
<text><location><page_10><loc_52><loc_27><loc_91><loc_39></location>The main purpose of the algorithm presented here is to calculate efficiently the moments of triaxial distribution functions. We have seen that the errors in the actions reported by the scheme can be large. However, when calculating moments of a distribution function, many action evaluations are required and there is scope for errors to substantially cancel, leaving the final value of the moment quite accurate. In this section, we demonstrate this phenomenon by for the first time constructing triaxial models from an analytic DF f ( J ) .</text>
<text><location><page_10><loc_55><loc_26><loc_89><loc_27></location>We adopt a simple distribution function (Posti et al. 2014)</text>
<formula><location><page_10><loc_52><loc_24><loc_91><loc_25></location>f ( x , v ) = f ( J ( x , v )) = ( J 0 + | J λ | + ζ | J µ | + η | J ν | ) p , (36)</formula>
<text><location><page_10><loc_52><loc_16><loc_91><loc_23></location>where J 0 = 10kms -1 kpc is a scale action, ζ controls whether the model is tangentially/radially biased and η controls the flattening in z of the model. We do not construct a self-consistent model but instead consider f to be a tracer population in the externally applied triaxial NFW potential of equation (35).</text>
<text><location><page_10><loc_55><loc_14><loc_91><loc_16></location>We set p = -3 , which, for our choice of potential, causes</text>
<text><location><page_10><loc_52><loc_7><loc_91><loc_12></location>1 If we wish to draw the consequent defined by x = 0 , y = 0 we have for | z | < ∆ 2 , λ = -α and µ = -β so z = √ ν + γ and p z = √ 4( ν + γ ) p ν , whilst for | z | > ∆ 2 , µ = -α and ν = -β so z = √ λ + γ and p z = √ 4( λ + γ ) p λ .</text>
<text><location><page_11><loc_9><loc_80><loc_48><loc_93></location>the density to go as r 0 in the centre and fall off as r -3 for large r . Note that the mass of this model diverges logarithmically. We set η = 1 . 88 and explore two values of ζ = 0 . 7 (tangential bias) and ζ = 3 . 28 (radial bias). Note that for the orbit classes to fill action space seamlessly, we must scale the radial action of the loop orbits by a factor of two (Binney & Spergel 1984). We proceed by calculating the moments of this distribution function in the test triaxial NFW potential at given spatial points, x . These non-zero moments are</text>
<formula><location><page_11><loc_9><loc_74><loc_48><loc_80></location>ρ ( x ) = ∫ d 3 v f ( x , v ) , σ 2 ij ( x ) = 1 ρ ∫ d 3 v v i v j f ( x , v ) . (37)</formula>
<text><location><page_11><loc_9><loc_44><loc_48><loc_73></location>Note that, as the potential is time-independent, the Hamiltonian is time-reversible and we need only integrate over half the velocity space and multiply the result by two. We integrate up to the maximum velocity, v max at x , given by v max = √ 2Φ( x ) . We will later calculate these moments extensively to demonstrate that the actionbased distribution functions obey the Jeans' equations. In Figs. 12 and 13 we plot the density of the radially-biased ( ζ = 3 . 28 ) and tangential-biased ( ζ = 0 . 7 ) models. We display contours of constant density in two planes along with the density along a line parallel to the x -axis decomposed into its contributions from each orbit class. The density is calculated using the adaptive Monte-Carlo Divonne routine in the CUBA package of Hahn (2005). The class of each orbit is determined by the limits of the motion in τ : λ -= -α , µ -= -β and ν -= -γ correspond to a box orbit, µ -= β , µ + = -α to a short-axis loop orbit, and ν -= -γ , ν + = -β to a long-axis loop. As we are calculating the density close to the x -axis, the long-axis loop orbits, which loop the x -axis, do not contribute significantly to the density integral. We see that for the radially-biased model the box orbits are the dominant contributors whilst for the tangentially-biased model the short-axis loop orbits are the major contributors.</text>
<text><location><page_11><loc_9><loc_42><loc_48><loc_44></location>We will now perform some checks to see whether our distribution functions are accurate.</text>
<section_header_level_1><location><page_11><loc_9><loc_38><loc_22><loc_39></location>5.1 Normalization</section_header_level_1>
<text><location><page_11><loc_9><loc_30><loc_48><loc_37></location>One quick check of our action estimation scheme is how accurately it recovers the normalization. To keep the normalization finite we set p = -3 . 5 for this section. We are able to calculate the normalization of our DF in two distinct ways. Firstly, we calculate the normalization analytically from the DF as</text>
<formula><location><page_11><loc_9><loc_20><loc_40><loc_29></location>M true = (2 π ) 3 ∫ d 3 J f ( J ) = (2 π ) 3 ∫ ∞ 0 d J λ ∫ ∞ 0 d J µ ∫ ∞ 0 d J ν f ( J ) = -(2 π ) 3 J p +3 0 ηζ ( p +1)( p +2)( p +3) .</formula>
<text><location><page_11><loc_9><loc_10><loc_48><loc_19></location>Note that for each J in the appropriate range there are two loop orbits - one circulating clockwise and one anti-clockwise. Therefore, we must multiply the normalization by two for these orbits. However, we have defined the 'radial' action to be four times the integral from τ -to τ + for these orbits so these factors cancel (Binney & Spergel 1984; de Zeeuw 1985). Additionally, we can calculate the mass as</text>
<formula><location><page_11><loc_9><loc_6><loc_36><loc_9></location>M est = 8 ∫ ( x,y,z ) > 0 d 3 x ∫ d 3 v f ( J ( x , v )) .</formula>
<text><location><page_11><loc_52><loc_80><loc_91><loc_92></location>For each spatial coordinate, we make the transformation u i = 1 / (1 + x i ) to make the integrand flatter. The limits of the integral are now u i = [0 , 1] . To reduce numerical noise, we split the integral such that we calculate the contribution near the axes separately. We perform the integral using the Monte Carlo Divonne routine. For the tangentially-biased model ( ζ = 0 . 7 ), we find M est ≈ 1 . 006 M true and for the radially-biased model ( ζ = 3 . 28 ) M est ≈ 1 . 007 M true so despite the often large errors in the actions the normalization of the model is well recovered.</text>
<section_header_level_1><location><page_11><loc_52><loc_76><loc_68><loc_77></location>5.2 The Jeans equation</section_header_level_1>
<text><location><page_11><loc_52><loc_73><loc_91><loc_75></location>Our distribution function must satisfy the collisionless Boltzmann equation</text>
<formula><location><page_11><loc_52><loc_69><loc_91><loc_72></location>d f d t = 0 . (38)</formula>
<text><location><page_11><loc_52><loc_66><loc_91><loc_68></location>In turn this means the distribution function must satisfy the Jeans equations (see equation (4.209) of Binney & Tremaine (2008))</text>
<formula><location><page_11><loc_52><loc_62><loc_91><loc_65></location>∂ ( ρσ 2 ij ) ∂ x i = -ρ ∂ Φ ∂ x j . (39)</formula>
<text><location><page_11><loc_52><loc_32><loc_91><loc_61></location>A simple test of our action-based distribution functions is checking whether they satisfy these equations. The right hand side is calculated from analytic differentiation of the potential and multiplying by the density. The left hand side is found by numerically differentiating the three-dimensional integrals ρσ 2 ij and summing the appropriate contributions. Numerical differentiation of an integral leads to significant noise. To combat this we use an adaptive vectorised integration-rule cubature scheme implemented in the CUBATURE package from Steven Johnson (http://abinitio.mit.edu/wiki/index.php/Cubature). Using a fixed-rule adaptive routine means the noise in the integrals is controlled such that the numerical derivatives are less noisy. In Figures 14 and 15 we show how accurately the Jeans equations are satisfied along several lines through the potential for our two models. We plot each side of each Jeans equation for a choice of j along a range of lines, along with the percentage error difference between the two sides of the equation. We avoid calculating the derivatives of the moments along the axes as the numerical differentiation is awkward there. In general, we find glyph[lessorsimilar] 10 per cent error for nearly all tested points with the majority having glyph[lessorsimilar] 4 per cent over a range of ∼ 8 orders of magnitude.</text>
<text><location><page_11><loc_52><loc_23><loc_91><loc_32></location>Despite the large errors introduced by the action estimation scheme, we have produced a distribution function that satisfies the Jeans equations to reasonable accuracy. Even for the heavily radially-biased model, which has large contributions from the box orbits, the Jeans equations are well satisfied. This gives us confidence that models based on triaxial distribution functions can be constructed using the scheme we have presented.</text>
<section_header_level_1><location><page_11><loc_52><loc_19><loc_69><loc_20></location>5.3 Error in the moments</section_header_level_1>
<text><location><page_11><loc_52><loc_7><loc_91><loc_17></location>When comparing smooth models to data, we are primarily interested in whether the action-estimation scheme produces accurate enough actions to reproduce the features of the data. If the broad features of the data are on scales larger than the errors in the individual actions, our method should be sufficiently accurate to recover these features from an appropriate f ( J ) . In this case, the error in the moments is more important than the error in the individual actions. Larger errors in the actions are expected to lead to larger</text>
<figure>
<location><page_12><loc_9><loc_70><loc_92><loc_92></location>
<caption>Figure 12. Density for the radially-biased model ( ζ = 3 . 28 ). The left panel shows equally-spaced contours of the logarithm of the density in the ( x, y ) plane and similarly for the central panel in the ( x, z ) plane. The outermost contour corresponds to log 10 ( ρ/ kpc -3 ) = 0 . 5 and the contours increase by 0 . 5 inwards. The right panel shows the total density in black along the line y = 1kpc , z = 1kpc as well as the contributions from the box orbits in blue, the short-axis loop orbits in red and the long-axis loop orbits in green.</caption>
</figure>
<figure>
<location><page_12><loc_9><loc_40><loc_35><loc_61></location>
<caption>Figure 13. Density for the tangentially-biased model ( ζ = 0 . 7 ). The left panel shows equally-spaced contours of the logarithm of the density in the ( x, y ) plane and similarly for the central panel in the ( x, z ) plane. The outermost contour corresponds to log 10 ( ρ/ kpc -3 ) = 1 . 0 in the left panel and log 10 ( ρ/ kpc -3 ) = 0 . 5 in the central panel, and the contours increase by 0 . 5 inwards. The right panel shows the total density in black along the line y = 1kpc , z = 1kpc as well as the contributions from the box orbits in blue, the short-axis loop orbits in red and the long-axis loop orbits in green.</caption>
</figure>
<figure>
<location><page_12><loc_37><loc_40><loc_65><loc_60></location>
</figure>
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<location><page_12><loc_66><loc_40><loc_92><loc_62></location>
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<text><location><page_12><loc_65><loc_53><loc_66><loc_53></location>c</text>
<text><location><page_12><loc_65><loc_53><loc_66><loc_53></location>p</text>
<text><location><page_12><loc_65><loc_52><loc_66><loc_53></location>k</text>
<text><location><page_12><loc_65><loc_52><loc_66><loc_52></location>5</text>
<text><location><page_12><loc_65><loc_52><loc_66><loc_52></location>.</text>
<text><location><page_12><loc_65><loc_51><loc_66><loc_52></location>0</text>
<text><location><page_12><loc_9><loc_18><loc_48><loc_30></location>errors in the moments but the relationship between the two is unclear. We have seen that, despite the presented method introducing large errors in some actions, the moments of the DF are well recovered such that the normalization is accurate to 0 . 6 per cent and the Jeans' equations are accurate to glyph[lessorsimilar] 4 per cent. In this section, we develop some understanding of how errors in the actions translate into errors in the moments. Initially, we can make some progress by considering the normalization of the DF: ∫ d 3 x d 3 v f ( x , v ) = ∫ d 3 J d 3 θ f ( J ) .</text>
<text><location><page_12><loc_9><loc_7><loc_48><loc_9></location>Suppose we have a set of angle-action variables ( J ' , θ ' ) that are not the true angle-action variables ( J , θ ) , we can relate the two</text>
<text><location><page_12><loc_52><loc_29><loc_82><loc_30></location>sets via the generating function, S ( J , θ ' ) , such that</text>
<formula><location><page_12><loc_52><loc_19><loc_91><loc_28></location>S ( J , θ ' ) = J · θ ' + ∑ n S n ( J ) sin n · θ ' , J ' = ∂ θ ' S = J + ∑ n n S n ( J ) cos n · θ ' , θ = ∂ J S = θ ' + ∑ n ∂ J S n ( J ) sin n · θ ' . (40)</formula>
<text><location><page_12><loc_52><loc_15><loc_91><loc_19></location>Now suppose we evaluate the normalization using J ' ( x , v ) . We transform the volume element d 3 J d 3 θ to d 3 J d 3 θ ' via the Jacobian</text>
<formula><location><page_12><loc_52><loc_8><loc_91><loc_14></location>det ( ∂ θ ∂ θ ' ) J = det ( I + ∑ n n ⊗ ∂ J S n cos n · θ ' ) glyph[similarequal] | 1 + ∑ n n · ∂ J S n cos n · θ ' | . (41)</formula>
<text><location><page_12><loc_52><loc_7><loc_91><loc_8></location>We assume that the approximate angle-action variables are suffi-</text>
<text><location><page_13><loc_14><loc_92><loc_22><loc_93></location>Radial bias,</text>
<text><location><page_13><loc_22><loc_92><loc_22><loc_93></location>ζ</text>
<text><location><page_13><loc_22><loc_92><loc_25><loc_93></location>=3</text>
<text><location><page_13><loc_25><loc_92><loc_25><loc_93></location>.</text>
<text><location><page_13><loc_25><loc_92><loc_27><loc_93></location>28</text>
<figure>
<location><page_13><loc_10><loc_49><loc_91><loc_93></location>
<caption>Figure 14. Accuracy of Jeans' equation calculation for the radially-biased model ( ζ = 3 . 28 ). In the top half of each panel we show -ρ ∂ Φ / ∂ x j as a series of black dots and ∂ ( ρσ 2 ij ) / ∂ x i as a red line. In the bottom half we show the percentage error difference between these quantities. Each panel shows a single component, i.e. a single j , along the line parametrized by the coordinate x k and given above the top-right corner of each panel. The bottom three panels all correspond to the same line. The grey dashed line is the zero error line.</caption>
</figure>
<text><location><page_13><loc_9><loc_36><loc_48><loc_40></location>ciently close to the true angle-action variables that the second term on the right-hand side is much less than unity. Therefore, the difference in the normalization is given by</text>
<formula><location><page_13><loc_9><loc_25><loc_48><loc_34></location>∫ d 3 J d 3 θ [ f ( J ) -f ( J ' )] = ∫ d 3 J d 3 θ ∂ J f · ( J -J ' ) glyph[similarequal] ∫ d 3 J d 3 θ ' (1 + ∑ n n · ∂ J S n cos n · θ ' ) × [ -∑ m ( m · ∂ J f ) S m ( J ) cos m · θ ' + · · · ] . (42)</formula>
<text><location><page_13><loc_9><loc_19><loc_48><loc_23></location>We see that, after integrating over θ ' , all terms with odd powers of trigonometric functions vanish, and the leading order error in the normalization is</text>
<formula><location><page_13><loc_9><loc_15><loc_34><loc_18></location>-4 π 3 ∫ d 3 J ∑ n ( n · ∂ J f )( n · ∂ J S n ) S n .</formula>
<text><location><page_13><loc_9><loc_7><loc_48><loc_13></location>This term is second order in the Fourier components of the generating function. Note that the sign of this term is unclear as both the S n and ∂ J S n can be positive and negative. We anticipate ∂ J f is negative such that the density falls with radius. From equation (40) we find that to leading order the errors in the angle-action variables</text>
<text><location><page_13><loc_52><loc_39><loc_67><loc_40></location>averaged over an orbit are</text>
<formula><location><page_13><loc_52><loc_34><loc_84><loc_36></location>∆ J i glyph[lessorequalslant] √ 1 2 ∑ n |S n n i | and ∆ θ i glyph[lessorequalslant] √ 1 2 ∑ n | ∂ J i S n | .</formula>
<text><location><page_13><loc_52><loc_15><loc_91><loc_31></location>Therefore, we can see that the error in the normalization is approximately first order in the error in the actions, ∆ J i , the error in the angles, ∆ θ i , and the gradient of the distribution function, ∂ J i f . Therefore, we anticipate that the relative error in the normalization will be small when ∆ J glyph[lessmuch] f ( ∂ J f ) -1 for all points in action space. This is essentially the expected result. Consider the distribution functions of Binney (2014): these are of the form f ( J z ) ∼ exp( -νJ z /σ 2 z ) such that ∆ J z glyph[lessmuch] σ 2 z /ν for a good estimate of the normalization, where ν is the vertical epicycle frequency. Near the Sun ν ≈ 0 . 1Myr -1 and σ z ≈ 30 km s -1 so ∆ J z glyph[lessmuch] 10 kpc km s -1 . For the distribution function considered in the previous section we require ∆ J glyph[lessmuch] max ( J, J 0 ) .</text>
<text><location><page_13><loc_52><loc_7><loc_91><loc_15></location>For the moments of the distribution function we expect similar results but we are not able to explicitly calculate the leading order errors in these quantities. Instead we briefly show how the error in the density changes with the error in the actions. We begin by calculating the density from a triaxial DF for which we know the true density - the perfect ellipsoid (de Zeeuw 1985), which has</text>
<text><location><page_14><loc_14><loc_92><loc_24><loc_93></location>Tangential bias,</text>
<text><location><page_14><loc_24><loc_92><loc_25><loc_93></location>ζ</text>
<text><location><page_14><loc_25><loc_92><loc_27><loc_93></location>=0</text>
<text><location><page_14><loc_27><loc_92><loc_28><loc_93></location>.</text>
<text><location><page_14><loc_28><loc_92><loc_28><loc_93></location>7</text>
<figure>
<location><page_14><loc_10><loc_48><loc_91><loc_93></location>
<caption>Figure 15. Accuracy of Jeans' equation calculation for the tangentially-biased model ( ζ = 0 . 7 ). In the top half of each panel we show -ρ ∂ Φ / ∂ x j as a series of black dots and ∂ ( ρσ 2 ij ) / ∂ x i as a red line. In the bottom half we show the percentage error difference between these quantities. Each panel shows a single component, i.e. a single j , along the line parametrized by the coordinate x k and given above the top-right corner of each panele. The bottom three panels all correspond to the same line. The grey dashed line is the zero error line.</caption>
</figure>
<text><location><page_14><loc_9><loc_39><loc_18><loc_40></location>density profile</text>
<formula><location><page_14><loc_9><loc_35><loc_48><loc_38></location>ρ ( x, y, z ) = ρ 0 (1 + m 2 ) 2 , (43)</formula>
<text><location><page_14><loc_9><loc_33><loc_13><loc_34></location>where</text>
<formula><location><page_14><loc_9><loc_30><loc_48><loc_33></location>m 2 ≡ x 2 x 2 P + y 2 y 2 P + z 2 z 2 P , x P glyph[greaterorequalslant] y P glyph[greaterorequalslant] z P glyph[greaterorequalslant] 0 . (44)</formula>
<text><location><page_14><loc_9><loc_14><loc_48><loc_29></location>We set ρ 0 = 7 . 2 × 10 8 M glyph[circledot] kpc -3 , x P = 5 . 5 kpc , y P = 4 . 5 kpc and z P = 1kpc . The actions in this potential can be found exactly using the scheme presented above by setting α = -x 2 P and β = -y 2 P . We use the tangentially-biased DF of the previous section. For four different Cartesian positions we calculate the true density, and then proceed to calculate the density when the logarithm of the actions are scattered normally by some fixed amount √ 〈 (∆ J ) 2 〉 . We plot the error in the density as a function of ∆ J in Fig. 16. We find that the relative error in the density goes as ∼ ∆ J 1 . 3 for the three densities near the axes, and is flatter for the density at ( x, y, z ) = (4 , 4 , 4) kpc .</text>
<text><location><page_14><loc_9><loc_7><loc_48><loc_13></location>This procedure is artificial as we have used non-canonical coordinates to evaluate the density. Therefore, for a fuller test we instead choose to calculate the density using the Stackel fudge scheme but changing α away from the truth. In this case, the error in the density is systematic. In the lower panel of Fig. 16 we plot the</text>
<text><location><page_14><loc_52><loc_32><loc_91><loc_40></location>density error from this procedure as a function of the distributionfunction-weighted RMS action errors. Again we find the density goes as ∼ √ 〈 (∆ J ) 2 〉 1 . 2 for all but the density on the z axis. As we are changing α this does not significantly affect the actions of the long-axis loop orbits, which dominate the density budget along the z axis.</text>
<section_header_level_1><location><page_14><loc_52><loc_25><loc_87><loc_26></location>6 EXTENDING THE SCOPE OF TORUS MAPPING</section_header_level_1>
<text><location><page_14><loc_52><loc_8><loc_91><loc_24></location>We have seen that the triaxial Stackel fudge can produce large errors in the actions for some orbits, but that the moments of an action-based DF are nevertheless well recovered using the method. However, even if one requires accurate actions, the Stackel fudge can be valuable because it enables one to construct a torus through a given point ( x , v ) by torus mapping rather than orbit integration (Sanders & Binney 2014). This option may be essential because torus mapping works even in a chaotic portion of phase space (Kaasalainen 1995), while the approach based on orbit integration is already problematic when resonantly trapped orbits take up a significant portion of phase space, and it breaks down with the onset of chaos.</text>
<text><location><page_14><loc_55><loc_7><loc_91><loc_8></location>Weproceed as follows. First we use the Stackel fudge to obtain</text>
<text><location><page_15><loc_12><loc_92><loc_13><loc_92></location>0</text>
<figure>
<location><page_15><loc_9><loc_47><loc_48><loc_92></location>
<caption>Figure 16. Relative error in the density as a function of distributionfunction-weighted RMS action error. The top panel shows the error when scattering by some fixed √ 〈 (∆ J ) 2 〉 and the bottom panel shows the error when using the wrong α value. Each line shows the density at a fixed Cartesian position shown in the legend along with the gradients of the lines, K .</caption>
</figure>
<text><location><page_15><loc_25><loc_46><loc_26><loc_49></location>√</text>
<text><location><page_15><loc_26><loc_46><loc_26><loc_49></location>〈</text>
<text><location><page_15><loc_30><loc_46><loc_30><loc_49></location>〉</text>
<section_header_level_1><location><page_15><loc_9><loc_34><loc_21><loc_35></location>approximate actions</section_header_level_1>
<formula><location><page_15><loc_9><loc_32><loc_18><loc_33></location>J = J St ( x , v ) .</formula>
<text><location><page_15><loc_9><loc_26><loc_48><loc_31></location>Then by torus mapping we obtain the torus with actions J . On account of errors, the given point ( x , v ) will not lie on the constructed torus, but one can identify the nearest point ( x ' , v ' ) that does lie on the torus. We find this point by minimising the tolerance</text>
<formula><location><page_15><loc_9><loc_23><loc_48><loc_25></location>η = | Ω | 2 | x ( θ ) -x | 2 + | v ( θ ) -v | 2 , (45)</formula>
<text><location><page_15><loc_9><loc_17><loc_48><loc_22></location>with respect to the angle, θ , on the torus. Ω is the frequency vector of the constructed torus. We use the Stackel fudge estimate of the angles as an initial guess for the minimisation. For the point ( x ' , v ' ) we know the true actions:</text>
<formula><location><page_15><loc_9><loc_15><loc_19><loc_16></location>J = J t ( x ' , v ' ) .</formula>
<text><location><page_15><loc_9><loc_11><loc_48><loc_14></location>Now we use the Stackel fudge to obtain approximate actions for this point</text>
<formula><location><page_15><loc_9><loc_9><loc_20><loc_10></location>J ' = J St ( x ' , v ' ) .</formula>
<text><location><page_15><loc_9><loc_7><loc_48><loc_8></location>If, as we expect, the errors in the Stackel fudge are systematic rather</text>
<text><location><page_15><loc_52><loc_91><loc_63><loc_92></location>than random, then</text>
<formula><location><page_15><loc_52><loc_89><loc_68><loc_90></location>J t ( x , v ) = J St ( x , v ) + ∆</formula>
<text><location><page_15><loc_52><loc_86><loc_91><loc_88></location>with ∆ a slowly varying function of phase-space position. So a better estimate of the true actions of the original point ( x , v ) is</text>
<formula><location><page_15><loc_52><loc_84><loc_77><loc_85></location>J '' = J + ∆ = J +( J -J ' ) = 2 J -J ' .</formula>
<text><location><page_15><loc_52><loc_76><loc_91><loc_82></location>If one is of a nervous disposition, one now uses torus mapping to construct the torus with actions J '' and seeks the point on this torus that is closest to the given point and applies the Stackel fudge there, and so on. This cycle can be repeated until the nearest point on the constructed torus satisfies some tolerance η = η ∗ .</text>
<text><location><page_15><loc_52><loc_40><loc_91><loc_75></location>In Fig. 17 we show an illustration of this procedure for the axisymmetric case. We use the axisymmetric Stackel fudge as given in Section 3.1 and the torus construction code as presented in McMillan & Binney (2008). For the axisymmetric Stackel fudge we set γ = -1 kpc 2 and α = -20 kpc 2 , such that the foci are at z = ± √ γ -α ≈ ± 4 . 4 kpc . We construct a torus of actions ( J r , L z , J z ) = (244 . 444 , 3422 . 213 , 488 . 887) kpc km s -1 in the 'best' potential from McMillan (2011). This potential is an axisymmetric multi-component Galactic potential consisting of two exponential discs representing the thin and thick discs, an axisymmetric bulge model from Bissantz & Gerhard (2002) and an NFW dark halo. The parameters of the mass model were chosen to satisfy recent observational constraints. We produce a series of ( x , v ) points on the constructed torus. The axisymmetric Stackel fudge gives errors of ∆ J i = 13kpckms -1 in the recovery of the actions for these points. After one iteration of the above procedure we find actions accurate to ∆ J i ≈ 0 . 3 kpc km s -1 , and after the procedure has converged to a tolerance η ∗ = (0 . 1 km s -1 ) 2 the actions are accurate to ∆ J i ≈ 0 . 07 kpc km s -1 . The majority of ( x , v ) points converge in less than five iterations. We limited the number of iterations to 20 and a few ( x , v ) did not converge within 20 iterations. This is due to non-linear behaviour of both the torus construction and Stackel code in small phase-space volumes. It is clear, however, that only one iteration is required for a substantial improvement in the actions, and it is hard to make a case for more iterations.</text>
<text><location><page_15><loc_52><loc_7><loc_91><loc_40></location>Finally, we demonstrate how the above method operates for a range of high-action orbits. We integrate a series of orbits in the 'best' potential from McMillan (2011) launched at 5 linearlyspaced points x 0 along the x -axis such that 4 kpc glyph[lessorequalslant] x 0 glyph[lessorequalslant] 12 kpc . We launch the orbits with velocity v = ( v 1 cos θ, v 0 , v 1 sin θ ) where v 0 = √ x 0 ∂ x Φ( x 0 , 0 , 0) and we choose 4 linearly-spaced values of v 1 such that 0 . 5 v 0 glyph[lessorequalslant] v 1 glyph[lessorequalslant] 0 . 8 v 0 and 5 linearly-spaced angles θ such that 0 . 2 rad glyph[lessorequalslant] θ glyph[lessorequalslant] 1 2 π rad . The range of radial and vertical actions for this collection of orbits is shown in the top panel of Fig. 18 and is approximately 1 kpc km s -1 glyph[lessorsimilar] J r glyph[lessorsimilar] 800 kpc km s -1 and 1 kpc km s -1 glyph[lessorsimilar] J z glyph[lessorsimilar] 800 kpc km s -1 (these are calculated as the mean of the fudge estimates along the orbit). For each orbit, we find the standard deviation of the action estimates from the axisymmetric Stackel fudge method and the iterative torus method for 10 widely time-separated phase-space points along the orbit. We use η ∗ = (0 . 1 km s -1 ) 2 , limit the maximum number of iterations to 5 and construct the tori with a relative error of ∼ 1 × 10 -4 . We plot the results in Fig. 18. We see the majority of orbits have lower iterative torus errors than fudge errors and follow a broad line that lies approximately two to three orders of magnitude beneath the 1:1 line. However, there are several orbits that lie close to the 1:1 line indicating the procedure has not converged to a greater accuracy than the initial accuracy produced by the Stackel fudge. These orbits are either near-resonant so re-</text>
<text><location><page_16><loc_22><loc_91><loc_42><loc_92></location>Stäckel, ∆JR = 13.34, ∆Jz = 12.56</text>
<unordered_list>
<list_item><location><page_16><loc_22><loc_89><loc_43><loc_91></location>First iteration, ∆JR = 0.26, ∆Jz = 0.31</list_item>
</unordered_list>
<text><location><page_16><loc_22><loc_88><loc_42><loc_89></location>Converged, ∆JR = 0.06, ∆Jz = 0.07</text>
<figure>
<location><page_16><loc_9><loc_57><loc_48><loc_87></location>
<caption>Figure 17. Illustration of the iterative torus scheme presented in Section 6. The black points show the initial estimate of the actions for a series of points along an orbit calculated using the axisymmetric Stackel fudge of Binney (2012a). The dark blue points show the improved estimate from a single iteration of the torus scheme, and the light green points show the estimate after the scheme was deemed to converge to an accuracy of η ∗ = (0 . 1 kms -1 ) 2 . The top-right inset shows a histogram of the number of iterations required to reach this accuracy. Above the plot we show the standard deviation of the action estimates for each set of points. The black dashed lines give the true action estimate. The bottom-left inset shows a zoom-in of the central region of the main plot.</caption>
</figure>
<text><location><page_16><loc_9><loc_30><loc_48><loc_38></location>quire more careful action assignment (Kaasalainen 1995), or have one action very much greater than the other (near radial or shell orbits) so require more accurate torus construction than our automated procedure has allowed. It is clear from the plot that none of the iterative procedures have diverged significantly as all points lie near or well below the 1:1 line.</text>
<text><location><page_16><loc_9><loc_7><loc_48><loc_30></location>The results for lower-action disc-type orbits are superior to those presented here. However, by and large the Stackel fudge action estimates for these orbits are sufficiently accurate for much scientific work (Piffl et al. 2014) so the iterative procedure is probably not required. One realistic application of the presented method is the modelling of tidal streams: Sanders (2014) used the expected angle and frequency structure of a stream in the correct potential to constrain the potential from a stream simulation. The frequencies of the collection of orbits explored here range from 10 to 120 kpc -1 kms -1 . For the majority of orbits the error in the frequencies recovered from the fudge are glyph[lessorsimilar] 10 per cent with the majority having errors of a few per cent, whilst the iterative torus approach reduces the errors to approximately 0 . 01 per cent for all orbits apart from those with large action errors discussed previously. The 10 4 M glyph[circledot] stream used in Sanders (2014) had a frequency width to absolute frequency ratio of ∼ 0 . 1 per cent so the iterative torus approach seems well suited to modelling of streams. In con-c</text>
<figure>
<location><page_16><loc_52><loc_50><loc_91><loc_92></location>
<caption>Figure 18. Actions of a selection of orbits shown in the top panel, along with the absolute error in the action for these orbits from the axisymmetric Stackel fudge of Binney (2012a) plotted against the absolute error from the iterative torus procedure for a range of orbits in the bottom panel. The black dots show the results for the radial action, whilst the red crosses show the results for the vertical action. The blue dotted line is the 1:1 line.</caption>
</figure>
<text><location><page_16><loc_52><loc_36><loc_91><loc_38></location>ion, we have demonstrated that the presented algorithm has the capability to produce accurate actions for a wide range of orbits.</text>
<section_header_level_1><location><page_16><loc_52><loc_31><loc_65><loc_32></location>7 CONCLUSIONS</section_header_level_1>
<text><location><page_16><loc_52><loc_7><loc_91><loc_30></location>We have presented a method for estimating the actions in a general triaxial potential using a Stackel approximation. The method is an extension of the Stackel fudge introduced by Binney (2012a) for the axisymmetric case. We have investigated the accuracy of the method for a range of orbits in an astrophysically-relevant triaxial potential. We have seen that the recovery of the actions is poorest for the box orbits, which probe a large radial range of the potential, and much better for the loop orbits, which are confined to a more limited radial range. The only parameters in the method are the choice of the focal positions ∆ i , which are selected for each input phase-space point. We have detailed a procedure for selecting these based on the energy of the input phase-space point. This choice is not optimal but, by adjusting ∆ i , we can, at best, increase the accuracy of the actions of a factor of two for the triaxial NFW potential considered. However, to achieve this accuracy requires additional computation for each input phase-space point (e.g. orbit integration). For general potentials the best action estimates will</text>
<text><location><page_17><loc_9><loc_86><loc_48><loc_92></location>be achieved when locally (over the region a given orbit probes) the potential is well approximated by some Stackel potential. Many potentials of interest are not well fitted globally by Stackel potentials so the accuracy of the action estimates will deteriorate for orbits with large radial actions.</text>
<text><location><page_17><loc_9><loc_66><loc_48><loc_85></location>The advantage of this method over other methods for estimating the actions in a triaxial potential is speed. Unlike the convergent method introduced by Sanders & Binney (2014), we obtain the actions without integrating an orbit - we only use the initial phasespace point. We have only to evaluate several algebraic expressions, find the limits of the orbits in the τ coordinate and perform Gaussian quadrature. These are all fast calculations. However, this speed comes at the expense of sometimes disappointing accuracy. If accurate results are required, the Stackel fudge can be combined with torus mapping to form a rapidly convergent scheme for the determination of J ( x , v ) . We demonstrated how such a scheme performed in the axisymmetric case and found a single torus construction provided a high level of accuracy that is not significantly improved by further torus constructions.</text>
<text><location><page_17><loc_9><loc_52><loc_48><loc_66></location>We went on to construct, for the first time, triaxial stellar systems from a specified DFs f ( J ) in Section 5. We demonstrated the mass of these models is well recovered using the Stackel fudge, and we showed how the error in the density of these models varies as a function of the action error. Notwithstanding the errors in individual actions, both a radially-biased model and a tangentially-biased model satisfy the Jeans equations to good accuracy. This is because individual errors largely cancel during integration over velocities when computing moments such as the density ρ ( x ) and the pressure tensor ρσ 2 ij ( x ) .</text>
<text><location><page_17><loc_9><loc_35><loc_48><loc_52></location>The results presented in this paper have focussed on a limited range of astrophysically-relevant models: we have used a single specific NFW potential and two simple distribution functions that depend on a linear sum of the actions. However, we anticipate that our results will extend to more general distribution functions. We have investigated analytically how the normalization of the distribution function varies with the error in the action estimates and shown that the normalization is well recovered provided the error in the actions is smaller than the action scale over which the distribution function varies significantly i.e. ∆ J glyph[lessmuch] f ( ∂ J f ) -1 . Therefore, the recovery of the moments is expected to be most accurate for distribution functions with shallow radial density profiles and to deteriorate with the steepness of the required profile.</text>
<text><location><page_17><loc_9><loc_21><loc_48><loc_34></location>Whilst the scheme presented here does not give accurate enough actions for working with streams (Sanders 2014) we have shown that it is an appropriate and powerful tool for constructing models from specified DFs f ( J ) . A key property of DFs of the form f ( J ) is that they can be trivially added to build up a multicomponent system. Hence the ability to extract observables from DFs of the form f ( J ) is likely to prove extremely useful for interpreting data on both external galaxies (Cappellari et al. 2011) and our Galaxy, in which components such as the stellar and dark haloes may be triaxial, and the bulge certainly is.</text>
<section_header_level_1><location><page_17><loc_9><loc_16><loc_25><loc_17></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_17><loc_9><loc_7><loc_48><loc_15></location>We thank Paul McMillan for provision of the torus construction machinery used in Section 6 and the Oxford Galactic Dynamics group for helpful comments. JLS acknowledges the support of STFC. JB was supported by STFC by grants R22138/GA001 and ST/K00106X/1. The research leading to these results has received funding from the European Research Council under the European</text>
<text><location><page_17><loc_52><loc_90><loc_91><loc_92></location>Union's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no. 321067.</text>
<section_header_level_1><location><page_17><loc_52><loc_86><loc_62><loc_87></location>REFERENCES</section_header_level_1>
<text><location><page_17><loc_53><loc_28><loc_91><loc_85></location>Aarseth S. J., Binney J., 1978, MNRAS , 185, 227 Allgood B., Flores R. A., Primack J. R., Kravtsov A. V., Wechsler R. H., Faltenbacher A., Bullock J. S., 2006, MNRAS , 367, 1781 Bailin J. et al., 2005, ApJL, 627, L17 Binney J., 2010, MNRAS , 401, 2318 Binney J., 2012a, MNRAS , 426, 1324 Binney J., 2012b, MNRAS , 426, 1328 Binney J., 2014, MNRAS , 440, 787 Binney J., Spergel D., 1984, MNRAS , 206, 159 Binney J., Tremaine S., 2008, Galactic Dynamics: Second Edition. Princeton University Press Bissantz N., Gerhard O., 2002, MNRAS , 330, 591 Cappellari M. et al., 2011, MNRAS , 413, 813 de Zeeuw T., 1985, MNRAS , 216, 273 Eyre A., Binney J., 2011, MNRAS , 413, 1852 Hahn T., 2005, Computer Physics Communications, 168, 78 Helmi A., White S. D. M., 1999, MNRAS , 307, 495 Jing Y. P., Suto Y., 2002, ApJ , 574, 538 Kaasalainen M., 1995, Physical Review E., 52, 1193 Kaasalainen M., Binney J., 1994, MNRAS , 268, 1033 Kazantzidis S., Kravtsov A. V., Zentner A. R., Allgood B., Nagai D., Moore B., 2004, ApJL, 611, L73 Law D. R., Majewski S. R., 2010, ApJ , 714, 229 McMillan P. J., 2011, MNRAS , 414, 2446 McMillan P. J., Binney J. J., 2008, MNRAS , 390, 429 Merritt D., Valluri M., 1999, AJ , 118, 1177 Navarro J. F., Frenk C. S., White S. D. M., 1997, ApJ , 490, 493 Ollongren A., 1962, Bull. Astr. Inst. Netherlands , 16, 241 Piffl T. et al., 2014, MNRAS , 445, 3133 Poincar'e H., 1892, Les methodes nouvelles de la mecanique celeste Posti L., Binney J., Nipoti C., Ciotti L., 2014, ArXiv e-prints Prendergast K. H., Tomer E., 1970, AJ , 75, 674 Rowley G., 1988, ApJ , 331, 124 Sanders J., 2012, MNRAS , 426, 128 Sanders J. L., 2014, MNRAS , 443, 423 Sanders J. L., 2015, in prep. Sanders J. L., Binney J., 2013, MNRAS , 433, 1826 Sanders J. L., Binney J., 2014, MNRAS , 441, 3284 Schonrich R., Binney J., 2009, MNRAS , 396, 203 Schwarzschild M., 1979, ApJ , 232, 236 Tremaine S., 1999, MNRAS , 307, 877 Valluri M., Debattista V. P., Quinn T., Moore B., 2010, MNRAS , 403, 525 Vera-Ciro C., Helmi A., 2013, ApJL, 773, L4 Vera-Ciro C. A., Sales L. V., Helmi A., Frenk C. S., Navarro J. F., Springel V., Vogelsberger M., White S. D. M., 2011, MNRAS , 416, 1377</text>
<text><location><page_17><loc_53><loc_27><loc_70><loc_28></location>Wilson C. P., 1975, AJ , 80, 175</text>
<section_header_level_1><location><page_17><loc_52><loc_22><loc_83><loc_23></location>APPENDIX A: ANGLES AND FREQUENCIES</section_header_level_1>
<text><location><page_17><loc_52><loc_18><loc_91><loc_21></location>With the framework presented in Section 3 we are also in a position to find the angles, θ , and frequencies, Ω . Following de Zeeuw (1985) we write</text>
<formula><location><page_17><loc_53><loc_7><loc_91><loc_17></location>∂E ∂E = 1 = ∑ τ = λ,µ,ν Ω τ ∂J τ ∂E , ∂E ∂a = 0 = ∑ τ = λ,µ,ν Ω τ ∂J τ ∂a , ∂E ∂b = 0 = ∑ Ω τ ∂J τ ∂b . (A1)</formula>
<formula><location><page_17><loc_60><loc_7><loc_64><loc_7></location>τ = λ,µ,ν</formula>
<text><location><page_18><loc_9><loc_91><loc_37><loc_92></location>Inversion of these equations gives, for instance,</text>
<formula><location><page_18><loc_9><loc_88><loc_48><loc_90></location>Ω λ = 1 Γ ∂ ( J µ , J ν ) ∂ ( a, b ) where Γ = ∂ ( J λ , J µ , J ν ) ∂ ( E,a,b ) , (A2)</formula>
<text><location><page_18><loc_9><loc_82><loc_48><loc_87></location>and Ω µ and Ω ν are given by cyclic permutation of { λ, µ, ν } . To find the derivatives of J τ with respect to the integrals we differentiate equation (13) under the integral sign at constant τ . From equation (17) we know p τ ( τ, E, A τ , B τ ) . We note that</text>
<formula><location><page_18><loc_10><loc_75><loc_48><loc_81></location>∂ ∂a ∣ ∣ ∣ τ = ∂A τ ∂a ∣ ∣ ∣ τ ∂ ∂A τ ∣ ∣ ∣ τ = ∂ ∂A τ ∣ ∣ ∣ τ , ∂ ∂b ∣ ∣ ∣ τ = ∂B τ ∂b ∣ ∣ ∣ τ ∂ ∂B τ ∣ ∣ ∣ τ = ∂ ∂B τ ∣ ∣ ∣ τ , (A3)</formula>
<text><location><page_18><loc_9><loc_72><loc_48><loc_75></location>as A τ = a -C τ and B τ = b + D τ where C τ and D τ are independent of τ . The required derivatives are</text>
<formula><location><page_18><loc_10><loc_63><loc_48><loc_71></location>∂p τ ∂E ∣ ∣ ∣ τ = τ 2 4 p τ ( τ + α )( τ + β )( τ + γ ) , ∂p τ ∂a ∣ ∣ ∣ τ = -1 τ ∂p τ ∂E ∣ ∣ ∣ τ , ∂p τ ∂b ∣ ∣ ∣ τ = 1 τ 2 ∂p τ ∂E ∣ ∣ ∣ τ . (A4)</formula>
<text><location><page_18><loc_9><loc_60><loc_48><loc_62></location>Note that p τ can vanish at the limits of integration. The change of variables</text>
<formula><location><page_18><loc_9><loc_57><loc_41><loc_59></location>τ = ˆ τ sin ϑ + ¯ τ ; ¯ τ = 1 2 ( τ -+ τ + ); ˆ τ = 1 2 ( τ + -τ -)</formula>
<text><location><page_18><loc_9><loc_52><loc_48><loc_56></location>causes the integrand to go smoothly to zero at the limits. To find the angles, we use the generating function, W ( λ, µ, ν, J λ , J µ , J ν ) , given by</text>
<formula><location><page_18><loc_9><loc_46><loc_49><loc_51></location>W = ∑ τ = λ,µ,ν W τ = ∑ τ = λ,µ,ν ∫ τ τ -d τ ' p ' τ + F τ ( p τ , x ) ∫ τ + τ -d τ ' | p ' τ | . (A5)</formula>
<text><location><page_18><loc_9><loc_40><loc_48><loc_45></location>F τ are factors included to remove the degeneracy in the τ coordinates such that θ τ covers the full range 0 to 2 π over one oscillation in the Cartesian coordinates. These factors can be written in the form</text>
<formula><location><page_18><loc_9><loc_31><loc_48><loc_39></location>F λ ( p λ , x ) = Π( λ -+ α )Θ( -x ) + Θ( -p λ ) , F µ ( p µ , x ) = Π( µ -+ β )[Θ( -y ) + Θ( -p µ )] +Π( ν + + β )Π( µ + + α )[ 1 2 +Θ( -x )] +Π( ν + + β )Π( λ -+ α )Θ( -p µ ) , F ν ( p ν , x ) = Θ( -z ) + Θ( -p ν ) , (A6)</formula>
<text><location><page_18><loc_9><loc_22><loc_48><loc_30></location>where Θ is the Heaviside step function and Π is one when its argument is zero and zero otherwise. The Π -function in F λ takes care of the cases when the orbit is a box or inner long-axis loop. The Π -functions in F µ take care of the cases when the orbit is a short-axis loop or a box, an outer long-axis loop, and an inner long-axis loop respectively. The angles are given by</text>
<formula><location><page_18><loc_9><loc_18><loc_48><loc_21></location>θ τ = ∂W ∂J τ = ∑ I = E,a,b ∂W ∂I ∂I ∂J τ . (A7)</formula>
<text><location><page_18><loc_9><loc_7><loc_48><loc_17></location>The first term on the right is, up to factors, the indefinite integral of the derivatives of J τ with respect to the integrals found previously, whilst the second term is found from inverting these derivatives. We have chosen the zero-point of θ τ to correspond to τ = τ -, p τ > 0 and ˙ x i glyph[greaterorequalslant] 0 for all i , except for the outer long-axis loop orbits which have θ µ = 0 at µ = -α , p τ > 0 and ˙ x i glyph[greaterorequalslant] 0 . Note that the angles are the 2 π modulus of the θ τ found from the above scheme.</text>
<text><location><page_18><loc_52><loc_76><loc_91><loc_92></location>In Fig. A1 we show the angles calculated from the Stackel fudge for the three orbits investigated in Section 4.2. We use the automatic choice of ∆ i for the box and short-axis loop orbit, and the choice that minimises the spread in actions for the long-axis loop orbit. The short-axis loop orbit shows the expected straightline structure in the angle coordinates, whilst for the long-axis loop and box orbits there is clear deviation from this expected straight line. We also show the angles calculated using the initial angle estimate and the average of the frequency estimates along the orbit. We see that they are well recovered but after approximately one period the error in the frequencies is sufficient for these angles to deviate from the angle estimates.</text>
<text><location><page_18><loc_52><loc_61><loc_91><loc_76></location>The standard deviations in the frequencies are reasonably large. For the box orbit, the mean frequencies are given by Ω = (18 . 1 , 20 . 3 , 24 . 3) kpc -1 kms -1 with errors ∆ Ω = (0 . 2 , 0 . 8 , 1 . 3) kpc -1 kms -1 . For the short-axis loop the mean frequencies are given by Ω = (34 . 9 , 21 . 9 , 25 . 0) kpc -1 kms -1 with errors ∆ Ω = (0 . 6 , 0 . 1 , 0 . 2) kpc -1 kms -1 . For the long-axis loop the mean frequencies are given by Ω = (36 . 9 , 22 . 4 , 24 . 0) kpc -1 kms -1 with errors ∆ Ω = (1 . 6 , 1 . 5 , 0 . 8) kpc -1 kms -1 . We note that the frequency errors are largest at the turning points of the orbits for the loop orbits or near the centre of the potential for the box orbit.</text>
<text><location><page_18><loc_52><loc_56><loc_91><loc_59></location>This paper has been typeset from a T E X/ L A T E X file prepared by the author.</text>
<figure>
<location><page_19><loc_9><loc_45><loc_49><loc_92></location>
<caption>Figure A1. Angles calculated using the triaxial Stackel fudge presented in this paper for three different orbits in the triaxial NFW potential. The solid red lines show the angles calculated from the initial angle estimate and the frequency estimates for approximately one period.</caption>
</figure>
</document> | [{"title": "ABSTRACT", "content": "We present an approach to approximating rapidly the actions in a general triaxial potential. The method is an extension of the axisymmetric approach presented by Binney (2012a), and operates by assuming that the true potential is locally sufficiently close to some Stackel potential. The choice of Stackel potential and associated ellipsoidal coordinates is tailored to each individual input phase-space point. We investigate the accuracy of the method when computing actions in a triaxial Navarro-Frenk-White potential. The speed of the algorithm comes at the expense of large errors in the actions, particularly for the box orbits. However, we show that the method can be used to recover the observables of triaxial systems from given distribution functions to sufficient accuracy for the Jeans equations to be satisfied. Consequently, such models could be used to build models of external galaxies as well as triaxial components of our own Galaxy. When more accurate actions are required, this procedure can be combined with torus mapping to produce a fast convergent scheme for action estimation. Key words: methods: numerical - Galaxy: kinematics and dynamics - galaxies: kinematics and dynamics", "pages": [1]}, {"title": "Jason L. Sanders 1 , 2 glyph[star] &James Binney 1", "content": "1 Rudolf Peierls Centre for Theoretical Physics, Keble Road, Oxford, OX1 3NP, UK 2 Institute of Astronomy, Madingley Road, Cambridge, CB3 0HA 10 June 2022", "pages": [1]}, {"title": "1 INTRODUCTION", "content": "The haloes that form in baryon-free cosmological simulations almost always have triaxial shapes (Jing & Suto 2002; Allgood et al. 2006; Vera-Ciro et al. 2011). When baryons are added to the simulations, many dark haloes become more spherical (Kazantzidis et al. 2004; Bailin et al. 2005; Valluri et al. 2010), but the most successful current models suggest our Galaxy's dark halo is triaxial (Law & Majewski 2010; Vera-Ciro & Helmi 2013). Moreover, there is considerable observational evidence that the so-called 'cored', slowly-rotating elliptical galaxies are generically triaxial (Cappellari et al. 2011). Hence dynamical models of triaxial stellar systems are of considerable astronomical interest. The first triaxial models were made by violent relaxation of an N -body model (Aarseth & Binney 1978), and these models prompted Schwarzschild (1979) to develop the technique of orbit superposition so triaxial models with prescribed density profiles could be constructed. Schwarzschild's work gave significant insight into how triaxial systems work for the first time, and this insight was enhanced by de Zeeuw (1985), who showed that Stackel potentials provided analytic models of orbits in a very interesting class of triaxial systems. The most important subsequent development in the study of triaxial systems was the demonstration by Merritt & Valluri (1999) that when a triaxial system lacks a homogeneous core, as real galaxies do, box orbits tend to become centrophobic resonant box orbits. Work on axisymmetric models in the context of our Galaxy has increased awareness of the value in stellar dynamics of the intimately related concepts of the Jeans' theorem and action integrals. Ollongren (1962) established that the space of quasi-periodic orbits in galactic potentials is three-dimensional. Jeans' theorem tells us that any non-negative function f on this space provides an equilibrium stellar system. The key to gaining access to the observable properties of this tantalising array of stellar systems, is finding a practical coordinate system for orbit space. A coordinate system for orbit space comprises a set of three functions I i ( x , v ) that are constant along any orbit in the gravitational potential \u03a6( x ) of the equilibrium system. A major difficulty is that in the case of a selfconsistent system \u03a6( x ) has to be determined from f ( I ) by computing the model's density, and the latter can be computed only when \u03a6( x ) is known. Hence the computation of \u03a6( x ) has to be done iteratively, and expressions are needed for the I i that are valid in any reasonable potential \u03a6( x ) , not merely the potential of the equilibrium model. If the system is axisymmetric, the energy E and component of angular momentum L z are integrals that are defined for any axisymmetric potential \u03a6( R,z ) , and equilibrium models of axisymmetric systems have been constructed from distribution functions (DFs) of the form f ( E,L z ) (Prendergast & Tomer 1970; Wilson 1975; Rowley 1988). However, these two-integral models are not generic, and they are much harder to construct than generic models when the DF is specified as a function f ( J ) of the actions (Binney 2014). Moreover, knowledge of the DF as a function of the actions is the key to Hamiltonian perturbation theory, and the ability to perturb models is crucial if we are to really understand how galaxies work, and evolve over time. Actions are also the key to modelling stellar streams, which are themselves promising probes of our Galaxy's distribution of dark matter (Tremaine 1999; Helmi &White 1999; Eyre & Binney 2011; Sanders & Binney 2013). Action integrals constitute uniquely advantageous coordinates for orbit space, which is often called action space because its natural Cartesian coordinates are the actions. Actions can be defined for any quasi-periodic orbit and, uniquely among isolating integrals, they can be complemented by canonically conjugate coordinates, the angle variables \u03b8 i . These have the convenient properties of (i) increasing linearly in time, so and (ii) being periodic such that any ordinary phase-space coordinate such as x satisfies x ( \u03b8 , J ) = x ( \u03b8 +2 \u03c0 m , J ) , where m is any triple of integers. The discussion above amply motivates the quest for algorithms that yield angle-action coordinates ( \u03b8 , J ) given ordinary phase-space coordinates ( x , v ) , and vice versa. These algorithms are usefully divided into convergent and non-convergent algorithms. Convergent algorithms yield approximations to the desired quantity that can achieve any desired accuracy given sufficient computational resource, whereas non-convergent algorithms provide, more cheaply, an approximation of uncontrolled accuracy. Torus mapping (Kaasalainen & Binney 1994; McMillan & Binney 2008) is a convergent algorithm that yields x ( \u03b8 , J ) and v ( \u03b8 , J ) , while Sanders & Binney (2014) introduced a convergent algorithm for \u03b8 ( x , v ) and J ( x , v ) . Both algorithms work by constructing the generating function for the canonical mapping of some 'toy' analytic system of angle-action variables into the real phase space. Torus mapping has been demonstrated only in two dimensions, but both axisymmetric and two-dimensional static barred potentials have been successfully handled, and there is no evident obstacle to generalising to the three-dimensional case. Sanders & Binney (2014) treated the triaxial case, but the restriction to lower dimensions and axisymmetry is trivial. These convergent algorithms are numerically costly, and, in the axisymmetric case, non-convergent algorithms have been used extensively, especially for extracting observables from a DF f ( J ) . These extractions require J to be evaluated at very many phasespace points, and speed is more important than accuracy. The adiabatic approximation (Binney 2010) has been extensively used in modelling the solar neighbourhood (e.g. Schonrich & Binney 2009) but its validity is restricted to orbits that keep close to the Galactic plane. Stackel fitting (Sanders 2012) has been successfully used to model stellar streams (Sanders 2014). This method estimates the actions as those in the best-fitting Stackel potential for the local region a give orbit probes. Sanders (2015, in prep.) shows that is is less cost-effective than the 'Stackel fudge' that was introduced by Binney (2012a). Binney (2012b) used the 'Stackel fudge' to model the solar neighbourhood and to explore the first family of self-consistent stellar systems with specified f ( J ) (Binney 2014). The Stackel fudge was recently used by Piffl et al. (2014) to place by far the strongest available constraints on the Galaxy's dark halo. In this paper we extend the Stackel fudge to triaxial systems. We begin in Section 2 by showing how to find the actions in a triaxial Stackel potential. In Section 3 we extend the Stackel fudge to general triaxial potentials. In Section 4 we apply this algorithm to a series of orbits in a triaxial Navarro-Frenk-White (NFW) potential, and in Section 5 we construct the first triaxial stellar systems with specified DFs f ( J ) , and demonstrate that, notwithstanding the uncontrolled nature of the fudge as an approximation, the models satisfy the Jeans equations to good accuracy. In Section 6 we de-sc be a new convergent algorithm for obtaining ( J , \u03b8 ) from ( x , v ) . Finally we conclude in Section 7.", "pages": [1, 2]}, {"title": "2 TRIAXIAL ST ACKEL POTENTIALS", "content": "In this section, we show how actions can be found in a triaxial Stackel potential. The presentation here follows that given by de Zeeuw (1985).", "pages": [2]}, {"title": "2.1 Ellipsoidal coordinates", "content": "Triaxial Stackel potentials are expressed in terms of ellipsoidal coordinates ( \u03bb, \u00b5, \u03bd ) . These coordinates are related to the Cartesian coordinates ( x, y, z ) as the three roots of the cubic in \u03c4 where \u03b1 , \u03b2 and \u03b3 are constants defining the coordinate system. For the potential explored later, we choose to set x as the major axis, y as the intermediate axis and z as the minor axis, such that -\u03b3 glyph[lessorequalslant] \u03bd glyph[lessorequalslant] -\u03b2 glyph[lessorequalslant] \u00b5 glyph[lessorequalslant] -\u03b1 glyph[lessorequalslant] \u03bb . Surfaces of constant \u03bb are ellipsoids, surfaces of constant \u00b5 are hyperboloids of one sheet (flared tubes of elliptical cross section that surround the x axis), and surfaces of constant \u03bd are hyperboloids of two sheets that have their extremal point on the z axis. In the plane z = 0 , lines of constant \u03bb are ellipses with foci at y = \u00b1 \u2206 1 \u2261 \u00b1 \u221a \u03b2 -\u03b1 , whilst, in the plane x = 0 , lines of constant \u00b5 are ellipses with foci at z = \u00b1 \u2206 2 \u2261 \u00b1 \u221a \u03b3 -\u03b2 . The expressions for the Cartesian coordinates as a function of the ellipsoidal coordinates are Note that a Cartesian coordinate ( x, y, z ) gives a unique ( \u03bb, \u00b5, \u03bd ) , whilst the point ( \u03bb, \u00b5, \u03bd ) corresponds to eight points in ( x, y, z ) . Therefore, we will only consider potentials with this symmetry i.e. triaxial potentials with axes aligned with the Cartesian axes. The generating function, S , to take us between Cartesian, ( x, y, z, p x , p y , p z ) , and ellipsoidal coordinates, ( \u03bb, \u00b5, \u03bd, p \u03bb , p \u00b5 , p \u03bd ) , is Using p \u03c4 = \u2202 S/ \u2202 \u03c4 we find, for instance, There are similar equations for p \u00b5 and p \u03bd . Inversion of these three equations gives us expressions for p x , p y and p z as functions of p \u03c4 and \u03c4 . For a general triaxial potential, \u03a6 , we can express the Hamiltonian, H , in terms of the ellipsoidal coordinates as where", "pages": [2, 3]}, {"title": "2.2 Stackel potentials", "content": "The most general triaxial Stackel potential, \u03a6 S , can be written as \u03a6 S is composed of three functions of one variable. Here we denote the three functions with the same letter, f , as their domains are distinct. Additionally, f ( \u03c4 ) must be differentiable everywhere and continuous at \u03c4 = -\u03b1 and \u03c4 = -\u03b2 for \u03a6 S to be finite at \u03bb = \u00b5 = -\u03b1 and \u00b5 = \u03bd = -\u03b2 . With this form for the potential we can solve the Hamilton-Jacobi equation (de Zeeuw 1985). We write p \u03c4 = \u2202 W/ \u2202 \u03c4 and equate the Hamiltonian to the total energy, E , in equation (5). We then multiply through by ( \u03bb -\u00b5 )( \u00b5 -\u03bd )( \u03bd -\u03bb ) to find (8) We make the Ansatz W = \u2211 \u03c4 W \u03c4 ( \u03c4 ) and define such that the Hamilton-Jacobi equation becomes Taking the second derivative of this expression with respect to \u03c4 = { \u03bb, \u00b5, \u03bd } we find that where a and b are constants. Therefore, the equations for the momenta can be written as For an initial phase-space point, ( x 0 , v 0 ) , we find \u03c4 0 ( x 0 , v 0 ) and p \u03c4 0 ( x 0 , v 0 ) using the coordinate transformations and can then find the integrals a and b by solving equation (12) (see de Zeeuw 1985, . for more details). These integrals are related to the classical integrals I 2 and I 3 in a simple way. As p \u03c4 is only a function of \u03c4 , the actions are then given by the 1D integrals where ( \u03c4 -, \u03c4 + ) are the roots of p \u03c4 ( \u03c4 ) = 0 , which we find by using Brent's method to find points where the right side of equation (12) vanishes. Note that for loop orbits we must divide the 'radial' action by two ( J \u03bb for the short-axis loops and outer long-axis loops, J \u00b5 for the inner long-axis loops). In Table 1, we give the limits ( \u03c4 -, \u03c4 + ) of the action integrals and the physical meaning of each of the actions for each of the four orbit classes. The approach to finding the actions presented here requires an explicit form for f . In the next section we will show how we can circumnavigate the need for this explicit form, which allows us to use the same equations for a general potential.", "pages": [3]}, {"title": "3 THE TRIAXIAL ST ACKEL FUDGE", "content": "We now show how we can use the insights from Stackel potentials to estimate actions in a more general potential. For a general triaxial potential, \u03a6 , we can attempt to find the actions by assuming that the general potential is close to a Stackel potential. Given a general potential we define the quantities where we have chosen a particular coordinate system, ( \u03b1, \u03b2, \u03b3 ) (see \u00a7 4.1). If \u03a6 were a Stackel potential, these quantities would be given by, for instance, Therefore, for a general potential, we can write where C \u03c4 and D \u03c4 are constants provided we always evaluate \u03c7 \u03c4 with two of the ellipsoidal coordinates fixed. For instance, we always evaluate \u03c7 \u03bb at fixed \u00b5 and \u03bd . 4 J. L. Sanders & J. Binney When we substitute these expressions into equation (12) we find For each \u03c4 coordinate there are two new integrals of motion given by A \u03c4 = a -C \u03c4 and B \u03c4 = b + D \u03c4 . Given an initial phase-space point, ( x 0 , v 0 ) , and a coordinate system, ( \u03b1, \u03b2, \u03b3 ) , we can calculate the ellipsoidal coordinates ( \u03bb 0 , \u00b5 0 , \u03bd 0 , p \u03bb 0 , p \u00b5 0 , p \u03bd 0 ) . Inserting this initial phase-space point into equation (17) gives us an expression for B \u03c4 as It remains to find an expression for A \u03c4 as a function of the initial phase-space point. To proceed we consider the derivative of the Hamiltonian with respect to \u03c4 . In a Stackel potential we can stay on the orbit while changing \u03c4 and p \u03c4 ( \u03c4 ) with all the other phase-space variables held constant. Therefore, in a Stackel potential \u2202 H/ \u2202 \u03c4 = 0 . Here we consider \u2202 H/ \u2202 \u03bb and will give the results for \u00b5 and \u03bd afterwards. Using equation (5) we write To evaluate \u2202 [ p 2 \u03bb /P 2 \u03bb ] / \u2202 \u03bb we use equation (18) to write such that where Upon substitution into equation (19) we note that the derivatives of \u03a6 cancel. Therefore, evaluating \u2202 H/ \u2202 \u03bb at the initial phase-space point we find This can be simplified further to Note that A \u03bb is independent of \u03bb 0 and p \u03bb 0 (except implicitly in the energy, E ) as P \u03c4 0 contains cancelling factors of ( \u03bb 0 -\u03c4 0 ) . Similarly For a true Stackel potential , given an initial phase-space point we can find 6 integrals of motion, ( A \u03bb , A \u00b5 , A \u03bd , B \u03bb , B \u00b5 , B \u03bd ) from equations (18), (24) and (25). Note that a general Stackel potential only admits three integrals of motion so the 6 derived integrals of motion are not independent. This procedure gives identical results to evaluating the integrals as in de Zeeuw (1985). Note that the expressions for these integrals do not explicitly involve the function f ( \u03c4 ) - they only involve the potential, \u03a6 . With the integrals of motion calculated we are in a position to find p \u03c4 ( \u03c4 ) and hence the actions from equation (13). For a general potential we may find six approximate integrals of motion using the same equations, and hence estimate the actions. In this case, although the potential may admit only three true integrals of motion, the 6 approximate integrals of motion are independent estimates of true integrals of motion. Again, as the expressions do not require f ( \u03c4 ) they can be evaluated for a general potential. In Appendix A we show how the angles and frequencies can be estimated using the same approach.", "pages": [3, 4]}, {"title": "3.1 Relation to axisymmetric case", "content": "The above procedure extends the work of Binney (2012a). Binney (2012a) constructed the 'Stackel fudge' algorithm for estimating actions in a general axisymmetric potential \u03a6( R,z ) , where R and z are the usual cylindrical polar coordinates. We now relate the procedure to that of Binney (2012a) to develop further understanding. Oblate axisymmetric Stackel potentials are associated with prolate elliptic coordinates ( \u03bb, \u03bd ) given by the roots for \u03c4 of where -\u03b3 glyph[lessorequalslant] \u03bd glyph[lessorequalslant] -\u03b1 glyph[lessorequalslant] \u03bb . Binney (2012a) uses the coordinates ( u, v ) which are related to ( \u03bb, \u03bd ) via An oblate axisymmetric Stackel potential can be written as and the equations for the momenta are given by (de Zeeuw 1985) For axisymmetric potentials I 2 = 1 2 L 2 z , where L z is the z -component of the angular momentum. For a general oblate axisymmetric potential, \u03a6 , we define If \u03a6 were a Stackel potential these quantities would be given by Therefore, for a general potential, we can write, -20 -10 0 10 20 x/kpc where D \u03c4 are constants provided we evaluate \u03c7 \u03bb at constant \u03bd and vice versa. We can write the equations for the momenta as where we have defined the integral of motion B \u03c4 = I 3 -D \u03c4 . B \u03c4 may be found given an initial phase-space point and we then integrate the equations for the momenta to find the actions. Note that in this case only two integrals of the motion, B \u03c4 , need to be found, as, in the axisymmetric case, we can find two exact integrals of motion, E and L z . This is the procedure followed in Binney (2012a) and, despite the differing conventions and presentation, this method gives identical results to that of Binney (2012a).", "pages": [4, 5]}, {"title": "4 TESTS", "content": "For the purposes of testing the above algorithm, we use a triaxial NFW halo (Navarro et al. 1997; Jing & Suto 2002): We set y s = 0 . 95 , z s = 0 . 85 , m 0 = 10kpc and GM s = (1109 km s -1 ) 2 kpc . In Fig. 1 we show the equipotential contours in the z = 0 and y = 0 planes. It is perhaps more conventional to include the triaxiality in the density (e.g. Jing & Suto 2002), but, for simplicity, we have chosen to include triaxiality in the potential. For our choice of parameters this leads to negative densities along the z -axis for z glyph[greaterorsimilar] 130 kpc . This is well outside the region we will probe in our experiments so we are not concerned that our model is unphysical at large z .", "pages": [5]}, {"title": "4.1 Selection of coordinate system", "content": "The accuracy of the above routine for a general potential will depend upon our choice of coordinate system, ( \u03b1, \u03b2, \u03b3 ) . Note that the potential is fixed and this coordinate system acts only as a set of parameters in the algorithm to find the actions. We can freely set \u03b3 = -1 kpc 2 as the coordinate system only depends on \u2206 1 = \u221a \u03b2 -\u03b1 and \u2206 2 = \u221a \u03b3 -\u03b2 . For each orbit we consider 20 10 0 -10 -20 y =0 we are in a position to choose different \u2206 i . Here we consider how we can choose suitable \u2206 i given an initial phase-space point. In Sanders (2012) the mixed derivative \u2202 \u03bb \u2202 \u03bd [( \u03bb -\u03bd )\u03a6] was used to select an appropriate coordinate system in an axisymmetric potential. For the triaxial case we could construct a similar quantity: \u2202 \u03bb \u2202 \u00b5 \u2202 \u03bd [( \u03bb -\u00b5 )( \u00b5 -\u03bd )( \u03bd -\u03bb )\u03a6] . However, this expression would involve third derivatives of the potential so is undesirable. Binney (2014) selected a coordinate system by fitting ellipses to shell orbits at each energy, E . We follow a similar procedure: we assume that the best choice of coordinate system is solely a function of E . In a Stackel potential the short-axis closed loops are ellipses confined to the plane z = 0 with foci at y = \u00b1 \u2206 1 = \u00b1 \u221a \u03b1 -\u03b2 , whilst the long-axis closed loops are confined to the plane x = 0 with foci at z = \u00b1 \u2206 2 = \u00b1 \u221a \u03b3 -\u03b2 . Additionally, for these closed loop orbits only one of the actions is non-zero ( J \u00b5 for the short-axis closed loop and J \u03bd for the long-axis closed loop). For a general potential we use these facts to select appropriate values for \u2206 i using a two step procedure: given a value for E we find the two closed loop orbits - one around the short axis and one around the long axis, and with these closed orbits found we alter the position of the foci to optimise the action estimates from our algorithm. Note that the structure of the closed orbits is independent of any choice of the foci positions such that the two steps of the procedure are distinct. First, to find the closed orbits with energy E , we select a point along the intermediate axis, y = y I , and launch an orbit with speed v = \u221a 2( E -\u03a6(0 , y I , 0)) in either the x (for the short-axis loop) or z direction (for the long-axis loop). The next time the orbit crosses the y -axis we note the y -intercept, y = y F and calculate | -y F -y I | . We repeat this procedure with a new y I until we have minimised | -y F -y I | using Brent's method. We only integrate half of the orbit and assume that the other half can be obtained by symmetry to avoid misidentifying fish-tail resonant orbits as closed loop orbits. With the closed orbits with energy E in our potential found, we turn to estimating the location of the foci. Using the long-axis closed loop orbit integration we find an estimate of \u2206 2 by minimising the standard deviation of the J \u03bd estimates from each time-step with respect to \u03b2 using Brent's method. The action estimates are found using the algorithm outlined in Section 3. This procedure is not sensitive to the choice of \u03b1 . Once we have found \u03b2 we perform a similar procedure for the short-axis loop: vary \u03b1 until we have minimised the standard deviation of J \u00b5 . We perform the above procedure for a range of energies from E min = \u03a6(0 , y min , 0) to E max = \u03a6(0 , y max , 0) , tabulating the found values of \u03b1 and \u03b2 for interpolation. For the NFW potential, we adopt y min = 0 . 05 kpc and y max = 60kpc . In Figs. 2 and 3 we plot the standard deviation of the actions of the closed loop orbits against \u2206 2 and \u2206 1 for the constant energy surface with E = \u03a6(0 , m 0 , 0) = -(290 km s -1 ) 2 . In both cases there is a clear minimum in the standard deviation. In Fig. 2 we show the standard deviation in J \u03bd as a function of \u2206 2 = \u221a \u03b3 -\u03b2 using two different values for \u03b1 . The results are indistinguishable. Provided we initially choose a sufficiently negative value of \u03b1 that the optimal \u03b2 satisfies \u03b2 > \u03b1 , we are free to first set \u2206 2 and then choose \u2206 1 .", "pages": [5]}, {"title": "4.1.1 Coordinate system procedure", "content": "For clarity, we now summarise the above procedure: (i) Given a general potential, create a regularly-spaced grid in energy, E , between some minimum and maximum energy. (ii) At each grid-point, E i , find the short-axis and long-axis closed loops by integrating orbits launched at (0 , y k , 0) with velocity \u221a 2( E i -\u03a6(0 , y k , 0)) in the direction of the long-axis or shortaxis respectively. The closed loops will cross the y -axis for the first time at (0 , -y k , 0) . We store the phase-space points ( x j , v j ) at each time sample t j . (iii) Minimise the standard deviation of the J \u03bd ( x j , v j ) from the long-axis closed loop orbit integration with respect to \u03b2 to find \u2206 2 . (iv) Minimise the standard deviation of the J \u00b5 ( x j , v j ) from the short-axis closed loop orbit integration with respect to \u03b1 to find \u2206 1 . We call the \u2206 1 and \u2206 2 found using this procedure the closed-loop estimates.", "pages": [5, 6]}, {"title": "4.1.2 Coordinate system results", "content": "In Fig. 4 we have plotted the closed-loop choice of \u2206 1 and \u2206 2 as a function of the energy. We see that for low energies (very centrally confined orbits) \u2206 i tends to zero. Due to the cusp at the centre of the NFW potential, loop orbits exist right down to the centre of the potential. The foci must lie within these loop orbits so \u2206 i must decrease as we go to lower energy. As we increase the energy \u2206 i increases with \u2206 1 < \u2206 2 . To check the closed-loop estimates we launch a series of orbits of constant energy E = \u03a6(0 , m 0 , 0) = -(290 km s -1 ) 2 at linearly-spaced intervals along the y -axis with velocity vectors in the ( x, z ) plane oriented at differing linearly-spaced angles, \u03b8 , to the x axis and integrate the orbits for approximately 10 . 3 Gyr storing phase-space points every 0 . 1 Gyr . Note again that the orbit integration is in the fixed NFW potential and so the structure of an orbit is independent of any choice of \u03b1 and \u03b2 . The choice of \u03b1 and \u03b2 only affects the recovery of the actions and we wish to find the optimal choice of \u03b1 and \u03b2 for each orbit i.e. the choice that makes the actions as constant in time as possible. Therefore, we minimise the sum of the variances of the actions with respect to \u03b1 and \u03b2 . The results of this procedure are shown in Fig. 5. We see that the major- its yield optimal \u2206 i similar to the closed-loop estimates. At the extremes of y \u2206 i deviates from this choice. These are the box orbits and they seem to favour lower \u2206 i . At fixed y the choice of \u2206 i is not so sensitive to \u03b8 . We could improve our choice of \u2206 1 and \u2206 2 by making the choice a function of an additional variable. For instance, we could make the choice a function of the total angular momentum, which is not an integral of motion. However, we will see that we cannot significantly improve the action recovery with a better choice of \u2206 i .", "pages": [6, 7]}, {"title": "4.2 Accuracy", "content": "We now briefly inspect the accuracy of the action recovery using the triaxial Stackel fudge. We take three orbits from the surface of constant energy explored in the previous section. The three orbits are a box orbit with y = 1 . 8234 kpc , \u03b8 = 0 . 6 rad (shown in Fig. 6), a short-axis loop orbit with y = 4 . 8234 kpc , \u03b8 = 0 . 4 rad (shown in Fig. 7), and a long-axis loop orbit with y = 3 . 8234 kpc , \u03b8 = 1 . 2 rad (shown in Fig. 8). The top row of each figure shows three projections of the orbit, while the three lower panels show the action estimates calculated at each point along the orbit using the closed-loop choice of \u2206 i in blue, and in green those obtained with the choice of \u2206 i that minimises the spread in the action estimates. Clearly no procedure for determining \u2206 i will give superior performance to that obtained with the latter, which is expensive to compute because it requires orbit integration. The intersection of the black lines in the bottom panels of Figs. 6 to 8 show the 'true' actions calculated with the method of Sanders & Binney (2014). The distributions of coloured points from the Stackel Fudge scatter around the true actions, as one would hope. The extent of the green distributions, obtained with the computationally costly values of \u2206 i , are at best a factor two smaller that the distributions of blue points, obtained with the cheap value of \u2206 i . From this experiment we conclude that there is not a great deal to be gained by devising a better way to evaluate the \u2206 i . In Appendix A we show how well the angle coordinates are recovered for these orbits. The actions of the box orbit are ( J \u03bb , J \u00b5 , J \u03bd ) = (686 , 192 , 137) kpc km s -1 and our method yields errors of (\u2206 J \u03bb , \u2206 J \u00b5 , \u2206 J \u03bd ) = (56 , 39 , 22) kpc km s -1 so approximately 10 -20 per cent. If we adjust \u2206 i to minimise the spread in the action estimates along the orbit, we find errors of (\u2206 J \u03bb , \u2206 J \u00b5 , \u2206 J \u03bd ) = (17 , 19 , 16) kpc km s -1 so approximately glyph[lessorsimilar] 10 per cent. We can achieve a factor of two improvement for J \u03bb and J \u00b5 . The actions of the short-axis loop orbit are ( J \u03bb , J \u00b5 , J \u03bd ) = (55 , 752 , 78) kpc km s -1 and our method yields errors of (\u2206 J \u03bb , \u2206 J \u00b5 , \u2206 J \u03bd ) = (2 , 3 , 1) kpc km s -1 so glyph[lessorsimilar] 4 per cent. If we adjust \u2206 i to minimise the spread in the action estimates along the orbit, we find errors of (\u2206 J \u03bb , \u2206 J \u00b5 , \u2206 J \u03bd ) = (0 . 8 , 2 . 0 , 0 . 9) kpc km s -1 . The actions of the long-axis loop orbit are ( J \u03bb , J \u00b5 , J \u03bd ) = (50 , 102 , 680) kpc km s -1 and our method yields errors of (\u2206 J \u03bb , \u2206 J \u00b5 , \u2206 J \u03bd ) = (4 , 5 , 6) kpc km s -1 so glyph[lessorsimilar] 8 per cent. If we adjust \u2206 i to minimise the spread in the action estimates along the orbit, we yield errors of (\u2206 J \u03bb , \u2206 J \u00b5 , \u2206 J \u03bd ) = (2 . 0 , 2 . 5 , 4 . 2) kpc km s -1 . For all the orbits shown in Fig. 5 (sampled from the constant energy surface E = \u03a6(0 , m 0 , 0) = -(290 km s -1 ) 2 ), we have plotted the logarithm of the fractional error in the actions in Fig. 9 (i.e. logarithm of the standard deviation of the action estimates around the orbit over the mean action estimate). We find the most accurate action recovery occurs for the orbits with the initial condition y \u2248 m 0 / 2 , where we have mostly loop orbits. For these loop orbits, J \u00b5 and J \u03bd are accurate to glyph[lessorsimilar] 1 per cent but the 'radial' action J \u03bb is small for these orbits so the relative error can be large. For the box orbits at the extremes of y , the relative error increases to \u223c 10 per cent but can be as large as order one in J \u00b5 for low y . In Fig. 10 we show the absolute errors in the actions as a function of action for the constant energy surface along with the orbit classification. These are again calculated as the standard deviation of action estimates around the orbit. Each phase-space point along the orbit is allocated a classification number based on the limits of \u03c4 found in the Stackel approximation (see Table 1): \u03bb -= -\u03b1, \u00b5 -= -\u03b2 and \u03bd -= -\u03b3 correspond to a box orbit (classification number 0), \u00b5 -= \u03b2 , \u00b5 + = -\u03b1 to a short-axis loop orbit (1), \u03bb -= -\u03b1 , \u03bd + = -\u03b2 to an inner long-axis loop (2), and \u00b5 + = -\u03b1 , \u03bd + = -\u03b2 to an outer long-axis loop (3). The orbit classification number is calculated as an average of these classifications along the orbit. With this scheme, orbits near the boundaries of the orbit classes that are chaotic or resonant are allocated noninteger orbit classification numbers. We see that the largest action errors occur at the interfaces between the orbit classes. In particular, \u2206 J \u03bb and \u2206 J \u00b5 are largest along the box-short-axis-loop interface, whilst \u2206 J \u03bd is largest at the box-long-axis-loop interface. It is at these boundaries that the orbits pass close to the foci so clearly our choice of foci affects the action recovery for these orbits. In general, we find that the action recovery for loop orbits is good, as these orbits probe a small radial range of the potential. For box orbits the recovery deteriorates as these orbits probe a larger central region of the potential. Additionally, we have seen that by altering \u2206 i we can achieve up to a factor of two improvement in the accuracy of the actions for both the loop and box orbits.", "pages": [7]}, {"title": "4.3 Surfaces of section", "content": "For understanding the behaviour of dynamical systems, Poincar'e (1892) introduced the concept of a surface of section. These diagrams simplify the motion of a high-dimensional dynamical system. A regular orbit in an integrable triaxial potential permits three loop 120 constants of the motion, thus confining the motion to a 3-torus. If we choose to only plot the series of points where the orbit passes through a 4-surface in phase-space, e.g. defined by y = 0 and z = 0 , the phase-space points will be confined to a line, or a consequent, which may be visualized clearly. We can test the Stackel approach outlined here by seeing how well it reproduces the surfaces of section. To produce the true surface of section we must integrate the orbit in the true potential and find the phase-space points where the orbit crosses our chosen 4surface. Here we use 4-surfaces defined by one of the spatial axes. The orbit will never pass through a given spatial axis in a finite time so we can only produce points arbitrarily close to the given axis. If we require the points where the orbit crosses the x -axis we integrate until the y and z steps have bracketed y = 0 and z = 0 . We then bisect the integration step N max times choosing the interval that brackets y = 0 and if the interval still brackets z = 0 after the N max bisections we store the final point. We integrate over our chosen step-size with a Dortmund-Prince 8th order adaptive integration scheme with a absolute accuracy of glyph[epsilon1] = 1 \u00d7 10 -10 . We choose a step-size of 0 . 005 kpc and set N max = 10 . This scheme | | | | | | produces points that are glyph[lessorsimilar] 0 . 001 kpc away from the x axis for the box orbit considered below. To produce the corresponding surface of section from the Stackel method we determine \u03c4 along our chosen spatial axis using equation (2) between the determined limits in \u03c4 , and use equation (17) to find the corresponding p \u03c4 . From p \u03c4 , we can use expres-s such as equation (4) to calculate p x , p y and p z : if we wish to draw the consequent defined by y = 0 , z = 0 we have that \u00b5 = -\u03b2 and \u03bd = -\u03b3 such that x = \u221a \u03bb + \u03b1 , and p x = \u221a 4( \u03bb + \u03b1 ) p \u03bb . If we wish to draw the consequent defined by x = 0 , z = 0 we have that for | y | > \u2206 1 , \u00b5 = -\u03b1 and \u03bd = -\u03b3 such that y = \u221a \u03bb + \u03b2 , and p y = \u221a 4( \u03bb + \u03b2 ) p \u03bb , whilst for | y | < \u2206 1 , \u03bb = -\u03b1 and \u03bd = -\u03b3 such that y = \u221a \u00b5 + \u03b2 and p y = \u221a 4( \u00b5 + \u03b2 ) p \u00b5 1 .", "pages": [7, 9, 10]}, {"title": "5 A TRIAXIAL MODEL WITH SPECIFIED DF", "content": "The main purpose of the algorithm presented here is to calculate efficiently the moments of triaxial distribution functions. We have seen that the errors in the actions reported by the scheme can be large. However, when calculating moments of a distribution function, many action evaluations are required and there is scope for errors to substantially cancel, leaving the final value of the moment quite accurate. In this section, we demonstrate this phenomenon by for the first time constructing triaxial models from an analytic DF f ( J ) . We adopt a simple distribution function (Posti et al. 2014) where J 0 = 10kms -1 kpc is a scale action, \u03b6 controls whether the model is tangentially/radially biased and \u03b7 controls the flattening in z of the model. We do not construct a self-consistent model but instead consider f to be a tracer population in the externally applied triaxial NFW potential of equation (35). We set p = -3 , which, for our choice of potential, causes 1 If we wish to draw the consequent defined by x = 0 , y = 0 we have for | z | < \u2206 2 , \u03bb = -\u03b1 and \u00b5 = -\u03b2 so z = \u221a \u03bd + \u03b3 and p z = \u221a 4( \u03bd + \u03b3 ) p \u03bd , whilst for | z | > \u2206 2 , \u00b5 = -\u03b1 and \u03bd = -\u03b2 so z = \u221a \u03bb + \u03b3 and p z = \u221a 4( \u03bb + \u03b3 ) p \u03bb . the density to go as r 0 in the centre and fall off as r -3 for large r . Note that the mass of this model diverges logarithmically. We set \u03b7 = 1 . 88 and explore two values of \u03b6 = 0 . 7 (tangential bias) and \u03b6 = 3 . 28 (radial bias). Note that for the orbit classes to fill action space seamlessly, we must scale the radial action of the loop orbits by a factor of two (Binney & Spergel 1984). We proceed by calculating the moments of this distribution function in the test triaxial NFW potential at given spatial points, x . These non-zero moments are Note that, as the potential is time-independent, the Hamiltonian is time-reversible and we need only integrate over half the velocity space and multiply the result by two. We integrate up to the maximum velocity, v max at x , given by v max = \u221a 2\u03a6( x ) . We will later calculate these moments extensively to demonstrate that the actionbased distribution functions obey the Jeans' equations. In Figs. 12 and 13 we plot the density of the radially-biased ( \u03b6 = 3 . 28 ) and tangential-biased ( \u03b6 = 0 . 7 ) models. We display contours of constant density in two planes along with the density along a line parallel to the x -axis decomposed into its contributions from each orbit class. The density is calculated using the adaptive Monte-Carlo Divonne routine in the CUBA package of Hahn (2005). The class of each orbit is determined by the limits of the motion in \u03c4 : \u03bb -= -\u03b1 , \u00b5 -= -\u03b2 and \u03bd -= -\u03b3 correspond to a box orbit, \u00b5 -= \u03b2 , \u00b5 + = -\u03b1 to a short-axis loop orbit, and \u03bd -= -\u03b3 , \u03bd + = -\u03b2 to a long-axis loop. As we are calculating the density close to the x -axis, the long-axis loop orbits, which loop the x -axis, do not contribute significantly to the density integral. We see that for the radially-biased model the box orbits are the dominant contributors whilst for the tangentially-biased model the short-axis loop orbits are the major contributors. We will now perform some checks to see whether our distribution functions are accurate.", "pages": [10, 11]}, {"title": "5.1 Normalization", "content": "One quick check of our action estimation scheme is how accurately it recovers the normalization. To keep the normalization finite we set p = -3 . 5 for this section. We are able to calculate the normalization of our DF in two distinct ways. Firstly, we calculate the normalization analytically from the DF as Note that for each J in the appropriate range there are two loop orbits - one circulating clockwise and one anti-clockwise. Therefore, we must multiply the normalization by two for these orbits. However, we have defined the 'radial' action to be four times the integral from \u03c4 -to \u03c4 + for these orbits so these factors cancel (Binney & Spergel 1984; de Zeeuw 1985). Additionally, we can calculate the mass as For each spatial coordinate, we make the transformation u i = 1 / (1 + x i ) to make the integrand flatter. The limits of the integral are now u i = [0 , 1] . To reduce numerical noise, we split the integral such that we calculate the contribution near the axes separately. We perform the integral using the Monte Carlo Divonne routine. For the tangentially-biased model ( \u03b6 = 0 . 7 ), we find M est \u2248 1 . 006 M true and for the radially-biased model ( \u03b6 = 3 . 28 ) M est \u2248 1 . 007 M true so despite the often large errors in the actions the normalization of the model is well recovered.", "pages": [11]}, {"title": "5.2 The Jeans equation", "content": "Our distribution function must satisfy the collisionless Boltzmann equation In turn this means the distribution function must satisfy the Jeans equations (see equation (4.209) of Binney & Tremaine (2008)) A simple test of our action-based distribution functions is checking whether they satisfy these equations. The right hand side is calculated from analytic differentiation of the potential and multiplying by the density. The left hand side is found by numerically differentiating the three-dimensional integrals \u03c1\u03c3 2 ij and summing the appropriate contributions. Numerical differentiation of an integral leads to significant noise. To combat this we use an adaptive vectorised integration-rule cubature scheme implemented in the CUBATURE package from Steven Johnson (http://abinitio.mit.edu/wiki/index.php/Cubature). Using a fixed-rule adaptive routine means the noise in the integrals is controlled such that the numerical derivatives are less noisy. In Figures 14 and 15 we show how accurately the Jeans equations are satisfied along several lines through the potential for our two models. We plot each side of each Jeans equation for a choice of j along a range of lines, along with the percentage error difference between the two sides of the equation. We avoid calculating the derivatives of the moments along the axes as the numerical differentiation is awkward there. In general, we find glyph[lessorsimilar] 10 per cent error for nearly all tested points with the majority having glyph[lessorsimilar] 4 per cent over a range of \u223c 8 orders of magnitude. Despite the large errors introduced by the action estimation scheme, we have produced a distribution function that satisfies the Jeans equations to reasonable accuracy. Even for the heavily radially-biased model, which has large contributions from the box orbits, the Jeans equations are well satisfied. This gives us confidence that models based on triaxial distribution functions can be constructed using the scheme we have presented.", "pages": [11]}, {"title": "5.3 Error in the moments", "content": "When comparing smooth models to data, we are primarily interested in whether the action-estimation scheme produces accurate enough actions to reproduce the features of the data. If the broad features of the data are on scales larger than the errors in the individual actions, our method should be sufficiently accurate to recover these features from an appropriate f ( J ) . In this case, the error in the moments is more important than the error in the individual actions. Larger errors in the actions are expected to lead to larger c p k 5 . 0 errors in the moments but the relationship between the two is unclear. We have seen that, despite the presented method introducing large errors in some actions, the moments of the DF are well recovered such that the normalization is accurate to 0 . 6 per cent and the Jeans' equations are accurate to glyph[lessorsimilar] 4 per cent. In this section, we develop some understanding of how errors in the actions translate into errors in the moments. Initially, we can make some progress by considering the normalization of the DF: \u222b d 3 x d 3 v f ( x , v ) = \u222b d 3 J d 3 \u03b8 f ( J ) . Suppose we have a set of angle-action variables ( J ' , \u03b8 ' ) that are not the true angle-action variables ( J , \u03b8 ) , we can relate the two sets via the generating function, S ( J , \u03b8 ' ) , such that Now suppose we evaluate the normalization using J ' ( x , v ) . We transform the volume element d 3 J d 3 \u03b8 to d 3 J d 3 \u03b8 ' via the Jacobian We assume that the approximate angle-action variables are suffi- Radial bias, \u03b6 =3 . 28 ciently close to the true angle-action variables that the second term on the right-hand side is much less than unity. Therefore, the difference in the normalization is given by We see that, after integrating over \u03b8 ' , all terms with odd powers of trigonometric functions vanish, and the leading order error in the normalization is This term is second order in the Fourier components of the generating function. Note that the sign of this term is unclear as both the S n and \u2202 J S n can be positive and negative. We anticipate \u2202 J f is negative such that the density falls with radius. From equation (40) we find that to leading order the errors in the angle-action variables averaged over an orbit are Therefore, we can see that the error in the normalization is approximately first order in the error in the actions, \u2206 J i , the error in the angles, \u2206 \u03b8 i , and the gradient of the distribution function, \u2202 J i f . Therefore, we anticipate that the relative error in the normalization will be small when \u2206 J glyph[lessmuch] f ( \u2202 J f ) -1 for all points in action space. This is essentially the expected result. Consider the distribution functions of Binney (2014): these are of the form f ( J z ) \u223c exp( -\u03bdJ z /\u03c3 2 z ) such that \u2206 J z glyph[lessmuch] \u03c3 2 z /\u03bd for a good estimate of the normalization, where \u03bd is the vertical epicycle frequency. Near the Sun \u03bd \u2248 0 . 1Myr -1 and \u03c3 z \u2248 30 km s -1 so \u2206 J z glyph[lessmuch] 10 kpc km s -1 . For the distribution function considered in the previous section we require \u2206 J glyph[lessmuch] max ( J, J 0 ) . For the moments of the distribution function we expect similar results but we are not able to explicitly calculate the leading order errors in these quantities. Instead we briefly show how the error in the density changes with the error in the actions. We begin by calculating the density from a triaxial DF for which we know the true density - the perfect ellipsoid (de Zeeuw 1985), which has Tangential bias, \u03b6 =0 . 7 density profile where We set \u03c1 0 = 7 . 2 \u00d7 10 8 M glyph[circledot] kpc -3 , x P = 5 . 5 kpc , y P = 4 . 5 kpc and z P = 1kpc . The actions in this potential can be found exactly using the scheme presented above by setting \u03b1 = -x 2 P and \u03b2 = -y 2 P . We use the tangentially-biased DF of the previous section. For four different Cartesian positions we calculate the true density, and then proceed to calculate the density when the logarithm of the actions are scattered normally by some fixed amount \u221a \u3008 (\u2206 J ) 2 \u3009 . We plot the error in the density as a function of \u2206 J in Fig. 16. We find that the relative error in the density goes as \u223c \u2206 J 1 . 3 for the three densities near the axes, and is flatter for the density at ( x, y, z ) = (4 , 4 , 4) kpc . This procedure is artificial as we have used non-canonical coordinates to evaluate the density. Therefore, for a fuller test we instead choose to calculate the density using the Stackel fudge scheme but changing \u03b1 away from the truth. In this case, the error in the density is systematic. In the lower panel of Fig. 16 we plot the density error from this procedure as a function of the distributionfunction-weighted RMS action errors. Again we find the density goes as \u223c \u221a \u3008 (\u2206 J ) 2 \u3009 1 . 2 for all but the density on the z axis. As we are changing \u03b1 this does not significantly affect the actions of the long-axis loop orbits, which dominate the density budget along the z axis.", "pages": [11, 12, 13, 14]}, {"title": "6 EXTENDING THE SCOPE OF TORUS MAPPING", "content": "We have seen that the triaxial Stackel fudge can produce large errors in the actions for some orbits, but that the moments of an action-based DF are nevertheless well recovered using the method. However, even if one requires accurate actions, the Stackel fudge can be valuable because it enables one to construct a torus through a given point ( x , v ) by torus mapping rather than orbit integration (Sanders & Binney 2014). This option may be essential because torus mapping works even in a chaotic portion of phase space (Kaasalainen 1995), while the approach based on orbit integration is already problematic when resonantly trapped orbits take up a significant portion of phase space, and it breaks down with the onset of chaos. Weproceed as follows. First we use the Stackel fudge to obtain 0 \u221a \u2329 \u232a", "pages": [14, 15]}, {"title": "approximate actions", "content": "Then by torus mapping we obtain the torus with actions J . On account of errors, the given point ( x , v ) will not lie on the constructed torus, but one can identify the nearest point ( x ' , v ' ) that does lie on the torus. We find this point by minimising the tolerance with respect to the angle, \u03b8 , on the torus. \u2126 is the frequency vector of the constructed torus. We use the Stackel fudge estimate of the angles as an initial guess for the minimisation. For the point ( x ' , v ' ) we know the true actions: Now we use the Stackel fudge to obtain approximate actions for this point If, as we expect, the errors in the Stackel fudge are systematic rather than random, then with \u2206 a slowly varying function of phase-space position. So a better estimate of the true actions of the original point ( x , v ) is If one is of a nervous disposition, one now uses torus mapping to construct the torus with actions J '' and seeks the point on this torus that is closest to the given point and applies the Stackel fudge there, and so on. This cycle can be repeated until the nearest point on the constructed torus satisfies some tolerance \u03b7 = \u03b7 \u2217 . In Fig. 17 we show an illustration of this procedure for the axisymmetric case. We use the axisymmetric Stackel fudge as given in Section 3.1 and the torus construction code as presented in McMillan & Binney (2008). For the axisymmetric Stackel fudge we set \u03b3 = -1 kpc 2 and \u03b1 = -20 kpc 2 , such that the foci are at z = \u00b1 \u221a \u03b3 -\u03b1 \u2248 \u00b1 4 . 4 kpc . We construct a torus of actions ( J r , L z , J z ) = (244 . 444 , 3422 . 213 , 488 . 887) kpc km s -1 in the 'best' potential from McMillan (2011). This potential is an axisymmetric multi-component Galactic potential consisting of two exponential discs representing the thin and thick discs, an axisymmetric bulge model from Bissantz & Gerhard (2002) and an NFW dark halo. The parameters of the mass model were chosen to satisfy recent observational constraints. We produce a series of ( x , v ) points on the constructed torus. The axisymmetric Stackel fudge gives errors of \u2206 J i = 13kpckms -1 in the recovery of the actions for these points. After one iteration of the above procedure we find actions accurate to \u2206 J i \u2248 0 . 3 kpc km s -1 , and after the procedure has converged to a tolerance \u03b7 \u2217 = (0 . 1 km s -1 ) 2 the actions are accurate to \u2206 J i \u2248 0 . 07 kpc km s -1 . The majority of ( x , v ) points converge in less than five iterations. We limited the number of iterations to 20 and a few ( x , v ) did not converge within 20 iterations. This is due to non-linear behaviour of both the torus construction and Stackel code in small phase-space volumes. It is clear, however, that only one iteration is required for a substantial improvement in the actions, and it is hard to make a case for more iterations. Finally, we demonstrate how the above method operates for a range of high-action orbits. We integrate a series of orbits in the 'best' potential from McMillan (2011) launched at 5 linearlyspaced points x 0 along the x -axis such that 4 kpc glyph[lessorequalslant] x 0 glyph[lessorequalslant] 12 kpc . We launch the orbits with velocity v = ( v 1 cos \u03b8, v 0 , v 1 sin \u03b8 ) where v 0 = \u221a x 0 \u2202 x \u03a6( x 0 , 0 , 0) and we choose 4 linearly-spaced values of v 1 such that 0 . 5 v 0 glyph[lessorequalslant] v 1 glyph[lessorequalslant] 0 . 8 v 0 and 5 linearly-spaced angles \u03b8 such that 0 . 2 rad glyph[lessorequalslant] \u03b8 glyph[lessorequalslant] 1 2 \u03c0 rad . The range of radial and vertical actions for this collection of orbits is shown in the top panel of Fig. 18 and is approximately 1 kpc km s -1 glyph[lessorsimilar] J r glyph[lessorsimilar] 800 kpc km s -1 and 1 kpc km s -1 glyph[lessorsimilar] J z glyph[lessorsimilar] 800 kpc km s -1 (these are calculated as the mean of the fudge estimates along the orbit). For each orbit, we find the standard deviation of the action estimates from the axisymmetric Stackel fudge method and the iterative torus method for 10 widely time-separated phase-space points along the orbit. We use \u03b7 \u2217 = (0 . 1 km s -1 ) 2 , limit the maximum number of iterations to 5 and construct the tori with a relative error of \u223c 1 \u00d7 10 -4 . We plot the results in Fig. 18. We see the majority of orbits have lower iterative torus errors than fudge errors and follow a broad line that lies approximately two to three orders of magnitude beneath the 1:1 line. However, there are several orbits that lie close to the 1:1 line indicating the procedure has not converged to a greater accuracy than the initial accuracy produced by the Stackel fudge. These orbits are either near-resonant so re- St\u00e4ckel, \u2206JR = 13.34, \u2206Jz = 12.56 Converged, \u2206JR = 0.06, \u2206Jz = 0.07 quire more careful action assignment (Kaasalainen 1995), or have one action very much greater than the other (near radial or shell orbits) so require more accurate torus construction than our automated procedure has allowed. It is clear from the plot that none of the iterative procedures have diverged significantly as all points lie near or well below the 1:1 line. The results for lower-action disc-type orbits are superior to those presented here. However, by and large the Stackel fudge action estimates for these orbits are sufficiently accurate for much scientific work (Piffl et al. 2014) so the iterative procedure is probably not required. One realistic application of the presented method is the modelling of tidal streams: Sanders (2014) used the expected angle and frequency structure of a stream in the correct potential to constrain the potential from a stream simulation. The frequencies of the collection of orbits explored here range from 10 to 120 kpc -1 kms -1 . For the majority of orbits the error in the frequencies recovered from the fudge are glyph[lessorsimilar] 10 per cent with the majority having errors of a few per cent, whilst the iterative torus approach reduces the errors to approximately 0 . 01 per cent for all orbits apart from those with large action errors discussed previously. The 10 4 M glyph[circledot] stream used in Sanders (2014) had a frequency width to absolute frequency ratio of \u223c 0 . 1 per cent so the iterative torus approach seems well suited to modelling of streams. In con-c ion, we have demonstrated that the presented algorithm has the capability to produce accurate actions for a wide range of orbits.", "pages": [15, 16]}, {"title": "7 CONCLUSIONS", "content": "We have presented a method for estimating the actions in a general triaxial potential using a Stackel approximation. The method is an extension of the Stackel fudge introduced by Binney (2012a) for the axisymmetric case. We have investigated the accuracy of the method for a range of orbits in an astrophysically-relevant triaxial potential. We have seen that the recovery of the actions is poorest for the box orbits, which probe a large radial range of the potential, and much better for the loop orbits, which are confined to a more limited radial range. The only parameters in the method are the choice of the focal positions \u2206 i , which are selected for each input phase-space point. We have detailed a procedure for selecting these based on the energy of the input phase-space point. This choice is not optimal but, by adjusting \u2206 i , we can, at best, increase the accuracy of the actions of a factor of two for the triaxial NFW potential considered. However, to achieve this accuracy requires additional computation for each input phase-space point (e.g. orbit integration). For general potentials the best action estimates will be achieved when locally (over the region a given orbit probes) the potential is well approximated by some Stackel potential. Many potentials of interest are not well fitted globally by Stackel potentials so the accuracy of the action estimates will deteriorate for orbits with large radial actions. The advantage of this method over other methods for estimating the actions in a triaxial potential is speed. Unlike the convergent method introduced by Sanders & Binney (2014), we obtain the actions without integrating an orbit - we only use the initial phasespace point. We have only to evaluate several algebraic expressions, find the limits of the orbits in the \u03c4 coordinate and perform Gaussian quadrature. These are all fast calculations. However, this speed comes at the expense of sometimes disappointing accuracy. If accurate results are required, the Stackel fudge can be combined with torus mapping to form a rapidly convergent scheme for the determination of J ( x , v ) . We demonstrated how such a scheme performed in the axisymmetric case and found a single torus construction provided a high level of accuracy that is not significantly improved by further torus constructions. We went on to construct, for the first time, triaxial stellar systems from a specified DFs f ( J ) in Section 5. We demonstrated the mass of these models is well recovered using the Stackel fudge, and we showed how the error in the density of these models varies as a function of the action error. Notwithstanding the errors in individual actions, both a radially-biased model and a tangentially-biased model satisfy the Jeans equations to good accuracy. This is because individual errors largely cancel during integration over velocities when computing moments such as the density \u03c1 ( x ) and the pressure tensor \u03c1\u03c3 2 ij ( x ) . The results presented in this paper have focussed on a limited range of astrophysically-relevant models: we have used a single specific NFW potential and two simple distribution functions that depend on a linear sum of the actions. However, we anticipate that our results will extend to more general distribution functions. We have investigated analytically how the normalization of the distribution function varies with the error in the action estimates and shown that the normalization is well recovered provided the error in the actions is smaller than the action scale over which the distribution function varies significantly i.e. \u2206 J glyph[lessmuch] f ( \u2202 J f ) -1 . Therefore, the recovery of the moments is expected to be most accurate for distribution functions with shallow radial density profiles and to deteriorate with the steepness of the required profile. Whilst the scheme presented here does not give accurate enough actions for working with streams (Sanders 2014) we have shown that it is an appropriate and powerful tool for constructing models from specified DFs f ( J ) . A key property of DFs of the form f ( J ) is that they can be trivially added to build up a multicomponent system. Hence the ability to extract observables from DFs of the form f ( J ) is likely to prove extremely useful for interpreting data on both external galaxies (Cappellari et al. 2011) and our Galaxy, in which components such as the stellar and dark haloes may be triaxial, and the bulge certainly is.", "pages": [16, 17]}, {"title": "ACKNOWLEDGMENTS", "content": "We thank Paul McMillan for provision of the torus construction machinery used in Section 6 and the Oxford Galactic Dynamics group for helpful comments. JLS acknowledges the support of STFC. JB was supported by STFC by grants R22138/GA001 and ST/K00106X/1. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no. 321067.", "pages": [17]}, {"title": "REFERENCES", "content": "Aarseth S. J., Binney J., 1978, MNRAS , 185, 227 Allgood B., Flores R. A., Primack J. R., Kravtsov A. V., Wechsler R. H., Faltenbacher A., Bullock J. S., 2006, MNRAS , 367, 1781 Bailin J. et al., 2005, ApJL, 627, L17 Binney J., 2010, MNRAS , 401, 2318 Binney J., 2012a, MNRAS , 426, 1324 Binney J., 2012b, MNRAS , 426, 1328 Binney J., 2014, MNRAS , 440, 787 Binney J., Spergel D., 1984, MNRAS , 206, 159 Binney J., Tremaine S., 2008, Galactic Dynamics: Second Edition. Princeton University Press Bissantz N., Gerhard O., 2002, MNRAS , 330, 591 Cappellari M. et al., 2011, MNRAS , 413, 813 de Zeeuw T., 1985, MNRAS , 216, 273 Eyre A., Binney J., 2011, MNRAS , 413, 1852 Hahn T., 2005, Computer Physics Communications, 168, 78 Helmi A., White S. D. M., 1999, MNRAS , 307, 495 Jing Y. P., Suto Y., 2002, ApJ , 574, 538 Kaasalainen M., 1995, Physical Review E., 52, 1193 Kaasalainen M., Binney J., 1994, MNRAS , 268, 1033 Kazantzidis S., Kravtsov A. V., Zentner A. R., Allgood B., Nagai D., Moore B., 2004, ApJL, 611, L73 Law D. R., Majewski S. R., 2010, ApJ , 714, 229 McMillan P. J., 2011, MNRAS , 414, 2446 McMillan P. J., Binney J. J., 2008, MNRAS , 390, 429 Merritt D., Valluri M., 1999, AJ , 118, 1177 Navarro J. F., Frenk C. S., White S. D. M., 1997, ApJ , 490, 493 Ollongren A., 1962, Bull. Astr. Inst. Netherlands , 16, 241 Piffl T. et al., 2014, MNRAS , 445, 3133 Poincar'e H., 1892, Les methodes nouvelles de la mecanique celeste Posti L., Binney J., Nipoti C., Ciotti L., 2014, ArXiv e-prints Prendergast K. H., Tomer E., 1970, AJ , 75, 674 Rowley G., 1988, ApJ , 331, 124 Sanders J., 2012, MNRAS , 426, 128 Sanders J. L., 2014, MNRAS , 443, 423 Sanders J. L., 2015, in prep. Sanders J. L., Binney J., 2013, MNRAS , 433, 1826 Sanders J. L., Binney J., 2014, MNRAS , 441, 3284 Schonrich R., Binney J., 2009, MNRAS , 396, 203 Schwarzschild M., 1979, ApJ , 232, 236 Tremaine S., 1999, MNRAS , 307, 877 Valluri M., Debattista V. P., Quinn T., Moore B., 2010, MNRAS , 403, 525 Vera-Ciro C., Helmi A., 2013, ApJL, 773, L4 Vera-Ciro C. A., Sales L. V., Helmi A., Frenk C. S., Navarro J. F., Springel V., Vogelsberger M., White S. D. M., 2011, MNRAS , 416, 1377 Wilson C. P., 1975, AJ , 80, 175", "pages": [17]}, {"title": "APPENDIX A: ANGLES AND FREQUENCIES", "content": "With the framework presented in Section 3 we are also in a position to find the angles, \u03b8 , and frequencies, \u2126 . Following de Zeeuw (1985) we write Inversion of these equations gives, for instance, and \u2126 \u00b5 and \u2126 \u03bd are given by cyclic permutation of { \u03bb, \u00b5, \u03bd } . To find the derivatives of J \u03c4 with respect to the integrals we differentiate equation (13) under the integral sign at constant \u03c4 . From equation (17) we know p \u03c4 ( \u03c4, E, A \u03c4 , B \u03c4 ) . We note that as A \u03c4 = a -C \u03c4 and B \u03c4 = b + D \u03c4 where C \u03c4 and D \u03c4 are independent of \u03c4 . The required derivatives are Note that p \u03c4 can vanish at the limits of integration. The change of variables causes the integrand to go smoothly to zero at the limits. To find the angles, we use the generating function, W ( \u03bb, \u00b5, \u03bd, J \u03bb , J \u00b5 , J \u03bd ) , given by F \u03c4 are factors included to remove the degeneracy in the \u03c4 coordinates such that \u03b8 \u03c4 covers the full range 0 to 2 \u03c0 over one oscillation in the Cartesian coordinates. These factors can be written in the form where \u0398 is the Heaviside step function and \u03a0 is one when its argument is zero and zero otherwise. The \u03a0 -function in F \u03bb takes care of the cases when the orbit is a box or inner long-axis loop. The \u03a0 -functions in F \u00b5 take care of the cases when the orbit is a short-axis loop or a box, an outer long-axis loop, and an inner long-axis loop respectively. The angles are given by The first term on the right is, up to factors, the indefinite integral of the derivatives of J \u03c4 with respect to the integrals found previously, whilst the second term is found from inverting these derivatives. We have chosen the zero-point of \u03b8 \u03c4 to correspond to \u03c4 = \u03c4 -, p \u03c4 > 0 and \u02d9 x i glyph[greaterorequalslant] 0 for all i , except for the outer long-axis loop orbits which have \u03b8 \u00b5 = 0 at \u00b5 = -\u03b1 , p \u03c4 > 0 and \u02d9 x i glyph[greaterorequalslant] 0 . Note that the angles are the 2 \u03c0 modulus of the \u03b8 \u03c4 found from the above scheme. In Fig. A1 we show the angles calculated from the Stackel fudge for the three orbits investigated in Section 4.2. We use the automatic choice of \u2206 i for the box and short-axis loop orbit, and the choice that minimises the spread in actions for the long-axis loop orbit. The short-axis loop orbit shows the expected straightline structure in the angle coordinates, whilst for the long-axis loop and box orbits there is clear deviation from this expected straight line. We also show the angles calculated using the initial angle estimate and the average of the frequency estimates along the orbit. We see that they are well recovered but after approximately one period the error in the frequencies is sufficient for these angles to deviate from the angle estimates. The standard deviations in the frequencies are reasonably large. For the box orbit, the mean frequencies are given by \u2126 = (18 . 1 , 20 . 3 , 24 . 3) kpc -1 kms -1 with errors \u2206 \u2126 = (0 . 2 , 0 . 8 , 1 . 3) kpc -1 kms -1 . For the short-axis loop the mean frequencies are given by \u2126 = (34 . 9 , 21 . 9 , 25 . 0) kpc -1 kms -1 with errors \u2206 \u2126 = (0 . 6 , 0 . 1 , 0 . 2) kpc -1 kms -1 . For the long-axis loop the mean frequencies are given by \u2126 = (36 . 9 , 22 . 4 , 24 . 0) kpc -1 kms -1 with errors \u2206 \u2126 = (1 . 6 , 1 . 5 , 0 . 8) kpc -1 kms -1 . We note that the frequency errors are largest at the turning points of the orbits for the loop orbits or near the centre of the potential for the box orbit. This paper has been typeset from a T E X/ L A T E X file prepared by the author.", "pages": [17, 18]}] |
2024arXiv241207652B | https://arxiv.org/pdf/2412.07652.pdf | <document>
<section_header_level_1><location><page_1><loc_10><loc_82><loc_90><loc_87></location>Mapping the spatial extent of H /i.pc -rich absorbers using Mg /i.pc/i.pc absorption along gravitational arcs</section_header_level_1>
<text><location><page_1><loc_6><loc_79><loc_94><loc_81></location>Trystyn A. M. Berg 1 , 2 , 3 , /star , Andrea Afruni 2 , 4 , 5 ,/star , Cédric Ledoux 3 , Sebastian Lopez 2 , Pasquier Noterdaeme 2 , 6 , 7 , Nicolas Tejos 8 , Joaquin Hernandez 9 , Felipe Barrientos 9 , and Evelyn J. Johnston 10</text>
<unordered_list>
<list_item><location><page_1><loc_11><loc_75><loc_86><loc_76></location>1 NRC Herzberg Astronomy and Astrophysics Research Centre, 5071 West Saanich Road, Victoria, B.C., Canada, V9E 2E7</list_item>
<list_item><location><page_1><loc_11><loc_74><loc_62><loc_75></location>2 Departamento de Astronomía, Universidad de Chile, Casilla 36-D, Santiago, Chile.</list_item>
<list_item><location><page_1><loc_11><loc_73><loc_67><loc_74></location>3 European Southern Observatory, Alonso de Cordova 3107, Casilla 19001, Santiago, Chile.</list_item>
<list_item><location><page_1><loc_11><loc_72><loc_79><loc_73></location>4 Kapteyn Astronomical Institute, University of Groningen, Landleven 12, 9747 AD Groningen, The Netherlands.</list_item>
<list_item><location><page_1><loc_11><loc_71><loc_82><loc_71></location>5 Dipartimento di Fisica e Astronomia, Università di Firenze, Via G. Sansone 1, 50019 Sesto Fiorentino, Firenze, Italy</list_item>
<list_item><location><page_1><loc_11><loc_69><loc_49><loc_70></location>6 French-Chilean Laboratory for Astronomy, IRL 3386, CNRS</list_item>
<list_item><location><page_1><loc_11><loc_68><loc_69><loc_69></location>7 Institut d'Astrophysique de Paris, CNRS-SU, UMR,7095, 98bis bd Arago, 75014 Paris, France</list_item>
<list_item><location><page_1><loc_11><loc_67><loc_70><loc_68></location>8 Instituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla 4059, Valparaíso, Chile</list_item>
<list_item><location><page_1><loc_11><loc_66><loc_86><loc_67></location>9 Instituto de Física, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, 7820436 Macul, Santiago, Chile</list_item>
<list_item><location><page_1><loc_10><loc_63><loc_90><loc_65></location>10 Instituto de Estudios Astrofísicos, Facultad de Ingeniería y Ciencias, Universidad Diego Portales, Av. Ejército Libertador 441, Santiago, Chile</list_item>
</unordered_list>
<text><location><page_1><loc_10><loc_61><loc_22><loc_62></location>December 11, 2024</text>
<section_header_level_1><location><page_1><loc_46><loc_58><loc_54><loc_59></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_10><loc_39><loc_90><loc_56></location>H /i.pc-rich absorbers seen within quasar spectra contain the bulk of neutral gas in the Universe. However, the spatial extent of these reservoirs are not extensively studied due to the pencil beam nature of quasar sightlines. Using two giant gravitational arc fields (at redshifts 1.17 and 2.06) as 2D background sources with known strong Mg /i.pc/i.pc absorption observed with the Multi Unit Spectroscopic Explorer integral field spectrograph (IFS), we investigated whether spatially mapped Mg /i.pc/i.pc absorption can predict the presence of strong H /i.pc systems, and determine both the physical extent and H /i.pc mass of the two absorbing systems. We created a simple model of an ensemble of gas clouds in order to simultaneously predict the H /i.pc column density and gas covering fraction of H /i.pc-rich absorbers based on observations of the Mg /i.pc/i.pc rest-frame equivalent width in IFS spaxels. We first test the model on the lensing field with H /i.pc observations already available from the literature, finding that we can recover H /i.pc column densities consistent with the previous estimates (although with large uncertainties). We then use our framework to simultaneously predict the gas covering fraction, H /i.pc column density and total H /i.pc gas mass ( M HI ) for both fields. We find that both of the observed strong systems have a covering fraction of ≈ 70 % and are likely damped Lyman α systems (DLAs) with M HI > 10 9 M /circledot . Our model shows that the typical Mg /i.pc/i.pc metrics used in the literature to identify the presence of DLAs are sensitive to the gas covering fraction. However, these Mg /i.pc/i.pc metrics are still sensitive to strong H /i.pc, and can be still applied to absorbers towards gravitational arcs or other spatially extended background sources. Based on our results, we speculate that the two strong absorbers are likely representative of a neutral inner circumgalactic medium and are a significant reservoir of fuel for star formation within the host galaxies.</text>
<text><location><page_1><loc_10><loc_37><loc_58><loc_38></location>Key words. galaxies: high redshift - galaxies: ISM - quasars: absorption lines</text>
<section_header_level_1><location><page_1><loc_6><loc_33><loc_19><loc_34></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_6><loc_19><loc_49><loc_32></location>The gas reservoirs of galaxies are critical for regulating galaxy formation and evolution; from the interstellar medium (ISM) where atomic H /i.pc cools into molecular gas to form stars (e.g., Saintonge et al. 2013), out into the galactic halos where the circumgalactic medium (CGM) hosts both accreted intergalactic gas and ejected material from galactic feedback processes which maycontinuetofeed the ISM (see Tumlinson et al. 2017; FaucherGiguère & Oh 2023, and references therein). Thus quantifying the properties of these galactic gas reservoirs across cosmic time is a necessity to constrain our understanding of galaxy evolution.</text>
<text><location><page_1><loc_6><loc_15><loc_49><loc_18></location>Quasar sightlines have proved to be a powerful probe of the gas reservoirs of galaxies, enabling a census of the amount of fuel for future star formation and to quantify the metallicity evolution</text>
<text><location><page_1><loc_51><loc_27><loc_94><loc_34></location>of these reservoirs across cosmic time. Whilst rare in the Universe, the strongest H /i.pc absorbers known as damped Ly α systems (DLAs; H /i.pc column densities of N(H /i.pc) ≥ 2 × 10 20 cm -2 ; Wolfe et al. 2005) contain the ≈ 80 % of neutral gas in the Universe (Lanzetta et al. 1995; Prochaska et al. 2005; Noterdaeme et al. 2009; Zafar et al. 2013; Sánchez-Ramírez et al. 2016).</text>
<text><location><page_1><loc_51><loc_14><loc_94><loc_26></location>Due to the UV cut-off from Earth's atmosphere, tracing Ly α absorption from galaxies below z ≲ 1 . 6 requires expensive spacebased observations. To get access to DLAs in this epoch of the Universe, low-ionization metal species have been used as DLA proxies. The most widely used metal tracer for H /i.pc-rich systems is Mg/i.pc/i.pc as it is accessible down to z ≈ 0 . 1 with ground-based observations (Rao & Turnshek 2000; Turnshek et al. 2015). There are various metrics in the literature for pre-selecting Mg /i.pc/i.pc absorbers at z ≈ 1 as potential DLAs based on the rest-frame equivalent widths (EWs) along quasar sightlines. These metrics include:</text>
<text><location><page_2><loc_9><loc_87><loc_49><loc_93></location>low-redshift DLAs tend to have EW 2796 ≥ 0 . 3 -0 . 6Å (Rao & Turnshek 2000; Rao et al. 2006; Turnshek et al. 2015). In this work, we focus on using the threshold of ≥ 0 . 3Å which tends to select all DLAs at a variety of redshifts (Matejek et al. 2013; Berg et al. 2017; Rao et al. 2017).</text>
<text><location><page_2><loc_7><loc_73><loc_49><loc_86></location>- the EWratio of Fe /i.pc/i.pc λ 2600/Mg/i.pc/i.pc λ 2796( EW 2600 EW 2796 ) /one.sup . For DLAs, EW 2600 EW 2796 is found to be between 0.5 and 1.0 (Rao et al. 2006). This metric is motivated by the self-shielding nature of DLAs, where the bulk of Fe and Mg will be in the singly-ionized phase. In lower logN(H /i.pc) column density absorbers where ionization effects have an effect, the ionizing field will preferentially ionize Fe outside the Fe /i.pc/i.pc state, leading to a smaller EW 2600 EW 2796 ratio. However, this ratio is also sensitive to the effects of nucleosynthesis (as it traces [ α /Fe]) and differential dust depletion (e.g., Dey et al. 2015; Lopez et al. 2020).</text>
<text><location><page_2><loc_7><loc_66><loc_49><loc_72></location>- the D -index (Ellison 2006). The D -index is computed as EW 2796 normalized by the width of integration bounds for the EW calculation. The minimum D -index for which a system is flagged as a potential DLA is set by the spectral resolution used.</text>
<text><location><page_2><loc_6><loc_28><loc_49><loc_64></location>While these three metrics are useful in identifying a potential DLA absorber along quasar sightlines, meeting the various cuts on these metrics is no guarantee that the absorption is indeed from a DLA. In Rao et al. (2017) and previous works, the EW 2796 and EW 2600 EW 2796 are typically very successful at finding DLAs at low redshifts. However, these metrics also identify other strong H /i.pc absorbers, such as subDLAs ( 19 ≤ logN(H /i.pc) < 20 . 3 cm -2 ) at high rates, leading to DLA samples with low purity (Rao et al. 2006; Bouché 2008). Furthermore, higher redshift DLAs that are metal-poor or have narrow velocity profiles sometimes do not pass these metrics (Ellison 2006; Berg et al. 2017; Rao et al. 2017). While the D -index metric is designed to take the velocity profile shape into account, the performance of the D -index is heavily dependent on the spectral resolution of the instrument, as the success rate of distinguishing a DLA from a sub-DLA decreases with decreasing spectral resolving power (Ellison 2006). For example, with the spectral resolving of a high resolution spectrograph (with a line spread function full-width half maximum [FWHM]of ≥ 1 . 4 Å), the D -index is very efficient at identifying DLAs with ≈ 90 % purity and a 5% false positive rate (Table 2 in Ellison 2006). However, for lower spectral resolutions, such as what is offered with current state-of-the-art integral field spectrograph facilities such as the Multi Unit Spectroscopic Explorer (MUSE Bacon et al. 2010, with an average spectral FWHM of 2.7Å), the D -index is expected to identify DLAs with a ≈ 84 % success rate and a false-positive rate of ≈ 16 %. Therefore, none of these three metrics on their own can guarantee that a given Mg /i.pc/i.pc absorber is a DLA.</text>
<text><location><page_2><loc_6><loc_16><loc_49><loc_27></location>Other than in a few specific cases (Fumagalli et al. 2015; Neeleman et al. 2016), very little is known about the morphology and the distribution of gas in DLAs, as quasar sightlines only provide 1D information of the gas. Whilst rare multiple close project pairs of (lensed) quasars (Lopez et al. 1999, 2005; Ellison et al. 2007; Tytler et al. 2009; Chen et al. 2014; Krogager et al. 2018; Rubin et al. 2018b) or faint extended galaxies as background sources (Bordoloi et al. 2014; Rubin et al. 2018a) have been used to assess the physical extent of DLAs or the CGM,</text>
<text><location><page_2><loc_51><loc_69><loc_94><loc_93></location>these techniques are still limited to 1D skewers of up to a handful of pointings through the same halo. After the advent of sensitive integral field spectrograph like MUSE and KCWI on 8-10m class telescopes, it is now possible to observe faint gravitational arcs as extended background sources to probe the spatial extent of gaseous reservoirs. This so-called gravitational arc tomography has demonstrated the ability to assess how the CGM material is distributed (Lopez et al. 2018, 2020; Mortensen et al. 2021; Tejos et al. 2021; Fernandez-Figueroa et al. 2022; Afruni et al. 2023) and in one case it has been used to put constraints on the mass of H /i.pc gas in a DLA (Bordoloi et al. 2022, further abbreviated as B22). However, based on the modest wavelength range of current telescope facilities combined with the typical redshifts of gravitational arcs, it is more difficult to study H /i.pc directly with gravitational arc tomography compared to neutral metal tracers such as Mg /i.pc/i.pc. As for the pencil-beam quasar sightlines, one possibility is to identify possible DLAs using the Mg /i.pc/i.pc metrics described above, but their accuracy has to date never been tested on extended sources.</text>
<text><location><page_2><loc_51><loc_38><loc_94><loc_68></location>In this paper, we investigate the use of the three commonlyused Mg /i.pc/i.pc metrics to identify DLAs along extended background sources, aiming to quantify the spatial extent, covering fractions, and H /i.pc gas mass of H /i.pc-rich absorption. We focus on MUSE observations of two gravitational arcs (one of them being the one analyzed in B22) with strong Mg /i.pc/i.pc absorption. We first verify that these commonly-used Mg /i.pc/i.pc metrics can be used for extended background sources. Next, based on the measured Mg /i.pc/i.pc metrics and using a novel model framework to reproduce the B22 results, we investigate the success of using Mg /i.pc/i.pc metrics to identify DLA absorbers along extended background sources. Based on this modeling, we estimate the covering fraction and the H /i.pc gas mass of the two absorbers studied. In Sect. 2, we report the data reduction and processing used to measure equivalent widths; in Sect. 3, we apply the Mg /i.pc/i.pc metrics to assess the extent of potential DLAs in the two fields; in Sect. 4, we describe our model, how we compared the model outputs with the observations and the main results obtained from this comparison; finally, in Sect. 5 we outline the limitations and assumptions of our modeling and we discuss the validity of the usual Mg /i.pc/i.pc metrics, while in Sect. 6 we summarize this work and we report our main conclusions. Throughout this work, we assume a flat Λ CDMcosmology with H 0 = 70 km s -1 Mpc -1 , Ω Λ = 0 . 7 and Ω M = 0 . 3 .</text>
<section_header_level_1><location><page_2><loc_51><loc_35><loc_81><loc_36></location>2. Observations and data processing</section_header_level_1>
<section_header_level_1><location><page_2><loc_51><loc_33><loc_73><loc_34></location>2.1. Targets and data reduction</section_header_level_1>
<text><location><page_2><loc_51><loc_10><loc_94><loc_32></location>We observed SGAS J0033+0242 and SGAS J1527+0652 (further referred to as J0033 and J1527, respectively) with the Multi Unit Spectroscopic Explorer (MUSE; Bacon et al. 2010)) at the Very Large Telescope, as part of programs 098.A-0459 and 0103.A-048, respectively (PI Lopez). These fields were originally selected for having indications of intervening Mg /i.pc/i.pc absorption at z ∼ 1 on top of the brightest arc knots (Rigby et al. 2018). The MUSE data revealed strong ( W 0 ≳ 3 Å) Mg /i.pc/i.pc and Fe /i.pc/i.pc systems at redshifts 1.16729 (J0033) and 2.05601 (J1527), making them good DLA candidates (e.g., Rao et al. 2017). In both cases the MUSE wide field mode was used, providing a field of view of 1 ' × 1 ' and a spatial sampling of 0.2 '' pix -1 . J0033 was observed without adaptive optics mode and the nominal wavelength range ( ≈ 4 700 -9 300 Å), whereas J1527 was observed with the adaptive optics mode and extended wavelength range mode ( 4 600 -9 300 Å). The on-target exposure times were 2 . 14 and 4 . 17 hours, respectively. Figure 1 shows a zoom-in of the MUSE white-light</text>
<figure>
<location><page_3><loc_11><loc_63><loc_91><loc_91></location>
<caption>Fig. 1. Lensing region towards the J0033 (left panel) and J1527 (right panel) fields. The whitelight image is shown in greyscale. Only a portion of the full MUSE cubes are shown for clarity. The colored magenta rectangles denote regions of spaxels (Table 1) that contain background sources which have detected Mg /i.pc/i.pc absorption. In both panels, the coordinates are relative to the center of the arc region in the respective field (Table 1). For J1527, the smaller cyan boxes denote the six apertures (A-F) from B22 whilst the red rectangle denotes the main region containing a nearby galaxy which contaminates a portion of the gravitational arc. The plate scale of the both images is 0.2 '' per pixel.</caption>
</figure>
<table>
<location><page_3><loc_14><loc_33><loc_86><loc_52></location>
<caption>Table 1. Rectangular regions of interest.</caption>
</table>
<text><location><page_3><loc_6><loc_26><loc_49><loc_29></location>image of the two fields. Both fields show strong gravitational arcs, whilst the J0033 field contains two lensing counterimages (labelled as CI1 and CI2) as well as a background quasar (QSO).</text>
<text><location><page_3><loc_6><loc_10><loc_49><loc_25></location>We reduced the MUSE data using the ESO MUSE pipeline (v2.6, Weilbacher et al. 2020) in the ESO Recipe Execution Tool (EsoRex) environment (ESO CPL Development Team 2015). Besides pre-processing (bias, flat-field, and vacuum-wavelength calibrations), the flux in each exposure was calibrated using standard star observations from the same nights as the science data, and the sky continuum was measured directly from the science exposures and subtracted off. Residual sky contamination was removed from the stacked cube using the Zurich Atmosphere Purge code (Soto et al. 2016). The final cubes were matched to the WCS of the Hubble Space Telescope (HST) images of J0033 (Fischer et al. 2019) and J1527 (Sharon et al. 2020). These cubes have</text>
<text><location><page_3><loc_51><loc_26><loc_94><loc_30></location>a point-spread-function of 0.84 '' and 0.78 '' , respectively and a spectral resolving power ranging from R /similarequal 1 770 at 4 800 Å to R /similarequal 3 590 at 9 300 Å. For further details see Lopez et al. (2024).</text>
<text><location><page_3><loc_51><loc_10><loc_94><loc_25></location>Table 1 contains the coordinates of the relevant regions we select in this work. These regions represent rectangles, centered at the Right Ascension ( α ) and Declination ( δ ), with a halfwidth of r RA and r Dec in the respective directions, and rotated by the position angle (measured north through east). Apertures A-F for regions in the J1527 field are the apertures used by B22 (R. Bordoloi, private communication). The outlines of the spaxels contained by all of these regions are shown as colored rectangles in Fig. 1. In addition, the last column of Table 1 provides the delensed projected distance ( ρ ⊥ ) from the center α and δ of each region relative to the center α and δ of the 'Arc' region of the respective field. The delensed projected distances are calculated</text>
<text><location><page_4><loc_6><loc_86><loc_49><loc_93></location>by delensing the image plane to the so called absorber plane (the plane of the absorbing gas), through parametric lens models built for each of the two fields with the software /l.pc/e.pc/n.pc/s.pc/t.pc/o.pc/o.pc/l.pc (Jullo et al. 2007) and the available HST imaging (we refer for more details to Sharon et al. 2020 for J1527 and to Fischer et al. 2019 for J0033).</text>
<section_header_level_1><location><page_4><loc_6><loc_82><loc_44><loc_83></location>2.2. Creating combined spectra from individual spaxels</section_header_level_1>
<text><location><page_4><loc_6><loc_49><loc_49><loc_81></location>For each region of interest within our data (Table 1), we generate flux-weighted combined spectra using all the native spaxels within the region. Spectra for spaxels within the region whose signal-to-noise ratio (S/N, next to spectral region with Mg /i.pc/i.pc absorption) was greater than 1.0 were used to form a combined spectrum. In order to match typical 1D analysis of quasars where the spectral trace is summed together, we first fit and normalize the continuum of the spaxel spectra (see Appendix A) and then we weight each spaxel's spectrum by the summed flux of the spectrum at all wavelengths (white-light image /two.sup ). This weighting is analogous to how pixels are combined along the spatial axis of the slit in 1D quasar sightline observations. We note that each region is larger than the point-spread-function of the observations, and thus the spatial binning from this combination process will minimize the overlapping information from combining the native spaxels. In order to obtain an estimate of the error in the continuum fitting and weighting procedure, we use a Monte Carlo approach by repeating this process 1000 times, where in each iteration the flux of each element of the MUSE cube was sampled from a Gaussian distribution centered on the flux with a standard deviation equal to the 1 σ error spectrum flux. Each of these 1000 spectra are then continuum normalized individually following the method outlined in Appendix A. We then produce a median spectrum by taking the median flux of each pixel, with the 25 th and 75 th percentiles to estimate the error.</text>
<text><location><page_4><loc_6><loc_19><loc_49><loc_48></location>It is clear from Fig. 1 that the regions (particularly the arc of J0033) contain additional background light in and around the selected region after sky subtraction (possibly from extended emission of nearby sources or the nearby bright star, or intra-cluster light). Because of the additional signal, pure-background spaxels can have a S/N ≥ 1 and thus are selected and included when creating the combined spectra as described above. By including this background, there is an additional flux within absorption lines that leads to lower EWs and potentially false signatures of emission or partial coverage. In order to remove this additional background prior to making the combined spectra, we first isolate the spaxels (including those with S/N < 1 ) which only contain this additional background flux. This is done by sigma-clipping out all spaxels from the entire region (Table 1) whose white-light image flux is above +1 σ the mean flux. The spectra of these background-only spaxels are then median combined into a background spectrum that is subtracted from all the spaxels of that region in order to correct for the additional signal. We note that the background signal we observe appears uniformly distributed spatially within the respective regions, and the median combined background spectrum is consistent with these values. Based on this information, we note that the non-uniform light from the galaxy nearby the J1527 arc likely adds additional flux in the</text>
<table>
<location><page_4><loc_53><loc_74><loc_92><loc_91></location>
<caption>Table 2. EWvelocity integration limits</caption>
</table>
<text><location><page_4><loc_51><loc_61><loc_94><loc_70></location>spectrum and is not fully subtracted from this method. While we exclude spaxels that contain the galaxy's spectrum from the analysis (i.e., those within the Galaxy region for field J1527 in Table 1), there may still be scattered light issues that add additional flux to the spectrum, particularly within the absorption trough of the key Ly series (KCWI data) and metal lines (MUSE data).</text>
<text><location><page_4><loc_51><loc_55><loc_94><loc_60></location>Figs B.1 - B.6 in Appendix B contain the median metal line profiles from the Monte Carlo analysis for apertures AF (respectively) in the field J1527 whilst Figs. C.1 - C.4 in Appendix C show the median velocity profiles for the regions within the J0033 field. Both appendices are available online.</text>
<section_header_level_1><location><page_4><loc_51><loc_51><loc_82><loc_52></location>2.3. Measuring rest-frame equivalent widths</section_header_level_1>
<text><location><page_4><loc_51><loc_34><loc_94><loc_50></location>EW measurements are made by integrating the continuumnormalized spectra (see Appendix A) between the velocity limits of the absorption feature (Table 2; where the zero-velocity is defined at the redshift of the strongest Mg /i.pc/i.pc absorption, z abs ). The velocity limits are determined by visually inspecting the absorption of all spaxels and selecting the limits where the strongest line reaches the continuum level. The same velocity limits are adopted for all metal absorption lines of the same system unless there is significant blending in a weak portion absorption profile (e.g., Ca /i.pc/i.pc λ 3934 in Fig. C.1 in Appendix C). We adopt the EW derived from the median of the 1000 Monte Carlo spectra, with errors on the EW coming from the the 25 th and 75 th percentile spectra.</text>
<text><location><page_4><loc_51><loc_22><loc_94><loc_33></location>Werequire that the rest-frame equivalent widths be measured at > 2 σ (based on the S/N of the median spectrum) to be considered as detected. Otherwise, 2 σ upper limits are adopted. Lines that are strongly blended use the measured equivalent width, but are flagged as upper limits. The median equivalent width measurements for the B22 apertures and full arc in the J1527 field as well as the J0033 regions are provided in Tables 3 and 4, respectively. The same equivalent width information is also provided online in Appendix B (J1527) and Appendix C (J0033).</text>
<section_header_level_1><location><page_4><loc_51><loc_18><loc_85><loc_19></location>3. DLA identification using Mg /i.pc/i.pc metrics</section_header_level_1>
<text><location><page_4><loc_51><loc_15><loc_93><loc_17></location>3.1. Verifying Mg /i.pc/i.pc DLA metrics on extended sources within the J1527 field</text>
<text><location><page_4><loc_51><loc_10><loc_94><loc_14></location>The Mg /i.pc/i.pc DLA identification metrics (i.e., EW 2796 , EW 2600 EW 2796 , and D -index; defined in the introduction) have been defined on quasar sightlines. It remains untested whether these three metrics can</text>
<table>
<location><page_5><loc_14><loc_81><loc_85><loc_91></location>
<caption>Table 3. Metal line rest-frame equivalent widths for regions of J1527 fieldTable 4. Metal line rest-frame equivalent widths for regions in J0033 field</caption>
</table>
<table>
<location><page_5><loc_6><loc_68><loc_97><loc_75></location>
<caption>Table 5. DLA metrics for B22 apertures of J1527 field</caption>
</table>
<table>
<location><page_5><loc_6><loc_48><loc_50><loc_61></location>
</table>
<text><location><page_5><loc_6><loc_46><loc_30><loc_47></location>Notes: a logN(H /i.pc) taken from B22.</text>
<table>
<location><page_5><loc_11><loc_32><loc_44><loc_41></location>
<caption>Table 6. DLA metrics for regions in J0033 field</caption>
</table>
<text><location><page_5><loc_6><loc_14><loc_49><loc_28></location>also be used for identifying DLAs in multiplexed data of extended sources such as gravitational arcs. We take advantage of testing these metrics towards the J1527 field, where existing KCWI observations have demonstrated that all six apertures (apertures A-F; Table 1) host a DLA whilst our MUSE data complement the KCWI data by providing coverage of several key metal lines, including the Mg /i.pc/i.pc doublet. Using these six aperture regions, we created combined spectra following the method outlined in Sect. 2.2. The EWs of the key metal lines are tabulated in Table 3 while the resulting DLA metrics are provided in Table 5 along with the H /i.pc column densities from B22.</text>
<text><location><page_5><loc_6><loc_9><loc_49><loc_13></location>For all six B22 apertures (which all host a DLA based on the logNH /i.pc measured in B22) and the full arc region, the EW 2796 and EW 2600 EW 2796 metrics tabulated in Table 5 are consistent with the</text>
<text><location><page_5><loc_51><loc_40><loc_94><loc_64></location>apertures containing a DLA using the standard quasar metric thresholds, suggesting that these two metrics from Rao & Turnshek (2000) can be used to identify DLAs. Based on the MUSE instrumental FWHM (2.54 Å) at the wavelength of Mg /i.pc/i.pc 2796Å, the expected D -index threshold for selecting a DLA is ≈ 3 . 6 (interpolating Table 2 of Ellison 2006). This D -index threshold would only select apertures A-C as hosting DLAs. However, we caution that at this low spectral resolution, the success rate of identifying DLAs is expected to be ≤ 84 per cent (Ellison 2006); the observed success rate of the D -index for these six apertures is 33%. All entries in Table 5 that would be flagged as a DLA based on the respective Mg /i.pc/i.pc metric have been bolded. Combining all the spaxels together from full arc in place of the individual six B22 apertures, we obtain similar results - both the EW 2796 and EW 2600 EW 2796 metrics are consistent with hosting DLAs, whilst the D -index ( 3 . 2 ± 0 . 1 ) is just below the threshold for selecting a DLA. We point out that the value of all three metrics for the combined full arc spectrum are effectively equal to the average of the values derived for the six B22 apertures.</text>
<text><location><page_5><loc_51><loc_21><loc_94><loc_39></location>As mentioned in Sect. 2.2, there is potential contamination in all apertures from galaxy light close in projection to the arc. As we are using weighting from the white-light image, the spaxels that contain more galaxy light than arc light are favored, and artificially weaken the Mg /i.pc/i.pc absorption in the combined spectrum by adding flux within the absorption trough. We therefore caution that EW measurements in the apertures (Table 3; particularly aperture F) are likely lower limits to the true value arising from pure Mg /i.pc/i.pc absorption. This may partially explain why the D -index is just below the threshold for selecting a DLA and is not as successful at identifying the DLA absorption in this particular field. It is possible that this contamination issue also affects the determination of the H /i.pc column density and the presence of partial coverage reported by B22.</text>
<section_header_level_1><location><page_5><loc_51><loc_17><loc_83><loc_18></location>3.2. Applying Mg /i.pc/i.pc metrics to the J0033 field</section_header_level_1>
<text><location><page_5><loc_51><loc_10><loc_94><loc_16></location>Following the same methodology for the J1527 field, we created combined spaxel spectra for all four regions and measured the EWs for the same metal lines (Table 4) and measured the Mg /i.pc/i.pc metrics for DLA identification (Table 6). Bolded entries in Table 6 represent regions where the given Mg /i.pc/i.pc metric would be</text>
<figure>
<location><page_6><loc_8><loc_63><loc_93><loc_93></location>
<caption>Fig. 2. Color maps of the lensed region in the J0033 field, showing the four different regions (Arc, QSO, CI1, and CI2) tracing Mg /i.pc/i.pc absorption. The top left ( D -index), top right (Mg /i.pc/i.pc λ 2796 rest-frame equivalent width), and bottom left (Fe /i.pc/i.pc λ 2600/Mg /i.pc/i.pc λ 2796 equivalent width ratio) panels show three different metrics used in the literature to identify if the detected Mg /i.pc/i.pc absorption is a potential DLA. The color bars are designed such that spaxels that are green and yellow colors in these three panels would be considered as potential DLA absorption. The bottom right panel shows the S/N at the position of the Mg /i.pc/i.pc 2796 Å absorption at each spaxel. In all four panels, the white-light image of the cube is shown for reference. Only spaxels with S/N ≥ 2 . 0 at the position of Mg /i.pc/i.pc 2796 Å and EW 2796 detected at ≥ 2 σ significance are colored in all four panels.</caption>
</figure>
<text><location><page_6><loc_6><loc_30><loc_49><loc_52></location>flagged as a DLA. In the combined spectra from three of the four background sources of the lensing system towards J0033 (i.e., the Arc, CI 1, and CI 2 regions in Table 1), there is very strong Mg /i.pc/i.pc λ 2796 absorption (rest-frame equivalent widths > 2 Å ; Figs. C.1 - C.3 in Appendix C) that suggests the presence of an H /i.pc-rich absorber based on the results from J1527. Unfortunately the combined spectrum for the QSO region has poor S/N (Fig. C.4 in Appendix C), so we exclude this background source from our analysis. What makes this particular field interesting is that large physical separations are probed by the arc and two counter images (separated by ≈ 14 -50 kpc), compared to the ≈ 10 kpc scales measured across the J1527 arc (see Table 1). Despite not having access to the Lyman series to measure the H /i.pc column density directly, the J0033 field provides an opportunity to study the spatial extent of a strong Mg /i.pc/i.pc absorber that is likely a DLA (or sub-DLA) based on the very strong Mg /i.pc/i.pc absorption ( EW 2796 ≳ 2 Å).</text>
<text><location><page_6><loc_6><loc_10><loc_49><loc_27></location>In Figs. C.1 - C.3 in Appendix C, we note there is no obvious difference in shape or centroid velocity of the Mg /i.pc/i.pc λ 2796 absorption profile between the four regions. The kinematics of this particular absorber will be discussed in more detail in a future paper (Ledoux et al., in prep.). While the spectral resolution of MUSE is insufficient to determine if sub-components of the velocity profile are in agreement between all three regions, the centroids are consistent within ≈ 50 km s -1 of each other (i.e., well within the spectral FWHM of MUSE), and (apart from the QSO region, which has low S/N) all regions appear to show a velocity profile with potentially at least two components. This would suggest that the stronger absorber has the same bulk kinematics across the entire ≈ 50 kpc extent traced by the background sources (Table 1).</text>
<section_header_level_1><location><page_6><loc_51><loc_51><loc_88><loc_52></location>3.3. Mapping Mg /i.pc/i.pc DLA metrics with MUSE spaxels</section_header_level_1>
<text><location><page_6><loc_51><loc_31><loc_94><loc_49></location>In order to study the physical extent of the two strong Mg /i.pc/i.pc absorbers, Figs. 2 and 3 show spaxels maps of each of these three metrics in the fields of J0033 and J1527 respectively. The spectra generated for each pixel (corresponding to a native spaxel) in these maps are created using the same method outlined in Sect. 2.2, and are combined using nine spaxels within a 3 × 3 spaxel grid centered at each native pixel. This procedure effectively corresponds to a 0.6 '' × 0.6 '' smoothing of the original map, averaging over spaxels contained within the point spread function of the observations ( ≈ 0 . 8 '' ). This smoothing thus minimizes effects from overlapping information within the native pixel scale. Despite this smoothing, we point out that there are only ≈ 30 and ≈ 44 spatially-independent measurements within the respective J0033 and J1527 fields, which we discuss in more detail in Sect. 5.1.</text>
<text><location><page_6><loc_51><loc_22><loc_94><loc_29></location>We note that there is a potential correlation between each of the three metrics and the measured S/N near the Mg /i.pc/i.pc doublet for spaxels with S/N < 1 and an EW detection significance < 2 σ in both fields. We thus proceed by implementing a quality control cut, and only analyze spaxels with a S/N ≥ 1 near the Mg /i.pc/i.pc doublet and a EW 2796 detected at ≥ 2 σ significance.</text>
<text><location><page_6><loc_51><loc_10><loc_94><loc_21></location>Table 7 provides the percentage of spaxels in Figs. 2 and 3 that pass the various Mg /i.pc/i.pc metric criterion typically used to flag an absorber as a potential DLA. In each of the three regions of J0033 and the full arc of J1527, 100% of spaxels that meet our quality control cut pass the EW 2796 ≥ 0 . 3 Å. This would suggest that this candidate DLA is completely extended over the background source. However combining with other metrics, (i) 44-89% of spaxels passed both the EW 2796 and EW 2600 EW 2796 criteria for potential DLA absorption, and (ii) 12%-34% of spaxels simultaneously</text>
<figure>
<location><page_7><loc_7><loc_67><loc_42><loc_93></location>
<caption>Fig. 3. Color maps of the lensed region in the J1527 field, showing the gravitational arc tracing Mg /i.pc/i.pc absorption. The notation is the same as in Fig. 2. Only spaxels with S/N ≥ 1 at the position of Mg /i.pc/i.pc 2796 Å and a EW 2796 detected at ≥ 2 σ significance are colored in all four panels.</caption>
</figure>
<figure>
<location><page_7><loc_7><loc_41><loc_44><loc_67></location>
</figure>
<figure>
<location><page_7><loc_44><loc_67><loc_79><loc_93></location>
</figure>
<figure>
<location><page_7><loc_44><loc_41><loc_77><loc_67></location>
<caption>Table 7. Percentage of spaxels that pass the Mg /i.pc/i.pc metric in each region</caption>
</figure>
<table>
<location><page_7><loc_20><loc_28><loc_80><loc_37></location>
</table>
<text><location><page_7><loc_6><loc_10><loc_49><loc_26></location>meet the D -index, EW 2796 and EW 2600 EW 2796 criteria for potential DLA absorption. We note that, of the 12% of spaxels that pass all three metric criteria within the CI2 region of the J0033 field, all spaxels are found on the outer edge of the region where the S/N is lowest (S/N ≈ 2 ). However, most of the spaxels that pass the three Mg /i.pc/i.pc metric criteria for the arc in both fields and CI1 region in J0033 are more evenly distributed across the region. Assuming a combination of all three of the Mg /i.pc/i.pc metrics is a definitive tracer for H /i.pc-rich absorption, the lower limit on the overall extent of H /i.pc gas across all three regions of the J0033 field is ≳ 30 %, and ≳ 16 % for the full arc towards J1527 (prior to accounting for the predicted false positive and success rates of</text>
<text><location><page_7><loc_51><loc_24><loc_94><loc_26></location>each of the three metrics; e.g., as seen for the D -index in Ellison 2006).</text>
<section_header_level_1><location><page_7><loc_51><loc_19><loc_69><loc_20></location>4. Modeling the fields</section_header_level_1>
<text><location><page_7><loc_51><loc_10><loc_94><loc_17></location>While in the previous sections we analyzed the direct observational properties of the strong Mg /i.pc/i.pc absorbers detected in the J1527 and J0033 fields, in the following we use a novel approach that makes use of Bayesian inference and parametric physical models to interpret the detected strong absorption. The main goals of this analysis are to i) infer the physical properties of the</text>
<figure>
<location><page_8><loc_8><loc_68><loc_47><loc_92></location>
<caption>Fig. 4. Schematics showing how we extract a pair of EW and D -index from a single spaxel, given a choice of the free parameters (highlighted in red) of our modeling. An ensemble of n clouds produces the absorption. The spaxel is only partially covered by gas, so that the flux is either equal to F los = F bg exp( -τ ) if the clouds are intercepted or to F bg = 1 if they are not. The total flux of the spaxel F spaxel depends on the value of the covering fraction C frac . For more details, see Sect. 4.1.1</caption>
</figure>
<text><location><page_8><loc_6><loc_54><loc_49><loc_56></location>absorbers and ii) test the robustness of Mg /i.pc/i.pc metrics and whether they can be applicable on extended sources (see Sect. 5.2).</text>
<text><location><page_8><loc_6><loc_39><loc_49><loc_53></location>In our modeling procedure we assume that the two Mg /i.pc/i.pc absorbing systems are physically represented by a population of cool clouds, from which we extract mock observations (specifically EWs and D -indices that one would extract from an ensemble of spaxels) that can be directly compared to the real data. These models are idealized and, while they are useful to make conclusions on the two dimensional properties of the absorbing gas (e.g., covering fraction), they do not have the pretense of depicting a realistic three dimensional cloud distribution. In Sect. 5.1 we discuss more in detail the various assumptions and limitations of the models.</text>
<section_header_level_1><location><page_8><loc_6><loc_36><loc_22><loc_37></location>4.1. Model description</section_header_level_1>
<section_header_level_1><location><page_8><loc_6><loc_34><loc_31><loc_35></location>4.1.1. Procedure for a single spaxel</section_header_level_1>
<text><location><page_8><loc_6><loc_28><loc_49><loc_32></location>We first assume that the flux detected by a spaxel is directly related to the gas covering fraction C frac within that spaxel (hence the fraction of the background source area covered by the gas) through the formula:</text>
<formula><location><page_8><loc_14><loc_25><loc_49><loc_26></location>F spaxel = C frac F los + F bg (1 -C frac ) . (1)</formula>
<text><location><page_8><loc_6><loc_10><loc_49><loc_24></location>Here, F bg is the background continuum flux, which we assume to be normalized to 1, while F los is the flux that would be detected by a single point-source line of sight that intercepts the absorbing gas. For Eq. (1) to be valid, we assume that the background flux is constant within the spaxel and that the absorbing gas, where present, has exactly the same properties within the same spaxel. This is clearly an approximation, which however should not significantly impact the general results of this paper. We emphasize that, in this paper, we define the covering fraction as the fraction of the area of the background source area (such as a spaxel) that is covered by foreground gas. This is different</text>
<text><location><page_8><loc_51><loc_84><loc_94><loc_93></location>to a more observationally-driven definition with the fraction of spaxels across an extended source that contain the same absorption feature, such as a DLA. The MUSE spaxels have a size of (0.2 '' × 0.2 '' ), resulting in typical physical sizes in both fields of 0.07 and 1.4 kpc along a side (after delensing). The output flux of the model is anyway independent of the spaxel size itself and is only affected by the covering fraction C frac (see Eq. 1).</text>
<text><location><page_8><loc_51><loc_80><loc_94><loc_84></location>Once a covering fraction is assumed, to calculate F spaxel one needs to calculate the value of F los = F bg exp( -τ ) , where the optical depth τ is defined as (see e.g., Liang & Kravtsov 2017):</text>
<formula><location><page_8><loc_61><loc_78><loc_94><loc_79></location>τ ( λ | N,b, v ) = Nσ 0 f osc Φ( λ | b, v ) , (2)</formula>
<text><location><page_8><loc_51><loc_61><loc_94><loc_77></location>where N is the gas column density, σ 0 is the cross section, f osc is the oscillator strength and Φ( λ | b, v ) is the Voigt profile function, which depends on the wavelength λ , on the Doppler parameter b and on the velocity v at which the transition takes place. We assume that the absorption is due to the superposition of n clouds along the line of sight, each of them contributing to the total optical depth. We first assume that each of these clouds has a neutral hydrogen column density N HI , c , so that the total HI column density along the point-source line of sight would be nN HI , c . We then define the spaxel HI column density N HI , spaxel as nN HI , c × C frac , given that the spaxel is only partially covered by gas. Assuming a value for N HI , spaxel , one can then solve for the number of clouds:</text>
<formula><location><page_8><loc_66><loc_57><loc_94><loc_60></location>n = N HI , spaxel C frac N HI , c . (3)</formula>
<text><location><page_8><loc_51><loc_25><loc_94><loc_56></location>The goal of this modeling is to find which HI column densities and covering fractions are needed to reproduce the observed Mg /i.pc/i.pc EWs and D -indices. To this purpose, we further assume that the clouds have a metallicity Z c and an intrinsic volumetric density n c /three.sup . We then use the photo-ionization code CLOUDY (Ferland et al. 2013) to infer the Mg /i.pc/i.pc column densities of the single clouds, assuming an ionizing flux given by the extragalactic UV background from Haardt & Madau (2012) at the redshift of the absorbers (see Sect. 2). At this high column densities most of the medium will be self-shielded and not affected by ionization. Models that consider completely neutral gas lead us to the same results and conclusions presented here (we discuss this in more detail in Sect. 5.1). We then assign to each of the n clouds a lineof-sight velocity extracted from a Gaussian distribution centered in zero and with a width equal to a velocity dispersion σ los and a Doppler parameter b equal to the sum in quadrature of a thermal ( b th ∼ 4 km s -1 , assuming a temperature T = 2 × 10 4 K) and a turbulent component, with b turb = 10 km s -1 (we discuss different values of b turb in Sect. 5.1). As we discuss below, σ los is one of the free parameters of our model and it determines the (simplistic) bulk kinematics of the cloud population and, in turn, the strength of the absorption. With all the above ingredients, one can solve Eqs. (2) and (1) to finally obtain the spaxel flux F spaxel , specifically for the Mg /i.pc/i.pc absorption lines.</text>
<text><location><page_8><loc_51><loc_20><loc_94><loc_24></location>As a final step, we convolve the spectrum with a Gaussian kernel assuming that the instrumental profile has a resolution (FWHM) of 150 km s -1 , which roughly resembles the resolution of MUSE at the redshift of the two absorbers. We then add</text>
<text><location><page_9><loc_6><loc_78><loc_49><loc_93></location>random Gaussian noise in the spectrum: the value of the final flux at each wavelength is extracted from a Gaussian centered on the original model flux value, with a width equal to the flux value divided by the observed signal-to-noise in the MUSE data. Once this procedure is complete, we can extract the Mg /i.pc/i.pc metrics of the spaxel, specifically the EW 2796 and the D -index. We point out that we do not include the ratio EW 2600 EW 2796 in this part of the analysis: this ratio might be dependent on our assumption of photo-ionization (which we discuss in Sect. 5.1) and is additionally also affected by dust and by chemical enrichment, which are not explicitly taken into account in our framework. The steps explained above are summarized in the schematics of Fig. 4.</text>
<section_header_level_1><location><page_9><loc_6><loc_73><loc_41><loc_75></location>4.1.2. Prediction from an ensemble of spaxels and comparison with the data</section_header_level_1>
<text><location><page_9><loc_6><loc_44><loc_49><loc_71></location>To compare with the observations outlined in the previous sections, we need to apply the procedure above to multiple spaxels. The Mg /i.pc/i.pc metrics extracted from this spaxel ensemble are then directly comparable to our data. In the following, we assume that for each model realization and across the same field all the clouds of our synthetic populations have the same metallicities and densities, with velocities extracted from the same Gaussian velocity distribution. However, in Sect. 3.3 we have seen that the observed properties of the fields vary across them and that for example a fraction of the spaxels do not pass the three metrics for potential DLA absorption. We therefore assume that the differences across the spaxels are due to differences in the spaxel covering fraction and in the total HI column density (i.e., different spaxels detect different numbers of clouds). We introduce the parameters N HI , spaxel , which is the mean observed neutral hydrogen column density of the spaxels (the column density of each spaxel is then drawn randomly from a range that goes from log( N HI , spaxel ) -0 . 3 to log( N HI , spaxel )+0 . 3 ) and C frac , which is the center of a normal distribution with a standard deviation equal to 0.2 and truncated at 0.1 and 1, from which the value of the covering fraction C frac of each spaxel is extracted.</text>
<text><location><page_9><loc_6><loc_26><loc_49><loc_44></location>A single model realization, hence a distribution of EW 2796 and D -indices (see Sect. 4.1.1), is uniquely defined by the choice of the five free parameters N HI , spaxel , C frac , Z c , N HI , c and σ los . Each distribution is composed of 200 values, a number consistent with the amount of spaxels in J1527 and J0033 (see Section 3.3). The idea is then to compare these distributions with the observed ones to find which choice of parameters better reproduce our data. In order to do this, we first select the spaxels in the data and in the model realizations using the same quality control cuts as used for the observations (i.e., S/N > 1 and an EW 2796 detection significance > 2 σ /four.sup ). We then quantitatively assess the consistency between the model predictions and the MUSE observations by performing a Kolmogorov-Smirnov (KS) test on the distributions of EWs and D -indices. Finally, we define our likelihood as:</text>
<formula><location><page_9><loc_19><loc_24><loc_49><loc_25></location>ln L = ln p EW +ln p D , (4)</formula>
<text><location><page_9><loc_6><loc_20><loc_49><loc_23></location>where p EW and p D are the p-values /five.sup obtained from the KS tests on respectively the EW and D -index distributions. We finally</text>
<text><location><page_9><loc_51><loc_84><loc_94><loc_93></location>use the likelihood expressed in Eq. (4) to perform two Bayesian analyses on the J1527 and the J0033 fields, whose results are presented in the next section. We point out that the results of our analysis (i.e., the recovered best-fit values of the five free parameters) are independent from any lensing model, as we are not assuming a priori the size and shape of the two fields or of the single spaxels.</text>
<section_header_level_1><location><page_9><loc_51><loc_81><loc_63><loc_82></location>4.2. Model results</section_header_level_1>
<text><location><page_9><loc_51><loc_68><loc_94><loc_79></location>In this section, we present the results of the Bayesian analysis performed to compare our models with our two observational fields, as explained in the previous section. We adopt the nested sampling method (Skilling 2004, 2006), using the /d.pc/y.pc/n.pc/e.pc/s.pc/t.pc/y.pc python package (Speagle 2020; Koposov et al. 2022). For both fields, we use flat priors for the five free parameters: 19 . 8 < log N HI , spaxel < 21 . 2 , 50 < σ los / (km s -1 ) < 200 , 0 . 1 < C frac < 1 , 18 < log N HI , c < 19 . 5 cm -2 , -3 < log Z c < 0 .</text>
<section_header_level_1><location><page_9><loc_51><loc_65><loc_60><loc_66></location>4.2.1. J1527</section_header_level_1>
<text><location><page_9><loc_51><loc_45><loc_94><loc_63></location>The results of the Bayesian analysis performed on the J1527 field are shown in the left-hand side of Fig. 5. We can first look at the intrinsic properties of the gas clouds, σ los , N HI , c and Z c . As expected, these three quantities are slightly degenerate with each other, as they all contribute to the strength of the absorption (Eq. 2). At fixed N HI , spaxel a lower value of N HI , c simply implies a larger number of clouds along the line of sight (Eq. 3). At the same time, our photo-ionization models predict that the ratio N MgII , c /N HI , c decreases with increasing N HI , c , so that higher N HI , c imply higher σ los and/or higher Z c , explaining these two degeneracies. The velocity dispersion of the clouds seems rather well constrained and it is around 100 km s -1 (we discuss this further in Sect. 5.1), while both the cloud column density and metallicity have very large uncertainties, implying that the choice of these two parameters does not strongly affect our outputs.</text>
<text><location><page_9><loc_51><loc_28><loc_94><loc_44></location>It is interesting to look at the recovered values of N HI , spaxel : we find that the posterior distribution is in agreement with the previous estimates of the H /i.pc column densities from B22 (see Table 5), shown as a black vertical line in Fig. 5 /six.sup . However this quantity is very poorly constrained with rather large uncertainties, due mainly to strong degeneracies with both the column density and the metallicity of the clouds. We conclude that we recover the main finding of B22 that this absorber is a DLA, but also that our predictions of the total HI column density are very uncertain, as can be expected for a method that is based exclusively on a comparison with Mg /i.pc/i.pc absorption lines. This also shows the limitations of the usual Mg /i.pc/i.pc metrics (which are much more simplistic than our modeling) in selecting DLAs.</text>
<text><location><page_9><loc_51><loc_17><loc_94><loc_27></location>The final and most important result is given by the covering fraction C frac : our analysis prefers C frac < 1 , with very tight constraints that are not degenerate with the other free parameters of the model. This result is consistent with our assumption that the DLA is composed by clouds and shows that these absorbers do not cover the entirety of the area traced by the gravitational arc. This is the first estimate of the structure of a DLA, which appears to be patchy on the scale of a MUSE spaxel (0.2 '' × 0.2 '' ).</text>
<figure>
<location><page_10><loc_7><loc_64><loc_93><loc_93></location>
<caption>Fig. 5. Posterior distributions resulting from the Bayesian analysis described in Sect. 4.2. The blue vertical dashed lines show the median and 2 σ uncertainties of each distribution and the black line marks the HI column density estimate of Bordoloi et al. (2022), which is consistent with our predictions.</caption>
</figure>
<section_header_level_1><location><page_10><loc_6><loc_56><loc_15><loc_57></location>4.2.2. J0033</section_header_level_1>
<text><location><page_10><loc_6><loc_48><loc_49><loc_55></location>The right-hand side of Fig. 5 shows the posterior distributions for the five free parameters in our analysis for the J0033 field. We can note that the intrinsic cloud kinematics, column density and metallicity ( σ los , N HI , c and Z c ) have similar values and trends with respect to the J1527 field, although in this case they seem to have slightly lower uncertainties.</text>
<text><location><page_10><loc_6><loc_38><loc_49><loc_47></location>The total neutral hydrogen column density, with a median value of the posterior distribution of about 10 20 . 6 cm -2 , seems to point to the presence of a DLA even in this field. The posterior distribution of this parameter has however very large uncertainties (as for the J1527 case), hence this result will need further confirmation. Despite this, with our analysis we have a hint, more robust with respect to the usual Mg /i.pc/i.pc metrics utilized in the literature, that this absorber is likely an extended DLA.</text>
<text><location><page_10><loc_6><loc_26><loc_49><loc_37></location>Very interestingly, the covering fraction of this extended absorber is again very well constrained and predicted to be lower than 1, with values that are even lower (but consistent within the uncertainties) with respect to the DLA in the J1527 field. This result seems hence to indicate that these two strong absorbers have similar properties and especially similar covering fractions, the most robust result of our Bayesian analysis. In both cases the gas distribution appears patchy, not covering completely the background extended source.</text>
<section_header_level_1><location><page_10><loc_6><loc_22><loc_17><loc_23></location>5. Discussion</section_header_level_1>
<section_header_level_1><location><page_10><loc_6><loc_20><loc_43><loc_21></location>5.1. Assumptions and limitations of model and data</section_header_level_1>
<text><location><page_10><loc_6><loc_14><loc_49><loc_18></location>The modeling framework presented in Sect. 4 is idealized and relies on a number of assumptions, some of which we already discussed above. In the following we summarize and discuss the most important of these assumptions.</text>
<text><location><page_10><loc_6><loc_10><loc_49><loc_13></location>First, we stress that the model is simplistic and it is not meant to represent a full 3D configuration of the absorbing gas. For this reason, we do not formulate predictions for the H /i.pc and Mg /i.pc/i.pc</text>
<text><location><page_10><loc_51><loc_44><loc_94><loc_57></location>maps to directly compare with the real observational data (see e.g., Figs. 2 and 3). We instead assume that the clouds have the same kinematics, metallicity and column densities everywhere across the fields, not taking into account of possible (likely) inhomogeneities, which we only attribute to different spaxels detecting different numbers of clouds and having therefore different covering fractions and total column densities. While this assumption is clearly an oversimplification, the results of Sect. 4.2 show how, by simply comparing the total distributions of observational diagnostics like the EW 2796 and the D -index, we can infer the general properties of the absorbing gas.</text>
<text><location><page_10><loc_51><loc_10><loc_94><loc_40></location>Another important approximation of the model is related to the cloud kinematics: we assume that the line-of-sight velocity distribution of the clouds is described by a Gaussian profile, which is most likely inaccurate. However, understanding the actual kinematics and dynamics of the absorbing gas is outside the scope of this work and the value of σ los is simply used to determine the strength of the absorption: a non Gaussian kinematics would not strongly affect the outputs of our model (and therefore our main results and conclusions), as long as the average line-ofsight velocity dispersion is similar to what we found here. We also assume that the Gaussian distribution is centered at 0 km s -1 in all the spaxels. This is justified by an ongoing kinematic analysis (Ledoux et al., in prep.) that shows that, at least for J0033, the bulk kinematics of the absorber is the same across the entire field. A variation of this center for different spaxels would anyway not impact our results, given that the strength of the absorption for each spaxel does not depend on the exact position of the velocity centroid but only on the velocity dispersion. However, we caution that the recovered value of σ los depends slightly on the choice of the Doppler parameter b , dominated by the turbulent component b turb , which we fixed to 10 km s -1 in our fiducial model. We find that higher or lower values of b turb , while leaving the general conclusions of this study unchanged, would lead to respectively lower and higher values of σ los .</text>
<text><location><page_11><loc_6><loc_61><loc_49><loc_93></location>Finally, we inferred the column densities of Mg /i.pc/i.pc using photo-ionized models (Ferland et al. 2013), while most of this gas is self-shielded and therefore in its neutral state. To investigate the impact of such choice on our results, we performed additional Bayesian analyses on the two fields assuming that the gas is entirely neutral and that the magnesium is all in the Mg /i.pc/i.pc state. The Mg /i.pc/i.pc cloud column density can therefore simply be obtained by assuming the magnesium solar abundance log(Mg / H) /circledot = -4 . 47 (Asplund et al. 2009) and a value for the metallicity Z c . The results of such model are perfectly consistent, for both fields, with those shown in Fig. 5, so we conclude that the photo-ionization assumption does not have an impact on our main findings, as expected given the high column densities of the absorbers (e.g., Dey et al. 2015). We note that the photo-ionization assumption might have an influence on EW 2600 EW 2796 , but we decided to exclude this diagnostic from our likelihood (Eq. 4). Models where we also include a comparison between the observed and the predicted EW 2600 EW 2796 distributions point towards the clouds having a low metallicity (median log Z c ≈ -2 . 6 ) and high HI column densities ( N HI , c ≳ 10 19 cm -2 ), but we choose to discard them, considering the uncertainties on the EW 2600 EW 2796 diagnostics (see Sect. 4.1). Interestingly, even in this case the recovered covering fraction C frac remains consistent with the fiducial values reported in Sect. 4.2.</text>
<text><location><page_11><loc_6><loc_30><loc_49><loc_60></location>The limitations of the data used may also impact the results of our modeling. As already mentioned in Sect. 3, our data might be contaminated by the galaxy light, resulting in an artificially weaker Mg /i.pc/i.pc absorption in our spectra. In the modeling, higher EW 2796 and D -indices could lead to slightly higher covering fractions with respect to what we obtained in Sect. 4.2. Moreover, due to the seeing, the MUSE native spaxels (even after the smoothing described in Sect. 3.3) are not spatially independent. The overall effect of the seeing is to smooth the properties of adjacent spaxels, so that we can expect that the overall impact on the distributions of EW 2796 and D -indices is to make them narrower than what they would originally be. Given that our model predicts average quantities of the absorbing material across the two fields, using wider distributions would likely not change significantly our findings. We note that the distribution of measured EW and D -indices from spatially independent spaxels (i.e., spaxels separated by the point spread function of the observations, every 3.5 spaxels, which correspond to ≈ 30 and ≈ 44 spatially-independent measurements within the respective J0033 and J1527 fields) are consistent with the full spaxel distribution (with KS test p-values of p ≈ 0 . 95 for J1527 and p ≈ 0 . 62 for J0033). We therefore opt to use all the spaxels to improve the sampling of the EW and D -index distributions without significantly impacting our modeling analysis.</text>
<section_header_level_1><location><page_11><loc_6><loc_25><loc_45><loc_27></location>5.2. Evaluating the success of DLA metrics in individual spaxels</section_header_level_1>
<text><location><page_11><loc_6><loc_10><loc_49><loc_24></location>Figure 6 encapsulates the dependence of EW 2796 and D -index on logN(H /i.pc) in the literature measurements taken from 1D spectra of quasar sightlines (pink errorbars; Ellison 2006; Rao et al. 2006; Berg et al. 2017, 2021) and the best-fit models (colored symbols, obtained by sampling the posterior distributions of σ los , C frac , N HI , c , Z c within the 32nd and 68th percentiles) for both J1527 (left panels) and J0033 (right panels). The colored symbols represent a range of logN(H /i.pc) that varies from the subDLA to the DLA regime. These distributions therefore are not intended to represent directly the distributions of the observed properties, but show instead the results of specific models as a</text>
<text><location><page_11><loc_51><loc_83><loc_94><loc_93></location>function of the HI column density. For reference, the distribution of observed EW 2796 and D -index for spaxels within the respective fields (i.e., spaxels in Figs. 2 and 3 with S/N ≥ 2 and EW 2796 measured at > 2 σ significance) are denoted by the hollow violins, and are centered on the median logN(H /i.pc) predicted by the models. The general agreement of the violins with the model points in the top two rows of Fig. 6 is a result of using these two distributions to constrain the models (i.e., Eq. 4).</text>
<text><location><page_11><loc_51><loc_64><loc_94><loc_82></location>The model is in general able to recover the bulk trends with logN(H /i.pc) seen in the 1D quasar sightlines for the two metrics in Fig. 6, but there are some notable differences. First, the normalization of the D -indices (bottom row) is different between models and data, but this is expected because of the lower spectral resolution of our MUSE data with respect to the literature data (see also below). Second, the EW 2796 (top row) tends to flatten in the 1D sightlines (in pink) for logN(H /i.pc) ≳ 20 . 3 cm -2 , while it increases monotonically to larger values in the model, especially for J1527 (left column). This is due to the fact that these models are calibrated to our tomographic observations, which exhibit EW 2796 significantly larger (see the position of the hollow violins) than the median values of the 1D data. To reproduce these high values, the EW 2796 needs to increase at larger HI column densities, instead of reaching a plateau.</text>
<text><location><page_11><loc_51><loc_26><loc_94><loc_63></location>The third difference is instead given by the EW 2796 scatter: whilst the scatter in the models appears similar (if not lower) to that of the 1D observations for sub-DLA HI column densities, the model scatter is instead much larger for logN(H /i.pc) ≳ 20 . 3 cm -2 . A larger scatter is in principle not expected, given that in the modeling all clouds have the same metallicity (and density) within a single absorber (Sect. 4.1), while the mass-metallicity relation of absorbers (e.g., Ledoux et al. 2006; Neeleman et al. 2013; Christensen et al. 2014) is expected to influence the width of the observed Mg /i.pc/i.pc absorption lines (e.g., Ellison 2006; Rao et al. 2006; Bouché 2008; Berg et al. 2017) and thus to increase the scatter of the respective Mg /i.pc/i.pc metrics. The large scatter in the model can be explained by the covering fraction, and a clear vertical gradient of covering fraction as a function of EW 2796 (top row) or D -index (bottom row) for a given logN(H /i.pc) is indeed visible in Fig. 6. As a result, it is clear that using the same Mg/i.pc/i.pc metric thresholds from 1D quasar sightlines on 2D extended spaxels may not be so straightforward to interpret. Given the small pencil-beam nature of quasar sightlines, the covering fraction across the quasar would only vary if the angular sizes of the gas clouds comprising the absorber are much smaller than the angular size of the continuum emission from the background quasar, which seems unlikely. However covering fraction should impact extended sources, where the coherence length of Mg /i.pc/i.pc absorption is expected to be ∼ 5 kpc (Afruni et al. 2023) with cloud sizes of ∼ 0 . 5 kpc (for both low, e.g., Mg /i.pc/i.pc, and intermediate, e.g., C /i.pc/v.pc, ionization gas; Faerman & Werk 2023; Lopez et al. 2024). For reference, as already mentioned, the range in spaxel sizes in both fields is 0.07 and 1.4 kpc along a side (after delensing).</text>
<text><location><page_11><loc_51><loc_10><loc_94><loc_25></location>Another way to assess the success of the DLA Mg /i.pc/i.pc metrics is to look at the fraction of the models that produce a DLA (f DLA ) for a given metric threshold. Table 8 provides the required threshold for EW 2796 and D -index to produce a given f DLA for the models of both fields. We note that f DLA is ≈ 20 % and ≈ 40 % using the typically adopted EW 2796 thresholds of 0.3Å and 0.6Å in the literature, whilst f DLA ≈ 90 %(i.e., similar to the purity found in Ellison 2006) using the D -index cut expected for MUSE-like resolution ( ≈ 3 . 6 ). The EW 2796 threshold obtained by the simulations is within a factor of two of the equivalent thresholds required to reproduce the same f DLA seen in the high redshift ( 2 < z < 4 . 5 ) XQ-100 survey (Berg et al. 2017, 2021),</text>
<figure>
<location><page_12><loc_10><loc_43><loc_64><loc_85></location>
<caption>Fig. 6. Model predictions of the DLA metrics EW 2796 (top panels) and D -index (bottom panels) as a function of logN(H /i.pc) in the two fields J1527 (left column) and J0033 (right column). The vertical dotted line denotes the DLA threshold logN(H /i.pc) = 20 . 3 cm -2 whilst the horizontal dashed line denote the minimum thresholds for a system to be considered a DLA for that metric; points in the top-right corner of each panel would be considered a DLA. The hollow violins show the distribution of EW 2796 and D -index for spaxels (with S/N ≥ 2 and EW 2796 measured at > 2 σ significance) within the field. The violins are arbitrarily centered at the median logN(H /i.pc) predicted by the modeling, whilst their widths are arbitrarily set for display purposes and do not represent the error in logN(H /i.pc). The pink errorbars show the median and 68 th percentile confidence interval of the literature measurements of single quasar sightlines (taken from Rao et al. 2006; Ellison 2006; Berg et al. 2017). The literature values are binned in increments of 0.5 dex in logN(H /i.pc). We note that the D -index measurements from the literature are obtained at a different spectral resolutions than the MUSE observations, and thus satisfy a different D -index threshold to be considered a DLA. These literature points are only shown to denote the dependence of D -index on logN(H /i.pc) for individual quasar sightlines. The colored symbols represent the range of models within the 32nd and 68th percentiles of the posteriors of the four free parameters ( σ, C frac , N HI , c , Z c ), and are colorcoded by the covering fraction C frac . We stress that, while the literature data are obtained from point-source quasar sightlines, our model framework is designed to reproduce the gas absorption properties on extended sources.</caption>
</figure>
<text><location><page_12><loc_6><loc_25><loc_49><loc_40></location>while the lower-redshift ( z < 1 . 65 ) sample from Rao et al. (2006) requires a consistently high EW 2796 threshold of ≳ 2 . 7 in order to reproduce the same f DLA . Whilst the scatter in the relations of Fig. 6 is larger in the modeled 2D spaxels in comparison to the 1D literature quasar sightlines, it appears that the purity of the sample (i.e., f DLA ) for a given EW 2796 threshold is roughly the same between the two types of background sources. The fact that our modeling is able to reproduce the observed trends and statistics for pencil-beam quasar sightlines validates that we are able to estimate the H /i.pc column density (albeit with large uncertainties) and the gas covering fraction based on Mg /i.pc/i.pc absorption towards extended background sources.</text>
<section_header_level_1><location><page_12><loc_6><loc_21><loc_29><loc_22></location>5.3. The nature of the absorbers</section_header_level_1>
<text><location><page_12><loc_6><loc_15><loc_49><loc_20></location>The results of Sect. 4 can be used to understand what is the nature of the two strong absorbers detected in the J1527 and J0033 fields and what is their potential role in the evolution of the galaxies with which they are associated.</text>
<text><location><page_12><loc_6><loc_10><loc_49><loc_14></location>Looking at the projected distances in the absorber planes reported in Table 1, we note that the two strong absorbers (both potentially DLAs) extend up to distances of a few tens of kpc. Compared to the typical sizes of galactic discs at redshift z > 1</text>
<text><location><page_12><loc_51><loc_35><loc_94><loc_40></location>(e.g., Shibuya et al. 2015), these absorbers seem to be significantly more extended. This indicates that this gas is likely more representative of the CGM, rather than of high-redshift rotating interstellar gas discs (see Neeleman et al. 2020; Kaur et al. 2024).</text>
<text><location><page_12><loc_51><loc_21><loc_94><loc_34></location>By summing the areas in the absorber plane (using the same lens models mentioned in Sect. 2) of all the MUSE spaxels, we can calculate the total area subtended by the extended source in the J1527 and J0033 fields, respectively 353 kpc 2 and 350 kpc 2 . Assuming a mean HI column density across the field equal to the median value of the two posterior distributions of N HI , spaxel (see Fig. 5) /seven.sup we then obtain M HI ≳ 2 × 10 9 M /circledot for J1527, consistent with the previous results of B22, and M HI ≳ 1 . 2 × 10 9 M /circledot for J0033. We stress that these two values represent strictly lower limits, given that the extension of the absorber could be larger than the extension of the background arc.</text>
<text><location><page_12><loc_51><loc_14><loc_94><loc_20></location>Our picture is consistent with the recent results of Stern et al. (2021), who using the FIRE simulations find that a predominantly neutral inner CGM, potentially giving rise to DLAs, is present in halos at z > 1 with virial masses of about 10 11 M /circledot , where the cooling time of the hot gas is shorter than the free-fall time</text>
<table>
<location><page_13><loc_10><loc_62><loc_45><loc_89></location>
<caption>Table 8. Fraction of modeled spaxels with logN(H /i.pc) ≥ 20 . 3 (f DLA ) for both J0033 and J1527 fields for a given threshold for EW 2796 and D -index.</caption>
</table>
<text><location><page_13><loc_6><loc_46><loc_49><loc_59></location>of the system. These authors find that in these halos the neutral gas can extend with large covering fractions ( 0 . 5 < C frac < 1 . 0 ) up to distances of tens of kpc from the host galaxy, in agreement with our findings. Moreover, in their simulations the cold gas is on average inflowing towards the galaxy (even though with relatively low infall velocities). If this scenario is correct, and we are detecting such material with our observations, this large amount of cold, neutral gas ( M HI ≳ 10 9 M /circledot ) would account for a substantial fraction of the baryons within the halo and would therefore represent the main source of fuel for the future star formation within the galaxy.</text>
<text><location><page_13><loc_6><loc_28><loc_49><loc_45></location>We conclude by emphasizing that, except for the above speculations, here we do not attempt to draw a full dynamical picture for this gas, especially since we can not determine which galaxies are responsible for these absorbers: for J1527, the MUSE spectral coverage does not allow us to detect galaxies in emission at the redshift of the absorber ( z abs /similarequal 2 . 06 ), while in the J0033 field we detect multiple galaxies (at z abs /similarequal 1 . 17 ) whose CGM could contribute to the absorption (Ledoux et al., in prep.). Disentangling whether this gas is a signature of galactic winds (e.g., Schroetter et al. 2019; Schneider et al. 2020; Fernandez-Figueroa et al. 2022), recycling material, or cold cosmological accretion (e.g., van de Voort et al. 2011; Bouché et al. 2013; Theuns 2021) is therefore outside the scope of the present study and is left for future work.</text>
<section_header_level_1><location><page_13><loc_6><loc_24><loc_30><loc_25></location>6. Summary and conclusions</section_header_level_1>
<text><location><page_13><loc_6><loc_10><loc_49><loc_22></location>In this paper, we analyzed two gravitational arc fields with known strong Mg /i.pc/i.pc absorption in order to predict the presence of a H /i.pcrich absorber (such as a DLA) and determine the spatial extent of the absorber. For the first time, we created 2D maps of strong Mg/i.pc/i.pc absorbers using three Mg /i.pc/i.pc metrics ( EW 2796 , D -index, and EW 2600 EW 2796 ) typically used in the literature for identifying DLAs (Rao & Turnshek 2000; Ellison 2006). These maps suggest that both gravitational arcs probe subDLAs or DLAs, and are extended over areas of ≈ 350 kpc 2 . In particular, one system (J0033) shows ≤ 50 km s -1 variations in the redshift of the observed</text>
<text><location><page_13><loc_51><loc_84><loc_94><loc_93></location>Mg/i.pc/i.pc absorption across ≈ 50 kpc of separation between sources, suggesting the same bulk gas is extended over such very large areas. The Mg /i.pc/i.pc metrics for the other field (J1527) suggest the presence of a DLA, which has been confirmed by KCWI data of the same system (Bordoloi et al. 2022). These Mg /i.pc/i.pc metrics can be successful for implying the strength of H /i.pc absorption along extended emission sources.</text>
<text><location><page_13><loc_51><loc_51><loc_94><loc_84></location>In order to quantify the success of these metrics, as well as provide an estimate of the H /i.pc column density towards the J0033 field, we developed a simple toy model in order to evaluate the robustness of the three Mg /i.pc/i.pc metrics used to identify DLAs in 1D quasar sightlines. The results from our toy model, which assumes a series of clouds in front of an extended background source, suggest that the typical covering fraction of gas clouds in front of the twogravitational arcs is ≈ 60 -80 %.Theresulting purity of these metrics in selecting DLA column densities in front of extended sources is similar to what has been observed in the literature for 1D quasar sightlines (Rao et al. 2006; Ellison 2006; Berg et al. 2017). The model also demonstrates that the covering fraction of gas can also influence both the EW 2796 and D -index metrics measured. As a result, choosing a threshold for these two metrics to identify DLAs towards extended sources is not straightforward. For EW 2796 , the threshold can depend on metallicity (Rao et al. 2006; Berg et al. 2017) and covering fraction. Whilst the D -index threshold depends on the spectral resolution of the observations (Ellison 2006) and covering fraction, we suspect that the covering fraction should not influence point-source, quasar observations as the typical cloud size is likely much larger than the background source. The typical spaxel size can vary between 0.07 and 1.4 kpc in our two fields after delensing. These results provide a cautionary tale of using Mg /i.pc/i.pc metrics to identify DLAs in front of extended background sources, despite the success at identifying the DLAs in both of these systems.</text>
<text><location><page_13><loc_51><loc_36><loc_94><loc_50></location>Despite the difficulty of using the Mg /i.pc/i.pc metrics, our model is able to predict for the first time the covering fraction of gas that gives rise to DLAs, providing unprecedented insight on the structure of this medium. Moreover, for the J0033 field we were able to estimate the previously unknown logN(H /i.pc) of the absorber, logN(H /i.pc) = 20 . 6 ± 0 . 3 , consistent with a DLA. Using the area of the delensed arc, we predict a lower limit of the total H /i.pc gas mass to be ≳ 10 9 M /circledot in both absorbers. We speculate that both absorbers are part of a neutral inner CGM and that, given the large amount of mass, they could be an essential source of fuel for future star formation in the host galaxies.</text>
<section_header_level_1><location><page_13><loc_51><loc_32><loc_65><loc_33></location>Data Availability</section_header_level_1>
<text><location><page_13><loc_51><loc_26><loc_94><loc_31></location>The median metal line profiles obtained for the different regions of the J1527 and J0033 fields are available online respectively in Appendix B ( https://zenodo.org/records/14225814 ) and Appendix C ( https://zenodo.org/records/14225839 ).</text>
<text><location><page_13><loc_51><loc_17><loc_94><loc_25></location>Acknowledgements. We thank both Rongmon Bordoloi and Sara Ellison for respectively providing us with the aperture information used in B22 and the D -index data from Ellison (2006). We also are grateful for Keren Sharon for creating the lens models for both of these systems. Finally, we thank the anonymous referee for a constructive and thorough report. S.L. acknowledges support by FONDECYT grant 1231187. This work is based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere under ESO programme(s) 098.A-0459(A) and 0103.A-0485(B).</text>
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<table>
<location><page_15><loc_21><loc_64><loc_34><loc_91></location>
<caption>Table A.1. Lower sigma clipping thresholds for average S/N of spectrum</caption>
</table>
<section_header_level_1><location><page_15><loc_6><loc_61><loc_32><loc_62></location>Appendix A: Continuum fitting</section_header_level_1>
<text><location><page_15><loc_6><loc_27><loc_49><loc_60></location>In order to continuum fit the spectra from every spaxel of the MUSE cubes and to propagate continuum fitting uncertainties within a Monte Carlo framework, we used a simple sigmaclipping method to automatically determine the continuum of a given spectrum. In summary, the continuum fitting method uses the /a.pc/s.pc/t.pc/r.pc/o.pc/p.pc/y.pc (Astropy Collaboration et al. 2018) sigma clipping algorithm within a window of the spectrum of width 1000Å (800 pixels). The algorithm determines the mean flux ( ¯ F ) within the window while rejecting all pixels with flux outside of the confidence interval ¯ F -n low σ and ¯ F + n high σ (where σ is the standard deviation of all flux measurements within the window). The continuum is built-up by repeating this across the entire spectrum by sliding the window in increments of 100Å (80 pixels). Once completed, a spline is fit to the points generated from the sliding window (centered in the window for each step), and the spline is then smoothed with a top-hat function of width of 300Å to remove strong variations in the flux from regions of the spectra with a low S/N. We note that the continuum of the background sources are free of emission lines and smooth, allowing us to use a large window and improve the statistics used in the sigma-clipping method. While we fix n high = 1 . 0 in the sigma clipping algorithm, we found the optimal choice of n low to distinguish between noise and absorption depends on the average S/N of the spectrum. We therefore adjust n low across each spectrum. Table A.1 shows the optimized n low adopted for a given S/N (per pixel) to exclude absorption in the continuum fitting process.</text>
</document> | [{"title": "ABSTRACT", "content": "H /i.pc-rich absorbers seen within quasar spectra contain the bulk of neutral gas in the Universe. However, the spatial extent of these reservoirs are not extensively studied due to the pencil beam nature of quasar sightlines. Using two giant gravitational arc fields (at redshifts 1.17 and 2.06) as 2D background sources with known strong Mg /i.pc/i.pc absorption observed with the Multi Unit Spectroscopic Explorer integral field spectrograph (IFS), we investigated whether spatially mapped Mg /i.pc/i.pc absorption can predict the presence of strong H /i.pc systems, and determine both the physical extent and H /i.pc mass of the two absorbing systems. We created a simple model of an ensemble of gas clouds in order to simultaneously predict the H /i.pc column density and gas covering fraction of H /i.pc-rich absorbers based on observations of the Mg /i.pc/i.pc rest-frame equivalent width in IFS spaxels. We first test the model on the lensing field with H /i.pc observations already available from the literature, finding that we can recover H /i.pc column densities consistent with the previous estimates (although with large uncertainties). We then use our framework to simultaneously predict the gas covering fraction, H /i.pc column density and total H /i.pc gas mass ( M HI ) for both fields. We find that both of the observed strong systems have a covering fraction of \u2248 70 % and are likely damped Lyman \u03b1 systems (DLAs) with M HI > 10 9 M /circledot . Our model shows that the typical Mg /i.pc/i.pc metrics used in the literature to identify the presence of DLAs are sensitive to the gas covering fraction. However, these Mg /i.pc/i.pc metrics are still sensitive to strong H /i.pc, and can be still applied to absorbers towards gravitational arcs or other spatially extended background sources. Based on our results, we speculate that the two strong absorbers are likely representative of a neutral inner circumgalactic medium and are a significant reservoir of fuel for star formation within the host galaxies. Key words. galaxies: high redshift - galaxies: ISM - quasars: absorption lines", "pages": [1]}, {"title": "Mapping the spatial extent of H /i.pc -rich absorbers using Mg /i.pc/i.pc absorption along gravitational arcs", "content": "Trystyn A. M. Berg 1 , 2 , 3 , /star , Andrea Afruni 2 , 4 , 5 ,/star , C\u00e9dric Ledoux 3 , Sebastian Lopez 2 , Pasquier Noterdaeme 2 , 6 , 7 , Nicolas Tejos 8 , Joaquin Hernandez 9 , Felipe Barrientos 9 , and Evelyn J. Johnston 10 December 11, 2024", "pages": [1]}, {"title": "1. Introduction", "content": "The gas reservoirs of galaxies are critical for regulating galaxy formation and evolution; from the interstellar medium (ISM) where atomic H /i.pc cools into molecular gas to form stars (e.g., Saintonge et al. 2013), out into the galactic halos where the circumgalactic medium (CGM) hosts both accreted intergalactic gas and ejected material from galactic feedback processes which maycontinuetofeed the ISM (see Tumlinson et al. 2017; FaucherGigu\u00e8re & Oh 2023, and references therein). Thus quantifying the properties of these galactic gas reservoirs across cosmic time is a necessity to constrain our understanding of galaxy evolution. Quasar sightlines have proved to be a powerful probe of the gas reservoirs of galaxies, enabling a census of the amount of fuel for future star formation and to quantify the metallicity evolution of these reservoirs across cosmic time. Whilst rare in the Universe, the strongest H /i.pc absorbers known as damped Ly \u03b1 systems (DLAs; H /i.pc column densities of N(H /i.pc) \u2265 2 \u00d7 10 20 cm -2 ; Wolfe et al. 2005) contain the \u2248 80 % of neutral gas in the Universe (Lanzetta et al. 1995; Prochaska et al. 2005; Noterdaeme et al. 2009; Zafar et al. 2013; S\u00e1nchez-Ram\u00edrez et al. 2016). Due to the UV cut-off from Earth's atmosphere, tracing Ly \u03b1 absorption from galaxies below z \u2272 1 . 6 requires expensive spacebased observations. To get access to DLAs in this epoch of the Universe, low-ionization metal species have been used as DLA proxies. The most widely used metal tracer for H /i.pc-rich systems is Mg/i.pc/i.pc as it is accessible down to z \u2248 0 . 1 with ground-based observations (Rao & Turnshek 2000; Turnshek et al. 2015). There are various metrics in the literature for pre-selecting Mg /i.pc/i.pc absorbers at z \u2248 1 as potential DLAs based on the rest-frame equivalent widths (EWs) along quasar sightlines. These metrics include: low-redshift DLAs tend to have EW 2796 \u2265 0 . 3 -0 . 6\u00c5 (Rao & Turnshek 2000; Rao et al. 2006; Turnshek et al. 2015). In this work, we focus on using the threshold of \u2265 0 . 3\u00c5 which tends to select all DLAs at a variety of redshifts (Matejek et al. 2013; Berg et al. 2017; Rao et al. 2017). - the EWratio of Fe /i.pc/i.pc \u03bb 2600/Mg/i.pc/i.pc \u03bb 2796( EW 2600 EW 2796 ) /one.sup . For DLAs, EW 2600 EW 2796 is found to be between 0.5 and 1.0 (Rao et al. 2006). This metric is motivated by the self-shielding nature of DLAs, where the bulk of Fe and Mg will be in the singly-ionized phase. In lower logN(H /i.pc) column density absorbers where ionization effects have an effect, the ionizing field will preferentially ionize Fe outside the Fe /i.pc/i.pc state, leading to a smaller EW 2600 EW 2796 ratio. However, this ratio is also sensitive to the effects of nucleosynthesis (as it traces [ \u03b1 /Fe]) and differential dust depletion (e.g., Dey et al. 2015; Lopez et al. 2020). - the D -index (Ellison 2006). The D -index is computed as EW 2796 normalized by the width of integration bounds for the EW calculation. The minimum D -index for which a system is flagged as a potential DLA is set by the spectral resolution used. While these three metrics are useful in identifying a potential DLA absorber along quasar sightlines, meeting the various cuts on these metrics is no guarantee that the absorption is indeed from a DLA. In Rao et al. (2017) and previous works, the EW 2796 and EW 2600 EW 2796 are typically very successful at finding DLAs at low redshifts. However, these metrics also identify other strong H /i.pc absorbers, such as subDLAs ( 19 \u2264 logN(H /i.pc) < 20 . 3 cm -2 ) at high rates, leading to DLA samples with low purity (Rao et al. 2006; Bouch\u00e9 2008). Furthermore, higher redshift DLAs that are metal-poor or have narrow velocity profiles sometimes do not pass these metrics (Ellison 2006; Berg et al. 2017; Rao et al. 2017). While the D -index metric is designed to take the velocity profile shape into account, the performance of the D -index is heavily dependent on the spectral resolution of the instrument, as the success rate of distinguishing a DLA from a sub-DLA decreases with decreasing spectral resolving power (Ellison 2006). For example, with the spectral resolving of a high resolution spectrograph (with a line spread function full-width half maximum [FWHM]of \u2265 1 . 4 \u00c5), the D -index is very efficient at identifying DLAs with \u2248 90 % purity and a 5% false positive rate (Table 2 in Ellison 2006). However, for lower spectral resolutions, such as what is offered with current state-of-the-art integral field spectrograph facilities such as the Multi Unit Spectroscopic Explorer (MUSE Bacon et al. 2010, with an average spectral FWHM of 2.7\u00c5), the D -index is expected to identify DLAs with a \u2248 84 % success rate and a false-positive rate of \u2248 16 %. Therefore, none of these three metrics on their own can guarantee that a given Mg /i.pc/i.pc absorber is a DLA. Other than in a few specific cases (Fumagalli et al. 2015; Neeleman et al. 2016), very little is known about the morphology and the distribution of gas in DLAs, as quasar sightlines only provide 1D information of the gas. Whilst rare multiple close project pairs of (lensed) quasars (Lopez et al. 1999, 2005; Ellison et al. 2007; Tytler et al. 2009; Chen et al. 2014; Krogager et al. 2018; Rubin et al. 2018b) or faint extended galaxies as background sources (Bordoloi et al. 2014; Rubin et al. 2018a) have been used to assess the physical extent of DLAs or the CGM, these techniques are still limited to 1D skewers of up to a handful of pointings through the same halo. After the advent of sensitive integral field spectrograph like MUSE and KCWI on 8-10m class telescopes, it is now possible to observe faint gravitational arcs as extended background sources to probe the spatial extent of gaseous reservoirs. This so-called gravitational arc tomography has demonstrated the ability to assess how the CGM material is distributed (Lopez et al. 2018, 2020; Mortensen et al. 2021; Tejos et al. 2021; Fernandez-Figueroa et al. 2022; Afruni et al. 2023) and in one case it has been used to put constraints on the mass of H /i.pc gas in a DLA (Bordoloi et al. 2022, further abbreviated as B22). However, based on the modest wavelength range of current telescope facilities combined with the typical redshifts of gravitational arcs, it is more difficult to study H /i.pc directly with gravitational arc tomography compared to neutral metal tracers such as Mg /i.pc/i.pc. As for the pencil-beam quasar sightlines, one possibility is to identify possible DLAs using the Mg /i.pc/i.pc metrics described above, but their accuracy has to date never been tested on extended sources. In this paper, we investigate the use of the three commonlyused Mg /i.pc/i.pc metrics to identify DLAs along extended background sources, aiming to quantify the spatial extent, covering fractions, and H /i.pc gas mass of H /i.pc-rich absorption. We focus on MUSE observations of two gravitational arcs (one of them being the one analyzed in B22) with strong Mg /i.pc/i.pc absorption. We first verify that these commonly-used Mg /i.pc/i.pc metrics can be used for extended background sources. Next, based on the measured Mg /i.pc/i.pc metrics and using a novel model framework to reproduce the B22 results, we investigate the success of using Mg /i.pc/i.pc metrics to identify DLA absorbers along extended background sources. Based on this modeling, we estimate the covering fraction and the H /i.pc gas mass of the two absorbers studied. In Sect. 2, we report the data reduction and processing used to measure equivalent widths; in Sect. 3, we apply the Mg /i.pc/i.pc metrics to assess the extent of potential DLAs in the two fields; in Sect. 4, we describe our model, how we compared the model outputs with the observations and the main results obtained from this comparison; finally, in Sect. 5 we outline the limitations and assumptions of our modeling and we discuss the validity of the usual Mg /i.pc/i.pc metrics, while in Sect. 6 we summarize this work and we report our main conclusions. Throughout this work, we assume a flat \u039b CDMcosmology with H 0 = 70 km s -1 Mpc -1 , \u2126 \u039b = 0 . 7 and \u2126 M = 0 . 3 .", "pages": [1, 2]}, {"title": "2.1. Targets and data reduction", "content": "We observed SGAS J0033+0242 and SGAS J1527+0652 (further referred to as J0033 and J1527, respectively) with the Multi Unit Spectroscopic Explorer (MUSE; Bacon et al. 2010)) at the Very Large Telescope, as part of programs 098.A-0459 and 0103.A-048, respectively (PI Lopez). These fields were originally selected for having indications of intervening Mg /i.pc/i.pc absorption at z \u223c 1 on top of the brightest arc knots (Rigby et al. 2018). The MUSE data revealed strong ( W 0 \u2273 3 \u00c5) Mg /i.pc/i.pc and Fe /i.pc/i.pc systems at redshifts 1.16729 (J0033) and 2.05601 (J1527), making them good DLA candidates (e.g., Rao et al. 2017). In both cases the MUSE wide field mode was used, providing a field of view of 1 ' \u00d7 1 ' and a spatial sampling of 0.2 '' pix -1 . J0033 was observed without adaptive optics mode and the nominal wavelength range ( \u2248 4 700 -9 300 \u00c5), whereas J1527 was observed with the adaptive optics mode and extended wavelength range mode ( 4 600 -9 300 \u00c5). The on-target exposure times were 2 . 14 and 4 . 17 hours, respectively. Figure 1 shows a zoom-in of the MUSE white-light image of the two fields. Both fields show strong gravitational arcs, whilst the J0033 field contains two lensing counterimages (labelled as CI1 and CI2) as well as a background quasar (QSO). We reduced the MUSE data using the ESO MUSE pipeline (v2.6, Weilbacher et al. 2020) in the ESO Recipe Execution Tool (EsoRex) environment (ESO CPL Development Team 2015). Besides pre-processing (bias, flat-field, and vacuum-wavelength calibrations), the flux in each exposure was calibrated using standard star observations from the same nights as the science data, and the sky continuum was measured directly from the science exposures and subtracted off. Residual sky contamination was removed from the stacked cube using the Zurich Atmosphere Purge code (Soto et al. 2016). The final cubes were matched to the WCS of the Hubble Space Telescope (HST) images of J0033 (Fischer et al. 2019) and J1527 (Sharon et al. 2020). These cubes have a point-spread-function of 0.84 '' and 0.78 '' , respectively and a spectral resolving power ranging from R /similarequal 1 770 at 4 800 \u00c5 to R /similarequal 3 590 at 9 300 \u00c5. For further details see Lopez et al. (2024). Table 1 contains the coordinates of the relevant regions we select in this work. These regions represent rectangles, centered at the Right Ascension ( \u03b1 ) and Declination ( \u03b4 ), with a halfwidth of r RA and r Dec in the respective directions, and rotated by the position angle (measured north through east). Apertures A-F for regions in the J1527 field are the apertures used by B22 (R. Bordoloi, private communication). The outlines of the spaxels contained by all of these regions are shown as colored rectangles in Fig. 1. In addition, the last column of Table 1 provides the delensed projected distance ( \u03c1 \u22a5 ) from the center \u03b1 and \u03b4 of each region relative to the center \u03b1 and \u03b4 of the 'Arc' region of the respective field. The delensed projected distances are calculated by delensing the image plane to the so called absorber plane (the plane of the absorbing gas), through parametric lens models built for each of the two fields with the software /l.pc/e.pc/n.pc/s.pc/t.pc/o.pc/o.pc/l.pc (Jullo et al. 2007) and the available HST imaging (we refer for more details to Sharon et al. 2020 for J1527 and to Fischer et al. 2019 for J0033).", "pages": [2, 3, 4]}, {"title": "2.2. Creating combined spectra from individual spaxels", "content": "For each region of interest within our data (Table 1), we generate flux-weighted combined spectra using all the native spaxels within the region. Spectra for spaxels within the region whose signal-to-noise ratio (S/N, next to spectral region with Mg /i.pc/i.pc absorption) was greater than 1.0 were used to form a combined spectrum. In order to match typical 1D analysis of quasars where the spectral trace is summed together, we first fit and normalize the continuum of the spaxel spectra (see Appendix A) and then we weight each spaxel's spectrum by the summed flux of the spectrum at all wavelengths (white-light image /two.sup ). This weighting is analogous to how pixels are combined along the spatial axis of the slit in 1D quasar sightline observations. We note that each region is larger than the point-spread-function of the observations, and thus the spatial binning from this combination process will minimize the overlapping information from combining the native spaxels. In order to obtain an estimate of the error in the continuum fitting and weighting procedure, we use a Monte Carlo approach by repeating this process 1000 times, where in each iteration the flux of each element of the MUSE cube was sampled from a Gaussian distribution centered on the flux with a standard deviation equal to the 1 \u03c3 error spectrum flux. Each of these 1000 spectra are then continuum normalized individually following the method outlined in Appendix A. We then produce a median spectrum by taking the median flux of each pixel, with the 25 th and 75 th percentiles to estimate the error. It is clear from Fig. 1 that the regions (particularly the arc of J0033) contain additional background light in and around the selected region after sky subtraction (possibly from extended emission of nearby sources or the nearby bright star, or intra-cluster light). Because of the additional signal, pure-background spaxels can have a S/N \u2265 1 and thus are selected and included when creating the combined spectra as described above. By including this background, there is an additional flux within absorption lines that leads to lower EWs and potentially false signatures of emission or partial coverage. In order to remove this additional background prior to making the combined spectra, we first isolate the spaxels (including those with S/N < 1 ) which only contain this additional background flux. This is done by sigma-clipping out all spaxels from the entire region (Table 1) whose white-light image flux is above +1 \u03c3 the mean flux. The spectra of these background-only spaxels are then median combined into a background spectrum that is subtracted from all the spaxels of that region in order to correct for the additional signal. We note that the background signal we observe appears uniformly distributed spatially within the respective regions, and the median combined background spectrum is consistent with these values. Based on this information, we note that the non-uniform light from the galaxy nearby the J1527 arc likely adds additional flux in the spectrum and is not fully subtracted from this method. While we exclude spaxels that contain the galaxy's spectrum from the analysis (i.e., those within the Galaxy region for field J1527 in Table 1), there may still be scattered light issues that add additional flux to the spectrum, particularly within the absorption trough of the key Ly series (KCWI data) and metal lines (MUSE data). Figs B.1 - B.6 in Appendix B contain the median metal line profiles from the Monte Carlo analysis for apertures AF (respectively) in the field J1527 whilst Figs. C.1 - C.4 in Appendix C show the median velocity profiles for the regions within the J0033 field. Both appendices are available online.", "pages": [4]}, {"title": "2.3. Measuring rest-frame equivalent widths", "content": "EW measurements are made by integrating the continuumnormalized spectra (see Appendix A) between the velocity limits of the absorption feature (Table 2; where the zero-velocity is defined at the redshift of the strongest Mg /i.pc/i.pc absorption, z abs ). The velocity limits are determined by visually inspecting the absorption of all spaxels and selecting the limits where the strongest line reaches the continuum level. The same velocity limits are adopted for all metal absorption lines of the same system unless there is significant blending in a weak portion absorption profile (e.g., Ca /i.pc/i.pc \u03bb 3934 in Fig. C.1 in Appendix C). We adopt the EW derived from the median of the 1000 Monte Carlo spectra, with errors on the EW coming from the the 25 th and 75 th percentile spectra. Werequire that the rest-frame equivalent widths be measured at > 2 \u03c3 (based on the S/N of the median spectrum) to be considered as detected. Otherwise, 2 \u03c3 upper limits are adopted. Lines that are strongly blended use the measured equivalent width, but are flagged as upper limits. The median equivalent width measurements for the B22 apertures and full arc in the J1527 field as well as the J0033 regions are provided in Tables 3 and 4, respectively. The same equivalent width information is also provided online in Appendix B (J1527) and Appendix C (J0033).", "pages": [4]}, {"title": "3. DLA identification using Mg /i.pc/i.pc metrics", "content": "3.1. Verifying Mg /i.pc/i.pc DLA metrics on extended sources within the J1527 field The Mg /i.pc/i.pc DLA identification metrics (i.e., EW 2796 , EW 2600 EW 2796 , and D -index; defined in the introduction) have been defined on quasar sightlines. It remains untested whether these three metrics can Notes: a logN(H /i.pc) taken from B22. also be used for identifying DLAs in multiplexed data of extended sources such as gravitational arcs. We take advantage of testing these metrics towards the J1527 field, where existing KCWI observations have demonstrated that all six apertures (apertures A-F; Table 1) host a DLA whilst our MUSE data complement the KCWI data by providing coverage of several key metal lines, including the Mg /i.pc/i.pc doublet. Using these six aperture regions, we created combined spectra following the method outlined in Sect. 2.2. The EWs of the key metal lines are tabulated in Table 3 while the resulting DLA metrics are provided in Table 5 along with the H /i.pc column densities from B22. For all six B22 apertures (which all host a DLA based on the logNH /i.pc measured in B22) and the full arc region, the EW 2796 and EW 2600 EW 2796 metrics tabulated in Table 5 are consistent with the apertures containing a DLA using the standard quasar metric thresholds, suggesting that these two metrics from Rao & Turnshek (2000) can be used to identify DLAs. Based on the MUSE instrumental FWHM (2.54 \u00c5) at the wavelength of Mg /i.pc/i.pc 2796\u00c5, the expected D -index threshold for selecting a DLA is \u2248 3 . 6 (interpolating Table 2 of Ellison 2006). This D -index threshold would only select apertures A-C as hosting DLAs. However, we caution that at this low spectral resolution, the success rate of identifying DLAs is expected to be \u2264 84 per cent (Ellison 2006); the observed success rate of the D -index for these six apertures is 33%. All entries in Table 5 that would be flagged as a DLA based on the respective Mg /i.pc/i.pc metric have been bolded. Combining all the spaxels together from full arc in place of the individual six B22 apertures, we obtain similar results - both the EW 2796 and EW 2600 EW 2796 metrics are consistent with hosting DLAs, whilst the D -index ( 3 . 2 \u00b1 0 . 1 ) is just below the threshold for selecting a DLA. We point out that the value of all three metrics for the combined full arc spectrum are effectively equal to the average of the values derived for the six B22 apertures. As mentioned in Sect. 2.2, there is potential contamination in all apertures from galaxy light close in projection to the arc. As we are using weighting from the white-light image, the spaxels that contain more galaxy light than arc light are favored, and artificially weaken the Mg /i.pc/i.pc absorption in the combined spectrum by adding flux within the absorption trough. We therefore caution that EW measurements in the apertures (Table 3; particularly aperture F) are likely lower limits to the true value arising from pure Mg /i.pc/i.pc absorption. This may partially explain why the D -index is just below the threshold for selecting a DLA and is not as successful at identifying the DLA absorption in this particular field. It is possible that this contamination issue also affects the determination of the H /i.pc column density and the presence of partial coverage reported by B22.", "pages": [4, 5]}, {"title": "3.2. Applying Mg /i.pc/i.pc metrics to the J0033 field", "content": "Following the same methodology for the J1527 field, we created combined spaxel spectra for all four regions and measured the EWs for the same metal lines (Table 4) and measured the Mg /i.pc/i.pc metrics for DLA identification (Table 6). Bolded entries in Table 6 represent regions where the given Mg /i.pc/i.pc metric would be flagged as a DLA. In the combined spectra from three of the four background sources of the lensing system towards J0033 (i.e., the Arc, CI 1, and CI 2 regions in Table 1), there is very strong Mg /i.pc/i.pc \u03bb 2796 absorption (rest-frame equivalent widths > 2 \u00c5 ; Figs. C.1 - C.3 in Appendix C) that suggests the presence of an H /i.pc-rich absorber based on the results from J1527. Unfortunately the combined spectrum for the QSO region has poor S/N (Fig. C.4 in Appendix C), so we exclude this background source from our analysis. What makes this particular field interesting is that large physical separations are probed by the arc and two counter images (separated by \u2248 14 -50 kpc), compared to the \u2248 10 kpc scales measured across the J1527 arc (see Table 1). Despite not having access to the Lyman series to measure the H /i.pc column density directly, the J0033 field provides an opportunity to study the spatial extent of a strong Mg /i.pc/i.pc absorber that is likely a DLA (or sub-DLA) based on the very strong Mg /i.pc/i.pc absorption ( EW 2796 \u2273 2 \u00c5). In Figs. C.1 - C.3 in Appendix C, we note there is no obvious difference in shape or centroid velocity of the Mg /i.pc/i.pc \u03bb 2796 absorption profile between the four regions. The kinematics of this particular absorber will be discussed in more detail in a future paper (Ledoux et al., in prep.). While the spectral resolution of MUSE is insufficient to determine if sub-components of the velocity profile are in agreement between all three regions, the centroids are consistent within \u2248 50 km s -1 of each other (i.e., well within the spectral FWHM of MUSE), and (apart from the QSO region, which has low S/N) all regions appear to show a velocity profile with potentially at least two components. This would suggest that the stronger absorber has the same bulk kinematics across the entire \u2248 50 kpc extent traced by the background sources (Table 1).", "pages": [5, 6]}, {"title": "3.3. Mapping Mg /i.pc/i.pc DLA metrics with MUSE spaxels", "content": "In order to study the physical extent of the two strong Mg /i.pc/i.pc absorbers, Figs. 2 and 3 show spaxels maps of each of these three metrics in the fields of J0033 and J1527 respectively. The spectra generated for each pixel (corresponding to a native spaxel) in these maps are created using the same method outlined in Sect. 2.2, and are combined using nine spaxels within a 3 \u00d7 3 spaxel grid centered at each native pixel. This procedure effectively corresponds to a 0.6 '' \u00d7 0.6 '' smoothing of the original map, averaging over spaxels contained within the point spread function of the observations ( \u2248 0 . 8 '' ). This smoothing thus minimizes effects from overlapping information within the native pixel scale. Despite this smoothing, we point out that there are only \u2248 30 and \u2248 44 spatially-independent measurements within the respective J0033 and J1527 fields, which we discuss in more detail in Sect. 5.1. We note that there is a potential correlation between each of the three metrics and the measured S/N near the Mg /i.pc/i.pc doublet for spaxels with S/N < 1 and an EW detection significance < 2 \u03c3 in both fields. We thus proceed by implementing a quality control cut, and only analyze spaxels with a S/N \u2265 1 near the Mg /i.pc/i.pc doublet and a EW 2796 detected at \u2265 2 \u03c3 significance. Table 7 provides the percentage of spaxels in Figs. 2 and 3 that pass the various Mg /i.pc/i.pc metric criterion typically used to flag an absorber as a potential DLA. In each of the three regions of J0033 and the full arc of J1527, 100% of spaxels that meet our quality control cut pass the EW 2796 \u2265 0 . 3 \u00c5. This would suggest that this candidate DLA is completely extended over the background source. However combining with other metrics, (i) 44-89% of spaxels passed both the EW 2796 and EW 2600 EW 2796 criteria for potential DLA absorption, and (ii) 12%-34% of spaxels simultaneously meet the D -index, EW 2796 and EW 2600 EW 2796 criteria for potential DLA absorption. We note that, of the 12% of spaxels that pass all three metric criteria within the CI2 region of the J0033 field, all spaxels are found on the outer edge of the region where the S/N is lowest (S/N \u2248 2 ). However, most of the spaxels that pass the three Mg /i.pc/i.pc metric criteria for the arc in both fields and CI1 region in J0033 are more evenly distributed across the region. Assuming a combination of all three of the Mg /i.pc/i.pc metrics is a definitive tracer for H /i.pc-rich absorption, the lower limit on the overall extent of H /i.pc gas across all three regions of the J0033 field is \u2273 30 %, and \u2273 16 % for the full arc towards J1527 (prior to accounting for the predicted false positive and success rates of each of the three metrics; e.g., as seen for the D -index in Ellison 2006).", "pages": [6, 7]}, {"title": "4. Modeling the fields", "content": "While in the previous sections we analyzed the direct observational properties of the strong Mg /i.pc/i.pc absorbers detected in the J1527 and J0033 fields, in the following we use a novel approach that makes use of Bayesian inference and parametric physical models to interpret the detected strong absorption. The main goals of this analysis are to i) infer the physical properties of the absorbers and ii) test the robustness of Mg /i.pc/i.pc metrics and whether they can be applicable on extended sources (see Sect. 5.2). In our modeling procedure we assume that the two Mg /i.pc/i.pc absorbing systems are physically represented by a population of cool clouds, from which we extract mock observations (specifically EWs and D -indices that one would extract from an ensemble of spaxels) that can be directly compared to the real data. These models are idealized and, while they are useful to make conclusions on the two dimensional properties of the absorbing gas (e.g., covering fraction), they do not have the pretense of depicting a realistic three dimensional cloud distribution. In Sect. 5.1 we discuss more in detail the various assumptions and limitations of the models.", "pages": [7, 8]}, {"title": "4.1.1. Procedure for a single spaxel", "content": "We first assume that the flux detected by a spaxel is directly related to the gas covering fraction C frac within that spaxel (hence the fraction of the background source area covered by the gas) through the formula: Here, F bg is the background continuum flux, which we assume to be normalized to 1, while F los is the flux that would be detected by a single point-source line of sight that intercepts the absorbing gas. For Eq. (1) to be valid, we assume that the background flux is constant within the spaxel and that the absorbing gas, where present, has exactly the same properties within the same spaxel. This is clearly an approximation, which however should not significantly impact the general results of this paper. We emphasize that, in this paper, we define the covering fraction as the fraction of the area of the background source area (such as a spaxel) that is covered by foreground gas. This is different to a more observationally-driven definition with the fraction of spaxels across an extended source that contain the same absorption feature, such as a DLA. The MUSE spaxels have a size of (0.2 '' \u00d7 0.2 '' ), resulting in typical physical sizes in both fields of 0.07 and 1.4 kpc along a side (after delensing). The output flux of the model is anyway independent of the spaxel size itself and is only affected by the covering fraction C frac (see Eq. 1). Once a covering fraction is assumed, to calculate F spaxel one needs to calculate the value of F los = F bg exp( -\u03c4 ) , where the optical depth \u03c4 is defined as (see e.g., Liang & Kravtsov 2017): where N is the gas column density, \u03c3 0 is the cross section, f osc is the oscillator strength and \u03a6( \u03bb | b, v ) is the Voigt profile function, which depends on the wavelength \u03bb , on the Doppler parameter b and on the velocity v at which the transition takes place. We assume that the absorption is due to the superposition of n clouds along the line of sight, each of them contributing to the total optical depth. We first assume that each of these clouds has a neutral hydrogen column density N HI , c , so that the total HI column density along the point-source line of sight would be nN HI , c . We then define the spaxel HI column density N HI , spaxel as nN HI , c \u00d7 C frac , given that the spaxel is only partially covered by gas. Assuming a value for N HI , spaxel , one can then solve for the number of clouds: The goal of this modeling is to find which HI column densities and covering fractions are needed to reproduce the observed Mg /i.pc/i.pc EWs and D -indices. To this purpose, we further assume that the clouds have a metallicity Z c and an intrinsic volumetric density n c /three.sup . We then use the photo-ionization code CLOUDY (Ferland et al. 2013) to infer the Mg /i.pc/i.pc column densities of the single clouds, assuming an ionizing flux given by the extragalactic UV background from Haardt & Madau (2012) at the redshift of the absorbers (see Sect. 2). At this high column densities most of the medium will be self-shielded and not affected by ionization. Models that consider completely neutral gas lead us to the same results and conclusions presented here (we discuss this in more detail in Sect. 5.1). We then assign to each of the n clouds a lineof-sight velocity extracted from a Gaussian distribution centered in zero and with a width equal to a velocity dispersion \u03c3 los and a Doppler parameter b equal to the sum in quadrature of a thermal ( b th \u223c 4 km s -1 , assuming a temperature T = 2 \u00d7 10 4 K) and a turbulent component, with b turb = 10 km s -1 (we discuss different values of b turb in Sect. 5.1). As we discuss below, \u03c3 los is one of the free parameters of our model and it determines the (simplistic) bulk kinematics of the cloud population and, in turn, the strength of the absorption. With all the above ingredients, one can solve Eqs. (2) and (1) to finally obtain the spaxel flux F spaxel , specifically for the Mg /i.pc/i.pc absorption lines. As a final step, we convolve the spectrum with a Gaussian kernel assuming that the instrumental profile has a resolution (FWHM) of 150 km s -1 , which roughly resembles the resolution of MUSE at the redshift of the two absorbers. We then add random Gaussian noise in the spectrum: the value of the final flux at each wavelength is extracted from a Gaussian centered on the original model flux value, with a width equal to the flux value divided by the observed signal-to-noise in the MUSE data. Once this procedure is complete, we can extract the Mg /i.pc/i.pc metrics of the spaxel, specifically the EW 2796 and the D -index. We point out that we do not include the ratio EW 2600 EW 2796 in this part of the analysis: this ratio might be dependent on our assumption of photo-ionization (which we discuss in Sect. 5.1) and is additionally also affected by dust and by chemical enrichment, which are not explicitly taken into account in our framework. The steps explained above are summarized in the schematics of Fig. 4.", "pages": [8, 9]}, {"title": "4.1.2. Prediction from an ensemble of spaxels and comparison with the data", "content": "To compare with the observations outlined in the previous sections, we need to apply the procedure above to multiple spaxels. The Mg /i.pc/i.pc metrics extracted from this spaxel ensemble are then directly comparable to our data. In the following, we assume that for each model realization and across the same field all the clouds of our synthetic populations have the same metallicities and densities, with velocities extracted from the same Gaussian velocity distribution. However, in Sect. 3.3 we have seen that the observed properties of the fields vary across them and that for example a fraction of the spaxels do not pass the three metrics for potential DLA absorption. We therefore assume that the differences across the spaxels are due to differences in the spaxel covering fraction and in the total HI column density (i.e., different spaxels detect different numbers of clouds). We introduce the parameters N HI , spaxel , which is the mean observed neutral hydrogen column density of the spaxels (the column density of each spaxel is then drawn randomly from a range that goes from log( N HI , spaxel ) -0 . 3 to log( N HI , spaxel )+0 . 3 ) and C frac , which is the center of a normal distribution with a standard deviation equal to 0.2 and truncated at 0.1 and 1, from which the value of the covering fraction C frac of each spaxel is extracted. A single model realization, hence a distribution of EW 2796 and D -indices (see Sect. 4.1.1), is uniquely defined by the choice of the five free parameters N HI , spaxel , C frac , Z c , N HI , c and \u03c3 los . Each distribution is composed of 200 values, a number consistent with the amount of spaxels in J1527 and J0033 (see Section 3.3). The idea is then to compare these distributions with the observed ones to find which choice of parameters better reproduce our data. In order to do this, we first select the spaxels in the data and in the model realizations using the same quality control cuts as used for the observations (i.e., S/N > 1 and an EW 2796 detection significance > 2 \u03c3 /four.sup ). We then quantitatively assess the consistency between the model predictions and the MUSE observations by performing a Kolmogorov-Smirnov (KS) test on the distributions of EWs and D -indices. Finally, we define our likelihood as: where p EW and p D are the p-values /five.sup obtained from the KS tests on respectively the EW and D -index distributions. We finally use the likelihood expressed in Eq. (4) to perform two Bayesian analyses on the J1527 and the J0033 fields, whose results are presented in the next section. We point out that the results of our analysis (i.e., the recovered best-fit values of the five free parameters) are independent from any lensing model, as we are not assuming a priori the size and shape of the two fields or of the single spaxels.", "pages": [9]}, {"title": "4.2. Model results", "content": "In this section, we present the results of the Bayesian analysis performed to compare our models with our two observational fields, as explained in the previous section. We adopt the nested sampling method (Skilling 2004, 2006), using the /d.pc/y.pc/n.pc/e.pc/s.pc/t.pc/y.pc python package (Speagle 2020; Koposov et al. 2022). For both fields, we use flat priors for the five free parameters: 19 . 8 < log N HI , spaxel < 21 . 2 , 50 < \u03c3 los / (km s -1 ) < 200 , 0 . 1 < C frac < 1 , 18 < log N HI , c < 19 . 5 cm -2 , -3 < log Z c < 0 .", "pages": [9]}, {"title": "4.2.1. J1527", "content": "The results of the Bayesian analysis performed on the J1527 field are shown in the left-hand side of Fig. 5. We can first look at the intrinsic properties of the gas clouds, \u03c3 los , N HI , c and Z c . As expected, these three quantities are slightly degenerate with each other, as they all contribute to the strength of the absorption (Eq. 2). At fixed N HI , spaxel a lower value of N HI , c simply implies a larger number of clouds along the line of sight (Eq. 3). At the same time, our photo-ionization models predict that the ratio N MgII , c /N HI , c decreases with increasing N HI , c , so that higher N HI , c imply higher \u03c3 los and/or higher Z c , explaining these two degeneracies. The velocity dispersion of the clouds seems rather well constrained and it is around 100 km s -1 (we discuss this further in Sect. 5.1), while both the cloud column density and metallicity have very large uncertainties, implying that the choice of these two parameters does not strongly affect our outputs. It is interesting to look at the recovered values of N HI , spaxel : we find that the posterior distribution is in agreement with the previous estimates of the H /i.pc column densities from B22 (see Table 5), shown as a black vertical line in Fig. 5 /six.sup . However this quantity is very poorly constrained with rather large uncertainties, due mainly to strong degeneracies with both the column density and the metallicity of the clouds. We conclude that we recover the main finding of B22 that this absorber is a DLA, but also that our predictions of the total HI column density are very uncertain, as can be expected for a method that is based exclusively on a comparison with Mg /i.pc/i.pc absorption lines. This also shows the limitations of the usual Mg /i.pc/i.pc metrics (which are much more simplistic than our modeling) in selecting DLAs. The final and most important result is given by the covering fraction C frac : our analysis prefers C frac < 1 , with very tight constraints that are not degenerate with the other free parameters of the model. This result is consistent with our assumption that the DLA is composed by clouds and shows that these absorbers do not cover the entirety of the area traced by the gravitational arc. This is the first estimate of the structure of a DLA, which appears to be patchy on the scale of a MUSE spaxel (0.2 '' \u00d7 0.2 '' ).", "pages": [9]}, {"title": "4.2.2. J0033", "content": "The right-hand side of Fig. 5 shows the posterior distributions for the five free parameters in our analysis for the J0033 field. We can note that the intrinsic cloud kinematics, column density and metallicity ( \u03c3 los , N HI , c and Z c ) have similar values and trends with respect to the J1527 field, although in this case they seem to have slightly lower uncertainties. The total neutral hydrogen column density, with a median value of the posterior distribution of about 10 20 . 6 cm -2 , seems to point to the presence of a DLA even in this field. The posterior distribution of this parameter has however very large uncertainties (as for the J1527 case), hence this result will need further confirmation. Despite this, with our analysis we have a hint, more robust with respect to the usual Mg /i.pc/i.pc metrics utilized in the literature, that this absorber is likely an extended DLA. Very interestingly, the covering fraction of this extended absorber is again very well constrained and predicted to be lower than 1, with values that are even lower (but consistent within the uncertainties) with respect to the DLA in the J1527 field. This result seems hence to indicate that these two strong absorbers have similar properties and especially similar covering fractions, the most robust result of our Bayesian analysis. In both cases the gas distribution appears patchy, not covering completely the background extended source.", "pages": [10]}, {"title": "5.1. Assumptions and limitations of model and data", "content": "The modeling framework presented in Sect. 4 is idealized and relies on a number of assumptions, some of which we already discussed above. In the following we summarize and discuss the most important of these assumptions. First, we stress that the model is simplistic and it is not meant to represent a full 3D configuration of the absorbing gas. For this reason, we do not formulate predictions for the H /i.pc and Mg /i.pc/i.pc maps to directly compare with the real observational data (see e.g., Figs. 2 and 3). We instead assume that the clouds have the same kinematics, metallicity and column densities everywhere across the fields, not taking into account of possible (likely) inhomogeneities, which we only attribute to different spaxels detecting different numbers of clouds and having therefore different covering fractions and total column densities. While this assumption is clearly an oversimplification, the results of Sect. 4.2 show how, by simply comparing the total distributions of observational diagnostics like the EW 2796 and the D -index, we can infer the general properties of the absorbing gas. Another important approximation of the model is related to the cloud kinematics: we assume that the line-of-sight velocity distribution of the clouds is described by a Gaussian profile, which is most likely inaccurate. However, understanding the actual kinematics and dynamics of the absorbing gas is outside the scope of this work and the value of \u03c3 los is simply used to determine the strength of the absorption: a non Gaussian kinematics would not strongly affect the outputs of our model (and therefore our main results and conclusions), as long as the average line-ofsight velocity dispersion is similar to what we found here. We also assume that the Gaussian distribution is centered at 0 km s -1 in all the spaxels. This is justified by an ongoing kinematic analysis (Ledoux et al., in prep.) that shows that, at least for J0033, the bulk kinematics of the absorber is the same across the entire field. A variation of this center for different spaxels would anyway not impact our results, given that the strength of the absorption for each spaxel does not depend on the exact position of the velocity centroid but only on the velocity dispersion. However, we caution that the recovered value of \u03c3 los depends slightly on the choice of the Doppler parameter b , dominated by the turbulent component b turb , which we fixed to 10 km s -1 in our fiducial model. We find that higher or lower values of b turb , while leaving the general conclusions of this study unchanged, would lead to respectively lower and higher values of \u03c3 los . Finally, we inferred the column densities of Mg /i.pc/i.pc using photo-ionized models (Ferland et al. 2013), while most of this gas is self-shielded and therefore in its neutral state. To investigate the impact of such choice on our results, we performed additional Bayesian analyses on the two fields assuming that the gas is entirely neutral and that the magnesium is all in the Mg /i.pc/i.pc state. The Mg /i.pc/i.pc cloud column density can therefore simply be obtained by assuming the magnesium solar abundance log(Mg / H) /circledot = -4 . 47 (Asplund et al. 2009) and a value for the metallicity Z c . The results of such model are perfectly consistent, for both fields, with those shown in Fig. 5, so we conclude that the photo-ionization assumption does not have an impact on our main findings, as expected given the high column densities of the absorbers (e.g., Dey et al. 2015). We note that the photo-ionization assumption might have an influence on EW 2600 EW 2796 , but we decided to exclude this diagnostic from our likelihood (Eq. 4). Models where we also include a comparison between the observed and the predicted EW 2600 EW 2796 distributions point towards the clouds having a low metallicity (median log Z c \u2248 -2 . 6 ) and high HI column densities ( N HI , c \u2273 10 19 cm -2 ), but we choose to discard them, considering the uncertainties on the EW 2600 EW 2796 diagnostics (see Sect. 4.1). Interestingly, even in this case the recovered covering fraction C frac remains consistent with the fiducial values reported in Sect. 4.2. The limitations of the data used may also impact the results of our modeling. As already mentioned in Sect. 3, our data might be contaminated by the galaxy light, resulting in an artificially weaker Mg /i.pc/i.pc absorption in our spectra. In the modeling, higher EW 2796 and D -indices could lead to slightly higher covering fractions with respect to what we obtained in Sect. 4.2. Moreover, due to the seeing, the MUSE native spaxels (even after the smoothing described in Sect. 3.3) are not spatially independent. The overall effect of the seeing is to smooth the properties of adjacent spaxels, so that we can expect that the overall impact on the distributions of EW 2796 and D -indices is to make them narrower than what they would originally be. Given that our model predicts average quantities of the absorbing material across the two fields, using wider distributions would likely not change significantly our findings. We note that the distribution of measured EW and D -indices from spatially independent spaxels (i.e., spaxels separated by the point spread function of the observations, every 3.5 spaxels, which correspond to \u2248 30 and \u2248 44 spatially-independent measurements within the respective J0033 and J1527 fields) are consistent with the full spaxel distribution (with KS test p-values of p \u2248 0 . 95 for J1527 and p \u2248 0 . 62 for J0033). We therefore opt to use all the spaxels to improve the sampling of the EW and D -index distributions without significantly impacting our modeling analysis.", "pages": [10, 11]}, {"title": "5.2. Evaluating the success of DLA metrics in individual spaxels", "content": "Figure 6 encapsulates the dependence of EW 2796 and D -index on logN(H /i.pc) in the literature measurements taken from 1D spectra of quasar sightlines (pink errorbars; Ellison 2006; Rao et al. 2006; Berg et al. 2017, 2021) and the best-fit models (colored symbols, obtained by sampling the posterior distributions of \u03c3 los , C frac , N HI , c , Z c within the 32nd and 68th percentiles) for both J1527 (left panels) and J0033 (right panels). The colored symbols represent a range of logN(H /i.pc) that varies from the subDLA to the DLA regime. These distributions therefore are not intended to represent directly the distributions of the observed properties, but show instead the results of specific models as a function of the HI column density. For reference, the distribution of observed EW 2796 and D -index for spaxels within the respective fields (i.e., spaxels in Figs. 2 and 3 with S/N \u2265 2 and EW 2796 measured at > 2 \u03c3 significance) are denoted by the hollow violins, and are centered on the median logN(H /i.pc) predicted by the models. The general agreement of the violins with the model points in the top two rows of Fig. 6 is a result of using these two distributions to constrain the models (i.e., Eq. 4). The model is in general able to recover the bulk trends with logN(H /i.pc) seen in the 1D quasar sightlines for the two metrics in Fig. 6, but there are some notable differences. First, the normalization of the D -indices (bottom row) is different between models and data, but this is expected because of the lower spectral resolution of our MUSE data with respect to the literature data (see also below). Second, the EW 2796 (top row) tends to flatten in the 1D sightlines (in pink) for logN(H /i.pc) \u2273 20 . 3 cm -2 , while it increases monotonically to larger values in the model, especially for J1527 (left column). This is due to the fact that these models are calibrated to our tomographic observations, which exhibit EW 2796 significantly larger (see the position of the hollow violins) than the median values of the 1D data. To reproduce these high values, the EW 2796 needs to increase at larger HI column densities, instead of reaching a plateau. The third difference is instead given by the EW 2796 scatter: whilst the scatter in the models appears similar (if not lower) to that of the 1D observations for sub-DLA HI column densities, the model scatter is instead much larger for logN(H /i.pc) \u2273 20 . 3 cm -2 . A larger scatter is in principle not expected, given that in the modeling all clouds have the same metallicity (and density) within a single absorber (Sect. 4.1), while the mass-metallicity relation of absorbers (e.g., Ledoux et al. 2006; Neeleman et al. 2013; Christensen et al. 2014) is expected to influence the width of the observed Mg /i.pc/i.pc absorption lines (e.g., Ellison 2006; Rao et al. 2006; Bouch\u00e9 2008; Berg et al. 2017) and thus to increase the scatter of the respective Mg /i.pc/i.pc metrics. The large scatter in the model can be explained by the covering fraction, and a clear vertical gradient of covering fraction as a function of EW 2796 (top row) or D -index (bottom row) for a given logN(H /i.pc) is indeed visible in Fig. 6. As a result, it is clear that using the same Mg/i.pc/i.pc metric thresholds from 1D quasar sightlines on 2D extended spaxels may not be so straightforward to interpret. Given the small pencil-beam nature of quasar sightlines, the covering fraction across the quasar would only vary if the angular sizes of the gas clouds comprising the absorber are much smaller than the angular size of the continuum emission from the background quasar, which seems unlikely. However covering fraction should impact extended sources, where the coherence length of Mg /i.pc/i.pc absorption is expected to be \u223c 5 kpc (Afruni et al. 2023) with cloud sizes of \u223c 0 . 5 kpc (for both low, e.g., Mg /i.pc/i.pc, and intermediate, e.g., C /i.pc/v.pc, ionization gas; Faerman & Werk 2023; Lopez et al. 2024). For reference, as already mentioned, the range in spaxel sizes in both fields is 0.07 and 1.4 kpc along a side (after delensing). Another way to assess the success of the DLA Mg /i.pc/i.pc metrics is to look at the fraction of the models that produce a DLA (f DLA ) for a given metric threshold. Table 8 provides the required threshold for EW 2796 and D -index to produce a given f DLA for the models of both fields. We note that f DLA is \u2248 20 % and \u2248 40 % using the typically adopted EW 2796 thresholds of 0.3\u00c5 and 0.6\u00c5 in the literature, whilst f DLA \u2248 90 %(i.e., similar to the purity found in Ellison 2006) using the D -index cut expected for MUSE-like resolution ( \u2248 3 . 6 ). The EW 2796 threshold obtained by the simulations is within a factor of two of the equivalent thresholds required to reproduce the same f DLA seen in the high redshift ( 2 < z < 4 . 5 ) XQ-100 survey (Berg et al. 2017, 2021), while the lower-redshift ( z < 1 . 65 ) sample from Rao et al. (2006) requires a consistently high EW 2796 threshold of \u2273 2 . 7 in order to reproduce the same f DLA . Whilst the scatter in the relations of Fig. 6 is larger in the modeled 2D spaxels in comparison to the 1D literature quasar sightlines, it appears that the purity of the sample (i.e., f DLA ) for a given EW 2796 threshold is roughly the same between the two types of background sources. The fact that our modeling is able to reproduce the observed trends and statistics for pencil-beam quasar sightlines validates that we are able to estimate the H /i.pc column density (albeit with large uncertainties) and the gas covering fraction based on Mg /i.pc/i.pc absorption towards extended background sources.", "pages": [11, 12]}, {"title": "5.3. The nature of the absorbers", "content": "The results of Sect. 4 can be used to understand what is the nature of the two strong absorbers detected in the J1527 and J0033 fields and what is their potential role in the evolution of the galaxies with which they are associated. Looking at the projected distances in the absorber planes reported in Table 1, we note that the two strong absorbers (both potentially DLAs) extend up to distances of a few tens of kpc. Compared to the typical sizes of galactic discs at redshift z > 1 (e.g., Shibuya et al. 2015), these absorbers seem to be significantly more extended. This indicates that this gas is likely more representative of the CGM, rather than of high-redshift rotating interstellar gas discs (see Neeleman et al. 2020; Kaur et al. 2024). By summing the areas in the absorber plane (using the same lens models mentioned in Sect. 2) of all the MUSE spaxels, we can calculate the total area subtended by the extended source in the J1527 and J0033 fields, respectively 353 kpc 2 and 350 kpc 2 . Assuming a mean HI column density across the field equal to the median value of the two posterior distributions of N HI , spaxel (see Fig. 5) /seven.sup we then obtain M HI \u2273 2 \u00d7 10 9 M /circledot for J1527, consistent with the previous results of B22, and M HI \u2273 1 . 2 \u00d7 10 9 M /circledot for J0033. We stress that these two values represent strictly lower limits, given that the extension of the absorber could be larger than the extension of the background arc. Our picture is consistent with the recent results of Stern et al. (2021), who using the FIRE simulations find that a predominantly neutral inner CGM, potentially giving rise to DLAs, is present in halos at z > 1 with virial masses of about 10 11 M /circledot , where the cooling time of the hot gas is shorter than the free-fall time of the system. These authors find that in these halos the neutral gas can extend with large covering fractions ( 0 . 5 < C frac < 1 . 0 ) up to distances of tens of kpc from the host galaxy, in agreement with our findings. Moreover, in their simulations the cold gas is on average inflowing towards the galaxy (even though with relatively low infall velocities). If this scenario is correct, and we are detecting such material with our observations, this large amount of cold, neutral gas ( M HI \u2273 10 9 M /circledot ) would account for a substantial fraction of the baryons within the halo and would therefore represent the main source of fuel for the future star formation within the galaxy. We conclude by emphasizing that, except for the above speculations, here we do not attempt to draw a full dynamical picture for this gas, especially since we can not determine which galaxies are responsible for these absorbers: for J1527, the MUSE spectral coverage does not allow us to detect galaxies in emission at the redshift of the absorber ( z abs /similarequal 2 . 06 ), while in the J0033 field we detect multiple galaxies (at z abs /similarequal 1 . 17 ) whose CGM could contribute to the absorption (Ledoux et al., in prep.). Disentangling whether this gas is a signature of galactic winds (e.g., Schroetter et al. 2019; Schneider et al. 2020; Fernandez-Figueroa et al. 2022), recycling material, or cold cosmological accretion (e.g., van de Voort et al. 2011; Bouch\u00e9 et al. 2013; Theuns 2021) is therefore outside the scope of the present study and is left for future work.", "pages": [12, 13]}, {"title": "6. Summary and conclusions", "content": "In this paper, we analyzed two gravitational arc fields with known strong Mg /i.pc/i.pc absorption in order to predict the presence of a H /i.pcrich absorber (such as a DLA) and determine the spatial extent of the absorber. For the first time, we created 2D maps of strong Mg/i.pc/i.pc absorbers using three Mg /i.pc/i.pc metrics ( EW 2796 , D -index, and EW 2600 EW 2796 ) typically used in the literature for identifying DLAs (Rao & Turnshek 2000; Ellison 2006). These maps suggest that both gravitational arcs probe subDLAs or DLAs, and are extended over areas of \u2248 350 kpc 2 . In particular, one system (J0033) shows \u2264 50 km s -1 variations in the redshift of the observed Mg/i.pc/i.pc absorption across \u2248 50 kpc of separation between sources, suggesting the same bulk gas is extended over such very large areas. The Mg /i.pc/i.pc metrics for the other field (J1527) suggest the presence of a DLA, which has been confirmed by KCWI data of the same system (Bordoloi et al. 2022). These Mg /i.pc/i.pc metrics can be successful for implying the strength of H /i.pc absorption along extended emission sources. In order to quantify the success of these metrics, as well as provide an estimate of the H /i.pc column density towards the J0033 field, we developed a simple toy model in order to evaluate the robustness of the three Mg /i.pc/i.pc metrics used to identify DLAs in 1D quasar sightlines. The results from our toy model, which assumes a series of clouds in front of an extended background source, suggest that the typical covering fraction of gas clouds in front of the twogravitational arcs is \u2248 60 -80 %.Theresulting purity of these metrics in selecting DLA column densities in front of extended sources is similar to what has been observed in the literature for 1D quasar sightlines (Rao et al. 2006; Ellison 2006; Berg et al. 2017). The model also demonstrates that the covering fraction of gas can also influence both the EW 2796 and D -index metrics measured. As a result, choosing a threshold for these two metrics to identify DLAs towards extended sources is not straightforward. For EW 2796 , the threshold can depend on metallicity (Rao et al. 2006; Berg et al. 2017) and covering fraction. Whilst the D -index threshold depends on the spectral resolution of the observations (Ellison 2006) and covering fraction, we suspect that the covering fraction should not influence point-source, quasar observations as the typical cloud size is likely much larger than the background source. The typical spaxel size can vary between 0.07 and 1.4 kpc in our two fields after delensing. These results provide a cautionary tale of using Mg /i.pc/i.pc metrics to identify DLAs in front of extended background sources, despite the success at identifying the DLAs in both of these systems. Despite the difficulty of using the Mg /i.pc/i.pc metrics, our model is able to predict for the first time the covering fraction of gas that gives rise to DLAs, providing unprecedented insight on the structure of this medium. Moreover, for the J0033 field we were able to estimate the previously unknown logN(H /i.pc) of the absorber, logN(H /i.pc) = 20 . 6 \u00b1 0 . 3 , consistent with a DLA. Using the area of the delensed arc, we predict a lower limit of the total H /i.pc gas mass to be \u2273 10 9 M /circledot in both absorbers. We speculate that both absorbers are part of a neutral inner CGM and that, given the large amount of mass, they could be an essential source of fuel for future star formation in the host galaxies.", "pages": [13]}, {"title": "Data Availability", "content": "The median metal line profiles obtained for the different regions of the J1527 and J0033 fields are available online respectively in Appendix B ( https://zenodo.org/records/14225814 ) and Appendix C ( https://zenodo.org/records/14225839 ). Acknowledgements. We thank both Rongmon Bordoloi and Sara Ellison for respectively providing us with the aperture information used in B22 and the D -index data from Ellison (2006). We also are grateful for Keren Sharon for creating the lens models for both of these systems. Finally, we thank the anonymous referee for a constructive and thorough report. S.L. acknowledges support by FONDECYT grant 1231187. This work is based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere under ESO programme(s) 098.A-0459(A) and 0103.A-0485(B).", "pages": [13]}, {"title": "References", "content": "Afruni, A., Lopez, S., Anshul, P., et al. 2023, A&A, 680, A112 Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, ARA&A, 47, 481 Astropy Collaboration, Price-Whelan, A. M., Sip\u0151cz, B. M., et al. 2018, AJ, 156, 123 Bacon, R., Accardo, M., Adjali, L., et al. 2010, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7735, Groundbased and Airborne Instrumentation for Astronomy III, ed. I. S. McLean, S. K. Ramsay, & H. Takami, 773508 Berg, T. A. M., Ellison, S. L., Prochaska, J. X., et al. 2017, MNRAS, 464, L56 Berg, T. A. M., Fumagalli, M., D'Odorico, V., et al. 2021, MNRAS, 502, 4009 Bordoloi, R., Lilly, S. J., Kacprzak, G. G., & Churchill, C. 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M., et al. 2015, MNRAS, 449, 1536 Tytler, D., Gleed, M., Melis, C., et al. 2009, MNRAS, 392, 1539 van de Voort, F., Schaye, J., Booth, C. M., Haas, M. R., & Dalla Vecchia, C. 2011, MNRAS, 414, 2458 Weilbacher, P. M., Palsa, R., Streicher, O., et al. 2020, A&A, 641, A28 Wolfe, A. M., Gawiser, E., & Prochaska, J. X. 2005, ARA&A, 43, 861 Zafar, T., P\u00e9roux, C., Popping, A., et al. 2013, A&A, 556, A141", "pages": [13, 14]}, {"title": "Appendix A: Continuum fitting", "content": "In order to continuum fit the spectra from every spaxel of the MUSE cubes and to propagate continuum fitting uncertainties within a Monte Carlo framework, we used a simple sigmaclipping method to automatically determine the continuum of a given spectrum. In summary, the continuum fitting method uses the /a.pc/s.pc/t.pc/r.pc/o.pc/p.pc/y.pc (Astropy Collaboration et al. 2018) sigma clipping algorithm within a window of the spectrum of width 1000\u00c5 (800 pixels). The algorithm determines the mean flux ( \u00af F ) within the window while rejecting all pixels with flux outside of the confidence interval \u00af F -n low \u03c3 and \u00af F + n high \u03c3 (where \u03c3 is the standard deviation of all flux measurements within the window). The continuum is built-up by repeating this across the entire spectrum by sliding the window in increments of 100\u00c5 (80 pixels). Once completed, a spline is fit to the points generated from the sliding window (centered in the window for each step), and the spline is then smoothed with a top-hat function of width of 300\u00c5 to remove strong variations in the flux from regions of the spectra with a low S/N. We note that the continuum of the background sources are free of emission lines and smooth, allowing us to use a large window and improve the statistics used in the sigma-clipping method. While we fix n high = 1 . 0 in the sigma clipping algorithm, we found the optimal choice of n low to distinguish between noise and absorption depends on the average S/N of the spectrum. We therefore adjust n low across each spectrum. Table A.1 shows the optimized n low adopted for a given S/N (per pixel) to exclude absorption in the continuum fitting process.", "pages": [15]}] |
2023PhLB..83937775C | https://arxiv.org/pdf/2303.08002.pdf | <document>
<section_header_level_1><location><page_1><loc_23><loc_77><loc_77><loc_79></location>A regular version of the extremal RN spacetime</section_header_level_1>
<text><location><page_1><loc_28><loc_70><loc_72><loc_75></location>Hristu Culetu, Ovidius University, Dept. of Physics and Electronics, Mamaia Avenue 124, 900527 Constanta, Romania, ∗</text>
<text><location><page_1><loc_44><loc_67><loc_56><loc_69></location>March 15, 2023</text>
<section_header_level_1><location><page_1><loc_47><loc_62><loc_53><loc_63></location>Abstract</section_header_level_1>
<text><location><page_1><loc_26><loc_47><loc_74><loc_61></location>A modified extremal Reissner-Nordstrom geometry, void of singularities, is proposed in this work, by means of an exponential factor depending on a positive constant k . All the metric coefficients are positive and finite and the spacetime has no any horizon. The curvature invariants are regular at the origin of coordinates and at infinity. The energy conditions for the stress tensor associate to the imperfect fluid are investigated. The gravitational field presents repulsive properties near the gravitational radius associated to the mass m . With the choice k = 1 /m , the Komar energy W K of the mass m changes its sign at r = λ ( λ is the Compton wavelength of m ), when the classical energy mc 2 equals the energy /planckover2pi1 c/r .</text>
<section_header_level_1><location><page_1><loc_22><loc_43><loc_40><loc_44></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_22><loc_34><loc_78><loc_41></location>Extremal black holes are distinguished from non-extremal ones, even at the classical level. This is in accordance with the view that extremal blach holes (EBHs) may be considered as solitons to be formed quantum-mechanically by pair production (and whose entropy would be thought to vanish) [1]. If so, some properties of EBHs might be different from near-extremal ones [2].</text>
<text><location><page_1><loc_22><loc_21><loc_78><loc_34></location>Carroll et al. [1] investigated also the entropy of EBHs. Keeping in mind that the surface gravity and the temperature vanish [3], the entropy of an EBH also vanishes [4] (Edery and Constantineau justified that due to the timeindependent geometry throughout, which corresponds to a single classical microstate). It is worth noting that, near the horizon, the EBH has an AdS 2 × S 2 structure. The entropy of the AdS 2 × S 2 compactification solution does not vanish [1]. However, the entropy calculation concerns the near-horizon region and not the original black hole solution. Therefore, Carroll et al. concluded that the EBHs should have zero entropy.</text>
<text><location><page_1><loc_22><loc_17><loc_78><loc_20></location>Bonanno and Reuter [5] argued that the renormalization group improved Schwarzschild spacetime is similar to a Reissner-Nordstrom (RN) black hole</text>
<text><location><page_2><loc_22><loc_69><loc_78><loc_84></location>and the critical quantum black hole mass M cr is of the order of the Planck mass. In their view, the Hawking evaporation process is 'switched off' once the mass approaches M cr . They also noticed that the near-horizon geometry of the critical BH is, to leading order, the Robinson-Bertotti line element for the product of a two-dimensional AdS with a two-sphere, AdS 2 × S 2 . Horowitz et al. [6] studied the near-horizon geometry of a four-dimensional extremal black hole. When the cosmological constant Λ is negative, they showed that the tidal forces on infalling particles diverge when the horizon is crossed, albeit all scalar curvature invariants remain finite. Similar effects appear when Λ is positive, but not when it vanishes.</text>
<text><location><page_2><loc_22><loc_51><loc_78><loc_69></location>Non-singular BHs in general have been firstly studied by Bardeen [7], who proposed a model of charged matter collapse with a charged matter core inside the black hole, with no central singularity. Hayward [8] considered a sphericallysymmetric regular BH with, apart from m , an extra parameter l . His metric has a de Sitter (deS) form when r = 0 is approached and becomes Schwarzschild for l = 0. More recently, Frolov [9] discussed useful generalizations of the Hayward spacetime (with finite curvature invariants), in four-dimensional and higher dimensional geometries. The additional parameter l determines the scale where modification of the solution of the Einstein equations becomes significant. Simpson and Visser [10], using a special exponential function (see also [11, 12]) and the increasingly important Lambert W -function, avoid rather messy cubic and quartic polynomial equations used in the previous models.</text>
<text><location><page_2><loc_22><loc_33><loc_78><loc_51></location>We propose in this work a modification of the RN extremal BH for rendering it void of singularities. For that goal one introduces an exponential factor in the metric components. The new geometry is regular throughout and all scalar invariants are finite at the origin of coordinates and at infinity. The timelike Killing vector is vanishing nowhere and the geometry has no any horizon, so the name 'black hole' is no longer appropriate. The spacetime can be considered in its own right even though we started with a RN black hole.We investigated the properties of the energy-momentum tensor acting as the source of the modified metric. We also found that the gravitational field presents repulsive properties close to the origin of coordinates, where the source is located. Moreover, the tidal forces on free falling particles are not divergent, thanks to the finiteness of all the components of the Riemann tensor.</text>
<text><location><page_2><loc_22><loc_30><loc_78><loc_33></location>Throughout the work geometrical units G = c = /planckover2pi1 = 1 are used, unless otherwise specified.</text>
<section_header_level_1><location><page_2><loc_22><loc_26><loc_52><loc_27></location>2 Modified extremal BH</section_header_level_1>
<text><location><page_2><loc_22><loc_23><loc_57><loc_24></location>Let us consider the Reissner-Nordstrom geometry</text>
<formula><location><page_2><loc_30><loc_17><loc_78><loc_22></location>ds 2 = -( 1 -2 m r + q 2 r 2 ) dt 2 + dr 2 ( 1 -2 m r + q 2 r 2 ) + r 2 d Ω 2 , (2.1)</formula>
<text><location><page_3><loc_22><loc_75><loc_78><loc_84></location>where d Ω 2 stands for the metric on the unit two-sphere and m and q are the mass and, respectively, the charge of the BH. As is well-known, when m 2 > q 2 , the RN black hole has two horizons: one at r + = m + √ m 2 -q 2 (event horizon) and r -= m -√ m 2 -q 2 (Cauchy horizon). If m 2 < q 2 , there is no any horizon and the surface r = 0 becomes a naked singularity. The case m 2 = q 2 represents an extremal BH, when the line-element (2.1) acquires the form</text>
<formula><location><page_3><loc_34><loc_69><loc_78><loc_74></location>ds 2 = -( 1 -m r ) 2 dt 2 + dr 2 ( 1 -m r ) 2 + r 2 d Ω 2 , (2.2)</formula>
<text><location><page_3><loc_22><loc_62><loc_78><loc_69></location>which is time independent even inside the BH horizon r = m . That is a consequence of the fact that the timelike Killing vector remains timelike everywhere (there is no a r -t signature flip when the event horizon r = m is crossed). However, the true singularity at the origin r = 0 survives when the BH becomes extremal.</text>
<text><location><page_3><loc_22><loc_57><loc_78><loc_62></location>To get rid of the singularity, we propose a modified version of the metric (2.1) by means of an exponential factor, by analogy with [11]. The modified spacetime is given by</text>
<formula><location><page_3><loc_24><loc_51><loc_78><loc_56></location>ds 2 = -[ 1 -( 2 m r -m 2 r 2 ) e -k r ] dt 2 + dr 2 1 -( 2 m r -m 2 r 2 ) e -k r + r 2 d Ω 2 , (2.3)</formula>
<text><location><page_3><loc_22><loc_45><loc_78><loc_52></location>with k a positive constant and we replaced the charge q with m . We take the geometry (2.3) in its own right, not mandatory related to the RN geometry. It is clear that (2.3) becomes the extremal BH (2.2) if we take k = 0 (or r >> k ), when the exponential factor is approximated to unity). The next step is to search the properties of the metric function</text>
<formula><location><page_3><loc_37><loc_40><loc_78><loc_44></location>f ( r ) ≡ -g tt = 1 -( 2 m r -m 2 r 2 ) e -k r (2.4)</formula>
<text><location><page_3><loc_22><loc_34><loc_78><loc_39></location>It tends to unity when r → 0 , r → ∞ or r = m/ 2, and has two extrema: a maximum at r 1 = ( m + k -√ m 2 + k 2 ) / 2 and a minimum at r 2 = ( m + k + √ m 2 + k 2 ) / 2. Noting that f ( r ) is always positive, because (2 m/r -m 2 /r 2 ) is smaller than unity for any r > 0.</text>
<text><location><page_3><loc_22><loc_23><loc_78><loc_33></location>As an example, consider a particular situation with k = m . In that case, the maximum is at r 1 = m (1 -√ 2 / 2) ≈ 0 . 3 m , with f ( r 1 ) = 1 . 16, and the minimum is at r 2 = m (1 + √ 2 / 2) ≈ 1 . 7 m , with f ( r 2 ) = 0 . 412 > 0. Therefore, f ( r ) is positive for any r , as we already noticed. It is worth observing that the extremal RN horizon at r = m is somewhere between r 1 and r 2 . Being f ( r ) always positive, the metric (2.3) has no any horizon and is regular at the origin of coordinates, where it becomes Minkowskian.</text>
<text><location><page_3><loc_22><loc_20><loc_78><loc_23></location>We now consider a static observer in the geometry (2.3) having a velocity vector field</text>
<formula><location><page_3><loc_31><loc_13><loc_78><loc_19></location>u b = 1 √ 1 -( 2 m r -m 2 r 2 ) e -k r , 0 , 0 , 0 , u b u b = -1 , (2.5)</formula>
<text><location><page_4><loc_22><loc_80><loc_78><loc_84></location>where the Latin indices take the values ( t, r, θ, φ ). The only nonzero component of the covariant acceleration a b = u a ∇ a u b is</text>
<formula><location><page_4><loc_34><loc_77><loc_78><loc_80></location>a r = m r 2 ( 1 -m r -k r + km r 2 ) e -k r = 1 2 f ' ( r ) , (2.6)</formula>
<text><location><page_4><loc_22><loc_67><loc_78><loc_76></location>where f ' ( r ) = df ( r ) /dr . The radial acceleration is vanishing when r → 0 and at infinity. If r >> k , we get a r ≈ ( m/r 2 )(1 -m/r ), namely the value corresponding to the extremal BH. Moreover, a r vanishes where f ' ( r ) = 0, i.e. at the values of r where f ( r ) has its extrema, namely at the previous r 1 and r 2 , being negative for r ∈ ( r 1 , r 2 ). This means the gravitational field felt by a static observer is repulsive in the aforementioned domain of r .</text>
<text><location><page_4><loc_22><loc_64><loc_79><loc_67></location>As long as the curvature invariants (the scalar curvature and the Kretschmann scalar) for the metric (2.3) are concerned, one obtains</text>
<formula><location><page_4><loc_38><loc_60><loc_78><loc_63></location>R a a = 2 km 2 r 5 ( 1 + k m -k 2 r ) e -k r (2.7)</formula>
<text><location><page_4><loc_22><loc_58><loc_24><loc_59></location>and</text>
<formula><location><page_4><loc_23><loc_50><loc_78><loc_57></location>K = 48 m 2 r 6 (1 -2( k + m ) r + 12 k 2 +24 km +7 m 2 6 r 2 -k (4 k 2 +18 km +11 m 2 ) 6 r 3 + k 2 ( k 2 +10 km +13 m 2 ) 12 r 4 -k 3 m ( k +3 m ) 12 r 5 + k 4 m 2 48 r 6 ) e -k r (2.8)</formula>
<text><location><page_4><loc_22><loc_45><loc_78><loc_49></location>It is clear from (2.7) and (2.8) that the two invariants are vanishing if r → 0 or r →∞ . Moreover, if k = 0, one obtains</text>
<formula><location><page_4><loc_39><loc_42><loc_78><loc_45></location>K = 48 m 2 r 6 ( 1 -2 m r + 7 m 2 6 r 2 ) , (2.9)</formula>
<text><location><page_4><loc_22><loc_40><loc_67><loc_41></location>a value corresponding to the standard extremal RN black hole.</text>
<section_header_level_1><location><page_4><loc_22><loc_36><loc_53><loc_37></location>3 Stress tensor properties</section_header_level_1>
<text><location><page_4><loc_22><loc_28><loc_78><loc_34></location>We look now for the source of curvature of the geometry (2.3), namely we need the components of the stress tensor to be inserted on the r.h.s. of Einstein's equations G ab = 8 πT ab for to get (2.3) as an exact solution. The only nonzero components of the Einstein tensor are given by</text>
<formula><location><page_4><loc_31><loc_21><loc_78><loc_28></location>G t t = G r r = -m 2 r 4 ( 1 + 2 k m -k r ) e -k r , G θ θ = G φ φ = m ( m +2 k ) r 4 [ 1 -k ( k +2 m ) ( m +2 k ) r + mk 2 2( m +2 k ) r 2 ] e -k r . (3.1)</formula>
<text><location><page_4><loc_22><loc_17><loc_78><loc_20></location>As the source of the geometry (2.3) we employ an energy-momentum tensor corresponding to an imperfect fluid [13, 14, 15]</text>
<formula><location><page_4><loc_35><loc_14><loc_78><loc_16></location>T ab = ( p t + ρ ) u a u b + p t g ab +( p r -p t ) n a n b , (3.2)</formula>
<text><location><page_5><loc_22><loc_78><loc_78><loc_84></location>where ρ is the energy density of the fluid, p r is the radial pressure, p t are the transversal pressures and n a = (0 , √ f ( r ) , 0 , 0) is a vector ortogonal to u a , with u a n a = 0 and n a n a = 1. Using the ansatz (2.5) for u a , the Einstein equations and the form (3.2) of the stress tensor, one finds that</text>
<formula><location><page_5><loc_31><loc_70><loc_78><loc_77></location>8 πρ = 8 πT a b u b u a = m 2 r 4 ( 1 + 2 k m -k r ) e -k r = -8 πp r , 8 πp t = m 2 r 4 ( 1 + 2 k m -k 2 mr -2 k r + k 2 2 r 2 ) e -k r . (3.3)</formula>
<text><location><page_5><loc_22><loc_59><loc_78><loc_69></location>One observes that ρ is negative for r < km/ ( m +2 k ), positive for r > km/ ( m + 2 k ) and is vanishing when r → 0, r →∞ and at r = km/ ( m +2 k ). In addition, the choice k = 0 leads to the well-known expressions ρ = -p r = p t = m 2 / 8 πr 4 . For the particular case k = m , ρ has a minimum 8 πρ min = -3888 /e 6 m 2 at ¯ r 1 = m/ 6, a maximum 8 πρ max = 16 /e 2 m 2 at ¯ r 2 = m/ 2 and is vanishing at ¯ r 0 = m/ 3. Note that the two extrema of ρ are located below the event horizon r = m of the RN extremal BH.</text>
<text><location><page_5><loc_22><loc_36><loc_78><loc_58></location>As Simpson and Visser [10] have noticed, to examine where the energy density is maximised is of interest, due to the exponential supression of the mass. They found that their ρ is maximised at r = a/ 4, where a plays the same role as k (= m ) in our situation. Our ρ is maximised at ¯ r 2 = m/ 2, a double value compared to theirs (anyway, of the same order of magnitude). In addition, when a > 2 m/e , their geometry has no horizons, which means no BH, a situation similar to ours. Another interesting analogy of our metric (2.3) with that of Simpson and Visser is the non-standard asymptotically Minkowskian core, when r → 0. As we shall see, what is new (to the best of our knowledge) is the regular geometry (2.3), the fact that it has no horizons and the application of our model in microphysics (see Sec.5). In addition, as a new task compared to the previous studies on the non-singular BHs, we computed the Komar energy, both for k = m and k = 1 /m (in Sec. 4 and 5), and studied its interesting properties, especially the competition between the classical and quantum contribution for the application in microphysics.</text>
<text><location><page_5><loc_22><loc_31><loc_78><loc_36></location>As far as the transversal pressures are concerned, they are negative for r ∈ ( r ∗ 1 , r ∗ 2 ), with r ∗ 1 , 2 = k ( k +2 m ∓ √ k 2 +2 m 2 ) / 2( m +2 k ) and positive outside this interval.</text>
<text><location><page_5><loc_22><loc_28><loc_78><loc_31></location>Let us check now the energy conditions corresponding to the stress tensor (3.2).</text>
<unordered_list>
<list_item><location><page_5><loc_24><loc_26><loc_70><loc_28></location>- WEC (weak energy condition, ρ ≥ 0 , ρ + p r ≥ 0 , ρ + p t ≥ 0)</list_item>
</unordered_list>
<text><location><page_5><loc_22><loc_22><loc_78><loc_27></location>We already found that ρ ≥ 0 when r ≥ km/ ( m +2 k ). The condition ρ + p t ≥ 0 gives us r ∈ (0 , r ' 1 ] ∪ [ r ' 2 , ∞ ), where r ' 1 , 2 = k ( k +3 m ∓ √ ( k -m ) 2 +4 m 2 ) / 4( m + 2 k ). The WEC is obeyed when r is within the above domains.</text>
<unordered_list>
<list_item><location><page_5><loc_24><loc_20><loc_63><loc_22></location>- NEC (null energy condition, ρ + p r ≥ 0 , ρ + p t ≥ 0)</list_item>
<list_item><location><page_5><loc_24><loc_17><loc_67><loc_19></location>- SEC (strong energy condition, NEC and ρ + p r +2 p t ≥ 0)</list_item>
</unordered_list>
<text><location><page_5><loc_24><loc_18><loc_68><loc_21></location>NEC is included in WEC, hence we have r ∈ (0 , r ' 1 ] ∪ [ r ' 2 , ∞ ).</text>
<unordered_list>
<list_item><location><page_5><loc_24><loc_15><loc_73><loc_18></location>The new condition leads to p t ≥ 0. That means r ∈ (0 , r ∗ 1 ] ∪ [ r ∗ 2 , ∞ ).</list_item>
<list_item><location><page_5><loc_24><loc_14><loc_62><loc_16></location>- DEC (dominant energy condition, ρ ≥ | p r | , ρ ≥ | p t | )</list_item>
</unordered_list>
<text><location><page_6><loc_22><loc_77><loc_78><loc_84></location>The first condition is satisfied if ρ ≥ 0, namely r ≥ km/ (2 k + m ). For the second we get two other conditions: r ≥ km/ 2( k + m ) and -ρ ≤ p t ≤ ρ , which leads to r ∈ (0 , r ' 1 ] ∪ [ r ' 2 , ∞ ). Noting that the inequality r ≥ km/ 2( k + m ) is not necessary, being included in r ≥ km/ (2 k + m ).</text>
<text><location><page_6><loc_22><loc_69><loc_78><loc_78></location>Let us make now a first choice of the length k . We have only one constant in the model, with units of distance: the source mass m , or more precisely, the half-gravitational radius of it. This is clearly appropriate for macroscopic masses, that is for masses m > m P = 10 -5 gr, where m P is the Planck mass. We consider Planck's mass to be the boundary between the macroscopic and microscopic masses.</text>
<text><location><page_6><loc_22><loc_66><loc_78><loc_69></location>The energy density, the radial pressure and the transversal pressures acquire the form</text>
<formula><location><page_6><loc_25><loc_62><loc_78><loc_65></location>ρ = -p r = 3 m 2 r 4 ( 1 -m 3 r ) e -m r , p t = 3 m 2 r 4 ( 1 -m r + m 2 6 r 2 ) e -m r . (3.4)</formula>
<text><location><page_6><loc_22><loc_52><loc_78><loc_61></location>We see that the ρ and p t expressions given above contain few correcting terms compared to those from [11] (Eqs.3.1). Therefore, we do not intend to show more details about the properties of the above quantities. Concerning the energy conditions, it can be shown that, with k = m , the WEC, NEC, SEC and DEC are observed provided r ≥ r ∗ 2 = ( m/ 2)(1 + √ 3 / 3) ≈ 0 . 78 m , which is smaller than the standard gravitational radius of the mass m .</text>
<section_header_level_1><location><page_6><loc_22><loc_48><loc_54><loc_49></location>4 Energetic considerations</section_header_level_1>
<text><location><page_6><loc_22><loc_45><loc_65><loc_46></location>It is instructive to compute the Komar mass-energy W K [16]</text>
<formula><location><page_6><loc_35><loc_40><loc_78><loc_44></location>W K = 2 ∫ V ( T ab -1 2 g ab T c c ) u a u b N √ hd 3 x, (4.1)</formula>
<text><location><page_6><loc_22><loc_35><loc_78><loc_39></location>where V is a 3-volume in our static metric, N 2 = -g tt is the lapse function, h = r 4 sin 2 θ/f ( r ) is the determinant of the spatial 3 - metric, h ab = g ab + u a u b , and d 3 x = dr dθ dφ . With u a from (2.5) and T ab from (3.2), we get</text>
<formula><location><page_6><loc_34><loc_31><loc_78><loc_34></location>( T ab -1 2 g ab T c c ) u a u b = p t , N √ h = r 2 sinθ. (4.2)</formula>
<text><location><page_6><loc_22><loc_28><loc_40><loc_30></location>Therefore, Eq.(4.1) yields</text>
<formula><location><page_6><loc_30><loc_24><loc_78><loc_27></location>W K = ∫ r 0 [ m ( m +2 k ) r ' 2 -mk ( k +2 m ) r ' 3 + m 2 k 2 2 r ' 4 ] e -k r ' dr ' , (4.3)</formula>
<text><location><page_6><loc_22><loc_21><loc_37><loc_23></location>Keeping in mind that</text>
<formula><location><page_6><loc_24><loc_16><loc_78><loc_20></location>∫ 1 r 3 e -k r dr = 1 k ( 1 k + 1 r ) e -k r , ∫ 1 r 4 e -k r dr = 2 k 3 ( 1 + k r + k 2 2 r 2 ) e -k r . (4.4)</formula>
<text><location><page_7><loc_22><loc_83><loc_30><loc_84></location>One obtains</text>
<formula><location><page_7><loc_23><loc_79><loc_78><loc_82></location>W K = [ m ( m +2 k ) k -m ( k +2 m )( 1 k + 1 r ) + km 2 ( 1 k 2 + 1 kr + 1 2 r 2 ) ] e -k r . (4.5)</formula>
<text><location><page_7><loc_22><loc_76><loc_49><loc_77></location>After a rearangement of terms, we get</text>
<formula><location><page_7><loc_37><loc_72><loc_78><loc_75></location>W K = m ( 1 -m r -k r + km 2 r 2 ) e -k r , (4.6)</formula>
<text><location><page_7><loc_22><loc_55><loc_78><loc_70></location>It is worth observing that, when k = 0, one obtains W K = m (1 -m/r ) = m -m 2 /r , a sum between the rest energy and the proper gravitational energy. In addition, if r →∞ , (4.6) gives us W K, ∞ = m , as expected; in contrast, W K is vanishing when r → 0. In terms of the radial acceleration (2.6), we have W K ( r ) = r 2 a r or, written differently, with fundamental constants included, , a r = GM ( r ) /r 2 = GW K ( r ) /c 2 r 2 . In other words, the radial acceleration at the distance r from the origin depends directly on the Komar mass. The sign of W K is given, of course, by the sign of a r or f ' ( r ) (see below (2.4)). Hence, W K is negative when r ∈ ( r 1 , r 2 ) (where the gravitational field is repulsive) and positive otherwise.</text>
<section_header_level_1><location><page_7><loc_22><loc_51><loc_48><loc_53></location>5 The choice k = 1 /m</section_header_level_1>
<text><location><page_7><loc_22><loc_42><loc_78><loc_50></location>This second choice of k leads to a semiclassical model of our system described by the spacetime (2.3): the usage of the Planck constant /planckover2pi1 , such that k equals the reduced Compton wavelength associated to the mass m , namely λ = /planckover2pi1 /mc . That means our paradigm has to be applied in microphysics, with m signifying the mass of an elementary particle, with, of course, m<<m P .</text>
<text><location><page_7><loc_22><loc_39><loc_78><loc_42></location>Our next task is to investigate the properties of some of the quantities analysed by now.</text>
<unordered_list>
<list_item><location><page_7><loc_24><loc_38><loc_58><loc_39></location>- The radial acceleration acquires now the form</list_item>
</unordered_list>
<formula><location><page_7><loc_38><loc_34><loc_78><loc_37></location>a r = m r 2 ( 1 -m r -λ r + l 2 P r 2 ) e -λ r , (5.1)</formula>
<text><location><page_7><loc_22><loc_26><loc_78><loc_32></location>where l P = √ G /planckover2pi1 /c 3 is the Planck length. For an elementary particle (say, a neutron), its gravitational radius is ≈ 10 -52 cm and λ ≈ 10 -13 cm , so that the second and the fourth terms in the parantheses may be neglected w.r.t. the third. Consequently, we can write</text>
<formula><location><page_7><loc_42><loc_22><loc_78><loc_25></location>a r ≈ m r 2 ( 1 -λ r ) e -λ r , (5.2)</formula>
<text><location><page_7><loc_22><loc_15><loc_78><loc_21></location>and the radial acceleration changes its sign at r = λ . If we calculate it exactly from (5.1) at that value of r , again for a neutron, one obtains a r = -( m/m P ) 4 (1 / 2 λ ) = -5 . 16 × 10 -42 cm/s 2 , which is indeed nearly zero ( m P is the Planck mass). That is a consequence of the fact that, with k = λ , we have</text>
<text><location><page_8><loc_22><loc_77><loc_78><loc_84></location>r 1 ≈ m/ 2 and r 2 ≈ λ , with a r < 0 for r ∈ ( r 1 , r 2 ) (see below (2.6)). But r 1 ≈ 0, so that, with a good approximation, a r < 0 for any r < λ . We have seen that a r is practically zero at r = λ but, however, it is not negligible at r = λ/ 2, where we have a r ≈ -1 . 3 · 10 -4 cm/s 2 .</text>
<text><location><page_8><loc_22><loc_63><loc_78><loc_78></location>Let us check the value of a r for r > λ , say r = 3 λ/ 2, using the approximate expression (5.2), by a comparison with the previous a r . We have a r (3 λ/ 2) = -a r ( λ/ 2) e 4 / 3 / 27 = 0 . 18 · 10 -4 cm/s 2 . One observes that the radial acceleration already changed its sign, and the gravitational field becomes attractive. As an extra example, one could find that a r ≈ 1 . 4 · 10 -8 cm/s 2 at r = 2 λ , an acceptable value. We see here something similar with the interaction between two molecules, due to the van der Waals forces. When they are approaching, the force between them is attractive but when the distance is decreased, repulsive forces come into play (in our situation, the second particle is a test particle found close to the neutron).</text>
<unordered_list>
<list_item><location><page_8><loc_24><loc_62><loc_56><loc_63></location>- The energy density when k = λ is given by</list_item>
</unordered_list>
<formula><location><page_8><loc_38><loc_57><loc_78><loc_60></location>8 πρ = m 2 r 4 ( 1 + 2 m 2 P m 2 -λ r ) e -λ r (5.3)</formula>
<text><location><page_8><loc_22><loc_48><loc_78><loc_56></location>For instance, at r = λ , we get ρ = 2 m 2 P /πeλ 4 = /planckover2pi1 c/ 4 πeλ 4 , which no longer depends on the Newton constant G , but only on /planckover2pi1 and c . We already found that ρ > 0 for any r ≥ km/ ( m +2 k ), which becomes now r ≥ l 2 P / ( m +2 λ ) ≈ l 2 P / 2 λ < l P . In other words, ρ is practically always positive with the above value of k .</text>
<text><location><page_8><loc_22><loc_36><loc_78><loc_48></location>By analogy with the macroscopic case, we are interested in the expression of the maximum value of the energy density. Working in the approximation m << m P , we find that ρ has two extrema: one minimum at r = R 1 ≈ m 2 λ/ 4 m 2 P , with negative ρ min , but very close to zero due to the exponential factor, and a maximum at r = R 2 ≈ λ/ 4, with ρ max = ρ ( R 2 ) = 32 m 2 P /πe 4 λ 4 . We have λ >> l P , such that the Markov [17] criterion is satisfied. We notice that Frolov [9] used Markov's recipe in his paper, as the limiting curvature conjecture (namely, the Plank value of curvature).</text>
<text><location><page_8><loc_22><loc_31><loc_78><loc_36></location>We proved that all the energy conditions are obeyed if r ≥ r ∗ 2 . With the above value of k and keeping in mind that λ >> m we get r ≥ λ/ 2. That means for r ≤ λ/ 2 some energy conditions are not satisfied.</text>
<unordered_list>
<list_item><location><page_8><loc_24><loc_30><loc_70><loc_32></location>- The Komar energy acquires the form, with the new value of k</list_item>
</unordered_list>
<formula><location><page_8><loc_36><loc_26><loc_78><loc_29></location>W K ( r ) = m ( 1 -m r -λ r + l 2 P 2 r 2 ) e -λ r . (5.4)</formula>
<text><location><page_8><loc_22><loc_22><loc_78><loc_25></location>But the second and the fourth terms inside the parantheses may be neglected w.r.t. the third. Therefore</text>
<formula><location><page_8><loc_40><loc_18><loc_78><loc_21></location>W K ( r ) ≈ m ( 1 -λ r ) e -λ r , (5.5)</formula>
<text><location><page_8><loc_22><loc_15><loc_78><loc_16></location>which changes sign at r = λ , as the radial acceleration. When the fundamental</text>
<text><location><page_9><loc_22><loc_83><loc_46><loc_84></location>constants are inserted, (5.5) yields</text>
<formula><location><page_9><loc_39><loc_78><loc_78><loc_81></location>W K ( r ) = ( mc 2 -/planckover2pi1 c r ) e -/planckover2pi1 mcr . (5.6)</formula>
<text><location><page_9><loc_22><loc_71><loc_78><loc_77></location>Firstly one observes that, with the above approximation, W K does not depend on G . Classically, when /planckover2pi1 → 0 or if r >> λ , one obtains W K = mc 2 , as expected. We notice that the two contributions, classical and quantum, are opposed, being equal at r = λ , when W K = 0.</text>
<section_header_level_1><location><page_9><loc_22><loc_67><loc_39><loc_69></location>6 Conclusions</section_header_level_1>
<text><location><page_9><loc_22><loc_57><loc_78><loc_66></location>As standard BHs, the extremal black holes are not regular at the origin of coordinates. Making them regular throughout might be an important task. It is exactly what we tried to do in the present paper. For that purpose we made use of an exponential factor rendering the spacetime void of singularities. The metric coefficients being positive, our spacetime has no horizons and the label 'black hole' is no longer appropriate.</text>
<text><location><page_9><loc_22><loc_48><loc_78><loc_56></location>The properties of the stress tensor of the imperfect fluid are studied, especially the energy conditions. An important property of our physical system is the Komar energy, which has been calculated and discussed. We chose for the constant k two values: k = m for the macroscopic case and k = 1 /m for the microscopic case, which is the semiclassical one, due to the Planck constant from the expression of the Compton wavelength.</text>
<section_header_level_1><location><page_9><loc_22><loc_43><loc_34><loc_45></location>References</section_header_level_1>
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<list_item><location><page_9><loc_23><loc_39><loc_78><loc_42></location>[1] S. Carroll, M. Johnson and L. Randall, J. High Energy Phys. 0911, 109 (2009) (arXiv: 0901.0931).</list_item>
<list_item><location><page_9><loc_23><loc_35><loc_78><loc_38></location>[2] G. Gibbons and R. Kallosh, Phys. Rev. D 51, 2839 (1995) (arXiv: hep-th/9407118).</list_item>
<list_item><location><page_9><loc_23><loc_33><loc_52><loc_34></location>[3] N. Cribiori et al., arXiv: 2207.04657.</list_item>
<list_item><location><page_9><loc_23><loc_29><loc_78><loc_31></location>[4] A. Edery and and B. Constantineau, Class. Quantum Grav. 28, 045003 (2011) (arXiv: 1010.5844).</list_item>
<list_item><location><page_9><loc_23><loc_25><loc_78><loc_27></location>[5] A. Bonanno and M. Reuter, Phys. Rev. D62, 043008 (2000) (arXiv: hep-th/0002196).</list_item>
<list_item><location><page_9><loc_23><loc_22><loc_70><loc_23></location>[6] G. Horowitz, M. Kolanovski and J. Santos, arXiv: 2210.02473.</list_item>
<list_item><location><page_9><loc_23><loc_18><loc_78><loc_21></location>[7] J. M. Bardeen, Proceedings of the International Conference GR5 (Tbilisi, USSR, 1968) p.174.</list_item>
<list_item><location><page_9><loc_23><loc_16><loc_78><loc_17></location>[8] S. A. Hayward, Phys. Rev. Lett. 96, 031103 (2006); arXiv: gr-qc/0506126.</list_item>
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<list_item><location><page_10><loc_23><loc_83><loc_72><loc_84></location>[9] V. P. Frolov, Phys. Rev. D 94, 104056 (2017); arXiv: 1609.01758.</list_item>
<list_item><location><page_10><loc_22><loc_80><loc_76><loc_81></location>[10] A. Simpson and M. Visser, Universe 6(1), 8 (2020); arXiv: 1911.01020.</list_item>
<list_item><location><page_10><loc_22><loc_78><loc_73><loc_79></location>[11] H. Culetu, Int. J. Theor. Phys. 54, 2855 (2015) (arXiv: 1408.3334).</list_item>
<list_item><location><page_10><loc_22><loc_75><loc_46><loc_76></location>[12] H. Culetu, arXiv: 1305.5964.</list_item>
<list_item><location><page_10><loc_22><loc_71><loc_78><loc_74></location>[13] M. Govender, K. P. Reddy and S. D. Maharaj, Int. J. Mod. Phys.D, vol.23, 1450013 (2014) (arXiv:1312.1546).</list_item>
<list_item><location><page_10><loc_22><loc_69><loc_74><loc_70></location>[14] H. Culetu, J. Phys. Soc. Jpn. 87, 014002 (2018) (arXiv: 1612.06009).</list_item>
<list_item><location><page_10><loc_22><loc_66><loc_77><loc_67></location>[15] H. Culetu, Class. Quantum Grav. 29, 235021 (2012) (arXiv: 1202.4296).</list_item>
<list_item><location><page_10><loc_22><loc_62><loc_78><loc_65></location>[16] T. Padmanabhan, Phys. Rev. D81 (2010) 124040, (arXiv: grqc/1003.5665).</list_item>
<list_item><location><page_10><loc_22><loc_60><loc_57><loc_61></location>[17] M. A. Markov, JETP Letters 36, 266 (1982).</list_item>
</unordered_list>
</document> | [] |
2020Univ....6..168M | https://arxiv.org/pdf/2010.02942.pdf | <document>
<figure>
<location><page_1><loc_12><loc_89><loc_33><loc_94></location>
</figure>
<text><location><page_1><loc_12><loc_86><loc_17><loc_87></location>Article</text>
<section_header_level_1><location><page_1><loc_12><loc_81><loc_78><loc_85></location>Using Unreal Engine to Visualize a Cosmological Volume</section_header_level_1>
<text><location><page_1><loc_13><loc_77><loc_49><loc_79></location>Christopher Marsden * and Francesco Shankar</text>
<text><location><page_1><loc_13><loc_74><loc_88><loc_76></location>Department of Physics and Astronomy, University of Southampton, Highfield, SO17 1BJ, UK; [email protected] * Correspondence: [email protected]</text>
<text><location><page_1><loc_13><loc_71><loc_61><loc_73></location>Received: 17 August 2020; Accepted: 30 September 2020; Published: date</text>
<text><location><page_1><loc_13><loc_46><loc_88><loc_70></location>Abstract: In this work we present 'Astera', a cosmological visualization tool that renders a mock universe in real time using Unreal Engine 4. The large scale structure of the cosmic web is hard to visualize in two dimensions, and a 3D real time projection of this distribution allows for an unprecedented view of the large scale universe, with visually accurate galaxies placed in a dynamic 3D world. The underlying data are based on empirical relations assigned using results from N-Body dark matter simulations, and are matched to galaxies with similar morphologies and sizes, images of which are extracted from the Sloan Digital Sky Survey. Within Unreal Engine 4, galaxy images are transformed into textures and dynamic materials (with appropriate transparency) that are applied to static mesh objects with appropriate sizes and locations. To ensure excellent performance, these static meshes are 'instanced' to utilize the full capabilities of a graphics processing unit. Additional components include a dynamic system for representing accelerated-time active galactic nuclei. The end result is a visually realistic large scale universe that can be explored by a user in real time, with accurate large scale structure. Astera is not yet ready for public release, but we are exploring options to make different versions of the code available for both research and outreach applications.</text>
<text><location><page_1><loc_13><loc_43><loc_46><loc_45></location>Keywords: galaxies; visualisation; cosmology</text>
<section_header_level_1><location><page_1><loc_13><loc_37><loc_24><loc_38></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_13><loc_12><loc_88><loc_36></location>The predominantly accepted cosmological paradigm, L CDM, predicts that structure in the universe forms via the collapse of dark matter into distinct filaments, voids and haloes [1,2]. This underlying substructure acts as a tracer for baryonic matter, which falls into the potential wells of dark matter haloes and forms galaxies. It is therefore predominantly agreed that the large scale, 3-Dimensional structure of the universe is dictated by the structure of underlying dark matter. This structure is notoriously difficult to visualize in two dimensions, and since the early days of cosmological research, numerous attempts have been made to visualize the ''cosmic web'. Initial attempts by, for example, [3] showed a representation of the cosmic web as a simple 2D ''slice', with simple-plotted points representing the density distribution. Since then, increasing advances in digital graphics have allowed for significantly more visually impressive images-e.g., [4] and [5]. Despite the impressive quality of these images, it is clear that a 2-Dimensional projection of the large scale structure of the universe cannot capture the high complexity its full 3D structure, so more elaborate projections in pseudo-3D or utilizing iso-surface density have been developed-e.g., in [6]. The advent of computer graphical rendering has allowed for truly breathtaking video representations of the large scale universe in projected 3D, such as [7], [8], [9], and [10].</text>
<text><location><page_1><loc_12><loc_9><loc_88><loc_12></location>Digital graphics can be broadly categorized into 'pre-rendered' graphics and 'real time' graphics. The former requires pre-processing of the digital assets (often at great computational expense, but only</text>
<figure>
<location><page_1><loc_80><loc_89><loc_88><loc_93></location>
</figure>
<text><location><page_2><loc_12><loc_74><loc_88><loc_89></location>once) into a series of frames that are assembled into a video format that is fixed in scope but can be replayed on virtually any device. This is most commonly seen in modern digital animation, commonly used in the entertainment industry. The latter, 'real time' rendering, requires digital assets to be processed 'on the fly', and must be processed quickly enough to ensure that the image can be recreated many times a second. This requires significant computational power, but crucially allows for user interactivity, with potentially unlimited scope for the user having dynamic control over the experience. Real-time rendering has, until recently, been restricted solely to video games, with visual quality notably worse that pre-rendered graphics. However, in recent years the quality of real time graphics has notably improved, allowing for their use in a wide variety of applications, such as on virtual film sets [11].</text>
<text><location><page_2><loc_12><loc_58><loc_88><loc_73></location>Real-time rendering depicting extragalactic scales have not yet been explored with full scientific accuracy. Several projects have come close; Celestia [12] is a free visualizer for many astronomical objects. Its open source nature has lead to numerous extensions, including planned cosmological visualizers. There are numerous sky visualizers that offer an extragalactic view of the universe based on astronomical imagery, such as WordWide Telescope [13], Google Sky [14] and others, focusing on real data. Universe Sandbox [15] is an interactive educational software application that simulates gravitational effects in various scales, including interactions between pairs of galaxies. SpaceEngine [16] is a remarkable achievement, and is capable of procedurally generating a vast universe the user can explore with scales ranging from individual planets to the extragalactic.</text>
<text><location><page_2><loc_12><loc_50><loc_88><loc_58></location>Each of these projects, mostly focused on small-scale bodies such as planets and stars, is limited by the compelling need to simulate a vast range of length scales, understandably focusing on the solar system and similar bodies. This methodology inevitably produces a lack of accuracy when simulating the large scale structure of the Universe. In this paper, we present our solution: 'Astera', a real-time cosmological visualization tool created using Unreal Engine 4.</text>
<text><location><page_2><loc_13><loc_41><loc_88><loc_49></location>This paper is structured as follows. In Section 2 we discuss the ''assets' behind Astera, which pertains to the creation of the underlying example galaxy catalogue that forms the theoretical foundation of Astera, and the astronomical images used to represent galaxies. This is followed (Section 3) by a description of the technical implementation of Astera itself within Unreal Engine 4. In Section 4 we present the 3D universe that Astera creates, and we conclude in Section 5.</text>
<section_header_level_1><location><page_2><loc_13><loc_38><loc_19><loc_40></location>2. Assets</section_header_level_1>
<text><location><page_2><loc_13><loc_31><loc_87><loc_37></location>Assets, as mentioned previously, are the reusable and replaceable components of a digital project. It is worth emphasizing that these components are able to be changed with a minimum of effort, so although integral parts of the experience they are not fixed parts of the software. This includes the underlying galaxy catalogue, and the astronomical images used to represent galaxies.</text>
<section_header_level_1><location><page_2><loc_13><loc_28><loc_26><loc_29></location>2.1. Galaxy Catalog</section_header_level_1>
<text><location><page_2><loc_12><loc_12><loc_87><loc_27></location>Mock galaxy catalogues are artificial datasets containing parameters for a large number of synthetic galaxies, extracted from simulations that utilize our best understanding of galaxy evolution. The comparison between mock catalogues and observational data is vital for probing underlying physical processes, but mock catalogues are also in demand for the calibration of the next generation of extra-galactic observations (e.g., Euclid, Athena). Mock catalogues are extracted from simulations that follow the evolution of galaxies over cosmic time, and from this a 'light cone' is normally constructed that represents the evolutionary state of objects at varying redshifts. In the case of Astera, we are interested in (at least initially) creating a catalogue that can be 'explored' by the user in real time, imitating non-physical superluminal speeds (or a universe 'frozen' with no relative motions of galaxies, no cosmological</text>
<text><location><page_3><loc_12><loc_86><loc_87><loc_89></location>expansion, etc). Therefore, the creation of a light cone is not necessary, and a simple cosmological volume will suffice.</text>
<text><location><page_3><loc_12><loc_77><loc_87><loc_85></location>It should be noted that Astera itself is Cosmology independent, as it simply presents coordinates and galaxy imagery. The underlying mock catalogue can be of any cosmology desired, and in this case we adopt a standard flat L CDMparadigm with H 0 = 70, W L = 0.7, W 0 = 0.3. In this section, we describe a simple 'recipe' that produces what the authors consider to be a reasonable mock catalogue for showcasing Astera, but in principle any catalogue could be used.</text>
<text><location><page_3><loc_12><loc_62><loc_88><loc_77></location>As appropriate for the L CDMparadigm, the foundation of our catalogue is a Dark Matter N-Body simulation. We use both the Bolshoi [17] ( 500 h -1 Mpc ) 3 and the Multi-Dark [18] ( 1000 h -1 Mpc ) 3 simulation catalogues 1 , depending on the volume desired (as larger volumes require more powerful hardware, although it should be noted that volumes can also be ''cropped' to manage performance). The most important variables in this catalogue are naturally the 3D coordinates (X, Y, Z), as they dictate the 3D positions of the haloes in the virtual world, but also vitally important are the virial masses of the haloes, upon which we use statistical ''empirical' relations to construct our model. We therefore assume that a galaxy exists at the centre of every dark matter halo resolved within the catalogue, and assign the stellar mass.</text>
<text><location><page_3><loc_13><loc_50><loc_88><loc_61></location>The importance of robust and self consistent relations between underlying dark matter and galaxy stellar masses cannot be underestimated. As demonstrated by [19] and [20], even small variations in the Halo Mass-Stellar Mass (HMSM) relation can yield large (and often non-physical) variations in the satellite accretion rate, required star formation rate and pair fractions. While many mock catalogues will undoubtedly have established Stellar Masses, we adopt the [19] HMSM relation, a parametric stellar mass to halo mass relation, a variation of the relation presented in [21]. For completeness, the full relation is shown in Equation (1), with tabulated parameters in Table 1, and a plot of the relation is shown in Figure 1.</text>
<formula><location><page_3><loc_30><loc_36><loc_87><loc_48></location>M ∗ ( M h , z ) = 2 M h N ( z ) [ ( M h Mn ( z ) ) -b ( z ) + ( M h Mn ( z ) ) g ( z ) ] -1 N ( z ) = N 0.1 + Nz ( z -0.1 z + 1 ) Mn ( z ) = M n , 0.1 + Mn , z ( z -0.1 z + 1 ) b ( z ) = b 0.1 + b z ( z -0.1 z + 1 ) g ( z ) = g 0.1 + g z ( z -0.1 z + 1 ) (1)</formula>
<table>
<location><page_3><loc_31><loc_25><loc_69><loc_31></location>
<caption>Table 1. Parameters for equation 1.</caption>
</table>
<text><location><page_3><loc_12><loc_16><loc_88><loc_24></location>where M ∗ represents the stellar mass of the galaxy M h represents the host halo mass. The associated parameters N , Mn , b and g are values set to describe the relation at varying reshifts. In table 1, the parameters are shown for central and satellite galaxies at z = 0.1 and evolving values (the subscript 0.1 referring to the value at z = 0.1, and the subscript z referring to the evolving value). This equation and table are reproduced from [19], wherein more details can be found.</text>
<figure>
<location><page_4><loc_27><loc_65><loc_70><loc_86></location>
<caption>Figure 1. Comparison of Halo Mass to Stellar Mass relations from [19] compared to [21].</caption>
</figure>
<text><location><page_4><loc_12><loc_38><loc_88><loc_61></location>Assigning stellar mass allows for a robust foundation upon which further properties can be built. Aside from 3D distribution, the most obvious property required for a visually realistic galaxy catalogue is obviously morphological classification. This was assigned using phenomenological relations derived from the Sloan Digital Sky Survey (SDSS) data [22]. Binning the SDSS by stellar mass allows for an approximate Stellar Mass-morphological type (TType) relation (see Figure 2 ( a )). In short, the catalogue of galaxies is binned in appropriately sized bins of stellar mass, and the mean TType and scatter for each bin are recorded, producing an (average) relation allowing transformation from stellar mass to TType. With the associated scatter, this relation can be applied to the simulation catalogue for a statistically comparable distribution of morphological types to the real universe. Each galaxy is therefore assigned a TType representing its morphological classification based on its stellar mass. A caveat here is that this distribution is only valid at low redshifts, limiting this technique to catalogues representing the universe in the 'present day'. This limitation is not insurmountable, as numerous semi-analytic models and semi-empirical models are capable of predicting morphological abundances at varying redshifts, and their datasets could be easily be used in Astera.</text>
<text><location><page_4><loc_12><loc_33><loc_88><loc_37></location>A galaxy's physical size (parametrised by the effective radius R eff ) is also naturally important for visual realism. These were assigned in identical fashion to morphological types, using the mean relations from the SDSS (again see Figure 2 ( b )).</text>
<text><location><page_4><loc_13><loc_14><loc_88><loc_32></location>Asimple validation to show that these components are 'working together' as expected is shown in Figure 3, where we show the average predicted 3D mass density profile for elliptical galaxies of stellar mass 11.3 < log 10 M ∗ / M glyph[circledot] < 11.7. Each elliptical galaxy in this mass range is additionally assigned a Sérsic index (again according to the mean relation in the SDSS), a de-projected stellar mass density profile according to the prescription of [23], and an NFW profile [24] with halo concentrations according to the model of [25]. Finally, the average mass density per bin of radius is calculated for the sample. This result is compared to the pure power law model of [26] r ( r ) GLYPH<181> r -g , where g = 2.2 at r ∼ Re , derived from 2D stellar kinematics and strong lensing measurements. Specifically, [26] inferred that 〈 g 〉 = 2.19 ± 0.03, valid in the range 0.1 Re to 4 Re and 10.2 < log 10 M ∗ / M glyph[circledot] < 11.7. This result is shown as the pink stripe in Figure 3, showing good agreement with the model at 〈 Re 〉 . The slopes at low radii were found to be very sensitive to the definition of Re , in this case showing a slight divergence from the data.</text>
<figure>
<location><page_5><loc_13><loc_66><loc_87><loc_88></location>
<caption>Figure 2. ( a ) Morphological type (TType) and ( b ) Size vs Stellar Mass relations within the SDSS, used to assign the respective parameters to the catalogue. The shaded regions show the 1 s uncertainty in these parameters.</caption>
</figure>
<figure>
<location><page_5><loc_25><loc_33><loc_72><loc_57></location>
<caption>Figure 3. Predicted (average) 3D density profile haloes hosting elliptical galaxies of mass 11.3 < log 10 M ∗ / M glyph[circledot] < 11.7 for the sample catalogue. The solid black line represents the combined density, whilst the blue (dot-dashed) and red (dashed) lines represents the average stellar (Sérsic) and dark matter (NFW) components, respectively. The pink shaded region represents the empirical fit from [26]. The grey shaded region shows the 1 s dispersion of the total density. The vertical blue dotted line shows the average half light radius, and the solid short black line shows (for comparison) the established slope of g = 2.2 at this radius.</caption>
</figure>
<text><location><page_5><loc_12><loc_9><loc_88><loc_19></location>Finally, active galaxies were also considered. These were assigned using a bespoke methodology that will be fully described in Marsden et al. (in preparation) and Allevato et al. (in preparation), but in general the central supermassive black hole mass was assigned from the stellar mass using the unbias relation from [27], with appropriate Eddington ratios and X-Ray luminosity assigned using a Schechter function with values chosen to fit the known X-Ray luminosity function and Eddington ratio distribution. The 'Duty Cycle' U or associated probability of a Supermassive Black Hole being active, was for now</text>
<text><location><page_6><loc_12><loc_86><loc_87><loc_89></location>set to U ∼ 0.1, although a more complete treatment of this process will be described in Marsden et al' (in preparation and Allevato et al. (in preparation)).</text>
<section_header_level_1><location><page_6><loc_13><loc_83><loc_26><loc_84></location>2.2. Galaxy Imagery</section_header_level_1>
<text><location><page_6><loc_12><loc_65><loc_88><loc_81></location>An obviously vital part of Astera is the galaxies. We elected to use actual astronomical imagery, and in section 3 we discuss this choice further. For now, we discuss the images themselves. Getting both good quality and a large variation in galaxy images will contribute greatly to the user experience, so various approaches were considered. There are relatively few high resolution images from the Hubble Space Telescope (HST), but much more diverse but lower resolution images from the Sloan Digital Sky Survey (SDSS), which may be more appropriate. Further work was also put into investigating the feasibility of creating our own artificial images based on a relatively small number of starting parameters; this is known as 'procedural generation'. This has been done before using Neural Networks and Hydrodynamical simulations [28], but a considerably simpler (at least initially) approach was desired here. We therefore focused on acquiring actual astronomical imagery.</text>
<text><location><page_6><loc_12><loc_41><loc_88><loc_64></location>The actual astronomical imagery from the SDSS is available on their website. Properly identifying likely target galaxies is challenging. A list of targets, identified by [29], was processed into a list of likely candidates. The most promising targets had a large angular diameter distance, allowing the most high quality galaxies to be quickly identified. Creating a visually appealing astronomical image from the data acquired in the appropriate bands is a somewhat subjective process. Of vital importance is the 'stretch function' applied to the data, which applies a mathematical transformation to the pixel values to (ideally) enhance brighter areas and saturate darker areas, eliminating noise. In this project, the stretch function was applied using the 'Fits Liberator' [30] software, which offers a GUI to allow the user to select and tweak the stretch function parameters on the fly. The fits Liberator can also export the image into the GIMP [31] image manipulation software as Red, Green and Blue (RGB) components, creating the image. Some additional tweaking was required at this point. Background stars and galaxies have to be carefully erased (GIMP offers many tools to do this, the best of which are actually designed to interpolate out blemishes from images of human skin, but equally applicable to 'blemishes' in the sky), and cubic interpolation to re-scale all the images to the same size.</text>
<text><location><page_6><loc_13><loc_29><loc_88><loc_41></location>Finally, one of the most important steps is the construction of an 'alpha' channel. This image layer contains values that dictate the transparency of the corresponding pixel in the image; an alpha value of one would signify a fully opaque pixel, whereas an alpha value of zero would be fully transparent. Setting this up properly is vital for Astera, as the transparency is the property that will 'soften' a geometric mesh into a believable diffuse galaxy. The alpha channel was assigned in this case using the sum of the RGB layers, appropriately normalized 'by eye' to appear visually realistic. The importance of this channel is demonstrated in Figure 4.</text>
<figure>
<location><page_7><loc_15><loc_63><loc_48><loc_87></location>
</figure>
<figure>
<location><page_7><loc_52><loc_62><loc_84><loc_87></location>
<caption>Figure 4. Example depiction of the importance of the alpha channel. When projected onto each other, the alpha channel allows the galaxies to appear as diffuse objects, hiding the sharp edges.</caption>
</figure>
<text><location><page_7><loc_13><loc_48><loc_88><loc_55></location>Using this method, a few hundred distinct galaxies were extracted and processed from the SDSS dataset. Some examples are shown in Figure 5. While these galaxies are not as high resolution as HST images, their diversity allows for a wide range of galaxy types and morphological classifications. The Stellar Mass and Morphological types of each galaxy was also recorded for later use.</text>
<figure>
<location><page_7><loc_12><loc_18><loc_88><loc_47></location>
<caption>Figure 5. Composite images of 45 spiral galaxies extracted from the SDSS, processed to be visually pleasing.</caption>
</figure>
<section_header_level_1><location><page_7><loc_13><loc_13><loc_25><loc_14></location>3. Unreal Engine</section_header_level_1>
<text><location><page_7><loc_13><loc_9><loc_88><loc_12></location>The Unreal Engine is a game creation engine. Technically, a complete suite of creation tools, Unreal is best known as a package containing a rendering engine, sound engine, physics engine, gameplay</text>
<text><location><page_8><loc_12><loc_77><loc_88><loc_89></location>framework, animation, artificial intelligence, networking, memory management and parallel processing support. These reusable software components act like a vast library of tools that can be utilized by the game developer to assemble their game. Strictly speaking, references to the Unreal Engine in this paper indicate Unreal Engine 4 (sometimes referred to as 'UE4'), the fourth release of the software. Unreal Engine 4 significantly overhauled many of the features of Unreal Engine 3 when it was released in 2014, so many of the tools and features discussed and utilized as part of this project may not be available in earlier versions of the software. Unreal is freely available for non-commercial use.</text>
<text><location><page_8><loc_13><loc_67><loc_88><loc_77></location>There are various possible approaches to render galaxies in the Unreal Engine. It is possible to render Galaxies in real time as a system of diffuse particles. These systems are often both computationally expensive and visually unrealistic, so a different approach was considered in the development of Astera. Because each galaxy will be relatively small on the scales that we are interested in viewing, a single geometric object with an applied material (sampling a texture based on actual astronomical imagery) will suffice, and free up resources to show a greater quantity of galaxies in the game world.</text>
<text><location><page_8><loc_12><loc_50><loc_88><loc_66></location>This is done as follows. Spiral Galaxies are generally disc-shaped, and Ellipticals are vaguely spherical. Although it is possible to construct 3D versions of these shapes, a more geometrically complex shape is both more expensive to render in bulk and harder to properly configure with a material. Galaxies are diffuse objects, but a simple approximation allows for a 'Static Mesh' object that defines the geometric shape of the galaxy to be used as a visual proxy. A static mesh will have a 'material' applied to it, which can contain colours and textures (in this case, the galaxy image), but also a large amount of additional complexity (such as transparency or varying textures in time, used for AGN activity). Based on this, every galaxy in Astera is based on a simple two polygon plane to keep the geometry simple (see Figure 6, with additional 'work' being done by the material applied to it). As the galaxies will be kept relatively small with respect to the camera, this should not be problematic.</text>
<figure>
<location><page_8><loc_12><loc_19><loc_88><loc_48></location>
<caption>Figure 6. Awireframe view of a small area with Astera. The instanced static mesh objects used in Astera to represent galaxies can be seen.</caption>
</figure>
<text><location><page_8><loc_13><loc_9><loc_87><loc_14></location>This is a fair approximation for spiral galaxies. On the other hand, Elliptical galaxies are spheroids, so must be represented differently. The obvious choice would, therefore, be a spheroidal mesh. However, this does not correctly represent the diffuse nature of an Elliptical. A far more realistic choice requires some</text>
<text><location><page_9><loc_12><loc_77><loc_88><loc_89></location>visual trickery; if the elliptical mesh always appears 'face on', then a moving camera will always perceive the object as a diffuse sphere. This requires all elliptical galaxies to (for now) be essentially spherical, but this is an acceptable limitation. It may be possible in the future to dynamically change the scaling of the mesh to a greater extent on one axis based on the camera's position, thereby enabling more irregular ellipticals. Lenticular galaxies are problematic (as they require a combination of these effects) and are therefore structurally treated as spirals in the first version of Astera. Unreal Engine's material system is powerful enough to allow this modification to take place within the material itself.</text>
<text><location><page_9><loc_12><loc_67><loc_88><loc_77></location>Rendering a large number of objects inevitably leads to a performance penalty. It should be noted at this point that performance in real time rendering, although measurable in many ways, is generally parametrised using 'Frames Per Second' (FPS), a number representing the mean number of updates to the screen per second (higher is better). Values between 30-60 FPS are generally considered acceptable. Naturally, a more complex scene requires more computational resources, and may therefore result in a lower FPS.</text>
<text><location><page_9><loc_12><loc_48><loc_88><loc_66></location>Extraordinary care must be therefore taken when rendering large numbers of distinct mesh objects in Unreal. The best approach in this case requires careful thought. Graphical Processing Units (GPUs) are very efficient at drawing polygons, but need to be 'fed' these data by the (comparatively slow) CPU. Whenever a new object is 'drawn' to the screen, it requires a separate CPU call (going through the graphics driver). An alternative is to combine these objects into a single mesh; requiring only one draw call. The only disadvantage to this approach is it requires all objects to have the same material and textures. As we are 'duplicating' our galaxies anyway, this is not a problem; we simply create one object per available galaxy image asset. Although we are performing more draw calls, the overall number of CPU calls is still significantly lower (hundreds as opposed to millions), so the performance is still excellent. A mesh created in this way is called an 'instanced' static mesh (ISM), and is the technology that makes a universe as large as Astera's possible.</text>
<text><location><page_9><loc_12><loc_38><loc_87><loc_48></location>Based on this technology, spawning the objects takes place as follows (this process is implemented in C++, using Unreal Engine's Application Programming Interface). For each galaxy image extracted for the SDSS, a parent unreal actor (with no mesh of its own) is spawned and the image and the properties of the imaged galaxy are assigned to it. Next, each galaxy in the catalogue is assigned to the object with its 'nearest' properties in TType-Stellar Mass space. Practically, this means that the galaxy is assigned to the object that minimizes A in the following relation:</text>
<formula><location><page_9><loc_42><loc_34><loc_87><loc_36></location>A 2 = a D T 2 type + b D M 2 ∗ (2)</formula>
<text><location><page_9><loc_12><loc_20><loc_88><loc_33></location>where D Ttype is the difference in TType between the galaxy and the object, D M ∗ is the difference in stellar mass. The constants a and b are adjustable normalization factors to ensure the relative weighting is appropriate. If D M ∗ was in units of log 10 M glyph[circledot] , then a = b = 1 was found to be a reasonable choice. The end result is that there will, of course, be many duplicates; individually assigning unique galaxy images to every galaxy in the catalogue is computationally unfeasible, and the aforementioned ISM technology requires duplicates. The aim here is to have a sufficient number of unique galaxies so that duplicates are not noticeable to the user, but sufficiently few to preserve performance. An appropriate compromise was a few hundred unique galaxy images, compared to the millions of galaxies in the catalogue.</text>
<text><location><page_9><loc_12><loc_10><loc_88><loc_19></location>Next, each object spawns its galaxies, each represented by a mesh with its galaxy image applied as a material. Spiral galaxies are assigned random orientations. At this point, different features can be activated, such as the technology to make the mesh follow the camera, approximating an elliptical. All galaxies have the capability to be AGN, where a central oscillating bright source was added with an active period equal to its duty cycle. The light curve of an AGN is not yet well constrained-e.g., in [32]-so the shape of the curve is approximated by the peak of a sine wave to give smooth variation in brightness.</text>
<text><location><page_10><loc_12><loc_82><loc_87><loc_89></location>Naturally the period of this oscillation requires some level of approximation as to the ''rate of time' at which the user perceives the universe, but keeping this period reasonably long ( ∼ 60 seconds) presented a visually pleasing result. Note that this is not entirely realistic, as on the timescales of AGN phases galaxies will themselves have moved and evolved. This is not something that Astera (yet) considers.</text>
<section_header_level_1><location><page_10><loc_13><loc_79><loc_20><loc_80></location>4. Results</section_header_level_1>
<text><location><page_10><loc_13><loc_72><loc_88><loc_78></location>Screenshots from Astera are shown in Figures 7, 8, 9, 10 and 11. Note that, in these images, galaxy brightness has been enhanced to improve visual clarity. Nonetheless, the experience Astera offers is hard to communicate in a document such as this. We strongly encourage the reader to watch the video available at https://astera.soton.ac.uk/AsteraVid.mp4 2 , to get a full experience of what Astera can offer.</text>
<text><location><page_10><loc_13><loc_63><loc_88><loc_72></location>Astera is capable of displaying the entire galaxy catalogue with volume(1000 h -1 Mpc) 3 at 60 FPS (on a NVidia Titan GPU), with each galaxy represented by a static mesh containing a galaxy image, effectively creating a ''universe' for the user to explore. The player has control of the camera facing, position and movement in real time. The finite nature of the catalogue means that a user can exit the cube and view it from the ''outside' (see Figure 11).</text>
<text><location><page_10><loc_12><loc_56><loc_88><loc_63></location>Astera has also received preliminary adaption for Virtual Reality using the Unreal Engine virtual reality tools, which ports Astera from projected 2D to simulated 3D when viewed through a VR headset. This is an entirely visual effect, but allows for some level of binocular vision, enabling the 3D large scale structure of the universe to be experienced in binocular 3D for the first time.</text>
<figure>
<location><page_10><loc_12><loc_22><loc_88><loc_55></location>
<caption>Figure 7. A screenshot from Astera, showing some nearby galaxies in the foreground and a dense cluster/filament in the background.</caption>
</figure>
<figure>
<location><page_11><loc_12><loc_56><loc_88><loc_89></location>
<caption>Figure 8. Ascreenshot of Astera showing a relatively dense region. The structure of the cosmic web is just visible.</caption>
</figure>
<figure>
<location><page_11><loc_12><loc_17><loc_88><loc_50></location>
<caption>Figure 9. Ascreenshot showing a distant view of four clusters joined by a filament.</caption>
</figure>
<figure>
<location><page_12><loc_12><loc_60><loc_88><loc_89></location>
<caption>Figure 10. Alarge scale screenshot showing many millions of galaxies within Astera.</caption>
</figure>
<figure>
<location><page_13><loc_16><loc_38><loc_84><loc_88></location>
<caption>Figure 11. A colour-inverted view of the full simulation volume. The large scale cosmic web is clearly visible</caption>
</figure>
<section_header_level_1><location><page_13><loc_13><loc_32><loc_35><loc_33></location>5. Discussion and Conclusions</section_header_level_1>
<text><location><page_13><loc_13><loc_14><loc_87><loc_31></location>This project's objective was to create a visualization of a mock galaxy catalogue, rendered in real time. This would one of the first attempts to execute a fully realized real time rendering of the large scale universe. To power this project, the Unreal Engine 4 game engine was selected. A framework was constructed using C++ to read the data, create mesh instances and place the galaxies. The primary problem behind this task was performance; how many static mesh objects (representing galaxies) could be drawn to the screen at an acceptable frame rate. Remarkably, through the use of Unreal Engine's Instanced Static Mesh Technology, this was achieved to the extent that every galaxy in a full frame of a catalogue representing a (1000 h -1 Mpc) 3 box could be shown simultaneously on hardware running a NVidia Titan GPUat a full 60 FPS. Less powerful hardware can still run large volumes, with a NVidia GTX 760 rendering a (300 h -1 Mpc) 3 box at 60 FPS.</text>
<text><location><page_13><loc_12><loc_9><loc_87><loc_13></location>This was attained through intelligent use of Unreal's powerful and efficient Instanced Static Mesh Technology. Another success of Astera is its compatibility with any dataset. Any galaxy catalogue could easily be imported (within hardware constraints), a feature that could allow a researcher to replace the</text>
<text><location><page_14><loc_12><loc_79><loc_88><loc_89></location>default data and explore their own universe. Exploring the universe within Astera reveals the large scale cosmic structure in a way that is vastly easier to understand than a 2D image or even potentially a video. The author noted that several ''wall' like structures become visible in the cosmic web, which are not apparent in 2D imagery, in agreement with [33]. The distribution of galaxy mythologies also seem to follow established trends (see Figure 12), such as elliptical galaxies occupying the central regions of clusters-e.g., in [34].</text>
<figure>
<location><page_14><loc_12><loc_39><loc_88><loc_77></location>
<caption>Figure 12. A cluster of galaxies within Astera, where the large elliptical galaxies have been circled. The elliptical galaxies preferentially occupy the centre of the cluster, in line with observations.</caption>
</figure>
<text><location><page_14><loc_12><loc_22><loc_88><loc_33></location>Astera is not yet ready for public release. The authors are exploring options for the most appropriate way to release Astera to astronomers and non-astronomers alike. Astera's 'scope' is also relatively narrow, so as a relatively easy to use piece of software, Astera can be understood by the non-astronomer reasonably quickly, a feature that many similar Astronomical Visualizers do not have. Astera will soon be exhibited at a local attraction in Southampton at the Winchester Science Centre. This will feature a modified version of Astera, 'gamified' to allow the user to scan and classify galaxies to gain points, with the aim of attracting a new generation to extra-galactic astronomy.</text>
<text><location><page_14><loc_12><loc_11><loc_87><loc_21></location>From an academic perspective, there are unique advantages to creating a mock universe that is rendered in real time. The first is simply the visual examination of Mock Galaxy catalogues (e.g., in preparation for large surveys); while not necessarily statistically robust, a quick visual examination can often reveal issues that would not otherwise be easy to detect. A more nuanced approach could be to reproduce the effects of a program, such as SkyMaker [35], where a mock galaxy catalogue is used to create a simulated astronomical image. Crucially, Astera would render this in real time, allowing for alternative</text>
<text><location><page_15><loc_13><loc_86><loc_87><loc_89></location>images to be rapidly explored, or even a simulated sequence of images over a period of time, which could simulate data from time-variable objects, such as AGN.</text>
<text><location><page_15><loc_16><loc_84><loc_56><loc_85></location>Future 'features' that the authors are exploring include:</text>
<unordered_list>
<list_item><location><page_15><loc_15><loc_79><loc_87><loc_82></location>· Dark Matter Viewer. The conspicuous absence of Dark Matter in Astera would be remedied by a view mode that would show the dark matter substructure.</list_item>
<list_item><location><page_15><loc_15><loc_71><loc_88><loc_79></location>· Time Evolution. An exciting option which would essentially integrate Astera with a semi-analytic model, the motions and evolution of galaxies would be visible in (accelerated) real time. The user would be able to, at the press of a button, watch the universe evolve in front of them. This would dramatically increase the strain on the hardware to perform this on real time, so the volume of this universe might be limited.</list_item>
<list_item><location><page_15><loc_15><loc_66><loc_87><loc_70></location>· Gravitational Lensing. An ambitious proposal, where the weak gravitational lensing of large clusters could be visually shown. Obviously solving the full equations from General Relativity would not be viable, but it might be possible to develop a ''lens' object that acts as a close approximation.</list_item>
<list_item><location><page_15><loc_15><loc_61><loc_87><loc_65></location>· Gamification. As previously mentioned, Astera is a potentially invaluable outreach tool for increasing public awareness of the large scale universe. Gamifing Astera by introducing elements that make exploring the cosmological volume fun and educational could increase this value even further.</list_item>
</unordered_list>
<text><location><page_15><loc_13><loc_56><loc_88><loc_59></location>More information, visual materials and videos can be found on the Astera website https://astera. soton.ac.uk. We will also update this website with any future details of Astera's public release.</text>
<text><location><page_15><loc_12><loc_51><loc_88><loc_55></location>Author Contributions: Conceptualization, methodology, software, visualization, writing-original draft preparation; C. Marsden; methodology, supervision, project administration, writing-review and editing, funding acquisition; F. Shankar. All authors have read and agreed to the published version of the manuscript.</text>
<text><location><page_15><loc_13><loc_46><loc_87><loc_50></location>Funding: C. Marsden acknowledges the ESPRC funding for his PhD. F. Shankar acknowledges partial support from a Leverhulme Trust Research Fellowship. This project has benefited from an STFC IAA Grant. We also acknowledge the benefit of an NVidia GPU grant.</text>
<text><location><page_15><loc_12><loc_38><loc_88><loc_45></location>Acknowledgments: Special thanks to Ciera Sargent, Oliwia Krupa and the Nuffield Foundation. Also special thanks to Mariangela Bernardi and Peter Berhoozi for discussions on this project. We thank Miguel Aragon for helpful discussions and his excellent video that provided the inspiration for this work. We thank the Bolshoi and MultiDark simulations for their data, and the Sloan Digital Sky Survey for their galaxy imagery. This project has made extensive use of Unreal Engine 4.13 and associated development tools. We acknowledge extensive use of the Python libraries astropy, matplotlib, numpy, pandas, and scipy.</text>
<text><location><page_15><loc_13><loc_33><loc_88><loc_37></location>Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.</text>
<section_header_level_1><location><page_15><loc_12><loc_30><loc_23><loc_31></location>Abbreviations</section_header_level_1>
<text><location><page_15><loc_12><loc_28><loc_49><loc_29></location>The following abbreviations are used in this manuscript:</text>
<text><location><page_16><loc_13><loc_87><loc_32><loc_89></location>L CDM L Cold Dark Matter</text>
<text><location><page_16><loc_13><loc_86><loc_41><loc_87></location>HMSM Halo Mass-Stellar Mass (relation)</text>
<text><location><page_16><loc_13><loc_84><loc_40><loc_85></location>AGN Active Galactic Nucleus/Nuclei</text>
<text><location><page_16><loc_13><loc_83><loc_16><loc_84></location>SDSS</text>
<text><location><page_16><loc_19><loc_83><loc_35><loc_84></location>Sloan Digital Sky Survey</text>
<text><location><page_16><loc_13><loc_81><loc_15><loc_82></location>HST</text>
<text><location><page_16><loc_19><loc_81><loc_35><loc_82></location>Hubble Space Telescope</text>
<text><location><page_16><loc_13><loc_80><loc_16><loc_81></location>NFW</text>
<text><location><page_16><loc_19><loc_80><loc_39><loc_81></location>Navarro-Frenk-White (profile)</text>
<text><location><page_16><loc_13><loc_78><loc_15><loc_79></location>GUI</text>
<text><location><page_16><loc_19><loc_78><loc_34><loc_79></location>Grapical User Interface</text>
<text><location><page_16><loc_13><loc_77><loc_30><loc_78></location>RBG Red, Green, Blue</text>
<text><location><page_16><loc_13><loc_75><loc_16><loc_76></location>GIMP</text>
<text><location><page_16><loc_19><loc_75><loc_42><loc_76></location>GNUImage Manipulator Program</text>
<text><location><page_16><loc_13><loc_73><loc_32><loc_75></location>FPS Frames Per Second</text>
<text><location><page_16><loc_13><loc_72><loc_35><loc_73></location>CPU Central Processing Unit</text>
<text><location><page_16><loc_13><loc_70><loc_36><loc_71></location>GPU Graphics Processing Unit</text>
<text><location><page_16><loc_13><loc_69><loc_30><loc_70></location>UE4 Unreal Engine 4</text>
<section_header_level_1><location><page_16><loc_13><loc_65><loc_21><loc_67></location>References</section_header_level_1>
<unordered_list>
<list_item><location><page_16><loc_13><loc_61><loc_87><loc_64></location>1. Zel'Dovich, Y.B. Reprint of 1970A&A.....5...84Z. Gravitational instability: an approximate theory for large density perturbations. Astronomy and Astrophysics 1970 , 500 , 13-18.</list_item>
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<text><location><page_18><loc_28><loc_13><loc_88><loc_17></location>©2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).</text>
</document> | [] |
2017arXiv171007906B | https://arxiv.org/pdf/1710.07906.pdf | <document>
<section_header_level_1><location><page_1><loc_19><loc_70><loc_76><loc_80></location>Nonlinear anisotropy growth in Bianchi-I spacetime in metric f ( R ) cosmology</section_header_level_1>
<text><location><page_1><loc_25><loc_56><loc_71><loc_67></location>Kaushik Bhattacharya, Saikat Chakraborty ∗ Department of Physics, Indian Institute of Technology, Kanpur 208016, India January 9, 2019</text>
<section_header_level_1><location><page_1><loc_44><loc_51><loc_52><loc_52></location>Abstract</section_header_level_1>
<text><location><page_1><loc_19><loc_24><loc_76><loc_50></location>The present work is related to anisotropic cosmological evolution in metric f ( R ) theory of gravity. The initial part of the paper develops the general cosmological dynamics of homogeneous anisotropic BianchiI spacetime in f ( R ) cosmology. The anisotropic spacetime is pervaded by a barotropic fluid which has isotropic pressure. The paper predicts nonlinear growth of anisotropy in such spacetimes. In the later part of the paper we display the predictive power of the nonlinear differential equation responsible for the cosmological anisotropy growth in various relevant cases. We present the exact solutions of anisotropy growth in Starobinsky inflation driven by quadratic gravity and exponential gravity theory. Semi-analytical results are presented for the contraction phase in quadratic gravity bounce. The various examples of anisotropy growth in Bianchi-I model universe shows the complex nature of the problem at hand.</text>
<section_header_level_1><location><page_1><loc_14><loc_18><loc_36><loc_20></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_14><loc_10><loc_81><loc_16></location>The issue related to stability of homogeneous and isotropic cosmological solutions with respect to small anisotropy has been studied intensely in theoretical cosmology [1], [2], [3], [4]. Behavior of small anisotropy has been studied in cosmological</text>
<text><location><page_2><loc_14><loc_49><loc_81><loc_86></location>models, using general relativity (GR), in the contexts of inflation [5], [6], [7], [8], [9] and pre-bounce ekpyrotic contraction phase [10], [11], [12], [13]. In the context of inflation the 'No-Hair' conjecture asserts that any pre-existing anisotropy must asymptotically die out in an inflating universe. Wald has been able to prove the conjecture for all the Bianchi models except Bianchi-IX [14] which requires a large cosmological constant to isotropize the spacetime . In a contracting universe, provided the universe is dominated by a matter component mimicking an ultra stiff barotropic fluid, growth of small anisotropy is suppressed with respect to that of the Hubble parameter. In absence of any such fluid in a contracting phase, small anisotropy grows large and dominates over all other matter components. This leads to the Belinsky-Khalatnikov-Lifshitz (BKL) instability [15], foiling the bounce. Mathematically, it can be shown that in presence of a slowly rolling scalar field the isotropic de-Sitter solution is an attractor for an expanding universe and in presence of a fast rolling scalar field the isotropic power law solution (for the scale-factor) is an attractor for contracting universe. Therefore an inflationary scenario is usually realized by a slowly rolling scalar field and an ekpyrotic scenario is usually realized by a fast rolling scalar field [16], [17], [18], [19], [20].</text>
<text><location><page_2><loc_14><loc_17><loc_81><loc_49></location>In the present work we have analyzed the evolution of spacetime anisotropy in f ( R ) gravity 1 , where the analysis becomes significantly more involved than that of models based on GR. Some attempts in modified, quadratic gravity [22,23] do discuss about anisotropic cosmologies while analyzing the past stages of a cosmological system near the singularity. In these works the authors show that near the singularity the universe may have an anisotropic mode of existence. The papers in general do not address a cosmological bounce scenario. Although previously there have been some progress in generalizing the no-hair theorem to incorporate higher order gravity theories [24], [25], [26], [27] and some applications of dynamical system analysis to understand anisotropic cosmology in higher order gravity [28], [29], [30], the previous attempts missed an important property of anisotropic cosmological dynamics related to nonlinear growth of anisotropy in the homogeneous and anisotropic Bianchi-I type of spacetime. In the present paper we first show analytically that in Starobinsky inflation any initial anisotropy will rapidly fade away.</text>
<text><location><page_2><loc_17><loc_14><loc_81><loc_16></location>It is then shown that in contraction phases in Bianchi-I metric where speci-</text>
<text><location><page_3><loc_14><loc_58><loc_81><loc_86></location>fying an unique scale-factor for contraction does not always yield a unique cosmological development. This result is possible in f ( R ) cosmology and in GR one cannot have this property. This non-uniqueness of cosmological development corresponding to a specific scale-factor opens up a new problem as cosmological evolution becomes more complex conceptually and as a consequence only simple f ( R ) models can be semi-analytically solved. Any f ( R ) model which is a higher order polynomial in R compared to the quadratic f ( R ) model requires a complete numerical solution for anisotropy growth. We show our result in quadratic f ( R ) model, which is gravitationally unstable if it has to accommodate a cosmological bounce. Our work may be taken as an effective toy model which is used to crack a formidable problem in cosmological dynamics. To our understanding the above mentioned topics are discussed for the first time in full generality in the present paper.</text>
<text><location><page_3><loc_14><loc_43><loc_81><loc_57></location>We also present some preliminary results of non-linear anisotropy growth in exponential gravity models. The exponential gravity model has two exact solutions. One exact solution is a bouncing solution in presence of matter and the other exact solution is a expanding universe solution at a de-Sitter point. In these cases the anisotropy generation equation turns out to be a transcendental equation. We present some simple solutions in this case probing the nature of growth of small initial anisotropy.</text>
<text><location><page_3><loc_14><loc_21><loc_81><loc_42></location>The material in the paper is organized in the following way. The second section discusses about the basics of Bianchi-I spacetimes and sets the notations and conventions followed throughout the paper. In section 3 we present the general formalism of homogeneous and anisotropic cosmological dynamics in metric f ( R ) cosmology. This part contains important results. In this section for the first time one comes across the complex nature of anisotropy development. In section 4 we present the results related to nonlinear anisotropy growth in quadratic f ( R ) theory induced inflation and cosmological bounce. Section 5 presents the results of anisotropy growth for some exact results in exponential gravity. The next section is the concluding section where we summarize the results obtained in the paper.</text>
<section_header_level_1><location><page_4><loc_14><loc_81><loc_81><loc_86></location>2 The anisotropic Bianchi-I metric and its properties</section_header_level_1>
<text><location><page_4><loc_14><loc_77><loc_69><loc_79></location>For our analysis, we have used the metric for Bianchi-I spacetime,</text>
<formula><location><page_4><loc_29><loc_73><loc_81><loc_75></location>ds 2 = -dt 2 + a 2 1 ( t ) dx 2 1 + a 2 2 ( t ) dx 2 2 + a 2 3 ( t ) dx 2 3 , (1)</formula>
<text><location><page_4><loc_14><loc_50><loc_81><loc_71></location>where a 1 ( t ), a 2 ( t ) and a 3 ( t ) are the different scale-factors, whose relative differences specify the amount of anisotropy in the evolving universe. Existence of such anisotropic cosmological models in higher order gravity theories have been extensively studied in literature [31], [32], [33], [34]. In most of the earlier attempts the authors have tried to find out the nature of anisotropic spacetimes using various forms of anisotropic metric and using various forms of gravitational Lagrangians. In the present paper we show that the previous attempts have missed a vital ingredient in anisotropic expansion/contraction. The effect we discuss is clearly visible in Bianchi-I spacetime, but we think similar effects may be present in other anisotropic cosmological models.</text>
<text><location><page_4><loc_14><loc_33><loc_81><loc_50></location>In this paper we will assume the presence of a perfect hydrodynamic fluid, in the Bianchi type-I spacetime, whose energy-momentum tensor (EMT)is given by T µν = ( ρ + P ) u µ u ν + Pg µν , where ρ is the energy-density and P specifies isotropic pressure of the perfect fluid. The 4-velocity of the fluid element is given by u µ , which being a time-like vector is normalized as u µ u µ = -1 . Although the spacetime metric is anisotropic the fluid which pervades the spacetime is assumed to be isotropic. In this paper we will assume a barotropic equation of state for the perfect fluid, P = ωρ, where ω specifies the barotropic ratio.</text>
<text><location><page_4><loc_14><loc_22><loc_81><loc_32></location>One can rewrite the form of the anisotropic metric, given in Eq. (1), in terms of the (geometric) average of the three scale-factors given by a ( t ) = [ a 1 ( t ) a 2 ( t ) a 3 ] 1 / 3 . The three different scale-factors in terms of the geometric average scale-factor can be written as, a i ( t ) = a ( t ) e β i ( t ) , where i = 1 , 2 , 3 2 . The time dependent functions β i ( t ) specify the anisotropy in the metric and they are constrained as</text>
<formula><location><page_4><loc_39><loc_18><loc_81><loc_20></location>β 1 + β 2 + β 3 = 0 . (2)</formula>
<text><location><page_4><loc_14><loc_15><loc_78><loc_16></location>Using the above relations one can now rewrite the metric given in Eq. (1) as</text>
<formula><location><page_4><loc_23><loc_11><loc_81><loc_12></location>ds 2 = -dt 2 + a 2 ( t ) [ e 2 β 1 ( t ) dx 2 1 + e 2 β 2 ( t ) dx 2 2 + e 2 β 3 ( t ) dx 2 3 ] . (3)</formula>
<text><location><page_5><loc_14><loc_82><loc_81><loc_86></location>In this notation one can define the Hubble parameter, as an arithmetic average, and its time-derivative as</text>
<formula><location><page_5><loc_25><loc_78><loc_81><loc_81></location>H ( t ) ≡ 1 3 ( ˙ a 1 a 1 + ˙ a 2 a 2 + ˙ a 3 a 3 ) = ˙ a a , ˙ H ( t ) = a a -˙ a 2 a 2 . (4)</formula>
<text><location><page_5><loc_14><loc_71><loc_81><loc_76></location>Mainly for the sake of brevity, henceforth in this article we will omit the word average (either geometric or arithmetic) before scale-factor or Hubble parameter. In presence of anisotropy the Ricci scalar turns out to be</text>
<formula><location><page_5><loc_32><loc_67><loc_81><loc_69></location>R = 6( ˙ H +2 H 2 ) + ( ˙ β 1 2 + ˙ β 2 2 + ˙ β 3 2 ) . (5)</formula>
<text><location><page_5><loc_14><loc_55><loc_81><loc_65></location>In the next section we formulate the anisotropic cosmological dynamics guided by metric f ( R ) theory. The fact that the derivatives of the anisotropy parameters are themselves present in the expression of the Ricci scalar will make anisotropic cosmological dynamics much more involved in f ( R ) gravity, compared to the general relativistic case.</text>
<section_header_level_1><location><page_5><loc_14><loc_46><loc_81><loc_52></location>3 Formulation of anisotropic cosmological dynamics guided by metric f ( R ) theory</section_header_level_1>
<text><location><page_5><loc_14><loc_43><loc_45><loc_44></location>The field equation in f ( R ) gravity is</text>
<formula><location><page_5><loc_34><loc_38><loc_81><loc_42></location>G µ ν = κ f ' ( R ) [ T µ ν + T µ ν (curv) ] , (6)</formula>
<text><location><page_5><loc_14><loc_25><loc_81><loc_37></location>where G µ ν is the Einstein tensor and T µ ν (curv) is the energy momentum tensor due to curvature. Here κ = 8 πG , where G is the Newton's gravitational constant and is related to the Planck mass M P via G = 1 /M 2 P . In the present paper we will approximately use M P ≈ 10 19 GeV. The prime on top right hand side of any function represents the ordinary derivative of that function with respect to the Ricci scalar R . In particular</text>
<formula><location><page_5><loc_18><loc_20><loc_81><loc_23></location>T µ ν (curv) ≡ 1 κ [ -( Rf ' ( R ) -f ( R ) 2 + glyph[square] f ' ( R ) ) δ µ ν + g µα D α D ν f ' ( R ) ] (7)</formula>
<text><location><page_5><loc_14><loc_12><loc_81><loc_19></location>where D µ A ν is the covariant derivative of the covariant 4-vector A ν and glyph[square] ≡ g αβ D α D β . The 0 -0 component of the field equation in f ( R ) theory in an anisotropic spacetime is then given as, G 0 0 = -κ f ' ( R ) ( ρ + ρ curv ), where</text>
<formula><location><page_5><loc_28><loc_7><loc_81><loc_11></location>ρ curv = 1 κ [ Rf ' ( R ) -f ( R ) 2 -3 H ˙ Rf '' ( R ) ] . (8)</formula>
<text><location><page_6><loc_14><loc_77><loc_81><loc_86></location>The other three equations, specifying the i -j components become, G i j = κ f ' ( R ) ( T i j + T i j (curv) ) , where i, j = 1 , 2 , 3. Here T i j = Pδ i j stands for pressure of the perfect hydrodynamic fluid(s) whose EMT(s) has(have) the same form as specified in section 2. The form of T i j (curv) is given as</text>
<formula><location><page_6><loc_17><loc_70><loc_81><loc_76></location>κT i j (curv) = -[ Rf ' ( R ) -f ( R ) 2 -Rf '' ( R ) -˙ R 2 f ''' ( R ) -2 H ˙ Rf '' ( R ) ] δ i j -B i j ˙ Rf '' ( R ) , (9)</formula>
<text><location><page_6><loc_14><loc_66><loc_59><loc_68></location>where the components of the tensor B i j are defined as</text>
<text><location><page_6><loc_64><loc_63><loc_64><loc_64></location>glyph[negationslash]</text>
<formula><location><page_6><loc_24><loc_63><loc_81><loc_65></location>B 1 1 = ˙ β 1 , B 2 2 = ˙ β 2 , B 3 3 = ˙ β 3 , B i j = 0 , if , i = j . (10)</formula>
<text><location><page_6><loc_14><loc_60><loc_57><loc_61></location>In terms of the above quantities one can now write,</text>
<formula><location><page_6><loc_29><loc_55><loc_81><loc_58></location>G i j = κ f ' ( R ) ( P + P curv ) δ i j -B i j ˙ Rf '' ( R ) , (11)</formula>
<text><location><page_6><loc_14><loc_52><loc_19><loc_53></location>where</text>
<formula><location><page_6><loc_29><loc_47><loc_81><loc_51></location>P curv = ˙ R 2 f ''' +2 H ˙ Rf '' + Rf '' κ -Rf ' -f 2 κ . (12)</formula>
<text><location><page_6><loc_14><loc_43><loc_81><loc_46></location>In terms of the Hubble parameter and the anisotropy parameter, the 0 -0 component of the field equation becomes</text>
<formula><location><page_6><loc_31><loc_37><loc_81><loc_41></location>H 2 = κ 3 f ' ( R ) ( ρ + ρ curv ) + 1 6 3 ∑ i =1 ˙ β 2 i , (13)</formula>
<text><location><page_6><loc_14><loc_34><loc_34><loc_35></location>while Eq. (11) becomes</text>
<formula><location><page_6><loc_17><loc_28><loc_81><loc_32></location>2 ˙ H +3 H 2 -3 H ˙ β i -¨ β i + 1 2 3 ∑ i =1 ˙ β 2 i = -κ f ' ( R ) ( P + P curv ) + ˙ β i ˙ R f '' ( R ) f ' ( R ) . (14)</formula>
<text><location><page_6><loc_14><loc_23><loc_81><loc_26></location>Adding the three equations, corresponding to each value of the index i in the above expression, one gets</text>
<formula><location><page_6><loc_27><loc_17><loc_81><loc_21></location>2 ˙ H +3 H 2 = -κ f ' ( R ) ( P + P curv ) -1 2 3 ∑ i =1 ˙ β 2 i . (15)</formula>
<text><location><page_6><loc_14><loc_13><loc_66><loc_15></location>If one uses Eq. (13) in the above equation then one gets ˙ H as</text>
<formula><location><page_6><loc_24><loc_7><loc_81><loc_12></location>˙ H = -κ 2 f ' ( R ) [(1 + ω ) ρ +( ρ curv + P curv )] -1 2 3 ∑ i =1 ˙ β 2 i , (16)</formula>
<text><location><page_7><loc_14><loc_82><loc_81><loc_86></location>where P = ωρ has been used. In the present case we define the anisotropy factor x as</text>
<formula><location><page_7><loc_39><loc_77><loc_81><loc_81></location>x 2 ( t ) ≡ 3 ∑ i =1 ˙ β 2 i ( t ) . (17)</formula>
<text><location><page_7><loc_14><loc_63><loc_81><loc_76></location>For an isotropic universe x 2 = 0 for all values of t , implying that all the β i 's are constant in time. In such a case one can appropriately make (time-independent) coordinate rescaling in an appropriate way to make the spacetime look exactly like the Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime. Using Eq. (2) and Eq. (14) and the above definition of the anisotropy factor one can show that x satisfies the differential equation:</text>
<formula><location><page_7><loc_35><loc_58><loc_81><loc_62></location>˙ x + ( 3 H + ˙ f ' ( R ) f ' ( R ) ) x = 0 , (18)</formula>
<text><location><page_7><loc_14><loc_55><loc_47><loc_56></location>whose (nontrivial) solution must be like</text>
<formula><location><page_7><loc_39><loc_50><loc_81><loc_53></location>x = b a 3 ( t ) f ' ( R ) , (19)</formula>
<text><location><page_7><loc_14><loc_32><loc_81><loc_49></location>where b is a real integration constant. The above equation contains the most important theoretical ingredient of the present paper. The Ricci scalar in the present case can be written as R = 6( ˙ H +2 H 2 ) + x 2 which depends on x and the last equation shows x is a function of R in f ( R ) gravity. As a consequence of the above relation in f ( R ) gravity, one cannot define an unique anisotropy dynamics. For any given f ( R ) gravity, in general multiple time evolutions of the anisotropy factor x is possible, each corresponding to a different equation of state for the barotropic fluid. In the present case the time derivative of the Ricci scalar is</text>
<formula><location><page_7><loc_35><loc_26><loc_81><loc_31></location>˙ R = 6( H +4 H ˙ H -Hx 2 ) ( 1 + 2 f '' ( R ) f ' ( R ) x 2 ) . (20)</formula>
<text><location><page_7><loc_14><loc_23><loc_45><loc_25></location>Working out similarly one can write,</text>
<formula><location><page_7><loc_17><loc_13><loc_81><loc_22></location>R = 6( ... H +4 H H +4 ˙ H 2 ) -2 [ ( 3 ˙ H + ˙ R 2 f ''' f ' -˙ f ' 2 f ' 2 ) -2 ( 3 H + ˙ f ' f ' ) 2 ] x 2 ( 1 + 2 x 2 f '' f ' ) . (21)</formula>
<text><location><page_7><loc_14><loc_8><loc_81><loc_11></location>The above equations show that once we know the form of x in terms of the scalefactor, we can write the values of R , ˙ R and R in terms of the scale-factor. The</text>
<text><location><page_8><loc_14><loc_82><loc_81><loc_86></location>cosmological dynamics of anisotropic f ( R ) theory is encoded in Eq. (16), Eq. (19) and the energy-momentum conservation equation</text>
<formula><location><page_8><loc_37><loc_79><loc_81><loc_80></location>˙ ρ +3 Hρ (1 + ω ) = 0 . (22)</formula>
<text><location><page_8><loc_14><loc_64><loc_81><loc_76></location>The specification of ω and this set of three equations and the initial conditions specifying a , ˙ a , a , ... a , b and initial ρ are enough to specify the anisotropic dynamics in f ( R ) cosmology if Eq. (19) has an unique root. If Eq. (19) does not have an unique root then the initial conditions must have to be enhanced. In the next section we will show when the above list of initial conditions require to be enhanced.</text>
<text><location><page_8><loc_14><loc_51><loc_82><loc_63></location>The first order differential equation in Eq. (18) specifies the growth of anisotropy in metric f ( R ) gravity models where spacetime is specified by a Bianchi-I model. From the form of the equation it is seen that the amount of nonlinearity in Eq. (18) depends upon time. It can be noted that x = 0 have some interesting properties. The first thing to note about this point is that at x = 0 one always has ˙ x = 0 . The other interesting properties about this point are as follows.</text>
<unordered_list>
<list_item><location><page_8><loc_16><loc_39><loc_81><loc_49></location>1. If the system resides at the point x = 0 then is is impossible to perturb the system to have non-zero values of x . The system can have x = 0 value only when b = 0, and b is specified by the initial condition. Consequently if the initial condition is such that x = 0 then there will be no anisotropy growth in the future.</list_item>
</unordered_list>
<text><location><page_8><loc_65><loc_35><loc_65><loc_37></location>glyph[negationslash]</text>
<unordered_list>
<list_item><location><page_8><loc_16><loc_31><loc_81><loc_36></location>2. On the other hand if the initial condition is such that b = 0 then the system will never reach x = 0 unless a 3 ( t ) f ' ( R ) diverges in finite time, signifying a cosmological singularity.</list_item>
</unordered_list>
<text><location><page_8><loc_50><loc_27><loc_50><loc_29></location>glyph[negationslash]</text>
<text><location><page_8><loc_14><loc_19><loc_81><loc_29></location>As x cannot be zero in the future if b = 0 in a non-singular cosmology, the important parameter which keeps track of effective anisotropy is given by the factor x 2 /H 2 . From Eq. (13), Eq. (15) and the expression of the Ricci scalar, R , it can be verified that when x 2 /H 2 glyph[lessmuch] 1 one can safely neglect the effect of anisotropy in the cosmological dynamics of Bianchi-I type models.</text>
<section_header_level_1><location><page_9><loc_14><loc_81><loc_88><loc_86></location>4 Evolution of the anisotropic factor x ( t ) in quadratic gravity</section_header_level_1>
<text><location><page_9><loc_14><loc_77><loc_55><loc_79></location>In this section we will focus on quadratic gravity</text>
<formula><location><page_9><loc_39><loc_74><loc_81><loc_75></location>f ( R ) = R + αR 2 , (23)</formula>
<text><location><page_9><loc_14><loc_55><loc_81><loc_71></location>where α is a real number. Although this is a simple form of f ( R ) but it can be used to model cosmological inflation as well as cosmological bounce for positive and negative signs of the constant α respectively 3 . In this section we will determine the evolution of x ( t ) in quadratic gravity. The technique of evolution of x ( t ) in higher order gravity will be similar but much more involved. For higher order polynomial gravity the order of the algebraic equation yielding the roots of x ( t ) may be five (or higher) and consequently there does not exist any general algebraic formalism yielding those roots.</text>
<text><location><page_9><loc_14><loc_50><loc_81><loc_54></location>From Eq. (19) one can easily verify that the algebraic equation specifying x ( t ) in quadratic gravity is a cubic equation of the form:</text>
<formula><location><page_9><loc_38><loc_46><loc_81><loc_48></location>x 3 + A 1 x + A 2 = 0 , (24)</formula>
<text><location><page_9><loc_14><loc_43><loc_19><loc_44></location>where</text>
<formula><location><page_9><loc_36><loc_38><loc_81><loc_42></location>A 1 = 6( ˙ H +2 H 2 ) + 1 2 α , (25)</formula>
<formula><location><page_9><loc_36><loc_34><loc_81><loc_38></location>A 2 = -b 2 αa 3 . (26)</formula>
<text><location><page_9><loc_14><loc_32><loc_76><loc_33></location>The discriminant, ∆, specifying the roots and their properties is given by</text>
<formula><location><page_9><loc_38><loc_28><loc_81><loc_30></location>∆ = -4 A 3 1 -27 A 2 2 . (27)</formula>
<text><location><page_9><loc_14><loc_20><loc_81><loc_26></location>If ∆ > 0 there will be three distinct real roots, if ∆ < 0 then there will be one real root (and two complex roots) and if ∆ = 0 there can be repeated real roots. The roots of Eq. (24) are as follows:</text>
<formula><location><page_9><loc_23><loc_13><loc_81><loc_19></location>x = { -(2 / 3) 1 / 3 A 1 ( -9 A 2 + √ -3∆) 1 / 3 + ( -9 A 2 + √ -3∆) 1 / 3 2 1 / 3 3 2 / 3 , (1 ± i √ 3) A 1 2 2 / 3 3 1 / 3 ( -9 A 2 + √ -3∆) 1 / 3 -(1 ∓ i √ 3)( -9 A 2 + √ -3∆) 1 / 3 2 4 / 3 3 2 / 3 . (28)</formula>
<text><location><page_10><loc_14><loc_84><loc_76><loc_86></location>In terms of trigonometric functions the above roots can be represented as,</text>
<formula><location><page_10><loc_26><loc_78><loc_81><loc_84></location>x = 2 √ -A 1 3 cos [ 1 3 tan -1 ( √ 3∆ -9 A 2 )] , -2 √ -A 1 3 cos [ 1 3 ( π ∓ tan -1 ( √ 3∆ -9 A 2 ))] . (29)</formula>
<text><location><page_10><loc_14><loc_60><loc_81><loc_76></location>In the numerical calculations we will use the above form of the roots as they are less cumbersome to handle when all the roots are real. If initially the three roots of Eq. (24) are all real then one does require a separate initial condition, specifying a particular initial root out of the three possible roots, to describe the anisotropic cosmological dynamics in f ( R ) gravity. On the other hand if there is only one real root then the added initial condition looses its significance and the initial conditions as specified in the last section is enough to describe the cosmological dynamics.</text>
<text><location><page_10><loc_14><loc_38><loc_81><loc_59></location>From Eq. (19) it is seen that the anisotropy factor depends upon a , ˙ a , a and the integration constant b . More over from the form of the cubic equation followed by x it can be easily seen that out of the three roots one tends to vanishes when b → 0, where as the other two roots in general do not tend to zero when b becomes arbitrarily small. If all the roots are real then the root which vanishes when b vanishes plays an important role as in this case one can tune the value of the initial anisotropy by tuning the value of b . When the system admits only one real root then this root always tends to zero when b tends to zero. In GR, when one deals with anisotropic Bianchi Type-I cosmology, the equation followed by the anisotropy factor is x ( t ) = b/a 3 ( t ) and hence no such complications arise.</text>
<section_header_level_1><location><page_10><loc_14><loc_33><loc_44><loc_34></location>4.1 Starobinsky inflation</section_header_level_1>
<text><location><page_10><loc_14><loc_8><loc_81><loc_31></location>We can now apply our formalism to get the first nontrivial result related to anisotropic cosmological dynamics in Starobinsky's model of inflation. In this model of inflation the universe inflates in absence of any hydrodynamic fluid. In quadratic gravity inflation the parameter α appearing in Eq. (23) is always positive which makes f ' ≡ df/dR > 1. In presence of anisotropy the inflating spacetime shows very fast growth in the average scale-factor a ( t ) while the average Hubble parameter satisfies the condition ˙ H = -glyph[epsilon1]H 2 where glyph[epsilon1] is a slow-roll parameter. During inflation glyph[epsilon1] glyph[lessmuch] 1 and this condition prevails until glyph[epsilon1] ∼ 1 at the end of inflation [36]. In the excellent review on Starobinsky inflation given in Ref. [36] the authors use the slow-roll approximation to derive the properties of the inflating FLRW spacetime. In this paper we will use the conventions of</text>
<text><location><page_11><loc_14><loc_76><loc_81><loc_86></location>the above reference but will not exactly apply slow-roll mechanism. We will use the full f ( R ) theory equations as discussed in the last section with specific inflationary initial condition which gives rise to a rapidly expanding universe. In a later publication we want to generalize slow-roll conditions in quadratic gravity inflation in anisotropic Bianchi-I spacetimes.</text>
<text><location><page_11><loc_14><loc_56><loc_81><loc_75></location>In this subsection we show that any kind of anisotropy, if present initially, will be damped during the inflationary phase in quadratic gravity. We analytically prove our result for large initial anisotropy, the proof remains the same for small initial anisotropy. If initially the anisotropy factor was large then inflation will successfully isotropize the universe and there will be no remaining anisotropy at the end of inflation. Although this fact is known to be true in inflationary models based on GR [14], in this article we show that similar outcome is also expected in quadratic theory of inflation. First we show that the maximum anisotropy allowed in quadratic gravity, during inflation, has a maximum bound:</text>
<formula><location><page_11><loc_41><loc_52><loc_81><loc_55></location>x ≤ √ 6 H, (30)</formula>
<text><location><page_11><loc_14><loc_36><loc_81><loc_50></location>consequently the maximum anisotropy which can be isotropized is related with the Hubble parameter. To prove the above assumption one must note that in quadratic gravity inflation, ρ = 0, and the inflationary phase is initiated by curvature energy density ρ curv . The anisotropy energy contribution, in Eq. (13), is non-negative and consequently for inflation to start initially (when ideally the anisotropy effect is maximum) ρ curv > 0. The fact that the curvature energydensity appearing in the constraint Eq. (13) as:</text>
<formula><location><page_11><loc_39><loc_31><loc_56><loc_34></location>H 2 = κρ curv 3 f ' ( R ) + x 2 6 ,</formula>
<text><location><page_11><loc_14><loc_28><loc_59><loc_29></location>cannot be negative during inflation justifies Eq. (30).</text>
<text><location><page_11><loc_14><loc_8><loc_81><loc_27></location>Even if the initial anisotropy present in the universe is given by the maximum bound of x in Eq. (30) the anisotropy factor rapidly fades away during quadratic inflation. To show this we first note that in Starobinsky inflation A 1 > 0, as | ˙ H | glyph[lessmuch] H 2 . As a result the discriminant ∆ < 0 implying that there is only one unique real root of Eq.(24). This root is given by the top right hand side term in Eq. (28). From the expression of A 2 one can see that it rapidly diminishes in an inflating universe and one can assume A 3 1 > A 2 2 after some time from the onset of inflation. Assuming that A 2 → 0 rapidly after inflation starts one can easily show that the relevant real root of Eq. (30) tends to zero during inflation.</text>
<figure>
<location><page_12><loc_29><loc_69><loc_66><loc_86></location>
<caption>Figure 1: Plot showing evolution of H and x in logarithmic scale during inflationary phase. Relatively large initial anisotropy x large corresponding to b = 10 -5 is shown by the dashed line. Small initial anisotropy x small corresponding to b = 10 -10 is plotted by the dotted line. The Hubble parameter remains approximately the same in both the cases. Time axis spans from 10 6 to 2 × 10 6 and initial H ∼ 2 × 10 -6 (for both the cases of large and small initial anisotropy). Planck units are used for time and H .</caption>
</figure>
<text><location><page_12><loc_14><loc_11><loc_81><loc_50></location>We will now present a numerical solution of the cosmological dynamics during inflation and point out that the anisotropy factor x 2 /H 2 always remains sufficiently smaller and at no stages of quadratic inflation x 2 ≈ H 2 . Here we assume that inflation starts at, t i = 10 6 , in Planck units. In this unit system the actual value of a quantity is obtained by multiplying the value of the physical quantity by a particular power of Plank mass M P . The specific power corresponds to the mass dimension of the physical quantity. In the present case the actual value of t i is t i M -1 P . In this article we will assume M P ≈ 10 19 GeV and consequently t i = 10 -13 GeV -1 expressed in energy units. Expressed in the seconds t i ≈ 10 -37 s and that is 10 6 times Planck time expressed in seconds. The value of α is chosen as α = 10 12 which in natural units will be 10 12 M -2 P . Phenomenologically one expects quadratic correction to Einstein gravity at a very early phase of the universe when H ∼ 10 12 -13 GeV or more. For this benchmark value of the H the Ricci scalar R ∼ 10 26 GeV 2 assuming x 2 < H 2 . If the quadratic correction αR 2 becomes effective at such a value of R then α ∼ 1 /R yielding α ∼ 10 -26 GeV -2 = 10 12 M -2 P , justifying our choice of α . The initial Hubble parameter is chosen to be H ( t i ) = 2 × 10 -6 in Planck units, its value in standard units is 10 13 GeV. Inflation ends at t f ≈ t i +70 × H ( t i ) -1 = 36 × 10 6 ,</text>
<figure>
<location><page_13><loc_30><loc_68><loc_65><loc_86></location>
<caption>Figure 2: Evolution of the x 2 /H 2 in logarithmic scale for the cases corresponding to relatively large anisotropy where b = 10 -5 , in dashed curve and relatively small anisotropy, where b = 10 -10 , in dotted curve. Time span and initial Hubble parameter value remains the same as specified in the caption of Fig. 1.</caption>
</figure>
<text><location><page_13><loc_14><loc_46><loc_81><loc_54></location>which corresponds to slightly more than 70 e-folds. The consistent inflationary 4 initial conditions are written in terms of the initial values of the slow-roll parameters. The initial slow-roll parameters are chosen as glyph[epsilon1] ( t i ) = -˙ H/H 2 ≈ 7 × 10 -3 and η ( t i ) = H/ ( H ˙ H ) ≈ 0. The other initial conditions are as:</text>
<formula><location><page_13><loc_35><loc_43><loc_81><loc_44></location>a ( t i ) = 1 , (31)</formula>
<formula><location><page_13><loc_35><loc_40><loc_81><loc_42></location>˙ a ( t i ) = H ( t i ) a ( t i ) , (32)</formula>
<formula><location><page_13><loc_35><loc_36><loc_81><loc_40></location>a ( t i ) = (1 -glyph[epsilon1] )˙ a ( t i ) 2 a ( t i ) , (33)</formula>
<formula><location><page_13><loc_35><loc_32><loc_81><loc_36></location>... a ( t i ) = (1 -ηglyph[epsilon1] -3 glyph[epsilon1] )˙ a ( t i ) 3 a ( t i ) 2 . (34)</formula>
<text><location><page_13><loc_14><loc_21><loc_81><loc_31></location>Although the initial conditions for inflation in the present section are written in terms of the slow-roll parameters glyph[epsilon1] and η at initial time we do not evolve glyph[epsilon1] or η with time or use the slow-roll parameters in any other place in our calculation. The calculation do not use slow-roll approximation and the results we present in this subsection are exact results.</text>
<text><location><page_13><loc_14><loc_14><loc_81><loc_20></location>To get a numerical solution, we plug the solution x ( a, ˙ a, a ) into the dynamical equation, Eq. (16). The resulting dynamical equation for a ( t ) is a fourth order ordinary differential equation for the case of quadratic f ( R ) gravity. Looking at</text>
<text><location><page_14><loc_14><loc_78><loc_81><loc_86></location>the structure of the roots of the cubic equation, presented in the initial part of the present section, it is seen that in the present case the coefficient A 1 > 0 and consequently there will be only one real root of the anisotropy factor x ( t ). This root continuously tends to zero as b tends to zero.</text>
<text><location><page_14><loc_14><loc_41><loc_81><loc_77></location>We have plotted the numerical results showing the growth of the Hubble parameter and x , for small and large relative initial anisotropy, in logarithmic scales in Fig. 1. Small anisotropy, x small , corresponds to b = 10 -10 and relatively large anisotropy, x large , corresponds to b = 10 -5 . For both small and large anisotropies the scale-factor and the Hubble parameter are approximately the same showing that the overall inflating nature of the system does not depend upon the initial anisotropy present in the system. The inflationary nature of the present system is shown by the near constant value of H in Fig. 1. Fig. 2 shows the growth of the anisotropy factor x 2 /H 2 in the two cases corresponding to the two b values as discussed above. This plot clearly shows that anisotropy rapidly dies in quadratic gravity inflation. We have numerically verified that anisotropy gets wiped out after the first two or three e-folds of inflation and consequently suppression of anisotropy happens very efficiently in quadratic inflation. Here it must be noted that we cannot arbitrarily increase b as when b > 10 -4 the consistency condition in Eq. (30) is violated and the system does not inflate any more. Our formalism shows both analytically and numerically that Starobinsky inflation is safe from initial anisotropy.</text>
<section_header_level_1><location><page_14><loc_14><loc_36><loc_73><loc_37></location>4.2 Contraction in toy model of quadratic bounce</section_header_level_1>
<text><location><page_14><loc_14><loc_15><loc_81><loc_34></location>In this subsection we discuss anisotropic contraction phase in a simple and partly unstable model, guided by quadratic gravity. The presentation in this subsection is more like a toy model analysis which shows the complexities of anisotropic contraction in polynomial f ( R ) gravity models. In general the solution of Eq. (18) becomes a polynomial equation in x and for higher polynomial orders (compared to quadratic order) the algebraic equations do not yield analytic solutions. The quadratic gravity bounce model, where α < 0 is the simplest polynomial bounce model, where the intricacies of anisotropy generation during the contraction phase can be semi-analytically shown.</text>
<text><location><page_14><loc_14><loc_9><loc_81><loc_14></location>The issue of anisotropy generation during a contraction phase is very important as anisotropy may get enhanced during this phase as it happens in GR based models of cosmological bounce. We want to see how anisotropy grows in Bianchi-</text>
<text><location><page_15><loc_14><loc_54><loc_81><loc_86></location>I models in the contraction phase in quadratic f ( R ) gravity. In the present case we will assume the existence of hydrodynamic matter and α < 0 as these conditions are required for a subsequent bounce [35]. Before we proceed we will like to make some remarks related to the choice of the sign of α . The negative sign of α implies that f ' ( R ) is not always positive. We can choose our dynamical system to be such that it satisfies f ' ( R ) > 0 for some range of R , as done in the present paper. The importance of negative α quadratic model is that one can have a cosmological bounce in this restricted regime of R , where R < 1 / (2 | α | ) for stability. There is another source of instability in the present case, related to the negative sign of α . In such models f '' < 0 which may lead to instabilities first proposed by Dolgov and Kawasaki [37] and later by V. Faraoni [38]. In the present model one cannot get rid of Dolgov and Kawasaki instability 5 , consequently in light of the stability issues we will like to interpret the present model of bounce as a toy model whose sole purpose is to describe the nonlinear growth of anisotropy. In the a later section we will apply or formalism in a stable gravitational model.</text>
<text><location><page_15><loc_14><loc_41><loc_81><loc_53></location>In GR it is known that anisotropy suppression during contraction phase requires the presence of an ultra-stiff matter component with ω (= P/ρ ) > 1. The presence of an ultra-stiff matter component can produce a slow contraction phase where preexisting anisotropy is suppressed 6 . In the present case we will see that a power law contraction phase may suppress initial anisotropy in quadratic f ( R ) cosmology.</text>
<text><location><page_15><loc_14><loc_36><loc_81><loc_40></location>We assume that during the contracting phase t < 0 and bounce occurs at t = 0. During the contracting phase the scale-factor decreases as</text>
<formula><location><page_15><loc_32><loc_33><loc_81><loc_34></location>a ( t ) ∝ ( -t ) n , where 0 < n < 1 , (35)</formula>
<text><location><page_15><loc_14><loc_29><loc_29><loc_30></location>and consequently</text>
<formula><location><page_15><loc_44><loc_26><loc_51><loc_29></location>H = n t .</formula>
<text><location><page_15><loc_14><loc_21><loc_81><loc_25></location>From physical considerations one can choose α = -10 12 in Planck units [35]. Eliminating ρ in Eq. (16) by using Eq. (13) we get,</text>
<formula><location><page_15><loc_31><loc_16><loc_81><loc_20></location>ω = (4 ˙ H +6 H 2 + x 2 ) f ' +2 κP curv ( x 2 -6 H 2 ) f ' +2 κρ curv . (36)</formula>
<figure>
<location><page_16><loc_28><loc_68><loc_67><loc_86></location>
<caption>Figure 3: Plot of initial anisotropy x with respect to parameter b at t = -10 10 . The dashed and the continuous parts correspond to the (second and third) roots with the minus/plus signs after π in the second line on the right hand side of Eq. (29). The dotted part corresponds to the (first) root on the first line on the right hand side of Eq. (29). All the three roots are real near b = 0.</caption>
</figure>
<text><location><page_16><loc_14><loc_49><loc_81><loc_53></location>In the present case the above equation yields the equation of state for the barotropic matter when one specifies the particular nature of the scale-factor.</text>
<text><location><page_16><loc_14><loc_10><loc_81><loc_48></location>Determining the form of the time evolution of anisotropy factor reduces to finding the root(s) of Eq. (19). One can have various phases of anisotropy development during a cosmological evolution depending upon the roots of Eq. (19). In this paper we will particularly focus on the contracting phase of the universe leading to a cosmic bounce. We present the results for the popular quadratic f ( R ) model which actually accommodates a cosmological bounce [35], [40], [41], [42]. The nature of the anisotropic contraction phase predicted in this model will give a glimpse of the interesting effects of f ( R ) models of anisotropic contraction in the Bianchi-I spacetime. The plot in Fig. 3 shows the nature of the roots at time t = -10 10 in Planck units. The time period of contraction is chosen in such a way that all the constraints as f ' ( R ) > 0 and ρ > 0 are maintained during this phase of contraction. As the power law contraction can never lead to a bounce the constraints compel us to terminate the power law contraction process some time before the bounce and in this paper we use the time interval -10 10 ≤ t ≤ -10 7 . The scale-factor during this time is assumed to be a ( t ) = ( -t/ 10 10 ) n such that a ( t = -10 10 ) = 1. The nature of the roots show that below a certain b value and above a certain b value there is only one real root. Near b = 0 the system admits three real roots of x ( t ). We have verified that the nature of the root structure, as</text>
<figure>
<location><page_17><loc_17><loc_61><loc_52><loc_87></location>
<caption>Figure 4: Region in the b -n plane giving rise to decreasing x 2 /H 2 . Here the abscissa is specified by the b values and the ordinate is specified by n values.</caption>
</figure>
<text><location><page_17><loc_14><loc_27><loc_81><loc_48></location>specified in Fig. 3, does feebly depend upon n in the interval 0 ≤ n ≤ 1. The plot in Fig. 3 shows three branches in three colors. The middle green (continuous line) branch smoothly matches to the blue (dotted) branch above and the red (dashed) one below. The continuous branch specifies a root of Eq. (24) which is real near b = 0 and gives rise to small values of anisotropy factor x 0 = x ( t = -10 10 ) initially. The dashed and dotted branches specify the other roots which are large for regions near b = 0. The connection of the three regions in the figure with the roots in Eq. (29) are specified in the caption of Fig. 3. As time evolves the nature of the plot in Fig. 3 changes but the general structure of the plot always remains qualitatively similar as the one plotted at the initial time.</text>
<text><location><page_17><loc_14><loc_10><loc_81><loc_26></location>The dynamics of anisotropy growth depends upon the parameters b and n . We can specify the region in the b -n plane which gives rise to decreasing anisotropy. The plot in Fig. 4 shows such a region in the b -n plane. The plot is done at t = -10 10 , the initial time, when the region is most constrained. In Fig. 5 we show how x 2 /H 2 varies in time if one uses any value of b, n in the shaded region in Fig.4. In particular we have chosen b = 10 -12 and n = 1 / 4. The above information shows that for some parameter values a power law contraction can indeed suppress small anisotropy in quadratic gravity. Physically anisotropy</text>
<figure>
<location><page_17><loc_56><loc_61><loc_82><loc_75></location>
<caption>Figure 5: Decrease of anisotropy in time when b, n lies in the shaded region in Fig. 4. The specific b, n value chosen for the plot is given in the text.</caption>
</figure>
<text><location><page_18><loc_14><loc_71><loc_81><loc_86></location>suppression for some regions in the b -n plane in the contracting phase is not surprising as both x and H do increase in time during contraction when b and n belongs to the shaded region in Fig. 4 (as expected) but H increases more than x in time and as a consequence x 2 /H 2 decreases with time. For some parameter space H can grow faster than x in time, when anisotropy factor decreases, and for other parameter values x increases more than H in time making the contracting universe completely anisotropic.</text>
<section_header_level_1><location><page_18><loc_14><loc_66><loc_76><loc_68></location>5 Anisotropy growth in exponential gravity</section_header_level_1>
<text><location><page_18><loc_14><loc_58><loc_81><loc_63></location>Recently it has been shown [43] that one can get bouncing solutions and expanding universe solutions in a unstable de-Sitter point in exponential gravity where</text>
<formula><location><page_18><loc_36><loc_54><loc_81><loc_57></location>f ( R ) = 1 α e αR , α > 0 , (37)</formula>
<text><location><page_18><loc_14><loc_39><loc_81><loc_53></location>where α is a dimensional, real constant. In the present case as α > 0 we have f ' ( R ) > 0 and f '' ( R ) > 0 and the theory remains stable for all values of R . It was shown in Ref. [43] that exponential gravity do admit some exact solutions. One exact solution is a bouncing solution and another one is an expanding universe solution with constant Hubble parameter which takes place at a de-Sitter point. As we know two exact solutions in exponential gravity we can investigate about the growth of anisotropy in these two cases.</text>
<text><location><page_18><loc_17><loc_37><loc_63><loc_38></location>In the present case the solution of anisotropy factor is,</text>
<formula><location><page_18><loc_29><loc_32><loc_81><loc_36></location>x = b f ' ( R ) = b a 3 e αR = b a 3 e 6 α ( ˙ H +2 H 2 ) e -αx 2 . (38)</formula>
<text><location><page_18><loc_14><loc_19><loc_81><loc_31></location>This is a transcendental equation in x ( t ). The form of the above equation also shows that there will be only one real solution at any given time, given graphically by the intersection of a straight line y = x and a Gaussian y = b a 3 e 6 α ( ˙ H +2 H 2 ) e -αx 2 . The small anisotropy solution can be obtained analytically. If we want to see how small anisotropy, defined by all the x values which satisfy x 2 glyph[lessmuch] H 2 , develop we may approximate the last equation as:</text>
<formula><location><page_18><loc_40><loc_15><loc_81><loc_18></location>x ∼ b a 3 e αR iso . (39)</formula>
<text><location><page_18><loc_14><loc_8><loc_81><loc_14></location>where R iso ≡ 6( ˙ H + 2 H 2 ) and a is the average scale-factor. We discuss the evolution of small anisotropy for the two exact solutions of exponential gravity which was extensively discussed in [43].</text>
<section_header_level_1><location><page_19><loc_14><loc_84><loc_41><loc_86></location>5.1 Bouncing solution</section_header_level_1>
<text><location><page_19><loc_14><loc_73><loc_81><loc_83></location>Exponential gravity has an exact bouncing solution, where the scale-factor is given by a ( t ) = e At 2 where A is a real constant. Bounce happens in the presence of matter at t = 0, and the conditions for an exact solution requires αA = 1 / 48 and the equation of state of matter ω = -4 / 3. In the present case R iso = 12 A (1 + 4 At 2 ) and consequently for small anisotropy we must have</text>
<formula><location><page_19><loc_38><loc_68><loc_81><loc_71></location>x ( t ) = b e 1 / 4 e -4 At 2 , (40)</formula>
<text><location><page_19><loc_14><loc_63><loc_81><loc_67></location>which shows that how the anisotropy factor changes with time. The real indicator of anisotropy is the ratio x 2 /H 2 and in our present case</text>
<formula><location><page_19><loc_37><loc_58><loc_81><loc_62></location>x 2 H 2 = b 2 4 A 2 √ e e -8 At 2 t 2 . (41)</formula>
<text><location><page_19><loc_14><loc_46><loc_81><loc_56></location>A small anisotropy ratio at t → -∞ remains smaller than one for some time but then after some finite time x 2 ∼ H 2 and cosmic dynamics is guided by the anisotropy factor leading to an instability. From our simple analysis we see that the specific bouncing scenario presented in this section is unstable under small values of anisotropy.</text>
<section_header_level_1><location><page_19><loc_14><loc_39><loc_81><loc_43></location>5.2 Expansion with constant Hubble parameter at the deSitter point</section_header_level_1>
<text><location><page_19><loc_14><loc_29><loc_81><loc_37></location>Exponential gravity has another exact, constant Hubble parameter solution at a de-Sitter point where R iso = 2 /α . The scale-factor of the universe at the de-Sitter point is a ( t ) = e Ht and this is a vacuum solution when H 2 = 1 / (6 α ) is satisfied. In the present case small anisotropy grows as</text>
<formula><location><page_19><loc_39><loc_24><loc_81><loc_28></location>x ( t ) = b e 2 e -3 Ht . (42)</formula>
<text><location><page_19><loc_14><loc_22><loc_29><loc_23></location>and consequently</text>
<formula><location><page_19><loc_37><loc_17><loc_81><loc_21></location>x 2 H 2 = 6 αb 2 e 4 e -√ (6 /α ) t . (43)</formula>
<text><location><page_19><loc_14><loc_8><loc_81><loc_16></location>In this case we see that small anisotropy decreases with time. This analysis is not complete as we do not know how large anisotropy behaves in these situations. To tackle the question of large anisotropy one has to purely rely on numerical methods.</text>
<section_header_level_1><location><page_20><loc_14><loc_84><loc_34><loc_86></location>6 Conclusion</section_header_level_1>
<text><location><page_20><loc_14><loc_48><loc_81><loc_82></location>This paper presents the general results for anisotropic cosmological development in Bianchi-I model in metric f ( R ) gravity. The initial part of the paper develops the formalism which can be used to track cosmological development in homogeneous and anisotropic Bianchi-I model. The formalism developed is dynamically complete and can predict the development of all the relevant cosmological and fluid parameters in cosmological time. The methods developed in this paper can be applied to expanding as well as contracting phase of the universe. As anisotropy reduces in the expanding phase in GR it does not mean that this rule will be generally followed in f ( R ) cosmology as the equation predicting anisotropy growth is non-linear in nature and may have surprises in store. Our preliminary calculations predicts that inflation in quadratic f ( R ) cosmology, in Bianchi-I spacetime, indeed suppresses anisotropy. The results related to inflationary models in anisotropic spacetimes in f ( R ) theory are presented in full details in the present paper. We first show that for Bianchi-I type models one can analytically prove that anisotropy fades away in quadratic gravity inflation. We numerically show the validity of our analytic proof.</text>
<text><location><page_20><loc_14><loc_9><loc_81><loc_47></location>As anisotropy development demands special attention in the contracting phase in cosmological models based on GR our aim was to see how the problem translates into f ( R ) cosmology. In this article we tried to verify whether anisotropy subsides in the f ( R ) theory driven contraction phase. The result we obtain is complex and opens up new areas of research. We have chosen quadratic f ( R ) theory to illustrate our results as in this case most of the calculations can be done analytically although the bouncing scenario is gravitationally unstable. For any other higher order polynomial f ( R ) one has to use numerical methods to determine the solutions of the differential equation predicting anisotropy dynamics. Our work shows the qualitative nature of the cosmological system, guided by quadratic gravity, undergoing anisotropic contraction and we expect qualitatively similar but quantitatively much more formidable results for other complicated, gravitationally stable polynomial f ( R ) cosmologies. Even in the case of quadratic gravity the various results coming out from our formalism is non-trivial. We have pointed out that even when we restrict the cosmological dynamics by enforcing conditions as f ' > 0 and ρ > 0 there appears various regions in the n, b plane which gives rise to different kind of anisotropy growth. For some possible cosmological evolutions we have shown that anisotropy reduces with time. There exists</text>
<text><location><page_21><loc_14><loc_82><loc_81><loc_86></location>other possibilities where anisotropy increases with time during the contraction phase in quadratic gravity.</text>
<text><location><page_21><loc_14><loc_63><loc_81><loc_81></location>As quadratic f ( R ) theory cosmological bounce is more like a toy model because in this case the cosmological dynamics is unstable we have tried to show the applicability of our result in stable exponential gravity model which admits an exact bouncing solution. In this case we have not presented a general result but has focussed on small anisotropy growth. Our result shows that the exact bouncing solution in exponential gravity model is unstable and consequently the cosmological system will tend towards an instability in the contraction phase. We have also showed that small anisotropy subsides in the expansion phase at the de-Sitter point in exponential gravity.</text>
<text><location><page_21><loc_14><loc_43><loc_81><loc_62></location>The present paper shows that the issue about anisotropy in Bianchi-I spacetimes in metric f ( R ) gravity is a nonlinear problem which may lead to very complex conditions in contracting regions of a bouncing model. For polynomial gravity theories the cosmological contraction process is much involved and requires full numerical simulation to find out meaningful results. For expanding cosmologies our theory has given expected results, the amount of anisotropy goes down with expansion. But whether anisotropy will reduce for all kinds of expansion processes requires a more general proof and we hope we will able to show more general and formal work in these lines in the near future.</text>
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</document> | [{"title": "Nonlinear anisotropy growth in Bianchi-I spacetime in metric f ( R ) cosmology", "content": "Kaushik Bhattacharya, Saikat Chakraborty \u2217 Department of Physics, Indian Institute of Technology, Kanpur 208016, India January 9, 2019", "pages": [1]}, {"title": "Abstract", "content": "The present work is related to anisotropic cosmological evolution in metric f ( R ) theory of gravity. The initial part of the paper develops the general cosmological dynamics of homogeneous anisotropic BianchiI spacetime in f ( R ) cosmology. The anisotropic spacetime is pervaded by a barotropic fluid which has isotropic pressure. The paper predicts nonlinear growth of anisotropy in such spacetimes. In the later part of the paper we display the predictive power of the nonlinear differential equation responsible for the cosmological anisotropy growth in various relevant cases. We present the exact solutions of anisotropy growth in Starobinsky inflation driven by quadratic gravity and exponential gravity theory. Semi-analytical results are presented for the contraction phase in quadratic gravity bounce. The various examples of anisotropy growth in Bianchi-I model universe shows the complex nature of the problem at hand.", "pages": [1]}, {"title": "1 Introduction", "content": "The issue related to stability of homogeneous and isotropic cosmological solutions with respect to small anisotropy has been studied intensely in theoretical cosmology [1], [2], [3], [4]. Behavior of small anisotropy has been studied in cosmological models, using general relativity (GR), in the contexts of inflation [5], [6], [7], [8], [9] and pre-bounce ekpyrotic contraction phase [10], [11], [12], [13]. In the context of inflation the 'No-Hair' conjecture asserts that any pre-existing anisotropy must asymptotically die out in an inflating universe. Wald has been able to prove the conjecture for all the Bianchi models except Bianchi-IX [14] which requires a large cosmological constant to isotropize the spacetime . In a contracting universe, provided the universe is dominated by a matter component mimicking an ultra stiff barotropic fluid, growth of small anisotropy is suppressed with respect to that of the Hubble parameter. In absence of any such fluid in a contracting phase, small anisotropy grows large and dominates over all other matter components. This leads to the Belinsky-Khalatnikov-Lifshitz (BKL) instability [15], foiling the bounce. Mathematically, it can be shown that in presence of a slowly rolling scalar field the isotropic de-Sitter solution is an attractor for an expanding universe and in presence of a fast rolling scalar field the isotropic power law solution (for the scale-factor) is an attractor for contracting universe. Therefore an inflationary scenario is usually realized by a slowly rolling scalar field and an ekpyrotic scenario is usually realized by a fast rolling scalar field [16], [17], [18], [19], [20]. In the present work we have analyzed the evolution of spacetime anisotropy in f ( R ) gravity 1 , where the analysis becomes significantly more involved than that of models based on GR. Some attempts in modified, quadratic gravity [22,23] do discuss about anisotropic cosmologies while analyzing the past stages of a cosmological system near the singularity. In these works the authors show that near the singularity the universe may have an anisotropic mode of existence. The papers in general do not address a cosmological bounce scenario. Although previously there have been some progress in generalizing the no-hair theorem to incorporate higher order gravity theories [24], [25], [26], [27] and some applications of dynamical system analysis to understand anisotropic cosmology in higher order gravity [28], [29], [30], the previous attempts missed an important property of anisotropic cosmological dynamics related to nonlinear growth of anisotropy in the homogeneous and anisotropic Bianchi-I type of spacetime. In the present paper we first show analytically that in Starobinsky inflation any initial anisotropy will rapidly fade away. It is then shown that in contraction phases in Bianchi-I metric where speci- fying an unique scale-factor for contraction does not always yield a unique cosmological development. This result is possible in f ( R ) cosmology and in GR one cannot have this property. This non-uniqueness of cosmological development corresponding to a specific scale-factor opens up a new problem as cosmological evolution becomes more complex conceptually and as a consequence only simple f ( R ) models can be semi-analytically solved. Any f ( R ) model which is a higher order polynomial in R compared to the quadratic f ( R ) model requires a complete numerical solution for anisotropy growth. We show our result in quadratic f ( R ) model, which is gravitationally unstable if it has to accommodate a cosmological bounce. Our work may be taken as an effective toy model which is used to crack a formidable problem in cosmological dynamics. To our understanding the above mentioned topics are discussed for the first time in full generality in the present paper. We also present some preliminary results of non-linear anisotropy growth in exponential gravity models. The exponential gravity model has two exact solutions. One exact solution is a bouncing solution in presence of matter and the other exact solution is a expanding universe solution at a de-Sitter point. In these cases the anisotropy generation equation turns out to be a transcendental equation. We present some simple solutions in this case probing the nature of growth of small initial anisotropy. The material in the paper is organized in the following way. The second section discusses about the basics of Bianchi-I spacetimes and sets the notations and conventions followed throughout the paper. In section 3 we present the general formalism of homogeneous and anisotropic cosmological dynamics in metric f ( R ) cosmology. This part contains important results. In this section for the first time one comes across the complex nature of anisotropy development. In section 4 we present the results related to nonlinear anisotropy growth in quadratic f ( R ) theory induced inflation and cosmological bounce. Section 5 presents the results of anisotropy growth for some exact results in exponential gravity. The next section is the concluding section where we summarize the results obtained in the paper.", "pages": [1, 2, 3]}, {"title": "2 The anisotropic Bianchi-I metric and its properties", "content": "For our analysis, we have used the metric for Bianchi-I spacetime, where a 1 ( t ), a 2 ( t ) and a 3 ( t ) are the different scale-factors, whose relative differences specify the amount of anisotropy in the evolving universe. Existence of such anisotropic cosmological models in higher order gravity theories have been extensively studied in literature [31], [32], [33], [34]. In most of the earlier attempts the authors have tried to find out the nature of anisotropic spacetimes using various forms of anisotropic metric and using various forms of gravitational Lagrangians. In the present paper we show that the previous attempts have missed a vital ingredient in anisotropic expansion/contraction. The effect we discuss is clearly visible in Bianchi-I spacetime, but we think similar effects may be present in other anisotropic cosmological models. In this paper we will assume the presence of a perfect hydrodynamic fluid, in the Bianchi type-I spacetime, whose energy-momentum tensor (EMT)is given by T \u00b5\u03bd = ( \u03c1 + P ) u \u00b5 u \u03bd + Pg \u00b5\u03bd , where \u03c1 is the energy-density and P specifies isotropic pressure of the perfect fluid. The 4-velocity of the fluid element is given by u \u00b5 , which being a time-like vector is normalized as u \u00b5 u \u00b5 = -1 . Although the spacetime metric is anisotropic the fluid which pervades the spacetime is assumed to be isotropic. In this paper we will assume a barotropic equation of state for the perfect fluid, P = \u03c9\u03c1, where \u03c9 specifies the barotropic ratio. One can rewrite the form of the anisotropic metric, given in Eq. (1), in terms of the (geometric) average of the three scale-factors given by a ( t ) = [ a 1 ( t ) a 2 ( t ) a 3 ] 1 / 3 . The three different scale-factors in terms of the geometric average scale-factor can be written as, a i ( t ) = a ( t ) e \u03b2 i ( t ) , where i = 1 , 2 , 3 2 . The time dependent functions \u03b2 i ( t ) specify the anisotropy in the metric and they are constrained as Using the above relations one can now rewrite the metric given in Eq. (1) as In this notation one can define the Hubble parameter, as an arithmetic average, and its time-derivative as Mainly for the sake of brevity, henceforth in this article we will omit the word average (either geometric or arithmetic) before scale-factor or Hubble parameter. In presence of anisotropy the Ricci scalar turns out to be In the next section we formulate the anisotropic cosmological dynamics guided by metric f ( R ) theory. The fact that the derivatives of the anisotropy parameters are themselves present in the expression of the Ricci scalar will make anisotropic cosmological dynamics much more involved in f ( R ) gravity, compared to the general relativistic case.", "pages": [4, 5]}, {"title": "3 Formulation of anisotropic cosmological dynamics guided by metric f ( R ) theory", "content": "The field equation in f ( R ) gravity is where G \u00b5 \u03bd is the Einstein tensor and T \u00b5 \u03bd (curv) is the energy momentum tensor due to curvature. Here \u03ba = 8 \u03c0G , where G is the Newton's gravitational constant and is related to the Planck mass M P via G = 1 /M 2 P . In the present paper we will approximately use M P \u2248 10 19 GeV. The prime on top right hand side of any function represents the ordinary derivative of that function with respect to the Ricci scalar R . In particular where D \u00b5 A \u03bd is the covariant derivative of the covariant 4-vector A \u03bd and glyph[square] \u2261 g \u03b1\u03b2 D \u03b1 D \u03b2 . The 0 -0 component of the field equation in f ( R ) theory in an anisotropic spacetime is then given as, G 0 0 = -\u03ba f ' ( R ) ( \u03c1 + \u03c1 curv ), where The other three equations, specifying the i -j components become, G i j = \u03ba f ' ( R ) ( T i j + T i j (curv) ) , where i, j = 1 , 2 , 3. Here T i j = P\u03b4 i j stands for pressure of the perfect hydrodynamic fluid(s) whose EMT(s) has(have) the same form as specified in section 2. The form of T i j (curv) is given as where the components of the tensor B i j are defined as glyph[negationslash] In terms of the above quantities one can now write, where In terms of the Hubble parameter and the anisotropy parameter, the 0 -0 component of the field equation becomes while Eq. (11) becomes Adding the three equations, corresponding to each value of the index i in the above expression, one gets If one uses Eq. (13) in the above equation then one gets \u02d9 H as where P = \u03c9\u03c1 has been used. In the present case we define the anisotropy factor x as For an isotropic universe x 2 = 0 for all values of t , implying that all the \u03b2 i 's are constant in time. In such a case one can appropriately make (time-independent) coordinate rescaling in an appropriate way to make the spacetime look exactly like the Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime. Using Eq. (2) and Eq. (14) and the above definition of the anisotropy factor one can show that x satisfies the differential equation: whose (nontrivial) solution must be like where b is a real integration constant. The above equation contains the most important theoretical ingredient of the present paper. The Ricci scalar in the present case can be written as R = 6( \u02d9 H +2 H 2 ) + x 2 which depends on x and the last equation shows x is a function of R in f ( R ) gravity. As a consequence of the above relation in f ( R ) gravity, one cannot define an unique anisotropy dynamics. For any given f ( R ) gravity, in general multiple time evolutions of the anisotropy factor x is possible, each corresponding to a different equation of state for the barotropic fluid. In the present case the time derivative of the Ricci scalar is Working out similarly one can write, The above equations show that once we know the form of x in terms of the scalefactor, we can write the values of R , \u02d9 R and R in terms of the scale-factor. The cosmological dynamics of anisotropic f ( R ) theory is encoded in Eq. (16), Eq. (19) and the energy-momentum conservation equation The specification of \u03c9 and this set of three equations and the initial conditions specifying a , \u02d9 a , a , ... a , b and initial \u03c1 are enough to specify the anisotropic dynamics in f ( R ) cosmology if Eq. (19) has an unique root. If Eq. (19) does not have an unique root then the initial conditions must have to be enhanced. In the next section we will show when the above list of initial conditions require to be enhanced. The first order differential equation in Eq. (18) specifies the growth of anisotropy in metric f ( R ) gravity models where spacetime is specified by a Bianchi-I model. From the form of the equation it is seen that the amount of nonlinearity in Eq. (18) depends upon time. It can be noted that x = 0 have some interesting properties. The first thing to note about this point is that at x = 0 one always has \u02d9 x = 0 . The other interesting properties about this point are as follows. glyph[negationslash] glyph[negationslash] As x cannot be zero in the future if b = 0 in a non-singular cosmology, the important parameter which keeps track of effective anisotropy is given by the factor x 2 /H 2 . From Eq. (13), Eq. (15) and the expression of the Ricci scalar, R , it can be verified that when x 2 /H 2 glyph[lessmuch] 1 one can safely neglect the effect of anisotropy in the cosmological dynamics of Bianchi-I type models.", "pages": [5, 6, 7, 8]}, {"title": "4 Evolution of the anisotropic factor x ( t ) in quadratic gravity", "content": "In this section we will focus on quadratic gravity where \u03b1 is a real number. Although this is a simple form of f ( R ) but it can be used to model cosmological inflation as well as cosmological bounce for positive and negative signs of the constant \u03b1 respectively 3 . In this section we will determine the evolution of x ( t ) in quadratic gravity. The technique of evolution of x ( t ) in higher order gravity will be similar but much more involved. For higher order polynomial gravity the order of the algebraic equation yielding the roots of x ( t ) may be five (or higher) and consequently there does not exist any general algebraic formalism yielding those roots. From Eq. (19) one can easily verify that the algebraic equation specifying x ( t ) in quadratic gravity is a cubic equation of the form: where The discriminant, \u2206, specifying the roots and their properties is given by If \u2206 > 0 there will be three distinct real roots, if \u2206 < 0 then there will be one real root (and two complex roots) and if \u2206 = 0 there can be repeated real roots. The roots of Eq. (24) are as follows: In terms of trigonometric functions the above roots can be represented as, In the numerical calculations we will use the above form of the roots as they are less cumbersome to handle when all the roots are real. If initially the three roots of Eq. (24) are all real then one does require a separate initial condition, specifying a particular initial root out of the three possible roots, to describe the anisotropic cosmological dynamics in f ( R ) gravity. On the other hand if there is only one real root then the added initial condition looses its significance and the initial conditions as specified in the last section is enough to describe the cosmological dynamics. From Eq. (19) it is seen that the anisotropy factor depends upon a , \u02d9 a , a and the integration constant b . More over from the form of the cubic equation followed by x it can be easily seen that out of the three roots one tends to vanishes when b \u2192 0, where as the other two roots in general do not tend to zero when b becomes arbitrarily small. If all the roots are real then the root which vanishes when b vanishes plays an important role as in this case one can tune the value of the initial anisotropy by tuning the value of b . When the system admits only one real root then this root always tends to zero when b tends to zero. In GR, when one deals with anisotropic Bianchi Type-I cosmology, the equation followed by the anisotropy factor is x ( t ) = b/a 3 ( t ) and hence no such complications arise.", "pages": [9, 10]}, {"title": "4.1 Starobinsky inflation", "content": "We can now apply our formalism to get the first nontrivial result related to anisotropic cosmological dynamics in Starobinsky's model of inflation. In this model of inflation the universe inflates in absence of any hydrodynamic fluid. In quadratic gravity inflation the parameter \u03b1 appearing in Eq. (23) is always positive which makes f ' \u2261 df/dR > 1. In presence of anisotropy the inflating spacetime shows very fast growth in the average scale-factor a ( t ) while the average Hubble parameter satisfies the condition \u02d9 H = -glyph[epsilon1]H 2 where glyph[epsilon1] is a slow-roll parameter. During inflation glyph[epsilon1] glyph[lessmuch] 1 and this condition prevails until glyph[epsilon1] \u223c 1 at the end of inflation [36]. In the excellent review on Starobinsky inflation given in Ref. [36] the authors use the slow-roll approximation to derive the properties of the inflating FLRW spacetime. In this paper we will use the conventions of the above reference but will not exactly apply slow-roll mechanism. We will use the full f ( R ) theory equations as discussed in the last section with specific inflationary initial condition which gives rise to a rapidly expanding universe. In a later publication we want to generalize slow-roll conditions in quadratic gravity inflation in anisotropic Bianchi-I spacetimes. In this subsection we show that any kind of anisotropy, if present initially, will be damped during the inflationary phase in quadratic gravity. We analytically prove our result for large initial anisotropy, the proof remains the same for small initial anisotropy. If initially the anisotropy factor was large then inflation will successfully isotropize the universe and there will be no remaining anisotropy at the end of inflation. Although this fact is known to be true in inflationary models based on GR [14], in this article we show that similar outcome is also expected in quadratic theory of inflation. First we show that the maximum anisotropy allowed in quadratic gravity, during inflation, has a maximum bound: consequently the maximum anisotropy which can be isotropized is related with the Hubble parameter. To prove the above assumption one must note that in quadratic gravity inflation, \u03c1 = 0, and the inflationary phase is initiated by curvature energy density \u03c1 curv . The anisotropy energy contribution, in Eq. (13), is non-negative and consequently for inflation to start initially (when ideally the anisotropy effect is maximum) \u03c1 curv > 0. The fact that the curvature energydensity appearing in the constraint Eq. (13) as: cannot be negative during inflation justifies Eq. (30). Even if the initial anisotropy present in the universe is given by the maximum bound of x in Eq. (30) the anisotropy factor rapidly fades away during quadratic inflation. To show this we first note that in Starobinsky inflation A 1 > 0, as | \u02d9 H | glyph[lessmuch] H 2 . As a result the discriminant \u2206 < 0 implying that there is only one unique real root of Eq.(24). This root is given by the top right hand side term in Eq. (28). From the expression of A 2 one can see that it rapidly diminishes in an inflating universe and one can assume A 3 1 > A 2 2 after some time from the onset of inflation. Assuming that A 2 \u2192 0 rapidly after inflation starts one can easily show that the relevant real root of Eq. (30) tends to zero during inflation. We will now present a numerical solution of the cosmological dynamics during inflation and point out that the anisotropy factor x 2 /H 2 always remains sufficiently smaller and at no stages of quadratic inflation x 2 \u2248 H 2 . Here we assume that inflation starts at, t i = 10 6 , in Planck units. In this unit system the actual value of a quantity is obtained by multiplying the value of the physical quantity by a particular power of Plank mass M P . The specific power corresponds to the mass dimension of the physical quantity. In the present case the actual value of t i is t i M -1 P . In this article we will assume M P \u2248 10 19 GeV and consequently t i = 10 -13 GeV -1 expressed in energy units. Expressed in the seconds t i \u2248 10 -37 s and that is 10 6 times Planck time expressed in seconds. The value of \u03b1 is chosen as \u03b1 = 10 12 which in natural units will be 10 12 M -2 P . Phenomenologically one expects quadratic correction to Einstein gravity at a very early phase of the universe when H \u223c 10 12 -13 GeV or more. For this benchmark value of the H the Ricci scalar R \u223c 10 26 GeV 2 assuming x 2 < H 2 . If the quadratic correction \u03b1R 2 becomes effective at such a value of R then \u03b1 \u223c 1 /R yielding \u03b1 \u223c 10 -26 GeV -2 = 10 12 M -2 P , justifying our choice of \u03b1 . The initial Hubble parameter is chosen to be H ( t i ) = 2 \u00d7 10 -6 in Planck units, its value in standard units is 10 13 GeV. Inflation ends at t f \u2248 t i +70 \u00d7 H ( t i ) -1 = 36 \u00d7 10 6 , which corresponds to slightly more than 70 e-folds. The consistent inflationary 4 initial conditions are written in terms of the initial values of the slow-roll parameters. The initial slow-roll parameters are chosen as glyph[epsilon1] ( t i ) = -\u02d9 H/H 2 \u2248 7 \u00d7 10 -3 and \u03b7 ( t i ) = H/ ( H \u02d9 H ) \u2248 0. The other initial conditions are as: Although the initial conditions for inflation in the present section are written in terms of the slow-roll parameters glyph[epsilon1] and \u03b7 at initial time we do not evolve glyph[epsilon1] or \u03b7 with time or use the slow-roll parameters in any other place in our calculation. The calculation do not use slow-roll approximation and the results we present in this subsection are exact results. To get a numerical solution, we plug the solution x ( a, \u02d9 a, a ) into the dynamical equation, Eq. (16). The resulting dynamical equation for a ( t ) is a fourth order ordinary differential equation for the case of quadratic f ( R ) gravity. Looking at the structure of the roots of the cubic equation, presented in the initial part of the present section, it is seen that in the present case the coefficient A 1 > 0 and consequently there will be only one real root of the anisotropy factor x ( t ). This root continuously tends to zero as b tends to zero. We have plotted the numerical results showing the growth of the Hubble parameter and x , for small and large relative initial anisotropy, in logarithmic scales in Fig. 1. Small anisotropy, x small , corresponds to b = 10 -10 and relatively large anisotropy, x large , corresponds to b = 10 -5 . For both small and large anisotropies the scale-factor and the Hubble parameter are approximately the same showing that the overall inflating nature of the system does not depend upon the initial anisotropy present in the system. The inflationary nature of the present system is shown by the near constant value of H in Fig. 1. Fig. 2 shows the growth of the anisotropy factor x 2 /H 2 in the two cases corresponding to the two b values as discussed above. This plot clearly shows that anisotropy rapidly dies in quadratic gravity inflation. We have numerically verified that anisotropy gets wiped out after the first two or three e-folds of inflation and consequently suppression of anisotropy happens very efficiently in quadratic inflation. Here it must be noted that we cannot arbitrarily increase b as when b > 10 -4 the consistency condition in Eq. (30) is violated and the system does not inflate any more. Our formalism shows both analytically and numerically that Starobinsky inflation is safe from initial anisotropy.", "pages": [10, 11, 12, 13, 14]}, {"title": "4.2 Contraction in toy model of quadratic bounce", "content": "In this subsection we discuss anisotropic contraction phase in a simple and partly unstable model, guided by quadratic gravity. The presentation in this subsection is more like a toy model analysis which shows the complexities of anisotropic contraction in polynomial f ( R ) gravity models. In general the solution of Eq. (18) becomes a polynomial equation in x and for higher polynomial orders (compared to quadratic order) the algebraic equations do not yield analytic solutions. The quadratic gravity bounce model, where \u03b1 < 0 is the simplest polynomial bounce model, where the intricacies of anisotropy generation during the contraction phase can be semi-analytically shown. The issue of anisotropy generation during a contraction phase is very important as anisotropy may get enhanced during this phase as it happens in GR based models of cosmological bounce. We want to see how anisotropy grows in Bianchi- I models in the contraction phase in quadratic f ( R ) gravity. In the present case we will assume the existence of hydrodynamic matter and \u03b1 < 0 as these conditions are required for a subsequent bounce [35]. Before we proceed we will like to make some remarks related to the choice of the sign of \u03b1 . The negative sign of \u03b1 implies that f ' ( R ) is not always positive. We can choose our dynamical system to be such that it satisfies f ' ( R ) > 0 for some range of R , as done in the present paper. The importance of negative \u03b1 quadratic model is that one can have a cosmological bounce in this restricted regime of R , where R < 1 / (2 | \u03b1 | ) for stability. There is another source of instability in the present case, related to the negative sign of \u03b1 . In such models f '' < 0 which may lead to instabilities first proposed by Dolgov and Kawasaki [37] and later by V. Faraoni [38]. In the present model one cannot get rid of Dolgov and Kawasaki instability 5 , consequently in light of the stability issues we will like to interpret the present model of bounce as a toy model whose sole purpose is to describe the nonlinear growth of anisotropy. In the a later section we will apply or formalism in a stable gravitational model. In GR it is known that anisotropy suppression during contraction phase requires the presence of an ultra-stiff matter component with \u03c9 (= P/\u03c1 ) > 1. The presence of an ultra-stiff matter component can produce a slow contraction phase where preexisting anisotropy is suppressed 6 . In the present case we will see that a power law contraction phase may suppress initial anisotropy in quadratic f ( R ) cosmology. We assume that during the contracting phase t < 0 and bounce occurs at t = 0. During the contracting phase the scale-factor decreases as and consequently From physical considerations one can choose \u03b1 = -10 12 in Planck units [35]. Eliminating \u03c1 in Eq. (16) by using Eq. (13) we get, In the present case the above equation yields the equation of state for the barotropic matter when one specifies the particular nature of the scale-factor. Determining the form of the time evolution of anisotropy factor reduces to finding the root(s) of Eq. (19). One can have various phases of anisotropy development during a cosmological evolution depending upon the roots of Eq. (19). In this paper we will particularly focus on the contracting phase of the universe leading to a cosmic bounce. We present the results for the popular quadratic f ( R ) model which actually accommodates a cosmological bounce [35], [40], [41], [42]. The nature of the anisotropic contraction phase predicted in this model will give a glimpse of the interesting effects of f ( R ) models of anisotropic contraction in the Bianchi-I spacetime. The plot in Fig. 3 shows the nature of the roots at time t = -10 10 in Planck units. The time period of contraction is chosen in such a way that all the constraints as f ' ( R ) > 0 and \u03c1 > 0 are maintained during this phase of contraction. As the power law contraction can never lead to a bounce the constraints compel us to terminate the power law contraction process some time before the bounce and in this paper we use the time interval -10 10 \u2264 t \u2264 -10 7 . The scale-factor during this time is assumed to be a ( t ) = ( -t/ 10 10 ) n such that a ( t = -10 10 ) = 1. The nature of the roots show that below a certain b value and above a certain b value there is only one real root. Near b = 0 the system admits three real roots of x ( t ). We have verified that the nature of the root structure, as specified in Fig. 3, does feebly depend upon n in the interval 0 \u2264 n \u2264 1. The plot in Fig. 3 shows three branches in three colors. The middle green (continuous line) branch smoothly matches to the blue (dotted) branch above and the red (dashed) one below. The continuous branch specifies a root of Eq. (24) which is real near b = 0 and gives rise to small values of anisotropy factor x 0 = x ( t = -10 10 ) initially. The dashed and dotted branches specify the other roots which are large for regions near b = 0. The connection of the three regions in the figure with the roots in Eq. (29) are specified in the caption of Fig. 3. As time evolves the nature of the plot in Fig. 3 changes but the general structure of the plot always remains qualitatively similar as the one plotted at the initial time. The dynamics of anisotropy growth depends upon the parameters b and n . We can specify the region in the b -n plane which gives rise to decreasing anisotropy. The plot in Fig. 4 shows such a region in the b -n plane. The plot is done at t = -10 10 , the initial time, when the region is most constrained. In Fig. 5 we show how x 2 /H 2 varies in time if one uses any value of b, n in the shaded region in Fig.4. In particular we have chosen b = 10 -12 and n = 1 / 4. The above information shows that for some parameter values a power law contraction can indeed suppress small anisotropy in quadratic gravity. Physically anisotropy suppression for some regions in the b -n plane in the contracting phase is not surprising as both x and H do increase in time during contraction when b and n belongs to the shaded region in Fig. 4 (as expected) but H increases more than x in time and as a consequence x 2 /H 2 decreases with time. For some parameter space H can grow faster than x in time, when anisotropy factor decreases, and for other parameter values x increases more than H in time making the contracting universe completely anisotropic.", "pages": [14, 15, 16, 17, 18]}, {"title": "5 Anisotropy growth in exponential gravity", "content": "Recently it has been shown [43] that one can get bouncing solutions and expanding universe solutions in a unstable de-Sitter point in exponential gravity where where \u03b1 is a dimensional, real constant. In the present case as \u03b1 > 0 we have f ' ( R ) > 0 and f '' ( R ) > 0 and the theory remains stable for all values of R . It was shown in Ref. [43] that exponential gravity do admit some exact solutions. One exact solution is a bouncing solution and another one is an expanding universe solution with constant Hubble parameter which takes place at a de-Sitter point. As we know two exact solutions in exponential gravity we can investigate about the growth of anisotropy in these two cases. In the present case the solution of anisotropy factor is, This is a transcendental equation in x ( t ). The form of the above equation also shows that there will be only one real solution at any given time, given graphically by the intersection of a straight line y = x and a Gaussian y = b a 3 e 6 \u03b1 ( \u02d9 H +2 H 2 ) e -\u03b1x 2 . The small anisotropy solution can be obtained analytically. If we want to see how small anisotropy, defined by all the x values which satisfy x 2 glyph[lessmuch] H 2 , develop we may approximate the last equation as: where R iso \u2261 6( \u02d9 H + 2 H 2 ) and a is the average scale-factor. We discuss the evolution of small anisotropy for the two exact solutions of exponential gravity which was extensively discussed in [43].", "pages": [18]}, {"title": "5.1 Bouncing solution", "content": "Exponential gravity has an exact bouncing solution, where the scale-factor is given by a ( t ) = e At 2 where A is a real constant. Bounce happens in the presence of matter at t = 0, and the conditions for an exact solution requires \u03b1A = 1 / 48 and the equation of state of matter \u03c9 = -4 / 3. In the present case R iso = 12 A (1 + 4 At 2 ) and consequently for small anisotropy we must have which shows that how the anisotropy factor changes with time. The real indicator of anisotropy is the ratio x 2 /H 2 and in our present case A small anisotropy ratio at t \u2192 -\u221e remains smaller than one for some time but then after some finite time x 2 \u223c H 2 and cosmic dynamics is guided by the anisotropy factor leading to an instability. From our simple analysis we see that the specific bouncing scenario presented in this section is unstable under small values of anisotropy.", "pages": [19]}, {"title": "5.2 Expansion with constant Hubble parameter at the deSitter point", "content": "Exponential gravity has another exact, constant Hubble parameter solution at a de-Sitter point where R iso = 2 /\u03b1 . The scale-factor of the universe at the de-Sitter point is a ( t ) = e Ht and this is a vacuum solution when H 2 = 1 / (6 \u03b1 ) is satisfied. In the present case small anisotropy grows as and consequently In this case we see that small anisotropy decreases with time. This analysis is not complete as we do not know how large anisotropy behaves in these situations. To tackle the question of large anisotropy one has to purely rely on numerical methods.", "pages": [19]}, {"title": "6 Conclusion", "content": "This paper presents the general results for anisotropic cosmological development in Bianchi-I model in metric f ( R ) gravity. The initial part of the paper develops the formalism which can be used to track cosmological development in homogeneous and anisotropic Bianchi-I model. The formalism developed is dynamically complete and can predict the development of all the relevant cosmological and fluid parameters in cosmological time. The methods developed in this paper can be applied to expanding as well as contracting phase of the universe. As anisotropy reduces in the expanding phase in GR it does not mean that this rule will be generally followed in f ( R ) cosmology as the equation predicting anisotropy growth is non-linear in nature and may have surprises in store. Our preliminary calculations predicts that inflation in quadratic f ( R ) cosmology, in Bianchi-I spacetime, indeed suppresses anisotropy. The results related to inflationary models in anisotropic spacetimes in f ( R ) theory are presented in full details in the present paper. We first show that for Bianchi-I type models one can analytically prove that anisotropy fades away in quadratic gravity inflation. We numerically show the validity of our analytic proof. As anisotropy development demands special attention in the contracting phase in cosmological models based on GR our aim was to see how the problem translates into f ( R ) cosmology. In this article we tried to verify whether anisotropy subsides in the f ( R ) theory driven contraction phase. The result we obtain is complex and opens up new areas of research. We have chosen quadratic f ( R ) theory to illustrate our results as in this case most of the calculations can be done analytically although the bouncing scenario is gravitationally unstable. For any other higher order polynomial f ( R ) one has to use numerical methods to determine the solutions of the differential equation predicting anisotropy dynamics. Our work shows the qualitative nature of the cosmological system, guided by quadratic gravity, undergoing anisotropic contraction and we expect qualitatively similar but quantitatively much more formidable results for other complicated, gravitationally stable polynomial f ( R ) cosmologies. Even in the case of quadratic gravity the various results coming out from our formalism is non-trivial. We have pointed out that even when we restrict the cosmological dynamics by enforcing conditions as f ' > 0 and \u03c1 > 0 there appears various regions in the n, b plane which gives rise to different kind of anisotropy growth. For some possible cosmological evolutions we have shown that anisotropy reduces with time. There exists other possibilities where anisotropy increases with time during the contraction phase in quadratic gravity. As quadratic f ( R ) theory cosmological bounce is more like a toy model because in this case the cosmological dynamics is unstable we have tried to show the applicability of our result in stable exponential gravity model which admits an exact bouncing solution. In this case we have not presented a general result but has focussed on small anisotropy growth. Our result shows that the exact bouncing solution in exponential gravity model is unstable and consequently the cosmological system will tend towards an instability in the contraction phase. We have also showed that small anisotropy subsides in the expansion phase at the de-Sitter point in exponential gravity. The present paper shows that the issue about anisotropy in Bianchi-I spacetimes in metric f ( R ) gravity is a nonlinear problem which may lead to very complex conditions in contracting regions of a bouncing model. For polynomial gravity theories the cosmological contraction process is much involved and requires full numerical simulation to find out meaningful results. For expanding cosmologies our theory has given expected results, the amount of anisotropy goes down with expansion. But whether anisotropy will reduce for all kinds of expansion processes requires a more general proof and we hope we will able to show more general and formal work in these lines in the near future.", "pages": [20, 21]}] |
2016arXiv160907294P | https://arxiv.org/pdf/1609.07294.pdf | <document>
<section_header_level_1><location><page_1><loc_18><loc_82><loc_80><loc_84></location>Entropy Product Formula for Gravitational Instanton</section_header_level_1>
<section_header_level_1><location><page_1><loc_35><loc_77><loc_62><loc_79></location>Parthapratim Pradhan 1</section_header_level_1>
<text><location><page_1><loc_28><loc_68><loc_69><loc_73></location>Department of Physics Hiralal Mazumdar Memorial College For Women Dakshineswar, Kolkata-700035, India.</text>
<section_header_level_1><location><page_1><loc_45><loc_63><loc_53><loc_64></location>Abstract</section_header_level_1>
<text><location><page_1><loc_17><loc_55><loc_81><loc_61></location>We investigate the entropy product formula for various gravitational instantons. We speculate that due to the mass-independent features of the said instantons they are universal as well as they are quantized . For isolated Euclidean Schwarzschild black hole, these properties simply fail .</text>
<section_header_level_1><location><page_1><loc_12><loc_50><loc_34><loc_52></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_12><loc_39><loc_86><loc_48></location>There has been a strong interest in microscopic interpretation of black hole (BH) entropy [2, 3, 4, 5, 6, 7, 8, 9] in terms of D -branes come due to the work by Strominger and Vapa [10]. In d -dimension Euclidean quantum gravity, this entropy is due to the ( d -2)dimensional fixed point sets of the imaginary time translation Killing vector. There are many fixed point sets which can also give rise to BH entropy.</text>
<text><location><page_1><loc_12><loc_30><loc_86><loc_39></location>Previously, Area (or Entropy) product formula evaluated for different class of BHs [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. In some cases, the product formula is not mass-independent (universal) and in some cases the product formula is indeed mass-independent i.e. universal. Upto the author's knowledge, there has been no attempt to compute the entropy product formula for gravitational instanton .</text>
<text><location><page_1><loc_12><loc_12><loc_86><loc_30></location>Thus in the present work, we wish to investigate the entropy product formula for various gravitational instantons. Instantons are non-singular and having imaginary time. They arises in quantum field thory (QFT) for evaluating the functional integral, in which the functional integral is Wick rotated and expressed as an integral over Euclidean field configuarations. They are the solutions of the Euclidean Einstein equations. They have the signature of the form (+ + ++). There are two types of instantons discoverd so far. One is an asymptotically locally flat (ALF), which was first discoverd by Hawking in 1977 [1] and the other is an asymptotically locally Euclidean (ALE), discovered by Gibbons and Hawking in 1978 [30]. The examples of ALE classes are Flat space, Eguchi-Hanson [31] and multi-instanton [30].</text>
<text><location><page_2><loc_12><loc_73><loc_86><loc_84></location>The ALF class of solutions are asymptotically flat (AF) in 3D sense, the fourth, imaginary -time, direction being periodic. The surfaces of large radii should be think of as an S 1 bundle over S 2 . The product bundle corresponds to the AF solutions which include the Euclidean Schwarzschild and Euclidean Kerr solutions [32, 33]. The twisted bundles correspond to the multi-Taub-NUT solution [1], and the Taub-Bolt solution was first discoverd by Page [34].</text>
<text><location><page_2><loc_12><loc_57><loc_86><loc_73></location>In our previous work [28, 29, 19], we investigated the properties of inner and outer horizon thermodynamics of Taub-NUT (Newman-Unti-Tamburino) BH, Kerr-Taub-NUT BH and Kerr-Newman-Taub-NUT BH in four dimensional Lorentzian geometry. The failure of First law of BH thermodynamics and Smarr-Gibbs-Duhem relation for TaubNUT and Kerr-Taub-NUT BH in the Lorentzian regime gives the motivation behind this work . What happens when one can go from Lorentzian geometry to Riemannian geometry? This is the prime aim in this work. By studing the properties of these instantons what should be the effects on the BH entropy product formula due to the non-trivial NUT parameter?</text>
<text><location><page_2><loc_12><loc_33><loc_86><loc_56></location>In general relativity, the non-trivial value for the BH entropy is due to the presence of the fixed point set of the periodic imaginary time Killing vector. The fixed point set here we considered actually is the BH horizons ( H ± ). Here H + is called event horizon and H -is called the Cauchy horizon. In four dimension, such fixed point sets are of two types, isolated points or zero dimensional which we call NUTs and two surfaces or two dimensional which we call Bolts. Thus one can thought Bolts as being the analogue of electric type mass- monopoles and the NUTs as being gravitational dyons endowed with a real electric type mass-monopole and an imaginary magnetic type mass-monopole. The presence of magnetic type mass introduces a Dirac string like singularity in the spacetime which is so called Misner string was first pointed out by Misner in his paper for the Lorentzian Taub-NUT spacetime [35]. A Misner string is a coordinate singularity which can be considered as a manifestation of a 'non-trivial topological twisting' [36] of the manifold ( M,g ab ). This twist is parametrized by a topological term, the NUT charge.</text>
<text><location><page_2><loc_12><loc_29><loc_86><loc_33></location>In our previous investigation [19], we have taken the metric in a Lorentzian spacetime in 3 + 1 split form as</text>
<formula><location><page_2><loc_32><loc_23><loc_86><loc_29></location>ds 2 = -F ( dt + w i dx i ) 2 + γ ij F dx i dx j . (1)</formula>
<text><location><page_2><loc_12><loc_21><loc_86><loc_24></location>In this work, we are interested to study the metric in a Riemannian spacetime which can be written in 3 + 1 split form:</text>
<formula><location><page_2><loc_32><loc_15><loc_86><loc_20></location>ds 2 = F ( dτ + w i dx i ) 2 + γ ij F dx i dx j . (2)</formula>
<text><location><page_2><loc_12><loc_8><loc_86><loc_16></location>Here all quantities are independent of t or τ . The Wick rotation that transforms from one case to other is by the transformation t ↦→ iτ and w i ↦→ iω i . F can be thought of as an electric type potential, and ω i or w i as a magnetic-type vector potential. The associated magnetic field is H ij = ∂ i ω j -∂ j ω i and it is gauge invariant.</text>
<text><location><page_3><loc_22><loc_79><loc_22><loc_82></location>/negationslash</text>
<text><location><page_3><loc_12><loc_64><loc_86><loc_84></location>This should be used to define a magnetic monopole moment called the NUT charge n [30]. If n = 0, the fibration should not be trivial. In the Lorentzian geometry, these fixed point sets are the two-dimensional Boyer bifurcation sets of event horizon [6, 37]. On the other hand in Riemannian geometry, these fixed point sets are of two types: zero dimensional point or NUTs, and two dimensional surfaces or Bolts [30]. A NUT possesses a pair of surface gravities κ 1 and κ 2 . p and q are a pair of co-prime integers such that κ 1 κ 2 = p q . If κ 1 κ 2 is irrational, p = q = 1. A NUT of type ( p, q ) has a NUT charge of n = β 8 πpq , where β is the period of the imaginary time coordinate. Moreover, n = Y β 8 π for a Bolt of self-intersection number Y . It should be noted that in the Riemannian case, the number of NUTs and Bolts are related to the Euler number χ and the Hirzebruch signature τ of the manifold M by</text>
<formula><location><page_3><loc_39><loc_58><loc_86><loc_63></location>χ = ∑ Bolts χ i + ∑ NUTs 1 , (3)</formula>
<text><location><page_3><loc_12><loc_56><loc_56><loc_58></location>where χ i is the Euler number for the i -th Bolt, and</text>
<formula><location><page_3><loc_28><loc_51><loc_86><loc_56></location>τ = ∑ Bolts Y i csc 2 θ -∑ NUTs cot p i θ cot q i θ + η (0 , θ ) . (4)</formula>
<text><location><page_3><loc_12><loc_44><loc_86><loc_52></location>The above Eq. (4) is valid for arbitrary θ . Y i is the self-interection number of the i -th Bolt, the i -th NUT is of type ( p i , q i ) and η (0 , θ ) is a correction term which depends solely on the boundary. It should be mentioned that for multi-Taub-NUT and multi-instanton solutions there are k NUTs of type (1 , 1) and τ = k -1, therefore</text>
<formula><location><page_3><loc_26><loc_39><loc_86><loc_44></location>τ = ∑ Bolts Y i csc 2 θ -∑ NUTs cot p i θ cot q i θ + k csc 2 θ -1 . (5)</formula>
<text><location><page_3><loc_12><loc_38><loc_63><loc_39></location>where the AF boundary conditions should be η (0 , θ ) = 0 [33].</text>
<text><location><page_3><loc_12><loc_23><loc_86><loc_37></location>The structure of the paper is as follows. In Sec. 2, we have considered the Euclidean Schwarzschild BH. In Sec. 3, we have investigated the properties of Self dual TaubNUT instantons. In Sec. 4, we have studied the mass-independent properties of TaubBoltiInstantons. In Sec. 5, we have described the properties of Eguchi-Hanson instantons. In Sec. 6, we have examined the properties of Taub-NUT-AdS Spacetime. In Sec. 7, we have studied the entropy product formula for Taub-Bolt-AdS spacetime and finally in Sec. 8, we have examined the product rules for Dyonic Taub-NUT-AdS and Taub-Bolt-AdS spacetime.</text>
<section_header_level_1><location><page_3><loc_12><loc_18><loc_62><loc_20></location>2 Euclidean Schwarzschild metric:</section_header_level_1>
<text><location><page_3><loc_12><loc_13><loc_86><loc_16></location>To give a warm up, let us first consider the Schwarzschild BH (where we have used units in G = c = 1) in Euclidean form as</text>
<formula><location><page_3><loc_24><loc_5><loc_86><loc_12></location>ds 2 = ( 1 -2 M r ) dτ 2 + dr 2 ( 1 -2 M r ) + r 2 ( dθ 2 +sin 2 θdφ 2 ) . (6)</formula>
<text><location><page_4><loc_12><loc_75><loc_86><loc_84></location>The apperent singularity at the event horizon r + = 2 M can be removed by identifying τ with a period β = 8 πM [33, 1]. The radial coordinate has the range 2 M ≤ r ≤ ∞ . Then the topology of the manifold is R 2 ×S 2 . The isometry group is O (2) ⊗ O (3), where the O (2) corresponds to translations in the periodically identified imaginary time τ and the O (3) corresponds to rotations of the θ and φ coordinates.</text>
<text><location><page_4><loc_12><loc_70><loc_86><loc_75></location>The Killing vector ∂ τ has unit magnitude at large radius and has a Bolt on the horizon r + = 2 M which is a 2-sphere S 2 of area</text>
<formula><location><page_4><loc_42><loc_68><loc_86><loc_70></location>A + = 16 πM 2 . (7)</formula>
<text><location><page_4><loc_12><loc_65><loc_38><loc_66></location>The surface gravity is given by</text>
<text><location><page_4><loc_12><loc_57><loc_34><loc_58></location>and the BH temperature is</text>
<formula><location><page_4><loc_40><loc_52><loc_86><loc_55></location>T + = κ + 2 π = 1 8 πM . (9)</formula>
<text><location><page_4><loc_12><loc_49><loc_75><loc_51></location>Thus for an isolated Euclidean Schwarzschild BH the area product becomes</text>
<formula><location><page_4><loc_42><loc_46><loc_86><loc_47></location>A + = 16 πM 2 . (10)</formula>
<text><location><page_4><loc_12><loc_41><loc_86><loc_44></location>which tells us that the product is dependent on mass parameter and thus it is not universal. Also it is not quantized.</text>
<text><location><page_4><loc_15><loc_39><loc_44><loc_40></location>The Euclidean action derived in [1]</text>
<formula><location><page_4><loc_38><loc_34><loc_86><loc_37></location>I = -ln Z = 4 πM 2 . (11)</formula>
<text><location><page_4><loc_82><loc_33><loc_86><loc_35></location>(12)</text>
<text><location><page_4><loc_12><loc_30><loc_50><loc_31></location>From that one can derive the entropy as in [1]</text>
<formula><location><page_4><loc_33><loc_24><loc_86><loc_28></location>S + = -( β ∂ ∂β -1 ) ln Z = 4 πM 2 . (13)</formula>
<text><location><page_4><loc_12><loc_22><loc_76><loc_23></location>Thus the entropy product for isolated Euclidean Schwarzschild BH should be</text>
<formula><location><page_4><loc_42><loc_18><loc_86><loc_20></location>S + = 4 πM 2 . (14)</formula>
<text><location><page_4><loc_12><loc_15><loc_58><loc_16></location>Indeed, it is not universal as well as it is not quantized.</text>
<formula><location><page_4><loc_40><loc_60><loc_86><loc_63></location>κ + = 2 π β = 1 4 M . (8)</formula>
<section_header_level_1><location><page_5><loc_12><loc_82><loc_62><loc_84></location>3 Self dual Taub-NUT Instantons:</section_header_level_1>
<text><location><page_5><loc_82><loc_69><loc_82><loc_71></location>/negationslash</text>
<text><location><page_5><loc_12><loc_62><loc_86><loc_80></location>In this section we shall calculate the entropy product and area product of four dimensional Taub-NUT spacetime. It is of ALF. ALF metrics have a NUT charge,or magnetic type mass, n , as well as the ordinary electric type mass, M . The NUT charge is βc 1 8 π , where c 1 is the first Chern number of the U (1) bundle over the sphere at infinity, in the orbit space Ξ. If c 1 = 0, then the boundary at infinity is S 1 ×S 2 and the spacetime is AF. The BH metrics are saddle points in the path integral for the partition function. Thus, if c 1 = 0, the boundary at infinity is a squashed S 3 , and the metric should not be analytically continued to a Lorentzian metric. The squashed S 3 is the three-dimensional space on which the boundary conformal field theory (CFT) will be compactified, with β identified with the inverse temperature i. e. T = 1 β .</text>
<text><location><page_5><loc_12><loc_59><loc_86><loc_62></location>Hawking [1] first given an examples of gravitational instanton was the self-dual TaubNUT metric described by</text>
<formula><location><page_5><loc_20><loc_52><loc_86><loc_57></location>ds 2 = F ( r ) ( dτ +2 n cos θdφ ) 2 + dr 2 F ( r ) + ( r 2 -n 2 ) ( dθ 2 +sin 2 θdφ 2 ) . (15)</formula>
<formula><location><page_5><loc_18><loc_50><loc_86><loc_53></location>F ( r ) = r -n r + n (16)</formula>
<text><location><page_5><loc_12><loc_42><loc_86><loc_49></location>It is ALF with a central NUT. The self-dual Taub-NUT instanton has M = n and the anti-self-dual instanton has M = -n . The value r = n is now a zero in F ( r ). The ( θ, φ ) two-sphere has a zero area at r = n , so the zero in F ( r ) is a zero-dimensional fixed point of ∂ φ , a NUT.</text>
<text><location><page_5><loc_12><loc_35><loc_86><loc_41></location>In order to make the solution regular, we take the region r ≥ n and let the period of τ be 8 πn . The metric has a NUT at r = n , with a Misner string running along the z -axis from the NUT out to infinity i.e. n ≤ r ≤ ∞ .</text>
<text><location><page_5><loc_12><loc_33><loc_86><loc_36></location>We know from the idea of path-integral formulation of quantum gravity which tells us that the Euclidean action derived in [1]</text>
<formula><location><page_5><loc_39><loc_28><loc_86><loc_31></location>I = -ln Z = 4 πn 2 . (17)</formula>
<text><location><page_5><loc_12><loc_26><loc_52><loc_28></location>where Z is the partition function of an ensemble</text>
<formula><location><page_5><loc_38><loc_20><loc_86><loc_25></location>Z = ∫ [ Dg ][ Dφ ] e -I ( g,φ ) . (18)</formula>
<text><location><page_5><loc_12><loc_17><loc_86><loc_20></location>with the path integral taken over all metrics g and matter field φ that are appropriately identified with the period β of τ . Therefore the entropy should be derived as</text>
<formula><location><page_5><loc_34><loc_11><loc_86><loc_15></location>S = -( β ∂ ∂β -1 ) ln Z = 4 πn 2 . (19)</formula>
<text><location><page_5><loc_12><loc_9><loc_57><loc_10></location>It is indeed mass-independent and thus it is universal.</text>
<text><location><page_6><loc_15><loc_82><loc_47><loc_84></location>The surface gravity is calculated to be</text>
<formula><location><page_6><loc_42><loc_78><loc_86><loc_81></location>κ = 2 π β = 1 4 n . (20)</formula>
<text><location><page_6><loc_12><loc_75><loc_47><loc_76></location>Thus the BH temperature should read off</text>
<formula><location><page_6><loc_41><loc_70><loc_86><loc_74></location>T = κ 2 π = 1 8 πn . (21)</formula>
<text><location><page_6><loc_15><loc_68><loc_86><loc_69></location>Now we see what happens the above results for another instantons that is Taub-Bolt.</text>
<section_header_level_1><location><page_6><loc_12><loc_63><loc_48><loc_65></location>4 Taub-Bolt Instantons:</section_header_level_1>
<text><location><page_6><loc_12><loc_59><loc_59><loc_61></location>The Taub-Bolt instanton is described by the metric [34]</text>
<formula><location><page_6><loc_20><loc_51><loc_86><loc_57></location>ds 2 = G ( r ) ( dτ +2 n cos θdφ ) 2 + dr 2 G ( r ) + ( r 2 -n 2 ) ( dθ 2 +sin 2 θdφ 2 ) . (22)</formula>
<formula><location><page_6><loc_19><loc_48><loc_86><loc_52></location>G ( r ) = ( r -2 n )( r -n 2 ) ( r 2 -n 2 ) = ( r -r + )( r -r -) ( r + n )( r -n ) (23)</formula>
<text><location><page_6><loc_12><loc_37><loc_86><loc_47></location>It is a non-self-dual, non-compact solution of the vacuum Euclidean Einstein equations. In order to make the solution regular we have restricted in the region r = r + ≥ 2 n , and the Euclidean time has period β = 8 πn . Asymptotically, the Taub-Bolt instanton behaves similar manner as the Taub-NUT, so it is ALF. Since we are setting the fixed point is at r = r + = 2 n , therefore the area of the S 2 does not vanish there and the fixed point set is 2-dimensional, thus it is a Bolt of area</text>
<formula><location><page_6><loc_42><loc_34><loc_86><loc_35></location>A + = 12 πn 2 . (24)</formula>
<text><location><page_6><loc_12><loc_31><loc_57><loc_32></location>Thus the area product for Taub-Bolt instanton will be</text>
<formula><location><page_6><loc_42><loc_27><loc_86><loc_29></location>A + = 12 πn 2 . (25)</formula>
<text><location><page_6><loc_12><loc_23><loc_86><loc_26></location>Thus the area product does independent of mass and also quantized. Similarly, the action was calculated in [38]</text>
<formula><location><page_6><loc_39><loc_18><loc_86><loc_21></location>I = -ln Z = πn 2 . (26)</formula>
<text><location><page_6><loc_12><loc_16><loc_58><loc_18></location>Thus the entropy was derived by the universal formula</text>
<formula><location><page_6><loc_34><loc_10><loc_86><loc_15></location>S + = -( β ∂ ∂β -1 ) ln Z = πn 2 . (27)</formula>
<text><location><page_6><loc_12><loc_9><loc_72><loc_10></location>It indicates that the entropy product should be universal and quantized.</text>
<section_header_level_1><location><page_7><loc_12><loc_82><loc_55><loc_84></location>5 Eguchi-Hanson Instantons:</section_header_level_1>
<text><location><page_7><loc_12><loc_77><loc_86><loc_80></location>A non-compact instanton which is a limiting case of the Taub-NUT solution is the EguchiHanson metric [31],</text>
<formula><location><page_7><loc_26><loc_69><loc_71><loc_76></location>ds 2 = ( 1 -n 4 r 4 ) ( r 8 n ) 2 ( dτ +4 n cos θdφ ) 2 + dr 2 ( 1 -n 4 r 4 )</formula>
<formula><location><page_7><loc_37><loc_63><loc_86><loc_68></location>+ r 2 4 ( dθ 2 +sin 2 θdφ 2 ) . (28)</formula>
<text><location><page_7><loc_12><loc_60><loc_86><loc_64></location>The instanton is regular if we consider the region r ≥ n , and let τ has period 8 πn . The metric is ALE type. There is a Bolt of area at r = n is given by</text>
<formula><location><page_7><loc_44><loc_57><loc_86><loc_59></location>A = πn 2 . (29)</formula>
<text><location><page_7><loc_12><loc_52><loc_86><loc_55></location>which gives rise to a Misner string along the z -axis. Thus the product is universal and should be quantized. The Euclidean action derived in [38]</text>
<formula><location><page_7><loc_44><loc_48><loc_86><loc_50></location>I = 0 . (30)</formula>
<text><location><page_7><loc_12><loc_45><loc_36><loc_47></location>Thus entropy corresponds to</text>
<formula><location><page_7><loc_37><loc_39><loc_86><loc_44></location>S = ( β ∂ ∂β -1 ) I = 0 . (31)</formula>
<text><location><page_7><loc_12><loc_28><loc_86><loc_38></location>We now turn our attention for the Taub-NUT and Taub-Bolt geometries in four dimensional locally AdS spacetime. The spacetimes have a global non-trivial topology due to the fact that one of the Killing vector has a zero dimensional fixed point set called NUT or a two-dimensional fixed point set called Bolt. Moreover, these four dimensional spacetimes have have Euclidean sections which can not be exactly matched to AdS spacetime at infinity.</text>
<section_header_level_1><location><page_7><loc_12><loc_23><loc_56><loc_25></location>6 Taub-NUT-AdS Spacetime:</section_header_level_1>
<text><location><page_7><loc_12><loc_16><loc_86><loc_21></location>In this section we shall consider the spacetime which are only locally asymptotically AdS and with non-trivial topology. The metric on the Euclidean section of this family of solutions could be written as [40, 41]</text>
<formula><location><page_7><loc_19><loc_7><loc_86><loc_13></location>ds 2 = H ( r ) ( dτ +2 n cos θdφ ) 2 + dr 2 H ( r ) + ( r 2 -n 2 ) ( dθ 2 +sin 2 θdφ 2 ) . (32)</formula>
<text><location><page_8><loc_12><loc_82><loc_17><loc_84></location>where</text>
<formula><location><page_8><loc_27><loc_76><loc_86><loc_81></location>H ( r ) = ( r 2 + n 2 ) -2 Mr + /lscript -2 ( r 4 -6 n 2 r 2 -3 n 4 ) r 2 -n 2 (33)</formula>
<text><location><page_8><loc_12><loc_67><loc_86><loc_76></location>and /lscript 2 = -3 Λ , with Λ < 0 being the cosmological constant. Here M is a (generalized) mass parameter and r is a radial coordinate. Also, τ , the analytically continued time i.e. Euclidean time, parametrizes a circle S 1 , which is fibered over the two-sphere S 2 , with coordinates θ, φ . The non-trivial fibration is a consequence of a non-vanishing NUT parameter n .</text>
<text><location><page_8><loc_12><loc_54><loc_86><loc_67></location>There are some restrictions [42] for existence of a regular NUT parameter. Firstly, in order to ensure that the fixed point set is zero dimensional, it is necessary that the Killing vector ∂ τ has a fixed point which occurs precisely when the area of the two-sphere is zero size. Secondly, in order for the Dirac-Misner string [35] to be unobservable, it is necessary that the period of τ be β = 8 πn . To avoid the conical singularity, we must check H ' ( r + = n ) = 1 2 n . Thirdly, the mass parameter M must be M = n -4 n 3 /lscript 2 . After simplifying the metric coefficients, we obtain</text>
<formula><location><page_8><loc_31><loc_48><loc_86><loc_53></location>H ( r ) = ( r -n r + n )[ 1 + ( r -n )( r +3 n ) /lscript 2 ] (34)</formula>
<text><location><page_8><loc_12><loc_44><loc_86><loc_48></location>and the range of the radial coordinate becomes n ≤ r ≤ ∞ . For our requirement, the Euclidean action for this spacetime was calculated in [43, 44]</text>
<formula><location><page_8><loc_34><loc_38><loc_86><loc_43></location>I = -ln Z = 4 πn 2 ( 1 -2 n 2 /lscript 2 ) . (35)</formula>
<text><location><page_8><loc_12><loc_36><loc_31><loc_37></location>and the entropy will be</text>
<formula><location><page_8><loc_31><loc_30><loc_86><loc_35></location>S + = ( β ∂ ∂β -1 ) I = 4 πn 2 ( 1 -6 n 2 /lscript 2 ) . (36)</formula>
<text><location><page_8><loc_12><loc_28><loc_42><loc_29></location>Thus the entropy product should be</text>
<formula><location><page_8><loc_38><loc_22><loc_86><loc_26></location>S + = 4 πn 2 ( 1 -6 n 2 /lscript 2 ) . (37)</formula>
<text><location><page_8><loc_12><loc_14><loc_86><loc_21></location>It is independence of mass parameter and does depend on NUT parameter and cosmological constant. Thus the entropy product is universal for Taub-NUT-AdS spacetime. The Hawking temperature T + = 1 8 πn is same as Taub-NUT BH. The first law of thermodynamics is also satisfied as dM = T + dS .</text>
<text><location><page_8><loc_12><loc_8><loc_86><loc_14></location>If we consider the extended phase space following our previous work [25] and in this framework, the cosmological constant (Λ) should be treated as thermodynamical pressure i.e. P = -Λ 8 π = 3 8 π/lscript 2 and its conjugate variable should be treated as thermodynamic</text>
<text><location><page_9><loc_12><loc_77><loc_86><loc_84></location>volume i.e. V + = 4 3 πr 3 + , where r + is the horizon radius. Then one should be interpreted the ADM mass M parameter not to be the energy rather it should be interpreted as enthalpy H = M = U + PV + of the gravitational thermodynamical system. Therefore the thermodynamic volume has been calculated in [39] for Taub-NUT-AdS spacetime:</text>
<formula><location><page_9><loc_37><loc_71><loc_86><loc_75></location>V + = ( ∂H ∂P ) S = -8 3 πn 3 . (38)</formula>
<text><location><page_9><loc_12><loc_63><loc_86><loc_70></location>One aspect, this is a peculiar result in a sense that the thermodynamic volume is negative and the other aspect is that the thermodynamic volume is universal because it is independent of the ADM mass parameter. It should be noted that the first law is fulfilled in this case and it yields</text>
<formula><location><page_9><loc_39><loc_59><loc_86><loc_61></location>dH = T + d S + V + dP . (39)</formula>
<text><location><page_9><loc_12><loc_56><loc_59><loc_58></location>Analogously, the Smarr-Gibbs-Duhem relation should be</text>
<formula><location><page_9><loc_39><loc_52><loc_86><loc_54></location>H = 2 T + S 2 PV + . (40)</formula>
<text><location><page_9><loc_12><loc_48><loc_86><loc_51></location>and another interesting result we first claimed that the internal energy for Taub-NUT-AdS BH is universal . It is given by</text>
<formula><location><page_9><loc_39><loc_42><loc_86><loc_46></location>U = n ( 1 -8 πPn 2 ) . (41)</formula>
<section_header_level_1><location><page_9><loc_12><loc_39><loc_55><loc_41></location>7 Taub-Bolt-AdS Spacetime:</section_header_level_1>
<text><location><page_9><loc_12><loc_29><loc_86><loc_38></location>For Taub-Bolt-AdS, the metric has the same form as in (32) but the fixed point set here is two dimensional or Bolt and with additional restrictions are the metric coefficients H ( r ) vanish at r = r b > n . In order to have a regular Bolt at r = r b , the following conditions must be satisfied: (i) H ( r b ) = 0, (ii) H ' ( r b ) = 1 2 n and the numerator of H ( r ) at r = r b being a single one. From the condition (i), we get the mass parameter at r = r b :</text>
<formula><location><page_9><loc_28><loc_23><loc_86><loc_27></location>M = M b = r 2 b + n 2 2 r b + 1 2 /lscript 2 ( r 3 b -6 n 2 r b -3 n 4 r b ) . (42)</formula>
<text><location><page_9><loc_12><loc_20><loc_26><loc_22></location>Then we find[42]</text>
<formula><location><page_9><loc_35><loc_14><loc_86><loc_19></location>H ' ( r b ) = 3 /lscript 2 ( r 2 b -n 2 + /lscript 2 / 3 r b ) (43)</formula>
<text><location><page_9><loc_12><loc_12><loc_72><loc_13></location>To satisfy the condition (ii) we must have the quadratic equation for r b :</text>
<formula><location><page_9><loc_35><loc_8><loc_86><loc_10></location>6 nr 2 b -/lscript 2 r b -6 n 3 +2 n/lscript 2 = 0 . (44)</formula>
<text><location><page_10><loc_12><loc_82><loc_51><loc_84></location>which gives the solution for r b in two branches</text>
<formula><location><page_10><loc_31><loc_76><loc_86><loc_81></location>r b ± = /lscript 2 12 n ( 1 ± √ 1 -48 n 2 /lscript 2 +144 n 4 /lscript 4 ) . (45)</formula>
<text><location><page_10><loc_12><loc_72><loc_86><loc_75></location>The discriminat of the above equation must be negative for r b to be real and for r b > n we obtain the following inequality for n :</text>
<formula><location><page_10><loc_37><loc_66><loc_86><loc_71></location>n ≤ n max = √ 1 6 -√ 3 12 /lscript . (46)</formula>
<text><location><page_10><loc_12><loc_63><loc_48><loc_65></location>The Euclidean action was computed in [44]</text>
<formula><location><page_10><loc_35><loc_57><loc_86><loc_62></location>I = 4 πn /lscript 2 ( M b /lscript 2 +3 n 2 r b -r 3 b ) . (47)</formula>
<text><location><page_10><loc_12><loc_56><loc_74><loc_58></location>Now it can be easily derive the entropy via the universal entropy formula:</text>
<formula><location><page_10><loc_27><loc_50><loc_86><loc_55></location>S + = ( β ∂ ∂β -1 ) I = 4 πn ( M b -3 n 2 r b /lscript 2 + r 3 b /lscript 2 ) . (48)</formula>
<text><location><page_10><loc_12><loc_48><loc_66><loc_50></location>Now substituating the values of M b , we find the value of entropy</text>
<formula><location><page_10><loc_26><loc_43><loc_86><loc_47></location>S + = 4 πn [ r 2 b + n 2 2 r b + 1 2 /lscript 2 ( 3 r 3 b -12 n 2 r b -3 n 4 r b )] . (49)</formula>
<text><location><page_10><loc_12><loc_41><loc_85><loc_42></location>Again putting the values of r b , we see that the entropy is universal as well as quantized.</text>
<section_header_level_1><location><page_10><loc_12><loc_33><loc_86><loc_38></location>8 Dyonic Taub-NUT-AdS and Taub-Bolt-AdS Spacetime:</section_header_level_1>
<text><location><page_10><loc_12><loc_28><loc_86><loc_31></location>The general form of the metric for dyonic Taub-NUT-AdS spacetime [45, 46, 47] is given by</text>
<text><location><page_10><loc_12><loc_20><loc_17><loc_22></location>where,</text>
<formula><location><page_10><loc_19><loc_21><loc_86><loc_27></location>ds 2 = N ( r ) ( dτ +2 n cos θdφ ) 2 + dr 2 N ( r ) + ( r 2 -n 2 ) ( dθ 2 +sin 2 θdφ 2 ) . (50)</formula>
<formula><location><page_10><loc_21><loc_15><loc_86><loc_19></location>N ( r ) = ( r 2 + n 2 +4 n 2 ν 2 -q 2 ) -2 Mr + /lscript -2 ( r 4 -6 n 2 r 2 -3 n 4 ) r 2 -n 2 (51)</formula>
<text><location><page_10><loc_12><loc_13><loc_32><loc_15></location>The gauge field reads off</text>
<formula><location><page_10><loc_24><loc_7><loc_86><loc_12></location>A ≡ A µ dx µ = ( qr r 2 -n 2 + ν r 2 + n 2 r 2 -n 2 ) ( dτ -2 n cos θdφ ) , (52)</formula>
<text><location><page_11><loc_12><loc_79><loc_86><loc_84></location>The conditions of smoothness of the Euclidean section implies that the parameter q is related to the parameter ν gives a deformation from the uncharged system. When these parameters go to to zero value, we obtain simply Taub-NUT-AdS spacetime.</text>
<text><location><page_11><loc_12><loc_73><loc_86><loc_78></location>In order to have a regular position of NUT or Bolt at r = r ± , we set N ( r ) = 0 and also the gauge field A must be regular at that point. Thus we obtain the mass parameter as</text>
<formula><location><page_11><loc_24><loc_67><loc_86><loc_72></location>M = r 2 ± + n 2 +4 n 2 ν 2 -ν 2 2 r ± + 1 2 /lscript 2 ( r 3 ± -6 n 2 r ± -3 n 4 r ± ) . (53)</formula>
<text><location><page_11><loc_12><loc_66><loc_15><loc_67></location>and</text>
<formula><location><page_11><loc_41><loc_61><loc_86><loc_65></location>q = -r 2 ± + n 2 r ± ν . (54)</formula>
<text><location><page_11><loc_12><loc_58><loc_61><loc_60></location>The electric charge and potential at infinity corresponds to</text>
<formula><location><page_11><loc_35><loc_52><loc_86><loc_56></location>Q = q φ ± = -q r ± r 2 ± + n 2 = ν . (55)</formula>
<text><location><page_11><loc_12><loc_47><loc_86><loc_51></location>Now the Euclidean action for the above spacetime was calculated in [48](in units where G = c = 1)</text>
<formula><location><page_11><loc_20><loc_41><loc_86><loc_46></location>I ± = -2 π [ r 4 ± -/lscript 2 r 2 ± + n 2 (3 n 2 -/lscript 2 ) ] r 2 ± -( r 4 ± +4 n 2 r 2 ± -n 4 ) /lscript 2 ν 2 (3 r 2 ± -3 n 2 + /lscript 2 ) r 2 ± +( r 2 ± -n 2 ) /lscript 2 ν 2 . (56)</formula>
<text><location><page_11><loc_12><loc_39><loc_38><loc_41></location>The entropy was calculated as:</text>
<formula><location><page_11><loc_16><loc_33><loc_86><loc_38></location>S ± = 2 π [ 3 r 4 ± +( /lscript 2 -12 n 2 ) r 2 ± + n 2 ( /lscript 2 -3 n 2 ) ] r 2 ± + ( r 4 ± +4 n 2 r 2 ± -n 4 ) /lscript 2 ν 2 (3 r 2 ± -3 n 2 + /lscript 2 ) r 2 ± +( r 2 ± -n 2 ) /lscript 2 ν 2 . (57)</formula>
<text><location><page_11><loc_12><loc_29><loc_86><loc_33></location>When we set r ± = r n = n , we get a dyonic NUT spacetime. For this spacetime the above calculations reduced to</text>
<formula><location><page_11><loc_32><loc_25><loc_86><loc_28></location>M = n -4 n 3 /lscript 2 , Q = -2 nν, φ ± = ν . (58)</formula>
<text><location><page_11><loc_12><loc_22><loc_15><loc_24></location>and</text>
<formula><location><page_11><loc_35><loc_17><loc_86><loc_21></location>I ± = 4 πn 2 ( 1 -2 n 2 /lscript 2 +2 ν 2 ) . (59)</formula>
<text><location><page_11><loc_12><loc_15><loc_38><loc_16></location>Finally, the entropy is given by</text>
<formula><location><page_11><loc_35><loc_7><loc_86><loc_12></location>S ± = 4 πn 2 ( 1 -6 n 2 /lscript 2 +2 ν 2 ) . (60)</formula>
<text><location><page_12><loc_12><loc_82><loc_61><loc_84></location>Thus the entropy product formula for dyonic Taub-NUT is</text>
<formula><location><page_12><loc_27><loc_76><loc_86><loc_81></location>S + S -= S 2 + = S 2 -= (4 πn 2 ) 2 ( 1 -6 n 2 /lscript 2 +2 ν 2 ) 2 . (61)</formula>
<text><location><page_12><loc_12><loc_74><loc_78><loc_75></location>The product formula is indeed independent of mass and depends on n , /lscript and ν .</text>
<text><location><page_12><loc_15><loc_72><loc_81><loc_73></location>For a dyonic Bolt, we set r ± = r b and satisfies the fourth order equation for r b :</text>
<formula><location><page_12><loc_26><loc_66><loc_86><loc_70></location>6 nr 4 b -/lscript 2 r 3 b +2 n ( /lscript 2 -3 n 2 + /lscript 2 ν 2 ) r 2 b -2 /lscript 2 ν 2 n 3 = 0 . (62)</formula>
<text><location><page_12><loc_12><loc_65><loc_81><loc_67></location>Let r b = r b ± be the solution of the equation. Then we find the action as previously</text>
<formula><location><page_12><loc_21><loc_58><loc_86><loc_62></location>I ± = -2 π [ r 4 b -/lscript 2 r 2 b + n 2 (3 n 2 -/lscript 2 )] r 2 b -( r 4 b +4 n 2 r 2 b -n 4 ) /lscript 2 ν 2 (3 r 2 b -3 n 2 + /lscript 2 ) r 2 b +( r 2 b -n 2 ) /lscript 2 ν 2 . (63)</formula>
<text><location><page_12><loc_12><loc_56><loc_39><loc_57></location>Similarly the entropy is given by</text>
<formula><location><page_12><loc_17><loc_49><loc_86><loc_54></location>S ± = 2 π [3 r 4 b +( /lscript 2 -12 n 2 ) r 2 b + n 2 ( /lscript 2 -3 n 2 )] r 2 b +( r 4 b +4 n 2 r 2 b -n 4 ) /lscript 2 ν 2 (3 r 2 b -3 n 2 + /lscript 2 ) r 2 b +( r 2 b -n 2 ) /lscript 2 ν 2 . (64)</formula>
<text><location><page_12><loc_12><loc_45><loc_86><loc_49></location>After substituating the value of r b ± in the entropy product formula, it seems that the product is independent of mass and depends on n, /lscript, ν for dyonic Taub-Bolt instanton.</text>
<section_header_level_1><location><page_12><loc_12><loc_40><loc_34><loc_42></location>9 Conclusion:</section_header_level_1>
<text><location><page_12><loc_12><loc_29><loc_86><loc_39></location>We have studied the mass-independent feature for various gravitational instantons. This universal feature gives us strong indication towards understanding the microscopic properties of BH entropy. It would be interesting if one considered the entropy product formula for other instantons like Multi-Taub NUT, Non-self dual Taub-NUT, S 4 , CP 2 , S 2 × S 2 and Twisted S 2 × S 2 . We expect these instantons also gives us universal features.</text>
<section_header_level_1><location><page_12><loc_12><loc_25><loc_27><loc_27></location>References</section_header_level_1>
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<list_item><location><page_14><loc_12><loc_9><loc_86><loc_12></location>[37] R. H. Boyer, 'Geodesic Killing Orbits and Bifurcate Killing Horizons', Proc. R. Soc. London A 311 , 245 (1968).</list_item>
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<list_item><location><page_15><loc_12><loc_66><loc_86><loc_69></location>[41] D. N. Page and C. Pope, 'Inhomogeneous Einstein Metrics On Complex Line Bundles', Class. Quantum Grav. 4 , 213 (1987).</list_item>
<list_item><location><page_15><loc_12><loc_61><loc_86><loc_64></location>[42] A. Chamblin et al., 'Large N phases, gravitational instantons and the nuts and bolts of AdS holography', Phys. Rev. D 59 , 064010 (1999).</list_item>
<list_item><location><page_15><loc_12><loc_58><loc_76><loc_59></location>[43] R. B. Mann, 'Misner string entropy', Phys. Rev. D 60 104047 (1999).</list_item>
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<list_item><location><page_15><loc_12><loc_48><loc_86><loc_51></location>[45] J. Plebanski, 'A class of solutions of Einstein Maxwell equations', Ann. Phys. NY 90 196-255 (1975).</list_item>
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</document> | [] |
2019arXiv190404029Z | https://arxiv.org/pdf/1904.04029.pdf | <document>
<section_header_level_1><location><page_1><loc_17><loc_79><loc_72><loc_81></location>The giants arcs as modeled by the superbubbles</section_header_level_1>
<section_header_level_1><location><page_1><loc_27><loc_72><loc_42><loc_73></location>Lorenzo Zaninetti</section_header_level_1>
<text><location><page_1><loc_27><loc_70><loc_49><loc_72></location>Physics Department, via P.Giuria 1, I-10125 Turin,Italy</text>
<text><location><page_1><loc_27><loc_68><loc_32><loc_69></location>E-mail:</text>
<text><location><page_1><loc_32><loc_68><loc_47><loc_69></location>[email protected]</text>
<text><location><page_1><loc_27><loc_57><loc_76><loc_65></location>Abstract. The giant arcs in the clusters of galaxies are modeled in the framework of the superbubbles. The density of the intracluster medium is assumed to follow a hyperbolic behavior. The analytical law of motion is function of the elapsed time and the polar angle. As a consequence the flux of kinetic energy in the expanding thin layer decreases with increasing polar angle making the giant arc invisible to the astronomical observations. In order to calibrate the arcsecparsec conversion three cosmologies are analyzed.</text>
<text><location><page_1><loc_16><loc_49><loc_76><loc_52></location>Keywords : galaxy groups, clusters, and superclusters; large scale structure of the Universe Cosmology</text>
<section_header_level_1><location><page_2><loc_16><loc_86><loc_28><loc_88></location>1. Introduction</section_header_level_1>
<text><location><page_2><loc_16><loc_55><loc_76><loc_85></location>The giant arcs in the cluster of galaxies start to be observed as narrow-like shape by [1, 2, 3] and the first theoretical explanation was the gravitational lensing, see [4, 5, 6, 7]. The determination of the statistical parameters of the giant arcs has been analyzed in order to derive the cosmological parameters, see [8], in connection with the ΛCDM cosmology, see [9], in the framework of the triaxiality and substructure of CDM halo, see [10], in connection with the Wilkinson Microwave Anisotropy Probe (WMAP) data, see [11], including the effects of baryon cooling in dark matter N-body simulations, see [12] and in order to derive the photometric properties of 105 giant arcs that in the Second Red-Sequence Cluster Survey (RCS-2), see [13]. The gravitational lensing is the most common theoretical explanation, we select some approaches among others: [14] evaluated the mass distribution inside distant clusters, [15] studied the statistics of giant arcs in flat cosmologies with and without a cosmological constant, [16] analyzed how the gravitational lensing influences the surface brightness of giant luminous arcs and [17] used the warm dark matter (WDM) cosmologies to explain the lensing in galaxy clusters. Another theoretical line of research explains the giant arcs as shells originated by the galaxies in the cluster: [18, 19] analyzed the the limbbrightened shell model, the gravitational lens model and the echo model, [20] suggested that the Gamma-ray burst (GRB) explosions are the sources of the shells with sizes of many kpc, and [21] suggested a connection between the Einstein ring associated to SDP.81 and the evolution of a superbubble (SB) in the intracluster medium.</text>
<text><location><page_2><loc_16><loc_43><loc_76><loc_55></location>This paper analyzes in Section 2 three cosmologies in order to calibrate the transversal distance which allows to convert the arcsec in pc. Section 3 is devoted to the evolution of a SB in the intracluster medium. Section 4 reports the observations of the giant arcs and the first phase of a SB. Section 5 reports the various steps which allow to reproduce the shape of the giant arc A2267 and the multiple arcs visible in the cluster of galaxies. Section 6 is dedicated to theory of the image: analytical formulae explain the hole in the central part of the SBs and numerical results reproduce the details of a giant arc.</text>
<section_header_level_1><location><page_2><loc_16><loc_39><loc_35><loc_41></location>2. Adopted cosmologies</section_header_level_1>
<text><location><page_2><loc_16><loc_37><loc_54><loc_38></location>In the following we review three cosmological theories.</text>
<section_header_level_1><location><page_2><loc_16><loc_33><loc_32><loc_35></location>2.1. Λ CDM cosmology</section_header_level_1>
<text><location><page_2><loc_16><loc_22><loc_76><loc_32></location>The basic parameters of ΛCDM cosmology are: the Hubble constant, H 0 , expressed in kms -1 Mpc -1 , the velocity of light, c , expressed in km s -1 , and the three numbers Ω M , Ω K , and Ω Λ , see [22] for more details. In the case of the Union 2.1 compilation, see [23], the parameters are H 0 = 69 . 81kms -1 Mpc -1 , Ω M = 0 . 239 and Ω Λ = 0 . 651. To have the luminosity distance, D L ( z ; H 0 , c, Ω M , Ω Λ ), as a function of the redshift only, we apply the minimax rational approximation, which is characterized by two parameters, p and q . The luminosity distance, D L, 3 , 2 , when p = 3 and q = 2</text>
<formula><location><page_2><loc_27><loc_17><loc_76><loc_21></location>D L, 3 , 2 = -7 . 761 -1788 . 53 z -3203 . 06 z 2 -65 . 8463 z 3 -0 . 438025 -0 . 334872 z +0 . 0203996 z 2 Mpc (1) for 0 . 001 < z < 4 .</formula>
<text><location><page_3><loc_16><loc_85><loc_76><loc_88></location>The transversal distance in ΛCDM cosmology, D T, 3 , 2 , which corresponds to the angle δ expressed in arcsec is</text>
<formula><location><page_3><loc_27><loc_81><loc_76><loc_84></location>D T, 3 , 2 = 4 . 84813 δ ( 2 . 328 + 502 . 067 z +113 . 03 z 2 ) 0 . 124085 + 0 . 149501 z +0 . 0932928 z 2 pc . (2)</formula>
<section_header_level_1><location><page_3><loc_16><loc_78><loc_30><loc_79></location>2.2. Flat Cosmology</section_header_level_1>
<text><location><page_3><loc_16><loc_74><loc_76><loc_77></location>The two parameters of the flat cosmology are H 0 , the Hubble constant expressed in kms -1 Mpc -1 , and Ω M which is</text>
<formula><location><page_3><loc_27><loc_71><loc_76><loc_74></location>Ω M = 8 π Gρ 0 3 H 2 0 , (3)</formula>
<text><location><page_3><loc_16><loc_66><loc_76><loc_70></location>where G is the Newtonian gravitational constant and ρ 0 is the mass density at the present time. In the case of m =2 and n =2 the minimax rational expression for the luminosity distance, d L,m, 2 , 2 , when H 0 = 70kms -1 Mpc -1 and Ω M = 0 . 277, is</text>
<formula><location><page_3><loc_27><loc_62><loc_76><loc_65></location>d L,m, 2 , 2 = 0 . 0889 + 748 . 555 z +5 . 58311 z 2 0 . 175804 + 0 . 206041 z +0 . 068685 z 2 Mpc . (4)</formula>
<text><location><page_3><loc_16><loc_59><loc_76><loc_62></location>The transversal distance in flat cosmology, D Tf, 3 , 2 , which corresponds to the angle δ expressed in arcsec is</text>
<formula><location><page_3><loc_27><loc_56><loc_76><loc_59></location>D Tf, 3 , 2 = 4 . 84813 δ ( 0 . 0889 + 748 . 555 z +5 . 58311 z 2 ) 0 . 175804 + 0 . 206041 z +0 . 0686854 z 2 pc (5)</formula>
<section_header_level_1><location><page_3><loc_16><loc_53><loc_32><loc_54></location>2.3. Modified tired light</section_header_level_1>
<text><location><page_3><loc_16><loc_49><loc_76><loc_52></location>In an Euclidean static framework the modified tired light (MTL) has been introduced in Section 2.2 in [24]. The distance in MTL is</text>
<formula><location><page_3><loc_27><loc_46><loc_76><loc_49></location>d = c H 0 ln(1 + z ) . (6)</formula>
<text><location><page_3><loc_19><loc_44><loc_61><loc_45></location>The distance modulus in the modified tired light (MTL) is</text>
<formula><location><page_3><loc_27><loc_40><loc_76><loc_43></location>m -M = 5 2 β ln ( z +1) ln (10) +5 1 ln (10) ln ( ln ( z +1) c H 0 ) +25 . (7)</formula>
<text><location><page_3><loc_16><loc_34><loc_76><loc_40></location>Here β is a parameter comprised between 1 and 3 which allows to match theory with observations. The number of free parameters in MTL is two: H 0 and β . The fit of the distance modulus with the data of Union 2.1 compilation gives β =2.37, H 0 = 69 . 32 ± 0 . 34, see [22], which means the following distance</text>
<formula><location><page_3><loc_27><loc_32><loc_76><loc_33></location>d = 4324 . 761 ln (1 + z ) (1 + z ) 1 . 185 Mpc . (8)</formula>
<text><location><page_3><loc_16><loc_28><loc_76><loc_31></location>The transversal distance in MTL, d T , which corresponds to the angle δ expressed in arcsec is</text>
<formula><location><page_3><loc_27><loc_26><loc_76><loc_28></location>d T = 20967 δ ln (1 + z ) (1 + z ) 1 . 185 pc (9)</formula>
<text><location><page_3><loc_16><loc_21><loc_76><loc_25></location>We report the angular distance for a fixed delta as function of redshift for the three cosmologies, see Figure 1. The angular distance in flat and ΛCDM cosmology does not increase with z, see [25], in contrast with the modified tired light .</text>
<section_header_level_1><location><page_3><loc_16><loc_18><loc_34><loc_19></location>3. The motion of a SB</section_header_level_1>
<text><location><page_3><loc_16><loc_14><loc_76><loc_16></location>We now summarize the adopted profile of density and the equation of motion for a SB.</text>
<figure>
<location><page_4><loc_31><loc_67><loc_55><loc_85></location>
<caption>Figure 1. Angular distances in kpc for the three cosmologies here considered when δ = 38 . 913456</caption>
</figure>
<section_header_level_1><location><page_4><loc_16><loc_42><loc_27><loc_43></location>3.1. The profile</section_header_level_1>
<text><location><page_4><loc_16><loc_38><loc_76><loc_41></location>The density is assumed to have the following hyperbolic dependence on Z which is the third Cartesian coordinate,</text>
<formula><location><page_4><loc_27><loc_34><loc_76><loc_38></location>ρ ( Z ; Z 0 , ρ 0 ) = { ρ 0 if z ≤ Z 0 ρ 0 Z 0 z if z > Z 0 (10)</formula>
<text><location><page_4><loc_16><loc_31><loc_76><loc_34></location>where the parameter Z 0 fixes the scale and ρ 0 is the density at Z = Z 0 . In spherical coordinates the dependence on the polar angle is</text>
<formula><location><page_4><loc_27><loc_27><loc_76><loc_30></location>ρ ( r ; θ, Z 0 , ρ 0 ) = { ρ 0 if cos( θ ) ≤ Z 0 ρ 0 Z 0 r cos( θ ) if r cos( θ ) > Z 0 (11)</formula>
<text><location><page_4><loc_16><loc_25><loc_63><loc_26></location>Given a solid angle ∆Ω the mass M 0 swept in the interval [0 , r 0 ] is</text>
<formula><location><page_4><loc_27><loc_22><loc_76><loc_24></location>M 0 = 1 3 ρ 0 r 0 3 ∆Ω . (12)</formula>
<text><location><page_4><loc_16><loc_20><loc_63><loc_21></location>The total mass swept, M ( r ; r 0 , Z 0 , α, θ, ρ 0 ), in the interval [0 , r ] is</text>
<formula><location><page_4><loc_27><loc_16><loc_76><loc_19></location>M ( r ; r 0 , Z 0 , α, θ, ρ 0 ) = ( 1 3 ρ 0 r 0 3 + 1 2 ρ 0 Z 0 ( r 2 -r 0 2 ) cos ( θ ) ) ∆Ω . (13)</formula>
<text><location><page_5><loc_16><loc_86><loc_49><loc_88></location>and its approximate value at high values of r is</text>
<formula><location><page_5><loc_27><loc_83><loc_76><loc_86></location>M ( r ; Z 0 , α, θ, ρ 0 ) ≈ 1 2 r 2 ρ 0 Z 0 cos ( θ ) ∆Ω . (14)</formula>
<text><location><page_5><loc_16><loc_78><loc_76><loc_82></location>The density ρ 0 can be obtained by introducing the number density, n 0 , expressed in particles cm -3 , the mass of hydrogen, m H , and a multiplicative factor f , which is chosen to be 1.4, see [26],</text>
<formula><location><page_5><loc_27><loc_75><loc_76><loc_77></location>ρ 0 = fm H n 0 . (15)</formula>
<text><location><page_5><loc_16><loc_72><loc_76><loc_75></location>The astrophysical version of the total approximate swept mass as given by equation (14), expressed in solar mass units, M glyph[circledot] , is</text>
<formula><location><page_5><loc_27><loc_68><loc_76><loc_71></location>M ( r pc ; Z 0 ,pc , n 0 , θ ) ≈ 0 . 0172 n 0 z 0 ,pc r pc 2 cos ( θ ) M glyph[circledot] ∆Ω , (16)</formula>
<text><location><page_5><loc_16><loc_66><loc_53><loc_67></location>where Z 0 ,pc , and r 0 ,pc are Z 0 , and r expressed in pc.</text>
<section_header_level_1><location><page_5><loc_16><loc_63><loc_36><loc_64></location>3.2. The equation of motion</section_header_level_1>
<text><location><page_5><loc_16><loc_59><loc_76><loc_62></location>The conservation of the classical momentum in spherical coordinates along the solid angle ∆Ω in the framework of the thin layer approximation states that</text>
<formula><location><page_5><loc_27><loc_57><loc_76><loc_58></location>M 0 ( r 0 ) v 0 = M ( r ) v , (17)</formula>
<text><location><page_5><loc_16><loc_51><loc_76><loc_56></location>where M 0 ( r 0 ) and M ( r ) are the swept masses at r 0 and r , and v 0 and v are the velocities of the thin layer at r 0 and r . This conservation law can be expressed as a differential equation of the first order by inserting v = dr dt :</text>
<formula><location><page_5><loc_27><loc_48><loc_76><loc_51></location>M ( r ) dr dt -M 0 v 0 = 0 . (18)</formula>
<text><location><page_5><loc_16><loc_46><loc_47><loc_47></location>The velocity as a function of the radius r is</text>
<text><location><page_5><loc_49><loc_44><loc_50><loc_45></location>r</text>
<text><location><page_5><loc_46><loc_43><loc_49><loc_44></location>cos (</text>
<text><location><page_5><loc_49><loc_43><loc_50><loc_44></location>θ</text>
<text><location><page_5><loc_50><loc_43><loc_50><loc_44></location>)</text>
<text><location><page_5><loc_51><loc_43><loc_52><loc_44></location>-</text>
<text><location><page_5><loc_52><loc_43><loc_53><loc_44></location>3</text>
<text><location><page_5><loc_53><loc_43><loc_54><loc_44></location>r</text>
<text><location><page_5><loc_42><loc_43><loc_43><loc_44></location>2</text>
<text><location><page_5><loc_43><loc_43><loc_44><loc_44></location>r</text>
<text><location><page_5><loc_44><loc_42><loc_45><loc_43></location>0</text>
<text><location><page_5><loc_45><loc_43><loc_45><loc_44></location>3</text>
<text><location><page_5><loc_53><loc_44><loc_56><loc_45></location>cos (</text>
<text><location><page_5><loc_56><loc_44><loc_57><loc_45></location>θ</text>
<text><location><page_5><loc_57><loc_44><loc_57><loc_45></location>)</text>
<text><location><page_5><loc_54><loc_42><loc_55><loc_43></location>0</text>
<text><location><page_5><loc_55><loc_43><loc_56><loc_44></location>2</text>
<text><location><page_5><loc_56><loc_43><loc_57><loc_44></location>Z</text>
<text><location><page_5><loc_57><loc_42><loc_57><loc_43></location>0</text>
<text><location><page_5><loc_58><loc_43><loc_60><loc_44></location>+3</text>
<text><location><page_5><loc_61><loc_43><loc_61><loc_44></location>r</text>
<text><location><page_5><loc_61><loc_43><loc_62><loc_44></location>2</text>
<text><location><page_5><loc_62><loc_43><loc_63><loc_44></location>Z</text>
<text><location><page_5><loc_63><loc_42><loc_64><loc_43></location>0</text>
<text><location><page_5><loc_16><loc_39><loc_76><loc_42></location>The differential equation which models the momentum conservation in the case of a hyperbolic profile is</text>
<formula><location><page_5><loc_27><loc_35><loc_76><loc_38></location> 1 3 r 0 3 + 1 2 Z 0 ( -r 0 2 +( r ( t )) 2 ) cos ( θ ) d d t r ( t ) -1 3 r 0 3 v 0 = 0 , (20)</formula>
<text><location><page_5><loc_16><loc_32><loc_60><loc_33></location>where the initial conditions are r = r 0 and v = v 0 when t = t 0 .</text>
<text><location><page_5><loc_19><loc_30><loc_72><loc_31></location>The variables can be separated and the radius as a function of the time is</text>
<formula><location><page_5><loc_27><loc_27><loc_76><loc_30></location>r ( t ; t 0 , r 0 , Z 0 , v 0 , θ ) = HN HD , (21)</formula>
<text><location><page_5><loc_16><loc_25><loc_20><loc_26></location>where</text>
<text><location><page_5><loc_16><loc_15><loc_19><loc_16></location>with</text>
<text><location><page_5><loc_50><loc_44><loc_50><loc_45></location>0</text>
<text><location><page_5><loc_50><loc_45><loc_51><loc_46></location>3</text>
<text><location><page_5><loc_51><loc_44><loc_52><loc_45></location>v</text>
<text><location><page_5><loc_52><loc_44><loc_52><loc_45></location>0</text>
<text><location><page_5><loc_27><loc_43><loc_28><loc_45></location>v</text>
<text><location><page_5><loc_28><loc_43><loc_29><loc_45></location>(</text>
<text><location><page_5><loc_29><loc_43><loc_30><loc_45></location>r</text>
<text><location><page_5><loc_30><loc_43><loc_30><loc_45></location>;</text>
<text><location><page_5><loc_30><loc_43><loc_31><loc_45></location>r</text>
<text><location><page_5><loc_31><loc_43><loc_32><loc_44></location>0</text>
<text><location><page_5><loc_32><loc_43><loc_34><loc_45></location>, Z</text>
<text><location><page_5><loc_34><loc_43><loc_34><loc_44></location>0</text>
<text><location><page_5><loc_34><loc_43><loc_36><loc_45></location>, v</text>
<text><location><page_5><loc_36><loc_43><loc_37><loc_44></location>0</text>
<text><location><page_5><loc_37><loc_43><loc_38><loc_45></location>, θ</text>
<text><location><page_5><loc_38><loc_43><loc_42><loc_45></location>) = 2</text>
<text><location><page_5><loc_66><loc_43><loc_66><loc_45></location>.</text>
<text><location><page_5><loc_73><loc_43><loc_76><loc_45></location>(19)</text>
<formula><location><page_5><loc_27><loc_22><loc_75><loc_25></location>HN = -3 √ 3 ( 2 cos( θ ) 3 √ 3 r 0 -3 3 √ 3 Z 0 -( -9 Z 0 3 / 2 +((9 t -9 t 0 ) v 0</formula>
<formula><location><page_5><loc_27><loc_18><loc_76><loc_20></location>+9 r 0 ) cos( θ ) √ Z 0 + √ 3 √ 27 √ AHN ) 2 / 3 ) r 0 , (22)</formula>
<formula><location><page_6><loc_27><loc_75><loc_76><loc_86></location>AHN = ( 8 (cos( θ )) 2 r 0 3 27 + Z 0 ( ( t -t 0 ) 2 v 0 2 +2 r 0 ( t -t 0 ) v 0 -1 3 r 0 2 ) cos( θ ) -2 v 0 Z 0 2 ( t -t 0 ) ) cos( θ ) (23)</formula>
<formula><location><page_6><loc_27><loc_67><loc_76><loc_71></location>HD = 3 √ Z 0 × 3 √ -9 Z 0 3 / 2 +((9 t -9 t 0 ) v 0 +9 r 0 ) cos ( θ ) √ Z 0 +9 √ BHD , (24)</formula>
<text><location><page_6><loc_16><loc_72><loc_18><loc_73></location>and</text>
<text><location><page_6><loc_16><loc_65><loc_19><loc_66></location>with</text>
<formula><location><page_6><loc_27><loc_54><loc_76><loc_65></location>BHD = ( 8 (cos( θ )) 2 r 0 3 27 + Z 0 ( ( t -t 0 ) 2 v 0 2 +2 r 0 ( t -t 0 ) v 0 -1 3 r 0 2 ) cos( θ ) -2 v 0 Z 0 2 ( t -t 0 ) ) cos( θ ) . (25)</formula>
<text><location><page_6><loc_16><loc_51><loc_55><loc_52></location>As a consequence the velocity as function of the time is</text>
<formula><location><page_6><loc_27><loc_47><loc_76><loc_50></location>v ( t ; t 0 , r 0 , Z 0 , v 0 , θ ) = dr ( t ; t 0 , r 0 , Z 0 , v 0 , θ ) dt . (26)</formula>
<text><location><page_6><loc_16><loc_43><loc_76><loc_47></location>More details as well the exploration of other profiles of density can be found in [27]. We now continue evaluating the flux of kinetic energy, F ek , in the thin emitting layer which is supposed to have density ρ l</text>
<formula><location><page_6><loc_27><loc_39><loc_76><loc_42></location>F ek ( t ; t 0 , r 0 , Z 0 , v 0 , θ ) = 1 2 ρ l 4 πr ( t ) 2 v ( t ) 3 . (27)</formula>
<text><location><page_6><loc_16><loc_37><loc_60><loc_39></location>The volume of the thin emitting layer, V l , is approximated by</text>
<formula><location><page_6><loc_27><loc_35><loc_76><loc_37></location>V l = 4 ∆ π r 2 , (28)</formula>
<text><location><page_6><loc_16><loc_30><loc_76><loc_35></location>where ∆ is thickness of the layer; as an example [26] quotes ∆ = r 12 . The two approximations for mass, equation (14), and volume, equation (28), allows to derive an approximate value for the density in the thin layer</text>
<formula><location><page_6><loc_27><loc_27><loc_76><loc_30></location>ρ l = 1 8 ρ 0 Z 0 f cos ( θ ) rπ . (29)</formula>
<text><location><page_6><loc_16><loc_23><loc_76><loc_26></location>Inserted in equation (27) the radius, velocity and density as given by equations (21), (26) and (29), we obtain</text>
<formula><location><page_6><loc_27><loc_20><loc_76><loc_22></location>F ek ( t ; t 0 , r 0 , Z 0 , v 0 , θ ) = FN FD , (30)</formula>
<text><location><page_7><loc_16><loc_86><loc_20><loc_88></location>where</text>
<formula><location><page_7><loc_27><loc_73><loc_76><loc_86></location>FN = , -√ 27 ( -3 √ 3 Z 0 3 / 2 + √ 27 √ F 1 cos ( θ ) + 3 F 5 ) 3 f × ( 2 3 √ 3 F 3 + ( -9 Z 0 3 / 2 + F 2 + √ 27 √ F 1 cos ( θ ) √ 3 ) 2 / 3 ) 3 r 0 4 × cos ( θ ) v0 3 √ Z 0 ( 2 3 √ 3 F 3 -( -9 Z 0 3 / 2 + F 2 + √ 27 √ F 1 cos( θ ) √ 3 ) 2 / 3 ) 3 √ 3 ρ 0 (31)</formula>
<formula><location><page_7><loc_27><loc_66><loc_76><loc_70></location>FD = 108 √ F 1 cos ( θ ) × ( -9 Z 0 3 / 2 + F 2 + √ 27 √ F 1 cos ( θ ) √ 3 ) 13 / 3 F 4 (32)</formula>
<formula><location><page_7><loc_27><loc_57><loc_76><loc_63></location>F 1 = 8 (cos ( θ )) 2 r 0 3 27 + ( ( t -t 0 ) 2 v0 2 +2 r 0 ( t -t 0 ) v0 -1 3 r 0 2 ) × Z 0 cos ( θ ) -2 v0 Z 0 2 ( t -t 0 ) , (33)</formula>
<formula><location><page_7><loc_27><loc_55><loc_76><loc_56></location>F 2 = ((9 t -9 t 0 ) v0 +9 r 0 ) cos ( θ ) √ Z 0 , (34)</formula>
<formula><location><page_7><loc_27><loc_52><loc_76><loc_53></location>F 3 = cos ( θ ) r 0 -3 / 2 Z 0 (35)</formula>
<text><location><page_7><loc_27><loc_49><loc_28><loc_50></location>F</text>
<text><location><page_7><loc_28><loc_49><loc_29><loc_49></location>4</text>
<text><location><page_7><loc_30><loc_49><loc_36><loc_50></location>= 8(cos(</text>
<text><location><page_7><loc_36><loc_49><loc_37><loc_50></location>θ</text>
<text><location><page_7><loc_37><loc_49><loc_38><loc_50></location>))</text>
<text><location><page_7><loc_39><loc_49><loc_39><loc_50></location>r</text>
<text><location><page_7><loc_39><loc_49><loc_40><loc_49></location>0</text>
<text><location><page_7><loc_41><loc_49><loc_44><loc_50></location>+27</text>
<text><location><page_7><loc_46><loc_49><loc_46><loc_50></location>(</text>
<text><location><page_7><loc_46><loc_49><loc_47><loc_50></location>t</text>
<text><location><page_7><loc_47><loc_49><loc_49><loc_50></location>-</text>
<text><location><page_7><loc_49><loc_49><loc_50><loc_50></location>t</text>
<text><location><page_7><loc_50><loc_49><loc_50><loc_49></location>0</text>
<text><location><page_7><loc_50><loc_49><loc_51><loc_50></location>)</text>
<text><location><page_7><loc_52><loc_49><loc_53><loc_50></location>v0</text>
<text><location><page_7><loc_55><loc_49><loc_57><loc_50></location>+2</text>
<text><location><page_7><loc_57><loc_49><loc_58><loc_50></location>r</text>
<text><location><page_7><loc_58><loc_49><loc_59><loc_49></location>0</text>
<text><location><page_7><loc_59><loc_49><loc_59><loc_50></location>(</text>
<text><location><page_7><loc_59><loc_49><loc_60><loc_50></location>t</text>
<text><location><page_7><loc_60><loc_49><loc_62><loc_50></location>-</text>
<text><location><page_7><loc_62><loc_49><loc_62><loc_50></location>t</text>
<text><location><page_7><loc_62><loc_49><loc_63><loc_49></location>0</text>
<text><location><page_7><loc_63><loc_49><loc_64><loc_50></location>)</text>
<text><location><page_7><loc_64><loc_49><loc_65><loc_50></location>v0</text>
<text><location><page_7><loc_66><loc_49><loc_67><loc_50></location>-</text>
<formula><location><page_7><loc_28><loc_45><loc_76><loc_48></location>1 3 r 0 2 ) Z 0 cos( θ ) -54 v0 Z 0 2 ( t -t 0 ) , (36)</formula>
<formula><location><page_7><loc_27><loc_42><loc_76><loc_45></location>F 5 = ( v0 ( t -t 0 ) + r 0 ) cos ( θ ) √ 3 √ Z 0 . (37)</formula>
<text><location><page_7><loc_16><loc_39><loc_76><loc_41></location>We now assumes that the amount of luminosity, L theo , reversed in the shocked emission is proportional to the flux of kinetic energy as given by equation (30)</text>
<formula><location><page_7><loc_27><loc_36><loc_76><loc_38></location>L theo ( t ; t 0 , r 0 , Z 0 , v 0 , θ ) ∝ F ek ( t ; t 0 , r 0 , Z 0 , v 0 , θ ) . (38)</formula>
<text><location><page_7><loc_16><loc_31><loc_76><loc_35></location>The theoretical luminosity is not equal along all the SB but is function of the polar angle θ . In this framework is useful to introduce the ratio, κ , between theoretical luminosity at θ and and that one at θ = 0,</text>
<formula><location><page_7><loc_27><loc_28><loc_76><loc_30></location>κ = L theo ( t ; t 0 , r 0 , Z 0 , v 0 , θ ) L theo ( t ; t 0 , r 0 , Z 0 , v 0 , θ = 0) . (39)</formula>
<text><location><page_7><loc_16><loc_24><loc_76><loc_27></location>The above model for the theoretical luminosity is independent from the image theory, see Section 6, and does not explains the hole of luminosity visible in the shells.</text>
<section_header_level_1><location><page_7><loc_16><loc_21><loc_40><loc_22></location>4. Astrophysical Environment</section_header_level_1>
<text><location><page_7><loc_16><loc_17><loc_76><loc_19></location>We now analyze the catalogue for the giant arcs, the two giant arcs SDP.81 and A2267 and the initial astrophysical conditions for the SBs.</text>
<text><location><page_7><loc_38><loc_49><loc_39><loc_50></location>2</text>
<text><location><page_7><loc_40><loc_49><loc_41><loc_50></location>3</text>
<text><location><page_7><loc_45><loc_50><loc_46><loc_50></location>(</text>
<text><location><page_7><loc_51><loc_49><loc_52><loc_50></location>2</text>
<text><location><page_7><loc_53><loc_49><loc_54><loc_50></location>2</text>
<text><location><page_7><loc_16><loc_71><loc_18><loc_72></location>and</text>
<text><location><page_7><loc_16><loc_63><loc_20><loc_65></location>being</text>
<figure>
<location><page_8><loc_23><loc_60><loc_66><loc_85></location>
<caption>Figure 2. Histogram of the radial distance from the arc center to the cluster center in Mpc in flat cosmology with conversion formula (5).</caption>
</figure>
<section_header_level_1><location><page_8><loc_16><loc_51><loc_29><loc_52></location>4.1. The catalogue</section_header_level_1>
<text><location><page_8><loc_16><loc_38><loc_76><loc_50></location>Some parameters of the giant arcs as detected as images by cluster lensing and the supernova survey with Hubble (CLASH) which is available as a catalogue at http://vizier.u-strasbg.fr/viz-bin/VizieR , see [28]. We are interested in the arc length which is given in arcsec, the arc length to width ratio, the photometric redshift, and in the radial distance from the arc center to the cluster center in arcsec. Table 1 reports the statistical parameters of the radial distance from the arc center to the cluster center in kpc and Figure 2, Figure 3, and Figure 4 the histogram of the frequencies in the framework of flat,ΛCDM and MTL cosmology respectively.</text>
<table>
<location><page_8><loc_22><loc_26><loc_70><loc_32></location>
<caption>Table 1. Statistical parameters of the radial distance from the arc center to the cluster center in kpc</caption>
</table>
<section_header_level_1><location><page_8><loc_16><loc_21><loc_31><loc_22></location>4.2. Single giant arcs</section_header_level_1>
<text><location><page_8><loc_16><loc_14><loc_76><loc_19></location>The ring associated with the galaxy SDP.81, see [29], is characterized by a foreground galaxy at z = 0 . 2999 and a background galaxy at z = 0 . 3042. This ring has been studied with the Atacama Large Millimeter/sub-millimeter Array (ALMA) by [30, 31, 32, 33, 34, 35] and has the observed parameters as in Table 4.2.</text>
<figure>
<location><page_9><loc_23><loc_60><loc_66><loc_84></location>
<caption>Figure 3. Histogram of the radial distance from the arc center to the cluster center in Mpc in ΛCDM cosmology with conversion formula (2).</caption>
</figure>
<figure>
<location><page_9><loc_23><loc_19><loc_66><loc_43></location>
<caption>Figure 4. Histogram of the radial distance from the arc center to the cluster center in Mpc in MTL cosmology with conversion formula (9).</caption>
</figure>
<table>
<location><page_10><loc_33><loc_80><loc_59><loc_85></location>
<caption>Table 2. Observed parameters of the giants arcs.</caption>
</table>
<text><location><page_10><loc_16><loc_71><loc_76><loc_76></location>Another giant arc is that in A2667 which is made by three pieces: A, B and C , see Figure 1 in [36]. The radius can be found from the equation of the circle given the three points A, B and C, see Table 3. The three pieces can be digitalized for a further comparison with a simulation, see empty red stars in Figure 8.</text>
<table>
<location><page_10><loc_34><loc_60><loc_58><loc_66></location>
<caption>Table 3. Radius of the giant arcs in kpc.</caption>
</table>
<section_header_level_1><location><page_10><loc_16><loc_54><loc_34><loc_55></location>4.3. The initial conditions</section_header_level_1>
<text><location><page_10><loc_16><loc_43><loc_76><loc_53></location>We review the starting equations for the evolution of the SB [37, 38, 39] which can be derived from the momentum conservation applied to a pyramidal section. The parameters of the thermal model are N ∗ , the number of SN explosions in 5 . 0 · 10 7 yr, Z OB , the distance of the OB associations from the galactic plane, E 51 , the energy in 10 51 erg usually chosen equal to one, v 0 , the initial velocity which is fixed by the bursting phase, t 0 , the initial time in yr which is equal to the bursting time, and t the proper time of the SB. With the above definitions the radius of the SB is</text>
<formula><location><page_10><loc_27><loc_39><loc_76><loc_42></location>R = 111 . 56 ( E 51 t 3 7 N ∗ n 0 ) 1 5 pc , (40)</formula>
<text><location><page_10><loc_16><loc_37><loc_27><loc_38></location>and its velocity</text>
<formula><location><page_10><loc_27><loc_33><loc_76><loc_36></location>V = 6 . 567 1 t 7 2 / 5 5 √ E 51 N ∗ n 0 km s . (41)</formula>
<text><location><page_10><loc_16><loc_30><loc_76><loc_32></location>In the following, we will assume that the bursting phase ends at t = t 7 , 0 (the bursting time is expressed in units of 10 7 yr) when N SN SN are exploded</text>
<formula><location><page_10><loc_27><loc_26><loc_76><loc_29></location>N SN = N ∗ t 7 , 0 · 10 7 5 · 10 7 . (42)</formula>
<text><location><page_10><loc_16><loc_23><loc_76><loc_25></location>The two following inverted formula allows to derive the parameters of the initial conditions for the SB with ours r 0 expressed in pc and v 0 expressed in kms -1 are</text>
<formula><location><page_10><loc_27><loc_19><loc_76><loc_22></location>t 7 , 0 = 0 . 05878095238 r 0 v 0 , (43)</formula>
<text><location><page_10><loc_16><loc_18><loc_18><loc_19></location>and</text>
<formula><location><page_10><loc_27><loc_14><loc_76><loc_17></location>N ∗ = 2 . 8289 10 -7 r 0 2 n 0 v 0 3 E 51 . (44)</formula>
<figure>
<location><page_11><loc_28><loc_60><loc_61><loc_87></location>
<caption>Figure 5. 3D surface of the SB connected with A2267, parameters as in Table 4 and axes in pc.</caption>
</figure>
<section_header_level_1><location><page_11><loc_16><loc_51><loc_38><loc_52></location>5. Astrophysical Simulation</section_header_level_1>
<text><location><page_11><loc_16><loc_46><loc_76><loc_49></location>We simulate a single giant arc, A2267, and then we simulate the statistics of many giant arcs.</text>
<section_header_level_1><location><page_11><loc_16><loc_43><loc_34><loc_44></location>5.1. Simulation of A2667</section_header_level_1>
<text><location><page_11><loc_16><loc_38><loc_76><loc_42></location>The final stage of the SB connected with A2267 is simulated with the parameters reported in Table 4 ; in particular Figure 5 displays the 3D shape and Figure 6 reports the 2D section.</text>
<table>
<location><page_11><loc_28><loc_18><loc_64><loc_33></location>
<caption>Table 4. Theoretical parameters of the SB connected with A2267.</caption>
</table>
<figure>
<location><page_12><loc_23><loc_61><loc_66><loc_84></location>
<caption>Figure 6. 2D section in the z = 0 plane of the SB connected with A2267, parameters as in Table 4 and axes in kpc.</caption>
</figure>
<text><location><page_12><loc_16><loc_48><loc_76><loc_52></location>Figure 7 reports the 2D section of the SB as well the three pieces of the giant arc connected with A2267. The similarity between the observed radius of curvature of the giant arc as well the theoretical one is reported in a zoom, see Figure 8.</text>
<text><location><page_12><loc_16><loc_31><loc_76><loc_47></location>We can understand the reason for which the giant arc A2267 has a limited angular extension of ≈ 31 · by plotting the ratio κ , equation (39), between the theoretical luminosity as function of θ and the theoretical luminosity at θ = 0 with parameters as in Table 4, see Figure 9. As a practical example at ≈ 31 · / 2, where the factor two arises from the symmetry of the framework, the theoretical luminosity is decreased of a factor κ = 0 . 987 in respect to the value at θ = 0. We now introduce the threshold luminosity, L tr , which is an observational parameter. The theoretical luminosity will scale as function of the polar angle as L theo ( θ ) ∝ L 0 ∗ r where L 0 is the theoretical luminosity at θ = 0 and κ has been defined in equation (39). When the inequality L theo < L tr is verified the giant arc is impossible to detect and only the zone characterized by low values of the polar angle will be detected.</text>
<text><location><page_12><loc_16><loc_27><loc_76><loc_31></location>In our model the velocity with parameters as in Table 4 is function of the polar angle, see Figure 10, and has range 37 km/s < v ( θ ) < 142 km/s . As a comparison a velocity 50 km/s < v < 75 km/s is measured in A2267, see Figure 5 in [36].</text>
<section_header_level_1><location><page_12><loc_16><loc_23><loc_41><loc_25></location>5.2. Simulation of many giants arcs</section_header_level_1>
<text><location><page_12><loc_16><loc_20><loc_76><loc_22></location>The presence of multiple giants arcs in the CLASH cluster, see as an example Figure 11 in [28], can be simulated adopting the following steps</text>
<unordered_list>
<list_item><location><page_12><loc_17><loc_16><loc_76><loc_19></location>· A given number of SBs, as an example 15, are generated with variable lifetime, t , see Figure 11</list_item>
<list_item><location><page_12><loc_17><loc_14><loc_76><loc_15></location>· For each SB we select a section around polar angle equal to zero characterized by</list_item>
</unordered_list>
<figure>
<location><page_13><loc_23><loc_61><loc_66><loc_84></location>
<caption>Figure 7. 2D section in the z = 0 plane of the SB connected with A2267, parameters as in Table 4 (full green points) and the three pieces of the giant arc in A2667 (empty red stars); axes in arcsec.</caption>
</figure>
<figure>
<location><page_13><loc_41><loc_20><loc_50><loc_43></location>
<caption>Figure 8. Enlarged view of the three pieces of the giant arc in A2667 (empty red stars) and the theoretical radius (full green points); axes in arcsec.</caption>
</figure>
<figure>
<location><page_14><loc_24><loc_61><loc_66><loc_84></location>
<caption>Figure 9. Normalized luminosity as function of the polar angle in deg, parameters as in Table 4</caption>
</figure>
<figure>
<location><page_14><loc_23><loc_19><loc_66><loc_42></location>
<caption>Figure 10. Velocity in km/s as function of the polar angle in deg.</caption>
</figure>
<figure>
<location><page_15><loc_29><loc_60><loc_60><loc_84></location>
<caption>Figure 11. Multiple sections of the SB with time, t comprised in [10 6 yr, 10 8 yr ] and other parameters as in Table 4.</caption>
</figure>
<text><location><page_15><loc_19><loc_49><loc_76><loc_52></location>a fixed angle of ≈ 31 · and we randomly rotate it around the origin, see Figure 12</text>
<unordered_list>
<list_item><location><page_15><loc_17><loc_46><loc_76><loc_49></location>· The centers of the SBs are randomly placed in a squared box with side of 300 kpc, see Figure 13</list_item>
</unordered_list>
<text><location><page_15><loc_16><loc_41><loc_76><loc_45></location>Table 5 reports the theoretical statistical parameters of the above simulation for the radial distance from the arc center to the cluster center in kpc. A comparison should be done with the astronomical parameters for the CLASH clusters of Table 1.</text>
<paragraph><location><page_15><loc_27><loc_36><loc_76><loc_38></location>Table 5. Statistical parameters of the radial distance from the theoretical arc center to the cluster center in kpc</paragraph>
<formula><location><page_15><loc_41><loc_34><loc_51><loc_35></location>50 165 349</formula>
<section_header_level_1><location><page_15><loc_16><loc_28><loc_35><loc_29></location>6. Theory of the image</section_header_level_1>
<text><location><page_15><loc_16><loc_24><loc_76><loc_26></location>We now review the theory of the image for the case of optically thin medium both from an analytical and an analytical point of view.</text>
<section_header_level_1><location><page_15><loc_16><loc_20><loc_32><loc_22></location>6.1. The elliptical shell</section_header_level_1>
<text><location><page_15><loc_16><loc_17><loc_76><loc_19></location>A real ellipsoid represents a first approximation of the asymmetric giants arcs and has equation</text>
<formula><location><page_15><loc_28><loc_13><loc_76><loc_16></location>z 2 a 2 + x 2 b 2 + y 2 d 2 = 1 , (45)</formula>
<figure>
<location><page_16><loc_23><loc_61><loc_66><loc_84></location>
<caption>Figure 12. Multiple sections of the SB as in Figure 11 with angular extension of the polar angle, θ , of ≈ 31 · and progressive rotation of the selected piece of section.</caption>
</figure>
<figure>
<location><page_16><loc_28><loc_21><loc_61><loc_45></location>
<caption>Figure 13. Multiple sections of SB as in Figure 12 with random shift of the origin of the selected SB (green empty stars). The random shift denotes the galaxies (red crosses).</caption>
</figure>
<figure>
<location><page_17><loc_30><loc_59><loc_58><loc_88></location>
<caption>Figure 14. Internal and external ellipses when a = 347 kpc , b = 237 kpc and c = a 12 kpc .</caption>
</figure>
<text><location><page_17><loc_16><loc_50><loc_41><loc_52></location>in which the polar axis is the z-axis.</text>
<text><location><page_17><loc_16><loc_47><loc_76><loc_50></location>We are interested in the section of the ellipsoid y = 0 which is defined by the following external ellipse</text>
<formula><location><page_17><loc_28><loc_44><loc_76><loc_47></location>z 2 a 2 + x 2 b 2 = 1 . (46)</formula>
<text><location><page_17><loc_16><loc_41><loc_76><loc_43></location>We assume that the emission takes place in a thin layer comprised between the external ellipse and the internal ellipse defined by</text>
<formula><location><page_17><loc_28><loc_37><loc_76><loc_40></location>z 2 ( a -c ) 2 + x 2 ( b -c ) 2 = 1 , (47)</formula>
<text><location><page_17><loc_16><loc_29><loc_76><loc_36></location>see Figure 14. We therefore assume that the number density C is constant and in particular rises from 0 at (0,a) to a maximum value C m , remains constant up to (0,ac) and then falls again to 0. The length of sight, when the observer is situated at the infinity of the x -axis, is the locus parallel to the x -axis which crosses the position z in a Cartesian x -z plane and terminates at the external ellipse. The locus length is</text>
<formula><location><page_17><loc_27><loc_19><loc_76><loc_24></location>l II = 2 √ a 2 -z 2 b a -2 √ a 2 -2 ac + c 2 -z 2 ( b -c ) a -c (49) when 0 ≤ z < ( a -c ) .</formula>
<formula><location><page_17><loc_27><loc_24><loc_76><loc_29></location>l I = 2 √ a 2 -z 2 b a (48) when ( a -c ) ≤ z < a</formula>
<text><location><page_17><loc_16><loc_17><loc_66><loc_18></location>In the case of optically thin medium, the intensity is split in two cases</text>
<formula><location><page_17><loc_27><loc_13><loc_76><loc_17></location>I I ( z ; a, b ) = I m × 2 √ a 2 -z 2 b a (50)</formula>
<figure>
<location><page_18><loc_23><loc_61><loc_66><loc_84></location>
<caption>Figure 15. The intensity profile along the z-axis when when a = 347 kpc , b = 237 kpc c = a 12 kpc and I m =1.</caption>
</figure>
<formula><location><page_18><loc_27><loc_44><loc_76><loc_52></location>when ( a -c ) ≤ z < a I II ( z ; a, , c ) = I m × ( 2 √ a 2 -z 2 b a -2 √ a 2 -2 ac + c 2 -z 2 ( b -c ) a -c ) (51) when 0 ≤ z < ( a -c ) ,</formula>
<text><location><page_18><loc_16><loc_37><loc_76><loc_43></location>where I m is a constant which allows to compare the theoretical intensity with the observed one. A typical profile in intensity along the z-axis is reported in Figure 15. The ratio, κ , between the theoretical intensity at the maximum, ( z = a -c ), and at the minimum, ( z = 0), is given by</text>
<formula><location><page_18><loc_28><loc_33><loc_76><loc_37></location>I I ( z = a -c ) I II ( z = 0) = κ = √ 2 a -cb √ ca . (52)</formula>
<text><location><page_18><loc_16><loc_29><loc_76><loc_33></location>As an example the values a = 6 kpc , b = 4 kpc , c = a 12 kpc gives κ = 3 . 19. The knowledge of the above ratio from the observations allows to deduce c once a and b are given by the observed morphology</text>
<formula><location><page_18><loc_27><loc_25><loc_76><loc_28></location>c = 2 ab 2 a 2 r 2 + b 2 . (53)</formula>
<text><location><page_18><loc_16><loc_22><loc_76><loc_25></location>The above analytical model explains the hole in luminosity visible in the astrophysical shells such as supernovae and SBs. More details can be found in [40].</text>
<section_header_level_1><location><page_18><loc_16><loc_19><loc_33><loc_20></location>6.2. The numerical shell</section_header_level_1>
<text><location><page_18><loc_16><loc_16><loc_71><loc_18></location>The source of luminosity is assumed here to be the flux of kinetic energy, L m ,</text>
<formula><location><page_18><loc_27><loc_13><loc_76><loc_16></location>L m = 1 2 ρAV 3 , (54)</formula>
<text><location><page_19><loc_16><loc_82><loc_76><loc_88></location>where A is the considered area, V is the velocity and ρ is the density. In our case A = r 2 ∆Ω, where ∆Ω is the considered solid angle and r ( θ ) the temporary radius along the chosen direction . The observed luminosity along a given direction can be expressed as</text>
<formula><location><page_19><loc_27><loc_80><loc_76><loc_81></location>L = glyph[epsilon1]L m , (55)</formula>
<text><location><page_19><loc_16><loc_76><loc_76><loc_79></location>where glyph[epsilon1] is a constant of conversion from the mechanical luminosity to the observed luminosity.</text>
<text><location><page_19><loc_19><loc_75><loc_65><loc_76></location>We review the algorithm that allows to build the image, see [41]:</text>
<unordered_list>
<list_item><location><page_19><loc_17><loc_73><loc_75><loc_74></location>· An empty memory grid M ( i, j, k ) which contains NDIM 3 pixels is considered</list_item>
<list_item><location><page_19><loc_17><loc_62><loc_76><loc_72></location>· We first generate an internal 3D surface of revolution by rotating the ideal image of 360 · around the polar direction and a second external surface of revolution at a fixed distance ∆ R from the first surface. As an example, we fixed ∆ R = R/ 12, where R is the momentary radius of expansion. The points on the memory grid which lie between the internal and external surfaces are memorized on M ( i, j, k ) by a variable integer number according to formula (54) and density ρ proportional to the swept mass.</list_item>
<list_item><location><page_19><loc_17><loc_58><loc_76><loc_61></location>· Each point of M ( i, j, k ) has spatial coordinates x, y, z which can be represented by the following 1 × 3 matrix, A ,</list_item>
</unordered_list>
<formula><location><page_19><loc_31><loc_52><loc_76><loc_58></location>A = x y z . (56)</formula>
<text><location><page_19><loc_19><loc_47><loc_76><loc_51></location>The orientation of the object is characterized by the Euler angles (Φ , Θ , Ψ) and therefore by a total 3 × 3 rotation matrix, E . The matrix point is represented by the following 1 × 3 matrix, B ,</text>
<formula><location><page_19><loc_31><loc_45><loc_76><loc_46></location>B = E · A . (57)</formula>
<unordered_list>
<list_item><location><page_19><loc_17><loc_41><loc_76><loc_44></location>· The intensity map is obtained by summing the points of the rotated images along a particular direction.</list_item>
</unordered_list>
<text><location><page_19><loc_19><loc_39><loc_72><loc_40></location>The image of A2267 built with the above algorithm is shown in Figure 16.</text>
<text><location><page_19><loc_19><loc_37><loc_40><loc_39></location>The threshold intensity, I tr , is</text>
<formula><location><page_19><loc_27><loc_35><loc_76><loc_37></location>I max κ = I max , (58)</formula>
<text><location><page_19><loc_16><loc_27><loc_76><loc_35></location>where I max , is the maximum value of intensity characterizing the ring and κ is a parameter which allows matching theory with observations and was previously defined in equation (39). A typical image with a hole is visible in Figure 17. The opening angle of the visible arc can be parametrized as function of the ratio κ , see Figure 18. An opening of ≈ 31 · is reached at κ ≈ 0 . 95.</text>
<section_header_level_1><location><page_19><loc_16><loc_24><loc_28><loc_25></location>7. Conclusions</section_header_level_1>
<section_header_level_1><location><page_19><loc_16><loc_21><loc_35><loc_22></location>The equation of motion</section_header_level_1>
<text><location><page_19><loc_16><loc_15><loc_76><loc_21></location>The giants arcs are connected with the visible part of the SBs which advance in the intracluster medium surrounding the host galaxies. The chosen profile of density is hyperbolic, see equation (10), and the momentum conservation along a given direction allows to derive the equation of motion as function of the polar angle, see equation (21).</text>
<text><location><page_19><loc_16><loc_14><loc_24><loc_15></location>The image</text>
<figure>
<location><page_20><loc_18><loc_56><loc_59><loc_83></location>
<caption>Figure 16. Contour map of I for A2267, the x and y axes are in arcsec . The three Euler angles characterizing the orientation are Φ=0 · , Θ=90 · and Ψ=90 · , and NDIM=400.</caption>
</figure>
<text><location><page_20><loc_16><loc_32><loc_76><loc_44></location>According to the theory here presented the giants arcs are the visible part of an advancing SB. An analytical explanation for the limited angular extent of the giant arcs is represented by the theoretical luminosity as function of the polar angle, see equation (39). An increase in the polar angle produces a decrease of the theoretical luminosity and the arc becomes invisible. Selecting a given numbers of SBs with variable lifetime and randomly inserting them in a cubic box of side ≈ 600 kpc is possible to simulate the giants arcs visible in the clusters of galaxies, see Figure 13 and relative statistical parameters in Table 5.</text>
<section_header_level_1><location><page_20><loc_16><loc_29><loc_30><loc_30></location>Acknowledgments</section_header_level_1>
<text><location><page_20><loc_16><loc_24><loc_76><loc_27></location>This research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France.</text>
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<caption>Figure 18. The opening angle of the visible simulated arc as function of κ ; parameters as in Figure 16.</caption>
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</document> | [] |
2023arXiv230713006M | https://arxiv.org/pdf/2307.13006.pdf | <document>
<section_header_level_1><location><page_1><loc_25><loc_92><loc_76><loc_93></location>The shadows of quantum gravity on Bell's inequality</section_header_level_1>
<text><location><page_1><loc_25><loc_89><loc_76><loc_90></location>Hooman Moradpour, 1 Shahram Jalalzadeh, 2, 3 and Hamid Tebyanian 4</text>
<text><location><page_1><loc_25><loc_87><loc_76><loc_88></location>1 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM),</text>
<text><location><page_1><loc_29><loc_86><loc_72><loc_87></location>University of Maragheh, P.O. Box 55136-553, Maragheh, Iran</text>
<text><location><page_1><loc_18><loc_82><loc_83><loc_86></location>2 Departamento de Fisica, Universidade Federal de Pernambuco, Recife, PE 50670-901, Brazil 3 Center for Theoretical Physics, Khazar University, 41 Mehseti Street, Baku, AZ1096, Azerbaijan 4 Department of Mathematics, University of York, Heslington, York, YO10 5DD, United Kingdom</text>
<text><location><page_1><loc_18><loc_69><loc_83><loc_81></location>This study delves into the validity of quantum mechanical operators in the context of quantum gravity, recognizing the potential need for their generalization. A primary objective is to investigate the repercussions of these generalizations on the inherent non-locality within quantum mechanics, as exemplified by Bell's inequality. Additionally, the study scrutinizes the consequences of introducing a non-zero minimal length into the established framework of Bell's inequality. The findings contribute significantly to our theoretical comprehension of the intricate interplay between quantum mechanics and gravity. Moreover, this research explores the impact of quantum gravity on Bell's inequality and its practical applications within quantum technologies, notably in the realms of device-independent protocols, quantum key distribution, and quantum randomness generation.</text>
<section_header_level_1><location><page_1><loc_20><loc_65><loc_37><loc_66></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_42><loc_49><loc_63></location>The quantum realm is governed by the Heisenberg uncertainty principle (HUP), which mandates that the Hamiltonian be written as the starting point, leading to the Schrodinger equation and, eventually, the eigenvalues and wave function of the quantum system under consideration. In Heisenberg's formulation of quantum mechanics (QM) in the Hilbert space, we encounter states rather than wave functions (although they are connected). In general, QM fails to produce satisfactory solutions for systems featuring the Newtonian gravitational potential in their Hamiltonian. Therefore, in conventional and widely accepted quantum mechanics, gravity is not accounted for in terms of its operators or corresponding Hilbert space (quantum states) carrying gravitational information.</text>
<text><location><page_1><loc_9><loc_8><loc_49><loc_41></location>The incompatibility of gravity and quantum mechanics is not limited to Newtonian gravity and persists even when general relativity is considered. On the other hand, the existence of gravity, even in a purely Newtonian regime, leads to a non-zero minimum (of the order of 10 -35 m (Planck length) [1]) for the uncertainty in position measurement [1-4]. Consistently, various scenarios of quantum gravity (QG), like String theory, also propose a non-zero minimal for the length measurement [3, 4]. The non-zero minimal length existence may affect the operators, and it leads to the generalization of HUP, called generalized uncertainty principle (GUP) [3, 4] that becomes significant at scales close to the Planck length and may even justify a modified gravity [5, 6]. This concept has profound implications for our understanding of space and time at the most fundamental level. It seems that minimal length is not merely a mathematical artifact of the theory but a physical reality that could have observable consequences. This is a crucial point, as it suggests that the effects of quantum gravity could be detected in experiments, a distinction subtle but essential, as it affects how we understand the physical implications of the GUP [7]. Furthermore, understanding the effects of</text>
<text><location><page_1><loc_52><loc_58><loc_92><loc_66></location>the GUP on various quantum mechanical phenomena is an important issue traced in diverse works like Refs. [8] where the GUP implications on i ) the behavior of various oscillators, ii ) the transformations of space and time, and iii ) the emergence of a cutoff in the energy spectrum of quantum systems have been investigated.</text>
<text><location><page_1><loc_52><loc_22><loc_92><loc_57></location>Operators and system states in QG may differ from those in QM. They are, in fact, functions of ordinary operators that appear in QM [4]. For instance, when considering the first order of the GUP parameter ( β ), we find that the momentum operator ˆ P can be expressed as ˆ p (1+ β ˆ p 2 ), where ˆ P and ˆ p represent momentum operators in QG and QM, respectively. In this representation, β is positive, the position operator remains unchanged [4], and GUP is written as ∆ˆ x ∆ ˆ P ≥ /planckover2pi1 2 [1 + β (∆ ˆ P ) 2 ]. Here, although β seems to be a positive parameter [3, 9, 10] related to a minimal length (of the order of the Planck length ( ≡ 10 -35 m)) as ∆ˆ x = /planckover2pi1 √ β , models including negative values for β have also been proposed [11]. Current experiments and theoretical ideas predict a large range for the upper bound of its value [4, 7, 12-14]. Therefore, it follows that gravity could impact our understanding of classical physics-based operator sets that have been established by QM [15, 16]. Consequently, it is possible to write ˆ O = ˆ o + β ˆ o p for some operators, where ˆ O and ˆ o are operators in QG and QM, respectively, and ˆ o p is the first-order correction obtained using perturbation theory [17]. It should also be noted that as the position operator does not change in the above mentioned representation [4], we have ˆ o p = 0 for this operator.</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_21></location>The discovery of quantum non-locality goes back to the famous thought experiment by Einstein, Podolsky and Rosen (EPR) designed to challenge the quantum mechanics [18]. It clearly shows the role of HUP in emerging quantum non-locality and thus, an advantage for QM versus classical mechanics [19, 20]. In order to establish a border between classical physics and quantum mechanical phenomena, J. S. Bell [21] introduces his inequality including the maximum possible correlation in clas-</text>
<table>
<location><page_2><loc_11><loc_87><loc_46><loc_94></location>
<caption>TABLE I: A comparison between QM and QG (up to the first order of β ). Here, | ψ 〉 and | ψ GUP 〉 denote the quantum states in QM and QG, respectively, and | ψ 〉 p is also calculable using the perturbation theory. It should be noted that for operators not affected by GUP (like the position operator in the above mentioned representation [4]), there is not any perturbation (ˆ o p = 0) meaning that their corresponding states remain unchanged.</caption>
</table>
<text><location><page_2><loc_9><loc_61><loc_49><loc_71></location>sical physics that respects the locality. In the presence of quantum non-locality, this inequality is violated [22], and the first experimental evidences of its violation (and thus, the existence of quantum non-locality) have been reported by Aspect et al. [23-25]. Interestingly enough, the quantum non-locality is also predicted in single particle systems [26, 27].</text>
<text><location><page_2><loc_9><loc_29><loc_49><loc_61></location>Motivated by the correlation between HUP and quantum non-locality (which is easily demonstrated in the square of Bell's inequality) [18-20], as well as the impact of GUP on operators, particularly angular momentum [28, 29], recent studies have revealed that minimal length can alter the square of Bell's operator [30]. Furthermore, GUP can affect the entanglement between energy and time, as evidenced by the results of a Franson experiment (which serves as a testing setup for time-energy entanglement) [14]. Table I clearly displays the generally expected modifications to operators and states resulting from minimal length. The term | ψ 〉 p indicates an increase in a quantum superposition, which is a probabilistic signal for entanglement enhancement [15, 16] and therefore, non-locality beyond quantum mechanics [31]. It is apparent that gravity impacts the information bound [17]. Indeed, studying the effects of gravity on quantum entanglement is a long-standing topic which will also establish ways to test the quantum aspects of gravity. In this regard, many efforts have been made based on the Newtonian gravity and its quantization and their effects on the quantum entanglement [32-37].</text>
<text><location><page_2><loc_9><loc_8><loc_49><loc_29></location>The inquiry into the influence of special and general relativity (SR and GR, respectively) on Bell's inequality (quantum non-locality) has been extensively studied over the years [38-42]. The existing research on the effects of SR on Bell's inequality can be classified into three general categories, depending on the method of applying Lorentz transformations: (i) the operators change while the states remain unchanged, (ii) only the states undergo the Lorentz transformation while the operators remain unaltered (the reverse of the previous one), and (iii) both the operators and states are affected by the Lorentz transformation [43-54]. In order to clarify the first two cases, consider a Lab frame, carrying a Bell state ( | φ 〉 ) and a Bell measurement apparatus ( B ), and a</text>
<text><location><page_2><loc_52><loc_60><loc_92><loc_93></location>moving frame (including a Bell measurement apparatus ( B ' )) so that they are connected to each other through the Lorentz transformation Λ. In this manner, the moving frame faces the Lorentz transformed Bell state | φ Λ 〉 , and whenever the Lab frame looks at the Bell measurement apparatus of the moving frame ( B ' ), its Lorentz transformed is seen ( B ' Λ ). Now, it is apparent that using the same directions for the Bell measurement, we find 〈 φ | B | φ 〉 /negationslash = 〈 φ Λ | B ' | φ Λ 〉 /negationslash = 〈 φ | B ' Λ | φ 〉 meaning that the maximum violation amount of Bell's inequality is reported by both observers at different measurement directions [43-47, 49-54]. In the third case, the moving observer is supposed to witness a Bell measurement done in the Lab frame. The moving frame sees | φ Λ 〉 and B Λ (the Lorentz transformed version of B ) leading to 〈 φ | B | φ 〉 = 〈 φ Λ | B Λ | φ Λ 〉 meaning that both the Lab observer and the moving viewer report the same amount for the Bell measurement, simultaneously [48]. Furthermore, certain implications of GR and non-inertial observers have also been addressed in Refs. [55-58]. Given the ongoing effort to bridge QG with QM [59], exploring the effects of QG on quantum non-locality is deemed inevitable and advantageous.</text>
<text><location><page_2><loc_52><loc_38><loc_92><loc_59></location>Bell's theorem suggests that certain experimental outcomes are constrained if the universe adheres to local realism. However, quantum entanglement, which seemingly allows distant particles to interact instantaneously, can breach these constraints [67]. This led to cryptographic solutions like quantum key distribution (QKD) [72] and quantum random number generation (QRNG) [65, 68]. However, classical noise can enter QKDs and QRNGs during implementation, which hackers can exploit to gain partial information. A device-independent (DI) method was developed to address this, ensuring security when a particular correlation is detected, irrespective of device noise. DI protocols often hinge on non-local game violations, like the CHSH inequality [61]. Section IV delves into the impacts of QG on these applications.</text>
<text><location><page_2><loc_52><loc_9><loc_92><loc_37></location>In this study, our primary goal is to explore the ramifications of QG on Bell's inequality, specifically by investigating the implications of minimal length (up to the first order of β ). To address this objective, we adopt a methodology analogous to the three scenarios previously examined concerning the effects of SR on quantum non-locality. To facilitate this exploration, we categorize the existing cases into three distinct groups, which we elaborate on in the following section. As the GUP effects become important at energy scales close to the Planck scale, the first case includes a quantum state produced at purely quantum mechanical situation (lowenergy) while the observer uses the Bell measurement apparatus prepared by employing the quantum aspects of gravity meaning that high-energy physics considerations have been employed to build the apparatus. Therefore, we face a high-energy affected observer (measurement), who tries to study the quantum non-locality stored in a low-energy state (a purely quantum mechanical state). The reversed situation is checked in the second case, and</text>
<text><location><page_3><loc_9><loc_80><loc_49><loc_93></location>the consequences of applying a quantum gravity-based Bell measurement, built by considering the effects of QG, on a state including the QG consideration are also investigated as the third case. The paper concludes by providing a comprehensive summary of our research findings, shedding light on the intricate interplay between quantum mechanics and gravity, elucidating the impact of QG on Bell's inequality, and exploring potential applications within various quantum-based systems.</text>
<section_header_level_1><location><page_3><loc_14><loc_74><loc_44><loc_76></location>II. BELL'S INEQUALITY AND THE IMPLICATIONS OF QG</section_header_level_1>
<text><location><page_3><loc_9><loc_62><loc_49><loc_71></location>In the framework of QM, assume two particles and four operators ˆ A, ˆ A ' , ˆ B, ˆ B ' with eigenvalues λ J ( J ∈ { ˆ A, ˆ A ' , ˆ B, ˆ B ' } ), while the first (second) two operators act on the first (second) particle. Now, operators ˆ j = ˆ J | λ J | ∈ { ˆ a, ˆ a ' , ˆ b, ˆ b ' } have eigenvalues ± 1, and Bell's inequality is defined as</text>
<formula><location><page_3><loc_16><loc_58><loc_49><loc_60></location>〈 ˆ B 〉 ≡ 〈 ˆ a ( ˆ b + ˆ b ' ) + ˆ a ' ( ˆ b -ˆ b ' ) 〉 ≤ 2 . (1)</formula>
<text><location><page_3><loc_9><loc_33><loc_49><loc_56></location>Taking into account the effects of QG (up to the first order), the operators are corrected as ˆ J GUP = ˆ J + β ˆ J p and ˆ j GUP = ˆ J + β ˆ J p | λ J GUP | where λ J GUP represents the eigenvalue of ˆ J GUP . Since QM should be recovered at the limit β → 0, one may expect λ J GUP /similarequal λ J + βλ J p . Moreover, as the βλ J p term is perturbative, it is reasonable to expect | β λ J p λ J | << 1 leading to | λ J + βλ J p | = | λ J | (1+ β λ J p λ J ). Applying modifications to the states, operators, or both in QG can result in three distinct situations. Similar studies conducted on the effects of SR on Bell's inequality have also revealed three cases [43-49, 54]. Therefore, it is necessary to consider the possibilities arising from these situations to understand the implications of quantum gravitational modifications. In the following paragraphs, we will examine these possibilities in depth.</text>
<section_header_level_1><location><page_3><loc_11><loc_26><loc_47><loc_28></location>1. Purely quantum mechanical entangled states in the presence of operators modified by QG</section_header_level_1>
<text><location><page_3><loc_9><loc_8><loc_49><loc_24></location>Firstly, let us contemplate the scenario in which an entangled state ( | ξ 〉 ) has been prepared away from the QG influences. This implies that the objective has been accomplished using purely quantum mechanical procedures. Furthermore, it is assumed that an observer utilizes Bell measurements that are constructed through the incorporation of operators containing the QG corrections ( ˆ j GUP ). In the framework of QM, the violation amount of inequality (1) depends on the directions of Bell's measurements. Here, we have ˆ j = ˆ j GUP + β ( λ J p λ J ˆ j GUP -ˆ J p | λ J | )</text>
<text><location><page_3><loc_52><loc_92><loc_73><loc_93></location>inserted into Eq. (1) to reach</text>
<formula><location><page_3><loc_51><loc_77><loc_92><loc_90></location>〈 ˆ B GUP 〉 ≡ (2) 〈 ˆ a GUP ( ˆ b GUP + ˆ b ' GUP ) +ˆ a ' GUP ( ˆ b GUP -ˆ b ' GUP )〉 ≤ 2 -〈 β ' a ˆ a GUP ( ˆ b GUP + ˆ b ' GUP ) + β ' a ' ˆ a ' GUP ( ˆ b GUP -ˆ b ' GUP )〉 -〈 ˆ a GUP ( β ' b ˆ b GUP + β ' b ' ˆ b ' GUP ) +ˆ a ' GUP ( β ' b ˆ b GUP -β ' b ' ˆ b ' GUP )〉 + β '' a 〈 ˆ A GUP ( ˆ b GUP + ˆ b ' GUP ) + ˆ A ' GUP ( ˆ b GUP -ˆ b ' GUP )〉 + β '' b 〈 ˆ a GUP ( ˆ B GUP + ˆ B ' GUP ) +ˆ a ' GUP ( ˆ B GUP -ˆ B ' GUP )〉 ,</formula>
<text><location><page_3><loc_76><loc_69><loc_76><loc_71></location>/negationslash</text>
<text><location><page_3><loc_52><loc_63><loc_92><loc_77></location>where β ' j = β λ J p λ J , β '' j = β | λ J | -1 and the last two expressions have been written using β '' a = β '' a ' and β '' b = β '' b ' . In this manner, it is clearly seen that although the state is unchanged, in general, 〈 ˆ B GUP 〉 = 〈 ˆ B 〉 as the operators are affected by quantum features of gravity [14, 29, 30]. In studying the effects of SR on Bell's inequality, whenever the states remain unchanged, and Lorentz transformations only affect Bell's operator, a similar situation is also obtained [43-49, 54].</text>
<section_header_level_1><location><page_3><loc_52><loc_57><loc_92><loc_60></location>2. Purely quantum mechanical measurements and quantum gravitational states</section_header_level_1>
<text><location><page_3><loc_52><loc_39><loc_92><loc_55></location>Now, let us consider the situation in which the Bell apparatus is built using purely quantum mechanical operators j , and the primary entangled state carries the Planck scale information, i.e., the quantum features of gravity. It means that the entangled state is made using the j GUP operators. A similar case in studies related to the effects of SR on Bell's inequality is the case where the Bell measurement does not go under the Lorentz transformation while the system state undergoes the Lorentz transformation [43-49, 54]. In this setup, we have | ξ GUP 〉 = | ξ 〉 + β | ξ 〉 p and thus</text>
<formula><location><page_3><loc_54><loc_33><loc_92><loc_38></location>〈 ξ GUP ∣ ∣ ˆ B ∣ ∣ ξ GUP 〉 ≡ 〈 ˆ B 〉 GUP = 〈 ˆ B 〉 +2 β 〈 ξ ∣ ∣ ˆ B ∣ ∣ ξ 〉 p ⇒ 〈 ˆ B 〉 GUP ≤ 2 ( 1 + β 〈 ξ ∣ ∣ ˆ B ∣ ∣ ξ 〉 p ) . (3)</formula>
<text><location><page_3><loc_52><loc_26><loc_92><loc_32></location>Correspondingly, if one considers a Bell measurement apparatus that yields 〈 ˆ B 〉 = 2 √ 2, then such an apparatus cannot lead 〈 ˆ B 〉 GUP to its maximum possible value whenever Lorentz symmetry is broken [60].</text>
<section_header_level_1><location><page_3><loc_54><loc_21><loc_90><loc_23></location>3. Bell's inequality in a purely quantum gravitational regime</section_header_level_1>
<text><location><page_3><loc_52><loc_9><loc_92><loc_19></location>In deriving Bell's inequality, it is a significant step to ensure that the operators' eigenvalues are only either ± 1, regardless of their origin, whether it be from QM or QG. If both the Bell measurement and the entangled state were prepared using the quantum gravitational operators, then it is evident that 〈 ξ GUP ∣ ∣ ˆ B GUP ∣ ∣ ξ GUP 〉 ≤ 2. This result indicates that, when considering the effects</text>
<text><location><page_4><loc_9><loc_83><loc_49><loc_93></location>of QG on both the state and the operators, Bell's inequality and the classical regime's limit (which is 2 in the inequality) remain unchanged compared to the previous setups. The same outcome is also achieved when it comes to the relationship between SR and Bell's inequality, provided that both the system state and Bell's measurement undergo a Lorentz transformation [48].</text>
<section_header_level_1><location><page_4><loc_23><loc_79><loc_35><loc_80></location>III. RESULTS</section_header_level_1>
<text><location><page_4><loc_9><loc_64><loc_49><loc_77></location>This section studies QG's implications on Bell's inequality, specifically within the contexts delineated earlier. The CHSH inequality, a specific form of Bell's inequality, provides a quantifiable limit on the correlations predicted by local hidden-variable theories [73]. A violation of the CHSH inequality underscores the inability of such approaches to account for the observed correlations in specific experiments with entangled quantum systems, as predicted by quantum mechanics [69].</text>
<text><location><page_4><loc_9><loc_58><loc_49><loc_64></location>Now, we define the scenario where there are two parties where an entangled pair is shared between them. The entangled state of two qubits can be represented by the Bell state:</text>
<formula><location><page_4><loc_20><loc_55><loc_49><loc_58></location>| ψ 〉 = 1 √ 2 ( | 00 〉 + | 11 〉 ) (4)</formula>
<text><location><page_4><loc_9><loc_42><loc_49><loc_54></location>Alice and Bob each measure their respective states. They can choose between two measurement settings: ˆ a, ˆ a ' for Alice and ˆ b, ˆ b ' for Bob. The measurement results can be either +1 or -1. The expected value of the CHSH game using the above quantum strategy and the Bell state is given in Eq. 1. Classically, the maximum value of 〈 ˆ B 〉 is 2. However, this value can reach 2 √ 2 with the quantum strategy, violating the CHSH inequality.</text>
<figure>
<location><page_4><loc_10><loc_23><loc_48><loc_40></location>
<caption>FIG. 1: The 2D plot of the CHSH inequality values as functions of detection angles θ 1 /π and θ 2 /π . Different colors indicate different 〈 ˆ B 〉 values, with a contour distinguishing the classical and quantum regions.</caption>
</figure>
<text><location><page_4><loc_9><loc_9><loc_49><loc_14></location>Fig. 1 illustrates that the CHSH inequality can be surpassed by judiciously selecting the appropriate detection angles, denoted as θ 1 and θ 2 . The color bar quantitatively represents the value of the inequality, highlighting</text>
<text><location><page_4><loc_52><loc_88><loc_92><loc_93></location>two distinct regions where the value exceeds the classical limit of 2. In Fig. 1, the simulation of Bell's inequality is conducted solely based on QM representations without incorporating QG impact.</text>
<text><location><page_4><loc_52><loc_48><loc_92><loc_87></location>Next, we consider the QG impact on Bell's inequality for various cases; better to say, we extend the well-known Bell inequality to account for the effects of QG. Equations 2 and 3 introduce new terms that are parameterized by β , a constant that quantifies the strength of quantum gravitational effects. These equations represent the modified Bell inequalities in the presence of QG. To explore the implications of these modifications, we plot, see Fig. 2, the degree of Bell inequality violation, denoted as 〈 ˆ B 〉 , as a function of θ for various angles β . Each sub-figure in Fig. 2 presents six curves representing simulations conducted on two different quantum computing platforms: IBM and Google. For IBM, the curves are colour-coded as blue, red, and green, corresponding to quantum mechanical predictions, first quantum gravitational corrections, and second quantum gravitational corrections, respectively. The Google platform uses cyan, pink, and grey to represent the same sequence of calculations. These simulations are repeated multiple times to account for noise, defects in quantum computer circuits, and errors in computation and simulation using the independent platforms of IBM and Google. The insets in each figure show that the results from both platforms are in agreement, confirming the reliability of the findings. The maximum violation observed in the presence of quantum gravity remains below 4, adhering to the theoretical limit set by the world box scenario [74].</text>
<text><location><page_4><loc_52><loc_37><loc_92><loc_48></location>The results notably indicate an escalating violation of the Bell inequality with the introduction of QG. As the parameter β increases, the violation surpasses the quantum mechanical limit of √ 8, signifying a more pronounced breach of the inequality. This implies that the presence of quantum gravitational effects could lead to a more pronounced violation of the Bell inequality than what is predicted by standard quantum mechanics.</text>
<section_header_level_1><location><page_4><loc_63><loc_32><loc_81><loc_33></location>IV. APPLICATIONS</section_header_level_1>
<text><location><page_4><loc_52><loc_12><loc_92><loc_30></location>QKD and QRNG represent two extensively researched and commercially implemented areas where the applications of quantum mechanics come to life. While quantum mechanics underpins the security of these systems, experimental imperfections can introduce vulnerabilities. To address this, DI protocols have been developed. These protocols harness the non-local correlations inherent in quantum entanglement. Importantly, they do not rely on an intricate understanding of the devices in use; their security is grounded solely in the observed violation of non-local correlations, such as the Bell inequalities. This approach offers a robust solution to the security challenges posed by device imperfections [62, 64].</text>
<text><location><page_4><loc_52><loc_9><loc_92><loc_11></location>In DI QKD, two distant parties share an entangled quantum state. They perform measurements on their</text>
<figure>
<location><page_5><loc_16><loc_61><loc_87><loc_91></location>
<caption>FIG. 2: Bell inequality values are plotted against the rotation angle θ , illustrating the effects of varying β values: 0.1, 0.2, 0.5, and 0.9. This plot comprehensively compares six curves, each representing simulations performed on two different quantum computing platforms: IBM and Google. For each platform, the curves are coloured distinctly blue, red, and green for IBM quantum computer simulations representing quantum mechanical predictions (QM), first quantum gravitational corrections (QG-1), and second quantum gravitational corrections (QG-2), respectively; similarly, cyan, pink, and grey represent the same sequence of calculations performed using a Google quantum simulator. The remarkable overlay of curves from the two platforms demonstrates consistent agreement, reinforcing the computational models' reliability. An inset within the figure provides a zoomed-in view to examine further the regions where the curves closely approach or reach the theoretical maximum violation. This feature is crucial for better comparing subtle differences between the curves and understanding the implications of each model. Notably, the maximum violation observed does not exceed the limit of 4, consistent with the boundaries set by the Boxworld theorem. This boundary is a crucial benchmark in general probabilistic theories. It indicates that while the quantum mechanical violations are significant, they do not exceed what is theoretically possible under models that assume no faster-than-light (superluminal) communication.</caption>
</figure>
<text><location><page_5><loc_9><loc_23><loc_49><loc_36></location>respective parts of the state, and due to the non-local nature of entanglement, the outcomes of these measurements are correlated in a way that disobeys classical explanation. These correlations serve as the foundation for key generation, with the security of the key guaranteed by the violation of Bell inequalities. Basically, any eavesdropper attempting to intercept or tamper with the quantum states would disrupt these correlations, making their presence detectable.</text>
<text><location><page_5><loc_9><loc_9><loc_49><loc_21></location>The security and randomness of DI QRNG do not depend on trusting the intrinsic workings of the devices. Traditional QRNGs require detailed models and assumptions about the device, but in DI QRNGs, as long as observed outcomes violate Bell inequalities, one can be assured of the randomness. With the rise of quantum computers, many cryptographic methods are at risk. Nevertheless, the unpredictability in DI QRNG is more than just computationally hard for quantum computers; it's</text>
<text><location><page_5><loc_52><loc_33><loc_92><loc_36></location>theoretically impossible to predict due to the inherent randomness of quantum processes [65, 68].</text>
<text><location><page_5><loc_52><loc_22><loc_92><loc_32></location>Incorporating the effects of QG in quantum information science and technology becomes an intellectual exercise and a practical necessity. Given the results in the previous section that QG effects can significantly enhance the violation of Bell inequalities, let us consider its implications for quantum information science and technology and its applications.</text>
<text><location><page_5><loc_52><loc_9><loc_92><loc_21></location>The security of QKD is guaranteed by the quantum mechanical violation of Bell inequalities; increasing the violation value of Bell's inequality makes QKD even more secure against attacks. This disturbance changes the quantum correlations between Alice's and Bob's measurements. In other words, if the eavesdropper is listening in, the observed violations of Bell's inequalities at Alice's and Bob's ends will reduce, moving closer to what would be expected classically. Thus, if you start</text>
<text><location><page_6><loc_9><loc_80><loc_49><loc_93></location>with a higher violation of Bell's inequalities (thanks to QG effects), you are raising the 'quantumness' of your initial state. The higher this initial level, the more sensitive your system becomes to any eavesdropping activities. A significant drop in the observed Bell inequality violation from this higher baseline would more quickly and definitively signal the presence of eavesdropping, thus enabling quicker and more reliable detection of any security breaches.</text>
<text><location><page_6><loc_9><loc_57><loc_49><loc_80></location>DI protocols prevent the need for trust in the hardware by utilizing Bell inequality violations the greater the violation, the higher the level of security. The introduction of QG effects adds an additional layer of robustness to DI protocols, fortifying them through quantum mechanical principles and integrating fundamental theories of nature. Similarly, for QRNGs, a heightened violation signifies a more quantum-coherent system, enhancing the quality of randomness, which comprises not merely an incremental advancement but a paradigmatic leap in the entropy of the generated random numbers. Consequently, this reduces the computational time required to achieve a given level of randomness and unpredictability, analogous to transitioning from conventional vehicular propulsion to advanced warp drives, all while adhering to the fundamental constraints of space-time.</text>
<text><location><page_6><loc_9><loc_46><loc_49><loc_57></location>More importantly, quantum gravity could offer richer quantum correlations in multipartite systems. Imagine a quantum network secured by quantum gravity effects each additional party would enhance not just the computational power but the security, generating what could be termed 'quantum gravity-secured entanglement.' Enabling a brand-new platform for multiparty quantum computations and secret sharing protocols.</text>
<text><location><page_6><loc_9><loc_38><loc_49><loc_45></location>In summary, enhanced violations of Bell inequalities render QKD virtually impregnable, elevate QRNGs to sources of high-entropy randomness, and establish DI protocols as the epitome of trust-free security mechanisms. Dismissing QG as a purely academic endeavor</text>
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<text><location><page_6><loc_52><loc_83><loc_92><loc_93></location>could overlook its potential as a critical element in safeguarding quantum data against even the most advanced computational threats. If quantum mechanics is considered the apex of security and efficiency, the advent of QG compels a reevaluation. It promises to redefine the boundaries of what is secure, efficient, and trustworthy in quantum technologies.</text>
<section_header_level_1><location><page_6><loc_64><loc_79><loc_79><loc_80></location>V. CONCLUSION</section_header_level_1>
<text><location><page_6><loc_52><loc_54><loc_92><loc_76></location>The study can be summarized by its two main components: i ) the origin of entangled states and ii ) Bell's measurement. Furthermore, the study has introduced the possibility of three outcomes depending on which cornerstone carries the quantum gravitational modifications. The first two scenarios suggest that if only one of the foundations stores the effects of QG, then a precise Bell measurement (depending on the value of β ) could detect the effects of QG. This is due to the differences between 〈 ˆ B 〉 , 〈 ˆ B GUP 〉 , and 〈 ˆ B 〉 GUP . In the third case, Bell's inequality remains invariant if we consider the quantum aspects of gravity on both the states and the operators. Moreover, the results demonstrate that the presence of QG enhances Bell's inequality violation, thereby offering avenues for improving the security and performance of DI QRNG and QKD protocols.</text>
<section_header_level_1><location><page_6><loc_65><loc_49><loc_79><loc_50></location>Acknowledgement</section_header_level_1>
<text><location><page_6><loc_52><loc_38><loc_92><loc_47></location>S.J. acknowledges financial support from the National Council for Scientific and Technological DevelopmentCNPq, Grant no. 308131/2022-3. H. T. acknowledge the Quantum Communications Hub of the UK Engineering and Physical Sciences Research Council (EPSRC) (Grant Nos. EP/M013472/1 and EP/T001011/1).</text>
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</document> | [{"title": "The shadows of quantum gravity on Bell's inequality", "content": "Hooman Moradpour, 1 Shahram Jalalzadeh, 2, 3 and Hamid Tebyanian 4 1 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), University of Maragheh, P.O. Box 55136-553, Maragheh, Iran 2 Departamento de Fisica, Universidade Federal de Pernambuco, Recife, PE 50670-901, Brazil 3 Center for Theoretical Physics, Khazar University, 41 Mehseti Street, Baku, AZ1096, Azerbaijan 4 Department of Mathematics, University of York, Heslington, York, YO10 5DD, United Kingdom This study delves into the validity of quantum mechanical operators in the context of quantum gravity, recognizing the potential need for their generalization. A primary objective is to investigate the repercussions of these generalizations on the inherent non-locality within quantum mechanics, as exemplified by Bell's inequality. Additionally, the study scrutinizes the consequences of introducing a non-zero minimal length into the established framework of Bell's inequality. The findings contribute significantly to our theoretical comprehension of the intricate interplay between quantum mechanics and gravity. Moreover, this research explores the impact of quantum gravity on Bell's inequality and its practical applications within quantum technologies, notably in the realms of device-independent protocols, quantum key distribution, and quantum randomness generation.", "pages": [1]}, {"title": "I. INTRODUCTION", "content": "The quantum realm is governed by the Heisenberg uncertainty principle (HUP), which mandates that the Hamiltonian be written as the starting point, leading to the Schrodinger equation and, eventually, the eigenvalues and wave function of the quantum system under consideration. In Heisenberg's formulation of quantum mechanics (QM) in the Hilbert space, we encounter states rather than wave functions (although they are connected). In general, QM fails to produce satisfactory solutions for systems featuring the Newtonian gravitational potential in their Hamiltonian. Therefore, in conventional and widely accepted quantum mechanics, gravity is not accounted for in terms of its operators or corresponding Hilbert space (quantum states) carrying gravitational information. The incompatibility of gravity and quantum mechanics is not limited to Newtonian gravity and persists even when general relativity is considered. On the other hand, the existence of gravity, even in a purely Newtonian regime, leads to a non-zero minimum (of the order of 10 -35 m (Planck length) [1]) for the uncertainty in position measurement [1-4]. Consistently, various scenarios of quantum gravity (QG), like String theory, also propose a non-zero minimal for the length measurement [3, 4]. The non-zero minimal length existence may affect the operators, and it leads to the generalization of HUP, called generalized uncertainty principle (GUP) [3, 4] that becomes significant at scales close to the Planck length and may even justify a modified gravity [5, 6]. This concept has profound implications for our understanding of space and time at the most fundamental level. It seems that minimal length is not merely a mathematical artifact of the theory but a physical reality that could have observable consequences. This is a crucial point, as it suggests that the effects of quantum gravity could be detected in experiments, a distinction subtle but essential, as it affects how we understand the physical implications of the GUP [7]. Furthermore, understanding the effects of the GUP on various quantum mechanical phenomena is an important issue traced in diverse works like Refs. [8] where the GUP implications on i ) the behavior of various oscillators, ii ) the transformations of space and time, and iii ) the emergence of a cutoff in the energy spectrum of quantum systems have been investigated. Operators and system states in QG may differ from those in QM. They are, in fact, functions of ordinary operators that appear in QM [4]. For instance, when considering the first order of the GUP parameter ( \u03b2 ), we find that the momentum operator \u02c6 P can be expressed as \u02c6 p (1+ \u03b2 \u02c6 p 2 ), where \u02c6 P and \u02c6 p represent momentum operators in QG and QM, respectively. In this representation, \u03b2 is positive, the position operator remains unchanged [4], and GUP is written as \u2206\u02c6 x \u2206 \u02c6 P \u2265 /planckover2pi1 2 [1 + \u03b2 (\u2206 \u02c6 P ) 2 ]. Here, although \u03b2 seems to be a positive parameter [3, 9, 10] related to a minimal length (of the order of the Planck length ( \u2261 10 -35 m)) as \u2206\u02c6 x = /planckover2pi1 \u221a \u03b2 , models including negative values for \u03b2 have also been proposed [11]. Current experiments and theoretical ideas predict a large range for the upper bound of its value [4, 7, 12-14]. Therefore, it follows that gravity could impact our understanding of classical physics-based operator sets that have been established by QM [15, 16]. Consequently, it is possible to write \u02c6 O = \u02c6 o + \u03b2 \u02c6 o p for some operators, where \u02c6 O and \u02c6 o are operators in QG and QM, respectively, and \u02c6 o p is the first-order correction obtained using perturbation theory [17]. It should also be noted that as the position operator does not change in the above mentioned representation [4], we have \u02c6 o p = 0 for this operator. The discovery of quantum non-locality goes back to the famous thought experiment by Einstein, Podolsky and Rosen (EPR) designed to challenge the quantum mechanics [18]. It clearly shows the role of HUP in emerging quantum non-locality and thus, an advantage for QM versus classical mechanics [19, 20]. In order to establish a border between classical physics and quantum mechanical phenomena, J. S. Bell [21] introduces his inequality including the maximum possible correlation in clas- sical physics that respects the locality. In the presence of quantum non-locality, this inequality is violated [22], and the first experimental evidences of its violation (and thus, the existence of quantum non-locality) have been reported by Aspect et al. [23-25]. Interestingly enough, the quantum non-locality is also predicted in single particle systems [26, 27]. Motivated by the correlation between HUP and quantum non-locality (which is easily demonstrated in the square of Bell's inequality) [18-20], as well as the impact of GUP on operators, particularly angular momentum [28, 29], recent studies have revealed that minimal length can alter the square of Bell's operator [30]. Furthermore, GUP can affect the entanglement between energy and time, as evidenced by the results of a Franson experiment (which serves as a testing setup for time-energy entanglement) [14]. Table I clearly displays the generally expected modifications to operators and states resulting from minimal length. The term | \u03c8 \u3009 p indicates an increase in a quantum superposition, which is a probabilistic signal for entanglement enhancement [15, 16] and therefore, non-locality beyond quantum mechanics [31]. It is apparent that gravity impacts the information bound [17]. Indeed, studying the effects of gravity on quantum entanglement is a long-standing topic which will also establish ways to test the quantum aspects of gravity. In this regard, many efforts have been made based on the Newtonian gravity and its quantization and their effects on the quantum entanglement [32-37]. The inquiry into the influence of special and general relativity (SR and GR, respectively) on Bell's inequality (quantum non-locality) has been extensively studied over the years [38-42]. The existing research on the effects of SR on Bell's inequality can be classified into three general categories, depending on the method of applying Lorentz transformations: (i) the operators change while the states remain unchanged, (ii) only the states undergo the Lorentz transformation while the operators remain unaltered (the reverse of the previous one), and (iii) both the operators and states are affected by the Lorentz transformation [43-54]. In order to clarify the first two cases, consider a Lab frame, carrying a Bell state ( | \u03c6 \u3009 ) and a Bell measurement apparatus ( B ), and a moving frame (including a Bell measurement apparatus ( B ' )) so that they are connected to each other through the Lorentz transformation \u039b. In this manner, the moving frame faces the Lorentz transformed Bell state | \u03c6 \u039b \u3009 , and whenever the Lab frame looks at the Bell measurement apparatus of the moving frame ( B ' ), its Lorentz transformed is seen ( B ' \u039b ). Now, it is apparent that using the same directions for the Bell measurement, we find \u3008 \u03c6 | B | \u03c6 \u3009 /negationslash = \u3008 \u03c6 \u039b | B ' | \u03c6 \u039b \u3009 /negationslash = \u3008 \u03c6 | B ' \u039b | \u03c6 \u3009 meaning that the maximum violation amount of Bell's inequality is reported by both observers at different measurement directions [43-47, 49-54]. In the third case, the moving observer is supposed to witness a Bell measurement done in the Lab frame. The moving frame sees | \u03c6 \u039b \u3009 and B \u039b (the Lorentz transformed version of B ) leading to \u3008 \u03c6 | B | \u03c6 \u3009 = \u3008 \u03c6 \u039b | B \u039b | \u03c6 \u039b \u3009 meaning that both the Lab observer and the moving viewer report the same amount for the Bell measurement, simultaneously [48]. Furthermore, certain implications of GR and non-inertial observers have also been addressed in Refs. [55-58]. Given the ongoing effort to bridge QG with QM [59], exploring the effects of QG on quantum non-locality is deemed inevitable and advantageous. Bell's theorem suggests that certain experimental outcomes are constrained if the universe adheres to local realism. However, quantum entanglement, which seemingly allows distant particles to interact instantaneously, can breach these constraints [67]. This led to cryptographic solutions like quantum key distribution (QKD) [72] and quantum random number generation (QRNG) [65, 68]. However, classical noise can enter QKDs and QRNGs during implementation, which hackers can exploit to gain partial information. A device-independent (DI) method was developed to address this, ensuring security when a particular correlation is detected, irrespective of device noise. DI protocols often hinge on non-local game violations, like the CHSH inequality [61]. Section IV delves into the impacts of QG on these applications. In this study, our primary goal is to explore the ramifications of QG on Bell's inequality, specifically by investigating the implications of minimal length (up to the first order of \u03b2 ). To address this objective, we adopt a methodology analogous to the three scenarios previously examined concerning the effects of SR on quantum non-locality. To facilitate this exploration, we categorize the existing cases into three distinct groups, which we elaborate on in the following section. As the GUP effects become important at energy scales close to the Planck scale, the first case includes a quantum state produced at purely quantum mechanical situation (lowenergy) while the observer uses the Bell measurement apparatus prepared by employing the quantum aspects of gravity meaning that high-energy physics considerations have been employed to build the apparatus. Therefore, we face a high-energy affected observer (measurement), who tries to study the quantum non-locality stored in a low-energy state (a purely quantum mechanical state). The reversed situation is checked in the second case, and the consequences of applying a quantum gravity-based Bell measurement, built by considering the effects of QG, on a state including the QG consideration are also investigated as the third case. The paper concludes by providing a comprehensive summary of our research findings, shedding light on the intricate interplay between quantum mechanics and gravity, elucidating the impact of QG on Bell's inequality, and exploring potential applications within various quantum-based systems.", "pages": [1, 2, 3]}, {"title": "II. BELL'S INEQUALITY AND THE IMPLICATIONS OF QG", "content": "In the framework of QM, assume two particles and four operators \u02c6 A, \u02c6 A ' , \u02c6 B, \u02c6 B ' with eigenvalues \u03bb J ( J \u2208 { \u02c6 A, \u02c6 A ' , \u02c6 B, \u02c6 B ' } ), while the first (second) two operators act on the first (second) particle. Now, operators \u02c6 j = \u02c6 J | \u03bb J | \u2208 { \u02c6 a, \u02c6 a ' , \u02c6 b, \u02c6 b ' } have eigenvalues \u00b1 1, and Bell's inequality is defined as Taking into account the effects of QG (up to the first order), the operators are corrected as \u02c6 J GUP = \u02c6 J + \u03b2 \u02c6 J p and \u02c6 j GUP = \u02c6 J + \u03b2 \u02c6 J p | \u03bb J GUP | where \u03bb J GUP represents the eigenvalue of \u02c6 J GUP . Since QM should be recovered at the limit \u03b2 \u2192 0, one may expect \u03bb J GUP /similarequal \u03bb J + \u03b2\u03bb J p . Moreover, as the \u03b2\u03bb J p term is perturbative, it is reasonable to expect | \u03b2 \u03bb J p \u03bb J | << 1 leading to | \u03bb J + \u03b2\u03bb J p | = | \u03bb J | (1+ \u03b2 \u03bb J p \u03bb J ). Applying modifications to the states, operators, or both in QG can result in three distinct situations. Similar studies conducted on the effects of SR on Bell's inequality have also revealed three cases [43-49, 54]. Therefore, it is necessary to consider the possibilities arising from these situations to understand the implications of quantum gravitational modifications. In the following paragraphs, we will examine these possibilities in depth.", "pages": [3]}, {"title": "1. Purely quantum mechanical entangled states in the presence of operators modified by QG", "content": "Firstly, let us contemplate the scenario in which an entangled state ( | \u03be \u3009 ) has been prepared away from the QG influences. This implies that the objective has been accomplished using purely quantum mechanical procedures. Furthermore, it is assumed that an observer utilizes Bell measurements that are constructed through the incorporation of operators containing the QG corrections ( \u02c6 j GUP ). In the framework of QM, the violation amount of inequality (1) depends on the directions of Bell's measurements. Here, we have \u02c6 j = \u02c6 j GUP + \u03b2 ( \u03bb J p \u03bb J \u02c6 j GUP -\u02c6 J p | \u03bb J | ) inserted into Eq. (1) to reach /negationslash where \u03b2 ' j = \u03b2 \u03bb J p \u03bb J , \u03b2 '' j = \u03b2 | \u03bb J | -1 and the last two expressions have been written using \u03b2 '' a = \u03b2 '' a ' and \u03b2 '' b = \u03b2 '' b ' . In this manner, it is clearly seen that although the state is unchanged, in general, \u2329 \u02c6 B GUP \u232a = \u2329 \u02c6 B \u232a as the operators are affected by quantum features of gravity [14, 29, 30]. In studying the effects of SR on Bell's inequality, whenever the states remain unchanged, and Lorentz transformations only affect Bell's operator, a similar situation is also obtained [43-49, 54].", "pages": [3]}, {"title": "2. Purely quantum mechanical measurements and quantum gravitational states", "content": "Now, let us consider the situation in which the Bell apparatus is built using purely quantum mechanical operators j , and the primary entangled state carries the Planck scale information, i.e., the quantum features of gravity. It means that the entangled state is made using the j GUP operators. A similar case in studies related to the effects of SR on Bell's inequality is the case where the Bell measurement does not go under the Lorentz transformation while the system state undergoes the Lorentz transformation [43-49, 54]. In this setup, we have | \u03be GUP \u3009 = | \u03be \u3009 + \u03b2 | \u03be \u3009 p and thus Correspondingly, if one considers a Bell measurement apparatus that yields \u2329 \u02c6 B \u232a = 2 \u221a 2, then such an apparatus cannot lead \u2329 \u02c6 B \u232a GUP to its maximum possible value whenever Lorentz symmetry is broken [60].", "pages": [3]}, {"title": "3. Bell's inequality in a purely quantum gravitational regime", "content": "In deriving Bell's inequality, it is a significant step to ensure that the operators' eigenvalues are only either \u00b1 1, regardless of their origin, whether it be from QM or QG. If both the Bell measurement and the entangled state were prepared using the quantum gravitational operators, then it is evident that \u2329 \u03be GUP \u2223 \u2223 \u02c6 B GUP \u2223 \u2223 \u03be GUP \u232a \u2264 2. This result indicates that, when considering the effects of QG on both the state and the operators, Bell's inequality and the classical regime's limit (which is 2 in the inequality) remain unchanged compared to the previous setups. The same outcome is also achieved when it comes to the relationship between SR and Bell's inequality, provided that both the system state and Bell's measurement undergo a Lorentz transformation [48].", "pages": [3, 4]}, {"title": "III. RESULTS", "content": "This section studies QG's implications on Bell's inequality, specifically within the contexts delineated earlier. The CHSH inequality, a specific form of Bell's inequality, provides a quantifiable limit on the correlations predicted by local hidden-variable theories [73]. A violation of the CHSH inequality underscores the inability of such approaches to account for the observed correlations in specific experiments with entangled quantum systems, as predicted by quantum mechanics [69]. Now, we define the scenario where there are two parties where an entangled pair is shared between them. The entangled state of two qubits can be represented by the Bell state: Alice and Bob each measure their respective states. They can choose between two measurement settings: \u02c6 a, \u02c6 a ' for Alice and \u02c6 b, \u02c6 b ' for Bob. The measurement results can be either +1 or -1. The expected value of the CHSH game using the above quantum strategy and the Bell state is given in Eq. 1. Classically, the maximum value of \u2329 \u02c6 B \u232a is 2. However, this value can reach 2 \u221a 2 with the quantum strategy, violating the CHSH inequality. Fig. 1 illustrates that the CHSH inequality can be surpassed by judiciously selecting the appropriate detection angles, denoted as \u03b8 1 and \u03b8 2 . The color bar quantitatively represents the value of the inequality, highlighting two distinct regions where the value exceeds the classical limit of 2. In Fig. 1, the simulation of Bell's inequality is conducted solely based on QM representations without incorporating QG impact. Next, we consider the QG impact on Bell's inequality for various cases; better to say, we extend the well-known Bell inequality to account for the effects of QG. Equations 2 and 3 introduce new terms that are parameterized by \u03b2 , a constant that quantifies the strength of quantum gravitational effects. These equations represent the modified Bell inequalities in the presence of QG. To explore the implications of these modifications, we plot, see Fig. 2, the degree of Bell inequality violation, denoted as \u2329 \u02c6 B \u232a , as a function of \u03b8 for various angles \u03b2 . Each sub-figure in Fig. 2 presents six curves representing simulations conducted on two different quantum computing platforms: IBM and Google. For IBM, the curves are colour-coded as blue, red, and green, corresponding to quantum mechanical predictions, first quantum gravitational corrections, and second quantum gravitational corrections, respectively. The Google platform uses cyan, pink, and grey to represent the same sequence of calculations. These simulations are repeated multiple times to account for noise, defects in quantum computer circuits, and errors in computation and simulation using the independent platforms of IBM and Google. The insets in each figure show that the results from both platforms are in agreement, confirming the reliability of the findings. The maximum violation observed in the presence of quantum gravity remains below 4, adhering to the theoretical limit set by the world box scenario [74]. The results notably indicate an escalating violation of the Bell inequality with the introduction of QG. As the parameter \u03b2 increases, the violation surpasses the quantum mechanical limit of \u221a 8, signifying a more pronounced breach of the inequality. This implies that the presence of quantum gravitational effects could lead to a more pronounced violation of the Bell inequality than what is predicted by standard quantum mechanics.", "pages": [4]}, {"title": "IV. APPLICATIONS", "content": "QKD and QRNG represent two extensively researched and commercially implemented areas where the applications of quantum mechanics come to life. While quantum mechanics underpins the security of these systems, experimental imperfections can introduce vulnerabilities. To address this, DI protocols have been developed. These protocols harness the non-local correlations inherent in quantum entanglement. Importantly, they do not rely on an intricate understanding of the devices in use; their security is grounded solely in the observed violation of non-local correlations, such as the Bell inequalities. This approach offers a robust solution to the security challenges posed by device imperfections [62, 64]. In DI QKD, two distant parties share an entangled quantum state. They perform measurements on their respective parts of the state, and due to the non-local nature of entanglement, the outcomes of these measurements are correlated in a way that disobeys classical explanation. These correlations serve as the foundation for key generation, with the security of the key guaranteed by the violation of Bell inequalities. Basically, any eavesdropper attempting to intercept or tamper with the quantum states would disrupt these correlations, making their presence detectable. The security and randomness of DI QRNG do not depend on trusting the intrinsic workings of the devices. Traditional QRNGs require detailed models and assumptions about the device, but in DI QRNGs, as long as observed outcomes violate Bell inequalities, one can be assured of the randomness. With the rise of quantum computers, many cryptographic methods are at risk. Nevertheless, the unpredictability in DI QRNG is more than just computationally hard for quantum computers; it's theoretically impossible to predict due to the inherent randomness of quantum processes [65, 68]. Incorporating the effects of QG in quantum information science and technology becomes an intellectual exercise and a practical necessity. Given the results in the previous section that QG effects can significantly enhance the violation of Bell inequalities, let us consider its implications for quantum information science and technology and its applications. The security of QKD is guaranteed by the quantum mechanical violation of Bell inequalities; increasing the violation value of Bell's inequality makes QKD even more secure against attacks. This disturbance changes the quantum correlations between Alice's and Bob's measurements. In other words, if the eavesdropper is listening in, the observed violations of Bell's inequalities at Alice's and Bob's ends will reduce, moving closer to what would be expected classically. Thus, if you start with a higher violation of Bell's inequalities (thanks to QG effects), you are raising the 'quantumness' of your initial state. The higher this initial level, the more sensitive your system becomes to any eavesdropping activities. A significant drop in the observed Bell inequality violation from this higher baseline would more quickly and definitively signal the presence of eavesdropping, thus enabling quicker and more reliable detection of any security breaches. DI protocols prevent the need for trust in the hardware by utilizing Bell inequality violations the greater the violation, the higher the level of security. The introduction of QG effects adds an additional layer of robustness to DI protocols, fortifying them through quantum mechanical principles and integrating fundamental theories of nature. Similarly, for QRNGs, a heightened violation signifies a more quantum-coherent system, enhancing the quality of randomness, which comprises not merely an incremental advancement but a paradigmatic leap in the entropy of the generated random numbers. Consequently, this reduces the computational time required to achieve a given level of randomness and unpredictability, analogous to transitioning from conventional vehicular propulsion to advanced warp drives, all while adhering to the fundamental constraints of space-time. More importantly, quantum gravity could offer richer quantum correlations in multipartite systems. Imagine a quantum network secured by quantum gravity effects each additional party would enhance not just the computational power but the security, generating what could be termed 'quantum gravity-secured entanglement.' Enabling a brand-new platform for multiparty quantum computations and secret sharing protocols. In summary, enhanced violations of Bell inequalities render QKD virtually impregnable, elevate QRNGs to sources of high-entropy randomness, and establish DI protocols as the epitome of trust-free security mechanisms. Dismissing QG as a purely academic endeavor could overlook its potential as a critical element in safeguarding quantum data against even the most advanced computational threats. If quantum mechanics is considered the apex of security and efficiency, the advent of QG compels a reevaluation. It promises to redefine the boundaries of what is secure, efficient, and trustworthy in quantum technologies.", "pages": [4, 5, 6]}, {"title": "V. CONCLUSION", "content": "The study can be summarized by its two main components: i ) the origin of entangled states and ii ) Bell's measurement. Furthermore, the study has introduced the possibility of three outcomes depending on which cornerstone carries the quantum gravitational modifications. The first two scenarios suggest that if only one of the foundations stores the effects of QG, then a precise Bell measurement (depending on the value of \u03b2 ) could detect the effects of QG. This is due to the differences between \u2329 \u02c6 B \u232a , \u2329 \u02c6 B GUP \u232a , and \u2329 \u02c6 B \u232a GUP . In the third case, Bell's inequality remains invariant if we consider the quantum aspects of gravity on both the states and the operators. Moreover, the results demonstrate that the presence of QG enhances Bell's inequality violation, thereby offering avenues for improving the security and performance of DI QRNG and QKD protocols.", "pages": [6]}, {"title": "Acknowledgement", "content": "S.J. acknowledges financial support from the National Council for Scientific and Technological DevelopmentCNPq, Grant no. 308131/2022-3. H. T. acknowledge the Quantum Communications Hub of the UK Engineering and Physical Sciences Research Council (EPSRC) (Grant Nos. EP/M013472/1 and EP/T001011/1). [gr-qc]]. [url:http://opg.optica.org/ol/abstract.cfm?URI=ol-46-12-2848]", "pages": [6, 8]}] |
2018arXiv181207773E | https://arxiv.org/pdf/1812.07773.pdf | <document>
<section_header_level_1><location><page_1><loc_15><loc_81><loc_85><loc_84></location>Dynamics in first-order mean motion resonances: analytical study of a simple model with stochastic behaviour</section_header_level_1>
<text><location><page_1><loc_15><loc_78><loc_35><loc_79></location>S. Efimov · V. Sidorenko</text>
<text><location><page_1><loc_15><loc_54><loc_86><loc_67></location>Abstract We examine a 2DOF Hamiltonian system, which arises in study of first-order mean motion resonance in spatial circular restricted three-body problem 'star-planet-asteroid', and point out some mechanisms of chaos generation. Phase variables of the considered system are subdivided into fast and slow ones: one of the fast variables can be interpreted as resonant angle, while the slow variables are parameters characterizing the shape and orientation of the asteroid's orbit. Averaging over the fast motion is applied to obtain evolution equations which describe the long-term behavior of the slow variables. These equations allowed us to provide a comprehensive classification of the slow variables' evolution paths. The bifurcation diagram showing changes in the topological structure of the phase portraits is constructed and bifurcation values of Hamiltonian are calculated. Finally, we study properties of the chaos emerging in the system.</text>
<text><location><page_1><loc_15><loc_52><loc_81><loc_53></location>Keywords Hamiltonian system · averaging method · mean motion resonance · chaotic dynamics</text>
<section_header_level_1><location><page_1><loc_15><loc_48><loc_26><loc_49></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_15><loc_32><loc_86><loc_46></location>The model system which will be considered below arises in studies of first-order mean motion resonances (MMR) in restricted three-body problem (R3BP) 'star-planet-asteroid'. If asteroid makes p ∈ N revolutions around the star in the same amount of time in which the planet makes p +1 revolutions, there is an exterior resonance of the first-order denoted as p : ( p +1). The term exterior refers to the fact that in this case the asteroid's semi-major axis is larger than semi-major axis of the planet. The interior MMR ( p + 1) : p takes place when asteroid makes p + 1 revolutions during the time in which planet makes p . The first-order MMRs are quite common and, therefore, intensively studied by many specialists. The related bibliography is given in Gallardo (2018) and Nesvorny et al. (2002). In particular, much effort has been spent to reveal why 2 : 1 resonance with Jupiter corresponds to one of the largest gaps in the main asteroid belt (so-called Hecuba gaps), whereas 3 : 2 MMR resonance is populated by numerous objects of Hilda group, and it is also very likely, that Thule group of objects in 4 : 3 MMR is rather large (Broz</text>
<text><location><page_1><loc_15><loc_29><loc_21><loc_30></location>S. Efimov</text>
<text><location><page_1><loc_15><loc_28><loc_43><loc_29></location>Moscow Institute of Physics and Technology</text>
<text><location><page_1><loc_15><loc_27><loc_64><loc_28></location>9 Institutskiy per., Dolgoprudny, Moscow Region, 141701, Russian Federation</text>
<text><location><page_1><loc_15><loc_26><loc_35><loc_27></location>E-mail: [email protected]</text>
<text><location><page_1><loc_15><loc_21><loc_49><loc_25></location>V. Sidorenko Keldysh Institute of Applied Mathematics Russian Academy of Sciences, Miusskaya Sq., 4, 125047 Moscow, Russian Federation</text>
<text><location><page_1><loc_15><loc_19><loc_17><loc_20></location>and</text>
<text><location><page_1><loc_15><loc_17><loc_43><loc_18></location>Moscow Institute of Physics and Technology</text>
<text><location><page_1><loc_15><loc_16><loc_64><loc_17></location>9 Institutskiy per., Dolgoprudny, Moscow Region, 141701, Russian Federation</text>
<text><location><page_2><loc_15><loc_83><loc_86><loc_88></location>and Vokrouhlicky 2008; Henrard 1996; Lemaitre and Henrard 1990). The discoveries of trans-Neptunian objects made it urgent to study the exterior resonances with Neptune: twotino (MMR 1 : 2) and plutino (MMR 2 : 3) form big subpopulations in the Kuiper belt (Li et al. 2014a,b; Nesvorny and Roig 2000, 2001).</text>
<text><location><page_2><loc_15><loc_68><loc_86><loc_82></location>It is possible to construct a model of dynamics in first-order MMR, taking into account only the leading terms in the Fourier series expansion of disturbing function (Sessin and Ferraz-Mello 1984; Wisdom 1986; Gerasimov and Mushailov 1990). However, studies of planar R3BP (Beaug'e 1994; Jancart et al. 2002) revealed, that some important characteristics of first-order MMRs are reproduced only when the secondorder Fourier terms are accounted for. In this paper we concentrate our attention on that part of a phase space, where eccentricities and inclinations are in relation e glyph[lessmuch] i glyph[lessmuch] 1 (this, in some sense, is a case opposite to the planar problem, for which the relation is e > i = 0). We intend to demonstrate, that in non-planar case second-order terms are no less important, as they make a model essentially stochastic. In contrast, the first-order models are proven to be integrable (Sessin and Ferraz-Mello 1984), and thus cannot reproduce chaotic dynamics found in multiple numerical studies of first-order resonances (Wisdom and Sussman 1988; Giffen 1973; Winter and Murray 1997a,b; Wisdom 1987),</text>
<text><location><page_2><loc_15><loc_53><loc_86><loc_67></location>There are different mechanisms for generating chaos in the dynamics of celestial bodies (Holmes 1990; Lissauer 1999; Morbidelli 2002). Presence of MMR may lead to the so-called adiabatic chaos (Wisdom 1985), which is caused, roughly speaking, by small quasi-random jumps between regular phase trajectories in certain parts of the phase space, where adiabatic approximation is violated. Applying systematically Wisdom's ideas to study of MMRs (Sidorenko 2006; Sidorenko et al. 2014; Sidorenko 2018), we found that adiabatic chaos often coexists with the quasi-probabilistic transitions between specific phase regions. Both phenomena occur in that part of the phase space, where the 'pendulum' or first-order Second Fundamental Model for Resonance (Henrard and Lamaitre 1983) approximations fail, because the first harmonic in the disturbing function Fourier series is not dominant. The goal of this paper is to carry out a comprehensive analysis of the introduced second-order model and investigate described mechanisms of chaotization, which, in our opinion, have not received proper attention in the past.</text>
<text><location><page_2><loc_15><loc_45><loc_86><loc_52></location>The paper is organized as follows. In Section 2 the model Hamiltonian system, which has a structure of slow-fast system, is introduced. In Section 3 the fast subsystem is studied. Equations of motion for slow subsystem are constructed in Section 4 and their solutions are analyzed in Section 5. Section 6 is devoted to different chaotic effects present in the discussed model. In Section 7 numerical evidence for existence of described phenomena is shown. The results are summarized in the last section. In Appendix A we reveal how the proposed model was derived. Details of the averaging procedure are elucidated in Appendix B.</text>
<section_header_level_1><location><page_2><loc_15><loc_41><loc_38><loc_41></location>2 Model Hamiltonian system</section_header_level_1>
<text><location><page_2><loc_15><loc_38><loc_72><loc_39></location>We are dealing with 2DOF Hamiltonian systems with specific symplectic structure:</text>
<formula><location><page_2><loc_41><loc_31><loc_86><loc_37></location>dϕ dτ = ∂Ξ ∂Φ , dΦ dτ = -∂Ξ ∂ϕ , dx dτ = ε ∂Ξ ∂y , dy dτ = -ε ∂Ξ ∂x . (1)</formula>
<text><location><page_2><loc_15><loc_29><loc_43><loc_30></location>The Hamiltonian Ξ in (1) is expressed by</text>
<formula><location><page_2><loc_26><loc_26><loc_86><loc_28></location>Ξ ( x, y, ϕ, Φ ) = Φ 2 2 + W ( x, y, ϕ ) , W ( x, y, ϕ ) = x cos ϕ + y sin ϕ +cos2 ϕ. (2)</formula>
<text><location><page_2><loc_15><loc_22><loc_86><loc_25></location>Appendix A describes in detail how the system (1)-(2) arises in studies of first-order MMR in threedimensional R3BP. Here we only note that</text>
<formula><location><page_2><loc_38><loc_20><loc_64><loc_21></location>ε ∼ µ 1 / 2 , x ∝ e cos ω, y ∝ -e sin ω,</formula>
<text><location><page_2><loc_15><loc_15><loc_86><loc_18></location>where µ glyph[lessmuch] 1 is the fraction of the planet's mass in the total mass of the system, e and ω denote the eccentricity and the argument of pericenter of asteroid's osculating orbit respectively. Further ε is treated as a small parameter of the problem. Since in general variables ϕ, Φ, x, y vary with different rates</text>
<text><location><page_3><loc_15><loc_85><loc_86><loc_88></location>( dϕ/dτ, dΦ/dτ ∼ 1, while dx/dτ, dy/dτ ∼ ε glyph[lessmuch] 1), we can distinguish in (1) fast subsystem (described by the first line of equations) and slow subsystem (the second line).</text>
<text><location><page_3><loc_15><loc_83><loc_86><loc_85></location>In limiting case ε = 0 equations of fast subsystem coincide with equations of motion for particle with unit mass in a field with potential W ( x, y, ϕ ), where x and y are treated as parameters. Let</text>
<formula><location><page_3><loc_42><loc_80><loc_86><loc_81></location>ϕ ( τ, x, y, ξ ) , Φ ( τ, x, y, ξ ) (3)</formula>
<text><location><page_3><loc_15><loc_75><loc_86><loc_79></location>be a solution of fast subsystem with fixed values of x , y , and value ξ of Hamiltonian Ξ , which is the first integral of system (1). In general the resonant angle ϕ in (3) can librate between two constant values or change monotonously through the whole interval [0 , 2 π ), i.e. circulate . In either case</text>
<formula><location><page_3><loc_37><loc_73><loc_64><loc_73></location>ϕ ( τ + T, x, y, ξ ) = ϕ ( τ, x, y, ξ ) mod (2 π ) ,</formula>
<text><location><page_3><loc_15><loc_70><loc_47><loc_71></location>where T ( x, y, ξ ) is a period of the solution (3).</text>
<text><location><page_3><loc_15><loc_66><loc_86><loc_70></location>Because fast variables vary much faster then the slow ones, the right-hand sides in differential equations of slow subsystem in (1) can be replaced by their average values along the solution (3). This yields the evolution equations , which describe secular variations of x and y in closed form:</text>
<formula><location><page_3><loc_39><loc_62><loc_86><loc_65></location>dx dτ = ε 〈 ∂Ξ ∂y 〉 , dy dτ = -ε 〈 ∂Ξ ∂x 〉 . (4)</formula>
<text><location><page_3><loc_15><loc_60><loc_18><loc_61></location>Here</text>
<formula><location><page_3><loc_35><loc_55><loc_86><loc_60></location>〈 Λ 〉 = 1 T ( x, y, ξ ) T ( x,y,ξ ) ∫ 0 Λ ( x, y, ϕ ( τ, x, y, ξ )) dτ. (5)</formula>
<text><location><page_3><loc_18><loc_54><loc_44><loc_54></location>The solution (3) has an action integral:</text>
<formula><location><page_3><loc_38><loc_48><loc_86><loc_52></location>J ( x, y, ξ ) = 1 2 π T ( x,y,ξ ) ∫ 0 Φ 2 ( τ, x, y, ξ ) dτ. (6)</formula>
<text><location><page_3><loc_19><loc_46><loc_19><loc_47></location>glyph[negationslash]</text>
<text><location><page_3><loc_15><loc_43><loc_86><loc_47></location>For ε = 0 function J ( x, y, ξ ) becomes an adiabatic invariant of slow-fast system (1). For a fixed ξ trajectories of averaged equations (4) on a phase plane ( x, y ) go along the lines with constant values of J . This allows to classify evolution equations (4) as adiabatic approximation (Neishtadt 1987a; Wisdom 1985).</text>
<text><location><page_3><loc_15><loc_40><loc_86><loc_42></location>In the next Sections we go through all the steps in construction of evolution equations via described approach and analyze in detail the behaviour of slow variables on different levels of Hamiltonian Ξ = ξ .</text>
<section_header_level_1><location><page_3><loc_15><loc_35><loc_83><loc_37></location>3 Properties of fast subsystem's solutions for different values of slow variables (limiting case ε = 0)</section_header_level_1>
<section_header_level_1><location><page_3><loc_15><loc_32><loc_66><loc_33></location>3.1 Partition of the plane ( x, y ) based on the number of librating solutions</section_header_level_1>
<text><location><page_3><loc_15><loc_26><loc_86><loc_30></location>Because the variables x and y change very slowly (1), they can be treated as constant parameters, when considering the motion of the fast subsystem. Then the potential W is just a 'two-harmonic' function of ϕ defined on circle S 1 , and the motion in such potential can be described in terms of elliptic functions.</text>
<text><location><page_3><loc_15><loc_22><loc_86><loc_26></location>There are different types of motion depending on the Hamiltonian level ξ at which it occurs (Fig. 1). For us it is important, that for some values of x and y two different librating solutions can exist on the same ξ level (Fig. 1c). This situation can take place when W ( ϕ ) has four extrema on S 1 .</text>
<text><location><page_3><loc_18><loc_21><loc_80><loc_22></location>A necessary condition for extremum ∂W / ∂ϕ = 0 after the replacement λ = tan ( ϕ/ 2) yields</text>
<formula><location><page_3><loc_38><loc_19><loc_86><loc_20></location>yλ 4 +2( x +4) λ 3 +2( x -4) λ -y = 0 . (7)</formula>
<text><location><page_3><loc_15><loc_15><loc_86><loc_17></location>Let A denote a region on the plane ( x, y ), in which W ( ϕ ) has four extrema. The equation (7) has four real roots inside this region and only two outside. Thus on the border of the region A the number of unique</text>
<figure>
<location><page_4><loc_15><loc_60><loc_73><loc_89></location>
<caption>Fig. 1 Levels ξ of Hamiltonian Ξ corresponding to different types of fast subsystem's motion: a. circulation, b. libration, c. two coexisting librating solutions, d. the motion is impossible</caption>
</figure>
<figure>
<location><page_4><loc_21><loc_34><loc_80><loc_54></location>
<caption>Fig. 2 Extremal surface of potential W ( x, y, ϕ ) (left), and astroid bounding the region A , where W has four extrema as a function of ϕ on S 1 (right). A 1 , . . . , A 4 - astroid's cusps</caption>
</figure>
<text><location><page_4><loc_15><loc_26><loc_86><loc_28></location>real roots is 3 (with the exception of finite number of points in which there is only one unique real root) and the discriminant of (7) is equal to zero. Therefore the equation for the border of the region A is</text>
<formula><location><page_4><loc_24><loc_24><loc_86><loc_25></location>x 6 +3 x 4 y 2 -48 x 4 +3 x 2 y 4 +336 x 2 y 2 +768 x 2 + y 6 -48 y 4 +768 y 2 -4096 = 0 . (8)</formula>
<text><location><page_4><loc_15><loc_20><loc_86><loc_22></location>By collecting the parts of this equation into perfect cube (8) is transformed to canonical algebraic equation of astroid (Fig. 2):</text>
<formula><location><page_4><loc_39><loc_18><loc_86><loc_20></location>( x 2 + y 2 -4 2 ) 3 +27 · 4 2 x 2 y 2 = 0 . (9)</formula>
<text><location><page_4><loc_15><loc_16><loc_37><loc_17></location>Which can be further reduced to</text>
<formula><location><page_4><loc_44><loc_15><loc_86><loc_16></location>x 2 / 3 + y 2 / 3 = 4 2 / 3 . (10)</formula>
<text><location><page_5><loc_15><loc_87><loc_68><loc_88></location>It is convenient to use this astroid for the reference on the phase plane ( x, y ).</text>
<text><location><page_5><loc_15><loc_83><loc_86><loc_84></location>3.2 Critical curve partitioning the plane ( x, y ) into regions with different types of fast subsystem' motion</text>
<text><location><page_5><loc_15><loc_75><loc_86><loc_81></location>Let us introduce the notations W min ( x, y ) and W max ( x, y ) for global minimum and maximum of function W on S 1 for given values of slow variables. If ( x, y ) ∈ A , then W has the second pair of minimum and maximum, which we shall denote W ∗ min ( x, y ) and W ∗ max ( x, y ) respectively. Using these auxiliary functions, we can partition the ( x, y ) plane for a given ξ into different regions based on the type of fast subsystem's motion:</text>
<formula><location><page_5><loc_31><loc_66><loc_71><loc_75></location>Q 0 = { ( x, y ) | ξ < W min } , Q 1 = { ( x, y ) / ∈ A ∣ ∣ ξ ∈ ( W min , W max )} ⋃ { ( x, y ) ∈ A ∣ ∣ ξ ∈ ( W min , W max ) \ ( W ∗ min , W ∗ max )} , Q 2 = { ( x, y ) ∈ A ∣ ∣ ξ ∈ ( W ∗ min , W ∗ max )} , Q 3 = { ( x, y ) | ξ > W max } .</formula>
<text><location><page_5><loc_15><loc_58><loc_86><loc_65></location>The region Q 0 ( ξ ) will be called a forbidden region , because inside of it Ξ < ξ for any values of fast variables and fast subsystem has no solutions (Fig. 1d). Region Q 1 ( ξ ) is the region with the single librating solution (Fig. 1b), Q 2 ( ξ ) is the region with two librating solutions at given level ξ (Fig. 1c), and finally the region Q 3 ( ξ ) is where fast subsystem's solution circulates (Fig. 1a). Illustrations for regions Q 0 ( ξ ) , . . . , Q 3 ( ξ ) will follow.</text>
<figure>
<location><page_5><loc_15><loc_27><loc_73><loc_56></location>
<caption>Fig. 3 Tangency of Hamiltonian level ξ and different extrema of W ( ϕ ), which occurs on the borders of regions Q 0 ( ξ ) , . . . , Q 3 ( ξ )</caption>
</figure>
<text><location><page_5><loc_15><loc_17><loc_86><loc_21></location>Before that let us consider a border Γ ( ξ ) between these regions. In every point of the border value of W is equal to ξ in one of its critical points (cf. Figures 1 and 3), which is why we shall call Γ ( ξ ) a critical curve . After replacement λ = tan ( ϕ/ 2) the equation W ( ϕ ) = ξ transforms into algebraic equation:</text>
<formula><location><page_5><loc_31><loc_15><loc_86><loc_16></location>(1 -ξ -x ) λ 4 +2 yλ 3 -2( ξ +3) λ 2 +2 yλ +( x +1 -ξ ) = 0 . (11)</formula>
<text><location><page_6><loc_15><loc_85><loc_86><loc_88></location>As Figure 3 demonstrates, the point ( x, y ) lies on the critical curve when equation (11) have at least one multiple real root, which is equivalent to discriminant of (11) being equal to zero:</text>
<formula><location><page_6><loc_17><loc_79><loc_86><loc_84></location>D ( x, y, ξ ) = 64 ξ 4 -128 ξ 2 -x 6 + ξ 2 x 4 -18 ξx 4 -3 x 4 y 2 -15 x 4 + +16 ξ 3 x 2 -80 ξ 2 x 2 -144 ξx 2 -3 x 2 y 4 +2 ξ 2 x 2 y 2 +78 x 2 y 2 -48 x 2 --y 6 + ξ 2 y 4 +18 ξy 4 -15 y 4 -16 ξ 3 y 2 -80 ξ 2 y 2 +144 ξy 2 -48 y 2 +64 = 0 . (12)</formula>
<text><location><page_6><loc_15><loc_71><loc_86><loc_78></location>Thus in the regions Q 0 ( ξ ), Q 2 ( ξ ), Q 3 ( ξ ) the discriminant D ( x, y, ξ ) > 0, while D ( x, y, ξ ) < 0 in Q 1 ( ξ ), and D ( x, y, ξ ) = 0 on the critical curve Γ ( ξ ). Figure 4 depicts Γ ( ξ ) on the plane of slow variables for different values of ξ . At | ξ | < 3 critical curve have cusps, which lie on astroid (10). Additionally this curve may have points of self-intersection. If -3 < ξ < -1, the curve intersects itself on axis x , with the x coordinates of self-intersection points being defined by equation</text>
<formula><location><page_6><loc_45><loc_69><loc_57><loc_70></location>x 2 +8( ξ +1) = 0 .</formula>
<text><location><page_6><loc_15><loc_67><loc_83><loc_68></location>If 1 < ξ < 3, points of self-intersection lie on y axis, and their y coordinates are defined by equation</text>
<formula><location><page_6><loc_45><loc_64><loc_57><loc_66></location>y 2 -8( ξ -1) = 0 .</formula>
<figure>
<location><page_6><loc_22><loc_40><loc_80><loc_60></location>
<caption>Fig. 4 Critical curve Γ ( ξ ): shape of the curve for different ξ values (left), and parametrization of the curve by glyph[slurabove] ϕ with arrows showing the direction in which the parameter increases (right)</caption>
</figure>
<text><location><page_6><loc_18><loc_33><loc_47><loc_34></location>The critical curve allows a parametrization</text>
<formula><location><page_6><loc_26><loc_30><loc_86><loc_32></location>Γ ( ξ ) = { x = cos glyph[slurabove] ϕ ( ξ +cos2 glyph[slurabove] ϕ -2) , y = sin glyph[slurabove] ϕ ( ξ +cos2 glyph[slurabove] ϕ +2) ∣ ∣ ∣ glyph[slurabove] ϕ ∈ S 1 } , (13)</formula>
<text><location><page_6><loc_15><loc_26><loc_86><loc_28></location>which is illustrated by Figure 4. The parameter glyph[slurabove] ϕ in (13) coincide with the critical points of potential W ( ϕ ) at given level ξ , as depicted in Figure 3, in respective points of ( x, y ) plane:</text>
<formula><location><page_6><loc_37><loc_21><loc_62><loc_25></location> W ( glyph[slurabove] ϕ, x ( glyph[slurabove] ϕ, ξ ) , y ( glyph[slurabove] ϕ, ξ )) = ξ, ∂ ∂ϕ W ( ϕ, x ( glyph[slurabove] ϕ, ξ ) , y ( glyph[slurabove] ϕ, ξ )) ∣ ∣ ∣ ϕ = glyph[slurabove] ϕ = 0 .</formula>
<text><location><page_6><loc_15><loc_15><loc_86><loc_20></location>The introduced parametric representation is convenient, in particular, for defining the location of cusps and self-intersection points. For cusps glyph[slurabove] ϕ = ϕ ∗ , where ϕ ∗ is obtained from the equation tan 2 ϕ ∗ = (3 + ξ )/(3 -ξ ), which have four roots on S 1 . We shall denote these cusps as Y 1 ,..., Y 4 with the lower index being the number of a quadrant, in which the respective value ϕ ∗ lies. Self-intersection points of</text>
<text><location><page_7><loc_15><loc_85><loc_86><loc_88></location>Γ ( ξ ) on x axis ( -3 < ξ < -1) we shall denote as B 1 and B 2 for right and left half-planes respectively. Self-intersection points on y axis (1 < ξ < 3) we shall denote as S 1 and S 2 for upper and lower half-planes.</text>
<text><location><page_7><loc_15><loc_81><loc_86><loc_85></location>Note: It can be demonstrated, that curves Γ ( ξ ) are the involutes of astroid (10) constructed with tethers of length 3 ± ξ extended from astroid's cusps. This makes Γ ( ξ ) also a family of equidistant curves with the distance | ξ a -ξ b | between any two curves Γ ( ξ a ) and Γ ( ξ b ).</text>
<text><location><page_7><loc_15><loc_77><loc_58><loc_78></location>3.3 Transformation of regions Q 0 ( ξ ) , ..., Q 3 ( ξ ) with change of ξ</text>
<text><location><page_7><loc_15><loc_72><loc_86><loc_75></location>It should be noted first, that region with a single librating solution Q 1 ( ξ ) is present on plane ( x, y ) for all values of ξ . Other regions appear and disappear, as ξ crosses several bifurcation values ξ i :</text>
<formula><location><page_7><loc_39><loc_70><loc_62><loc_71></location>ξ 1 = -3 , ξ 2 = -1 , ξ 3 = 1 , ξ 4 = 3 .</formula>
<text><location><page_7><loc_15><loc_67><loc_60><loc_68></location>We shall describe, how the regions are transformed, as ξ increases.</text>
<unordered_list>
<list_item><location><page_7><loc_15><loc_64><loc_86><loc_67></location>If ξ < ξ 1 , there exists a forbidden region Q 0 ( ξ ) around the point (0 , 0), with the rest of ( x, y ) plane being the Q 1 ( ξ ) region.</list_item>
<list_item><location><page_7><loc_15><loc_62><loc_86><loc_64></location>At ξ = ξ 1 on the right and on the left from region Q 0 ( ξ ) two parts of region Q 2 ( ξ ) (region with two librating solutions) appear (Fig. 5).</list_item>
</unordered_list>
<text><location><page_7><loc_18><loc_60><loc_70><loc_61></location>At ξ = ξ 2 the region Q 0 ( ξ ) disappears and Q 2 ( ξ ) becomes connected (Fig. 6).</text>
<unordered_list>
<list_item><location><page_7><loc_15><loc_57><loc_86><loc_60></location>At ξ = ξ 3 region Q 2 ( ξ ) is separated into two parts again by appearing region Q 3 ( ξ ) (region with circulating resonant angle) around the point (0 , 0) as seen in Figure 7.</list_item>
</unordered_list>
<text><location><page_7><loc_15><loc_54><loc_86><loc_57></location>At ξ = ξ 4 region Q 2 ( ξ ) disappears (Fig. 8). For ξ > ξ 4 there exists only region Q 3 ( ξ ) surrounded by Q 1 ( ξ ).</text>
<figure>
<location><page_7><loc_22><loc_38><loc_79><loc_52></location>
<caption>Fig. 5 Bifurcation at ξ = ξ 1 : appearance of region Q 2 . Here and further region Q 0 is colored dark gray, region Q 2 - orange. The rest blank space corresponds to region Q 1</caption>
</figure>
<figure>
<location><page_7><loc_22><loc_17><loc_79><loc_30></location>
<caption>Fig. 6 Bifurcation at ξ = ξ 2 : vanishing of forbidden region Q 0</caption>
</figure>
<figure>
<location><page_8><loc_22><loc_75><loc_79><loc_89></location>
<caption>Fig. 7 Bifurcation at ξ = ξ 3 : appearance of region Q 3 (here and further colored green)</caption>
</figure>
<figure>
<location><page_8><loc_22><loc_54><loc_79><loc_68></location>
<caption>Fig. 8 Bifurcation at ξ = ξ 4 : vanishing of region Q 2</caption>
</figure>
<text><location><page_8><loc_15><loc_44><loc_86><loc_49></location>Let us now describe how borders of regions Q 0 ( ξ ) , ..., Q 3 ( ξ ) transforms with increase of ξ . The border between Q 0 ( ξ ) and Q 1 ( ξ ) we shall call the existence curve and denote it as Γ 0 ( ξ ). It corresponds to the part of the critical curve Γ ( ε ) in which W min ( x, y ) = ξ . For ξ < ξ 1 curve Γ ( ε ) ≡ Γ 0 ( ε ). For ξ 1 < ξ < ξ 2 curve Γ 0 ( ε ) consists of two intervals of Γ ( ε ), lying between points of self-intersection B 1 and B 2 .</text>
<text><location><page_8><loc_15><loc_36><loc_86><loc_44></location>We shall adopt the traditional terminology common in studies of slow-fast systems (Wisdom 1985; Neishtadt 1987a) with modifications made to better represent the specifics of the discussed problem. The border between regions Q 1 ( ξ ) and Q 3 ( ξ ) we shall call an uncertainty curve of the first kind and use a notation Γ 1 ( ξ ) for it. Points of the uncertainty curve of the first kind are defined by condition W max ( x, y ) = ξ . If ξ > ξ 4 , then Γ 1 ( ξ ) ≡ Γ ( ξ ). For ξ 3 < ξ < ξ 4 , the curve Γ 1 ( ξ ) consists of Γ ( ξ ) parts, which are contained between points of self-intersection S 1 and S 2 .</text>
<text><location><page_8><loc_15><loc_28><loc_86><loc_36></location>The part of the border between Q 1 ( ξ ) and Q 3 ( ξ ), along which holds the equality W ∗ max ( x, y ) = ξ , we shall call an uncertainty curve of the second kind and denote it Γ 2 ( ξ ). For ξ 1 < ξ < ξ 3 the Γ 2 ( ξ ) = Y 1 Y 3 ∪ Y 2 Y 4 , where Y 1 Y 3 and Y 2 Y 4 are segments of Γ ( ξ ), which lie between corresponding cusps. If ξ 1 < ξ < ξ 2 then curve Γ 2 ( ξ ) = S 1 Y 1 ∪ S 1 Y 2 ∪ S 2 Y 3 ∪ S 2 Y 4 . For the rest part of the border between Q 1 ( ξ ) and Q 3 ( ξ ) holds W ∗ min ( x, y ) = ξ . As no dynamical effects of interest are happening on this segment, we shall not refer to it further.</text>
<text><location><page_8><loc_15><loc_24><loc_86><loc_28></location>Figure 9 depicts a diagram, that shows values of glyph[slurabove] ϕ defining positions of cusps and self-intersection points on Γ ( ξ ), as well as the segments which correspond to existence curve and two uncertainty curves. Due to the symmetry, it is sufficient to consider only glyph[slurabove] ϕ ∈ [0 , π/ 2] (Fig. 4).</text>
<section_header_level_1><location><page_8><loc_15><loc_20><loc_57><loc_21></location>3.4 Three-dimensional representation of the set of curves Γ ( ξ )</section_header_level_1>
<text><location><page_8><loc_15><loc_15><loc_86><loc_18></location>Curves Γ ( ξ ) can be interpreted as cross sections of some surface F in the space xyξ by equi-Hamiltonian planes ξ = const (Fig. 10a,b). In this space for fixed value of glyph[slurabove] ϕ the equations (13) define a straight line, which means that the surface F is ruled (Fig. 10c).</text>
<figure>
<location><page_9><loc_34><loc_69><loc_65><loc_88></location>
<caption>Fig. 9 Diagram showing the partition of critical curve into existence curve Γ 0 ( ξ ) and critical curves Γ 1 , 2 ( ξ )</caption>
</figure>
<figure>
<location><page_9><loc_21><loc_51><loc_78><loc_64></location>
<caption>Fig. 10 Surface F composed of curves Γ ( ξ ) in xyξ space: a. general representation of the surface, b. surface's cross sections by equi-Hamiltonian planes, c. rulings of surface F</caption>
</figure>
<text><location><page_9><loc_15><loc_41><loc_86><loc_45></location>The same surface F defined by the equation analogous to (12) also appears in a completely different problem studied by Batkhin (2012), where it partitions the parametric space of some mechanical system into regions with different stability properties.</text>
<section_header_level_1><location><page_9><loc_15><loc_37><loc_42><loc_38></location>4 Evolution equations construction</section_header_level_1>
<text><location><page_9><loc_15><loc_34><loc_47><loc_35></location>4.1 Averaging along fast subsystem's solutions</text>
<text><location><page_9><loc_15><loc_31><loc_26><loc_32></location>Considering that</text>
<formula><location><page_9><loc_42><loc_29><loc_60><loc_31></location>∂Ξ ∂x = cos ϕ, ∂Ξ ∂y = sin ϕ,</formula>
<text><location><page_9><loc_15><loc_27><loc_74><loc_28></location>construction of the evolution equations (4) require calculating two averaged properties:</text>
<formula><location><page_9><loc_36><loc_19><loc_86><loc_25></location>〈 sin ϕ 〉 = 1 T ( x,y,ξ ) T ( x,y,ξ ) ∫ 0 sin ϕ ( τ, x, y, ξ ) dτ, 〈 cos ϕ 〉 = 1 T ( x,y,ξ ) T ( x,y,ξ ) ∫ 0 cos ϕ ( τ, x, y, ξ ) dτ. (14)</formula>
<text><location><page_9><loc_15><loc_17><loc_24><loc_18></location>The equality</text>
<formula><location><page_9><loc_40><loc_14><loc_61><loc_16></location>dϕ dτ = Φ = ± √ 2 [ ξ -W ( x, y, ϕ )]</formula>
<text><location><page_10><loc_15><loc_85><loc_86><loc_88></location>allows finding a period of fast subsystem's solution at Hamiltonian level Ξ = ξ and calculating (after proper change of variables) values of integrals on the right-hand side of (14), e.g. for librating solutions:</text>
<formula><location><page_10><loc_34><loc_78><loc_86><loc_84></location>T ( x, y, ξ ) = 2 ϕ ∗ ∫ ϕ ∗ dϕ √ 2[ ξ -W ( x,y,ϕ )] , T ( x,y,ξ ) ∫ 0 f ( ϕ ( t, x, y, ξ )) dτ = 2 ϕ ∗ ∫ ϕ ∗ f ( ϕ ) dϕ √ 2[ ξ -W ( x,y,ϕ )] . (15)</formula>
<text><location><page_10><loc_15><loc_74><loc_86><loc_77></location>Here ϕ ∗ and ϕ ∗ denote minimum and maximum values of angle ϕ in librating solution, along which the averaging is being performed.</text>
<text><location><page_10><loc_15><loc_70><loc_86><loc_74></location>For the system (2) period T ( x, y, ξ ) and integrals (14) can be expressed in terms of complete elliptical integrals of the first and the third kind. Further the concise description of this transformation is given, using the case</text>
<formula><location><page_10><loc_44><loc_69><loc_86><loc_70></location>-π ≤ ϕ ∗ < ϕ ∗ ≤ π (16)</formula>
<text><location><page_10><loc_15><loc_67><loc_61><loc_68></location>as an example. After standard substitution λ = tan ( ϕ/ 2) we obtain</text>
<formula><location><page_10><loc_34><loc_56><loc_86><loc_66></location>T ( x, y, ξ ) = 4 λ ∗ ∫ λ ∗ dλ √ 2 R 4 ( λ ) , 〈 sin ϕ 〉 = ϕ ∗ ∫ ϕ ∗ sin ϕdϕ √ 2[ ξ -W ( x,y,ϕ )] =4 λ ∗ ∫ λ ∗ λdλ (1+ λ 2 ) √ 2 R 4 ( λ ) , 〈 cos ϕ 〉 = ϕ ∗ ∫ ϕ ∗ cos ϕdϕ √ 2[ ξ -W ( x,y,ϕ )] =2 λ ∗ ∫ λ ∗ ( 1 -λ 2 ) dλ (1+ λ 2 ) √ 2 R 4 ( λ ) . (17)</formula>
<text><location><page_10><loc_15><loc_54><loc_74><loc_55></location>Here λ ∗ = tan ( ϕ ∗ / 2), λ ∗ = tan ( ϕ ∗ / 2). Function R 4 ( λ ) in (17) is a quartic polynomial</text>
<formula><location><page_10><loc_37><loc_51><loc_64><loc_53></location>R 4 ( λ ) = d 0 λ 4 + d 1 λ 3 + d 2 λ 2 + d 3 λ + d 4</formula>
<text><location><page_10><loc_15><loc_49><loc_26><loc_50></location>with coefficients</text>
<formula><location><page_10><loc_28><loc_47><loc_73><loc_48></location>d 0 = ξ -1 + x, d 1 = d 3 = -2 y, d 2 = 2 ξ +6 , d 4 = ξ -1 -x.</formula>
<text><location><page_10><loc_18><loc_44><loc_63><loc_45></location>Integrals on the right-hand side in (17) can be rewritten as follows:</text>
<formula><location><page_10><loc_36><loc_35><loc_86><loc_43></location>T ( x, y, ξ ) = 4 √ 2 | d 0 | I 0 , 0 , ϕ ∗ ∫ ϕ ∗ sin ϕdϕ √ 2[ ξ -W ( x,y,ϕ )] = 4 √ 2 | d 0 | I 1 , 1 , ϕ ∗ ∫ ϕ ∗ cos ϕdϕ √ 2[ ξ -W ( x,y,ϕ )] = 2 √ 2 | d 0 | ( 2 I 1 , 0 -I 0 , 0 ) , (18)</formula>
<text><location><page_10><loc_15><loc_33><loc_42><loc_34></location>where notation I k,r is used for integrals:</text>
<formula><location><page_10><loc_31><loc_27><loc_86><loc_32></location>I k,r = λ ∗ ∫ λ ∗ λ r dλ ( λ 2 +1) k √ ± ( λ -a 1 )( λ -a 2 )( λ -a 3 )( λ -a 4 ) . (19)</formula>
<text><location><page_10><loc_15><loc_23><loc_86><loc_26></location>a k - roots of polinomial R 4 ( λ ), and the sign in the square root is determined by sign of coefficient d 0 . Integrals (19) can be presented as linear combinations of elliptic integrals of the first and the third kind:</text>
<formula><location><page_10><loc_35><loc_18><loc_86><loc_22></location>I 0 , 0 = c 0 , 0 K ( k ) , I 1 , 0 = c 1 , 1 K ( k ) + c 1 , 3 Π ( h, k ) + ¯ c 1 , 3 Π ( ¯ h, k ) , I 1 , 1 = g 1 , 1 K ( k ) + g 1 , 3 Π ( h, k ) + ¯ g 1 , 3 Π ( ¯ h, k ) . (20)</formula>
<text><location><page_10><loc_15><loc_15><loc_86><loc_17></location>Analytical expressions for coefficients c m,l , g m,l , moduli k , and parameters h in (20) depend on integration interval and properties of R 4 ( λ ) roots. These expressions are gathered in the Appendix B.</text>
<formula><location><page_11><loc_33><loc_83><loc_68><loc_84></location>cΠ ( h, k ) + ¯ cΠ ( ¯ h, k ) ( c, h ∈ C , k ∈ R , 0 < k 2 < 1)</formula>
<text><location><page_11><loc_15><loc_77><loc_86><loc_80></location>can be further reduced to linear combinations of complete elliptic integrals of the first kind and the third kind with real parameter (Byrd and Friedman 1954; Lang and Stevens 1960). This, however, results in more complicated expressions, which is why we use representation (20).</text>
<text><location><page_11><loc_15><loc_72><loc_86><loc_76></location>When averaging along librating solutions of fast subsystem, which violate the condition (16), after substitution λ = tan( ϕ/ 2) the integration in (17) is carried over two semi-infinite intervals. E.g., for ϕ ∗ < π, ϕ ∗ > π, ϕ ∗ -ϕ ∗ < 2 π the expression for the period T is</text>
<formula><location><page_11><loc_34><loc_65><loc_67><loc_70></location>T ( x, y, ξ ) = 4 λ ∗ ∫ -∞ dλ √ 2 R 4 ( λ ) + + ∞ ∫ λ ∗ dλ √ 2 R 4 ( λ ) .</formula>
<text><location><page_11><loc_15><loc_62><loc_72><loc_63></location>In these cases it is implied that all I k,r in (18) are the sum of two integrals as well.</text>
<text><location><page_11><loc_18><loc_60><loc_71><loc_61></location>After all described transformations evolution equations (4) take a simple form</text>
<formula><location><page_11><loc_39><loc_55><loc_86><loc_58></location>dx dτ = ε 2 I 1 , 1 I 0 , 0 , dy dτ = ε [ 1 -2 I 1 , 0 I 0 , 0 ] . (21)</formula>
<text><location><page_11><loc_15><loc_47><loc_86><loc_53></location>It should be noted, that there is an ambiguity in calculation of the right-hand side parts of evolution equations in the region Q 2 ( ξ ): the result depends on the choice of the fast subsystem's solution, and there are two different librating solutions in Q 2 ( ξ ). Consequently, phase portraits of (4) shall contain two sets of trajectories in Q 2 ( ξ ), which correspond to two possible variants of slow variables' evolution.</text>
<text><location><page_11><loc_15><loc_42><loc_86><loc_47></location>When describing the crossing of uncertainty curves Γ i ( ξ ) by the projection ζ ( τ ) = ( x ( τ ) , y ( τ )) T of the phase point z ( τ ) = ( ϕ ( τ ) , Φ ( τ ) , x ( τ ) , y ( τ )) T on the plane ( x, y ), we shall confine ourselves to formal continuation of averaged system's trajectories, lying on opposite sides of Γ i . The detailed analysis of these events is given in Neishtadt (1987a), Neishtadt and Sidorenko (2004), and Sidorenko et al. (2014).</text>
<section_header_level_1><location><page_11><loc_15><loc_36><loc_62><loc_37></location>4.2 Fast subsystem's action variable - integral of evolution equations</section_header_level_1>
<text><location><page_11><loc_15><loc_30><loc_86><loc_34></location>As was mentioned in the Section 2, adiabatic invariant J ( x, y, ξ ) of slow-fast system is a first integral of evolution equations (4). Formula (6) for J ( x, y, ξ ) can be expressed as a linear combination of integrals I k,r defined by (19):</text>
<formula><location><page_11><loc_17><loc_19><loc_86><loc_27></location>J ( x, y, ξ ) = 1 π ϕ ∗ ∫ ϕ ∗ Φdϕ = 1 π ϕ ∗ ∫ ϕ ∗ √ 2 [ ξ -W ( x, y, ϕ )] dϕ = 2 π λ ∗ ∫ λ ∗ √ 2 R 4 ( λ ) dλ (1 + λ 2 ) 2 = = 2 √ 2 π √ | d 0 | [ d 0 I 0 , 0 + d 1 I 1 , 1 +( d 2 -2 d 0 ) I 1 , 0 +( d 4 + d 0 -d 2 ) I 2 , 0 ] . (22)</formula>
<text><location><page_11><loc_15><loc_15><loc_86><loc_17></location>Analytical expression for I 2 , 0 is presented in the Appendix B alongside the rest of the integrals previously shown in (20).</text>
<figure>
<location><page_12><loc_15><loc_26><loc_74><loc_89></location>
<caption>Fig. 11 Phase portraits of evolution equations: a. ξ = -4, b. ξ = -1 . 5, c. ξ = -0 . 1, d. ξ = 1 . 45, e. ξ = 3 . 4, f. ξ = 8 . 0</caption>
</figure>
<section_header_level_1><location><page_13><loc_15><loc_87><loc_63><loc_88></location>5 Study of slow variables' behavior using evolution equations</section_header_level_1>
<text><location><page_13><loc_15><loc_84><loc_56><loc_85></location>5.1 Phase portraits of evolution equations. Stationary points</text>
<text><location><page_13><loc_15><loc_72><loc_86><loc_82></location>To analyze solutions of the slow subsystem (4), we build its phase portraits. For values ξ < ξ 1 the structure of phase portrait is simple - all phase trajectories are represented by closed loops encircling the forbidden region Q 0 ( ξ ) (Fig. 11a). Figure 11b depicts a typical phase portrait for ξ ∈ ( ξ 1 , ξ 2 ) - two symmetrical parts of region Q 2 ( ξ ) adjoining the central region Q 0 ( ξ ) contain two layers of phase trajectories. For ξ ∈ ( ξ 2 , ξ 3 ) there are only two regions on the phase plane, i.e. Q 0 ( ξ ) and Q 2 ( ξ ) (Fig. 11c). In addition to presented in Figure 11c,d change of phase portrait's global structure, the behaviour of phase trajectories near the uncertainty curves Γ 1 , 2 ( ξ ) at ξ ∈ ( ξ 2 , ξ 4 ) have some specific qualitative differences at different ξ values. The detailed description of that is given in the end of this section.</text>
<text><location><page_13><loc_15><loc_66><loc_86><loc_71></location>Phase portraits at ξ > ξ 3 have five stationary points: the origin of xy -plane is the stable point of the center type, two more center points are symmetrically located on the y -axis, and two unstable saddle points are symmetrically located on the x -axis (Fig. 11d-f). Phase portraits depicted in Figures 11e and 11f are differ in relative positions of heteroclinic trajectories and uncertainty curve Γ 1 .</text>
<text><location><page_13><loc_18><loc_64><loc_77><loc_65></location>Ordinates y of the center points above and below plane's origin are defined by equation</text>
<formula><location><page_13><loc_44><loc_61><loc_86><loc_62></location>K ( m ) = 2 Π ( n | m ) , (23)</formula>
<text><location><page_13><loc_15><loc_57><loc_19><loc_58></location>where</text>
<formula><location><page_13><loc_30><loc_54><loc_71><loc_56></location>m = U + U -, n = U + ξ +1 -| y | , U ± = ξ -3 ± √ y 2 +8(1 -ξ ) .</formula>
<text><location><page_13><loc_15><loc_51><loc_80><loc_52></location>Abscissae x of the saddle points are defined by the same relation (23) with different parameters:</text>
<formula><location><page_13><loc_28><loc_47><loc_73><loc_49></location>m = Q + -Q --Q ++ Q -+ , n = Q + -Q --, Q ±± = √ x 2 +8(1 + ξ ) ± 4 ±| x | .</formula>
<text><location><page_13><loc_15><loc_42><loc_86><loc_44></location>Solutions of (23) are plotted in Figure 12 as functions of ξ for both types of stationary points. Note, that top and bottom center points are located in Q 2 at ξ < ξ 5 = 2 and in Q 1 at ξ > ξ 5 (Fig. 13).</text>
<figure>
<location><page_13><loc_21><loc_19><loc_79><loc_38></location>
<caption>Fig. 12 Coordinates of evolution equations' stationary points, which lie on axis x (left), and on axis y (right). Values of coordinates as functions of ξ are represented by violet lines. The rest of the coloring is consistent with Figure 11 in denoting the regions Q i and points from different parts of the critical curve</caption>
</figure>
<figure>
<location><page_14><loc_36><loc_66><loc_65><loc_88></location>
<caption>Fig. 13 Transition of the top center point from Q 2 ( ξ ) to Q 1 ( ξ ) at ξ = ξ 5 = 2</caption>
</figure>
<text><location><page_14><loc_15><loc_59><loc_43><loc_60></location>5.2 Limiting points on uncertainty curves</text>
<text><location><page_14><loc_15><loc_54><loc_86><loc_58></location>To conclude the description of how phase portraits' topology changes with ξ value, points of uncertainty curves' intersections with limiting trajectories, that are limiting cases for different families of trajectories, should be considered. We shall refer to them as limiting points .</text>
<figure>
<location><page_14><loc_31><loc_24><loc_70><loc_52></location>
<caption>Fig. 14 Limiting points R i and I i at ξ ≈ 0</caption>
</figure>
<text><location><page_14><loc_15><loc_15><loc_86><loc_20></location>Several types of limiting points are depicted in Figure 14. Trajectories, which go in Q 1 between two points on uncertainty curve Γ 2 , can be divided into two families: the trajectories that intersect x axis, and those that do not. Thus there is limiting trajectory that separates these two families (in Figure 14 it is colored red). We shall denote the limiting points corresponding to this trajectory as R 1 , ..., R 4 (Fig. 14).</text>
<text><location><page_15><loc_15><loc_84><loc_86><loc_88></location>The subset of trajectories, which do not cross x -axis in Q 1 as another limiting case contain trajectories, which come to uncertainty curve from the side of Q 2 and then reflect back without exiting to Q 1 . In Figure 14 these trajectories are colored purple, and the corresponding limiting points are denoted I 1 , ..., I 4 .</text>
<figure>
<location><page_15><loc_21><loc_40><loc_80><loc_81></location>
<caption>Fig. 15 Limiting points K i , M i and V i ( ξ ∼ 1 . 5)</caption>
</figure>
<text><location><page_15><loc_15><loc_24><loc_86><loc_35></location>More limiting points are depicted in figure 15. Points K 1 , ..., K 4 ∈ Γ 2 are connected to cusps Y 1 , ..., Y 4 by limiting phase trajectories, which separate the family of trajectories lying to the one side of uncertainty curve Γ 2 from those which in Q 1 connect symmetric points on left and right sides of uncertainty curves Γ 1 , 2 . Limiting points M 1 , ..., M 4 divides each of four Γ 2 segments (located in each of four quadrants of xy plane) into two sections: one section has both trajectories, which adjoin from Q 2 side, going in the same direction, while trajectories adjoining the other section go in opposite directions (also see Figure 19). Points V 1 , ..., V 4 of Γ 1 curve's intersections with mentioned earlier separatrices constitute the last type of limiting points.</text>
<text><location><page_15><loc_15><loc_15><loc_86><loc_24></location>All differences between the phase portraits (Fig. 11) emerge from changes in position of limiting points on the critical curve. By adding these points to diagram in Figure 9 we obtain a bifurcation diagram (Fig. 16), from which all changes in topological structure of phase portraits can be understood. Let us describe, what is happening to different limiting points by going successively from low to high values of ξ . The first limiting points R i and I i - appear at ξ = ξ 6 ≈ -0 . 22073. At ξ = ξ 7 ≈ 0 . 27704 points R i merge with cusps Y i , and at the same time points K i appear. Points M i and V i emerge simultaneously with region Q 3 and self-intersections points S i of the critical curve at ξ = ξ 3 . All points on the Γ 2 part of</text>
<text><location><page_16><loc_15><loc_85><loc_86><loc_88></location>the critical curve I i , K i , and M i , as well as the ends Y i , S i of Γ 2 itself - disappear at ξ = ξ 4 merging with the astroid's cusps A 2 and A 4 . Finally, at ξ = ξ 8 ≈ 5 . 57954 points V i disappear by merging pairwise.</text>
<figure>
<location><page_16><loc_32><loc_61><loc_70><loc_80></location>
<caption>Fig. 16 Bifurcation diagram showing the dependance of limiting points position on the critical curve Γ on ξ .</caption>
</figure>
<text><location><page_16><loc_15><loc_48><loc_86><loc_53></location>Bifurcation Hamiltonian values ξ 1 , . . . , ξ 8 partition the whole range of ξ into nine intervals, meaning that there is a total number of nine different types of phase portraits for slow subsystem. The bifurcation at ξ 5 (Fig. 13) is omitted from Figure 16 because it does not bear any significance for the dynamical effects considered further.</text>
<section_header_level_1><location><page_16><loc_15><loc_34><loc_33><loc_35></location>6 Quasi-random effects</section_header_level_1>
<text><location><page_16><loc_15><loc_31><loc_85><loc_32></location>6.1 Probabilistic change of fast subsystem's motion regime on the uncertainty curve of the second kind</text>
<text><location><page_16><loc_15><loc_15><loc_86><loc_28></location>In each point of Γ 2 ( ξ ) three guiding trajectories , i.e. trajectories of averaged system (4), meet: two on the side of Q 2 ( ξ ) region and one on the Q 1 ( ξ ) side. In the case, when two out of these three trajectories are outgoing, the transition of the phase point to either one of them can be considered as a probabilistic event. In the original system (1) initial values of fast variables corresponding to two different outcomes are strongly mixed in the phase space. Therefore even small variation of initial conditions z (0) = ( ϕ (0) , Φ (0) , x (0) , y (0)) T can lead to qualitative change in system's evolution. As an example, Figure 17 depicts projections on the plane ( x, y ) of two trajectories γ 1 , 2 , obtained as solutions of the system (1) with close initial conditions. Both trajectories approach uncertainty curve along the same guiding trajectory, but diverge after that γ 1 exits Q 2 ( ξ ) and follows the outgoing guiding trajectory in Q 1 ( ξ ), while γ 2 turns and goes back along the other guiding trajectory in Q 2 ( ξ ).</text>
<figure>
<location><page_17><loc_43><loc_72><loc_70><loc_88></location>
<caption>Fig. 17 Two phase trajectories starting from very close initial conditions may diverge at uncertainty curve Γ 2 , as γ 1 and γ 2 do. Green and red lines show two guiding trajectories departing from the same point of Γ 2 , to which the blue one arrives. Trajectory γ of non-averaged system may go along either one of them after reaching the border between Q 1 and Q 2</caption>
</figure>
<text><location><page_17><loc_15><loc_57><loc_86><loc_64></location>In deterministic systems with strongly entangled trajectories the probability of a specific outcome is determined by the fraction of phase volume occupied by corresponding initial conditions (formal definition can be found in Arnold (1963) and Neishtadt (1987b)). In order to find the probabilities of transitions to different outgoing trajectories in some point ( x ∗ , y ∗ ) on Γ 2 , two auxiliary parameters must be calculated first (Neishtadt 1987b; Artemyev et al. 2013):</text>
<formula><location><page_17><loc_31><loc_52><loc_86><loc_56></location>Θ 1 , 2 = + ∞ ∫ -∞ ( ∂W ∗ max ∂x ∂W ∂y -∂W ∗ max ∂y ∂W ∂x ) ϕ s 1 , 2 ( t,x ∗ ,y ∗ ,ξ ) dτ. (24)</formula>
<text><location><page_17><loc_15><loc_47><loc_92><loc_51></location>These parameters have a meaning of rates with which areas bounded by separatrices ( ϕ s 1 ( τ, x ∗ , y ∗ , ξ ) , Φ s 1 ( τ, x ∗ , y ∗ , ξ )) and ( ϕ s 2 ( τ, x ∗ , y ∗ , ξ ) , Φ s 2 ( τ, x ∗ , y ∗ , ξ )) in fast variables' phase space change. After substitution of specific potential function (2) into (24) and change of integration variable to ϕ we obtain:</text>
<formula><location><page_17><loc_24><loc_41><loc_86><loc_46></location>Θ 1 = 2 glyph[slurabove] ϕ ∫ ϕ s min sin( ϕ -glyph[slurabove] ϕ ) √ 2 ( ξ -W ( x ∗ , y ∗ , ϕ )) dϕ, Θ 2 = 2 ϕ s max ∫ glyph[slurabove] ϕ sin( ϕ -glyph[slurabove] ϕ ) √ 2 ( ξ -W ( x ∗ , y ∗ , ϕ )) dϕ. (25)</formula>
<text><location><page_17><loc_15><loc_34><loc_86><loc_40></location>Here glyph[slurabove] ϕ is the coordinate ϕ of the saddle point in the fast subsystem's phase portrait (it coincides with the value glyph[slurabove] ϕ corresponding to point ( x ∗ , y ∗ ) in (13)); ϕ s min and ϕ s max are the minimal and the maximal values of ϕ in homoclinic trajectories to the left and to the right of the saddle point respectively. Applying the substitution λ = cot[( ϕ -glyph[slurabove] ϕ ) / 2] to (25) we obtain:</text>
<formula><location><page_17><loc_36><loc_29><loc_86><loc_33></location>Θ 1 , 2 = 4 √ A λ max ∫ λ min λ dλ (1 + λ 2 ) √ ( λ -a )( λ -b ) , (26)</formula>
<text><location><page_17><loc_15><loc_27><loc_19><loc_28></location>where</text>
<formula><location><page_17><loc_35><loc_24><loc_66><loc_28></location>a = B -√ B 2 -AC A , b = B + √ B 2 -AC A ,</formula>
<text><location><page_17><loc_32><loc_23><loc_69><loc_24></location>A = ξ +3cos(2 glyph[slurabove] ϕ ) , B = 2sin(2 glyph[slurabove] ϕ ) , C = ξ -cos(2 glyph[slurabove] ϕ ) .</text>
<text><location><page_17><loc_15><loc_18><loc_86><loc_22></location>It should be noted, that inequalities A > 0 and B 2 > AC hold for all points of Γ 2 . Integration in (26) is carried out over the intervals ( -∞ , a ) and ( b, + ∞ ) for Θ 1 and Θ 1 respectively. The result can be obtained by using Cauchy's residue theorem:</text>
<formula><location><page_17><loc_34><loc_14><loc_67><loc_17></location>Θ 1 = 4 √ A Re ( L -iπ αβ ) , Θ 2 = 4 √ A Re ( L αβ ) .</formula>
<text><location><page_18><loc_15><loc_87><loc_18><loc_88></location>Here</text>
<formula><location><page_18><loc_34><loc_84><loc_67><loc_87></location>L = ln ( b -a ( α -β ) 2 ) , α = √ a + i, β = √ b + i,</formula>
<text><location><page_18><loc_15><loc_82><loc_83><loc_83></location>and the branches of multifunctions are selected in such way, that Im( L ), arg( α ), and arg( β ) ∈ (0 , π ).</text>
<text><location><page_18><loc_15><loc_77><loc_86><loc_82></location>Now we can write down the expressions for probabilities of different evolution scenarios for a phase point on Γ 2 ( ξ ). Let us denote the probability of point going to region Q 1 ( ξ ) as P 0 , and probabilities corresponding to two trajectories going inside Q 2 ( ξ ) as P 1 and P 2 . The resulting formulae (Artemyev et al. 2013) take a form:</text>
<formula><location><page_18><loc_28><loc_74><loc_86><loc_75></location>P 0 = 1 -P 1 -P 2 , P 1 = max( Θ 1 , 0) /Θ Σ , P 2 = max( Θ 2 , 0) /Θ Σ , (27)</formula>
<text><location><page_18><loc_15><loc_72><loc_19><loc_73></location>where</text>
<formula><location><page_18><loc_33><loc_70><loc_69><loc_71></location>Θ Σ = max( Θ 1 , 0) + max( Θ 2 , 0) + max( -Θ 1 -Θ 2 , 0) .</formula>
<text><location><page_18><loc_15><loc_65><loc_86><loc_69></location>In Figure 18 and Figure 19 change of probabilities (27) along Γ 2 is shown alongside with phase portraits at corresponding values of Hamiltonian and limiting points, which mark the change of sign in Θ 1 , 2 or Θ 1 + Θ 2 .</text>
<figure>
<location><page_18><loc_21><loc_45><loc_79><loc_62></location>
<caption>Fig. 18 Change of P i along Y 4 Y 1 segment of Γ 2 at ξ = 0. The graph for Y 2 Y 3 can be reconstructed by the symmetry, changing the y sign and swapping red and blue plots. In I 1 , . . . , I 4 the sum Θ 1 + Θ 2 = 0, which results in P 1 plot sticking to 0 on one side from these points</caption>
</figure>
<figure>
<location><page_18><loc_21><loc_18><loc_79><loc_36></location>
<caption>Fig. 19 Change of P i along segments Y 4 S 2 and S 1 Y 1 of Γ 2 at ξ = 2 . 4. In M 1 , 2 parameter Θ 1 = 0, and in M 3 , 4 parameter Θ 2 = 0, which results in singularities of probabilities plots in these limiting points</caption>
</figure>
<section_header_level_1><location><page_19><loc_15><loc_87><loc_28><loc_88></location>6.2 Adiabatic chaos</section_header_level_1>
<text><location><page_19><loc_15><loc_79><loc_86><loc_85></location>Adiabatic chaos emerges due to non-applicability of adiabatic approximation near the uncertainty curve. As a result the projection of phase point ζ ( τ ) = ( x ( τ ) , y ( τ )) T leaves the vicinity of uncertainty curve along the guiding trajectory, which slightly differs from the direct continuation of approach trajectory (Fig. 20). The resulting offset between incoming and outgoing guiding trajectories can be treated as a quasi-random jump with order of magnitude ε | ln ε | (Tennyson et al. 1986; Neishtadt 1987b,a).</text>
<text><location><page_19><loc_15><loc_75><loc_86><loc_78></location>As a result of persistent jumps any trajectory that crosses the uncertainty curve after a long time will fill a whole region of phase plane, which we will call an adiabatic chaos region (Fig. 21). This region consists of points belonging to trajectories, which cross the uncertainty curve.</text>
<figure>
<location><page_19><loc_31><loc_55><loc_70><loc_72></location>
<caption>Fig. 20 The jump of trajectory γ from guiding trajectory γ 1 to γ 2 upon crossing the uncertainty curve</caption>
</figure>
<figure>
<location><page_19><loc_32><loc_22><loc_69><loc_48></location>
<caption>Fig. 21 Adiabatic chaos region at ξ = 1 . 45</caption>
</figure>
<text><location><page_19><loc_15><loc_15><loc_86><loc_17></location>At ξ ∈ ( ξ 3 , ξ 8 ) the uncertainty curve is crossed by the separatrices, which connect two saddle points. A phase point projection moving along a guiding trajectory close to separatrix, when crossing the uncertainty</text>
<text><location><page_20><loc_15><loc_79><loc_86><loc_88></location>curve, may jump over the separatrix and begin to move along the other guiding trajectory belonging to completely different family. Thus the properties of long-term evolution are suddenly changed on a qualitative level. E.g., in Figure 21 the motion of a phase point projection ζ ( τ ) circling around the coordinate origin in the central part of adiabatic chaos region by crossing the separatrix may transform into circulation in opposite direction around one of two center points (23), which lie on y -axis in upper or lower half-plane. This event can be interpreted as a capture into Kozai-Lidov resonance and it is accompanied by decrease of average inclination value, about which the long-term oscillations occur.</text>
<section_header_level_1><location><page_20><loc_15><loc_74><loc_34><loc_75></location>7 Numerical simulations</section_header_level_1>
<text><location><page_20><loc_15><loc_69><loc_86><loc_72></location>Construction of the discussed analytical model involved several assumptions that may seem loose. It is thus required to test whether the model can be applied to orbital dynamics of real life objects and these assumptions were not overly restrictive.</text>
<text><location><page_20><loc_15><loc_59><loc_86><loc_68></location>For this purpose we used Mercury integrator (Chambers 1999) and carried out several numerical simulations of the Solar system composed of the Sun, the four giant planets, and about 700 known Kuiper belt objects (KBO) near 1 : 2, 2 : 3, and 3 : 4 resonances with Neptune represented by test particles (masses of four inner planets were added to the Sun in order to facilitate the integration). Total time of integration 15 Myr is one order of magnitude larger then the characteristic time T N /ε 2 ≈ 1 . 6 Myr of slow variables evolution (given the orbital period of Neptune T N ≈ 160 y and Neptune/Sun mass ratio defining the small parameter ε 2 ∼ 10 -4 ).</text>
<figure>
<location><page_20><loc_21><loc_35><loc_79><loc_56></location>
<caption>Fig. 22 Comparison of the Solar system's numerical integration with analytical model. Several Kuiper belt objects with Hamiltonian values close to ξ ≈ -0 . 5 (left) and ξ ≈ 23 (right) are plotted on the plane of slow variables. Phase trajectories of the model plotted by dashed lines</caption>
</figure>
<text><location><page_20><loc_15><loc_26><loc_86><loc_28></location>For interpretation of the simulation results we shall utilize a scaled version of previously used slow variables:</text>
<formula><location><page_20><loc_42><loc_24><loc_60><loc_25></location>x ∝ e cos ω, y ∝ -e sin ω.</formula>
<text><location><page_20><loc_15><loc_20><loc_86><loc_22></location>For exact relations between ( e, ω ) and ( x, y ), as well as the expression for small parameter ε , see Appendix A.</text>
<text><location><page_20><loc_15><loc_15><loc_86><loc_20></location>Phase trajectories of several objects are plotted in Figure 22. Main sources of difference between analytical phase portraits and numerical ones are non-zero eccentricity of Neptune ( e ' ≈ 0 . 01), high eccentricity of KBOs (up to 0 . 3 on the right side of Figure 22), and presence of other planets, which introduce additional disturbances distorting the phase plane. Nevertheless, it is clear that the overall</text>
<text><location><page_21><loc_15><loc_84><loc_86><loc_88></location>topological structure of phase portraits is reproduced in analytical model. Note, that in restricted threebody problem all solutions of Sessin and Ferraz-Mello (1984) on the same phase plane would be represented by concentric circles with e = const and ω changing linearly with time.</text>
<text><location><page_21><loc_15><loc_74><loc_86><loc_84></location>The principal difference between our model and the one introduced by Sessin and Ferraz-Mello (1984) is the expansion of disturbing function past the first term in the Fourier series. Thus we can assert, that the region of the complete phase space, where the second term influences the dynamics in a substantial way, is significantly large. Indeed about 10% of objects in our simulations deviated from circular trajectories in projection on the plane of slow variables. The rest correspond to very high or very low values of ξ , at which all trajectories in Q 1 and Q 3 , as well as the curve Γ separating them, in our model are likewise very close to circles.</text>
<figure>
<location><page_21><loc_22><loc_58><loc_79><loc_72></location>
<caption>Fig. 23 Resonant angle of 2011 UG 411 vs time, demonstrating jumps between two different librating solutions in Q 2</caption>
</figure>
<text><location><page_21><loc_15><loc_39><loc_86><loc_52></location>Some effects, that can be derived from the analytical model, are also observed in numerical simulations within the pool of selected KBOs. E.g. resonant angle of 2011 UG 411 shows jumps between two different librating regimes characteristic to motion in region Q 2 (Fig. 23), while 2007 JJ 43 demonstrates the intermittent behaviour (Fig. 24) with the resonance angle constantly switching between libration and circulation. Phase trajectory of 2007 JJ 43 on the plane of slow variables is presented in Figure 25, showing that changes in resonant angle behaviour are conditioned by secular trajectory crossing of the critical curve Γ . Similarly, analytical trajectory γ 0 goes between regions of libration and circulation of resonant angle (Fig. 25). To compensate for angle ω precession on this kind of trajectories the modified resonant angle θ = ϕ + ω was used in previous plots 1 . Same as ϕ , resonant angle θ circulates in Q 3 and librates in Q 1 and Q 2 .</text>
<figure>
<location><page_21><loc_22><loc_23><loc_79><loc_36></location>
<caption>Fig. 24 Resonant angle of 2007 JJ 43 vs time. Intervals of circulation and libration colored green and gray respectively</caption>
</figure>
<figure>
<location><page_22><loc_21><loc_68><loc_79><loc_88></location>
<caption>Fig. 25 Phase trajectory of 2007 JJ 43 alongside the analogous trajectory γ 0 on the analytical phase portrait. Intervals of resonant angle circulation and libration on the left panel are colored green and gray respectively. It is seen, that the green segments of the trajectory concentrate inside the region bound by Γ , while the gray ones mostly lie outside of it</caption>
</figure>
<section_header_level_1><location><page_22><loc_15><loc_58><loc_25><loc_59></location>8 Conclusion</section_header_level_1>
<text><location><page_22><loc_15><loc_48><loc_86><loc_56></location>In this paper, using the averaging technique, we study Hamiltonian system that approximately describes the dynamics of a three-body system in first-order MMR (within the restricted circular problem). Our model incorporates harmonics of the Fourier series expansion of disturbing function up to the second order. Thus it can accurately describe the dynamics in that part of the phase space where first two harmonics have comparable magnitudes, and where the well known integrable model of first-order MMR is not applicable. This is the region, from which chaos emerges.</text>
<text><location><page_22><loc_15><loc_43><loc_86><loc_48></location>The nonintegrability of our model does not become an obstacle for a detailed analytical investigation of its properties. In particular, we have constructed bifurcation diagrams and phase portraits characterizing the long-term dynamics on different level sets defined by the system's Hamiltonian and obtained expressions for probabilities of quasi-random transitions between different phase trajectories.</text>
<text><location><page_22><loc_15><loc_37><loc_86><loc_42></location>The important question is the scope of the correctness of proposed model. We will try to answer it in subsequent studies, using Wisdom's approach to investigate several first-order MMRs without truncating the averaged disturbing function. Nevertheless, even now we can note that the secular evolution of some Kuiper belt objects qualitatively resembles what our simple model predicts.</text>
<text><location><page_22><loc_15><loc_35><loc_86><loc_37></location>We hope also that our investigation outlines an approach which can be applied for analysis of similar degeneracy of the averaged disturbing function in the case of other MMRs (e.g., Sidorenko (2006)).</text>
<section_header_level_1><location><page_22><loc_15><loc_29><loc_86><loc_31></location>Appendix A: Constructing a model system, that reveals the origin of chaos in first-order MMR</section_header_level_1>
<text><location><page_22><loc_15><loc_20><loc_86><loc_27></location>We shall confine ourselves to a case of exterior resonance p : ( p + 1) in restricted three-body problem. The interior resonance ( p +1) : p can be reduced to the same model using similar approach. The distance between the two major bodies, i.e. a star and a planet, and the sum of their masses are taken here as units of length and mass. The unit of time is chosen such that the orbital period of major bodies' rotation about barycenter is equal to 2 π . The mass of the planet µ is considered to be a small parameter of the problem.</text>
<text><location><page_22><loc_18><loc_18><loc_62><loc_19></location>Equations of motion for minor body (asteroid) in canonical form:</text>
<formula><location><page_22><loc_33><loc_14><loc_86><loc_17></location>d ( L, G, H ) dt = -∂ K ∂ ( l, g, h ) , d ( l, g, h ) dt = ∂ K ∂ ( L, G, H ) , (28)</formula>
<text><location><page_23><loc_15><loc_85><loc_86><loc_88></location>where L, G, H, l, g, h are the Delaunay variables (Murray and Dermott 2000). They can be expressed in terms of Keplerian elements a, e, i, Ω, ω as</text>
<formula><location><page_23><loc_28><loc_83><loc_73><loc_84></location>L = √ (1 -µ ) a, G = L √ 1 -e 2 , H = G cos i, g = ω, h = Ω.</formula>
<text><location><page_23><loc_15><loc_81><loc_49><loc_82></location>The last variable l is the asteroid's mean anomaly.</text>
<text><location><page_23><loc_18><loc_80><loc_34><loc_81></location>Hamiltonian K in (28) is</text>
<formula><location><page_23><loc_36><loc_76><loc_86><loc_79></location>K = -(1 -µ ) 2 2 L 2 -µR ( L, G, H, l, g, h -λ ' ) . (29)</formula>
<text><location><page_23><loc_15><loc_70><loc_86><loc_75></location>Here R is disturbing function in restricted circular three-body problem. Mean anomaly λ ' appearing in (29) depends linearly on time: λ ' = t + λ ' 0 . Therefore it is convenient to use variable ˜ h = h -λ ' instead of h , as it enables writing down the equations of motion in autonomous form as canonical equations with Hamiltonian</text>
<formula><location><page_23><loc_41><loc_69><loc_60><loc_70></location>˜ K = K ( L, G, H, l, g, ˜ h ) -H.</formula>
<text><location><page_23><loc_15><loc_66><loc_87><loc_68></location>Weintroduce resonant angle ¯ ϕ using the canonical transformation ( L, G, H, l, g, ˜ h ) → ( P ϕ , P g , P h , ¯ ϕ, ¯ g, ¯ h ) defined by generating function</text>
<formula><location><page_23><loc_27><loc_63><loc_74><loc_65></location>S = ( p +1) P ϕ l + [ P h + p ( P ϕ -P ∗ ϕ )] ˜ h + [ P g +( p +1) ( P ϕ -P ∗ ϕ )] g,</formula>
<text><location><page_23><loc_15><loc_60><loc_86><loc_62></location>where P ∗ ϕ = L ∗ / ( p +1), while L ∗ = 3 √ ( p +1) /p is the value of L corresponding to exact p : ( p +1) MMR in unperturbed problem ( µ = 0). The new variables are related to the old ones as follows:</text>
<formula><location><page_23><loc_30><loc_56><loc_71><loc_59></location>L = ∂S ∂l = ( p +1) P ϕ , G = ∂S ∂g = P g +( p +1) ( P ϕ -P ∗ ϕ ) ,</formula>
<formula><location><page_23><loc_40><loc_53><loc_61><loc_56></location>H = ∂S ∂ ˜ h = P h + p ( P ϕ -P ∗ ϕ ) ,</formula>
<formula><location><page_23><loc_27><loc_51><loc_74><loc_53></location>¯ ϕ = ∂S ∂P ϕ = ( p +1) l + p ˜ h +( p +1) g, ¯ g = ∂S ∂P g = g, ¯ h = ∂S ∂P h = ˜ h.</formula>
<text><location><page_23><loc_15><loc_49><loc_56><loc_50></location>The resonant angle can also be expressed in traditional form</text>
<formula><location><page_23><loc_43><loc_47><loc_59><loc_48></location>¯ ϕ = ( p +1) λ -pλ ' -Ω.</formula>
<text><location><page_23><loc_15><loc_44><loc_27><loc_45></location>New Hamiltonian</text>
<formula><location><page_23><loc_35><loc_42><loc_67><loc_45></location>˜ K = -(1 -µ ) 2 2( p +1) 2 P 2 ϕ -[ P h + p ( P ϕ -P ∗ ϕ ) ] -µR.</formula>
<text><location><page_23><loc_18><loc_40><loc_86><loc_41></location>The resonant case we are interested in corresponds to region R of phase space, selected by the condition</text>
<formula><location><page_23><loc_43><loc_38><loc_59><loc_39></location>∣ ∣ pn ' -( p +1) n ∣ ∣ glyph[lessorsimilar] µ 1 / 2 .</formula>
<text><location><page_23><loc_15><loc_36><loc_81><loc_37></location>Here n ' = 1 and n are mean motions of planet and asteroid respectively. It is also true in R that</text>
<formula><location><page_23><loc_45><loc_34><loc_86><loc_35></location>∣ ∣ P ϕ -P ∗ ϕ ∣ ∣ glyph[lessorsimilar] µ 1 / 2 . (30)</formula>
<text><location><page_23><loc_15><loc_30><loc_86><loc_33></location>In the resonant case, i.e. when the previous inequation holds, variables can be divided into fast , semi-fast and slow . Fast and semi-fast variables in R are ¯ h and ¯ ϕ respectively:</text>
<formula><location><page_23><loc_43><loc_27><loc_58><loc_29></location>d ¯ h dt ∼ 1 , d ¯ ϕ dt ∼ µ 1 / 2 .</formula>
<text><location><page_23><loc_15><loc_25><loc_64><loc_26></location>Slow variables, which vary with a rate of order µ , are P ϕ , P g , P h , and ¯ g .</text>
<text><location><page_23><loc_15><loc_22><loc_86><loc_25></location>To study secular effects the averaging over the fast variable ¯ h is performed, which results in equations of motion taking a canonical form with Hamiltonian</text>
<formula><location><page_23><loc_17><loc_20><loc_31><loc_21></location>¯ K ( P ϕ , P g , P h , ¯ ϕ, ¯ g ) =</formula>
<formula><location><page_23><loc_37><loc_15><loc_85><loc_19></location>= 1 2 π ( p +1) 2 π ( p +1) ∫ 0 ˜ K ( L ( P ϕ ) , G ( P g , P ϕ ) , H ( P h , P ϕ ) , l ( ¯ ϕ, ¯ g, ¯ h ) , ¯ g, ¯ h ) d ¯ h,</formula>
<text><location><page_24><loc_15><loc_87><loc_19><loc_88></location>where</text>
<formula><location><page_24><loc_38><loc_84><loc_64><loc_87></location>l ( ¯ ϕ, ¯ g, ¯ h ) = 1 k +1 [ ¯ ϕ -( k +1)¯ g -k ¯ h ] .</formula>
<text><location><page_24><loc_15><loc_80><loc_86><loc_84></location>After such averaging the fast variable ¯ h vanishes, and the term 'fast' is exempted. Thus in the rest of the paper we adopt name fast for denoting variables, which vary with the rate µ 1 / 2 , instead of referring to them as semi-fast, when the distinction of three different time scales was needed.</text>
<text><location><page_24><loc_15><loc_76><loc_86><loc_80></location>Moreover, because there is no longer ¯ h in the Hamiltonian ¯ K , the conjugate momentum P h is a constant in considered approximation and can be treated as a parameter of the problem. Thus ¯ K is the Hamiltonian of a system with two degrees of freedom. Further instead of P h we shall use parameter</text>
<formula><location><page_24><loc_46><loc_71><loc_56><loc_74></location>σ = √ 1 -P 2 h L 2 ∗ .</formula>
<text><location><page_24><loc_15><loc_67><loc_86><loc_70></location>Inequation e ≤ σ defines region S in phase space, to which the motion of the system is bound by the Kozai-Lidov integral (Sidorenko et al. 2014).</text>
<text><location><page_24><loc_15><loc_65><loc_86><loc_67></location>Next standard step in analysis of system's dynamics in resonant region R is the scaling transformation (Arnold et al. 2006):</text>
<formula><location><page_24><loc_41><loc_63><loc_86><loc_64></location>¯ τ = ¯ εt, ¯ Φ = ( P ∗ ϕ -P ϕ ) / ¯ ε, (31)</formula>
<text><location><page_24><loc_15><loc_59><loc_86><loc_62></location>where ¯ ε = µ 1 / 2 is a new small parameter. Using variables (31), the equations of motion can be rewritten in a form of slow-fast system without loss of accuracy:</text>
<formula><location><page_24><loc_41><loc_53><loc_86><loc_58></location>d ¯ ϕ d ¯ τ = χ ¯ Φ, d ¯ Φ d ¯ τ = -∂ ¯ W ∂ ¯ ϕ , dP g = ¯ ε ∂ ¯ W , d ¯ g = -¯ ε ∂ ¯ W . (32)</formula>
<formula><location><page_24><loc_41><loc_53><loc_60><loc_54></location>d ¯ τ ∂ ¯ g d ¯ τ ∂P g</formula>
<text><location><page_24><loc_15><loc_50><loc_18><loc_51></location>Here</text>
<formula><location><page_24><loc_25><loc_44><loc_77><loc_50></location>χ = 3 p 4 / 3 ( p +1) 2 / 3 , ¯ W ( ¯ ϕ, ¯ g, P g ; σ ) = 1 2 π ( k +1) 2 π ( k +1) ∫ 0 R ( L ∗ , P g , L ∗ √ 1 -σ 2 , l ( ¯ ϕ, ¯ g, ¯ h ) , ¯ g, ¯ h ) d ¯ h.</formula>
<text><location><page_24><loc_15><loc_41><loc_86><loc_43></location>For σ glyph[lessmuch] 1 the approximate expression for ¯ W ( ¯ ϕ, ¯ g, P g ; σ ) can be obtained as a series expansion (Murray and Dermott 2000):</text>
<formula><location><page_24><loc_17><loc_38><loc_44><loc_39></location>¯ W ( ¯ ϕ, ¯ g, P g ; σ ) ≈ W 0 + W 1 e cos( ¯ ϕ -¯ g )+</formula>
<formula><location><page_24><loc_45><loc_36><loc_86><loc_37></location>+ e 2 [ W 10 + W 11 cos 2( ¯ ϕ -¯ g )] + i 2 [ W 20 + W 21 cos 2 ¯ ϕ ] , (33)</formula>
<text><location><page_24><loc_15><loc_33><loc_19><loc_34></location>where</text>
<formula><location><page_24><loc_33><loc_30><loc_86><loc_33></location>e 2 ≈ 1 -P 2 g L 2 ∗ , i 2 ≈ 2 ( 1 -P h L ∗ √ 1 -e 2 ) ≈ σ 2 -e 2 . (34)</formula>
<text><location><page_24><loc_15><loc_27><loc_86><loc_29></location>Expressions (34) are obtained taking into account resonance condition (30), and that the values e and i are limited by σ glyph[lessmuch] 1, leading to e, i glyph[lessmuch] 1.</text>
<text><location><page_24><loc_18><loc_26><loc_69><loc_27></location>Coefficients W 0 , W 1 , W 10 , W 11 , W 20 , W 21 in (33) are calculated as follows:</text>
<formula><location><page_24><loc_22><loc_14><loc_86><loc_24></location>W 0 = α 2 b (0) 1 / 2 , W 1 = α 2 [ (2 p +1) b ( p ) 1 / 2 + α db ( p ) 1 / 2 dα ] -δ 1 p 2 α , W 10 = α 2 8 ( 2 db (0) 1 / 2 dα + α d 2 b (0) 1 / 2 dα 2 ) , W 11 = α 8 ( [ 2 -14( p +1) + 16( p +1) 2 ] b (2 p ) 1 / 2 + α [8( p +1) -2] db (2 p ) 1 / 2 dα + α 2 d 2 b (2 p ) 1 / 2 dα 2 ) , W 20 = -W 10 , W 21 = α 2 8 b (2 p +1) 3 / 2 . (35)</formula>
<text><location><page_25><loc_15><loc_85><loc_86><loc_88></location>Here α = ( p/ ( p +1)) 2 / 3 , δ mn is the Kronecker delta, and b ( n ) 1 / 2 ( α ), b ( n ) 3 / 2 ( α ) are Laplace coefficients. Numerical values of coefficients (35) for several resonances are gathered in Table 1.</text>
<table>
<location><page_25><loc_33><loc_71><loc_68><loc_81></location>
<caption>Table 1 Numerical values of coefficients in series expansion of averaged disturbing function</caption>
</table>
<text><location><page_25><loc_15><loc_63><loc_86><loc_67></location>Out of the whole region S we are interested in the part with small eccentricities. Thus we further assume e/σ glyph[lessorsimilar] σ , which leads to e glyph[lessorsimilar] σ 2 . Taking into account (34) we can write down expression for averaged disturbing function up to the terms of order σ 2 :</text>
<formula><location><page_25><loc_29><loc_61><loc_72><loc_62></location>¯ W ( ¯ ϕ, ¯ g, P g ; σ ) ≈ W 0 + W 1 e cos( ¯ ϕ -¯ g ) + σ 2 [ W 20 + W 21 cos 2 ¯ ϕ ] .</formula>
<text><location><page_25><loc_15><loc_59><loc_34><loc_60></location>By introducing the variables</text>
<formula><location><page_25><loc_33><loc_57><loc_68><loc_58></location>¯ x = √ 2( L ∗ -P g ) cos ¯ g, ¯ y = -√ 2( L ∗ -P g ) sin ¯ g,</formula>
<text><location><page_25><loc_15><loc_55><loc_61><loc_56></location>the equations of motion (32) can be reduced to a Hamiltonian form</text>
<formula><location><page_25><loc_42><loc_48><loc_60><loc_54></location>d ¯ ϕ d ¯ τ = ∂ ¯ Ξ ∂ ¯ Φ , d ¯ Φ d ¯ τ = -∂ ¯ Ξ ∂ ¯ ϕ , d ¯ x d ¯ τ = ¯ ε ∂ ¯ Ξ ∂ ¯ y , d ¯ y d ¯ τ = -¯ ε ∂ ¯ Ξ ∂ ¯ x</formula>
<text><location><page_25><loc_15><loc_46><loc_30><loc_47></location>with the Hamiltonian</text>
<formula><location><page_25><loc_31><loc_43><loc_70><loc_46></location>¯ Ξ = χ ¯ Φ 2 2 + W 1 √ L ∗ cos ¯ ϕ · ¯ x -W 1 √ L ∗ sin ¯ ϕ · ¯ y + σ 2 W 21 cos 2 ¯ ϕ.</formula>
<text><location><page_25><loc_15><loc_41><loc_36><loc_42></location>The final rescaling of variables</text>
<formula><location><page_25><loc_37><loc_31><loc_86><loc_40></location>ϕ = -¯ ϕ, Φ = -√ χ σ 2 W 21 ¯ Φ, x = W 1 σ 2 W 21 √ L ∗ ¯ x, y = W 1 σ 2 W 21 √ L ∗ ¯ y, τ = σ √ χW 21 ¯ τ, ε = W 1 2 χ 1 / 2 L ∗ σ 3 W 3 / 2 21 ¯ ε (36)</formula>
<text><location><page_25><loc_15><loc_29><loc_39><loc_30></location>results in the model system (1)-(2).</text>
<section_header_level_1><location><page_25><loc_15><loc_24><loc_86><loc_26></location>Appendix B: Analytical expressions for integrals, emerging during the averaging over fast subsystem's period</section_header_level_1>
<text><location><page_25><loc_15><loc_20><loc_86><loc_22></location>Right-hand side parts of evolution equations (4), as well as adiabatic invariant formula (22), can be expressed in terms of integrals</text>
<formula><location><page_25><loc_31><loc_14><loc_70><loc_19></location>I k,r = λ ∗ ∫ λ ∗ λ r dλ ( λ 2 +1) k √ ± ( λ -a 1 )( λ -a 2 )( λ -a 3 )( λ -a 4 ) ,</formula>
<text><location><page_26><loc_15><loc_85><loc_86><loc_88></location>where k = 0 , 1 , 2; r = 0 , 1, and integration limits λ ∗ , λ ∗ can be either real numbers or ±∞ . These integrals can be reduced to the linear combinations of elliptic integrals of the first, the second and the third kind:</text>
<formula><location><page_26><loc_29><loc_77><loc_86><loc_84></location>I 0 , 0 = c 0 , 0 K ( k ) , I 1 , 0 = c 1 , 1 K ( k ) + c 1 , 3 Π ( h, k ) + ¯ c 1 , 3 Π ( ¯ h, k ) , I 1 , 1 = g 1 , 1 K ( k ) + g 1 , 3 Π ( h, k ) + ¯ g 1 , 3 Π ( ¯ h, k ) , I 2 , 0 = 1 2 I 1 , 0 + c 2 , 1 K ( k ) + c 2 , 2 E ( k ) + c 2 , 3 Π ( h, k ) + ¯ c 2 , 3 Π ( ¯ h, k ) , I 2 , 1 = g 2 , 1 K ( k ) + g 2 , 2 E ( k ) + g 2 , 3 Π ( h, k ) + ¯ g 2 , 3 Π ( ¯ h, k ) . (37)</formula>
<text><location><page_26><loc_15><loc_75><loc_86><loc_77></location>Further the formulae for coefficients c m,l , g m,l , moduli k and parameters h are gathered for all necessary cases.</text>
<formula><location><page_26><loc_15><loc_71><loc_34><loc_72></location>The case a j ∈ R 1 ( j = 1 , 4)</formula>
<text><location><page_26><loc_15><loc_68><loc_60><loc_69></location>We will further assume, that a j are numbered in ascending order:</text>
<formula><location><page_26><loc_44><loc_66><loc_57><loc_67></location>a 1 < a 2 < a 3 < a 4 .</formula>
<text><location><page_26><loc_15><loc_64><loc_46><loc_65></location>Four different instances should be considered:</text>
<unordered_list>
<list_item><location><page_26><loc_15><loc_62><loc_34><loc_64></location>A. Integration over ( a 1 , a 2 );</list_item>
<list_item><location><page_26><loc_15><loc_61><loc_34><loc_62></location>B. Integration over ( a 2 , a 3 );</list_item>
<list_item><location><page_26><loc_15><loc_60><loc_34><loc_61></location>C. Integration over ( a 3 , a 4 );</list_item>
<list_item><location><page_26><loc_15><loc_58><loc_43><loc_60></location>D. Integration over ( -∞ , a 1 ) ⋃ ( a 4 , + ∞ ).</list_item>
</unordered_list>
<text><location><page_26><loc_15><loc_56><loc_22><loc_57></location>Instance A</text>
<text><location><page_26><loc_15><loc_53><loc_70><loc_54></location>Here λ ∗ = a 1 and λ ∗ = a 2 . We shall also use the following auxiliary parameters:</text>
<formula><location><page_26><loc_21><loc_50><loc_80><loc_52></location>A 0 = 2 √ ( a 4 -a 2 )( a 3 -a 1 ) , C 0 = 1 a 2 -i , α 2 = a 2 -a 1 a 3 -a 1 , α 2 1 = ( a 2 -a 1 )( i -a 3 ) ( a 3 -a 1 )( i -a 2 ) .</formula>
<text><location><page_26><loc_15><loc_48><loc_47><loc_49></location>The value of elliptic integrals' modulus in (37):</text>
<formula><location><page_26><loc_42><loc_44><loc_60><loc_47></location>k = √ ( a 4 -a 3 )( a 2 -a 1 ) ( a 4 -a 2 )( a 3 -a 1 ) ,</formula>
<text><location><page_26><loc_15><loc_42><loc_31><loc_44></location>and parameter h = α 2 1 .</text>
<text><location><page_26><loc_15><loc_40><loc_86><loc_42></location>In order to find coefficients in (37), we considered such linear combinations of I k,r , that are reduced to integral (253.39) from (Byrd and Friedman 1954):</text>
<formula><location><page_26><loc_29><loc_33><loc_86><loc_39></location>I 1 , 1 + iI 1 , 0 = a 2 ∫ a 1 dλ ( λ -i ) √ -( λ -a 1 )( λ -a 2 )( λ -a 3 )( λ -a 4 ) , I 2 , 0 -iI 2 , 1 -1 2 I 1 , 0 = -1 2 a 2 ∫ a 1 dλ ( λ -i ) 2 √ -( λ -a 1 )( λ -a 2 )( λ -a 3 )( λ -a 4 ) . (38)</formula>
<text><location><page_26><loc_15><loc_31><loc_45><loc_32></location>After some simple calculations, one can find:</text>
<formula><location><page_26><loc_29><loc_19><loc_86><loc_31></location>g 1 , 1 = Re C 1 , c 1 , 1 = Im C 1 , C 1 = A 0 C 0 α 2 α 2 1 , g 1 , 3 = A 0 C 0 2 ( 1 -α 2 α 2 1 ) , c 1 , 3 = -ig 1 , 3 , c 2 , 1 = -Re C 2 , g 2 , 1 = Im C 2 , C 2 = A 0 C 2 0 2 α 4 1 [ α 4 + ( α 2 1 -α 2 ) 2 2( α 2 1 -1) ] , c 2 , 2 = -Re C 3 , g 2 , 2 = Im C 3 , C 3 = A 0 C 2 0 ( α 2 1 -α 2 ) 2 4 α 2 1 ( α 2 1 -1)( k 2 -α 2 1 ) , c 2 , 3 = -A 0 C 2 0 ( α 2 1 -α 2 ) 4 α 4 1 [ 2 α 2 + (2 α 2 1 k 2 +2 α 2 1 -α 4 1 -3 k 2 )( α 2 1 -α 2 ) 2( α 2 1 -1)( k 2 -α 2 1 ) ] , g 2 , 3 = ic 2 , 3 . (39)</formula>
<text><location><page_26><loc_15><loc_17><loc_58><loc_18></location>Using (253.00) from Byrd and Friedman (1954) we also obtain:</text>
<formula><location><page_26><loc_47><loc_15><loc_86><loc_16></location>c 0 , 0 = A 0 . (40)</formula>
<text><location><page_27><loc_15><loc_17><loc_19><loc_18></location>where</text>
<section_header_level_1><location><page_27><loc_15><loc_87><loc_22><loc_88></location>Instance B</section_header_level_1>
<text><location><page_27><loc_15><loc_83><loc_86><loc_85></location>For integrals over the interval ( a 2 , a 3 ) the only difference is the parameters α 2 , α 2 1 and modulus k . Thus in (39) the values</text>
<formula><location><page_27><loc_27><loc_79><loc_74><loc_81></location>α 2 = a 3 -a 2 a 3 -a 1 , α 2 1 = ( a 3 -a 2 )( i -a 1 ) ( a 3 -a 1 )( i -a 2 ) , k = √ ( a 3 -a 2 )( a 4 -a 1 ) ( a 4 -a 2 )( a 3 -a 1 )</formula>
<text><location><page_27><loc_15><loc_76><loc_26><loc_77></location>should be used.</text>
<text><location><page_27><loc_15><loc_73><loc_22><loc_74></location>Instance C</text>
<text><location><page_27><loc_15><loc_70><loc_46><loc_71></location>For integrals over the interval ( a 3 , a 4 ) in (39):</text>
<formula><location><page_27><loc_32><loc_66><loc_70><loc_69></location>C 0 = 1 a 3 -i , α 2 = a 4 -a 3 a 4 -a 2 , α 2 1 = ( a 4 -a 3 )( i -a 2 ) ( a 4 -a 2 )( i -a 3 ) .</formula>
<text><location><page_27><loc_15><loc_64><loc_63><loc_65></location>The modulus k and parameter A 0 are the same as for interval ( a 1 , a 2 ).</text>
<text><location><page_27><loc_15><loc_56><loc_86><loc_64></location>Note: The equality of k and c 0 , 0 in instances A and C means, that when the potential (2) has two minima, the periods of librations T about them on the same energy level are equal. This also holds true for local minima corresponding to instances B and D , as the substitution λ = tan[( ϕ -˜ ϕ ) / 2] always allows to get rid of semi-infinite intervals of integration. Moreover, the averaged values of cos in (18) are also the same for librations around two local minima. The averaged values of sin in (18) for librations about two local minima differ by π , and values of adiabatic invariant (22) differ by y/ √ 2.</text>
<section_header_level_1><location><page_27><loc_15><loc_53><loc_22><loc_53></location>Instance D</section_header_level_1>
<text><location><page_27><loc_15><loc_50><loc_70><loc_51></location>For integrals over two semi-infinite intervals the values of parameters in (39) are</text>
<formula><location><page_27><loc_32><loc_46><loc_70><loc_48></location>C 0 = 1 a 4 -i , α 2 = a 4 -a 1 a 3 -a 1 , α 2 1 = ( a 4 -a 1 )( i -a 3 ) ( a 3 -a 1 )( i -a 4 ) .</formula>
<text><location><page_27><loc_15><loc_44><loc_79><loc_45></location>The modulus k is the same as for integration over ( a 2 , a 3 ), and A 0 is the same as for ( a 1 , a 2 ).</text>
<formula><location><page_27><loc_15><loc_39><loc_52><loc_41></location>The case a 1 , a 2 ∈ R 1 ( a 1 < a 2 ), a 3 , a 4 ∈ C 1 ( a 3 = ¯ a 4 )</formula>
<text><location><page_27><loc_15><loc_37><loc_38><loc_38></location>Here there are only two instances:</text>
<formula><location><page_27><loc_42><loc_33><loc_43><loc_34></location>).</formula>
<formula><location><page_27><loc_15><loc_33><loc_42><loc_35></location>A. Integration over ( a 1 , a 2 ); B. Integration over ( -∞ , a 1 ) ⋃ ( a 2 , + ∞</formula>
<text><location><page_27><loc_15><loc_30><loc_22><loc_31></location>Instance A</text>
<text><location><page_27><loc_15><loc_27><loc_73><loc_28></location>Here the following auxiliary parameters in expressions for c m,l and g m,l will be used:</text>
<formula><location><page_27><loc_33><loc_23><loc_68><loc_26></location>α = ( a 1 A 2 -a 2 A 1 ) -( A 2 -A 1 ) i ( a 1 A 2 + a 2 A 1 ) -( A 2 + A 1 ) i , α 1 = A 2 -A 1 A 2 + A 1 ,</formula>
<formula><location><page_27><loc_39><loc_19><loc_63><loc_22></location>C 0 = A 1 + A 2 ( A 2 a 1 -A 1 a 2 ) -( A 2 -A 1 ) i ,</formula>
<formula><location><page_27><loc_27><loc_14><loc_74><loc_18></location>A 1 = √ ( a 1 -a 0 ) 2 + b 2 0 , A 2 = √ ( a 2 -a 0 ) 2 + b 2 0 , A 0 = 1 / √ A 1 A 2 , a 0 = Re a 3 = Re a 4 , b 0 = Im a 3 = -Im a 4 .</formula>
<text><location><page_28><loc_15><loc_87><loc_81><loc_88></location>Using (38) and formulae (259.04), (341.01)-(341.04) 2 from (Byrd and Friedman 1954), we obtain</text>
<formula><location><page_28><loc_28><loc_77><loc_86><loc_86></location>c 0 , 0 = 2 A 0 , g 1 , 1 = Re C 1 , c 1 , 1 = Im C 1 , C 1 = 2 A 0 C 0 α 1 , c 1 , 3 = -i ( α -α 1 ) (1 -α 2 ) A 0 C 0 , g 1 , 3 = ic 1 , 3 , c 2 , 1 = -Re C 2 , g 2 , 1 = Im C 2 , C 2 = A 0 C 2 0 [ α 2 1 + ( α -α 1 ) 2 α 2 -1 ] , c 2 , 2 = -Re C 3 , g 2 , 2 = Im C 3 , C 3 = A 0 C 2 0 α 2 ( α -α 1 ) 2 (1 -α 2 )( k 2 + α 2 k ' 2 ) , c 2 , 3 = A 0 C 2 0 2 ( α -α 1 ) ( α 2 -1) [ 2 α 1 + ( α -α 1 ) [ α 2 (2 k -1) -2 k 2 ] ( α 2 -1)( k 2 + α 2 k ' 2 ) ] , g 2 , 3 = ic 2 , 3 . (41)</formula>
<text><location><page_28><loc_15><loc_74><loc_62><loc_75></location>Modulus and parameter of elliptic integrals are calculated as follows:</text>
<formula><location><page_28><loc_35><loc_70><loc_67><loc_74></location>k = √ ( a 2 -a 1 ) 2 -( A 2 -A 1 ) 2 4 A 1 A 2 , h = α 2 α 2 -1 .</formula>
<text><location><page_28><loc_15><loc_68><loc_22><loc_69></location>Instance B</text>
<text><location><page_28><loc_15><loc_65><loc_54><loc_66></location>Expressions (41) stay the same. Auxiliary parameters are</text>
<formula><location><page_28><loc_33><loc_62><loc_68><loc_64></location>α = ( a 2 A 1 + a 1 A 2 ) -( A 2 + A 1 ) i ( a 2 A 1 -a 1 A 2 ) + ( A 2 -A 1 ) i , α 1 = A 2 + A 1 A 1 -A 2 ,</formula>
<formula><location><page_28><loc_39><loc_59><loc_63><loc_61></location>C 0 = A 1 -A 2 ( A 2 a 1 + A 1 a 2 ) -( A 2 + A 1 ) i .</formula>
<text><location><page_28><loc_15><loc_57><loc_50><loc_58></location>The modulus k of elliptic integrals is also different:</text>
<formula><location><page_28><loc_40><loc_53><loc_62><loc_56></location>k = √ ( A 2 + A 1 ) 2 -( a 2 -a 1 ) 2 4 A 1 A 2 .</formula>
<text><location><page_28><loc_15><loc_49><loc_34><loc_51></location>The case a j ∈ C 1 ( j = 1 , 4)</text>
<text><location><page_28><loc_15><loc_47><loc_64><loc_48></location>Let us denote real parts of roots a j as p 1 , 2 , and imaginary parts as b 1 , 2 :</text>
<formula><location><page_28><loc_40><loc_44><loc_62><loc_46></location>a 1 , 2 = p 1 ± b 1 , a 3 , 4 = p 2 ± b 2 .</formula>
<text><location><page_28><loc_15><loc_41><loc_86><loc_44></location>Without loss of generality let us assume, that p 1 < p 2 , b 1 > 0, and b 2 > 0 (if p 1 = p 2 , then also b 1 < b 2 ). In expressions for c m,l and g m,l the following auxiliary parameters will be used</text>
<formula><location><page_28><loc_21><loc_34><loc_80><loc_40></location>A 1 = √ ( p 2 -p 1 ) 2 +( b 2 -b 1 ) 2 , A 2 = √ ( p 2 -p 1 ) 2 +( b 2 + b 1 ) 2 , A 0 = 2 / ( A 1 + A 2 ) , g 1 = √ 4 b 2 1 -( A 2 -A 1 ) 2 ( A 2 + A 1 ) 2 -4 b 2 1 , α = b 1 + g 1 ( p 1 -i ) p 1 -b 1 g 1 -i , C 0 = 1 b 1 + g 1 ( p 1 -i ) .</formula>
<text><location><page_28><loc_15><loc_33><loc_37><loc_34></location>Similar to previous case we find:</text>
<formula><location><page_28><loc_26><loc_22><loc_75><loc_32></location>c 0 , 0 = 2 A 0 , c 1 , 1 = Im C 1 , g 1 , 1 = Re C 1 , C 1 = 2 α (1+ g 1 α ) 1+ α 2 A 0 C 0 , g 1 , 3 = α 2 ( α -g 1 ) A 0 C 0 1+ α 2 , c 1 , 3 = -ig 1 , 3 , c 2 , 1 = -Re C 2 , g 2 , 1 = Im C 2 , C 2 = A 0 C 2 0 [ g 2 1 + 2 g 1 ( α -g 1 ) 1+ α 2 + ( α -g 1 ) 2 (1+ α 2 )( α 2 + k ' 2 ) ( 2 k ' 2 +2 α 2 -α 2 k 2 1+ α 2 -k ' 2 )] , c 2 , 2 = Re C 3 , g 2 , 2 = -Im C 3 , C 3 = A 0 C 2 0 ( α -g 1 ) 2 α 2 ( α 2 +1)( α 2 + k ' 2 ) , c 2 , 3 = -A 0 C 2 0 2 ( α -g 1 ) α 2 ( α 2 +1) [ 2 g 1 + ( α -g 1 )(2 k ' 2 +2 α 2 -α 2 k 2 ) ( α 2 +1)( α 2 + k ' 2 ) ] , g 2 , 3 = ic 2 , 3 .</formula>
<text><location><page_28><loc_15><loc_21><loc_45><loc_21></location>Modulus and parameter of elliptic integrals:</text>
<formula><location><page_28><loc_41><loc_17><loc_60><loc_20></location>k = 2 √ A 1 A 2 A 1 + A 2 , h = α 2 +1 .</formula>
<text><location><page_29><loc_15><loc_84><loc_86><loc_88></location>Acknowledgements The work was supported by the Presidium of the Russian Academy of Sciences (Program 28 'Space: investigations of the fundamental processes and their interrelationships'). We are grateful to A.I.Neishtadt, A.Correia, A.Morbidelli and J.Wisdom for useful discussions. We would also like to thank D.A.Pritykin for proofreading the manuscript.</text>
<section_header_level_1><location><page_29><loc_15><loc_78><loc_23><loc_79></location>References</section_header_level_1>
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2016PhRvD..94d4042G | https://arxiv.org/pdf/1603.09655.pdf | <document>
<section_header_level_1><location><page_1><loc_26><loc_90><loc_77><loc_91></location>From Smooth Curves to Universal Metrics</section_header_level_1>
<text><location><page_1><loc_23><loc_87><loc_79><loc_88></location>Metin Gürses, 1, ∗ Tahsin Çağrı Şişman, 2, † and Bayram Tekin 3, ‡</text>
<text><location><page_1><loc_31><loc_85><loc_72><loc_86></location>1 Department of Mathematics, Faculty of Sciences</text>
<text><location><page_1><loc_34><loc_83><loc_69><loc_84></location>Bilkent University, 06800 Ankara, Turkey</text>
<text><location><page_1><loc_23><loc_74><loc_80><loc_83></location>2 Department of Astronautical Engineering, University of Turkish Aeronautical Association, 06790 Ankara, Turkey 3 Department of Physics, Middle East Technical University, 06800 Ankara, Turkey (Dated: November 2, 2021)</text>
<text><location><page_1><loc_18><loc_58><loc_85><loc_73></location>Aspecial class of metrics, called universal metrics, solve all gravity theories defined by covariant field equations purely based on the metric tensor. Since we currently lack the knowledge of what the full of quantum-corrected field equations of gravity are at a given microscopic length scale, these metrics are particularly important in understanding quantum fields in curved backgrounds in a consistent way. But, finding explicit universal metrics has been a hard problem as there does not seem to be a procedure for it. In this work, we overcome this difficulty and give a construction of universal metrics of d -dimensional spacetime from curves constrained to live in a ( d -1)-dimensional Minkowski spacetime or a Euclidean space.</text>
<section_header_level_1><location><page_1><loc_41><loc_52><loc_62><loc_53></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_12><loc_20><loc_90><loc_50></location>There is a non-ignorable problem in high energy gravity: we do not know the full field equations and the microscopic degrees of freedom responsible for gravity. What we know is that Einstein's theory is an effective one which will be modified with powers of curvature and its derivatives, (most probably) in a diffeomorphism invariant way, as long as the Riemannian spacetime model remains intact as a valid description of gravity. At this stage, there is no compelling reason to suspect that such a description ceases to make sense well below the Planck scale. One might be deterred to say anything about high energy gravity, in the absence of what the theory is, but the situation is not that bleak as there are certain types of spacetimes that solve any metric-based equations. This approach to high energy gravity is a remarkable one which started long ago [1, 2] not exactly in this language, but developed [35] over the years and culminated in a rather nice summary [6], where the notion of universal metrics with further refinements was made, see also the more recent discussion in [7, 8]. Note that our definition of a universal metric is somewhat different from the one defined in the previous literature: namely, for us, a universal metric is a metric that solves all gravity theories defined by covariant field equations purely based on the metric tensor. (We shall not go into that distinction here and also not distinguish 'critical' versus 'non-critical' metrics, where the former extremize an action while the latter solve a covariantly conserved field equation not necessarily coming from an action.)</text>
<text><location><page_2><loc_12><loc_73><loc_90><loc_91></location>The interest in the universal metrics is actually two-fold: these are valuable on their own as they are solutions to putative low energy quantum gravity at any order in the curvature. But, as importantly, when one does quantum field theory at high energies, working about these solutions will provide a better, self-consistent, picture as gravity also plays a role. Of course, such metrics are hard to find, as we are not given what the equations are. Therefore, one will be hard-pressed to find, in the literature, examples of these metrics save the examples given in the papers noted above. Perhaps, more important is the fact that there is really no well-defined procedure of finding these solutions except trial and error: namely, given a rather symmetric metric, one can compute all possible curvature invariants and hope that they vanish or at best they are constants and all conserved second rank tensors built from the Riemann tensor and its derivatives are proportional to the metric and the Ricci tensor.</text>
<text><location><page_2><loc_12><loc_59><loc_90><loc_73></location>In this work, we shall show that there is a proper way to find universal metrics in d -dimensions using curves in one less dimensions. This, not so obvious, solution-generation that we shall lay out here, came as a serendipitous surprise in our rather intense excursion to the universal metric territory in the following works: we have shown that the plane wave and spherical wave metrics, built on the anti-de Sitter seeds, solve generic gravity theories [9-11], modulo the assumption that the Lagrangian is solely composed of the curvature, covariant derivatives of the curvature and the metric tensor in a Lorentz-invariant way (or the field equation is a covariantly conserved two-tensor built from the metric). All of these solutions are in the form of the Kerr-Schild-Kundt metrics</text>
<formula><location><page_2><loc_42><loc_56><loc_90><loc_57></location>g µν = ¯ g µν +2 V λ µ λ ν , (1)</formula>
<text><location><page_2><loc_12><loc_50><loc_90><loc_54></location>where the seed ¯ g µν metrics are maximally symmetric, whose explicit forms will be dictated by the curves that will generate the solutions. The other ingredients of (1) will be discussed below. We first discuss the curves.</text>
<section_header_level_1><location><page_2><loc_33><loc_45><loc_70><loc_46></location>II. CURVES IN FLAT SPACETIMES</section_header_level_1>
<text><location><page_2><loc_12><loc_38><loc_90><loc_43></location>Let z µ ( τ ) define a smooth curve C in R d , with the metric η µν . Here τ is the parameter of the curve. From an arbitrary point P ( x µ ) not on the curve, there are two null lines intersecting the curve at two points as shown in the Figure. 1</text>
<text><location><page_2><loc_12><loc_33><loc_90><loc_37></location>These intersection points are called the retarded ( τ 0 ) and advanced ( τ 1 ) times [12, 13]. Let Ω be the distance between the points P ( x µ ) and Q ( z µ ), then since the spacetime is flat, it is simply given as</text>
<formula><location><page_2><loc_36><loc_31><loc_90><loc_33></location>Ω 2 = η µν ( x µ -z µ ( τ ) ) ( x ν -z ν ( τ ) ) , (2)</formula>
<text><location><page_2><loc_12><loc_27><loc_90><loc_30></location>which vanishes for the retarded and advanced times. There is a natural null vector ∂ µ τ 0 that one can obtain by differentiating Ω ( τ 0 ) = 0 with respect to x µ as</text>
<formula><location><page_2><loc_40><loc_22><loc_90><loc_26></location>/lscript µ ≡ ∂ µ τ 0 = x µ -z µ ( τ 0 ) R , (3)</formula>
<text><location><page_2><loc_12><loc_18><loc_90><loc_22></location>where R is the retarded distance: R ≡ ˙ z α ( τ 0 ) ( x α -z α ( τ 0 ) ) with ˙ z α ( τ 0 ) ≡ ∂ τ 0 z α ( τ 0 ). We have chosen to work with the retarded time τ 0 , but we could equally have worked with the</text>
<figure>
<location><page_3><loc_36><loc_70><loc_67><loc_91></location>
<caption>Figure 1: Two null lines stretching from an arbitrary point P ( x ) outside the curve meet the curve C at the points corresponding to the retarded and advanced times, that is τ 0 and τ 1 , respectively. Q ( z ( τ )) represents an arbitrary point on the curve.</caption>
</figure>
<text><location><page_3><loc_12><loc_56><loc_90><loc_60></location>advanced time τ 1 and the ensuing results would not change. Moreover, in what follows, for notational simplicity, we omit the subscript 0 from τ 0 and use τ instead. Taking one more partial derivative of the null vector, one has</text>
<formula><location><page_3><loc_31><loc_51><loc_90><loc_54></location>∂ ν /lscript µ = 1 R ( η µν -˙ z µ /lscript ν -˙ z ν /lscript µ -( A -/epsilon1 ) /lscript µ /lscript ν ) , (4)</formula>
<text><location><page_3><loc_12><loc_47><loc_90><loc_50></location>with A ≡ z µ ( x µ -z µ ) and /epsilon1 ≡ ˙ z µ ˙ z µ , ( /epsilon1 = ± 1 , 0), and the argument of z µ and its derivatives is always the retarded time.</text>
<section_header_level_1><location><page_3><loc_37><loc_42><loc_66><loc_44></location>III. UNIVERSAL METRICS</section_header_level_1>
<text><location><page_3><loc_12><loc_34><loc_90><loc_40></location>The above has been a generic discussion of the curves in flat backgrounds. Now comes the remarkable part of employing these curves to generate solutions of generic gravity theories. Let us assume that the spacetime metric is of the form (1). Then, one can show that the following relations hold for the metrics belonging to the Kerr-Schild-Kundt class [9, 10]</text>
<formula><location><page_3><loc_25><loc_31><loc_90><loc_33></location>λ µ λ µ = 0 , ∇ µ λ ν ≡ ξ ( µ λ ν ) , ξ µ λ µ = 0 , λ µ ∂ µ V = 0 . (5)</formula>
<text><location><page_3><loc_12><loc_18><loc_90><loc_29></location>It is important to note that a new vector ξ µ appears, besides the two defining ingredients of the metric, the profile function V and the vector λ µ . The first three relations describe Kerr-Schild metrics belonging to the Kundt class and the last relation is an assumption which puts a further restriction on this class of metrics. However, this last relation is crucial in proving the universality of KSK metrics [11]. The covariant derivative of λ µ satisfies ∇ µ λ ν = ¯ ∇ µ λ ν where ¯ ∇ µ is the covariant derivative of the seed metric. Then, for the AdS metric in the conformally flat coordinates</text>
<formula><location><page_3><loc_35><loc_13><loc_90><loc_16></location>d ¯ s 2 = /lscript 2 z 2 ( -dt 2 + d -2 ∑ m =1 ( dx m ) 2 + dz 2 ) , (6)</formula>
<text><location><page_4><loc_12><loc_90><loc_35><loc_91></location>∇ µ λ ν can be calculated as</text>
<formula><location><page_4><loc_33><loc_85><loc_90><loc_89></location>∇ µ λ ν = ∂ µ λ ν -1 z η µν λ z + 1 z ( λ µ δ z ν + λ ν δ z µ ) . (7)</formula>
<text><location><page_4><loc_12><loc_83><loc_79><loc_84></location>On the other hand, for the dS seed metric in the conformally flat coordinates</text>
<formula><location><page_4><loc_38><loc_78><loc_90><loc_82></location>d ¯ s 2 = /lscript 2 t 2 ( -dt 2 + d -1 ∑ m =1 ( dx m ) 2 ) , (8)</formula>
<text><location><page_4><loc_12><loc_75><loc_19><loc_77></location>one has</text>
<formula><location><page_4><loc_33><loc_72><loc_90><loc_76></location>∇ µ λ ν = ∂ µ λ ν -1 t η µν λ t + 1 t ( λ µ δ t ν + λ ν δ t µ ) . (9)</formula>
<text><location><page_4><loc_12><loc_69><loc_90><loc_72></location>By using these results and the defining expression ∇ µ λ ν = ξ ( µ λ ν ) from (5), the partial derivative of λ µ can be written, collectively for the AdS and dS, as</text>
<formula><location><page_4><loc_32><loc_64><loc_90><loc_67></location>∂ ν λ µ = aη µν + λ µ ( 1 2 ξ ν -ζ ν ) + λ ν ( 1 2 ξ µ -ζ µ ) (10)</formula>
<text><location><page_4><loc_78><loc_60><loc_90><loc_61></location>, can be found,</text>
<text><location><page_4><loc_12><loc_58><loc_84><loc_63></location>where a = λ z z , ζ ν = 1 z δ z ν for the AdS seed [9] and a = -λ t t , ζ ν = 1 t δ t ν for the dS seed. The traceless-Ricci tensor, S µν ≡ R µν -R d g µν , and the Weyl tensor, C µανβ after some tedious computation, as [10]</text>
<formula><location><page_4><loc_36><loc_55><loc_90><loc_57></location>S µν = ρλ µ λ ν , C µανβ = 4 λ [ µ Ω α ][ β λ ν ] , (11)</formula>
<text><location><page_4><loc_12><loc_51><loc_90><loc_54></location>where the square brackets denote anti-symmetrization with a 1 / 2 factor and the scalar function ρ is given as</text>
<formula><location><page_4><loc_29><loc_46><loc_90><loc_49></location>ρ = -( glyph[square] +2 ξ µ ∂ µ + 1 2 ξ µ ξ µ -2 ( d -2) /lscript 2 ) V ≡ -Q V. (12)</formula>
<text><location><page_4><loc_12><loc_40><loc_90><loc_45></location>The second equality defines the operator Q which will play a role in the field equations of the generic theory below. The symmetric tensor Ω αβ , that appears in the Weyl tensor, can be compactly written as</text>
<formula><location><page_4><loc_23><loc_35><loc_90><loc_39></location>Ω αβ ≡ -[ ∇ α ∂ β + ξ ( α ∂ β ) + 1 2 ξ α ξ β -1 d -2 g αβ ( Q + 2 ( d -2) /lscript 2 )] V. (13)</formula>
<text><location><page_4><loc_12><loc_32><loc_79><loc_34></location>For the seed metric, there are three possible choices whose explicit forms are:</text>
<formula><location><page_4><loc_29><loc_27><loc_90><loc_31></location>d ¯ s 2 = /lscript 2 cos 2 θ -du 2 +2 dudr r 2 + dθ 2 +sin 2 θ dω 2 , (14)</formula>
<formula><location><page_4><loc_29><loc_23><loc_90><loc_26></location>d ¯ s 2 = /lscript 2 z 2 ( du 2 +2 dudr + dx 2 + · · · + dz 2 ) , (15)</formula>
<formula><location><page_4><loc_29><loc_19><loc_90><loc_23></location>d ¯ s 2 = /lscript 2 cosh 2 θ du 2 +2 dudr r 2 + dθ 2 +sinh 2 θ dω 2 , (16)</formula>
<text><location><page_4><loc_12><loc_12><loc_90><loc_17></location>where /lscript is related to the cosmological constant and dω 2 is the metric of the ( d -3) unitsphere. The first and the second metrics are AdS metrics, while the third one is the dS metric.</text>
<text><location><page_5><loc_12><loc_83><loc_90><loc_91></location>Recently [10], we have shown that the AdS-plane wave and the pp -wave metrics in the Kerr-Schild form, and more generally all Kerr-Schild-Kundt metrics are universal. The seed is the flat Minkowski metric for the pp -waves, it is the AdS metric for the AdS-plane and AdS-spherical waves, and it is the dS metric for the dS-hyperbolic wave. Referring to [10, 11] for the full proof, let us briefly recapitulate how this works.</text>
<text><location><page_5><loc_12><loc_75><loc_90><loc_83></location>Let the most general gravity theory be a (2 N +2)-derivative theory. As examples, for Einstein's gravity (and Einstein-Gauss-Bonnet gravity) N = 0, for quadratic and f (Riemann) theories N = 1, and for higher order theories N ≥ 2. We have shown that the equations of the most general (2 N +2)-derivative gravity theory reduce, when evaluated for these metrics, to a rather compact form</text>
<formula><location><page_5><loc_40><loc_70><loc_90><loc_74></location>eg µν + N ∑ n =0 a n glyph[square] n S µν = 0 , (17)</formula>
<text><location><page_5><loc_12><loc_62><loc_90><loc_69></location>where e and a n s are constants which are functions of the parameters of the theory. Here, the constant e determines the possible effective cosmological constants in terms of the parameters of the theory. After some algebraic manipulations, the traceless part of (17) reduces to a scalar equation of the metric function V :</text>
<formula><location><page_5><loc_41><loc_57><loc_90><loc_61></location>N ∏ n =1 ( Qm 2 n ) Q V = 0 . (18)</formula>
<text><location><page_5><loc_12><loc_53><loc_90><loc_56></location>The generic solution is V = V E + ∑ N n =1 V n where the Einsteinian part ( V E ) and the other (massive) parts satisfy the following equations, respectively,</text>
<formula><location><page_5><loc_36><loc_49><loc_90><loc_51></location>Q V E = 0 , ( Qm 2 n ) V n = 0 , (19)</formula>
<text><location><page_5><loc_12><loc_42><loc_90><loc_48></location>provided that all m n 's are different and none is zero. If any two or more m n 's coincide and or equal to zero, then the second equation in (19) changes in the following way: let r be the number (multiplicity) of m n 's that are equal to m r , then the corresponding V r satisfies an irreducibly higher derivative equation</text>
<formula><location><page_5><loc_44><loc_38><loc_90><loc_40></location>( Qm 2 r ) r V r = 0 , (20)</formula>
<text><location><page_5><loc_12><loc_34><loc_90><loc_37></location>with new branches, so called log-solutions, appear. In that case, the general solution becomes V = V E + V r + ∑ N -r n =0 V n and V r contains log r -1 terms.</text>
<text><location><page_5><loc_12><loc_23><loc_90><loc_34></location>Let us now get back to the issue of constructing these solutions from the curves in flat space discussed in the previous section. The structural similarity of the partial derivative of /lscript µ in (4) and the partial derivative of λ µ in (10) suggests the following procedure of generating Kerr-Schild-Kundt class metrics: First, one takes the vectors /lscript µ and λ µ in (4) and (10) to be equal and derives the corresponding vector ξ µ ; and secondly, sets λ µ ξ µ = 0 to satisfy the third condition in (5) and to obtain the constraint on z µ ( τ ). The second step constrains z µ ( τ ) curves to live in one less dimension.</text>
<text><location><page_5><loc_12><loc_19><loc_90><loc_22></location>Let us execute this procedure: when the seed metric is AdS as given in (6), equating (4) and (10), one finds</text>
<formula><location><page_5><loc_35><loc_16><loc_90><loc_19></location>ξ µ = -2 R ( ˙ z µ + 1 2 ( A -/epsilon1 ) λ µ ) + 2 z δ z µ . (21)</formula>
<text><location><page_5><loc_12><loc_12><loc_90><loc_16></location>To satisfy λ µ ξ µ = 0, we must have λ z = z R and z z = 0. Hence, all these curves live in a ( d -1)-dimensional Minkowski spacetime. In this case, we have only timelike and</text>
<text><location><page_6><loc_12><loc_85><loc_90><loc_91></location>null curves. We can have spacelike curves, but the metrics generated by these curves are equivalent to the metrics generated from timelike curves via diffeomorphisms and possibly via complex transformations. On the other hand, when the seed metric is the dS metric as given in (8), we find</text>
<formula><location><page_6><loc_35><loc_82><loc_90><loc_85></location>ξ µ = -2 R ( ˙ z µ + 1 2 ( A -/epsilon1 ) λ µ ) + 2 t δ t µ . (22)</formula>
<text><location><page_6><loc_12><loc_76><loc_90><loc_81></location>To satisfy λ µ ξ µ = 0, we must have λ t = -t R and z t = 0. Hence, the curve C in this case lives in a ( d -1)-dimensional Euclidean space where we can have only spacelike curves. Let us turn to some explicit examples.</text>
<section_header_level_1><location><page_6><loc_37><loc_72><loc_65><loc_73></location>IV. EXPLICIT EXAMPLES</section_header_level_1>
<text><location><page_6><loc_12><loc_65><loc_90><loc_70></location>We have infinitely many metrics characterized by the curves either in the ( d -1)dimensional Minkowski space or the ( d -1)-dimensional Euclidean space. In the examples below, for the sake of simplicity, we set d = 4.</text>
<text><location><page_6><loc_13><loc_63><loc_90><loc_65></location>Example 1 : (Timelike case). Let z µ = τ δ 0 µ , then τ = t ± r . Choose τ = t -r , then R = -r</text>
<text><location><page_6><loc_12><loc_61><loc_26><loc_62></location>and so one finds</text>
<formula><location><page_6><loc_32><loc_58><loc_90><loc_61></location>λ µ = ( 1 , /vectorx r ) , ξ µ = 2 r ( δ 0 µ -1 2 λ µ ) + 2 z δ z µ . (23)</formula>
<text><location><page_6><loc_12><loc_54><loc_90><loc_57></location>This gives the AdS-spherical wave solution together with the profile function V solving the corresponding equations. For the explicit form of V , see [9].</text>
<text><location><page_6><loc_13><loc_53><loc_90><loc_54></location>Example 2 : (Null case). If the curve is null, then /epsilon1 = 0. Let z µ = τ n µ where η µν n µ n ν = 0,</text>
<text><location><page_6><loc_12><loc_50><loc_27><loc_51></location>then we arrive at</text>
<formula><location><page_6><loc_17><loc_46><loc_90><loc_49></location>τ = x 2 2 n µ x µ , x 2 = η µν x µ x ν , λ µ = x µ -τ n µ R , R = n µ x µ , A = 0 , λ µ n µ = 1 . (24)</formula>
<text><location><page_6><loc_12><loc_41><loc_90><loc_44></location>Furthermore, one finds ξ µ = -2 R n µ + 2 z δ µ z . Choosing n µ = (1 , 1 , 0 , 0) and performing a couple of coordinate transformations, one obtains the AdS-plane wave metric</text>
<formula><location><page_6><loc_31><loc_37><loc_90><loc_40></location>ds 2 = /lscript 2 ρ 2 ( 2 dτdv + dσ 2 + dρ 2 ) +2 V ( τ, σ, ρ ) dτ 2 . (25)</formula>
<text><location><page_6><loc_13><loc_34><loc_90><loc_35></location>Example 3 : (Spacelike Case). When the curve is spacelike, the AdS seed is not allowed.</text>
<text><location><page_6><loc_12><loc_28><loc_90><loc_33></location>However, de Sitter seed is possible, i.e. , ¯ g µν = ( /lscript 2 /t 2 ) η µν . Then, for this case, the vector ξ µ takes the form (22). Let z µ = τδ µ x , then we find τ = x ± √ t 2 -y 2 -z 2 and R = x -τ = ∓ √ t 2 -y 2 -z 2 . Letting r = √ t 2 -y 2 -z 2 and choosing the + sign, we get</text>
<formula><location><page_6><loc_42><loc_24><loc_90><loc_27></location>λ µ = ( t r , 1 , -y r , -z r ) . (26)</formula>
<text><location><page_6><loc_12><loc_21><loc_85><loc_23></location>Letting t = r cosh( θ ), y = r sinh θ cos φ , z = r sinh θ sin φ , the metric takes the form</text>
<formula><location><page_6><loc_20><loc_17><loc_90><loc_20></location>ds 2 = /lscript 2 r 2 cosh 2 θ ( du 2 +2 dudr + r 2 ( dθ 2 +sinh 2 θ dφ 2 ) ) +2 V ( u, θ, φ ) du 2 , (27)</formula>
<text><location><page_6><loc_12><loc_12><loc_90><loc_16></location>for λ µ = δ 0 µ and ξ µ = 1 r δ 0 µ +2tanh( θ ) δ 2 µ . The above metric is the dS-hyperbolic wave metric given in (16) which was noticed very recently [14].</text>
<section_header_level_1><location><page_7><loc_14><loc_88><loc_89><loc_91></location>V. KSK METRICS IN ROBINSON-TRAUTMAN COORDINATES IN FOUR DIMENSIONS</section_header_level_1>
<text><location><page_7><loc_12><loc_75><loc_90><loc_86></location>The metric form (1) in the coordinates ( t, x, y, z ) gives a very complicated expression for the operator Q . With this form, it is highly difficult to solve the equations in (19) for the metric function V . In addition, we must also satisfy λ µ ∂ µ V = 0. For this purpose, one should search for new coordinates where both the metric and the operator Q take simpler forms. Two of the new coordinates are (the natural coordinates) τ and R . They are defined through Ω ( τ ) = 0 and R = ˙ z α ( τ ) ( x α -z α ( τ ) ) . The coordinate transformation can be given as [15]</text>
<formula><location><page_7><loc_32><loc_73><loc_90><loc_74></location>x µ = Rλ µ ( τ, θ, φ ) + z µ ( τ ) , µ = 0 , 1 , 2 , 3 . (28)</formula>
<text><location><page_7><loc_12><loc_69><loc_90><loc_72></location>Here, the null vector λ µ does not depend on the new coordinate R [15]. In these new coordinates, we have</text>
<formula><location><page_7><loc_38><loc_65><loc_90><loc_69></location>∂ R V = ∂x µ ∂R ∂ µ V = λ µ ∂ µ V = 0 . (29)</formula>
<text><location><page_7><loc_12><loc_62><loc_90><loc_65></location>Hence, the metric function is independent of the new coordinate R . Furthermore, in the new coordinates, λ µ dx µ = dτ . Hence, we have</text>
<formula><location><page_7><loc_39><loc_59><loc_90><loc_60></location>ds 2 = d ¯ s 2 +2 V ( τ, θ, φ ) dτ 2 , (30)</formula>
<text><location><page_7><loc_12><loc_54><loc_90><loc_57></location>where d ¯ s 2 is the background line element. The new form of the metric in the new coordinates is called the Robinson-Trautman (RT) metrics.</text>
<text><location><page_7><loc_12><loc_48><loc_90><loc_54></location>In four dimensions, to introduce the KSK metrics in the coordinates of RT metrics, we first need the parametrizations of the two-dimensional unit sphere and the two-dimensional unit hyperboloid. A parametrization of the two-dimensional unit sphere, ( X 1 ) 2 +( X 2 ) 2 +( X 3 ) 2 = 1, is given by the spherical coordinates</text>
<formula><location><page_7><loc_28><loc_44><loc_90><loc_46></location>X 1 = sin θ sin φ, X 2 = sin θ cos φ, X 3 = cos θ. (31)</formula>
<text><location><page_7><loc_12><loc_40><loc_90><loc_43></location>Similarly, the parametrization of a two-dimensional hyperboloid, -( Y 0 ) 2 +( Y 1 ) 2 +( Y 2 ) 2 = -1, is given by</text>
<formula><location><page_7><loc_27><loc_37><loc_90><loc_38></location>Y 1 = sinh θ sin φ, Y 2 = sinh θ cos φ, Y 0 = cosh θ. (32)</formula>
<section_header_level_1><location><page_7><loc_39><loc_32><loc_63><loc_33></location>A. The AdS Background</section_header_level_1>
<text><location><page_7><loc_15><loc_29><loc_88><loc_30></location>Following [15], one can write the KSK metrics (1) for AdS seed in the following form:</text>
<formula><location><page_7><loc_21><loc_24><loc_90><loc_27></location>ds 2 = 1 f 2 ( Hdτ 2 +2 dτdr + r 2 P 2 ( dθ 2 +sin 2 θdφ 2 ) ) +2 V ( τ, θ, φ ) dτ 2 , (33)</formula>
<text><location><page_7><loc_12><loc_21><loc_38><loc_22></location>where the metric functions are</text>
<formula><location><page_7><loc_30><loc_17><loc_90><loc_20></location>H = /epsilon1 -2 r∂ τ log P, f = r /lscriptP cos θ, (34)</formula>
<formula><location><page_7><loc_31><loc_15><loc_90><loc_17></location>P = -˙ z 0 ( τ ) + ˙ z 1 ( τ ) X 1 + ˙ z 2 ( τ ) X 2 , (35)</formula>
<formula><location><page_7><loc_31><loc_13><loc_90><loc_15></location>/epsilon1 = -( ˙ z 0 ( τ )) 2 +(˙ z 1 ( τ )) 2 +(˙ z 2 ( τ )) 2 +(˙ z 3 ( τ )) 2 . (36)</formula>
<text><location><page_8><loc_12><loc_85><loc_90><loc_91></location>Here, z µ = ( z 0 ( τ ) , z 1 ( τ ) , z 2 ( τ ) , z 3 ( τ )) is the parametrization of an arbitrary curve C satisfying (36) with /epsilon1 = -1 , 0 , 1, and X i 's ( i = 1 , 2 , 3) are defined in (31). As a result of the discussion above, the curve z ( τ ) lives in one less dimension since z 3 ( τ ) = 0. The Ricci tensor takes the form</text>
<formula><location><page_8><loc_41><loc_82><loc_90><loc_85></location>R µν = -3 /lscript 2 g µν + ρλ µ λ ν , (37)</formula>
<text><location><page_8><loc_12><loc_78><loc_90><loc_82></location>where λ µ = δ 0 µ and the function ρ has the form given in (12). To calculate ρ explicitly, one needs to find ξ µ from its defining relation</text>
<formula><location><page_8><loc_41><loc_75><loc_90><loc_77></location>∇ µ λ ν = ¯ ∇ µ λ ν = ξ ( µ λ ν ) . (38)</formula>
<text><location><page_8><loc_12><loc_72><loc_82><loc_74></location>Here, ¯ ∇ is the covariant derivative of the AdS seed which can be put in the form</text>
<formula><location><page_8><loc_31><loc_67><loc_90><loc_71></location>d ¯ s 2 = 1 f 2 ( Hdτ 2 +2 dτdr ) + /lscript 2 cos 2 θ g mn dy m dy n , (39)</formula>
<text><location><page_8><loc_12><loc_64><loc_80><loc_66></location>with the metric of the two-dimensional unit sphere g mn . Since λ µ = δ 0 µ , one has</text>
<formula><location><page_8><loc_40><loc_61><loc_90><loc_63></location>¯ ∇ µ λ ν = -¯ Γ α µν λ α = -¯ Γ 0 µν , (40)</formula>
<text><location><page_8><loc_12><loc_58><loc_51><loc_59></location>and from this relation, ξ µ can be calculated as</text>
<formula><location><page_8><loc_32><loc_53><loc_90><loc_56></location>ξ µ = 2 ∂ µ log f -2 δ r µ ∂ r log f + 1 2 λ µ f 2 ∂ r ( H f 2 ) . (41)</formula>
<text><location><page_8><loc_12><loc_50><loc_80><loc_51></location>Using this result in (12), after a long calculation, the function ρ is found to be</text>
<formula><location><page_8><loc_17><loc_45><loc_90><loc_49></location>ρ = -Q V = -( ¯ g mn ¯ ∇ m ∂ n +2¯ g mn ∂ m log f∂ n +2¯ g mn ∂ m log f∂ n log f -4 /lscript 2 ) V, (42)</formula>
<text><location><page_8><loc_12><loc_39><loc_90><loc_44></location>where ¯ g mn ≡ /lscript 2 cos 2 θ g mn . Clearly, the operator Q contains derivatives only with respect to the angular coordinates and can be found once one has the explicit form of the curve is given. Then, one can solve the massless and massive wave equations given in (19).</text>
<section_header_level_1><location><page_8><loc_40><loc_35><loc_63><loc_36></location>B. The dS Background</section_header_level_1>
<text><location><page_8><loc_12><loc_28><loc_90><loc_33></location>Since the dS case follows similar to the AdS case (albeit with subtle differences) above, without much ado let us give the metric and the relevant results. First, the KSK metrics (1) for dS seed becomes</text>
<formula><location><page_8><loc_21><loc_23><loc_90><loc_26></location>ds 2 = 1 f 2 ( Hdτ 2 +2 dτdr + r 2 P 2 ( dθ 2 +sinh 2 θ dφ 2 ) ) +2 V ( τ, θ, φ ) dτ 2 , (43)</formula>
<text><location><page_8><loc_12><loc_20><loc_38><loc_21></location>where the metric functions are</text>
<formula><location><page_8><loc_34><loc_16><loc_90><loc_19></location>H = 1 -2 r∂ τ log P, f = r /lscriptP cosh θ, (44)</formula>
<formula><location><page_8><loc_34><loc_14><loc_90><loc_16></location>P = ˙ z 1 ( τ ) Y 1 + ˙ z 2 ( τ ) Y 2 + ˙ z 3 ( τ ) , (45)</formula>
<formula><location><page_8><loc_34><loc_12><loc_90><loc_14></location>1 = (˙ z 1 ( τ )) 2 +(˙ z 2 ( τ )) 2 +(˙ z 3 ( τ )) 2 . (46)</formula>
<text><location><page_9><loc_12><loc_90><loc_87><loc_91></location>Here, Y i 's ( i = 1 , 2) are defined in (32) and z 0 ( τ ) = 0. The Ricci tensor takes the form</text>
<formula><location><page_9><loc_42><loc_85><loc_90><loc_88></location>R µν = 3 /lscript 2 g µν + ρλ µ λ ν , (47)</formula>
<text><location><page_9><loc_12><loc_82><loc_56><loc_84></location>where λ µ = δ 0 µ . The dS seed can be put in the form</text>
<formula><location><page_9><loc_31><loc_77><loc_90><loc_81></location>d ¯ s 2 = 1 f 2 ( Hdτ 2 +2 dτdr ) + /lscript 2 cosh 2 θ g mn dy m dy n , (48)</formula>
<text><location><page_9><loc_12><loc_73><loc_90><loc_76></location>with the metric of the two-dimensional unit hyperboloid g mn . Again, to find the function ρ , one needs to find ξ µ which takes the same form (41). Then, the function ρ in (12) becomes</text>
<formula><location><page_9><loc_22><loc_68><loc_90><loc_71></location>ρ = -( ¯ g mn ¯ ∇ m ∂ n +2¯ g mn ∂ m log f∂ n +2¯ g mn ∂ m log f∂ n log f + 4 /lscript 2 ) V, (49)</formula>
<text><location><page_9><loc_12><loc_65><loc_32><loc_67></location>where ¯ g mn ≡ /lscript 2 cosh 2 θ g mn .</text>
<section_header_level_1><location><page_9><loc_42><loc_61><loc_61><loc_62></location>VI. CONCLUSION</section_header_level_1>
<text><location><page_9><loc_12><loc_41><loc_90><loc_59></location>We have given a way of constructing the Kerr-Schild-Kundt type of metrics which we have shown previously to be universal metrics of generic purely metric-based theories of gravity. It is highly interesting that we have three families of curves generating all these universal metrics. When the seed metric is the AdS spacetime, we have two families corresponding to timelike and null curves. They generate the AdS-plane wave and AdS-spherical wave families. For the dS seed metric, only the spacelike curves generate the dS-hyperbolic wave family. Hence, we obtain, in principle, an infinite number of Kerr-Schild-Kundt type of metrics where the AdS-plane wave, AdS-spherical wave [9], and dS-hyperbolic wave metrics [14] correspond to the straight lines and the rest of the family offer an exciting new territory of investigation. Using the Robinson-Trautman coordinates, we recast the KSK metrics in a convenient form which is suitable for studying explicit solutions.</text>
<section_header_level_1><location><page_9><loc_43><loc_37><loc_59><loc_38></location>Acknowledgment</section_header_level_1>
<text><location><page_9><loc_12><loc_30><loc_90><loc_34></location>This work is partially supported by TUBITAK. M. G. and B. T. are supported by the TUBITAK grant 113F155. T. C. S. is supported by the Science Academy's Young Scientist Program (BAGEP 2015).</text>
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</document> | [] |
2014ApJ...791...84W | https://arxiv.org/pdf/1407.4004.pdf | <document>
<section_header_level_1><location><page_1><loc_14><loc_82><loc_86><loc_86></location>Magnetic Field Restructuring Associated with Two Successive Solar Eruptions</section_header_level_1>
<text><location><page_1><loc_24><loc_77><loc_76><loc_79></location>Rui Wang 1 , Ying D. Liu 1 , Zhongwei Yang 1 and Huidong Hu 1</text>
<text><location><page_1><loc_20><loc_72><loc_27><loc_74></location>Received</text>
<text><location><page_1><loc_48><loc_72><loc_49><loc_74></location>;</text>
<text><location><page_1><loc_52><loc_72><loc_59><loc_74></location>accepted</text>
<section_header_level_1><location><page_2><loc_44><loc_85><loc_56><loc_86></location>ABSTRACT</section_header_level_1>
<text><location><page_2><loc_17><loc_47><loc_83><loc_80></location>We examine two successive flare eruptions (X5.4 and X1.3) on 2012 March 7 in the NOAA active region 11429 and investigate the magnetic field reconfiguration associated with the two eruptions. Using an advanced non-linear force-free field (NLFFF) extrapolation method based on the SDO/HMI vector magnetograms, we obtain a stepwise decrease in the magnetic free energy during the eruptions, which is roughly 20% -30% of the energy of the pre-flare phase. We also calculate the magnetic helicity, and suggest that the changes of the sign of the helicity injection rate might be associated with the eruptions. Through the investigation of the magnetic field evolution, we find that the appearance of the 'implosion' phenomenon has a strong relationship with the occurrence of the first X-class flare. Meanwhile, the magnetic field changes of the successive eruptions with implosion and without implosion were well observed.</text>
<text><location><page_2><loc_17><loc_42><loc_71><loc_43></location>Subject headings: Sun: activity - Sun: flares - Sun: magnetic fields</text>
<section_header_level_1><location><page_3><loc_42><loc_85><loc_58><loc_86></location>1. Introduction</section_header_level_1>
<text><location><page_3><loc_12><loc_48><loc_88><loc_81></location>It is commonly believed that the coronal magnetic field plays a very important role during the eruptions of solar flares and coronal mass ejections (CMEs). The magnetic free energy and helicity often change prominently during the process of these transient phenomena. Flares and CMEs derive their energy stored in the magnetic field, and in general the energy is released from an active region (AR). The energy released is just the magnetic free energy, which is the energy exceeding the potential field energy. A potential magnetic field structure is a minimal energy configuration of the magnetic fields. The causes for an AR non-potential configuration mainly include twisting and shearing of the magnetic field produced by the footpoint motion of the magnetic field lines on the surface of the photosphere, and flux emergence from underneath the photosphere. Sometimes even the newly emerged flux itself is non-potential. No matter what the cause is, it is a disturbance to the magnetic field and the magnetic energy and helicity may change accordingly.</text>
<text><location><page_3><loc_12><loc_26><loc_88><loc_45></location>During big flares the magnetic field often shows a rapid, irreversible change (e.g., Liu et al. 2012; Wang et al. 2012; Petrie 2013a; Wang et al. 2013). The magnetic field becomes more horizontal after the eruption than before. This is explained by an 'implosion' theory (Hudson 2000). Specifically, during the eruption part of the magnetic field shrinks or collapses ('implodes') so that there is an overall decrease in the magnetic energy in the region of eruption. The contraction behavior of the magnetic field will happen when it loses the energy supporting its configuration, so as to achieve a new force balance.</text>
<text><location><page_3><loc_12><loc_11><loc_88><loc_24></location>The AR NOAA 11429 spawned a powerful X5.4 flare on 2012 March 7, which is the second largest flare eruption event since 2010. It is associated with a wide and fast CME of more than 2000 km s -1 around 00:24 UT on March 7 (Liu et al. 2013, 2014). Of particular interest is that there were two successive eruptions, both of which are X-class flares. The second flare reached X1.3 less than 1 hr after the first X5.4 flare eruption. Related work has</text>
<text><location><page_4><loc_12><loc_73><loc_88><loc_86></location>been done about AR 11429. Wang et al. (2012) and Petrie (2013b) both derive a stepwise increase in the magnetic field on the photosphere during the eruptions. Also, Wang et al. (2012) compute the Lorentz force and find that the stepwise decrease in the Lorentz force has a positive correlation with the peak soft X-ray flux, so they suggest that the CME mass can be estimated by the Lorentz force change.</text>
<text><location><page_4><loc_12><loc_51><loc_88><loc_70></location>In this paper, we use data from the Helioseismic and Magnetic Imager (HMI; Schou et al. 2012) on board the Solar Dynamics Observatory (SDO) to investigate the magnetic free energy and magnetic helicity of this AR. In Section 2, we show data analysis and use an advanced coronal magnetic field extrapolation method to understand the the coronal magnetic field topology. We also discuss the evolution of the magnetic free energy and magnetic helicity. In Section 3 we discuss the magnetic field reconfiguration during the successive eruptions. Section 4 summaries the conclusions.</text>
<section_header_level_1><location><page_4><loc_33><loc_44><loc_67><loc_46></location>2. Observations and Data Analysis</section_header_level_1>
<text><location><page_4><loc_12><loc_11><loc_88><loc_41></location>The HMI instrument provides high time-resolution vector magnetic field data for NOAA AR 11429 with a 12 minute cadence and 0' . 5 pixel size. In this investigation, we adopt the Lambert Cylindrical Equal Area (CEA) projected and remapped vector magnetic field data (Gary & Hagyard 1990; Calabretta & Greisen 2002; Thompson 2006), as shown in Figure 1, to investigate the evolution of the magnetic free energy and helicity of the two successive X-class flares on 2012 March 7. The X5.4 flare started at 00:02 UT, peaked at 00:24 UT, ended at 00:40 UT, and was companied by a CME of more than 2000 km s -1 ; the X1.3 flare started at 01:05 UT, peaked at 01:14 UT, ended at 01:23 UT, and was accompanied by another CME of about 1800 km s -1 (Liu et al. 2013). The 180 · azimuthal ambiguity in the transverse field of the data is resolved by a minimum energy algorithm (Metcalf 1994; Leka et al. 2009). We also adopt the HMI vector magnetic field data to</text>
<text><location><page_5><loc_12><loc_76><loc_86><loc_86></location>calculate the magnetic helicity. In addition, we extract ten slices from the extrapolated coronal magnetic field data cube, which have the same area (72.5 × 72.5 Mm), and their positions are shown in Figure 1. With the ten slices we can compare the magnetic flux changes in different locations.</text>
<section_header_level_1><location><page_5><loc_22><loc_69><loc_78><loc_70></location>2.1. Evolution of Photospheric Magnetic Field Near PIL</section_header_level_1>
<text><location><page_5><loc_12><loc_52><loc_86><loc_66></location>We investigate the evolution of the photospheric magnetic fields near the polarity inversion line (PIL). Figure 2 shows the temporal profiles of the magnetic field changes. These plots of temporal changes are derived by calculating area integrals of the field components over the chosen photospheric areas along the PIL in 176 12-minute images, from 12:00 UT on March 6 to 23:48 UT on March 7, i.e.,</text>
<formula><location><page_5><loc_42><loc_47><loc_88><loc_51></location>F PIL = ∫ A PIL BdA, (1)</formula>
<text><location><page_5><loc_12><loc_10><loc_88><loc_46></location>where B is the magnetic field on the photosphere, and A PIL is the area marked by a black rectangle corresponding to the region near the PIL. The top panel of Figure 2 shows the average vertical magnetic intensity, and the cyan/purple lines represent positive/negative intensity, respectively. They have a similar trend. The average positive vertical intensity is a little higher than the negative one. During the eruptions, they seem to have an opposite change but not so obvious. Before 08:00 UT, there was a transient drop of the vertical and horizontal intensity. We think the data is fake. As we can see there was an interval of data missing before 08:00 UT. So the observation values are not real, just in a transition period, namely the value increases from zero to normal level. The error bars of the average magnetic intensity shown in Figure 2 are given by 3 σ where σ is the standard deviation of the HMI data in the region of the black rectangle. The bottom panel shows the horizontal fields. There was an obvious increase in the horizontal average magnetic intensity during the eruptions, and the most prominent part of the increase occurred during the first</text>
<text><location><page_6><loc_12><loc_64><loc_88><loc_86></location>larger flare. Compared with the horizontal fields, the vertical fields had no such abrupt increase during the eruptions, and also the field strength is relatively weaker. Similar rapid enhancements of transverse fields during big flare eruptions are also found in other works (Wang 1992; Wang & Liu 2010; Liu et al. 2012; Wang et al. 2012; Petrie 2013a; Wang et al. 2013). According to Hudson et al. (2008), the reason that the magnetic field becomes more horizontal is that the coronal magnetic field contracts downward. The magnetic contraction could be explained by the 'implosion' theory of Hudson (2000), and this will be discussed further in Section 3.</text>
<section_header_level_1><location><page_6><loc_30><loc_57><loc_70><loc_58></location>2.2. Evolution in Magnetic Free Energy</section_header_level_1>
<text><location><page_6><loc_12><loc_23><loc_88><loc_54></location>Here, we obtain the coronal magnetic fields by adopting the Non-Linear Force-Free Field (NLFFF) method as proposed by Wheatland et al. (2000) and extended by Wiegelmann (2004) and Wiegelmann and Inhester (2010). We use the latest version of the NLFFF optimization code improved in 2011 (Wiegelmann et al. 2012) to extrapolate the coronal field from the observed vector magnetograms in a Cartesian domain. A preprocessing procedure has been used to remove most of the net force and torque from the data, so the boundary can be more consistent with the force-free assumption. Also, we obtain a potential field (PF) configuration from the same observation using the vertical component of the fields with the help of a Fourier representation based on Green's function (Seehafer 1978). The magnetic free energy ( E free ) can be inferred by subtracting the PF energy ( E pot ) from the NLFFF energy ( E nff ),</text>
<formula><location><page_6><loc_36><loc_18><loc_88><loc_22></location>E free = ∫ V B 2 nff 8 π dV -∫ V B 2 pot 8 π dV, (2)</formula>
<text><location><page_6><loc_12><loc_10><loc_87><loc_17></location>where the energy is computed from the field strength within a certain volume V , and the subscripts nff and pot denote NLFFF and PF, respectively. The time resolution of the free energy time series is 1 hr, but there is a higher resolution of 12 min before and after</text>
<text><location><page_7><loc_12><loc_85><loc_68><loc_86></location>the eruptions from 23:00 UT on March 6 to 02:00 UT on March 7.</text>
<text><location><page_7><loc_12><loc_49><loc_88><loc_82></location>A set of sample field lines from the resulting NLFFF extrapolation are displayed in Figure 3, over the 193 ˚ A channel images from the Atmospheric Imaging Assembly (AIA; Lemen et al. 2012). We have chosen three different times, namely before the first flare eruption (FE1), after FE1 but before the second flare eruption (FE2), and after FE2, respectively. The AIA images indicate that FE1 occurred in the east part of the AR and FE2 in the west part. The extrapolated field lines seem to have good alignment with the EUV background pattern. Contours of ± 500, ± 1300 G of B z are also overlaid on the EUV images in order to identify the footpoints of the field lines. In the left panel, the brightest structure on the east side of the AR seems to correspond to a flux rope structure, and our extrapolated field lines appear to confirm this. There are some pre-flare arcades overlying the brightened structures. In the middle and right panels, a post-flare 'arcade' gradually formed and expanded.</text>
<text><location><page_7><loc_12><loc_15><loc_88><loc_46></location>Figure 4 shows the magnetic free energy calculated using Eq 2. The magnetic free energy was increasing before the eruptions. During the pre-flare phase from 12:00 UT on March 6 to 00:00 UT on March 7, although there were some C- and M- class flares as shown by the GOES X-ray flux, we do not see obvious decreases in the magnetic free energy. When the largest X-flare occurred at 00:02 UT on March 7, the free energy shows a dramatic decrease. The decrease in the magnetic free energy ends after the peak time of FE1, which indicates that most of the free energy was released by FE1. The amount of the energy drop during the first eruption is 3 . 0 × 10 32 erg, accounting for about 20% -30% of the pre-flare free energy. During FE2, we observe a small increase rather than a decrease in the magnetic free energy. This is interesting as FE2 was also an X-class flare. After the two eruptions, the curve of the free energy became relatively flat.</text>
<section_header_level_1><location><page_8><loc_28><loc_85><loc_72><loc_86></location>2.3. Evolution in Magnetic Helicity Injection</section_header_level_1>
<text><location><page_8><loc_12><loc_68><loc_87><loc_81></location>There has been increasing observational evidence that the helicity of magnetic fields holds an important clue to solar flare eruptions. According to the work of Berger & Field (1984), the measure of the magnetic helicity change in the solar coronal can provide physically reasonable results for torsional motions on the boundary plane. Berger & Field (1984) derived the Poynting theorem for the helicity in an open volume:</text>
<formula><location><page_8><loc_30><loc_63><loc_88><loc_66></location>dH dt = ∮ 2( B t · A p ) v z dS + ∮ -2( v t · A p ) B z dS, (3)</formula>
<text><location><page_8><loc_12><loc_13><loc_88><loc_61></location>where A p is the vector potential of the magnetic field, which is calculated by means of a fast Fourier transform method as implemented by Chae (2001), v t is the tangential velocity of the real motion of the plasma on the photosphere. In Eq. 3, the first term 2( B t · A p ) v z represents the helicity injection rate via the passage of the helical field lines through the photospheric surface (i.e., emergence of new flux), and the second term -2( v t · A p ) B z represents the helicity injection rate via the shuffling horizontal motion of the field lines on the surface (i.e., shearing or twisting motions). In the work of Chae (2001), the horizontal velocity v t is equal to the tracking velocity from the Local Correlation Tracking (LCT) method (Laurence & George 1988). However, Kusano et al. (2002; 2003) indicated that the tracking velocity obtained from LCT is an 'image motion' of the magnetic footpoints rather than the material motion v t . Later, D'emoulin & Berger (2003) pointed out the use of v z deduced from Doppler measurements would only duplicate part of the helicity injection rate already included in the tracking velocity. They deduced and proved that all the helicity injection rate only from the emergence of new flux can be present in the second term of Eq. 3 determined by the tracking method. Here, the track velocity is determined by -v z B t /B z . On the other hand, the tracking method does not just measure the vertical plasma motion v z , but also the horizontal motion v t . Therefore, we can present the tracking</text>
<text><location><page_9><loc_12><loc_85><loc_45><loc_86></location>velocity u by the sum of both velocities:</text>
<formula><location><page_9><loc_43><loc_79><loc_88><loc_83></location>u = v t -v z B z B t . (4)</formula>
<text><location><page_9><loc_12><loc_67><loc_88><loc_77></location>Namely, the tracking method provides the transverse velocities including both of the effects of the shearing motion and the vertical motion. This method can only compute the total helicity injection rate across the photosphere as the equation below (Demoulin & Berger 2003):</text>
<formula><location><page_9><loc_39><loc_63><loc_88><loc_67></location>dH dt = -2 ∮ ( u · A p ) B z dS. (5)</formula>
<text><location><page_9><loc_12><loc_37><loc_88><loc_62></location>We determine the tracking velocity u of the magnetic fields at their photospheric footpoints using a Fourier Local Correlation Tracking (FLCT) method (Fisher & Welsch 2008), which is the upgraded version of LCT. When we use FLCT to measure the tracking velocity, some appropriate parameter settings will make the results more accurate. In order to calculate the tracking velocity, we need to give the time interval ∆ t between two magnetograms for FLCT. Here it is 720 s. We also use a Gaussian windowing function as weighting function for sub-images in the tracking method. The parameter σ as the width of Gaussian function is 15, which has been proved to be the best setting in the numerical experiment of Welsch et al. (2007).</text>
<text><location><page_9><loc_12><loc_10><loc_88><loc_35></location>Figure 5 shows the helicity injection rate as a function of time, which is determined by Eq. 5. The rate is determined every 12 minutes for 36 hrs. The rate shows a considerable fluctuation. Before FE1, the AR accumulated the negative helicity. Then the negative injection rate decreased around FE1 and changed its sign. Until FE2 the helicity injection rate reached to its positive peak, about -4 . 0 × 10 41 Mx 2 hr -1 . After FE2, the helicity injection decreased and changed its sign again to negative and kept this balanced stage with a considerable fluctuation. It suggests some possibility that the changes of the sign of the helicity injection rate might be associated with the eruptions. Similar results can be found in the work of Kusano et al. (2003).</text>
<text><location><page_10><loc_12><loc_64><loc_88><loc_86></location>Figure 6 shows the accumulated change of the magnetic helicity, which was obtained by integrating the measured dH/dt from the start of the observing run to the specified time. The black curve represents the accumulated change of the magnetic helicity from the helicity injection rate of Eq. 5. The negative accumulated change of the injection rate kept an approximately constant increase until FE1, then because of the change of the sign of the injection rate (see Figure 5), it stopped increasing. After FE2, it increased again in a relative slower growth rate. The increase of the magnetic helicity should be attributed to the effects of both the emergence and the shearing motions.</text>
<section_header_level_1><location><page_10><loc_21><loc_57><loc_79><loc_58></location>3. Coronal Magnetic Field Restructuring due to Implosion</section_header_level_1>
<text><location><page_10><loc_12><loc_38><loc_88><loc_54></location>A prominent feature during the eruptions is the magnetic implosion phenomenon. This scenario was proposed by Hudson (2000) for the first time. The energy for coronal transient phenomena comes from the stressed coronal magnetic field. According to Hudson (2000), the energy release corresponds to a reduction in the magnetic pressure, and the unbalanced forces in the magnetic field will cause the magnetic structure to contract, i.e., the appearance of the implosion phenomenon.</text>
<text><location><page_10><loc_12><loc_11><loc_88><loc_35></location>Figure 7 shows the evolution of the NLFFF extrapolated magnetic field component perpendicular to the slice as marked in Figure 1. An obvious contraction of the magnetic fields before FE1 can be seen in the upper panels corresponding to the slice CS1. This is consistent with the change in the photospheric horizontal field shown in Figure 2. Note that the times selected for CS1 are before the peak time of FE1. The contraction of the magnetic field may be caused by the relaxation of the twisted magnetic field, and is likely part of the energy release process during FE1. The evolution of the magnetic fields is consistent with the conjecture of Hudson (2000). The lower panels of Figure 7, however, indicate that magnetic contraction did not occur during FE2.</text>
<text><location><page_11><loc_12><loc_20><loc_88><loc_86></location>In order to find out more details about the magnetic field change during the two flares, we calculate the magnetic flux for the ten cross sections (or slices) of the extrapolated coronal magnetic field data cube (as shown by the white, red and blue lines in Figure 1). The magnetic flux F ⊥ perpendicular to the cross sections and the flux F ‖ parallel to the cross sections are shown in Figure 8 and Figure 9, respectively. Here we define the flux F ‖ as the integration of the absolute value of the parallel magnetic field, which consists of the north-south and vertical components of the magnetic field vector, over the cross section. Looking at all plots of these slices in Figure 8 and 9, the most prominent changes of the magnetic field, all happened in plots (c), (d), and (e), which should correspond to the source region of FE1 (see Figure 1) and show the probable position and range of the source region. During FE1, the decrease in F ⊥ (see Figure 8) in this region may be owing to the ejection of the flux rope structure associated with the first CME. This process is so much like the 'tether-cutting' model (Moore et al. 2001), i.e., the flux rope expanded upward and two-ribbon flare near the solar surface formed at the same time. This may interpret why F ⊥ decreased and F ‖ increased (see Figure 9) in plots (c), (d), and (e). On the other hand, the magnetic flux through each slice increased during FE1. Since the computation region is in a relatively low altitude, we suggest that it reflects the magnetic contraction during FE1, i.e., the field lines decrease in length from the higher to lower altitudes, which increases the magnetic field density in the lower corona as suggested by Hudson (2000). This can be visually seen in the upper panels of Figure 7. Compared with the field changes during FE1, the magnetic flux changes during FE2 were small (see Figures 8 and 9). Specifically, no implosion phenomenon occurred during FE2. This is consistent with the lower panels of Figure 7.</text>
<text><location><page_11><loc_12><loc_11><loc_86><loc_18></location>From Figure 8 and 9, the difference of the variations of the magnetic field between the eruptions with implosion (FE1) and without implosion (FE2) can be well observed. During FE1, generally speaking, the energy was released (see Figure 4) for radiation or</text>
<text><location><page_12><loc_12><loc_64><loc_88><loc_86></location>other observation phenomena, the magnetic flux should decrease. However, contrary to what we think, the total magnetic flux near the source region increased (see Figure 9). This is interesting if we use the implosion theory to explain it, i.e., if part of the coronal field expand to higher corona, a further compensating implosion in the lower corona will simultaneously take place, and the magnetic pressure inward may make the lower magnetic field become more compact and the magnetic flux in lower corona increase (seen as (c), (d), and (e) in Figure 9). By contrast, there was no implosion phenomenon during FE2 and the magnetic flux changed as usual.</text>
<section_header_level_1><location><page_12><loc_42><loc_57><loc_58><loc_58></location>4. Conclusions</section_header_level_1>
<text><location><page_12><loc_12><loc_32><loc_88><loc_54></location>We have analyzed in detail 36 hours of 12-minute SDO/HMI vector magnetic field observations covering the X5.4 and X1.3 successive flares at 00:24 UT and 01:14 UT on 2012 March 7, respectively. By means of an advanced NLFFF extrapolation method, we derived the coronal magnetic fields and magnetic free energy from the preprocessed boundary conditions. The magnetic helicity was computed using the HMI vector magnetograms. Through the extrapolated coronal magnetic field, magnetic free energy and helicity, we investigated the magnetic field restructuring associated with the two prominent successive eruptions. The main conclusions are:</text>
<unordered_list>
<list_item><location><page_12><loc_12><loc_22><loc_88><loc_30></location>1. Near the PIL region, the photospheric vector fields became more horizontal after the first flare than the preflare state. It shows a stepwise increase in the photospheric horizontal field component.</list_item>
<list_item><location><page_12><loc_12><loc_13><loc_87><loc_20></location>2. The magnetic free energy shows a stepwise decrease during the first flare while no apparent change during the second one. The amount of the energy drop during the first eruption is 3 . 0 × 10 32 erg, accounting for about 20% -30% of the pre-flare free energy.</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_13><loc_12><loc_79><loc_86><loc_86></location>3. The helicity injection rate changed the sign from negative to positive, reaching its positive peak about -4 . 0 × 10 41 Mx 2 hr -1 during the eruptions. It suggests that the changes of the sign of the helicity injection rate might be associated with the eruptions.</list_item>
<list_item><location><page_13><loc_12><loc_66><loc_87><loc_76></location>4. The extrapolated coronal magnetic field shows a contraction behavior during the first eruption. This is consistent with the implosion process suggested by Hudson (2000). Meanwhile, the magnetic field changes of the successive eruptions with implosion and without implosion were well observed in the same AR.</list_item>
</unordered_list>
<text><location><page_13><loc_12><loc_46><loc_87><loc_62></location>We are grateful to Dr. Thomas Wiegelmann for his generous sharing of his latest version of the NLFFF optimization code. The work was supported by the Specialized Research Fund for State Key Laboratories of China, the Recruitment Program of Global Experts of China under grant Y3B0Z1A840, the Strategic Priority Research Program on Space Science from the Chinese Academy of Sciences (XDA04060801), and the CAS Key Research Program KZZD-EW-01. We acknowledge the use of data from SDO.</text>
<section_header_level_1><location><page_14><loc_43><loc_85><loc_58><loc_86></location>REFERENCES</section_header_level_1>
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<figure>
<location><page_16><loc_19><loc_40><loc_75><loc_81></location>
<caption>Fig. 1.- Remapped HMI vector magnetogram for the region of AR 11429 as viewed from overhead. The vertical field ( B z ) is plotted as the background. The black (white) arrows indicate the horizontal field ( B h ) with positive (negative) vertical field components. The blue, green and black contours are plotted at -500, -1000, and -2000 G; the purple, pink, and red contours are plotted at 500, 1000 and 2000 G, respectively. The positions of ten uniform slices are marked as the white solid lines, and red (CS1) and blue (CS2) dashed lines. The black rectangle marks a region near the polarity inversion line, which is used in the subsequent analysis. The coordinate is in arc sec. The origin is relative to the center of the solar disk.</caption>
</figure>
<figure>
<location><page_17><loc_22><loc_35><loc_79><loc_76></location>
<caption>Fig. 2.- Average vertical magnetic intensity (upper panel) and horizontal field (bottom panel) near the neutral line as a function of time. The cyan and purple lines represent positive and negative field components in the upper panel. The thicker and thinner vertical red lines represent the first and second GOES flare peak times, respectively. The uncertainties of the average magnetic field are plotted as error bars in 3 σ level.</caption>
</figure>
<figure>
<location><page_18><loc_11><loc_51><loc_37><loc_71></location>
<caption>Fig. 3.- AIA images of AR 11429 in 193 ˚ A channel observed at 00:00 UT (left), 00:36 UT (middle), and 01:24 UT (right) on March 7, which correspond to the times before the flares, post the first flare but before the second one, and post the second flare, respectively. Overplotted are some arbitrarily chosen field lines from the NLFFF model. Contours of ± 500, ± 1300 G of B z are also overlaid on the EUV images and are marked in cyan (negative) and pink (positive), respectively. The rough positions of the first and second eruptions, flux rope and pre-flare arcades are marked as FE1, FE2, FR and PA in the left panel, respectively. The rough positions of CS1 and CS2 are also overplotted in the left panel.</caption>
</figure>
<text><location><page_18><loc_51><loc_52><loc_59><loc_53></location>2012-03-07T00:36:10.23Z</text>
<text><location><page_18><loc_76><loc_52><loc_84><loc_53></location>2012-03-07T01:24:10.40Z</text>
<figure>
<location><page_19><loc_21><loc_43><loc_79><loc_73></location>
<caption>Fig. 4.- Evolution of the magnetic free energy of AR 11429 from 12:00 UT on March 6 to 23:48 UT on March 7. The solid black line corresponds to the magnetic free energy, and the purple curve corresponds to the GOES soft-X ray flux (1-8 ˚ A channel). Vertical blue, green and red lines denote the peak times of C-, M-, and X-class flares, respectively, with their thickness roughly corresponding to the magnitude of the flare class. The vertical yellow and green squares in both panels correspond to the intervals of the first and second eruptions, respectively.</caption>
</figure>
<figure>
<location><page_20><loc_21><loc_65><loc_80><loc_84></location>
<caption>Fig. 5.- Helicity injection rate as a function of time, which is determined by Eq. 5. The vertical lines and squares have the same meaning as in Figure 4.</caption>
</figure>
<figure>
<location><page_20><loc_21><loc_19><loc_80><loc_53></location>
<caption>Fig. 6.- Accumulated change of the magnetic helicity as a function of time. The vertical lines and squares have the same meaning as in Figure 4.</caption>
</figure>
<figure>
<location><page_21><loc_21><loc_40><loc_81><loc_70></location>
<caption>Fig. 7.- Distributions of the horizontal NLFFF field ( B h ) in two vertical cross sections (CS1 and CS2 as shown in Figure 1). Only the component perpendicular to the cross section is shown. a, b, c: Distribution of the horizontal field component in CS1 at 23:48 UT on March 6, 00:12 UT and 00:24 UT on March 7, respectively. d, e, f: Distribution of the horizontal field component in CS2 at 01:00 UT, 01:24 UT, and 02:00 UT on March 7, respectively.</caption>
</figure>
<figure>
<location><page_22><loc_21><loc_33><loc_80><loc_79></location>
<caption>Fig. 8.- The magnetic flux of the field components perpendicular to the cross sections from 23:00 UT on March 6 to 02:00 UT on March 7. The ten panels from (a) to (j) correspond to the ten cross sections from left to right in Figure 1, respectively. The two gray squares indicate the times of the first and second eruptions from their beginning to end, and the vertical lines mark the peak times of FE1 and FE2.</caption>
</figure>
<figure>
<location><page_23><loc_21><loc_29><loc_80><loc_75></location>
<caption>Fig. 9.- Similar to Figure 8, but for the magnetic flux of the field components parallel to the cross sections.</caption>
</figure>
</document> | [] |
2021SCPMA..6447411Z | https://arxiv.org/pdf/2012.03274.pdf | <document>
<section_header_level_1><location><page_1><loc_25><loc_92><loc_76><loc_93></location>Bulk Superconductivity in the Dirac Semimetal TlSb</section_header_level_1>
<text><location><page_1><loc_19><loc_87><loc_81><loc_90></location>Yuxing Zhou, 1 Bin Li, 2 Zhefeng Lou, 1 Huancheng Chen, 1 Qin Chen, 1 Binjie Xu, 1 Chunxiang Wu, 1 Jianhua Du, 3 Jinhu Yang, 4 Hangdong Wang, 4 and Minghu Fang 1, 5,</text>
<text><location><page_1><loc_15><loc_79><loc_86><loc_86></location>Department of Physics, Zhejiang University, Hangzhou 310027 , China 2 New Energy Technology Engineering Laboratory of Jiangsu Province and School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023 , China 3 Department of Applied Physics, China Jiliang University, Hangzhou 310018 , China 4 Department of Physics, Hangzhou Normal University, Hangzhou 310036 , China 5 Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093 , China</text>
<text><location><page_1><loc_26><loc_86><loc_82><loc_89></location>∗ 1</text>
<text><location><page_1><loc_41><loc_78><loc_59><loc_79></location>(Dated: December 8, 2020)</text>
<text><location><page_1><loc_18><loc_63><loc_83><loc_76></location>A feasible strategy to realize the Majorana fermions is searching for a simple compound with both bulk superconductivity and Dirac surface states. In this paper, we performed calculations of electronic band structure, the Fermi surface and surface states, as well as measured the resistivity, magnetization, specific heat for TlSb compound with a CsCl-type structure. The band structure calculations show that TlSb is a Dirac semimetal when spin-orbit coupling is taken into account. Meanwhile, we first found that TlSb is a type-II superconductor with T c = 4.38 K, H c 1 (0) = 148 Oe, H c 2 (0) = 1.12 T and κ GL = 10.6, and confirmed it to be a moderately coupled s -wave superconductor. Although we can not determine which bands near the Fermi level E F to be responsible for superconductivity, its coexistence with the topological surface states implies that TlSb compound may be a simple material platform to realize the fault-tolerant quantum computations.</text>
<section_header_level_1><location><page_1><loc_21><loc_59><loc_37><loc_60></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_8><loc_49><loc_56></location>Topological superconductors host Majorana fermions described by a real wave function, providing protection for quantum computations [1]. So, realizing topological superconductivity (TSC) has became one of the most interesting topics in the condensed matter physics in the past decades. According to the discussion in Ref. [2], there are intrinsic and artificial engineered topological superconductors. For the intrinsic, the topological nontrivial gap function naturally shows up. Sr 2 RuO 4 [3-6] is the first proposed topological superconductor although the existence of chiral p -wave superconductivity (SC) is still under debate. Cu x Bi 2 Se 3 [7], as the first material to show SC ( T c ∼ 4 K) upon doping charge carrier into a topological insulator (TI), is a promising ground to look for two dimensional (2D) TSC due to the topological surface states surviving in TI even when carriers are doped. Many experiments, such as the conductance spectroscopy [8], nuclear manetic resonance (NMR) measurements of the Knight-shift [9], and specific heat in applied magnetic fields [10], have already given evidences for TSC seem emerging in Cu x Bi 2 Se 3 . The nematic SC discovered in Cu x Bi 2 Se 3 [9, 10] was also observed in the similar superconductors derived from Bi 2 Se 3 , such as in Sr x Bi 2 Se 3 [11], Nb x Cu 2 Se 3 [12]. Sn 1 -x In x Te [13] is another superconductor upon doping charge carriers into a topological crystalline insulator. In the cleanest sample ( x ∼ 0.04) with the lowest T c (1.2 K), a pronounced zero-bias conductance peak (ZBCP) similar to that in Cu x Bi 2 Se 3 has been observed by point contact spectroscopy [13]. Another is the artificial engineered TSC in hybrid structures. According to the idea proposed by Fu and Kane [14], if s -wave pairing is imposed on the topological sur-</text>
<text><location><page_1><loc_52><loc_47><loc_92><loc_61></location>states of a three dimensional (3D) TI through superconducting proximity effect, the resulting superconducting state should be a 2D p -wave SC harboring a Majorana zero mode in the vortex core. Experimentally, proximityinduced SC on the surface of 3D TIs has been studied by many groups[15-21]. The observation [22] of 4 π -period Josephson supercurrent in 3D HgTe TI is encouraging, although, it is difficult to elucidate the topological nature of the induced 2D SC.</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_46></location>However, there are controversies about TSC emerging and the observed ZBCP being a Majorana zero-energy mode (MZM) in the doped topological material. To realize TSC through a superconducting proximity effect in hybrid structures has many engineering challenges. A feasible way is to realize TSC in a simple compound, in which both the topological surface state and the bulk SC coexist, thus Majorana fermions emerge at the edge of superconductor. Recently, the observations of MZM in the core of vortex [23, 24], at the end of the atomic defect line [25], and near the Bi islands [26] and the interstitial Fe atoms [27] in the simplest Fe-based superconductor Fe 1+ y Te 0 . 5 Se 0 . 5 ( T c = 14 K) [28] motivate us to search for the similar system. TlSb crystallizes in a cubic CsCl structure with space group Pm ¯ 3 m (No. 221) (as shown in the inset of Fig. 1), from this structure a large number of topological semimetal/metals (TMs) were designed [29], ranging from triple nodal points, type-I nodal lines, and critical type nodal lines to hybrid nodal lines. For example, CaTe is a typical type-I nodal line and Dirac TM[30]; YIr is a typical triple-nodal-point TM [29], YMg possesses multiple types of band crossing [29]. Therefore, we tried to grow TlSb crystals for studying its topological natures and SC, unfortunately, only polycrystalline TlSb samples were obtained.</text>
<figure>
<location><page_2><loc_11><loc_72><loc_47><loc_94></location>
<caption>Figure 1. (a) Polycrystalline XRD pattern with its refinement profile at room temperature of TlSb. The inset shows the schematic structure of TlSb, thallium atoms are in gray while the antimony atoms are brown.</caption>
</figure>
<text><location><page_2><loc_9><loc_35><loc_49><loc_62></location>In this paper, we performed calculations of the electronic band structure, the Fermi surface and the surface states on (001) plane, as well as measured resistivity, magnetization and specific heat for the polycrystalline TlSb sample. The band structure calculations show TlSb is a Dirac semimetal with 4-fold degenerate nodes near Γ and R points. It is also found that TlSb is a typeII superconductor with the superconducting transition temperature T c = 4.38 K, the lower critical field H c 1 (0) = 148 Oe, and the upper critical field H c 2 (0) = 1.12 T and the Ginzburg-Landau (GL) parameter κ GL = 10.6. The obtained specific heat jump, ∆ C el / γ n T c ∼ 1.42, indicates that TlSb is a conventional phonon-mediated superconductor with s -wave superconducting symmetry. These results indicate that both s -wave SC and surface states coexist in TlSb, whether the Majorana fermions emerge or not on the edges is needed to confirm in the future.</text>
<section_header_level_1><location><page_2><loc_21><loc_30><loc_37><loc_31></location>II. EXPERIMENTAL</section_header_level_1>
<text><location><page_2><loc_9><loc_9><loc_49><loc_28></location>Polycrystalline TlSb samples were synthesized by a peritectic reaction method. The mixture of stoichiometric high purity Tl (99.99%) chunk and Sb (99.999%) powder was placed in an alumina crucible, sealed in an evacuated quartz tube and heated at 450 · C for 20 hrs, then decreases to 195 · C waiting for the mixture melting completely. To avoid the decomposition of obtain TlSb phase at 191 · C [31], the quartz tube was quenched to room temperature at 195 · C, however, the obtained TlSb samples always contain a small amount of unreacted Sb impurities due to the precipitation of Sb before the peritectic reaction. The obtained TlSb alloy is easily to cut for the subsequent structure characterizations and property</text>
<text><location><page_2><loc_52><loc_59><loc_92><loc_93></location>measurements. Polycrystalline x-ray diffraction (XRD) was carried out on a PANalytical diffractometer equiped with CuK α radiation. The TlSb XRD pattern is shown in Fig. 1, in which the main peaks can be fitted by the CsCl-type structure with space group Pm ¯ 3 m . The lattice parameters a = b = c = 3.86(5) ˚ A were obtained by the Rietveld refinement by using general structure analysis system (GSAS) [32]. A rectangular bar of the sample was cut for the magnetization and resistivity mesurements, which were performed on a magnetic property measurement system (Quantum Design, MPMS - 7 T) and a physical property measurement system (Quantum Design, PPMS - 9 T), respectively. The band structure was calculated by using density function theory (DFT) with the WIEN2k package [33]. Generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) [34] was employed for the exchange correlation potential calculations. A cutoff energy of 520 eV and a 13 × 15 × 15 k -point mesh were used to perform the bulk calculations. The Fermi surface (FS) was performed with WannierTools [35] package which is based on the maximally localized Wannier function tight-binding model [36-38] constructed by using the Wannier90 [39] package.</text>
<section_header_level_1><location><page_2><loc_58><loc_54><loc_86><loc_55></location>III. RESULTS AND DISCUSSIONS</section_header_level_1>
<text><location><page_2><loc_52><loc_9><loc_92><loc_52></location>We first discuss the electronic band structure without considering spin-orbit coupling (SOC). As shown in Fig. 2(a), both conduction (red) and valence (blue) band cross the Fermi level E F . At the high symmetry Γ and R points, there are three bands crossing, with a threefold degenerate at 0.8 eV and 2 eV below E F , respectively. However, when SOC is taken into account [see Fig. 2(b)], gaps open and leaves two twofold degenerate bands at both high symmetry points. Since both time-reversal and inversion symmetries are present, no spin-splitting occurs and then the twofold degenerated bands come together to fourfold degenerate points, indicating TlSb is a Dirac semimetal. We also calculated the density of state (DOS), as shown in the right panel of Fig. 2(b), the DOS at E F is mainly contributed by Sb orbits. To further clarify the band structure of TlSb, we calculated its 3D bulk FS of the first Brillouin zone (BZ) as shown in Fig. 2(d), exhibiting very complex 3D characteristics. Figure 2(e) presents the FS on the k z = 0 plane, which is the cross section passing the Γ point of the 3D FS. Figure 2(f) displays the energy dispersion in the k x -k y plane, in which the Dirac dispersion is clearly seen, demonstrating further that TlSb is a Dirac semimetal. Then we calculated the surface states on (001) plane by using a surface Green's function method [40]. As shown in Fig. 3(a), the projected Dirac points are hidden in the continuous bulk states, the surface states are shown as the red curves. The (001) surface energy contour is shown in Fig. 3(b), (c) and (d) with E = E F , E = -1.5 eV and E = -4 eV,</text>
<figure>
<location><page_3><loc_10><loc_56><loc_47><loc_93></location>
<caption>Figure 2. The electronic band structures of TlSb without (a) and with (b) SOC. (c) 3D bulk Fermi surfaces and color-coded Fermi velocities (red is high velocity). (d) Calculated Fermi surfaces cross section at the k z = 0 plane. (e) Calculated energy distribution at Dirac point in k x -k y plane.</caption>
</figure>
<figure>
<location><page_3><loc_10><loc_16><loc_48><loc_42></location>
<caption>Figure 3. (a) Surface band structure for (001) plane along projected high symmetry points. The surface spectra of (001) plane with (b) E = E F , (c) E = -1.5 eV and (d) E = -4 eV.</caption>
</figure>
<figure>
<location><page_3><loc_53><loc_62><loc_91><loc_93></location>
<caption>Figure 4. Wilson loops of six time-reversal invariant planes at (a) k 1 = 0.0, (b) k 1 = 0.5, (c) k 2 = 0.0, (d) k 2 = 0.5, (e) k 3 = 0.0, (f) k 3 = 0.5, where k 1 , k 2 , k 3 are in units of the reciprocal lattice vectors.</caption>
</figure>
<text><location><page_3><loc_52><loc_36><loc_92><loc_52></location>respectively. The surface band at E =-4 eV being deeply below E F can be ignored due to negligible contribution to the electronic properties of material. Due to TlSb having inversion-symmetry, its topology can be described by one strong topological index ν 0 and three weak indices ν 1 , ν 2 , ν 3 [41]. Thus we calculated the Wilson loops on six time-reversal invariant planes using WannierTools [39], and the results are shown in Fig. 4. According the the definition of Wilson loops [42, 43], the topological indices are (1;000) indicating that TlSb is a strong topological material.</text>
<text><location><page_3><loc_52><loc_12><loc_92><loc_35></location>Second, we focus on the SC emerging in the Dirac semimetal TlSb discovered first by us. Figure 5(a) displays the temperature dependence of resistivity, ρ ( T ), measured at zero field. With decreasing temperature, ρ xx decreases leisurely, exhibiting a poor metallic behavior, then drops to zero at 4.32 K, a superconductivity transition occurring with the mid-temperature T mid c = 4.38 K, ∆ T c =0.15 K. This superconducting transition is also confirmed by the susceptibility measurement. Figure 5(b) presents the temperature dependence of susceptibility, χ ( T ), measured at H = 5 Oe with both zero-field cooling (ZFC) and field cooling (FC) process. It is clear that a sharp diamagnetic transition emerges at 4.3 K, and the complete diamagnetism (4 πχ ∼ -1) below T c indicates the bulk superconductivity being from TlSb, since Sb element is non-superconducting at ambient pressure.</text>
<text><location><page_3><loc_52><loc_9><loc_92><loc_11></location>Figure 6(a) shows the field dependence of magnetization, M ( H ), measured at 2 K for a TlSb sample, exhibit-</text>
<text><location><page_4><loc_12><loc_93><loc_14><loc_95></location>(a)</text>
<text><location><page_4><loc_10><loc_88><loc_13><loc_88></location>)</text>
<text><location><page_4><loc_10><loc_87><loc_13><loc_88></location>m</text>
<text><location><page_4><loc_10><loc_86><loc_13><loc_87></location>c</text>
<text><location><page_4><loc_10><loc_83><loc_13><loc_83></location>(</text>
<text><location><page_4><loc_10><loc_82><loc_14><loc_82></location>/s114</text>
<text><location><page_4><loc_8><loc_64><loc_13><loc_65></location>/s99</text>
<text><location><page_4><loc_10><loc_64><loc_13><loc_64></location>π</text>
<text><location><page_4><loc_9><loc_63><loc_12><loc_64></location>4</text>
<figure>
<location><page_4><loc_11><loc_52><loc_47><loc_93></location>
<caption>Figure 5. (a) The temperature dependence of resistivity, ρ ( T ), of TlSb. The inset: enlarged ρ ( T ) near the superconducting transition. (b) The temperature dependence of magnetic susceptibility, χ ( T ), measured at H = 5 Oe.</caption>
</figure>
<text><location><page_4><loc_9><loc_29><loc_49><loc_42></location>ing a hysteresis, which indicates that TlSb is a type-II superconductor. Then we measured the M ( H ) at various temperatures below 4.5 K, as shown in Fig. 6(b). The lower critical field H c 1 ( T ) can be estimated by the field, at which M ( H ) curve starts to deviate from the linear relationship. The obtained H c 1 ( T ) is shown in the inset of Fig. 6(b), the lower critical field at zero temperature H c 1 (0) = 148 Oe was obtained by the fitting using the GL relationship:</text>
<formula><location><page_4><loc_19><loc_25><loc_49><loc_28></location>H c 1 ( T ) = H c 1 (0)(1 -( T T c ) 2 ) (1)</formula>
<text><location><page_4><loc_9><loc_8><loc_49><loc_23></location>In order to get the upper critical field H c 2 (0), we measured the superconducting transition temperature ( T mid c ) at various applied magnetic fields. As shown in the inset of Fig. 7, the T c decreases, and the transition width ∆ T c increases with increasing magnetic field. By using the GL formula H c 2 ( T ) = H c 2 (0)(1t 2 )/(1+ t 2 ), where t is the normalized temperature t = T / T c , to fit the H c 2 ( T ) data, the zero temperature upper critical field H c 2 (0) = 1.12 T was obtained, which is much lower than the Pauli limit field H P c 2 (0) = 1.86 T c = 8.18</text>
<text><location><page_4><loc_55><loc_92><loc_57><loc_94></location>(a)</text>
<text><location><page_4><loc_53><loc_87><loc_56><loc_87></location>)</text>
<text><location><page_4><loc_53><loc_86><loc_56><loc_87></location>g</text>
<text><location><page_4><loc_53><loc_85><loc_56><loc_86></location>/</text>
<text><location><page_4><loc_53><loc_85><loc_56><loc_85></location>u</text>
<text><location><page_4><loc_53><loc_84><loc_56><loc_85></location>m</text>
<text><location><page_4><loc_53><loc_83><loc_56><loc_84></location>e</text>
<text><location><page_4><loc_53><loc_82><loc_56><loc_83></location>(</text>
<text><location><page_4><loc_53><loc_81><loc_56><loc_82></location>M</text>
<text><location><page_4><loc_52><loc_63><loc_54><loc_63></location>)</text>
<text><location><page_4><loc_52><loc_62><loc_54><loc_63></location>g</text>
<text><location><page_4><loc_52><loc_62><loc_54><loc_62></location>/</text>
<text><location><page_4><loc_52><loc_61><loc_54><loc_62></location>u</text>
<text><location><page_4><loc_52><loc_60><loc_54><loc_61></location>m</text>
<text><location><page_4><loc_52><loc_59><loc_54><loc_60></location>e</text>
<text><location><page_4><loc_52><loc_59><loc_54><loc_59></location>(</text>
<text><location><page_4><loc_52><loc_57><loc_54><loc_58></location>M</text>
<figure>
<location><page_4><loc_54><loc_46><loc_90><loc_93></location>
<caption>Figure 6. (a) Field dependence of magnetization M ( H ) measured at 2 K. (b) The low field magnetization of TlSb at different temperatures. The dashed line indicates the initial linear magnetization curve. The inset shows the temperature dependence of lower critical field, H c 1 ( T ) determined by the magnetization curve deviating from linear. The red line is the H c 1 ( T ) fitted by GL relation</caption>
</figure>
<text><location><page_4><loc_52><loc_23><loc_92><loc_31></location>T. Then, the GL coherence length ξ GL (0) = 15.3 nm was estimated by using the formula H c 2 (0) = Φ 0 /2 πξ 2 GL , where Φ 0 is the quantum flux ( h /2 e ). The penetration depth λ GL (0) = 162 nm was estimated by using the formula H c 1 (0) = (Φ 0 /4 πλ 2 GL (0))ln( λ GL (0)/ ξ GL (0)), and the GL parameter κ GL = λ GL (0)/ ξ GL (0) = 10.6.</text>
<text><location><page_4><loc_52><loc_9><loc_92><loc_22></location>We also measured the specific heat as a function of temperature, C p ( T ), for TlSb in the temperature range of 0.5 - 5 K at both zero field and 3 T, respectively, as shown in Fig. 8. It is clear that the zero-field C p ( T ), compared with the C p ( T ) measured at 3 T ( > H c 2 , in this case bulk SC is completely suppressed), exhibits a small and broad peak near T c , corresponding to the superconducting transition. No other anomaly was observed except for the peak near T c = 4.38 K, indicating</text>
<text><location><page_4><loc_58><loc_93><loc_59><loc_95></location>3</text>
<text><location><page_5><loc_9><loc_87><loc_12><loc_88></location>)</text>
<text><location><page_5><loc_9><loc_86><loc_12><loc_87></location>T</text>
<text><location><page_5><loc_9><loc_86><loc_12><loc_86></location>(</text>
<text><location><page_5><loc_9><loc_83><loc_12><loc_85></location>H</text>
<text><location><page_5><loc_8><loc_82><loc_13><loc_83></location>/s109</text>
<figure>
<location><page_5><loc_11><loc_71><loc_47><loc_93></location>
<caption>Figure 7. The temperature dependence of upper critical field H c 2 determined from resistivity measurements. The inset shows the low temperature ρ ( T ) curves measured at various magnetic fields.</caption>
</figure>
<text><location><page_5><loc_14><loc_62><loc_17><loc_64></location>150</text>
<figure>
<location><page_5><loc_11><loc_38><loc_47><loc_62></location>
<caption>Figure 8. The temperature dependence of specific heat of TlSb measured at 0 T (black circles) and 3 T (red circles) plotted as C p versus T 2 . The solid blue line is a fit by the Debye model. The inset shows normalized electronic specific heat C el / γ n T versus T / T c at zero field.</caption>
</figure>
<text><location><page_5><loc_9><loc_58><loc_12><loc_58></location>)</text>
<text><location><page_5><loc_9><loc_57><loc_11><loc_58></location>2</text>
<text><location><page_5><loc_9><loc_57><loc_11><loc_57></location>-</text>
<text><location><page_5><loc_9><loc_56><loc_12><loc_57></location>K</text>
<text><location><page_5><loc_9><loc_55><loc_11><loc_55></location>1</text>
<text><location><page_5><loc_9><loc_54><loc_11><loc_55></location>-</text>
<text><location><page_5><loc_9><loc_54><loc_12><loc_54></location>l</text>
<text><location><page_5><loc_9><loc_53><loc_12><loc_54></location>o</text>
<text><location><page_5><loc_9><loc_52><loc_12><loc_53></location>m</text>
<text><location><page_5><loc_9><loc_51><loc_12><loc_52></location>J</text>
<text><location><page_5><loc_9><loc_50><loc_12><loc_51></location>m</text>
<text><location><page_5><loc_9><loc_49><loc_12><loc_50></location>(</text>
<text><location><page_5><loc_9><loc_48><loc_12><loc_49></location>T</text>
<text><location><page_5><loc_9><loc_47><loc_12><loc_48></location>/</text>
<text><location><page_5><loc_9><loc_46><loc_12><loc_47></location>C</text>
<text><location><page_5><loc_9><loc_20><loc_49><loc_27></location>that no Tl impurities ( T c = 2.39 K) emerge in our sample although a small amount of non-superconducting Sb impurities was detected in the XRD. We fitted the low temperature C p ( T ) data measured at 3 T using the Debye model:</text>
<formula><location><page_5><loc_19><loc_17><loc_49><loc_19></location>C p / T = γ n + β 3 T 2 + β 5 T 4 (2)</formula>
<text><location><page_5><loc_9><loc_8><loc_49><loc_16></location>where γ n is the Sommerfeld Coefficient, both the β 3 T 3 and β 5 T 5 are the phonon contributions to specific heat. The parameters γ n = 5.56 mJ mol -1 K -2 , β 3 = 1.39 mJ mol -1 K -4 , β 5 = 0.44 mJ mol -1 K -6 were obtained. The inset of Fig. 8 shows the normalized ∆ C p γ n T</text>
<text><location><page_5><loc_13><loc_93><loc_16><loc_95></location>1.6</text>
<table>
<location><page_5><loc_54><loc_73><loc_89><loc_91></location>
<caption>TABLE I. Superconducting parameters of TlSb</caption>
</table>
<text><location><page_5><loc_52><loc_46><loc_92><loc_71></location>= C p (0T) -C p (3T) γ n T as a function of the normalized temperature t = T / T c . The bulk superconducting temperature T c = 4.3 K was estimated by a entropy-balance method, consistent with the results from the resistivity and susceptibility measurements mentioned above. The normalized specific heat jump ∆ C el / γ n T c =1.42 was estimated, almost the same with the predicted value (1.43) by the Bardeen-Cooper-Schrieffer (BCS) theory [44], indicating that TlSb is a s -wave phonon-mediated superconductor. The Debye temperature Θ D = 141 K was estimated by using the formula Θ D = (12 π 4 NR /5 β 3 ) 1 3 , where N = 2 is the number of atoms in an unit cell and the R = 8.314 J mol -1 K -1 is the molar gas constant. Using the obtained Θ D and T c values, we calculated the electron-phonon coupling constant λ ep by using the McMillan formula [45]:</text>
<formula><location><page_5><loc_57><loc_42><loc_92><loc_46></location>λ ep = 1 . 04 + µ ∗ ln(Θ D / 1 . 45 T c ) (1 -0 . 62 µ ∗ ) ln(Θ D / 1 . 45 T c ) -1 . 04 (3)</formula>
<text><location><page_5><loc_52><loc_30><loc_92><loc_42></location>where µ ∗ = 0.13 is a typical value of the Coulomb repulsion pseudopotential for the intermetallic superconductors. The obtained λ ep = 0.78 value is comparable to that of other superconductors such as PbTaSe 2 ( λ ep = 0.74) [46] and Nb 0 . 18 Re 0 . 82 ( λ ep = 0.73) [47], suggesting TlSb is a moderately coupled superconductor. The obtained superconducting parameters are summarised in Table I.</text>
<section_header_level_1><location><page_5><loc_64><loc_26><loc_79><loc_27></location>IV. CONCLUSION</section_header_level_1>
<text><location><page_5><loc_52><loc_9><loc_92><loc_23></location>In summary, the calculations of the electronic band structure, the FS and the surface states show that TlSb with a CsCl-type structure is a Dirac semimetal. We measured the resistivity, magnetization, specific heat for the polycrystalline TlSb sample. We first found that TlSb is a type-II superconductor with T c = 4.38 K, H c 1 (0) = 148 Oe, H c 2 = 1.12 T and κ GL = 10.6. The specific heat results demonstrate it to be a moderately coupled s -wave superconductor. Although we can not determine which bands near E F to be responsible for</text>
<text><location><page_6><loc_9><loc_87><loc_49><loc_93></location>SC, the coexistence of bulk SC with s -wave symmetry and the Dirac fermions on the surface in a single TlSb compound provides an opportunity to realize Majorana zero energy mode.</text>
<section_header_level_1><location><page_6><loc_19><loc_82><loc_39><loc_83></location>ACKNOWLEDGEMENTS</section_header_level_1>
<text><location><page_6><loc_9><loc_70><loc_49><loc_80></location>This research is supported by the National Key Program of China under Grant No. 2016YFA0300402 and the National Natural Science Foundation of China (Grants No. NSFC-12074335 and 11974095) the Fundamental Research Funds for the Central Universities, an open program from the National Lab of Solid State Microstructures of Nanjing University (Grant No. M32025).</text>
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</document> | [{"title": "Bulk Superconductivity in the Dirac Semimetal TlSb", "content": "Yuxing Zhou, 1 Bin Li, 2 Zhefeng Lou, 1 Huancheng Chen, 1 Qin Chen, 1 Binjie Xu, 1 Chunxiang Wu, 1 Jianhua Du, 3 Jinhu Yang, 4 Hangdong Wang, 4 and Minghu Fang 1, 5, Department of Physics, Zhejiang University, Hangzhou 310027 , China 2 New Energy Technology Engineering Laboratory of Jiangsu Province and School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023 , China 3 Department of Applied Physics, China Jiliang University, Hangzhou 310018 , China 4 Department of Physics, Hangzhou Normal University, Hangzhou 310036 , China 5 Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093 , China \u2217 1 (Dated: December 8, 2020) A feasible strategy to realize the Majorana fermions is searching for a simple compound with both bulk superconductivity and Dirac surface states. In this paper, we performed calculations of electronic band structure, the Fermi surface and surface states, as well as measured the resistivity, magnetization, specific heat for TlSb compound with a CsCl-type structure. The band structure calculations show that TlSb is a Dirac semimetal when spin-orbit coupling is taken into account. Meanwhile, we first found that TlSb is a type-II superconductor with T c = 4.38 K, H c 1 (0) = 148 Oe, H c 2 (0) = 1.12 T and \u03ba GL = 10.6, and confirmed it to be a moderately coupled s -wave superconductor. Although we can not determine which bands near the Fermi level E F to be responsible for superconductivity, its coexistence with the topological surface states implies that TlSb compound may be a simple material platform to realize the fault-tolerant quantum computations.", "pages": [1]}, {"title": "I. INTRODUCTION", "content": "Topological superconductors host Majorana fermions described by a real wave function, providing protection for quantum computations [1]. So, realizing topological superconductivity (TSC) has became one of the most interesting topics in the condensed matter physics in the past decades. According to the discussion in Ref. [2], there are intrinsic and artificial engineered topological superconductors. For the intrinsic, the topological nontrivial gap function naturally shows up. Sr 2 RuO 4 [3-6] is the first proposed topological superconductor although the existence of chiral p -wave superconductivity (SC) is still under debate. Cu x Bi 2 Se 3 [7], as the first material to show SC ( T c \u223c 4 K) upon doping charge carrier into a topological insulator (TI), is a promising ground to look for two dimensional (2D) TSC due to the topological surface states surviving in TI even when carriers are doped. Many experiments, such as the conductance spectroscopy [8], nuclear manetic resonance (NMR) measurements of the Knight-shift [9], and specific heat in applied magnetic fields [10], have already given evidences for TSC seem emerging in Cu x Bi 2 Se 3 . The nematic SC discovered in Cu x Bi 2 Se 3 [9, 10] was also observed in the similar superconductors derived from Bi 2 Se 3 , such as in Sr x Bi 2 Se 3 [11], Nb x Cu 2 Se 3 [12]. Sn 1 -x In x Te [13] is another superconductor upon doping charge carriers into a topological crystalline insulator. In the cleanest sample ( x \u223c 0.04) with the lowest T c (1.2 K), a pronounced zero-bias conductance peak (ZBCP) similar to that in Cu x Bi 2 Se 3 has been observed by point contact spectroscopy [13]. Another is the artificial engineered TSC in hybrid structures. According to the idea proposed by Fu and Kane [14], if s -wave pairing is imposed on the topological sur- states of a three dimensional (3D) TI through superconducting proximity effect, the resulting superconducting state should be a 2D p -wave SC harboring a Majorana zero mode in the vortex core. Experimentally, proximityinduced SC on the surface of 3D TIs has been studied by many groups[15-21]. The observation [22] of 4 \u03c0 -period Josephson supercurrent in 3D HgTe TI is encouraging, although, it is difficult to elucidate the topological nature of the induced 2D SC. However, there are controversies about TSC emerging and the observed ZBCP being a Majorana zero-energy mode (MZM) in the doped topological material. To realize TSC through a superconducting proximity effect in hybrid structures has many engineering challenges. A feasible way is to realize TSC in a simple compound, in which both the topological surface state and the bulk SC coexist, thus Majorana fermions emerge at the edge of superconductor. Recently, the observations of MZM in the core of vortex [23, 24], at the end of the atomic defect line [25], and near the Bi islands [26] and the interstitial Fe atoms [27] in the simplest Fe-based superconductor Fe 1+ y Te 0 . 5 Se 0 . 5 ( T c = 14 K) [28] motivate us to search for the similar system. TlSb crystallizes in a cubic CsCl structure with space group Pm \u00af 3 m (No. 221) (as shown in the inset of Fig. 1), from this structure a large number of topological semimetal/metals (TMs) were designed [29], ranging from triple nodal points, type-I nodal lines, and critical type nodal lines to hybrid nodal lines. For example, CaTe is a typical type-I nodal line and Dirac TM[30]; YIr is a typical triple-nodal-point TM [29], YMg possesses multiple types of band crossing [29]. Therefore, we tried to grow TlSb crystals for studying its topological natures and SC, unfortunately, only polycrystalline TlSb samples were obtained. In this paper, we performed calculations of the electronic band structure, the Fermi surface and the surface states on (001) plane, as well as measured resistivity, magnetization and specific heat for the polycrystalline TlSb sample. The band structure calculations show TlSb is a Dirac semimetal with 4-fold degenerate nodes near \u0393 and R points. It is also found that TlSb is a typeII superconductor with the superconducting transition temperature T c = 4.38 K, the lower critical field H c 1 (0) = 148 Oe, and the upper critical field H c 2 (0) = 1.12 T and the Ginzburg-Landau (GL) parameter \u03ba GL = 10.6. The obtained specific heat jump, \u2206 C el / \u03b3 n T c \u223c 1.42, indicates that TlSb is a conventional phonon-mediated superconductor with s -wave superconducting symmetry. These results indicate that both s -wave SC and surface states coexist in TlSb, whether the Majorana fermions emerge or not on the edges is needed to confirm in the future.", "pages": [1, 2]}, {"title": "II. EXPERIMENTAL", "content": "Polycrystalline TlSb samples were synthesized by a peritectic reaction method. The mixture of stoichiometric high purity Tl (99.99%) chunk and Sb (99.999%) powder was placed in an alumina crucible, sealed in an evacuated quartz tube and heated at 450 \u00b7 C for 20 hrs, then decreases to 195 \u00b7 C waiting for the mixture melting completely. To avoid the decomposition of obtain TlSb phase at 191 \u00b7 C [31], the quartz tube was quenched to room temperature at 195 \u00b7 C, however, the obtained TlSb samples always contain a small amount of unreacted Sb impurities due to the precipitation of Sb before the peritectic reaction. The obtained TlSb alloy is easily to cut for the subsequent structure characterizations and property measurements. Polycrystalline x-ray diffraction (XRD) was carried out on a PANalytical diffractometer equiped with CuK \u03b1 radiation. The TlSb XRD pattern is shown in Fig. 1, in which the main peaks can be fitted by the CsCl-type structure with space group Pm \u00af 3 m . The lattice parameters a = b = c = 3.86(5) \u02da A were obtained by the Rietveld refinement by using general structure analysis system (GSAS) [32]. A rectangular bar of the sample was cut for the magnetization and resistivity mesurements, which were performed on a magnetic property measurement system (Quantum Design, MPMS - 7 T) and a physical property measurement system (Quantum Design, PPMS - 9 T), respectively. The band structure was calculated by using density function theory (DFT) with the WIEN2k package [33]. Generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) [34] was employed for the exchange correlation potential calculations. A cutoff energy of 520 eV and a 13 \u00d7 15 \u00d7 15 k -point mesh were used to perform the bulk calculations. The Fermi surface (FS) was performed with WannierTools [35] package which is based on the maximally localized Wannier function tight-binding model [36-38] constructed by using the Wannier90 [39] package.", "pages": [2]}, {"title": "III. RESULTS AND DISCUSSIONS", "content": "We first discuss the electronic band structure without considering spin-orbit coupling (SOC). As shown in Fig. 2(a), both conduction (red) and valence (blue) band cross the Fermi level E F . At the high symmetry \u0393 and R points, there are three bands crossing, with a threefold degenerate at 0.8 eV and 2 eV below E F , respectively. However, when SOC is taken into account [see Fig. 2(b)], gaps open and leaves two twofold degenerate bands at both high symmetry points. Since both time-reversal and inversion symmetries are present, no spin-splitting occurs and then the twofold degenerated bands come together to fourfold degenerate points, indicating TlSb is a Dirac semimetal. We also calculated the density of state (DOS), as shown in the right panel of Fig. 2(b), the DOS at E F is mainly contributed by Sb orbits. To further clarify the band structure of TlSb, we calculated its 3D bulk FS of the first Brillouin zone (BZ) as shown in Fig. 2(d), exhibiting very complex 3D characteristics. Figure 2(e) presents the FS on the k z = 0 plane, which is the cross section passing the \u0393 point of the 3D FS. Figure 2(f) displays the energy dispersion in the k x -k y plane, in which the Dirac dispersion is clearly seen, demonstrating further that TlSb is a Dirac semimetal. Then we calculated the surface states on (001) plane by using a surface Green's function method [40]. As shown in Fig. 3(a), the projected Dirac points are hidden in the continuous bulk states, the surface states are shown as the red curves. The (001) surface energy contour is shown in Fig. 3(b), (c) and (d) with E = E F , E = -1.5 eV and E = -4 eV, respectively. The surface band at E =-4 eV being deeply below E F can be ignored due to negligible contribution to the electronic properties of material. Due to TlSb having inversion-symmetry, its topology can be described by one strong topological index \u03bd 0 and three weak indices \u03bd 1 , \u03bd 2 , \u03bd 3 [41]. Thus we calculated the Wilson loops on six time-reversal invariant planes using WannierTools [39], and the results are shown in Fig. 4. According the the definition of Wilson loops [42, 43], the topological indices are (1;000) indicating that TlSb is a strong topological material. Second, we focus on the SC emerging in the Dirac semimetal TlSb discovered first by us. Figure 5(a) displays the temperature dependence of resistivity, \u03c1 ( T ), measured at zero field. With decreasing temperature, \u03c1 xx decreases leisurely, exhibiting a poor metallic behavior, then drops to zero at 4.32 K, a superconductivity transition occurring with the mid-temperature T mid c = 4.38 K, \u2206 T c =0.15 K. This superconducting transition is also confirmed by the susceptibility measurement. Figure 5(b) presents the temperature dependence of susceptibility, \u03c7 ( T ), measured at H = 5 Oe with both zero-field cooling (ZFC) and field cooling (FC) process. It is clear that a sharp diamagnetic transition emerges at 4.3 K, and the complete diamagnetism (4 \u03c0\u03c7 \u223c -1) below T c indicates the bulk superconductivity being from TlSb, since Sb element is non-superconducting at ambient pressure. Figure 6(a) shows the field dependence of magnetization, M ( H ), measured at 2 K for a TlSb sample, exhibit- (a) ) m c ( /s114 /s99 \u03c0 4 ing a hysteresis, which indicates that TlSb is a type-II superconductor. Then we measured the M ( H ) at various temperatures below 4.5 K, as shown in Fig. 6(b). The lower critical field H c 1 ( T ) can be estimated by the field, at which M ( H ) curve starts to deviate from the linear relationship. The obtained H c 1 ( T ) is shown in the inset of Fig. 6(b), the lower critical field at zero temperature H c 1 (0) = 148 Oe was obtained by the fitting using the GL relationship: In order to get the upper critical field H c 2 (0), we measured the superconducting transition temperature ( T mid c ) at various applied magnetic fields. As shown in the inset of Fig. 7, the T c decreases, and the transition width \u2206 T c increases with increasing magnetic field. By using the GL formula H c 2 ( T ) = H c 2 (0)(1t 2 )/(1+ t 2 ), where t is the normalized temperature t = T / T c , to fit the H c 2 ( T ) data, the zero temperature upper critical field H c 2 (0) = 1.12 T was obtained, which is much lower than the Pauli limit field H P c 2 (0) = 1.86 T c = 8.18 (a) ) g / u m e ( M ) g / u m e ( M T. Then, the GL coherence length \u03be GL (0) = 15.3 nm was estimated by using the formula H c 2 (0) = \u03a6 0 /2 \u03c0\u03be 2 GL , where \u03a6 0 is the quantum flux ( h /2 e ). The penetration depth \u03bb GL (0) = 162 nm was estimated by using the formula H c 1 (0) = (\u03a6 0 /4 \u03c0\u03bb 2 GL (0))ln( \u03bb GL (0)/ \u03be GL (0)), and the GL parameter \u03ba GL = \u03bb GL (0)/ \u03be GL (0) = 10.6. We also measured the specific heat as a function of temperature, C p ( T ), for TlSb in the temperature range of 0.5 - 5 K at both zero field and 3 T, respectively, as shown in Fig. 8. It is clear that the zero-field C p ( T ), compared with the C p ( T ) measured at 3 T ( > H c 2 , in this case bulk SC is completely suppressed), exhibits a small and broad peak near T c , corresponding to the superconducting transition. No other anomaly was observed except for the peak near T c = 4.38 K, indicating 3 ) T ( H /s109 150 ) 2 - K 1 - l o m J m ( T / C that no Tl impurities ( T c = 2.39 K) emerge in our sample although a small amount of non-superconducting Sb impurities was detected in the XRD. We fitted the low temperature C p ( T ) data measured at 3 T using the Debye model: where \u03b3 n is the Sommerfeld Coefficient, both the \u03b2 3 T 3 and \u03b2 5 T 5 are the phonon contributions to specific heat. The parameters \u03b3 n = 5.56 mJ mol -1 K -2 , \u03b2 3 = 1.39 mJ mol -1 K -4 , \u03b2 5 = 0.44 mJ mol -1 K -6 were obtained. The inset of Fig. 8 shows the normalized \u2206 C p \u03b3 n T 1.6 = C p (0T) -C p (3T) \u03b3 n T as a function of the normalized temperature t = T / T c . The bulk superconducting temperature T c = 4.3 K was estimated by a entropy-balance method, consistent with the results from the resistivity and susceptibility measurements mentioned above. The normalized specific heat jump \u2206 C el / \u03b3 n T c =1.42 was estimated, almost the same with the predicted value (1.43) by the Bardeen-Cooper-Schrieffer (BCS) theory [44], indicating that TlSb is a s -wave phonon-mediated superconductor. The Debye temperature \u0398 D = 141 K was estimated by using the formula \u0398 D = (12 \u03c0 4 NR /5 \u03b2 3 ) 1 3 , where N = 2 is the number of atoms in an unit cell and the R = 8.314 J mol -1 K -1 is the molar gas constant. Using the obtained \u0398 D and T c values, we calculated the electron-phonon coupling constant \u03bb ep by using the McMillan formula [45]: where \u00b5 \u2217 = 0.13 is a typical value of the Coulomb repulsion pseudopotential for the intermetallic superconductors. The obtained \u03bb ep = 0.78 value is comparable to that of other superconductors such as PbTaSe 2 ( \u03bb ep = 0.74) [46] and Nb 0 . 18 Re 0 . 82 ( \u03bb ep = 0.73) [47], suggesting TlSb is a moderately coupled superconductor. The obtained superconducting parameters are summarised in Table I.", "pages": [2, 3, 4, 5]}, {"title": "IV. CONCLUSION", "content": "In summary, the calculations of the electronic band structure, the FS and the surface states show that TlSb with a CsCl-type structure is a Dirac semimetal. We measured the resistivity, magnetization, specific heat for the polycrystalline TlSb sample. We first found that TlSb is a type-II superconductor with T c = 4.38 K, H c 1 (0) = 148 Oe, H c 2 = 1.12 T and \u03ba GL = 10.6. The specific heat results demonstrate it to be a moderately coupled s -wave superconductor. Although we can not determine which bands near E F to be responsible for SC, the coexistence of bulk SC with s -wave symmetry and the Dirac fermions on the surface in a single TlSb compound provides an opportunity to realize Majorana zero energy mode.", "pages": [5, 6]}, {"title": "ACKNOWLEDGEMENTS", "content": "This research is supported by the National Key Program of China under Grant No. 2016YFA0300402 and the National Natural Science Foundation of China (Grants No. NSFC-12074335 and 11974095) the Fundamental Research Funds for the Central Universities, an open program from the National Lab of Solid State Microstructures of Nanjing University (Grant No. M32025). 336 , 52 (2012).", "pages": [6]}] |
2017PhRvD..96f4022C | https://arxiv.org/pdf/1708.04715.pdf | <document>
<section_header_level_1><location><page_1><loc_31><loc_92><loc_69><loc_93></location>Static phantom wormholes of finite size</section_header_level_1>
<text><location><page_1><loc_44><loc_89><loc_57><loc_90></location>Mauricio Cataldo ∗</text>
<text><location><page_1><loc_34><loc_84><loc_66><loc_89></location>Departamento de F'ısica, Facultad de Ciencias, Universidad del B'ıo-B'ıo, Avenida Collao 1202, Casilla 15-C, Concepci'on, Chile, and Grupo de Cosmolog'ıa y Gravitaci'on-UBB.</text>
<section_header_level_1><location><page_1><loc_44><loc_80><loc_57><loc_82></location>Fabian Orellana †</section_header_level_1>
<text><location><page_1><loc_32><loc_74><loc_68><loc_80></location>Departamento de F'ısica, Universidad de Concepci'on, Casilla 160-C, Concepci'on, Chile; and Facultad de Ingenier'ıa y Tecnolog'ıa, Universidad San Sebasti'an, Lientur 1457, Concepci'on 4080871, Chile.</text>
<text><location><page_1><loc_41><loc_71><loc_60><loc_72></location>(Dated: September 30, 2018)</text>
<text><location><page_1><loc_18><loc_60><loc_83><loc_70></location>In this paper we derive new static phantom traversable wormholes by assuming a shape function with a quadratic dependence on the radial coordinate r . We mainly focus our study on wormholes sustained by exotic matter with positive energy density (as seen by any static observer) and a variable equation of state p r /ρ < -1, dubbed phantom matter. Among phantom wormhole spacetimes extending to infinity, we show that a quadratic shape function allows us to construct static spacetimes of finite size, composed by a phantom wormhole connected to an anisotropic spherically symmetric distribution of dark energy. The wormhole part of the full spacetime does not fulfill the dominant energy condition, while the dark energy part does.</text>
<text><location><page_1><loc_18><loc_57><loc_45><loc_58></location>PACS numbers: 04.20.Jb, 04.70.Dy,11.10.Kk</text>
<section_header_level_1><location><page_1><loc_20><loc_53><loc_37><loc_54></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_20><loc_49><loc_50></location>The accelerated expansion of the universe is one of the most exciting and significant discoveries in modern cosmology. In the framework of general relativity, dark energy, which has an equation of state satisfying the relation -1 < p/ρ < -1 / 3, is the most accepted hypothesis to explain the observed acceleration. However, there are observational evidences that the cosmic fluid leading to the acceleration of the universe may satisfy also an equation of state of the form p/ρ < -1, with positive energy density. A cosmic fluid characterized by such an equation of state is dubbed phantom energy, and has received increased attention among theorists in cosmology, since if this type of source dominates the cosmic expansion, the universe may end in a Big Rip singularity [1] (the positive energy density becomes infinite in finite time, as well as the pressure). This phantom energy has a very strong negative pressure and violates the dominant energy condition (DEC), which can be written as ρ ≥ 0 and -ρ ≤ p ≤ ρ . In such a way, late cosmological evidences cast a serious doubt on the validity of the energy conditions.</text>
<text><location><page_1><loc_9><loc_15><loc_49><loc_19></location>Although the cosmic phantom energy is a time dependent matter source, conceptually it can be also used in study of static gravitational configurations. An interest-</text>
<text><location><page_1><loc_52><loc_39><loc_92><loc_54></location>ing and useful application is the construction of static wormhole spacetimes, which need to be sustained by non-standard matter fields, which violates DEC. Note that the cosmic phantom energy is a homogeneous field with isotropic pressure. Since, wormholes are inhomogeneous spacetimes, an extension of phantom energy must be carried out. Specifically, for static wormholes the phantom matter is considered as an inhomogeneous and anisotropic fluid, with radial pressure satisfying the relation ω = p r /ρ < -1.</text>
<text><location><page_1><loc_52><loc_32><loc_92><loc_39></location>In general, in spherically symmetric spacetimes the radial pressure and the lateral one are different, so one must require the model to satisfy the DEC specified by ρ ≥ 0 and -ρ ≤ p i ≤ ρ , where p i are the radial and lateral pressures.</text>
<text><location><page_1><loc_52><loc_16><loc_92><loc_32></location>The study of phantom wormholes involves mainly asymptotically flat phantom wormhole solutions [2], which extend from the throat to infinity. Non asymptotically flat phantom wormholes also have been studied. In Ref. [3] such wormholes are considered, and spacetimes extend from the throat to infinity, so they are glued to the external Schwarzschild solution. Asymptotically and non asymptotically flat phantom wormholes are also discussed in Ref. [4]. Non asymptotically flat phantom wormholes also have been studied in three dimensions [5]. All these spacetimes also extend to spatial infinity.</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_15></location>In Ref. [6] the notion of phantom energy is also extended to inhomogeneous and anisotropic spherically symmetric spacetimes. The author finds an exact wormhole solution and shows that a spatial distribution of the phantom energy is mainly limited to the vicinity of the</text>
<text><location><page_2><loc_9><loc_92><loc_21><loc_93></location>wormhole throat.</text>
<text><location><page_2><loc_9><loc_82><loc_49><loc_92></location>It is interesting to note that evolving wormholes supported by phantom energy also have been studied in the presence of a cosmological constant [7] and without it [8]. In both cases the equation of state of the radial pressure has the form ω r = p r ( t, r ) /ρ ( t, r ) < -1, with constant state parameter ω r (see also Ref. [9] for a slight generalization of dynamical phantom equation of state).</text>
<text><location><page_2><loc_9><loc_73><loc_49><loc_82></location>This paper presents phantom traversable wormholes by resorting to a quadratic shape function. For constructing them we use the conventional approach of Morris and Thorne based on the assumption of particular forms of the shape function b ( r ), and the redshift function φ ( r ), in the metric [10]</text>
<formula><location><page_2><loc_10><loc_67><loc_49><loc_72></location>ds 2 = e 2 φ ( r ) dt 2 -dr 2 1 -b ( r ) r -r 2 ( dθ 2 +sin 2 θdϕ 2 ) . (1)</formula>
<text><location><page_2><loc_9><loc_61><loc_49><loc_67></location>We assume that the matter source threading the wormhole is described by a single anisotropic fluid characterized by T ν µ = diag ( ρ, -p r , -p l , -p l ). Therefore, the Einstein field equations are given by</text>
<formula><location><page_2><loc_34><loc_57><loc_49><loc_60></location>κρ ( r ) = b ' r 2 , (2)</formula>
<formula><location><page_2><loc_22><loc_53><loc_49><loc_57></location>κp r ( r ) = 2 ( 1 -b r ) φ ' r -b r 3 , (3)</formula>
<formula><location><page_2><loc_13><loc_46><loc_49><loc_50></location>[ φ '' + φ ' 2 -b ' r + b -2 r 2 r ( r -b ) φ ' -b ' r -b 2 r 2 ( r -b ) ] , (4)</formula>
<formula><location><page_2><loc_28><loc_50><loc_43><loc_53></location>κp l ( r ) = ( 1 -b r ) ×</formula>
<text><location><page_2><loc_9><loc_41><loc_49><loc_46></location>where ρ is the energy density, p r and p l are the radial and lateral pressures respectively, and the prime denotes the derivative d/dr .</text>
<text><location><page_2><loc_9><loc_33><loc_49><loc_41></location>The paper is organized as follows. In Sec. II we study Morris-Thorne wormholes characterized by a quadratic shape function. In Sec. III we discuss energy conditions and the positivity of energy density. In Sec. IV we construct phantom wormholes of finite size. In Sec. V we conclude with some remarks.</text>
<section_header_level_1><location><page_2><loc_9><loc_27><loc_48><loc_30></location>II. WORMHOLES WITH QUADRATIC SHAPE FUNCTIONS</section_header_level_1>
<text><location><page_2><loc_9><loc_23><loc_49><loc_25></location>Now we will study Morris-Thorne wormholes by using a quadratic shape function in the form</text>
<formula><location><page_2><loc_21><loc_20><loc_49><loc_21></location>b ( r ) = a 1 r 2 + a 2 r + a 3 (5)</formula>
<text><location><page_2><loc_9><loc_11><loc_49><loc_19></location>where a 1 , a 2 and a 3 are constant parameters. In order to have a wormhole we must impose the Morris-Thorne constraints on the shape function, so a 1 , a 2 and a 3 are not all free parameters, and they must satisfy specific constraints which we will now discuss.</text>
<text><location><page_2><loc_9><loc_8><loc_49><loc_11></location>First of all, the wormhole must have a minimum radius r = r 0 , where the wormhole throat is located. This</text>
<text><location><page_2><loc_52><loc_92><loc_77><loc_93></location>requirement is expressed by [10, 11]</text>
<formula><location><page_2><loc_67><loc_89><loc_92><loc_90></location>b ( r 0 ) = r 0 , (6)</formula>
<text><location><page_2><loc_52><loc_84><loc_92><loc_88></location>and r 0 is the minimum value of the radial coordinate r . On the other hand, the shape function must satisfy the condition</text>
<formula><location><page_2><loc_68><loc_80><loc_92><loc_83></location>b ( r ) r ≤ 1 , (7)</formula>
<text><location><page_2><loc_52><loc_77><loc_91><loc_79></location>in order to the metric (1) remains Lorentzian ( g rr ≤ 0).</text>
<text><location><page_2><loc_52><loc_75><loc_92><loc_77></location>Evaluating the shape function (5) at the throat, i.e. imposing the fulfilment requirement (6), we obtain</text>
<formula><location><page_2><loc_61><loc_70><loc_92><loc_73></location>b ( r ) = ( r -r 0 ) ( a 1 r -a 3 r 0 ) + r, (8)</formula>
<text><location><page_2><loc_52><loc_68><loc_76><loc_69></location>and the metric (1) takes the form</text>
<formula><location><page_2><loc_56><loc_58><loc_92><loc_67></location>ds 2 = e 2 φ ( r ) dt 2 -dr 2 ( 1 -r 0 r ) ( a 3 r 0 -a 1 r ) -r 2 ( dθ 2 +sin 2 θdφ 2 ) . (9)</formula>
<text><location><page_2><loc_52><loc_55><loc_92><loc_59></location>It becomes clear that traversable versions of Schwarzschild wormholes are obtained if a 1 = 0 and a 3 = r 0 .</text>
<text><location><page_2><loc_52><loc_51><loc_92><loc_55></location>Now for having a wormhole geometry the shape function must satisfy the flare-out condition, which is given by [10]</text>
<formula><location><page_2><loc_67><loc_46><loc_75><loc_50></location>b -b ' r 2 b 2 > 0 .</formula>
<text><location><page_2><loc_52><loc_44><loc_73><loc_46></location>From this equation we obtain</text>
<formula><location><page_2><loc_66><loc_41><loc_92><loc_43></location>a 3 -a 1 r 2 > 0 . (10)</formula>
<text><location><page_2><loc_52><loc_38><loc_92><loc_40></location>This condition allows us to classify and construct three classes of wormhole solutions. Namely:</text>
<section_header_level_1><location><page_2><loc_53><loc_36><loc_59><loc_37></location>Case1:</section_header_level_1>
<formula><location><page_2><loc_66><loc_33><loc_92><loc_35></location>a 1 < 0 , a 3 > 0 , (11)</formula>
<text><location><page_2><loc_52><loc_30><loc_81><loc_32></location>and wormhole exists for 0 < r 0 ≤ r < ∞ .</text>
<text><location><page_2><loc_53><loc_30><loc_59><loc_31></location>Case 2:</text>
<formula><location><page_2><loc_66><loc_27><loc_92><loc_28></location>a 1 < 0 , a 3 < 0 , (12)</formula>
<text><location><page_2><loc_52><loc_24><loc_69><loc_25></location>and wormhole exists for</text>
<formula><location><page_2><loc_64><loc_19><loc_92><loc_23></location>√ a 3 a 1 < r 0 < r < ∞ . (13)</formula>
<text><location><page_2><loc_53><loc_18><loc_59><loc_19></location>Case 3:</text>
<formula><location><page_2><loc_66><loc_15><loc_92><loc_16></location>a 1 > 0 , a 3 > 0 , (14)</formula>
<text><location><page_2><loc_52><loc_12><loc_69><loc_14></location>and wormhole exists for</text>
<formula><location><page_2><loc_64><loc_8><loc_92><loc_11></location>0 < r 0 < r < √ a 3 a 1 . (15)</formula>
<text><location><page_3><loc_9><loc_89><loc_49><loc_93></location>For a 1 > 0 and a 3 < 0 we have static spherically symmetric gravitational configurations which are not wormholes.</text>
<text><location><page_3><loc_9><loc_78><loc_49><loc_89></location>It should be noted that these constraints should be compatible with the condition (7), which is expressed in the form ( 1 -r 0 r ) ( a 3 r 0 -a 1 r ) > 0, as we can see from the radial metric component of Eq. (9). Since we have that r ≥ r 0 for the radial coordinate in a wormhole geometry, we conclude that also it is necessary to satisfy the inequation</text>
<formula><location><page_3><loc_23><loc_74><loc_49><loc_77></location>a 3 r 0 -a 1 r > 0 . (16)</formula>
<text><location><page_3><loc_9><loc_70><loc_49><loc_73></location>For the case 1 this constraint is satisfied automatically. For the case 2 we obtain that Eq. (16) implies that</text>
<formula><location><page_3><loc_24><loc_66><loc_49><loc_70></location>r > a 3 a 1 r 0 , (17)</formula>
<text><location><page_3><loc_9><loc_64><loc_34><loc_65></location>while for the case 3 we obtain that</text>
<formula><location><page_3><loc_24><loc_61><loc_49><loc_64></location>r < a 3 a 1 r 0 . (18)</formula>
<text><location><page_3><loc_9><loc_52><loc_49><loc_60></location>It is interesting to note that, in principle, we can make that ranges allowed by the flare-out condition coincide with ranges imposed by Eq. (16). This can be performed by requiring that √ a 3 a 1 = a 3 a 1 r 0 . Then we have that</text>
<formula><location><page_3><loc_24><loc_50><loc_49><loc_52></location>a 3 = a 1 r 2 0 . (19)</formula>
<text><location><page_3><loc_9><loc_41><loc_49><loc_49></location>If we put this expression into the radial metric component of the line element (9) we obtain that g -1 rr = a 1 ( r -r 0 ) 2 r -1 . This implies that the relation (19) may be applied only for the case 2, since for the case 3 we have that a 1 > 0 and the line element (9) becomes nonLorentzian.</text>
<section_header_level_1><location><page_3><loc_10><loc_35><loc_48><loc_37></location>III. THE POSITIVITY OF ENERGY DENSITY AND ENERGY CONDITIONS</section_header_level_1>
<text><location><page_3><loc_9><loc_22><loc_49><loc_33></location>Since we are interested in finding wormholes supported by phantom energy, we need to require the positivity of energy density. Physically, this requirement implies that everywhere any static observer will measure a positive energy density. Therefore, we shall study conditions which must satisfy the relevant parameters a 1 and a 3 in order to have a positive energy density. For the considered shape function (8) the energy density is given by</text>
<formula><location><page_3><loc_18><loc_17><loc_49><loc_20></location>ρ = a 1 r 0 (2 r -r 0 ) + r 0 -a 3 r 0 r 2 . (20)</formula>
<text><location><page_3><loc_9><loc_8><loc_49><loc_16></location>Case 1: We consider first the case a 1 < 0 and a 3 > 0 Note that for large values of radial coordinate we have that ρ ≈ 2 a 1 r , so if r → ∞ then ρ → -0. The expression (20) vanishes at r 3 = r 0 -a 3 + | a 1 | r 2 0 2 | a 1 | r 0 . In such a way, if r 3 > r 0 , then the energy density is positive for</text>
<figure>
<location><page_3><loc_61><loc_71><loc_85><loc_93></location>
<caption>FIG. 1: The figure shows the qualitative behavior of energy density for a 1 < 0 and a 3 > 0. For all plots the throat is located at r 0 . The dashed line describes the case for which ρ ( r 0 ) ≥ 0. The solid and dotted lines represent the cases where ρ ( r 0 ) = 0 and ρ ( r 0 ) ≤ 0, respectively. We can see that in the cases of solid and dotted line the energy density is everywhere negative, while for the dashed line the energy density is positive for r 0 ≤ r < r 3 and becomes negative for r > r 3 > r 0 .</caption>
</figure>
<text><location><page_3><loc_52><loc_49><loc_92><loc_54></location>r 0 < r < r 3 , while ρ ≤ 0 for r ≤ r 3 . The energy density is everywhere negative for r ≥ r 0 if a 3 > r 0 - | a 1 | r 2 0 , and vanishes at r 0 if a 3 = r 0 - | a 1 | r 2 0 .</text>
<text><location><page_3><loc_52><loc_47><loc_92><loc_50></location>Case 2: Now, Eq. (20) implies that if a 1 < 0 and a 3 < 0 we have two possibilities to be considered: if</text>
<formula><location><page_3><loc_64><loc_44><loc_92><loc_46></location>| a 3 |≤ r 2 0 | a 1 | -r 0 (21)</formula>
<text><location><page_3><loc_52><loc_41><loc_92><loc_43></location>then ρ ( r 0 ) ≤ 0 and the energy density is negative everywhere for r > r 0 , while if</text>
<formula><location><page_3><loc_64><loc_37><loc_92><loc_40></location>| a 3 | > r 2 0 | a 1 | -r 0 (22)</formula>
<text><location><page_3><loc_52><loc_33><loc_92><loc_37></location>then ρ ( r 0 ) > 0, and we obtain ρ ( r ) > 0 for r 0 < r < r 1 , and ρ ( r ) ≤ 0 for r ≥ r 1 , where r 1 = | a 3 | + r 0 + r 2 0 | a 1 | 2 r 0 | a 1 | .</text>
<text><location><page_3><loc_52><loc_31><loc_92><loc_34></location>Case 3: Lastly, for a 1 > 0 and a 3 > 0 we have also two possibilities to be considered: if</text>
<formula><location><page_3><loc_66><loc_28><loc_92><loc_30></location>a 3 ≤ r 2 0 a 1 + r 0 (23)</formula>
<text><location><page_3><loc_52><loc_25><loc_92><loc_27></location>then ρ ( r 0 ) ≥ 0 and the energy density ρ ( r ) > 0 for r > r 0 , while if</text>
<formula><location><page_3><loc_66><loc_22><loc_92><loc_24></location>a 3 > r 2 0 a 1 + r 0 (24)</formula>
<text><location><page_3><loc_52><loc_17><loc_92><loc_21></location>then ρ ( r 0 ) < 0 and we obtain ρ ( r ) < 0 for r 0 < r < r 2 , and ρ ≥ 0 for r ≥ r 2 , where r 2 = a 3 + r 2 0 a 1 -r 0 2 a 1 r 0 .</text>
<text><location><page_3><loc_52><loc_12><loc_92><loc_18></location>In conclusion, phantom wormholes may be constructed for all cases a 1 < 0, a 3 > 0; a 1 < 0, a 3 < 0 and a 1 > 0, a 3 > 0. In Figs. 1, 2 and 3 we show the qualitative behavior of the energy density for these cases.</text>
<text><location><page_3><loc_52><loc_8><loc_92><loc_12></location>Note that the energy density (20) may be rewritten in the form ρ = 2 a 1 r -a 3 + a 1 r 2 0 -r 0 r 0 r 2 . From this expression it</text>
<figure>
<location><page_4><loc_16><loc_72><loc_43><loc_94></location>
<caption>FIG. 2: The figure shows the qualitative behavior of energy density for a 1 < 0 and a 3 < 0. The throat is located at r 0 . The dotted line describes the case | a 3 | = r 2 0 | a 1 | -r 0 for which ρ ( r 0 ) = 0. The solid and dashed lines represent the cases | a 3 | ≤ r 2 0 | a 1 | -r 0 and | a 3 | ≥ r 2 0 | a 1 | -r 0 , for which we have at the throat ρ ( r 0 ) < 0 and ρ ( r 0 ) > 0, respectively. We can see that in the case of solid line the energy density is everywhere negative, while for the dashed line the energy density is positive for r 0 ≤ r < r 1 and becomes negative for r > r 1 .</caption>
</figure>
<figure>
<location><page_4><loc_16><loc_35><loc_42><loc_55></location>
<caption>FIG. 3: The figure shows the qualitative behavior of energy density for a 1 > 0 and a 3 > 0. The throat is located at r 0 . The dotted line represents the case a 3 = r 2 0 a 1 + r 0 for which ρ ( r 0 ) = 0. The solid and dashed lines represent the cases a 3 ≤ r 2 0 a 1 + r 0 and a 3 ≥ r 2 0 a 1 + r 0 , for which we have at the throat ρ ( r 0 ) > 0 and ρ ( r 0 ) < 0, respectively. We can see that in the case of solid line the energy density is everywhere positive, while for the dashed line the energy density is negative for r 0 ≤ r < r 2 and becomes positive for r > r 2 .</caption>
</figure>
<text><location><page_4><loc_9><loc_15><loc_49><loc_18></location>becomes clear that for positive a 1 and a 3 we have always a positive energy density by requiring</text>
<formula><location><page_4><loc_21><loc_12><loc_49><loc_14></location>a 3 + a 1 r 2 0 -r 0 ≤ 0 . (25)</formula>
<text><location><page_4><loc_9><loc_9><loc_49><loc_11></location>If this inequation is not satisfied then the energy density vanishes at some r , changing its sign, as described by</text>
<text><location><page_4><loc_52><loc_92><loc_58><loc_93></location>Eq. (24).</text>
<text><location><page_4><loc_52><loc_89><loc_92><loc_91></location>Now some words about the energy conditions. It is well known that the violation of the DEC</text>
<formula><location><page_4><loc_61><loc_85><loc_92><loc_87></location>ρ ≥ 0 , ρ + p r ≥ 0 , ρ + p l ≥ 0 (26)</formula>
<text><location><page_4><loc_52><loc_77><loc_92><loc_84></location>is a necessary condition for a static wormhole to exist. It is interesting to note that for the Morris-Thorne metric (1), if φ ( r ) = const , the strong energy condition ρ + p total ≥ 0 is satisfied, since the relation ρ + p r +2 p l = 0 is everywhere valid.</text>
<text><location><page_4><loc_52><loc_69><loc_92><loc_76></location>In order to discuss DEC, for simplicity, we shall consider the zero-tidal-force wormhole version of these quadratic wormholes (i.e. φ ( r ) = const ). For a such wormhole the energy density is defined by Eq. (20), and the pressures are given by</text>
<formula><location><page_4><loc_60><loc_63><loc_92><loc_67></location>p r = -( r -r 0 ) ( a 1 r -a 3 r 0 ) + r r 3 , (27)</formula>
<formula><location><page_4><loc_60><loc_60><loc_92><loc_63></location>p l = a 3 -a 1 r 2 2 r 3 . (28)</formula>
<text><location><page_4><loc_52><loc_48><loc_92><loc_59></location>By rewriting the radial pressure as p r = -a 1 r -a 3 r 3 + a 3 + a 1 r 2 0 -r 0 r 0 r 2 we conclude that for positive a 1 and a 3 , if Eq. (25) is fulfilled, the radial pressure is everywhere negative. The lateral pressure vanishes at r = √ a 3 /a 1 , and p l > 0 for r 0 ≤ r < √ a 3 /a 1 , while p l < 0 for r ≥ a 3 /a 1 .</text>
<text><location><page_4><loc_52><loc_46><loc_92><loc_50></location>√ Let us now consider the behavior of ρ + p r and ρ + p l . For the first expression we have that</text>
<formula><location><page_4><loc_64><loc_41><loc_92><loc_44></location>ρ + p r = a 1 r 2 -a 3 r 3 . (29)</formula>
<text><location><page_4><loc_52><loc_22><loc_92><loc_39></location>At the throat this relation gives ρ + p r = 1 r 0 ( a 1 -a 3 r 2 0 ) . It should be noted that the expression (29) vanishes at r = √ a 3 /a 1 . For negative values of a 1 and a 3 we obtain that ρ + p r ≤ 0 for r ≥ r 0 . Now, for positive a 1 and a 3 , from Eq. (15) we have that √ a 3 /a 1 > r 0 , implying that a 1 < a 3 /r 2 0 , and then at the throat ρ + p r ≤ 0. Since the expression (29) vanishes at r = √ a 3 /a 1 , the weak energy condition is violated at r 0 ≤ r < √ a 3 /a 1 as we should expect. For the range √ a 3 /a 1 ≤ r ≤ a 3 /a 1 we have that ρ + p r ≥ 0, so DEC may be fulfilled in this range.</text>
<text><location><page_4><loc_53><loc_21><loc_81><loc_23></location>For the expression ρ + p l we have that</text>
<formula><location><page_4><loc_57><loc_17><loc_92><loc_20></location>ρ + p l = 3 a 1 2 r + a 3 2 r 3 -a 3 + a 1 r 2 0 -r 0 r 0 r 2 , (30)</formula>
<text><location><page_4><loc_52><loc_9><loc_92><loc_15></location>and at the throat ρ + p l = a 1 r 2 0 -a 3 +2 r 0 2 r 3 0 , which in general can be positive as well as negative. However, for positive a 1 and a 3 if Eq. (25) is fulfilled then the expression in Eq. (30) is everywhere positive.</text>
<section_header_level_1><location><page_5><loc_9><loc_91><loc_49><loc_93></location>IV. PHANTOM WORMHOLES OF FINITE SIZE AND THEIR EMBEDDINGS</section_header_level_1>
<text><location><page_5><loc_9><loc_83><loc_49><loc_89></location>Now let us consider embeddings diagrams of the studied wormholes. The embedding of two dimensional slices t = const, θ = π 2 of the metric (9) is performed by using the embedding function z ( r ) in equation</text>
<formula><location><page_5><loc_19><loc_77><loc_49><loc_81></location>dz ( r ) dr = ( r b ( r ) -1 ) -1 / 2 , (31)</formula>
<text><location><page_5><loc_9><loc_75><loc_24><loc_77></location>which takes the form</text>
<formula><location><page_5><loc_15><loc_67><loc_49><loc_74></location>dz ( r ) dr = √ √ √ √ √ ( r -r 0 ) ( a 1 r -a 3 r 0 ) + r r ( 1 -r 0 r ) ( a 3 r 0 -a 1 r ) . (32)</formula>
<text><location><page_5><loc_9><loc_56><loc_49><loc_68></location>It becomes clear that, the embedding exists in a Euclidean space if the expression under the square root is positive. The Eqs. (16)-(18) imply that the denominator is positive, so we must require the positivity of the numerator, i.e. b ( r ) > 0. The shape function (8) is quadratic in r , so the numerator under the square root in Eq. (32) may have two roots, one root, or no roots. The roots of Eq. (8) are given by</text>
<formula><location><page_5><loc_15><loc_50><loc_49><loc_56></location>r ± = 1 2 a 1 r 0 ( a 3 + a 1 r 2 0 -r 0 ) ± √ ∆ 2 a 1 , (33)</formula>
<text><location><page_5><loc_9><loc_45><loc_49><loc_50></location>where ∆ = ( a 1 r 0 + a 3 /r 0 -1) 2 -4 a 1 a 3 . The existence of real roots depends on values of ∆, therefore the existence of a wormhole embedding in the Euclidean space depends on values of ∆.</text>
<text><location><page_5><loc_9><loc_40><loc_49><loc_44></location>In the following we will focus our attention on the study of wormholes with finite size (case 3), therefore we shall consider wormholes with positive a 1 and a 3 .</text>
<text><location><page_5><loc_9><loc_27><loc_49><loc_40></location>Case ∆ = 0 : Let us first consider the case where ∆ = 0 (i.e. there exists a unique root, and b ( r ) ≥ 0). This condition gives a 3 = r 0 + a 1 r 2 0 ± 2 r 0 √ a 1 r 0 , obtaining two branches for the root (33): r + = r -= r 0 ± √ r 0 /a 1 . In this case, the shape function is given by b ( r ) = r 0 + ( r 2 + r 2 0 -2 rr 0 ) a 1 -2( r -r 0 ) √ r 1 a 1 and the wormhole extends from r 0 to r max = √ r 0 + a 1 r 2 0 ± 2 r 0 √ a 1 r 0 a 1 . Notice that, for the negative branch we have that</text>
<formula><location><page_5><loc_17><loc_22><loc_49><loc_26></location>r 0 < √ r 0 + a 1 r 2 0 -2 r 0 √ a 1 r 0 a 1 (34)</formula>
<text><location><page_5><loc_9><loc_19><loc_41><loc_21></location>if a 1 < 1 / (4 r 0 ), while for the positive branch</text>
<formula><location><page_5><loc_17><loc_14><loc_49><loc_18></location>r 0 < √ r 0 + a 1 r 2 0 +2 r 0 √ a 1 r 0 a 1 (35)</formula>
<text><location><page_5><loc_9><loc_11><loc_19><loc_13></location>for any a 1 > 0.</text>
<text><location><page_5><loc_9><loc_8><loc_49><loc_11></location>Case ∆ < 0 : In this case there are not roots, and b ( r ) > 0. Since a 1 > 0, the requirement ∆ < 0 implies</text>
<text><location><page_5><loc_52><loc_90><loc_92><loc_93></location>that the embedding always exist if the parameter a 3 satisfies the constraints</text>
<formula><location><page_5><loc_52><loc_86><loc_93><loc_90></location>r 0 (1 + a 1 r 0 -2 √ a 1 r 0 ) < a 3 < r 0 (1 + a 1 r 0 +2 √ a 1 r 0 ) . (36)</formula>
<text><location><page_5><loc_52><loc_77><loc_92><loc_85></location>In other words, if positive a 1 and a 3 satisfy Eq. (36), then the obtained finite size wormhole can be entirely embedded into the Euclidean space: i.e. the wormhole spacetime, and simultaneously its embedding in a Euclidean space, extend from r 0 to r max = a 3 a 1 r 0 .</text>
<text><location><page_5><loc_52><loc_60><loc_92><loc_77></location>Case ∆ > 0 : If condition (36) is not satisfied, then roots r + and r -of Eq. (33) are real, and due to the positivity of a 1 , the shape function is positive in the intervals ( -∞ , r -) ∪ ( r + , + ∞ ). Therefore, the embedding of a constructed finite size wormhole may partially exists in a Euclidean space. Specifically, an equatorial slice θ = π 2 can be embedded into the Euclidean space in those ranges obtained from the intersection of intervals ( -∞ , r -) ∪ ( r + , + ∞ ) with the extension of the wormhole spacetime [ r 0 , a 3 a 1 r 0 ] .</text>
<section_header_level_1><location><page_5><loc_61><loc_58><loc_83><loc_59></location>A. Constructing wormholes</section_header_level_1>
<text><location><page_5><loc_52><loc_44><loc_92><loc_56></location>For constructing explicit examples of phantom wormholes of finite size we shall use the condition (25) discussed above. As we have shown, for positive a 1 and a 3 the fulfilment of the condition (25) ensures that the energy density and ρ + p l are everywhere positive, while the radial pressure p r is everywhere negative. By locating the throat at r 0 = 1 the condition (25) becomes a 1 + a 3 ≤ 1. For simplicity, we shall use the equality a 1 + a 3 = 1.</text>
<text><location><page_5><loc_52><loc_39><loc_92><loc_44></location>Wormhole with positive ρ : Let us first consider the parameter set a 1 = 1 / 5 and a 3 = 4 / 5. From Eq. (15) we conclude that the wormhole extends from r 0 = 1 to r max = 2. The energy density and pressures are given by</text>
<formula><location><page_5><loc_58><loc_33><loc_92><loc_36></location>ρ = 2 5 r , p r = -r 2 +4 5 r 3 , p l = 4 -r 2 10 r 3 . (37)</formula>
<text><location><page_5><loc_52><loc_22><loc_92><loc_32></location>It becomes clear that everywhere the energy density is positive and the radial pressure is negative, while the lateral pressure is negative for 1 ≤ r < 2 and positive for r > 2. From Eq. (37) we have that ρ + p r = r 2 -4 5 r 3 and ρ + p l = 3 r 2 +4 10 r 2 , therefore in the range 1 < r < 2 we have that ρ + p r < 0 and DEC is violated. Notice that the metric is given by</text>
<formula><location><page_5><loc_52><loc_15><loc_92><loc_21></location>ds 2 = dt 2 -dr 2 ( 1 -1 r ) ( 4 5 -r 5 ) -r 2 ( dθ 2 +sin 2 θdϕ 2 ) . (38)</formula>
<text><location><page_5><loc_52><loc_9><loc_92><loc_14></location>For r > 4 the spacetime ceases to be Lorentzian, then the spacetime extends from the throat at r 0 = 1 to r = 4. So, the whole spacetime is of finite size, characterized by a wormhole part connected to a dark energy distribution,</text>
<figure>
<location><page_6><loc_17><loc_72><loc_42><loc_93></location>
<caption>FIG. 4: The figure depicts the embedding of the spacetime (38). The embedding function is given by z ( r ) = ∫ √ r 2 +4 5 r -r 2 -4 dr , and its second derivative vanishes at r = 2. The whole spacetime is finite and extends from r 0 = 1 to r = 4. The phantom wormhole configuration extends from the throat to r = 2, where an inflection point is present. From r = 2 to r = 4 a dark energy distribution is located, to which is connected the phantom wormhole. For r < 2 DEC is not satisfied, while for r ≥ 2 it does.</caption>
</figure>
<text><location><page_6><loc_9><loc_37><loc_49><loc_54></location>which extends from r = 2 to r = 4. In this case the shape function is given by b ( r ) = ( r -1)( r/ 5 -4 / 5) + r , and it is positive for any value of radial coordinate r , implying that the embedding exists for the whole spacetime (38) as shown in Fig. 4. Notice that the spacetime in the range 2 ≤ r ≤ 4 is supported by a dark energy distribution, which satisfies DEC. It is interesting to discuss the behavior of the variable equation of state p r /ρ . For 1 ≤ r ≤ 2 we have the phantom behavior -2 . 5 ≤ p r /ρ ≤ -1, while for 2 ≤ r ≤ 4 we have that -1 ≤ p r /ρ ≤ -0 . 625, as we would expect since the wormhole is connected to a distribution of dark energy (see Figs. 4-6).</text>
<text><location><page_6><loc_9><loc_18><loc_49><loc_37></location>Microscopic wormhole: It is relevant to note that the wormhole part of the spacetime can be made arbitrarily small. For doing this the parameters a 1 > 0 and a 3 > 0 must be chosen in such a way that √ a 3 /a 1 ≈ r 0 . By using the condition (25) we may construct microscopic wormholes by imposing the equality a 3 + a 1 r 2 0 -r 0 = 0, implying that the relations a 1 ≈ 1 2 r 0 and a 3 ≈ r 0 2 must be required. As an example, let us consider the case r 0 = 1. Then we can construct an arbitrarily small wormhole by making a 1 = 1 2 -δ , a 3 = 1 2 + δ , where δ ≈ 0. Let us put δ = 0 . 01. Then the wormhole extends from r 0 = 1 to r = 1 . 0202, and the whole spacetime to r = 1 . 04082. For energy density and pressures we have that</text>
<formula><location><page_6><loc_10><loc_13><loc_49><loc_17></location>ρ = 0 . 98 r , p r = -0 . 51 + 0 . 49 r 2 r 3 , p l = 0 . 51 -0 . 49 r 2 2 r 3 . (39)</formula>
<text><location><page_6><loc_9><loc_8><loc_49><loc_11></location>Clearly, ρ > 0 and p r < 0 everywhere. On the other hand, ρ + p r = -0 . 51+0 . 49 r 2 r 3 , ρ + p l = 0 . 51+1 . 47 r 2 2 r 3 > 0,</text>
<figure>
<location><page_6><loc_59><loc_73><loc_86><loc_93></location>
<caption>FIG. 5: The figure compares the behavior of the energy density (dash-dotted line), radial (solid line) and lateral (dotted line) pressures; and the equation of state p r /ρ (dashed line) for r 0 = 1, a 1 = 1 / 5 and a 3 = 4 / 5. Everywhere the energy density is positive and radial pressure negative; and p l > 0 for 1 ≤ r < 2, and p l < 0 for 2 < r ≤ 4. On the other hand, for 1 ≤ r ≤ 2 the equation of state has a phantom character defined by -2 . 5 ≤ p r /ρ ≤ -1, while at 2 ≤ r ≤ 4 the equation of state behaves as dark energy, since -1 ≤ p r /ρ ≤ -0 . 625. In such a way, the wormhole is connected to a distribution of dark energy.</caption>
</figure>
<figure>
<location><page_6><loc_68><loc_37><loc_81><loc_53></location>
<caption>FIG. 6: The figure shows the three dimensional embedding diagram of the wormhole of finite size (38) with the energy distribution.</caption>
</figure>
<text><location><page_6><loc_52><loc_17><loc_92><loc_28></location>-g -1 rr = (1 -1 /r )(0 . 51 -0 . 49 r ) ≥ 0 at 1 ≤ r ≤ 1 . 04082, and b ( r ) = ( r -1)(0 . 49 r -0 . 51) + r > 0 for r ≥ 1. The relation ρ + p r is negative at 1 ≤ r < 1 . 0202, while ρ + p r ≥ 0 at 1 . 0202 ≤ r ≤ 1 . 04082. In this case, for the wormhole part we have that -1 . 0204 ≤ p r /ρ ≤ -1, and for the dark energy distribution part we have -1 ≤ p r /ρ ≤ -0 . 9804.</text>
<text><location><page_6><loc_52><loc_7><loc_92><loc_13></location>√ √ a 3 a 1 . This condition is satisfied by requiring that a 1 =</text>
<text><location><page_6><loc_52><loc_11><loc_92><loc_17></location>Wormhole with negative ρ : We can construct also a finite size wormhole with a negative energy density. In order to do this we can require that the energy density vanishes at r = a 3 /a 1 . This implies that a 3 + a 1 r 2 0 -r 0 2 r 0 a 1 =</text>
<figure>
<location><page_7><loc_17><loc_73><loc_43><loc_93></location>
<caption>FIG. 8: The figure compares plots of energy density (dotted line), radial pressure (dashed line) and the radial equation of state (solid line) for the case r 0 = 1. a 1 = 1 and a 3 = 4. The energy density is negative at the wormhole part, while out of it ρ is positive. In the range 2 ≤ r < 4 we have that 0 ≤ p r /ρ ≤ -1 / 4, so DEC is fulfilled.</caption>
</figure>
<figure>
<location><page_7><loc_59><loc_73><loc_86><loc_93></location>
<caption>FIG. 7: The figure compares the behavior of the energy density (solid line), radial (dotted line) and lateral (dashed line) pressures, ρ + p r (dash-dotted line) and ρ + p l (long-dashed line) for r 0 = 1, a 1 = 1 and a 3 = 4. It becomes clear that the energy density is negative for 1 ≤ r < 2 and positive for 2 < r ≤ 4, while p r ≤ 0 for 1 ≤ r ≤ 4 and p l > 0 for 1 ≤ r < 2, and p l < 0 for 2 < r ≤ 4. On the other hand, ρ + p r and ρ + p l are negative for 1 ≤ r < 2 and positive for 2 < r ≤ 4.</caption>
</figure>
<text><location><page_7><loc_9><loc_53><loc_49><loc_57></location>a 3 + r 0 ± 2 √ r 0 a 3 r 2 0 . This allows us to write the energy density and pressures as (for positive and negative branches)</text>
<formula><location><page_7><loc_12><loc_48><loc_49><loc_52></location>ρ = -2( √ a 3 ± √ r 0 )( r 0 √ a 3 -( √ a 3 ± √ r 0 ) r ) r 2 0 r 2 , (40)</formula>
<formula><location><page_7><loc_22><loc_41><loc_49><loc_45></location>p l = -( √ r 0 ± √ a 3 ) 2 r 2 + r 2 0 a 3 2 r 2 0 r 3 . (42)</formula>
<formula><location><page_7><loc_20><loc_45><loc_49><loc_49></location>p r = -(( √ r 0 ± √ a 3 ) r ∓ r 0 √ a 3 ) 2 r 2 0 r 3 , (41)</formula>
<text><location><page_7><loc_10><loc_39><loc_46><loc_40></location>These relations give for the radial state parameter</text>
<formula><location><page_7><loc_17><loc_33><loc_49><loc_38></location>p r ρ = -( √ r 0 ± √ a 3 ) r ∓ r 0 √ a 3 2 r ( √ r 0 ± √ a 3 ) (43)</formula>
<text><location><page_7><loc_9><loc_30><loc_49><loc_33></location>For an explicit example we take r 0 = 1, a 1 = 1, a 3 = 4. From expressions of the negative branch we get that</text>
<formula><location><page_7><loc_13><loc_26><loc_49><loc_29></location>ρ = 2 r -4 r 2 , p r = -( r -2) 2 r 3 , p l = 4 -r 2 2 r 3 , (44)</formula>
<text><location><page_7><loc_9><loc_12><loc_49><loc_25></location>while ρ + p r = r 2 -4 r 3 , ρ + p l = (3 r -2)( r -2) 2 R 3 . The general behavior of these relevant physical magnitudes are shown in Fig. 7. It becomes clear that due to ρ < 0 at 1 ≤ r < 2, DEC is violated where the wormhole is located. At r = 2 this wormhole is connected to an anisotropic spherically symmetric distribution respecting DEC. In this case for the radial equation of state we have that 0 ≤ p r /ρ ≤ -1 / 4 for 2 ≤ r < 4 (see Fig. 8).</text>
<text><location><page_7><loc_9><loc_9><loc_49><loc_13></location>Wormhole with mixed energy dependence: Now we are interested in construction of finite wormholes with an energy density exhibiting a mixed dependence, in the</text>
<text><location><page_7><loc_52><loc_45><loc_92><loc_60></location>sense that the energy density changes its sign at some sphere of radius r 0 < r 2 ≤ a 3 /a 1 , where ρ ( r 2 ) = 0, and r 2 = a 3 + r 2 0 a 1 -r 0 2 a 1 r 0 . We have discussed above that if energy density vanishes at some r , then at the throat always ρ ( r 0 ) < 0. Therefore, if r 0 < r 2 < √ a 3 /a 1 we have that ρ ( r ) < 0 in this range, and for r 2 < r < a 3 /a 1 the energy density becomes positive. In this way, the requirement r 0 < a 3 + r 2 0 a 1 -r 0 2 a 1 r 0 < √ a 3 /a 1 implies that the parameter a 1 satisfies</text>
<text><location><page_7><loc_52><loc_40><loc_78><loc_42></location>while the parameter a 3 the condition</text>
<formula><location><page_7><loc_63><loc_41><loc_80><loc_46></location>0 < a 1 < 1 4 ( 1 + √ 17 )</formula>
<formula><location><page_7><loc_52><loc_35><loc_93><loc_39></location>-a 2 1 + a 1 +2 a 1 < a 3 < -a 4 1 + a 3 1 +2 a 3 1 +2 √ -a 4 1 -a 3 1 -1 a 6 1 .</formula>
<text><location><page_7><loc_52><loc_31><loc_92><loc_34></location>For example, we may construct such a wormhole for r 0 = 1, a 1 = 1 and a 3 = 3, obtaining for the relevant quantities</text>
<formula><location><page_7><loc_55><loc_27><loc_86><loc_30></location>ρ = 2 r -3 r 2 , p r = -r 2 +3 r -3 r 3 , p l = 3 -r 2 2 r 3 ,</formula>
<text><location><page_7><loc_52><loc_24><loc_80><loc_26></location>and ρ + p r = r 2 -3 r 3 , ρ + p l = 3( r 2 -2 r +1) 2 r 3 .</text>
<text><location><page_7><loc_52><loc_14><loc_92><loc_24></location>The change of sign of the energy density may also occur for a some radius between √ a 3 /a 1 and a 3 /a 1 . In this case we must require that a 3 > a 1 + 2 √ a 1 + 1 for any positive a 1 . This implies that for the wormhole structure the exotic energy density is always negative, while for the anisotropic spherically symmetric distribution of matter respecting DEC, the energy density changes its sign.</text>
<text><location><page_7><loc_52><loc_8><loc_92><loc_14></location>Lastly, we may construct finite wormhole solutions for which the energy density is negative for the whole spacetime, i.e. at r 0 ≤ r ≤ a 3 /a 1 . By requiring that the energy density vanishes at a 3 /a 1 we assure that ρ ( r ) < 0</text>
<text><location><page_8><loc_11><loc_84><loc_11><loc_86></location>/negationslash</text>
<text><location><page_8><loc_9><loc_80><loc_49><loc_93></location>everywhere. From the condition a 3 + r 2 0 a 1 -r 0 2 a 1 r 0 = a 3 a 1 , we obtain that if r 0 = 1 2 we must require a 1 = 2 and a 3 > 2. If the throat is located at 0 < r 0 < 1 2 we must require that 0 < a 1 < r 0 r 2 0 -2 r 0 +1 , while for 1 2 < r 0 ≤ a 3 /a 1 (with r 0 = 1) the condition a 1 > r 0 r 2 0 -2 r 0 +1 must be satisfied. In the last two cases we have that a 3 = a 1 r 2 0 -r 0 2 r 0 -1 . Notice that for r 0 = 1 is not possible to construct such a spacetime with ρ ( a 3 /a 1 ) = 0.</text>
<section_header_level_1><location><page_8><loc_21><loc_76><loc_37><loc_77></location>V. CONCLUSIONS</section_header_level_1>
<text><location><page_8><loc_9><loc_55><loc_49><loc_73></location>In this paper we derived new static spherically symmetric traversable wormholes by assuming a shape function with a quadratic dependence on the radial coordinate r , and it is shown that there exist wormhole spacetimes sustained by phantom energy. In order to do this, we specify the equation of state of the radial pressure for the distribution of the energy density threading the wormhole by imposing on it a phantom equation of state of the form p r /ρ < -1. It should be noted that for a quadratic shape function we have an equation of state p r /ρ with a variable character. We mainly focus our study on wormholes sustained by exotic matter with positive energy density, as seen by any static observer.</text>
<text><location><page_8><loc_9><loc_37><loc_49><loc_55></location>An important feature of the wormhole description with a quadratic shape function is that it includes phantom wormhole spacetimes extending to infinity, as well as static spacetimes of finite size, composed by a phantom wormhole connected to an inhomogeneous and anisotropic spherically symmetric distribution of dark energy. For latter wormhole types we can construct solutions with phantom matter confined to a finite region around the throat, which is connected to the dark energy distribution. The wormhole part does not fulfill the dominant energy condition, while the dark energy distribution part does.</text>
<text><location><page_8><loc_9><loc_26><loc_49><loc_37></location>Summarizing, in general for finite wormholes ( a 1 > 0 and a 3 > 0) the exotic matter threading the phantom wormhole extends from the throat at r 0 to the sphere of radius r max = √ a 3 /a 1 , and the whole spacetime extends to the square of this r max . The matter source of the gravitational configuration at √ a 3 /a 1 ≤ r ≤ a 3 /a 1 is of dark energy, so it satisfies DEC.</text>
<text><location><page_8><loc_9><loc_20><loc_49><loc_27></location>Finally, let us note that in general the spacetime (9) is not asymptotically flat. As a result, the matter distribution for wormholes extending to infinity must be cut off at some radius r = r ∗ > r 0 and joined to an exterior asymptotically flat spacetime, such as, for example, the vacuum</text>
<text><location><page_8><loc_52><loc_74><loc_92><loc_93></location>Schwarzschild spacetime without cosmological constant (note that the studied wormholes (9) satisfy Einstein equations in the absence of cosmological constant). On the other hand, for wormholes with finite dimensions, in which the phantom matter distribution extends from the throat r 0 to the radius r max = √ a 3 /a 1 , and the dark energy distribution extends from √ a 3 /a 1 to r = a 3 /a 1 , the matching to the exterior vacuum Schwarzschild spacetime can be performed at r ∗ = √ a 3 /a 1 > r 0 or r ∗ = a 3 /a 1 > r 0 . In other words, the discussed here wormhole spacetimes can be considered as an interior solution, which must be matched to an exterior solution, such as the Schwarzschild geometry, at some radius r ∗ > r 0 .</text>
<text><location><page_8><loc_52><loc_66><loc_92><loc_74></location>The procedure of construction of traversable wormholes through matching an interior wormhole solution to the exterior Schwarzschild solutions is discussed by authors in Ref. [3]. In order to do this matching one must apply the junction conditions that follow from the theory of general relativity.</text>
<text><location><page_8><loc_52><loc_60><loc_92><loc_65></location>Due to the spherical symmetry of the spacetime, the components g θθ and g φφ are already continuous [3], so one needs to impose continuity only on the remaining metric components g tt and g rr at r = r ∗ , i.e.</text>
<formula><location><page_8><loc_63><loc_55><loc_78><loc_58></location>g W tt ( r ∗ ) = g Schw tt ( r ∗ ) , g W rr ( r ∗ ) = g Schw rr ( r ∗ ) .</formula>
<text><location><page_8><loc_52><loc_51><loc_92><loc_54></location>These requirements, in turn, lead to following restrictions for the redshift and shape functions</text>
<formula><location><page_8><loc_63><loc_46><loc_78><loc_49></location>φ W ( r ∗ ) = φ Schw ( r ∗ ) , b W ( r ∗ ) = b Schw ( r ∗ ) .</formula>
<text><location><page_8><loc_52><loc_42><loc_92><loc_45></location>In such a way, the exterior and interior solutions become identical at the sphere boundary r = r ∗ .</text>
<text><location><page_8><loc_52><loc_26><loc_92><loc_42></location>It is interesting to note that for spherically symmetric spacetimes, one can use directly the field equations to perform the match at the boundary r ∗ . Einstein equations allow us to determine the energy density and stresses of the surface r = r ∗ necessary to have a match between the interior and exterior spacetimes. If there are no surface stress-energy terms at the surface r ∗ , the junction is called a boundary surface. On the other hand, if surface stress-energy terms are present, the junction is called a thin shell (see Lemos et al. [3] for a nice review of this issue).</text>
<text><location><page_8><loc_52><loc_20><loc_92><loc_26></location>Acknowledgements: This work was supported by Direcci'on de Investigaci'on de la Universidad del B'ıoB'ıo through grants N 0 DIUBB 140708 4/R and N 0 GI121407/VBC.</text>
<text><location><page_8><loc_55><loc_9><loc_92><loc_15></location>90 , no. 11, 1319 (2016); Y. Heydarzade, N. Riazi and H. Moradpour, Can. J. Phys. 93 , no. 12, 1523 (2015); F. S. N. Lobo, F. Parsaei and N. Riazi, Phys. Rev. D 87 , no. 8, 084030 (2013); J. A. Gonzalez, F. S. Guzman,</text>
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<list_item><location><page_9><loc_10><loc_77><loc_43><loc_78></location>[6] S. V. Sushkov, Phys. Rev. D 71 , 043520 (2005).</list_item>
<list_item><location><page_9><loc_10><loc_69><loc_49><loc_77></location>[7] M. Cataldo and P. Meza, Phys. Rev. D 87 , no. 6, 064012 (2013); M. Cataldo and S. del Campo, Phys. Rev. D 85 , 104010 (2012); M. Cataldo, P. Meza and P. Minning, Phys. Rev. D 83 , 044050 (2011); M. Cataldo, S. del Campo, P. Minning and P. Salgado, Phys. Rev. D 79 , 024005 (2009).</list_item>
<list_item><location><page_9><loc_10><loc_67><loc_49><loc_69></location>[8] M. Cataldo, P. Labrana, S. del Campo, J. Crisostomo and P. Salgado, Phys. Rev. D 78 , 104006 (2008).</list_item>
<list_item><location><page_9><loc_10><loc_64><loc_49><loc_67></location>[9] M. Cataldo, F. Arstica and S. Bahamonde, Eur. Phys. J. C 73 , no. 8, 2517 (2013).</list_item>
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<list_item><location><page_9><loc_52><loc_87><loc_92><loc_89></location>[11] M. Visser, Lorentzian Wormholes: From Einstein to Hawking, (AIP, New York, 1995).</list_item>
<list_item><location><page_9><loc_52><loc_81><loc_92><loc_86></location>[12] S. H. Mazharimousavi and M. Halilsoy, Mod. Phys. Lett. A 31 , no. 34, 1650192 (2016); M. Jamil and M. U. Farooq, Int. J. Theor. Phys. 49 , 835 (2010); R. A. Konoplya and A. Zhidenko, Phys. Rev. D 81 , 124036 (2010).</list_item>
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</document> | [] |
2015PhLB..746...73R | https://arxiv.org/pdf/1406.3045.pdf | <document>
<section_header_level_1><location><page_1><loc_19><loc_83><loc_75><loc_88></location>Wormhole inspired by non-commutative geometry</section_header_level_1>
<section_header_level_1><location><page_1><loc_20><loc_74><loc_75><loc_78></location>Farook Rahaman a , Sreya Karmakar b , Indrani Karar c , Saibal Ray d</section_header_level_1>
<text><location><page_1><loc_15><loc_70><loc_79><loc_73></location>a Department of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India</text>
<text><location><page_1><loc_17><loc_66><loc_77><loc_69></location>b Department of Physics, Calcutta Institute of Engineering and Management, Kolkata 700040, West Bengal, India</text>
<unordered_list>
<list_item><location><page_1><loc_17><loc_62><loc_78><loc_65></location>c Department of Mathematics, Saroj Mohan Institute of Technology, Guptipara, West Bengal, India</list_item>
</unordered_list>
<text><location><page_1><loc_19><loc_58><loc_76><loc_61></location>d Department of Physics, Government College of Engineering & Ceramic Technology, Kolkata 700010, West Bengal, India</text>
<section_header_level_1><location><page_1><loc_15><loc_52><loc_23><loc_53></location>Abstract</section_header_level_1>
<text><location><page_1><loc_15><loc_35><loc_80><loc_50></location>In the present letter we search for a new wormhole solution inspired by noncommutative geometry with the additional condition of allowing conformal Killing vectors (CKV). A special aspect of noncommutative geometry is that it replaces point-like structures of gravitational sources with smeared objects under Gaussian distribution. However, the purpose of this paper is to obtain wormhole solutions with noncommutative geometry as a background where we consider a point-like structure of gravitational object without smearing effect. It is found through this investigation that wormhole solutions exist in this Lorentzian distribution with viable physical properties.</text>
<text><location><page_1><loc_15><loc_31><loc_70><loc_33></location>Key words: General Relativity; noncommutative geometry; wormholes</text>
<section_header_level_1><location><page_1><loc_15><loc_22><loc_36><loc_24></location>1 INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_15><loc_15><loc_79><loc_19></location>A wormhole , which is similar to a tunnel with two ends each in separate points in spacetime or two connecting black holes, was conjectured first by Weyl [1]</text>
<text><location><page_2><loc_15><loc_86><loc_79><loc_91></location>and later on by [2]. In a more concrete physical definition it is essentially some kind of hypothetical topological feature of spacetime which may acts as shortcut through spacetime topology.</text>
<text><location><page_2><loc_15><loc_70><loc_79><loc_84></location>It is argued by Morris et al. [3,4] and others [5,6,7] that in principle a wormhole would allow travel in time as well as in space and can be shown explicitly how to convert a wormhole traversing space into one traversing time. However, there are other types of wormholes available in the literature where the traversing path does not pass through a region of exotic matter [8,9]. Following the work of Visser [8], a new type of thin-shell wormhole, which was constructed by applying the cut-and-paste technique to two copies of a charged black hole [10], is of special mention in this regard.</text>
<text><location><page_2><loc_15><loc_53><loc_79><loc_68></location>Thus a traversable wormhole, tunnel-like structure connecting different regions of our Universe or of different universes altogether, has been an issue of special investigation under Einstein's general theory of relativity [11]. It is argued by Rahaman et al. [12] that although just as good a prediction of Einstein's theory as black holes, wormholes have so far eluded detection. As one of the peculiar features a wormhole requires the violation of the null energy condition (NEC) [4]. One can note that phantom dark energy also violates the NEC and hence could have deep connection to in formation of wormholes [13,14].</text>
<text><location><page_2><loc_15><loc_34><loc_79><loc_52></location>It is believed that some perspective of quantum gravity can be explored mathematically in a better way with the help of non-commutative geometry. This is based on the non-commutativity of the coordinates encoded in the commutator, [ x µ , x ν ] = θ µν , where θ µν is an anti-symmetric and real second-ordered matrix which determines the fundamental cell discretization of spacetime [15,16,17,18,19]. We also invoke the inheritance symmetry of the spacetime under conformal Killing vectors (CKV). Basically CKVs are motions along which the metric tensor of a spacetime remains invariant up to a certain scale factor. In a given spacetime manifold M , one can define a globally smooth conformal vector field ξ , such that for the metric g ab it can be written as</text>
<formula><location><page_2><loc_19><loc_30><loc_79><loc_32></location>ξ a ; b = ψg ab + F ab , (1)</formula>
<text><location><page_2><loc_15><loc_24><loc_79><loc_27></location>where ψ : M → R is the smooth conformal function of ξ and F ab is the conformal bivector of ξ . This is equivalent to the following form:</text>
<formula><location><page_2><loc_19><loc_21><loc_79><loc_22></location>L ξ g ik = ξ i ; k + ξ k ; i = ψg ik , (2)</formula>
<text><location><page_2><loc_15><loc_16><loc_61><loc_18></location>where L signifies the Lie derivatives along the CKV ξ α .</text>
<text><location><page_2><loc_15><loc_9><loc_79><loc_14></location>In favor of the prescription of this mathematical technique CKV we find out the following features: (1) it provides a deeper insight into the spacetime geometry and facilitates the generation of exact solutions to the Einstein field</text>
<text><location><page_3><loc_15><loc_78><loc_79><loc_91></location>equations in a more comprehensive forms, (2) the study of this particular symmetry in spacetime is physically very important as it plays a crucial role of discovering conservation laws and to devise spacetime classification schemes, and (3) because of the highly non-linearity of the Einstein field equations one can reduce easily the partial differential equations to ordinary differential equations by using CKV. Interested readers may look at the recent works on CKV technique available in the literature [20,21,22].</text>
<text><location><page_3><loc_15><loc_55><loc_79><loc_76></location>In this letter therefore we search for some new solutions of wormhole admitting conformal motion of Killing Vectors. It is a formal practice to consider the inheritance symmetry to establish a natural relationship between spacetime geometry and matter-energy distribution for a astrophysical system. Thus our main goal here is to examine the solutions of Einstein field equations by admitting CKV under non-commutative geometry. The scheme of the investigation is as follows: in the Sec. 2 we provide the mathematical formalism and Einstein's field equations under the framework of this technique. A specific matter-energy density profile has been employed in Sec. 3 to obtain various physical features of the wormhole under consideration by addressing the issues like the conservation equation, stability of the system, active gravitational mass and gravitational energy. Sec. 4 is devoted for some concluding remarks.</text>
<section_header_level_1><location><page_3><loc_15><loc_47><loc_80><loc_49></location>2 CONFORMALKILLINGVECTORANDBASICEQUATIONS</section_header_level_1>
<text><location><page_3><loc_15><loc_42><loc_73><loc_43></location>We take the static spherically symmetric metric in the following form</text>
<formula><location><page_3><loc_19><loc_37><loc_79><loc_40></location>ds 2 = e ν ( r ) dt 2 -e λ ( r ) dr 2 -r 2 ( dθ 2 + sin 2 θdφ 2 ) , (3)</formula>
<text><location><page_3><loc_15><loc_31><loc_79><loc_34></location>where r is the radial coordinate. Here ν and λ are the metric potentials which have functional dependence on r only.</text>
<text><location><page_3><loc_15><loc_26><loc_79><loc_29></location>Thus, the only survived Einstein's field equations in their explicit forms (rendering G = c = 1) are</text>
<formula><location><page_3><loc_19><loc_20><loc_79><loc_24></location>e -λ [ λ ' r -1 r 2 ] + 1 r 2 = 8 πρ, (4)</formula>
<formula><location><page_3><loc_19><loc_14><loc_79><loc_18></location>e -λ [ 1 r 2 + ν ' r ] -1 r 2 = 8 πp r , (5)</formula>
<formula><location><page_3><loc_19><loc_8><loc_79><loc_12></location>1 2 e -λ [ 1 2 ( ν ' ) 2 + ν '' -1 2 λ ' ν ' + 1 r ( ν ' -λ ' ) ] = 8 πp t , (6)</formula>
<text><location><page_4><loc_15><loc_86><loc_79><loc_91></location>where ρ , p r and p t are matter-energy density, radial pressure and transverse pressure respectively for the fluid distribution. Here ' over ν and λ denotes partial derivative w.r.t. radial coordinate r .</text>
<text><location><page_4><loc_15><loc_80><loc_79><loc_84></location>The conformal Killing equations, as mentioned in Eqs. (2), then yield as follows:</text>
<formula><location><page_4><loc_15><loc_77><loc_23><loc_79></location>ξ 1 ν ' = ψ ,</formula>
<formula><location><page_4><loc_15><loc_74><loc_32><loc_75></location>ξ 4 = C 1 = constant ,</formula>
<formula><location><page_4><loc_15><loc_70><loc_22><loc_72></location>ξ 1 = ψr 2 ,</formula>
<formula><location><page_4><loc_15><loc_67><loc_28><loc_69></location>ξ 1 λ ' +2 ξ 1 , 1 = ψ ,</formula>
<text><location><page_4><loc_15><loc_62><loc_79><loc_65></location>where ξ α are the conformal 4-vectors and ψ is the conformal function as mentioned earlier.</text>
<text><location><page_4><loc_15><loc_57><loc_79><loc_60></location>This set of equations, in a straight forward way, imply the following simple forms:</text>
<formula><location><page_4><loc_19><loc_53><loc_79><loc_55></location>e ν = C 2 2 r 2 , (7)</formula>
<formula><location><page_4><loc_19><loc_49><loc_79><loc_52></location>e λ = C 3 2 ψ 2 , (8)</formula>
<formula><location><page_4><loc_19><loc_43><loc_79><loc_47></location>ξ i = C 1 δ i 4 + ( ψr 2 ) δ i 1 , (9)</formula>
<text><location><page_4><loc_15><loc_37><loc_79><loc_41></location>where C 2 and C 3 are integration constants. Here the non-zero components of the conformal Killing vector ξ a are ξ 0 and ξ 1 .</text>
<text><location><page_4><loc_15><loc_32><loc_79><loc_36></location>Now using solutions (7) and (8), the equations (3)-(5) take the following form as</text>
<formula><location><page_4><loc_19><loc_27><loc_79><loc_31></location>1 r 2 [ 1 -ψ 2 C 2 3 ] -2 ψψ ' C 2 3 r = 8 πρ, (10)</formula>
<formula><location><page_4><loc_19><loc_22><loc_79><loc_26></location>1 r 2 [ 1 -3 ψ 2 C 2 3 ] = -8 πp r , (11)</formula>
<formula><location><page_4><loc_19><loc_17><loc_79><loc_21></location>[ ψ 2 C 2 3 r 2 ] + 2 ψψ ' C 2 3 r = 8 πp t . (12)</formula>
<text><location><page_4><loc_15><loc_9><loc_79><loc_14></location>These are the equations forming master set which has all the information of the fluid distribution under the framework of Einstein's general theory of relativity with the associated non-commutative geometry and conformal Killing vectors.</text>
<section_header_level_1><location><page_5><loc_15><loc_88><loc_80><loc_91></location>3 THE MATTER-ENERGY DENSITY PROFILE AND PHYSICAL FEATURES OF THE WORMHOLE</section_header_level_1>
<text><location><page_5><loc_15><loc_64><loc_79><loc_84></location>As stated by Rahaman et al. [11], the necessary ingredients that supply fuel to construct wormholes remain an elusive goal for theoretical physicists and there are several proposals that have been put forward by different authors [23,24,25,26,27,28]. However, in our present work we consider cosmic fluid as source and thus have provided a new class of wormhole solutions. Keeping the essential aspects of the noncommutativity approach which are specifically sensitive to the Gaussian nature of the smearing as employed by Nicolini et al. [18], we rather get inspired by the work of Mehdipour [29] to search for a new fluid model admitting conformal motion. Therefore, we assume a Lorentzian distribution of particle-like gravitational source and hence the energy density profile as given in Ref. [29] as follows:</text>
<formula><location><page_5><loc_19><loc_58><loc_79><loc_63></location>ρ ( r ) = M √ φ π 2 ( r 2 + φ ) 2 , (13)</formula>
<text><location><page_5><loc_15><loc_51><loc_79><loc_54></location>where φ is the noncommutativity parameter and M is the smeared mass distribution.</text>
<text><location><page_5><loc_15><loc_48><loc_44><loc_49></location>Now, solving equation (10) we get</text>
<formula><location><page_5><loc_19><loc_42><loc_79><loc_46></location>ψ 2 = C 2 3 -( 4 C 2 3 M πr )[ tan -1 ( r √ φ ) -r √ φ r 2 + φ ] + D 1 r , (14)</formula>
<text><location><page_5><loc_15><loc_37><loc_67><loc_38></location>where D 1 is an integration constant and can be taken as zero.</text>
<text><location><page_5><loc_15><loc_31><loc_79><loc_35></location>The parameters, like the radial pressure, tangential pressure and metric potentials, are found as</text>
<formula><location><page_5><loc_19><loc_26><loc_79><loc_30></location>p r = 1 8 π [ 2 r 2 -12 M πr 3 { tan -1 ( r √ φ ) -r √ φ r 2 + φ }] , (15)</formula>
<formula><location><page_5><loc_19><loc_20><loc_79><loc_24></location>p t = 1 8 π [ 1 r 2 -8 π ( M √ φ π 2 ( r 2 + φ ) 2 )] , (16)</formula>
<formula><location><page_5><loc_19><loc_16><loc_79><loc_18></location>e ν = C 2 2 r 2 , (17)</formula>
<formula><location><page_5><loc_19><loc_8><loc_79><loc_14></location>e λ = 1 [ 1 -( 4 M πr ) ( tan -1 ( r √ φ ) -r √ φ r 2 + φ )] . (18)</formula>
<figure>
<location><page_6><loc_26><loc_57><loc_69><loc_90></location>
<caption>Fig. 1. The throat of the wormhole is located at r √ φ = r 0 √ φ (maximum root), where</caption>
</figure>
<text><location><page_6><loc_15><loc_40><loc_79><loc_52></location>b ( r √ φ ) -r √ φ cuts the r √ φ -axis. For M √ φ < 2 . 2, there exists no root, and therefore no throats. For M √ φ > 2 . 2, however we have two roots: (i) for M √ φ = 3, the location of the external root i.e. throat of the wormhole is r 0 √ φ = 4 . 275, and (ii) for M √ φ = 2 . 2, we have one and only one solution and this corresponds to the situation when two roots coincide and it can be interpreted as an extreme situation</text>
<text><location><page_6><loc_15><loc_36><loc_79><loc_39></location>Let us now write down the metric potential conveniently in terms of the shape function b ( r ) as follows:</text>
<formula><location><page_6><loc_19><loc_29><loc_79><loc_34></location>e λ = 1 1 -b ( r ) r , (19)</formula>
<text><location><page_6><loc_15><loc_25><loc_33><loc_26></location>where b ( r ) is given by</text>
<formula><location><page_6><loc_19><loc_19><loc_79><loc_24></location>b ( r ) = ( 4 M π ) [ tan -1 ( r √ φ ) -r √ φ r 2 + φ ] . (20)</formula>
<text><location><page_6><loc_15><loc_9><loc_80><loc_16></location>Now, we will discuss the behavioral effects of different aspects of the above shape function b ( r ) and its derivative. The throat location of the wormhole is obtained by imposing the equation b ( r 0 ) = r 0 . One can note that the appearance of a throat depends on the parameter M and φ . However, the larger root</text>
<figure>
<location><page_7><loc_16><loc_73><loc_86><loc_91></location>
<caption>Fig. 2. (Left) Diagram of the shape function of the wormhole for the specific value of the parameter M √ φ = 3. (Middle) Diagram of the asymptotic behaviour of shape function. (Right) Diagram of the derivative of the shape function of the wormhole.</caption>
</figure>
<text><location><page_7><loc_15><loc_41><loc_79><loc_60></location>of the equation b ( r 0 √ φ ) = r 0 √ φ , where r 0 √ φ is dimensionless, gives the throat which depends only one parameter M √ φ . Figure 1 shows that the throat of the wormhole is located at r √ φ = r 0 √ φ (maximum root), where r √ φ -b ( r 0 √ φ ) cuts the r √ φ -axis. One can note that position of the throat is increasing with the increase of smeared mass distribution M. For M √ φ < 2 . 2 no throat exists. From the above analysis, we notice that we may get feasible wormholes for M √ φ > 2 . 2. For the sake of brevity, we assume M √ φ = 3 for the rest of the study.</text>
<text><location><page_7><loc_15><loc_34><loc_79><loc_39></location>From the left panel of Fig. 2, we observe that shape function is increasing, therefore, b ' ( r √ φ ) > 0. From Fig. 1, one can also note that for ( r √ φ ) ></text>
<text><location><page_7><loc_15><loc_24><loc_79><loc_29></location>is an essential requirement for a shape function. Right panel of figure 2 indicates that the flare-out condition b ' ( r √ φ ) < 1 for ( r √ φ ) > ( r 0 √ φ ) is satisfied.</text>
<formula><location><page_7><loc_15><loc_28><loc_79><loc_34></location>( r 0 √ φ ) , ( r √ φ ) -b ( r √ φ ) > 0. This immediately implies that b ( r √ φ ) ( r √ φ ) < 1 which</formula>
<text><location><page_7><loc_15><loc_15><loc_79><loc_22></location>We also observe the asymptotic behaviour from the middle panel of Fig. 2 such that b ( ( r √ φ ) ) ( r √ φ ) → 0 as ( r √ φ ) → ∞ . Unfortunately, this has the similar</text>
<text><location><page_7><loc_15><loc_9><loc_79><loc_16></location>explanation as done in Ref. [20] that the redshift function does not approach zero as r → ∞ due to the conformal symmetry. This means the wormhole spacetime is not asymptotically flat and hence will have to be cut off at some radial distance which smoothly joins to an exterior vacuum solution.</text>
<figure>
<location><page_8><loc_28><loc_60><loc_68><loc_90></location>
<caption>Fig. 3. The violation of null energy condition is shown against ( r √ φ ) .</caption>
</figure>
<text><location><page_8><loc_15><loc_50><loc_79><loc_54></location>One can now find out the redshift function f ( r ), where e 2 f ( r ) = e ν ( r ) . Using Eq. (7) we find</text>
<formula><location><page_8><loc_19><loc_47><loc_79><loc_48></location>f ( r ) = ln ( C 2 r ) . (21)</formula>
<text><location><page_8><loc_15><loc_40><loc_79><loc_43></location>It can be observed from the above expression that the wormhole presented here is traversable one as redshift function remains finite.</text>
<text><location><page_8><loc_15><loc_33><loc_79><loc_38></location>The above solution should be matched with the exterior vacuum spacetime of the Schwarzschild type at some junction interface with radius R . Using this matching condition, one can easily find the value of unknown constant C 2 as</text>
<formula><location><page_8><loc_19><loc_27><loc_79><loc_31></location>C 2 = e f ( R ) R , (22)</formula>
<text><location><page_8><loc_15><loc_23><loc_58><loc_24></location>so that the redshift function now explicitly becomes</text>
<formula><location><page_8><loc_19><loc_17><loc_79><loc_21></location>f ( r ) = ln [ re f ( R ) R ] . (23)</formula>
<text><location><page_8><loc_15><loc_9><loc_79><loc_14></location>The redshift function is therefore finite in the region r 0 < r < R , as required because this will prevent an event horizon. According to Fig. 3, φ ( ρ + p r ) < 0, therefore, the null energy condition is violated to hold a wormhole open.</text>
<section_header_level_1><location><page_9><loc_15><loc_89><loc_69><loc_91></location>3.1 THE TOLMAN-OPPENHEIMER-VOLKOFF EQUATION</section_header_level_1>
<text><location><page_9><loc_15><loc_82><loc_83><loc_86></location>Following the suggestion of Ponce de Leon [30], we write the Tolman-OppenheimerVolkoff (TOV) equation in the following form</text>
<formula><location><page_9><loc_19><loc_77><loc_79><loc_80></location>-M G ( ρ + p r ) r 2 e λ -ν 2 -dp r dr + 2 r ( p t -p r ) = 0 , (24)</formula>
<text><location><page_9><loc_15><loc_70><loc_79><loc_74></location>where M G = M G ( r ) is the effective gravitational mass within the region from r 0 up to the radius r and is given by</text>
<formula><location><page_9><loc_19><loc_65><loc_79><loc_69></location>M G ( r ) = 1 2 r 2 e ν -λ 2 ν ' . (25)</formula>
<text><location><page_9><loc_15><loc_55><loc_79><loc_62></location>Equation (24) expresses the equilibrium condition for matter distribution comprising the wormhole subject to the gravitational force F g , hydrostatic force F h plus another force F a due to anisotropic pressure. Now, the above Eq. (24) can be easily written as</text>
<formula><location><page_9><loc_19><loc_52><loc_79><loc_53></location>F g + F h + F a = 0 , (26)</formula>
<text><location><page_9><loc_15><loc_47><loc_20><loc_48></location>where</text>
<formula><location><page_9><loc_19><loc_36><loc_79><loc_44></location>F g = -ν ' 2 ( ρ + p r ) = -1 4 πr 3 -M √ φ π 2 r ( r 2 + φ ) 2 -3 M √ φ 2 π 2 r 3 ( r 2 + φ ) + 3 M 2 π 2 r 4 tan -1 ( r √ φ ) , (27)</formula>
<formula><location><page_9><loc_19><loc_23><loc_79><loc_31></location>F h = -dp r dr = 1 2 πr 3 + 3 M √ φ π 2 r 3 ( r 2 + φ ) + 3 M √ φ π 2 r ( r 2 + φ ) 2 -9 M 2 π 2 r 4 tan -1 ( r √ φ ) + 3 M 2 π 2 r 3 ( r 2 + φ ) , (28)</formula>
<formula><location><page_9><loc_19><loc_9><loc_79><loc_18></location>F a = 2 r ( p t -p r ) = -1 4 πr 3 -2 M √ φ π 2 r ( r 2 + φ ) 2 -3 M √ φ π 2 r 3 ( r 2 + φ ) + 3 M π 2 r 4 tan -1 ( r √ φ ) . (29)</formula>
<figure>
<location><page_10><loc_16><loc_64><loc_51><loc_90></location>
<caption>Fig. 4. The variation of φ (3 / 2) × forces are shown against r √ φ .</caption>
</figure>
<text><location><page_10><loc_15><loc_54><loc_79><loc_57></location>From the Fig. 4 it can be observed that stability of the system has been attained by gravitational and anisotropic forces against hydrostatic force.</text>
<section_header_level_1><location><page_10><loc_15><loc_49><loc_50><loc_50></location>3.2 ACTIVE GRAVITATIONAL MASS</section_header_level_1>
<text><location><page_10><loc_15><loc_41><loc_79><loc_45></location>The active gravitational mass within the region from the throat r 0 up to the radius R can be found as</text>
<formula><location><page_10><loc_19><loc_35><loc_79><loc_40></location>M active = 4 π R ∫ r 0 + ρr 2 dr = 2 M π [ tan -1 ( r √ φ ) -r √ φ r 2 + φ ] R r 0 + . (30)</formula>
<text><location><page_10><loc_15><loc_25><loc_79><loc_32></location>We observe here that the active gravitational mass M active of the wormhole is positive under the constraint tan -1 ( r √ φ ) > r √ φ r 2 + φ and also the nature of variation is physically acceptable as can be seen from Fig. 5.</text>
<section_header_level_1><location><page_10><loc_15><loc_20><loc_52><loc_21></location>3.3 TOTAL GRAVITATIONAL ENERGY</section_header_level_1>
<text><location><page_10><loc_15><loc_13><loc_79><loc_16></location>Using the prescription given by Lyndell-Bell et al. [31] and Nandi et al. [32], we calculate the total gravitational energy of the wormhole as</text>
<formula><location><page_10><loc_19><loc_8><loc_79><loc_11></location>E g = Mc 2 -E M , (31)</formula>
<figure>
<location><page_11><loc_16><loc_63><loc_51><loc_91></location>
<caption>Fig. 5. The variation of M active √ φ is shown against r √ φ .</caption>
</figure>
<figure>
<location><page_11><loc_30><loc_29><loc_65><loc_57></location>
<caption>Fig. 6. The variation of E g √ φ is shown against r √ φ . Fig. 1 indicates that for M √ φ = 3, r 0 √ φ takes the value 4.275.</caption>
</figure>
<text><location><page_11><loc_15><loc_16><loc_79><loc_21></location>where, Mc 2 = 1 2 ∫ R r 0 + T 0 0 r 2 dr + r 0 2 is the total energy and E M = 1 2 ∫ R r 0 + √ g rr ρr 2 dr is the total mechanical energy. Note that here 4 π 8 π yields the factor 1 2 .</text>
<text><location><page_11><loc_15><loc_9><loc_79><loc_15></location>The range of the integration is considered here from the throat r 0 √ φ to the embedded radial space of the wormhole geometry. We have solved the above Eq. (31) numerically.</text>
<table>
<location><page_12><loc_35><loc_54><loc_59><loc_86></location>
<caption>Table 1 Data for plotting Fig. 6</caption>
</table>
<text><location><page_12><loc_15><loc_34><loc_79><loc_52></location>In Fig. 6 we have considered M √ φ = 3, throat radius r 0 √ φ = 4 . 275, the upper limit R √ φ is varying from 4.275+ to 7. We have prepared a data sheet in Table 1 for plotting Fig. 6. Here our observations are as follows: (1) When we are taking r 0 √ φ less than 4 . 275, the value of the integration become complex; (2) The numerical value of the integration becomes real from the range of lower limit 4 . 275+. These real and positive values imply E g > 0, which at once indicates that there is a repulsion around the throat. Obviously this result is expected for construction of a physically valid wormhole to maintain stability of the fluid distribution.</text>
<section_header_level_1><location><page_12><loc_15><loc_27><loc_46><loc_28></location>4 CONCLUDING REMARKS</section_header_level_1>
<text><location><page_12><loc_15><loc_9><loc_79><loc_23></location>In the present letter we have considered anisotropic real matter source for constructing new wormhole solutions. The background geometry is inspired by noncommutativity along with conformal Killing vectors to constrain the form of the metric tensor. Speciality of this noncommutative geometry is to replace point-like structure of gravitational source by smeared distribution of the energy density under Gaussian distribution. Notably, in this work we consider a point-like structure of gravitational object without smearing effect where matter-energy density is of the form provided by Mehdipour [29].</text>
<text><location><page_13><loc_15><loc_86><loc_79><loc_91></location>Our investigation indicates that traversable wormhole solutions exist in this Lorentzian distribution with physically interesting properties under appropriate conditions.</text>
<text><location><page_13><loc_15><loc_82><loc_79><loc_83></location>The main observational highlights of the present study therefore are as follows:</text>
<unordered_list>
<list_item><location><page_13><loc_15><loc_71><loc_79><loc_80></location>(1) The stability of the matter distribution comprising of the wormhole has been attained in the present model. For this we have calculated the TOV equation which expresses the equilibrium condition for matter distribution subject to the gravitational force F g , hydrostatic force F h plus another force F a due to anisotropic pressure.</list_item>
<list_item><location><page_13><loc_15><loc_62><loc_79><loc_69></location>(2) The active gravitational mass M active of the wormhole is positive under the constraint tan -1 ( r √ φ ) > r √ φ r 2 + φ as is expected from the physical point of view.</list_item>
<list_item><location><page_13><loc_15><loc_57><loc_79><loc_60></location>(3) Since the total gravitational energy, E g > 0, there is a repulsion around the throat which is expected usually for stable configuration of a wormhole.</list_item>
</unordered_list>
<text><location><page_13><loc_15><loc_35><loc_79><loc_54></location>As stated earlier in the text, in the present letter we employ energy density given by Mehdipour [29] instead of Nicolini-Smailagic-Spallucci type [18]. However, our overall observation is that in our present approach the solutions and properties of the model are physically valid and interesting as much as in the former approach. As a special mention we would like to look at the Fig. 2 where we observe that the shape function is increasing instead of monotone increase as in the former case (Fig. 2 of Ref. [20]]. Further, the redshift function does not approach zero as r > r 0 due to the conformal symmetry in both the approaches. So, exploration can be done with some other rigorous studies between the two approaches, i.e. Refs. [18] and [29], which can be sought for in a future project.</text>
<section_header_level_1><location><page_13><loc_15><loc_26><loc_32><loc_27></location>Acknowledgments</section_header_level_1>
<text><location><page_13><loc_15><loc_9><loc_79><loc_21></location>F.R.and S.R.are thankful to the Inter-University Centre for Astronomy and Astrophysics (IUCAA), India for providing Visiting Research Associateship under which a part this work was carried out. IK is also thankful to IUCAA for research facilities. F.R.is grateful to UGC, India for financial support under its Research Award Scheme (Reference No.: F.30-43/2011 (SA-II) ). We are very grateful to an anonymous referee for his/her insightful comments that have led to significant improvements, particularly on the interpretational aspects.</text>
<section_header_level_1><location><page_14><loc_15><loc_89><loc_25><loc_91></location>References</section_header_level_1>
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</document> | [{"title": "Farook Rahaman a , Sreya Karmakar b , Indrani Karar c , Saibal Ray d", "content": "a Department of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India b Department of Physics, Calcutta Institute of Engineering and Management, Kolkata 700040, West Bengal, India d Department of Physics, Government College of Engineering & Ceramic Technology, Kolkata 700010, West Bengal, India", "pages": [1]}, {"title": "Abstract", "content": "In the present letter we search for a new wormhole solution inspired by noncommutative geometry with the additional condition of allowing conformal Killing vectors (CKV). A special aspect of noncommutative geometry is that it replaces point-like structures of gravitational sources with smeared objects under Gaussian distribution. However, the purpose of this paper is to obtain wormhole solutions with noncommutative geometry as a background where we consider a point-like structure of gravitational object without smearing effect. It is found through this investigation that wormhole solutions exist in this Lorentzian distribution with viable physical properties. Key words: General Relativity; noncommutative geometry; wormholes", "pages": [1]}, {"title": "1 INTRODUCTION", "content": "A wormhole , which is similar to a tunnel with two ends each in separate points in spacetime or two connecting black holes, was conjectured first by Weyl [1] and later on by [2]. In a more concrete physical definition it is essentially some kind of hypothetical topological feature of spacetime which may acts as shortcut through spacetime topology. It is argued by Morris et al. [3,4] and others [5,6,7] that in principle a wormhole would allow travel in time as well as in space and can be shown explicitly how to convert a wormhole traversing space into one traversing time. However, there are other types of wormholes available in the literature where the traversing path does not pass through a region of exotic matter [8,9]. Following the work of Visser [8], a new type of thin-shell wormhole, which was constructed by applying the cut-and-paste technique to two copies of a charged black hole [10], is of special mention in this regard. Thus a traversable wormhole, tunnel-like structure connecting different regions of our Universe or of different universes altogether, has been an issue of special investigation under Einstein's general theory of relativity [11]. It is argued by Rahaman et al. [12] that although just as good a prediction of Einstein's theory as black holes, wormholes have so far eluded detection. As one of the peculiar features a wormhole requires the violation of the null energy condition (NEC) [4]. One can note that phantom dark energy also violates the NEC and hence could have deep connection to in formation of wormholes [13,14]. It is believed that some perspective of quantum gravity can be explored mathematically in a better way with the help of non-commutative geometry. This is based on the non-commutativity of the coordinates encoded in the commutator, [ x \u00b5 , x \u03bd ] = \u03b8 \u00b5\u03bd , where \u03b8 \u00b5\u03bd is an anti-symmetric and real second-ordered matrix which determines the fundamental cell discretization of spacetime [15,16,17,18,19]. We also invoke the inheritance symmetry of the spacetime under conformal Killing vectors (CKV). Basically CKVs are motions along which the metric tensor of a spacetime remains invariant up to a certain scale factor. In a given spacetime manifold M , one can define a globally smooth conformal vector field \u03be , such that for the metric g ab it can be written as where \u03c8 : M \u2192 R is the smooth conformal function of \u03be and F ab is the conformal bivector of \u03be . This is equivalent to the following form: where L signifies the Lie derivatives along the CKV \u03be \u03b1 . In favor of the prescription of this mathematical technique CKV we find out the following features: (1) it provides a deeper insight into the spacetime geometry and facilitates the generation of exact solutions to the Einstein field equations in a more comprehensive forms, (2) the study of this particular symmetry in spacetime is physically very important as it plays a crucial role of discovering conservation laws and to devise spacetime classification schemes, and (3) because of the highly non-linearity of the Einstein field equations one can reduce easily the partial differential equations to ordinary differential equations by using CKV. Interested readers may look at the recent works on CKV technique available in the literature [20,21,22]. In this letter therefore we search for some new solutions of wormhole admitting conformal motion of Killing Vectors. It is a formal practice to consider the inheritance symmetry to establish a natural relationship between spacetime geometry and matter-energy distribution for a astrophysical system. Thus our main goal here is to examine the solutions of Einstein field equations by admitting CKV under non-commutative geometry. The scheme of the investigation is as follows: in the Sec. 2 we provide the mathematical formalism and Einstein's field equations under the framework of this technique. A specific matter-energy density profile has been employed in Sec. 3 to obtain various physical features of the wormhole under consideration by addressing the issues like the conservation equation, stability of the system, active gravitational mass and gravitational energy. Sec. 4 is devoted for some concluding remarks.", "pages": [1, 2, 3]}, {"title": "2 CONFORMALKILLINGVECTORANDBASICEQUATIONS", "content": "We take the static spherically symmetric metric in the following form where r is the radial coordinate. Here \u03bd and \u03bb are the metric potentials which have functional dependence on r only. Thus, the only survived Einstein's field equations in their explicit forms (rendering G = c = 1) are where \u03c1 , p r and p t are matter-energy density, radial pressure and transverse pressure respectively for the fluid distribution. Here ' over \u03bd and \u03bb denotes partial derivative w.r.t. radial coordinate r . The conformal Killing equations, as mentioned in Eqs. (2), then yield as follows: where \u03be \u03b1 are the conformal 4-vectors and \u03c8 is the conformal function as mentioned earlier. This set of equations, in a straight forward way, imply the following simple forms: where C 2 and C 3 are integration constants. Here the non-zero components of the conformal Killing vector \u03be a are \u03be 0 and \u03be 1 . Now using solutions (7) and (8), the equations (3)-(5) take the following form as These are the equations forming master set which has all the information of the fluid distribution under the framework of Einstein's general theory of relativity with the associated non-commutative geometry and conformal Killing vectors.", "pages": [3, 4]}, {"title": "3 THE MATTER-ENERGY DENSITY PROFILE AND PHYSICAL FEATURES OF THE WORMHOLE", "content": "As stated by Rahaman et al. [11], the necessary ingredients that supply fuel to construct wormholes remain an elusive goal for theoretical physicists and there are several proposals that have been put forward by different authors [23,24,25,26,27,28]. However, in our present work we consider cosmic fluid as source and thus have provided a new class of wormhole solutions. Keeping the essential aspects of the noncommutativity approach which are specifically sensitive to the Gaussian nature of the smearing as employed by Nicolini et al. [18], we rather get inspired by the work of Mehdipour [29] to search for a new fluid model admitting conformal motion. Therefore, we assume a Lorentzian distribution of particle-like gravitational source and hence the energy density profile as given in Ref. [29] as follows: where \u03c6 is the noncommutativity parameter and M is the smeared mass distribution. Now, solving equation (10) we get where D 1 is an integration constant and can be taken as zero. The parameters, like the radial pressure, tangential pressure and metric potentials, are found as b ( r \u221a \u03c6 ) -r \u221a \u03c6 cuts the r \u221a \u03c6 -axis. For M \u221a \u03c6 < 2 . 2, there exists no root, and therefore no throats. For M \u221a \u03c6 > 2 . 2, however we have two roots: (i) for M \u221a \u03c6 = 3, the location of the external root i.e. throat of the wormhole is r 0 \u221a \u03c6 = 4 . 275, and (ii) for M \u221a \u03c6 = 2 . 2, we have one and only one solution and this corresponds to the situation when two roots coincide and it can be interpreted as an extreme situation Let us now write down the metric potential conveniently in terms of the shape function b ( r ) as follows: where b ( r ) is given by Now, we will discuss the behavioral effects of different aspects of the above shape function b ( r ) and its derivative. The throat location of the wormhole is obtained by imposing the equation b ( r 0 ) = r 0 . One can note that the appearance of a throat depends on the parameter M and \u03c6 . However, the larger root of the equation b ( r 0 \u221a \u03c6 ) = r 0 \u221a \u03c6 , where r 0 \u221a \u03c6 is dimensionless, gives the throat which depends only one parameter M \u221a \u03c6 . Figure 1 shows that the throat of the wormhole is located at r \u221a \u03c6 = r 0 \u221a \u03c6 (maximum root), where r \u221a \u03c6 -b ( r 0 \u221a \u03c6 ) cuts the r \u221a \u03c6 -axis. One can note that position of the throat is increasing with the increase of smeared mass distribution M. For M \u221a \u03c6 < 2 . 2 no throat exists. From the above analysis, we notice that we may get feasible wormholes for M \u221a \u03c6 > 2 . 2. For the sake of brevity, we assume M \u221a \u03c6 = 3 for the rest of the study. From the left panel of Fig. 2, we observe that shape function is increasing, therefore, b ' ( r \u221a \u03c6 ) > 0. From Fig. 1, one can also note that for ( r \u221a \u03c6 ) > is an essential requirement for a shape function. Right panel of figure 2 indicates that the flare-out condition b ' ( r \u221a \u03c6 ) < 1 for ( r \u221a \u03c6 ) > ( r 0 \u221a \u03c6 ) is satisfied. We also observe the asymptotic behaviour from the middle panel of Fig. 2 such that b ( ( r \u221a \u03c6 ) ) ( r \u221a \u03c6 ) \u2192 0 as ( r \u221a \u03c6 ) \u2192 \u221e . Unfortunately, this has the similar explanation as done in Ref. [20] that the redshift function does not approach zero as r \u2192 \u221e due to the conformal symmetry. This means the wormhole spacetime is not asymptotically flat and hence will have to be cut off at some radial distance which smoothly joins to an exterior vacuum solution. One can now find out the redshift function f ( r ), where e 2 f ( r ) = e \u03bd ( r ) . Using Eq. (7) we find It can be observed from the above expression that the wormhole presented here is traversable one as redshift function remains finite. The above solution should be matched with the exterior vacuum spacetime of the Schwarzschild type at some junction interface with radius R . Using this matching condition, one can easily find the value of unknown constant C 2 as so that the redshift function now explicitly becomes The redshift function is therefore finite in the region r 0 < r < R , as required because this will prevent an event horizon. According to Fig. 3, \u03c6 ( \u03c1 + p r ) < 0, therefore, the null energy condition is violated to hold a wormhole open.", "pages": [5, 6, 7, 8]}, {"title": "3.1 THE TOLMAN-OPPENHEIMER-VOLKOFF EQUATION", "content": "Following the suggestion of Ponce de Leon [30], we write the Tolman-OppenheimerVolkoff (TOV) equation in the following form where M G = M G ( r ) is the effective gravitational mass within the region from r 0 up to the radius r and is given by Equation (24) expresses the equilibrium condition for matter distribution comprising the wormhole subject to the gravitational force F g , hydrostatic force F h plus another force F a due to anisotropic pressure. Now, the above Eq. (24) can be easily written as where From the Fig. 4 it can be observed that stability of the system has been attained by gravitational and anisotropic forces against hydrostatic force.", "pages": [9, 10]}, {"title": "3.2 ACTIVE GRAVITATIONAL MASS", "content": "The active gravitational mass within the region from the throat r 0 up to the radius R can be found as We observe here that the active gravitational mass M active of the wormhole is positive under the constraint tan -1 ( r \u221a \u03c6 ) > r \u221a \u03c6 r 2 + \u03c6 and also the nature of variation is physically acceptable as can be seen from Fig. 5.", "pages": [10]}, {"title": "3.3 TOTAL GRAVITATIONAL ENERGY", "content": "Using the prescription given by Lyndell-Bell et al. [31] and Nandi et al. [32], we calculate the total gravitational energy of the wormhole as where, Mc 2 = 1 2 \u222b R r 0 + T 0 0 r 2 dr + r 0 2 is the total energy and E M = 1 2 \u222b R r 0 + \u221a g rr \u03c1r 2 dr is the total mechanical energy. Note that here 4 \u03c0 8 \u03c0 yields the factor 1 2 . The range of the integration is considered here from the throat r 0 \u221a \u03c6 to the embedded radial space of the wormhole geometry. We have solved the above Eq. (31) numerically. In Fig. 6 we have considered M \u221a \u03c6 = 3, throat radius r 0 \u221a \u03c6 = 4 . 275, the upper limit R \u221a \u03c6 is varying from 4.275+ to 7. We have prepared a data sheet in Table 1 for plotting Fig. 6. Here our observations are as follows: (1) When we are taking r 0 \u221a \u03c6 less than 4 . 275, the value of the integration become complex; (2) The numerical value of the integration becomes real from the range of lower limit 4 . 275+. These real and positive values imply E g > 0, which at once indicates that there is a repulsion around the throat. Obviously this result is expected for construction of a physically valid wormhole to maintain stability of the fluid distribution.", "pages": [10, 11, 12]}, {"title": "4 CONCLUDING REMARKS", "content": "In the present letter we have considered anisotropic real matter source for constructing new wormhole solutions. The background geometry is inspired by noncommutativity along with conformal Killing vectors to constrain the form of the metric tensor. Speciality of this noncommutative geometry is to replace point-like structure of gravitational source by smeared distribution of the energy density under Gaussian distribution. Notably, in this work we consider a point-like structure of gravitational object without smearing effect where matter-energy density is of the form provided by Mehdipour [29]. Our investigation indicates that traversable wormhole solutions exist in this Lorentzian distribution with physically interesting properties under appropriate conditions. The main observational highlights of the present study therefore are as follows: As stated earlier in the text, in the present letter we employ energy density given by Mehdipour [29] instead of Nicolini-Smailagic-Spallucci type [18]. However, our overall observation is that in our present approach the solutions and properties of the model are physically valid and interesting as much as in the former approach. As a special mention we would like to look at the Fig. 2 where we observe that the shape function is increasing instead of monotone increase as in the former case (Fig. 2 of Ref. [20]]. Further, the redshift function does not approach zero as r > r 0 due to the conformal symmetry in both the approaches. So, exploration can be done with some other rigorous studies between the two approaches, i.e. Refs. [18] and [29], which can be sought for in a future project.", "pages": [12, 13]}, {"title": "Acknowledgments", "content": "F.R.and S.R.are thankful to the Inter-University Centre for Astronomy and Astrophysics (IUCAA), India for providing Visiting Research Associateship under which a part this work was carried out. IK is also thankful to IUCAA for research facilities. F.R.is grateful to UGC, India for financial support under its Research Award Scheme (Reference No.: F.30-43/2011 (SA-II) ). We are very grateful to an anonymous referee for his/her insightful comments that have led to significant improvements, particularly on the interpretational aspects.", "pages": [13]}] |
2015PhRvD..92b4041M | https://arxiv.org/pdf/1503.06651.pdf | <document>
<section_header_level_1><location><page_1><loc_25><loc_92><loc_75><loc_93></location>Is cosmic censorship restored in higher dimensions?</section_header_level_1>
<text><location><page_1><loc_24><loc_89><loc_76><loc_90></location>M. D. Mkenyeleye, 1, ∗ Rituparno Goswami, 1, † and Sunil D. Maharaj 1, ‡</text>
<text><location><page_1><loc_27><loc_85><loc_74><loc_88></location>1 Astrophysics and Cosmology Research Unit, School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa.</text>
<text><location><page_1><loc_18><loc_72><loc_83><loc_83></location>In this paper we extend the analysis of gravitational collapse of spherically symmetric generalised Vaidya spacetimes to higher dimensions, in the context of the Cosmic Censorship Conjecture. We present the sufficient conditions on the generalised Vaidya mass function, that will generate a locally naked singular end state. Our analysis here generalises all the earlier works on collapsing higher dimensional generalised Vaidya spacetimes. With specific examples, we show the existence of classes of mass functions that lead to a naked singularity in four dimensions, which gets covered on transition to higher dimensions. Hence for these classes of mass function Cosmic Censorship gets restored in higher dimensions and the transition to higher dimensions restricts the set of initial data that results in a naked singularity.</text>
<text><location><page_1><loc_18><loc_69><loc_40><loc_70></location>PACS numbers: 04.20.Cv , 04.20.Dw</text>
<section_header_level_1><location><page_1><loc_20><loc_65><loc_37><loc_66></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_37><loc_49><loc_63></location>The singularity theorems predict the occurrence of spacetime singularities for a wide class of theories of gravity under very generic conditions, namely the attractive nature of gravity, existence of closed trapped surfaces and no violations of causality in the spacetime [1]. However these theorems do not say anything about the causal nature of these singularities, that is, if it is possible for future directed null geodesics from the close vicinity of these singular points, to escape to infinity. To avoid such scenarios where a naked singularity exists that can causally influence the future infinities, the Cosmic Censorship Conjecture (CCC) was proposed by Penrose [2]. This states that spacetime singularities produced by the gravitational collapse of physically realistic matter fields are always covered by trapped surfaces. Hence the final state of continual gravitational collapse always leads to a black hole, where the singularity is shielded from any external observer.</text>
<text><location><page_1><loc_9><loc_16><loc_49><loc_37></location>Though the general proof of this conjecture still remains elusive, there are a number of important counterexamples that show otherwise. Investigations of spherically symmetric dynamical collapse models in General Relativity for large classes of matter fields, in four dimensional spacetimes, indicate that there exist sets of initial data of non-zero measure, at the epoch of the commencement of the collapse, that lead to the formation of a locally naked singularity. In these cases the trapped surfaces do not form early enough to shield the singularity (or the spacetime fireball) from external observers. It is also shown in these studies that families of future outgoing non-spacelike geodesics emerge from such a naked singularity, providing a non-zero measure set of trajecto-</text>
<text><location><page_1><loc_52><loc_58><loc_92><loc_67></location>ries escaping away [3-5]. Though these counter examples are mainly presented in case of spherical symmetry (with a few exceptions of non-spherical models), they suffice to be relevant because if the censorship is one of key aspect of gravitation theory, it should not depend on symmetries of spacetime.</text>
<section_header_level_1><location><page_1><loc_65><loc_54><loc_79><loc_55></location>A. The question</section_header_level_1>
<text><location><page_1><loc_52><loc_46><loc_92><loc_52></location>To avoid the unpleasantries of nudity, the obvious question that arises (influenced by higher dimensional and emergent theories of gravity - e.g string theory or braneworld models), is as follows:</text>
<text><location><page_1><loc_52><loc_38><loc_92><loc_43></location>Question. Does the transition to higher dimensional spacetimes (with compact or non-compact extra dimensions) restricts the above mentioned set of initial data that leads to a naked singularity?</text>
<text><location><page_1><loc_52><loc_19><loc_92><loc_36></location>In other words, how does the number of spacetime dimensions dictate the dynamics of trapped regions in the spacetime? This question is important as most of the proofs of the key theorems of black hole dynamics and thermodynamics demand the spacetimes to be future asymptotically simple, which is not possible if the censorship is violated [1]. If the locally naked singularities in 4-dimensional spacetime are naturally absent in higher dimensions, then that will be an argument in favour of higher dimensional (or emergent theories) of gravity, as in those cases the important results of black hole dynamics and thermodynamics would be more relevant.</text>
<section_header_level_1><location><page_1><loc_65><loc_15><loc_79><loc_16></location>B. Earlier works</section_header_level_1>
<text><location><page_1><loc_52><loc_8><loc_92><loc_10></location>important result [6, 7]: The naked singularities occurring</text>
<text><location><page_2><loc_9><loc_80><loc_49><loc_93></location>in dust collapse from smooth initial data (which include those discovered by Eardley and Smarr [8], Christodoulou [9], and Newman [10]) are eliminated when we make transition to higher dimensional spacetimes. The cosmic censorship is then restored for dust collapse which will always produce a black hole as the collapse end state for dimensions D ≥ 6, under conditions such as the smoothness of initial data from which the collapse develops, which follows from physical grounds.</text>
<text><location><page_2><loc_9><loc_70><loc_49><loc_80></location>The physical reason behind the above result is that higher dimensional spacetimes favour trapped surface formation and the formation of horizons advance in time. Hence for dimensions greater than five, the vicinity of the singularity always gets trapped even before the singularity is formed, and hence the singularity is causally cut-off from any external observer.</text>
<text><location><page_2><loc_9><loc_59><loc_49><loc_70></location>Several other works on higher dimensional radiation collapse and perfect fluid collapse has been done [1117], where the matter field is taken to be of a specific form (for example: perfect fluids with linear equation of state, pure radiation, charged radiation etc.). All of these studies give an indication that higher dimensions do favour trapping and hence the epoch of trapped surface formation advances as we go to higher dimensions.</text>
<section_header_level_1><location><page_2><loc_20><loc_55><loc_38><loc_56></location>C. The present paper</section_header_level_1>
<text><location><page_2><loc_9><loc_19><loc_49><loc_52></location>The main criticism of the dustlike models or pure perfect fluid models is that they are far too idealised. For any realistic massive astrophysical body, which is undergoing gravitational collapse, the pressure and the radiative processes must play an important role together. One of the known spacetimes that can closely mimic such a collapse scenario is the generalised Vaidya spacetime, where the matter field is a specific combination of Type I matter (whose energy momentum tensor has one timelike and three spacelike eigenvectors) that moves along timelike trajectories and Type II matter (whose energy momentum tensor has double null eigenvectors) that moves along null trajectories. Thus, a collapsing generalised Vaidya spacetime depicts the collapse of usual perfect fluid combined with radiation. Therefore the collapse scenario here is much closer to what is expected for the collapse of a realistic astrophysical star. In our earlier paper [18], we investigated the gravitational collapse of generalised Vaidya spacetimes in four dimensions and developed a general mathematical framework to study the conditions on the mass function such that future directed non-spacelike geodesics can terminate at the singularity in the past. In this paper:</text>
<unordered_list>
<list_item><location><page_2><loc_11><loc_9><loc_49><loc_19></location>1. We extend the earlier results to any arbitrary N -dimensional spacetimes. Though the general mathematical framework remains similar, the conditions on the mass function and it's derivatives for the collapse leading to a locally naked singularity, change as we make a transition to higher dimensional spacetimes.</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_2><loc_54><loc_73><loc_92><loc_93></location>2. Using explicit examples we show that there exist classes of mass functions, that lead the collapsing star to a naked singularity in four dimensions, will necessarily end in a black hole end state in dimensions greater than four. The reason for this remains the same as in dust models: formation of trapped surfaces is favoured in higher dimensions, and hence the vicinity of the central singularity gets trapped even before the singularity is formed. This gives a definite indication that the dynamics of trapped regions do depend on the spacetime dimensions for a large class of matter fields and the occurrence of trapped surfaces advance in time in higher dimensions.</list_item>
</unordered_list>
<text><location><page_2><loc_52><loc_62><loc_92><loc_72></location>Unless otherwise specified, we use natural units ( c = 8 πG = 1) throughout this paper, Latin indices run from 0 to N -1. The symbol ∇ represents the usual covariant derivative and ∂ corresponds to partial differentiation. We use the ( -, + , + , + , + , · · · ) signature and the Ricci tensor is obtained by contracting the first and the third indices of the Riemann tensor</text>
<formula><location><page_2><loc_54><loc_58><loc_92><loc_61></location>R a bcd = Γ a bd,c -Γ a bc,d +Γ e bd Γ a ce -Γ e bc Γ a de , (1)</formula>
<text><location><page_2><loc_52><loc_55><loc_92><loc_58></location>The Hilbert-Einstein action in the presence of matter is given by</text>
<formula><location><page_2><loc_59><loc_50><loc_92><loc_54></location>S = 1 2 ∫ d 4 x √ -g [ R -2Λ -2 L m ] , (2)</formula>
<text><location><page_2><loc_52><loc_48><loc_91><loc_50></location>variation of which gives the Einstein field equations as</text>
<formula><location><page_2><loc_65><loc_46><loc_92><loc_47></location>G ab +Λ g ab = T ab . (3)</formula>
<section_header_level_1><location><page_2><loc_53><loc_40><loc_90><loc_42></location>II. HIGHER DIMENSIONAL GENERALISED VAIDYA SPACETIME</section_header_level_1>
<text><location><page_2><loc_52><loc_35><loc_92><loc_38></location>The spherically symmetric line element for an N -dimensional generalised Vaidya spacetime is given as</text>
<formula><location><page_2><loc_53><loc_30><loc_92><loc_34></location>ds 2 = -( 1 -2 m ( v, r ) r ( N -3) ) dv 2 +2 dvdr + r 2 d Ω 2 ( N -2) , (4)</formula>
<text><location><page_2><loc_52><loc_28><loc_56><loc_30></location>where</text>
<formula><location><page_2><loc_59><loc_21><loc_92><loc_27></location>d Ω 2 ( N -2) = N -2 ∑ i =1 i -1 ∏ j =1 sin 2 ( θ j ) ( dθ i ) 2 , (5)</formula>
<text><location><page_2><loc_52><loc_9><loc_92><loc_21></location>is the metric on the ( N -2) sphere in polar coordinates with θ i being spherical coordinates. m ( v, r ) is the generalized mass function related to the gravitational energy within a given radius r [19], which can be carefully defined so that the energy conditions are satisfied. The coordinate v represents the Eddington advanced time where r is decreasing towards the future along a ray v = Const. (ingoing). When N = 4, the line element reduces to the generalized Vaidya solution [20] in 4-dimensions.</text>
<text><location><page_3><loc_10><loc_92><loc_34><loc_93></location>Defining the following quantities</text>
<formula><location><page_3><loc_13><loc_88><loc_44><loc_91></location>˙ m ( v, r ) ≡ ∂m ( v, r ) ∂v , m ' ( v, r ) ≡ ∂m ( v, r ) ∂r ,</formula>
<text><location><page_3><loc_9><loc_84><loc_49><loc_87></location>we can write the non-vanishing components of the Ricci tensor as</text>
<formula><location><page_3><loc_14><loc_81><loc_49><loc_84></location>R v v = R r r = m '' ( v, r ) r ( N -3) -( N -4) m ' ( v, r ) r ( N -2) , (6a)</formula>
<formula><location><page_3><loc_14><loc_78><loc_49><loc_81></location>R θ 1 θ 1 = R θ 2 θ 2 = · · · = R θ ( N -2) θ ( N -2) = 2 m ' ( v, r ) r ( N -2) . (6b)</formula>
<text><location><page_3><loc_9><loc_76><loc_29><loc_77></location>The Ricci scalar is given by</text>
<formula><location><page_3><loc_19><loc_72><loc_49><loc_75></location>R = 2 m '' ( v, r ) r ( N -3) + 4 m ' ( v, r ) r ( N -2) , (7)</formula>
<text><location><page_3><loc_9><loc_68><loc_49><loc_71></location>while the non-vanishing components of the Einstein tensor are given by</text>
<formula><location><page_3><loc_14><loc_64><loc_49><loc_67></location>G v v = G r r = -( N -2) m ' ( v, r ) r ( N -2) , (8a)</formula>
<formula><location><page_3><loc_14><loc_61><loc_49><loc_64></location>G r v = ( N -2) ˙ m ( v, r ) r ( N -2) , (8b)</formula>
<formula><location><page_3><loc_14><loc_58><loc_49><loc_61></location>G θ 1 θ 1 = G θ 2 θ 2 = · · · = G θ ( N -2) θ ( N -2) = -m '' ( v, r ) r ( N -3) . (8c)</formula>
<text><location><page_3><loc_9><loc_54><loc_49><loc_57></location>The Energy Momentum Tensor (EMT) can be written in the form [21]</text>
<formula><location><page_3><loc_22><loc_52><loc_49><loc_53></location>T µν = T ( n ) µν + T ( m ) µν , (9)</formula>
<text><location><page_3><loc_9><loc_49><loc_13><loc_51></location>where</text>
<formula><location><page_3><loc_16><loc_47><loc_49><loc_48></location>T ( n ) µν = µl µ l ν , (10a)</formula>
<formula><location><page_3><loc_15><loc_45><loc_49><loc_46></location>T ( m ) µν = ( ρ + /rho1 )( l µ n ν + l ν n µ ) + /rho1g µν . (10b)</formula>
<text><location><page_3><loc_9><loc_42><loc_18><loc_44></location>In the above,</text>
<formula><location><page_3><loc_12><loc_35><loc_49><loc_41></location>µ = ( N -2) ˙ m ( v, r ) r ( N -2) , ρ = ( N -2) m ' ( v, r ) r ( N -2) , (11) /rho1 = -m '' ( v, r ) r ( N -3) ,</formula>
<text><location><page_3><loc_9><loc_33><loc_36><loc_34></location>with l µ and n µ being two null vectors,</text>
<formula><location><page_3><loc_13><loc_28><loc_49><loc_32></location>l µ = δ 0 µ , n µ = 1 2 [ 1 -2 m ( v, r ) r ( N -3) ] δ 0 µ -δ 1 µ , (12)</formula>
<text><location><page_3><loc_9><loc_26><loc_37><loc_28></location>where l µ l µ = n µ n µ = 0 and l µ n µ = -1.</text>
<text><location><page_3><loc_9><loc_16><loc_49><loc_27></location>Eq. (9) is taken as a generalized Energy-Momentum Tensor for the generalized Vaidya spacetime, with the component T ( n ) µν being considered as the matter field that moves along the null hypersurfaces v = constant, while T ( m ) µν describes the matter moving along timelike trajectories. If the EMT of Eq. (9) is projected to the orthonormal basis, defined by the vectors,</text>
<formula><location><page_3><loc_10><loc_8><loc_49><loc_15></location>E µ (0) = l µ + n µ √ 2 , E µ (1) = l µ -n µ √ 2 , E µ (2) = 1 r δ µ 2 , E µ ( N ) = 1 r sin θ 1 sin θ 2 sin θ 3 · · · sin θ ( N -2) δ µ N , (13)</formula>
<text><location><page_3><loc_52><loc_90><loc_92><loc_93></location>it can be found [20], that the symmetric EMT can be given as the N × N matrix,</text>
<formula><location><page_3><loc_58><loc_80><loc_92><loc_89></location>[ T ( µ )( ν ) ] = µ 2 + ρ µ 2 0 · · · 0 µ 2 µ 2 -ρ 0 0 0 0 0 /rho1 0 0 . . . · · · 0 /rho1 . . . 0 0 0 · · · /rho1 . (14)</formula>
<text><location><page_3><loc_52><loc_78><loc_89><loc_79></location>For this fluid the energy conditions are given as [1],</text>
<text><location><page_3><loc_54><loc_76><loc_85><loc_77></location>1. The Weak and Strong energy conditions :</text>
<formula><location><page_3><loc_60><loc_72><loc_92><loc_74></location>µ ≥ 0 , ρ ≥ 0 , /rho1 ≥ 0 , ( µ = 0) . (15)</formula>
<text><location><page_3><loc_80><loc_72><loc_80><loc_74></location>/negationslash</text>
<text><location><page_3><loc_54><loc_69><loc_79><loc_71></location>2. The Dominant energy condition :</text>
<formula><location><page_3><loc_61><loc_66><loc_92><loc_68></location>µ ≥ 0 , ρ ≥ /rho1 ≥ 0 , ( µ = 0) . (16)</formula>
<text><location><page_3><loc_79><loc_66><loc_79><loc_68></location>/negationslash</text>
<text><location><page_3><loc_52><loc_62><loc_92><loc_64></location>These energy conditions can be satisfied by suitable choices of the mass function m ( v, r ).</text>
<section_header_level_1><location><page_3><loc_55><loc_56><loc_89><loc_58></location>III. HIGHER DIMENSIONAL COLLAPSE MODEL</section_header_level_1>
<text><location><page_3><loc_52><loc_46><loc_92><loc_54></location>In this section, we examine the gravitational collapse of a collapsing matter field in the generalized Vaidya spacetime when a spherically symmetric configuration of Type I and Type II matter collapses at the centre of symmetry in an otherwise empty universe which is asymptotically flat far away [22].</text>
<text><location><page_3><loc_52><loc_41><loc_92><loc_45></location>If K µ is the tangent to non-spacelike geodesics with K µ = dx µ dk , where k is the affine parameter, then K µ ; ν K ν = 0 and</text>
<formula><location><page_3><loc_66><loc_38><loc_92><loc_40></location>g µν K µ K ν = β, (17)</formula>
<text><location><page_3><loc_52><loc_30><loc_92><loc_37></location>where β is a constant that characterizes different classes of geodesics with β = 0 for null geodesic vectors, β < 0 for timelike geodesics and β > 0 for spacelike geodesics [22]. Here we consider the case of null geodesics, that is, β = 0.</text>
<text><location><page_3><loc_52><loc_25><loc_92><loc_30></location>We calculate the equations dK v /dk and dK r /dk using the Lagrange equations given by L = 1 2 g µν dx µ dk dx ν dk and Euler-Lagrange equations</text>
<formula><location><page_3><loc_64><loc_20><loc_92><loc_24></location>∂L ∂x a -d dk ( ∂L ∂x a ,k ) = 0 , (18)</formula>
<text><location><page_3><loc_52><loc_16><loc_92><loc_19></location>In the case of the higher dimensional generalised Vaidya spacetime, these equations are given by</text>
<formula><location><page_3><loc_52><loc_8><loc_92><loc_15></location>dK v dk + ( ( N -3) m ( v, r ) r ( N -2) -m ' ( v, r ) r ( N -3) ) ( K v ) 2 = 0 , (19a) dK r dk + ˙ m ( v, r ) r ( N -3) ( K v ) 2 = 0 . (19b)</formula>
<text><location><page_4><loc_9><loc_90><loc_49><loc_93></location>All other components are considered to be 0. If we follow [23] and write K v as</text>
<formula><location><page_4><loc_24><loc_86><loc_49><loc_89></location>K v = P ( v, r ) r , (20)</formula>
<text><location><page_4><loc_9><loc_84><loc_29><loc_85></location>then using K µ K ν = 0 we get</text>
<formula><location><page_4><loc_17><loc_79><loc_49><loc_82></location>K v = dv dk = P ( v, r ) r , (21a)</formula>
<formula><location><page_4><loc_17><loc_75><loc_49><loc_79></location>K r = dr dk = P 2 r ( 1 -2 m ( v, r ) r ( N -3) ) . (21b)</formula>
<section_header_level_1><location><page_4><loc_11><loc_70><loc_47><loc_73></location>IV. CONDITIONS FOR LOCALLY NAKED SINGULARITY</section_header_level_1>
<text><location><page_4><loc_9><loc_63><loc_49><loc_68></location>The nature (a locally naked singularity or a black hole) of the collapsing solutions can be characterized by the existence of radial null geodesics coming out of the singularity [15, 22].</text>
<text><location><page_4><loc_9><loc_58><loc_49><loc_62></location>The radial null geodesics of the line element (4) can be calculated using Eqs. (21a) and (21b). These geodesics are given by the equation</text>
<formula><location><page_4><loc_20><loc_53><loc_49><loc_57></location>dv dr = 2 r ( N -3) r ( N -3) -2 m ( v, r ) . (22)</formula>
<text><location><page_4><loc_9><loc_47><loc_49><loc_52></location>This differential equation has a singularity at r = 0 , v = 0. Using the same techniques utilised in [18, 24, 25], Eq. (22) can be re-written near the singular point as</text>
<formula><location><page_4><loc_16><loc_41><loc_49><loc_45></location>dv dr = 2( N -3) r ( N -3) ( N -3) r ( N -3) -2 m ' 0 r -2 ˙ m 0 v , (23)</formula>
<text><location><page_4><loc_9><loc_39><loc_13><loc_41></location>where</text>
<formula><location><page_4><loc_19><loc_36><loc_49><loc_38></location>m 0 = lim v → 0 ,r → 0 m ( v, r ) , (24a)</formula>
<formula><location><page_4><loc_19><loc_32><loc_49><loc_35></location>˙ m 0 = lim v → 0 ,r → 0 ∂ ∂v m ( v, r ) , (24b)</formula>
<formula><location><page_4><loc_19><loc_29><loc_49><loc_32></location>m ' 0 = lim v → 0 ,r → 0 ∂ ∂r m ( v, r ) . (24c)</formula>
<section_header_level_1><location><page_4><loc_10><loc_25><loc_48><loc_26></location>A. Existence of outgoing nonspacelike geodesics</section_header_level_1>
<text><location><page_4><loc_9><loc_9><loc_49><loc_23></location>We can clearly see that Eq. (23) has a singularity at v = 0 , r = 0. The classification of the tangents of both radial and non-radial outgoing non-spacelike geodesics terminating at the singularity in the past can be given by the limiting values at v = 0 , r = 0. The conditions for the existence for such geodesics have been described in detail in [18] using the concept of contraction mappings. The existence of these radial null geodesics also characterizes the nature (a naked singularity or a black hole) of the collapsing solutions. If we let X to be the limiting value</text>
<text><location><page_4><loc_52><loc_90><loc_92><loc_93></location>at r = 0 , v = 0, we can determine the nature of this limiting value on a singular geodesic as</text>
<formula><location><page_4><loc_61><loc_86><loc_92><loc_89></location>X 0 = lim v → 0 ,r → 0 X = lim v → 0 ,r → 0 v r . (25)</formula>
<text><location><page_4><loc_52><loc_75><loc_92><loc_85></location>Using a suitably chosen mass function, Eq. (23) and l'Hopital's rule, we can explicitly find the expression for the tangent values X 0 which governs the behaviour of the null geodesics near the singular point. Thus, the nature of the singularity can then be determined by studying the solution of this algebraic equation. This expression can be calculated as</text>
<formula><location><page_4><loc_55><loc_67><loc_92><loc_74></location>X 0 = lim v → 0 ,r → 0 dv dr = lim v → 0 ,r → 0 2( N -3) r ( N -4) ( N -3) r ( N -4) -2 m ' 0 -2 ˙ m 0 X 0 . (26)</formula>
<section_header_level_1><location><page_4><loc_63><loc_63><loc_80><loc_64></location>B. Apparent Horizon</section_header_level_1>
<text><location><page_4><loc_52><loc_50><loc_92><loc_61></location>The existence of the apparent horizon, which is the boundary of the trapped surface region in the spacetime also determines the nature of the singularity. If at least one value of the limiting positive values X 0 is less than the slope of the apparent horizon at the central singularity, then the central singularity is locally naked with the outgoing radial null geodesics escaping from the past to the future.</text>
<text><location><page_4><loc_52><loc_47><loc_92><loc_50></location>For the generalized higher dimensional Vaidya spacetime, the apparent horizon is defined by</text>
<formula><location><page_4><loc_65><loc_44><loc_92><loc_46></location>2 m ( v, r ) = r ( N -3) . (27)</formula>
<text><location><page_4><loc_52><loc_40><loc_92><loc_43></location>The slope of the apparent horizon can be calculated as follows:</text>
<formula><location><page_4><loc_62><loc_36><loc_92><loc_39></location>2 dm ( v, r ) dr = ( N -3) r ( N -4) , (28a)</formula>
<formula><location><page_4><loc_54><loc_30><loc_92><loc_34></location>2 ( ∂m ∂v )( dv dr ) AH +2 ∂m ∂r = ( N -3) r ( N -4) . (28b)</formula>
<text><location><page_4><loc_52><loc_26><loc_92><loc_29></location>Thus, the slope of the apparent horizon at the central singularity is given by</text>
<formula><location><page_4><loc_54><loc_20><loc_92><loc_25></location>X AH = ( dv dr ) AH = lim v → 0 ,r → 0 ( N -3) r ( N -4) -2 m ' 0 2 ˙ m 0 , (29)</formula>
<section_header_level_1><location><page_4><loc_62><loc_16><loc_81><loc_17></location>C. Sufficient conditions</section_header_level_1>
<text><location><page_4><loc_52><loc_9><loc_92><loc_14></location>We can now write the sufficient conditions for the existence of a locally naked central singularity for a collapsing generalised Vaidya spacetime in arbitrary dimensions N , which we state in the following proposition:</text>
<table>
<location><page_5><loc_9><loc_84><loc_92><loc_91></location>
<caption>TABLE I: Algebraic equations for X 0 and X AH for different values of n and N</caption>
</table>
<text><location><page_5><loc_9><loc_74><loc_49><loc_81></location>Proposition 1. Consider a collapsing N-dimensional generalised Vaidya spacetime from a regular epoch, with a mass function m ( v, r ) , that obeys all physically reasonable energy conditions and is differentiable in the entire spacetime. If the following conditions are satisfied :</text>
<unordered_list>
<list_item><location><page_5><loc_11><loc_70><loc_49><loc_73></location>1. The limits of the partial derivatives of the mass function m ( v, r ) exist at the central singularity,</list_item>
<list_item><location><page_5><loc_11><loc_66><loc_49><loc_69></location>2. There exist one or more positive real roots X 0 of the equation (26),</list_item>
<list_item><location><page_5><loc_11><loc_62><loc_49><loc_65></location>3. At least one of the positive real roots of X 0 is less than the smallest root of equation (29),</list_item>
</unordered_list>
<text><location><page_5><loc_9><loc_58><loc_49><loc_61></location>then the central singularity is locally naked with outgoing C 1 radial null geodesics escaping to the future.</text>
<text><location><page_5><loc_9><loc_52><loc_49><loc_57></location>We emphasise here, that all the previous works of higher dimensional generalised Vaidya collapse [15-17] are special cases of the general analysis presented above. In the next section, we give a specific example to trans-</text>
<text><location><page_5><loc_52><loc_78><loc_92><loc_81></location>parently demonstrate the effect of transition to higher dimensions on the nature of the central singularity.</text>
<section_header_level_1><location><page_5><loc_54><loc_73><loc_90><loc_75></location>V. A GENERAL LAURENT EXPANDABLE MASS FUNCTION</section_header_level_1>
<text><location><page_5><loc_52><loc_67><loc_92><loc_71></location>We consider here a Laurent expandable mass function of the generalized Vaidya spacetime in higher dimensions in the general form as</text>
<formula><location><page_5><loc_53><loc_62><loc_92><loc_65></location>2 m ( v, r ) = λ 1 m 1 ( v ) -λ 2 m 2 ( v ) r ( N -3) -λ 3 m 3 ( v ) r ( N -2) + · · · , (30)</formula>
<text><location><page_5><loc_52><loc_60><loc_56><loc_61></location>where</text>
<text><location><page_5><loc_52><loc_56><loc_92><loc_59></location>m n ( v ) = v (2 N + n -8) , n = 1 , 2 , · · · and λ n 's are constants .</text>
<text><location><page_5><loc_52><loc_51><loc_92><loc_56></location>Using Eq. (26) and (29), we get the expression of the tangent to null geodesics X 0 and tangent to the apparent horizon X AH in higher dimensions as</text>
<formula><location><page_5><loc_17><loc_40><loc_92><loc_46></location>X 0 = 2( N -3) ( N -3) -(2 N -7) λ 1 X (2 N -7) 0 +( N -3) ( λ 2 X 2 N -6 0 + λ 3 X (2 N -5) 0 + λ 4 X (2 N -4) 0 + · · · ) . (31)</formula>
<formula><location><page_5><loc_19><loc_35><loc_92><loc_39></location>X AH = ( N -3) -( N -3) λ 2 X (2 N -6) AH -( N -2) λ 3 X (2 N -5) AH -( N -1) λ 4 X (2 N -4) AH -··· λ 1 (2 N -7) X (2 N -8) AH -(2 N -6) λ 2 X (2 N -7) AH -(2 N -5) λ 3 X (2 N -6) AH -··· , (32)</formula>
<text><location><page_5><loc_9><loc_30><loc_49><loc_31></location>These expressions can be written in the general form as</text>
<formula><location><page_5><loc_9><loc_24><loc_49><loc_29></location>∞ ∑ n =1 ( f n ( N,λ i ) X (2 N + n -7) 0 ) +( N -3) X 0 -2( N -3) = 0 , (33)</formula>
<text><location><page_5><loc_9><loc_22><loc_11><loc_23></location>and</text>
<formula><location><page_5><loc_15><loc_16><loc_49><loc_21></location>∞ ∑ n =1 g n ( N,λ i ) X (2 N + n -8) AH -( N -3) = 0 . (34)</formula>
<text><location><page_5><loc_9><loc_13><loc_49><loc_16></location>where f n ( N,λ i ) and g n ( N,λ i ) are some functions of N and the λ i 's.</text>
<text><location><page_5><loc_9><loc_9><loc_49><loc_13></location>These expressions can explicitly be solved for X 0 and X AH using some specific values of n , N and λ i 's (see Table I) and then one can make conclusions about the</text>
<text><location><page_5><loc_52><loc_29><loc_92><loc_31></location>nature of the singularity by using the following conditions:</text>
<unordered_list>
<list_item><location><page_5><loc_53><loc_22><loc_92><loc_28></location>(i) If there is no positive real solution for X 0 , then there are no outgoing null geodesics from the singularity and the singularity is causally cut off from the external observer.</list_item>
<list_item><location><page_5><loc_53><loc_15><loc_92><loc_21></location>(ii) If there is no real solution for X AH , then there are no trapped surfaces and the singularity is globally naked, provided there is at least one positive real root of X 0 .</list_item>
<list_item><location><page_5><loc_52><loc_9><loc_92><loc_14></location>(iii) If there are one or multiple real solutions for X AH with the smallest solution less than X 0 , then it can be concluded that the collapse results in a black hole end state.</list_item>
</unordered_list>
<table>
<location><page_6><loc_9><loc_83><loc_92><loc_91></location>
<caption>TABLE II: Values of X 0 and Min[ X AH ] for different dimensions for λ 1 = 5 . 0, λ 2 = 0 . 01, λ 3 = 2 . 3, λ 4 = 0 . 05.</caption>
</table>
<text><location><page_6><loc_9><loc_74><loc_49><loc_80></location>(iv) If the smallest solution Min[ X AH ] is greater than any one of the positive solutions of X 0 , then there will be future directed null geodesics from the singularity and hence the singularity is locally naked.</text>
<text><location><page_6><loc_9><loc_65><loc_49><loc_73></location>We can easily see from Table I, that the general expression obtained here contains the expressions for X 0 and X AH corresponding to Vaidya collapse in 4-D ( n = 1 , N = 4) [22, 23], charged Vaidya-de Sitter in 4-D ( n = 2 , N = 4) [17] and charged Vaidya in 5-D ( n = 2 , N = 5) [16].</text>
<section_header_level_1><location><page_6><loc_11><loc_59><loc_47><loc_62></location>A. Example: Class of naked singularity in 4D being eliminated in higher dimensions</section_header_level_1>
<text><location><page_6><loc_9><loc_49><loc_49><loc_57></location>In this section we will consider a specific example, that can be easily generalised to an open set, to show explicitly how a naked singularity in four dimensions gets covered in higher dimensions. Let us consider a scenario where n = 4. In this case the expression for X 0 and X AH become</text>
<formula><location><page_6><loc_10><loc_43><loc_49><loc_48></location>(2 N -7) λ 1 X 2 N -6 0 -( N -3) λ 2 X 2 N -5 0 -( N -3) λ 3 X 2 N -4 0 -( N -3) λ 4 X 2 N -3 0 -( N -3) X 0 +2( N -3) = 0 , (35)</formula>
<text><location><page_6><loc_9><loc_40><loc_11><loc_42></location>and</text>
<formula><location><page_6><loc_10><loc_35><loc_49><loc_39></location>(2 N -7) λ 1 X 2 N -7 AH -( N -3) λ 2 X 2 N -6 AH -( N -3) λ 3 X 2 N -5 AH -( N -3) λ 4 X 2 N -4 AH -( N -3) = 0 . (36)</formula>
<text><location><page_6><loc_9><loc_9><loc_49><loc_34></location>We can solve these equations numerically to get the values of X 0 and X AH in different dimensions. For our calculations we took λ 1 = 5 . 0, λ 2 = 0 . 01, λ 3 = 2 . 3, λ 4 = 0 . 05. From Table II we can easily see that in 4 dimensions, this class of mass function leads to a naked singularity, as the trapped surfaces do not form early enough to shield the singularity from outside observers. However when we make the transition to higher dimensions we see that the value of the tangent to the outgoing null geodesic from the central singularity is greater than the slope of the apparent horizon curve at the central singularity. In this case the outgoing null direction is within the trapped region and hence the singularity is causally cut off from the external observer. By the continuity of the mass function considered above, this can be easily converted to a open set in the mass function space, where this scenario continues to be true and we shall explicitly prove this in the following subsection.</text>
<section_header_level_1><location><page_6><loc_52><loc_77><loc_92><loc_80></location>B. Proof of existence of open set of mass functions with the above properties</section_header_level_1>
<text><location><page_6><loc_52><loc_61><loc_92><loc_75></location>Having found out a specific example of a mass function for which the naked singularities in 4D is eliminated when we go to higher dimension, we would now require to prove that such a mass function is generic in the sense that there exists a open set of such mass functions in the function space. Since this problem of deducing the nature of the central singularity is reduced to finding and comparing real roots of polynomials (35) and (36), all we need to show here is the real roots of these polynomials are continuous functions of the coefficients.</text>
<text><location><page_6><loc_52><loc_51><loc_92><loc_61></location>To do this, first of all we observe that the roots that are given in the Table II are all of multiplicity one. This can be easily seen by differentiating the LHS of (35) and (36) and substituting the roots to find non-zero values. Now, for any complex polynomial p ( z ) of degree n ≥ 1 with m distinct roots { α 1 , · · · , α m } , (1 ≤ m ≤ n ), let us define the quantity R 0 ( p ) as follows:</text>
<formula><location><page_6><loc_53><loc_45><loc_92><loc_49></location>R 0 ( p ) = { 1 2 , if m = 1 . 1 2 min | α i -α j | , i ≤ j ≤ m, if m> 1 . (37)</formula>
<text><location><page_6><loc_52><loc_41><loc_92><loc_44></location>We now state the well known result of complex analysis [26]:</text>
<text><location><page_6><loc_52><loc_31><loc_92><loc_40></location>Theorem 1. Let p ( z ) be a polynomial of degree n ≥ 1 , with real coefficients { µ k } . Suppose α be a real root of p ( z ) of multiplicity one. Then for any /epsilon1 with 0 ≤ /epsilon1 ≤ R 0 ( p ) , there exists a δ ( /epsilon1 ) > 0 such that any polynomial q ( z ) with real coefficients ν k and | µ k -ν k | ≤ δ , has a real root β with | α -β | ≤ /epsilon1 .</text>
<text><location><page_6><loc_52><loc_19><loc_92><loc_30></location>The above theorem shows that if a polynomial p ( z ) with real coefficients has a real root α of multiplicity one, then any polynomial q ( z ) obtained by small (real) perturbations to the coefficients of p ( z ) will also have a real root in a neighbourhood of α . That is, not only the root depends continuously on coefficients, but also it remains real, under sufficiently small perturbations of coefficients.</text>
<text><location><page_6><loc_52><loc_9><loc_92><loc_19></location>This results directly translates to our problem of open set of mass functions in the mass function space. Once we have a specific example as shown in Table II, any perturbations around that will have the same outcome as far as the nature of the singularities are concerned. Hence this class of mass functions is not fine tuned, but quite generic and the outcome is stable under perturbations.</text>
<table>
<location><page_7><loc_9><loc_81><loc_92><loc_89></location>
<caption>TABLE III: Range for X 0 and Min[ X AH ] for different dimensions: { 4 . 8 < λ 1 < 5 . 25 , 0 . 009 < λ 2 < 0 . 012 , 2 . 25 < λ 3 < 2 . 38 , λ 4 = 0 . 05 }TABLE IV: Range for X 0 and Min[ X AH ] for different dimensions: { λ 1 = 5 . 2 , 0 . 009 < λ 2 < 0 . 012 , λ 3 = 2 . 3 , 0 ≤ λ 4 < 0 . 4 }</caption>
</table>
<table>
<location><page_7><loc_9><loc_67><loc_92><loc_76></location>
</table>
<section_header_level_1><location><page_7><loc_19><loc_62><loc_39><loc_63></location>C. Numerical verification</section_header_level_1>
<text><location><page_7><loc_9><loc_37><loc_49><loc_60></location>We would now like to verify explicitly, with the aid of numerical calculations, the results in the previous subsection. We numerically solve equations (35) and (36) to get the values of X 0 and X AH in different dimensions to show that there exists a set of parameter intervals for which the mass function leads to a naked singularity in 4 dimensions and a black hole in higher dimensions. For example, some of the intervals are { 4 . 8 < λ 1 < 5 . 25 , 0 . 009 < λ 2 < 0 . 012 , 2 . 25 < λ 3 < 2 . 38 , λ 4 = 0 . 05 } with the range values shown in Table III and { λ 1 = 5 . 2 , 0 . 009 < λ 2 < 0 . 012 , λ 3 = 2 . 3 , 0 ≤ λ 4 < 0 . 4 } as shown in Table IV, we can easily see that in 4 dimensions, these classes of mass function leads to a naked singularity, as the trapped surfaces do not form early enough. However when we make the transition to higher dimensions, the final outcome is a black hole.</text>
<section_header_level_1><location><page_7><loc_23><loc_33><loc_35><loc_34></location>D. The result</section_header_level_1>
<text><location><page_7><loc_9><loc_27><loc_49><loc_31></location>As a result of our detailed analytical and numerical investigations of the previous subsections, we can state the following proposition:</text>
<text><location><page_7><loc_9><loc_18><loc_49><loc_25></location>Proposition 2. There exist classes of mass function in generalised Vaidya spacetimes, that produce a locally naked central singularity in 4 dimensions, but these naked singularity gets eliminated in higher dimensions due to temporal advancement of trapped surface formation.</text>
<section_header_level_1><location><page_7><loc_18><loc_14><loc_40><loc_15></location>VI. SUMMING IT ALL UP</section_header_level_1>
<text><location><page_7><loc_9><loc_8><loc_49><loc_12></location>In this paper we extended our analysis of the gravitational collapse of generalised Vaidya spacetime in 4dimensions [18], to spacetimes of arbitrary dimensions,</text>
<text><location><page_7><loc_52><loc_52><loc_92><loc_63></location>in the context of the Cosmic Censorship Conjecture. We found the sufficient conditions on the generalised Vaidya mass function, that generates a locally naked central singularity that can causally communicate with an external observer. We carefully investigated the effect of the number of dimensions on the dynamics of the trapped regions, by studying the slope of the apparent horizon curve at the central singularity.</text>
<text><location><page_7><loc_52><loc_34><loc_92><loc_52></location>By considering specific examples, we showed that there exist classes of mass functions for which a naked singularity in 4-dimensions gets covered as we make the transition to higher dimensional spacetime. Interestingly, the reason for this is same as in the case of dust collapse. From our analysis here, we can easily see that for a wide class of matter fields, a transition to higher dimensions favours trapped surface formation and the epoch of trapping advances as we go to higher dimensions. This makes the vicinity of the central singularity trapped even before the singularity is formed, and hence it is necessarily covered.</text>
<text><location><page_7><loc_52><loc_27><loc_92><loc_34></location>Therefore, we can safely conclude that for a large class of matter fields, which include both Type I and Type II matter, transition to higher dimensions does indeed restrict the set of physically realistic initial data, that leads to the formation of a locally naked singularity.</text>
<section_header_level_1><location><page_7><loc_59><loc_23><loc_84><loc_24></location>VII. ACKNOWLEDGEMENTS</section_header_level_1>
<text><location><page_7><loc_52><loc_8><loc_92><loc_21></location>We are indebted to the National Research Foundation and the University of KwaZulu-Natal for financial support. SDM acknowledges that this work is based upon research supported by the South African Research Chair Initiative of the Department of Science and Technology. MDM extends his appreciation to the University of Dodoma in Tanzania for a study leave. We also wish to thank the anonymous referee for his constructive comments on this paper.</text>
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</document> | [{"title": "Is cosmic censorship restored in higher dimensions?", "content": "M. D. Mkenyeleye, 1, \u2217 Rituparno Goswami, 1, \u2020 and Sunil D. Maharaj 1, \u2021 1 Astrophysics and Cosmology Research Unit, School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa. In this paper we extend the analysis of gravitational collapse of spherically symmetric generalised Vaidya spacetimes to higher dimensions, in the context of the Cosmic Censorship Conjecture. We present the sufficient conditions on the generalised Vaidya mass function, that will generate a locally naked singular end state. Our analysis here generalises all the earlier works on collapsing higher dimensional generalised Vaidya spacetimes. With specific examples, we show the existence of classes of mass functions that lead to a naked singularity in four dimensions, which gets covered on transition to higher dimensions. Hence for these classes of mass function Cosmic Censorship gets restored in higher dimensions and the transition to higher dimensions restricts the set of initial data that results in a naked singularity. PACS numbers: 04.20.Cv , 04.20.Dw", "pages": [1]}, {"title": "I. INTRODUCTION", "content": "The singularity theorems predict the occurrence of spacetime singularities for a wide class of theories of gravity under very generic conditions, namely the attractive nature of gravity, existence of closed trapped surfaces and no violations of causality in the spacetime [1]. However these theorems do not say anything about the causal nature of these singularities, that is, if it is possible for future directed null geodesics from the close vicinity of these singular points, to escape to infinity. To avoid such scenarios where a naked singularity exists that can causally influence the future infinities, the Cosmic Censorship Conjecture (CCC) was proposed by Penrose [2]. This states that spacetime singularities produced by the gravitational collapse of physically realistic matter fields are always covered by trapped surfaces. Hence the final state of continual gravitational collapse always leads to a black hole, where the singularity is shielded from any external observer. Though the general proof of this conjecture still remains elusive, there are a number of important counterexamples that show otherwise. Investigations of spherically symmetric dynamical collapse models in General Relativity for large classes of matter fields, in four dimensional spacetimes, indicate that there exist sets of initial data of non-zero measure, at the epoch of the commencement of the collapse, that lead to the formation of a locally naked singularity. In these cases the trapped surfaces do not form early enough to shield the singularity (or the spacetime fireball) from external observers. It is also shown in these studies that families of future outgoing non-spacelike geodesics emerge from such a naked singularity, providing a non-zero measure set of trajecto- ries escaping away [3-5]. Though these counter examples are mainly presented in case of spherical symmetry (with a few exceptions of non-spherical models), they suffice to be relevant because if the censorship is one of key aspect of gravitation theory, it should not depend on symmetries of spacetime.", "pages": [1]}, {"title": "A. The question", "content": "To avoid the unpleasantries of nudity, the obvious question that arises (influenced by higher dimensional and emergent theories of gravity - e.g string theory or braneworld models), is as follows: Question. Does the transition to higher dimensional spacetimes (with compact or non-compact extra dimensions) restricts the above mentioned set of initial data that leads to a naked singularity? In other words, how does the number of spacetime dimensions dictate the dynamics of trapped regions in the spacetime? This question is important as most of the proofs of the key theorems of black hole dynamics and thermodynamics demand the spacetimes to be future asymptotically simple, which is not possible if the censorship is violated [1]. If the locally naked singularities in 4-dimensional spacetime are naturally absent in higher dimensions, then that will be an argument in favour of higher dimensional (or emergent theories) of gravity, as in those cases the important results of black hole dynamics and thermodynamics would be more relevant.", "pages": [1]}, {"title": "B. Earlier works", "content": "important result [6, 7]: The naked singularities occurring in dust collapse from smooth initial data (which include those discovered by Eardley and Smarr [8], Christodoulou [9], and Newman [10]) are eliminated when we make transition to higher dimensional spacetimes. The cosmic censorship is then restored for dust collapse which will always produce a black hole as the collapse end state for dimensions D \u2265 6, under conditions such as the smoothness of initial data from which the collapse develops, which follows from physical grounds. The physical reason behind the above result is that higher dimensional spacetimes favour trapped surface formation and the formation of horizons advance in time. Hence for dimensions greater than five, the vicinity of the singularity always gets trapped even before the singularity is formed, and hence the singularity is causally cut-off from any external observer. Several other works on higher dimensional radiation collapse and perfect fluid collapse has been done [1117], where the matter field is taken to be of a specific form (for example: perfect fluids with linear equation of state, pure radiation, charged radiation etc.). All of these studies give an indication that higher dimensions do favour trapping and hence the epoch of trapped surface formation advances as we go to higher dimensions.", "pages": [1, 2]}, {"title": "C. The present paper", "content": "The main criticism of the dustlike models or pure perfect fluid models is that they are far too idealised. For any realistic massive astrophysical body, which is undergoing gravitational collapse, the pressure and the radiative processes must play an important role together. One of the known spacetimes that can closely mimic such a collapse scenario is the generalised Vaidya spacetime, where the matter field is a specific combination of Type I matter (whose energy momentum tensor has one timelike and three spacelike eigenvectors) that moves along timelike trajectories and Type II matter (whose energy momentum tensor has double null eigenvectors) that moves along null trajectories. Thus, a collapsing generalised Vaidya spacetime depicts the collapse of usual perfect fluid combined with radiation. Therefore the collapse scenario here is much closer to what is expected for the collapse of a realistic astrophysical star. In our earlier paper [18], we investigated the gravitational collapse of generalised Vaidya spacetimes in four dimensions and developed a general mathematical framework to study the conditions on the mass function such that future directed non-spacelike geodesics can terminate at the singularity in the past. In this paper: Unless otherwise specified, we use natural units ( c = 8 \u03c0G = 1) throughout this paper, Latin indices run from 0 to N -1. The symbol \u2207 represents the usual covariant derivative and \u2202 corresponds to partial differentiation. We use the ( -, + , + , + , + , \u00b7 \u00b7 \u00b7 ) signature and the Ricci tensor is obtained by contracting the first and the third indices of the Riemann tensor The Hilbert-Einstein action in the presence of matter is given by variation of which gives the Einstein field equations as", "pages": [2]}, {"title": "II. HIGHER DIMENSIONAL GENERALISED VAIDYA SPACETIME", "content": "The spherically symmetric line element for an N -dimensional generalised Vaidya spacetime is given as where is the metric on the ( N -2) sphere in polar coordinates with \u03b8 i being spherical coordinates. m ( v, r ) is the generalized mass function related to the gravitational energy within a given radius r [19], which can be carefully defined so that the energy conditions are satisfied. The coordinate v represents the Eddington advanced time where r is decreasing towards the future along a ray v = Const. (ingoing). When N = 4, the line element reduces to the generalized Vaidya solution [20] in 4-dimensions. Defining the following quantities we can write the non-vanishing components of the Ricci tensor as The Ricci scalar is given by while the non-vanishing components of the Einstein tensor are given by The Energy Momentum Tensor (EMT) can be written in the form [21] where In the above, with l \u00b5 and n \u00b5 being two null vectors, where l \u00b5 l \u00b5 = n \u00b5 n \u00b5 = 0 and l \u00b5 n \u00b5 = -1. Eq. (9) is taken as a generalized Energy-Momentum Tensor for the generalized Vaidya spacetime, with the component T ( n ) \u00b5\u03bd being considered as the matter field that moves along the null hypersurfaces v = constant, while T ( m ) \u00b5\u03bd describes the matter moving along timelike trajectories. If the EMT of Eq. (9) is projected to the orthonormal basis, defined by the vectors, it can be found [20], that the symmetric EMT can be given as the N \u00d7 N matrix, For this fluid the energy conditions are given as [1], 1. The Weak and Strong energy conditions : /negationslash 2. The Dominant energy condition : /negationslash These energy conditions can be satisfied by suitable choices of the mass function m ( v, r ).", "pages": [2, 3]}, {"title": "III. HIGHER DIMENSIONAL COLLAPSE MODEL", "content": "In this section, we examine the gravitational collapse of a collapsing matter field in the generalized Vaidya spacetime when a spherically symmetric configuration of Type I and Type II matter collapses at the centre of symmetry in an otherwise empty universe which is asymptotically flat far away [22]. If K \u00b5 is the tangent to non-spacelike geodesics with K \u00b5 = dx \u00b5 dk , where k is the affine parameter, then K \u00b5 ; \u03bd K \u03bd = 0 and where \u03b2 is a constant that characterizes different classes of geodesics with \u03b2 = 0 for null geodesic vectors, \u03b2 < 0 for timelike geodesics and \u03b2 > 0 for spacelike geodesics [22]. Here we consider the case of null geodesics, that is, \u03b2 = 0. We calculate the equations dK v /dk and dK r /dk using the Lagrange equations given by L = 1 2 g \u00b5\u03bd dx \u00b5 dk dx \u03bd dk and Euler-Lagrange equations In the case of the higher dimensional generalised Vaidya spacetime, these equations are given by All other components are considered to be 0. If we follow [23] and write K v as then using K \u00b5 K \u03bd = 0 we get", "pages": [3, 4]}, {"title": "IV. CONDITIONS FOR LOCALLY NAKED SINGULARITY", "content": "The nature (a locally naked singularity or a black hole) of the collapsing solutions can be characterized by the existence of radial null geodesics coming out of the singularity [15, 22]. The radial null geodesics of the line element (4) can be calculated using Eqs. (21a) and (21b). These geodesics are given by the equation This differential equation has a singularity at r = 0 , v = 0. Using the same techniques utilised in [18, 24, 25], Eq. (22) can be re-written near the singular point as where", "pages": [4]}, {"title": "A. Existence of outgoing nonspacelike geodesics", "content": "We can clearly see that Eq. (23) has a singularity at v = 0 , r = 0. The classification of the tangents of both radial and non-radial outgoing non-spacelike geodesics terminating at the singularity in the past can be given by the limiting values at v = 0 , r = 0. The conditions for the existence for such geodesics have been described in detail in [18] using the concept of contraction mappings. The existence of these radial null geodesics also characterizes the nature (a naked singularity or a black hole) of the collapsing solutions. If we let X to be the limiting value at r = 0 , v = 0, we can determine the nature of this limiting value on a singular geodesic as Using a suitably chosen mass function, Eq. (23) and l'Hopital's rule, we can explicitly find the expression for the tangent values X 0 which governs the behaviour of the null geodesics near the singular point. Thus, the nature of the singularity can then be determined by studying the solution of this algebraic equation. This expression can be calculated as", "pages": [4]}, {"title": "B. Apparent Horizon", "content": "The existence of the apparent horizon, which is the boundary of the trapped surface region in the spacetime also determines the nature of the singularity. If at least one value of the limiting positive values X 0 is less than the slope of the apparent horizon at the central singularity, then the central singularity is locally naked with the outgoing radial null geodesics escaping from the past to the future. For the generalized higher dimensional Vaidya spacetime, the apparent horizon is defined by The slope of the apparent horizon can be calculated as follows: Thus, the slope of the apparent horizon at the central singularity is given by", "pages": [4]}, {"title": "C. Sufficient conditions", "content": "We can now write the sufficient conditions for the existence of a locally naked central singularity for a collapsing generalised Vaidya spacetime in arbitrary dimensions N , which we state in the following proposition: Proposition 1. Consider a collapsing N-dimensional generalised Vaidya spacetime from a regular epoch, with a mass function m ( v, r ) , that obeys all physically reasonable energy conditions and is differentiable in the entire spacetime. If the following conditions are satisfied : then the central singularity is locally naked with outgoing C 1 radial null geodesics escaping to the future. We emphasise here, that all the previous works of higher dimensional generalised Vaidya collapse [15-17] are special cases of the general analysis presented above. In the next section, we give a specific example to trans- parently demonstrate the effect of transition to higher dimensions on the nature of the central singularity.", "pages": [4, 5]}, {"title": "V. A GENERAL LAURENT EXPANDABLE MASS FUNCTION", "content": "We consider here a Laurent expandable mass function of the generalized Vaidya spacetime in higher dimensions in the general form as where m n ( v ) = v (2 N + n -8) , n = 1 , 2 , \u00b7 \u00b7 \u00b7 and \u03bb n 's are constants . Using Eq. (26) and (29), we get the expression of the tangent to null geodesics X 0 and tangent to the apparent horizon X AH in higher dimensions as These expressions can be written in the general form as and where f n ( N,\u03bb i ) and g n ( N,\u03bb i ) are some functions of N and the \u03bb i 's. These expressions can explicitly be solved for X 0 and X AH using some specific values of n , N and \u03bb i 's (see Table I) and then one can make conclusions about the nature of the singularity by using the following conditions: (iv) If the smallest solution Min[ X AH ] is greater than any one of the positive solutions of X 0 , then there will be future directed null geodesics from the singularity and hence the singularity is locally naked. We can easily see from Table I, that the general expression obtained here contains the expressions for X 0 and X AH corresponding to Vaidya collapse in 4-D ( n = 1 , N = 4) [22, 23], charged Vaidya-de Sitter in 4-D ( n = 2 , N = 4) [17] and charged Vaidya in 5-D ( n = 2 , N = 5) [16].", "pages": [5, 6]}, {"title": "A. Example: Class of naked singularity in 4D being eliminated in higher dimensions", "content": "In this section we will consider a specific example, that can be easily generalised to an open set, to show explicitly how a naked singularity in four dimensions gets covered in higher dimensions. Let us consider a scenario where n = 4. In this case the expression for X 0 and X AH become and We can solve these equations numerically to get the values of X 0 and X AH in different dimensions. For our calculations we took \u03bb 1 = 5 . 0, \u03bb 2 = 0 . 01, \u03bb 3 = 2 . 3, \u03bb 4 = 0 . 05. From Table II we can easily see that in 4 dimensions, this class of mass function leads to a naked singularity, as the trapped surfaces do not form early enough to shield the singularity from outside observers. However when we make the transition to higher dimensions we see that the value of the tangent to the outgoing null geodesic from the central singularity is greater than the slope of the apparent horizon curve at the central singularity. In this case the outgoing null direction is within the trapped region and hence the singularity is causally cut off from the external observer. By the continuity of the mass function considered above, this can be easily converted to a open set in the mass function space, where this scenario continues to be true and we shall explicitly prove this in the following subsection.", "pages": [6]}, {"title": "B. Proof of existence of open set of mass functions with the above properties", "content": "Having found out a specific example of a mass function for which the naked singularities in 4D is eliminated when we go to higher dimension, we would now require to prove that such a mass function is generic in the sense that there exists a open set of such mass functions in the function space. Since this problem of deducing the nature of the central singularity is reduced to finding and comparing real roots of polynomials (35) and (36), all we need to show here is the real roots of these polynomials are continuous functions of the coefficients. To do this, first of all we observe that the roots that are given in the Table II are all of multiplicity one. This can be easily seen by differentiating the LHS of (35) and (36) and substituting the roots to find non-zero values. Now, for any complex polynomial p ( z ) of degree n \u2265 1 with m distinct roots { \u03b1 1 , \u00b7 \u00b7 \u00b7 , \u03b1 m } , (1 \u2264 m \u2264 n ), let us define the quantity R 0 ( p ) as follows: We now state the well known result of complex analysis [26]: Theorem 1. Let p ( z ) be a polynomial of degree n \u2265 1 , with real coefficients { \u00b5 k } . Suppose \u03b1 be a real root of p ( z ) of multiplicity one. Then for any /epsilon1 with 0 \u2264 /epsilon1 \u2264 R 0 ( p ) , there exists a \u03b4 ( /epsilon1 ) > 0 such that any polynomial q ( z ) with real coefficients \u03bd k and | \u00b5 k -\u03bd k | \u2264 \u03b4 , has a real root \u03b2 with | \u03b1 -\u03b2 | \u2264 /epsilon1 . The above theorem shows that if a polynomial p ( z ) with real coefficients has a real root \u03b1 of multiplicity one, then any polynomial q ( z ) obtained by small (real) perturbations to the coefficients of p ( z ) will also have a real root in a neighbourhood of \u03b1 . That is, not only the root depends continuously on coefficients, but also it remains real, under sufficiently small perturbations of coefficients. This results directly translates to our problem of open set of mass functions in the mass function space. Once we have a specific example as shown in Table II, any perturbations around that will have the same outcome as far as the nature of the singularities are concerned. Hence this class of mass functions is not fine tuned, but quite generic and the outcome is stable under perturbations.", "pages": [6]}, {"title": "C. Numerical verification", "content": "We would now like to verify explicitly, with the aid of numerical calculations, the results in the previous subsection. We numerically solve equations (35) and (36) to get the values of X 0 and X AH in different dimensions to show that there exists a set of parameter intervals for which the mass function leads to a naked singularity in 4 dimensions and a black hole in higher dimensions. For example, some of the intervals are { 4 . 8 < \u03bb 1 < 5 . 25 , 0 . 009 < \u03bb 2 < 0 . 012 , 2 . 25 < \u03bb 3 < 2 . 38 , \u03bb 4 = 0 . 05 } with the range values shown in Table III and { \u03bb 1 = 5 . 2 , 0 . 009 < \u03bb 2 < 0 . 012 , \u03bb 3 = 2 . 3 , 0 \u2264 \u03bb 4 < 0 . 4 } as shown in Table IV, we can easily see that in 4 dimensions, these classes of mass function leads to a naked singularity, as the trapped surfaces do not form early enough. However when we make the transition to higher dimensions, the final outcome is a black hole.", "pages": [7]}, {"title": "D. The result", "content": "As a result of our detailed analytical and numerical investigations of the previous subsections, we can state the following proposition: Proposition 2. There exist classes of mass function in generalised Vaidya spacetimes, that produce a locally naked central singularity in 4 dimensions, but these naked singularity gets eliminated in higher dimensions due to temporal advancement of trapped surface formation.", "pages": [7]}, {"title": "VI. SUMMING IT ALL UP", "content": "In this paper we extended our analysis of the gravitational collapse of generalised Vaidya spacetime in 4dimensions [18], to spacetimes of arbitrary dimensions, in the context of the Cosmic Censorship Conjecture. We found the sufficient conditions on the generalised Vaidya mass function, that generates a locally naked central singularity that can causally communicate with an external observer. We carefully investigated the effect of the number of dimensions on the dynamics of the trapped regions, by studying the slope of the apparent horizon curve at the central singularity. By considering specific examples, we showed that there exist classes of mass functions for which a naked singularity in 4-dimensions gets covered as we make the transition to higher dimensional spacetime. Interestingly, the reason for this is same as in the case of dust collapse. From our analysis here, we can easily see that for a wide class of matter fields, a transition to higher dimensions favours trapped surface formation and the epoch of trapping advances as we go to higher dimensions. This makes the vicinity of the central singularity trapped even before the singularity is formed, and hence it is necessarily covered. Therefore, we can safely conclude that for a large class of matter fields, which include both Type I and Type II matter, transition to higher dimensions does indeed restrict the set of physically realistic initial data, that leads to the formation of a locally naked singularity.", "pages": [7]}, {"title": "VII. ACKNOWLEDGEMENTS", "content": "We are indebted to the National Research Foundation and the University of KwaZulu-Natal for financial support. SDM acknowledges that this work is based upon research supported by the South African Research Chair Initiative of the Department of Science and Technology. MDM extends his appreciation to the University of Dodoma in Tanzania for a study leave. We also wish to thank the anonymous referee for his constructive comments on this paper.", "pages": [7]}] |
2021arXiv211102656G | https://arxiv.org/pdf/2111.02656.pdf | <document>
<section_header_level_1><location><page_1><loc_18><loc_77><loc_82><loc_85></location>An Enlargeability Obstruction for Spacetimes with both Big Bang and Big Crunch</section_header_level_1>
<text><location><page_1><loc_41><loc_73><loc_59><loc_74></location>Jonathan Glöckle</text>
<text><location><page_1><loc_41><loc_69><loc_59><loc_71></location>November 5, 2021</text>
<text><location><page_1><loc_43><loc_48><loc_43><loc_51></location>/negationslash</text>
<text><location><page_1><loc_20><loc_43><loc_80><loc_65></location>Given a spacelike hypersurface M of a time-oriented Lorentzian manifold ( M,g ), the pair ( g, k ) consisting of the induced Riemannian metric g and the second fundamental form k is known as initial data set. In this article, we study the space of all initial data sets ( g, k ) on a fixed closed manifold M that are subject to a strict version of the dominant energy condition. Whereas the pairs of the form ( g, τg ) and ( g, -τg ), for a sufficiently large τ > 0, belong to the same path-component of this space when M admits a positive scalar curvature metric, it was observed in a previous work [11] that this is not the case when the existence of a positive scalar curvature metric on M is obstructed by α ( M ) = 0. In the present article we extend this nonconnectedness result to Gromov-Lawson's enlargeability obstruction, which covers many examples, also in dimension 3. In the context of relativity theory, this result may be interpreted as excluding the existence of certain globally hyperbolic spacetimes with both a big bang and a big crunch singularity.</text>
<section_header_level_1><location><page_1><loc_15><loc_37><loc_32><loc_39></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_15><loc_19><loc_85><loc_34></location>The interplay of curvature restrictions and the topology of a manifold has always been a central topic in differential geometry. Curvature restrictions also have important applications in general relativity. Namely, the famous singularity theorems of Hawking and Penrose (cf. [17, Sec. 8.2]) rely on the fact that all matter is assumed to satisfy certain energy conditions, which, by the Einstein field equations, correspond to curvature conditions for the Lorentzian space-time manifold. For example, the dominant energy condition - derived from the physical assumption that every observer should experience non-negative mass density - translates into a 'non-negativity' condition for the Einstein curvature Ein = ric -1 2 scal g .</text>
<text><location><page_2><loc_15><loc_79><loc_85><loc_90></location>A usual strategy for studying solutions of the Einstein equations is to look at time slices: Let M be a spacelike hypersurface of a time-oriented Lorentzian manifold ( M,g ). The Lorentzian metric g induces on M a Riemannian metric g and a symmetric 2tensor k ∈ Γ( T ∗ M ⊗ T ∗ M ), the second fundamental form of M with respect to the future-directed unit normal e 0 . The Gauß-Codazzi equations imply that the components ρ = Ein( e 0 , e 0 ) and j = Ein( e 0 , -) ∈ Ω 1 ( M ) of the Einstein curvature may be computed via the so-called constraint equations</text>
<formula><location><page_2><loc_40><loc_74><loc_85><loc_78></location>2 ρ = scal g +(tr k ) 2 -‖ k ‖ 2 j = div k -dtr k. (1)</formula>
<text><location><page_2><loc_15><loc_58><loc_85><loc_74></location>By the Einstein equations, ρ and j are determined by the distribution of matter: ρ is the energy density and j gets the interpretation of a momentum density . If g satisfies the dominant energy condition, then ρ ≥ ‖ j ‖ everywhere on M . At least a partial converse of this is also true: Suppose that a pair ( g, k ) of metric and symmetric 2-tensor on M is such that ρ and j - determined by (1) - satisfy the strict inequality ρ > ‖ j ‖ on all of M . Then M embeds as a spacelike hypersurface into some time-oriented Lorentzian manifold ( M,g ) subject to the dominant energy condition such that g is the induced metric and k is the induced second fundamental form. Therefore, we say that an initial data set ( g, k ) satisfies the dominant energy condition (dec) if ρ ≥ ‖ j ‖ . This condition plays a major role in the positive mass theorem.</text>
<text><location><page_2><loc_15><loc_45><loc_85><loc_56></location>If M ⊆ M is totally geodesic, i. e. if k ≡ 0, then the dominant energy condition reduces to the condition that g has non-negative scalar curvature. Non-negative and even more so positive scalar curvature (psc) has been a vast field of study over the last decades, see e. g. [22]. Apart from minimal hypersurface techniques, most of these results were obtained by Dirac operator methods. For example, it was shown in [5] and [7] that the C ∞ -space R > ( M ) of psc metrics on a fixed closed spin manifold M of dimension n ≥ 6 has infinitely many non-trivial homotopy groups if it is non-empty.</text>
<text><location><page_2><loc_15><loc_29><loc_85><loc_43></location>In [11], the author was able to transfer some of these psc results to the dominant energy setting. Namely, the non-trivial elements in π k ( R > ( M )) mentioned above give rise to non-trivial elements in π k +1 ( I > ( M )), where I > ( M ) denotes the space of initial data sets ( g, k ) on M subject to the strict version ρ > ‖ j ‖ of the dominant energy condition. The proof relies on a suspension construction - associating to a psc metric a path of initial data sets strictly satisfying dec - and the computation of a family index of (Cliffordlinear) Dirac-Witten operators . These operators are a certain zero-order perturbation of Dirac operators. Their classical version dates back to Witten's spinorial proof of the positive mass theorem [24] (cf. [21] for a more detailed account of the proof).</text>
<text><location><page_2><loc_15><loc_18><loc_85><loc_27></location>In a some way, the comparison arguments between π k ( R > ( M )) and π k +1 ( I > ( M )) still apply in the case k = -1. Here, statement is that I > ( M ) is not path-connected if M is a closed spin manifold with non-zero α -index α ( M ). More precisely, the path-component C + that contains all data sets ( g, τg ) ∈ I > ( M ) for sufficiently large τ > 0 and the path-component C -that contains ( g, -τg ) for large τ > 0 are distinct in this case.</text>
<text><location><page_3><loc_28><loc_79><loc_28><loc_82></location>/negationslash</text>
<text><location><page_3><loc_15><loc_69><loc_85><loc_90></location>This has a direct application to general relativity. Near a big bang singularity the initial data sets induced on spacelike hypersurfaces are expected to lie in C + , whereas those near a big crunch singularity would belong to C -. Recall that a globally hyperbolic spacetime ( M,g ) admits a foliation M ∼ = M × /CA into spacelike (Cauchy) hypersurfaces M × { t } by a famous theorem by Bernal and Sánchez [4]. Taking this together, we see that α ( M ) = 0 is not only an obstruction to psc but also an obstruction to the existence of a globally hyperbolic spacetime ( M ∼ = M × /CA , g ) strictly satisfying dec and having both a big bang and a big crunch singularity. The slight drawback here - that we require ( M,g ) to strictly satisfy dec meaning that on every spacelike hypersurface ρ > ‖ j ‖ holds - is overcome in the recent article [1] of Ammann and the author, where sufficient conditions for the removal of the strictness assumption are deduced. This passage from strict dec to dec is similar to the passage from positive scalar curvature to non-negative scalar curvature performed by Schick and Wraith [23].</text>
<text><location><page_3><loc_15><loc_35><loc_85><loc_67></location>Unfortunately, precisely in the physically relevant spatial dimension n = 3, we always have α ( M ) = 0 as it is an element of KO -3 ( {∗} ) = 0. It is the purpose of the present article to extend the result of [11] to another (index theoretic) obstruction to positive scalar curvature that is also of use in the 3-dimensional case: enlargeability. The concept of enlargeability was first introduced by Gromov and Lawson [13]. There exist various versions, the most basic ones being the following. A closed Riemannian n -manifold ( M,g ) is called enlargeable if for every R > 0 there exists a Riemannian covering of ( M,g ) that is 'large' in the following sense: There exists a distance-non-increasing map to a standard sphere S n of radius R that is of non-zero degree. Despite being geometrically defined, the notion turns out to be homotopy invariant (in particular, independent of g ), and thus provides a link between geometry and topology. Moreover, the class of enlargeable manifolds is rich. It contains all manifolds that carry a metric of non-positive sectional curvature, especially tori, and further examples may be constructed through products and direct sums. In the aforementioned article [13] Gromov and Lawson show that spin manifolds do not carry a psc metric when they are compactly enlargeable , i. e. the coverings in the definition of enlargeability may be chosen to be finite-sheeted. Later they extended this both to (not necessarily compactly) enlargeable spin manifolds and, in dimensions n ≤ 7, to compactly enlargeable (not necessarily spin) manifolds (cf. [14]). In the present article we prove an initial data version of Gromov and Lawson's result:</text>
<text><location><page_3><loc_15><loc_29><loc_85><loc_32></location>Main Theorem (Theorem 5.3) . If M is an enlargeable spin manifold, then I > ( M ) is not path-connected. More precisely, the path-components C + and C -are distinct.</text>
<text><location><page_3><loc_15><loc_19><loc_85><loc_27></location>In fact, we prove this for the slighly more general notion of ˆ A -area-enlargeability, cf. Definition 5.1. Similar to what has been stated above, this has the following consequence to general relativity: If M is an enlargeable spin manifold and ( M ∼ = M × /CA , g ) a globally hyperbolic spacetime that strictly satisfies dec in the sense that ρ > ‖ j ‖ on every</text>
<text><location><page_4><loc_15><loc_82><loc_85><loc_90></location>spacelike hypersurface, then ( M,g ) cannot have both a big bang and a big crunch singularity. In particular, this applies when M is a quotient of a (3-dimesional) torus, which is still considered to be within the range of physical observations of our universe, cf. [10]. Probably, the strictness condition can be removed under certain additional assumptions with techniques as in [1] and [23], but this is beyond the scope of this article.</text>
<text><location><page_4><loc_34><loc_71><loc_34><loc_74></location>/negationslash</text>
<text><location><page_4><loc_15><loc_59><loc_85><loc_80></location>Let us briefly discuss the idea of the proof in the case where M is even-dimensional and compactly enlargeable. From the (compact) enlargeability Gromov and Lawson construct a sequence of (finite-sheeted) coverings M i → M and complex vector bundles E i → M i with hermitian metric and metric connection such that the curvatures R E i -→ 0 for i -→ ∞ and ˆ A ( M i , E i ) = 0 for all i ∈ /C6 . Thereby ˆ A ( M i , E i ) = ∫ M i ˆ A ( TM i ) ∧ ch( E i ) is the E i -twisted ˆ A -genus, which is equal to the index of the E i -twisted Dirac operator D E i by the Atiyah-Singer index theorem. Note that the existence of such a sequence of coverings and bundles is also the starting point of Hanke and Schick's proof that enlargeability implies non-triviality of the Rosenberg index α /CA max ( M ) ∈ KO -n ( C ∗ max , /CA π 1 ( M )) [15, 16]. In our initial data setting, we need a suitable analog of D E i . This is provided by the twisted Dirac-Witten operator D E i = D E i -1 2 tr( k ) e 0 · defined on a certain E i -twisted spinor bundle. This operator is associated to an initial data set ( g, k ) on M i (in our case it is pulled back from M ) and satisfies the Schrödinger-Lichnerowicz type formula</text>
<formula><location><page_4><loc_32><loc_54><loc_68><loc_58></location>( D E i ) 2 ψ = ∇ ∗ ∇ ψ + 1 2 ( ρ -e 0 · j /sharp · ) ψ + R E i ψ.</formula>
<text><location><page_4><loc_64><loc_43><loc_64><loc_45></location>/negationslash</text>
<text><location><page_4><loc_15><loc_41><loc_85><loc_53></location>Hence, if the strict dominant energy condition ρ > ‖ j ‖ holds, then D E i is invertible for large enough i ∈ /C6 . In particular, given a metric g and a sufficiently large τ > 0 the twisted index difference ind-diff E i (( g, -τg ) , ( g, τg )), a spectral-flow-like invariant associated to a path γ of initial data sets from ( g, -τg ) to ( g, τg ), vanishes for large i ∈ /C6 if the path γ may be chosen to lie within I > ( M ). On the other hand, an index theorem shows that ind-diff E i (( g, -τg ) , ( g, τg )) = ˆ A ( M i , E i ) = 0 for all i ∈ /C6 . Thus it is not possible to connect ( g, -τg ) and ( g, τg ) by a path in I > ( M ).</text>
<text><location><page_4><loc_15><loc_26><loc_85><loc_40></location>The article is structured as follows: We start off by describing how Dirac-Witten operators arise in a Lorentzian setup. In the second section, we construct the twisted index difference of Dirac-Witten operators using a framework laid out by Ebert [9]. This is a bit technical, as we also deal with non-compact manifolds so that we can later apply it to infinite covers M i → M as well. In fact, we construct a relative twisted index difference , relative meaning that it depends on the 'difference' of two twist bundles E (0) i and E (1) i that coincide outside a compact set. The third section is devoted to the proof of the (relative) index theorem ind-diff E i (( g, -τg ) , ( g, τg )) = ˆ A ( M i , E i ). In the last section we put the arguments together to prove the main theorem.</text>
<section_header_level_1><location><page_5><loc_15><loc_88><loc_32><loc_90></location>Acknowledgements</section_header_level_1>
<text><location><page_5><loc_15><loc_81><loc_85><loc_86></location>I would like express my gratitude to Bernd Ammann, who came up with the idea for this project, for his ongoing support and encouragement. During its execution, I was supported by the SFB 1085 'Higher Invariants' funded by the DFG.</text>
<section_header_level_1><location><page_5><loc_15><loc_73><loc_80><loc_77></location>2. Lorentzian manifolds, initial data sets and Dirac-Witten operators</section_header_level_1>
<text><location><page_5><loc_15><loc_62><loc_85><loc_70></location>Within this section, we want to recall several notions from Lorentzian geometry and thereby fix certain notations. Although the rest of the article could be understood to a very large extent without knowledge of the Lorentzian setup, the objects that we will be considering appear more naturally in this context, thus providing a better understanding.</text>
<text><location><page_5><loc_15><loc_49><loc_85><loc_60></location>Let us consider a Lorentzian manifold ( M,g ) of dimension n +1. We use the signature convention such that a generalized orthonormal basis e 0 , e 1 , . . . , e n satisfies g ( e 0 , e 0 ) = -1 and g ( e i , e i ) = 1 for 1 ≤ i ≤ n . In particular, the induced metric g on a spacelike hypersurface M ⊆ M will be Riemannian. In this context, we will denote by e 0 a (generalized) unit normal on M . Usually, we will assume that M is time-oriented and then we agree that e 0 is future-pointing. Apart from the induced metric M will carry an induced second fundamental form k with respect to e 0 that we define by</text>
<formula><location><page_5><loc_39><loc_46><loc_61><loc_48></location>∇ X Y -∇ X Y = k ( X,Y ) e 0</formula>
<text><location><page_5><loc_15><loc_38><loc_85><loc_45></location>for all vectors fields X,Y ∈ Γ( TM ). Thereby, ∇ denotes the Levi-Civita connection of ( M,g ) and ∇ is the one of the hypersurface ( M,g ). Pairs ( g, k ) consisting of a Riemannian metric g and a symmetric 2-tensor k will be called initial data set . Hence the above procedure provides an induced initial data set ( g, k ) on a spacelike hypersurface of a time-oriented Lorentzian manifold.</text>
<text><location><page_5><loc_15><loc_25><loc_85><loc_36></location>The dominant energy condition is the condition that for all future-causal vectors V, W of ( M,g ), the Einstein tensor Ein = ric -1 2 scal g satisfies Ein( V, W ) ≥ 0. Equivalently, for every future-causal vector V the metric dual of Ein( V, -) is required to be past-causal. Applying this to the future-pointing unit normal e 0 of a spacelike hypersurface M , we get ρ ≥ ‖ j ‖ for ρ = Ein( e 0 , e 0 ) and j = Ein( e 0 , -) ∈ Ω 1 ( M ). As explained in the introduction, ρ and j are completely determined by the induced initial data set ( g, k ) on M via (1).</text>
<text><location><page_5><loc_15><loc_18><loc_85><loc_22></location>Definition 2.1. An initial data set ( g, k ) on a manifold M is said to satisfy the dominant energy condition if ρ ≥ ‖ j ‖ for ρ and j defined by (1). It satisfies the strict dominant</text>
<text><location><page_6><loc_15><loc_84><loc_85><loc_90></location>energy condition if ρ > ‖ j ‖ . When M is compact, we denote by I ( M ) the space of all initial data sets ( g, k ) on M , equipped with the C ∞ -topology of uniform convergence, and by I > ( M ) the subspace of those initial data sets strictly satisfying the dominant energy condition.</text>
<text><location><page_6><loc_15><loc_75><loc_85><loc_81></location>It should be noted that when M is compact of dimension n ≥ 2 and g is any metric on M , then ( g, τg ) and ( g, -τg ) satisfy the strict dominant energy condition once the function τ > 0 is large enough. In fact, in this case ρ > ‖ j ‖ reduces to</text>
<formula><location><page_6><loc_35><loc_72><loc_85><loc_75></location>n ( n -1) 2 τ 2 > -1 2 scal g +( n -1) ‖ d τ ‖ , (2)</formula>
<text><location><page_6><loc_50><loc_60><loc_50><loc_62></location>↦</text>
<text><location><page_6><loc_15><loc_58><loc_85><loc_71></location>which can always be achieved by adding a constant to τ that continuously depends on g (in C ∞ -topology). As, moreover, for fixed g the convex combination between some τ > 0 and the constant max( τ ) keeps satisfying (2), all pairs of the form ( g, τg ) for τ > 0 belong to the same path-component C + of I > ( M ), likewise ( g, -τg ) all belong to the same path-component C -. When there exists a positive scalar curvature metric g on M , then the path [ -1 , 1] →I > ( M ) , t → ( g, tg ) shows that C + = C -. For future reference, we also note that the question of whether C + = C -satisfies a certain stability property:</text>
<text><location><page_6><loc_15><loc_50><loc_85><loc_55></location>Lemma 2.2. Let M be a compact manifold such that the path-components C + and C -of I > ( M ) coincide. Then for any compact manifold N the path-components C + and C -of I > ( M × N ) agree as well.</text>
<text><location><page_6><loc_15><loc_44><loc_85><loc_48></location>Proof. Let us first consider pairs of product form, i. e. ( g M + g N , k M + k N ) ∈ I ( M × N ) for ( g M , k M ) ∈ I ( M ) and ( g N , k N ) ∈ I ( N ). For these we compute</text>
<formula><location><page_6><loc_23><loc_39><loc_77><loc_43></location>2 ρ = scal g M +scal g N +(tr g M ( k M ) + tr g N ( k N )) 2 -‖ k M ‖ 2 g M -‖ k N ‖ 2 g N = 2 ρ M +2 ρ N +2tr g M ( k M ) tr g N ( k N )</formula>
<text><location><page_6><loc_15><loc_37><loc_18><loc_38></location>and</text>
<formula><location><page_6><loc_24><loc_32><loc_70><loc_35></location>j = div g M ( k M ) + div g N ( k N ) -d (tr g M ( k M )) -d (tr g N ( k N )) = j M + j N .</formula>
<text><location><page_6><loc_15><loc_29><loc_29><loc_30></location>Hence, we obtain</text>
<formula><location><page_6><loc_26><loc_25><loc_65><loc_28></location>‖ j ‖ 2 = ‖ j M ‖ 2 g M + ‖ j N ‖ 2 g N ≤ ( ‖ j M ‖ g M + ‖ j N ‖ g N ) 2</formula>
<text><location><page_6><loc_15><loc_23><loc_22><loc_24></location>and thus</text>
<formula><location><page_6><loc_24><loc_19><loc_85><loc_22></location>ρ -‖ j ‖ ≥ ( ρ M -‖ j M ‖ g M ) + ( ρ N -‖ j N ‖ g N ) + tr g M ( k M ) tr g N ( k N ) . (3)</formula>
<text><location><page_7><loc_35><loc_86><loc_35><loc_88></location>↦</text>
<text><location><page_7><loc_15><loc_73><loc_85><loc_79></location>˜ The required path from ( g M + g N , -τ ( g M + g N )) to ( g M + g N , τ ( g M + g N )) can be easily pieced together from three segments:</text>
<text><location><page_7><loc_15><loc_75><loc_85><loc_90></location>Now, by assumption, for sufficiently large τ > 0 the pairs ( g M , -τg M ) and ( g M , τ g M ) can be connected by a path t → γ M ( t ) = ( g M ( t ) , k M ( t )) in I > ( M ). By the discussion following Definition 2.1, we may assume that τ is a constant. As the interval [ -1 , 1] is compact, ρ M - ‖ j M ‖ g M attains a positive minimum. Replacing γ M by ˜ γ M = ( C -2 g M , C -1 k M ) for some suitably chosen C > 0, we may assume that this minimum is larger than -1 2 min p ∈ N scal g N ( p ). This is due to the fact, that in this rescaling ˜ ρ M = C 2 ρ M and ∥ ∥ ∥ ˜ j M ∥ ∥ ∥ g M = C 2 ‖ j M ‖ g M , indicating the rescaled quantities by ˜ · .</text>
<text><location><page_7><loc_32><loc_65><loc_32><loc_68></location>↦</text>
<text><location><page_7><loc_15><loc_59><loc_85><loc_63></location>In the first section, both tr g M ( k M ) and tr g N ( k N ) are non-positive, so its product is non-negative. Furthermore, for the pair ( g N , (2 t +1) g N ) with t ∈ [ -1 , -1 2 ] we have</text>
<formula><location><page_7><loc_26><loc_62><loc_74><loc_72></location>[ -1 , 1] -→ I > ( M × N ) t -→ ( g M + g N , -τg M +(2 t +1) τg N ) t ∈ [ -1 , -1 2 ] ( g M (2 t ) + g N , k M (2 t )) t ∈ [ -1 2 , 1 2 ] ( g M + g N , τ g M +(2 t -1) τg N ) t ∈ [ 1 2 , 1] .</formula>
<formula><location><page_7><loc_18><loc_56><loc_82><loc_59></location>ρ N -‖ j N ‖ g N = 1 2 (scal g N +dim( N )(dim( N ) -1)(2 t +1) 2 τ 2 ) ≥ 1 2 min p ∈ N scal g N ( p ) .</formula>
<text><location><page_7><loc_15><loc_46><loc_85><loc_55></location>By choice of the rescaling, we have that ( g M , -τg M ) = γ M ( -1) satisfies ρ M -‖ j M ‖ g M > -1 2 min p ∈ N scal g N ( p ), and so (3) shows that the first section lies in I > ( M × N ). The same argument applies for the last section. In the middle section, the last term in (3) is zero, ρ N -‖ j N ‖ g N ≥ 1 2 min p ∈ N scal g N ( p ), and so by our choice of rescaling, the pair is in I > ( M × N ) for all t ∈ [ -1 2 , 1 2 ].</text>
<text><location><page_7><loc_15><loc_28><loc_85><loc_44></location>We conclude this section by a brief discussion of the Dirac-Witten operator. Assume that ( M,g ) is a space- and time-oriented Lorentzian spin manifold. Let Σ M → M be the classical spinor bundle of ( M,g ), i. e. the spinor bundle associated to an irreducible representation of /BV l n, 1 . A short summary of spin geometry in the semi-Riemannian setting can be found in [3]. Restricting this bundle to the spacelike hypersurface M yields the induced hypersurface spinor bundle Σ M | M → M . The Levi-Civita connection of ( M,g ) induces a connection ∇ on Σ M | M , and the associated Dirac type operator Dψ = ∑ n i =1 e i · ∇ e i ψ , where ( e 1 , . . . , e n ) is an local orthonormal frame of TM , is known as Dirac-Witten operator . It was first observed by Witten [24] that it is linked to the dominant energy condition by a Schrödinger-Lichnerowicz type formula:</text>
<formula><location><page_7><loc_37><loc_25><loc_85><loc_28></location>D 2 ψ = ∇ ∗ ∇ ψ + 1 2 ( ρ -e 0 · j /sharp · ) ψ (4)</formula>
<text><location><page_7><loc_15><loc_19><loc_85><loc_24></location>Here, ψ is any smooth section of Σ M and -∗ denotes the formal adjoint. The formula shows that the dominant energy condition implies that D is positive and hence invertible.</text>
<text><location><page_8><loc_15><loc_82><loc_85><loc_90></location>It now turns out, that D does not depend on the whole Lorentzian manifold ( M,g ), but only on induced initial data set ( g, k ) on M . In fact, denoting by e 0 the futurepointing unit normal as above, we obtain an induced Spin( n )-principal bundle on M by pulling back the Spin 0 ( n, 1)-principal bundle P Spin 0 ( n, 1) M | M along the inclusion of frame bundles</text>
<formula><location><page_8><loc_38><loc_76><loc_62><loc_80></location>P SO( n ) M -→ P SO 0 ( n, 1) M | M ( e 1 , . . . , e n ) -→ ( e 0 , e 1 , . . . , e n ) .</formula>
<text><location><page_8><loc_47><loc_76><loc_47><loc_78></location>↦</text>
<text><location><page_8><loc_15><loc_65><loc_85><loc_75></location>When n = 2 m is even, there are two irreducible (ungraded) /BV l n, 1 -representations and either restricts to the unique irreducible (ungraded) representation of /BV l n along the inclusion 1 /BV l n ↪ → /BV l n, 1 . On associated bundles, this yields an isomorphism Σ M | M ∼ = Σ M , where Σ M → M is the classical spinor bundle on M . The difference between the two representations results in the fact that in one case multiplication by e 0 is given by ω = i m e 1 · · · e n , whereas in the other case it is given by -ω .</text>
<text><location><page_8><loc_15><loc_60><loc_85><loc_65></location>Apart from ∇ , there is another canonical connection on Σ M | M ∼ = Σ M : the one induced by the Levi-Civita connection of ( M,g ), called ∇ . Those two connections differ by a term depending on the second fundamental form, namely</text>
<formula><location><page_8><loc_37><loc_55><loc_63><loc_59></location>∇ X ψ = ∇ X ψ -1 2 e 0 · k ( X, -) /sharp · ψ</formula>
<text><location><page_8><loc_15><loc_52><loc_63><loc_55></location>for ψ ∈ Γ(Σ M ) and X ∈ TM . As a consequence, we obtain</text>
<formula><location><page_8><loc_40><loc_49><loc_60><loc_52></location>Dψ = Dψ -1 2 tr( k ) e 0 · ψ</formula>
<text><location><page_8><loc_15><loc_46><loc_67><loc_48></location>for all ψ ∈ Γ(Σ M ), where D denotes the Dirac operator on Σ M .</text>
<text><location><page_8><loc_15><loc_29><loc_85><loc_45></location>Unlike the Dirac operator D , which - for even n = 2 m - is odd with respect to the /CI / 2 /CI -grading defined by the volume element ω of Σ M , there is no natural grading which is compatible with the Dirac-Witten operator in this case. As for index theory, however, such gradings are very useful, we consider a /CI / 2 /CI -graded version instead. We do so by replacing Σ M with Σ M , the bundle associated to the unique irreducible /CI / 2 /CI -graded /BV l n, 1 -representation 2 . This representation is obtained by summing the two irreducible ungraded /BV l n, 1 -representations and taking an appropriate grading. More precisely, starting with the irreducible representation with i m e 0 · · · e n = 1 , the other one can be obtained by replacing e 0 · with -e 0 · , keeping the multiplication by the other basis vectors the same. Then the grading on the sum can be chosen as</text>
<formula><location><page_8><loc_45><loc_24><loc_85><loc_29></location>ι = ( 0 ω ω 0 ) (5)</formula>
<text><location><page_9><loc_15><loc_86><loc_85><loc_90></location>for ω = i m e 1 · · · e n as above. As a consequence, there is an isomorphism Σ M | M ∼ = Σ M ⊕ Σ M such that the grading is given by (5) and multiplication by e 0 is given by</text>
<formula><location><page_9><loc_43><loc_82><loc_85><loc_87></location>e 0 · = ( ω 0 0 -ω ) . (6)</formula>
<text><location><page_9><loc_15><loc_80><loc_82><loc_81></location>It should be noted that doubling the spinor bundle creates an additional symmetry:</text>
<formula><location><page_9><loc_44><loc_75><loc_85><loc_80></location>c 1 = ( i 0 0 -i ) (7)</formula>
<text><location><page_9><loc_15><loc_70><loc_85><loc_74></location>defines an odd /BV l 1 -action on Σ M | M , which has the property that it commutes with the Dirac-Witten operator. For this reason, the Dirac-Witten operator on this bundle is called /BV l 1 -linear Dirac-Witten operator .</text>
<section_header_level_1><location><page_9><loc_15><loc_64><loc_80><loc_66></location>3. The twisted index difference for Dirac-Witten operators</section_header_level_1>
<text><location><page_9><loc_15><loc_45><loc_85><loc_61></location>Let M be a connected spin manifold of even dimension n = 2 m . We will not assume that M is compact. The necessity for also considering non-compact manifolds - although the main result is only concerned with compact ones - strives from the fact, that we will later be looking at coverings and we do not want to assume them to be finite. The aim of this section is to define a homotopy invariant for a path γ : ( I, ∂I ) → ( I ( M ) , I > R E + c ( M )), where I = [ -1 , 1] and c > 0. This invariant will have the property of being zero if a representative of its homotopy class takes values only in I > R E + c ( M ). The way, it is constucted, is quite similar to the α -difference in [11], yet it differs in the way that instead of real Clifford-linear spinors, it uses complex spinors with coefficients in a twist bundle. This is needed to make use of enlargeability.</text>
<text><location><page_9><loc_15><loc_33><loc_85><loc_43></location>To be able to define this also when M is non-compact, we need some extra care in the definition of I ( M ): It will be the space of pairs ( g, k ), where g is a complete Riemannian metric and k ∈ Γ( T ∗ M ⊗ T ∗ M ) is symmetric. It carries the topology of uniform convergence of all derivatives on compact sets. I > R E + c ( M ) denotes the subspace of those pairs satisfying the dominant energy condition in the stricter sense 1 2 ( ρ -‖ j ‖ ) > ‖R E ‖ + c (cf. (1)), where R E is the curvature endomorphism of a twist bundle E .</text>
<text><location><page_9><loc_15><loc_24><loc_85><loc_32></location>The twist bundle E arises as a direct sum of two complex vector bundles E 0 , E 1 → M with hermitian metrics and metric connections, such that outside a compactum K they can be identified by an isometric and connection preserving bundle isomorphism Ψ: E 0 | M \ K → E 1 | M \ K . If M is already compact, then, of course, we may take K = M and the compatibility condition becomes void.</text>
<text><location><page_9><loc_15><loc_19><loc_85><loc_22></location>The constuction of the desired homotopy invariant begins as follows: For a chosen (complete) Riemannian metric g on M , let Σ M be the classical complex spinor bundle asso-</text>
<text><location><page_10><loc_15><loc_80><loc_85><loc_90></location>iated to the (topological) spin structure on M . We consider the double spinor bundle Σ M := Σ M ⊕ Σ M with its direct sum hermitian metric. This carries a (self-adjoint) /CI / 2 /CI -grading ι and a (skew-adjoint) odd /BV l 1 -action c 1 given by (5) and (7), respectively. Moreover, Σ M admits an operator e 0 · defined by (6), which is self-adjoint, odd and commutes with the /BV l 1 -action. If D is the Dirac operator of the double spinor bundle, then the /BV l 1 -linear Dirac-Witten operator</text>
<formula><location><page_10><loc_42><loc_76><loc_58><loc_79></location>D = D -1 2 tr g ( k ) e 0 ·</formula>
<text><location><page_10><loc_15><loc_72><loc_85><loc_75></location>is formally self-adjoint, odd and commutes with the /BV l 1 -action. We mean to associate a suitable (relative) index to it.</text>
<text><location><page_10><loc_15><loc_67><loc_85><loc_70></location>We now bring in the twist bundles. From E 0 and E 1 , we form the sum E = E 0 ⊕ E 1 , which we endow with the /CI / 2 /CI -grading</text>
<formula><location><page_10><loc_42><loc_62><loc_58><loc_67></location>η = ( 1 E 0 0 0 -1 E 1 ) .</formula>
<text><location><page_10><loc_15><loc_53><loc_85><loc_61></location>On the twist bundle Σ E M := Σ M ⊗ E , we have a /CI / 2 /CI -grading ι ⊗ η and an odd /BV l 1 -action c 1 ⊗ η . The connections on Σ M , E 0 and E 1 define a connection on Σ E M , giving rise to a twisted Dirac operator D E and a twisted /BV l 1 -linear Dirac-Witten operator D E = D E -1 2 tr g ( k ) e 0 · ⊗ 1 E , which is again formally self-adjoint, odd and commutes with the /BV l 1 -action.</text>
<text><location><page_10><loc_15><loc_45><loc_85><loc_51></location>The assumption that E 0 and E 1 agree outside a compact set K leads to even more structure on the twisted bundle. Namely, we choose a smooth cut-off function ϑ : M → [0 , 1] with compact support such that ϑ ≡ 1 on K . Then</text>
<formula><location><page_10><loc_35><loc_41><loc_65><loc_46></location>T = ι ⊗ ( 0 -(1 -ϑ ) i Ψ -1 (1 -ϑ ) i Ψ 0 )</formula>
<text><location><page_10><loc_15><loc_35><loc_86><loc_40></location>is a self-adjoint, odd /BV l 1 -linear operator that anti-commutes with D E away from supp(d ϑ ). Hence, for any σ > 0, the operator D E + σT is again formally self-adjoint, odd and /BV l 1 -linear and satisfies</text>
<formula><location><page_10><loc_36><loc_30><loc_85><loc_34></location>( D E + σT ) 2 = ( D E ) 2 + σ 2 T 2 ≥ σ 2 (8)</formula>
<text><location><page_10><loc_15><loc_27><loc_85><loc_30></location>outside of supp( ϑ ). Note that if M itself is compact, then ϑ ≡ 1 is a possible choice, in which case T = 0.</text>
<text><location><page_10><loc_15><loc_19><loc_85><loc_25></location>We now consider a family of such operators. Here, a technical difficulty arises: The spinor bundle depends on the metric, so the operators act on different bundles. In order to link these bundles we employ a construction similar to the method of generalized cylinders [3] and its futher development, the universal spinor bundle [20]. We start by recalling from</text>
<text><location><page_11><loc_15><loc_78><loc_85><loc_90></location>[3] that a topological spin structure is given by a double covering P ˜ GL + ( n ) M → P GL + ( n ) M of the principal bundle of positively oriented frames. Moreover, the map associating to a given basis the scalar product, for which this basis is orthonormal, gives rise to an SO( n )-principal bundle P GL + ( n ) M → ⊙ 2 + T ∗ M , where ⊙ 2 + T ∗ M is the bundle of symmetric positive definite bilinear forms. Now, denoting by g = ( g t ) t ∈ I the family of complete Riemannian metrics obtained by looking at the first component of the path γ : I →I ( M ), we form the pullback squares</text>
<text><location><page_11><loc_49><loc_66><loc_49><loc_67></location>↦</text>
<figure>
<location><page_11><loc_35><loc_65><loc_66><loc_77></location>
</figure>
<text><location><page_11><loc_39><loc_64><loc_40><loc_67></location>×</text>
<text><location><page_11><loc_15><loc_56><loc_85><loc_63></location>Notice that the so-defined principal bundles, the SO( n )-principal bundle P SO( n ) ( g · ) → M × I and the Spin( n )-principal bundle P Spin( n ) ( g · ) → M × I , are in general just continuous, not smooth, as we assumed the path γ : I → I ( M ) only to be continuous. However, when we restrict to a certain parameter t ∈ I , these give back the (smooth) principal bundles associated to the metric g t , i. e. the following is a pullback diagram:</text>
<text><location><page_11><loc_63><loc_53><loc_63><loc_54></location>)</text>
<text><location><page_11><loc_57><loc_63><loc_59><loc_67></location>⊙</text>
<figure>
<location><page_11><loc_37><loc_43><loc_63><loc_54></location>
</figure>
<text><location><page_11><loc_58><loc_42><loc_60><loc_44></location>×</text>
<text><location><page_11><loc_48><loc_44><loc_48><loc_45></location>↦</text>
<text><location><page_11><loc_15><loc_32><loc_85><loc_40></location>By associating to P Spin( n ) ( g · ) the double of the irreducible /BV l n -representation, we obtain a continuous bundle Σ g · → M × I that restricts for each t ∈ I to the double spinor bundle Σ( M,g t ) → M for the metric g t . Moreover, the twisted bundle Σ E g · := Σ g · ⊗ p ∗ E , where p : M × I → M is the canonical projection, restricts for fixed t ∈ I to the twisted bundle Σ E ( M,g t ) considered above.</text>
<text><location><page_11><loc_15><loc_20><loc_85><loc_30></location>A single differential operator on a vector bundle is often best considered as an unbounded operator acting on the L 2 -space of sections of that bundle. The corresponding notion for families of operators acting on a family of vector bundles is that of a densely defined operator family on a continuous field of Hilbert spaces. This notion was developed by Ebert [9] building on work by Dixmier and Douady [8] and in the following we stick to his notation.</text>
<text><location><page_12><loc_15><loc_81><loc_85><loc_90></location>We start by constructing a continuous field of Hilbert spaces with /BV l 1 -structure from the bundle Σ E g · → M × I . Roughly speaking, this consists of the spaces of L 2 -sections of Σ E g t → M , parametrized over t ∈ I , together with the datum of when a family ( u t ) t ∈ I of L 2 -sections is continuous. It will be obtained as completion of an appropriate field of pre-Hilbert spaces: For each t ∈ I let</text>
<text><location><page_12><loc_15><loc_76><loc_70><loc_77></location>be the space of smooth compactly supported sections and denote by</text>
<formula><location><page_12><loc_24><loc_77><loc_45><loc_81></location>V t = C ∞ c ( M, Σ E ( M,g t ) )</formula>
<formula><location><page_12><loc_25><loc_70><loc_76><loc_75></location>Λ = { u ∈ C 0 c ( M × I, Σ E g · )∣ ∣ ∣ u | M ×{ t } ∈ V t for all t ∈ I } ⊆ ∏ t ∈ I V t</formula>
<text><location><page_12><loc_15><loc_66><loc_85><loc_70></location>the subset of those families of such sections that assemble into a compactly supported continuous section of the bundle Σ E g · → M × I .</text>
<text><location><page_12><loc_15><loc_61><loc_85><loc_65></location>Lemma 3.1. Together with the L 2 -scalar product on V t , the pair (( V t ) t ∈ I , Λ) defines a continuous field of pre-Hilbert spaces.</text>
<text><location><page_12><loc_15><loc_57><loc_45><loc_59></location>Proof. The first thing to show is that</text>
<text><location><page_12><loc_38><loc_51><loc_38><loc_53></location>↦</text>
<formula><location><page_12><loc_36><loc_50><loc_64><loc_56></location>I -→ /CA t -→ ∫ M 〈 u ( x, t ) , v ( x, t ) 〉 dvol g t ( x )</formula>
<text><location><page_12><loc_15><loc_44><loc_85><loc_50></location>is continuous for u, v ∈ Λ. By the definition of Λ, the functions ( x, t ) → ‖ u ( x, t ) ‖ and hence x → max t ∈ I ‖ u ( x, t ) ‖ are continuous and compactly supported, similarly for v . Thus, using the Cauchy-Schwarz inequality, the theorem of dominated convergence implies the desired continuity.</text>
<text><location><page_12><loc_66><loc_39><loc_66><loc_42></location>↦</text>
<text><location><page_12><loc_75><loc_48><loc_75><loc_50></location>↦</text>
<text><location><page_12><loc_25><loc_46><loc_25><loc_48></location>↦</text>
<text><location><page_12><loc_15><loc_37><loc_85><loc_42></location>Secondly, we have to see that the restriction map Λ → V t , u → u | M ×{ t } is dense for all t ∈ I . In fact, we will show surjectivity. So let u t ∈ V t be given. This defines a commutative diagram</text>
<figure>
<location><page_12><loc_42><loc_30><loc_58><loc_37></location>
</figure>
<text><location><page_12><loc_15><loc_17><loc_85><loc_29></location>whereby Σ E denotes the twisted double spinor bundle associated to the spin structure P ˜ GL + ( n ) M → ⊙ 2 + T ∗ M . The twist is given by the pull back of E along ⊙ 2 + T ∗ M → M . We wish to extend u t to a smooth compactly supported section ˜ u , as this gives rise to a section u ∈ Λ restricting to u t . As u t is compactly supported, we can turn any smooth extension ˜ u into a compactly supported one by multiplying with a suitable cutoff function. We construct ˜ u by gluing local pieces using a partition of unity of ⊙ 2 + T ∗ M :</text>
<formula><location><page_12><loc_42><loc_28><loc_58><loc_37></location>Σ E M ⊙ 2 + T ∗ M, u t g t ˜ u</formula>
<text><location><page_13><loc_15><loc_80><loc_85><loc_90></location>For x ∈ M , let U ⊆ ⊙ 2 + T ∗ M be an open neighborhood of g t ( x ) with the property that there exists a section ε of P ˜ GL + ( n ) M | U → U . Possibly restricting U , we may assume that { g t ( x ) | x ∈ π ( U ) } ⊆ U , where π : ⊙ 2 + T ∗ M → M is the canonical projection. We now obtain an extension of u t on U by taking the coefficients with respect to ε in the associated bundle Σ E | U → U to be constant along the fibers of π .</text>
<text><location><page_13><loc_15><loc_70><loc_85><loc_78></location>We will denote by ( L 2 (Σ E g · ) , Λ) the continuous field of Hilbert spaces obtained as completion of (( V t ) t ∈ I , Λ). It is clear that it carries a /BV l 1 -structure induced by the /CI / 2 /CI -grading ι ⊗ η and the Clifford multiplication c 1 ⊗ η . Next, we want to see how D E + σT defines an unbounded operator family on ( L 2 (Σ E g · ) , Λ) and establish its main analytic properties with the goal to associate a suitable index to it.</text>
<text><location><page_13><loc_15><loc_64><loc_85><loc_68></location>Lemma 3.2. The operators D E t + σ t T : V t → V t for t ∈ I assemble to a densely defined operator family on ( L 2 (Σ E g · ) , Λ) with initial domain (( V t ) t ∈ I , Λ) .</text>
<text><location><page_13><loc_15><loc_45><loc_85><loc_61></location>Proof. We have to show that D E · + σT maps sections u ∈ Λ to sections in Λ. We will show that ( D E · + σT ) u ∈ Λ ⊆ Λ. The only thing that is not clear here is that ( D E · + σT ) u is a continuous section of Σ E g · → M × I . This boils down to showing that D E · u is a continuous section, where D E · is fiberwise the twisted Dirac operator as above. As continuity may be checked locally, we can restrict our attention to an open subset of M × I , where there exists a continuous section ε of P Spin( n ) ( g · ) → M × I . The associated orthonormal frame will be called ( e 1 , . . . , e n ). Assuming that u can be written as a tensor product Ψ ⊗ e with Ψ a section of Σ g · → M × I and e a section of p ∗ E → M × I (in general, u will be a sum of such) and expressing Ψ = [ ε, ψ ], we have</text>
<text><location><page_13><loc_15><loc_36><loc_85><loc_40></location>This expression is continuous as we assumed that the family ( g t ) t ∈ I and its derivatives to be uniformly continuous on all compact sets, which particularly implies that the Christoffel symbols are uniformly continuous on compact sets.</text>
<formula><location><page_13><loc_16><loc_40><loc_84><loc_46></location>D E · u = n ∑ i =1 ε , e i · ∂ e i ψ + n ∑ j,k =1 1 2 g · ( ∇ g · e i e j , e k ) e i · e j · e k · ψ ⊗ e + n ∑ i =1 ( e i · Ψ) ⊗∇ E e i e.</formula>
<text><location><page_13><loc_15><loc_29><loc_85><loc_33></location>The closure of this operator family will be denoted by D E · + σT : dom( D E · + σT ) → ( L 2 (Σ E g · ) , Λ).</text>
<text><location><page_13><loc_15><loc_25><loc_72><loc_27></location>Lemma 3.3. The unbounded operator family D E · + σT is self-adjoint.</text>
<text><location><page_13><loc_15><loc_18><loc_85><loc_23></location>Proof. By definition, this is the case if and only if the operators D E t + σT are self-adjoint and regular for each t ∈ I . A sufficient criterion for this is the existence of a coercive,</text>
<text><location><page_14><loc_15><loc_86><loc_85><loc_90></location>i. e. bounded below and proper, smooth function h t : M → /CA , such that the commutator [ D E t + σ t T, h t ] is bounded, cf. [9, Thm. 1.14].</text>
<text><location><page_14><loc_15><loc_70><loc_85><loc_85></location>For some fixed x 0 ∈ M , we consider the distance function d t ( x ) = dist g t ( x 0 , x ). This is bounded below by 0. As g t is complete, by the theorem of Hopf-Rinow, d t is proper. Yet d t is not smooth, only a Lipschitz function, with Lipschitz constant 1. The desired function h t can now be obtained by suitably smoothing d t out. For example, using the main theorem of [2], we obtain the existence of a smooth function h t with sup x ∈ M | h t ( x ) -d t ( x ) | ≤ ε and Lipschitz constant 1 + ε (for any ε > 0). The first condition ensures that h t is also bounded below and proper, whereas the second one implies that [ D E t + σ t T, h t ] is bounded by 1 + ε , as the principal symbol of D E t + σ t T is given by Clifford multiplication.</text>
<text><location><page_14><loc_15><loc_64><loc_85><loc_67></location>Proposition 3.4. The self-adjoint unbounded operator family D E · + σT is a Fredholm family.</text>
<text><location><page_14><loc_15><loc_47><loc_85><loc_61></location>Proof. The first thing to note is that D E · + σT arises as closure of a formally self-adjoint elliptic differential operator of order 1 on the bundle Σ E g · → M × I . In view of [9, Thm. 2.41], the statement is basically a consequence of the fact that ( D E · + σT ) 2 ≥ σ 2 outside the compact set supp( ϑ ) × I . However, we are not precisely in the setting of Ebert's article. Namely, the bundle Σ E g · → M × I is only continuous and not smooth; but this lower regularity does not affect the proofs. Moreover, we have not yet established the existence of a smooth coercive function h : M × I → /CA such that [ D E · + σT,h ] is bounded - and we will not do so.</text>
<text><location><page_14><loc_15><loc_36><loc_85><loc_46></location>Instead, we note that h serves only two purposes. Firstly, it (again) shows that the operator family is self-adjoint, as the functions h ( -, t ) can play the role of the h t above. Secondly, it serves as a basis for constructing a compactly supported smooth function f : M × I → /CA such that ( D E · + σT ) 2 + f 2 ≥ σ 2 everywhere on M × I and ‖ [ D E · + σT,f ] ‖ ≤ σ 2 2 , which is needed in the proof of Fredholmness. So we may just construct such a function f directly.</text>
<text><location><page_14><loc_15><loc_25><loc_85><loc_34></location>For any t ∈ I , let R t be chosen such that supp( ϑ ) ⊆ B g t R t ( x 0 ) and h t a function as above (for ε ≤ 1 3 ). Furthermore, we choose a smooth cut-off function Ψ: /CA → [0 , 1] with Ψ( r ) = σ for r ≤ 1 and | Ψ ' ( r ) | ≤ σ 2 3 for all r ∈ /CA . Denote by L a number such that Ψ ≡ 0 on [ L, ∞ ). Now, let f t ( x ) = Ψ( h t ( x ) -R t ). Note that f t ≡ σ on supp( ϑ ), as h t ( x ) ≤ d t ( x ) + ε < R t +1 for all x ∈ supp( ϑ ).</text>
<text><location><page_14><loc_15><loc_20><loc_85><loc_25></location>Continuity of ( g s ) s ∈ I allows us to choose δ t > 0 such that ‖ g -1 s ‖ g t ≤ ( 9 8 ) 2 for all</text>
<text><location><page_15><loc_15><loc_87><loc_81><loc_90></location>s ∈ U t := ( t -δ t , t + δ t ) on B g t R t + L ( x 0 ). Using ‖ d f t ‖ 2 g s ≤ ‖ g -1 s ‖ g t ‖ d f t ‖ 2 g t , we obtain</text>
<formula><location><page_15><loc_33><loc_79><loc_85><loc_87></location>‖ [ D E s + σT,f t ] ‖ = ‖ d f t ‖ g s ≤ √ ‖ g -1 s ‖ g t ‖ Ψ ' ‖ ∞ ‖ d h t ‖ g t (9) ≤ 9 8 · σ 2 3 · (1 + ε ) ≤ σ 2 2</formula>
<text><location><page_15><loc_15><loc_77><loc_26><loc_79></location>for all s ∈ U t .</text>
<text><location><page_15><loc_15><loc_68><loc_85><loc_76></location>Now, there exists a finite collection t 1 , . . . , t n such that U t 1 , . . . , U t n cover I and a smooth partition of unity ψ 1 , . . . ψ n subordinate to this open cover. We define f ( x, t ) = ∑ n i =1 ψ i ( t ) f t ( x ). Then f ≡ σ on supp( ϑ ), which implies ( D E · + σT ) 2 + f 2 ≥ σ 2 everywhere on M × I . Moreover, the second property of f immediately follows from (9).</text>
<text><location><page_15><loc_15><loc_63><loc_85><loc_67></location>It is important to know when the Fredholm family D E · + σT is invertible. The next lemma provides a criterion for this.</text>
<section_header_level_1><location><page_15><loc_15><loc_59><loc_39><loc_61></location>Lemma 3.5. Let A ⊆ I with</section_header_level_1>
<text><location><page_15><loc_15><loc_48><loc_85><loc_56></location>Then there is a σ ' > 0 such that D E · + σT is invertible over A for all 0 < σ < σ ' . Here, R E t denotes the curvature endomorphism defined by R E t ( φ ⊗ e ) = ∑ i<j e i · e j · φ ⊗ R E ( e i , e j ) e for φ ⊗ e ∈ (Σ p M ⊕ Σ p M ) ⊗ /BV E p and an orthonormal basis ( e 1 , . . . , e n ) of T p M , p ∈ M , with respect to g t .</text>
<formula><location><page_15><loc_36><loc_56><loc_85><loc_60></location>inf t ∈ A, x ∈ M ( ρ t -‖ j t ‖ g t -2 ‖R E t ‖ ) > 0 . (10)</formula>
<text><location><page_15><loc_15><loc_42><loc_85><loc_47></location>Proof. We first consider the situation for some fixed metric g . The twisted Dirac-Witten operator associated to g satisfies the following Schrödinger-Lichnerowicz type formula, the proof of which is deferred to the appendix:</text>
<formula><location><page_15><loc_30><loc_37><loc_70><loc_42></location>( D E ) 2 ψ = ( ∇ E ) ∗ ∇ E ψ + 1 2 ( ρ -e 0 · j /sharp · ) ψ + R E ψ.</formula>
<text><location><page_15><loc_15><loc_35><loc_85><loc_38></location>Here, ∇ E denotes the connection on Σ E M induced by ∇ and the connection on E , and the star indicates the formal adjoint. Together with</text>
<text><location><page_15><loc_15><loc_30><loc_24><loc_31></location>this implies</text>
<formula><location><page_15><loc_25><loc_30><loc_75><loc_35></location>( D E + σT ) 2 ψ = ( D E ) 2 ψ + σ ( D E T + TD E ) ψ + σ 2 (1 -ϑ ) 2 ψ</formula>
<formula><location><page_15><loc_16><loc_19><loc_83><loc_30></location>( ( D E + σT ) 2 ψ, ψ ) L 2 ≥ ( ( D E ) 2 ψ, ψ ) L 2 -σ ‖ d ϑ ‖ g ‖ ψ ‖ 2 L 2 + σ 2 (1 -ϑ ) 2 ‖ ψ ‖ 2 L 2 ≥ ‖∇ E ψ ‖ 2 L 2 + ( 1 2 ( ρ -‖ j ‖ g ) -‖R E ‖ ) ‖ ψ ‖ 2 L 2 -σ ‖ d ϑ ‖ g ‖ ψ ‖ 2 L 2 ≥ ( 1 2 ( ρ -‖ j ‖ g ) -‖R E ‖ ) ‖ ψ ‖ 2 L 2 -σ ‖ d ϑ ‖ g ‖ ψ ‖ 2 L 2 .</formula>
<text><location><page_16><loc_15><loc_87><loc_85><loc_90></location>for any compactly supported smooth section ψ (norms without subscript L 2 denote pointwise norms).</text>
<text><location><page_16><loc_15><loc_82><loc_85><loc_85></location>As ‖ dϑ ‖ g t is a continuous compactly supported function on M × I and (10) holds, we may choose σ ' > 0 such that</text>
<formula><location><page_16><loc_29><loc_77><loc_71><loc_82></location>inf t ∈ A,x ∈ M ( ρ t -‖ j t ‖ g t -2 ‖R E t ‖ ) ≥ σ ' sup t ∈ A,x ∈ M ‖ dϑ ‖ g t .</formula>
<text><location><page_16><loc_15><loc_72><loc_85><loc_77></location>Then for all 0 < σ < σ ' , there exists some constant c > 0 with ( D E t + σT ) 2 ≥ c for all t ∈ A . From this, the statement follows immediately (cf. [9][Prop. 1.21, Lem. 2.6]).</text>
<text><location><page_16><loc_15><loc_66><loc_85><loc_70></location>Proposition 3.6. The operator family D E · + σT is odd with respect to ι ⊗ η and /BV l 1 -linear with respect to c 1 ⊗ η . For suitably small σ , it defines an element</text>
<text><location><page_16><loc_15><loc_55><loc_85><loc_62></location>that is independent of the choices of K,ϑ, Ψ and σ (as long as they fulfill the assumed requirements) and depends only on the relative homotopy class of γ : ( I, ∂I ) → ( I ( M ) , I > R E + c ( M )) . Moreover, this class is zero if γ is homotopic to a path I → I > R E + c ( M ) .</text>
<formula><location><page_16><loc_27><loc_62><loc_73><loc_66></location>[ ( L 2 (Σ E g · ) , Λ) , ι ⊗ η, -iιc 1 ⊗ 1 E , D E · + σT ] ∈ K 1 ( I, ∂I ) ,</formula>
<text><location><page_16><loc_15><loc_50><loc_83><loc_53></location>Note that, as I ( M ) is convex, the homotopy class of γ just depends on its endpoints.</text>
<text><location><page_16><loc_15><loc_43><loc_85><loc_49></location>Definition 3.7. For ( g -1 , k -1 ) , ( g 1 , k 1 ) ∈ I > R E + c ( M ), their E -relative index difference ind-diff E (( g -1 , k -1 ) , ( g 1 , k 1 )) ∈ K 1 ( I, ∂I ) is the class defined in Proposition 3.6 using some path γ : ( I, ∂I ) → ( I ( M ) , I > R E + c ( M )) connecting these two pairs.</text>
<text><location><page_16><loc_15><loc_38><loc_85><loc_41></location>If M is compact, then E 0 and E 1 need not be isomorphic anywhere. In particular, we may take E 0 = /BV and E 1 = 0, which gives the untwisted index difference.</text>
<text><location><page_16><loc_15><loc_30><loc_85><loc_35></location>Definition 3.8. If M is compact, the index difference of ( g -1 , k -1 ) and ( g 1 , k 1 ) ∈ I + ( M ) is ind-diff(( g -1 , k -1 ) , ( g 1 , k 1 )) := ind-diff E (( g -1 , k -1 ) , ( g 1 , k 1 )) ∈ K 1 ( I, ∂I ) for the trivial bundles E 0 = /BV and E 1 = 0.</text>
<text><location><page_16><loc_15><loc_19><loc_85><loc_28></location>Proof. That D E · + σT is odd and /BV l 1 -linear follows from the fact that this holds for D E t + σT for all t ∈ I . As D E · + σT is, moreover, an unbounded Fredholm family that is by Lemma 3.5 invertible over ∂I , we get an element in the K-theory of ( I, ∂I ). Note that the K-theoretic model described in [9][Ch. 3], which we are using here, requires a /BV l 1 -antilinear operator rather than a /BV l 1 -linear one. But this is no problem as a /BV l 1 -linear</text>
<text><location><page_17><loc_15><loc_86><loc_85><loc_90></location>operator is /BV l 1 -antilinear with respect to the /BV l 1 -structure defined by i ( ι ⊗ η )( c 1 ⊗ η ) = -iιc 1 ⊗ 1 .</text>
<text><location><page_17><loc_15><loc_71><loc_85><loc_85></location>We now show independence of the choices starting with σ . Let σ 0 > 0 and σ 1 > 0 be two admissible values, i. e. smaller than σ ' from Lemma 3.5 applied to A = ∂I . We consider the pullback of the ( ( L 2 (Σ E g · ) , Λ) , ι ⊗ η, -iιc 1 ⊗ 1 ) along the canonical projection I × [0 , 1] → I . This continuous field of /CI / 2 /CI -graded Hilbert spaces with /BV l 1 -structure carries the odd /BV l 1 -antilinear Fredholm family D E t +((1 -s ) σ 0 + sσ 1 ) T for t ∈ I and s ∈ [0 , 1], which is invertible over ∂I × [0 , 1]. Thus, we have a concordance between the cycles ( ( L 2 (Σ E g · ) , Λ) , ι ⊗ η, -iιc 1 ⊗ 1 , D E · + σT ) for σ = σ 0 and σ = σ 1 .</text>
<text><location><page_17><loc_15><loc_62><loc_85><loc_71></location>Independence of ϑ , K and Ψ are slightly connected, as we have to have ϑ ≡ 1 on K and Ψ is defined on the complement of K . Given two such triples ( ϑ 0 , K 0 , Ψ 0 ) and ( ϑ 1 , K 1 , Ψ 1 ), we first show that we can first replace the ϑ 0 by some ϑ with ϑ ≡ 1 on K = K 0 ∪ K 1 without changing the K 1 -class. Then noting that ( ϑ, K 0 , Ψ 0 ) and ( ϑ, K, Ψ 0 | M \ K ) even define the same operator family, it just remains to show that the K 1 -class is independent of Ψ for fixed ϑ and K .</text>
<text><location><page_17><loc_15><loc_55><loc_85><loc_60></location>Concering the replacement of ϑ 0 by ϑ (similarly for ϑ 1 by ϑ ), we use the same argumentation as for σ with the difference that this time the operator family is given by D E t + σT s with</text>
<formula><location><page_17><loc_24><loc_49><loc_76><loc_54></location>T s = ι ⊗ ( 0 -(1 -(1 -s ) ϑ 0 -sϑ ) i Ψ -1 0 (1 -(1 -s ) ϑ 0 -sϑ ) i Ψ 0 0 ) .</formula>
<text><location><page_17><loc_15><loc_39><loc_85><loc_47></location>For changing Ψ 0 | M \ K to Ψ 1 | M \ K , we note that the requirement that these bundle isomorphisms preserve hermitian metric and connection implies that they differ by a single element of U ( k ) on every connected component of M \ K , where k is the rank of E 0 (and E 1 ). As U ( k ) is connected, there exists a homotopy (Ψ s ) s ∈ [0 , 1] connecting these. Again, the operator family D E t + σT s defines a concordance, where this time</text>
<formula><location><page_17><loc_34><loc_33><loc_66><loc_38></location>T s = ι ⊗ ( 0 -(1 -ϑ ) i Ψ -1 s (1 -ϑ ) i Ψ s 0 ) .</formula>
<text><location><page_17><loc_15><loc_24><loc_86><loc_31></location>Now assume that we are given a homotopy H : ( I × [0 , 1] , ∂I × [0 , 1]) → ( I ( M ) , I > R E + c ( M )) between γ 0 = H ( -, 0) and γ 1 = H ( -, 1). In this case we can construct a concordance between the K 1 -cycles associated to γ 0 and γ 1 as follows. Let ( g t,s ) t ∈ I, s ∈ [0 , 1] be the first components of the pairs ( H ( t, s )) t ∈ I, s ∈ [0 , 1] . Similarly to before, we may form the</text>
<text><location><page_18><loc_15><loc_88><loc_22><loc_90></location>pullback</text>
<text><location><page_18><loc_51><loc_78><loc_51><loc_78></location>↦</text>
<figure>
<location><page_18><loc_34><loc_76><loc_67><loc_88></location>
</figure>
<text><location><page_18><loc_15><loc_67><loc_85><loc_75></location>obtain a bundle Σ E g · , · → M × I × [0 , 1] and a continuous field of /CI / 2 /CI -graded Hilbert spaces ( L 2 (Σ E g · , · ) , Λ) with /BV l 1 -structure. The operators ( D E t,s + σT ) t ∈ I, s ∈ [0 , 1] constitute an unbounded Fredholm family on this continuous field of Hilbert spaces. For suitably small σ , by an analogous statement to Lemma 3.5, this is invertible over ∂I × [0 , 1]. Together, these provide the required concordance.</text>
<text><location><page_18><loc_15><loc_58><loc_85><loc_65></location>For the last statement assume that the image of γ is contained in I > R E + c . In this case, D E · + σT is invertible on all of I by Lemma 3.5, as long as σ is chosen suitably small. Therefore, the K 1 -class under consideration is in the image of the restriction homomorphism K 1 ( I, I ) → K 1 ( I, ∂I ). But as K 1 ( I, I ) = 0, this class has to be zero.</text>
<section_header_level_1><location><page_18><loc_15><loc_53><loc_73><loc_54></location>4. An index theorem for the twisted index difference</section_header_level_1>
<text><location><page_18><loc_15><loc_42><loc_85><loc_50></location>The purpose of this section is to calculate the E -relative index difference for between pairs of the form ( g, τg ) and ( g, -τg ) for some τ > 0. This is done in two steps. The first one is to express the relative index difference as relative index of a suitable Dirac type operator. In the second step, a relative index theorem identifies this index with a twisted version of the ˆ A -genus.</text>
<text><location><page_18><loc_15><loc_34><loc_85><loc_40></location>The first thing we realize is the following: If for ( g -1 , k -1 ) and ( g 1 , k 1 ) there exists some σ ' > 0 such that D E · + σT is invertible for all 0 < σ < σ ' , then it makes sense to speak of their E -relative index difference, even if they are not contained in some I > R E + c ( M ).</text>
<text><location><page_18><loc_15><loc_28><loc_85><loc_33></location>Let g be some complete metric on M . Such a metric always exists by a classical result of Greene [12], it is even the case that every conformal class contains such a metric by [19]. For τ > 0, we consider the pairs ( g, τg ) and ( g, -τg ). Using</text>
<formula><location><page_18><loc_32><loc_23><loc_72><loc_28></location>( D E ) 2 = ( D E ± 1 2 τe 0 · ⊗ 1 E ) 2 = ( D E ) 2 + n 2 4 τ 2</formula>
<text><location><page_19><loc_15><loc_88><loc_23><loc_90></location>we obtain</text>
<formula><location><page_19><loc_28><loc_80><loc_72><loc_88></location>( D E + σT ) 2 = ( D E ) 2 + σ ( D E T + TD E ) + σ 2 (1 -ϑ ) 2 ≥ ( D E ) 2 + n 2 4 τ 2 -σ ‖ d ϑ ‖ g ,</formula>
<text><location><page_19><loc_15><loc_76><loc_85><loc_80></location>which shows that for these pairs D E + σT is invertible for sufficiently small σ > 0. Thus it makes sense to speak of ind-diff E (( g, -τg ) , ( g, τg )) ∈ K 1 ( I, ∂I ).</text>
<text><location><page_19><loc_15><loc_73><loc_81><loc_75></location>We denote by D E 0 the Dirac operator on Σ E M := Σ M ⊗ E for the metric g . With</text>
<formula><location><page_19><loc_34><loc_68><loc_66><loc_73></location>T 0 = ω ⊗ ( 0 -(1 -ϑ ) i Ψ -1 (1 -ϑ ) i Ψ 0 ) ,</formula>
<text><location><page_19><loc_15><loc_63><loc_85><loc_67></location>this gives rise to a Fredholm operator D E 0 + σT 0 , for σ > 0, which is odd with respect to the /CI / 2 /CI -grading ω ⊗ η . The proof of Fredholmness uses that g is complete. It is similar to Lemma 3.3 and Proposition 3.4, but much less delicate.</text>
<text><location><page_19><loc_15><loc_56><loc_85><loc_60></location>Proposition 4.1. The isomorphism K 1 ( I, ∂I ) ∼ = K 0 ( {∗} ) ∼ = /CI , induced by the Bott map, maps the class ind-diff E (( g, -τg ) , ( g, τg )) to ind( D E 0 + σT 0 ) .</text>
<text><location><page_19><loc_15><loc_48><loc_85><loc_54></location>Proof. We start from the class [ L 2 (Σ E M ) , ω ⊗ η, D E 0 + σT 0 ] ∈ K 0 ( {∗} ) corresponding to the integer value ind( D E 0 + σT 0 ). Using the conventions of Ebert [9], the Bott map K 0 ( {∗} ) ∼ → K 1 ( /CA , /CA \ { 0 } ) sends this class to</text>
<formula><location><page_19><loc_15><loc_44><loc_85><loc_49></location>[ /BV 2 ⊗ p ∗ L 2 (Σ E M ) , ( ω ⊗ η 0 0 -ω ⊗ η ) , ( 0 -ω ⊗ η ω ⊗ η 0 ) , ( D E 0 + σT 0 tω ⊗ η tω ⊗ η D E 0 + σT 0 )] ,</formula>
<text><location><page_19><loc_15><loc_37><loc_85><loc_43></location>where p : /CA → {∗} is the projection and t is the /CA -coordinate. As the inclusion ( I, ∂I ) → ( /CA , /CA \ { 0 } ) induces an isomorphism in K -theory, the same formula defines the corresponding element in K 1 ( I, ∂I ); now assuming p : I →{∗} and t ∈ I .</text>
<text><location><page_19><loc_15><loc_31><loc_85><loc_37></location>Note that we may identify ( L 2 (Σ E g · ) , Λ) = /BV 2 ⊗ p ∗ L 2 (Σ E M ) for the constant family g t = g . Using the automorphism of ( L 2 (Σ E g · ) , Λ ) given by</text>
<formula><location><page_19><loc_26><loc_27><loc_74><loc_32></location>( 1 Σ M 0 0 1 Σ M ) ⊗ ( 1 E 0 0 0 0 ) + ( 1 Σ M 0 0 -1 Σ M ) ⊗ ( 0 0 0 1 E 1 ) ,</formula>
<text><location><page_19><loc_15><loc_24><loc_37><loc_26></location>the K 1 -class translates into</text>
<formula><location><page_19><loc_19><loc_19><loc_81><loc_24></location>[ ( L 2 (Σ E g · ) , Λ ) , ( ω 0 0 -ω ) ⊗ η, ( 0 -ω ω 0 ) ⊗ 1 E , ( D E 0 + σT 0 tω ⊗ 1 E tω ⊗ 1 E D E 0 -σT 0 )] .</formula>
<text><location><page_20><loc_15><loc_88><loc_48><loc_90></location>Applying furthermore the automorphism</text>
<formula><location><page_20><loc_31><loc_83><loc_69><loc_88></location>1 √ 2 (( 1 Σ M 0 0 1 Σ M ) + ( 0 1 Σ M -1 Σ M 0 )) ⊗ 1 E ,</formula>
<text><location><page_20><loc_15><loc_81><loc_22><loc_82></location>this gets</text>
<formula><location><page_20><loc_17><loc_72><loc_83><loc_80></location>[ ( L 2 (Σ E g · ) , Λ ) , ( 0 ω ω 0 ) ⊗ η, ( 0 -ω ω 0 ) ⊗ 1 E , ( D E 0 -tω ⊗ 1 E σT 0 σT 0 D E 0 + tω ⊗ 1 E )] = [( L 2 (Σ E g · ) , Λ ) , ι ⊗ η, -iιc 1 ⊗ 1 E , D E -te 0 · ⊗ 1 E + σT ] .</formula>
<text><location><page_20><loc_15><loc_64><loc_85><loc_72></location>Bearing in mind that D E = D E -1 2 tr( k ) e 0 · ⊗ 1 E , this is the class defined by the DiracWitten operators associated to the straight unit speed path from ( g, -2 g ) to ( g, 2 g ). For general τ > 0, the result follows by rescaling, e. g. replacing the inclusion ( I, ∂I ) → ( /CA , /CA \ { 0 } ) above with the map t → τ 2 t .</text>
<text><location><page_20><loc_43><loc_64><loc_43><loc_67></location>↦</text>
<text><location><page_20><loc_15><loc_59><loc_85><loc_63></location>It now remains to determine ind( D E 0 + σT 0 ). This is done by the relative index theorem going back to [14].</text>
<text><location><page_20><loc_15><loc_55><loc_49><loc_56></location>Theorem 4.2 (Relative Index Theorem) .</text>
<formula><location><page_20><loc_25><loc_50><loc_75><loc_55></location>ind( D E 0 + σT 0 ) = ∫ M ˆ A ( TM ) ∧ (ch( E 1 ) -ch( E 0 )) =: ˆ A ( M,E ) .</formula>
<text><location><page_20><loc_15><loc_44><loc_85><loc_49></location>Proof. Although probably well-known, there seems not to be an easily citable reference matching the setup here. We therefore provide a proof using cut-and-paste-techniques from [6].</text>
<text><location><page_20><loc_15><loc_36><loc_85><loc_42></location>First, we note that the /CI / 2 /CI -graded index of D E 0 + σT 0 is by definition just the usual Fredholm index of its "positive" part ( D E 0 ) + + σT + 0 : Γ((Σ E M ) + ) → Γ((Σ E M ) -) mapping from positive to negative half-spinors. The operator ( D E 0 ) + + σT + 0 is of Dirac type, so we may use the decomposition theorem from [6].</text>
<text><location><page_20><loc_15><loc_26><loc_85><loc_34></location>In order to do so, let M = M 1 ∪ M 2 be a decomposition into two smooth manifolds with boundary ∂M 1 = ∂M 2 such that M 1 is compact and supp ϑ ⊆ M 1 . Let B ⊆ H 1 2 ( ∂M 1 , (Σ E M ) + ) be an elliptic boundary condition and denote by B ⊥ its L 2 -orthogonal complement. For example, B could be Atiyah-Patodi-Singer boundary conditions. The decomposition theorem states that</text>
<formula><location><page_20><loc_17><loc_21><loc_83><loc_25></location>Ind ( ( D E 0 ) + + σT + 0 ) = Ind ( (( D E 0 ) + + σT + 0 ) | M 1 , B ) +Ind ( (( D E 0 ) + + σT + 0 ) | M 2 , B ⊥ ) .</formula>
<text><location><page_21><loc_15><loc_88><loc_41><loc_90></location>The first summand computes to</text>
<formula><location><page_21><loc_23><loc_78><loc_85><loc_88></location>Ind ( (( D E 0 ) + + σT + 0 ) | M 1 , B ) = Ind ( ( D E 0 ) + | M 1 , B ) = Ind ( ( D E 0 0 ) + | M 1 , B ) -Ind ( ( D E 1 0 ) + | M 1 , B ) = ∫ M ˆ A ∧ (ch( E 1 ) -ch( E 0 )) . (11)</formula>
<text><location><page_21><loc_60><loc_76><loc_60><loc_78></location>↦</text>
<text><location><page_21><loc_15><loc_75><loc_85><loc_78></location>Here, in the first step, we used the homotopy [0 , 1] /owner t → ( D E 0 ) + + tσT + 0 . The second step follows from the decomposition</text>
<formula><location><page_21><loc_27><loc_68><loc_73><loc_74></location>( D E 0 ) + = ( ( D E 0 0 ) + 0 0 ( D E 1 0 ) -) = ( ( D E 0 0 ) + 0 0 ( ( D E 1 0 ) + ) ∗ ) .</formula>
<text><location><page_21><loc_15><loc_67><loc_84><loc_68></location>The last step is (contained in the proof of) the relative index theorem [6, Thm. 1.21].</text>
<text><location><page_21><loc_15><loc_54><loc_85><loc_65></location>It remains to show that the second summand is zero. To see this, we note that there exists a bundle ˜ E 1 → M admitting a metric and connection preserving isomorphism ˜ Ψ: E 0 → ˜ E 1 , such that ˜ E 1 | M 2 = E 1 | M 2 and ˜ Ψ | M 2 = Ψ | M 2 . For instance, such a bundle can be obtained by gluing E 0 | M 1 and E 1 | M 2 . Similarly as before, we denote by D ˜ E 0 the Dirac operator on Σ ˜ E M for ˜ E = E 0 ⊕ E 1 and define</text>
<text><location><page_21><loc_15><loc_46><loc_85><loc_50></location>Notice, that we are allowed to take ϑ ≡ 0, as ˜ Ψ is defined on all of M . Again, we have a decomposition</text>
<formula><location><page_21><loc_40><loc_49><loc_60><loc_58></location>˜ ˜ T 0 = ω ⊗ ( 0 -i ˜ Ψ -1 i ˜ Ψ 0 ) .</formula>
<formula><location><page_21><loc_16><loc_41><loc_84><loc_47></location>Ind ( ( D ˜ E 0 ) + + σ ˜ T + 0 ) = Ind ( (( D ˜ E 0 ) + + σ ˜ T + 0 ) | M 1 , B ) +Ind ( (( D ˜ E 0 ) + + σ ˜ T + 0 ) | M 2 , B ⊥ ) .</formula>
<text><location><page_21><loc_15><loc_37><loc_85><loc_42></location>In this case, the calculation (11) shows that the first summand is zero, as ch( ˜ E 1 ) = ch( E 0 ). The second summand is the second summand from above as the bundles and operators are equal on M 2 . Thus, we obtain</text>
<formula><location><page_21><loc_26><loc_30><loc_73><loc_37></location>Ind ( (( D E 0 ) + + σT + 0 ) | M 2 , B ⊥ ) = Ind ( (( D ˜ E 0 ) + + σ ˜ T + 0 ) | M 2 , B ⊥ = Ind ( D E 0 ) + + σ T + 0 = 0 ,</formula>
<text><location><page_21><loc_15><loc_24><loc_70><loc_28></location>where we used that D ˜ E 0 + σ ˜ T 0 is invertible as ( D ˜ E 0 + σ ˜ T 0 ) 2 ≥ σ 2 > 0.</text>
<formula><location><page_21><loc_54><loc_28><loc_75><loc_37></location>) ( ˜ ˜ )</formula>
<text><location><page_21><loc_15><loc_18><loc_85><loc_24></location>Corollary 4.3 (Relative index theorem for the twisted index difference) . The isomorphism K 1 ( I, ∂I ) ∼ = K 0 ( {∗} ) ∼ = /CI sends ind-diff E (( g, -τg ) , ( g, τg )) to ˆ A ( M,E ) . If M is compact, ind-diff(( g, -τg ) , ( g, τg )) is sent to ˆ A ( M ) .</text>
<text><location><page_22><loc_15><loc_84><loc_85><loc_90></location>The relative index theorem allows to obtain an obstruction to the path-connectedness of the space of initial data sets that strictly satisfy the dominant energy condition. The following corollary illustrates the general strategy and turns out to be a special case of the enlargeability obstruction Theorem 5.3 that we discuss in the remaining section.</text>
<text><location><page_22><loc_66><loc_77><loc_66><loc_79></location>/negationslash</text>
<text><location><page_22><loc_15><loc_75><loc_85><loc_81></location>Corollary 4.4. Let M be a compact manifold, g a metric on M and τ > 0 be chosen such that ( g, -τg ) , ( g, τg ) ∈ I > ( M ) . If M is spin and ˆ A ( M ) = 0 , then ( g, -τg ) and ( g, τg ) belong to different path-components of I > ( M ) .</text>
<text><location><page_22><loc_15><loc_69><loc_85><loc_73></location>Proof. If there were a path γ : I → I > ( M ) from ( g, -τg ) to ( g, τg ), then, by Proposition 3.6, ind-diff(( g, -τg ) , ( g, τg )) would be zero. But by Corollary 4.3 it is mapped to ˆ A ( M ) = 0.</text>
<text><location><page_22><loc_20><loc_68><loc_20><loc_70></location>/negationslash</text>
<text><location><page_22><loc_15><loc_58><loc_85><loc_66></location>Remark 4.5. The statement of Corollary 4.4 also follows from the main result in [11]. This is due to the fact that, in even dimension n , Hitchin's α -index is mapped to the ˆ A -genus under the complexification map KO -n ( {∗} ) → K -n ( {∗} ) ∼ = /CI . In fact, also the proof is the same as complexification turns the α -difference defined in [11] into the (untwisted) index-difference considered in this article, up to Bott periodicity.</text>
<section_header_level_1><location><page_22><loc_15><loc_52><loc_68><loc_54></location>5. Enlargeability obstruction for initial data sets</section_header_level_1>
<text><location><page_22><loc_15><loc_40><loc_85><loc_49></location>Gromov-Lawson's enlargeability obstruction gives a major source of examples of manifolds that do not admit a positive scalar curvature metric. In this section, we prove that enlargeability is also an obstruction to path-connectedness of the space of initial data sets strictly satisfying the dominant energy condition. There are many versions of enlargeability. In what follows, we will always understand enlargeability in the sense of ˆ A -area-enlargeability:</text>
<text><location><page_22><loc_15><loc_26><loc_85><loc_37></location>Definition 5.1. A smooth map f : ( M,g ) → ( N,h ) between Riemannian manifolds is ε -area-contracting for some ε > 0 if the induced map f ∗ : Λ 2 TM → Λ 2 TN satisfies ‖ f ∗ ‖ ≤ ε . A compact Riemannian manifold ( M,g ) of dimension n is called area-enlargeable in dimension k if for all ε > 0 there exists a Riemannian covering ( M ' , g ' ) → ( M,g ) admitting an ε -area-contracting map ( M ' , g ' ) → ( S k , g Std ) that is constant outside a compact set and of non-zero ˆ A -degree. It is called ˆ A -area-enlargeable if it is area-enlargeable in some dimension k .</text>
<text><location><page_22><loc_15><loc_20><loc_85><loc_23></location>Recall that the ˆ A -degree of a smooth map f : X → Y , where Y is compact and connected and f is constant outside a compact set, may be defined by the requirement that</text>
<text><location><page_23><loc_15><loc_79><loc_85><loc_91></location>∫ X ˆ A ( TX ) ∧ f ∗ ( ω ) = ˆ A-deg( f ) ∫ Y ω for all top dimensional forms ω ∈ Ω dim( Y ) ( Y ). If Y is non-connected, there is one such number for every connected component of Y and the ˆ A -degree is the vector consisting of these. From this definition we see that it can only be non-zero if dim( Y ) ≤ dim( X ) and dim( Y ) ≡ dim( X ) mod 4. The ˆ A -degree can be thought of as interpolating between the following two special cases: If dim( Y ) = dim( X ), the ˆ A -degree is just the usual degree. If dim( Y ) = 0 and Y is connected, it is the ˆ A -genus ˆ A ( X ) of X .</text>
<text><location><page_23><loc_15><loc_63><loc_85><loc_77></location>Although the definition of enlargeability uses a Riemannian metric, the property itself is independent of this choice. Manifolds that are enlargeable in dimension 0 are precisely the ones having non-zero ˆ A -genus. Another main example is the torus T n = /CA n / /CI n , which is enlargeable in the top dimesion n . Furthermore, every compact manifold that admits a metric of non-positive sectional curvature is enlargeable (in the top dimension) by the Cartan-Hadarmard theorem. If M is enlargeable then the direct sum M # N with another manifold N is again enlargeable. Furthermore, for an enlargeable manifold M the product M × S 1 with a circle is again enlargeable. This, and much more, is discussed in great detail in [18, Sec. IV.5].</text>
<text><location><page_23><loc_15><loc_56><loc_85><loc_61></location>The proof that enlargeable spin manifolds do not admit psc metrics [14, Thm. 5.21] can be split into two parts. The first part consists of using the enlargeability condition to construct a suitable family of complex vector bundles over coverings of the manifold:</text>
<text><location><page_23><loc_15><loc_48><loc_85><loc_54></location>Theorem 5.2 (Gromov-Lawson) . Let ( M,g ) be an ˆ A -area-enlargeable manifold of even dimension. Then there exists a sequence of coverings M i → M and /CI / 2 /CI -graded hermitian vector bundles E i = E (0) i ⊕ E (1) i → M i with compatible connection, such that</text>
<unordered_list>
<list_item><location><page_23><loc_17><loc_44><loc_85><loc_47></location>· for all i ∈ /C6 the bundles E (0) i → M i and E (1) i → M i are isometrically isomorphic in a connection preserving way outside a compactum K i ,</list_item>
</unordered_list>
<text><location><page_23><loc_59><loc_39><loc_59><loc_41></location>/negationslash</text>
<unordered_list>
<list_item><location><page_23><loc_17><loc_39><loc_76><loc_43></location>· ˆ A ( M i , E i ) = ∫ M i ˆ A ( TM i ) ∧ (ch( E (1) i ) -ch( E (0) i )) = 0 for all i ∈ /C6 and</list_item>
<list_item><location><page_23><loc_17><loc_35><loc_41><loc_38></location>· ‖ R E i ‖ ∞ -→ 0 for i -→ ∞ .</list_item>
</unordered_list>
<text><location><page_23><loc_67><loc_31><loc_67><loc_33></location>/negationslash</text>
<text><location><page_23><loc_15><loc_22><loc_85><loc_34></location>Roughly speaking, the construction is the following. When ˆ A ( M ) = 0, then one can just take the constant sequence consisting of the identity M → M as covering and the trivial /CI / 2 /CI -graded bundle /BV ⊕ 0 → M as bundle. Else, if ˆ A ( M ) = 0, one may take a sequence M i → M such that M i admits a 1 i -area-contracting map M i → S 2 /lscript , where 2 /lscript = k > 0 is chosen as in the definition of enlargeability. The bundles are obtained by pulling back a bundle E (0) ⊕ E (1) → S 2 /lscript , where E (0) → S 2 /lscript satisfies c /lscript ( E (0) ) = 0 and E (1) → S 2 /lscript is a trivial bundle of the same rank.</text>
<text><location><page_23><loc_65><loc_23><loc_65><loc_25></location>/negationslash</text>
<text><location><page_23><loc_15><loc_19><loc_85><loc_20></location>The second part consists of calculating the index of the Dirac operator on the twisted</text>
<text><location><page_24><loc_40><loc_86><loc_40><loc_88></location>/negationslash</text>
<text><location><page_24><loc_15><loc_82><loc_85><loc_90></location>spinor bundle Σ M i ⊗ E i in two different ways. On the one hand, by the the relative index theorem, its index is ˆ A ( M i , E i ) = 0. On the other hand, assuming that M carries a psc metric, the twisted Schrödinger-Lichnerowicz formula implies that this Dirac operator is invertible and thus has index zero, for large i ∈ /C6 . As we are interested in initial data sets, we replace this second step and obtain our main theorem:</text>
<text><location><page_24><loc_15><loc_72><loc_85><loc_80></location>Theorem 5.3 (Main Theorem) . Let M be a compact spin manifold that is ˆ A -areaenlargeable. Then the path-components C -and C + of I > ( M ) do not agree, i. e. if g is a metric on M and τ > 0 is chosen so large that ( g, -τg ) , ( g, τg ) ∈ I > ( M ) , then ( g, -τg ) and ( g, τg ) belong to different path-components of I > ( M ) .</text>
<text><location><page_24><loc_82><loc_62><loc_82><loc_64></location>/negationslash</text>
<text><location><page_24><loc_15><loc_61><loc_85><loc_71></location>Proof. We first consider the case where the dimension of M is even. Let g be a metric on M and τ > 0 be large enough that ( g, -τg ) , ( g, τg ) ∈ I > ( M ). We choose a sequence of complex vector bundles E i → M i as in Theorem 5.2 and denote by g i the pull-back metric of g on M i . From the relative index theorem Corollary 4.3, we obtain that for all i ∈ /C6 the twisted index difference ind-diff E i (( g i , -τg i ) , ( g i , τ g i )) corresponds to ˆ A ( M i , E i ) = 0 under the isomorphism K 1 ( I, ∂I ) ∼ = /CI , in particular it is non-zero.</text>
<text><location><page_24><loc_22><loc_55><loc_22><loc_58></location>↦</text>
<text><location><page_24><loc_15><loc_52><loc_85><loc_59></location>We now assume for contradiction that ( g, -τg ) and ( g, τg ) are connected in I > ( M ) by a path t → ( g ( t ) , k ( t )). As the interval I is compact, there is a constant c > 0, such that ρ ( t ) -‖ j ( t ) ‖ ≥ 4 c for all t ∈ I . Of course, this holds as well for the pulled-back path in I > ( M i ), with the same constant. Since for any φ ∈ Σ M i and e ∈ E i</text>
<formula><location><page_24><loc_24><loc_48><loc_76><loc_53></location>‖R E i ( φ ⊗ e ) ‖ ≤ ∑ j<k ‖ φ ‖‖ R E i ( e j , e k ) e ‖ ≤ n ( n -1) 2 ‖ R E i ‖‖ φ ⊗ e ‖ .</formula>
<text><location><page_24><loc_50><loc_44><loc_50><loc_46></location>↦</text>
<text><location><page_24><loc_15><loc_42><loc_85><loc_48></location>and ‖ R E i ‖ ∞ -→ 0 for i -→ ∞ , we have ‖R E i ‖ < c as long as i ∈ /C6 is large enough. Hence for large i ∈ /C6 the pulled back path t → ( g i ( t ) , k i ( t )) lies entirely in I > R E i + c ( M i ). Thus by Proposition 3.6 ind-diff E i (( g i , -τg i ) , ( g i , τ g i )) = 0, which is the desired contradiction.</text>
<text><location><page_24><loc_78><loc_36><loc_78><loc_38></location>/negationslash</text>
<text><location><page_24><loc_15><loc_34><loc_85><loc_40></location>In the odd-dimensional case, we replace M by M × S 1 , which will again be ˆ A -areaenlargeable and spin, and is of even dimension. Thus, we conclude that C + = C -in I > ( M × S 1 ). By Lemma 2.2, the same holds for I > ( M ).</text>
<section_header_level_1><location><page_24><loc_15><loc_27><loc_73><loc_31></location>A. Schrödinger-Lichnerowicz formula for the twisted Dirac-Witten operator</section_header_level_1>
<formula><location><page_24><loc_15><loc_18><loc_68><loc_24></location>Theorem A.1. For all ψ ∈ Γ((Σ M ⊕ Σ M ) ⊗ /BV E ) ( D E ) 2 ψ = ∇ ∗ ∇ ψ + 1 2 ( ρ -e 0 · j /sharp · ) ψ + R E ψ,</formula>
<text><location><page_25><loc_15><loc_84><loc_85><loc_90></location>where ρ and j are defined as in (1) in terms of the pair ( g, k ) and R E ( φ ⊗ e ) = ∑ i<j e i · e j · φ ⊗ R E ( e i , e j ) e for φ ⊗ e ∈ (Σ p M ⊕ Σ p M ) ⊗ /BV E p and an orthonormal basis ( e 1 , . . . , e n ) of T p M , p ∈ M .</text>
<text><location><page_25><loc_15><loc_75><loc_85><loc_82></location>Proof. We show how to reduce the formula to the Schödinger-Lichnerowicz type formula in the untwisted case (4), using a local calculation. For this, let ( e 1 , . . . , e n ) be a local orthonormal frame. Without loss of generality, we may assume that ψ can be written locally as φ ⊗ e as everything is linear. Then</text>
<formula><location><page_25><loc_15><loc_61><loc_71><loc_76></location>( D E ) 2 ( φ ⊗ e ) = ∑ i,j e i · ∇ e i ( e j · ∇ e j φ ) ⊗ e + ∑ i,j e i · ∇ e i ( e j · φ ) ⊗∇ E e j e + ∑ i,j e i · e j · ( ∇ e j φ ) ⊗∇ E e i e + ∑ i,j e i · e j · φ ⊗∇ E e i ∇ E e j e = ( D 2 φ ) ⊗ e + ∑ i,j e i · ( ∇ e i e j ) · φ ⊗∇ E e j e -2 ∑ i ( ∇ e i φ ) ⊗∇ E e i e + ∑ i,j e i · e j · φ ⊗∇ E e i ∇ E e j e</formula>
<text><location><page_25><loc_15><loc_59><loc_18><loc_60></location>and</text>
<formula><location><page_25><loc_17><loc_35><loc_85><loc_59></location>∇ ∗ ∇ ( φ ⊗ e ) = ∑ i ∇ ∗ ( e ∗ i ⊗∇ e i ( φ ⊗ e )) = -∑ i ∇ e i ∇ e i ( φ ⊗ e ) -∑ i e 0 · k ( e i , -) /sharp · ∇ e i ( φ ⊗ e ) + ∑ i ∇ ∇ e i e i ( φ ⊗ e ) = -∑ i ( ∇ e i ∇ e i φ ) ⊗ e -2 ∑ i ( ∇ e i φ ) ⊗∇ E e i e -∑ i φ ⊗∇ E e i ∇ E e i e -∑ i e 0 · k ( e i , -) /sharp · ( ∇ e i φ ) ⊗ e -∑ i e 0 · k ( e i , -) /sharp · φ ⊗∇ E e i e + ∑ i ( ∇ ∇ e i e i φ ) ⊗ e + ∑ i φ ⊗∇ E ∇ e i e i e = ( ∇ ∗ ∇ φ ) ⊗ e -2 ∑ i ( ∇ e i φ ) ⊗∇ E e i e -∑ i φ ⊗∇ E e i ∇ E e i e -∑ i e 0 · k ( e i , -) /sharp · φ ⊗∇ E e i e + ∑ i φ ⊗∇ E ∇ e i e i e</formula>
<text><location><page_25><loc_24><loc_31><loc_24><loc_33></location>↦</text>
<formula><location><page_25><loc_15><loc_18><loc_87><loc_30></location>∑ i,j e i · ( ∇ e i e j ) · φ ⊗∇ E e j e = ∑ i,j,k g ( ∇ e i e j , e k ) e i · e k · φ ⊗∇ E e j e + ∑ i,j k ( e i , e j ) e i · e 0 · φ ⊗∇ E e j e = -∑ i,j,k g ( e j , ∇ e i e k ) e i · e k · φ ⊗∇ E e j e + ∑ j k ( -, e j ) /sharp · e 0 · φ ⊗∇ E e j e = -∑ i,j e i · e j · φ ⊗∇ E ∇ e i e j e -∑ i e 0 · k ( e i , -) /sharp · φ ⊗∇ E e i e</formula>
<text><location><page_25><loc_15><loc_29><loc_85><loc_35></location>using ∇ X ψ = ∇ X ψ -1 2 e 0 · k ( X, -) /sharp · ψ and that the formal adjoint of ∇ is given by ∇ ∗ : α ⊗ ψ → -∑ j ∇ e j ( α ⊗ ψ )( e j ) = -∑ j ∇ e j ( α ( e j ) ψ ) + ∑ j α ( ∇ e j e j ) ψ , α ∈ Ω 1 ( M ). Noting that</text>
<text><location><page_26><loc_15><loc_88><loc_24><loc_90></location>this implies</text>
<text><location><page_26><loc_65><loc_84><loc_65><loc_85></location>/negationslash</text>
<formula><location><page_26><loc_17><loc_80><loc_83><loc_88></location>( D E ) 2 ( φ ⊗ e ) -∇ ∗ ∇ ( φ ⊗ e ) = ( D 2 φ ) ⊗ e -( ∇ ∗ ∇ φ ) ⊗ e -∑ i = j e i · e j · φ ⊗∇ E ∇ e i e j e + ∑ i = j e i · e j · φ ⊗∇ E e i ∇ E e j e.</formula>
<text><location><page_26><loc_45><loc_81><loc_45><loc_82></location>/negationslash</text>
<text><location><page_26><loc_15><loc_75><loc_85><loc_80></location>Using the untwisted Schrödinger-Lichnerowicz type formula (4), the first two terms compute to 1 2 ( ρ -e 0 · j /sharp · ) φ ⊗ e . Thus it remains to identify the remaining terms with R E ( φ ⊗ e ):</text>
<text><location><page_26><loc_35><loc_65><loc_35><loc_66></location>/negationslash</text>
<formula><location><page_26><loc_23><loc_64><loc_77><loc_76></location>R E ( φ ⊗ e ) = ∑ i<j e i · e j · φ ⊗ R E ( e i , e j ) e = ∑ i<j e i · e j · φ ⊗ ( ∇ E e i ∇ E e j -∇ E e j ∇ E e i -∇ E ∇ e i e j + ∇ E ∇ e j e i ) e = ∑ i = j e i · e j · φ ⊗ ( ∇ E e i ∇ E e j -∇ E ∇ e i e j ) e.</formula>
<section_header_level_1><location><page_26><loc_15><loc_59><loc_27><loc_61></location>References</section_header_level_1>
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<list_item><location><page_26><loc_16><loc_53><loc_85><loc_56></location>[1] Bernd Ammann and Jonathan Glöckle. Dominant energy condition and spinors on Lorentzian manifolds. ArXiV:2103.11032, 2021.</list_item>
<list_item><location><page_26><loc_16><loc_47><loc_85><loc_51></location>[2] Daniel Azagra, Juan Ferrera, Fernando López-Mesas, and Yenny Rangel. Smooth approximation of Lipschitz functions on Riemannian manifolds. J. Math. Anal. Appl. , 326(2):1370-1378, 2007.</list_item>
<list_item><location><page_26><loc_16><loc_42><loc_85><loc_45></location>[3] Christian Bär, Paul Gauduchon, and Andrei Moroianu. Generalized cylinders in semi-Riemannian and spin geometry. Math. Z. , 249(3):545-580, 3 2005.</list_item>
<list_item><location><page_26><loc_16><loc_35><loc_85><loc_40></location>[4] Antonio N. Bernal and Miguel Sánchez. Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes. Comm. Math. Phys. , 257(1):4350, 2005.</list_item>
<list_item><location><page_26><loc_16><loc_30><loc_85><loc_33></location>[5] Boris Botvinnik, Johannes Ebert, and Oscar Randal-Williams. Infinite loop spaces and positive scalar curvature. Invent. Math. , 209(3):749-835, 2017.</list_item>
<list_item><location><page_26><loc_16><loc_26><loc_85><loc_29></location>[6] Christian Bär and Werner Ballmann. Boundary value problems for elliptic differential operators of first order. Surv. Differ. Geom. , 17, 01 2011.</list_item>
<list_item><location><page_26><loc_16><loc_19><loc_85><loc_24></location>[7] Diarmuid Crowley, Thomas Schick, and Wolfgang Steimle. Harmonic spinors and metrics of positive curvature via the Gromoll filtration and Toda brackets. J. Topol. , 11(4):1077-1099, 2018.</list_item>
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<location><page_27><loc_15><loc_19><loc_85><loc_90></location>
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<list_item><location><page_28><loc_15><loc_85><loc_85><loc_90></location>[22] Jonathan Rosenberg. Manifolds of positive scalar curvature: a progress report. In Surveys in differential geometry. Vol. XI , volume 11 of Surv. Differ. Geom. , pages 259-294. Int. Press, Somerville, MA, 2007.</list_item>
<list_item><location><page_28><loc_15><loc_80><loc_85><loc_83></location>[23] Thomas Schick and David J. Wraith. Non-negative versus positive scalar curvature. J. Math. Pures Appl. , 146:218-232, 2021.</list_item>
<list_item><location><page_28><loc_15><loc_75><loc_85><loc_79></location>[24] Edward Witten. A new proof of the positive energy theorem. Comm. Math. Phys. , 80:381-402, 1981.</list_item>
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</document> | [] |
2019MNRAS.490.3799M | https://arxiv.org/pdf/1910.09042.pdf | <document>
<section_header_level_1><location><page_1><loc_7><loc_83><loc_80><loc_85></location>Periodic orbits of the retrograde coorbital problem</section_header_level_1>
<section_header_level_1><location><page_1><loc_8><loc_79><loc_41><loc_81></location>M.H.M. Morais 1 ? F. Namouni, 2</section_header_level_1>
<text><location><page_1><loc_8><loc_76><loc_94><loc_79></location>1 Universidade Estadual Paulista (UNESP), Instituto de Geociˆencias e Ciˆencias Exatas, Av. 24-A, 1515, 13506-900 Rio Claro, SP, Brazil 2 Universit'e Cˆote d'Azur, CNRS, Observatoire de la Cˆote d'Azur, CS 24229, 06304 Nice, France</text>
<text><location><page_1><loc_7><loc_72><loc_40><loc_73></location>Accepted XXX. Received YYY; in original form ZZZ</text>
<section_header_level_1><location><page_1><loc_29><loc_69><loc_39><loc_70></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_29><loc_52><loc_91><loc_68></location>Asteroid (514107) Ka'epaoka'awela is the first example of an object in the 1/1 mean motion resonance with Jupiter with retrograde motion around the Sun. Its orbit was shown to be stable over the age of the Solar System which implies that it must have been captured from another star when the Sun was still in its birth cluster. Ka'epaoka'awela orbit is also located at the peak of the capture probability in the coorbital resonance. Identifying the periodic orbits that Ka'epaoka'awela and similar asteroids followed during their evolution is an important step towards precisely understanding their capture mechanism. Here, we find the families of periodic orbits in the two-dimensional retrograde coorbital problem and analyze their stability and bifurcations into three-dimensional periodic orbits. Our results explain the radical differences observed in 2D and 3D coorbital capture simulations. In particular, we find that analytical and numerical results obtained for planar motion are not always valid at infinitesimal deviations from the plane.</text>
<text><location><page_1><loc_29><loc_50><loc_79><loc_51></location>Key words: celestial mechanics - minor planets, asteroids: general</text>
<section_header_level_1><location><page_1><loc_7><loc_44><loc_24><loc_45></location>1 INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_7><loc_21><loc_47><loc_43></location>The Solar System contains only one known asteroid in coorbital resonance with a planet, Jupiter, that moves with a retrograde motion around the Sun: asteroid (514107) Ka'epaoka'awela (Wiegert et al. 2017; Morais & Namouni 2017). Large scale numerical integrations of its past orbital evolution, including perturbations from the four giant planets and the Galactic tide, have shown that it has been at its current location since the end of planet formation 4.5 Gyr in the past. Since a retrograde orbit could not have formed from the material of the Sun's protoplanetary disk at that early epoch, Ka'epaoka'awela must have been belonged to a different star system and was captured by our own when the Sun was still in its birth cluster (Namouni & Morais 2018b). Ka'epaoka'awela is thus the first known example of an interstellar long-term resident in the Solar system. Understanding exactly how it reached its current location is particularly important.</text>
<text><location><page_1><loc_7><loc_12><loc_47><loc_21></location>Coorbital retrograde resonance has been studied in the framework of the restricted three-body problem. Several stable planar and three-dimensional coorbital configurations or modes are known to exist (Morais & Namouni 2013, 2016). Simulations of retrograde asteroids radially and adiabatically drifting towards Jupiter's orbit showed that when retrograde motion is almost coplanar, capture occurs in the</text>
<unordered_list>
<list_item><location><page_1><loc_7><loc_7><loc_37><loc_8></location>? E-mail: [email protected] (MHMM)</list_item>
</unordered_list>
<text><location><page_1><loc_51><loc_35><loc_91><loc_45></location>coorbital mode that corresponds to Ka'epaoka'awela's current orbit (Morais & Namouni 2016; Namouni & Morais 2018a). If motion is exactly coplanar, capture occurs in a distinct coorbital mode (Morais & Namouni 2016; Namouni & Morais 2018a). In order to understand such differences and characterize the path that Ka'epaoka'awela followed in its capture by Jupiter, we aim to identify the periodic orbits of retrograde coorbital motion in the three-body problem.</text>
<text><location><page_1><loc_51><loc_14><loc_91><loc_34></location>The importance of periodic orbits (POs) in the study of a dynamical system has been recognized since the seminal work of Poincar'e. Stable POs are surrounded by islands of regular (quasiperiodic) motion, whereas chaos appears at the location of unstable POs (Hadjedemetriou 2006). In the three-body problem, POs are the solutions such that the relative distances between the bodies repeat over a period T (Henon 1974). They form continuous families and may be classified as (linearly) stable or unstable (Henon 1974; Hadjedemetriou 2006). In the circular restricted three-body problem (CR3BP) with a dominant central mass, these families may be resonant or non-resonant. The former correspond to commensurabilities between the orbital frequencies, whereas the latter correspond to circular solutions of the unperturbed (two-body) problem (Hadjedemetriou 2006).</text>
<text><location><page_1><loc_51><loc_7><loc_91><loc_13></location>In this article, we report on our search of periodic orbits in the CR3BP with a mass ratio GLYPH<22> = 10 GLYPH<0> 3 . In Section 2, we explain how we compute the families of periodic orbits and study their stability. In Section 3, we describe the families that exist in the 2D-configuration and the bifurcations</text>
<text><location><page_2><loc_7><loc_83><loc_47><loc_88></location>from planar families to the 3D-configuration. In Section 4, we discuss how these results explain the differences observed in the 2D and 3D capture simulations. The conclusions of this study are presented in Section 5.</text>
<section_header_level_1><location><page_2><loc_7><loc_77><loc_44><loc_78></location>2 COMPUTATION OF PERIODIC ORBITS</section_header_level_1>
<text><location><page_2><loc_7><loc_64><loc_47><loc_76></location>From the periodicity theorem of Roy & Ovenden (1955), symmetric periodic orbits (SPOs) in the n-body problem must fullfill two mirror configurations. In the CR3BP the possible mirror configurations are: (a) perpendicular intersection of the ' x ; z ' plane; (b) perpendicular intersection of the x axis. The SPOs may be classified according to the combinations of mirror configurations: (a)-(a); (b)-(b); (a)(b) (Zagouras & Markellos 1977). Planar SPOs intersect the x -axis perpendicularly at times T GLYPH<157> 2 and T (Hadjedemetriou 2006).</text>
<unordered_list>
<list_item><location><page_2><loc_7><loc_56><loc_47><loc_63></location>Morais & Namouni (2013) showed that planar periodic orbits associated with the retrograde 1/1 resonance are symmetric and have multiplicity 2, i.e. they intersect the surface of section y = 0 , perpendicularly ( GLYPH<219> x = 0 ) and with the same sign for GLYPH<219> y , at t = T GLYPH<157> 2 and t = T . We use the following standard algorithm to find planar POs:</list_item>
<list_item><location><page_2><loc_7><loc_51><loc_47><loc_55></location>(i) A guess initial condition ' x 0 ; 0 ; 0 ; GLYPH<219> y 0 ' is followed until the 2nd intersection with the surface of section occurs within j y j < GLYPH<15> 0 at time T .</list_item>
<list_item><location><page_2><loc_7><loc_46><loc_47><loc_51></location>(ii) If j GLYPH<219> x j < GLYPH<15> then the initial conditions correspond to a PO with period T . A new search is started varying x 0 or GLYPH<219> y 0 . Otherwise, a differential correction is applied to x 0 or GLYPH<219> y 0 and the procedure is repeated.</list_item>
</unordered_list>
<text><location><page_2><loc_7><loc_28><loc_47><loc_45></location>The variational equations have the general solution ¯ GLYPH<24> ' t ' = GLYPH<1> ' t ' ¯ GLYPH<24> ' 0 ' where ¯ GLYPH<24> ' t ' is the phase-space displacement vector at time t and GLYPH<1> ' t ' is the 6-dimensional state transition matrix. The eigenvalues of GLYPH<1> ' T ' indicate if the PO with period T is linearly stable or unstable. Due to the symplectic property of the equations of motion these eigenvalues appear as reciprocal pairs GLYPH<21> i GLYPH<21> GLYPH<3> i = 1 ( i = 1 ; 2 ; 3 ) and they may be real or complex conjugate. The periodicity condition implies that one pair of eigenvalues is GLYPH<21> 1 = GLYPH<21> GLYPH<3> 1 = 1 . The PO is linearly stable if the remaining pairs of eigenvalues are complex conjugate on the unit circle and unstable otherwise with instability increasing with the largest eigenvalue's absolute value (Hadjedemetriou 2006).</text>
<text><location><page_2><loc_7><loc_20><loc_47><loc_28></location>For planar motion, the state transition matrix, GLYPH<1> 2 ' t ' , is 4 dimensional. The horizontal stability index is k 2 = GLYPH<21> 2 + GLYPH<21> GLYPH<3> 2 where GLYPH<21> 2 and GLYPH<21> GLYPH<3> 2 are the non trivial eigenvalues of GLYPH<1> 2 ' T ' . Stable planar periodic orbits have GLYPH<0> 2 < k 2 < 2 . Change of stability occurs when j k 2 j = 2 (or GLYPH<21> 2 = GLYPH<21> GLYPH<3> 2 = GLYPH<6> 1 ) which is often associated with bifurcation of a new family of POs.</text>
<text><location><page_2><loc_7><loc_7><loc_47><loc_20></location>The variational equations for displacements out of the plane and in the plane of motion are decoupled when z = GLYPH<219> z = 0 (H'enon 1973) hence the evolution of ¯ GLYPH<24> z ' t ' = ' dz ; d GLYPH<219> z ' is described by a 2 dimensional state transition matrix, GLYPH<1> 3 ' t ' . The vertical stability index is k 3 = GLYPH<21> 3 + GLYPH<21> GLYPH<3> 3 where GLYPH<21> 3 and GLYPH<21> GLYPH<3> 3 are the eigenvalues of GLYPH<1> 3 ' T ' . Motion around stable 2D periodic orbits is maintained when there are small deviations out of the plane only if the vertical stability index GLYPH<0> 2 < k 3 < 2 . When this stability index reaches the critical value 2 (vertical critical orbit or vco ) a bifurcation into a new family</text>
<text><location><page_2><loc_51><loc_85><loc_91><loc_88></location>of 3D periodic orbits with the same multiplicity may occur (H'enon 1973; Ichtiaroglou & Michalodimitrakis 1980).</text>
<unordered_list>
<list_item><location><page_2><loc_51><loc_83><loc_91><loc_85></location>To find 3D SPOs that bifurcate from vcos we follow a procedure similar to Zagouras & Markellos (1977):</list_item>
<list_item><location><page_2><loc_51><loc_77><loc_91><loc_82></location>(i) Initial conditions corresponding to mirror configurations: (a): ' x 0 ; 0 ; z 0 ; 0 ; GLYPH<219> y 0 ; 0 ' ; or (b): ' x 0 ; 0 ; 0 0 ; 0 ; GLYPH<219> y 0 ; GLYPH<219> z 0 ' ; are followed until the the 2nd intersection with the surface of section occurs within j y j < GLYPH<15> 0 at time T .</list_item>
<list_item><location><page_2><loc_51><loc_70><loc_91><loc_77></location>(ii) If j GLYPH<219> x j < GLYPH<15> and j GLYPH<219> z j < GLYPH<15> (a) or j GLYPH<219> x j < GLYPH<15> and j z j < GLYPH<15> (b) then the initial conditions correspond to a PO with period T and a new search is started. Otherwise, a differential correction is applied to 2 components of the initial condition vector and the procedure is repeated.</list_item>
</unordered_list>
<text><location><page_2><loc_51><loc_55><loc_91><loc_70></location>For 3D motion, the pairs of eigenvalues GLYPH<21> i , GLYPH<21> GLYPH<3> i ( i = 2 ; 3 ) of GLYPH<1> ' T ' are the roots of the characteristic polynomial GLYPH<21> 4 + GLYPH<11> GLYPH<21> 3 + GLYPH<12> GLYPH<21> 2 + GLYPH<11> GLYPH<21> + 1 with GLYPH<11> = 2 GLYPH<0> Tr ' GLYPH<1> ' T '' , 2 GLYPH<12> = GLYPH<11> 2 + 2 GLYPH<0> Tr ' GLYPH<1> ' T ' 2 ' (Bray & Goudas 1967). They are complex conjugate on the unit circle (linear stability) if GLYPH<14> = ' GLYPH<11> 2 GLYPH<0> 4 ' GLYPH<12> GLYPH<0> 2 '' > 0 and j p j = j' GLYPH<11> + p GLYPH<14> 'jGLYPH<157> 2 < 2 , j q j = j' GLYPH<11> GLYPH<0> p GLYPH<14> 'jGLYPH<157> 2 < 2 (Zagouras & Markellos 1977). Change of stability with possible bifurcation into a new family of POs occurs when pairs of eigenvalues coallesce on the real axis while complex instability occurs when they collaesce on the unit circle and then move away from it (Heggie 1985).</text>
<text><location><page_2><loc_51><loc_43><loc_91><loc_55></location>The numerical integration of the CR3BP equations of motion and associated variational equations were done using the Bulirsch-Stoer algorithm with per step accuracy 10 GLYPH<0> 13 . Distance and time were scaled by Jupiter's semimajor axis and orbital period. The computations for an individual test particle were stopped when the distance to a massive body was within its physical radius (taken equal to the Sun's and Jupiter's radius). They were also stopped when the heliocentric distance exceeded 3 times Jupiter's semi-major axis.</text>
<text><location><page_2><loc_51><loc_26><loc_91><loc_43></location>The thresholds for deciding if an orbit is periodic were chosen so that the differential correction procedure converges for each specific type of PO. We used GLYPH<15> 0 = 10 GLYPH<0> 11 and GLYPH<15> = 10 GLYPH<0> 10 to find planar and 3D POs. In general, lower (sometimes unfeasible) values are necessary to follow unstable families, as expected due to the exponential divergence of solutions close to unstable POs. To monitor the POs computations we checked that j GLYPH<1> ' T 'j = 1 with at least 11 significant digits. Stability and bifurcation points were further checked by explicitly computing the eigenvalues of GLYPH<1> ' T ' . Unstable critical motion (near the transition to stability) was confirmed by computing the chaos indicator MEGNO (Cincotta & Giordano 2006).</text>
<section_header_level_1><location><page_2><loc_51><loc_21><loc_89><loc_23></location>3 THE 2D FAMILIES AND BIFURCATIONS INTO 3D</section_header_level_1>
<text><location><page_2><loc_51><loc_7><loc_91><loc_20></location>Morais & Namouni (2013) showed that the relevant resonant argument for planar retrograde coorbitals in the CR3BP is GLYPH<30> GLYPH<3> = GLYPH<21> GLYPH<0> GLYPH<21> p GLYPH<0> 2 ! where GLYPH<21> and ! are the test particle's mean longitude and argument of pericenter, GLYPH<21> p is the mean longitude of the planet. There are 3 types of retrograde coorbitals: mode1 which corresponds to libration of GLYPH<30> GLYPH<3> around 0 and occurs at a wide range of eccentricities; modes 2 and 3 which correspond to libration of GLYPH<30> GLYPH<3> around 180 GLYPH<14> and occur, respectively, at small eccentricity (mode 3) and large eccentricity (mode 2). These modes are retrieved in a 2D</text>
<text><location><page_3><loc_7><loc_85><loc_47><loc_88></location>model for retrograde coorbital resonance based on the averaged Hamiltonian (Huang et al. 2018).</text>
<section_header_level_1><location><page_3><loc_7><loc_82><loc_21><loc_83></location>3.1 Planar SPOs</section_header_level_1>
<text><location><page_3><loc_7><loc_78><loc_47><loc_81></location>We show how the families of SPOs associated with mode 1 (Fig. 1) and modes 2 and 3 (Fig. 2) evolve with the Jacobi constant, C .</text>
<text><location><page_3><loc_7><loc_72><loc_47><loc_77></location>Mode 1 resonant POs are horizontally stable when C > GLYPH<0> 1 : 2256 and vertically stable when C > GLYPH<0> 1 : 0507 ( a < 1 : 0380 , e > 0 : 1125 ). The family ends by collision with the star when e GLYPH<25> 1 .</text>
<text><location><page_3><loc_7><loc_62><loc_47><loc_72></location>Inner nearly circular non-resonant POs are stable if C > GLYPH<0> 0 : 8429 ( a < 0 : 9265 ). At C = GLYPH<0> 0 : 9562 ( a = 0 : 9801 ) there is a bifurcation into a stable inner resonant PO. This family is stable up to C = GLYPH<0> 0 : 8643 ( a = 0 : 9864 , e = 0 : 3208 : inner mode 3) and stable again from C = GLYPH<0> 0 : 3349 ( a = 0 : 9976 , e = 0 : 7411 : mode 2). Therefore, mode 2 and inner mode 3 resonant POs form a single family which is always vertically stable. The family ends by collision with the star when e GLYPH<25> 1 .</text>
<text><location><page_3><loc_7><loc_54><loc_47><loc_62></location>Outer nearly circular non-resonant POs are vertically unstable when GLYPH<0> 1 : 0395 > C > GLYPH<0> 1 : 1499 ( 1 : 0218 < a < 1 : 0804 ). At C = GLYPH<0> 1 : 0387 ( a = 1 : 0215 ) there is a bifurcation into a pair of stable (outer mode 3) and unstable (nearly circular) POs. Outer mode 3 resonant POs are stable up to C = GLYPH<0> 0 : 9553 ( a = 1 : 0151 , e = 0 : 2636 ).</text>
<section_header_level_1><location><page_3><loc_7><loc_51><loc_27><loc_52></location>3.2 Bifurcations into 3D</section_header_level_1>
<text><location><page_3><loc_7><loc_44><loc_47><loc_50></location>Morais & Namouni (2013, 2016) showed that in the 3D coorbital problem the relevant resonant angles are GLYPH<30> = GLYPH<21> GLYPH<0> GLYPH<21> p and GLYPH<30> GLYPH<3> = GLYPH<21> GLYPH<0> GLYPH<21> p GLYPH<0> 2 ! . The 3D retrograde coorbital modes correspond to: GLYPH<30> librating around 180 GLYPH<14> (mode 4); GLYPH<30> GLYPH<3> librating around 0 (mode 1) or 180 GLYPH<14> (modes 2 and 3).</text>
<text><location><page_3><loc_7><loc_34><loc_47><loc_43></location>Planar retrograde modes 1 and 2 are horizontally and vertically stable when C > GLYPH<0> 1 : 0507 and C > GLYPH<0> 0 : 3349 , respectively, hence the associated POs are surrounded by quasiperiodic orbits in the 3D problem. In particular, quasiperiodic mode 1 and mode 2 orbits may extend down to inclinations i = 90 GLYPH<14> and i = 120 GLYPH<14> , respectively (Morais & Namouni 2016).</text>
<text><location><page_3><loc_7><loc_26><loc_47><loc_34></location>The vertical critical orbits ( vcos ) occur: on mode 1 family at C = GLYPH<0> 1 : 0507 (Fig. 1); on the outer circular family at C = GLYPH<0> 1 : 1499 and C = GLYPH<0> 1 : 0395 (Fig. 2). At the vcos there are bifurcations into new families of 3D periodic orbits which we show in Fig. 3. The mode 2, mode 3 outer and inner families have no vcos as the vertical stability index, k 3 < 2 .</text>
<text><location><page_3><loc_7><loc_7><loc_47><loc_26></location>At C = GLYPH<0> 1 : 1499 there is a bifurcation of a nearly circular 2D outer PO into a stable 3D resonant PO on configuration (b) which corresponds to mode 4 (libration center GLYPH<30> = 180 GLYPH<14> ). This family reaches critical stability at C = GLYPH<0> 1 : 0321 when i GLYPH<25> 173 GLYPH<14> . Fig. 4 shows a PO at this point on the family. Initially, GLYPH<30> = 180 GLYPH<14> with ! circulating fast similarly to mode 4 stable branch (Fig. 4: left). The peaks in a and e occur twice per period, at the encounters with the planet. After t = 8 GLYPH<2> 10 3 , chaotic diffusion is obvious (MEGNO increases linearly with time) and from t = 1 : 8 GLYPH<2> 10 4 there are transitions between libration around ! = 90 GLYPH<14> ; 270 GLYPH<14> at small e when GLYPH<30> = 180 GLYPH<14> and circulation around the Kozai centers ! = 0 ; 180 GLYPH<14> with eccentricity oscillations up to e = 0 : 14 (Fig. 4: right) when GLYPH<30> circulates. The Kozai cicles around ! = 0 ; 180 GLYPH<14> raise the eccentricity and shift the libration center to GLYPH<30> GLYPH<3> = 0 .</text>
<text><location><page_3><loc_51><loc_74><loc_91><loc_88></location>At C = GLYPH<0> 1 : 0507 there is a bifurcation of planar mode 1 into a 3D PO on configuration (a). This family is unstable but nearly critical. It has a v-shape with lower / upper branches corresponding the intersection with the surface of section at the apocentric / pericentric encounters (Fig. 6: left). There is a bifurcation at C = GLYPH<0> 1 : 0321 coinciding with the bifurcation on the mode 4 family. Fig. 5 shows a PO at this bifurcation point. Initially, GLYPH<30> = 180 GLYPH<14> with ! circulating fast (Fig. 5: left). After t = 7 GLYPH<2> 10 3 , chaotic diffusion is again obvious (Fig. 5: right) with the same qualitative behaviour observed in Fig. 4.</text>
<text><location><page_3><loc_51><loc_57><loc_91><loc_73></location>A shift of t = 0 : 25 between the time series in Figs. 4,5 (left) causes overlap of orbital elements. Further inspection shows that they correspond to the same PO of symmetry type (a)-(b) at different intersections with the surface of section (Fig. 6: right). Hence, the 3D families bifurcating from the vcos at C = GLYPH<0> 1 : 1499 and C = GLYPH<0> 1 : 0507 join at C = GLYPH<0> 1 : 0321 generating a single unstable circular family which could be continued to i GLYPH<25> 8 GLYPH<14> and a = 0 : 999 . Since instability on this family increases sharply with decreasing inclination the differential correction scheme stops converging preventing further continuation. We suspect that termination occurs at the Lagrangian point L3 when i = 0 which is further supported by the shape of the last computed PO in the rotating frame.</text>
<text><location><page_3><loc_51><loc_41><loc_91><loc_56></location>At C = GLYPH<0> 1 : 0395 , near the end of the stable branch of the 2D nearly circular outer family, there is a bifurcation into an unstable 3D PO on configuration (a). This 3D family corresponds to an unstable fixed point of the coorbital resonance Hamiltonian ( GLYPH<30> = 0 and GLYPH<30> ? = 180 GLYPH<14> ). It could be continued to i GLYPH<25> 88 GLYPH<14> , e GLYPH<25> 0 : 80 and a = 1 : 001 at which point the family is approaching critical stability. However, the long integration of the initial conditions that approximate the last computed periodic orbit (PO) shows that the eccentricity increases sharply towards unity around t = 5 GLYPH<2> 10 3 thus leading to collision with the star. The proximity of the collision singularity prevents further continuation of the family.</text>
<section_header_level_1><location><page_3><loc_51><loc_35><loc_87><loc_37></location>4 COORBITAL CAPTURE IN 2D AND 3D CASES</section_header_level_1>
<text><location><page_3><loc_51><loc_25><loc_91><loc_34></location>In the planar problem, outer orbits slowly approaching the planet follow the nearly circular non-resonant family which bifurcates into a resonant SPO at C = GLYPH<0> 1 : 0387 when a = 1 : 0215 (outer mode 3). Capture into outer mode 3 occurs with probability 1 in agreement with Namouni & Morais (2018a) but the family becomes unstable at C = GLYPH<0> 0 : 9553 when a = 1 : 0151 and e = 0 : 2636 .</text>
<text><location><page_3><loc_51><loc_7><loc_91><loc_25></location>However, Fig. 3 shows that the behaviour in the (real) 3D problem at infinitesimal deviations from the plane is radically different. The nearly circular non-resonant family is vertically unstable between GLYPH<0> 1 : 0395 > C > GLYPH<0> 1 : 1499 . The vco at C = GLYPH<0> 1 : 1499 ( a = 1 : 0804 ) bifurcates into a resonant mode 4 stable 3D family. Hence, outer circular orbits slowly approaching the planet still follow initially the non-resonant family which bifurcates into the 3D mode 4 family. The inclination then decreases and at i GLYPH<25> 173 GLYPH<14> mode 4 family becomes unstable. At this point, mode 4 family connects with the critical 3D family which bifurcates from the vco at C = GLYPH<0> 1 : 0507 ( a = 1 : 0380 , e = 0 : 1125 ) on the stable branch of the mode 1 planar family. Chaotic transition between the Kozai centers located at ! = 90 GLYPH<14> ; 270 GLYPH<14> and the separatrices</text>
<figure>
<location><page_4><loc_13><loc_46><loc_90><loc_88></location>
<caption>Figure 1. Family of SPOs corresponding to retrograde mode 1 with respect to the Jacobi constant, C . Top panel: 2D (black) and 3D (gray) stability indexes. Low panel: semi-major axis a and eccentricity e (horizontally stable (red) and unstable (orange)).</caption>
</figure>
<text><location><page_4><loc_7><loc_32><loc_47><loc_39></location>around ! = 0 ; 180 GLYPH<14> are accompanied by eccentricity oscillations up to 0.14 and a shift of the libration center towards GLYPH<30> GLYPH<3> = 0 . Exit of this chaotic region due to a slow decrease in semi-major axis allows permanent capture into a quasiperiodic mode 1 orbit, in agreement with the simulations by Morais & Namouni (2016); Namouni & Morais (2018a).</text>
<text><location><page_4><loc_7><loc_20><loc_47><loc_31></location>Inner circular orbits slowly approaching the planet near the plane become horizontally unstable at C = GLYPH<0> 0 : 8429 when a = 0 : 9265 . Therefore, the inner mode 3 resonant family cannot be reached (Morais & Namouni 2016). The resonant family starts at C = GLYPH<0> 0 : 9562 ( a = 0 : 9801 : inner mode 3), becomes horizontally unstable at C = GLYPH<0> 0 : 8643 ( a = 0 : 9864 , e = 0 : 3208 ) and is stable again when C > GLYPH<0> 0 : 3349 ( a > 0 : 9976 , e > 0 : 7411 : mode 2). There are no vcos on the inner families.</text>
<section_header_level_1><location><page_4><loc_7><loc_16><loc_22><loc_17></location>5 CONCLUSION</section_header_level_1>
<text><location><page_4><loc_7><loc_7><loc_47><loc_15></location>We showed how the families of periodic orbits for the planar retrograde coorbital problem and their bifurcations into 3D explain the radical differences seen in our capture simulations, namely why 2D orbits are captured into mode 3 while 3D orbits are captured into mode 1 (Morais & Namouni 2016). In the planar problem, outer circular orbits</text>
<text><location><page_4><loc_51><loc_20><loc_91><loc_39></location>slowly drifting towards the planet follow the non-resonant family which bifurcates into a resonant mode 3 family at C = GLYPH<0> 1 : 0387 ( a = 1 : 0215 ). This family becomes unstable at C = GLYPH<0> 0 : 9553 ( a = 1 : 0151 , e = 0 : 2636 ) . However, in the (real) 3D problem, mode 3 orbits are never reached. The nearly circular 2D family becomes vertically unstable ( vco ) at C = GLYPH<0> 1 : 1499 when a = 1 : 0804 where a bifurcation into a 3D resonant family corresponding to mode 4 ( GLYPH<30> = 180 GLYPH<14> ) occurs. This family becomes unstable when i GLYPH<25> 173 GLYPH<14> as it connects with a 3D family bifurcating from the vco on mode 1 ( GLYPH<30> GLYPH<3> = 0 ). Chaotic transitions between the libration centers GLYPH<30> = 180 GLYPH<14> and GLYPH<30> GLYPH<3> = 0 are associated with motion in the vicinity of Kozai separatrices. As the semi-major axis decreases due to dissipation there is capture on a mode 1 inclined quasiperiodic orbit, similar to that of Ka'epaoka'awela .</text>
<text><location><page_4><loc_51><loc_7><loc_91><loc_18></location>Our results explain why mode 1 is the likely end state for objects on retrograde outer circular orbits slowly drifting towards the planet. If the planet migrated inwards, retrograde inner nearly circular orbits become horizontally unstable at C = GLYPH<0> 0 : 8427 when a = 0 : 9265 hence capture into inner mode 3 is not possible. However, eccentric inner retrograde orbits could be captured directly into mode 2 if the relative semi-major axis evolved in discrete steps. Similarly, eccentric outer orbits may be captured directly into mode 1. This</text>
<figure>
<location><page_5><loc_13><loc_46><loc_90><loc_88></location>
<caption>Figure 2. Family of SPOs corresponding to the outer and inner circular families and retrograde modes 2 and 3 with respect to the Jacobi constant, C . Top panel: 2D (black) and 3D (gray) stability indexes. Low panel: semi-major axis a and eccentricity e (horizontally stable (red) and unstable (orange)).</caption>
</figure>
<text><location><page_5><loc_7><loc_30><loc_47><loc_38></location>could occur e.g. if the semi-major axis evolves stochastically due to planetary close approaches (Carusi et al. 1990). In the early solar system the latter mechanism (outer eccentric capture) is more likely to occur than the former (inner eccentric capture) and this could explain how Ka'epaoka'awela arrived at the current location.</text>
<text><location><page_5><loc_7><loc_19><loc_47><loc_29></location>Analytical and numerical results obtained in 2D models are often thought to be valid when the motion is almost coplanar. Here, we showed that such extrapolation is not valid for the retrograde coorbital problem. This is due to the vertical instability of the nearly circular 2D family of POs. A similar mechanism has been reported for the 2/1 and 3/1 prograde resonances in the planetary (non-restricted) 3body problem (Voyatzis et al. 2014).</text>
<text><location><page_5><loc_7><loc_7><loc_47><loc_18></location>Searches for 3D POs typically show that families end by collision with one of the massive bodies or otherwise exist over the entire inclination range 0 GLYPH<20> i GLYPH<20> 180 GLYPH<14> (Kotoulas & Voyatzis 2005; Antoniadou & Libert 2019). Here, we computed the families that originate at the vcos of the planar retrograde coorbital problem. The family corresponding to an unstable fixed point of the coorbital Hamiltonian ( GLYPH<30> = 0 , GLYPH<30> ? = 180 GLYPH<14> ) could be continued until it becomes a nearly polar orbit in the vicinity of an instability that leads to col-</text>
<text><location><page_5><loc_51><loc_32><loc_91><loc_38></location>lision with the star. The unstable doubly-symmetric circular family corresponding to the libration center GLYPH<30> = 180 GLYPH<14> seems to end at the colinear Lagrangian point L3. However, as this family becomes increasingly unstable as the inclination approaches zero its exact termination could not be ascertained.</text>
<section_header_level_1><location><page_5><loc_51><loc_27><loc_72><loc_28></location>ACKNOWLEDGEMENTS</section_header_level_1>
<text><location><page_5><loc_51><loc_21><loc_91><loc_26></location>Bibliography access was provided by CAPES-Brazil. M.H.M. Morais research had financial support from S˜ao Paulo Research Foundation (FAPESP/2018/08620-1) and CNPQ-Brazil (PQ2/304037/2018-4) .</text>
<section_header_level_1><location><page_5><loc_51><loc_17><loc_63><loc_18></location>REFERENCES</section_header_level_1>
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<text><location><page_5><loc_51><loc_7><loc_91><loc_14></location>Carusi A., Valsecchi G. B., Greenberg R., 1990, Celestial Mechan- ics and Dynamical Astronomy, 49, 111 Cincotta P. M., Giordano M., 2006 Hadjedemetriou J. D., 2006, p. 43 Heggie D. C., 1985, Celestial Mechanics, 35, 357 H´enon M., 1973, Celestial Mechanics, 8, 269</text>
<figure>
<location><page_6><loc_9><loc_26><loc_90><loc_88></location>
<caption>Figure 3. Families of 3D periodic orbits bifurcating from the vcos at C = GLYPH<0> 1 : 1499 and C = GLYPH<0> 1 : 0395 on the outer nearly circular family, C = GLYPH<0> 1 : 0507 on the planar mode 1 family. These vcos are labeled 1, 2 and 3, respectively. Top panel: inclination i . Mid panel: eccentricity e . Low panel: semi-major axis a . The families are coloured blue (gray) when stable (unstable). The 2D stable families from which the 3D families bifurcate are coloured red.</caption>
</figure>
<section_header_level_1><location><page_6><loc_7><loc_15><loc_35><loc_16></location>Henon M., 1974, Celestial Mechanics, 10, 375</section_header_level_1>
<text><location><page_6><loc_7><loc_14><loc_39><loc_15></location>Huang Y., Li M., Li J., Gong S., 2018, AJ, 155, 262</text>
<text><location><page_6><loc_7><loc_12><loc_43><loc_13></location>Ichtiaroglou S., Michalodimitrakis M., 1980, A&A, 81, 30</text>
<text><location><page_6><loc_7><loc_11><loc_38><loc_12></location>Kotoulas T. A., Voyatzis G., 2005, A&A, 441, 807</text>
<text><location><page_6><loc_7><loc_8><loc_47><loc_10></location>Morais M. H. M., Namouni F., 2013, Celestial Mechanics and Dynamical Astronomy, 117, 405</text>
<text><location><page_6><loc_7><loc_7><loc_47><loc_8></location>Morais M. H. M., Namouni F., 2016, Celestial Mechanics and</text>
<section_header_level_1><location><page_6><loc_54><loc_15><loc_73><loc_16></location>Dynamical Astronomy, 125, 91</section_header_level_1>
<text><location><page_6><loc_51><loc_11><loc_91><loc_15></location>Morais M. H. M., Namouni F., 2017, Nature, 543, 635 Namouni F., Morais M. H. M., 2018a, J. Comp. App. Math., 37, 65</text>
<text><location><page_6><loc_51><loc_10><loc_88><loc_11></location>Namouni F., Morais M. H. M., 2018b, MNRAS, 477, L117</text>
<text><location><page_6><loc_51><loc_8><loc_84><loc_9></location>Roy A. E., Ovenden M. W., 1955, MNRAS, 115, 296</text>
<text><location><page_6><loc_51><loc_7><loc_91><loc_8></location>Voyatzis G., Antoniadou K. I., Tsiganis K., 2014, Celestial Me-</text>
<figure>
<location><page_7><loc_9><loc_58><loc_89><loc_88></location>
<caption>Figure 4. Evolution of PO at the critical point ( C = GLYPH<0> 1 : 0321 ) on the 3D mode 4 family. From top to bottom panels: semi-major axis / MEGNO; eccentricity, cosine of inclination, argument of pericenter ! (black) and longitude of ascending node GLYPH<10> (gray); resonant angles GLYPH<30> (black) and GLYPH<30> GLYPH<3> (gray).</caption>
</figure>
<figure>
<location><page_7><loc_9><loc_21><loc_89><loc_51></location>
<caption>Figure 5. Evolution of PO at the critical point ( C = GLYPH<0> 1 : 0321 ) on the 3D family which bifurcates from the vco on the planar mode 1 family. Same panels as Fig. 4.</caption>
</figure>
<section_header_level_1><location><page_7><loc_9><loc_15><loc_37><loc_16></location>chanics and Dynamical Astronomy, 119, 221</section_header_level_1>
<text><location><page_7><loc_7><loc_12><loc_44><loc_14></location>Wiegert P., Connors M., Veillet C., 2017, Nature, 543, 687 Zagouras C., Markellos V. V., 1977, A&A, 59, 79</text>
<text><location><page_7><loc_7><loc_8><loc_47><loc_10></location>This paper has been typeset from a T E X/L A T E X file prepared by the author.</text>
<figure>
<location><page_8><loc_14><loc_68><loc_43><loc_88></location>
<caption>Figure 6. POs in rotating frame: (left) mode 1 vco has encounters and intersections with the surface of section at pericenter or apocenter; (right) 3D bifurcation at C = GLYPH<0> 1 : 0321 (mirror configurations (a) and (b) are shifted by t = 0 : 25 ).</caption>
</figure>
<text><location><page_8><loc_63><loc_74><loc_64><loc_74></location>1</text>
<text><location><page_8><loc_67><loc_73><loc_68><loc_74></location>-1</text>
<text><location><page_8><loc_73><loc_73><loc_74><loc_74></location>(a)</text>
<text><location><page_8><loc_50><loc_79><loc_50><loc_80></location>z</text>
<text><location><page_8><loc_52><loc_83><loc_54><loc_83></location>0.2</text>
<text><location><page_8><loc_52><loc_81><loc_54><loc_82></location>0.1</text>
<text><location><page_8><loc_53><loc_79><loc_54><loc_80></location>0</text>
<text><location><page_8><loc_54><loc_79><loc_55><loc_79></location>-1</text>
<text><location><page_8><loc_52><loc_78><loc_54><loc_78></location>-0.1</text>
<text><location><page_8><loc_52><loc_76><loc_54><loc_77></location>-0.2</text>
<text><location><page_8><loc_55><loc_77><loc_57><loc_78></location>-0.5</text>
<text><location><page_8><loc_56><loc_76><loc_56><loc_76></location>x</text>
<text><location><page_8><loc_58><loc_76><loc_59><loc_77></location>0</text>
<text><location><page_8><loc_60><loc_75><loc_61><loc_76></location>0.5</text>
<text><location><page_8><loc_64><loc_81><loc_65><loc_82></location>(b)</text>
<text><location><page_8><loc_70><loc_74><loc_72><loc_75></location>-0.5</text>
<text><location><page_8><loc_74><loc_75><loc_75><loc_75></location>0</text>
<text><location><page_8><loc_76><loc_74><loc_77><loc_74></location>y</text>
<text><location><page_8><loc_78><loc_75><loc_79><loc_76></location>0.5</text>
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2021PhRvD.104f4048M | https://arxiv.org/pdf/2012.11209.pdf | <document>
<section_header_level_1><location><page_1><loc_29><loc_92><loc_72><loc_93></location>Spherically symmetric black holes in metric gravity</section_header_level_1>
<text><location><page_1><loc_35><loc_89><loc_65><loc_90></location>Sebastian Murk 1, 2, ∗ and Daniel R. Terno 1, †</text>
<text><location><page_1><loc_19><loc_86><loc_82><loc_88></location>1 Department of Physics and Astronomy, Macquarie University, Sydney, New South Wales 2109, Australia 2 Sydney Quantum Academy, Sydney, New South Wales 2006, Australia</text>
<text><location><page_1><loc_18><loc_76><loc_83><loc_85></location>The existence of black holes is one of the key predictions of general relativity (GR) and therefore a basic consistency test for modified theories of gravity. In the case of spherical symmetry in GR the existence of an apparent horizon and its regularity is consistent with only two distinct classes of physical black holes. Here we derive constraints that any self-consistent modified theory of gravity must satisfy to be compatible with their existence. We analyze their properties and illustrate characteristic features using the Starobinsky model. Both of the GR solutions can be regarded as zeroth-order terms in perturbative solutions of this model. We also show how to construct nonperturbative solutions without a well-defined GR limit.</text>
<section_header_level_1><location><page_1><loc_22><loc_72><loc_36><loc_73></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_53><loc_49><loc_70></location>General Relativity (GR), one of the two pillars of modern physics, is the simplest member of the family of metric theories of gravity. It is the only theory that is derived from an invariant that is linear in second derivatives of the metric. However, interpretations of astrophysical and cosmological data as well as theoretical considerations [1, 2] encourage us to consider GR as the low-energy limit of some effective theory of quantum gravity [3-5]. Extended theories of gravity, such as metric theories that involve higher-order invariants of the Riemann tensor, metric-affine theories, and theories with torsion, include additional terms in the action functional. Here we focus on metric modified theories of gravity (MTG).</text>
<text><location><page_1><loc_9><loc_28><loc_49><loc_52></location>A prerequisite for the validity of any proposed generalization of GR is that it must be compatible with current astrophysical and cosmological data. In particular, a viable candidate theory must provide a model to describe the observed astrophysical black hole candidates. Popular contemporary models describe them as ultra-compact objects with or without a horizon [6]. While there is a considerable diversity of opinions on what exactly constitutes a black hole, the presence of a trapped region - a domain of spacetime from which nothing can escape - is its most commonly accepted characteristic [7]. A trapped spacetime region that is externally bounded by an apparent horizon is referred to as physical black hole (PBH) [8]. A PBH may contain other features of black hole solutions of classical GR, such as an event horizon or singularity, or it may be a singularity-free regular black hole. To be of physical relevance, the apparent horizon must form in finite time according to a distant observer [9].</text>
<text><location><page_1><loc_9><loc_16><loc_49><loc_27></location>It is commonly accepted that curvature invariants, such as the Ricci and Kretschmann scalar, are finite at the apparent horizon. When expressed mathematically, the requirements of regularity and finite formation time provide the basis for a self-consistent analysis of black holes. In spherical symmetry (to which we restrict our considerations here), this allows for a comprehensive classification of the near-horizon geometries. There are only two classes of solutions labeled</text>
<text><location><page_1><loc_52><loc_63><loc_92><loc_73></location>by k = 0 and k = 1 , where the value of k reflects the scaling behavior of particular functions of the components of the energy-momentum tensor (EMT) near the apparent horizon. The properties of the near-horizon geometry lead to the identification of a unique scenario for black hole formation [9, 10] that involves both types of PBH solutions. We summarize its main results in Sec. III.</text>
<text><location><page_1><loc_52><loc_50><loc_92><loc_63></location>Understanding the true nature of the observed ultracompact objects requires detailed knowledge of the black hole models, their alternatives, as well as the observational signatures of both classes of solutions in GR and extended theories of gravity [6, 11]. Vacuum black hole solutions exist in a variety of MTG [1, 2, 12]. On the other hand, these theories are also used to construct models of horizonless ultra-compact objects. A generic property among some of them is the absence of horizon formation in the final stage of the collapse [13].</text>
<text><location><page_1><loc_52><loc_27><loc_92><loc_50></location>Even the simplest MTG require perturbative treatment due to the mathematical complexity inherent to the higher-order nature of the equations [2, 14, 15]. We briefly review the relevant formalism and its relationship to the self-consistent approach in Sec. II. In Sec. IV, we derive a set of conditions necessary for the existence of a PBH in an arbitrary metric MTG. The solutions are presented as expansions in the coordinate distance from the apparent horizon and do not require a GR solution as the zeroth-order perturbative solution of a MTG. Using the Starobinsky model [2, 16] (Sec. V) we demonstrate the application of the general results, illustrating the well-known features of matching solutions of systems of partial differential equations of different orders [2, 14, 15]: we find that the two classes of GR solutions can be regarded as zeroth-order perturbative solutions of this MTG, and identify a MTG solution without a well-defined GR limit.</text>
<section_header_level_1><location><page_1><loc_55><loc_21><loc_89><loc_23></location>II. MODIFIED GRAVITY FIELD EQUATIONS IN SPHERICAL SYMMETRY</section_header_level_1>
<section_header_level_1><location><page_1><loc_63><loc_18><loc_80><loc_19></location>A. General considerations</section_header_level_1>
<text><location><page_1><loc_52><loc_8><loc_92><loc_15></location>We work in the framework of semiclassical gravity, use classical notions (e.g. metric, horizons, trajectories), and describe dynamics via the modified Einstein equations. We do not make any assumptions about the underlying reason for modifications of the bulk part of the gravitational Lagrangian</text>
<text><location><page_2><loc_9><loc_89><loc_49><loc_93></location>density, but organize it according to powers of derivatives of the metric as commonly done in effective field theories [3, 4, 17], i.e.</text>
<formula><location><page_2><loc_9><loc_80><loc_49><loc_88></location>L g √ -g = M 2 P 16 π ( R + λ F ( g µν , R µνρσ ) ) = M 2 P 16 π R + a 1 R µν R µν + a 2 R 2 + a 3 R µνρσ R µνρσ + (1)</formula>
<text><location><page_2><loc_9><loc_67><loc_49><loc_79></location>where M P is the Planck mass that we set to one in what follows, the cosmological constant was omitted, and the coefficients a 1 , a 2 , a 3 are dimensionless. The dimensionless parameter λ is used to organize the perturbative analysis and set to one at the end of the calculations. Many popular models belong to the class of f ( R ) theories, where L g √ -g = f ( R ) . The prototypical example is the Starobinsky model with F = ςR 2 , ς = 16 πa 2 /M 2 P .</text>
<text><location><page_2><loc_10><loc_66><loc_37><loc_67></location>Varying the gravitational action results in</text>
<formula><location><page_2><loc_21><loc_63><loc_49><loc_64></location>G µν + λ E µν = 8 πT µν , (2)</formula>
<text><location><page_2><loc_9><loc_55><loc_49><loc_61></location>where G µν is the Einstein tensor, the terms E µν result from the variation of F ( g µν , R µνρσ ) , and T µν ≡ 〈 ˆ T µν 〉 ω denotes the expectation value of the renormalized EMT. We do not make any specific assumptions about the state ω .</text>
<text><location><page_2><loc_9><loc_47><loc_49><loc_55></location>In fact, apart from imposing spherical symmetry, we assume only that (i) an apparent horizon is formed in finite time of a distant observer; (ii) it is regular, i.e. the scalars T := T µ µ = R/ 8 π + O ( λ ) and T := T µν T µν = R µν R µν / 64 π 2 + O ( λ 2 ) are finite at the horizon.</text>
<text><location><page_2><loc_9><loc_45><loc_49><loc_47></location>A general spherically symmetric metric in Schwarzschild coordinates is given by</text>
<formula><location><page_2><loc_11><loc_41><loc_49><loc_43></location>ds 2 = -e 2 h ( t,r ) f ( t, r ) dt 2 + f ( t, r ) -1 dr 2 + r 2 d Ω , (3)</formula>
<text><location><page_2><loc_9><loc_37><loc_49><loc_40></location>where r denotes the areal radius. The Misner-Sharp mass [18, 19] C ( t, r ) is invariantly defined via</text>
<formula><location><page_2><loc_20><loc_34><loc_49><loc_36></location>1 -C ( t, r ) /r := ∂ µ r∂ µ r, (4)</formula>
<text><location><page_2><loc_9><loc_23><loc_49><loc_33></location>and thus the function f ( t, r ) = 1 -C ( t, r ) /r is invariant under general coordinate transformations. For a Schwarzschild black hole C = 2 M . We use the definition of Eq. (4) for consistency with the description of solutions in higherdimensional versions of GR. The apparent horizon is located at the Schwarzschild radius r g ( t ) that is the largest root of f ( t, r ) = 0 [19].</text>
<text><location><page_2><loc_10><loc_22><loc_47><loc_23></location>The Misner-Sharp mass of a PBH can be represented as</text>
<formula><location><page_2><loc_20><loc_18><loc_49><loc_20></location>C = r g ( t ) + W ( t, r -r g ) , (5)</formula>
<text><location><page_2><loc_9><loc_16><loc_43><loc_17></location>where the definition of the apparent horizon implies</text>
<formula><location><page_2><loc_19><loc_13><loc_49><loc_14></location>W ( t, 0) = 0 , W ( t, x ) < x, (6)</formula>
<text><location><page_2><loc_9><loc_9><loc_49><loc_11></location>and x := r -r g is the coordinate distance from the apparent horizon.</text>
<text><location><page_2><loc_50><loc_81><loc_52><loc_83></location>· · ·</text>
<text><location><page_2><loc_52><loc_82><loc_53><loc_83></location>,</text>
<text><location><page_2><loc_53><loc_92><loc_84><loc_93></location>The modified Einstein equations take the form</text>
<formula><location><page_2><loc_58><loc_89><loc_92><loc_91></location>fr -2 e 2 h ∂ r C + λ E tt = 8 πT tt , (7)</formula>
<formula><location><page_2><loc_58><loc_87><loc_92><loc_89></location>r -2 ∂ t C + λ E r t = 8 πT r t , (8)</formula>
<formula><location><page_2><loc_58><loc_84><loc_92><loc_87></location>2 f 2 r -1 ∂ r h -fr -2 ∂ r C + λ E rr = 8 πT rr . (9)</formula>
<text><location><page_2><loc_53><loc_83><loc_62><loc_84></location>The notation</text>
<formula><location><page_2><loc_55><loc_80><loc_92><loc_81></location>τ t := e -2 h T tt , τ r t := e -h T r t , τ r := T rr (10)</formula>
<text><location><page_2><loc_52><loc_77><loc_89><loc_79></location>is useful in dealing with equations in both GR and MTG.</text>
<text><location><page_2><loc_52><loc_70><loc_92><loc_77></location>Regularity of the apparent horizon is expressed as a set of conditions on the potentially divergent parts of the scalars T and T . In spherical symmetry T θ θ ≡ T φ φ and we assume that it is finite as in GR [9]. The constraints can therefore be represented mathematically as</text>
<formula><location><page_2><loc_54><loc_67><loc_92><loc_69></location>T = ( τ r -τ t ) /f → g 1 ( t ) f k 1 , (11)</formula>
<text><location><page_2><loc_52><loc_58><loc_92><loc_64></location>for some g 1 , 2 ( t ) and k 1 , 2 /greaterorequalslant 0 . There are a priori infinitely many solutions that satisfy these constraints. After reviewing the special case of GR and presenting the two admissible solutions we discuss this behavior in Sec. IV.</text>
<formula><location><page_2><loc_54><loc_63><loc_92><loc_67></location>T = ( ( τ t ) 2 -2( τ r t ) 2 +( τ r ) 2 ) /f 2 → g 2 ( t ) f k 2 , (12)</formula>
<text><location><page_2><loc_52><loc_51><loc_92><loc_58></location>Many useful results can be obtained by means of comparison of various quantities written in Schwarzschild coordinates ( t, r ) with their counterpart expressions written using the ingoing v or outgoing u null coordinate and the same areal radius r . Using ( v, r ) coordinates,</text>
<formula><location><page_2><loc_63><loc_48><loc_92><loc_50></location>dt = e -h ( e h + dv -f -1 dr ) , (13)</formula>
<text><location><page_2><loc_52><loc_45><loc_92><loc_47></location>is particularly fruitful. EMT components in ( v, r ) and ( t, r ) coordinates are related via</text>
<formula><location><page_2><loc_60><loc_42><loc_92><loc_43></location>θ v := e -2 h + Θ vv = τ t , (14)</formula>
<formula><location><page_2><loc_60><loc_39><loc_92><loc_41></location>θ vr := e -h + Θ vr = ( τ r t -τ t ) /f, (15)</formula>
<formula><location><page_2><loc_60><loc_37><loc_92><loc_39></location>θ r := Θ rr = ( τ r + τ t -2 τ r t ) /f 2 , (16)</formula>
<text><location><page_2><loc_52><loc_35><loc_89><loc_36></location>where Θ µν labels EMT components in ( v, r ) coordinates.</text>
<section_header_level_1><location><page_2><loc_63><loc_31><loc_81><loc_32></location>B. Perturbative expansion</section_header_level_1>
<text><location><page_2><loc_52><loc_26><loc_92><loc_29></location>From a formal perspective the pure GR case can be described as a system of field equations [20]</text>
<formula><location><page_2><loc_68><loc_24><loc_92><loc_25></location>E (¯ g , ¯ T ) = 0 , (17)</formula>
<text><location><page_2><loc_52><loc_18><loc_92><loc_23></location>where the EMT ¯ T and metric ¯ gnear the apparent horizon are described in a spherically symmetric setting in Sec. III. It is then usually assumed that any solution</text>
<formula><location><page_2><loc_67><loc_15><loc_92><loc_17></location>E λ ( g λ , T λ ) = 0 (18)</formula>
<text><location><page_2><loc_52><loc_9><loc_92><loc_14></location>of the MTG belongs to a one-parameter family of analytic solutions [14, 15]. The EMT T λ depends on λ through the metric g λ , and potentially also through effective corrections resulting from perturbative corrections to the modified field</text>
<text><location><page_3><loc_9><loc_83><loc_49><loc_93></location>equations Eqs. (7)-(9). The self-consistent approach is based on the assumption of at least continuity of the curvature invariants, but uses the Schwarzschild coordinate system where the metric is discontinuous [9, 10]. Imposing the requirement of regularity then allows to identify the valid black hole solutions, whose analytic properties become apparent once they are written in their 'natural' coordinate system [21].</text>
<text><location><page_3><loc_9><loc_72><loc_49><loc_83></location>The field equations are supplemented by a set of initial and boundary conditions or constraints. Higher-order terms in the action lead to higher-order equations. Even f ( R ) theories already result in systems with fourth-order metric derivatives. However, it is worth pointing out that the unperturbed solution may not satisfy the boundary conditions since its corresponding equations do not involve the higher-order derivatives [15, 22].</text>
<text><location><page_3><loc_9><loc_57><loc_49><loc_71></location>For our purposes it suffices to restrict all considerations to first-order perturbation theory. In any given theory higherorder contributions can be successfully evaluated. There are methods to produce a consistent hierarchy of the higher-order terms and deal with additional degrees of freedom that result from the presence of derivatives of order higher than two. Nevertheless, including terms of order O ( λ 2 ) and higher may not be justified without detailed knowledge of the relative importance of all possible terms in the effective Lagrangian and the cut-off scale that is used to derive it.</text>
<text><location><page_3><loc_9><loc_50><loc_49><loc_57></location>Spherical symmetry prescribes the form of the metric for all values of λ . We assume that there is a solution of Eq. (2) with the two metric functions C λ and h λ . To avoid spurious divergences we use the physical value of r g ( t ) that corresponds to the perturbed metric g λ , C λ ( r g , t ) = r g. We set</text>
<formula><location><page_3><loc_17><loc_47><loc_49><loc_49></location>C λ =: r g ( t ) + ¯ W ( t, r ) + λ Σ( t, r ) , (19)</formula>
<formula><location><page_3><loc_17><loc_45><loc_49><loc_47></location>h λ =: ¯ h ( t, r ) + λ Ω( t, r ) , (20)</formula>
<text><location><page_3><loc_9><loc_41><loc_49><loc_44></location>and define ¯ C := r g + ¯ W . Similarly, the EMT T λ ≡ T is decomposed as</text>
<formula><location><page_3><loc_22><loc_38><loc_49><loc_40></location>T µν =: ¯ T µν + λ ˜ T, (21)</formula>
<text><location><page_3><loc_9><loc_33><loc_49><loc_37></location>where ¯ T is extracted from E ( ¯ g [ r g , ¯ W, ¯ h ] , ¯ T ) = 0 . The perturbative terms must satisfy the boundary conditions</text>
<formula><location><page_3><loc_20><loc_30><loc_49><loc_31></location>Σ( t, 0) = 0 , (22)</formula>
<formula><location><page_3><loc_19><loc_27><loc_49><loc_30></location>lim r → r g Ω( t, r ) / ¯ h ( t, r ) = O (1) , (23)</formula>
<text><location><page_3><loc_9><loc_19><loc_49><loc_26></location>where the first condition follows from the definition of the Schwarzschild radius, and the perturbation can be treated as small only if the divergence of Ω is not stronger than that of ¯ h . Substituting C λ and h λ into Eq. (2) and keeping only the first-order terms in λ results in</text>
<formula><location><page_3><loc_13><loc_14><loc_49><loc_18></location>¯ G µν + λ ˜ G µν + λ ¯ E µν = 8 π ( ¯ T µν + λ ˜ T µν ) , (24)</formula>
<text><location><page_3><loc_9><loc_9><loc_49><loc_15></location>where ¯ G µν ≡ G µν [ r g , ¯ W, ¯ h ] , ˜ G µν is the first-order term in the Taylor expansion in λ where each monomial involves either Σ or Ω , and ¯ E µν ≡ E µν [ r g , ¯ W, ¯ h ] , i.e. the modified gravity terms are functions of the unperturbed solutions.</text>
<text><location><page_3><loc_52><loc_90><loc_92><loc_93></location>The explicit form of the equations can be obtained as follows. First note that</text>
<formula><location><page_3><loc_61><loc_87><loc_92><loc_90></location>e 2 h = e 2 ¯ h (1 + 2 λ Ω) + O ( λ 2 ) . (25)</formula>
<text><location><page_3><loc_52><loc_84><loc_92><loc_87></location>We introduce the splitting τ = ¯ τ + λ ˜ τ such that, for instance, the EMT terms of the tt equation can be written as</text>
<formula><location><page_3><loc_54><loc_81><loc_92><loc_83></location>¯ T tt + λ ˜ T tt = e 2 ¯ h (1 + 2 λ Ω)(¯ τ t + λ ˜ τ t ) (26)</formula>
<formula><location><page_3><loc_62><loc_77><loc_92><loc_81></location>= e 2 ¯ h ( ¯ τ t + λ (2Ω¯ τ t + ˜ τ t ) ) + O ( λ 2 ) , (27)</formula>
<text><location><page_3><loc_52><loc_72><loc_92><loc_78></location>with T r t and T rr expanded analogously. The regularity conditions Eqs. (11) and (12) imply that ˜ τ terms should either have the same behavior as their ¯ τ counterparts when r → r g, or go to zero faster.</text>
<text><location><page_3><loc_53><loc_71><loc_85><loc_72></location>Consequently, the schematic of Eq. (24) implies</text>
<formula><location><page_3><loc_54><loc_65><loc_92><loc_70></location>¯ G tt = e 2 ¯ h r 3 ( r -¯ C ) ∂ r ¯ C, (28)</formula>
<text><location><page_3><loc_52><loc_61><loc_77><loc_62></location>and thus the explicit form of Eq. (7) is</text>
<formula><location><page_3><loc_54><loc_61><loc_92><loc_66></location>˜ G tt = e 2 ¯ h r 3 [ -Σ ∂ r ¯ C + ( r -¯ C ) ( 2Ω ∂ r ¯ C + ∂ r Σ )] , (29)</formula>
<formula><location><page_3><loc_52><loc_55><loc_92><loc_60></location>-Σ ∂ r ¯ C + ( r -¯ C ) ∂ r Σ+ r 3 e -2 ¯ h ¯ E tt = 8 πr 3 ˜ τ t . (30) Similarly, Eqs. (8) and (9) can be written explicitly as</formula>
<formula><location><page_3><loc_57><loc_52><loc_92><loc_54></location>∂ t Σ+ r 2 ¯ E r t = 8 πr 2 e ¯ h (Ω¯ τ r t + ˜ τ r t ) , (31)</formula>
<formula><location><page_3><loc_57><loc_51><loc_79><loc_52></location>Σ ∂ r ¯ C ( r ¯ C )(4Σ ∂ r ¯ h + ∂ r Σ)</formula>
<formula><location><page_3><loc_62><loc_48><loc_92><loc_52></location>--+2( r -¯ C ) 2 ∂ r Ω+ r 3 ¯ E rr = 8 πr 3 ˜ τ r . (32)</formula>
<section_header_level_1><location><page_3><loc_57><loc_45><loc_87><loc_46></location>III. SELF-CONSISTENT SOLUTIONS IN GR</section_header_level_1>
<text><location><page_3><loc_52><loc_33><loc_92><loc_42></location>Here we give a brief summary of the relevant properties of the self-consistent solutions in GR [9, 10, 21]. In accord with the previous section (and in anticipation of the notation we use in Sec. IV), we label functions of pure classical GR (i.e. λ = 0 ) with a bar, e.g. the metric functions ¯ C and ¯ h . The Einstein field equations for ¯ G tt , ¯ G r t , and ¯ G rr are expressed in terms of the metric functions ¯ C and ¯ h as follows:</text>
<formula><location><page_3><loc_63><loc_30><loc_92><loc_32></location>∂ r ¯ C = 8 πr 2 ¯ τ t / ¯ f, (33)</formula>
<formula><location><page_3><loc_63><loc_28><loc_92><loc_30></location>∂ t ¯ C = 8 πr 2 e ¯ h ¯ τ r t , (34)</formula>
<formula><location><page_3><loc_63><loc_26><loc_92><loc_27></location>∂ r ¯ h = 4 πr (¯ τ t + ¯ τ r ) / ¯ f 2 . (35)</formula>
<text><location><page_3><loc_52><loc_20><loc_92><loc_24></location>Only two distinct classes of dynamic solutions are possible [21]. With respect to the regularity conditions of Eqs. (11) and (12), they correspond to the values k = 0 and k = 1 .</text>
<section_header_level_1><location><page_3><loc_63><loc_16><loc_81><loc_17></location>A. k = 0 class of solutions</section_header_level_1>
<text><location><page_3><loc_52><loc_12><loc_92><loc_14></location>In the k = 0 class of solutions, the limiting form of the reduced EMT components is given by</text>
<formula><location><page_3><loc_53><loc_8><loc_92><loc_11></location>¯ τ t →-¯ Υ 2 ( t ) , ¯ τ r →-¯ Υ 2 ( t ) , ¯ τ r t →± ¯ Υ 2 ( t ) , (36)</formula>
<text><location><page_4><loc_9><loc_90><loc_49><loc_93></location>for some function ¯ Υ( t ) . The leading terms of the metric functions are</text>
<formula><location><page_4><loc_18><loc_87><loc_49><loc_90></location>¯ C = r g -4 πr 3 / 2 g ¯ Υ √ x + O ( x ) , (37)</formula>
<formula><location><page_4><loc_19><loc_84><loc_49><loc_87></location>¯ h = -1 2 ln x ¯ ξ + O ( √ x ) , (38)</formula>
<text><location><page_4><loc_9><loc_77><loc_49><loc_83></location>where ¯ ξ ( t ) is determined by the asymptotic properties of the solution. Higher-order terms depend on the higher-order terms in the EMT expansion and will be discussed in Sec. IV. Consistency of the Einstein equations implies</text>
<formula><location><page_4><loc_23><loc_72><loc_49><loc_76></location>r ' g = ± 4 ¯ Υ √ πr g ¯ ξ. (39)</formula>
<text><location><page_4><loc_9><loc_67><loc_49><loc_73></location>The null energy condition requires T µν l µ l ν /greaterorequalslant 0 for all null vectors l µ [20, 23]. It is violated by radial vectors l ˆ a = (1 , ∓ 1 , 0 , 0) for both the evaporating and accreting solutions, respectively.</text>
<text><location><page_4><loc_9><loc_54><loc_49><loc_67></location>The accreting solution r ' g ( t ) > 0 leads to a firewall: energy density, pressure and flux experienced by an infalling observer diverge at the apparent horizon [21]. The resulting averaged negative energy density in the reference frame of a geodesic observer violates a particular quantum energy inequality [23, 24]. Unless we accept that semiclassical physics breaks down already at the horizon scale, this contradiction implies that a PBH cannot grow after its formation [21]. Hence we consider only evaporating r ' g ( t ) < 0 PBHs in what follows.</text>
<text><location><page_4><loc_9><loc_48><loc_49><loc_54></location>Matching our results with the standard semiclassical results on black hole evaporation (and accepting that the metric is sufficiently close to the ingoing Vaidya metric with decreasing mass, see [25] for details) results in</text>
<formula><location><page_4><loc_26><loc_44><loc_49><loc_48></location>¯ ξ ∼ α r g , (40)</formula>
<text><location><page_4><loc_9><loc_37><loc_49><loc_43></location>where the black hole evaporates according to r ' g ( t ) = -α/r 2 g [20, 26]. Outside of the apparent horizon the geometry differs from the Schwarzschild metric at least on the scale r -r g =: x ∼ ¯ ξ .</text>
<section_header_level_1><location><page_4><loc_23><loc_34><loc_35><loc_35></location>B. k = 1 solution</section_header_level_1>
<text><location><page_4><loc_9><loc_21><loc_49><loc_31></location>In the second class of solutions k = 1 and the limiting form of the EMT expansion is given by functions ¯ τ a ∝ ¯ f . Again, accretion leads to a firewall and thus we will consider only evaporating solutions. It has been shown that dynamic solutions are consistent only in a single case [10], where in the Schwarzschild frame the energy density ρ ( r g ) = ¯ E and pressure p ( r g ) = ¯ P at the apparent horizon are given by</text>
<formula><location><page_4><loc_21><loc_18><loc_49><loc_20></location>¯ E = -¯ P = 1 / (8 πr 2 g ) . (41)</formula>
<text><location><page_4><loc_9><loc_15><loc_49><loc_18></location>Since this is their maximal possible value this k = 1 solution is referred to as extreme [10]. The k = 1 metric functions are</text>
<formula><location><page_4><loc_20><loc_11><loc_49><loc_14></location>¯ C = r -c 32 x 3 / 2 + O ( x 2 ) , (42)</formula>
<formula><location><page_4><loc_20><loc_9><loc_49><loc_12></location>¯ h = -3 2 ln x ¯ ξ + O ( √ x ) . (43)</formula>
<text><location><page_4><loc_52><loc_92><loc_85><loc_93></location>Consistency of the Einstein equations then implies</text>
<formula><location><page_4><loc_66><loc_88><loc_92><loc_91></location>r ' g = -c 32 ¯ ξ 3 / 2 /r g . (44)</formula>
<text><location><page_4><loc_52><loc_85><loc_92><loc_88></location>For future reference we note here that for the k = 1 solution the Ricci scalar is given by</text>
<formula><location><page_4><loc_66><loc_81><loc_92><loc_84></location>¯ R = 2 /r 2 g + O ( x ) . (45)</formula>
<text><location><page_4><loc_52><loc_64><loc_92><loc_81></location>Evaporating black holes are conveniently represented in ( v, r ) coordinates, and the limiting form of the k = 0 solution as r → r g is a Vaidya metric with decreasing Misner-Sharp mass C + ( v ) ' < 0 [25]. Using ( v, r ) coordinates to describe geometry at the formation of the first marginally trapped surface reveals how the two classes of solutions are connected (see Ref. [10] for details): at its formation, a PBH is described by a k = 1 solution with ¯ E = -¯ P = 1 / (8 πr 2 g ) . It immediately switches to the k = 0 solution. However, the abrupt transition from f 1 to f 0 behavior does not lead to discontinuities in the curvature scalars or other physical quantities that could potentially be measured by a local or quasilocal observer.</text>
<section_header_level_1><location><page_4><loc_56><loc_60><loc_88><loc_61></location>IV. SELF-CONSISTENT SOLUTIONS IN MTG</section_header_level_1>
<text><location><page_4><loc_52><loc_39><loc_92><loc_58></location>To describe perturbative PBH solutions in MTG the equations must satisfy the same consistency relations as their GR counterparts. Taking the GR solutions as the zeroth-order approximation, we express the functions describing the MTG metric g λ = ¯ g + λ ˜ g and thus represent the modified Einstein equations as series in integer and half-integer powers of x := r -r g. Their order-by-order solution results in formal expressions for Σ( t, r ) and Ω( t, r ) . However, we also obtain a number of consistency conditions that must be satisfied identically in order for a given theory to admit formation of a PBH. The GR solutions with k ∈ { 0 , 1 } are sufficiently different to merit a separate treatment provided in Subsec. IV A and IV B, respectively.</text>
<text><location><page_4><loc_52><loc_30><loc_92><loc_39></location>In both instances, power expansions in various expression have to match up to allow for self-consistent solutions of the modified Einstein equations. Moreover, the relations between the EMT components that are given by Eqs. (14)-(16) must hold separately for both the unperturbed terms and the perturbations.</text>
<text><location><page_4><loc_52><loc_9><loc_92><loc_30></location>For a given MTG (that is defined by the set of parameters { a 1 , a 2 , a 3 , · · · } in Eq. (1)) these constraints may conceivably lead to several outcomes: first, it is possible that some of the terms in the Lagrangian Eq. (1) contribute terms to ¯ E µν such that their expansions around x = 0 lead to terms that diverge stronger than any other terms in Eqs. (30)-(32). If only one higher-order curvature term is responsible for such behavior, then such a theory cannot produce perturbative PBH solutions, and only nonperturbative solutions may be possible or the corresponding coefficient a i ≡ 0 . If the divergences originate from several terms, they can either cancel if a particular relationship exists between their coefficients a i , a i ' , . . . , or not. In the former case the existence of perturbative PBH solutions imposes a constraint, not on the form of the available terms, but on the relationships between their coefficients.</text>
<text><location><page_5><loc_9><loc_76><loc_49><loc_93></location>It is also possible that, as it happens in the Starobinsky model (Sec. V), divergences of the terms ¯ E µν match the divergences of the GR terms. The constraints can then be satisfied (i) identically (providing us with no additional information); (ii) only for a particular combination of the coefficients a i , thereby constraining the possible classes of MTG; (iii) only in the presence of particular higher-order terms, irrespective of the coefficients, and only for certain unperturbed solutions. In the last scenario, where only certain GR solutions are consistent with a small perturbation, this should be interpreted as an argument against the presence of that particular term in the Lagrangian of Eq. (1).</text>
<text><location><page_5><loc_9><loc_66><loc_49><loc_76></location>There is a priori no reason why ¯ g µν /greatermuch λ ˜ g µν should hold in some boundary layer around r g [15, 22]. If this condition is not satisfied, then the classification scheme of the GR solutions and a mandatory violation of the null energy condition are not necessarily true. We discuss some of the properties of the solutions without a GR limit and derive the necessary conditions for their existence in Sec. IV C.</text>
<text><location><page_5><loc_9><loc_56><loc_49><loc_66></location>Throughout this section we use the letter j ∈ Z 1 2 to label integer and half-integer coefficients and powers of x in series expansions and /lscript to refer to generic coefficients. Since we give explicit expressions only for the first few terms in each expression, we write c 12 instead of c 1 / 2 , h 12 instead of h 1 / 2 , and similarly for higher orders and coefficients of the EMT expansion.</text>
<section_header_level_1><location><page_5><loc_18><loc_51><loc_40><loc_52></location>A. Black holes of the k = 0 type</section_header_level_1>
<text><location><page_5><loc_9><loc_44><loc_49><loc_49></location>For the k = 0 class of solutions the leading terms in the metric functions of classical GR are given as series in powers of x := r -r g as</text>
<formula><location><page_5><loc_17><loc_39><loc_38><loc_44></location>¯ C = r g -c 12 √ x + ∞ ∑ 1 /lessorequalslant j ∈ Z 1 2 c j x j</formula>
<formula><location><page_5><loc_17><loc_29><loc_49><loc_41></location>= r g -c 12 √ x + c 1 x + O ( x 3 / 2 ) , (46) ¯ h = -1 2 ln x ¯ ξ + ∞ ∑ 1 2 /lessorequalslant j ∈ Z 1 2 h j x j = -1 2 ln x ¯ ξ + h 12 √ x + O ( x ) , (47)</formula>
<text><location><page_5><loc_9><loc_27><loc_13><loc_28></location>where</text>
<formula><location><page_5><loc_12><loc_22><loc_49><loc_26></location>c 12 = 4 √ πr 3 / 2 g ¯ Υ , c 1 = 1 3 + 4 √ πr 3 / 2 g (¯ τ t ) 12 3 ¯ Υ , (48)</formula>
<formula><location><page_5><loc_12><loc_18><loc_49><loc_23></location>h 12 = 2 ¯ Υ+ √ πr 3 / 2 g (3(¯ τ r ) 12 -(¯ τ t ) 12 ) 6 √ πr 3 / 2 g ¯ Υ 2 , (49)</formula>
<text><location><page_5><loc_9><loc_14><loc_49><loc_17></location>and higher-order coefficients of the metric functions are related to higher-order terms in the EMT expansion</text>
<formula><location><page_5><loc_19><loc_8><loc_49><loc_13></location>¯ τ a = -¯ Υ 2 + ∞ ∑ 1 2 /lessorequalslant j ∈ Z 1 2 (¯ τ a ) j x j , (50)</formula>
<text><location><page_5><loc_52><loc_90><loc_92><loc_93></location>where a ∈ { t , r t , r } ≡ { tt , r t , rr } . We omit the explicit specification j ∈ Z 1 2 from the summation range in what follows.</text>
<text><location><page_5><loc_52><loc_85><loc_92><loc_90></location>Regularity of the metric at the apparent horizon and consistency of the Einstein equations establish algebraic and differential relations between various coefficients. In particular, using Eqs. (14)-(16) and Eq. (34), we find</text>
<formula><location><page_5><loc_63><loc_82><loc_92><loc_84></location>(¯ τ t ) 12 +(¯ τ r ) 12 = 2(¯ τ r t ) 12 , (51)</formula>
<text><location><page_5><loc_52><loc_80><loc_54><loc_81></location>and</text>
<formula><location><page_5><loc_66><loc_75><loc_92><loc_78></location>r ' g = -4 ¯ Υ √ πr g ¯ ξ. (52)</formula>
<text><location><page_5><loc_52><loc_71><loc_92><loc_75></location>The expansion of e 2 h that is given by Eq. (25) is obtained as follows: separating the logarithmically divergent part of h ( t, x ) from the rest, Eq. (23) allows to write</text>
<formula><location><page_5><loc_64><loc_66><loc_92><loc_70></location>e 2 h = ¯ ξ + λ ˜ ξ x e 2¯ χ +2 λω (53)</formula>
<text><location><page_5><loc_52><loc_61><loc_92><loc_65></location>for some ˜ ξ ( t ) , where ¯ χ = ∑ j h j x j and ω = ∑ j ω j x j are convergent functions. First-order expansion in λ then leads to Eq. (25) with</text>
<formula><location><page_5><loc_67><loc_56><loc_92><loc_60></location>Ω = ˜ ξ 2 ¯ ξ + ω. (54)</formula>
<text><location><page_5><loc_52><loc_53><loc_92><loc_55></location>Therefore, the first-order corrections of Eqs. (19)-(20) to the metric functions of Eqs. (46)-(47) are given by the series</text>
<formula><location><page_5><loc_55><loc_47><loc_92><loc_52></location>Σ = ∞ ∑ j /greaterorequalslant 1 2 σ j x j = σ 12 x 1 / 2 + σ 1 x + O ( x 3 / 2 ) , (55)</formula>
<formula><location><page_5><loc_55><loc_42><loc_92><loc_47></location>Ω = ˜ ξ 2 ¯ ξ + ∞ ∑ j /greaterorequalslant 1 2 ω j x j = ˜ ξ 2 ¯ ξ + ω 12 x 1 / 2 + O ( x ) . (56)</formula>
<text><location><page_5><loc_52><loc_29><loc_92><loc_41></location>These two functions can be expressed in terms of the unperturbed solution and corrections ˜ τ a to the EMT from the series expansion of Eqs. (30)-(32). These equations contain various divergent expressions. For example, the term Σ ∂ r ¯ C as well as all other terms apart from e -2 ¯ h ¯ E tt in Eq. (30) are finite when x → 0 . Then Eq. (47) implies that the series expansion of ¯ E tt starts with a term that is proportional to 1 /x . Performing the same analysis for the two remaining Einstein equations Eqs. (31)-(32) yields the decompositions</text>
<formula><location><page_5><loc_59><loc_23><loc_92><loc_28></location>¯ E tt = æ ¯ 1 x + æ 12 √ x +æ 0 x 0 + ∞ ∑ j /greaterorequalslant 1 2 æ j x j , (57)</formula>
<formula><location><page_5><loc_59><loc_18><loc_92><loc_23></location>¯ E r t = œ 12 √ x +œ 0 x 0 + ∞ ∑ j /greaterorequalslant 1 2 œ j x j , (58)</formula>
<formula><location><page_5><loc_59><loc_13><loc_92><loc_18></location>¯ E rr = ø 0 + ∞ ∑ j /greaterorequalslant 1 2 ø j x j , (59)</formula>
<text><location><page_5><loc_52><loc_9><loc_92><loc_13></location>of the modified gravity terms that should hold for any F ( g µν , R µνρσ ) , where indices of coefficients of negative exponents of x are labeled by a bar.</text>
<text><location><page_6><loc_9><loc_90><loc_49><loc_93></location>From the requirement that the Ricci scalar R [ g λ ] be finite at the horizon, we obtain the condition</text>
<formula><location><page_6><loc_18><loc_86><loc_49><loc_90></location>σ 12 | R = ˜ ξ 2 ¯ ξ c 12 = 2 √ πr 3 / 2 g ˜ ξ ¯ Υ ¯ ξ . (60)</formula>
<text><location><page_6><loc_9><loc_81><loc_49><loc_85></location>We use additional subscripts (e.g. ' | R ' in the expression above) to indicate what equation was used to derive the explicit expression.</text>
<text><location><page_6><loc_9><loc_68><loc_49><loc_81></location>The perturbative contributions Eqs. (55)-(56) to the metric functions Eqs. (46)-(47) are obtained order-by-order from the series solutions of Eqs. (30)-(32). Expressions for every expansion coefficient can be obtained separately from each equation. Matching of the expressions then allows to identify the coefficients æ /lscript , œ /lscript , ø /lscript of the modified gravity terms Eqs. (57)-(59). Expressions for σ 12 for instance are obtained from the lowest-order coefficients of Eqs. (30)-(32). As a result, we obtain three independent constraints</text>
<formula><location><page_6><loc_18><loc_65><loc_49><loc_67></location>σ 12 | R = σ 12 | tt = σ 12 | tr = σ 12 | rr . (61)</formula>
<text><location><page_6><loc_9><loc_63><loc_45><loc_64></location>They are simultaneously satisfied (see Appendix A 1) if</text>
<formula><location><page_6><loc_10><loc_56><loc_49><loc_62></location>æ ¯ 1 = -8 π ˜ ξ ¯ Υ 2 , œ 12 = -8 π ˜ ξ ¯ Υ 2 √ ¯ ξ , ø 0 = -8 π ˜ ξ ¯ Υ 2 ¯ ξ . (62)</formula>
<text><location><page_6><loc_9><loc_53><loc_49><loc_57></location>These three equations not only identify the function ˜ ξ ( t ) in terms of unperturbed quantities, but also establish the two relations</text>
<formula><location><page_6><loc_22><loc_48><loc_49><loc_52></location>æ ¯ 1 = √ ¯ ξ œ 12 = ¯ ξ ø 0 (63)</formula>
<text><location><page_6><loc_9><loc_43><loc_49><loc_49></location>between the leading expansion coefficients of the MTG terms. Similarly, the next-highest order coefficients of Eqs. (30)-(32) allow to obtain expressions for σ 1 , see Appendix A 1. Comparison of</text>
<formula><location><page_6><loc_21><loc_41><loc_49><loc_42></location>σ 1 | tr ( ω 12 ) = σ 1 | rr ( ω 12 ) (64)</formula>
<text><location><page_6><loc_9><loc_35><loc_49><loc_40></location>gives an expression for ω 12 . Substitution into Eq. (64) and subsequent comparison of σ 1 | tr = σ 1 | rr with σ 1 | tt gives a relation between the next-highest order coefficients, namely</text>
<formula><location><page_6><loc_21><loc_31><loc_49><loc_34></location>æ 12 = 2 √ ¯ ξ œ 0 -¯ ξ ø 12 . (65)</formula>
<text><location><page_6><loc_9><loc_19><loc_49><loc_31></location>The modified gravity terms ¯ E µν must adhere to the expansion structures in Eqs. (57)-(59), and the relations Eqs. (63) and (65) between their coefficients must be satisfied identically. Otherwise the MTG solutions do not exist. For terms in the metric functions of order O ( x 3 / 2 ) and higher the three equations Eqs. (30)-(32) for the tt , tr , and rr component contain three additional independent variables (¯ τ a ) j /greaterorequalslant 3 / 2 and will therefore not lead to any additional constraints.</text>
<text><location><page_6><loc_9><loc_15><loc_49><loc_19></location>It is worth pointing out that the analogs of Eqs. (63) and (65) are also satisfied by the coefficients of the corresponding metric tensor and Ricci tensor components themselves, i.e.</text>
<formula><location><page_6><loc_12><loc_10><loc_49><loc_14></location>(¯ g tt ) ¯ 1 = √ ¯ ξ (¯ g r t ) 12 = ¯ ξ (¯ g rr ) 0 = 0 , (66)</formula>
<formula><location><page_6><loc_11><loc_7><loc_49><loc_11></location>( ¯ R tt ) ¯ 1 = √ ¯ ξ ( ¯ R r t ) 12 = ¯ ξ ( ¯ R rr ) 0 = -c 2 12 ¯ ξ/ (2 r 3 g ) , (67)</formula>
<text><location><page_6><loc_52><loc_92><loc_54><loc_93></location>and</text>
<text><location><page_6><loc_63><loc_19><loc_63><loc_20></location>2</text>
<formula><location><page_6><loc_55><loc_87><loc_92><loc_91></location>(¯ g tt ) 12 = 2 √ ¯ ξ (¯ g r t ) 0 -¯ ξ (¯ g rr ) 12 = -c 12 ¯ ξ/r g , (68)</formula>
<text><location><page_6><loc_52><loc_78><loc_92><loc_85></location>where Eq. (66) is satisfied trivially and Eq. (68) simplifies to (¯ g tt ) 12 = -¯ ξ (¯ g rr ) 12 due to the diagonal form of the metric tensor ¯ g r t = 0 (see Eq. (3)). Explicit expressions for the coefficients of the Ricci tensor components in Eq. (69) are provided in Appendix B, see Eqs. (B1)-(B3).</text>
<formula><location><page_6><loc_55><loc_84><loc_92><loc_88></location>( ¯ R tt ) 12 = 2 √ ¯ ξ ( ¯ R r t ) 0 -¯ ξ ( ¯ R rr ) 12 , (69)</formula>
<section_header_level_1><location><page_6><loc_61><loc_74><loc_83><loc_75></location>B. Black holes of the k = 1 type</section_header_level_1>
<text><location><page_6><loc_52><loc_68><loc_92><loc_72></location>The EMT expansion for the k = 1 solution is given in terms of x := r -r g by</text>
<formula><location><page_6><loc_56><loc_64><loc_92><loc_68></location>τ t = ¯ τ t + λ ˜ τ t = ¯ f ( ¯ E + λ ˜ E ) + ∑ j /greaterorequalslant 2 e j x j , (70)</formula>
<formula><location><page_6><loc_56><loc_56><loc_92><loc_60></location>τ r = ¯ τ r + λ ˜ τ r = ¯ f ( ¯ P + λ ˜ P ) + ∑ j /greaterorequalslant 2 p j x j , (72)</formula>
<formula><location><page_6><loc_55><loc_60><loc_92><loc_64></location>τ r t = ¯ τ r t + λ ˜ τ r t = ¯ f ( ¯ Φ+ λ ˜ Φ ) + ∑ j /greaterorequalslant 2 φ j x j , (71)</formula>
<text><location><page_6><loc_52><loc_47><loc_92><loc_56></location>where ¯ E = -¯ P = 1 / (8 πr 2 g ) and ¯ Φ = 0 . To improve readability and clarify the connection to physical quantities (energy, pressure, flux) we set (¯ τ t ) j =: ¯ e j , (¯ τ r ) j =: ¯ p j , and (¯ τ r t ) j =: ¯ φ j , and analogously for the perturbative coefficients (˜ τ a ) j . Additional relations between the coefficients are obtained from Eqs. (14)-(15), i.e.</text>
<formula><location><page_6><loc_60><loc_44><loc_92><loc_46></location>˜ E + ˜ P = 2 ˜ Φ , (73)</formula>
<formula><location><page_6><loc_60><loc_42><loc_92><loc_44></location>¯ e 2 = ¯ p 2 = ¯ φ 2 , ¯ e 52 + ¯ p 52 = 2 ¯ φ 52 , (74)</formula>
<formula><location><page_6><loc_60><loc_40><loc_92><loc_42></location>˜ e 2 + ˜ p 2 = 2 ˜ φ 2 , ˜ e 52 + ˜ p 52 = 2 ˜ φ 52 , (75)</formula>
<text><location><page_6><loc_52><loc_34><loc_92><loc_39></location>for the two next-highest orders j = 2 , 5 2 . Recall that (cf. Eqs. (42)-(43)) the leading terms in the metric functions of classical GR are given as series in powers of x := r -r g as</text>
<formula><location><page_6><loc_61><loc_31><loc_92><loc_34></location>¯ C = r g + x -c 32 x 3 / 2 + O ( x 2 ) , (76)</formula>
<formula><location><page_6><loc_61><loc_28><loc_92><loc_32></location>¯ h = -3 2 ln x ¯ ξ + h 12 √ x + O ( x ) , (77)</formula>
<text><location><page_6><loc_52><loc_26><loc_63><loc_27></location>with coefficients</text>
<formula><location><page_6><loc_56><loc_21><loc_92><loc_25></location>c 32 = 4 r 3 / 2 g √ -π ¯ e 2 / 3 , (78) 3 4 3¯ e 2 /π</formula>
<text><location><page_6><loc_57><loc_20><loc_58><loc_21></location>12</text>
<text><location><page_6><loc_60><loc_20><loc_63><loc_21></location>14¯</text>
<text><location><page_6><loc_62><loc_20><loc_63><loc_21></location>e</text>
<text><location><page_6><loc_64><loc_19><loc_65><loc_23></location>(</text>
<text><location><page_6><loc_66><loc_19><loc_68><loc_23></location>√</text>
<text><location><page_6><loc_68><loc_20><loc_69><loc_23></location>-</text>
<text><location><page_6><loc_68><loc_19><loc_69><loc_20></location>r</text>
<text><location><page_6><loc_69><loc_20><loc_69><loc_21></location>5</text>
<text><location><page_6><loc_69><loc_20><loc_70><loc_21></location>/</text>
<text><location><page_6><loc_70><loc_20><loc_71><loc_21></location>2</text>
<text><location><page_6><loc_69><loc_19><loc_69><loc_20></location>g</text>
<text><location><page_6><loc_79><loc_19><loc_80><loc_22></location>-</text>
<text><location><page_6><loc_83><loc_19><loc_85><loc_23></location>)</text>
<text><location><page_6><loc_52><loc_14><loc_92><loc_18></location>Higher-order coefficients are obtained from higher-order terms of the EMT expansion using consistency of the Einstein equations, e.g. the next-highest order coefficient of Eq. (76) is</text>
<formula><location><page_6><loc_61><loc_8><loc_92><loc_13></location>c 2 = 4 7 r g ( 1 + r 5 / 2 g √ 3 π ¯ e 52 √ -¯ e 2 ) . (80)</formula>
<text><location><page_6><loc_77><loc_20><loc_78><loc_21></location>52</text>
<text><location><page_6><loc_82><loc_20><loc_83><loc_21></location>52</text>
<text><location><page_6><loc_56><loc_20><loc_57><loc_22></location>h</text>
<text><location><page_6><loc_58><loc_20><loc_60><loc_22></location>=</text>
<text><location><page_6><loc_74><loc_20><loc_77><loc_22></location>+5¯</text>
<text><location><page_6><loc_76><loc_20><loc_77><loc_22></location>e</text>
<text><location><page_6><loc_80><loc_20><loc_82><loc_22></location>7¯</text>
<text><location><page_6><loc_81><loc_20><loc_82><loc_22></location>p</text>
<text><location><page_6><loc_85><loc_20><loc_85><loc_22></location>.</text>
<text><location><page_6><loc_89><loc_20><loc_92><loc_22></location>(79)</text>
<text><location><page_7><loc_9><loc_90><loc_49><loc_93></location>In addition, consistency of the Einstein equations requires Eq. (44) and</text>
<formula><location><page_7><loc_21><loc_86><loc_49><loc_90></location>¯ p 52 = 2 √ -¯ e 2 √ 3 πr 5 / 2 g + ¯ e 52 . (81)</formula>
<text><location><page_7><loc_9><loc_84><loc_46><loc_85></location>Substituting Eq. (81) into Eq. (79) we obtain the identity</text>
<formula><location><page_7><loc_25><loc_81><loc_49><loc_82></location>c 2 = c 32 h 12 , (82)</formula>
<text><location><page_7><loc_9><loc_74><loc_49><loc_80></location>which leads to many simplifying cancellations, e.g. the absence of the √ x term in the Ricci scalar ¯ R (cf. Eq. (45)) due to R 12 ∝ c 32 h 12 -c 2 , where R 12 denotes the √ x coefficient of ¯ R .</text>
<text><location><page_7><loc_9><loc_70><loc_49><loc_74></location>Again, the expansion of e 2 h that is given by Eq. (25) is obtained by separating the logarithmic part of h ( t, x ) from the rest. From the expansion</text>
<formula><location><page_7><loc_19><loc_65><loc_49><loc_69></location>e 2 h = ( ¯ ξ + λ ˜ ξ x ) 3 e 2¯ χ +2 λω , (83)</formula>
<text><location><page_7><loc_9><loc_63><loc_27><loc_64></location>we then obtain Eq. (25) with</text>
<formula><location><page_7><loc_24><loc_58><loc_49><loc_62></location>Ω = 3 ˜ ξ 2 ¯ ξ + ω. (84)</formula>
<text><location><page_7><loc_9><loc_54><loc_49><loc_57></location>The series expansions of the perturbative corrections of Eqs. (19)-(20) are therefore given by the power series</text>
<formula><location><page_7><loc_10><loc_49><loc_49><loc_53></location>Σ = ∞ ∑ j /greaterorequalslant 3 2 σ j x j = σ 32 x 3 / 2 + σ 2 x 2 + O ( x 5 / 2 ) , (85)</formula>
<formula><location><page_7><loc_10><loc_43><loc_49><loc_49></location>Ω = 3 ˜ ξ 2 ¯ ξ + ∞ ∑ j /greaterorequalslant 1 2 ω j x j = 3 ˜ ξ 2 ¯ ξ + ω 12 √ x + ω 1 x + O ( x 3 / 2 ) . (86)</formula>
<text><location><page_7><loc_9><loc_39><loc_49><loc_42></location>Finiteness of the Ricci scalar at the horizon requires Eq. (44) and</text>
<formula><location><page_7><loc_17><loc_35><loc_49><loc_39></location>σ 32 | R = 3 ˜ ξ 2 ¯ ξ c 32 = 2 r 3 / 2 g ˜ ξ √ -3 π ¯ e 2 ¯ ξ . (87)</formula>
<text><location><page_7><loc_9><loc_31><loc_49><loc_34></location>The expansion structure of the modified gravity terms ¯ E µν is obtained analogous to Sec. IV A. We find</text>
<formula><location><page_7><loc_13><loc_25><loc_49><loc_30></location>¯ E tt = æ 32 x 3 / 2 + æ ¯ 1 x + æ 12 √ x +æ 0 + ∞ ∑ j /greaterorequalslant 1 2 æ j x j , (88)</formula>
<formula><location><page_7><loc_13><loc_21><loc_49><loc_25></location>¯ E r t = œ 0 + ∞ ∑ j /greaterorequalslant 1 2 œ j x j , (89)</formula>
<formula><location><page_7><loc_13><loc_16><loc_49><loc_21></location>¯ E rr = ∞ ∑ j /greaterorequalslant 3 2 ø j x j . (90)</formula>
<text><location><page_7><loc_9><loc_13><loc_49><loc_15></location>The equation for the x 0 coefficient of the tr component Eq. (31) allows to identify</text>
<formula><location><page_7><loc_23><loc_9><loc_49><loc_12></location>˜ E | tr = œ 0 r g 8 π ¯ ξ 3 / 2 c 32 . (91)</formula>
<text><location><page_7><loc_52><loc_40><loc_54><loc_41></location>and</text>
<text><location><page_7><loc_52><loc_89><loc_92><loc_93></location>Substitution of Eq. (91) into the expression σ 32 | tt ( ˜ E ) obtained from Eq. (30) and subsequent comparison with the expression σ 32 | rr obtained from Eq. (32) establishes the relation</text>
<formula><location><page_7><loc_64><loc_85><loc_92><loc_88></location>æ 32 = 2 ¯ ξ 3 / 2 œ 0 -¯ ξ 3 ø 32 (92)</formula>
<text><location><page_7><loc_52><loc_73><loc_92><loc_84></location>between the lowest-order coefficients of the MTG terms, see Appendix A 2. Similarly, by substituting Eq. (91) into the expression for σ 2 | tt , and ˜ ξ | tr obtained from the √ x coefficient of Eq. (31) into the expression Eq. (87) for σ 32 | R , we can derive two distinct expressions for the sum ˜ e 2 +˜ p 2 by comparison of σ 2 | tt and σ 2 | rr obtained from Eqs. (30) and (32), respectively, as well as comparison of σ 32 | R and σ 32 | tt . Their identification establishes the additional relation</text>
<formula><location><page_7><loc_54><loc_69><loc_92><loc_72></location>æ ¯ 1 = 2 ¯ ξ 3 / 2 ( h 12 œ 0 +œ 12 ) -¯ ξ 3 (2 h 12 ø 32 +ø 2 ) (93)</formula>
<text><location><page_7><loc_52><loc_53><loc_92><loc_69></location>between the modified gravity coefficients of Eqs. (88)-(90). A detailed derivation with explicit expressions is provided in Appendix A 2. Analogous to the class of k = 0 black hole solutions discussed in Sec. IV A, the modified gravity terms ¯ E µν of any self-consistent MTG must follow the expansion structures prescribed by Eqs. (88)-(90) and identically satisfy the two relations Eqs. (92)-(93) to be compatible with black hole solutions of the k = 1 type. Again, consideration of higher-order coefficients in Eqs. (30)-(32) introduces new independent variables and will thus not yield any additional constraints.</text>
<text><location><page_7><loc_52><loc_47><loc_92><loc_53></location>Once more, the analogs of the MTG coefficient relations Eqs. (92)-(93) are also satisfied by the coefficients of the corresponding metric tensor and Ricci tensor components themselves, i.e.</text>
<formula><location><page_7><loc_59><loc_44><loc_92><loc_46></location>(¯ g tt ) 32 = -¯ ξ 3 (¯ g rr ) 32 = -c 32 ¯ ξ 3 /r g , (94)</formula>
<formula><location><page_7><loc_58><loc_42><loc_92><loc_44></location>( ¯ R tt ) 32 = 2 ¯ ξ 3 / 2 ( ¯ R r t ) 0 -¯ ξ 3 ( ¯ R rr ) 32 = 0 , (95)</formula>
<formula><location><page_7><loc_58><loc_34><loc_92><loc_39></location>(¯ g tt ) ¯ 1 = -¯ ξ 3 ( 2 h 12 (¯ g rr ) 32 +(¯ g rr ) 2 ) = -c 32 h 12 ¯ ξ 3 /r g , (96)</formula>
<text><location><page_7><loc_52><loc_22><loc_92><loc_27></location>where Eq. (82) was used to simplify the expressions, ( ¯ R tt ) 32 = ( ¯ R r t ) 0 = ( ¯ R rr ) 32 = 0 , ( ¯ R rr ) 2 = -3 c 2 32 / (2 r 3 g ) , and Eqs. (94) and (96) simplify due to the diagonal form of the metric tensor ¯ g r t = 0 (see Eq. (3)).</text>
<formula><location><page_7><loc_58><loc_27><loc_92><loc_35></location>( ¯ R tt ) ¯ 1 = 2 ¯ ξ 3 / 2 ( h 12 ( ¯ R r t ) 0 +( ¯ R r t ) 12 ) -¯ ξ 3 ( 2 h 12 ( ¯ R rr ) 32 +( ¯ R rr ) 2 ) = -3 c 2 32 ¯ ξ 3 / ( 2 r 3 g ) , (97)</formula>
<section_header_level_1><location><page_7><loc_64><loc_18><loc_80><loc_19></location>C. λ -expanded solutions</section_header_level_1>
<text><location><page_7><loc_52><loc_9><loc_92><loc_16></location>We now consider solutions where the leading reduced components of the EMT are not dominated by terms of order O ( λ 0 ) . To obtain mathematically consistent expressions we have to extend the expansion to terms of order O ( λ 2 ) as higher-order terms, if needed, are obtained analogously.</text>
<text><location><page_8><loc_9><loc_90><loc_49><loc_93></location>The k = 0 solution without GR limit has the following properties: the EMT expansion</text>
<formula><location><page_8><loc_12><loc_81><loc_49><loc_88></location>τ a = λ ˜ Ξ + λ 2 ˜ Ξ (2) + ∞ ∑ j /greaterorequalslant 1 2 [ (¯ τ a ) j + λ (˜ τ a ) j + λ 2 ( ˜ τ (2) a ) j ] x j (98)</formula>
<text><location><page_8><loc_9><loc_74><loc_49><loc_80></location>corresponds to the case where as r → r g, lim τ t = lim τ r = lim τ r t . The equations below are trivially extendable to the case where the leading term in τ r t = -λ ˜ Ξ .</text>
<text><location><page_8><loc_10><loc_73><loc_41><loc_74></location>In either case, the metric functions are given by</text>
<formula><location><page_8><loc_11><loc_66><loc_49><loc_71></location>C = r g -λσ 12 √ x + ∞ ∑ j /greaterorequalslant 1 2 ( ζ j + λσ j + λ 2 σ (2) j ) x j , (99)</formula>
<formula><location><page_8><loc_11><loc_62><loc_49><loc_66></location>h = -1 2 ln x ξ + ∞ ∑ j /greaterorequalslant 1 2 ( η j + λω j + λ 2 ω (2) j ) x j , (100)</formula>
<text><location><page_8><loc_9><loc_52><loc_49><loc_60></location>similar to the k = 0 perturbative solution. Here, the structure of the metric function h was simplified by redefining the time, and the coefficient c 12 = ζ 12 + λσ 12 + λ 2 σ (2) 12 was simplified by taking into account the requirement that the Ricci scalar must be finite at the apparent horizon, i.e.</text>
<formula><location><page_8><loc_21><loc_47><loc_49><loc_50></location>c 12 → λσ 12 = -r ' g r g √ ξ . (101)</formula>
<text><location><page_8><loc_9><loc_34><loc_49><loc_44></location>Unlike in GR, the sign of ˜ Ξ (and ˜ Ξ (2) ) cannot be determined solely from the requirements of existence and consistency of the modified Einstein equations. It is therefore unclear whether or not violation of the null energy condition is a prerequisite for the formation of a PBH. This is in contrast to GR, where such a violation has been shown to be mandatory in a variety of settings [9, 21, 26, 27].</text>
<text><location><page_8><loc_9><loc_29><loc_49><loc_33></location>The expansion structure of the non-GR terms E µν remains the same as in the perturbative k = 0 scenario discussed in Subsec. IV A, that is</text>
<formula><location><page_8><loc_14><loc_22><loc_49><loc_27></location>E tt = æ ¯ 1 x + æ 12 √ x +æ 0 x 0 + ∞ ∑ j /greaterorequalslant 1 2 æ j x j , (102)</formula>
<formula><location><page_8><loc_14><loc_18><loc_49><loc_23></location>E r t = œ 12 √ x +œ 0 x 0 + ∞ ∑ j /greaterorequalslant 1 2 œ j x j , (103)</formula>
<formula><location><page_8><loc_14><loc_13><loc_49><loc_18></location>E rr = ø 0 + ∞ ∑ j /greaterorequalslant 1 2 ø j x j , (104)</formula>
<text><location><page_8><loc_9><loc_8><loc_49><loc_11></location>where æ /lscript := ¯æ /lscript + λ ˜ æ /lscript , and similarly for the coefficients œ /lscript , ø /lscript of the non-GR terms E r t and E rr . Substitution into the</text>
<text><location><page_8><loc_52><loc_92><loc_87><loc_93></location>generic modified Einstein equations Eqs. (7)-(9) gives</text>
<formula><location><page_8><loc_52><loc_80><loc_92><loc_91></location>O ( x 0 ) terms of Eq. (7) ( ¯ æ ¯ 1 ξ -8 π ˜ Ξ ) λ = 0 , (105) ( ˜ æ ¯ 1 ξ +8 π ˜ Ξ (2) -σ 2 12 2 r 3 g ) λ 2 = 0 , (106)</formula>
<formula><location><page_8><loc_54><loc_73><loc_73><loc_77></location> x 0 ¯ ø 0 8 π ˜ Ξ λ = 0 ,</formula>
<formula><location><page_8><loc_52><loc_74><loc_92><loc_83></location>O ( x -1 / 2 ) terms of Eq. (8) ( ¯ œ 12 -8 π √ ξ ˜ Ξ ) λ = 0 , (107) [ ˜ œ 12 + √ ξ 2 ( 16 π ˜ Ξ (2) -σ 2 12 r 3 g )] λ 2 = 0 , (108)</formula>
<text><location><page_8><loc_52><loc_59><loc_92><loc_70></location> at the respective leading orders of x , and leads to the following constraints: first, the expansion coefficients ( τ a ) 12 = 0 ∀ a , where ( τ a ) /lscript := (¯ τ a ) /lscript + λ (˜ τ a ) /lscript + λ 2 (˜ τ (2) a ) /lscript , see Eq. (98). At the leading expansion order (which is O ( λ ) since the non-GR terms appear as λ E µν in Eqs. (7)-(9)) the lowest-order x coefficients satisfy</text>
<formula><location><page_8><loc_52><loc_66><loc_92><loc_75></location>O ( ) terms of Eq. (9) ( -) (109) ( ˜ ø 0 +8 π ˜ Ξ (2) -σ 2 12 2 r 3 g ) λ 2 = 0 , (110)</formula>
<formula><location><page_8><loc_62><loc_54><loc_92><loc_57></location>¯ æ ¯ 1 = √ ξ ¯ œ 12 = ξ ¯ ø 0 = 8 π ˜ Ξ ξ, (111)</formula>
<text><location><page_8><loc_52><loc_50><loc_92><loc_55></location>which is analogous to Eq. (63), but in this case identifies the leading reduced term in the EMT. Similarly, the next-order O ( λ 2 ) expansion coefficients satisfy</text>
<formula><location><page_8><loc_56><loc_45><loc_92><loc_50></location>˜ æ ¯ 1 = √ ξ ˜ œ 12 = ξ ˜ ø 0 = -8 π ˜ Ξ (2) ξ + σ 2 12 ξ 2 r 3 g . (112)</formula>
<section_header_level_1><location><page_8><loc_54><loc_42><loc_90><loc_43></location>V. BLACK HOLES IN THE STAROBINSKY MODEL</section_header_level_1>
<text><location><page_8><loc_52><loc_22><loc_92><loc_40></location>Numerous modifications of GR have been proposed, including theories that involve higher-order curvature invariants. Apopular class among these are so-called f ( R ) theories [2], in which the gravitational Lagrangian density L g is an arbitrary function of the Ricci scalar R . In this section, we consider the Starobinsky model [16] with F = ςR 2 , ς = 16 πa 2 /M 2 P (see Eq. (1)). It is a straightforward extension of GR with quadratic corrections in the Ricci scalar that is of relevance in cosmological contexts. In particular, it is the first selfconsistent model of inflation. New horizonless solutions in this model have been identified recently in an analysis [28] of static, spherically symmetric, and asymptotically flat vacuum solutions.</text>
<section_header_level_1><location><page_8><loc_62><loc_18><loc_82><loc_19></location>A. Modified Einstein equations</section_header_level_1>
<text><location><page_8><loc_52><loc_13><loc_92><loc_15></location>In f ( R ) theories, the relevant equations have a relatively simple form. For the action</text>
<formula><location><page_8><loc_56><loc_7><loc_92><loc_12></location>S = 1 16 π ∫ ( f ( R ) + L m ) √ -g d 4 x + S b , (113)</formula>
<text><location><page_9><loc_9><loc_89><loc_49><loc_93></location>where the gravitational Lagrangian L g = f ( R ) , the matter Lagrangian is represented by L m , and S b denotes the boundary term, the field equations for the metric g µν are given by</text>
<formula><location><page_9><loc_11><loc_83><loc_49><loc_88></location>f ' R µν -1 2 f g µν + ( g µν /square -∇ µ ∇ ν ) f ' = 8 πT µν , (114)</formula>
<text><location><page_9><loc_9><loc_80><loc_49><loc_85></location>where f ' := ∂ f ( R ) /∂R and /square := g µν ∇ µ ∇ ν . It is convenient to set f ( R ) =: R + λ F ( R ) . The modified Einstein equations are then</text>
<formula><location><page_9><loc_9><loc_75><loc_51><loc_80></location>G µν + λ ( F ' R µν -1 2 F g µν + ( g µν /square -∇ µ ∇ ν ) F ' ) = 8 πT µν . (115)</formula>
<text><location><page_9><loc_9><loc_70><loc_49><loc_74></location>Performing the expansion in λ and only keeping terms up to the first order we obtain expressions for the modified gravity terms ¯ E µν , i.e.</text>
<formula><location><page_9><loc_11><loc_65><loc_49><loc_69></location>¯ E µν = F ' ¯ R µν -1 2 F ¯ g µν + ( ¯ g µν ¯ /square -¯ ∇ µ ¯ ∇ ν ) F ' , (116)</formula>
<formula><location><page_9><loc_11><loc_57><loc_49><loc_61></location>/square F ' = [ ∂ t ∂ t + ∂ r ∂ r +( ∂ t h ) ∂ t +( ∂ r h +2 /r ) ∂ r ] F ' . (117)</formula>
<text><location><page_9><loc_9><loc_60><loc_49><loc_65></location>where all objects labeled by the bar are evaluated with respect to the unperturbed metric ¯ g, and F ≡ F ( ¯ R ) . In spherical symmetry the d'Alembertian is given by</text>
<text><location><page_9><loc_9><loc_54><loc_49><loc_56></location>Second-order covariant derivatives of a scalar function can be expressed in terms of partial derivatives, i.e.</text>
<formula><location><page_9><loc_18><loc_51><loc_49><loc_53></location>∇ µ ∇ ν F ' = ∂ µ ∂ ν F ' -Γ ζ µν ∂ ζ F ' . (118)</formula>
<text><location><page_9><loc_9><loc_48><loc_49><loc_51></location>In the Starobinsky model F ( R ) = ς ¯ R 2 + O ( λ ) and Eqs. (30)(32) become</text>
<formula><location><page_9><loc_10><loc_39><loc_49><loc_47></location>¯ E tt / ( λς ) = 2 ¯ R ¯ R tt -1 2 ¯ R 2 ¯ g tt +2 [ ¯ g tt ( ∂ t ∂ t + ∂ r ∂ r +( ∂ t ¯ h ) ∂ t +( ∂ r ¯ h +2 r -1 ) ∂ r ) -∂ t ∂ t +Γ t tt ∂ t +Γ r tt ∂ r ] ¯ R, (119) ¯ E r t / ( λς ) = 2 ¯ R ¯ R r t -2 ∂ t ∂ r +Γ r tt ∂ t +Γ r tr ∂ r ¯ R, (120)</formula>
<formula><location><page_9><loc_10><loc_33><loc_49><loc_41></location>( ) ¯ E rr / ( λς ) = 2 ¯ R ¯ R rr -1 2 ¯ R 2 ¯ g rr +2¯ g rr [ ∂ t ∂ t +( ∂ t ¯ h -Γ r rt ) ∂ t +( ∂ r ¯ h +2 r -1 -Γ r rr ) ∂ r ] ¯ R. (121)</formula>
<section_header_level_1><location><page_9><loc_9><loc_31><loc_48><loc_32></location>B. Compatibility with the k = 0 class of black hole solutions</section_header_level_1>
<text><location><page_9><loc_9><loc_18><loc_49><loc_29></location>With the k = 0 metric functions Eqs. (46)-(47), the constraint Eq. (52) that is obtained from the requirement that the Ricci scalar be non-divergent leads to cancellations in the Ricci tensor components ¯ R tt and ¯ R rr which ensures that the MTG terms Eqs. (119)-(121) of the ˜ f ( ¯ R ) = ς ¯ R 2 Starobinsky model conform to the structures of Eqs. (57)-(59). We find that both of the two constraints posed by Eq. (63) are satisfied, i.e.</text>
<formula><location><page_9><loc_10><loc_9><loc_49><loc_17></location>æ ¯ 1 = √ ¯ ξ œ 12 = ¯ ξ ø 0 = c 2 12 ¯ ξ ( -2 ( R 0 + r g R 1 ) + h 12 r g R 12 ) -c 12 r 2 g √ ¯ ξR ' 0 2 r 3 g , (122)</formula>
<text><location><page_9><loc_52><loc_87><loc_92><loc_93></location>where R j is used to denote coefficients of the Ricci scalar ¯ R = ∑ j R j x j = R 0 + R 12 √ x + R 1 x + O ( x 3 / 2 ) . Similarly, the next-highest order coefficients satisfy the constraint of Eq. (65), see App. B, Eqs. (B4)-(B6).</text>
<section_header_level_1><location><page_9><loc_58><loc_83><loc_85><loc_84></location>C. Compatibility with the k = 1 solution</section_header_level_1>
<text><location><page_9><loc_52><loc_70><loc_92><loc_81></location>Similar to the k = 0 case, the k = 1 constraint on the evolution of the horizon radius Eq. (44) that is required to ensure consistency of the Einstein equations and finiteness of the Ricci scalar leads to cancellations in ¯ R tt , and the k = 1 Starobinsky MTG terms of Eqs. (119)-(121) follow the structures prescribed by Eqs. (88)-(90). Using the k = 1 metric functions Eqs. (76)-(77) we obtain the lowest-order coefficients</text>
<formula><location><page_9><loc_54><loc_67><loc_92><loc_68></location>æ 32 = 2 c 32 ¯ ξ 3 /r 5 g , (123)</formula>
<formula><location><page_9><loc_55><loc_62><loc_92><loc_66></location>æ ¯ 1 = -c 32 ¯ ξ 3 ( -2 h 12 +3 c 32 ( 4 + r 3 g R 1 )) /r 5 g , (124)</formula>
<text><location><page_9><loc_52><loc_61><loc_90><loc_63></location>of ¯ E tt from Eq. (119). Similarly, we obtain the coefficients</text>
<formula><location><page_9><loc_55><loc_56><loc_92><loc_60></location>œ 0 = 0 , œ 12 = -3 c 2 32 ¯ ξ 3 / 2 ( 4 + r 3 g R 1 ) /r 5 g , (125) ø 32 = -2 c 32 /r 5 g , (126)</formula>
<text><location><page_9><loc_52><loc_45><loc_92><loc_53></location>of ¯ E r t and ¯ E rr from Eqs. (120)-(121), where R 1 denotes the x coefficient of the Ricci scalar ¯ R = ∑ j R j x j = 2 /r 2 g + R 1 x + O ( x 3 / 2 ) and R 12 = 0 , cf. Eq. (45). With the expressions given in Eqs. (123)-(127), it is easy to verify that both k = 1 constraints Eq. (92) and Eq. (93) are satisfied identically.</text>
<formula><location><page_9><loc_56><loc_52><loc_92><loc_56></location>ø 2 = c 32 ( 2 h 12 -3 c 32 ( 4 + r 3 g R 1 ) /r 5 g ) . (127)</formula>
<section_header_level_1><location><page_9><loc_53><loc_40><loc_91><loc_42></location>D. Compatibility with the λ -expanded k = 0 class of black hole solutions</section_header_level_1>
<text><location><page_9><loc_52><loc_21><loc_92><loc_38></location>Equality of the coefficients in Eqs. (111) and (112) follows in exactly the same fashion as in Sec. V B. Explicit calculation confirms that the coefficients of the MTG terms in the Starobinsky model Eqs. (119)-(121) obtained using the EMT expansion of Eq. (98) and metric functions Eqs. (99)-(100) coincide with those of Eq. (122) at the leading expansion order O ( λ ) . Terms of order O ( λ 0 ) vanish in accordance with Eq. (122) (note that c 12 ∝ O ( λ ) ). This confirms that the Starobinsky solution is consistent with the generic form of PBH solutions. However, since ˜ Ξ (2) is undetermined in the self-consistent approach, Eq. (112) does not impose any constraints on the function ξ .</text>
<section_header_level_1><location><page_9><loc_66><loc_16><loc_78><loc_17></location>VI. DISCUSSION</section_header_level_1>
<text><location><page_9><loc_52><loc_9><loc_92><loc_14></location>We have analyzed the properties of metric MTG and derived several constraints that they must satisfy to be compatible with the existence of an apparent horizon. Since we have not specified the origin of the deviations from GR, the results</text>
<text><location><page_10><loc_9><loc_90><loc_49><loc_93></location>presented here are generic and apply to all conceivable selfconsistent metric MTG.</text>
<text><location><page_10><loc_9><loc_66><loc_49><loc_90></location>Constraints on a perturbative solution in a particular metric MTG arise from two sources: first, the series expansions of the modified gravity terms ¯ E µν in terms of the distance x := r -r g from the horizon must follow a particular structure that is prescribed by the modified Einstein equations with terms that diverge in the limit r → r g. Second, a general spherically symmetric metric allows for two independent functions C and h that must satisfy three Einstein equations. The resulting relations between coefficients σ /lscript , ω /lscript of their perturbative corrections translate into relationships between the coefficients c /lscript and h /lscript , and eventually components of the unperturbed EMT. These constraints must be satisfied identically. Otherwise, a valid solution of GR cannot be perturbatively extended to a solution of a MTG. Identities that must be satisfied for the existence of the perturbative k = 0 solutions are given by Eqs. (63) and (65), and for the k = 1 solution by Eqs. (92)-(93).</text>
<text><location><page_10><loc_9><loc_58><loc_49><loc_65></location>On the other hand, there are nonperturbative solutions that do not have a well-defined GR limit. In this case, the constraints on a MTG that are imposed by the existence of a regular apparent horizon formed in finite time of a distant observer are given by Eqs. (111)-(112).</text>
<text><location><page_10><loc_9><loc_43><loc_49><loc_58></location>Using the Starobinsky R 2 model, arguably the simplest possible MTG, we identify both perturbative and nonperturbative solutions. However, this is not the only theory that should be investigated: in a future article [29], we will consider generic f ( R ) theories of the form f ( R ) = R + λ F ( R ) , where F ( R ) = ςR q and q, ς ∈ R . In particular, this includes the case q = 1 / 2 (i.e. f ( R ) = R + λς √ R ) considered in Ref. [30], as well as the case of negative exponents q < 0 considered in Ref. [31]. More general MTG (e.g. those involving higher-order curvature invariants) will also be considered.</text>
<section_header_level_1><location><page_10><loc_21><loc_37><loc_37><loc_38></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_10><loc_9><loc_23><loc_49><loc_34></location>We thank Eleni Kontou, Robert Mann, Shin'ichi Nojiri, Vasilis Oikonomou, and Christian Steinwachs for useful discussions and helpful comments. SM is supported by an International Macquarie University Research Excellence Scholarship and a Sydney Quantum Academy Scholarship. The work of DRT was supported in part by the Southern University of Science and Technology, Shenzhen, China, and by the ARC Discovery project grant DP210101279.</text>
<section_header_level_1><location><page_10><loc_12><loc_17><loc_45><loc_18></location>Appendix A: Coefficients of perturbative corrections</section_header_level_1>
<section_header_level_1><location><page_10><loc_19><loc_14><loc_39><loc_15></location>1. k = 0 black hole solutions</section_header_level_1>
<text><location><page_10><loc_9><loc_9><loc_49><loc_11></location>With the metric functions Eqs. (46)-(47) of the k = 0 solutions we obtain the following coefficients for the perturbative</text>
<text><location><page_10><loc_52><loc_92><loc_81><loc_93></location>correction Σ of Eq. (55) from Eqs. (30)-(32):</text>
<formula><location><page_10><loc_53><loc_87><loc_92><loc_90></location>σ 12 | tt = -r 3 / 2 g æ ¯ 1 4 √ π ¯ ξ ¯ Υ , (A1)</formula>
<formula><location><page_10><loc_53><loc_82><loc_92><loc_86></location>σ 12 | tr = -r 3 / 2 g ( 4 π ˜ ξ ¯ Υ 2 + √ ¯ ξ œ 12 ) 2 √ π ¯ ξ ¯ Υ , (A2)</formula>
<formula><location><page_10><loc_53><loc_78><loc_92><loc_82></location>σ 12 | rr = -r 3 / 2 g ø 0 4 √ π ¯ Υ , (A3)</formula>
<formula><location><page_10><loc_54><loc_70><loc_92><loc_78></location>σ 1 | tt = 4 ¯ Υæ ¯ 1 +6 √ πr 3 / 2 g ¯ Υ 2 36 π ¯ ξ ¯ Υ 3 ( -æ 12 +8 π ¯ ξ (˜ τ t ) 12 + √ πr 3 / 2 g æ ¯ 1 ( 6(¯ τ r ) 12 -5(¯ τ t ) 12 ) ) , (A4)</formula>
<formula><location><page_10><loc_54><loc_63><loc_93><loc_71></location>σ 1 | tr = -1 12 √ π ¯ ξ ¯ Υ [ ˜ ξ ( -4 √ π ¯ Υ+8 πr 3 / 2 g (¯ τ t ) 12 ) +3 r 3 / 2 g × √ ¯ ξ ( -œ 0 +4 π √ ¯ ξ ( (˜ τ t ) 12 +(˜ τ r ) 12 -2 ¯ Υ 2 ω 12 ) ) ] , (A5)</formula>
<formula><location><page_10><loc_54><loc_55><loc_94><loc_63></location>σ 1 | rr = 1 12 π ¯ Υ 3 ( -2 ¯ Υø 0 +6 √ πr 3 / 2 g ¯ Υ 2 ( -ø 12 +8 π (˜ τ r ) 12 ) + √ πr 3 / 2 g ø 0 ( (¯ τ t ) 12 -6(¯ τ r ) 12 ) -96 π 3 / 2 r 3 / 2 g ¯ Υ 4 ω 12 ) . (A6)</formula>
<text><location><page_10><loc_52><loc_51><loc_92><loc_54></location>Via comparison of Eqs. (A5) and (A6) we can identify the coefficient</text>
<formula><location><page_10><loc_52><loc_41><loc_92><loc_50></location>ω 12 = 1 72 π ¯ ξ ¯ Υ 2 r 3 / 2 g ( 20 √ π ˜ ξ ¯ Υ+3 r 3 / 2 g √ ¯ ξ œ 0 -6 r 3 / 2 g ¯ ξ ( ø 12 +2 π (˜ τ t ) 12 -6 π (˜ τ r ) 12 ) +16 r 3 / 2 g π ˜ ξ ( 3(¯ τ r ) 12 -(¯ τ t ) 12 ) ) (A7)</formula>
<text><location><page_10><loc_52><loc_39><loc_81><loc_40></location>for the perturbative correction Ω of Eq. (56).</text>
<text><location><page_10><loc_53><loc_37><loc_92><loc_38></location>Substitution of Eq. (A7) into Eqs. (A5) and (A6) then yields</text>
<formula><location><page_10><loc_53><loc_24><loc_92><loc_35></location>σ 1 | tr = σ 1 | rr = 1 18 √ π ¯ ξ ¯ Υ [ 3 r 3 / 2 g √ ¯ ξ ( -2œ 0 + √ ¯ ξ ( ø 12 +8 π (˜ τ t ) 12 ) ) -4 ˜ ξ ( 4 √ π ¯ Υ+ πr 3 / 2 g ( 6(¯ τ r ) 12 -5(¯ τ t ) 12 ) ) ] . (A8)</formula>
<text><location><page_10><loc_52><loc_19><loc_92><loc_23></location>Subsequent comparison of Eq. (A8) and (A4) establishes the relation Eq. (65) between the coefficients æ 12 , œ 0 , and ø 12 .</text>
<section_header_level_1><location><page_10><loc_62><loc_14><loc_81><loc_15></location>2. k = 1 black hole solution</section_header_level_1>
<text><location><page_10><loc_52><loc_9><loc_92><loc_11></location>With the metric functions Eqs. (76)-(77) of the k = 1 solution we obtain the following coefficients for the perturbative</text>
<text><location><page_11><loc_9><loc_92><loc_41><loc_93></location>correction Σ of Eq. (85) from Eqs. (30) and (32):</text>
<formula><location><page_11><loc_10><loc_87><loc_49><loc_91></location>σ 32 | tt = r 2 g ( æ 32 r g ¯ ξ 3 -8 πc 32 ˜ E ) , (A9)</formula>
<formula><location><page_11><loc_10><loc_85><loc_49><loc_87></location>σ 32 | rr = -ø 32 r 3 g +8 πc 32 r 2 g ˜ P, (A10)</formula>
<formula><location><page_11><loc_11><loc_82><loc_37><loc_85></location>σ 2 | tt = r g ¯ ξ 3 æ 32 r g (3 c 32 -2 h 12 ) + æ ¯ 1 r g</formula>
<formula><location><page_11><loc_10><loc_71><loc_49><loc_78></location>σ 2 | rr = -r 2 g ( ø 2 r g -c 32 ( 3ø 32 r g -8 π (3 c 32 + h 12 ) ˜ P ) -8 πr g ˜ p 2 ) . (A12)</formula>
<formula><location><page_11><loc_17><loc_77><loc_49><loc_85></location>2 [ -8 π ¯ ξ 3 ( c 32 (3 c 32 -h 12 ) ˜ E + r g ˜ e 2 )] , (A11)</formula>
<text><location><page_11><loc_10><loc_71><loc_46><loc_73></location>From the x 0 and √ x coefficients of Eq. (31) we obtain</text>
<formula><location><page_11><loc_11><loc_67><loc_49><loc_70></location>˜ E | tr = œ 0 r g 8 πc 32 ¯ ξ 3 / 2 , (A13)</formula>
<formula><location><page_11><loc_11><loc_60><loc_49><loc_66></location>˜ ξ | tr = -2 ¯ ξσ 32 3 c 32 -4 r 3 g 9 c 2 32 √ ¯ ξ ( œ 12 -8 π ¯ ξ 3 / 2 ˜ p 2 ) , (A14)</formula>
<text><location><page_11><loc_9><loc_57><loc_49><loc_61></location>where ˜ Φ = ( ˜ E + ˜ P ) / 2 and ˜ φ 2 = (˜ e 2 + ˜ p 2 ) / 2 , see Eqs. (73) and (75). By substitution of Eq. (A13) into Eq. (A11) we obtain</text>
<formula><location><page_11><loc_10><loc_48><loc_49><loc_56></location>˜ e 2 + ˜ p 2 = 6 c 2 32 ˜ Φ r g + 1 8 π ¯ ξ 3 [ æ ¯ 1 +æ 32 (3 c 32 -2 h 12 ) + ¯ ξ 3 (ø 2 -3 c 32 ø 32 ) + 2 ¯ ξ 3 / 2 œ 0 ( h 12 -3 c 32 ) ] . (A15)</formula>
<text><location><page_11><loc_52><loc_88><loc_92><loc_93></location>from the comparison σ 2 | tt -σ 2 | rr = 0 . Similarly, substitution of Eq. (A14) into Eq. (87) and subsequent comparison of σ 32 | R -σ 32 | tt = 0 yields</text>
<formula><location><page_11><loc_54><loc_79><loc_92><loc_83></location>˜ e 2 + ˜ p 2 = 6 c 2 32 ˜ Φ r g + 3æ 32 c 32 -¯ ξ 3 / 2 (6 c 32 -œ 0 œ 12 ) 4 π ¯ ξ 3 . (A16)</formula>
<text><location><page_11><loc_52><loc_70><loc_92><loc_73></location>Subtracting Eq. (A15) from Eq. (A16) and subsequent multiplication by 8 π ¯ ξ 3 r g yields</text>
<formula><location><page_11><loc_52><loc_58><loc_92><loc_64></location>-r g [ æ ¯ 1 -æ 32 (3 c 32 +2 h 12 ) + ¯ ξ 3 / 2 ( ¯ ξ 3 / 2 (ø 2 -3 c 32 ø 32 ) +6 c 32 œ 0 +2 h 12 œ 0 -2œ 12 )] = 0 . (A17)</formula>
<text><location><page_11><loc_52><loc_50><loc_92><loc_52></location>Lastly, substituting æ 32 from Eq. (92) into (A17) and rearranging gives Eq. (93).</text>
<paragraph><location><page_11><loc_27><loc_45><loc_74><loc_46></location>Appendix B: Additional explicit expressions for k = 0 black hole solutions</paragraph>
<text><location><page_11><loc_10><loc_40><loc_47><loc_41></location>Explicit expressions for the individual terms in Eq. (69):</text>
<formula><location><page_11><loc_16><loc_23><loc_92><loc_37></location>( ¯ R tt ) 12 = 2 √ ¯ ξ ( ¯ R r t ) 0 -¯ ξ ( ¯ R rr ) 12 = -1 24 c 12 r 3 g [ -2 c 3 12 ¯ ξ ( h 3 12 r g +6 h 32 r g + h 12 (9 h 1 r g -6) ) 6 r g √ ¯ ξ ( c 1 -1) ( √ ¯ ξ ( c 1 -1) 2 -2 r g c ' 12 ) × [ 4 c 32 r g ¯ ξ ( c 1 -1) + r g √ ¯ ξ ( 2 r g c ' 1 -h 12 ( √ ¯ ξ ( c 1 -1) 2 -2 r g c ' 12 ))] +6 c 2 12 ( ¯ ξ ( 3 -2 h 1 r g + c 32 h 12 r g -h 2 12 r g + c 1 ( -3 + 2 h 1 r g + h 2 12 r g ) ) + r 2 g √ ¯ ξh ' 12 ) ] . (B1)</formula>
<formula><location><page_11><loc_15><loc_9><loc_92><loc_24></location>( ¯ R r t ) 0 = c 12 √ ¯ ξ ( c 1 -1) /r 3 g . (B2) ( ¯ R rr ) 12 = 1 24 c 12 r 3 g √ ¯ ξ [ -2 c 3 12 √ ¯ ξ ( h 3 12 r g +6 h 32 r g + h 12 (9 h 1 r g -6) ) +6 r g ( c 1 -1) ( √ ¯ ξ ( c 1 -1) 2 -2 r g c ' 12 ) +3 c 12 r g [ 4 c 32 √ ¯ ξ ( c 1 -1) + 2 r g c ' 1 -h 12 ( √ ¯ ξ ( c 1 -1) -2 r g c ' 12 )] +6 c 2 12 [ √ ¯ ξ ( h 12 c 32 r g -2 h 1 r g -h 2 12 r g + c 1 ( 5 + 2 h 1 r g + h 2 12 r g ) -5 ) + r 2 g h ' 12 ] ] . (B3)</formula>
<text><location><page_12><loc_10><loc_92><loc_92><loc_93></location>Explicit expressions for the MTG coefficients æ 12 , œ 0 , ø 12 in the Starobinsky model of the k = 0 solution (see Subsec. V B):</text>
<formula><location><page_12><loc_13><loc_71><loc_92><loc_89></location>æ 12 = 1 12 c 12 r 5 g [ -12 r g ¯ ξR 0 ( c 1 -1) 3 +2 c 3 12 ¯ ξ ( 2 R 0 ( -6 h 12 + r g (9 h 12 h 1 + h 3 12 +6 h 32 ) ) -6 h 12 r 3 g R 1 +3 r 2 g R 12 × ( -6 + r g ( h 2 12 + h 1 ) ) +24 r 2 g R 0 √ ¯ ξc ' 12 ( c 1 -1) -3 c 12 r g √ ¯ ξ [ 8 c 32 √ ¯ ξR 0 ( c 1 -1) + 2 h 12 R 0 × ( 2 r g c ' 12 √ ¯ ξ ( c 1 -1) 2 ) + r g ( 4 R 0 c ' 1 +2 r 2 g R 12 c ' 12 -4 R ' 0 ( c 1 -1) -r g √ ¯ ξR 12 ( c 1 -1) 2 ) ] +6 c 2 12 [ -2 ¯ ξR 0 ( 5 + r g ( 2 h 1 ( c 1 -1) + h 12 ( c 32 + h 12 ( c 1 -1) ) ) -5 c 1 ) +4 ¯ ξR 2 0 + r 3 g ¯ ξ ( R 12 ( h 12 -h 12 c 1 + c 32 ) + 4 R 1 ( c 1 -1) ) -2 r 2 g √ ¯ ξR 0 h ' 12 -r 4 g √ ¯ ξR ' 12 ] ] . (B4)</formula>
<formula><location><page_12><loc_13><loc_46><loc_92><loc_64></location>ø 12 = 1 12 c 12 r 5 g √ ¯ ξ [ -2 c 3 12 √ ¯ ξ ( 2 R 0 ( r g ( 9 h 12 h 1 + h 3 12 +6 h 32 ) -30 h 12 ) -18 h 12 r 3 g R 1 +3 ( -3 h 1 + h 2 12 ) ) +12 r g R 1 ( c 1 -1) ( √ ¯ ξ ( c 1 -1) 2 -2 r g c ' 12 ) -3 c 12 r g [ -8 c 32 √ ¯ ξR 0 ( c 1 -1) + 2 h 12 R 0 × ( √ ¯ ξ ( c 1 -1) 2 -2 r g c ' 12 ) r g ( r g √ ¯ ξR 12 ( c 1 -1) 2 -4 R 0 c ' 1 +2 r 2 g R 12 c ' 12 +4 R ' 0 ( c 1 -1) )] -6 c 2 12 [ -2 √ ¯ ξR 0 ( 3 ( c 1 -1) ) + r g ( 2 h 1 ( c 1 -1) + h 12 ( c 32 + h 12 ( c 1 -1) ) ) +4 √ ¯ ξR 2 0 -2 r 2 g R 0 h ' 12 + r 2 g ( c 32 r g √ ¯ ξR 12 + h 12 ( r g √ ¯ ξR 12 ( c 1 -1) -8 R ' 0 ) +5 r 2 g R ' 12 )] ] . (B6)</formula>
<formula><location><page_12><loc_13><loc_63><loc_92><loc_71></location>œ 0 = 1 2 r 5 g [ c 2 12 √ ¯ ξ ( 2 h 12 ( 4 R 0 + r 3 g R 1 ) + r 2 g (2 h 1 r g -3) ) -r 4 g R 12 c ' 12 + c 12 ( 8 √ ¯ ξR 0 ( c 1 -1) + r 2 g ( r g √ ¯ ξ ( c 1 -1) (2 R 1 -h 12 R 12 ) + 4 h 12 R ' 0 -3 r 2 g R 2 12 ) ) ] . (B5)</formula>
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</document> | [{"title": "Spherically symmetric black holes in metric gravity", "content": "Sebastian Murk 1, 2, \u2217 and Daniel R. Terno 1, \u2020 1 Department of Physics and Astronomy, Macquarie University, Sydney, New South Wales 2109, Australia 2 Sydney Quantum Academy, Sydney, New South Wales 2006, Australia The existence of black holes is one of the key predictions of general relativity (GR) and therefore a basic consistency test for modified theories of gravity. In the case of spherical symmetry in GR the existence of an apparent horizon and its regularity is consistent with only two distinct classes of physical black holes. Here we derive constraints that any self-consistent modified theory of gravity must satisfy to be compatible with their existence. We analyze their properties and illustrate characteristic features using the Starobinsky model. Both of the GR solutions can be regarded as zeroth-order terms in perturbative solutions of this model. We also show how to construct nonperturbative solutions without a well-defined GR limit.", "pages": [1]}, {"title": "I. INTRODUCTION", "content": "General Relativity (GR), one of the two pillars of modern physics, is the simplest member of the family of metric theories of gravity. It is the only theory that is derived from an invariant that is linear in second derivatives of the metric. However, interpretations of astrophysical and cosmological data as well as theoretical considerations [1, 2] encourage us to consider GR as the low-energy limit of some effective theory of quantum gravity [3-5]. Extended theories of gravity, such as metric theories that involve higher-order invariants of the Riemann tensor, metric-affine theories, and theories with torsion, include additional terms in the action functional. Here we focus on metric modified theories of gravity (MTG). A prerequisite for the validity of any proposed generalization of GR is that it must be compatible with current astrophysical and cosmological data. In particular, a viable candidate theory must provide a model to describe the observed astrophysical black hole candidates. Popular contemporary models describe them as ultra-compact objects with or without a horizon [6]. While there is a considerable diversity of opinions on what exactly constitutes a black hole, the presence of a trapped region - a domain of spacetime from which nothing can escape - is its most commonly accepted characteristic [7]. A trapped spacetime region that is externally bounded by an apparent horizon is referred to as physical black hole (PBH) [8]. A PBH may contain other features of black hole solutions of classical GR, such as an event horizon or singularity, or it may be a singularity-free regular black hole. To be of physical relevance, the apparent horizon must form in finite time according to a distant observer [9]. It is commonly accepted that curvature invariants, such as the Ricci and Kretschmann scalar, are finite at the apparent horizon. When expressed mathematically, the requirements of regularity and finite formation time provide the basis for a self-consistent analysis of black holes. In spherical symmetry (to which we restrict our considerations here), this allows for a comprehensive classification of the near-horizon geometries. There are only two classes of solutions labeled by k = 0 and k = 1 , where the value of k reflects the scaling behavior of particular functions of the components of the energy-momentum tensor (EMT) near the apparent horizon. The properties of the near-horizon geometry lead to the identification of a unique scenario for black hole formation [9, 10] that involves both types of PBH solutions. We summarize its main results in Sec. III. Understanding the true nature of the observed ultracompact objects requires detailed knowledge of the black hole models, their alternatives, as well as the observational signatures of both classes of solutions in GR and extended theories of gravity [6, 11]. Vacuum black hole solutions exist in a variety of MTG [1, 2, 12]. On the other hand, these theories are also used to construct models of horizonless ultra-compact objects. A generic property among some of them is the absence of horizon formation in the final stage of the collapse [13]. Even the simplest MTG require perturbative treatment due to the mathematical complexity inherent to the higher-order nature of the equations [2, 14, 15]. We briefly review the relevant formalism and its relationship to the self-consistent approach in Sec. II. In Sec. IV, we derive a set of conditions necessary for the existence of a PBH in an arbitrary metric MTG. The solutions are presented as expansions in the coordinate distance from the apparent horizon and do not require a GR solution as the zeroth-order perturbative solution of a MTG. Using the Starobinsky model [2, 16] (Sec. V) we demonstrate the application of the general results, illustrating the well-known features of matching solutions of systems of partial differential equations of different orders [2, 14, 15]: we find that the two classes of GR solutions can be regarded as zeroth-order perturbative solutions of this MTG, and identify a MTG solution without a well-defined GR limit.", "pages": [1]}, {"title": "A. General considerations", "content": "We work in the framework of semiclassical gravity, use classical notions (e.g. metric, horizons, trajectories), and describe dynamics via the modified Einstein equations. We do not make any assumptions about the underlying reason for modifications of the bulk part of the gravitational Lagrangian density, but organize it according to powers of derivatives of the metric as commonly done in effective field theories [3, 4, 17], i.e. where M P is the Planck mass that we set to one in what follows, the cosmological constant was omitted, and the coefficients a 1 , a 2 , a 3 are dimensionless. The dimensionless parameter \u03bb is used to organize the perturbative analysis and set to one at the end of the calculations. Many popular models belong to the class of f ( R ) theories, where L g \u221a -g = f ( R ) . The prototypical example is the Starobinsky model with F = \u03c2R 2 , \u03c2 = 16 \u03c0a 2 /M 2 P . Varying the gravitational action results in where G \u00b5\u03bd is the Einstein tensor, the terms E \u00b5\u03bd result from the variation of F ( g \u00b5\u03bd , R \u00b5\u03bd\u03c1\u03c3 ) , and T \u00b5\u03bd \u2261 \u3008 \u02c6 T \u00b5\u03bd \u3009 \u03c9 denotes the expectation value of the renormalized EMT. We do not make any specific assumptions about the state \u03c9 . In fact, apart from imposing spherical symmetry, we assume only that (i) an apparent horizon is formed in finite time of a distant observer; (ii) it is regular, i.e. the scalars T := T \u00b5 \u00b5 = R/ 8 \u03c0 + O ( \u03bb ) and T := T \u00b5\u03bd T \u00b5\u03bd = R \u00b5\u03bd R \u00b5\u03bd / 64 \u03c0 2 + O ( \u03bb 2 ) are finite at the horizon. A general spherically symmetric metric in Schwarzschild coordinates is given by where r denotes the areal radius. The Misner-Sharp mass [18, 19] C ( t, r ) is invariantly defined via and thus the function f ( t, r ) = 1 -C ( t, r ) /r is invariant under general coordinate transformations. For a Schwarzschild black hole C = 2 M . We use the definition of Eq. (4) for consistency with the description of solutions in higherdimensional versions of GR. The apparent horizon is located at the Schwarzschild radius r g ( t ) that is the largest root of f ( t, r ) = 0 [19]. The Misner-Sharp mass of a PBH can be represented as where the definition of the apparent horizon implies and x := r -r g is the coordinate distance from the apparent horizon. \u00b7 \u00b7 \u00b7 , The modified Einstein equations take the form The notation is useful in dealing with equations in both GR and MTG. Regularity of the apparent horizon is expressed as a set of conditions on the potentially divergent parts of the scalars T and T . In spherical symmetry T \u03b8 \u03b8 \u2261 T \u03c6 \u03c6 and we assume that it is finite as in GR [9]. The constraints can therefore be represented mathematically as for some g 1 , 2 ( t ) and k 1 , 2 /greaterorequalslant 0 . There are a priori infinitely many solutions that satisfy these constraints. After reviewing the special case of GR and presenting the two admissible solutions we discuss this behavior in Sec. IV. Many useful results can be obtained by means of comparison of various quantities written in Schwarzschild coordinates ( t, r ) with their counterpart expressions written using the ingoing v or outgoing u null coordinate and the same areal radius r . Using ( v, r ) coordinates, is particularly fruitful. EMT components in ( v, r ) and ( t, r ) coordinates are related via where \u0398 \u00b5\u03bd labels EMT components in ( v, r ) coordinates.", "pages": [1, 2]}, {"title": "B. Perturbative expansion", "content": "From a formal perspective the pure GR case can be described as a system of field equations [20] where the EMT \u00af T and metric \u00af gnear the apparent horizon are described in a spherically symmetric setting in Sec. III. It is then usually assumed that any solution of the MTG belongs to a one-parameter family of analytic solutions [14, 15]. The EMT T \u03bb depends on \u03bb through the metric g \u03bb , and potentially also through effective corrections resulting from perturbative corrections to the modified field equations Eqs. (7)-(9). The self-consistent approach is based on the assumption of at least continuity of the curvature invariants, but uses the Schwarzschild coordinate system where the metric is discontinuous [9, 10]. Imposing the requirement of regularity then allows to identify the valid black hole solutions, whose analytic properties become apparent once they are written in their 'natural' coordinate system [21]. The field equations are supplemented by a set of initial and boundary conditions or constraints. Higher-order terms in the action lead to higher-order equations. Even f ( R ) theories already result in systems with fourth-order metric derivatives. However, it is worth pointing out that the unperturbed solution may not satisfy the boundary conditions since its corresponding equations do not involve the higher-order derivatives [15, 22]. For our purposes it suffices to restrict all considerations to first-order perturbation theory. In any given theory higherorder contributions can be successfully evaluated. There are methods to produce a consistent hierarchy of the higher-order terms and deal with additional degrees of freedom that result from the presence of derivatives of order higher than two. Nevertheless, including terms of order O ( \u03bb 2 ) and higher may not be justified without detailed knowledge of the relative importance of all possible terms in the effective Lagrangian and the cut-off scale that is used to derive it. Spherical symmetry prescribes the form of the metric for all values of \u03bb . We assume that there is a solution of Eq. (2) with the two metric functions C \u03bb and h \u03bb . To avoid spurious divergences we use the physical value of r g ( t ) that corresponds to the perturbed metric g \u03bb , C \u03bb ( r g , t ) = r g. We set and define \u00af C := r g + \u00af W . Similarly, the EMT T \u03bb \u2261 T is decomposed as where \u00af T is extracted from E ( \u00af g [ r g , \u00af W, \u00af h ] , \u00af T ) = 0 . The perturbative terms must satisfy the boundary conditions where the first condition follows from the definition of the Schwarzschild radius, and the perturbation can be treated as small only if the divergence of \u2126 is not stronger than that of \u00af h . Substituting C \u03bb and h \u03bb into Eq. (2) and keeping only the first-order terms in \u03bb results in where \u00af G \u00b5\u03bd \u2261 G \u00b5\u03bd [ r g , \u00af W, \u00af h ] , \u02dc G \u00b5\u03bd is the first-order term in the Taylor expansion in \u03bb where each monomial involves either \u03a3 or \u2126 , and \u00af E \u00b5\u03bd \u2261 E \u00b5\u03bd [ r g , \u00af W, \u00af h ] , i.e. the modified gravity terms are functions of the unperturbed solutions. The explicit form of the equations can be obtained as follows. First note that We introduce the splitting \u03c4 = \u00af \u03c4 + \u03bb \u02dc \u03c4 such that, for instance, the EMT terms of the tt equation can be written as with T r t and T rr expanded analogously. The regularity conditions Eqs. (11) and (12) imply that \u02dc \u03c4 terms should either have the same behavior as their \u00af \u03c4 counterparts when r \u2192 r g, or go to zero faster. Consequently, the schematic of Eq. (24) implies and thus the explicit form of Eq. (7) is", "pages": [2, 3]}, {"title": "III. SELF-CONSISTENT SOLUTIONS IN GR", "content": "Here we give a brief summary of the relevant properties of the self-consistent solutions in GR [9, 10, 21]. In accord with the previous section (and in anticipation of the notation we use in Sec. IV), we label functions of pure classical GR (i.e. \u03bb = 0 ) with a bar, e.g. the metric functions \u00af C and \u00af h . The Einstein field equations for \u00af G tt , \u00af G r t , and \u00af G rr are expressed in terms of the metric functions \u00af C and \u00af h as follows: Only two distinct classes of dynamic solutions are possible [21]. With respect to the regularity conditions of Eqs. (11) and (12), they correspond to the values k = 0 and k = 1 .", "pages": [3]}, {"title": "A. k = 0 class of solutions", "content": "In the k = 0 class of solutions, the limiting form of the reduced EMT components is given by for some function \u00af \u03a5( t ) . The leading terms of the metric functions are where \u00af \u03be ( t ) is determined by the asymptotic properties of the solution. Higher-order terms depend on the higher-order terms in the EMT expansion and will be discussed in Sec. IV. Consistency of the Einstein equations implies The null energy condition requires T \u00b5\u03bd l \u00b5 l \u03bd /greaterorequalslant 0 for all null vectors l \u00b5 [20, 23]. It is violated by radial vectors l \u02c6 a = (1 , \u2213 1 , 0 , 0) for both the evaporating and accreting solutions, respectively. The accreting solution r ' g ( t ) > 0 leads to a firewall: energy density, pressure and flux experienced by an infalling observer diverge at the apparent horizon [21]. The resulting averaged negative energy density in the reference frame of a geodesic observer violates a particular quantum energy inequality [23, 24]. Unless we accept that semiclassical physics breaks down already at the horizon scale, this contradiction implies that a PBH cannot grow after its formation [21]. Hence we consider only evaporating r ' g ( t ) < 0 PBHs in what follows. Matching our results with the standard semiclassical results on black hole evaporation (and accepting that the metric is sufficiently close to the ingoing Vaidya metric with decreasing mass, see [25] for details) results in where the black hole evaporates according to r ' g ( t ) = -\u03b1/r 2 g [20, 26]. Outside of the apparent horizon the geometry differs from the Schwarzschild metric at least on the scale r -r g =: x \u223c \u00af \u03be .", "pages": [3, 4]}, {"title": "B. k = 1 solution", "content": "In the second class of solutions k = 1 and the limiting form of the EMT expansion is given by functions \u00af \u03c4 a \u221d \u00af f . Again, accretion leads to a firewall and thus we will consider only evaporating solutions. It has been shown that dynamic solutions are consistent only in a single case [10], where in the Schwarzschild frame the energy density \u03c1 ( r g ) = \u00af E and pressure p ( r g ) = \u00af P at the apparent horizon are given by Since this is their maximal possible value this k = 1 solution is referred to as extreme [10]. The k = 1 metric functions are Consistency of the Einstein equations then implies For future reference we note here that for the k = 1 solution the Ricci scalar is given by Evaporating black holes are conveniently represented in ( v, r ) coordinates, and the limiting form of the k = 0 solution as r \u2192 r g is a Vaidya metric with decreasing Misner-Sharp mass C + ( v ) ' < 0 [25]. Using ( v, r ) coordinates to describe geometry at the formation of the first marginally trapped surface reveals how the two classes of solutions are connected (see Ref. [10] for details): at its formation, a PBH is described by a k = 1 solution with \u00af E = -\u00af P = 1 / (8 \u03c0r 2 g ) . It immediately switches to the k = 0 solution. However, the abrupt transition from f 1 to f 0 behavior does not lead to discontinuities in the curvature scalars or other physical quantities that could potentially be measured by a local or quasilocal observer.", "pages": [4]}, {"title": "IV. SELF-CONSISTENT SOLUTIONS IN MTG", "content": "To describe perturbative PBH solutions in MTG the equations must satisfy the same consistency relations as their GR counterparts. Taking the GR solutions as the zeroth-order approximation, we express the functions describing the MTG metric g \u03bb = \u00af g + \u03bb \u02dc g and thus represent the modified Einstein equations as series in integer and half-integer powers of x := r -r g. Their order-by-order solution results in formal expressions for \u03a3( t, r ) and \u2126( t, r ) . However, we also obtain a number of consistency conditions that must be satisfied identically in order for a given theory to admit formation of a PBH. The GR solutions with k \u2208 { 0 , 1 } are sufficiently different to merit a separate treatment provided in Subsec. IV A and IV B, respectively. In both instances, power expansions in various expression have to match up to allow for self-consistent solutions of the modified Einstein equations. Moreover, the relations between the EMT components that are given by Eqs. (14)-(16) must hold separately for both the unperturbed terms and the perturbations. For a given MTG (that is defined by the set of parameters { a 1 , a 2 , a 3 , \u00b7 \u00b7 \u00b7 } in Eq. (1)) these constraints may conceivably lead to several outcomes: first, it is possible that some of the terms in the Lagrangian Eq. (1) contribute terms to \u00af E \u00b5\u03bd such that their expansions around x = 0 lead to terms that diverge stronger than any other terms in Eqs. (30)-(32). If only one higher-order curvature term is responsible for such behavior, then such a theory cannot produce perturbative PBH solutions, and only nonperturbative solutions may be possible or the corresponding coefficient a i \u2261 0 . If the divergences originate from several terms, they can either cancel if a particular relationship exists between their coefficients a i , a i ' , . . . , or not. In the former case the existence of perturbative PBH solutions imposes a constraint, not on the form of the available terms, but on the relationships between their coefficients. It is also possible that, as it happens in the Starobinsky model (Sec. V), divergences of the terms \u00af E \u00b5\u03bd match the divergences of the GR terms. The constraints can then be satisfied (i) identically (providing us with no additional information); (ii) only for a particular combination of the coefficients a i , thereby constraining the possible classes of MTG; (iii) only in the presence of particular higher-order terms, irrespective of the coefficients, and only for certain unperturbed solutions. In the last scenario, where only certain GR solutions are consistent with a small perturbation, this should be interpreted as an argument against the presence of that particular term in the Lagrangian of Eq. (1). There is a priori no reason why \u00af g \u00b5\u03bd /greatermuch \u03bb \u02dc g \u00b5\u03bd should hold in some boundary layer around r g [15, 22]. If this condition is not satisfied, then the classification scheme of the GR solutions and a mandatory violation of the null energy condition are not necessarily true. We discuss some of the properties of the solutions without a GR limit and derive the necessary conditions for their existence in Sec. IV C. Throughout this section we use the letter j \u2208 Z 1 2 to label integer and half-integer coefficients and powers of x in series expansions and /lscript to refer to generic coefficients. Since we give explicit expressions only for the first few terms in each expression, we write c 12 instead of c 1 / 2 , h 12 instead of h 1 / 2 , and similarly for higher orders and coefficients of the EMT expansion.", "pages": [4, 5]}, {"title": "A. Black holes of the k = 0 type", "content": "For the k = 0 class of solutions the leading terms in the metric functions of classical GR are given as series in powers of x := r -r g as where and higher-order coefficients of the metric functions are related to higher-order terms in the EMT expansion where a \u2208 { t , r t , r } \u2261 { tt , r t , rr } . We omit the explicit specification j \u2208 Z 1 2 from the summation range in what follows. Regularity of the metric at the apparent horizon and consistency of the Einstein equations establish algebraic and differential relations between various coefficients. In particular, using Eqs. (14)-(16) and Eq. (34), we find and The expansion of e 2 h that is given by Eq. (25) is obtained as follows: separating the logarithmically divergent part of h ( t, x ) from the rest, Eq. (23) allows to write for some \u02dc \u03be ( t ) , where \u00af \u03c7 = \u2211 j h j x j and \u03c9 = \u2211 j \u03c9 j x j are convergent functions. First-order expansion in \u03bb then leads to Eq. (25) with Therefore, the first-order corrections of Eqs. (19)-(20) to the metric functions of Eqs. (46)-(47) are given by the series These two functions can be expressed in terms of the unperturbed solution and corrections \u02dc \u03c4 a to the EMT from the series expansion of Eqs. (30)-(32). These equations contain various divergent expressions. For example, the term \u03a3 \u2202 r \u00af C as well as all other terms apart from e -2 \u00af h \u00af E tt in Eq. (30) are finite when x \u2192 0 . Then Eq. (47) implies that the series expansion of \u00af E tt starts with a term that is proportional to 1 /x . Performing the same analysis for the two remaining Einstein equations Eqs. (31)-(32) yields the decompositions of the modified gravity terms that should hold for any F ( g \u00b5\u03bd , R \u00b5\u03bd\u03c1\u03c3 ) , where indices of coefficients of negative exponents of x are labeled by a bar. From the requirement that the Ricci scalar R [ g \u03bb ] be finite at the horizon, we obtain the condition We use additional subscripts (e.g. ' | R ' in the expression above) to indicate what equation was used to derive the explicit expression. The perturbative contributions Eqs. (55)-(56) to the metric functions Eqs. (46)-(47) are obtained order-by-order from the series solutions of Eqs. (30)-(32). Expressions for every expansion coefficient can be obtained separately from each equation. Matching of the expressions then allows to identify the coefficients \u00e6 /lscript , \u0153 /lscript , \u00f8 /lscript of the modified gravity terms Eqs. (57)-(59). Expressions for \u03c3 12 for instance are obtained from the lowest-order coefficients of Eqs. (30)-(32). As a result, we obtain three independent constraints They are simultaneously satisfied (see Appendix A 1) if These three equations not only identify the function \u02dc \u03be ( t ) in terms of unperturbed quantities, but also establish the two relations between the leading expansion coefficients of the MTG terms. Similarly, the next-highest order coefficients of Eqs. (30)-(32) allow to obtain expressions for \u03c3 1 , see Appendix A 1. Comparison of gives an expression for \u03c9 12 . Substitution into Eq. (64) and subsequent comparison of \u03c3 1 | tr = \u03c3 1 | rr with \u03c3 1 | tt gives a relation between the next-highest order coefficients, namely The modified gravity terms \u00af E \u00b5\u03bd must adhere to the expansion structures in Eqs. (57)-(59), and the relations Eqs. (63) and (65) between their coefficients must be satisfied identically. Otherwise the MTG solutions do not exist. For terms in the metric functions of order O ( x 3 / 2 ) and higher the three equations Eqs. (30)-(32) for the tt , tr , and rr component contain three additional independent variables (\u00af \u03c4 a ) j /greaterorequalslant 3 / 2 and will therefore not lead to any additional constraints. It is worth pointing out that the analogs of Eqs. (63) and (65) are also satisfied by the coefficients of the corresponding metric tensor and Ricci tensor components themselves, i.e. and 2 where Eq. (66) is satisfied trivially and Eq. (68) simplifies to (\u00af g tt ) 12 = -\u00af \u03be (\u00af g rr ) 12 due to the diagonal form of the metric tensor \u00af g r t = 0 (see Eq. (3)). Explicit expressions for the coefficients of the Ricci tensor components in Eq. (69) are provided in Appendix B, see Eqs. (B1)-(B3).", "pages": [5, 6]}, {"title": "B. Black holes of the k = 1 type", "content": "The EMT expansion for the k = 1 solution is given in terms of x := r -r g by where \u00af E = -\u00af P = 1 / (8 \u03c0r 2 g ) and \u00af \u03a6 = 0 . To improve readability and clarify the connection to physical quantities (energy, pressure, flux) we set (\u00af \u03c4 t ) j =: \u00af e j , (\u00af \u03c4 r ) j =: \u00af p j , and (\u00af \u03c4 r t ) j =: \u00af \u03c6 j , and analogously for the perturbative coefficients (\u02dc \u03c4 a ) j . Additional relations between the coefficients are obtained from Eqs. (14)-(15), i.e. for the two next-highest orders j = 2 , 5 2 . Recall that (cf. Eqs. (42)-(43)) the leading terms in the metric functions of classical GR are given as series in powers of x := r -r g as with coefficients 12 14\u00af e ( \u221a - r 5 / 2 g - ) Higher-order coefficients are obtained from higher-order terms of the EMT expansion using consistency of the Einstein equations, e.g. the next-highest order coefficient of Eq. (76) is 52 52 h = +5\u00af e 7\u00af p . (79) In addition, consistency of the Einstein equations requires Eq. (44) and Substituting Eq. (81) into Eq. (79) we obtain the identity which leads to many simplifying cancellations, e.g. the absence of the \u221a x term in the Ricci scalar \u00af R (cf. Eq. (45)) due to R 12 \u221d c 32 h 12 -c 2 , where R 12 denotes the \u221a x coefficient of \u00af R . Again, the expansion of e 2 h that is given by Eq. (25) is obtained by separating the logarithmic part of h ( t, x ) from the rest. From the expansion we then obtain Eq. (25) with The series expansions of the perturbative corrections of Eqs. (19)-(20) are therefore given by the power series Finiteness of the Ricci scalar at the horizon requires Eq. (44) and The expansion structure of the modified gravity terms \u00af E \u00b5\u03bd is obtained analogous to Sec. IV A. We find The equation for the x 0 coefficient of the tr component Eq. (31) allows to identify and Substitution of Eq. (91) into the expression \u03c3 32 | tt ( \u02dc E ) obtained from Eq. (30) and subsequent comparison with the expression \u03c3 32 | rr obtained from Eq. (32) establishes the relation between the lowest-order coefficients of the MTG terms, see Appendix A 2. Similarly, by substituting Eq. (91) into the expression for \u03c3 2 | tt , and \u02dc \u03be | tr obtained from the \u221a x coefficient of Eq. (31) into the expression Eq. (87) for \u03c3 32 | R , we can derive two distinct expressions for the sum \u02dc e 2 +\u02dc p 2 by comparison of \u03c3 2 | tt and \u03c3 2 | rr obtained from Eqs. (30) and (32), respectively, as well as comparison of \u03c3 32 | R and \u03c3 32 | tt . Their identification establishes the additional relation between the modified gravity coefficients of Eqs. (88)-(90). A detailed derivation with explicit expressions is provided in Appendix A 2. Analogous to the class of k = 0 black hole solutions discussed in Sec. IV A, the modified gravity terms \u00af E \u00b5\u03bd of any self-consistent MTG must follow the expansion structures prescribed by Eqs. (88)-(90) and identically satisfy the two relations Eqs. (92)-(93) to be compatible with black hole solutions of the k = 1 type. Again, consideration of higher-order coefficients in Eqs. (30)-(32) introduces new independent variables and will thus not yield any additional constraints. Once more, the analogs of the MTG coefficient relations Eqs. (92)-(93) are also satisfied by the coefficients of the corresponding metric tensor and Ricci tensor components themselves, i.e. where Eq. (82) was used to simplify the expressions, ( \u00af R tt ) 32 = ( \u00af R r t ) 0 = ( \u00af R rr ) 32 = 0 , ( \u00af R rr ) 2 = -3 c 2 32 / (2 r 3 g ) , and Eqs. (94) and (96) simplify due to the diagonal form of the metric tensor \u00af g r t = 0 (see Eq. (3)).", "pages": [6, 7]}, {"title": "C. \u03bb -expanded solutions", "content": "We now consider solutions where the leading reduced components of the EMT are not dominated by terms of order O ( \u03bb 0 ) . To obtain mathematically consistent expressions we have to extend the expansion to terms of order O ( \u03bb 2 ) as higher-order terms, if needed, are obtained analogously. The k = 0 solution without GR limit has the following properties: the EMT expansion corresponds to the case where as r \u2192 r g, lim \u03c4 t = lim \u03c4 r = lim \u03c4 r t . The equations below are trivially extendable to the case where the leading term in \u03c4 r t = -\u03bb \u02dc \u039e . In either case, the metric functions are given by similar to the k = 0 perturbative solution. Here, the structure of the metric function h was simplified by redefining the time, and the coefficient c 12 = \u03b6 12 + \u03bb\u03c3 12 + \u03bb 2 \u03c3 (2) 12 was simplified by taking into account the requirement that the Ricci scalar must be finite at the apparent horizon, i.e. Unlike in GR, the sign of \u02dc \u039e (and \u02dc \u039e (2) ) cannot be determined solely from the requirements of existence and consistency of the modified Einstein equations. It is therefore unclear whether or not violation of the null energy condition is a prerequisite for the formation of a PBH. This is in contrast to GR, where such a violation has been shown to be mandatory in a variety of settings [9, 21, 26, 27]. The expansion structure of the non-GR terms E \u00b5\u03bd remains the same as in the perturbative k = 0 scenario discussed in Subsec. IV A, that is where \u00e6 /lscript := \u00af\u00e6 /lscript + \u03bb \u02dc \u00e6 /lscript , and similarly for the coefficients \u0153 /lscript , \u00f8 /lscript of the non-GR terms E r t and E rr . Substitution into the generic modified Einstein equations Eqs. (7)-(9) gives \uf8f3 at the respective leading orders of x , and leads to the following constraints: first, the expansion coefficients ( \u03c4 a ) 12 = 0 \u2200 a , where ( \u03c4 a ) /lscript := (\u00af \u03c4 a ) /lscript + \u03bb (\u02dc \u03c4 a ) /lscript + \u03bb 2 (\u02dc \u03c4 (2) a ) /lscript , see Eq. (98). At the leading expansion order (which is O ( \u03bb ) since the non-GR terms appear as \u03bb E \u00b5\u03bd in Eqs. (7)-(9)) the lowest-order x coefficients satisfy which is analogous to Eq. (63), but in this case identifies the leading reduced term in the EMT. Similarly, the next-order O ( \u03bb 2 ) expansion coefficients satisfy", "pages": [7, 8]}, {"title": "V. BLACK HOLES IN THE STAROBINSKY MODEL", "content": "Numerous modifications of GR have been proposed, including theories that involve higher-order curvature invariants. Apopular class among these are so-called f ( R ) theories [2], in which the gravitational Lagrangian density L g is an arbitrary function of the Ricci scalar R . In this section, we consider the Starobinsky model [16] with F = \u03c2R 2 , \u03c2 = 16 \u03c0a 2 /M 2 P (see Eq. (1)). It is a straightforward extension of GR with quadratic corrections in the Ricci scalar that is of relevance in cosmological contexts. In particular, it is the first selfconsistent model of inflation. New horizonless solutions in this model have been identified recently in an analysis [28] of static, spherically symmetric, and asymptotically flat vacuum solutions.", "pages": [8]}, {"title": "A. Modified Einstein equations", "content": "In f ( R ) theories, the relevant equations have a relatively simple form. For the action where the gravitational Lagrangian L g = f ( R ) , the matter Lagrangian is represented by L m , and S b denotes the boundary term, the field equations for the metric g \u00b5\u03bd are given by where f ' := \u2202 f ( R ) /\u2202R and /square := g \u00b5\u03bd \u2207 \u00b5 \u2207 \u03bd . It is convenient to set f ( R ) =: R + \u03bb F ( R ) . The modified Einstein equations are then Performing the expansion in \u03bb and only keeping terms up to the first order we obtain expressions for the modified gravity terms \u00af E \u00b5\u03bd , i.e. where all objects labeled by the bar are evaluated with respect to the unperturbed metric \u00af g, and F \u2261 F ( \u00af R ) . In spherical symmetry the d'Alembertian is given by Second-order covariant derivatives of a scalar function can be expressed in terms of partial derivatives, i.e. In the Starobinsky model F ( R ) = \u03c2 \u00af R 2 + O ( \u03bb ) and Eqs. (30)(32) become", "pages": [8, 9]}, {"title": "B. Compatibility with the k = 0 class of black hole solutions", "content": "With the k = 0 metric functions Eqs. (46)-(47), the constraint Eq. (52) that is obtained from the requirement that the Ricci scalar be non-divergent leads to cancellations in the Ricci tensor components \u00af R tt and \u00af R rr which ensures that the MTG terms Eqs. (119)-(121) of the \u02dc f ( \u00af R ) = \u03c2 \u00af R 2 Starobinsky model conform to the structures of Eqs. (57)-(59). We find that both of the two constraints posed by Eq. (63) are satisfied, i.e. where R j is used to denote coefficients of the Ricci scalar \u00af R = \u2211 j R j x j = R 0 + R 12 \u221a x + R 1 x + O ( x 3 / 2 ) . Similarly, the next-highest order coefficients satisfy the constraint of Eq. (65), see App. B, Eqs. (B4)-(B6).", "pages": [9]}, {"title": "C. Compatibility with the k = 1 solution", "content": "Similar to the k = 0 case, the k = 1 constraint on the evolution of the horizon radius Eq. (44) that is required to ensure consistency of the Einstein equations and finiteness of the Ricci scalar leads to cancellations in \u00af R tt , and the k = 1 Starobinsky MTG terms of Eqs. (119)-(121) follow the structures prescribed by Eqs. (88)-(90). Using the k = 1 metric functions Eqs. (76)-(77) we obtain the lowest-order coefficients of \u00af E tt from Eq. (119). Similarly, we obtain the coefficients of \u00af E r t and \u00af E rr from Eqs. (120)-(121), where R 1 denotes the x coefficient of the Ricci scalar \u00af R = \u2211 j R j x j = 2 /r 2 g + R 1 x + O ( x 3 / 2 ) and R 12 = 0 , cf. Eq. (45). With the expressions given in Eqs. (123)-(127), it is easy to verify that both k = 1 constraints Eq. (92) and Eq. (93) are satisfied identically.", "pages": [9]}, {"title": "D. Compatibility with the \u03bb -expanded k = 0 class of black hole solutions", "content": "Equality of the coefficients in Eqs. (111) and (112) follows in exactly the same fashion as in Sec. V B. Explicit calculation confirms that the coefficients of the MTG terms in the Starobinsky model Eqs. (119)-(121) obtained using the EMT expansion of Eq. (98) and metric functions Eqs. (99)-(100) coincide with those of Eq. (122) at the leading expansion order O ( \u03bb ) . Terms of order O ( \u03bb 0 ) vanish in accordance with Eq. (122) (note that c 12 \u221d O ( \u03bb ) ). This confirms that the Starobinsky solution is consistent with the generic form of PBH solutions. However, since \u02dc \u039e (2) is undetermined in the self-consistent approach, Eq. (112) does not impose any constraints on the function \u03be .", "pages": [9]}, {"title": "VI. DISCUSSION", "content": "We have analyzed the properties of metric MTG and derived several constraints that they must satisfy to be compatible with the existence of an apparent horizon. Since we have not specified the origin of the deviations from GR, the results presented here are generic and apply to all conceivable selfconsistent metric MTG. Constraints on a perturbative solution in a particular metric MTG arise from two sources: first, the series expansions of the modified gravity terms \u00af E \u00b5\u03bd in terms of the distance x := r -r g from the horizon must follow a particular structure that is prescribed by the modified Einstein equations with terms that diverge in the limit r \u2192 r g. Second, a general spherically symmetric metric allows for two independent functions C and h that must satisfy three Einstein equations. The resulting relations between coefficients \u03c3 /lscript , \u03c9 /lscript of their perturbative corrections translate into relationships between the coefficients c /lscript and h /lscript , and eventually components of the unperturbed EMT. These constraints must be satisfied identically. Otherwise, a valid solution of GR cannot be perturbatively extended to a solution of a MTG. Identities that must be satisfied for the existence of the perturbative k = 0 solutions are given by Eqs. (63) and (65), and for the k = 1 solution by Eqs. (92)-(93). On the other hand, there are nonperturbative solutions that do not have a well-defined GR limit. In this case, the constraints on a MTG that are imposed by the existence of a regular apparent horizon formed in finite time of a distant observer are given by Eqs. (111)-(112). Using the Starobinsky R 2 model, arguably the simplest possible MTG, we identify both perturbative and nonperturbative solutions. However, this is not the only theory that should be investigated: in a future article [29], we will consider generic f ( R ) theories of the form f ( R ) = R + \u03bb F ( R ) , where F ( R ) = \u03c2R q and q, \u03c2 \u2208 R . In particular, this includes the case q = 1 / 2 (i.e. f ( R ) = R + \u03bb\u03c2 \u221a R ) considered in Ref. [30], as well as the case of negative exponents q < 0 considered in Ref. [31]. More general MTG (e.g. those involving higher-order curvature invariants) will also be considered.", "pages": [9, 10]}, {"title": "ACKNOWLEDGMENTS", "content": "We thank Eleni Kontou, Robert Mann, Shin'ichi Nojiri, Vasilis Oikonomou, and Christian Steinwachs for useful discussions and helpful comments. SM is supported by an International Macquarie University Research Excellence Scholarship and a Sydney Quantum Academy Scholarship. The work of DRT was supported in part by the Southern University of Science and Technology, Shenzhen, China, and by the ARC Discovery project grant DP210101279.", "pages": [10]}, {"title": "1. k = 0 black hole solutions", "content": "With the metric functions Eqs. (46)-(47) of the k = 0 solutions we obtain the following coefficients for the perturbative correction \u03a3 of Eq. (55) from Eqs. (30)-(32): Via comparison of Eqs. (A5) and (A6) we can identify the coefficient for the perturbative correction \u2126 of Eq. (56). Substitution of Eq. (A7) into Eqs. (A5) and (A6) then yields Subsequent comparison of Eq. (A8) and (A4) establishes the relation Eq. (65) between the coefficients \u00e6 12 , \u0153 0 , and \u00f8 12 .", "pages": [10]}, {"title": "2. k = 1 black hole solution", "content": "With the metric functions Eqs. (76)-(77) of the k = 1 solution we obtain the following coefficients for the perturbative correction \u03a3 of Eq. (85) from Eqs. (30) and (32): From the x 0 and \u221a x coefficients of Eq. (31) we obtain where \u02dc \u03a6 = ( \u02dc E + \u02dc P ) / 2 and \u02dc \u03c6 2 = (\u02dc e 2 + \u02dc p 2 ) / 2 , see Eqs. (73) and (75). By substitution of Eq. (A13) into Eq. (A11) we obtain from the comparison \u03c3 2 | tt -\u03c3 2 | rr = 0 . Similarly, substitution of Eq. (A14) into Eq. (87) and subsequent comparison of \u03c3 32 | R -\u03c3 32 | tt = 0 yields Subtracting Eq. (A15) from Eq. (A16) and subsequent multiplication by 8 \u03c0 \u00af \u03be 3 r g yields Lastly, substituting \u00e6 32 from Eq. (92) into (A17) and rearranging gives Eq. (93). Explicit expressions for the individual terms in Eq. (69): Explicit expressions for the MTG coefficients \u00e6 12 , \u0153 0 , \u00f8 12 in the Starobinsky model of the k = 0 solution (see Subsec. V B): cepts and New Developments (Springer, Dordrecht, 1998).", "pages": [10, 11, 12, 13]}] |
2020MNRAS.491.3155K | https://arxiv.org/pdf/1910.11893.pdf | <document>
<section_header_level_1><location><page_1><loc_7><loc_79><loc_85><loc_85></location>Stellar Dynamos with Solar and Anti-solar Differential Rotations: Implications to Magnetic Cycles of Slowly Rotating Stars</section_header_level_1>
<section_header_level_1><location><page_1><loc_8><loc_74><loc_69><loc_75></location>Bidya Binay Karak ? , Aparna Tomar and Vindya Vashishth</section_header_level_1>
<text><location><page_1><loc_8><loc_73><loc_71><loc_74></location>1 Department of Physics, Indian Institute of Technology (Banaras Hindu University), Varanasi, India</text>
<text><location><page_1><loc_7><loc_68><loc_40><loc_69></location>Accepted XXX. Received YYY; in original form ZZZ</text>
<section_header_level_1><location><page_1><loc_29><loc_64><loc_39><loc_65></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_29><loc_38><loc_91><loc_64></location>Simulations of magnetohydrodynamics convection in slowly rotating stars predict antisolar differential rotation (DR) in which the equator rotates slower than poles. This anti-solar DR in the usual GLYPH<11> GLYPH<10> dynamo model does not produce polarity reversal. Thus, the features of large-scale magnetic fields in slowly rotating stars are expected to be different than stars having solar-like DR. In this study, we perform mean-field kinematic dynamo modelling of different stars at different rotation periods. We consider anti-solar DR for the stars having rotation period larger than 30 days and solar-like DR otherwise. We show that with particular GLYPH<11> profiles, the dynamo model produces magnetic cycles with polarity reversals even with the anti-solar DR provided, the DR is quenched when the toroidal field grows considerably high and there is a sufficiently strong GLYPH<11> for the generation of toroidal field. Due to the anti-solar DR, the model produces an abrupt increase of magnetic field exactly when the DR profile is changed from solar-like to anti-solar. This enhancement of magnetic field is in good agreement with the stellar observational data as well as some global convection simulations. In the solar-like DR branch, with the decreasing rotation period, we find the magnetic field strength increases while the cycle period shortens. Both of these trends are in general agreement with observations. Our study provides additional support for the possible existence of anti-solar DR in slowly rotating stars and the presence of unusually enhanced magnetic fields and possibly cycles which are prone to production of superflare.</text>
<text><location><page_1><loc_29><loc_34><loc_91><loc_37></location>Key words: Sun: activity, dynamo, magnetic fields - stars: solar-type, rotation, activity.</text>
<section_header_level_1><location><page_1><loc_7><loc_29><loc_24><loc_30></location>1 INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_7><loc_11><loc_47><loc_28></location>Many sun-like low-main sequence stars show magnetic cycles which are usually studied by measuring the chromospheric emissions in Ca II H & K line cores and the coronal X-ray (Baliunas et al. 1995; Pallavicini et al. 1981). These measurements show that the periods of magnetic cycles vary in different stars and there is a weak trend of the period becoming shorter with the increase of rotation rates (Noyes et al. 1984b; Su'arez Mascare˜no et al. 2016; Boro Saikia et al. 2018). Some recent observations, however, do not find this trend in the so-called active branch; see Fig. 21 of Su'arez Mascare˜no et al. (2016). On the other hand, the stellar magnetic activity or the field strength increases with the increase of rotation rate in the small rotation range and then it tends</text>
<text><location><page_1><loc_51><loc_9><loc_91><loc_30></location>to saturate in the rapid rotation limit. This behaviour is often represented with respect to the Coriolis number (or the inverse of Rossby number) which is a ratio of the convective turnover time to the rotation period (Noyes et al. 1984a; Wright et al. 2011; Wright & Drake 2016). However, Reiners et al. (2014) show that rotation period alone better represents the data. In the slowly rotating stars with rotation rates below about the solar value, an interesting behaviour has been recognised. Giampapa et al. (2006, 2017) have found an increase of magnetic activity with a decrease of GLYPH<28> GLYPH<157> P rot (i.e., increase of rotation period). Another exciting aspect of stellar activity is that the magnetic cycles of Sun-like stars show the Waldmeier effect (the stronger cycles rise faster than the weaker ones) (Garg et al. 2019), which is popularly known for the Sun (Waldmeier 1935; Karak & Choudhuri 2011).</text>
<text><location><page_1><loc_54><loc_7><loc_91><loc_8></location>Starting from the pioneering work of Gilman (1977,</text>
<section_header_level_1><location><page_2><loc_7><loc_90><loc_34><loc_91></location>2 Karak, Tomar & Vashishth</section_header_level_1>
<text><location><page_2><loc_7><loc_52><loc_47><loc_88></location>1983), numerous global magnetohydrodynamics (MHD) convection simulations in spherical geometry have been performed to study the magnetic fields and flows in stars. These simulations in some parameter ranges produce large-scale magnetic fields and even cycles. In general, it has been observed that in the rapidly rotating stars, magnetic cycles and polarity reversals are preferred, while in the slowly rotating ones, simulations rarely produce reversals (Brun & Browning 2017; Warnecke 2018). Most of the simulations near the solar rotation rate (or Rossby number around one) find a transition of differential rotation (hereafter DR) from solar-like to the so-called anti-solar profile, in which the equator rotates slower than high latitudes (Gilman 1977; Guerrero et al. 2013; Gastine et al. 2014; Kapyla et al. 2014; Fan & Fang 2014; Karak et al. 2015; Featherstone & Miesch 2015; Karak et al. 2018a). In most of the cases, when simulations produce anti-solar DR, they do not produce magnetic field reversals (Warnecke 2018). It is not difficult to understand that the anti-solar DR in GLYPH<11> GLYPH<10> dynamo model, does not allow polarity reversal. The anti-solar DR generates a toroidal field in such a way that the poloidal field produced through the GLYPH<11> effect (positive in the northern hemisphere) from this toroidal field is in the same direction as that of the old poloidal field. In Figure 1, we show how a Babcock-Leighton type GLYPH<11> GLYPH<10> dynamo with anti-solar DR produces poloidal field in the same direction as that of the original field and thereby does not offer reversal of the poloidal field at the end of a dynamo cycle.</text>
<text><location><page_2><loc_7><loc_10><loc_47><loc_51></location>Based on the available data of the observed polar field, DR and other surface features, it is expected that the solar dynamo is primarily of GLYPH<11> GLYPH<10> type (Cameron & Schussler 2015). In recent years, it has been realized that it is the Babcock-Leighton process which acts like an GLYPH<11> effect to generate the poloidal field through the decay and dispersal of tilted bipolar magnetic regions (BMRs) (DasiEspuig et al. 2010; Kitchatinov & Olemskoy 2011a; Mu˜nozJaramillo et al. 2013; Priyal et al. 2014). Although in the Sun, DR is the dominating source of the toroidal field, a weak toroidal field might be generated through the GLYPH<11> effect. With the increase of stellar rotation rate, GLYPH<11> increases (Krause & Radler 1980), while the DR ( GLYPH<10> ) does not have a strong dependency with the rotation (Kitchatinov & Rudiger 1999). Thus essentially, the stellar dynamo is of GLYPH<11> 2 GLYPH<10> type and this type of dynamo can result in polarity reversal depending on the profiles of GLYPH<11> and GLYPH<10> . Therefore, in the simulations with anti-solar DR, the magnetic field reversal is subtle as the toroidal field can be produced through the GLYPH<11> effect in addition to GLYPH<10> . Incidentally, Karak et al. (2015) found little reversal of magnetic field although it does not occur globally in all latitudes; see their Fig. 9 and 10, Runs ABC. They also found some irregular cycles in anti-solar DR regime. Recently, Viviani et al. (2018) found noticeable polarity reversal in anti-solar DR regime, however, Warnecke (2018) found almost no signature of polarity reversals in this regime. In summary, there is no consensus in the polarity reversal of the large-scale magnetic field in the simulations of the slowly-rotating stars producing anti-solar DR. In this study, we shall explore this and understand how the magnetic polarity reversal and cycles can be possible in this regime.</text>
<text><location><page_2><loc_7><loc_7><loc_47><loc_9></location>The most interesting feature is the change of magnetic field strength in the slowly rotating stars. As referenced</text>
<figure>
<location><page_2><loc_54><loc_74><loc_86><loc_88></location>
<caption>Figure 1. Pictorial representation of the Babcock-Leighton dynamo model with anti-solar DR. (a) Initial poloidal field line. (b) and (c) This field is stretched by the anti-solar DR (equator rotates slower than poles) to produce a toroidal field. (d) The toroidal field rises to the surface and form tilted BMRs with loop structures. (e) Opposite polarity of sunspots connects near the equator and create a big poloidal loop (dashed line). (f) The large-scale poloidal field developed (dashed line) is in the same orientation as that of the original one.</caption>
</figure>
<text><location><page_2><loc_51><loc_19><loc_91><loc_55></location>above, observations of slowly rotating stars show an increase of magnetic activity with the increase of rotation period which is possibly caused by the transition of DR to antisolar from the solar-like profile as proposed by Brandenburg &Giampapa (2018). Indeed, Karak et al. (2015) found an increase of the magnetic field in the anti-solar DR regime. Simulations of Warnecke (2018) in different parameter regimes seem to show a little increase of the magnetic field, in contrast, Viviani et al. (2018) did not find significant increase. This remains a curiosity in the community whether the magnetic field indeed increases in the anti-solar DR regime and if so, then whether this is actually caused by the anti-solar DR. The answers to these are not obvious because the strength of the dynamo, as determined by the GLYPH<11> and shear, do not necessarily increase in the anti-solar DR regime (slowly rotating stars). In fact, we expect the strength of GLYPH<11> to decrease with the decrease of rotation rate. Well, in simulations it has been observed that the shear is much strong in the anti-solar DR regime (see Table 1 of Karak et al. (2015) and Viviani et al. (2018)). But this cannot be the reason for the increased magnetic activity in slowly rotating stars because otherwise both Viviani et al. (2018) and Warnecke (2018) would also find an abrupt increase of magnetic field. Therefore, we shall understand the behaviour of the stellar dynamo with antisolar DR which is found in the slowly rotating stars. We shall explore whether the increase of magnetic field as seen in observation is possible in the dynamo model with anti-solar DR and what causes this increase.</text>
<text><location><page_2><loc_51><loc_7><loc_91><loc_18></location>In our study, without going through the complexity of the global MHD convection simulations and saving the computational resources, we shall develop a simple kinematic dynamo model by specifying GLYPH<11> and GLYPH<10> for the sources of magnetic fields and make some clean simulations to explore the physics behind the magnetic cycles and polarity reversal in stars. We shall explore how the magnetic field strength varies when the DR profile changes from solar to anti-solar. We shall also present how other features of magnetic cycles,</text>
<text><location><page_3><loc_7><loc_85><loc_47><loc_88></location>namely cycle period and the ratio of poloidal to toroidal fields change with the stellar rotation.</text>
<section_header_level_1><location><page_3><loc_7><loc_80><loc_16><loc_81></location>2 MODEL</section_header_level_1>
<text><location><page_3><loc_7><loc_69><loc_47><loc_79></location>In our study, we develop a kinematic mean-field dynamo model by considering only the diagonal terms of the GLYPH<11> coefficients and an isotropic turbulent diffusion GLYPH<17> . We further assume magnetic field to be axisymmetric, thus writing magnetic field B = r GLYPH<2> » A ' r ; GLYPH<18>; t ' ˆ GLYPH<30> … + B ' r ; GLYPH<18>; t ' ˆ GLYPH<30> , where A is vector potential for poloidal field B p GLYPH<17> ' B r ˆ r ; B GLYPH<18> ˆ GLYPH<18> ' , B is toroidal field, and GLYPH<18> is co-latitude. Subsequently, the equations for A and B can be derived as</text>
<formula><location><page_3><loc_7><loc_65><loc_47><loc_67></location>@ A @ t + 1 s ' v p GLYPH<1> r '' sA ' = GLYPH<17> GLYPH<18> r 2 GLYPH<0> 1 s 2 GLYPH<19> A + GLYPH<11> GLYPH<30> GLYPH<30> B (1)</formula>
<formula><location><page_3><loc_10><loc_57><loc_47><loc_63></location>@ B @ t + 1 r GLYPH<20> @ @ r ' r v r B ' + @ @GLYPH<18> ' v GLYPH<18> B ' GLYPH<21> = GLYPH<17> GLYPH<18> r 2 GLYPH<0> 1 s 2 GLYPH<19> B + s ' B p GLYPH<1> r ' GLYPH<10> + S Tor GLYPH<11> ' r ; GLYPH<18> ' + 1 r d GLYPH<17> dr @ ' rB ' @ r (2)</formula>
<text><location><page_3><loc_7><loc_54><loc_32><loc_55></location>where s = r sin GLYPH<18>; v p GLYPH<17> ' v r ˆ r ; v GLYPH<18> ˆ GLYPH<18> '' , and</text>
<formula><location><page_3><loc_11><loc_44><loc_47><loc_53></location>S Tor GLYPH<11> ' r ; GLYPH<18> ' = GLYPH<0> GLYPH<11> GLYPH<18> GLYPH<18> GLYPH<18> 2 r @ A @ r + @ 2 A @ r 2 GLYPH<19> + GLYPH<11> rr r 2 GLYPH<18> A sin 2 GLYPH<18> GLYPH<0> cot GLYPH<18> @ A @GLYPH<18> GLYPH<0> @ 2 A @GLYPH<18> 2 GLYPH<19> GLYPH<0> 1 r @GLYPH<11> rr @GLYPH<18> GLYPH<18> A cot GLYPH<18> r + 1 r @ A @GLYPH<18> GLYPH<19> GLYPH<0> @GLYPH<11> GLYPH<18> GLYPH<18> @ r GLYPH<18> A r + @ A @ r GLYPH<19> (3)</formula>
<text><location><page_3><loc_7><loc_36><loc_47><loc_43></location>with GLYPH<11> rr ; GLYPH<11> GLYPH<18> GLYPH<18> and GLYPH<11> GLYPH<30> GLYPH<30> being the diagonal components of GLYPH<11> tensor. In our model, we shall ignore the off-diagonal components (which are responsible for magnetic pumping) because of the limited knowledge of their profiles in the solar and stellar convection zones (CZs) and to keep our model and the interpretation of the results tractable.</text>
<text><location><page_3><loc_7><loc_17><loc_47><loc_35></location>Numerical simulations of magneto-convection in local Cartesian domain (e.g., Kapyla & Brandenburg 2009; Karak et al. 2014b) and global solar/stellar CZs (e.g., Simard et al. 2013, 2016; Warnecke et al. 2018) provide some guidance, although the simulation carried out are still far from the real Sun (in terms of its fundamental parameters). These simulations show that the GLYPH<11> components are highly inhomogeneous across radius and latitudes. They change sign at the equator and have no resemblance with each other. We also note that there is some amount of uncertainty in the measurement of GLYPH<11> coefficients, and so far, we do not have a well-tested method for their measurement. Thus keeping general features of the GLYPH<11> coefficients as obtained through theory and simulations, we begin with the following simple profiles for GLYPH<11> rr and GLYPH<11> GLYPH<18> GLYPH<18> .</text>
<formula><location><page_3><loc_7><loc_14><loc_47><loc_17></location>GLYPH<11> rr = GLYPH<11> 0 1 2 GLYPH<20> 1 + erf GLYPH<18> r GLYPH<0> 0 : 7 R s 0 : 01 R s GLYPH<19> GLYPH<21> cos GLYPH<18>; (4)</formula>
<formula><location><page_3><loc_7><loc_10><loc_47><loc_12></location>GLYPH<11> GLYPH<18> GLYPH<18> = GLYPH<11> rr ; (5)</formula>
<text><location><page_3><loc_7><loc_7><loc_47><loc_9></location>where R s is the solar radius; see Figure 2 top panels for the variations of this profile.</text>
<figure>
<location><page_3><loc_51><loc_75><loc_91><loc_88></location>
<caption>Figure 2. Left: Radial variations of GLYPH<11> rr (Equation (4)) and GLYPH<11> GLYPH<30> GLYPH<30> (Equation (6)) at 45 GLYPH<14> latitude for GLYPH<11> 0 = 1 m s GLYPH<0> 1 . Right: Latitudinal variations of the same at r = 0 : 95 R s .</caption>
</figure>
<text><location><page_3><loc_51><loc_65><loc_91><loc_67></location>For GLYPH<11> GLYPH<30> GLYPH<30> , we use a different profile (Figure 2 bottom panels) and it is given by</text>
<formula><location><page_3><loc_53><loc_60><loc_91><loc_64></location>GLYPH<11> GLYPH<30> GLYPH<30> = GLYPH<11> 0 1 4 GLYPH<20> 1 + erf GLYPH<18> r GLYPH<0> 0 : 95 R s 0 : 05 R s GLYPH<19> GLYPH<21> GLYPH<20> 1 GLYPH<0> erf GLYPH<18> r GLYPH<0> R s 0 : 01 R s GLYPH<19> GLYPH<21> GLYPH<2> f s cos GLYPH<18> sin GLYPH<18>; (6)</formula>
<text><location><page_3><loc_51><loc_39><loc_91><loc_59></location>where f s is a function that takes care of suppressing GLYPH<11> GLYPH<30> GLYPH<30> above GLYPH<6> 45 GLYPH<14> latitudes and thus f s = 1 GLYPH<157>» 1 + exp f 30 ' GLYPH<25> GLYPH<157> 4 GLYPH<0> GLYPH<18> 'g… for GLYPH<18> < GLYPH<25> GLYPH<157> 2 and 1 GLYPH<157>» 1 + exp f 30 ' GLYPH<18> GLYPH<0> 3 GLYPH<25> GLYPH<157> 4 'g… for otherwise. We take this factor f s in GLYPH<11> GLYPH<30> GLYPH<30> to restrict the strong toroidal field and thus the band of formation of sunspots in the low latitudes, which is a common practice in the flux transport dynamo models (Kuker et al. 2001; Dikpati et al. 2004; Guerrero & de Gouveia Dal Pino 2007; Hotta & Yokoyama 2010; Karak & Cameron 2016). Further, we choose GLYPH<11> GLYPH<30> GLYPH<30> to operate only near the surface so that the source for the poloidal field is slightly segregated from the source for the toroidal field. This, in turn, helps in producing longer cycle period of 11 years. For GLYPH<11> rr and GLYPH<11> GLYPH<18> GLYPH<18> we keep the profiles simple so that they both are non-zero in the whole CZ and have a cos GLYPH<18> dependence. Later in § 4, we shall also use different GLYPH<11> profiles and explore the robustness of our results.</text>
<text><location><page_3><loc_51><loc_35><loc_91><loc_38></location>To limit the growth of magnetic field in our kinematic dynamo model, we consider following simple nonlinear quenching</text>
<formula><location><page_3><loc_51><loc_31><loc_91><loc_34></location>GLYPH<11> ii = GLYPH<11> ii 1 + GLYPH<16> B B 0 GLYPH<17> 2 (7)</formula>
<text><location><page_3><loc_51><loc_25><loc_91><loc_30></location>where i = r ; GLYPH<18>; GLYPH<30> and B 0 is the saturation field strength which is fixed at 4 GLYPH<2> 10 4 G in all the simulations. Because of this nonlinear saturation, we always present the magnetic field from our model with respect to B 0 .</text>
<text><location><page_3><loc_51><loc_23><loc_91><loc_25></location>For DR, we take following profile which reasonably fits the helioseismology data</text>
<formula><location><page_3><loc_51><loc_19><loc_91><loc_22></location>GLYPH<10> ' r ; GLYPH<18> ' = GLYPH<10> RZ + 1 2 GLYPH<20> 1 + erf GLYPH<18> r GLYPH<0> 0 : 7 R s 0 : 025 R s GLYPH<19> GLYPH<21> ' GLYPH<10> CZ GLYPH<0> GLYPH<10> RZ ' ; (8)</formula>
<text><location><page_3><loc_51><loc_13><loc_91><loc_19></location>where GLYPH<10> RZ GLYPH<157> 2 GLYPH<25> = 432 : 8 nHz, and GLYPH<10> CZ GLYPH<157> 2 GLYPH<25> = 460 : 7 GLYPH<0> 62 : 69 cos 2 GLYPH<18> GLYPH<0> 67 : 13 cos 4 GLYPH<18> nHz. When we make the DR antisolar, we take GLYPH<10> CZ GLYPH<157> 2 GLYPH<25> = 460 : 7 + 62 : 69 cos 2 GLYPH<18> + 67 : 13 cos 4 GLYPH<18> nHz. Note that this will allow GLYPH<10> to increase with latitudes and that is the anti-solar DR which is shown in Figure 3.</text>
<text><location><page_3><loc_51><loc_10><loc_91><loc_12></location>The turbulent magnetic diffusivity GLYPH<17> has the following form:</text>
<formula><location><page_3><loc_51><loc_7><loc_95><loc_9></location>GLYPH<17> ' r ' = GLYPH<17> RZ + GLYPH<17> SCZ 2 GLYPH<20> 1 + erf GLYPH<18> r GLYPH<0> 0 : 7 R s 0 : 02 R s GLYPH<19> GLYPH<21> + GLYPH<17> surf 2 GLYPH<20> 1 + erf GLYPH<18> r GLYPH<0> 0 : 9 R s 0 : 02 R s GLYPH<19> GLYPH<21> ;</formula>
<figure>
<location><page_4><loc_20><loc_71><loc_36><loc_87></location>
<caption>Figure 3. Angular frequency GLYPH<10> GLYPH<157> 2 GLYPH<25> for the anti-solar DR; see Equation (8).</caption>
</figure>
<text><location><page_4><loc_45><loc_63><loc_47><loc_64></location>(9)</text>
<text><location><page_4><loc_7><loc_54><loc_47><loc_62></location>where GLYPH<17> RZ = 5 GLYPH<2> 10 8 cm 2 s GLYPH<0> 1 , GLYPH<17> SCZ = 5 GLYPH<2> 10 10 cm 2 s GLYPH<0> 1 , and GLYPH<17> surf = 2 GLYPH<2> 10 12 cm 2 s GLYPH<0> 1 . Meridional circulation ( v r and v GLYPH<18> ) profile is the same as given in Hotta & Yokoyama (2010); also see Dikpati et al. (2004); Karak et al. (2014a); Karak & Petrovay (2013). For the sake of completeness, we write these profiles and boundary conditions in Appendix A.</text>
<section_header_level_1><location><page_4><loc_7><loc_49><loc_18><loc_50></location>3 RESULTS</section_header_level_1>
<text><location><page_4><loc_7><loc_39><loc_47><loc_48></location>We start all simulations by specifying some initial magnetic field, and we analyse the results only after running the code for several diffusion times so that the magnetic field reaches a statistically stationary state. We first present the result of an GLYPH<11> 2 model by setting GLYPH<10> = 0 . As the results for such a dynamo model for the Sun with parameters given above has not been presented before, we shall first show the result.</text>
<section_header_level_1><location><page_4><loc_7><loc_35><loc_44><loc_36></location>3.1 Solar dynamo without differential rotation</section_header_level_1>
<text><location><page_4><loc_7><loc_16><loc_47><loc_34></location>The critical GLYPH<11> 0 for the GLYPH<11> 2 dynamo model without DR for the Sun is about 10 m s GLYPH<0> 1 . However, the prominent dynamo cycles with polarity reversals are seen only when GLYPH<11> 0 is approximately above 18 m s GLYPH<0> 1 . Butterfly diagram for GLYPH<11> 0 = 20 ms GLYPH<0> 1 is shown in Figure 4. We notice that even this model without DR produces some basic features of the solar cycle, namely, (i) the polarity reversal, (ii) 11-year periodicity, (iii) equatorward migration of toroidal field at the base of the CZ, and (iv) the poleward migration of surface poloidal field. Feature (i) is due to the selected inhomogeneous profile of GLYPH<11> taken in our model. The other properties are largely determined by the inclusion of meridional flow which is poleward near the surface and equatorward near the base of the CZ and a relatively weak diffusivity in the bulk of the CZ.</text>
<text><location><page_4><loc_7><loc_7><loc_47><loc_16></location>The oscillatory magnetic field in GLYPH<11> 2 dynamo in spherical geometry is not new. It was realized that the GLYPH<11> 2 dynamo with certain GLYPH<11> profiles (Baryshnikova & Shukurov 1987; Stefani & Gerbeth 2003; Elstner & Rudiger 2007), and/or boundary conditions (Mitra et al. 2010; Jabbari et al. 2017) produces an oscillatory solution. We also note that the mean-field model constructed using GLYPH<11> profiles obtained from the global</text>
<figure>
<location><page_4><loc_53><loc_68><loc_90><loc_86></location>
<caption>Figure 4. Result of GLYPH<11> 2 dynamo model with meridional circulation. Time-latitude diagrams of (a) toroidal magnetic field B at r = 0 : 7 R s and surface radial field B r . All fields are measured with respect to B 0 .</caption>
</figure>
<text><location><page_4><loc_51><loc_56><loc_91><loc_58></location>MHDsimulation also finds oscillatory solution (Simard et al. 2013).</text>
<section_header_level_1><location><page_4><loc_51><loc_52><loc_90><loc_53></location>3.2 Stellar dynamo with solar and anti-solar DR</section_header_level_1>
<text><location><page_4><loc_51><loc_35><loc_91><loc_51></location>Now we perform dynamo simulations of different stars with rotation period 1, 5, 10, 17, 25.38 (solar value), 30, 32, 40 and 50 days. As discussed in the introduction that the slowly rotating stars are expected to have anti-solar DR, which is, at least, confirmed in the global MHD simulations. This transition from solar to anti-solar DR is possibly happening below the solar rotation with Rossby number not too far from unity. Therefore, we assume that all stars with rotation periods up to 30 days are having solar-like DR, while stars of periods longer than 30 days have anti-solar DR. Further, the amount of internal DR is expected to change with the rotation rate. Hence, we choose it to scale in the following way.</text>
<formula><location><page_4><loc_51><loc_31><loc_91><loc_34></location>GLYPH<10> ' r ; GLYPH<18> ' = GLYPH<18> T s T GLYPH<19> n GLYPH<10> s ' r ; GLYPH<18> ' ; (10)</formula>
<text><location><page_4><loc_51><loc_10><loc_91><loc_30></location>where T s = 25 : 38 days (solar rotation period), T is the rotation period of star, and GLYPH<10> s ' r ; GLYPH<18> ' is the internal angular frequency of the Sun as given in Equation (8). Some numerical simulations suggest n is about 0.3 (Ballot et al. 2007; Brown et al. 2008), while the recent work of Viviani et al. (2018) in a more wider range finds n GLYPH<25> GLYPH<0> 0 : 08 for the rotation rates up to 5 times solar value and n = GLYPH<0> 0 : 96 for rotation rate of 5-31 times the solar rotation. Surface observations find somewhat similar results, for example, Barnes et al. (2005), Reinhold & Gizon (2015), Donahue et al. (1996), and Lehtinen et al. (2016) respectively give n = 0 : 15 ; 0 : 29 ; 0 : 7 ; and GLYPH<0> 0 : 36 . In our study, we perform two sets of simulations by considering two values, namely n = 0 : 7 , i.e., the DR increases with the increase of rotation rate with an exponent of 0 : 7 and n = 0 : 0 , i.e. no change in DR. These two sets of simulations are labelled as A1-A50 and B1-B50; see Table 1.</text>
<text><location><page_4><loc_51><loc_7><loc_91><loc_9></location>The amplitude of GLYPH<11> is also expected to increase with the increase of rotation rate (Krause & Radler 1980). As we</text>
<table>
<location><page_5><loc_7><loc_22><loc_49><loc_82></location>
<caption>Table 1. Summary of simulations. Here, T is the rotation period of star in days, P cyc is the mean magnetic cycle period, SL and AS stand for solar and anti-solar DR, respectively.</caption>
</table>
<text><location><page_5><loc_7><loc_16><loc_47><loc_19></location>do not know how exactly GLYPH<11> scales with the rotation rate, we choose it to increase linearly. Thus, in Equations (4-6), we take</text>
<formula><location><page_5><loc_7><loc_13><loc_47><loc_15></location>GLYPH<11> 0 = T s T GLYPH<11> 0 ; s ; (11)</formula>
<text><location><page_5><loc_7><loc_10><loc_47><loc_12></location>where GLYPH<11> 0 ; s is the value of GLYPH<11> 0 for the solar case, in which we take GLYPH<11> 0 ; s = 80 m s GLYPH<0> 1 .</text>
<text><location><page_5><loc_7><loc_7><loc_47><loc_9></location>The meridional circulation is another important ingredient in the model which is expected to vary with the rotation</text>
<figure>
<location><page_5><loc_52><loc_55><loc_90><loc_87></location>
<caption>Figure 5. Butterfly diagram of the toroidal field averaged over a thickness of 0 : 04 R s centered at 0 : 71 R s from stars of rotation periods 5, 10, 25.38 (Sun), 32, and 50 days (top to bottom); Runs A5, A10, A25, A32, and A50.</caption>
</figure>
<text><location><page_5><loc_51><loc_32><loc_91><loc_45></location>rate. Our knowledge of meridional circulation even for the sun is very limited. Mean-field models and global convection simulations all suggest that its profile as well as amplitude changes with the rotation rate (Kitchatinov & Olemskoy 2011b; Featherstone & Miesch 2015; Karak et al. 2015, 2018a). Simulations of Brown et al. (2008) showed that the kinetic energy of meridional circulation approximately scales as GLYPH<10> GLYPH<0> 0 : 9 . Therefore in one set of simulations (Runs C1-C50 in Table 1), we shall change meridional circulation along with other parameters in the following way.</text>
<formula><location><page_5><loc_51><loc_29><loc_91><loc_31></location>v 0 = GLYPH<18> T s T GLYPH<19> GLYPH<0> 0 : 45 v 0 ; s ; (12)</formula>
<text><location><page_5><loc_51><loc_24><loc_91><loc_28></location>where v 0 ; s is the amplitude of meridional circulation of sun, for which we have taken 10 m s GLYPH<0> 1 . We do not make any other changes in the model.</text>
<text><location><page_5><loc_51><loc_8><loc_91><loc_24></location>Interestingly, all the stars, including ones with anti-solar DR (rotation periods 32-50 days), show prominent dynamo cycles and polarity reversals except for the very rapidly rotating star of rotation period 1 day. The butterfly diagrams of stars with rotation periods 5, 10, 25.38, 32, and 50 days (from Runs A1-A50) are shown in Figures 5 and 6. We observe that the 5 days rotating star (Run A5) does not show magnetic cycle at the base of the CZ, but it does some cycles in low latitudes of mid-CZ. The magnetic fields are largely anti-symmetric across the equator in all the cases. The equatorward migration is largely due to the return meridional flow.</text>
<text><location><page_5><loc_54><loc_7><loc_91><loc_8></location>To quantify how the magnetic field strength and cy-</text>
<section_header_level_1><location><page_6><loc_7><loc_90><loc_34><loc_91></location>6 Karak, Tomar & Vashishth</section_header_level_1>
<figure>
<location><page_6><loc_8><loc_55><loc_45><loc_87></location>
<caption>Figure 6. Same as Figure 5 but obtained from the toroidal field at 0 : 85 R s .</caption>
</figure>
<text><location><page_6><loc_7><loc_40><loc_47><loc_47></location>cle period change with different stars, we compute the rootmean-square ( rms ) field strength B rms = r D ' B 2 r + B 2 GLYPH<18> + B 2 GLYPH<30> ' E , the ratio of poloidal to toroidal field Bpol/Btor = q GLYPH<10> ' B 2 r + B 2 GLYPH<18> ' GLYPH<11> GLYPH<157> r D B 2 GLYPH<30> E (where angular brackets denote the av-</text>
<text><location><page_6><loc_7><loc_34><loc_47><loc_39></location>erage over the whole domain), and the mean period of the polarity reversals of the toroidal field. These quantities are listed in Table 1 and the variations with rotation period are shown in Figure 7.</text>
<text><location><page_6><loc_7><loc_7><loc_47><loc_34></location>In the rapidly rotating regime, the magnetic field strength increases with the decrease of rotation period, which is in agreement with observations (Petit et al. 2008). This behaviour in fact is not very surprising because the magnetic field sources ( GLYPH<11> and GLYPH<10> ) are increased with the rotation rates through equations (11) and (10). However, what is surprising in Figure 7(a) is that the magnetic field abruptly increased just above the rotation period of 30 days. That is the point where the DR pattern is changed from solar-like to anti-solar. Thus the anti-solar DR causes a sudden increase of magnetic field in the slowly rotating stars, which is in good agreement with the stellar observational data (Giampapa et al. 2006, 2017; Brandenburg & Giampapa 2018) and also with global MHD convection simulations (Karak et al. 2015). Further, we note that even the simulations (Run B1-B50; red asterisks in Figure 7a) in which GLYPH<10> does not change with the rotation rate (i.e., n = 0 in Equation (10)) also produces a similar variation of magnetic field. As the shear is not decreased in this case, the field in the anti-solar branch is little stronger in comparison to Set A (black points). The Set C, in which meridional flow is increased with the rotation period</text>
<figure>
<location><page_6><loc_52><loc_46><loc_90><loc_87></location>
<caption>Figure 7. Top-bottom: Dependences of the magnetic field, the ratio of toroidal to poloidal fields, and cycle periods with rotation periods of the stars. Circular and asterisk points are obtained from simulations in which DR decreases with rotation period through Equation (10) (Runs A1-A50) and DR does not change (Runs B1-B50), respectively. Triangular points come from the same simulations as that of circular points but the meridional circulation is increased with rotation period through Equation (12) (Runs C1-C50). The shaded area represents the solar-like DR regime, while the white space is anti-solar.</caption>
</figure>
<text><location><page_6><loc_51><loc_25><loc_91><loc_29></location>(Equation (12)), also show a similar variation of magnetic field (blue triangles in Figure 7a), which is unusual in the GLYPH<11> GLYPH<10> type flux transport dynamo models (Karak 2010).</text>
<text><location><page_6><loc_51><loc_7><loc_91><loc_25></location>To understand the enhancement of the magnetic field in the anti-solar DR regime, we perform the following experiment. We consider the simulation of the 30-day rotation period having solar-like DR (Run A30). We first make sure that this model is producing a steady dynamo solution. Then, when the toroidal field reverses, that is when the toroidal field is at a minimum, we stop the code. The vertical line at t = 10 years in Figure 8 shows this time. Then we make the DR anti-solar and run the code for four years (up to the second vertical line in Figure 8). Finally, we revert the DR to the solar-like profile and continue the run for another 10 years. We see a significant increase of magnetic field in Figure 8(a) after the DR is made anti-solar. This increase of field must be caused by the change in DR</text>
<figure>
<location><page_7><loc_9><loc_70><loc_48><loc_87></location>
<caption>Figure 8. Simulation result of a star with rotation period 30 days in which the DR is made anti-solar during 10-14 years (identified by two vertical lines). Butterfly diagram of (a) the toroidal field at 0 : 85 R s and (b) the same but averaged over a thickness of 0 : 04 R s centered at 0 : 71 R s .</caption>
</figure>
<text><location><page_7><loc_7><loc_48><loc_47><loc_59></location>from solar to anti-solar profile. We know that during the reversal, the anti-solar DR produces toroidal field of the same sign as that of the previous cycle. As seen in Figure 8(a), when the toroidal field in the northern hemisphere is mostly negative during the phase of anti-solar DR ( t = 10-14 years), the dynamo needs a positive field to reverse it. However, the anti-solar DR gives negative polarity field, and thus, it tries to enhance the old field (negative in the northern hemisphere). So essentially what is happening is the following:</text>
<text><location><page_7><loc_7><loc_36><loc_47><loc_48></location>Pol( + ) GLYPH<0>GLYPH<0>GLYPH<0>GLYPH<0>GLYPH<0>GLYPH<0>GLYPH<0>GLYPH<0>! ' Solar GLYPH<10> ' Tor( GLYPH<0> ) GLYPH<0> ! GLYPH<11> Pol( GLYPH<0> ) GLYPH<0>GLYPH<0>GLYPH<0>GLYPH<0>GLYPH<0>GLYPH<0>GLYPH<0>GLYPH<0>GLYPH<0>GLYPH<0>GLYPH<0>GLYPH<0>! ' Anti GLYPH<0> sol GLYPH<10> ' Tor( GLYPH<0> ). Note in the last phase, the toroidal field is produced in the same polarity as that of the previous cycle and thus the field is enhanced; also see Figure 1. We may mention that instead of anti-solar DR if the GLYPH<11> is reversed, then also the same effect will be seen. In fact, following this idea, recently Karak et al. (2018b) explained the sudden increase of the poloidal field and appearance of double peaks in the sunspot cycles using a momentarily reversed GLYPH<11> due to wrong BMR tilt.</text>
<text><location><page_7><loc_7><loc_27><loc_47><loc_35></location>Later, when the DR is changed to solar-like, this increasing effect of anti-solar DR is stopped and the model succeed to reverse the field (at around t = 16 years in Figure 8). We note that the increase of magnetic field is not immediately seen in the averaged field at the base of the CZ in Figure 8(b) because of the cancellation and finite transport time, but it is clearly seen in the next cycle.</text>
<text><location><page_7><loc_7><loc_7><loc_47><loc_26></location>Above discussion shows that the anti-solar DR amplifies the old toroidal field by supplying toroidal field of the same polarity. Thus we do not expect polarity reversal if the antisolar DR acts for all the time and if there is no other source for the toroidal field generation. In our model, we certainly have GLYPH<11> effect for the generation of the toroidal field in addition to GLYPH<10> . However, this may not be sufficient to reverse the field. We realized that when the toroidal field has reached a very high value, the DR is needed to be suppressed otherwise the GLYPH<11> effect is not able to reverse the toroidal field and the cyclic dynamo is not possible. When the toroidal field is sufficiently high due to anti-solar DR, it is expected that this strong field acts back on the DR through Lorentz forces and tries to suppress the shear. The suppression mechanism could be complicated as the strong magnetic field can give</text>
<figure>
<location><page_7><loc_52><loc_61><loc_91><loc_87></location>
<caption>Figure 9. Results from Run A32 computed over the whole CZ in the northern hemisphere: Temporal variations of (a) the toroidal field, (b) the mean effective latitudinal shear 1 r @ GLYPH<10> @GLYPH<18> GLYPH<157> GLYPH<16> 1 + ' B GLYPH<157> B 0 ' 2 GLYPH<17> (in unit of 10 GLYPH<0> 9 nHz deg GLYPH<0> 1 ), (c) the mean source term due to GLYPH<11> effect for the toroidal field ( S Tor GLYPH<11> in Equation (2) in unit of 10 GLYPH<0> 6 s GLYPH<0> 1 ), and (d) the same as (a), but obtained from simulations in which there is no quenching in the shear (red dashed line) and GLYPH<11> 0 = 20 m s GLYPH<0> 1 (solid blue), instead of 80 m s GLYPH<0> 1 which is used in all Runs A1-A50.</caption>
</figure>
<text><location><page_7><loc_51><loc_11><loc_91><loc_44></location>Lorentz forces on the large-scale flows as well as suppress the convective angular momentum transport (see e.g, Karak et al. 2015; Kapyla et al. 2016). The bottom line is that the shear must be decreased once the toroidal field reaches a sufficiently high value. This, in fact, is seen in the simulations of Karak et al. (2015) that the shear parameters are quenched when the magnetic field is strong; see their Fig. 13(e-f). We also note that even in the Sun, there is an indication of the reduction of shear during the solar maximum; see Fig. 6-7 of Antia et al. (2008) and also Barekat et al. (2016). Motivated by these, we have introduced a nonlinear magnetic field dependent quenching f q = 1 GLYPH<157> GLYPH<16> 1 + ' B GLYPH<157> B 0 ' 2 GLYPH<17> in the shear such that r GLYPH<10> in Equation (2) is replaced by f q r GLYPH<10> . Through this nonlinear quenching, the toroidal field tries to reduce the shear when it exceeds B 0 . In Figure 9(b), we observe that the shear is strong during cycle minimum and thus, it produces a strong toroidal field. Then the strong field quenches the shear and the toroidal field cannot grow rapidly. Interestingly, the toroidal field generation due to GLYPH<11> effect S Tor GLYPH<11> (see Equation (3)) is not negligible (Figure 9(c)) and it is this process which eventually dominates and reverses the toroidal field slowly. Essentially, it is the nonlinear competition between the toroidal field generation through shear and GLYPH<11> , which makes the polarity reversal possible even with the anti-solar DR.</text>
<text><location><page_7><loc_51><loc_7><loc_91><loc_11></location>To further support above conclusion, we show that if we do not include the quenching (i.e., f q = 1 ) in the shear, then the dynamo fails to reverse the magnetic field, rather</text>
<text><location><page_8><loc_7><loc_78><loc_47><loc_88></location>the toroidal field increases in time; see red/dashed line in Figure 9(d). On the other hand, if we keep the quenching in shear but reduce the strength of GLYPH<11> by 75% , then also, dynamo fails to reverse the cycle and the magnetic field remains steady; see blue/solid line in Figure 9(d). In conclusion, to obtain the polarity reversal with anti-solar DR, there must be a sufficiently strong GLYPH<11> and the shear must be reduced when the toroidal field becomes very strong.</text>
<text><location><page_8><loc_7><loc_38><loc_47><loc_77></location>Returning to Figure 7(b), we observe that for all the stars, the mean poloidal field is about one order of magnitude smaller than that of the toroidal field. The ratio Bpol/Btor has a non-monotonous behaviour. It is maximum at the solar rotation and decreases on both sides. This implies that the toroidal field dominates over the poloidal field both for rapidly and slowly rotating stars. With the increase of rotation rate, the toroidal field must increase faster than the poloidal component because there are two sources of toroidal field ( GLYPH<11> and shear) and both increase with the rotation rate. In the anti-solar branch, Bpol/Btor decreases because toroidal field generation is stronger with the anti-solar DR. The increase of poloidal contribution with the increase of cycle period in the solar-like DR branch is in agreement with the observational findings (Petit et al. 2008). These observational results also show that the magnetic energy of the toroidal field dominates below the rotation period of 12 days, while in our simulations the toroidal energy is always greater than the poloidal field. We should not forget that observational results are based on the field what was detected and it could be that most of the toroidal field in observations was not be detected. Therefore, instead of comparing the actual value of magnetic energy, we should see how the ratio of poloidal to toroidal field changes with the rotation rate and this is in somewhat agreement with observation (compare Figure 6 of Petit et al. (2008) with the left branch of our Figure 7(b)). Unfortunately, Petit et al. (2008) do not have data in the slowly rotating branch above the solar rotation period and thus we cannot compare this behaviour with observations.</text>
<text><location><page_8><loc_7><loc_7><loc_47><loc_37></location>Finally, the cycle period again shows a non-monotonous behaviour. It decreases with the increase of rotation rate, which is in general agreement with observations (Noyes et al. 1984b; Su'arez Mascare˜no et al. 2016; Boro Saikia et al. 2018), although the observational trend is messy. The slow decrease of rotation period in the anti-solar DR branch is not apparent in the observed data; see, for example, Fig. 9 of Boro Saikia et al. (2018). In our model, the decrease of cycle period with the increase of rotation rate is because the dynamo becomes stronger with the rotation rate which makes the conversion between poloidal and toroidal faster and reduces the cycle period. In global MHD simulations of stellar dynamos, Guerrero et al. (2018) also find a decrease of period with the increase of rotation rate, while Warnecke (2018) find this trend only at slow rotation and then an increasing trend in the rapid rotation. In latter simulations, the increase of period with rotation rate is due to the decrease of shear. In our study, when the DR is anti-solar, it tries to produce the same polarity field as that of the previous cycle and thus GLYPH<11> takes longer time to reverse the field-causing the cycle longer. This effect decreases when the rotation period is too long because the strength of the anti-solar DR decreases and thus the polarity reversal be-</text>
<text><location><page_8><loc_51><loc_85><loc_91><loc_88></location>mes faster. This causes the cycle period to shorten with the increase of rotation period in the anti-solar branch.</text>
<text><location><page_8><loc_51><loc_69><loc_91><loc_85></location>As the meridional flow transports the magnetic fields from source regions, the cycle duration tends to be longer with the decrease of meridional flow (Dikpati & Charbonneau 1999; Karak 2010). This was usually seen in previous flux transport dynamo models of stellar cycles (Nandy 2004; Jouve et al. 2010; Karak et al. 2014a), but not in the turbulent pumping-dominated regime (Karak & Cameron 2016; Hazra et al. 2019). Therefore, in Set C in which meridional circulation is decreased with the rotation rate following Equation (12), we expected an increase of cycle period. However, we see a reverse trend (see blue points in Figure 7(c)), because the dynamo becomes stronger and this tries to make the cycle shorter.</text>
<section_header_level_1><location><page_8><loc_51><loc_64><loc_82><loc_65></location>4 ROBUSTNESS OF OUR RESULTS</section_header_level_1>
<text><location><page_8><loc_51><loc_48><loc_91><loc_63></location>We have above seen that the results are only little sensitive to the change in GLYPH<10> and meridional circulation. Notably, the enhancement of the magnetic field at the transition point from solar to anti-solar DR is prominent. To explore the robustness of these results further, we consider different profiles for GLYPH<11> . In principle, we can consider countless different profiles for GLYPH<11> as the actual profiles are not known even for the Sun. Different profiles will make the dynamo solutions different but not necessarily the variations of magnetic field with rotation period. To demonstrate this, we present the results for two more sets of simulations, namely, Set D (Runs D1D50) and set E (Runs E1-E50); see Table 1.</text>
<text><location><page_8><loc_54><loc_47><loc_86><loc_48></location>In Set D, we change GLYPH<11> rr and GLYPH<11> GLYPH<18> GLYPH<18> to as follows:</text>
<formula><location><page_8><loc_51><loc_44><loc_91><loc_46></location>GLYPH<11> rr = GLYPH<11> 0 sin GLYPH<20> 2 GLYPH<25> GLYPH<18> r GLYPH<0> 0 : 7 R s 0 : 3 R s GLYPH<19> GLYPH<21> cos GLYPH<18> sin 2 GLYPH<18>; (13)</formula>
<formula><location><page_8><loc_51><loc_40><loc_91><loc_42></location>GLYPH<11> GLYPH<18> GLYPH<18> = GLYPH<11> 0 sin GLYPH<20> 2 GLYPH<25> GLYPH<18> r GLYPH<0> 0 : 7 R s 0 : 3 R s GLYPH<19> GLYPH<21> cos GLYPH<18>; (14)</formula>
<text><location><page_8><loc_51><loc_35><loc_91><loc_39></location>both for r GLYPH<21> 0 : 7 R s , while GLYPH<11> rr = GLYPH<11> GLYPH<18> GLYPH<18> = 0 for r < 0 : 7 R s and GLYPH<11> 0 = 100 m s GLYPH<0> 1 ; see Figure A1 for these profiles. No other parameters are changed in this Set.</text>
<text><location><page_8><loc_51><loc_33><loc_91><loc_35></location>On the other hand, in Set E, we change all three GLYPH<11> profiles to followings.</text>
<formula><location><page_8><loc_51><loc_29><loc_91><loc_32></location>GLYPH<11> rr = GLYPH<11> GLYPH<18> GLYPH<18> = GLYPH<11> GLYPH<30> GLYPH<30> = GLYPH<11> 0 1 2 GLYPH<20> 1 + erf GLYPH<18> r GLYPH<0> 0 : 7 R s 0 : 01 R s GLYPH<19> GLYPH<21> cos GLYPH<18> : (15)</formula>
<text><location><page_8><loc_51><loc_14><loc_91><loc_29></location>With these new GLYPH<11> , we, however, do not get polarity reversal in a wide range of GLYPH<11> 0 both in solar and anti-solar DR cases. Interestingly, this new GLYPH<11> profile gives polarity reversals if we increase the diffusivity in the bulk of the CZ. That is, when we take GLYPH<17> SCZ = 1 GLYPH<2> 10 12 cm 2 s GLYPH<0> 1 , GLYPH<17> surf = 1 : 05 GLYPH<2> 10 12 cm 2 s GLYPH<0> 1 (see Figure A2 dashed line) and GLYPH<11> 0 = 80 ms GLYPH<0> 1 , we obtain polarity reversals in all runs except Run E1; see Figure A3. It could be that in this new GLYPH<11> profile, unless we increase the diffusion in the deeper CZ, the toroidal field generation due to shear overpowers the same due to GLYPH<11> effect. This higher diffusion makes the equatorward migration of toroidal field less important and the magnetic field in run E50 quadrupolar.</text>
<text><location><page_8><loc_51><loc_7><loc_91><loc_13></location>We note that with these new GLYPH<11> profiles, we again obtain a clear increase of magnetic field in both the sets D and E; see Table 1, although the increase is relatively small. We note that in both sets, GLYPH<11> and GLYPH<10> are decreased with the increase of rotation period as before following equations (11) and (10).</text>
<section_header_level_1><location><page_9><loc_7><loc_87><loc_22><loc_88></location>5 CONCLUSION</section_header_level_1>
<text><location><page_9><loc_7><loc_73><loc_47><loc_85></location>In this study, using a mean-field kinematic dynamo model we have explored the features of large-scale magnetic fields of solar-like stars rotating at different rotation periods and having the same internal structures as that of the Sun. Our main motivation is to understand the large-scale magnetic field generation in the slowly rotating stars with rotation period larger than the solar value. This region is of particular interest because these stars possibly possess anti-solar DR and thus the operation of dynamo can be fundamentally different than stars having the usual solar-like DR.</text>
<text><location><page_9><loc_7><loc_37><loc_47><loc_72></location>By carrying out simulations for different stars at different model parameters, we show that the fundamental features of stellar magnetic field change with rotation period. In the solar-like DR branch, we find the magnetic field strength increases with the decrease of rotation period. This result is, in general, agreement with the observational findings (Noyes et al. 1984a; Petit et al. 2008; Wright et al. 2011; Wright & Drake 2016), the global MHD convection simulations (Viviani et al. 2018; Warnecke 2018), and mean-field dynamo modellings (Jouve et al. 2010; Karak et al. 2014a; Kitchatinov & Olemskoy 2015; Hazra et al. 2019). However, our model does not produce the observed saturation of magnetic field in the very rapidly rotating stars, which in the kinematic models, requires some additional dynamo saturation (Karak et al. 2014a; Kitchatinov & Olemskoy 2015). In the slowly rotating stars, we see a sudden jump of magnetic field strength at the point where the DR profile changes to anti-solar from solar. This result is in agreement with the stellar observations (Giampapa et al. 2006, 2017; Brandenburg & Giampapa 2018) and also with the MHD convection simulations of Karak et al. (2015) and Warnecke (2018). The abrupt increase of magnetic field is due to the anti-solar DR which amplifies the existing toroidal field by supplying the same polarity field. The idea of the enhancement of magnetic field due to the change of DR was already proposed by Brandenburg & Giampapa (2018), however, no detailed dynamo modelling was performed.</text>
<text><location><page_9><loc_7><loc_22><loc_47><loc_37></location>We further show that with particular GLYPH<11> profiles, the polarity reversal of the large-scale magnetic field is possible even in the slowly rotating stars with anti-solar DR rotation provided, (i) there is a sufficiently strong GLYPH<11> for the generation of toroidal field and (ii) the anti-solar DR is nonlinearly modulated with the magnetic field such that when the toroidal field becomes strong, it quenches the shear. Our conclusion of polarity reversal in general supports the work of Viviani et al. (2019), who showed that the polarity in their global MHD convection with anti-solar DR is possible as the magnetic field generation through GLYPH<11> effect is comparable to that of GLYPH<10> effect.</text>
<text><location><page_9><loc_7><loc_7><loc_47><loc_21></location>One may argue that the global MHD convection simulations of stellar CZs are still far from the real stars and there is always a question to what extent the results from these simulations hold to the real stars. Interestingly, one robust result of these simulations is that they all produce anti-solar DR in the slowly rotating stars with rotation period somewhere above the solar value with Rossby number around one. Available techniques are still insufficient to confirm the existence of anti-solar DR in solar-like dwarfs; see Reinhold & Arlt (2015). However, this has been confirmed in some K-giants (Strassmeier et al. 2003; Weber et al. 2005;</text>
<text><location><page_9><loc_51><loc_74><loc_91><loc_88></location>K"ov'ari et al. 2017) and subgiants (Harutyunyan et al. 2016). The enhancement of magnetic activity in the slowly rotating stars and its generation through the anti-solar DR give another support for the existence of anti-solar DR in slowly rotating dwarfs. Furthermore, this study along with previous observational results and global simulations suggest that the slowly rotating stars possess strong large-scale magnetic fields and possibly polarity reversals and cycles. These slowly rotating solar-like stars may also be prone to produce superflares (Maehara et al. 2012), which was also suggested by Katsova et al. (2018).</text>
<section_header_level_1><location><page_9><loc_51><loc_70><loc_72><loc_71></location>ACKNOWLEDGEMENTS</section_header_level_1>
<text><location><page_9><loc_51><loc_53><loc_91><loc_69></location>We thank the anonymous referee for carefully checking the manuscript and raising interesting questions which particularly helped us to correct an error that we made in the earlier draft. We further thank Gopal Hazra and Sudip Mandal for discussion on various aspects of the stellar dynamo. We sincerely acknowledge financial support from Department of Science and Technology (SERB/DST), India through the Ramanujan Fellowship awarded to B.B.K. (project no SB/S2/RJN-017/2018). BBK appreciates gracious hospitality at Indian Institute of Astrophysics, Bangalore during the last phase of this project. V.V. acknowledges financial support from DST through INSPIRE Fellowship.</text>
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<section_header_level_1><location><page_10><loc_51><loc_56><loc_91><loc_57></location>APPENDIX A: SUPPLEMENTARY MATERIAL</section_header_level_1>
<text><location><page_10><loc_51><loc_53><loc_91><loc_55></location>The meridional flow is obtained from the following analytical form.</text>
<formula><location><page_10><loc_51><loc_48><loc_93><loc_52></location>v r ' r ; GLYPH<18> ' = v 0 GLYPH<18> R s r GLYPH<19> 2 GLYPH<20> GLYPH<0> 1 m + 1 + c 1 2 m + 1 GLYPH<24> m GLYPH<0> c 2 2 m + p + 1 GLYPH<24> m + p GLYPH<21> GLYPH<24> h 2 cos 2 GLYPH<18> GLYPH<0> sin 2 GLYPH<18> i (A1)</formula>
<formula><location><page_10><loc_51><loc_42><loc_91><loc_45></location>v GLYPH<18> ' r ; GLYPH<18> ' = v 0 GLYPH<18> R s r GLYPH<19> 3 GLYPH<2> GLYPH<0> 1 + c 1 GLYPH<24> m GLYPH<0> c 2 GLYPH<24> m + p GLYPH<3> sin GLYPH<18> cos GLYPH<18>; (A2)</formula>
<text><location><page_10><loc_51><loc_40><loc_54><loc_41></location>with</text>
<formula><location><page_10><loc_55><loc_34><loc_91><loc_40></location>GLYPH<24> ' r ' = R s r GLYPH<0> 1 ; c 1 = ' 2 m + 1 '' m + p ' ' m + 1 ' p GLYPH<24> GLYPH<0> m p ; c 2 = ' 2 m + p + 1 ' m ' m + 1 ' p GLYPH<24> GLYPH<0>' m + p ' p ; and GLYPH<24> p = R s r p GLYPH<0> 1 : (A3)</formula>
<text><location><page_10><loc_51><loc_28><loc_91><loc_34></location>Here m = 0 : 5 , p = 0 : 25 , v 0 = 10 m s GLYPH<0> 1 , and r p = 0 : 62 R s . The boundary conditions are exactly taken from Chatterjee et al. (2004) and thus we do not repeat those here. For the initial magnetic field, we take</text>
<formula><location><page_10><loc_59><loc_23><loc_91><loc_27></location>A = 0 and B = B 0 sin ' 2 GLYPH<18> ' sin GLYPH<20> GLYPH<25> ' r GLYPH<0> 0 : 55 R s ' ' R s GLYPH<0> 0 : 55 R s ' GLYPH<21> : (A4)</formula>
<text><location><page_10><loc_51><loc_19><loc_91><loc_21></location>This paper has been typeset from a T E X/L A T E X file prepared by the author.</text>
<figure>
<location><page_11><loc_7><loc_75><loc_47><loc_88></location>
<caption>Figure A1. Left: Radial variations of GLYPH<11> rr and GLYPH<11> GLYPH<18> GLYPH<18> in m s GLYPH<0> 1 at 45 GLYPH<14> latitude. Right: Latitudinal variations of the same at r = 0 : 95 R s . These are the profiles used in Set D; Equation (14) with GLYPH<11> 0 = 1 m s GLYPH<0> 1 .</caption>
</figure>
<figure>
<location><page_11><loc_13><loc_51><loc_40><loc_65></location>
<caption>Figure A2. Solid line: diffusivity profile GLYPH<17> used in all the sets of simulations, except Set E for which the profile shown by the dashed line is used.</caption>
</figure>
<figure>
<location><page_11><loc_52><loc_55><loc_89><loc_87></location>
<caption>Figure A3. Same as Figure 5, but obtained from Runs E5, E17, E25, E32, E50 (top to bottom).</caption>
</figure>
</document> | [{"title": "ABSTRACT", "content": "Simulations of magnetohydrodynamics convection in slowly rotating stars predict antisolar differential rotation (DR) in which the equator rotates slower than poles. This anti-solar DR in the usual GLYPH<11> GLYPH<10> dynamo model does not produce polarity reversal. Thus, the features of large-scale magnetic fields in slowly rotating stars are expected to be different than stars having solar-like DR. In this study, we perform mean-field kinematic dynamo modelling of different stars at different rotation periods. We consider anti-solar DR for the stars having rotation period larger than 30 days and solar-like DR otherwise. We show that with particular GLYPH<11> profiles, the dynamo model produces magnetic cycles with polarity reversals even with the anti-solar DR provided, the DR is quenched when the toroidal field grows considerably high and there is a sufficiently strong GLYPH<11> for the generation of toroidal field. Due to the anti-solar DR, the model produces an abrupt increase of magnetic field exactly when the DR profile is changed from solar-like to anti-solar. This enhancement of magnetic field is in good agreement with the stellar observational data as well as some global convection simulations. In the solar-like DR branch, with the decreasing rotation period, we find the magnetic field strength increases while the cycle period shortens. Both of these trends are in general agreement with observations. Our study provides additional support for the possible existence of anti-solar DR in slowly rotating stars and the presence of unusually enhanced magnetic fields and possibly cycles which are prone to production of superflare. Key words: Sun: activity, dynamo, magnetic fields - stars: solar-type, rotation, activity.", "pages": [1]}, {"title": "Bidya Binay Karak ? , Aparna Tomar and Vindya Vashishth", "content": "1 Department of Physics, Indian Institute of Technology (Banaras Hindu University), Varanasi, India Accepted XXX. Received YYY; in original form ZZZ", "pages": [1]}, {"title": "1 INTRODUCTION", "content": "Many sun-like low-main sequence stars show magnetic cycles which are usually studied by measuring the chromospheric emissions in Ca II H & K line cores and the coronal X-ray (Baliunas et al. 1995; Pallavicini et al. 1981). These measurements show that the periods of magnetic cycles vary in different stars and there is a weak trend of the period becoming shorter with the increase of rotation rates (Noyes et al. 1984b; Su'arez Mascare\u02dcno et al. 2016; Boro Saikia et al. 2018). Some recent observations, however, do not find this trend in the so-called active branch; see Fig. 21 of Su'arez Mascare\u02dcno et al. (2016). On the other hand, the stellar magnetic activity or the field strength increases with the increase of rotation rate in the small rotation range and then it tends to saturate in the rapid rotation limit. This behaviour is often represented with respect to the Coriolis number (or the inverse of Rossby number) which is a ratio of the convective turnover time to the rotation period (Noyes et al. 1984a; Wright et al. 2011; Wright & Drake 2016). However, Reiners et al. (2014) show that rotation period alone better represents the data. In the slowly rotating stars with rotation rates below about the solar value, an interesting behaviour has been recognised. Giampapa et al. (2006, 2017) have found an increase of magnetic activity with a decrease of GLYPH<28> GLYPH<157> P rot (i.e., increase of rotation period). Another exciting aspect of stellar activity is that the magnetic cycles of Sun-like stars show the Waldmeier effect (the stronger cycles rise faster than the weaker ones) (Garg et al. 2019), which is popularly known for the Sun (Waldmeier 1935; Karak & Choudhuri 2011). Starting from the pioneering work of Gilman (1977,", "pages": [1]}, {"title": "2 Karak, Tomar & Vashishth", "content": "1983), numerous global magnetohydrodynamics (MHD) convection simulations in spherical geometry have been performed to study the magnetic fields and flows in stars. These simulations in some parameter ranges produce large-scale magnetic fields and even cycles. In general, it has been observed that in the rapidly rotating stars, magnetic cycles and polarity reversals are preferred, while in the slowly rotating ones, simulations rarely produce reversals (Brun & Browning 2017; Warnecke 2018). Most of the simulations near the solar rotation rate (or Rossby number around one) find a transition of differential rotation (hereafter DR) from solar-like to the so-called anti-solar profile, in which the equator rotates slower than high latitudes (Gilman 1977; Guerrero et al. 2013; Gastine et al. 2014; Kapyla et al. 2014; Fan & Fang 2014; Karak et al. 2015; Featherstone & Miesch 2015; Karak et al. 2018a). In most of the cases, when simulations produce anti-solar DR, they do not produce magnetic field reversals (Warnecke 2018). It is not difficult to understand that the anti-solar DR in GLYPH<11> GLYPH<10> dynamo model, does not allow polarity reversal. The anti-solar DR generates a toroidal field in such a way that the poloidal field produced through the GLYPH<11> effect (positive in the northern hemisphere) from this toroidal field is in the same direction as that of the old poloidal field. In Figure 1, we show how a Babcock-Leighton type GLYPH<11> GLYPH<10> dynamo with anti-solar DR produces poloidal field in the same direction as that of the original field and thereby does not offer reversal of the poloidal field at the end of a dynamo cycle. Based on the available data of the observed polar field, DR and other surface features, it is expected that the solar dynamo is primarily of GLYPH<11> GLYPH<10> type (Cameron & Schussler 2015). In recent years, it has been realized that it is the Babcock-Leighton process which acts like an GLYPH<11> effect to generate the poloidal field through the decay and dispersal of tilted bipolar magnetic regions (BMRs) (DasiEspuig et al. 2010; Kitchatinov & Olemskoy 2011a; Mu\u02dcnozJaramillo et al. 2013; Priyal et al. 2014). Although in the Sun, DR is the dominating source of the toroidal field, a weak toroidal field might be generated through the GLYPH<11> effect. With the increase of stellar rotation rate, GLYPH<11> increases (Krause & Radler 1980), while the DR ( GLYPH<10> ) does not have a strong dependency with the rotation (Kitchatinov & Rudiger 1999). Thus essentially, the stellar dynamo is of GLYPH<11> 2 GLYPH<10> type and this type of dynamo can result in polarity reversal depending on the profiles of GLYPH<11> and GLYPH<10> . Therefore, in the simulations with anti-solar DR, the magnetic field reversal is subtle as the toroidal field can be produced through the GLYPH<11> effect in addition to GLYPH<10> . Incidentally, Karak et al. (2015) found little reversal of magnetic field although it does not occur globally in all latitudes; see their Fig. 9 and 10, Runs ABC. They also found some irregular cycles in anti-solar DR regime. Recently, Viviani et al. (2018) found noticeable polarity reversal in anti-solar DR regime, however, Warnecke (2018) found almost no signature of polarity reversals in this regime. In summary, there is no consensus in the polarity reversal of the large-scale magnetic field in the simulations of the slowly-rotating stars producing anti-solar DR. In this study, we shall explore this and understand how the magnetic polarity reversal and cycles can be possible in this regime. The most interesting feature is the change of magnetic field strength in the slowly rotating stars. As referenced above, observations of slowly rotating stars show an increase of magnetic activity with the increase of rotation period which is possibly caused by the transition of DR to antisolar from the solar-like profile as proposed by Brandenburg &Giampapa (2018). Indeed, Karak et al. (2015) found an increase of the magnetic field in the anti-solar DR regime. Simulations of Warnecke (2018) in different parameter regimes seem to show a little increase of the magnetic field, in contrast, Viviani et al. (2018) did not find significant increase. This remains a curiosity in the community whether the magnetic field indeed increases in the anti-solar DR regime and if so, then whether this is actually caused by the anti-solar DR. The answers to these are not obvious because the strength of the dynamo, as determined by the GLYPH<11> and shear, do not necessarily increase in the anti-solar DR regime (slowly rotating stars). In fact, we expect the strength of GLYPH<11> to decrease with the decrease of rotation rate. Well, in simulations it has been observed that the shear is much strong in the anti-solar DR regime (see Table 1 of Karak et al. (2015) and Viviani et al. (2018)). But this cannot be the reason for the increased magnetic activity in slowly rotating stars because otherwise both Viviani et al. (2018) and Warnecke (2018) would also find an abrupt increase of magnetic field. Therefore, we shall understand the behaviour of the stellar dynamo with antisolar DR which is found in the slowly rotating stars. We shall explore whether the increase of magnetic field as seen in observation is possible in the dynamo model with anti-solar DR and what causes this increase. In our study, without going through the complexity of the global MHD convection simulations and saving the computational resources, we shall develop a simple kinematic dynamo model by specifying GLYPH<11> and GLYPH<10> for the sources of magnetic fields and make some clean simulations to explore the physics behind the magnetic cycles and polarity reversal in stars. We shall explore how the magnetic field strength varies when the DR profile changes from solar to anti-solar. We shall also present how other features of magnetic cycles, namely cycle period and the ratio of poloidal to toroidal fields change with the stellar rotation.", "pages": [2, 3]}, {"title": "2 MODEL", "content": "In our study, we develop a kinematic mean-field dynamo model by considering only the diagonal terms of the GLYPH<11> coefficients and an isotropic turbulent diffusion GLYPH<17> . We further assume magnetic field to be axisymmetric, thus writing magnetic field B = r GLYPH<2> \u00bb A ' r ; GLYPH<18>; t ' \u02c6 GLYPH<30> \u2026 + B ' r ; GLYPH<18>; t ' \u02c6 GLYPH<30> , where A is vector potential for poloidal field B p GLYPH<17> ' B r \u02c6 r ; B GLYPH<18> \u02c6 GLYPH<18> ' , B is toroidal field, and GLYPH<18> is co-latitude. Subsequently, the equations for A and B can be derived as where s = r sin GLYPH<18>; v p GLYPH<17> ' v r \u02c6 r ; v GLYPH<18> \u02c6 GLYPH<18> '' , and with GLYPH<11> rr ; GLYPH<11> GLYPH<18> GLYPH<18> and GLYPH<11> GLYPH<30> GLYPH<30> being the diagonal components of GLYPH<11> tensor. In our model, we shall ignore the off-diagonal components (which are responsible for magnetic pumping) because of the limited knowledge of their profiles in the solar and stellar convection zones (CZs) and to keep our model and the interpretation of the results tractable. Numerical simulations of magneto-convection in local Cartesian domain (e.g., Kapyla & Brandenburg 2009; Karak et al. 2014b) and global solar/stellar CZs (e.g., Simard et al. 2013, 2016; Warnecke et al. 2018) provide some guidance, although the simulation carried out are still far from the real Sun (in terms of its fundamental parameters). These simulations show that the GLYPH<11> components are highly inhomogeneous across radius and latitudes. They change sign at the equator and have no resemblance with each other. We also note that there is some amount of uncertainty in the measurement of GLYPH<11> coefficients, and so far, we do not have a well-tested method for their measurement. Thus keeping general features of the GLYPH<11> coefficients as obtained through theory and simulations, we begin with the following simple profiles for GLYPH<11> rr and GLYPH<11> GLYPH<18> GLYPH<18> . where R s is the solar radius; see Figure 2 top panels for the variations of this profile. For GLYPH<11> GLYPH<30> GLYPH<30> , we use a different profile (Figure 2 bottom panels) and it is given by where f s is a function that takes care of suppressing GLYPH<11> GLYPH<30> GLYPH<30> above GLYPH<6> 45 GLYPH<14> latitudes and thus f s = 1 GLYPH<157>\u00bb 1 + exp f 30 ' GLYPH<25> GLYPH<157> 4 GLYPH<0> GLYPH<18> 'g\u2026 for GLYPH<18> < GLYPH<25> GLYPH<157> 2 and 1 GLYPH<157>\u00bb 1 + exp f 30 ' GLYPH<18> GLYPH<0> 3 GLYPH<25> GLYPH<157> 4 'g\u2026 for otherwise. We take this factor f s in GLYPH<11> GLYPH<30> GLYPH<30> to restrict the strong toroidal field and thus the band of formation of sunspots in the low latitudes, which is a common practice in the flux transport dynamo models (Kuker et al. 2001; Dikpati et al. 2004; Guerrero & de Gouveia Dal Pino 2007; Hotta & Yokoyama 2010; Karak & Cameron 2016). Further, we choose GLYPH<11> GLYPH<30> GLYPH<30> to operate only near the surface so that the source for the poloidal field is slightly segregated from the source for the toroidal field. This, in turn, helps in producing longer cycle period of 11 years. For GLYPH<11> rr and GLYPH<11> GLYPH<18> GLYPH<18> we keep the profiles simple so that they both are non-zero in the whole CZ and have a cos GLYPH<18> dependence. Later in \u00a7 4, we shall also use different GLYPH<11> profiles and explore the robustness of our results. To limit the growth of magnetic field in our kinematic dynamo model, we consider following simple nonlinear quenching where i = r ; GLYPH<18>; GLYPH<30> and B 0 is the saturation field strength which is fixed at 4 GLYPH<2> 10 4 G in all the simulations. Because of this nonlinear saturation, we always present the magnetic field from our model with respect to B 0 . For DR, we take following profile which reasonably fits the helioseismology data where GLYPH<10> RZ GLYPH<157> 2 GLYPH<25> = 432 : 8 nHz, and GLYPH<10> CZ GLYPH<157> 2 GLYPH<25> = 460 : 7 GLYPH<0> 62 : 69 cos 2 GLYPH<18> GLYPH<0> 67 : 13 cos 4 GLYPH<18> nHz. When we make the DR antisolar, we take GLYPH<10> CZ GLYPH<157> 2 GLYPH<25> = 460 : 7 + 62 : 69 cos 2 GLYPH<18> + 67 : 13 cos 4 GLYPH<18> nHz. Note that this will allow GLYPH<10> to increase with latitudes and that is the anti-solar DR which is shown in Figure 3. The turbulent magnetic diffusivity GLYPH<17> has the following form: (9) where GLYPH<17> RZ = 5 GLYPH<2> 10 8 cm 2 s GLYPH<0> 1 , GLYPH<17> SCZ = 5 GLYPH<2> 10 10 cm 2 s GLYPH<0> 1 , and GLYPH<17> surf = 2 GLYPH<2> 10 12 cm 2 s GLYPH<0> 1 . Meridional circulation ( v r and v GLYPH<18> ) profile is the same as given in Hotta & Yokoyama (2010); also see Dikpati et al. (2004); Karak et al. (2014a); Karak & Petrovay (2013). For the sake of completeness, we write these profiles and boundary conditions in Appendix A.", "pages": [3, 4]}, {"title": "3 RESULTS", "content": "We start all simulations by specifying some initial magnetic field, and we analyse the results only after running the code for several diffusion times so that the magnetic field reaches a statistically stationary state. We first present the result of an GLYPH<11> 2 model by setting GLYPH<10> = 0 . As the results for such a dynamo model for the Sun with parameters given above has not been presented before, we shall first show the result.", "pages": [4]}, {"title": "3.1 Solar dynamo without differential rotation", "content": "The critical GLYPH<11> 0 for the GLYPH<11> 2 dynamo model without DR for the Sun is about 10 m s GLYPH<0> 1 . However, the prominent dynamo cycles with polarity reversals are seen only when GLYPH<11> 0 is approximately above 18 m s GLYPH<0> 1 . Butterfly diagram for GLYPH<11> 0 = 20 ms GLYPH<0> 1 is shown in Figure 4. We notice that even this model without DR produces some basic features of the solar cycle, namely, (i) the polarity reversal, (ii) 11-year periodicity, (iii) equatorward migration of toroidal field at the base of the CZ, and (iv) the poleward migration of surface poloidal field. Feature (i) is due to the selected inhomogeneous profile of GLYPH<11> taken in our model. The other properties are largely determined by the inclusion of meridional flow which is poleward near the surface and equatorward near the base of the CZ and a relatively weak diffusivity in the bulk of the CZ. The oscillatory magnetic field in GLYPH<11> 2 dynamo in spherical geometry is not new. It was realized that the GLYPH<11> 2 dynamo with certain GLYPH<11> profiles (Baryshnikova & Shukurov 1987; Stefani & Gerbeth 2003; Elstner & Rudiger 2007), and/or boundary conditions (Mitra et al. 2010; Jabbari et al. 2017) produces an oscillatory solution. We also note that the mean-field model constructed using GLYPH<11> profiles obtained from the global MHDsimulation also finds oscillatory solution (Simard et al. 2013).", "pages": [4]}, {"title": "3.2 Stellar dynamo with solar and anti-solar DR", "content": "Now we perform dynamo simulations of different stars with rotation period 1, 5, 10, 17, 25.38 (solar value), 30, 32, 40 and 50 days. As discussed in the introduction that the slowly rotating stars are expected to have anti-solar DR, which is, at least, confirmed in the global MHD simulations. This transition from solar to anti-solar DR is possibly happening below the solar rotation with Rossby number not too far from unity. Therefore, we assume that all stars with rotation periods up to 30 days are having solar-like DR, while stars of periods longer than 30 days have anti-solar DR. Further, the amount of internal DR is expected to change with the rotation rate. Hence, we choose it to scale in the following way. where T s = 25 : 38 days (solar rotation period), T is the rotation period of star, and GLYPH<10> s ' r ; GLYPH<18> ' is the internal angular frequency of the Sun as given in Equation (8). Some numerical simulations suggest n is about 0.3 (Ballot et al. 2007; Brown et al. 2008), while the recent work of Viviani et al. (2018) in a more wider range finds n GLYPH<25> GLYPH<0> 0 : 08 for the rotation rates up to 5 times solar value and n = GLYPH<0> 0 : 96 for rotation rate of 5-31 times the solar rotation. Surface observations find somewhat similar results, for example, Barnes et al. (2005), Reinhold & Gizon (2015), Donahue et al. (1996), and Lehtinen et al. (2016) respectively give n = 0 : 15 ; 0 : 29 ; 0 : 7 ; and GLYPH<0> 0 : 36 . In our study, we perform two sets of simulations by considering two values, namely n = 0 : 7 , i.e., the DR increases with the increase of rotation rate with an exponent of 0 : 7 and n = 0 : 0 , i.e. no change in DR. These two sets of simulations are labelled as A1-A50 and B1-B50; see Table 1. The amplitude of GLYPH<11> is also expected to increase with the increase of rotation rate (Krause & Radler 1980). As we do not know how exactly GLYPH<11> scales with the rotation rate, we choose it to increase linearly. Thus, in Equations (4-6), we take where GLYPH<11> 0 ; s is the value of GLYPH<11> 0 for the solar case, in which we take GLYPH<11> 0 ; s = 80 m s GLYPH<0> 1 . The meridional circulation is another important ingredient in the model which is expected to vary with the rotation rate. Our knowledge of meridional circulation even for the sun is very limited. Mean-field models and global convection simulations all suggest that its profile as well as amplitude changes with the rotation rate (Kitchatinov & Olemskoy 2011b; Featherstone & Miesch 2015; Karak et al. 2015, 2018a). Simulations of Brown et al. (2008) showed that the kinetic energy of meridional circulation approximately scales as GLYPH<10> GLYPH<0> 0 : 9 . Therefore in one set of simulations (Runs C1-C50 in Table 1), we shall change meridional circulation along with other parameters in the following way. where v 0 ; s is the amplitude of meridional circulation of sun, for which we have taken 10 m s GLYPH<0> 1 . We do not make any other changes in the model. Interestingly, all the stars, including ones with anti-solar DR (rotation periods 32-50 days), show prominent dynamo cycles and polarity reversals except for the very rapidly rotating star of rotation period 1 day. The butterfly diagrams of stars with rotation periods 5, 10, 25.38, 32, and 50 days (from Runs A1-A50) are shown in Figures 5 and 6. We observe that the 5 days rotating star (Run A5) does not show magnetic cycle at the base of the CZ, but it does some cycles in low latitudes of mid-CZ. The magnetic fields are largely anti-symmetric across the equator in all the cases. The equatorward migration is largely due to the return meridional flow. To quantify how the magnetic field strength and cy-", "pages": [4, 5]}, {"title": "6 Karak, Tomar & Vashishth", "content": "cle period change with different stars, we compute the rootmean-square ( rms ) field strength B rms = r D ' B 2 r + B 2 GLYPH<18> + B 2 GLYPH<30> ' E , the ratio of poloidal to toroidal field Bpol/Btor = q GLYPH<10> ' B 2 r + B 2 GLYPH<18> ' GLYPH<11> GLYPH<157> r D B 2 GLYPH<30> E (where angular brackets denote the av- erage over the whole domain), and the mean period of the polarity reversals of the toroidal field. These quantities are listed in Table 1 and the variations with rotation period are shown in Figure 7. In the rapidly rotating regime, the magnetic field strength increases with the decrease of rotation period, which is in agreement with observations (Petit et al. 2008). This behaviour in fact is not very surprising because the magnetic field sources ( GLYPH<11> and GLYPH<10> ) are increased with the rotation rates through equations (11) and (10). However, what is surprising in Figure 7(a) is that the magnetic field abruptly increased just above the rotation period of 30 days. That is the point where the DR pattern is changed from solar-like to anti-solar. Thus the anti-solar DR causes a sudden increase of magnetic field in the slowly rotating stars, which is in good agreement with the stellar observational data (Giampapa et al. 2006, 2017; Brandenburg & Giampapa 2018) and also with global MHD convection simulations (Karak et al. 2015). Further, we note that even the simulations (Run B1-B50; red asterisks in Figure 7a) in which GLYPH<10> does not change with the rotation rate (i.e., n = 0 in Equation (10)) also produces a similar variation of magnetic field. As the shear is not decreased in this case, the field in the anti-solar branch is little stronger in comparison to Set A (black points). The Set C, in which meridional flow is increased with the rotation period (Equation (12)), also show a similar variation of magnetic field (blue triangles in Figure 7a), which is unusual in the GLYPH<11> GLYPH<10> type flux transport dynamo models (Karak 2010). To understand the enhancement of the magnetic field in the anti-solar DR regime, we perform the following experiment. We consider the simulation of the 30-day rotation period having solar-like DR (Run A30). We first make sure that this model is producing a steady dynamo solution. Then, when the toroidal field reverses, that is when the toroidal field is at a minimum, we stop the code. The vertical line at t = 10 years in Figure 8 shows this time. Then we make the DR anti-solar and run the code for four years (up to the second vertical line in Figure 8). Finally, we revert the DR to the solar-like profile and continue the run for another 10 years. We see a significant increase of magnetic field in Figure 8(a) after the DR is made anti-solar. This increase of field must be caused by the change in DR from solar to anti-solar profile. We know that during the reversal, the anti-solar DR produces toroidal field of the same sign as that of the previous cycle. As seen in Figure 8(a), when the toroidal field in the northern hemisphere is mostly negative during the phase of anti-solar DR ( t = 10-14 years), the dynamo needs a positive field to reverse it. However, the anti-solar DR gives negative polarity field, and thus, it tries to enhance the old field (negative in the northern hemisphere). So essentially what is happening is the following: Pol( + ) GLYPH<0>GLYPH<0>GLYPH<0>GLYPH<0>GLYPH<0>GLYPH<0>GLYPH<0>GLYPH<0>! ' Solar GLYPH<10> ' Tor( GLYPH<0> ) GLYPH<0> ! GLYPH<11> Pol( GLYPH<0> ) GLYPH<0>GLYPH<0>GLYPH<0>GLYPH<0>GLYPH<0>GLYPH<0>GLYPH<0>GLYPH<0>GLYPH<0>GLYPH<0>GLYPH<0>GLYPH<0>! ' Anti GLYPH<0> sol GLYPH<10> ' Tor( GLYPH<0> ). Note in the last phase, the toroidal field is produced in the same polarity as that of the previous cycle and thus the field is enhanced; also see Figure 1. We may mention that instead of anti-solar DR if the GLYPH<11> is reversed, then also the same effect will be seen. In fact, following this idea, recently Karak et al. (2018b) explained the sudden increase of the poloidal field and appearance of double peaks in the sunspot cycles using a momentarily reversed GLYPH<11> due to wrong BMR tilt. Later, when the DR is changed to solar-like, this increasing effect of anti-solar DR is stopped and the model succeed to reverse the field (at around t = 16 years in Figure 8). We note that the increase of magnetic field is not immediately seen in the averaged field at the base of the CZ in Figure 8(b) because of the cancellation and finite transport time, but it is clearly seen in the next cycle. Above discussion shows that the anti-solar DR amplifies the old toroidal field by supplying toroidal field of the same polarity. Thus we do not expect polarity reversal if the antisolar DR acts for all the time and if there is no other source for the toroidal field generation. In our model, we certainly have GLYPH<11> effect for the generation of the toroidal field in addition to GLYPH<10> . However, this may not be sufficient to reverse the field. We realized that when the toroidal field has reached a very high value, the DR is needed to be suppressed otherwise the GLYPH<11> effect is not able to reverse the toroidal field and the cyclic dynamo is not possible. When the toroidal field is sufficiently high due to anti-solar DR, it is expected that this strong field acts back on the DR through Lorentz forces and tries to suppress the shear. The suppression mechanism could be complicated as the strong magnetic field can give Lorentz forces on the large-scale flows as well as suppress the convective angular momentum transport (see e.g, Karak et al. 2015; Kapyla et al. 2016). The bottom line is that the shear must be decreased once the toroidal field reaches a sufficiently high value. This, in fact, is seen in the simulations of Karak et al. (2015) that the shear parameters are quenched when the magnetic field is strong; see their Fig. 13(e-f). We also note that even in the Sun, there is an indication of the reduction of shear during the solar maximum; see Fig. 6-7 of Antia et al. (2008) and also Barekat et al. (2016). Motivated by these, we have introduced a nonlinear magnetic field dependent quenching f q = 1 GLYPH<157> GLYPH<16> 1 + ' B GLYPH<157> B 0 ' 2 GLYPH<17> in the shear such that r GLYPH<10> in Equation (2) is replaced by f q r GLYPH<10> . Through this nonlinear quenching, the toroidal field tries to reduce the shear when it exceeds B 0 . In Figure 9(b), we observe that the shear is strong during cycle minimum and thus, it produces a strong toroidal field. Then the strong field quenches the shear and the toroidal field cannot grow rapidly. Interestingly, the toroidal field generation due to GLYPH<11> effect S Tor GLYPH<11> (see Equation (3)) is not negligible (Figure 9(c)) and it is this process which eventually dominates and reverses the toroidal field slowly. Essentially, it is the nonlinear competition between the toroidal field generation through shear and GLYPH<11> , which makes the polarity reversal possible even with the anti-solar DR. To further support above conclusion, we show that if we do not include the quenching (i.e., f q = 1 ) in the shear, then the dynamo fails to reverse the magnetic field, rather the toroidal field increases in time; see red/dashed line in Figure 9(d). On the other hand, if we keep the quenching in shear but reduce the strength of GLYPH<11> by 75% , then also, dynamo fails to reverse the cycle and the magnetic field remains steady; see blue/solid line in Figure 9(d). In conclusion, to obtain the polarity reversal with anti-solar DR, there must be a sufficiently strong GLYPH<11> and the shear must be reduced when the toroidal field becomes very strong. Returning to Figure 7(b), we observe that for all the stars, the mean poloidal field is about one order of magnitude smaller than that of the toroidal field. The ratio Bpol/Btor has a non-monotonous behaviour. It is maximum at the solar rotation and decreases on both sides. This implies that the toroidal field dominates over the poloidal field both for rapidly and slowly rotating stars. With the increase of rotation rate, the toroidal field must increase faster than the poloidal component because there are two sources of toroidal field ( GLYPH<11> and shear) and both increase with the rotation rate. In the anti-solar branch, Bpol/Btor decreases because toroidal field generation is stronger with the anti-solar DR. The increase of poloidal contribution with the increase of cycle period in the solar-like DR branch is in agreement with the observational findings (Petit et al. 2008). These observational results also show that the magnetic energy of the toroidal field dominates below the rotation period of 12 days, while in our simulations the toroidal energy is always greater than the poloidal field. We should not forget that observational results are based on the field what was detected and it could be that most of the toroidal field in observations was not be detected. Therefore, instead of comparing the actual value of magnetic energy, we should see how the ratio of poloidal to toroidal field changes with the rotation rate and this is in somewhat agreement with observation (compare Figure 6 of Petit et al. (2008) with the left branch of our Figure 7(b)). Unfortunately, Petit et al. (2008) do not have data in the slowly rotating branch above the solar rotation period and thus we cannot compare this behaviour with observations. Finally, the cycle period again shows a non-monotonous behaviour. It decreases with the increase of rotation rate, which is in general agreement with observations (Noyes et al. 1984b; Su'arez Mascare\u02dcno et al. 2016; Boro Saikia et al. 2018), although the observational trend is messy. The slow decrease of rotation period in the anti-solar DR branch is not apparent in the observed data; see, for example, Fig. 9 of Boro Saikia et al. (2018). In our model, the decrease of cycle period with the increase of rotation rate is because the dynamo becomes stronger with the rotation rate which makes the conversion between poloidal and toroidal faster and reduces the cycle period. In global MHD simulations of stellar dynamos, Guerrero et al. (2018) also find a decrease of period with the increase of rotation rate, while Warnecke (2018) find this trend only at slow rotation and then an increasing trend in the rapid rotation. In latter simulations, the increase of period with rotation rate is due to the decrease of shear. In our study, when the DR is anti-solar, it tries to produce the same polarity field as that of the previous cycle and thus GLYPH<11> takes longer time to reverse the field-causing the cycle longer. This effect decreases when the rotation period is too long because the strength of the anti-solar DR decreases and thus the polarity reversal be- mes faster. This causes the cycle period to shorten with the increase of rotation period in the anti-solar branch. As the meridional flow transports the magnetic fields from source regions, the cycle duration tends to be longer with the decrease of meridional flow (Dikpati & Charbonneau 1999; Karak 2010). This was usually seen in previous flux transport dynamo models of stellar cycles (Nandy 2004; Jouve et al. 2010; Karak et al. 2014a), but not in the turbulent pumping-dominated regime (Karak & Cameron 2016; Hazra et al. 2019). Therefore, in Set C in which meridional circulation is decreased with the rotation rate following Equation (12), we expected an increase of cycle period. However, we see a reverse trend (see blue points in Figure 7(c)), because the dynamo becomes stronger and this tries to make the cycle shorter.", "pages": [6, 7, 8]}, {"title": "4 ROBUSTNESS OF OUR RESULTS", "content": "We have above seen that the results are only little sensitive to the change in GLYPH<10> and meridional circulation. Notably, the enhancement of the magnetic field at the transition point from solar to anti-solar DR is prominent. To explore the robustness of these results further, we consider different profiles for GLYPH<11> . In principle, we can consider countless different profiles for GLYPH<11> as the actual profiles are not known even for the Sun. Different profiles will make the dynamo solutions different but not necessarily the variations of magnetic field with rotation period. To demonstrate this, we present the results for two more sets of simulations, namely, Set D (Runs D1D50) and set E (Runs E1-E50); see Table 1. In Set D, we change GLYPH<11> rr and GLYPH<11> GLYPH<18> GLYPH<18> to as follows: both for r GLYPH<21> 0 : 7 R s , while GLYPH<11> rr = GLYPH<11> GLYPH<18> GLYPH<18> = 0 for r < 0 : 7 R s and GLYPH<11> 0 = 100 m s GLYPH<0> 1 ; see Figure A1 for these profiles. No other parameters are changed in this Set. On the other hand, in Set E, we change all three GLYPH<11> profiles to followings. With these new GLYPH<11> , we, however, do not get polarity reversal in a wide range of GLYPH<11> 0 both in solar and anti-solar DR cases. Interestingly, this new GLYPH<11> profile gives polarity reversals if we increase the diffusivity in the bulk of the CZ. That is, when we take GLYPH<17> SCZ = 1 GLYPH<2> 10 12 cm 2 s GLYPH<0> 1 , GLYPH<17> surf = 1 : 05 GLYPH<2> 10 12 cm 2 s GLYPH<0> 1 (see Figure A2 dashed line) and GLYPH<11> 0 = 80 ms GLYPH<0> 1 , we obtain polarity reversals in all runs except Run E1; see Figure A3. It could be that in this new GLYPH<11> profile, unless we increase the diffusion in the deeper CZ, the toroidal field generation due to shear overpowers the same due to GLYPH<11> effect. This higher diffusion makes the equatorward migration of toroidal field less important and the magnetic field in run E50 quadrupolar. We note that with these new GLYPH<11> profiles, we again obtain a clear increase of magnetic field in both the sets D and E; see Table 1, although the increase is relatively small. We note that in both sets, GLYPH<11> and GLYPH<10> are decreased with the increase of rotation period as before following equations (11) and (10).", "pages": [8]}, {"title": "5 CONCLUSION", "content": "In this study, using a mean-field kinematic dynamo model we have explored the features of large-scale magnetic fields of solar-like stars rotating at different rotation periods and having the same internal structures as that of the Sun. Our main motivation is to understand the large-scale magnetic field generation in the slowly rotating stars with rotation period larger than the solar value. This region is of particular interest because these stars possibly possess anti-solar DR and thus the operation of dynamo can be fundamentally different than stars having the usual solar-like DR. By carrying out simulations for different stars at different model parameters, we show that the fundamental features of stellar magnetic field change with rotation period. In the solar-like DR branch, we find the magnetic field strength increases with the decrease of rotation period. This result is, in general, agreement with the observational findings (Noyes et al. 1984a; Petit et al. 2008; Wright et al. 2011; Wright & Drake 2016), the global MHD convection simulations (Viviani et al. 2018; Warnecke 2018), and mean-field dynamo modellings (Jouve et al. 2010; Karak et al. 2014a; Kitchatinov & Olemskoy 2015; Hazra et al. 2019). However, our model does not produce the observed saturation of magnetic field in the very rapidly rotating stars, which in the kinematic models, requires some additional dynamo saturation (Karak et al. 2014a; Kitchatinov & Olemskoy 2015). In the slowly rotating stars, we see a sudden jump of magnetic field strength at the point where the DR profile changes to anti-solar from solar. This result is in agreement with the stellar observations (Giampapa et al. 2006, 2017; Brandenburg & Giampapa 2018) and also with the MHD convection simulations of Karak et al. (2015) and Warnecke (2018). The abrupt increase of magnetic field is due to the anti-solar DR which amplifies the existing toroidal field by supplying the same polarity field. The idea of the enhancement of magnetic field due to the change of DR was already proposed by Brandenburg & Giampapa (2018), however, no detailed dynamo modelling was performed. We further show that with particular GLYPH<11> profiles, the polarity reversal of the large-scale magnetic field is possible even in the slowly rotating stars with anti-solar DR rotation provided, (i) there is a sufficiently strong GLYPH<11> for the generation of toroidal field and (ii) the anti-solar DR is nonlinearly modulated with the magnetic field such that when the toroidal field becomes strong, it quenches the shear. Our conclusion of polarity reversal in general supports the work of Viviani et al. (2019), who showed that the polarity in their global MHD convection with anti-solar DR is possible as the magnetic field generation through GLYPH<11> effect is comparable to that of GLYPH<10> effect. One may argue that the global MHD convection simulations of stellar CZs are still far from the real stars and there is always a question to what extent the results from these simulations hold to the real stars. Interestingly, one robust result of these simulations is that they all produce anti-solar DR in the slowly rotating stars with rotation period somewhere above the solar value with Rossby number around one. Available techniques are still insufficient to confirm the existence of anti-solar DR in solar-like dwarfs; see Reinhold & Arlt (2015). However, this has been confirmed in some K-giants (Strassmeier et al. 2003; Weber et al. 2005; K\"ov'ari et al. 2017) and subgiants (Harutyunyan et al. 2016). The enhancement of magnetic activity in the slowly rotating stars and its generation through the anti-solar DR give another support for the existence of anti-solar DR in slowly rotating dwarfs. Furthermore, this study along with previous observational results and global simulations suggest that the slowly rotating stars possess strong large-scale magnetic fields and possibly polarity reversals and cycles. These slowly rotating solar-like stars may also be prone to produce superflares (Maehara et al. 2012), which was also suggested by Katsova et al. (2018).", "pages": [9]}, {"title": "ACKNOWLEDGEMENTS", "content": "We thank the anonymous referee for carefully checking the manuscript and raising interesting questions which particularly helped us to correct an error that we made in the earlier draft. We further thank Gopal Hazra and Sudip Mandal for discussion on various aspects of the stellar dynamo. We sincerely acknowledge financial support from Department of Science and Technology (SERB/DST), India through the Ramanujan Fellowship awarded to B.B.K. (project no SB/S2/RJN-017/2018). BBK appreciates gracious hospitality at Indian Institute of Astrophysics, Bangalore during the last phase of this project. V.V. acknowledges financial support from DST through INSPIRE Fellowship.", "pages": [9]}, {"title": "REFERENCES", "content": "Guerrero G., Smolarkiewicz P. K., Kosovichev A. G., Mansour N. N., 2013, ApJ, 779, 176 Harutyunyan G., Strassmeier K. G., Kunstler A., Carroll T. A., Weber M., 2016, A&A, 592, A117 Karak B. B., Rheinhardt M., Brandenburg A., Kapyla P. J., Kapyla M. J., 2014b, ApJ, 795, 16 Karak B. B., Kapyla P. J., Kapyla M. J., Brandenburg A., Olspert N., Pelt J., 2015, A&A, 576, A26 Karak B. B., Mandal S., Banerjee D., 2018b, ApJ, 866, 17 Katsova M. M., Kitchatinov L. L., Livshits M. A., Moss D. L., Sokoloff D. D., Usoskin I. G., 2018, Astronomy Reports, 62, 72 Kitchatinov L. L., Olemskoy S. V., 2011b, MNRAS, 411, 1059 Kitchatinov L. L., Olemskoy S. V., 2015, Research in Astronomy and Astrophysics, 15, 1801 Kitchatinov L. L., Rudiger G., 1999, A&A, 344, 911 Krause F., Radler K. H., 1980, Mean-field magnetohydrodynamics and dynamo theory. Oxford: Pergamon Press Kuker M., Rudiger G., Schultz M., 2001, A&A, 374, 301 Lehtinen J., Jetsu L., Hackman T., Kajatkari P., Henry G. W., 2016, A&A, 588, A38 Maehara H., et al., 2012, Nature, 485, 478 Nandy D., 2004, Sol. Phys., 224, 161 Noyes R. W., Weiss N. O., Vaughan A. H., 1984b, ApJ, 287, 769 Pallavicini R., Golub L., Rosner R., Vaiana G. S., Ayres T., Linsky J. L., 1981, ApJ, 248, 279 Petit P., et al., 2008, MNRAS, 388, 80 Reiners A., Schussler M., Passegger V. M., 2014, ApJ, 794, 144 Reinhold T., Arlt R., 2015, A&A, 576, A15 Simard C., Charbonneau P., Bouchat A., 2013, ApJ, 768, 16 Warnecke J., 2018, A&A, 616, A72 ApJ, 743, 48", "pages": [10]}, {"title": "APPENDIX A: SUPPLEMENTARY MATERIAL", "content": "The meridional flow is obtained from the following analytical form. with Here m = 0 : 5 , p = 0 : 25 , v 0 = 10 m s GLYPH<0> 1 , and r p = 0 : 62 R s . The boundary conditions are exactly taken from Chatterjee et al. (2004) and thus we do not repeat those here. For the initial magnetic field, we take This paper has been typeset from a T E X/L A T E X file prepared by the author.", "pages": [10]}] |
2017PhLB..764..300B | https://arxiv.org/pdf/1610.09952.pdf | <document>
<section_header_level_1><location><page_1><loc_30><loc_80><loc_70><loc_85></location>Remarks on the Taub-NUT solution in Chern-Simons modified gravity</section_header_level_1>
<text><location><page_1><loc_36><loc_77><loc_64><loc_78></location>Yves Brihaye † and Eugen Radu ‡</text>
<text><location><page_1><loc_26><loc_73><loc_73><loc_75></location>† Physique-Math'ematique, Universite de Mons-Hainaut, Mons, Belgium</text>
<text><location><page_1><loc_28><loc_70><loc_72><loc_73></location>‡ Departamento de F'ısica da Universidade de Aveiro and CIDMA, Campus de Santiago, 3810-183 Aveiro, Portugal</text>
<text><location><page_1><loc_44><loc_67><loc_56><loc_68></location>July 21, 2021</text>
<section_header_level_1><location><page_1><loc_47><loc_62><loc_53><loc_63></location>Abstract</section_header_level_1>
<text><location><page_1><loc_16><loc_50><loc_84><loc_61></location>We discuss the generalization of the NUT spacetime in General Relativity (GR) within the framework of the (dynamical) Einstein-Chern-Simons (ECS) theory with a massless scalar field. These configurations approach asymptotically the NUT spacetime and are characterized by the 'electric' and 'magnetic' mass parameters and a scalar 'charge'. The solutions are found both analytically and numerically. The analytical approach is perturbative around the Einstein gravity background. Our results indicate that the ECS configurations share all basic properties of the NUT spacetime in GR. However, when considering the solutions inside the event horizon, we find that in contrast to the GR case, the spacetime curvature grows (apparently) without bound.</text>
<section_header_level_1><location><page_1><loc_12><loc_46><loc_30><loc_48></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_12><loc_37><loc_88><loc_45></location>The Einstein-Chern-Simons (ECS) theory [1] is one of the most interesting generalizations of the General Relativity (GR) [2]. In its dynamical version, this model possesses a (real) scalar field φ , with an axionic-type coupling with the Pontryagin density [3]. As such, its action contains extra-terms quadratic in the curvature which can potentially lead to new effects in the strong-field regime. Moreover, this model is motivated by string theory results [4] and occurs also in the framework of loop quantum gravity [5], [6].</text>
<text><location><page_1><loc_12><loc_27><loc_88><loc_37></location>In contrast to its Einstein-Gauss-Bonnet counterpart (in which case φ couples to the Gauss-Bonnet scalar), it can be shown that any static spherically symmetric solution of GR is also a solution of ECS gravity. Therefore this model is almost unique, as it leads to different results only in the presence of a parity-odd source such as rotation. However, despite the presence in the literature of some partial results [7], [8], [9], the generalizations of the (astrophysically relevant) Kerr solution in ECS theory is still unknown, presumably due to the complexity of the problem. Therefore the study of ECS generalizations of known GR rotating solutions is a pertinent task which, ultimately, could lead to some progress in the Kerr problem.</text>
<text><location><page_1><loc_12><loc_11><loc_88><loc_26></location>One of the most intriguing solutions of GR has been found in 1963 by Newman, Tamburino and Unti (NUT) [10]. This is a generalization of the Schwarzschild solution which solves the Einstein vacuum field equations, possessing in addition to the mass parameter M an extra-parameter-the NUT charge n . In its usual interpretation, it describes a gravitational dyon with both ordinary and magnetic mass. The NUT charge n plays a dual role to ordinary ADM mass M , in the same way that electric and magnetic charges are dual within Maxwell theory [11]. This solution has a number of unusual properties, becoming renowned for being 'a counter-example to almost anything' [12]. For example, the NUT spacetime is not asymptotically flat in the usual sense although it does obey the required fall-off conditions, and, moreover, contains closed timelike curves. As such, it is cannot be taken as a realistic model for a macroscopic object, although its Euclideanized version might play a role in the context of quantum gravity [14].</text>
<text><location><page_2><loc_12><loc_81><loc_88><loc_90></location>For the purposes of this work, the NUT metric is interesting from another point of view: its line-element can be taken as Kerr-like, in the sense that it has a crossed metric component g ϕt , see (2.7) bellow. This term does not produce an ergoregion but it leads to an effect similar to the dragging of inertial frames [15]. Moreover, one can say that a NUT spacetime consists of two counter-rotating regions, with a vanishing total angular momentum [16], [17]. Therefore, the study of its generalization in the framework of ECS theory is a legitimate task.</text>
<text><location><page_2><loc_12><loc_76><loc_88><loc_80></location>Also, one should mention that the NUT solution has been generalized already in various models. For example, nutty solutions with gauge fields have been has been found in [18], [19], [20]. The low-energy string theory possess also nontrivial solutions with NUT charge (see e.g. [21]).</text>
<text><location><page_2><loc_12><loc_66><loc_88><loc_75></location>The paper is structured as follows: in the next Section we review the basic framework of the model which includes the metric and scalar field Ansatz. Some properties of general nutty solutions are also discussed there. In Section 3 we present the results of a perturbative construction of solutions as a power series in the CS coupling constant. The basic properties of the non-perturbative configurations are discussed in Section 4. We conclude with Section 5 where the results are compiled. There we present also our results for the Taub region of the solutions and give arguments that the solution is divergent there.</text>
<section_header_level_1><location><page_2><loc_12><loc_62><loc_33><loc_64></location>2 The framework</section_header_level_1>
<section_header_level_1><location><page_2><loc_12><loc_59><loc_51><loc_61></location>2.1 The Chern-Simons modified gravity</section_header_level_1>
<text><location><page_2><loc_12><loc_57><loc_57><loc_58></location>The action of the dynamical CS modified gravity is provided by</text>
<formula><location><page_2><loc_26><loc_52><loc_88><loc_56></location>I = ∫ d 4 x √ -g ( κR + α 4 φ ∗ RR -1 2 g ab ( ∇ a φ )( ∇ b φ ) -V ( φ ) ) , (2.1)</formula>
<text><location><page_2><loc_12><loc_48><loc_88><loc_51></location>where g is the determinant of the metric g µν , R is the Ricci scalar and we note κ -1 = 16 πG . The quantity ∗ RR is the Pontryagin density, defined via</text>
<formula><location><page_2><loc_30><loc_44><loc_88><loc_47></location>∗ RR = ∗ R a b cd R b acd , with ∗ R a b cd = 1 2 /epsilon1 cdef R a bef , (2.2)</formula>
<text><location><page_2><loc_12><loc_42><loc_80><loc_43></location>(where /epsilon1 cdef is the 4-dimensional Levi-Civita tensor). The gravity equations for this model read</text>
<formula><location><page_2><loc_28><loc_37><loc_88><loc_40></location>R ab -1 2 g ab R = 1 2 κ T ( eff ) ab , with T ( eff ) ab = T ( φ ) ab -2 αC ab , (2.3)</formula>
<text><location><page_2><loc_12><loc_35><loc_16><loc_36></location>where</text>
<formula><location><page_2><loc_32><loc_32><loc_88><loc_34></location>C ab = ( ∇ c φ ) /epsilon1 cde ( a ∇ e R b ) d +( ∇ c ∇ d φ ) ∗ R d ( ab ) c , (2.4)</formula>
<text><location><page_2><loc_12><loc_29><loc_54><loc_31></location>and T ( φ ) ab is the energy-momentum tensor of the scalar field,</text>
<formula><location><page_2><loc_30><loc_25><loc_88><loc_28></location>T ( φ ) ab = ( ∇ a φ ) ( ∇ b φ ) -[ 1 2 g ab ( ∇ c φ ) ( ∇ c φ ) + g ab V ( φ ) ] . (2.5)</formula>
<text><location><page_2><loc_12><loc_21><loc_88><loc_24></location>The scalar field solves the Klein-Gordon equation in the presence of a source term given by the Pontryagin density,</text>
<formula><location><page_2><loc_42><loc_18><loc_88><loc_21></location>∇ 2 φ = dV dφ -α 4 ∗ RR. (2.6)</formula>
<text><location><page_2><loc_12><loc_14><loc_88><loc_17></location>To simplify the picture, in this work we shall report results for a massless, non-selfinteracting scalar only, V ( φ ) = 0.</text>
<section_header_level_1><location><page_3><loc_12><loc_88><loc_28><loc_90></location>2.2 The Ansatz</section_header_level_1>
<text><location><page_3><loc_12><loc_86><loc_74><loc_87></location>We consider a NUT-charged spacetime whose metric can be written locally in the form</text>
<formula><location><page_3><loc_24><loc_82><loc_88><loc_85></location>ds 2 = dr 2 N ( r ) + g ( r )( dθ 2 +sin 2 θdϕ 2 ) -N ( r ) σ 2 ( r )( dt +4 n sin 2 θ 2 dϕ ) 2 , (2.7)</formula>
<text><location><page_3><loc_12><loc_76><loc_88><loc_81></location>while the scalar field depends on the r -coordinate only, φ = φ ( r ). Here θ and ϕ are the standard angles parametrizing an S 2 with the usual range. As usual, we define the NUT parameter 1 n (with n ≥ 0, without any loss of generality), in terms of the coefficient appearing in the differential dt +4 n sin 2 θ 2 dϕ .</text>
<text><location><page_3><loc_12><loc_69><loc_88><loc_77></location>The form of N ( r ) , σ ( r ) and g ( r ) emerges as result of demanding the metric to be a solution of the ECS equations (2.3) (note the existence of a metric gauge freedom in (2.7), which is fixed later by convenience). The equations satisfied by these functions (and the corresponding one for φ ( r )) are rather complicated and we shall not not include them here. However, we notice that they can also be derived from the effective action</text>
<formula><location><page_3><loc_37><loc_66><loc_88><loc_69></location>L eff = L E + κ ( α 4 L CS + L φ ) , (2.8)</formula>
<text><location><page_3><loc_12><loc_64><loc_16><loc_65></location>where</text>
<formula><location><page_3><loc_15><loc_57><loc_89><loc_64></location>L E = 2 σ [ 1 + ( N ' 2 N + g ' 4 g + σ ' σ ) Ng ' + σ 2 N g n 2 ] , L φ = -1 2 Nσgφ ' 2 , L CS = 8 n Nσ 2 g [ ( N ' N -g ' g + 2 σ ' σ )(1 + 4 n 2 Nσ 2 g ) φ + ( N ' 2 4 N 2 + g ' 2 4 g 2 + σ ' 2 σ 2 -g ' N ' 2 gN -g ' σ ' gσ + N ' σ ' Nσ ) Ngφ ' ] ,</formula>
<text><location><page_3><loc_12><loc_52><loc_88><loc_56></location>(where a prime denotes a derivative w.r.t. the radial coordinate r ). Remarkably, one can see that, due to the factorization of the angular dependence for the metric Ansatz (2.7), all functions solve second order equations of motion 2 .</text>
<text><location><page_3><loc_12><loc_47><loc_88><loc_51></location>The reduced action (2.8) makes transparent the scaling symmetries of the problem. For example, to simplify the analysis, it is convenient to work with conventions where κ = 1 (this is obtained by rescaling the scalar field and the coupling constant α ). Then the system still has a residual scaling symmetry</text>
<formula><location><page_3><loc_32><loc_44><loc_88><loc_46></location>α → αλ 2 , r → λr, n → λn, and g → λ 2 g, (2.9)</formula>
<text><location><page_3><loc_12><loc_43><loc_44><loc_44></location>which can be used to fix the value of α or n .</text>
<text><location><page_3><loc_12><loc_40><loc_88><loc_42></location>Finally, we note that the NUT solution is found for α = 0, φ = 0, being usually written for a gauge choice with</text>
<formula><location><page_3><loc_28><loc_36><loc_88><loc_39></location>σ ( r ) = 1 and N ( r ) = 1 -2( Mr + n 2 ) r 2 + n 2 , g ( r ) = r 2 + n 2 , (2.10)</formula>
<text><location><page_3><loc_12><loc_34><loc_44><loc_35></location>possessing a nonvanishing Pontryagin density</text>
<formula><location><page_3><loc_20><loc_29><loc_88><loc_33></location>∗ RR = 96 n 2 ( r 2 + n 2 ) 6 ( n 2 ( n 2 -3 r 2 ) + Mr (3 n 2 -r 2 ) ) ( n 2 ( M -3 r ) + r 2 ( r -3 M ) ) , (2.11)</formula>
<text><location><page_3><loc_12><loc_26><loc_88><loc_29></location>(and thus it cannot be promoted to a solution of the ECS model). This metric has an (outer) horizon located at 3</text>
<formula><location><page_3><loc_39><loc_23><loc_88><loc_26></location>r H = M + √ M 2 + n 2 > 0 . (2.12)</formula>
<text><location><page_3><loc_12><loc_19><loc_88><loc_23></location>Here, similar to the Schwarzschild limit, N ( r H ) = 0 is only a coordinate singularity where all curvature invariants are finite. In fact, a nonsingular extension across this null surface can be found just as at the event horizon of a black hole.</text>
<section_header_level_1><location><page_4><loc_12><loc_88><loc_35><loc_90></location>2.3 General properties</section_header_level_1>
<text><location><page_4><loc_44><loc_79><loc_44><loc_81></location>/negationslash</text>
<text><location><page_4><loc_12><loc_69><loc_88><loc_87></location>Some basic properties of the line element (2.7) are generic, independent on the specific details of the considered gravity model. As a result, the general nutty configurations always share the same troubles exhibited by the original NUT solution in GR. For example, the Killing symmetries of (2.7) are time translation and SO (3) rotations. However, spherical symmetry in a conventional sense is lost, since the rotations act on the time coordinate as well. Moreover, for n = 0, the metric (2.7) has a singular symmetry axis. However, following the discussion in [12] for the GR limit, these singularities can be removed by appropriate identifications and changes in the topology of the spacetime manifold, which imply a periodic time coordinate. Then such a configuration cannot be interpreted properly as black hole. In fact, the pathology of closed timelike curves is not special to the NUT solution in GR but afflicts all solutions with a 'dual' magnetic mass in general [22]. As discussed in [23], this condition emerges only from the asymptotic form of the fields. Therefore, it is not sensitive to the precise details of the nature of the source, or the precise nature of the theory of gravity at short distances.</text>
<text><location><page_4><loc_12><loc_64><loc_88><loc_69></location>In our approach we are interested in solutions whose far field asymptotics are similar, to leading order, to those of the Einstein gravity solution (2.10), with N ( r ) → 1, g ( r ) → r 2 , σ ( r ) → 1 and φ ( r ) → 0 as r →∞ . The solution will posses also an horizon at r = r H > 0, where N ( r H ) = 0, and g ( r ), σ ( r ) strictly positive.</text>
<text><location><page_4><loc_12><loc_60><loc_88><loc_64></location>In the absence of a global Cauchy surface, the thermodynamical description of (Lorentzian signature) nutty solutions is still poorly understood. However, one can still define a temperature of solutions via the surface gravity associated with the Killing vector ∂/∂t ,</text>
<formula><location><page_4><loc_40><loc_56><loc_88><loc_59></location>T H = 1 4 π N ' ( r H ) σ ( r H ) , (2.13)</formula>
<text><location><page_4><loc_12><loc_54><loc_36><loc_56></location>and also an even horizon area [24]</text>
<formula><location><page_4><loc_32><loc_49><loc_88><loc_54></location>A H = ∫ π 0 dθ ∫ 2 π 0 dϕ √ g θθ g ϕϕ ∣ ∣ r = r H = 4 πg ( r H ) . (2.14)</formula>
<text><location><page_4><loc_12><loc_44><loc_88><loc_50></location>The mass of the solutions can be computed by employing the quasilocal formalism in conjuction with the boundary counterterm method [25]. A direct computation shows that, similar to the Einstein gravity case, the mass of the solutions is identified with the constant M in the far field expansion of the metric function g tt ,</text>
<formula><location><page_4><loc_40><loc_40><loc_88><loc_43></location>g tt = -1 + 2 M r + . . . . (2.15)</formula>
<section_header_level_1><location><page_4><loc_12><loc_37><loc_44><loc_39></location>3 A perturbative approach</section_header_level_1>
<text><location><page_4><loc_12><loc_31><loc_88><loc_36></location>An exact solution of the equations (2.3), (2.6) can be found in the limit of small α , by treating the ECS configurations as perturbations around the Einstein gravity background. Here we have found convenient to work in a gauge with</text>
<formula><location><page_4><loc_43><loc_29><loc_88><loc_31></location>g ( r ) = r 2 + n 2 . (3.16)</formula>
<text><location><page_4><loc_12><loc_27><loc_44><loc_28></location>Then we consider a perturbative Ansatz with</text>
<formula><location><page_4><loc_18><loc_25><loc_88><loc_26></location>N ( r ) = N 0 ( r )(1 + α 2 N 2 ( r ) + . . . ) , σ ( r ) = 1 + α 2 σ 2 ( r ) + . . . , φ ( r ) = αφ 1 ( r ) + . . . , (3.17)</formula>
<text><location><page_4><loc_12><loc_22><loc_74><loc_24></location>where N 0 = 1 -2( M 0 r + n 2 ) / ( r 2 + n 2 ) corresponds to the solution in Einstein gravity.</text>
<text><location><page_4><loc_14><loc_21><loc_78><loc_23></location>To this order, one arrives at the following system of linear ordinary differential equations</text>
<formula><location><page_4><loc_15><loc_10><loc_92><loc_21></location>rN ' 2 + 1 N 0 N 2 -6 n 2 g σ 2 = 2 n g 2 ( r ( r 2 -3 n 2 ) + M 0 ( n 2 -3 r 2 ) )( φ '' 1 -r ( r 2 -3 n 2 ) + M 0 ( n 2 -3 r 2 ) N 0 g 2 φ ' ) -1 4 gφ ' 2 1 , rσ ' 2 + 2 n 2 g σ 2 = 1 4 gφ ' 2 1 -n g 2 ( r ( r 2 -3 n 2 ) + M 0 ( n 2 -3 r 2 ) ) φ '' 1 , (3.18) φ '' 1 -2( M 0 -r ) N 0 g φ ' = 24 n N 0 g 6 ( M 0 r ( r 2 -3 n 2 ) -n 2 ( n 2 -3 r 2 ) )( r ( r 2 -3 n 2 ) + M 0 ( n 2 -3 r 2 ) ) .</formula>
<text><location><page_5><loc_12><loc_84><loc_88><loc_90></location>When solving them, there are four integration constants. These constants are chosen such that the corrected NUT metric is still smooth at r = r H and approaches a background with N ( r ) → 1 and σ ( r ) → 1 asymptotically, while φ ( r ) → 0. Then, to lowest order, the solution has the generic structure</text>
<formula><location><page_5><loc_27><loc_80><loc_88><loc_84></location>F = P 0 ( r ) + P 1 ( r ) arctan ( n r ) + P 2 ( r ) log ( ( n 2 + r 2 ) r 2 H ( n 2 + rr H ) 2 ) , (3.19)</formula>
<text><location><page_5><loc_12><loc_77><loc_88><loc_80></location>with F = { N 2 , σ 2 , φ 1 } . The functions P 0 , P 1 and P 2 are ratio of polynomials, possessing a simple form for φ 1 only, with</text>
<formula><location><page_5><loc_27><loc_69><loc_88><loc_76></location>P 0 = n 2 ( r 2 + n 2 ) 3 ( ( n 2 -r 2 H ) nr H ( n 2 + ( r 2 -n 2 ) 2 4 n 2 ) + 4 rn ) -r 2 n ( r 2 + n 2 ) , (3.20) P 1 = 1 n 2 , P 2 = -r 2 n ( r 2 + n 2 ) -n 2 -r 2 H 4 nr H ,</formula>
<text><location><page_5><loc_12><loc_65><loc_88><loc_68></location>the corresponding expressions for N 2 , σ 2 being too complicated to display here. To this order in perturbation theory, one finds to following far field expression of the scalar field</text>
<formula><location><page_5><loc_29><loc_60><loc_88><loc_64></location>φ 1 ( r ) = q r -n ( n 2 -r 2 H ) 4 r 3 H 1 r 2 + . . . , with q = n 2 r 2 H > 0 , (3.21)</formula>
<text><location><page_5><loc_12><loc_58><loc_51><loc_59></location>while the mass parameter has the following expression</text>
<formula><location><page_5><loc_12><loc_53><loc_90><loc_57></location>M = M 0 + α 2 M 2 , with M 2 = 1 64 n 5 r 5 H ( U 0 ( n, r H ) + U 1 ( n, r H ) arctan( n r H ) + U 2 ( n, r H ) log( r 2 H n 2 + r 2 H ) ) , (3.22)</formula>
<text><location><page_5><loc_12><loc_50><loc_36><loc_52></location>where M 0 = ( r 2 H -n 2 ) / (2 r H ), and</text>
<formula><location><page_5><loc_25><loc_46><loc_63><loc_50></location>U 0 = n 210 ( 429 n 6 +2716 n 4 r 2 H -2555 n 2 r 4 H -3570 r 6 H ) ,</formula>
<formula><location><page_5><loc_25><loc_44><loc_78><loc_47></location>U 1 = -r H ( n 2 + r 2 H )(11 n 4 +5 r 2 r 2 H -22 r 4 H ) , U 2 = 1 n ( r 4 H -n 4 )(5 r 4 H -n 4 ) .</formula>
<text><location><page_5><loc_12><loc_42><loc_57><loc_43></location>The same type of expression is found for the temperature, with</text>
<formula><location><page_5><loc_17><loc_33><loc_88><loc_41></location>T H = 1 4 πr H [ 1 + α 2 6720 n 2 r 4 H ( n 2 + r 2 H ) 2 ( n 2 (429 n 8 +5951 n 6 r 2 H +343 n 4 r 4 H -3115 n 2 r 6 H -1680 r 8 H ) -210( n 2 -r 2 H )( n 2 + r 2 H ) 3 ( 11 nr H arctan( n r H ) -( n 2 -3 r 2 H ) log( r 2 H n 2 + r 2 H ) ) )] . (3.23)</formula>
<text><location><page_5><loc_12><loc_28><loc_88><loc_33></location>An inspection of the (3.22) shows that M 2 is a strictly negative quantity. However, the CS correction to T H has no definite sign. For a given n , it is negative for small r H and becomes strictly positive for large enough r H (in particular for r H > n ).</text>
<text><location><page_5><loc_12><loc_24><loc_88><loc_28></location>This approach can be extended to higher order in α . Unfortunately, the resulting equations are too complicated for an analytical treatment. Although they can be solved numerically, we have preferred to consider instead a fully nonperturbative approach.</text>
<section_header_level_1><location><page_5><loc_12><loc_20><loc_36><loc_21></location>4 Numerical results</section_header_level_1>
<text><location><page_5><loc_12><loc_10><loc_88><loc_18></location>The nonperturbative solutions are constructed by solving numerically the ECS eqs. (2.3), (2.6), as a boundary value problem. In this approach, it is convenient to employ the same metric gauge as in Einstein gravity, and take σ ( r ) = 1. Then we consider solutions in the domain r H ≤ r < ∞ (with r H > 0), smoothly interpolating between the following boundary values: N ( r H ) = 0, g ( r H ) = g 0 > 0, φ ( r H ) = φ 0 and N = 1, g = r 2 , φ = 0 as r → ∞ . An approximate expression of the solutions compatible with these asymptotics can easily be</text>
<figure>
<location><page_6><loc_13><loc_70><loc_49><loc_89></location>
<caption>Figure 1: Left : The profiles of r 2 N ' / 2 and g ' / (2 r ) are shown for several values of α . The solutions have r h = 1, n = 0 . 1. Right: The same for the scalar field φ and the Ricci scalar R .</caption>
</figure>
<text><location><page_6><loc_69><loc_70><loc_70><loc_71></location>log</text>
<text><location><page_6><loc_55><loc_78><loc_56><loc_82></location>event horizon</text>
<text><location><page_6><loc_56><loc_71><loc_57><loc_72></location>0</text>
<text><location><page_6><loc_63><loc_71><loc_65><loc_72></location>0.25</text>
<text><location><page_6><loc_70><loc_71><loc_72><loc_72></location>0.5</text>
<text><location><page_6><loc_77><loc_71><loc_79><loc_72></location>0.75</text>
<text><location><page_6><loc_85><loc_71><loc_86><loc_72></location>1</text>
<text><location><page_6><loc_71><loc_70><loc_72><loc_71></location>r</text>
<text><location><page_6><loc_70><loc_70><loc_71><loc_71></location>10</text>
<text><location><page_6><loc_12><loc_59><loc_37><loc_61></location>found. Its first terms as r → r H are</text>
<formula><location><page_6><loc_20><loc_53><loc_88><loc_59></location>N ( r ) = N 1 ( r -r H ) -1 g 0 g 2 0 +3 N 1 n 2 α 2 g 2 0 -3 N 1 n 2 α 2 ( r -r H ) 2 + . . . , (4.24) g ( r ) = g 0 + 1 N 2 g 2 0 g 2 3 N n 2 α 2 ( r -r H ) + . . . , φ ( r ) = φ 0 -6 nα g 2 3 N n 2 α 2 ( r -r H ) + . . . ,</formula>
<formula><location><page_6><loc_30><loc_52><loc_69><loc_54></location>1 0 -1 0 -1</formula>
<text><location><page_6><loc_12><loc_50><loc_85><loc_52></location>{ N 1 , g 0 , φ 0 } being three undetermined parameters, while the leading order expansion in the far field is</text>
<formula><location><page_6><loc_28><loc_40><loc_88><loc_49></location>N ( r ) = 1 -2 M r -2 n 2 r 2 +2 M ( n 2 -q 2 12 ) 1 r 3 + . . . , g ( r ) = r 2 +( n 2 -q 2 4 ) -Mq 2 3 r -q 6 (3 M 2 q + n ( nq -2 α )) 1 r 2 + . . . , (4.25) φ ( r ) = q r + Mq r 2 +(4 M 2 + n 2 + q 2 4 ) q 3 r 3 + . . . ,</formula>
<text><location><page_6><loc_12><loc_36><loc_88><loc_39></location>containing the parameters M and q fixed by numerics. These constants are identified with the mass and the scalar 'charge' of the solutions.</text>
<text><location><page_6><loc_12><loc_23><loc_88><loc_36></location>The ECS equations have been solved by using a solver which employs a Newton-Raphson method with an adaptive mesh selection procedure [26], the input parameters being { r H , n ; α } . Starting with the GR solutions and slowly increasing α , we have found numerical evidence that the NUT metric possesses nonperturbative generalizations in ECS theory. For all considered solutions, the metric functions N ( r ) , g ( r ) are qualitatively very similar to their α = 0 counterparts, while the scalar field smoothly interpolate 4 between the asymptotic expansions (4.24), (4.25). To reveal the effects of the CS term, we show in Figure 1 (left) the function r 2 N ' / 2 (whose asymptotic value corresponds to the mass M ) together with the function g ' / (2 r ) (whose values is one in GR). The corresponding scalar field φ and the Ricci scalar R are shown on the right hand panel of the figure. The solutions there have r H = 1, n = 0 . 1 and several values of α .</text>
<text><location><page_6><loc_12><loc_15><loc_88><loc_23></location>The determination of the domain of existence of the solutions would be a complicated task. In this work we will only report partial results in this direction, by analyzing the pattern of several classes of solutions only. Typical results of the numerical integration are shown 5 in Figure 2 as a function of α (left) and for a varying horizon size (right). Note that all displayed quantities are expressed in units set by the NUT charge n , being invariant under the transformation (2.9).</text>
<text><location><page_6><loc_52><loc_86><loc_54><loc_87></location>0.3</text>
<text><location><page_6><loc_50><loc_80><loc_52><loc_81></location>φ</text>
<text><location><page_6><loc_52><loc_79><loc_54><loc_80></location>0.15</text>
<text><location><page_6><loc_53><loc_73><loc_54><loc_73></location>0</text>
<text><location><page_6><loc_58><loc_73><loc_59><loc_74></location>α</text>
<text><location><page_6><loc_60><loc_76><loc_61><loc_77></location>=1</text>
<text><location><page_6><loc_60><loc_76><loc_60><loc_77></location>α</text>
<text><location><page_6><loc_59><loc_73><loc_60><loc_74></location>=0</text>
<text><location><page_6><loc_63><loc_78><loc_65><loc_79></location>=1.5</text>
<text><location><page_6><loc_62><loc_78><loc_63><loc_79></location>α</text>
<text><location><page_6><loc_65><loc_83><loc_66><loc_84></location>R</text>
<text><location><page_6><loc_67><loc_87><loc_69><loc_88></location>0.04</text>
<text><location><page_6><loc_67><loc_84><loc_69><loc_84></location>0.02</text>
<text><location><page_6><loc_68><loc_80><loc_69><loc_81></location>0</text>
<text><location><page_6><loc_70><loc_79><loc_71><loc_80></location>0</text>
<text><location><page_6><loc_76><loc_79><loc_78><loc_80></location>0.1</text>
<text><location><page_6><loc_83><loc_79><loc_85><loc_80></location>0.2</text>
<text><location><page_6><loc_75><loc_78><loc_76><loc_79></location>log</text>
<text><location><page_6><loc_77><loc_78><loc_78><loc_79></location>r</text>
<text><location><page_6><loc_76><loc_78><loc_77><loc_79></location>10</text>
<text><location><page_6><loc_83><loc_75><loc_84><loc_76></location>=1</text>
<text><location><page_6><loc_83><loc_75><loc_83><loc_75></location>H</text>
<text><location><page_6><loc_75><loc_82><loc_76><loc_82></location>=1</text>
<text><location><page_6><loc_74><loc_82><loc_75><loc_83></location>α</text>
<text><location><page_6><loc_78><loc_85><loc_80><loc_86></location>=1.5</text>
<text><location><page_6><loc_77><loc_85><loc_78><loc_86></location>α</text>
<text><location><page_6><loc_79><loc_75><loc_83><loc_76></location>n=0.1 r</text>
<figure>
<location><page_7><loc_14><loc_70><loc_49><loc_89></location>
</figure>
<figure>
<location><page_7><loc_52><loc_70><loc_87><loc_89></location>
<caption>Figure 2: Left: Some parameters of the ECS solutions are shown as a function of α (left) and of the horizon area (right).</caption>
</figure>
<text><location><page_7><loc_12><loc_52><loc_88><loc_61></location>As stated above, the ECS solutions smoothly emerge from the α = 0 GR ones. At the same time, the numerical results suggest that, for given ( r H , n ), the value of the parameter α cannot be arbitrary large. It turns out that, when the Chern-Simons parameter becomes too large, the scalar field becomes very peaked at the horizon, with large values of the Ricci scalar there, and the overall numerical accuracy strongly decreases. Also, in agreement with the perturbation theory results, the mass M decreases with α , while the scalar 'charge' q is strictly positive, increasing with α .</text>
<text><location><page_7><loc_12><loc_46><loc_88><loc_52></location>When varying instead the horizon size for fixed { α ; n } (Figure 2 (right)), we notice the existence of a minimal value of A H , a feature shared with the GR solution. For a given n , this minimal value decreases as α increases. Also, the scalar field vanishes gradually for large size of the horizon and becomes peaked at the horizon as the minimal A H is approached.</text>
<section_header_level_1><location><page_7><loc_12><loc_42><loc_67><loc_44></location>5 Further remarks. The issue of Taub solution</section_header_level_1>
<text><location><page_7><loc_12><loc_33><loc_88><loc_41></location>The main purpose of this work was to investigate the basic properties of the Lorentzian NUT solution in Einstein-Chern-Simons (ECS) theory, viewed as a toy model for a rotating configuration. Even if the primary interest is in the ECS generalization of the Kerr metric (which would possess usual asymptotics and no causal pathologies), we hope that, by widening the context to solutions with NUT charge, one may achieve a deeper appreciation of the model.</text>
<text><location><page_7><loc_12><loc_26><loc_88><loc_33></location>The problem has been approached from two different directions: using an expansion in powers of α (the CS coupling constant) around the GR solution, and solving the problem numerically. As expected, our results indicate that the basic properties (in particular the pathologies) of the NUT solution persist for ECS configurations, without spectacular new features. One interesting aspect which deserves further investigation is the possible existence of a maximal value of α , as suggested by the numerical results.</text>
<text><location><page_7><loc_12><loc_19><loc_88><loc_25></location>This work can be continued in various directions. For example, once the geometry is known, one can study the effects of the CS term on the geodesic motion. In the GR limit, α = 0, this problem has been extensively discussed in the literature, see e.g. [15], [27]-[32]. Restricting to null circular orbits, one can shown that, for σ ( r ) = 1, the radius r = r 0 > r H of the photon sphere is a solution of the equation</text>
<formula><location><page_7><loc_41><loc_15><loc_88><loc_18></location>( N ' g -Ng ' ) | r = r 0 = 0 , (5.26)</formula>
<text><location><page_7><loc_67><loc_13><loc_67><loc_15></location>/negationslash</text>
<text><location><page_7><loc_12><loc_12><loc_88><loc_15></location>which in the GR case, reduces to r 3 0 -3 Mr 2 0 -3 n 2 r 0 + Mn 2 = 0. For α = 0, the solution of (5.26) is found numerically. Our results indicate that for a given n , the ratio r c /M increases with α (although for all</text>
<text><location><page_8><loc_67><loc_71><loc_68><loc_72></location>1</text>
<figure>
<location><page_8><loc_32><loc_70><loc_67><loc_89></location>
<caption>Figure 3: The Ricci scalar R and the derivative of the scalar field φ ' are shown as a function of r , inside and outside the horizon, for two values of α and r H = 1, n = 0 . 1.</caption>
</figure>
<text><location><page_8><loc_12><loc_58><loc_88><loc_61></location>solutions we have considered from this direction, the differences w.r.t. the GR case are at the level of a few percents). It would be interesting to extend this study and to compute e.g. the shadow of the ECS solutions.</text>
<text><location><page_8><loc_12><loc_49><loc_88><loc_57></location>Returning to the GR solution (2.10), one remarks that the NUT metric is interesting from yet another point of view. By continuing it through its horizon at r = r H one arrives in the Taub universe, which may be interpreted as a homogeneous, non-isotropic cosmology with an S 3 spatial topology. (In fact, as discussed by Misner in [13], the NUT spacetime can be joined analytically to the Taub spacetime as a single Taub-NUT spacetime.) Whereas the Schwarzschild solution has a curvature singularity at r = 0, this is not the case for n = 0 and the radius coordinate in Taub-NUT (TN) solution may range on the whole real axis.</text>
<text><location><page_8><loc_12><loc_40><loc_88><loc_48></location>Since the regularity of the TN solution over the whole space-time is somehow exceptional, it is natural to address the question of the behaviour of the ECS solutions inside the horizon. Starting again with a perturbative approach, we remark that the solution derived in Section 3 holds also for r < r H . Then one can show that the corrections N 2 ( r ) and σ 2 ( r ) to the TN solution diverge 6 as 1 /r 2 as r → 0. As expected, this divergence manifests itself also in the curvature invariants, leading to a divergent character of the solutions, at least to lowest order in perturbation theory.</text>
<text><location><page_8><loc_13><loc_48><loc_13><loc_50></location>/negationslash</text>
<text><location><page_8><loc_12><loc_14><loc_88><loc_39></location>A similar conclusion is reached when considering a non-perturbative construction of solutions inside the horizon. This is a feasible problem, since we have obtained already the solutions at r = r H . This set is used as initial data to integrate inwards, on an interval [ r I , r H ], by decreasing progressively r I . The results of the numerical (non-perturbative) integration can be summarized as follows. For all values of the parameters which we have considered, the integration inside can be performed only for r ∈ ] r c , r H ] with 0 < r c < r H . The minimal value r c depends on the choice of the parameters { r H , n ; α } . In particular, the Ricci scalar increases considerably in the limit r → r c , as shown by Figs. 3 (note that a similar picture holds for the Kretschmann invariant K ). These results strongly suggest that all ECS solutions present an essential singularity at r = r c . Unfortunately, we failed to find an analytical argument explaining this feature. However, inspecting the different functions entering in the equations, it turns out that, for the chosen metric gauge, | φ ' ( r ) | strongly increases as r → r c (see Fig. 3). This induces strong variations of the functions g ' , g '' and likely leads to the divergence of R and K . Finally, let us stress that -in agreement with the perturbative analysis- the critical radius r c decreases towards zero when α decreases. At the same time, its value increases with α . Moreover, the existing results suggest that this critical value reaches the horizon radius, r c → r H , as the maximal value of α (noticed in the previous Section) is approached, which would imply a singular horizon in that limit. However, a clarification of these aspects seems to require another parametrization of the problem and possibly a different numerical approach.</text>
<text><location><page_9><loc_12><loc_85><loc_88><loc_90></location>One should mention that we have also constructed ECS solutions with a massive scalar field, V ( φ ) = µ 2 φ 2 / 2. However, all qualitative features of the massless solutions are recovered in that case. In particular, the solution inside the horizon still appears to possess a singularity for a critical value of r .</text>
<text><location><page_9><loc_12><loc_82><loc_88><loc_85></location>Finally, we remark that it would be interesting to find how a (dynamical) CS term affects the properties of the Euclideanized Taub-NUT solution.</text>
<section_header_level_1><location><page_9><loc_14><loc_79><loc_29><loc_80></location>Acknowledgement</section_header_level_1>
<text><location><page_9><loc_12><loc_75><loc_88><loc_79></location>E. R. acknowledges funding from the FCT-IF programme. This work was also partially supported by the H2020-MSCA-RISE-2015 Grant No. StronGrHEP-690904, and by the CIDMA project UID/MAT/04106/2013.</text>
<section_header_level_1><location><page_9><loc_12><loc_70><loc_24><loc_72></location>References</section_header_level_1>
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</document> | [{"title": "Remarks on the Taub-NUT solution in Chern-Simons modified gravity", "content": "Yves Brihaye \u2020 and Eugen Radu \u2021 \u2020 Physique-Math'ematique, Universite de Mons-Hainaut, Mons, Belgium \u2021 Departamento de F'\u0131sica da Universidade de Aveiro and CIDMA, Campus de Santiago, 3810-183 Aveiro, Portugal July 21, 2021", "pages": [1]}, {"title": "Abstract", "content": "We discuss the generalization of the NUT spacetime in General Relativity (GR) within the framework of the (dynamical) Einstein-Chern-Simons (ECS) theory with a massless scalar field. These configurations approach asymptotically the NUT spacetime and are characterized by the 'electric' and 'magnetic' mass parameters and a scalar 'charge'. The solutions are found both analytically and numerically. The analytical approach is perturbative around the Einstein gravity background. Our results indicate that the ECS configurations share all basic properties of the NUT spacetime in GR. However, when considering the solutions inside the event horizon, we find that in contrast to the GR case, the spacetime curvature grows (apparently) without bound.", "pages": [1]}, {"title": "1 Introduction", "content": "The Einstein-Chern-Simons (ECS) theory [1] is one of the most interesting generalizations of the General Relativity (GR) [2]. In its dynamical version, this model possesses a (real) scalar field \u03c6 , with an axionic-type coupling with the Pontryagin density [3]. As such, its action contains extra-terms quadratic in the curvature which can potentially lead to new effects in the strong-field regime. Moreover, this model is motivated by string theory results [4] and occurs also in the framework of loop quantum gravity [5], [6]. In contrast to its Einstein-Gauss-Bonnet counterpart (in which case \u03c6 couples to the Gauss-Bonnet scalar), it can be shown that any static spherically symmetric solution of GR is also a solution of ECS gravity. Therefore this model is almost unique, as it leads to different results only in the presence of a parity-odd source such as rotation. However, despite the presence in the literature of some partial results [7], [8], [9], the generalizations of the (astrophysically relevant) Kerr solution in ECS theory is still unknown, presumably due to the complexity of the problem. Therefore the study of ECS generalizations of known GR rotating solutions is a pertinent task which, ultimately, could lead to some progress in the Kerr problem. One of the most intriguing solutions of GR has been found in 1963 by Newman, Tamburino and Unti (NUT) [10]. This is a generalization of the Schwarzschild solution which solves the Einstein vacuum field equations, possessing in addition to the mass parameter M an extra-parameter-the NUT charge n . In its usual interpretation, it describes a gravitational dyon with both ordinary and magnetic mass. The NUT charge n plays a dual role to ordinary ADM mass M , in the same way that electric and magnetic charges are dual within Maxwell theory [11]. This solution has a number of unusual properties, becoming renowned for being 'a counter-example to almost anything' [12]. For example, the NUT spacetime is not asymptotically flat in the usual sense although it does obey the required fall-off conditions, and, moreover, contains closed timelike curves. As such, it is cannot be taken as a realistic model for a macroscopic object, although its Euclideanized version might play a role in the context of quantum gravity [14]. For the purposes of this work, the NUT metric is interesting from another point of view: its line-element can be taken as Kerr-like, in the sense that it has a crossed metric component g \u03d5t , see (2.7) bellow. This term does not produce an ergoregion but it leads to an effect similar to the dragging of inertial frames [15]. Moreover, one can say that a NUT spacetime consists of two counter-rotating regions, with a vanishing total angular momentum [16], [17]. Therefore, the study of its generalization in the framework of ECS theory is a legitimate task. Also, one should mention that the NUT solution has been generalized already in various models. For example, nutty solutions with gauge fields have been has been found in [18], [19], [20]. The low-energy string theory possess also nontrivial solutions with NUT charge (see e.g. [21]). The paper is structured as follows: in the next Section we review the basic framework of the model which includes the metric and scalar field Ansatz. Some properties of general nutty solutions are also discussed there. In Section 3 we present the results of a perturbative construction of solutions as a power series in the CS coupling constant. The basic properties of the non-perturbative configurations are discussed in Section 4. We conclude with Section 5 where the results are compiled. There we present also our results for the Taub region of the solutions and give arguments that the solution is divergent there.", "pages": [1, 2]}, {"title": "2.1 The Chern-Simons modified gravity", "content": "The action of the dynamical CS modified gravity is provided by where g is the determinant of the metric g \u00b5\u03bd , R is the Ricci scalar and we note \u03ba -1 = 16 \u03c0G . The quantity \u2217 RR is the Pontryagin density, defined via (where /epsilon1 cdef is the 4-dimensional Levi-Civita tensor). The gravity equations for this model read where and T ( \u03c6 ) ab is the energy-momentum tensor of the scalar field, The scalar field solves the Klein-Gordon equation in the presence of a source term given by the Pontryagin density, To simplify the picture, in this work we shall report results for a massless, non-selfinteracting scalar only, V ( \u03c6 ) = 0.", "pages": [2]}, {"title": "2.2 The Ansatz", "content": "We consider a NUT-charged spacetime whose metric can be written locally in the form while the scalar field depends on the r -coordinate only, \u03c6 = \u03c6 ( r ). Here \u03b8 and \u03d5 are the standard angles parametrizing an S 2 with the usual range. As usual, we define the NUT parameter 1 n (with n \u2265 0, without any loss of generality), in terms of the coefficient appearing in the differential dt +4 n sin 2 \u03b8 2 d\u03d5 . The form of N ( r ) , \u03c3 ( r ) and g ( r ) emerges as result of demanding the metric to be a solution of the ECS equations (2.3) (note the existence of a metric gauge freedom in (2.7), which is fixed later by convenience). The equations satisfied by these functions (and the corresponding one for \u03c6 ( r )) are rather complicated and we shall not not include them here. However, we notice that they can also be derived from the effective action where (where a prime denotes a derivative w.r.t. the radial coordinate r ). Remarkably, one can see that, due to the factorization of the angular dependence for the metric Ansatz (2.7), all functions solve second order equations of motion 2 . The reduced action (2.8) makes transparent the scaling symmetries of the problem. For example, to simplify the analysis, it is convenient to work with conventions where \u03ba = 1 (this is obtained by rescaling the scalar field and the coupling constant \u03b1 ). Then the system still has a residual scaling symmetry which can be used to fix the value of \u03b1 or n . Finally, we note that the NUT solution is found for \u03b1 = 0, \u03c6 = 0, being usually written for a gauge choice with possessing a nonvanishing Pontryagin density (and thus it cannot be promoted to a solution of the ECS model). This metric has an (outer) horizon located at 3 Here, similar to the Schwarzschild limit, N ( r H ) = 0 is only a coordinate singularity where all curvature invariants are finite. In fact, a nonsingular extension across this null surface can be found just as at the event horizon of a black hole.", "pages": [3]}, {"title": "2.3 General properties", "content": "/negationslash Some basic properties of the line element (2.7) are generic, independent on the specific details of the considered gravity model. As a result, the general nutty configurations always share the same troubles exhibited by the original NUT solution in GR. For example, the Killing symmetries of (2.7) are time translation and SO (3) rotations. However, spherical symmetry in a conventional sense is lost, since the rotations act on the time coordinate as well. Moreover, for n = 0, the metric (2.7) has a singular symmetry axis. However, following the discussion in [12] for the GR limit, these singularities can be removed by appropriate identifications and changes in the topology of the spacetime manifold, which imply a periodic time coordinate. Then such a configuration cannot be interpreted properly as black hole. In fact, the pathology of closed timelike curves is not special to the NUT solution in GR but afflicts all solutions with a 'dual' magnetic mass in general [22]. As discussed in [23], this condition emerges only from the asymptotic form of the fields. Therefore, it is not sensitive to the precise details of the nature of the source, or the precise nature of the theory of gravity at short distances. In our approach we are interested in solutions whose far field asymptotics are similar, to leading order, to those of the Einstein gravity solution (2.10), with N ( r ) \u2192 1, g ( r ) \u2192 r 2 , \u03c3 ( r ) \u2192 1 and \u03c6 ( r ) \u2192 0 as r \u2192\u221e . The solution will posses also an horizon at r = r H > 0, where N ( r H ) = 0, and g ( r ), \u03c3 ( r ) strictly positive. In the absence of a global Cauchy surface, the thermodynamical description of (Lorentzian signature) nutty solutions is still poorly understood. However, one can still define a temperature of solutions via the surface gravity associated with the Killing vector \u2202/\u2202t , and also an even horizon area [24] The mass of the solutions can be computed by employing the quasilocal formalism in conjuction with the boundary counterterm method [25]. A direct computation shows that, similar to the Einstein gravity case, the mass of the solutions is identified with the constant M in the far field expansion of the metric function g tt ,", "pages": [4]}, {"title": "3 A perturbative approach", "content": "An exact solution of the equations (2.3), (2.6) can be found in the limit of small \u03b1 , by treating the ECS configurations as perturbations around the Einstein gravity background. Here we have found convenient to work in a gauge with Then we consider a perturbative Ansatz with where N 0 = 1 -2( M 0 r + n 2 ) / ( r 2 + n 2 ) corresponds to the solution in Einstein gravity. To this order, one arrives at the following system of linear ordinary differential equations When solving them, there are four integration constants. These constants are chosen such that the corrected NUT metric is still smooth at r = r H and approaches a background with N ( r ) \u2192 1 and \u03c3 ( r ) \u2192 1 asymptotically, while \u03c6 ( r ) \u2192 0. Then, to lowest order, the solution has the generic structure with F = { N 2 , \u03c3 2 , \u03c6 1 } . The functions P 0 , P 1 and P 2 are ratio of polynomials, possessing a simple form for \u03c6 1 only, with the corresponding expressions for N 2 , \u03c3 2 being too complicated to display here. To this order in perturbation theory, one finds to following far field expression of the scalar field while the mass parameter has the following expression where M 0 = ( r 2 H -n 2 ) / (2 r H ), and The same type of expression is found for the temperature, with An inspection of the (3.22) shows that M 2 is a strictly negative quantity. However, the CS correction to T H has no definite sign. For a given n , it is negative for small r H and becomes strictly positive for large enough r H (in particular for r H > n ). This approach can be extended to higher order in \u03b1 . Unfortunately, the resulting equations are too complicated for an analytical treatment. Although they can be solved numerically, we have preferred to consider instead a fully nonperturbative approach.", "pages": [4, 5]}, {"title": "4 Numerical results", "content": "The nonperturbative solutions are constructed by solving numerically the ECS eqs. (2.3), (2.6), as a boundary value problem. In this approach, it is convenient to employ the same metric gauge as in Einstein gravity, and take \u03c3 ( r ) = 1. Then we consider solutions in the domain r H \u2264 r < \u221e (with r H > 0), smoothly interpolating between the following boundary values: N ( r H ) = 0, g ( r H ) = g 0 > 0, \u03c6 ( r H ) = \u03c6 0 and N = 1, g = r 2 , \u03c6 = 0 as r \u2192 \u221e . An approximate expression of the solutions compatible with these asymptotics can easily be log event horizon 0 0.25 0.5 0.75 1 r 10 found. Its first terms as r \u2192 r H are { N 1 , g 0 , \u03c6 0 } being three undetermined parameters, while the leading order expansion in the far field is containing the parameters M and q fixed by numerics. These constants are identified with the mass and the scalar 'charge' of the solutions. The ECS equations have been solved by using a solver which employs a Newton-Raphson method with an adaptive mesh selection procedure [26], the input parameters being { r H , n ; \u03b1 } . Starting with the GR solutions and slowly increasing \u03b1 , we have found numerical evidence that the NUT metric possesses nonperturbative generalizations in ECS theory. For all considered solutions, the metric functions N ( r ) , g ( r ) are qualitatively very similar to their \u03b1 = 0 counterparts, while the scalar field smoothly interpolate 4 between the asymptotic expansions (4.24), (4.25). To reveal the effects of the CS term, we show in Figure 1 (left) the function r 2 N ' / 2 (whose asymptotic value corresponds to the mass M ) together with the function g ' / (2 r ) (whose values is one in GR). The corresponding scalar field \u03c6 and the Ricci scalar R are shown on the right hand panel of the figure. The solutions there have r H = 1, n = 0 . 1 and several values of \u03b1 . The determination of the domain of existence of the solutions would be a complicated task. In this work we will only report partial results in this direction, by analyzing the pattern of several classes of solutions only. Typical results of the numerical integration are shown 5 in Figure 2 as a function of \u03b1 (left) and for a varying horizon size (right). Note that all displayed quantities are expressed in units set by the NUT charge n , being invariant under the transformation (2.9). 0.3 \u03c6 0.15 0 \u03b1 =1 \u03b1 =0 =1.5 \u03b1 R 0.04 0.02 0 0 0.1 0.2 log r 10 =1 H =1 \u03b1 =1.5 \u03b1 n=0.1 r As stated above, the ECS solutions smoothly emerge from the \u03b1 = 0 GR ones. At the same time, the numerical results suggest that, for given ( r H , n ), the value of the parameter \u03b1 cannot be arbitrary large. It turns out that, when the Chern-Simons parameter becomes too large, the scalar field becomes very peaked at the horizon, with large values of the Ricci scalar there, and the overall numerical accuracy strongly decreases. Also, in agreement with the perturbation theory results, the mass M decreases with \u03b1 , while the scalar 'charge' q is strictly positive, increasing with \u03b1 . When varying instead the horizon size for fixed { \u03b1 ; n } (Figure 2 (right)), we notice the existence of a minimal value of A H , a feature shared with the GR solution. For a given n , this minimal value decreases as \u03b1 increases. Also, the scalar field vanishes gradually for large size of the horizon and becomes peaked at the horizon as the minimal A H is approached.", "pages": [5, 6, 7]}, {"title": "5 Further remarks. The issue of Taub solution", "content": "The main purpose of this work was to investigate the basic properties of the Lorentzian NUT solution in Einstein-Chern-Simons (ECS) theory, viewed as a toy model for a rotating configuration. Even if the primary interest is in the ECS generalization of the Kerr metric (which would possess usual asymptotics and no causal pathologies), we hope that, by widening the context to solutions with NUT charge, one may achieve a deeper appreciation of the model. The problem has been approached from two different directions: using an expansion in powers of \u03b1 (the CS coupling constant) around the GR solution, and solving the problem numerically. As expected, our results indicate that the basic properties (in particular the pathologies) of the NUT solution persist for ECS configurations, without spectacular new features. One interesting aspect which deserves further investigation is the possible existence of a maximal value of \u03b1 , as suggested by the numerical results. This work can be continued in various directions. For example, once the geometry is known, one can study the effects of the CS term on the geodesic motion. In the GR limit, \u03b1 = 0, this problem has been extensively discussed in the literature, see e.g. [15], [27]-[32]. Restricting to null circular orbits, one can shown that, for \u03c3 ( r ) = 1, the radius r = r 0 > r H of the photon sphere is a solution of the equation /negationslash which in the GR case, reduces to r 3 0 -3 Mr 2 0 -3 n 2 r 0 + Mn 2 = 0. For \u03b1 = 0, the solution of (5.26) is found numerically. Our results indicate that for a given n , the ratio r c /M increases with \u03b1 (although for all 1 solutions we have considered from this direction, the differences w.r.t. the GR case are at the level of a few percents). It would be interesting to extend this study and to compute e.g. the shadow of the ECS solutions. Returning to the GR solution (2.10), one remarks that the NUT metric is interesting from yet another point of view. By continuing it through its horizon at r = r H one arrives in the Taub universe, which may be interpreted as a homogeneous, non-isotropic cosmology with an S 3 spatial topology. (In fact, as discussed by Misner in [13], the NUT spacetime can be joined analytically to the Taub spacetime as a single Taub-NUT spacetime.) Whereas the Schwarzschild solution has a curvature singularity at r = 0, this is not the case for n = 0 and the radius coordinate in Taub-NUT (TN) solution may range on the whole real axis. Since the regularity of the TN solution over the whole space-time is somehow exceptional, it is natural to address the question of the behaviour of the ECS solutions inside the horizon. Starting again with a perturbative approach, we remark that the solution derived in Section 3 holds also for r < r H . Then one can show that the corrections N 2 ( r ) and \u03c3 2 ( r ) to the TN solution diverge 6 as 1 /r 2 as r \u2192 0. As expected, this divergence manifests itself also in the curvature invariants, leading to a divergent character of the solutions, at least to lowest order in perturbation theory. /negationslash A similar conclusion is reached when considering a non-perturbative construction of solutions inside the horizon. This is a feasible problem, since we have obtained already the solutions at r = r H . This set is used as initial data to integrate inwards, on an interval [ r I , r H ], by decreasing progressively r I . The results of the numerical (non-perturbative) integration can be summarized as follows. For all values of the parameters which we have considered, the integration inside can be performed only for r \u2208 ] r c , r H ] with 0 < r c < r H . The minimal value r c depends on the choice of the parameters { r H , n ; \u03b1 } . In particular, the Ricci scalar increases considerably in the limit r \u2192 r c , as shown by Figs. 3 (note that a similar picture holds for the Kretschmann invariant K ). These results strongly suggest that all ECS solutions present an essential singularity at r = r c . Unfortunately, we failed to find an analytical argument explaining this feature. However, inspecting the different functions entering in the equations, it turns out that, for the chosen metric gauge, | \u03c6 ' ( r ) | strongly increases as r \u2192 r c (see Fig. 3). This induces strong variations of the functions g ' , g '' and likely leads to the divergence of R and K . Finally, let us stress that -in agreement with the perturbative analysis- the critical radius r c decreases towards zero when \u03b1 decreases. At the same time, its value increases with \u03b1 . Moreover, the existing results suggest that this critical value reaches the horizon radius, r c \u2192 r H , as the maximal value of \u03b1 (noticed in the previous Section) is approached, which would imply a singular horizon in that limit. However, a clarification of these aspects seems to require another parametrization of the problem and possibly a different numerical approach. One should mention that we have also constructed ECS solutions with a massive scalar field, V ( \u03c6 ) = \u00b5 2 \u03c6 2 / 2. However, all qualitative features of the massless solutions are recovered in that case. In particular, the solution inside the horizon still appears to possess a singularity for a critical value of r . Finally, we remark that it would be interesting to find how a (dynamical) CS term affects the properties of the Euclideanized Taub-NUT solution.", "pages": [7, 8, 9]}, {"title": "Acknowledgement", "content": "E. R. acknowledges funding from the FCT-IF programme. This work was also partially supported by the H2020-MSCA-RISE-2015 Grant No. StronGrHEP-690904, and by the CIDMA project UID/MAT/04106/2013.", "pages": [9]}] |
2024arXiv240918854C | https://arxiv.org/pdf/2409.18854.pdf | <document>
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<location><page_1><loc_8><loc_92><loc_32><loc_98></location>
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<section_header_level_1><location><page_1><loc_8><loc_85><loc_80><loc_90></location>New Insights into Supradense Matter from Dissecting Scaled Stellar Structure Equations</section_header_level_1>
<text><location><page_1><loc_8><loc_82><loc_33><loc_84></location>Bao-Jun Cai 1, ∗ , Bao-An Li 2,</text>
<text><location><page_1><loc_33><loc_83><loc_34><loc_84></location>∗</text>
<text><location><page_1><loc_8><loc_75><loc_79><loc_82></location>1 Quantum Machine Learning Laboratory, Shadow Creator Inc., Shanghai 201208, People's Republic of China 2 Department of Physics and Astronomy, Texas A & M University-Commerce, Commerce, TX 75429-3011, USA</text>
<text><location><page_1><loc_8><loc_73><loc_24><loc_75></location>Correspondence*:</text>
<text><location><page_1><loc_8><loc_72><loc_16><loc_73></location>Bao-An Li</text>
<text><location><page_1><loc_8><loc_70><loc_25><loc_72></location>[email protected]</text>
<section_header_level_1><location><page_1><loc_8><loc_65><loc_16><loc_67></location>Abstract</section_header_level_1>
<text><location><page_1><loc_8><loc_44><loc_92><loc_65></location>The strong-field gravity in General Relativity (GR) realized in neutron stars (NSs) renders the Equation of State (EOS) P ( ε ) of supradense neutron star (NS) matter to be essentially nonlinear and refines the upper bound for φ ≡ P / ε to be much smaller than the Special Relativity (SR) requirement with linear EOSs, where P and ε are respectively the pressure and energy density of the system considered. Specifically, a tight bound φ ≲ 0.374 is obtained by anatomizing perturbatively the intrinsic structures of the scaled TolmanOppenheimer-Volkoff (TOV) equations without using any input nuclear EOS. New insights gained from this novel analysis provide EOS-model independent constraints on properties (e.g., density profiles of the sound speed squared s 2 = d P /d ε and trace anomaly ∆ = 1/3 -φ ) of cold supradense matter in NS cores. Using the gravity-matter duality in theories describing NSs, we investigate the impact of gravity on supradense matter EOS in NSs. In particular, we show that the NS mass M NS , radius R and its compactness ξ ≡ M NS / R scale with certain combinations of its central pressure and energy density (encapsulating its central EOS). Thus, observational data on these properties of NSs can straightforwardly constrain NS central EOSs without relying on any specific nuclear EOS-model.</text>
<text><location><page_1><loc_8><loc_39><loc_92><loc_42></location>Keywords: Equation of State, Supradense Matter, Neutron Star, Tolman-Oppenheimer-Volkoff Equations, Principle of Causality, Special Relativity, Speed of Sound, Generality Relativity, Neutron-rich Matter, Gravity-matter Duality</text>
<section_header_level_1><location><page_1><loc_8><loc_35><loc_24><loc_37></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_8><loc_24><loc_92><loc_34></location>The speed of sound squared (SSS) s 2 = d P /d ε (Landau and Lifshitz, 1987) quantifies the stiffness of the Equation of State (EOS) expressed in terms of the relationship P ( ε ) between the pressure P and energy density ε of the system considered. The Principle of Causality of Special Relativity (SR) requires the speed of sound of any signal to stay smaller than the speed of light c ≡ 1, i.e., s ≤ 1. For a linear EOS of the form P = w ε with w being some constant, the condition s 2 ≤ 1 is globally equivalent to φ = P / ε ≤ 1. For such EOSs, the causality condition can be equivalently written as:</text>
<formula><location><page_1><loc_19><loc_20><loc_92><loc_22></location>Principle of Causality of SR with linear EOS implies P ≤ ε ↔ φ ≡ P / ε ≤ 1. (1)</formula>
<text><location><page_1><loc_8><loc_10><loc_92><loc_18></location>The indicated equivalence between s 2 ≤ 1 and φ ≤ 1 could be demonstrated straightforwardly as follows: If P could be greater than ε somewhere then the curve of P ( ε ) may unavoidably across the line P = ε from below to above, indicating the slope at the crossing point is necessarily larger than 1, as illustrated in FIG. 1. In the following, we use the above causality requirement on φ with linear EOSs as a reference in discussing properties of supradense matter in strong-field gravity.</text>
<figure>
<location><page_2><loc_34><loc_70><loc_64><loc_90></location>
<caption>FIG. 1. (Color Online). An illustration of the equivalence between s 2 ≤ 1 with a linear EOS and φ ≤ 1: If P could be greater than ε somewhere then the curve of P ( ε ) have to across the line P = ε from below to above, indicating that s 2 = d P /d ε > 1 at the crossing point.</caption>
</figure>
<text><location><page_2><loc_8><loc_24><loc_92><loc_61></location>The EOS of nuclear matter may be strongly nonlinear depending on both the internal interactions and the external environment/constraint of the system; this means that φ ≤ 1 is necessary but not sufficient to ensure supradense matter in all NSs always stay casual. For example, the EOS of noninteracting degenerate Fermions (e.g., electrons) can be written in the polytropic form P = K ε β (Shapiro and Teukolsky, 1983) where β = 5/3 for non-relativistic and β = 4/3 for extremely relativistic electrons; consequently φ ≤ β -1 < 1. Similarly, a long time ago Zel'dovich considered the EOS of an isolated ultra-dense system of baryons interacting through a vector field (Zel'dovich, 1961). In this case, P = ε ∼ ρ 2 , here ρ is the baryon number density. Consequently, P / ε ≤ 1 is obtained. The EOS of dense nuclear matter where nucleons interact through both the σ -meson and ω -meson in the Walecka model (Walecka, 1974) is an example of this type. In particular, the ω -field scales at asymptotically large density as ω ∼ ρ while the σ -field scales σ ∼ ρ s with the scalar density ρ s approaching some constant for ρ →∞ (Cai and Li, 2016); therefore the vector field dominates at these densities. More generally, however, going beyond the vector field, the baryon density dependence of either P ( ρ ) or ε ( ρ ) could be very complicated and nontrivial. The resulting EOS P ( ε ) could also be significantly nonlinear. The EOS of supradense matter under intense gravity of NSs could be forced to be nonlinear as the equilibrium state of NSs is determined by extremizing the total action of the matter-gravity system through the Hamilton's variational principle. It is well known that the strong-field gravity in General Relativity (GR) is fundamentally nonlinear, the EOS of NS matter especially in its core is thus also expected to be nonlinear. Therefore, the causality condition s 2 ≤ 1 may be appreciably different from φ ≤ 1, and it may also effectively render the upper bound for φ to be smaller than 1. Determining accurately an upper bound of φ (equivalently a lower bound of the dimensionless trace anomaly ∆ = 1/3 -φ ) will thus help constrain properties of supradense matter in strong-field gravity.</text>
<text><location><page_2><loc_8><loc_10><loc_92><loc_23></location>The upper bound for φ is a fundamental quantity encapsulating essentially the strong-field properties of gravity in GR. Its accurate determination may help improve our understanding about the nature of gravity. The latter is presently least known among the four fundamental forces despite being the first one discovered in Nature (Hoyle, 2003). An upper bound on φ substantially different from 1 then vividly characterizes how GR affects the supradense matter existing in NSs. In some physical senses, this is similar to the effort in determining the Bertsch parameter. The latter was introduced as the ratio E UFG / E FFG of the EOS of a unitary Fermi gas (UFG) over that of the free Fermi gas (FFG) E FFG (Giorgini et al., 2008); here E FFG and E UFG are the energies per</text>
<text><location><page_3><loc_8><loc_85><loc_92><loc_91></location>particle in the two systems considered. It characterizes the strong interactions among Fermions under the unitary condition. Extensive theoretical and experimental efforts have been made to constrain/fix the Bertsch parameter. Indeed, its accurate determination has already made strong impact on understanding strongly-interacting Fermions (Giorgini et al., 2008; Bloch et al., 2008).</text>
<text><location><page_3><loc_8><loc_51><loc_92><loc_83></location>There are fundamental physics issues regarding both strong-field gravity and supradense matter EOS as well as their couplings. What is gravity? Is a new theory of light and matter needed to explain what happens at very high energies and temperatures? These are among the eleven greatest unanswered physics questions for this century identified in 2003 by the National Research Council of the US National Academies (Committee on the Physics of the Universe, 2003). Compact stars provide far more extreme conditions necessary to test possible answers to these questions than terrestrial laboratories. A gravity-matter duality exists in theories describing NS properties, see, e.g., Refs. (Psaltis, 2008; Shao, 2019) for recent reviews. Neutron stars are natural testing grounds of our knowledge on these issues. Some of their observational properties may help break the gravity-matter duality, see, e.g., Refs. (DeDeo and Psaltis, 2003; Wen et al., 2009; Lin et al., 2014; He et al., 2015; Yang et al., 2020). Naturally, these issues are intertwined and one may gain new insights about the EOS of supradense matter from analyzing features of strong-field gravity or vice versa. The matter-gravity duality reflects the deep connection between microscopic physics of supradense matter and the powerful gravity effects of NSs. They both have to be fully understand to unravel mysteries associated with compact objects in the Universe. In this brief review, we summarize the main physics motivation, formalism and results of our recent efforts to gain new insights into the EOS of supradense matter in NS cores by dissecting perturbatively the intrinsic structures of the Tolman-Oppehnheimer-Volkoff (TOV) equations(Tolman, 1939; Oppenheimer and Volkoff, 1939) without using any input nuclear EOS. For more details, we refer the readers to our original publications in Refs. (Cai et al., 2023b,a; Cai and Li, 2024a,b).</text>
<text><location><page_3><loc_8><loc_34><loc_92><loc_50></location>The rest of this paper is organized as follows: First of all, to be complete and easy of our following presentation, in Section 2 we make a few remarks about some existing constraints on the EOS of supradense NS matter. Section 3 introduces the scaled TOV equations starting from which one can execute an effective perturbative expansion; the central SSS is obtained in Section 4, we then infer an upper bound for the ratio X ≡ φ c = P c / ε c of central pressure P c over central energy density ε c for NSs at the maximum-mass configuration along the M-R curve. The generalization for the upper bound of P / ε is also studied in Section 4. In Section 5, we compare our prediction on the lower bound of ∆ = 1/3 -P / ε with existing predictions in the literature. We summarize in Section 6 and give some perspectives for future studies along this line. In the Appendix, we discuss an effective correction to s 2 c obtained in Section 4.</text>
<section_header_level_1><location><page_3><loc_8><loc_29><loc_83><loc_31></location>2 Remarks on Some Existing Constraints on Supradense NS Matter</section_header_level_1>
<text><location><page_3><loc_8><loc_10><loc_92><loc_28></location>Understanding the EOS of supradense matter has long been an important issue in both nuclear physics and astrophysics (Walecka, 1974; Chin, 1977; Freedman and McLerran, 1977; Baluni, 1978; Wiringa et al., 1988; Akmal et al., 1998; Migdal, 1978; Morley and Kislinger, 1979; Shuryak, 1980; Bailin and Love, 1984; Lattimer and Prakash, 2001; Danielewicz et al., 2002; Steiner et al., 2005; Lattimer and Prakash, 2007; Alford et al., 2008; Li et al., 2008; Watts et al., 2016; Özel and Freire, 2016; Oertel et al., 2017; Vidaña, 2018). In fact, it has been an outstanding science driver at many research facilities in both fields. For example, finding the EOS of densest visible matter existing in our Universe is an ultimate goal of astrophysics in the era of highprecision multimessenger astronomy (Sathyaprakash et al., 2019). However, despite of much effort and progress made during the last few decades using various observational data and models especially since the discovery of GW170817 (Abbott et al., 2017a, 2018), GW190425 (Abbott et al.,</text>
<text><location><page_4><loc_8><loc_59><loc_92><loc_91></location>2020a), GW190814(Abbott et al., 2020b) and the recent NASA's NICER (Neutron Star Interior Composition Explorer) mass-radius measurements for PSR J0740+6620 (Fonseca et al., 2021; Riley et al., 2021; Miller et al., 2021; Salmi et al., 2022; Dittmann et al., 2024; Salmi et al., 2024), PSR J0030+0451(Riley et al., 2019; Miller et al., 2019; Vinciguerra et al., 2024) and PSR J04374715(Choudhury et al., 2024; Reardon et al., 2024), knowledge about NS core EOS remains ambiguous and quite elusive, see, e.g., Refs. (Bose et al., 2018; De et al., 2018; Fattoyev et al., 2018; Lim and Holt, 2018; Most et al., 2018; Radice et al., 2018; Tews et al., 2018; Zhang et al., 2018; Bauswein et al., 2019, 2020; Baym et al., 2019; McLerran and Reddy, 2019; Most et al., 2019; Annala et al., 2020, 2023; Sedrakian et al., 2020; Zhao and Lattimer, 2020; Weih et al., 2020; Xie and Li, 2019, 2020, 2021; Drischler et al., 2020, 2021a; Li et al., 2020; Bombaci et al., 2021; Al-Mamun et al., 2021; Nathanail et al., 2021; Raaijmakers et al., 2021; Altiparmak et al., 2022; Breschi et al., 2022; Komoltsev and Kurkela, 2022; Perego et al., 2022; Huang et al., 2022; Tan et al., 2022a,b; Brandes et al., 2023b,a; Gorda et al., 2023; Han et al., 2023; Jiang et al., 2023; Ofengeim et al., 2023; Mroczek et al., 2023; Raithel and Most, 2023; Somasundaram et al., 2023; Zhang and Li, 2020, 2021, 2023b,a; Pang et al., 2023; Fujimoto et al., 2024; Providência et al., 2024; Rutherford et al., 2024). For more discussions, see recent reviews, e.g., Refs. (Baym et al., 2018; Baiotti, 2019; Li et al., 2019; Orsaria et al., 2019; Blaschke et al., 2020; Capano et al., 2020; Chatziioannou, 2020; Burgio et al., 2021; Dexheimer et al., 2021; Drischler et al., 2021b; Lattimer, 2021; Li et al., 2021; Lovato et al., 2022; Sedrakian et al., 2023; Kumar et al., 2024; Sorensen et al., 2024; Tsang et al., 2024).</text>
<text><location><page_4><loc_8><loc_49><loc_92><loc_57></location>Extensive theoretical investigations about the EOS of supradense NS matter have been done and many interesting predictions have been made. For example, the realization of approximate conformal symmetry of quark matter at extremely high densities ρ ≳ 40 ρ 0 with ρ 0 ≡ ρ sat the nuclear saturation density implies the corresponding EOS approaches that of an ultra-relativistic Fermi gas (URFG) from below, namely(Bjorken, 1983; Kurkela et al., 2010)</text>
<formula><location><page_4><loc_26><loc_46><loc_92><loc_47></location>URFG: P ≲ ε /3 ↔ φ ≲ 1/3, at extremely high densities. (2)</formula>
<text><location><page_4><loc_8><loc_40><loc_92><loc_43></location>For the URFG, 3 P ≈ ε ∼ ρ 4/3 . Therefore φ = P / ε is at least upper bounded to be below 1/3 at these densities, equivalently a lower bound on the dimensionless trace anomaly emerges:</text>
<formula><location><page_4><loc_27><loc_36><loc_92><loc_38></location>∆ ≡ 1/3 -P / ε ≳ 0, at extremely high densities ρ ≳ 40 ρ 0 . (3)</formula>
<text><location><page_4><loc_8><loc_10><loc_92><loc_34></location>This prompts the question whether the bound φ ≤ 1/3 holds globally for dense matter or some other bound(s) on φ may exist. In this sense, massive NSs like PSR J1614-2230 (Demorest et al., 2010; Arzoumanian et al., 2018), PSR J0348+0432(Antoniadis et al., 2013), PSR J0740+6620 (Fonseca et al., 2021; Riley et al., 2021; Miller et al., 2021; Salmi et al., 2022; Dittmann et al., 2024; Salmi et al., 2024) and PSR J2215+5135(Sullivan and Romani, 2024) provide an ideal testing bed for exploring such quantity. A sizable φ ≳ O (0.1) arises for NSs but not for ordinary stars or lowdensity nuclear matter (Cai and Li, 2024a). For example, considering stars such as white dwarfs (WDs), one has P ≲ 10 22-23 dynes/cm 2 ≈ 10 -(11-10) MeV/fm 3 and ε ≲ 10 8-9 kg/m 3 ∼ 10 -6 MeV/fm 3 , thus φ ≲ 10 -(5-4) . The φ could be even smaller for main-sequence stars like the sun. Specifically, the pressure and energy density in the solar core are about 10 -16 MeV/fm 3 and 10 -10 MeV/fm 3 , respectively, and therefore φ ≈ 10 -6 . These stars are Newtonian in the sense that GR effects are almost absent. Similarly, for NS matter around nuclear saturation density ρ 0 = ρ sat ≈ 0.16fm -3 , the pressure is estimated to be P ( ρ 0 ) ≈ P 0 ( ρ 0 ) + P sym ( ρ 0 ) δ 2 ≈ 3 -1 L ρ 0 δ 2 ≲ 3MeV/fm 3 . Its isospindependent part is P sym ( ρ 0 ) = 3 -1 L ρ 0 with L ≈ 60MeV(Li et al., 2018, 2021) being the slope parameter of nuclear symmetry energy E sym ( ρ ) at ρ 0 , δ is the isospin asymmetry of the system</text>
<text><location><page_5><loc_8><loc_85><loc_92><loc_91></location>( δ 2 ≲ 1), and P 0 ( ρ 0 ) = 0 is the pressure of symmetric nuclear matter (SNM) at ρ 0 . The energy density at ρ 0 is similarly estimated as ε ( ρ 0 ) ≈ [ E 0 ( ρ 0 ) + E sym ( ρ 0 ) δ 2 + M N ] ρ 0 ≈ 150MeV/fm 3 with M N ≈ 939MeV the nucleon static mass, E 0 ( ρ 0 ) ≈-16MeV the binding energy at ρ 0 for SNM and E sym ( ρ 0 ) ≈ 32MeV(Li, 2017), leading to φ ≲ 0.02.</text>
<text><location><page_5><loc_8><loc_80><loc_92><loc_83></location>Based on the dimensional analysis and the definition of sound speed, we may write out the SSS generally as (we use the units in which c = 1)</text>
<formula><location><page_5><loc_41><loc_77><loc_92><loc_78></location>s 2 = φ f ( φ ), φ = P / ε , (4)</formula>
<text><location><page_5><loc_8><loc_51><loc_92><loc_75></location>where f ( φ ) is dimensionless. For low-density matter, such as those in ordinary stars and WDs or the nuclear matter around saturation density ρ 0 , the ratio φ is also small (as estimated in the last paragraph), indicating that f ( φ ) could be expanded around φ = 0 as f ( φ ) ≈ f 0 + f 1 φ + f 2 φ 2 +··· , where f 0 > 0 (to guarantee the stability condition s 2 ≥ 0). Keeping the first leading-order term f 0 enables us to obtain s 2 ≈ f 0 φ , so s 2 has a similar value of φ if f 0 ∼ O (1) and the EOS does not take a linear form (except for f 0 = 1). Moreover, the causality principle requires φ ≲ f -1 0 . The s 2 ≈ 0.03 ∼ φ ≲ 0.02 at ρ 0 from chiral effective field calculations (Essick et al., 2021) confirms our order-of-magnitude estimate on s 2 . If the next-leading-order term f 1 is small and positive, then the upper bound for φ becomes φ ≲ f -1 0 (1 -f 1 / f 2 0 ) which is even reduced compared with f -1 0 . The exact form of f ( φ ) should be worked out/analyzed by the general-relativistic structure equations for NSs (Tolman, 1939; Oppenheimer and Volkoff, 1939). By doing that, we demonstrated earlier that φ is upper bounded as φ ≲ 0.374 near the centers of stable NSs (Cai et al., 2023b,a; Cai and Li, 2024a,b). The corresponding trace anomaly ∆ in NS cores is thus bounded to be above -0.04. In the next sections, we first show the main steps leading to these conclusions and then discuss their ramifications in comparison with existing predictions on ∆ in the literature.</text>
<section_header_level_1><location><page_5><loc_8><loc_44><loc_91><loc_46></location>3 Analyzing Scaled TOV Equations, Mass/Radius Scalings and Central SSS</section_header_level_1>
<text><location><page_5><loc_8><loc_38><loc_92><loc_42></location>The TOV equations describe the radial evolution of pressure P ( r ) and mass M ( r ) of a NS under static hydrodynamic equilibrium conditions (Tolman, 1939; Oppenheimer and Volkoff, 1939). In particular, we have (adopting c = 1)</text>
<formula><location><page_5><loc_24><loc_33><loc_92><loc_36></location>d P d r =-GM ε r 2 GLYPH<181> 1 + P ε ¶GLYPH<181> 1 + 4 π r 3 P M ¶GLYPH<181> 1 -2 GM r ¶ -1 , d M d r = 4 π r 2 ε , (5)</formula>
<text><location><page_5><loc_8><loc_25><loc_92><loc_31></location>here the mass M = M ( r ), pressure P = P ( r ) and energy density ε = ε ( r ) are functions of the distance r from NS center. The central energy density ε c is a specific and important quantity, which straightforwardly connects the central pressure P c via the EOS P c = P ( ε c ). Using ε c , we can construct a mass scale W and a length scale Q :</text>
<formula><location><page_5><loc_28><loc_20><loc_92><loc_24></location>W = 1 G 1 p 4 π G ε c = 1 π 4 πε c , Q = 1 p 4 π G ε c = 1 π 4 πε c , (6)</formula>
<text><location><page_5><loc_8><loc_16><loc_92><loc_19></location>respectively, here the second relations follow with G = 1. Using W and Q , we can rewrite the TOV equations in the following dimensionless form (Cai et al., 2023b,a; Cai and Li, 2024a,b),</text>
<formula><location><page_5><loc_31><loc_9><loc_92><loc_13></location>d b P d b r =-b ε c M b r 2 (1 + b P / b ε )(1 + b r 3 b P / c M ) 1 -2 c M / b r , d c M d b r = b r 2 b ε , (7)</formula>
<text><location><page_6><loc_8><loc_89><loc_70><loc_91></location>where b P = P / ε c , b ε = ε / ε c , b r = r / Q and c M = M / W . The general smallness of</text>
<formula><location><page_6><loc_42><loc_85><loc_92><loc_87></location>X ≡ φ c ≡ b P c ≡ P c / ε c , (8)</formula>
<text><location><page_6><loc_8><loc_82><loc_34><loc_83></location>together with the smallness of</text>
<formula><location><page_6><loc_43><loc_79><loc_92><loc_81></location>µ ≡ b ε -b ε c = b ε -1, (9)</formula>
<text><location><page_6><loc_8><loc_75><loc_92><loc_78></location>near NS centers enable us to develop effective/controllable expansion of a relevant quantity U over X and µ as(Cai et al., 2023b,a; Cai and Li, 2024a,b)</text>
<formula><location><page_6><loc_40><loc_70><loc_92><loc_73></location>U / U c ≈ 1 + X i + j ≥ 1 u i j X i µ j , (10)</formula>
<text><location><page_6><loc_8><loc_61><loc_92><loc_68></location>here U c is the quantity U at the center. Since both GR and its Newtonian counterpart with small φ and X are nonlinear, the TOV equations are also nonlinear. Due to the more involved nonlinearity of the TOV equations, one often solves them by adopting numerical algorithms via a selected ε c and an input dense matter EOS(Cai and Li, 2016; Li et al., 2022) as well as the termination condition:</text>
<formula><location><page_6><loc_41><loc_58><loc_92><loc_60></location>P ( R ) = 0 ↔ b P ( b R ) = 0, (11)</formula>
<text><location><page_6><loc_8><loc_55><loc_57><loc_57></location>which defines the NS radius R . The NS mass is given as</text>
<formula><location><page_6><loc_29><loc_50><loc_92><loc_54></location>M NS = c M NS W , with c M NS ≡ c M ( b R ) = Z b R 0 d b r b r 2 b ε ( b r ). (12)</formula>
<text><location><page_6><loc_8><loc_41><loc_92><loc_46></location>Starting from the scaled TOV equations of (7), we can show that both b P and b ε are even under the transformation b r ↔-b r while c M is odd (Cai and Li, 2024a). Therefore, we can write down the general expansions for b ε , b P and c M near b r = 0:</text>
<formula><location><page_6><loc_34><loc_37><loc_92><loc_39></location>b ε ( b r ) ≈ 1 + a 2 b r 2 + a 4 b r 4 + a 6 b r 6 +··· , (13)</formula>
<formula><location><page_6><loc_33><loc_35><loc_92><loc_37></location>b P ( b r ) ≈ X + b 2 b r 2 + b 4 b r 4 + b 6 b r 6 +··· , (14)</formula>
<formula><location><page_6><loc_33><loc_31><loc_92><loc_34></location>c M ( b r ) ≈ 1 3 b r 3 + 1 5 a 2 b r 5 + 1 7 a 4 b r 7 + 1 9 a 6 b r 9 +··· , (15)</formula>
<text><location><page_6><loc_8><loc_23><loc_92><loc_29></location>the expansion for c M follows directly from that for b ε . As a direct consequence, we find that s 2 ( b r ) = s 2 ( -b r ), i.e., there would be no odd terms in b r in the expansion of s 2 over b r . The relationships between { a j } and { b j } are determined by the scaled TOV equations of (7); and the results are (Cai et al., 2023b)</text>
<formula><location><page_6><loc_17><loc_18><loc_92><loc_21></location>b 2 =-1 6 ¡ 1 + 3 b P 2 c + 4 b P c ¢ , (16)</formula>
<formula><location><page_6><loc_17><loc_14><loc_92><loc_17></location>b 4 = b P c 12 ¡ 1 + 3 b P 2 c + 4 b P c ¢ -a 2 30 ¡ 4 + 9 b P c ¢ , (17)</formula>
<formula><location><page_6><loc_17><loc_10><loc_92><loc_14></location>b 6 =-1 216 ¡ 1 + 9 b P 2 c ¢¡ 1 + 3 b P 2 c + 4 b P c ¢ -a 2 2 30 + GLYPH<181> 2 15 b P 2 c + 1 45 b P c -1 54 ¶ a 2 -5 + 12 b P c 63 a 4 , (18)</formula>
<text><location><page_7><loc_8><loc_88><loc_92><loc_91></location>etc., and all the odd terms of { b j } and { a j } are zero. The coefficient a 2 can be expressed in terms of b 2 via the SSS, because</text>
<formula><location><page_7><loc_35><loc_84><loc_92><loc_88></location>s 2 = d b P d b ε = d b P d b r · d b r d b ε = b 2 + 2 b 4 b r 2 +··· a 2 + 2 a 4 b r 2 +··· . (19)</formula>
<text><location><page_7><loc_8><loc_80><loc_92><loc_84></location>Evaluating it at b r = 0 gives s 2 c = b 2 / a 2 , or inversely a 2 = b 2 / s 2 c . Since s 2 c > 0 and b 2 < 0, we find a 2 < 0, i.e., the energy density is a monotonically decreasing function of b r near b r ≈ 0.</text>
<text><location><page_7><loc_8><loc_75><loc_92><loc_78></location>According to the definition of NS radius given in Eq. (11), we obtain from the truncated equation X + b 2 b R 2 ≈ 0 that b R ≈ ( -X/ b 2 ) 1/2 = [6X/(1 + 3X 2 + 4X)] 1/2 and therefore the radius R (Cai et al., 2023b):</text>
<formula><location><page_7><loc_26><loc_69><loc_92><loc_72></location>R = b RQ ≈ GLYPH<181> 3 2 π G ¶ 1/2 ν c , with ν c ≡ 1 π ε c GLYPH<181> X 1 + 3X 2 + 4X ¶ 1/2 . (20)</formula>
<text><location><page_7><loc_8><loc_65><loc_51><loc_67></location>Similarly, the NS mass scales as (Cai et al., 2023b)</text>
<formula><location><page_7><loc_19><loc_59><loc_92><loc_63></location>M NS ≈ 1 3 b R 3 b ε c W = 1 3 b R 3 W ≈ GLYPH<181> 6 π G 3 ¶ 1/2 Γ c , with Γ c ≡ 1 π ε c GLYPH<181> X 1 + 3X 2 + 4X ¶ 3/2 . (21)</formula>
<text><location><page_7><loc_8><loc_56><loc_64><loc_57></location>Consequently, the NS compactness ξ scales as (Cai and Li, 2024b)</text>
<formula><location><page_7><loc_25><loc_50><loc_92><loc_54></location>ξ ≡ M NS R ≈ 2 G X 1 + 3X 2 + 4X = 2 Π c G , with Π c ≡ X 1 + 3X 2 + 4X . (22)</formula>
<text><location><page_7><loc_8><loc_39><loc_92><loc_48></location>For small X (Newtonian limit), ξ ≈ 2X. The relation (22) implies that X is the source and also a measure of NS compactness(Cai and Li, 2024b). The correlation between X and ξ was studied and fitted numerically in the form of lnX ≈ P i z i ξ i using varius EOS models(Saes and Mendes, 2022). Such fitting schemes become eventually effective as enough parameters z i 's are used. However, the real correlation between X and ξ is somehow lost. In particular, our correlation tells that ξ ∼ τ 0 + τ 1 X + τ 2 X 2 +··· with τ 0 ≈ 0 and τ 1 ≈ 2.</text>
<figure>
<location><page_7><loc_31><loc_16><loc_68><loc_35></location>
<caption>FIG. 2. (Color Online). An illustration of the TOV configuration on a typical mass-radius sequence. The cores of NSs at the TOV configuration contain the densest visible matter existing in our Universe, the compactness ξ for such NSs is the largest among all stable NSs.</caption>
</figure>
<text><location><page_8><loc_8><loc_77><loc_92><loc_91></location>The maximum-mass configuration (or the TOV configuration) along the NS M-R curve is a special point. Consider a typical NS M-R curve near the TOV configuration from right to left, the radius R (mass M NS ) eventually decreases (increases), the compactness ξ = M NS / R correspondingly increases and reaches its maximum value at the TOV configuration. When going to the left along the M-R curve even further, the stars becomes unstable and then may collapse into black holes (BHs). So the NS at the TOV configuration is denser than its surroundings and the cores of such NSs contain the stable densest visible matter existing in the Universe. The TOV configuration is indicated on a typical M-R sequence in FIG. 2. Mathematically, the TOV configuration is described as,</text>
<formula><location><page_8><loc_39><loc_73><loc_92><loc_76></location>d M NS d ε c fl fl fl fl M NS = M max NS = M TOV = 0. (23)</formula>
<text><location><page_8><loc_8><loc_71><loc_50><loc_72></location>Using the NS mass scaling of Eq. (21), we obtain</text>
<formula><location><page_8><loc_24><loc_66><loc_92><loc_69></location>d M NS d ε c = 1 2 M NS ε c · 3 GLYPH<181> s 2 c X -1 ¶ 1 -3X 2 1 + 3X 2 + 4X -1 , , where s 2 c ≡ d P c d ε c . (24)</formula>
<text><location><page_8><loc_8><loc_63><loc_88><loc_64></location>Inversely, we obtain the expression for the central SSS (Cai et al., 2023a; Cai and Li, 2024a),</text>
<formula><location><page_8><loc_23><loc_57><loc_92><loc_61></location>for stable NSs along M-R curve: s 2 c = X GLYPH<181> 1 + 1 + Ψ 3 1 + 3X 2 + 4X 1 -3X 2 ¶ , (25)</formula>
<text><location><page_8><loc_8><loc_54><loc_13><loc_55></location>where</text>
<formula><location><page_8><loc_42><loc_51><loc_92><loc_54></location>Ψ = 2 dln M NS dln ε c ≥ 0. (26)</formula>
<text><location><page_8><loc_8><loc_49><loc_89><loc_50></location>We see that the SSS really has the form of Eq. (4). For NSs at the TOV configuration, we have</text>
<formula><location><page_8><loc_24><loc_43><loc_92><loc_46></location>for NSs at the TOV configuration: s 2 c = X GLYPH<181> 1 + 1 3 1 + 3X 2 + 4X 1 -3X 2 ¶ . (27)</formula>
<text><location><page_8><loc_8><loc_37><loc_92><loc_41></location>since now Ψ = 0. Using the s 2 c of Eq. (27) for NSs at the TOV configuration, we can calculate the derivative of NS radius R with respective to ε c around the TOV point, that is (Cai et al., 2023b)</text>
<formula><location><page_8><loc_25><loc_32><loc_92><loc_36></location>d R d ε c ∼ d d ε c GLYPH<181> b R π ε c ¶ R max ↔ M max NS = GLYPH<181> s 2 c X -1 ¶ 1 -3X 2 1 + 3X 2 + 4X -1 =-2 3 , (28)</formula>
<text><location><page_8><loc_8><loc_26><loc_92><loc_31></location>i.e., as ε c increases, the radius R decreases (self-gravitating property), as expected. On the other hand, for stable NSs along the M-R curve with a nonzero Ψ , we have d R /d ε c ∼ ( Ψ -2)/3; this means if Ψ is around 2, the dependence of the radius on ε c would be weak.</text>
<text><location><page_8><loc_8><loc_10><loc_92><loc_25></location>For verification, the scaling R max -ν c (panel (a)) of Eq. (20) and the scaling M max NS -Γ c (panel (b)) of Eq.(21) are shown in FIG. 3 by using 87 phenomenological and 17 extra microscopic NS EOSs with and/or without considering hadron-quark phase transitions and hyperons by solving numerically the original TOV equations, see Ref. Cai et al. (2023b) for more details on these EOS samples. The observed strong linear correlations demonstrate vividly that the R max -ν c and M max NS -Γ c scalings are nearly universal. While it is presently unclear where the mass threshold for massive NSs to collapse into BHs is located, the TOV configuration is the closest to it theoretically. It is also well known that certain properties of BHs are universal and only depend on quantities like mass, charge and angular momentum. One thus expects the NS mass and radius scalings near the TOV</text>
<figure>
<location><page_9><loc_26><loc_61><loc_74><loc_91></location>
<caption>FIG. 3. (Color Online). Panel (a): the R max -ν c correlation using 104 EOS samples (colored symbols), see Ref. Cai et al. (2023b) for more detailed descriptions on these EOSs, the constraints on the mass (Fonseca et al., 2021) and radius (Riley et al., 2021) of PSR J0740+6620 are shown by the pink hatched bands. Panel (b): similar as the left panel but fot M max NS -Γ c . The orange arrows and captions nearby in each panel indicate the ν c and Γ c defined in Eq. (20) and Eq. (21), respectively. Figures taken from Ref. Cai et al. (2023b).</caption>
</figure>
<text><location><page_9><loc_8><loc_40><loc_92><loc_45></location>configuration to be more EOS-independent compared to light NSs. It is also particularly interesting to notice that EOSs allowing phase transitions and/or hyperon formations predict consistently the same scalings.</text>
<text><location><page_9><loc_8><loc_31><loc_92><loc_34></location>By performing linear fits of the results obtained from the EOS samples, the quantitative scaling relations are (Cai et al., 2023b,a; Cai and Li, 2024a)</text>
<formula><location><page_9><loc_29><loc_26><loc_92><loc_30></location>R max /km ≈ 1050 + 30 -30 × GLYPH<181> ν c fm 3/2 /MeV 1/2 ¶ + 0.64 + 0.25 -0.25 , (29)</formula>
<formula><location><page_9><loc_28><loc_22><loc_92><loc_26></location>M max NS / M ⊙ ≈ 1730 + 30 -30 × GLYPH<181> Γ c fm 3/2 /MeV 1/2 ¶ -0.106 + 0.035 -0.035 , (30)</formula>
<text><location><page_9><loc_8><loc_10><loc_92><loc_21></location>with their Pearson's coefficients about 0.958 and 0.986, respectively, here ν c and Γ c are measured in fm 3/2 /MeV 1/2 . In addition, the standard errors for the radius and mass fittings are about 0.031 and 0.003 for these EOS samples. In FIG. 3, the condition M max NS ≳ 1.2 M ⊙ used is necessary to mitigate influences of uncertainties in modeling the crust EOS (Baym et al., 1971; Iida and Sato, 1997; Xu et al., 2009) for low-mass NSs. For the heavier NSs studied here, it is reassuring to see that although the above 104 EOSs predicted quite different crust properties, they all fall closely around the same scaling lines consistently, especially for the M max NS -Γ c relation.</text>
<section_header_level_1><location><page_10><loc_8><loc_87><loc_92><loc_91></location>4 Gravitational Upper Bound on X ≡ φ c = P c / ε c , its Generalizations and the Impact on Supradense NS Matter EOS</section_header_level_1>
<text><location><page_10><loc_9><loc_84><loc_92><loc_85></location>Based on Eq. (27) and the Principle of Causality of SR, we obtain immediately (Cai et al., 2023b)</text>
<formula><location><page_10><loc_38><loc_79><loc_92><loc_81></location>s 2 c ≤ 1 ↔ X = b P c ≲ 0.374 ≡ X GR + . (31)</formula>
<text><location><page_10><loc_8><loc_74><loc_92><loc_77></location>Although the causality condition requires apparently b P c ≤ 1, the supradense nature of core NS matter indicated by the nonlinear dependence of s 2 c on b P c essentially renders it to be much smaller.</text>
<text><location><page_10><loc_8><loc_56><loc_92><loc_72></location>Asmall X < 1 was in fact studied/indicated earlier in the literature (Koranda et al., 1997; Saes and Mendes, 2022). For example, in Ref. Koranda et al. (1997), the minimum-period EOS of the form P ( ε ) = 0 for ε < ε f and P ( ε ) = ε -ε f for ε ≥ ε f was adopted; here ε f is a free parameter of the model. Such EOS is simplified and unrealistic in the sense: (1) both the parameter ε f ≈ 2.156 × 10 15 g/cm 3 ≈ 8.1 ε 0 and the central energy density ε c ≈ 4.778 × 10 15 g/cm 3 ≈ 17.9 ε 0 are unrealistically large for a 1.442 M ⊙ NS(Koranda et al., 1997); the consequent ratio X in this model is X = 1 -ε f / ε c ≈ 0.55; (2) the central SSS of 1 of such model is basically inconsistent with Eq. (27). Actually, only with X = 1 -ε f / ε c ≈ 0.374 or ε f / ε c ≈ 0.626 one can make this EOS model consistent with Eq. (27), i.e., the parameter space for ε f is limited; however a vanishing pressure up to ε f / ε c ≈ 0.626 is fundamentally unsatisfactory. Therefore, X ≈ 0.55 is only qualitatively meaningful.</text>
<text><location><page_10><loc_8><loc_50><loc_92><loc_54></location>The bound (31) is obtained under the specific condition that it gives the upper limit for φ = P / ε at the center of NSs at TOV configurations. In order to bound a general φ = P / ε = b P / b ε , we need to take three generalizations of X ≲ 0.374 obtained from Eq. (31) by asking (Cai et al., 2023a),</text>
<unordered_list>
<list_item><location><page_10><loc_8><loc_46><loc_82><loc_48></location>(a) How does φ = b P / b ε behave at a finite b r for the maximum-mass configuration M max NS ?</list_item>
<list_item><location><page_10><loc_8><loc_43><loc_92><loc_46></location>(b) How does the limit X ≲ 0.374 modify when considering stable NSs on the M-R curve away from the TOV configuration?</list_item>
<list_item><location><page_10><loc_8><loc_39><loc_92><loc_42></location>(c) By combining (a) and (b), how does φ behave for stable NSs at finite distances b r away from their centers?</list_item>
</unordered_list>
<text><location><page_10><loc_8><loc_33><loc_92><loc_37></location>For the first question, since the pressure b P and b ε are both decreasing functions of b r , i.e., b P ≈ b P c + b 2 b r 2 < b P c and b ε ≈ 1 + s -2 c b 2 b r 2 < 1 (notice b ε c = 1 and a 2 = b 2 / s 2 c ), we obtain by taking their ratio:</text>
<formula><location><page_10><loc_17><loc_29><loc_92><loc_32></location>φ = P / ε = b P / b ε ≈ b P c / b ε c + GLYPH<181> 1 -b P c s 2 c ¶ b 2 b r 2 = b P c + GLYPH<181> 1 -b P c s 2 c ¶ b 2 b r 2 ≈ b P c -GLYPH<181> 1 + 7 b P c 24 ¶ b r 2 < b P c . (32)</formula>
<text><location><page_10><loc_8><loc_17><loc_92><loc_27></location>Generally, 1 -b P c / s 2 c > 0, the smallb P c expansions of s 2 c of Eq. (27) and b 2 of Eq. (16) are used in the last step. This means that not only b P and b ε decrease for finite b r , but also does their ratio b P / b ε . Therefore for NSs at the TOV configuration of the M-R curves, we have φ = b P / b ε ≤ b P c ≲ 0.374. Considering the second question and for stable NSs on the M-R curve, one has Ψ > 0 (of Eq. (26)) and Eq.(25) induces an even smaller upper bound for X compared with 0.374. Furthermore, for the last question (c), the inequality (32) still holds and is slightly modified for small b P c as,</text>
<formula><location><page_10><loc_26><loc_13><loc_92><loc_16></location>φ = b P / b ε ≈ b P c -1 24 1 + Ψ (1 + Ψ /4) 2 · 1 + 7 b P c + Ψ GLYPH<181> b P c + 1 4 ¶, b r 2 < b P c , (33)</formula>
<text><location><page_10><loc_8><loc_10><loc_54><loc_11></location>which implies φ = b P / b ε for Ψ , 0 also decreases with b r .</text>
<text><location><page_11><loc_9><loc_90><loc_46><loc_91></location>Combining the above three aspects, we find</text>
<formula><location><page_11><loc_16><loc_85><loc_92><loc_87></location>for stable NSs along M-R curve near/at the centers: φ = P / ε = b P / b ε ≤ X ≲ 0.374. (34)</formula>
<text><location><page_11><loc_8><loc_54><loc_92><loc_83></location>Nevertheless, the validity of this conclusion is limited to small b r due to the perturbtive nature of the expansions of b P ( b r ) and b ε ( b r ). Whether φ = P / ε could exceed such upper limit at even larger distances away from the centers depends on the joint analysis of s 2 and P / ε , e.g., by including more higher order contributions of the expansions (Cai et al., 2023a). The upper bound P / ε ≲ 0.374 (at least near the NS centers) is an intrinsic property of the TOV equations, which embody the strongfield aspects of gravity in GR, especially the strong self-gravitating nature. In this sense, there is no guarantee a prior that this bound is consistent with all microscopic nuclear EOSs (either relativistic or non-relativistic). This is mainly because the latter were conventionally constructed without considering the strong-field ingredients of gravity. The robustness of such upper bound for φ = P / ε can be checked only by observable astrophysical quantities/processes involving strongfield aspects of gravity such as NS M-R data, NS-NS mergers and/or NS-BH mergers (Baumgarte and Shapiro, 2010; Shibata, 2015; Baiotti and Rezzolla, 2017; Kyutoku et al., 2021). As mentioned earlier, in the NS matter-gravity inseparable system, it is the total action that determines the matter state and NS structure. Thus, to our best knowledge, there is no physics requirement that the EOS of supradense matter created in vacuum from high-energy heavy-ion collisions or other laboratory experiments where effects of gravity can be neglected to be the same as that in NSs as nuclear matter in the two situations are in very different environments. Nevertheless, ramifications of the above findings and logical arguments should be further investigated.</text>
<figure>
<location><page_11><loc_14><loc_36><loc_85><loc_52></location>
<caption>FIG. 4. (Color Online). An illustration of gravitational effects on supradense matter EOS in NSs: The nonlinearity of Newtonian gravity reduces the upper bound for φ from 1 (obtained by requiring s 2 ≤ 1 in SR via a linear EOS of the form P = const. × ε for supradense matter in vacuum) to 3/4=0.75 and the even stronger nonlinearity of the gravity in GR further refines it to be about 0.374.</caption>
</figure>
<text><location><page_11><loc_22><loc_35><loc_24><loc_38></location>≤</text>
<text><location><page_11><loc_26><loc_35><loc_28><loc_38></location>↔</text>
<text><location><page_11><loc_30><loc_35><loc_31><loc_38></location>≤</text>
<text><location><page_11><loc_8><loc_22><loc_92><loc_27></location>Next, we consider the Newtonian limit where φ and X are small, then we can neglect 3X 2 + 4X in the coefficient b 2 , consequently b 2 =-1/6 is obtained (Chandrasekhar, 2010). In such case, we shall obtain from Eq. (27):</text>
<formula><location><page_11><loc_38><loc_20><loc_92><loc_22></location>Newtonian limit: s 2 c ≈ 4X/3, (35)</formula>
<text><location><page_11><loc_8><loc_10><loc_92><loc_18></location>and the principle of causality requires X ≤ 3/4 = 0.75 ≡ X N + . The latter can be applied to nuclear matter created in laboratory experiments where effects of gravity can be neglected. Turning on gravity in NSs, we see that the nonlinearity of Newtonian gravity has already reduced the upper bound for φ from 1 obtained by requiring s 2 ≤ 1 in SR via a linear EOS of the form P = const. × ε to 3/4, the even stronger nonlinearity of the gravity in GR reduces it further. These effects of</text>
<text><location><page_12><loc_8><loc_83><loc_92><loc_91></location>gravity on φ are illustrated in FIG. 4. It is seen that the strong-field gravity in GR brings a relative reduction on the upper bound for φ by about 100%. Though the φ or X in Newtonian gravity is generally smaller, the upper bound for φ or X is however larger than its GR counterpart. The index s 2 c /X being greater than 1 in both Newtonian gravity and in GR imply that the central EOS in NSs once considering the gravity effect could not be linear or conformal.</text>
<text><location><page_12><loc_8><loc_68><loc_92><loc_80></location>We emphasize that all of the analyses above based on SR and GR are general from analyzing perturbatively analytical solutions of the scaled TOV equations without using any specific nuclear EOS. Because the TOV equations are results of a hydrodynamical equilibrium of NS matter in the environment of a strong-field gravity from extremizing the total action of the matter-gravity system, features revealed above from SR and GR inherent in the TOV equations must be matched by the nuclear EOS. This requirement can then put strong constraints on the latter. In particular, the upper bound for φ as φ ≲ X GR + ≈ 0.374 of Eq. (31) enables us to limit the density dependence of nuclear EOS relevant for NS modelings.</text>
<figure>
<location><page_12><loc_27><loc_37><loc_73><loc_64></location>
<caption>FIG. 5. (Color Online). Gravitational impact on the EOS of supradense matter and the underlying strong interaction in NSs: the general X GR + -dependence of σ large and σ small of Eq. (38), based on the nuclear EOS model of Eq. (36); here B FFG ≈ 35MeV, M N ≈ 939MeV and ℓ = ρ / ρ 0 ≈ 6.</caption>
</figure>
<text><location><page_12><loc_8><loc_22><loc_92><loc_27></location>In the following, we provide an example illustrating how the strong-field gravity can restrict the behavior of superdense matter in NSs. For simplicity, we assume that the energy per baryon takes the following form</text>
<formula><location><page_12><loc_37><loc_19><loc_92><loc_22></location>E ( ρ ) = B FFG GLYPH<181> ρ ρ 0 ¶ 2/3 + B GLYPH<181> ρ ρ 0 ¶ σ , (36)</formula>
<text><location><page_12><loc_8><loc_10><loc_92><loc_18></location>where the first term is the kinetic energy of a FFG of neutrons in NSs with B FFG ≈ 35 MeV being its known value at ρ 0 and the second term is the contribution from interactions described with the parameters B and σ . The pressure and the energy density are obtained from P ( ρ ) = ρ 2 d E /d ρ and ε ( ρ ) = [ E ( ρ ) + M N ] ρ , respectively. The ratio φ = P / ε and the SSS s 2 = d P /d ε could be obtained correspondingly. Denote the reduced density ρ / ρ 0 where s 2 → 1 and φ → X → X GR + as ℓ (e.g., ℓ ≲ 8</text>
<text><location><page_13><loc_8><loc_90><loc_69><loc_91></location>for realistic NSs), the following constraining equation for σ is obtained:</text>
<formula><location><page_13><loc_24><loc_85><loc_92><loc_88></location>σ ‡ X GR + σ -1 · + ℓ 2/3 3 GLYPH<181> B FFG M N ¶GLYPH<181> σ -2 3 ¶ h (3 σ + 2)X GR + -2 σ -3 i = 0. (37)</formula>
<text><location><page_13><loc_8><loc_78><loc_92><loc_83></location>Thus, X GR + effectively restricts the index σ characterizing the stiffness of nuclear EOS. There are two solutions of Eq. (37) with one being greater than 1 (denoted as σ large ) and the other smaller than 1 (denoted as σ small ). They can be explicitly written as</text>
<formula><location><page_13><loc_10><loc_72><loc_92><loc_77></location>σ = 1 2 GLYPH<181> X GR + + Λ GLYPH<181> X GR + -2 3 ¶¶ -1 1 + 5 9 Λ ± v u u t 1 + 16 Λ 9 "ˆ X GR,2 + -3X GR + 2 + 5 8 ! + Λ GLYPH<181> X GR + -13 12 ¶ 2 # , (38)</formula>
<text><location><page_13><loc_8><loc_69><loc_13><loc_71></location>where</text>
<text><location><page_13><loc_8><loc_64><loc_40><loc_65></location>The expression for the coefficient B is</text>
<formula><location><page_13><loc_40><loc_59><loc_92><loc_63></location>B = GLYPH<181> 1 + 5 Λ /9 σ 2 -1 1 ℓ σ ¶ M N , (40)</formula>
<text><location><page_13><loc_8><loc_52><loc_92><loc_58></location>which depends on X GR + through σ . As a numerical example, using M N ≈ 939MeV, B FFG ≈ 35MeV and ℓ ≈ 6 leads to σ large ≈ 3.1 and B large ≈ 0.45MeV or σ small ≈ 0.06 and B small ≈ -906MeV (this second solution is unphysical since B > 0 is necessarily required to make E ( ρ ) > 0 at NS densities). If one takes artificially X GR + = 1, then the two solutions (38) approach</text>
<formula><location><page_13><loc_18><loc_46><loc_92><loc_50></location>σ small → 2 3 1 1 + 3/ Λ = 2 3 GLYPH<181> 1 + 3 ℓ 2/3 GLYPH<181> M N B FFG ¶¶ -1 ≪ 1, and σ large → 1 from above. (41)</formula>
<text><location><page_13><loc_8><loc_37><loc_92><loc_45></location>Now, neither solution is physical since B small < 0 for σ small while B large →+∞ for σ large → 1 from above, according to Eq. (40). The general X GR + -dependence of σ large and σ small of Eq. (38) is shown in FIG. 5. It is seen that only as X GR + → 1 the EOS approaches a linear form E ( ρ ) ≈ B ρ / ρ 0 ∼ ρ (so P ≈ B ρ 2 / ρ 0 and ε ≈ B ρ 2 / ρ 0 + M N ρ ) at large densities (magenta line), being consistent with our general analyses and expectation.</text>
<text><location><page_13><loc_8><loc_30><loc_92><loc_35></location>Since the parameterization (36) is over-simplified, for general cases more density-dependent terms should be included, i.e., B ( ρ / ρ 0 ) σ → P J j = 1 B j ( ρ / ρ 0 ) σ j . We may then obtain two related equations from φ → X → X GR + and s 2 → 1 as (for either X GR + = 1 or X GR + , 1):</text>
<formula><location><page_13><loc_26><loc_23><loc_92><loc_27></location>J X j = 1 GLYPH<181> B j M N ¶ ‡ σ j -X GR + · ℓ σ j + ℓ 2/3 GLYPH<181> B FFG M N ¶GLYPH<181> 2 3 -X GR + ¶ -X GR + = 0, (42)</formula>
<formula><location><page_13><loc_30><loc_19><loc_92><loc_23></location>J X j = 1 GLYPH<181> B j M N ¶ ‡ 1 -σ 2 j · ℓ 1/3 + σ j -ℓ 1/3 GLYPH<181> 1 + 5 9 ℓ 2/3 GLYPH<181> B FFG M N ¶¶ = 0. (43)</formula>
<text><location><page_13><loc_8><loc_10><loc_92><loc_16></location>These constraints for B j and σ j should be taken appropriately into account when writing an effective NS EOS based on density expansions. For example, when extending Eq. (36) to be E ( ρ ) = B FFG ( ρ / ρ 0 ) 2/3 + B 1 ( ρ / ρ 0 ) σ 1 + B 2 ( ρ / ρ 0 ) σ 2 under two conditions E ( ρ 0 , δ ) ≈ E 0 ( ρ 0 ) + E sym ( ρ 0 ) δ 2 ≈ 15MeV for pure neutron matter with δ = 1 and P ( ρ 0 ) ≈ 3MeV/fm 3 , using ℓ ≈ 6 together with X GR + ≈ 0.374,</text>
<formula><location><page_13><loc_41><loc_66><loc_92><loc_69></location>Λ ≡ ℓ 2/3 GLYPH<181> B FFG M N ¶ ≪ 1. (39)</formula>
<text><location><page_14><loc_8><loc_86><loc_92><loc_91></location>we may obtain σ 1 ≈ 0.3 and σ 2 ≈ 3.0 (as well as B 1 ≈-20.5MeV and B 2 ≈ 0.5MeV), respectively. From this example, one can see quantitatively that the gravitational bound naturally leads to a constraint on the nuclear EOS and the underlying interactions in NSs.</text>
<section_header_level_1><location><page_14><loc_8><loc_82><loc_92><loc_84></location>5 Gravitational Lower Bound on Trace Anomaly ∆ in Supradense NS Matter</section_header_level_1>
<text><location><page_14><loc_8><loc_76><loc_92><loc_81></location>After the above general demonstration on the gravitational upper limit for φ near NS centers given by (31) or (34), we equivalently obtain a lower limit on the dimensionless trace anomaly ∆ = 1/3 -φ as</text>
<formula><location><page_14><loc_43><loc_74><loc_92><loc_75></location>∆ ≥ ∆ GR ≈-0.04. (44)</formula>
<text><location><page_14><loc_8><loc_61><loc_92><loc_72></location>It is very interesting to notice that such GR bound on ∆ is very close to the one predicted by perturbative QCD (pQCD) at extremely high densities owning to the realization of approximate conformal symmetry of quark matter(Bjorken, 1983; Fujimoto et al., 2022), as shown in FIG. 6 using certain NS modelings. A possible negative ∆ in NSs was first pointed out in Ref. Fujimoto et al. (2022), since then several studies have been made on this issue. In the following, we summarize the main findings of these studies by others and compare with what we found above when it is possible.</text>
<figure>
<location><page_14><loc_25><loc_35><loc_75><loc_58></location>
<caption>FIG. 6. (Color Online). Trace anomaly ∆ as a function of energy density ε / ε 0 , here the ∆ in NSs tends to be negative although the pQCD prediction on it approaches zero, ε 0 ≈ 150MeV/fm 3 is the energy density at nuclear saturation density. Figure taken from Ref. Fujimoto et al. (2022).</caption>
</figure>
<text><location><page_14><loc_8><loc_10><loc_92><loc_26></location>The analysis in Ref. Ecker and Rezzolla (2022) using an agnostic EOS showed that ∆ is very close to zero for M TOV ≳ 2.18 ∼ 2.35 M ⊙ and may be slightly negative for even more massive NSs (e.g., ∆ ≳ -0.021 + 0.039 -0.136 for M TOV ≳ 2.52 M ⊙ ); the radial dependence of ∆ is shown in the upper panel of FIG.7 from which one finds the ∆ for NS at the TOV configuration is much deeper than that in a canonical NS. Moreover, incorporating the pQCD effects ( ∆ pQCD → 0) was found to effectively increase the inference on ∆ . An updated analysis of Ref. Ecker and Rezzolla (2022) was given in Ref. Musolino et al. (2024), where ∆ ≳ -0.059 + 0.162 -0.158 or ∆ ≳ 0.019 + 0.100 -0.129 was obtained under the constraint M TOV ≳ 2.35 M ⊙ without or with considering the pQCD effects, see the lower panel of FIG. 7 for the PDFs. Similarly, if M TOV ≳ 2.20 M ⊙ was required, these two limits become ∆ ≳ -0.046 + 0.167 -0.166 and ∆ ≳ 0.029 + 0.108 -0.133 (Musolino et al., 2024), respectively. In Ref. Takátsy et al. (2023),</text>
<figure>
<location><page_15><loc_35><loc_64><loc_67><loc_87></location>
</figure>
<figure>
<location><page_15><loc_22><loc_28><loc_78><loc_58></location>
<caption>FIG. 7. (Color Online). Upper panel: radial dependence of ∆ with the constraint M TOV / M ⊙ ≳ 2.35. Figure taken from Ref. Ecker and Rezzolla (2022). Lower panel: PDF for ∆ with/without considering the pQCD limit at extremely high densities. The first (second) line in the lower panel is for non-rotating (Kepler rotating) NSs. Figure taken from Ref. Musolino et al. (2024).</caption>
</figure>
<text><location><page_15><loc_8><loc_10><loc_92><loc_18></location>the central minimum value of ∆ is found to be about 0.04 using the NICER data together with the tidal deformability from GW170817, and a value of ∆ min ≈-0.04 + 0.11 -0.09 was inferred considering additionally the second component of GW190814 as a NS with mass about 2.59 M ⊙ (Abbott et al., 2020b) using two hadronic EOS models(Takátsy et al., 2023), see the upper panel of FIG. 8. By incorporating the constraints from AT2017gfo (Abbott et al., 2017b), it was found (Pang et al., 2024)</text>
<figure>
<location><page_16><loc_31><loc_67><loc_71><loc_90></location>
</figure>
<figure>
<location><page_16><loc_27><loc_39><loc_73><loc_64></location>
<caption>FIG. 8. (Color Online). Two typical inferences on the energy density (or baryon density) dependence of ∆ . Figures taken from Ref. Takátsy et al. (2023) (upper panel) and Ref. Pang et al. (2024) (lower panel).</caption>
</figure>
<text><location><page_16><loc_8><loc_16><loc_92><loc_32></location>that the minimum of ∆ is very close to zero (about -0.03 to 0.05), as shown in the lower panel of FIG. 8. Using similar low-density nuclear constraints as well as astrophysical data especially including the black widow pulsar PSR J0952-0607 (Romani et al., 2022), Ref. Brandes et al. (2023a) predicted ∆ ≳ -0.086 + 0.07 -0.07 taken at ε ≈ 1GeV/fm 3 . Another analysis within the Bayesian framework considering the state-of-the-art theoretical calculations showed that ∆ ≳ -0.01 (Annala et al., 2023) (where M TOV ≈ 2.27 + 0.11 -0.11 M ⊙ is assumed). Furthermore, by considering the slope and curvature of energy per particle in NSs, Ref. Marczenko et al. (2024) showed that ∆ is lower bounded for M TOV to be about -0.02 + 0.03 -0.03 . In addition, Ref. Cao and Chen (2023) found that the ∆ should be roughly larger than about -0.04 + 0.08 -0.09 in self-bound quark stars while that in a normal NS is generally greater than zero.</text>
<text><location><page_16><loc_8><loc_10><loc_92><loc_15></location>A very recent study classified the EOSs by using the local and/or global derivative d M NS /d R of the resulting mass-radius sequences (Ferreira and Providência, 2024). Limiting the sign of d M NS /d R to positive on the M-R curve for NS masses between about 1 M ⊙ and M TOV , it was</text>
<figure>
<location><page_17><loc_15><loc_67><loc_42><loc_89></location>
</figure>
<figure>
<location><page_17><loc_48><loc_64><loc_82><loc_91></location>
</figure>
<figure>
<location><page_17><loc_19><loc_32><loc_75><loc_59></location>
<caption>FIG. 9. (Color Online). Upper left panel: density dependence of ∆ inferred under the constraint d M NS /d R < 0 for all NS masses (blue) or d M NS /d R ≥ 0 for a certain mass range (red); inference in the bottom figure with astrophysical constraints. Figure taken from Ref. Ferreira and Providência (2024). Upper right panel: two types of M-R curves classified by using the derivative d M NS /d R for NS masses between about 1 M ⊙ and M TOV to help understand the behavior of trace anomaly ∆ 's shown in the left panel. Bottom: The trace anomaly for twin stars satisfying static and dynamic stability conditions. Figure taken from Ref. Jiménez et al. (2024).</caption>
</figure>
<text><location><page_17><loc_8><loc_10><loc_92><loc_15></location>found that ∆ ≳ 0.008 + 0.133 -0.160 (Ferreira and Providência, 2024). On the other hand, if d M NS /d R < 0 is required for all NS masses then ∆ ≳ -0.057 + 0.119 -0.119 is found, see the upper left panel of FIG. 9. Our understanding on this behavior goes as follows: A negative slope d M NS /d R along the whole</text>
<figure>
<location><page_18><loc_17><loc_60><loc_83><loc_91></location>
</figure>
<figure>
<location><page_18><loc_22><loc_37><loc_79><loc_56></location>
<caption>FIG. 10. (Color Online). Upper panel: Summary of current constraints on the lower bound of trace anomaly ∆ in NSs from different analyses with respect to the pQCD (dot-dashed line) and GR (black solid line) predictions. See the text for details. Lower panel: Sketch of two imagined patterns for ∆ = 1/3 -P / ε in NSs. The ∆ is well constrained around the fiducial density ε 0 ≈ 150MeV/fm 3 by low-energy nuclear theories and is predicted to vanish due to the approximate conformality of the matter at ε ≳ 50 ε 0 (or equivalently ρ ≳ 40 ρ 0 ) using pQCD theories. The magenta curve is based on the assumption that causality limit is reached in the most massive NS observed where ε / ε 0 being roughly around 4 ∼ 8. Figure taken from Ref. Cai et al. (2023a).</caption>
</figure>
<text><location><page_18><loc_8><loc_10><loc_92><loc_21></location>M-R curve with M NS / M ⊙ ≳ 1(Ferreira and Providência, 2024) implies the radius of NS at the TOV configuration is relatively smaller compared with the one with a positive d M NS /d R on a certain M-R segment, as indicated in the upper right panel of FIG. 9. Thus the NS compactness ξ in the former case is relatively larger, which induces a larger X via Eq. (22) and correspondingly a smaller ∆ (Cai and Li, 2024b); the smaller radius also implies that the NS is much denser so the maximum baryon density is correspondingly larger (Ferreira and Providência, 2024). In another very recent study, the dense matter trace anomaly in twin stars satisfying relevant static and dynamic stability</text>
<text><location><page_19><loc_8><loc_83><loc_92><loc_91></location>conditions was studied (Jiménez et al., 2024). The ∆ was found to be deeply bounded roughly as ∆ ≳ -0.035 (Jiménez et al., 2024), as shown in the bottom panel of FIG. 9. A deep negative ∆ implies a large φ or X, so the compactness is correspondingly large according to the relation (22). We notice that the radii obtained in Ref. Jiménez et al. (2024) for certain NS masses (e.g., around 2 M ⊙ ) may be small compared with the observational data, e.g., PSR J0740+6620 (Riley et al., 2021).</text>
<text><location><page_19><loc_8><loc_69><loc_92><loc_81></location>The above constraints on the lower limit of ∆ (realized in NSs) are summarized in the upper panel of FIG. 10. Clearly, assuming all results are equally reliable within their individual errors indicated, there is a strong indication that the lower bound of ∆ is negative in NSs. Moreover, except the prediction of Ref. Jiménez et al. (2024), the lower bounds of ∆ from various analyses are very close to the pQCD ( ∆ pQCD = 0) or GR limit ( ∆ GR ≈ -0.04). It is interesting to note that the ∆ GR and ∆ pQCD have no inner-relation to our best knowledge currently. However, we speculate that the matter-gravity duality in massive NSs mentioned earlier may be at work here. Certainly, this speculation deserves further studies.</text>
<text><location><page_19><loc_8><loc_34><loc_92><loc_67></location>How relevant are the GR or pQCD limit for understanding the trace anomaly ∆ in NSs? The ∆ and its energy density dependence are crucial for studying the s 2 in NSs(Fujimoto et al., 2022). For instance, one can explore whether there would be a peaked structure in the density/radius profile of s 2 or not in NSs. Sketched in the lower panel of FIG. 10 (Cai et al., 2023a) are two imagined ∆ functions versus the reduced energy density ε / ε 0 , here ε 0 ≈ 150MeV/fm 3 around which the lowenergy nuclear theories constrain the ∆ quite well. We notice that these two functions are educated guesses certainly with biases. In fact, it has been pointed out that applying a particular EOS in extracting ∆ from observational data may influence the conclusion (Musolino et al., 2024). In the literature, there have been different imaginations/predictions/speculations on how the ∆ at finite energy density may vary and finally reach its pQCD limit of ∆ = 0 at very large energy densities ε ≳ 50 ε 0 ≈ 7.5GeV/fm 3 (Fujimoto et al., 2022; Kurkela et al., 2010) or equivalently ρ ≳ 40 ρ 0 . The latter is far larger than the energy density reachable in the most massive NSs reported so far based on our present knowledge. The pQCD limit on ∆ is thus possibly relevant (Zhou, 2024) but not fundamental for explaining the inferred φ = P / ε ≳ 1/3 from NS observational data based on various microscopic and/or phenomenological models. On the other hand, we also have no confirmation in any way that the causality limit is reached in any NS. The magenta curve is based on the assumption that the causality limit under GR is reached in the most massive NSs observed so far. Based on most model calculations, in the cores of these NSs the ε / ε 0 is roughly around 4 ∼ 8. However, if the matter-gravity in massive NSs is indeed at work, we have no reason to expect that the GR limit is reached at an energy density lower than the one where the pQCD is applicable.</text>
<text><location><page_19><loc_8><loc_10><loc_92><loc_33></location>Keeping a positive attitude in our exploration of a completely uncharted area, we make below a few more comments on how the trace anomaly may reach the pQCD limit. As a negative ∆ is unlikely to be observed in ordinary NSs, the evolution of ∆ is probably more like the green curve in the lower panel of Fig. 10. An (unconventional) exception may come from light but very compact NSs, e.g., a 1.7 M ⊙ NS at the TOV configuration with radius about 9.3 km has its ∆ c ≈ -0.02, since ε c ≈ 1.86GeV/fm 3 together with P c ≈ 654MeV/fm 3 should be obtained via the mass and radius scalings of (30) and (29) and so X = b P c ≈ 0.351. On the other hand, massive and compact NSs (masses ≳ 2 M ⊙ ) are most relevant to observing a negative ∆ (as indicated by the magenta curves) and how it evolves to the pQCD bound, thus revealing more about properties of supradense matter(Cai et al., 2023a). Interestingly, both the green and magenta curves for the ∆ pattern are closely connected with the density-dependence of the SSS using the trace anomaly decomposition of s 2 (Fujimoto et al., 2022) (we do not dive into detailed discussions on these interesting topics in the current review). Unfortunately, the region with ε / ε 0 ≳ 8 is largely inaccessible in NSs due to their self-gravitating nature.</text>
<section_header_level_1><location><page_20><loc_8><loc_89><loc_49><loc_91></location>6 Summary and Future Perspectives</section_header_level_1>
<text><location><page_20><loc_8><loc_71><loc_92><loc_87></location>In summary, perturbative analyses of the scaled TOV equations reveal interesting new insights into properties of supradense matter in NS cores without using any input nuclear EOS. In specific, the ratio φ = P / ε of pressure P over energy density ε (the corresponding trace anomaly ∆ = 1/3 -φ ) in NS cores is bounded to be below 0.374 (above -0.04) by the causality condition under GR independent of the nuclear EOS. Moreover, we demonstrate that the NS mass M NS , radius R and compactness ξ = M NS / R strongly correlate with Γ c = ε -1/2 c Π 3/2 c , ν c = ε -1/2 c Π 1/2 c and Π c = X/(1 + 3X 2 + 4X) with X ≡ φ c = P c / ε c , respectively; therefore observational data on M NS and R as well as on ξ via red-shift measurements can directly constrain the central EOS P c = P c ( ε c ) in a model-independent manner. Besides the topics we have already investigated (Cai et al., 2023b,a; Cai and Li, 2024a,b), there are interesting issues to be further explored in this direction. Particularly, we notice:</text>
<unordered_list>
<list_item><location><page_20><loc_8><loc_62><loc_92><loc_70></location>1. The upper limit for φ = P / ε near NS cores is obtained by truncating the perturbative expansion of P and ε to low orders in reduced radius b r . While the results are quite consistent with existing constraints from state-of-the-art simulations/inferences, refinement by including even higherorder b r terms would be important for studying the radius profile of φ or ∆ in NSs. In the Appendix we estimate such an effective correction.</list_item>
<list_item><location><page_20><loc_8><loc_51><loc_92><loc_60></location>2. Ironically, the upper bound φ = P / ε ≲ 0.374 from GR is very close to that ( P / ε ≲ 1/3) from pQCD at extremely high densities(Bjorken, 1983; Kurkela et al., 2010; Fujimoto et al., 2022). While we speculated that the well-known matter-gravity duality in massive NSs may be at work, it is currently unclear to us if there is a fundamental connection between them. Efforts in understanding their relations may provide useful hints for developing a unified theory for strong-field gravity and elementary particles in supradense matter.</list_item>
</unordered_list>
<section_header_level_1><location><page_20><loc_8><loc_45><loc_29><loc_47></location>Acknowledgments</section_header_level_1>
<text><location><page_20><loc_8><loc_37><loc_92><loc_43></location>We would like to thank James Lattimer and Zhen Zhang for helpful discussions. This work was supported in part by the U.S. Department of Energy, Office of Science, under Award Number DESC0013702, the CUSTIPEN (China-U.S. Theory Institute for Physics with Exotic Nuclei) under the US Department of Energy Grant No. DE-SC0009971.</text>
<section_header_level_1><location><page_20><loc_8><loc_31><loc_65><loc_33></location>Appendix: Estimate on an Effective Correction to s 2 c</section_header_level_1>
<text><location><page_20><loc_8><loc_21><loc_92><loc_29></location>In this appendix, we estimate an effective correction to s 2 c given in Eq. (27) for NSs at the TOV configuration(Cai et al., 2023a). When writing down M NS in Eq. (21), we adopt M NS = 3 -1 b R 3 W which only includes the first term in the systematic expansion (15); necessarily we may include higher order terms from (15) in M NS . As an effective correction we now include 5 -1 a 2 b R 5 from (15) to the NS mass, which modifies Eq. (21) as,</text>
<formula><location><page_20><loc_16><loc_16><loc_92><loc_20></location>M NS ≈ GLYPH<181> 1 3 b R 3 + 1 5 a 2 b R 5 ¶ W = 1 3 b R 3 W GLYPH<181> 1 + 3 5 a 2 b R 2 ¶ = 1 3 b R 3 W GLYPH<181> 1 -3 5 X s 2 c ¶ ∼ Γ c GLYPH<181> 1 -3 5 X s 2 c ¶ , (A1)</formula>
<text><location><page_20><loc_8><loc_10><loc_92><loc_15></location>where b R is given by Eq. (20) through X + b 2 b R 2 ≈ 0, the coefficient Γ c ∼ b R 3 W is defined in Eq. (21) and the general relation a 2 = b 2 / s 2 c is used to write 3 a 2 b R 2 /5 =-3X/5 s 2 c . The factor '1 + 3 a 2 b R 2 /5' is actually the averaged reduced energy density 〈 b ε 〉 by including the a 2 -term in b ε of Eq. (13), namely</text>
<text><location><page_21><loc_8><loc_89><loc_28><loc_91></location>M NS / W ≈ 3 -1 b R 3 〈 b ε 〉 with</text>
<formula><location><page_21><loc_25><loc_84><loc_92><loc_88></location>〈 b ε 〉 = Z b R 0 d b r b r 2 b ε ( b r ) , Z b R 0 d b r b r 2 = 1 + 3 5 a 2 b R 2 , b ε ( b r ) ≈ 1 + a 2 b r 2 . (A2)</formula>
<text><location><page_21><loc_8><loc_78><loc_92><loc_82></location>Moreover, the s 2 c in Eq. (A1) is now not given by Eq. (27), but should include corrections due to including of the a 2 -term in b ε ( b r ). Generally, we write it as:</text>
<formula><location><page_21><loc_23><loc_74><loc_92><loc_77></location>s 2 c ≈ X GLYPH<181> 1 + 1 3 1 + 3X 2 + 4X 1 -3X 2 ¶ (1 + κ 1 X) ≈ 4 3 X + 4 3 (1 + κ 1) X 2 + O (X 3 ), (A3)</formula>
<text><location><page_21><loc_8><loc_68><loc_92><loc_73></location>where κ 1 is a coefficient to be determined. In addition, we have 1 -3X/5 s 2 c ≈ (11/20)[1 + 9(1 + κ 1 )X/11] using the s 2 c of Eq. (A3); taking d M NS /d ε c = 0 with M NS given by Eq. (A1) gives the expression for s 2 c (which is quite complicated), we then expanding the latter over X to order X 2 to give</text>
<formula><location><page_21><loc_35><loc_63><loc_92><loc_66></location>s 2 c ≈ 4 3 X + 1 11 GLYPH<181> 38 3 -2 κ 1 ¶ X 2 + O (X 3 ). (A4)</formula>
<text><location><page_21><loc_8><loc_55><loc_92><loc_62></location>Matching the two expressions (A3) and (A4) at order X 2 gives κ 1 =-3/25. After that, we determine X ≲ 0.381 via s 2 c ≤ 1, which is close to and consistent with 0.374 obtained in the main text; and similarly ∆ ≳ -0.048. The magnitude of the correction ' + κ 1 X' in s 2 c is smaller than 5% while the corresponding correction on X GR + is smaller than 2%. In addition, the NS mass now scales as</text>
<formula><location><page_21><loc_33><loc_49><loc_92><loc_53></location>M NS ∼ 1 π ε c GLYPH<181> X 1 + 3X 2 + 4X ¶ 3/2 · GLYPH<181> 1 + 18 25 X ¶ . (A5)</formula>
<text><location><page_21><loc_8><loc_37><loc_92><loc_46></location>In order to obtain the corrections to s 2 c more self-consistently and improve the accuracy on X GR + , one may include more terms in the expansion of b P over b R of Eq. (14) (i.e., b 2 -term, b 4 -term and b 6 -term, etc.), the expansion of c M over b R of Eq. (15) (i.e., a 2 -term, a 4 -term, a 6 -term, etc.) and in the mean while introduce corrections '1 + κ 1 X + κ 2 X 2 + κ 3 X 3 +··· ' in s 2 c as we did in Eq. (A3); then determine the coefficients κ 1 , κ 2 and κ 3 , etc. The procedure becomes eventually involved as more terms are included.</text>
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</document> | [] |
2024arXiv241205262N | https://arxiv.org/pdf/2412.05262.pdf | <document>
<section_header_level_1><location><page_1><loc_17><loc_90><loc_83><loc_93></location>New analysis of SNeIa Pantheon Catalog: Variable speed of light as an alternative to dark energy</section_header_level_1>
<text><location><page_1><loc_43><loc_87><loc_58><loc_89></location>Hoang Ky Nguyen ∗</text>
<text><location><page_1><loc_24><loc_85><loc_77><loc_87></location>Department of Physics, Babeş-Bolyai University, Cluj-Napoca 400084, Romania (Dated: December 17, 2024)</text>
<text><location><page_1><loc_18><loc_54><loc_83><loc_82></location>In A&A 412, 35 (2003) Blanchard, Douspis, Rowan-Robinson, and Sarkar (BDRS) slightly modified the primordial fluctuation spectrum and produced an excellent fit to WMAP's CMB power spectrum for an Einstein-de Sitter (EdS) universe, bypassing dark energy. Curiously, they obtained a Hubble value of H 0 ≈ 46, in sharp conflict with the canonical range ∼ 67-73. However, we will demonstrate that the reduced value of H 0 ≈ 46 achieved by BDRS is fully compatible with the use of variable speed of light in analyzing the late-time cosmic acceleration observed in Type Ia supernovae (SNeIa). In arXiv:2412.04257 [gr-qc] we considered a generic class of scale-invariant actions that allow matter to couple non-minimally with gravity via a dilaton field χ . We discovered a hidden aspect of these actions: the dynamics of the dilaton can induce a variation in the speed of light c as c ∝ χ 1 / 2 , thereby causing c to vary alongside χ across spacetime. For an EdS universe with varying c , besides the effects of cosmic expansion, light waves emitted from distant SNeIa are further subject to a refraction effect, which alters the Lemaître redshift relation to 1 + z = a -3 / 2 . Based on this new formula, we achieve a fit to the SNeIa Pantheon Catalog exceeding the quality of the ΛCDM model. Crucially, our approach does not require dark energy and produces H 0 = 47 . 2 ± 0 . 4 (95% CL) in strong alignment with the BDRS finding of H 0 ≈ 46. The reduction in H 0 in our work, compared with the canonical range ∼ 67-73, arises due to the 3 / 2 -exponent in the modified Lemaître redshift formula. Hence, BDRS's analysis of the (early-time) CMB power spectrum and our variablec analysis of the (late-time) Hubble diagram of SNeIa fully agree on two counts: (i) the dark energy hypothesis is avoided , and (ii) H 0 is reduced to ∼ 47, which also yields an age t 0 = 2 / (3 H 0 ) = 13 . 8 Gy for an EdS universe, without requiring dark energy. Most importantly, we will demonstrate that the late-time acceleration can be attributed to the declining speed of light in an expanding EdS universe , rather than to a dark energy component.</text>
<section_header_level_1><location><page_1><loc_22><loc_50><loc_36><loc_51></location>I. MOTIVATION</section_header_level_1>
<text><location><page_1><loc_9><loc_27><loc_49><loc_48></location>The ΛCDM model serves as the standard framework for modern cosmology, efficiently accounting for a wide range of astronomical observations. While the model is widely regarded as successful, it faces significant challenges [1]. Notably, ongoing tensions in the determination of the Hubble constant H 0 and the amplitude of matter fluctuations σ 8 raise questions about the underpinning principles of the model [2]. Moreover, an integral component of this model is dark energy (DE), which constitutes approximately 70% of the total energy budget of the universe. The nature of DE itself-along with its finetuning and coincidence problems-poses profound challenges both in cosmology and in the broader context of field theories [3].</text>
<text><location><page_1><loc_9><loc_15><loc_49><loc_27></location>In 2003, Blanchard, Douspis, Rowan-Robinson, and Sarkar (BDRS) proposed a novel approach to mitigate the need for DE when analyzing the cosmic microwave background (CMB) power spectrum [4]. They relied on the Einstein-de Sitter universe, which corresponds to the flat ΛCDM model with Ω Λ = 0. Instead of the conventional single-power primordial fluctuation spectrum, P ( k )= Ak n , they employed a double-power form</text>
<formula><location><page_1><loc_63><loc_46><loc_92><loc_49></location>P ( k ) = { A 1 k n 1 k ⩽ k ∗ A 2 k n 2 k ⩾ k ∗ (1)</formula>
<text><location><page_1><loc_52><loc_32><loc_92><loc_45></location>with continuity imposed across the breakpoint k ∗ . Remarkably, this modest modification produced an excellent fit to the CMB power spectrum without invoking DE. In [5], Hunt and Sarkar further developed a supergravity-based inflation scenario to validate the double-power form given in Eq. (1) and also attained an excellent fit while avoiding DE. The works by BDRS and the Hunt-Sarkar teamif correct -would seriously undermine the viability of the DE hypothesis.</text>
<text><location><page_1><loc_52><loc_16><loc_92><loc_31></location>Surprisingly, the fit by BDRS yielded a new value of H 0 ≈ 46, while the fit by Hunt and Sarkar produced a comparable value of H 0 ≈ 44. Obviously, these values are at odds with the value of H 0 ∼ 70 derived from the Hubble diagram of Type Ia supernovae (SNeIa), based on the ΛCDM model with Ω Λ ≈ 0 . 7. Since DE has been regarded as the driving force of the late-time cosmic acceleration, interest in the works by BDRS and the Hunt-Sarkar team has largely diminished in favor of the standard ΛCDM model.</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_16></location>Amid this backdrop, we will reexamine the SNeIa data in the context of a cosmology that supports a varying speed of light c on an expanding EdS cosmic background, rather than the ΛCDM model. Recently, the theoretical foundation for a varying c in spacetime has been</text>
<text><location><page_2><loc_9><loc_80><loc_49><loc_94></location>derived, using a class of scale-invariant actions that enable non-minimal coupling of matter with gravity via a dilaton field [6, 7]. In cosmology, a varying c should impact the propagation of light rays from distant SNeIa to an Earth-based observer, fundamentally altering the distance-vs-redshift relationship. This modification necessitates a re-evaluation of the H 0 value derived from the Hubble diagram of SNeIa , potentially replacing the canonical value ∼ 70 that relies on the ΛCDM model.</text>
<text><location><page_2><loc_9><loc_70><loc_49><loc_80></location>The purpose of our paper is two-fold: (i) to investigate the viability of the variable speed of light (VSL) theory developed in Refs. [6, 7] in accounting for the late-time cosmic acceleration while bypassing DE, and (ii) to determine whether-and how-the finding of H 0 ≈ 46 by BDRS for the CMB can be reconciled with our reexamination of SNeIa in the VSL context.</text>
<text><location><page_2><loc_10><loc_67><loc_43><loc_69></location>Our paper is organized into four major parts:</text>
<unordered_list>
<list_item><location><page_2><loc_9><loc_62><loc_49><loc_67></location>∗ Foundation of VSL: Section II covers the history of VSL and provides a recap of our mechanism for generating varying c , as presented in [6, 7].</list_item>
<list_item><location><page_2><loc_9><loc_52><loc_49><loc_62></location>∗ Cosmography of VSL: Sections III and IV prepare the foundational material necessary for cosmography in the presence of varying c . Section V develops various modified redshift relations-Lemaître, distance vs. z , and luminosity distance vs. z -by incorporating variations in the speed of light. Importantly, we derive a modified Hubble law, applicable to our VSL scheme .</list_item>
<list_item><location><page_2><loc_9><loc_45><loc_49><loc_51></location>∗ Cosmology of VSL: Section VI presents our analysis of the Combined Pantheon Sample of SNeIa data using our modified luminosity distance vs. redshift formula derived in the preceding sections.</list_item>
<list_item><location><page_2><loc_9><loc_30><loc_49><loc_45></location>∗ Consequences of VSL: We aim for four objectives: (i) Section VII presents a new interpretation of the accelerating expansion through varying c instead of DE; (ii) Section VIII revisits BDRS's analysis of the CMB power spectrum without requiring DE and reconcile it with our findings from the VSL-based analysis of SNeIa; (iii) Section IX resolves the age problem without using DE; and (iv) Section X offers a potential resolution to the H 0 tension from an astronomical origin while avoiding dynamical DE.</list_item>
</unordered_list>
<text><location><page_2><loc_9><loc_27><loc_49><loc_30></location>Section XI discusses and summarizes our findings, and the appendices contain technical supplements.</text>
<section_header_level_1><location><page_2><loc_11><loc_21><loc_46><loc_23></location>II. A NEW MECHANISM TO GENERATE VARYING c FROM DILATON DYNAMICS</section_header_level_1>
<text><location><page_2><loc_9><loc_9><loc_49><loc_19></location>The variability in the speed of light was first recognized by Einstein in 1911 during his pursuit for a generally covariant theory of gravitation, which ultimately culminated in the theory of General Relativity (GR) in 1915. In Ref. [8] he explicitly allowed the gravitational field Φ to influence the value of c in spacetime. In particular, he proposed that c = c 0 ( 1 + Φ /c 2 ) , where c 0 is</text>
<text><location><page_2><loc_52><loc_66><loc_92><loc_94></location>the speed of light at a reference point where Φ vanishes. Notably, he conceived this radical idea six years after his formulation of Special Relativity (SR). As Einstein emphasized in [9, 10], a variation in c does not contradict the principle of the constancy of c under Lorentz transformations, an underpinning requirement of SR. This is because Lorentz invariance, confirmed by the MichelsonMorley (MM) experiment, is only required to hold in local inertial frames and does not necessitate its global validity in curved spacetimes. More concretely, in a region vicinity to a given point x ∗ , the tangent frames to the spacetime manifold possess the Lorentz symmetry with a common value of c applicable only to that region . Yet, in a spacetime influenced by a gravitational field, different regions can-in principle-correspond to different values of c . Utilizing the language of Riemannian geometry, the speed of light can be promoted to a scalar field: while c is an invariant (i.e., unaffected upon diffeomorphism), it can nonetheless be position-dependent , viz. c ( x ∗ ).</text>
<text><location><page_2><loc_52><loc_48><loc_92><loc_66></location>Einstein's pioneering concept of VSL, nevertheless, was quickly overshadowed by the success of his GR and subsequently fell into dormancy for several decades. The variability of c was briefly rediscovered by Dicke in 1957 [11], prior to his own development of Brans-Dicke gravity [12], which instead allowed Newton's gravitational constant G to vary. In the 1990s, the idea of VSL was independently revived by Moffat [13] and by Albrecht and Magueijo [14] in the context of early-time cosmology. Their proposals aimed to resolve the horizon puzzle while avoiding the need for cosmic inflation. Since then, several researchers actively explore various aspects of VSL [15-86].</text>
<text><location><page_2><loc_52><loc_39><loc_92><loc_48></location>In a recent report [6], we considered a scale-invariant action that facilitates non-minimal coupling of matter with gravity via a dilaton field χ . We uncovered a hidden mechanism that induces a dependence of c and ℏ on the dilaton field χ , thereby causing c and ℏ to vary alongside χ in spacetime. Below is a recap of our mechanism.</text>
<section_header_level_1><location><page_2><loc_58><loc_34><loc_85><loc_35></location>The essence of our VSL mechanism</section_header_level_1>
<text><location><page_2><loc_53><loc_31><loc_78><loc_33></location>Let us consider a prototype action:</text>
<formula><location><page_2><loc_55><loc_28><loc_92><loc_30></location>S = ∫ d 4 x √ -g [ L grav + L mat ] (2)</formula>
<formula><location><page_2><loc_52><loc_25><loc_92><loc_27></location>L grav = χ 2 R4 ωg µν ∂ µ χ∂ ν χ (3)</formula>
<formula><location><page_2><loc_52><loc_20><loc_92><loc_25></location>L mat = i ¯ ψγ µ ∇ µ ψ + √ α ¯ ψγ µ A µ ψ + µχ ¯ ψψ -1 4 F µν F µν (4)</formula>
<text><location><page_2><loc_52><loc_9><loc_92><loc_19></location>The gravitational sector L grav is equivalent to the wellknown Brans-Dicke theory, L BD = φ Rω φ g µν ∂ µ φ∂ ν φ , upon substituting φ := χ 2 [12]. The matter Lagrangian L mat describes quantum electrodynamics (QED) for an electron field ψ , coupled with an electromagnetic field A µ (with the field tensor defined as F µν := ∂ µ A ν -∂ ν A µ ) and embedded in a curved spacetime characterized by</text>
<text><location><page_3><loc_9><loc_86><loc_49><loc_94></location>the metric g µν . The Dirac gamma matrices satisfy γ µ γ ν + γ ν γ µ = 2 g µν , and the spacetime covariant derivative ∇ µ acts on the spinor via vierbein and spin connection. However, the electron field couples non-minimally with gravity via the dilaton field, viz. χ ¯ ψψ .</text>
<text><location><page_3><loc_9><loc_81><loc_49><loc_86></location>All parameters α , µ , and ω are dimensionless. The full action is scale invariant, viz. unchanged under the global Weyl rescaling</text>
<formula><location><page_3><loc_10><loc_79><loc_49><loc_81></location>g µν → Ω 2 g µν ; χ → Ω -1 χ ; ψ → Ω -3 2 ψ ; A µ → A µ (5)</formula>
<text><location><page_3><loc_9><loc_74><loc_49><loc_78></location>It has been established in [87, 88] that a scale-invariant action, such as the one described in Eqs. (6)-(8), can evade observational constraints on the fifth force.</text>
<text><location><page_3><loc_9><loc_67><loc_49><loc_73></location>Next, let us revisit the 'canonical' QED action for an electron field ψ carrying a U (1) gauge charge e and inertial mass m , coupled with an electromagnetic field A µ and embedded in an Einstein-Hilbert spacetime</text>
<formula><location><page_3><loc_11><loc_63><loc_49><loc_65></location>S 0 = ∫ d 4 x √ -g [ L EH + L QED ] (6)</formula>
<formula><location><page_3><loc_10><loc_59><loc_49><loc_62></location>L EH = c 3 16 π ℏ G R (7)</formula>
<formula><location><page_3><loc_9><loc_54><loc_49><loc_59></location>L QED = i ¯ ψγ µ ∇ µ ψ + e √ ℏ c ¯ ψγ µ A µ ψ + m c ℏ ¯ ψψ -1 4 F µν F µν (8)</formula>
<text><location><page_3><loc_9><loc_49><loc_49><loc_54></location>In these expressions, the quantum of action ℏ , speed of light c , and Newton's gravitational parameter G are explicitly restored.</text>
<text><location><page_3><loc_9><loc_44><loc_49><loc_49></location>Excluding the kinetic term g µν ∂ µ χ∂ ν χ of the dilaton in Eq. (3), the action S can be brought into the form S 0 via the following identification:</text>
<formula><location><page_3><loc_14><loc_40><loc_49><loc_43></location>c 3 16 π ℏ G := χ 2 ; e √ ℏ c := √ α ; m c ℏ := µχ (9)</formula>
<text><location><page_3><loc_9><loc_35><loc_49><loc_40></location>These identities link the charge e and inertial mass m of the electron with the three 'fundamental constants' c , ℏ , and G , as well as the dilaton field χ .</text>
<text><location><page_3><loc_9><loc_23><loc_49><loc_35></location>To proceed, we require that the intrinsic properties of the electron-namely, its charge e and inertial mass m -remain independent of the background spacetime, particularly the dilation field χ which belongs to the gravitational sector. Consequently, based on the last two identities of Eq. (9), both the speed of light c and the quantum of action ℏ must be treated as scalar fields related to χ . The following assignments capture this relationship:</text>
<formula><location><page_3><loc_16><loc_19><loc_49><loc_22></location>c χ := ˆ c ( χ ˆ χ ) 1 / 2 ; ℏ χ := ˆ ℏ ( χ ˆ χ ) -1 / 2 (10)</formula>
<text><location><page_3><loc_41><loc_14><loc_41><loc_15></location>/negationslash</text>
<text><location><page_3><loc_9><loc_13><loc_49><loc_19></location>Here, the subscript χ signifies the dependence of c and ℏ on χ , while ˆ c and ˆ ℏ represent the values of c χ and ℏ χ at a reference point where χ = ˆ χ (with ˆ χ = 0). It is straightforward to derive from Eqs. (9) and (10) that</text>
<formula><location><page_3><loc_13><loc_8><loc_49><loc_12></location>e = ( α ˆ ℏ ˆ c ) 1 / 2 ; m = µ ˆ ℏ ˆ χ ˆ c ; G = ˆ c 3 16 π ˆ ℏ ˆ χ 2 (11)</formula>
<text><location><page_3><loc_52><loc_90><loc_92><loc_94></location>This confirms that e and m are indeed constants 1 . Furthermore, G is also constant.</text>
<text><location><page_3><loc_52><loc_83><loc_92><loc_90></location>As the dilaton χ varies in spacetime as a component of the gravitational sector, the scalar fields c χ and ℏ χ , defined in Eq. (10), also vary in spacetime. Therefore, the dynamics of the dilaton χ induces variations in c χ and ℏ χ on the spacetime manifold.</text>
<section_header_level_1><location><page_3><loc_57><loc_79><loc_87><loc_80></location>Comments on Brans-Dicke's variable G</section_header_level_1>
<text><location><page_3><loc_52><loc_62><loc_92><loc_77></location>Traditionally, Brans-Dicke (BD) gravity is associated with variable Newton's gravitational constant G [12]. It should be noted that Brans and Dicke only allowed matter to couple minimally with gravity, namely, through the 4-volume element √ -g ; in this case, the matter action is not scale invariant. To achieve scale invariance, matter must couple with gravity in a non-minimal way, such as the Lagrangian given in Eq. (4). In this case, if one presumes that c and ℏ are constants , then the mass parameters of (massive) fields also become variable [89-92].</text>
<text><location><page_3><loc_52><loc_59><loc_92><loc_62></location>Indeed, under the assumption of constant c and ℏ , Eq. (9) readily produces</text>
<formula><location><page_3><loc_54><loc_55><loc_92><loc_58></location>e = ( α ℏ c ) 1 / 2 ; m χ = µ ℏ c χ ; G χ = c 3 16 π ℏ χ -2 (12)</formula>
<text><location><page_3><loc_52><loc_41><loc_92><loc_54></location>Here, the subscript χ signifies the dependence of m and G on χ . In [6], we referred to these results as 'the FujiiWetterich scheme', since these authors appear to be the first to report results (in [89-92]) essentially equivalent to Eq. (12). In this scheme, while m is associated with χ , the charge e remains independent of χ , rendering an unequal treatment of e and m . Moreover, whereas χ affects the electron's mass per Eq. (12), massless particles, such as photons, remain unaffected .</text>
<text><location><page_3><loc_52><loc_32><loc_92><loc_41></location>Our mechanism thus represents a significant departure from the variable G (and mass) approach. Importantly, it allows the dilaton χ -through its influence on the speed of light c χ and quantum of action ℏ χ -to govern the propagation and quantization of all fields -viz. electron and photon-on a universal and equal basis.</text>
<text><location><page_3><loc_52><loc_23><loc_92><loc_32></location>While both approaches-(i) variable G and m versus (ii) variable c and ℏ -are mathematically permissible, they are not physically equivalent [6], and the validity of each approach should be determined through empirical evidence, including predictions, experiments and observations.</text>
<text><location><page_3><loc_52><loc_16><loc_92><loc_23></location>Our mechanism leads to a direct and immediate consequence in cosmology, however. Specifically, the aspect of our mechanism where the dynamical χ induces a variation in c χ , which in turn governs massless field (viz.</text>
<text><location><page_4><loc_9><loc_83><loc_49><loc_94></location>the light quanta) has significant implications. A varying c influences the propagation of light rays emitted from distant sources toward an Earth-based observer, thereby affecting the Hubble diagram of these light sources, particularly for SNeIa. This intuition serves as the underpinning for the analysis presented in the remainder of this paper.</text>
<section_header_level_1><location><page_4><loc_11><loc_79><loc_46><loc_80></location>Scaling properties of length, time, and energy</section_header_level_1>
<text><location><page_4><loc_9><loc_70><loc_49><loc_78></location>In [6] we further deduced that at a given point x ∗ , the prevailing value of the dilaton χ ( x ∗ ) determines the lengthscale, timescale, and energy scale for physical processes occurring at that point. The lengthscale l and energy scale E are dependent on χ as follows</text>
<formula><location><page_4><loc_22><loc_67><loc_49><loc_69></location>l ∝ χ -1 ; E ∝ χ. (13)</formula>
<text><location><page_4><loc_9><loc_63><loc_49><loc_67></location>However, the most important outcome is that the timescale τ behaves in an anisotropic fashion, as</text>
<formula><location><page_4><loc_25><loc_61><loc_49><loc_63></location>τ ∝ χ -3 / 2 (14)</formula>
<text><location><page_4><loc_9><loc_58><loc_20><loc_60></location>or, equivalently</text>
<formula><location><page_4><loc_26><loc_56><loc_49><loc_57></location>τ ∝ l 3 / 2 . (15)</formula>
<text><location><page_4><loc_9><loc_47><loc_49><loc_55></location>This leads to a novel time dilation effect induced by the dilaton, representing a concrete prediction of our mechanism. Moreover, the 3 / 2-exponent in this time scaling law plays a crucial role in the Hubble diagram of SNeIa, as we will explore in the following sections.</text>
<text><location><page_4><loc_9><loc_43><loc_49><loc_47></location>A detailed exposition of our mechanism and the new time dilation effect is presented in Ref. [6].</text>
<section_header_level_1><location><page_4><loc_13><loc_37><loc_45><loc_39></location>III. IMPACTS OF VARYING c IN AN EINSTEIN-DE SITTER UNIVERSE</section_header_level_1>
<text><location><page_4><loc_9><loc_22><loc_49><loc_35></location>In a cosmology accommodating VSL, as a lightwave travels from a distant SNeIa toward an Earth-based observer, a varying speed of light along its trajectory induces a refraction effect akin to that experienced by a physical wave traveling through an inhomogenous medium with varying wave speed. The alteration of the wavelength results in a new set of cosmographic formulae, including a modified Hubble law and a modified relationship between redshift and luminosity distance.</text>
<section_header_level_1><location><page_4><loc_9><loc_18><loc_49><loc_19></location>A. A drawback in previous VSL analyses of SNeIa</section_header_level_1>
<text><location><page_4><loc_9><loc_9><loc_49><loc_16></location>It is important to note that since the revival of VSL by Moffat and the Albrecht-Magueijo team in the 1990s, several authors have applied VSL to late-time cosmology, particularly in the analysis of the Hubble diagram of SNeIa. However, these attempts have not</text>
<text><location><page_4><loc_52><loc_62><loc_92><loc_94></location>met with much success [15, 16, 33-37]. A common theme among these analyses is the assumption that c varies as a function of the global cosmic factor a of the Friedmann-Lemaître-Roberson-Walker (FLRW) metric (e.g., in the form c ∝ a -ζ first proposed by Barrow [15]). These works generally conclude that, despite the dependence of c on a , VSL does not alter the classic Lemaître redshift formula 1 + z = a -1 and, therefore, cannot play any role in the Hubble diagram of SNeIa. However, upon closer scrutiny into these works, we identify a significant oversight: they implicitly assumed that c is a function solely of cosmic time t , through the dependence of a on t in the FLRW metric. This assumption is not valid in our VSL framework, where c -through its dependence on the dilaton field χ -varies in both space and time, rather than time alone. In this section, as well as Sections IV and V, we will demonstrate that the variation of c as a function of the dilaton field χ , rather than merely as a function of the cosmic factor (viz. a ) as assumed in prior VSL works, fundamentally alters the Lemaître redshift formula and necessitates a re-analysis of SNeIa data.</text>
<section_header_level_1><location><page_4><loc_60><loc_57><loc_84><loc_58></location>B. The modified FLRW metric</section_header_level_1>
<text><location><page_4><loc_52><loc_53><loc_92><loc_56></location>The FLRW metric for the isotropic and homogeneous intergalactic space reads</text>
<formula><location><page_4><loc_58><loc_48><loc_92><loc_51></location>ds 2 = c 2 dt 2 -a 2 ( t ) [ dr 2 1 -κr 2 + r 2 d Ω 2 ] (16)</formula>
<text><location><page_4><loc_52><loc_44><loc_92><loc_47></location>where a ( t ) is the global cosmic scale factor that evolves with cosmic time t .</text>
<text><location><page_4><loc_52><loc_36><loc_92><loc_44></location>Our goal is to investigate whether an Einstein-de Sitter universe, when supplemented with a varying c , can account for the Hubble diagram of SNeIa as provided by the Pantheon Catalog. We will make three working assumptions:</text>
<text><location><page_4><loc_52><loc_31><loc_92><loc_36></location>Assumption #1: The FLRW universe is spatially flat, corresponding to κ = 0. There is robust observational evidence supporting this assumption.</text>
<text><location><page_4><loc_53><loc_29><loc_92><loc_30></location>Assumption #2: The cosmic scale factor evolves as</text>
<formula><location><page_4><loc_66><loc_24><loc_92><loc_27></location>a = a 0 ( t t 0 ) 2 / 3 (17)</formula>
<text><location><page_4><loc_52><loc_9><loc_92><loc_23></location>Justification: In our VSL mechanism, the timescale τ and lengthscale l of a given physical process are related by τ ∝ l 3 / 2 , as expressed in Eq. (15). Regarding the evolution of the FLRW metric, its timescale and lengthscale can be identified with t and a , respectively. The growth law given in Eq. (17) is therefore justified. Note: This growth is identical to the evolution of an EdS universe, viz. a spatially flat, expanding universe consisting solely of matter, with no contribution from DE or a cosmological constant.</text>
<text><location><page_5><loc_9><loc_90><loc_49><loc_94></location>Assumption #3: The dilaton field in the cosmic background depends on the cosmic factor in the form</text>
<formula><location><page_5><loc_26><loc_88><loc_49><loc_89></location>χ ∝ a -1 . (18)</formula>
<text><location><page_5><loc_9><loc_79><loc_49><loc_87></location>Justification: In our VSL mechanism, the lengthscale of a given physical process is inversely proportional to the dilaton field, per Eq. (13). Given that the cosmic factor a plays the role of the lengthscale for the FLRW metric, the dependency expressed in Eq. (18) is therefore justified.</text>
<text><location><page_5><loc_9><loc_75><loc_49><loc_79></location>Combining Eqs. (10) and (18) then renders c ∝ a -1 / 2 , or more explicitly</text>
<formula><location><page_5><loc_23><loc_71><loc_49><loc_74></location>c = c 0 ( a a 0 ) -1 / 2 (19)</formula>
<text><location><page_5><loc_9><loc_57><loc_49><loc_70></location>Here, a 0 is the current cosmic scale factor (often set equal 1), and c 0 is the speed of light measured at our current time in the intergalactic space. We should emphasize that the value of c 0 is not identical with the one measured inside the Milky Way , which is equal to 300 , 000 km/s. This is because the Milky Way is a gravitationally bound structure whereas the intergalactic space is regions subject to cosmic expansion. This issue will be explained in Section IV.</text>
<text><location><page_5><loc_9><loc_53><loc_49><loc_56></location>Combining Eqs. (16) and (19), and setting κ = 0, we then obtain the modified FLRW metric</text>
<formula><location><page_5><loc_16><loc_48><loc_49><loc_51></location>ds 2 = c 2 0 a 0 a ( t ) dt 2 -a 2 ( t ) [ dr 2 + r 2 d Ω 2 ] (20)</formula>
<text><location><page_5><loc_9><loc_44><loc_49><loc_48></location>which describes an EdS universe with a declining speed of light, per c ∝ a -1 / 2 .</text>
<section_header_level_1><location><page_5><loc_21><loc_40><loc_36><loc_41></location>C. Frequency shift</section_header_level_1>
<text><location><page_5><loc_9><loc_34><loc_49><loc_39></location>For the modified FLRW metric derived above, the null geodesic ( ds 2 = 0) for a lightwave traveling from a distant emitter toward Earth (viz. d Ω = 0) is</text>
<formula><location><page_5><loc_22><loc_30><loc_49><loc_33></location>c 0 a 1 / 2 0 dt a 3 / 2 ( t ) = dr (21)</formula>
<text><location><page_5><loc_9><loc_21><loc_49><loc_29></location>Hereafter, we will use the subscripts ' em ' and ' ob ' for 'emission' and 'observation' respectively. Denote t em and t ob the emission and observation time points of the lightwave, and r em the co-moving distance of the emitter from Earth. From (21), we have:</text>
<formula><location><page_5><loc_20><loc_17><loc_49><loc_20></location>c 0 a 1 / 2 0 ∫ t ob t em dt a 3 / 2 ( t ) = r em (22)</formula>
<text><location><page_5><loc_9><loc_13><loc_49><loc_16></location>The next wavecrest to leave the emitter at t em + δt em and arrive at Earth at t ob + δt ob satisfies:</text>
<formula><location><page_5><loc_18><loc_8><loc_49><loc_12></location>c 0 a 1 / 2 0 ∫ t ob + δt ob t em + δt em dt a 3 / 2 ( t ) = r em (23)</formula>
<text><location><page_5><loc_52><loc_92><loc_80><loc_94></location>Subtracting these two equations yields:</text>
<formula><location><page_5><loc_64><loc_88><loc_92><loc_91></location>δt ob a 3 / 2 ( t ob ) = δt em a 3 / 2 ( t em ) (24)</formula>
<text><location><page_5><loc_52><loc_84><loc_92><loc_87></location>which leads to the ratio between the emitted frequency and the observed frequency:</text>
<formula><location><page_5><loc_58><loc_79><loc_92><loc_83></location>ν ob ν em = δt em δt ob = a 3 / 2 ( t em ) a 3 / 2 ( t ob ) = ( a em a ob ) 3 / 2 (25)</formula>
<text><location><page_5><loc_52><loc_72><loc_92><loc_78></location>This contrasts with the standard relation, ν ob ν em = a em a ob . To derive a Lemaître formula applicable for VSL, further consideration is needed. This task will be carried out in the next section.</text>
<section_header_level_1><location><page_5><loc_55><loc_65><loc_88><loc_68></location>IV. IMPACTS OF VARYING c ACROSS BOUNDARIES OF GALAXIES</section_header_level_1>
<text><location><page_5><loc_52><loc_59><loc_92><loc_64></location>This section presents the pivotal elements that enable the 3 / 2-exponent in the frequency ratio, as expressed in Eq. (25), to manifest in observations.</text>
<section_header_level_1><location><page_5><loc_55><loc_55><loc_89><loc_56></location>A. The loss of validity of Lemaître formula</section_header_level_1>
<text><location><page_5><loc_52><loc_49><loc_92><loc_54></location>Let us first revisit the drawback in previous VSL works alluded to in Section III A. The frequency ratio given by Eq. (25) can be converted into the wavelength ratio</text>
<formula><location><page_5><loc_58><loc_44><loc_92><loc_48></location>λ ob λ em = c ob c em . ν em ν ob = a -1 / 2 ob a -1 / 2 em . a 3 / 2 ob a 3 / 2 em = a ob a em (26)</formula>
<text><location><page_5><loc_52><loc_37><loc_92><loc_43></location>This expression is exactly identical to that in standard cosmology, viz. where c is non-varying. At first, it may seem tempting to relate the redshift z with λ ob -λ em λ em , namely</text>
<formula><location><page_5><loc_62><loc_33><loc_92><loc_36></location>1 + z ? = λ ob λ em = a ob a em = a -1 (27)</formula>
<text><location><page_5><loc_52><loc_23><loc_92><loc_33></location>in which a ob is set equal 1 and a em is denoted as a . In Refs. [16, 33-37], based on Eq. (27), it was concluded that the classic Lemaître redshift formula, 1 + z = a -1 , remained valid. Subsequently, virtually all empirical VSL works continued using the classic Lemaître formula to analyze the Hubble diagram of SNeIa.</text>
<text><location><page_5><loc_52><loc_9><loc_92><loc_23></location>However, the formula in Eq. (27) is incorrect . One key reason is that λ ob , representing the wavelength in the intergalactic space enclosing the Milky Way, is not what the Earth-based astronomer directly measures. For the light wave to reach the astronomer's telescope, it must pass through the gravitationally-bound Milky Way, which has its own local scale ˆ a ob differing from the current global cosmic scale because the matter-populated Milky Way resists cosmic expansion. This crucial point will be clarified shortly in the section below. In brief, a</text>
<figure>
<location><page_6><loc_7><loc_71><loc_92><loc_93></location>
<caption>Figure 1. A lightwave from an SNeIa explosion (shown on the far left) makes three transits to reach the Earth-based astronomer (shown on the far right). During Transit #1, the lightwave exits the (gravitationally-bound) host galaxy to enter the surrounding intergalactic region; the wavecrest gets compressed as light speed decreases at this juncture. During Transit #2, the lightwave travels in the intergalactic space which undergoes cosmic expansion; accordingly, the wavecrest expands. During Transit #3, the lightwave enters the (gravitationally-bound) Milky Way; the wavecrest expands further as light speed increases at this juncture. The Earth-based astronomer measures the wavelength ˆ λ ob and compares it with the 'benchmark' wavelength λ ∗ (see text for explanation) to calculate the redshift z (shown in the lower right corner box).</caption>
</figure>
<text><location><page_6><loc_9><loc_50><loc_49><loc_56></location>change in scale (from global to local) across the boundary of the Milky Way induces a corresponding change in the speed of light. This effect alters the wavelength from λ ob to ˆ λ ob which is then measured by the astronomer.</text>
<section_header_level_1><location><page_6><loc_12><loc_44><loc_46><loc_47></location>B. Refractive effect due to varying c across boundaries of galaxies</section_header_level_1>
<text><location><page_6><loc_9><loc_13><loc_49><loc_43></location>The global cosmic scale factor a grows with comic time t , leading to the stretching of wavelength of light from λ em to λ ob . However, the Solar System is not subject to cosmic expansion, which is a crucial condition so that the Earth-based observer can detect the redshift of distant emission sources. It is well understood that if the Solar System expanded along with the intergalactic space, the observer's instruments would also expand in sync with the wavelength of the light ray emitted from a distant supernova, making the detection of any redshift impossible. More generally, mature galaxies-those hosting distant SNeIa, and the Milky Way where the Earth-based observer resides, are gravitationally bound and resist cosmic expansion. Despite the expansion of intergalactic space, matured galaxies maintain their relatively stable size primarily through gravitational attraction, counterbalanced by the rotational motion of the matter within them. Consequently, each galaxy has a stable local scale ˆ a that remains relatively constant over time, despite increases in the global scale a .</text>
<text><location><page_6><loc_9><loc_9><loc_49><loc_13></location>As discussed in Section III B, the dilaton field χ in intergalactic space is inversely proportional to the global scale factor a . Similarly, within a galaxy, the dilaton</text>
<text><location><page_6><loc_52><loc_50><loc_92><loc_56></location>field is inversely proportional to the galaxy's local scale ˆ a . Since a grows over time whereas ˆ a remains relatively stable, the dilaton field declines in the intergalactic space while it remains largely unchanged within galaxies.</text>
<text><location><page_6><loc_52><loc_31><loc_92><loc_50></location>For simplicity, we model the local scale ˆ a as homogeneous within a galaxy, and allow it to merge with the global scale a at the galaxy's boundary. Importantly, the local scale of the Milky Way may differ from the local scale of the galaxy hosting a specific SNeIa being observed. This is because a gravitationally bound galaxy lives on an FLRW cosmic background that is expanding, rather than static. As a result, its local scale ˆ a might, in principle, experience modest growth in response to increases in the global scale a . Therefore, it is reasonable to model the local scale ˆ a of a galaxy as a universal function (to be determined) of the redshift z of the galaxy, supplemented by a negligible idiosyncratic component.</text>
<text><location><page_6><loc_52><loc_21><loc_92><loc_30></location>Consequently, as the dilaton field χ varies across the boundaries of galaxies, the speed of light also varies at the boundaries due to the relationship c ∝ χ 1 / 2 . Figures 1 depicts an intuitive schematic of a lightwave emitted from an SNeIa as it propagates to the Earth-based observer. On its journey, the lightwave undergoes 3 transits:</text>
<unordered_list>
<list_item><location><page_6><loc_52><loc_14><loc_92><loc_21></location>* Transit #1: The lightwave emitted from an SNeIa residing 'inside host galaxy', which is gravitationally bound and characterized by a local scale ˆ a em , must first exit into the surrounding intergalactic space, characterized by a global scale a em .</list_item>
<list_item><location><page_6><loc_52><loc_9><loc_92><loc_13></location>* Transit #2: The lightwave then traverses the null geodesic of the FLRW metric and expands along with the cosmic scale factor a ( t ) of the intergalactic space until it</list_item>
</unordered_list>
<text><location><page_7><loc_9><loc_90><loc_49><loc_94></location>reaches the outskirts of the Milky Way, where the global scale is a ob .</text>
<unordered_list>
<list_item><location><page_7><loc_9><loc_84><loc_49><loc_90></location>* Transit #3: The lightwave enters the Milky Way, which is gravitationally bound and characterized by a local scale ˆ a ob , and finally reaches the Earth-based astronomer's telescope.</list_item>
</unordered_list>
<text><location><page_7><loc_9><loc_73><loc_49><loc_84></location>While the middle stage of this journey, Transit #2, is well understood in standard cosmology, the first and last stages have been overlooked in previous VSL studies, seriously undermining their analyses and conclusions. In the context of VSL, these stages are crucial due to the additional refraction effects that occur at the boundaries of the host galaxy and the Milky Way.</text>
<text><location><page_7><loc_9><loc_60><loc_49><loc_72></location>Figure 2 illustrates the typical behavior of χ -1 , c , and the wavelength λ along a lightwave trajectory. In the top panel, it can be expected that a em > ˆ a em (since the host galaxy resists cosmic expansion), a ob > a em (due to the expansion of intergalactic space), and ˆ a ob < a ob (since the Milky Way also resists cosmic expansion). Quantitatively, we can deduce the variation of wavelength during the three transits as follows:</text>
<unordered_list>
<list_item><location><page_7><loc_11><loc_55><loc_49><loc_58></location>· The emission event: an SNeIa radiates a wavetrain with a specific wavelength ˆ λ em .</list_item>
<list_item><location><page_7><loc_11><loc_47><loc_49><loc_54></location>· Transit #1: The wavetrain exits the host galaxy to enter the surrounding intergalactic space. During this transition, its wavelength is compressed to λ em due to the reduction in the speed of light from ˆ c em to c em across the host galaxy's boundary, viz.</list_item>
</unordered_list>
<formula><location><page_7><loc_22><loc_42><loc_49><loc_45></location>λ em ˆ λ em = c em ˆ c em = a -1 / 2 em ˆ a -1 / 2 em (28)</formula>
<text><location><page_7><loc_13><loc_36><loc_49><loc_41></location>Appendix A summarizes the components involved in the refraction that is induced by variations in the velocity of wavetrains.</text>
<unordered_list>
<list_item><location><page_7><loc_11><loc_27><loc_49><loc_35></location>· Transit #2: The wavetrain follows the null geodesics of the FLRW metric. As it approaches the outskirts of the Milky Way, its wavelength has expanded from λ em to λ ob , as given in Eq. (26), viz.</list_item>
</unordered_list>
<formula><location><page_7><loc_24><loc_23><loc_49><loc_26></location>λ ob λ em = a ob a em (see Eq. (26))</formula>
<unordered_list>
<list_item><location><page_7><loc_11><loc_14><loc_49><loc_21></location>· Transit #3: The wavetrain enters the Milky Way and reaches the astronomer's telescope. Its wavelength is further prolonged due to an increase in the speed of light from c ob to ˆ c ob across the Milky Way's boundary, viz.</list_item>
</unordered_list>
<formula><location><page_7><loc_22><loc_8><loc_49><loc_13></location>ˆ λ ob λ ob = ˆ c ob c ob = ˆ a -1 / 2 ob a -1 / 2 ob . (29)</formula>
<figure>
<location><page_7><loc_51><loc_41><loc_92><loc_93></location>
<caption>Figure 2. Variations of inverse dilaton (upper panel), speed of light (middle panel), and wavelength (lower panel) along the lightwave trajectory from emission to observation.</caption>
</figure>
<section_header_level_1><location><page_7><loc_59><loc_30><loc_85><loc_31></location>C. The 'benchmark' wavelength</section_header_level_1>
<text><location><page_7><loc_52><loc_9><loc_92><loc_28></location>There is one more crucial element to consider. In calculating the redshift of an SNeIa, it would be incorrect to directly compare the observed wavelength ˆ λ ob with the emitted wavelength ˆ λ em . This is because ˆ λ em is associated with the emission event occurring inside the host galaxy, and the observer cannot directly measure ˆ λ em since she is located within the Milky Way. If the SNeIa were situated inside the Milky Way, it would emit a wavelength λ ∗ that differs from ˆ λ em , as the two galaxies can have different values of local scales, ˆ a em versus ˆ a ob . The wavelength λ ∗ , which the observer can measure, is the 'benchmark' wavelength to be compared with the observed ˆ λ ob in calculating the redshift.</text>
<text><location><page_8><loc_9><loc_77><loc_49><loc_94></location>To illustrate this issue, let us recall that the lengthscale of any physical process is inversely proportional to the dilaton field, according to Eq. (13) in Section II. For a galaxy, the dilaton field is in turn inversely proportional to the local scale of that galaxy. Consider two identical atoms, one located inside the host galaxy and the other within the Milky Way. If the atom in the host galaxy emits a lightwave with wavelength ˆ λ em , its counterpart in the Milky Way emits an identical lightwave but with wavelength λ ∗ adjusted to the Milky Way's local scale. The following equality holds</text>
<formula><location><page_8><loc_25><loc_73><loc_49><loc_77></location>λ ∗ ˆ λ em = ˆ a ob ˆ a em (30)</formula>
<text><location><page_8><loc_9><loc_65><loc_49><loc_72></location>As shown in the right side of Figure 1, the observer must compare ˆ λ ob with her 'benchmark' wavelength λ ∗ . Finally, the observer calculates the redshift z as the relative change between the observed wavelength and the 'benchmark' wavelength, given by</text>
<formula><location><page_8><loc_24><loc_60><loc_49><loc_64></location>z := ˆ λ ob -λ ∗ λ ∗ . (31)</formula>
<text><location><page_8><loc_9><loc_55><loc_49><loc_60></location>We should note that allowing ˆ a ob to differ from ˆ a em creates a potential pathway to resolving the H 0 tension, a topic that will be discussed in Section X.</text>
<section_header_level_1><location><page_8><loc_9><loc_50><loc_48><loc_52></location>V. MODIFYING REDSHIFT FORMULAE AND HUBBLE LAW USING VARYING c</section_header_level_1>
<text><location><page_8><loc_9><loc_45><loc_49><loc_48></location>We are now fully equipped to derive cosmographic formulae applicable to our VSL cosmology.</text>
<section_header_level_1><location><page_8><loc_12><loc_41><loc_46><loc_42></location>A. Modifying the Lemaître redshift formula</section_header_level_1>
<text><location><page_8><loc_9><loc_30><loc_49><loc_40></location>What is remarkable in the demonstration depicted in Figure 1 is that the stretching of the wavecrest during Transit #3 does not cancel out the compression of the wavecrest during Transit #1. The net effect of the two transits increases the value of z and results in a new formula for the redshift. Below is our derivation.</text>
<text><location><page_8><loc_9><loc_27><loc_49><loc_30></location>Combining Eq. (31) with Eqs. (28), (26), and (29), we obtain</text>
<formula><location><page_8><loc_11><loc_22><loc_49><loc_27></location>1 + z = ˆ λ ob λ ∗ = ˆ λ ob λ ob . λ ob λ em . λ em ˆ λ em . ˆ λ em λ ∗ = a 3 / 2 ob a 3 / 2 em . ˆ a 3 / 2 em ˆ a 3 / 2 ob (32)</formula>
<text><location><page_8><loc_9><loc_18><loc_49><loc_22></location>Defining the ratio of local scales as a function of redshift:</text>
<formula><location><page_8><loc_24><loc_15><loc_49><loc_18></location>ˆ a em ˆ a ob := F ( z ) (33)</formula>
<text><location><page_8><loc_9><loc_12><loc_32><loc_14></location>where F ( z = 0) = 1, and setting</text>
<formula><location><page_8><loc_26><loc_8><loc_49><loc_11></location>a := a em a ob (34)</formula>
<text><location><page_8><loc_52><loc_92><loc_89><loc_94></location>we arrive at the modified Lemaître redshift formula :</text>
<formula><location><page_8><loc_64><loc_89><loc_92><loc_91></location>1 + z = a -3 / 2 F 3 / 2 ( z ) (35)</formula>
<text><location><page_8><loc_52><loc_84><loc_92><loc_88></location>If F ( z ) ≡ 1 ∀ z , viz. all galaxies have the same local scale, the modified Lemaître redshift formula simplifies to:</text>
<formula><location><page_8><loc_67><loc_81><loc_92><loc_83></location>1 + z = a -3 / 2 (36)</formula>
<text><location><page_8><loc_52><loc_74><loc_92><loc_80></location>These formulae are decisively different from the classic Lemaître redshift formula, 1 + z = a -1 . The 3 / 2exponent in the modified Lemaître formulae arises as a result of the anisotropic time scaling in Eq. (15).</text>
<text><location><page_8><loc_52><loc_60><loc_92><loc_74></location>It is essential to emphasize that the alteration in wavelength-due to the refraction effect across boundaries of galaxies-is instrumental in enabling the VSL effects to manifest in the modified Lemaître redshift formula. To the best of our knowledge, existing VSL analyses in the literature have not considered this wavelength alteration. This omission hinders theirs ability to detect the effects of VSL on the Hubble diagram of SNeIa and late-time cosmic acceleration.</text>
<section_header_level_1><location><page_8><loc_55><loc_54><loc_89><loc_56></location>B. Modifying the Hubble law: An emergent multiplicative factor of 3 / 2</section_header_level_1>
<text><location><page_8><loc_53><loc_51><loc_90><loc_52></location>The current-time Hubble constant H 0 is defined as</text>
<formula><location><page_8><loc_66><loc_46><loc_92><loc_50></location>H 0 := 1 a da dt | t = t 0 (37)</formula>
<text><location><page_8><loc_52><loc_44><loc_80><loc_45></location>For a lowz emission source, this yields</text>
<formula><location><page_8><loc_63><loc_41><loc_92><loc_42></location>a = 1 + H 0 ( t -t 0 ) + . . . (38)</formula>
<text><location><page_8><loc_52><loc_32><loc_92><loc_39></location>Let d = c 0 . ( t 0 -t ) represent the distance from Earth to the emission source, and note that F ( z ) /similarequal 1 for low z . For small z and d , the Taylor expansion for the modified Lemaî redshift formula obtained in Eq. (35) produces the modified Hubble law :</text>
<formula><location><page_8><loc_68><loc_27><loc_92><loc_31></location>z = 3 2 H 0 d c 0 (39)</formula>
<text><location><page_8><loc_52><loc_23><loc_92><loc_26></location>In comparison to the classic Hubble law, where the speed of light is explicitly restored:</text>
<formula><location><page_8><loc_66><loc_18><loc_92><loc_21></location>z (classic) = H 0 d c (40)</formula>
<text><location><page_8><loc_52><loc_9><loc_92><loc_18></location>the modified Hubble law acquires a multiplicative prefactor of 3/2. A significant consequence of this adjustment is a (re)-evaluation of the Hubble constant H 0 , which has implications for BDRS's CMB analysis and the age problem-topics that will be discussed in Sections VIII and IX.</text>
<section_header_level_1><location><page_9><loc_12><loc_92><loc_46><loc_93></location>C. Modifying the distance-redshift formula</section_header_level_1>
<text><location><page_9><loc_9><loc_87><loc_49><loc_90></location>Using the evolution a ∝ t 2 / 3 per Eq. (17), we can derive that</text>
<formula><location><page_9><loc_18><loc_83><loc_49><loc_87></location>H ( t ) = 1 a da dt = 2 3 t = H 0 a -3 / 2 (41)</formula>
<text><location><page_9><loc_9><loc_79><loc_49><loc_83></location>The modified Lemaître redshift formula, Eq. (35), can be recast as</text>
<formula><location><page_9><loc_21><loc_75><loc_49><loc_79></location>ln (1 + z ) 2 / 3 F = -ln a (42)</formula>
<text><location><page_9><loc_9><loc_73><loc_36><loc_75></location>and, with the aid of Eq. (41), renders</text>
<formula><location><page_9><loc_18><loc_69><loc_49><loc_72></location>d ln (1 + z ) 2 / 3 F = -H 0 a -3 / 2 dt (43)</formula>
<text><location><page_9><loc_9><loc_65><loc_49><loc_68></location>For the modified FLRW metric described in Eq. (20), the coordinate distance in flat space is</text>
<formula><location><page_9><loc_22><loc_60><loc_49><loc_64></location>r = c 0 ∫ t ob t em dt a 3 / 2 ( t ) (44)</formula>
<text><location><page_9><loc_9><loc_55><loc_49><loc_59></location>From Eqs. (43) and (44), and noting that F ( z = 0) = 1, we obtain the modified distance-redshift formula in a compact expression</text>
<formula><location><page_9><loc_21><loc_50><loc_49><loc_54></location>r c 0 = 1 H 0 ln (1 + z ) 2 / 3 F . (45)</formula>
<section_header_level_1><location><page_9><loc_11><loc_45><loc_47><loc_47></location>D. Modifying the luminosity distance-redshift formula</section_header_level_1>
<text><location><page_9><loc_9><loc_39><loc_49><loc_44></location>In standard cosmology, the luminosity distance d L is defined via the absolute luminosity L and the apparent luminosity J :</text>
<formula><location><page_9><loc_25><loc_35><loc_49><loc_38></location>d 2 L = L 4 πJ (46)</formula>
<text><location><page_9><loc_9><loc_31><loc_49><loc_34></location>The absolute luminosity L and the apparent luminosity J are related as</text>
<formula><location><page_9><loc_22><loc_27><loc_49><loc_31></location>4 πr 2 J = L ˆ λ em ˆ λ ob . ˆ λ em ˆ λ ob (47)</formula>
<text><location><page_9><loc_9><loc_19><loc_49><loc_26></location>In the right hand side of Eq. (47), the first term ˆ λ em / ˆ λ ob represents the 'loss' in energy of the redshifted photon known as the 'Doppler theft' 2 . The second (identical) term ˆ λ em / ˆ λ ob arises from the dilution factor in photon</text>
<text><location><page_9><loc_52><loc_86><loc_92><loc_94></location>density, as the same number of photons is distributed over a prolonged wavecrest in the radial direction (i.e., along the light ray). The 4 πr 2 in the left hand side of Eq. (47) accounts for the spherical dilution in flat space. From (46) and (47), we obtain</text>
<formula><location><page_9><loc_68><loc_82><loc_92><loc_86></location>d L = r ˆ λ ob ˆ λ em (48)</formula>
<text><location><page_9><loc_52><loc_77><loc_92><loc_81></location>Using the definitions of redshift and the 'benchmark' wavelength, Eqs. (31) and (30) respectively, the luminosity distance becomes</text>
<formula><location><page_9><loc_60><loc_72><loc_92><loc_76></location>d L = r ˆ λ ob λ /star . λ /star ˆ λ em = r (1 + z ) ˆ a ob ˆ a em (49)</formula>
<text><location><page_9><loc_52><loc_70><loc_67><loc_72></location>or, by including (33):</text>
<formula><location><page_9><loc_65><loc_66><loc_92><loc_70></location>d L = r (1 + z ) 1 F ( z ) (50)</formula>
<text><location><page_9><loc_52><loc_61><loc_92><loc_66></location>Due to the refraction effect during Transit #3, the apparent luminosity distance observed by the Earth-based astronomer ˆ d L differs from d L by the factor ˆ c ob /c ob , viz.</text>
<formula><location><page_9><loc_69><loc_56><loc_92><loc_59></location>ˆ d L ˆ c ob = d L c ob (51)</formula>
<text><location><page_9><loc_52><loc_52><loc_92><loc_55></location>Finally, combining Eqs. (45), (50), and (51), we arrive at the modified luminosity distance-redshift relation:</text>
<formula><location><page_9><loc_62><loc_48><loc_92><loc_52></location>ˆ d L ˆ c ob = 1 + z H 0 F ( z ) ln (1 + z ) 2 / 3 F ( z ) (52)</formula>
<text><location><page_9><loc_52><loc_38><loc_92><loc_47></location>where ˆ d L is the luminosity distance observed by the Earth-based astronomer and ˆ c ob the speed of light measured in the Milky Way (i.e., 300 , 000 km/s). Formula (52) contains a single parameters H 0 and involves a function F ( z ) that captures the evolution of the local scale of galaxies as a function of redshift.</text>
<section_header_level_1><location><page_9><loc_53><loc_32><loc_91><loc_35></location>VI. RE-ANALYZING PANTHEON CATALOG USING VARYING c</section_header_level_1>
<text><location><page_9><loc_52><loc_15><loc_92><loc_31></location>This section applies the new formula, Eq. (52), to the Combined Pantheon Sample of SNeIa. In [93], Scolnic and collaborators produced a dataset of apparent magnitudes for 1 , 048 SNeIa with redshift z ranging from 0 . 01 to 2 . 25, accessible in [94]. For each SNeIa i th , the catalog provides the redshift z i , the apparent magnitude m Pantheon B,i together with its error bar σ Pantheon i . We apply the absolute magnitude M = -19 . 35 to compute the distance modulus, µ Pantheon := m Pantheon B -M . The distance modulus is then converted to the luminosity distance d L using the following relation:</text>
<formula><location><page_9><loc_63><loc_12><loc_92><loc_14></location>µ = 5log 10 ( d L / Mpc) + 25 (53)</formula>
<text><location><page_9><loc_52><loc_9><loc_92><loc_12></location>The Pantheon data, along with their error bars, are displayed in the Hubble diagram shown in Fig. 3.</text>
<section_header_level_1><location><page_10><loc_11><loc_91><loc_47><loc_93></location>A. Λ CDM and standard EdS as benchmarking models</section_header_level_1>
<text><location><page_10><loc_9><loc_83><loc_49><loc_89></location>For benchmarking purposes, we first fit the Pantheon Catalog with the flat ΛCDM model. The luminosity distance-redshift relation for this model is a wellestablished result (where Ω M +Ω Λ = 1)</text>
<formula><location><page_10><loc_16><loc_79><loc_49><loc_82></location>d L c = 1 + z H 0 ∫ z 0 dz ' √ Ω M (1 + z ' ) 3 +Ω Λ (54)</formula>
<text><location><page_10><loc_9><loc_76><loc_39><loc_78></location>Our fit will minimize the normalized error</text>
<formula><location><page_10><loc_16><loc_71><loc_49><loc_75></location>χ 2 := 1 N N ∑ j =1 ( µ model j -µ Pantheon j σ Pantheon j ) 2 (55)</formula>
<text><location><page_10><loc_9><loc_62><loc_49><loc_71></location>with the sum taken over all N = 1 , 048 Pantheon data points. The best fit for the ΛCDM model yields H 0 = 70 . 2 km/s/Mpc, Ω M = 0 . 285, Ω Λ = 0 . 715, with the minimum error χ 2 min (ΛCDM) = 0 . 98824. The d L -z curve for the ΛCDM model is depicted by the dashed line in Fig. 3.</text>
<text><location><page_10><loc_9><loc_54><loc_49><loc_61></location>Also for benchmarking purposes, we consider a 'fiducial' model: the standard EdS universe (i.e. with constant speed of light). The luminosity distance-redshift formula for this fiducial model can be obtained by setting Ω Λ = 0 and Ω M = 1 in Eq. (54), yielding</text>
<formula><location><page_10><loc_19><loc_50><loc_49><loc_53></location>d L c = 2 1 + z H 0 ( 1 -1 √ 1 + z ) (56)</formula>
<text><location><page_10><loc_9><loc_32><loc_49><loc_49></location>Figure 3 displays the d l -z curve as a dotted line for the fiducial EdS model (using the H 0 = 70 . 2 value obtained above for the ΛCDM model). This curve fits well with the Pantheon data for low z but fails to capture the data for high z . The Pantheon data with z ≳ 0 . 1 show an excess in the distance modulus compared with the baseline EdS model, meaning that highredshift SNeIa appear dimmer than predicted by the fiducial EdS model. As a result, this discrepancy necessitated the introduction of the Λ component, commonly referred to as dark energy, characterized by an equation of state w = -1 and an energy density of Ω Λ ≈ 0 . 7.</text>
<section_header_level_1><location><page_10><loc_12><loc_28><loc_46><loc_29></location>B. Fitting with VSL model: Disabling F ( z )</section_header_level_1>
<text><location><page_10><loc_9><loc_20><loc_49><loc_26></location>In this subsection, we will disable the evolution of the local scale of galaxies by setting F ( z ) ≡ 1 in Formula (52). This means that the fit is carried out with respect to a simplified formula with one adjustable parameter H 0 :</text>
<formula><location><page_10><loc_21><loc_14><loc_49><loc_18></location>ˆ d L ˆ c ob = 1 + z 3 2 H 0 ln(1 + z ) (57)</formula>
<text><location><page_10><loc_9><loc_9><loc_49><loc_14></location>Hereafter, the luminosity distance ˆ d L observed by the Earth-based astronomer will be used in the conversion described by Eq. (53).</text>
<figure>
<location><page_10><loc_54><loc_69><loc_89><loc_94></location>
<caption>Figure 3. Fitting Pantheon using Formula (57). Open circles: 1,048 data points with error bars. Solid line: our Formula (57) with H 0 = 44 . 4. Dashed line: ΛCDM Formula (54) with H 0 = 70 . 2, Ω Λ = 0 . 715. Dotted line: EdS Formula (56) with H 0 = 70 . 2.</caption>
</figure>
<text><location><page_10><loc_52><loc_48><loc_92><loc_59></location>The best fit of the Pantheon data to this formula yields H 0 = 44 . 4 km/s/Mpc, corresponding to χ 2 min = 1 . 25366. Figure 3 displays our fit as the solid line. Although this fit performs worse than the ΛCDM model, which has χ 2 min (ΛCDM) = 0 . 98824, it substantially reduces the excess in distance moduli for z ≳ 0 . 1 compared with the 'fiducial' EdS model, as shown in Fig. 3.</text>
<text><location><page_10><loc_52><loc_36><loc_92><loc_48></location>We must emphasize that both Formulae (56) and (57) are one-parameter models. Both models are based on an EdS universe, but our VSL model accommodates varying speed of light, whereas the 'fiducial' EdS model operates under the assumption of a constant speed of light. Therefore, we can conclude that varying speed of light is responsible for the improved performance of our VSL model compared to the 'fiducial' EdS model.</text>
<text><location><page_10><loc_52><loc_33><loc_92><loc_36></location>This aspect can be explained as follows. In the high z limit, Formula (56) of the 'fiducial' EdS model yields</text>
<formula><location><page_10><loc_69><loc_30><loc_92><loc_32></location>d L /similarequal z (58)</formula>
<text><location><page_10><loc_52><loc_28><loc_85><loc_29></location>whereas Formula (57) of our VSL model gives</text>
<formula><location><page_10><loc_66><loc_25><loc_92><loc_27></location>d L ∝ ˆ d L /similarequal z ln z (59)</formula>
<text><location><page_10><loc_52><loc_15><loc_92><loc_24></location>The additional ln z term in Eq. (59) compared to Eq. (58) induces a steeper slope in the highz portion of the d L -z curve, which translates to an excess in distance modulus at high redshift. Notably, our VSL model does not require dark energy whatsoever to account for this behavior.</text>
<text><location><page_10><loc_52><loc_9><loc_92><loc_15></location>The performance of our VSL model can be improved by enabling the function F ( z ), which involves allowing the local scales of galaxies to evolve. This task will be carried out in the following subsections.</text>
<figure>
<location><page_11><loc_11><loc_69><loc_46><loc_93></location>
<caption>Figure 4. Fitting Pantheon using Formula (60). Open circles: 1,048 data points with error bars. Solid line: our Formula (60) with H 0 = 46 . 6 and the F ( i ) values given in Table I. Dashed line: ΛCDM Formula (54) with H 0 = 70 . 2, Ω Λ = 0 . 715. Dotted line: EdS Formula (56) with H 0 = 70 . 2.</caption>
</figure>
<table>
<location><page_11><loc_13><loc_41><loc_46><loc_59></location>
</table>
<section_header_level_1><location><page_11><loc_13><loc_35><loc_44><loc_36></location>C. Enabling F ( z ) : A binning approach</section_header_level_1>
<text><location><page_11><loc_9><loc_19><loc_49><loc_34></location>In this subsection, we will incorporate the variation in the local scales of galaxies, as characterized by F ( z ). Developing model for F ( z ) would require knowledge of galactic formation and structures. Here, we avoid that complexity by extracting F ( z ) directly from the Pantheon data. To do so, we spit the Pantheon dataset into 10 bins ordered by increasing redshift. Bins #1 to #9 each contains 105 data points, while Bin #10 contains 103 data points, totaling 1,048 data points. The range of redshift for each bin is given in Table I.</text>
<text><location><page_11><loc_9><loc_13><loc_49><loc_19></location>All Pantheon data points in Bin # i are treated as having a common value of F ( i ) for the function F ( z ). For a data point # j that belongs to Bin # i , Formula (52) reads</text>
<formula><location><page_11><loc_19><loc_8><loc_49><loc_12></location>ˆ d L,j ˆ c ob = 1 + z j H 0 F ( i ) ln (1 + z j ) 2 / 3 F ( i ) (60)</formula>
<figure>
<location><page_11><loc_52><loc_77><loc_90><loc_93></location>
<caption>Figure 5. The variation of F ( z ) as functions of redshift (left panel) and cosmic scale factor (right panel). Solid staircase lines are the result obtained in Section VI C. Dashed lines are the result obtained in Section VI D.</caption>
</figure>
<text><location><page_11><loc_52><loc_56><loc_92><loc_68></location>Instead of fitting each bin separately, we impose one common value for H 0 across all bins. The fit thus involves H 0 and 10 values for { F ( i ) , i = 1 .. 10 } . Figure 4 displays our fit to Formula (60). The best fit yields H 0 = 46 . 6 km/s/Mpc, and the values of F ( i ) are given in the last column of Table I. The minimum error is χ 2 min = 0 . 97803. The staircase lines in Fig. 5 depict F ( i ) as function of z and the cosmic factor a .</text>
<text><location><page_11><loc_52><loc_41><loc_92><loc_55></location>The values F ( i ) reveal a monotonic decrease with respect to redshift, or equivalently, a monotonic increase in terms of a . This behavior indicates that the local scales of galaxies gradually grow during the course of cosmic expansion, implying that galaxies cannot fully resist this expansion. From z /similarequal 2 to the present time, galaxies have slightly expanded by about 3%, during which process the cosmic scale factor has approximately doubled, i.e. a | z /similarequal 2 /similarequal 0 . 5 with F ( z /similarequal 2) ≈ 0 . 969.</text>
<section_header_level_1><location><page_11><loc_53><loc_37><loc_90><loc_38></location>D. Enabling F ( z ) : A functional form approach</section_header_level_1>
<text><location><page_11><loc_52><loc_29><loc_92><loc_34></location>The steady decline of F ( i ) across the 10 bins with respect to redshift suggests adopting the following functional form for F ( z ):</text>
<formula><location><page_11><loc_58><loc_26><loc_92><loc_28></location>F ( z ) = 1 -(1 -F ∞ ) ( 1 -(1 + z ) -b 1 ) b 2 (61)</formula>
<text><location><page_11><loc_52><loc_19><loc_92><loc_25></location>with b 1 ∈ R + , b 2 ∈ R + , and F ∞ ∈ [0 , 1], supporting a monotonic interpolation from F ( z = 0) = 1 to F ( z → ∞ ) = F ∞ . After some experimentation, we find that setting b 1 = b 2 = 2 offers good overall performance.</text>
<text><location><page_11><loc_52><loc_9><loc_92><loc_19></location>We will apply Formula (52) in conjunction with (61) (with b 1 = b 2 = 2) to fit the Pantheon dataset. The best fit is displayed in Fig. 6, yielding H 0 = 47 . 22 and F ∞ = 0 . 931. The minimum error is χ 2 min = 0 . 98556, a performance that is competitive with-if not exceedingthat of the ΛCDM model, which has χ 2 min (ΛCDM) = 0 . 98824.</text>
<figure>
<location><page_12><loc_11><loc_69><loc_46><loc_94></location>
<caption>Figure 6. Fitting Pantheon using Formula (52). Open circles: 1,048 data points with error bars. Solid line: our Formula (52) with H 0 = 47 . 22, and F ( z ) given in Eq. (61) with b 1 = b 2 = 2 and F ∞ = 0 . 931. Dashed line: ΛCDM Formula (54) with H 0 = 70 . 2, Ω Λ = 0 . 715. Dotted line: EdS Formula (56) with H 0 = 70 . 2.</caption>
</figure>
<text><location><page_12><loc_9><loc_43><loc_49><loc_55></location>Figure 5 shows the variation of F in dashed lines with respect to both redshift and the cosmic factor. We also produce the joint distribution for H 0 and F ∞ , as shown in Figure 7, yielding H 0 = 47 . 21 ± 0 . 4 km/s/Mpc (95% CL) and F ∞ = 0 . 931 ± 0 . 008 (95% CL). This value of F ∞ indicates that the local scales of galaxies have increased by approximately 7% since the formation of the first stable galaxies (i.e., those at the largest redshift).</text>
<section_header_level_1><location><page_12><loc_9><loc_39><loc_49><loc_41></location>Comparison of VSL approach with Λ CDM model</section_header_level_1>
<text><location><page_12><loc_9><loc_22><loc_49><loc_37></location>Our VSL fit, in effect, involves two parameters: H 0 and F ∞ -the same number of parameters as the ΛCDM model ( H 0 and Ω Λ ). However, the parameter F ∞ has a well-defined astrophysical meaning; it denotes the local scales of the first stable galaxies in comparison to the local scale of the Milky Way. Moreover, the function F ( z ), which captures the evolution of the local scales of galaxies during cosmic expansion, plays a role in a potential resolution of the H 0 tension-a topic that will be discussed in Section X.</text>
<text><location><page_12><loc_9><loc_9><loc_49><loc_20></location>In contrast, the ΛCDM model requires a Λ component, the nature of which is still not understood. Its energy density value Ω Λ ≈ 0 . 7 also raises a coincidence problem. Furthermore, the ΛCDM model currently encounters the H 0 tension. If the Λ component is treated as dynamicalan approach explored in several ongoing efforts to resolve the H 0 tension-this would introduce an array of new parameters to the ΛCDM model.</text>
<figure>
<location><page_12><loc_57><loc_72><loc_86><loc_94></location>
<caption>Figure 7. Joint distribution of H 0 and F ∞ , showing 68% CL and 95% CL regions. Peak occurs at H 0 = 47 . 22 , F ∞ = 0 . 931.</caption>
</figure>
<section_header_level_1><location><page_12><loc_55><loc_62><loc_89><loc_63></location>E. The cause for the reduction in H 0 value</section_header_level_1>
<text><location><page_12><loc_52><loc_57><loc_92><loc_60></location>In the z → 0 limit, Eq. (61) with b 1 = b 2 = 2 can be approximated as</text>
<formula><location><page_12><loc_61><loc_54><loc_92><loc_56></location>F ( z ) /similarequal 1 -4(1 -F ∞ ) z 2 + . . . (62)</formula>
<text><location><page_12><loc_52><loc_51><loc_77><loc_52></location>From this, Formula (52) then gives</text>
<formula><location><page_12><loc_64><loc_45><loc_92><loc_49></location>ˆ d L ˆ c ob = 2 3 H 0 z + O ( z 2 ) (63)</formula>
<text><location><page_12><loc_52><loc_42><loc_77><loc_44></location>leading to the modified Hubble law :</text>
<formula><location><page_12><loc_67><loc_37><loc_92><loc_41></location>z = 3 2 H 0 ˆ d L ˆ c ob (64)</formula>
<text><location><page_12><loc_52><loc_30><loc_92><loc_36></location>This result aligns with Eq. (39) derived in Section V B based on the modified Lemaître redshift formula, Eq. (35). In contrast to the classic Hubble law, where the speed of light is constant:</text>
<formula><location><page_12><loc_61><loc_25><loc_92><loc_28></location>z (classic) = H 0 d L c (see Eq. (40))</formula>
<text><location><page_12><loc_52><loc_16><loc_92><loc_24></location>the modified Hubble law acquires a multiplicative factor of 3/2. Hence, in our VSL cosmology, low-redshift emission sources exhibit a linear relationship between z and the luminosity distance, but characterized by a coefficient of 3 2 H 0 rather than H 0 .</text>
<text><location><page_12><loc_52><loc_9><loc_92><loc_16></location>Since a linear-line fit of z on d L for low-redshift emission sources is known to yield a slope of approximately 70, the resulting value of H 0 obtained through our VSL approach is thus only 2 / 3 of this value, specifically H 0 ≈ 47, rather than H 0 ≈ 70 as predicted by standard cosmology.</text>
<section_header_level_1><location><page_13><loc_9><loc_89><loc_49><loc_93></location>VII. A NEW INTERPRETATION: VARIABLE SPEED OF LIGHT AS AN ALTERNATIVE TO DARK ENERGY AND COSMIC ACCELERATION</section_header_level_1>
<text><location><page_13><loc_9><loc_63><loc_49><loc_88></location>The Hubble diagram of SNeIa has been interpreted as a definitive hallmark of late-time accelerated expansion, providing the (only) direct evidence for dark energy. These stellar explosions serve as 'standard candles' due to their consistent peak brightness, allowing astronomers to determine distances to the galaxies in which they reside. In the late 1990s, two independent teams, the HighZ Supernova Search Team [95] and the Supernova Cosmology Project [96], measured the apparent brightness of distant SNeIa, finding them dimmer than expected based on the EdS model, which describes a flat, expanding universe dominated by matter. This can be seen in the Hubble diagram of SNeIa (see Fig. (4)), in which the section with z ≳ 0 . 1 exhibits an distance modulus greater than that predicted by the EdS model. This behavior has been interpreted as indicating that the expansion of the universe is accelerating rather than decelerating.</text>
<text><location><page_13><loc_9><loc_55><loc_49><loc_63></location>However, our quantitative analysis of SNeIa in the preceding section offers a new interpretation as an alternative to late-time acceleration. Mathematically, as explained in Section VI B, in the high z limit, the EdS universe yields</text>
<formula><location><page_13><loc_24><loc_53><loc_49><loc_54></location>d L /similarequal z (see Eq. (58))</formula>
<text><location><page_13><loc_9><loc_50><loc_37><loc_52></location>whereas our VSL-based formula renders</text>
<formula><location><page_13><loc_24><loc_48><loc_49><loc_49></location>d L /similarequal z ln z (see Eq. (59))</formula>
<text><location><page_13><loc_9><loc_39><loc_49><loc_47></location>Thus, high-redshift SNeIa acquire an additional factor of ln z compared with the EdS model. This results in an further upward slope relative to that of the EdS model, successfully capturing the behavior of SNeIa in the high-z section.</text>
<text><location><page_13><loc_9><loc_11><loc_49><loc_39></location>Physical intuition: There is a fundamental reasonbased on our VSL framework-behind this excess in distance modulus that we will explain below. Consider two supernovae A and B at distances d A and d B away from the Earth, such that d B = 2 d A . In standard cosmology, their redshift values z A and z B are related by z A ≈ 2 z A (to first-order approximation). However, this relation breaks down in the VSL context. In a VSL cosmology which accommodates variation in the speed of light in the formï¿œ c ∝ a -1 / 2 , light traveled faster in the distant past (when the cosmic factor a /lessmuch 1) than in the more recent epoch (when a ≲ 1). Therefore, the photon emitted from supernova B was able to cover twice the distance in less than twice the time required for the photon emitted from supernova A. Having spent less time in transit than what standard cosmology would require, the B-photon experienced less cosmic expansion than expected and thus a lower redshift than what the classic Lemaître formula would dictate. Namely:</text>
<formula><location><page_13><loc_19><loc_9><loc_49><loc_10></location>z B < 2 z A for d B = 2 d A (65)</formula>
<text><location><page_13><loc_52><loc_80><loc_92><loc_94></location>Conversely, consider a supernova C with z C = 2 z A . For the C-photon to experience twice the redshift of the Aphoton, it must travel a distance greater than twice that of the A-photon, viz.: d C > 2 d A . This is because since the C-photon traveled faster at the beginning of its journey toward Earth, it must originate from a farther distance (thus appearing fainter than expected) to experience enough cosmic expansion and therefore the requisite amount of redshift. Namely:</text>
<formula><location><page_13><loc_62><loc_78><loc_92><loc_79></location>d C > 2 d A for z C = 2 z A (66)</formula>
<text><location><page_13><loc_52><loc_74><loc_92><loc_77></location>Consequently, the SNeIa data exhibit an additional upward slope in their Hubble diagram in the highz section.</text>
<text><location><page_13><loc_52><loc_55><loc_92><loc_72></location>Conclusion: Hence, a declining speed of light presents a viable alternative to cosmic acceleration , eliminating the need for the Λ component and dissolving its finetuning and coincidence problems. In our VSL framework, during the universe expansion, the dilaton field χ in intergalactic space decreases and leads to a decline in c (per c ∝ χ 1 / 2 ∝ a -1 / 2 ), affecting the propagation of lightwaves from distant SNeIa to the observer. While the impact of a declining c on lightwaves is negligible within the Solar System and on galactic scales, it accumulates on the cosmic scale and makes high-redshift SNeIa appear dimmer than predicted by the standard EdS model.</text>
<text><location><page_13><loc_52><loc_45><loc_92><loc_54></location>The VSL framework, therefore, offers a significant shift in perspective: rather than supporting a Λ CDM universe undergoing late-time acceleration, the Hubble diagram of SNeIa should be reinterpreted as evidence for a declining speed of light in an expanding Einstein-de Sitter universe.</text>
<section_header_level_1><location><page_13><loc_52><loc_38><loc_91><loc_42></location>VIII. RETHINKING BLANCHARD-DOUSPISROWAN-ROBINSON-SARKAR'S 2003 CMB ANALYSIS AND H 0 ≈ 46</section_header_level_1>
<text><location><page_13><loc_52><loc_31><loc_92><loc_37></location>Let us now turn our discussion to a remarkable proposal advanced by Blanchard, Douspis, Rowan-Robinson, and Sarkar (BDRS) in 2003 and its relation to our SNeIa analysis.</text>
<text><location><page_13><loc_52><loc_9><loc_92><loc_31></location>It is well established that the ΛCDM model, augmented by the primordial fluctuation spectrum (presumably arising from inflation) in the form P ( k ) = Ak n , successfully accounts for the observed anisotropies in the cosmic microwave background (CMB). This model predicts a dark energy density of Ω Λ ≈ 0 . 7, a Hubble constant of H 0 ≈ 67, and a spectral index n ≈ 0 . 96 [97, 98]. Yet, in [4] BDRS reanalyzed the CMB, available at the time from the Wilkinson Microwave Anisotropy Probe (WMAP), in a new perspective. These authors deliberately relied on the EdS model, which corresponds to a flat ΛCDM model with Ω Λ = 0. Rather than invoking the Λ component, they adopted a slightly modified form for the primordial fluctuation spectrum. They reasoned that, since the spectral index n is scale-dependent for</text>
<text><location><page_14><loc_9><loc_87><loc_49><loc_94></location>any polynomial potential of the inflaton and is constant only for an exponential potential, it is reasonable to consider a double-power form for the spectrum of primordial fluctuations</text>
<formula><location><page_14><loc_20><loc_83><loc_49><loc_86></location>P ( k ) = { A 1 k n 1 k ⩽ k ∗ A 2 k n 2 k ⩾ k ∗ (67)</formula>
<text><location><page_14><loc_9><loc_78><loc_49><loc_81></location>with a continuity condition ( A 1 k n 1 ∗ = A 2 k n 2 ∗ ) across the breakpoint k ∗ .</text>
<text><location><page_14><loc_9><loc_57><loc_49><loc_77></location>Using this new function, BDRS produced an excellent fit to the CMB power spectrum, resulting in the following parameters: H 0 = 46 km/s/Mpc, ω baryon := Ω baryon ( H 0 / 100) 2 = 0 . 019, τ = 0 . 16 (the optical depth to last scattering), k ∗ = 0 . 0096 Mpc -1 , n 1 = 1 . 015, and n 2 = 0 . 806. The most remarkable outcome of the BDRS work is the 'low' value of H 0 = 46, representing a 34% reduction from the accepted value of H 0 ∼ 70. A detailed follow-up study by Hunt and Sarkar [5], based on a supergravity-induced multiple inflation scenario, yielded a comparable value of H 0 ≈ 44. Notably, around the same time, Shanks argued that a value of H 0 ≲ 50 might permit a simpler inflationary model with Ω baryon = 1, i.e. without invoking dark energy or cold dark matter [99].</text>
<text><location><page_14><loc_9><loc_42><loc_49><loc_56></location>The success achieved by BDRS in reproducing the CMB power spectrum can be interpreted as indicating a degeneracy in the parameter space { Ω Λ , H 0 } . Specifically, the BDRS pair { Ω Λ = 0 , H 0 = 46 } is 'nearly degenerate' with the canonical pair { Ω Λ ≈ 0 . 7 , H 0 ≈ 70 } , insofar as the CMB data is concerned. Importantly, BDRS's modest modification in the primordial fluctuation spectrum can make Ω Λ redundant. In other words, the Λ component is vulnerable to other exogenous underlying assumptions that supplement the Λ CDM model .</text>
<text><location><page_14><loc_9><loc_23><loc_49><loc_41></location>Notably, strong degeneracies in the parameter space related to the CMB have been reported recently. In [100] Alestas et al found that the best-fit value of H 0 obtained from the CMB power spectrum is degenerate with a constant equation of state (EoS) parameter w ; the relationship is approximately linear, given by H 0 + 30 . 93 w -36 . 47 = 0 (with H 0 in km/s/Mpc). Although this finding is not directly related to the BDRS work, the H 0 -vsw degeneracy reinforces the general conclusion regarding the sensitivity of H 0 to other exogenous underlying assumptions that supplement the ΛCDM model-in the case of Alestas et al, the EoS parameter w .</text>
<text><location><page_14><loc_9><loc_13><loc_49><loc_23></location>While a drastically low value of H 0 ≈ 46 at first seems to be 'a steep price to pay', we have demonstrated in the preceding sections that this new value is fully compatible with the H 0 = 47 . 2 obtained from the Hubble diagram of SNeIa data when analyzed within the context of VSL cosmology. Consequently, the Λ component becomes redundant not only for the CMB but also for SNeIa.</text>
<text><location><page_14><loc_9><loc_9><loc_49><loc_12></location>The alignment of our findings with those of BDRS is especially remarkable for several reasons:</text>
<unordered_list>
<list_item><location><page_14><loc_54><loc_80><loc_92><loc_94></location>· The Hubble diagram of SNeIa and the CMB power spectrum are two 'orthogonal' datasets. SNeIa data relates to observations along the time direction, while the CMB captures a two-dimensional snapshot across space at the recombination event. Furthermore, they correspond to two separate epochs-one representing late time (SNeIa) and the other representing early time (CMB)- each characterized by distinct relevant physics.</list_item>
<list_item><location><page_14><loc_54><loc_66><loc_92><loc_80></location>· There is no a priori reason to expect the doublepower primordial fluctuation spectrum used in the BDRS work to result in a reduction in H 0 rather than an enhancement. Moreover, there is no inherent indication of the 34% change in H 0 . The strength of our VSL analysis of SNeIa is in its capability to explain both the direction and magnitude of the change in H 0 through the 3 / 2-factor in the modified Hubble law; see Section VI E.</list_item>
<list_item><location><page_14><loc_54><loc_52><loc_92><loc_65></location>· Our VSL framework is inspired from theoretical consideration of scale-invariant actions (see Section II herein and Ref. [6]) and does not rely on prior knowledge of BDRS's analysis. It was not deliberately designed to address BDRS's surprise finding of H 0 ≈ 46. In this regard, our findings should be viewed as a retrodiction of BDRS's results, supporting H 0 ∼ 46-47 and bypassing the need for the Λ component.</list_item>
</unordered_list>
<text><location><page_14><loc_52><loc_42><loc_92><loc_51></location>Together with our SNeIa analysis, the work of BDRS eliminates the need for the Λ component regarding the two 'orthogonal' datasets-the CMB and SNeIa. Future applications of our VSL framework to gravitational lensing, Baryonic Acoustic Oscillations (BAO), and other areas are worthwhile.</text>
<section_header_level_1><location><page_14><loc_55><loc_37><loc_89><loc_39></location>What caused BDRS to abandon their H 0 ≈ 46 finding?</section_header_level_1>
<text><location><page_14><loc_52><loc_15><loc_92><loc_35></location>In 2006 BDRS revisited their 2003 A&A proposal by applying it to the Sloan Digital Sky Survey (SDSS) of luminous red galaxies (LRG) which became available in [101]. In their 2006 follow-up work [102], BDRS claimed that the 'low' value of H 0 ≈ 46 was unable to produce an acceptable fit to the two-point correlation function of LRG in observed (redshift) space. The upper panel of Fig. 8 reproduces their finding, showing that the SDSS data (and their error bars in red segments) largely aligns with the ΛCDM model (dotted line), while the BDRS model (dashed-dotted line) is significantly off. This discrepancy eventually forced BDRS to abandon their 2003 proposal in its entirety (although Hunt and Sarkar continued with their follow-up study shortly thereafter [5]).</text>
<text><location><page_14><loc_52><loc_9><loc_92><loc_15></location>However, we believe that BDRS's 2006 SDSS analysis contained an oversight, in light of our VSL cosmology. The standard Lemaître redshift formula and the conventional Hubble law are not applicable in the presence of</text>
<figure>
<location><page_15><loc_12><loc_71><loc_45><loc_94></location>
</figure>
<figure>
<location><page_15><loc_12><loc_47><loc_45><loc_70></location>
<caption>Figure 8. The correlation function in observed (redshift) space, as reproduced from BDRS's 2006 SDSS study [102]. Upper panel: BDRS's original result. Lower panel: the SDSS data are corrected by reducing s and s 2 ξ by factors of 3 / 2 and (3 / 2) 2 , respectively.</caption>
</figure>
<text><location><page_15><loc_9><loc_14><loc_49><loc_34></location>varying speed of light. As discussed in Sections V A and VB, these expressions are modified by a factor of 3 / 2 due to varying c . Therefore, the SDSS would need a reevaluation to incorporate this VSL-induced adjustment. Here, we tentatively make a rudimentary fix: we correct the comoving distance s (measured in multiples of h , defined as H 0 / (100 km/s/Mpc)), downward by a factor of 3 / 2. In the lower panel of Fig. 8, we adjust the SDSS data (and their error bars) by reducing s by a factor of 3 / 2 and s 2 ξ by a factor of (3 / 2) 2 . Upon these adjustments, the peak (at s /similarequal 75) and trough (at s /similarequal 60) of the SDSS become aligned with those of the BDRS's model (dasheddotted line), thereby lessening the discrepancy issue that led to BDRS's abandonment of their original proposal.</text>
<text><location><page_15><loc_9><loc_9><loc_49><loc_13></location>We conclude that it was premature for BDRS to abandon their 2003 CMB study and the finding of H 0 , ≈ 46. Rather, their proposal should be revived and applied to</text>
<text><location><page_15><loc_52><loc_85><loc_92><loc_94></location>the upgraded Planck dataset for the CMB [97]. We should also note that since the CMB data is a twodimensional snapshot of the sky at the time at recombination, VSL is not expected to be a dominant player in the CMB. Nevertheless, potential impacts of VSL on the CMB are an interesting avenue for future research.</text>
<section_header_level_1><location><page_15><loc_56><loc_79><loc_88><loc_80></location>IX. RESOLVING THE AGE PROBLEM</section_header_level_1>
<text><location><page_15><loc_52><loc_73><loc_92><loc_77></location>From the definition of the Hubble constant, H ( t ) := 1 a da dt , and the evolution, a ∝ t 2 / 3 , the age of an EdS universe is related to the current-time H 0 value by</text>
<formula><location><page_15><loc_67><loc_68><loc_92><loc_72></location>t EdS 0 = 2 3 H 0 (68)</formula>
<text><location><page_15><loc_52><loc_62><loc_92><loc_68></location>A value of H 0 ∼ 70, would result in an age of 9.3 billion years which would be too short to accommodate the existence of the oldest stars-a paradox commonly referred to as the age problem.</text>
<text><location><page_15><loc_52><loc_54><loc_92><loc_61></location>Standard cosmology resolves the age problem by invoking the Λ component which induces an acceleration phase following a deceleration phase. The spatially flat ΛCDM model is known to give the age formula in an analytical form (with Ω M +Ω Λ = 1 and Ω Λ > 0) [103]</text>
<formula><location><page_15><loc_60><loc_50><loc_92><loc_53></location>t ΛCDM 0 = 2 3 √ Ω Λ H 0 arcsinh √ Ω Λ Ω M . (69)</formula>
<text><location><page_15><loc_52><loc_43><loc_92><loc_49></location>which restores Eq. (68) when Ω M → 1 and Ω Λ → 0. For positive Ω Λ , the age exceeds 2 / (3 H 0 ). With H 0 = 70 . 2, Ω M = 0 . 285, Ω Λ = 0 . 715, it yields an age of 13 . 6 billion years, an accepted figure in standard cosmology.</text>
<text><location><page_15><loc_52><loc_32><loc_92><loc_42></location>However, our VSL cosmology naturally overcomes the age problem without invoking the Λ component. The reason is that H 0 is reduced by a factor of 3 / 2, as detailed in Section VI E. The reduced value H 0 = 47 . 22 ± 0 . 4 (95% CL) promptly yields t 0 = 13 . 82 ± 0 . 11 billion years (95% CL), consistent with the accepted age value, thereby successfully resolving the age problem.</text>
<section_header_level_1><location><page_15><loc_53><loc_25><loc_91><loc_27></location>X. TOWARD A NEW RESOLUTION OF THE H 0 TENSION</section_header_level_1>
<text><location><page_15><loc_52><loc_9><loc_92><loc_23></location>Galaxies are gravitationally bound structures, stabilized by gravitational attraction and rotational motion of matter within them. However, they are embedded in a cosmic background that is not static, but rather expanding over time. As such, stable galaxies in principle may adjust to the growth in the scale of the 'ambient' intergalactic space surrounding them; viz., their local scales may increase in response to the cosmic expansion. This growth in the local scales of galaxies-if it exists-would be of astronomical nature. To investigate</text>
<text><location><page_16><loc_9><loc_82><loc_49><loc_94></location>this phenomenon, one could explore the evolution of a spinning disc-shaped distribution of matter (serving as a simplified model for a galaxy) on an expanding cosmic background within the scale-invariant theory mentioned in Section II, although such an exploration lies beyond the scope of this paper. We should note that recent observational studies have reported evidence of galaxies experiencing growth in size [104, 105].</text>
<text><location><page_16><loc_9><loc_61><loc_49><loc_82></location>For our purposes, in Section IV, we have modeled the local scale ˆ a of an individual galaxy as a function of its redshift z , supplemented with a negligible idiosyncratic component that randomly varies from one galaxy to another. The function F ( z ), defined in Eq. (33) as the ratio of the local scale of galaxies at redshift z to the local scale of the Milky Way, captures the evolution of the local scale over cosmic time. In Section VI C, F ( z ) was empirically extracted from the Pantheon Catalog, with Fig. 5 displaying F ( z ) as a function of the redshift and of cosmic scale factor, respectively. In accordance with our expectation, the local scale ˆ a of galaxies gradually increases in response to the growth of the global cosmic factor a over cosmic time.</text>
<section_header_level_1><location><page_16><loc_22><loc_57><loc_36><loc_58></location>A 'running' H 0 ( z )</section_header_level_1>
<text><location><page_16><loc_9><loc_51><loc_49><loc_55></location>The function F ( z ) can be absorbed into an 'effective' Hubble constant H 0 ( z ) which depends on redshift z . Specifically, Formula (52) can be rewritten as</text>
<formula><location><page_16><loc_21><loc_46><loc_49><loc_50></location>ˆ d L ˆ c ob = 1 + z 3 2 H 0 ( z ) ln(1 + z ) (70)</formula>
<text><location><page_16><loc_9><loc_43><loc_48><loc_45></location>where the newly introduced function H 0 ( z ) is given by</text>
<formula><location><page_16><loc_16><loc_39><loc_49><loc_43></location>H 0 ( z ) = H 0 F ( z ) ( 1 -3 2 ln F ( z ) ln(1 + z ) ) . (71)</formula>
<text><location><page_16><loc_9><loc_21><loc_49><loc_38></location>Formulae (70) and (71) thus allow for a current-time H 0 ( z ) 'running' as a function of the redshift of the data that are used to estimate it. With the function F ( z ) parametrized in Eq. (61) with b 1 = b 2 = 2 and F ∞ = 0 . 931 as produced in Section VI D, H 0 ( z ) can be computed using Eq. (71), as displayed in Fig. 9. At first, H 0 ( z ) decreases from 47 . 2 (at z = 0) to 41 . 5 (at z /similarequal 2), experiencing a 12% reduction. For z ≳ 2, H 0 ( z ) slowly rerises. At z → ∞ , with F ∞ = 0 . 931 and H 0 ( z = 0) = 47 . 2, per Eq. (71), H 0 ( z ) asymptotically approaches F ∞ H 0 = 43 . 95, representing an 7% reduction from H 0 ( z = 0).</text>
<text><location><page_16><loc_10><loc_18><loc_38><loc_20></location>Two immediate remarks can be made:</text>
<unordered_list>
<list_item><location><page_16><loc_11><loc_9><loc_49><loc_18></location>1. Interestingly, the overall 7% reduction in the H 0 estimate at highestz SNeIa data is of the comparable magnitude with the discrepancy in H 0 reported in standard cosmology, which observes a decreases of H 0 from 73 (using SNeIa) to 67 (using the Planck CMB), an 8% reduction.</list_item>
</unordered_list>
<figure>
<location><page_16><loc_51><loc_76><loc_92><loc_93></location>
<caption>Figure 9. The variation of H 0 ( z ) as functions of redshift (left panel) and of cosmic scale factor (right panel).</caption>
</figure>
<unordered_list>
<list_item><location><page_16><loc_54><loc_63><loc_92><loc_69></location>2. Remarkably, the asymptotic value H 0 ( z → ∞ ) = 43 . 95 that we just derived agrees surprisingly well with the H 0 ≈ 44 value obtained by Hunt and Sarkar in their follow-up study of the CMB [5].</list_item>
</unordered_list>
<text><location><page_16><loc_52><loc_56><loc_92><loc_62></location>The 'running' phenomenon of H 0 ( z ) arises because astronomical objects-either the CMB or SNeIa-are subject to their local scale which gradually grows during the cosmic expansion.</text>
<section_header_level_1><location><page_16><loc_53><loc_49><loc_90><loc_50></location>Hints at an astronomical origin of the H 0 tension</section_header_level_1>
<text><location><page_16><loc_52><loc_27><loc_92><loc_48></location>We have, therefore, linked the 'running' current-time H 0 ( z ) with the function F ( z ). Since F ( z ) captures the evolution of galaxies' local scales in response to the growth of the global scale of intergalactic space, the 'running' H 0 ( z ) is thus of astronomical origin. The empirical evaluation for F ( z ) from the Pantheon Catalog, as detailed in Sections VI C and VI D, demonstrates that the local scale gradually increases with cosmic time, indicating that galaxies cannot fully resist cosmic expansion. As mentioned earlier, understanding the growth in F ( z ) would require an in-depth examination of a spinning disc-shaped distribution of matter in an expanding cosmic background within a scale-invariant theory of gravity and matter, a task that is left for future investigation.</text>
<text><location><page_16><loc_52><loc_19><loc_92><loc_26></location>We should also note similar works along this line of 'running' H 0 ( z ) [106-108]. For example, in [108], Dainotti et al considered an extension of the flat w 0 w a CDM . They proposed the following luminosity distance-redshift formula</text>
<formula><location><page_16><loc_54><loc_9><loc_92><loc_18></location>d L c = (1 + z ) × ∫ z 0 dz ' H 0 ( z ' ) √ Ω 0 M (1 + z ' ) 3 +Ω 0Λ e 3 ∫ z ' 0 du 1+ w ( u ) 1+ u (72)</formula>
<text><location><page_17><loc_9><loc_72><loc_49><loc_94></location>with the parametrization H 0 ( z ) = ˜ H 0 (1 + z ) -α and an evolutionary equation of state for the Λ component w ( z ) = w 0 + w a z/ (1 + z ). In this formula, H 0 ( z ) can be interpreted as a 'running' current-time Hubble value, which depends on the redshift of the data used to estimate it. These authors are able to bring the value of H 0 at z =1,100 within 1 σ of the Planck measurements, hence effectively removing the H 0 tension. However, unlike our approach, where the function F ( z ) has a well-defined astrophysical interpretation, the use of H 0 ( z ) and w ( z ) in Ref. [108] represents ad hoc parametrizations, with their underlying nature remaining unknown. Additionally, the H 0 ( z ) in Ref. [108] is likely of a cosmic origin, whereas the equation of state w ( z ) of the Λ component is of a field theoretical origin.</text>
<text><location><page_17><loc_9><loc_63><loc_49><loc_71></location>In closing of this section, our study offers a potential resolution to the H 0 tension. Furthermore, it suggests that this tension has an astronomical origin, arising from the growth in the local scale of gravitationally-bound galaxies over cosmic time.</text>
<section_header_level_1><location><page_17><loc_14><loc_57><loc_44><loc_59></location>XI. DISCUSSIONS AND SUMMARY</section_header_level_1>
<text><location><page_17><loc_9><loc_53><loc_49><loc_56></location>This paper was inspired by three separate lines of development:</text>
<text><location><page_17><loc_9><loc_39><loc_49><loc_52></location>1. In 2003, Blanchard et al (BDRS) proposed a novel CMB analysis that avoids the Λ component [4]. Based solely on the EdS model (i.e., Ω Λ = 0) and adopting a double-power primordial fluctuation spectrum, BDRS achieved an excellent fit to WMAP's CMB power spectrum. Surprisingly, they obtained a new value H 0 ≈ 46, representing a 34% reduction compared to the accepted value H 0 ∼ 70 that relies on the flat ΛCDM model with Ω Λ ≈ 0 . 7.</text>
<text><location><page_17><loc_9><loc_16><loc_49><loc_38></location>As independently reported more recently in [100], there exists a strong degeneracy inherent in the parameter space concerning the CMB data. Drawn from this observation, we can interpret BDRS's findings as indicating that within the flat Λ CDM model, the parameter pairs { Ω Λ = 0 , H 0 ≈ 46 } and { Ω Λ ≈ 0 . 7 , H 0 ≈ 70 } are 'nearly degerenate' insofar as the CMB power spectrum is concerned. With a modest modification to the primordial fluctuation spectrum, the BDRS parameter pair { Ω Λ = 0 , H 0 ≈ 46 } becomes advantageous over the ΛCDM pair { Ω Λ ≈ 0 . 7 , H 0 ≈ 70 } . While the cost of this modification is not prohibitive, as BDRS provided justifications in support of a double-power primordial fluctuation spectrum, the benefit is profound in that the DE hypothesis is rendered unnecessary.</text>
<text><location><page_17><loc_9><loc_9><loc_49><loc_16></location>This perspective raises an intriguing possibility that the parameter pairs { Ω Λ = 0 , H 0 ≈ 46 } and { Ω Λ ≈ 0 . 7 , H 0 ≈ 70 } may also be 'nearly degenerate' insofar as the Hubble diagram of SNeIa is concerned. To materialize this possibility, one must first seek an alternative</text>
<text><location><page_17><loc_52><loc_86><loc_92><loc_94></location>approach to late-time acceleration that does not invoke DE. We should emphasize that such an alternative-if it exists-must not only eliminate the role of Ω Λ but also reduce the H 0 value from ∼ 70 to ∼ 46. This presents a stringent requirement to be met.</text>
<unordered_list>
<list_item><location><page_17><loc_52><loc_68><loc_92><loc_85></location>2. A recent theoretical approach developed by the present author [6] induces a variation in the speed of light c (and a variation in the quantum of action ℏ ) from a dynamical dilaton χ . The derivation applies to a class of scale-invariant actions that allow matter to couple with the dilaton. The dynamics of c (and ℏ ) stems parsimoniously from the dilaton, rather than as serving as an auxiliary addition to the action. The dependencies are determined to be c ∝ χ 1 / 2 and ℏ ∝ χ -1 / 2 . It was also found that the timescale τ and lengthscale l of a given physical process are related in the an anisotropic fashion, τ ∝ l 3 / 2 . See Section II.</list_item>
<list_item><location><page_17><loc_52><loc_51><loc_92><loc_67></location>3. Existing efforts in the literature to apply variable speed of light (VSL) theories to the Hubble diagram of SNeIa have been impeded by a detrimental oversight. All available VSL analyses to date have relied on the standard Lemaître redshift relation, 1 + z = a -1 , leading to a flawed consensus that VSL plays no role in late-time acceleration. This error stems from the assumption that c is solely a function of cosmic time t , overlooking the possibility that c can vary across the boundaries of galaxies, where a gravitationally-bound galactic region merges with the expanding intergalactic space surrounding it.</list_item>
</unordered_list>
<text><location><page_17><loc_52><loc_35><loc_92><loc_50></location>In this paper, we build upon the VSL theory referenced in Point #2, correct the error mentioned in Point #3, and reanalyze the Pantheon Catalog. The effects of VSL modify the Lemaîitre formula to 1 + z = a -3 / 2 , with the 3 / 2-exponent arising from the anisotropic time scaling referenced earlier, τ ∝ l 3 / 2 . Intuitively, this factor 3/2 influences the evaluation of H 0 , resulting in a reduction from the canonical value of H 0 ∼ 70 by a factor of 3 / 2 to H 0 = 47 . 2. The new value is compatible with BDRS's findings for the CMB mentioned in Point #1.</text>
<text><location><page_17><loc_52><loc_31><loc_92><loc_34></location>Our derivation and analysis: The logical steps of our work are as follows.</text>
<text><location><page_17><loc_52><loc_26><loc_92><loc_31></location>(i) Modifying the FLRW metric. The universe is modeled as an EdS spacetime supporting a varying c as (see Eq. (20))</text>
<formula><location><page_17><loc_56><loc_20><loc_87><loc_25></location> ds 2 = c 2 ( a ) dt 2 -a 2 ( t ) [ dr 2 + r 2 d Ω 2 ] c ( a ) = c 0 ( a a 0 ) -1 / 2</formula>
<text><location><page_17><loc_52><loc_14><loc_92><loc_19></location>with the expansion obeying the growth law a ( t ) = a 0 ( t/t 0 ) 2 / 3 , see Eq. (17). Justifications for this model are provided in Section III B.</text>
<unordered_list>
<list_item><location><page_17><loc_52><loc_9><loc_92><loc_13></location>(ii) Modifying the Lemaître redshift formula. Across the boundaries of galaxies, c also varies, leading to a refraction on the lightwaves. Due to this effect, we find</list_item>
</unordered_list>
<text><location><page_18><loc_9><loc_89><loc_49><loc_94></location>that the classic Lemaître redshift formula 1 + z = a -1 is inapplicable for the VSL cosmology, and is replaced by the modified Lemaître redshift formula (see Eq. (35))</text>
<formula><location><page_18><loc_21><loc_86><loc_37><loc_88></location>1 + z = a -3 / 2 F 3 / 2 ( z )</formula>
<text><location><page_18><loc_9><loc_81><loc_49><loc_85></location>with a new exponent of 3 / 2 and F ( z ) measuring the relative change in the local scale of galaxies. See Sections III C, IV and V A.</text>
<text><location><page_18><loc_9><loc_75><loc_49><loc_80></location>(iii) Modifying the Hubble law. The 3 / 2-exponent in the modified Lemaître redshift formula above leads to the modified Hubble law (see Eq. (39))</text>
<formula><location><page_18><loc_25><loc_71><loc_33><loc_75></location>z = 3 2 H 0 d c 0</formula>
<text><location><page_18><loc_9><loc_66><loc_49><loc_71></location>This new Hubble law differs from the classic Hubble law by a multiplicative factor of 3 / 2, resulting in a reduction in the H 0 estimate by a factor of 3 / 2. See Section V B.</text>
<text><location><page_18><loc_9><loc_62><loc_49><loc_66></location>(iv) Modifying the luminosity distance-vsz formula: This formula is the centerpiece of our study (see Eq. (52))</text>
<formula><location><page_18><loc_19><loc_58><loc_39><loc_62></location>ˆ d L ˆ c ob = 1 + z H 0 F ( z ) ln (1 + z ) 2 / 3 F ( z )</formula>
<text><location><page_18><loc_9><loc_56><loc_38><loc_57></location>See Sections V C and V D for derivation.</text>
<text><location><page_18><loc_9><loc_46><loc_49><loc_55></location>(v) A re-analysis of the Pantheon data based on VSL: In Section VI, we apply the Formulae above to the Combined Pantheon Sample of SNeIa. We produce an excellent fit without invoking the Λ component; the fit is as robust as that obtained from the Λ CDM model. The optimal values for the parameters are:</text>
<unordered_list>
<list_item><location><page_18><loc_11><loc_41><loc_49><loc_45></location>· The Hubble constant H 0 = 47 . 2 ± 0 . 4 (95% CL). This value of consistent with the 3 / 2 reduction referenced in Point (iii) above.</list_item>
<list_item><location><page_18><loc_11><loc_35><loc_49><loc_40></location>· The local scale of galaxies decreases with respect to redshift as F ( z ) = 1 -(1 -F ∞ ) . ( 1 -(1 + z ) -2 ) 2 , with F ∞ = 0 . 931 ± 0 . 11 (95% CL).</list_item>
</unordered_list>
<text><location><page_18><loc_9><loc_26><loc_49><loc_34></location>Our modified Lemaître redshift formula, Eq. (35), can also effectively viewed as a form of 'redshift remapping', a technique advocated in Refs. [109-111]. Interestingly, our value of H 0 = 47 . 2 ± 0 . 4 aligns with the result H 0 = 48 ± 2 reported in [111].</text>
<text><location><page_18><loc_9><loc_22><loc_49><loc_25></location>Implications: Four important findings emerge from our analysis.</text>
<unordered_list>
<list_item><location><page_18><loc_9><loc_9><loc_49><loc_20></location>I) Declining speed of light as an alternative interpretation of the Hubble diagram of SNeIa . At high redshift, the luminosity distance in an EdS universe behaves as d L ∝ z , whereas in our VSL cosmology, it behaves as d L ∝ z ln z . Due to VSL, high-redshift SNeIa thus benefit from the additional ln z term, making them appear dimmer than predicted by the EdS model. Another intuitive way to understand this behavior is to note</list_item>
</unordered_list>
<text><location><page_18><loc_52><loc_80><loc_92><loc_94></location>that since c ∝ a -1 / 2 , light traveled faster in the past than in later epochs. As a result, lightwaves from distant SNeIa require less time to traverse the earlier sections of their trajectories, hence experiencing less cosmic expansion (and redshift) than the EdS model predicts. Hence, the highz section of the Hubble diagram of SNeIa can be explained-qualitatively and quantitativelyby a declining speed of light rather than a recent cosmic acceleration . A detailed exposition is given in Section VII.</text>
<text><location><page_18><loc_52><loc_60><loc_92><loc_79></location>II) Reviving BDRS's work on the CMB, avoiding dark energy. Despite the very different natures of the data involved, our VSL-based analysis of SNeIa and BDRS's work on the CMB fully agree on two aspects: (i) the universe obeys the EdS model (i.e. Ω Λ = 0), and (ii) H 0 is reduced to 46-47. The BDRS parameter pair { Ω Λ = 0 , H 0 ≈ 46 } is advantageous over the ΛCDM pair { Ω Λ ≈ 0 . 7 , H 0 ≈ 70 } regarding both the CMB and SNeIa, which are 'orthogonal' datasets. Detailed discussions are presented in Section VIII. Together, our current work and BDRS's 2003 analysis challenge the existence of dark energy-one of the foundational assumptions of the cosmological concordance model.</text>
<text><location><page_18><loc_52><loc_49><loc_92><loc_60></location>III) Resolving the age problem. The age of an EdS universe is given by: t 0 = 2 / (3 H 0 ). Using the reduced value of H 0 = 47 . 22 ± 0 . 4 km/s/Mpc, one obtains t 0 = 13 . 82 ± 0 . 11 billion years. The age problem is thus resolved through the reduction in H 0 , without requiring a recent acceleration phase induced by dark energy. See Section IX.</text>
<text><location><page_18><loc_52><loc_40><loc_92><loc_49></location>IV) Addressing the H 0 tension. Utilizing the function F ( z ), we recast the current-time Hubble constant as a function H 0 ( z ) of redshift. Between z = 0 and z →∞ , the 'running' H 0 ( z ) exhibits a 7% decrease, a reduction in similar magnitude with the ongoing H 0 tension between the CMB and SNeIa. See Section X.</text>
<section_header_level_1><location><page_18><loc_53><loc_33><loc_91><loc_36></location>On the cosmological time dilation extracted from the Dark Energy Survey (DES)</section_header_level_1>
<text><location><page_18><loc_52><loc_14><loc_92><loc_32></location>A recent paper [112] using DES supernova light curves showed no deviation from the relation ∆ t obs = ∆ t em (1 + z ). However, this finding does not contradict our modified Lemaitre redshift formula, 1 + z = a -3 / 2 F ( z ). This is because the result in Ref. [112] only verifies that the speed of light inside the galaxies hosting the supernovae and that inside the Milky Way are approximately the same. Galaxies are gravitationally bound and thus not subject to cosmic expansion. Reference [112] does not deal with the speed of light in the intergalactic space, the expansion of which causes c to decline over time. We clarified this distinction in Section IV.</text>
<section_header_level_1><location><page_19><loc_21><loc_92><loc_37><loc_93></location>XII. CONCLUSION</section_header_level_1>
<text><location><page_19><loc_9><loc_80><loc_49><loc_90></location>The nearly identical agreement of the CMB and SNeIa regarding the reduced value of H 0 ∼ 46-47 is highly encouraging. This alignment points toward a consistent cosmological framework based on the Einstein-de Sitter model with a variable speed of light, thus eliminating the need for energy and dissolving its fine-tuning and coincidence problems.</text>
<text><location><page_19><loc_9><loc_72><loc_49><loc_79></location>Importantly, we have built a case for an alternative perspective: rather than supporting a Λ CDM universe undergoing late-time acceleration, the Hubble diagram of SNeIa can be reinterpreted as evidence for a declining speed of light in an expanding Einstein-de Sitter universe .</text>
<text><location><page_19><loc_9><loc_56><loc_49><loc_71></location>Finally, we note that the observational bounds established in the literature in support of a constant speed of light have predominantly relied on standard cosmology [33-37, 41-45, 60, 61, 65, 67, 68, 71, 82]. However, our new Lemaître redshift formula represents a critical departure from this conventional framework. Therefore, the consensus regarding the absence of variation in c in observational cosmology must be reconsidered in light of our findings, prompting a comprehensive reanalysis of these constraints.</text>
<section_header_level_1><location><page_19><loc_19><loc_52><loc_39><loc_53></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_19><loc_9><loc_45><loc_49><loc_51></location>I thank Clifford Burgess, Tiberiu Harko, Robert Mann, Anne-Christine Davis and Eoin Ó Colgáin for their constructive and supportive comments during the development of this work.</text>
<text><location><page_19><loc_28><loc_41><loc_30><loc_43></location>∞</text>
<text><location><page_19><loc_20><loc_41><loc_28><loc_43></location>-----</text>
<text><location><page_19><loc_30><loc_41><loc_38><loc_43></location>-----</text>
<section_header_level_1><location><page_19><loc_17><loc_37><loc_41><loc_38></location>Appendix A: Refraction effect</section_header_level_1>
<text><location><page_19><loc_9><loc_28><loc_49><loc_35></location>Let us start with a well-understood phenomenon: the behavior of a wavetrain in a medium with varying speed of wave. It is well established that the wavelength of the wavetrain at a given location is proportional to the speed of wave at that location:</text>
<formula><location><page_19><loc_27><loc_26><loc_49><loc_27></location>λ ∝ v (A1)</formula>
<text><location><page_19><loc_9><loc_12><loc_49><loc_25></location>Figure 10 illustrates the change in wavelength as a wave travels at varying speed. In the upper panel, as the speed increases, the front end of the wavecrest will rush forward leaving its back end behind thus stretching out the wavecrest. In the lower panel, the reverse situation occurs: as the speed decreases, the front end of the wavecrest will slow down while its back end continues its course thus compressing the wavecrest. In either situation, the wavelength and the speed of wave are directly proportional:</text>
<formula><location><page_19><loc_26><loc_8><loc_49><loc_11></location>λ 2 λ 1 = v 2 v 1 (A2)</formula>
<figure>
<location><page_19><loc_56><loc_66><loc_87><loc_93></location>
<caption>Figure 10. Change in wavelength as a wavetrain travels in a medium with varying speed of wave. Upper panel: wavelength doubles as its speed doubles. Lower panel: wavelength halves as its speed halves. In either case, wavelength and speed are proportional: λ 2 /v 2 = λ 1 /v 1 .</caption>
</figure>
<text><location><page_19><loc_52><loc_51><loc_92><loc_54></location>Note that the details of how the variation of v does not participate in formula above.</text>
<section_header_level_1><location><page_19><loc_55><loc_46><loc_89><loc_48></location>Appendix B: An equivalent derivation of the modified Lemaître redshift formula</section_header_level_1>
<text><location><page_19><loc_52><loc_39><loc_92><loc_44></location>We produce an alternative route by way of frequency transformation to modifying Lemaître's redshift formula (35). We have derived in Eq. (25) that</text>
<formula><location><page_19><loc_68><loc_35><loc_92><loc_38></location>ν ob ν em = a 3 / 2 em a 3 / 2 ob (B1)</formula>
<text><location><page_19><loc_52><loc_30><loc_92><loc_34></location>For transits between local regions to global regions (i.e., Transit #1 and Transit #3 in Fig. 1 in Page 6), since λ ∝ c , the frequency is:</text>
<formula><location><page_19><loc_66><loc_26><loc_92><loc_29></location>ν = c λ = const (B2)</formula>
<text><location><page_19><loc_52><loc_23><loc_92><loc_26></location>This means that the frequency of the lightwave does not change during Transit #1 and Transit #3, viz.</text>
<formula><location><page_19><loc_64><loc_21><loc_92><loc_22></location>ˆ ν em = ν em ; ˆ ν ob = ν ob (B3)</formula>
<text><location><page_19><loc_52><loc_18><loc_59><loc_20></location>Given that</text>
<formula><location><page_19><loc_56><loc_15><loc_92><loc_18></location>ˆ λ ob = ˆ c ob ˆ ν ob ; ˆ λ em = ˆ c em ˆ ν em ; λ ∗ ˆ λ em = ˆ a ob ˆ a em (B4)</formula>
<text><location><page_19><loc_52><loc_12><loc_54><loc_14></location>and</text>
<formula><location><page_19><loc_67><loc_8><loc_92><loc_12></location>ˆ c ob ˆ c em = ˆ a -1 / 2 ob ˆ a -1 / 2 em (B5)</formula>
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</document> | [{"title": "New analysis of SNeIa Pantheon Catalog: Variable speed of light as an alternative to dark energy", "content": "Hoang Ky Nguyen \u2217 Department of Physics, Babe\u015f-Bolyai University, Cluj-Napoca 400084, Romania (Dated: December 17, 2024) In A&A 412, 35 (2003) Blanchard, Douspis, Rowan-Robinson, and Sarkar (BDRS) slightly modified the primordial fluctuation spectrum and produced an excellent fit to WMAP's CMB power spectrum for an Einstein-de Sitter (EdS) universe, bypassing dark energy. Curiously, they obtained a Hubble value of H 0 \u2248 46, in sharp conflict with the canonical range \u223c 67-73. However, we will demonstrate that the reduced value of H 0 \u2248 46 achieved by BDRS is fully compatible with the use of variable speed of light in analyzing the late-time cosmic acceleration observed in Type Ia supernovae (SNeIa). In arXiv:2412.04257 [gr-qc] we considered a generic class of scale-invariant actions that allow matter to couple non-minimally with gravity via a dilaton field \u03c7 . We discovered a hidden aspect of these actions: the dynamics of the dilaton can induce a variation in the speed of light c as c \u221d \u03c7 1 / 2 , thereby causing c to vary alongside \u03c7 across spacetime. For an EdS universe with varying c , besides the effects of cosmic expansion, light waves emitted from distant SNeIa are further subject to a refraction effect, which alters the Lema\u00eetre redshift relation to 1 + z = a -3 / 2 . Based on this new formula, we achieve a fit to the SNeIa Pantheon Catalog exceeding the quality of the \u039bCDM model. Crucially, our approach does not require dark energy and produces H 0 = 47 . 2 \u00b1 0 . 4 (95% CL) in strong alignment with the BDRS finding of H 0 \u2248 46. The reduction in H 0 in our work, compared with the canonical range \u223c 67-73, arises due to the 3 / 2 -exponent in the modified Lema\u00eetre redshift formula. Hence, BDRS's analysis of the (early-time) CMB power spectrum and our variablec analysis of the (late-time) Hubble diagram of SNeIa fully agree on two counts: (i) the dark energy hypothesis is avoided , and (ii) H 0 is reduced to \u223c 47, which also yields an age t 0 = 2 / (3 H 0 ) = 13 . 8 Gy for an EdS universe, without requiring dark energy. Most importantly, we will demonstrate that the late-time acceleration can be attributed to the declining speed of light in an expanding EdS universe , rather than to a dark energy component.", "pages": [1]}, {"title": "I. MOTIVATION", "content": "The \u039bCDM model serves as the standard framework for modern cosmology, efficiently accounting for a wide range of astronomical observations. While the model is widely regarded as successful, it faces significant challenges [1]. Notably, ongoing tensions in the determination of the Hubble constant H 0 and the amplitude of matter fluctuations \u03c3 8 raise questions about the underpinning principles of the model [2]. Moreover, an integral component of this model is dark energy (DE), which constitutes approximately 70% of the total energy budget of the universe. The nature of DE itself-along with its finetuning and coincidence problems-poses profound challenges both in cosmology and in the broader context of field theories [3]. In 2003, Blanchard, Douspis, Rowan-Robinson, and Sarkar (BDRS) proposed a novel approach to mitigate the need for DE when analyzing the cosmic microwave background (CMB) power spectrum [4]. They relied on the Einstein-de Sitter universe, which corresponds to the flat \u039bCDM model with \u2126 \u039b = 0. Instead of the conventional single-power primordial fluctuation spectrum, P ( k )= Ak n , they employed a double-power form with continuity imposed across the breakpoint k \u2217 . Remarkably, this modest modification produced an excellent fit to the CMB power spectrum without invoking DE. In [5], Hunt and Sarkar further developed a supergravity-based inflation scenario to validate the double-power form given in Eq. (1) and also attained an excellent fit while avoiding DE. The works by BDRS and the Hunt-Sarkar teamif correct -would seriously undermine the viability of the DE hypothesis. Surprisingly, the fit by BDRS yielded a new value of H 0 \u2248 46, while the fit by Hunt and Sarkar produced a comparable value of H 0 \u2248 44. Obviously, these values are at odds with the value of H 0 \u223c 70 derived from the Hubble diagram of Type Ia supernovae (SNeIa), based on the \u039bCDM model with \u2126 \u039b \u2248 0 . 7. Since DE has been regarded as the driving force of the late-time cosmic acceleration, interest in the works by BDRS and the Hunt-Sarkar team has largely diminished in favor of the standard \u039bCDM model. Amid this backdrop, we will reexamine the SNeIa data in the context of a cosmology that supports a varying speed of light c on an expanding EdS cosmic background, rather than the \u039bCDM model. Recently, the theoretical foundation for a varying c in spacetime has been derived, using a class of scale-invariant actions that enable non-minimal coupling of matter with gravity via a dilaton field [6, 7]. In cosmology, a varying c should impact the propagation of light rays from distant SNeIa to an Earth-based observer, fundamentally altering the distance-vs-redshift relationship. This modification necessitates a re-evaluation of the H 0 value derived from the Hubble diagram of SNeIa , potentially replacing the canonical value \u223c 70 that relies on the \u039bCDM model. The purpose of our paper is two-fold: (i) to investigate the viability of the variable speed of light (VSL) theory developed in Refs. [6, 7] in accounting for the late-time cosmic acceleration while bypassing DE, and (ii) to determine whether-and how-the finding of H 0 \u2248 46 by BDRS for the CMB can be reconciled with our reexamination of SNeIa in the VSL context. Our paper is organized into four major parts: Section XI discusses and summarizes our findings, and the appendices contain technical supplements.", "pages": [1, 2]}, {"title": "II. A NEW MECHANISM TO GENERATE VARYING c FROM DILATON DYNAMICS", "content": "The variability in the speed of light was first recognized by Einstein in 1911 during his pursuit for a generally covariant theory of gravitation, which ultimately culminated in the theory of General Relativity (GR) in 1915. In Ref. [8] he explicitly allowed the gravitational field \u03a6 to influence the value of c in spacetime. In particular, he proposed that c = c 0 ( 1 + \u03a6 /c 2 ) , where c 0 is the speed of light at a reference point where \u03a6 vanishes. Notably, he conceived this radical idea six years after his formulation of Special Relativity (SR). As Einstein emphasized in [9, 10], a variation in c does not contradict the principle of the constancy of c under Lorentz transformations, an underpinning requirement of SR. This is because Lorentz invariance, confirmed by the MichelsonMorley (MM) experiment, is only required to hold in local inertial frames and does not necessitate its global validity in curved spacetimes. More concretely, in a region vicinity to a given point x \u2217 , the tangent frames to the spacetime manifold possess the Lorentz symmetry with a common value of c applicable only to that region . Yet, in a spacetime influenced by a gravitational field, different regions can-in principle-correspond to different values of c . Utilizing the language of Riemannian geometry, the speed of light can be promoted to a scalar field: while c is an invariant (i.e., unaffected upon diffeomorphism), it can nonetheless be position-dependent , viz. c ( x \u2217 ). Einstein's pioneering concept of VSL, nevertheless, was quickly overshadowed by the success of his GR and subsequently fell into dormancy for several decades. The variability of c was briefly rediscovered by Dicke in 1957 [11], prior to his own development of Brans-Dicke gravity [12], which instead allowed Newton's gravitational constant G to vary. In the 1990s, the idea of VSL was independently revived by Moffat [13] and by Albrecht and Magueijo [14] in the context of early-time cosmology. Their proposals aimed to resolve the horizon puzzle while avoiding the need for cosmic inflation. Since then, several researchers actively explore various aspects of VSL [15-86]. In a recent report [6], we considered a scale-invariant action that facilitates non-minimal coupling of matter with gravity via a dilaton field \u03c7 . We uncovered a hidden mechanism that induces a dependence of c and \u210f on the dilaton field \u03c7 , thereby causing c and \u210f to vary alongside \u03c7 in spacetime. Below is a recap of our mechanism.", "pages": [2]}, {"title": "The essence of our VSL mechanism", "content": "Let us consider a prototype action: The gravitational sector L grav is equivalent to the wellknown Brans-Dicke theory, L BD = \u03c6 R\u03c9 \u03c6 g \u00b5\u03bd \u2202 \u00b5 \u03c6\u2202 \u03bd \u03c6 , upon substituting \u03c6 := \u03c7 2 [12]. The matter Lagrangian L mat describes quantum electrodynamics (QED) for an electron field \u03c8 , coupled with an electromagnetic field A \u00b5 (with the field tensor defined as F \u00b5\u03bd := \u2202 \u00b5 A \u03bd -\u2202 \u03bd A \u00b5 ) and embedded in a curved spacetime characterized by the metric g \u00b5\u03bd . The Dirac gamma matrices satisfy \u03b3 \u00b5 \u03b3 \u03bd + \u03b3 \u03bd \u03b3 \u00b5 = 2 g \u00b5\u03bd , and the spacetime covariant derivative \u2207 \u00b5 acts on the spinor via vierbein and spin connection. However, the electron field couples non-minimally with gravity via the dilaton field, viz. \u03c7 \u00af \u03c8\u03c8 . All parameters \u03b1 , \u00b5 , and \u03c9 are dimensionless. The full action is scale invariant, viz. unchanged under the global Weyl rescaling It has been established in [87, 88] that a scale-invariant action, such as the one described in Eqs. (6)-(8), can evade observational constraints on the fifth force. Next, let us revisit the 'canonical' QED action for an electron field \u03c8 carrying a U (1) gauge charge e and inertial mass m , coupled with an electromagnetic field A \u00b5 and embedded in an Einstein-Hilbert spacetime In these expressions, the quantum of action \u210f , speed of light c , and Newton's gravitational parameter G are explicitly restored. Excluding the kinetic term g \u00b5\u03bd \u2202 \u00b5 \u03c7\u2202 \u03bd \u03c7 of the dilaton in Eq. (3), the action S can be brought into the form S 0 via the following identification: These identities link the charge e and inertial mass m of the electron with the three 'fundamental constants' c , \u210f , and G , as well as the dilaton field \u03c7 . To proceed, we require that the intrinsic properties of the electron-namely, its charge e and inertial mass m -remain independent of the background spacetime, particularly the dilation field \u03c7 which belongs to the gravitational sector. Consequently, based on the last two identities of Eq. (9), both the speed of light c and the quantum of action \u210f must be treated as scalar fields related to \u03c7 . The following assignments capture this relationship: /negationslash Here, the subscript \u03c7 signifies the dependence of c and \u210f on \u03c7 , while \u02c6 c and \u02c6 \u210f represent the values of c \u03c7 and \u210f \u03c7 at a reference point where \u03c7 = \u02c6 \u03c7 (with \u02c6 \u03c7 = 0). It is straightforward to derive from Eqs. (9) and (10) that This confirms that e and m are indeed constants 1 . Furthermore, G is also constant. As the dilaton \u03c7 varies in spacetime as a component of the gravitational sector, the scalar fields c \u03c7 and \u210f \u03c7 , defined in Eq. (10), also vary in spacetime. Therefore, the dynamics of the dilaton \u03c7 induces variations in c \u03c7 and \u210f \u03c7 on the spacetime manifold.", "pages": [2, 3]}, {"title": "Comments on Brans-Dicke's variable G", "content": "Traditionally, Brans-Dicke (BD) gravity is associated with variable Newton's gravitational constant G [12]. It should be noted that Brans and Dicke only allowed matter to couple minimally with gravity, namely, through the 4-volume element \u221a -g ; in this case, the matter action is not scale invariant. To achieve scale invariance, matter must couple with gravity in a non-minimal way, such as the Lagrangian given in Eq. (4). In this case, if one presumes that c and \u210f are constants , then the mass parameters of (massive) fields also become variable [89-92]. Indeed, under the assumption of constant c and \u210f , Eq. (9) readily produces Here, the subscript \u03c7 signifies the dependence of m and G on \u03c7 . In [6], we referred to these results as 'the FujiiWetterich scheme', since these authors appear to be the first to report results (in [89-92]) essentially equivalent to Eq. (12). In this scheme, while m is associated with \u03c7 , the charge e remains independent of \u03c7 , rendering an unequal treatment of e and m . Moreover, whereas \u03c7 affects the electron's mass per Eq. (12), massless particles, such as photons, remain unaffected . Our mechanism thus represents a significant departure from the variable G (and mass) approach. Importantly, it allows the dilaton \u03c7 -through its influence on the speed of light c \u03c7 and quantum of action \u210f \u03c7 -to govern the propagation and quantization of all fields -viz. electron and photon-on a universal and equal basis. While both approaches-(i) variable G and m versus (ii) variable c and \u210f -are mathematically permissible, they are not physically equivalent [6], and the validity of each approach should be determined through empirical evidence, including predictions, experiments and observations. Our mechanism leads to a direct and immediate consequence in cosmology, however. Specifically, the aspect of our mechanism where the dynamical \u03c7 induces a variation in c \u03c7 , which in turn governs massless field (viz. the light quanta) has significant implications. A varying c influences the propagation of light rays emitted from distant sources toward an Earth-based observer, thereby affecting the Hubble diagram of these light sources, particularly for SNeIa. This intuition serves as the underpinning for the analysis presented in the remainder of this paper.", "pages": [3, 4]}, {"title": "Scaling properties of length, time, and energy", "content": "In [6] we further deduced that at a given point x \u2217 , the prevailing value of the dilaton \u03c7 ( x \u2217 ) determines the lengthscale, timescale, and energy scale for physical processes occurring at that point. The lengthscale l and energy scale E are dependent on \u03c7 as follows However, the most important outcome is that the timescale \u03c4 behaves in an anisotropic fashion, as or, equivalently This leads to a novel time dilation effect induced by the dilaton, representing a concrete prediction of our mechanism. Moreover, the 3 / 2-exponent in this time scaling law plays a crucial role in the Hubble diagram of SNeIa, as we will explore in the following sections. A detailed exposition of our mechanism and the new time dilation effect is presented in Ref. [6].", "pages": [4]}, {"title": "III. IMPACTS OF VARYING c IN AN EINSTEIN-DE SITTER UNIVERSE", "content": "In a cosmology accommodating VSL, as a lightwave travels from a distant SNeIa toward an Earth-based observer, a varying speed of light along its trajectory induces a refraction effect akin to that experienced by a physical wave traveling through an inhomogenous medium with varying wave speed. The alteration of the wavelength results in a new set of cosmographic formulae, including a modified Hubble law and a modified relationship between redshift and luminosity distance.", "pages": [4]}, {"title": "A. A drawback in previous VSL analyses of SNeIa", "content": "It is important to note that since the revival of VSL by Moffat and the Albrecht-Magueijo team in the 1990s, several authors have applied VSL to late-time cosmology, particularly in the analysis of the Hubble diagram of SNeIa. However, these attempts have not met with much success [15, 16, 33-37]. A common theme among these analyses is the assumption that c varies as a function of the global cosmic factor a of the Friedmann-Lema\u00eetre-Roberson-Walker (FLRW) metric (e.g., in the form c \u221d a -\u03b6 first proposed by Barrow [15]). These works generally conclude that, despite the dependence of c on a , VSL does not alter the classic Lema\u00eetre redshift formula 1 + z = a -1 and, therefore, cannot play any role in the Hubble diagram of SNeIa. However, upon closer scrutiny into these works, we identify a significant oversight: they implicitly assumed that c is a function solely of cosmic time t , through the dependence of a on t in the FLRW metric. This assumption is not valid in our VSL framework, where c -through its dependence on the dilaton field \u03c7 -varies in both space and time, rather than time alone. In this section, as well as Sections IV and V, we will demonstrate that the variation of c as a function of the dilaton field \u03c7 , rather than merely as a function of the cosmic factor (viz. a ) as assumed in prior VSL works, fundamentally alters the Lema\u00eetre redshift formula and necessitates a re-analysis of SNeIa data.", "pages": [4]}, {"title": "B. The modified FLRW metric", "content": "The FLRW metric for the isotropic and homogeneous intergalactic space reads where a ( t ) is the global cosmic scale factor that evolves with cosmic time t . Our goal is to investigate whether an Einstein-de Sitter universe, when supplemented with a varying c , can account for the Hubble diagram of SNeIa as provided by the Pantheon Catalog. We will make three working assumptions: Assumption #1: The FLRW universe is spatially flat, corresponding to \u03ba = 0. There is robust observational evidence supporting this assumption. Assumption #2: The cosmic scale factor evolves as Justification: In our VSL mechanism, the timescale \u03c4 and lengthscale l of a given physical process are related by \u03c4 \u221d l 3 / 2 , as expressed in Eq. (15). Regarding the evolution of the FLRW metric, its timescale and lengthscale can be identified with t and a , respectively. The growth law given in Eq. (17) is therefore justified. Note: This growth is identical to the evolution of an EdS universe, viz. a spatially flat, expanding universe consisting solely of matter, with no contribution from DE or a cosmological constant. Assumption #3: The dilaton field in the cosmic background depends on the cosmic factor in the form Justification: In our VSL mechanism, the lengthscale of a given physical process is inversely proportional to the dilaton field, per Eq. (13). Given that the cosmic factor a plays the role of the lengthscale for the FLRW metric, the dependency expressed in Eq. (18) is therefore justified. Combining Eqs. (10) and (18) then renders c \u221d a -1 / 2 , or more explicitly Here, a 0 is the current cosmic scale factor (often set equal 1), and c 0 is the speed of light measured at our current time in the intergalactic space. We should emphasize that the value of c 0 is not identical with the one measured inside the Milky Way , which is equal to 300 , 000 km/s. This is because the Milky Way is a gravitationally bound structure whereas the intergalactic space is regions subject to cosmic expansion. This issue will be explained in Section IV. Combining Eqs. (16) and (19), and setting \u03ba = 0, we then obtain the modified FLRW metric which describes an EdS universe with a declining speed of light, per c \u221d a -1 / 2 .", "pages": [4, 5]}, {"title": "C. Frequency shift", "content": "For the modified FLRW metric derived above, the null geodesic ( ds 2 = 0) for a lightwave traveling from a distant emitter toward Earth (viz. d \u2126 = 0) is Hereafter, we will use the subscripts ' em ' and ' ob ' for 'emission' and 'observation' respectively. Denote t em and t ob the emission and observation time points of the lightwave, and r em the co-moving distance of the emitter from Earth. From (21), we have: The next wavecrest to leave the emitter at t em + \u03b4t em and arrive at Earth at t ob + \u03b4t ob satisfies: Subtracting these two equations yields: which leads to the ratio between the emitted frequency and the observed frequency: This contrasts with the standard relation, \u03bd ob \u03bd em = a em a ob . To derive a Lema\u00eetre formula applicable for VSL, further consideration is needed. This task will be carried out in the next section.", "pages": [5]}, {"title": "IV. IMPACTS OF VARYING c ACROSS BOUNDARIES OF GALAXIES", "content": "This section presents the pivotal elements that enable the 3 / 2-exponent in the frequency ratio, as expressed in Eq. (25), to manifest in observations.", "pages": [5]}, {"title": "A. The loss of validity of Lema\u00eetre formula", "content": "Let us first revisit the drawback in previous VSL works alluded to in Section III A. The frequency ratio given by Eq. (25) can be converted into the wavelength ratio This expression is exactly identical to that in standard cosmology, viz. where c is non-varying. At first, it may seem tempting to relate the redshift z with \u03bb ob -\u03bb em \u03bb em , namely in which a ob is set equal 1 and a em is denoted as a . In Refs. [16, 33-37], based on Eq. (27), it was concluded that the classic Lema\u00eetre redshift formula, 1 + z = a -1 , remained valid. Subsequently, virtually all empirical VSL works continued using the classic Lema\u00eetre formula to analyze the Hubble diagram of SNeIa. However, the formula in Eq. (27) is incorrect . One key reason is that \u03bb ob , representing the wavelength in the intergalactic space enclosing the Milky Way, is not what the Earth-based astronomer directly measures. For the light wave to reach the astronomer's telescope, it must pass through the gravitationally-bound Milky Way, which has its own local scale \u02c6 a ob differing from the current global cosmic scale because the matter-populated Milky Way resists cosmic expansion. This crucial point will be clarified shortly in the section below. In brief, a change in scale (from global to local) across the boundary of the Milky Way induces a corresponding change in the speed of light. This effect alters the wavelength from \u03bb ob to \u02c6 \u03bb ob which is then measured by the astronomer.", "pages": [5, 6]}, {"title": "B. Refractive effect due to varying c across boundaries of galaxies", "content": "The global cosmic scale factor a grows with comic time t , leading to the stretching of wavelength of light from \u03bb em to \u03bb ob . However, the Solar System is not subject to cosmic expansion, which is a crucial condition so that the Earth-based observer can detect the redshift of distant emission sources. It is well understood that if the Solar System expanded along with the intergalactic space, the observer's instruments would also expand in sync with the wavelength of the light ray emitted from a distant supernova, making the detection of any redshift impossible. More generally, mature galaxies-those hosting distant SNeIa, and the Milky Way where the Earth-based observer resides, are gravitationally bound and resist cosmic expansion. Despite the expansion of intergalactic space, matured galaxies maintain their relatively stable size primarily through gravitational attraction, counterbalanced by the rotational motion of the matter within them. Consequently, each galaxy has a stable local scale \u02c6 a that remains relatively constant over time, despite increases in the global scale a . As discussed in Section III B, the dilaton field \u03c7 in intergalactic space is inversely proportional to the global scale factor a . Similarly, within a galaxy, the dilaton field is inversely proportional to the galaxy's local scale \u02c6 a . Since a grows over time whereas \u02c6 a remains relatively stable, the dilaton field declines in the intergalactic space while it remains largely unchanged within galaxies. For simplicity, we model the local scale \u02c6 a as homogeneous within a galaxy, and allow it to merge with the global scale a at the galaxy's boundary. Importantly, the local scale of the Milky Way may differ from the local scale of the galaxy hosting a specific SNeIa being observed. This is because a gravitationally bound galaxy lives on an FLRW cosmic background that is expanding, rather than static. As a result, its local scale \u02c6 a might, in principle, experience modest growth in response to increases in the global scale a . Therefore, it is reasonable to model the local scale \u02c6 a of a galaxy as a universal function (to be determined) of the redshift z of the galaxy, supplemented by a negligible idiosyncratic component. Consequently, as the dilaton field \u03c7 varies across the boundaries of galaxies, the speed of light also varies at the boundaries due to the relationship c \u221d \u03c7 1 / 2 . Figures 1 depicts an intuitive schematic of a lightwave emitted from an SNeIa as it propagates to the Earth-based observer. On its journey, the lightwave undergoes 3 transits: reaches the outskirts of the Milky Way, where the global scale is a ob . While the middle stage of this journey, Transit #2, is well understood in standard cosmology, the first and last stages have been overlooked in previous VSL studies, seriously undermining their analyses and conclusions. In the context of VSL, these stages are crucial due to the additional refraction effects that occur at the boundaries of the host galaxy and the Milky Way. Figure 2 illustrates the typical behavior of \u03c7 -1 , c , and the wavelength \u03bb along a lightwave trajectory. In the top panel, it can be expected that a em > \u02c6 a em (since the host galaxy resists cosmic expansion), a ob > a em (due to the expansion of intergalactic space), and \u02c6 a ob < a ob (since the Milky Way also resists cosmic expansion). Quantitatively, we can deduce the variation of wavelength during the three transits as follows: Appendix A summarizes the components involved in the refraction that is induced by variations in the velocity of wavetrains.", "pages": [6, 7]}, {"title": "C. The 'benchmark' wavelength", "content": "There is one more crucial element to consider. In calculating the redshift of an SNeIa, it would be incorrect to directly compare the observed wavelength \u02c6 \u03bb ob with the emitted wavelength \u02c6 \u03bb em . This is because \u02c6 \u03bb em is associated with the emission event occurring inside the host galaxy, and the observer cannot directly measure \u02c6 \u03bb em since she is located within the Milky Way. If the SNeIa were situated inside the Milky Way, it would emit a wavelength \u03bb \u2217 that differs from \u02c6 \u03bb em , as the two galaxies can have different values of local scales, \u02c6 a em versus \u02c6 a ob . The wavelength \u03bb \u2217 , which the observer can measure, is the 'benchmark' wavelength to be compared with the observed \u02c6 \u03bb ob in calculating the redshift. To illustrate this issue, let us recall that the lengthscale of any physical process is inversely proportional to the dilaton field, according to Eq. (13) in Section II. For a galaxy, the dilaton field is in turn inversely proportional to the local scale of that galaxy. Consider two identical atoms, one located inside the host galaxy and the other within the Milky Way. If the atom in the host galaxy emits a lightwave with wavelength \u02c6 \u03bb em , its counterpart in the Milky Way emits an identical lightwave but with wavelength \u03bb \u2217 adjusted to the Milky Way's local scale. The following equality holds As shown in the right side of Figure 1, the observer must compare \u02c6 \u03bb ob with her 'benchmark' wavelength \u03bb \u2217 . Finally, the observer calculates the redshift z as the relative change between the observed wavelength and the 'benchmark' wavelength, given by We should note that allowing \u02c6 a ob to differ from \u02c6 a em creates a potential pathway to resolving the H 0 tension, a topic that will be discussed in Section X.", "pages": [7, 8]}, {"title": "V. MODIFYING REDSHIFT FORMULAE AND HUBBLE LAW USING VARYING c", "content": "We are now fully equipped to derive cosmographic formulae applicable to our VSL cosmology.", "pages": [8]}, {"title": "A. Modifying the Lema\u00eetre redshift formula", "content": "What is remarkable in the demonstration depicted in Figure 1 is that the stretching of the wavecrest during Transit #3 does not cancel out the compression of the wavecrest during Transit #1. The net effect of the two transits increases the value of z and results in a new formula for the redshift. Below is our derivation. Combining Eq. (31) with Eqs. (28), (26), and (29), we obtain Defining the ratio of local scales as a function of redshift: where F ( z = 0) = 1, and setting we arrive at the modified Lema\u00eetre redshift formula : If F ( z ) \u2261 1 \u2200 z , viz. all galaxies have the same local scale, the modified Lema\u00eetre redshift formula simplifies to: These formulae are decisively different from the classic Lema\u00eetre redshift formula, 1 + z = a -1 . The 3 / 2exponent in the modified Lema\u00eetre formulae arises as a result of the anisotropic time scaling in Eq. (15). It is essential to emphasize that the alteration in wavelength-due to the refraction effect across boundaries of galaxies-is instrumental in enabling the VSL effects to manifest in the modified Lema\u00eetre redshift formula. To the best of our knowledge, existing VSL analyses in the literature have not considered this wavelength alteration. This omission hinders theirs ability to detect the effects of VSL on the Hubble diagram of SNeIa and late-time cosmic acceleration.", "pages": [8]}, {"title": "B. Modifying the Hubble law: An emergent multiplicative factor of 3 / 2", "content": "The current-time Hubble constant H 0 is defined as For a lowz emission source, this yields Let d = c 0 . ( t 0 -t ) represent the distance from Earth to the emission source, and note that F ( z ) /similarequal 1 for low z . For small z and d , the Taylor expansion for the modified Lema\u00ee redshift formula obtained in Eq. (35) produces the modified Hubble law : In comparison to the classic Hubble law, where the speed of light is explicitly restored: the modified Hubble law acquires a multiplicative prefactor of 3/2. A significant consequence of this adjustment is a (re)-evaluation of the Hubble constant H 0 , which has implications for BDRS's CMB analysis and the age problem-topics that will be discussed in Sections VIII and IX.", "pages": [8]}, {"title": "C. Modifying the distance-redshift formula", "content": "Using the evolution a \u221d t 2 / 3 per Eq. (17), we can derive that The modified Lema\u00eetre redshift formula, Eq. (35), can be recast as and, with the aid of Eq. (41), renders For the modified FLRW metric described in Eq. (20), the coordinate distance in flat space is From Eqs. (43) and (44), and noting that F ( z = 0) = 1, we obtain the modified distance-redshift formula in a compact expression", "pages": [9]}, {"title": "D. Modifying the luminosity distance-redshift formula", "content": "In standard cosmology, the luminosity distance d L is defined via the absolute luminosity L and the apparent luminosity J : The absolute luminosity L and the apparent luminosity J are related as In the right hand side of Eq. (47), the first term \u02c6 \u03bb em / \u02c6 \u03bb ob represents the 'loss' in energy of the redshifted photon known as the 'Doppler theft' 2 . The second (identical) term \u02c6 \u03bb em / \u02c6 \u03bb ob arises from the dilution factor in photon density, as the same number of photons is distributed over a prolonged wavecrest in the radial direction (i.e., along the light ray). The 4 \u03c0r 2 in the left hand side of Eq. (47) accounts for the spherical dilution in flat space. From (46) and (47), we obtain Using the definitions of redshift and the 'benchmark' wavelength, Eqs. (31) and (30) respectively, the luminosity distance becomes or, by including (33): Due to the refraction effect during Transit #3, the apparent luminosity distance observed by the Earth-based astronomer \u02c6 d L differs from d L by the factor \u02c6 c ob /c ob , viz. Finally, combining Eqs. (45), (50), and (51), we arrive at the modified luminosity distance-redshift relation: where \u02c6 d L is the luminosity distance observed by the Earth-based astronomer and \u02c6 c ob the speed of light measured in the Milky Way (i.e., 300 , 000 km/s). Formula (52) contains a single parameters H 0 and involves a function F ( z ) that captures the evolution of the local scale of galaxies as a function of redshift.", "pages": [9]}, {"title": "VI. RE-ANALYZING PANTHEON CATALOG USING VARYING c", "content": "This section applies the new formula, Eq. (52), to the Combined Pantheon Sample of SNeIa. In [93], Scolnic and collaborators produced a dataset of apparent magnitudes for 1 , 048 SNeIa with redshift z ranging from 0 . 01 to 2 . 25, accessible in [94]. For each SNeIa i th , the catalog provides the redshift z i , the apparent magnitude m Pantheon B,i together with its error bar \u03c3 Pantheon i . We apply the absolute magnitude M = -19 . 35 to compute the distance modulus, \u00b5 Pantheon := m Pantheon B -M . The distance modulus is then converted to the luminosity distance d L using the following relation: The Pantheon data, along with their error bars, are displayed in the Hubble diagram shown in Fig. 3.", "pages": [9]}, {"title": "A. \u039b CDM and standard EdS as benchmarking models", "content": "For benchmarking purposes, we first fit the Pantheon Catalog with the flat \u039bCDM model. The luminosity distance-redshift relation for this model is a wellestablished result (where \u2126 M +\u2126 \u039b = 1) Our fit will minimize the normalized error with the sum taken over all N = 1 , 048 Pantheon data points. The best fit for the \u039bCDM model yields H 0 = 70 . 2 km/s/Mpc, \u2126 M = 0 . 285, \u2126 \u039b = 0 . 715, with the minimum error \u03c7 2 min (\u039bCDM) = 0 . 98824. The d L -z curve for the \u039bCDM model is depicted by the dashed line in Fig. 3. Also for benchmarking purposes, we consider a 'fiducial' model: the standard EdS universe (i.e. with constant speed of light). The luminosity distance-redshift formula for this fiducial model can be obtained by setting \u2126 \u039b = 0 and \u2126 M = 1 in Eq. (54), yielding Figure 3 displays the d l -z curve as a dotted line for the fiducial EdS model (using the H 0 = 70 . 2 value obtained above for the \u039bCDM model). This curve fits well with the Pantheon data for low z but fails to capture the data for high z . The Pantheon data with z \u2273 0 . 1 show an excess in the distance modulus compared with the baseline EdS model, meaning that highredshift SNeIa appear dimmer than predicted by the fiducial EdS model. As a result, this discrepancy necessitated the introduction of the \u039b component, commonly referred to as dark energy, characterized by an equation of state w = -1 and an energy density of \u2126 \u039b \u2248 0 . 7.", "pages": [10]}, {"title": "B. Fitting with VSL model: Disabling F ( z )", "content": "In this subsection, we will disable the evolution of the local scale of galaxies by setting F ( z ) \u2261 1 in Formula (52). This means that the fit is carried out with respect to a simplified formula with one adjustable parameter H 0 : Hereafter, the luminosity distance \u02c6 d L observed by the Earth-based astronomer will be used in the conversion described by Eq. (53). The best fit of the Pantheon data to this formula yields H 0 = 44 . 4 km/s/Mpc, corresponding to \u03c7 2 min = 1 . 25366. Figure 3 displays our fit as the solid line. Although this fit performs worse than the \u039bCDM model, which has \u03c7 2 min (\u039bCDM) = 0 . 98824, it substantially reduces the excess in distance moduli for z \u2273 0 . 1 compared with the 'fiducial' EdS model, as shown in Fig. 3. We must emphasize that both Formulae (56) and (57) are one-parameter models. Both models are based on an EdS universe, but our VSL model accommodates varying speed of light, whereas the 'fiducial' EdS model operates under the assumption of a constant speed of light. Therefore, we can conclude that varying speed of light is responsible for the improved performance of our VSL model compared to the 'fiducial' EdS model. This aspect can be explained as follows. In the high z limit, Formula (56) of the 'fiducial' EdS model yields whereas Formula (57) of our VSL model gives The additional ln z term in Eq. (59) compared to Eq. (58) induces a steeper slope in the highz portion of the d L -z curve, which translates to an excess in distance modulus at high redshift. Notably, our VSL model does not require dark energy whatsoever to account for this behavior. The performance of our VSL model can be improved by enabling the function F ( z ), which involves allowing the local scales of galaxies to evolve. This task will be carried out in the following subsections.", "pages": [10]}, {"title": "C. Enabling F ( z ) : A binning approach", "content": "In this subsection, we will incorporate the variation in the local scales of galaxies, as characterized by F ( z ). Developing model for F ( z ) would require knowledge of galactic formation and structures. Here, we avoid that complexity by extracting F ( z ) directly from the Pantheon data. To do so, we spit the Pantheon dataset into 10 bins ordered by increasing redshift. Bins #1 to #9 each contains 105 data points, while Bin #10 contains 103 data points, totaling 1,048 data points. The range of redshift for each bin is given in Table I. All Pantheon data points in Bin # i are treated as having a common value of F ( i ) for the function F ( z ). For a data point # j that belongs to Bin # i , Formula (52) reads Instead of fitting each bin separately, we impose one common value for H 0 across all bins. The fit thus involves H 0 and 10 values for { F ( i ) , i = 1 .. 10 } . Figure 4 displays our fit to Formula (60). The best fit yields H 0 = 46 . 6 km/s/Mpc, and the values of F ( i ) are given in the last column of Table I. The minimum error is \u03c7 2 min = 0 . 97803. The staircase lines in Fig. 5 depict F ( i ) as function of z and the cosmic factor a . The values F ( i ) reveal a monotonic decrease with respect to redshift, or equivalently, a monotonic increase in terms of a . This behavior indicates that the local scales of galaxies gradually grow during the course of cosmic expansion, implying that galaxies cannot fully resist this expansion. From z /similarequal 2 to the present time, galaxies have slightly expanded by about 3%, during which process the cosmic scale factor has approximately doubled, i.e. a | z /similarequal 2 /similarequal 0 . 5 with F ( z /similarequal 2) \u2248 0 . 969.", "pages": [11]}, {"title": "D. Enabling F ( z ) : A functional form approach", "content": "The steady decline of F ( i ) across the 10 bins with respect to redshift suggests adopting the following functional form for F ( z ): with b 1 \u2208 R + , b 2 \u2208 R + , and F \u221e \u2208 [0 , 1], supporting a monotonic interpolation from F ( z = 0) = 1 to F ( z \u2192 \u221e ) = F \u221e . After some experimentation, we find that setting b 1 = b 2 = 2 offers good overall performance. We will apply Formula (52) in conjunction with (61) (with b 1 = b 2 = 2) to fit the Pantheon dataset. The best fit is displayed in Fig. 6, yielding H 0 = 47 . 22 and F \u221e = 0 . 931. The minimum error is \u03c7 2 min = 0 . 98556, a performance that is competitive with-if not exceedingthat of the \u039bCDM model, which has \u03c7 2 min (\u039bCDM) = 0 . 98824. Figure 5 shows the variation of F in dashed lines with respect to both redshift and the cosmic factor. We also produce the joint distribution for H 0 and F \u221e , as shown in Figure 7, yielding H 0 = 47 . 21 \u00b1 0 . 4 km/s/Mpc (95% CL) and F \u221e = 0 . 931 \u00b1 0 . 008 (95% CL). This value of F \u221e indicates that the local scales of galaxies have increased by approximately 7% since the formation of the first stable galaxies (i.e., those at the largest redshift).", "pages": [11, 12]}, {"title": "Comparison of VSL approach with \u039b CDM model", "content": "Our VSL fit, in effect, involves two parameters: H 0 and F \u221e -the same number of parameters as the \u039bCDM model ( H 0 and \u2126 \u039b ). However, the parameter F \u221e has a well-defined astrophysical meaning; it denotes the local scales of the first stable galaxies in comparison to the local scale of the Milky Way. Moreover, the function F ( z ), which captures the evolution of the local scales of galaxies during cosmic expansion, plays a role in a potential resolution of the H 0 tension-a topic that will be discussed in Section X. In contrast, the \u039bCDM model requires a \u039b component, the nature of which is still not understood. Its energy density value \u2126 \u039b \u2248 0 . 7 also raises a coincidence problem. Furthermore, the \u039bCDM model currently encounters the H 0 tension. If the \u039b component is treated as dynamicalan approach explored in several ongoing efforts to resolve the H 0 tension-this would introduce an array of new parameters to the \u039bCDM model.", "pages": [12]}, {"title": "E. The cause for the reduction in H 0 value", "content": "In the z \u2192 0 limit, Eq. (61) with b 1 = b 2 = 2 can be approximated as From this, Formula (52) then gives leading to the modified Hubble law : This result aligns with Eq. (39) derived in Section V B based on the modified Lema\u00eetre redshift formula, Eq. (35). In contrast to the classic Hubble law, where the speed of light is constant: the modified Hubble law acquires a multiplicative factor of 3/2. Hence, in our VSL cosmology, low-redshift emission sources exhibit a linear relationship between z and the luminosity distance, but characterized by a coefficient of 3 2 H 0 rather than H 0 . Since a linear-line fit of z on d L for low-redshift emission sources is known to yield a slope of approximately 70, the resulting value of H 0 obtained through our VSL approach is thus only 2 / 3 of this value, specifically H 0 \u2248 47, rather than H 0 \u2248 70 as predicted by standard cosmology.", "pages": [12]}, {"title": "VII. A NEW INTERPRETATION: VARIABLE SPEED OF LIGHT AS AN ALTERNATIVE TO DARK ENERGY AND COSMIC ACCELERATION", "content": "The Hubble diagram of SNeIa has been interpreted as a definitive hallmark of late-time accelerated expansion, providing the (only) direct evidence for dark energy. These stellar explosions serve as 'standard candles' due to their consistent peak brightness, allowing astronomers to determine distances to the galaxies in which they reside. In the late 1990s, two independent teams, the HighZ Supernova Search Team [95] and the Supernova Cosmology Project [96], measured the apparent brightness of distant SNeIa, finding them dimmer than expected based on the EdS model, which describes a flat, expanding universe dominated by matter. This can be seen in the Hubble diagram of SNeIa (see Fig. (4)), in which the section with z \u2273 0 . 1 exhibits an distance modulus greater than that predicted by the EdS model. This behavior has been interpreted as indicating that the expansion of the universe is accelerating rather than decelerating. However, our quantitative analysis of SNeIa in the preceding section offers a new interpretation as an alternative to late-time acceleration. Mathematically, as explained in Section VI B, in the high z limit, the EdS universe yields whereas our VSL-based formula renders Thus, high-redshift SNeIa acquire an additional factor of ln z compared with the EdS model. This results in an further upward slope relative to that of the EdS model, successfully capturing the behavior of SNeIa in the high-z section. Physical intuition: There is a fundamental reasonbased on our VSL framework-behind this excess in distance modulus that we will explain below. Consider two supernovae A and B at distances d A and d B away from the Earth, such that d B = 2 d A . In standard cosmology, their redshift values z A and z B are related by z A \u2248 2 z A (to first-order approximation). However, this relation breaks down in the VSL context. In a VSL cosmology which accommodates variation in the speed of light in the form\u00ef\u00bf\u0153 c \u221d a -1 / 2 , light traveled faster in the distant past (when the cosmic factor a /lessmuch 1) than in the more recent epoch (when a \u2272 1). Therefore, the photon emitted from supernova B was able to cover twice the distance in less than twice the time required for the photon emitted from supernova A. Having spent less time in transit than what standard cosmology would require, the B-photon experienced less cosmic expansion than expected and thus a lower redshift than what the classic Lema\u00eetre formula would dictate. Namely: Conversely, consider a supernova C with z C = 2 z A . For the C-photon to experience twice the redshift of the Aphoton, it must travel a distance greater than twice that of the A-photon, viz.: d C > 2 d A . This is because since the C-photon traveled faster at the beginning of its journey toward Earth, it must originate from a farther distance (thus appearing fainter than expected) to experience enough cosmic expansion and therefore the requisite amount of redshift. Namely: Consequently, the SNeIa data exhibit an additional upward slope in their Hubble diagram in the highz section. Conclusion: Hence, a declining speed of light presents a viable alternative to cosmic acceleration , eliminating the need for the \u039b component and dissolving its finetuning and coincidence problems. In our VSL framework, during the universe expansion, the dilaton field \u03c7 in intergalactic space decreases and leads to a decline in c (per c \u221d \u03c7 1 / 2 \u221d a -1 / 2 ), affecting the propagation of lightwaves from distant SNeIa to the observer. While the impact of a declining c on lightwaves is negligible within the Solar System and on galactic scales, it accumulates on the cosmic scale and makes high-redshift SNeIa appear dimmer than predicted by the standard EdS model. The VSL framework, therefore, offers a significant shift in perspective: rather than supporting a \u039b CDM universe undergoing late-time acceleration, the Hubble diagram of SNeIa should be reinterpreted as evidence for a declining speed of light in an expanding Einstein-de Sitter universe.", "pages": [13]}, {"title": "VIII. RETHINKING BLANCHARD-DOUSPISROWAN-ROBINSON-SARKAR'S 2003 CMB ANALYSIS AND H 0 \u2248 46", "content": "Let us now turn our discussion to a remarkable proposal advanced by Blanchard, Douspis, Rowan-Robinson, and Sarkar (BDRS) in 2003 and its relation to our SNeIa analysis. It is well established that the \u039bCDM model, augmented by the primordial fluctuation spectrum (presumably arising from inflation) in the form P ( k ) = Ak n , successfully accounts for the observed anisotropies in the cosmic microwave background (CMB). This model predicts a dark energy density of \u2126 \u039b \u2248 0 . 7, a Hubble constant of H 0 \u2248 67, and a spectral index n \u2248 0 . 96 [97, 98]. Yet, in [4] BDRS reanalyzed the CMB, available at the time from the Wilkinson Microwave Anisotropy Probe (WMAP), in a new perspective. These authors deliberately relied on the EdS model, which corresponds to a flat \u039bCDM model with \u2126 \u039b = 0. Rather than invoking the \u039b component, they adopted a slightly modified form for the primordial fluctuation spectrum. They reasoned that, since the spectral index n is scale-dependent for any polynomial potential of the inflaton and is constant only for an exponential potential, it is reasonable to consider a double-power form for the spectrum of primordial fluctuations with a continuity condition ( A 1 k n 1 \u2217 = A 2 k n 2 \u2217 ) across the breakpoint k \u2217 . Using this new function, BDRS produced an excellent fit to the CMB power spectrum, resulting in the following parameters: H 0 = 46 km/s/Mpc, \u03c9 baryon := \u2126 baryon ( H 0 / 100) 2 = 0 . 019, \u03c4 = 0 . 16 (the optical depth to last scattering), k \u2217 = 0 . 0096 Mpc -1 , n 1 = 1 . 015, and n 2 = 0 . 806. The most remarkable outcome of the BDRS work is the 'low' value of H 0 = 46, representing a 34% reduction from the accepted value of H 0 \u223c 70. A detailed follow-up study by Hunt and Sarkar [5], based on a supergravity-induced multiple inflation scenario, yielded a comparable value of H 0 \u2248 44. Notably, around the same time, Shanks argued that a value of H 0 \u2272 50 might permit a simpler inflationary model with \u2126 baryon = 1, i.e. without invoking dark energy or cold dark matter [99]. The success achieved by BDRS in reproducing the CMB power spectrum can be interpreted as indicating a degeneracy in the parameter space { \u2126 \u039b , H 0 } . Specifically, the BDRS pair { \u2126 \u039b = 0 , H 0 = 46 } is 'nearly degenerate' with the canonical pair { \u2126 \u039b \u2248 0 . 7 , H 0 \u2248 70 } , insofar as the CMB data is concerned. Importantly, BDRS's modest modification in the primordial fluctuation spectrum can make \u2126 \u039b redundant. In other words, the \u039b component is vulnerable to other exogenous underlying assumptions that supplement the \u039b CDM model . Notably, strong degeneracies in the parameter space related to the CMB have been reported recently. In [100] Alestas et al found that the best-fit value of H 0 obtained from the CMB power spectrum is degenerate with a constant equation of state (EoS) parameter w ; the relationship is approximately linear, given by H 0 + 30 . 93 w -36 . 47 = 0 (with H 0 in km/s/Mpc). Although this finding is not directly related to the BDRS work, the H 0 -vsw degeneracy reinforces the general conclusion regarding the sensitivity of H 0 to other exogenous underlying assumptions that supplement the \u039bCDM model-in the case of Alestas et al, the EoS parameter w . While a drastically low value of H 0 \u2248 46 at first seems to be 'a steep price to pay', we have demonstrated in the preceding sections that this new value is fully compatible with the H 0 = 47 . 2 obtained from the Hubble diagram of SNeIa data when analyzed within the context of VSL cosmology. Consequently, the \u039b component becomes redundant not only for the CMB but also for SNeIa. The alignment of our findings with those of BDRS is especially remarkable for several reasons: Together with our SNeIa analysis, the work of BDRS eliminates the need for the \u039b component regarding the two 'orthogonal' datasets-the CMB and SNeIa. Future applications of our VSL framework to gravitational lensing, Baryonic Acoustic Oscillations (BAO), and other areas are worthwhile.", "pages": [13, 14]}, {"title": "What caused BDRS to abandon their H 0 \u2248 46 finding?", "content": "In 2006 BDRS revisited their 2003 A&A proposal by applying it to the Sloan Digital Sky Survey (SDSS) of luminous red galaxies (LRG) which became available in [101]. In their 2006 follow-up work [102], BDRS claimed that the 'low' value of H 0 \u2248 46 was unable to produce an acceptable fit to the two-point correlation function of LRG in observed (redshift) space. The upper panel of Fig. 8 reproduces their finding, showing that the SDSS data (and their error bars in red segments) largely aligns with the \u039bCDM model (dotted line), while the BDRS model (dashed-dotted line) is significantly off. This discrepancy eventually forced BDRS to abandon their 2003 proposal in its entirety (although Hunt and Sarkar continued with their follow-up study shortly thereafter [5]). However, we believe that BDRS's 2006 SDSS analysis contained an oversight, in light of our VSL cosmology. The standard Lema\u00eetre redshift formula and the conventional Hubble law are not applicable in the presence of varying speed of light. As discussed in Sections V A and VB, these expressions are modified by a factor of 3 / 2 due to varying c . Therefore, the SDSS would need a reevaluation to incorporate this VSL-induced adjustment. Here, we tentatively make a rudimentary fix: we correct the comoving distance s (measured in multiples of h , defined as H 0 / (100 km/s/Mpc)), downward by a factor of 3 / 2. In the lower panel of Fig. 8, we adjust the SDSS data (and their error bars) by reducing s by a factor of 3 / 2 and s 2 \u03be by a factor of (3 / 2) 2 . Upon these adjustments, the peak (at s /similarequal 75) and trough (at s /similarequal 60) of the SDSS become aligned with those of the BDRS's model (dasheddotted line), thereby lessening the discrepancy issue that led to BDRS's abandonment of their original proposal. We conclude that it was premature for BDRS to abandon their 2003 CMB study and the finding of H 0 , \u2248 46. Rather, their proposal should be revived and applied to the upgraded Planck dataset for the CMB [97]. We should also note that since the CMB data is a twodimensional snapshot of the sky at the time at recombination, VSL is not expected to be a dominant player in the CMB. Nevertheless, potential impacts of VSL on the CMB are an interesting avenue for future research.", "pages": [14, 15]}, {"title": "IX. RESOLVING THE AGE PROBLEM", "content": "From the definition of the Hubble constant, H ( t ) := 1 a da dt , and the evolution, a \u221d t 2 / 3 , the age of an EdS universe is related to the current-time H 0 value by A value of H 0 \u223c 70, would result in an age of 9.3 billion years which would be too short to accommodate the existence of the oldest stars-a paradox commonly referred to as the age problem. Standard cosmology resolves the age problem by invoking the \u039b component which induces an acceleration phase following a deceleration phase. The spatially flat \u039bCDM model is known to give the age formula in an analytical form (with \u2126 M +\u2126 \u039b = 1 and \u2126 \u039b > 0) [103] which restores Eq. (68) when \u2126 M \u2192 1 and \u2126 \u039b \u2192 0. For positive \u2126 \u039b , the age exceeds 2 / (3 H 0 ). With H 0 = 70 . 2, \u2126 M = 0 . 285, \u2126 \u039b = 0 . 715, it yields an age of 13 . 6 billion years, an accepted figure in standard cosmology. However, our VSL cosmology naturally overcomes the age problem without invoking the \u039b component. The reason is that H 0 is reduced by a factor of 3 / 2, as detailed in Section VI E. The reduced value H 0 = 47 . 22 \u00b1 0 . 4 (95% CL) promptly yields t 0 = 13 . 82 \u00b1 0 . 11 billion years (95% CL), consistent with the accepted age value, thereby successfully resolving the age problem.", "pages": [15]}, {"title": "X. TOWARD A NEW RESOLUTION OF THE H 0 TENSION", "content": "Galaxies are gravitationally bound structures, stabilized by gravitational attraction and rotational motion of matter within them. However, they are embedded in a cosmic background that is not static, but rather expanding over time. As such, stable galaxies in principle may adjust to the growth in the scale of the 'ambient' intergalactic space surrounding them; viz., their local scales may increase in response to the cosmic expansion. This growth in the local scales of galaxies-if it exists-would be of astronomical nature. To investigate this phenomenon, one could explore the evolution of a spinning disc-shaped distribution of matter (serving as a simplified model for a galaxy) on an expanding cosmic background within the scale-invariant theory mentioned in Section II, although such an exploration lies beyond the scope of this paper. We should note that recent observational studies have reported evidence of galaxies experiencing growth in size [104, 105]. For our purposes, in Section IV, we have modeled the local scale \u02c6 a of an individual galaxy as a function of its redshift z , supplemented with a negligible idiosyncratic component that randomly varies from one galaxy to another. The function F ( z ), defined in Eq. (33) as the ratio of the local scale of galaxies at redshift z to the local scale of the Milky Way, captures the evolution of the local scale over cosmic time. In Section VI C, F ( z ) was empirically extracted from the Pantheon Catalog, with Fig. 5 displaying F ( z ) as a function of the redshift and of cosmic scale factor, respectively. In accordance with our expectation, the local scale \u02c6 a of galaxies gradually increases in response to the growth of the global cosmic factor a over cosmic time.", "pages": [15, 16]}, {"title": "A 'running' H 0 ( z )", "content": "The function F ( z ) can be absorbed into an 'effective' Hubble constant H 0 ( z ) which depends on redshift z . Specifically, Formula (52) can be rewritten as where the newly introduced function H 0 ( z ) is given by Formulae (70) and (71) thus allow for a current-time H 0 ( z ) 'running' as a function of the redshift of the data that are used to estimate it. With the function F ( z ) parametrized in Eq. (61) with b 1 = b 2 = 2 and F \u221e = 0 . 931 as produced in Section VI D, H 0 ( z ) can be computed using Eq. (71), as displayed in Fig. 9. At first, H 0 ( z ) decreases from 47 . 2 (at z = 0) to 41 . 5 (at z /similarequal 2), experiencing a 12% reduction. For z \u2273 2, H 0 ( z ) slowly rerises. At z \u2192 \u221e , with F \u221e = 0 . 931 and H 0 ( z = 0) = 47 . 2, per Eq. (71), H 0 ( z ) asymptotically approaches F \u221e H 0 = 43 . 95, representing an 7% reduction from H 0 ( z = 0). Two immediate remarks can be made: The 'running' phenomenon of H 0 ( z ) arises because astronomical objects-either the CMB or SNeIa-are subject to their local scale which gradually grows during the cosmic expansion.", "pages": [16]}, {"title": "Hints at an astronomical origin of the H 0 tension", "content": "We have, therefore, linked the 'running' current-time H 0 ( z ) with the function F ( z ). Since F ( z ) captures the evolution of galaxies' local scales in response to the growth of the global scale of intergalactic space, the 'running' H 0 ( z ) is thus of astronomical origin. The empirical evaluation for F ( z ) from the Pantheon Catalog, as detailed in Sections VI C and VI D, demonstrates that the local scale gradually increases with cosmic time, indicating that galaxies cannot fully resist cosmic expansion. As mentioned earlier, understanding the growth in F ( z ) would require an in-depth examination of a spinning disc-shaped distribution of matter in an expanding cosmic background within a scale-invariant theory of gravity and matter, a task that is left for future investigation. We should also note similar works along this line of 'running' H 0 ( z ) [106-108]. For example, in [108], Dainotti et al considered an extension of the flat w 0 w a CDM . They proposed the following luminosity distance-redshift formula with the parametrization H 0 ( z ) = \u02dc H 0 (1 + z ) -\u03b1 and an evolutionary equation of state for the \u039b component w ( z ) = w 0 + w a z/ (1 + z ). In this formula, H 0 ( z ) can be interpreted as a 'running' current-time Hubble value, which depends on the redshift of the data used to estimate it. These authors are able to bring the value of H 0 at z =1,100 within 1 \u03c3 of the Planck measurements, hence effectively removing the H 0 tension. However, unlike our approach, where the function F ( z ) has a well-defined astrophysical interpretation, the use of H 0 ( z ) and w ( z ) in Ref. [108] represents ad hoc parametrizations, with their underlying nature remaining unknown. Additionally, the H 0 ( z ) in Ref. [108] is likely of a cosmic origin, whereas the equation of state w ( z ) of the \u039b component is of a field theoretical origin. In closing of this section, our study offers a potential resolution to the H 0 tension. Furthermore, it suggests that this tension has an astronomical origin, arising from the growth in the local scale of gravitationally-bound galaxies over cosmic time.", "pages": [16, 17]}, {"title": "XI. DISCUSSIONS AND SUMMARY", "content": "This paper was inspired by three separate lines of development: 1. In 2003, Blanchard et al (BDRS) proposed a novel CMB analysis that avoids the \u039b component [4]. Based solely on the EdS model (i.e., \u2126 \u039b = 0) and adopting a double-power primordial fluctuation spectrum, BDRS achieved an excellent fit to WMAP's CMB power spectrum. Surprisingly, they obtained a new value H 0 \u2248 46, representing a 34% reduction compared to the accepted value H 0 \u223c 70 that relies on the flat \u039bCDM model with \u2126 \u039b \u2248 0 . 7. As independently reported more recently in [100], there exists a strong degeneracy inherent in the parameter space concerning the CMB data. Drawn from this observation, we can interpret BDRS's findings as indicating that within the flat \u039b CDM model, the parameter pairs { \u2126 \u039b = 0 , H 0 \u2248 46 } and { \u2126 \u039b \u2248 0 . 7 , H 0 \u2248 70 } are 'nearly degerenate' insofar as the CMB power spectrum is concerned. With a modest modification to the primordial fluctuation spectrum, the BDRS parameter pair { \u2126 \u039b = 0 , H 0 \u2248 46 } becomes advantageous over the \u039bCDM pair { \u2126 \u039b \u2248 0 . 7 , H 0 \u2248 70 } . While the cost of this modification is not prohibitive, as BDRS provided justifications in support of a double-power primordial fluctuation spectrum, the benefit is profound in that the DE hypothesis is rendered unnecessary. This perspective raises an intriguing possibility that the parameter pairs { \u2126 \u039b = 0 , H 0 \u2248 46 } and { \u2126 \u039b \u2248 0 . 7 , H 0 \u2248 70 } may also be 'nearly degenerate' insofar as the Hubble diagram of SNeIa is concerned. To materialize this possibility, one must first seek an alternative approach to late-time acceleration that does not invoke DE. We should emphasize that such an alternative-if it exists-must not only eliminate the role of \u2126 \u039b but also reduce the H 0 value from \u223c 70 to \u223c 46. This presents a stringent requirement to be met. In this paper, we build upon the VSL theory referenced in Point #2, correct the error mentioned in Point #3, and reanalyze the Pantheon Catalog. The effects of VSL modify the Lema\u00eeitre formula to 1 + z = a -3 / 2 , with the 3 / 2-exponent arising from the anisotropic time scaling referenced earlier, \u03c4 \u221d l 3 / 2 . Intuitively, this factor 3/2 influences the evaluation of H 0 , resulting in a reduction from the canonical value of H 0 \u223c 70 by a factor of 3 / 2 to H 0 = 47 . 2. The new value is compatible with BDRS's findings for the CMB mentioned in Point #1. Our derivation and analysis: The logical steps of our work are as follows. (i) Modifying the FLRW metric. The universe is modeled as an EdS spacetime supporting a varying c as (see Eq. (20)) with the expansion obeying the growth law a ( t ) = a 0 ( t/t 0 ) 2 / 3 , see Eq. (17). Justifications for this model are provided in Section III B. that the classic Lema\u00eetre redshift formula 1 + z = a -1 is inapplicable for the VSL cosmology, and is replaced by the modified Lema\u00eetre redshift formula (see Eq. (35)) with a new exponent of 3 / 2 and F ( z ) measuring the relative change in the local scale of galaxies. See Sections III C, IV and V A. (iii) Modifying the Hubble law. The 3 / 2-exponent in the modified Lema\u00eetre redshift formula above leads to the modified Hubble law (see Eq. (39)) This new Hubble law differs from the classic Hubble law by a multiplicative factor of 3 / 2, resulting in a reduction in the H 0 estimate by a factor of 3 / 2. See Section V B. (iv) Modifying the luminosity distance-vsz formula: This formula is the centerpiece of our study (see Eq. (52)) See Sections V C and V D for derivation. (v) A re-analysis of the Pantheon data based on VSL: In Section VI, we apply the Formulae above to the Combined Pantheon Sample of SNeIa. We produce an excellent fit without invoking the \u039b component; the fit is as robust as that obtained from the \u039b CDM model. The optimal values for the parameters are: Our modified Lema\u00eetre redshift formula, Eq. (35), can also effectively viewed as a form of 'redshift remapping', a technique advocated in Refs. [109-111]. Interestingly, our value of H 0 = 47 . 2 \u00b1 0 . 4 aligns with the result H 0 = 48 \u00b1 2 reported in [111]. Implications: Four important findings emerge from our analysis. that since c \u221d a -1 / 2 , light traveled faster in the past than in later epochs. As a result, lightwaves from distant SNeIa require less time to traverse the earlier sections of their trajectories, hence experiencing less cosmic expansion (and redshift) than the EdS model predicts. Hence, the highz section of the Hubble diagram of SNeIa can be explained-qualitatively and quantitativelyby a declining speed of light rather than a recent cosmic acceleration . A detailed exposition is given in Section VII. II) Reviving BDRS's work on the CMB, avoiding dark energy. Despite the very different natures of the data involved, our VSL-based analysis of SNeIa and BDRS's work on the CMB fully agree on two aspects: (i) the universe obeys the EdS model (i.e. \u2126 \u039b = 0), and (ii) H 0 is reduced to 46-47. The BDRS parameter pair { \u2126 \u039b = 0 , H 0 \u2248 46 } is advantageous over the \u039bCDM pair { \u2126 \u039b \u2248 0 . 7 , H 0 \u2248 70 } regarding both the CMB and SNeIa, which are 'orthogonal' datasets. Detailed discussions are presented in Section VIII. Together, our current work and BDRS's 2003 analysis challenge the existence of dark energy-one of the foundational assumptions of the cosmological concordance model. III) Resolving the age problem. The age of an EdS universe is given by: t 0 = 2 / (3 H 0 ). Using the reduced value of H 0 = 47 . 22 \u00b1 0 . 4 km/s/Mpc, one obtains t 0 = 13 . 82 \u00b1 0 . 11 billion years. The age problem is thus resolved through the reduction in H 0 , without requiring a recent acceleration phase induced by dark energy. See Section IX. IV) Addressing the H 0 tension. Utilizing the function F ( z ), we recast the current-time Hubble constant as a function H 0 ( z ) of redshift. Between z = 0 and z \u2192\u221e , the 'running' H 0 ( z ) exhibits a 7% decrease, a reduction in similar magnitude with the ongoing H 0 tension between the CMB and SNeIa. See Section X.", "pages": [17, 18]}, {"title": "On the cosmological time dilation extracted from the Dark Energy Survey (DES)", "content": "A recent paper [112] using DES supernova light curves showed no deviation from the relation \u2206 t obs = \u2206 t em (1 + z ). However, this finding does not contradict our modified Lemaitre redshift formula, 1 + z = a -3 / 2 F ( z ). This is because the result in Ref. [112] only verifies that the speed of light inside the galaxies hosting the supernovae and that inside the Milky Way are approximately the same. Galaxies are gravitationally bound and thus not subject to cosmic expansion. Reference [112] does not deal with the speed of light in the intergalactic space, the expansion of which causes c to decline over time. We clarified this distinction in Section IV.", "pages": [18]}, {"title": "XII. CONCLUSION", "content": "The nearly identical agreement of the CMB and SNeIa regarding the reduced value of H 0 \u223c 46-47 is highly encouraging. This alignment points toward a consistent cosmological framework based on the Einstein-de Sitter model with a variable speed of light, thus eliminating the need for energy and dissolving its fine-tuning and coincidence problems. Importantly, we have built a case for an alternative perspective: rather than supporting a \u039b CDM universe undergoing late-time acceleration, the Hubble diagram of SNeIa can be reinterpreted as evidence for a declining speed of light in an expanding Einstein-de Sitter universe . Finally, we note that the observational bounds established in the literature in support of a constant speed of light have predominantly relied on standard cosmology [33-37, 41-45, 60, 61, 65, 67, 68, 71, 82]. However, our new Lema\u00eetre redshift formula represents a critical departure from this conventional framework. Therefore, the consensus regarding the absence of variation in c in observational cosmology must be reconsidered in light of our findings, prompting a comprehensive reanalysis of these constraints.", "pages": [19]}, {"title": "ACKNOWLEDGMENTS", "content": "I thank Clifford Burgess, Tiberiu Harko, Robert Mann, Anne-Christine Davis and Eoin \u00d3 Colg\u00e1in for their constructive and supportive comments during the development of this work. \u221e ----- -----", "pages": [19]}, {"title": "Appendix A: Refraction effect", "content": "Let us start with a well-understood phenomenon: the behavior of a wavetrain in a medium with varying speed of wave. It is well established that the wavelength of the wavetrain at a given location is proportional to the speed of wave at that location: Figure 10 illustrates the change in wavelength as a wave travels at varying speed. In the upper panel, as the speed increases, the front end of the wavecrest will rush forward leaving its back end behind thus stretching out the wavecrest. In the lower panel, the reverse situation occurs: as the speed decreases, the front end of the wavecrest will slow down while its back end continues its course thus compressing the wavecrest. In either situation, the wavelength and the speed of wave are directly proportional: Note that the details of how the variation of v does not participate in formula above.", "pages": [19]}, {"title": "Appendix B: An equivalent derivation of the modified Lema\u00eetre redshift formula", "content": "We produce an alternative route by way of frequency transformation to modifying Lema\u00eetre's redshift formula (35). We have derived in Eq. (25) that For transits between local regions to global regions (i.e., Transit #1 and Transit #3 in Fig. 1 in Page 6), since \u03bb \u221d c , the frequency is: This means that the frequency of the lightwave does not change during Transit #1 and Transit #3, viz. Given that and it is straightforward to verify that a relation that is in perfect agreement with Eq. (35). an invariant energy scale , Phys. Rev. Lett. 88 , 190403 (2002), arXiv:hep-th/0112090", "pages": [19, 20]}, {"title": "arXiv:1704.07368 [gr-qc]", "content": "Rays above 100 TeV , Phys. Rev. Lett. 124 (2020) no.13, 131101, arXiv:1911.08070 [astroph.HE]", "pages": [22]}] |
2024APh...15802951T | https://arxiv.org/pdf/2008.05899.pdf | <document>
<section_header_level_1><location><page_1><loc_9><loc_91><loc_62><loc_93></location>The exact solution approach to warm inflation</section_header_level_1>
<text><location><page_1><loc_9><loc_88><loc_21><loc_90></location>Oem Trivedi a , ∗</text>
<text><location><page_1><loc_9><loc_85><loc_86><loc_86></location>a International Centre for Space and Cosmology, School of Arts and Sciences, Ahmedabad University, Navrangpura, Ahmedabad, 380009, India</text>
<section_header_level_1><location><page_1><loc_9><loc_82><loc_23><loc_83></location>ARTICLE INFO</section_header_level_1>
<text><location><page_1><loc_9><loc_78><loc_22><loc_81></location>Keywords : early universe cosmology</text>
<text><location><page_1><loc_9><loc_76><loc_22><loc_78></location>warm inflation modified gravity theories</text>
<section_header_level_1><location><page_1><loc_38><loc_82><loc_48><loc_83></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_38><loc_65><loc_91><loc_81></location>The theory of cosmic inflation has received a great amount of deserved attention in recent years due to it's stunning predictions about the early universe. Alongside the usual cold inflation paradigm, warm inflation has garnered a huge amount of interest in modern inflationary studies. It's peculiar features and specifically different predictions from cold inflation have led to a substantial amount of literature about it. Various modified cosmological scenarios have also been studied in the warm inflationary regime. In this work, we introduce the exact solution approach for warm inflation. This approach allows one to directly study warm inflationary regime in a variety of modified cosmological scenarios. We begin by outlining our method and show that it generalizes the modified Friedmann approach of Del Campo , and reduces to the well known Hamilton-Jacobi formalism for inflation in particular limits. We also find the perturbation spectra for cosmological and tensor perturbations in the early universe, and then apply our method to study warm inflation in a Tsallis entropy modified Friedmann universe. We end our paper with some concluding remarks on the domain of applicability of our work.</text>
<section_header_level_1><location><page_1><loc_9><loc_61><loc_22><loc_62></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_9><loc_33><loc_48><loc_60></location>Some of the most captivating problems in modern cosmology concern the workings of the very early Universe. A lot of the physics of how our Universe was initially is not very well understood and this was evident ever since fine tuning problems were discovered in the traditional big bang cosmology model itself. The most prominent of these problems were the flatness, horizon and monopole problems, which were very promisingly solved by the introduction of Cosmic inflation [24, 3, 38, 47, 20, 42, 17, 44, 43, 52]. Cosmic inflation is the idea that the very early Universe went through a period of rapid accelerated expansion. The dynamics of the traditional inflationary models generally consider a real scalar field, the "inflaton" field, as the predominant contributor to the universal energy density at that time. While there are other inflationary models which consider more exotic scenarios [22, 49, 35], there is still a very high amount of evident interest in single real scalar field driven inflationary regimes [5, 36, 13, 37].</text>
<text><location><page_1><loc_9><loc_12><loc_48><loc_32></location>The conventional models of inflation by a single scalar field consist of a period of rapid accelerated expansion. The sub horizon scale quantum fluctuations during this period of expansion, lead to cosmological perturbations which give us a spectrum of scalar and tensor perturbations ( vector perturbations are usually ignored). This generates the large scale structure of our Universe which we see today. Also , after the period of expansion sets off a period of " reheating" [3] , where the energy of the inflaton field decays to form radiation energy densities. These models do not take into account the dissipation of inflaton energy to form radiation while the inflationary period itself is going on and hence, these are known popularly now as " cold inflation" scenarios.</text>
<text><location><page_1><loc_13><loc_9><loc_24><loc_10></location>[email protected]</text>
<text><location><page_1><loc_24><loc_9><loc_31><loc_10></location>(O. Trivedi)</text>
<text><location><page_1><loc_12><loc_8><loc_17><loc_9></location>ORCID(s):</text>
<text><location><page_1><loc_52><loc_48><loc_91><loc_63></location>On the other hand, the class of inflationary regimes which do take into account this dissipation effect are known as " warm inflation" regimes [6, 10, 46, 9]. Warm inflation includes inflaton interactions with other fields throughout the inflationary epoch instead of confining such interactions to a distinct reheating era. Consequently, these inflationary models do not need a separate period of reheating after inflation to make the universe radiation dominated, and hence this provides a solution to the graceful exit problem of inflation [7].</text>
<text><location><page_1><loc_52><loc_42><loc_91><loc_46></location>Recently a lot of interest has been weighed towards cosmological models which modify the structure of the usual Friedmann Equation</text>
<formula><location><page_1><loc_56><loc_37><loc_91><loc_40></location>𝐻 2 = 8 𝜋 3 𝑚 2 𝑝 𝜌 (1)</formula>
<text><location><page_1><loc_52><loc_18><loc_91><loc_36></location>where we have used the units 𝑐 = ℏ = 1 and 𝑚 2 𝑝 is the usual reduced Planck mass. 𝜌 is the energy density in the Universe, which in the context of usual Inflationary Models refers to the energy density of the inflaton field. Various different cosmological scenarios bring about a change in this usual Friedmann equation . Superstring and M-Theory bring the possibility of considering our universe as a domain wall embedded in a higher dimensional space. In this scenario the standard model of particle is confined to the brane, while gravitation propagate into the bulk spacetime. The effect of extra dimensions induces a well known change in the Friedmann equation [51] ,</text>
<formula><location><page_1><loc_56><loc_13><loc_91><loc_17></location>𝐻 2 = 8 𝜋 3 𝑚 2 𝑝 𝜌 (1 + 𝜌 2 𝜆 ) (2)</formula>
<text><location><page_1><loc_52><loc_11><loc_80><loc_12></location>where 𝜆 is a measure of the brane tension.</text>
<text><location><page_1><loc_52><loc_6><loc_91><loc_9></location>It is also possible to consider quantum corrected area entropy relations and study universal evolution in such scenarios. For</text>
<text><location><page_2><loc_9><loc_92><loc_47><loc_93></location>example, [14] considered the quantum corrected Entropy</text>
<formula><location><page_2><loc_13><loc_87><loc_48><loc_90></location>𝑆 = 𝑚 2 𝑝 𝐴 4 -̃ 𝛼 ln( 𝑚 2 𝑝 𝐴 4 ) (3)</formula>
<text><location><page_2><loc_9><loc_79><loc_48><loc_86></location>where A is the area of the apparent horizon and ̃ 𝛼 is dimensionless positive constant determined by conformal anolmaly of the fields.This conformal anomaly is interpreted as a quantum correction to the entropy of the apparent horizon. The resulting Friedmann equation is of the form</text>
<formula><location><page_2><loc_13><loc_75><loc_48><loc_78></location>𝐻 2 -𝛽𝐻 4 = 8 𝜋 3 𝑚 2 𝑝 𝜌 (4)</formula>
<text><location><page_2><loc_9><loc_60><loc_48><loc_73></location>, where 𝛽 is some positive constant in the units of inverse mass squared. This type of Friedmann equation can also be arrived at by using a Chern-Simons modification of gravity [23] . Finally, there has been a surge in interest recently in the form of Black hole Entropy due to Tsallis and Cirto [55] , where they argued that argued that the microscopic mathematical expression of the thermodynamical entropy of a black hole does not obey the area law and can be modified as</text>
<formula><location><page_2><loc_13><loc_57><loc_48><loc_59></location>𝑆 = 𝛾𝐴 𝜅 (5)</formula>
<text><location><page_2><loc_9><loc_44><loc_48><loc_56></location>, where 𝛾 is an unknown constant , A is the Black Hole Horizon area and 𝜅 is a real parameter which quantifies the degree of nonextensivity, known as the Tsallis parameter. Considering the apparent horizon entropy of an FLRW type universe to be the Tsallis type, [50] was able to derive a Friedmann equation for the universe while also taking into account and modifying Padmanabhan's emergent gravity proposal [45]. The concerned Friedmann equation is [50] ,</text>
<formula><location><page_2><loc_13><loc_40><loc_48><loc_43></location>( 𝐻 2 ) 2𝜅 = 8 𝜋 3 𝑚 2 𝑝 𝜌 (6)</formula>
<text><location><page_2><loc_9><loc_25><loc_48><loc_38></location>Inflation has been studied in a lot of the popular braneworld scenarios , both in the traditional supercooled regime [18, 29, 31] and in the warm inflation regime [32, 19, 21, 15]. [15] also studied Chern-Simons type modified gravity of [23] in the cold inflation regime. A crucial point , as noted in [15], which emerges from the studies of inflationary studies in different cosmological scenarios is that the concerned Friedmann equation of investigation can be often taken of the form</text>
<formula><location><page_2><loc_13><loc_21><loc_48><loc_24></location>𝐹 ( 𝐻 ) = 8 𝜋 3 𝑚 2 𝑝 𝜌 (7)</formula>
<text><location><page_2><loc_9><loc_6><loc_48><loc_19></location>, where 𝐹 ( 𝐻 ) is some function of the Hubble parameter H. A very useful approach to studying different cold inflation scenarios was discussed in [15]. We would now like to study different cosmological scenarios using (7) as the primary equation of investigation. We hence structure our paper as follows. In section 2, we outline the exact solution approach for Warm Inflation and describe the dynamical aspects of the method while in Section 3 we discuss Perturbation spectra and derive crucial observational quantities like the scalar and</text>
<text><location><page_2><loc_52><loc_85><loc_91><loc_93></location>tensor spectral index in concern to the approach. In Section 4 we apply the method to study Warm Inflation in the high dissipative regime in a Tsallis entropy type Universe of [50] . Then finally, we summarize our results in section 5 in the Concluding remarks of the paper.</text>
<section_header_level_1><location><page_2><loc_52><loc_80><loc_77><loc_82></location>2. The exact solution method</section_header_level_1>
<text><location><page_2><loc_52><loc_74><loc_91><loc_79></location>In warm inflationary approach, there is consideration given to both the inflaton field and the radiation being created due to it's dissipation during Inflation itself. Hence, the usual Friedmann equation of warm inflation is [6]</text>
<formula><location><page_2><loc_56><loc_69><loc_91><loc_73></location>𝐻 2 = 8 𝜋 3 𝑚 2 𝑝 ( 𝜌 𝜙 + 𝜌 𝑟 ) (8)</formula>
<text><location><page_2><loc_52><loc_64><loc_91><loc_68></location>where the energy density 𝜌 consists of both inflaton and radiation energy densities, 𝜌 𝜙 and 𝜌 𝑟 respectively. But during inflation , 𝜌 𝜙 >> 𝜌 𝑟 [6], hence (8) becomes</text>
<formula><location><page_2><loc_56><loc_59><loc_91><loc_62></location>𝐻 2 = 8 𝜋 3 𝑚 2 𝑝 ( 𝜌 𝜙 ) (9)</formula>
<text><location><page_2><loc_52><loc_56><loc_67><loc_58></location>, where 𝜌 𝜙 is given by ,</text>
<formula><location><page_2><loc_56><loc_52><loc_91><loc_55></location>𝜌 𝜙 = ̇ 𝜙 2 2 + 𝑉 ( 𝜙 ) (10)</formula>
<text><location><page_2><loc_52><loc_47><loc_91><loc_51></location>with 𝑉 ( 𝜙 ) being the potential under which the inflaton field is. The inflaton field equation , the inflaton energy density and radiation energy density evolution equation is given by,</text>
<formula><location><page_2><loc_56><loc_44><loc_91><loc_46></location>̈ 𝜙 +(3 𝐻 +Γ) ̇ 𝜙 + 𝑉 ' ( 𝜙 ) = 0 (11)</formula>
<formula><location><page_2><loc_56><loc_38><loc_91><loc_40></location>̇ 𝜌 𝜙 +3 𝐻 ( 𝜌 𝜙 + 𝑝 𝜙 ) = -Γ ̇ 𝜙 2 (12)</formula>
<formula><location><page_2><loc_56><loc_33><loc_91><loc_35></location>̇ 𝜌 𝑟 +4 𝐻𝜌 𝑟 = Γ ̇ 𝜙 2 (13)</formula>
<text><location><page_2><loc_52><loc_28><loc_91><loc_32></location>Here , Γ is the dissipation coefficient of inflaton energy. Defining a quantity 𝑄 = Γ 3 𝐻 , we can rewrite the above equations as</text>
<formula><location><page_2><loc_56><loc_25><loc_91><loc_26></location>̈ 𝜙 +3 𝐻 (1 + 𝑄 ) ̇ 𝜙 + 𝑉 ' ( 𝜙 ) = 0 (14)</formula>
<formula><location><page_2><loc_56><loc_19><loc_91><loc_21></location>̇ 𝜌 𝜙 +3 𝐻 ( 𝜌 𝜙 + 𝑝 𝜙 ) = -3 𝐻𝑄 ̇ 𝜙 2 (15)</formula>
<formula><location><page_2><loc_56><loc_14><loc_91><loc_15></location>̇ 𝜌 𝑟 +4 𝐻𝜌 𝑟 = 3 𝐻𝑄 ̇ 𝜙 2 (16)</formula>
<text><location><page_2><loc_52><loc_10><loc_91><loc_12></location>During inflation, we can take both ̇ 𝜌 𝑟 to be approximately zero [6] . This gives us (16)</text>
<formula><location><page_2><loc_56><loc_6><loc_91><loc_8></location>𝜌 𝑟 = 3 𝑄 4 ̇ 𝜙 2 (17)</formula>
<text><location><page_3><loc_9><loc_87><loc_48><loc_93></location>This completes a small review of the basics of warm inflation which we will now freely refer to throughout the entirety of paper. Consider now the following general form of the Friedmann equation</text>
<formula><location><page_3><loc_13><loc_82><loc_48><loc_87></location>𝐹 ( 𝐻 ) = 8 𝜋 3 𝑚 2 𝑝 𝜌 𝜙 = 8 𝜋 3 𝑚 2 𝑝 ( ̇ 𝜙 2 2 + 𝑉 ( 𝜙 ) ) (18)</formula>
<text><location><page_3><loc_9><loc_77><loc_48><loc_81></location>The famous slow roll conditions of cold inflation [5], hold well similarly in warm inflation as well [6] and so during warm inflation,</text>
<formula><location><page_3><loc_13><loc_74><loc_48><loc_76></location>̇ 𝜙 2 << 𝑉 ( 𝜙 ) (19)</formula>
<text><location><page_3><loc_9><loc_68><loc_48><loc_73></location>This gives us the energy density to be 𝜌 𝜙 ≈ 𝑉 ( 𝜙 ) Hence the general Friedmann equation(18) during Inflation, takes the form</text>
<formula><location><page_3><loc_13><loc_64><loc_48><loc_67></location>𝐹 ( 𝐻 ) ≈ 8 𝜋 3 𝑚 2 𝑝 𝑉 ( 𝜙 ) (20)</formula>
<text><location><page_3><loc_9><loc_45><loc_48><loc_63></location>Now, it was shown in [33] that treating 𝜙 itself as the explicitly dependent variable for H can lead to a very apt method of studying inflationary models, which is referred to as the " Hamilton-Jacobi approach to inflation". [48] studied the Hamilton-Jacobi approach with respect to warm inflation and found that treating the Hubble parameter and the disspiation coefficients in terms of 𝜙 is indeed very helpful for warm inflationary regimes. Hence, now we treat the Hubble parameter ( and hence the scale factor as well) primarily in terms of the field variable 𝜙 . So 𝐻 = 𝐻 ( 𝜙 ) . Further the field variable will generally be time dependent, hence the Hubble parameter stays implicitly time dependent.</text>
<text><location><page_3><loc_9><loc_41><loc_48><loc_43></location>With these considerations, we take the derivative of (20) with respect to 𝜙 ,</text>
<formula><location><page_3><loc_13><loc_36><loc_48><loc_40></location>𝐹 ' 𝐻 𝐻 ' ( 𝜙 ) = 8 𝜋 3 𝑚 2 𝑝 𝑉 ' ( 𝜙 ) (21)</formula>
<text><location><page_3><loc_9><loc_29><loc_48><loc_35></location>where , following the convention in [15], 𝐹 ' 𝐻 is the partial derivative of F with respect to H , and the prime in the superscript denotes derivative with respect to 𝜙 . In the slow roll approximation (11) becomes,</text>
<formula><location><page_3><loc_13><loc_27><loc_48><loc_28></location>3 𝐻 (1 + 𝑄 ) ̇ 𝜙 = -𝑉 ' ( 𝜙 ) (22)</formula>
<text><location><page_3><loc_9><loc_24><loc_45><loc_26></location>Using the expression for 𝑉 ' ( 𝜙 ) from (21) we have ̇ 𝜙 as</text>
<formula><location><page_3><loc_13><loc_19><loc_48><loc_23></location>̇ 𝜙 = -𝑚 2 𝑝 8 𝜋 𝐹 ' 𝐻 𝐻 ' 𝐻 (1 + 𝑄 ) (23)</formula>
<text><location><page_3><loc_9><loc_16><loc_48><loc_18></location>Using (23) , we can get a different expression for 𝜌 𝑟 by taking into account (17) ,</text>
<formula><location><page_3><loc_13><loc_11><loc_48><loc_15></location>𝜌 𝑟 = 3 𝑄 4 ( 𝑚 2 𝑝 8 𝜋 𝐹 ' 𝐻 𝐻 ' 𝐻 (1 + 𝑄 ) ) 2 (24)</formula>
<text><location><page_3><loc_9><loc_6><loc_48><loc_9></location>Hence, we now have expression for the field velocity given 𝐹 ( 𝐻 ) ,an ansatz for H in terms of 𝜙 and a form of the</text>
<text><location><page_3><loc_52><loc_85><loc_91><loc_93></location>dissipation coefficient. As mentioned in [40, 10], dissipation coefficient is popularly used in power laws in terms of both 𝜙 and temperature T. The temperature plays an especially important role for high dissipative regimes, which is characterized by</text>
<formula><location><page_3><loc_63><loc_84><loc_80><loc_85></location>𝑄 >> 1 ⟹ Γ >> 3 𝐻</formula>
<text><location><page_3><loc_52><loc_77><loc_91><loc_83></location>and temperature itself plays a very crucial in the generation of quantum fluctuations during Inflation [7, 10]. Using (24) we can get an expression for 𝜌 𝑟 in terms of 𝜌 𝜙 and it's easy to check that</text>
<formula><location><page_3><loc_56><loc_72><loc_91><loc_77></location>𝜌 𝑟 = 𝑄𝜌 𝜙 𝑚 2 𝑝 32 𝜋𝐹 ( 𝐹 ' 𝐻 𝐻 ' 𝐻 (1 + 𝑄 ) ) 2 (25)</formula>
<text><location><page_3><loc_52><loc_69><loc_88><loc_71></location>We also note that as 𝜌 𝑟 is pure radiation , we can write</text>
<formula><location><page_3><loc_56><loc_66><loc_91><loc_68></location>𝜌 𝑟 = 𝛼𝑇 4 (26)</formula>
<text><location><page_3><loc_52><loc_62><loc_91><loc_65></location>where T is the radiation temperature and 𝛼 is the Stefan Boltzmann Constant. Now, we have from (17) and (24),</text>
<formula><location><page_3><loc_56><loc_57><loc_91><loc_62></location>𝛼𝑇 4 = 3 𝑄 4 ( 𝑚 2 𝑝 8 𝜋 𝐹 ' 𝐻 𝐻 ' 𝐻 (1 + 𝑄 ) ) 2 (27)</formula>
<text><location><page_3><loc_52><loc_54><loc_61><loc_55></location>which gives us</text>
<formula><location><page_3><loc_56><loc_49><loc_91><loc_54></location>𝑇 = ( 3 𝑄 4 𝛼 ) 1/4 ( 𝑚 2 𝑝 8 𝜋 𝐹 ' 𝐻 𝐻 ' 𝐻 (1 + 𝑄 ) ) 1/2 (28)</formula>
<text><location><page_3><loc_52><loc_46><loc_80><loc_47></location>We note from the Friedmann equation that</text>
<formula><location><page_3><loc_56><loc_41><loc_91><loc_45></location>3 𝑚 2 𝑝 8 𝜋 𝐹 -1 2 ̇ 𝜙 2 = 𝑉 ( 𝜙 ) (29)</formula>
<text><location><page_3><loc_52><loc_37><loc_91><loc_40></location>Using the expression for ̇ 𝜙 (23), we have the potential given as</text>
<formula><location><page_3><loc_56><loc_33><loc_91><loc_37></location>𝑉 ( 𝜙 ) = 3 𝑚 2 𝑝 8 𝜋 𝐹 -1 2 ( 𝑚 2 𝑝 8 𝜋 𝐹 ' 𝐻 𝐻 ' 𝐻 (1 + 𝑄 ) ) 2 (30)</formula>
<text><location><page_3><loc_52><loc_30><loc_56><loc_31></location>Also as</text>
<formula><location><page_3><loc_67><loc_26><loc_75><loc_30></location>𝜌 𝜙 = 3 𝑚 2 𝑝 8 𝜋 𝐹</formula>
<text><location><page_3><loc_52><loc_23><loc_91><loc_26></location>, we can also write the potential in terms of the inflaton energy density</text>
<formula><location><page_3><loc_56><loc_17><loc_91><loc_22></location>𝑉 ( 𝜙 ) = 𝜌 𝜙 ⎛ ⎜ ⎜ ⎝ 1 𝑚 2 𝑝 24 𝜋 ( 𝐹 ' 𝐻 𝐻 ' 𝐻 (1 + 𝑄 ) ) 2 ⎞ ⎟ ⎟ ⎠ (31)</formula>
<text><location><page_3><loc_52><loc_12><loc_91><loc_15></location>The number of e-folds is a very important quantity to measure the amount of Inflation. The number of e-folds is [[5] ],</text>
<formula><location><page_3><loc_56><loc_6><loc_91><loc_10></location>𝑁 = ∫ 𝜙 𝑒 𝜙 𝑜 𝐻 ̇ 𝜙 𝑑𝜙 (32)</formula>
<text><location><page_4><loc_9><loc_88><loc_48><loc_93></location>where 𝜙 𝑒 and 𝜙 are the field values at the end and beginning of Inflation, respectively. Once again using (23), we write N as</text>
<formula><location><page_4><loc_13><loc_84><loc_48><loc_87></location>𝑁 = ∫ 𝜙 𝜙 𝑒 8 𝜋 𝑚 2 𝑝 𝐻 2 (1 + 𝑄 ) 𝐹 ' 𝐻 𝐻 ' 𝑑𝜙 (33)</formula>
<text><location><page_4><loc_9><loc_81><loc_30><loc_82></location>Now, by the definition , we have</text>
<formula><location><page_4><loc_13><loc_77><loc_48><loc_80></location>𝑑𝑁 = 𝑑𝑎 𝑎 (34)</formula>
<text><location><page_4><loc_9><loc_75><loc_18><loc_76></location>which gives us</text>
<formula><location><page_4><loc_13><loc_70><loc_48><loc_75></location>𝑎 ( 𝜙 ) = 𝑎 ( 𝜙 𝑜 ) exp ( -∫ 𝜙 𝜙 𝑒 8 𝜋 𝑚 2 𝑝 𝐻 2 (1 + 𝑄 ) 𝐹 ' 𝐻 𝐻 ' 𝑑𝜙 ) (35)</formula>
<text><location><page_4><loc_9><loc_64><loc_48><loc_69></location>Another important quantity for inflationary models are the slow roll parameters and from the basic definition of the 𝜖 slow roll parameter, one can write</text>
<formula><location><page_4><loc_13><loc_61><loc_48><loc_63></location>⃛ 𝑎 𝑎 = 𝐻 2 + ̇ 𝐻 = 𝐻 2 (1 𝜖 ) (36)</formula>
<text><location><page_4><loc_9><loc_58><loc_47><loc_59></location>where 𝜖 is the "first" Hubble slow roll parameter given by</text>
<formula><location><page_4><loc_13><loc_54><loc_48><loc_58></location>𝜖 = 𝑚 𝑝 2 8 𝜋 𝐹 ' 𝐻 𝐻 (1 + 𝑄 ) ( 𝐻 ' 𝐻 ) 2 (37)</formula>
<text><location><page_4><loc_9><loc_51><loc_43><loc_52></location>This can be arrived at directly by first realizing that</text>
<formula><location><page_4><loc_13><loc_46><loc_48><loc_50></location>̇ 𝐻 = 𝐻 ' ̇ 𝜙 = -𝑚 2 𝑝 8 𝜋 𝐹 ' 𝐻 𝐻 ' 𝐻 (1 + 𝑄 ) (38)</formula>
<text><location><page_4><loc_9><loc_44><loc_17><loc_45></location>This implies</text>
<formula><location><page_4><loc_13><loc_39><loc_48><loc_43></location>-̇ 𝐻 𝐻 2 = 𝑚 𝑝 2 8 𝜋 𝐹 ' 𝐻 𝐻 (1 + 𝑄 ) ( 𝐻 ' 𝐻 ) 2 (39)</formula>
<text><location><page_4><loc_9><loc_36><loc_44><loc_37></location>𝜖 is also realized using the concurrent definition [39]</text>
<formula><location><page_4><loc_13><loc_31><loc_48><loc_35></location>𝜖 = -𝑑 ln 𝐻 𝑑 ln 𝑎 = 𝑚 𝑝 2 8 𝜋 𝐹 ' 𝐻 𝐻 (1 + 𝑄 ) ( 𝐻 ' 𝐻 ) 2 (40)</formula>
<text><location><page_4><loc_9><loc_27><loc_48><loc_30></location>The second Hubble Slow Roll Parameter 𝜂 is similarly defined to be</text>
<formula><location><page_4><loc_13><loc_23><loc_48><loc_26></location>𝜂 = -𝑑 ln 𝐻 ' 𝑑 ln 𝑎 = 𝑚 𝑝 2 8 𝜋 𝐹 ' 𝐻 𝐻 (1 + 𝑄 ) 𝐻 '' 𝐻 (41)</formula>
<text><location><page_4><loc_9><loc_6><loc_48><loc_21></location>We now take a brief detour and discuss a bit of subtlety surrounding the definition of slow roll parameters for Warm Inflation. While the 𝜖 and 𝜂 slow roll parameters above is derived as it comes from the basic definition (38) and (39), there still exists conflicting views in literature about the appropriate definition for the parameter in the context of Warm Inflation. While some authors [27, 56] like to define the 𝜖 and 𝜂 parameters in their usual supercooled inflation forms , others [21, 48] like to " absorb" the dependence of the damping function Q as it comes into the slow roll</text>
<text><location><page_4><loc_52><loc_66><loc_91><loc_93></location>parameters from the basic definitions (38) and (39) . The proponents of the former approach feel that defining the first and second slow roll parameters in the usual cold inflation form allows them to have a more relaxed constraint in warm inflationary scenarios [27] . The author of this paper is also in harmony with this line of thought, but however feels the latter approach of defining the first two Hubble parameters in warm inflation allows for them to be evidently more general then their supercooled inflation form. This allows the reader to more smoothly see the transition of the parameters from warm to cold inflation in the extremely low dissipation regime. But we are certainly of the opinion that while both approaches of defining the parameters may appear different on the level of substance, they carry virtually the same essence overall. So now we just note for the sake of completeness that we can equivalently define the 𝜖 and 𝜂 parameters suitable for the approach [27, 56] as (which we will call 𝜖 𝑎 and 𝜂 𝑎 )</text>
<formula><location><page_4><loc_56><loc_62><loc_91><loc_66></location>𝜖 𝑎 = 𝑚 𝑝 2 8 𝜋 𝐹 ' 𝐻 𝐻 ( 𝐻 ' 𝐻 ) 2 (42)</formula>
<formula><location><page_4><loc_56><loc_56><loc_91><loc_59></location>𝜂 𝑎 = 𝑚 𝑝 2 8 𝜋 𝐹 ' 𝐻 𝐻 𝐻 '' 𝐻 (43)</formula>
<text><location><page_4><loc_52><loc_35><loc_91><loc_55></location>which is just the form of the parameters as shown in [15] . The second form of the parameters , which is in line with the approach of [21, 48], are just the 𝜂 and 𝜖 parameters defined in (38) and (39). We readily see that (42) and (43) are just (38) and (39) respectively, in the 𝑄 << 1 approximation. Alongside the usual slow roll parameters 𝜂 and 𝜖 , [56] showed that for appropriately studying warm inflationary paradigms some other parameters with the derivatives of Γ would be very useful. These parameters were derived by the using the slow roll conditions for warm inflation. Hence we would now like to define some new slow roll parameters for our model, in order to better cater to the needs of warm inflation scenarios. One of the primary slow roll conditions reads</text>
<formula><location><page_4><loc_56><loc_31><loc_91><loc_34></location>-̈ 𝐻 ̇ 𝐻𝐻 << 1 (44)</formula>
<text><location><page_4><loc_52><loc_25><loc_91><loc_30></location>We would now evaluate the quantity -̈ 𝐻 ̇ 𝐻𝐻 to arrive at our new slow roll parameters taking lead from [48] . We begin by noting that</text>
<formula><location><page_4><loc_56><loc_21><loc_91><loc_25></location>̈ 𝐻 = 𝑑 𝑑𝑡 ( 𝐻 ' ̇ 𝜙 ) (45)</formula>
<text><location><page_4><loc_52><loc_19><loc_91><loc_21></location>which can be realized using (38) . Now this allows us to write</text>
<formula><location><page_4><loc_56><loc_16><loc_91><loc_17></location>̈ 𝐻 = ̇ 𝜙 2 𝐻 '' + ̈ 𝜙𝐻 ' (46)</formula>
<text><location><page_4><loc_52><loc_7><loc_91><loc_14></location>To evaluate (44) , we need to first have an expression for ̈ 𝜙 appropriate for our use. This can be done by using (23) supplemented by the fact that ̇ 𝐹 ' 𝐻 = ̇ 𝐻𝐹 ' 𝐻𝐻 where ( 𝐹 ' 𝐻𝐻 = 𝜕 2 𝐹 𝜕𝐻 2 ) . A little bit of algebra leads us to</text>
<formula><location><page_5><loc_13><loc_84><loc_48><loc_94></location>̈ 𝜙 = -3 𝑚 2 𝑝 8 𝜋 ( ( 𝐻 ' 2 𝐹 ' 𝐻𝐻 + 𝐹 ' 𝐻 𝐻 '' ) (3 𝐻 +Γ) (3 𝐻 +Γ) 2 -(3 𝐻 ' +Γ ' ) 𝐹 ' 𝐻 𝐻 ' (3 𝐻 +Γ) 2 ) (47)</formula>
<text><location><page_5><loc_9><loc_82><loc_24><loc_83></location>This allows us to write</text>
<formula><location><page_5><loc_13><loc_72><loc_49><loc_81></location>-̈ 𝐻 ̇ 𝐻𝐻 = -𝜂 -𝑚 2 𝑝 8 𝜋𝐻 2 ( ( 𝐻 ' 2 𝐹 ' 𝐻𝐻 + 𝐹 ' 𝐻 𝐻 '' ) (1 + 𝑄 ) (1 + 𝑄 ) 2 -1 (1 + 𝑄 ) 2 ( 3 𝐻 ' +Γ ' ) 𝐹 ' 𝐻 𝐻 ' 3 𝐻 ) (48)</formula>
<text><location><page_5><loc_9><loc_69><loc_23><loc_70></location>And further we write</text>
<formula><location><page_5><loc_13><loc_58><loc_48><loc_68></location>-̈ 𝐻 ̇ 𝐻𝐻 = 2 𝜂 + 𝑚 2 𝑝 𝐻 ' 2 8 𝜋𝐻 2 𝐹 ' 𝐻𝐻 1 + 𝑄 -𝑄 1 + 𝑄 ( 𝑚 2 𝑝 8 𝜋 𝐹 ' 𝐻 𝐻 Γ ' 𝐻 ' Γ 𝐻 + 3 𝑚 2 𝑝 𝐹 ' 𝐻 8 𝜋𝐻 𝐻 ' Γ 𝐻 (1 + 𝑄 ) ) (49)</formula>
<text><location><page_5><loc_9><loc_55><loc_40><loc_57></location>this leads us to define 𝛽 parameter [56, 48] as ,</text>
<formula><location><page_5><loc_13><loc_51><loc_48><loc_54></location>𝛽 = 𝑚 2 𝑝 8 𝜋 𝐹 ' 𝐻 𝐻 Γ ' 𝐻 ' Γ 𝐻 1 1 + 𝑄 (50)</formula>
<text><location><page_5><loc_9><loc_45><loc_48><loc_49></location>In addition to these slow roll parameters, we define other parameters which will be helpful for our perturbation spectra studies. These are ,</text>
<formula><location><page_5><loc_13><loc_41><loc_48><loc_44></location>𝜒 = 𝑚 2 𝑝 𝐹 ' 𝐻 8 𝜋𝐻 𝐻 ' Γ 𝐻 (1 + 𝑄 ) (51)</formula>
<text><location><page_5><loc_9><loc_38><loc_11><loc_39></location>and</text>
<formula><location><page_5><loc_13><loc_34><loc_48><loc_37></location>𝛾 = 𝑚 2 𝑝 𝐻 ' 2 8 𝜋𝐻 2 𝐹 ' 𝐻𝐻 1 + 𝑄 (52)</formula>
<text><location><page_5><loc_9><loc_31><loc_35><loc_32></location>which now allows us to express (46) as</text>
<formula><location><page_5><loc_13><loc_27><loc_48><loc_30></location>-̈ 𝐻 ̇ 𝐻𝐻 = 2 𝜂 + 𝜒 -𝑄 1 + 𝑄 ( 𝛽 +3 𝛾 ) (53)</formula>
<text><location><page_5><loc_9><loc_23><loc_48><loc_26></location>In addition to these we define more parameters which will be helpful for us in our perturbation spectra analysis</text>
<formula><location><page_5><loc_13><loc_18><loc_48><loc_23></location>𝛿 = 𝑚 2 𝑝 8 𝜋 ( 𝐹 ' 𝐻 (1 + 𝑄 ) 𝐻 ) 2 Γ '' 𝐻 ' 2 Γ 𝐻 2 (54)</formula>
<formula><location><page_5><loc_13><loc_12><loc_48><loc_16></location>𝜎 = 𝑚 2 𝑝 8 𝜋 1 1 + 𝑄 𝐹 ' 𝐻 𝐻 𝐻 ''' 𝐻 ' (55)</formula>
<formula><location><page_5><loc_13><loc_6><loc_48><loc_9></location>𝜓 = 𝑚 2 𝑝 8 𝜋 𝐹 ' 𝐻𝐻𝐻 (1 + 𝑄 ) 𝐻 ' 2 𝐻 (56)</formula>
<text><location><page_5><loc_52><loc_78><loc_91><loc_93></location>This completes the proper dynamical outline of the Exact Solution approach for warm inflation. It is quick to check that this approach reduces to the usual cold inflation approach for exact solutions [15] in the extremely low dissipation regime 𝑄 << 1 . It also further reduces to the Hamilton-Jacobi method for warm inflation for 𝐹 ( 𝐻 ) = 𝐻 2 [48], and further to the usual cold inflation Hamilton-Jacobi approach [33] in the extremely low dissipation regime for the same 𝐹 ( 𝐻 ) . With the dynamical aspects covered, we shall now explore the perturbation spectra of warm inflation in this approach.</text>
<section_header_level_1><location><page_5><loc_52><loc_74><loc_80><loc_75></location>3. Perturbation spectra analysis</section_header_level_1>
<text><location><page_5><loc_52><loc_45><loc_91><loc_73></location>Cosmological density and gravitational wave perturbations in the inflationary scenario arise as quantum fluctuations which redshift to long wavelengths due to rapid cosmological expansion during Inflation [5, 28, 53, 25]. In warm inflation, only the density perturbations couple strongly with the thermal background and hence, the scalar density spectra is the one which looks more evidently different from it's usual supercooled inflationary counterpart [54] . Tensor perturbations do not couple strongly to the thermal background and so gravitational waves are only generated by quantum fluctuations, as in standard cold inflation. In addition to the usual adiabatic perturbations in cold inflation, Isocurvature perturbations also are generated in the warm inflationary era due to thermal fluctuations in the radiation field. These perturbations can be characterised by fluctuations in the entropy of the particle species undergoing thermal fluctuations relative to the number density of photons. But in this paper, we will limit our focus to the study of only adiabatic perturbations.</text>
<text><location><page_5><loc_52><loc_40><loc_91><loc_43></location>The square of the amplitude of adiabatic perturbations is calculated in a similar way to cold inflation [54, 10, 48] ,</text>
<formula><location><page_5><loc_56><loc_34><loc_91><loc_39></location>𝑃 𝑠 ( 𝑘 ) 2 = 4 25 ( 𝐻 ̇ | 𝜙 | ) 2 𝑑𝜙 2 (57)</formula>
<text><location><page_5><loc_52><loc_30><loc_91><loc_33></location>where 𝑑𝜙 2 for the high dissipative regime ( 𝑄 >> 1 )is given by [8] ,</text>
<formula><location><page_5><loc_56><loc_26><loc_91><loc_29></location>𝑑𝜙 2 = 𝑘 𝐹 𝑇 2 𝜋 (58)</formula>
<text><location><page_5><loc_52><loc_23><loc_87><loc_25></location>where 𝑘 𝐹 is the so called freeze out number given by,</text>
<formula><location><page_5><loc_56><loc_20><loc_91><loc_23></location>𝑘 𝐹 = √ Γ 𝐻 (59)</formula>
<text><location><page_5><loc_52><loc_7><loc_91><loc_19></location>The definition of the freeze out number 𝑘 𝐹 is not changed by considering a general Friedmann equation of the form (18). It is so because the definition of the freeze out number stems primarily from the field equation of 𝜙 (11) , in particular from the evolution equation of the fluctuations 𝛿𝜙 ( 𝑥, 𝑡 ) ( where 𝜙 (x,t) = 𝜙 𝑜 ( 𝑥, 𝑡 ) + 𝛿𝜙 ( 𝑥, 𝑡 ) ) [54, 8].This remains unchanged by the consideration of (18) , takes the usual form with the inclusion of the spatial Laplacian and an additional</text>
<text><location><page_6><loc_9><loc_92><loc_30><loc_93></location>white noise random force term ,</text>
<formula><location><page_6><loc_12><loc_87><loc_48><loc_91></location>Γ 𝑑𝛿𝜙 ( 𝑘, 𝑡 ) 𝑑𝑡 = -[ ( 𝑘 2 + 𝑉 '' ( 𝜙 𝑜 ) ) ] 𝛿𝜙 ( 𝑘, 𝑡 ) + 𝜁 ( 𝑘, 𝑡 ) (60)</formula>
<text><location><page_6><loc_9><loc_80><loc_48><loc_86></location>where we have Fourier transformed to the mpmentum space and 𝜁 ( 𝑘, 𝑡 ) is the white noise term. In a similar way to [54] , we reach at 𝑘 𝐹 = √ Γ 𝐻 . Now using the expression for T (26), and the definition of 𝜖 (40) , we can write 𝑃 𝑠 ( 𝑘 ) as</text>
<formula><location><page_6><loc_11><loc_75><loc_48><loc_79></location>𝑃 𝑠 ( 𝑘 ) 2 = 2 25 𝜋 √ Γ 𝐻 ( 3 𝑄 4 𝜎 ) 1 4 ( 𝜖𝐻 2 𝐻 ' ) 1 2 ( 𝐻 ' 𝜖𝐻 ) 2 (61)</formula>
<text><location><page_6><loc_9><loc_73><loc_24><loc_74></location>which leads us to write</text>
<formula><location><page_6><loc_13><loc_66><loc_48><loc_72></location>𝑃 𝑠 ( 𝑘 ) = ⎛ ⎜ ⎜ ⎝ ( 48 1562500 𝜎𝜋 4 ) 1 2 Γ 3 2 𝐻 -3 2 𝜖 -3 𝐻 ' 3 ⎞ ⎟ ⎟ ⎠ 1 2 (62)</formula>
<text><location><page_6><loc_9><loc_62><loc_48><loc_64></location>The scalar spectral index is defined by the well known equation</text>
<formula><location><page_6><loc_13><loc_57><loc_48><loc_60></location>𝑛 𝑠 -1 = 𝑑 ln 𝑃 𝑠 ( 𝑘 ) 2 𝑑 ln 𝑘 (63)</formula>
<text><location><page_6><loc_9><loc_53><loc_48><loc_56></location>where 𝑑 ln 𝑘 is given as the negative of the differential of the number of e-folds</text>
<formula><location><page_6><loc_13><loc_49><loc_48><loc_52></location>𝑑 ln 𝑘 = -𝑑𝑁 = 8 𝜋 3 𝑚 2 𝑝 𝐻 Γ 𝐹 ' 𝐻 𝐻 ' 𝑑𝜙 (64)</formula>
<text><location><page_6><loc_9><loc_45><loc_48><loc_48></location>For proceeding further, we note that the derivative of the 𝜖 parameter is given by</text>
<formula><location><page_6><loc_13><loc_41><loc_48><loc_44></location>𝜖 ' = 𝑑𝜖 𝑑𝜙 = 𝐻 ' 𝐻 ( 𝛾 +2 𝜂 -2 𝜖 -𝛽 ) (65)</formula>
<text><location><page_6><loc_9><loc_38><loc_24><loc_39></location>This allows us to write</text>
<formula><location><page_6><loc_13><loc_34><loc_48><loc_37></location>𝑑𝜖 𝑑 ln 𝑘 = 𝜖 ( 𝛾 +2 𝜂 -2 𝜖 -𝛽 ) (66)</formula>
<formula><location><page_6><loc_12><loc_28><loc_48><loc_32></location>𝑑 ln 𝑃 𝑠 ( 𝑘 ) 2 𝑑 ln 𝑘 = 3 𝛽 4 -3 𝜖 4 + 3 𝜂 2 -3 2 𝜖 ' 𝜖 3 𝑚 2 𝑝 8 𝜋 𝐹 ' 𝐻 𝐻 ' 𝐻 (Γ) (67)</formula>
<formula><location><page_6><loc_13><loc_22><loc_48><loc_25></location>𝑑 ln 𝑃 𝑠 ( 𝑘 ) 2 𝑑 ln 𝑘 = 9 4 𝛽 + 9 4 𝜖 -3 2 𝛾 -3 2 𝜂 (68)</formula>
<text><location><page_6><loc_9><loc_18><loc_48><loc_21></location>This finally allows to us to express the scalar spectral index as</text>
<formula><location><page_6><loc_13><loc_14><loc_48><loc_17></location>𝑛 𝑠 = 1 + 9 4 𝛽 + 9 4 𝜖 -3 2 𝛾 -3 2 𝜂 (69)</formula>
<text><location><page_6><loc_9><loc_10><loc_48><loc_13></location>This is the scalar spectral index for the high dissipative regime.</text>
<text><location><page_6><loc_9><loc_9><loc_42><loc_10></location>For the low dissipative regime ( 𝑄 << 1 ) we have</text>
<formula><location><page_6><loc_13><loc_6><loc_48><loc_7></location>𝑑𝜙 = 𝐻𝑇 (70)</formula>
<text><location><page_6><loc_52><loc_90><loc_91><loc_93></location>which allows us to write the power spectrum for the low dissipative regime as</text>
<formula><location><page_6><loc_56><loc_86><loc_91><loc_89></location>𝑃 ∗ 𝑠 ( 𝑘 ) 2 = 4 25 𝐻 ' 2 𝜖 2 𝑇 2 (71)</formula>
<text><location><page_6><loc_52><loc_82><loc_91><loc_85></location>Using the expression for the temperature derived above (26) , we have</text>
<formula><location><page_6><loc_56><loc_78><loc_91><loc_81></location>𝑃 ∗ 𝑠 ( 𝑘 ) 2 = ( 4 625 𝛼 ( 𝐻 ' ) 2 𝐻 3 Γ ) 1 2 (72)</formula>
<text><location><page_6><loc_52><loc_74><loc_91><loc_77></location>Again , the definition of the scalar spectral index in this case is the same as for the high dissipation scenario</text>
<formula><location><page_6><loc_56><loc_70><loc_91><loc_73></location>𝑛 ∗ 𝑠 -1 = 𝑑 ln 𝑃 ∗ 𝑠 ( 𝑘 ) 2 𝑑 ln 𝑘 (73)</formula>
<text><location><page_6><loc_52><loc_66><loc_91><loc_69></location>Pursuing a similar analysis as for the previous case, we arrive at</text>
<formula><location><page_6><loc_56><loc_63><loc_91><loc_65></location>𝑛 ∗ 𝑠 = 1 + 3 2 𝜖 + 𝜂 + 𝛽 2 (74)</formula>
<text><location><page_6><loc_52><loc_40><loc_91><loc_61></location>One of the more exciting findings of the observational data from the Planck, WMAP and COBE experiments is that there is a significant running of the scalar spectral index as well. Traditionally it was taken to be negligible but these experimental findings make them a very important observational quantity. Alongside the Running of the scalar spectral index and the index itself, the tensor to scalar ratio is another important quantity of observational relevance. We will now focus more on the high dissipative regime and calculate the running of the scalar spectral index and the tensor-to-scalar ratio in that limit. We will not calculate the same in the low dissipative regime but one can calculate them in that limit by pursuing a similar method as we do in the following for the high dissipative regime.</text>
<text><location><page_6><loc_52><loc_35><loc_91><loc_38></location>The running of the spectral index is defined by it's usual definition</text>
<formula><location><page_6><loc_56><loc_32><loc_91><loc_35></location>𝛼 𝑠 = 𝑑𝑛 𝑠 𝑑 ln 𝑘 (75)</formula>
<text><location><page_6><loc_52><loc_28><loc_91><loc_31></location>In a similar way as we calculated 𝑑𝜖 𝑑 ln 𝑘 in (66), we arrive at the following differentials</text>
<formula><location><page_6><loc_56><loc_24><loc_91><loc_27></location>𝑑𝜂 𝑑 ln 𝑘 = 𝜂𝛾 + 𝜎𝜖 -𝜂𝛽 -𝜂𝜖 (76)</formula>
<formula><location><page_6><loc_56><loc_19><loc_91><loc_21></location>𝑑𝛽 𝑑 ln 𝑘 = 𝛽𝛾 + 𝛿 + 𝛽𝜂 -𝛽𝜖 -2 𝛽 2 (77)</formula>
<formula><location><page_6><loc_56><loc_13><loc_91><loc_16></location>𝑑𝛾 𝑑 ln 𝑘 = 𝜓𝜖 + 𝛾𝜂 -𝛾𝛽 -𝜖𝛾 (78)</formula>
<text><location><page_6><loc_52><loc_10><loc_91><loc_12></location>The above expressions allow us to write the running of scalar spectral index using (67) as ,</text>
<formula><location><page_6><loc_52><loc_6><loc_91><loc_8></location>𝛼 𝑠 = 15 4 ( 𝛽𝛾 + 𝜂𝛽 + 𝛾𝜖 + 𝜂𝜖 )-9 2 ( 𝜖𝛽 + 𝜖 2 + 𝛽 2 + 𝜂𝛾 )-3 2 ( 𝜓𝜖 )+ 9 4 𝛿</formula>
<text><location><page_7><loc_46><loc_92><loc_48><loc_93></location>(79)</text>
<text><location><page_7><loc_9><loc_87><loc_48><loc_90></location>Further, the squared tensor perturbation power spectrum amplitude is defined as [34]</text>
<formula><location><page_7><loc_13><loc_83><loc_48><loc_86></location>𝑃 𝑇 ( 𝑘 ) 2 = 32 75 𝑚 4 𝑝 𝑉 ( 𝜙 ) (80)</formula>
<text><location><page_7><loc_9><loc_78><loc_48><loc_82></location>During inflation, 𝑉 ≈ 3 𝑚 2 𝑝 𝐹 8 𝜋 which is clear by (20). The definition of the tensor-to-scalar ratio is</text>
<formula><location><page_7><loc_13><loc_73><loc_48><loc_77></location>𝑟 = 𝑃 𝑇 ( 𝑘 ) 2 𝑃 𝑠 ( 𝑘 ) 2 (81)</formula>
<text><location><page_7><loc_9><loc_69><loc_48><loc_72></location>We see that (78) now allows us to write the tensor-to-scalar ratio as</text>
<formula><location><page_7><loc_13><loc_64><loc_48><loc_69></location>𝑟 = 2 𝐹 𝑚 2 𝑝 ( 4 𝛼 3 ) 1 4 ( Γ -3 2 𝐻 ' -3 2 𝜖 3 𝐻 3 ) 1 2 (82)</formula>
<text><location><page_7><loc_9><loc_57><loc_48><loc_63></location>Again we remark that the above formulations of 𝑟 and 𝛼 𝑠 are specifically for the high dissipative regime 𝑄 >> 1 . One can easily formulate the same for the low dissipative regime 𝑄 << 1 using the same procedure we have shown above.</text>
<text><location><page_7><loc_9><loc_51><loc_48><loc_57></location>Now, we have completed all the theoretical basis of our approach. We will now apply it on a Tsallis entropy modified universe. We will study aspects of Warm Inflation in this model in the high dissipative regime.</text>
<section_header_level_1><location><page_7><loc_9><loc_46><loc_48><loc_49></location>4. Warm inflation in Tsallis entropy modified universe</section_header_level_1>
<text><location><page_7><loc_9><loc_41><loc_48><loc_45></location>A Tsallis entropy modified universe has the Friedmann equation of the form (6). This allows us to write 𝐹 ( 𝐻 ) for this model as</text>
<formula><location><page_7><loc_13><loc_38><loc_48><loc_39></location>𝐹 ( 𝐻 ) = 𝐻 2(2𝜅 ) (83)</formula>
<text><location><page_7><loc_9><loc_14><loc_48><loc_36></location>Moving forward we would need to ascertain two more quantities for studying warm inflation in this model, which are the Hubble parameter 𝐻 and the dissipation coefficient Γ . Both of them will be taken as functions for 𝜙 . While this statement can be understood in a straightforward way for 𝐻 , for Γ the answer could have been different. Usually, Γ is taken as a function of only 𝜙 or of both 𝜙 and the radiation temperature 𝑇 [48, 11, 26, 12] . This is because temperature plays a crucial role in the dissipative scenario of warm inflation. However, temperature in general is written in terms of 𝜙 eventually and the dissipation function in cases with Γ = Γ( 𝜙, 𝑇 ) turns into a function of only 𝜙 [16, 40, 4, 30] . So in our case we will treat both H and Γ as 𝐻 = 𝐻 ( 𝜙 ) and Γ = Γ( 𝜙 ) . Further, we consider both H and Γ to be power law functions of 𝜙 , as</text>
<formula><location><page_7><loc_13><loc_11><loc_48><loc_13></location>𝐻 ( 𝜙 ) = 𝐻 𝑜 𝜙 𝑛 (84)</formula>
<formula><location><page_7><loc_13><loc_6><loc_48><loc_8></location>Γ( 𝜙 ) = Γ 𝑜 𝜙 𝑚 (85)</formula>
<text><location><page_7><loc_52><loc_72><loc_91><loc_93></location>where 𝐻 𝑜 , Γ 𝑜 are some constants. The powers n and m are left undetermined here as we will use the Planck data to find out which power laws best fit with our concerned model. The reason for choosing power law form for the Hubble parameter is because they seem to be a good fit with the latest Planck Data [2] , while the consideration of a supersymmetric interaction of the authors [11, 12] lead to the dissipation coefficient being a linear function of 𝜙 . While other authors from several distinct considerations have been led to power law forms of the dissipation coefficient [16, 30, 48]. This suggests to us that power law forms can indeed be very viable and general forms of the dissipation coefficients. Hence, we take the coefficient to be in a power law form of the field variable.</text>
<text><location><page_7><loc_52><loc_63><loc_91><loc_72></location>With all the preliminaries cleared up, we now move towards concrete calculations. As stated previously, we will focus on warm inflation in the high dissipative regime for this case. To make progress, we would need the form of the field 𝜙 at the time of horizon exit and at the end of inflation. It is straightforward to get the latter by setting 𝜖 = 1 in (40),</text>
<formula><location><page_7><loc_56><loc_58><loc_91><loc_62></location>1 = 3 𝑚 2 𝑝 2(2 𝜅 ) 𝑛 2 𝐻 3-2 𝜅 𝑜 8 𝜋 Γ 𝑜 𝜙 (3-2 𝜅 ) 𝑛 -𝑚 -2 𝑒 (86)</formula>
<formula><location><page_7><loc_56><loc_52><loc_91><loc_55></location>3 𝑚 2 𝑝 2(2 𝜅 ) 𝑛 2 𝐻 3-2 𝜅 𝑜 8 𝜋 Γ 𝑜 = 𝜙 𝑚 +2𝑛 (3-2 𝜅 ) 𝑒 (87)</formula>
<text><location><page_7><loc_52><loc_46><loc_91><loc_51></location>In order to get the field at the time of horizon exit, we take the help of the number of e-folds. For our model, it is given by</text>
<formula><location><page_7><loc_56><loc_41><loc_91><loc_45></location>𝑁 = ∫ 𝜙 𝜙 𝑒 8 𝜋 3 𝑚 2 𝑝 𝐻 𝑜 Γ 𝑜 𝜙 𝑛 𝜙 𝑚 2(2 𝜅 ) 𝐻 3-2 𝜅 𝑜 𝜙 𝑛 (3-2 𝜅 ) 𝑛𝐻 𝑜 𝜙 𝑛 -1 (88)</formula>
<text><location><page_7><loc_52><loc_39><loc_60><loc_40></location>This leads to</text>
<formula><location><page_7><loc_56><loc_31><loc_92><loc_38></location>3 𝑚 2 𝑝 2(2 𝜅 ) 𝑛𝐻 3-2 𝜅 𝑜 8 𝜋 Γ 𝑜 𝑁 = 1 𝑚 +2𝑛 (3 - 2 𝜅 ) ( 𝜙 𝑚 +2𝑛 (3-2 𝜅 ) -𝜙 𝑚 +2𝑛 (3-2 𝜅 ) 𝑁 𝑒 ) (89)</formula>
<text><location><page_7><loc_52><loc_27><loc_91><loc_30></location>This finally allows us to write the field at the time of horizon exit as</text>
<formula><location><page_7><loc_52><loc_21><loc_92><loc_27></location>𝜙 𝑚 +2𝑛 (3-2 𝜅 ) = 𝜙 𝑚 +2𝑛 (3-2 𝜅 ) 𝑒 ( 1 + ( 𝑚 +2𝑛 (3 - 2 𝜅 )) 𝑁 𝑛 ) (90)</formula>
<text><location><page_7><loc_52><loc_18><loc_71><loc_19></location>Now, using (23) we can write</text>
<formula><location><page_7><loc_56><loc_13><loc_91><loc_17></location>𝑑𝜙 𝑑𝑡 = -3 𝑚 2 𝑝 2(2 𝜅 ) 𝐻 3-2 𝜅 𝑜 𝜙 𝑛 (3-2 𝜅 ) 𝑛𝐻 𝑜 𝜙 𝑛 -1 8 𝜋 Γ 𝑜 𝜙 𝑚 (91)</formula>
<text><location><page_7><loc_52><loc_10><loc_80><loc_11></location>Integrating from 𝑡 𝑜 to some t , we arrive at</text>
<formula><location><page_7><loc_56><loc_6><loc_97><loc_8></location>𝜙 𝑚 +2𝑛 (4-2 𝜅 ) ( 𝑡 ) = (4 - 2 𝜅 ) 𝑛 -𝑚 -2 𝑛 𝜙 𝑚 +2𝑛 (3-2 𝜅 ) 𝑒 𝐻 𝑜 ( 𝑡 -𝑡 𝑜 )+</formula>
<text><location><page_8><loc_9><loc_86><loc_13><loc_87></location>where</text>
<formula><location><page_8><loc_14><loc_88><loc_48><loc_94></location>( 𝜙 𝑚 +2𝑛 (3-2 𝜅 ) 𝑒 ( 1 + ( 𝑚 +2𝑛 (3 - 2 𝜅 )) 𝑁 𝑛 )) 𝑔 (92)</formula>
<formula><location><page_8><loc_20><loc_83><loc_37><loc_86></location>𝑔 = ( 𝑚 +2𝑛 (4 - 2 𝜅 )) 𝑚 +2𝑛 (3 - 2 𝜅 )</formula>
<text><location><page_8><loc_9><loc_80><loc_48><loc_82></location>This immediately allows us to write the Hubble parameter and the dissipation coefficient as functions of time</text>
<formula><location><page_8><loc_13><loc_70><loc_50><loc_79></location>𝐻 ( 𝑡 ) = 𝐻 𝑜 [ (4 - 2 𝜅 ) 𝑛 -𝑚 -2 𝑛 𝜙 𝑚 +2𝑛 (3-2 𝜅 ) 𝑒 𝐻 𝑜 ( 𝑡 -𝑡 𝑜 )+ ( 𝜙 𝑚 +2𝑛 (3-2 𝜅 ) 𝑒 ( 1 + ( 𝑚 +2𝑛 (3 - 2 𝜅 )) 𝑁 𝑛 )) 𝑔 ] 𝑛𝑤 (93)</formula>
<formula><location><page_8><loc_13><loc_60><loc_49><loc_67></location>Γ( 𝑡 ) = Γ 𝑜 [ (4 - 2 𝜅 ) 𝑛 -𝑚 -2 𝑛 𝜙 𝑚 +2𝑛 (3-2 𝜅 ) 𝑒 𝐻 𝑜 ( 𝑡 -𝑡 𝑜 )+ ( 𝜙 𝑚 +2𝑛 (3-2 𝜅 ) 𝑒 ( 1 + ( 𝑚 +2𝑛 (3 - 2 𝜅 )) 𝑁 𝑛 )) 𝑔 ] 𝑚𝑤</formula>
<formula><location><page_8><loc_20><loc_53><loc_37><loc_56></location>𝑤 = 1 ( 𝑚 +2𝑛 (4 - 2 𝜅 ))</formula>
<text><location><page_8><loc_52><loc_92><loc_89><loc_93></location>Doing a little bit of algebra on this formula, we arrive at</text>
<formula><location><page_8><loc_56><loc_87><loc_91><loc_90></location>𝜌 𝑟 ( 𝜙 ) = 𝑚 2 𝑝 (2 𝜅 ) 𝜙 𝑚 +2𝑛 (3-2 𝜅 ) 𝑒 𝜌 𝜙 16 𝜋𝜙 𝑚 +2𝑛 (3-2 𝜅 ) (98)</formula>
<text><location><page_8><loc_52><loc_83><loc_91><loc_85></location>Using (92) , we can further write the above expression in terms of time ,</text>
<formula><location><page_8><loc_56><loc_70><loc_92><loc_82></location>𝜌 𝑟 ( 𝑡 ) = 𝑚 2 𝑝 (2 𝜅 ) 𝜙 𝑚 +2𝑛 (3-2 𝜅 ) 𝑒 𝜌 𝜙 16 𝜋 [ (4 - 2 𝜅 ) 𝑛 -𝑚 -2 𝑛 𝜙 𝑚 +2𝑛 (3-2 𝜅 ) 𝑒 𝐻 𝑜 ( 𝑡 -𝑡 𝑜 ) + ( 𝜙 𝑚 +2𝑛 (3-2 𝜅 ) 𝑒 ( 1+ ( 𝑚 +2𝑛 (3 - 2 𝜅 )) 𝑁 𝑛 )) 𝑔 ] 𝑔 (99)</formula>
<text><location><page_8><loc_52><loc_68><loc_91><loc_69></location>Wenote further that at the time of horizon exit, (98) becomes</text>
<formula><location><page_8><loc_54><loc_62><loc_91><loc_66></location>𝜌 𝑟 = 𝑚 2 𝑝 (2 𝜅 ) 𝜌 𝜙 16 𝜋 ( 𝑛 𝑛 +( 𝑚 +2𝑛 (3 - 2 𝜅 )) 𝑁 ) (100)</formula>
<text><location><page_8><loc_48><loc_55><loc_91><loc_61></location>(94) The above expression tells us that in the high dissipation regime at the time of horizon exit, only the free parameters 𝜅 , 𝑚 and 𝑛 and the e-fold number N determines the relationship between 𝜌 𝜙 and 𝜌 𝑟 .</text>
<text><location><page_8><loc_9><loc_55><loc_13><loc_57></location>where</text>
<text><location><page_8><loc_9><loc_49><loc_48><loc_52></location>The inflationary potential corresponding to this cosmology is given by (30)</text>
<formula><location><page_8><loc_13><loc_40><loc_48><loc_48></location>𝑉 ( 𝜙 ) = 3 𝑚 2 𝑝 𝐻 4-2 𝜅 𝑜 𝜙 4-2 𝜅 8 𝜋 -1 2 ( 3 𝑚 2 𝑝 2(2 𝜅 ) 𝐻 4-2 𝜅 𝑜 𝑛𝜙 (3-2 𝜅 ) 𝑛 -𝑚 -1 8 𝜋 Γ 𝑜 ) 2 (95)</formula>
<text><location><page_8><loc_9><loc_36><loc_48><loc_38></location>By the definition of the Hubble parameter we have the scale factor as a function of time as</text>
<formula><location><page_8><loc_13><loc_19><loc_48><loc_35></location>𝑎 ( 𝑡 ) = 𝑎 𝑜 exp ( 𝑛𝐻 𝑜 𝜙 𝑚 +2𝑛 (3-2 𝜅 ) 𝑒 ( 𝑚 +2-(3-2 𝜅 ) 𝑛 ) 𝑁 [ 𝜙 𝑚 +2𝑛 (3-2 𝜅 ) 𝑒 ( 1 + 𝑚 +2𝑛 (3 - 2 𝜅 ) 𝑛 ) -[ (4 - 2 𝜅 ) 𝑛 -𝑚 -2 𝑛 𝜙 𝑚 +2𝑛 (3-2 𝜅 ) 𝑒 𝐻 𝑜 ( 𝑡 -𝑡 𝑜 ) + ( 𝜙 𝑚 +2𝑛 (3-2 𝜅 ) 𝑒 ( 1 + ( 𝑚 +2𝑛 (3 - 2 𝜅 )) 𝑁 𝑛 ) ) 𝑔 ] 𝑔 ]) (96)</formula>
<text><location><page_8><loc_9><loc_11><loc_48><loc_17></location>We can also get a relationship between the radiation energy density and the inflaton energy density, as a function of 𝜙 and consequently of time. Using (25) in the high dissipation regime we have,</text>
<formula><location><page_8><loc_13><loc_6><loc_48><loc_10></location>𝜌 𝑟 ( 𝜙 ) = 3 𝑚 2 𝑝 𝜌 𝜙 32 𝜋𝐹 [ ( 𝐹 ' 𝐻 𝐻 ' ) 2 Γ 𝐻 ] (97)</formula>
<text><location><page_8><loc_52><loc_36><loc_91><loc_53></location>Now in order to fully get the details of Warm Inflation in the high dissipation regime in this cosmology, we would like to have appropriate values of the free parameters 𝜅 , 𝑛 and 𝑚 which fit with the observational data [[2] , [1]]. We would like to have expressions for important observational quantities like the scalar spectral index and the running of the scalar spectral index at the time of horizon exit. This can be done by evaluating all the relevant slow roll and other cosmological parameters defined previously at the time of horizon exit . Using (90), we can evaluate the slow roll parameters at the time of horizon exit. The 𝜖 slow roll parameter in particular is</text>
<formula><location><page_8><loc_56><loc_32><loc_91><loc_35></location>𝜖 = 𝑛 𝑛 +( 𝑚 +2𝑛 (3 - 2 𝜅 )) 𝑁 (101)</formula>
<text><location><page_8><loc_52><loc_25><loc_91><loc_31></location>We have emphasized about the 𝜖 parameter here because it is possible to express all the other parameters which we have mentioned before in terms of this parameter. Evaluating all the parameters at horizon exit, we get</text>
<formula><location><page_8><loc_56><loc_22><loc_91><loc_24></location>𝛽 = 𝑚 𝑛 𝜖 (102)</formula>
<formula><location><page_8><loc_56><loc_16><loc_91><loc_19></location>𝜂 = 𝑛 -1 𝑛 𝜖 (103)</formula>
<formula><location><page_8><loc_56><loc_12><loc_91><loc_13></location>𝛾 = (3 - 2 𝜅 ) 𝜖 (104)</formula>
<formula><location><page_8><loc_56><loc_6><loc_91><loc_8></location>𝜎 = ( 𝑛 -1)( 𝑛 -2) 𝑛 𝜖 (105)</formula>
<formula><location><page_9><loc_13><loc_88><loc_48><loc_91></location>𝛿 = ( 𝑚 )( 𝑚 -1) 𝑛 2 𝜖 2 (106)</formula>
<formula><location><page_9><loc_13><loc_83><loc_48><loc_84></location>𝜓 = (3 - 2 𝜅 )(2 - 2 𝜅 ) 𝜖 (107)</formula>
<text><location><page_9><loc_9><loc_79><loc_48><loc_82></location>Using these definitions and (70), we arrive at the following expressions for the scalar spectral index,</text>
<formula><location><page_9><loc_13><loc_75><loc_48><loc_78></location>𝑛 𝑠 = 1 + 3(4 𝜅𝑛 +3 𝑚 -𝑛 -2) 4 𝑁 (2( 𝜅 -1) 𝑛 + 𝑚 +2) (108)</formula>
<text><location><page_9><loc_9><loc_73><loc_48><loc_74></location>And the running of the scalar spectral index is given by (82),</text>
<formula><location><page_9><loc_10><loc_67><loc_48><loc_70></location>𝛼 𝑠 = -3( 𝑛 (2( 𝜅 (4 𝜅 -11) + 8) 𝑛 +42 𝜅 -55) + 51) 4((2 𝜅 -3) 𝑛𝑁 + 𝑛 +5 𝑁 ) 2 (109)</formula>
<text><location><page_9><loc_9><loc_48><loc_48><loc_66></location>With that, we have now studied all the analytical aspects of warm inflation in this scenario.To get more insight into the paradigm of warm inflation in this model, we will have to take note of the latest observational data available from the Planck satellite experiment [1] and see which values of the free parameters in this model most suitably fit the data available on the spectral index and it's running. Another important quantity in this excursion of ours is the efold Number. For inflation to solve cosmological problems and contribute in large scale structure formation, the e-fold number can conveniently between around 60 [5] , so we will henceforth set 𝑁 = 60 .</text>
<text><location><page_9><loc_9><loc_44><loc_48><loc_48></location>Using (70) and (80) , we find that a suitable choice of the free parameters which satisfies the inflationary Requirements is ( 𝑚, 𝑛, 𝜅 ) = (3 , -5 , 1 . 4) . For these values ,</text>
<formula><location><page_9><loc_13><loc_41><loc_48><loc_42></location>𝑛 𝑠 ≈ 0 . 964912 (110)</formula>
<formula><location><page_9><loc_13><loc_36><loc_48><loc_37></location>𝛼 𝑠 ≈ -0 . 003 (111)</formula>
<text><location><page_9><loc_9><loc_21><loc_48><loc_34></location>Which is in perfect agreement with the Planck 2018 data [2] of 𝑛 𝑠 = 0 . 9649 ± 0 . 0042 ( 68% 𝐶𝐿 , Planck 𝑇𝑇,𝑇𝐸,𝐸𝐸 + 𝑙𝑜𝑤𝐸 + 𝑙𝑒𝑛𝑠𝑖𝑛𝑔 ) and the negligible running of the spectral index. Putting these values in the required equations derived above would give one full details of Warm Inflation in this cosmology.One can also further use the constraints on the tensor-to-scalar ratio provided by the data to bound the constants 𝐻 𝑜 and Γ 𝑜 in a similar way as done in [48] , but we do not pursue that here.</text>
<section_header_level_1><location><page_9><loc_9><loc_17><loc_22><loc_19></location>5. Conclusions</section_header_level_1>
<text><location><page_9><loc_9><loc_6><loc_48><loc_16></location>In this paper, we have introduced the exact solution approach for Warm onflation. We started off by showing how many modified cosmological scenarios like braneworld cosmologies , modified gravity Cosmologies and various modified entropy cosmologies have a similar form of the Friedmann equation which can be used to consider a generalized Friedmann equation with a general function of</text>
<text><location><page_9><loc_52><loc_58><loc_91><loc_93></location>the Hubble parameter . We then began the description of our method with a light review of the basics dynamics of warm inflation. After that, we described our approach and showed how various important quantities for warm inflationary regimes like the Hubble parameter, the dissipation coefficient, the e-fold number, the inflaton field function etc. can be derived using this approach. We further explored scalar and tensorial inflationary perturbations in this method and derived important Inflationary Parameters like the scalar spectral index, it's running, the tensor-to-scalar ratio etc. Finally, we applied this method to study high dissipation warm inflation in a Tsallis modified entropy universe. We here point to one peculiarity of our model which we have not yet touched upon. In obtaining the equation of the inflaton field we have assumed that the matter, specified by the inflaton scalar field, enters into the action Lagrangian in such a way that its variation in a FLRW background metric leads to the Klein-Gordon equation, expressed by (11). Therefore our method is only applicable to theories where the background metric alongside the perturbations, are not modified.This means that Horava-Lifshitz theories of gravity [41] or theories of similar plight are beyond the scope of our approach.</text>
<section_header_level_1><location><page_9><loc_52><loc_52><loc_68><loc_53></location>Acknowledgements</section_header_level_1>
<text><location><page_9><loc_52><loc_47><loc_91><loc_51></location>I would like to thank the referee for their insightful comments on the work, which have increased the depth of the work multi folds.</text>
<section_header_level_1><location><page_9><loc_52><loc_43><loc_61><loc_44></location>References</section_header_level_1>
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</document> | [{"title": "ABSTRACT", "content": "The theory of cosmic inflation has received a great amount of deserved attention in recent years due to it's stunning predictions about the early universe. Alongside the usual cold inflation paradigm, warm inflation has garnered a huge amount of interest in modern inflationary studies. It's peculiar features and specifically different predictions from cold inflation have led to a substantial amount of literature about it. Various modified cosmological scenarios have also been studied in the warm inflationary regime. In this work, we introduce the exact solution approach for warm inflation. This approach allows one to directly study warm inflationary regime in a variety of modified cosmological scenarios. We begin by outlining our method and show that it generalizes the modified Friedmann approach of Del Campo , and reduces to the well known Hamilton-Jacobi formalism for inflation in particular limits. We also find the perturbation spectra for cosmological and tensor perturbations in the early universe, and then apply our method to study warm inflation in a Tsallis entropy modified Friedmann universe. We end our paper with some concluding remarks on the domain of applicability of our work.", "pages": [1]}, {"title": "The exact solution approach to warm inflation", "content": "Oem Trivedi a , \u2217 a International Centre for Space and Cosmology, School of Arts and Sciences, Ahmedabad University, Navrangpura, Ahmedabad, 380009, India", "pages": [1]}, {"title": "ARTICLE INFO", "content": "Keywords : early universe cosmology warm inflation modified gravity theories", "pages": [1]}, {"title": "1. Introduction", "content": "Some of the most captivating problems in modern cosmology concern the workings of the very early Universe. A lot of the physics of how our Universe was initially is not very well understood and this was evident ever since fine tuning problems were discovered in the traditional big bang cosmology model itself. The most prominent of these problems were the flatness, horizon and monopole problems, which were very promisingly solved by the introduction of Cosmic inflation [24, 3, 38, 47, 20, 42, 17, 44, 43, 52]. Cosmic inflation is the idea that the very early Universe went through a period of rapid accelerated expansion. The dynamics of the traditional inflationary models generally consider a real scalar field, the \"inflaton\" field, as the predominant contributor to the universal energy density at that time. While there are other inflationary models which consider more exotic scenarios [22, 49, 35], there is still a very high amount of evident interest in single real scalar field driven inflationary regimes [5, 36, 13, 37]. The conventional models of inflation by a single scalar field consist of a period of rapid accelerated expansion. The sub horizon scale quantum fluctuations during this period of expansion, lead to cosmological perturbations which give us a spectrum of scalar and tensor perturbations ( vector perturbations are usually ignored). This generates the large scale structure of our Universe which we see today. Also , after the period of expansion sets off a period of \" reheating\" [3] , where the energy of the inflaton field decays to form radiation energy densities. These models do not take into account the dissipation of inflaton energy to form radiation while the inflationary period itself is going on and hence, these are known popularly now as \" cold inflation\" scenarios. [email protected] (O. Trivedi) ORCID(s): On the other hand, the class of inflationary regimes which do take into account this dissipation effect are known as \" warm inflation\" regimes [6, 10, 46, 9]. Warm inflation includes inflaton interactions with other fields throughout the inflationary epoch instead of confining such interactions to a distinct reheating era. Consequently, these inflationary models do not need a separate period of reheating after inflation to make the universe radiation dominated, and hence this provides a solution to the graceful exit problem of inflation [7]. Recently a lot of interest has been weighed towards cosmological models which modify the structure of the usual Friedmann Equation where we have used the units \ud835\udc50 = \u210f = 1 and \ud835\udc5a 2 \ud835\udc5d is the usual reduced Planck mass. \ud835\udf0c is the energy density in the Universe, which in the context of usual Inflationary Models refers to the energy density of the inflaton field. Various different cosmological scenarios bring about a change in this usual Friedmann equation . Superstring and M-Theory bring the possibility of considering our universe as a domain wall embedded in a higher dimensional space. In this scenario the standard model of particle is confined to the brane, while gravitation propagate into the bulk spacetime. The effect of extra dimensions induces a well known change in the Friedmann equation [51] , where \ud835\udf06 is a measure of the brane tension. It is also possible to consider quantum corrected area entropy relations and study universal evolution in such scenarios. For example, [14] considered the quantum corrected Entropy where A is the area of the apparent horizon and \u0303 \ud835\udefc is dimensionless positive constant determined by conformal anolmaly of the fields.This conformal anomaly is interpreted as a quantum correction to the entropy of the apparent horizon. The resulting Friedmann equation is of the form , where \ud835\udefd is some positive constant in the units of inverse mass squared. This type of Friedmann equation can also be arrived at by using a Chern-Simons modification of gravity [23] . Finally, there has been a surge in interest recently in the form of Black hole Entropy due to Tsallis and Cirto [55] , where they argued that argued that the microscopic mathematical expression of the thermodynamical entropy of a black hole does not obey the area law and can be modified as , where \ud835\udefe is an unknown constant , A is the Black Hole Horizon area and \ud835\udf05 is a real parameter which quantifies the degree of nonextensivity, known as the Tsallis parameter. Considering the apparent horizon entropy of an FLRW type universe to be the Tsallis type, [50] was able to derive a Friedmann equation for the universe while also taking into account and modifying Padmanabhan's emergent gravity proposal [45]. The concerned Friedmann equation is [50] , Inflation has been studied in a lot of the popular braneworld scenarios , both in the traditional supercooled regime [18, 29, 31] and in the warm inflation regime [32, 19, 21, 15]. [15] also studied Chern-Simons type modified gravity of [23] in the cold inflation regime. A crucial point , as noted in [15], which emerges from the studies of inflationary studies in different cosmological scenarios is that the concerned Friedmann equation of investigation can be often taken of the form , where \ud835\udc39 ( \ud835\udc3b ) is some function of the Hubble parameter H. A very useful approach to studying different cold inflation scenarios was discussed in [15]. We would now like to study different cosmological scenarios using (7) as the primary equation of investigation. We hence structure our paper as follows. In section 2, we outline the exact solution approach for Warm Inflation and describe the dynamical aspects of the method while in Section 3 we discuss Perturbation spectra and derive crucial observational quantities like the scalar and tensor spectral index in concern to the approach. In Section 4 we apply the method to study Warm Inflation in the high dissipative regime in a Tsallis entropy type Universe of [50] . Then finally, we summarize our results in section 5 in the Concluding remarks of the paper.", "pages": [1, 2]}, {"title": "2. The exact solution method", "content": "In warm inflationary approach, there is consideration given to both the inflaton field and the radiation being created due to it's dissipation during Inflation itself. Hence, the usual Friedmann equation of warm inflation is [6] where the energy density \ud835\udf0c consists of both inflaton and radiation energy densities, \ud835\udf0c \ud835\udf19 and \ud835\udf0c \ud835\udc5f respectively. But during inflation , \ud835\udf0c \ud835\udf19 >> \ud835\udf0c \ud835\udc5f [6], hence (8) becomes , where \ud835\udf0c \ud835\udf19 is given by , with \ud835\udc49 ( \ud835\udf19 ) being the potential under which the inflaton field is. The inflaton field equation , the inflaton energy density and radiation energy density evolution equation is given by, Here , \u0393 is the dissipation coefficient of inflaton energy. Defining a quantity \ud835\udc44 = \u0393 3 \ud835\udc3b , we can rewrite the above equations as During inflation, we can take both \u0307 \ud835\udf0c \ud835\udc5f to be approximately zero [6] . This gives us (16) This completes a small review of the basics of warm inflation which we will now freely refer to throughout the entirety of paper. Consider now the following general form of the Friedmann equation The famous slow roll conditions of cold inflation [5], hold well similarly in warm inflation as well [6] and so during warm inflation, This gives us the energy density to be \ud835\udf0c \ud835\udf19 \u2248 \ud835\udc49 ( \ud835\udf19 ) Hence the general Friedmann equation(18) during Inflation, takes the form Now, it was shown in [33] that treating \ud835\udf19 itself as the explicitly dependent variable for H can lead to a very apt method of studying inflationary models, which is referred to as the \" Hamilton-Jacobi approach to inflation\". [48] studied the Hamilton-Jacobi approach with respect to warm inflation and found that treating the Hubble parameter and the disspiation coefficients in terms of \ud835\udf19 is indeed very helpful for warm inflationary regimes. Hence, now we treat the Hubble parameter ( and hence the scale factor as well) primarily in terms of the field variable \ud835\udf19 . So \ud835\udc3b = \ud835\udc3b ( \ud835\udf19 ) . Further the field variable will generally be time dependent, hence the Hubble parameter stays implicitly time dependent. With these considerations, we take the derivative of (20) with respect to \ud835\udf19 , where , following the convention in [15], \ud835\udc39 ' \ud835\udc3b is the partial derivative of F with respect to H , and the prime in the superscript denotes derivative with respect to \ud835\udf19 . In the slow roll approximation (11) becomes, Using the expression for \ud835\udc49 ' ( \ud835\udf19 ) from (21) we have \u0307 \ud835\udf19 as Using (23) , we can get a different expression for \ud835\udf0c \ud835\udc5f by taking into account (17) , Hence, we now have expression for the field velocity given \ud835\udc39 ( \ud835\udc3b ) ,an ansatz for H in terms of \ud835\udf19 and a form of the dissipation coefficient. As mentioned in [40, 10], dissipation coefficient is popularly used in power laws in terms of both \ud835\udf19 and temperature T. The temperature plays an especially important role for high dissipative regimes, which is characterized by and temperature itself plays a very crucial in the generation of quantum fluctuations during Inflation [7, 10]. Using (24) we can get an expression for \ud835\udf0c \ud835\udc5f in terms of \ud835\udf0c \ud835\udf19 and it's easy to check that We also note that as \ud835\udf0c \ud835\udc5f is pure radiation , we can write where T is the radiation temperature and \ud835\udefc is the Stefan Boltzmann Constant. Now, we have from (17) and (24), which gives us We note from the Friedmann equation that Using the expression for \u0307 \ud835\udf19 (23), we have the potential given as Also as , we can also write the potential in terms of the inflaton energy density The number of e-folds is a very important quantity to measure the amount of Inflation. The number of e-folds is [[5] ], where \ud835\udf19 \ud835\udc52 and \ud835\udf19 are the field values at the end and beginning of Inflation, respectively. Once again using (23), we write N as Now, by the definition , we have which gives us Another important quantity for inflationary models are the slow roll parameters and from the basic definition of the \ud835\udf16 slow roll parameter, one can write where \ud835\udf16 is the \"first\" Hubble slow roll parameter given by This can be arrived at directly by first realizing that This implies \ud835\udf16 is also realized using the concurrent definition [39] The second Hubble Slow Roll Parameter \ud835\udf02 is similarly defined to be We now take a brief detour and discuss a bit of subtlety surrounding the definition of slow roll parameters for Warm Inflation. While the \ud835\udf16 and \ud835\udf02 slow roll parameters above is derived as it comes from the basic definition (38) and (39), there still exists conflicting views in literature about the appropriate definition for the parameter in the context of Warm Inflation. While some authors [27, 56] like to define the \ud835\udf16 and \ud835\udf02 parameters in their usual supercooled inflation forms , others [21, 48] like to \" absorb\" the dependence of the damping function Q as it comes into the slow roll parameters from the basic definitions (38) and (39) . The proponents of the former approach feel that defining the first and second slow roll parameters in the usual cold inflation form allows them to have a more relaxed constraint in warm inflationary scenarios [27] . The author of this paper is also in harmony with this line of thought, but however feels the latter approach of defining the first two Hubble parameters in warm inflation allows for them to be evidently more general then their supercooled inflation form. This allows the reader to more smoothly see the transition of the parameters from warm to cold inflation in the extremely low dissipation regime. But we are certainly of the opinion that while both approaches of defining the parameters may appear different on the level of substance, they carry virtually the same essence overall. So now we just note for the sake of completeness that we can equivalently define the \ud835\udf16 and \ud835\udf02 parameters suitable for the approach [27, 56] as (which we will call \ud835\udf16 \ud835\udc4e and \ud835\udf02 \ud835\udc4e ) which is just the form of the parameters as shown in [15] . The second form of the parameters , which is in line with the approach of [21, 48], are just the \ud835\udf02 and \ud835\udf16 parameters defined in (38) and (39). We readily see that (42) and (43) are just (38) and (39) respectively, in the \ud835\udc44 << 1 approximation. Alongside the usual slow roll parameters \ud835\udf02 and \ud835\udf16 , [56] showed that for appropriately studying warm inflationary paradigms some other parameters with the derivatives of \u0393 would be very useful. These parameters were derived by the using the slow roll conditions for warm inflation. Hence we would now like to define some new slow roll parameters for our model, in order to better cater to the needs of warm inflation scenarios. One of the primary slow roll conditions reads We would now evaluate the quantity -\u0308 \ud835\udc3b \u0307 \ud835\udc3b\ud835\udc3b to arrive at our new slow roll parameters taking lead from [48] . We begin by noting that which can be realized using (38) . Now this allows us to write To evaluate (44) , we need to first have an expression for \u0308 \ud835\udf19 appropriate for our use. This can be done by using (23) supplemented by the fact that \u0307 \ud835\udc39 ' \ud835\udc3b = \u0307 \ud835\udc3b\ud835\udc39 ' \ud835\udc3b\ud835\udc3b where ( \ud835\udc39 ' \ud835\udc3b\ud835\udc3b = \ud835\udf15 2 \ud835\udc39 \ud835\udf15\ud835\udc3b 2 ) . A little bit of algebra leads us to This allows us to write And further we write this leads us to define \ud835\udefd parameter [56, 48] as , In addition to these slow roll parameters, we define other parameters which will be helpful for our perturbation spectra studies. These are , and which now allows us to express (46) as In addition to these we define more parameters which will be helpful for us in our perturbation spectra analysis This completes the proper dynamical outline of the Exact Solution approach for warm inflation. It is quick to check that this approach reduces to the usual cold inflation approach for exact solutions [15] in the extremely low dissipation regime \ud835\udc44 << 1 . It also further reduces to the Hamilton-Jacobi method for warm inflation for \ud835\udc39 ( \ud835\udc3b ) = \ud835\udc3b 2 [48], and further to the usual cold inflation Hamilton-Jacobi approach [33] in the extremely low dissipation regime for the same \ud835\udc39 ( \ud835\udc3b ) . With the dynamical aspects covered, we shall now explore the perturbation spectra of warm inflation in this approach.", "pages": [2, 3, 4, 5]}, {"title": "3. Perturbation spectra analysis", "content": "Cosmological density and gravitational wave perturbations in the inflationary scenario arise as quantum fluctuations which redshift to long wavelengths due to rapid cosmological expansion during Inflation [5, 28, 53, 25]. In warm inflation, only the density perturbations couple strongly with the thermal background and hence, the scalar density spectra is the one which looks more evidently different from it's usual supercooled inflationary counterpart [54] . Tensor perturbations do not couple strongly to the thermal background and so gravitational waves are only generated by quantum fluctuations, as in standard cold inflation. In addition to the usual adiabatic perturbations in cold inflation, Isocurvature perturbations also are generated in the warm inflationary era due to thermal fluctuations in the radiation field. These perturbations can be characterised by fluctuations in the entropy of the particle species undergoing thermal fluctuations relative to the number density of photons. But in this paper, we will limit our focus to the study of only adiabatic perturbations. The square of the amplitude of adiabatic perturbations is calculated in a similar way to cold inflation [54, 10, 48] , where \ud835\udc51\ud835\udf19 2 for the high dissipative regime ( \ud835\udc44 >> 1 )is given by [8] , where \ud835\udc58 \ud835\udc39 is the so called freeze out number given by, The definition of the freeze out number \ud835\udc58 \ud835\udc39 is not changed by considering a general Friedmann equation of the form (18). It is so because the definition of the freeze out number stems primarily from the field equation of \ud835\udf19 (11) , in particular from the evolution equation of the fluctuations \ud835\udeff\ud835\udf19 ( \ud835\udc65, \ud835\udc61 ) ( where \ud835\udf19 (x,t) = \ud835\udf19 \ud835\udc5c ( \ud835\udc65, \ud835\udc61 ) + \ud835\udeff\ud835\udf19 ( \ud835\udc65, \ud835\udc61 ) ) [54, 8].This remains unchanged by the consideration of (18) , takes the usual form with the inclusion of the spatial Laplacian and an additional white noise random force term , where we have Fourier transformed to the mpmentum space and \ud835\udf01 ( \ud835\udc58, \ud835\udc61 ) is the white noise term. In a similar way to [54] , we reach at \ud835\udc58 \ud835\udc39 = \u221a \u0393 \ud835\udc3b . Now using the expression for T (26), and the definition of \ud835\udf16 (40) , we can write \ud835\udc43 \ud835\udc60 ( \ud835\udc58 ) as which leads us to write The scalar spectral index is defined by the well known equation where \ud835\udc51 ln \ud835\udc58 is given as the negative of the differential of the number of e-folds For proceeding further, we note that the derivative of the \ud835\udf16 parameter is given by This allows us to write This finally allows to us to express the scalar spectral index as This is the scalar spectral index for the high dissipative regime. For the low dissipative regime ( \ud835\udc44 << 1 ) we have which allows us to write the power spectrum for the low dissipative regime as Using the expression for the temperature derived above (26) , we have Again , the definition of the scalar spectral index in this case is the same as for the high dissipation scenario Pursuing a similar analysis as for the previous case, we arrive at One of the more exciting findings of the observational data from the Planck, WMAP and COBE experiments is that there is a significant running of the scalar spectral index as well. Traditionally it was taken to be negligible but these experimental findings make them a very important observational quantity. Alongside the Running of the scalar spectral index and the index itself, the tensor to scalar ratio is another important quantity of observational relevance. We will now focus more on the high dissipative regime and calculate the running of the scalar spectral index and the tensor-to-scalar ratio in that limit. We will not calculate the same in the low dissipative regime but one can calculate them in that limit by pursuing a similar method as we do in the following for the high dissipative regime. The running of the spectral index is defined by it's usual definition In a similar way as we calculated \ud835\udc51\ud835\udf16 \ud835\udc51 ln \ud835\udc58 in (66), we arrive at the following differentials The above expressions allow us to write the running of scalar spectral index using (67) as , (79) Further, the squared tensor perturbation power spectrum amplitude is defined as [34] During inflation, \ud835\udc49 \u2248 3 \ud835\udc5a 2 \ud835\udc5d \ud835\udc39 8 \ud835\udf0b which is clear by (20). The definition of the tensor-to-scalar ratio is We see that (78) now allows us to write the tensor-to-scalar ratio as Again we remark that the above formulations of \ud835\udc5f and \ud835\udefc \ud835\udc60 are specifically for the high dissipative regime \ud835\udc44 >> 1 . One can easily formulate the same for the low dissipative regime \ud835\udc44 << 1 using the same procedure we have shown above. Now, we have completed all the theoretical basis of our approach. We will now apply it on a Tsallis entropy modified universe. We will study aspects of Warm Inflation in this model in the high dissipative regime.", "pages": [5, 6, 7]}, {"title": "4. Warm inflation in Tsallis entropy modified universe", "content": "A Tsallis entropy modified universe has the Friedmann equation of the form (6). This allows us to write \ud835\udc39 ( \ud835\udc3b ) for this model as Moving forward we would need to ascertain two more quantities for studying warm inflation in this model, which are the Hubble parameter \ud835\udc3b and the dissipation coefficient \u0393 . Both of them will be taken as functions for \ud835\udf19 . While this statement can be understood in a straightforward way for \ud835\udc3b , for \u0393 the answer could have been different. Usually, \u0393 is taken as a function of only \ud835\udf19 or of both \ud835\udf19 and the radiation temperature \ud835\udc47 [48, 11, 26, 12] . This is because temperature plays a crucial role in the dissipative scenario of warm inflation. However, temperature in general is written in terms of \ud835\udf19 eventually and the dissipation function in cases with \u0393 = \u0393( \ud835\udf19, \ud835\udc47 ) turns into a function of only \ud835\udf19 [16, 40, 4, 30] . So in our case we will treat both H and \u0393 as \ud835\udc3b = \ud835\udc3b ( \ud835\udf19 ) and \u0393 = \u0393( \ud835\udf19 ) . Further, we consider both H and \u0393 to be power law functions of \ud835\udf19 , as where \ud835\udc3b \ud835\udc5c , \u0393 \ud835\udc5c are some constants. The powers n and m are left undetermined here as we will use the Planck data to find out which power laws best fit with our concerned model. The reason for choosing power law form for the Hubble parameter is because they seem to be a good fit with the latest Planck Data [2] , while the consideration of a supersymmetric interaction of the authors [11, 12] lead to the dissipation coefficient being a linear function of \ud835\udf19 . While other authors from several distinct considerations have been led to power law forms of the dissipation coefficient [16, 30, 48]. This suggests to us that power law forms can indeed be very viable and general forms of the dissipation coefficients. Hence, we take the coefficient to be in a power law form of the field variable. With all the preliminaries cleared up, we now move towards concrete calculations. As stated previously, we will focus on warm inflation in the high dissipative regime for this case. To make progress, we would need the form of the field \ud835\udf19 at the time of horizon exit and at the end of inflation. It is straightforward to get the latter by setting \ud835\udf16 = 1 in (40), In order to get the field at the time of horizon exit, we take the help of the number of e-folds. For our model, it is given by This leads to This finally allows us to write the field at the time of horizon exit as Now, using (23) we can write Integrating from \ud835\udc61 \ud835\udc5c to some t , we arrive at where This immediately allows us to write the Hubble parameter and the dissipation coefficient as functions of time Doing a little bit of algebra on this formula, we arrive at Using (92) , we can further write the above expression in terms of time , Wenote further that at the time of horizon exit, (98) becomes (94) The above expression tells us that in the high dissipation regime at the time of horizon exit, only the free parameters \ud835\udf05 , \ud835\udc5a and \ud835\udc5b and the e-fold number N determines the relationship between \ud835\udf0c \ud835\udf19 and \ud835\udf0c \ud835\udc5f . where The inflationary potential corresponding to this cosmology is given by (30) By the definition of the Hubble parameter we have the scale factor as a function of time as We can also get a relationship between the radiation energy density and the inflaton energy density, as a function of \ud835\udf19 and consequently of time. Using (25) in the high dissipation regime we have, Now in order to fully get the details of Warm Inflation in the high dissipation regime in this cosmology, we would like to have appropriate values of the free parameters \ud835\udf05 , \ud835\udc5b and \ud835\udc5a which fit with the observational data [[2] , [1]]. We would like to have expressions for important observational quantities like the scalar spectral index and the running of the scalar spectral index at the time of horizon exit. This can be done by evaluating all the relevant slow roll and other cosmological parameters defined previously at the time of horizon exit . Using (90), we can evaluate the slow roll parameters at the time of horizon exit. The \ud835\udf16 slow roll parameter in particular is We have emphasized about the \ud835\udf16 parameter here because it is possible to express all the other parameters which we have mentioned before in terms of this parameter. Evaluating all the parameters at horizon exit, we get Using these definitions and (70), we arrive at the following expressions for the scalar spectral index, And the running of the scalar spectral index is given by (82), With that, we have now studied all the analytical aspects of warm inflation in this scenario.To get more insight into the paradigm of warm inflation in this model, we will have to take note of the latest observational data available from the Planck satellite experiment [1] and see which values of the free parameters in this model most suitably fit the data available on the spectral index and it's running. Another important quantity in this excursion of ours is the efold Number. For inflation to solve cosmological problems and contribute in large scale structure formation, the e-fold number can conveniently between around 60 [5] , so we will henceforth set \ud835\udc41 = 60 . Using (70) and (80) , we find that a suitable choice of the free parameters which satisfies the inflationary Requirements is ( \ud835\udc5a, \ud835\udc5b, \ud835\udf05 ) = (3 , -5 , 1 . 4) . For these values , Which is in perfect agreement with the Planck 2018 data [2] of \ud835\udc5b \ud835\udc60 = 0 . 9649 \u00b1 0 . 0042 ( 68% \ud835\udc36\ud835\udc3f , Planck \ud835\udc47\ud835\udc47,\ud835\udc47\ud835\udc38,\ud835\udc38\ud835\udc38 + \ud835\udc59\ud835\udc5c\ud835\udc64\ud835\udc38 + \ud835\udc59\ud835\udc52\ud835\udc5b\ud835\udc60\ud835\udc56\ud835\udc5b\ud835\udc54 ) and the negligible running of the spectral index. Putting these values in the required equations derived above would give one full details of Warm Inflation in this cosmology.One can also further use the constraints on the tensor-to-scalar ratio provided by the data to bound the constants \ud835\udc3b \ud835\udc5c and \u0393 \ud835\udc5c in a similar way as done in [48] , but we do not pursue that here.", "pages": [7, 8, 9]}, {"title": "5. Conclusions", "content": "In this paper, we have introduced the exact solution approach for Warm onflation. We started off by showing how many modified cosmological scenarios like braneworld cosmologies , modified gravity Cosmologies and various modified entropy cosmologies have a similar form of the Friedmann equation which can be used to consider a generalized Friedmann equation with a general function of the Hubble parameter . We then began the description of our method with a light review of the basics dynamics of warm inflation. After that, we described our approach and showed how various important quantities for warm inflationary regimes like the Hubble parameter, the dissipation coefficient, the e-fold number, the inflaton field function etc. can be derived using this approach. We further explored scalar and tensorial inflationary perturbations in this method and derived important Inflationary Parameters like the scalar spectral index, it's running, the tensor-to-scalar ratio etc. Finally, we applied this method to study high dissipation warm inflation in a Tsallis modified entropy universe. We here point to one peculiarity of our model which we have not yet touched upon. In obtaining the equation of the inflaton field we have assumed that the matter, specified by the inflaton scalar field, enters into the action Lagrangian in such a way that its variation in a FLRW background metric leads to the Klein-Gordon equation, expressed by (11). Therefore our method is only applicable to theories where the background metric alongside the perturbations, are not modified.This means that Horava-Lifshitz theories of gravity [41] or theories of similar plight are beyond the scope of our approach.", "pages": [9]}, {"title": "Acknowledgements", "content": "I would like to thank the referee for their insightful comments on the work, which have increased the depth of the work multi folds.", "pages": [9]}, {"title": "References", "content": "[29] Hawkins, R.M., Lidsey, J.E., 2001. Inflation on a single brane: Exact solutions. Physical Review D 63, 041301.", "pages": [10]}] |
2021RAA....21...90S | https://arxiv.org/pdf/2007.06472.pdf | <document>
<text><location><page_1><loc_15><loc_85><loc_85><loc_86></location>Explaining recently studied intermediate luminosity optical transients (ILOTs) with jet powering</text>
<text><location><page_1><loc_38><loc_83><loc_62><loc_84></location>Noam Soker 1, 2 and Noa Kaplan 1</text>
<text><location><page_1><loc_19><loc_79><loc_80><loc_82></location>1 Department of Physics, Technion, Haifa, 3200003, Israel; [email protected] 2 Guangdong Technion Israel Institute of Technology, Shantou 515069, Guangdong Province, China</text>
<section_header_level_1><location><page_1><loc_45><loc_76><loc_55><loc_77></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_54><loc_86><loc_75></location>We apply the jet-powered ILOT scenario to two recently studied intermediate luminosity optical transients (ILOTs), and find the relevant shell mass and jets' energy that might account for the outbursts of these ILOTs. In the jet-powered ILOT scenario accretion disk around one of the stars of a binary system launches jets. The interaction of the jets with a previously ejected slow shell converts kinetic energy to thermal energy, part of which is radiated away. We apply two models of the jetpowered ILOT scenario. In the spherical shell model the jets accelerate a spherical shell, while in the cocoon toy model the jets penetrate into the shell and inflate hot bubbles, the cocoons. We find consistent results. For the ILOT (ILRT: intermediate luminosity red transient) SNhunt120 we find the shell mass and jets' energy to be M s glyph[similarequal] 0 . 5 -1 M glyph[circledot] and E 2j glyph[similarequal] 5 × 10 47 erg, respectively. The jets' half opening angle is α j glyph[similarequal] 30 · -60 · . For the second peak of the ILOT (luminous red nova) AT 2014ej we find these quantities to be M s glyph[similarequal] 1 -2 M glyph[circledot] and E 2j glyph[similarequal] 1 . 5 × 10 48 erg, with α j glyph[similarequal] 20 · -30 · . The models cannot tell whether these ILOTs were powered by a stellar merger that leaves one star, or by mass transfer where both stars survived. In both cases the masses of the shells and energies of the jets suggest that the binary progenitor system was massive, with a combined mass of M 1 + M 2 glyph[greaterorsimilar] 10 M glyph[circledot] .</text>
<text><location><page_1><loc_14><loc_50><loc_61><loc_51></location>Keywords: binaries: close - stars: jets - stars: variables: general</text>
<section_header_level_1><location><page_1><loc_20><loc_47><loc_36><loc_48></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_26><loc_48><loc_46></location>The transient events with peak luminosities above those of classical novae and below those of typical supernovae might differ from each other by one or more properties, like the number of peaks in the light curve, total power, progenitor masses, and powering mechanism (e.g. Mould et al. 1990; Bond et al. 2003; Rau et al. 2007; Ofek et al. 2008; Mason et al. 2010; Kasliwal 2011; Tylenda et al. 2013; Kasliwal et al. 2012; Blagorodnova et al. 2017; Kaminski et al. 2018; Pastorello et al. 2018; Boian & Groh 2019; Cai et al. 2019; Jencson et al. 2019; Kashi et al. 2019; Pastorello et al. 2019; Howitt et al. 2020; Jones 2020; Klencki et al. 2020). They form a heterogeneous group of 'gap transients'.</text>
<text><location><page_1><loc_8><loc_11><loc_48><loc_26></location>We study those transients that are powered by an accretion process that releases gravitational energy. The accretion process might be a mass transfer from one star to another, or an extreme case of stellar merger, where either one star destroys another, or one star (or a planet; Retter & Marom 2003; Bear et al. 2011; Kashi & Soker 2017; Kashi et al. 2019) enters the envelope of a larger star to start a common envelope evolution (e.g., Tylenda et al. 2011; Ivanova et al. 2013; Nandez et al. 2014; Kami'nski et al. 2015; MacLeod et al. 2017; Segev</text>
<text><location><page_1><loc_52><loc_44><loc_92><loc_48></location>et al. 2019; Schrøder et al. 2020; MacLeod & Loeb 2020; Soker 2020b). We refer to all these systems as intermediate luminosity optical transients (ILOTs).</text>
<text><location><page_1><loc_52><loc_19><loc_92><loc_44></location>In cases where both stars survive and stay detached the binary system can experience more than one outburst, and can have several separated peaks in its light curve. This is the case for example in the grazing envelope evolution (Soker 2016). The same holds when the binary system forms a temporary common envelope. Namely, the more compact companion enters the envelope and then gets out. An example of the later process is the repeating common envelope jets supernova (CEJSN) impostor scenario (Gilkis et al. 2019). In a CEJSN impostor event a neutron star (or a black hole) gets into the envelope of a giant massive star, accretes mass and launches jets that power an ILOT event (that might be classified as a supernova impostor), and then gets out of the envelope (Soker & Gilkis 2018; Gilkis et al. 2019; Yalinewich & Matzner 2019).</text>
<text><location><page_1><loc_52><loc_9><loc_92><loc_18></location>Mass outflow accompanies the bright outbursts of ILOTs. Many studies attribute the powering of ILOTs, both the kinetic energy of the outflow and the radiation, to stellar binary interaction processes (e.g., Soker & Tylenda 2003; Tylenda & Soker 2006; Kashi et al. 2010; Mcley & Soker 2014; Pejcha et al. 2016a,b; Soker</text>
<text><location><page_2><loc_8><loc_65><loc_48><loc_91></location>2016; MacLeod et al. 2018; Michaelis et al. 2018; Pastorello et al. 2019). As a fast outflow hits a previously ejected slower outflow, the collision channels kinetic energy to radiation. There are two types of binary scenarios in that respect, those that take the main collision to take place in and near the equatorial plane (e.g., Pejcha et al. 2016a,b; Metzger & Pejcha 2017; Hubov'a, & Pejcha 2019), and those that attribute the main collision to fast polar outflow, i.e., jets. In most of the cases with high mass accretion rates that power ILOTs, the high-accretion-powered ILOT (HAPI) model (Kashi & Soker 2016; Soker & Kashi 2016), the accretion of mass is likely to be through an accretion disk. This accretion disk is very likely to launch two opposite jets. If the jets collide with a previously ejected slow shell an efficient conversion of kinetic energy to radiation might take place. This is the jet-powered ILOT scenario .</text>
<text><location><page_2><loc_8><loc_46><loc_48><loc_65></location>In a recent study Soker (2020a) argues that the jetsshell interaction of the jet-powered ILOT scenario is more efficient in converting kinetic energy to radiation than collision of equatorial ejecta. He further applies a simple jet-shell interaction model to three ILOTs, the Great Eruption of Eta Carinae (Davidson, & Humphreys 1997), which is a luminous blue variable (LBV), to V838 Mon (Munari et al. 2002) that is a stellar merger (also termed luminous red nova; LRN), and to the ILOT V4332 Sgr that has a bipolar structure (Kaminski et al. 2018). We apply this simple spherical shell model to two other ILOTs (Sections 2.2 and 2.3).</text>
<text><location><page_2><loc_8><loc_23><loc_48><loc_46></location>As said, in this study we use the term ILOT (Berger et al. 2009; Kashi & Soker 2016; Muthukrishna et al. 2019). There are different classifications of the heterogeneous class of transients, like the one by Kashi & Soker (2016) 1 , the one by Pastorello et al. (2019) and Pastorello & Fraser (2019), and also by Jencson et al. (2019). Some refer to transients from stellar merger by LRNe and to outbursts that involve a massive giant star by intermediate luminosity red transients (ILRTs). We simply refer to all transients that are powered by gravitational energy of mass transfer (or merger), the HAPI model, as ILOTs. This saves us the need to classify a specific event by its unknown progenitors. We are mainly interested in the roles of jets, that might play a role in all types of ILOTs (although not in all ILOTs).</text>
<text><location><page_2><loc_8><loc_15><loc_48><loc_22></location>Two recent studies of two ILOTs support two crucial ingredients of the jet-powered ILOT scenario. Blagorodnova et al. (2020) study the ILOT M31-LRN-2015 that is possibly a merger remnant (some earlier studies related to this ILOT include Williams et al. 2015; Lipunov et</text>
<text><location><page_2><loc_52><loc_81><loc_92><loc_91></location>al. 2017; MacLeod et al. 2017; Metzger & Pejcha 2017). Blagorodnova et al. (2020) estimate the primary mass to be M 1 glyph[similarequal] 5 M glyph[circledot] and deduce that during the two years pre-outburst activity the system lost a mass of about > 0 . 14 M glyph[circledot] . Such a pre-outburst formation of a shell (circumbinary matter) is an important ingredient in the jet-powered ILOT scenario.</text>
<text><location><page_2><loc_52><loc_59><loc_92><loc_80></location>In other recent papers Kaminski et al. (2020, 2021) study in details the ILOT (stellar-merger candidate) Nova 1670 (CK Vulpeculae). This 350-years old nebula has a bipolar structure (Shara et al. 1985) with an 'S' shape along the long axis (Kaminski et al. 2020, 2021). This is an extremely strong indication of shaping by precessing jets. We take it to imply that the jet-powered ILOT scenario accounts for Nova 1670. The intervals from the first to second peak and from the second to third peak in the triple-peaks light curve are about equal at about 1 year (Shara et al. 1985). We take it to imply a multiple jets-launching episodes, or more likely in this case, a variability in jets' launching power as the jets precess.</text>
<text><location><page_2><loc_52><loc_35><loc_92><loc_58></location>These two recent studies, and in particular the clear demonstration of an 'S' shape morphology of the ILOT Nova 1670 (Kaminski et al. 2020, 2021) motivate us to apply the jet-powered ILOT scenario to two recently studied ILOTs. We emphasise that our main aim is to find plausible parameters for these two recently studied ILOTs in the frame of the jet-driven model, as the formation of jets in binary merger can be very common (e.g., L'opez-C'amara et al. 2020 and references therein). In section 2 we describe the basic features of the jetpowered ILOT scenario and apply it in a simple way to the ILOTs SNhunt120 and AT 2014ej. In section 3 we build a more sophisticated toy model to describe the jet-powered ILOT scenario and apply it to these two ILOTs. We summarise in section 4.</text>
<section_header_level_1><location><page_2><loc_56><loc_32><loc_88><loc_33></location>2. THE JET-POWERED ILOT SCENARIO</section_header_level_1>
<section_header_level_1><location><page_2><loc_57><loc_30><loc_87><loc_31></location>2.1. Features of the spherical shell model</section_header_level_1>
<text><location><page_2><loc_52><loc_16><loc_92><loc_29></location>The basic flow structure of the jet-powered ILOT scenario is as follows (Soker 2020a). A binary interaction leads to the ejection of a shell, spherical or not, at velocities of tens to hundreds of km s -1 . The shell ejection period can last from few weeks to several years. In a delay of about days to several months (or even a few years) the binary system launches two opposite jets. The jets collide with the shell, an interaction that converts kinetic energy, mainly of the jets, to radiation.</text>
<text><location><page_2><loc_52><loc_9><loc_92><loc_15></location>There are two types of evolutionary channels to launch jets. (1) The more compact secondary star accretes mass from the primary star and launches the jets, as in the jet-powered ILOT scenario of the Great Eruption of Eta</text>
<text><location><page_3><loc_8><loc_76><loc_48><loc_91></location>Carinae (e.g., Soker 2007; Kashi & Soker 2010a). The binary stellar system might stay detached, might experience the grazing envelope evolution, and/or enters a common envelope evolution. In this case the binary systems might experience several jets-launching episodes. (2) The primary star gravitationally destroys the secondary star to form an accretion disk around the primary star, and this accretion disk launches the jets. In this case there is one jets-launching episode, although the jets' intensity can very with time.</text>
<text><location><page_3><loc_8><loc_57><loc_48><loc_76></location>Soker (2020a) obtains the following approximate relations for jets that interact with a slower spherically symmetric shell and power an ILOT. We refer to this model as the spherical shell model. Soker (2020a) considers jets-shell interaction that (1) transfers a large fraction of the kinetic energy of the outflow to radiation, and (2) radiates much more energy than what recombination of the outflowing gas can supply. Soker (2020a) considers two opposite fast jets that hit a uniform spherical shell and accelerate the entire shell. In section 3 we build a toy model where the jets penetrate into the shell and interact with shell's material only in the polar directions.</text>
<text><location><page_3><loc_8><loc_48><loc_48><loc_57></location>In the simple flow structure that Soker (2020a) considers the relevant properties of the jets are their half opening angle α j glyph[greaterorsimilar] 10 · , velocity v j ≈ 10 3 km s -1 , and their total mass M 2j ≈ 0 . 01 -1 M glyph[circledot] . With a conversion efficiency of jets' kinetic energy to radiation f rad , the total energy in radiation is</text>
<formula><location><page_3><loc_9><loc_42><loc_48><loc_45></location>E rad , j = 10 48 f rad ( M 2j 0 . 1 M glyph[circledot] ) ( v j 1000 km s -1 ) 2 erg . (1)</formula>
<text><location><page_3><loc_8><loc_37><loc_48><loc_40></location>The relevant properties of the spherical shell are its velocity v s glyph[lessmuch] v j , mass M s , radius r s , and width ∆ r s .</text>
<text><location><page_3><loc_8><loc_20><loc_48><loc_37></location>The jet-shell interaction converts kinetic energy, mainly of the jets, to thermal energy. The hot bubbles that the jets inflate lose their energy adiabatically by accelerating the shell and non-adiabatically by radiation. The adiabatic cooling proceeds on a typical time scale that is the expansion time t exp , while energy losses to radiation occurs during a typical photon-diffusion time out t diff . Namely, the relative rates, ˙ E/E , of adiabatic and radiative energy losses are t -1 exp and t -1 diff , respectively. This implies that the fraction of energy that ends in radiation is</text>
<formula><location><page_3><loc_15><loc_14><loc_48><loc_18></location>f rad glyph[similarequal] t -1 diff t -1 diff + t -1 exp = ( 1 + t diff t exp ) -1 . (2)</formula>
<text><location><page_3><loc_8><loc_9><loc_48><loc_12></location>For the simple spherically symmetric geometry he assumes, Soker (2020a) estimates the two time scales to</text>
<text><location><page_3><loc_52><loc_90><loc_54><loc_91></location>be</text>
<text><location><page_3><loc_52><loc_80><loc_55><loc_82></location>and</text>
<formula><location><page_3><loc_56><loc_73><loc_92><loc_79></location>t diff glyph[similarequal] 3 τ ∆ r s c glyph[similarequal] 55 ( M s 1 M glyph[circledot] )( κ 0 . 1 cm 2 g -1 ) × ( r s 10 14 cm ) -1 ( ∆ r s 0 . 3 r s ) days , (4)</formula>
<text><location><page_3><loc_52><loc_67><loc_92><loc_72></location>where τ = ρ s κ ∆ r s is the optical depth of the shell, κ is the opacity, and c is the light speed. The relevant ratio to substitute in equation (2) is</text>
<formula><location><page_3><loc_53><loc_59><loc_92><loc_66></location>t diff t exp ≈ 0 . 75 ( M s 1 M glyph[circledot] )( κ 0 . 1 cm 2 g -1 ) ( v j 1000 km s -1 ) × ( r s 10 14 cm ) -2 ( ∆ r s 0 . 3 r s )[ M 2j 0 . 1( M 2j + M s ) ] 1 / 2 . (5)</formula>
<text><location><page_3><loc_52><loc_47><loc_92><loc_58></location>We emphasise that we do not assume any value for the jets' energy E 2j . We rather take the jets' velocity from observations, and use the time scale of the ILOT together with an assumed opacity to find the mass in the shell (equation 4). We then calculate the efficiency f rad together with the mass in the jets (or their energy) to fit the total radiated energy (equations 2 and 5).</text>
<text><location><page_3><loc_52><loc_12><loc_92><loc_47></location>Soker (2020a) applies this spherical shell model of the jet-powered ILOT radiation to the ILOT (LRN) V838 Mon, to the Great Eruption of Eta Carinae which is an LBV, and to the ILOT V4332 Sgr. He could find plausible set of shell and jets parameters that might explain these ILOTs. Here we apply the spherical shell model to the ILOT (ILRT) SNhunt120 and to the ILOT (LRN) AT 2014ej. We summarise the plausible physical parameters of the ILOT events in Table 1, and explain their derivation in sections 2.2 and 2.3. We emphasise that due to the very simple model we apply here, e.g., we use a spherical shell and we keep the opacity and shell thickness constant, the properties of the jets and shells we derive are very crude, and might even not be unique. Nonetheless, they demonstrate the potential of the jet-powered ILOT scenario to account for many ILOTs. The opacity of a fully ionised gas that is appropriate for ILOTs is κ glyph[similarequal] 0 . 3 cm 2 g -1 (e.g., Ivanova et al. 2013; Soker & Kashi 2016). We expect that in the outer parts of the shell hydrogen is partially neutral, and that opacity is therefore lower. Therefore, we scale with κ = 0 . 1 cm 2 g -1 .</text>
<text><location><page_3><loc_62><loc_10><loc_82><loc_11></location>2.2. The ILOT SNhunt120</text>
<formula><location><page_3><loc_58><loc_83><loc_92><loc_89></location>t exp ≈ 73 ( r s 10 14 cm )( v j 1000 km s -1 ) -1 × [ M 2j 0 . 1( M 2j + M s ) ] -1 / 2 days , (3)</formula>
<table>
<location><page_4><loc_8><loc_68><loc_48><loc_92></location>
<caption>Table 1. Summary of plausible approximate values of parameters in the spherical-shell ILOT model of Soker (2020a) for the ILOTs SNhunt120 and AT 2014ej. We assume that AT 2014ej was powered by two jet-launching episodes, each accounting for one of the two peaks in the light curve. The symbol '[O]' in the first column implies a quantity we take from observations, while '[J]' indicates that we derive the plausible parameter. We derive these parameters under the assumption of a constant opacity of κ = 0 . 1 cm 2 g -1 and a constant shell width of ∆ r s = 0 . 3 r s . In both ILOTs observation suggest jets' velocity of v j glyph[similarequal] 1000 km s -1 which we also use here.</caption>
</table>
<text><location><page_4><loc_8><loc_28><loc_48><loc_48></location>Stritzinger et al. (2020b) study the ILOT (ILRT) SNhunt120 and find the following relevant properties. The velocities of different emission lines are in the range of glyph[similarequal] 300 -1800 km s -1 , with a typical velocity of ≈ 10 3 km s -1 . The typical photospheric radius is R BB glyph[similarequal] 2 × 10 14 cm. The time to double the luminosity at rise is about 10 days, and the decline time to half the maximum luminosity is about 20 days. The total energy in radiation is E rad glyph[similarequal] 4 × 10 47 erg. Stritzinger et al. (2020b) further find that existing electron capture supernova models over-predict the energy in radiation. We do not consider this event to be a supernova, but rather an ILOT.</text>
<text><location><page_4><loc_8><loc_15><loc_48><loc_27></location>Following these parameters we scale the parameters for SNhunt120 with v j glyph[similarequal] 1000 km s -1 and r s glyph[similarequal] 2 × 10 14 cm. To get a photon diffusion time of about the decline time of 20 days, we find from equation (4) for κ = 0 . 1 cm 2 g -1 and ∆ r s = 0 . 3 r s that M s ≈ 0 . 7 M glyph[circledot] . For an opacity of κ = 0 . 3 cm 2 g -1 and somewhat a thicker shell with ∆ r s = 0 . 5 R s , the required shell mass is only M s ≈ 0 . 15 M glyph[circledot]</text>
<text><location><page_4><loc_8><loc_12><loc_48><loc_15></location>Equation (5) gives then t diff /t exp ≈ 0 . 1, and from equation (2) f rad glyph[similarequal] 0 . 9. To account for the emitted</text>
<text><location><page_4><loc_52><loc_89><loc_92><loc_91></location>energy, we find from equation (1) that the mass in the two jets is M 2j ≈ 0 . 045 M glyph[circledot] ( v j / 1000 km s -1 ) -2 .</text>
<text><location><page_4><loc_52><loc_73><loc_92><loc_88></location>In this analysis there is no reference to the shell velocity, beside that it should be much lower than the jets' velocity. This implies here 100 km s -1 glyph[lessorsimilar] v s glyph[lessorsimilar] 500 km s -1 . To reach a distance of r s = 2 × 10 14 the binary system ejected the shell about ∆ t s glyph[similarequal] 0 . 6( v s / 100 km s -1 ) -1 yr before detection. The kinetic energy in the shell for these parameters of M s glyph[similarequal] 0 . 7 M glyph[circledot] and v s glyph[similarequal] 100 km s -1 is about 15% of the jets' energy. In any case, most of the kientic enrgy of the shell does not convert to radiation.</text>
<text><location><page_4><loc_52><loc_43><loc_92><loc_73></location>In case that the secondary star launches the jets with a mass of M 2j glyph[similarequal] 0 . 045 M glyph[circledot] , it should accrete a mass of M acc , 2 glyph[similarequal] 0 . 45 M glyph[circledot] from a more evolved primary star, possibly a giant. This implies that the secondary star should be a massive star itself. We are therefore considering a massive binary system. Alternatively, it is possible that the primary star destroyed the secondary star of mass M 2 glyph[similarequal] 0 . 3 -1 M glyph[circledot] to form an accretion disk that launched the jets. The primary is then a massive main sequence star, and the secondary is not yet settled on the main sequence, such that its average density is lower than that of the primary star (as in the merger model of V838 Mon; Tylenda & Soker 2006). In any case, the primary star mass can be in the range of M 1 ≈ 10 M glyph[circledot] , similar to the range that Stritzinger et al. (2020b) consider. Since there is only one jets-launching episode, the jet-powered ILOT scenario does not directly refer in the case of SNhunt120 to the question of which of these two evolutionary routes apply here.</text>
<section_header_level_1><location><page_4><loc_63><loc_38><loc_82><loc_39></location>2.3. The ILOT AT 2014ej</section_header_level_1>
<text><location><page_4><loc_52><loc_30><loc_92><loc_37></location>Stritzinger et al. (2020a) study the ILOT (LRN) AT 2014ej. They find that the light curve of models of equatorial collision (Metzger & Pejcha 2017; section 1), under-predict the luminosity. We therefore consider powering by jets, i.e., polar collision.</text>
<text><location><page_4><loc_52><loc_9><loc_92><loc_29></location>Stritzinger et al. (2020a) find that AT 2014ej has slow component(s) moving at ≈ 100 km s -1 and fast component(s) moving at ≈ 1000 km s -1 . The total radiated energy is E rad ≈ 2 × 10 48 erg, with two large peaks in the light curve. From discovery to first minimum 20 days later, the luminosity decreased from L 0 = 3 . 2 × 10 41 erg s -1 to L min , 1 = 1 . 2 × 10 41 erg s -1 . Over the next 35 days the luminosity increased to L AT ≡ L peak , 2 glyph[similarequal] 2 . 6 × 10 41 erg s -1 , after which the luminosity decreased over a time scale of several weeks. The photosphere was hotter in the first peak than in the second one. The photosphere (black body surface) moderately followed the behavior of the luminosity, and first</text>
<text><location><page_5><loc_8><loc_89><loc_48><loc_91></location>declined somewhat and then increased somewhat. Its approximate average value is R BB glyph[similarequal] 2 . 5 × 10 14 cm.</text>
<text><location><page_5><loc_8><loc_73><loc_48><loc_88></location>In the jet-powered ILOT scenario such multiple-peaks can be accounted for by multiple jet-launching episodes. From Stritzinger et al. (2020a) we find that the radiated energy from detection to first minimum (0 to 20 days) is glyph[similarequal] 4 × 10 47 erg. If we take a similar energy at rise, the energy in the first peak is E rad , 1p ≈ 10 48 erg. The energy in the second peak, from 20 to about 95 days, is E rad , 2p ≈ 1 . 4 × 10 48 erg. The outburst of V838 Mon has a similar qualitative behavior with three peaks and three declines in the photospheric radius (Tylenda 2005).</text>
<text><location><page_5><loc_8><loc_51><loc_48><loc_73></location>In AT 2014ej the two peaks have about the same energy (under our assumption), but the second peak is slower by a factor of about two. From equation (4) the mass in the shell should be larger in the second peak by a factor of two, glyph[similarequal] 2 M glyph[circledot] instead of glyph[similarequal] 1 M glyph[circledot] . We do not expect the system to lose much more slow mass in that short time. The difference in the time scales of the two peaks might come from different values of κ and/or ∆ r s between the two peaks, rather than from different shell masses. This can also be accompanied by precessing jets, i.e., the jets' axes in the two jet-launching episodes have different directions. In the present study we use a simple model and do not calculate the opacity, and so we simply take for both peaks M s glyph[similarequal] 1 . 5 M glyph[circledot] .</text>
<text><location><page_5><loc_8><loc_46><loc_48><loc_51></location>From the equations of section 2.1 we derive the crude plausible values of the shell mass, jets' energy, and emission efficiency for the two peaks, as we list in Table 1.</text>
<text><location><page_5><loc_8><loc_37><loc_48><loc_46></location>According to the jet-powered ILOT scenario the two distinguished peaks result from two jets-launching episode. This suggests that the secondary star, possibly in an eccentric orbit, accreted mass and launches the jets. Most likely, the secondary star survived the interaction.</text>
<section_header_level_1><location><page_5><loc_17><loc_34><loc_39><loc_35></location>3. A BIPOLAR TOY-MODEL</section_header_level_1>
<section_header_level_1><location><page_5><loc_19><loc_32><loc_37><loc_33></location>3.1. The cocoon toy-model</section_header_level_1>
<text><location><page_5><loc_8><loc_9><loc_48><loc_31></location>In the simple spherical-shell model that we apply in section 2 the jets interact with the entire shell (Soker 2020a). We now turn to a more realistic toy model where the jets interact only with the shell segments along the polar directions. In this 'cocoon toy model' the jet-shell interaction inflates a 'cocoon', i.e., a relatively hot bubble composed of the post-shock shell material and postshock jet's material. We further simplify the interaction by assuming that the jets' activity time period is short, such that we can treat the jet-shell interaction that creates the cocoon as a 'mini explosion'. We base the cocoon toy model on our usage of this model to account for peaks in the light curves of core collapse supernovae (Kaplan & Soker 2020; for the geometry of a</text>
<text><location><page_5><loc_52><loc_79><loc_92><loc_91></location>jet-ejecta interaction in core collapse supernova see the three-dimensional simulations of Akashi & Soker 2020). In the cocoon toy model we only calculate the timescale of the emission peak (eruption) and its maximum luminosity (or total energy). We do not calculate the shape of the light curve, but rather assume a simple shape for the light curve. We then calculate the total radiated energy by integrating the luminosity over time.</text>
<text><location><page_5><loc_52><loc_59><loc_92><loc_79></location>We assume that each mini-explosion that results from jet-shell interaction is spherically symmetric around the jet-shell interaction point (Akashi & Soker 2020), and that cooling is due to photon diffusion and adiabatic expansion. These assumptions allow us to determine the luminosity and the time scale of each mini-explosion. As we deal with ILOTs where the total radiated energy is larger than the recombination energy of the outflowing gas, we neglect the recombination energy. Like Kaplan & Soker (2020) we use equations (4) from Kasen & Woosley (2009) to calculate the time of maximum luminosity t j and the maximum luminosity L j for one jet. These expressions read</text>
<formula><location><page_5><loc_59><loc_51><loc_92><loc_57></location>t j = ( 3 2 5 / 2 π 2 c ) 1 / 2 E -1 / 4 j M 3 / 4 js κ 1 / 2 c , L j = 2 πc 3 sin α j β M -1 js E j κ -1 c R BB , (6)</formula>
<text><location><page_5><loc_52><loc_28><loc_92><loc_50></location>where E j , M js , κ c , α j , β and R BB are the energy that one jet deposits into the shell, the mass in the interaction region of one jet with the shell, the opacity in the cocoon, the half opening angle of the jet, the distance of the jetejecta interaction relative to the shell's outer edge (the photosphere radius R BB ), and the photospheric radius of the shell, respectively. Namely, in this model the mini explosion takes place at a radius (measured from the center of the binary system) of r me = βR BB . There is one mini-explosion on each of the two polar regions. The value of r me is constant and does not change with time. What increases with time is the radius of the cocoon itself, a c , that is measured from the place of the mini explosion.</text>
<text><location><page_5><loc_52><loc_9><loc_92><loc_28></location>We build the light curve of the jet as follows. We assume that the shape of the rise of the peak to maximum luminosity is similar to the rise to maximum of the light curve of a core collapse supernova (based on photometric data of SN 2008ax, taken from The Open Supernova Catalog Guillochon et al. 2017). Since the light curve of the jet does not have a tail powered by radioactive processes and recombination, we take the decline of the mini-explosion from maximum to be symmetric to its rise. Again, we do not try to fit the light curve. We rather only derive the properties of the jets that might lead to an event that has the same timescale, luminosity</text>
<text><location><page_6><loc_8><loc_87><loc_48><loc_91></location>and radiated energy. We assume a light curve, but our results are not sensitive to the exact shape of the light curve we assume.</text>
<text><location><page_6><loc_8><loc_81><loc_48><loc_87></location>We turn to estimate the jets' properties that according to the cocoon toy model might fit the eruption times and luminosities of the ILOTs SNhunt120 (section 3.2) and AT 2014ej (section 3.3).</text>
<section_header_level_1><location><page_6><loc_12><loc_77><loc_44><loc_78></location>3.2. The cocoon toy model fit of SNhunt120</section_header_level_1>
<text><location><page_6><loc_8><loc_47><loc_48><loc_76></location>First we extend the observed light curve of SNhunt120 (Stritzinger et al. 2020b; thick-red line in Fig. 1) by taking a linear fit before discovery and beyond t = 30 days after discovery, in both sides down to L = 0. This is the solid-blue line in Fig. 1. The observed light curve of this ILOT has a break at about 40 days post-discovery, where the decline becomes shallower. This might result from a second and weaker jet-launching episode or from matter collision in the equatorial plane. We are interested here only in the light curve around the maximum, so we continue the steep decline beyond 30 days post-discovery down to L = 0. We then find the radiated energy of SNhunt120 of our fit to the peak to be E rad , hunt = 3 . 8 × 10 47 erg. As we explained in section 3.1, we then build a toy-model symmetric light curve that has the same maximum luminosity as SNhunt120, L hunt = 1 . 4 × 10 41 erg s -1 , and the same total radiated energy. This is the green line in Fig. 1 (for Case 1 that we describe next).</text>
<text><location><page_6><loc_8><loc_23><loc_48><loc_47></location>We calculate the energy of one jet E j and the mass in the region of interaction of one jet with the shell (the cocoon), M js , as follows. We build a symmetric toy model light curve (one example is the green line in Fig. 1) that is characterised by its maximum luminosity L j and by its timescale from start to maximum t j by equations (6). We then calculate the total radiated energy according to this light curve (area under the green light curve). We iterate the values of E j and M js until we obtain the luminosity due to the two jets together of L 2j = L hunt = 1 . 4 × 10 41 erg s -1 , and the total radiated energy from the two jets is E rad , 2j = E rad , hunt = 3 . 8 × 10 47 erg. We note that the cocoon toy model is not sensitive to the expansion velocities of the shell and of the jets, as long as the v j glyph[greatermuch] v s .</text>
<text><location><page_6><loc_8><loc_9><loc_48><loc_23></location>We do not vary the values of the photosphere radius R BB = 2 × 10 14 cm that we take from Stritzinger et al. (2020b), and of β = 0 . 7 in equations (6). We do vary the values of the jet's half opening angle α j and of the opacity κ c . We continue with the wide jets that we discussed in section 2 (Soker 2020a) and scale with α j = 60 · , but we consider narrower jets as well. We scale the opacity with κ c = 0 . 1 cm 2 g -1 but examine also κ c = 0 . 05 cm 2 g -1 and κ c = 0 . 3 cm 2 g -1 to</text>
<figure>
<location><page_6><loc_53><loc_67><loc_91><loc_91></location>
<caption>Figure 1. The light curve of SNhunt120 (thick-red line) from Stritzinger et al. (2020b), our extension of the peak of light curve (blue line), and a light curve of the cocoon toy model (green line for case 1). We constrain the green light curve to fit the total radiated energy of the peak E rad , hunt = 3 . 8 × 10 47 erg and its maximum luminosity L hunt = 1 . 4 × 10 41 erg s -1 . The parameters of this fit (Case 1) are the opacity κ c , the jets' half opening angle α j , and the radius of the jet-shell interaction region βR BB , where R BB is the photosphere radius. We calculate the values of the combined energy of the two jets E 2j and the combined masses in the jets-shell interaction regions M 2js . Note that we do not try to fit the shape of the light curves, but rather only try to explain the amount of radiated energy and maximum luminosity of the peak.</caption>
</figure>
<text><location><page_6><loc_71><loc_66><loc_82><loc_67></location>√]⌉˜(⋃√⊎</text>
<text><location><page_6><loc_52><loc_37><loc_92><loc_42></location>demonstrate the model sensitivity to opacity. The relevant scaling of equations (6) for SNhunt120, (for one jet) read</text>
<formula><location><page_6><loc_58><loc_29><loc_92><loc_36></location>t j = 22 . 7 ( E j 2 × 10 47 erg ) -1 / 4 × ( M js 0 . 1 M glyph[circledot] ) 3 / 4 ( κ c 0 . 1 cm 2 g -1 ) 1 / 2 d , (7)</formula>
<text><location><page_6><loc_52><loc_26><loc_55><loc_28></location>and</text>
<formula><location><page_6><loc_55><loc_15><loc_92><loc_25></location>L j = 7 . 3 × 10 40 ( sin α j 0 . 87 )( β 0 . 7 ) × ( M js 0 . 1 M glyph[circledot] ) -3 / 2 ( E j 2 × 10 47 erg ) 3 / 2 × ( κ c 0 . 1 cm 2 g -1 ) -1 ( R BB 2 × 10 14 cm ) erg s -1 . (8)</formula>
<text><location><page_6><loc_52><loc_9><loc_92><loc_13></location>As with the spherical shell model, we do not assume the energy of the jets. The input variables to the fitting process are the light curve, the half opening angle of the jets,</text>
<text><location><page_7><loc_8><loc_81><loc_48><loc_91></location>the opacity, and the values of β and sin α j . We take the radius of the continuum black body photosphere from observations. We then substitute in equations (9) and (10) the observed ILOT's (or one peak of the ILOT) duration t j and the energy radiated from one jet-shell interaction L j , and solve for the one jet's energy E j and the mass of the shell that one jet interacts with M js .</text>
<text><location><page_7><loc_8><loc_71><loc_48><loc_80></location>In Table 2 we present six sets of values in the cocoon toy model for SNhunt120. We emphasise that we do not try to fit the shape of the light curves, and only try to explain the amount of radiated energy, the timescale, and the maximum luminosity of the peak. In Fig. 1 we show by the green line Case 1.</text>
<table>
<location><page_7><loc_10><loc_56><loc_45><loc_70></location>
<caption>Table 2. Six different sets of parameters that fit the peak of the light curve and the total radiated energy of the ILOT SNhunt120 in the frame of the cocoon toy model. The opacity κ c and the jets' half opening angle α j are input parameters of the modelling. Other parameters are as in equations (7) and (8). We calculate from these equations (see text) the combined energy of the two jets E 2j and the combined mass in the interaction regions of the two jets with the shell M 2js . In the last column we list the emission efficiency f rad = E rad /E 2j .</caption>
</table>
<text><location><page_7><loc_8><loc_30><loc_48><loc_39></location>The energy of the jets and the mass they interact with vary between the cases. The energy range is E 2j glyph[similarequal] 4 × 10 47 erg -11 × 10 47 erg. In the spherical-shell model of section 2.2 the jets' energy is 4 . 5 × 10 47 erg. From the cases of tables 1 and 2 we crudely take the jets' energy for this ILOT to be E 2j (SNhunt120) glyph[similarequal] 5 × 10 47 erg.</text>
<text><location><page_7><loc_8><loc_9><loc_48><loc_29></location>In the cocoon toy model the jets interact with a fraction of the shell. After the 'mini-explosion' the assumed spherical interaction zone (cocoon) expands from its initial cocoon-radius a c , 0 = sin α j βR BB to larger radii. The mass in the interaction zone is then M 2js > (1 -cos α j ) M s . Namely, the shell mass is M s < M 2js / (1 -cos α j ). From Table 2 we find the shell masses of the different cases to be M s (Case2) < 0 . 3 M glyph[circledot] to M s (Case6) < 2 . 2 M glyph[circledot] . In the spherical shell model the shell mass is 0 . 7 M glyph[circledot] (table 1). We crudely take for this ILOT M s (SNhunt120) glyph[similarequal] 0 . 5 -1 M glyph[circledot] , but we note that the model can accommodate somewhat lower shell masses. As we discussed in section 2.2 the progenitor bi-</text>
<text><location><page_7><loc_52><loc_89><loc_92><loc_91></location>system of this ILOT might have a combined mass of M 1 + M 2 ≈ 10 M glyph[circledot] .</text>
<section_header_level_1><location><page_7><loc_57><loc_86><loc_88><loc_88></location>3.3. The cocoon toy model fit of AT 2014ej</section_header_level_1>
<text><location><page_7><loc_52><loc_48><loc_92><loc_86></location>Because at discovery AT 2014ej was already in its decline from the first peak in its light curve, we try to fit the maximum luminosity and the radiated energy of the second peak only. In Fig. 2 we plot by the thick-red line the black-body light curve of AT 2014ej as Stritzinger et al. (2020a) estimate (their figure 4). The maximum luminosity of the second peak is L AT = 2 . 6 × 10 41 erg s -1 . In our cocoon toy model this value implies L j = L 2j / 2 = L AT / 2 = 1 . 3 × 10 41 erg s -1 . We examine only the time near maximum luminosity before the break around t glyph[similarequal] 70 days. We therefore extend the black-body light curve near maximum (solid-blue line in Fig. 2) by taking a linear fit before t = 42 days and beyond t = 67 days after discovery, in both sides down to L AT = 1 . 2 × 10 41 , which is the minimum in the light curve between the two peaks. We find that the total energy that this ILOT radiated in its second peak according to our fit (solid-blue line in Fig. 2) is E rad , AT = 1 . 1 × 10 48 erg. We note that in section 2.3 where we apply the spherical shell model we include the 'hump' at t glyph[similarequal] 90 days, and therefore the radiated energy is somewhat larger. The hump can result from a weak third jet-launching episode or from mass collision in the equatorial plane.</text>
<text><location><page_7><loc_52><loc_34><loc_92><loc_48></location>We recall that our cocoon toy model does not fit a light curve, but rather fit only the maximum luminosity and total radiated energy (or timescale). We rather assume a symmetric light curve (green line in Fig. 2 for Case 1). We proceed as in section 3.2 and solve iterativelly equations (6) for several combinations of the input parameters α j and κ c . We can scale equations (6) with typical values for AT 2014ej (Case 1). The scaled equations read</text>
<formula><location><page_7><loc_58><loc_26><loc_92><loc_33></location>t j = 31 ( E j 1 . 14 × 10 48 erg ) -1 / 4 × ( M js 0 . 46 M glyph[circledot] ) 3 / 4 ( κ c 0 . 1 cm 2 g -1 ) 1 / 2 d , (9)</formula>
<text><location><page_7><loc_52><loc_24><loc_55><loc_25></location>and</text>
<formula><location><page_7><loc_55><loc_13><loc_92><loc_23></location>L j = 6 . 9 × 10 40 ( sin α j 0 . 5 )( β 0 . 7 ) × ( M js 0 . 46 M glyph[circledot] ) -3 / 2 ( E j 1 . 14 × 10 48 erg ) 3 / 2 × ( κ c 0 . 1 cm 2 g -1 ) -1 ( R BB 2 . 5 × 10 14 cm ) erg s -1 . (10)</formula>
<text><location><page_7><loc_52><loc_9><loc_92><loc_12></location>We examine four cases with different values of α j and κ c that we summarise in Table 3.</text>
<figure>
<location><page_8><loc_9><loc_67><loc_48><loc_91></location>
<caption>Figure 2. Similar to Fig. 1 but for AT 2014ej. We show the light curve of AT 2014ej (thick-red line; from Stritzinger et al. 2020a), our fit to the light curve of the second peak of AT 2014ej (blue line), and the assumed light curve of the cocoon toy model (green line). We fit the radiated energy of the second peak E rad , AT = 1 . 1 × 10 48 erg and the maximum luminosity L AT = 2 . 6 × 10 41 erg s -1 . The relevant scaledequations are (9) and (10).</caption>
</figure>
<table>
<location><page_8><loc_10><loc_40><loc_45><loc_51></location>
<caption>Table 3. Similar to Table 2 but for the second peak of the ILOT AT 2014ej (Fig. 2), and with the scaling of equations (9) and (10).</caption>
</table>
<text><location><page_8><loc_8><loc_17><loc_48><loc_32></location>We find that we can better fit the second peak in the light curve of AT 2014ej with moderately wide jets α j glyph[similarequal] 20 -30 · . Fitting with wide jets do not give acceptable results. For the parameters we list in Table 3 the jets' energies range is E 2j glyph[similarequal] 1 . 14 × 10 48 -3 × 10 48 erg. In the spherical shell model for the second peak we found this energy to be 1 . 6 × 10 48 erg (Table 1). We take the jets' energy for this ILOT to be E 2j (AT 2014ej) ≈ 1 . 5 × 10 48 erg. For jets' velocity of v j = 1000 km s -1 the mass in the jets is then M 2j glyph[similarequal] 0 . 15 M glyph[circledot] .</text>
<text><location><page_8><loc_8><loc_9><loc_48><loc_17></location>We proceed as in section 3.2 to put an upper limit on the shell mass M s < M 2js / (1 -cos α j ). We calculate from Table 3 M s < 2 M glyph[circledot] , 1 M glyph[circledot] , 3 . 7 M glyph[circledot] , 1 . 2 M glyph[circledot] for Cases 1, 2, 3, 4 respectively. In the spherical shell model we crudely estimate (Table 1) the shell mass to be M s ≈</text>
<text><location><page_8><loc_52><loc_82><loc_92><loc_91></location>1 . 5 M glyph[circledot] . We take the slow shell mass for this ILOT to crudely be M s (AT 2014ej) ≈ 1 -2 M glyph[circledot] . If this shell mass holds, then the progenitor binary system of this ILOT cannot be a low mass system, and requires the combined mass to be M 1 + M 2 > 5 M glyph[circledot] , and more likely M 1 + M 2 glyph[greaterorsimilar] 10 M glyph[circledot] .</text>
<section_header_level_1><location><page_8><loc_67><loc_79><loc_77><loc_80></location>4. SUMMARY</section_header_level_1>
<text><location><page_8><loc_52><loc_53><loc_92><loc_78></location>We apply the jet-power ILOT scenario to two recently studied ILOTs, SNhunt120 (Stritzinger et al. 2020b) and AT 2014ej (Stritzinger et al. 2020a). In section 2 we apply the spherical shell model (Soker 2020a), and in section 3 we apply the cocoon toy model that we have used to explain some peaks in the light curve of core collapse supernovae (Kaplan & Soker 2020). In both these models of the jet-power ILOT scenario fast jets catch up with a slower and older shell and collide with it. The collision converts kinetic energy to thermal energy. The post-shock shell and jets gases form a hot bubble, the cocoon. The cocoon cools by photon diffusion that turns to radiation, and by adiabatic expansion. The competition between these processes determine the efficiency of converting kinetic energy, mainly of the jets, to radiation.</text>
<text><location><page_8><loc_52><loc_34><loc_92><loc_53></location>These two models are very crude because we neither conduct hydrodynamic simulations of the interaction nor radiative transfer calculations. As well, we take some parameters to have constant values, in particular the opacity. Even if one does conduct these numerical calculations, the parameter space of the model is very large. Namely, we have no knowledge of the properties of the shell and of the jets, in particular the distribution of the momentum flux of the shell and of the jets with direction and time. Nonetheless, we did reach our main goal, which is to show that the jet-powered ILOT scenario can account for these two ILOTs.</text>
<text><location><page_8><loc_52><loc_14><loc_92><loc_34></location>We found the following properties of the jet-powered ILOT scenario for these ILOTs. For SNhunt120 (Table 2) we found that we need to use moderately-wide, α j glyph[similarequal] 30 · , to wide, α j glyph[similarequal] 60 · , jets. For wider jets the assumptions of the model break (like that the cocoon has time to expand), and for narrower jets the shell becomes too massive. The typical jets' energy that might explain the peak of SNhunt120 is E 2j (SNhunt120) glyph[similarequal] 5 × 10 47 erg (Tables 1 and 2). For jets' velocity of v j = 1000 km s -1 the mass in the jets is then M 2j glyph[similarequal] 0 . 05 M glyph[circledot] . The mass of the shell is less certain, and it is sensitive to the parameters of the models. We crudely estimated M s (SNhunt120) glyph[similarequal] 0 . 5 -1 M glyph[circledot] .</text>
<text><location><page_8><loc_52><loc_9><loc_92><loc_13></location>For the second peak of AT 2014ej we had to use moderately wide jets (Table 3). The jets' energy is E 2j (AT 2014ej) ≈ 1 . 5 × 10 48 erg (Tables 1 and 3). For</text>
<text><location><page_9><loc_8><loc_87><loc_48><loc_92></location>jets' velocity of v j = 1000 km s -1 the mass in the jets is then M 2j glyph[similarequal] 0 . 15 M glyph[circledot] . We crudely estimated M s (AT 2014ej) ≈ 1 -2 M glyph[circledot] .</text>
<text><location><page_9><loc_8><loc_75><loc_48><loc_87></location>To launch jets with a mass of glyph[similarequal] 0 . 1 M glyph[circledot] the star that launches the jets should accrete M acc glyph[similarequal] 10 M 2j glyph[similarequal] 1 M glyph[circledot] . An example for such a case is a very young massive star of ≈ 10 M glyph[circledot] that tidally destroys a pre-main sequence star of glyph[similarequal] M glyph[circledot] and accretes most of its mass. This high value of accreted mass and the massive shell M s ≈ 1 M glyph[circledot] , suggest that the binary system progenitors of these two ILOTs are massive, namely M 1 + M 2 glyph[greaterorsimilar] 10 M glyph[circledot] .</text>
<text><location><page_9><loc_52><loc_84><loc_92><loc_91></location>Future studies should include more accurate numerical simulations of the jet-shell interaction and of radiative transfer. A parallel line of studies should examine which type of binary systems can lead to such high mass transfer and mass loss rates.</text>
<section_header_level_1><location><page_9><loc_62><loc_81><loc_82><loc_82></location>ACKNOWLEDGEMENTS</section_header_level_1>
<text><location><page_9><loc_52><loc_73><loc_92><loc_80></location>We thank Ari Laor for useful discussions and Amit Kashi and an anonymous referee for helpful comments. This research was supported by a grant from the Israel Science Foundation (420/16 and 769/20) and a grant from the Asher Space Research Fund at the Technion.</text>
<section_header_level_1><location><page_9><loc_44><loc_69><loc_56><loc_70></location>REFERENCES</section_header_level_1>
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</document> | [{"title": "ABSTRACT", "content": "We apply the jet-powered ILOT scenario to two recently studied intermediate luminosity optical transients (ILOTs), and find the relevant shell mass and jets' energy that might account for the outbursts of these ILOTs. In the jet-powered ILOT scenario accretion disk around one of the stars of a binary system launches jets. The interaction of the jets with a previously ejected slow shell converts kinetic energy to thermal energy, part of which is radiated away. We apply two models of the jetpowered ILOT scenario. In the spherical shell model the jets accelerate a spherical shell, while in the cocoon toy model the jets penetrate into the shell and inflate hot bubbles, the cocoons. We find consistent results. For the ILOT (ILRT: intermediate luminosity red transient) SNhunt120 we find the shell mass and jets' energy to be M s glyph[similarequal] 0 . 5 -1 M glyph[circledot] and E 2j glyph[similarequal] 5 \u00d7 10 47 erg, respectively. The jets' half opening angle is \u03b1 j glyph[similarequal] 30 \u00b7 -60 \u00b7 . For the second peak of the ILOT (luminous red nova) AT 2014ej we find these quantities to be M s glyph[similarequal] 1 -2 M glyph[circledot] and E 2j glyph[similarequal] 1 . 5 \u00d7 10 48 erg, with \u03b1 j glyph[similarequal] 20 \u00b7 -30 \u00b7 . The models cannot tell whether these ILOTs were powered by a stellar merger that leaves one star, or by mass transfer where both stars survived. In both cases the masses of the shells and energies of the jets suggest that the binary progenitor system was massive, with a combined mass of M 1 + M 2 glyph[greaterorsimilar] 10 M glyph[circledot] . Keywords: binaries: close - stars: jets - stars: variables: general", "pages": [1]}, {"title": "1. INTRODUCTION", "content": "The transient events with peak luminosities above those of classical novae and below those of typical supernovae might differ from each other by one or more properties, like the number of peaks in the light curve, total power, progenitor masses, and powering mechanism (e.g. Mould et al. 1990; Bond et al. 2003; Rau et al. 2007; Ofek et al. 2008; Mason et al. 2010; Kasliwal 2011; Tylenda et al. 2013; Kasliwal et al. 2012; Blagorodnova et al. 2017; Kaminski et al. 2018; Pastorello et al. 2018; Boian & Groh 2019; Cai et al. 2019; Jencson et al. 2019; Kashi et al. 2019; Pastorello et al. 2019; Howitt et al. 2020; Jones 2020; Klencki et al. 2020). They form a heterogeneous group of 'gap transients'. We study those transients that are powered by an accretion process that releases gravitational energy. The accretion process might be a mass transfer from one star to another, or an extreme case of stellar merger, where either one star destroys another, or one star (or a planet; Retter & Marom 2003; Bear et al. 2011; Kashi & Soker 2017; Kashi et al. 2019) enters the envelope of a larger star to start a common envelope evolution (e.g., Tylenda et al. 2011; Ivanova et al. 2013; Nandez et al. 2014; Kami'nski et al. 2015; MacLeod et al. 2017; Segev et al. 2019; Schr\u00f8der et al. 2020; MacLeod & Loeb 2020; Soker 2020b). We refer to all these systems as intermediate luminosity optical transients (ILOTs). In cases where both stars survive and stay detached the binary system can experience more than one outburst, and can have several separated peaks in its light curve. This is the case for example in the grazing envelope evolution (Soker 2016). The same holds when the binary system forms a temporary common envelope. Namely, the more compact companion enters the envelope and then gets out. An example of the later process is the repeating common envelope jets supernova (CEJSN) impostor scenario (Gilkis et al. 2019). In a CEJSN impostor event a neutron star (or a black hole) gets into the envelope of a giant massive star, accretes mass and launches jets that power an ILOT event (that might be classified as a supernova impostor), and then gets out of the envelope (Soker & Gilkis 2018; Gilkis et al. 2019; Yalinewich & Matzner 2019). Mass outflow accompanies the bright outbursts of ILOTs. Many studies attribute the powering of ILOTs, both the kinetic energy of the outflow and the radiation, to stellar binary interaction processes (e.g., Soker & Tylenda 2003; Tylenda & Soker 2006; Kashi et al. 2010; Mcley & Soker 2014; Pejcha et al. 2016a,b; Soker 2016; MacLeod et al. 2018; Michaelis et al. 2018; Pastorello et al. 2019). As a fast outflow hits a previously ejected slower outflow, the collision channels kinetic energy to radiation. There are two types of binary scenarios in that respect, those that take the main collision to take place in and near the equatorial plane (e.g., Pejcha et al. 2016a,b; Metzger & Pejcha 2017; Hubov'a, & Pejcha 2019), and those that attribute the main collision to fast polar outflow, i.e., jets. In most of the cases with high mass accretion rates that power ILOTs, the high-accretion-powered ILOT (HAPI) model (Kashi & Soker 2016; Soker & Kashi 2016), the accretion of mass is likely to be through an accretion disk. This accretion disk is very likely to launch two opposite jets. If the jets collide with a previously ejected slow shell an efficient conversion of kinetic energy to radiation might take place. This is the jet-powered ILOT scenario . In a recent study Soker (2020a) argues that the jetsshell interaction of the jet-powered ILOT scenario is more efficient in converting kinetic energy to radiation than collision of equatorial ejecta. He further applies a simple jet-shell interaction model to three ILOTs, the Great Eruption of Eta Carinae (Davidson, & Humphreys 1997), which is a luminous blue variable (LBV), to V838 Mon (Munari et al. 2002) that is a stellar merger (also termed luminous red nova; LRN), and to the ILOT V4332 Sgr that has a bipolar structure (Kaminski et al. 2018). We apply this simple spherical shell model to two other ILOTs (Sections 2.2 and 2.3). As said, in this study we use the term ILOT (Berger et al. 2009; Kashi & Soker 2016; Muthukrishna et al. 2019). There are different classifications of the heterogeneous class of transients, like the one by Kashi & Soker (2016) 1 , the one by Pastorello et al. (2019) and Pastorello & Fraser (2019), and also by Jencson et al. (2019). Some refer to transients from stellar merger by LRNe and to outbursts that involve a massive giant star by intermediate luminosity red transients (ILRTs). We simply refer to all transients that are powered by gravitational energy of mass transfer (or merger), the HAPI model, as ILOTs. This saves us the need to classify a specific event by its unknown progenitors. We are mainly interested in the roles of jets, that might play a role in all types of ILOTs (although not in all ILOTs). Two recent studies of two ILOTs support two crucial ingredients of the jet-powered ILOT scenario. Blagorodnova et al. (2020) study the ILOT M31-LRN-2015 that is possibly a merger remnant (some earlier studies related to this ILOT include Williams et al. 2015; Lipunov et al. 2017; MacLeod et al. 2017; Metzger & Pejcha 2017). Blagorodnova et al. (2020) estimate the primary mass to be M 1 glyph[similarequal] 5 M glyph[circledot] and deduce that during the two years pre-outburst activity the system lost a mass of about > 0 . 14 M glyph[circledot] . Such a pre-outburst formation of a shell (circumbinary matter) is an important ingredient in the jet-powered ILOT scenario. In other recent papers Kaminski et al. (2020, 2021) study in details the ILOT (stellar-merger candidate) Nova 1670 (CK Vulpeculae). This 350-years old nebula has a bipolar structure (Shara et al. 1985) with an 'S' shape along the long axis (Kaminski et al. 2020, 2021). This is an extremely strong indication of shaping by precessing jets. We take it to imply that the jet-powered ILOT scenario accounts for Nova 1670. The intervals from the first to second peak and from the second to third peak in the triple-peaks light curve are about equal at about 1 year (Shara et al. 1985). We take it to imply a multiple jets-launching episodes, or more likely in this case, a variability in jets' launching power as the jets precess. These two recent studies, and in particular the clear demonstration of an 'S' shape morphology of the ILOT Nova 1670 (Kaminski et al. 2020, 2021) motivate us to apply the jet-powered ILOT scenario to two recently studied ILOTs. We emphasise that our main aim is to find plausible parameters for these two recently studied ILOTs in the frame of the jet-driven model, as the formation of jets in binary merger can be very common (e.g., L'opez-C'amara et al. 2020 and references therein). In section 2 we describe the basic features of the jetpowered ILOT scenario and apply it in a simple way to the ILOTs SNhunt120 and AT 2014ej. In section 3 we build a more sophisticated toy model to describe the jet-powered ILOT scenario and apply it to these two ILOTs. We summarise in section 4.", "pages": [1, 2]}, {"title": "2.1. Features of the spherical shell model", "content": "The basic flow structure of the jet-powered ILOT scenario is as follows (Soker 2020a). A binary interaction leads to the ejection of a shell, spherical or not, at velocities of tens to hundreds of km s -1 . The shell ejection period can last from few weeks to several years. In a delay of about days to several months (or even a few years) the binary system launches two opposite jets. The jets collide with the shell, an interaction that converts kinetic energy, mainly of the jets, to radiation. There are two types of evolutionary channels to launch jets. (1) The more compact secondary star accretes mass from the primary star and launches the jets, as in the jet-powered ILOT scenario of the Great Eruption of Eta Carinae (e.g., Soker 2007; Kashi & Soker 2010a). The binary stellar system might stay detached, might experience the grazing envelope evolution, and/or enters a common envelope evolution. In this case the binary systems might experience several jets-launching episodes. (2) The primary star gravitationally destroys the secondary star to form an accretion disk around the primary star, and this accretion disk launches the jets. In this case there is one jets-launching episode, although the jets' intensity can very with time. Soker (2020a) obtains the following approximate relations for jets that interact with a slower spherically symmetric shell and power an ILOT. We refer to this model as the spherical shell model. Soker (2020a) considers jets-shell interaction that (1) transfers a large fraction of the kinetic energy of the outflow to radiation, and (2) radiates much more energy than what recombination of the outflowing gas can supply. Soker (2020a) considers two opposite fast jets that hit a uniform spherical shell and accelerate the entire shell. In section 3 we build a toy model where the jets penetrate into the shell and interact with shell's material only in the polar directions. In the simple flow structure that Soker (2020a) considers the relevant properties of the jets are their half opening angle \u03b1 j glyph[greaterorsimilar] 10 \u00b7 , velocity v j \u2248 10 3 km s -1 , and their total mass M 2j \u2248 0 . 01 -1 M glyph[circledot] . With a conversion efficiency of jets' kinetic energy to radiation f rad , the total energy in radiation is The relevant properties of the spherical shell are its velocity v s glyph[lessmuch] v j , mass M s , radius r s , and width \u2206 r s . The jet-shell interaction converts kinetic energy, mainly of the jets, to thermal energy. The hot bubbles that the jets inflate lose their energy adiabatically by accelerating the shell and non-adiabatically by radiation. The adiabatic cooling proceeds on a typical time scale that is the expansion time t exp , while energy losses to radiation occurs during a typical photon-diffusion time out t diff . Namely, the relative rates, \u02d9 E/E , of adiabatic and radiative energy losses are t -1 exp and t -1 diff , respectively. This implies that the fraction of energy that ends in radiation is For the simple spherically symmetric geometry he assumes, Soker (2020a) estimates the two time scales to be and where \u03c4 = \u03c1 s \u03ba \u2206 r s is the optical depth of the shell, \u03ba is the opacity, and c is the light speed. The relevant ratio to substitute in equation (2) is We emphasise that we do not assume any value for the jets' energy E 2j . We rather take the jets' velocity from observations, and use the time scale of the ILOT together with an assumed opacity to find the mass in the shell (equation 4). We then calculate the efficiency f rad together with the mass in the jets (or their energy) to fit the total radiated energy (equations 2 and 5). Soker (2020a) applies this spherical shell model of the jet-powered ILOT radiation to the ILOT (LRN) V838 Mon, to the Great Eruption of Eta Carinae which is an LBV, and to the ILOT V4332 Sgr. He could find plausible set of shell and jets parameters that might explain these ILOTs. Here we apply the spherical shell model to the ILOT (ILRT) SNhunt120 and to the ILOT (LRN) AT 2014ej. We summarise the plausible physical parameters of the ILOT events in Table 1, and explain their derivation in sections 2.2 and 2.3. We emphasise that due to the very simple model we apply here, e.g., we use a spherical shell and we keep the opacity and shell thickness constant, the properties of the jets and shells we derive are very crude, and might even not be unique. Nonetheless, they demonstrate the potential of the jet-powered ILOT scenario to account for many ILOTs. The opacity of a fully ionised gas that is appropriate for ILOTs is \u03ba glyph[similarequal] 0 . 3 cm 2 g -1 (e.g., Ivanova et al. 2013; Soker & Kashi 2016). We expect that in the outer parts of the shell hydrogen is partially neutral, and that opacity is therefore lower. Therefore, we scale with \u03ba = 0 . 1 cm 2 g -1 . 2.2. The ILOT SNhunt120 Stritzinger et al. (2020b) study the ILOT (ILRT) SNhunt120 and find the following relevant properties. The velocities of different emission lines are in the range of glyph[similarequal] 300 -1800 km s -1 , with a typical velocity of \u2248 10 3 km s -1 . The typical photospheric radius is R BB glyph[similarequal] 2 \u00d7 10 14 cm. The time to double the luminosity at rise is about 10 days, and the decline time to half the maximum luminosity is about 20 days. The total energy in radiation is E rad glyph[similarequal] 4 \u00d7 10 47 erg. Stritzinger et al. (2020b) further find that existing electron capture supernova models over-predict the energy in radiation. We do not consider this event to be a supernova, but rather an ILOT. Following these parameters we scale the parameters for SNhunt120 with v j glyph[similarequal] 1000 km s -1 and r s glyph[similarequal] 2 \u00d7 10 14 cm. To get a photon diffusion time of about the decline time of 20 days, we find from equation (4) for \u03ba = 0 . 1 cm 2 g -1 and \u2206 r s = 0 . 3 r s that M s \u2248 0 . 7 M glyph[circledot] . For an opacity of \u03ba = 0 . 3 cm 2 g -1 and somewhat a thicker shell with \u2206 r s = 0 . 5 R s , the required shell mass is only M s \u2248 0 . 15 M glyph[circledot] Equation (5) gives then t diff /t exp \u2248 0 . 1, and from equation (2) f rad glyph[similarequal] 0 . 9. To account for the emitted energy, we find from equation (1) that the mass in the two jets is M 2j \u2248 0 . 045 M glyph[circledot] ( v j / 1000 km s -1 ) -2 . In this analysis there is no reference to the shell velocity, beside that it should be much lower than the jets' velocity. This implies here 100 km s -1 glyph[lessorsimilar] v s glyph[lessorsimilar] 500 km s -1 . To reach a distance of r s = 2 \u00d7 10 14 the binary system ejected the shell about \u2206 t s glyph[similarequal] 0 . 6( v s / 100 km s -1 ) -1 yr before detection. The kinetic energy in the shell for these parameters of M s glyph[similarequal] 0 . 7 M glyph[circledot] and v s glyph[similarequal] 100 km s -1 is about 15% of the jets' energy. In any case, most of the kientic enrgy of the shell does not convert to radiation. In case that the secondary star launches the jets with a mass of M 2j glyph[similarequal] 0 . 045 M glyph[circledot] , it should accrete a mass of M acc , 2 glyph[similarequal] 0 . 45 M glyph[circledot] from a more evolved primary star, possibly a giant. This implies that the secondary star should be a massive star itself. We are therefore considering a massive binary system. Alternatively, it is possible that the primary star destroyed the secondary star of mass M 2 glyph[similarequal] 0 . 3 -1 M glyph[circledot] to form an accretion disk that launched the jets. The primary is then a massive main sequence star, and the secondary is not yet settled on the main sequence, such that its average density is lower than that of the primary star (as in the merger model of V838 Mon; Tylenda & Soker 2006). In any case, the primary star mass can be in the range of M 1 \u2248 10 M glyph[circledot] , similar to the range that Stritzinger et al. (2020b) consider. Since there is only one jets-launching episode, the jet-powered ILOT scenario does not directly refer in the case of SNhunt120 to the question of which of these two evolutionary routes apply here.", "pages": [2, 3, 4]}, {"title": "2.3. The ILOT AT 2014ej", "content": "Stritzinger et al. (2020a) study the ILOT (LRN) AT 2014ej. They find that the light curve of models of equatorial collision (Metzger & Pejcha 2017; section 1), under-predict the luminosity. We therefore consider powering by jets, i.e., polar collision. Stritzinger et al. (2020a) find that AT 2014ej has slow component(s) moving at \u2248 100 km s -1 and fast component(s) moving at \u2248 1000 km s -1 . The total radiated energy is E rad \u2248 2 \u00d7 10 48 erg, with two large peaks in the light curve. From discovery to first minimum 20 days later, the luminosity decreased from L 0 = 3 . 2 \u00d7 10 41 erg s -1 to L min , 1 = 1 . 2 \u00d7 10 41 erg s -1 . Over the next 35 days the luminosity increased to L AT \u2261 L peak , 2 glyph[similarequal] 2 . 6 \u00d7 10 41 erg s -1 , after which the luminosity decreased over a time scale of several weeks. The photosphere was hotter in the first peak than in the second one. The photosphere (black body surface) moderately followed the behavior of the luminosity, and first declined somewhat and then increased somewhat. Its approximate average value is R BB glyph[similarequal] 2 . 5 \u00d7 10 14 cm. In the jet-powered ILOT scenario such multiple-peaks can be accounted for by multiple jet-launching episodes. From Stritzinger et al. (2020a) we find that the radiated energy from detection to first minimum (0 to 20 days) is glyph[similarequal] 4 \u00d7 10 47 erg. If we take a similar energy at rise, the energy in the first peak is E rad , 1p \u2248 10 48 erg. The energy in the second peak, from 20 to about 95 days, is E rad , 2p \u2248 1 . 4 \u00d7 10 48 erg. The outburst of V838 Mon has a similar qualitative behavior with three peaks and three declines in the photospheric radius (Tylenda 2005). In AT 2014ej the two peaks have about the same energy (under our assumption), but the second peak is slower by a factor of about two. From equation (4) the mass in the shell should be larger in the second peak by a factor of two, glyph[similarequal] 2 M glyph[circledot] instead of glyph[similarequal] 1 M glyph[circledot] . We do not expect the system to lose much more slow mass in that short time. The difference in the time scales of the two peaks might come from different values of \u03ba and/or \u2206 r s between the two peaks, rather than from different shell masses. This can also be accompanied by precessing jets, i.e., the jets' axes in the two jet-launching episodes have different directions. In the present study we use a simple model and do not calculate the opacity, and so we simply take for both peaks M s glyph[similarequal] 1 . 5 M glyph[circledot] . From the equations of section 2.1 we derive the crude plausible values of the shell mass, jets' energy, and emission efficiency for the two peaks, as we list in Table 1. According to the jet-powered ILOT scenario the two distinguished peaks result from two jets-launching episode. This suggests that the secondary star, possibly in an eccentric orbit, accreted mass and launches the jets. Most likely, the secondary star survived the interaction.", "pages": [4, 5]}, {"title": "3.1. The cocoon toy-model", "content": "In the simple spherical-shell model that we apply in section 2 the jets interact with the entire shell (Soker 2020a). We now turn to a more realistic toy model where the jets interact only with the shell segments along the polar directions. In this 'cocoon toy model' the jet-shell interaction inflates a 'cocoon', i.e., a relatively hot bubble composed of the post-shock shell material and postshock jet's material. We further simplify the interaction by assuming that the jets' activity time period is short, such that we can treat the jet-shell interaction that creates the cocoon as a 'mini explosion'. We base the cocoon toy model on our usage of this model to account for peaks in the light curves of core collapse supernovae (Kaplan & Soker 2020; for the geometry of a jet-ejecta interaction in core collapse supernova see the three-dimensional simulations of Akashi & Soker 2020). In the cocoon toy model we only calculate the timescale of the emission peak (eruption) and its maximum luminosity (or total energy). We do not calculate the shape of the light curve, but rather assume a simple shape for the light curve. We then calculate the total radiated energy by integrating the luminosity over time. We assume that each mini-explosion that results from jet-shell interaction is spherically symmetric around the jet-shell interaction point (Akashi & Soker 2020), and that cooling is due to photon diffusion and adiabatic expansion. These assumptions allow us to determine the luminosity and the time scale of each mini-explosion. As we deal with ILOTs where the total radiated energy is larger than the recombination energy of the outflowing gas, we neglect the recombination energy. Like Kaplan & Soker (2020) we use equations (4) from Kasen & Woosley (2009) to calculate the time of maximum luminosity t j and the maximum luminosity L j for one jet. These expressions read where E j , M js , \u03ba c , \u03b1 j , \u03b2 and R BB are the energy that one jet deposits into the shell, the mass in the interaction region of one jet with the shell, the opacity in the cocoon, the half opening angle of the jet, the distance of the jetejecta interaction relative to the shell's outer edge (the photosphere radius R BB ), and the photospheric radius of the shell, respectively. Namely, in this model the mini explosion takes place at a radius (measured from the center of the binary system) of r me = \u03b2R BB . There is one mini-explosion on each of the two polar regions. The value of r me is constant and does not change with time. What increases with time is the radius of the cocoon itself, a c , that is measured from the place of the mini explosion. We build the light curve of the jet as follows. We assume that the shape of the rise of the peak to maximum luminosity is similar to the rise to maximum of the light curve of a core collapse supernova (based on photometric data of SN 2008ax, taken from The Open Supernova Catalog Guillochon et al. 2017). Since the light curve of the jet does not have a tail powered by radioactive processes and recombination, we take the decline of the mini-explosion from maximum to be symmetric to its rise. Again, we do not try to fit the light curve. We rather only derive the properties of the jets that might lead to an event that has the same timescale, luminosity and radiated energy. We assume a light curve, but our results are not sensitive to the exact shape of the light curve we assume. We turn to estimate the jets' properties that according to the cocoon toy model might fit the eruption times and luminosities of the ILOTs SNhunt120 (section 3.2) and AT 2014ej (section 3.3).", "pages": [5, 6]}, {"title": "3.2. The cocoon toy model fit of SNhunt120", "content": "First we extend the observed light curve of SNhunt120 (Stritzinger et al. 2020b; thick-red line in Fig. 1) by taking a linear fit before discovery and beyond t = 30 days after discovery, in both sides down to L = 0. This is the solid-blue line in Fig. 1. The observed light curve of this ILOT has a break at about 40 days post-discovery, where the decline becomes shallower. This might result from a second and weaker jet-launching episode or from matter collision in the equatorial plane. We are interested here only in the light curve around the maximum, so we continue the steep decline beyond 30 days post-discovery down to L = 0. We then find the radiated energy of SNhunt120 of our fit to the peak to be E rad , hunt = 3 . 8 \u00d7 10 47 erg. As we explained in section 3.1, we then build a toy-model symmetric light curve that has the same maximum luminosity as SNhunt120, L hunt = 1 . 4 \u00d7 10 41 erg s -1 , and the same total radiated energy. This is the green line in Fig. 1 (for Case 1 that we describe next). We calculate the energy of one jet E j and the mass in the region of interaction of one jet with the shell (the cocoon), M js , as follows. We build a symmetric toy model light curve (one example is the green line in Fig. 1) that is characterised by its maximum luminosity L j and by its timescale from start to maximum t j by equations (6). We then calculate the total radiated energy according to this light curve (area under the green light curve). We iterate the values of E j and M js until we obtain the luminosity due to the two jets together of L 2j = L hunt = 1 . 4 \u00d7 10 41 erg s -1 , and the total radiated energy from the two jets is E rad , 2j = E rad , hunt = 3 . 8 \u00d7 10 47 erg. We note that the cocoon toy model is not sensitive to the expansion velocities of the shell and of the jets, as long as the v j glyph[greatermuch] v s . We do not vary the values of the photosphere radius R BB = 2 \u00d7 10 14 cm that we take from Stritzinger et al. (2020b), and of \u03b2 = 0 . 7 in equations (6). We do vary the values of the jet's half opening angle \u03b1 j and of the opacity \u03ba c . We continue with the wide jets that we discussed in section 2 (Soker 2020a) and scale with \u03b1 j = 60 \u00b7 , but we consider narrower jets as well. We scale the opacity with \u03ba c = 0 . 1 cm 2 g -1 but examine also \u03ba c = 0 . 05 cm 2 g -1 and \u03ba c = 0 . 3 cm 2 g -1 to \u221a]\u2309\u02dc(\u22c3\u221a\u228e demonstrate the model sensitivity to opacity. The relevant scaling of equations (6) for SNhunt120, (for one jet) read and As with the spherical shell model, we do not assume the energy of the jets. The input variables to the fitting process are the light curve, the half opening angle of the jets, the opacity, and the values of \u03b2 and sin \u03b1 j . We take the radius of the continuum black body photosphere from observations. We then substitute in equations (9) and (10) the observed ILOT's (or one peak of the ILOT) duration t j and the energy radiated from one jet-shell interaction L j , and solve for the one jet's energy E j and the mass of the shell that one jet interacts with M js . In Table 2 we present six sets of values in the cocoon toy model for SNhunt120. We emphasise that we do not try to fit the shape of the light curves, and only try to explain the amount of radiated energy, the timescale, and the maximum luminosity of the peak. In Fig. 1 we show by the green line Case 1. The energy of the jets and the mass they interact with vary between the cases. The energy range is E 2j glyph[similarequal] 4 \u00d7 10 47 erg -11 \u00d7 10 47 erg. In the spherical-shell model of section 2.2 the jets' energy is 4 . 5 \u00d7 10 47 erg. From the cases of tables 1 and 2 we crudely take the jets' energy for this ILOT to be E 2j (SNhunt120) glyph[similarequal] 5 \u00d7 10 47 erg. In the cocoon toy model the jets interact with a fraction of the shell. After the 'mini-explosion' the assumed spherical interaction zone (cocoon) expands from its initial cocoon-radius a c , 0 = sin \u03b1 j \u03b2R BB to larger radii. The mass in the interaction zone is then M 2js > (1 -cos \u03b1 j ) M s . Namely, the shell mass is M s < M 2js / (1 -cos \u03b1 j ). From Table 2 we find the shell masses of the different cases to be M s (Case2) < 0 . 3 M glyph[circledot] to M s (Case6) < 2 . 2 M glyph[circledot] . In the spherical shell model the shell mass is 0 . 7 M glyph[circledot] (table 1). We crudely take for this ILOT M s (SNhunt120) glyph[similarequal] 0 . 5 -1 M glyph[circledot] , but we note that the model can accommodate somewhat lower shell masses. As we discussed in section 2.2 the progenitor bi- system of this ILOT might have a combined mass of M 1 + M 2 \u2248 10 M glyph[circledot] .", "pages": [6, 7]}, {"title": "3.3. The cocoon toy model fit of AT 2014ej", "content": "Because at discovery AT 2014ej was already in its decline from the first peak in its light curve, we try to fit the maximum luminosity and the radiated energy of the second peak only. In Fig. 2 we plot by the thick-red line the black-body light curve of AT 2014ej as Stritzinger et al. (2020a) estimate (their figure 4). The maximum luminosity of the second peak is L AT = 2 . 6 \u00d7 10 41 erg s -1 . In our cocoon toy model this value implies L j = L 2j / 2 = L AT / 2 = 1 . 3 \u00d7 10 41 erg s -1 . We examine only the time near maximum luminosity before the break around t glyph[similarequal] 70 days. We therefore extend the black-body light curve near maximum (solid-blue line in Fig. 2) by taking a linear fit before t = 42 days and beyond t = 67 days after discovery, in both sides down to L AT = 1 . 2 \u00d7 10 41 , which is the minimum in the light curve between the two peaks. We find that the total energy that this ILOT radiated in its second peak according to our fit (solid-blue line in Fig. 2) is E rad , AT = 1 . 1 \u00d7 10 48 erg. We note that in section 2.3 where we apply the spherical shell model we include the 'hump' at t glyph[similarequal] 90 days, and therefore the radiated energy is somewhat larger. The hump can result from a weak third jet-launching episode or from mass collision in the equatorial plane. We recall that our cocoon toy model does not fit a light curve, but rather fit only the maximum luminosity and total radiated energy (or timescale). We rather assume a symmetric light curve (green line in Fig. 2 for Case 1). We proceed as in section 3.2 and solve iterativelly equations (6) for several combinations of the input parameters \u03b1 j and \u03ba c . We can scale equations (6) with typical values for AT 2014ej (Case 1). The scaled equations read and We examine four cases with different values of \u03b1 j and \u03ba c that we summarise in Table 3. We find that we can better fit the second peak in the light curve of AT 2014ej with moderately wide jets \u03b1 j glyph[similarequal] 20 -30 \u00b7 . Fitting with wide jets do not give acceptable results. For the parameters we list in Table 3 the jets' energies range is E 2j glyph[similarequal] 1 . 14 \u00d7 10 48 -3 \u00d7 10 48 erg. In the spherical shell model for the second peak we found this energy to be 1 . 6 \u00d7 10 48 erg (Table 1). We take the jets' energy for this ILOT to be E 2j (AT 2014ej) \u2248 1 . 5 \u00d7 10 48 erg. For jets' velocity of v j = 1000 km s -1 the mass in the jets is then M 2j glyph[similarequal] 0 . 15 M glyph[circledot] . We proceed as in section 3.2 to put an upper limit on the shell mass M s < M 2js / (1 -cos \u03b1 j ). We calculate from Table 3 M s < 2 M glyph[circledot] , 1 M glyph[circledot] , 3 . 7 M glyph[circledot] , 1 . 2 M glyph[circledot] for Cases 1, 2, 3, 4 respectively. In the spherical shell model we crudely estimate (Table 1) the shell mass to be M s \u2248 1 . 5 M glyph[circledot] . We take the slow shell mass for this ILOT to crudely be M s (AT 2014ej) \u2248 1 -2 M glyph[circledot] . If this shell mass holds, then the progenitor binary system of this ILOT cannot be a low mass system, and requires the combined mass to be M 1 + M 2 > 5 M glyph[circledot] , and more likely M 1 + M 2 glyph[greaterorsimilar] 10 M glyph[circledot] .", "pages": [7, 8]}, {"title": "4. SUMMARY", "content": "We apply the jet-power ILOT scenario to two recently studied ILOTs, SNhunt120 (Stritzinger et al. 2020b) and AT 2014ej (Stritzinger et al. 2020a). In section 2 we apply the spherical shell model (Soker 2020a), and in section 3 we apply the cocoon toy model that we have used to explain some peaks in the light curve of core collapse supernovae (Kaplan & Soker 2020). In both these models of the jet-power ILOT scenario fast jets catch up with a slower and older shell and collide with it. The collision converts kinetic energy to thermal energy. The post-shock shell and jets gases form a hot bubble, the cocoon. The cocoon cools by photon diffusion that turns to radiation, and by adiabatic expansion. The competition between these processes determine the efficiency of converting kinetic energy, mainly of the jets, to radiation. These two models are very crude because we neither conduct hydrodynamic simulations of the interaction nor radiative transfer calculations. As well, we take some parameters to have constant values, in particular the opacity. Even if one does conduct these numerical calculations, the parameter space of the model is very large. Namely, we have no knowledge of the properties of the shell and of the jets, in particular the distribution of the momentum flux of the shell and of the jets with direction and time. Nonetheless, we did reach our main goal, which is to show that the jet-powered ILOT scenario can account for these two ILOTs. We found the following properties of the jet-powered ILOT scenario for these ILOTs. For SNhunt120 (Table 2) we found that we need to use moderately-wide, \u03b1 j glyph[similarequal] 30 \u00b7 , to wide, \u03b1 j glyph[similarequal] 60 \u00b7 , jets. For wider jets the assumptions of the model break (like that the cocoon has time to expand), and for narrower jets the shell becomes too massive. The typical jets' energy that might explain the peak of SNhunt120 is E 2j (SNhunt120) glyph[similarequal] 5 \u00d7 10 47 erg (Tables 1 and 2). For jets' velocity of v j = 1000 km s -1 the mass in the jets is then M 2j glyph[similarequal] 0 . 05 M glyph[circledot] . The mass of the shell is less certain, and it is sensitive to the parameters of the models. We crudely estimated M s (SNhunt120) glyph[similarequal] 0 . 5 -1 M glyph[circledot] . For the second peak of AT 2014ej we had to use moderately wide jets (Table 3). The jets' energy is E 2j (AT 2014ej) \u2248 1 . 5 \u00d7 10 48 erg (Tables 1 and 3). For jets' velocity of v j = 1000 km s -1 the mass in the jets is then M 2j glyph[similarequal] 0 . 15 M glyph[circledot] . We crudely estimated M s (AT 2014ej) \u2248 1 -2 M glyph[circledot] . To launch jets with a mass of glyph[similarequal] 0 . 1 M glyph[circledot] the star that launches the jets should accrete M acc glyph[similarequal] 10 M 2j glyph[similarequal] 1 M glyph[circledot] . An example for such a case is a very young massive star of \u2248 10 M glyph[circledot] that tidally destroys a pre-main sequence star of glyph[similarequal] M glyph[circledot] and accretes most of its mass. This high value of accreted mass and the massive shell M s \u2248 1 M glyph[circledot] , suggest that the binary system progenitors of these two ILOTs are massive, namely M 1 + M 2 glyph[greaterorsimilar] 10 M glyph[circledot] . Future studies should include more accurate numerical simulations of the jet-shell interaction and of radiative transfer. A parallel line of studies should examine which type of binary systems can lead to such high mass transfer and mass loss rates.", "pages": [8, 9]}, {"title": "ACKNOWLEDGEMENTS", "content": "We thank Ari Laor for useful discussions and Amit Kashi and an anonymous referee for helpful comments. This research was supported by a grant from the Israel Science Foundation (420/16 and 769/20) and a grant from the Asher Space Research Fund at the Technion.", "pages": [9]}] |
2015MNRAS.448.2362P | https://arxiv.org/pdf/1402.4362.pdf | <document>
<section_header_level_1><location><page_1><loc_13><loc_83><loc_90><loc_87></location>A call for a paradigm shift from neutrino-driven to jet-driven core-collapse supernova mechanisms</section_header_level_1>
<text><location><page_1><loc_31><loc_80><loc_72><loc_81></location>Oded Papish 1 Jason Nordhaus 2 , 3 and Noam Soker 1</text>
<section_header_level_1><location><page_1><loc_46><loc_75><loc_57><loc_76></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_17><loc_48><loc_86><loc_72></location>Three-dimensional (3D) simulations in recent years have shown severe difficulties producing 10 51 erg explosions of massive stars with neutrino based mechanisms while on the other hand demonstrated the large potential of mechanical effects, such as winds and jets in driving explosions. In this paper we study the typical time-scale and energy for accelerating gas by neutrinos in core-collapse supernovae (CCSNe) and find that under the most extremely favorable (and probably unrealistic) conditions, the energy of the ejected mass can reach at most 5 × 10 50 erg. More typical conditions yield explosion energies an order-of-magnitude below the observed 10 51 erg explosions. On the other hand, non-spherical effects with directional outflows hold promise to reach the desired explosion energy and beyond. Such directional outflows, which in some simulations are produced by numerical effects of 2D grids, can be attained by angular momentum and jet launching. Our results therefore call for a paradigm shift from neutrino-based explosions to jet-driven explosions for CCSNe.</text>
<section_header_level_1><location><page_1><loc_42><loc_42><loc_61><loc_43></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_12><loc_29><loc_91><loc_40></location>Eighty years after Baade & Zwicky (1934) first suggested that supernovae (SNe) are powered by stars collapsing into neutron stars (NS), the processes by which part of this gravitational energy is channelled to explosion remains controversial. Wilson (1985) and Bethe & Wilson (1985) refined the neutrino mechanism (Colgate & White 1966) into the delayed-neutrino mechanism, whereby neutrinos emitted within a period of ∼ 1 s after the bounce of the collapsed core heat material in the gain region ( r ≈ 100 -200 km ). This subsequent neutrino-heating was thought</text>
<text><location><page_1><loc_12><loc_15><loc_91><loc_18></location>3 Center for Computational Relativity and Gravitation, Department of Mathematics, Rochester Institute of Technology, Rochester, NY, USA</text>
<text><location><page_2><loc_12><loc_83><loc_91><loc_87></location>to revive the stalled shock thereby exploding the star and producing a canonical core-collapse supernova (CCSN) with an observed energy of E exp /greaterorsimilar 1 foe , where 1 foe ≡ 10 51 erg .</text>
<text><location><page_2><loc_12><loc_54><loc_97><loc_82></location>In the last three decades, sophisticated multidimensional simulations with increasing capabilities were used to study the delayed-neutrino mechanism (e.g., Bethe & Wilson 1985; Burrows & Lattimer 1985; Burrows et al. 1995; Fryer & Warren 2002; Buras et al. 2003; Ott et al. 2008; Marek & Janka 2009; Nordhaus et al. 2010; Brandt et al. 2011; Hanke et al. 2012; Kuroda et al. 2012; Hanke et al. 2012; Mueller et al. 2012; Bruenn et al. 2013; Muller & Janka2014; Mezzacappa et al. 2014; Bruenn et al. 2014). The outcome of such numerical experiments varied widely with many failing to revive the stalled shock while others produced tepid explosions with energies less than 1 foe . Historically, in spherically symmetric calculations (1D), the vast majority of progenitors can not even explode (Burrows et al. 1995; Rampp & Janka 2000; Mezzacappa et al. 2001; Liebendorfer et al. 2005). The exception being the 8.8M /circledot progenitor of Nomoto & Hashimoto (1988) which resulted in a ∼ 3 × 10 49 erg neutrino-driven-wind explosion due to the rarified stellar envelope (Kitaura et al. 2006). Extension to axisymmetric calculations (2D) yielded similar outcomes over their 1D counterparts despite the inclusion of instabilities such as neutrino-driven convection and the standingaccretion-shock instability (SASI) (Burrows et al. 1995; Janka & Mueller 1996; Buras et al. 2006a,b; Ott et al. 2008; Marek & Janka 2009).</text>
<text><location><page_2><loc_12><loc_36><loc_91><loc_53></location>It should be noted that while many of the current numerical experiments incorporate multidimensional hydrodynamics, performing 3D radiation is currently prohibitive computationally (Zhang et al. 2013). Many groups utilize multi-group-flux-limited diffusion (MGFLD) in the 1D 'ray-by-ray' transport approximation. This is a reasonable approach to core-collapse simulations both because of the limitation of current computational resources and because the results for multiangle transport are similar to those for MGFLD except in the cases of extremely rapid rotation (Ott et al. 2008). Thus, it's unlikely that future simulations that incorporate 3D transport will yield fundamental differences over current state-of-the-art calculations in terms of the viability of neutrino mechanism.</text>
<text><location><page_2><loc_12><loc_17><loc_91><loc_35></location>Recently, a number of groups have published 3D core-collapse simulations with differing computational approaches and various levels of sophistication (Nordhaus et al. 2010; Janka 2013; Couch 2013; Dolence et al. 2013; Takiwaki et al. 2014; Dolence et al. 2014; Hanke et al. 2012, 2013; Couch & O'Connor 2014; Mezzacappa et al. 2014). Some groups find that the extra-degree of freedom available in 3D simulations makes it easier to achieve shock revival over their axisymmetric counterparts (Nordhaus et al. 2010; Dolence et al. 2013). On the other hand, several groups have found the opposite; namely that explosions are harder to achieve in 3D than 2D (Janka 2013; Couch 2013; Takiwaki et al. 2014; Hanke et al. 2012, 2013; Couch & O'Connor 2014). If that's the case, then it may well be that the delayed-neutrino mechanism categorically fails and alternative mechanisms should be investigated.</text>
<text><location><page_3><loc_12><loc_74><loc_91><loc_87></location>In one recent case, axisymmetric calculations of 12-, 15-, 20-, and 25M /circledot progenitors successfully revived the shock with explosion energy estimates of ∼ 0 . 3 -0 . 9 foe (Bruenn et al. 2013; Mezzacappa et al. 2014; Bruenn et al. 2014). Their energy is supplied primarily by an enthalpy flux. This is actually a wind, mainly along the imposed symmetry axis, i.e., a collimated wind. This wind is driven by the inflowing (accreted) gas. Winds were suggested to power CCSN in the past (e.g., Burrows & Goshy 1993; Burrows et al. 1995), but were found to have limited contribution to the explosion for a more massive than 8 . 8 M /circledot starts.</text>
<text><location><page_3><loc_12><loc_51><loc_91><loc_73></location>Many CCSNe, e.g., some recent Type Ic SNe (Roy et al. 2013; Takaki et al. 2013) explode with kinetic energy of /greaterorsimilar 10 foe . Neutrino based mechanisms cannot account for such energies even under favorable conditions. For example, Ugliano et al. (2012) performed a set of simulations where the energy was artificially scaled to that of SN 1987A, and found that even if neutrino explosions do work for some CCSNe, no explosions with kinetic energy of > 2 foe are achieved. This scaling was achieved by artificially setting the inner boundary luminosity to obtain an explosion with an energy equal to that of SN 1987A. The delayed-neutrino mechanism must be shown to produce robust explosions with canonical supernova energies for a range of progenitors if it is to continue to be a contender in core-collapse theory. Despite decades of effort with the most sophisticated physics to date, no current simulation has produced a successful 10 51 erg supernova. It is this fact that leads us to argue that the delayed-neutrino mechanism has a generic character that prevents it from exploding the star with an observed energy of 1 foe .</text>
<text><location><page_3><loc_12><loc_29><loc_91><loc_49></location>The delicate and problematic nature of neutrino-driven mechanisms were already revealed with 1D simulations, such that even the most sophisticated neutrino transport calculations were unable to explode stars for progenitor masses /greaterorsimilar 12 M /circledot (e.g., Liebendorfer et al. 2001). Multidimensional effects were then seen as necessary for triggering an explosion. The most common multi-dimensional processes that have been studied as a rescue for the delayed-neutrino mechanism were neutrino-driven convection (e.g., Burrows et al. 1995) and hydrodynamic instabilities, such as the SASI (Blondin et al. 2003). These axisymmetic (2D) simulations have shown mixed and contradicting results. Most do not get an 'explosion' at all, while others obtain explosions with very little energy, i.e., /lessmuch 1 foe (e.g., Suwa 2014; Suwa et al. 2010). In most of these cases where an 'explosion' is claimed, it is actually only shock revival and not a typical explosion, as the energy is much too low to explain most observed CCSNe.</text>
<text><location><page_3><loc_12><loc_17><loc_91><loc_27></location>In the past few years, the regime of 3D flow structures have been explored in more detail (e. g. Nordhaus et al. 2010). The simulations have not reached any consensus on the outcome. While some show that it is easier to revive the shock in 3D than in 2D (e.g., Nordhaus et al. 2010; Dolence et al. 2013), others showed the opposite (e.g., Hanke et al. 2013; Couch 2013; Couch & O'Connor 2014; Takiwaki et al. 2014). Even in 3D simulations that successfully revive the shock, the energy is significantly lower than 1 foe . Recently, turbulence from convective</text>
<text><location><page_4><loc_12><loc_84><loc_91><loc_87></location>burning in the Si/O shell were shown to aid shock revival (Couch & Ott 2014; Mueller & Janka 2014).</text>
<text><location><page_4><loc_12><loc_69><loc_91><loc_82></location>A recent demonstration of outcome sensitivity on initial setting are the two 3D studies by Nakamura et al. (2014) and Mosta et al. (2014). Nakamura et al. (2014) find an explosion energy of ∼ 1 foe for a case with a rapid core rotation. For a rotation velocity of 0 . 2 times that rapid rotation, the explosion energy was only ∼ 0 . 1 foe . They did not include magnetic fields. Mosta et al. (2014) included very strong magnetic fields in the pre-collapse core as well as a very rapid rotation, about twice as large as the rapid rotation case of Nakamura et al. (2014). Mosta et al. (2014) obtained jets but did not manage to revive the stalled shock and did not obtain any explosion.</text>
<text><location><page_4><loc_12><loc_55><loc_91><loc_68></location>The structure of this paper is as follows. In section 2, we expand upon the argument presented in Papish & Soker (2012a) that the delayed-neutrino mechanism cannot achieve canonical supernova energies. We consider the limitation of the delayed-neutrino mechanism from another perspective in section 3. In section 4 we discuss the role of progenitor perturbations and why contradicting results are common among the groups simulating neutrino-based mechanisms, and in section 5 we discuss the energy available from recombination of free nucleons. A discussion of the collimated-wind obtained by Bruenn et al. (2014) and our summary are in section 6.</text>
<section_header_level_1><location><page_4><loc_34><loc_49><loc_69><loc_51></location>2. TIME-SCALE CONSIDERATIONS</section_header_level_1>
<text><location><page_4><loc_12><loc_40><loc_91><loc_47></location>We start with simple time-scale considerations during the revival of the shock in a spherically symmetric outflow. The 'gain region' of the delayed neutrino mechanism, i.e. where neutrino heating outweighs neutrino cooling, typically occurs in the region r /similarequal 100 -400 km (Janka 2001).</text>
<text><location><page_4><loc_12><loc_17><loc_91><loc_39></location>For an explosion to be initiated the advection timescale τ adv should be larger than the heating timescale τ heat . This advection timescale is the time needed for material to cross the gain region during accretion. Most core-collapse simulations fail when this condition is not fulfilled. When this condition is met the internal energy can increase until there is enough energy to unbind the material and an explosion is initiated. At this point the total energy of the gas in the gain region is very close to zero. From this time the net heating adds up to the positive explosion energy. After the gas reaches large radii, /greaterorsimilar 1000 km , heating becomes inefficient. It is true that some gas expands at a lower velocity and it is closer to the center. However, density decreases and so does the neutrino optical depth that decreases below its initial value, such that neutrino heating becomes even less efficient. Material near the neutrinosphere has, by definition, a large optical depth. It can in principle absorb energy and expand. But this process is a neutrino-driven wind, which is not part of the delayed-neutrino mechanism, and was found to have limited contribution to the explosion</text>
<text><location><page_5><loc_12><loc_82><loc_91><loc_87></location>(e.g., Burrows & Goshy 1993; Burrows et al. 1995). The time from the start of acceleration to the end of efficient heating is marked t esc . From simulations t est /similarequal 50 ms (Bruenn et al. 2013, 2014; Marek & Janka 2009). In section 3 we find a similar time from a simple analytical estimate.</text>
<text><location><page_5><loc_12><loc_54><loc_91><loc_80></location>In figure 2 of Bruenn et al. (2013) the shock is starting to expand and an explosion is initiated at time t /similarequal 200 ms . At this time the total positive energy is close to zero (figure 4 in Bruenn et al. 2013). At that time the shock is at a distance of r s /similarequal 400 km . This shows that during the time the shock moves from 200 km to 400 km the total energy increased from a negative value to about zero. We take the time of zero energy to be the starting point of postive energy accumulation, and use it to estimate the explosion energy. In the simulations of Bruenn et al. (2013) at time t = 300 ms the shock is already at a distance of r s /similarequal 1000 -1500 km . Some material is closer to the center, but its density is lower than at earlier times, opacity is lower, and heating is inefficient. We note again the long duration of energy increase in the work of Bruenn et al. (2013, 2014) and Mezzacappa et al. (2014), where energy increases linearly with time for over a second, a time when the shock is already at a distance of r s /similarequal 10 , 000 km . This linear growth of the energy can be explained by a strong neutrino driven wind from the proto-neutron star. In the new 3D case presented by Mezzacappa et al. (2014) the shock radius position is similar to their results of 1D simulations where no explosion have been obtained.</text>
<text><location><page_5><loc_12><loc_38><loc_91><loc_53></location>A similar dynamic can be seen in figure 4 of the 2D simulation of Marek & Janka (2009), where at time t = 524 ms the shock is at a radius of r s /similarequal 200 km . The shock moves outward to 400 km at t = 610 ms , but then at time t = 650 ms the shock radius decrease back to 200 km . This shows that at that time the energy is about zero and is not positive. The acceleration time can be inferred from figure 6 where the average shock moves from 400 km to 700 km during ∼ 50 ms . In each direction the acceleration time lasts for ∼ 50 ms . However, as the acceleration occurs at different times at different directions, the behavior of the average shock radius gives the impression that the acceleration phase is longer than 50 ms .</text>
<text><location><page_5><loc_12><loc_34><loc_91><loc_37></location>For a neutrinoshpere at r ν /similarequal 50 km (e.g., Couch & O'Connor 2014) the neutrino 'optical depth' from r to infinity is given by</text>
<formula><location><page_5><loc_42><loc_31><loc_91><loc_33></location>τ ν /similarequal 0 . 1( r/ 100 km) -3 (1)</formula>
<text><location><page_5><loc_12><loc_24><loc_91><loc_30></location>(Janka 2001), where the typical electron neutrino luminosity is L ν /similarequal L ¯ ν /similarequal 5 × 10 52 erg s -1 (e.g., Mueller et al. 2012). Over all, if the interaction occurs near a radius r in the gain region, the energy that can be acquired by the expanding gas is</text>
<formula><location><page_5><loc_22><loc_20><loc_91><loc_24></location>E shell /similarequal t esc τL ν /similarequal 0 . 25 ( t esc 50 ms )( L ν 5 × 10 52 erg s -1 ) ( r 100 km ) -3 foe . (2)</formula>
<text><location><page_5><loc_12><loc_15><loc_91><loc_18></location>Using a more typical radius of ∼ 200 km for the acceleration region, reduces the total energy to 0 . 03 foe . Non-spherical flows that allow some simultaneous inflow-outflow structure, might</text>
<text><location><page_6><loc_12><loc_33><loc_14><loc_34></location>or</text>
<formula><location><page_6><loc_26><loc_24><loc_91><loc_32></location>t a /similarequal ( 9 8 ) 1 / 3 ( r a -r 0 ) 2 / 3 ( M a fL ν ) 1 / 3 = 0 . 05 ( r a -r 0 500 km ) 2 / 3 ( M a 0 . 1 M /circledot ) 1 / 3 ( L ν 5 × 10 52 erg s -1 ) -1 / 3 ( f 0 . 1 ) -1 / 3 s . (5)</formula>
<text><location><page_6><loc_12><loc_21><loc_91><loc_24></location>Asimilar acceleration time is estimated from numerical results as we discussed in section 2, where this time is marked t esc . Under these assumptions, the energy of the ejected mass is</text>
<formula><location><page_6><loc_15><loc_15><loc_91><loc_19></location>E a /similarequal t a fL ν /similarequal 0 . 24 ( r a -r 0 500 km ) 2 / 3 ( M a 0 . 1 M /circledot ) 1 / 3 ( L ν 5 × 10 52 erg s -1 ) 2 / 3 ( f 0 . 1 ) 2 / 3 foe . (6)</formula>
<text><location><page_6><loc_12><loc_78><loc_91><loc_87></location>under favorable conditions be expected to increase the energy by a factor of few to ∼ 0 . 1 -0 . 3 foe . This is consistent with numerical simulation results of the delayed neutrino mechanism summarized in section 1. It is interesting to note that Bethe & Wilson (1985) found an explosion energy limit of 0 . 4 foe . This was based on their simulations and not on any physical reason why the neutrino mechanism fails.</text>
<section_header_level_1><location><page_6><loc_36><loc_72><loc_67><loc_73></location>3. ENERGY CONSIDERATIONS</section_header_level_1>
<text><location><page_6><loc_12><loc_59><loc_91><loc_70></location>We examine the situation by considering in more detail the acceleration from the delayedneutrino mechanism. Consider a mass M a that is accelerated and ejected by absorbing a fraction f of the neutrino energy. The mass starts at radius r 0 with zero energy. Namely the sum of internal and gravitational energy is zero. This is an optimistic assumption, as the internal energy itself also needs to be supplied by neutrinos. Neutrino losses can be absorbed into the parameter f . After an acceleration time t the energy of the mass is fL ν t and its velocity is</text>
<formula><location><page_6><loc_41><loc_54><loc_91><loc_58></location>v = dr dt /similarequal ( 2 fL ν t M a ) 1 / 2 . (3)</formula>
<text><location><page_6><loc_12><loc_43><loc_91><loc_52></location>Here we assume that most of the energy is transferred to kinetic energy. Initially, more energy can be stored as thermal energy. However, not much thermal energy can be stored after the gas energy becomes positive, as it starts to accelerate outward and thermal energy is converted to kinetic energy on a dynamical time scale. The thermal energy acts to overcome gravity. We calculate here the extra energy that goes to gas outward motion.</text>
<text><location><page_6><loc_16><loc_40><loc_82><loc_42></location>Let the acceleration be effective to radius r a at time t a . Integrating over time gives</text>
<formula><location><page_6><loc_39><loc_35><loc_91><loc_39></location>r a -r 0 /similarequal 2 3 ( 2 fL ν M a ) 1 / 2 t 3 / 2 a , (4)</formula>
<text><location><page_7><loc_12><loc_78><loc_91><loc_87></location>In these calculations, we assumed a constant neutrino luminosity. As the neutrino luminosity decreases with time (e.g., Fischer et al. 2012), the term fL ν , in equation (6) actually overestimates the available energy. More typical values for acceleration over ∼ 500 km are f < 0 . 1 due to the low neutrino opacity (eq. 1), and lower accelerated mass. These values give E a < 0 . 2 foe as in Equation 2.</text>
<section_header_level_1><location><page_7><loc_29><loc_72><loc_74><loc_73></location>4. THE ALMOST UNBOUND STALLED SHOCK</section_header_level_1>
<text><location><page_7><loc_12><loc_51><loc_91><loc_70></location>The energy of the immediately post-shocked gas falling from thousands of km to hundreds of kmis close to zero before there is much neutrino cooling. Whether the shocked gas falls or expands is a question of whether a small amount of energy is added to revive the shock. When there are departures from spherical symmetry, like the perturbations introduced by Couch & Ott (2013) or instabilities in the post-shock region, in some areas the extra energy comes at the expense of other areas. For example, a vortex can add a positive velocity in the region of the flow where the flow goes out. Even if the shock is revived, the energy limitations given in Sections 2 and 3 apply. The SASI itself is a manifestation of the process where one region of the stalled shock can go out in expense of other regions. The extra energy from neutrino heating can even revive the entire sphere. However, the energy gained by neutrino heating is limited.</text>
<text><location><page_7><loc_12><loc_20><loc_91><loc_50></location>A recent attempt to revive the stalled shock is that of Couch & Ott (2013), who introduced perturbations to the Si/O layers, and found them to enable shock revival under certain conditions. What Couch & Ott (2013) term a successful explosion is actually a revival of the stalled shock. They did not obtain the desired ∼ 1 foe explosion. As with many other simulations, small changes in the initial conditions determine whether shock revival occurs or not. For example, Couch & Ott (2013) find shock revival when their neutrino heat factor is 1.02, but not when it is 1. They present their average shock position until it reaches a radius of 430 km at t = 0 . 32 s . Examining their successful revival run presented in their figure 3, we find the average shock outward velocity in the last part they show, 370 to 430 km, to be 〈 v shock 〉 = 8000 km s -1 . This is less than 0.3 times the escape velocity at that radius. The shock does not seem to accelerate in the last 50 km. Within ∆ t /similarequal 0 . 04 s the shock will reach a radius of about 700 km, where no more energy gain is possible (Janka 2001). At 400 km the neutrino optical depth is very small, τ < 0 . 1 . Indeed, at an average shock radius of 350 km the heating efficiency in their simulation η , defined as net heating rate divided by L ν e + L ¯ ν e , drops below 0.1. This implies that the gained energy will be very small, ∆ E < τL ν ∆ t < 0 . 2 foe . We therefore estimate that even the perturbations introduced by Couch & Ott (2013) will not bring the explosion, if occurs, close to 1 foe .</text>
<text><location><page_7><loc_12><loc_16><loc_91><loc_19></location>Let us quantify the statement of energy close-to-zero. We can make the following estimations based on the models of Woosley et al. (2002) of massive stars prior to the collapse. The gas at</text>
<text><location><page_8><loc_12><loc_72><loc_91><loc_87></location>2000 km has a specific gravitational energy of e G 0 = -10 18 erg g -1 and a specific internal energy of e I 0 = 5 . 5 × 10 17 erg g -1 . After mass loss to neutrinos from the core, the inner mass reduces by ∼ 10% . However, by that time the shell that starts at few × 1000 km has been accelerated inward. So we take the total specific energy to be as the pre-collapse energy. As an example, we take the stalled shock to be at r s = 200 km . When reaching r s = 200 km the specific total energy e t = e I 0 + e G 0 , stays the same. The specific gravitational energy is e Gs /similarequal 10 e G 0 = -10 19 erg g -1 , and the specific internal (thermal + kinetic+nuclear) energy, is e Is = e t -10 e G 0 /similarequal 9 . 5 × 10 18 erg g -1 . The net specific energy relative to gravitational energy in this demonstrative example is</text>
<formula><location><page_8><loc_44><loc_66><loc_91><loc_71></location>ξ s ≡ ∣ ∣ ∣ e Is e Gs ∣ ∣ ∣ /similarequal 0 . 95 . (7)</formula>
<text><location><page_8><loc_12><loc_61><loc_91><loc_68></location>∣ ∣ The mass is very close to be unbound. Small amount of net heating can revive the shock. For a typical mass in the gain region of M g /lessorsimilar 0 . 05 M /circledot (e.g., Couch 2013), an extra energy of ∆ E = 5 × 10 49 erg = 0 . 05 foe will revive the shock.</text>
<section_header_level_1><location><page_8><loc_27><loc_55><loc_76><loc_56></location>5. ENERGY AVAILABLE FROM RECOMBINATION</section_header_level_1>
<text><location><page_8><loc_12><loc_40><loc_91><loc_53></location>Adding nuclear energy of free nucleons does not change the above property of an almost unbound stalled shock, and the conclusion of low 'explosion' energy. Consider the scenario where disintegration of nuclei form free nucleons beyond the stalled shock, and the available nuclear energy is reused later after the free nucleons are accelerated outwards by neutrinos (Janka et al. 2012). When the nucleons recombine to form heavy nuclei an energy of up to 9 MeV per nucleon can in principle be used to explode the star (Janka et al. 2012). A mass of 0 . 06 M /circledot in the gain region can then release in principle ∼ 10 51 erg (Scheck et al. 2006).</text>
<text><location><page_8><loc_12><loc_28><loc_91><loc_39></location>However, the recombination of free nucleons to alpha particles, a process that uses 7 MeV from the 9 MeV available in forming silicon, starts when the reviving post-shock gas reaches r ∼ 250 km (Fern'andez & Thompson 2009). The energy released by recombination accelerates the material (Fern'andez & Thompson 2009), which results in a shorter acceleration time than given in Equation (5). This further lowers the energy that can be supplied by neutrinos below that given in Equations (2) and (6).</text>
<text><location><page_8><loc_12><loc_15><loc_97><loc_26></location>The energy available from recombination is limited as well. From Figure 5 of Fern'andez & Thompson (2009), we find the total fraction of α particles in the gas inside the shock when the shock radius is 500 km to be X α /lessorsimilar 0 . 5 ; the fraction just behind the shock front at 500 km is X α /similarequal 0 . 9 . In the results of Fern'andez & Thompson (2009) the fraction of α particles increases as the shock moves outward. For this fraction, the average energy available from recombination is 5 MeV/ nucleaon (Janka et al. 2012). However, the shock is only at 500 km and a large fraction of the mass is much</text>
<text><location><page_9><loc_12><loc_74><loc_91><loc_87></location>deeper. As the shock expands further, the fraction of α particles will increase and the available energy will decrease. Taking a mass in the gain region of M gain /lessorsimilar 0 . 05 M /circledot (e.g., Couch 2013), we find the 'explosion' energy to be E nuc /lessorsimilar 0 . 5 foe . In some 2D simulations the mass in the gain layer is /greaterorsimilar 0 . 05 M /circledot , but in 3D simulations the gain layer has lower mass than in 2D simulations (e.g., Couch 2013). Over all, the available energy without neutrino winds or jets is < 0 . 5 foe . This value is an upper limit and consistent with many of the simulations summarized in section 1 that achieve much lower energies or do not revive the shock at all.</text>
<text><location><page_9><loc_12><loc_60><loc_91><loc_73></location>It should be emphasized that the recombination is not a new energy source, as the thermal energy of the shocked gas is used to disintegrate the nuclei. The recombination is the re-usage of this energy. The extra energy must come from neutrinos that lift the free nucleons to larger radii. The total available energy from recombination is proportional to the mass of the free nucleons that are lifted from small r /lessorsimilar 150 km to large radii r /greaterorsimilar 500 km . However, the amount of mass that can be accumulated at small radii is limited because if the density is too high then cooling overcomes neutrino heating, and the shock will not be revive.</text>
<text><location><page_9><loc_12><loc_50><loc_91><loc_58></location>Yamamoto et al. (2013) preformed 1D and 2D simulations of shock revival and examined explosion energy including recombination and shock nuclear burning. They tuned the neutrino luminosity to a critical value that gives successful explosions. Their successful runs have shock relaunch times of 0 . 3 -0 . 4 s in 2D flows. The explosion energy in these runs is in the range of 0 . 6 -1 . 5 foe . We note the following regarding their tuned calculations:</text>
<unordered_list>
<list_item><location><page_9><loc_12><loc_42><loc_91><loc_49></location>(1) Yamamoto et al. (2013) assume that neutrino heating alone revives the stalled shock. Then they can use the entire recombination energy to explode the rest of the star. The more realistic calculations of Fern'andez & Thompson (2009) show that at least half the recombination energy is required to help revive the shock.</list_item>
</unordered_list>
<text><location><page_9><loc_12><loc_25><loc_91><loc_41></location>(2) The above assumption implies the need for high neutrino luminosity. Indeed, in Yamamoto et al. (2013) successful 2D runs the required critical neutrino luminosities are L ν,c = L ¯ ν,c = 4 . 8 × 10 52 erg s -1 and 4 . 5 × 10 52 erg s -1 for shock relaunching times of 0 . 3 s and 0 . 4 s , respectively. These neutrino luminosities are ∼ 50% higher than what most realistic numerical simulations find, e.g., Fischer et al. (2012), and ∼ 30% higher than the neutrino luminosities obtained by Mueller et al. (2012) who included general relativistic effects. Interestingly, Mueller et al. (2012) find for their 11 M /circledot model that recombination of nucleons and α -particles in the ejecta would provide an additional energy of E rec /similarequal 0 . 02 foe . For their 15 M /circledot model they argue that burning in the shock will add on the order of 0 . 1 -0 . 2 foe or more.</text>
<text><location><page_9><loc_12><loc_21><loc_91><loc_25></location>(3) The contribution of nuclear and recombination energies to the diagnostic explosion energy of Yamamoto et al. (2013) are very similar to the contribution of neutrino heating.</text>
<text><location><page_9><loc_12><loc_17><loc_91><loc_20></location>Based on these points we can use a more realistic value of neutrino heating, E ν < 0 . 2 foe , and conclude that the combined explosion energy in realistic simulations will be E exp < 0 . 5 foe .</text>
<text><location><page_10><loc_12><loc_78><loc_91><loc_87></location>Again we reach the conclusion that including recombination energy will at most bring the explosion energy to E exp < 0 . 5 foe . Although close to the canonical 1 foe value, one must keep in mind that this value is obtained with very favorable conditions, and in scaled, rather than realistic, simulations. In more realistic simulations the recombination energy is found to be E rec /lessorsimilar 0 . 2 foe , e.g., Mueller et al. (2012).</text>
<section_header_level_1><location><page_10><loc_36><loc_72><loc_67><loc_73></location>6. DISCUSSION AND SUMMARY</section_header_level_1>
<text><location><page_10><loc_12><loc_61><loc_91><loc_70></location>Using simple estimates of a spherically-symmetric mass ejection by neutrino flux in corecollapse supernovae (CCSNe), we found that in the delayed-neutrino mechanism (Bethe & Wilson 1985), where the main energy source of the explosion is due to neutrino heating in the gain region, the explosion energy is limited to E /lessorsimilar 0 . 5 foe , with a more likely limit of 0 . 3 foe (eq. 2 and 6). This falls short of what is required in most CCSNe.</text>
<text><location><page_10><loc_12><loc_45><loc_91><loc_59></location>Although our simple analytical estimates are limited to spherically symmetric outflows, they none-the-less catch the essence of the delayed-neutrino mechanism. In a non-spherical flow, instabilities, such as neutrino-driven convection and the standing accretion shock instability (SASI), play a major role (e.g., Hanke et al. 2013). Such instabilities allow inflow and outflow to occur simultaneously. Still, recent and highly sophisticated 3D simulations with enough details to resolve such instabilities do not obtain enough energy to revive the stalled shock, (e.g., Janka 2013). The energy that can be used from the neutrino flux might, under favorite conditions, revive the stalled shock, but cannot lead to explosions with energies of E e /greaterorsimilar 0 . 3 foe .</text>
<text><location><page_10><loc_12><loc_29><loc_91><loc_43></location>Our conclusion holds as long as no substantially new ingredient is added to the delayedneutrino mechanism. Such an ingredient can be a strong wind, as was applied by artificial energy deposition by Nordhaus et al. (2010, 2012). In their 2.5D simulations, Scheck et al. (2006) achieved explosion that was mainly driven by a continuous wind. The problem we see with winds is that they are less efficient than jets. Indeed, in order to obtain an explosion the winds in the simulations of Scheck et al. (2006) had to be massive. For that, in cases where they obtained energetic enough explosions the final mass of the NS was low ( M NS < 1 . 3 M /circledot ) . Such a wind must be active while accretion takes place; the accretion is required to supply the energy (Marek & Janka 2009).</text>
<text><location><page_10><loc_12><loc_17><loc_91><loc_27></location>With the severe problems encountered by neutrino heating, research groups have turned to study dynamical processes. Couch & Ott (2013), Couch & Ott (2014), and Mueller & Janka (2014) argued that the effective turbulent ram pressure exerted on the stalled shock allows shock revival with less neutrino heating than 1D models. However, Abdikamalov et al. (2014) found that increasing the numerical resolution allows cascade of turbulent energy to smaller scales, and the shock revival becomes harder to achieve at high numerical resolutions.</text>
<text><location><page_11><loc_12><loc_59><loc_91><loc_87></location>Another dynamical process is a collimated wind blown by the newly formed NS. Bruenn et al. (2014) performed 2D simulations up to 1 . 4 s post-bounce, and obtained an explosion energy of 0 . 3 -0 . 9 foe , depending on the stellar model (initial mass without rotation). They find the main energy source to be by what they term an 'enthalpy flux'. This is actually a wind, mainly along the imposed symmetry axis, i.e., a collimated wind. This wind is driven by the inflowing (accreted) gas. At some instant, their results show jet-like outflows along the symmetry axis. It seems that the collimated wind is induced by the numerical grid. Contrast that to their corresponding 3D simulations (Mezzacappa et al. 2014) which show no such explosion. Mezzacappa et al. (2014) present one new result of a 3D run for their 15 M /circledot model at t = 267 ms post-bounce. We estimate the average shock radius at that time to be ∼ 220 km . This is very similar to their 1D results (Bruenn et al. 2013), where the shock radius is much smaller than in their 2D simulations, and where no explosions occur. Non-the-less, the results of Bruenn et al. (2014) show the great potential of an inflow-outflow mechanism in exploding CCSNe. An inflow-outflow situation with collimated outflows over a relatively long time naturally occurs with jets launched by accretion disks, without the numerically induced symmetry axis in 2D grids.</text>
<text><location><page_11><loc_12><loc_20><loc_92><loc_58></location>For the above, the lack of persisting success, and possibly failure, of the delayed-neutrino mechanism calls for a paradigm shift. As well, the rich variety of CCSN properties (e.g., Arcavi et al. 2012) further emphasizes the need to study alternative models for CCSN explosions, some of which are based on jet-driven explosions (Janka 2012). In CCSNe simulations jets have been shown to be launched when the pre-collapsing core posses both a rapid rotation and a very strong magnetic field (e.g. LeBlanc & Wilson 1970; Meier et al. 1976; Bisnovatyi-Kogan et al. 1976; Khokhlov et al. 1999; MacFadyen et al. 2001; Hoflich et al. 2001; Woosley & Janka 2005; Burrows et al. 2007; Couch et al. 2009, 2011; Lazzati et al. 2012; Takiwaki & Kotake 2011). However, these jets do not explode the core via a feedback mechanism, such that they too often give extreme cases as gamma ray bursts, or they fail to explode the star, e.g., Mosta et al. (2014). Recent observations (e.g. Milisavljevic et al. 2013; Lopez et al. 2013) suggest that jets might play a role in at least some CCSNe. Another motivation to consider jet-driven explosion mechanisms is that jets might supply the site for the r-process (Winteler et al. 2012; Papish & Soker 2012b). The question is whether the accreted mass possesses sufficient specific angular momentum to form an accretion disk. Persistent accretion disk requires the pre-collapsing core to rotate fast, as in the magnetohydrodynamics class of models ( e.g. LeBlanc & Wilson 1970; Meier et al. 1976; Bisnovatyi-Kogan et al. 1976; Khokhlov et al. 1999; MacFadyen et al. 2001; Hoflich etal. 2001; Woosley & Janka 2005; Burrows et al. 2007; Couch et al. 2009, 2011; Lazzati et al. 2012). Most massive stars reach the core-collapse phase with a too slow core rotation for the magnetorotational mechanism to be significant.</text>
<text><location><page_11><loc_12><loc_16><loc_91><loc_19></location>One alternative to the delayed-neutrino mechanism which overcomes the angular momentum barrier is the so-called 'jittering-jet' mechanism of Papish & Soker (2011). The jittering-jet mech-</text>
<text><location><page_12><loc_12><loc_65><loc_91><loc_87></location>anism overcomes the requirement for rapid core rotation, and was introduced as a mechanism to explode all CCSNe (Papish & Soker 2011, 2012b, 2014). The angular momentum source is the convective regions in the core (Gilkis & Soker 2014), and/or instabilities in the shocked region of the collapsing core. Blondin & Mezzacappa (2007), Fern'andez (2010), and Rantsiou et al. (2011) suggested that the source of the angular momentum of pulsars is the spiral mode of the SASI. In the jittering-jet mechanism there is no need to revive the accretion shock, and it is a mechanism based on a negative feedback cycle. As long as the core was not exploded, the accretion continues. After an energy equals several times the core binding energy is deposited to the core by the jets, the star explodes. This energy amounts to ∼ 1 foe . If the feedback is less efficient, more accretion is required to accumulate the required energy. If the efficiency is very low, the accreted mass onto the NS brings it to collapse to a black hole and launch relativistic jets. Namely, in general, the less efficient the feedback mechanism is, the more violent the explosion is (Gilkis & Soker 2014).</text>
<text><location><page_12><loc_12><loc_56><loc_91><loc_63></location>We thank the anonymous referee for helpful comments. This research was supported by the Asher Fund for Space Research at the Technion, and a generous grant from the president of the Technion Prof. Peretz Lavie. OP is supported by the Gutwirth Fellowship. JN is supported by an NSF award AST-1102738 and by NASA HST grant AR-12146.01-A.</text>
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</document> | [] |
2019arXiv191001699C | https://arxiv.org/pdf/1910.01699.pdf | <document>
<section_header_level_1><location><page_1><loc_38><loc_80><loc_72><loc_81></location>King's College London</section_header_level_1>
<figure>
<location><page_1><loc_23><loc_38><loc_87><loc_77></location>
</figure>
<text><location><page_1><loc_52><loc_35><loc_57><loc_36></location>in the</text>
<text><location><page_1><loc_36><loc_31><loc_74><loc_33></location>Theoretical Particle Physics and Cosmology</text>
<text><location><page_1><loc_45><loc_29><loc_65><loc_30></location>Department of Physics</text>
<text><location><page_1><loc_49><loc_23><loc_61><loc_25></location>1st April 2019</text>
<figure>
<location><page_1><loc_45><loc_10><loc_64><loc_20></location>
</figure>
<section_header_level_1><location><page_2><loc_19><loc_71><loc_61><loc_73></location>Declaration of Authorship</section_header_level_1>
<unordered_list>
<list_item><location><page_2><loc_19><loc_64><loc_90><loc_68></location>I, Rhiannon Cuttell , declare that this thesis titled, 'Deformed general relativity' and the work presented in it are my own. I confirm that:</list_item>
<list_item><location><page_2><loc_22><loc_58><loc_90><loc_62></location>· This work was done wholly or mainly while in candidature for a research degree at this University.</list_item>
<list_item><location><page_2><loc_22><loc_52><loc_90><loc_56></location>· Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated.</list_item>
<list_item><location><page_2><loc_22><loc_48><loc_90><loc_50></location>· Where I have consulted the published work of others, this is always clearly attributed.</list_item>
<list_item><location><page_2><loc_22><loc_42><loc_90><loc_46></location>· Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work.</list_item>
<list_item><location><page_2><loc_22><loc_39><loc_60><loc_40></location>· I have acknowledged all main sources of help.</list_item>
<list_item><location><page_2><loc_22><loc_33><loc_90><loc_37></location>· Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself.</list_item>
</unordered_list>
<text><location><page_2><loc_19><loc_29><loc_25><loc_30></location>Signed:</text>
<text><location><page_2><loc_19><loc_23><loc_23><loc_24></location>Date:</text>
<text><location><page_2><loc_32><loc_23><loc_44><loc_24></location>1st April 2019</text>
<figure>
<location><page_2><loc_32><loc_29><loc_47><loc_32></location>
</figure>
<text><location><page_3><loc_19><loc_68><loc_67><loc_69></location>' I really don't know what I'm doing... I don't. It's terrible...'</text>
<text><location><page_3><loc_75><loc_63><loc_90><loc_64></location>Leonardo DiCaprio</text>
<section_header_level_1><location><page_4><loc_42><loc_82><loc_67><loc_83></location>KING'S COLLEGE LONDON</section_header_level_1>
<section_header_level_1><location><page_4><loc_49><loc_75><loc_60><loc_77></location>Abstract</section_header_level_1>
<section_header_level_1><location><page_4><loc_37><loc_70><loc_73><loc_71></location>School of Natural and Mathematical Sciences</section_header_level_1>
<text><location><page_4><loc_46><loc_67><loc_64><loc_68></location>Department of Physics</text>
<text><location><page_4><loc_46><loc_62><loc_63><loc_63></location>Doctor of Philosophy</text>
<section_header_level_1><location><page_4><loc_42><loc_57><loc_67><loc_58></location>Deformed general relativity</section_header_level_1>
<text><location><page_4><loc_45><loc_53><loc_64><loc_54></location>by Rhiannon Cuttell</text>
<text><location><page_4><loc_19><loc_43><loc_90><loc_49></location>In this thesis, I investigate how to construct a self-consistent model of deformed general relativity using canonical methods and metric variables. The specific deformation of general covariance is predicted by some studies into loop quantum cosmology.</text>
<text><location><page_4><loc_19><loc_16><loc_90><loc_41></location>I firstly find the minimally-deformed model for a scalar-tensor theory, thereby establishing a classical reference point, and investigate the cosmological effects of a non-minimal coupled scalar field. By treating the deformation perturbatively, I derive the deformed gravitational action which includes the nearest order of curvature corrections. Then working more generally, I derive the deformed scalar-tensor constraint to all orders and I find that the momenta and spatial derivatives from gravity and matter must combine in a very specific form. It suggests that the deformation should be equally affected by matter field derivatives as it is by gravitational curvature. Finally, I derive the deformed gravitational action to all orders, and find how intrinsic and extrinsic curvatures differently affect the deformation. The deformation seems to be required to satisfy a non-linear equation usually found in</text>
<text><location><page_4><loc_19><loc_14><loc_32><loc_15></location>fluid mechanics.</text>
<section_header_level_1><location><page_5><loc_42><loc_86><loc_68><loc_88></location>Acknowledgements</section_header_level_1>
<text><location><page_5><loc_19><loc_80><loc_90><loc_84></location>Thanks to my supervisor, Mairi, who patiently facilitated and enabled this, and helped guide me away from dead ends.</text>
<text><location><page_5><loc_19><loc_74><loc_90><loc_78></location>Thanks to Martin, who helped me find and correct a serious error in my methodology before it was too late.</text>
<text><location><page_5><loc_19><loc_68><loc_90><loc_72></location>Thanks to those in the physics department who have helped so much with navigating through difficult situations, especially Jean, Julia and Rowena.</text>
<text><location><page_5><loc_19><loc_62><loc_90><loc_66></location>Thanks to Marc, Brinda, Agnes, Ruth and Gwyn, the medical and mental health professionals that helped me keep afloat.</text>
<text><location><page_5><loc_19><loc_56><loc_90><loc_60></location>Thanks to my parents Jeff and Liz, who gave enough encouragement for me to be doing this and on whom I have depended too much.</text>
<text><location><page_5><loc_19><loc_50><loc_90><loc_54></location>Thanks to Joanna, for being the perfect big sister I don't deserve, and thanks to Jim for being there for her in turn.</text>
<text><location><page_5><loc_19><loc_44><loc_90><loc_48></location>Thanks to Naomi, for always being there with love, support and a goofy joke. I couldn't imagine managing to finish this without you. ( ϕ ω ϕ ) ∼</text>
<text><location><page_5><loc_19><loc_41><loc_84><loc_42></location>Thanks to Xin Xin, Jack, Pebbles, Muffin and Tuxedo Kamen for just being you.</text>
<section_header_level_1><location><page_6><loc_19><loc_80><loc_36><loc_82></location>Contents</section_header_level_1>
<table>
<location><page_6><loc_19><loc_10><loc_90><loc_74></location>
</table>
<table>
<location><page_7><loc_19><loc_28><loc_91><loc_88></location>
</table>
<section_header_level_1><location><page_8><loc_19><loc_80><loc_47><loc_82></location>List of Figures</section_header_level_1>
<table>
<location><page_8><loc_22><loc_49><loc_90><loc_74></location>
</table>
<section_header_level_1><location><page_9><loc_19><loc_80><loc_49><loc_82></location>List of Symbols</section_header_level_1>
<table>
<location><page_9><loc_29><loc_17><loc_80><loc_74></location>
</table>
<text><location><page_10><loc_49><loc_60><loc_60><loc_61></location>For Naomi</text>
<section_header_level_1><location><page_11><loc_19><loc_79><loc_33><loc_81></location>Chapter 1</section_header_level_1>
<section_header_level_1><location><page_11><loc_19><loc_70><loc_43><loc_73></location>Introduction</section_header_level_1>
<text><location><page_11><loc_19><loc_49><loc_90><loc_64></location>In this thesis I investigate deformed general relativity, which is a semi-classical model attempting to capture the leading effects of a correction to general relativity predicted in some studies of loop quantum gravity. It uses the methods of canonical gravity but with space-time covariance deformed by a phase-space function. By assuming a general deformation, I find the general models which are consistent with it, demonstrating multiple routes which can be taken to find them.</text>
<text><location><page_11><loc_19><loc_43><loc_90><loc_47></location>Before going into more depth on this, I must first discuss the motivations for this investigation.</text>
<section_header_level_1><location><page_11><loc_19><loc_37><loc_74><loc_39></location>1.1 The need for a theory of quantum gravity</section_header_level_1>
<text><location><page_11><loc_19><loc_20><loc_90><loc_34></location>It is known that matter fields are quantised due to the remarkable agreement of experimental results with quantum field theory [1-3]. There have been some attempts to allow for classical gravity to couple to quantum fields at a fundamental level [4, 5], and some interesting phenomena have been discovered from considering effective models of quantum fields on a curved space-time [6-8]. However, it is generally expected that gravity must be quantised too [9,10].</text>
<text><location><page_11><loc_24><loc_11><loc_86><loc_17></location>The gravitational field, like all other fields, therefore must be quantized, or else the logical structure of quantum field theory must be profoundly altered, or both. [11, B. DeWitt]</text>
<text><location><page_12><loc_19><loc_69><loc_90><loc_88></location>Besides gravity being known to couple to quantum fields, there are known limitations to the current common understanding. General relativity predicts its own demise due to singularities arising in the equations describing black holes and the very early universe [12]. They are known to exist due to robust experimental observations supporting the existence of black holes [13] and supporting an early universe which closely matches what is predicted of a hot big bang [14]. These phenomena exist at the intersection of general relativity and quantum mechanics since they involve both massive systems and small scales. It seems they cannot be fully understood without a framework which consistently bridges the gap.</text>
<text><location><page_12><loc_19><loc_47><loc_90><loc_66></location>As a precedent for the singularity problem, classical mechanics could not sufficiently account for experimental results showing that atoms contained small, massive nuclei orbited by electrons (the Nagaoka-Rutherford model). This is due to accelerating point charges (electric field singularities) being known to emit radiation as per the Landau formula, and therefore an electron orbit should radiatively decay, causing atoms to be unstable. However, the development of quantum mechanics resolved this by introducing discrete and stationary orbitals in the Bohr model. The hope is that quantising gravity will similarly cure it of some of its pathologies.</text>
<text><location><page_12><loc_19><loc_33><loc_90><loc_45></location>One might not want to jettison all that is good about general relativity in pursuit of a quantised theory. The key underlying idea, equivalence of all frames, is considered a philosophically and aesthetically satisfying aspect. Conversely, the requirement in the orthodox interpretation of quantum mechanics for an external observer is considered troubling, hence why Einstein spent much of the latter part of his career challenging it [15].</text>
<text><location><page_12><loc_19><loc_22><loc_90><loc_31></location>One crucial sticking point in reconciling general relativity and quantum mechanics is the problem of time [16,17]. In quantum mechanics time is a fixed external parameter, in general relativity it is internal to the system and is not uniquely defined. These are seemingly incommensurable differences, and to bridge the gap requires significant compromise.</text>
<text><location><page_12><loc_19><loc_11><loc_90><loc_20></location>The solution in canonical gravity for reconciling the two is to split space-time at the formal level, but include symmetry requirements so that the full general covariance is kept implicitly [10, 18, 19]. One is left with a description of a spatial slice evolving through time rather than one of a static and eternal bulk. These methods are often required for</text>
<text><location><page_13><loc_19><loc_84><loc_90><loc_88></location>numerically simulating general relativity due to the necessity of specifying a time coordinate when setting up an evolution simulation.</text>
<text><location><page_13><loc_19><loc_68><loc_90><loc_82></location>This introduces on each spatial manifold a conserved quantity or 'constraint' given by φ I → 0 for each dimension of time and space, analogous to a generalisation of the conservation of energy and momentum. These constraints form an algebra which contains important information about the geometric nature of space-time, and is of the form { φ I , φ J } = f K IJ φ K [10, 20]. This is a Lie algebroid which describes the relationships between the constraints and generates transformations between different choices of coordinates [21,22].</text>
<text><location><page_13><loc_19><loc_59><loc_90><loc_66></location>The important { C, C } part of this algebra ensures that the spatial manifold evolving through time is equivalent to a stack of spatial manifolds embedded in a geometric spacetime manifold.</text>
<text><location><page_13><loc_24><loc_48><loc_86><loc_57></location>In this more general case of gravitation in interaction with other fields, [the equation 1 ] not only guarantees the embeddability of the 3-geometries in a spacetime but also ensures that these additional fields evolve consistently within this space-time. [23, C. Teitelboim]</text>
<text><location><page_13><loc_19><loc_42><loc_90><loc_46></location>This part of the algebra is what I am going to consider to be deformed, but where does this hypothesis come from?</text>
<section_header_level_1><location><page_13><loc_19><loc_36><loc_51><loc_37></location>1.2 Loop quantum gravity</section_header_level_1>
<text><location><page_13><loc_19><loc_16><loc_90><loc_32></location>Though there are several candidates for a theory of quantum gravity, I am going to only consider loop quantum gravity [24,25]. There are other somewhat related theories which also deal directly with quantising gravity, such as: causal dynamical triangulations [26]; causal set theory [27]; group field theory [28]; and asymptotically safe gravity [29]. The main alternative candidate is string theory and its variants, which prioritises bringing gravity into the established framework for quantum particles in order to create a unified theory [30,31].</text>
<text><location><page_14><loc_19><loc_71><loc_90><loc_88></location>Loop quantum gravity focuses on maintaining some key concepts from general relativity such as background independence and local dynamics throughout the process of combining gravity and quantum mechanics. It describes space-time as not being a continuous manifold, but instead being a network of nodes connected by ordered links with quantum numbers for geometrical quantities such as volume. Such a network is not merely embedded in space but is space itself . As such, due to the quantisation of geometry, one cannot shrink the length of a link between nodes to being infinitesimal as in the classical case.</text>
<text><location><page_14><loc_19><loc_60><loc_90><loc_69></location>If general relativity is truly the classical limit of loop quantum gravity, then there should be a semi-classical limit where the dynamics are well approximated by general relativity with minor quantum corrections. These should become larger at small scales and in regions of high curvature.</text>
<text><location><page_14><loc_19><loc_44><loc_90><loc_58></location>A closely related theory is loop quantum cosmology, which uses concepts and techniques from loop quantum gravity and applies them directly at the cosmological level by using midi-superspace models [32, 33]. That is, by quantising a universe which already has certain symmetries assumed such as isotropy to simplify the process. There has been some progress towards proving that loop quantum gravity can be symmetry-reduced to loop quantum cosmology, but as yet this has not been shown definitively [34,35].</text>
<text><location><page_14><loc_19><loc_22><loc_90><loc_41></location>For models of loop quantum cosmology to be self-consistent and anomaly-free while including some of the interesting effects from the discrete geometry, it seems that the algebra of constraints must be deformed. Specifically, some of the structure functions become more dependent on the phase space variables through a deformation function f K IJ ( q ) → β ( q, p ) f K IJ ( q ) [36-42]. Deforming rather than breaking the algebra in principle maintains general covariance but the transformations between different choices of coordinates become highly non-linear [43]. It becomes less clear to what extent one can still interpret space-time geometrically, at least in terms of classical notions of geometry.</text>
<text><location><page_14><loc_19><loc_11><loc_90><loc_20></location>However, there is ambiguity in the correct choice of variables used for loop quantum gravity. The results cited in the previous paragraph are for real variables for which there has been significant difficulty including matter and local degrees of freedom [44]. The main alternative, self-dual variables, have had some positive results for including those degrees</text>
<text><location><page_15><loc_19><loc_84><loc_90><loc_88></location>of freedom without deforming the constraint algebra [45], but might not have the desired quality of resolving curvature singularities [46].</text>
<text><location><page_15><loc_19><loc_73><loc_90><loc_82></location>Interesting predictions coming from loop quantum gravity include: a bouncing universe [47]; black hole singularity resolution and transition to white holes [48]; and signature change of the effective metric [41]. Some of these predictions are closely associated with a deformation of classical symmetries in regions of high energy density.</text>
<section_header_level_1><location><page_15><loc_19><loc_67><loc_72><loc_68></location>1.3 Why study deformed general relativity?</section_header_level_1>
<text><location><page_15><loc_19><loc_44><loc_90><loc_63></location>Deformed general relativity builds directly from the idea that the constraint algebra is deformed [49]. It is constructed by taking the deformed constraint algebra, and finding a corresponding model which includes local degrees of freedom a priori . This can be done because, if one starts from an algebra and makes some reasonable assumptions, one can deduce the general form of all the constraints [21,50]. This should provide a more intuitive understanding of how the deformation affects dynamics and may provide a guide for how to include the problematic degrees of freedom when working with real variables in loop quantum gravity.</text>
<text><location><page_15><loc_19><loc_33><loc_90><loc_42></location>The constraint algebra is important because, as said previously, it closely relates to the structure of space-time [23]. Quantum geometry will behave differently to classical geometry, and deformed general relativity attempts to capture some of the effects in a semiclassical model which is more amenable to phenomenological investigations.</text>
<text><location><page_15><loc_19><loc_19><loc_90><loc_31></location>Phenomenological models which are comparable to deformed general relativity, such as deformed special relativity [51] and rainbow gravity [52], struggle to go beyond describing individual particles coupled to an energy-dependent metric. They can suffer from a breakdown of causality [53], or find it difficult to describe multi-particle states [54]. Deformed general relativity does not suffer from these problems by construction.</text>
<section_header_level_1><location><page_16><loc_19><loc_87><loc_52><loc_88></location>1.4 Overview of this thesis</section_header_level_1>
<text><location><page_16><loc_19><loc_56><loc_90><loc_83></location>The main focus of this thesis is to investigate how to construct a self-consistent model of deformed general relativity using canonical methods and metric variables. I review important concepts and methodology in chapter 2. In chapter 3, I find the minimallydeformed model for a scalar-tensor theory, establishing a classical reference point. Then in chapter 4, I derive the deformed gravitational action which includes the lowest non-trivial order of perturbative curvature corrections coming from the deformation. In chapter 5, I derive the deformed scalar-tensor constraint to all orders and I find that the momenta and space derivatives must combine in a specific form. Finally, in chapter 6, I find the deformed gravitational action to all orders, and find how intrinsic and extrinsic curvatures differently affect the deformation. I identify some of the cosmological consequences for the significant results of each chapter.</text>
<text><location><page_16><loc_19><loc_37><loc_90><loc_54></location>There are several research questions which I attempt to answer in this thesis. How are the form of the deformation function and the form of the model related? In particular, what is the deformed scalar-tensor Hamiltonian and what is the deformed gravitational Lagrangian, using either perturbative or non-perturbative methods? How do they relate to the classical limit and to each other? How can matter fields be incorporated in deformed models? How does the deformation function depend on curvature, and is it different for intrinsic and extrinsic curvatures?</text>
<text><location><page_16><loc_19><loc_29><loc_90><loc_35></location>The research chapters 3 and 4 are adapted from the previously published papers [55] and [56], respectively. The other research chapters, 5 and 6, were recently submitted for publication [57,58]</text>
<section_header_level_1><location><page_16><loc_19><loc_22><loc_41><loc_24></location>1.5 Wider impact</section_header_level_1>
<text><location><page_16><loc_19><loc_13><loc_90><loc_19></location>This study is directly motivated by the prediction of a deformed constraint algebra appearing in loop quantum cosmology [36-42]. As such it should provide insight into the lingering questions of how matter and local degrees of freedom need to be incorporated</text>
<text><location><page_17><loc_19><loc_84><loc_90><loc_88></location>into the motivating theory in the presence of a deformation, and how spatial and time derivatives are differently affected.</text>
<text><location><page_17><loc_19><loc_68><loc_90><loc_82></location>There are also potentially wider implications for this study. For example, it has been shown that taking the deformed constraint algebra to the flat-space limit gives a deformed version of the Poincar'e algebra, which leads to a modified dispersion relation [46,59]. This might indicate something such as a variable speed of light or an observer-independent energy scale. In this respect it is similar to the phenomenological models of deformed special relativity [51] and rainbow gravity [52].</text>
<text><location><page_17><loc_19><loc_49><loc_90><loc_66></location>The deformation might indicate a non-commutative character to geometry [60,61] although apparently not a multifractional one [62]. It might represent a variable dimensionality of space-time and a running of the spectral dimension [63]. The deformation function may change sign, as suggested in the motivating studies [41]. This makes the hyperbolic equations become elliptical and implies a phase transition from classical Lorentzian spacetime to an effectively Euclidean quantum regime [22,64]. It therefore may be a potential mechanism for the Hartle-Hawking no-boundary proposal [65].</text>
<section_header_level_1><location><page_18><loc_19><loc_79><loc_33><loc_81></location>Chapter 2</section_header_level_1>
<section_header_level_1><location><page_18><loc_19><loc_70><loc_44><loc_73></location>Methodology</section_header_level_1>
<text><location><page_18><loc_19><loc_60><loc_90><loc_64></location>In this thesis I am primarily building on preceding work done by others [21, 49, 50] and elaborating on previously published material [55,56].</text>
<section_header_level_1><location><page_18><loc_19><loc_54><loc_55><loc_55></location>2.1 Space-time decomposition</section_header_level_1>
<text><location><page_18><loc_19><loc_33><loc_90><loc_50></location>Quantum mechanics naturally works in the canonical or Hamiltonian framework. The canonical framework takes variables defined at a certain time and evolves them through time. That evolution defines a canonical momentum for each variable. To make general relativity more amenable to quantum mechanics, one must likewise make a distinction between the time dimension and the spatial dimensions. So I foliate the bulk space-time manifold M into a stack of labelled spatial hypersurfaces, Σ t . I assume it is globally hyperbolic, so topologically M = Σ × R [10, 18, 19].</text>
<text><location><page_18><loc_19><loc_20><loc_90><loc_31></location>A future-pointing vector normal to the spatial hypersurface Σ t is defined such that g ab n a n b = -1 . The spatial slices Σ t are themselves Riemannian manifolds with an induced metric q ab = g ab + n a n b , such that q ab n b = 0 . The spatial metric has an inverse defined as q ab = g ab + n a n b , so that q b a := q ac q bc = δ b a + n a n b acts as a spatial 1 projection tensor.</text>
<text><location><page_18><loc_19><loc_14><loc_90><loc_17></location>If the spatial foliation, and therefore the spatial coordinates, are arbitrary, the timeevolution vector field t a cannot be uniquely determined by the time function t . One can</text>
<text><location><page_19><loc_19><loc_84><loc_90><loc_88></location>project it into its normal and spatial components, defining the lapse function N = -n a t a , and the spatial shift vector N a = q a b t b . Therefore, t a = Nn a + N a .</text>
<text><location><page_19><loc_19><loc_76><loc_90><loc_82></location>Since the coordinates are arbitrary, it is convenient to take the normal to the spatial surface as the time-like direction for defining velocities rather than using the time-vector itself. Therefore,</text>
<formula><location><page_19><loc_27><loc_70><loc_90><loc_73></location>v ab := L n q ab = 1 N ( ˙ q ab -2 ∇ ( a N b ) ) , ν I := L n ψ I = 1 N ( ˙ ψ I -N a ∂ a ψ I ) , (2.1)</formula>
<text><location><page_19><loc_19><loc_63><loc_90><loc_67></location>where ˙ X := L t X , and the extrinsic curvature of the spatial slice is related to this by K ab = 1 2 v ab .</text>
<section_header_level_1><location><page_19><loc_19><loc_57><loc_49><loc_59></location>2.2 Canonical formalism</section_header_level_1>
<text><location><page_19><loc_19><loc_50><loc_90><loc_54></location>I take a general first-order action for a model with dynamical fields ψ I , and non-dynamical fields λ I ,</text>
<formula><location><page_19><loc_44><loc_47><loc_90><loc_49></location>S = ∫ d 4 xL ( ψ I , ∂ a ψ I , λ I ) , (2.2)</formula>
<text><location><page_19><loc_19><loc_40><loc_90><loc_45></location>where ∂ a ψ I := ∂ψ I ∂x a =: ψ I,a . Varying the action with respect to each field, fixing the variation at the boundaries, and imposing the principle of least action,</text>
<formula><location><page_19><loc_47><loc_35><loc_90><loc_38></location>δS δψ I ≈ 0 , δS δλ I ≈ 0 , (2.3)</formula>
<text><location><page_19><loc_19><loc_31><loc_56><loc_32></location>gives the Euler-Lagrange equations of motion,</text>
<formula><location><page_19><loc_44><loc_25><loc_90><loc_28></location>0 ≈ ∂L ∂ψ I -∂ a ( ∂L ∂ ( ∂ a ψ I ) ) , (2.4a)</formula>
<formula><location><page_19><loc_51><loc_22><loc_90><loc_25></location>0 ≈ ∂L ∂λ I . (2.4b)</formula>
<text><location><page_19><loc_19><loc_10><loc_90><loc_19></location>The approximation symbol is used to indicate something that is true in the dynamical regime, or 'on-shell', rather than something that is true kinematically, or 'off-shell'. The non-dynamical fields λ I can be seen to produce constraints on the system given by (2.4b), they are also known as Lagrange multipliers.</text>
<text><location><page_20><loc_19><loc_84><loc_90><loc_88></location>Making a space-time decomposition as in section 2.1, one can define the canonical momenta of each field,</text>
<formula><location><page_20><loc_39><loc_80><loc_90><loc_84></location>π I ψ := δS δ ˙ ψ I = ∂L ∂ ˙ ψ I , π I λ := δS δ ˙ λ I = ∂L ∂ ˙ λ I . (2.5)</formula>
<text><location><page_20><loc_19><loc_72><loc_90><loc_79></location>Since L does not depend on ˙ λ I , one can see that π I λ ≈ 0 are primary constraints on the system. If the matrix ∂ 2 L ∂ ˙ ψ I ∂ ˙ ψ J is non-degenerate, then the above equation can be inverted to find ˙ ψ I = ˙ ψ I ( ψ J , π J ψ , λ J ) , and so one can replace the time derivatives in the action.</text>
<text><location><page_20><loc_19><loc_70><loc_83><loc_71></location>Making a Legendre transform to find the Hamiltonian associated to this action,</text>
<formula><location><page_20><loc_38><loc_64><loc_90><loc_68></location>H = ∫ d t d 3 x ( ∑ I ˙ ψ I π I ψ + ∑ I µ λ I π I λ ) -S, (2.6)</formula>
<text><location><page_20><loc_19><loc_58><loc_90><loc_62></location>where µ λ I is a coefficient which acts like a Lagrange multiplier. The Poisson bracket of a quantity with the Hamiltonian equals the time derivative of that quantity on-shell,</text>
<formula><location><page_20><loc_22><loc_51><loc_90><loc_55></location>˙ F ≈ { F, H } = ∫ d 3 x { ∑ I δF δψ I ( x ) δH δπ I ψ ( x ) + ∑ I δF δλ I ( x ) δH δπ I λ ( x ) } -( F ↔ H ) , (2.7)</formula>
<text><location><page_20><loc_19><loc_40><loc_90><loc_49></location>and if F ≈ 0 should be true at all times, then ˙ F ≈ 0 must also be true [20]. Therefore, evaluating { π I λ , H } either gives back a function of the primary constraints π J λ , produces a secondary constraint φ I ( ψ J , π J ψ , λ J ) ≈ 0 , or gives a specific form for the coefficients of the constraints µ I . The equations (2.4b) appear here as secondary constraints.</text>
<text><location><page_20><loc_19><loc_31><loc_90><loc_38></location>I repeat the process with { φ I , H } until I have found all the constraints on the system, at which point there is no need to differentiate between primary and secondary constraints, and I have found the generalised Hamiltonian,</text>
<formula><location><page_20><loc_35><loc_25><loc_90><loc_29></location>H glyph[star] = ∫ d t d 3 x ( ∑ I ˙ ψ I π I ψ + ∑ I µ I φ I ) -S ≈ H. (2.8)</formula>
<text><location><page_20><loc_19><loc_22><loc_62><loc_23></location>The set of constraints has a Poisson bracket structure</text>
<formula><location><page_20><loc_39><loc_17><loc_90><loc_19></location>{ φ I , φ J } = f K IJ φ K + α IJ , α IJ / ∈ { φ K } , (2.9)</formula>
<text><location><page_20><loc_27><loc_13><loc_27><loc_14></location>glyph[negationslash]</text>
<text><location><page_20><loc_19><loc_12><loc_90><loc_14></location>and if α IJ = 0 then some of φ I are what are called 'second-class' constraints, in which case</text>
<text><location><page_21><loc_19><loc_74><loc_90><loc_88></location>some of the coefficients µ I are uniquely determined. If α IJ = 0 then all of φ I are 'firstclass', in which case the constraints not only restrict the values of the dynamical fields, but also generate gauge transformations [10,20]. This is because, in general the evolution (2.7) will depend on µ I . For an undetermined µ I to influence the mathematics but not the physical observables, a change of its value must correspond to a gauge transformation generated by the relevant first-class constraint.</text>
<text><location><page_21><loc_19><loc_63><loc_90><loc_72></location>For classical general relativity, the action does not depend on ˙ N or ˙ N a (up to boundary terms) and is only linearly dependent on N and N a . 2 As such, there are primary constraints given by π N and π N a , which generate secondary constraints known as the Hamiltonian constraint and diffeomorphism constraint respectively,</text>
<formula><location><page_21><loc_34><loc_57><loc_90><loc_60></location>C := δH δN = { H,π N } , D a := δH δN a = { H,π N a } , (2.10)</formula>
<text><location><page_21><loc_19><loc_45><loc_90><loc_55></location>which are all first-class constraints. This means that N and N a are gauge functions which do not affect the observables, and therefore the spatial slicing does not affect the dynamics. The theory is background independent and the constraints generate gauge transformations 3 ,</text>
<formula><location><page_21><loc_37><loc_43><loc_90><loc_44></location>{ F, C [ N ] } = N L n F, { F, D a [ N a ] } = L N F. (2.11)</formula>
<text><location><page_21><loc_19><loc_39><loc_88><loc_40></location>The Hamiltonian can be rewritten as a sum of the constraints up to a boundary term,</text>
<formula><location><page_21><loc_36><loc_34><loc_90><loc_36></location>H = ∫ d t d 3 x ( NC + N a D a + µ N π N + µ a N π N a ) . (2.12)</formula>
<text><location><page_21><loc_19><loc_26><loc_90><loc_30></location>Considering the Poisson bracket structure of these constraints, given by (2.9) with φ I ∈ { C, D a } , one finds that they form a Lie algebroid 4 [22],</text>
<formula><location><page_22><loc_34><loc_82><loc_90><loc_84></location>{ D a [ N a 1 ] , D b [ N b 2 ] } = D a [ L N 2 N a 1 ] , (2.13a)</formula>
<formula><location><page_22><loc_35><loc_79><loc_90><loc_81></location>{ C [ N 1 ] , D a [ N a 2 ] } = C [ L N 2 N 1 ] , (2.13b)</formula>
<formula><location><page_22><loc_36><loc_76><loc_90><loc_78></location>{ C [ N 1 ] , C [ N 2 ] } = D a [ q ab ( N 1 ∂ b N 2 -∂ b N 1 N 2 ) ] . (2.13c)</formula>
<text><location><page_22><loc_19><loc_61><loc_90><loc_73></location>where ( N 1 , N a 1 ) and ( N 2 , N a 2 ) each represent the lapse and shift of two different hypersurface transformations. As interpreted in ref. [23], (2.13a) shows that D a is the generator of spatial morphisms, (2.13b) shows that C is a scalar density of weight one (as defined in appendix B) and (2.13c) specifies the form of C such that it ensures the embeddability of the spatial slices in space-time geometry.</text>
<section_header_level_1><location><page_22><loc_19><loc_55><loc_47><loc_56></location>2.3 Choice of variables</section_header_level_1>
<text><location><page_22><loc_19><loc_45><loc_90><loc_51></location>Classical canonical general relativity can be formulated equivalently using different variables. There is geometrodynamics, which uses the spatial metric and its canonical momentum ( q ab , p cd ) , the latter of which is directly related to extrinsic curvature,</text>
<formula><location><page_22><loc_44><loc_40><loc_90><loc_42></location>p ab = ω 2 √ q ( K ab -Kq ab ) , (2.14)</formula>
<text><location><page_22><loc_19><loc_28><loc_90><loc_37></location>where q := det q ab and ω is the gravitational coupling. An alternative is connection dynamics, which uses the Ashtekar-Barbero connection and densitised triads ( A I a , E b J ) , where capital letters signify internal indices rather than coordinate indices [66, 67]. This can be related to geometrodynamics by using the equations [10],</text>
<formula><location><page_22><loc_48><loc_23><loc_90><loc_25></location>q δ IJ = q ab E a I E b J , (2.15a)</formula>
<formula><location><page_22><loc_48><loc_20><loc_90><loc_22></location>A I a = Γ I a + γ BI K I a , (2.15b)</formula>
<formula><location><page_22><loc_42><loc_16><loc_90><loc_19></location>Γ I a = 1 2 √ q q bc glyph[epsilon1] IJK E b J ∇ a ( E c K √ q ) , (2.15c)</formula>
<formula><location><page_22><loc_47><loc_13><loc_90><loc_16></location>K I a = 1 √ q δ IJ K ab E b J , (2.15d)</formula>
<text><location><page_23><loc_19><loc_84><loc_90><loc_88></location>where γ BI is the Barbero-Immirzi parameter and glyph[epsilon1] IJK is the covariant Levi-Civita tensor. The exact value of γ BI should not affect the dynamics [68].</text>
<text><location><page_23><loc_19><loc_68><loc_90><loc_82></location>The other alternative I mention here is loop dynamics, which uses holonomies of the connection and gravitational flux ( h glyph[lscript] [ A ] , F I glyph[lscript] [ E ]) . Classically, h glyph[lscript] [ A ] is given by the path-ordered exponential of the connection integrated along a curve glyph[lscript] and F I glyph[lscript] [ E ] is the flux of the densitised triad through a surface that the curve glyph[lscript] intersects. If glyph[lscript] is taken to be infinitesimal, one can easily relate loop dynamics and connection dynamics because then h glyph[lscript] = 1 + A ( ˙ glyph[lscript] ) + O ( | glyph[lscript] | 2 ) [25, p. 21].</text>
<text><location><page_23><loc_19><loc_54><loc_90><loc_66></location>When each set of variables is quantised, they are no longer equivalent, for example the value of γ BI does now affect the dynamics [46,69]. For complex γ BI , care has to be taken to make sure the classical limit is real general relativity, rather than complex general relativity. Significantly, quantising loop variables (loop quantum gravity) discretises geometry, and so glyph[lscript] cannot be taken to be infinitesimal [25, p. 105].</text>
<text><location><page_23><loc_19><loc_45><loc_90><loc_52></location>In this work, I choose to use metric variables to build a semi-classical model of gravity. This is because the comparison to other modified gravity models should be clearer, and there is no ambiguity arising from γ BI .</text>
<section_header_level_1><location><page_23><loc_19><loc_39><loc_61><loc_41></location>2.4 Higher order models of gravity</section_header_level_1>
<text><location><page_23><loc_19><loc_35><loc_82><loc_36></location>In four dimensions, the Einstein-Hilbert action for general relativity is given by</text>
<formula><location><page_23><loc_46><loc_29><loc_90><loc_32></location>S = ω 2 ∫ d 4 x √ -g (4) R. (2.16)</formula>
<text><location><page_23><loc_19><loc_17><loc_90><loc_26></location>where ω = 1 / 8 π · G is the gravitational coupling and g := det g ab . The integrand is the four dimensional Ricci curvature scalar which is contracted from the Riemann curvature tensor (4) R := (4) R a bac g bc . For any Riemannian manifold, this is defined using the commutator of two covariant derivatives of an arbitrary vector,</text>
<formula><location><page_23><loc_42><loc_13><loc_90><loc_14></location>∇ c ∇ d A a -∇ d ∇ c A a = R a bcd A b . (2.17)</formula>
<text><location><page_24><loc_19><loc_63><loc_90><loc_88></location>There are many reasons why theoretical physicists seek to find models of gravity which go beyond the Einstein-Hilbert action. For instance, mysteries known as dark matter [70] and dark energy [71] may originate with gravity behaving differently than expected rather than being due to unknown dark substances [72]. The indication that there was a period of inflationary expansion in the early universe has also caused a search for relevant models [73, 74]. Moreover, the classical equations of gravity predict their own demise in extraordinary circumstances such as in a black hole or at a hot big bang. A theory of gravity that solves these problems to which classical general relativity is the low-curvature, largescale limit may have a semi-classical regime where corrections appear, at leading orders, similar to these theories of modified gravity [73,75,76].</text>
<text><location><page_24><loc_19><loc_57><loc_90><loc_61></location>One way of attempting to find alternative models of gravity is by constructing actions from higher order combinations of the Riemann tensor, so you instead have the general action</text>
<formula><location><page_24><loc_42><loc_52><loc_90><loc_55></location>S = ω 2 ∫ d 4 x √ -gF ( (4) R a bcd ) . (2.18)</formula>
<text><location><page_24><loc_19><loc_43><loc_90><loc_49></location>To bring this in line with the space-time split, I replace the determinant, g = -N 2 q . The Riemann tensor must be decomposed by projecting it along its normal and tangential components relative to the spatial slice,</text>
<formula><location><page_24><loc_32><loc_37><loc_90><loc_40></location>q e a q f b q g c q h d (4) R efgh = 1 4 v ac v bd -1 4 v ad v bc + (3) R abcd , (2.19a)</formula>
<formula><location><page_24><loc_32><loc_34><loc_90><loc_37></location>q e a q f b q g c n h (4) R efgh = 1 2 ∇ a v bc -1 2 ∇ b v ac , (2.19b)</formula>
<formula><location><page_24><loc_32><loc_31><loc_90><loc_34></location>q e a n f q g b n h (4) R efgh = -1 2 L n v ab + 1 4 q bc v ac v bd + 1 N ∇ ( a ∇ b ) N. (2.19c)</formula>
<text><location><page_24><loc_19><loc_17><loc_90><loc_28></location>These identities are respectively known as the Gauss equation, the Codazzi equation, and the Ricci equation [10, 77]. All other projections vanish due to the tensor's antisymmetry. As can be seen from (2.19c), there are second order time derivatives included in the Riemann tensor. Including second order time derivatives in an action is problematic because it may introduce the Ostrogradsky instability [78]. To demonstrate what this</text>
<text><location><page_25><loc_19><loc_87><loc_56><loc_88></location>means, I take a one dimensional model action,</text>
<formula><location><page_25><loc_47><loc_82><loc_90><loc_84></location>S = ∫ d tL ( q, ˙ q, q ) , (2.20)</formula>
<text><location><page_25><loc_19><loc_75><loc_90><loc_79></location>I cannot find the associated Hamiltonian when there are time derivatives higher than second order, and the Euler-Lagrange equations may involve fourth order time derivatives,</text>
<formula><location><page_25><loc_41><loc_69><loc_90><loc_72></location>0 ≈ ∂L ∂q -d d t ( ∂L ∂ ˙ q ) + d 2 d t 2 ( ∂L ∂ q ) , (2.21)</formula>
<text><location><page_25><loc_25><loc_65><loc_25><loc_66></location>glyph[negationslash]</text>
<text><location><page_25><loc_19><loc_62><loc_90><loc_67></location>if ∂ 2 L ∂ q 2 = 0 . So I must introduce an additional variable to absorb the higher order terms. The Ostrogradsky method [79] is to replace ˙ q with an independent variable v .</text>
<formula><location><page_25><loc_41><loc_57><loc_90><loc_59></location>S = ∫ d t { L ( q, v, ˙ v ) + ψ ( v -˙ q ) } , (2.22)</formula>
<text><location><page_25><loc_19><loc_48><loc_90><loc_54></location>however, I instead do this slightly differently for reasons which will be apparent later. Following the method used in ref. [77, 80] and using variables like in ref. [81], I instead replace q with an auxiliary variable a ,</text>
<formula><location><page_25><loc_41><loc_43><loc_90><loc_45></location>S = ∫ d t { L ( q, ˙ q, a ) + ψ (q -a ) } , (2.23)</formula>
<text><location><page_25><loc_19><loc_36><loc_90><loc_39></location>and integrate by parts to move the second order time derivative to the Lagrange multiplier ψ , promoting it to a dynamical variable,</text>
<formula><location><page_25><loc_41><loc_31><loc_90><loc_33></location>S = ∫ d t { L ( q, ˙ q, a ) -˙ q ˙ ψ -ψa } , (2.24)</formula>
<text><location><page_25><loc_19><loc_26><loc_48><loc_27></location>which gives the canonical momenta,</text>
<formula><location><page_25><loc_33><loc_20><loc_90><loc_24></location>p := δS δ ˙ q = ∂L ∂ ˙ q -˙ ψ, π := δS δ ˙ ψ = -˙ q, π a := δS δ ˙ a = 0 . (2.25)</formula>
<text><location><page_26><loc_19><loc_84><loc_90><loc_88></location>So I can invert these definitions to find the velocities in terms of the momenta. Then make a Legendre transform to find the Hamiltonian,</text>
<formula><location><page_26><loc_37><loc_76><loc_90><loc_82></location>H = ∫ d t ( ˙ qp + ˙ ψπ + µ a π a ) -S, = ∫ d t {-pπ + µ a π a -L ( q, π, a ) + ψa } , (2.26)</formula>
<text><location><page_26><loc_42><loc_64><loc_42><loc_66></location>glyph[negationslash]</text>
<text><location><page_26><loc_19><loc_62><loc_90><loc_73></location>where µ a is a Lagrange multiplier. The equation of motion for a produces the secondary constraint φ = ∂L ∂a -ψ ≈ 0 . Finding { φ, H } ≈ 0 produces an equation for µ a and therefore φ is a second-class constraint and a is uniquely determined. The constraint can be solved for a ( q, ψ, π ) as long as ∂ 2 L ∂a 2 = 0 and this can be substituted into the Hamiltonian without incident, in which case I find,</text>
<formula><location><page_26><loc_37><loc_57><loc_90><loc_59></location>H = ∫ d t {-pπ -L ( q, ψ, π ) + ψa ( q, ψ, π ) } (2.27)</formula>
<text><location><page_26><loc_19><loc_45><loc_90><loc_54></location>which is only linear in p . This means that the energy is unbounded from below and above, and so the model may be unstable [79]. For specific models of this kind rather than this simple example, I can find a well behaved Hamiltonian if there are sufficient restrictions on the values that ψ can take [81].</text>
<text><location><page_26><loc_19><loc_39><loc_90><loc_42></location>If I do have a well behaved Hamiltonian, it is clear that the higher order derivative action L ( q, ˙ q, q ) contains an additional degree of freedom, which has been absorbed by ψ .</text>
<section_header_level_1><location><page_26><loc_19><loc_33><loc_76><loc_35></location>2.4.1 Non-minimally coupled scalar from F ( (4) R ) gravity</section_header_level_1>
<text><location><page_26><loc_19><loc_16><loc_90><loc_31></location>In ref. [77, 80], it was shown how to find the Hamiltonian form of any F ( (4) R a bcd ) action. The Riemann tensor is split into its normal and tangential components (2.19), and auxiliary tensors are introduced as in (2.23). The tensor which is the Lagrange multiplier of (2.19c) becomes dynamical by integrating by parts. This turns the action into being first order in time derivatives, and therefore one can find the associated Hamiltonian. This field contains the additional degrees of freedom allowed by the higher order derivatives.</text>
<text><location><page_26><loc_19><loc_10><loc_90><loc_14></location>To include tensor contractions such as (4) R ab (4) R ab and (4) R abcd (4) R abcd produces several additional degrees of freedom, and requires considering spatial derivatives of velocity or</text>
<text><location><page_27><loc_19><loc_82><loc_90><loc_88></location>momenta because of (2.19b). For the sake of simplicity, in this chapter and throughout the thesis, I will only consider models which are comparable with F ( (4) R ) . So the action is given by,</text>
<formula><location><page_27><loc_37><loc_78><loc_90><loc_81></location>S = ω 2 ∫ d t d 3 xN √ q { F ( ρ ) + ψ ( (4) R -ρ )} . (2.28)</formula>
<text><location><page_27><loc_19><loc_75><loc_52><loc_76></location>I decompose the Ricci scalar using (2.19),</text>
<formula><location><page_27><loc_31><loc_70><loc_90><loc_72></location>(4) R = R + q ab L n v ab + 1 4 v 2 -3 2 v ab v ab -2 N ∆ N, R = (3) R, (2.29)</formula>
<text><location><page_27><loc_19><loc_63><loc_90><loc_67></location>where ∆ := q ab ∇ a ∇ b . Then integrate the action (2.28) by parts to move the second order time derivative to ψ ,</text>
<formula><location><page_27><loc_29><loc_58><loc_90><loc_61></location>S = ω 2 ∫ d t d 3 xN √ q { F ( ρ ) + ψ ( R -K2 N ∆ N -ρ ) -νv } , (2.30)</formula>
<text><location><page_27><loc_19><loc_51><loc_90><loc_55></location>where q := det q ab , ν := L n ψ , and K := ( v 2 -v ab v ab ) / 4 is the standard extrinsic curvature contraction. The conjugate momenta are,</text>
<formula><location><page_27><loc_37><loc_39><loc_90><loc_49></location>p ab := δS δ ˙ q ab = 1 N δS δv ab = ω 2 √ q { ψ 2 v cd ( Q abcd -q ab q cd ) -νq ab } , (2.31a) π := δS δ ˙ ψ = 1 N δS δν = -ω 2 √ q v, (2.31b)</formula>
<text><location><page_27><loc_19><loc_35><loc_72><loc_37></location>where Q abcd := q a ( c q d ) b for convenience. I can invert these to find,</text>
<formula><location><page_27><loc_34><loc_30><loc_90><loc_33></location>v ab = 2 ω √ q ( 2 ψ p T ab -q ab π ) , ν = 2 3 ω √ q ( ψπ -p ) . (2.32)</formula>
<text><location><page_27><loc_19><loc_26><loc_79><loc_27></location>where I have separated the trace and the traceless parts of the momentum,</text>
<formula><location><page_27><loc_48><loc_20><loc_90><loc_23></location>p ab = p ab T + 1 3 q ab p. (2.33)</formula>
<text><location><page_28><loc_19><loc_87><loc_73><loc_88></location>I Legendre transform the action to find the associated Hamiltonian,</text>
<formula><location><page_28><loc_32><loc_79><loc_90><loc_85></location>H = ∫ d 3 x ( ˙ q ab p ab + ˙ ψπ + µ ρ π ρ + µ N π N + µ N a π a N ) -S, = ∫ d 3 x ( NC + N a D a + µ ρ π ρ + µ N π N + µ N a π a N ) , (2.34)</formula>
<text><location><page_28><loc_19><loc_75><loc_57><loc_76></location>with the corresponding Hamiltonian constraint,</text>
<formula><location><page_28><loc_22><loc_69><loc_90><loc_73></location>C := δH δN = 2 ω √ q ( 1 ψ P 1 3 pπ + ψ 6 π 2 ) + ω √ q 2 ( ψρ -ψR -F ( ρ ) + 2∆ ψ ) , (2.35)</formula>
<text><location><page_28><loc_19><loc_65><loc_71><loc_67></location>where P := p T ab p ab T . Finding { π ρ , H } gives a secondary constraint,</text>
<formula><location><page_28><loc_42><loc_60><loc_90><loc_63></location>φ ρ = ω 2 N √ q ( ψ -F ' ( ρ ) ) ≈ 0 , (2.36)</formula>
<text><location><page_28><loc_70><loc_56><loc_70><loc_57></location>glyph[negationslash]</text>
<text><location><page_28><loc_19><loc_51><loc_90><loc_58></location>which is second-class. It can be solved to find ρ ( ψ ) as long as F '' = 0 , in which case we can find the Hamiltonian constraint in terms of only the metric and the scalar field ψ . This leaves me with a term depending on ψ which acts like a scalar field potential,</text>
<formula><location><page_28><loc_23><loc_45><loc_90><loc_48></location>U geo ( ψ ) = ω 2 ( ψρ ( ψ ) -F ( ρ ( ψ ) ) ) = ω 2 { ψ ( F ' ) -1 ( ψ ) -F ( ( F ' ) -1 ( ψ ) )} , (2.37)</formula>
<text><location><page_28><loc_19><loc_34><loc_90><loc_43></location>which I call the geometric scalar potential. As I will further elaborate in section 3, this scalar-tensor model I have derived from an F ( (4) R ) model of gravitation is equivalent to letting the gravitational coupling in the Einstein-Hilbert action become dynamical, ω → ωψ .</text>
<text><location><page_28><loc_19><loc_23><loc_90><loc_32></location>So models of gravity that have an action which is an arbitrary function of the spacetime curvature scalar (4) R can be converted into a scalar-tensor theory in the Hamiltonian formalism. The structure of general covariance underlying general relativity should be preserved in these models, though they do contain an additional degree of freedom.</text>
<section_header_level_1><location><page_29><loc_19><loc_87><loc_58><loc_88></location>2.5 Deformed constraint algebra</section_header_level_1>
<text><location><page_29><loc_19><loc_59><loc_90><loc_83></location>As previously mentioned in section 1.2, loop quantum cosmology predicts that the symmetries of general relativity should be deformed in a specific way in the semi-classical limit [36-42]. This appears from incorporating loop variables in a mini-superspace model, but specifying that all anomalies α IJ in (2.9) vanish while allowing counter-terms to deform the classical form of the algebra. This ensures that the constraints are first-class, retaining the gauge invariance of the theory and of the arbitrariness of the lapse and shift. If anomalous terms were to appear in the constraint algebra, then the gauge invariance would be broken and the constraints could only be solved at all times for specific N or N a . This means that there would a privileged frame of reference, and therefore no general covariance.</text>
<text><location><page_29><loc_19><loc_53><loc_90><loc_57></location>In the referenced studies, it is strongly indicated that the bracket of two Hamiltonian constraints (2.13c) is deformed by a phase space function β ,</text>
<formula><location><page_29><loc_36><loc_48><loc_90><loc_50></location>{ C [ N 1 ] , C [ N 2 ] } = D a [ βq ab ( N 1 ∂ b N 2 -∂ b N 1 N 2 )] . (2.38)</formula>
<text><location><page_29><loc_19><loc_33><loc_90><loc_45></location>This has not been shown generally, but has been shown for several models independently. There are no anomalies in the constraint algebra, so a form of general covariance is preserved. However, it may be that the interpretation of a spatial manifold evolving with time being equivalent to a foliation of space-time (also known as 'embeddability') is no longer valid.</text>
<text><location><page_29><loc_30><loc_17><loc_30><loc_18></location>glyph[negationslash]</text>
<text><location><page_29><loc_19><loc_14><loc_90><loc_31></location>These deformations only appear to be necessary for models when the Barbero-Immirzi parameter γ BI is real. For self-dual models, when γ BI = ± i , this deformation does not appear necessary [45]. However, self-dual variables are not desirable in other ways. They do not seem to resolve curvature singularities as hoped, and obtaining the correct classical limit is non-trivial [46]. Because of this, even though I use metric variables in this work, considering β = 1 and ensuring the correct classical limit means there should be relevance to the models of loop quantum cosmology with real γ BI .</text>
<section_header_level_1><location><page_30><loc_19><loc_87><loc_70><loc_88></location>2.6 Derivation of the distribution equation</section_header_level_1>
<text><location><page_30><loc_19><loc_74><loc_90><loc_83></location>From the constraint algebra, I am able to find the specific form of the Hamiltonian constraint C for a given deformation β . The diffeomorphism constraint D a is not affected when the deformation is a weightless scalar 5 and so is completely determined as shown in appendix B. With D a and β as inputs, I can find C by manipulating (2.38).</text>
<text><location><page_30><loc_19><loc_68><loc_90><loc_72></location>Firstly, I must find the unsmeared form of the deformed algebra. At this point I do not need to specify my canonical variables, and leave them merely as ( q I , p I ) ,</text>
<formula><location><page_30><loc_26><loc_63><loc_90><loc_65></location>0 = { C [ N 1 ] , C [ N 2 ] } -D a [ βq ab ( N 1 ∂ b N 2 -∂ b N 1 N 2 )] , (2.39a)</formula>
<formula><location><page_30><loc_27><loc_59><loc_90><loc_63></location>= ∫ d 3 z { ∑ I δC [ N 1 ] δq I ( z ) δC [ N 2 ] δp I ( z ) -( D a βN 1 ∂ a N 2 ) z } -( N 1 ↔ N 2 ) . (2.39b)</formula>
<text><location><page_30><loc_19><loc_55><loc_71><loc_56></location>Take the functional derivatives with respect to N 1 ( x ) and N 2 ( y ) ,</text>
<formula><location><page_30><loc_31><loc_49><loc_90><loc_53></location>0 = ∑ I ∫ d 3 z δC ( x ) δq I ( z ) δC ( y ) δp I ( z ) -( D a β∂ a ) x δ ( x, y ) -( x ↔ y ) , (2.40)</formula>
<text><location><page_30><loc_19><loc_43><loc_90><loc_47></location>where δ ( x, y ) is the three dimensional Dirac delta distribution 6 . If I note that I will only consider constraints without spatial derivatives on momenta, this simplifies,</text>
<formula><location><page_30><loc_34><loc_37><loc_90><loc_40></location>0 = ∑ I δC ( x ) δq I ( y ) ∂C ∂p I ∣ ∣ ∣ ∣ y -( βD a ∂ a ) x δ ( x, y ) -( x ↔ y ) . (2.41)</formula>
<text><location><page_30><loc_19><loc_30><loc_90><loc_34></location>For when I wish to derive the action instead of the constraint, I can transform the equation by noting that,</text>
<formula><location><page_30><loc_41><loc_26><loc_90><loc_29></location>δC [ N ] δq I = -δL [ N ] δq I , Nv I = δC [ N ] δp I , (2.42)</formula>
<text><location><page_30><loc_19><loc_21><loc_90><loc_25></location>where v I := L n q I and the Lagrangian is here defined such that S = ∫ d t d 3 xNL . I substitute these into (2.39b), then take the functional derivatives to remove N 1 and N 2 ,</text>
<formula><location><page_30><loc_34><loc_15><loc_90><loc_18></location>0 = ∑ I δL ( x ) δq I ( y ) v I ( y ) + ( βD a ∂ a ) x δ ( x, y ) -( x ↔ y ) . (2.43)</formula>
<text><location><page_31><loc_19><loc_87><loc_90><loc_88></location>To find a useful form for this, I need to use a specific form for the diffeomorphism constraint.</text>
<text><location><page_31><loc_19><loc_84><loc_67><loc_85></location>Because it depends on momenta, I must replace them using,</text>
<formula><location><page_31><loc_46><loc_78><loc_90><loc_82></location>p I := δS δ ˙ q I = 1 N δL [ N ] δv I , (2.44)</formula>
<text><location><page_31><loc_19><loc_72><loc_90><loc_76></location>and, as before, if I note that I will only consider actions without spatial derivatives of momenta this simplifies to</text>
<formula><location><page_31><loc_51><loc_69><loc_90><loc_72></location>p I = ∂L ∂v I . (2.45)</formula>
<text><location><page_31><loc_19><loc_61><loc_90><loc_67></location>Therefore, substituting the diffeomorphism constraint found in appendix B and momenta (2.45) into (2.43), I find the distribution equation which can be used for restricting the form of the deformed action.</text>
<text><location><page_31><loc_19><loc_55><loc_90><loc_58></location>So, the key equations I use as a basis for finding the action or constraint for deformed general relativity are (2.41) and (2.43).</text>
<section_header_level_1><location><page_31><loc_19><loc_48><loc_77><loc_50></location>2.7 Order of the deformed action and constraint</section_header_level_1>
<text><location><page_31><loc_19><loc_41><loc_90><loc_45></location>I can determine the relationship between the order of the deformation function and the order of the associated constraint (or action) by comparing orders of momenta (or velocity).</text>
<section_header_level_1><location><page_31><loc_19><loc_36><loc_44><loc_37></location>2.7.1 Hamiltonian route</section_header_level_1>
<text><location><page_31><loc_19><loc_32><loc_80><loc_33></location>As an example, take the distribution equation (2.41) with only a scalar field,</text>
<formula><location><page_31><loc_35><loc_26><loc_90><loc_29></location>0 = δC ( x ) δψ ( y ) ∂C ∂π ∣ ∣ ∣ ∣ y -( βπ∂ a ψ∂ a ) x δ ( x, y ) -( x ↔ y ) , (2.46)</formula>
<text><location><page_31><loc_19><loc_14><loc_90><loc_23></location>where I have used the diffeomorphism constraint (B.6). I take a simplified model with two spatial derivatives represented by ∆ , only taking even orders of derivatives because of assuming spatial parity. I take the distribution equation (2.46) and put it into schematic form,</text>
<formula><location><page_31><loc_47><loc_11><loc_90><loc_14></location>0 = ∂C ∂ ∆ ∂C ∂π -β π. (2.47)</formula>
<text><location><page_32><loc_19><loc_82><loc_90><loc_88></location>so that I can consider orders of π in a way analogous to dimensional analysis. This equation must be satisfied independently at each order of momenta, so I isolate the coefficient of π n ,</text>
<formula><location><page_32><loc_40><loc_78><loc_90><loc_82></location>0 = n C ∑ m =1 m ∂C ( n -m +1) ∂ ∆ C ( m ) -β ( n -1) , (2.48)</formula>
<text><location><page_32><loc_19><loc_75><loc_63><loc_76></location>where I have expanded the constraint and deformation,</text>
<formula><location><page_32><loc_40><loc_69><loc_90><loc_73></location>C = n C ∑ m =0 C ( m ) π m , β = n β ∑ m =0 β ( m ) π m . (2.49)</formula>
<text><location><page_32><loc_19><loc_50><loc_90><loc_67></location>The highest order contribution to (2.48) comes when m = n C and n -m +1 = n C , in which case n = 2 n C -1 . This is the highest order at which β won't automatically be constrained to vanish, so I find its highest order of momenta to be n β = 2 n C -2 . However, this result does not take into account the fact that the combined order of momenta and spatial derivatives may be restricted. If this is the case (as is found in chapter 5), then the highest order contribution to the (2.48) will be when n -m +1 = n C -2 , in which case I find the relation</text>
<formula><location><page_32><loc_49><loc_47><loc_90><loc_49></location>2 n C -n β = 4 . (2.50)</formula>
<text><location><page_32><loc_19><loc_33><loc_90><loc_45></location>I see that a deformed second order constraint only requires considering a zeroth order deformation as I do in chapter 3, but a fourth order constraint requires considering a fourth order deformation. I consider the constraint to general order in chapter 5. Note that this relation suggests there are higher order deformations which allow for constraints given by finite order polynomials.</text>
<section_header_level_1><location><page_32><loc_19><loc_28><loc_43><loc_29></location>2.7.2 Lagrangian route</section_header_level_1>
<text><location><page_32><loc_19><loc_24><loc_71><loc_25></location>Consider the distribution equation (2.43) with only a scalar field,</text>
<formula><location><page_32><loc_34><loc_18><loc_90><loc_21></location>0 = δL ( x ) δψ ( y ) ν ( y ) + ( β ∂L ∂ν ∂ a ψ∂ a ) x δ ( x, y ) -( x ↔ y ) , (2.51)</formula>
<text><location><page_32><loc_19><loc_14><loc_90><loc_15></location>where I have used the diffeomorphism constraint (B.6) and the momentum definition (2.45).</text>
<text><location><page_32><loc_19><loc_11><loc_90><loc_12></location>Let me consider a simplified model to match the derivative orders for the deformation and</text>
<text><location><page_33><loc_19><loc_79><loc_90><loc_88></location>the derivative orders for the Lagrangian in a way analogous to dimensional analysis. First order time derivatives are given by ν and two orders of spatial derivatives are given by ∆ . I can collect terms in the distribution equation of the same order of time derivatives as they are linearly independent. Schematically, the distribution equation is given by,</text>
<formula><location><page_33><loc_48><loc_73><loc_90><loc_76></location>0 = ∂L ∂ ∆ ν + ∂L ∂ν β, (2.52)</formula>
<text><location><page_33><loc_19><loc_70><loc_70><loc_71></location>and expanding the Lagrangian and deformation in powers of ν ,</text>
<formula><location><page_33><loc_40><loc_63><loc_90><loc_68></location>L = n L ∑ m =0 L ( m ) ν m , β = n β ∑ m =0 β ( m ) ν m , (2.53)</formula>
<text><location><page_33><loc_19><loc_60><loc_48><loc_61></location>the coefficient of ν n is then given by,</text>
<formula><location><page_33><loc_36><loc_54><loc_90><loc_58></location>0 = ∂L ( n -1) ∂ ∆ + n β ∑ m =0 ( n -m +1) L ( n -m +1) β ( m ) . (2.54)</formula>
<text><location><page_33><loc_19><loc_48><loc_90><loc_52></location>I can relabel and rearrange to find a schematic solution for the highest order of L appearing here,</text>
<formula><location><page_33><loc_34><loc_44><loc_90><loc_48></location>L ( n ) = -1 nβ (0) { ∂L ( n -2) ∂ ∆ + n β ∑ m =1 ( n -m ) β ( m ) L ( n -m ) } . (2.55)</formula>
<text><location><page_33><loc_19><loc_31><loc_90><loc_43></location>I can see that if n β > 0 , then this equation is recursive and n L →∞ because there is no natural cut-off, suggesting that L is required to be non-polynomial. If I wish to truncate the action at some order, then it must be treated as an perturbative approximation. I consider a perturbative fourth order action in chapter 4, and the completely general action in chapter 6.</text>
<section_header_level_1><location><page_33><loc_19><loc_25><loc_38><loc_26></location>2.8 Cosmology</section_header_level_1>
<text><location><page_33><loc_19><loc_18><loc_90><loc_21></location>Since the main motivations for this study centre around cosmological implications of the deformed constraint algebra, I need to lay out how I find the cosmological dynamics of a</text>
<text><location><page_34><loc_19><loc_84><loc_90><loc_88></location>model. I restrict to an isotropic and homogeneous space, using the Friedmann-Lema ˆ itreRobertson-Walker metric (FLRW),</text>
<formula><location><page_34><loc_37><loc_79><loc_90><loc_81></location>q ab = a 2 ( t )Σ ab , a = (det q ab ) 1 / 6 N a = 0 , (2.56)</formula>
<text><location><page_34><loc_19><loc_70><loc_90><loc_76></location>where Σ ab is time-independent and describes a three dimensional spatial slice with constant curvature k . When space is flat, k = 0 , this is given by Σ ab = δ ab . The normal derivative of the spatial metric is given by,</text>
<formula><location><page_34><loc_38><loc_64><loc_90><loc_67></location>v ab = 2 N a ˙ a Σ ab , ∴ K = 6˙ a 2 a 2 N 2 =: 6 N 2 H 2 , (2.57)</formula>
<text><location><page_34><loc_19><loc_60><loc_84><loc_61></location>where H is the Hubble expansion rate, and the Ricci curvature scalar is given by,</text>
<formula><location><page_34><loc_51><loc_55><loc_90><loc_58></location>R = 6 k a 2 . (2.58)</formula>
<text><location><page_34><loc_19><loc_51><loc_74><loc_52></location>When using canonical coordinates, the metric momentum is given by</text>
<formula><location><page_34><loc_42><loc_46><loc_90><loc_48></location>p ab = ¯ p Σ ab , ¯ p = ( det p ab ) 1 / 3 , (2.59)</formula>
<text><location><page_34><loc_19><loc_41><loc_59><loc_43></location>which changes the metric's commutation relation,</text>
<formula><location><page_34><loc_31><loc_36><loc_90><loc_39></location>{ q ab ( x ) , p cd ( y ) } = δ cd ab ( x ) δ ( x, y ) → { a ( x ) , ¯ p ( y ) } = δ ( x, y ) 6 a ( x ) , (2.60)</formula>
<text><location><page_34><loc_19><loc_27><loc_90><loc_33></location>where δ cd ab := δ c ( a δ d b ) . The spatial derivatives of matter fields vanish, ∂ a ψ I = 0 . One may couple a perfect fluid to the metric by including the energy density ρ in the constraint or the action [82],</text>
<formula><location><page_34><loc_46><loc_24><loc_90><loc_26></location>C ⊃ a 3 ρ, L ⊃ -a 3 ρ, (2.61)</formula>
<text><location><page_34><loc_19><loc_21><loc_54><loc_22></location>which must satisfy the continuity equation,</text>
<formula><location><page_34><loc_46><loc_16><loc_90><loc_17></location>˙ ρ +3 H ρ (1 + w ρ ) = 0 , (2.62)</formula>
<text><location><page_35><loc_19><loc_84><loc_90><loc_88></location>where w ρ is the perfect fluid's cosmological equation of state, the ratio of the pressure density to the energy density.</text>
<text><location><page_35><loc_19><loc_73><loc_90><loc_82></location>For investigations into whether there are implications for the hypothesised inflationary period in the very early universe, I must define what is considered to be a period of inflation. The simple definition is when the finite scale factor is both expanding and accelerating, ˙ a > 0 and a > 0 .</text>
<text><location><page_35><loc_19><loc_57><loc_90><loc_71></location>As said above, loop quantum cosmology with real variables seems to predict a big bounce instead of a big bang or crunch. In this thesis, I take the very literal interpretation of this (as found in ref. [83]) and define a bounce as a turning point for a finite scale factor, a > 0 , ˙ a = 0 and a > 0 . This definition may be usable, but it is not ideal. If a bounce does indeed happen when β < 0 , as predicted in the literature, then this is when the effective metric signature is Euclidean, when ˙ a may be a complex number.</text>
<text><location><page_35><loc_19><loc_43><loc_90><loc_54></location>Ideally, I would like to extract cosmological observables such as the primordial scalar index to find phenemenological constraints [84]. However, to calculate the power spectra of primordial fluctuations would require adapting the cosmological perturbation theory formalism to ensure it is valid for deformed covariance, something which would probably be highly non-trivial. Unfortunately, there was not enough time to investigate this.</text>
<section_header_level_1><location><page_36><loc_19><loc_79><loc_33><loc_81></location>Chapter 3</section_header_level_1>
<section_header_level_1><location><page_36><loc_19><loc_64><loc_83><loc_73></location>Second order scalar-tensor model and the classical limit</section_header_level_1>
<text><location><page_36><loc_19><loc_41><loc_90><loc_58></location>In this chapter, I derive the general form of a minimally-deformed, non-minimally-coupled scalar-tensor model which includes up to two orders in momenta or time derivatives. This allows me to demonstrate that the higher order gravity model derived in section 2.4.1 does not deform the constraint algebra or general covariance, and therefore show how the deformed models derived in subsequent chapters are distinct. For those later chapters, this minimally-deformed model provides a useful reference point. This chapter is adapted from work I previously published in ref. [55].</text>
<text><location><page_36><loc_19><loc_22><loc_90><loc_39></location>I find the form of the model by deriving restrictions on the constraint using (2.41) and then transform to find the action. It would be completely equivalent to derive the action first, because the minimally deformed case maintains a linear relationship between velocities and momenta, meaning that the transformation between the action and constraint is trivial. After finding the constraint and action, I look at some of the cosmological implications in section 3.3, especially the interesting influence of the non-minimal coupling of the scalar field.</text>
<text><location><page_36><loc_19><loc_14><loc_90><loc_20></location>I use the structure of the scalar-tensor constraint which is a parameterisation of F ( (4) R ) , (2.35), to guide the structure of my general ansatz for a spatial metric coupled to several scalar fields. I include spatially covariant terms up to second order in momenta or spatial</text>
<text><location><page_37><loc_19><loc_87><loc_58><loc_88></location>derivatives, and ignore terms linear in momenta,</text>
<formula><location><page_37><loc_35><loc_79><loc_90><loc_84></location>C = C ∅ + C ( R ) R + C ( p 2 ) abcd p ab p cd + C ( pπ I ) pπ I + C ( ψ ' I ψ ' J ) ∂ a ψ I ∂ a ψ J + C ( ψ '' I ) ∆ ψ I + C ( π I π J ) π I π J , (3.1)</formula>
<text><location><page_37><loc_19><loc_57><loc_90><loc_77></location>with summation over I and J implied. I have included C ( ψ ' I ψ ' J ) because it appears in the constraint for minimally coupled scalar fields [10, p. 62]. I aimed to define the most general ansatz for a scalar-tensor constraint containing up to two orders in derivatives which is covariant under general spatial diffeomorphisms, as well as under time reversal, and preserves spatial parity. Each coefficient is potentially a function of q and ψ I , allowing for non-minimal coupling. The spatial indices of C ( p 2 ) abcd only represent different combinations of the metric. The zeroth order term might include terms such as scalar field potentials or perfect fluids, and it behaves as a generalised potential C ∅ = √ q U ( q, ψ I ) .</text>
<section_header_level_1><location><page_37><loc_19><loc_51><loc_63><loc_53></location>3.1 Solving the distribution equation</section_header_level_1>
<text><location><page_37><loc_19><loc_41><loc_90><loc_48></location>I substitute into the distribution equation (2.41) my ansatz for a second order constraint (3.1), the diffeomorphism constraint from (B.6) and (B.11), and a zeroth order deformation β ( q, ψ ) ,</text>
<formula><location><page_37><loc_27><loc_33><loc_90><loc_39></location>0 = δC 0 ( x ) δq ab ( y ) ( 2 p cd C ( p 2 ) abcd + πq ab C ( pπ ) ) y + δC 0 ( x ) δψ ( y ) ( p C ( pπ ) +2 π C ( π 2 ) ) y + { 2 β ( ∂ b p ab +Γ a bc p bc ) -β∂ a ψπ } x ∂ a ( x ) δ ( x, y ) -( x ↔ y ) , (3.2)</formula>
<text><location><page_37><loc_19><loc_24><loc_90><loc_31></location>where C 0 is the part of the constraint without momenta. From here there are two routes to solution, by focusing on either the p ab and π components. I must do both to find all consistency conditions on the coefficients of the Hamiltonian constraint.</text>
<section_header_level_1><location><page_38><loc_19><loc_87><loc_35><loc_88></location>3.1.1 p ab sector</section_header_level_1>
<text><location><page_38><loc_19><loc_80><loc_90><loc_84></location>To proceed to the metric momentum sector, I take (3.2) and find the functional derivative with respect to p ab ( z ) ,</text>
<formula><location><page_38><loc_28><loc_72><loc_90><loc_78></location>0 = ( 2 δC 0 ( x ) δq cd ( y ) C ( p 2 ) abcd ( y ) + δC 0 ( x ) δψ ( y ) C ( pπ ) ab ( y ) ) δ ( z, y ) +2 β ( x ) { ( δ c ( a ∂ b ) ) x δ ( z, x ) + Γ c ab ( x ) δ ( z, x ) } ∂ c ( x ) δ ( x, y ) -( x ↔ y ) , (3.3)</formula>
<text><location><page_38><loc_19><loc_62><loc_90><loc_69></location>where I explicitly show the coordinate of the partial derivative as ∂ a ( y ) := ∂ ∂y a because the distinction is important when integrating by parts. I then proceed by moving derivatives away from δ ( z, y ) terms and discarding total derivatives,</text>
<formula><location><page_38><loc_27><loc_54><loc_90><loc_60></location>0 = ( 2 δC 0 ( x ) δq cd ( y ) C ( p 2 ) abcd ( y ) + δC 0 ( x ) δψ ( y ) C ( pπ ) ab ( y ) + 2 ∂ c ( y ) [ ( βδ c ( a ∂ b ) ) y δ ( y, x ) ] -2 ( β Γ c ab ∂ c ) y δ ( y, x ) ) δ ( z, y ) -( x ↔ y ) , (3.4)</formula>
<text><location><page_38><loc_19><loc_50><loc_40><loc_51></location>which can be rewritten as,</text>
<formula><location><page_38><loc_40><loc_45><loc_90><loc_46></location>0 = A ab ( x, y ) δ ( z, y ) -A ab ( y, x ) δ ( z, x ) . (3.5)</formula>
<text><location><page_38><loc_19><loc_38><loc_90><loc_41></location>Integrating over y , I find that part of the equation can be combined into a tensor dependent only on x ,</text>
<formula><location><page_38><loc_29><loc_30><loc_90><loc_35></location>0 = A ab ( x, z ) -δ ( z, x ) ∫ d 3 yA ab ( y, x ) , = A ab ( x, z ) -δ ( z, x ) A ab ( x ) , where A ab ( x ) = ∫ d 3 yA ab ( y, x ) . (3.6)</formula>
<text><location><page_38><loc_19><loc_26><loc_65><loc_27></location>Substituting in the definition of A ab ( x, z ) then relabelling,</text>
<formula><location><page_38><loc_28><loc_17><loc_90><loc_23></location>0 = 2 δC 0 ( x ) δq cd ( y ) C ( p 2 ) abcd ( y ) + δC 0 ( x ) δψ ( y ) C ( pπ ) ab ( y ) + 2 ∂ c ( y ) [ ( βδ c ( a ∂ b ) ) y δ ( y, x ) ] -2 ( β Γ c ab ∂ c ) y δ ( y, x ) -A ab ( x ) δ ( y, x ) . (3.7)</formula>
<text><location><page_39><loc_19><loc_87><loc_87><loc_88></location>Multiplying by an arbitrary test tensor θ ab ( y ) , then integrating by parts over y , I get</text>
<formula><location><page_39><loc_28><loc_74><loc_90><loc_84></location>0 = θ ab ( · · · ) ab + ∂ c θ ab { 2 C ( p 2 ) abde ∂C 0 ∂q de,c +4 ∂ d C ( p 2 ) abef ∂C 0 ∂q ef,cd + C ( pπ ) ab ∂C 0 ∂ψ ,c +2 ∂ d C ( pπ ) ab ∂C 0 ∂ψ ,cd +2 δ c ( a ∂ b ) β +2 β Γ c ab } + ∂ cd θ ab { 2 C ( p 2 ) abef ∂C 0 ∂q ef,cd + C ( pπ ) ab ∂C 0 ∂ψ ,cd +2 βδ cd ab } , (3.8)</formula>
<text><location><page_39><loc_19><loc_65><loc_90><loc_72></location>where I do not need to consider the zeroth derivative terms because they do not produce restrictions on the form of the constraint. Since θ ab is arbitrary beyond the symmetry of its indices, each unique contraction of it forms a linearly independent equation.</text>
<text><location><page_39><loc_19><loc_57><loc_90><loc_63></location>To calculate the derivatives of C 0 , I must use the decomposition of the Riemann tensor (A.6) and the second covariant derivative of the metric variation expressed in terms of partial derivatives (A.9). This gives,</text>
<formula><location><page_39><loc_23><loc_48><loc_90><loc_55></location>∂C 0 ∂ψ ,ab = C ( ψ '' ) q ab , ∂C 0 ∂ψ ,a = 2 C ( ψ ' 2 ) ∂ a ψ -C ( ψ '' ) Γ a , ∂C 0 ∂q ab,cd = C ( R ) Φ abcd , ∂C 0 ∂q ab,c = C ( ψ '' ) ( 1 2 q ab ∂ c ψ -q c ( a ∂ b ) ψ ) -C ( R ) Φ defg ( Γ c fg δ ab de +4 δ ( a ( d Γ b ) e )( f δ c g ) ) , (3.9)</formula>
<text><location><page_39><loc_19><loc_38><loc_90><loc_45></location>where Φ abcd = Q abcd -q ab q cd as found in (A.8). Note that Q abcd := q a ( c q d ) b and δ ab de := δ ( a d δ b ) e . I evaluate the coefficient of ∂ dc θ ab and find the linearly independent components,</text>
<formula><location><page_39><loc_33><loc_30><loc_90><loc_32></location>q ab ∂ 2 θ ab : 0 = -2 C ( R ) ( 2 C ( p 2 ‖ ) + C ( p 2 x ) ) + C ( ψ '' ) C ( pπ ) , (3.10a)</formula>
<formula><location><page_39><loc_35><loc_27><loc_90><loc_29></location>∂ ab θ ab : 0 = C ( R ) C ( p 2 x ) + β, (3.10b)</formula>
<text><location><page_40><loc_19><loc_84><loc_90><loc_88></location>where I have decomposed the constraint coefficient C ( p 2 ) abcd = q ab q cd C ( p 2 ‖ ) + Q abcd C ( p 2 x ) . Then evaluating similarly for ∂ c θ ab ,</text>
<formula><location><page_40><loc_24><loc_77><loc_90><loc_81></location>q ab ∂ c ψ∂ c θ ab : 0 = 2 ( C ( ψ ' 2 ) + C ( ψ '' ) ∂ ψ ) C ( pπ ) + ( C ( ψ '' ) -8 C ( R ) ∂ ψ ) C ( p 2 ‖ ) + ( C ( ψ '' ) -4 C ( R ) ∂ ψ ) C ( p 2 x ) , (3.11a)</formula>
<formula><location><page_40><loc_26><loc_74><loc_90><loc_75></location>∂ b ψ∂ a θ ab : 0 = ( -C ( ψ '' ) +2 C ( R ) ∂ ψ ) C ( p 2 x ) + ∂ ψ β, (3.11b)</formula>
<formula><location><page_40><loc_27><loc_71><loc_90><loc_72></location>X b ∂ a θ ab : 0 = C ( R ) (1 + 2 ∂ q ) C ( p 2 x ) + ∂ q β, (3.11c)</formula>
<formula><location><page_40><loc_24><loc_68><loc_90><loc_70></location>X c q ab ∂ c θ ab : 0 = -2 C ( R ) (1 + 4 ∂ q ) ( 2 C ( p 2 ‖ ) + C ( p 2 x ) ) (3.11d)</formula>
<formula><location><page_40><loc_36><loc_64><loc_90><loc_66></location>+ C ( ψ '' ) (1 + 4 ∂ q ) C ( pπ ) , (3.11e)</formula>
<text><location><page_40><loc_19><loc_57><loc_90><loc_62></location>where ∂ ψ := ∂ ∂ψ , ∂ q := ∂ ∂ log q and X a := q bc ∂ a q bc . Note that the equations for ∂ c q ab ∂ c θ ab , ∂ a q bc ∂ c θ ab and q ab ∂ d q cd ∂ c θ ab are not included because they are identical to (3.10).</text>
<text><location><page_40><loc_19><loc_54><loc_77><loc_56></location>Using (3.10b) to solve for C ( p 2 x ) , then substituting it into (3.11c), I find,</text>
<formula><location><page_40><loc_43><loc_48><loc_90><loc_51></location>∂ log C ( R ) ∂ log q = 1 2 ( 1 + ∂ log β ∂ log q ) , (3.12)</formula>
<text><location><page_40><loc_19><loc_36><loc_90><loc_45></location>which is solved by C ( R ) ( q, ψ ) = f ( ψ ) √ q | β ( q, ψ ) | , where f ( ψ ) is some unknown function. If I solve (3.10) for C ( p 2 ‖ ) and C ( p 2 x ) , then substitute them into (3.11e), I find a similar equation to the one above for C ( R ) , and therefore C ( ψ '' ) ( q, ψ ) = f ( ψ '' ) ( ψ ) √ q | β ( q, ψ ) | . Taking (3.11b) then substituting in for C ( p 2 x ) , C ( R ) and C ( ψ '' ) , I find that f ( ψ '' ) ( ψ ) = -2 ∂ ψ f ( ψ ) ,</text>
<formula><location><page_40><loc_29><loc_28><loc_90><loc_29></location>C ( R ) = f √ q | β | , C ( ψ '' ) = -2 ∂ ψ f √ q | β | , (3.13a)</formula>
<formula><location><page_40><loc_28><loc_23><loc_90><loc_27></location>C ( p 2 x ) = -σ β f √ | β | q , C ( p 2 ‖ ) = σ β 2 f √ | β | q -∂ ψ f 2 f C ( pπ ) , (3.13b)</formula>
<text><location><page_40><loc_19><loc_14><loc_90><loc_21></location>where σ β := sgn( β ) , which is all the conditions which can be obtained from the metric momentum sector of the distribution equation. The remaining conditions must be found in the scalar momentum sector.</text>
<section_header_level_1><location><page_41><loc_19><loc_87><loc_34><loc_88></location>3.1.2 π sector</section_header_level_1>
<text><location><page_41><loc_19><loc_80><loc_90><loc_84></location>Similar to subsection 3.1.1 above, I take the functional derivative of (3.2) with respect to π ( z ) ,</text>
<formula><location><page_41><loc_35><loc_72><loc_90><loc_78></location>0 = ( δC 0 ( x ) δq ab ( y ) C ( pπ ) ab ( y ) + 2 δC 0 ( x ) δψ ( y ) C ( π 2 ) ( y ) ) δ ( z, y ) -( β∂ a ψ∂ a ) x δ ( x, y ) δ ( z, x ) -( x ↔ y ) , (3.14)</formula>
<text><location><page_41><loc_19><loc_68><loc_61><loc_69></location>then exchange terms to find the coefficient of δ ( z, y ) ,</text>
<formula><location><page_41><loc_38><loc_60><loc_90><loc_66></location>0 = ( δC 0 ( x ) δq ab ( y ) C ( pπ ) ab ( y ) + 2 δC 0 ( x ) δψ ( y ) C ( π 2 ) ( y ) +( β∂ a ψ∂ a ) y δ ( y, x ) ) δ ( z, y ) -( x ↔ y ) , (3.15)</formula>
<text><location><page_41><loc_19><loc_56><loc_40><loc_57></location>which can be rewritten as,</text>
<formula><location><page_41><loc_32><loc_51><loc_90><loc_53></location>0 = A ( x, y ) δ ( z, y ) -A ( y, x ) δ ( z, x ) , (3.16a)</formula>
<formula><location><page_41><loc_32><loc_49><loc_90><loc_51></location>0 = A ( x, z ) -δ ( z, x ) ∫ d 3 yA ( y, x ) , (3.16b)</formula>
<formula><location><page_41><loc_33><loc_45><loc_90><loc_47></location>= A ( x, z ) -δ ( z, x ) A ( x ) , where A ( x ) = ∫ d 3 yA ( y, x ) , (3.16c)</formula>
<text><location><page_41><loc_19><loc_40><loc_27><loc_42></location>leading to</text>
<formula><location><page_41><loc_23><loc_35><loc_90><loc_38></location>0 = δC 0 ( x ) δq ab ( y ) C ( pπ ) ab ( y ) + 2 δC 0 ( x ) δψ ( y ) C ( π 2 ) ( y ) + ( β∂ a ψ∂ a ) y δ ( y, x ) -A ( x ) δ ( y, x ) . (3.17)</formula>
<text><location><page_41><loc_19><loc_31><loc_87><loc_32></location>Multiplying by an arbitrary test function η ( y ) , then integrating by parts over y , I get</text>
<formula><location><page_41><loc_20><loc_22><loc_90><loc_28></location>0 = η ( · · · ) + ∂ ab η ( C ( pπ ) cd ∂C 0 ∂q cd,ab +2 C ( π 2 ) ∂C 0 ∂ψ ,ab ) + ∂ a η ( C ( pπ ) bc ∂C 0 ∂q bc,a +2 ∂ b C ( pπ ) cd ∂C 0 ∂q cd,ab +2 C ( π 2 ) ∂C 0 ∂ψ ,a +4 ∂ b C ( π 2 ) ∂C 0 ∂ψ ,ab -β∂ a ψ ) . (3.18)</formula>
<text><location><page_42><loc_19><loc_87><loc_74><loc_88></location>I then substitute in (3.9) to find the linearly independent conditions,</text>
<formula><location><page_42><loc_26><loc_82><loc_90><loc_84></location>∂ 2 η : 0 = C ( R ) C ( pπ ) -C ( ψ '' ) C ( π 2 ) , (3.19a)</formula>
<formula><location><page_42><loc_22><loc_78><loc_90><loc_81></location>∂ a ψ∂ a η : 0 = ( 1 2 C ( ψ '' ) -4 C ( R ) ∂ ψ ) C ( pπ ) +4 ( C ( ψ ' 2 ) + C ( ψ '' ) ∂ ψ ) C ( π 2 ) -β, (3.19b)</formula>
<formula><location><page_42><loc_23><loc_75><loc_90><loc_77></location>X a ∂ a η : 0 = C ( R ) (1 + 4 ∂ q ) C ( pπ ) -C ( ψ '' ) (1 + 4 ∂ q ) C ( π 2 ) . (3.19c)</formula>
<text><location><page_42><loc_19><loc_71><loc_84><loc_72></location>Note that there is another condition from ∂ b q ab ∂ a η , but it is identical to (3.19a).</text>
<text><location><page_42><loc_19><loc_67><loc_90><loc_69></location>I can solve (3.19a) for C ( pπ ) = C ( ψ '' ) C ( π 2 ) /C ( R ) , and then substitute into (3.19b) to find,</text>
<formula><location><page_42><loc_30><loc_62><loc_90><loc_65></location>0 = C ( π 2 ) { C ( ψ ' 2 ) -∂ ψ C ( ψ '' ) + C ( ψ '' ) C ( R ) ( ∂ ψ C ( R ) + C ( ψ '' ) 8 )} -β 4 , (3.20)</formula>
<text><location><page_42><loc_19><loc_53><loc_90><loc_59></location>which I can solve for C ( π 2 ) , and is the same conclusion I get from (3.11a) (though I did not explicitly write it above because it is simpler to write it here). The condition (3.19c) is solved when I substitute in all my results so far,</text>
<formula><location><page_42><loc_35><loc_47><loc_90><loc_50></location>C ( π 2 ) = σ β 4 √ | β | q { C ( ψ ' 2 ) √ q | β | +2 f '' -3 f ' 2 2 f } -1 , (3.21a)</formula>
<formula><location><page_42><loc_35><loc_42><loc_90><loc_46></location>C ( pπ ) = -σ β f ' 2 f √ | β | q { C ( ψ ' 2 ) √ q | β | +2 f '' -3 f ' 2 2 f } -1 , (3.21b)</formula>
<text><location><page_42><loc_19><loc_38><loc_76><loc_39></location>and if I collect all of the coefficients, I find the Hamiltonian constraint,</text>
<formula><location><page_42><loc_22><loc_29><loc_90><loc_36></location>C = √ q | β | ( fR -2 f ' ∆ ψ ) + C ( ψ ' 2 ) ∂ a ψ∂ a ψ + C ∅ + σ β √ | β | q 1 f ( p 2 6 -P ) + 1 4 ( π -f ' f p ) 2 ( C ( ψ ' 2 ) √ q | β | +2 f '' -3 f ' 2 2 f ) -1 , (3.22)</formula>
<text><location><page_42><loc_19><loc_12><loc_96><loc_26></location>so the freedom in any (3+1) dimensional scalar-tensor theory with time symmetry and minimally deformed general covariance comes down to the choice of f ( ψ ) , β ( q, ψ ) , C ( ψ ' 2 ) ( q, ψ ) and the zeroth order term C ∅ ( q, ψ ) . It is convenient to make a redefinition, C ( ψ ' 2 ) = g ( q, ψ ) √ q | β | , where I have made the scalar weight and expected dependence on β explicit. It is worth remembering that this is an assumption, and that g could be a function of β . It is also convenient to treat the zeroth order term as a general potential, and to extract the scalar</text>
<text><location><page_43><loc_19><loc_87><loc_40><loc_89></location>density, C ∅ = √ q U ( q, ψ ) .</text>
<text><location><page_43><loc_19><loc_81><loc_90><loc_85></location>I find the effective Lagrangian associated with this Hamiltonian constraint by performing a Legendre transformation,</text>
<formula><location><page_43><loc_31><loc_71><loc_90><loc_79></location>L = √ q | β | { f ( K β -R ) + f ' ( νv β +2∆ ψ ) + ( g +2 f '' ) ν 2 β -g ∂ a ψ∂ a ψ -U √ | β | } . (3.23)</formula>
<text><location><page_43><loc_19><loc_60><loc_90><loc_69></location>Integrating by parts at the level of the action does not affect the dynamics because it only eliminates boundary terms. This allows me to find the effective form of the Lagrangian, with a space-time decomposition and without second order time derivatives. I can also do this in the opposite direction to find the covariant form of the above effective Lagrangian,</text>
<formula><location><page_43><loc_30><loc_55><loc_90><loc_57></location>L cov = √ q | β | ( -f (4 ,β ) R -( g +2 f '' ) ∂ (4 ,β ) µ ψ∂ µ (4 ,β ) ψ ) -√ q U, (3.24)</formula>
<text><location><page_43><loc_19><loc_51><loc_87><loc_52></location>where the deformed four dimensional Ricci scalar and partial derivative are given by,</text>
<formula><location><page_43><loc_24><loc_44><loc_80><loc_48></location>(4 ,β ) R = R + σ β √ | β | q ab L n ( v ab √ | β | ) + 1 4 β v 2 -3 4 β v ab v ab -2∆ ( √ | β | N ) √ | β | N ,</formula>
<formula><location><page_43><loc_42><loc_40><loc_68><loc_43></location>∂ (4 ,β ) µ ψ∂ µ (4 ,β ) ψ = ∂ a ψ∂ a ψ -β ν 2 .</formula>
<formula><location><page_43><loc_64><loc_41><loc_90><loc_46></location>(3.25a) 1 (3.25b)</formula>
<text><location><page_43><loc_19><loc_29><loc_90><loc_38></location>If this is compared to (2.29), I see that the deformation seems to have transformed the effective lapse function N → √ | β | N , and transformed the effective normalisation of the normal vector to g µν n µ n ν = -σ β . Here is where I see the effective signature change which comes from the deformation.</text>
<text><location><page_43><loc_19><loc_20><loc_90><loc_27></location>It is useful to take the Lagrangian in covariant form and use it to redefine the coupling functions so that minimal coupling is when the functions are equal to unity, f = -1 2 ω R and g = -1 2 ω ψ + ω '' R ,</text>
<formula><location><page_43><loc_24><loc_15><loc_90><loc_18></location>L cov = 1 2 √ q | β | ( ω R ( ψ ) (4 ,β ) R -ω ψ ( q, ψ ) ∂ (4 ,β ) µ ψ∂ µ (4 ,β ) ψ ) -√ q U ( q, ψ ) , (3.26)</formula>
<text><location><page_44><loc_19><loc_87><loc_73><loc_88></location>so the effective forms of the constraint and Lagrangian are given by,</text>
<formula><location><page_44><loc_23><loc_81><loc_68><loc_85></location>L = 1 2 √ q | β | { ω R ( R -K β ) -ω ' R ( νv β +2∆ ψ ) + ω ψ ν 2 β</formula>
<formula><location><page_44><loc_23><loc_70><loc_90><loc_82></location>-( ω ψ +2 ω '' R ) ∂ a ψ∂ a ψ } -√ q U, (3.27a) C = √ q | β | { 2 σ β qω R ( P p 2 6 ) -ω R 2 R + σ β 2 q ( π -ω ' R ω R p ) 2 ( ω ψ + 3 ω ' 2 R 2 ω R ) -1 + ω ' R ∆ ψ + ( ω ψ 2 + ω '' R ) ∂ a ψ∂ a ψ } + √ q U, (3.27b)</formula>
<text><location><page_44><loc_19><loc_67><loc_69><loc_69></location>which is the main result of this section in its most useful form.</text>
<text><location><page_44><loc_19><loc_58><loc_90><loc_65></location>Since I have non-minimal coupling, I am working in the Jordan frame. I can get to the Einstein frame by making a specific conformal transformation which absorbs the coupling ω R by setting q ab = ω R ˜ q ab and N = ω -1 / 2 R ˜ N ,</text>
<formula><location><page_44><loc_21><loc_53><loc_90><loc_56></location>˜ L = 1 2 √ ˜ q | β | {( ˜ R -˜ K β ) + ( ω ψ ω R + 3 ω ' 2 R 2 ω 2 R )( ˜ ν 2 β -˜ q ab ∂ a ψ∂ b ψ ) } -√ ˜ q ( U ω 2 R ) , (3.28)</formula>
<text><location><page_44><loc_19><loc_46><loc_90><loc_50></location>where variables with tildes are Einstein-frame quantities. So the Einstein frame couplings are given by ˜ ω R = 1 , ˜ ω ψ = ( ω ψ ω R +3 ω ' 2 R / 2 ) /ω 2 R , and the potential by ˜ U = U/ω 2 R .</text>
<text><location><page_44><loc_19><loc_35><loc_90><loc_44></location>When the term 'Einstein frame' is used elsewhere in the literature, it often refers to an action which is transformed further so that the effective scalar coupling is also unity. I can make this transformation to a minimally coupled scalar ϕ by solving the differential equation,</text>
<formula><location><page_44><loc_44><loc_32><loc_90><loc_35></location>∂ϕ ∂ψ = √ ω ψ ω R + 3 2 ( ∂ ψ ω R ω R ) 2 , (3.29)</formula>
<text><location><page_44><loc_19><loc_22><loc_90><loc_30></location>for example, when ω ψ = 0 , this is solved by ϕ ( ψ ) = √ 3 2 log ω R ( ψ ) sgn( ∂ ψ log ω R ( ψ )) . For the parameterisation of F ( (4) R ) given in section 2.4.1, ω R = ωψ , and the transformation is given by ψ ( ϕ ) ∝ e ϕ √ 2 / 3 as long as ψ > 0 .</text>
<section_header_level_1><location><page_45><loc_19><loc_87><loc_49><loc_88></location>3.2 Multiple scalar fields</section_header_level_1>
<text><location><page_45><loc_19><loc_74><loc_90><loc_83></location>Consider the case of multiple scalar fields. I start from the distribution equation as before, but label the scalar field variables with an index. Proceeding like in section 3.1.1 by taking functional derivatives with respect to p ab and then integrating by parts with test function θ ab , I obtain the conditions,</text>
<formula><location><page_45><loc_28><loc_69><loc_90><loc_71></location>∂ ab θ ab : 0 = C ( R ) C ( p 2 x ) + β, (3.30a)</formula>
<formula><location><page_45><loc_26><loc_65><loc_90><loc_68></location>q ab ∂ 2 θ ab : 0 = -2 C ( R ) ( 2 C ( p 2 ‖ ) + C ( p 2 x ) ) + ∑ I C ( ψ '' I ) C ( pπ I ) , (3.30b)</formula>
<formula><location><page_45><loc_26><loc_62><loc_90><loc_64></location>X b ∂ a θ ab : 0 = C ( R ) (1 + 2 ∂ q ) C ( p 2 x ) + ∂ q β, (3.30c)</formula>
<formula><location><page_45><loc_25><loc_59><loc_90><loc_61></location>∂ b ψ I ∂ a θ ab : 0 = ( C ( ψ '' I ) -2 C ( R ) ∂ ψ I ) C ( p 2 x ) -∂ ψ I β, (3.30d)</formula>
<formula><location><page_45><loc_22><loc_53><loc_90><loc_58></location>q ab ∂ c ψ I ∂ c θ ab : 0 = ( C ( ψ '' I ) -8 C ( R ) ∂ ψ I ) C ( p 2 ‖ ) + ( C ( ψ '' I ) -4 C ( R ) ∂ ψ I ) C ( p 2 x ) +2 ( C ( ψ ' 2 I ) + C ( ψ '' I ) ∂ ψ I ) C ( pπ I ) + ∑ ( C ( ψ ' I ψ ' J ) + C ( ψ '' J ) ∂ ψ I ) C ( pπ J ) . (3.30e)</formula>
<formula><location><page_45><loc_53><loc_52><loc_56><loc_53></location>J = I</formula>
<text><location><page_45><loc_54><loc_52><loc_54><loc_53></location>glyph[negationslash]</text>
<text><location><page_45><loc_19><loc_43><loc_90><loc_50></location>I note that there are other independent terms, but they do not produce any extra conditions. Likewise, if I follow the route taken in section 3.1.2, taking the functional derivative with respect to π I then integrating by parts with test function η I , I find the conditions,</text>
<formula><location><page_45><loc_26><loc_38><loc_71><loc_41></location>∂ 2 η I : 0 = C ( R ) C ( pπ I ) -C ( ψ '' I ) C ( π 2 I ) -1 2 ∑ C ( ψ '' J ) C ( π I π J ) ,</formula>
<text><location><page_45><loc_37><loc_30><loc_37><loc_31></location>glyph[negationslash]</text>
<text><location><page_45><loc_58><loc_37><loc_58><loc_38></location>glyph[negationslash]</text>
<formula><location><page_45><loc_20><loc_14><loc_90><loc_40></location>J = I (3.31a) X a ∂ a η I : 0 = C ( R ) (1 + 4 ∂ q ) C ( pπ I ) -C ( ψ '' I ) (1 + 4 ∂ q ) C ( π 2 I ) -1 2 ∑ J = I C ( ψ '' J ) (1 + 4 ∂ q ) C ( π I π J ) , (3.31b) ∂ a ψ I ∂ a η I : 0 = ( 1 2 C ( ψ '' I ) -4 C ( R ) ∂ ψ I ) C ( pπ I ) +4 ( C ( ψ ' 2 I ) + C ( ψ '' I ) ∂ ψ I ) C ( π 2 I ) + ∑ J = I ( C ( ψ ' I ψ ' J ) +2 C ( ψ '' J ) ∂ ψ I ) C ( π I π J ) -β, (3.31c) ∂ a ψ J = I ∂ a η I : 0 = ( 1 2 C ( ψ '' J ) -2 C ( R ) ∂ ψ J ) C ( pπ I ) +2 ( C ( ψ ' I ψ ' J ) +2 C ( ψ '' I ) ∂ ψ J ) C ( π 2 I ) +2 ( C ( ψ ' 2 J ) + C ( ψ '' J ) ∂ ψ J ) C ( π I π J ) + ∑ K = I,J ( C ( ψ ' J ψ ' K ) +2 C ( ψ '' K ) ∂ J ) C ( π I π K ) , (3.31d)</formula>
<text><location><page_45><loc_24><loc_19><loc_24><loc_20></location>glyph[negationslash]</text>
<text><location><page_45><loc_35><loc_22><loc_35><loc_23></location>glyph[negationslash]</text>
<text><location><page_45><loc_52><loc_14><loc_52><loc_15></location>glyph[negationslash]</text>
<text><location><page_46><loc_19><loc_84><loc_90><loc_88></location>and similar to above, there are other independent terms which do no produce any unique conditions.</text>
<text><location><page_46><loc_19><loc_70><loc_90><loc_82></location>To solve this system of equations I must make assumptions, in particular about the relationship between the scalar fields. One choice might be to assume an O ( N ) symmetry, where the coupling and deformation would only depend on the absolute value of the scalar field multiplet | ψ | = √ ∑ I ψ 2 I , and relationships between the C ( ψ ' I ψ ' J ) coefficients could be assumed.</text>
<text><location><page_46><loc_19><loc_59><loc_90><loc_68></location>However, I instead choose to take one non-minimally coupled field ( ψ, π ψ ) and one minimally coupled field ( ϕ, π ϕ ) with no cross-terms in the spatial derivative sector, C ( ϕ ' ψ ' ) = 0 . The minimally coupled field only appears in terms other than the potential U ( q, ψ, ϕ ) through the deformation function β ( q, ψ, ϕ ) . For example, C ( R ) = C ( R ) ( q, ψ, β ) .</text>
<text><location><page_46><loc_19><loc_56><loc_49><loc_57></location>Solving (3.30a) and (3.30c) gives me,</text>
<formula><location><page_46><loc_31><loc_50><loc_90><loc_54></location>C ( R ) = f ( ψ ) √ q | β ( q, ψ, ϕ ) | , C ( p 2 x ) = -1 f ( ψ ) √ | β ( q, ψ, ϕ ) | q , (3.32)</formula>
<text><location><page_46><loc_19><loc_46><loc_70><loc_47></location>as before. Substituting these into (3.30b) and (3.30d) gives me,</text>
<formula><location><page_46><loc_26><loc_40><loc_90><loc_44></location>C ( ψ '' ) = -2 f ' √ q | β | , C ( ϕ '' ) = 0 , C ( p 2 ‖ ) = σ β 2 f √ | β | q -f ' 2 f C ( pπ ψ ) , (3.33)</formula>
<text><location><page_46><loc_19><loc_36><loc_46><loc_38></location>and the remaining conditions are,</text>
<formula><location><page_46><loc_35><loc_30><loc_74><loc_34></location>C ( pπ ψ ) = -σ β f ' 2 f √ | β | q { C ( ψ ' 2 ) √ q | β | +2 f '' -3 f ' 2 2 f } -1</formula>
<formula><location><page_46><loc_27><loc_23><loc_77><loc_30></location>C ( π ϕ π ψ ) = -∂ ϕ β ∂ ψ f 4 C ( ϕ ' 2 ) 2 C ( ψ ' 2 ) √ q | β | ( 1 -∂ log C ( ψ ' 2 ) ∂ log β ) +2 f '' -3 f ' 2 2 f [ C ( ψ ' 2 ) √ q | β | +2 f '' -3 f ' 2 2 f ] 2 ,</formula>
<formula><location><page_46><loc_38><loc_19><loc_71><loc_22></location>C ( π 2 ϕ ) = β 4 C ( ϕ ' 2 ) , C ( pπ ϕ ) = -f f C ( π ϕ π ψ ) .</formula>
<formula><location><page_46><loc_64><loc_20><loc_90><loc_33></location>(3.34a) (3.34b) ' (3.34c)</formula>
<text><location><page_46><loc_19><loc_12><loc_90><loc_17></location>I note that the constraint is significantly simpler if I assume C ( ϕ ' 2 ) = g ϕ ( ψ ) √ q | β | and C ( ψ ' 2 ) = g ψ ( ψ ) √ q | β | , where g ϕ and g ψ are arbitrary functions. In this case the whole</text>
<text><location><page_47><loc_19><loc_87><loc_39><loc_88></location>Hamiltonian constraint is</text>
<formula><location><page_47><loc_22><loc_77><loc_90><loc_85></location>C = √ q | β | ( fR -2 f ' ∆ ψ + g ϕ ∂ a ϕ∂ a ϕ + g ψ ∂ a ψ∂ a ψ ) + √ q U + σ β √ | β | q π 2 ϕ 4 g ϕ + 1 f ( p 2 6 -P ) + ( π ψ -f ' f p )( π ψ -f ' f p -f ' ∂ ϕ β βg ϕ π ϕ ) 4 ( g ψ +2 f '' -3 f ' 2 2 f ) , (3.35)</formula>
<text><location><page_47><loc_19><loc_73><loc_51><loc_74></location>and the associated Lagrangian density is</text>
<formula><location><page_47><loc_26><loc_68><loc_78><loc_71></location>L = √ q | β | { f ( K β -R ) + f ' ( ν ψ v β +2∆ ψ ) + ( ˆ g ψ h + 3 f ' 2 2 f ) ν 2 ψ β</formula>
<formula><location><page_47><loc_37><loc_60><loc_73><loc_63></location>ˆ g ψ = g ψ +2 f '' -3 f ' 2 2 f , h = 1 -f ' 2 ∂ ϕ β 2 4 g ϕ ˆ g ψ β 2 .</formula>
<formula><location><page_47><loc_28><loc_61><loc_90><loc_68></location>-g ψ ∂ a ψ∂ a ψ + g ϕ hβ ν 2 ϕ -g ϕ ∂ a ϕ∂ a ϕ + f ' ∂ ϕ β hβ ν ϕ ν ψ } -√ q U, (3.36a) (3.36b)</formula>
<text><location><page_47><loc_19><loc_54><loc_90><loc_57></location>If β does not depend on ϕ , then this can be simplified greatly, in which case the effective and covariant forms of the Lagrangian are given by,</text>
<formula><location><page_47><loc_24><loc_44><loc_90><loc_51></location>L = 1 2 √ q | β | { ω R ( R -K β ) -ω ' R ( ν ψ v β +2∆ ψ ) + ω ϕ ( ν 2 ϕ β -∂ a ϕ∂ a ϕ ) + ω ψ ν 2 ψ β -( ω ψ +2 ω '' R ) ∂ a ψ∂ a ψ } -√ q U, (3.37a)</formula>
<formula><location><page_47><loc_22><loc_40><loc_90><loc_43></location>L cov = 1 2 √ q | β | ( ω R (4 ,β ) R -ω ψ ∂ (4 ,β ) µ ψ∂ µ (4 ,β ) ψ -ω ϕ ∂ (4 ,β ) µ ϕ∂ µ (4 ,β ) ϕ ) -√ q U, (3.37b)</formula>
<text><location><page_47><loc_19><loc_23><loc_90><loc_37></location>where ω R = -2 f , ω ψ = 2( g ψ +2 f '' ) , ω ϕ = 2 g ϕ . Therefore, when I assume that the minimally coupled scalar field can also be considered to be minimally coupled to the deformation function, I find that the action simplifies to the expected form. It would be interesting to see what effects appear for scalar field multiplets, especially for non-Abelian symmetries, but that is beyond the scope of this study. Instead, I now turn to studying the cosmological dynamics of my results.</text>
<section_header_level_1><location><page_48><loc_19><loc_87><loc_38><loc_88></location>3.3 Cosmology</section_header_level_1>
<text><location><page_48><loc_19><loc_74><loc_90><loc_83></location>To find the cosmological dynamics, I restrict to a flat, homogeneous, and isotropic metric in proper time ( N = 1 ). I also assume that β does not depend on the minimally coupled scalar field ϕ for the sake of simplicity. From (3.37), I find the Friedmann equation, which can be written in two equivalent forms,</text>
<formula><location><page_48><loc_22><loc_67><loc_23><loc_68></location>(</formula>
<formula><location><page_48><loc_23><loc_65><loc_90><loc_72></location>H ( ω R H + ω ' R ˙ ψ ) = 1 3 ( ω ψ 2 ˙ ψ 2 + ω ϕ 2 ˙ ϕ 2 + σ β √ | β | U ) , (3.38a) ω R H + 1 2 ω ' R ˙ ψ ) 2 = 1 3 [ 1 2 ( ω R ω ψ + 3 2 ω ' 2 R ) ˙ ψ 2 + ω R ω ϕ 2 ˙ ϕ 2 + σ β ω R √ | β | U ] . (3.38b)</formula>
<text><location><page_48><loc_19><loc_51><loc_90><loc_63></location>From (3.38b) I see that ω R ω ψ +3 ω ' 2 R / 2 ≥ 0 and ω R ω ϕ ≥ 0 are necessary when U → 0 to ensure real-valued fields. If I compare this condition to the Einstein frame Lagrangian (3.28), I can see that it is also the condition which follows from insisting that the scalar field ψ is not ghost-like in that frame. Similarly, I see that σ β ω R > 0 is necessary when ˙ ψ, ˙ ϕ → 0 .</text>
<text><location><page_48><loc_19><loc_37><loc_90><loc_49></location>For the reasonable assumption that the minimally coupled field ϕ does not affect the deformation function β , the only way that field is modified is through a variable maximum phase speed c 2 ϕ = β . Due to this minimal modification, it does not produce any of the cosmological phenomena I am interested in (bounce, inflation) through any novel mechanism. Therefore, I will ignore this field for the rest of the chapter.</text>
<text><location><page_48><loc_19><loc_31><loc_90><loc_35></location>I find the equations of motion by varying the Lagrangian (3.37) with respect to the fields. For the simple undeformed case β = 1 the equations are given by,</text>
<formula><location><page_48><loc_26><loc_15><loc_90><loc_29></location>( ω R ω ψ + 3 2 ω ' 2 R ) ¨ ψ = -3 ˙ ψ H ( ω R ω ψ + ω ' 2 R ) -ω R ∂ ψ U + 3 2 ω R ω ' R H 2 -1 2 ˙ ψ 2 ( ω R ω ' ψ + 3 2 ω ' R ω ψ +3 ω ' R ω '' R ) + 3 2 ω ' R ( 1 + a 3 ∂ ∂a ) U, (3.39a) ( ω R ω ψ + 3 2 ω ' 2 R ) a a = -1 2 H 2 ( ω R ω ψ +3 ω ' 2 R ) + ω ψ 2 ( 1 + a 3 ∂ ∂a ) U -1 4 ˙ ψ 2 ( ω 2 ψ +2 ω ψ ω '' R -ω ' ψ ω ' R ) + 1 2 ω ' R ω ψ ˙ ψ Hω ' R 2 ∂ ψ U, (3.39b)</formula>
<text><location><page_49><loc_19><loc_84><loc_90><loc_88></location>where I can see from the equations of motion that the model breaks down if ω R ω ψ +3 ω ' 2 R / 2 → 0 because it will tend to cause | ¨ ψ | → ∞ and | a | → ∞ .</text>
<section_header_level_1><location><page_49><loc_19><loc_79><loc_33><loc_80></location>3.3.1 Bounce</section_header_level_1>
<text><location><page_49><loc_19><loc_57><loc_90><loc_76></location>I will address the question of whether there are conditions under which there can be a big bounce as defined in section 2.8. I find in chapter 4 (and in ref. [56]) that a deformation function which depends on curvature terms can generate a bounce. Elsewhere in the literature on loop quantum cosmology the bounce happens in a regime when β < 0 because the terms depending on curvature or energy density overpower the zeroth order terms [40, 41]. However, I am not including derivatives in the deformation here so the effect would have to come from the non-minimal coupling of the scalar field or the zeroth order deformation.</text>
<text><location><page_49><loc_19><loc_51><loc_90><loc_54></location>I take ˙ a = 0 for finite a , include a deformation and I ignore the minimally coupled field for simplicity. From the Friedmann equation (3.38) I find,</text>
<formula><location><page_49><loc_46><loc_45><loc_90><loc_48></location>0 = ω ψ 2 ˙ ψ 2 + σ β √ | β | U, (3.40)</formula>
<text><location><page_49><loc_19><loc_36><loc_90><loc_42></location>which implies that σ β ω ψ < 0 for a bounce because otherwise the equation cannot balance for U > 0 and ψ ∈ R . Substituting (3.40) into the full equation of motion for the scale factor, and demanding that a > 0 to make it a turning point, I find the following conditions,</text>
<formula><location><page_49><loc_51><loc_31><loc_90><loc_32></location>σ β ω ψ < 0 , (3.41a)</formula>
<formula><location><page_49><loc_48><loc_27><loc_90><loc_30></location>ω R ω ψ + 3 2 ω ' 2 R > 0 , (3.41b)</formula>
<formula><location><page_49><loc_24><loc_24><loc_90><loc_27></location>σ β √ | β | ( ω ψ +2 ω '' R ) U -σ β ω ' R 2 ω ψ ∂ ψ ( √ | β | ω ψ U ) + aβ 6 ∂ ∂a ( ω ψ U √ | β | ) > 0 , (3.41c)</formula>
<text><location><page_49><loc_19><loc_17><loc_90><loc_21></location>from which I can determine what the coupling functions, deformation and potential must be for a bounce. For example, if I look at the minimally coupled case, when ω R = ω ψ = 1 ,</text>
<text><location><page_50><loc_19><loc_87><loc_72><loc_88></location>and assume that U > 0 , I can see that the conditions are given by,</text>
<formula><location><page_50><loc_41><loc_81><loc_90><loc_85></location>σ β < 0 , ∂ log ( | β | -1 / 2 U ) ∂ log a < -6 . (3.42)</formula>
<text><location><page_50><loc_19><loc_66><loc_90><loc_78></location>Since I must have β → 1 in the classical limit and σ β < 0 at the moment of the bounce, then β must change sign at some point. Therefore, a universe which bounces purely due to a zeroth order deformation must have effective signature change. Another example is obtained by assuming scale independence and choosing β = 1 and U > 0 . In this case the bounce conditions become,</text>
<formula><location><page_50><loc_29><loc_61><loc_90><loc_64></location>ω ψ < 0 , ω ψ ω R + 3 2 ω ' 2 R > 0 , ω ψ +2 ω '' R -1 2 ω ' R ∂ ψ log ( ω ψ U ) > 0 , (3.43)</formula>
<text><location><page_50><loc_19><loc_54><loc_90><loc_58></location>which I can use to find a model which bounces purely due to a scale-independent nonminimally coupled scalar. I present this model in subsection 3.3.5.</text>
<section_header_level_1><location><page_50><loc_19><loc_49><loc_34><loc_50></location>3.3.2 Inflation</section_header_level_1>
<text><location><page_50><loc_19><loc_40><loc_90><loc_46></location>Now consider the inflationary dynamics. For simplicity I assume that inflation will come from a scenario similar to slow-roll inflation with possible enhancements coming from the non-minimal coupling or the deformation. The conditions for slow-roll inflation are,</text>
<formula><location><page_50><loc_40><loc_34><loc_90><loc_37></location>˙ ψ 2 glyph[lessmuch] U, ∣ ∣ ∣ ¨ ψ ∣ ∣ ∣ glyph[lessmuch] ∣ ∣ ∣ ˙ ψ H ∣ ∣ ∣ , ∣ ∣ ∣ ˙ H ∣ ∣ ∣ glyph[lessmuch] H 2 , (3.44)</formula>
<text><location><page_50><loc_19><loc_28><loc_90><loc_32></location>assuming the couplings, potential and deformation are scale independent and the deformation is positive, I get the following slow roll equations,</text>
<formula><location><page_50><loc_40><loc_22><loc_90><loc_26></location>H glyph[similarequal] √ β 1 / 2 U 3 ω R , (3.45a)</formula>
<formula><location><page_50><loc_40><loc_16><loc_90><loc_21></location>˙ ψ glyph[similarequal] -√ β 1 / 2 U 3 ω R ∂ ψ log ( U β 1 / 2 ω 2 R ) ω ψ ω R + ω ' 2 R ω 2 R + β ' ω ' R 2 βω R , (3.45b)</formula>
<text><location><page_51><loc_19><loc_87><loc_48><loc_88></location>and define the slow-roll parameters,</text>
<formula><location><page_51><loc_41><loc_81><loc_90><loc_84></location>glyph[epsilon1] := -˙ H H 2 , η := -H H ˙ H , ζ := -¨ ψ ˙ ψ H , (3.46)</formula>
<text><location><page_51><loc_19><loc_77><loc_56><loc_79></location>which, under slow-roll conditions are given by,</text>
<formula><location><page_51><loc_40><loc_70><loc_90><loc_75></location>glyph[epsilon1] glyph[similarequal] ∂ ψ log ( β 1 / 2 U ω R ) ∂ ψ log ( U β 1 / 2 ω 2 R ) 2 ( ω ψ ω R + ω ' 2 R ω 2 R + β ' ω ' R 2 βω R ) (3.47a)</formula>
<formula><location><page_51><loc_39><loc_64><loc_90><loc_70></location>η glyph[similarequal] ∂ ψ log ( U β 1 / 2 ω 2 R ) ω ψ ω R + ω ' 2 R ω 2 R + β ' ω ' R 2 βω R ∂ ψ log glyph[epsilon1] +2 glyph[epsilon1], (3.47b)</formula>
<formula><location><page_51><loc_40><loc_59><loc_90><loc_64></location>ζ glyph[similarequal] ∂ ψ ∂ ψ log ( U β 1 / 2 ω 2 R ) ω ψ ω R + ω ' 2 R ω 2 R + β ' ω ' R 2 βω R + glyph[epsilon1], (3.47c)</formula>
<text><location><page_51><loc_19><loc_53><loc_90><loc_57></location>where a prime indicates a partial derivative with respect to ψ , i.e. β ' = ∂ ψ β . The slow-roll regime ends when the absolute value of any of these three parameters approaches unity.</text>
<text><location><page_51><loc_19><loc_47><loc_90><loc_51></location>Defining N to mean the number of e-folds from the end of inflation, a ( t ) = a end e -N ( t ) , I find that,</text>
<formula><location><page_51><loc_42><loc_43><loc_90><loc_46></location>N = -∫ t t end d t H = -∫ ψ ψ end d ψ H ˙ ψ , (3.48)</formula>
<text><location><page_51><loc_19><loc_40><loc_50><loc_42></location>and using the slow-roll approximation,</text>
<formula><location><page_51><loc_42><loc_33><loc_90><loc_38></location>N glyph[similarequal] ∫ ψ ψ end d ψ ω ψ ω R + ω ' 2 R ω 2 R + β ' ω ' R 2 βω R ∂ ψ log ( U β 1 / 2 ω 2 R ) , (3.49)</formula>
<text><location><page_51><loc_19><loc_17><loc_90><loc_31></location>which can be solved once I specify the form of the couplings, deformation and potential. I cannot find equations for observables such as the spectral index n s because it would require investigating how the cosmological perturbation theory is modified in the presence of non-minimal coupling and deformed general covariance. Beyond this, it is difficult to make general statements about the dynamics unless I restrict to a given model, so I will now consider some models and discuss their specific dynamics.</text>
<section_header_level_1><location><page_52><loc_19><loc_87><loc_49><loc_88></location>3.3.3 Geometric scalar model</section_header_level_1>
<text><location><page_52><loc_19><loc_64><loc_90><loc_84></location>As demonstrated in the previous chapter, section 2.4.1, the geometric scalar model comes from parameterising F ( (4) R ) gravity so that the additional degree of freedom of the scalar curvature is instead embodied in a non-minimally coupled scalar field ψ [77,80]. Its couplings are given by ω R = ψ and ω ψ = 0 . This model is a special case of the Brans-Dicke model, which has ω ψ = ω 0 /ψ , when the Dicke coupling constant ω 0 vanishes. I can add in a minimally coupled scalar field with ω ϕ = 1 and thereby see the effect of this scalar-tensor gravity on the matter sector. However, I set ω ϕ = 0 because it does not significantly affect my results.</text>
<text><location><page_52><loc_19><loc_61><loc_56><loc_62></location>The effective action for this model is given by,</text>
<formula><location><page_52><loc_31><loc_55><loc_78><loc_58></location>L geo = 1 2 √ q | β | { ψ ( R -K β ) -ν ψ v β -2∆ ψ } -√ q U ( ψ ) ,</formula>
<formula><location><page_52><loc_37><loc_53><loc_72><loc_55></location>U ( ψ ) = ψ ( F ' ) -1 ( ψ ) -1 F ( ( F ' ) -1 ( ψ ) ) ,</formula>
<formula><location><page_52><loc_45><loc_52><loc_90><loc_57></location>(3.50a) 2 2 (3.50b)</formula>
<text><location><page_52><loc_19><loc_46><loc_90><loc_49></location>where F refers to the F ( (4) R ) function which has been parameterised. The equations of motion when β → 1 are given by,</text>
<formula><location><page_52><loc_47><loc_40><loc_90><loc_43></location>H ( ψ H + ˙ ψ ) = 1 3 U, (3.51a)</formula>
<formula><location><page_52><loc_48><loc_37><loc_90><loc_40></location>a a = -H 2 + 1 3 ∂U ∂ψ , (3.51b)</formula>
<formula><location><page_52><loc_37><loc_33><loc_90><loc_36></location>¨ ψ = -2 ˙ ψ H + ψ H 2 + ( 1 + a 3 ∂ ∂a -2 ψ 3 ∂ ∂ψ ) U, (3.51c)</formula>
<text><location><page_52><loc_19><loc_24><loc_90><loc_31></location>from which I can see that the scalar field has very different dynamics compared to minimally coupled scalars. This reflects its origin as a geometric degree of freedom rather than a purely matter field.</text>
<text><location><page_52><loc_19><loc_18><loc_90><loc_22></location>Looking at inflation, the geometric scalar model with a potential corresponding to the Starobinsky model,</text>
<formula><location><page_52><loc_33><loc_13><loc_90><loc_16></location>F ( (4) R ) = (4) R + 1 2 M 2 (4) R 2 → U = M 2 4 ( ψ -1) 2 , (3.52)</formula>
<figure>
<location><page_53><loc_27><loc_74><loc_84><loc_87></location>
</figure>
<figure>
<location><page_53><loc_42><loc_58><loc_68><loc_71></location>
<caption>Figure 3.1: Inflation from the geometric scalar model version of the Starobinsky model through slow-roll of the non-minimally coupled scalar field. For the scale factor, I compare the Jordan and Einstein frames because the coupling causes the former to oscillate unusually. Initial conditions, a = 1 , ψ = 20 , ˙ ψ = 0 , M = 1 .</caption>
</figure>
<text><location><page_53><loc_19><loc_35><loc_90><loc_47></location>can indeed cause inflation through a slow-roll of the scalar field down its potential. The non-minimal coupling of the scalar to the metric also causes the scale factor to oscillate unusually, however. It is interesting to compare in Fig. 3.1 the scale factor in the Jordan frame, a , and the conformally transformed scale factor in the Einstein frame, ˜ a = a √ ω R . Assuming ψ > 1 during inflation, the slow-roll parameters (3.47) are given by,</text>
<formula><location><page_53><loc_38><loc_30><loc_90><loc_33></location>glyph[epsilon1] glyph[similarequal] ψ +1 ( ψ -1) 2 , η glyph[similarequal] -2 ψ 2 -1 , ζ glyph[similarequal] 1 ψ -1 , (3.53)</formula>
<text><location><page_53><loc_19><loc_19><loc_90><loc_26></location>so the slow-roll regime of inflation ends at ψ ≈ 3 when glyph[epsilon1] → 1 . The equation for the number of e-folds of inflation in the slow-roll regime (3.49) is given by N glyph[similarequal] 1 2 ( ψ -ψ end -log ψ ψ end ) .</text>
<figure>
<location><page_54><loc_26><loc_66><loc_84><loc_88></location>
<caption>Figure 3.2: A contour plot of ω R ω ψ +3 ω ' 2 R / 2 for the non-minimally enhanced scalar model is shown in (a). In (b), the red region is when the metric becomes ghost-like (when ω R < 0 ). In both, the white regions are forbidden because it is where ω R ω ψ +3 ω ' 2 R / 2 < 0 , implying imaginary fields. The green region is the region of well-behaved evolution.</caption>
</figure>
<section_header_level_1><location><page_54><loc_19><loc_53><loc_64><loc_55></location>3.3.4 Non-minimally enhanced scalar model</section_header_level_1>
<text><location><page_54><loc_19><loc_36><loc_90><loc_50></location>Unlike the geometric scalar model considered above, the non-minimally enhanced scalar model (NES) from [85], takes a scalar field from the matter sector and introduces a nonminimal coupling rather than extracting a degree of freedom from the gravity sector. The coupling functions are given by ω R = 1 + ξψ 2 , ω ψ = 1 and ω ϕ = 0 . The strength of the quadratic non-minimal coupling is determined by the constant ξ . The deformed effective Lagrangian for this model is given by,</text>
<formula><location><page_54><loc_30><loc_27><loc_90><loc_34></location>L NES = √ q | β | { 1 2 ( 1 + ξψ 2 ) ( R -K β ) + 1 2 ( ν 2 ψ β -∂ a ψ∂ a ψ ) -2 ξ ( ψν ψ v 2 β + ψ ∆ ψ + ∂ a ψ∂ a ψ )} -√ q U ( ψ ) . (3.54)</formula>
<text><location><page_54><loc_19><loc_21><loc_90><loc_24></location>For some negative values of ξ , there are values of ψ which are forbidden if I am to keep my variables real, shown in Fig. 3.2.</text>
<text><location><page_55><loc_19><loc_87><loc_79><loc_88></location>The equations of motion for this model when it is undeformed are given by,</text>
<formula><location><page_55><loc_39><loc_81><loc_71><loc_84></location>1 + ξψ 2 ) H 2 +2 ξψ ˙ ψ H = 1 3 ( 1 2 ˙ ψ 2 + U ) ,</formula>
<formula><location><page_55><loc_24><loc_68><loc_80><loc_73></location>( 1 + (1 + 6 ξ ) ξψ 2 ) ¨ ψ = -3 ˙ ψ H ( 1 + (1 + 4 ξ ) ξψ 2 ) -( 1 + ξψ 2 ) ∂ ψ U +3 ξψ ( ( 1 + ξψ 2 ) H 2 -1 + 4 ξ 2 ˙ ψ 2 + U + a 3 ∂U ∂a ) .</formula>
<formula><location><page_55><loc_27><loc_70><loc_90><loc_83></location>( (3.55a) ( 1 + (1 + 6 ξ ) ξψ 2 ) a a = -1 2 H 2 ( 1 + (1 + 12 ξ ) ξψ 2 ) -1 + 4 ξ 4 ˙ ψ 2 + ξψ ˙ ψ H + 1 2 ( 1 + a 3 ∂ ∂a ) U + ξψ∂ ψ U, (3.55b) (3.55c)</formula>
<text><location><page_55><loc_19><loc_58><loc_90><loc_65></location>and I proceed to use them to consider this model's inflationary dynamics. For a power-law potential U = λ n | ψ | n and ξ > 0 , the slow-roll parameter which reaches unity first is glyph[epsilon1] at ψ end glyph[similarequal] ± n √ 2 + n (6 -n ) ξ . The number of e-folds from the end of inflation is given by,</text>
<formula><location><page_55><loc_35><loc_52><loc_90><loc_56></location>N NES ( ψ ) glyph[similarequal] ∫ ψ ψ end d ϕ ϕ ( 1 + (1 + 4 ξ ) ξϕ 2 ) (1 + ξϕ 2 ) ( n +( n -4) ξϕ 2 ) , (3.56)</formula>
<text><location><page_55><loc_19><loc_49><loc_45><loc_50></location>and if I specify that n = 4 , I find</text>
<formula><location><page_55><loc_35><loc_43><loc_90><loc_46></location>N NES glyph[similarequal] 1 + 4 ξ 8 ψ 2 -1 + 1 2 log 1 + 12 ξ (1 + 4 ξ ) (1 + ξψ 2 ) , (3.57)</formula>
<text><location><page_55><loc_19><loc_34><loc_90><loc_40></location>and the presence of ξ in the dominant first term shows how the non-minimal coupling enhances the amount of inflation. If I compare this result to numerical solutions in Fig. 3.3, I see this effect.</text>
<text><location><page_55><loc_19><loc_26><loc_90><loc_32></location>The slow-roll approximation works less well as ξ increases. I can see this when I look at Fig. 3.3(b) where I compare the slow-roll approximation to when I numerically determine the end of inflation, i.e. when glyph[epsilon1] = -˙ H / H 2 = 1 .</text>
<text><location><page_55><loc_19><loc_12><loc_90><loc_23></location>I must be wary when dealing with this model, because the coupling can produce an effective potential which is not bounded from below. If I substitute the Friedmann equation (3.55a) into (3.55b) and (3.55c) I can find effective potential terms. These terms are those which do not vanish when all time derivatives are set to zero, and I can infer what bare potential they effectively behave like. If the bare potential is U = λψ 2 / 2 , then the effective potential</text>
<figure>
<location><page_56><loc_20><loc_69><loc_90><loc_88></location>
<caption>Figure 3.3: For the non-minimally enhanced scalar model with U = ψ 4 / 4 , (a) shows numerical solutions of inflation for different coupling strengths. Initial conditions, ψ = 20 , ˙ ψ = 0 , H > 0 . In (b), N for ψ = 20 is compared for the numerical solutions (red crosses) and the analytical solution in the slow-roll approximation (3.57) (blue line).</caption>
</figure>
<text><location><page_56><loc_19><loc_57><loc_51><loc_58></location>term in the scalar equation behaves like</text>
<formula><location><page_56><loc_34><loc_51><loc_90><loc_54></location>U ψ = -λψ 2 2 (1 + 6 ξ ) + λ (1 + 3 ξ ) ξ (1 + 6 ξ ) 2 log ( 1 + (1 + 6 ξ ) ξψ 2 ) , (3.58)</formula>
<text><location><page_56><loc_19><loc_42><loc_90><loc_49></location>which is not bounded from below when ξ > 0 and λ > 0 and is therefore unstable. More generally, there are local maxima in the effective potential at ψ = ± √ n ξ (4 -n ) , so for ξ > 0 the model is stable for bare potentials which are of quartic order or higher.</text>
<section_header_level_1><location><page_56><loc_19><loc_37><loc_48><loc_38></location>3.3.5 Bouncing scalar model</section_header_level_1>
<text><location><page_56><loc_19><loc_22><loc_90><loc_34></location>As I said in subsection 3.3.1, I have taken the bounce conditions and constructed a model which bounces purely from the non-minimal coupling. This model consists of a nonminimally coupled scalar with periodic symmetry. My couplings are given by ω R = cos ψ and ω ψ = 1 + b cos ψ 1 + b , where b is some real constant, and for simplicity I ignore deformations and the minimally coupled scalar field. The bouncing scalar model Lagrangian in</text>
<figure>
<location><page_57><loc_21><loc_71><loc_89><loc_88></location>
<caption>Figure 3.4: Cosmological bounce generated by non-minimally coupled scalar field with b = 2 and U = sin 2 ( ψ/ 2) . Initial conditions, ψ = 0 , ˙ ψ = 1 / 25 , H < 0</caption>
</figure>
<figure>
<location><page_57><loc_39><loc_55><loc_71><loc_70></location>
</figure>
<text><location><page_57><loc_45><loc_53><loc_65><loc_54></location>(c) Scalar coupling (zoomed)</text>
<text><location><page_57><loc_19><loc_43><loc_53><loc_44></location>covariant and effective forms are given by,</text>
<formula><location><page_57><loc_26><loc_30><loc_90><loc_40></location>L BS , cov = √ q ( cos ψ 2 (4) R -1 + b cos ψ 2 (1 + b ) ∂ µ ψ∂ µ ψ -U ) , (3.59a) L BS = √ q 2 ( cos ψ ( R -K ) + sin ψ ( νv +2∆ ψ ) + ( 1 + b cos ψ 1 + b ) ν 2 + ( (2 + b ) cos ψ -1 1 + b ) ∂ a ψ∂ a ψ -2 U ) . (3.59b)</formula>
<text><location><page_57><loc_19><loc_16><loc_90><loc_28></location>As confirmed by numerically evolving the equations of motion, I know from the bouncing conditions (3.41) that this model will bounce when b > 1 because then there is a value of ψ for which ω ψ < 0 . As I show in Fig. 3.4, the collapsing universe excites the scalar field so much that it 'tunnels' through to another minima of the potential. The bounce happens when the field becomes momentarily ghost-like, when ω ψ < 0 .</text>
<text><location><page_57><loc_19><loc_10><loc_90><loc_14></location>I can construct other models which produce a bounce purely through non-minimal coupling by having any U ( ψ ) with multiple minima and couplings of the approximate form</text>
<text><location><page_58><loc_19><loc_76><loc_99><loc_88></location>ω ∼ 1 -U . However, to ensure the scalar does not attempt to tunnel through the potential to infinity and thereby not prevent collapse, the coupling functions must become negative only for values of ψ between stable minima. For example, for the Z 2 potential U ( ψ ) = λ ( ψ 2 -1 ) 2 , couplings which are guaranteed to produce a bounce are ω R ( ψ ) = ω ψ ( ψ ) = 1 -e when λ > 1 .</text>
<section_header_level_1><location><page_58><loc_19><loc_70><loc_36><loc_72></location>3.4 Summary</section_header_level_1>
<text><location><page_58><loc_19><loc_53><loc_90><loc_67></location>In this chapter I have presented my calculation of the most general action for a second-order non-minimally coupled scalar-tensor model which satisfies a minimally deformed general covariance. I presented a similar calculation which involves multiple scalar fields. I showed how the magnitude of the deformation can be removed by a transformation of the lapse function, but the sign of the deformation and the associated effective signature change cannot be removed.</text>
<text><location><page_58><loc_19><loc_36><loc_90><loc_50></location>I explored the background dynamics of the action, in particular showing the conditions required for either a big bounce or a period of slow-roll inflation. By specifying the free functions I showed how to regain well-known models from my general action. In particular I discussed the geometric scalar model, which is a parameterisation of F ( (4) R ) gravity and related to the Brans-Dicke model; and I discussed the non-minimally enhanced scalar model of a conventional scalar field with quadratic non-minimal coupling to the curvature.</text>
<text><location><page_58><loc_19><loc_20><loc_90><loc_34></location>I presented a model which produces a cosmological bounce purely through non-minimal coupling of a periodic scalar field to gravity. I also provided the general method of producing similar models without a periodic symmetry. I did not consider in detail the effect that the deformation has on the cosmological dynamics. However, I did show that a big bounce which is purely due to a zeroth order deformation necessarily involves effective signature change.</text>
<text><location><page_58><loc_19><loc_14><loc_90><loc_18></location>Perhaps most importantly, I have established the minimally-deformed low-curvature limit that the subsequent chapters refer to.</text>
<section_header_level_1><location><page_59><loc_19><loc_79><loc_33><loc_81></location>Chapter 4</section_header_level_1>
<section_header_level_1><location><page_59><loc_19><loc_64><loc_69><loc_73></location>Fourth order perturbative gravitational action</section_header_level_1>
<text><location><page_59><loc_19><loc_46><loc_90><loc_58></location>As I showed in section 2.7.2, the deformed action doesn't seem to naturally have a cut-off for higher powers of derivatives, and it must either be considered completely in general or treated perturbatively as a polynomial expansion. In this chapter I will treat it perturbatively in order to find the lowest order corrections which are non-trivial. This chapter is mostly adapted from a previously published paper [56].</text>
<text><location><page_59><loc_19><loc_33><loc_90><loc_44></location>Firstly, I solve the distribution equation for the deformed gravitational action in section 4.1. Then I specify the variables used to construct the action and thereby find the conditions restricting its form in section 4.2. Afterwards, I progressively restrict the action when it is perturbatively expanded to fourth order in derivatives section 4.3. Finally, I investigate the cosmological consequences of the results in section 4.4.</text>
<section_header_level_1><location><page_59><loc_19><loc_26><loc_73><loc_28></location>4.1 Solving the action's distribution equation</section_header_level_1>
<text><location><page_59><loc_19><loc_22><loc_78><loc_23></location>The general deformed action must satisfy the distribution equation (2.43),</text>
<formula><location><page_59><loc_28><loc_15><loc_90><loc_19></location>0 = δL ( x ) δq ab ( y ) v ab ( y ) + ∑ I δL ( x ) δψ I ( y ) ν I ( y ) + ( βD a ∂ a ) x δ ( x, y ) -( x ↔ y ) . (4.1)</formula>
<text><location><page_60><loc_19><loc_84><loc_90><loc_88></location>I restrict to the case when there is only a metric field, for which the diffeomorphism constraint is given by (B.11),</text>
<formula><location><page_60><loc_38><loc_78><loc_90><loc_82></location>D a = -2 ∇ b p ab = -2 ( δ a ( b ∂ c ) +Γ a bc ) ∂L ∂v bc . (4.2)</formula>
<text><location><page_60><loc_19><loc_72><loc_90><loc_76></location>Firstly, I integrate (4.1) by parts to move spatial derivatives from L and onto the delta functions. I discard the surface term and find,</text>
<formula><location><page_60><loc_37><loc_63><loc_90><loc_70></location>0 = δL ( x ) δq ab ( y ) v ab ( y ) -2 ( β ∂L ∂v bc Γ a bc ∂ a ) x δ ( x, y ) +2 ( ∂L ∂v ab ∂ b ) x [( β∂ a ) x δ ( x, y )] -( x ↔ y ) , (4.3)</formula>
<text><location><page_60><loc_19><loc_57><loc_90><loc_61></location>from this I take the functional derivative with respect to v ab ( z ) (after relabelling the other indices),</text>
<formula><location><page_60><loc_22><loc_44><loc_90><loc_55></location>0 = δL ( x ) δq ab ( y ) δ ( y, z ) + { δ∂L ( x ) δq cd ( y ) ∂v ab ( x ) v cd ( y ) +2 [ ∂ ∂v ab ( ∂ d β ∂L ∂v cd -β ∂L ∂v de Γ c de ) ∂ c + ∂ ∂v ab ( β ∂L ∂v cd ) ∂ cd ] x δ ( x, y ) } δ ( x, z ) +2 ( ∂β ,d ∂v ab,e ∂L ∂v cd ) x ∂ c ( x ) δ ( x, y ) ∂ d ( x ) δ ( x, z ) -( x ↔ y ) . (4.4)</formula>
<text><location><page_60><loc_19><loc_38><loc_90><loc_42></location>I move the derivative from δ ( x, z ) and exchange some terms using the ( x ↔ y ) symmetry to find it in the form,</text>
<formula><location><page_60><loc_40><loc_34><loc_90><loc_35></location>0 = A ab ( x, y ) δ ( y, z ) -A ab ( y, x ) δ ( x, z ) , (4.5)</formula>
<text><location><page_60><loc_19><loc_29><loc_24><loc_30></location>where,</text>
<formula><location><page_60><loc_21><loc_20><loc_90><loc_27></location>A ab ( x, y ) = δL ( x ) δq ab ( y ) -v cd ( x ) δ∂L ( y ) δq cd ( x ) ∂v ab ( y ) +2 { ∂ ∂v ab ( β ∂L ∂v de Γ c de -∂ d β ∂L ∂v cd ) ∂ c -∂ ∂v ab ( β ∂L ∂v cd ) ∂ cd + ∂ e ( ∂β ,d ∂v ab,e ∂L ∂v cd ) ∂ c } y δ ( y, x ) . (4.6)</formula>
<text><location><page_61><loc_19><loc_84><loc_90><loc_88></location>Integrating over y , I find that part of the equation can be combined into a tensor dependent only on x ,</text>
<formula><location><page_61><loc_29><loc_76><loc_90><loc_82></location>0 = A ab ( x, z ) -δ ( z, x ) ∫ d 3 yA ab ( y, x ) , = A ab ( x, z ) -δ ( z, x ) A ab ( x ) , where A ab ( x ) = ∫ d 3 yA ab ( y, x ) . (4.7)</formula>
<text><location><page_61><loc_19><loc_72><loc_65><loc_73></location>Substituting in the definition of A ab ( x, z ) then relabelling,</text>
<formula><location><page_61><loc_24><loc_63><loc_90><loc_70></location>0 = δL ( x ) δq ab ( y ) -v cd ( x ) δ∂L ( y ) δq cd ( x ) ∂v ab ( y ) +2 { ∂ ∂v ab ( β ∂L ∂v de Γ c de -∂ d β ∂L ∂v cd ) ∂ c -∂ ∂v ab ( β ∂L ∂v cd ) ∂ cd + ∂ e ( ∂β ,d ∂v ab,e ∂L ∂v cd ) ∂ c } y δ ( y, x ) -A ab ( x ) δ ( x, y ) . (4.8)</formula>
<text><location><page_61><loc_19><loc_57><loc_90><loc_60></location>To find this in terms of one independent variable, I multiply by the test tensor θ ab ( y ) and integrate by parts over y ,</text>
<formula><location><page_61><loc_29><loc_40><loc_90><loc_55></location>0 = ∂L ∂q ab θ ab + ∂L ∂q ab,c ∂ c θ ab + ∂L ∂q ab,cd ∂ cd θ ab -v cd ∂ 2 L ∂q cd ∂v ab θ ab + v cd ∂ e ( ∂ 2 L ∂q cd,e ∂v ab θ ab ) -v cd ∂ ef ( ∂ 2 L ∂q cd,ef ∂v ab θ ab ) +2 ∂ c { θ ab ∂ ∂v ab ( ∂ d β ∂L ∂v cd -β ∂L ∂v de Γ c de ) -θ ab ∂ e ( ∂β ,d ∂v ab,e ∂L ∂v cd )} +2 ∂ cd { θ ab ∂β ,e ∂v ab, ( c ∂L ∂v d ) e -θ ab ∂ ∂v ab ( β ∂L ∂v cd )} -A ab θ ab . (4.9)</formula>
<text><location><page_61><loc_19><loc_37><loc_46><loc_38></location>Then collecting derivatives of θ ab ,</text>
<formula><location><page_61><loc_20><loc_24><loc_90><loc_35></location>0 = θ ab ( · · · ) ab + ∂ c θ ab { ∂L ∂q ab,c + v de ∂ 2 L ∂q de,c ∂v ab -2 v ef ∂ d ( ∂ 2 L ∂q ef,cd ∂v ab ) +2 ∂ ∂v ab ( ∂ d β ∂L ∂v cd -β ∂L ∂v de Γ c de ) -4 ∂ d [ ∂ ∂v ab ( β ∂L ∂v cd )] +2 ∂ e ( ∂β ,d ∂v ab,c ∂L ∂v de )} + ∂ cd θ ab { ∂L ∂q ab,cd -v ef ∂ 2 L ∂q ef,cd ∂v ab -2 ∂ ∂v ab ( β ∂L ∂v cd ) +2 ∂β ,e ∂v ab, ( c ∂L ∂v d ) e } , (4.10)</formula>
<text><location><page_61><loc_19><loc_18><loc_90><loc_22></location>where I have discarded the terms containing θ ab without derivatives, because they do not provide any restrictions on the form of the action. This is simplified by noting that ∂ c and</text>
<text><location><page_62><loc_19><loc_85><loc_81><loc_88></location>∂ ∂v ab commute, and that ∂β ,e ∂v ab,c = δ c e ∂β ∂v ab . Therefore, the solution is given by,</text>
<formula><location><page_62><loc_23><loc_72><loc_90><loc_83></location>0 = θ ab ( · · · ) ab + ∂ c θ ab { ∂L ∂q ab,c + v de ∂ 2 L ∂q de,c ∂v ab -2 v ef ∂ d ( ∂ 2 L ∂q ef,cd ∂v ab ) -2Γ c de ∂ ∂v ab ( β ∂L ∂v de ) -2 ∂ d β ∂ 2 L ∂v ab ∂v cd -4 β∂ d ( ∂ 2 L ∂v ab ∂v cd ) -2 ∂β ∂v ab ∂ d ( ∂L ∂v cd )} + ∂ cd θ ab { ∂L ∂q ab,cd -v ef ∂ 2 L ∂q ef,cd ∂v ab -2 β ∂ 2 L ∂v ab ∂v cd } . (4.11)</formula>
<text><location><page_62><loc_19><loc_66><loc_90><loc_70></location>At this point I need to make some assumptions about the form of the action before I can use this equation to restrict its form.</text>
<section_header_level_1><location><page_62><loc_19><loc_60><loc_67><loc_62></location>4.2 Finding the conditions on the action</section_header_level_1>
<text><location><page_62><loc_19><loc_35><loc_90><loc_57></location>Firstly, the variables used for the action and deformation must be determined. I am considering only the spatial metric field q ab and its normal derivative v ab , and for simplicity I am only considering tensor contractions which contain up to second order in derivatives, as previously stated in section 2.4.1. The only covariant quantities I can form up to second order in derivatives from the spatial metric are the determinant q = det q ab and the Ricci curvature scalar R . The normal derivative can be split into its trace and traceless components, v ab = v T ab + 1 3 vq ab , so it can form scalars from the trace v and a variety of contractions of the traceless tensor v T ab . However, to second order I only need to consider w := Q abcd v T ab v T cd = v T ab v ab T .</text>
<text><location><page_62><loc_19><loc_24><loc_90><loc_33></location>Substituting these variables into (4.11), the resulting equation contains a series of unique tensor combinations. The test tensor θ ab is completely arbitrary so the coefficient of each unique tensor contraction with it must independently vanish if the whole equation is to be satisfied.</text>
<text><location><page_62><loc_19><loc_18><loc_90><loc_21></location>Firstly, I focus on the terms depending on the second order derivative ∂ cd θ ab . I evaluate each individual term in appendix C. Substituting (C.3) into (4.11), I find the following</text>
<text><location><page_63><loc_19><loc_87><loc_38><loc_88></location>independent conditions,</text>
<formula><location><page_63><loc_35><loc_81><loc_90><loc_84></location>q ab ∂ 2 θ ab : 0 = ∂L ∂R -2 v 3 ∂ 2 L ∂R∂v +2 β ( ∂ 2 L ∂v 2 -2 3 ∂L ∂w ) , (4.12a)</formula>
<formula><location><page_63><loc_33><loc_78><loc_90><loc_81></location>Q abcd ∂ cd θ ab : 0 = ∂L ∂R -4 β ∂L ∂w , (4.12b)</formula>
<formula><location><page_63><loc_32><loc_74><loc_90><loc_77></location>q ab v cd T ∂ cd θ ab : 0 = ∂ 2 L ∂R∂v +4 β ∂ 2 L ∂w∂v , (4.12c)</formula>
<formula><location><page_63><loc_35><loc_71><loc_90><loc_74></location>v ab T ∂ 2 θ ab : 0 = v 3 ∂ 2 L ∂R∂w -β ∂ 2 L ∂v∂w , (4.12d)</formula>
<formula><location><page_63><loc_32><loc_68><loc_90><loc_71></location>v ab T v cd T ∂ cd θ ab : 0 = ∂ 2 L ∂R∂w +4 β ∂ 2 L ∂w 2 . (4.12e)</formula>
<text><location><page_63><loc_19><loc_48><loc_90><loc_65></location>Before I analyse these equations, I will find the conditions from the first order derivative part of (4.11). There are many complicated tensor combinations that need to be considered, so for convenience I define X a := q bc ∂ a q bc and Y a := q bc ∂ c q ab . I evaluate the individual terms in appendix C. When I substitute the results (C.4) into (4.11), I once again find a series of unique tensor combinations with their own coefficient which vanishes independently. Most of these conditions are the same as those found in (4.12) so I won't bother duplicating them again here. However, I do find the following new conditions,</text>
<formula><location><page_63><loc_27><loc_43><loc_90><loc_46></location>X a ∂ b θ ab : 0 = ∂L ∂R -4 ( ∂ q β +2 β∂ q ) ∂L ∂w , (4.13a)</formula>
<formula><location><page_63><loc_25><loc_36><loc_90><loc_43></location>q ab X c ∂ c θ ab : 0 = -1 2 ∂L ∂R + v 3 (4 ∂ q -1) ∂ 2 L ∂v∂R + ∂β ∂v (1 -2 ∂ q ) ∂L ∂v +( β -2 ∂ q β -4 β∂ q ) ( ∂ 2 L ∂v 2 -2 3 ∂L ∂w ) , (4.13b)</formula>
<formula><location><page_63><loc_25><loc_32><loc_90><loc_36></location>v ab T X c ∂ c θ ab : 0 = v 3 (4 ∂ q -1) ∂ 2 L ∂w∂R + ∂β ∂w (1 -2 ∂ q ) ∂L ∂v (4.13c)</formula>
<formula><location><page_63><loc_37><loc_29><loc_59><loc_32></location>+( β -2 ∂ q β -4 β∂ q ) ∂ 2 L ∂v∂w ,</formula>
<formula><location><page_63><loc_23><loc_26><loc_90><loc_29></location>q ab v cd T X d ∂ c θ ab : 0 = (1 -2 ∂ q ) ∂ 2 L ∂v∂R -4 ( ∂ q β +2 β∂ q ) ∂ 2 L ∂v∂w -4 ∂β ∂v ∂ q ∂L ∂w , (4.13d)</formula>
<formula><location><page_63><loc_23><loc_23><loc_90><loc_26></location>v ab T v cd T X d ∂ c θ ab : 0 = (1 -2 ∂ q ) ∂ 2 L ∂w∂R -4 ( ∂ q β +2 β∂ q ) ∂ 2 L ∂w 2 -4 ∂β ∂w ∂ q ∂L ∂w , (4.13e)</formula>
<formula><location><page_63><loc_23><loc_19><loc_90><loc_22></location>q ab v cd T Y d ∂ c θ ab : 0 = 2 β ∂ 2 L ∂v∂w + ∂β ∂v ∂L ∂w , (4.13f)</formula>
<formula><location><page_63><loc_23><loc_16><loc_90><loc_19></location>v ab T v cd T Y d ∂ c θ ab : 0 = 2 β ∂ 2 L ∂w 2 + ∂β ∂w ∂L ∂w , (4.13g)</formula>
<formula><location><page_64><loc_28><loc_85><loc_90><loc_88></location>∂ a F∂ b θ ab : 0 = ( ∂β ∂F +2 β ∂ ∂F ) ∂L ∂w , (4.13h)</formula>
<formula><location><page_64><loc_26><loc_78><loc_90><loc_85></location>q ab ∂ c F∂ c θ ab : 0 = 2 v 3 ∂ 3 L ∂F∂v∂R -∂β ∂v ∂ 2 L ∂F∂v -( ∂β ∂F +2 β ∂ ∂F )( ∂ 2 L ∂v 2 -2 3 ∂L ∂w ) , (4.13i)</formula>
<formula><location><page_64><loc_26><loc_75><loc_90><loc_78></location>v ab T ∂ c F∂ c θ ab : 0 = 2 v 3 ∂ 3 L ∂F∂w∂R -∂β ∂w ∂ 2 L ∂F∂v -( ∂β ∂F +2 β ∂ ∂F ) ∂ 2 L ∂v∂w , (4.13j)</formula>
<formula><location><page_64><loc_23><loc_71><loc_90><loc_74></location>q ab v cd T ∂ d F∂ c θ ab : 0 = 1 2 ∂ 3 L ∂F∂v∂R + ∂β ∂v ∂ 2 L ∂F∂w + ( ∂β ∂F +2 β ∂ ∂F ) ∂ 2 L ∂v∂w , (4.13k)</formula>
<formula><location><page_64><loc_23><loc_67><loc_90><loc_70></location>v ab T v cd T ∂ d F∂ c θ ab : 0 = 1 2 ∂ 3 L ∂F∂w∂R + ∂β ∂w ∂ 2 L ∂F∂w + ( ∂β ∂F +2 β ∂ ∂F ) ∂ 2 L ∂w 2 , (4.13l)</formula>
<text><location><page_64><loc_19><loc_64><loc_36><loc_65></location>where F ∈ { v, w, R } .</text>
<text><location><page_64><loc_19><loc_58><loc_90><loc_61></location>By this point, I have accumulated all conditions on the form of the Lagrangian for my choice of variables. The next step is to try and consolidate them.</text>
<section_header_level_1><location><page_64><loc_19><loc_51><loc_81><loc_53></location>4.3 Evaluating the fourth order perturbative action</section_header_level_1>
<text><location><page_64><loc_19><loc_31><loc_90><loc_48></location>For this section, I construct an ansatz for the action and deformation that is explicit in being a perturbative expansion. For each time derivative above the classical solution, I include the small parameter ε , and consider up to O ( ε 2 ) . I consider two orders because in models of loop quantum cosmology which have deformed covariance, the holonomy corrections to the action expand into even powers of time derivatives [39, 42]. Therefore, considering a fourth order action and a second order deformation should include the nearest higher-order terms in an expansion of those holonomy functions. Therefore I write,</text>
<formula><location><page_64><loc_58><loc_26><loc_90><loc_28></location>2 3 (4.14a)</formula>
<formula><location><page_64><loc_31><loc_24><loc_78><loc_28></location>L = L 0 + L ( v ) v + L ( w ) w + L ( v 2 ) v + ε ( L ( vw ) vw + L ( v 3 ) v ) + ε 2 ( L ( w 2 ) w 2 + L ( v 2 w ) v 2 w + L ( v 4 ) v 4 ) + O ( ε 3 ) ,</formula>
<formula><location><page_64><loc_32><loc_21><loc_90><loc_22></location>β = β 0 + εβ ( v ) v + ε 2 ( β ( v 2 ) v 2 + β ( w ) w ) + O ( ε 3 ) , (4.14b)</formula>
<text><location><page_64><loc_19><loc_16><loc_66><loc_17></location>where each coefficient is potentially a function of q and R .</text>
<text><location><page_65><loc_19><loc_84><loc_90><loc_88></location>I take the condition from Q abcd ∂ cd θ ab , (4.12b) and truncate to O ( ε 2 ) . Separating different powers of v and w , it gives the following conditions for the Lagrangian coefficients,</text>
<formula><location><page_65><loc_27><loc_80><loc_27><loc_81></location>ε</formula>
<formula><location><page_65><loc_27><loc_68><loc_90><loc_81></location>2 w 2 : ∂ R L ( w 2 ) = 0 , ε 2 v 2 w : ∂ R L ( v 2 w ) = 0 , ε 2 v 4 : ∂ R L ( v 4 ) = 0 , εvw : ∂ R L ( vw ) = 0 , εv 3 : ∂ R L ( v 3 ) = 0 , (4.15a) w : ∂ R L ( w ) = 4 ε 2 ( β ( w ) L ( w ) +2 β 0 L ( w 2 ) ) , v 2 : ∂ R L ( v 2 ) = 4 ε 2 ( β ( v 2 ) L ( w ) + β ( v ) L ( vw ) + β 0 L ( v 2 w ) ) , v : ∂ R L ( v ) = 4 ε ( β ( v ) L ( w ) + β 0 L ( vw ) ) . (4.15b)</formula>
<text><location><page_65><loc_19><loc_51><loc_90><loc_65></location>So from the five conditions in (4.15a), one can see that terms with three or four time derivatives must not contain any spatial derivatives. From the three conditions in (4.15b), one can see that including R in these coefficients requires including a factor of ε for every combined derivative order above two. Therefore, the spatial derivatives must be treated equally with time derivatives when one is performing a perturbative expansion, as expected. So I can now further expand the ansatz to include explicit factors of R ,</text>
<formula><location><page_65><loc_25><loc_38><loc_90><loc_48></location>L = L ∅ + L ( v ) v + L ( w ) w + L ( v 2 ) v 2 + L ( R ) R + ε ( L ( vw ) vw + L ( v 3 ) v 3 + L ( vR ) vR ) + ε 2 ( L ( w 2 ) w 2 + L ( v 2 w ) v 2 w + L ( v 4 ) v 4 + L ( wR ) wR + L ( v 2 R ) v 2 R + L ( R 2 ) R 2 ) + O ( ε 3 ) , (4.16a) β = β ∅ + εβ ( v ) v + ε 2 ( β ( v 2 ) v 2 + β ( w ) w + β ( R ) R ) + O ( ε 3 ) , (4.16b)</formula>
<text><location><page_65><loc_19><loc_28><loc_90><loc_34></location>where each coefficient is potentially a function of q . I now substitute this ansatz into the conditions found for the action so that its form can be progressively restricted. Looking once again at the condition from Q abcd ∂ cd θ ab (4.12b), one finds it is satisfied by the following</text>
<text><location><page_66><loc_19><loc_87><loc_27><loc_88></location>solutions,</text>
<formula><location><page_66><loc_26><loc_81><loc_90><loc_84></location>∅ : L ( w ) = L ( R ) 4 β ∅ , (4.17a)</formula>
<formula><location><page_66><loc_25><loc_78><loc_90><loc_81></location>εv : L ( vw ) = 1 4 β 2 ∅ ( β ∅ L ( vR ) -4 β ( v ) L ( R ) ) , (4.17b)</formula>
<formula><location><page_66><loc_24><loc_74><loc_90><loc_77></location>ε 2 R : L ( wR ) = 1 4 β 2 ∅ ( 2 β ∅ L ( R 2 ) -2 β ( R ) L ( R ) ) , (4.17c)</formula>
<formula><location><page_66><loc_24><loc_70><loc_90><loc_73></location>ε 2 v 2 : L ( v 2 w ) = 1 4 β 3 ∅ { β 2 ∅ L ( v 2 R ) -β ∅ β ( v ) L ( vR ) + ( β 2 ( v ) -β ∅ β ( v 2 ) ) L ( R ) } , (4.17d)</formula>
<formula><location><page_66><loc_24><loc_67><loc_90><loc_70></location>ε 2 w : L ( w 2 ) = 1 32 β 3 ∅ { 2 β ∅ L ( R 2 ) -( β ( R ) +4 β ∅ β ( w ) ) L ( R ) } , (4.17e)</formula>
<text><location><page_66><loc_19><loc_63><loc_65><loc_65></location>and then looking at the condition from v ab T ∂ 2 θ ab , (4.12d),</text>
<formula><location><page_66><loc_34><loc_58><loc_90><loc_61></location>ε : L ( vR ) = β ( v ) L ( R ) β ∅ , (4.18a)</formula>
<formula><location><page_66><loc_32><loc_54><loc_90><loc_57></location>ε 2 v : L ( v 2 R ) = 1 6 β 2 ∅ { 2 β ∅ L ( R 2 ) + ( 6 β ∅ β ( v 2 ) -β ( R ) ) L ( R ) } , (4.18b)</formula>
<text><location><page_66><loc_19><loc_47><loc_90><loc_51></location>where (4.18a) and (4.17b) combine to give L ( vw ) = 0 . Then looking at the condition from q ab v cd T ∂ cd θ ab , (4.12c)</text>
<formula><location><page_66><loc_42><loc_43><loc_90><loc_44></location>ε : β ( v ) = 0 , (4.19a)</formula>
<formula><location><page_66><loc_40><loc_39><loc_90><loc_42></location>ε 2 v : L ( R 2 ) = L ( R ) 2 β ∅ ( β ( R ) -3 β ∅ β ( v 2 ) ) , (4.19b)</formula>
<text><location><page_66><loc_19><loc_33><loc_90><loc_36></location>one can see that L ( vR ) = 0 and therefore all the third order terms all vanish. Looking at the condition from q ab ∂ 2 θ ab , (4.12a),</text>
<formula><location><page_66><loc_47><loc_27><loc_90><loc_30></location>∅ : L ( v 2 ) = -L ( R ) 6 β ∅ , (4.20a)</formula>
<formula><location><page_66><loc_46><loc_24><loc_90><loc_26></location>εv : L ( v 3 ) = 0 , (4.20b)</formula>
<formula><location><page_66><loc_45><loc_21><loc_90><loc_24></location>ε 2 w : β ( v 2 ) = -2 3 β ( w ) , (4.20c)</formula>
<formula><location><page_66><loc_45><loc_17><loc_90><loc_21></location>ε 2 v 2 : L ( v 4 ) = -β ( w ) L ( R ) 36 β 2 ∅ , (4.20d)</formula>
<text><location><page_67><loc_19><loc_87><loc_45><loc_88></location>and then from X a ∂ b θ ab , (4.13a),</text>
<formula><location><page_67><loc_47><loc_82><loc_90><loc_84></location>∅ : L ( R ) = f √ q | β ∅ | , (4.21a)</formula>
<formula><location><page_67><loc_46><loc_78><loc_90><loc_81></location>ε 2 R : β ( w ) = b -β ( R ) 4 β ∅ , (4.21b)</formula>
<text><location><page_67><loc_19><loc_74><loc_76><loc_76></location>where f and b arise as integration constants. From q ab X c ∂ c θ ab , (4.13b),</text>
<formula><location><page_67><loc_48><loc_69><loc_90><loc_72></location>εv : L ( v ) = ξ √ q, (4.22)</formula>
<text><location><page_67><loc_19><loc_62><loc_90><loc_66></location>where ξ is also an integration constant. Finally, the condition from ∂ a R∂ b θ ab , (4.13h), means that</text>
<formula><location><page_67><loc_51><loc_60><loc_90><loc_61></location>ε : b = 0 . (4.23)</formula>
<text><location><page_67><loc_19><loc_53><loc_90><loc_57></location>From this point on the remaining equations don't provide any new conditions on the Lagrangian coefficients.</text>
<text><location><page_67><loc_19><loc_42><loc_90><loc_51></location>To make sure the classical limit of the result matches the action found in chapter 3, I set f = ω/ 2 , and replace the normal derivatives with the standard extrinsic curvature contraction K = v 2 6 -w 4 . Therefore, the fourth order perturbative gravitational action is given by,</text>
<formula><location><page_67><loc_23><loc_37><loc_90><loc_40></location>L = L ∅ + ξv √ q + ω 2 √ q | β ∅ | { R -K β ∅ -ε 2 β ( R ) 4 β ∅ ( R + K β ∅ ) 2 } + O ( ε 3 ) , (4.24)</formula>
<text><location><page_67><loc_19><loc_33><loc_45><loc_34></location>with the associated deformation</text>
<formula><location><page_67><loc_40><loc_27><loc_90><loc_30></location>β = β ∅ + ε 2 β ( R ) ( R + K β ∅ ) + O ( ε 3 ) . (4.25)</formula>
<text><location><page_67><loc_19><loc_18><loc_90><loc_24></location>So the remaining freedom in the action comes down to the constants ξ and ω , the functions β ∅ and β ( R ) . There is also a term which doesn't affect the kinematic structure and acts like a generalised notion of a potential, so can be rewritten as L ∅ ( q ) = -√ q U ( q ) .</text>
<section_header_level_1><location><page_68><loc_19><loc_87><loc_38><loc_88></location>4.4 Cosmology</section_header_level_1>
<text><location><page_68><loc_19><loc_77><loc_90><loc_83></location>In this section I find the cosmological implications of the nearest order corrections coming from the deformation to general covariance. Since it is a perturbative expansion, the results when the corrections become large should be taken to be indicative rather than predictive.</text>
<text><location><page_68><loc_19><loc_74><loc_58><loc_75></location>I restrict to a flat FLRW metric as in section 2.8,</text>
<formula><location><page_68><loc_32><loc_68><loc_90><loc_71></location>L = -a 3 U ( a ) -3 σ ∅ ωa 3 N 2 √ | β ∅ | H 2 ( 1 + 3 ε 2 β 2 2 N 2 β ∅ H 2 ) + O ( ε 3 ) , (4.26)</formula>
<text><location><page_68><loc_19><loc_61><loc_90><loc_65></location>where a is the scale factor, H = ˙ a/a is the Hubble expansion rate, σ ∅ := sgn( β ∅ ) , and β 2 = β ( R ) /β ∅ is the coefficient of K in the deformation.</text>
<text><location><page_68><loc_19><loc_53><loc_90><loc_59></location>I couple this to matter with energy density ρ and pressure density P = w ρ ρ . I Legendre transform the effective Lagrangian to find the Hamiltonian. Imposing the Hamiltonian constraint C ≈ 0 gives us</text>
<formula><location><page_68><loc_39><loc_47><loc_90><loc_51></location>1 N 2 H 2 ( 1 + 9 ε 2 β 2 2 β ∅ N 2 H 2 ) = σ ∅ 3 ω √ | β ∅ | U, (4.27)</formula>
<text><location><page_68><loc_19><loc_44><loc_68><loc_45></location>which can be solved to find the modified Friedmann equation,</text>
<formula><location><page_68><loc_46><loc_38><loc_90><loc_41></location>1 N 2 H 2 = 2 σ ∅ √ | β ∅ | 3 ω (1 + α ) U, (4.28)</formula>
<text><location><page_68><loc_19><loc_34><loc_42><loc_35></location>where the correction factor is</text>
<formula><location><page_68><loc_46><loc_28><loc_90><loc_32></location>α := √ 1 + 6 ε 2 β 2 ω √ | β ∅ | U. (4.29)</formula>
<text><location><page_68><loc_19><loc_20><loc_90><loc_26></location>Going back to the effective Lagrangian, and varying it with respect to the scale factor, I find the Euler-Lagrange equation of motion. When I substitute in Eq. (4.28), I get the acceleration equation</text>
<formula><location><page_68><loc_24><loc_11><loc_90><loc_17></location>a aN 2 = σ ∅ √ | β ∅ | 6 α U { 2 + ∂ log U ∂ log a +2 ∂ log N ∂ log a + 1 2 ∂ log β ∅ ∂ log a +2 ( α -1 α +1 )[ 1 + ∂ log N ∂ log a -1 2 ∂ ∂ log a log ( β 2 β ∅ )]} . (4.30)</formula>
<text><location><page_69><loc_19><loc_84><loc_90><loc_88></location>If I take a perfect fluid, then U = ρ , where ρ is the fluid's energy density, which satisfies the continuity equation</text>
<formula><location><page_69><loc_46><loc_81><loc_90><loc_83></location>˙ ρ +3 H ρ (1 + w ρ ) = 0 . (4.31)</formula>
<text><location><page_69><loc_19><loc_70><loc_90><loc_79></location>where w ρ is the perfect fluid's equation of state. Note that there are corrections to the matter sector due to the modified constraint algebra [86,87], as shown for scalar fields in other chapters. However, these have not been included here, as it is not known how the deformation would affect a perfect fluid.</text>
<text><location><page_69><loc_19><loc_67><loc_71><loc_68></location>Since ε is a small parameter, it can be used to expand Eq. (4.28),</text>
<formula><location><page_69><loc_36><loc_61><loc_90><loc_64></location>1 N 2 H 2 = σ ∅ √ | β ∅ | 3 ω ρ ( 1 + 3 ε 2 β 2 ω √ | β ∅ | ρ ) + O ( ε 3 ) , (4.32)</formula>
<text><location><page_69><loc_19><loc_54><loc_90><loc_58></location>and expanding the bracket in Eq. (4.30) to first order, it can be seen that a/a > 0 when w ρ < w a , where</text>
<formula><location><page_69><loc_28><loc_48><loc_90><loc_52></location>w a = -1 3 { 1 -1 2 ∂ log β ∅ ∂ log a + 6 ε 2 β 2 ω √ | β ∅ | ρ [ 1 -1 2 ∂ ∂ log a log ( β 2 β ∅ )] } , (4.33)</formula>
<text><location><page_69><loc_19><loc_45><loc_63><loc_46></location>having set N = 1 , so this is applicable for cosmic time.</text>
<text><location><page_69><loc_19><loc_36><loc_90><loc_43></location>When β 2 < 0 , the modified Friedmann equation (4.32) suggests a big bounce rather than a big bang at high energy density, since ˙ a → 0 when a > 0 and a > 0 is possible when ρ → ρ c where</text>
<formula><location><page_69><loc_49><loc_33><loc_90><loc_36></location>ρ c = ω √ | β ∅ | 6 ε 2 | β 2 | . (4.34)</formula>
<text><location><page_69><loc_19><loc_30><loc_90><loc_31></location>This requires either ρ c to be constant, or for it to diverge at a slower rate than ρ as a → 0 .</text>
<text><location><page_69><loc_19><loc_11><loc_90><loc_28></location>Let me emphasise that the bounce is found considering only holonomy corrections manifesting as higher-order powers of of second-order derivatives and not considering ignoring higher-order derivatives. The equations (4.32) and (4.33) have been expanded to leading order in β 2 , so I should be cautious about the regime of their validity. Note that the Lagrangian is also an expansion; β 2 is a coefficient of the fourth order term and appears only linearly, I conclude that there is no good reason why I should have more trust in equations such as (4.28) or (4.30) simply because they contain higher orders. In Ref. [47],</text>
<text><location><page_70><loc_19><loc_76><loc_90><loc_88></location>Ashtekar, Pawlowski and Singh write their effective Friedmann equation with leading order corrections (which is the same as (4.32)) and say that it holds surprisingly well even for ρ ≈ ρ c , the regime when the perturbative expansion should break down (I should note that their work refers only to the case where w ρ = 1 , and does not say whether this is true generally).</text>
<section_header_level_1><location><page_70><loc_19><loc_71><loc_56><loc_72></location>4.4.1 Linking the β function to LQC</section_header_level_1>
<text><location><page_70><loc_19><loc_57><loc_90><loc_68></location>I need to know β ∅ ( a ) and β 2 ( a ) in order to make progress beyond this point, so I compare my results to those found in previous investigations. In Ref. [42], Cailleteau, Linsefors and Barrau have found information about the correction function when inverse-volume and holonomy effects are both included in a perturbed FLRW system. Their equation (Eq. (5 . 18) in Ref. [42]) gives (rewritten slightly)</text>
<formula><location><page_70><loc_32><loc_51><loc_90><loc_54></location>β ( a, ˙ a ) = f ( a )Σ( a, ˙ a ) ∂ 2 ∂ ˙ a 2 { γ ∅ ( a, ˙ a ) ( sin[ γ BI µ ( a )˙ a ] γ BI µ ( a ) ) 2 } , (4.35)</formula>
<text><location><page_70><loc_19><loc_31><loc_90><loc_48></location>where γ BI is the Barbero-Immirzi parameter, γ ∅ is the function which contains information about inverse-volume corrections, Σ( a, ˙ a ) depends on the form of γ ∅ , and f ( a ) is left unspecified. I just consider the case where γ ∅ = γ ∅ ( a ) , in which case Σ = 1 / ( 2 √ γ ∅ ) and µ = a δ -1 √ γ ∅ ♦ . The constant ♦ is usually interpreted as being the 'area gap' derived in loop quantum gravity. I leave δ unspecified for now, because different quantisations of loop quantum cosmology give it equal to different values in the range [0 , 1] . Equation (4.35) now becomes</text>
<formula><location><page_70><loc_42><loc_29><loc_90><loc_31></location>β = f √ γ ∅ cos ( 2 γ BI √ γ ∅ ♦ a δ H ) , (4.36)</formula>
<text><location><page_70><loc_19><loc_20><loc_90><loc_26></location>The 'old dynamics' or ' µ 0 scheme' corresponds to δ = 1 , and the favoured 'improved dynamics' or ' ¯ µ scheme' corresponds to δ = 0 [88, 89]. In the semi-classical regime, H √ ♦ glyph[lessmuch] 1 , so I can Taylor expand this equation for the correction function to get</text>
<formula><location><page_70><loc_37><loc_15><loc_90><loc_17></location>β = f √ γ ∅ -2 γ 2 BI ♦ a 2 δ f ( γ ∅ ) 3 / 2 H 2 + O ( ♦ 2 ) . (4.37)</formula>
<text><location><page_71><loc_19><loc_74><loc_90><loc_88></location>The way that γ ∅ is defined is that it multiplies the background gravitational term in the Hamiltonian constraint relative to the classical form. Since I am assuming γ ∅ = γ ∅ ( a ) , I can isolate it by taking the Lagrangian (4.26) and setting β 2 = 0 . If I then Legendre transform to find a Hamiltonian expressed in terms of the momentum of the scale factor, I find that it is proportional to √ | β ∅ | . Thus, I conclude that γ ∅ = √ | β ∅ | when γ ∅ is just a function of the scale factor. Using this to compare (4.37) to (4.25),</text>
<formula><location><page_71><loc_43><loc_69><loc_90><loc_71></location>β = β ∅ +6 ε 2 β 2 H 2 + O ( ε 3 ) , (4.38)</formula>
<text><location><page_71><loc_19><loc_62><loc_90><loc_66></location>I find that f = σ ∅ | β ∅ | 3 / 4 , and therefore f = σ ∅ γ 3 / 2 ∅ . From this, I can now deduce the form of the coefficient for the higher-order corrections,</text>
<formula><location><page_71><loc_45><loc_56><loc_90><loc_59></location>ε 2 β 2 = -σ ∅ 3 γ 2 BI ♦ a 2 δ γ 3 ∅ . (4.39)</formula>
<text><location><page_71><loc_19><loc_50><loc_90><loc_54></location>The exact form of γ ∅ ( a ) is uncertain, and the possible forms that have been found also contain quantisation ambiguities. The form given by Bojowald in Ref. [90] is</text>
<formula><location><page_71><loc_23><loc_44><loc_90><loc_47></location>γ ∅ = 3 r 1 -l 2 l { ( r +1) l +2 -| r -1 | l +2 l +2 -r ( r +1) l +1 -sgn( r -1) | r -1 | l +1 l +1 } , (4.40)</formula>
<text><location><page_71><loc_19><loc_35><loc_90><loc_42></location>where l ∈ (0 , 1) , r = a 2 /a 2 glyph[star] and a glyph[star] is the characteristic scale of the inverse-volume corrections, related to the discreteness scale. I will only use the asymptotic expansions of this function, namely</text>
<formula><location><page_71><loc_36><loc_28><loc_90><loc_35></location>γ ∅ ≈ 1 + (2 -l )(1 -l ) 10 ( a a glyph[star] ) -4 , if a glyph[greatermuch] a glyph[star] 3 1 + l ( a a glyph[star] ) 2(2 -l ) , if a glyph[lessmuch] a glyph[star] (4.41)</formula>
<text><location><page_71><loc_19><loc_20><loc_90><loc_27></location>and even then I will only take γ ∅ ≈ 1 for a glyph[greatermuch] a glyph[star] , since the correction quickly becomes vanishingly small. I replace the area gap with a dimensionless parameter ˜ ♦ = ♦ ω which is of order unity. The modified Friedmann equation (4.32) is now given by</text>
<formula><location><page_71><loc_40><loc_15><loc_90><loc_18></location>H 2 = σ ∅ γ ∅ 3 ω ρ ( 1 -σ ∅ γ 2 BI ˜ ♦ 3 ω 2 a 2 δ γ 2 ∅ ρ ) , (4.42)</formula>
<text><location><page_72><loc_19><loc_84><loc_90><loc_88></location>which I need to compare for different types of matter. First of all I will consider a perfect fluid, and then I will consider a scalar field with a power-law potential.</text>
<section_header_level_1><location><page_72><loc_19><loc_79><loc_38><loc_80></location>4.4.2 Perfect fluid</section_header_level_1>
<text><location><page_72><loc_19><loc_72><loc_90><loc_76></location>I consider the simple case of a perfect fluid. Solving the continuity equation (2.62) gives us the energy density as a function of the scale factor,</text>
<formula><location><page_72><loc_47><loc_67><loc_90><loc_69></location>ρ ( a ) = ρ 0 a -3(1+ w ρ ) . (4.43)</formula>
<text><location><page_72><loc_19><loc_60><loc_90><loc_64></location>To investigate whether there can be a big bounce, I insert this into Eq. (4.42), which becomes of the form</text>
<formula><location><page_72><loc_41><loc_57><loc_90><loc_60></location>H 2 ∝ a -3(1+ w ρ ) ( 1 -γ 2 BI ˜ ♦ 3 ω 2 ρ 0 a Θ ) , (4.44)</formula>
<text><location><page_72><loc_19><loc_54><loc_69><loc_55></location>where Θ depends on which regime of (4.41) we are in, namely</text>
<formula><location><page_72><loc_37><loc_46><loc_90><loc_51></location>Θ = 2 δ -3(1 + w ρ ) , if a glyph[greatermuch] a glyph[star] , 2 δ +4(2 -l ) -3(1 + w ρ ) , if a glyph[lessmuch] a glyph[star] , (4.45)</formula>
<text><location><page_72><loc_19><loc_29><loc_90><loc_44></location>and I simply ignored the constant coefficients for a glyph[lessmuch] a glyph[star] . Whether a bounce happens depends on whether H → 0 when a > 0 , which would happen if the higher-order correction in the modified Friedmann equation became dominant for small values of a , i.e. if Θ < 0 , which is also required to match the classical limit. The reason this is required is because ρ needs to diverge faster than ρ c as a → 0 in order for there to be a bounce. This will happen when w ρ > w b , where</text>
<formula><location><page_72><loc_39><loc_21><loc_90><loc_27></location>w b = -1 + 2 3 δ, if a glyph[greatermuch] a glyph[star] -1 + 2 3 δ + 4 3 (2 -l ) , if a glyph[lessmuch] a glyph[star] (4.46)</formula>
<text><location><page_72><loc_19><loc_13><loc_90><loc_19></location>which means that, if the bounce does not happen in the a glyph[greatermuch] a glyph[star] regime, the inverse-volume corrections make the bounce less likely to happen. If I use the favoured value of δ = 0 , and assume l = 1 , then w b = 1 / 3 and so w ρ still needs to be greater than that found</text>
<text><location><page_73><loc_19><loc_84><loc_90><loc_88></location>for radiation in order for there to be a bounce. A possible candidate for this would be a massless (or kinetic-dominated) scalar field, where w ρ = 1 .</text>
<text><location><page_73><loc_19><loc_78><loc_90><loc_82></location>Another aspect to investigate is whether the conditions for inflation are modified. Taking (4.33), I see that acceleration happens when w ρ < w a , where</text>
<formula><location><page_73><loc_34><loc_69><loc_90><loc_76></location>w a = -1 3 + 2 γ 2 BI ˜ ♦ 9 ω 2 (1 -δ ) ρ 0 a Θ , if a glyph[greatermuch] a glyph[star] 1 -2 l 3 -2 γ 2 BI ˜ ♦ ω 2 a 4(2 -l ) glyph[star] 1 + δ -l (1 + l ) 2 ρ 0 a Θ , if a glyph[lessmuch] a glyph[star] (4.47)</formula>
<text><location><page_73><loc_19><loc_53><loc_90><loc_67></location>so the range of values of w ρ which can cause accelerated expansion is indeed modified. Holonomy-type corrections increase the range since Θ ≤ 0 , and so may inverse-volume corrections. However, the latter also seems to include a cut-off when the last term of Eq. (4.47) in the a glyph[lessmuch] a glyph[star] regime dominates. Since a bounce requires ˙ a = 0 and a > 0 , the condition w b < w ρ < w a must be satisfied and so it must happen before the cut-off dominates if it is to happen at all.</text>
<section_header_level_1><location><page_73><loc_19><loc_47><loc_37><loc_48></location>4.4.3 Scalar field</section_header_level_1>
<text><location><page_73><loc_19><loc_38><loc_90><loc_44></location>I now investigate the effects that the inverse-volume and holonomy corrections can have when I couple gravity to an undeformed scalar field. In this case, the energy and pressure densities are given by</text>
<formula><location><page_73><loc_39><loc_32><loc_90><loc_35></location>ρ = 1 2 ˙ ϕ 2 + U ( ϕ ) , P = 1 2 ˙ ϕ 2 -U ( ϕ ) , (4.48)</formula>
<text><location><page_73><loc_19><loc_28><loc_82><loc_30></location>and the continuity equation gives us the equation of motion for the scalar field,</text>
<formula><location><page_73><loc_47><loc_24><loc_90><loc_25></location>¨ ϕ +3 H ˙ ϕ + U ' = 0 , (4.49)</formula>
<formula><location><page_73><loc_19><loc_18><loc_32><loc_21></location>where U ' = ∂U ∂ϕ .</formula>
<text><location><page_74><loc_19><loc_84><loc_90><loc_88></location>Let us investigate the era of slow-roll inflation. Using the assumptions | ¨ ϕ/U ' | glyph[lessmuch] 1 and 1 2 ˙ ϕ 2 glyph[lessmuch] U , I have the slow-roll equations,</text>
<formula><location><page_74><loc_40><loc_79><loc_90><loc_82></location>˙ ϕ = -U ' 3 H , (4.50a)</formula>
<formula><location><page_74><loc_40><loc_75><loc_90><loc_78></location>H 2 = σ ∅ γ ∅ 3 ω U ( 1 -σ ∅ γ 2 BI ˜ ♦ 3 ω 2 a 2 δ γ 2 ∅ U ) . (4.50b)</formula>
<text><location><page_74><loc_19><loc_68><loc_90><loc_72></location>If I substitute (4.50b) into (4.50a), take the derivative with respect to time and substitute in (4.50b) and (4.50a) again, I find</text>
<formula><location><page_74><loc_45><loc_63><loc_90><loc_66></location>¨ ϕ U ' = 1 3 η, ˙ ϕ 2 2 U = 1 3 glyph[epsilon1], (4.51)</formula>
<text><location><page_74><loc_19><loc_59><loc_47><loc_60></location>where the slow-roll parameters are</text>
<formula><location><page_74><loc_38><loc_54><loc_90><loc_57></location>η := 1 1 -ς ( ω γ ∅ U '' U -(1 -2 ς ) glyph[epsilon1] + χ -δς ) , (4.52a)</formula>
<formula><location><page_74><loc_38><loc_50><loc_90><loc_53></location>glyph[epsilon1] := 1 1 -ς ω 2 γ ∅ ( U ' U ) 2 , (4.52b)</formula>
<formula><location><page_74><loc_38><loc_46><loc_90><loc_49></location>χ := 1 -3 ς 2 ∂ log γ ∅ ∂ log a (4.52c)</formula>
<formula><location><page_74><loc_38><loc_42><loc_90><loc_46></location>ς := γ 2 BI ˜ ♦ 3 ω 2 a 2 δ γ 2 ∅ U, (4.52d)</formula>
<text><location><page_74><loc_19><loc_39><loc_54><loc_40></location>and the conditions for slow-roll inflation are</text>
<formula><location><page_74><loc_40><loc_34><loc_90><loc_35></location>| η | glyph[lessmuch] 1 , glyph[epsilon1] glyph[lessmuch] 1 , | χ | glyph[lessmuch] 1 , | ς | glyph[lessmuch] 1 . (4.53)</formula>
<text><location><page_74><loc_19><loc_24><loc_90><loc_30></location>I would like to investigate how these semi-classical effects affect the number of e-folds of the scale factor during inflation. The number of e-folds before the end of inflation N ( ϕ ) is defined by a ( ϕ ) = a end e -N ( ϕ ) , where</text>
<formula><location><page_74><loc_31><loc_18><loc_90><loc_22></location>N ( ϕ ) = -∫ ϕ ϕ end d ϕ H ˙ ϕ = ∫ ϕ ϕ end d ϕ γ ∅ U ωU ' ( 1 -γ 2 BI ˜ ♦ 3 ω 2 a 2 δ γ 2 ∅ U ) . (4.54)</formula>
<text><location><page_74><loc_19><loc_12><loc_90><loc_16></location>If I remove the explicit dependence on a from the integral by setting δ = 0 and γ ∅ = 1 (i.e. taking only a certain form of holonomy corrections and ignoring inverse-volume</text>
<text><location><page_75><loc_19><loc_87><loc_56><loc_88></location>corrections), and choose a power-law potential</text>
<formula><location><page_75><loc_44><loc_81><loc_90><loc_84></location>U ( ϕ ) = λ n ϕ n = ˜ λ n ϕ n ω 2 -n 2 , (4.55)</formula>
<text><location><page_75><loc_19><loc_77><loc_85><loc_79></location>where ˜ λ > 0 and n/ 2 ∈ N , then the number of e-folds before the end of inflation is</text>
<formula><location><page_75><loc_30><loc_72><loc_90><loc_75></location>N ( ϕ ) = 1 2 nω ( ϕ 2 -ϕ 2 end ) -γ 2 BI ˜ ♦ ˜ λ 3 n 2 ( n +2) ω 1+ n 2 ( ϕ 2+ n -ϕ 2+ n end ) . (4.56)</formula>
<text><location><page_75><loc_19><loc_63><loc_90><loc_69></location>If I take the approximation that slow-roll inflation is valid beyond the regime specified by (4.53), then I can calculate a value for the maximum number of e-folds by starting inflation at the big bounce,</text>
<formula><location><page_75><loc_33><loc_51><loc_90><loc_61></location>N max = 1 2 n ( 3 n γ 2 BI ˜ ♦ ˜ λ ) 2 n -ϕ 2 end ω -γ 2 BI ˜ ♦ ˜ λ 3 n 2 ( n +2) ( 3 n γ 2 BI ˜ ♦ ˜ λ ) 1+ 2 n -( ϕ 2 end ω ) 1+ n 2 , (4.57)</formula>
<text><location><page_75><loc_19><loc_47><loc_49><loc_48></location>and if I can assume ϕ 2 end /ω glyph[lessmuch] 1 , then</text>
<formula><location><page_75><loc_42><loc_40><loc_90><loc_45></location>N max = 1 2( n +2) ( 3 n γ 2 BI ˜ ♦ ˜ λ ) 2 /n . (4.58)</formula>
<text><location><page_75><loc_19><loc_34><loc_90><loc_38></location>Let us now find the attractor solutions for slow-roll inflation. Substituting the Hubble parameter (4.42) into the equation of motion for the scalar field (4.49), I obtain</text>
<formula><location><page_75><loc_26><loc_28><loc_90><loc_31></location>¨ ϕ + ˙ ϕ √ √ √ √ 3 γ ∅ ω ( 1 2 ˙ ϕ 2 + U ) { 1 -γ 2 BI ˜ ♦ 3 ω 2 a 2 δ γ 2 ∅ ( 1 2 ˙ ϕ 2 + U ) } + U ' = 0 . (4.59)</formula>
<text><location><page_75><loc_19><loc_18><loc_90><loc_25></location>I can remove the explicit scale-factor dependence of the equation by setting δ = 0 and γ ∅ = 1 (the same assumptions as I used to find N ). Then substituting in the power-law potential (4.55) I get</text>
<formula><location><page_75><loc_25><loc_12><loc_90><loc_16></location>¨ ϕ + ˙ ϕ √ √ √ √ 3 ω ( 1 2 ˙ ϕ 2 + λ n ϕ n ) { 1 -γ 2 BI ˜ ♦ 3 ω 2 ( 1 2 ˙ ϕ 2 + λ n ϕ n ) } + λϕ n -1 = 0 , (4.60)</formula>
<text><location><page_76><loc_19><loc_87><loc_76><loc_88></location>which is applicable only for the region where ρ is below a critical value,</text>
<formula><location><page_76><loc_43><loc_81><loc_90><loc_84></location>1 -γ 2 BI ˜ ♦ 3 ω 2 ( 1 2 ˙ ϕ 2 + λ n ϕ n ) > 0 , (4.61)</formula>
<text><location><page_76><loc_19><loc_75><loc_90><loc_79></location>otherwise H and ˙ ϕ are complex. I use this equation to plot phase space trajectories in Fig. 4.1.</text>
<text><location><page_76><loc_19><loc_71><loc_81><loc_73></location>I can find the slow-roll attractor solution for | ¨ ϕϕ 1 -n /λ | glyph[lessmuch] 1 and 1 2 ˙ ϕ 2 glyph[lessmuch] λ n ϕ n ,</text>
<formula><location><page_76><loc_38><loc_65><loc_90><loc_69></location>˙ ϕ ≈ -√ nλω 3 ϕ n 2 -1 ( 1 -γ 2 BI ˜ ♦ λ 3 nω 2 ϕ n ) -1 / 2 , (4.62)</formula>
<text><location><page_76><loc_19><loc_56><loc_90><loc_63></location>where the term in the bracket is the correction to the classical solution. Looking at Fig. 4.1(b) and 4.1(d), I conclude that the attractor solutions diverge from a linear relationship as they approach the boundary.</text>
<text><location><page_76><loc_19><loc_53><loc_72><loc_54></location>The condition for acceleration for the case I am considering here is</text>
<formula><location><page_76><loc_37><loc_47><loc_90><loc_51></location>w ρ < w a = -1 3 { 1 -2 γ 2 BI ˜ ♦ 3 ω 2 ( 1 2 ˙ ϕ 2 + λ n ϕ n ) } (4.63)</formula>
<text><location><page_76><loc_19><loc_30><loc_90><loc_45></location>where we can define the effective equation of state as w ρ = P ( ϕ ) /ρ ( ϕ ) using (4.48). I plot in Fig. 4.2 this region on the phase space of the scalar field to see how accelerated expansion can happen in a wider range than in the classical case. In order to be able to solve the equations and make plots, I have neglected non-zero values of δ and non-unity values of γ ∅ . It may be that in these cases the big bounce and inflation are no longer inevitable, as was found for the perfect fluid.</text>
<section_header_level_1><location><page_76><loc_19><loc_24><loc_37><loc_26></location>4.5 Discussion</section_header_level_1>
<text><location><page_76><loc_19><loc_14><loc_90><loc_21></location>In this chapter, I calculated the general conditions on a deformed action which has been formed from the variables ( q, v, w, R ) . I then found the nearest-order curvature corrections coming from the deformation by solving these conditions for a fourth order action. I found</text>
<text><location><page_77><loc_19><loc_84><loc_90><loc_88></location>that these corrections can act as a repulsive gravitational effect which may produce a big bounce.</text>
<text><location><page_77><loc_19><loc_63><loc_90><loc_82></location>When coupling gravity to a perfect fluid, the effects that the quantum corrections have depend on the equation of state, but inflation and a big bounce are possible. I coupled deformed gravity to an undeformed scalar in this preliminary investigation into higher order curvature corrections. I investigated slow-roll inflation and a big bounce in the presence of this scalar field. In chapter 5, I find that scalar fields must be deformed in much the same way as the metric. Therefore, these results might be interesting on some level, but cannot be taken too literally. Unfortunately, there was simply not enough time to research the fully deformed cases, hence why this material remains.</text>
<figure>
<location><page_78><loc_26><loc_58><loc_55><loc_79></location>
<caption>Figure 4.1: Line integral convolution plots showing trajectories in phase space for a scalar field with potential λϕ n /n with holonomy corrections. The hue at each point indicates the magnitude of the vector ( ˙ ϕ, ¨ ϕ ) , with blue indicating low values. The trajectories do not extend outside of the region (4.61). The attractor solution (the trajectory approached by a wide range of inital conditions) is well approximated by (4.62), corresponding to slow-roll inflation. I use ˜ λ = (8 π · ) (4 -n ) / 2 , ˜ ♦ = √ 3 γ BI / 4 , δ = 0 , γ ∅ = 1 . Plots are in Planck units, ω = 1 / 8 π ·</caption>
</figure>
<figure>
<location><page_78><loc_56><loc_58><loc_84><loc_78></location>
<caption>(a) Full phase space for U ( ϕ ) = λϕ 2 / 2</caption>
</figure>
<figure>
<location><page_78><loc_26><loc_34><loc_54><loc_55></location>
<caption>(b) Attractor solution for U ( ϕ ) = λϕ 2 / 2</caption>
</figure>
<figure>
<location><page_78><loc_56><loc_34><loc_84><loc_54></location>
<caption>(c) Full phase space for U ( ϕ ) = λϕ 4 / 4</caption>
</figure>
<paragraph><location><page_78><loc_56><loc_33><loc_83><loc_34></location>(d) Attractor solution for U ( ϕ ) = λϕ 4 / 4</paragraph>
<figure>
<location><page_79><loc_22><loc_44><loc_88><loc_68></location>
<caption>(a) Accelerating values of w ρ for U ( ϕ ) = λϕ 2 / 2 (b) Accelerating values of w ρ for U ( ϕ ) = λϕ 4 / 4</caption>
</figure>
<paragraph><location><page_79><loc_27><loc_30><loc_82><loc_41></location>Figure 4.2: Contour plots showing the region in scalar phase space satisfying the condition for accelerated expansion when holonomy corrections are included (4.63). The dashed line indicates the classical acceleration condition w a = -1 / 3 and the dotted line indicates the bounce boundary. The white line indicates the slow-roll solution (4.62). The contours indicate the value of w ρ by their colour, and the most blue contour is for w ρ ≈ 0 . 2 . I use ˜ λ = (8 π · ) (4 -n ) / 2 , ˜ ♦ = √ 3 γ BI / 4 , δ = 0 , γ ∅ = 1 . Plots are in Planck units, ω = 1 / 8 π · .</paragraph>
<section_header_level_1><location><page_80><loc_19><loc_79><loc_33><loc_81></location>Chapter 5</section_header_level_1>
<section_header_level_1><location><page_80><loc_19><loc_64><loc_89><loc_73></location>Deformed scalar-tensor constraint to all orders</section_header_level_1>
<text><location><page_80><loc_19><loc_49><loc_90><loc_58></location>In this chapter I find the most general gravitational constraint which satisfies the deformed constraint algebra. To find the constraint is easier than finding the action, so I also include a non-minimally coupled scalar field in order to find the most general deformed scalartensor constraint. This material has not been previously published.</text>
<text><location><page_80><loc_19><loc_27><loc_90><loc_47></location>As stated in chapter 2, I am not looking for models with degrees of freedom beyond a simple scalar-tensor model. Since actions which contain Riemann tensor squared contractions introduce additional tensor degrees of freedom [77], I automatically do not consider such terms here. This means I only need to expand the constraint using variables which are tensor contractions containing up to two orders of spatial derivatives or up to two in momenta. It also means I do not need to consider spatial derivatives of momenta in the constraint. Therefore, for a metric tensor field ( q ab , p cd ) and a scalar field ( ψ, π ) , I expand the constraint into the following variables,</text>
<formula><location><page_80><loc_22><loc_20><loc_90><loc_25></location>q = det q ab , p = q ab p ab , P = Q abcd p ab T p cd T , R, ψ, π, ∆ := q ab ∇ a ∇ b ψ = ∂ 2 ψ -q ab Γ c ab ∂ c ψ, γ := q ab ∇ a ψ ∇ b ψ = ∂ a ψ∂ a ψ, (5.1)</formula>
<text><location><page_80><loc_19><loc_13><loc_90><loc_17></location>where p ab T := p ab -1 3 pq ab is the traceless part of the metric momentum. Therefore, I start with the constraint given by C = C ( q, p, P , R, ψ, π, ∆ , γ ) . I must solve the distribution</text>
<text><location><page_81><loc_19><loc_82><loc_90><loc_88></location>equation again to find the equations which restrict the form of the constraint. The calculations in this chapter generalise those presented in chapter 3 where the minimally deformed scalar-tensor constraint was regained from the constraint algebra.</text>
<section_header_level_1><location><page_81><loc_19><loc_75><loc_63><loc_77></location>5.1 Solving the distribution equation</section_header_level_1>
<text><location><page_81><loc_19><loc_68><loc_90><loc_72></location>Starting from (2.41), I have the general distribution equation for a Hamiltonian constraint, without derivatives of the momenta, which depends on a metric tensor and a scalar field,</text>
<formula><location><page_81><loc_29><loc_62><loc_90><loc_66></location>0 = δC ( x ) δq ab ( y ) ∂C ∂p ab ∣ ∣ ∣ ∣ y + δC ( x ) δψ ( y ) ∂C ∂π ∣ ∣ ∣ ∣ y -( βD a ∂ a ) x δ ( x, y ) -( x ↔ y ) . (5.2)</formula>
<text><location><page_81><loc_19><loc_48><loc_90><loc_60></location>To solve this I will take the functional derivative with respect to a momentum variable, manipulate a few steps and then integrate with a test tensor to find several equations which the constraint must satisfy. Since I have two fields, I must do this procedure twice. The first route I consider will be where I take the derivative with respect to the metric momentum.</text>
<section_header_level_1><location><page_81><loc_19><loc_43><loc_34><loc_45></location>5.1.1 p ab route</section_header_level_1>
<text><location><page_81><loc_19><loc_36><loc_90><loc_40></location>Starting from the distribution equation (5.2), relabel indices, then take the functional derivative with respect to p ab ( z ) ,</text>
<formula><location><page_81><loc_30><loc_22><loc_90><loc_34></location>0 = δC ( x ) δq cd ( y ) ∂ 2 C ∂p ab ∂p cd ∣ ∣ ∣ ∣ y δ ( z, y ) + δ∂C ( x ) δq cd ( y ) ∂p ab ( x ) ∂C ∂p cd ∣ ∣ ∣ ∣ y δ ( z, x ) + δC ( x ) δψ ( y ) ∂ 2 C ∂p ab ∂π ∣ ∣ ∣ ∣ y δ ( z, y ) + δ∂C ( x ) δψ ( y ) ∂p ab ( x ) ∂C ∂π ∣ ∣ ∣ ∣ y δ ( z, x ) -∂ c ( x ) δ ( x, y ) ( ∂ ( βD c ) ∂p ab + β ∂D c ∂p ab ,d ∂ d ) x δ ( z, x ) -( x ↔ y ) . (5.3)</formula>
<text><location><page_81><loc_19><loc_19><loc_85><loc_20></location>Move derivatives and discard surface terms so that it is reorganised into the form,</text>
<formula><location><page_81><loc_40><loc_14><loc_90><loc_15></location>0 = A ab ( x, y ) δ ( z, y ) -A ab ( y, x ) δ ( z, x ) , (5.4)</formula>
<text><location><page_82><loc_19><loc_87><loc_24><loc_88></location>where,</text>
<formula><location><page_82><loc_29><loc_72><loc_90><loc_85></location>A ab ( x, y ) = δC ( x ) δq cd ( y ) ∂ 2 C ∂p ab ∂p cd ∣ ∣ ∣ ∣ y -δ∂C ( y ) δq cd ( x ) ∂p ab ( y ) ∂C ∂p cd ∣ ∣ ∣ ∣ x + δC ( x ) δψ ( y ) ∂ 2 C ∂p ab ∂π ∣ ∣ ∣ ∣ y -δ∂C ( y ) δψ ( x ) ∂p ab ( y ) ∂C ∂π ∣ ∣ ∣ ∣ x + ( ∂ ( βD c ) ∂p ab ∂ c ) y δ ( y, x ) -∂ d ( y ) ( β ∂D c ∂p ab ,d ∂ c ) y δ ( y, x ) . (5.5)</formula>
<text><location><page_82><loc_19><loc_66><loc_90><loc_70></location>If I take (5.4) and integrate over y , I can find A ab ( x, y ) in terms of a function dependent on only a single independent variable,</text>
<formula><location><page_82><loc_29><loc_61><loc_90><loc_63></location>0 = A ab ( x, z ) -A ab ( x ) δ ( z, x ) , where , A ab ( x ) = ∫ d 3 yA ab ( y, x ) . (5.6)</formula>
<text><location><page_82><loc_19><loc_54><loc_90><loc_58></location>I then multiply this by an arbitrary, symmetric test tensor θ ab ( z ) , integrate over z , and separate out different orders of derivatives of θ ab ,</text>
<formula><location><page_82><loc_29><loc_33><loc_90><loc_52></location>0 = θ ab ( · · · ) ab + ∂ c θ ab { ∂C ∂q ef,c ∂ 2 C ∂p ab ∂p ef +2 ∂ 2 C ∂q ef,cd ∂ d ( ∂ 2 C ∂p ab p ef ) + ∂C ∂p ef ∂ 2 C ∂q ef,c ∂p ab -2 ∂C ∂p ef ∂ d ( ∂ 2 C ∂q ef,cd ∂p ab ) + ∂C ∂ψ ,c ∂ 2 C ∂p ab ∂π +2 ∂C ∂ψ ,cd ∂ d ( ∂ 2 C ∂p ab ∂π ) + ∂C ∂π ∂ 2 C ∂ψ ,c ∂p ab -2 ∂C ∂π ∂ d ( ∂ 2 C ∂ψ ,cd ∂p ab ) -∂ ( βD c ) ∂p ab -∂ d ( β ∂D d ∂p ab ,c )} + ∂ cd θ ab { ∂C ∂q ef,cd ∂ 2 C ∂p ab ∂p ef -∂C ∂p ef ∂ 2 C ∂q ef,cd ∂p ab + ∂C ∂ψ ,cd ∂ 2 C ∂p ab ∂π -∂C ∂π ∂ 2 C ∂ψ ,cd ∂p ab -β ∂D c ∂p ab ,d } . (5.7)</formula>
<text><location><page_82><loc_19><loc_27><loc_90><loc_31></location>As done in previous chapters, I disregard the term zeroth order derivative of θ ab because it does not provide useful information.</text>
<text><location><page_82><loc_19><loc_13><loc_90><loc_25></location>Before I can attempt to interpret this equation, I must first separate out all the different tensor combinations that there are. Because θ ab is arbitrary, the coefficients of each unique tensor combination must vanish independently. When I substitute in C = C ( q, p, P , R, ψ, π, ∆ , γ ) , there are many complicated tensor combinations that need to be considered, so for convenience I define X a := q bc ∂ a q bc .</text>
<text><location><page_83><loc_19><loc_84><loc_90><loc_88></location>I evaluate each term in the ∂ cd θ ab bracket, and write them in (D.2), in appendix D. So the linearly independent terms depending on ∂ cd θ ab produce the following conditions,</text>
<formula><location><page_83><loc_29><loc_79><loc_48><loc_82></location>∂ ab θ ab : 0 = ∂C ∂C + β,</formula>
<formula><location><page_83><loc_24><loc_61><loc_90><loc_81></location>∂R ∂ P (5.8a) q ab ∂ 2 θ ab : 0 = ∂C ∂p ∂ 2 C ∂p∂R -∂C ∂R ( ∂ 2 C ∂p 2 + 1 3 ∂C ∂ P ) + 1 2 ∂C ∂ ∆ ∂ 2 C ∂π∂p -1 2 ∂C ∂π ∂ 2 C ∂p∂ ∆ , (5.8b) q ab p cd T ∂ cd θ ab : 0 = ∂C ∂R ∂ 2 C ∂p∂ P -∂C ∂ P ∂ 2 C ∂p∂R , (5.8c) p T ab ∂ 2 θ ab : 0 = ∂C ∂p ∂ 2 C ∂ P ∂R -∂C ∂R ∂ 2 C ∂p∂ P + 1 2 ∂C ∂ ∆ ∂ 2 C ∂π∂ P -1 2 ∂C ∂π ∂ 2 C ∂ P ∂ ∆ , (5.8d) p T ab p cd T ∂ cd θ ab : 0 = ∂C ∂R ∂ 2 C ∂ P 2 -∂C ∂ P ∂ 2 C ∂ P ∂R . (5.8e)</formula>
<text><location><page_83><loc_19><loc_49><loc_90><loc_58></location>I then evaluate each term in the ∂ c θ ab bracket of (5.7) and write them in (D.3). There are many unique terms which should be considered here, but in this case most of these are already solved by a constraint which satisfies (5.8). So the equations containing new information are,</text>
<formula><location><page_83><loc_27><loc_44><loc_62><loc_47></location>∂ a ψ∂ b θ ab : 0 = ( 2 ∂C ∂R ∂ ψ -∂C ∂ ∆ ) ∂C ∂ P + ∂ ψ β,</formula>
<formula><location><page_83><loc_22><loc_18><loc_81><loc_21></location>p T ab p cd T ∂ d ψ∂ c θ ab : 0 = ( 2 ∂C ∂R ∂ ψ -∂C ∂ ∆ ) ∂ 2 C ∂ P 2 -∂C ∂ P ( 2 ∂ ψ ∂ 2 C ∂ P ∂R + ∂ 2 C ∂ P ∂ ∆ )</formula>
<formula><location><page_83><loc_22><loc_19><loc_90><loc_46></location>(5.9a) q ab ∂ c ψ∂ c θ ab : 0 = ( 1 2 ∂C ∂ ∆ -4 ∂C ∂R ∂ ψ )( ∂ 2 C ∂p 2 + 1 3 ∂C ∂ P ) + 1 2 ∂C ∂ ∆ ∂C ∂ P + ∂C ∂p ( 1 2 ∂ 2 C ∂p∂ ∆ +4 ∂ ψ ∂ 2 C ∂p∂R ) +2 ( ∂C ∂γ + ∂C ∂ ∆ ∂ ψ ) ∂ 2 C ∂π∂p +2 ∂C ∂π ( ∂ 2 C ∂p∂γ -∂ ψ ∂ 2 C ∂p∂ ∆ ) -π ∂β ∂p , (5.9b) p T ab ∂ c ψ∂ c θ ab : 0 = ( 1 2 ∂C ∂ ∆ -4 ∂C ∂R ) ∂ 2 C ∂p∂ P + ∂C ∂p ( 1 2 ∂ 2 C ∂ P ∂ ∆ +4 ∂ ψ ∂ 2 C ∂ P ∂R ) +2 ( ∂C ∂γ + ∂C ∂ ∆ ∂ ψ ) ∂ 2 C ∂π∂ P +2 ∂C ∂π ( ∂ 2 C ∂ P ∂γ -∂ ψ ∂ 2 C ∂ P ∂ ∆ ) -π ∂β ∂ P , (5.9c) q ab p cd T ∂ d ψ∂ c θ ab : 0 = ( 2 ∂C ∂R ∂ ψ -∂C ∂ ∆ ) ∂ 2 C ∂p∂ P -∂C ∂ P ( 2 ∂ ψ ∂ 2 C ∂p∂R + ∂ 2 C ∂p∂ ∆ ) (5.9d) (5.9e)</formula>
<formula><location><page_84><loc_27><loc_85><loc_90><loc_88></location>X a ∂ b θ ab : 0 = ∂C ∂R (1 + 2 ∂ q ) ∂C ∂ P + ∂ q β, (5.9f)</formula>
<formula><location><page_84><loc_24><loc_82><loc_80><loc_85></location>q ab X c ∂ c θ ab : 0 = ∂C ∂p (4 ∂ q -1) ∂ 2 C ∂p∂R -∂C ∂R (4 ∂ q +1) ( ∂ 2 C ∂p 2 + 1 3 ∂C ∂ P )</formula>
<formula><location><page_84><loc_36><loc_78><loc_90><loc_82></location>+ 1 2 ∂C ∂π (1 -4 ∂ q ) ∂ 2 C ∂ ∆ ∂p + 1 2 ∂C ∂ ∆ (1 + 4 ∂ q ) ∂ 2 C ∂π∂p -1 3 p ∂β ∂p , (5.9g)</formula>
<formula><location><page_84><loc_24><loc_75><loc_72><loc_78></location>p T ab X c ∂ c θ ab : 0 = ∂C ∂p (4 ∂ q -1) ∂ 2 C ∂ P ∂R -∂C ∂R (4 ∂ q +1) ∂ 2 C ∂ P ∂p</formula>
<formula><location><page_84><loc_36><loc_71><loc_90><loc_75></location>+ 1 2 ∂C ∂π (1 -4 ∂ q ) ∂ 2 C ∂ P ∂ ∆ + 1 2 ∂C ∂ ∆ (1 + 4 ∂ q ) ∂ 2 C ∂ P ∂π -1 3 p ∂β ∂ P , (5.9h)</formula>
<formula><location><page_84><loc_22><loc_68><loc_90><loc_71></location>q ab p cd T X d ∂ c θ ab : 0 = ∂C ∂R (1 + 2 ∂ q ) ∂ 2 C ∂p∂ P + ∂C ∂ P (1 -2 ∂ q ) ∂ 2 C ∂p∂R , (5.9i)</formula>
<formula><location><page_84><loc_22><loc_64><loc_90><loc_67></location>p T ab p cd T X d ∂ c θ ab : 0 = ∂C ∂R (1 + 2 ∂ q ) ∂ 2 C ∂ P 2 + ∂C ∂ P (1 -2 ∂ q ) ∂ 2 C ∂ P ∂R , (5.9j)</formula>
<formula><location><page_84><loc_34><loc_57><loc_90><loc_60></location>∂ a F∂ b θ ab : 0 = 2 ∂C ∂R ∂ 2 C ∂F∂ P + ∂β ∂F , (5.9k)</formula>
<formula><location><page_84><loc_32><loc_53><loc_90><loc_57></location>q ab ∂ c F∂ c θ ab : 0 = 2 ∂C ∂p ∂ 3 C ∂F∂p∂R -2 ∂C ∂R ∂ ∂F ( ∂ 2 C ∂p 2 + 1 3 ∂C ∂ P ) (5.9l)</formula>
<text><location><page_84><loc_44><loc_51><loc_46><loc_52></location>+</text>
<text><location><page_84><loc_46><loc_52><loc_49><loc_53></location>∂C</text>
<text><location><page_84><loc_46><loc_50><loc_47><loc_51></location>∂</text>
<text><location><page_84><loc_48><loc_50><loc_49><loc_51></location>∆</text>
<text><location><page_84><loc_51><loc_52><loc_52><loc_53></location>∂</text>
<text><location><page_84><loc_52><loc_53><loc_53><loc_53></location>3</text>
<text><location><page_84><loc_53><loc_52><loc_54><loc_53></location>C</text>
<text><location><page_84><loc_49><loc_50><loc_56><loc_51></location>∂F∂p∂π</text>
<text><location><page_84><loc_57><loc_51><loc_58><loc_52></location>-</text>
<text><location><page_84><loc_59><loc_52><loc_61><loc_53></location>∂C</text>
<text><location><page_84><loc_59><loc_50><loc_61><loc_51></location>∂π</text>
<text><location><page_84><loc_64><loc_52><loc_65><loc_53></location>∂</text>
<text><location><page_84><loc_65><loc_53><loc_65><loc_53></location>3</text>
<text><location><page_84><loc_65><loc_52><loc_67><loc_53></location>C</text>
<text><location><page_84><loc_62><loc_50><loc_67><loc_51></location>∂F∂p∂</text>
<text><location><page_84><loc_67><loc_50><loc_69><loc_51></location>∆</text>
<formula><location><page_84><loc_54><loc_49><loc_68><loc_50></location>3 3</formula>
<text><location><page_84><loc_69><loc_51><loc_71><loc_52></location>+</text>
<text><location><page_84><loc_71><loc_52><loc_72><loc_53></location>1</text>
<text><location><page_84><loc_71><loc_50><loc_72><loc_51></location>3</text>
<text><location><page_84><loc_73><loc_51><loc_73><loc_52></location>δ</text>
<text><location><page_84><loc_73><loc_52><loc_74><loc_53></location>p</text>
<text><location><page_84><loc_73><loc_51><loc_74><loc_52></location>F</text>
<text><location><page_84><loc_75><loc_52><loc_77><loc_53></location>∂β</text>
<text><location><page_84><loc_75><loc_50><loc_77><loc_51></location>∂p</text>
<text><location><page_84><loc_77><loc_51><loc_78><loc_52></location>,</text>
<formula><location><page_84><loc_32><loc_46><loc_90><loc_50></location>p T ab ∂ c F∂ c θ ab : 0 = 2 ∂C ∂p ∂ C ∂F∂ P ∂R -2 ∂C ∂R ∂ C ∂F∂p∂ P (5.9m)</formula>
<text><location><page_84><loc_44><loc_44><loc_46><loc_45></location>+</text>
<text><location><page_84><loc_46><loc_45><loc_49><loc_46></location>∂C</text>
<text><location><page_84><loc_46><loc_43><loc_47><loc_44></location>∂</text>
<text><location><page_84><loc_48><loc_43><loc_49><loc_44></location>∆</text>
<text><location><page_84><loc_51><loc_45><loc_52><loc_46></location>∂</text>
<text><location><page_84><loc_52><loc_45><loc_53><loc_46></location>3</text>
<text><location><page_84><loc_49><loc_43><loc_53><loc_44></location>∂F∂</text>
<text><location><page_84><loc_53><loc_43><loc_54><loc_44></location>P</text>
<text><location><page_84><loc_54><loc_43><loc_57><loc_44></location>∂π</text>
<text><location><page_84><loc_57><loc_44><loc_59><loc_45></location>-</text>
<text><location><page_84><loc_59><loc_45><loc_62><loc_46></location>∂C</text>
<text><location><page_84><loc_59><loc_43><loc_62><loc_44></location>∂π</text>
<text><location><page_84><loc_64><loc_45><loc_65><loc_46></location>∂</text>
<text><location><page_84><loc_65><loc_45><loc_66><loc_46></location>3</text>
<text><location><page_84><loc_62><loc_43><loc_66><loc_44></location>∂F∂</text>
<text><location><page_84><loc_66><loc_43><loc_67><loc_44></location>P</text>
<text><location><page_84><loc_67><loc_43><loc_68><loc_44></location>∂</text>
<text><location><page_84><loc_68><loc_43><loc_70><loc_44></location>∆</text>
<text><location><page_84><loc_70><loc_44><loc_72><loc_45></location>+</text>
<text><location><page_84><loc_72><loc_45><loc_73><loc_46></location>1</text>
<text><location><page_84><loc_72><loc_43><loc_73><loc_44></location>3</text>
<text><location><page_84><loc_74><loc_44><loc_74><loc_45></location>δ</text>
<text><location><page_84><loc_74><loc_45><loc_75><loc_45></location>p</text>
<text><location><page_84><loc_74><loc_44><loc_75><loc_44></location>F</text>
<text><location><page_84><loc_76><loc_45><loc_78><loc_46></location>∂β</text>
<text><location><page_84><loc_76><loc_43><loc_77><loc_44></location>∂</text>
<text><location><page_84><loc_77><loc_43><loc_78><loc_44></location>P</text>
<text><location><page_84><loc_78><loc_44><loc_79><loc_45></location>,</text>
<formula><location><page_84><loc_30><loc_40><loc_90><loc_43></location>q ab p cd T ∂ d F∂ c θ ab : 0 = ∂C ∂R ∂ 3 C ∂F∂p∂ P -∂C ∂ P ∂ 3 C ∂F∂p∂R , (5.9n)</formula>
<formula><location><page_84><loc_31><loc_36><loc_90><loc_39></location>p T ab p cd T ∂ c Fθ ab : 0 = ∂C ∂R ∂ 3 C ∂F∂ P 2 -∂C ∂ P ∂ 3 C ∂F∂ P ∂R , (5.9o)</formula>
<text><location><page_84><loc_19><loc_27><loc_90><loc_34></location>where F ∈ { p, P , R, ∆ , γ } . These conditions strongly restrict the form of the constraint, but before I attempt to consolidate them I must find the conditions coming from the scalar field.</text>
<text><location><page_84><loc_53><loc_45><loc_55><loc_46></location>C</text>
<text><location><page_84><loc_66><loc_45><loc_68><loc_46></location>C</text>
<section_header_level_1><location><page_85><loc_19><loc_87><loc_33><loc_88></location>5.1.2 π route</section_header_level_1>
<text><location><page_85><loc_19><loc_80><loc_90><loc_84></location>Similar to the calculation using the metric momentum, I return to the distribution equation (5.2) and take the functional derivative with respect to π ( z ) ,</text>
<formula><location><page_85><loc_31><loc_66><loc_90><loc_78></location>0 = δC ( x ) δq ab ( y ) ∂ 2 C ∂π∂p ab ∣ ∣ ∣ ∣ y δ ( z, y ) + δ∂C ( x ) δq ab ( y ) ∂π ( x ) ∂C ∂p ab ∣ ∣ ∣ ∣ y δ ( z, x ) + δC ( x ) δψ ( y ) ∂ 2 C ∂π 2 ∣ ∣ ∣ ∣ y δ ( z, y ) + δ∂C ( x ) δψ ( y ) ∂π ( x ) ∂C ∂π ∣ ∣ ∣ ∣ y δ ( z, x ) -δ ( z, x ) ( ∂ ( βD a ) ∂π ∂ a ) x δ ( x, y ) -( x ↔ y ) , (5.10)</formula>
<text><location><page_85><loc_19><loc_63><loc_40><loc_64></location>which can be rewritten as,</text>
<formula><location><page_85><loc_41><loc_59><loc_90><loc_60></location>0 = A ( x, y ) δ ( z, y ) -A ( y, x ) δ ( z, x ) , (5.11)</formula>
<text><location><page_85><loc_19><loc_54><loc_24><loc_55></location>where,</text>
<formula><location><page_85><loc_26><loc_44><loc_90><loc_52></location>A ( x, y ) = δC ( x ) δq ab ( y ) ∂ 2 C ∂π∂p ab ∣ ∣ ∣ ∣ y -δ∂C ( y ) δq ab ( x ) ∂π ( y ) ∂C ∂p ab ∣ ∣ ∣ ∣ x + δC ( x ) δψ ( y ) ∂ 2 C ∂π 2 ∣ ∣ ∣ ∣ y -δ∂C ( y ) δψ ( x ) ∂π ( y ) ∂C ∂π ∣ ∣ ∣ ∣ x + ( ∂ ( βD a ) ∂π ∂ a ) y δ ( y, x ) , (5.12)</formula>
<text><location><page_85><loc_19><loc_38><loc_90><loc_42></location>and similar to above, (5.11) can be solved to find 0 = A ( x, z ) -A ( x ) δ ( x, z ) . Multiply this by a test scalar field η ( z ) and integrate over z ,</text>
<formula><location><page_85><loc_23><loc_22><loc_90><loc_36></location>0 = η ( · · · ) + ∂ a η { ∂C ∂q cd,a ∂ 2 C ∂π∂p cd +2 ∂C ∂q cd,ab ∂ b ( ∂ 2 C ∂π∂p cd ) + ∂C ∂p cd ∂ 2 C ∂q cd,a ∂π -2 ∂C ∂p cd ∂ 2 C ∂q cd,ab ∂π + ∂C ∂ψ ,a ∂ 2 C ∂π 2 +2 ∂C ∂ψ ,ab ∂ b ( ∂ 2 C ∂π 2 ) + ∂C ∂π ∂ 2 C ∂ψ ,a ∂π -2 ∂C ∂π ∂ b ( ∂ 2 C ∂ψ ,ab ∂π ) -∂ ( βD a ) ∂π } + ∂ ab η { ∂C ∂q cd,ab ∂ 2 C ∂π∂p cd -∂C ∂p cd ∂ 2 C ∂q cd,ab ∂π + ∂C ∂ψ ,ab ∂ 2 C ∂π 2 -∂C ∂π ∂ 2 C ∂ψ ,ab ∂π } . (5.13)</formula>
<text><location><page_86><loc_19><loc_84><loc_90><loc_88></location>I evaluate each of the terms for ∂ ab η , and write them in (D.4). From these, I find the independent equations,</text>
<formula><location><page_86><loc_29><loc_79><loc_90><loc_82></location>∂ 2 η : 0 = ∂C ∂p ∂ 2 C ∂π∂R -∂C ∂R ∂ 2 C ∂π∂p + 1 2 ∂C ∂ ∆ ∂ 2 C ∂π 2 -1 2 ∂C ∂π ∂ 2 C ∂ ∆ ∂π , (5.14a)</formula>
<formula><location><page_86><loc_27><loc_75><loc_90><loc_78></location>p ab T ∂ ab η : 0 = ∂C ∂R ∂ 2 C ∂π∂ P -∂C ∂ P ∂ 2 C ∂π∂R . (5.14b)</formula>
<text><location><page_86><loc_19><loc_69><loc_90><loc_72></location>Then, I evaluate all the terms for ∂ a η , and write them in (D.5). Therefore, ignoring terms solved by (5.14), the equations I get from ∂ a η are,</text>
<formula><location><page_86><loc_25><loc_60><loc_82><loc_66></location>∂ a ψ∂ a η : 0 = ( 1 2 ∂C ∂ ∆ -4 ∂C ∂R ∂ ψ ) ∂ 2 C ∂π∂p + ∂C ∂p ( 1 2 ∂ 2 C ∂ ∆ ∂π +4 ∂ ψ ∂ 2 C ∂R∂π ) +2 ( ∂C ∂γ + ∂C ∂ ∆ ∂ ψ ) ∂ 2 C ∂π 2 +2 ∂C ∂π ( ∂ 2 C ∂γ∂π -∂ ψ ∂ 2 C ∂ ∆ ∂π ) -( β + π ∂β ∂π ) ,</formula>
<formula><location><page_86><loc_22><loc_35><loc_90><loc_63></location>(5.15a) p ab T ∂ b ψ∂ a η : 0 = ( ∂C ∂R ∂ ψ -1 2 ∂C ∂ ∆ ) ∂ 2 C ∂π∂ P -∂C ∂ P ( ∂ ψ ∂ 2 C ∂π∂R + 1 2 ∂ 2 C ∂π∂ ∆ ) , (5.15b) X a ∂ a η : 0 = ∂C ∂p (4 ∂ q -1) ∂ 2 C ∂π∂R -∂C ∂R (4 ∂ q +1) ∂ 2 C ∂π∂p + 1 2 ∂C ∂ ∆ (1 + 4 ∂ q ) ∂ 2 C ∂π 2 + 1 2 ∂C ∂π (1 -4 ∂ q ) ∂ 2 C ∂π∂ ∆ -1 3 p ∂β ∂π , (5.15c) p ab T X b ∂ a η : 0 = ∂C ∂R (1 + 2 ∂ q ) ∂ 2 C ∂π∂ P + ∂C ∂ P (1 -2 ∂ q ) ∂ 2 C ∂π∂R , (5.15d) ∂ a F∂ a η : 0 = ∂C ∂p ∂ 3 C ∂F∂π∂R -∂C ∂R ∂ 3 C ∂F∂π∂R + 1 2 ∂C ∂ ∆ ∂ 3 C ∂F∂π 2 -1 2 ∂C ∂π ∂ 3 C ∂F∂π∂ ∆ + 1 6 δ p F ∂β ∂π , (5.15e) p ab T ∂ b F∂ a η : 0 = ∂C ∂R ∂ 3 C ∂F∂π∂ P -∂C ∂ P ∂ 3 C ∂F∂π∂R , (5.15f)</formula>
<text><location><page_86><loc_19><loc_29><loc_90><loc_33></location>where F ∈ { p, P , R, ∆ , γ } . Now that I have all of the conditions restricting the form of the constraint, I can move on to consolidating and interpreting them.</text>
<section_header_level_1><location><page_86><loc_19><loc_23><loc_55><loc_24></location>5.2 Solving for the constraint</section_header_level_1>
<text><location><page_86><loc_19><loc_15><loc_90><loc_19></location>Now I have the full list of equations, I seek to find the restrictions on the form of C they impose. Firstly, I use the condition from ∂ ab θ ab , (5.8a) to find</text>
<formula><location><page_86><loc_47><loc_10><loc_90><loc_13></location>∂C ∂R = -β ( ∂C ∂ P ) -1 , (5.16)</formula>
<text><location><page_87><loc_19><loc_86><loc_68><loc_88></location>which I substitute into the equation from p T ab p cd T ∂ cd θ ab , (5.8e),</text>
<formula><location><page_87><loc_43><loc_74><loc_90><loc_84></location>0 = ∂C ∂R ∂ 2 C ∂ P 2 -∂C ∂ P ∂ 2 C ∂ P ∂R = -2 β ( ∂C ∂ P ) -1 ∂ 2 C ∂ P 2 + ∂β ∂ P = β ∂ ∂ P log { β ( ∂C ∂ P ) -2 } , (5.17)</formula>
<text><location><page_87><loc_19><loc_70><loc_86><loc_71></location>and because β → 1 in the classical limit and so cannot vanish generally, I find that,</text>
<formula><location><page_87><loc_41><loc_64><loc_90><loc_68></location>β = b 1 ( ∂C ∂ P ) 2 , where ∂b 1 ∂ P = 0 . (5.18)</formula>
<text><location><page_87><loc_19><loc_58><loc_90><loc_62></location>Substituting this back into (5.16) gives me ∂C ∂R = -b 1 ∂C ∂ P , and from this I can find the first restriction on the form of the constraint,</text>
<formula><location><page_87><loc_36><loc_49><loc_90><loc_54></location>C ( q, p, P , R, ψ, π, ∆ , γ ) = C 1 ( q, p, ψ, π, ∆ , γ, χ 1 ) , where χ 1 := P -∫ R 0 b 1 ( q, p, x, ψ, π, ∆ , ψ )d x. (5.19)</formula>
<text><location><page_87><loc_19><loc_45><loc_70><loc_47></location>Substituting this into the condition from ∂ a F∂ b θ ab , (5.9k), gives</text>
<formula><location><page_87><loc_36><loc_40><loc_90><loc_43></location>0 = ∂b 1 ∂F ( ∂C 1 ∂χ 1 ) 2 , for F ∈ { p, P , R, π, ∆ , γ } , (5.20)</formula>
<text><location><page_87><loc_19><loc_33><loc_90><loc_37></location>and therefore b 1 must only be a function of q and ψ . Substituting this into (5.19) leads to χ 1 = P b 1 R . Turning to the condition from X a ∂ b θ ab , (5.9f), I find</text>
<formula><location><page_87><loc_45><loc_28><loc_90><loc_31></location>0 = ( ∂C 1 ∂χ 1 ) 2 ( ∂ q -1) b 1 , (5.21)</formula>
<text><location><page_87><loc_19><loc_19><loc_90><loc_25></location>which is solved by b 1 ( q, ψ ) = q b 2 ( ψ ) . This is as expected because it means both terms in χ 1 have a density weight of two. From this I see that the condition coming from ∂ a ψ∂ b θ ab , (5.9a), gives</text>
<formula><location><page_87><loc_44><loc_15><loc_90><loc_18></location>0 = ∂C 1 ∂χ 1 ( qb ' 2 ∂C 1 ∂χ 1 -∂C 1 ∂ ∆ ) (5.22)</formula>
<text><location><page_88><loc_19><loc_87><loc_71><loc_88></location>which provides further restrictions on the form of the constraint,</text>
<formula><location><page_88><loc_34><loc_82><loc_90><loc_84></location>C = C 2 ( q, p, ψ, π, γ, χ 2 ) , χ 2 := P q ( b 2 R -b ' 2 ∆ ) . (5.23)</formula>
<text><location><page_88><loc_19><loc_77><loc_54><loc_79></location>Look at the condition from p T ab ∂ 2 θ ab , (5.8d),</text>
<formula><location><page_88><loc_23><loc_64><loc_90><loc_75></location>0 = ∂C ∂p ∂ 2 C ∂ P ∂R -∂C ∂R ∂ 2 C ∂p∂ P + 1 2 ∂C ∂ ∆ ∂ 2 C ∂π∂ P -1 2 ∂C ∂π ∂ 2 C ∂ P ∂ ∆ , = qb 2 { -∂C 2 ∂p ∂ 2 C 2 ∂χ 2 2 + ∂C 2 ∂χ 2 ∂ 2 C ∂p∂χ 2 + b ' 2 2 b 2 ( ∂C 2 ∂χ 2 ∂ 2 C 2 ∂π∂χ 2 -∂C 2 ∂π ∂ 2 C 2 ∂χ 2 2 )} = qb 2 ∂C 2 ∂χ 2 ( ∂C 2 ∂p + b ' 2 2 b 2 ∂C 2 ∂π ) ∂ ∂χ 2 log { ( ∂C 2 ∂p + b ' 2 2 b 2 ∂C 2 ∂π )( ∂C 2 ∂χ 2 ) -1 } , (5.24)</formula>
<text><location><page_88><loc_19><loc_60><loc_90><loc_62></location>and because b 2 is a non-zero constant in the classical limit, this can be integrated to find</text>
<formula><location><page_88><loc_40><loc_55><loc_90><loc_58></location>∂C 2 ∂p + b ' 2 2 b 2 ∂C 2 ∂π = g 1 ( q, p, ψ, π, γ ) ∂C 2 ∂χ 2 , (5.25)</formula>
<text><location><page_88><loc_19><loc_49><loc_90><loc_52></location>where g 1 is a unknown function arising as an integration constant, and needs to be determined. This provides a further restriction on the form of the constraint,</text>
<formula><location><page_88><loc_31><loc_40><loc_90><loc_46></location>C = C 3 ( q, ψ, γ, Π , χ 3 ) , Π := π -b ' 2 2 b 2 p, χ 3 := P -q ( b 2 R -b ' 2 ∆ ) + ∫ p 0 g 1 ( q, x, ψ, Π+ b ' 2 2 b 2 x, γ ) d x. (5.26)</formula>
<text><location><page_88><loc_19><loc_36><loc_67><loc_38></location>Substituting this into the condition from ∂ 2 η , (5.14a), gives</text>
<formula><location><page_88><loc_40><loc_31><loc_90><loc_34></location>0 = q b 2 ( ∂C 3 ∂χ 3 ) 2 ∂ ∂π g 1 ( q, p, ψ, π, γ ) , (5.27)</formula>
<text><location><page_88><loc_19><loc_27><loc_30><loc_28></location>and therefore,</text>
<formula><location><page_88><loc_36><loc_23><loc_90><loc_27></location>χ 3 = P -q ( b 2 R -b ' 2 ∆ ) + ∫ p 0 g 1 ( q, x, ψ, γ ) d x. (5.28)</formula>
<text><location><page_88><loc_19><loc_20><loc_61><loc_22></location>Evaluating the condition from q ab ∂ 2 θ ab , (5.8b), gives</text>
<formula><location><page_88><loc_37><loc_15><loc_90><loc_18></location>0 = 1 3 qb 2 ( ∂C 3 ∂χ 3 ) 2 ( 1 + 3 ∂ ∂p ) g 1 ( q, p, ψ, γ ) , (5.29)</formula>
<text><location><page_89><loc_19><loc_87><loc_86><loc_88></location>which can be integrated to find g 1 = g 2 ( q, ψ, γ ) -p/ 3 and therefore (5.28) becomes,</text>
<formula><location><page_89><loc_40><loc_81><loc_90><loc_84></location>χ 3 = P 1 6 p 2 + g 2 p -q ( b 2 R -b ' 2 ∆ ) . (5.30)</formula>
<text><location><page_89><loc_19><loc_77><loc_81><loc_79></location>Then look at the condition from p T ab X c ∂ c θ ab , (5.9h), from which can be found</text>
<formula><location><page_89><loc_42><loc_71><loc_90><loc_75></location>0 = 2 qb 2 ∂C 3 ∂χ 3 ∂ 2 C 3 ∂χ 2 3 (2 ∂ q -1) g 2 , (5.31)</formula>
<text><location><page_89><loc_79><loc_67><loc_79><loc_68></location>glyph[negationslash]</text>
<text><location><page_89><loc_19><loc_61><loc_90><loc_69></location>which can be solved by, g 2 ( q, ψ, γ ) = √ q g 3 ( ψ, γ ) if we assume that ∂ 2 C 3 ∂χ 2 3 = 0 generally, which is true for any deformation dependent on curvature ∂β ∂χ 3 = 0 . This is what is expected for the density weight of each term in χ 3 to match.</text>
<text><location><page_89><loc_69><loc_64><loc_69><loc_65></location>glyph[negationslash]</text>
<text><location><page_89><loc_19><loc_58><loc_77><loc_59></location>I now look at the condition for p T ab ∂ c γ∂ c θ ab , which is (5.9m) with F = γ ,</text>
<formula><location><page_89><loc_45><loc_52><loc_90><loc_56></location>0 = q 3 / 2 b 2 ∂g 3 ∂γ ∂C 3 ∂χ 3 ∂ 2 C 3 ∂χ 2 3 , (5.32)</formula>
<text><location><page_89><loc_19><loc_49><loc_44><loc_50></location>which is true when g 3 = g 3 ( ψ ) .</text>
<text><location><page_89><loc_19><loc_43><loc_90><loc_46></location>At this point it gets harder to progress further as I have done so far. To review, I have restricted the constraint and deformation to the forms,</text>
<formula><location><page_89><loc_25><loc_34><loc_90><loc_41></location>C ( q, p, P , R, ψ, π, ∆ , γ ) = C 3 ( q, ψ, Π , γ, χ 3 ) , β = q b 2 ( ψ ) ( ∂C 3 ∂χ 3 ) 2 , Π = π -b ' 2 2 b 2 p, χ 3 = P 1 6 p 2 + p √ q g 3 ( ψ ) -q ( b 2 R -b ' 2 ∆ ) , (5.33)</formula>
<text><location><page_89><loc_19><loc_25><loc_90><loc_31></location>which satisfies all the conditions in (5.8), (5.14), (5.15) and (5.9) apart from the conditions for q ab ∂ c ψ∂ c θ ab , (5.9b), and ∂ a ψ∂ a η , (5.15a). As it stands, these conditions are not easy to solve.</text>
<section_header_level_1><location><page_90><loc_19><loc_87><loc_87><loc_88></location>5.2.1 Solving the fourth order constraint to inform the general case</section_header_level_1>
<text><location><page_90><loc_19><loc_80><loc_90><loc_84></location>To break this impasse, I use a test ansatz for the constraint which contains up to four orders in momenta,</text>
<formula><location><page_90><loc_35><loc_72><loc_90><loc_77></location>C 3 → C 0 + C (Π) Π+ C (Π 2 ) Π 2 + C (Π 3 ) Π 3 + C (Π 4 ) Π 4 + C ( χ ) χ 3 + C ( χ 2 ) χ 2 3 + C (Π χ ) Π χ 3 + C (Π 2 χ ) Π 2 χ 3 , (5.34)</formula>
<text><location><page_90><loc_19><loc_64><loc_90><loc_70></location>where each coefficient is an unknown function to be determined dependent on q , ψ and γ . There is an asymmetric term included in χ 3 determined by the function g 3 ( ψ ) , so I do not restrict myself to only even orders of momenta, unlike section 3.</text>
<text><location><page_90><loc_19><loc_42><loc_90><loc_61></location>Substituting this into (5.15a), I can separate out the multiplier of each unique combination of variables as an independent equation. For each of the terms which are the multipliers of 5 or 6 orders of momenta, I find a condition specifying that the constraint coefficients for terms 3 or 4 orders of momenta must not depend on γ , e.g. ∂ ∂γ C ( χ 2 ) = 0 , ∂ ∂γ C (Π 3 ) = 0 . Since γ depends on two spatial derivatives, I see that each term in the constraint must not depend on a higher order of spatial derivatives than it does momenta. If I include higher orders of spatial derivatives in the ansatz, I quickly find them ruled out in a similar fashion. Therefore, I use this information to further expand my ansatz,</text>
<formula><location><page_90><loc_25><loc_32><loc_90><loc_39></location>C 3 → C ∅ + C ( γ ) γ + C ( γ 2 ) γ 2 + C (Π) Π+ C (Π γ ) Π γ + C (Π 2 ) Π 2 + C (Π 2 γ ) Π 2 γ + C (Π 3 ) Π 3 + C (Π 4 ) Π 4 + C ( χ ) χ 3 + C ( χγ ) χ 3 γ + C ( χ 2 ) χ 2 3 + C (Π χ ) Π χ 3 + C (Π 2 χ ) Π 2 χ 3 , (5.35)</formula>
<text><location><page_90><loc_19><loc_28><loc_69><loc_29></location>where each coefficient is now an unknown function of q and ψ .</text>
<text><location><page_90><loc_19><loc_19><loc_90><loc_26></location>One can find all the necessary conditions from (5.15a), for which the solution also satisfies (5.9b). I will show a route which can taken to progressively restrict C . The condition coming from P 2 is solved if</text>
<formula><location><page_90><loc_36><loc_13><loc_90><loc_17></location>C (Π 2 χ ) = 1 2 C ( χ 2 ) ( C ( χγ ) 2 qb 2 C ( χ 2 ) + 7 b ' 2 2 8 b 2 2 -b '' 2 b 2 ) -1 , (5.36)</formula>
<text><location><page_91><loc_19><loc_87><loc_47><loc_88></location>the condition from γ 2 is solved by,</text>
<formula><location><page_91><loc_37><loc_81><loc_90><loc_85></location>C (Π 2 γ ) = 1 4 C ( χγ ) ( 2 C ( γ 2 ) b 2 C ( χγ ) + 7 b ' 2 2 8 b 2 2 -b '' 2 b 2 ) -1 , (5.37)</formula>
<text><location><page_91><loc_19><loc_77><loc_47><loc_79></location>the condition from γ P is solved by,</text>
<formula><location><page_91><loc_49><loc_72><loc_90><loc_75></location>C ( γ 2 ) = C 2 ( χγ ) 4 C ( χ 2 ) , (5.38)</formula>
<text><location><page_91><loc_19><loc_68><loc_47><loc_69></location>the condition from π 4 is solved by,</text>
<formula><location><page_91><loc_36><loc_62><loc_90><loc_66></location>C (Π 4 ) = 1 16 C ( χ 2 ) ( C ( χγ ) 2 qb 2 C ( χ 2 ) + 7 b ' 2 2 8 b 2 2 -b '' 2 b 2 ) -1 , (5.39)</formula>
<text><location><page_91><loc_19><loc_56><loc_90><loc_59></location>and all the other conditions coming from four momenta are solved. Turning to the third order, the condition from π P is solved by,</text>
<formula><location><page_91><loc_30><loc_45><loc_90><loc_54></location>C (Π 3 ) = 1 12 { C (Π χ ) [ 1 qb 2 ( 3 C ( χγ ) 2 C ( χ 2 ) -2 C (Π γ ) C (Π χ ) ) + 7 b ' 2 2 8 b 2 2 -b '' 2 b 2 ] -√ qC ( χ 2 ) ( 4 g ' 3 -3 g 3 b ' 2 b 2 )} ( C ( χγ ) 2 qb 2 C ( χ 2 ) + 7 b ' 2 2 8 b 2 2 -b '' 2 b 2 ) -2 , (5.40)</formula>
<text><location><page_91><loc_19><loc_42><loc_50><loc_43></location>and the condition from πγ is solved by,</text>
<formula><location><page_91><loc_47><loc_36><loc_90><loc_39></location>C (Π γ ) = C (Π χ ) C ( χγ ) 2 C ( χ 2 ) , (5.41)</formula>
<text><location><page_91><loc_19><loc_32><loc_50><loc_34></location>and the condition from π 3 is solved by,</text>
<formula><location><page_91><loc_29><loc_26><loc_90><loc_30></location>C (Π χ ) = -1 2 √ qC ( χ 2 ) ( 4 g ' 3 -3 g 3 b ' 2 b 2 ) ( C ( χγ ) 2 qb 2 C ( χ 2 ) + 7 b ' 2 2 8 b 2 2 -b '' 2 b 2 ) -1 , (5.42)</formula>
<text><location><page_91><loc_19><loc_20><loc_90><loc_24></location>which completes all the terms from third order. The only new condition coming from second order is solved by,</text>
<formula><location><page_91><loc_29><loc_10><loc_90><loc_18></location>C (Π 2 ) = { 1 4 C (Π χ ) [ 1 qb 2 ( C ( χγ ) C ( χ 2 ) -C ( γ ) C ( χ ) ) + 7 b ' 2 2 8 b 2 2 -b '' 2 b 2 ] + 1 16 qC ( χ 2 ) ( 4 g ' 3 -3 g 3 b ' 2 b 2 ) 2 }( C ( χγ ) 2 qb 2 C ( χ 2 ) + 7 b ' 2 2 8 b 2 2 -b '' 2 b 2 ) -2 , (5.43)</formula>
<text><location><page_92><loc_19><loc_87><loc_70><loc_88></location>and the only new condition coming from first order is solved by,</text>
<formula><location><page_92><loc_24><loc_76><loc_90><loc_85></location>C (Π) = -1 4 √ qC ( χ ) ( 4 g ' 3 -3 g 3 b ' 2 b 2 ) { 1 qb 2 ( C ( χγ ) C ( χ 2 ) -C ( γ ) C ( χ ) ) + 7 b ' 2 2 8 b 2 2 -b '' 2 b 2 } × ( C ( χγ ) 2 qb 2 C ( χ 2 ) + 7 b ' 2 2 8 b 2 2 -b '' 2 b 2 ) -2 , (5.44)</formula>
<text><location><page_92><loc_19><loc_73><loc_40><loc_74></location>and from the zeroth order,</text>
<formula><location><page_92><loc_47><loc_69><loc_90><loc_72></location>C ( χγ ) = 2 C ( γ ) C ( χ 2 ) C ( χ ) , (5.45)</formula>
<text><location><page_92><loc_19><loc_65><loc_90><loc_67></location>When all of these terms are combined, I find the solution for the fourth order constraint,</text>
<formula><location><page_92><loc_20><loc_60><loc_90><loc_63></location>C = C ∅ + C ( χ ) ( χ 3 + Π(Π -Ξ) 4Ω + C ( γ ) C ( χ ) γ ) + C ( χ 2 ) ( χ 3 + Π(Π -Ξ) 4Ω + C ( γ ) C ( χ ) γ ) 2 , (5.46)</formula>
<text><location><page_92><loc_19><loc_56><loc_24><loc_57></location>where</text>
<formula><location><page_92><loc_33><loc_52><loc_90><loc_56></location>Ω = C ( γ ) b 2 qC ( χ ) + 7 b ' 2 2 8 b 2 2 -b '' 2 b 2 , Ξ = √ q ( 4 g ' 3 -3 g 3 b ' 2 b 2 ) . (5.47)</formula>
<text><location><page_92><loc_19><loc_49><loc_53><loc_50></location>If this solution is generalised to all orders,</text>
<formula><location><page_92><loc_20><loc_43><loc_90><loc_46></location>C = C 4 ( q, ψ, χ 4 ) , χ 4 = P1 6 p 2 + √ qpg 3 -q ( b 2 R -b ' 2 ∆ ) + Π(Π -Ξ) 4Ω + C ( γ ) C ( χ ) γ, (5.48)</formula>
<text><location><page_92><loc_19><loc_32><loc_90><loc_41></location>one can check that it satisfies all the conditions from (5.8), (5.14), (5.15) and (5.9). It is possible that directly generalising from the fourth order constraint rather than continuing to work generally means that this is not the most general solution. However, at least I now know a form of the constraint which can solve all the conditions.</text>
<text><location><page_92><loc_19><loc_23><loc_90><loc_29></location>Now that I have a form for the general constraint, I seek to compare it to the low-curvature limit, when C → χ 4 C χ + C ∅ , and match terms with that found previously (3.27) in chapter 3 and [55]. I find that,</text>
<formula><location><page_92><loc_36><loc_14><loc_90><loc_21></location>b 2 = σ β ω 2 R 4 , σ β := sgn( β ) = sgn( β ∅ ) . C χ = 2 σ β ω R √ | β ∅ | q , C γ = √ q | β ∅ | ( ω ψ 2 + ω '' R ) , (5.49)</formula>
<text><location><page_92><loc_19><loc_10><loc_90><loc_11></location>For convenience, I redefine the function determining the asymmetry, g 3 = ξ/ 2 , and I expand</text>
<text><location><page_93><loc_19><loc_84><loc_90><loc_88></location>the constraint in terms of the weightless (or 'de-densitised') scalar R := χ 4 /q . This means that the general form of the deformed constraint is given by,</text>
<formula><location><page_93><loc_40><loc_79><loc_69><loc_82></location>C = C ( q, ψ, R ) , β = σ β q ( ∂C ∂ R ) 2 ,</formula>
<formula><location><page_93><loc_22><loc_71><loc_90><loc_81></location>(5.50a) R := 2 σ β qω R ( P 1 6 p 2 ) -ω R 2 R + ω ' R ∆ ψ + ( ω ψ 2 + ω '' R ) ∂ a ψ∂ a ψ, + σ β ω R ω ψ ω R + 3 2 ω ' 2 R { 1 2 q ( π -ω ' R ω R p ) 2 + ξ √ q [ ω ψ ω R p + 3 ω ' R 2 ω R π -ξ ' ξ ( π -ω ' R ω R p )] } . (5.50b)</formula>
<text><location><page_93><loc_19><loc_65><loc_90><loc_68></location>It is probably more appropriate to see the deformation function itself as the driver of deformations to the constraint, so I rearrange (5.50a),</text>
<formula><location><page_93><loc_49><loc_59><loc_90><loc_62></location>∂C ∂ R = √ q | β | , (5.51)</formula>
<text><location><page_93><loc_19><loc_55><loc_45><loc_56></location>which can be integrated to find,</text>
<formula><location><page_93><loc_40><loc_49><loc_90><loc_53></location>C = ∫ R 0 √ q | β ( q, ψ, r ) | d r + C ∅ ( q, ψ ) . (5.52)</formula>
<text><location><page_93><loc_19><loc_38><loc_93><loc_47></location>From either form of the general solution (5.50a) or (5.52), one can now understand the meaning of (2.50), which relates the order of the constraint and the deformation, 2 n C -n β = 4 . The differential form (5.50a) is like n β = 2( n C -2) , and the integral form (5.52) is like n C = 2 + n β / 2 .</text>
<text><location><page_93><loc_19><loc_30><loc_90><loc_36></location>From the integral form of the solution (5.52), I can now check a few examples of what constraint corresponds to certain deformations. Here are a few examples of easily integrable functions with the appropriate limit,</text>
<text><location><page_93><loc_76><loc_22><loc_76><loc_23></location>glyph[negationslash]</text>
<formula><location><page_93><loc_24><loc_14><loc_90><loc_27></location>β = β ∅ (1 + β 2 R ) n → C = C ∅ + 2 √ q | β ∅ | ( n +2) β 2 { sgn(1 + β 2 R ) | 1 + β 2 R| n +2 2 -1 } , n = -2 , √ q | β ∅ | β 2 sgn(1 + β 2 R ) log | 1 + β 2 R| , n = -2 , glyph[similarequal] C ∅ + √ q | β ∅ | { R + nβ 2 4 R 2 + · · · } . (5.53)</formula>
<formula><location><page_94><loc_32><loc_82><loc_90><loc_89></location>β = β ∅ e β 2 R → C = C ∅ + 2 √ q | β ∅ | β 2 ( e β 2 R / 2 -1 ) glyph[similarequal] C ∅ + √ q | β ∅ | ( R + β 2 4 R 2 + · · · ) , (5.54)</formula>
<formula><location><page_94><loc_29><loc_74><loc_90><loc_80></location>β = β ∅ sech 2 ( β 2 R ) → C = C ∅ + √ q | β ∅ | β 2 gd( β 2 R ) , glyph[similarequal] C ∅ + √ q | β ∅ | ( Rβ 2 2 6 R 3 + · · · ) , (5.55)</formula>
<text><location><page_94><loc_19><loc_63><loc_90><loc_72></location>where gd( x ) := ∫ x 0 d t sech( t ) is the Gudermannian function. Most other deformation functions would need to be integrated numerically to find the constraint. As can be seen from the small R expansions, it would be possible to constrain β ∅ and β 2 phenomenologically but the asymptotic behaviour of β would be difficult to determine.</text>
<text><location><page_94><loc_19><loc_57><loc_90><loc_61></location>The simplest constraint that can be expressed as a polynomial of R that contains higher orders than the classical solution is given by,</text>
<formula><location><page_94><loc_32><loc_52><loc_90><loc_55></location>β = β ∅ (1 + β 2 R ) 2 → C = C ∅ + √ q | β ∅ | ( R + β 2 2 R 2 ) , (5.56)</formula>
<text><location><page_94><loc_19><loc_48><loc_71><loc_49></location>which is equivalent to the fourth order constraint found in (5.46).</text>
<section_header_level_1><location><page_94><loc_19><loc_41><loc_70><loc_43></location>5.3 Looking back at the constraint algebra</section_header_level_1>
<text><location><page_94><loc_19><loc_29><loc_90><loc_38></location>For this deformed constraint to mean anything, it must not reduce to the undeformed constraint through a simple transformation. If I write the constraint as a function of the undeformed vacuum constraint ¯ C = √ q R , I see that the deformation in the constraint algebra can be absorbed by a redefinition of the lapse functions,</text>
<formula><location><page_94><loc_31><loc_24><loc_90><loc_26></location>{ C [ N ] , C [ M ] } = ∫ d x d yN ( x ) M ( y ) { C ( x ) , C ( y ) } , (5.57a)</formula>
<formula><location><page_94><loc_42><loc_20><loc_90><loc_23></location>= ∫ d x d y ( N ∂C ∂ ¯ C ) x ( M ∂C ∂ ¯ C ) y { ¯ C ( x ) , ¯ C ( y ) } , (5.57b)</formula>
<formula><location><page_94><loc_42><loc_17><loc_90><loc_19></location>= ∫ d x d y ( σ ∂C ¯ N ) x ( σ ∂C ¯ M ) y { ¯ C ( x ) , ¯ C ( y ) } , (5.57c)</formula>
<formula><location><page_94><loc_42><loc_14><loc_90><loc_15></location>= { ¯ C [ σ ∂C ¯ N ] , ¯ C [ σ ∂C ¯ M ] } , (5.57d)</formula>
<text><location><page_95><loc_19><loc_84><loc_90><loc_88></location>where ¯ N := N ∣ ∣ ∂C/∂ ¯ C ∣ ∣ , ¯ M := M ∣ ∣ ∂C/∂ ¯ C ∣ ∣ and σ ∂C := sgn( ∂C/∂ ¯ C ) , because the lapse functions should remain positive. The other side of the equality,</text>
<formula><location><page_95><loc_25><loc_69><loc_90><loc_82></location>D a [ βq ab ( N∂ b M -∂ b NM )] , = ∫ d xD a βq ab ( N∂ b M -∂ b NM ) = ∫ d xD a σ β ( ∂C ∂ ¯ C ) 2 ( N∂ b M -∂ b NM ) = ∫ d xD a σ β ( ¯ N∂ b ¯ M -∂ b ¯ N ¯ M ) , = D a [ σ β q ab ( ¯ N∂ b ¯ M -∂ b ¯ N ¯ M )] , (5.58)</formula>
<text><location><page_95><loc_19><loc_65><loc_84><loc_67></location>which I can combine to show the that the following two equations are equivalent,</text>
<formula><location><page_95><loc_40><loc_61><loc_90><loc_62></location>{ C [ N ] , C [ M ] } = D a [ βq ab ( N∂ b M -∂ b NM )] , (5.59a)</formula>
<formula><location><page_95><loc_34><loc_58><loc_90><loc_59></location>{ ¯ C [ σ ∂C ¯ N ] , ¯ C [ σ ∂C ¯ M ] } = D a [ σ β q ab ( ¯ N∂ b ¯ M -∂ b ¯ N ¯ M )] . (5.59b)</formula>
<text><location><page_95><loc_19><loc_40><loc_90><loc_54></location>The two σ ∂C on the left side should cancel out, but they are included here to show the limit to the redefinition of the lapse functions. While it may seem like I have regained the undeformed constraint algebra up to the sign σ β with a simple transformation, it shouldn't be taken to mean that this is actually the algebra of constraints. That is, the above equation doesn't ensure that ¯ C ≈ 0 instead of C ≈ 0 when on-shell. The surfaces in phase space described by ¯ C = 0 and C = 0 are different in general.</text>
<section_header_level_1><location><page_95><loc_19><loc_34><loc_38><loc_35></location>5.4 Cosmology</section_header_level_1>
<text><location><page_95><loc_19><loc_24><loc_90><loc_30></location>I restrict to an isotropic and homogeneous space to find the background cosmological dynamics, following the definitions in section 2.8. Writing the constraint as C = C ( a, ψ, R ) where R = R ( a, ψ, ¯ p, π ) , the equations of motion are given by,</text>
<formula><location><page_95><loc_36><loc_15><loc_90><loc_22></location>˙ a N = 1 6 a ∂ R ∂ ¯ p ∂C ∂ R , ˙ ¯ p N = -1 6 a ( ∂C ∂a + ∂ R ∂a ∂C ∂ R ) , ˙ ψ N = ∂ R ∂π ∂C ∂ R , ˙ π N = -∂C ∂ψ -∂ R ∂ψ ∂C ∂ R , (5.60)</formula>
<text><location><page_96><loc_19><loc_83><loc_90><loc_88></location>into which I can substitute ∂C ∂ R = a 3 √ | β | . When I assume minimal coupling ( ω ' R = 0 , ω ' ψ = 0 ) and time-symmetry ( ξ = 0 ), the equations of motion become,</text>
<formula><location><page_96><loc_35><loc_70><loc_90><loc_81></location>R→ -3 σ β ¯ p 2 ω R a 2 -3 kω R a 2 + σ β π 2 2 ω ψ a 6 , ˙ a N = -σ β ¯ p ω R √ | β | , ˙ ψ N = σ β π ω ψ a 3 √ | β | , ˙ π N = -∂C ∂ψ , ˙ ¯ p N = -1 6 a ∂C ∂a -a √ | β | ( σ β ¯ p 2 ω R a 2 + kω R a 2 -σ β π 2 2 ω ψ a 6 ) . (5.61)</formula>
<text><location><page_96><loc_19><loc_67><loc_88><loc_68></location>To find the Friedmann equation, find the equation for H 2 /N 2 , and substitute in for R ,</text>
<formula><location><page_96><loc_37><loc_61><loc_90><loc_64></location>H 2 N 2 = | β | ¯ p 2 ω 2 R = β ( -R 3 ω R -k a 2 + σ β π 2 6 ω R ω ψ a 6 ) , (5.62)</formula>
<text><location><page_96><loc_19><loc_57><loc_83><loc_59></location>and when the constraint is solved, C ≈ 0 , then R can be found in terms of C ∅ .</text>
<section_header_level_1><location><page_96><loc_19><loc_52><loc_56><loc_53></location>5.4.1 Cosmology with a perfect fluid</section_header_level_1>
<text><location><page_96><loc_19><loc_43><loc_90><loc_49></location>I here find the deformed Friedmann equations for various forms of the deformation. For simplicity, I ignore the scalar field and include a perfect fluid C ∅ = a 3 ρ ( a ) . From the deformation function β = β ∅ (1 + β 2 R ) n , solving the constraint (5.53) gives</text>
<text><location><page_96><loc_71><loc_37><loc_71><loc_39></location>glyph[negationslash]</text>
<formula><location><page_96><loc_33><loc_32><loc_90><loc_40></location>R = σ 2 β 2 { σ 2 -( n +2) σ 2 β 2 ρ 2 √ | β ∅ | } n +2 2 -1 β 2 , n = -2 , σ 2 β 2 exp ( -σ 2 β 2 ρ √ | β ∅ | ) -1 β 2 , n = -2 , (5.63)</formula>
<text><location><page_96><loc_19><loc_26><loc_90><loc_29></location>where σ ∅ := sgn( β ∅ ) and σ 2 := sgn(1 + β 2 R ) . When I simplify by assuming σ 2 = 1 , the Friedmann equation is given by,</text>
<text><location><page_96><loc_79><loc_21><loc_79><loc_22></location>glyph[negationslash]</text>
<formula><location><page_96><loc_21><loc_15><loc_90><loc_23></location>H 2 N 2 = ( β ∅ 3 ω R β 2 [ 1 -( 1 -ρ ρ c ( n ) ) 2 n +2 ] -kβ ∅ a 2 ) ( 1 -ρ ρ c ( n ) ) 2 n n +2 , n = -2 , ( β ∅ 3 ω R β 2 [ 1 -exp ( -β 2 ρ √ | β ∅ | )] -kβ ∅ a 2 ) exp ( 2 β 2 ρ √ | β ∅ | ) n = -2 , (5.64)</formula>
<figure>
<location><page_97><loc_23><loc_71><loc_87><loc_88></location>
<caption>Figure 5.1: Behaviour of the Friedmann equation for various deformation functions β ( R ) when k = 0 .</caption>
</figure>
<text><location><page_97><loc_19><loc_49><loc_90><loc_64></location>where ρ c ( n ) = 2 √ | β ∅ | β 2 ( n +2) . To see the behaviour of the modified Friedmann equation for different values of n , look at Fig. 5.1(a). For n > 0 , the Hubble rate vanishes as the universe approaches the critical energy density, this indicates that a collapsing universe reaches a turning point at which point the repulsive effect causes a bounce. For 0 > n > -2 , there appears a sudden singularity in H at finite ρ (therefore finite a ). In the ρ →∞ limit, H 2 ∼ e 2 ρ when n = -2 and H 2 ∼ ρ 2 n n +2 when n < -2 .</text>
<text><location><page_97><loc_19><loc_38><loc_90><loc_47></location>The singularities for 0 > n > -2 appear to be similar to sudden future singularities characterised in [83, 91]. However, the singularities here might instead be called sudden 'past' singularities as they happen when a is small (but non-zero) and ρ is large. Moreover, they happen for any perfect fluid with w > -1 , i.e. including matter and radiation.</text>
<text><location><page_97><loc_19><loc_34><loc_90><loc_36></location>For the deformation function β = β ∅ exp( β 2 R ) from (5.54), solving the constraint gives,</text>
<formula><location><page_97><loc_44><loc_29><loc_90><loc_32></location>R = 2 β 2 log ( 1 -β 2 ρ 2 √ | β ∅ | ) , (5.65)</formula>
<text><location><page_97><loc_19><loc_25><loc_51><loc_26></location>and the Friedmann equation is given by,</text>
<formula><location><page_97><loc_31><loc_19><loc_90><loc_23></location>H 2 N 2 = { -2 β ∅ 3 ω R β 2 log ( 1 -β 2 ρ 2 √ | β ∅ | ) -kβ 2 a 2 }( 1 -β 2 ρ 2 √ | β ∅ | ) 2 . (5.66)</formula>
<text><location><page_97><loc_19><loc_13><loc_57><loc_16></location>and a critical density appears for ρ → 2 √ | β ∅ | β 2 .</text>
<text><location><page_98><loc_19><loc_87><loc_90><loc_88></location>For the deformation function β = β ∅ sech 2 ( β 2 R ) from (5.55), solving the constraint gives,</text>
<formula><location><page_98><loc_45><loc_81><loc_90><loc_84></location>R = -1 β 2 gd -1 ( β 2 ρ √ | β ∅ | ) . (5.67)</formula>
<text><location><page_98><loc_19><loc_77><loc_66><loc_78></location>Substituting this back into the deformation function gives,</text>
<formula><location><page_98><loc_46><loc_72><loc_90><loc_75></location>β = β ∅ cos 2 ( β 2 ρ √ | β ∅ | ) (5.68)</formula>
<text><location><page_98><loc_19><loc_68><loc_51><loc_69></location>and the Friedmann equation is given by</text>
<formula><location><page_98><loc_33><loc_62><loc_90><loc_66></location>H 2 N 2 = { β ∅ 3 ω R β 2 gd -1 ( β 2 ρ √ | β ∅ | ) -kβ ∅ a 2 } cos 2 ( β 2 ρ √ | β ∅ | ) . (5.69)</formula>
<text><location><page_98><loc_19><loc_56><loc_58><loc_60></location>where there is a critical density 1 , ρ → π · √ | β ∅ | 2 β 2 .</text>
<text><location><page_98><loc_19><loc_49><loc_90><loc_55></location>These exponential-type deformation functions that I consider all predict a upper limit on energy density. To illustrate this, I plot the modified Friedmann equations for these functions in Fig. 5.1(b).</text>
<section_header_level_1><location><page_98><loc_19><loc_43><loc_74><loc_45></location>5.4.2 Cosmology with a minimally coupled scalar field</section_header_level_1>
<text><location><page_98><loc_19><loc_21><loc_90><loc_40></location>Since the metric and scalar kinetic terms must combine into one quantity, R , a deformation function should not affect the relative structure between fields. To illustrate this, take a free scalar field (without a potential) which is minimally coupled to gravity, and assume no perfect fluid component. This means that the generalised potential term C ∅ will vanish, in which case solving the constraint, C ≈ 0 , merely implies R = 0 . Consequently, since the deformation function β is a function of R , the only deformation remaining will be the zeroth order term β = β ∅ ( q, ψ ) . Combining the equations of motion (5.61) allows me to find the Friedmann equation,</text>
<formula><location><page_98><loc_46><loc_15><loc_90><loc_19></location>H 2 N 2 = ω ψ ˙ ψ 2 6 ω R N 2 -kβ ∅ a 2 , (5.70)</formula>
<text><location><page_99><loc_55><loc_87><loc_55><loc_88></location>glyph[negationslash]</text>
<text><location><page_99><loc_19><loc_79><loc_90><loc_88></location>that is, the minimally-deformed case. For β = β ∅ , it is required that R must not vanish, which itself requires that C ∅ must be non-zero. Therefore, for the dynamics to depend on a deformation which is a function of curvature, there must be a non-zero potential term which acts as a background against which the fields are deformed.</text>
<section_header_level_1><location><page_99><loc_19><loc_74><loc_54><loc_75></location>5.4.3 Deformation correspondence</section_header_level_1>
<text><location><page_99><loc_19><loc_59><loc_90><loc_71></location>As discussed in the perturbative action chapter 4, the form of the deformation used in the literature which includes holonomy effects is given by the cosine of the extrinsic curvature [40-42]. Of particular importance to this is that the deformation vanishes and changes sign for high values of extrinsic curvature. Since the extrinsic curvature is proportional to the Hubble expansion rate, write the deformation (4.36) here as,</text>
<formula><location><page_99><loc_48><loc_54><loc_90><loc_56></location>β = β ∅ cos ( β k H ) . (5.71)</formula>
<text><location><page_99><loc_19><loc_42><loc_90><loc_51></location>I wish to find C ( R ) and β ( R ) associated with this deformation of form β ( K ) . To do so, I need to find the relationship between the Hubble parameter H = ˙ a/a and the momentum ¯ p , and thereby infer the form of β ( R ) . Then, using (5.52) I can find the constraint C ( R ) . So, using the equations of motion (5.61), I find</text>
<formula><location><page_99><loc_31><loc_36><loc_90><loc_39></location>h = r √ | cos h | , where , h := β k H , r := -Nσ β ¯ p ω R a β k √ | β ∅ | , (5.72)</formula>
<text><location><page_99><loc_19><loc_30><loc_90><loc_34></location>this is an implicit equation which cannot be solved analytically for h ( r ) , and so must be solved numerically.</text>
<text><location><page_99><loc_19><loc_11><loc_90><loc_28></location>For the general relation h = r √ | β ( h ) | , there are similar β functions which can be transformed analytically. One example is β ( h ) = 1 -4 π -2 · h 2 , which also has the same limits of β (0) = 1 and β ( h →± π · / 2) = 0 , and can be transformed to find β ( r ) = ( 1 + 4 π -2 · r 2 ) -1 . In Fig. 5.2, I plot β ( h ) and h ( r ) in the region | h | ≤ π · / 2 . After making the transformation, I find β ( r ) . Note that, unlike for h , β does not vanish for finite r . So it seems that a deformation which vanishes for finite extrinsic curvature does not necessarily vanish for finite intrinsic curvature or metric momenta (at least not in the isotropic and homogeneous</text>
<figure>
<location><page_100><loc_26><loc_56><loc_84><loc_88></location>
<caption>Figure 5.2: Plot showing the process of starting from a deformation β ( h ) (a), transforming h ( r ) (b), finding the new form of the deformation β ( r ) (c), and finding the kinetic part of the constraint C k ( r ) (d). I include the function β = 1 -4 π -2 · h 2 (blue dashed line) because it has the same limits as β = cos h (red solid line) for the region | h | ≤ π · / 2 but the transformation can be done analytically</caption>
</figure>
<text><location><page_100><loc_19><loc_39><loc_90><loc_43></location>case). In this respect, it matches the dynamics found for exponential-form deformations in Fig. 5.1.</text>
<text><location><page_100><loc_19><loc_33><loc_90><loc_37></location>Returning to the solution for the constraint, (5.52), reducing it to depending on only a and ¯ p gives</text>
<formula><location><page_100><loc_38><loc_30><loc_90><loc_33></location>C = -6 a ω R ∫ ¯ p 0 d p ' σ β p ' √ | β ( a, p ' ) | + C ∅ ( a ) , (5.73)</formula>
<text><location><page_100><loc_19><loc_24><loc_90><loc_28></location>and transforming from ¯ p to r as defined in (5.72), while making the assumptions σ β = 1 , N = 1 , β ∅ = 1 , and β k ∼ constant , this becomes</text>
<formula><location><page_100><loc_32><loc_18><loc_90><loc_22></location>C = -6 ω R a 3 β 2 k C k ( r ) + C ∅ ( a ) , C k ( r ) := ∫ r 0 d r ' r ' √ β ( r ' ) . (5.74)</formula>
<text><location><page_100><loc_19><loc_12><loc_90><loc_16></location>I numerically integrate the solution for β ( r ) found for when β = cos( h ) . I plot the function C k ( r ) in Fig. 5.2(d).</text>
<figure>
<location><page_101><loc_26><loc_72><loc_84><loc_88></location>
<caption>Figure 5.3: Plot showing transformations for the deformations given by β ( h ) = cos h 2 (red solid line) and β ( h ) = 1 -4 π -2 · h 4 (blue dashed line).</caption>
</figure>
<text><location><page_101><loc_19><loc_48><loc_90><loc_65></location>If instead of the extrinsic curvature itself, the deformation is a cosine of the standard extrinsic curvature contraction, β = cos β k K ∼ cos h 2 , it still cannot be transformed analytically. However, it does match the function β ( h ) = 1 -4 π -2 · h 4 well, as I have plotted in Fig. 5.3. However, numerically finding the constraint for these two deformations, then considering the low R limit, I see that C ∼ R 2 + C ∅ . Therefore, this deformation can be ruled out if C ∼ R + C ∅ is known to be the low curvature limit of the Hamiltonian constraint.</text>
<text><location><page_101><loc_19><loc_40><loc_90><loc_46></location>Considering the function β ( h ) = 1 -4 π -2 · h 2 in Fig. 5.2, transforming from h to K and from r to R to R , we can see the correspondence between different limits of the deformation function,</text>
<formula><location><page_101><loc_36><loc_36><loc_90><loc_39></location>β ( K , 0) = 1 -β 2 K , → β (0 , R ) = 1 1 + β 2 R . (5.75)</formula>
<text><location><page_101><loc_19><loc_23><loc_90><loc_34></location>This is what I found in chapter 6, where the general form of this particular deformation is actually the product of these two limits. However, for non-linear deformation functions, β ( K , R ) cannot be determined so easily from β ( K , 0) and β (0 , R ) . That being said, given β ( R ) , the dependence on K could be found by simply solving and evolving the equations of motion.</text>
<section_header_level_1><location><page_102><loc_19><loc_87><loc_37><loc_88></location>5.5 Discussion</section_header_level_1>
<text><location><page_102><loc_19><loc_64><loc_90><loc_83></location>In this chapter, I have found the general form that a deformed constraint can take for non-minimally coupled scalar-tensor variables. The momenta and spatial derivatives for all fields must maintain the same relative structure in how they appear compared to the minimally-deformed constraint. This means that the constraint is a function of the fields and the general kinetic term R . The freedom within this kinetic term comes down to the coupling functions. While a lapse function transformation can apparently take the constraint algebra back to the undeformed form, this seems to be merely a cosmetic change as it does not in fact alter the Hamiltonian constraint itself.</text>
<text><location><page_102><loc_19><loc_48><loc_90><loc_62></location>I have shown how to obtain the cosmological equations of motion, and given a few simple examples of how they are modified. For some deformation functions, a upper bound on energy density appears, which probably generates a cosmological bounce. For other deformation functions, a sudden singularity in the expansion appears when the deformation diverges for high densities. I have shown that deformations to the field dynamics requires a background general potential against which the deformation must be contrasted.</text>
<text><location><page_102><loc_19><loc_31><loc_90><loc_45></location>Using the cosmological equations of motion, I made contact with the holonomy-generated deformation which is a cosine of the extrinsic curvature. Through this, I have demonstrated how the relationship of momenta and extrinsic curvature becomes non-linear with a nontrivial deformation. It seems that when the deformation produces an upper bound on extrinsic curvature, there does not seem to be an upper bound on intrinsic curvature or momenta.</text>
<section_header_level_1><location><page_103><loc_19><loc_79><loc_33><loc_81></location>Chapter 6</section_header_level_1>
<section_header_level_1><location><page_103><loc_19><loc_64><loc_88><loc_73></location>Deformed gravitational action to all orders</section_header_level_1>
<text><location><page_103><loc_19><loc_46><loc_90><loc_58></location>As shown in section 2.7, the deformed action must be calculated either perturbatively, as has been done in chapter 4, or completely generally. It appears that this is because it does not permit a closed polynomial solution when the deformation depends on curvature. In this chapter I attempt this general calculation. This material has been subsequently published in ref. [57].</text>
<text><location><page_103><loc_19><loc_38><loc_90><loc_44></location>Take the equations (4.12) and (4.13), which solve the distribution equation for the gravitational action when I expand it in terms of the variables ( q, v, w, R ) , and see what can be deduced about the action when it is treated non-perturbatively.</text>
<text><location><page_103><loc_19><loc_32><loc_90><loc_36></location>Start with the equation for ∂ a F∂ b θ ab where F ∈ { v, w, R } , (4.13h), this can be rewritten as</text>
<formula><location><page_103><loc_40><loc_28><loc_90><loc_32></location>0 = β ( ∂L ∂w ) 2 ∂ ∂F log { β ( ∂L ∂w ) 2 } , (6.1)</formula>
<text><location><page_103><loc_19><loc_25><loc_34><loc_26></location>which implies that</text>
<formula><location><page_103><loc_47><loc_22><loc_90><loc_25></location>β ( ∂L ∂w ) 2 = λ 1 ( q ) , (6.2)</formula>
<text><location><page_103><loc_19><loc_17><loc_59><loc_20></location>and so I can solve up to a sign, σ L := sgn ( ∂L ∂w ) ,</text>
<formula><location><page_103><loc_48><loc_11><loc_90><loc_15></location>∂L ∂w = σ L √ ∣ ∣ ∣ ∣ λ 1 β ∣ ∣ ∣ ∣ . (6.3)</formula>
<text><location><page_104><loc_19><loc_87><loc_50><loc_88></location>Then, from Q abcd ∂ cd θ ab , (4.12b), I find</text>
<formula><location><page_104><loc_43><loc_81><loc_90><loc_84></location>∂L ∂R = 4 β ∂L ∂w = 4 σ L σ β √ | λ 1 β | , (6.4)</formula>
<text><location><page_104><loc_19><loc_72><loc_90><loc_79></location>where σ β := sgn( β ( q, v, w, R )) . If I then compare the second derivative of the action, ∂ 2 L ∂w∂R , using both equations, I find a nonlinear partial differential equation for the deformation function,</text>
<formula><location><page_104><loc_48><loc_69><loc_90><loc_72></location>0 = ∂β ∂R +4 β ∂β ∂w , (6.5)</formula>
<text><location><page_104><loc_19><loc_61><loc_90><loc_67></location>which is the same form as Burgers' equation for a fluid with vanishing viscosity [92]. However, before I attempt to interpret this, I will find further restrictions on the action and deformation.</text>
<text><location><page_104><loc_19><loc_54><loc_90><loc_58></location>I now seek to find how the trace of the metric's normal derivative, v , appears. Take the condition for v ab T ∂ 2 θ ab , (4.12d)</text>
<formula><location><page_104><loc_34><loc_49><loc_90><loc_52></location>0 = v 3 ∂ 2 L ∂R∂w -β ∂ 2 L ∂v∂w = σ L 2 √ ∣ ∣ ∣ ∣ λ 1 β ∣ ∣ ∣ ∣ ( 4 v 3 ∂β ∂w + ∂β ∂v ) (6.6)</formula>
<text><location><page_104><loc_19><loc_37><loc_90><loc_46></location>which I can solve to find that β = β ( q, ¯ w,R ) , where ¯ w = w -2 v 2 / 3 . So in the deformation, the trace v must always be paired with the traceless tensor squared w like this. I can see that this is related to the standard extrinsic curvature contraction by ¯ w = -4 K . To find how the trace appears in the action, I look at the condition from q ab ∂ 2 θ ab , (4.12a),</text>
<formula><location><page_104><loc_38><loc_32><loc_90><loc_35></location>0 = ∂L ∂R -2 v 3 ∂ 2 L ∂v∂R +2 β ( ∂ 2 L ∂v 2 -2 3 ∂L ∂w ) (6.7)</formula>
<text><location><page_104><loc_19><loc_25><loc_90><loc_29></location>inputting my solutions so far, I can solve for the second derivative with respect to the trace,</text>
<formula><location><page_104><loc_41><loc_21><loc_90><loc_25></location>∂ 2 L ∂v 2 = -4 σ L 3 √ ∣ ∣ ∣ ∣ λ 1 β ∣ ∣ ∣ ∣ ( 1 -v 2 ∂β ∂v ) . (6.8)</formula>
<text><location><page_104><loc_19><loc_18><loc_54><loc_20></location>I integrate over v to find the first derivative,</text>
<formula><location><page_104><loc_32><loc_13><loc_90><loc_16></location>∂L ∂v = -4 vσ L 3 √ ∣ ∣ ∣ ∣ λ 1 β ∣ ∣ ∣ ∣ + ξ 1 ( q, w, R ) = -4 v 3 ∂L ∂w + ξ 1 ( q, w, R ) . (6.9)</formula>
<text><location><page_105><loc_19><loc_79><loc_90><loc_88></location>To make sure that the solutions (6.3), (6.4) and (6.9) match for the second derivatives ∂ 2 L ∂v∂R and ∂ 2 L ∂v∂w , I find that ξ 1 = ξ 1 ( q ) . Therefore, from this I can see that the action should have the metric normal derivatives appear in the combined form ¯ w apart from a single linear term L ⊃ vξ 1 ( q ) .</text>
<text><location><page_105><loc_19><loc_73><loc_90><loc_77></location>I now just have to see what conditions there are on how the metric determinant appears in the action. First I have the condition from X a ∂ b θ ab , (4.13a),</text>
<formula><location><page_105><loc_42><loc_64><loc_90><loc_71></location>0 = ∂L ∂R -4 ( ∂ q β +2 β∂ q ) ∂L ∂w , = 4 σ L σ β √ | λ 1 β | ( 1 -∂ q λ 1 λ 1 ) , (6.10)</formula>
<formula><location><page_105><loc_44><loc_62><loc_55><loc_63></location>∴ λ 1 ( q ) = qλ 2 ,</formula>
<text><location><page_105><loc_19><loc_57><loc_65><loc_59></location>and second I have the condition from v ab T X c ∂ c θ ab , (4.13c),</text>
<formula><location><page_105><loc_24><loc_49><loc_90><loc_55></location>0 = v 3 (4 ∂ q -1) ∂ 2 L ∂w∂R + ∂β ∂w (1 -2 ∂ q ) ∂L ∂v +( β -2 ∂ q β -4 β∂ q ) ∂ 2 L ∂v∂w , = ∂β ∂w ( ξ 1 -2 ∂ q ξ 1 ) , ∴ ξ 1 ( q ) = ξ 2 √ q, (6.11)</formula>
<text><location><page_105><loc_19><loc_43><loc_90><loc_47></location>and both these results show that my action will indeed have the correct density weight when β → 1 , that is L ∝ √ q .</text>
<text><location><page_105><loc_19><loc_32><loc_90><loc_41></location>All the remaining conditions from the distribution equation that have not been explicitly referenced are solved by what I have found so far, so to make progress I must now attempt to consolidate my equations to find an explicit form for the action. If I integrate (6.3), I find</text>
<formula><location><page_105><loc_37><loc_29><loc_90><loc_32></location>L = σ L √ | qλ 2 | ∫ ¯ w 0 d x √ | β ( q, x, R ) | + f 1 ( q, v, R ) , (6.12)</formula>
<text><location><page_105><loc_19><loc_23><loc_90><loc_27></location>and then if I match the derivative of this with respect to v with (6.9), I find the v part of the second term,</text>
<formula><location><page_105><loc_43><loc_21><loc_90><loc_23></location>f 1 ( q, v, R ) = vξ 2 √ q + f 2 ( q, R ) . (6.13)</formula>
<text><location><page_105><loc_19><loc_17><loc_82><loc_18></location>If I then match the derivative of (6.12) with respect to R with (6.4), I see that</text>
<formula><location><page_105><loc_24><loc_11><loc_90><loc_14></location>∂L ∂R = 4 σ L σ β √ | qλ 2 β | = ∂f 2 ∂R -σ L 2 √ | qλ 2 | ∫ ¯ w 0 σ β d x | β ( q, x, R ) | 3 / 2 ∂ ∂R β ( q, x, R ) (6.14)</formula>
<text><location><page_106><loc_19><loc_87><loc_55><loc_88></location>and using (6.5) to change the derivative of β ,</text>
<formula><location><page_106><loc_26><loc_81><loc_90><loc_84></location>4 σ L σ β √ | qλ 2 β | = ∂f 2 ∂R +2 σ L √ | qλ 2 | ∫ ¯ w 0 d x √ | β ( q, x, R ) | ∂ ∂x β ( q, x, R ) , (6.15)</formula>
<text><location><page_106><loc_19><loc_77><loc_55><loc_79></location>and so I can change the integration variable,</text>
<formula><location><page_106><loc_34><loc_72><loc_90><loc_75></location>4 σ L σ β √ | qλ 2 β | = ∂f 2 ∂R +2 σ L √ | qλ 2 | ∫ β ( q, ¯ w,R ) β ( q, 0 ,R ) d b √ | b | , (6.16)</formula>
<text><location><page_106><loc_19><loc_68><loc_89><loc_69></location>the upper integration limit cancels with the left hand side of the equality, and therefore</text>
<formula><location><page_106><loc_38><loc_62><loc_90><loc_65></location>∂f 2 ∂R = 4 σ L sgn( β ( q, 0 , R )) √ | qλ 2 β ( q, 0 , R ) | . (6.17)</formula>
<text><location><page_106><loc_19><loc_59><loc_42><loc_60></location>Then integrating this over R ,</text>
<formula><location><page_106><loc_30><loc_53><loc_90><loc_56></location>f 2 ( q, R ) = 4 σ L √ | qλ 2 | ∫ R 0 sgn( β ( q, 0 , r )) √ | β ( q, 0 , r ) | d r + f 3 ( q ) , (6.18)</formula>
<text><location><page_106><loc_19><loc_49><loc_72><loc_50></location>which means that finally I have my solution for the general action,</text>
<formula><location><page_106><loc_23><loc_40><loc_90><loc_47></location>L = σ L √ | qλ 2 | ( ∫ ¯ w 0 d x √ | β ( q, x, R ) | +4 ∫ R 0 sgn( β ( q, 0 , r )) √ | β ( q, 0 , r ) | d r ) + vξ 2 √ q + f 3 ( q ) . (6.19)</formula>
<text><location><page_106><loc_19><loc_34><loc_90><loc_38></location>Now, I test this with a zeroth order deformation so I can match terms with my previous results. Using β = β ∅ ( q ) ,</text>
<formula><location><page_106><loc_30><loc_28><loc_90><loc_32></location>L = σ L √ | qλ 2 | ( ¯ w √ | β ∅ | +4 R sgn( β ∅ ) √ | β ∅ | ) + vξ 2 √ q + f 3 ( q ) , (6.20)</formula>
<text><location><page_106><loc_19><loc_25><loc_62><loc_26></location>comparing this to (4.24) and using ¯ w = -4 K leads to</text>
<formula><location><page_106><loc_35><loc_19><loc_90><loc_22></location>σ L = σ β , √ | λ 2 | = ω 8 , ξ 2 = ξ, f 3 = -√ qV ( q ) , (6.21)</formula>
<text><location><page_107><loc_19><loc_87><loc_53><loc_88></location>and therefore, the full solution is given by,</text>
<formula><location><page_107><loc_25><loc_79><loc_90><loc_85></location>L = ωσ β √ q 2 ( ∫ R 0 sgn( β ( q, 0 , r )) √ | β ( q, 0 , r ) | d r -∫ K 0 d k √ | β ( q, k, R ) | ) √ (6.22)</formula>
<formula><location><page_107><loc_27><loc_78><loc_41><loc_79></location>+ q ( vξ -V ( q )) ,</formula>
<text><location><page_107><loc_19><loc_74><loc_87><loc_75></location>and the deformation function must satisfy the non-linear partial differential equation,</text>
<formula><location><page_107><loc_48><loc_69><loc_90><loc_72></location>0 = ∂β ∂R -β ∂β ∂ K . (6.23)</formula>
<text><location><page_107><loc_19><loc_61><loc_90><loc_65></location>By performing a Legendre transform, I can see that the Hamiltonian constraint associated with this action is given by,</text>
<formula><location><page_107><loc_35><loc_51><loc_90><loc_60></location>C = ωσ β √ q 2 { ∫ K 0 d k √ | β ( q, k, R ) | -2 K √ | β ( q, K , R ) | -∫ R 0 sgn( β ( q, 0 , r )) √ | β ( q, 0 , r ) | d r } + √ q V, (6.24)</formula>
<section_header_level_1><location><page_107><loc_19><loc_47><loc_57><loc_49></location>6.1 Solving for the deformation</section_header_level_1>
<text><location><page_107><loc_19><loc_37><loc_90><loc_44></location>The nonlinear partial differential equation for the deformation function is an unexpected result, and invites a comparison to a very different area of physics. I can compare it to Burgers' equation for nonlinear diffusion, [92],</text>
<formula><location><page_107><loc_47><loc_32><loc_90><loc_35></location>∂u ∂t + u ∂u ∂x = η ∂ 2 u ∂x 2 , (6.25)</formula>
<text><location><page_107><loc_19><loc_18><loc_90><loc_29></location>(where u is a density function), and see that the deformation equation is very similar to the limit of vanishing viscosity η → 0 . This equation is not trivial to solve because it can develop discontinuities where the equation breaks down, termed 'shock waves'. Returning to my own equation (6.23), I analyse its characteristics. It implies that there are trajectories parameterised by s given by</text>
<formula><location><page_107><loc_39><loc_12><loc_90><loc_15></location>d q d s = 0 , d R d s = 1 , d K d s = -β ( q, K , R ) , (6.26)</formula>
<text><location><page_108><loc_19><loc_87><loc_75><loc_88></location>along which β is constant. These trajectories have gradients given by,</text>
<formula><location><page_108><loc_48><loc_81><loc_90><loc_84></location>d R d K = -1 β ( q, K , R ) (6.27)</formula>
<text><location><page_108><loc_19><loc_67><loc_90><loc_79></location>and because β is constant along the trajectories, they are a straight line in the ( K , R ) plane. I must have an 'initial' condition in order to solve the equation, and because R is here the analogue of -t in (6.25) I define the initial function when R = 0 , given by β ( q, K , 0) =: α ( q, K ) . Since there are trajectories along which β is constant, I can use α to solve for R ( K ) along those curves, given an initial value K 0 ,</text>
<formula><location><page_108><loc_49><loc_61><loc_90><loc_64></location>R = K 0 -K α ( K 0 ) . (6.28)</formula>
<text><location><page_108><loc_19><loc_55><loc_90><loc_59></location>Reorganising to get, K 0 = K + Rα ( K 0 ) , and then substituting into β , this leads to the implicit relation,</text>
<formula><location><page_108><loc_40><loc_52><loc_90><loc_54></location>β ( q, K , R ) = α ( q, K + Rβ ( q, K , R )) . (6.29)</formula>
<text><location><page_108><loc_19><loc_49><loc_76><loc_50></location>I invoke the implicit function theorem to calculate the derivatives of β ,</text>
<formula><location><page_108><loc_41><loc_43><loc_90><loc_46></location>∂β ∂ K = α ' 1 -Rα ' , ∂β ∂R = -βα ' 1 -Rα ' , (6.30)</formula>
<text><location><page_108><loc_19><loc_34><loc_90><loc_41></location>which show that a discontinuity develops when Rα ' → 1 . This is the point where the characteristic trajectories along which β is constant converge to form a caustic. Beyond this point, β seems to become a multi-valued function.</text>
<text><location><page_108><loc_19><loc_31><loc_62><loc_32></location>An analytic solution to β only exists when α is linear,</text>
<formula><location><page_108><loc_37><loc_25><loc_90><loc_28></location>α = α 1 ( q ) + α 2 ( q ) K , β = α 1 ( q ) + α 2 ( q ) K 1 -α 2 ( q ) R , (6.31)</formula>
<text><location><page_108><loc_19><loc_21><loc_62><loc_22></location>and when α 2 ( q ) is small, I can expand β into a series,</text>
<formula><location><page_108><loc_41><loc_15><loc_90><loc_19></location>β glyph[similarequal] α 1 + α 2 ( K + α 1 R ) ∞ ∑ n =0 R n α n 2 , (6.32)</formula>
<text><location><page_108><loc_19><loc_12><loc_90><loc_13></location>and by comparing this to the perturbative deformation found previously, (4.25), I can see</text>
<figure>
<location><page_109><loc_21><loc_75><loc_88><loc_88></location>
<caption>Figure 6.1: Numerically solved deformation function for initial function α = tanh( ω K ) . The numerical evolution breaks for R > ω because a discontinuity has developed. The initial function is indicated by the black line. The plots are in ω = 1 units.</caption>
</figure>
<text><location><page_109><loc_19><loc_56><loc_90><loc_65></location>the correspondence α 1 = β ∅ and α 2 = ε 2 β ( R ) /β ∅ = ε 2 β 2 . For other initial functions, I must numerically solve the deformation. As a test, in Fig. 6.1, I numerically solve for β when α = tanh( ω K ) . I see that, as R increases, the positive gradient in K intensifies to form a discontinuity, and softens as R decreases.</text>
<text><location><page_109><loc_19><loc_39><loc_90><loc_53></location>I have also numerically solved for the deformation when the initial function is given by α = cos ( ω K ) , shown in Fig. 6.2. This function is motivated by loop quantum cosmology models with holonomy corrections [40-42]. As with the tanh numerical solution in Fig. 6.1, I see the positive gradient intensify and the negative gradient soften. I could not evolve the equations past the formation of the shock wave so I cannot say for certain whether a periodicity emerges in R , but I can compare the cross sections for β in Fig. 6.2(d).</text>
<text><location><page_109><loc_19><loc_31><loc_90><loc_37></location>This cross section appears to match what was found in section 5.4.3 when I attempted to find the correspondence between β ( K , 0) and β ( R ) . It would seem that β (0 , R ) should be a non-vanishing function of the shape as shown in Fig. 5.2(c).</text>
<text><location><page_109><loc_19><loc_17><loc_90><loc_28></location>When the inviscid Burgers' equation is being simulated in the context of fluid dynamics, a choice must be made on how to model the shock wave [92]. The direct continuation of the equation means that the density function u becomes multi-valued, and the physical intepretation of it as a density breaks down. The alternative is to propagate the shock wave as a singular object, which requires a modification to the equations.</text>
<text><location><page_109><loc_19><loc_11><loc_90><loc_15></location>Considering my case of the deformation function, allowing a shock wave to propagate does not seem to make sense. It might require being able to interpret β as a density</text>
<figure>
<location><page_110><loc_30><loc_57><loc_80><loc_88></location>
<caption>Figure 6.2: Numerically solved deformation function with an initial function α = cos ( ω K ) and periodic boundary conditions. The numerical evolution breaks for | R | > ω because discontinuities have developed. The initial function is indicated by the black line. The plots are in ω = 1 units.</caption>
</figure>
<text><location><page_110><loc_19><loc_36><loc_90><loc_47></location>function and the space of ( K , R ) to be interpreted as a medium. Whether or not the shock wave remains singular or becomes multi-valued, the most probable interpretation is that it represents a disconnection between different branches of curvature configurations. That is, for a universe to transition from one side of the discontinuity to the other may require taking an indirect path through the phase space.</text>
<section_header_level_1><location><page_110><loc_19><loc_30><loc_47><loc_31></location>6.2 Linear deformation</section_header_level_1>
<text><location><page_110><loc_19><loc_20><loc_90><loc_26></location>If I take the analytic solution for the deformation function when its initial condition is linear (6.31), I can substitute it into the general form for the gravitational action (6.22). If I assume I am in a region where 1 -α 2 R > 0 , I get the solution,</text>
<formula><location><page_110><loc_22><loc_14><loc_90><loc_18></location>L = ω √ q α 2 { sgn ( 1 + α 2 K α 1 ) √ | α 1 | -√ | α 1 + α 2 K| √ | 1 -α 2 R | } + √ q ( vξ -V ) , (6.33)</formula>
<text><location><page_111><loc_19><loc_87><loc_83><loc_88></location>and expanding in series for small α 2 when I am in a region where | α 1 | glyph[greatermuch] | α 2 K| ,</text>
<formula><location><page_111><loc_25><loc_81><loc_90><loc_84></location>L = ω 2 √ q | α 1 | ( R -K α 1 -α 2 4 ( R + K α 1 ) 2 + O ( α 3 2 ) ) + √ q ( vξ -V ) , (6.34)</formula>
<text><location><page_111><loc_19><loc_69><loc_90><loc_78></location>which matches exactly the fourth order perturbative action I found previously (4.24). The Hamiltonian constraint associated with the non-perturbative action can be found from (6.24), and then I can solve for K when the constraint vanishes (as long as I specify that it must be finite in the limit α 2 → 0 ),</text>
<formula><location><page_111><loc_24><loc_63><loc_90><loc_67></location>K = { 2 ω sgn( α 1 ) √ | α 1 | V ( 1 -α 2 V 2 ω √ | α 1 | ) -α 1 R }( 1 -α 2 V ω √ | α 1 | ) -2 , (6.35)</formula>
<text><location><page_111><loc_19><loc_57><loc_90><loc_61></location>and if I restrict to the FLRW metric and a perfect fluid as in section 2.8, I find the modified Friedmann equation,</text>
<formula><location><page_111><loc_25><loc_51><loc_90><loc_55></location>H 2 N 2 = { sgn( α 1 ) √ | α 1 | 3 ω ρ ( 1 -α 2 ρ 2 ω √ | α 1 | ) -α 1 k a 2 }( 1 -α 2 ρ ω √ | α 1 | ) -2 . (6.36)</formula>
<text><location><page_111><loc_19><loc_40><loc_90><loc_49></location>There is a correction term similar to that found for the fourth order perturbative action which suggests there could be a bounce when ρ → 2 ω √ | α 1 | /α 2 . However, there is also an additional factor which causes H to diverge when ρ → ω √ | α 1 | /α 2 , which is before that potential bounce.</text>
<text><location><page_111><loc_19><loc_28><loc_90><loc_37></location>This is directly comparable to the modified Friedmann equation found for the deformation function β ( R ) = β ∅ (1 + β 2 R ) -1 , (5.64) investigated in section 5.4.1, with α 1 = β ∅ and α 2 = ωβ 2 / 2 . As is found here, those results suggested a sudden singularity where H diverges when a and ρ remain finite.</text>
<section_header_level_1><location><page_111><loc_19><loc_22><loc_37><loc_24></location>6.3 Discussion</section_header_level_1>
<text><location><page_111><loc_19><loc_12><loc_90><loc_19></location>I have found the general form of the deformed gravitation action when considering tensor combinations of derivatives up to second order. The way in which the deformation, and thereby the action, depends on the extrinsic and intrinsic curvature was found to be highly</text>
<text><location><page_112><loc_19><loc_84><loc_90><loc_88></location>non-linear. Curiously, its form matches an equation found in fluid dynamics. The meaning of this comparison is far from clear.</text>
<text><location><page_112><loc_19><loc_76><loc_90><loc_82></location>For different initial functions, I numerically solved for the deformation function until a discontinuity formed. The meaning of this discontinuity is not clear, but might manifest as a barrier across which paths through phase space cannot cross.</text>
<section_header_level_1><location><page_113><loc_19><loc_79><loc_33><loc_81></location>Chapter 7</section_header_level_1>
<section_header_level_1><location><page_113><loc_19><loc_70><loc_42><loc_73></location>Conclusions</section_header_level_1>
<text><location><page_113><loc_19><loc_55><loc_90><loc_64></location>I have attempted to thoroughly investigate the effects that a quantum-motivated deformation to the hypersurface deformation algebra of general relativity has in the semi-classical limit. Starting from the algebra, I have shown how to regain a deformed gravitational action or a deformed scalar-tensor constraint.</text>
<text><location><page_113><loc_19><loc_38><loc_90><loc_52></location>Finding the minimally-deformed version of a non-minimally coupled scalar-tensor model, I was able to establish the classical low-curvature reference point. I was able to show how the higher-order curvature terms arising from a deformation are qualitatively different from conventional higher-order terms which can absorbed by a non-minimally coupled scalar field. I also investigated some of the interesting effects which non-minimal coupling has on cosmology.</text>
<text><location><page_113><loc_19><loc_22><loc_90><loc_36></location>As a first step towards including higher-order curvature terms coming from a deformation, I derived the fourth order gravitational action perturbatively. The nearest order corrections demonstrate a change in the relative structure between time and space since the higher order curvature terms appear with a different sign. I investigated the cosmological implications of the higher order terms, albeit while using the assumption that the action found perturbatively could be extended beyond the perturbative regime.</text>
<text><location><page_113><loc_19><loc_11><loc_90><loc_20></location>In attempting to find the deformed scalar-tensor constraint to any order, I was able to show how the momenta and spatial derivatives maintain the same relative kinetic structure. Interestingly, the way the scalar field and gravitational kinetic terms combine must also be unchanged. That is to say that higher order gravitational terms are necessarily</text>
<text><location><page_114><loc_19><loc_74><loc_90><loc_88></location>accompanied by higher order scalar terms of the same form. The main consequence of this seems to be that a potential term (in a general sense) must be present for a deformation of the kinetic terms to affect the dynamics. By testing different deformation functions, I was able to show what kinds of cosmological effects should be expected. Interestingly, the deformations which cause a big bounce seem to be required to vanish, but are not required to change sign.</text>
<text><location><page_114><loc_19><loc_55><loc_90><loc_72></location>For the final chapter, I derived the general deformed gravitational action. The way the deformation function is differently affected by extrinsic and intrinsic curvature (or, equivalently, by time and space derivatives) was found to be similar to a differential equation which usually appears in fluid mechanics. Discontinuities in the deformation function seem to be inevitable, but the interpretation of what they mean is not clear. By checking the nearest order perturbative corrections, I was able to validate the perturbative action derived in an earlier chapter.</text>
<text><location><page_114><loc_19><loc_28><loc_90><loc_53></location>One of the original motivations of this study was to provide insight into the problem of incorporating spatial derivatives, local degrees of freedom and matter fields into models of loop quantum cosmology which deform space-time covariance. From my results, it would seem that the problem comes from considering the kinetic terms as separable, or as differently deformed. The kinetic term, when constructed with canonical variables, cannot have its internal structure deformed beyond a sign. The deformation can only be a function of the combined term, which means that matter field derivatives deform the space-time covariance in a similar way to curvature. This may strike at the heart of the way the loop quantisation project, which attempts to first find a quantum theory of gravity, typically adds in matter as an afterthought.</text>
<text><location><page_114><loc_19><loc_12><loc_90><loc_26></location>That being said, there are important caveats to this work which must be kept in mind. The fact that I used metric variables rather than the preferred connection or loop variables might limit the applicability of my results when comparing to the motivating theory. Moreover, the deformation of the constraint algebra is only predicted for real values of γ BI . I also only considered combinations of derivatives or momenta that were a maximum of two orders, when higher order combinations and higher order derivatives are likely to</text>
<text><location><page_115><loc_19><loc_87><loc_48><loc_88></location>appear in true quantum corrections.</text>
<text><location><page_115><loc_19><loc_68><loc_90><loc_85></location>As said in the introduction, 1, there are potentially wider implications for this study. The deformation can lead to a modified dispersion relation, possibly indicating a variable speed of light or an invariant energy scale. It might be related to non-classical geometric qualities such a non-commutativity or scale-dependent dimensionality. In the literature, it is indicated that the deformation function may change sign, implying a transition from a Lorentzian to a Euclidean geometry at high densities. In such a way, it might be a potential mechanism for the Hartle-Hawking no-boundary proposal.</text>
<section_header_level_1><location><page_116><loc_19><loc_79><loc_36><loc_81></location>Appendix A</section_header_level_1>
<section_header_level_1><location><page_116><loc_19><loc_70><loc_72><loc_73></location>Decomposing the curvature</section_header_level_1>
<text><location><page_116><loc_19><loc_60><loc_90><loc_64></location>In our calculations, we need to decompose the three dimensional Riemann curvature frequently, so we collect the relevant identities in this appendix.</text>
<text><location><page_116><loc_19><loc_56><loc_90><loc_58></location>The Riemann tensor is defined as the commutator of two covariant derivatives of a vector</text>
<formula><location><page_116><loc_42><loc_51><loc_90><loc_53></location>∇ c ∇ d A a -∇ d ∇ c A a = R a bcd A b , (A.1)</formula>
<text><location><page_116><loc_19><loc_47><loc_61><loc_48></location>and can be given in terms of the Christoffel symbols,</text>
<formula><location><page_116><loc_38><loc_42><loc_90><loc_44></location>R a bcd = ∂ c Γ a db -∂ d Γ a cb +Γ a ce Γ e db -Γ a de Γ e cb , (A.2)</formula>
<text><location><page_116><loc_19><loc_38><loc_34><loc_39></location>which are given by</text>
<formula><location><page_116><loc_45><loc_34><loc_90><loc_37></location>Γ a bc = q ad ∂ ( b q c ) d -1 2 ∂ a q bc , (A.3)</formula>
<text><location><page_116><loc_19><loc_30><loc_75><loc_32></location>The variation of the Riemann tensor is given by the Palatini equation,</text>
<formula><location><page_116><loc_44><loc_25><loc_90><loc_27></location>δR a bcd = ∇ c δ Γ a db -∇ d δ Γ a cb , (A.4)</formula>
<text><location><page_116><loc_19><loc_21><loc_51><loc_22></location>where the variation of the connection is</text>
<formula><location><page_116><loc_43><loc_15><loc_90><loc_18></location>δ Γ a bc = q ad ∇ ( b δq c ) d -1 2 ∇ a δq bc , (A.5)</formula>
<text><location><page_117><loc_19><loc_87><loc_42><loc_88></location>from which we can calculate,</text>
<formula><location><page_117><loc_39><loc_82><loc_90><loc_84></location>δR a bcd = Θ a ef bcd δq ef +Φ a efgh bcd ∇ ef δq gh (A.6)</formula>
<text><location><page_117><loc_19><loc_77><loc_50><loc_79></location>where we've defined the useful tensors,</text>
<formula><location><page_117><loc_28><loc_72><loc_90><loc_75></location>Θ a ef bcd = -1 2 ( q a ( e R f ) bcd + δ ( e b R f ) a cd ) , (A.7a)</formula>
<formula><location><page_117><loc_26><loc_69><loc_90><loc_72></location>Φ a efgh bcd = 1 2 ( q a ( e δ f ) d δ gh bc + q a ( g δ h ) d δ ef bc -q a ( e δ f ) c δ gh bd -q a ( g δ h ) c δ ef bd ) , (A.7b)</formula>
<text><location><page_117><loc_19><loc_65><loc_58><loc_66></location>but contracted versions of these are more useful,</text>
<formula><location><page_117><loc_23><loc_59><loc_90><loc_62></location>Θ cd ab := δ ef ab Θ g cd egf = 1 2 ( Q cdef R e ( ab ) f + δ ( c ( a R d ) b ) ) , q cd Θ cd ab = 0 , q ab Θ cd ab = 0 , (A.8a)</formula>
<formula><location><page_117><loc_21><loc_56><loc_90><loc_59></location>Φ cdef ab := δ gh ab Φ i cdef gih = 1 2 ( q c ( e δ f ) d ab + q d ( e δ f ) c ab -q cd δ ef ab -q ef δ cd ab ) , (A.8b)</formula>
<formula><location><page_117><loc_22><loc_54><loc_90><loc_55></location>Φ abcd := q ef Φ abcd ef = Q abcd -q ab q cd . (A.8c)</formula>
<text><location><page_117><loc_19><loc_47><loc_90><loc_50></location>To decompose the Riemann tensor in terms of partial derivatives, use this formula for decomposing the second covariant derivative of the variation of the metric,</text>
<formula><location><page_117><loc_24><loc_39><loc_90><loc_44></location>∇ d ∇ c δq ab = ∂ d ∂ c δq ab + ∂ g δq ef ( -Γ g dc δ ef ab -4 δ ( e ( a Γ f ) b )( c δ g d ) ) + δq ef ( -2 ∂ d Γ ( e c ( a δ f ) b ) +2Γ g dc Γ ( e g ( a δ f ) b ) +2Γ g d ( a δ ( e b ) Γ f ) cg +2Γ ( e d ( a Γ f ) b ) c ) . (A.9)</formula>
<text><location><page_117><loc_19><loc_32><loc_90><loc_35></location>The two equations we need most are the derivative of the Ricci scalar with respect to the first and second spatial derivative of the metric, and we can find these from combining the</text>
<text><location><page_118><loc_19><loc_87><loc_32><loc_88></location>above equations,</text>
<formula><location><page_118><loc_25><loc_67><loc_90><loc_84></location>∂R ∂ ( ∂ d ∂ c q ab ) = ∂ ( ∇ h ∇ g q ef ) ∂ ( ∂ d ∂ c q ab ) ∂R ∂ ( ∇ h ∇ g q ef ) = δ d h δ c g δ ab ef Φ efgh = Φ abcd ∴ ∂R ∂q ab,cd = Φ abcd = Q abcd -q ab q cd , (A.10a) ∂R ∂ ( ∂ c q ab ) = ∂ ( ∇ h ∇ g q ef ) ∂ ( ∂ c q ab ) ∂R ∂ ( ∇ h ∇ g q ef ) = ( -Γ c gh δ ab ef -4 δ ( a ( e Γ b ) f )( g δ c h ) ) Φ efgh , ∴ ∂R ∂q ab,c = 3 2 Q abde ∂ c q de -Q edc ( a ∂ b ) q de + q ab Y c -2 q c ( b Y a ) -1 2 q ab X c + q c ( b X a ) , (A.10b)</formula>
<text><location><page_118><loc_19><loc_63><loc_59><loc_65></location>where X a := q bc ∂ a q bc and Y a := q bc ∂ ( c q b ) a = ∂ b q ba .</text>
<section_header_level_1><location><page_119><loc_19><loc_79><loc_36><loc_81></location>Appendix B</section_header_level_1>
<section_header_level_1><location><page_119><loc_19><loc_64><loc_72><loc_73></location>The general diffeomorphism constraint</section_header_level_1>
<text><location><page_119><loc_19><loc_44><loc_90><loc_58></location>I start from the assumption that the equal-time slices of our foliation are internally diffeomorphism covariant. That is to say that spatial transformations and distortions are not deformed by the deformation of the constraint algebra. As such, the Hamiltonian constraint is susceptible to deformation and the diffeomorphism constraint is not. Therefore I need to consider what form the diffeomorphism constraint has. In the hyperspace deformation algebra (2.13), the diffeomorphism constraint forms a closed sub-algebra,</text>
<formula><location><page_119><loc_42><loc_39><loc_90><loc_40></location>{ D a [ N a ] , D b [ M b ] } = D a [ L M N a ] . (B.1)</formula>
<text><location><page_119><loc_19><loc_32><loc_90><loc_35></location>This equation shows that the diffeomorphism constraint is the generator of spatial diffeomorphisms (hence the name),</text>
<formula><location><page_119><loc_46><loc_29><loc_90><loc_31></location>{ F, D a [ N a ] } = L N F, (B.2)</formula>
<text><location><page_119><loc_19><loc_23><loc_90><loc_26></location>for any phase space function F . Using this relation, I can determine the unique form of the constraint for any field content.</text>
<text><location><page_119><loc_19><loc_17><loc_90><loc_20></location>For these calculations, I must include the concept of a tensor density, which does not transform under a change of coordinates as a tensor does. A tensor density of weight</text>
<text><location><page_120><loc_19><loc_87><loc_57><loc_88></location>w Ψ ∈ R transforms under the change x a → x ' a ' ,</text>
<formula><location><page_120><loc_29><loc_81><loc_90><loc_85></location>Ψ ' b ' 1 ...b ' i a ' 1 ...a ' j = ∣ ∣ ∣ ∣ det ( ∂x c ∂x ' c ' )∣ ∣ ∣ ∣ w Ψ Ψ b 1 ...b i a 1 ...a j ∂x ' b ' 1 ∂x b 1 · · · ∂x ' b ' i ∂x b i ∂x a 1 ∂x ' a ' 1 · · · ∂x a j ∂x ' a ' j , (B.3)</formula>
<text><location><page_120><loc_19><loc_64><loc_90><loc_79></location>and one can 'de-densitise' to find a tensor 1 by multiplying it by q -w Ψ / 2 , because √ q is a scalar density of weight one [10, p. ˜ 276]. The integration measure d 3 x has a weight of -1 , so for an integral to be appropriately tensorial, the integrand must have a weight of +1 , e.g. ∫ d 3 x √ q . Since making a Legendre transformation requires using the term ∫ d 3 x ˙ ψπ for a conjugate pair ( ψ, π ) , when the variable ψ is of weight w ψ , the momentum π is of weight 1 -w ψ .</text>
<section_header_level_1><location><page_120><loc_19><loc_58><loc_76><loc_60></location>B.1 Diffeomorphism constraint for a scalar field</section_header_level_1>
<text><location><page_120><loc_19><loc_54><loc_63><loc_55></location>I consider a scalar field ( ψ, π ) . Take (B.2) with F = ψ ,</text>
<formula><location><page_120><loc_23><loc_35><loc_90><loc_51></location>{ ψ ( x ) , D a [ N a ] } = ∫ d 3 yN a ( y ) δD a ( y ) δπ ( x ) , = N a ∂D a ∂π -∂ b ( N a ∂D a ∂π ,b ) + ∂ bc ( N a ∂D a ∂π ,bc ) + . . . = N a { ∂D a ∂π -∂ b ( ∂D a ∂π ,b ) + ∂ bc ( ∂D a ∂π ,bc )} + ∂ b N a { -∂D a ∂π ,b +2 ∂ c ( ∂D a ∂π ,bc )} + ∂ bc N a ( ∂D a ∂π ,bc ) + . . . , (B.4a) L N ψ = N a ∂ a ψ, (B.4b)</formula>
<text><location><page_120><loc_19><loc_30><loc_63><loc_31></location>comparing these two equations, one can easily see that</text>
<formula><location><page_120><loc_40><loc_24><loc_90><loc_27></location>∂D a ∂π = ∂ a ψ, ∂D a ∂π ,b = 0 , ∂D a ∂π ,bc = 0 . (B.5)</formula>
<text><location><page_120><loc_19><loc_18><loc_90><loc_22></location>Checking what result I get for F = π merely produces the same equations and therefore the diffeomorphism constraint for a scalar field is given by,</text>
<formula><location><page_120><loc_50><loc_13><loc_90><loc_15></location>D a = π∂ a ψ. (B.6)</formula>
<text><location><page_121><loc_19><loc_82><loc_90><loc_88></location>I considered up to second order spatial derivatives here as a demonstration, but no diffeomorphism constraint goes beyond first order, so I will not bother with them for further equations below.</text>
<section_header_level_1><location><page_121><loc_19><loc_75><loc_71><loc_77></location>B.2 Diffeomorphism constraint for a vector</section_header_level_1>
<text><location><page_121><loc_19><loc_71><loc_82><loc_72></location>I consider a weightless contravariant vector ( A a , P b ) . Take (B.2) with F = A a ,</text>
<formula><location><page_121><loc_25><loc_55><loc_90><loc_68></location>{ A a ( x ) , D b [ N b ] } = ∫ d 3 yN b ( y ) δD b ( y ) δP a ( x ) , = N b ∂D b ∂P a -∂ c ( N b ∂D b ∂P a,c ) + . . . = N b { ∂D b ∂P a -∂ c ( ∂D b ∂P a,c )} + ∂ c N b ( -∂D b ∂P a,c ) + . . . (B.7a) L N A a = N b ∂ b A a -A b ∂ b N a , (B.7b)</formula>
<text><location><page_121><loc_19><loc_46><loc_90><loc_52></location>looking at the derivative of N a , I can see that ∂D b ∂P a,c = δ a b A c , and substituting this back into the equation I find, ∂D b ∂P a = δ a b ∂ c A c + ∂ b A a . If I check with F = P a I find the same equations, leading us to the diffeomorphism constraint</text>
<formula><location><page_121><loc_44><loc_41><loc_90><loc_42></location>D a = P b ∂ a A b + ∂ b ( P a A a ) . (B.8)</formula>
<section_header_level_1><location><page_121><loc_19><loc_35><loc_71><loc_36></location>B.3 Diffeomorphism constraint for a tensor</section_header_level_1>
<text><location><page_121><loc_19><loc_27><loc_90><loc_31></location>I consider a rank-2 tensor defined on a three dimensional spatial manifold ( q ab , p cd ) . I use the example of the metric, but our result is general. Test (B.2) using F = q ab ,</text>
<formula><location><page_121><loc_25><loc_11><loc_90><loc_25></location>{ q ab ( x ) , D c [ N c ] } = ∫ d 3 yN c ( y ) δD c ( y ) δp ab ( x ) , = N c ∂D c ∂p ab -∂ d ( N c ∂D c ∂p ab ,d ) + . . . = N c { ∂D c ∂p ab -∂ d ( ∂D c ∂p ab ,d )} + ∂ d N c ( -∂D c ∂p ab ,d ) + . . . (B.9a) L N q ab = N c ∂ c q ab +2 q c ( b ∂ a ) N c , (B.9b)</formula>
<text><location><page_122><loc_19><loc_82><loc_90><loc_88></location>looking at the derivative of N a , I can see that ∂D c ∂p ab ,d = -2 q c ( b δ d a ) , and substituting this back into the equation I find, ∂D c ∂p ab = ∂ c q ab -2 ∂ ( a q b ) c . If I check with F = p ab I find the same equations, leading us to the diffeomorphism constraint</text>
<formula><location><page_122><loc_42><loc_77><loc_90><loc_79></location>D a = p bc ∂ a q bc -2 ∂ ( c ( q b ) a p bc ) , (B.10)</formula>
<text><location><page_122><loc_19><loc_72><loc_65><loc_73></location>and for the specific example of the metric, this reduces to</text>
<formula><location><page_122><loc_48><loc_67><loc_90><loc_69></location>D a = -2 q ab ∇ c p bc . (B.11)</formula>
<section_header_level_1><location><page_122><loc_19><loc_61><loc_80><loc_63></location>B.4 Diffeomorphism constraint for a tensor density</section_header_level_1>
<text><location><page_122><loc_19><loc_51><loc_90><loc_58></location>For the general case of a tensor density with n covariant indices, m contravariant indices and weight w Ψ , ( Ψ b 1 ··· b m a 1 ··· a n , Π c 1 ··· c n d 1 ··· d m ) where the canonical momentum has weight 1 -w Ψ , the associated diffeomorphism constraint is given by,</text>
<formula><location><page_122><loc_31><loc_43><loc_90><loc_49></location>D a = Π b 1 ··· b n c 1 ··· c m ∂ a Ψ c 1 ··· c m b 1 ··· b n -w Ψ ∂ a ( Π b 1 ··· b n c 1 ··· c m Ψ c 1 ··· c m b 1 ··· b n ) -n∂ ( b 1 ( Ψ c 1 ··· c m b 2 ··· b n ) a Π b 1 ··· b n c 1 ··· c m ) + m∂ ( c 1 ( Π b 1 ··· b n c 2 ··· c m ) a Ψ c 1 ··· c m b 1 ··· b n ) . (B.12)</formula>
<section_header_level_1><location><page_123><loc_19><loc_79><loc_36><loc_81></location>Appendix C</section_header_level_1>
<section_header_level_1><location><page_123><loc_19><loc_64><loc_72><loc_73></location>Fourth order perturbative gravitational action: Extras</section_header_level_1>
<text><location><page_123><loc_19><loc_57><loc_49><loc_58></location>For convenience, I use the definitions,</text>
<formula><location><page_123><loc_24><loc_52><loc_90><loc_53></location>X a = q bc ∂ a q bc , Y a = q bc ∂ c q ba = ∂ b q ab , Z a = v bc T ∂ a q bc , W a = v bc T ∂ c q ba . (C.1)</formula>
<text><location><page_123><loc_19><loc_47><loc_87><loc_48></location>Evaluating each term in the ∂ cd θ ab bracket of (4.11), by substituting in the variables</text>
<formula><location><page_123><loc_29><loc_42><loc_90><loc_45></location>q := det q ab , v := q ab v ab , w := v ab v ab -1 3 v 2 , R := q bc R a bac (C.2)</formula>
<text><location><page_123><loc_19><loc_38><loc_73><loc_39></location>and using the equations derived for decomposing R in appendix A,</text>
<formula><location><page_123><loc_31><loc_21><loc_90><loc_35></location>∂L ∂q ab,cd = ( Q abcd -q ab q cd ) ∂L ∂R , (C.3a) v ef ∂ 2 L ∂q ef,cd ∂v ab = ( v cd T -2 3 vq cd )( q ab ∂ 2 L ∂v∂R +2 v ab T ∂ 2 L ∂w∂R ) , (C.3b) ∂ 2 L ∂v ab ∂v cd = q ab q cd ( ∂ 2 L ∂v 2 -2 3 ∂L ∂w ) +2 Q abcd ∂L ∂w +2 ( q ab v cd T + v ab T q cd ) ∂ 2 L ∂v∂w +4 v ab T v cd T ∂ 2 L ∂w 2 . (C.3c)</formula>
<text><location><page_124><loc_19><loc_87><loc_60><loc_88></location>Evaluating each term in the ∂ c θ ab bracket of (4.11),</text>
<formula><location><page_124><loc_36><loc_78><loc_90><loc_84></location>∂L ∂q ab,c = ∂L ∂R ( 3 2 Q abde ∂ c q de -q c ( d q e )( a ∂ b ) q de + q ab Y c -1 2 q ab X c -2 q c ( a Y b ) + q c ( a X b ) ) , (C.4a)</formula>
<formula><location><page_124><loc_33><loc_69><loc_90><loc_76></location>v ef ∂ 2 L ∂q ef,c ∂v ab = ( 3 2 Z c -W c -2 v cd T Y d + v cd T X d + v 3 X c ) × ( q ab ∂ 2 L ∂v∂R +2 v ab T ∂ 2 L ∂w∂R ) , (C.4b)</formula>
<formula><location><page_124><loc_25><loc_57><loc_90><loc_67></location>v ef ∂ d ( ∂ 2 L ∂q ef,cd ∂v ab ) = ( v cd T -2 v 3 q cd ){ ( q ab ∂ d -Q abef ∂ d q ef ) ∂ 2 L ∂v∂R +2 ( v ab T ∂ d + Q abef ∂ d v T ef -2 v e ( a T q b ) f ∂ d q ef ) ∂ 2 L ∂w∂R } + ( Z c -W c + v 3 X c + v 3 Y c -v cd T Y d ) ( q ab ∂ 2 L ∂v∂R +2 v ab T ∂ 2 L ∂w∂R ) , (C.4c)</formula>
<formula><location><page_124><loc_26><loc_45><loc_90><loc_55></location>Γ c de ∂ 2 L ∂v ab ∂v de = ( Y c -1 2 X c ){ q ab ( ∂ 2 L ∂v 2 -2 3 ∂L ∂w ) +2 v ab T ∂ 2 L ∂v∂w } + ( 2 q cd q e ( a ∂ b ) q de -Q abde ∂ c q de ) ∂L ∂w +(2 W c -Z c ) ( q ab ∂ 2 L ∂v∂w +2 v ab T ∂ 2 L ∂w 2 ) , (C.4d)</formula>
<formula><location><page_124><loc_22><loc_41><loc_90><loc_44></location>Γ c de ∂β ∂v ab ∂L ∂v cd = ( q ab ∂β ∂v +2 v ab T ∂β ∂w ){( Y c -1 2 X c ) ∂L ∂v +(2 W c -Z c ) ∂L ∂w } , (C.4e)</formula>
<formula><location><page_124><loc_24><loc_33><loc_90><loc_40></location>∂ d β ∂ 2 L ∂v ab ∂v cd = ∂ c β { q ab ( ∂ 2 L ∂v 2 -2 3 ∂L ∂w ) +2 v ab T ∂ 2 L ∂v∂w } +2 q c ( a ∂ b ) β ∂L ∂w +2 v cd T ∂ d β ( q ab ∂ 2 L ∂v∂w +2 v ab T ∂ 2 L ∂w 2 ) , (C.4f)</formula>
<formula><location><page_124><loc_23><loc_12><loc_90><loc_31></location>∂ d ( ∂ 2 L ∂v ab ∂v cd ) = ( q ab ∂ c -q ab Y c -Q abef ∂ c q ef ) ( ∂ 2 L ∂v 2 -2 3 ∂L ∂w ) +2 ( q c ( a ∂ b ) -q c ( a Y b ) -q c ( e q f )( a ∂ b ) q ef ) ∂L ∂w +2 { q ab ( v cd T ∂ d -v cd T Y d -W c + q cd ∂ e v T de ) + v ab T ∂ c -v ab T Y c + Q abef ( ∂ c v T ef -v cd T ∂ d q ef ) -2 v e ( a T q b ) f ∂ c q ef } ∂ 2 L ∂v∂w +4 { v ab T ( v cd T ∂ d -W c -v cd T Y d + q cd ∂ e v T de ) + Q abef v cd T ∂ d v T ef -2 v e ( a T q b ) f v cd T ∂ d q ef } ∂ 2 L ∂w 2 , (C.4g)</formula>
<formula><location><page_125><loc_21><loc_82><loc_90><loc_89></location>∂β ∂v ab ∂ d ( ∂L ∂v cd ) = ( q ab ∂β ∂v +2 v ab T ∂β ∂w ) × { ( ∂ c -Y c ) ∂L ∂v +2 ( v cd T ∂ d + q cd ∂ e v T de -v cd T Y d -W c ) ∂L ∂w } . (C.4h)</formula>
<section_header_level_1><location><page_126><loc_19><loc_79><loc_36><loc_81></location>Appendix D</section_header_level_1>
<section_header_level_1><location><page_126><loc_19><loc_64><loc_89><loc_73></location>Deformed scalar-tensor constraint to all orders: Extras</section_header_level_1>
<text><location><page_126><loc_19><loc_57><loc_55><loc_58></location>Use the following definitions for convenience,</text>
<formula><location><page_126><loc_24><loc_52><loc_90><loc_53></location>X a = q bc ∂ a q bc , Y a = q bc ∂ c q ba = ∂ b q ab , Z a = p bc T ∂ a q bc , W a = p bc T ∂ c q ba . (D.1)</formula>
<text><location><page_126><loc_19><loc_47><loc_60><loc_49></location>Evaluating each term in the ∂ cd θ ab bracket of (5.7),</text>
<formula><location><page_126><loc_26><loc_38><loc_90><loc_45></location>∂C ∂q ef,cd ∂ 2 C ∂p ab ∂p cd = 2 δ cd ab ∂C ∂R ∂C ∂ P -2 q ab q cd ∂C ∂R ( ∂ 2 C ∂p 2 + 1 3 ∂C ∂ P ) +2 ( q ab p cd T -2 p T ab q cd ) ∂C ∂R ∂ 2 C ∂p∂ P +4 p T ab p cd T ∂C ∂R ∂ 2 C ∂ P 2 (D.2a)</formula>
<formula><location><page_126><loc_25><loc_33><loc_90><loc_36></location>-∂C ∂p ef ∂ 2 C ∂q ef,cd ∂p ab = 2 ( q cd ∂C ∂p -p cd T ∂C ∂ P )( q ab ∂ 2 C ∂p∂R +2 p T ab ∂ 2 C ∂ P ∂R ) (D.2b)</formula>
<formula><location><page_126><loc_36><loc_28><loc_90><loc_31></location>∂C ∂ψ ,cd ∂ 2 C ∂p ab ∂π = q cd ∂C ∂ ∆ ( q ab ∂ 2 C ∂p∂π +2 p T ab ∂ 2 C ∂ P ∂π ) (D.2c)</formula>
<formula><location><page_126><loc_33><loc_24><loc_90><loc_27></location>-∂C ∂π ∂ 2 C ∂ψ ,cd ∂p ab = -∂C ∂π q cd ( q ab ∂ 2 C ∂p∂ ∆ +2 p T ab ∂ 2 C ∂ P ∂ ∆ ) (D.2d)</formula>
<formula><location><page_126><loc_48><loc_19><loc_90><loc_22></location>-β ∂D c ∂p ab ,d = 2 βδ cd ab , (D.2e)</formula>
<text><location><page_127><loc_19><loc_87><loc_59><loc_88></location>Evaluating each term in the ∂ c θ ab bracket of (5.7),</text>
<formula><location><page_127><loc_24><loc_67><loc_81><loc_85></location>∂C ∂q ef,c ∂ 2 C ∂p ab ∂p ef = ∂C ∂ ∆ { q ab [ ∂ c ψ ( 1 2 ∂ 2 C ∂p 2 + 2 3 ∂C ∂ P ) -2 p cd T ∂ d ψ ∂ 2 C ∂p∂ P ] -2 δ c ( a ∂ b ) ψ ∂C ∂ P + p T ab [ ∂ c ψ ∂ 2 C ∂p∂ P -4 p cd T ∂ d ψ ∂ 2 C ∂ P 2 ]} + ∂C ∂R { ∂C ∂ P [ 3 ∂ c q ab -2 q cd ∂ ( a q b ) d +2 q ab Y c -q ab X c -4 δ c ( a Y b ) +2 δ c ( a X b ) ] + q ab X c ( ∂ 2 C ∂p 2 -2 3 ∂C ∂ P ) +2 p T ab X c ∂ 2 C ∂p∂ P + ( 3 Z c -2 W c -4 p cd T Y d +2 p cd T X d ) ( q ab ∂ 2 C ∂p∂ P +2 p T ab ∂ 2 C ∂ P 2 )} ,</formula>
<formula><location><page_127><loc_22><loc_59><loc_24><loc_63></location>∂q +2</formula>
<text><location><page_127><loc_22><loc_21><loc_24><loc_22></location>+2</text>
<formula><location><page_127><loc_23><loc_16><loc_82><loc_19></location>∂C ∂ψ ,c ∂ 2 C ∂π∂p ab = { 2 ∂ c ψ ∂C ∂γ + ( 1 2 X c -Y c ) ∂C ∂ ∆ }( q ab ∂C ∂π∂p +2 p T ab ∂C ∂π∂ P ) ,</formula>
<formula><location><page_127><loc_23><loc_17><loc_90><loc_76></location>(D.3a) ∂ 2 C ef,cd ∂ d ( ∂ 2 C ∂p ab ∂p ef ) = ∂C ∂R { [ q ab ( Y c -X c -2 ∂ c ) -2 ∂ c q ab ] ( ∂ 2 C ∂p 2 -2 3 ∂C ∂ P ) ( δ c ( a ∂ b ) -q ab ∂ c + q cd ∂ ( a q b ) d + δ c ( a Y b ) -2 ∂ c q ab ) ∂C ∂ P +2 [ q ab ( p cd T ∂ d + ∂ d p cd T + W c -Z c + p cd T Y d ) + p cd T ∂ d q ab + p T ab ( Y c -X c -2 ∂ c ) -2 Q abde ∂ c p de T -4 ∂ c q d ( a p T d b ) ] ∂ 2 C ∂p∂ P +4 [ Q abef p cd T ∂ d p ef T +2 p cd T ∂ d q e ( a p T e b ) + p T ab ( ∂ d p cd T + W c -Z c + p cd T Y d + p cd T ∂ d )] ∂ 2 C ∂ P 2 } , (D.3b) ∂C ∂p ef ∂ 2 C ∂q ef,c ∂p ab = { ∂C ∂p [ X c ∂ ∂R + 1 2 ∂ c ψ ∂ ∂ ∆ ] + ∂C ∂ P [ ( 3 Z c -2 W c -4 p cd T Y d ) ∂ ∂R -2 p cd T ∂ d ψ ∂ ∂ ∆ ]}( q ab ∂C ∂p +2 p T ab ∂C ∂ P ) +2 p cd T ∂C ∂ P { q ab ∂ d ∂ 2 C ∂p∂R +2 p T ab ∂ d ∂ 2 C ∂ P ∂R } , (D.3c) ∂C ∂p ef ∂ d ( ∂ 2 C ∂q ef,cd ∂p ab ) = ∂C ∂p { [ q ab ( X c + Y c -2 ∂ c ) -2 ∂ c q ab ] ∂ 2 C ∂p∂R +2 [ p T ab ( X c + Y c -2 ∂ c ) -2 Q abef ∂ c p ef T -4 ∂ c q d ( a p T d b ) ] ∂ 2 C ∂ P ∂R } +2 ∂C ∂ P { [ q ab ( Z c -W c -p cd T Y d + p cd T ∂ d ) + p cd T ∂ d q ab ] ∂ 2 C ∂p∂R [ p T ab ( Z c -W c -p cd T Y d + p cd T ∂ d ) + Q abef p cd T ∂ d p ef T +2 p cd T ∂ d q e ( a p T e b ) ] ∂ 2 C ∂ P ∂R } , (D.3d) (D.3e)</formula>
<formula><location><page_128><loc_25><loc_82><loc_79><loc_89></location>∂C ∂ψ ,cd ∂ d ( ∂ 2 C ∂π∂p ab ) = ∂C ∂ ∆ { ( ∂ c q ab + q ab ∂ c ) ∂ 2 C ∂π∂p +2 ( Q abef ∂ c p ef T +2 ∂ c q d ( a p T d a ) + p T ab ∂ c ) ∂ 2 C ∂π∂ P } ,</formula>
<formula><location><page_128><loc_32><loc_74><loc_78><loc_80></location>∂C ∂π ∂ d ( ∂ 2 C ∂ψ ,cd ∂p ab ) = ∂C ∂π { ( q ab ∂ c + ∂ c q ab -q ab Y c ) ∂ 2 C ∂p∂ ∆ +2 ( Q abef ∂ c p ef T +2 ∂ c q d ( a p T d b ) + p T ab ∂ c -p T ab Y c ) ∂ 2 C ∂ P ∂ ∆ } ,</formula>
<formula><location><page_128><loc_32><loc_66><loc_78><loc_72></location>∂ ( βD c ) ∂p ab = β ( ∂ c q ab -2 q cd ∂ ( a q b ) d ) + ( q ab ∂β ∂p +2 p T ab ∂β ∂ P ) × ( π∂ c ψ -2 ∂ d p cd T -2 3 ∂ c p -2 W c + Z c + 1 3 pX c ) ,</formula>
<formula><location><page_128><loc_44><loc_60><loc_65><loc_64></location>∂ d ( β ∂D c ∂p ab ,d ) = -2 δ c ( a ∂ b ) β,</formula>
<formula><location><page_128><loc_85><loc_62><loc_90><loc_86></location>(D.3f) (D.3g) (D.3h) (D.3i)</formula>
<text><location><page_128><loc_19><loc_54><loc_61><loc_56></location>Evaluating each term in the ∂ cd η ab bracket of (5.13),</text>
<formula><location><page_128><loc_35><loc_49><loc_90><loc_52></location>∂C ∂q cd,ab ∂ 2 C ∂π∂p cd = ∂C ∂R ( -2 q ab ∂ 2 C ∂π∂p +2 p ab T ∂ 2 C ∂π∂ P ) , (D.4a)</formula>
<formula><location><page_128><loc_36><loc_43><loc_90><loc_46></location>∂C ∂p cd ∂ 2 C ∂q cd,ab ∂π = ∂ 2 C ∂R∂π ( -2 q ab ∂C ∂p +2 p ab T ∂C ∂ P ) , (D.4b)</formula>
<formula><location><page_128><loc_32><loc_39><loc_90><loc_42></location>∂C ∂ψ ,ab ∂ 2 C ∂π 2 -∂C ∂π ∂ 2 C ∂ψ ,ab ∂π = q ab ( ∂C ∂ ∆ ∂ 2 C ∂π 2 -∂C ∂π ∂ 2 C ∂ ∆ ∂π ) , (D.4c)</formula>
<text><location><page_128><loc_19><loc_32><loc_60><loc_34></location>Evaluating each term in the ∂ c η ab bracket of (5.13),</text>
<formula><location><page_128><loc_22><loc_23><loc_90><loc_30></location>∂C ∂q cd,a ∂ 2 C ∂π∂p cd = ∂C ∂R { X a ∂ 2 C ∂π∂p + ( 3 Z a -2 W a -4 p ab T Y b +2 p ab T ∂ b ) ∂ 2 C ∂π∂ P } + ∂C ∂ ∆ { 1 2 ∂ a ψ ∂ 2 C ∂π∂p -2 p ab T ∂ b ψ ∂ 2 C ∂π∂ P } , (D.5a)</formula>
<formula><location><page_128><loc_25><loc_14><loc_90><loc_21></location>∂C ∂q cd,ab ∂ b ( ∂ 2 C ∂π∂p cd ) = ∂C ∂R { ( Y a -X a -2 ∂ a ) ∂ 2 C ∂π∂p +2 ( ∂ b p ab T + W a -Z a + p ab T Y b + p ab T ∂ b ) ∂ 2 C ∂π∂ P } , (D.5b)</formula>
<formula><location><page_129><loc_26><loc_81><loc_90><loc_88></location>∂C ∂p cd ∂ 2 C ∂q cd,a ∂π = ∂C ∂p { X a ∂ 2 C ∂R∂π + 1 2 ∂ a ψ ∂ 2 C ∂ ∆ ∂π } + ∂C ∂ P { ( 3 Z a -2 W a -4 p ab T Y b +2 p ab T X b ) ∂ 2 C ∂R∂π -2 p ab T ∂ b ψ ∂ 2 C ∂ ∆ ∂π } , (D.5c)</formula>
<formula><location><page_129><loc_27><loc_73><loc_90><loc_80></location>∂C ∂p cd ∂ b ( ∂ 2 C ∂q cd,ab ∂π ) = { ∂C ∂p ( X a + Y a -2 ∂ a ) +2 ∂C ∂ P ( Z a -W a -p ab T Y b + p ab T ∂ b ) } ∂ 2 C ∂R∂π , (D.5d)</formula>
<formula><location><page_129><loc_26><loc_65><loc_90><loc_72></location>∂C ∂ψ ,a ∂ 2 C ∂π 2 + ∂C ∂π ∂ 2 C ∂ψ ,a ∂π = 2 ∂ a ψ ( ∂C ∂γ ∂ 2 C ∂π 2 + ∂C ∂π ∂ 2 C ∂π∂γ ) + ( 1 2 X a + Y a )( ∂C ∂ ∆ ∂ 2 C ∂π 2 + ∂C ∂π ∂ 2 C ∂π∂ ∆ ) , (D.5e)</formula>
<formula><location><page_129><loc_21><loc_60><loc_90><loc_64></location>∂C ∂ψ ,ab ∂ b ( ∂ 2 C ∂π 2 ) -∂C ∂π ∂ b ( ∂ 2 C ∂ψ ,ab ∂π ) = ∂C ∂ ∆ ∂ a ( ∂ 2 C ∂π 2 ) + ∂C ∂π ( Y a -∂ a ) ∂ 2 C ∂π∂ ∆ , (D.5f)</formula>
<formula><location><page_129><loc_22><loc_56><loc_90><loc_59></location>∂ ( βD a ) ∂π = ∂β ∂π ( -2 ∂ b p ab T -2 3 ∂ a p +2 W a -Z a -1 3 pX a ) + ∂ a ψ ( β + π ∂β ∂π ) . (D.5g)</formula>
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</document> | [{"title": "King's College London", "content": "in the Theoretical Particle Physics and Cosmology Department of Physics 1st April 2019", "pages": [1]}, {"title": "Declaration of Authorship", "content": "Signed: Date: 1st April 2019 ' I really don't know what I'm doing... I don't. It's terrible...' Leonardo DiCaprio", "pages": [2, 3]}, {"title": "School of Natural and Mathematical Sciences", "content": "Department of Physics Doctor of Philosophy", "pages": [4]}, {"title": "Deformed general relativity", "content": "by Rhiannon Cuttell In this thesis, I investigate how to construct a self-consistent model of deformed general relativity using canonical methods and metric variables. The specific deformation of general covariance is predicted by some studies into loop quantum cosmology. I firstly find the minimally-deformed model for a scalar-tensor theory, thereby establishing a classical reference point, and investigate the cosmological effects of a non-minimal coupled scalar field. By treating the deformation perturbatively, I derive the deformed gravitational action which includes the nearest order of curvature corrections. Then working more generally, I derive the deformed scalar-tensor constraint to all orders and I find that the momenta and spatial derivatives from gravity and matter must combine in a very specific form. It suggests that the deformation should be equally affected by matter field derivatives as it is by gravitational curvature. Finally, I derive the deformed gravitational action to all orders, and find how intrinsic and extrinsic curvatures differently affect the deformation. The deformation seems to be required to satisfy a non-linear equation usually found in fluid mechanics.", "pages": [4]}, {"title": "Acknowledgements", "content": "Thanks to my supervisor, Mairi, who patiently facilitated and enabled this, and helped guide me away from dead ends. Thanks to Martin, who helped me find and correct a serious error in my methodology before it was too late. Thanks to those in the physics department who have helped so much with navigating through difficult situations, especially Jean, Julia and Rowena. Thanks to Marc, Brinda, Agnes, Ruth and Gwyn, the medical and mental health professionals that helped me keep afloat. Thanks to my parents Jeff and Liz, who gave enough encouragement for me to be doing this and on whom I have depended too much. Thanks to Joanna, for being the perfect big sister I don't deserve, and thanks to Jim for being there for her in turn. Thanks to Naomi, for always being there with love, support and a goofy joke. I couldn't imagine managing to finish this without you. ( \u03d5 \u03c9 \u03d5 ) \u223c Thanks to Xin Xin, Jack, Pebbles, Muffin and Tuxedo Kamen for just being you.", "pages": [5]}, {"title": "List of Symbols", "content": "For Naomi", "pages": [10]}, {"title": "Introduction", "content": "In this thesis I investigate deformed general relativity, which is a semi-classical model attempting to capture the leading effects of a correction to general relativity predicted in some studies of loop quantum gravity. It uses the methods of canonical gravity but with space-time covariance deformed by a phase-space function. By assuming a general deformation, I find the general models which are consistent with it, demonstrating multiple routes which can be taken to find them. Before going into more depth on this, I must first discuss the motivations for this investigation.", "pages": [11]}, {"title": "1.1 The need for a theory of quantum gravity", "content": "It is known that matter fields are quantised due to the remarkable agreement of experimental results with quantum field theory [1-3]. There have been some attempts to allow for classical gravity to couple to quantum fields at a fundamental level [4, 5], and some interesting phenomena have been discovered from considering effective models of quantum fields on a curved space-time [6-8]. However, it is generally expected that gravity must be quantised too [9,10]. The gravitational field, like all other fields, therefore must be quantized, or else the logical structure of quantum field theory must be profoundly altered, or both. [11, B. DeWitt] Besides gravity being known to couple to quantum fields, there are known limitations to the current common understanding. General relativity predicts its own demise due to singularities arising in the equations describing black holes and the very early universe [12]. They are known to exist due to robust experimental observations supporting the existence of black holes [13] and supporting an early universe which closely matches what is predicted of a hot big bang [14]. These phenomena exist at the intersection of general relativity and quantum mechanics since they involve both massive systems and small scales. It seems they cannot be fully understood without a framework which consistently bridges the gap. As a precedent for the singularity problem, classical mechanics could not sufficiently account for experimental results showing that atoms contained small, massive nuclei orbited by electrons (the Nagaoka-Rutherford model). This is due to accelerating point charges (electric field singularities) being known to emit radiation as per the Landau formula, and therefore an electron orbit should radiatively decay, causing atoms to be unstable. However, the development of quantum mechanics resolved this by introducing discrete and stationary orbitals in the Bohr model. The hope is that quantising gravity will similarly cure it of some of its pathologies. One might not want to jettison all that is good about general relativity in pursuit of a quantised theory. The key underlying idea, equivalence of all frames, is considered a philosophically and aesthetically satisfying aspect. Conversely, the requirement in the orthodox interpretation of quantum mechanics for an external observer is considered troubling, hence why Einstein spent much of the latter part of his career challenging it [15]. One crucial sticking point in reconciling general relativity and quantum mechanics is the problem of time [16,17]. In quantum mechanics time is a fixed external parameter, in general relativity it is internal to the system and is not uniquely defined. These are seemingly incommensurable differences, and to bridge the gap requires significant compromise. The solution in canonical gravity for reconciling the two is to split space-time at the formal level, but include symmetry requirements so that the full general covariance is kept implicitly [10, 18, 19]. One is left with a description of a spatial slice evolving through time rather than one of a static and eternal bulk. These methods are often required for numerically simulating general relativity due to the necessity of specifying a time coordinate when setting up an evolution simulation. This introduces on each spatial manifold a conserved quantity or 'constraint' given by \u03c6 I \u2192 0 for each dimension of time and space, analogous to a generalisation of the conservation of energy and momentum. These constraints form an algebra which contains important information about the geometric nature of space-time, and is of the form { \u03c6 I , \u03c6 J } = f K IJ \u03c6 K [10, 20]. This is a Lie algebroid which describes the relationships between the constraints and generates transformations between different choices of coordinates [21,22]. The important { C, C } part of this algebra ensures that the spatial manifold evolving through time is equivalent to a stack of spatial manifolds embedded in a geometric spacetime manifold. In this more general case of gravitation in interaction with other fields, [the equation 1 ] not only guarantees the embeddability of the 3-geometries in a spacetime but also ensures that these additional fields evolve consistently within this space-time. [23, C. Teitelboim] This part of the algebra is what I am going to consider to be deformed, but where does this hypothesis come from?", "pages": [11, 12, 13]}, {"title": "1.2 Loop quantum gravity", "content": "Though there are several candidates for a theory of quantum gravity, I am going to only consider loop quantum gravity [24,25]. There are other somewhat related theories which also deal directly with quantising gravity, such as: causal dynamical triangulations [26]; causal set theory [27]; group field theory [28]; and asymptotically safe gravity [29]. The main alternative candidate is string theory and its variants, which prioritises bringing gravity into the established framework for quantum particles in order to create a unified theory [30,31]. Loop quantum gravity focuses on maintaining some key concepts from general relativity such as background independence and local dynamics throughout the process of combining gravity and quantum mechanics. It describes space-time as not being a continuous manifold, but instead being a network of nodes connected by ordered links with quantum numbers for geometrical quantities such as volume. Such a network is not merely embedded in space but is space itself . As such, due to the quantisation of geometry, one cannot shrink the length of a link between nodes to being infinitesimal as in the classical case. If general relativity is truly the classical limit of loop quantum gravity, then there should be a semi-classical limit where the dynamics are well approximated by general relativity with minor quantum corrections. These should become larger at small scales and in regions of high curvature. A closely related theory is loop quantum cosmology, which uses concepts and techniques from loop quantum gravity and applies them directly at the cosmological level by using midi-superspace models [32, 33]. That is, by quantising a universe which already has certain symmetries assumed such as isotropy to simplify the process. There has been some progress towards proving that loop quantum gravity can be symmetry-reduced to loop quantum cosmology, but as yet this has not been shown definitively [34,35]. For models of loop quantum cosmology to be self-consistent and anomaly-free while including some of the interesting effects from the discrete geometry, it seems that the algebra of constraints must be deformed. Specifically, some of the structure functions become more dependent on the phase space variables through a deformation function f K IJ ( q ) \u2192 \u03b2 ( q, p ) f K IJ ( q ) [36-42]. Deforming rather than breaking the algebra in principle maintains general covariance but the transformations between different choices of coordinates become highly non-linear [43]. It becomes less clear to what extent one can still interpret space-time geometrically, at least in terms of classical notions of geometry. However, there is ambiguity in the correct choice of variables used for loop quantum gravity. The results cited in the previous paragraph are for real variables for which there has been significant difficulty including matter and local degrees of freedom [44]. The main alternative, self-dual variables, have had some positive results for including those degrees of freedom without deforming the constraint algebra [45], but might not have the desired quality of resolving curvature singularities [46]. Interesting predictions coming from loop quantum gravity include: a bouncing universe [47]; black hole singularity resolution and transition to white holes [48]; and signature change of the effective metric [41]. Some of these predictions are closely associated with a deformation of classical symmetries in regions of high energy density.", "pages": [13, 14, 15]}, {"title": "1.3 Why study deformed general relativity?", "content": "Deformed general relativity builds directly from the idea that the constraint algebra is deformed [49]. It is constructed by taking the deformed constraint algebra, and finding a corresponding model which includes local degrees of freedom a priori . This can be done because, if one starts from an algebra and makes some reasonable assumptions, one can deduce the general form of all the constraints [21,50]. This should provide a more intuitive understanding of how the deformation affects dynamics and may provide a guide for how to include the problematic degrees of freedom when working with real variables in loop quantum gravity. The constraint algebra is important because, as said previously, it closely relates to the structure of space-time [23]. Quantum geometry will behave differently to classical geometry, and deformed general relativity attempts to capture some of the effects in a semiclassical model which is more amenable to phenomenological investigations. Phenomenological models which are comparable to deformed general relativity, such as deformed special relativity [51] and rainbow gravity [52], struggle to go beyond describing individual particles coupled to an energy-dependent metric. They can suffer from a breakdown of causality [53], or find it difficult to describe multi-particle states [54]. Deformed general relativity does not suffer from these problems by construction.", "pages": [15]}, {"title": "1.4 Overview of this thesis", "content": "The main focus of this thesis is to investigate how to construct a self-consistent model of deformed general relativity using canonical methods and metric variables. I review important concepts and methodology in chapter 2. In chapter 3, I find the minimallydeformed model for a scalar-tensor theory, establishing a classical reference point. Then in chapter 4, I derive the deformed gravitational action which includes the lowest non-trivial order of perturbative curvature corrections coming from the deformation. In chapter 5, I derive the deformed scalar-tensor constraint to all orders and I find that the momenta and space derivatives must combine in a specific form. Finally, in chapter 6, I find the deformed gravitational action to all orders, and find how intrinsic and extrinsic curvatures differently affect the deformation. I identify some of the cosmological consequences for the significant results of each chapter. There are several research questions which I attempt to answer in this thesis. How are the form of the deformation function and the form of the model related? In particular, what is the deformed scalar-tensor Hamiltonian and what is the deformed gravitational Lagrangian, using either perturbative or non-perturbative methods? How do they relate to the classical limit and to each other? How can matter fields be incorporated in deformed models? How does the deformation function depend on curvature, and is it different for intrinsic and extrinsic curvatures? The research chapters 3 and 4 are adapted from the previously published papers [55] and [56], respectively. The other research chapters, 5 and 6, were recently submitted for publication [57,58]", "pages": [16]}, {"title": "1.5 Wider impact", "content": "This study is directly motivated by the prediction of a deformed constraint algebra appearing in loop quantum cosmology [36-42]. As such it should provide insight into the lingering questions of how matter and local degrees of freedom need to be incorporated into the motivating theory in the presence of a deformation, and how spatial and time derivatives are differently affected. There are also potentially wider implications for this study. For example, it has been shown that taking the deformed constraint algebra to the flat-space limit gives a deformed version of the Poincar'e algebra, which leads to a modified dispersion relation [46,59]. This might indicate something such as a variable speed of light or an observer-independent energy scale. In this respect it is similar to the phenomenological models of deformed special relativity [51] and rainbow gravity [52]. The deformation might indicate a non-commutative character to geometry [60,61] although apparently not a multifractional one [62]. It might represent a variable dimensionality of space-time and a running of the spectral dimension [63]. The deformation function may change sign, as suggested in the motivating studies [41]. This makes the hyperbolic equations become elliptical and implies a phase transition from classical Lorentzian spacetime to an effectively Euclidean quantum regime [22,64]. It therefore may be a potential mechanism for the Hartle-Hawking no-boundary proposal [65].", "pages": [16, 17]}, {"title": "Methodology", "content": "In this thesis I am primarily building on preceding work done by others [21, 49, 50] and elaborating on previously published material [55,56].", "pages": [18]}, {"title": "2.1 Space-time decomposition", "content": "Quantum mechanics naturally works in the canonical or Hamiltonian framework. The canonical framework takes variables defined at a certain time and evolves them through time. That evolution defines a canonical momentum for each variable. To make general relativity more amenable to quantum mechanics, one must likewise make a distinction between the time dimension and the spatial dimensions. So I foliate the bulk space-time manifold M into a stack of labelled spatial hypersurfaces, \u03a3 t . I assume it is globally hyperbolic, so topologically M = \u03a3 \u00d7 R [10, 18, 19]. A future-pointing vector normal to the spatial hypersurface \u03a3 t is defined such that g ab n a n b = -1 . The spatial slices \u03a3 t are themselves Riemannian manifolds with an induced metric q ab = g ab + n a n b , such that q ab n b = 0 . The spatial metric has an inverse defined as q ab = g ab + n a n b , so that q b a := q ac q bc = \u03b4 b a + n a n b acts as a spatial 1 projection tensor. If the spatial foliation, and therefore the spatial coordinates, are arbitrary, the timeevolution vector field t a cannot be uniquely determined by the time function t . One can project it into its normal and spatial components, defining the lapse function N = -n a t a , and the spatial shift vector N a = q a b t b . Therefore, t a = Nn a + N a . Since the coordinates are arbitrary, it is convenient to take the normal to the spatial surface as the time-like direction for defining velocities rather than using the time-vector itself. Therefore, where \u02d9 X := L t X , and the extrinsic curvature of the spatial slice is related to this by K ab = 1 2 v ab .", "pages": [18, 19]}, {"title": "2.2 Canonical formalism", "content": "I take a general first-order action for a model with dynamical fields \u03c8 I , and non-dynamical fields \u03bb I , where \u2202 a \u03c8 I := \u2202\u03c8 I \u2202x a =: \u03c8 I,a . Varying the action with respect to each field, fixing the variation at the boundaries, and imposing the principle of least action, gives the Euler-Lagrange equations of motion, The approximation symbol is used to indicate something that is true in the dynamical regime, or 'on-shell', rather than something that is true kinematically, or 'off-shell'. The non-dynamical fields \u03bb I can be seen to produce constraints on the system given by (2.4b), they are also known as Lagrange multipliers. Making a space-time decomposition as in section 2.1, one can define the canonical momenta of each field, Since L does not depend on \u02d9 \u03bb I , one can see that \u03c0 I \u03bb \u2248 0 are primary constraints on the system. If the matrix \u2202 2 L \u2202 \u02d9 \u03c8 I \u2202 \u02d9 \u03c8 J is non-degenerate, then the above equation can be inverted to find \u02d9 \u03c8 I = \u02d9 \u03c8 I ( \u03c8 J , \u03c0 J \u03c8 , \u03bb J ) , and so one can replace the time derivatives in the action. Making a Legendre transform to find the Hamiltonian associated to this action, where \u00b5 \u03bb I is a coefficient which acts like a Lagrange multiplier. The Poisson bracket of a quantity with the Hamiltonian equals the time derivative of that quantity on-shell, and if F \u2248 0 should be true at all times, then \u02d9 F \u2248 0 must also be true [20]. Therefore, evaluating { \u03c0 I \u03bb , H } either gives back a function of the primary constraints \u03c0 J \u03bb , produces a secondary constraint \u03c6 I ( \u03c8 J , \u03c0 J \u03c8 , \u03bb J ) \u2248 0 , or gives a specific form for the coefficients of the constraints \u00b5 I . The equations (2.4b) appear here as secondary constraints. I repeat the process with { \u03c6 I , H } until I have found all the constraints on the system, at which point there is no need to differentiate between primary and secondary constraints, and I have found the generalised Hamiltonian, The set of constraints has a Poisson bracket structure glyph[negationslash] and if \u03b1 IJ = 0 then some of \u03c6 I are what are called 'second-class' constraints, in which case some of the coefficients \u00b5 I are uniquely determined. If \u03b1 IJ = 0 then all of \u03c6 I are 'firstclass', in which case the constraints not only restrict the values of the dynamical fields, but also generate gauge transformations [10,20]. This is because, in general the evolution (2.7) will depend on \u00b5 I . For an undetermined \u00b5 I to influence the mathematics but not the physical observables, a change of its value must correspond to a gauge transformation generated by the relevant first-class constraint. For classical general relativity, the action does not depend on \u02d9 N or \u02d9 N a (up to boundary terms) and is only linearly dependent on N and N a . 2 As such, there are primary constraints given by \u03c0 N and \u03c0 N a , which generate secondary constraints known as the Hamiltonian constraint and diffeomorphism constraint respectively, which are all first-class constraints. This means that N and N a are gauge functions which do not affect the observables, and therefore the spatial slicing does not affect the dynamics. The theory is background independent and the constraints generate gauge transformations 3 , The Hamiltonian can be rewritten as a sum of the constraints up to a boundary term, Considering the Poisson bracket structure of these constraints, given by (2.9) with \u03c6 I \u2208 { C, D a } , one finds that they form a Lie algebroid 4 [22], where ( N 1 , N a 1 ) and ( N 2 , N a 2 ) each represent the lapse and shift of two different hypersurface transformations. As interpreted in ref. [23], (2.13a) shows that D a is the generator of spatial morphisms, (2.13b) shows that C is a scalar density of weight one (as defined in appendix B) and (2.13c) specifies the form of C such that it ensures the embeddability of the spatial slices in space-time geometry.", "pages": [19, 20, 21, 22]}, {"title": "2.3 Choice of variables", "content": "Classical canonical general relativity can be formulated equivalently using different variables. There is geometrodynamics, which uses the spatial metric and its canonical momentum ( q ab , p cd ) , the latter of which is directly related to extrinsic curvature, where q := det q ab and \u03c9 is the gravitational coupling. An alternative is connection dynamics, which uses the Ashtekar-Barbero connection and densitised triads ( A I a , E b J ) , where capital letters signify internal indices rather than coordinate indices [66, 67]. This can be related to geometrodynamics by using the equations [10], where \u03b3 BI is the Barbero-Immirzi parameter and glyph[epsilon1] IJK is the covariant Levi-Civita tensor. The exact value of \u03b3 BI should not affect the dynamics [68]. The other alternative I mention here is loop dynamics, which uses holonomies of the connection and gravitational flux ( h glyph[lscript] [ A ] , F I glyph[lscript] [ E ]) . Classically, h glyph[lscript] [ A ] is given by the path-ordered exponential of the connection integrated along a curve glyph[lscript] and F I glyph[lscript] [ E ] is the flux of the densitised triad through a surface that the curve glyph[lscript] intersects. If glyph[lscript] is taken to be infinitesimal, one can easily relate loop dynamics and connection dynamics because then h glyph[lscript] = 1 + A ( \u02d9 glyph[lscript] ) + O ( | glyph[lscript] | 2 ) [25, p. 21]. When each set of variables is quantised, they are no longer equivalent, for example the value of \u03b3 BI does now affect the dynamics [46,69]. For complex \u03b3 BI , care has to be taken to make sure the classical limit is real general relativity, rather than complex general relativity. Significantly, quantising loop variables (loop quantum gravity) discretises geometry, and so glyph[lscript] cannot be taken to be infinitesimal [25, p. 105]. In this work, I choose to use metric variables to build a semi-classical model of gravity. This is because the comparison to other modified gravity models should be clearer, and there is no ambiguity arising from \u03b3 BI .", "pages": [22, 23]}, {"title": "2.4 Higher order models of gravity", "content": "In four dimensions, the Einstein-Hilbert action for general relativity is given by where \u03c9 = 1 / 8 \u03c0 \u00b7 G is the gravitational coupling and g := det g ab . The integrand is the four dimensional Ricci curvature scalar which is contracted from the Riemann curvature tensor (4) R := (4) R a bac g bc . For any Riemannian manifold, this is defined using the commutator of two covariant derivatives of an arbitrary vector, There are many reasons why theoretical physicists seek to find models of gravity which go beyond the Einstein-Hilbert action. For instance, mysteries known as dark matter [70] and dark energy [71] may originate with gravity behaving differently than expected rather than being due to unknown dark substances [72]. The indication that there was a period of inflationary expansion in the early universe has also caused a search for relevant models [73, 74]. Moreover, the classical equations of gravity predict their own demise in extraordinary circumstances such as in a black hole or at a hot big bang. A theory of gravity that solves these problems to which classical general relativity is the low-curvature, largescale limit may have a semi-classical regime where corrections appear, at leading orders, similar to these theories of modified gravity [73,75,76]. One way of attempting to find alternative models of gravity is by constructing actions from higher order combinations of the Riemann tensor, so you instead have the general action To bring this in line with the space-time split, I replace the determinant, g = -N 2 q . The Riemann tensor must be decomposed by projecting it along its normal and tangential components relative to the spatial slice, These identities are respectively known as the Gauss equation, the Codazzi equation, and the Ricci equation [10, 77]. All other projections vanish due to the tensor's antisymmetry. As can be seen from (2.19c), there are second order time derivatives included in the Riemann tensor. Including second order time derivatives in an action is problematic because it may introduce the Ostrogradsky instability [78]. To demonstrate what this means, I take a one dimensional model action, I cannot find the associated Hamiltonian when there are time derivatives higher than second order, and the Euler-Lagrange equations may involve fourth order time derivatives, glyph[negationslash] if \u2202 2 L \u2202 q 2 = 0 . So I must introduce an additional variable to absorb the higher order terms. The Ostrogradsky method [79] is to replace \u02d9 q with an independent variable v . however, I instead do this slightly differently for reasons which will be apparent later. Following the method used in ref. [77, 80] and using variables like in ref. [81], I instead replace q with an auxiliary variable a , and integrate by parts to move the second order time derivative to the Lagrange multiplier \u03c8 , promoting it to a dynamical variable, which gives the canonical momenta, So I can invert these definitions to find the velocities in terms of the momenta. Then make a Legendre transform to find the Hamiltonian, glyph[negationslash] where \u00b5 a is a Lagrange multiplier. The equation of motion for a produces the secondary constraint \u03c6 = \u2202L \u2202a -\u03c8 \u2248 0 . Finding { \u03c6, H } \u2248 0 produces an equation for \u00b5 a and therefore \u03c6 is a second-class constraint and a is uniquely determined. The constraint can be solved for a ( q, \u03c8, \u03c0 ) as long as \u2202 2 L \u2202a 2 = 0 and this can be substituted into the Hamiltonian without incident, in which case I find, which is only linear in p . This means that the energy is unbounded from below and above, and so the model may be unstable [79]. For specific models of this kind rather than this simple example, I can find a well behaved Hamiltonian if there are sufficient restrictions on the values that \u03c8 can take [81]. If I do have a well behaved Hamiltonian, it is clear that the higher order derivative action L ( q, \u02d9 q, q ) contains an additional degree of freedom, which has been absorbed by \u03c8 .", "pages": [23, 24, 25, 26]}, {"title": "2.4.1 Non-minimally coupled scalar from F ( (4) R ) gravity", "content": "In ref. [77, 80], it was shown how to find the Hamiltonian form of any F ( (4) R a bcd ) action. The Riemann tensor is split into its normal and tangential components (2.19), and auxiliary tensors are introduced as in (2.23). The tensor which is the Lagrange multiplier of (2.19c) becomes dynamical by integrating by parts. This turns the action into being first order in time derivatives, and therefore one can find the associated Hamiltonian. This field contains the additional degrees of freedom allowed by the higher order derivatives. To include tensor contractions such as (4) R ab (4) R ab and (4) R abcd (4) R abcd produces several additional degrees of freedom, and requires considering spatial derivatives of velocity or momenta because of (2.19b). For the sake of simplicity, in this chapter and throughout the thesis, I will only consider models which are comparable with F ( (4) R ) . So the action is given by, I decompose the Ricci scalar using (2.19), where \u2206 := q ab \u2207 a \u2207 b . Then integrate the action (2.28) by parts to move the second order time derivative to \u03c8 , where q := det q ab , \u03bd := L n \u03c8 , and K := ( v 2 -v ab v ab ) / 4 is the standard extrinsic curvature contraction. The conjugate momenta are, where Q abcd := q a ( c q d ) b for convenience. I can invert these to find, where I have separated the trace and the traceless parts of the momentum, I Legendre transform the action to find the associated Hamiltonian, with the corresponding Hamiltonian constraint, where P := p T ab p ab T . Finding { \u03c0 \u03c1 , H } gives a secondary constraint, glyph[negationslash] which is second-class. It can be solved to find \u03c1 ( \u03c8 ) as long as F '' = 0 , in which case we can find the Hamiltonian constraint in terms of only the metric and the scalar field \u03c8 . This leaves me with a term depending on \u03c8 which acts like a scalar field potential, which I call the geometric scalar potential. As I will further elaborate in section 3, this scalar-tensor model I have derived from an F ( (4) R ) model of gravitation is equivalent to letting the gravitational coupling in the Einstein-Hilbert action become dynamical, \u03c9 \u2192 \u03c9\u03c8 . So models of gravity that have an action which is an arbitrary function of the spacetime curvature scalar (4) R can be converted into a scalar-tensor theory in the Hamiltonian formalism. The structure of general covariance underlying general relativity should be preserved in these models, though they do contain an additional degree of freedom.", "pages": [26, 27, 28]}, {"title": "2.5 Deformed constraint algebra", "content": "As previously mentioned in section 1.2, loop quantum cosmology predicts that the symmetries of general relativity should be deformed in a specific way in the semi-classical limit [36-42]. This appears from incorporating loop variables in a mini-superspace model, but specifying that all anomalies \u03b1 IJ in (2.9) vanish while allowing counter-terms to deform the classical form of the algebra. This ensures that the constraints are first-class, retaining the gauge invariance of the theory and of the arbitrariness of the lapse and shift. If anomalous terms were to appear in the constraint algebra, then the gauge invariance would be broken and the constraints could only be solved at all times for specific N or N a . This means that there would a privileged frame of reference, and therefore no general covariance. In the referenced studies, it is strongly indicated that the bracket of two Hamiltonian constraints (2.13c) is deformed by a phase space function \u03b2 , This has not been shown generally, but has been shown for several models independently. There are no anomalies in the constraint algebra, so a form of general covariance is preserved. However, it may be that the interpretation of a spatial manifold evolving with time being equivalent to a foliation of space-time (also known as 'embeddability') is no longer valid. glyph[negationslash] These deformations only appear to be necessary for models when the Barbero-Immirzi parameter \u03b3 BI is real. For self-dual models, when \u03b3 BI = \u00b1 i , this deformation does not appear necessary [45]. However, self-dual variables are not desirable in other ways. They do not seem to resolve curvature singularities as hoped, and obtaining the correct classical limit is non-trivial [46]. Because of this, even though I use metric variables in this work, considering \u03b2 = 1 and ensuring the correct classical limit means there should be relevance to the models of loop quantum cosmology with real \u03b3 BI .", "pages": [29]}, {"title": "2.6 Derivation of the distribution equation", "content": "From the constraint algebra, I am able to find the specific form of the Hamiltonian constraint C for a given deformation \u03b2 . The diffeomorphism constraint D a is not affected when the deformation is a weightless scalar 5 and so is completely determined as shown in appendix B. With D a and \u03b2 as inputs, I can find C by manipulating (2.38). Firstly, I must find the unsmeared form of the deformed algebra. At this point I do not need to specify my canonical variables, and leave them merely as ( q I , p I ) , Take the functional derivatives with respect to N 1 ( x ) and N 2 ( y ) , where \u03b4 ( x, y ) is the three dimensional Dirac delta distribution 6 . If I note that I will only consider constraints without spatial derivatives on momenta, this simplifies, For when I wish to derive the action instead of the constraint, I can transform the equation by noting that, where v I := L n q I and the Lagrangian is here defined such that S = \u222b d t d 3 xNL . I substitute these into (2.39b), then take the functional derivatives to remove N 1 and N 2 , To find a useful form for this, I need to use a specific form for the diffeomorphism constraint. Because it depends on momenta, I must replace them using, and, as before, if I note that I will only consider actions without spatial derivatives of momenta this simplifies to Therefore, substituting the diffeomorphism constraint found in appendix B and momenta (2.45) into (2.43), I find the distribution equation which can be used for restricting the form of the deformed action. So, the key equations I use as a basis for finding the action or constraint for deformed general relativity are (2.41) and (2.43).", "pages": [30, 31]}, {"title": "2.7 Order of the deformed action and constraint", "content": "I can determine the relationship between the order of the deformation function and the order of the associated constraint (or action) by comparing orders of momenta (or velocity).", "pages": [31]}, {"title": "2.7.1 Hamiltonian route", "content": "As an example, take the distribution equation (2.41) with only a scalar field, where I have used the diffeomorphism constraint (B.6). I take a simplified model with two spatial derivatives represented by \u2206 , only taking even orders of derivatives because of assuming spatial parity. I take the distribution equation (2.46) and put it into schematic form, so that I can consider orders of \u03c0 in a way analogous to dimensional analysis. This equation must be satisfied independently at each order of momenta, so I isolate the coefficient of \u03c0 n , where I have expanded the constraint and deformation, The highest order contribution to (2.48) comes when m = n C and n -m +1 = n C , in which case n = 2 n C -1 . This is the highest order at which \u03b2 won't automatically be constrained to vanish, so I find its highest order of momenta to be n \u03b2 = 2 n C -2 . However, this result does not take into account the fact that the combined order of momenta and spatial derivatives may be restricted. If this is the case (as is found in chapter 5), then the highest order contribution to the (2.48) will be when n -m +1 = n C -2 , in which case I find the relation I see that a deformed second order constraint only requires considering a zeroth order deformation as I do in chapter 3, but a fourth order constraint requires considering a fourth order deformation. I consider the constraint to general order in chapter 5. Note that this relation suggests there are higher order deformations which allow for constraints given by finite order polynomials.", "pages": [31, 32]}, {"title": "2.7.2 Lagrangian route", "content": "Consider the distribution equation (2.43) with only a scalar field, where I have used the diffeomorphism constraint (B.6) and the momentum definition (2.45). Let me consider a simplified model to match the derivative orders for the deformation and the derivative orders for the Lagrangian in a way analogous to dimensional analysis. First order time derivatives are given by \u03bd and two orders of spatial derivatives are given by \u2206 . I can collect terms in the distribution equation of the same order of time derivatives as they are linearly independent. Schematically, the distribution equation is given by, and expanding the Lagrangian and deformation in powers of \u03bd , the coefficient of \u03bd n is then given by, I can relabel and rearrange to find a schematic solution for the highest order of L appearing here, I can see that if n \u03b2 > 0 , then this equation is recursive and n L \u2192\u221e because there is no natural cut-off, suggesting that L is required to be non-polynomial. If I wish to truncate the action at some order, then it must be treated as an perturbative approximation. I consider a perturbative fourth order action in chapter 4, and the completely general action in chapter 6.", "pages": [32, 33]}, {"title": "2.8 Cosmology", "content": "Since the main motivations for this study centre around cosmological implications of the deformed constraint algebra, I need to lay out how I find the cosmological dynamics of a model. I restrict to an isotropic and homogeneous space, using the Friedmann-Lema \u02c6 itreRobertson-Walker metric (FLRW), where \u03a3 ab is time-independent and describes a three dimensional spatial slice with constant curvature k . When space is flat, k = 0 , this is given by \u03a3 ab = \u03b4 ab . The normal derivative of the spatial metric is given by, where H is the Hubble expansion rate, and the Ricci curvature scalar is given by, When using canonical coordinates, the metric momentum is given by which changes the metric's commutation relation, where \u03b4 cd ab := \u03b4 c ( a \u03b4 d b ) . The spatial derivatives of matter fields vanish, \u2202 a \u03c8 I = 0 . One may couple a perfect fluid to the metric by including the energy density \u03c1 in the constraint or the action [82], which must satisfy the continuity equation, where w \u03c1 is the perfect fluid's cosmological equation of state, the ratio of the pressure density to the energy density. For investigations into whether there are implications for the hypothesised inflationary period in the very early universe, I must define what is considered to be a period of inflation. The simple definition is when the finite scale factor is both expanding and accelerating, \u02d9 a > 0 and a > 0 . As said above, loop quantum cosmology with real variables seems to predict a big bounce instead of a big bang or crunch. In this thesis, I take the very literal interpretation of this (as found in ref. [83]) and define a bounce as a turning point for a finite scale factor, a > 0 , \u02d9 a = 0 and a > 0 . This definition may be usable, but it is not ideal. If a bounce does indeed happen when \u03b2 < 0 , as predicted in the literature, then this is when the effective metric signature is Euclidean, when \u02d9 a may be a complex number. Ideally, I would like to extract cosmological observables such as the primordial scalar index to find phenemenological constraints [84]. However, to calculate the power spectra of primordial fluctuations would require adapting the cosmological perturbation theory formalism to ensure it is valid for deformed covariance, something which would probably be highly non-trivial. Unfortunately, there was not enough time to investigate this.", "pages": [33, 34, 35]}, {"title": "Second order scalar-tensor model and the classical limit", "content": "In this chapter, I derive the general form of a minimally-deformed, non-minimally-coupled scalar-tensor model which includes up to two orders in momenta or time derivatives. This allows me to demonstrate that the higher order gravity model derived in section 2.4.1 does not deform the constraint algebra or general covariance, and therefore show how the deformed models derived in subsequent chapters are distinct. For those later chapters, this minimally-deformed model provides a useful reference point. This chapter is adapted from work I previously published in ref. [55]. I find the form of the model by deriving restrictions on the constraint using (2.41) and then transform to find the action. It would be completely equivalent to derive the action first, because the minimally deformed case maintains a linear relationship between velocities and momenta, meaning that the transformation between the action and constraint is trivial. After finding the constraint and action, I look at some of the cosmological implications in section 3.3, especially the interesting influence of the non-minimal coupling of the scalar field. I use the structure of the scalar-tensor constraint which is a parameterisation of F ( (4) R ) , (2.35), to guide the structure of my general ansatz for a spatial metric coupled to several scalar fields. I include spatially covariant terms up to second order in momenta or spatial derivatives, and ignore terms linear in momenta, with summation over I and J implied. I have included C ( \u03c8 ' I \u03c8 ' J ) because it appears in the constraint for minimally coupled scalar fields [10, p. 62]. I aimed to define the most general ansatz for a scalar-tensor constraint containing up to two orders in derivatives which is covariant under general spatial diffeomorphisms, as well as under time reversal, and preserves spatial parity. Each coefficient is potentially a function of q and \u03c8 I , allowing for non-minimal coupling. The spatial indices of C ( p 2 ) abcd only represent different combinations of the metric. The zeroth order term might include terms such as scalar field potentials or perfect fluids, and it behaves as a generalised potential C \u2205 = \u221a q U ( q, \u03c8 I ) .", "pages": [36, 37]}, {"title": "3.1 Solving the distribution equation", "content": "I substitute into the distribution equation (2.41) my ansatz for a second order constraint (3.1), the diffeomorphism constraint from (B.6) and (B.11), and a zeroth order deformation \u03b2 ( q, \u03c8 ) , where C 0 is the part of the constraint without momenta. From here there are two routes to solution, by focusing on either the p ab and \u03c0 components. I must do both to find all consistency conditions on the coefficients of the Hamiltonian constraint.", "pages": [37]}, {"title": "3.1.1 p ab sector", "content": "To proceed to the metric momentum sector, I take (3.2) and find the functional derivative with respect to p ab ( z ) , where I explicitly show the coordinate of the partial derivative as \u2202 a ( y ) := \u2202 \u2202y a because the distinction is important when integrating by parts. I then proceed by moving derivatives away from \u03b4 ( z, y ) terms and discarding total derivatives, which can be rewritten as, Integrating over y , I find that part of the equation can be combined into a tensor dependent only on x , Substituting in the definition of A ab ( x, z ) then relabelling, Multiplying by an arbitrary test tensor \u03b8 ab ( y ) , then integrating by parts over y , I get where I do not need to consider the zeroth derivative terms because they do not produce restrictions on the form of the constraint. Since \u03b8 ab is arbitrary beyond the symmetry of its indices, each unique contraction of it forms a linearly independent equation. To calculate the derivatives of C 0 , I must use the decomposition of the Riemann tensor (A.6) and the second covariant derivative of the metric variation expressed in terms of partial derivatives (A.9). This gives, where \u03a6 abcd = Q abcd -q ab q cd as found in (A.8). Note that Q abcd := q a ( c q d ) b and \u03b4 ab de := \u03b4 ( a d \u03b4 b ) e . I evaluate the coefficient of \u2202 dc \u03b8 ab and find the linearly independent components, where I have decomposed the constraint coefficient C ( p 2 ) abcd = q ab q cd C ( p 2 \u2016 ) + Q abcd C ( p 2 x ) . Then evaluating similarly for \u2202 c \u03b8 ab , where \u2202 \u03c8 := \u2202 \u2202\u03c8 , \u2202 q := \u2202 \u2202 log q and X a := q bc \u2202 a q bc . Note that the equations for \u2202 c q ab \u2202 c \u03b8 ab , \u2202 a q bc \u2202 c \u03b8 ab and q ab \u2202 d q cd \u2202 c \u03b8 ab are not included because they are identical to (3.10). Using (3.10b) to solve for C ( p 2 x ) , then substituting it into (3.11c), I find, which is solved by C ( R ) ( q, \u03c8 ) = f ( \u03c8 ) \u221a q | \u03b2 ( q, \u03c8 ) | , where f ( \u03c8 ) is some unknown function. If I solve (3.10) for C ( p 2 \u2016 ) and C ( p 2 x ) , then substitute them into (3.11e), I find a similar equation to the one above for C ( R ) , and therefore C ( \u03c8 '' ) ( q, \u03c8 ) = f ( \u03c8 '' ) ( \u03c8 ) \u221a q | \u03b2 ( q, \u03c8 ) | . Taking (3.11b) then substituting in for C ( p 2 x ) , C ( R ) and C ( \u03c8 '' ) , I find that f ( \u03c8 '' ) ( \u03c8 ) = -2 \u2202 \u03c8 f ( \u03c8 ) , where \u03c3 \u03b2 := sgn( \u03b2 ) , which is all the conditions which can be obtained from the metric momentum sector of the distribution equation. The remaining conditions must be found in the scalar momentum sector.", "pages": [38, 39, 40]}, {"title": "3.1.2 \u03c0 sector", "content": "Similar to subsection 3.1.1 above, I take the functional derivative of (3.2) with respect to \u03c0 ( z ) , then exchange terms to find the coefficient of \u03b4 ( z, y ) , which can be rewritten as, leading to Multiplying by an arbitrary test function \u03b7 ( y ) , then integrating by parts over y , I get I then substitute in (3.9) to find the linearly independent conditions, Note that there is another condition from \u2202 b q ab \u2202 a \u03b7 , but it is identical to (3.19a). I can solve (3.19a) for C ( p\u03c0 ) = C ( \u03c8 '' ) C ( \u03c0 2 ) /C ( R ) , and then substitute into (3.19b) to find, which I can solve for C ( \u03c0 2 ) , and is the same conclusion I get from (3.11a) (though I did not explicitly write it above because it is simpler to write it here). The condition (3.19c) is solved when I substitute in all my results so far, and if I collect all of the coefficients, I find the Hamiltonian constraint, so the freedom in any (3+1) dimensional scalar-tensor theory with time symmetry and minimally deformed general covariance comes down to the choice of f ( \u03c8 ) , \u03b2 ( q, \u03c8 ) , C ( \u03c8 ' 2 ) ( q, \u03c8 ) and the zeroth order term C \u2205 ( q, \u03c8 ) . It is convenient to make a redefinition, C ( \u03c8 ' 2 ) = g ( q, \u03c8 ) \u221a q | \u03b2 | , where I have made the scalar weight and expected dependence on \u03b2 explicit. It is worth remembering that this is an assumption, and that g could be a function of \u03b2 . It is also convenient to treat the zeroth order term as a general potential, and to extract the scalar density, C \u2205 = \u221a q U ( q, \u03c8 ) . I find the effective Lagrangian associated with this Hamiltonian constraint by performing a Legendre transformation, Integrating by parts at the level of the action does not affect the dynamics because it only eliminates boundary terms. This allows me to find the effective form of the Lagrangian, with a space-time decomposition and without second order time derivatives. I can also do this in the opposite direction to find the covariant form of the above effective Lagrangian, where the deformed four dimensional Ricci scalar and partial derivative are given by, If this is compared to (2.29), I see that the deformation seems to have transformed the effective lapse function N \u2192 \u221a | \u03b2 | N , and transformed the effective normalisation of the normal vector to g \u00b5\u03bd n \u00b5 n \u03bd = -\u03c3 \u03b2 . Here is where I see the effective signature change which comes from the deformation. It is useful to take the Lagrangian in covariant form and use it to redefine the coupling functions so that minimal coupling is when the functions are equal to unity, f = -1 2 \u03c9 R and g = -1 2 \u03c9 \u03c8 + \u03c9 '' R , so the effective forms of the constraint and Lagrangian are given by, which is the main result of this section in its most useful form. Since I have non-minimal coupling, I am working in the Jordan frame. I can get to the Einstein frame by making a specific conformal transformation which absorbs the coupling \u03c9 R by setting q ab = \u03c9 R \u02dc q ab and N = \u03c9 -1 / 2 R \u02dc N , where variables with tildes are Einstein-frame quantities. So the Einstein frame couplings are given by \u02dc \u03c9 R = 1 , \u02dc \u03c9 \u03c8 = ( \u03c9 \u03c8 \u03c9 R +3 \u03c9 ' 2 R / 2 ) /\u03c9 2 R , and the potential by \u02dc U = U/\u03c9 2 R . When the term 'Einstein frame' is used elsewhere in the literature, it often refers to an action which is transformed further so that the effective scalar coupling is also unity. I can make this transformation to a minimally coupled scalar \u03d5 by solving the differential equation, for example, when \u03c9 \u03c8 = 0 , this is solved by \u03d5 ( \u03c8 ) = \u221a 3 2 log \u03c9 R ( \u03c8 ) sgn( \u2202 \u03c8 log \u03c9 R ( \u03c8 )) . For the parameterisation of F ( (4) R ) given in section 2.4.1, \u03c9 R = \u03c9\u03c8 , and the transformation is given by \u03c8 ( \u03d5 ) \u221d e \u03d5 \u221a 2 / 3 as long as \u03c8 > 0 .", "pages": [41, 42, 43, 44]}, {"title": "3.2 Multiple scalar fields", "content": "Consider the case of multiple scalar fields. I start from the distribution equation as before, but label the scalar field variables with an index. Proceeding like in section 3.1.1 by taking functional derivatives with respect to p ab and then integrating by parts with test function \u03b8 ab , I obtain the conditions, glyph[negationslash] I note that there are other independent terms, but they do not produce any extra conditions. Likewise, if I follow the route taken in section 3.1.2, taking the functional derivative with respect to \u03c0 I then integrating by parts with test function \u03b7 I , I find the conditions, glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] and similar to above, there are other independent terms which do no produce any unique conditions. To solve this system of equations I must make assumptions, in particular about the relationship between the scalar fields. One choice might be to assume an O ( N ) symmetry, where the coupling and deformation would only depend on the absolute value of the scalar field multiplet | \u03c8 | = \u221a \u2211 I \u03c8 2 I , and relationships between the C ( \u03c8 ' I \u03c8 ' J ) coefficients could be assumed. However, I instead choose to take one non-minimally coupled field ( \u03c8, \u03c0 \u03c8 ) and one minimally coupled field ( \u03d5, \u03c0 \u03d5 ) with no cross-terms in the spatial derivative sector, C ( \u03d5 ' \u03c8 ' ) = 0 . The minimally coupled field only appears in terms other than the potential U ( q, \u03c8, \u03d5 ) through the deformation function \u03b2 ( q, \u03c8, \u03d5 ) . For example, C ( R ) = C ( R ) ( q, \u03c8, \u03b2 ) . Solving (3.30a) and (3.30c) gives me, as before. Substituting these into (3.30b) and (3.30d) gives me, and the remaining conditions are, I note that the constraint is significantly simpler if I assume C ( \u03d5 ' 2 ) = g \u03d5 ( \u03c8 ) \u221a q | \u03b2 | and C ( \u03c8 ' 2 ) = g \u03c8 ( \u03c8 ) \u221a q | \u03b2 | , where g \u03d5 and g \u03c8 are arbitrary functions. In this case the whole Hamiltonian constraint is and the associated Lagrangian density is If \u03b2 does not depend on \u03d5 , then this can be simplified greatly, in which case the effective and covariant forms of the Lagrangian are given by, where \u03c9 R = -2 f , \u03c9 \u03c8 = 2( g \u03c8 +2 f '' ) , \u03c9 \u03d5 = 2 g \u03d5 . Therefore, when I assume that the minimally coupled scalar field can also be considered to be minimally coupled to the deformation function, I find that the action simplifies to the expected form. It would be interesting to see what effects appear for scalar field multiplets, especially for non-Abelian symmetries, but that is beyond the scope of this study. Instead, I now turn to studying the cosmological dynamics of my results.", "pages": [45, 46, 47]}, {"title": "3.3 Cosmology", "content": "To find the cosmological dynamics, I restrict to a flat, homogeneous, and isotropic metric in proper time ( N = 1 ). I also assume that \u03b2 does not depend on the minimally coupled scalar field \u03d5 for the sake of simplicity. From (3.37), I find the Friedmann equation, which can be written in two equivalent forms, From (3.38b) I see that \u03c9 R \u03c9 \u03c8 +3 \u03c9 ' 2 R / 2 \u2265 0 and \u03c9 R \u03c9 \u03d5 \u2265 0 are necessary when U \u2192 0 to ensure real-valued fields. If I compare this condition to the Einstein frame Lagrangian (3.28), I can see that it is also the condition which follows from insisting that the scalar field \u03c8 is not ghost-like in that frame. Similarly, I see that \u03c3 \u03b2 \u03c9 R > 0 is necessary when \u02d9 \u03c8, \u02d9 \u03d5 \u2192 0 . For the reasonable assumption that the minimally coupled field \u03d5 does not affect the deformation function \u03b2 , the only way that field is modified is through a variable maximum phase speed c 2 \u03d5 = \u03b2 . Due to this minimal modification, it does not produce any of the cosmological phenomena I am interested in (bounce, inflation) through any novel mechanism. Therefore, I will ignore this field for the rest of the chapter. I find the equations of motion by varying the Lagrangian (3.37) with respect to the fields. For the simple undeformed case \u03b2 = 1 the equations are given by, where I can see from the equations of motion that the model breaks down if \u03c9 R \u03c9 \u03c8 +3 \u03c9 ' 2 R / 2 \u2192 0 because it will tend to cause | \u00a8 \u03c8 | \u2192 \u221e and | a | \u2192 \u221e .", "pages": [48, 49]}, {"title": "3.3.1 Bounce", "content": "I will address the question of whether there are conditions under which there can be a big bounce as defined in section 2.8. I find in chapter 4 (and in ref. [56]) that a deformation function which depends on curvature terms can generate a bounce. Elsewhere in the literature on loop quantum cosmology the bounce happens in a regime when \u03b2 < 0 because the terms depending on curvature or energy density overpower the zeroth order terms [40, 41]. However, I am not including derivatives in the deformation here so the effect would have to come from the non-minimal coupling of the scalar field or the zeroth order deformation. I take \u02d9 a = 0 for finite a , include a deformation and I ignore the minimally coupled field for simplicity. From the Friedmann equation (3.38) I find, which implies that \u03c3 \u03b2 \u03c9 \u03c8 < 0 for a bounce because otherwise the equation cannot balance for U > 0 and \u03c8 \u2208 R . Substituting (3.40) into the full equation of motion for the scale factor, and demanding that a > 0 to make it a turning point, I find the following conditions, from which I can determine what the coupling functions, deformation and potential must be for a bounce. For example, if I look at the minimally coupled case, when \u03c9 R = \u03c9 \u03c8 = 1 , and assume that U > 0 , I can see that the conditions are given by, Since I must have \u03b2 \u2192 1 in the classical limit and \u03c3 \u03b2 < 0 at the moment of the bounce, then \u03b2 must change sign at some point. Therefore, a universe which bounces purely due to a zeroth order deformation must have effective signature change. Another example is obtained by assuming scale independence and choosing \u03b2 = 1 and U > 0 . In this case the bounce conditions become, which I can use to find a model which bounces purely due to a scale-independent nonminimally coupled scalar. I present this model in subsection 3.3.5.", "pages": [49, 50]}, {"title": "3.3.2 Inflation", "content": "Now consider the inflationary dynamics. For simplicity I assume that inflation will come from a scenario similar to slow-roll inflation with possible enhancements coming from the non-minimal coupling or the deformation. The conditions for slow-roll inflation are, assuming the couplings, potential and deformation are scale independent and the deformation is positive, I get the following slow roll equations, and define the slow-roll parameters, which, under slow-roll conditions are given by, where a prime indicates a partial derivative with respect to \u03c8 , i.e. \u03b2 ' = \u2202 \u03c8 \u03b2 . The slow-roll regime ends when the absolute value of any of these three parameters approaches unity. Defining N to mean the number of e-folds from the end of inflation, a ( t ) = a end e -N ( t ) , I find that, and using the slow-roll approximation, which can be solved once I specify the form of the couplings, deformation and potential. I cannot find equations for observables such as the spectral index n s because it would require investigating how the cosmological perturbation theory is modified in the presence of non-minimal coupling and deformed general covariance. Beyond this, it is difficult to make general statements about the dynamics unless I restrict to a given model, so I will now consider some models and discuss their specific dynamics.", "pages": [50, 51]}, {"title": "3.3.3 Geometric scalar model", "content": "As demonstrated in the previous chapter, section 2.4.1, the geometric scalar model comes from parameterising F ( (4) R ) gravity so that the additional degree of freedom of the scalar curvature is instead embodied in a non-minimally coupled scalar field \u03c8 [77,80]. Its couplings are given by \u03c9 R = \u03c8 and \u03c9 \u03c8 = 0 . This model is a special case of the Brans-Dicke model, which has \u03c9 \u03c8 = \u03c9 0 /\u03c8 , when the Dicke coupling constant \u03c9 0 vanishes. I can add in a minimally coupled scalar field with \u03c9 \u03d5 = 1 and thereby see the effect of this scalar-tensor gravity on the matter sector. However, I set \u03c9 \u03d5 = 0 because it does not significantly affect my results. The effective action for this model is given by, where F refers to the F ( (4) R ) function which has been parameterised. The equations of motion when \u03b2 \u2192 1 are given by, from which I can see that the scalar field has very different dynamics compared to minimally coupled scalars. This reflects its origin as a geometric degree of freedom rather than a purely matter field. Looking at inflation, the geometric scalar model with a potential corresponding to the Starobinsky model, can indeed cause inflation through a slow-roll of the scalar field down its potential. The non-minimal coupling of the scalar to the metric also causes the scale factor to oscillate unusually, however. It is interesting to compare in Fig. 3.1 the scale factor in the Jordan frame, a , and the conformally transformed scale factor in the Einstein frame, \u02dc a = a \u221a \u03c9 R . Assuming \u03c8 > 1 during inflation, the slow-roll parameters (3.47) are given by, so the slow-roll regime of inflation ends at \u03c8 \u2248 3 when glyph[epsilon1] \u2192 1 . The equation for the number of e-folds of inflation in the slow-roll regime (3.49) is given by N glyph[similarequal] 1 2 ( \u03c8 -\u03c8 end -log \u03c8 \u03c8 end ) .", "pages": [52, 53]}, {"title": "3.3.4 Non-minimally enhanced scalar model", "content": "Unlike the geometric scalar model considered above, the non-minimally enhanced scalar model (NES) from [85], takes a scalar field from the matter sector and introduces a nonminimal coupling rather than extracting a degree of freedom from the gravity sector. The coupling functions are given by \u03c9 R = 1 + \u03be\u03c8 2 , \u03c9 \u03c8 = 1 and \u03c9 \u03d5 = 0 . The strength of the quadratic non-minimal coupling is determined by the constant \u03be . The deformed effective Lagrangian for this model is given by, For some negative values of \u03be , there are values of \u03c8 which are forbidden if I am to keep my variables real, shown in Fig. 3.2. The equations of motion for this model when it is undeformed are given by, and I proceed to use them to consider this model's inflationary dynamics. For a power-law potential U = \u03bb n | \u03c8 | n and \u03be > 0 , the slow-roll parameter which reaches unity first is glyph[epsilon1] at \u03c8 end glyph[similarequal] \u00b1 n \u221a 2 + n (6 -n ) \u03be . The number of e-folds from the end of inflation is given by, and if I specify that n = 4 , I find and the presence of \u03be in the dominant first term shows how the non-minimal coupling enhances the amount of inflation. If I compare this result to numerical solutions in Fig. 3.3, I see this effect. The slow-roll approximation works less well as \u03be increases. I can see this when I look at Fig. 3.3(b) where I compare the slow-roll approximation to when I numerically determine the end of inflation, i.e. when glyph[epsilon1] = -\u02d9 H / H 2 = 1 . I must be wary when dealing with this model, because the coupling can produce an effective potential which is not bounded from below. If I substitute the Friedmann equation (3.55a) into (3.55b) and (3.55c) I can find effective potential terms. These terms are those which do not vanish when all time derivatives are set to zero, and I can infer what bare potential they effectively behave like. If the bare potential is U = \u03bb\u03c8 2 / 2 , then the effective potential term in the scalar equation behaves like which is not bounded from below when \u03be > 0 and \u03bb > 0 and is therefore unstable. More generally, there are local maxima in the effective potential at \u03c8 = \u00b1 \u221a n \u03be (4 -n ) , so for \u03be > 0 the model is stable for bare potentials which are of quartic order or higher.", "pages": [54, 55, 56]}, {"title": "3.3.5 Bouncing scalar model", "content": "As I said in subsection 3.3.1, I have taken the bounce conditions and constructed a model which bounces purely from the non-minimal coupling. This model consists of a nonminimally coupled scalar with periodic symmetry. My couplings are given by \u03c9 R = cos \u03c8 and \u03c9 \u03c8 = 1 + b cos \u03c8 1 + b , where b is some real constant, and for simplicity I ignore deformations and the minimally coupled scalar field. The bouncing scalar model Lagrangian in (c) Scalar coupling (zoomed) covariant and effective forms are given by, As confirmed by numerically evolving the equations of motion, I know from the bouncing conditions (3.41) that this model will bounce when b > 1 because then there is a value of \u03c8 for which \u03c9 \u03c8 < 0 . As I show in Fig. 3.4, the collapsing universe excites the scalar field so much that it 'tunnels' through to another minima of the potential. The bounce happens when the field becomes momentarily ghost-like, when \u03c9 \u03c8 < 0 . I can construct other models which produce a bounce purely through non-minimal coupling by having any U ( \u03c8 ) with multiple minima and couplings of the approximate form \u03c9 \u223c 1 -U . However, to ensure the scalar does not attempt to tunnel through the potential to infinity and thereby not prevent collapse, the coupling functions must become negative only for values of \u03c8 between stable minima. For example, for the Z 2 potential U ( \u03c8 ) = \u03bb ( \u03c8 2 -1 ) 2 , couplings which are guaranteed to produce a bounce are \u03c9 R ( \u03c8 ) = \u03c9 \u03c8 ( \u03c8 ) = 1 -e when \u03bb > 1 .", "pages": [56, 57, 58]}, {"title": "3.4 Summary", "content": "In this chapter I have presented my calculation of the most general action for a second-order non-minimally coupled scalar-tensor model which satisfies a minimally deformed general covariance. I presented a similar calculation which involves multiple scalar fields. I showed how the magnitude of the deformation can be removed by a transformation of the lapse function, but the sign of the deformation and the associated effective signature change cannot be removed. I explored the background dynamics of the action, in particular showing the conditions required for either a big bounce or a period of slow-roll inflation. By specifying the free functions I showed how to regain well-known models from my general action. In particular I discussed the geometric scalar model, which is a parameterisation of F ( (4) R ) gravity and related to the Brans-Dicke model; and I discussed the non-minimally enhanced scalar model of a conventional scalar field with quadratic non-minimal coupling to the curvature. I presented a model which produces a cosmological bounce purely through non-minimal coupling of a periodic scalar field to gravity. I also provided the general method of producing similar models without a periodic symmetry. I did not consider in detail the effect that the deformation has on the cosmological dynamics. However, I did show that a big bounce which is purely due to a zeroth order deformation necessarily involves effective signature change. Perhaps most importantly, I have established the minimally-deformed low-curvature limit that the subsequent chapters refer to.", "pages": [58]}, {"title": "Fourth order perturbative gravitational action", "content": "As I showed in section 2.7.2, the deformed action doesn't seem to naturally have a cut-off for higher powers of derivatives, and it must either be considered completely in general or treated perturbatively as a polynomial expansion. In this chapter I will treat it perturbatively in order to find the lowest order corrections which are non-trivial. This chapter is mostly adapted from a previously published paper [56]. Firstly, I solve the distribution equation for the deformed gravitational action in section 4.1. Then I specify the variables used to construct the action and thereby find the conditions restricting its form in section 4.2. Afterwards, I progressively restrict the action when it is perturbatively expanded to fourth order in derivatives section 4.3. Finally, I investigate the cosmological consequences of the results in section 4.4.", "pages": [59]}, {"title": "4.1 Solving the action's distribution equation", "content": "The general deformed action must satisfy the distribution equation (2.43), I restrict to the case when there is only a metric field, for which the diffeomorphism constraint is given by (B.11), Firstly, I integrate (4.1) by parts to move spatial derivatives from L and onto the delta functions. I discard the surface term and find, from this I take the functional derivative with respect to v ab ( z ) (after relabelling the other indices), I move the derivative from \u03b4 ( x, z ) and exchange some terms using the ( x \u2194 y ) symmetry to find it in the form, where, Integrating over y , I find that part of the equation can be combined into a tensor dependent only on x , Substituting in the definition of A ab ( x, z ) then relabelling, To find this in terms of one independent variable, I multiply by the test tensor \u03b8 ab ( y ) and integrate by parts over y , Then collecting derivatives of \u03b8 ab , where I have discarded the terms containing \u03b8 ab without derivatives, because they do not provide any restrictions on the form of the action. This is simplified by noting that \u2202 c and \u2202 \u2202v ab commute, and that \u2202\u03b2 ,e \u2202v ab,c = \u03b4 c e \u2202\u03b2 \u2202v ab . Therefore, the solution is given by, At this point I need to make some assumptions about the form of the action before I can use this equation to restrict its form.", "pages": [59, 60, 61, 62]}, {"title": "4.2 Finding the conditions on the action", "content": "Firstly, the variables used for the action and deformation must be determined. I am considering only the spatial metric field q ab and its normal derivative v ab , and for simplicity I am only considering tensor contractions which contain up to second order in derivatives, as previously stated in section 2.4.1. The only covariant quantities I can form up to second order in derivatives from the spatial metric are the determinant q = det q ab and the Ricci curvature scalar R . The normal derivative can be split into its trace and traceless components, v ab = v T ab + 1 3 vq ab , so it can form scalars from the trace v and a variety of contractions of the traceless tensor v T ab . However, to second order I only need to consider w := Q abcd v T ab v T cd = v T ab v ab T . Substituting these variables into (4.11), the resulting equation contains a series of unique tensor combinations. The test tensor \u03b8 ab is completely arbitrary so the coefficient of each unique tensor contraction with it must independently vanish if the whole equation is to be satisfied. Firstly, I focus on the terms depending on the second order derivative \u2202 cd \u03b8 ab . I evaluate each individual term in appendix C. Substituting (C.3) into (4.11), I find the following independent conditions, Before I analyse these equations, I will find the conditions from the first order derivative part of (4.11). There are many complicated tensor combinations that need to be considered, so for convenience I define X a := q bc \u2202 a q bc and Y a := q bc \u2202 c q ab . I evaluate the individual terms in appendix C. When I substitute the results (C.4) into (4.11), I once again find a series of unique tensor combinations with their own coefficient which vanishes independently. Most of these conditions are the same as those found in (4.12) so I won't bother duplicating them again here. However, I do find the following new conditions, where F \u2208 { v, w, R } . By this point, I have accumulated all conditions on the form of the Lagrangian for my choice of variables. The next step is to try and consolidate them.", "pages": [62, 63, 64]}, {"title": "4.3 Evaluating the fourth order perturbative action", "content": "For this section, I construct an ansatz for the action and deformation that is explicit in being a perturbative expansion. For each time derivative above the classical solution, I include the small parameter \u03b5 , and consider up to O ( \u03b5 2 ) . I consider two orders because in models of loop quantum cosmology which have deformed covariance, the holonomy corrections to the action expand into even powers of time derivatives [39, 42]. Therefore, considering a fourth order action and a second order deformation should include the nearest higher-order terms in an expansion of those holonomy functions. Therefore I write, where each coefficient is potentially a function of q and R . I take the condition from Q abcd \u2202 cd \u03b8 ab , (4.12b) and truncate to O ( \u03b5 2 ) . Separating different powers of v and w , it gives the following conditions for the Lagrangian coefficients, So from the five conditions in (4.15a), one can see that terms with three or four time derivatives must not contain any spatial derivatives. From the three conditions in (4.15b), one can see that including R in these coefficients requires including a factor of \u03b5 for every combined derivative order above two. Therefore, the spatial derivatives must be treated equally with time derivatives when one is performing a perturbative expansion, as expected. So I can now further expand the ansatz to include explicit factors of R , where each coefficient is potentially a function of q . I now substitute this ansatz into the conditions found for the action so that its form can be progressively restricted. Looking once again at the condition from Q abcd \u2202 cd \u03b8 ab (4.12b), one finds it is satisfied by the following solutions, and then looking at the condition from v ab T \u2202 2 \u03b8 ab , (4.12d), where (4.18a) and (4.17b) combine to give L ( vw ) = 0 . Then looking at the condition from q ab v cd T \u2202 cd \u03b8 ab , (4.12c) one can see that L ( vR ) = 0 and therefore all the third order terms all vanish. Looking at the condition from q ab \u2202 2 \u03b8 ab , (4.12a), and then from X a \u2202 b \u03b8 ab , (4.13a), where f and b arise as integration constants. From q ab X c \u2202 c \u03b8 ab , (4.13b), where \u03be is also an integration constant. Finally, the condition from \u2202 a R\u2202 b \u03b8 ab , (4.13h), means that From this point on the remaining equations don't provide any new conditions on the Lagrangian coefficients. To make sure the classical limit of the result matches the action found in chapter 3, I set f = \u03c9/ 2 , and replace the normal derivatives with the standard extrinsic curvature contraction K = v 2 6 -w 4 . Therefore, the fourth order perturbative gravitational action is given by, with the associated deformation So the remaining freedom in the action comes down to the constants \u03be and \u03c9 , the functions \u03b2 \u2205 and \u03b2 ( R ) . There is also a term which doesn't affect the kinematic structure and acts like a generalised notion of a potential, so can be rewritten as L \u2205 ( q ) = -\u221a q U ( q ) .", "pages": [64, 65, 66, 67]}, {"title": "4.4 Cosmology", "content": "In this section I find the cosmological implications of the nearest order corrections coming from the deformation to general covariance. Since it is a perturbative expansion, the results when the corrections become large should be taken to be indicative rather than predictive. I restrict to a flat FLRW metric as in section 2.8, where a is the scale factor, H = \u02d9 a/a is the Hubble expansion rate, \u03c3 \u2205 := sgn( \u03b2 \u2205 ) , and \u03b2 2 = \u03b2 ( R ) /\u03b2 \u2205 is the coefficient of K in the deformation. I couple this to matter with energy density \u03c1 and pressure density P = w \u03c1 \u03c1 . I Legendre transform the effective Lagrangian to find the Hamiltonian. Imposing the Hamiltonian constraint C \u2248 0 gives us which can be solved to find the modified Friedmann equation, where the correction factor is Going back to the effective Lagrangian, and varying it with respect to the scale factor, I find the Euler-Lagrange equation of motion. When I substitute in Eq. (4.28), I get the acceleration equation If I take a perfect fluid, then U = \u03c1 , where \u03c1 is the fluid's energy density, which satisfies the continuity equation where w \u03c1 is the perfect fluid's equation of state. Note that there are corrections to the matter sector due to the modified constraint algebra [86,87], as shown for scalar fields in other chapters. However, these have not been included here, as it is not known how the deformation would affect a perfect fluid. Since \u03b5 is a small parameter, it can be used to expand Eq. (4.28), and expanding the bracket in Eq. (4.30) to first order, it can be seen that a/a > 0 when w \u03c1 < w a , where having set N = 1 , so this is applicable for cosmic time. When \u03b2 2 < 0 , the modified Friedmann equation (4.32) suggests a big bounce rather than a big bang at high energy density, since \u02d9 a \u2192 0 when a > 0 and a > 0 is possible when \u03c1 \u2192 \u03c1 c where This requires either \u03c1 c to be constant, or for it to diverge at a slower rate than \u03c1 as a \u2192 0 . Let me emphasise that the bounce is found considering only holonomy corrections manifesting as higher-order powers of of second-order derivatives and not considering ignoring higher-order derivatives. The equations (4.32) and (4.33) have been expanded to leading order in \u03b2 2 , so I should be cautious about the regime of their validity. Note that the Lagrangian is also an expansion; \u03b2 2 is a coefficient of the fourth order term and appears only linearly, I conclude that there is no good reason why I should have more trust in equations such as (4.28) or (4.30) simply because they contain higher orders. In Ref. [47], Ashtekar, Pawlowski and Singh write their effective Friedmann equation with leading order corrections (which is the same as (4.32)) and say that it holds surprisingly well even for \u03c1 \u2248 \u03c1 c , the regime when the perturbative expansion should break down (I should note that their work refers only to the case where w \u03c1 = 1 , and does not say whether this is true generally).", "pages": [68, 69, 70]}, {"title": "4.4.1 Linking the \u03b2 function to LQC", "content": "I need to know \u03b2 \u2205 ( a ) and \u03b2 2 ( a ) in order to make progress beyond this point, so I compare my results to those found in previous investigations. In Ref. [42], Cailleteau, Linsefors and Barrau have found information about the correction function when inverse-volume and holonomy effects are both included in a perturbed FLRW system. Their equation (Eq. (5 . 18) in Ref. [42]) gives (rewritten slightly) where \u03b3 BI is the Barbero-Immirzi parameter, \u03b3 \u2205 is the function which contains information about inverse-volume corrections, \u03a3( a, \u02d9 a ) depends on the form of \u03b3 \u2205 , and f ( a ) is left unspecified. I just consider the case where \u03b3 \u2205 = \u03b3 \u2205 ( a ) , in which case \u03a3 = 1 / ( 2 \u221a \u03b3 \u2205 ) and \u00b5 = a \u03b4 -1 \u221a \u03b3 \u2205 \u2666 . The constant \u2666 is usually interpreted as being the 'area gap' derived in loop quantum gravity. I leave \u03b4 unspecified for now, because different quantisations of loop quantum cosmology give it equal to different values in the range [0 , 1] . Equation (4.35) now becomes The 'old dynamics' or ' \u00b5 0 scheme' corresponds to \u03b4 = 1 , and the favoured 'improved dynamics' or ' \u00af \u00b5 scheme' corresponds to \u03b4 = 0 [88, 89]. In the semi-classical regime, H \u221a \u2666 glyph[lessmuch] 1 , so I can Taylor expand this equation for the correction function to get The way that \u03b3 \u2205 is defined is that it multiplies the background gravitational term in the Hamiltonian constraint relative to the classical form. Since I am assuming \u03b3 \u2205 = \u03b3 \u2205 ( a ) , I can isolate it by taking the Lagrangian (4.26) and setting \u03b2 2 = 0 . If I then Legendre transform to find a Hamiltonian expressed in terms of the momentum of the scale factor, I find that it is proportional to \u221a | \u03b2 \u2205 | . Thus, I conclude that \u03b3 \u2205 = \u221a | \u03b2 \u2205 | when \u03b3 \u2205 is just a function of the scale factor. Using this to compare (4.37) to (4.25), I find that f = \u03c3 \u2205 | \u03b2 \u2205 | 3 / 4 , and therefore f = \u03c3 \u2205 \u03b3 3 / 2 \u2205 . From this, I can now deduce the form of the coefficient for the higher-order corrections, The exact form of \u03b3 \u2205 ( a ) is uncertain, and the possible forms that have been found also contain quantisation ambiguities. The form given by Bojowald in Ref. [90] is where l \u2208 (0 , 1) , r = a 2 /a 2 glyph[star] and a glyph[star] is the characteristic scale of the inverse-volume corrections, related to the discreteness scale. I will only use the asymptotic expansions of this function, namely and even then I will only take \u03b3 \u2205 \u2248 1 for a glyph[greatermuch] a glyph[star] , since the correction quickly becomes vanishingly small. I replace the area gap with a dimensionless parameter \u02dc \u2666 = \u2666 \u03c9 which is of order unity. The modified Friedmann equation (4.32) is now given by which I need to compare for different types of matter. First of all I will consider a perfect fluid, and then I will consider a scalar field with a power-law potential.", "pages": [70, 71, 72]}, {"title": "4.4.2 Perfect fluid", "content": "I consider the simple case of a perfect fluid. Solving the continuity equation (2.62) gives us the energy density as a function of the scale factor, To investigate whether there can be a big bounce, I insert this into Eq. (4.42), which becomes of the form where \u0398 depends on which regime of (4.41) we are in, namely and I simply ignored the constant coefficients for a glyph[lessmuch] a glyph[star] . Whether a bounce happens depends on whether H \u2192 0 when a > 0 , which would happen if the higher-order correction in the modified Friedmann equation became dominant for small values of a , i.e. if \u0398 < 0 , which is also required to match the classical limit. The reason this is required is because \u03c1 needs to diverge faster than \u03c1 c as a \u2192 0 in order for there to be a bounce. This will happen when w \u03c1 > w b , where which means that, if the bounce does not happen in the a glyph[greatermuch] a glyph[star] regime, the inverse-volume corrections make the bounce less likely to happen. If I use the favoured value of \u03b4 = 0 , and assume l = 1 , then w b = 1 / 3 and so w \u03c1 still needs to be greater than that found for radiation in order for there to be a bounce. A possible candidate for this would be a massless (or kinetic-dominated) scalar field, where w \u03c1 = 1 . Another aspect to investigate is whether the conditions for inflation are modified. Taking (4.33), I see that acceleration happens when w \u03c1 < w a , where so the range of values of w \u03c1 which can cause accelerated expansion is indeed modified. Holonomy-type corrections increase the range since \u0398 \u2264 0 , and so may inverse-volume corrections. However, the latter also seems to include a cut-off when the last term of Eq. (4.47) in the a glyph[lessmuch] a glyph[star] regime dominates. Since a bounce requires \u02d9 a = 0 and a > 0 , the condition w b < w \u03c1 < w a must be satisfied and so it must happen before the cut-off dominates if it is to happen at all.", "pages": [72, 73]}, {"title": "4.4.3 Scalar field", "content": "I now investigate the effects that the inverse-volume and holonomy corrections can have when I couple gravity to an undeformed scalar field. In this case, the energy and pressure densities are given by and the continuity equation gives us the equation of motion for the scalar field, Let us investigate the era of slow-roll inflation. Using the assumptions | \u00a8 \u03d5/U ' | glyph[lessmuch] 1 and 1 2 \u02d9 \u03d5 2 glyph[lessmuch] U , I have the slow-roll equations, If I substitute (4.50b) into (4.50a), take the derivative with respect to time and substitute in (4.50b) and (4.50a) again, I find where the slow-roll parameters are and the conditions for slow-roll inflation are I would like to investigate how these semi-classical effects affect the number of e-folds of the scale factor during inflation. The number of e-folds before the end of inflation N ( \u03d5 ) is defined by a ( \u03d5 ) = a end e -N ( \u03d5 ) , where If I remove the explicit dependence on a from the integral by setting \u03b4 = 0 and \u03b3 \u2205 = 1 (i.e. taking only a certain form of holonomy corrections and ignoring inverse-volume corrections), and choose a power-law potential where \u02dc \u03bb > 0 and n/ 2 \u2208 N , then the number of e-folds before the end of inflation is If I take the approximation that slow-roll inflation is valid beyond the regime specified by (4.53), then I can calculate a value for the maximum number of e-folds by starting inflation at the big bounce, and if I can assume \u03d5 2 end /\u03c9 glyph[lessmuch] 1 , then Let us now find the attractor solutions for slow-roll inflation. Substituting the Hubble parameter (4.42) into the equation of motion for the scalar field (4.49), I obtain I can remove the explicit scale-factor dependence of the equation by setting \u03b4 = 0 and \u03b3 \u2205 = 1 (the same assumptions as I used to find N ). Then substituting in the power-law potential (4.55) I get which is applicable only for the region where \u03c1 is below a critical value, otherwise H and \u02d9 \u03d5 are complex. I use this equation to plot phase space trajectories in Fig. 4.1. I can find the slow-roll attractor solution for | \u00a8 \u03d5\u03d5 1 -n /\u03bb | glyph[lessmuch] 1 and 1 2 \u02d9 \u03d5 2 glyph[lessmuch] \u03bb n \u03d5 n , where the term in the bracket is the correction to the classical solution. Looking at Fig. 4.1(b) and 4.1(d), I conclude that the attractor solutions diverge from a linear relationship as they approach the boundary. The condition for acceleration for the case I am considering here is where we can define the effective equation of state as w \u03c1 = P ( \u03d5 ) /\u03c1 ( \u03d5 ) using (4.48). I plot in Fig. 4.2 this region on the phase space of the scalar field to see how accelerated expansion can happen in a wider range than in the classical case. In order to be able to solve the equations and make plots, I have neglected non-zero values of \u03b4 and non-unity values of \u03b3 \u2205 . It may be that in these cases the big bounce and inflation are no longer inevitable, as was found for the perfect fluid.", "pages": [73, 74, 75, 76]}, {"title": "4.5 Discussion", "content": "In this chapter, I calculated the general conditions on a deformed action which has been formed from the variables ( q, v, w, R ) . I then found the nearest-order curvature corrections coming from the deformation by solving these conditions for a fourth order action. I found that these corrections can act as a repulsive gravitational effect which may produce a big bounce. When coupling gravity to a perfect fluid, the effects that the quantum corrections have depend on the equation of state, but inflation and a big bounce are possible. I coupled deformed gravity to an undeformed scalar in this preliminary investigation into higher order curvature corrections. I investigated slow-roll inflation and a big bounce in the presence of this scalar field. In chapter 5, I find that scalar fields must be deformed in much the same way as the metric. Therefore, these results might be interesting on some level, but cannot be taken too literally. Unfortunately, there was simply not enough time to research the fully deformed cases, hence why this material remains.", "pages": [76, 77]}, {"title": "Deformed scalar-tensor constraint to all orders", "content": "In this chapter I find the most general gravitational constraint which satisfies the deformed constraint algebra. To find the constraint is easier than finding the action, so I also include a non-minimally coupled scalar field in order to find the most general deformed scalartensor constraint. This material has not been previously published. As stated in chapter 2, I am not looking for models with degrees of freedom beyond a simple scalar-tensor model. Since actions which contain Riemann tensor squared contractions introduce additional tensor degrees of freedom [77], I automatically do not consider such terms here. This means I only need to expand the constraint using variables which are tensor contractions containing up to two orders of spatial derivatives or up to two in momenta. It also means I do not need to consider spatial derivatives of momenta in the constraint. Therefore, for a metric tensor field ( q ab , p cd ) and a scalar field ( \u03c8, \u03c0 ) , I expand the constraint into the following variables, where p ab T := p ab -1 3 pq ab is the traceless part of the metric momentum. Therefore, I start with the constraint given by C = C ( q, p, P , R, \u03c8, \u03c0, \u2206 , \u03b3 ) . I must solve the distribution equation again to find the equations which restrict the form of the constraint. The calculations in this chapter generalise those presented in chapter 3 where the minimally deformed scalar-tensor constraint was regained from the constraint algebra.", "pages": [80, 81]}, {"title": "5.1 Solving the distribution equation", "content": "Starting from (2.41), I have the general distribution equation for a Hamiltonian constraint, without derivatives of the momenta, which depends on a metric tensor and a scalar field, To solve this I will take the functional derivative with respect to a momentum variable, manipulate a few steps and then integrate with a test tensor to find several equations which the constraint must satisfy. Since I have two fields, I must do this procedure twice. The first route I consider will be where I take the derivative with respect to the metric momentum.", "pages": [81]}, {"title": "5.1.1 p ab route", "content": "Starting from the distribution equation (5.2), relabel indices, then take the functional derivative with respect to p ab ( z ) , Move derivatives and discard surface terms so that it is reorganised into the form, where, If I take (5.4) and integrate over y , I can find A ab ( x, y ) in terms of a function dependent on only a single independent variable, I then multiply this by an arbitrary, symmetric test tensor \u03b8 ab ( z ) , integrate over z , and separate out different orders of derivatives of \u03b8 ab , As done in previous chapters, I disregard the term zeroth order derivative of \u03b8 ab because it does not provide useful information. Before I can attempt to interpret this equation, I must first separate out all the different tensor combinations that there are. Because \u03b8 ab is arbitrary, the coefficients of each unique tensor combination must vanish independently. When I substitute in C = C ( q, p, P , R, \u03c8, \u03c0, \u2206 , \u03b3 ) , there are many complicated tensor combinations that need to be considered, so for convenience I define X a := q bc \u2202 a q bc . I evaluate each term in the \u2202 cd \u03b8 ab bracket, and write them in (D.2), in appendix D. So the linearly independent terms depending on \u2202 cd \u03b8 ab produce the following conditions, I then evaluate each term in the \u2202 c \u03b8 ab bracket of (5.7) and write them in (D.3). There are many unique terms which should be considered here, but in this case most of these are already solved by a constraint which satisfies (5.8). So the equations containing new information are, + \u2202C \u2202 \u2206 \u2202 3 C \u2202F\u2202p\u2202\u03c0 - \u2202C \u2202\u03c0 \u2202 3 C \u2202F\u2202p\u2202 \u2206 + 1 3 \u03b4 p F \u2202\u03b2 \u2202p , + \u2202C \u2202 \u2206 \u2202 3 \u2202F\u2202 P \u2202\u03c0 - \u2202C \u2202\u03c0 \u2202 3 \u2202F\u2202 P \u2202 \u2206 + 1 3 \u03b4 p F \u2202\u03b2 \u2202 P , where F \u2208 { p, P , R, \u2206 , \u03b3 } . These conditions strongly restrict the form of the constraint, but before I attempt to consolidate them I must find the conditions coming from the scalar field. C C", "pages": [81, 82, 83, 84]}, {"title": "5.1.2 \u03c0 route", "content": "Similar to the calculation using the metric momentum, I return to the distribution equation (5.2) and take the functional derivative with respect to \u03c0 ( z ) , which can be rewritten as, where, and similar to above, (5.11) can be solved to find 0 = A ( x, z ) -A ( x ) \u03b4 ( x, z ) . Multiply this by a test scalar field \u03b7 ( z ) and integrate over z , I evaluate each of the terms for \u2202 ab \u03b7 , and write them in (D.4). From these, I find the independent equations, Then, I evaluate all the terms for \u2202 a \u03b7 , and write them in (D.5). Therefore, ignoring terms solved by (5.14), the equations I get from \u2202 a \u03b7 are, where F \u2208 { p, P , R, \u2206 , \u03b3 } . Now that I have all of the conditions restricting the form of the constraint, I can move on to consolidating and interpreting them.", "pages": [85, 86]}, {"title": "5.2 Solving for the constraint", "content": "Now I have the full list of equations, I seek to find the restrictions on the form of C they impose. Firstly, I use the condition from \u2202 ab \u03b8 ab , (5.8a) to find which I substitute into the equation from p T ab p cd T \u2202 cd \u03b8 ab , (5.8e), and because \u03b2 \u2192 1 in the classical limit and so cannot vanish generally, I find that, Substituting this back into (5.16) gives me \u2202C \u2202R = -b 1 \u2202C \u2202 P , and from this I can find the first restriction on the form of the constraint, Substituting this into the condition from \u2202 a F\u2202 b \u03b8 ab , (5.9k), gives and therefore b 1 must only be a function of q and \u03c8 . Substituting this into (5.19) leads to \u03c7 1 = P b 1 R . Turning to the condition from X a \u2202 b \u03b8 ab , (5.9f), I find which is solved by b 1 ( q, \u03c8 ) = q b 2 ( \u03c8 ) . This is as expected because it means both terms in \u03c7 1 have a density weight of two. From this I see that the condition coming from \u2202 a \u03c8\u2202 b \u03b8 ab , (5.9a), gives which provides further restrictions on the form of the constraint, Look at the condition from p T ab \u2202 2 \u03b8 ab , (5.8d), and because b 2 is a non-zero constant in the classical limit, this can be integrated to find where g 1 is a unknown function arising as an integration constant, and needs to be determined. This provides a further restriction on the form of the constraint, Substituting this into the condition from \u2202 2 \u03b7 , (5.14a), gives and therefore, Evaluating the condition from q ab \u2202 2 \u03b8 ab , (5.8b), gives which can be integrated to find g 1 = g 2 ( q, \u03c8, \u03b3 ) -p/ 3 and therefore (5.28) becomes, Then look at the condition from p T ab X c \u2202 c \u03b8 ab , (5.9h), from which can be found glyph[negationslash] which can be solved by, g 2 ( q, \u03c8, \u03b3 ) = \u221a q g 3 ( \u03c8, \u03b3 ) if we assume that \u2202 2 C 3 \u2202\u03c7 2 3 = 0 generally, which is true for any deformation dependent on curvature \u2202\u03b2 \u2202\u03c7 3 = 0 . This is what is expected for the density weight of each term in \u03c7 3 to match. glyph[negationslash] I now look at the condition for p T ab \u2202 c \u03b3\u2202 c \u03b8 ab , which is (5.9m) with F = \u03b3 , which is true when g 3 = g 3 ( \u03c8 ) . At this point it gets harder to progress further as I have done so far. To review, I have restricted the constraint and deformation to the forms, which satisfies all the conditions in (5.8), (5.14), (5.15) and (5.9) apart from the conditions for q ab \u2202 c \u03c8\u2202 c \u03b8 ab , (5.9b), and \u2202 a \u03c8\u2202 a \u03b7 , (5.15a). As it stands, these conditions are not easy to solve.", "pages": [86, 87, 88, 89]}, {"title": "5.2.1 Solving the fourth order constraint to inform the general case", "content": "To break this impasse, I use a test ansatz for the constraint which contains up to four orders in momenta, where each coefficient is an unknown function to be determined dependent on q , \u03c8 and \u03b3 . There is an asymmetric term included in \u03c7 3 determined by the function g 3 ( \u03c8 ) , so I do not restrict myself to only even orders of momenta, unlike section 3. Substituting this into (5.15a), I can separate out the multiplier of each unique combination of variables as an independent equation. For each of the terms which are the multipliers of 5 or 6 orders of momenta, I find a condition specifying that the constraint coefficients for terms 3 or 4 orders of momenta must not depend on \u03b3 , e.g. \u2202 \u2202\u03b3 C ( \u03c7 2 ) = 0 , \u2202 \u2202\u03b3 C (\u03a0 3 ) = 0 . Since \u03b3 depends on two spatial derivatives, I see that each term in the constraint must not depend on a higher order of spatial derivatives than it does momenta. If I include higher orders of spatial derivatives in the ansatz, I quickly find them ruled out in a similar fashion. Therefore, I use this information to further expand my ansatz, where each coefficient is now an unknown function of q and \u03c8 . One can find all the necessary conditions from (5.15a), for which the solution also satisfies (5.9b). I will show a route which can taken to progressively restrict C . The condition coming from P 2 is solved if the condition from \u03b3 2 is solved by, the condition from \u03b3 P is solved by, the condition from \u03c0 4 is solved by, and all the other conditions coming from four momenta are solved. Turning to the third order, the condition from \u03c0 P is solved by, and the condition from \u03c0\u03b3 is solved by, and the condition from \u03c0 3 is solved by, which completes all the terms from third order. The only new condition coming from second order is solved by, and the only new condition coming from first order is solved by, and from the zeroth order, When all of these terms are combined, I find the solution for the fourth order constraint, where If this solution is generalised to all orders, one can check that it satisfies all the conditions from (5.8), (5.14), (5.15) and (5.9). It is possible that directly generalising from the fourth order constraint rather than continuing to work generally means that this is not the most general solution. However, at least I now know a form of the constraint which can solve all the conditions. Now that I have a form for the general constraint, I seek to compare it to the low-curvature limit, when C \u2192 \u03c7 4 C \u03c7 + C \u2205 , and match terms with that found previously (3.27) in chapter 3 and [55]. I find that, For convenience, I redefine the function determining the asymmetry, g 3 = \u03be/ 2 , and I expand the constraint in terms of the weightless (or 'de-densitised') scalar R := \u03c7 4 /q . This means that the general form of the deformed constraint is given by, It is probably more appropriate to see the deformation function itself as the driver of deformations to the constraint, so I rearrange (5.50a), which can be integrated to find, From either form of the general solution (5.50a) or (5.52), one can now understand the meaning of (2.50), which relates the order of the constraint and the deformation, 2 n C -n \u03b2 = 4 . The differential form (5.50a) is like n \u03b2 = 2( n C -2) , and the integral form (5.52) is like n C = 2 + n \u03b2 / 2 . From the integral form of the solution (5.52), I can now check a few examples of what constraint corresponds to certain deformations. Here are a few examples of easily integrable functions with the appropriate limit, glyph[negationslash] where gd( x ) := \u222b x 0 d t sech( t ) is the Gudermannian function. Most other deformation functions would need to be integrated numerically to find the constraint. As can be seen from the small R expansions, it would be possible to constrain \u03b2 \u2205 and \u03b2 2 phenomenologically but the asymptotic behaviour of \u03b2 would be difficult to determine. The simplest constraint that can be expressed as a polynomial of R that contains higher orders than the classical solution is given by, which is equivalent to the fourth order constraint found in (5.46).", "pages": [90, 91, 92, 93, 94]}, {"title": "5.3 Looking back at the constraint algebra", "content": "For this deformed constraint to mean anything, it must not reduce to the undeformed constraint through a simple transformation. If I write the constraint as a function of the undeformed vacuum constraint \u00af C = \u221a q R , I see that the deformation in the constraint algebra can be absorbed by a redefinition of the lapse functions, where \u00af N := N \u2223 \u2223 \u2202C/\u2202 \u00af C \u2223 \u2223 , \u00af M := M \u2223 \u2223 \u2202C/\u2202 \u00af C \u2223 \u2223 and \u03c3 \u2202C := sgn( \u2202C/\u2202 \u00af C ) , because the lapse functions should remain positive. The other side of the equality, which I can combine to show the that the following two equations are equivalent, The two \u03c3 \u2202C on the left side should cancel out, but they are included here to show the limit to the redefinition of the lapse functions. While it may seem like I have regained the undeformed constraint algebra up to the sign \u03c3 \u03b2 with a simple transformation, it shouldn't be taken to mean that this is actually the algebra of constraints. That is, the above equation doesn't ensure that \u00af C \u2248 0 instead of C \u2248 0 when on-shell. The surfaces in phase space described by \u00af C = 0 and C = 0 are different in general.", "pages": [94, 95]}, {"title": "5.4 Cosmology", "content": "I restrict to an isotropic and homogeneous space to find the background cosmological dynamics, following the definitions in section 2.8. Writing the constraint as C = C ( a, \u03c8, R ) where R = R ( a, \u03c8, \u00af p, \u03c0 ) , the equations of motion are given by, into which I can substitute \u2202C \u2202 R = a 3 \u221a | \u03b2 | . When I assume minimal coupling ( \u03c9 ' R = 0 , \u03c9 ' \u03c8 = 0 ) and time-symmetry ( \u03be = 0 ), the equations of motion become, To find the Friedmann equation, find the equation for H 2 /N 2 , and substitute in for R , and when the constraint is solved, C \u2248 0 , then R can be found in terms of C \u2205 .", "pages": [95, 96]}, {"title": "5.4.1 Cosmology with a perfect fluid", "content": "I here find the deformed Friedmann equations for various forms of the deformation. For simplicity, I ignore the scalar field and include a perfect fluid C \u2205 = a 3 \u03c1 ( a ) . From the deformation function \u03b2 = \u03b2 \u2205 (1 + \u03b2 2 R ) n , solving the constraint (5.53) gives glyph[negationslash] where \u03c3 \u2205 := sgn( \u03b2 \u2205 ) and \u03c3 2 := sgn(1 + \u03b2 2 R ) . When I simplify by assuming \u03c3 2 = 1 , the Friedmann equation is given by, glyph[negationslash] where \u03c1 c ( n ) = 2 \u221a | \u03b2 \u2205 | \u03b2 2 ( n +2) . To see the behaviour of the modified Friedmann equation for different values of n , look at Fig. 5.1(a). For n > 0 , the Hubble rate vanishes as the universe approaches the critical energy density, this indicates that a collapsing universe reaches a turning point at which point the repulsive effect causes a bounce. For 0 > n > -2 , there appears a sudden singularity in H at finite \u03c1 (therefore finite a ). In the \u03c1 \u2192\u221e limit, H 2 \u223c e 2 \u03c1 when n = -2 and H 2 \u223c \u03c1 2 n n +2 when n < -2 . The singularities for 0 > n > -2 appear to be similar to sudden future singularities characterised in [83, 91]. However, the singularities here might instead be called sudden 'past' singularities as they happen when a is small (but non-zero) and \u03c1 is large. Moreover, they happen for any perfect fluid with w > -1 , i.e. including matter and radiation. For the deformation function \u03b2 = \u03b2 \u2205 exp( \u03b2 2 R ) from (5.54), solving the constraint gives, and the Friedmann equation is given by, and a critical density appears for \u03c1 \u2192 2 \u221a | \u03b2 \u2205 | \u03b2 2 . For the deformation function \u03b2 = \u03b2 \u2205 sech 2 ( \u03b2 2 R ) from (5.55), solving the constraint gives, Substituting this back into the deformation function gives, and the Friedmann equation is given by where there is a critical density 1 , \u03c1 \u2192 \u03c0 \u00b7 \u221a | \u03b2 \u2205 | 2 \u03b2 2 . These exponential-type deformation functions that I consider all predict a upper limit on energy density. To illustrate this, I plot the modified Friedmann equations for these functions in Fig. 5.1(b).", "pages": [96, 97, 98]}, {"title": "5.4.2 Cosmology with a minimally coupled scalar field", "content": "Since the metric and scalar kinetic terms must combine into one quantity, R , a deformation function should not affect the relative structure between fields. To illustrate this, take a free scalar field (without a potential) which is minimally coupled to gravity, and assume no perfect fluid component. This means that the generalised potential term C \u2205 will vanish, in which case solving the constraint, C \u2248 0 , merely implies R = 0 . Consequently, since the deformation function \u03b2 is a function of R , the only deformation remaining will be the zeroth order term \u03b2 = \u03b2 \u2205 ( q, \u03c8 ) . Combining the equations of motion (5.61) allows me to find the Friedmann equation, glyph[negationslash] that is, the minimally-deformed case. For \u03b2 = \u03b2 \u2205 , it is required that R must not vanish, which itself requires that C \u2205 must be non-zero. Therefore, for the dynamics to depend on a deformation which is a function of curvature, there must be a non-zero potential term which acts as a background against which the fields are deformed.", "pages": [98, 99]}, {"title": "5.4.3 Deformation correspondence", "content": "As discussed in the perturbative action chapter 4, the form of the deformation used in the literature which includes holonomy effects is given by the cosine of the extrinsic curvature [40-42]. Of particular importance to this is that the deformation vanishes and changes sign for high values of extrinsic curvature. Since the extrinsic curvature is proportional to the Hubble expansion rate, write the deformation (4.36) here as, I wish to find C ( R ) and \u03b2 ( R ) associated with this deformation of form \u03b2 ( K ) . To do so, I need to find the relationship between the Hubble parameter H = \u02d9 a/a and the momentum \u00af p , and thereby infer the form of \u03b2 ( R ) . Then, using (5.52) I can find the constraint C ( R ) . So, using the equations of motion (5.61), I find this is an implicit equation which cannot be solved analytically for h ( r ) , and so must be solved numerically. For the general relation h = r \u221a | \u03b2 ( h ) | , there are similar \u03b2 functions which can be transformed analytically. One example is \u03b2 ( h ) = 1 -4 \u03c0 -2 \u00b7 h 2 , which also has the same limits of \u03b2 (0) = 1 and \u03b2 ( h \u2192\u00b1 \u03c0 \u00b7 / 2) = 0 , and can be transformed to find \u03b2 ( r ) = ( 1 + 4 \u03c0 -2 \u00b7 r 2 ) -1 . In Fig. 5.2, I plot \u03b2 ( h ) and h ( r ) in the region | h | \u2264 \u03c0 \u00b7 / 2 . After making the transformation, I find \u03b2 ( r ) . Note that, unlike for h , \u03b2 does not vanish for finite r . So it seems that a deformation which vanishes for finite extrinsic curvature does not necessarily vanish for finite intrinsic curvature or metric momenta (at least not in the isotropic and homogeneous case). In this respect, it matches the dynamics found for exponential-form deformations in Fig. 5.1. Returning to the solution for the constraint, (5.52), reducing it to depending on only a and \u00af p gives and transforming from \u00af p to r as defined in (5.72), while making the assumptions \u03c3 \u03b2 = 1 , N = 1 , \u03b2 \u2205 = 1 , and \u03b2 k \u223c constant , this becomes I numerically integrate the solution for \u03b2 ( r ) found for when \u03b2 = cos( h ) . I plot the function C k ( r ) in Fig. 5.2(d). If instead of the extrinsic curvature itself, the deformation is a cosine of the standard extrinsic curvature contraction, \u03b2 = cos \u03b2 k K \u223c cos h 2 , it still cannot be transformed analytically. However, it does match the function \u03b2 ( h ) = 1 -4 \u03c0 -2 \u00b7 h 4 well, as I have plotted in Fig. 5.3. However, numerically finding the constraint for these two deformations, then considering the low R limit, I see that C \u223c R 2 + C \u2205 . Therefore, this deformation can be ruled out if C \u223c R + C \u2205 is known to be the low curvature limit of the Hamiltonian constraint. Considering the function \u03b2 ( h ) = 1 -4 \u03c0 -2 \u00b7 h 2 in Fig. 5.2, transforming from h to K and from r to R to R , we can see the correspondence between different limits of the deformation function, This is what I found in chapter 6, where the general form of this particular deformation is actually the product of these two limits. However, for non-linear deformation functions, \u03b2 ( K , R ) cannot be determined so easily from \u03b2 ( K , 0) and \u03b2 (0 , R ) . That being said, given \u03b2 ( R ) , the dependence on K could be found by simply solving and evolving the equations of motion.", "pages": [99, 100, 101]}, {"title": "5.5 Discussion", "content": "In this chapter, I have found the general form that a deformed constraint can take for non-minimally coupled scalar-tensor variables. The momenta and spatial derivatives for all fields must maintain the same relative structure in how they appear compared to the minimally-deformed constraint. This means that the constraint is a function of the fields and the general kinetic term R . The freedom within this kinetic term comes down to the coupling functions. While a lapse function transformation can apparently take the constraint algebra back to the undeformed form, this seems to be merely a cosmetic change as it does not in fact alter the Hamiltonian constraint itself. I have shown how to obtain the cosmological equations of motion, and given a few simple examples of how they are modified. For some deformation functions, a upper bound on energy density appears, which probably generates a cosmological bounce. For other deformation functions, a sudden singularity in the expansion appears when the deformation diverges for high densities. I have shown that deformations to the field dynamics requires a background general potential against which the deformation must be contrasted. Using the cosmological equations of motion, I made contact with the holonomy-generated deformation which is a cosine of the extrinsic curvature. Through this, I have demonstrated how the relationship of momenta and extrinsic curvature becomes non-linear with a nontrivial deformation. It seems that when the deformation produces an upper bound on extrinsic curvature, there does not seem to be an upper bound on intrinsic curvature or momenta.", "pages": [102]}, {"title": "Deformed gravitational action to all orders", "content": "As shown in section 2.7, the deformed action must be calculated either perturbatively, as has been done in chapter 4, or completely generally. It appears that this is because it does not permit a closed polynomial solution when the deformation depends on curvature. In this chapter I attempt this general calculation. This material has been subsequently published in ref. [57]. Take the equations (4.12) and (4.13), which solve the distribution equation for the gravitational action when I expand it in terms of the variables ( q, v, w, R ) , and see what can be deduced about the action when it is treated non-perturbatively. Start with the equation for \u2202 a F\u2202 b \u03b8 ab where F \u2208 { v, w, R } , (4.13h), this can be rewritten as which implies that and so I can solve up to a sign, \u03c3 L := sgn ( \u2202L \u2202w ) , Then, from Q abcd \u2202 cd \u03b8 ab , (4.12b), I find where \u03c3 \u03b2 := sgn( \u03b2 ( q, v, w, R )) . If I then compare the second derivative of the action, \u2202 2 L \u2202w\u2202R , using both equations, I find a nonlinear partial differential equation for the deformation function, which is the same form as Burgers' equation for a fluid with vanishing viscosity [92]. However, before I attempt to interpret this, I will find further restrictions on the action and deformation. I now seek to find how the trace of the metric's normal derivative, v , appears. Take the condition for v ab T \u2202 2 \u03b8 ab , (4.12d) which I can solve to find that \u03b2 = \u03b2 ( q, \u00af w,R ) , where \u00af w = w -2 v 2 / 3 . So in the deformation, the trace v must always be paired with the traceless tensor squared w like this. I can see that this is related to the standard extrinsic curvature contraction by \u00af w = -4 K . To find how the trace appears in the action, I look at the condition from q ab \u2202 2 \u03b8 ab , (4.12a), inputting my solutions so far, I can solve for the second derivative with respect to the trace, I integrate over v to find the first derivative, To make sure that the solutions (6.3), (6.4) and (6.9) match for the second derivatives \u2202 2 L \u2202v\u2202R and \u2202 2 L \u2202v\u2202w , I find that \u03be 1 = \u03be 1 ( q ) . Therefore, from this I can see that the action should have the metric normal derivatives appear in the combined form \u00af w apart from a single linear term L \u2283 v\u03be 1 ( q ) . I now just have to see what conditions there are on how the metric determinant appears in the action. First I have the condition from X a \u2202 b \u03b8 ab , (4.13a), and second I have the condition from v ab T X c \u2202 c \u03b8 ab , (4.13c), and both these results show that my action will indeed have the correct density weight when \u03b2 \u2192 1 , that is L \u221d \u221a q . All the remaining conditions from the distribution equation that have not been explicitly referenced are solved by what I have found so far, so to make progress I must now attempt to consolidate my equations to find an explicit form for the action. If I integrate (6.3), I find and then if I match the derivative of this with respect to v with (6.9), I find the v part of the second term, If I then match the derivative of (6.12) with respect to R with (6.4), I see that and using (6.5) to change the derivative of \u03b2 , and so I can change the integration variable, the upper integration limit cancels with the left hand side of the equality, and therefore Then integrating this over R , which means that finally I have my solution for the general action, Now, I test this with a zeroth order deformation so I can match terms with my previous results. Using \u03b2 = \u03b2 \u2205 ( q ) , comparing this to (4.24) and using \u00af w = -4 K leads to and therefore, the full solution is given by, and the deformation function must satisfy the non-linear partial differential equation, By performing a Legendre transform, I can see that the Hamiltonian constraint associated with this action is given by,", "pages": [103, 104, 105, 106, 107]}, {"title": "6.1 Solving for the deformation", "content": "The nonlinear partial differential equation for the deformation function is an unexpected result, and invites a comparison to a very different area of physics. I can compare it to Burgers' equation for nonlinear diffusion, [92], (where u is a density function), and see that the deformation equation is very similar to the limit of vanishing viscosity \u03b7 \u2192 0 . This equation is not trivial to solve because it can develop discontinuities where the equation breaks down, termed 'shock waves'. Returning to my own equation (6.23), I analyse its characteristics. It implies that there are trajectories parameterised by s given by along which \u03b2 is constant. These trajectories have gradients given by, and because \u03b2 is constant along the trajectories, they are a straight line in the ( K , R ) plane. I must have an 'initial' condition in order to solve the equation, and because R is here the analogue of -t in (6.25) I define the initial function when R = 0 , given by \u03b2 ( q, K , 0) =: \u03b1 ( q, K ) . Since there are trajectories along which \u03b2 is constant, I can use \u03b1 to solve for R ( K ) along those curves, given an initial value K 0 , Reorganising to get, K 0 = K + R\u03b1 ( K 0 ) , and then substituting into \u03b2 , this leads to the implicit relation, I invoke the implicit function theorem to calculate the derivatives of \u03b2 , which show that a discontinuity develops when R\u03b1 ' \u2192 1 . This is the point where the characteristic trajectories along which \u03b2 is constant converge to form a caustic. Beyond this point, \u03b2 seems to become a multi-valued function. An analytic solution to \u03b2 only exists when \u03b1 is linear, and when \u03b1 2 ( q ) is small, I can expand \u03b2 into a series, and by comparing this to the perturbative deformation found previously, (4.25), I can see the correspondence \u03b1 1 = \u03b2 \u2205 and \u03b1 2 = \u03b5 2 \u03b2 ( R ) /\u03b2 \u2205 = \u03b5 2 \u03b2 2 . For other initial functions, I must numerically solve the deformation. As a test, in Fig. 6.1, I numerically solve for \u03b2 when \u03b1 = tanh( \u03c9 K ) . I see that, as R increases, the positive gradient in K intensifies to form a discontinuity, and softens as R decreases. I have also numerically solved for the deformation when the initial function is given by \u03b1 = cos ( \u03c9 K ) , shown in Fig. 6.2. This function is motivated by loop quantum cosmology models with holonomy corrections [40-42]. As with the tanh numerical solution in Fig. 6.1, I see the positive gradient intensify and the negative gradient soften. I could not evolve the equations past the formation of the shock wave so I cannot say for certain whether a periodicity emerges in R , but I can compare the cross sections for \u03b2 in Fig. 6.2(d). This cross section appears to match what was found in section 5.4.3 when I attempted to find the correspondence between \u03b2 ( K , 0) and \u03b2 ( R ) . It would seem that \u03b2 (0 , R ) should be a non-vanishing function of the shape as shown in Fig. 5.2(c). When the inviscid Burgers' equation is being simulated in the context of fluid dynamics, a choice must be made on how to model the shock wave [92]. The direct continuation of the equation means that the density function u becomes multi-valued, and the physical intepretation of it as a density breaks down. The alternative is to propagate the shock wave as a singular object, which requires a modification to the equations. Considering my case of the deformation function, allowing a shock wave to propagate does not seem to make sense. It might require being able to interpret \u03b2 as a density function and the space of ( K , R ) to be interpreted as a medium. Whether or not the shock wave remains singular or becomes multi-valued, the most probable interpretation is that it represents a disconnection between different branches of curvature configurations. That is, for a universe to transition from one side of the discontinuity to the other may require taking an indirect path through the phase space.", "pages": [107, 108, 109, 110]}, {"title": "6.2 Linear deformation", "content": "If I take the analytic solution for the deformation function when its initial condition is linear (6.31), I can substitute it into the general form for the gravitational action (6.22). If I assume I am in a region where 1 -\u03b1 2 R > 0 , I get the solution, and expanding in series for small \u03b1 2 when I am in a region where | \u03b1 1 | glyph[greatermuch] | \u03b1 2 K| , which matches exactly the fourth order perturbative action I found previously (4.24). The Hamiltonian constraint associated with the non-perturbative action can be found from (6.24), and then I can solve for K when the constraint vanishes (as long as I specify that it must be finite in the limit \u03b1 2 \u2192 0 ), and if I restrict to the FLRW metric and a perfect fluid as in section 2.8, I find the modified Friedmann equation, There is a correction term similar to that found for the fourth order perturbative action which suggests there could be a bounce when \u03c1 \u2192 2 \u03c9 \u221a | \u03b1 1 | /\u03b1 2 . However, there is also an additional factor which causes H to diverge when \u03c1 \u2192 \u03c9 \u221a | \u03b1 1 | /\u03b1 2 , which is before that potential bounce. This is directly comparable to the modified Friedmann equation found for the deformation function \u03b2 ( R ) = \u03b2 \u2205 (1 + \u03b2 2 R ) -1 , (5.64) investigated in section 5.4.1, with \u03b1 1 = \u03b2 \u2205 and \u03b1 2 = \u03c9\u03b2 2 / 2 . As is found here, those results suggested a sudden singularity where H diverges when a and \u03c1 remain finite.", "pages": [110, 111]}, {"title": "6.3 Discussion", "content": "I have found the general form of the deformed gravitation action when considering tensor combinations of derivatives up to second order. The way in which the deformation, and thereby the action, depends on the extrinsic and intrinsic curvature was found to be highly non-linear. Curiously, its form matches an equation found in fluid dynamics. The meaning of this comparison is far from clear. For different initial functions, I numerically solved for the deformation function until a discontinuity formed. The meaning of this discontinuity is not clear, but might manifest as a barrier across which paths through phase space cannot cross.", "pages": [111, 112]}, {"title": "Conclusions", "content": "I have attempted to thoroughly investigate the effects that a quantum-motivated deformation to the hypersurface deformation algebra of general relativity has in the semi-classical limit. Starting from the algebra, I have shown how to regain a deformed gravitational action or a deformed scalar-tensor constraint. Finding the minimally-deformed version of a non-minimally coupled scalar-tensor model, I was able to establish the classical low-curvature reference point. I was able to show how the higher-order curvature terms arising from a deformation are qualitatively different from conventional higher-order terms which can absorbed by a non-minimally coupled scalar field. I also investigated some of the interesting effects which non-minimal coupling has on cosmology. As a first step towards including higher-order curvature terms coming from a deformation, I derived the fourth order gravitational action perturbatively. The nearest order corrections demonstrate a change in the relative structure between time and space since the higher order curvature terms appear with a different sign. I investigated the cosmological implications of the higher order terms, albeit while using the assumption that the action found perturbatively could be extended beyond the perturbative regime. In attempting to find the deformed scalar-tensor constraint to any order, I was able to show how the momenta and spatial derivatives maintain the same relative kinetic structure. Interestingly, the way the scalar field and gravitational kinetic terms combine must also be unchanged. That is to say that higher order gravitational terms are necessarily accompanied by higher order scalar terms of the same form. The main consequence of this seems to be that a potential term (in a general sense) must be present for a deformation of the kinetic terms to affect the dynamics. By testing different deformation functions, I was able to show what kinds of cosmological effects should be expected. Interestingly, the deformations which cause a big bounce seem to be required to vanish, but are not required to change sign. For the final chapter, I derived the general deformed gravitational action. The way the deformation function is differently affected by extrinsic and intrinsic curvature (or, equivalently, by time and space derivatives) was found to be similar to a differential equation which usually appears in fluid mechanics. Discontinuities in the deformation function seem to be inevitable, but the interpretation of what they mean is not clear. By checking the nearest order perturbative corrections, I was able to validate the perturbative action derived in an earlier chapter. One of the original motivations of this study was to provide insight into the problem of incorporating spatial derivatives, local degrees of freedom and matter fields into models of loop quantum cosmology which deform space-time covariance. From my results, it would seem that the problem comes from considering the kinetic terms as separable, or as differently deformed. The kinetic term, when constructed with canonical variables, cannot have its internal structure deformed beyond a sign. The deformation can only be a function of the combined term, which means that matter field derivatives deform the space-time covariance in a similar way to curvature. This may strike at the heart of the way the loop quantisation project, which attempts to first find a quantum theory of gravity, typically adds in matter as an afterthought. That being said, there are important caveats to this work which must be kept in mind. The fact that I used metric variables rather than the preferred connection or loop variables might limit the applicability of my results when comparing to the motivating theory. Moreover, the deformation of the constraint algebra is only predicted for real values of \u03b3 BI . I also only considered combinations of derivatives or momenta that were a maximum of two orders, when higher order combinations and higher order derivatives are likely to appear in true quantum corrections. As said in the introduction, 1, there are potentially wider implications for this study. The deformation can lead to a modified dispersion relation, possibly indicating a variable speed of light or an invariant energy scale. It might be related to non-classical geometric qualities such a non-commutativity or scale-dependent dimensionality. In the literature, it is indicated that the deformation function may change sign, implying a transition from a Lorentzian to a Euclidean geometry at high densities. In such a way, it might be a potential mechanism for the Hartle-Hawking no-boundary proposal.", "pages": [113, 114, 115]}, {"title": "Decomposing the curvature", "content": "In our calculations, we need to decompose the three dimensional Riemann curvature frequently, so we collect the relevant identities in this appendix. The Riemann tensor is defined as the commutator of two covariant derivatives of a vector and can be given in terms of the Christoffel symbols, which are given by The variation of the Riemann tensor is given by the Palatini equation, where the variation of the connection is from which we can calculate, where we've defined the useful tensors, but contracted versions of these are more useful, To decompose the Riemann tensor in terms of partial derivatives, use this formula for decomposing the second covariant derivative of the variation of the metric, The two equations we need most are the derivative of the Ricci scalar with respect to the first and second spatial derivative of the metric, and we can find these from combining the above equations, where X a := q bc \u2202 a q bc and Y a := q bc \u2202 ( c q b ) a = \u2202 b q ba .", "pages": [116, 117, 118]}, {"title": "The general diffeomorphism constraint", "content": "I start from the assumption that the equal-time slices of our foliation are internally diffeomorphism covariant. That is to say that spatial transformations and distortions are not deformed by the deformation of the constraint algebra. As such, the Hamiltonian constraint is susceptible to deformation and the diffeomorphism constraint is not. Therefore I need to consider what form the diffeomorphism constraint has. In the hyperspace deformation algebra (2.13), the diffeomorphism constraint forms a closed sub-algebra, This equation shows that the diffeomorphism constraint is the generator of spatial diffeomorphisms (hence the name), for any phase space function F . Using this relation, I can determine the unique form of the constraint for any field content. For these calculations, I must include the concept of a tensor density, which does not transform under a change of coordinates as a tensor does. A tensor density of weight w \u03a8 \u2208 R transforms under the change x a \u2192 x ' a ' , and one can 'de-densitise' to find a tensor 1 by multiplying it by q -w \u03a8 / 2 , because \u221a q is a scalar density of weight one [10, p. \u02dc 276]. The integration measure d 3 x has a weight of -1 , so for an integral to be appropriately tensorial, the integrand must have a weight of +1 , e.g. \u222b d 3 x \u221a q . Since making a Legendre transformation requires using the term \u222b d 3 x \u02d9 \u03c8\u03c0 for a conjugate pair ( \u03c8, \u03c0 ) , when the variable \u03c8 is of weight w \u03c8 , the momentum \u03c0 is of weight 1 -w \u03c8 .", "pages": [119, 120]}, {"title": "B.1 Diffeomorphism constraint for a scalar field", "content": "I consider a scalar field ( \u03c8, \u03c0 ) . Take (B.2) with F = \u03c8 , comparing these two equations, one can easily see that Checking what result I get for F = \u03c0 merely produces the same equations and therefore the diffeomorphism constraint for a scalar field is given by, I considered up to second order spatial derivatives here as a demonstration, but no diffeomorphism constraint goes beyond first order, so I will not bother with them for further equations below.", "pages": [120, 121]}, {"title": "B.2 Diffeomorphism constraint for a vector", "content": "I consider a weightless contravariant vector ( A a , P b ) . Take (B.2) with F = A a , looking at the derivative of N a , I can see that \u2202D b \u2202P a,c = \u03b4 a b A c , and substituting this back into the equation I find, \u2202D b \u2202P a = \u03b4 a b \u2202 c A c + \u2202 b A a . If I check with F = P a I find the same equations, leading us to the diffeomorphism constraint", "pages": [121]}, {"title": "B.3 Diffeomorphism constraint for a tensor", "content": "I consider a rank-2 tensor defined on a three dimensional spatial manifold ( q ab , p cd ) . I use the example of the metric, but our result is general. Test (B.2) using F = q ab , looking at the derivative of N a , I can see that \u2202D c \u2202p ab ,d = -2 q c ( b \u03b4 d a ) , and substituting this back into the equation I find, \u2202D c \u2202p ab = \u2202 c q ab -2 \u2202 ( a q b ) c . If I check with F = p ab I find the same equations, leading us to the diffeomorphism constraint and for the specific example of the metric, this reduces to", "pages": [121, 122]}, {"title": "B.4 Diffeomorphism constraint for a tensor density", "content": "For the general case of a tensor density with n covariant indices, m contravariant indices and weight w \u03a8 , ( \u03a8 b 1 \u00b7\u00b7\u00b7 b m a 1 \u00b7\u00b7\u00b7 a n , \u03a0 c 1 \u00b7\u00b7\u00b7 c n d 1 \u00b7\u00b7\u00b7 d m ) where the canonical momentum has weight 1 -w \u03a8 , the associated diffeomorphism constraint is given by,", "pages": [122]}, {"title": "Fourth order perturbative gravitational action: Extras", "content": "For convenience, I use the definitions, Evaluating each term in the \u2202 cd \u03b8 ab bracket of (4.11), by substituting in the variables and using the equations derived for decomposing R in appendix A, Evaluating each term in the \u2202 c \u03b8 ab bracket of (4.11),", "pages": [123, 124]}, {"title": "Deformed scalar-tensor constraint to all orders: Extras", "content": "Use the following definitions for convenience, Evaluating each term in the \u2202 cd \u03b8 ab bracket of (5.7), Evaluating each term in the \u2202 c \u03b8 ab bracket of (5.7), +2 Evaluating each term in the \u2202 cd \u03b7 ab bracket of (5.13), Evaluating each term in the \u2202 c \u03b7 ab bracket of (5.13),", "pages": [126, 127, 128]}, {"title": "Bibliography", "content": "[92] J. Smoller, Springer-Verlag, New Shock Waves and Reaction-Diffusion Equations . York, 2nd ed., 1994.", "pages": [138]}] |
2017cgrc.book..119X | https://arxiv.org/pdf/1601.05607.pdf | <document>
<text><location><page_1><loc_19><loc_79><loc_41><loc_81></location>International Journal of Modern Physics D c © World Scientific Publishing Company</text>
<section_header_level_1><location><page_1><loc_34><loc_70><loc_62><loc_71></location>Strange Matter: a state before black hole</section_header_level_1>
<text><location><page_1><loc_46><loc_65><loc_47><loc_66></location>∗</text>
<text><location><page_1><loc_28><loc_63><loc_68><loc_66></location>Renxin Xu and Yanjun Guo School of Physics and KIAA, Peking University, Beijing 100871, P. R. China;</text>
<text><location><page_1><loc_35><loc_62><loc_61><loc_63></location>[email protected], [email protected]</text>
<text><location><page_1><loc_22><loc_47><loc_74><loc_59></location>Normal baryonic matter inside an evolved massive star can be intensely compressed by gravity after a supernova. General relativity predicts formation of a black hole if the core material is compressed into a singularity, but the real state of such compressed baryonic matter (CBM) before an event horizon of black hole appears is not yet well understood because of the non-perturbative nature of the fundamental strong interaction. Certainly, the rump left behind after a supernova explosion could manifest as a pulsar if its mass is less than the unknown maximum mass, M max. It is conjectured that pulsarlike compact stars are made of strange matter (i.e., with 3-flavour symmetry), where quarks are still localized as in the case of nuclear matter. In principle, di ff erent manifestations of pulsar-like objects could be explained in the regime of this conjecture. Besides compact stars, strange matter could also be manifested in the form of cosmic rays and even dark matter.</text>
<text><location><page_1><loc_22><loc_44><loc_64><loc_45></location>Keywords : Dense matter; Pulsars; Elementary particles, Cosmic ray, Dark matter.</text>
<text><location><page_1><loc_19><loc_16><loc_77><loc_42></location>The baryonic part of the Universe is well-understood in the standard model of particle physics (consolidated enormously by the discovery of Higgs Boson), where quark masses are key parameters to make a judgment on the quark-flavour degrees of freedom at a certain energy scale. Unlike the leptons, quarks could be described with mass parameters to be measured indirectly through their influence on hadronic properties since they are confined inside hadrons rather than free particles. The masses of both up and down quarks are only a few MeV while the strange quark is a little bit heavier, with an averaged mass of up and down quarks, mud = (3 . 40 ± 0 . 25) MeV, as well as the strange quark mass of ms = (93 . 5 ± 2 . 5) MeV obtained from lattice QCD (quantum chromo-dynamics) simulations. 1 For nuclei or nuclear matter, the separation between quarks is ∆ x ∼ 0 . 5 fm, and the energy scale is then order of E nucl ∼ 400 MeV according to Heisenberg's relation ∆ x · pc ∼ /planckover2pi1 c /similarequal 200 MeV · fm. One may then superficially understand why nuclei are of two (i.e., u and d ) flavours as these two flavours of quarks are the lightest. However, because the nuclear energy scale is much larger than the mass di ff erences between strange and up / down quarks, E nucl /greatermuch ( ms -mud ), why is the valence strangeness degree of freedom absolutely missing in stable nuclei?</text>
<text><location><page_1><loc_19><loc_11><loc_77><loc_16></location>We argue and explain in this paper that 3-flavour ( u , d and s ) symmetry would be restored if the strong-interaction matter at low temperature is very big, with a length scale /greatermuch the electron Compton wavelength λ e = h / ( mec ) /similarequal 0 . 024 Å. We call this kind of matter</text>
<section_header_level_1><location><page_2><loc_19><loc_80><loc_26><loc_81></location>2 Xu & Guo</section_header_level_1>
<text><location><page_2><loc_19><loc_67><loc_77><loc_78></location>as strange matter too, but it is worth noting that quarks are still localized with this definition (in analogy to 2-flavour symmetric nuclei) because the energy scale here (larger than but still around E nucl) is still much smaller than the perturbative scale of QCD dynamics, Λ χ > 1 GeV. We know that normal nuclei are relatively small, with length scale (1 ∼ 10) fm /lessmuch λ e, and it is very di ffi cult for us to gather up huge numbers ( > 10 9 ) of nuclei together because of the Coulomb barrier between them in laboratory. Then, where could one find a large nucleus with possible 3-flavour symmetry (i.e., strange matter)?</text>
<text><location><page_2><loc_19><loc_50><loc_77><loc_66></location>Such kind of strange matter can only be created through extremely astrophysical events. A good candidate of strange matter could be the supernova-produced rump left behind after core-collapsing of an evolved massive star, where normal micro -nuclei are intensely compressed by gravity to form a single gigantic nucleus (also called as compressed baryonic matter, CBM), the prototype of which was speculated and discussed firstly by Lev Landau. 2 The strange matter object could manifest the behaviors of pulsar-like compact stars if its mass is less than M max, the maximum mass being dependent on the equation of state of strange matter, but it could soon collapse further into a black hole if its mass > M max. We may then conclude that strange matter could be the state of gravity-controlled CBM before an event horizon comes out (i.e., a black hole forms).</text>
<text><location><page_2><loc_19><loc_34><loc_77><loc_50></location>This paper is organized as follows. In § 1, the gravity-compressed dense matter (a particular form of CBM), a topic relevant to Einstein's general relativity, is introduced in order to make sense of realistic CBM / strange matter in astrophysics. We try to convince the reader that such kind of astrophysical CBM should be in a state of strange matter, which would be distinguished significantly from the previous version of strange quark matter, in § 2. Cold strange matter would be in a solid state due to strong color interaction there, but the solution of a solid star with su ffi cient rigidity is still a challenge in general relativity. Nevertheless, the structure of solid strange star is presented ( § 3) in the very simple case for static and spherically symmetric objects. Di ff erent manifestations and astrophysical implications of strange matter are broadly discussed in § 4. Finally, § 5 is a brief summary.</text>
<section_header_level_1><location><page_2><loc_19><loc_30><loc_46><loc_31></location>1. Dense matter compressed by gravity</section_header_level_1>
<text><location><page_2><loc_19><loc_13><loc_77><loc_29></location>As the first force recognized among the four fundamental interactions, gravity is mysterious and fascinating because of its unique feature. Gravity is universal, which is well known from the epoch of Newton's theory. Nothing could escape the control of gravity, from the falling of apple towards the Earth, to the motion of moon in the sky. In Einstein's general relativity, gravity is related to the geometry of curved spacetime. This beautiful and elegant idea significantly influences our world view. Spacetime is curved by matter / energy, while the motion of object is along the 'straight' line (geodesic) of the curved spacetime. General relativity has passed all experimental tests up to now. However, there is intrinsic conflict between quantum theory and general relativity. Lots of e ff orts have been made to quantize gravity, but no success has been achieved yet.</text>
<text><location><page_2><loc_19><loc_8><loc_77><loc_12></location>Gravity is extremely weak compared to the other fundamental forces, so it is usually ignored in micro-physics. Nonetheless, on the scale of universe, things are mostly controlled by gravity because it is long-range and has no screening e ff ect. One century has passed</text>
<text><location><page_3><loc_77><loc_80><loc_77><loc_81></location>3</text>
<text><location><page_3><loc_19><loc_60><loc_77><loc_78></location>since Einstein established general relativity, but only a few solutions to the field equation have been found, among which three solutions are most famous and useful. The most simple case is for static and spherical spacetime, and the solution was derived by Schwarzschild just one month after Einstein's field equation. The Schwarzschild solution indicates also the existence of black hole, where everything is doomed to fall towards the center after passing through the event horizon. Consider a non-vancum case with ideal fluid as source, the field equation could be transformed to Tolman-Oppenheimer-Voko ff equation 3 , which could be applied to the interior of pulsar-like compact stars. Based on the so-called cosmological Copernicus principle, Friedmann equation can be derived with the assumption of homogenous and isotropic universe, which sets the foundation of cosmology. These three solutions of Einstein's field equation represent the most frontier topics in modern astrophysics.</text>
<text><location><page_3><loc_19><loc_55><loc_77><loc_60></location>At the late stage of stellar evolution, how does the core of massive star collapse to a black hole? Or equivalently, how is normal baryonic matter squeezed into the singularity? What's the state of compressed baryonic matter (CBM) before collapsing into a black hole?</text>
<text><location><page_3><loc_19><loc_44><loc_77><loc_55></location>We are focusing on these questions in this chapter. In the standard model of particle physics, there are totally six flavours of quarks. Among them, three ( u , d and s ) are light, with masses < 10 2 MeV, while other three flavours ( c , t and b ), with mass > 10 3 MeV, are too heavy to be excited in the nuclear energy scale, E nucl /similarequal 400 MeV. However, the ordinary matter in our world is built from u and d quarks only, and the numbers of these two flavours tend to be balanced in a stable nucleus. It is then interesting to think philosophically about the fact that our baryonic matter is 2-flavour symmetric.</text>
<text><location><page_3><loc_19><loc_18><loc_78><loc_43></location>An explanation could be: micro-nuclei are too small to have 3-flavour symmetry, but bigger is di ff erent. In fact, rational thinking about stable strangeness dates back to 1970s 4 , in which Bodmer speculated that so-called 'collapsed nuclei' with strangeness could be energetically favored if baryon number A > A min, but without quantitative estimation of the minimal number A min. Bulk matter composed of almost free quarks ( u , d , and s ) was then focused on 5 , 6 , even for astrophysical manifestations 7 , 8 . CBM can manifest as a pulsar if the mass is not large enough to form a black hole, and strange quark matter could possibly exist in compact stars, either in the core of neutron star (i.e., mixed or hybrid stars 9 ) or as the whole star (strange quark star 5 , 7 , 8 ). Although the asymptotic freedom is well recognized, one essential point is whether the color coupling between quarks is still perturbative in astrophysical CBM so that quarks are itinerant there. In case of non-perturbative coupling, the strong force there might render quarks grouped in so-called quark-clusters , forming a nucleus-like strange object 10 with 3-flavour symmetry, when CBM is big enough that relativistic electrons are inside (i.e., A > A min /similarequal 10 9 ). Anyway, we could simply call 3-flavour baryonic matter as strange matter , in which the constituent quarks could be either itinerant or localized.</text>
<text><location><page_3><loc_19><loc_14><loc_77><loc_17></location>Why is big CBM strange? This is actually a conjecture to be extensively discussed in the next section, but it could be reasonable.</text>
<section_header_level_1><location><page_4><loc_19><loc_80><loc_26><loc_81></location>4 Xu & Guo</section_header_level_1>
<section_header_level_1><location><page_4><loc_19><loc_77><loc_50><loc_78></location>2. A Bodmer-Witten's conjecture generalized</section_header_level_1>
<text><location><page_4><loc_19><loc_64><loc_77><loc_75></location>Besides being meaningful for understanding the nature of sub-nucleon at a deeper level, strangeness would also have a consequence of the physics of super-dense matter. The discovery of strangeness (a general introduction to Murray Gell-Mann and his strangeness could be found in the biography by Johnson 11 ) is known as a milestone in particle physics since our normal baryons are non-strange. Nonetheless, condensed matter with strangeness should be worth exploring as the energy scale E nucl /greatermuch ms at the nuclear and even supranuclear densities.</text>
<text><location><page_4><loc_19><loc_50><loc_77><loc_64></location>Previously, bulk strange object (suggested to be 3-flavour quark matter) is speculated to be the absolutely stable ground state of strong-interaction matter, which is known as the Bodmer-Witten's conjecture. 4 , 6 But we are discussing a general conjecture in next subsections, arguing that quarks might not be necessarily free in stable strange matter, and would still be hadron-like localized as in a nucleus if non-perturbative QCD e ff ects are significant (i.e., E nucl < Λ χ ) and the repulsive core keeps to work in both cases of 2-flavour (nonstrange) nuclear matter and 3-flavour (strange) matter. In this sense, protons and neutrons are of 2-flavour quark clusters, while strange matter could be condensed matter of quark clusters with strangeness (i.e., strange quark-clusters).</text>
<text><location><page_4><loc_19><loc_42><loc_77><loc_49></location>Summarily, it is well known that micro-nuclei are non-strange, but macro-nuclei in the form of CBM could be strange. Therefore, astrophysical CBM and nucleus could be very similar, but only with a simple change from non-strange to strange: '2' → '3'. We are explaining two approaches to this strange quark-cluster matter state, bottom-up and topdown, respectively as following.</text>
<section_header_level_1><location><page_4><loc_19><loc_38><loc_56><loc_39></location>2.1. Macro-nuclei with 3-flavour symmetry: bottom-up</section_header_level_1>
<text><location><page_4><loc_19><loc_33><loc_77><loc_37></location>Micro-nucleus is made up of protons and neutrons, and there is an observed tendency to have equal numbers of protons ( Z ) and neutrons ( N ). In liquid-drop model, the mass formula of a nucleus with atomic number A ( = Z + N ) consists of five terms,</text>
<formula><location><page_4><loc_26><loc_29><loc_77><loc_32></location>E ( Z , N ) = a v A -a s A 2 / 3 -a sym ( N -Z ) 2 A -a c Z ( Z -1) A 1 / 3 + a p ∆ ( N , Z ) A 1 / 2 , (1)</formula>
<text><location><page_4><loc_19><loc_11><loc_77><loc_29></location>where the third term is for the symmetry energy, which vanishes with equal number of protons and neutrons. This nuclear symmetry energy represents a symmetry between proton and neutron in the nucleon degree of freedom, and is actually that of u and d quarks in the quark degree 12 . The underlying physics of symmetry energy is not well understood yet. If the nucleons are treated as Fermi gas, there is a term with the same form as symmetry energy in the formula of Fermi energy, known as the kinetic term of nuclear symmetry energy. But the interaction is not negligible, and the potential term of symmetry energy would be significant. Recent scattering experiments show that, because of short-range interactions, the neutron-proton pairs are nearly 20 times as prevalent as proton-proton (and neutronneutron by inference) pairs, 13 , 14 which hints that the potential term would dominate in the symmetry energy.</text>
<text><location><page_4><loc_19><loc_8><loc_77><loc_11></location>Since the electric charges of u and d quarks is + 2 / 3 are -1 / 3 respectively, 2-flavour symmetric strong-interaction matter should be positively charged, and electrons are needed</text>
<text><location><page_5><loc_19><loc_65><loc_77><loc_78></location>to keep electric neutrality. The possibility of electrons inside a nucleus is negligible because the nuclear radius is much smaller than the Compton wavelength λ e ∼ 10 3 fm, and the lepton degree of freedom would then be not significant for nucleus. Therefore, electrons contribute negligible energy for micro-nuclei, as the coupling constant of electromagnetic interaction ( α em) is much less than that of strong interaction ( α s). The kinematic motion of electrons is bound by electromagnetic interaction, so p 2 / m e ∼ e 2 / l . From Heisenberg's relation, p · l ∼ /planckover2pi1 , combining the above two equations, we have l ∼ 1 α em /planckover2pi1 c m e c 2 , and the interaction energy is order of e 2 / l ∼ α 2 em m e c 2 ∼ 10 -5 MeV.</text>
<text><location><page_5><loc_19><loc_42><loc_77><loc_64></location>However, bigger is di ff erent, and there might be 3-flavour symmetry in gigantic / macronuclei, as electrons are inside a gigantic nucleus. With the number of nucleons A > 10 9 , the scale of macro-nuclei should be larger than the Compton wavelength of electrons λ e ∼ 10 3 fm. If 2-flavour symmetry keeps, macro-nucleus will become a huge Thomson atom with electrons evenly distributed there. Though Coulomb energy could not be significant, the Fermi energy of electrons are not negligible, to be E F ∼ /planckover2pi1 cn 1 / 3 ∼ 10 2 MeV. However, the situation becomes di ff erent if strangeness is included: no electrons exist if the matter is composed by equal numbers of light quarks of u , d , and s in chemical equilibrium. In this case, the 3-flavor symmetry, an analogy of the symmetry of u and d in nucleus, may results in a ground state of matter for gigantic nuclei. Certainly the mass di ff erence between u / d and s quarks would also break the 3-flavour symmetry, but the interaction between quarks could lower the e ff ect of mass di ff erences and favor the restoration of 3-flavour symmetry. If macro-nuclei are almost 3-flavour symmetric, the contribution of electrons would be negligible, with n e /lessmuch n q and E F ∼ 10 MeV.</text>
<text><location><page_5><loc_19><loc_31><loc_77><loc_42></location>The new degree of freedom (strangeness) is also possible to be excited, according to an order-of-magnitude estimation from either Heisenberg's relation (localized quarks) or Fermi energy (free quarks). For quarks localized with length scale l , from Heisenberg's uncertainty relation, the kinetic energy would be of ∼ p 2 / m q ∼ /planckover2pi1 2 / ( m q l 2 ), which has to be comparable to the color interaction energy of E ∼ α s /planckover2pi1 c / l in order to have a bound state, with an assumption of Coulomb-like strong interaction. One can then have if quarks are dressed,</text>
<formula><location><page_5><loc_32><loc_27><loc_77><loc_30></location>l ∼ 1 α s /planckover2pi1 c m q c 2 /similarequal 1 α s fm , E ∼ α 2 s m q c 2 /similarequal 300 α 2 s MeV . (2)</formula>
<text><location><page_5><loc_19><loc_21><loc_77><loc_26></location>As α s may well be close to and even greater than 1 at several times the nuclear density, the energy scale would be approaching and even larger than ∼ 400 MeV. A further calculation of Fermi energy also gives,</text>
<formula><location><page_5><loc_36><loc_17><loc_77><loc_20></location>E NR F ≈ /planckover2pi1 2 2 m q (3 π 2 ) 2 / 3 · n 2 / 3 = 380 MeV , (3)</formula>
<text><location><page_5><loc_19><loc_15><loc_54><loc_16></location>if quarks are considered moving non-relativistically, or</text>
<formula><location><page_5><loc_36><loc_12><loc_77><loc_13></location>E ER F ≈ /planckover2pi1 c (3 π 2 ) 1 / 3 · n 1 / 3 = 480 MeV , (4)</formula>
<text><location><page_5><loc_19><loc_8><loc_77><loc_11></location>if quarks are considered moving extremely relativistically. So we have the energy scale ∼ 400 MeV by either Heisenberg's relation or Fermi energy, which could certainly be</text>
<section_header_level_1><location><page_6><loc_19><loc_80><loc_26><loc_81></location>6 Xu & Guo</section_header_level_1>
<text><location><page_6><loc_19><loc_68><loc_77><loc_78></location>larger the mass di ff erence ( ∼ 100 MeV) between s and u / d quarks, However, for micronuclei where electron contributes also negligible energy, there could be 2-flavour (rather than 3-flavour) symmetry because s quark mass is larger than u / d quark masses. We now understand that it is more economical to have 2-flavour micro-nuclei because of massive s -quark and negligible electron kinematic energy, whereas macro / gigantic-nuclei might be 3-flavour symmetric.</text>
<text><location><page_6><loc_19><loc_62><loc_77><loc_68></location>The 2-flavour micro-nucleus consists of u and d quarks grouped in nucleons, while the 3-flavour macro-nucleus is made up of u , d and s quarks grouped in so-called strange quarkclusters. Such macro-nucleus with 3-flavour symmetry can be named as strange quarkcluster matter, or simply strange matter.</text>
<section_header_level_1><location><page_6><loc_19><loc_58><loc_55><loc_59></location>2.2. Macro-nuclei with 3-flavour symmetry: top-down</section_header_level_1>
<text><location><page_6><loc_19><loc_52><loc_77><loc_57></location>Besides this bottom-up scenario (an approach from the hadronic state), we could also start from deconfined quark state with the inclusion of stronger and stronger interaction between quarks (a top-down scenario).</text>
<text><location><page_6><loc_19><loc_39><loc_77><loc_52></location>The underlying theory of the elementary strong interaction is believed to be quantum chromodynamics (QCD), a non-Abelian SU (3) gauge theory. In QCD, the e ff ective coupling between quarks decreases with energy (the asymptotic freedom ) 15 , 16 . Quark matter (or quark-gluon plasma), the soup of deconfined quarks and gluons, is a direct consequence of asymptotic freedom when temperature or baryon density are extremely enough. Hot quark matter could be reproduced in the experiments of relativistic heavy ion collisions. Ultra-high chemical potential is required to create cold quark matter, and it can only exist in rare astrophysical conditions, the compact stars.</text>
<text><location><page_6><loc_19><loc_18><loc_77><loc_39></location>What kind of cold matter can we expect from QCD theory, in e ff ective models, or even based on phenomenology? This is a question too hard to answer because of (i) the non-perturbative e ff ects of strong interaction between quarks at low energy scales and (ii) the many-body problem due to vast assemblies of interacting particles. A colorsuperconductivity (CSC) state is focused on in QCD-based models, as well as in phenomenological ones 17 . The ground state of extremely dense quark matter could certainly be that of an ideal Fermi gas at an extremely high density. Nevertheless, it has been found that the highly degenerate Fermi surface could be unstable against the formation of quark Cooper pairs, which condense near the Fermi surface due to the existence of color-attractive channels between the quarks. A BCS-like color superconductivity, similar to electric superconductivity, has been formulated within perturbative QCD at ultra-high baryon densities. It has been argued, based on QCD-like e ff ective models, that color superconductivity could also occur even at the more realistic baryon densities of pulsar-like compact stars 17 .</text>
<text><location><page_6><loc_19><loc_11><loc_77><loc_17></location>Can the realistic stellar densities be high enough to justify the use of perturbative QCD? It is surely a challenge to calculate the coupling constant, α s , from first principles. Nevertheless, there are some approaches to the non-perturbative e ff ects of QCD, one of which uses the Dyson-Schwinger equations tried by Fischer et al., 18 , 19 who formulated,</text>
<formula><location><page_6><loc_38><loc_7><loc_77><loc_10></location>α s ( x ) = α s (0) ln( e + a 1 x a 2 + b 1 x b 2 ) , (5)</formula>
<text><location><page_7><loc_19><loc_57><loc_77><loc_78></location>where a 1 = 5 . 292 GeV -2 a 2 , a 2 = 2 . 324, b 1 = 0 . 034 GeV -2 b 2 , b 2 = 3 . 169, x = p 2 with p the typical momentum in GeV, and that α s freezes at α s (0) = 2 . 972. For our case of assumed dense quark matter at ∼ 3 ρ 0, the chemical potential is ∼ 0 . 4 GeV, and then p 2 /similarequal 0 . 16 GeV 2 . Thus, it appears that the coupling in realistic dense quark matter should be greater than 2, being close to 3 in the Fischer's estimate presented in Eq.(5). Therefore, this surely means that a weakly coupling treatment could be dangerous for realistic cold quark matter (the interaction energy ∼ 300 α 2 s MeV could even be much larger than the Fermi energy), i.e., the non-perturbative e ff ect in QCD should not be negligible if we try to know the real state of compact stars. It is also worth noting that the dimensionless electromagnetic coupling constant (i.e., the fine-structure constant) is 1 / 137 < 0 . 01, which makes QED tractable. That is to say, a weakly coupling strength comparable with that of QED is possible in QCD only if the density is unbelievably and unrealistically high ( n B > 10 123 n 0! with n 0 = 0 . 16 fm -3 the baryon density of nuclear matter).</text>
<text><location><page_7><loc_19><loc_34><loc_77><loc_56></location>Quark-clusters may form in relatively low temperature quark matter due to the strong interaction (i.e., large α s ), and the clusters could locate in periodic lattices (normal solid) when temperature becomes su ffi ciently low. Although it is hitherto impossible to know if quark-clusters could form in cold quark matter via calculation from first principles, there could be a few points that favor clustering. Experimentally , though quark matter is argued to be weakly coupled at high energy and thus deconfined, it is worth noting that, as revealed by the recent achievements in relativistic heavy ion collision experiments, the interaction between quarks in a fireball of quarks and gluons is still very strong (i.e. the strongly coupled quark-gluon plasma, sQGP 20 ). The strong coupling between quarks may naturally render quarks grouped in clusters, i.e., a condensation in position space rather than in momentum space. Theoretically , the baryon-like particles in quarkyonic matter 21 might be grouped further due to residual color interaction if the baryon density is not extremely high, and quark-clusters would form then at only a few nuclear density. Certainly, more elaborate research work is necessary.</text>
<text><location><page_7><loc_19><loc_14><loc_77><loc_33></location>For cold quark matter at 3 n 0 density, the distance between quarks is ∼ fm /greatermuch the Planck scale ∼ 10 -20 fm, so quarks and electrons can well be approximated as point-like particles. If Q α -like clusters are created in the quark matter 10 , the distance between clusters are ∼ 2 fm. The length scale l and color interaction energy of quark-clusters has been estimated by the uncertainty relation, assuming quarks are as dressed (the constituent quark mass is m q ∼ 300 MeV) and move non-relativistically in a cluster. We have l ∼ /planckover2pi1 c / ( α s m q c 2 ) /similarequal 1 fm if α s ∼ 1, and the color interaction energy ∼ α 2 s m q c 2 could be greater than the baryon chemical potential if α s /greaterorsimilar 1. The strong coupling could render quarks grouped in position space to form clusters, forming a nucleus-like strange object 10 with 3-flavour symmetry, if it is big enough that relativistic electrons are inside (i.e., A > A min /similarequal 10 9 ). Quark-clusters could be considered as classical particles in cold quark-cluster matter and would be in lattices at a lower temperature.</text>
<text><location><page_7><loc_19><loc_9><loc_77><loc_14></location>In conclusion, quark-clusters could emerge in cold dense matter because of the strong coupling between quarks. The quark-clustering phase has high density and the strong interaction is still dominant, so it is di ff erent from the usual hadron phase, and on the other hand,</text>
<section_header_level_1><location><page_8><loc_19><loc_80><loc_26><loc_81></location>8 Xu & Guo</section_header_level_1>
<text><location><page_8><loc_19><loc_72><loc_77><loc_78></location>the quark-clustering phase is also di ff erent from the conventional quark matter phase which is composed of relativistic and weakly interacting quarks. The quark-clustering phase could be considered as an intermediate state between hadron phase and free-quark phase, with deconfined quarks grouped into quark-clusters.</text>
<section_header_level_1><location><page_8><loc_19><loc_64><loc_53><loc_65></location>2.3. Comparison of micro-nuclei and macro-nuclei</section_header_level_1>
<text><location><page_8><loc_19><loc_60><loc_77><loc_63></location>In summary, there could be some similarities and di ff erences between micro-nuclei and macro-nuclei, listed as following.</text>
<text><location><page_8><loc_19><loc_52><loc_77><loc_60></location>Similarity 1: Both micro-nuclei and macro-nuclei are self-bound by the strong color interaction, in which quarks are localized in groups called generally as quark-clusters. We are sure that there are two kinds of quark-clusters inside micro-nuclei, the proton (with structure uud ) and neutron ( udd ), but don't know well the clusters in macro-nuclei due to the lack of detailed experiments related.</text>
<text><location><page_8><loc_19><loc_44><loc_77><loc_52></location>Similarity 2: Since the strong interaction might not be very sensitive to flavour, the interaction between general quark-clusters should be similar to that of nucleon, which is found to be Lennard-Jones-like by both experiment and modeling. Especially, one could then expect a hard core 22 (or repulsive core) of the interaction potential between strange quark-clusters though no direct experiment now hints this existence.</text>
<text><location><page_8><loc_19><loc_39><loc_77><loc_43></location>Di ff erence 1: The most crucial di ff erence is the change of flavour degree of freedom, from two ( u and d ) in micro-nuclei to three ( u , d and s ) in macro / gigantic-nuclei. We could thus have following di ff erent aspects derived.</text>
<text><location><page_8><loc_19><loc_29><loc_77><loc_38></location>Di ff erence 2: The number of quarks in a quark-cluster is 3 for micro nuclei, but could be 6, 9, 12, and even 18 for macro nuclei, since the interaction between Λ -particles could be attractive 23 , 24 so that no positive pressure can support a gravitational star of Λ -cluster matter. We therefore call proton / neutron as light quark-clusters, while the strange quarkcluster as heavy clusters because of (1) massive s -quark and (2) large number of quarks inside.</text>
<text><location><page_8><loc_19><loc_23><loc_77><loc_29></location>Di ff erence 3: A micro nucleus could be considered as a quantum system so that one could apply quantum mean-field theory, whereas the heavy clusters in strange matter may be classical particles since the quantum wavelength of massive clusters might be even smaller than the mean distance between them.</text>
<text><location><page_8><loc_19><loc_14><loc_77><loc_23></location>Di ff erence 4: The equation of state (EoS) of strange matter would be sti ff er 25 than that of nuclear matter because the clusters in former should be non-relativistic but relativistic in latter. The kinematic energy of a cluster in both micro- and macro-nuclei could be ∼ 0 . 5 GeV, which is much smaller than the rest mass (generally /greaterorsimilar 2 GeV) of a strange quarkcluster.</text>
<text><location><page_8><loc_19><loc_8><loc_77><loc_14></location>Di ff erence 5: Condensed matter of strange quark-clusters could be in a solid state at low temperature much smaller than the interaction energy between clusters. We could then expect solid pulsars 10 in nature although an idea of solid nucleus was also addressed 26 a long time ago.</text>
<text><location><page_9><loc_77><loc_80><loc_77><loc_81></location>9</text>
<section_header_level_1><location><page_9><loc_19><loc_77><loc_50><loc_78></location>2.4. A general conjecture of flavour symmetry</section_header_level_1>
<text><location><page_9><loc_19><loc_68><loc_77><loc_75></location>The 3-flavour symmetry may hint the nature of strong interaction at low-energy scale. Let's tell a story of science fiction about flavour symmetry. Our protagonist is a fairy who is an expert in QCD at high energy scale (i.e., perturbative QCD) but knows little about spectacular non-perturbative e ff ects. There is a conversation between the fairy and God about strong-interaction matter.</text>
<text><location><page_9><loc_19><loc_64><loc_77><loc_67></location>God: 'I know six flavours of quarks, but how many flavours could there exist in stable strong-interaction matter?'</text>
<text><location><page_9><loc_19><loc_60><loc_77><loc_64></location>Fairy: 'It depends ... how dense is the matter? (aside: the nuclear saturation density arises from the short-distance repulsive core, a consequence of non-perturbative QCD e ff ect she may not know much.)'</text>
<text><location><page_9><loc_21><loc_58><loc_74><loc_59></location>God: 'Hum ... I am told that quark number density is about 0.48 fm -3 (3 n 0) there.'</text>
<text><location><page_9><loc_19><loc_55><loc_77><loc_57></location>Fairy: 'Ah, in this energy scale of ∼ 0 . 5 GeV, there could only be light flavours (i.e., u , d and s ) in a stable matter if quarks are free.'</text>
<text><location><page_9><loc_21><loc_53><loc_53><loc_54></location>God: 'Two flavours ( u and d ) or three flavours?'</text>
<text><location><page_9><loc_19><loc_48><loc_77><loc_53></location>Fairy: 'There could be two flavours of free quarks if strong-interaction matter is very small ( /lessmuch λ e), but would be three flavours for bulk strong-interaction matter (aside: the Bodmer-Witten's conjecture).'</text>
<text><location><page_9><loc_19><loc_45><loc_77><loc_48></location>God: 'Small 2-flavour strong-interaction matter is very useful, and I can make life and mankind with huge numbers of the pieces. We can call them atoms.'</text>
<text><location><page_9><loc_21><loc_43><loc_64><loc_44></location>Fairy: 'Thanks, God! I can also help mankind to have better life.'</text>
<text><location><page_9><loc_21><loc_42><loc_49><loc_43></location>God: 'But ... are quarks really free there?'</text>
<text><location><page_9><loc_19><loc_35><loc_77><loc_41></location>Fairy: 'Hum ... there could be clustered quarks in both two and three flavour (aside: a Bodmer-Witten's conjecture generalized) cases if the interaction between quarks is so strong that quarks are grouped together. You name a piece of small two flavour matter atom , what would we call a 3-flavour body in bulk?'</text>
<text><location><page_9><loc_21><loc_33><loc_60><loc_35></location>God: Oh ... simply, a strange object because of strangeness.</text>
<section_header_level_1><location><page_9><loc_19><loc_30><loc_47><loc_31></location>3. Solid strange star in general relativity</section_header_level_1>
<text><location><page_9><loc_19><loc_18><loc_77><loc_29></location>Cold strange matter with 3-flavour symmetry could be in a solid state because of (1) a relatively small quantum wave packet of quark-cluster (wave length λ q < a , where a is the separation between quark-clusters) and (2) low temperature T < (10 -1 -10 -2 ) U ( U is the interaction energy between quark-clusters). The packet scales λ q ∼ h / ( mqc ) for free quark-cluster, with mq the rest mass of a quark cluster, but could be much smaller if it is constrained in a potential with depth of > /planckover2pi1 c / a /similarequal 100 MeV (2 fm / a ). A star made of strange matter would then be a solid star.</text>
<text><location><page_9><loc_19><loc_8><loc_77><loc_17></location>It is very fundamental to study static and spherically symmetric gravitational sources in general relativity, especially for the interior solutions. The TOV solution 3 is only for perfect fluid. However, for solid strange stars, since the local press could be anisotropic in elastic matter, the radial pressure gradient could be partially balanced by the tangential shear force although a general understanding of relativistic, elastic bodies has unfortunately not been achieved 27 . The origin of this local anisotropic force in solid quark stars could</text>
<text><location><page_10><loc_19><loc_73><loc_77><loc_78></location>be from the development of elastic energy as a star (i) spins down (its ellipticity decreases) and (ii) cools (it may shrink). Release of the elastic as well as the gravitational energies would be not negligible, and may have significant astrophysical implications.</text>
<text><location><page_10><loc_19><loc_62><loc_77><loc_73></location>The structure of solid quark stars can be numerically calculated as following. For the sack of simplicity, only spherically symmetric sources are dealt with, in order to make sense of possible astrophysical consequence of solid quark stars. By introducing respectively radial and tangential pressures, P and P ⊥ , the stellar equilibrium equation of static anisotropic matter in Newtonian gravity is 28 : d P / d r = -Gm ( r ) ρ/ r 2 + 2( P ⊥ -P ) / r , where ρ and G denote mass density and the gravitational constant, respectively, and m ( r ) = ∫ r 0 4 π r 2 ρ ( x )d x . However, in Einstein's gravity, this equilibrium equation is modified to be 29 ,</text>
<formula><location><page_10><loc_33><loc_56><loc_77><loc_61></location>d P d r = -Gm ( r ) ρ r 2 (1 + P ρ c 2 )(1 + 4 π r 3 P m ( r ) c 2 ) 1 -2 Gm ( r ) rc 2 + 2 /epsilon1 r P , (6)</formula>
<text><location><page_10><loc_19><loc_46><loc_77><loc_56></location>where P ⊥ = (1 + /epsilon1 ) P is introduced. In case of isotropic pressure, /epsilon1 = 0, Eq. (6) turns out to be the TOV equation. It is evident from Eq. (6) that the radial pressure gradient, | d P / d r | , decreases if P ⊥ > P , which may result in a higher maximum mass of compact stars. One can also see that a sudden decrease of P ⊥ (equivalently of elastic force) in a star may cause substantial energy release, since the star's radius decreases and the absolute gravitational energy increases.</text>
<text><location><page_10><loc_19><loc_32><loc_77><loc_46></location>Starquakes may result in a sudden change of /epsilon1 , with an energy release of the gravitational energy as well as the tangential strain energy. Generally, it is evident that the di ff erences of radius, gravitational energy, and moment of inertia increase proportionally to stellar mass and the parameter /epsilon1 . This means that an event should be more important for a bigger change of /epsilon1 in a quark star with higher mass. Typical energy of 10 44 ∼ 47 erg is released during superflares of SGRs, and a giant starquake with /epsilon1 /lessorsimilar 10 -4 could produce such a flare 29 . A sudden change of /epsilon1 can also result in a jump of spin frequency, ∆Ω / Ω = -∆ I / I . Glitches with ∆Ω / Ω ∼ 10 -10 ∼-4 could occur for parameters of M = (0 . 1 ∼ 1 . 4) M /circledot and /epsilon1 = 10 -9 ∼-4 . It is suggestive that a giantflare may accompany a high-amplitude glitch.</text>
<section_header_level_1><location><page_10><loc_19><loc_28><loc_53><loc_29></location>4. Astrophysical manifestations of strange matter</section_header_level_1>
<text><location><page_10><loc_19><loc_19><loc_77><loc_27></location>Howto create macro-nuclei (even gigantic) in the Universe? Besides a collapse event where normal baryonic matter is intensely compressed by gravity, strange matter could also be produced after cosmic hadronization. 6 Strange matter may manifest itself as a variety of objects with a broad mass spectrum, including compact objects, cosmic rays and even dark matter.</text>
<section_header_level_1><location><page_10><loc_19><loc_15><loc_69><loc_16></location>4.1. Pulsar-like compact star: compressed baryonic matter after supernova</section_header_level_1>
<text><location><page_10><loc_19><loc_8><loc_77><loc_14></location>In 1932, soon after Chandrasekhar found a unique mass (the mass limit of white dwarfs), Landau speculated a state of matter, the density of which 'becomes so great that atomic nuclei come in close contact, forming one gigantic nucleus ' 2 . A star composed mostly of such matter is called a 'neutron' star, and Baade and Zwicky even suggested in 1934</text>
<text><location><page_11><loc_19><loc_72><loc_77><loc_78></location>that neutron stars (NSs) could be born after supernovae. NSs theoretically predicted were finally discovered when Hewish and his collaborators detected radio pulsars in 1967 30 . More kinds of pulsar-like stars, such as X-ray pulsars and X-ray bursts in binary systems, were also discovered later, and all of them are suggested to be NSs.</text>
<text><location><page_11><loc_19><loc_55><loc_77><loc_71></location>In gigantic nucleus, protons and electrons combined to form the neutronic state, which involves weak equilibrium between protons and neutrons. However, the simple and beautiful idea proposed by Landau and others had one flaw at least: nucleons (neutrons and protons) are in fact not structureless point-like particles although they were thought to be elementary particles in 1930s. A success in the classification of hadrons discovered in cosmic rays and in accelerators leaded Gell-Mann to coin ' quark ' with fraction charges ( ± 1 / 3 , ∓ 2 / 3) in mathematical description, rather than in reality 31 . All the six flavors of quarks ( u , d , c , s , t , b ) have experimental evidence (the evidence for the last one, top quark, was reported in 1995). Is weak equilibrium among u , d and s quarks possible, instead of simply that between u and d quarks?</text>
<text><location><page_11><loc_19><loc_37><loc_77><loc_55></location>At the late stage of stellar evolution, normal baryonic matter is intensely compressed by gravity in the core of massive star during supernova. The Fermi energy of electrons are significant in CBM, and it is very essential to cancel the electrons by weak interaction in order to make lower energy state. There are two ways to kill electrons as shown in Fig 1: one is via neutronization , e -+ p → n + ν e, where the fundamental degrees of freedom could be nucleons; the other is through strangenization , where the degrees of freedom are quarks. While neutronization works for removing electrons, strangenization has both the advantages of minimizing the electron's contribution of kinetic energy and maximizing the flavour number, the later could be related to the flavour symmetry of strong-interaction matter. These two ways to kill electrons are relevant to the nature of pulsar, to be neutron star or strange star, as summarized in Fig. 1.</text>
<figure>
<location><page_11><loc_21><loc_19><loc_75><loc_33></location>
<caption>Fig. 1. Neutronization and Strangenization are two competing ways to cancel energetic electrons.</caption>
</figure>
<text><location><page_11><loc_19><loc_8><loc_77><loc_11></location>There are many speculations about the nature of pulsar due to unknown nonperturbative QCD at low energy. Among di ff erent pulsar models, hadron star and hy-</text>
<section_header_level_1><location><page_12><loc_19><loc_80><loc_26><loc_81></location>12 Xu & Guo</section_header_level_1>
<text><location><page_12><loc_19><loc_62><loc_77><loc_78></location>brid / mixed star are conventional neutron stars, while quark star and quark-cluster star are strange stars with light flavour symmetry. In hadron star model, quarks are confined in hadrons such as neutron / proton and hyperon, while a quark star is dominated by deconfined free quarks. A hybrid / mixed star, with quark matter in its cores, is a mixture of hadronic and quark states. However, a quark-cluster star, in which strong coupling causes individual quarks grouped in clusters, is neither a hadron star nor a quark star. As an analog of neutrons, quark-clusters are bound states of several quarks, so to this point of view a quark-cluster star is more similar to a real giant nucleus of self-bound (not that of Landau), rather than a 'giant hadron' which describes traditional quark stars. Di ff erent models of pulsar's inner structure are illustrated in Fig. 2.</text>
<figure>
<location><page_12><loc_25><loc_48><loc_36><loc_58></location>
</figure>
<figure>
<location><page_12><loc_37><loc_48><loc_47><loc_58></location>
</figure>
<figure>
<location><page_12><loc_50><loc_51><loc_59><loc_58></location>
</figure>
<text><location><page_12><loc_50><loc_50><loc_59><loc_51></location>Hybrid/mixed star:</text>
<text><location><page_12><loc_50><loc_49><loc_59><loc_50></location>quarks de-con./con.</text>
<text><location><page_12><loc_51><loc_48><loc_57><loc_48></location>gravity-bound</text>
<figure>
<location><page_12><loc_61><loc_48><loc_71><loc_58></location>
<caption>Fig. 2. Di ff erent models of pulsar's nature. Hadron star and hybrid / mixed star are of conventional neutron stars, while strangeness plays an important role for quark star and quark-cluster star (simply strange star ) as a result of three light-flavour ( u , d and s ) symmetry.</caption>
</figure>
<text><location><page_12><loc_19><loc_11><loc_77><loc_38></location>It is shown in Fig. 2 that conventional neutron stars (hadron star and hybrid / mixed star) are gravity-bound, while strange stars (strange quark star and strange quark-cluster star) are self-bound on surface by strong force. This feature di ff erence is very useful to identify observationally. In the neutron star picture, the inner and outer cores and the crust keep chemical equilibrium at each boundary, so neutron star is bound by gravity. The core should have a boundary and is in equilibrium with the ordinary matter because the star has a surface composed of ordinary matter. There is, however, no clear observational evidence for a neutron star's surface, although most of authors still take it for granted that there should be ordinary matter on surface, and consequently a neutron star has di ff erent components from inner to outer parts. Being similar to traditional quark stars, quark-cluster stars have almost the same composition from the center to the surface, and the quark matter surface could be natural for understanding some di ff erent observations. It is also worth noting that, although composed of quark-clusters, quark-cluster stars are self-bound by the residual interaction between quark-clusters. This is di ff erent from but similar to the traditional MIT bag scenario. The interaction between quark-clusters could be strong enough to make condensed matter, and on the surface, the quark-clusters are just in the potential well of the interaction, leading to non-vanishing density but vanishing pressure.</text>
<text><location><page_12><loc_19><loc_8><loc_77><loc_11></location>Observations of pulsar-like compact stars, including surface and global properties, could provide hints for the state of CBM, as discussed in the following.</text>
<section_header_level_1><location><page_13><loc_19><loc_77><loc_35><loc_78></location>4.1.1. Surface properties</section_header_level_1>
<text><location><page_13><loc_19><loc_53><loc_77><loc_75></location>Dirfting subpulses. Although pulsar-like stars have many di ff erent manifestations, they are populated by radio pulsars. Among the magnetospheric emission models for pulsar radio radiative process, the user-friendly nature of Ruderman-Sutherland 32 model is a virtue not shared by others, and clear drifting sub-pulses were first explained. In the seminal paper, a vacuum gap was suggested above the polar cap of a pulsar. The sparks produced by the inner-gap breakdown result in the subpulses, and the observed drifting feature is caused by E × B . However, that model can only work in strict conditions for conventional neutron stars: strong magnetic field and low temperature on surfaces of pulsars with Ω · B < 0, while calculations showed, unfortunately, that these conditions usually cannot be satisfied there. The above model encounters the so-called 'binding energy problem'. Calculations have shown that the binding energy of Fe at the neutron star surface is < 1 keV 33 , 34 , which is not su ffi cient to reproduce the vacuum gap. These problems might be alleviated within a partially screened inner gap model 35 for NSs with Ω · B < 0, but could be naturally solved for any Ω · B in the bare strange (quark-cluster) star scenario.</text>
<text><location><page_13><loc_19><loc_30><loc_77><loc_53></location>The magnetospheric activity of bare quark-cluster star was investigated in quantitative details 36 . Since quarks on the surface are confined by strong color interaction, the binding energy of quarks can be even considered as infinity compared to electromagnetic interaction. As for electrons on the surface, on one hand the potential barrier of the vacuum gap prevents electrons from streaming into the magnetosphere, on the other hand the total energy of electrons on the Fermi surface is none-zero. Therefore, the binding energy of electrons is determined by the di ff erence between the height of the potential barrier in the vacuum gap and the total energy of electrons. Calculations have shown that the huge potential barrier built by the electric field in the vacuum gap above the polar cap can usually prevent electrons from streaming into the magnetosphere, unless the electric potential of a pulsar is su ffi ciently lower than that at the infinite interstellar medium. In the bare quark-cluster star model, both positively and negatively charged particles on the surface are usually bound strongly enough to form a vacuum gap above its polar cap, and the drifting (even bi-drifting) subpulses can be understood naturally 37 , 38 .</text>
<text><location><page_13><loc_19><loc_17><loc_77><loc_30></location>X-ray spectral lines. In conventional neutron star (NS) / crusted strange star models, an atmosphere exists above the surface of a central star. Many theoretical calculations, first developed by Romani 39 , predicted the existence of atomic features in the thermal X-ray emission of NS (also for crusted strange star) atmospheres, and advanced facilities of Chandra and XMM-Newton were then proposed to be constructed for detecting those lines. One expects to know the chemical composition and magnetic field of the atmosphere through such observations, and eventually to constrain stellar mass and radius according to the redshift and pressure broadening of spectral lines.</text>
<text><location><page_13><loc_19><loc_9><loc_77><loc_17></location>However, unfortunately, none of the expected spectral features has been detected with certainty up to now, and this negative test may hint a fundamental weakness of the NS models. Although conventional NS models cannot be completely ruled out by only nonatomic thermal spectra since modified NS atmospheric models with very strong surface magnetic fields 40 , 41 might reproduce a featureless spectrum too, a natural suggestion to</text>
<text><location><page_14><loc_19><loc_75><loc_77><loc_78></location>understand the general observation could be that pulsars are actually bare strange (quark or quark-cluster) star 42 , almost without atoms there on the surfaces.</text>
<text><location><page_14><loc_19><loc_62><loc_77><loc_74></location>More observations, however, did show absorption lines of PSR-like stars, and the best absorption features were detected for the central compact object (CCO) 1E 1207.4-5209 in the center of supernova remnant PKS 1209-51 / 52, at ∼ 0 . 7 keV and ∼ 1 . 4 keV 43 , 44 , 45 . Although initially these features were thought to be due to atomic transitions of ionized helium in an atmosphere with a strong magnetic field, soon thereafter it was noted that these lines might be of electron-cyclotron origin, and 1E 1207 could be a bare strange star with surface field of ∼ 10 11 G 46 . Further observations of both spectra feature 45 and precise timing 47 favor the electron-cyclotron model of 1E 1207.</text>
<text><location><page_14><loc_19><loc_41><loc_78><loc_61></location>But this simple single particle approximation might not be reliable due to high electron density in strange stars, and Xu et al. investigated the global motion of the electron seas on the magnetized surfaces 48 . It is found that hydrodynamic surface fluctuations of the electron sea would be greatly a ff ected by the magnetic field, and an analysis shows that the seas may undergo hydrocyclotron oscillations whose eigen frequencies are given by ω ( l ) = ω c / [ l ( l + 1)], where l = 1 , 2 , 3 , ... and ω c = eB / mc is the cyclotron frequency. The fact that the absorption feature of 1E 1207.4-5209 at 0 . 7 keV is not much stronger than that at 1 . 4 keV could be understood in this hydrocyclotron oscillations model, because these two lines with l and l + 1 could have nearly equal intensity, while the strength of the first harmonic is much smaller than that of the fundamental in the electron-cyclotron model. Besides the absorption in 1E 1207.4-5209, the detected lines around (17 . 5 , 11 . 2 , 7 . 5 , 5 . 0) keV in the burst spectrum of SGR 1806-20 and those in other dead pulsars (e.g., radio quiet compact objects) would also be of hydrocyclotron origin 48 .</text>
<text><location><page_14><loc_19><loc_14><loc_77><loc_40></location>Planck-like continue spectra. The X-ray spectra from some sources (e.g., RX J1856) are well fitted by blackbody, especially with high-energy tails surprisingly close to Wien's formula: decreasing exponentially ( ∝ e -ν ). Because there is an atmosphere above the surface of neutron star / crusted strange stars, the spectrum determined by the radiative transfer in atmosphere should di ff er substantially from Planck-like one, depending on the chemical composition, magnetic field, etc. 49 Can the thermal spectrum of quark-cluster star be well described by Planck's radiation law? In bag models where quarks are nonlocal, one limitation is that bare strange stars are generally supposed to be poor radiators in thermal X-ray as a result of their high plasma frequency, ∼ 10 MeV. Nonetheless, if quarks are localized to form quark-clusters in cold quark matter due to very strong interactions, a regular lattice of the clusters (i.e., similar to a classical solid state) emerges as a consequence of the residual interaction between clusters 10 . In this latter case, the metal-like solid quark matter would induce a metal-like radiative spectrum, with which the observed thermal X-ray data of RX J1856 can be fitted 50 . Alternatively, other radiative mechanism in the electrosphere (e.g., electron bremsstrahlung in the strong electric field 51 and even of negligible ions above the sharp surface) may also reproduce a Planck-like continue spectrum.</text>
<text><location><page_14><loc_19><loc_8><loc_77><loc_14></location>Supernova and gamma-ray bursts. It is well known that the radiation fireballs of gamma-ray bursts (GRBs) and supernovae as a whole move towards the observer with a high Lorentz factor. 52 The bulk Lorentz factor of the ultrarelativistic fireball of GRBs is estimated to be order 53 of Γ ∼ 10 2 -10 3 . For such an ultra-relativistic fireball, the total</text>
<text><location><page_15><loc_19><loc_60><loc_77><loc_78></location>mass of baryons can not be too high, otherwise baryons would carry out too much energy of the central engine, the so-called 'baryon contamination'. For conventional neutron stars as the central engine, the number of baryons loaded with the fireball is unlikely to be small, since neutron stars are gravity-confined and the luminosity of fireball is extremely high. However, the baryon contamination problem can be solved naturally if the central compact objects are strange quark-cluster stars. The bare and chromatically confined surface of quark-cluster stars separates baryonic matter from the photon and lepton dominated fireball. Inside the star, baryons are in quark-cluster phase and can not escape due to strong color interaction, but e ± -pairs, photons and neutrino pairs can escape from the surface. Thus, the surface of quark-cluster stars automatically generates a low baryon condition for GRBs as well as supernovae. 54 , 55 , 56</text>
<text><location><page_15><loc_19><loc_32><loc_77><loc_60></location>It is still an unsolved problem to simulate supernovae successfully in the neutrinodriven explosion models of neutron stars. Nevertheless, in the quark-cluster star scenario, the bare quark surfaces could be essential for successful explosions of both core and accretion-induced collapses 57 . A nascent quark-cluster star born in the center of GRB or supernova would radiate thermal emission due to its ultrahigh surface temperature 58 , and the photon luminosity is not constrained by the Eddington limit since the surface of quark-cluster stars could be bare and chromatically confined. Therefore, in this photondriven scenario, 59 the strong radiation pressure caused by enormous thermal emissions from quark-cluster stars might play an important role in promoting core-collapse supernovae. Calculations have shown that the radiation pressure due to such strong thermal emission can push the overlying mantle away through photon-electron scattering with energy as much as ∼ 10 51 ergs. Such photon-driven mechanism in core-collapse supernovae by forming a quark-cluster star inside the collapsing core is promising to alleviate the current di ffi culty in core-collapse supernovae. The recent discovery of highly super-luminous supernova ASASSN-15lh, with a total observed energy (1 . 1 ± 0 . 2) × 10 52 ergs, 60 might also be understood in this regime if a very massive strange quark-cluster star, with mass smaller than but approaching M max, forms.</text>
<section_header_level_1><location><page_15><loc_19><loc_28><loc_35><loc_29></location>4.1.2. Global properties.</section_header_level_1>
<text><location><page_15><loc_19><loc_9><loc_77><loc_27></location>Free or torque-induced precession. Rigid body precesses naturally when spinning, either freely or by torque, but fluid one can hardly. The observation of possible precession or even free precession of B1821-11 61 and others could suggest a global solid structure for pulsar-like stars. Low-mass quark stars with masses of /lessorsimilar 10 -2 M /circledot and radii of a few kilometers are gravitationally force-free, and their surfaces could then be irregular (i.e., asteroidlike). Therefore, free or torque-induced precession may easily be excited and expected with larger amplitude in low-mass quark stars. The masses of AXPs / SGRs (anomalous Xray pulsars / soft gamma-ray repeaters) could be approaching the mass-limit ( > 1 . 5 M /circledot ) in the AIQ (accretion-induced quake) model 62 ; these objects could then manifest no or weak precession as observed, though they are more likely than CCOs / DTNs (eg., RX J1856) to be surrounded by dust disks because of their higher masses (thus stronger gravity).</text>
<text><location><page_15><loc_21><loc_8><loc_77><loc_9></location>Normal and slow glitches. Abig disadvantage that one believes that pulsars are strange</text>
<text><location><page_16><loc_19><loc_59><loc_77><loc_78></location>quark stars lies in the fact that the observation of pulsar glitches conflicts with the hypothesis of conventional quark stars in fluid states 63 , 64 (e.g., in MIT bag models). That problem could be solved in a solid quark-cluster star model since a solid stellar object would inevitably result in star-quakes when strain energy develops to a critical value. Huge energy should be released, and thus large spin-change occurs, after a quake of a solid quark star. Star-quakes could then be a simple and intuitional mechanism for pulsars to have glitches frequently with large amplitudes. In the regime of solid quark star, by extending the model for normal glitches 65 , one can also model pulsar's slow glitches 66 not well understood in NS models. In addition, both types of glitches without (Vela-like, Type I) and with (AXP / SGR-like, Type II) X-ray enhancement could be naturally understood in the star-quake model of solid strange star, 67 since the energy release during a type I (for fast rotators) and a type II (for slow rotators) starquake are very di ff erent.</text>
<text><location><page_16><loc_19><loc_39><loc_77><loc_58></location>Energy budget . The substantial free energy released after star-quakes, both elastic and gravitational, would power some extreme events detected in AXPs / SGRs and during GRBs. Besides persistent pulsed X-ray emission with luminosity well in excess of the spin-down power, AXPs / SGRs show occasional bursts (associated possibly with glitches), even superflares with isotropic energy ∼ 10 44 -46 erg and initial peak luminosity ∼ 10 6 -9 times of the Eddington one. They are speculated to be magnetars, with the energy reservoir of magnetic fields /greaterorsimilar 10 14 G (to be still a matter of debate about the origin since the dynamo action might not be so e ff ective and the strong magnetic field could decay e ff ectively), but failed predictions are challenges to the model. 68 However, AXPs / SGRs could also be solid quark stars with surface magnetic fields similar to that of radio pulsars. Star-quakes are responsible to both bursts / flares and glitches in the latter scenario, 62 and kinematic oscillation energy could e ff ectively power the magnetospheric activity. 69</text>
<text><location><page_16><loc_19><loc_31><loc_77><loc_38></location>The most conspicuous asteroseismic manifestion of solid phase of quark stars is their capability of sustaining torsional shear oscillations induced by SGR's starquake 70 . In addition, there are more and more authors who are trying to connect the GRB central engines to SGRs' flares, in order to understand di ff erent GRB light-curves observed, especially the internal-plateau X-ray emission. 71 , 72</text>
<text><location><page_16><loc_19><loc_21><loc_77><loc_30></location>Mass and radius of compact star . The EoS of quark-cluster matter would be sti ff er than that of nuclear matter, because (1) quark-cluster should be non-relativistic particle for its large mass, and (2) there could be strong short-distance repulsion between quark-clusters. Besides, both the problems of hyperon puzzle and quark-confinement do not exist in quarkcluster star. Sti ff EoS implies high maximum mass, while low mass is a direct consequence of self-bound surface.</text>
<text><location><page_16><loc_19><loc_8><loc_77><loc_20></location>It has been addressed that quark-cluster stars could have high maximum masses ( > 2 M /circledot ) as well as very low masses ( < 10 -2 M /circledot ). 73 Later radio observations of PSR J16142230, a binary millisecond pulsar with a strong Shapiro delay signature, imply that the pulsar mass is 1.97 ± 0.04 M /circledot 74 , which indicates a sti ff EoS for CBM. Another 2 M /circledot pulsar is also discovered afterwards 75 . It is conventionally thought that the state of dense matter softens and thus cannot result in high maximum mass if pulsars are quark stars, and that the discovery of massive 2 M /circledot pulsar may make pulsars unlikely to be quark stars. However, quark-cluster star could not be ruled out by massive pulsars, and the observations of pulsars</text>
<text><location><page_17><loc_19><loc_70><loc_77><loc_78></location>with higher mass (e.g. > 2 . 5 M /circledot ) , would even be a strong support to quark-cluster star model, and give further constraints to the parameters. The mass and radius of 4U 1746-37 could be constrained by PRE (photospheric radius expansion) bursts, on the assumption that the touchdown flux corresponds to Eddington luminosity and the obscure e ff ect is included. 76 It turns out that 4U 1746-37 could be a strange star with small radius.</text>
<text><location><page_17><loc_19><loc_36><loc_77><loc_70></location>There could be other observational hints of low-mass strange stars. Thermal radiation components from some PSR-like stars are detected, the radii of which are usually much smaller than 10 km in blackbody models where one fits spectral data by Planck spectrum, 77 and Pavlov and Luna 78 find no pulsations with periods longer than ∼ 0 . 68 s in the CCO of Cas A, and constrain stellar radius and mass to be R = (4 ∼ 5 . 5) km and M /lessorsimilar 0 . 8 M /circledot in hydrogen NS atmosphere models. Two kinds of e ff orts are made toward an understanding of the fact in conventional NS models. (1) The emissivity of NS's surface isn't simply of blackbody or of hydrogen-like atmospheres. The CCO in Cas A is suggested to covered by a carbon atmosphere 79 . However, the spectra from some sources (e.g., RX J1856) are still puzzling, being well fitted by blackbody. (2) The small emission areas would represent hot spots on NS's surfaces, i.e., to fit the X-ray spectra with at least two blackbodies, but this has three points of weakness in NS models. a , about P and ˙ P . No or very weak pulsation has been detected in some of thermal component-dominated sources (e.g., the Cas A CCO 78 ), and the inferred magnetic field from ˙ P seems not to be consistent with the atmosphere models at least for RX J1856 80 . b , fitting of thermal X-ray spectra (e.g., PSR J1852 + 0040) with two blackbodies finds two small emitting radii (significantly smaller than 10 km), which are not yet understood 81 . c , the blackbody temperature of the entire surface of some PSR-like stars are much lower than those predicted by the standard NS cooling models, 82 even provided that hot spots exist. Nevertheless, besides that two above, a natural idea could be that the detected small thermal regions ( if being global) of CCOs and others may reflect their small radii (and thus low masses in quark-cluster star scenario). 57</text>
<text><location><page_17><loc_19><loc_18><loc_77><loc_35></location>Another low-mass strange (quark-cluster) star could be 4U 1700 + 24. Because of strangeness barrier existing above a quark-cluster surface, a strange star may be surrounded by a hot corona or an atmosphere, or even a crust for di ff erent accretion rates. Both the redshifted O VIII Lyα emission line (only z = 0 . 009) and the change in the blackbody radiation area (with an inferred scale of ∼ (10 -10 2 ) m) could naturally be understood if 4U 1700 + 24 is a low-mass quark-cluster star which exhibits weak wind accretion. 83 Additionally, the mass function via observing the G-type red giant company is only fo = (1 . 8 ± 0 . 9) × 10 -5 M /circledot , 84 from which the derived mass of compact star should be much lower than 1 M /circledot unless there is geometrically fine-tuning (inclination angle i < 2 o , see Fig. 3). All these three independent observations (redshift, hot spot and mass function) may point to the fact that 4U 1700 + 24 could be a low mass strange quark-cluster star.</text>
<text><location><page_17><loc_19><loc_8><loc_77><loc_17></location>Future observations with more advanced facilities, such as FAST and SKA, could provide more observational hints for the nature of CBM. Pulsar mass measurement could help us find more massive pulsar, while measurement of the momentum of inertia may give information on the radius. Searching sub-millisecond pulsars could be an expected way to provide clear evidence for (low-mass) quark stars. Normal neutron stars can not spin with periods less than ∼ 0 . 5 M 1 / 2 1 R -3 / 2 6 ms ( R 6 = R / 10 6 cm), as the rotation is limited by Kepler</text>
<section_header_level_1><location><page_18><loc_19><loc_80><loc_26><loc_81></location>18 Xu & Guo</section_header_level_1>
<figure>
<location><page_18><loc_32><loc_64><loc_63><loc_77></location>
<caption>Fig. 3. The compact star mass as a function of orbital inclination for di ff erent values of mass function.</caption>
</figure>
<text><location><page_18><loc_19><loc_53><loc_77><loc_59></location>frequency. But low-mass bare strange stars has no such limitation on the spin period, which could be even less than 1 ms. We need thus a much short sampling time, and would deal with then a huge amount of data in order to find a sub-millisecond pulsars. Besides, the pulse profile of pulsar is helpful for the understanding of its magnetospheric activity.</text>
<section_header_level_1><location><page_18><loc_19><loc_49><loc_62><loc_50></location>4.2. Strange matter in cosmic rays and as dark matter candidate</section_header_level_1>
<text><location><page_18><loc_19><loc_43><loc_77><loc_48></location>Strange quark-nuggets, in the form of cosmic rays, could be ejected during the birth of central compact star, 85 or during collision of strange stars in a binary system spiraling towards each other due to loss of orbital energy via gravitational waves. 86</text>
<text><location><page_18><loc_19><loc_32><loc_77><loc_43></location>A strangelet with mass per baryon < 940 MeV (i.e., binding energy per baryon /greaterorsimilar 100 MeV)could be stable in cosmic rays, and would decay finally into nucleons when collisioninduced decrease of baryon number make it unstable due to the increase of surface energy. When a stable strangelet bombards the atmosphere of the Earth, its fragmented nuggets may decay quickly into Λ -particles by strong interaction and further into nucleons by weak interaction. What if a strange nugget made of quark clusters bombards the Earth? It is interesting and necessary to investigate.</text>
<text><location><page_18><loc_19><loc_17><loc_77><loc_31></location>In the early Universe ( ∼ 10 µ s), quark-gluon plasma condenses to form hadron gas during the QCD phase transition. If the cosmological QCD transition is first-order, bubbles of hadron gas are nucleated and grow until they merge and fill up the whole Universe. A separation of phases during the coexistence of the hadronic and the quark phases could gather a large number of baryons in strange nuggets. 6 If quark clustering occurs, evaporation and boiling may be suppressed, and strange nuggets may survive and contribute to the dark matter today. Strange nuggets as cold quark matter may favor the formation of seed black holes in primordial halos, alleviating the current di ffi culty of quasars at redshift as high as z ∼ 6, 87 and the small pulsar glitches detected may hint the role of strange nuggets. 88</text>
<section_header_level_1><location><page_18><loc_19><loc_13><loc_29><loc_15></location>5. Conclusions</section_header_level_1>
<text><location><page_18><loc_19><loc_8><loc_77><loc_12></location>Although normal micro-nuclei are 2-flavour symmetric, we argue that 3-flavour symmetry would be restored in macro / gigantic-nuclei compressed by gravity during a supernova. Strange matter is conjectured to be condensed matter of 3-flavour quark-clusters, and fu-</text>
<text><location><page_19><loc_19><loc_73><loc_77><loc_78></location>ture advanced facilities (e.g., FAST, SKA) would provide clear evidence for strange stars. Strange nuggets manifested in the form of cosmic rays and even dark matter have significant astrophysical consequences, to be tested observationally.</text>
<text><location><page_19><loc_19><loc_64><loc_77><loc_72></location>Acknowledgements. This work is supported by the National Basic Research Program of China (No. 2012CB821801) and NNSFC (No. 11225314). The FAST FELLOWSHIP is supported by the Special Funding for Advanced Users, budgeted and administrated by Center for Astronomical Mega-Science, Chinese Academy of Sciences (CAS). We would like to thank Ms. Yong Su for reading and checking § 2.4.</text>
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<section_header_level_1><location><page_21><loc_53><loc_80><loc_77><loc_81></location>Strange Matter: a state before black hole 21</section_header_level_1>
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</document> | [{"title": "ABSTRACT", "content": "International Journal of Modern Physics D c \u00a9 World Scientific Publishing Company", "pages": [1]}, {"title": "Strange Matter: a state before black hole", "content": "\u2217 Renxin Xu and Yanjun Guo School of Physics and KIAA, Peking University, Beijing 100871, P. R. China; [email protected], [email protected] Normal baryonic matter inside an evolved massive star can be intensely compressed by gravity after a supernova. General relativity predicts formation of a black hole if the core material is compressed into a singularity, but the real state of such compressed baryonic matter (CBM) before an event horizon of black hole appears is not yet well understood because of the non-perturbative nature of the fundamental strong interaction. Certainly, the rump left behind after a supernova explosion could manifest as a pulsar if its mass is less than the unknown maximum mass, M max. It is conjectured that pulsarlike compact stars are made of strange matter (i.e., with 3-flavour symmetry), where quarks are still localized as in the case of nuclear matter. In principle, di ff erent manifestations of pulsar-like objects could be explained in the regime of this conjecture. Besides compact stars, strange matter could also be manifested in the form of cosmic rays and even dark matter. Keywords : Dense matter; Pulsars; Elementary particles, Cosmic ray, Dark matter. The baryonic part of the Universe is well-understood in the standard model of particle physics (consolidated enormously by the discovery of Higgs Boson), where quark masses are key parameters to make a judgment on the quark-flavour degrees of freedom at a certain energy scale. Unlike the leptons, quarks could be described with mass parameters to be measured indirectly through their influence on hadronic properties since they are confined inside hadrons rather than free particles. The masses of both up and down quarks are only a few MeV while the strange quark is a little bit heavier, with an averaged mass of up and down quarks, mud = (3 . 40 \u00b1 0 . 25) MeV, as well as the strange quark mass of ms = (93 . 5 \u00b1 2 . 5) MeV obtained from lattice QCD (quantum chromo-dynamics) simulations. 1 For nuclei or nuclear matter, the separation between quarks is \u2206 x \u223c 0 . 5 fm, and the energy scale is then order of E nucl \u223c 400 MeV according to Heisenberg's relation \u2206 x \u00b7 pc \u223c /planckover2pi1 c /similarequal 200 MeV \u00b7 fm. One may then superficially understand why nuclei are of two (i.e., u and d ) flavours as these two flavours of quarks are the lightest. However, because the nuclear energy scale is much larger than the mass di ff erences between strange and up / down quarks, E nucl /greatermuch ( ms -mud ), why is the valence strangeness degree of freedom absolutely missing in stable nuclei? We argue and explain in this paper that 3-flavour ( u , d and s ) symmetry would be restored if the strong-interaction matter at low temperature is very big, with a length scale /greatermuch the electron Compton wavelength \u03bb e = h / ( mec ) /similarequal 0 . 024 \u00c5. We call this kind of matter", "pages": [1]}, {"title": "2 Xu & Guo", "content": "as strange matter too, but it is worth noting that quarks are still localized with this definition (in analogy to 2-flavour symmetric nuclei) because the energy scale here (larger than but still around E nucl) is still much smaller than the perturbative scale of QCD dynamics, \u039b \u03c7 > 1 GeV. We know that normal nuclei are relatively small, with length scale (1 \u223c 10) fm /lessmuch \u03bb e, and it is very di ffi cult for us to gather up huge numbers ( > 10 9 ) of nuclei together because of the Coulomb barrier between them in laboratory. Then, where could one find a large nucleus with possible 3-flavour symmetry (i.e., strange matter)? Such kind of strange matter can only be created through extremely astrophysical events. A good candidate of strange matter could be the supernova-produced rump left behind after core-collapsing of an evolved massive star, where normal micro -nuclei are intensely compressed by gravity to form a single gigantic nucleus (also called as compressed baryonic matter, CBM), the prototype of which was speculated and discussed firstly by Lev Landau. 2 The strange matter object could manifest the behaviors of pulsar-like compact stars if its mass is less than M max, the maximum mass being dependent on the equation of state of strange matter, but it could soon collapse further into a black hole if its mass > M max. We may then conclude that strange matter could be the state of gravity-controlled CBM before an event horizon comes out (i.e., a black hole forms). This paper is organized as follows. In \u00a7 1, the gravity-compressed dense matter (a particular form of CBM), a topic relevant to Einstein's general relativity, is introduced in order to make sense of realistic CBM / strange matter in astrophysics. We try to convince the reader that such kind of astrophysical CBM should be in a state of strange matter, which would be distinguished significantly from the previous version of strange quark matter, in \u00a7 2. Cold strange matter would be in a solid state due to strong color interaction there, but the solution of a solid star with su ffi cient rigidity is still a challenge in general relativity. Nevertheless, the structure of solid strange star is presented ( \u00a7 3) in the very simple case for static and spherically symmetric objects. Di ff erent manifestations and astrophysical implications of strange matter are broadly discussed in \u00a7 4. Finally, \u00a7 5 is a brief summary.", "pages": [2]}, {"title": "1. Dense matter compressed by gravity", "content": "As the first force recognized among the four fundamental interactions, gravity is mysterious and fascinating because of its unique feature. Gravity is universal, which is well known from the epoch of Newton's theory. Nothing could escape the control of gravity, from the falling of apple towards the Earth, to the motion of moon in the sky. In Einstein's general relativity, gravity is related to the geometry of curved spacetime. This beautiful and elegant idea significantly influences our world view. Spacetime is curved by matter / energy, while the motion of object is along the 'straight' line (geodesic) of the curved spacetime. General relativity has passed all experimental tests up to now. However, there is intrinsic conflict between quantum theory and general relativity. Lots of e ff orts have been made to quantize gravity, but no success has been achieved yet. Gravity is extremely weak compared to the other fundamental forces, so it is usually ignored in micro-physics. Nonetheless, on the scale of universe, things are mostly controlled by gravity because it is long-range and has no screening e ff ect. One century has passed 3 since Einstein established general relativity, but only a few solutions to the field equation have been found, among which three solutions are most famous and useful. The most simple case is for static and spherical spacetime, and the solution was derived by Schwarzschild just one month after Einstein's field equation. The Schwarzschild solution indicates also the existence of black hole, where everything is doomed to fall towards the center after passing through the event horizon. Consider a non-vancum case with ideal fluid as source, the field equation could be transformed to Tolman-Oppenheimer-Voko ff equation 3 , which could be applied to the interior of pulsar-like compact stars. Based on the so-called cosmological Copernicus principle, Friedmann equation can be derived with the assumption of homogenous and isotropic universe, which sets the foundation of cosmology. These three solutions of Einstein's field equation represent the most frontier topics in modern astrophysics. At the late stage of stellar evolution, how does the core of massive star collapse to a black hole? Or equivalently, how is normal baryonic matter squeezed into the singularity? What's the state of compressed baryonic matter (CBM) before collapsing into a black hole? We are focusing on these questions in this chapter. In the standard model of particle physics, there are totally six flavours of quarks. Among them, three ( u , d and s ) are light, with masses < 10 2 MeV, while other three flavours ( c , t and b ), with mass > 10 3 MeV, are too heavy to be excited in the nuclear energy scale, E nucl /similarequal 400 MeV. However, the ordinary matter in our world is built from u and d quarks only, and the numbers of these two flavours tend to be balanced in a stable nucleus. It is then interesting to think philosophically about the fact that our baryonic matter is 2-flavour symmetric. An explanation could be: micro-nuclei are too small to have 3-flavour symmetry, but bigger is di ff erent. In fact, rational thinking about stable strangeness dates back to 1970s 4 , in which Bodmer speculated that so-called 'collapsed nuclei' with strangeness could be energetically favored if baryon number A > A min, but without quantitative estimation of the minimal number A min. Bulk matter composed of almost free quarks ( u , d , and s ) was then focused on 5 , 6 , even for astrophysical manifestations 7 , 8 . CBM can manifest as a pulsar if the mass is not large enough to form a black hole, and strange quark matter could possibly exist in compact stars, either in the core of neutron star (i.e., mixed or hybrid stars 9 ) or as the whole star (strange quark star 5 , 7 , 8 ). Although the asymptotic freedom is well recognized, one essential point is whether the color coupling between quarks is still perturbative in astrophysical CBM so that quarks are itinerant there. In case of non-perturbative coupling, the strong force there might render quarks grouped in so-called quark-clusters , forming a nucleus-like strange object 10 with 3-flavour symmetry, when CBM is big enough that relativistic electrons are inside (i.e., A > A min /similarequal 10 9 ). Anyway, we could simply call 3-flavour baryonic matter as strange matter , in which the constituent quarks could be either itinerant or localized. Why is big CBM strange? This is actually a conjecture to be extensively discussed in the next section, but it could be reasonable.", "pages": [2, 3]}, {"title": "2. A Bodmer-Witten's conjecture generalized", "content": "Besides being meaningful for understanding the nature of sub-nucleon at a deeper level, strangeness would also have a consequence of the physics of super-dense matter. The discovery of strangeness (a general introduction to Murray Gell-Mann and his strangeness could be found in the biography by Johnson 11 ) is known as a milestone in particle physics since our normal baryons are non-strange. Nonetheless, condensed matter with strangeness should be worth exploring as the energy scale E nucl /greatermuch ms at the nuclear and even supranuclear densities. Previously, bulk strange object (suggested to be 3-flavour quark matter) is speculated to be the absolutely stable ground state of strong-interaction matter, which is known as the Bodmer-Witten's conjecture. 4 , 6 But we are discussing a general conjecture in next subsections, arguing that quarks might not be necessarily free in stable strange matter, and would still be hadron-like localized as in a nucleus if non-perturbative QCD e ff ects are significant (i.e., E nucl < \u039b \u03c7 ) and the repulsive core keeps to work in both cases of 2-flavour (nonstrange) nuclear matter and 3-flavour (strange) matter. In this sense, protons and neutrons are of 2-flavour quark clusters, while strange matter could be condensed matter of quark clusters with strangeness (i.e., strange quark-clusters). Summarily, it is well known that micro-nuclei are non-strange, but macro-nuclei in the form of CBM could be strange. Therefore, astrophysical CBM and nucleus could be very similar, but only with a simple change from non-strange to strange: '2' \u2192 '3'. We are explaining two approaches to this strange quark-cluster matter state, bottom-up and topdown, respectively as following.", "pages": [4]}, {"title": "2.1. Macro-nuclei with 3-flavour symmetry: bottom-up", "content": "Micro-nucleus is made up of protons and neutrons, and there is an observed tendency to have equal numbers of protons ( Z ) and neutrons ( N ). In liquid-drop model, the mass formula of a nucleus with atomic number A ( = Z + N ) consists of five terms, where the third term is for the symmetry energy, which vanishes with equal number of protons and neutrons. This nuclear symmetry energy represents a symmetry between proton and neutron in the nucleon degree of freedom, and is actually that of u and d quarks in the quark degree 12 . The underlying physics of symmetry energy is not well understood yet. If the nucleons are treated as Fermi gas, there is a term with the same form as symmetry energy in the formula of Fermi energy, known as the kinetic term of nuclear symmetry energy. But the interaction is not negligible, and the potential term of symmetry energy would be significant. Recent scattering experiments show that, because of short-range interactions, the neutron-proton pairs are nearly 20 times as prevalent as proton-proton (and neutronneutron by inference) pairs, 13 , 14 which hints that the potential term would dominate in the symmetry energy. Since the electric charges of u and d quarks is + 2 / 3 are -1 / 3 respectively, 2-flavour symmetric strong-interaction matter should be positively charged, and electrons are needed to keep electric neutrality. The possibility of electrons inside a nucleus is negligible because the nuclear radius is much smaller than the Compton wavelength \u03bb e \u223c 10 3 fm, and the lepton degree of freedom would then be not significant for nucleus. Therefore, electrons contribute negligible energy for micro-nuclei, as the coupling constant of electromagnetic interaction ( \u03b1 em) is much less than that of strong interaction ( \u03b1 s). The kinematic motion of electrons is bound by electromagnetic interaction, so p 2 / m e \u223c e 2 / l . From Heisenberg's relation, p \u00b7 l \u223c /planckover2pi1 , combining the above two equations, we have l \u223c 1 \u03b1 em /planckover2pi1 c m e c 2 , and the interaction energy is order of e 2 / l \u223c \u03b1 2 em m e c 2 \u223c 10 -5 MeV. However, bigger is di ff erent, and there might be 3-flavour symmetry in gigantic / macronuclei, as electrons are inside a gigantic nucleus. With the number of nucleons A > 10 9 , the scale of macro-nuclei should be larger than the Compton wavelength of electrons \u03bb e \u223c 10 3 fm. If 2-flavour symmetry keeps, macro-nucleus will become a huge Thomson atom with electrons evenly distributed there. Though Coulomb energy could not be significant, the Fermi energy of electrons are not negligible, to be E F \u223c /planckover2pi1 cn 1 / 3 \u223c 10 2 MeV. However, the situation becomes di ff erent if strangeness is included: no electrons exist if the matter is composed by equal numbers of light quarks of u , d , and s in chemical equilibrium. In this case, the 3-flavor symmetry, an analogy of the symmetry of u and d in nucleus, may results in a ground state of matter for gigantic nuclei. Certainly the mass di ff erence between u / d and s quarks would also break the 3-flavour symmetry, but the interaction between quarks could lower the e ff ect of mass di ff erences and favor the restoration of 3-flavour symmetry. If macro-nuclei are almost 3-flavour symmetric, the contribution of electrons would be negligible, with n e /lessmuch n q and E F \u223c 10 MeV. The new degree of freedom (strangeness) is also possible to be excited, according to an order-of-magnitude estimation from either Heisenberg's relation (localized quarks) or Fermi energy (free quarks). For quarks localized with length scale l , from Heisenberg's uncertainty relation, the kinetic energy would be of \u223c p 2 / m q \u223c /planckover2pi1 2 / ( m q l 2 ), which has to be comparable to the color interaction energy of E \u223c \u03b1 s /planckover2pi1 c / l in order to have a bound state, with an assumption of Coulomb-like strong interaction. One can then have if quarks are dressed, As \u03b1 s may well be close to and even greater than 1 at several times the nuclear density, the energy scale would be approaching and even larger than \u223c 400 MeV. A further calculation of Fermi energy also gives, if quarks are considered moving non-relativistically, or if quarks are considered moving extremely relativistically. So we have the energy scale \u223c 400 MeV by either Heisenberg's relation or Fermi energy, which could certainly be", "pages": [4, 5]}, {"title": "6 Xu & Guo", "content": "larger the mass di ff erence ( \u223c 100 MeV) between s and u / d quarks, However, for micronuclei where electron contributes also negligible energy, there could be 2-flavour (rather than 3-flavour) symmetry because s quark mass is larger than u / d quark masses. We now understand that it is more economical to have 2-flavour micro-nuclei because of massive s -quark and negligible electron kinematic energy, whereas macro / gigantic-nuclei might be 3-flavour symmetric. The 2-flavour micro-nucleus consists of u and d quarks grouped in nucleons, while the 3-flavour macro-nucleus is made up of u , d and s quarks grouped in so-called strange quarkclusters. Such macro-nucleus with 3-flavour symmetry can be named as strange quarkcluster matter, or simply strange matter.", "pages": [6]}, {"title": "2.2. Macro-nuclei with 3-flavour symmetry: top-down", "content": "Besides this bottom-up scenario (an approach from the hadronic state), we could also start from deconfined quark state with the inclusion of stronger and stronger interaction between quarks (a top-down scenario). The underlying theory of the elementary strong interaction is believed to be quantum chromodynamics (QCD), a non-Abelian SU (3) gauge theory. In QCD, the e ff ective coupling between quarks decreases with energy (the asymptotic freedom ) 15 , 16 . Quark matter (or quark-gluon plasma), the soup of deconfined quarks and gluons, is a direct consequence of asymptotic freedom when temperature or baryon density are extremely enough. Hot quark matter could be reproduced in the experiments of relativistic heavy ion collisions. Ultra-high chemical potential is required to create cold quark matter, and it can only exist in rare astrophysical conditions, the compact stars. What kind of cold matter can we expect from QCD theory, in e ff ective models, or even based on phenomenology? This is a question too hard to answer because of (i) the non-perturbative e ff ects of strong interaction between quarks at low energy scales and (ii) the many-body problem due to vast assemblies of interacting particles. A colorsuperconductivity (CSC) state is focused on in QCD-based models, as well as in phenomenological ones 17 . The ground state of extremely dense quark matter could certainly be that of an ideal Fermi gas at an extremely high density. Nevertheless, it has been found that the highly degenerate Fermi surface could be unstable against the formation of quark Cooper pairs, which condense near the Fermi surface due to the existence of color-attractive channels between the quarks. A BCS-like color superconductivity, similar to electric superconductivity, has been formulated within perturbative QCD at ultra-high baryon densities. It has been argued, based on QCD-like e ff ective models, that color superconductivity could also occur even at the more realistic baryon densities of pulsar-like compact stars 17 . Can the realistic stellar densities be high enough to justify the use of perturbative QCD? It is surely a challenge to calculate the coupling constant, \u03b1 s , from first principles. Nevertheless, there are some approaches to the non-perturbative e ff ects of QCD, one of which uses the Dyson-Schwinger equations tried by Fischer et al., 18 , 19 who formulated, where a 1 = 5 . 292 GeV -2 a 2 , a 2 = 2 . 324, b 1 = 0 . 034 GeV -2 b 2 , b 2 = 3 . 169, x = p 2 with p the typical momentum in GeV, and that \u03b1 s freezes at \u03b1 s (0) = 2 . 972. For our case of assumed dense quark matter at \u223c 3 \u03c1 0, the chemical potential is \u223c 0 . 4 GeV, and then p 2 /similarequal 0 . 16 GeV 2 . Thus, it appears that the coupling in realistic dense quark matter should be greater than 2, being close to 3 in the Fischer's estimate presented in Eq.(5). Therefore, this surely means that a weakly coupling treatment could be dangerous for realistic cold quark matter (the interaction energy \u223c 300 \u03b1 2 s MeV could even be much larger than the Fermi energy), i.e., the non-perturbative e ff ect in QCD should not be negligible if we try to know the real state of compact stars. It is also worth noting that the dimensionless electromagnetic coupling constant (i.e., the fine-structure constant) is 1 / 137 < 0 . 01, which makes QED tractable. That is to say, a weakly coupling strength comparable with that of QED is possible in QCD only if the density is unbelievably and unrealistically high ( n B > 10 123 n 0! with n 0 = 0 . 16 fm -3 the baryon density of nuclear matter). Quark-clusters may form in relatively low temperature quark matter due to the strong interaction (i.e., large \u03b1 s ), and the clusters could locate in periodic lattices (normal solid) when temperature becomes su ffi ciently low. Although it is hitherto impossible to know if quark-clusters could form in cold quark matter via calculation from first principles, there could be a few points that favor clustering. Experimentally , though quark matter is argued to be weakly coupled at high energy and thus deconfined, it is worth noting that, as revealed by the recent achievements in relativistic heavy ion collision experiments, the interaction between quarks in a fireball of quarks and gluons is still very strong (i.e. the strongly coupled quark-gluon plasma, sQGP 20 ). The strong coupling between quarks may naturally render quarks grouped in clusters, i.e., a condensation in position space rather than in momentum space. Theoretically , the baryon-like particles in quarkyonic matter 21 might be grouped further due to residual color interaction if the baryon density is not extremely high, and quark-clusters would form then at only a few nuclear density. Certainly, more elaborate research work is necessary. For cold quark matter at 3 n 0 density, the distance between quarks is \u223c fm /greatermuch the Planck scale \u223c 10 -20 fm, so quarks and electrons can well be approximated as point-like particles. If Q \u03b1 -like clusters are created in the quark matter 10 , the distance between clusters are \u223c 2 fm. The length scale l and color interaction energy of quark-clusters has been estimated by the uncertainty relation, assuming quarks are as dressed (the constituent quark mass is m q \u223c 300 MeV) and move non-relativistically in a cluster. We have l \u223c /planckover2pi1 c / ( \u03b1 s m q c 2 ) /similarequal 1 fm if \u03b1 s \u223c 1, and the color interaction energy \u223c \u03b1 2 s m q c 2 could be greater than the baryon chemical potential if \u03b1 s /greaterorsimilar 1. The strong coupling could render quarks grouped in position space to form clusters, forming a nucleus-like strange object 10 with 3-flavour symmetry, if it is big enough that relativistic electrons are inside (i.e., A > A min /similarequal 10 9 ). Quark-clusters could be considered as classical particles in cold quark-cluster matter and would be in lattices at a lower temperature. In conclusion, quark-clusters could emerge in cold dense matter because of the strong coupling between quarks. The quark-clustering phase has high density and the strong interaction is still dominant, so it is di ff erent from the usual hadron phase, and on the other hand,", "pages": [6, 7]}, {"title": "8 Xu & Guo", "content": "the quark-clustering phase is also di ff erent from the conventional quark matter phase which is composed of relativistic and weakly interacting quarks. The quark-clustering phase could be considered as an intermediate state between hadron phase and free-quark phase, with deconfined quarks grouped into quark-clusters.", "pages": [8]}, {"title": "2.3. Comparison of micro-nuclei and macro-nuclei", "content": "In summary, there could be some similarities and di ff erences between micro-nuclei and macro-nuclei, listed as following. Similarity 1: Both micro-nuclei and macro-nuclei are self-bound by the strong color interaction, in which quarks are localized in groups called generally as quark-clusters. We are sure that there are two kinds of quark-clusters inside micro-nuclei, the proton (with structure uud ) and neutron ( udd ), but don't know well the clusters in macro-nuclei due to the lack of detailed experiments related. Similarity 2: Since the strong interaction might not be very sensitive to flavour, the interaction between general quark-clusters should be similar to that of nucleon, which is found to be Lennard-Jones-like by both experiment and modeling. Especially, one could then expect a hard core 22 (or repulsive core) of the interaction potential between strange quark-clusters though no direct experiment now hints this existence. Di ff erence 1: The most crucial di ff erence is the change of flavour degree of freedom, from two ( u and d ) in micro-nuclei to three ( u , d and s ) in macro / gigantic-nuclei. We could thus have following di ff erent aspects derived. Di ff erence 2: The number of quarks in a quark-cluster is 3 for micro nuclei, but could be 6, 9, 12, and even 18 for macro nuclei, since the interaction between \u039b -particles could be attractive 23 , 24 so that no positive pressure can support a gravitational star of \u039b -cluster matter. We therefore call proton / neutron as light quark-clusters, while the strange quarkcluster as heavy clusters because of (1) massive s -quark and (2) large number of quarks inside. Di ff erence 3: A micro nucleus could be considered as a quantum system so that one could apply quantum mean-field theory, whereas the heavy clusters in strange matter may be classical particles since the quantum wavelength of massive clusters might be even smaller than the mean distance between them. Di ff erence 4: The equation of state (EoS) of strange matter would be sti ff er 25 than that of nuclear matter because the clusters in former should be non-relativistic but relativistic in latter. The kinematic energy of a cluster in both micro- and macro-nuclei could be \u223c 0 . 5 GeV, which is much smaller than the rest mass (generally /greaterorsimilar 2 GeV) of a strange quarkcluster. Di ff erence 5: Condensed matter of strange quark-clusters could be in a solid state at low temperature much smaller than the interaction energy between clusters. We could then expect solid pulsars 10 in nature although an idea of solid nucleus was also addressed 26 a long time ago. 9", "pages": [8, 9]}, {"title": "2.4. A general conjecture of flavour symmetry", "content": "The 3-flavour symmetry may hint the nature of strong interaction at low-energy scale. Let's tell a story of science fiction about flavour symmetry. Our protagonist is a fairy who is an expert in QCD at high energy scale (i.e., perturbative QCD) but knows little about spectacular non-perturbative e ff ects. There is a conversation between the fairy and God about strong-interaction matter. God: 'I know six flavours of quarks, but how many flavours could there exist in stable strong-interaction matter?' Fairy: 'It depends ... how dense is the matter? (aside: the nuclear saturation density arises from the short-distance repulsive core, a consequence of non-perturbative QCD e ff ect she may not know much.)' God: 'Hum ... I am told that quark number density is about 0.48 fm -3 (3 n 0) there.' Fairy: 'Ah, in this energy scale of \u223c 0 . 5 GeV, there could only be light flavours (i.e., u , d and s ) in a stable matter if quarks are free.' God: 'Two flavours ( u and d ) or three flavours?' Fairy: 'There could be two flavours of free quarks if strong-interaction matter is very small ( /lessmuch \u03bb e), but would be three flavours for bulk strong-interaction matter (aside: the Bodmer-Witten's conjecture).' God: 'Small 2-flavour strong-interaction matter is very useful, and I can make life and mankind with huge numbers of the pieces. We can call them atoms.' Fairy: 'Thanks, God! I can also help mankind to have better life.' God: 'But ... are quarks really free there?' Fairy: 'Hum ... there could be clustered quarks in both two and three flavour (aside: a Bodmer-Witten's conjecture generalized) cases if the interaction between quarks is so strong that quarks are grouped together. You name a piece of small two flavour matter atom , what would we call a 3-flavour body in bulk?' God: Oh ... simply, a strange object because of strangeness.", "pages": [9]}, {"title": "3. Solid strange star in general relativity", "content": "Cold strange matter with 3-flavour symmetry could be in a solid state because of (1) a relatively small quantum wave packet of quark-cluster (wave length \u03bb q < a , where a is the separation between quark-clusters) and (2) low temperature T < (10 -1 -10 -2 ) U ( U is the interaction energy between quark-clusters). The packet scales \u03bb q \u223c h / ( mqc ) for free quark-cluster, with mq the rest mass of a quark cluster, but could be much smaller if it is constrained in a potential with depth of > /planckover2pi1 c / a /similarequal 100 MeV (2 fm / a ). A star made of strange matter would then be a solid star. It is very fundamental to study static and spherically symmetric gravitational sources in general relativity, especially for the interior solutions. The TOV solution 3 is only for perfect fluid. However, for solid strange stars, since the local press could be anisotropic in elastic matter, the radial pressure gradient could be partially balanced by the tangential shear force although a general understanding of relativistic, elastic bodies has unfortunately not been achieved 27 . The origin of this local anisotropic force in solid quark stars could be from the development of elastic energy as a star (i) spins down (its ellipticity decreases) and (ii) cools (it may shrink). Release of the elastic as well as the gravitational energies would be not negligible, and may have significant astrophysical implications. The structure of solid quark stars can be numerically calculated as following. For the sack of simplicity, only spherically symmetric sources are dealt with, in order to make sense of possible astrophysical consequence of solid quark stars. By introducing respectively radial and tangential pressures, P and P \u22a5 , the stellar equilibrium equation of static anisotropic matter in Newtonian gravity is 28 : d P / d r = -Gm ( r ) \u03c1/ r 2 + 2( P \u22a5 -P ) / r , where \u03c1 and G denote mass density and the gravitational constant, respectively, and m ( r ) = \u222b r 0 4 \u03c0 r 2 \u03c1 ( x )d x . However, in Einstein's gravity, this equilibrium equation is modified to be 29 , where P \u22a5 = (1 + /epsilon1 ) P is introduced. In case of isotropic pressure, /epsilon1 = 0, Eq. (6) turns out to be the TOV equation. It is evident from Eq. (6) that the radial pressure gradient, | d P / d r | , decreases if P \u22a5 > P , which may result in a higher maximum mass of compact stars. One can also see that a sudden decrease of P \u22a5 (equivalently of elastic force) in a star may cause substantial energy release, since the star's radius decreases and the absolute gravitational energy increases. Starquakes may result in a sudden change of /epsilon1 , with an energy release of the gravitational energy as well as the tangential strain energy. Generally, it is evident that the di ff erences of radius, gravitational energy, and moment of inertia increase proportionally to stellar mass and the parameter /epsilon1 . This means that an event should be more important for a bigger change of /epsilon1 in a quark star with higher mass. Typical energy of 10 44 \u223c 47 erg is released during superflares of SGRs, and a giant starquake with /epsilon1 /lessorsimilar 10 -4 could produce such a flare 29 . A sudden change of /epsilon1 can also result in a jump of spin frequency, \u2206\u2126 / \u2126 = -\u2206 I / I . Glitches with \u2206\u2126 / \u2126 \u223c 10 -10 \u223c-4 could occur for parameters of M = (0 . 1 \u223c 1 . 4) M /circledot and /epsilon1 = 10 -9 \u223c-4 . It is suggestive that a giantflare may accompany a high-amplitude glitch.", "pages": [9, 10]}, {"title": "4. Astrophysical manifestations of strange matter", "content": "Howto create macro-nuclei (even gigantic) in the Universe? Besides a collapse event where normal baryonic matter is intensely compressed by gravity, strange matter could also be produced after cosmic hadronization. 6 Strange matter may manifest itself as a variety of objects with a broad mass spectrum, including compact objects, cosmic rays and even dark matter.", "pages": [10]}, {"title": "4.1. Pulsar-like compact star: compressed baryonic matter after supernova", "content": "In 1932, soon after Chandrasekhar found a unique mass (the mass limit of white dwarfs), Landau speculated a state of matter, the density of which 'becomes so great that atomic nuclei come in close contact, forming one gigantic nucleus ' 2 . A star composed mostly of such matter is called a 'neutron' star, and Baade and Zwicky even suggested in 1934 that neutron stars (NSs) could be born after supernovae. NSs theoretically predicted were finally discovered when Hewish and his collaborators detected radio pulsars in 1967 30 . More kinds of pulsar-like stars, such as X-ray pulsars and X-ray bursts in binary systems, were also discovered later, and all of them are suggested to be NSs. In gigantic nucleus, protons and electrons combined to form the neutronic state, which involves weak equilibrium between protons and neutrons. However, the simple and beautiful idea proposed by Landau and others had one flaw at least: nucleons (neutrons and protons) are in fact not structureless point-like particles although they were thought to be elementary particles in 1930s. A success in the classification of hadrons discovered in cosmic rays and in accelerators leaded Gell-Mann to coin ' quark ' with fraction charges ( \u00b1 1 / 3 , \u2213 2 / 3) in mathematical description, rather than in reality 31 . All the six flavors of quarks ( u , d , c , s , t , b ) have experimental evidence (the evidence for the last one, top quark, was reported in 1995). Is weak equilibrium among u , d and s quarks possible, instead of simply that between u and d quarks? At the late stage of stellar evolution, normal baryonic matter is intensely compressed by gravity in the core of massive star during supernova. The Fermi energy of electrons are significant in CBM, and it is very essential to cancel the electrons by weak interaction in order to make lower energy state. There are two ways to kill electrons as shown in Fig 1: one is via neutronization , e -+ p \u2192 n + \u03bd e, where the fundamental degrees of freedom could be nucleons; the other is through strangenization , where the degrees of freedom are quarks. While neutronization works for removing electrons, strangenization has both the advantages of minimizing the electron's contribution of kinetic energy and maximizing the flavour number, the later could be related to the flavour symmetry of strong-interaction matter. These two ways to kill electrons are relevant to the nature of pulsar, to be neutron star or strange star, as summarized in Fig. 1. There are many speculations about the nature of pulsar due to unknown nonperturbative QCD at low energy. Among di ff erent pulsar models, hadron star and hy-", "pages": [10, 11]}, {"title": "12 Xu & Guo", "content": "brid / mixed star are conventional neutron stars, while quark star and quark-cluster star are strange stars with light flavour symmetry. In hadron star model, quarks are confined in hadrons such as neutron / proton and hyperon, while a quark star is dominated by deconfined free quarks. A hybrid / mixed star, with quark matter in its cores, is a mixture of hadronic and quark states. However, a quark-cluster star, in which strong coupling causes individual quarks grouped in clusters, is neither a hadron star nor a quark star. As an analog of neutrons, quark-clusters are bound states of several quarks, so to this point of view a quark-cluster star is more similar to a real giant nucleus of self-bound (not that of Landau), rather than a 'giant hadron' which describes traditional quark stars. Di ff erent models of pulsar's inner structure are illustrated in Fig. 2. Hybrid/mixed star: quarks de-con./con. gravity-bound It is shown in Fig. 2 that conventional neutron stars (hadron star and hybrid / mixed star) are gravity-bound, while strange stars (strange quark star and strange quark-cluster star) are self-bound on surface by strong force. This feature di ff erence is very useful to identify observationally. In the neutron star picture, the inner and outer cores and the crust keep chemical equilibrium at each boundary, so neutron star is bound by gravity. The core should have a boundary and is in equilibrium with the ordinary matter because the star has a surface composed of ordinary matter. There is, however, no clear observational evidence for a neutron star's surface, although most of authors still take it for granted that there should be ordinary matter on surface, and consequently a neutron star has di ff erent components from inner to outer parts. Being similar to traditional quark stars, quark-cluster stars have almost the same composition from the center to the surface, and the quark matter surface could be natural for understanding some di ff erent observations. It is also worth noting that, although composed of quark-clusters, quark-cluster stars are self-bound by the residual interaction between quark-clusters. This is di ff erent from but similar to the traditional MIT bag scenario. The interaction between quark-clusters could be strong enough to make condensed matter, and on the surface, the quark-clusters are just in the potential well of the interaction, leading to non-vanishing density but vanishing pressure. Observations of pulsar-like compact stars, including surface and global properties, could provide hints for the state of CBM, as discussed in the following.", "pages": [12]}, {"title": "4.1.1. Surface properties", "content": "Dirfting subpulses. Although pulsar-like stars have many di ff erent manifestations, they are populated by radio pulsars. Among the magnetospheric emission models for pulsar radio radiative process, the user-friendly nature of Ruderman-Sutherland 32 model is a virtue not shared by others, and clear drifting sub-pulses were first explained. In the seminal paper, a vacuum gap was suggested above the polar cap of a pulsar. The sparks produced by the inner-gap breakdown result in the subpulses, and the observed drifting feature is caused by E \u00d7 B . However, that model can only work in strict conditions for conventional neutron stars: strong magnetic field and low temperature on surfaces of pulsars with \u2126 \u00b7 B < 0, while calculations showed, unfortunately, that these conditions usually cannot be satisfied there. The above model encounters the so-called 'binding energy problem'. Calculations have shown that the binding energy of Fe at the neutron star surface is < 1 keV 33 , 34 , which is not su ffi cient to reproduce the vacuum gap. These problems might be alleviated within a partially screened inner gap model 35 for NSs with \u2126 \u00b7 B < 0, but could be naturally solved for any \u2126 \u00b7 B in the bare strange (quark-cluster) star scenario. The magnetospheric activity of bare quark-cluster star was investigated in quantitative details 36 . Since quarks on the surface are confined by strong color interaction, the binding energy of quarks can be even considered as infinity compared to electromagnetic interaction. As for electrons on the surface, on one hand the potential barrier of the vacuum gap prevents electrons from streaming into the magnetosphere, on the other hand the total energy of electrons on the Fermi surface is none-zero. Therefore, the binding energy of electrons is determined by the di ff erence between the height of the potential barrier in the vacuum gap and the total energy of electrons. Calculations have shown that the huge potential barrier built by the electric field in the vacuum gap above the polar cap can usually prevent electrons from streaming into the magnetosphere, unless the electric potential of a pulsar is su ffi ciently lower than that at the infinite interstellar medium. In the bare quark-cluster star model, both positively and negatively charged particles on the surface are usually bound strongly enough to form a vacuum gap above its polar cap, and the drifting (even bi-drifting) subpulses can be understood naturally 37 , 38 . X-ray spectral lines. In conventional neutron star (NS) / crusted strange star models, an atmosphere exists above the surface of a central star. Many theoretical calculations, first developed by Romani 39 , predicted the existence of atomic features in the thermal X-ray emission of NS (also for crusted strange star) atmospheres, and advanced facilities of Chandra and XMM-Newton were then proposed to be constructed for detecting those lines. One expects to know the chemical composition and magnetic field of the atmosphere through such observations, and eventually to constrain stellar mass and radius according to the redshift and pressure broadening of spectral lines. However, unfortunately, none of the expected spectral features has been detected with certainty up to now, and this negative test may hint a fundamental weakness of the NS models. Although conventional NS models cannot be completely ruled out by only nonatomic thermal spectra since modified NS atmospheric models with very strong surface magnetic fields 40 , 41 might reproduce a featureless spectrum too, a natural suggestion to understand the general observation could be that pulsars are actually bare strange (quark or quark-cluster) star 42 , almost without atoms there on the surfaces. More observations, however, did show absorption lines of PSR-like stars, and the best absorption features were detected for the central compact object (CCO) 1E 1207.4-5209 in the center of supernova remnant PKS 1209-51 / 52, at \u223c 0 . 7 keV and \u223c 1 . 4 keV 43 , 44 , 45 . Although initially these features were thought to be due to atomic transitions of ionized helium in an atmosphere with a strong magnetic field, soon thereafter it was noted that these lines might be of electron-cyclotron origin, and 1E 1207 could be a bare strange star with surface field of \u223c 10 11 G 46 . Further observations of both spectra feature 45 and precise timing 47 favor the electron-cyclotron model of 1E 1207. But this simple single particle approximation might not be reliable due to high electron density in strange stars, and Xu et al. investigated the global motion of the electron seas on the magnetized surfaces 48 . It is found that hydrodynamic surface fluctuations of the electron sea would be greatly a ff ected by the magnetic field, and an analysis shows that the seas may undergo hydrocyclotron oscillations whose eigen frequencies are given by \u03c9 ( l ) = \u03c9 c / [ l ( l + 1)], where l = 1 , 2 , 3 , ... and \u03c9 c = eB / mc is the cyclotron frequency. The fact that the absorption feature of 1E 1207.4-5209 at 0 . 7 keV is not much stronger than that at 1 . 4 keV could be understood in this hydrocyclotron oscillations model, because these two lines with l and l + 1 could have nearly equal intensity, while the strength of the first harmonic is much smaller than that of the fundamental in the electron-cyclotron model. Besides the absorption in 1E 1207.4-5209, the detected lines around (17 . 5 , 11 . 2 , 7 . 5 , 5 . 0) keV in the burst spectrum of SGR 1806-20 and those in other dead pulsars (e.g., radio quiet compact objects) would also be of hydrocyclotron origin 48 . Planck-like continue spectra. The X-ray spectra from some sources (e.g., RX J1856) are well fitted by blackbody, especially with high-energy tails surprisingly close to Wien's formula: decreasing exponentially ( \u221d e -\u03bd ). Because there is an atmosphere above the surface of neutron star / crusted strange stars, the spectrum determined by the radiative transfer in atmosphere should di ff er substantially from Planck-like one, depending on the chemical composition, magnetic field, etc. 49 Can the thermal spectrum of quark-cluster star be well described by Planck's radiation law? In bag models where quarks are nonlocal, one limitation is that bare strange stars are generally supposed to be poor radiators in thermal X-ray as a result of their high plasma frequency, \u223c 10 MeV. Nonetheless, if quarks are localized to form quark-clusters in cold quark matter due to very strong interactions, a regular lattice of the clusters (i.e., similar to a classical solid state) emerges as a consequence of the residual interaction between clusters 10 . In this latter case, the metal-like solid quark matter would induce a metal-like radiative spectrum, with which the observed thermal X-ray data of RX J1856 can be fitted 50 . Alternatively, other radiative mechanism in the electrosphere (e.g., electron bremsstrahlung in the strong electric field 51 and even of negligible ions above the sharp surface) may also reproduce a Planck-like continue spectrum. Supernova and gamma-ray bursts. It is well known that the radiation fireballs of gamma-ray bursts (GRBs) and supernovae as a whole move towards the observer with a high Lorentz factor. 52 The bulk Lorentz factor of the ultrarelativistic fireball of GRBs is estimated to be order 53 of \u0393 \u223c 10 2 -10 3 . For such an ultra-relativistic fireball, the total mass of baryons can not be too high, otherwise baryons would carry out too much energy of the central engine, the so-called 'baryon contamination'. For conventional neutron stars as the central engine, the number of baryons loaded with the fireball is unlikely to be small, since neutron stars are gravity-confined and the luminosity of fireball is extremely high. However, the baryon contamination problem can be solved naturally if the central compact objects are strange quark-cluster stars. The bare and chromatically confined surface of quark-cluster stars separates baryonic matter from the photon and lepton dominated fireball. Inside the star, baryons are in quark-cluster phase and can not escape due to strong color interaction, but e \u00b1 -pairs, photons and neutrino pairs can escape from the surface. Thus, the surface of quark-cluster stars automatically generates a low baryon condition for GRBs as well as supernovae. 54 , 55 , 56 It is still an unsolved problem to simulate supernovae successfully in the neutrinodriven explosion models of neutron stars. Nevertheless, in the quark-cluster star scenario, the bare quark surfaces could be essential for successful explosions of both core and accretion-induced collapses 57 . A nascent quark-cluster star born in the center of GRB or supernova would radiate thermal emission due to its ultrahigh surface temperature 58 , and the photon luminosity is not constrained by the Eddington limit since the surface of quark-cluster stars could be bare and chromatically confined. Therefore, in this photondriven scenario, 59 the strong radiation pressure caused by enormous thermal emissions from quark-cluster stars might play an important role in promoting core-collapse supernovae. Calculations have shown that the radiation pressure due to such strong thermal emission can push the overlying mantle away through photon-electron scattering with energy as much as \u223c 10 51 ergs. Such photon-driven mechanism in core-collapse supernovae by forming a quark-cluster star inside the collapsing core is promising to alleviate the current di ffi culty in core-collapse supernovae. The recent discovery of highly super-luminous supernova ASASSN-15lh, with a total observed energy (1 . 1 \u00b1 0 . 2) \u00d7 10 52 ergs, 60 might also be understood in this regime if a very massive strange quark-cluster star, with mass smaller than but approaching M max, forms.", "pages": [13, 14, 15]}, {"title": "4.1.2. Global properties.", "content": "Free or torque-induced precession. Rigid body precesses naturally when spinning, either freely or by torque, but fluid one can hardly. The observation of possible precession or even free precession of B1821-11 61 and others could suggest a global solid structure for pulsar-like stars. Low-mass quark stars with masses of /lessorsimilar 10 -2 M /circledot and radii of a few kilometers are gravitationally force-free, and their surfaces could then be irregular (i.e., asteroidlike). Therefore, free or torque-induced precession may easily be excited and expected with larger amplitude in low-mass quark stars. The masses of AXPs / SGRs (anomalous Xray pulsars / soft gamma-ray repeaters) could be approaching the mass-limit ( > 1 . 5 M /circledot ) in the AIQ (accretion-induced quake) model 62 ; these objects could then manifest no or weak precession as observed, though they are more likely than CCOs / DTNs (eg., RX J1856) to be surrounded by dust disks because of their higher masses (thus stronger gravity). Normal and slow glitches. Abig disadvantage that one believes that pulsars are strange quark stars lies in the fact that the observation of pulsar glitches conflicts with the hypothesis of conventional quark stars in fluid states 63 , 64 (e.g., in MIT bag models). That problem could be solved in a solid quark-cluster star model since a solid stellar object would inevitably result in star-quakes when strain energy develops to a critical value. Huge energy should be released, and thus large spin-change occurs, after a quake of a solid quark star. Star-quakes could then be a simple and intuitional mechanism for pulsars to have glitches frequently with large amplitudes. In the regime of solid quark star, by extending the model for normal glitches 65 , one can also model pulsar's slow glitches 66 not well understood in NS models. In addition, both types of glitches without (Vela-like, Type I) and with (AXP / SGR-like, Type II) X-ray enhancement could be naturally understood in the star-quake model of solid strange star, 67 since the energy release during a type I (for fast rotators) and a type II (for slow rotators) starquake are very di ff erent. Energy budget . The substantial free energy released after star-quakes, both elastic and gravitational, would power some extreme events detected in AXPs / SGRs and during GRBs. Besides persistent pulsed X-ray emission with luminosity well in excess of the spin-down power, AXPs / SGRs show occasional bursts (associated possibly with glitches), even superflares with isotropic energy \u223c 10 44 -46 erg and initial peak luminosity \u223c 10 6 -9 times of the Eddington one. They are speculated to be magnetars, with the energy reservoir of magnetic fields /greaterorsimilar 10 14 G (to be still a matter of debate about the origin since the dynamo action might not be so e ff ective and the strong magnetic field could decay e ff ectively), but failed predictions are challenges to the model. 68 However, AXPs / SGRs could also be solid quark stars with surface magnetic fields similar to that of radio pulsars. Star-quakes are responsible to both bursts / flares and glitches in the latter scenario, 62 and kinematic oscillation energy could e ff ectively power the magnetospheric activity. 69 The most conspicuous asteroseismic manifestion of solid phase of quark stars is their capability of sustaining torsional shear oscillations induced by SGR's starquake 70 . In addition, there are more and more authors who are trying to connect the GRB central engines to SGRs' flares, in order to understand di ff erent GRB light-curves observed, especially the internal-plateau X-ray emission. 71 , 72 Mass and radius of compact star . The EoS of quark-cluster matter would be sti ff er than that of nuclear matter, because (1) quark-cluster should be non-relativistic particle for its large mass, and (2) there could be strong short-distance repulsion between quark-clusters. Besides, both the problems of hyperon puzzle and quark-confinement do not exist in quarkcluster star. Sti ff EoS implies high maximum mass, while low mass is a direct consequence of self-bound surface. It has been addressed that quark-cluster stars could have high maximum masses ( > 2 M /circledot ) as well as very low masses ( < 10 -2 M /circledot ). 73 Later radio observations of PSR J16142230, a binary millisecond pulsar with a strong Shapiro delay signature, imply that the pulsar mass is 1.97 \u00b1 0.04 M /circledot 74 , which indicates a sti ff EoS for CBM. Another 2 M /circledot pulsar is also discovered afterwards 75 . It is conventionally thought that the state of dense matter softens and thus cannot result in high maximum mass if pulsars are quark stars, and that the discovery of massive 2 M /circledot pulsar may make pulsars unlikely to be quark stars. However, quark-cluster star could not be ruled out by massive pulsars, and the observations of pulsars with higher mass (e.g. > 2 . 5 M /circledot ) , would even be a strong support to quark-cluster star model, and give further constraints to the parameters. The mass and radius of 4U 1746-37 could be constrained by PRE (photospheric radius expansion) bursts, on the assumption that the touchdown flux corresponds to Eddington luminosity and the obscure e ff ect is included. 76 It turns out that 4U 1746-37 could be a strange star with small radius. There could be other observational hints of low-mass strange stars. Thermal radiation components from some PSR-like stars are detected, the radii of which are usually much smaller than 10 km in blackbody models where one fits spectral data by Planck spectrum, 77 and Pavlov and Luna 78 find no pulsations with periods longer than \u223c 0 . 68 s in the CCO of Cas A, and constrain stellar radius and mass to be R = (4 \u223c 5 . 5) km and M /lessorsimilar 0 . 8 M /circledot in hydrogen NS atmosphere models. Two kinds of e ff orts are made toward an understanding of the fact in conventional NS models. (1) The emissivity of NS's surface isn't simply of blackbody or of hydrogen-like atmospheres. The CCO in Cas A is suggested to covered by a carbon atmosphere 79 . However, the spectra from some sources (e.g., RX J1856) are still puzzling, being well fitted by blackbody. (2) The small emission areas would represent hot spots on NS's surfaces, i.e., to fit the X-ray spectra with at least two blackbodies, but this has three points of weakness in NS models. a , about P and \u02d9 P . No or very weak pulsation has been detected in some of thermal component-dominated sources (e.g., the Cas A CCO 78 ), and the inferred magnetic field from \u02d9 P seems not to be consistent with the atmosphere models at least for RX J1856 80 . b , fitting of thermal X-ray spectra (e.g., PSR J1852 + 0040) with two blackbodies finds two small emitting radii (significantly smaller than 10 km), which are not yet understood 81 . c , the blackbody temperature of the entire surface of some PSR-like stars are much lower than those predicted by the standard NS cooling models, 82 even provided that hot spots exist. Nevertheless, besides that two above, a natural idea could be that the detected small thermal regions ( if being global) of CCOs and others may reflect their small radii (and thus low masses in quark-cluster star scenario). 57 Another low-mass strange (quark-cluster) star could be 4U 1700 + 24. Because of strangeness barrier existing above a quark-cluster surface, a strange star may be surrounded by a hot corona or an atmosphere, or even a crust for di ff erent accretion rates. Both the redshifted O VIII Ly\u03b1 emission line (only z = 0 . 009) and the change in the blackbody radiation area (with an inferred scale of \u223c (10 -10 2 ) m) could naturally be understood if 4U 1700 + 24 is a low-mass quark-cluster star which exhibits weak wind accretion. 83 Additionally, the mass function via observing the G-type red giant company is only fo = (1 . 8 \u00b1 0 . 9) \u00d7 10 -5 M /circledot , 84 from which the derived mass of compact star should be much lower than 1 M /circledot unless there is geometrically fine-tuning (inclination angle i < 2 o , see Fig. 3). All these three independent observations (redshift, hot spot and mass function) may point to the fact that 4U 1700 + 24 could be a low mass strange quark-cluster star. Future observations with more advanced facilities, such as FAST and SKA, could provide more observational hints for the nature of CBM. Pulsar mass measurement could help us find more massive pulsar, while measurement of the momentum of inertia may give information on the radius. Searching sub-millisecond pulsars could be an expected way to provide clear evidence for (low-mass) quark stars. Normal neutron stars can not spin with periods less than \u223c 0 . 5 M 1 / 2 1 R -3 / 2 6 ms ( R 6 = R / 10 6 cm), as the rotation is limited by Kepler", "pages": [15, 16, 17]}, {"title": "18 Xu & Guo", "content": "frequency. But low-mass bare strange stars has no such limitation on the spin period, which could be even less than 1 ms. We need thus a much short sampling time, and would deal with then a huge amount of data in order to find a sub-millisecond pulsars. Besides, the pulse profile of pulsar is helpful for the understanding of its magnetospheric activity.", "pages": [18]}, {"title": "4.2. Strange matter in cosmic rays and as dark matter candidate", "content": "Strange quark-nuggets, in the form of cosmic rays, could be ejected during the birth of central compact star, 85 or during collision of strange stars in a binary system spiraling towards each other due to loss of orbital energy via gravitational waves. 86 A strangelet with mass per baryon < 940 MeV (i.e., binding energy per baryon /greaterorsimilar 100 MeV)could be stable in cosmic rays, and would decay finally into nucleons when collisioninduced decrease of baryon number make it unstable due to the increase of surface energy. When a stable strangelet bombards the atmosphere of the Earth, its fragmented nuggets may decay quickly into \u039b -particles by strong interaction and further into nucleons by weak interaction. What if a strange nugget made of quark clusters bombards the Earth? It is interesting and necessary to investigate. In the early Universe ( \u223c 10 \u00b5 s), quark-gluon plasma condenses to form hadron gas during the QCD phase transition. If the cosmological QCD transition is first-order, bubbles of hadron gas are nucleated and grow until they merge and fill up the whole Universe. A separation of phases during the coexistence of the hadronic and the quark phases could gather a large number of baryons in strange nuggets. 6 If quark clustering occurs, evaporation and boiling may be suppressed, and strange nuggets may survive and contribute to the dark matter today. Strange nuggets as cold quark matter may favor the formation of seed black holes in primordial halos, alleviating the current di ffi culty of quasars at redshift as high as z \u223c 6, 87 and the small pulsar glitches detected may hint the role of strange nuggets. 88", "pages": [18]}, {"title": "5. Conclusions", "content": "Although normal micro-nuclei are 2-flavour symmetric, we argue that 3-flavour symmetry would be restored in macro / gigantic-nuclei compressed by gravity during a supernova. Strange matter is conjectured to be condensed matter of 3-flavour quark-clusters, and fu- ture advanced facilities (e.g., FAST, SKA) would provide clear evidence for strange stars. Strange nuggets manifested in the form of cosmic rays and even dark matter have significant astrophysical consequences, to be tested observationally. Acknowledgements. This work is supported by the National Basic Research Program of China (No. 2012CB821801) and NNSFC (No. 11225314). The FAST FELLOWSHIP is supported by the Special Funding for Advanced Users, budgeted and administrated by Center for Astronomical Mega-Science, Chinese Academy of Sciences (CAS). We would like to thank Ms. Yong Su for reading and checking \u00a7 2.4.", "pages": [18, 19]}] |
2014CMaPh.329..919C | https://arxiv.org/pdf/1302.5321.pdf | <document>
<section_header_level_1><location><page_1><loc_22><loc_81><loc_72><loc_84></location>MINIMIZING PROPERTIES OF CRITICAL POINTS OF QUASI-LOCAL ENERGY</section_header_level_1>
<text><location><page_1><loc_26><loc_77><loc_68><loc_78></location>PO-NING CHEN, MU-TAO WANG, AND SHING-TUNG YAU</text>
<text><location><page_1><loc_18><loc_60><loc_76><loc_74></location>Abstract. In relativity, the energy of a moving particle depends on the observer, and the rest mass is the minimal energy seen among all observers. The Wang-Yau quasilocal mass for a surface in spacetime introduced in [7] and [8] is defined by minimizing quasi-local energy associated with admissible isometric embeddings of the surface into the Minkowski space. A critical point of the quasi-local energy is an isometric embedding satisfying the Euler-Lagrange equation. In this article, we prove results regarding both local and global minimizing properties of critical points of the Wang-Yau quasi-local energy. In particular, under a condition on the mean curvature vector we show a critical point minimizes the quasi-local energy locally. The same condition also implies that the critical point is globally minimizing among all axially symmetric embedding provided the image of the associated isometric embedding lies in a totally geodesic Euclidean 3-space.</text>
<section_header_level_1><location><page_1><loc_40><loc_53><loc_54><loc_54></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_12><loc_43><loc_82><loc_51></location>Let Σ be a closed embedded spacelike 2-surface in a spacetime N . We assume Σ is a topological 2-sphere and the mean curvature vector field H of Σ in N is a spacelike vector field. The mean curvature vector field defines a connection one-form of the normal bundle α H = 〈∇ N ( · ) e 3 , e 4 〉 , where e 3 = -H | H | and e 4 is the future unit timelike normal vector that is orthogonal to e 3 . Let σ be the induced metric on Σ.</text>
<text><location><page_1><loc_12><loc_26><loc_82><loc_43></location>We shall consider isometric embeddings of σ into the Minkowski space R 3 , 1 . We recall that this means an embedding X : Σ → R 3 , 1 such that the induced metric on the image is σ . Throughout this paper, we shall fix a constant unit timelike vector T 0 in R 3 , 1 . The time function τ on the image of the isometric embedding X is defined to be τ = -X · T 0 . The existence of such an isometric embedding is guaranteed by a convexity condition on σ and τ (Theorem 3.1 in [8]). The condition is equivalent to that the metric σ + dτ ⊗ dτ has positive Gaussian curvature. Let ̂ Σ be the projection of the image of X , X (Σ), onto the orthogonal complement of T 0 , a totally geodesic Euclidean 3-space in R 3 , 1 . ̂ Σ is a convex 2-surface in the Euclidean 3-space. Then the isometric embedding of Σ into R 3 , 1 with time function τ exists and is unique up to an isometry of the orthogonal complement of T 0 .</text>
<text><location><page_1><loc_12><loc_18><loc_82><loc_23></location>Part of this work was carried out while all three authors were visiting Department of Mathematics of National Taiwan University and Taida Institute for Mathematical Sciences in Taipei, Taiwan. M.-T. Wang is supported by NSF grant DMS-1105483 and S.-T. Yau is supported by NSF grants DMS-0804454 and PHY-07146468.</text>
<text><location><page_2><loc_18><loc_77><loc_88><loc_88></location>In [7] and [8], Wang and Yau define a quasi-local energy for a surface Σ with spacelike mean curvature vector H in a spacetime N , with respect to an isometric embedding X of Σ into R 3 , 1 . The definition relies on the physical data on Σ which consist of the induced metric σ , the norm of the mean curvature vector | H | > 0, and the connection one-form α H . The definition also relies on the reference data from the isometric embedding X : Σ → R 3 , 1 whose induced metric is the same as σ . In terms of τ = -X · T 0 , the quasi-local energy is defined to be 1</text>
<text><location><page_2><loc_18><loc_63><loc_88><loc_72></location>where ∇ and ∆ are the covariant derivative and Laplace operator with respect to σ , and θ is defined by sinh θ = -∆ τ | H | √ 1+ |∇ τ | 2 . Finally, ̂ H is the mean curvature of ̂ Σ in R 3 . We note that in E (Σ , τ ), the first argument Σ represents a physical surface in spacetime with the data ( σ, | H | , α H ), while the second argument τ indicates an isometric embedding of the induced metric into R 3 , 1 with time function τ with respect to the fixed T 0 .</text>
<formula><location><page_2><loc_18><loc_70><loc_84><loc_77></location>E (Σ , τ ) = ∫ ̂ Σ ̂ Hdv ̂ Σ -∫ Σ [ √ 1 + |∇ τ | 2 cosh θ | H | - ∇ τ · ∇ θ -α H ( ∇ τ ) ] dv Σ , (1.1)</formula>
<text><location><page_2><loc_18><loc_53><loc_88><loc_63></location>The Wang-Yau quasi-local mass for the surface Σ in N is defined to be the minimum of E (Σ , τ ) among all 'admissible' time functions τ (or isometric embeddings). This admissible condition is given in Definition 5.1 of [8] (see also section 3.1). This condition for τ implies that E (Σ , τ ) is non-negative if N satisfies the dominant energy condition. We recall the Euler-Lagrange equation for the functional E (Σ , τ ) of τ , which is derived in [8]. Of course, a critical point τ of E (Σ , τ ) satisfies this equation.</text>
<text><location><page_2><loc_18><loc_46><loc_88><loc_52></location>Definition 1. Given the physical data ( σ, | H | , α H ) on a 2-surface Σ . We say that a smooth function τ is a solution to the optimal embedding equation for ( σ, | H | , α H ) if the metric σ = σ + dτ ⊗ dτ can be isometrically embedded into R 3 with image Σ such that</text>
<text><location><page_2><loc_18><loc_32><loc_88><loc_42></location>where ∇ and ∆ are the covariant derivative and Laplace operator with respect to σ , θ is defined by sinh θ = -∆ τ | H | √ 1+ |∇ τ | 2 . ̂ h ab and ̂ H are the second fundamental form and the mean curvature of ̂ Σ , respectively. It is natural to ask the following questions:</text>
<formula><location><page_2><loc_18><loc_40><loc_86><loc_49></location>̂ ̂ (1.2) -( ̂ H ˆ σ ab -ˆ σ ac ˆ σ bd ˆ h cd ) ∇ b ∇ a τ √ 1 + |∇ τ | 2 + div σ ( ∇ τ √ 1 + |∇ τ | 2 cosh θ | H | - ∇ θ -α H ) = 0</formula>
<unordered_list>
<list_item><location><page_2><loc_21><loc_31><loc_72><loc_32></location>(1) How do we find solutions to the optimal embedding equation?</list_item>
<list_item><location><page_2><loc_21><loc_28><loc_88><loc_31></location>(2) Does a solution of the optimal embedding equation minimize E (Σ , τ ), either locally or globally?</list_item>
</unordered_list>
<text><location><page_2><loc_18><loc_24><loc_88><loc_27></location>Before addressing these questions, let us fix the notation on the space of isometric embeddings of (Σ , σ ) into R 3 , 1 .</text>
<text><location><page_3><loc_12><loc_80><loc_82><loc_88></location>Notation 1. T 0 is a fixed constant future timelike unit vector throughout the paper, X τ will denote an isometric embedding of σ into R 3 , 1 with time function τ = -X · T 0 . We denote the image of X τ by Σ τ , the mean curvature vector of Σ τ by H τ and the connection one-form of the normal bundle of Σ τ determined by H τ by α H τ . We denote the projection of Σ τ onto the orthogonal complement of T 0 by Σ τ and the mean curvature of Σ τ by H τ .</text>
<text><location><page_3><loc_12><loc_74><loc_82><loc_81></location>̂ ̂ ̂ In particular, when σ has positive Gauss curvature, X 0 will denote an isometric embedding of σ into the orthogonal complement of T 0 . Σ 0 denotes the image of X 0 , The mean curvature H 0 of Σ 0 can be viewed as a positive function by the assumption.</text>
<text><location><page_3><loc_12><loc_59><loc_82><loc_74></location>In [2], the authors studied the above questions at spatial or null infinity of asymptotically flat manifolds and proved that a series solution exists for the optimal embedding equation, the solution minimizes the quasi-local energy locally and the mass it achieves agrees with the ADM or Bondi mass at infinity for asymptotically flat manifolds. On the other hand, when α H is divergence free, τ = 0 is a solution to the optimal embedding equation. Miao-Tam-Xie [4] and Miao-Tam [5] studied the time-symmetric case and found several conditions such that τ = 0 is a local minimum. In particular, this holds if H 0 > | H | > 0 where H 0 is the mean curvature of the isometric embedding X 0 of Σ into R 3 (i.e. with time function τ = 0).</text>
<text><location><page_3><loc_12><loc_56><loc_82><loc_59></location>For theorems proved in this paper, we impose the following assumption on the physical surface Σ.</text>
<text><location><page_3><loc_12><loc_50><loc_82><loc_55></location>Assumption 1. Let Σ be a closed embedded spacelike 2-surface in a spacetime N satisfying the dominant energy condition. We assume that Σ is a topological 2-sphere and the mean curvature vector field H of Σ in N is a spacelike vector field.</text>
<text><location><page_3><loc_14><loc_48><loc_74><loc_49></location>We first prove the following comparison theorem among quasi-local energies.</text>
<text><location><page_3><loc_12><loc_44><loc_82><loc_47></location>Theorem 1. Suppose Σ satisfies Assumption 1 and τ 0 is a is a critical point of the quasilocal energy functional E (Σ , τ ) . Assume further that</text>
<formula><location><page_3><loc_43><loc_40><loc_51><loc_43></location>| H τ 0 | > | H |</formula>
<text><location><page_3><loc_12><loc_36><loc_82><loc_41></location>where H τ 0 is the mean curvature vector of the isometric embedding of Σ into R 3 , 1 with time function τ 0 . Then, for any time function τ such that σ + dτ ⊗ dτ has positive Gaussian curvature, we have</text>
<formula><location><page_3><loc_35><loc_33><loc_59><loc_35></location>E (Σ , τ ) ≥ E (Σ , τ 0 ) + E (Σ τ 0 , τ ) .</formula>
<text><location><page_3><loc_12><loc_31><loc_59><loc_33></location>Moreover, equality holds if and only if τ -τ 0 is a constant .</text>
<text><location><page_3><loc_12><loc_28><loc_82><loc_31></location>As a corollary of Theorem 1, we prove the following theorem about local minimizing property of an arbitrary, non-time-symmetric, solution to the optimal embedding equation.</text>
<text><location><page_3><loc_12><loc_24><loc_82><loc_27></location>Theorem 2. Suppose Σ satisfies Assumption 1 and τ 0 is a critical point of the quasi-local energy functional E (Σ , τ ) . Assume further that</text>
<formula><location><page_3><loc_41><loc_21><loc_53><loc_23></location>| H τ 0 | > | H | > 0</formula>
<text><location><page_3><loc_12><loc_18><loc_82><loc_21></location>where H τ 0 is the mean curvature vector of the isometric embedding X τ 0 of Σ into R 3 , 1 with time function τ 0 . Then, τ 0 is a local minimum for E (Σ , τ ) .</text>
<text><location><page_4><loc_18><loc_78><loc_88><loc_88></location>The special case when τ 0 = 0 was proved by Miao-Tam-Xie [4]. They estimate the second variation of quasi-local energy around the critical point τ = 0 by linearizing the optimal embedding equation near the critical point and then applying a generalization of Reilly's formula. As we allow τ 0 to be an arbitrary solution of the optimal isometric embedding equation, a different method is devised to deal with the fully nonlinear nature of the equation.</text>
<text><location><page_4><loc_18><loc_72><loc_88><loc_78></location>In general, the space of admissible isometric embeddings as solutions of a fully nonlinear elliptic system is very complicated and global knowledge of the quasi-local energy is difficult to obtained. However, we are able to prove a global minimizing result in the axially symmetric case.</text>
<text><location><page_4><loc_18><loc_66><loc_88><loc_71></location>Theorem 3. Let Σ satisfy Assumption 1. Suppose that the induced metric σ of Σ is axially symmetric with positive Gauss curvature, τ = 0 is a solution to the optimal embedding equation for Σ in N , and</text>
<formula><location><page_4><loc_47><loc_64><loc_58><loc_66></location>H 0 > | H | > 0 .</formula>
<text><location><page_4><loc_18><loc_61><loc_88><loc_64></location>Then for any axially symmetric time function τ such that σ + dτ ⊗ dτ has positive Gauss curvature,</text>
<formula><location><page_4><loc_46><loc_58><loc_60><loc_61></location>E (Σ , τ ) ≥ E (Σ , 0) .</formula>
<text><location><page_4><loc_18><loc_57><loc_61><loc_59></location>Moreover, equality holds if and only if τ is a constant .</text>
<text><location><page_4><loc_18><loc_47><loc_88><loc_56></location>This theorem will have applications in studying quasi-local energy in the Kerr spacetime, or more generally an axially symmetric spacetime. It is very likely that the global minimum of quasilocal energy of an axially symmetric datum is achieved at an axially symmetric isometric embedding into the Minkowski, though we cannot prove it at this moment. In [4], Miao, Tam and Xie described several situations where the condition H 0 > | H | holds. In particular, this includes large spheres in Kerr spacetime.</text>
<text><location><page_4><loc_18><loc_35><loc_88><loc_47></location>In section 2, we prove Theorem 1 using the nonlinear structure of the quasi-local energy. In section 3, we prove the admissibility of the time function τ for the surface Σ τ 0 in R 3 , 1 when τ is close to τ 0 . The positivity of quasi-local mass follows from the admissibility of the time function. Combining with Theorem 1, this proves Theorem 2. In section 4, we prove Theorem 3. Instead of using admissibility, we prove the necessary positivity of quasi-local energy using variation of quasi-local energy and a point-wise mean curvature inequality.</text>
<section_header_level_1><location><page_4><loc_31><loc_33><loc_75><loc_34></location>2. A comparison theorem for quasi-local energy</section_header_level_1>
<text><location><page_4><loc_20><loc_30><loc_48><loc_32></location>In this section, we prove Theorem 1.</text>
<text><location><page_4><loc_18><loc_21><loc_88><loc_29></location>Proof. We start with a metric σ and consider an isometric embedding into R 3 , 1 with time function τ 0 . The image is an embedded space-like 2-surface Σ τ 0 in R 3 , 1 . The corresponding data on Σ τ 0 are denoted as | H τ 0 | and α H τ 0 . We consider the quasi-local energy of Σ τ 0 as a physical surface in the spacetime R 3 , 1 with respect to another isometric embedding X τ into R 3 , 1 with time function τ . We recall that</text>
<formula><location><page_4><loc_18><loc_15><loc_88><loc_22></location>E (Σ τ 0 , τ ) = ∫ ̂ Σ τ ̂ Hdv ̂ Σ τ -∫ Σ [ √ 1 + |∇ τ | 2 cosh θ ( τ,τ 0 ) | H τ 0 | - ∇ τ · ∇ θ ( τ,τ 0 ) -α H τ 0 ( ∇ τ ) ] dv Σ</formula>
<text><location><page_5><loc_12><loc_84><loc_53><loc_88></location>where θ ( τ,τ 0 ) is defined by sinh θ ( τ,τ 0 ) = -∆ τ | H τ 0 | √ 1+ |∇ τ | 2 . Using E (Σ τ 0 , τ ), E (Σ , τ ) can be expressed as</text>
<formula><location><page_5><loc_12><loc_76><loc_78><loc_82></location>E (Σ , τ ) = ∫ ̂ Σ ̂ Hdv ̂ Σ -∫ Σ [ √ 1 + |∇ τ | 2 cosh θ | H | - ∇ τ · ∇ θ -α H ( ∇ τ ) ] dv Σ = E (Σ τ 0 , τ ) + A (2.1)</formula>
<text><location><page_5><loc_12><loc_72><loc_17><loc_74></location>where</text>
<formula><location><page_5><loc_12><loc_62><loc_74><loc_71></location>A = ∫ Σ [ √ 1 + |∇ τ | 2 cosh θ ( τ,τ 0 ) | H τ 0 | - ∇ τ · ∇ θ ( τ,τ 0 ) -α H τ 0 ( ∇ τ ) ] dv Σ -∫ Σ [ √ 1 + |∇ τ | 2 cosh θ | H | - ∇ τ · ∇ θ -α H ( ∇ τ ) ] dv Σ . (2.2)</formula>
<text><location><page_5><loc_12><loc_58><loc_51><loc_61></location>In the following, we shall show that A ≥ E (Σ , τ 0 ).</text>
<text><location><page_5><loc_12><loc_56><loc_82><loc_59></location>One can rewrite div σ α H and div σ α H τ 0 using the optimal embedding equation. First, τ 0 is a solution to the original optimal embedding equation. We have</text>
<text><location><page_5><loc_12><loc_52><loc_16><loc_53></location>(2.3)</text>
<formula><location><page_5><loc_13><loc_40><loc_81><loc_53></location>div σ α H = -( ̂ H ˆ σ ab -ˆ σ ac ˆ σ bd ˆ h cd ) ∇ b ∇ a τ 0 √ 1 + |∇ τ 0 | 2 + div σ [ ∇ τ 0 √ 1 + |∇ τ 0 | 2 √ H 2 + (∆ τ 0 ) 2 1 + |∇ τ 0 | 2 ] +∆ [ sinh -1 ∆ τ 0 | H | √ 1 + |∇ τ 0 | 2 ]</formula>
<text><location><page_5><loc_12><loc_35><loc_82><loc_40></location>where ˆ h ab and ̂ H are the second fundamental form and mean curvature of ̂ Σ τ 0 , respectively. On the other hand, τ = τ 0 locally minimizes E (Σ τ 0 , τ ) by the positivity of quasi-local energy. Hence,</text>
<text><location><page_5><loc_12><loc_31><loc_16><loc_33></location>(2.4)</text>
<formula><location><page_5><loc_12><loc_20><loc_82><loc_32></location>div σ α H τ 0 = -( ̂ H ˆ σ ab -ˆ σ ac ˆ σ bd ˆ h cd ) ∇ b ∇ a τ 0 √ 1 + |∇ τ 0 | 2 + div σ [ ∇ τ 0 √ 1 + |∇ τ 0 | 2 √ H 2 τ 0 + (∆ τ 0 ) 2 1 + |∇ τ 0 | 2 ] +∆ [ sinh -1 ∆ τ 0 | H τ 0 | √ 1 + |∇ τ 0 | 2 ]</formula>
<text><location><page_5><loc_12><loc_14><loc_82><loc_19></location>where ˆ h ab and ̂ H are the same as in equation (2.3) (this can be verified directly as in [9]).</text>
<text><location><page_6><loc_20><loc_86><loc_52><loc_88></location>Using equations (2.3) and (2.4), we have</text>
<text><location><page_6><loc_18><loc_65><loc_88><loc_69></location>With a new variable x = ∆ τ √ 1+ |∇ τ | 2 and x 0 = ∆ τ 0 √ 1+ |∇ τ 0 | 2 , the first six terms of the integrand is simply</text>
<text><location><page_6><loc_18><loc_56><loc_21><loc_58></location>Let</text>
<formula><location><page_6><loc_25><loc_67><loc_81><loc_87></location>A = ∫ Σ √ (1 + |∇ τ | 2 ) | H τ 0 | 2 +(∆ τ ) 2 -√ (1 + |∇ τ | 2 ) | H | 2 +(∆ τ ) 2 -∆ τ sinh -1 ( ∆ τ | H τ 0 | √ 1 + |∇ τ | 2 ) + ∆ τ sinh -1 ( ∆ τ | H | √ 1 + |∇ τ | 2 ) +∆ τ sinh -1 ( ∆ τ 0 | H τ 0 | √ 1 + |∇ τ 0 | 2 ) -∆ τ sinh -1 ( ∆ τ 0 | H | √ 1 + |∇ τ 0 | 2 ) -∇ τ 0 · ∇ τ √ 1 + |∇ τ 0 | 2 [ √ | H τ 0 | 2 + (∆ τ 0 ) 2 1 + |∇ τ 0 | 2 -√ | H | 2 + (∆ τ 0 ) 2 1 + |∇ τ 0 | 2 ] .</formula>
<formula><location><page_6><loc_24><loc_57><loc_82><loc_65></location>√ 1 + |∇ τ | 2 [ √ | H τ 0 | 2 + x 2 -√ | H | 2 + x 2 -x ( sinh -1 x | H τ 0 | -sinh -1 x | H | -sinh -1 x 0 | H τ 0 | +sinh -1 x 0 | H | ) ] .</formula>
<formula><location><page_6><loc_26><loc_49><loc_80><loc_57></location>f ( x ) = √ | H τ 0 | 2 + x 2 -√ | H | 2 + x 2 -x [ sinh -1 x | H τ 0 | -sinh -1 x | H | -sinh -1 x 0 | H τ 0 | +sinh -1 x 0 | H | ] .</formula>
<text><location><page_6><loc_18><loc_48><loc_42><loc_50></location>Direct computation shows that</text>
<formula><location><page_6><loc_28><loc_44><loc_78><loc_48></location>f ' ( x ) = sinh -1 x | H | -sinh -1 x | H τ 0 | +sinh -1 x 0 | H τ 0 | -sinh -1 x 0 | H | .</formula>
<text><location><page_6><loc_18><loc_28><loc_85><loc_41></location>A ≥ ∫ Σ ( √ 1 + |∇ τ | 2 -∇ τ 0 · ∇ τ √ 1 + |∇ τ 0 | 2 ) [ √ | H τ 0 | 2 + (∆ τ 0 ) 2 1 + |∇ τ 0 | 2 -√ | H | 2 + (∆ τ 0 ) 2 1 + |∇ τ 0 | 2 ] ≥ ∫ Σ ( 1 √ 1 + |∇ τ 0 | 2 ) [ √ | H τ 0 | 2 + (∆ τ 0 ) 2 1 + |∇ τ 0 | 2 -√ | H | 2 + (∆ τ 0 ) 2 1 + |∇ τ 0 | 2 ] The last inequality follows simply from</text>
<text><location><page_6><loc_18><loc_40><loc_88><loc_45></location>If | H τ 0 | > | H | , f ( x 0 ) = √ | H τ 0 | 2 + x 2 0 -√ | H | 2 + x 2 0 is the global minimum for f (x) and equality holds if and only if x = x 0 . Hence,</text>
<formula><location><page_6><loc_23><loc_22><loc_83><loc_28></location>( √ 1 + |∇ τ | 2 -∇ τ 0 · ∇ τ √ 1 + |∇ τ 0 | 2 ) ≥ ( √ 1 + |∇ τ | 2 -|∇ τ 0 ||∇ τ | √ 1 + |∇ τ 0 | 2 ) ≥ 1 √ 1 + |∇ τ 0 | 2</formula>
<text><location><page_6><loc_18><loc_21><loc_68><loc_24></location>and equality holds if and only if ∇ τ = ∇ τ 0 . On the other hand,</text>
<formula><location><page_6><loc_24><loc_15><loc_82><loc_23></location>E (Σ , τ 0 ) = ∫ Σ ( 1 √ 1 + |∇ τ 0 | 2 ) [ √ | H τ 0 | 2 + (∆ τ 0 ) 2 1 + |∇ τ 0 | 2 -√ | H | 2 + (∆ τ 0 ) 2 1 + |∇ τ 0 | 2 ]</formula>
<text><location><page_7><loc_12><loc_86><loc_82><loc_88></location>if one evaluates div σ α H and div σ α H τ 0 using equations (2.3) and (2.4). /square</text>
<section_header_level_1><location><page_7><loc_15><loc_81><loc_79><loc_83></location>3. Local minimizing property of critical points of quasi-local energy</section_header_level_1>
<text><location><page_7><loc_12><loc_69><loc_82><loc_80></location>In this section, we start with a metric σ and consider an isometric embedding into R 3 , 1 with time function τ 0 . The image is an embedded space-like 2-surface Σ τ 0 in R 3 , 1 . The corresponding data on Σ τ 0 are denoted as | H τ 0 | and α H τ 0 . We consider the quasi-local energy of Σ τ 0 as a physical surface in the spacetime R 3 , 1 with respect to another isometric embedding X τ into R 3 , 1 with time function τ . In this section, we prove that for τ close to τ 0 , τ is admissible with respect to the surface Σ τ 0 . This shows that, for τ close to τ 0 , we have</text>
<formula><location><page_7><loc_42><loc_65><loc_52><loc_68></location>E (Σ τ 0 , τ ) ≥ 0</formula>
<unordered_list>
<list_item><location><page_7><loc_12><loc_61><loc_82><loc_64></location>3.1. Admissible isometric embeddings. We recall definitions and theorems related to admissible isometric embeddings from [8].</list_item>
</unordered_list>
<text><location><page_7><loc_12><loc_55><loc_82><loc_59></location>Definition 2. Suppose i : Σ ↪ → N is an embedded spacelike two-surface in a spacetime N . Given a smooth function τ on Σ and a spacelike unit normal e 3 , the generalized mean curvature associated with these data is defined to be</text>
<formula><location><page_7><loc_29><loc_49><loc_65><loc_54></location>h (Σ , i, τ, e 3 ) = -√ 1 + |∇ τ | 2 〈 H,e 3 〉 -α e 3 ( ∇ τ )</formula>
<text><location><page_7><loc_12><loc_47><loc_82><loc_50></location>where, as before, H is the mean curvature vector of Σ in N and α e 3 is the connection form of the normal bundle determined by e 3 .</text>
<text><location><page_7><loc_12><loc_41><loc_82><loc_45></location>Recall, in Definition 5.1 of [8], given a physical surface Σ in spacetime N with induced metric σ and mean curvature vector H , there are three conditions for a function τ to be admissible.</text>
<unordered_list>
<list_item><location><page_7><loc_15><loc_37><loc_63><loc_40></location>(1) The metric, σ + dτ ⊗ dτ , has positive Gaussian curvature.</list_item>
<list_item><location><page_7><loc_15><loc_35><loc_82><loc_38></location>(2) Σ bounds a hypersurface Ω in N where Jang's equation with Dirichlet boundary condition τ is solvable on Ω . Let the solution be f .</list_item>
<list_item><location><page_7><loc_15><loc_32><loc_82><loc_35></location>(3) The generalized mean curvature h (Σ , i, τ, e ' 3 ) is positive where e ' 3 is determined by the solution f of Jang's equation as follows:</list_item>
</unordered_list>
<formula><location><page_7><loc_38><loc_29><loc_56><loc_30></location>e ' 3 = cosh θe 3 +sinh θe 4</formula>
<text><location><page_7><loc_18><loc_23><loc_82><loc_27></location>where sinh θ = e 3 ( f ) √ 1+ |∇ τ | 2 and e 3 is the outward unit spacelike normal of Σ in Ω, e 4 is the future timelike unit normal of Ω in N .</text>
<text><location><page_7><loc_12><loc_18><loc_82><loc_22></location>We recall that from [8] if τ corresponds to an admissible isometric embedding and Σ is a 2-surface in spacetime N that satisfies Assumption 1, then the quasi-local energy E (Σ , τ ) is non-negative.</text>
<text><location><page_8><loc_18><loc_78><loc_88><loc_88></location>3.2. Solving Jang's equation. Let (Ω , g ij ) be a Riemannian manifold with boundary ∂ Ω = Σ. Let p ij be a symmetric 2-tensor on Ω. The Jang's equation asks for a hypersurface ˜ Ω in Ω × R , defined as a graph of a function f over Ω, such that the mean curvature of ˜ Ω is the same as the trace of the restriction of p to ˜ Ω . In a local coordinate x i on Ω, Jang's equation takes the following form:</text>
<text><location><page_8><loc_18><loc_65><loc_88><loc_73></location>where D is the covariant derivative with respect to the metric g ij . When p ij = 0, Jang's equation becomes the equation for minimal graph. Equation of of minimal surface type may have blow-up solutions. In [6], it is shown by Schoen-Yau that solutions of Jang's equation can only blow-up at marginally trapped surface in Ω. Namely, surfaces S in Ω such that</text>
<formula><location><page_8><loc_35><loc_72><loc_71><loc_78></location>3 ∑ i,j =1 ( g ij -f i f j 1 + | Df | 2 )( D i D j f √ 1 + | Df | 2 -p ij ) = 0</formula>
<formula><location><page_8><loc_47><loc_62><loc_59><loc_64></location>H S ± tr S p = 0 .</formula>
<text><location><page_8><loc_18><loc_59><loc_88><loc_62></location>Following the analysis of Jang's equation in [6] and [8], we prove the following theorem for the existence of solution to the Dirichlet problem of Jang's equation.</text>
<text><location><page_8><loc_18><loc_53><loc_88><loc_57></location>Theorem 4. Let (Ω , g ij ) be a Riemannian manifold with boundary ∂ Ω = Σ , p ij be a symmetric 2-tensor on Ω , and τ be a function on Σ . Then Jang's equation with Dirichlet boundary data τ is solvable on Ω if</text>
<formula><location><page_8><loc_48><loc_49><loc_58><loc_52></location>H Σ > | tr Σ p | ,</formula>
<text><location><page_8><loc_18><loc_47><loc_58><loc_49></location>and there is no marginally trapped surface inside Ω .</text>
<text><location><page_8><loc_18><loc_43><loc_88><loc_46></location>Following the approach in [6] and Section 4.3 of [8], it suffices to control the boundary gradient of the solution to Jang's equation. We have the following theorem:</text>
<text><location><page_8><loc_18><loc_39><loc_88><loc_42></location>Theorem 5. The normal derivative of a solution of the Dirichlet problem of Jang's equation is bounded if</text>
<formula><location><page_8><loc_48><loc_36><loc_58><loc_38></location>H Σ > | tr Σ p | .</formula>
<text><location><page_8><loc_18><loc_31><loc_88><loc_35></location>Remark 1. In [1] , Andersson, Eichmair and Metzger proved a similar result about boundedness of boundary gradient of solutions to Jang's equation in order to study existence of marginally trapped surfaces.</text>
<text><location><page_8><loc_18><loc_23><loc_88><loc_29></location>Proof. We follow the approach used in Theorem 4.2 of [8] but with a different form for the sub and super solutions. Let g ij and p ij be the induced metric and second fundamental form of the hypersurface Ω. Let Σ be the boundary of Ω with induced metric σ and mean curvature H Σ in Ω. We consider the following operator for Jang's equation.</text>
<formula><location><page_8><loc_34><loc_15><loc_72><loc_22></location>Q ( f ) = 3 ∑ i,j =1 ( g ij -f i f j 1 + | Df | 2 )( D i D j f √ 1 + | Df | 2 -p ij ) .</formula>
<text><location><page_9><loc_12><loc_44><loc_33><loc_46></location>As /epsilon1 approaches 0, we have</text>
<text><location><page_9><loc_12><loc_85><loc_82><loc_88></location>We extend the boundary data τ to the interior of the hypersurface Ω. We still denote the extension by τ . Consider the following test function</text>
<formula><location><page_9><loc_42><loc_81><loc_52><loc_84></location>f = Ψ( d ) /epsilon1 + τ</formula>
<text><location><page_9><loc_12><loc_79><loc_65><loc_80></location>where d is the distance function to the boundary of Ω. We compute</text>
<formula><location><page_9><loc_31><loc_71><loc_63><loc_78></location>D i f = 1 /epsilon1 Ψ ' d i + D i τ D i D j f = 1 /epsilon1 (Ψ '' d i d j +Ψ ' D i D j d ) + D i D j τ</formula>
<text><location><page_9><loc_12><loc_69><loc_20><loc_71></location>Therefore,</text>
<formula><location><page_9><loc_19><loc_54><loc_75><loc_69></location>Q ( f ) =( g ij -f i f j 1 + | Df | 2 )( D i D j f √ 1 + | Df | 2 -p ij ) = 1 /epsilon1 ( g ij -f i f j 1 + | Df | 2 ) Ψ '' d i d j √ 1 + | Df | 2 + 1 /epsilon1 ( g ij -f i f j 1 + | Df | 2 ) Ψ ' D i D j d √ 1 + | Df | 2 +( g ij -f i f j 1 + | Df | 2 ) D i D j τ √ 1 + | Df | 2 +( g ij -f i f j 1 + | Df | 2 ) p ij</formula>
<text><location><page_9><loc_12><loc_52><loc_82><loc_55></location>At the boundary of Ω, it is convenient to use a frame { e a , e 3 } where e a are tangent to the boundary and e 3 is normal to the boundary.</text>
<formula><location><page_9><loc_40><loc_47><loc_54><loc_52></location>D 3 f = 1 /epsilon1 Ψ ' + D n τ D a f = D a τ.</formula>
<formula><location><page_9><loc_35><loc_38><loc_59><loc_44></location>/epsilon1 √ 1 + | Df | 2 = | Ψ ' | + O ( /epsilon1 ) ( g ij f i f j 1 + Df 2 ) = σ ab + O ( /epsilon1 ) .</formula>
<formula><location><page_9><loc_38><loc_37><loc_47><loc_40></location>-| |</formula>
<text><location><page_9><loc_12><loc_35><loc_62><loc_37></location>Moreover, the distance function d to the boundary of Ω satisfies</text>
<formula><location><page_9><loc_31><loc_33><loc_63><loc_35></location>σ ab d a d b = 0 and σ ab D a D b d = H Σ .</formula>
<text><location><page_9><loc_14><loc_30><loc_33><loc_32></location>Hence, we conclude that</text>
<formula><location><page_9><loc_35><loc_25><loc_59><loc_30></location>Q ( f ) = Ψ ' | Ψ ' | H Σ + tr Σ p + O ( /epsilon1 )</formula>
<text><location><page_9><loc_12><loc_22><loc_82><loc_25></location>As a result, sub and super solutions exist when H Σ > | tr Σ p | . One can then use Perron method to find the solution between the sub and super solution. /square</text>
<text><location><page_9><loc_12><loc_18><loc_82><loc_21></location>Remark 2. Here we proved the result when the dimension of Ω is 3. The result holds in higher dimension as well.</text>
<section_header_level_1><location><page_10><loc_18><loc_86><loc_40><loc_88></location>3.3. Proof of Theorem 2.</section_header_level_1>
<text><location><page_10><loc_18><loc_82><loc_88><loc_86></location>Proof. It suffices to show that, for τ close to τ 0 , τ is admissible with respect to Σ τ 0 in R 3 , 1 if H τ 0 is spacelike.</text>
<text><location><page_10><loc_18><loc_79><loc_88><loc_82></location>For such a τ , the Gauss curvature of σ + dτ ⊗ dτ is close to that of σ + dτ 0 ⊗ dτ 0 . In particular, it remains positive. This verifies the first condition for admissibility.</text>
<text><location><page_10><loc_18><loc_72><loc_88><loc_79></location>We apply Theorem 4 in the case where Σ bounds a spacelike hypersurface Ω in R 3 , 1 and g ij and p ij be the induced metric and second fundamental form on Ω, respectively. Moreover, the projection of Σ onto the orthogonal complement of T 0 is convex since the Gauss curvature of σ + dτ ⊗ dτ is positive.</text>
<text><location><page_10><loc_18><loc_64><loc_88><loc_72></location>That the mean curvature of Σ is spacelike implies that | H Σ | > | tr Σ p | . Moreover, that the projection ̂ Σ is convex implies H Σ > 0. It follows that H Σ > | tr Σ p | . We recall there is no marginally trapped surface in R 3 , 1 (see for example, [3]). As a result, Jang's equation with the Dirichlet boundary data τ is solvable as long as Σ = ∂ Ω has spacelike mean curvature vector. This verifies the second condition for admissibility.</text>
<text><location><page_10><loc_18><loc_61><loc_88><loc_64></location>To verify the last condition, it suffices to check for τ 0 since it is an open condition. Namely, it suffices to prove that</text>
<formula><location><page_10><loc_45><loc_58><loc_61><loc_60></location>h (Σ , X τ 0 , τ 0 , e ' 3 ) > 0 .</formula>
<text><location><page_10><loc_18><loc_50><loc_88><loc_58></location>Lemma 4 in the appendix implies that the generalized mean curvature h (Σ , X τ 0 , τ 0 , e ' 3 ) is the same as the generalized mean curvature h (Σ , X τ 0 , τ 0 , ˘ e 3 (Σ τ 0 )) where ˘ e 3 (Σ τ 0 ) is the vector field on Σ τ 0 obtained by parallel translation of the outward unit normal of ̂ Σ τ 0 along T 0 . The last condition for τ 0 now follows from Proposition 3.1 of [8], which states that for ˘ e 3 (Σ τ 0 ),</text>
<formula><location><page_10><loc_37><loc_43><loc_88><loc_50></location>h (Σ , X τ 0 , τ 0 , ˘ e 3 (Σ τ 0 )) = ̂ H √ 1 + |∇ τ 0 | 2 > 0 . /square</formula>
<section_header_level_1><location><page_10><loc_31><loc_41><loc_75><loc_43></location>4. Global minimum in the axially symmetric case</section_header_level_1>
<text><location><page_10><loc_18><loc_36><loc_88><loc_40></location>In proving Theorem 3, we need three Lemmas concerning a spacelike 2-surface Σ τ in R 3 , 1 with time function τ . First we introduce a new energy functional which depends on a physical gauge.</text>
<text><location><page_10><loc_18><loc_30><loc_88><loc_35></location>Definition 3. Let Σ be closed embedded spacelike 2-surface in spacetime N with induced metric σ . Let e 3 be a spacelike normal vector field along Σ in N . For any f such that the isometric embedding of σ into R 3 , 1 with time function f exists, we define</text>
<formula><location><page_10><loc_18><loc_24><loc_74><loc_30></location>(4.1) ˜ E (Σ , e 3 , f ) = ∫ ̂ Σ f ̂ H f dv ̂ Σ f -∫ Σ h (Σ , i, f, e 3 ) dv Σ where h (Σ , i, f, e 3 ) is the generalized mean curvature (see Definition 2).</formula>
<text><location><page_10><loc_18><loc_20><loc_88><loc_23></location>This functional is less nonlinear than E (Σ , f ). Provided the mean curvature vector H of Σ is spacelike, the following relation holds</text>
<formula><location><page_10><loc_43><loc_17><loc_63><loc_19></location>E (Σ , f ) = ˜ E (Σ , e can 3 ( f ) , f )</formula>
<text><location><page_11><loc_12><loc_86><loc_38><loc_88></location>where e can 3 ( f ) is chosen such that</text>
<formula><location><page_11><loc_35><loc_78><loc_59><loc_84></location>〈 H,e can 4 ( f ) 〉 = -∆ f | H | √ 1 + |∇ f | 2</formula>
<text><location><page_11><loc_12><loc_75><loc_82><loc_78></location>In addition, the first variation of ˜ E (Σ , e 3 , f ) with respect to f can be computed as in [8]:</text>
<formula><location><page_11><loc_22><loc_67><loc_72><loc_73></location>( ̂ H ˆ σ ab -ˆ σ ac ˆ σ bd ˆ h cd ) ∇ b ∇ a f √ 1 + |∇ f | 2 + div σ ( 〈 H,e 3 〉∇ f √ 1 + |∇ f | 2 ) + div σ α e 3 .</formula>
<text><location><page_11><loc_12><loc_54><loc_82><loc_57></location>Lemma 1. For a spacelike 2-surface Σ τ in R 3 , 1 with time function τ , f = τ is a critical point of the functional ˜ E (Σ τ , ˘ e 3 (Σ τ ) , f ) .</text>
<text><location><page_11><loc_12><loc_55><loc_82><loc_67></location>where ˆ h ab and ̂ H are the second fundamental form and mean curvature of ̂ Σ f , respectively. We recall that for Σ τ , assuming the projection onto the orthonormal complement of T 0 is an embedded surface, there is a unique outward normal spacelike unit vector field ˘ e 3 (Σ τ ) which is orthogonal to T 0 . Indeed, ˘ e 3 (Σ τ ) can be obtained by parallel translating the unit outward normal vector of ̂ Σ τ , ˆ ν , along T 0 .</text>
<text><location><page_11><loc_12><loc_50><loc_56><loc_52></location>Proof. The first variation of ˜ E (Σ τ , ˘ e 3 (Σ τ ) , f ) at f = τ is</text>
<formula><location><page_11><loc_23><loc_42><loc_71><loc_48></location>( ̂ H ˆ σ ab -ˆ σ ac ˆ σ bd ˆ h cd ) ∇ b ∇ a τ √ 1 + |∇ τ | 2 + div σ ( h ∇ τ √ 1 + |∇ τ | 2 ) + div σ α.</formula>
<text><location><page_11><loc_12><loc_33><loc_82><loc_42></location>where h = 〈 H τ , ˘ e 3 (Σ τ ) 〉 and α = α ˘ e 3 (Σ τ ) are data on Σ τ with respect to the gauge ˘ e 3 (Σ τ ) and ˆ h ab and ̂ H are the second fundamental form and mean curvature of ̂ Σ τ . Denote the covariant derivative on ̂ Σ with respect to the induced metric ˆ σ by ˆ ∇ . We wish to show that the first variation is 0. Recall from [8], we have</text>
<formula><location><page_11><loc_39><loc_26><loc_55><loc_32></location>̂ H = -h -α ( ∇ τ ) 1 + |∇ τ | 2 .</formula>
<formula><location><page_11><loc_19><loc_17><loc_75><loc_21></location>ˆ σ ab = σ ab -τ a τ b 1 + |∇ τ | 2 , ˆ σ ab ˆ ∇ a τ = τ b 1 + |∇ τ | 2 , and ˆ ∇ a ˆ ∇ b τ = ∇ a ∇ b τ 1 + |∇ τ | 2 .</formula>
<text><location><page_11><loc_12><loc_20><loc_82><loc_26></location>Moreover, we have the following relation between metric and covariant derivative of Σ τ and ̂ Σ τ :</text>
<text><location><page_12><loc_20><loc_85><loc_87><loc_88></location>In addition, for a tangent vector field W a , ˆ ∇ a W a = ∇ a W a + ( ∇ b ∇ c τ ) τ c W b 1+ |∇ τ | 2 . As a result,</text>
<formula><location><page_12><loc_20><loc_79><loc_44><loc_85></location>( ̂ H ˆ σ ab -ˆ σ ac ˆ σ bd ˆ h cd ) ∇ b ∇ a τ √ 1 + |∇ τ | 2</formula>
<formula><location><page_12><loc_20><loc_69><loc_50><loc_75></location>-( ̂ H ˆ σ ab -ˆ σ ac ˆ σ bd ˆ h cd ) ˆ ∇ a τ τ e ∇ b ∇ e τ √ 1 + |∇ τ | 2 b [( H ˆ σ ab ˆ σ ac ˆ σ bd ˆ h cd ) 1 + τ 2 ˆ a</formula>
<formula><location><page_12><loc_18><loc_73><loc_89><loc_82></location>= ˆ ∇ b [( ̂ H ˆ σ ab -ˆ σ ac ˆ σ bd ˆ h cd ) √ 1 + |∇ τ | 2 ˆ ∇ a τ ] -( ̂ H ˆ σ ab -ˆ σ ac ˆ σ bd ˆ h cd )( ˆ ∇ a τ )( ˆ ∇ b √ 1 + |∇ τ | 2 ) = ∇ b [( ̂ H ˆ σ ab -ˆ σ ac ˆ σ bd ˆ h cd ) √ 1 + |∇ τ | 2 ˆ ∇ a τ ] + τ e ( ∇ b ∇ e τ ) 1 + |∇ τ | 2 ( ̂ H ˆ σ ab -ˆ σ ac ˆ σ bd ˆ h cd ) √ 1 + |∇ τ | 2 ˆ ∇ a τ</formula>
<text><location><page_12><loc_18><loc_69><loc_19><loc_70></location>=</text>
<text><location><page_12><loc_20><loc_68><loc_21><loc_70></location>∇</text>
<text><location><page_12><loc_18><loc_67><loc_23><loc_68></location>Hence,</text>
<text><location><page_12><loc_24><loc_66><loc_25><loc_71></location>̂</text>
<text><location><page_12><loc_38><loc_67><loc_39><loc_72></location>√</text>
<text><location><page_12><loc_18><loc_48><loc_44><loc_49></location>On the other hand, by definition,</text>
<text><location><page_12><loc_18><loc_41><loc_23><loc_43></location>Hence,</text>
<text><location><page_12><loc_18><loc_35><loc_30><loc_36></location>This shows that</text>
<text><location><page_12><loc_28><loc_68><loc_29><loc_70></location>-</text>
<text><location><page_12><loc_42><loc_68><loc_44><loc_70></location>|∇</text>
<text><location><page_12><loc_45><loc_68><loc_46><loc_70></location>|</text>
<text><location><page_12><loc_47><loc_68><loc_48><loc_70></location>∇</text>
<text><location><page_12><loc_49><loc_69><loc_50><loc_70></location>τ</text>
<text><location><page_12><loc_50><loc_69><loc_50><loc_70></location>]</text>
<text><location><page_12><loc_50><loc_69><loc_51><loc_70></location>.</text>
<formula><location><page_12><loc_28><loc_48><loc_78><loc_66></location>( ̂ H ˆ σ ab -ˆ σ ac ˆ σ bd ˆ h cd ) ∇ b ∇ a τ √ 1 + |∇ τ | 2 + div Σ ( h ∇ τ √ 1 + |∇ τ | 2 ) + div σ α = ∇ b [( ̂ H ˆ σ ab -ˆ σ ac ˆ σ bd ˆ h cd ) √ 1 + |∇ τ | 2 ˆ ∇ a τ + hτ b √ 1 + |∇ τ | 2 + α b ] = ∇ b [ -ˆ σ ac ˆ σ bd ˆ h cd √ 1 + |∇ τ | 2 ˆ ∇ a τ -τ b 1 + |∇ τ | 2 α ( ∇ τ ) + α b ] = ∇ b [ -ˆ σ bd ˆ h cd τ c √ 1 + |∇ τ | 2 -τ b 1 + |∇ τ | 2 α ( ∇ τ ) + α b ] .</formula>
<formula><location><page_12><loc_33><loc_41><loc_73><loc_47></location>α a = 1 √ 1 + |∇ τ | 2 〈∇ ∂ ∂x a ˆ ν, T 0 + ∇ τ 〉 = ˆ h ac τ c √ 1 + |∇ τ | 2 .</formula>
<formula><location><page_12><loc_22><loc_34><loc_84><loc_41></location>-τ b 1 + |∇ τ | 2 α ( ∇ τ ) + α b = σ ab ˆ h ac τ c √ 1 + |∇ τ | 2 -τ b 1 + |∇ τ | 2 ˆ h ac τ a τ c √ 1 + |∇ τ | 2 = ˆ σ ab ˆ h ac τ c √ 1 + |∇ τ | 2 .</formula>
<text><location><page_12><loc_18><loc_29><loc_48><loc_31></location>and completes the proof of the lemma.</text>
<formula><location><page_12><loc_37><loc_29><loc_69><loc_35></location>-ˆ σ bd ˆ h cd τ c √ 1 + |∇ τ | 2 -τ b 1 + |∇ τ | 2 α ( ∇ τ ) + α b = 0</formula>
<text><location><page_12><loc_87><loc_30><loc_88><loc_31></location>/square</text>
<text><location><page_12><loc_18><loc_20><loc_88><loc_28></location>Lemma 2. Let X sτ , 0 ≤ s ≤ 1 be a family of isometric embeddings of σ into R 3 , 1 with time function sτ . Suppose Σ 0 , the image of X 0 , lies in a totally geodesic Euclidean 3-space, E 3 , and Σ sτ , the image of X sτ , projects to an embedded surface in E 3 for 0 ≤ s ≤ 1 . Assume further that Σ 0 is mean convex and H 2 0 ≥ 〈 H sτ , H sτ 〉 for 0 ≤ s ≤ 1 . Regarding Σ 0 as a physical surface in the spacetime R 3 , 1 , then</text>
<formula><location><page_12><loc_48><loc_17><loc_58><loc_19></location>E (Σ 0 , τ ) ≥ 0 .</formula>
<text><location><page_13><loc_12><loc_85><loc_82><loc_88></location>Proof. Instead of proving admissibility of τ , we will use variation of the quasi-local energy and the point-wise inequality of the mean curvatures.</text>
<text><location><page_13><loc_14><loc_83><loc_29><loc_85></location>Hence, we consider</text>
<formula><location><page_13><loc_40><loc_80><loc_54><loc_82></location>F ( s ) = E (Σ 0 , sτ )</formula>
<text><location><page_13><loc_12><loc_76><loc_82><loc_79></location>for 0 ≤ s ≤ 1. F (0) = 0 and we shall prove that F (1) is non-negative by deriving a differential inequality for F ( s ).</text>
<text><location><page_13><loc_14><loc_74><loc_17><loc_75></location>Let</text>
<text><location><page_13><loc_12><loc_65><loc_61><loc_68></location>For any 0 ≤ s 0 ≤ 1, G ( s ) is related to ˜ E (Σ s 0 τ , ˘ e 3 (Σ s 0 τ ) , sτ ) by (4.2)</text>
<formula><location><page_13><loc_38><loc_67><loc_56><loc_74></location>G ( s ) = ∫ ̂ Σ sτ ̂ H sτ dv ̂ Σ sτ .</formula>
<formula><location><page_13><loc_12><loc_60><loc_85><loc_66></location>G ( s ) = ˜ E (Σ s 0 τ , ˘ e 3 (Σ s 0 τ ) , sτ )+ ∫ Σ s 0 τ ( -〈 H s 0 τ , ˘ e 3 (Σ s 0 τ ) 〉 √ 1 + |∇ sτ | 2 -α ˘ e 3 (Σ s 0 τ ) ( ∇ sτ ) ) dv Σ s 0 τ .</formula>
<text><location><page_13><loc_14><loc_58><loc_45><loc_59></location>As a consequence of Lemma 1, we have</text>
<text><location><page_13><loc_12><loc_50><loc_26><loc_51></location>By equation (4.2),</text>
<formula><location><page_13><loc_34><loc_50><loc_60><loc_57></location>∂ ∂s ˜ E (Σ s 0 τ , ˘ e 3 (Σ s 0 τ ) , sτ ) ∣ ∣ ∣ s = s 0 = 0 .</formula>
<formula><location><page_13><loc_17><loc_37><loc_77><loc_49></location>G ' ( s 0 ) = ∫ Σ s 0 τ ( -〈 H s 0 τ , ˘ e 3 (Σ s 0 τ ) 〉 s 0 |∇ τ | 2 √ 1 + s 2 0 |∇ τ | 2 -α ˘ e 3 (Σ s 0 τ ) ( ∇ τ ) ) dv Σ s 0 τ = G ( s 0 ) s 0 -1 s 0 ∫ Σ s 0 τ 1 √ 1 + s 2 0 |∇ τ | 2 √ 〈 H s 0 τ , H s 0 τ 〉 + ( s 0 ∆ τ ) 2 1 + | s 0 ∇ τ | 2 dv Σ s 0 τ .</formula>
<text><location><page_13><loc_12><loc_36><loc_51><loc_37></location>Recall that by the definition of quasi-local energy:</text>
<formula><location><page_13><loc_13><loc_28><loc_81><loc_35></location>F ( s ) = G ( s ) -∫ Σ 0 ( √ (1 + | s ∇ τ | 2 ) H 2 0 +( s ∆ τ ) 2 -s ∆ τ sinh -1 ( s ∆ τ H 0 √ 1 + | s ∇ τ | 2 ) ) dv Σ 0 .</formula>
<text><location><page_13><loc_14><loc_27><loc_49><loc_28></location>We write the integrand of the last integral as</text>
<formula><location><page_13><loc_16><loc_19><loc_78><loc_26></location>√ 1 + s 2 |∇ τ | 2 (√ H 2 0 + ( s ∆ τ ) 2 1 + | s ∇ τ | 2 -( s ∆ τ ) √ 1 + | s ∇ τ | 2 sinh -1 ( s ∆ τ H 0 √ 1 + | s ∇ τ | 2 ) ) .</formula>
<text><location><page_13><loc_12><loc_18><loc_66><loc_19></location>Differentiate this expression with respect to the variable s , we obtain</text>
<formula><location><page_14><loc_22><loc_65><loc_84><loc_87></location>s |∇ τ | 2 √ 1 + s 2 |∇ τ | 2 ( √ H 2 0 + ( s ∆ τ ) 2 1 + | s ∇ τ | 2 -( s ∆ τ ) √ 1 + | s ∇ τ | 2 sinh -1 ( s ∆ τ H 0 √ 1 + | s ∇ τ | 2 ) ) -√ 1 + s 2 |∇ τ | 2 [ s ∆ τ (1 + | s ∇ τ | 2 ) 3 / 2 sinh -1 ( s ∆ τ H 0 √ 1 + | s ∇ τ | 2 ) ] = 1 s [ √ (1 + | s ∇ τ | 2 ) H 0 2 +( s ∆ τ ) 2 -s ∆ τ sinh -1 ( s ∆ τ H 0 √ 1 + | s ∇ τ | 2 ) ] -1 s 1 √ 1 + s 2 |∇ τ | 2 √ H 2 0 + ( s ∆ τ ) 2 1 + | s ∇ τ | 2 .</formula>
<formula><location><page_14><loc_24><loc_56><loc_82><loc_64></location>F ' ( s ) = F ( s ) s + 1 s ∫ Σ 0 √ H 2 0 + ( s ∆ τ ) 2 1+ | s ∇ τ | 2 -√ 〈 H s τ , H s τ 〉 + ( s ∆ τ ) 2 1+ | s ∇ τ | 2 √ 1 + s 2 |∇ τ | 2 dv Σ 0 .</formula>
<text><location><page_14><loc_18><loc_64><loc_88><loc_67></location>Since the induced metrics on Σ s 0 τ and Σ 0 are the same, we can evaluate all integrals on the surface Σ 0 . This leads to</text>
<text><location><page_14><loc_20><loc_55><loc_82><loc_57></location>The assumption H 2 0 ≥ 〈 H sτ , H sτ 〉 implies the last term is non-negative and thus</text>
<formula><location><page_14><loc_48><loc_52><loc_58><loc_55></location>F ' ( s ) ≥ F ( s ) s .</formula>
<text><location><page_14><loc_18><loc_48><loc_88><loc_51></location>As F (0) = F ' (0) = 0, the positivity of F ( s ) follows from a simple comparison result for ordinary differential equation. /square</text>
<text><location><page_14><loc_20><loc_46><loc_64><loc_47></location>The last lemma specializes to axially symmetric metrics.</text>
<text><location><page_14><loc_18><loc_37><loc_88><loc_45></location>Lemma 3. Suppose the isometric embedding X 0 of an axially symmetric metric σ = P 2 dθ 2 + Q 2 sin 2 θdφ 2 into R 3 is given by the coordinates ( u sin φ, u cos φ, v ) where P , Q , u , and v are functions of θ . Let τ = τ ( θ ) be an axially symmetric function and X τ be the isometric embedding of σ in R 3 , 1 with time function τ . The following identity holds for the mean curvature vector H τ of Σ τ in R 3 , 1 .</text>
<formula><location><page_14><loc_40><loc_32><loc_66><loc_36></location>〈 H τ , H τ 〉 = H 2 0 -( v θ ∆ τ -τ θ ∆ v ) 2 v 2 θ + τ 2 θ</formula>
<text><location><page_14><loc_18><loc_30><loc_47><loc_32></location>where ∆ is the Laplace operator of σ .</text>
<text><location><page_14><loc_18><loc_25><loc_90><loc_29></location>Proof. The isometric embedding for an axially symmetric metric is reduced to solving ordinary differential equations. The isometric embedding of σ into R 3 is given by ( u sin φ, u cos φ, v ) where</text>
<formula><location><page_14><loc_37><loc_23><loc_69><loc_25></location>u 2 = Q 2 sin 2 θ and v 2 θ + u 2 θ = P 2 .</formula>
<text><location><page_14><loc_18><loc_19><loc_88><loc_23></location>The isometric embedding of the metric σ + dτ ⊗ dτ into R 3 is given by ( u sin φ, u cos φ, ˜ v ) where</text>
<formula><location><page_14><loc_46><loc_17><loc_60><loc_19></location>˜ v 2 θ + u 2 θ = P 2 + τ 2 θ .</formula>
<text><location><page_15><loc_12><loc_86><loc_16><loc_88></location>Thus,</text>
<text><location><page_15><loc_12><loc_82><loc_49><loc_83></location>Differentiating one more time with respect to θ ,</text>
<text><location><page_15><loc_12><loc_75><loc_22><loc_76></location>and therefore</text>
<formula><location><page_15><loc_41><loc_82><loc_53><loc_87></location>˜ v θ = √ v 2 θ + τ 2 θ .</formula>
<formula><location><page_15><loc_39><loc_75><loc_55><loc_81></location>˜ v θθ = v θ v θθ + τ θ τ θθ √ v 2 θ + τ 2 θ ,</formula>
<formula><location><page_15><loc_35><loc_69><loc_59><loc_75></location>∆˜ v = 1 √ v 2 θ + τ 2 θ ( v θ ∆ v + τ θ ∆ τ ) .</formula>
<text><location><page_15><loc_12><loc_69><loc_38><loc_70></location>For the mean curvature, we have</text>
<formula><location><page_15><loc_25><loc_63><loc_69><loc_68></location>H 2 0 =(∆( u sin φ )) 2 +(∆( u cos φ )) 2 +(∆ v ) 2 , and 〈 H τ , H τ 〉 =(∆( u sin φ )) 2 +(∆( u cos φ )) 2 +(∆˜ v ) 2 -(∆ τ ) 2 .</formula>
<text><location><page_15><loc_12><loc_62><loc_55><loc_63></location>Taking the difference and completing square, we obtain</text>
<formula><location><page_15><loc_31><loc_56><loc_63><loc_61></location>H 2 0 -〈 H τ , H τ 〉 =(∆ v ) 2 +(∆ τ ) 2 -(∆˜ v ) 2 = 1 .</formula>
<formula><location><page_15><loc_45><loc_55><loc_62><loc_58></location>v 2 θ + τ 2 θ [ v θ ∆ τ -τ θ ∆ v ] 2</formula>
<text><location><page_15><loc_81><loc_54><loc_82><loc_55></location>/square</text>
<text><location><page_15><loc_14><loc_51><loc_45><loc_52></location>Let's recall the statement of Theorem 3.</text>
<text><location><page_15><loc_12><loc_44><loc_82><loc_49></location>Theorem 3 Let Σ satisfy Assumption 1. Suppose that the induced metric σ of Σ is axially symmetric with positive Gauss curvature, τ = 0 is a solution to the optimal embedding equation for Σ in N , and</text>
<formula><location><page_15><loc_41><loc_41><loc_52><loc_44></location>H 0 > | H | > 0 .</formula>
<text><location><page_15><loc_12><loc_38><loc_82><loc_41></location>Then for any axially symmetric time function τ such that σ + dτ ⊗ dτ has positive Gauss curvature,</text>
<formula><location><page_15><loc_40><loc_36><loc_54><loc_38></location>E (Σ , τ ) ≥ E (Σ , 0) .</formula>
<text><location><page_15><loc_12><loc_35><loc_55><loc_36></location>Moreover, equality holds if and only if τ is a constant .</text>
<text><location><page_15><loc_12><loc_28><loc_82><loc_33></location>Proof. By Theorem 1, it suffices to show that E (Σ 0 , τ ) ≥ 0. First, we show that for any 0 ≤ s ≤ 1, the isometric embedding with time function sτ exists. Recall the Gaussian curvature for the metric σ + dτ ⊗ dτ is</text>
<text><location><page_15><loc_12><loc_18><loc_82><loc_24></location>where K is the Gaussian curvature for the metric σ . Since K and K +(1+ |∇ τ | 2 ) -1 det ( ∇ 2 τ ) are both positive, we conclude that σ + d ( sτ ) ⊗ d ( sτ ) has positive Gaussian curvature for all 0 ≤ s ≤ 1. By Lemma 3, we have H 2 0 ≥ 〈 H sτ , H sτ 〉 . The theorem now follows from Lemma 2. /square</text>
<formula><location><page_15><loc_32><loc_23><loc_62><loc_28></location>1 1 + |∇ τ | 2 [ K +(1 + |∇ τ | 2 ) -1 det ( ∇ 2 τ ) ]</formula>
<section_header_level_1><location><page_16><loc_39><loc_86><loc_67><loc_88></location>Appendix A. Proof of Lemma 4</section_header_level_1>
<text><location><page_16><loc_18><loc_82><loc_88><loc_85></location>Here we present the proof of Lemma 4 used in the proof of Theorem 2. Since the lemma is a general statement for any time function τ 0 , we use τ instead of τ 0 .</text>
<text><location><page_16><loc_18><loc_78><loc_88><loc_81></location>Lemma 4. For a surface Σ τ in R 3 , 1 which bounds a spacelike hypersurface Ω , let f be the solution of Jang's equation on Ω with boundary value τ . Then</text>
<formula><location><page_16><loc_39><loc_75><loc_67><loc_77></location>h (Σ , X τ , τ, e ' 3 ) = h (Σ , X τ , τ, ˘ e 3 (Σ τ ))</formula>
<text><location><page_16><loc_18><loc_71><loc_88><loc_75></location>Proof. It suffices to show that e ' 3 = ˘ e 3 (Σ τ ). For simplicity, denote ˘ e 3 (Σ τ ) by ˘ e 3 in this proof.</text>
<text><location><page_16><loc_18><loc_64><loc_88><loc_71></location>Let ̂ Ω and ̂ Σ denote the projection of Ω and its boundary, Σ τ , to the complement of T 0 . Let ˆ ∇ be the covriant derivative on ̂ Σ and ˆ D be the covariant derivative on ̂ Ω. Let ∇ be the covriant derivative on Σ.</text>
<text><location><page_16><loc_18><loc_58><loc_88><loc_63></location>We choose an orthonormal frame { ˆ e a } for T ̂ Σ. Let ˆ e 3 be the outward normal of ̂ Σ in ̂ Ω. { ˆ e a , ˆ e 3 , T 0 } forms an orthonormal frame of the tangent space of R 3 , 1 . The frame is extend along T 0 direction by parallel translation to a frame of the tangent space of R 3 , 1 on Σ.</text>
<text><location><page_16><loc_18><loc_61><loc_88><loc_66></location>Write Ω as the graph over ̂ Ω of the function f . f can be viewed as a function on Ω as well. f is precisely the solution to Jang's equation on Ω with Dirichlet boundary data τ .</text>
<text><location><page_16><loc_18><loc_52><loc_88><loc_58></location>Let { e 3 , e 4 } denote the frame of the normal bundle of Σ such that e 3 is the unit outward normal of Σ in Ω and e 4 is the furture directed unit normal of Ω in R 3 , 1 . In terms of the frame { ˆ e a , ˆ e 3 , T 0 } ,</text>
<text><location><page_16><loc_18><loc_38><loc_88><loc_41></location>Let { e ' 3 , e ' 4 } denote the frame determined by Jang's equation. By Definition 5.1 of [8], it is chosen such that</text>
<formula><location><page_16><loc_18><loc_40><loc_76><loc_53></location>e 3 = 1 √ 1 -| ˆ Df | 2 √ 1 -| ˆ ∇ τ | 2 ˆ e 3 + ˆ e 3 ( f ) √ 1 -| ˆ ∇ τ | 2 ( T 0 + ˆ ∇ τ ) e 4 = 1 √ 1 -| ˆ Df | 2 ( T 0 + ˆ Df ) (A.1)</formula>
<text><location><page_16><loc_18><loc_32><loc_35><loc_34></location>Using equation (A.1),</text>
<text><location><page_16><loc_18><loc_25><loc_27><loc_26></location>As a result,</text>
<formula><location><page_16><loc_44><loc_32><loc_61><loc_38></location>〈 e 3 , e ' 4 〉 = -e 3 ( f ) √ 1 + |∇ τ | 2</formula>
<formula><location><page_16><loc_29><loc_25><loc_76><loc_31></location>〈 e 3 , e ' 4 〉 = -e 3 ( f ) √ 1 + |∇ τ | 2 = -ˆ e 3 ( f ) √ 1 + |∇ τ | 2 √ 1 -| ˆ Df | 2 √ 1 -| ˆ ∇ τ | 2</formula>
<text><location><page_16><loc_18><loc_17><loc_43><loc_19></location>since (1 -| ˆ ∇ τ | 2 )(1 + |∇ τ | 2 ) = 1 .</text>
<formula><location><page_16><loc_44><loc_18><loc_61><loc_24></location>〈 e 3 , e ' 4 〉 = -ˆ e 3 ( f ) √ 1 -| ˆ Df | 2</formula>
<text><location><page_17><loc_12><loc_84><loc_82><loc_88></location>On the other hand, the frame { ˘ e 3 , ˘ e 4 } is the frame of normal bundle such that ˘ e 3 = ˆ e 3 . In terms of the frame { ˆ e a , ˆ e 3 , T 0 } ,</text>
<text><location><page_17><loc_12><loc_77><loc_29><loc_79></location>Using equation (A.1),</text>
<text><location><page_17><loc_12><loc_69><loc_21><loc_71></location>As a result,</text>
<formula><location><page_17><loc_40><loc_78><loc_53><loc_84></location>˘ e 4 = T 0 + ˆ ∇ τ √ 1 -| ˆ ∇ τ | 2</formula>
<formula><location><page_17><loc_22><loc_70><loc_72><loc_78></location>〈 e 3 , ˘ e 4 〉 = 1 √ 1 -| ˆ ∇ τ | 2 ˆ e 3 ( f )( -1 + | ˆ ∇ τ | 2 ) √ 1 -| ˆ Df | 2 √ 1 -| ˆ ∇ τ | 2 = -ˆ e 3 ( f ) √ 1 -| ˆ Df | 2</formula>
<formula><location><page_17><loc_40><loc_67><loc_54><loc_69></location>〈 e 3 , ˘ e 4 〉 = 〈 e 3 , e ' 4 〉</formula>
<text><location><page_17><loc_12><loc_65><loc_82><loc_68></location>Hence, { e ' 3 , e ' 4 } and { ˘ e 3 , ˘ e 4 } are the same frame for the normal bundle of Σ /square</text>
<section_header_level_1><location><page_17><loc_42><loc_63><loc_52><loc_64></location>References</section_header_level_1>
<unordered_list>
<list_item><location><page_17><loc_12><loc_60><loc_82><loc_62></location>[1] L. Andersson, M. Eichmair, and J. Metzger, Jang's equation and its applications to marginally trapped surfaces Complex analysis and dynamical systems IV. Part 2, 13-45.</list_item>
<list_item><location><page_17><loc_12><loc_57><loc_82><loc_59></location>[2] P.-N. Chen, M.-T. Wang, and S.-T. Yau, Evaluating quasilocal energy and solving optimal embedding equation at null infinity. Comm. Math. Phys. 308 (2011), no. 3, 845-863.</list_item>
<list_item><location><page_17><loc_12><loc_54><loc_82><loc_57></location>[3] M. Khuri, Note on the nonexistence of generalized apparent horizons in Minkowski space. Classical Quantum Gravity 26 (2009), no. 7, 078001,</list_item>
<list_item><location><page_17><loc_12><loc_51><loc_82><loc_54></location>[4] P. Miao, L.-F. Tam, and N.Q. Xie, Critical points of Wang-Yau quasi-local energy. Ann. Henri Poincar'e. 12 (5): 987-1017, 2011.</list_item>
<list_item><location><page_17><loc_12><loc_50><loc_81><loc_51></location>[5] P. Miao and L.-F. Tam, On second variation of Wang-Yau quasi-local energy. arXiv:math/1301.4656.</list_item>
<list_item><location><page_17><loc_12><loc_47><loc_82><loc_50></location>[6] R. Schoen and S.-T. Yau, Proof of the positive mass theorem. II. Comm. Math. Phys. 79 (1981), no. 2, 231-260.</list_item>
<list_item><location><page_17><loc_12><loc_44><loc_82><loc_47></location>[7] M.-T. Wang and S.-T. Yau, Quasilocal mass in general relativity. Phys. Rev. Lett. 102 (2009), no. 2, no. 021101.</list_item>
<list_item><location><page_17><loc_12><loc_42><loc_82><loc_44></location>[8] M.-T. Wang and S.-T. Yau, Isometric embeddings into the Minkowski space and new quasi-local mass. Comm. Math. Phys. 288 (2009), no. 3, 919-942.</list_item>
<list_item><location><page_17><loc_12><loc_40><loc_40><loc_41></location>[9] Y.-K. Wang, personal communication.</list_item>
</document> | [{"title": "MINIMIZING PROPERTIES OF CRITICAL POINTS OF QUASI-LOCAL ENERGY", "content": "PO-NING CHEN, MU-TAO WANG, AND SHING-TUNG YAU Abstract. In relativity, the energy of a moving particle depends on the observer, and the rest mass is the minimal energy seen among all observers. The Wang-Yau quasilocal mass for a surface in spacetime introduced in [7] and [8] is defined by minimizing quasi-local energy associated with admissible isometric embeddings of the surface into the Minkowski space. A critical point of the quasi-local energy is an isometric embedding satisfying the Euler-Lagrange equation. In this article, we prove results regarding both local and global minimizing properties of critical points of the Wang-Yau quasi-local energy. In particular, under a condition on the mean curvature vector we show a critical point minimizes the quasi-local energy locally. The same condition also implies that the critical point is globally minimizing among all axially symmetric embedding provided the image of the associated isometric embedding lies in a totally geodesic Euclidean 3-space.", "pages": [1]}, {"title": "1. Introduction", "content": "Let \u03a3 be a closed embedded spacelike 2-surface in a spacetime N . We assume \u03a3 is a topological 2-sphere and the mean curvature vector field H of \u03a3 in N is a spacelike vector field. The mean curvature vector field defines a connection one-form of the normal bundle \u03b1 H = \u3008\u2207 N ( \u00b7 ) e 3 , e 4 \u3009 , where e 3 = -H | H | and e 4 is the future unit timelike normal vector that is orthogonal to e 3 . Let \u03c3 be the induced metric on \u03a3. We shall consider isometric embeddings of \u03c3 into the Minkowski space R 3 , 1 . We recall that this means an embedding X : \u03a3 \u2192 R 3 , 1 such that the induced metric on the image is \u03c3 . Throughout this paper, we shall fix a constant unit timelike vector T 0 in R 3 , 1 . The time function \u03c4 on the image of the isometric embedding X is defined to be \u03c4 = -X \u00b7 T 0 . The existence of such an isometric embedding is guaranteed by a convexity condition on \u03c3 and \u03c4 (Theorem 3.1 in [8]). The condition is equivalent to that the metric \u03c3 + d\u03c4 \u2297 d\u03c4 has positive Gaussian curvature. Let \u0302 \u03a3 be the projection of the image of X , X (\u03a3), onto the orthogonal complement of T 0 , a totally geodesic Euclidean 3-space in R 3 , 1 . \u0302 \u03a3 is a convex 2-surface in the Euclidean 3-space. Then the isometric embedding of \u03a3 into R 3 , 1 with time function \u03c4 exists and is unique up to an isometry of the orthogonal complement of T 0 . Part of this work was carried out while all three authors were visiting Department of Mathematics of National Taiwan University and Taida Institute for Mathematical Sciences in Taipei, Taiwan. M.-T. Wang is supported by NSF grant DMS-1105483 and S.-T. Yau is supported by NSF grants DMS-0804454 and PHY-07146468. In [7] and [8], Wang and Yau define a quasi-local energy for a surface \u03a3 with spacelike mean curvature vector H in a spacetime N , with respect to an isometric embedding X of \u03a3 into R 3 , 1 . The definition relies on the physical data on \u03a3 which consist of the induced metric \u03c3 , the norm of the mean curvature vector | H | > 0, and the connection one-form \u03b1 H . The definition also relies on the reference data from the isometric embedding X : \u03a3 \u2192 R 3 , 1 whose induced metric is the same as \u03c3 . In terms of \u03c4 = -X \u00b7 T 0 , the quasi-local energy is defined to be 1 where \u2207 and \u2206 are the covariant derivative and Laplace operator with respect to \u03c3 , and \u03b8 is defined by sinh \u03b8 = -\u2206 \u03c4 | H | \u221a 1+ |\u2207 \u03c4 | 2 . Finally, \u0302 H is the mean curvature of \u0302 \u03a3 in R 3 . We note that in E (\u03a3 , \u03c4 ), the first argument \u03a3 represents a physical surface in spacetime with the data ( \u03c3, | H | , \u03b1 H ), while the second argument \u03c4 indicates an isometric embedding of the induced metric into R 3 , 1 with time function \u03c4 with respect to the fixed T 0 . The Wang-Yau quasi-local mass for the surface \u03a3 in N is defined to be the minimum of E (\u03a3 , \u03c4 ) among all 'admissible' time functions \u03c4 (or isometric embeddings). This admissible condition is given in Definition 5.1 of [8] (see also section 3.1). This condition for \u03c4 implies that E (\u03a3 , \u03c4 ) is non-negative if N satisfies the dominant energy condition. We recall the Euler-Lagrange equation for the functional E (\u03a3 , \u03c4 ) of \u03c4 , which is derived in [8]. Of course, a critical point \u03c4 of E (\u03a3 , \u03c4 ) satisfies this equation. Definition 1. Given the physical data ( \u03c3, | H | , \u03b1 H ) on a 2-surface \u03a3 . We say that a smooth function \u03c4 is a solution to the optimal embedding equation for ( \u03c3, | H | , \u03b1 H ) if the metric \u03c3 = \u03c3 + d\u03c4 \u2297 d\u03c4 can be isometrically embedded into R 3 with image \u03a3 such that where \u2207 and \u2206 are the covariant derivative and Laplace operator with respect to \u03c3 , \u03b8 is defined by sinh \u03b8 = -\u2206 \u03c4 | H | \u221a 1+ |\u2207 \u03c4 | 2 . \u0302 h ab and \u0302 H are the second fundamental form and the mean curvature of \u0302 \u03a3 , respectively. It is natural to ask the following questions: Before addressing these questions, let us fix the notation on the space of isometric embeddings of (\u03a3 , \u03c3 ) into R 3 , 1 . Notation 1. T 0 is a fixed constant future timelike unit vector throughout the paper, X \u03c4 will denote an isometric embedding of \u03c3 into R 3 , 1 with time function \u03c4 = -X \u00b7 T 0 . We denote the image of X \u03c4 by \u03a3 \u03c4 , the mean curvature vector of \u03a3 \u03c4 by H \u03c4 and the connection one-form of the normal bundle of \u03a3 \u03c4 determined by H \u03c4 by \u03b1 H \u03c4 . We denote the projection of \u03a3 \u03c4 onto the orthogonal complement of T 0 by \u03a3 \u03c4 and the mean curvature of \u03a3 \u03c4 by H \u03c4 . \u0302 \u0302 \u0302 In particular, when \u03c3 has positive Gauss curvature, X 0 will denote an isometric embedding of \u03c3 into the orthogonal complement of T 0 . \u03a3 0 denotes the image of X 0 , The mean curvature H 0 of \u03a3 0 can be viewed as a positive function by the assumption. In [2], the authors studied the above questions at spatial or null infinity of asymptotically flat manifolds and proved that a series solution exists for the optimal embedding equation, the solution minimizes the quasi-local energy locally and the mass it achieves agrees with the ADM or Bondi mass at infinity for asymptotically flat manifolds. On the other hand, when \u03b1 H is divergence free, \u03c4 = 0 is a solution to the optimal embedding equation. Miao-Tam-Xie [4] and Miao-Tam [5] studied the time-symmetric case and found several conditions such that \u03c4 = 0 is a local minimum. In particular, this holds if H 0 > | H | > 0 where H 0 is the mean curvature of the isometric embedding X 0 of \u03a3 into R 3 (i.e. with time function \u03c4 = 0). For theorems proved in this paper, we impose the following assumption on the physical surface \u03a3. Assumption 1. Let \u03a3 be a closed embedded spacelike 2-surface in a spacetime N satisfying the dominant energy condition. We assume that \u03a3 is a topological 2-sphere and the mean curvature vector field H of \u03a3 in N is a spacelike vector field. We first prove the following comparison theorem among quasi-local energies. Theorem 1. Suppose \u03a3 satisfies Assumption 1 and \u03c4 0 is a is a critical point of the quasilocal energy functional E (\u03a3 , \u03c4 ) . Assume further that where H \u03c4 0 is the mean curvature vector of the isometric embedding of \u03a3 into R 3 , 1 with time function \u03c4 0 . Then, for any time function \u03c4 such that \u03c3 + d\u03c4 \u2297 d\u03c4 has positive Gaussian curvature, we have Moreover, equality holds if and only if \u03c4 -\u03c4 0 is a constant . As a corollary of Theorem 1, we prove the following theorem about local minimizing property of an arbitrary, non-time-symmetric, solution to the optimal embedding equation. Theorem 2. Suppose \u03a3 satisfies Assumption 1 and \u03c4 0 is a critical point of the quasi-local energy functional E (\u03a3 , \u03c4 ) . Assume further that where H \u03c4 0 is the mean curvature vector of the isometric embedding X \u03c4 0 of \u03a3 into R 3 , 1 with time function \u03c4 0 . Then, \u03c4 0 is a local minimum for E (\u03a3 , \u03c4 ) . The special case when \u03c4 0 = 0 was proved by Miao-Tam-Xie [4]. They estimate the second variation of quasi-local energy around the critical point \u03c4 = 0 by linearizing the optimal embedding equation near the critical point and then applying a generalization of Reilly's formula. As we allow \u03c4 0 to be an arbitrary solution of the optimal isometric embedding equation, a different method is devised to deal with the fully nonlinear nature of the equation. In general, the space of admissible isometric embeddings as solutions of a fully nonlinear elliptic system is very complicated and global knowledge of the quasi-local energy is difficult to obtained. However, we are able to prove a global minimizing result in the axially symmetric case. Theorem 3. Let \u03a3 satisfy Assumption 1. Suppose that the induced metric \u03c3 of \u03a3 is axially symmetric with positive Gauss curvature, \u03c4 = 0 is a solution to the optimal embedding equation for \u03a3 in N , and Then for any axially symmetric time function \u03c4 such that \u03c3 + d\u03c4 \u2297 d\u03c4 has positive Gauss curvature, Moreover, equality holds if and only if \u03c4 is a constant . This theorem will have applications in studying quasi-local energy in the Kerr spacetime, or more generally an axially symmetric spacetime. It is very likely that the global minimum of quasilocal energy of an axially symmetric datum is achieved at an axially symmetric isometric embedding into the Minkowski, though we cannot prove it at this moment. In [4], Miao, Tam and Xie described several situations where the condition H 0 > | H | holds. In particular, this includes large spheres in Kerr spacetime. In section 2, we prove Theorem 1 using the nonlinear structure of the quasi-local energy. In section 3, we prove the admissibility of the time function \u03c4 for the surface \u03a3 \u03c4 0 in R 3 , 1 when \u03c4 is close to \u03c4 0 . The positivity of quasi-local mass follows from the admissibility of the time function. Combining with Theorem 1, this proves Theorem 2. In section 4, we prove Theorem 3. Instead of using admissibility, we prove the necessary positivity of quasi-local energy using variation of quasi-local energy and a point-wise mean curvature inequality.", "pages": [1, 2, 3, 4]}, {"title": "2. A comparison theorem for quasi-local energy", "content": "In this section, we prove Theorem 1. Proof. We start with a metric \u03c3 and consider an isometric embedding into R 3 , 1 with time function \u03c4 0 . The image is an embedded space-like 2-surface \u03a3 \u03c4 0 in R 3 , 1 . The corresponding data on \u03a3 \u03c4 0 are denoted as | H \u03c4 0 | and \u03b1 H \u03c4 0 . We consider the quasi-local energy of \u03a3 \u03c4 0 as a physical surface in the spacetime R 3 , 1 with respect to another isometric embedding X \u03c4 into R 3 , 1 with time function \u03c4 . We recall that where \u03b8 ( \u03c4,\u03c4 0 ) is defined by sinh \u03b8 ( \u03c4,\u03c4 0 ) = -\u2206 \u03c4 | H \u03c4 0 | \u221a 1+ |\u2207 \u03c4 | 2 . Using E (\u03a3 \u03c4 0 , \u03c4 ), E (\u03a3 , \u03c4 ) can be expressed as where In the following, we shall show that A \u2265 E (\u03a3 , \u03c4 0 ). One can rewrite div \u03c3 \u03b1 H and div \u03c3 \u03b1 H \u03c4 0 using the optimal embedding equation. First, \u03c4 0 is a solution to the original optimal embedding equation. We have (2.3) where \u02c6 h ab and \u0302 H are the second fundamental form and mean curvature of \u0302 \u03a3 \u03c4 0 , respectively. On the other hand, \u03c4 = \u03c4 0 locally minimizes E (\u03a3 \u03c4 0 , \u03c4 ) by the positivity of quasi-local energy. Hence, (2.4) where \u02c6 h ab and \u0302 H are the same as in equation (2.3) (this can be verified directly as in [9]). Using equations (2.3) and (2.4), we have With a new variable x = \u2206 \u03c4 \u221a 1+ |\u2207 \u03c4 | 2 and x 0 = \u2206 \u03c4 0 \u221a 1+ |\u2207 \u03c4 0 | 2 , the first six terms of the integrand is simply Let Direct computation shows that A \u2265 \u222b \u03a3 ( \u221a 1 + |\u2207 \u03c4 | 2 -\u2207 \u03c4 0 \u00b7 \u2207 \u03c4 \u221a 1 + |\u2207 \u03c4 0 | 2 ) [ \u221a | H \u03c4 0 | 2 + (\u2206 \u03c4 0 ) 2 1 + |\u2207 \u03c4 0 | 2 -\u221a | H | 2 + (\u2206 \u03c4 0 ) 2 1 + |\u2207 \u03c4 0 | 2 ] \u2265 \u222b \u03a3 ( 1 \u221a 1 + |\u2207 \u03c4 0 | 2 ) [ \u221a | H \u03c4 0 | 2 + (\u2206 \u03c4 0 ) 2 1 + |\u2207 \u03c4 0 | 2 -\u221a | H | 2 + (\u2206 \u03c4 0 ) 2 1 + |\u2207 \u03c4 0 | 2 ] The last inequality follows simply from If | H \u03c4 0 | > | H | , f ( x 0 ) = \u221a | H \u03c4 0 | 2 + x 2 0 -\u221a | H | 2 + x 2 0 is the global minimum for f (x) and equality holds if and only if x = x 0 . Hence, and equality holds if and only if \u2207 \u03c4 = \u2207 \u03c4 0 . On the other hand, if one evaluates div \u03c3 \u03b1 H and div \u03c3 \u03b1 H \u03c4 0 using equations (2.3) and (2.4). /square", "pages": [4, 5, 6, 7]}, {"title": "3. Local minimizing property of critical points of quasi-local energy", "content": "In this section, we start with a metric \u03c3 and consider an isometric embedding into R 3 , 1 with time function \u03c4 0 . The image is an embedded space-like 2-surface \u03a3 \u03c4 0 in R 3 , 1 . The corresponding data on \u03a3 \u03c4 0 are denoted as | H \u03c4 0 | and \u03b1 H \u03c4 0 . We consider the quasi-local energy of \u03a3 \u03c4 0 as a physical surface in the spacetime R 3 , 1 with respect to another isometric embedding X \u03c4 into R 3 , 1 with time function \u03c4 . In this section, we prove that for \u03c4 close to \u03c4 0 , \u03c4 is admissible with respect to the surface \u03a3 \u03c4 0 . This shows that, for \u03c4 close to \u03c4 0 , we have Definition 2. Suppose i : \u03a3 \u21aa \u2192 N is an embedded spacelike two-surface in a spacetime N . Given a smooth function \u03c4 on \u03a3 and a spacelike unit normal e 3 , the generalized mean curvature associated with these data is defined to be where, as before, H is the mean curvature vector of \u03a3 in N and \u03b1 e 3 is the connection form of the normal bundle determined by e 3 . Recall, in Definition 5.1 of [8], given a physical surface \u03a3 in spacetime N with induced metric \u03c3 and mean curvature vector H , there are three conditions for a function \u03c4 to be admissible. where sinh \u03b8 = e 3 ( f ) \u221a 1+ |\u2207 \u03c4 | 2 and e 3 is the outward unit spacelike normal of \u03a3 in \u2126, e 4 is the future timelike unit normal of \u2126 in N . We recall that from [8] if \u03c4 corresponds to an admissible isometric embedding and \u03a3 is a 2-surface in spacetime N that satisfies Assumption 1, then the quasi-local energy E (\u03a3 , \u03c4 ) is non-negative. 3.2. Solving Jang's equation. Let (\u2126 , g ij ) be a Riemannian manifold with boundary \u2202 \u2126 = \u03a3. Let p ij be a symmetric 2-tensor on \u2126. The Jang's equation asks for a hypersurface \u02dc \u2126 in \u2126 \u00d7 R , defined as a graph of a function f over \u2126, such that the mean curvature of \u02dc \u2126 is the same as the trace of the restriction of p to \u02dc \u2126 . In a local coordinate x i on \u2126, Jang's equation takes the following form: where D is the covariant derivative with respect to the metric g ij . When p ij = 0, Jang's equation becomes the equation for minimal graph. Equation of of minimal surface type may have blow-up solutions. In [6], it is shown by Schoen-Yau that solutions of Jang's equation can only blow-up at marginally trapped surface in \u2126. Namely, surfaces S in \u2126 such that Following the analysis of Jang's equation in [6] and [8], we prove the following theorem for the existence of solution to the Dirichlet problem of Jang's equation. Theorem 4. Let (\u2126 , g ij ) be a Riemannian manifold with boundary \u2202 \u2126 = \u03a3 , p ij be a symmetric 2-tensor on \u2126 , and \u03c4 be a function on \u03a3 . Then Jang's equation with Dirichlet boundary data \u03c4 is solvable on \u2126 if and there is no marginally trapped surface inside \u2126 . Following the approach in [6] and Section 4.3 of [8], it suffices to control the boundary gradient of the solution to Jang's equation. We have the following theorem: Theorem 5. The normal derivative of a solution of the Dirichlet problem of Jang's equation is bounded if Remark 1. In [1] , Andersson, Eichmair and Metzger proved a similar result about boundedness of boundary gradient of solutions to Jang's equation in order to study existence of marginally trapped surfaces. Proof. We follow the approach used in Theorem 4.2 of [8] but with a different form for the sub and super solutions. Let g ij and p ij be the induced metric and second fundamental form of the hypersurface \u2126. Let \u03a3 be the boundary of \u2126 with induced metric \u03c3 and mean curvature H \u03a3 in \u2126. We consider the following operator for Jang's equation. As /epsilon1 approaches 0, we have We extend the boundary data \u03c4 to the interior of the hypersurface \u2126. We still denote the extension by \u03c4 . Consider the following test function where d is the distance function to the boundary of \u2126. We compute Therefore, At the boundary of \u2126, it is convenient to use a frame { e a , e 3 } where e a are tangent to the boundary and e 3 is normal to the boundary. Moreover, the distance function d to the boundary of \u2126 satisfies Hence, we conclude that As a result, sub and super solutions exist when H \u03a3 > | tr \u03a3 p | . One can then use Perron method to find the solution between the sub and super solution. /square Remark 2. Here we proved the result when the dimension of \u2126 is 3. The result holds in higher dimension as well.", "pages": [7, 8, 9]}, {"title": "3.3. Proof of Theorem 2.", "content": "Proof. It suffices to show that, for \u03c4 close to \u03c4 0 , \u03c4 is admissible with respect to \u03a3 \u03c4 0 in R 3 , 1 if H \u03c4 0 is spacelike. For such a \u03c4 , the Gauss curvature of \u03c3 + d\u03c4 \u2297 d\u03c4 is close to that of \u03c3 + d\u03c4 0 \u2297 d\u03c4 0 . In particular, it remains positive. This verifies the first condition for admissibility. We apply Theorem 4 in the case where \u03a3 bounds a spacelike hypersurface \u2126 in R 3 , 1 and g ij and p ij be the induced metric and second fundamental form on \u2126, respectively. Moreover, the projection of \u03a3 onto the orthogonal complement of T 0 is convex since the Gauss curvature of \u03c3 + d\u03c4 \u2297 d\u03c4 is positive. That the mean curvature of \u03a3 is spacelike implies that | H \u03a3 | > | tr \u03a3 p | . Moreover, that the projection \u0302 \u03a3 is convex implies H \u03a3 > 0. It follows that H \u03a3 > | tr \u03a3 p | . We recall there is no marginally trapped surface in R 3 , 1 (see for example, [3]). As a result, Jang's equation with the Dirichlet boundary data \u03c4 is solvable as long as \u03a3 = \u2202 \u2126 has spacelike mean curvature vector. This verifies the second condition for admissibility. To verify the last condition, it suffices to check for \u03c4 0 since it is an open condition. Namely, it suffices to prove that Lemma 4 in the appendix implies that the generalized mean curvature h (\u03a3 , X \u03c4 0 , \u03c4 0 , e ' 3 ) is the same as the generalized mean curvature h (\u03a3 , X \u03c4 0 , \u03c4 0 , \u02d8 e 3 (\u03a3 \u03c4 0 )) where \u02d8 e 3 (\u03a3 \u03c4 0 ) is the vector field on \u03a3 \u03c4 0 obtained by parallel translation of the outward unit normal of \u0302 \u03a3 \u03c4 0 along T 0 . The last condition for \u03c4 0 now follows from Proposition 3.1 of [8], which states that for \u02d8 e 3 (\u03a3 \u03c4 0 ),", "pages": [10]}, {"title": "4. Global minimum in the axially symmetric case", "content": "In proving Theorem 3, we need three Lemmas concerning a spacelike 2-surface \u03a3 \u03c4 in R 3 , 1 with time function \u03c4 . First we introduce a new energy functional which depends on a physical gauge. Definition 3. Let \u03a3 be closed embedded spacelike 2-surface in spacetime N with induced metric \u03c3 . Let e 3 be a spacelike normal vector field along \u03a3 in N . For any f such that the isometric embedding of \u03c3 into R 3 , 1 with time function f exists, we define This functional is less nonlinear than E (\u03a3 , f ). Provided the mean curvature vector H of \u03a3 is spacelike, the following relation holds where e can 3 ( f ) is chosen such that In addition, the first variation of \u02dc E (\u03a3 , e 3 , f ) with respect to f can be computed as in [8]: Lemma 1. For a spacelike 2-surface \u03a3 \u03c4 in R 3 , 1 with time function \u03c4 , f = \u03c4 is a critical point of the functional \u02dc E (\u03a3 \u03c4 , \u02d8 e 3 (\u03a3 \u03c4 ) , f ) . where \u02c6 h ab and \u0302 H are the second fundamental form and mean curvature of \u0302 \u03a3 f , respectively. We recall that for \u03a3 \u03c4 , assuming the projection onto the orthonormal complement of T 0 is an embedded surface, there is a unique outward normal spacelike unit vector field \u02d8 e 3 (\u03a3 \u03c4 ) which is orthogonal to T 0 . Indeed, \u02d8 e 3 (\u03a3 \u03c4 ) can be obtained by parallel translating the unit outward normal vector of \u0302 \u03a3 \u03c4 , \u02c6 \u03bd , along T 0 . Proof. The first variation of \u02dc E (\u03a3 \u03c4 , \u02d8 e 3 (\u03a3 \u03c4 ) , f ) at f = \u03c4 is where h = \u3008 H \u03c4 , \u02d8 e 3 (\u03a3 \u03c4 ) \u3009 and \u03b1 = \u03b1 \u02d8 e 3 (\u03a3 \u03c4 ) are data on \u03a3 \u03c4 with respect to the gauge \u02d8 e 3 (\u03a3 \u03c4 ) and \u02c6 h ab and \u0302 H are the second fundamental form and mean curvature of \u0302 \u03a3 \u03c4 . Denote the covariant derivative on \u0302 \u03a3 with respect to the induced metric \u02c6 \u03c3 by \u02c6 \u2207 . We wish to show that the first variation is 0. Recall from [8], we have Moreover, we have the following relation between metric and covariant derivative of \u03a3 \u03c4 and \u0302 \u03a3 \u03c4 : In addition, for a tangent vector field W a , \u02c6 \u2207 a W a = \u2207 a W a + ( \u2207 b \u2207 c \u03c4 ) \u03c4 c W b 1+ |\u2207 \u03c4 | 2 . As a result, = \u2207 Hence, \u0302 \u221a On the other hand, by definition, Hence, This shows that - |\u2207 | \u2207 \u03c4 ] . and completes the proof of the lemma. /square Lemma 2. Let X s\u03c4 , 0 \u2264 s \u2264 1 be a family of isometric embeddings of \u03c3 into R 3 , 1 with time function s\u03c4 . Suppose \u03a3 0 , the image of X 0 , lies in a totally geodesic Euclidean 3-space, E 3 , and \u03a3 s\u03c4 , the image of X s\u03c4 , projects to an embedded surface in E 3 for 0 \u2264 s \u2264 1 . Assume further that \u03a3 0 is mean convex and H 2 0 \u2265 \u3008 H s\u03c4 , H s\u03c4 \u3009 for 0 \u2264 s \u2264 1 . Regarding \u03a3 0 as a physical surface in the spacetime R 3 , 1 , then Proof. Instead of proving admissibility of \u03c4 , we will use variation of the quasi-local energy and the point-wise inequality of the mean curvatures. Hence, we consider for 0 \u2264 s \u2264 1. F (0) = 0 and we shall prove that F (1) is non-negative by deriving a differential inequality for F ( s ). Let For any 0 \u2264 s 0 \u2264 1, G ( s ) is related to \u02dc E (\u03a3 s 0 \u03c4 , \u02d8 e 3 (\u03a3 s 0 \u03c4 ) , s\u03c4 ) by (4.2) As a consequence of Lemma 1, we have By equation (4.2), Recall that by the definition of quasi-local energy: We write the integrand of the last integral as Differentiate this expression with respect to the variable s , we obtain Since the induced metrics on \u03a3 s 0 \u03c4 and \u03a3 0 are the same, we can evaluate all integrals on the surface \u03a3 0 . This leads to The assumption H 2 0 \u2265 \u3008 H s\u03c4 , H s\u03c4 \u3009 implies the last term is non-negative and thus As F (0) = F ' (0) = 0, the positivity of F ( s ) follows from a simple comparison result for ordinary differential equation. /square The last lemma specializes to axially symmetric metrics. Lemma 3. Suppose the isometric embedding X 0 of an axially symmetric metric \u03c3 = P 2 d\u03b8 2 + Q 2 sin 2 \u03b8d\u03c6 2 into R 3 is given by the coordinates ( u sin \u03c6, u cos \u03c6, v ) where P , Q , u , and v are functions of \u03b8 . Let \u03c4 = \u03c4 ( \u03b8 ) be an axially symmetric function and X \u03c4 be the isometric embedding of \u03c3 in R 3 , 1 with time function \u03c4 . The following identity holds for the mean curvature vector H \u03c4 of \u03a3 \u03c4 in R 3 , 1 . where \u2206 is the Laplace operator of \u03c3 . Proof. The isometric embedding for an axially symmetric metric is reduced to solving ordinary differential equations. The isometric embedding of \u03c3 into R 3 is given by ( u sin \u03c6, u cos \u03c6, v ) where The isometric embedding of the metric \u03c3 + d\u03c4 \u2297 d\u03c4 into R 3 is given by ( u sin \u03c6, u cos \u03c6, \u02dc v ) where Thus, Differentiating one more time with respect to \u03b8 , and therefore For the mean curvature, we have Taking the difference and completing square, we obtain /square Let's recall the statement of Theorem 3. Theorem 3 Let \u03a3 satisfy Assumption 1. Suppose that the induced metric \u03c3 of \u03a3 is axially symmetric with positive Gauss curvature, \u03c4 = 0 is a solution to the optimal embedding equation for \u03a3 in N , and Then for any axially symmetric time function \u03c4 such that \u03c3 + d\u03c4 \u2297 d\u03c4 has positive Gauss curvature, Moreover, equality holds if and only if \u03c4 is a constant . Proof. By Theorem 1, it suffices to show that E (\u03a3 0 , \u03c4 ) \u2265 0. First, we show that for any 0 \u2264 s \u2264 1, the isometric embedding with time function s\u03c4 exists. Recall the Gaussian curvature for the metric \u03c3 + d\u03c4 \u2297 d\u03c4 is where K is the Gaussian curvature for the metric \u03c3 . Since K and K +(1+ |\u2207 \u03c4 | 2 ) -1 det ( \u2207 2 \u03c4 ) are both positive, we conclude that \u03c3 + d ( s\u03c4 ) \u2297 d ( s\u03c4 ) has positive Gaussian curvature for all 0 \u2264 s \u2264 1. By Lemma 3, we have H 2 0 \u2265 \u3008 H s\u03c4 , H s\u03c4 \u3009 . The theorem now follows from Lemma 2. /square", "pages": [10, 11, 12, 13, 14, 15]}, {"title": "Appendix A. Proof of Lemma 4", "content": "Here we present the proof of Lemma 4 used in the proof of Theorem 2. Since the lemma is a general statement for any time function \u03c4 0 , we use \u03c4 instead of \u03c4 0 . Lemma 4. For a surface \u03a3 \u03c4 in R 3 , 1 which bounds a spacelike hypersurface \u2126 , let f be the solution of Jang's equation on \u2126 with boundary value \u03c4 . Then Proof. It suffices to show that e ' 3 = \u02d8 e 3 (\u03a3 \u03c4 ). For simplicity, denote \u02d8 e 3 (\u03a3 \u03c4 ) by \u02d8 e 3 in this proof. Let \u0302 \u2126 and \u0302 \u03a3 denote the projection of \u2126 and its boundary, \u03a3 \u03c4 , to the complement of T 0 . Let \u02c6 \u2207 be the covriant derivative on \u0302 \u03a3 and \u02c6 D be the covariant derivative on \u0302 \u2126. Let \u2207 be the covriant derivative on \u03a3. We choose an orthonormal frame { \u02c6 e a } for T \u0302 \u03a3. Let \u02c6 e 3 be the outward normal of \u0302 \u03a3 in \u0302 \u2126. { \u02c6 e a , \u02c6 e 3 , T 0 } forms an orthonormal frame of the tangent space of R 3 , 1 . The frame is extend along T 0 direction by parallel translation to a frame of the tangent space of R 3 , 1 on \u03a3. Write \u2126 as the graph over \u0302 \u2126 of the function f . f can be viewed as a function on \u2126 as well. f is precisely the solution to Jang's equation on \u2126 with Dirichlet boundary data \u03c4 . Let { e 3 , e 4 } denote the frame of the normal bundle of \u03a3 such that e 3 is the unit outward normal of \u03a3 in \u2126 and e 4 is the furture directed unit normal of \u2126 in R 3 , 1 . In terms of the frame { \u02c6 e a , \u02c6 e 3 , T 0 } , Let { e ' 3 , e ' 4 } denote the frame determined by Jang's equation. By Definition 5.1 of [8], it is chosen such that Using equation (A.1), As a result, since (1 -| \u02c6 \u2207 \u03c4 | 2 )(1 + |\u2207 \u03c4 | 2 ) = 1 . On the other hand, the frame { \u02d8 e 3 , \u02d8 e 4 } is the frame of normal bundle such that \u02d8 e 3 = \u02c6 e 3 . In terms of the frame { \u02c6 e a , \u02c6 e 3 , T 0 } , Using equation (A.1), As a result, Hence, { e ' 3 , e ' 4 } and { \u02d8 e 3 , \u02d8 e 4 } are the same frame for the normal bundle of \u03a3 /square", "pages": [16, 17]}] |
2021PhRvD.103d6016G | https://arxiv.org/pdf/2010.04738.pdf | <document>
<section_header_level_1><location><page_1><loc_15><loc_89><loc_85><loc_91></location>Quantum gravitational onset of Starobinsky inflation in a closed universe</section_header_level_1>
<text><location><page_1><loc_27><loc_86><loc_72><loc_88></location>Lucia Gordon 1 , ∗ Bao-Fei Li 2 , † and Parampreet Singh 2 ‡</text>
<text><location><page_1><loc_22><loc_80><loc_77><loc_84></location>1 Department of Physics, Harvard University, Cambridge, MA 02138, USA 2 Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA</text>
<text><location><page_1><loc_17><loc_50><loc_82><loc_78></location>Recent cosmic microwave background observations favor low energy scale inflationary models in a closed universe. However, onset of inflation in such models for a closed universe is known to be severely problematic. In particular, such a universe recollapses within a few Planck seconds and encounters a big crunch singularity when initial conditions are given in the Planck regime. We show that this problem of onset of inflation in low energy scale inflationary models can be successfully overcome in a quantum gravitational framework where the big bang/big crunch singularities are resolved and a non-singular cyclic evolution exists prior to inflation. As an example we consider a model in loop quantum cosmology and demonstrate that the successful onset of low energy scale inflation in a closed universe is possible for the Starobinsky inflationary model starting from a variety of initial conditions where it is impossible in the classical theory. For comparison we also investigate the onset of inflation in the φ 2 inflationary model under highly unfavorable conditions and find similar results. Our numerical investigation including the phase space analysis shows that the pre-inflationary phase with quantum gravity effects is composed of non-identical cycles of bounces and recollapses resulting in a hysteresis-like phenomenon, which plays an important role in creating suitable conditions for inflation to occur after some number of non-singular cycles. Our analysis shows that the tension in the classical theory amounting to the unsuitability of closed FLRW universes with respect to the onset of low energy scale inflation can be successfully resolved in loop quantum cosmology.</text>
<section_header_level_1><location><page_1><loc_41><loc_44><loc_59><loc_45></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_12><loc_14><loc_88><loc_41></location>Cosmic microwave background (CMB) observations suggest that inflationary models with an inflaton in a plateau potential have been favored since Planck 2013 [1-3]. The inflationary models that are consistent with the Planck constraints on the spectral index and the tensor-to-scalar ratio for the pivot mode include the Starobinsky model [4], the Higgs inflationary model [5, 6] and a broad class of cosmological attractor models [7-10]. In these models, inflation takes place at an energy scale far below the Planck density. For this reason, they are also called the low energy scale inflationary models. But if the universe is closed, then the onset of inflation in such models is severely problematic [11-13]. This problem, which is tied to the recollapse of the universe and the big crunch singularity, can be avoided in spatially-flat and open universes [11, 12]. However, the recent Planck Legacy 2018 (PL 2018) release has confirmed the presence of an enhanced lensing amplitude in the CMB power spectra [14, 15], which favors a closed universe over a spatially-flat one [16]. Consequently, a pertinent question is the following: How does a closed universe starting from the Planck regime lead to successful low scale inflation? To answer this question, one needs a framework which goes beyond the classical description of spacetime and allows a robust resolution of the big crunch singularity while alleviating the problem of initial conditions for low scale inflation in a closed universe.</text>
<text><location><page_2><loc_12><loc_57><loc_88><loc_91></location>While the low energy inflationary model in a closed universe starting from the Planck regime typically ends in a big crunch singularity in a few Planck seconds, the onset of inflation in a closed universe is not as problematic for the chaotic inflationary models such as φ 2 inflation which, however, is not favored by the Planck data. In these models, where inflation can occur at higher energy scales, the initial conditions of the inflaton starting from a single Planck sized domain can be such that the kinetic and gradient energy of the scalar field are smaller than its potential energy which is order U ∼ 1 (in Planck units). Since the potential energy is dominant, the universe can avoid the recollapse and the subsequent big crunch singularity, and the dynamical evolution successfully results in a phase of inflation shortly after the initial time. Note that this is true only for those initial conditions where potential energy is dominant. Of course, if the inflaton starts with kinetic energy domination then the fate of such a universe even in a φ 2 inflationary model can be similar to that of low energy scale inflation in the sense that the universe encounters a big crunch singularity before a phase of inflation can start. Previous studies show that the probability of the quantum creation of a closed universe in which inflation takes place at the energy scale U glyph[lessmuch] 1 is exponentially suppressed [17-21]. However, all of the above arguments exclude the role of non-perturbative quantum gravity effects in the Planck regime. Further, an important problem irrespective of the spatial curvature of the universe is that inflationary models are past-incomplete in the classical theory [22, 23]. Since quantum gravity effects are expected to resolve spacetime singularities, it is quite possible that the above situation changes dramatically when one considers a quantum gravitational version of the model.</text>
<text><location><page_2><loc_12><loc_41><loc_88><loc_56></location>The goal of this paper is to show that the above problem of the onset of inflation in a closed universe can be successfully resolved in a setting motivated by loop quantum gravity where big bang and big crunch singularities are replaced by a big bounce due to quantum geometric effects. In our analysis we consider the Starobinsky inflationary model as an example of a low energy inflationary model and study the onset of inflation for a variety of initial conditions. For comparison we also investigate the φ 2 model for initial conditions which are not favorable in the classical theory for the onset of inflation. While our analysis uses techniques of loop quantum cosmology (LQC) for the chaotic and Starobinsky potentials, the essential idea and results can be replicated for any bouncing model and other inflationary potentials.</text>
<text><location><page_2><loc_12><loc_7><loc_88><loc_41></location>Our framework is based on a loop quantization of a closed universe performed in LQC which is a non-perturbative quantization of cosmological models based on loop quantum gravity [24]. Unlike the Wheeler-DeWitt quantum cosmology where the spacetime manifold is differentiable, LQC is based on a discrete quantum geometry predicted by loop quantum gravity. The quantum evolution is dictated by a non-singular quantum difference equation which results in a resolution of the big bang singularity by replacing it with a quantum bounce when the spacetime curvature becomes Planckian [25-27], including in the presence of inflationary potentials [28, 29], anisotropies and inhomogenities [24]. Using the consistent histories formalism one can also compute the probability of the bounce which turns out to be unity [30]. For a closed universe, loop quantum gravitational effects resolve both the past and future singularities, resulting in a cyclic universe [31-33]. It turns out that the quantum evolution can be very well approximated by the effective dynamics obtained from an effective spacetime description [26, 31, 34, 35]. Using this description singularity resolution has been established to be a generic feature for various isotropic and anisotropic models in LQC [36, 37], and various phenomenologically interesting consequences for the early universe and resulting signatures in the CMB have been discussed [38]. For the spatially-flat model it was found that the inflationary dynamics is an attractor after the bounce for φ 2 , power-law inflation and the Starobinsky potential [39-47], and the probability for inflation to occur is very large [48-50]. For closed universes, the dynamics with the φ 2 inflationary potential in the presence of spatial curvature has been previously studied in LQC [51], which revealed some novel features in comparison to the spatially-flat case. It was found that due to the recollapse caused by the spatial curvature and the</text>
<text><location><page_3><loc_12><loc_69><loc_88><loc_91></location>bounce caused by quantum geometry, the evolution of a closed universe filled with a homogeneous scalar field in an inflationary potential is usually characterized by a number of cycles in the preinflationary era with a hysteresis-like phenomenon due to the asymmetry in the equation of state (or equivalently the asymmetry in the pressure of the scalar field) in each cycle. It was found that quantum geometric modifications in LQC enhance this hysteresis-like phenomenon in comparison to previously studied bouncing models where this phenomenon was noted earlier [52-54]. Owing to this hysteresis-like behavior, even when starting from unfavorable initial conditions for inflation, the universe comes out of non-inflationary cyclic evolution and enters into a phase of inflation. Once the universe is in this phase, the scalar field starts to play a dominant role and the effects of the spatial curvature become negligible due to the exponential expansion of the universe, implying that no further cycles occur. Though the work in Ref. [51] found evidence of inflation occurring after hysteresis in φ 2 inflation, the robustness of the existence of the inflationary phase and its possible generalization to low energy scale inflation models was not studied.</text>
<text><location><page_3><loc_12><loc_41><loc_88><loc_68></location>The aim of this paper is to reconcile the closed universe scenario with low scale inflation in the framework of LQC for the Starobinsky potential. 1 Our strategy follows the encouraging results for φ 2 inflation in the closed model of LQC [51] with an aim to understand the onset of inflation with the Starobinsky potential and also to gain further insights by comparing it with φ 2 inflation. Using the effective Hamilton's equations resulting from the holonomy quantization for a closed LQC universe [31], we consider a single scalar field minimally coupled to gravity with the initial conditions set at the maximum energy density at some initial volume. We explore the background evolution of the universe starting with different types of initial conditions, with varying ratios of kinetic and potential energy, some of which are unfavorable for inflation to start in the classical theory, and perform numerical simulations for the φ 2 and Starobinsky potentials up through the onset of inflation. To investigate some aspects of the qualitative behavior we also study two-dimensional phase space portraits and find inflationary separatrices and other cosmological attractors in the plots. Our results show that both φ 2 inflation and Starobinsky inflation can take place under a variety of initial conditions starting from the Planck regime in the closed model. The primary reason for this is the hysteresis-like phenomena which are found to be much weaker in Starobinsky inflation. Due to this, the onset of inflation is delayed in comparison to φ 2 inflation.</text>
<text><location><page_3><loc_12><loc_16><loc_88><loc_41></location>This manuscript is organized as follows. In Sec. II, we briefly review the effective dynamics of a closed FLRW universe using the holonomy quantization in LQC. We obtain the necessary equations which include the Hamilton's equations and the modified Friedmann equation for the purpose of a detailed analysis of the background dynamics of the universe when gravity is minimally coupled to a single scalar field with an inflationary potential. In Sec. III, we consider φ 2 inflation in a closed FLRW universe in LQC with the initial conditions given in the Planck regime. We address the general properties of the pre-inflationary dynamics and present phase space portraits with trajectories starting from various initial conditions that all end up with a reheating phase, corresponding to the spiral at the center of the plots. In Sec. IV, we study the Starobinsky potential in a closed universe in LQC. We show that after taking into account quantum gravitational effects, many initial conditions that are unfavorable for inflation in classical cosmology do in fact lead to inflation at late times in LQC. The phase space portraits will be presented in order to show the qualitative behavior of numerical solutions starting with various initial conditions. Finally, in Sec. V, we summarize the main results concerning φ 2 and Starobinsky inflation in a closed universe in LQC and discuss the differences and similarities between these two inflationary models.</text>
<text><location><page_3><loc_14><loc_14><loc_88><loc_15></location>In our paper, we use Planck units with glyph[planckover2pi1] = c = 1 while keeping Newton's constant G explicit</text>
<text><location><page_4><loc_12><loc_89><loc_62><loc_91></location>in our formulas. In the numerical analysis, G is also set to unity.</text>
<section_header_level_1><location><page_4><loc_15><loc_84><loc_85><loc_86></location>II. EFFECTIVE DYNAMICS OF A CLOSED UNIVERSE IN LOOP QUANTUM COSMOLOGY</section_header_level_1>
<text><location><page_4><loc_12><loc_54><loc_88><loc_81></location>In LQC, the quantization of the homogeneous and isotropic FLRW universe is carried out in terms of the Ashtekar-Barbero connection A i a and its conjugate triad E i a . The non-perturbative modifications to the classical Hamiltonian constraint arise from the regularization of the field strength of the connection and the inverse volume terms. It turns out that the latter are not dominant compared to the field strength modifications for singularity resolution [31] and quickly decay for volumes greater than the Planck volume. For this reason we will only focus on the effective dynamics incorporating modifications to the field strength of the connection. In the literature, the regularization of the field strength in a closed FLRW universe has been explored in two different ways. One is based on the quantization of the holonomy of the connection over closed loops [31], the other is the connection based quantization [33]. Both of these approaches, which can be viewed as quantization ambiguities, lead to singularity resolution and give similar qualitative behavior away from the bounce regime [56]. For the φ 2 potential a comparative analysis of these two quantizations was performed in Ref. [51], which revealed the robustness of the qualitative features of hysteresis and the subsequent inflationary phase. In the following, we will study the holonomy based quantization and we expect the main results to hold for the connection based quantization as well.</text>
<text><location><page_4><loc_12><loc_39><loc_88><loc_54></location>In a cosmological setting, due to the homogeneity and isotropy of the universe, the AshtekarBarbero connection and its conjugate triad can be symmetry-reduced to the canonical pair of c and p [31]. This set of canonical variables is equivalent to a new set of variables, namely b and v that are commonly used in the ¯ µ scheme [26]. In a closed universe, the physical volume of the unit sphere spatial manifold is given by v = | p | 3 / 2 = 2 π 2 a 3 with a representing the scale factor of the universe. The conjugate variable b is defined via b = c | p | -1 / 2 . Besides the gravitational degrees of freedom, in order to initiate the onset of inflation, one also needs degrees of freedom in the matter sector which are the scalar field φ and its conjugate momentum p φ . These fundamental canonical pairs in the phase space satisfy the Poisson brackets:</text>
<formula><location><page_4><loc_38><loc_36><loc_88><loc_38></location>{ b, v } = 4 πGγ, { φ, p φ } = 1 , (2.1)</formula>
<text><location><page_4><loc_12><loc_32><loc_88><loc_35></location>where γ is the Barbero-Immirzi parameter fixed by black hole thermodynamics in LQG. As is usual in LQC, we take the value of γ ≈ 0 . 2375.</text>
<text><location><page_4><loc_12><loc_29><loc_88><loc_31></location>In terms of the canonical variables introduced above, the effective Hamiltonian constraint of a closed FLRW universe for the holonomy quantization takes the form [31]</text>
<formula><location><page_4><loc_25><loc_24><loc_88><loc_27></location>H eff = -3 v 8 πGγ 2 λ 2 [ sin 2 ( λb -D ) -sin 2 D +(1 + γ 2 ) D 2 ] + H m , (2.2)</formula>
<text><location><page_4><loc_12><loc_22><loc_29><loc_23></location>where D is defined by</text>
<formula><location><page_4><loc_43><loc_17><loc_88><loc_21></location>D = λ ( 2 π 2 v ) 1 / 3 , (2.3)</formula>
<text><location><page_4><loc_12><loc_11><loc_88><loc_17></location>and λ (= 2 √ √ 3 πγ ) is the minimum area eigenvalue in LQG. Since we only consider a single massive scalar field coupled to gravity, the matter sector of the Hamiltonian constraint denoted by H m is given by</text>
<formula><location><page_4><loc_44><loc_6><loc_88><loc_10></location>H m = p 2 φ 2 v + v U, (2.4)</formula>
<text><location><page_5><loc_12><loc_86><loc_88><loc_91></location>where U refers to the potential of the scalar field. From the total Hamiltonian constraint (2.2), one can derive in a straightforward way the equations of motion for each canonical variable, which turn out to be</text>
<formula><location><page_5><loc_36><loc_82><loc_88><loc_85></location>˙ v = 3 v λγ sin ( λb -D ) cos ( λb -D ) , (2.5)</formula>
<formula><location><page_5><loc_36><loc_79><loc_88><loc_81></location>˙ b = -4 πGγ [ ρ -ρ 1 + P ] , (2.6)</formula>
<formula><location><page_5><loc_36><loc_75><loc_88><loc_79></location>˙ φ = p φ v , ˙ p φ = -v U ,φ , (2.7)</formula>
<text><location><page_5><loc_12><loc_70><loc_88><loc_74></location>here U ,φ denotes the derivative of the potential with respect to the scalar field. The energy density ρ and the pressure P are defined respectively by ρ = H m /v and P = -∂ H m /∂v . In terms of the scalar field and its momentum, they are explicitly given by</text>
<formula><location><page_5><loc_38><loc_65><loc_88><loc_68></location>ρ = p 2 φ 2 v 2 + U, P = p 2 φ 2 v 2 -U. (2.8)</formula>
<text><location><page_5><loc_12><loc_62><loc_41><loc_63></location>Moreover, in Eq. (2.6), ρ 1 is given by</text>
<formula><location><page_5><loc_29><loc_58><loc_88><loc_61></location>ρ 1 = Dρ crit 3 [ 2(1 + γ 2 ) D -sin (2 λb -2 D ) -sin (2 D ) ] , (2.9)</formula>
<text><location><page_5><loc_12><loc_50><loc_88><loc_56></location>here ρ crit = 3 / 8 πGγ 2 λ 2 . It turns out that this is the maximum energy density allowed in a spatiallyflat FLRW universe by LQC where the bounce occurs. In the spatially closed model, the bounce density can be different. From Eq. (2.5) and the vanishing of the total Hamiltonian constraint, it is straightforward to find the modified Friedmann equation which is</text>
<formula><location><page_5><loc_32><loc_45><loc_88><loc_48></location>H 2 = ˙ v 2 9 v 2 = 8 πG 3 ( ρ -ρ min ) ( 1 -ρ -ρ min ρ crit ) , (2.10)</formula>
<text><location><page_5><loc_12><loc_42><loc_16><loc_43></location>where</text>
<formula><location><page_5><loc_36><loc_39><loc_88><loc_41></location>ρ min = ρ crit [ ( 1 + γ 2 ) D 2 -sin 2 D ] . (2.11)</formula>
<text><location><page_5><loc_12><loc_34><loc_88><loc_37></location>Since the right-hand side of the Friedmann equation is non-negative, the energy density at any moment during the evolution has to satisfy the condition</text>
<formula><location><page_5><loc_43><loc_31><loc_88><loc_32></location>ρ min ≤ ρ ≤ ρ max , (2.12)</formula>
<text><location><page_5><loc_12><loc_28><loc_31><loc_29></location>where ρ max is defined by</text>
<formula><location><page_5><loc_42><loc_25><loc_88><loc_26></location>ρ max = ρ min + ρ crit . (2.13)</formula>
<text><location><page_5><loc_12><loc_10><loc_88><loc_23></location>It should be noted that unlike ρ crit , the maximum and minimum energy densities in a closed universe depend explicitly on the volume of the universe and thus do not have fixed values. The bounces and recollapses of a closed universe happen at the turning points when the Hubble rate vanishes, which is equivalent to the condition ρ = ρ min or ρ = ρ max . The character of the turning point, namely whether it is a bounce point or a recollapse point, depends on the second derivative of the volume. To be specific, a turning point is a bounce point when v > 0 and a recollapse point when v < 0. From the Hamilton's equations (2.5)-(2.6), it is straightforward to find that when ρ = ρ min</text>
<formula><location><page_5><loc_38><loc_7><loc_88><loc_8></location>v | ρ min = -12 πGv ( ρ + P -ρ 2 ) , (2.14)</formula>
<text><location><page_6><loc_12><loc_89><loc_27><loc_91></location>and when ρ = ρ max</text>
<text><location><page_6><loc_12><loc_83><loc_16><loc_84></location>where</text>
<formula><location><page_6><loc_36><loc_79><loc_88><loc_82></location>ρ 2 = Dρ crit 3 [ 2(1 + γ 2 ) D -sin(2 D ) ] . (2.16)</formula>
<text><location><page_6><loc_12><loc_70><loc_88><loc_78></location>As a result, depending on the initial conditions, the bounces and recollapses can occur at either the maximum or the minimum energy density. Furthermore, at the turning point when the Hubble rate vanishes, from the equation of motion (2.5), one can find sin(2 λb -2 D ) also vanishes. Therefore, ρ 1 = ρ 2 at a bounce or recollapse point which indicates that v and ˙ b have the same sign at the minimum energy density and opposite signs at the maximum energy density.</text>
<text><location><page_6><loc_12><loc_64><loc_88><loc_69></location>In addition, it can be shown in a straightforward way that the modified Friedmann equation has the right classical limit. In Eq. (2.11), if we take the limit v glyph[greatermuch] 1, then ρ min ≈ ρ crit γ 2 D 2 = 3 8 πGa 2 and the Friedmann equation (2.10) reduces to</text>
<formula><location><page_6><loc_27><loc_59><loc_88><loc_63></location>H 2 = 8 πG 3 ( ρ -ρ 2 ρ crit +2 ργ 2 D 2 -ρ crit γ 2 D 2 -ρ crit γ 4 D 4 ) . (2.17)</formula>
<text><location><page_6><loc_12><loc_57><loc_85><loc_58></location>Considering the classical limit ρ glyph[lessmuch] 1 and a glyph[greatermuch] 1, the above equation can be further reduced to</text>
<formula><location><page_6><loc_34><loc_48><loc_88><loc_55></location>H 2 = 8 πG 3 ρ -8 πG 3 γ 2 D 2 ρ crit , = 8 πG 3 ρ -( 2 π 2 v ) 2 / 3 = 8 πG 3 ρ -1 a 2 , (2.18)</formula>
<text><location><page_6><loc_12><loc_39><loc_88><loc_47></location>which is exactly the classical Friedmann equation for a closed universe. Since loop quantization is only applied to the geometrical sector of the classical phase space, the equations of motion in the matter sector are not changed by the quantum geometrical effects. By using the Hamilton's equations of the scalar field in (2.7), it is straightforward to show that the Klein-Gordon equation and thus the continuity equation also hold in a closed universe of LQC.</text>
<text><location><page_6><loc_12><loc_32><loc_88><loc_38></location>Finally, we would like to briefly review the hysteresis-like phenomenon in a cyclic universe filled with a homogeneous scalar field [51, 53]. Assuming the evolution of the cyclic universe is adiabatic, the work done by the scalar field during each contraction-expansion cycle can be explicitly computed as</text>
<formula><location><page_6><loc_32><loc_27><loc_88><loc_31></location>W = ∮ Pdv = ∫ contraction Pdv + ∫ expansion Pdv. (2.19)</formula>
<text><location><page_6><loc_12><loc_23><loc_88><loc_26></location>On the other hand, the change in the total energy of the scalar field at two consecutive recollapse points is simply given by</text>
<formula><location><page_6><loc_39><loc_19><loc_88><loc_21></location>δM = ρ ( i ) rec v ( i ) rec -ρ ( i -1) rec v ( i -1) rec , (2.20)</formula>
<text><location><page_6><loc_12><loc_8><loc_88><loc_18></location>where ρ ( i ) rec / v ( i ) rec denotes the energy density/volume at the i th recollapse point. Using the energy conservation law W + δM = 0, one can relate the change in the maximum volume of the universe at two successive turnarounds to the net work done by the scalar field in one complete cycle. This relationship in general is model-dependent and also determined by the character of the turnarounds in each model. In the current case with the effective dynamics determined by the modified Friedmann equation (2.10), assuming the energy densities at the recollapse points are given by the</text>
<formula><location><page_6><loc_38><loc_86><loc_88><loc_88></location>v | ρ max = 12 πGv ( ρ + P -ρ 2 ) , (2.15)</formula>
<text><location><page_7><loc_12><loc_88><loc_88><loc_91></location>minimum density ρ min , then when v ( i ) is much greater than unity, the difference in the volumes of two consecutive recollapses can easily be shown to be</text>
<formula><location><page_7><loc_40><loc_83><loc_88><loc_87></location>δv 1 / 3 rec = -∮ Pdv (2 π 2 ) 2 / 3 ρ crit γ 2 λ 2 . (2.21)</formula>
<text><location><page_7><loc_12><loc_73><loc_88><loc_82></location>As a result, there is a change in the maximum volume in each cycle which is directly related to the asymmetry of the pressure of the scalar field during the contraction and expansion phases of each cycle. This results in hysteresis-like behavior which becomes evident via the plots of the equation of state [51, 53]. We find that an increase in the maximum volume and a decrease in the equation of state in each cycle will play an important role in the onset of inflation even with initial conditions which are unfavorable in the classical theory.</text>
<section_header_level_1><location><page_7><loc_23><loc_68><loc_77><loc_70></location>III. φ 2 INFLATION IN A CLOSED FLRW UNIVERSE IN LQC</section_header_level_1>
<text><location><page_7><loc_12><loc_39><loc_88><loc_66></location>Before we discuss the Starobinsky potential case in the next section, we study the occurrence of φ 2 inflation in a closed LQC universe that is sourced by a single scalar field. We focus on initial conditions which in the classical theory do not lead to an inflationary spacetime. The initial conditions are imposed in the Planck regime considering a small homogeneous patch of the universe. For such a patch the spatial curvature term plays an important role and inflation can only take place when the initial conditions are selected such that the potential energy of the scalar field is dominant at the initial time and is large enough to overcome the spatial curvature of the closed universe. When the initial energy density is dominated by the kinetic energy of the inflaton field, the universe recollapses before inflation can occur due to the curvature of a closed universe. This results in a big crunch singularity following the recollapse. One may be tempted to consider initial conditions corresponding to a very large initial volume such that the effect of the spatial curvature becomes so small that the pre-inflationary branch has no recollapse. However, this requires assuming an unnaturally large initial homogeneous patch of the universe in order to set initial conditions for inflation in the Planck regime. Typical initial conditions in the Planck regime start with patches which are not assumed to be homogeneous at macroscopic scales and these are the ones considered in our analysis.</text>
<text><location><page_7><loc_12><loc_27><loc_88><loc_39></location>In the following, we present two representative sets of initial conditions in the parameter space. The first set of initial conditions has the potential energy dominant at the bounce but does not allow for inflation in the classical theory because the potential energy is unable to overcome the classical recollapse caused by the spatial curvature term. The second representative set of initial conditions corresponds to initially dominant kinetic energy which again results in a classical recollapse and a big crunch singularity. For both sets we find that because of quantum gravitational effects, big crunch singularities are avoided and inflation occurs after a few cycles of expansion and contraction.</text>
<text><location><page_7><loc_12><loc_14><loc_88><loc_27></location>In our numerical analysis, the fundamental equations of motion are the effective Hamilton's equations (2.5)-(2.7) introduced in Sec. II. The initial conditions are chosen such that the universe has the highest allowed density given its volume at t = 0. In general, the initial conditions that must be specified are the values of the phase space variables v , b , φ and p φ at t = 0. (The initial conditions will be labelled by the subscript '0'.) In our simulations, we choose the initial volume v 0 and the initial value of the scalar field φ 0 as the two initial free parameters. Since the initial conditions are chosen at a point where the energy density is maximal, the conjugate momentum of the scalar field can be determined by</text>
<formula><location><page_7><loc_38><loc_11><loc_88><loc_13></location>p φ, 0 = ± v 0 √ 2 ρ max -2 U ( φ 0 ) , (3.1)</formula>
<text><location><page_7><loc_12><loc_7><loc_88><loc_10></location>where the ' ± ' reflects the two possible signs of the initial velocity of the scalar field. Finally, the momentum b 0 is fixed by the vanishing of the Hamiltonian constraint (2.2), given v 0 , φ 0</text>
<figure>
<location><page_8><loc_13><loc_53><loc_87><loc_91></location>
<caption>FIG. 1: In this figure, the time evolution of the configuration variables, the equation of state and the energy densities are depicted for the initial conditions (3.3). The initial energy density is chosen to be the maximum allowed density and is completely dominated by the potential energy. All the variables except the scalar field exhibit oscillatory behavior as the universe undergoes a series of bounces and recollapses. The energy density reaches its maximum value at each bounce and recollapse. Inflation happens in the Planck regime when the energy density is about 0 . 41 as the inflaton slowly rolls down the right wing of the chaotic potential.</caption>
</figure>
<text><location><page_8><loc_12><loc_33><loc_88><loc_40></location>and p φ, 0 . Of the possible solutions for b 0 , we used the positive solution for all the plots shown below. Finally, all our simulations were performed using a combination of the StiffnessSwitching, ExplicitRungeKutta, and Automatic numerical integration solving methods in Mathematica with the precision and accuracy goals set to 11.</text>
<section_header_level_1><location><page_8><loc_35><loc_29><loc_65><loc_30></location>A. Representative initial conditions</section_header_level_1>
<text><location><page_8><loc_12><loc_17><loc_88><loc_27></location>Below we discuss two representative cases for the inflationary potential U = 1 2 m 2 φ 2 , one with the potential energy dominant at an initial time in the Planck regime and the other with the kinetic energy dominant. Since we are interested in cases in which inflation does eventually take place, the mass of the inflaton field for the chaotic potential will be fixed as in the standard inflationary paradigm using the scalar power spectrum A s and the scalar spectral index n s for the pivot mode [3]</text>
<formula><location><page_8><loc_21><loc_14><loc_88><loc_15></location>ln(10 10 A s ) = 3 . 044 ± 0 . 014 (68%CL) , n s = 0 . 9649 ± 0 . 0042 (68%CL) . (3.2)</formula>
<text><location><page_8><loc_12><loc_11><loc_79><loc_12></location>Using these the mass of the scalar field is fixed to m = 1 . 23 × 10 -6 in this subsection.</text>
<text><location><page_8><loc_14><loc_9><loc_88><loc_10></location>The first set of representative initial conditions we would like to discuss is presented in Fig. 1.</text>
<text><location><page_9><loc_12><loc_89><loc_27><loc_91></location>This corresponds to</text>
<formula><location><page_9><loc_38><loc_86><loc_88><loc_88></location>v 0 = 10 7 , φ 0 = 7 . 33 × 10 5 , (3.3)</formula>
<text><location><page_9><loc_12><loc_68><loc_88><loc_85></location>in Planck units. With these initial conditions, at initial time t 0 = 0, the initial velocity of the scalar field is zero. Though the inflaton starts with all its energy in potential energy, such a universe soon recollapses and encounters a big crunch singularity in GR. In contrast, we find that in LQC the universe undergoes a series of bounces and recollapses. These cycles occur before the conditions become favorable for inflation to begin. We find that both the maxima and minima of the volume of the universe increase with each bounce following the initial recollapse at t = 0. Note that this case provides an example of the type of universe discussed in Sec. II where the maximal energy density corresponds to a recollapse rather than a bounce. In the current case, since the universe is initially dominated by potential energy, the equation of state is close to negative unity and thus at the maximum energy density,</text>
<formula><location><page_9><loc_42><loc_65><loc_88><loc_67></location>v | ρ max ≈ -12 πGvρ 2 . (3.4)</formula>
<text><location><page_9><loc_12><loc_49><loc_88><loc_64></location>As a result, depending on the sign of ρ 2 , the turning point at the maximum energy density can either be a bounce or a recollapse. From the ρ max -ρ and b plots of Fig. 1, we see that the bounces and recollapses occur at the maximum energy density allowed at that time. It is evident by comparing the ρ max -ρ plot to the volume plot above it that this difference vanishes whenever the universe reaches a turnaround point, corresponding to the maximum density being achieved. Moreover, since v and ˙ b have the opposite signs at ρ = ρ max , the sign of v changes between each bounce and recollapse, which is consistent with the oscillating behavior of the momentum b . The peaks of b also show a noticeable decreasing hysteresis-like behavior [51]. This behavior is also seen in the increase in the maxima (also the minima) of the volume in subsequent cycles.</text>
<text><location><page_9><loc_12><loc_32><loc_88><loc_48></location>When the universe is in the cyclic phase, the equation of state w = P/ρ also changes periodically. It is interesting to note that although the universe is initially dominated by the potential energy of the scalar field, corresponding to w = -1, inflation does not take place immediately after the first bounce but rather at around t = 10 4 (in Planck seconds). The cyclic phase of the universe is accompanied by oscillatory behavior in the equation of state, which attains a local maximum at each bounce and a local minimum at each recollapse. Furthermore, as seen from the behavior of the energy density, we find that inflation starts when the energy density is almost Planckian. Thus, for the above initial conditions we find a successful onset of inflation because of loop quantum gravitational effects, even though this universe is unable to inflate in the classical theory due to the big crunch singularity.</text>
<text><location><page_9><loc_14><loc_30><loc_74><loc_31></location>The second example is depicted in Fig. 2 with the initial conditions given by</text>
<formula><location><page_9><loc_40><loc_27><loc_88><loc_28></location>v 0 = 10 4 , φ 0 = 7 . 36 , (3.5)</formula>
<text><location><page_9><loc_12><loc_7><loc_88><loc_25></location>in Planck units. These correspond to a negative initial velocity for the scalar field. As in the previous case, the positive curvature slows the expansion of the universe after t = 0, which in this case is a bounce, leading to a recollapse which is followed by a non-singular bounce in LQC. Initially the universe undergoes cycles of expansion and contraction with little change in the maximum volume. It is at time t = 300 that the recollapse volume begins to significantly increase with each cycle. During this phase, with each subsequent recollapse, the potential energy fraction gets higher, corresponding to the equation of state reaching a lower minimum value. The time it takes to complete a cycle also tends to increase with time, with the next recollapse happening a little later after the bounce than the one before it. A noticeable hysteresis-like phenomenon can be observed in the w plot where all the troughs correspond to recollapse points whose values decrease with time, while all the peaks correspond to bounce points, which remain at unity throughout the</text>
<figure>
<location><page_10><loc_13><loc_54><loc_87><loc_91></location>
<caption>FIG. 2: With the initial conditions given in (3.5) and a kinetic energy dominated initial state, the volume, the equation of state, the energy density and the momentum b are shown from the initial bounce through the first few cycles. The inset plots in these subfigures show the behavior of the respective variables near the onset of inflation at around 1 . 91 × 10 5 Planck seconds. For the plot of the energy density, its evolution at late times is displayed. In this figure, the bounces occur at the maximum energy density while the recollapses occur at the minimum energy density.</caption>
</figure>
<text><location><page_10><loc_12><loc_36><loc_88><loc_41></location>pre-inflationary evolution. We see that the minimum of the equation of state decreases during each cycle until w reaches -1 / 3 after one final bounce and then inflation takes place, preventing any further recollapses.</text>
<text><location><page_10><loc_12><loc_26><loc_88><loc_35></location>In the figure, we only display the volume, equation of state, and momentum b corresponding to the first several cycles. The behavior of these variables at the onset of inflation is depicted in the inset plots. Note that the momentum b behaves differently than in the first case. In this case, b is monotonically decreasing. As a result, v is always negative when ρ = ρ min and positive when ρ = ρ max . That is, in contrast to Fig. 1, each bounce happens at the maximum energy density while each recollapse happens at the minimum energy density.</text>
<section_header_level_1><location><page_10><loc_39><loc_21><loc_61><loc_22></location>B. Phase space portraits</section_header_level_1>
<text><location><page_10><loc_12><loc_9><loc_88><loc_19></location>In this subsection, we present phase space portraits which are used to understand some aspects of the qualitative behavior of the numerical solutions for a variety of distinct initial conditions. These phase space portraits are based on a set of first-order ordinary differential equations that are equivalent to the Hamilton's equations for the scalar field in (2.7). More specifically, in order to make sure that all the initial conditions are selected from the unit circle representing the bounce point at the initial time, the following phase space variables are used in the phase space portraits,</text>
<figure>
<location><page_11><loc_13><loc_46><loc_86><loc_91></location>
<caption>FIG. 3: Phase space portrait along with volume and density plots for the chaotic potential with m = 0 . 50 and initial volume v 0 = 100 . 00. The upper left plot shows the entire phase space region. The solid black curve corresponds to the energy density at t = 0 where the initial conditions are set, and the dashed red curve represents the maximum energy density that is achieved during the time evolution of any of these initial conditions, which just slightly exceeds the initial energy density. The upper right plot zooms in on the spiral structure. The two lower plots display the volume and energy density, respectively, for the initial conditions ( X 0 = 1 , Y 0 = 0), corresponding to the blue solid curve in the phase space plot. The dashed red straight line in the bottom right panel represents the energy density at which the initial bounce takes place.</caption>
</figure>
<text><location><page_11><loc_12><loc_28><loc_18><loc_29></location>namely,</text>
<formula><location><page_11><loc_36><loc_24><loc_88><loc_27></location>X = mφ √ 2 ρ max , 0 , Y = ˙ φ √ 2 ρ max , 0 . (3.6)</formula>
<text><location><page_11><loc_12><loc_21><loc_47><loc_22></location>At the initial time these satisfy the condition</text>
<formula><location><page_11><loc_45><loc_18><loc_88><loc_20></location>X 2 0 + Y 2 0 = 1 , (3.7)</formula>
<text><location><page_11><loc_12><loc_16><loc_38><loc_17></location>and obey the equations of motion</text>
<formula><location><page_11><loc_42><loc_13><loc_88><loc_15></location>˙ X = mY, (3.8)</formula>
<formula><location><page_11><loc_42><loc_11><loc_88><loc_13></location>˙ Y = -mX -3 HY. (3.9)</formula>
<text><location><page_11><loc_12><loc_7><loc_88><loc_10></location>It should be noted that the variables X and Y and their respective dynamical equations (3.8)-(3.9) do not form a closed system as the Hubble rate H in a closed universe cannot be expressed solely</text>
<text><location><page_12><loc_12><loc_74><loc_88><loc_91></location>as a function of X and Y because of the presence of spatial curvature. As a result, the Friedmann equation (2.10) should be added to form a closed system described by the variables X , Y and v . Since we are interested in understanding inflationary attractors, we focus on the phase space portraits in the subspace spanned by X and Y . Furthermore, since the purpose of this section is to investigate the qualitative behavior of the solutions, in order to achieve a faster convergence of the solutions to the attractors in the plots, a fictitious mass that is much larger than the actual mass (determined from the inflationary scenario and CMB data) is used. For chaotic inflation, we show two representative phase space portraits which are Figs. 3-4. In the former figure, the universe undergoes a few bounces before inflation takes place, while in the latter, there is a greater number of bounces and recollapses before inflation occurs.</text>
<text><location><page_12><loc_12><loc_43><loc_88><loc_73></location>Fig. 3 displays the time evolution of eight distinct solutions, which are shown in different colors and styles from the initial time to the reheating phase. Since the initial maximum energy density is determined by the initial volume (fixed to v 0 = 100), the solutions in the figure start from the same density and hence the same solid black circle but differ in the initial value of the scalar field and its time derivative. There are trajectories originating from a potential energy dominated bounce/recollapse, such as the green dot-dashed and blue solid curves, and trajectories originating from a kinetic-dominated bounce, such as the brown dashed and yellow dotted curves. Other trajectories such as the dot-dash-dashed red and dot-dash-dash-dashed purple curves start from a bounce where the potential energy and the kinetic energy are comparable in magnitude. All these trajectories have qualitatively similar behavior in the sense that after a few bounces and recollapses, the curves starting from each half of the circle merge into horizontal lines (these are almost parallel to the x axis and are known as inflationary separatrices). Subsequently, the two horizontal lines merge in a spiral at the center of the portrait. The spiral structure corresponds to the reheating phase where all the trajectories overlap with one another as shown in the right panel of Fig. 3. In this regime, the scalar field behaves like a damped harmonic oscillator, and solutions starting from different initial conditions in the Planck regime result in the same classical evolution. The origin ( X 0 = 0 , Y 0 = 0) is a fixed point of the system as it is a static solution of the dynamical equations (3.8)-(3.9).</text>
<text><location><page_12><loc_12><loc_26><loc_88><loc_43></location>In the bottom panels, we have explicitly shown the time evolution of the volume and energy density for the set of initial conditions with ( X 0 = 1 , Y 0 = 0). This describes a universe that starts from a potential energy dominated state and after two bounces quickly enters into a phase of inflation when the energy density of the inflaton field is still in the Planck regime. We find that initially, the universe is in a state of contraction. When the volume of the universe reaches about 50 (in Planck volume), a quantum bounce takes place and the universe enters an expanding phase. The bottom left panel of Fig. 3 explicitly shows the first and second bounces after the initial recollapse. The volume at the second bounce is around 90. Since the volumes at the first and second bounces are smaller than the initial volume, the energy density at these bounces exceeds the initial energy density as is depicted in the bottom right panel of the figure.</text>
<text><location><page_12><loc_12><loc_7><loc_88><loc_26></location>The next example we want to analyze is Fig. 4 where four distinct solutions are shown. Two of the solutions with ( X 0 = ± 1 , Y 0 = 0) start from potential energy domination, while the other two with ( X 0 = 0 , Y 0 = ± 1) correspond to kinetic energy domination. All four solutions undergo a number of bounces and recollapses before inflation takes place. The qualitative dynamics of the solutions after the onset of inflation in Fig. 4 is the same as for those in Fig. 3. In Fig. 4, we also observe inflationary separatrices in the top left panel as well as a spiral near the origin in the top right panel where four distinct trajectories converge, showing that all the solutions result in a classical reheating phase. Distinctive features in the pre-inflationary regime can be observed in the bottom panels where we focus on the time evolution of the blue solid curve after the initial recollapse. In the bottom left panel, we see that there are altogether 12 bounces after the initial recollapse. As compared with Fig. 3, the change in the maximum energy density from one bounce</text>
<figure>
<location><page_13><loc_13><loc_46><loc_86><loc_91></location>
<caption>FIG. 4: Phase space portrait along with volume and density plots for the chaotic potential with m = 0 . 05 and initial volume v 0 = 100. The upper left plot shows the entire phase space region with the dashed red circle showing the maximum energy density that is achieved during the evolution of two of the solutions, which is significantly larger than the initial energy density represented by the thick black circle. The upper right plot zooms in on the reheating phase. The two lower plots display the volume and energy density of the blue solid curve in the phase space plot. In the volume plot on the lower left there are twelve bounces before the inflationary phase begins. In the density plot on the lower right the energy density during many of the bounces exceeds the initial energy density, represented by the red dashed line. These bounces correspond to peaks outside of the black circle in the top left phase space plot. The universes corresponding to the red (dot-dashed) and light green (dotted) curves also undergo several bounces, but during these bounces the energy density is near the maximal value, unlike for the blue and brown curves.</caption>
</figure>
<text><location><page_13><loc_12><loc_13><loc_88><loc_25></location>to the next is much larger in Fig. 4, which makes the red dashed circle in the top left panel be separated from the black circle. It can also be observed from the phase space portraits that the inflationary period is much longer for the potential energy dominated initial conditions than for the kinetic energy dominated ones. The trajectories with the former initial conditions merge into inflationary separatrices earlier than those with the latter initial conditions. When the kinetic energy dominates initially, the bounces along the red dot-dashed and green dotted curves happen mostly near the black circle in the phase space portraits, indicating a weak hysteresis.</text>
<figure>
<location><page_14><loc_14><loc_54><loc_86><loc_91></location>
<caption>FIG. 5: This figure shows the volume, equation of state, density, and scalar field as a function of time for the Starobinsky potential with the initial conditions specified by (4.2). The initial conditions are set at the bounce, which takes place at the maximum allowed density. The initial energy density is all in the form of potential energy. This universe undergoes a single bounce before entering inflation. The inflaton rolls down the left wing of the potential, up the right wing, and then inflation occurs as it rolls down the right wing.</caption>
</figure>
<section_header_level_1><location><page_14><loc_17><loc_41><loc_83><loc_42></location>IV. STAROBINSKY INFLATION IN A CLOSED FLRW UNIVERSE IN LQC</section_header_level_1>
<text><location><page_14><loc_12><loc_30><loc_88><loc_38></location>In the previous section we considered the case of chaotic inflation and found that quantum gravity effects assist the onset of inflation in a closed universe. In this section we will study Starobinsky inflation in the same setting. Unlike in classical cosmology, in LQC Starobinsky inflation is not obtained from an R 2 term in the action, 2 rather one generally takes as given the Starobinsky potential in effective dynamics [42, 44, 45, 57], whose form is explicitly given by</text>
<formula><location><page_14><loc_38><loc_26><loc_88><loc_29></location>U = 3 m 2 32 πG ( 1 -e -√ 16 πG 3 φ ) 2 . (4.1)</formula>
<text><location><page_14><loc_12><loc_11><loc_88><loc_25></location>The mass m is fixed to 2 . 44 × 10 -6 from the scalar power spectrum and the spectral index given in (3.2). To determine the value of the mass we assume that the pre-inflationary dynamics would not change the scalar power spectrum in a significant way. Note that unlike chaotic inflation, which can take place in the Planck regime, Starobinsky inflation can only occur on the right wing of the potential, which corresponds to an energy scale that is 10 13 orders of magnitude lower than the Planck scale. As a result, in the classical theory, when starting from the Planck regime the universe inevitably recollapses before inflation sets in, resulting in a big crunch singularity. We now study how the dynamics change in LQC.</text>
<figure>
<location><page_15><loc_13><loc_54><loc_86><loc_91></location>
<caption>FIG. 6: This figure shows the volume, equation of state, density, and scalar field as a function of time for the Starobinsky potential with the initial conditions specified by (4.3). The initial conditions are set at the bounce, which takes place at the maximum allowed density. The initial energy density is dominated by the kinetic energy. This universe undergoes three bounces before entering inflation. The inflaton rolls down the right wing of the potential, up and down the left wing, back up the right wing, and then inflation occurs as it rolls down the right wing.</caption>
</figure>
<section_header_level_1><location><page_15><loc_32><loc_39><loc_67><loc_40></location>A. Some representative initial conditions</section_header_level_1>
<text><location><page_15><loc_12><loc_34><loc_88><loc_37></location>The first example we would like to discuss corresponds to Fig. 5, which results from the initial conditions (in Planck units)</text>
<formula><location><page_15><loc_40><loc_31><loc_88><loc_32></location>v 0 = 10 7 , φ 0 = -3 . 48 . (4.2)</formula>
<text><location><page_15><loc_12><loc_12><loc_88><loc_29></location>These initial conditions are chosen at a bounce with energy density ρ 0 = 0 . 41. In this case, the initial bounce is completely dominated by the potential energy of the scalar field and the inflaton is released from rest on the left wing of the potential, rolls down, and then climbs up the right wing until reaching the turnaround point. Inflation takes place at ρ ≈ 10 -13 when the inflaton slowly rolls down the right wing of potential. Hence the scalar field behaves in the same way as in a spatially-flat universe. However, the behavior of other dynamical variables is quite different. From the volume plot in Fig. 5, one can see that before the onset of inflation at around t = 8 . 39 × 10 6 , the universe undergoes a bounce at t = 1 . 46 × 10 6 where the energy density reaches the maximum allowed value and a recollapse at around t = 6 . 47 × 10 5 with the minimum allowed energy density at that time.</text>
<text><location><page_15><loc_12><loc_7><loc_88><loc_12></location>The equation of state in the upper right panel of Fig. 5 shows explicitly that the initial bounce is dominated by potential energy with w ≈ -1. Meanwhile the single bounce that takes place during the evolution is dominated by the kinetic energy of the scalar field as it corresponds to the last</text>
<text><location><page_16><loc_12><loc_77><loc_88><loc_91></location>peak in the w plot before the onset of inflation. The second to last peak in the w plot corresponds to the moment when the inflaton crosses the origin, where the potential energy vanishes, forcing w = 1. It should be noted that even though the potential energy is initially dominant, inflation cannot occur on the left wing of the potential because it is too steep for the slow-roll to occur. Only the right wing of the Starobinsky potential can drive inflation in both closed and spatiallyflat universes. This is also manifest in the phase space portrait shown later where only a single inflationary separatrix is observed. Since this universe only has a single bounce after the initial one, we do not see oscillatory behavior in w before the onset of inflation.</text>
<text><location><page_16><loc_12><loc_72><loc_88><loc_77></location>The second example is given in Fig. 6, corresponding to a universe initially dominated by the kinetic energy of the scalar field. The initial conditions for the numerical simulation are set at the big bounce at t = 0, given explicitly by</text>
<formula><location><page_16><loc_38><loc_69><loc_88><loc_71></location>v 0 = 3 . 75 × 10 7 , φ 0 = 5 . 00 , (4.3)</formula>
<text><location><page_16><loc_12><loc_32><loc_88><loc_67></location>which implies that the inflaton is initially released from the right wing of the potential with a large initial velocity. From the bottom right panel of Fig. 6, we see that the inflaton first rolls down the right wing of the potential, then climbs up the left wing and momentarily stops at the turnaround point. Afterwards, it rolls down the left wing of the potential, climbs up the right wing and then reaches another turnaround point. Finally, slow-roll inflation takes place when the inflaton slowly rolls down the right wing of the potential for the second time. The energy density at which inflation occurs is about 10 13 orders of magnitude below the Planck energy. Fig. 6 shows that as compared with the previous example, the current case is richer in terms of the pre-inflationary dynamics. Note that the three recollapses associated with the three bounces in this example all occur at an energy scale far below the Planck energy, while all the bounces happen at a similar energy density that is around ρ crit . From the figure one notes that the third cycle of expansion and contraction is highly asymmetrical, which is due to the asymmetry of the Starobinsky potential itself. For this particular cycle the expanding (contracting) phase takes place on the left (right) wing of the potential. From the w plot in the top right panel, one can easily identify several important moments featuring the behavior of the scalar field before the onset of inflation. For example, the first trough at w = -1 before t = 10 4 corresponds to the turnaround point of the inflaton on the left wing of the potential. While the first two bounces take place before this trough, the effective dynamics following the first trough are qualitatively similar to what happened in Fig. 5. This is confirmed by the similar behavior of the w plots in these two cases in two particular regions: around the second to last peak where the inflaton crosses the origin and around the final peak where the universe undergoes a bounce.</text>
<text><location><page_16><loc_14><loc_30><loc_75><loc_31></location>The next example concerns Fig. 7, which corresponds to the initial conditions</text>
<formula><location><page_16><loc_37><loc_27><loc_88><loc_29></location>v 0 = 3 . 11 × 10 8 , φ 0 = -0 . 50 . (4.4)</formula>
<text><location><page_16><loc_12><loc_7><loc_88><loc_25></location>In this case, the initial bounce at t = 0 is dominated by the kinetic energy of the scalar field. The universe undergoes a longer series of bounces and recollapses before the onset of inflation as is depicted in the top left panel of Fig. 7. With a large initial velocity, the inflaton, which starts from the left wing, gets much higher up on the right wing than in the two previous cases. Hence it takes longer for the inflaton to turn around, leading to many more cycles in the pre-inflationary phase. All the bounces in the pre-inflationary phase happen at the Planck energy density around ρ crit as is shown for the first four bounces. Meanwhile, all the recollapses occur at the minimum energy density far below the Planck scale. Similar to the previous two cases, following the turnaround point on the right wing of the potential, inflation takes place at an energy density of around 10 -13 times the Planck density. In addition to the volume of the universe and its energy density, the equation of state, which is shown in the top right panel, also oscillates with the same period as the</text>
<text><location><page_17><loc_15><loc_83><loc_15><loc_83></location>✈</text>
<figure>
<location><page_17><loc_14><loc_55><loc_86><loc_91></location>
<caption>FIG. 7: This figure shows the volume, equation of state, density, and scalar field as a function of time for the Starobinsky potential with the initial conditions specified by (4.4). The initial conditions are set at the bounce, which takes place at the maximum allowed density. The initial energy density is dominated by the kinetic energy. This universe undergoes many bounces before entering inflation. The inflaton rolls up and down the left wing of the potential, up the right wing, and then inflation occurs as it rolls down the right wing.</caption>
</figure>
<text><location><page_17><loc_12><loc_21><loc_88><loc_41></location>volume. At each bounce, the equation of state reaches a maximum, while with each consecutive recollapse in the pre-inflationary phase it reaches a lower minimum value, slowly approaching -1 / 3, marked by the upper red dashed line in the plot. We see that even though the equation of state has a value close to -1 / 3 during the first recollapse, it subsequently takes many cycles for it to cross this value and inflation to begin. This is due to the fact that these initial conditions give rise to a weak hysteresis, which is also evident from the volume plot. The volume at which the universe recollapses increases in consecutive cycles, resulting in a decrease in the minimum energy density at each recollapse. With each cycle the curvature effect is slightly weaker, and hence each cycle brings the universe closer to the right conditions for inflation to begin. Though weaker in strength than in previous cases, the increasing hysteresis prevents the universe from undergoing infinitely many cycles of expansion and contraction, instead facilitating the onset of an inflationary period in a closed LQC universe after a finite number of cycles.</text>
<text><location><page_17><loc_12><loc_18><loc_88><loc_21></location>Fig. 8 is an example of a case in which inflation does not appear in our simulation, which ran through time 10 8 . The initial conditions are given by</text>
<formula><location><page_17><loc_38><loc_15><loc_88><loc_17></location>v 0 = 2 . 50 × 10 7 , φ 0 = 5 . 00 . (4.5)</formula>
<text><location><page_17><loc_12><loc_7><loc_88><loc_13></location>In this case, the inflaton is released from the right wing of the potential with a large initial velocity, climbs up the left wing of the potential and then turns around, finally climbing up the right wing. The volume plot shows a very large number of bounces and recollapses. Similar to the previous cases, all the bounces occur at the maximum energy density in the Planck regime while all the</text>
<figure>
<location><page_18><loc_13><loc_54><loc_86><loc_91></location>
<caption>FIG. 8: This figure shows the volume, equation of state, density, and scalar field as a function of time for the Starobinsky potential with the initial conditions specified by Eq. (4.5). The initial energy density at the bounce is dominated by the kinetic energy. This universe undergoes many bounces and we are unable to see the onset of inflation up through the time we have plotted. The inflaton rolls down the right wing of the potential, up and down the left wing, and then rolls up the right wing, not turning around in the time shown here.</caption>
</figure>
<text><location><page_18><loc_12><loc_17><loc_88><loc_41></location>recollapses take place at the minimum energy density, which is far below the Planck energy. For the time range of the simulation, which is 10 8 (in Planck seconds), the volume plot shows a long cyclic phase and no inflation. However, as in the previous case where we did observe the onset of inflation, an increasing hysteresis can be seen in the w plot, which shows the minima of the equation of state decrease with each recollapse. The decrease with each cycle is very small, and since towards the end of the evolution shown the minima have only reached -0.08, it is evident that far more time is needed before they can cross -1/3. Note that the decrease in the minimum of w with each cycle was also small in the previous case, but because at the beginning of the oscillatory phase the minima were already close to -1/3, we were able to see inflation before time 10 8 . The minima of w being higher in this case at the start of the oscillatory phase reflects the fact that these initial conditions are less favorable for inflation than in the previous cases, where either the minima of w were close to -1/3 at the start of the oscillatory phase or the initial conditions were such that the hysteresis effect was stronger and so an extremely long series of cycles was not necessary to bring about the onset of inflation.</text>
<section_header_level_1><location><page_19><loc_39><loc_89><loc_61><loc_91></location>B. Phase space portraits</section_header_level_1>
<text><location><page_19><loc_12><loc_84><loc_88><loc_87></location>To understand the qualitative dynamics of general solutions with various initial conditions, we present phase space portraits for the Starobinsky potential using</text>
<formula><location><page_19><loc_32><loc_79><loc_88><loc_83></location>X = χ 0 ( 1 -e -√ 16 πG 3 φ ) , Y = ˙ φ √ 2 ρ max , 0 , (4.6)</formula>
<text><location><page_19><loc_12><loc_73><loc_88><loc_77></location>where χ 0 = √ 3 m 2 32 πGρ max , 0 and ρ max , 0 stands for the initial maximum energy density. These variables obey the following set of first-order differential equations</text>
<formula><location><page_19><loc_37><loc_70><loc_88><loc_72></location>˙ X = mY (1 -X/χ 0 ) , (4.7)</formula>
<formula><location><page_19><loc_37><loc_68><loc_88><loc_70></location>˙ Y = -3 HY -mX (1 -X/χ 0 ) . (4.8)</formula>
<text><location><page_19><loc_12><loc_43><loc_88><loc_66></location>Together with (2.10), the above equations (4.7)-(4.8) form a complete set of dynamical equations for a system that is described by X , Y , and v . Since the main goal of our numerical analysis is to determine whether the inflationary phase is a local attractor for a variety of initial conditions, we focus on a 2-dimensional phase space plot in the subspace spanned by X and Y . From Eqs. (4.7)-(4.8), it follows that there are two fixed points in the system, which are ( X = 0 , Y = 0) and ( X = χ 0 , Y = 0). The first fixed point corresponds to the end of the reheating phase where the energy density of the inflaton field is very small and the volume of the universe is very large. The second fixed point is located along the X = χ 0 boundary separating the region with real values of the scalar field from the region where the scalar field is complex [42]. In the following, we will only focus on the region in which the scalar field is real (to the left of the boundary line). As for the φ 2 potential, in order to make the various solutions converge in a short time, we use a larger value for the mass associated with the scalar field in the phase space portraits. The use of such a value does not change the qualitative behavior of the solutions, including the existence of inflationary separatrices and cosmological attractors, which are the properties of interest.</text>
<text><location><page_19><loc_12><loc_7><loc_88><loc_42></location>The first phase space portrait is presented in Fig. 9. The upper left plot shows the entire phase space region which includes trajectories for six distinct initial conditions. For the Starobinsky potential only the trajectories confined to the left of the vertical black line X = χ 0 correspond to real values of the scalar field. Thus, these are the only cases we consider in the phase space plots. The phase space evolution of a generic solution with one bounce is represented by the solid blue curve in the phase space portrait, which corresponds to the initial conditions ( φ 0 = 0 . 23 , p φ 0 = -930) at v 0 = 100. The trajectory starts from the black solid circle which corresponds to the initial bounce at the maximum energy density. It then moves towards the origin as the scalar field loses energy. When the blue curve is close to the origin, a recollapse occurs and the universe enters into a contracting phase, increasing the energy density of the scalar field and making the blue curve move away from the origin. Afterwards, as the energy density of the scalar field increases, a bounce takes place when the blue curve hits the dashed red circle and the maximum energy density at that time is reached. After the bounce, the universe re-enters a state of expansion and the scalar field starts to lose energy again. The blue curve then moves towards the origin and merges into the inflationary separatrix (the short, curved, horizontal line traveling left towards the origin, during which inflation takes place) before finally falling into the spiral at the center of the plot. The other trajectories have similar qualitative behavior to the blue solid curve. The difference lies in the number of bounces in the pre-inflationary phase, which for this plot is either none or one, as well as the value of the maximum energy density at which the bounce occurs. For example, the dot dashed green curve starts from the top and moves directly towards the inflationary separatrix without any further bounces, while the red dotted, yellow dot-dashed, and green dot-dot-dashed</text>
<figure>
<location><page_20><loc_13><loc_46><loc_86><loc_91></location>
<caption>FIG. 9: Phase space portrait, volume, and density plots for the Starobinsky potential with m = 0 . 62 and initial volume v 0 = 1000. There are six trajectories shown, each corresponding to a distinct set of initial conditions. For all the curves, the initial maximum density is ρ max , 0 = 0 . 44 and χ 0 = 0 . 16. The red dashed circle in the phase space portrait represents the maximum density achieved during any of the cycles shown. The red dashed line in the density plot represents the maximum density at the initial time.</caption>
</figure>
<text><location><page_20><loc_12><loc_17><loc_88><loc_34></location>curves all experience a single bounce that happens at the highest allowed energy density, which is close to the initial energy density. Compared with φ 2 inflation, where we saw relatively long and straight horizontal lines heading towards the origin, the inflationary separatrix in Starobinsky inflation is significantly less noticeable. For this reason, we show the upper right panel which zooms in on the inflationary separatrix, where all the curves merge, and the attractor at the origin. Finally, to show the details of the evolution of a generic solution in the phase space portrait, the bottom panels show the behavior of the volume and energy density for the solid blue curve. We see that in the volume plot on the bottom left, the single bounce happens at around t = 228, while in the density plot on the bottom right we see that the energy density at this bounce exceeds the initial energy density, represented by the dashed red line in the graph.</text>
<text><location><page_20><loc_12><loc_9><loc_88><loc_17></location>The second phase space portrait presented in Fig. 10 includes trajectories corresponding to three distinct initial conditions. All of them describe universes that undergo bounces before inflation. The upper left plot shows the entire phase space region. For the universes corresponding to these initial conditions, the energy density never exceeds the initial density. As a result, all the trajectories are confined within the initial black unit circle. The representative solution is again</text>
<figure>
<location><page_21><loc_13><loc_46><loc_86><loc_91></location>
<caption>FIG. 10: Phase space portrait along with volume and density plots for the Starobinsky potential with m = 0 . 79 and initial volume v 0 = 50. For all the curves, the initial maximum density is ρ max , 0 = 1 . 21 and χ 0 = 0 . 12. The red dashed line in the density plot shows the maximum density at the initial time.</caption>
</figure>
<text><location><page_21><loc_12><loc_18><loc_88><loc_37></location>displayed as the solid blue curve, which is analyzed further in the bottom panels. The initial conditions for the blue curve are ( φ 0 , p φ 0 ) = (8 . 97 , -77) at v 0 = 50. As shown in the bottom panels, the universe starts at a bounce and then undergoes a series of cycles with alternating contracting and expanding phases until inflation begins. All the bounces happen at the maximum energy density with a volume no larger than the initial volume, while all the recollapses happen at the minimum energy density for each cycle, all of which are also in the Planck regime as can be seen in the ρ plot. In the top right panel, we zoom in on the inflationary separatrix and the attractor at the center. Since all the recollapses happen in the Planck regime, there are no curves that approach the origin before merging into the separatrix. We can clearly see from the plot that trajectories starting from different initial conditions have the same late-time dynamics consisting of an inflationary phase followed by a reheating phase.</text>
<section_header_level_1><location><page_21><loc_43><loc_14><loc_57><loc_15></location>V. SUMMARY</section_header_level_1>
<text><location><page_21><loc_12><loc_7><loc_88><loc_12></location>The goal of this manuscript is to understand the onset of inflation in closed universes for low energy scale inflationary models, such as Starobinsky inflation. Starting inflation in such cases has remained a long-standing problem because of the recollapse caused by the spatial curvature</text>
<text><location><page_22><loc_12><loc_77><loc_88><loc_91></location>and the big crunch singularity that are unavoidable in the classical theory[11, 12]. We explored a resolution of this problem in the setting of LQC where big bang/big crunch singularities are robustly resolved due to non-perturbative quantum gravity effects and the pre-inflationary phase is in general characterized by a series of bounces and recollapses of the universe. For comparison we first considered the case of φ 2 inflation and demonstrated that non-singular cyclic evolution in the pre-inflationary phase sets the stage for inflation to begin even for very unfavorable initial conditions. The analysis was then repeated for Starobinsky inflation, where the problem is far more severe, yielding similar results.</text>
<text><location><page_22><loc_12><loc_31><loc_88><loc_77></location>For the φ 2 potential, inflation can take place at different energy scales ranging from the Planck regime to an energy density that is 10 12 orders of magnitude below the Planck density. Inflation can also take place on both sides of the potential, resulting in two inflationary separatrices in the phase space portraits. On the other hand, for the Starobinsky potential, with the mass parameter fixed by observations, inflation can only take place on the right wing of the potential at an energy density that is around 10 13 orders of magnitude below the Planck density. Due to the asymmetry of the potential, there is only one inflationary separatrix in the phase space portraits. We found different features in terms of the pre-inflationary dynamics, which is when the cycles take place. In the φ 2 model, both the bounces as well as the recollapses can occur at the maximum energy density, which means that t = 0 where the energy density takes on the maximum allowed value can correspond to either a bounce or a recollapse. In the Starobinsky model, on the other hand, the bounces always happen at the maximum energy density and the recollapses happen at the minimum energy density, which means that t = 0 always corresponds to a bounce. Furthermore, for the Starobinsky potential it tends to take longer for the first recollapse to occur, and there also tends to be more time between subsequent cycles than for the φ 2 potential. This contributes to the delay in the onset of inflation in the Starobinsky model as compared with the φ 2 model. With respect to which initial conditions give rise to inflation, we found that the evolution of the universes in both models differ most in those cases where the oscillatory behavior of w begins with the minima of w not close to -1 / 3. In such cases, with each cycle the minimum of w decreases noticeably for the φ 2 potential, so that after some relatively small, finite number of cycles w crosses -1/3 and inflation begins. For the Starobinsky potential, however, the minima of w decrease very slowly with each cycle. This means that when starting with initial conditions for which the minima of w are not close to -1/3 when the oscillatory behavior in pre-inflationary epoch begins, then it can take an extremely long time for the minima of w to decrease enough to cross -1/3. Thus, in comparison to φ 2 potential, the mechanism resulting from non-singular cyclic evolution and leading to the onset of inflation in the Starobinsky potential is weaker but nevertheless strong enough to overcome problems encountered in the classical theory.</text>
<text><location><page_22><loc_12><loc_7><loc_88><loc_31></location>In summary, we have shown that the problem of the onset of inflation for low energy scale models, such as the Starobinsky potential, in a closed universe can be successfully resolved by quantum gravity effects. The primary reason for the initiation of the inflationary phase is a progressive decrease in the value of the equation of state with each cycle captured in the hysteresislike phenomena seen earlier for the φ 2 potential in LQC [51] and other bouncing models [53, 54]. When the equation of state becomes less than -1 / 3 in a particular cycle, recollapses no longer occur and inflation starts. While for the φ 2 potential we found inflation to occur in all the considered cases, for the Starobinsky inflation we found some cases in which the hysteresis-like phenomenon is so weak that the onset of inflation, though expected, is delayed. It would be interesting to understand physical phenomena that can make the onset of inflation even in such extreme cases more favorable, an example of which will be discussed in a future work [58]. Apart from such cases, thanks to the singularity resolution due to non-perturbative quantum gravity effects, the Starobinsky potential results in inflation in a short time even when starting from initial conditions which are highly unfavorable for inflation in the classical theory.</text>
<section_header_level_1><location><page_23><loc_42><loc_89><loc_58><loc_91></location>Acknowledgements</section_header_level_1>
<text><location><page_23><loc_12><loc_82><loc_88><loc_87></location>This work is supported by the NSF grants PHY-1454832 and NSF PHY-1852356. L.G. thanks the REU program of the Department of Physics and Astronomy at LSU during which most of this work was carried out.</text>
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<text><location><page_25><loc_15><loc_86><loc_47><loc_88></location>M. Motaharfar, P. Singh, To appear (2020).</text>
</document> | [{"title": "Quantum gravitational onset of Starobinsky inflation in a closed universe", "content": "Lucia Gordon 1 , \u2217 Bao-Fei Li 2 , \u2020 and Parampreet Singh 2 \u2021 1 Department of Physics, Harvard University, Cambridge, MA 02138, USA 2 Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA Recent cosmic microwave background observations favor low energy scale inflationary models in a closed universe. However, onset of inflation in such models for a closed universe is known to be severely problematic. In particular, such a universe recollapses within a few Planck seconds and encounters a big crunch singularity when initial conditions are given in the Planck regime. We show that this problem of onset of inflation in low energy scale inflationary models can be successfully overcome in a quantum gravitational framework where the big bang/big crunch singularities are resolved and a non-singular cyclic evolution exists prior to inflation. As an example we consider a model in loop quantum cosmology and demonstrate that the successful onset of low energy scale inflation in a closed universe is possible for the Starobinsky inflationary model starting from a variety of initial conditions where it is impossible in the classical theory. For comparison we also investigate the onset of inflation in the \u03c6 2 inflationary model under highly unfavorable conditions and find similar results. Our numerical investigation including the phase space analysis shows that the pre-inflationary phase with quantum gravity effects is composed of non-identical cycles of bounces and recollapses resulting in a hysteresis-like phenomenon, which plays an important role in creating suitable conditions for inflation to occur after some number of non-singular cycles. Our analysis shows that the tension in the classical theory amounting to the unsuitability of closed FLRW universes with respect to the onset of low energy scale inflation can be successfully resolved in loop quantum cosmology.", "pages": [1]}, {"title": "I. INTRODUCTION", "content": "Cosmic microwave background (CMB) observations suggest that inflationary models with an inflaton in a plateau potential have been favored since Planck 2013 [1-3]. The inflationary models that are consistent with the Planck constraints on the spectral index and the tensor-to-scalar ratio for the pivot mode include the Starobinsky model [4], the Higgs inflationary model [5, 6] and a broad class of cosmological attractor models [7-10]. In these models, inflation takes place at an energy scale far below the Planck density. For this reason, they are also called the low energy scale inflationary models. But if the universe is closed, then the onset of inflation in such models is severely problematic [11-13]. This problem, which is tied to the recollapse of the universe and the big crunch singularity, can be avoided in spatially-flat and open universes [11, 12]. However, the recent Planck Legacy 2018 (PL 2018) release has confirmed the presence of an enhanced lensing amplitude in the CMB power spectra [14, 15], which favors a closed universe over a spatially-flat one [16]. Consequently, a pertinent question is the following: How does a closed universe starting from the Planck regime lead to successful low scale inflation? To answer this question, one needs a framework which goes beyond the classical description of spacetime and allows a robust resolution of the big crunch singularity while alleviating the problem of initial conditions for low scale inflation in a closed universe. While the low energy inflationary model in a closed universe starting from the Planck regime typically ends in a big crunch singularity in a few Planck seconds, the onset of inflation in a closed universe is not as problematic for the chaotic inflationary models such as \u03c6 2 inflation which, however, is not favored by the Planck data. In these models, where inflation can occur at higher energy scales, the initial conditions of the inflaton starting from a single Planck sized domain can be such that the kinetic and gradient energy of the scalar field are smaller than its potential energy which is order U \u223c 1 (in Planck units). Since the potential energy is dominant, the universe can avoid the recollapse and the subsequent big crunch singularity, and the dynamical evolution successfully results in a phase of inflation shortly after the initial time. Note that this is true only for those initial conditions where potential energy is dominant. Of course, if the inflaton starts with kinetic energy domination then the fate of such a universe even in a \u03c6 2 inflationary model can be similar to that of low energy scale inflation in the sense that the universe encounters a big crunch singularity before a phase of inflation can start. Previous studies show that the probability of the quantum creation of a closed universe in which inflation takes place at the energy scale U glyph[lessmuch] 1 is exponentially suppressed [17-21]. However, all of the above arguments exclude the role of non-perturbative quantum gravity effects in the Planck regime. Further, an important problem irrespective of the spatial curvature of the universe is that inflationary models are past-incomplete in the classical theory [22, 23]. Since quantum gravity effects are expected to resolve spacetime singularities, it is quite possible that the above situation changes dramatically when one considers a quantum gravitational version of the model. The goal of this paper is to show that the above problem of the onset of inflation in a closed universe can be successfully resolved in a setting motivated by loop quantum gravity where big bang and big crunch singularities are replaced by a big bounce due to quantum geometric effects. In our analysis we consider the Starobinsky inflationary model as an example of a low energy inflationary model and study the onset of inflation for a variety of initial conditions. For comparison we also investigate the \u03c6 2 model for initial conditions which are not favorable in the classical theory for the onset of inflation. While our analysis uses techniques of loop quantum cosmology (LQC) for the chaotic and Starobinsky potentials, the essential idea and results can be replicated for any bouncing model and other inflationary potentials. Our framework is based on a loop quantization of a closed universe performed in LQC which is a non-perturbative quantization of cosmological models based on loop quantum gravity [24]. Unlike the Wheeler-DeWitt quantum cosmology where the spacetime manifold is differentiable, LQC is based on a discrete quantum geometry predicted by loop quantum gravity. The quantum evolution is dictated by a non-singular quantum difference equation which results in a resolution of the big bang singularity by replacing it with a quantum bounce when the spacetime curvature becomes Planckian [25-27], including in the presence of inflationary potentials [28, 29], anisotropies and inhomogenities [24]. Using the consistent histories formalism one can also compute the probability of the bounce which turns out to be unity [30]. For a closed universe, loop quantum gravitational effects resolve both the past and future singularities, resulting in a cyclic universe [31-33]. It turns out that the quantum evolution can be very well approximated by the effective dynamics obtained from an effective spacetime description [26, 31, 34, 35]. Using this description singularity resolution has been established to be a generic feature for various isotropic and anisotropic models in LQC [36, 37], and various phenomenologically interesting consequences for the early universe and resulting signatures in the CMB have been discussed [38]. For the spatially-flat model it was found that the inflationary dynamics is an attractor after the bounce for \u03c6 2 , power-law inflation and the Starobinsky potential [39-47], and the probability for inflation to occur is very large [48-50]. For closed universes, the dynamics with the \u03c6 2 inflationary potential in the presence of spatial curvature has been previously studied in LQC [51], which revealed some novel features in comparison to the spatially-flat case. It was found that due to the recollapse caused by the spatial curvature and the bounce caused by quantum geometry, the evolution of a closed universe filled with a homogeneous scalar field in an inflationary potential is usually characterized by a number of cycles in the preinflationary era with a hysteresis-like phenomenon due to the asymmetry in the equation of state (or equivalently the asymmetry in the pressure of the scalar field) in each cycle. It was found that quantum geometric modifications in LQC enhance this hysteresis-like phenomenon in comparison to previously studied bouncing models where this phenomenon was noted earlier [52-54]. Owing to this hysteresis-like behavior, even when starting from unfavorable initial conditions for inflation, the universe comes out of non-inflationary cyclic evolution and enters into a phase of inflation. Once the universe is in this phase, the scalar field starts to play a dominant role and the effects of the spatial curvature become negligible due to the exponential expansion of the universe, implying that no further cycles occur. Though the work in Ref. [51] found evidence of inflation occurring after hysteresis in \u03c6 2 inflation, the robustness of the existence of the inflationary phase and its possible generalization to low energy scale inflation models was not studied. The aim of this paper is to reconcile the closed universe scenario with low scale inflation in the framework of LQC for the Starobinsky potential. 1 Our strategy follows the encouraging results for \u03c6 2 inflation in the closed model of LQC [51] with an aim to understand the onset of inflation with the Starobinsky potential and also to gain further insights by comparing it with \u03c6 2 inflation. Using the effective Hamilton's equations resulting from the holonomy quantization for a closed LQC universe [31], we consider a single scalar field minimally coupled to gravity with the initial conditions set at the maximum energy density at some initial volume. We explore the background evolution of the universe starting with different types of initial conditions, with varying ratios of kinetic and potential energy, some of which are unfavorable for inflation to start in the classical theory, and perform numerical simulations for the \u03c6 2 and Starobinsky potentials up through the onset of inflation. To investigate some aspects of the qualitative behavior we also study two-dimensional phase space portraits and find inflationary separatrices and other cosmological attractors in the plots. Our results show that both \u03c6 2 inflation and Starobinsky inflation can take place under a variety of initial conditions starting from the Planck regime in the closed model. The primary reason for this is the hysteresis-like phenomena which are found to be much weaker in Starobinsky inflation. Due to this, the onset of inflation is delayed in comparison to \u03c6 2 inflation. This manuscript is organized as follows. In Sec. II, we briefly review the effective dynamics of a closed FLRW universe using the holonomy quantization in LQC. We obtain the necessary equations which include the Hamilton's equations and the modified Friedmann equation for the purpose of a detailed analysis of the background dynamics of the universe when gravity is minimally coupled to a single scalar field with an inflationary potential. In Sec. III, we consider \u03c6 2 inflation in a closed FLRW universe in LQC with the initial conditions given in the Planck regime. We address the general properties of the pre-inflationary dynamics and present phase space portraits with trajectories starting from various initial conditions that all end up with a reheating phase, corresponding to the spiral at the center of the plots. In Sec. IV, we study the Starobinsky potential in a closed universe in LQC. We show that after taking into account quantum gravitational effects, many initial conditions that are unfavorable for inflation in classical cosmology do in fact lead to inflation at late times in LQC. The phase space portraits will be presented in order to show the qualitative behavior of numerical solutions starting with various initial conditions. Finally, in Sec. V, we summarize the main results concerning \u03c6 2 and Starobinsky inflation in a closed universe in LQC and discuss the differences and similarities between these two inflationary models. In our paper, we use Planck units with glyph[planckover2pi1] = c = 1 while keeping Newton's constant G explicit in our formulas. In the numerical analysis, G is also set to unity.", "pages": [1, 2, 3, 4]}, {"title": "II. EFFECTIVE DYNAMICS OF A CLOSED UNIVERSE IN LOOP QUANTUM COSMOLOGY", "content": "In LQC, the quantization of the homogeneous and isotropic FLRW universe is carried out in terms of the Ashtekar-Barbero connection A i a and its conjugate triad E i a . The non-perturbative modifications to the classical Hamiltonian constraint arise from the regularization of the field strength of the connection and the inverse volume terms. It turns out that the latter are not dominant compared to the field strength modifications for singularity resolution [31] and quickly decay for volumes greater than the Planck volume. For this reason we will only focus on the effective dynamics incorporating modifications to the field strength of the connection. In the literature, the regularization of the field strength in a closed FLRW universe has been explored in two different ways. One is based on the quantization of the holonomy of the connection over closed loops [31], the other is the connection based quantization [33]. Both of these approaches, which can be viewed as quantization ambiguities, lead to singularity resolution and give similar qualitative behavior away from the bounce regime [56]. For the \u03c6 2 potential a comparative analysis of these two quantizations was performed in Ref. [51], which revealed the robustness of the qualitative features of hysteresis and the subsequent inflationary phase. In the following, we will study the holonomy based quantization and we expect the main results to hold for the connection based quantization as well. In a cosmological setting, due to the homogeneity and isotropy of the universe, the AshtekarBarbero connection and its conjugate triad can be symmetry-reduced to the canonical pair of c and p [31]. This set of canonical variables is equivalent to a new set of variables, namely b and v that are commonly used in the \u00af \u00b5 scheme [26]. In a closed universe, the physical volume of the unit sphere spatial manifold is given by v = | p | 3 / 2 = 2 \u03c0 2 a 3 with a representing the scale factor of the universe. The conjugate variable b is defined via b = c | p | -1 / 2 . Besides the gravitational degrees of freedom, in order to initiate the onset of inflation, one also needs degrees of freedom in the matter sector which are the scalar field \u03c6 and its conjugate momentum p \u03c6 . These fundamental canonical pairs in the phase space satisfy the Poisson brackets: where \u03b3 is the Barbero-Immirzi parameter fixed by black hole thermodynamics in LQG. As is usual in LQC, we take the value of \u03b3 \u2248 0 . 2375. In terms of the canonical variables introduced above, the effective Hamiltonian constraint of a closed FLRW universe for the holonomy quantization takes the form [31] where D is defined by and \u03bb (= 2 \u221a \u221a 3 \u03c0\u03b3 ) is the minimum area eigenvalue in LQG. Since we only consider a single massive scalar field coupled to gravity, the matter sector of the Hamiltonian constraint denoted by H m is given by where U refers to the potential of the scalar field. From the total Hamiltonian constraint (2.2), one can derive in a straightforward way the equations of motion for each canonical variable, which turn out to be here U ,\u03c6 denotes the derivative of the potential with respect to the scalar field. The energy density \u03c1 and the pressure P are defined respectively by \u03c1 = H m /v and P = -\u2202 H m /\u2202v . In terms of the scalar field and its momentum, they are explicitly given by Moreover, in Eq. (2.6), \u03c1 1 is given by here \u03c1 crit = 3 / 8 \u03c0G\u03b3 2 \u03bb 2 . It turns out that this is the maximum energy density allowed in a spatiallyflat FLRW universe by LQC where the bounce occurs. In the spatially closed model, the bounce density can be different. From Eq. (2.5) and the vanishing of the total Hamiltonian constraint, it is straightforward to find the modified Friedmann equation which is where Since the right-hand side of the Friedmann equation is non-negative, the energy density at any moment during the evolution has to satisfy the condition where \u03c1 max is defined by It should be noted that unlike \u03c1 crit , the maximum and minimum energy densities in a closed universe depend explicitly on the volume of the universe and thus do not have fixed values. The bounces and recollapses of a closed universe happen at the turning points when the Hubble rate vanishes, which is equivalent to the condition \u03c1 = \u03c1 min or \u03c1 = \u03c1 max . The character of the turning point, namely whether it is a bounce point or a recollapse point, depends on the second derivative of the volume. To be specific, a turning point is a bounce point when v > 0 and a recollapse point when v < 0. From the Hamilton's equations (2.5)-(2.6), it is straightforward to find that when \u03c1 = \u03c1 min and when \u03c1 = \u03c1 max where As a result, depending on the initial conditions, the bounces and recollapses can occur at either the maximum or the minimum energy density. Furthermore, at the turning point when the Hubble rate vanishes, from the equation of motion (2.5), one can find sin(2 \u03bbb -2 D ) also vanishes. Therefore, \u03c1 1 = \u03c1 2 at a bounce or recollapse point which indicates that v and \u02d9 b have the same sign at the minimum energy density and opposite signs at the maximum energy density. In addition, it can be shown in a straightforward way that the modified Friedmann equation has the right classical limit. In Eq. (2.11), if we take the limit v glyph[greatermuch] 1, then \u03c1 min \u2248 \u03c1 crit \u03b3 2 D 2 = 3 8 \u03c0Ga 2 and the Friedmann equation (2.10) reduces to Considering the classical limit \u03c1 glyph[lessmuch] 1 and a glyph[greatermuch] 1, the above equation can be further reduced to which is exactly the classical Friedmann equation for a closed universe. Since loop quantization is only applied to the geometrical sector of the classical phase space, the equations of motion in the matter sector are not changed by the quantum geometrical effects. By using the Hamilton's equations of the scalar field in (2.7), it is straightforward to show that the Klein-Gordon equation and thus the continuity equation also hold in a closed universe of LQC. Finally, we would like to briefly review the hysteresis-like phenomenon in a cyclic universe filled with a homogeneous scalar field [51, 53]. Assuming the evolution of the cyclic universe is adiabatic, the work done by the scalar field during each contraction-expansion cycle can be explicitly computed as On the other hand, the change in the total energy of the scalar field at two consecutive recollapse points is simply given by where \u03c1 ( i ) rec / v ( i ) rec denotes the energy density/volume at the i th recollapse point. Using the energy conservation law W + \u03b4M = 0, one can relate the change in the maximum volume of the universe at two successive turnarounds to the net work done by the scalar field in one complete cycle. This relationship in general is model-dependent and also determined by the character of the turnarounds in each model. In the current case with the effective dynamics determined by the modified Friedmann equation (2.10), assuming the energy densities at the recollapse points are given by the minimum density \u03c1 min , then when v ( i ) is much greater than unity, the difference in the volumes of two consecutive recollapses can easily be shown to be As a result, there is a change in the maximum volume in each cycle which is directly related to the asymmetry of the pressure of the scalar field during the contraction and expansion phases of each cycle. This results in hysteresis-like behavior which becomes evident via the plots of the equation of state [51, 53]. We find that an increase in the maximum volume and a decrease in the equation of state in each cycle will play an important role in the onset of inflation even with initial conditions which are unfavorable in the classical theory.", "pages": [4, 5, 6, 7]}, {"title": "III. \u03c6 2 INFLATION IN A CLOSED FLRW UNIVERSE IN LQC", "content": "Before we discuss the Starobinsky potential case in the next section, we study the occurrence of \u03c6 2 inflation in a closed LQC universe that is sourced by a single scalar field. We focus on initial conditions which in the classical theory do not lead to an inflationary spacetime. The initial conditions are imposed in the Planck regime considering a small homogeneous patch of the universe. For such a patch the spatial curvature term plays an important role and inflation can only take place when the initial conditions are selected such that the potential energy of the scalar field is dominant at the initial time and is large enough to overcome the spatial curvature of the closed universe. When the initial energy density is dominated by the kinetic energy of the inflaton field, the universe recollapses before inflation can occur due to the curvature of a closed universe. This results in a big crunch singularity following the recollapse. One may be tempted to consider initial conditions corresponding to a very large initial volume such that the effect of the spatial curvature becomes so small that the pre-inflationary branch has no recollapse. However, this requires assuming an unnaturally large initial homogeneous patch of the universe in order to set initial conditions for inflation in the Planck regime. Typical initial conditions in the Planck regime start with patches which are not assumed to be homogeneous at macroscopic scales and these are the ones considered in our analysis. In the following, we present two representative sets of initial conditions in the parameter space. The first set of initial conditions has the potential energy dominant at the bounce but does not allow for inflation in the classical theory because the potential energy is unable to overcome the classical recollapse caused by the spatial curvature term. The second representative set of initial conditions corresponds to initially dominant kinetic energy which again results in a classical recollapse and a big crunch singularity. For both sets we find that because of quantum gravitational effects, big crunch singularities are avoided and inflation occurs after a few cycles of expansion and contraction. In our numerical analysis, the fundamental equations of motion are the effective Hamilton's equations (2.5)-(2.7) introduced in Sec. II. The initial conditions are chosen such that the universe has the highest allowed density given its volume at t = 0. In general, the initial conditions that must be specified are the values of the phase space variables v , b , \u03c6 and p \u03c6 at t = 0. (The initial conditions will be labelled by the subscript '0'.) In our simulations, we choose the initial volume v 0 and the initial value of the scalar field \u03c6 0 as the two initial free parameters. Since the initial conditions are chosen at a point where the energy density is maximal, the conjugate momentum of the scalar field can be determined by where the ' \u00b1 ' reflects the two possible signs of the initial velocity of the scalar field. Finally, the momentum b 0 is fixed by the vanishing of the Hamiltonian constraint (2.2), given v 0 , \u03c6 0 and p \u03c6, 0 . Of the possible solutions for b 0 , we used the positive solution for all the plots shown below. Finally, all our simulations were performed using a combination of the StiffnessSwitching, ExplicitRungeKutta, and Automatic numerical integration solving methods in Mathematica with the precision and accuracy goals set to 11.", "pages": [7, 8]}, {"title": "A. Representative initial conditions", "content": "Below we discuss two representative cases for the inflationary potential U = 1 2 m 2 \u03c6 2 , one with the potential energy dominant at an initial time in the Planck regime and the other with the kinetic energy dominant. Since we are interested in cases in which inflation does eventually take place, the mass of the inflaton field for the chaotic potential will be fixed as in the standard inflationary paradigm using the scalar power spectrum A s and the scalar spectral index n s for the pivot mode [3] Using these the mass of the scalar field is fixed to m = 1 . 23 \u00d7 10 -6 in this subsection. The first set of representative initial conditions we would like to discuss is presented in Fig. 1. This corresponds to in Planck units. With these initial conditions, at initial time t 0 = 0, the initial velocity of the scalar field is zero. Though the inflaton starts with all its energy in potential energy, such a universe soon recollapses and encounters a big crunch singularity in GR. In contrast, we find that in LQC the universe undergoes a series of bounces and recollapses. These cycles occur before the conditions become favorable for inflation to begin. We find that both the maxima and minima of the volume of the universe increase with each bounce following the initial recollapse at t = 0. Note that this case provides an example of the type of universe discussed in Sec. II where the maximal energy density corresponds to a recollapse rather than a bounce. In the current case, since the universe is initially dominated by potential energy, the equation of state is close to negative unity and thus at the maximum energy density, As a result, depending on the sign of \u03c1 2 , the turning point at the maximum energy density can either be a bounce or a recollapse. From the \u03c1 max -\u03c1 and b plots of Fig. 1, we see that the bounces and recollapses occur at the maximum energy density allowed at that time. It is evident by comparing the \u03c1 max -\u03c1 plot to the volume plot above it that this difference vanishes whenever the universe reaches a turnaround point, corresponding to the maximum density being achieved. Moreover, since v and \u02d9 b have the opposite signs at \u03c1 = \u03c1 max , the sign of v changes between each bounce and recollapse, which is consistent with the oscillating behavior of the momentum b . The peaks of b also show a noticeable decreasing hysteresis-like behavior [51]. This behavior is also seen in the increase in the maxima (also the minima) of the volume in subsequent cycles. When the universe is in the cyclic phase, the equation of state w = P/\u03c1 also changes periodically. It is interesting to note that although the universe is initially dominated by the potential energy of the scalar field, corresponding to w = -1, inflation does not take place immediately after the first bounce but rather at around t = 10 4 (in Planck seconds). The cyclic phase of the universe is accompanied by oscillatory behavior in the equation of state, which attains a local maximum at each bounce and a local minimum at each recollapse. Furthermore, as seen from the behavior of the energy density, we find that inflation starts when the energy density is almost Planckian. Thus, for the above initial conditions we find a successful onset of inflation because of loop quantum gravitational effects, even though this universe is unable to inflate in the classical theory due to the big crunch singularity. The second example is depicted in Fig. 2 with the initial conditions given by in Planck units. These correspond to a negative initial velocity for the scalar field. As in the previous case, the positive curvature slows the expansion of the universe after t = 0, which in this case is a bounce, leading to a recollapse which is followed by a non-singular bounce in LQC. Initially the universe undergoes cycles of expansion and contraction with little change in the maximum volume. It is at time t = 300 that the recollapse volume begins to significantly increase with each cycle. During this phase, with each subsequent recollapse, the potential energy fraction gets higher, corresponding to the equation of state reaching a lower minimum value. The time it takes to complete a cycle also tends to increase with time, with the next recollapse happening a little later after the bounce than the one before it. A noticeable hysteresis-like phenomenon can be observed in the w plot where all the troughs correspond to recollapse points whose values decrease with time, while all the peaks correspond to bounce points, which remain at unity throughout the pre-inflationary evolution. We see that the minimum of the equation of state decreases during each cycle until w reaches -1 / 3 after one final bounce and then inflation takes place, preventing any further recollapses. In the figure, we only display the volume, equation of state, and momentum b corresponding to the first several cycles. The behavior of these variables at the onset of inflation is depicted in the inset plots. Note that the momentum b behaves differently than in the first case. In this case, b is monotonically decreasing. As a result, v is always negative when \u03c1 = \u03c1 min and positive when \u03c1 = \u03c1 max . That is, in contrast to Fig. 1, each bounce happens at the maximum energy density while each recollapse happens at the minimum energy density.", "pages": [8, 9, 10]}, {"title": "B. Phase space portraits", "content": "To understand the qualitative dynamics of general solutions with various initial conditions, we present phase space portraits for the Starobinsky potential using where \u03c7 0 = \u221a 3 m 2 32 \u03c0G\u03c1 max , 0 and \u03c1 max , 0 stands for the initial maximum energy density. These variables obey the following set of first-order differential equations Together with (2.10), the above equations (4.7)-(4.8) form a complete set of dynamical equations for a system that is described by X , Y , and v . Since the main goal of our numerical analysis is to determine whether the inflationary phase is a local attractor for a variety of initial conditions, we focus on a 2-dimensional phase space plot in the subspace spanned by X and Y . From Eqs. (4.7)-(4.8), it follows that there are two fixed points in the system, which are ( X = 0 , Y = 0) and ( X = \u03c7 0 , Y = 0). The first fixed point corresponds to the end of the reheating phase where the energy density of the inflaton field is very small and the volume of the universe is very large. The second fixed point is located along the X = \u03c7 0 boundary separating the region with real values of the scalar field from the region where the scalar field is complex [42]. In the following, we will only focus on the region in which the scalar field is real (to the left of the boundary line). As for the \u03c6 2 potential, in order to make the various solutions converge in a short time, we use a larger value for the mass associated with the scalar field in the phase space portraits. The use of such a value does not change the qualitative behavior of the solutions, including the existence of inflationary separatrices and cosmological attractors, which are the properties of interest. The first phase space portrait is presented in Fig. 9. The upper left plot shows the entire phase space region which includes trajectories for six distinct initial conditions. For the Starobinsky potential only the trajectories confined to the left of the vertical black line X = \u03c7 0 correspond to real values of the scalar field. Thus, these are the only cases we consider in the phase space plots. The phase space evolution of a generic solution with one bounce is represented by the solid blue curve in the phase space portrait, which corresponds to the initial conditions ( \u03c6 0 = 0 . 23 , p \u03c6 0 = -930) at v 0 = 100. The trajectory starts from the black solid circle which corresponds to the initial bounce at the maximum energy density. It then moves towards the origin as the scalar field loses energy. When the blue curve is close to the origin, a recollapse occurs and the universe enters into a contracting phase, increasing the energy density of the scalar field and making the blue curve move away from the origin. Afterwards, as the energy density of the scalar field increases, a bounce takes place when the blue curve hits the dashed red circle and the maximum energy density at that time is reached. After the bounce, the universe re-enters a state of expansion and the scalar field starts to lose energy again. The blue curve then moves towards the origin and merges into the inflationary separatrix (the short, curved, horizontal line traveling left towards the origin, during which inflation takes place) before finally falling into the spiral at the center of the plot. The other trajectories have similar qualitative behavior to the blue solid curve. The difference lies in the number of bounces in the pre-inflationary phase, which for this plot is either none or one, as well as the value of the maximum energy density at which the bounce occurs. For example, the dot dashed green curve starts from the top and moves directly towards the inflationary separatrix without any further bounces, while the red dotted, yellow dot-dashed, and green dot-dot-dashed curves all experience a single bounce that happens at the highest allowed energy density, which is close to the initial energy density. Compared with \u03c6 2 inflation, where we saw relatively long and straight horizontal lines heading towards the origin, the inflationary separatrix in Starobinsky inflation is significantly less noticeable. For this reason, we show the upper right panel which zooms in on the inflationary separatrix, where all the curves merge, and the attractor at the origin. Finally, to show the details of the evolution of a generic solution in the phase space portrait, the bottom panels show the behavior of the volume and energy density for the solid blue curve. We see that in the volume plot on the bottom left, the single bounce happens at around t = 228, while in the density plot on the bottom right we see that the energy density at this bounce exceeds the initial energy density, represented by the dashed red line in the graph. The second phase space portrait presented in Fig. 10 includes trajectories corresponding to three distinct initial conditions. All of them describe universes that undergo bounces before inflation. The upper left plot shows the entire phase space region. For the universes corresponding to these initial conditions, the energy density never exceeds the initial density. As a result, all the trajectories are confined within the initial black unit circle. The representative solution is again displayed as the solid blue curve, which is analyzed further in the bottom panels. The initial conditions for the blue curve are ( \u03c6 0 , p \u03c6 0 ) = (8 . 97 , -77) at v 0 = 50. As shown in the bottom panels, the universe starts at a bounce and then undergoes a series of cycles with alternating contracting and expanding phases until inflation begins. All the bounces happen at the maximum energy density with a volume no larger than the initial volume, while all the recollapses happen at the minimum energy density for each cycle, all of which are also in the Planck regime as can be seen in the \u03c1 plot. In the top right panel, we zoom in on the inflationary separatrix and the attractor at the center. Since all the recollapses happen in the Planck regime, there are no curves that approach the origin before merging into the separatrix. We can clearly see from the plot that trajectories starting from different initial conditions have the same late-time dynamics consisting of an inflationary phase followed by a reheating phase.", "pages": [19, 20, 21]}, {"title": "IV. STAROBINSKY INFLATION IN A CLOSED FLRW UNIVERSE IN LQC", "content": "In the previous section we considered the case of chaotic inflation and found that quantum gravity effects assist the onset of inflation in a closed universe. In this section we will study Starobinsky inflation in the same setting. Unlike in classical cosmology, in LQC Starobinsky inflation is not obtained from an R 2 term in the action, 2 rather one generally takes as given the Starobinsky potential in effective dynamics [42, 44, 45, 57], whose form is explicitly given by The mass m is fixed to 2 . 44 \u00d7 10 -6 from the scalar power spectrum and the spectral index given in (3.2). To determine the value of the mass we assume that the pre-inflationary dynamics would not change the scalar power spectrum in a significant way. Note that unlike chaotic inflation, which can take place in the Planck regime, Starobinsky inflation can only occur on the right wing of the potential, which corresponds to an energy scale that is 10 13 orders of magnitude lower than the Planck scale. As a result, in the classical theory, when starting from the Planck regime the universe inevitably recollapses before inflation sets in, resulting in a big crunch singularity. We now study how the dynamics change in LQC.", "pages": [14]}, {"title": "A. Some representative initial conditions", "content": "The first example we would like to discuss corresponds to Fig. 5, which results from the initial conditions (in Planck units) These initial conditions are chosen at a bounce with energy density \u03c1 0 = 0 . 41. In this case, the initial bounce is completely dominated by the potential energy of the scalar field and the inflaton is released from rest on the left wing of the potential, rolls down, and then climbs up the right wing until reaching the turnaround point. Inflation takes place at \u03c1 \u2248 10 -13 when the inflaton slowly rolls down the right wing of potential. Hence the scalar field behaves in the same way as in a spatially-flat universe. However, the behavior of other dynamical variables is quite different. From the volume plot in Fig. 5, one can see that before the onset of inflation at around t = 8 . 39 \u00d7 10 6 , the universe undergoes a bounce at t = 1 . 46 \u00d7 10 6 where the energy density reaches the maximum allowed value and a recollapse at around t = 6 . 47 \u00d7 10 5 with the minimum allowed energy density at that time. The equation of state in the upper right panel of Fig. 5 shows explicitly that the initial bounce is dominated by potential energy with w \u2248 -1. Meanwhile the single bounce that takes place during the evolution is dominated by the kinetic energy of the scalar field as it corresponds to the last peak in the w plot before the onset of inflation. The second to last peak in the w plot corresponds to the moment when the inflaton crosses the origin, where the potential energy vanishes, forcing w = 1. It should be noted that even though the potential energy is initially dominant, inflation cannot occur on the left wing of the potential because it is too steep for the slow-roll to occur. Only the right wing of the Starobinsky potential can drive inflation in both closed and spatiallyflat universes. This is also manifest in the phase space portrait shown later where only a single inflationary separatrix is observed. Since this universe only has a single bounce after the initial one, we do not see oscillatory behavior in w before the onset of inflation. The second example is given in Fig. 6, corresponding to a universe initially dominated by the kinetic energy of the scalar field. The initial conditions for the numerical simulation are set at the big bounce at t = 0, given explicitly by which implies that the inflaton is initially released from the right wing of the potential with a large initial velocity. From the bottom right panel of Fig. 6, we see that the inflaton first rolls down the right wing of the potential, then climbs up the left wing and momentarily stops at the turnaround point. Afterwards, it rolls down the left wing of the potential, climbs up the right wing and then reaches another turnaround point. Finally, slow-roll inflation takes place when the inflaton slowly rolls down the right wing of the potential for the second time. The energy density at which inflation occurs is about 10 13 orders of magnitude below the Planck energy. Fig. 6 shows that as compared with the previous example, the current case is richer in terms of the pre-inflationary dynamics. Note that the three recollapses associated with the three bounces in this example all occur at an energy scale far below the Planck energy, while all the bounces happen at a similar energy density that is around \u03c1 crit . From the figure one notes that the third cycle of expansion and contraction is highly asymmetrical, which is due to the asymmetry of the Starobinsky potential itself. For this particular cycle the expanding (contracting) phase takes place on the left (right) wing of the potential. From the w plot in the top right panel, one can easily identify several important moments featuring the behavior of the scalar field before the onset of inflation. For example, the first trough at w = -1 before t = 10 4 corresponds to the turnaround point of the inflaton on the left wing of the potential. While the first two bounces take place before this trough, the effective dynamics following the first trough are qualitatively similar to what happened in Fig. 5. This is confirmed by the similar behavior of the w plots in these two cases in two particular regions: around the second to last peak where the inflaton crosses the origin and around the final peak where the universe undergoes a bounce. The next example concerns Fig. 7, which corresponds to the initial conditions In this case, the initial bounce at t = 0 is dominated by the kinetic energy of the scalar field. The universe undergoes a longer series of bounces and recollapses before the onset of inflation as is depicted in the top left panel of Fig. 7. With a large initial velocity, the inflaton, which starts from the left wing, gets much higher up on the right wing than in the two previous cases. Hence it takes longer for the inflaton to turn around, leading to many more cycles in the pre-inflationary phase. All the bounces in the pre-inflationary phase happen at the Planck energy density around \u03c1 crit as is shown for the first four bounces. Meanwhile, all the recollapses occur at the minimum energy density far below the Planck scale. Similar to the previous two cases, following the turnaround point on the right wing of the potential, inflation takes place at an energy density of around 10 -13 times the Planck density. In addition to the volume of the universe and its energy density, the equation of state, which is shown in the top right panel, also oscillates with the same period as the \u2708 volume. At each bounce, the equation of state reaches a maximum, while with each consecutive recollapse in the pre-inflationary phase it reaches a lower minimum value, slowly approaching -1 / 3, marked by the upper red dashed line in the plot. We see that even though the equation of state has a value close to -1 / 3 during the first recollapse, it subsequently takes many cycles for it to cross this value and inflation to begin. This is due to the fact that these initial conditions give rise to a weak hysteresis, which is also evident from the volume plot. The volume at which the universe recollapses increases in consecutive cycles, resulting in a decrease in the minimum energy density at each recollapse. With each cycle the curvature effect is slightly weaker, and hence each cycle brings the universe closer to the right conditions for inflation to begin. Though weaker in strength than in previous cases, the increasing hysteresis prevents the universe from undergoing infinitely many cycles of expansion and contraction, instead facilitating the onset of an inflationary period in a closed LQC universe after a finite number of cycles. Fig. 8 is an example of a case in which inflation does not appear in our simulation, which ran through time 10 8 . The initial conditions are given by In this case, the inflaton is released from the right wing of the potential with a large initial velocity, climbs up the left wing of the potential and then turns around, finally climbing up the right wing. The volume plot shows a very large number of bounces and recollapses. Similar to the previous cases, all the bounces occur at the maximum energy density in the Planck regime while all the recollapses take place at the minimum energy density, which is far below the Planck energy. For the time range of the simulation, which is 10 8 (in Planck seconds), the volume plot shows a long cyclic phase and no inflation. However, as in the previous case where we did observe the onset of inflation, an increasing hysteresis can be seen in the w plot, which shows the minima of the equation of state decrease with each recollapse. The decrease with each cycle is very small, and since towards the end of the evolution shown the minima have only reached -0.08, it is evident that far more time is needed before they can cross -1/3. Note that the decrease in the minimum of w with each cycle was also small in the previous case, but because at the beginning of the oscillatory phase the minima were already close to -1/3, we were able to see inflation before time 10 8 . The minima of w being higher in this case at the start of the oscillatory phase reflects the fact that these initial conditions are less favorable for inflation than in the previous cases, where either the minima of w were close to -1/3 at the start of the oscillatory phase or the initial conditions were such that the hysteresis effect was stronger and so an extremely long series of cycles was not necessary to bring about the onset of inflation.", "pages": [15, 16, 17, 18]}, {"title": "V. SUMMARY", "content": "The goal of this manuscript is to understand the onset of inflation in closed universes for low energy scale inflationary models, such as Starobinsky inflation. Starting inflation in such cases has remained a long-standing problem because of the recollapse caused by the spatial curvature and the big crunch singularity that are unavoidable in the classical theory[11, 12]. We explored a resolution of this problem in the setting of LQC where big bang/big crunch singularities are robustly resolved due to non-perturbative quantum gravity effects and the pre-inflationary phase is in general characterized by a series of bounces and recollapses of the universe. For comparison we first considered the case of \u03c6 2 inflation and demonstrated that non-singular cyclic evolution in the pre-inflationary phase sets the stage for inflation to begin even for very unfavorable initial conditions. The analysis was then repeated for Starobinsky inflation, where the problem is far more severe, yielding similar results. For the \u03c6 2 potential, inflation can take place at different energy scales ranging from the Planck regime to an energy density that is 10 12 orders of magnitude below the Planck density. Inflation can also take place on both sides of the potential, resulting in two inflationary separatrices in the phase space portraits. On the other hand, for the Starobinsky potential, with the mass parameter fixed by observations, inflation can only take place on the right wing of the potential at an energy density that is around 10 13 orders of magnitude below the Planck density. Due to the asymmetry of the potential, there is only one inflationary separatrix in the phase space portraits. We found different features in terms of the pre-inflationary dynamics, which is when the cycles take place. In the \u03c6 2 model, both the bounces as well as the recollapses can occur at the maximum energy density, which means that t = 0 where the energy density takes on the maximum allowed value can correspond to either a bounce or a recollapse. In the Starobinsky model, on the other hand, the bounces always happen at the maximum energy density and the recollapses happen at the minimum energy density, which means that t = 0 always corresponds to a bounce. Furthermore, for the Starobinsky potential it tends to take longer for the first recollapse to occur, and there also tends to be more time between subsequent cycles than for the \u03c6 2 potential. This contributes to the delay in the onset of inflation in the Starobinsky model as compared with the \u03c6 2 model. With respect to which initial conditions give rise to inflation, we found that the evolution of the universes in both models differ most in those cases where the oscillatory behavior of w begins with the minima of w not close to -1 / 3. In such cases, with each cycle the minimum of w decreases noticeably for the \u03c6 2 potential, so that after some relatively small, finite number of cycles w crosses -1/3 and inflation begins. For the Starobinsky potential, however, the minima of w decrease very slowly with each cycle. This means that when starting with initial conditions for which the minima of w are not close to -1/3 when the oscillatory behavior in pre-inflationary epoch begins, then it can take an extremely long time for the minima of w to decrease enough to cross -1/3. Thus, in comparison to \u03c6 2 potential, the mechanism resulting from non-singular cyclic evolution and leading to the onset of inflation in the Starobinsky potential is weaker but nevertheless strong enough to overcome problems encountered in the classical theory. In summary, we have shown that the problem of the onset of inflation for low energy scale models, such as the Starobinsky potential, in a closed universe can be successfully resolved by quantum gravity effects. The primary reason for the initiation of the inflationary phase is a progressive decrease in the value of the equation of state with each cycle captured in the hysteresislike phenomena seen earlier for the \u03c6 2 potential in LQC [51] and other bouncing models [53, 54]. When the equation of state becomes less than -1 / 3 in a particular cycle, recollapses no longer occur and inflation starts. While for the \u03c6 2 potential we found inflation to occur in all the considered cases, for the Starobinsky inflation we found some cases in which the hysteresis-like phenomenon is so weak that the onset of inflation, though expected, is delayed. It would be interesting to understand physical phenomena that can make the onset of inflation even in such extreme cases more favorable, an example of which will be discussed in a future work [58]. Apart from such cases, thanks to the singularity resolution due to non-perturbative quantum gravity effects, the Starobinsky potential results in inflation in a short time even when starting from initial conditions which are highly unfavorable for inflation in the classical theory.", "pages": [21, 22]}, {"title": "Acknowledgements", "content": "This work is supported by the NSF grants PHY-1454832 and NSF PHY-1852356. L.G. thanks the REU program of the Department of Physics and Astronomy at LSU during which most of this work was carried out. with an inflationary potential , arXiv:2007.06597. M. Motaharfar, P. Singh, To appear (2020).", "pages": [23, 24, 25]}] |
2013arXiv1307.1450H | https://arxiv.org/pdf/1307.1450.pdf | <document>
<section_header_level_1><location><page_1><loc_9><loc_90><loc_92><loc_93></location>TeV-PeV neutrinos over the atmospheric background: originating from two groups of sources?</section_header_level_1>
<text><location><page_1><loc_26><loc_87><loc_75><loc_88></location>Hao-Ning He, 1 Rui-Zhi Yang, 1 Yi-Zhong Fan ∗ , 1 and Da-Ming Wei 1</text>
<text><location><page_1><loc_31><loc_85><loc_69><loc_87></location>1 Key Laboratory of Dark Matter and Space Astronomy,</text>
<text><location><page_1><loc_22><loc_84><loc_79><loc_85></location>Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China</text>
<text><location><page_1><loc_18><loc_70><loc_83><loc_83></location>In addition to the two ∼ 1 PeV neutrinos, the IceCube Collaboration recently reported a detection of 26 neutrino candidates at energies from 30 TeV to 250 TeV, implying a confidence level of 4 . 3 σ over the atmospheric background. We suggest that these TeV-PeV non-atmospheric neutrinos may originate from two groups of sources, motivated by the non-detection of neutrinos in the energy range 250 TeV -1 PeV in current data. If intrinsic, the non-detection of 250 TeV -1 PeV neutrinos disfavors the single power-law spectrum model for the TeV -PeV non-atmospheric neutrinos at a confidence level of ∼ 2 σ . We then interpret the current neutrino data with a two-component spectrum model. One has a flat spectrum with a cutoff at the energy ∼ 250 TeV and the other has a sharp peak at ∼ 1 PeV. The former is likely via pp collision while the latter may be generated by the photomeson interaction.</text>
<text><location><page_1><loc_18><loc_67><loc_45><loc_68></location>PACS numbers: 95.85.Ry, 95.85.Hp, 98.70.Sa</text>
<text><location><page_1><loc_9><loc_42><loc_49><loc_65></location>The construction of IceCube was completed in 2010, and data were collected from May 2010 to May 2011 with 79 strings and an effective livetime of 285.8 days, and from May 2011 to May 2012 with the full 86-string detector and a livetime of 330.1 days. In a search for very-high energy neutrinos, i.e., GZK neutrinos produced via ultra-high energy cosmic rays interacting with cosmic microwave background (CMB) photons and extragalactic background light (EBL) photons [1-3], the IceCube Collaboration discovered two cascade events with energy ∼ 1 PeV [4, 5], which is 2 . 8 σ beyond the atmospheric background. The energies of these two events are the highest so far but lower than that expected for GZK neutrinos. However, if assuming an unbroken E -2 power-law spectrum, there should be about 8 -9 more events observed above 2 PeV.</text>
<text><location><page_1><loc_9><loc_27><loc_49><loc_42></location>Quite recently, they carry on the high-energy contained vertex search to analysis the 662 days data from May 2010 to May 2012, which is sensitive at energy above 50 TeV, and at 1PeV, three times as sensitive as GZK search. At the IceCube Particle Astrophysics Symposium (IPA-2013), they reported an observation of 28 neutrino candidates, including 26 events with energy from 30 TeV to 250 TeV, and the two ∼ 1 PeV neutrinos detected previously, implying a significance level of 4 . 3 σ over the atmospheric background of 10 . 6 +4 . 5 -3 . 9 events [6].</text>
<text><location><page_1><loc_9><loc_8><loc_49><loc_27></location>The non-detection of events above ∼ 2 PeV and the excess of events over atmospheric background is the two momentous features of the new IceCube data. In favor of the astrophysical origins, the non-detection above ∼ 2 PeV may indicate a broken or peaked spectrum with a break/cutoff or a peak in the PeV band [6, 7], or a soft unbroken spectrum with index larger than 2.3 [8, 9]. He et al. [10] calculated the neutrino spectrum from pp collision in ultra-luminous infrared galaxies (ULIRGs), and suggested a cutoff of neutrinos spectrum at several PeV which is governed by the abilities of hypernovae (HNe) accelerating protons and ULIRGs confining high energy protons. However, the predicted PeV neutrino flux is</text>
<text><location><page_1><loc_52><loc_37><loc_92><loc_65></location>so low that can not account for the IceCube data unless the HNe rate in ULIRGs are ∼ tens times higher than current estimate [10]. In the literature, the suggested sources for PeV neutrinos include active galactic neucleis (AGNs, [11]), star forming galaxies (SFGs, [7]), clusters of galaxies[12], hypernovae (HNe, [10, 13]), gamma-ray bursts (GRBs, see [14], but see [15, 16]), and cosmogenic neutrinos (see [17, 18], but see [19, 20]). In some literature, a single power-law spectrum of the TeVPeV non-atmospheric neutrinos has been assumed (e.g., [8, 9]). It is however not clear how robust such an assumption is since different astrophysical sources can produce neutrino emission in quite different energy ranges (e.g., [7, 10, 11, 21]) and the neutrino spectrum may display some wiggle-like structure. The main purpose of this work is to examine whether there is some tentative evidence for wiggle-like structure in current nonatmospheric neutrino data and then discuss the possible sources.</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_37></location>To estimate the TeV-PeV non-atmospheric neutrino spectrum, Anchordoqui et al. [8] partitioned the observation data into three energy bins, i.e., 50 TeV -1 PeV, 1 -2 PeV and 2 -10 PeV. Such a fit to the data suggests a single power-law spectrum index Γ ∼ 2 . 3. We, however, notice that no events have been reliably detected between 250 TeV and 1 PeV, but about 16 events observed over 10 atmospheric background neutrinos in the energy range of 30 -250 TeV, and 2 events in the energy range of 1 -2 PeV [6]. If the observed 18 non-atmospheric neutrinos have the same origin and can be described by a single power law spectrum with index of Γ, adopting the exposure of the full IceCube's 662 days operations shown in [8], we can constrain the index via the flux in the two energy bins of 30 -250 TeV and 1 -2 PeV(The two data points are shown in Fig. 1). The fit to the data yields a Γ ∼ 2 . 2, which suggests ∼ 9 neutrinos in the energy range of 250 TeV - 1 PeV, while the data analysis found no reliable events in the same energy band. Such a fact favors the hypothesis that the TeV-PeV non-atmospheric</text>
<text><location><page_2><loc_9><loc_69><loc_49><loc_93></location>neutrino spectrum may be not a single power-law, instead it may have structure. For the single power law spectrum fits within 2 σ error region, the lowest induced neutrino counts are 3.6. Therefore, if the non-detection in the energy range of 250 TeV -1 PeV is intrinsic (i.e., neither due to the instrument effect nor limited by the data analysis method), a single power-law spectrum model can be excluded at a confidence level of ≈ 2 σ . As a result, the TeV-PeV non-atmospheric neutrino spectrum may consist of two components, one has a cutoff at ∼ 250 TeV and the other may have a sharp peak at ∼ 1 PeV. We would like to caution that the above speculation is based on a small sample consisting of only ∼ 18 events. Since IceCube continues to collect data, our speculation will be directly confirmed or ruled out in the near future . In the following discussion we focus on the two-component spectrum model and discuss the possible sources.</text>
<text><location><page_2><loc_9><loc_13><loc_49><loc_68></location>There are two fundamental processes to produce TeVPeV neutrinos, i.e., the pp collision and the pγ interaction. These two processes both produce charged pions and convert 20% of the proton energy into pions. The charged pions then decay into neutrinos ( π + → µ + + ν µ → e + + ν e + ¯ ν µ + ν µ and π -→ µ -+ ¯ ν µ → e -+¯ ν e + ν µ +¯ ν µ ). Usually people assume that the decay products share the energy equally, then 5% of the proton energy will be converted into neutrinos. Since the cross section of pp collision changes very slowly with the energy of the ultra-relativistic protons, the resulting neutrino spectrum traces that of the ultra-relativistic protons and the cutoff energy E ν, cut of the neutrino spectrum mainly depends on the maximum energy E p, max of accelerated protons, i.e., E ν, cut /similarequal 0 . 05 E p, max . As a result of the ∆ resonance, the cross section of the pγ interaction peaks at the photon energy E ' γ /similarequal 0 . 3 GeV in the proton-rest frame. Therefore, the resulting neutrino spectrum sensitively depends on not only the proton spectrum but also the photon spectrum. The neutrino spectrum will peak at energy of 1 . 5 × 10 15 Γ 2 /E γ, p eV, where E γ, p is the peak energy of photons in the observer's frame and Γ is the Lorentz factor of the emitting region[15]. Therefore, sources like SNe, which can accelerate protons up to 5 -10 PeV in the observer's frame, with dense circum medium surrounding, for example, in the SFGs, may account for the 30 -250 TeV component via pp collision. And sources producing sufficient photons peaking at energy of 5 eV, i.e., around the ultraviolet (UV) band, are possible to explain the PeV neutrino excess via pγ process. What's more, the thermal emission from the accretion disk of AGN and the EBL photons in UV band can be the target photons. The observed peak energy of the target photons can be higher if considering the relativistic effect, for instance, X-ray photons interacting with protons in a region with a Lorentz factor of tens can also produce an observed neutrino spectrum peaking around PeV.</text>
<text><location><page_2><loc_9><loc_9><loc_49><loc_13></location>The 30 -250 TeV component. SNe are widely suggested to be the dominant sources for cosmic rays (CRs) at energies below the 'knee' at ∼ 3 × 10 15 eV, most</text>
<text><location><page_2><loc_52><loc_76><loc_92><loc_93></location>probably through the diffusive shock acceleration mechanism [22]. The maximum energy of protons accelerated at the shock front of a SN expanding into the uniform dense interstellar medium (ISM) can be estimated as [23], ε p , max ≈ 10 16 eV( V 3 × 10 8 cm s -1 ) 2 ( n 10 3 cm -3 ) 1 / 6 ( M SN 10 M /circledot ) 1 / 3 , where n is the number density of ISM, and M SN is the rest mass of the SN ejecta. with a typical kinetic energy E k ∼ 0 . 5 -5 × 10 51 erg and a typical velocity V ∼ 3 × 10 8 cm s -1 ( E k / 10 51 erg) 1 / 2 ( M SN / 10 M /circledot ) -1 / 2 . The accelerated protons would lose energy into γ -ray photons, electrons and positrons, and neutrinos, through pp collisions when injected into the ISM.</text>
<text><location><page_2><loc_52><loc_64><loc_92><loc_76></location>The energy loss time of protons is τ loss = (0 . 5 nσ pp c ) -1 , where the factor 0 . 5 is inelasticity [25], and σ pp is the inelastic nuclear collision cross section, which is ∼ 70mb for protons at energies ε ' p ∼ 1 -10 PeV in the rest frame that is of our great interest [26]. Introducing a parameter Σ gas ≡ m p nl as the surface mass density of gas, with l as the scale of the dense region, the energy loss time reads [27]</text>
<formula><location><page_2><loc_57><loc_59><loc_92><loc_63></location>τ loss = 1 . 4 × 10 4 yr l 100pc ( Σ gas 1 g cm -2 ) -1 . (1)</formula>
<text><location><page_2><loc_52><loc_55><loc_92><loc_58></location>The confinement time can be rewritten as (i.e., Eq. (8) of [10])</text>
<formula><location><page_2><loc_55><loc_51><loc_92><loc_54></location>τ conf ≈ 2 × 10 5 yr ( ε ' p 10PeV ) -0 . 5 ( Σ gas 1gcm -2 ) 0 . 5 . (2)</formula>
<text><location><page_2><loc_52><loc_41><loc_92><loc_49></location>The fraction of the energy that protons lose into pions is f π = 1 -exp( -τ conf /τ loss ), which is close to 1 as long as τ conf /greaterorequalslant τ loss . As a result, the protons with energy ε ' p lose almost all of their energy via pp collision before escaping from the dense region as long as τ loss ≤ τ conf , which constrains the critical gas surface density Σ crit as</text>
<formula><location><page_2><loc_52><loc_35><loc_92><loc_39></location>Σ gas /greaterorsimilar Σ crit = 0 . 17 g cm -2 ( ε ' p 10PeV ) 1 / 3 ( l 100pc ) 2 / 3 . (3)</formula>
<text><location><page_2><loc_52><loc_23><loc_92><loc_34></location>With the aim to explain neutrinos with energy of 30 -250 TeV, we assume that the neutrinos spectrum cuts off at ∼ 200 TeV. To confine protons with the maximum energy of 4 × 10 15 eV, the source should have gas surface density ≥ 0 . 13 g cm -2 if its typical scale is about 100 pc. Most starburst galaxies (SGs) satisfy such a request and hence are optimal candidates for accelerating protons and producing sub-PeV neutrinos.</text>
<text><location><page_2><loc_52><loc_9><loc_92><loc_23></location>As shown in [10], ULIRGs with the gas surface density about 1 . 0 g cm -2 can accelerate protons with energy up to 100 PeV by hypernovae (HNe), confine the high energy protons and produce neutrinos with energy up to 5 PeV. In contrast with ULIRGs, the star formation rate (SFR) density of SGs is ∼ 5 times larger [29, 30]. Consequently, the Type II SN rate is ∼ 5 times larger than that estimated in [10] since the Type II SN rate is proportional to the SFR [28]. On the other hand, the ratio of the HN rate to the Type II SN rate is ∼ 0.01 and the</text>
<text><location><page_3><loc_9><loc_80><loc_49><loc_93></location>typical ratio of the HN energy to the Type II SN energy is ∼ 10, hence the energy budget of SGs for the PeV cosmic rays is ∼ 50 times larger than that of ULIRGs for higher-energy protons. As a result, the flux of sub-PeV neutrinos from SGs can be larger than that from ULIRGs by a factor of ∼ 50, suggesting a sub-PeV-neutrino flux from SGs ∼ 10 -7 GeVcm -2 s -1 sr -1 , consistent with the flux level of 6 . 25 ± 1 . 56 × 10 -8 GeVcm -2 s -1 sr -1 observed by IceCube in 30 -250 TeV.</text>
<text><location><page_3><loc_9><loc_57><loc_49><loc_80></location>The neutrinos from SGs or SFGs have also been discussed in [21] and [7]. In [21], these authors attributed the GHz radio emission of SGs to synchrotron emission of secondary electrons produced via pp collisions. Then they normalized the flux of neutrinos at GeV via the total energy observed in GHz band being contributed by GeV secondary electrons, since the energy of protons converted into electrons and neutrinos are similar in the pion productions. Depending on the spectral slopes, the extrapolated < 250 TeV neutrino flux is in the range of 4 × 10 -9 -10 -7 GeV cm -2 s -1 sr -1 (see Fig.1 of [21]), which may be able to explain the new IceCube data, consistent with our result. Murase et al. [7] found out that SFGs can provide the necessary energy budget to explain the neutrino flux in PeV band, but with the acceleration and confinement of ∼ 100 PeV protons as the challenges.</text>
<text><location><page_3><loc_9><loc_37><loc_49><loc_57></location>The PeV component. Stecker et al. [33] proposed a model to produce a neutrino spectrum peaking at 1 -10 PeV, via protons accelerated by shocks in the cores of AGNs interacting with photons of the 'big blue bump' of thermal emission from the accretion disk [31]. They estimated the value of the muon neutrino flux [32] ∼ E 2 ν Φ( E ν ) ∼ 10 -8 GeVcm -2 s -1 sr -1 at 100 TeV and ∼ 6 × 10 -8 GeVcm -2 s -1 sr -1 at ∼ 1 PeV. The uncertainty of the flux lies in the uncertainty of the AGN rate, the energy budget of AGNs and the uncertainty of other model parameters, such as the spectrum of the accelerated protons. So it's possible to explain the observed flux of (3 . 28 ± 2 . 28) × 10 -8 GeVcm -2 s -1 sr -1 in the energy range 1 -2 PeV [11].</text>
<text><location><page_3><loc_9><loc_21><loc_49><loc_37></location>Another model with neutrino spectrum peaking at PeV is proposed in [18], which interprets the PeV neutrinos via the EeV protons accelerated by AGNs interacting with the EBL photons. By adopting the EBL model with higher photons around 1 µm and softer spectrum of accelerated protons with spectral index of α = 2 . 6, they predict the flux of PeV neutrinos to be (0 . 5 -2) × 10 -8 GeVcm -2 s -1 sr -1 , consistent with the observed flux at PeV of (3 . 28 ± 2 . 28) × 10 -8 GeVcm -2 s -1 sr -1 . In Fig.1, we adopt such a scenario to account for the PeV excess.</text>
<text><location><page_3><loc_9><loc_11><loc_49><loc_21></location>Summary and Discussions. Very recently the IceCube collaboration has carried out the high-energy contained vertex search to analysis the 662 days data from May 2010 to May 2012. Such a search is sensitive at energies above 50 TeV. The main finding is 26 events with energy from 30 TeV to 250 TeV besides two ∼ 1 PeV neutrinos, while the atmospheric background is only expected to be</text>
<text><location><page_3><loc_52><loc_47><loc_92><loc_93></location>10 . 6 +4 . 5 -3 . 9 events. The TeV-PeV non-atmospheric neutrino detection has a confidence level of 4 . 3 σ [6]. The physical origin of the TeV-PeV non-atmospheric neutrino emission has been widely discussed in recent literature. In this work we examine the possibility that the TeV-PeV non-atmospheric neutrino emission may originate from two groups of sources, motivated by the non-detection of neutrinos in the energy range 250 TeV -1 PeV in current data. If intrinsic, the non-detection of neutrinos in the energy range 250 TeV -1 PeV disfavors the single power-law spectrum model for the TeV-PeV nonatmospheric neutrinos at a confidence level of ∼ 2 σ . If the TeV-PeV non-atmospheric neutrino spectrum does consist two components, one likely has a flat spectrum with a sharp cutoff at the energy ∼ 250 TeV while the other peaks at ∼ 1 PeV. Interestingly, the former may be related to the Type II SNe which can accelerate protons to energies ≤ 10 PeV and these protons interact with the surrounding dense medium (via pp collision), and produce charged pions which then decay into neutrinos. The flat energy spectrum of the sub-PeV neutrinos can be straightforwardly understood since the cross section of pp collision changes very slowly with the energy of the ultra-relativistic protons. We show that the starburst galaxies with high Type II SN rate and rich gas are the promising sources of producing sub-PeV neutrinos. The PeV neutrino excess component is hard to be interpreted within the pp collision process. Instead, the pγ process can produce neutrino spectrum with a sharp peak if the target photons have a very narrow energy distribution (e.g., [3, 33]) and has been adopted to interpret the PeV neutrino excess [18].</text>
<text><location><page_3><loc_52><loc_31><loc_92><loc_46></location>Our speculation that the TeV-PeV non-atmospheric neutrino emission may originate from two (or more) groups of sources is based on a small sample consisting of only ∼ 18 events. Much more data are needed to test whether it is the case. Since IceCube continues to collect data, our speculation will be directly confirmed or ruled out in the near future. If confirmed in the future, the wiggle-like structure of the TeV-PeV nonatmospheric neutrino spectrum will shed valuable light on the underlying physical processes and the astrophysical sources.</text>
<text><location><page_3><loc_52><loc_16><loc_92><loc_30></location>Acknowledgments. This work was supported in part by 973 Program of China under grant 2013CB837000, National Natural Science of China under grants 11173064 and 11273063, and by China Postdoctoral science foundation under grant 2012M521137 and 2013T60569, and Jiangsu Province Postdoctoral science foundation under grant 1202052C. YZF is also supported by the 100 Talents program of Chinese Academy of Sciences and the Foundation for Distinguished Young Scholars of Jiangsu Province, China (No. BK2012047).</text>
<text><location><page_3><loc_52><loc_11><loc_92><loc_15></location>∗ Corresponding author. Electric addresses: [email protected], [email protected], [email protected], [email protected]</text>
<figure>
<location><page_4><loc_13><loc_72><loc_49><loc_91></location>
<caption>FIG. 1: A possible two-component spectrum model for the new IceCube data. The convenience atmospheric muon neutrino emission is denoted by demand symbols, the prompt atmospheric muon neutrino emission is denoted by dotted line [6] and the data for the non-atmospheric neutrinos observed by IceCube are denoted by plus symbols. The solid line represents the lower energy component with a flat spectrum followed by an exponential cutoff at the energy of 150 TeV, while the dashed line represents the neutrino spectrum produced via protons accelerated by AGNs interacting with the EBL (adopted from [18]).</caption>
</figure>
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</document> | [{"title": "TeV-PeV neutrinos over the atmospheric background: originating from two groups of sources?", "content": "Hao-Ning He, 1 Rui-Zhi Yang, 1 Yi-Zhong Fan \u2217 , 1 and Da-Ming Wei 1 1 Key Laboratory of Dark Matter and Space Astronomy, Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China In addition to the two \u223c 1 PeV neutrinos, the IceCube Collaboration recently reported a detection of 26 neutrino candidates at energies from 30 TeV to 250 TeV, implying a confidence level of 4 . 3 \u03c3 over the atmospheric background. We suggest that these TeV-PeV non-atmospheric neutrinos may originate from two groups of sources, motivated by the non-detection of neutrinos in the energy range 250 TeV -1 PeV in current data. If intrinsic, the non-detection of 250 TeV -1 PeV neutrinos disfavors the single power-law spectrum model for the TeV -PeV non-atmospheric neutrinos at a confidence level of \u223c 2 \u03c3 . We then interpret the current neutrino data with a two-component spectrum model. One has a flat spectrum with a cutoff at the energy \u223c 250 TeV and the other has a sharp peak at \u223c 1 PeV. The former is likely via pp collision while the latter may be generated by the photomeson interaction. PACS numbers: 95.85.Ry, 95.85.Hp, 98.70.Sa The construction of IceCube was completed in 2010, and data were collected from May 2010 to May 2011 with 79 strings and an effective livetime of 285.8 days, and from May 2011 to May 2012 with the full 86-string detector and a livetime of 330.1 days. In a search for very-high energy neutrinos, i.e., GZK neutrinos produced via ultra-high energy cosmic rays interacting with cosmic microwave background (CMB) photons and extragalactic background light (EBL) photons [1-3], the IceCube Collaboration discovered two cascade events with energy \u223c 1 PeV [4, 5], which is 2 . 8 \u03c3 beyond the atmospheric background. The energies of these two events are the highest so far but lower than that expected for GZK neutrinos. However, if assuming an unbroken E -2 power-law spectrum, there should be about 8 -9 more events observed above 2 PeV. Quite recently, they carry on the high-energy contained vertex search to analysis the 662 days data from May 2010 to May 2012, which is sensitive at energy above 50 TeV, and at 1PeV, three times as sensitive as GZK search. At the IceCube Particle Astrophysics Symposium (IPA-2013), they reported an observation of 28 neutrino candidates, including 26 events with energy from 30 TeV to 250 TeV, and the two \u223c 1 PeV neutrinos detected previously, implying a significance level of 4 . 3 \u03c3 over the atmospheric background of 10 . 6 +4 . 5 -3 . 9 events [6]. The non-detection of events above \u223c 2 PeV and the excess of events over atmospheric background is the two momentous features of the new IceCube data. In favor of the astrophysical origins, the non-detection above \u223c 2 PeV may indicate a broken or peaked spectrum with a break/cutoff or a peak in the PeV band [6, 7], or a soft unbroken spectrum with index larger than 2.3 [8, 9]. He et al. [10] calculated the neutrino spectrum from pp collision in ultra-luminous infrared galaxies (ULIRGs), and suggested a cutoff of neutrinos spectrum at several PeV which is governed by the abilities of hypernovae (HNe) accelerating protons and ULIRGs confining high energy protons. However, the predicted PeV neutrino flux is so low that can not account for the IceCube data unless the HNe rate in ULIRGs are \u223c tens times higher than current estimate [10]. In the literature, the suggested sources for PeV neutrinos include active galactic neucleis (AGNs, [11]), star forming galaxies (SFGs, [7]), clusters of galaxies[12], hypernovae (HNe, [10, 13]), gamma-ray bursts (GRBs, see [14], but see [15, 16]), and cosmogenic neutrinos (see [17, 18], but see [19, 20]). In some literature, a single power-law spectrum of the TeVPeV non-atmospheric neutrinos has been assumed (e.g., [8, 9]). It is however not clear how robust such an assumption is since different astrophysical sources can produce neutrino emission in quite different energy ranges (e.g., [7, 10, 11, 21]) and the neutrino spectrum may display some wiggle-like structure. The main purpose of this work is to examine whether there is some tentative evidence for wiggle-like structure in current nonatmospheric neutrino data and then discuss the possible sources. To estimate the TeV-PeV non-atmospheric neutrino spectrum, Anchordoqui et al. [8] partitioned the observation data into three energy bins, i.e., 50 TeV -1 PeV, 1 -2 PeV and 2 -10 PeV. Such a fit to the data suggests a single power-law spectrum index \u0393 \u223c 2 . 3. We, however, notice that no events have been reliably detected between 250 TeV and 1 PeV, but about 16 events observed over 10 atmospheric background neutrinos in the energy range of 30 -250 TeV, and 2 events in the energy range of 1 -2 PeV [6]. If the observed 18 non-atmospheric neutrinos have the same origin and can be described by a single power law spectrum with index of \u0393, adopting the exposure of the full IceCube's 662 days operations shown in [8], we can constrain the index via the flux in the two energy bins of 30 -250 TeV and 1 -2 PeV(The two data points are shown in Fig. 1). The fit to the data yields a \u0393 \u223c 2 . 2, which suggests \u223c 9 neutrinos in the energy range of 250 TeV - 1 PeV, while the data analysis found no reliable events in the same energy band. Such a fact favors the hypothesis that the TeV-PeV non-atmospheric neutrino spectrum may be not a single power-law, instead it may have structure. For the single power law spectrum fits within 2 \u03c3 error region, the lowest induced neutrino counts are 3.6. Therefore, if the non-detection in the energy range of 250 TeV -1 PeV is intrinsic (i.e., neither due to the instrument effect nor limited by the data analysis method), a single power-law spectrum model can be excluded at a confidence level of \u2248 2 \u03c3 . As a result, the TeV-PeV non-atmospheric neutrino spectrum may consist of two components, one has a cutoff at \u223c 250 TeV and the other may have a sharp peak at \u223c 1 PeV. We would like to caution that the above speculation is based on a small sample consisting of only \u223c 18 events. Since IceCube continues to collect data, our speculation will be directly confirmed or ruled out in the near future . In the following discussion we focus on the two-component spectrum model and discuss the possible sources. There are two fundamental processes to produce TeVPeV neutrinos, i.e., the pp collision and the p\u03b3 interaction. These two processes both produce charged pions and convert 20% of the proton energy into pions. The charged pions then decay into neutrinos ( \u03c0 + \u2192 \u00b5 + + \u03bd \u00b5 \u2192 e + + \u03bd e + \u00af \u03bd \u00b5 + \u03bd \u00b5 and \u03c0 -\u2192 \u00b5 -+ \u00af \u03bd \u00b5 \u2192 e -+\u00af \u03bd e + \u03bd \u00b5 +\u00af \u03bd \u00b5 ). Usually people assume that the decay products share the energy equally, then 5% of the proton energy will be converted into neutrinos. Since the cross section of pp collision changes very slowly with the energy of the ultra-relativistic protons, the resulting neutrino spectrum traces that of the ultra-relativistic protons and the cutoff energy E \u03bd, cut of the neutrino spectrum mainly depends on the maximum energy E p, max of accelerated protons, i.e., E \u03bd, cut /similarequal 0 . 05 E p, max . As a result of the \u2206 resonance, the cross section of the p\u03b3 interaction peaks at the photon energy E ' \u03b3 /similarequal 0 . 3 GeV in the proton-rest frame. Therefore, the resulting neutrino spectrum sensitively depends on not only the proton spectrum but also the photon spectrum. The neutrino spectrum will peak at energy of 1 . 5 \u00d7 10 15 \u0393 2 /E \u03b3, p eV, where E \u03b3, p is the peak energy of photons in the observer's frame and \u0393 is the Lorentz factor of the emitting region[15]. Therefore, sources like SNe, which can accelerate protons up to 5 -10 PeV in the observer's frame, with dense circum medium surrounding, for example, in the SFGs, may account for the 30 -250 TeV component via pp collision. And sources producing sufficient photons peaking at energy of 5 eV, i.e., around the ultraviolet (UV) band, are possible to explain the PeV neutrino excess via p\u03b3 process. What's more, the thermal emission from the accretion disk of AGN and the EBL photons in UV band can be the target photons. The observed peak energy of the target photons can be higher if considering the relativistic effect, for instance, X-ray photons interacting with protons in a region with a Lorentz factor of tens can also produce an observed neutrino spectrum peaking around PeV. The 30 -250 TeV component. SNe are widely suggested to be the dominant sources for cosmic rays (CRs) at energies below the 'knee' at \u223c 3 \u00d7 10 15 eV, most probably through the diffusive shock acceleration mechanism [22]. The maximum energy of protons accelerated at the shock front of a SN expanding into the uniform dense interstellar medium (ISM) can be estimated as [23], \u03b5 p , max \u2248 10 16 eV( V 3 \u00d7 10 8 cm s -1 ) 2 ( n 10 3 cm -3 ) 1 / 6 ( M SN 10 M /circledot ) 1 / 3 , where n is the number density of ISM, and M SN is the rest mass of the SN ejecta. with a typical kinetic energy E k \u223c 0 . 5 -5 \u00d7 10 51 erg and a typical velocity V \u223c 3 \u00d7 10 8 cm s -1 ( E k / 10 51 erg) 1 / 2 ( M SN / 10 M /circledot ) -1 / 2 . The accelerated protons would lose energy into \u03b3 -ray photons, electrons and positrons, and neutrinos, through pp collisions when injected into the ISM. The energy loss time of protons is \u03c4 loss = (0 . 5 n\u03c3 pp c ) -1 , where the factor 0 . 5 is inelasticity [25], and \u03c3 pp is the inelastic nuclear collision cross section, which is \u223c 70mb for protons at energies \u03b5 ' p \u223c 1 -10 PeV in the rest frame that is of our great interest [26]. Introducing a parameter \u03a3 gas \u2261 m p nl as the surface mass density of gas, with l as the scale of the dense region, the energy loss time reads [27] The confinement time can be rewritten as (i.e., Eq. (8) of [10]) The fraction of the energy that protons lose into pions is f \u03c0 = 1 -exp( -\u03c4 conf /\u03c4 loss ), which is close to 1 as long as \u03c4 conf /greaterorequalslant \u03c4 loss . As a result, the protons with energy \u03b5 ' p lose almost all of their energy via pp collision before escaping from the dense region as long as \u03c4 loss \u2264 \u03c4 conf , which constrains the critical gas surface density \u03a3 crit as With the aim to explain neutrinos with energy of 30 -250 TeV, we assume that the neutrinos spectrum cuts off at \u223c 200 TeV. To confine protons with the maximum energy of 4 \u00d7 10 15 eV, the source should have gas surface density \u2265 0 . 13 g cm -2 if its typical scale is about 100 pc. Most starburst galaxies (SGs) satisfy such a request and hence are optimal candidates for accelerating protons and producing sub-PeV neutrinos. As shown in [10], ULIRGs with the gas surface density about 1 . 0 g cm -2 can accelerate protons with energy up to 100 PeV by hypernovae (HNe), confine the high energy protons and produce neutrinos with energy up to 5 PeV. In contrast with ULIRGs, the star formation rate (SFR) density of SGs is \u223c 5 times larger [29, 30]. Consequently, the Type II SN rate is \u223c 5 times larger than that estimated in [10] since the Type II SN rate is proportional to the SFR [28]. On the other hand, the ratio of the HN rate to the Type II SN rate is \u223c 0.01 and the typical ratio of the HN energy to the Type II SN energy is \u223c 10, hence the energy budget of SGs for the PeV cosmic rays is \u223c 50 times larger than that of ULIRGs for higher-energy protons. As a result, the flux of sub-PeV neutrinos from SGs can be larger than that from ULIRGs by a factor of \u223c 50, suggesting a sub-PeV-neutrino flux from SGs \u223c 10 -7 GeVcm -2 s -1 sr -1 , consistent with the flux level of 6 . 25 \u00b1 1 . 56 \u00d7 10 -8 GeVcm -2 s -1 sr -1 observed by IceCube in 30 -250 TeV. The neutrinos from SGs or SFGs have also been discussed in [21] and [7]. In [21], these authors attributed the GHz radio emission of SGs to synchrotron emission of secondary electrons produced via pp collisions. Then they normalized the flux of neutrinos at GeV via the total energy observed in GHz band being contributed by GeV secondary electrons, since the energy of protons converted into electrons and neutrinos are similar in the pion productions. Depending on the spectral slopes, the extrapolated < 250 TeV neutrino flux is in the range of 4 \u00d7 10 -9 -10 -7 GeV cm -2 s -1 sr -1 (see Fig.1 of [21]), which may be able to explain the new IceCube data, consistent with our result. Murase et al. [7] found out that SFGs can provide the necessary energy budget to explain the neutrino flux in PeV band, but with the acceleration and confinement of \u223c 100 PeV protons as the challenges. The PeV component. Stecker et al. [33] proposed a model to produce a neutrino spectrum peaking at 1 -10 PeV, via protons accelerated by shocks in the cores of AGNs interacting with photons of the 'big blue bump' of thermal emission from the accretion disk [31]. They estimated the value of the muon neutrino flux [32] \u223c E 2 \u03bd \u03a6( E \u03bd ) \u223c 10 -8 GeVcm -2 s -1 sr -1 at 100 TeV and \u223c 6 \u00d7 10 -8 GeVcm -2 s -1 sr -1 at \u223c 1 PeV. The uncertainty of the flux lies in the uncertainty of the AGN rate, the energy budget of AGNs and the uncertainty of other model parameters, such as the spectrum of the accelerated protons. So it's possible to explain the observed flux of (3 . 28 \u00b1 2 . 28) \u00d7 10 -8 GeVcm -2 s -1 sr -1 in the energy range 1 -2 PeV [11]. Another model with neutrino spectrum peaking at PeV is proposed in [18], which interprets the PeV neutrinos via the EeV protons accelerated by AGNs interacting with the EBL photons. By adopting the EBL model with higher photons around 1 \u00b5m and softer spectrum of accelerated protons with spectral index of \u03b1 = 2 . 6, they predict the flux of PeV neutrinos to be (0 . 5 -2) \u00d7 10 -8 GeVcm -2 s -1 sr -1 , consistent with the observed flux at PeV of (3 . 28 \u00b1 2 . 28) \u00d7 10 -8 GeVcm -2 s -1 sr -1 . In Fig.1, we adopt such a scenario to account for the PeV excess. Summary and Discussions. Very recently the IceCube collaboration has carried out the high-energy contained vertex search to analysis the 662 days data from May 2010 to May 2012. Such a search is sensitive at energies above 50 TeV. The main finding is 26 events with energy from 30 TeV to 250 TeV besides two \u223c 1 PeV neutrinos, while the atmospheric background is only expected to be 10 . 6 +4 . 5 -3 . 9 events. The TeV-PeV non-atmospheric neutrino detection has a confidence level of 4 . 3 \u03c3 [6]. The physical origin of the TeV-PeV non-atmospheric neutrino emission has been widely discussed in recent literature. In this work we examine the possibility that the TeV-PeV non-atmospheric neutrino emission may originate from two groups of sources, motivated by the non-detection of neutrinos in the energy range 250 TeV -1 PeV in current data. If intrinsic, the non-detection of neutrinos in the energy range 250 TeV -1 PeV disfavors the single power-law spectrum model for the TeV-PeV nonatmospheric neutrinos at a confidence level of \u223c 2 \u03c3 . If the TeV-PeV non-atmospheric neutrino spectrum does consist two components, one likely has a flat spectrum with a sharp cutoff at the energy \u223c 250 TeV while the other peaks at \u223c 1 PeV. Interestingly, the former may be related to the Type II SNe which can accelerate protons to energies \u2264 10 PeV and these protons interact with the surrounding dense medium (via pp collision), and produce charged pions which then decay into neutrinos. The flat energy spectrum of the sub-PeV neutrinos can be straightforwardly understood since the cross section of pp collision changes very slowly with the energy of the ultra-relativistic protons. We show that the starburst galaxies with high Type II SN rate and rich gas are the promising sources of producing sub-PeV neutrinos. The PeV neutrino excess component is hard to be interpreted within the pp collision process. Instead, the p\u03b3 process can produce neutrino spectrum with a sharp peak if the target photons have a very narrow energy distribution (e.g., [3, 33]) and has been adopted to interpret the PeV neutrino excess [18]. Our speculation that the TeV-PeV non-atmospheric neutrino emission may originate from two (or more) groups of sources is based on a small sample consisting of only \u223c 18 events. Much more data are needed to test whether it is the case. Since IceCube continues to collect data, our speculation will be directly confirmed or ruled out in the near future. If confirmed in the future, the wiggle-like structure of the TeV-PeV nonatmospheric neutrino spectrum will shed valuable light on the underlying physical processes and the astrophysical sources. Acknowledgments. This work was supported in part by 973 Program of China under grant 2013CB837000, National Natural Science of China under grants 11173064 and 11273063, and by China Postdoctoral science foundation under grant 2012M521137 and 2013T60569, and Jiangsu Province Postdoctoral science foundation under grant 1202052C. YZF is also supported by the 100 Talents program of Chinese Academy of Sciences and the Foundation for Distinguished Young Scholars of Jiangsu Province, China (No. BK2012047). \u2217 Corresponding author. Electric addresses: [email protected], [email protected], [email protected], [email protected]", "pages": [1, 2, 3]}] |
2017arXiv170508585D | https://arxiv.org/pdf/1705.08585.pdf | <document>
<section_header_level_1><location><page_1><loc_20><loc_92><loc_81><loc_93></location>Elliptic function solutions in Jackiw-Teitelboim dilaton gravity</section_header_level_1>
<text><location><page_1><loc_34><loc_89><loc_67><loc_90></location>Jennie D'Ambroise 1 and Floyd L. Williams 2</text>
<text><location><page_1><loc_18><loc_77><loc_83><loc_85></location>We present a new family of solutions for the Jackiw-Teitelboim model of two-dimensional gravity with a negative cosmological constant. Here, a metric of constant Ricci scalar curvature is constructed, and explicit linearly independent solutions of the corresponding dilaton field equations are determined. The metric is transformed to a black hole metric, and the dilaton solutions are expressed in terms of Jacobi elliptic functions. Using these solutions we compute, for example, Killing vectors for the metric.</text>
<text><location><page_1><loc_18><loc_75><loc_51><loc_76></location>PACS numbers: 02.30.Jr, 02.40.Ky, 04.20.Jb, 04.70.Bw</text>
<section_header_level_1><location><page_1><loc_42><loc_68><loc_59><loc_69></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_59><loc_92><loc_66></location>It is well-known that the Einstein gravitational field equations for a vacuum (with a zero matter tensor) are automatically solved by any metric g on a two-dimensional space-time M . A proof of this fact is given in section 2 of [1], for example. A non-trivial theory of gravity for such an M was worked out in 1984 by R. Jackiw and C. Teitelboim (J-T). This involves in addition to g a scalar field Φ on M called a dilaton field; see [2],[3]. The pair ( g, Φ) is subject to the equations of motion</text>
<formula><location><page_1><loc_39><loc_56><loc_92><loc_59></location>R ( g ) = 2 l 2 , ∇ i ∇ j Φ = g ij Φ l 2 (1.1)</formula>
<text><location><page_1><loc_9><loc_54><loc_31><loc_55></location>derived from the action integral</text>
<formula><location><page_1><loc_32><loc_49><loc_92><loc_53></location>S ( g, Φ) = constant · ∫ M d 2 x √ | det g | Φ ( R ( g ) -2 l 2 ) (1.2)</formula>
<text><location><page_1><loc_9><loc_46><loc_92><loc_49></location>where R ( g ) is the constant Ricci scalar curvature of g , and where the (negative) cosmological constant is Λ = -1 /l 2 . In local coordinates ( x 1 , x 2 ) on M , the Hessian in (1.1) is given by</text>
<formula><location><page_1><loc_34><loc_41><loc_92><loc_46></location>∇ i ∇ j Φ = ∂ 2 Φ ∂x i ∂x j -2 ∑ k =1 Γ k ij ∂ Φ ∂x k , 1 ≤ i, j ≤ 2 , (1.3)</formula>
<text><location><page_1><loc_9><loc_38><loc_92><loc_41></location>where Γ k ij are the Christoffel symbols (of the second kind) of g [1]. The J-T theory has, for example, the (Lorentzian) black hole solution</text>
<formula><location><page_1><loc_35><loc_34><loc_92><loc_38></location>g : ds 2 = -( m 2 r 2 -M ) dT 2 + dr 2 m 2 r 2 -M , (1.4)</formula>
<text><location><page_1><loc_9><loc_33><loc_38><loc_34></location>with coordinates ( x 1 , x 2 ) = ( T, r ), where</text>
<formula><location><page_1><loc_35><loc_29><loc_92><loc_32></location>Λ = -m 2 , R ( g ) = 2 m 2 , Φ( T, r ) def. = mr, (1.5)</formula>
<text><location><page_1><loc_9><loc_26><loc_92><loc_29></location>with M a black hole mass parameter. Here and throughout we note that our sign convention for scalar curvature is the negative of that used in [2],[3], and by others in the literature.</text>
<text><location><page_1><loc_58><loc_24><loc_58><loc_26></location>/negationslash</text>
<text><location><page_1><loc_9><loc_23><loc_92><loc_26></location>The purpose of this paper is the following. For real numbers a, b = 0 and for a soliton velocity parameter v we consider the following metric in the variables ( x 1 , x 2 ) = ( τ, ρ ):</text>
<formula><location><page_1><loc_28><loc_15><loc_92><loc_23></location>ds 2 def. = a 2 b 2 dn 2 ( ρ, κ ) [( a 2 κ 4 sn 2 ( ρ, κ )cn 2 ( ρ, κ ) dn 2 ( ρ, κ ) -v 2 4 ) dτ 2 -κ 4 sn 2 ( ρ, κ )cn 2 ( ρ, κ ) dn 2 ( ρ, κ ) ( a 2 κ 4 sn 2 ( ρ, κ )cn 2 ( ρ, κ ) dn 2 ( ρ, κ ) -v 2 4 ) -1 dρ 2 ] (1.6)</formula>
<text><location><page_2><loc_9><loc_90><loc_92><loc_93></location>where sn( x, κ ) , cn( x, κ ) , dn( x, κ ) are the standard Jacobi elliptic functions with modulus κ ; 0 ≤ κ ≤ 1 [4]. We will generally assume that</text>
<text><location><page_2><loc_44><loc_87><loc_44><loc_89></location>/negationslash</text>
<formula><location><page_2><loc_42><loc_85><loc_92><loc_90></location>κ = 0 , ∣ ∣ v 2 aκ 2 ∣ ∣ > 1 . (1.7)</formula>
<text><location><page_2><loc_9><loc_83><loc_92><loc_88></location>∣ ∣ As will be seen later, this metric is the diagonalization of a metric constructed from solutions r ( x, t ) , s ( x, t ) of the reaction diffusion system</text>
<formula><location><page_2><loc_42><loc_76><loc_92><loc_82></location>r t -r xx + 2 b 2 r 2 s = 0 (1.8) s t + s xx -2 b 2 rs 2 = 0 .</formula>
<text><location><page_2><loc_9><loc_70><loc_92><loc_76></location>We will explicate the solutions r ( x, t ) , s ( x, t ) in terms of the elliptic function dn( x, κ ). Remarkably, the metric in (1.6) has constant scalar curvature R ( g ) = 8 /b 2 so that the first equation in (1.1) holds. The main work of the paper then is to solve the corresponding system of partial differential equations (the dilaton field equations) in (1.1), which for g in (1.6) are</text>
<formula><location><page_2><loc_34><loc_66><loc_92><loc_69></location>∇ i ∇ j Φ = R ( g ) 2 g ij Φ = 4 b 2 g ij Φ , 1 ≤ i, j ≤ 2 . (1.9)</formula>
<text><location><page_2><loc_9><loc_63><loc_41><loc_65></location>Here the cosmological constant is Λ = -4 /b 2 .</text>
<text><location><page_2><loc_9><loc_56><loc_92><loc_64></location>Given the complicated nature of our g , the system (1.9) is necessarily quite difficult to solve directly. Our method is to construct a series of transformations of variables so that g in (1.6) is transformed to g in (1.4). Then we can use the simple solution Φ( T, r ) = mr in (1.5), and other known solutions, to work backwards through these transformations of variables to construct Φ( τ, ρ ) that satisfies (1.9). The various details involved, with further remarks that lead to (1.6), will be the business of sections 2, 3, and 4.</text>
<text><location><page_2><loc_9><loc_52><loc_92><loc_56></location>In the end, we obtain the following main result: The metric in (1.6) solves the first J-T equation of motion (1.1). Namely R ( g ) = 8 /b 2 , as we have remarked. Also three linearly independent solutions of the field equations in (1.1), namely of the system of equations (1.9), are given by</text>
<text><location><page_2><loc_9><loc_39><loc_11><loc_40></location>for</text>
<formula><location><page_2><loc_24><loc_40><loc_92><loc_51></location>Φ (1) ( τ, ρ ) = 2 a 2 dn 2 ( ρ, κ ) + v 2 4 -a 2 (2 -κ 2 ) (1.10) Φ (2) ( τ, ρ ) = dn( ρ, κ ) sinh ( √ Aτ ) √ v 2 4 -a 2 κ 4 sn 2 ( ρ, κ )cn 2 ( ρ, κ ) / dn 2 ( ρ, κ ) Φ (3) ( τ, ρ ) = dn( ρ, κ ) cosh ( √ Aτ ) √ v 2 4 -a 2 κ 4 sn 2 ( ρ, κ )cn 2 ( ρ, κ ) / dn 2 ( ρ, κ )</formula>
<formula><location><page_2><loc_38><loc_35><loc_92><loc_38></location>A def. = v 4 16 -v 2 a 2 2 (2 -κ 2 ) + a 4 κ 4 , (1.11)</formula>
<text><location><page_2><loc_9><loc_28><loc_92><loc_35></location>which we assume is non-zero. Given (1.7) we shall see in section 4 that A = 0 only for a = ± (1 -√ 1 -κ 2 ) v/ 2 κ 2 and moreover that the second expression under the radical (i.e. v 2 / 4 - · · · ) in (1.10) is positive. For κ = 1, A = ( v 2 / 4 -a 2 ) 2 > 0, but we can have A < 0 for some κ < 1. Also for κ = 1 the solutions in (1.10) reduce to those given in (4.19), with (1.6) given by (4.20).</text>
<section_header_level_1><location><page_2><loc_15><loc_24><loc_85><loc_25></location>II. REACTION DIFFUSION SYSTEMS AND DERIVATION OF THE METRIC IN (1.6)</section_header_level_1>
<text><location><page_2><loc_9><loc_20><loc_92><loc_22></location>Since the metric (1.6) is one of the main objects of interest we indicate in this section its derivation. For a constant B consider the system of partial differential equations</text>
<formula><location><page_2><loc_43><loc_14><loc_92><loc_19></location>r t -r xx + Br 2 s = 0 (2.1) s t + s xx -Brs 2 = 0</formula>
<text><location><page_2><loc_9><loc_13><loc_83><loc_14></location>in the variables ( x, t ). This system is a special case of the more general reaction diffusion system (RDS)</text>
<formula><location><page_2><loc_43><loc_8><loc_92><loc_12></location>r t = d r r xx + F ( r, s ) (2.2) s t = d s s xx + G ( r, s )</formula>
<text><location><page_3><loc_10><loc_13><loc_24><loc_14></location>Apply this to (2.7):</text>
<text><location><page_3><loc_9><loc_89><loc_92><loc_93></location>that occurs in chemistry, physics, or biology, for example, where d r , d s are diffusion constants, and F, G are growth and interaction functions. The key point for us is that from solutions r ( x, t ) , s ( x, t ) of (2.1) one can construct a metric g of constant Ricci scalar curvature R ( g ) = 4 B by the following prescription [5],[6],[7]:</text>
<formula><location><page_3><loc_30><loc_83><loc_92><loc_88></location>g 11 def. = -r x s x , g 12 def. = 1 2 ( sr x -rs x ) , g 22 def. = rs ds 2 def. = g 11 dt 2 +2 g 12 dtdx + g 22 dx 2 . (2.3)</formula>
<text><location><page_3><loc_9><loc_77><loc_92><loc_81></location>One could also simply start with the definitions in (2.3), apart from the preceding references that employ Cartan's zweibein formalism [8], and use a Maple program (tensor), for example, to check directly that indeed R ( g ) = 4 B . Our interest is in the choice B = 2 /b 2 , where for real a, b, v , with a, b = 0 as in section 1, r ( x, t ) , s ( x, t ) given by</text>
<text><location><page_3><loc_55><loc_76><loc_55><loc_78></location>/negationslash</text>
<formula><location><page_3><loc_26><loc_68><loc_92><loc_76></location>r ( x, t ) def. = ab dn( a ( x -vt ) , κ )exp ([ v 2 4 + a 2 (2 -κ 2 ) ] t -vx 2 ) (2.4) s ( x, t ) def. = -ab dn( a ( x -vt ) , κ )exp ( -[ v 2 4 + a 2 (2 -κ 2 ) ] t + vx 2 )</formula>
<text><location><page_3><loc_9><loc_64><loc_92><loc_68></location>are solutions the system (2.1), which also could be checked directly by Maple. For B = 2 /b 2 , (2.1) is system (1.8) with solutions (2.4) promised in section 1, and g in (2.3) has the scalar curvature 4 B = 8 /b 2 discussed in section 1. From [4] various formulas like</text>
<formula><location><page_3><loc_27><loc_54><loc_92><loc_62></location>sn 2 ( x, κ ) + cn 2 ( x, κ ) = 1 , dn 2 ( x, κ ) + κ 2 sn 2 ( x, κ ) = 1 d dx sn( x, κ ) = cn( x, κ )dn( x, κ ) , d dx cn( x, κ ) = -sn( x, κ )dn( x, κ ) d dx dn( x, κ ) = -κ 2 sn( x, κ )cn( x, κ ) (2.5)</formula>
<text><location><page_3><loc_9><loc_52><loc_52><loc_53></location>are available. Using the prescription (2.3) one computes that</text>
<formula><location><page_3><loc_24><loc_44><loc_92><loc_51></location>g 11 = a 2 b 2 [ a 2 κ 4 sn 2 ( a ( x -vt ) , κ )cn 2 ( a ( x -vt ) , κ ) -v 2 4 dn 2 ( a ( x -vt ) , κ ) ] g 12 = a 2 b 2 v 2 dn 2 ( a ( x -vt ) , κ ) , g 22 = -a 2 b 2 dn 2 ( a ( x -vt ) , κ ) . (2.6)</formula>
<text><location><page_3><loc_9><loc_40><loc_71><loc_43></location>For ρ def. = a ( x -vt ), so that dρ = a ( dx -vdt ), g can be expressed more conveniently as</text>
<formula><location><page_3><loc_24><loc_36><loc_92><loc_40></location>ds 2 = a 2 b 2 dn 2 ( ρ, κ ) [( a 2 κ 4 sn 2 ( ρ, κ )cn 2 ( ρ, κ ) dn 2 ( ρ, κ ) -v 2 4 ) dt 2 -v a dtdρ -dρ 2 a 2 ] . (2.7)</formula>
<text><location><page_3><loc_9><loc_31><loc_92><loc_35></location>The goal now is to set up a change of variables ( t, ρ ) -→ ( τ, ρ ) so that g in (2.7) is transformed to (1.6) - where the cross term dτdρ does not appear, in comparison with the term dtdρ appearing in (2.7). For this purpose note first, in general, that for</text>
<formula><location><page_3><loc_37><loc_28><loc_92><loc_30></location>h = A ( ρ ) dt 2 + C 1 ( ρ ) dρdt + C 2 ( ρ ) dρ 2 (2.8)</formula>
<text><location><page_3><loc_9><loc_25><loc_81><loc_27></location>the change of variables τ = t + φ ( ρ ) gives dt = dτ -φ ' ( ρ ) dρ , dt 2 = dτ 2 -2 φ ' ( ρ ) dτdρ + φ ' ( ρ ) 2 dρ 2 , and</text>
<formula><location><page_3><loc_20><loc_21><loc_92><loc_24></location>h = A ( ρ ) dτ 2 +[ -2 φ ' ( ρ ) A ( ρ ) + C 1 ( ρ )] dτdρ + [ A ( ρ ) φ ' ( ρ ) 2 -C 1 ( ρ ) φ ' ( ρ ) + C 2 ( ρ ) ] dρ 2 . (2.9)</formula>
<text><location><page_3><loc_9><loc_20><loc_71><loc_21></location>The condition that the cross term dτdρ does not appear is therefore that φ ( ρ ) satisifies</text>
<formula><location><page_3><loc_45><loc_16><loc_92><loc_19></location>φ ' ( ρ ) = C 1 ( ρ ) 2 A ( ρ ) . (2.10)</formula>
<formula><location><page_3><loc_33><loc_8><loc_92><loc_12></location>φ ' ( ρ ) = -v 2 a · ( a 2 κ 4 sn 2 ( ρ, κ )cn 2 ( ρ, κ ) dn 2 ( ρ, κ ) -v 2 4 ) -1 . (2.11)</formula>
<text><location><page_4><loc_9><loc_84><loc_92><loc_93></location>Now by (2.5), dn 2 ( x, κ ) = 1 -κ 2 sn 2 ( x, κ ) = sn 2 ( x, κ )+cn 2 ( x, κ ) -κ 2 sn 2 ( x, κ ) = cn 2 ( x, κ )+(1 -κ 2 )sn 2 ( x, κ ) ≥ cn 2 ( x, κ ) and sn 2 ( x, κ ) ≤ sn 2 ( x, κ ) + cn 2 ( x, κ ) = 1 ⇒ sn 2 ( x, κ )cn 2 ( x, κ ) / dn 2 ( x, κ ) ≤ 1. If the term in parenthesis in (2.11) were zero, this would therefore force the inequality v 2 / 4 a 2 κ 4 ≤ 1. That is, if ∣ ∣ v/ 2 aκ 2 ∣ ∣ > 1, which is the assumption in (1.7), then v 2 / 4 a 2 κ 4 > 1 and therefore the denominator term in parenthesis in (2.11) is non-zero, which means that φ ' ( ρ ) is a continuous function and (2.11) therefore indeed has a solution φ ( ρ ), with the assumption (1.7) imposed. Also, the coefficient of dρ 2 in (2.9) is</text>
<formula><location><page_4><loc_28><loc_80><loc_92><loc_83></location>a 2 b 2 dn 2 Q v 2 4 a 2 Q 2 -ab 2 v dn 2 v 2 aQ -b 2 dn 2 = -b 2 v 2 dn 2 4 Q -b 2 dn 2 (2.12)</formula>
<text><location><page_4><loc_9><loc_77><loc_66><loc_79></location>where for convenience we write sn,cn,dn for sn( ρ, κ ) , cn( ρ, κ ) , dn( ρ, κ ) and Q for</text>
<formula><location><page_4><loc_36><loc_73><loc_92><loc_76></location>Q ( ρ, κ ) def. = a 2 κ 4 sn 2 ( ρ, κ )cn 2 ( ρ, κ ) dn 2 ( ρ, κ ) -v 2 4 . (2.13)</formula>
<text><location><page_4><loc_9><loc_70><loc_13><loc_72></location>Then</text>
<formula><location><page_4><loc_38><loc_59><loc_92><loc_70></location>Q + v 2 4 = a 2 κ 4 sn 2 cn 2 dn 2 ⇒ 1 + v 2 4 Q = a 2 κ 4 sn 2 cn 2 dn 2 Q ⇒ (2.14) -b 2 dn 2 -b 2 v 2 dn 2 4 Q = -a 2 b 2 κ 4 sn 2 cn 2 Q ,</formula>
<text><location><page_4><loc_9><loc_57><loc_62><loc_58></location>which is the coefficient of dρ 2 in (2.9) by (2.12). Then by (2.10),(2.9) reads</text>
<formula><location><page_4><loc_37><loc_52><loc_92><loc_56></location>h = a 2 b 2 dn 2 Qdτ 2 -a 2 b 2 κ 4 sn 2 cn 2 Q dρ 2 , (2.15)</formula>
<text><location><page_4><loc_9><loc_47><loc_92><loc_51></location>which is (1.6). That is, we have verified that the change of variables τ = t + φ ( ρ ) with φ ( ρ ) subject to condition (2.11) (which in fact renders φ ' ( ρ ) a continuous function, again assuming (1.7)) transforms the reaction diffusion metric in (2.7) to the diagonal metric in (1.6).</text>
<text><location><page_4><loc_10><loc_46><loc_47><loc_47></location>In the special case when the elliptic modulus κ = 1</text>
<formula><location><page_4><loc_33><loc_43><loc_92><loc_44></location>sn( x, 1) = tanh( x ) , cn( x, 1) = dn( x, 1) = sech(x) (2.16)</formula>
<text><location><page_4><loc_9><loc_40><loc_26><loc_42></location>and (2.7), (1.6) simplify:</text>
<formula><location><page_4><loc_22><loc_31><loc_92><loc_39></location>ds 2 = a 2 b 2 sech 2 ρ [( a 2 tanh 2 ρ -v 2 4 ) dt 2 -v a dtd ρ -d ρ 2 a 2 ] , (2.17) ds 2 = a 2 b 2 sech 2 ρ [ ( a 2 tanh 2 ρ -v 2 4 ) dt 2 -tanh 2 ρ ( a 2 tanh 2 ρ -v 2 4 ) -1 d ρ 2 ] ,</formula>
<text><location><page_4><loc_9><loc_25><loc_92><loc_30></location>which are the line elements (3.12), (3.14), respectively, in [6]; a here corresponds to the notation k there. Also the cosmological constant Λ 0 in [6] corresponds to our 2Λ = -8 /b 2 : b 2 = 8 / ( -Λ 0 ). Similarly, r and s in (2.4) reduce to the dissipative soliton solutions q + and q -, respectively, in (2.32) of [6], apart from the factor b . One can also explicitly determine φ ( ρ ) in (2.11).</text>
<section_header_level_1><location><page_4><loc_15><loc_21><loc_86><loc_22></location>III. TRANSFORMATION OF THE METRIC IN (1.6) TO A J-T BLACK HOLE METRIC</section_header_level_1>
<text><location><page_4><loc_9><loc_9><loc_92><loc_19></location>Now that the existence of the metric in (1.6) has been described in the context of a reaction diffusion system (namely (1.8)), the strategy of this section is to set up a series of changes of variables, as indicated in the introduction, that transforms it to the simpler J-T form (1.4). Other applications, of independent interest, can flow from this - apart from our main focus to solve the system (1.9). A general method to go from (1.6) to (1.4) has been developed by the first named author. Alternatively, one can generalize part of the argument in [6] that leads at least to a Schwarzschild form, as we do here, and then argue a bit more to obtain the J-T form - the final result being expressed by equations (3.11)-(3.13) below.</text>
<text><location><page_5><loc_10><loc_91><loc_82><loc_93></location>Start with the change of variables r = | a | dn( ρ, κ ) so that dr = -κ 2 | a | sn( ρ, κ ) · cn( ρ, κ ) dρ by (2.5) ⇒</text>
<formula><location><page_5><loc_40><loc_87><loc_92><loc_91></location>dr 2 r 2 = κ 4 sn 2 ( ρ, κ )cn 2 ( ρ, κ ) dρ 2 dn 2 ( ρ, κ ) . (3.1)</formula>
<text><location><page_5><loc_9><loc_85><loc_18><loc_86></location>Also by (2.5)</text>
<text><location><page_5><loc_9><loc_52><loc_11><loc_54></location>for</text>
<formula><location><page_5><loc_43><loc_48><loc_92><loc_51></location>r 2 0 def. = v 2 4 -a 2 κ 4 > 0 . (3.3)</formula>
<text><location><page_5><loc_9><loc_45><loc_88><loc_47></location>Again by (1.7), v 2 / 4 a 2 κ 4 > 1 ⇒ v 2 / 4 > a 2 κ 4 ⇒ indeed r 2 0 > 0. By (3.1), (3.2) we see that we can write (1.6) as</text>
<formula><location><page_5><loc_28><loc_37><loc_92><loc_45></location>g = b 2 [ -r 2 ( r 2 + r 2 0 + a 2 ( κ 4 + κ 2 -2) + a 4 (1 -κ 2 ) r 2 ) dτ 2 + ( r 2 + r 2 0 + a 2 ( κ 4 + κ 2 -2) + a 4 (1 -κ 2 ) r 2 ) -1 dr 2 ] . (3.4)</formula>
<text><location><page_5><loc_9><loc_33><loc_92><loc_36></location>Next let x def. = (2 r 2 + r 2 0 ) /r 4 0 , as in (3.18) of [6] but where our r 2 0 in (3.3) generalizes their r 2 0 , and for convenience let</text>
<formula><location><page_5><loc_36><loc_29><loc_92><loc_31></location>α def. = a 2 ( κ 4 + κ 2 -2) , β def. = a 4 (1 -κ 2 ) (3.5)</formula>
<text><location><page_5><loc_9><loc_27><loc_39><loc_28></location>in (3.4). Then g in (3.4) assumes the form</text>
<formula><location><page_5><loc_31><loc_18><loc_68><loc_26></location>g = b 2 [( -r 4 0 4 ( r 4 0 x 2 -1) -αr 2 0 2 ( r 2 0 x -1) -β ) dτ 2 + 1 16 r 8 0 ( r 4 0 4 ( r 4 0 x 2 -1) + α 2 r 2 0 ( r 2 0 x -1) + β ) -1 dx 2 ]</formula>
<formula><location><page_5><loc_32><loc_8><loc_92><loc_16></location>= b 2 r 4 0 4 [ -( r 4 0 x 2 -1 + 2 α r 2 0 ( r 2 0 x -1) + 4 β r 4 0 ) dτ 2 + ( r 4 0 x 2 -1 + 2 α r 2 0 ( r 2 0 x -1) + 4 β r 4 0 ) -1 dx 2 ] (3.6)</formula>
<formula><location><page_5><loc_30><loc_55><loc_92><loc_82></location>a 2 κ 4 sn 2 ( ρ, κ )cn 2 ( ρ, κ ) dn 2 ( ρ, κ ) = a 2 κ 2 (1 -dn 2 ( ρ, κ ))(1 -sn 2 ( ρ, κ )) dn 2 ( ρ, κ ) = ( a 2 -a 2 dn 2 ( ρ, κ ))( κ 2 -κ 2 sn 2 ( ρ, κ )) dn 2 ( ρ, κ ) = ( a 2 -r 2 )( κ 2 +dn 2 ( ρ, κ ) -1) r 2 /a 2 = ( a 2 -r 2 )( κ 2 -1 + r 2 /a 2 ) r 2 /a 2 (3.2) = ( a 2 -r 2 ) r 2 [ a 2 ( κ 2 -1) + r 2 ] ⇒ a 2 κ 4 sn 2 ( ρ, κ )cn 2 ( ρ, κ ) dn 2 ( ρ, κ ) -v 2 4 = ( a 2 -r 2 ) r 2 [ a 2 ( κ 2 -1) + r 2 ] -v 2 4 = a 4 ( κ 2 -1) r 2 +2 a 2 -a 2 κ 2 -r 2 -v 2 4 = a 4 ( κ 2 -1) r 2 +2 a 2 -a 2 κ 2 -r 2 -r 2 0 -a 2 κ 4</formula>
<text><location><page_6><loc_9><loc_92><loc_81><loc_93></location>which generalizes the Schwarzchild form (3.19) of [6] since for κ = 1 we have that α = β = 0 in (3.5).</text>
<text><location><page_6><loc_10><loc_89><loc_84><loc_92></location>For the change of variables t = A 0 τ , r -= A 0 x with A 0 def. = | b | r 2 0 / 2, the Schwarzschild g in (3.6) goes to</text>
<formula><location><page_6><loc_33><loc_81><loc_92><loc_89></location>g = -[ 4 b 2 r 2 --1 + 2 α r 2 0 ( 2 | b | r --1 ) + 4 β r 4 0 ] dt 2 + [ 4 b 2 r 2 --1 + 2 α r 2 0 ( 2 | b | r --1 ) + 4 β r 4 0 ] -1 dr 2 -, (3.7)</formula>
<text><location><page_6><loc_9><loc_79><loc_24><loc_80></location>which in turn goes to</text>
<formula><location><page_6><loc_32><loc_70><loc_92><loc_78></location>g = -[ 4 b 2 r 2 1 -b 2 + 2 α r 2 0 ( 2 b | b | r 1 -b 2 ) + 4 βb 2 r 4 0 ] dT 2 (3.8) + [ 4 r 2 1 b 2 -b 2 + 2 α r 2 0 ( 2 b | b | r 1 -b 2 ) + 4 βb 2 r 4 0 ] -1 dr 2 1</formula>
<text><location><page_6><loc_9><loc_68><loc_91><loc_70></location>by way of the change of variables t = bT , r -= r 1 /b . We need one final observation: In general a metric of the form</text>
<text><location><page_6><loc_14><loc_62><loc_14><loc_64></location>/negationslash</text>
<formula><location><page_6><loc_29><loc_63><loc_92><loc_67></location>g 1 = -[ A 1 x 2 + B 1 x + C 1 ] dT 2 + [ A 1 x 2 + B 1 x + C 1 ] -1 dx 2 , (3.9)</formula>
<text><location><page_6><loc_9><loc_62><loc_53><loc_64></location>say A 1 = 0 can be transformed to the J-T form (1.4), namely</text>
<formula><location><page_6><loc_29><loc_57><loc_92><loc_61></location>g 1 = -[ A 1 r 2 + C 1 -B 2 1 4 A 1 ] dT 2 + [ A 1 r 2 + C 1 -B 2 1 4 A 1 ] -1 dr 2 , (3.10)</formula>
<text><location><page_6><loc_9><loc_55><loc_81><loc_56></location>by way of the change of variables r = x + B 1 2 A 1 . Apply this to (3.8) with x playing the role of r 1 there:</text>
<formula><location><page_6><loc_29><loc_49><loc_92><loc_53></location>g = -[ A 1 r 2 + C 1 -B 2 1 4 A 1 ] dT 2 + [ A 1 r 2 + C 1 -B 2 1 4 A 1 ] -1 dr 2 , (3.11)</formula>
<text><location><page_6><loc_9><loc_47><loc_11><loc_49></location>for</text>
<formula><location><page_6><loc_30><loc_42><loc_92><loc_47></location>A 1 def. = 4 b 2 , B 1 def. = 4 αb r 2 0 | b | , C 1 def. = -b 2 -2 αb 2 r 2 0 + 4 βb 2 r 4 0 . (3.12)</formula>
<text><location><page_6><loc_9><loc_39><loc_76><loc_42></location>Using definition (3.5) for α, β and r 2 0 = v 2 4 -a 2 κ 4 , which is definition (3.3), one computes that</text>
<formula><location><page_6><loc_34><loc_35><loc_92><loc_39></location>C 1 -B 2 1 4 A 1 = -b 2 r 4 0 [ v 4 16 -a 2 v 2 2 (2 -κ 2 ) + a 4 κ 4 ] (3.13)</formula>
<text><location><page_6><loc_9><loc_33><loc_15><loc_34></location>in (3.11).</text>
<section_header_level_1><location><page_6><loc_17><loc_29><loc_84><loc_30></location>IV. DERIVATION OF THE SOLUTIONS (1.10) OF THE FIELD EQUATIONS (1.9)</section_header_level_1>
<text><location><page_6><loc_9><loc_23><loc_92><loc_27></location>The main result is derived in this section. Namely, we indicate how the series of changes of variables in section 3 (according to remarks in the introduction) lead to the linearly independent solutions Φ ( j ) ( τ, ρ ), j = 1 , 2 , 3 in (1.10) of the dilaton field equations in (1.9). There the metric elements g ij are given by (1.6): For Q ( ρ, κ ) in (2.13)</text>
<formula><location><page_6><loc_30><loc_15><loc_92><loc_21></location>g 11 def. = a 2 b 2 dn 2 ( ρ, κ ) Q ( ρ, κ ) , g 12 = g 21 = 0 , g 22 = a 2 b 2 dn 2 ( ρ, κ ) ( -κ 4 sn 2 ( ρ, κ )cn 2 ( ρ, κ ) dn 2 ( ρ, κ ) ) Q ( ρ, κ ) -1 , (4.1)</formula>
<text><location><page_6><loc_9><loc_9><loc_92><loc_15></location>and the ∇ i ∇ j Φ are given by (1.3) for ( x 1 , x 2 ) = ( τ, ρ ). The Christoffel symbols Γ k ij in (1.3) (which could be computed by Maple, for example) will not be needed for the derivation of (1.10), although they could be used to verify these solutions. Obviously any dilaton solution could be replaced by any non-zero multiple of itself. In the following then, we can disregard such multiples if we wish to.</text>
<text><location><page_7><loc_9><loc_90><loc_92><loc_93></location>In addition to the dilaton solution Φ (1) ( T, r ) def. = mr in (1.5) for the metric (1.4) in the variables ( T, r ), there are solutions</text>
<formula><location><page_7><loc_35><loc_83><loc_92><loc_90></location>Φ (2) ( T, r ) def. = √ m 2 r 2 -M sinh( m √ MT ) Φ (3) ( T, r ) def. = √ m 2 r 2 -M cosh( m √ MT ) . (4.2)</formula>
<text><location><page_7><loc_9><loc_81><loc_92><loc_84></location>We work backwards the changes of variables in section 3 for Φ (1) ( T, r ) , Φ (2) ( T, r ), for example, to see how one arrives at the first two solutions Φ (1) ( τ, ρ ) , Φ (2) ( τ, ρ ) in (1.10), in the variables ( τ, ρ ).</text>
<text><location><page_7><loc_9><loc_75><loc_92><loc_81></location>Starting with the (3.11) version of (1.4), we have m 2 = A 1 = 4 /b 2 by (3.12), with M = -( C 1 -B 2 1 / 4 A 1 ) given by (3.13). Here m √ M = √ B 2 1 -4 A 1 C 1 / 2 (for m = 2 / | b | ) ⇒</text>
<formula><location><page_7><loc_22><loc_72><loc_92><loc_76></location>Φ (1) ( T, r ) = 2 | b | r, Φ (2) ( T, r ) = √ A 1 r 2 + C 1 -B 2 1 4 A 1 sinh ( √ B 2 1 -4 A 1 C 1 2 T ) . (4.3)</formula>
<text><location><page_7><loc_9><loc_69><loc_83><loc_71></location>By the final change of variables r = r 1 + B 1 2 A 1 in section 3, we see that A 1 r 2 = A 1 r 2 1 + B 1 r 1 + B 2 1 / 4 A 1 ⇒</text>
<formula><location><page_7><loc_28><loc_60><loc_92><loc_68></location>Φ (1) ( T, r 1 ) = 2 | b | ( r 1 + B 1 2 A 1 ) , (4.4) Φ (2) ( T, r 1 ) = √ A 1 r 2 1 + B 1 r 1 + C 1 sinh ( √ B 2 1 -4 A 1 C 1 2 T ) .</formula>
<text><location><page_7><loc_9><loc_57><loc_72><loc_59></location>The change of variables t = bT , r -= r 1 /b preceded the change r = r 1 + B 1 / 2 A 1 , so that</text>
<formula><location><page_7><loc_29><loc_49><loc_92><loc_57></location>Φ (1) ( t, r -) = 2 b | b | r -+ B 1 | b | A 1 , (4.5) Φ (2) ( t, r -) = √ 4 r 2 -+ B 1 br -+ C 1 sinh ( √ B 2 1 -4 A 1 C 1 2 b t )</formula>
<text><location><page_7><loc_9><loc_45><loc_63><loc_48></location>since A 1 b 2 def. = 4. We had t = A 0 τ , r -= A 0 x for A 0 def. = | b | r 2 0 / 2, which gives</text>
<formula><location><page_7><loc_24><loc_37><loc_92><loc_45></location>Φ (1) ( τ, x ) = br 2 0 x + B 1 | b | A 1 , | b | b Φ (1) ( τ, x ) = | b | r 2 0 x + B 1 bA 1 , (4.6) Φ (2) ( τ, x ) = √ b 2 r 4 0 x 2 + B 1 b | b | 2 r 2 0 x + C 1 sinh ( √ B 2 1 -4 A 1 C 1 4 · | b | r 2 0 b τ ) ,</formula>
<text><location><page_7><loc_9><loc_35><loc_69><loc_36></location>for the Schwarzchild version of our metric in (3.6). Next let x = (2 r 2 + r 2 0 ) /r 4 0 to get</text>
<formula><location><page_7><loc_21><loc_26><loc_92><loc_33></location>Φ (1) ( τ, r ) = | b | (2 r 2 + r 2 0 ) /r 2 0 + B 1 bA 1 , (4.7) Φ (2) ( τ, r ) = √ b 2 (2 r 2 + r 2 0 ) 2 r 4 0 + B 1 b | b | 2 (2 r 2 + r 2 0 ) r 2 0 + C 1 sinh ( √ B 2 1 -4 A 1 C 1 4 r 2 0 τ )</formula>
<text><location><page_7><loc_9><loc_21><loc_92><loc_25></location>where we have disregarded the multiple | b | /b = ± 1 in (4.6) and have used that sinh( | b | x/b ) = ( | b | /b ) sinh( x ). Finally, the first change of variables r = | a | dn( ρ, κ ) in section 3 gives</text>
<formula><location><page_7><loc_35><loc_16><loc_92><loc_21></location>Φ (1) ( τ, ρ ) = | b | r 2 0 ( 2 a 2 dn 2 ( ρ, κ ) + r 2 0 ) + α | b | r 2 0 , (4.8)</formula>
<text><location><page_7><loc_9><loc_13><loc_92><loc_16></location>by definition (3.12). If we disregard the multiple | b | /r 2 0 in (4.8) and use that r 2 0 + α def. = v 2 / 4+ a 2 ( κ 2 -2) by definitions (3.3), (3.5) we obtain from (4.8) the first solution</text>
<formula><location><page_7><loc_35><loc_9><loc_92><loc_12></location>Φ (1) ( τ, ρ ) = 2 a 2 dn 2 ( ρ, κ ) + v 2 4 + a 2 ( κ 2 -2) (4.9)</formula>
<text><location><page_8><loc_9><loc_92><loc_64><loc_93></location>in (1.10). More work is required of course to obtain the second solution there.</text>
<text><location><page_8><loc_10><loc_90><loc_36><loc_92></location>First we note that by (3.12), (3.13)</text>
<formula><location><page_8><loc_23><loc_82><loc_92><loc_89></location>B 2 1 -4 A 1 C 1 = -4 A 1 ( C 1 -B 2 1 4 A 1 ) = 16 r 4 0 [ v 4 16 -a 2 v 2 2 (2 -κ 2 ) + a 4 κ 4 ] ⇒ √ B 2 1 -4 A 1 C 1 4 r 2 0 τ = √ v 4 16 -a 2 v 2 2 (2 -κ 2 ) + a 4 κ 4 · τ, (4.10)</formula>
<text><location><page_8><loc_9><loc_79><loc_87><loc_82></location>which is the √ Aτ in (1.10). Also for r = | a | dn, dn = dn( ρ, κ ), the quantity under the other radical in (4.7) is</text>
<formula><location><page_8><loc_29><loc_71><loc_92><loc_78></location>b 2 r 4 0 ( 2 a 2 dn 2 + r 2 0 ) 2 + B 1 b | b | 2 r 2 0 (2 a 2 dn 2 + r 2 0 ) + C 1 = (4.11) 4 a 4 b 2 r 4 0 dn 4 + ( 4 a 2 b 2 r 2 0 + B 1 b | b | a 2 r 2 0 ) dn 2 + b 2 + B 1 b | b | 2 + C 1 ,</formula>
<text><location><page_8><loc_9><loc_69><loc_27><loc_70></location>where by definition (3.12)</text>
<formula><location><page_8><loc_30><loc_53><loc_92><loc_68></location>B 1 b | b | a 2 r 2 0 = 4 a 2 b 2 α r 4 0 , b 2 + B 1 b | b | 2 + C 1 = b 2 + 2 b 2 α r 2 0 -b 2 -2 αb 2 r 2 0 + 4 βb 2 r 4 0 = 4 βb 2 r 4 0 ⇒ (4.12) 4 a 2 b 2 r 2 0 + B 1 b | b | a 2 r 2 0 = 4 r 2 0 a 2 b 2 +4 a 2 b 2 α r 4 0 = 4 a 2 b 2 r 4 0 ( r 2 0 + α ) = 4 a 2 b 2 r 4 0 [ v 2 4 + a 2 ( κ 2 -2) ] ,</formula>
<text><location><page_8><loc_9><loc_49><loc_92><loc_52></location>again by definitions (3.3), (3.5). That is, since β = a 4 (1 -κ 2 ) by definition (3.5) the quantity in (4.11) (which is under the radical in (4.7) for r = | a | dn) is</text>
<formula><location><page_8><loc_28><loc_41><loc_92><loc_48></location>4 a 4 b 2 r 4 0 dn 4 + 4 a 2 b 2 r 4 0 [ v 2 4 + a 2 ( κ 2 -2) ] dn 2 + 4 a 4 b 2 (1 -κ 2 ) r 4 0 = (4.13) 4 a 2 b 2 r 4 0 [ a 2 dn 4 + ( v 2 4 + a 2 ( κ 2 -2) ) dn 2 + a 2 (1 -κ 2 ) ] .</formula>
<text><location><page_8><loc_9><loc_34><loc_92><loc_40></location>We let B ( ρ ) denote the latter bracket here. By (4.7), (4.10), (4.13) we see that (for now) Φ (2) ( τ, ρ ) = √ B ( ρ ) sinh ( √ Aτ ) , if we disregard the multiple √ 4 a 2 b 2 /r 4 0 = 2 | a || b | /r 2 0 . We find an alternate expression for B ( ρ ), which is simpler and which shows that B ( ρ ) > 0, given (1.7). Again we</text>
<text><location><page_9><loc_9><loc_92><loc_60><loc_93></location>write sn, cn, dn for sn( ρ, κ ) , cn( ρ, κ ) , dn( ρ, κ ), and we make use of (2.5).</text>
<formula><location><page_9><loc_28><loc_58><loc_92><loc_91></location>B ( ρ ) def. = dn 2 [ a 2 dn 2 + v 2 4 -2 a 2 + a 2 κ 2 + a 2 (1 -κ 2 ) dn 2 ] = dn 2 [ a 2 (1 -κ 2 sn 2 ) + v 2 4 -2 a 2 + a 2 κ 2 + a 2 (1 -κ 2 ) dn 2 ] = dn 2 [ -a 2 κ 2 sn 2 + v 2 4 -a 2 (1 -κ 2 ) + a 2 (1 -κ 2 ) dn 2 ] = dn 2 [ v 2 4 -a 2 κ 2 sn 2 + a 2 (1 -κ 2 ) dn 2 (1 -dn 2 ) ] (4.14) = dn 2 [ v 2 4 -a 2 κ 2 sn 2 + a 2 (1 -κ 2 ) dn 2 κ 2 sn 2 ] = dn 2 [ v 2 4 -a 2 κ 2 sn 2 (dn 2 -1 + κ 2 ) dn 2 ] = dn 2 [ v 2 4 -a 2 κ 2 sn 2 dn 2 ( -κ 2 sn 2 + κ 2 ) ] = dn 2 [ v 2 4 -a 2 κ 4 sn 2 dn 2 (1 -sn 2 ) ] = dn 2 [ v 2 4 -a 2 κ 4 sn 2 cn 2 dn 2 ] ,</formula>
<text><location><page_9><loc_9><loc_55><loc_49><loc_57></location>where we noted in section 2 that sn 2 cn 2 / dn 2 ≤ 1. Hence</text>
<formula><location><page_9><loc_38><loc_51><loc_92><loc_55></location>v 2 4 -a 2 κ 4 sn 2 cn 2 dn 2 ≥ v 2 4 -a 2 κ 4 > 0 (4.15)</formula>
<text><location><page_9><loc_59><loc_48><loc_59><loc_50></location>/negationslash</text>
<text><location><page_9><loc_9><loc_39><loc_92><loc_44></location>The quartic equation A = 0 has roots a = ± (1 + √ 1 -κ 2 ) v/ 2 κ 2 , ± (1 -√ 1 -κ 2 ) v/ 2 κ 2 with a 2 = (2 -κ 2 + 2 √ 1 -κ 2 ) v 2 / 4 κ 4 , (2 -κ 2 -2 √ 1 -κ 2 ) v 2 / 4 κ 4 respectively. (1.7) requires that a 2 < v 2 / 4 κ 4 , which forces the inequalities</text>
<text><location><page_9><loc_9><loc_43><loc_92><loc_50></location>by (1.7), again as in (3.3), and we see that B ( ρ ) > 0 since dn( ρ, κ ) = 0 for ρ a real number. Moreover we have established the desired expression for Φ (2) ( τ, ρ ) in (1.10). Clearly one can replace the hyperbolic sine in the preceding discussion by the hyperbolic cosine in (4.2) to obtain the third solution Φ (3) ( τ, ρ ) in (1.10). To finish other claims made in section 1 we check that in (1.11) A = 0 only for a = ± (1 -√ 1 -κ 2 ) v/ 2 κ 2 . We continue to assume (1.7) of course.</text>
<formula><location><page_9><loc_33><loc_35><loc_92><loc_39></location>2 -κ 2 +2 √ 1 -κ 2 < 1 , 2 -κ 2 -2 √ 1 -κ 2 < 1 , (4.16)</formula>
<text><location><page_9><loc_9><loc_31><loc_92><loc_36></location>of which the first one reads 1 -κ 2 +2 √ 1 -κ 2 < 0, with the left hand side here being ≥ 0 - a contradiction. That is, we cannot have a = ± (1 + √ 1 -κ 2 ) v/ 2 κ 2 which means that a = ± (1 -√ 1 -κ 2 ) v/ 2 κ 2 . Also we check that the solutions are linearly independent: Assume for constants c 1 , c 2 , c 3 that</text>
<formula><location><page_9><loc_35><loc_28><loc_92><loc_30></location>c 1 Φ (1) ( τ, ρ ) + c 2 Φ (2) ( τ, ρ ) + c 3 Φ (3) ( τ, ρ ) = 0 . (4.17)</formula>
<text><location><page_9><loc_9><loc_26><loc_64><loc_27></location>Differentiate this equation with respect to τ and evaluate the result at ( τ, 0):</text>
<formula><location><page_9><loc_32><loc_22><loc_92><loc_25></location>c 2 √ A cosh( √ Aτ ) | v | / 2 + c 3 √ A sinh( √ Aτ ) | v | / 2 = 0 (4.18)</formula>
<text><location><page_9><loc_69><loc_19><loc_69><loc_21></location>/negationslash</text>
<text><location><page_9><loc_9><loc_17><loc_92><loc_21></location>since dn(0 , κ ) = 1 , sn(0 , κ ) = 0. The choice τ = 0 then gives c 2 = 0, since A,v = 0, and differentiation of the equation c 3 √ A sinh( √ Aτ ) | v | / 2 = 0 and at τ = 0 gives c 3 = 0. Using again that dn(0 , κ ) = 1 we see by (1.10) that Φ (1) ( τ, 0) def. = v 2 / 4 + a 2 κ 2 > 0 and hence also c 1 = 0.</text>
<text><location><page_9><loc_9><loc_13><loc_92><loc_17></location>Note that if v = a = 2 and κ = 1 / 2, for example, then even though v/ 2 aκ 2 = 2 > 1 (so that (1.7) is satisfied), we have that A = -12 < 0.</text>
<text><location><page_9><loc_9><loc_8><loc_92><loc_14></location>Again in the special case when the elliptic modulus κ = 1, we have in (1.11) that A = ( v 2 / 4 -a 2 ) 2 > 0 and √ A = v 2 / 4 -a 2 > 0 (by (1.7) or (3.3)), and B ( ρ ) = sech 2 ρ [ v 2 / 4 -a 2 tanh 2 ρ ] = sech 2 ρ [ v 2 -4a 2 tanh 2 ρ ] / 4, by</text>
<text><location><page_10><loc_9><loc_89><loc_92><loc_93></location>(2.16), (4.14) ⇒ √ B ( ρ ) = 1 2 sech ρ √ v 2 -4a 2 tanh 2 ρ. Here (directly) tanh 2 ρ ≤ 1 ⇒ v 2 -4 a 2 tanh 2 ρ > 0, again as v 2 > 4 a 2 . Thus by (1.10) and (2.17)</text>
<formula><location><page_10><loc_28><loc_79><loc_92><loc_90></location>Φ (1) ( τ, ρ ) = 2 a 2 sech 2 ρ + v 2 4 -a 2 , Φ (2) ( τ, ρ ) = ( sinh ( v 2 4 -a 2 ) τ ) (sech ρ ) √ v 2 -4 a 2 tanh 2 ρ, (4.19) Φ (3) ( τ, ρ ) = ( cosh ( v 2 4 -a 2 ) τ ) (sech ρ ) √ v 2 -4 a 2 tanh 2 ρ</formula>
<formula><location><page_10><loc_9><loc_72><loc_92><loc_79></location>(where we have disregarded the multiple 1 / 2 in √ B ( ρ )) are dilaton field solutions for the metric ds 2 = a 2 b 2 sech 2 ρ [ ( a 2 tanh 2 ρ -v 2 4 ) d τ 2 -( tanh 2 ρ ) ( a 2 tanh 2 ρ -v 2 4 ) -1 d ρ 2 ] . (4.20)</formula>
<text><location><page_10><loc_9><loc_71><loc_35><loc_72></location>The solutions in (4.19) are also new.</text>
<section_header_level_1><location><page_10><loc_23><loc_67><loc_78><loc_68></location>V. KILLING VECTOR FIELDS FOR THE SOLUTIONS (1.6), (1.10)</section_header_level_1>
<text><location><page_10><loc_9><loc_62><loc_92><loc_65></location>Recall that a smooth vector field Y on an n -dimensional Riemannian manifold ( M,g ) is called a Killing vector field (or an infinitesimal motion of M ) if for arbitrary smooth vector fields X,Z on M</text>
<formula><location><page_10><loc_35><loc_60><loc_92><loc_61></location>Y g ( X,Z ) = g ([ Y, X ] , Z ) + g ( X, [ Y, Z ]) = 0 . (5.1)</formula>
<text><location><page_10><loc_9><loc_54><loc_92><loc_59></location>If Y = n ∑ i =1 Y i ∂ ∂x i is an expression of Y in terms of local coordinates ( x 1 , . . . , x n ) on M , then (5.1) is equivalent to the</text>
<formula><location><page_10><loc_38><loc_49><loc_92><loc_53></location>n ∑ i =1 g ki ∂Y i ∂x j + g ji ∂Y i ∂x k + ∂g jk ∂x i Y i = 0 (5.2)</formula>
<text><location><page_10><loc_9><loc_54><loc_23><loc_55></location>system of equations</text>
<text><location><page_10><loc_9><loc_46><loc_92><loc_49></location>for 1 ≤ j, k ≤ n [8], [9]. In the special (diagonal) case with g ij = 0 for i = j , and with n = 2, the Killing equations (5.2) simplify to the following three equations:</text>
<text><location><page_10><loc_61><loc_47><loc_61><loc_49></location>/negationslash</text>
<formula><location><page_10><loc_38><loc_36><loc_92><loc_45></location>2 g 11 ∂Y 1 ∂x 1 + ∂g 11 ∂x 1 Y 1 + ∂g 11 ∂x 2 Y 2 = 0 g 11 ∂Y 1 ∂x 2 + g 22 ∂Y 2 ∂x 1 = 0 (5.3) ∂g 22 ∂x 1 Y 1 +2 g 22 ∂Y 2 ∂x 2 + ∂g 22 ∂x 2 Y 2 = 0</formula>
<text><location><page_10><loc_9><loc_32><loc_92><loc_35></location>As was shown in [10], every solution ( g, Φ) of the field equations in (1.1) gives rise to a corresponding Killing vector field Y = Y ( g, Φ) by way of the local prescription</text>
<formula><location><page_10><loc_44><loc_28><loc_92><loc_32></location>Y i = l/epsilon1 ij | det g | ∂ Φ ∂x j (5.4)</formula>
<text><location><page_10><loc_9><loc_19><loc_92><loc_30></location>√ with /epsilon1 ij = a permutation symbol. Y preserves both g and Φ. For g in (1.4) and for the fields Φ in (1.5) and (4.2), the corresponding Killing vector fields are given in equations (16), (17), (18) of [11], for example. Our interest of course is in the case of the three solutions ( g, Φ ( j ) ) in (1.10) with g given by (1.6). By (4.1), √ | det g | = a 2 b 2 κ 2 | sncn | dn. Since Y i could be replaced by a scalar multiple of itself (for example, -Y i ), we shall disregard the absolute value of sncn here, and given (1.9) we shall take l = b 2 (instead of | b | 2 ). For /epsilon1 11 = /epsilon1 22 = 0 , /epsilon1 12 = -1 = -/epsilon1 21 (5.4) then assumes the generic form</text>
<formula><location><page_10><loc_32><loc_8><loc_92><loc_18></location>Y 1 = [ 2 a 2 bκ 2 sn( ρ, κ )cn( ρ, κ )dn( ρ, κ ) ] -1 ( -∂ Φ ∂ρ ) (5.5) Y 2 = [ 2 a 2 bκ 2 sn( ρ, κ )cn( ρ, κ )dn( ρ, κ ) ] -1 ∂ Φ ∂τ ; Y = Y 1 ∂ ∂τ + Y 2 ∂ ∂ρ ,</formula>
<text><location><page_11><loc_9><loc_92><loc_37><loc_93></location>where we take ( x 1 , x 2 ) = ( τ, ρ ) in (5.3).</text>
<text><location><page_11><loc_10><loc_90><loc_25><loc_92></location>For the first solution</text>
<formula><location><page_11><loc_35><loc_86><loc_92><loc_90></location>Φ (1) ( τ, ρ ) = 2 a 2 dn( ρ, κ ) + v 4 4 -a 2 (2 -κ 2 ) (5.6)</formula>
<text><location><page_11><loc_9><loc_83><loc_92><loc_85></location>in (1.10), the computation of the corresponding Killing vector field Y is trivial: By (2.5) and (5.5) Y 1 = 2 /b , and of course Y 2 = 0. Since ∂g 11 /∂τ = ∂g 22 /∂τ = 0 by (4.1), the Killing equations in (5.3) are satisfied and we see that</text>
<formula><location><page_11><loc_47><loc_79><loc_92><loc_82></location>Y = 2 b ∂ ∂τ (5.7)</formula>
<text><location><page_11><loc_9><loc_75><loc_92><loc_78></location>for ( g, Φ (1) ). Computations for the other two solutions Φ (2) , Φ (3) in (1.10) are more involved. The result is the following, where again</text>
<text><location><page_11><loc_9><loc_67><loc_23><loc_70></location>in definition (1.11). For Φ (2) ( τ, ρ )</text>
<formula><location><page_11><loc_35><loc_69><loc_92><loc_74></location>A = 1 16 ( v 4 -16 v 2 a 2 +8 v 2 a 2 κ 2 +16 a 4 κ 4 ) (5.8)</formula>
<formula><location><page_11><loc_19><loc_57><loc_92><loc_66></location>Y 1 = 1 4 ( sinh( √ Aτ ) ) [ v 2 +( -v 2 κ 2 -4 a 2 κ 4 -8 a 2 κ 2 )sn 2 ( ρ, κ ) + 8 a 2 κ 4 sn 4 ( ρ, κ ) + 4 a 2 κ 2 ] a 2 b dn 2 ( ρ, κ ) √ v 2 +( -v 2 κ 2 -4 a 2 κ 4 )sn 2 ( ρ, κ ) + 4 a 2 κ 4 sn 4 ( ρ, κ ) Y 2 = √ A 4 ( cosh( √ Aτ ) ) √ v 2 +( -v 2 κ 2 -4 a 2 κ 4 )sn 2 ( ρ, κ ) + 4 a 2 κ 4 sn 4 ( ρ, κ ) a 2 bκ 2 sn( ρ, κ )cn( ρ, κ )dn( ρ, κ ) . (5.9)</formula>
<text><location><page_11><loc_9><loc_51><loc_92><loc_56></location>For Φ (3) ( τ, ρ ) one has quite similar formulas for Y 1 , Y 2 except (as expected) the roles of the hyperbolic sine and hyperbolic cosine in (5.9) are interchanged: the factor sinh( √ Aτ ) for Y 1 in (5.9) is replaced by cosh( √ Aτ ), and, similarly, cosh( √ Aτ ) for Y 2 in (5.9) is replaced by sinh( √ Aτ ).</text>
<text><location><page_11><loc_10><loc_50><loc_87><loc_51></location>One can also find the following alternative expressions for the Killing vector field components for Φ (2) ( τ, ρ ):</text>
<formula><location><page_11><loc_28><loc_40><loc_92><loc_49></location>Y 1 = ( v 2 +4 a 2 κ 2 cn 2 ( ρ, κ ) -4 a 2 κ 2 sn 2 ( ρ, κ ) ) sinh( √ Aτ ) 4 a 2 b dn( ρ, κ ) √ v 2 -4 a 2 κ 4 cn 2 ( ρ, κ )sn 2 ( ρ, κ )dn -2 ( ρ, κ ) (5.10) Y 2 = 4 √ A cosh( √ Aτ ) √ v 2 -4 a 2 κ 4 cn 2 ( ρ, κ )sn 2 ( ρ, κ )dn -2 ( ρ, κ ) 16 a 2 bκ 2 cn( ρ, κ )sn( ρ, κ )</formula>
<text><location><page_11><loc_9><loc_34><loc_92><loc_38></location>for A in (1.11). Corresponding alternative expressions for Φ (3) ( τ, ρ ) are similar to (5.10) except the roles of the hyperbolic sine and hyperbolic cosine are interchanged. By a direct check one sees that the dilaton fields computed in (1.10) are indeed invariant along the corresponding Killing directions. That is, they satisfy</text>
<formula><location><page_11><loc_42><loc_30><loc_92><loc_33></location>∂ Φ ( i ) ∂τ Y 1 + ∂ Φ ( i ) ∂ρ Y 2 = 0 (5.11)</formula>
<text><location><page_11><loc_9><loc_27><loc_80><loc_29></location>for each of i = 1 , 2 , 3 as we indicated in the sentence following equation (5.4) about Y preserving Φ.</text>
<section_header_level_1><location><page_11><loc_37><loc_23><loc_64><loc_24></location>VI. SOME CLOSING REMARKS</section_header_level_1>
<text><location><page_11><loc_9><loc_14><loc_92><loc_21></location>For the metric g in (1.6), whose derivation was discussed in section 2, we have obtained as a main result explicit linearly independent solutions Φ ( j ) , j = 1 , 2 , 3, in (1.10) of the corresponding system of dilaton field equations in (1.9). We have also computed the associated Killing vector fields Y ( g, Φ ( j ) ) that leave both g and the Φ ( j ) invariant; see (5.7), (5.9), (5.10) and the remarks that follow (5.9) and (5.10). The dilaton fields simplify to the expressions given in (4.19) in the special case when the elliptic modulus κ is 1,and g simplifies to the expression given in (4.20).</text>
<text><location><page_11><loc_9><loc_10><loc_92><loc_14></location>For Q ( ρ, κ ) defined in (2.13) it was shown in the short argument following (2.11) that if Q ( ρ, κ ) = 0 for some ρ then necessarily v 2 / 4 a 2 κ 4 ≤ 1:</text>
<formula><location><page_11><loc_46><loc_8><loc_92><loc_10></location>| v | ≤ 2 | a | κ 2 (6.1)</formula>
<text><location><page_12><loc_9><loc_90><loc_92><loc_93></location>in contrast to the standing assumption (1.7). To better understand the meaning of this inequality note first by (4.1) that Q ( ρ, κ ) = 0 ⇒ g 11 = 0 so g exhibits a horizon singularity at</text>
<formula><location><page_12><loc_35><loc_83><loc_92><loc_89></location>a 2 κ 4 sn 2 ( ρ, κ )cn 2 ( ρ, κ ) / dn 2 ( ρ, κ ) = v 2 4 : sn( ρ, κ )cn( ρ, κ ) / dn( ρ, κ ) = ± v 2 aκ 2 , (6.2)</formula>
<text><location><page_12><loc_9><loc_75><loc_92><loc_83></location>again by (2.13). Keep in mind that v is a velocity parameter - of a dissipative soliton (also called a dissipaton ) as in (2.4) for example, especially for κ = 1 as we have remarked at the end of section 2. The inequality (6.1) is the statement therefore that for an arbitrary elliptic modulus κ , with 0 < κ ≤ 1, the velocity of a black hole dissipaton cannot exceed the limiting value | v max | def. = 2 | a | κ 2 . This statement was deduced in [6], [7], for example, in the special (but important) case of κ = 1.</text>
<text><location><page_12><loc_9><loc_68><loc_92><loc_75></location>In section 3, by a series of explicit transformations of variables, g moreover was transformed to a Jackiw-Teitelboim black hole metric g J -T of the simple form (1.4) - namely to g J -T given by (3.11), with accompanying data given by (3.12), (3.13). Here again assumption (1.7) was imposed. An advantage of the parameterization (3.11) is that, for example, simple formulas exist [10], [12] for thermodynamic quantities such as the Hawking temperature T H and black hole entropy S .</text>
<text><location><page_12><loc_9><loc_62><loc_92><loc_68></location>We point out, for the record, that the general solutions of all 2d dilaton gravity models are known. For example, see section 3 of the paper [13] of T.Klosch and T.Stroble. However (again) we have constructed very explicit elliptic solutions for the specific model of interest - solutions that we feel do not follow directly at all as a corollary of the results of [13].</text>
<text><location><page_12><loc_9><loc_58><loc_92><loc_62></location>In view of an interesting suggestion of the referee, whom we thank (and also for the references [14], [15]), we add some final remarks that provide a brief review of a connection of the J-T model to cold plasma physics. This connection is facilitated by way of a resonant nonlinear Schrodinger (RNLS) equation.</text>
<text><location><page_12><loc_9><loc_51><loc_92><loc_58></location>The authors in [14] consider a system of nonlinear equations that describe the dynamics of two-component cold collision-less plasma in the presence of an external magnetic field B . For uni-axial plasma propagation, this system is reduced to a system that describes the propagation of nonlinear magneto-acoustic waves in a cold plasma with a transverse magnetic field. By way of a shallow water approximation of the latter system, a reduction of it to a RNLS equation of the form</text>
<formula><location><page_12><loc_35><loc_46><loc_92><loc_50></location>i ∂ψ ∂t ' + ∂ 2 ψ ∂x ' 2 -1 2 | ψ | 2 ψ = (1 + β 2 ) 1 | ψ | ∂ 2 ψ ∂x ' 2 ψ (6.3)</formula>
<text><location><page_12><loc_9><loc_41><loc_92><loc_46></location>is achieved. Here x ' = βx , t ' = βt are rescaled space and time variables, and B has an expression in terms of a suitable power series expansion in the parameter β 2 . ψ has the form ψ = √ ρ e -iS where S ( x ' , t ' ) is a velocity potential and ρ is the mass density of the plasma. Also note the remarks in [15]. A key point of interest for us is that for</text>
<formula><location><page_12><loc_37><loc_35><loc_92><loc_40></location>r def. = √ ρ e S/β > 0 , def. = -√ ρ e S/β < 0 , B def. = 1 2 β 2 = 2 b 2 , b def. = 2 β, (6.4)</formula>
<text><location><page_12><loc_9><loc_30><loc_92><loc_34></location>in the variables ( x ' , τ def. = βt ' ), the reaction diffusion (RD) system (2.1) is satisfied; r, s are denoted by e (+) , e ( -) in [14]. On page 186 of [1] it is shown that, conversely, given solutions r > 0 , s < 0 of the RD system (2.1), one can naturally construct a RNLS solution. By (2.4), with b = 2 β by (6.4), we can take</text>
<formula><location><page_12><loc_25><loc_21><loc_92><loc_29></location>r ( x ' , τ ) = 2 αβ dn( a ( x ' -vτ ) , κ )exp ([ v 2 4 + a 2 (2 -κ 2 ) ] τ -vx ' 2 ) (6.5) s ( x ' , τ ) = -2 αβ dn( a ( x ' -vτ ) , κ )exp ( -[ v 2 4 + a 2 (2 -κ 2 ) ] τ + vx ' 2 ) .</formula>
<text><location><page_12><loc_9><loc_13><loc_92><loc_21></location>All of this means that we can apply the prescription (2.3) to construct a metric g plasma of constant Ricci scalar curvature R = 4 B def. = 8 b 2 def. = 2 β 2 , as we did in (2.7) where the notation t, ρ there is now taken to mean τ, a ( x ' -vτ ). Moreover our results show that g plasma can be transformed to a J-T black hole metric of the form (1.4). Thus we can account for a J-T black hole connection in cold plasma physics. Our results also provide for elliptic solutions of the corresponding dilaton field equations.</text>
<text><location><page_12><loc_10><loc_9><loc_78><loc_10></location>The authors declare that there is no conflict of interest regarding the publication of this paper.</text>
<section_header_level_1><location><page_13><loc_46><loc_92><loc_54><loc_93></location>References</section_header_level_1>
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</document> | [{"title": "Elliptic function solutions in Jackiw-Teitelboim dilaton gravity", "content": "Jennie D'Ambroise 1 and Floyd L. Williams 2 We present a new family of solutions for the Jackiw-Teitelboim model of two-dimensional gravity with a negative cosmological constant. Here, a metric of constant Ricci scalar curvature is constructed, and explicit linearly independent solutions of the corresponding dilaton field equations are determined. The metric is transformed to a black hole metric, and the dilaton solutions are expressed in terms of Jacobi elliptic functions. Using these solutions we compute, for example, Killing vectors for the metric. PACS numbers: 02.30.Jr, 02.40.Ky, 04.20.Jb, 04.70.Bw", "pages": [1]}, {"title": "I. INTRODUCTION", "content": "It is well-known that the Einstein gravitational field equations for a vacuum (with a zero matter tensor) are automatically solved by any metric g on a two-dimensional space-time M . A proof of this fact is given in section 2 of [1], for example. A non-trivial theory of gravity for such an M was worked out in 1984 by R. Jackiw and C. Teitelboim (J-T). This involves in addition to g a scalar field \u03a6 on M called a dilaton field; see [2],[3]. The pair ( g, \u03a6) is subject to the equations of motion derived from the action integral where R ( g ) is the constant Ricci scalar curvature of g , and where the (negative) cosmological constant is \u039b = -1 /l 2 . In local coordinates ( x 1 , x 2 ) on M , the Hessian in (1.1) is given by where \u0393 k ij are the Christoffel symbols (of the second kind) of g [1]. The J-T theory has, for example, the (Lorentzian) black hole solution with coordinates ( x 1 , x 2 ) = ( T, r ), where with M a black hole mass parameter. Here and throughout we note that our sign convention for scalar curvature is the negative of that used in [2],[3], and by others in the literature. /negationslash The purpose of this paper is the following. For real numbers a, b = 0 and for a soliton velocity parameter v we consider the following metric in the variables ( x 1 , x 2 ) = ( \u03c4, \u03c1 ): where sn( x, \u03ba ) , cn( x, \u03ba ) , dn( x, \u03ba ) are the standard Jacobi elliptic functions with modulus \u03ba ; 0 \u2264 \u03ba \u2264 1 [4]. We will generally assume that /negationslash \u2223 \u2223 As will be seen later, this metric is the diagonalization of a metric constructed from solutions r ( x, t ) , s ( x, t ) of the reaction diffusion system We will explicate the solutions r ( x, t ) , s ( x, t ) in terms of the elliptic function dn( x, \u03ba ). Remarkably, the metric in (1.6) has constant scalar curvature R ( g ) = 8 /b 2 so that the first equation in (1.1) holds. The main work of the paper then is to solve the corresponding system of partial differential equations (the dilaton field equations) in (1.1), which for g in (1.6) are Here the cosmological constant is \u039b = -4 /b 2 . Given the complicated nature of our g , the system (1.9) is necessarily quite difficult to solve directly. Our method is to construct a series of transformations of variables so that g in (1.6) is transformed to g in (1.4). Then we can use the simple solution \u03a6( T, r ) = mr in (1.5), and other known solutions, to work backwards through these transformations of variables to construct \u03a6( \u03c4, \u03c1 ) that satisfies (1.9). The various details involved, with further remarks that lead to (1.6), will be the business of sections 2, 3, and 4. In the end, we obtain the following main result: The metric in (1.6) solves the first J-T equation of motion (1.1). Namely R ( g ) = 8 /b 2 , as we have remarked. Also three linearly independent solutions of the field equations in (1.1), namely of the system of equations (1.9), are given by for which we assume is non-zero. Given (1.7) we shall see in section 4 that A = 0 only for a = \u00b1 (1 -\u221a 1 -\u03ba 2 ) v/ 2 \u03ba 2 and moreover that the second expression under the radical (i.e. v 2 / 4 - \u00b7 \u00b7 \u00b7 ) in (1.10) is positive. For \u03ba = 1, A = ( v 2 / 4 -a 2 ) 2 > 0, but we can have A < 0 for some \u03ba < 1. Also for \u03ba = 1 the solutions in (1.10) reduce to those given in (4.19), with (1.6) given by (4.20).", "pages": [1, 2]}, {"title": "II. REACTION DIFFUSION SYSTEMS AND DERIVATION OF THE METRIC IN (1.6)", "content": "Since the metric (1.6) is one of the main objects of interest we indicate in this section its derivation. For a constant B consider the system of partial differential equations in the variables ( x, t ). This system is a special case of the more general reaction diffusion system (RDS) Apply this to (2.7): that occurs in chemistry, physics, or biology, for example, where d r , d s are diffusion constants, and F, G are growth and interaction functions. The key point for us is that from solutions r ( x, t ) , s ( x, t ) of (2.1) one can construct a metric g of constant Ricci scalar curvature R ( g ) = 4 B by the following prescription [5],[6],[7]: One could also simply start with the definitions in (2.3), apart from the preceding references that employ Cartan's zweibein formalism [8], and use a Maple program (tensor), for example, to check directly that indeed R ( g ) = 4 B . Our interest is in the choice B = 2 /b 2 , where for real a, b, v , with a, b = 0 as in section 1, r ( x, t ) , s ( x, t ) given by /negationslash are solutions the system (2.1), which also could be checked directly by Maple. For B = 2 /b 2 , (2.1) is system (1.8) with solutions (2.4) promised in section 1, and g in (2.3) has the scalar curvature 4 B = 8 /b 2 discussed in section 1. From [4] various formulas like are available. Using the prescription (2.3) one computes that For \u03c1 def. = a ( x -vt ), so that d\u03c1 = a ( dx -vdt ), g can be expressed more conveniently as The goal now is to set up a change of variables ( t, \u03c1 ) -\u2192 ( \u03c4, \u03c1 ) so that g in (2.7) is transformed to (1.6) - where the cross term d\u03c4d\u03c1 does not appear, in comparison with the term dtd\u03c1 appearing in (2.7). For this purpose note first, in general, that for the change of variables \u03c4 = t + \u03c6 ( \u03c1 ) gives dt = d\u03c4 -\u03c6 ' ( \u03c1 ) d\u03c1 , dt 2 = d\u03c4 2 -2 \u03c6 ' ( \u03c1 ) d\u03c4d\u03c1 + \u03c6 ' ( \u03c1 ) 2 d\u03c1 2 , and The condition that the cross term d\u03c4d\u03c1 does not appear is therefore that \u03c6 ( \u03c1 ) satisifies Now by (2.5), dn 2 ( x, \u03ba ) = 1 -\u03ba 2 sn 2 ( x, \u03ba ) = sn 2 ( x, \u03ba )+cn 2 ( x, \u03ba ) -\u03ba 2 sn 2 ( x, \u03ba ) = cn 2 ( x, \u03ba )+(1 -\u03ba 2 )sn 2 ( x, \u03ba ) \u2265 cn 2 ( x, \u03ba ) and sn 2 ( x, \u03ba ) \u2264 sn 2 ( x, \u03ba ) + cn 2 ( x, \u03ba ) = 1 \u21d2 sn 2 ( x, \u03ba )cn 2 ( x, \u03ba ) / dn 2 ( x, \u03ba ) \u2264 1. If the term in parenthesis in (2.11) were zero, this would therefore force the inequality v 2 / 4 a 2 \u03ba 4 \u2264 1. That is, if \u2223 \u2223 v/ 2 a\u03ba 2 \u2223 \u2223 > 1, which is the assumption in (1.7), then v 2 / 4 a 2 \u03ba 4 > 1 and therefore the denominator term in parenthesis in (2.11) is non-zero, which means that \u03c6 ' ( \u03c1 ) is a continuous function and (2.11) therefore indeed has a solution \u03c6 ( \u03c1 ), with the assumption (1.7) imposed. Also, the coefficient of d\u03c1 2 in (2.9) is where for convenience we write sn,cn,dn for sn( \u03c1, \u03ba ) , cn( \u03c1, \u03ba ) , dn( \u03c1, \u03ba ) and Q for Then which is the coefficient of d\u03c1 2 in (2.9) by (2.12). Then by (2.10),(2.9) reads which is (1.6). That is, we have verified that the change of variables \u03c4 = t + \u03c6 ( \u03c1 ) with \u03c6 ( \u03c1 ) subject to condition (2.11) (which in fact renders \u03c6 ' ( \u03c1 ) a continuous function, again assuming (1.7)) transforms the reaction diffusion metric in (2.7) to the diagonal metric in (1.6). In the special case when the elliptic modulus \u03ba = 1 and (2.7), (1.6) simplify: which are the line elements (3.12), (3.14), respectively, in [6]; a here corresponds to the notation k there. Also the cosmological constant \u039b 0 in [6] corresponds to our 2\u039b = -8 /b 2 : b 2 = 8 / ( -\u039b 0 ). Similarly, r and s in (2.4) reduce to the dissipative soliton solutions q + and q -, respectively, in (2.32) of [6], apart from the factor b . One can also explicitly determine \u03c6 ( \u03c1 ) in (2.11).", "pages": [2, 3, 4]}, {"title": "III. TRANSFORMATION OF THE METRIC IN (1.6) TO A J-T BLACK HOLE METRIC", "content": "Now that the existence of the metric in (1.6) has been described in the context of a reaction diffusion system (namely (1.8)), the strategy of this section is to set up a series of changes of variables, as indicated in the introduction, that transforms it to the simpler J-T form (1.4). Other applications, of independent interest, can flow from this - apart from our main focus to solve the system (1.9). A general method to go from (1.6) to (1.4) has been developed by the first named author. Alternatively, one can generalize part of the argument in [6] that leads at least to a Schwarzschild form, as we do here, and then argue a bit more to obtain the J-T form - the final result being expressed by equations (3.11)-(3.13) below. Start with the change of variables r = | a | dn( \u03c1, \u03ba ) so that dr = -\u03ba 2 | a | sn( \u03c1, \u03ba ) \u00b7 cn( \u03c1, \u03ba ) d\u03c1 by (2.5) \u21d2 Also by (2.5) for Again by (1.7), v 2 / 4 a 2 \u03ba 4 > 1 \u21d2 v 2 / 4 > a 2 \u03ba 4 \u21d2 indeed r 2 0 > 0. By (3.1), (3.2) we see that we can write (1.6) as Next let x def. = (2 r 2 + r 2 0 ) /r 4 0 , as in (3.18) of [6] but where our r 2 0 in (3.3) generalizes their r 2 0 , and for convenience let in (3.4). Then g in (3.4) assumes the form which generalizes the Schwarzchild form (3.19) of [6] since for \u03ba = 1 we have that \u03b1 = \u03b2 = 0 in (3.5). For the change of variables t = A 0 \u03c4 , r -= A 0 x with A 0 def. = | b | r 2 0 / 2, the Schwarzschild g in (3.6) goes to which in turn goes to by way of the change of variables t = bT , r -= r 1 /b . We need one final observation: In general a metric of the form /negationslash say A 1 = 0 can be transformed to the J-T form (1.4), namely by way of the change of variables r = x + B 1 2 A 1 . Apply this to (3.8) with x playing the role of r 1 there: for Using definition (3.5) for \u03b1, \u03b2 and r 2 0 = v 2 4 -a 2 \u03ba 4 , which is definition (3.3), one computes that in (3.11).", "pages": [4, 5, 6]}, {"title": "IV. DERIVATION OF THE SOLUTIONS (1.10) OF THE FIELD EQUATIONS (1.9)", "content": "The main result is derived in this section. Namely, we indicate how the series of changes of variables in section 3 (according to remarks in the introduction) lead to the linearly independent solutions \u03a6 ( j ) ( \u03c4, \u03c1 ), j = 1 , 2 , 3 in (1.10) of the dilaton field equations in (1.9). There the metric elements g ij are given by (1.6): For Q ( \u03c1, \u03ba ) in (2.13) and the \u2207 i \u2207 j \u03a6 are given by (1.3) for ( x 1 , x 2 ) = ( \u03c4, \u03c1 ). The Christoffel symbols \u0393 k ij in (1.3) (which could be computed by Maple, for example) will not be needed for the derivation of (1.10), although they could be used to verify these solutions. Obviously any dilaton solution could be replaced by any non-zero multiple of itself. In the following then, we can disregard such multiples if we wish to. In addition to the dilaton solution \u03a6 (1) ( T, r ) def. = mr in (1.5) for the metric (1.4) in the variables ( T, r ), there are solutions We work backwards the changes of variables in section 3 for \u03a6 (1) ( T, r ) , \u03a6 (2) ( T, r ), for example, to see how one arrives at the first two solutions \u03a6 (1) ( \u03c4, \u03c1 ) , \u03a6 (2) ( \u03c4, \u03c1 ) in (1.10), in the variables ( \u03c4, \u03c1 ). Starting with the (3.11) version of (1.4), we have m 2 = A 1 = 4 /b 2 by (3.12), with M = -( C 1 -B 2 1 / 4 A 1 ) given by (3.13). Here m \u221a M = \u221a B 2 1 -4 A 1 C 1 / 2 (for m = 2 / | b | ) \u21d2 By the final change of variables r = r 1 + B 1 2 A 1 in section 3, we see that A 1 r 2 = A 1 r 2 1 + B 1 r 1 + B 2 1 / 4 A 1 \u21d2 The change of variables t = bT , r -= r 1 /b preceded the change r = r 1 + B 1 / 2 A 1 , so that since A 1 b 2 def. = 4. We had t = A 0 \u03c4 , r -= A 0 x for A 0 def. = | b | r 2 0 / 2, which gives for the Schwarzchild version of our metric in (3.6). Next let x = (2 r 2 + r 2 0 ) /r 4 0 to get where we have disregarded the multiple | b | /b = \u00b1 1 in (4.6) and have used that sinh( | b | x/b ) = ( | b | /b ) sinh( x ). Finally, the first change of variables r = | a | dn( \u03c1, \u03ba ) in section 3 gives by definition (3.12). If we disregard the multiple | b | /r 2 0 in (4.8) and use that r 2 0 + \u03b1 def. = v 2 / 4+ a 2 ( \u03ba 2 -2) by definitions (3.3), (3.5) we obtain from (4.8) the first solution in (1.10). More work is required of course to obtain the second solution there. First we note that by (3.12), (3.13) which is the \u221a A\u03c4 in (1.10). Also for r = | a | dn, dn = dn( \u03c1, \u03ba ), the quantity under the other radical in (4.7) is where by definition (3.12) again by definitions (3.3), (3.5). That is, since \u03b2 = a 4 (1 -\u03ba 2 ) by definition (3.5) the quantity in (4.11) (which is under the radical in (4.7) for r = | a | dn) is We let B ( \u03c1 ) denote the latter bracket here. By (4.7), (4.10), (4.13) we see that (for now) \u03a6 (2) ( \u03c4, \u03c1 ) = \u221a B ( \u03c1 ) sinh ( \u221a A\u03c4 ) , if we disregard the multiple \u221a 4 a 2 b 2 /r 4 0 = 2 | a || b | /r 2 0 . We find an alternate expression for B ( \u03c1 ), which is simpler and which shows that B ( \u03c1 ) > 0, given (1.7). Again we write sn, cn, dn for sn( \u03c1, \u03ba ) , cn( \u03c1, \u03ba ) , dn( \u03c1, \u03ba ), and we make use of (2.5). where we noted in section 2 that sn 2 cn 2 / dn 2 \u2264 1. Hence /negationslash The quartic equation A = 0 has roots a = \u00b1 (1 + \u221a 1 -\u03ba 2 ) v/ 2 \u03ba 2 , \u00b1 (1 -\u221a 1 -\u03ba 2 ) v/ 2 \u03ba 2 with a 2 = (2 -\u03ba 2 + 2 \u221a 1 -\u03ba 2 ) v 2 / 4 \u03ba 4 , (2 -\u03ba 2 -2 \u221a 1 -\u03ba 2 ) v 2 / 4 \u03ba 4 respectively. (1.7) requires that a 2 < v 2 / 4 \u03ba 4 , which forces the inequalities by (1.7), again as in (3.3), and we see that B ( \u03c1 ) > 0 since dn( \u03c1, \u03ba ) = 0 for \u03c1 a real number. Moreover we have established the desired expression for \u03a6 (2) ( \u03c4, \u03c1 ) in (1.10). Clearly one can replace the hyperbolic sine in the preceding discussion by the hyperbolic cosine in (4.2) to obtain the third solution \u03a6 (3) ( \u03c4, \u03c1 ) in (1.10). To finish other claims made in section 1 we check that in (1.11) A = 0 only for a = \u00b1 (1 -\u221a 1 -\u03ba 2 ) v/ 2 \u03ba 2 . We continue to assume (1.7) of course. of which the first one reads 1 -\u03ba 2 +2 \u221a 1 -\u03ba 2 < 0, with the left hand side here being \u2265 0 - a contradiction. That is, we cannot have a = \u00b1 (1 + \u221a 1 -\u03ba 2 ) v/ 2 \u03ba 2 which means that a = \u00b1 (1 -\u221a 1 -\u03ba 2 ) v/ 2 \u03ba 2 . Also we check that the solutions are linearly independent: Assume for constants c 1 , c 2 , c 3 that Differentiate this equation with respect to \u03c4 and evaluate the result at ( \u03c4, 0): /negationslash since dn(0 , \u03ba ) = 1 , sn(0 , \u03ba ) = 0. The choice \u03c4 = 0 then gives c 2 = 0, since A,v = 0, and differentiation of the equation c 3 \u221a A sinh( \u221a A\u03c4 ) | v | / 2 = 0 and at \u03c4 = 0 gives c 3 = 0. Using again that dn(0 , \u03ba ) = 1 we see by (1.10) that \u03a6 (1) ( \u03c4, 0) def. = v 2 / 4 + a 2 \u03ba 2 > 0 and hence also c 1 = 0. Note that if v = a = 2 and \u03ba = 1 / 2, for example, then even though v/ 2 a\u03ba 2 = 2 > 1 (so that (1.7) is satisfied), we have that A = -12 < 0. Again in the special case when the elliptic modulus \u03ba = 1, we have in (1.11) that A = ( v 2 / 4 -a 2 ) 2 > 0 and \u221a A = v 2 / 4 -a 2 > 0 (by (1.7) or (3.3)), and B ( \u03c1 ) = sech 2 \u03c1 [ v 2 / 4 -a 2 tanh 2 \u03c1 ] = sech 2 \u03c1 [ v 2 -4a 2 tanh 2 \u03c1 ] / 4, by (2.16), (4.14) \u21d2 \u221a B ( \u03c1 ) = 1 2 sech \u03c1 \u221a v 2 -4a 2 tanh 2 \u03c1. Here (directly) tanh 2 \u03c1 \u2264 1 \u21d2 v 2 -4 a 2 tanh 2 \u03c1 > 0, again as v 2 > 4 a 2 . Thus by (1.10) and (2.17) The solutions in (4.19) are also new.", "pages": [6, 7, 8, 9, 10]}, {"title": "V. KILLING VECTOR FIELDS FOR THE SOLUTIONS (1.6), (1.10)", "content": "Recall that a smooth vector field Y on an n -dimensional Riemannian manifold ( M,g ) is called a Killing vector field (or an infinitesimal motion of M ) if for arbitrary smooth vector fields X,Z on M If Y = n \u2211 i =1 Y i \u2202 \u2202x i is an expression of Y in terms of local coordinates ( x 1 , . . . , x n ) on M , then (5.1) is equivalent to the system of equations for 1 \u2264 j, k \u2264 n [8], [9]. In the special (diagonal) case with g ij = 0 for i = j , and with n = 2, the Killing equations (5.2) simplify to the following three equations: /negationslash As was shown in [10], every solution ( g, \u03a6) of the field equations in (1.1) gives rise to a corresponding Killing vector field Y = Y ( g, \u03a6) by way of the local prescription \u221a with /epsilon1 ij = a permutation symbol. Y preserves both g and \u03a6. For g in (1.4) and for the fields \u03a6 in (1.5) and (4.2), the corresponding Killing vector fields are given in equations (16), (17), (18) of [11], for example. Our interest of course is in the case of the three solutions ( g, \u03a6 ( j ) ) in (1.10) with g given by (1.6). By (4.1), \u221a | det g | = a 2 b 2 \u03ba 2 | sncn | dn. Since Y i could be replaced by a scalar multiple of itself (for example, -Y i ), we shall disregard the absolute value of sncn here, and given (1.9) we shall take l = b 2 (instead of | b | 2 ). For /epsilon1 11 = /epsilon1 22 = 0 , /epsilon1 12 = -1 = -/epsilon1 21 (5.4) then assumes the generic form where we take ( x 1 , x 2 ) = ( \u03c4, \u03c1 ) in (5.3). For the first solution in (1.10), the computation of the corresponding Killing vector field Y is trivial: By (2.5) and (5.5) Y 1 = 2 /b , and of course Y 2 = 0. Since \u2202g 11 /\u2202\u03c4 = \u2202g 22 /\u2202\u03c4 = 0 by (4.1), the Killing equations in (5.3) are satisfied and we see that for ( g, \u03a6 (1) ). Computations for the other two solutions \u03a6 (2) , \u03a6 (3) in (1.10) are more involved. The result is the following, where again in definition (1.11). For \u03a6 (2) ( \u03c4, \u03c1 ) For \u03a6 (3) ( \u03c4, \u03c1 ) one has quite similar formulas for Y 1 , Y 2 except (as expected) the roles of the hyperbolic sine and hyperbolic cosine in (5.9) are interchanged: the factor sinh( \u221a A\u03c4 ) for Y 1 in (5.9) is replaced by cosh( \u221a A\u03c4 ), and, similarly, cosh( \u221a A\u03c4 ) for Y 2 in (5.9) is replaced by sinh( \u221a A\u03c4 ). One can also find the following alternative expressions for the Killing vector field components for \u03a6 (2) ( \u03c4, \u03c1 ): for A in (1.11). Corresponding alternative expressions for \u03a6 (3) ( \u03c4, \u03c1 ) are similar to (5.10) except the roles of the hyperbolic sine and hyperbolic cosine are interchanged. By a direct check one sees that the dilaton fields computed in (1.10) are indeed invariant along the corresponding Killing directions. That is, they satisfy for each of i = 1 , 2 , 3 as we indicated in the sentence following equation (5.4) about Y preserving \u03a6.", "pages": [10, 11]}, {"title": "VI. SOME CLOSING REMARKS", "content": "For the metric g in (1.6), whose derivation was discussed in section 2, we have obtained as a main result explicit linearly independent solutions \u03a6 ( j ) , j = 1 , 2 , 3, in (1.10) of the corresponding system of dilaton field equations in (1.9). We have also computed the associated Killing vector fields Y ( g, \u03a6 ( j ) ) that leave both g and the \u03a6 ( j ) invariant; see (5.7), (5.9), (5.10) and the remarks that follow (5.9) and (5.10). The dilaton fields simplify to the expressions given in (4.19) in the special case when the elliptic modulus \u03ba is 1,and g simplifies to the expression given in (4.20). For Q ( \u03c1, \u03ba ) defined in (2.13) it was shown in the short argument following (2.11) that if Q ( \u03c1, \u03ba ) = 0 for some \u03c1 then necessarily v 2 / 4 a 2 \u03ba 4 \u2264 1: in contrast to the standing assumption (1.7). To better understand the meaning of this inequality note first by (4.1) that Q ( \u03c1, \u03ba ) = 0 \u21d2 g 11 = 0 so g exhibits a horizon singularity at again by (2.13). Keep in mind that v is a velocity parameter - of a dissipative soliton (also called a dissipaton ) as in (2.4) for example, especially for \u03ba = 1 as we have remarked at the end of section 2. The inequality (6.1) is the statement therefore that for an arbitrary elliptic modulus \u03ba , with 0 < \u03ba \u2264 1, the velocity of a black hole dissipaton cannot exceed the limiting value | v max | def. = 2 | a | \u03ba 2 . This statement was deduced in [6], [7], for example, in the special (but important) case of \u03ba = 1. In section 3, by a series of explicit transformations of variables, g moreover was transformed to a Jackiw-Teitelboim black hole metric g J -T of the simple form (1.4) - namely to g J -T given by (3.11), with accompanying data given by (3.12), (3.13). Here again assumption (1.7) was imposed. An advantage of the parameterization (3.11) is that, for example, simple formulas exist [10], [12] for thermodynamic quantities such as the Hawking temperature T H and black hole entropy S . We point out, for the record, that the general solutions of all 2d dilaton gravity models are known. For example, see section 3 of the paper [13] of T.Klosch and T.Stroble. However (again) we have constructed very explicit elliptic solutions for the specific model of interest - solutions that we feel do not follow directly at all as a corollary of the results of [13]. In view of an interesting suggestion of the referee, whom we thank (and also for the references [14], [15]), we add some final remarks that provide a brief review of a connection of the J-T model to cold plasma physics. This connection is facilitated by way of a resonant nonlinear Schrodinger (RNLS) equation. The authors in [14] consider a system of nonlinear equations that describe the dynamics of two-component cold collision-less plasma in the presence of an external magnetic field B . For uni-axial plasma propagation, this system is reduced to a system that describes the propagation of nonlinear magneto-acoustic waves in a cold plasma with a transverse magnetic field. By way of a shallow water approximation of the latter system, a reduction of it to a RNLS equation of the form is achieved. Here x ' = \u03b2x , t ' = \u03b2t are rescaled space and time variables, and B has an expression in terms of a suitable power series expansion in the parameter \u03b2 2 . \u03c8 has the form \u03c8 = \u221a \u03c1 e -iS where S ( x ' , t ' ) is a velocity potential and \u03c1 is the mass density of the plasma. Also note the remarks in [15]. A key point of interest for us is that for in the variables ( x ' , \u03c4 def. = \u03b2t ' ), the reaction diffusion (RD) system (2.1) is satisfied; r, s are denoted by e (+) , e ( -) in [14]. On page 186 of [1] it is shown that, conversely, given solutions r > 0 , s < 0 of the RD system (2.1), one can naturally construct a RNLS solution. By (2.4), with b = 2 \u03b2 by (6.4), we can take All of this means that we can apply the prescription (2.3) to construct a metric g plasma of constant Ricci scalar curvature R = 4 B def. = 8 b 2 def. = 2 \u03b2 2 , as we did in (2.7) where the notation t, \u03c1 there is now taken to mean \u03c4, a ( x ' -v\u03c4 ). Moreover our results show that g plasma can be transformed to a J-T black hole metric of the form (1.4). Thus we can account for a J-T black hole connection in cold plasma physics. Our results also provide for elliptic solutions of the corresponding dilaton field equations. The authors declare that there is no conflict of interest regarding the publication of this paper.", "pages": [11, 12]}] |
2014IJMPD..2350084B | https://arxiv.org/pdf/1405.0764.pdf | <document>
<section_header_level_1><location><page_1><loc_19><loc_92><loc_81><loc_93></location>Stellar oscillations induced by the passage of a fast stellar object</section_header_level_1>
<text><location><page_1><loc_34><loc_89><loc_67><loc_90></location>C.A. Bertulani, M. Naizer, and W. Newton</text>
<text><location><page_1><loc_12><loc_86><loc_89><loc_88></location>Department of Physics and Astronomy, Texas A & M University - Commerce, Commerce, TX 75429, USA (Dated: January 16, 2021)</text>
<text><location><page_1><loc_18><loc_76><loc_83><loc_85></location>We investigate induced oscillations by the gravitational field of a fast stellar object, such as a neutron star or a black-hole in a near miss collision with another star. Non-adiabatic collision conditions may lead to large amplitude oscillations in the star. We show that for a solar-type star a resonant condition can be achieved by a fast moving stellar object with velocity in the range of 100 km/s to 1000 km/s, passing at a distance of a few multiples of the star radius. Although such collisions are rare, they are more frequent than head-on collisions, and their effects could be observed through a visible change of the star luminosity occurring within a few hours.</text>
<text><location><page_1><loc_18><loc_73><loc_40><loc_74></location>PACS numbers: 97,98.10.+z,98.35.Df</text>
<text><location><page_1><loc_57><loc_71><loc_57><loc_72></location>!</text>
<figure>
<location><page_1><loc_52><loc_55><loc_92><loc_70></location>
<caption>FIG. 1. Tidal oscillations induced in a star by the passage of a fast stellar object.</caption>
</figure>
<text><location><page_1><loc_52><loc_42><loc_92><loc_46></location>star. The tidal force is best described by expanding the gravitational field of the FSO into multipoles, yielding at a position x inside the star [14]</text>
<formula><location><page_1><loc_53><loc_36><loc_92><loc_41></location>V ( x , t ) = -GM ∑ lm 4 π 2 l +1 Y lm ( ˆ R ( t )) x l R l +1 ( t ) Y ∗ lm (ˆ x ) . (1)</formula>
<text><location><page_1><loc_52><loc_18><loc_92><loc_35></location>In the center of mass of the star, the force on a mass element at x is obtained from the derivative of Eq. (1) with respect to x . The distance R ( t ) is a function of time, and t = 0 is taken when the two stars are at the periapsis, or distance of closest approach. In the frame of reference of the star, the tidal force acts in opposite sides from its center, trying to elongate it and leading to a time-dependent ellipsoidal shaped oscillation. To lowest order, the passage of a FSO will induce quadrupole shaped vibrations, as seen in Figure 1. Higher multipole vibrations such as octupole oscillations are also possible, but are orders of magnitude smaller and have been neglected here.</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_17></location>The time dependence of R ( t ) and θ ( t ) is described by a hyperbolic Kepler trajectory, parametrized by an orbital eccentricity glyph[epsilon1] > 1, where θ ( t ) is the angular position of the FSO measured from the center of mass of the system and with respect to the line joining it to the star so that, at t = 0, θ = 0, and R = a , the distance at the periapsis.</text>
<text><location><page_1><loc_9><loc_50><loc_49><loc_70></location>Stellar oscillations, commonly known as pulsations, are understood in terms of modulation geared by the interaction of radiation with matter on its way from the center of the star [1]. Little is known about how other kinds of oscillations can be generated in stellar collisions, although an effort has been made in studying tidal oscillations due to the gravitational interaction with a companion in a binary system (see, e.g., [2-4]). Stellar collisions are often investigated in the context of gravitational waves which could, in principle, be detected by ground and spacebased laser interferometers [5]. Gamma-ray bursts arising from tidal disruption of neutron stars (NS) in NS-NS or NS-Black Hole (BH) collisions have also attracted interest [6-10].</text>
<text><location><page_1><loc_9><loc_22><loc_49><loc_49></location>The kinetic energy of a star with m glyph[circledot] and velocity v = 1000 km/s is about 10 49 ergs, which is enough energy to power the luminosity of the Sun for one billion years. Acentral collision between such objects would be nothing less than spectacular. But if only a small fraction of this energy is transformed into the internal energy of a star during a relatively short time, the consequences would also be dramatic. This can be achieved in near miss collisions, with impact parameters larger than the sum of the radii of the collisional partners. There is plenty of available space for near misses, but very little room for central collisions and far collisions are much more frequent. It is therefore relevant to find out what are their consequences. Such processes have been previously studied for the specific purpose of assessing tidal oscillations in NS [2, 11]. Here we show that a resonant condition arises for solar-like stars within the range of possible velocities, leading to effects within our observational reach.</text>
<text><location><page_1><loc_9><loc_8><loc_49><loc_22></location>We consider the internal response of a star due to the passage of a fast stellar object (FSO), e.g., a NS or a BH, at large impact parameters. This response can be modeled by considering the tidal force on a mass element dm s of the star, roughly given by [12, 13] dF ∼ 2 GMdm s x/R 3 , where M is the mass of the FSO, G the gravitational constant, R is the distance between the center of mass of the FSO and the star, and x is the distance of the mass element dm s from the center of the</text>
<figure>
<location><page_2><loc_11><loc_75><loc_46><loc_93></location>
<caption>FIG. 2. The amplitudes | ˜ x i ( ω ) | 2 multiplied by 10 -22 (m 2 s 2 units) as a function of the ratio of the frequency ω and the natural oscillation frequency ω 0 , for longitudinal (dashed line) and transverse (solid line) oscillations.</caption>
</figure>
<text><location><page_2><loc_9><loc_55><loc_49><loc_64></location>For a collision with impact parameter b one has a = 2 b/ ( α + √ α 2 +4) where α = GMm s / ( Eb ), E = µv 2 / 2 is the collision energy and µ = Mm s / ( m s + M ) is the reduced mass. The relation between the angular position and time can be obtained solving coupled equations for R and t along the trajectory (see, e.g., Ref [15]).</text>
<text><location><page_2><loc_9><loc_36><loc_49><loc_55></location>Non-radial stellar oscillation modes can be described with hydrodynamical models to high accuracy (see, e.g., [16]). We have no reason to use such models, as the physical situation we consider here has never been observed before. We adopt a simple model including only (a) a single inertia parameter, (b) a linear restoring force, and (c) a damping parameter. The simplest model of this kind was developed by Lord Kelvin, described, e.g., in Refs. [2, 17-20]. For quadrupole oscillations in a spherically homogeneous self-gravitating star with radius r s , mass m s , and average density ρ 0 = m s / (4 πr 3 s / 3), the assumption of an incompressible fluid yields a natural oscillation frequency</text>
<formula><location><page_2><loc_18><loc_32><loc_49><loc_35></location>ω 2 0 = K 2 M 2 = 16 πGρ 0 15 = 4 Gm s 5 r 3 s , (2)</formula>
<text><location><page_2><loc_9><loc_24><loc_49><loc_31></location>where M 2 and K 2 are the respective quadrupole inertia and stiffness parameters. For small amplitude quadrupole oscillations the inertia parameter has been deduced in Ref. [21] from which one also obtains the stiffness parameters, namely,</text>
<formula><location><page_2><loc_15><loc_20><loc_49><loc_23></location>M 2 = 3 m s r 2 s 10 , and K 2 = 6 Gm 2 s 25 r s . (3)</formula>
<text><location><page_2><loc_9><loc_9><loc_49><loc_19></location>The stiffness arises from increase of gravitational energy due to the quadrupole deformation from a spherical star shape. Stellar oscillation damping is difficult to model as it can arise from 'gas', 'radiation', and 'turbulence' contributions, each of them varying wildly over temperature, density, and other properties of the stellar interior. For a gas the viscosity varies as γ g ∼ T 5 / 2 whereas</text>
<text><location><page_2><loc_52><loc_81><loc_92><loc_93></location>for radiation γ r ∼ T 4 . In the presence of turbulence, γ t = R e γ r / 3, where R e is the Reynolds number and the viscosity is several orders of magnitude larger than the radiative (or Jeans) viscosity [22]. In the absence of turbulence, radiation damping dominates over gas viscosity. To avoid dealing with specific stellar conditions, we assume a friction coefficient of the form γ = A γ M 2 ω 0 , with A γ taken as a free parameter.</text>
<text><location><page_2><loc_52><loc_57><loc_92><loc_81></location>The velocity distribution of nearby stars ( glyph[lessorsimilar] 100 pc), obtained with the Hypparcos satellite, shows a nonnegligible number of stars moving at speeds in excess of 100 km/s [23]. Hypervelocity stars, with v glyph[greaterorsimilar] 1000 km/s are rare, and able to escape the galaxy, but have already been observed [24]. To maximize the effect we are looking after we consider an FSO moving at a high speed, v = 1000 km/s, relative to the star. For a collision with an impact parameter b , the 'collision time', i.e. the time during which the gravitational force is most effective, is t coll ∼ b/v . For a collision with b = 5 r glyph[circledot] and v = 1000 km/s, one gets t coll ∼ 1 h. The period of oscillations associated with Eq. (2) for a solar-type star is t osc = 2 π/ω 0 ∼ 3 h. Hence, we expect a resonating response of oscillations in this system for impact parameters in the range of a few times r s .</text>
<text><location><page_2><loc_52><loc_39><loc_92><loc_56></location>The stellar oscillations can be disentangled into a mixture of transverse and longitudinal oscillations, as displayed in Figure 1. For collisions with impact parameters equal to 5 r s and larger and for velocities v ∼ 1000 km/s, the orbital eccentricity is large for solar-type stars, and the hyperbolic orbits become nearly straight lines. Therefore, we can safely consider transverse ( t ) and longitudinal ( l ) oscillations as being those transverse and along the asymptotic velocity, respectively. The equations of motion for small forced harmonic oscillations can be derived from Eq. (1) in terms of the inertia and stiffness parameters of Eqs. (2) and (3), yielding</text>
<formula><location><page_2><loc_57><loc_35><loc_92><loc_37></location>f i ( t ) = x i ( t ) + β ˙ x i ( t ) + ω 2 0 x i ( t ) , i = t, l, (4)</formula>
<text><location><page_2><loc_52><loc_20><loc_92><loc_33></location>along the two directions, where f i ( t ) is the driving tidal force per unit mass, and β = A γ ω 0 . For a straight line trajectory ( R 2 = b 2 + v 2 t 2 ) with no coupling among the orthogonal oscillations, this problem is solvable in analytical form. For hyperbolic trajectories with large eccentricities, our simulations show that accurate results can be obtained replacing the impact parameter b by b ' = a in the analytical solutions below, with a equal to the distance of closest approach at the periapsis.</text>
<text><location><page_2><loc_52><loc_14><loc_92><loc_20></location>The solution of Eq. (4) is expressed in terms of the Fourier transform x i ( t ) = (2 π ) -1 / 2 ∫ ˜ x i ( ω ) exp( iωt ) dω . For a straight line trajectory with effective impact parameter a , the amplitudes ˜ x i ( ω ) are given by</text>
<formula><location><page_2><loc_54><loc_8><loc_92><loc_12></location>˜ x t ( ω ) = ( 8 π ) 1 / 2 GM av ξK 1 ( ξ ) [( ω 2 0 -ω 2 ) 2 +4 β 2 ω 2 ] 1 / 2 , (5)</formula>
<figure>
<location><page_3><loc_11><loc_75><loc_46><loc_93></location>
<caption>FIG. 3. Oscillation amplitudes in units of the stellar radius as a function of time. The same parameters were used as in Figure 2.</caption>
</figure>
<figure>
<location><page_3><loc_10><loc_46><loc_49><loc_65></location>
<caption>FIG. 4. Energy, in ergs, transferred to oscillations in a solartype star along ( l - longitudinal) and perpendicular ( t - transverse) the direction of incidence of a fast stellar object as a function of the impact parameter and in units of the star radius. The thin lines represent calculations obtained with parametrized hyperbolic trajectories. The inset shows the ratio between the two transferred energies as a function of the same impact parameter measure.</caption>
</figure>
<text><location><page_3><loc_9><loc_28><loc_49><loc_29></location>where K 1 is the first order modified Bessel function, and</text>
<formula><location><page_3><loc_11><loc_23><loc_49><loc_27></location>˜ x l ( ω ) = i ( 8 π ) 1 / 2 GM av ξK 0 ( ξ ) [( ω 2 0 -ω 2 ) 2 +4 β 2 ω 2 ] 1 / 2 , (6)</formula>
<text><location><page_3><loc_9><loc_10><loc_49><loc_22></location>with K 0 the corresponding zeroth order modified Bessel function. The 'adiabacity' parameter ξ = ωa/v measures the degree to which the star responds adiabatically to the driving tidal force. The function ξK 1 ( ξ ) is nearly constant ( K 1 ∼ 1 /ξ ) for ξ < 1, and decays exponentially K 1 ∼ exp( -ξ ) for ξ > 1. Hence, oscillation modes with frequencies up to ω ∼ v/a will be preferred and those with larger frequencies will be suppressed exponentially.</text>
<text><location><page_3><loc_10><loc_9><loc_49><loc_10></location>In Figure 2 we plot the amplitudes | ˜ x i ( ω ) | 2 (multiplied</text>
<text><location><page_3><loc_52><loc_65><loc_92><loc_93></location>by 10 -22 m 2 s 2 ) as a function of frequency, for longitudinal (dashed line) and transverse (solid line) oscillations. We used M = 2 m glyph[circledot] , b = 5 r glyph[circledot] , v = 1000 km/s, m s = m glyph[circledot] , r s = r glyph[circledot] , and A γ = 0 . 1. Radii and masses are taken in units of the solar mass, m glyph[circledot] , and radius, r glyph[circledot] , respectively. We observe a remarkable resonant condition for these choice of parameters. As expected, the resonance peak decreases as the stellar viscosity increases (increasing A γ ). The resonance peak also decreases with the stellar radius because the star gets stiffer as the radius decreases, if its mass is kept constant. In this case the natural oscillation frequency ω 0 becomes large and the resonance matching condition b/v ∼ 1 /ω 0 does not take place, except for very high FSO velocities, beyond reasonable expectations from present observations. Keeping the same parameters above but varying r s , we conclude that for white dwarfs (WD) ( r s ∼ r glyph[circledot] / 10 2 ) and neutron stars (NS) ( r s ∼ r glyph[circledot] / 10 5 ) the natural oscillation frequency is too high to match the resonant condition.</text>
<text><location><page_3><loc_52><loc_51><loc_92><loc_64></location>Notice that the collision mechanism discussed here is different than the stellar tidal disruption or breakup in a head-on collision of either a black-hole or a neutron star with another neutron star [6], or those induced in mergers in binary systems [26, 29]. A distant collision with a FSO (unless its mass is very large) is unable to yield a tidal disruption of either a WD or a NS, unless maybe for very small impact parameters (see below and also Ref. [11]).</text>
<text><location><page_3><loc_52><loc_20><loc_92><loc_50></location>In Figure 3 we show the time-dependent oscillation displacements from equilibrium in units of the stellar radius with the same parameters used in Figure 2. At the periapsis the oscillation amplitudes can reach 10% of the star radius. This is a large amplitude oscillation, unprecedented by any known observation. Evidently, for large amplitudes one expects a non-linear behavior of the oscillations, requiring a more sophisticated model than adopted here. The stellar oscillations start well before the FSO reaches the periapsis ( t = 0) and are largest at t = 0. The results displayed in Figure 3 are close to resonance. An even larger effect would be obtained for a grazing impact parameter, when the stars nearly touch each other at the periapsis. As expected, induced longitudinal oscillations are smaller than transverse ones, but not by much. The difference between oscillations along the two directions increases for conditions off the resonance region. By increasing the star radius by a factor of 10, resonance conditions can be achieved even for a distant collision, b = 100 r s and larger.</text>
<text><location><page_3><loc_52><loc_9><loc_92><loc_19></location>The energy transferred to the stellar oscillations can be obtained from ∆ E = ∑ j = l,t ∫ ∞ -∞ F j ( t ) ˙ x j ( t ) or ∆ E = -2 R e ∑ j ∫ ∞ 0 iω ˜ x j ( ω ) ˜ F ∗ j ( ω ) dω. The momentum transferred to the recoil, or center of mass motion, of the star is approximately given by ∆ p cm = 2 GMm s /bv and the recoil energy by ∆ E cm = (∆ p ) 2 cm / 2 m s . For a collision with M = 2 m glyph[circledot] , b = 5 r glyph[circledot] , v = 1000 km/s, m s = m glyph[circledot] ,</text>
<text><location><page_4><loc_9><loc_66><loc_49><loc_93></location>r s = r glyph[circledot] , we get ∆ E cm = 5 . 5 × 10 46 ergs, e.g., 0.3% of the FSO bombarding energy is transferred to recoil. A much smaller energy is transferred to tidal oscillations. Using A γ = 0 . 1, one obtains 0 . 49 × 10 41 ergs and 1 . 6 × 10 41 ergs transferred to longitudinal and transverse stellar oscillations, respectively. This is larger than energies emitted in X-ray bursts from accretion in binary systems. However, this energy is transferred to the star (and possibly released in form of radiation) in a much larger time scale: a few hours instead of seconds as in X-ray bursts. Only a fraction 2 . 8 × 10 -6 of the recoil energy goes into internal excitation of the star. But assuming this star radiates all this energy in form of light with the sun's luminosity (3 . 9 × 10 26 W), it would be enough for 1.3 years of steady solar luminosity. An appreciable amount of this energy may be emitted in long wavelength radiation of long duration, i.e., within few hours. The characteristics of this radiation depend on many intrinsic stellar properties.</text>
<text><location><page_4><loc_9><loc_48><loc_49><loc_66></location>Figure 4 shows the energy in ergs transferred to longitudinal (dashed line) and transverse (solid line) oscillations in a solar mass star as a function of the impact parameter in units of the star radius. We use the same parameters for M , r s , m s and A γ as in Figure 2. The thin lines show the results obtained with exact hyperbolic trajectories. Only for small impact parameters there is a visible deviation from the results using straight-line trajectories with recoil correction. The inset shows that the ratio between the two energies increases in the same impact parameter range. For large b the longitudinal contribution becomes as relevant as the transverse one.</text>
<table>
<location><page_4><loc_10><loc_40><loc_48><loc_47></location>
<caption>TABLE I. The longitudinal and transverse energy transferred to a neutron star (NS), white dwarf (WD) and a solar-type (ST) star, all with masses m s = 1 . 4 m glyph[circledot] , due to a collision with a fast stellar object with mass M = 2 m glyph[circledot] passing by an impact parameter b = 5 r glyph[circledot] . The first column lists the assumed radius for the star. The last column gives the maximum tidal displacement in units of the assumed stellar radius.</caption>
</table>
<text><location><page_4><loc_9><loc_9><loc_49><loc_26></location>The resonant conditions for induced oscillations by a FSO are ideal for solar-type stars. But it is worthwhile to investigate what happens in the case of a neutron star (NS) or a white dwarf (WD). In Table I we show the longitudinal and transverse energy transferred to a NS, WD and a solar-type star, all with masses m s = 1 . 4 m glyph[circledot] , due to a collision with a fast stellar object with mass M = 2 m glyph[circledot] passing by an impact parameter b = 5 r glyph[circledot] . The first column lists the assumed radius for the star. The last column gives the maximum tidal displacement in units of the assumed stellar radius. One observes a dramatic change in the energy transfer due to the smaller</text>
<text><location><page_4><loc_52><loc_87><loc_92><loc_93></location>star size in contrast to a solar-type star. For neutron stars the energy transfer is negligible. The larger stiffness of a compact star corresponds to a large natural frequency, thus quenching the aforementioned resonant condition.</text>
<text><location><page_4><loc_52><loc_65><loc_92><loc_87></location>The results in Table I are for b = 5 r glyph[circledot] . But compact stars also allow closer collisions if the FSO is a WD, a NS, or a BH. Table II shows the same as in Table I but for closer encounters of the FSO with a WD and a NS. The collision impact parameter b is measured in units of 5 times the WD (rows 2 and 3) radius, or 5 times the NS radius (row 4). In these cases, the trajectories are significantly modified by the gravitational attraction and we solve Eq. (1) parametrized by a hyperbolic trajectory. A close collision of a FSO and a NS might require the solution of general relativity equations for the trajectory, which we do not consider. Our results show that the energy emitted over a few hours is well below those of known cosmic cataclysmic events, such as gamma-ray bursts [28], but not worthless more investigation.</text>
<table>
<location><page_4><loc_52><loc_56><loc_92><loc_63></location>
<caption>TABLE II. Same as in Table I but for closer encounter collisions of the FSO with an WD and a NS. The collision impact parameter b is measured in units of 5 times the WD (rows 2 and 3) radius, or 5 times the NS radius (row 4).</caption>
</table>
<text><location><page_4><loc_52><loc_29><loc_92><loc_46></location>In the case of NS-NS collisions, our calculated energy transfer is ∼ 10 46 ergs at b = 5 r NS . Notice that we do not explore the equation of state of nuclear matter, relying solely on the physics of an incompressible fluid. According to Ref. [29], this energy would induce high frequency seismic oscillations in the NS which can couple to the magnetic field and spark a particle fireball burst. For solar-like stars and WDs, a close encounter with an ultrafast and ultramassive FSO can lead to stellar fission , similar to those occurring in a stretched water droplet. Although rare, such phenomena would be amenable to observation.</text>
<text><location><page_4><loc_52><loc_21><loc_92><loc_28></location>We thank beneficial discussions with Seung-Hoon Cha and Kurtis Williams. C.B. and W.N. also acknowledge support under U.S. DOE Grant DDE- FG02- 08ER41533, the NASA Astrophysics Theory Program, Grant 10ATP10-0095 and the Cottrell College Science Awards.</text>
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</document> | [] |
2017ApJ...841...25G | https://arxiv.org/pdf/1708.07553.pdf | <document>
<section_header_level_1><location><page_1><loc_17><loc_86><loc_83><loc_87></location>ELECTRON EXCITATION OF HIGH DIPOLE MOMENT MOLECULES REEXAMINED</section_header_level_1>
<text><location><page_1><loc_34><loc_83><loc_65><loc_85></location>Paul F. Goldsmith 1 and Jens Kauffmann 2 Draft version September 21, 2021</text>
<section_header_level_1><location><page_1><loc_45><loc_80><loc_55><loc_81></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_58><loc_86><loc_80></location>Emission from high-dipole moment molecules such as HCN allows determination of the density in molecular clouds, and is often considered to trace the 'dense' gas available for star formation. We assess the importance of electron excitation in various environments. The ratio of the rate coefficients for electrons and H 2 molecules, glyph[similarequal] 10 5 for HCN, yields the requirements for electron excitation to be of practical importance if n (H 2 ) ≤ 10 5 . 5 cm -3 and X (e -) ≥ 10 -5 , where the numerical factors reflect critical values n c (H 2 ) and X ∗ (e -). This indicates that in regions where a large fraction of carbon is ionized, X (e -) will be large enough to make electron excitation significant. The situation is in general similar for other 'high density tracers', including HCO + , CN, and CS. But there are significant differences in the critical electron fractional abundance, X ∗ (e -), defined by the value required for equal effect from collisions with H 2 and e -. Electron excitation is, for example, unimportant for CO and C + . Electron excitation may be responsible for the surprisingly large spatial extent of the emission from dense gas tracers in some molecular clouds (Pety et al. 2017; Kauffmann, Goldsmith et al. 2017). The enhanced estimates for HCN abundances and HCN/CO and HCN/HCO + ratios observed in the nuclear regions of luminous galaxies may be in part a result of electron excitation of high dipole moment tracers. The importance of electron excitation will depend on detailed models of the chemistry, which may well be non-steady state and non-static.</text>
<section_header_level_1><location><page_1><loc_21><loc_55><loc_36><loc_56></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_22><loc_48><loc_54></location>The possible importance of excitation of the rotational levels of molecules by collisions with electrons, and consideration of the effect of such collisions on observed line ratios is not at all new (e.g. Dickinson et al. 1977). However, relatively early observations and modeling of molecular ions in dense clouds showed that in well-shielded regions, the fractional abundance of electrons is very low, 10 -7 - 10 -8 (Guelin et al. 1977; Wootten et al. 1979). Values of X (e -) in this range would make electron excitation insignificant, although the situation in clouds with lower extinction is dramatically different, with X (e -) ≥ 10 -5 making electron collisions the dominant excitation mechanism for CS (Drdla et al. 1989) and for CN (Black & van Dishoeck 1991). The early calculations of electron excitation rate coefficients (e.g. Dickinson & Richards 1975; Dickinson et al. 1977) were forced to make a variety of approximations, but suggested that the excitation rates scale as the square of the molecule's permanent electric dipole moment. Thus, electron excitation would be important for the widely-observed CO molecule only under very exceptional circumstances, but the question of the possible importance of electron excitation in varied regions of the interstellar medium has not been examined.</text>
<text><location><page_1><loc_8><loc_14><loc_48><loc_22></location>In studies of star formation in other galaxies, emission from the high-dipole moment molecule HCN has been used as a measure of the 'dense' gas in which star formation takes place (Gao & Solomon 2004). The question of the H 2 density that characterizes the regions responsible for this emission thus arises. Since this emission is</text>
<unordered_list>
<list_item><location><page_1><loc_10><loc_9><loc_48><loc_13></location>1 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena CA, 91109, USA; [email protected]</list_item>
<list_item><location><page_1><loc_10><loc_7><loc_48><loc_9></location>2 Max Planck Institut fur Radioastronomie, Auf dem Hugel 69, D-53121 Bonn, Germany</list_item>
</unordered_list>
<text><location><page_1><loc_52><loc_36><loc_92><loc_56></location>generally not spatially resolved, excitation by electrons could be contributing, especially in the outer regions of clouds subject to high radiation fields. Kauffmann et al. (2017) have studied the density associated with HCN emission in the Orion molecular cloud, and find that a large fraction of the flux is produced in regions having n (H 2 ) ≈ 10 3 cm -3 , well below the range ≥ 3 × 10 4 cm -3 assumed by Gao & Solomon (2004). Pety et al. (2017) studied a variety of molecules including HCN in Orion B, and found that the spatial extent of their emission was not correlated with the density of H 2 required for collisional excitation. One possible explanation is electron excitation in the outer regions of the cloud, making reexamination of the possible role of electron excitation appropriate 3 .</text>
<text><location><page_1><loc_52><loc_16><loc_92><loc_36></location>In this paper, we review the recent rate calculations for HCN, HCO + , CS, and CN in § 2. Their influence on the excitation of molecules-with a particular focus on HCN-is summarized in § 3, which utilizes the threelevel model developed in Appendix A together with multilevel statistical equilibrium calculations. This section ends with a more general discussion that extends the argument to transitions of HCO + , CS, and CN ( § 3.4). Clouds of different types in different environments are examined using a PDR code to determine their electron density distribution in § 4. In this section we also discuss the question of the abundance of molecules in the highelectron density regions including diffuse and translucent clouds, molecular cloud edges, and the central regions of active galaxies. We summarize our conclusions in § 5.</text>
<section_header_level_1><location><page_1><loc_57><loc_14><loc_87><loc_15></location>2. COLLISION RATE COEFFICIENTS</section_header_level_1>
<text><location><page_1><loc_52><loc_11><loc_92><loc_13></location>We first discuss the HCN molecule as perhaps the premier example of a high-dipole moment molecule for</text>
<text><location><page_1><loc_52><loc_7><loc_92><loc_9></location>3 We acknowledge appreciatively the suggestion by Simon Glover to anayze the possible role of electron excitation.</text>
<text><location><page_2><loc_8><loc_60><loc_48><loc_92></location>which electron excitation can be relatively important. Along with presenting their quantum collision rate coefficients for CS, Varambhia et al. (2010) make a brief comparison of excitation by electrons and H 2 , concluding that for X (e -) ≥ 10 -5 , the latter should not be ignored. To a reasonable approximation the collision cross sections for electron excitation will be dominated by long-range forces and scale as the square of the permanent electric dipole moment, µ e (Dickinson & Richards 1975; Dickinson et al. 1977; Varambhia et al. 2010). Thus the rates for the CS molecule having µ e =1.958 D (Winnewisser & Cook 1968) or HF with µ e = 1.827 D (Muenter & Klemperer 1970) would be glyph[similarequal] 50% of those of HCN having µ e = 2.985 D (Ebenstein & Muenter 1984). An extremely polar molecule such as LiH, having µ e = 5.88 D (Buldakov et al. 2004) would have electron collision rates almost a factor of 4 greater than those of HCN. But overall, rates for high-dipole moment molecules are fairly well confined within about an order of magnitude. The obvious outlier is CO, with µ e = 0.11 D (Goorvitch 1994), thus having electron collision rate coefficients glyph[similarequal] 0.003 of those for high-dipole moment molecules. In the following section we focus on HCN, given its observational importance. We also consider HCO + , CS, and CN in § 2.3-2.5.</text>
<section_header_level_1><location><page_2><loc_16><loc_57><loc_41><loc_58></location>2.1. HCN Excitation by Electrons</section_header_level_1>
<text><location><page_2><loc_8><loc_45><loc_48><loc_57></location>The calculation of electronic excitation of the lower rotational levels of HCN and isotopologues by Faure et al. (2007) includes treatment of the hyperfine levels, and considered the HNC molecule and isotopologues as well. Here, we do not consider the issue of the hyperfine populations, which although observable and informative in dark clouds with relatively narrow line widths, are not an issue for study of GMCs, especially large-scale imaging in the Milky Way and other galaxies.</text>
<text><location><page_2><loc_8><loc_19><loc_48><loc_45></location>Faure et al. (2007) present their results in the form of polynomial coefficients for the deexcitation ( J final < J initial ) rate coefficients as a function of the kinetic temperature. We have calculated rates for a number of temperatures and give the results for some of the lowest rotational transitions in Table 1. We include here as well the analogous results for HCO + , CS, and CN, which are discussed in § 2.3-2.5. We see that the temperature dependence of the deexcitation rate coefficients is quite weak, and that in common with previous analyses, | ∆ J | = 1 (dipole-like) transitions are strongly favored for electron excitation of neutrals, but less strongly so for electron excitation of ions. For any transition, the collision rate is the product of the collision rate coefficient and the density of collision partners (e.g. electrons or H 2 molecules); C (s -1 ) = R (cm 3 s -1 ) n (e -or H 2 ; cm -3 ). The full set of deexcitation rate coefficients is available on the LAMBDA website (Schoier et al. (2005); http://home.strw.leidenuniv.nl/ ~ moldata/ ).</text>
<section_header_level_1><location><page_2><loc_16><loc_17><loc_41><loc_18></location>2.2. HCN Excitation by H and H 2</section_header_level_1>
<text><location><page_2><loc_8><loc_7><loc_48><loc_16></location>The calculation of rate coefficients for collisions between HCN and H 2 molecules started with Green & Thaddeus (1974), who considered HCN as having only rotational levels and included He as the collision partner, representing H 2 in its ground para-H 2 (I = 0) state with antiparallel nuclear spins. Interest in the nonLTE ratio of HCN hyperfine components led Monteiro</text>
<table>
<location><page_2><loc_52><loc_72><loc_91><loc_85></location>
<caption>TABLE 1 Electron Deexcitation Rate Coefficients for Lower Rotational Transitions of HCN 1 , HCO +2 , and CS 3 (Units are 10 -6 cm 3 s -1 )</caption>
</table>
<list_item><location><page_2><loc_52><loc_71><loc_58><loc_72></location>1 see § 2.1</list_item>
<list_item><location><page_2><loc_52><loc_70><loc_58><loc_71></location>2 see § 2.3</list_item>
<list_item><location><page_2><loc_52><loc_69><loc_58><loc_70></location>3 see § 2.4</list_item>
<text><location><page_2><loc_52><loc_56><loc_92><loc_66></location>& Stutzki (1986) to include the hyperfine levels separately. They found that the individual excitation rate coefficients summed together to give total rotational excitation rate coefficients very similar to those found by Green & Thaddeus (1974). Sarrasin et al. (2010) again considered He as the collision partner, while focusing on differences between the excitation rates for HCN and HNC.</text>
<text><location><page_2><loc_52><loc_41><loc_92><loc_56></location>Dumouchel et al. (2010) employed a new potential energy surface (PES), while still considering the collision partner to be He. The rate coefficients are not very different from those found previously, but they do confirm the difference between HCN and HNC. Ben Abdallah et al. (2012) treat the colliding H 2 molecule as having internal structure, but average over H 2 orientations, considering effectively only molecular hydrogen in the j = 0 level (we employ lower case j to denote the rotational level of H 2 in order to avoid confusion with the rotational level of HCN).</text>
<text><location><page_2><loc_52><loc_17><loc_92><loc_41></location>Vera et al. (2014) have recently calculated collisions between HCN and H 2 molecules, considering for the first time the latter in individual rotational states. They find that there is a significant difference between collisions with the H 2 in the j = 0 level, compared to being in higher rotational levels. The deexcitation rate coefficients for the lower levels of HCN for H 2 ( j ≥ 1) are quite similar, and are 3 to 9 times greater than those for H 2 ( j = 0), rather than, for example, there being a systematic difference for ortho- and para-H 2 rates. The HCN deexcitation rates for H 2 ( j = 0) are generally similar in magnitude to those of Dumouchel et al. (2010) and Ben Abdallah et al. (2012); the deexcitation rates for ∆ J = -1 and -2 transitions are comparable, in contrast to those for H 2 ( j ≥ 1), which show a significant propensity for ∆ J = -2. The numerical results for many of these calculations (often not given in the published articles) are available on the website http://basecol.obspm.fr .</text>
<text><location><page_2><loc_52><loc_7><loc_92><loc_17></location>The major difference between the rates for H 2 ( j = 0) and H 2 ( j ≥ 1) adds a significant complication to the analysis of HCN excitation since it implies a dependence on the H 2 ortho to para ratio, which is itself poorlyknown, and likely varies considerably as a function of environment (e.g. Neufeld et al. 2006; Maret et al. 2009). If the H 2 ortho to para ratio is close to equilibrium at the local kinetic temperature, then in all but the most</text>
<text><location><page_3><loc_8><loc_89><loc_48><loc_92></location>excited regions of molecular clouds, only the collision rate coefficients with H 2 ( j = 0) will be significant.</text>
<section_header_level_1><location><page_3><loc_24><loc_86><loc_33><loc_88></location>2.3. HCO +</section_header_level_1>
<text><location><page_3><loc_8><loc_59><loc_48><loc_86></location>Faure & Tennyson (2001) considered electron excitation of the HCO + ion. This work provided coefficients for evaluating the rates only for the three lowest rotational states, but the rates themselves are comparable to those of HCN, and thus are reasonably consistent with a scaling following µ 2 e . This calculation was supplemented by one including more levels with improved accuracy at low temperatures described by Fuente et al. (2008) and kindly provided to us by A. Faure. The newer deexcitation rate coefficients for low-J transitions are a factor glyph[similarequal] 3 larger at 10 K, but the difference drops rapidly for higher kinetic temperatures and is only 10 to 20% for T k = 100 K. We include deexcitation rate coefficients in Table 1. Collision rates for HCO + excitation by H 2 have been calculated by Flower (1999). The deexcitation rate coefficients for collisions are a factor of 3 to 25 times larger than those for HCN. The HCO + ∆ J = -1 rate coefficients are larger than those for ∆ J = -2 collisions, unlike the case for HCN, for which the inverse relationship holds.</text>
<section_header_level_1><location><page_3><loc_26><loc_57><loc_31><loc_58></location>2.4. CS</section_header_level_1>
<text><location><page_3><loc_8><loc_35><loc_48><loc_56></location>Electron collisions for CS have been analyzed by Varambhia et al. (2010), who found deexcitation rate coefficients a factor of 2 smaller than those for HCN, and with an exceptionally strong propensity rule favoring ∆ J = -1 collisions. A selection of the lowest transitions are included in Table 1. The various calculations for electron excitation of the lower transitions of CS by various methods vary by less than 30%, and less for temperatures ≥ 50 K (Varambhia et al. 2010), a typical accuracy that is probably characteristic of the electron collision rate coefficients for other molecules. Deexcitation rate coefficients for collisions with H 2 are from Lique et al. (2006), but these results do not differ appreciably from those of Turner et al. 1992) and are glyph[similarequal] 3 × 10 -11 cm 3 s -1 , comparable to those for HCN for ∆ J = -2, but a factor 2 to 3 larger for ∆ J = -1.</text>
<section_header_level_1><location><page_3><loc_26><loc_32><loc_32><loc_33></location>2.5. CN</section_header_level_1>
<text><location><page_3><loc_8><loc_18><loc_48><loc_31></location>The spin-rotation coupling for CN complicates the energy level structure and makes accurate comparisons with simple rotors difficult. Excitation rate coefficients for collisions with electrons were calculated by Allison & Dalgarno (1971) for the lowest few levels, and extended by Black & van Dishoeck (1991). From the Lambda Leiden Molecular Data Base ( http://home.strw. leidenuniv.nl/ ~ moldata/datafiles/cn.dat ) we find a characteristic rate coefficient at 20 K for N =1-0 equal to 5.7 × 10 -7 cm 3 s -1 .</text>
<section_header_level_1><location><page_3><loc_12><loc_14><loc_45><loc_16></location>3. EXCITATION BY ELECTRONS AND H 2 MOLECULES</section_header_level_1>
<text><location><page_3><loc_8><loc_7><loc_48><loc_13></location>In this section we discuss how molecules are excited in collisions with electrons and H 2 molecules. Throughout this section we often use HCN as a reference case, given the high astrophysical importance of the molecule and its high sensitivity to collisions with electrons.</text>
<table>
<location><page_3><loc_58><loc_76><loc_85><loc_85></location>
<caption>TABLE 2 Critical Densities and Critical Electron Fractional abundances for the J = 1-0 Transitions 1 of Different Species at 20 K</caption>
</table>
<text><location><page_3><loc_58><loc_73><loc_85><loc_76></location>1 The entries for CN are characteristic values for the N = 1-0 transitions, and are somewhat approximate</text>
<section_header_level_1><location><page_3><loc_56><loc_70><loc_88><loc_71></location>3.1. Critical Electron Fractional Abundance</section_header_level_1>
<text><location><page_3><loc_52><loc_51><loc_92><loc_69></location>The critical density of a molecule is often used to indicate how a molecular species depends on the environmental density. Appendix A provides a discussion of such trends. For our discussion it is important to realize that we are dealing with two critical densities per molecule: one critical value that describes the excitation with electrons, n c (e -), and one that describes the excitation in collisions with H 2 molecules, n c (H 2 ). These densities can then be used to gauge the relative importance of electron excitation for the excitation of a given molecule. In the context of the simplified 3-level model described in Appendix A, the critical fractional abundance of electrons required to have the electron collision rate be equal to the H 2 collision rate is</text>
<formula><location><page_3><loc_61><loc_47><loc_92><loc_50></location>X ∗ (e -) = R e 10 (H 2 ) R e 10 (e -) = n c (e -) n c (H 2 ) . (1)</formula>
<text><location><page_3><loc_52><loc_25><loc_92><loc_46></location>Table 2 gives the critical densities and critical fractional abundance of electrons for the J = 1-0 transitions of HCN, HCO + , CS, and CN. The entries for CN are representative values for the N = 1-0 transitions (near 113 GHz) and these must be regarded as somewhat more uncertain due to more complex molecular structure and less detailed calculations, as discussed above. HCN has relatively small rate coefficients for collisions with H 2 and consequently large n c (H 2 ), while only modestly smaller rate coefficients for collisions with electrons. The result is that the critical electron fraction for HCN is lower than for the other high-dipole moment species considered. CN follows, with CS having a somewhat higher value of X ∗ (e -). HCO + has a rather significantly higher value yet, making this species less likely to be impacted by electron excitation than the others.</text>
<section_header_level_1><location><page_3><loc_57><loc_22><loc_87><loc_23></location>3.2. Emission and Excitation Temperature</section_header_level_1>
<text><location><page_3><loc_52><loc_10><loc_92><loc_22></location>A complementary approach is to consider how the integrated intensity of different species is affected by electron excitation. For optically thin emission, the integrated antenna temperature is just proportional to the upper level column density which for a uniform cloud this is proportional to the upper level density. In the low density limit with no background radiation, from equations A4 and A6 we can write (for excitation by any combination of collision partners)</text>
<formula><location><page_3><loc_60><loc_7><loc_92><loc_9></location>∫ T a dv ∝ N 1 A 10 = N (HCN) C t 01 , (2)</formula>
<text><location><page_4><loc_8><loc_89><loc_48><loc_92></location>where N 1 is the column density of HCN in the J =1 state, N (HCN) is the total column density of the molecule, and</text>
<formula><location><page_4><loc_19><loc_87><loc_48><loc_88></location>C t 01 = C e 01 (e -) + C e 01 (H 2 ) (3)</formula>
<text><location><page_4><loc_8><loc_82><loc_48><loc_86></location>is the total collisional excitation rate from J =0to J =1. The total deexcitation rate is determined from equation3 through detailed balance.</text>
<text><location><page_4><loc_8><loc_75><loc_48><loc_82></location>Using the relationship between the collision rates and collision rate coefficients for the electrons and H 2 molecules and the critical densities for each species (from equations A9 and A10), we can express the integrated intensity as</text>
<formula><location><page_4><loc_11><loc_71><loc_48><loc_74></location>∫ T a dv ∝ N (HCN) A 10 ( n (e -) n c (e -) + n (H 2 ) n c (H 2 ) ) . (4)</formula>
<text><location><page_4><loc_8><loc_65><loc_48><loc_70></location>A measure of the degree of excitation of the J = 1-0 transition is its excitation temperature, which is defined by the ratio of molecules per statistical weight in the upper and lower levels. With T ∗ 10 = ∆ E/k B we can write</text>
<formula><location><page_4><loc_11><loc_60><loc_48><loc_64></location>T ex 10 = T ∗ 10 ln ( N 0 g 1 N 1 g 0 ) = 4 . 25 K ln ( 3 n 0 n 1 ) = 4 . 25 K ln ( 3 A 10 C t 01 ) , (5)</formula>
<text><location><page_4><loc_8><loc_58><loc_27><loc_59></location>where g 0 = 1 and g 1 = 3.</text>
<text><location><page_4><loc_8><loc_46><loc_48><loc_58></location>The importance of electron collisions for the excitation of HCN (or other high-dipole moment molecules) does depend on the rate of neutral particle excitation that is present, as shown in Figure 1. The thermalization parameter Y = C t 10 /A 10 gives the total deexcitation rate relative to the spontaneous decay rate. Each of the curves in Figure 1 is for a given value of Y , with small values of Y indicating subthermal excitation and Y glyph[greatermuch] 1 indicating thermalization.</text>
<text><location><page_4><loc_8><loc_36><loc_48><loc_46></location>In the area where the curves are essentially vertical, the H 2 density is sufficient to provide the specified value of Y and the fractional electron abundance is sufficiently small that electrons are unimportant collision partners. In the area in which the curves run diagonally, the value of Y increases linearly as a function of X (e -) and n (H 2 ), indicating that electrons are the dominant collision partners.</text>
<text><location><page_4><loc_8><loc_21><loc_48><loc_35></location>In order that electron collisions be of practical importance, we must satisfy two conditions. First, the H 2 density must be insufficient to thermalize the excitation temperature, meaning that n (H 2 ) ≤ n c (H 2 ). For HCN J = 1-0 this implies n (H 2 ) ≤ 10 5 . 5 . Second, X (e -) must be sufficiently large that electrons are the dominant collision partner. This means that we must be in or near to the area of the diagonal curves, which in combination with requirement 1 for that molecular transition we take as defined by X (e -) ≥ X ∗ (e -). For HCN J = 1-0 this means X (e -) ≥ 10 -5 .</text>
<section_header_level_1><location><page_4><loc_17><loc_18><loc_40><loc_19></location>3.3. Multilevel Results for HCN</section_header_level_1>
<text><location><page_4><loc_8><loc_7><loc_48><loc_17></location>In Figure 2 we show the results for purely electron excitation of the HCN J = 1-0 transition from a 10 level calculation using RADEX (van der Tak et al. 2007) for the indicated conditions. In the upper panel we show the excitation temperature as a function of the electron density for the cases of a background temperature equal to 2.7 K (blue symbols) and equal to 0.0 K (red triangles). Given that the equivalent temperature difference</text>
<figure>
<location><page_4><loc_52><loc_62><loc_92><loc_92></location>
<caption>Fig. 1.Curves showing the value of the electron fractional abundance as function of the H 2 density required to achieve the indicated values of the the thermalization parameter Y = C t 10 /A 10 . In the area where the curves are vertical, electron excitation is unimportant, and in the area where the curves are diagonal, the electrons are the major contributor to the total collision rate.</caption>
</figure>
<text><location><page_4><loc_52><loc_42><loc_92><loc_52></location>between the upper and lower levels is 4.25 K, the higher background temperature corresponds to a significant excitation rate, and T ex rises above the background only for an electron density of a few tenths cm -3 . In the case of no background, however, there is no 'competition' for the collisional excitation, and T ex increases monotonically starting from the lowest values of the electron density.</text>
<text><location><page_4><loc_52><loc_31><loc_92><loc_41></location>In the lower panel we show the integrated intensity of the J = 1-0 line. With or without background, the emission increases linearly with collision rate as expected, as long as n (e -) is well below n c (e -) = 6.5 cm -3 (Equation A9). The nonzero background temperature reduces the integrated intensity by a constant factor due to the reduced population in the J = 0 level available for excitation and emission of photons (Linke et al. 1977).</text>
<text><location><page_4><loc_52><loc_7><loc_92><loc_31></location>In Figure 3 we present the results of a multilevel calculation for the conditions indicated. The H 2 excitation provides a certain level of excitation and emission; this is seen most clearly by comparing the left and right lower panels showing the integrated intensity. The excitation temperature for low electron densities is dominated by the background radiation and for both of the H 2 densities considered, the collisional excitation rate is not sufficient to increase T ex significantly above 2.7 K. In the presence of the background, the integrated emission is a more sensitive reflection of electron excitation than the excitation temperature. In agreement with the previous approximate analysis, the effect of the electron collisions becomes significant when the electron fractional abundance X (e -) reaches 10 -5 (for n (H 2 ) = 10 3 or 10 4 cm -3 ), at which point the emission has increased by 50%. Electron excitation is dominant for X (e -) = 10 -4 , with the integrated intensity increasing by a factor of 5.5 for</text>
<figure>
<location><page_5><loc_9><loc_63><loc_48><loc_92></location>
<caption>Fig. 2.Purely electron excitation of the HCN J = 1-0 transition. The kinetic temperature is 20 K, the HCN column density is 10 13 cm -2 , and the line width is 2.5 km s -1 . Upper panel: excitation temperature; Lower panel: Integrated line intensity. The points in blue are for a background temperature equal to 2.7 K and those in red for no background.</caption>
</figure>
<text><location><page_5><loc_8><loc_43><loc_48><loc_53></location>n (H 2 ) = 10 3 cm -3 , and by a factor of 4.5 for n (H 2 ) = 10 4 cm -3 . For an H 2 density equal to 10 4 cm -3 the H 2 excitation is significantly greater due to the order of magnitude greater density, but the electron density required to reach a level of emission significantly greater than that produced by the H 2 alone (e.g. ∫ T ∗ A dv = 3 K kms -1 ) is independent of the H 2 density.</text>
<section_header_level_1><location><page_5><loc_15><loc_41><loc_42><loc_42></location>3.4. Extension to HCO + , CS, and CN</section_header_level_1>
<text><location><page_5><loc_8><loc_23><loc_48><loc_40></location>It is more difficult for electron excitation to play a role for HCO + than for HCN, since a much higher fractional electron abundance is required in order that the electron rate be comparable to or exceed that for H 2 . This is shown by the offset in the electron fractional abundances of the diamond symbols in Figure 4 which show the value of X(e -) required to double the intensity of the J = 1-0 transition relative to that produced by collisions with H 2 . A fractional abundance of electrons glyph[similarequal] 15 times greater is required for HCO + relative to that for HCN, largely due to the far larger H 2 deexcitation rate coefficients for HCO + more than outweighing its only somewhat larger e -deexcitation rate coefficients.</text>
<text><location><page_5><loc_8><loc_11><loc_48><loc_23></location>Figure 5 compares HCN and CS excitation as a function of electron density for a H 2 density of 3 × 10 3 ; this lower density is appropriate to ensure subthermal excitation. We consider the J = 2-1 transition of CS and the J = 1-0 transition of HCN in order to ensure comparable spontaneous decay rates. We see that an electron density glyph[similarequal] 7 times greater for CS than for HCN is required to produce a factor of 2 enhancement in the integrated intensity.</text>
<text><location><page_5><loc_8><loc_7><loc_48><loc_11></location>The conclusion from comparison of CS and HCO + with HCN is that for the latter molecule, a significantly lower fractional abundance of electrons can result in doubling</text>
<text><location><page_5><loc_52><loc_83><loc_92><loc_92></location>the integrated intensity of the emission. Thus, if electron excitation is significant, we might expect enhanced HCN to HCO + ratio, more extended HCN emission, or both, and the same, though to a lesser degree, relative to CS. However these conclusions are highly dependent on the chemistry that is determining the abundances of these species.</text>
<section_header_level_1><location><page_5><loc_53><loc_79><loc_91><loc_81></location>4. CLOUD MODELS AND AND THE ELECTRON ABUNDANCE</section_header_level_1>
<section_header_level_1><location><page_5><loc_59><loc_77><loc_85><loc_78></location>4.1. Diffuse and Translucent Clouds</section_header_level_1>
<text><location><page_5><loc_52><loc_52><loc_92><loc_76></location>Diffuse and translucent clouds have modest total extinction ( A v ≤ 2 mag.) and densities typically 50-100 cm -3 . In consequence, carbon is largely ionized and the electron fractional abundance is on the order of 10 -4 . Thus, as mentioned in § 1, electron excitation of highdipole moment molecules will be very significant. We have used the Meudon PDR code (Le Petit et al. 2006) to calculate the thermal and chemical structure of this cloud and show the results in Figure 6. Hydrogen is largely molecular except in the outer 0.25 mag. of the cloud and the electron abundance of 0.01 cm -3 results in n (e -)/ n (H 2 ) = 2 × 10 -4 throughout most of the cloud 4 . Excitation of high-dipole moment molecules will thus predominantly be the result of collisions with electrons. However, as seen in the Figure, the density of HCN is only glyph[similarequal] 10 -10 cm -3 in the center of this cloud, corresponding to a fractional abundance relative to H 2 of 3 × 10 -13 . X (HCN) falls rapidly below this value for A v ≤ 0.2 mag.</text>
<text><location><page_5><loc_52><loc_36><loc_92><loc_52></location>For C + itself, the situation is quite different. The deexcitation rate coefficients for collisions with electrons (Wilson & Bell 2002) are glyph[similarequal] 350 times larger than those for collisions with atomic hydrogen (Barinovs et al. 2005), and glyph[similarequal] 100 times greater than those for collisions with H 2 with an ortho to para ratio of unity (Wiesenfeld & Goldsmith 2014). Thus, even with atomic carbon totally ionized, collisions with electrons will be unimportant compared to those with hydrogen, whether in atomic or molecular form. In fully ionized gas, on the other hand, excitation of C + will be via collisions with electrons.</text>
<section_header_level_1><location><page_5><loc_57><loc_34><loc_88><loc_35></location>4.2. Electron Density in GMC Cloud Edges</section_header_level_1>
<text><location><page_5><loc_52><loc_23><loc_92><loc_33></location>Giant molecular clouds (GMCs) exist in a large range of masses, sizes, and radiation environments, making it difficult to draw specific conclusions about the electron density within them, which varies significantly as function of position. We are interested primarily in the outer portion of the cloud where we expect the electron fractional abundance to be maximum. Such regions are, in fact the Photon Dominated Region (PDR) that borders</text>
<text><location><page_5><loc_52><loc_7><loc_92><loc_21></location>4 All of the Meudon PDR code results presented in this paper assume a carbon to hydrogen ratio equal to 1.3 × 10 -4 . This is somewhat lower than the value 1.6 × 10 -4 obtained for four sources by Sofia et al. (2004) using UV observations of C + absorption, and the value 1.4 × 10 -4 adopted for analysis of the [C ii ] 158 µ m fine structure line by Gerin et al. (2015). Measurements of carbon and oxygen abundances in ionized regions compiled by Esteban et al. (2013) suggest a significant gradient in the [C]/[H] ratio, which they determine to be 6.3 × 10 -4 at a galactocentric distance of 6 kpc and 2.5 × 10 -4 at 10.5 kpc. A higher carbon abundance translates to higher electron abundance where carbon is ionized, so that we may be underestimating the importance of electron excitation, but by an amount that likely depends on environment and location.</text>
<figure>
<location><page_6><loc_21><loc_47><loc_79><loc_92></location>
<caption>Fig. 3.Effect of electron excitation on the J = 1-0 transition of HCN. The kinetic temperature is 20 K, the HCN column density is 10 13 cm -2 , the line width is 2.5 km s -1 , and the background temperature is 2.7 K. The left hand panels are for H 2 density equal to 10 3 cm -3 and the right hand panels for H 2 density equal to 10 4 cm -3 . In each column the upper panel shows the excitation temperature and the lower panel the integrated line intensity.</caption>
</figure>
<text><location><page_6><loc_8><loc_29><loc_48><loc_38></location>every such cloud. As an illustrative example, we consider a slab cloud with a thickness equal to 5 × 10 18 cm and a Gaussian density distribution with central proton density equal to 1 × 10 5 cm -3 and 1/e radius 2.35 × 10 18 cm, leading to an edge density equal to 1.1 × 10 3 cm -3 . The total cloud column density measured normal to the surface is 4.2 × 10 23 cm -2 .</text>
<text><location><page_6><loc_8><loc_11><loc_48><loc_29></location>The results from the Meudon PDR code are shown in Figure 7. The solid curves are for a radiation field a factor of 10 4 greater than the standard ISRF. In the outer portion of the cloud shown, the transition from H to H 2 occurs at a density of ∼ 10 3 cm -3 and an extinction of glyph[similarequal] 1.2 mag, as a result of the relatively high external radiation field incident on the surface of the cloud. This level of radiation field is not unreasonably large for a cloud in the vicinity of a massive young stars. Using the model of Stacey et al. (1993), the front surface of the Orion cloud within a radius of ∼ 0.9 pc of the Trapezium cluster is subject to a radiation field of this or greater intensity.</text>
<text><location><page_6><loc_8><loc_7><loc_48><loc_11></location>In the outer portion of the cloud, the electron density is essentially equal to that of C + , and the fractional abundance X (e -) glyph[similarequal] 2 × 10 -4 in the outermost 1.2 mag.,</text>
<text><location><page_6><loc_52><loc_29><loc_92><loc_38></location>where atomic hydrogen is dominant, and remains at this value to the point where A v = 3 mag. The electron density drops significantly moving inward from this point, falling to 10 -5 at A v = 4 mag. From Figure 3, we see that the electrons increase the emission in HCN J = 1-0 by an order of magnitude relative that from H 2 at the point where n (H 2 ) glyph[similarequal] 10 3 cm -3 .</text>
<text><location><page_6><loc_52><loc_18><loc_92><loc_29></location>The dotted and dashed curves show the electron density for radiation fields increased by factors 10 3 and 10 2 , respectively, relative to the standard ISRF. The lower radiation fields reduce the thickness of the layer of high electron density, but only slightly affect the density there. Reduction in the radiation field by a factor of 100 reduces the thickness of the layer by glyph[similarequal] a factor of 2 in terms of extinction.</text>
<text><location><page_6><loc_52><loc_8><loc_92><loc_18></location>The results from this modeling suggest that regions of significant size can have electron densities sufficient to increase the excitation of any high-dipole moment molecules that may be present by a factor glyph[similarequal] 5. An important requirement for this to be of observational significance is that the density of the species in question be sufficient in the region of enhanced electron density. This issue is discussed in the following section.</text>
<figure>
<location><page_7><loc_9><loc_60><loc_45><loc_87></location>
<caption>Fig. 5.Comparison of HCN and CS excitation by H 2 molecules and by electrons. The excitation temperature (upper panel) and integrated intensity (lower panel) of the J = 1-0 transition of HCN (blue) and the J = 2-1 transition of CS (green) are shown as a function of the electron density for a H 2 density of 3 × 10 3 cm -3 and a kinetic temperature of 10 K. The diamonds indicate the electron density for which the integrated intensity is doubled as a result of increased collisional excitation by electrons.</caption>
</figure>
<figure>
<location><page_7><loc_53><loc_60><loc_89><loc_87></location>
<caption>Fig. 4.Comparison of HCN and HCO + excitation by H 2 molecules and by electrons. The excitation temperature (upper panel) and integrated intensity (lower panel) of the J = 1-0 transition of HCN (blue) and HCO + (red) are shown as a function of the electron density for a H 2 density of 10 4 cm -3 . The diamonds indicate the electron density for which the integrated intensity is doubled as a result of increased collisional excitation by electrons.</caption>
</figure>
<section_header_level_1><location><page_7><loc_11><loc_39><loc_46><loc_40></location>4.3. Molecular Abundances in GMC Cloud Edges</section_header_level_1>
<text><location><page_7><loc_8><loc_7><loc_48><loc_39></location>Standard (e.g. the Meudon PDR code utilized in § 4.2) models of the chemistry in low-extinction portions of interstellar clouds predict that the density and fractional abundance of HCN will be quite low, as illustrated in Figure 6 discussed in § 4.1. A result for a more extended, higher density region, also obtained using the Meudon PDR code, is shown in Figure 8, which focuses on the outer region of a cloud with uniform proton density = 10 5 cm -3 and total extinction = 50 mag. The incident radiation field is the standard ISRF. Within 2 mag of the cloud boundary we find X (HCN) ∼ 4 × 10 -9 , a factor glyph[similarequal] 40 less than that in the region with A v ≥ 4 mag. This is sufficiently small to make the emission per unit area from the outer portion of the cloud, even with electron excitation, relatively weak relative to that in the better-shielded portion of the cloud. However, depending on the structure of the cloud and the geometry of any nearby sources enhancing the external radiation field, the electron excitation could significantly increase the total high-dipole moment molecular emission from the cloud. As indicated in Figure 8, an increased cosmic ray ionization rate does increase the HCN abundance in portions of the cloud characterized by A v ≥ 1 mag. The larger rate here is a reasonable upper limit for the Milky Way,</text>
<text><location><page_7><loc_52><loc_37><loc_92><loc_40></location>so this effect is likely to be limited, but not necessarily in other galaxies (e.g. Bisbas et al. 2015).</text>
<text><location><page_7><loc_52><loc_23><loc_92><loc_37></location>The abundance of high-dipole moment molecules (e.g. HCN and CS) and that of electrons, as enhanced by either UV or cosmic rays, are to a significant degree anticorrelated in standard PDR chemistry. This is illustrated in Table 3 which gives some results for PDR models of clouds of different densities with total visual extinction equal to 50 mag, illuminated from both sides by standard ISRF, and experiencing a cosmic ray ionization rate equal to 5 × 10 -16 s -1 . This enhanced rate is responsible for the relatively large densities of atomic hydrogen in the well-shielded portions of the cloud.</text>
<text><location><page_7><loc_52><loc_7><loc_92><loc_23></location>We see that only for the two lowest densities and for visual extinction less than 1 mag is the density of electrons high enough to significantly increase the excitation rate. However, the HCN density under these conditions corresponds to a fractional abundance only 1/1000 of that which can be reached in the well-shielded portions of clouds. Thus, the effect of the electron enhancement of the collision rate would be very difficult to discern. At high hydrogen densities, the density of electrons increases but their abundance relative to H 2 , the dominant form of hydrogen is quite low. Thus, even though the HCN density increases almost as the square of the total den-</text>
<table>
<location><page_8><loc_33><loc_61><loc_67><loc_87></location>
<caption>TABLE 3 Densities of Different Species in Clouds of Different Densities</caption>
</table>
<figure>
<location><page_8><loc_8><loc_31><loc_48><loc_60></location>
<caption>Fig. 6.One half of a diffuse cloud modeled as a uniform density slab having total extinction = 2.0 mag., irradiated on both sides, with parameters indicated. The electrons present come primarily from ionized carbon, except in the outer 0.05 mag. of the cloud.</caption>
</figure>
<text><location><page_8><loc_8><loc_20><loc_48><loc_23></location>sity at low extinctions in this model, the HCN emission when strong, will be produced by collisions with H 2 .</text>
<text><location><page_8><loc_8><loc_7><loc_48><loc_20></location>The issue of the fractional abundance of all molecules in all regions may not yet be treated completely by such models. For example, the abundance of CO in diffuse clouds is well known to be much greater relative to standard models, and a variety of processes involving transient high temperatures have been proposed (Elitzur & Watson 1978; Federman et al. 1996; Zsarg'o & Federman 2003); see also Section 4.4 of Goldsmith (2013). Possible mechanisms responsible for the elevated temperature include shocks, Alfv'en waves, and turbulent dissipation</text>
<figure>
<location><page_8><loc_52><loc_31><loc_92><loc_60></location>
<caption>Fig. 7.Edge region of a cloud with Gaussian density distribution having parameters given in the Figure. The solid curves are for radiation field enhanced by a factor of 10 4 relative to standard ISRF. Ionized carbon is the dominant source of electrons, and this region encompasses the outer 3 magnitudes of the cloud; X (e -) glyph[similarequal] 2 × 10 -4 for A v ≤ 2.6 mag., and drops to glyph[similarequal] 10 -5 at A v = 4 mag. The dotted and dashed curves describe the electron density for radiation field enhancements relative to the standard ISRF of factors 10 3 and 10 2 , respectively, and show that the region of large n (e -) becomes more limited as the incident radiation field intensity is reduced.</caption>
</figure>
<section_header_level_1><location><page_8><loc_52><loc_15><loc_67><loc_16></location>(Godard et al. 2009).</section_header_level_1>
<text><location><page_8><loc_52><loc_7><loc_92><loc_15></location>The situation in GMCs is even less clear as their range of densities and other physical conditions makes determination of abundance of a specific species at a particular position in a cloud very difficult. However, if any or all of the above processes suggested to operate in diffuse clouds also are present in the outer regions (or possibly the en-</text>
<figure>
<location><page_9><loc_8><loc_63><loc_48><loc_92></location>
<caption>Fig. 8.Effect of cosmic ray ionization rate on the outer portion of a cloud having uniform proton density = 10 5 cm -3 and total extinction = 50 mag. The incident radiation field is the standard ISRF. The dotted curves are for a standard cosmic ray ionization rate of 5 × 10 -17 s -1 , while the solid curves are for a rate a factor of 10 higher. The higher cosmic ray ionization rate has very little effect on the electron density in the outer few mag, where it is very close to the abundance of C + . An enhanced cosmic ray ionization rate does increase the HCN abundance by a factor of a few for A v ≥ 1 mag.</caption>
</figure>
<text><location><page_9><loc_8><loc_36><loc_48><loc_48></location>tire volume) of GMCs, molecular abundances may also be significantly different than would expected from models with chemistry determined by the local kinetic temperature. The possibility of significant additional energy input to the gas in the boundary of the Taurus molecular cloud, a region with relatively low radiative flux, is suggested by the detection of emission in the rotational transitions of H 2 , indicating that temperatures of several hundred K are present (Goldsmith et al. 2010).</text>
<text><location><page_9><loc_8><loc_19><loc_48><loc_36></location>Questions such as the apparent high abundance of atomic carbon throughout the volume of clouds (Plume et al. 2000; Howe et al. 2000), in contrast to what is predicted by chemical models with smoothly-varying density (e.g. Tielens & Hollenbach 1985), has motivated creation of highly-inhomogeneous 'clumpy' cloud models (Meixner & Tielens 1993; Stoerzer et al. 1996; Rollig et al. 2006; Cubick et al. 2008). In this picture, UV photons can permeate a large fraction of the clouds' volume, producing PDR regions on the clumps distributed throughout the cloud. Thus, the regions of high electron density which are located adjacent to where the C + transitions to C, are also distributed throughout the cloud.</text>
<text><location><page_9><loc_8><loc_7><loc_48><loc_19></location>An entirely different class of models involves largescale circulation of condensations between the outer regions of clouds and their interiors (Boland & de Jong 1982; Chieze & Pineau Des Forets 1989). The effect of the circulation depends on many parameters, in particular the characteristic timescale, which is not very well determined. Yet another effect that may be significant is turbulent diffusion, which can significantly affect the radial distribution of abundances if the diffusion coef-</text>
<text><location><page_9><loc_52><loc_83><loc_92><loc_92></location>ficient is sufficiently rapid (Xie et al. 1995). HCN and electrons are not included specifically in the results these authors present, but given the nature of the mechanism, it is likely that the distribution of these species, with the former centrally concentrated and the latter greater at the edge in the absence of turbulent diffusion, will be made more uniform.</text>
<text><location><page_9><loc_52><loc_69><loc_92><loc_82></location>If the abundance of HCN (and other high density tracers) is reasonably large in the outer portion of molecular clouds, and the electron fractional abundance approaches or exceeds 10 -4 there, the total mass of the 'high density region' may be overestimated. This could have an impact on using such molecular transitions as tracers of the gas available for the formation of new stars (e.g. Gao & Solomon 2004), and the possible role of electron excitation in enhancing the size of the high-dipole moment molecular emission should be considered.</text>
<section_header_level_1><location><page_9><loc_60><loc_67><loc_84><loc_68></location>4.4. Extreme Cloud Environments</section_header_level_1>
<text><location><page_9><loc_52><loc_49><loc_92><loc_66></location>The central regions of both starburst and AGN galaxies are extreme environments, with dramatically enhanced energy inputs compared to the 'normal' ISM of the Milky Way and normal galactic disks. Determining the conditions in these regions is naturally challenging, but with the increasing availability of interferometers such as ALMA, there has been heightened interest in unravelling the physical conditions in the central nuclear concentration(s) as well as the surrounding tori that are seen in some galaxies. The density is one of the most important parameters, and lowJ transitions of HCN and its intensity relative to other species, are one of the most often-used probes.</text>
<text><location><page_9><loc_52><loc_25><loc_92><loc_49></location>The ratio of HCN to CO and HCN to HCO + in the lowest rotational transition of each were observed in a number of nearby Seyfert galaxies by Kohno et al. (2001), who proposed that the observed enhancement of R = I (HCN)/ I (CO) in those dominated by an AGN could be produced by enhanced X-ray irradiation of the central region, based on the modeling of Lepp & Dalgarno (1996). This interpretation was used to explain observations of the AGN NGC1068 by Usero et al. (2004). This diagnostic was extended to the J =3-2 transitions of HCN and HCO + in NGC1097 by Hsieh et al. (2012), who found that the enhanced ratio of these two molecules was consistent with X-ray ionization, using the model of Maloney et al. (1996). More detailed models of X-ray dominated regions have since been developed by Meijerink & Spaans (2005) and Meijerink et al. (2007), which indicate a different effect on the HCN/HCO + ratio than found by Maloney et al. (1996).</text>
<text><location><page_9><loc_52><loc_7><loc_92><loc_25></location>However, the X-ray ionization and heating is not the only mechanism proposed to explain enhanced HCN emission. Heating alone, if sufficient to accelerate the reaction CN + H 2 → HCN + H (∆ E/k = 820 K), can increase the abundance of HCN. Aalto et al. (2012) proposed that shocks could be compressing and heating the gas in the outflow associated with AGN Mrk 231. Izumi et al. (2013) observed the J =4-3 transition of HCN and HCO + , along with other molecules, in the nucleus of AGN NGC1097. Combining their data with others (their Table 7) indicates that the HCN enhancement is greater in AGN than in Starburst galaxies. Their chemical modeling suggests a dramatically enhanced HCN abundance based solely on having the gas temperature exceed 500</text>
<text><location><page_10><loc_8><loc_84><loc_48><loc_92></location>K. These authors reject UV and X-rays as the explanation and favor mechanical heating, possibly from a (so far unobserved) AGN jet. Mart'ın et al. (2015) also favor non-X-ray heating to explain their observations of the galaxy Arp 220, employing the chemical models of Harada et al. (2010) and Harada et al. (2013).</text>
<text><location><page_10><loc_8><loc_72><loc_48><loc_84></location>In the context of these observations and proposed models, the relevance of electron excitation of high dipole moment molecules such as HCN is that the regions of the enhanced HCN abundance, whether produced by Xrays, shocks, or UV, could well include substantial electron densities as well. We discussed previously that the integrated intensities of HCN emission could be substantially enhanced if the fractional abundance of electrons is on the order of 10 -4 .</text>
<text><location><page_10><loc_8><loc_68><loc_48><loc_72></location>For subthermal excitation and optically thin emission, the J = 1-0 integrated intensity (equation 4) can be written</text>
<formula><location><page_10><loc_9><loc_64><loc_48><loc_67></location>∫ T a dv ∝ N (molecule) A 10 n (H 2 ) n c (H 2 ) ( 1 + X (e -) X ∗ (e -) ) . (6)</formula>
<text><location><page_10><loc_8><loc_50><loc_48><loc_63></location>If we consider a given molecular species in a region of specified H 2 density, the effect of the electron excitation is contained in the second term in brackets. As examples, for a fractional abundance of 10 -5 , the HCN and CN emission will be approximately doubled, while that of HCO + and CS will be only slightly enhanced. For a fractional abundance of 10 -4 , the emission from HCN and CN will be increased by approximately an order of magnitude, while that of CS and HCO + by factors glyph[similarequal] 2.3 and 3.6, respectively.</text>
<text><location><page_10><loc_8><loc_39><loc_48><loc_50></location>Electron excitation could thus be responsible at least in part for the enhanced HCN/HCO + ratio reported by Kohno et al. (2001) in some Seyfert nucleii. For CO ( § 2), the electron excitation rates are dramatically smaller than those for HCN so that X(e -) ≥ 10 -5 will dramatically enhance HCN emission relative to that of CO, which Kohno et al. (2001) report to be correlated with enhanced HCN/HCO + .</text>
<text><location><page_10><loc_8><loc_7><loc_48><loc_39></location>Izumi et al. (2013) employed the ratio of different HCN transitions to determine volume densities of H 2 and properties of the HCN-emitting region. Electron excitation can produce different ratios than does H 2 excitation as a result of the different J -dependence of the collision rate coefficients (Appendix A and Dickinson et al. (1977)). Figure 9 shows the ratio of integrated intensities of different transitions J -J -1 relative to that of the 1-0 transition. A 10-level calculation using RADEX was employed. The presence of a fractional abundance of electrons equal to 10 -4 reduces the H 2 density to achieve a specified integrated intensity ratio. Observed ratios yield H 2 densities in the range 10 5 to 10 6 cm -3 . X (e -) = 10 -4 reduces the required H 2 density by a factor 3 to 4, which would have a significant impact on characterizing the central regions of galaxies. How important and prevalent the effect of electron excitation is depends on having more reliable models of the radiation field, ionization and chemistry in the central regions of these luminous galaxies. Many effects that may be playing a role. IR pumping is likely important in some sources, as indicated by the detection of vibrationally excited HCN by Aalto et al. (2015a,b). Bisbas et al. (2015) studied the effect of cosmic ray ionization rates up to a factor of 10 3</text>
<text><location><page_10><loc_52><loc_84><loc_92><loc_92></location>greater than standard (as compared to the modest factor of 10 considered in Figure 8) on the CO/H 2 ratio. The CO/H 2 ratio is reduced, but the magnitude of the effect depends on the local density. This study suggests that turbulent mixing, although not included in the modeling, is potentially important.</text>
<figure>
<location><page_10><loc_52><loc_52><loc_92><loc_82></location>
<caption>Fig. 9.Ratio of integrated antenna temperatures of higher rotational transitions of HCN to that of the J =1-0 transition. The different colors are for the three higher transitions indicated. The kinetic temperature is 20 K, the column density of HCN is 1 × 10 13 cm -2 , and the line width is 2.5 km s -1 . The solid lines give the ratio when collisions are exclusively with H 2 . The dotted curves give the ratio when the H 2 is accompanied by a fractional abundance of electrons X (e -) = 1 × 10 -4 . The presence of the electrons reduces the H 2 density required to achieve a moderate ratio, 0.5 - 2.0, by a factor of 2 to 4 depending on the higher transition in question.</caption>
</figure>
<section_header_level_1><location><page_10><loc_65><loc_36><loc_79><loc_37></location>5. CONCLUSIONS</section_header_level_1>
<text><location><page_10><loc_52><loc_16><loc_92><loc_35></location>We have used quantum calculations of collisional excitation of the rotational levels of HCN, HCO + , CN, and CS by electrons and H 2 molecules to evaluate the relative importance of electron excitation. The collisional deexcitation rate coefficients at the temperatures of molecular clouds are close to 10 5 times larger for electrons than for H 2 molecules ( § 3.1). The electron deexcitation rate coefficients scale as the square of the permanent electric dipole moment of the target molecule, so this effect is unimportant for the widely-used tracer CO. For subthermal excitation, the integrated intensity of the J = 1-0 transition is proportional to the sum of the electron and H 2 densities each normalized to the appropriate critical density (Eq. 4).</text>
<text><location><page_10><loc_52><loc_7><loc_92><loc_16></location>The requirements for electron excitation to be of practical importance are n (H 2 ) ≤ n c (H 2 ) and X (e -) ≥ X ∗ (e -) ( § 3.2; also see Eq. 6). For the J = 1-0 transition of HCN this implies n (H 2 ) ≤ 10 5 . 5 cm -3 and X (e -) ≥ 10 -5 . In regions where carbon is largely ionized but hydrogen is molecular, the fractional abundance of electrons, X (e -) = n (e -)/ n (H 2 ) can exceed</text>
<text><location><page_11><loc_8><loc_69><loc_48><loc_92></location>10 -4 , making electrons dominant for the excitation of HCN ( § 4.2). The situation for CN is similar, although somwhat more uncertain due to less complete collision rate calculations. For HCO + , the rate coefficients for collisions with H 2 are more than an order of magnitude larger than those for HCN, more than outweighing the somewhat larger rate coefficients for electron collisions, and demanding a factor 7-10 of higher electron fractional abundance for electron excitation to be significant. For CS, the rate coefficients for electrons are a factor of 2 smaller than those for HCN, while the H 2 rate coefficients are a factor glyph[similarequal] 3 larger, with the combination results in requirement of a factor of 6 larger electron fractional abundance for electron excitation to be significant. Thus, HCN (and to slightly lesser degree CN) appears to be an unusually sensitive probe of electron excitation (Table 2).</text>
<text><location><page_11><loc_8><loc_53><loc_48><loc_69></location>Conditions favoring high X (e -) can occur in low extinction regions such as diffuse and translucent clouds ( § 4.1), and the outer parts of almost any molecular cloud, especially in regions with enhanced UV flux ( § 4.2). Thus, the excitation in the HCN-emitting region may not necessarily be controlled by the high H 2 density generally assumed. The central regions of luminous galaxies often show enhanced HCN emission, which could be in part a result of electron excitation, although the explanation is not certain, with enhanced UV, X-rays, cosmic rays, and mechanical heating having all been proposed as responsible for increasing the abundance of HCN ( § 4.4).</text>
<text><location><page_11><loc_52><loc_80><loc_92><loc_92></location>Accurate determination of the possible importance of electron excitation will depend on having much improved models of the chemistry and dynamics of these regions, including the effects of transient heating and enhanced transport due to turbulence ( § 4.3). Significant additional theoretical work is therefore needed before a satisfying explanation can be given for the extended emission from molecules like HCN in low density environments (Kauffmann et al. 2017).</text>
<text><location><page_11><loc_52><loc_53><loc_92><loc_78></location>We thank Simon Glover for suggesting that we consider electron excitation in the outer parts of molecular clouds. We thank the anonymous reviewer for suggestions that significantly broadened and improved the present investigation. We appreciate Floris van der Tak's critical help in entering data into and using RADEX, and thank Franck LePetit for valuable information concerning the Meudon PDR code. The authors appreciate information and pointers received from Susanne Aalto about molecules in Active Galactic Nucleii. Alexandre Faure graciously provided the full unpublished HCO + -electron deexcitation rate coefficients from unpublished work by Faure and Tennyson. We thank Bill Langer and Kostas Tassis for a number of suggestions that improved this paper. This research was conducted in part at the Jet Propulsion Laboratory, which is operated by the California Institute of Technology under contract with the National Aeronautics and Space Administration (NASA). c © 2016 California Institute of Technology.</text>
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<section_header_level_1><location><page_12><loc_46><loc_53><loc_55><loc_54></location>APPENDIX</section_header_level_1>
<section_header_level_1><location><page_12><loc_33><loc_51><loc_67><loc_51></location>SIMPLIFIED MODELS OF EXCITATION</section_header_level_1>
<text><location><page_12><loc_8><loc_42><loc_92><loc_50></location>In this Appendix we outline a very simplified model with which to give an idea of the relative importance of collisional excitation by electrons and by molecular hydrogen in the limit of low collision rates. We again adopt HCN as a representative high-dipole moment molecule. We adopt the rate coefficients for collisions with H 2 from Dumouchel et al. (2010), understanding that while the result of Ben Abdallah et al. (2012) are quite similar, the situation could be quite different if a large fraction of the H 2 were in states having j ≥ 1 and the results of Vera et al. (2014) discussed above obtain.</text>
<text><location><page_12><loc_8><loc_34><loc_92><loc_42></location>It is instructive to consider only a three level system (levels and rotational quantum numbers J = 0, 1, and 2) with no background radiation and optically thin transitions. The collision rates C ij (s -1 ) are equal to the collisional rate coefficients R ij (cm 3 s -1 ) multiplied by the density of colliding particles (electrons or H 2 molecules, cm -3 ). In general we must consider upwards and downwards collisions, but in the limit of low excitation, with the spontaneous downwards radiative rate, A ul , much larger than the corresponding downwards collision rate, C ul , downwards collisions can be neglected. The rate equations for the densities of HCN, n 0 , n 1 and n 2 in levels J = 0, 1 and 2, respectively, are</text>
<formula><location><page_12><loc_41><loc_31><loc_92><loc_33></location>n 0 ( C 01 + C 02 ) = n 1 A 10 , (A1)</formula>
<formula><location><page_12><loc_41><loc_28><loc_92><loc_29></location>n 1 A 10 = n 0 C 01 + n 2 A 21 , (A2)</formula>
<text><location><page_12><loc_8><loc_25><loc_11><loc_27></location>and</text>
<formula><location><page_12><loc_41><loc_24><loc_92><loc_25></location>n 2 A 21 = n 0 C 02 + n 1 C 12 , (A3)</formula>
<text><location><page_12><loc_8><loc_22><loc_88><loc_23></location>where C lu denotes the upward collision rate from level l to level u . Equation A1 gives us immediately the ratio</text>
<formula><location><page_12><loc_44><loc_18><loc_92><loc_21></location>n 1 n 0 = C 01 + C 02 A 10 . (A4)</formula>
<text><location><page_12><loc_8><loc_13><loc_92><loc_17></location>For more than three levels in the low excitation limit, it is appropriate to consider the total upwards collision rate out of J = 0 when analyzing the excitation of the J = 1 to J = 0 transition, as every such collisional excitation results in emission of a J = 1 to J = 0 photon. We define an effective excitation rate</text>
<formula><location><page_12><loc_39><loc_11><loc_92><loc_12></location>C e 01 (H 2 ) = C 01 (H 2 ) + C 02 (H 2 ) , (A5)</formula>
<text><location><page_12><loc_8><loc_9><loc_20><loc_10></location>for 3 levels, and</text>
<formula><location><page_12><loc_41><loc_7><loc_92><loc_8></location>C e 01 (H 2 ) = Σ k = N k =1 C 0 k (H 2 ) (A6)</formula>
<text><location><page_13><loc_8><loc_88><loc_92><loc_92></location>for N levels, since collisions with H 2 can result in | ∆ J | > 1. We can express this as well in terms of effective rate coefficients since we have only to divide by the density of collision partners, and for the deexcitation rate coefficients we have R e 10 (H 2 ) = C e 10 (H 2 ) /n (H 2 ). From detailed balance for collisions with any partner</text>
<formula><location><page_13><loc_40><loc_84><loc_92><loc_87></location>R e 10 = R e 01 g 0 g 1 exp( T ∗ 10 /T k ) , (A7)</formula>
<text><location><page_13><loc_8><loc_80><loc_92><loc_84></location>where the g ' s are the statistical weights, T ∗ 10 is the equivalent temperature of the J = 1 to J = 0 transition (∆ E/k B = 4.25 K for HCN), and T k is the kinetic temperature. Published calculations generally give the downwards rate coefficients, and the upwards rate coefficients must be calculated individually using detailed balance.</text>
<text><location><page_13><loc_8><loc_74><loc_92><loc_80></location>For collisions with H 2 at a kinetic temperature of 20 K, Dumouchel et al. (2010) give R 10 = 1.41 × 10 -11 cm 3 s -1 and R 20 = 2.1 × 10 -11 cm 3 s -1 . R 30 is 100 times smaller than these rate coefficients, while R 40 = 2.7 × 10 -12 cm 3 s -1 is marginally significant. From each of these we calculate the upwards rate coefficient, and the effective downward rate coefficient is R e 10 (H 2 ) = 3.8 × 10 -11 cm 3 s -1 and R e 01 (H 2 ) = 9.2 × 10 -11 cm 3 s -1 .</text>
<text><location><page_13><loc_10><loc_73><loc_57><loc_74></location>For electrons, since we consider only dipole-like collisions, we have</text>
<formula><location><page_13><loc_37><loc_71><loc_92><loc_72></location>C e 10 (e -) = C 10 (e -) = R 10 (e -) n (e -) , (A8)</formula>
<text><location><page_13><loc_8><loc_67><loc_92><loc_70></location>where R 10 (e -) is just the value from Table 1 at the appropriate kinetic temperature, which is 3.7 × 10 -6 cm 3 s -1 at 20 K.</text>
<text><location><page_13><loc_8><loc_65><loc_92><loc_67></location>The critical density n c is the density of colliding partners at which the downwards collision rate is equal to the spontaneous decay rate. This gives us for the J = 1-0 transition of HCN</text>
<formula><location><page_13><loc_39><loc_61><loc_92><loc_64></location>n c (e -) = A 10 R e 10 (e -) = 6 . 5 cm -3 , (A9)</formula>
<text><location><page_13><loc_8><loc_59><loc_11><loc_60></location>and</text>
<formula><location><page_13><loc_36><loc_56><loc_92><loc_59></location>n c (H 2 ) = A 10 R e 10 (H 2 ) = 6 . 5 × 10 5 cm -3 . (A10)</formula>
<text><location><page_13><loc_8><loc_54><loc_37><loc_55></location>Table 2 gives values for other molecules.</text>
</document> | [] |
2020JHEP...11..102M | https://arxiv.org/pdf/2008.12109.pdf | <document>
<section_header_level_1><location><page_1><loc_17><loc_87><loc_81><loc_89></location>Remarks on infinite towers of gravitational memories</section_header_level_1>
<section_header_level_1><location><page_1><loc_44><loc_79><loc_55><loc_81></location>Pujian Mao</section_header_level_1>
<text><location><page_1><loc_21><loc_72><loc_78><loc_75></location>Center for Joint Quantum Studies and Department of Physics, School of Science, Tianjin University, 135 Yaguan Road, Tianjin 300350, China</text>
<text><location><page_1><loc_18><loc_54><loc_80><loc_66></location>ABSTRACT. An infinite tower of gravitational memories was proposed in [1] by considering the matter-induced vacuum transition in the impulsive limit. We give an alternative realization of the infinite towers of gravitational memories in Newman-Penrose formalism. We also demonstrate that the memories at each order can be associated to the same supertranslation instead of infinite towers of supertranslations or superrotations.</text>
<section_header_level_1><location><page_1><loc_15><loc_49><loc_34><loc_51></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_15><loc_29><loc_84><loc_47></location>Gravitational memory [2-5] obtained rewed attentions in recent years since its intrinsic connections to BMS supertranslations and soft graviton theorem were revealed [6]. The gravitational memory formula is a Fourier transformation of soft graviton theorem. While the memory effect is a consequence of the fact that the gravitational radiation induces transitions among two different vacua that are connected by BMS supertranslations. The triangle relation [7] has accumulated considerable evidence in its favor in particular in discovering new features of gravitational memory. In [1], it was reported that infinite towers of gravitational memory effects can be generated by matter-induced transitions (see also [8] from the soft theorem side).</text>
<text><location><page_1><loc_15><loc_5><loc_84><loc_27></location>In this paper, we show that the infinite towers of gravitational memories can be derived in the Newman-Penrose (NP) formalism [9]. The gravitational wave is characterized by the asymptotic shear σ 0 and σ 0 in the NP formalism. The infinite towers of gravitational memories are derived from the evolution of the components of the Weyl tensor. The memory effect at each order is encoded in different choices of σ 0 and σ 0 . The infinite towers of gravitational memories can be associated to the unique BMS supertranslation rather than infinite towers of supertranslations and superrotations proposed in [1]. There are infinite conserved quantities in NP formalism at each order [10] which can be associated to the unique supertranslation charge [11]. The gravitational memories are equivalent to soft graviton theorems at the first two orders [6, 12]. We show that the derivation of equivalence for the first two orders is not valid at the third order. Thus the connection beyond</text>
<text><location><page_2><loc_15><loc_83><loc_84><loc_89></location>the second order is still obscure. In the present work, we are restricted in linearized gravitational theory. However the nonlinearities can be captured from the contribution of the gravitational wave burst's gravitons [13].</text>
<section_header_level_1><location><page_2><loc_15><loc_74><loc_84><loc_79></location>2 Infinite towers of gravitational memories in NewmanPenrose formalism</section_header_level_1>
<text><location><page_2><loc_15><loc_57><loc_84><loc_71></location>The gravitational memory effect is a relative displacement of nearby observers. The displacement of nearby observers can be derived from the geodesic deviation [14]. Consider a geodesic x µ ' z µ p τ q , where τ is the proper time, with four velocity u µ p τ q . Suppose there is a nearby geodesic x µ p τ q ' z µ p τ q' L µ p τ q , where L µ p τ q is small and purely spatial L µ u µ ' 0 . The coordinate displacement L µ p τ q can be considered as a vector on the first geodesic to first order in L µ p τ q . Space-time curvature causes the separation vector L µ p τ q to change with time with an acceleration determined by the geodesic deviation equation</text>
<formula><location><page_2><loc_35><loc_52><loc_84><loc_55></location>u µ ∇ µ p u ν ∇ ν L α q ' ' R α µνβ u µ L ν u β . (1)</formula>
<text><location><page_2><loc_15><loc_41><loc_84><loc_51></location>The infinite towers of memory observable in [1] is a displacement memory effect that derived from the geodesic deviation equation at each order in the 1 r expansion. In the NP formalism, σ and σ are the complex shear of the null geodesic generator V ' B B r and measure its geodesic deviation. In the linearized theory, they are controlled by the radial NP equation 1</text>
<text><location><page_2><loc_15><loc_36><loc_20><loc_38></location>Hence</text>
<text><location><page_2><loc_15><loc_30><loc_40><loc_31></location>where Ψ 0 is given as initial data 2</text>
<formula><location><page_2><loc_42><loc_38><loc_84><loc_42></location>B r σ ' ' 2 σ r ' Ψ 0 . (2)</formula>
<formula><location><page_2><loc_39><loc_31><loc_84><loc_37></location>σ ' σ 0 r 2 ' 8 ÿ n ' 0 1 n ' 2 Ψ n 0 r n ' 4 , (3)</formula>
<formula><location><page_2><loc_43><loc_23><loc_84><loc_28></location>Ψ 0 ' 8 ÿ n ' 0 Ψ n 0 r n ' 5 . (4)</formula>
<text><location><page_3><loc_15><loc_85><loc_84><loc_89></location>Considering the empty-space case, the evolution of the components of the Weyl tensor at each order are [10]</text>
<formula><location><page_3><loc_29><loc_80><loc_84><loc_83></location>B u Ψ 0 2 ' ' ð 2 B u σ 0 , (5)</formula>
<formula><location><page_3><loc_29><loc_76><loc_84><loc_78></location>B u Ψ 0 0 ' ' ð Ψ 0 1 , (7)</formula>
<formula><location><page_3><loc_29><loc_78><loc_84><loc_81></location>B u Ψ 0 1 ' ' ð Ψ 0 2 , (6)</formula>
<formula><location><page_3><loc_29><loc_71><loc_84><loc_76></location>p n ' 1 qB u Ψ n ' 1 0 ' ' ' ðð 'p n ' 5 q n % Ψ n 0 p n ě 0 q , (8)</formula>
<text><location><page_3><loc_15><loc_67><loc_84><loc_72></location>where Ψ 0 i p i ' 1 , 2 q are at order O p r i ' 5 q . We use p u, r, z, ¯ z q coordinates where z ' e iφ cot θ 2 , ¯ z ' e ' iφ cot θ 2 are the standard stereographic coordinates. The operators ð and ð are defiend in Appendix A.</text>
<text><location><page_3><loc_15><loc_60><loc_84><loc_66></location>In the NP formalism, the gravitational wave is characterized by σ 0 and σ 0 . The time evolution of the components of the Weyl tensor in (5)-(8) are completely determined by σ 0 and its time integrations</text>
<formula><location><page_3><loc_23><loc_55><loc_84><loc_58></location>Ψ 0 2 ' ' ð 2 σ 0 , (9)</formula>
<formula><location><page_3><loc_23><loc_47><loc_84><loc_52></location>Ψ 0 0 ' ' ð 4 ż d u 1 d u 2 σ 0 , (11)</formula>
<formula><location><page_3><loc_23><loc_51><loc_84><loc_56></location>Ψ 0 1 ' ð 3 ż d u 1 σ 0 , (10)</formula>
<formula><location><page_3><loc_23><loc_42><loc_84><loc_48></location>Ψ n ' 1 0 ' p'q n n ź k ' 0 ' 1 k ' 1 r ðð 'p k ' 5 q k s ż d u k ' 1 ð 4 ż d u 1 d u 2 σ 0 . (12)</formula>
<text><location><page_3><loc_15><loc_24><loc_84><loc_42></location>The memories at different orders are completely determined by the same σ 0 . The independence of each memory is encoded in different choice of σ 0 . For instance, σ 0 ' Θ p u q f p z, ¯ z q will induce a permanent change in σ 0 which will lead to a leading memory. 3 Then σ 0 ' δ p u q f p z, ¯ z q will lead to a subleading memory where a permanent change happens in ş d u 1 σ 0 instead of σ 0 . Similarly, the nth order memory can be associated to σ 0 ' d Θ p u q d n ' 1 u f p z, ¯ z q . The memories derived from (11) and (12) are nothing but the higher order terms in (3). However the memory derived from (10) is somewhat different since there is a gap in the expansion in (3) and Ψ 0 1 is not in the expansion. Nevertheless the observational effect of the subleading memory is a time delay [12].</text>
<section_header_level_1><location><page_3><loc_15><loc_18><loc_58><loc_20></location>3 The relation to supertranslations</section_header_level_1>
<text><location><page_3><loc_15><loc_9><loc_84><loc_15></location>Memories are associated to asymptotic symmetries. The infinite towers of gravitational memories in [1] are associated to infinite towers of supertranslations and superrotations. However there is only one unique supertranslation [11, 16] in the NP formalism in the</text>
<text><location><page_4><loc_15><loc_83><loc_84><loc_89></location>broadly used Newman-Unti gauge [17]. Nonetheless the infinite towers of gravitational memories defined in (9)-(12) can be associated to the unique supertranslation. The action of the supertranslation on σ 0 is</text>
<formula><location><page_4><loc_41><loc_78><loc_84><loc_81></location>δ st σ 0 ' ' ð 2 T p θ, φ q . (13)</formula>
<text><location><page_4><loc_15><loc_74><loc_84><loc_77></location>where T p θ, φ q defines the supertranslation. The leading memory is encoded in the transition</text>
<formula><location><page_4><loc_43><loc_71><loc_84><loc_73></location>σ 0 ' Θ p u q δ st σ 0 . (14)</formula>
<text><location><page_4><loc_15><loc_69><loc_36><loc_70></location>The subleading transition is</text>
<formula><location><page_4><loc_43><loc_66><loc_84><loc_68></location>σ 0 ' δ p u q δ st σ 0 . (15)</formula>
<text><location><page_4><loc_15><loc_64><loc_77><loc_65></location>This transition can be understood as two supertranslations in the following form</text>
<formula><location><page_4><loc_33><loc_58><loc_84><loc_62></location>σ 0 ' lim δu Ñ 0 ˆ Θ p u q δu ' Θ p u ' δu q δu ˙ δ st σ 0 . (16)</formula>
<text><location><page_4><loc_15><loc_56><loc_39><loc_57></location>Then the nth order transition is</text>
<formula><location><page_4><loc_42><loc_51><loc_84><loc_55></location>σ 0 ' d Θ p u q d n ' 1 u δ st σ 0 , (17)</formula>
<text><location><page_4><loc_15><loc_48><loc_54><loc_50></location>which can be considered as 2 n ' 1 supertranslations.</text>
<text><location><page_4><loc_15><loc_25><loc_84><loc_47></location>The physical processes that create the impulsive transitions of σ 0 and Ψ i 0 are similar to the transitions in [1]. They are given by shock wave induced by matter fields at different orders. 4 The common interpretation of the characteristic initial data problem in NP formalism is that B u σ 0 and B u σ 0 are the news functions whose time evolutions are not constrained from Einstein equation. Then the time evolutions of all orders of Ψ i 0 are determined once σ 0 is given as what we applied previously. However this procedure can also go backwards once Ψ i 0 is given with an impulsive transition Ψ i 0 ' Θ p u q δ st σ 0 . For Ψ j 0 that j ą i , their time evolutions are determined since Ψ i 0 is given. For σ 0 and Ψ k 0 that k ă i , they are completely determined through the time evolution equation (8). Finally, the transition σ 0 ' δσ 0 will be determined by</text>
<formula><location><page_4><loc_26><loc_19><loc_84><loc_25></location>p'q i ' 1 i ' 1 ź k ' 0 ' 1 k ' 1 r ðð 'p k ' 5 q k s ð 4 δσ 0 ' d Θ p u q d i ' 2 u δ st σ 0 . (18)</formula>
<section_header_level_1><location><page_4><loc_15><loc_16><loc_61><loc_18></location>4 Infinite towers of conservation laws</section_header_level_1>
<text><location><page_4><loc_15><loc_10><loc_84><loc_13></location>Memories have their intrinsic connections to conserved quantities. For instance, the leading memory is related to Bondi mass aspect [6] while the subleading memory is related to</text>
<text><location><page_5><loc_15><loc_79><loc_84><loc_89></location>angular momentum flux [12]. The infinite towers of gravitational memories in [1] are related to the Noether charges of the infinite towers of supertranslations and superrotations. In the NP formalism, there are infinite conserved quantities 5 derived from the evolution equations of the components of the Weyl tensor at each order [10]. Equation (5) leads to the conservation of mass</text>
<formula><location><page_5><loc_27><loc_73><loc_84><loc_78></location>B u ż d z d ¯ z γ z ¯ z 0 Y 0 , 0 Ψ 0 2 ' 'B u ż d z d ¯ z γ z ¯ z 0 Y 0 , 0 ð 2 σ 0 ' 0 , (19)</formula>
<text><location><page_5><loc_15><loc_69><loc_84><loc_73></location>where the relations in (36) have been used. Equation (8) yields the infinite amount of subleading conservation laws as</text>
<formula><location><page_5><loc_31><loc_55><loc_84><loc_68></location>p n ' 1 qB u ż d z d ¯ z γ z ¯ z 2 Y n ' k ' 2 ,m Ψ n ' 1 0 '' ż d z d ¯ z γ z ¯ z 2 Y n ' k ' 2 ,m ' ðð 'p n ' 5 q n ˘ Ψ n 0 '' ż d z d ¯ z γ z ¯ z 2 Y n ' k ' 2 ,m k p 2 n ' k ' 5 q Ψ n 0 , (20)</formula>
<text><location><page_5><loc_15><loc_53><loc_19><loc_55></location>hence</text>
<text><location><page_5><loc_15><loc_46><loc_84><loc_49></location>when applying the relations in (36). Equations (6) and (7) only induce identically vanishing quantities</text>
<formula><location><page_5><loc_36><loc_49><loc_84><loc_54></location>B u ż d z d ¯ z γ z ¯ z 2 Y n ' 2 ,m Ψ n ' 1 0 ' 0 , (21)</formula>
<formula><location><page_5><loc_26><loc_36><loc_84><loc_44></location>B u ż d z d ¯ z γ z ¯ z 0 Y 0 , 0 ð Ψ 0 1 ' ' ż d z d ¯ z γ z ¯ z 0 Y 0 , 0 ðð Ψ 0 2 ' 0 , (22) B u ż d z d ¯ z γ z ¯ z 0 Y 0 , 0 ð 2 Ψ 0 0 ' ' ż d z d ¯ z γ z ¯ z 0 Y 0 , 0 ð 2 ð Ψ 0 1 ' 0 . (23)</formula>
<text><location><page_5><loc_15><loc_29><loc_84><loc_35></location>The conserved quantities can be reorganized as a unique charge in expansion (see, for instance, [18-21] for relevant developments). The supertranslation charge at each order proposed in [11] is given by 6</text>
<formula><location><page_5><loc_16><loc_17><loc_84><loc_27></location>Q st ' ż d z d ¯ z γ z ¯ z T p z, ¯ z q ' Ψ 0 2 ' Ψ 0 2 ' ð Ψ 0 1 ' ð Ψ 0 1 3 r ' ð 2 Ψ 0 0 ' ð 2 Ψ 0 0 12 r 2 ' 8 ÿ m ' 1 1 p m ' 1 qp m ' 4 q ð 2 Ψ p m q 0 ' ð 2 Ψ p m q 0 r m ' 2 . (24)</formula>
<text><location><page_5><loc_15><loc_13><loc_84><loc_17></location>All the conserved quantities in equations (19)-(23) can be recovered from certain order of the supertranslation charge (24). It is worth to point out that the charge (24) was</text>
<text><location><page_6><loc_15><loc_62><loc_84><loc_89></location>proposed from the connections to soft graviton theorems. It is not derived from action principle of the theory. Recently, BMS charges at different orders were studied carefully by covariant phase space methods [19]. At the leading order, the charge in [19] agrees with (24), while the subleading order charge in [19] is absent for vacuum gravity. At the third order, the charge in [19] is much more complicated than that in (24). Although the subleading charges in [19] are different from (24), they do have intrinsic connections to the Newman-Penrose conserved quantities reviewed previously in this section. Since the charge (24) can be understood as a recast from soft theorems in the low energy expansion to a charge in the 1 r expansion. It is possible that the NP conserved quantities, i.e. the specially selected supertranslation charge, are the essence to recover the soft theorems rather than the full supertranslation charge. Nonetheless, defining conserved charges to asymptotic symmetries is a well-established issue [22,23], extending those charges to the subleading orders is still a tricky question.</text>
<section_header_level_1><location><page_6><loc_15><loc_56><loc_69><loc_58></location>5 Comment on the relation to soft theorems</section_header_level_1>
<text><location><page_6><loc_15><loc_33><loc_84><loc_53></location>Memories and soft theorems are mathematically equivalent in the context of the triangle relation [7]. However such relation is not easy to verify beyond the subsubleading order since the soft graviton theorems beyond the subsubleading order do not have an universal factorization property [8, 24]. Correspondingly, the higher order memory formulas (12) are in more complicated forms. The first two orders of memories are shown to be equivalent to the leading soft graviton theorems [6] and subleading soft graviton theorem [12]. The key observation of the equivalence is the fact that the soft factors, after Fourier transform and projection on the null infinity, can be considered as classical fields that satisfy classical equations of motion, namely the Einstein equation. Hence the equivalence between the first two orders of memories and soft theorems can be interpreted as [6,12]</text>
<formula><location><page_6><loc_35><loc_27><loc_84><loc_31></location>S p 0 q zz ' γ z ¯ z σ 0 , S p 1 q zz ' ż d u 1 γ z ¯ z σ 0 . (25)</formula>
<text><location><page_6><loc_15><loc_23><loc_84><loc_26></location>Including local stress-energy tensor, the classical equations of motion 7 that the leading factor and the subleading soft factor satisfy are</text>
<formula><location><page_6><loc_32><loc_16><loc_84><loc_21></location>Ψ 0 2 ' Ψ 0 2 ' ð 2 B u σ 0 ' ð 2 B u σ 0 ' 1 2 ż d u 1 T 0 uu , (26)</formula>
<text><location><page_6><loc_15><loc_15><loc_41><loc_16></location>which is the real (electric) part of</text>
<formula><location><page_6><loc_39><loc_10><loc_84><loc_13></location>B u Ψ 0 2 ' ð 2 B u σ 0 ' 1 4 T 0 uu , (27)</formula>
<text><location><page_7><loc_15><loc_87><loc_17><loc_89></location>and</text>
<formula><location><page_7><loc_19><loc_82><loc_84><loc_86></location>ð Ψ 0 1 ' ð Ψ 0 1 ' ðð ż d u 1 p ð 2 σ 0 ' ð 2 σ 0 q ' ż d u 1 ˆ ð T 0 u ¯ z 2 ? γ z ¯ z ' ð T 0 uz 2 ? γ z ¯ z ˙ , (28)</formula>
<text><location><page_7><loc_15><loc_80><loc_48><loc_81></location>which is the imaginary (magnetic) part of 8</text>
<formula><location><page_7><loc_39><loc_75><loc_84><loc_79></location>B u Ψ 0 1 ' ð Ψ 0 2 ' 1 2 ? γ z ¯ z T 0 u ¯ z . (29)</formula>
<text><location><page_7><loc_15><loc_68><loc_84><loc_73></location>Regarding to the third memory, it is supposed to be equivalent to the subsubleading soft factor discovered in [24]. After Fourier transform and projection on the null infinity, the subsubleading soft factor should be interpreted as classical field</text>
<formula><location><page_7><loc_40><loc_62><loc_84><loc_67></location>S p 2 q zz ' ż d u 1 d u 2 γ z ¯ z σ 0 , (30)</formula>
<text><location><page_7><loc_15><loc_61><loc_84><loc_62></location>and the classical equation of motion for the subsubleading soft factor should be related to</text>
<formula><location><page_7><loc_39><loc_56><loc_84><loc_60></location>B u Ψ 0 0 ' ð Ψ 0 1 ' 3 4 γ z ¯ z T 0 ¯ z ¯ z . (31)</formula>
<text><location><page_7><loc_15><loc_54><loc_43><loc_55></location>Equation (31) can be re-organized as</text>
<formula><location><page_7><loc_16><loc_39><loc_84><loc_52></location>ð 2 Ψ 0 0 ' ð 2 Ψ 0 0 ' ð 2 ð 2 ż d u 1 d u 2 p ð 2 σ 0 ' ð 2 σ 0 q ' ż d u 1 d u 2 ˆ ð 2 ð T 0 u ¯ z 2 ? γ z ¯ z ' ð 2 ð T 0 uz 2 ? γ z ¯ z ˙ ' ż d u 1 ˆ ð 2 3 T 0 ¯ z ¯ z 4 ? γ z ¯ z ' ð 2 3 T 0 zz 4 ? γ z ¯ z ˙ , (32)</formula>
<text><location><page_7><loc_15><loc_8><loc_84><loc_39></location>to connect to the expression ş d u 1 d u 2 γ z ¯ z σ 0 . However, considering stress-energy tensor from massless particles or localized wave packets which puncture the null infinity at points p u k , z k , ¯ z k q as in [12], direct computation shows that equation (32) does not fulfill when ş d u 1 d u 2 γ z ¯ z σ 0 is replaced by the subsubleading soft factor. The connection at the third order seems to be still of a mystery. However the failure of demonstrating the equivalence between gravitational memories and soft graviton theorems beyond subleading order may not be surprising in the sense that the soft graviton theorems beyond subleading order may have different natures. On the one hand, the soft graviton theorems beyond subsubleading order do not have a universal property [8] in contrast to the first three orders. On the other hand, the double copy relation [25] indicates that a gravity amplitude can be expressed as a sum of the square of color-ordered Yang-Mills amplitudes. The first three soft factors in gravity satisfy precisely the double copy relation [26, 27]. From this point of view, the third order gravitational memory should have intrinsic relation to the first two orders. Unfortunately, we are not able to uncover that in the present work. It is definitely an interesting question that should be addressed elsewhere.</text>
<section_header_level_1><location><page_8><loc_15><loc_86><loc_32><loc_87></location>6 Conclusion</section_header_level_1>
<text><location><page_8><loc_15><loc_71><loc_84><loc_83></location>We demonstrated that there are infinite towers of gravitational memories which can be associated to the same supertranslation in the NP formalism. An infinite number of conserved quantities can be obtained from each order of the supertranslation charge. We also commented on the relation between the infinite towers of gravitational memories and the infinite towers of soft graviton theorems. The equivalence of those two subjects is not yet clear beyond the second order.</text>
<text><location><page_8><loc_15><loc_64><loc_84><loc_70></location>As a final remark, it is worthwhile to point out that the analysis performed in gravitational theory can be easily extended to electromagnetism by applying the results in [10] and [28].</text>
<section_header_level_1><location><page_8><loc_15><loc_58><loc_37><loc_60></location>Acknowledgments</section_header_level_1>
<text><location><page_8><loc_15><loc_50><loc_84><loc_55></location>The author thanks Geoffrey Comp'ere for useful discussions. This work is supported in part by the National Natural Science Foundation of China under Grants No. 11905156 and No. 11935009.</text>
<section_header_level_1><location><page_8><loc_15><loc_44><loc_62><loc_46></location>A Spin-weighted derivative operators</section_header_level_1>
<text><location><page_8><loc_15><loc_35><loc_84><loc_41></location>The operators ð and ð were originally introduced in [29] to replace the covariant derivative on a sphere with metric d S 2 ' 2 γ z ¯ z d z d ¯ z . The choice in [10] is γ z ¯ z ' 1 p 1 ' z ¯ z q 2 which is not the unit sphere. The definitions of ð and ð on a field η are</text>
<formula><location><page_8><loc_31><loc_31><loc_84><loc_34></location>ð η ' γ s ' 1 2 z ¯ z B ¯ z p ηγ ' s 2 z ¯ z q , ð η ' γ ' s ' 1 2 z ¯ z B z p ηγ s 2 z ¯ z q , (33)</formula>
<text><location><page_8><loc_15><loc_27><loc_84><loc_30></location>where s is the spin weight of η . The spin weight of the relevant fields are listed in Table 1.</text>
<paragraph><location><page_8><loc_41><loc_23><loc_58><loc_24></location>Table 1: Spin weights</paragraph>
<formula><location><page_8><loc_32><loc_16><loc_65><loc_21></location>σ 0 σ 0 B u σ 0 Ψ 0 2 Ψ 0 1 Ψ 0 0 Ψ p m q 0 s 2 ' 2 ' 2 0 1 2 2</formula>
<text><location><page_8><loc_15><loc_9><loc_84><loc_12></location>The operator ð ( ð ) will increase (decrease) the spin weight. These two operators do not commute in general. Their commutation relation is</text>
<formula><location><page_8><loc_42><loc_4><loc_84><loc_8></location>r ð , ð s η ' s 2 R S η , (34)</formula>
<text><location><page_9><loc_15><loc_87><loc_48><loc_89></location>where R S is the Ricci scalar of the sphere.</text>
<text><location><page_9><loc_15><loc_81><loc_84><loc_86></location>Spin s spherical harmonics are defined by acting ð and ð on the spherical harmonics Y l,m p l ' 0 , 1 , 2 , . . . ; m ' ' l, . . . , l q . They are given by</text>
<formula><location><page_9><loc_28><loc_69><loc_84><loc_81></location>s Y l,m ' $ ' ' ' ' & ' ' ' ' d p l ' s q ! p l ' s q ! ð s Y l,m p 0 ď s ď l q p' 1 q s d p l ' s q ! p l ' s q ! ð ' s Y l,m p' l ď s ď 0 q . (35)</formula>
<text><location><page_9><loc_15><loc_68><loc_75><loc_73></location>% From the definition of spin weighted spherical harmonics, one can obtain that</text>
<formula><location><page_9><loc_34><loc_52><loc_84><loc_67></location>ż d z d ¯ z γ z ¯ z s Y l,m ð l ' s ' 1 η ' 0 , ż d z d ¯ z γ z ¯ z s ¯ Y l,m ð l ' s ' 1 ζ ' 0 , ðð s Y l,m ' 'p l ' s qp l ' s ' 1 q s Y l,m , ż d z d ¯ z γ z ¯ z A ð B ' ' ż d z d ¯ z γ z ¯ z B ð A. (36)</formula>
<text><location><page_9><loc_15><loc_48><loc_84><loc_52></location>where η and ζ are with spin weight ' l ' 1 and l ' 1 respectively, and the expression A ð B has spin weight zero.</text>
<section_header_level_1><location><page_9><loc_15><loc_42><loc_28><loc_44></location>References</section_header_level_1>
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</document> | [{"title": "Pujian Mao", "content": "Center for Joint Quantum Studies and Department of Physics, School of Science, Tianjin University, 135 Yaguan Road, Tianjin 300350, China ABSTRACT. An infinite tower of gravitational memories was proposed in [1] by considering the matter-induced vacuum transition in the impulsive limit. We give an alternative realization of the infinite towers of gravitational memories in Newman-Penrose formalism. We also demonstrate that the memories at each order can be associated to the same supertranslation instead of infinite towers of supertranslations or superrotations.", "pages": [1]}, {"title": "1 Introduction", "content": "Gravitational memory [2-5] obtained rewed attentions in recent years since its intrinsic connections to BMS supertranslations and soft graviton theorem were revealed [6]. The gravitational memory formula is a Fourier transformation of soft graviton theorem. While the memory effect is a consequence of the fact that the gravitational radiation induces transitions among two different vacua that are connected by BMS supertranslations. The triangle relation [7] has accumulated considerable evidence in its favor in particular in discovering new features of gravitational memory. In [1], it was reported that infinite towers of gravitational memory effects can be generated by matter-induced transitions (see also [8] from the soft theorem side). In this paper, we show that the infinite towers of gravitational memories can be derived in the Newman-Penrose (NP) formalism [9]. The gravitational wave is characterized by the asymptotic shear \u03c3 0 and \u03c3 0 in the NP formalism. The infinite towers of gravitational memories are derived from the evolution of the components of the Weyl tensor. The memory effect at each order is encoded in different choices of \u03c3 0 and \u03c3 0 . The infinite towers of gravitational memories can be associated to the unique BMS supertranslation rather than infinite towers of supertranslations and superrotations proposed in [1]. There are infinite conserved quantities in NP formalism at each order [10] which can be associated to the unique supertranslation charge [11]. The gravitational memories are equivalent to soft graviton theorems at the first two orders [6, 12]. We show that the derivation of equivalence for the first two orders is not valid at the third order. Thus the connection beyond the second order is still obscure. In the present work, we are restricted in linearized gravitational theory. However the nonlinearities can be captured from the contribution of the gravitational wave burst's gravitons [13].", "pages": [1, 2]}, {"title": "2 Infinite towers of gravitational memories in NewmanPenrose formalism", "content": "The gravitational memory effect is a relative displacement of nearby observers. The displacement of nearby observers can be derived from the geodesic deviation [14]. Consider a geodesic x \u00b5 ' z \u00b5 p \u03c4 q , where \u03c4 is the proper time, with four velocity u \u00b5 p \u03c4 q . Suppose there is a nearby geodesic x \u00b5 p \u03c4 q ' z \u00b5 p \u03c4 q' L \u00b5 p \u03c4 q , where L \u00b5 p \u03c4 q is small and purely spatial L \u00b5 u \u00b5 ' 0 . The coordinate displacement L \u00b5 p \u03c4 q can be considered as a vector on the first geodesic to first order in L \u00b5 p \u03c4 q . Space-time curvature causes the separation vector L \u00b5 p \u03c4 q to change with time with an acceleration determined by the geodesic deviation equation The infinite towers of memory observable in [1] is a displacement memory effect that derived from the geodesic deviation equation at each order in the 1 r expansion. In the NP formalism, \u03c3 and \u03c3 are the complex shear of the null geodesic generator V ' B B r and measure its geodesic deviation. In the linearized theory, they are controlled by the radial NP equation 1 Hence where \u03a8 0 is given as initial data 2 Considering the empty-space case, the evolution of the components of the Weyl tensor at each order are [10] where \u03a8 0 i p i ' 1 , 2 q are at order O p r i ' 5 q . We use p u, r, z, \u00af z q coordinates where z ' e i\u03c6 cot \u03b8 2 , \u00af z ' e ' i\u03c6 cot \u03b8 2 are the standard stereographic coordinates. The operators \u00f0 and \u00f0 are defiend in Appendix A. In the NP formalism, the gravitational wave is characterized by \u03c3 0 and \u03c3 0 . The time evolution of the components of the Weyl tensor in (5)-(8) are completely determined by \u03c3 0 and its time integrations The memories at different orders are completely determined by the same \u03c3 0 . The independence of each memory is encoded in different choice of \u03c3 0 . For instance, \u03c3 0 ' \u0398 p u q f p z, \u00af z q will induce a permanent change in \u03c3 0 which will lead to a leading memory. 3 Then \u03c3 0 ' \u03b4 p u q f p z, \u00af z q will lead to a subleading memory where a permanent change happens in \u015f d u 1 \u03c3 0 instead of \u03c3 0 . Similarly, the nth order memory can be associated to \u03c3 0 ' d \u0398 p u q d n ' 1 u f p z, \u00af z q . The memories derived from (11) and (12) are nothing but the higher order terms in (3). However the memory derived from (10) is somewhat different since there is a gap in the expansion in (3) and \u03a8 0 1 is not in the expansion. Nevertheless the observational effect of the subleading memory is a time delay [12].", "pages": [2, 3]}, {"title": "3 The relation to supertranslations", "content": "Memories are associated to asymptotic symmetries. The infinite towers of gravitational memories in [1] are associated to infinite towers of supertranslations and superrotations. However there is only one unique supertranslation [11, 16] in the NP formalism in the broadly used Newman-Unti gauge [17]. Nonetheless the infinite towers of gravitational memories defined in (9)-(12) can be associated to the unique supertranslation. The action of the supertranslation on \u03c3 0 is where T p \u03b8, \u03c6 q defines the supertranslation. The leading memory is encoded in the transition The subleading transition is This transition can be understood as two supertranslations in the following form Then the nth order transition is which can be considered as 2 n ' 1 supertranslations. The physical processes that create the impulsive transitions of \u03c3 0 and \u03a8 i 0 are similar to the transitions in [1]. They are given by shock wave induced by matter fields at different orders. 4 The common interpretation of the characteristic initial data problem in NP formalism is that B u \u03c3 0 and B u \u03c3 0 are the news functions whose time evolutions are not constrained from Einstein equation. Then the time evolutions of all orders of \u03a8 i 0 are determined once \u03c3 0 is given as what we applied previously. However this procedure can also go backwards once \u03a8 i 0 is given with an impulsive transition \u03a8 i 0 ' \u0398 p u q \u03b4 st \u03c3 0 . For \u03a8 j 0 that j \u0105 i , their time evolutions are determined since \u03a8 i 0 is given. For \u03c3 0 and \u03a8 k 0 that k \u0103 i , they are completely determined through the time evolution equation (8). Finally, the transition \u03c3 0 ' \u03b4\u03c3 0 will be determined by", "pages": [3, 4]}, {"title": "4 Infinite towers of conservation laws", "content": "Memories have their intrinsic connections to conserved quantities. For instance, the leading memory is related to Bondi mass aspect [6] while the subleading memory is related to angular momentum flux [12]. The infinite towers of gravitational memories in [1] are related to the Noether charges of the infinite towers of supertranslations and superrotations. In the NP formalism, there are infinite conserved quantities 5 derived from the evolution equations of the components of the Weyl tensor at each order [10]. Equation (5) leads to the conservation of mass where the relations in (36) have been used. Equation (8) yields the infinite amount of subleading conservation laws as hence when applying the relations in (36). Equations (6) and (7) only induce identically vanishing quantities The conserved quantities can be reorganized as a unique charge in expansion (see, for instance, [18-21] for relevant developments). The supertranslation charge at each order proposed in [11] is given by 6 All the conserved quantities in equations (19)-(23) can be recovered from certain order of the supertranslation charge (24). It is worth to point out that the charge (24) was proposed from the connections to soft graviton theorems. It is not derived from action principle of the theory. Recently, BMS charges at different orders were studied carefully by covariant phase space methods [19]. At the leading order, the charge in [19] agrees with (24), while the subleading order charge in [19] is absent for vacuum gravity. At the third order, the charge in [19] is much more complicated than that in (24). Although the subleading charges in [19] are different from (24), they do have intrinsic connections to the Newman-Penrose conserved quantities reviewed previously in this section. Since the charge (24) can be understood as a recast from soft theorems in the low energy expansion to a charge in the 1 r expansion. It is possible that the NP conserved quantities, i.e. the specially selected supertranslation charge, are the essence to recover the soft theorems rather than the full supertranslation charge. Nonetheless, defining conserved charges to asymptotic symmetries is a well-established issue [22,23], extending those charges to the subleading orders is still a tricky question.", "pages": [4, 5, 6]}, {"title": "5 Comment on the relation to soft theorems", "content": "Memories and soft theorems are mathematically equivalent in the context of the triangle relation [7]. However such relation is not easy to verify beyond the subsubleading order since the soft graviton theorems beyond the subsubleading order do not have an universal factorization property [8, 24]. Correspondingly, the higher order memory formulas (12) are in more complicated forms. The first two orders of memories are shown to be equivalent to the leading soft graviton theorems [6] and subleading soft graviton theorem [12]. The key observation of the equivalence is the fact that the soft factors, after Fourier transform and projection on the null infinity, can be considered as classical fields that satisfy classical equations of motion, namely the Einstein equation. Hence the equivalence between the first two orders of memories and soft theorems can be interpreted as [6,12] Including local stress-energy tensor, the classical equations of motion 7 that the leading factor and the subleading soft factor satisfy are which is the real (electric) part of and which is the imaginary (magnetic) part of 8 Regarding to the third memory, it is supposed to be equivalent to the subsubleading soft factor discovered in [24]. After Fourier transform and projection on the null infinity, the subsubleading soft factor should be interpreted as classical field and the classical equation of motion for the subsubleading soft factor should be related to Equation (31) can be re-organized as to connect to the expression \u015f d u 1 d u 2 \u03b3 z \u00af z \u03c3 0 . However, considering stress-energy tensor from massless particles or localized wave packets which puncture the null infinity at points p u k , z k , \u00af z k q as in [12], direct computation shows that equation (32) does not fulfill when \u015f d u 1 d u 2 \u03b3 z \u00af z \u03c3 0 is replaced by the subsubleading soft factor. The connection at the third order seems to be still of a mystery. However the failure of demonstrating the equivalence between gravitational memories and soft graviton theorems beyond subleading order may not be surprising in the sense that the soft graviton theorems beyond subleading order may have different natures. On the one hand, the soft graviton theorems beyond subsubleading order do not have a universal property [8] in contrast to the first three orders. On the other hand, the double copy relation [25] indicates that a gravity amplitude can be expressed as a sum of the square of color-ordered Yang-Mills amplitudes. The first three soft factors in gravity satisfy precisely the double copy relation [26, 27]. From this point of view, the third order gravitational memory should have intrinsic relation to the first two orders. Unfortunately, we are not able to uncover that in the present work. It is definitely an interesting question that should be addressed elsewhere.", "pages": [6, 7]}, {"title": "6 Conclusion", "content": "We demonstrated that there are infinite towers of gravitational memories which can be associated to the same supertranslation in the NP formalism. An infinite number of conserved quantities can be obtained from each order of the supertranslation charge. We also commented on the relation between the infinite towers of gravitational memories and the infinite towers of soft graviton theorems. The equivalence of those two subjects is not yet clear beyond the second order. As a final remark, it is worthwhile to point out that the analysis performed in gravitational theory can be easily extended to electromagnetism by applying the results in [10] and [28].", "pages": [8]}, {"title": "Acknowledgments", "content": "The author thanks Geoffrey Comp'ere for useful discussions. This work is supported in part by the National Natural Science Foundation of China under Grants No. 11905156 and No. 11935009.", "pages": [8]}, {"title": "A Spin-weighted derivative operators", "content": "The operators \u00f0 and \u00f0 were originally introduced in [29] to replace the covariant derivative on a sphere with metric d S 2 ' 2 \u03b3 z \u00af z d z d \u00af z . The choice in [10] is \u03b3 z \u00af z ' 1 p 1 ' z \u00af z q 2 which is not the unit sphere. The definitions of \u00f0 and \u00f0 on a field \u03b7 are where s is the spin weight of \u03b7 . The spin weight of the relevant fields are listed in Table 1. The operator \u00f0 ( \u00f0 ) will increase (decrease) the spin weight. These two operators do not commute in general. Their commutation relation is where R S is the Ricci scalar of the sphere. Spin s spherical harmonics are defined by acting \u00f0 and \u00f0 on the spherical harmonics Y l,m p l ' 0 , 1 , 2 , . . . ; m ' ' l, . . . , l q . They are given by % From the definition of spin weighted spherical harmonics, one can obtain that where \u03b7 and \u03b6 are with spin weight ' l ' 1 and l ' 1 respectively, and the expression A \u00f0 B has spin weight zero.", "pages": [8, 9]}] |
2016arXiv160906817H | https://arxiv.org/pdf/1609.06817.pdf | <document>
<section_header_level_1><location><page_1><loc_12><loc_86><loc_88><loc_91></location>Fast Evolution and Waveform Generator for Extreme-Mass-Ratio Inspirals in Equatorial-Circular Orbits</section_header_level_1>
<text><location><page_1><loc_43><loc_82><loc_57><loc_83></location>Wen-Biao Han ∗</text>
<text><location><page_1><loc_22><loc_76><loc_78><loc_80></location>Shanghai Astronomical Observatory, Chinese Academy of Sciences 80 Nandan Road, Shanghai, China P.R., 200030</text>
<section_header_level_1><location><page_1><loc_45><loc_73><loc_54><loc_75></location>Abstract</section_header_level_1>
<text><location><page_1><loc_12><loc_43><loc_88><loc_71></location>In this paper we discuss the development of a fast and accurate waveform model for the quasicircular orbital evolution of extreme-mass-ratio-inspirals (EMRIs). This model simply employs the data of a few numerical Teukoulsky-based energy fluxes and waveforms to fit out a set of polynomials for the entire fluxes and waveforms. These obtained polynomials are accurate enough in the entire evolution domain, and much more accurate than the resummation post-Newtonian (PN) energy fluxes and waveforms, especially when the spin of a black hole becomes large. The dynamical equation we adopted for orbital revolution is the effective-one-body (EOB) formalism. Because of the simplified expressions, the efficiency of calculating the orbital evolution with our polynomials is also better than the traditional method which uses the resummed PN analytical fluxes. Our model should be useful in calculation of waveform templates of EMRIs for the gravitational wave detectors such as the evolved Laser Interferometer Space Antenna (eLISA).</text>
<section_header_level_1><location><page_2><loc_12><loc_89><loc_32><loc_91></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_2><loc_12><loc_61><loc_88><loc_86></location>An extreme-mass-ratio inspiral (EMRI) arises following the capture of a compact star with stellar mass (white dwarf, neutron star or black hole) by a supermassive black hole. The orbital radius of such EMRI is about or less than O (10 1 ) Schwarzschild radius of the supermassive black hole. Because of gravitational radiation, the orbit of the small body shrinks toward the central black hole in a long time scale. EMRIs are very important for revealing the properties of supermassive black holes since people may observe the gravitational signals with many wave cycles from the region near the horizon of the black hole. EMRIs are potential sources for eLISA (evolved Laser Interferometer Space Antenna), a space gravitational wave observatory supported by the European Space Agency now [1]. A pathfinder has been launched in 2015 to pave the way for eLISA.</text>
<text><location><page_2><loc_12><loc_23><loc_88><loc_59></location>Due to the match-filter technology employed in gravitational wave detection, people must have a huge amount of theoretical waveform templates with enough accuracy in a very large parameter space. Up to now, there are usually three methods to compute the theoretical waveforms: the first one is the post-Newtonian (PN) approximation, the second one is numerical relativity, and the last one is black hole perturbation theory. As analyzed in a great deal of literatures (for example see [2]), for EMRIs, the PN expansion would lose accuracy greatly in such a highly relativistic region, and even the factorized-resummed PN waveforms in an effective-one-body (EOB-an analytical approach which aims at providing an accurate description of the motion and radiation of coalescing binary black holes with arbitrary mass ratio, see review [3] for details) frame also do not have a good performances for spinning black holes (see for example Ref. [4] and references insides); Numerical relativity still can not handle the binary black hole systems with extreme mass ratio. An recent paper documents the simulation of a case of mass ratio 1:100 (without spin) with only two orbits' evolution [5].</text>
<text><location><page_2><loc_12><loc_7><loc_88><loc_22></location>Therefore, for such small mass-ratio binaries, the black hole perturbation theory is a good tool to study EMRIs (mass-ratio ∼ 10 -7 -10 -4 ) [6-14, 16, 17] and even IMRIs (mass-ratio ∼ 10 -3 -10 -2 ) [15, 18-22]. It means that the small body can be treated as a perturbation of the background field of the central supermassive black hole. The black hole perturbation theory was built by Regge, Wheeler and Zerilli in Schwarzschild spacetime [23, 24] and by Teukolsky in Kerr background [25, 26].</text>
<text><location><page_3><loc_12><loc_47><loc_88><loc_91></location>Usually, the eLISA observes wave-signals over a span of several years. This means that one needs to model the waveforms of EMRIs over a length of several years. Though the cost of calculation of the Teukolsky equation is much less than numerical relativity, it is still unaffordable for evolving O (10 5 -10 6 ) orbits in various parameters. Therefore, researchers are challenged about how to greatly reduce the CPU time of the simulation of EMRI waveform, while at the same time maintaining enough accuracy. For the quasi-circular orbits, Yunes et al used Teukolsky-based energy fluxes to fit higher order pN fluxes [27, 28]. Using the self-force data of Schwarzschild black hole, Lackeos and Burko fitted polynomials mixed PN expressions to do the orbital evolution of IMRIs, but they still numerically solved the Teukolsky equation to get the waveforms [29]. Fujita gave out the expressions of gravitational radiation up to 14th PN order by computing the Teukolsky equation analytically [33]. However, in this paper, we completely abandon using the analytical PN expansions. We use the Teukolsky-based numerical data to directly fit out a set of polynomials for energy fluxes and waveforms. This method is very simple and efficient, and the fitted polynomials can give very highly accurate energy-fluxes and waveforms. In principle, these polynomials are also a kind of PN expansions, but all the coefficients of such 'PN expansions' are obtained numerically from the fitting of the Teukolsky-based data.</text>
<text><location><page_3><loc_12><loc_34><loc_88><loc_46></location>In the next section, we introduce our EOB-Teukolsky codes (ET codes) shortly. The details of our fitting polynomial method are presented in the section 3. In section 4, results and comparisons are shown. Finally, conclusions and remarks are given in section 5. Throughout the paper, we use units G = c = 1 and the metric signature ( -, + , + , +). Distance and time are measured by the central black-hole mass M .</text>
<section_header_level_1><location><page_3><loc_12><loc_27><loc_65><loc_29></location>II. EOB-TEUKOLSKY FREQUENCY-DOMAIN CODES</section_header_level_1>
<text><location><page_3><loc_12><loc_10><loc_88><loc_24></location>Our ET codes include two main parts: one is the EOB dynamics driver, the other is the Teukolsky equation solver. The EOB part gives the orbital parameters to the Teukolsky equation, and then the later one produces waveforms and energy-fluxes. Next the Teukolskybased energy-fluxes source the EOB dynamics to drive the orbital evolution of the small body around the supermassive black hole [21, 34]. A detailed introduction of the EOB dynamics will be presented in the section 4.</text>
<text><location><page_3><loc_14><loc_7><loc_88><loc_8></location>The Teukolsky equation in ET codes can be solved in frequency-domain for inspiralling</text>
<text><location><page_4><loc_12><loc_81><loc_88><loc_91></location>phase and in time-domain for plunge and merge states. In the present work, we focus on only inspiralling process, we just need the frequency-domain Teukolsky calculation. We employ a semi-analytical method to solve the frequency-domain equation which was developed in [36-39] to replace the previous numerical integration method [40].</text>
<text><location><page_4><loc_14><loc_79><loc_83><loc_80></location>After decomposing the Weyl curvature (complex) scalar ψ 4 in a Fourier series [25],</text>
<formula><location><page_4><loc_30><loc_73><loc_88><loc_77></location>ψ 4 = ρ 4 ∫ + ∞ -∞ dω ∑ lm R lmω ( r ) -2 S aω lm ( θ ) e imφ e -iωt , (1)</formula>
<text><location><page_4><loc_12><loc_67><loc_88><loc_71></location>where ρ = -1 / ( r -ia cos θ ), the Teukolsky equation is divided into two parts. One is the radial master equation</text>
<formula><location><page_4><loc_31><loc_62><loc_88><loc_65></location>∆ 2 d dr ( 1 ∆ dR lmω dr ) -V ( r ) R lmω = -T lmω ( r ) , (2)</formula>
<text><location><page_4><loc_12><loc_56><loc_88><loc_60></location>where T lmω ( r ) is the source term which is connected with the energy-momentum tensor of the test particle around a black hole, and the potential is</text>
<formula><location><page_4><loc_33><loc_51><loc_88><loc_54></location>V ( r ) = -K 2 +4 i ( r -M ) K ∆ +8 iωr + λ, (3)</formula>
<text><location><page_4><loc_12><loc_46><loc_88><loc_49></location>where K = ( r 2 + a 2 ) ω -ma, λ = E lm + a 2 ω 2 -2 amw -2. The other is the angular equation</text>
<formula><location><page_4><loc_21><loc_38><loc_88><loc_46></location>1 sin θ d dθ ( sin θ d -2 S aω lm dθ ) + [ ( aω ) 2 cos 2 θ +4 aω cos θ -( m 2 -4 m cos θ +4 sin 2 θ ) + E lm ] -2 S aω lm = 0 , (4)</formula>
<text><location><page_4><loc_12><loc_35><loc_57><loc_37></location>where -2 S aω lm ( θ ) is the spin-weighted angular function.</text>
<text><location><page_4><loc_12><loc_22><loc_88><loc_34></location>If T lmω ( r ) = 0, Eq. (2) becomes a homogeneous equation, then can be solved quickly and accurately by a semi-analytical numerical method developed by Fujita and Tagoshi [38, 39]. The homogeneous solutions of Teukolsky radial equation are expressed in terms of two kinds of series of special functions. The first one consists of series of hypergeometric functions and is convergent at the horizon</text>
<formula><location><page_4><loc_30><loc_16><loc_88><loc_19></location>R H lmω = e iglyph[epsilon1]κx ( -x ) -s -i ( glyph[epsilon1] + τ ) / 2 (1 -x ) i ( glyph[epsilon1] -τ ) / 2 p in ( x ) , (5)</formula>
<text><location><page_4><loc_12><loc_13><loc_65><loc_15></location>where p in is expanded in a series of hypergeometric functions as</text>
<formula><location><page_4><loc_22><loc_7><loc_88><loc_12></location>p in ( x ) = ∞ ∑ n = -∞ a n F ( n + ν +1 -iτ, -n -ν -iτ ; 1 -s -iglyph[epsilon1] -iτ ; x ) , (6)</formula>
<text><location><page_5><loc_12><loc_87><loc_88><loc_92></location>and x = ω ( r + -r ) /glyph[epsilon1]κ , glyph[epsilon1] = 2 Mω , κ = √ 1 -a 2 , τ = ( glyph[epsilon1] -ma ) /κ . The hypergeometric function F ( α, β ; γ ; x ) can be found in mathematic handbooks.</text>
<text><location><page_5><loc_12><loc_81><loc_88><loc_85></location>The second one consists of series of Coulomb wave functions which is convergent at infinity. The homogeneous solution of Teukolsky equation is</text>
<formula><location><page_5><loc_34><loc_77><loc_88><loc_79></location>R C = z -1 -s (1 -glyph[epsilon1]κ/z ) -s -i ( glyph[epsilon1] + τ ) / 2 f ν ( z ) , (7)</formula>
<text><location><page_5><loc_12><loc_74><loc_67><loc_76></location>where f ν ( z ) is expressed in a series of Coulamb wave functions as</text>
<formula><location><page_5><loc_27><loc_68><loc_88><loc_73></location>f ν ( z ) = ∞ ∑ -∞ ( -i ) n ( ν +1+ s -iglyph[epsilon1] ) n ( ν +1 -s + iglyph[epsilon1] ) n a n F n + ν ( -is -glyph[epsilon1], z ) , (8)</formula>
<text><location><page_5><loc_12><loc_63><loc_88><loc_68></location>and z = ω ( r -r -) , ( a ) n = Γ( a + n ) / Γ( a ), F N ( η, z ) is a Coulomb wave function. The outgoing homogeneous solution can be expressed in Coulomb wave functions as,</text>
<formula><location><page_5><loc_39><loc_60><loc_88><loc_62></location>R ∞ lmω = A ν -z -1 -2 s e i ( z + glyph[epsilon1] ln z ) , (9)</formula>
<text><location><page_5><loc_12><loc_56><loc_34><loc_58></location>where the coefficient A ν -is</text>
<formula><location><page_5><loc_24><loc_50><loc_88><loc_55></location>A ν -= 2 -1 -s + iglyph[epsilon1] e -πglyph[epsilon1]/ 2 e -iπ ( ν +1 -s ) / 2 + ∞ ∑ -∞ ( -1) n ( ν +1+ s -iglyph[epsilon1] ) n ( ν +1 -s + iglyph[epsilon1] ) n f ν n . (10)</formula>
<text><location><page_5><loc_12><loc_43><loc_88><loc_50></location>Note that Eqs.(6-10) involve a parameter ν , the so-called renormalized angular momentum. The key part of this semi-analytical method is to search the renormalized angular momentum numerically (see [38, 39] for details).</text>
<text><location><page_5><loc_12><loc_38><loc_88><loc_42></location>With the R ∞ lmω ( r ) and R H lmω ( r ) in hand, take into the source term T lmω (for an equatorialcircular case, see our previous work [40]),</text>
<formula><location><page_5><loc_21><loc_29><loc_88><loc_36></location>T lmω ( r ) = ∫ dt ∆ 2 { ( A nn 0 + A n ¯ m 0 + A ¯ m ¯ m 0 ) δ ( r -r 0 )+ ∂ r [( A n ¯ m 1 + A ¯ m ¯ m 1 ) δ ( r -r 0 )] + ∂ 2 r [ A ¯ m ¯ m 2 δ ( r -r 0 )] } . (11)</formula>
<text><location><page_5><loc_14><loc_27><loc_21><loc_28></location>we have</text>
<formula><location><page_5><loc_37><loc_22><loc_88><loc_26></location>Z H lm = π iω m B in lmω T r I H ( r 0 ) , (12)</formula>
<formula><location><page_5><loc_37><loc_18><loc_88><loc_22></location>Z ∞ lm = πB hole iω m B in lmω D ∞ lmω T r I ∞ ( r 0 ) , (13)</formula>
<text><location><page_5><loc_47><loc_13><loc_50><loc_14></location>dR</text>
<text><location><page_5><loc_50><loc_14><loc_51><loc_15></location>H,</text>
<text><location><page_5><loc_51><loc_14><loc_53><loc_15></location>∞</text>
<text><location><page_5><loc_50><loc_13><loc_52><loc_13></location>lmω</text>
<text><location><page_5><loc_49><loc_11><loc_51><loc_12></location>dt</text>
<text><location><page_5><loc_69><loc_14><loc_70><loc_14></location>2</text>
<text><location><page_5><loc_68><loc_13><loc_69><loc_14></location>d</text>
<text><location><page_5><loc_70><loc_13><loc_72><loc_14></location>R</text>
<text><location><page_5><loc_72><loc_14><loc_73><loc_15></location>H,</text>
<text><location><page_5><loc_73><loc_14><loc_75><loc_15></location>∞</text>
<text><location><page_5><loc_72><loc_13><loc_74><loc_13></location>lmω</text>
<text><location><page_5><loc_72><loc_11><loc_73><loc_12></location>2</text>
<text><location><page_5><loc_70><loc_11><loc_72><loc_12></location>dt</text>
<text><location><page_5><loc_79><loc_14><loc_81><loc_15></location>]</text>
<text><location><page_5><loc_14><loc_15><loc_19><loc_17></location>where</text>
<text><location><page_5><loc_20><loc_14><loc_21><loc_15></location>[</text>
<text><location><page_5><loc_13><loc_11><loc_14><loc_13></location>I</text>
<text><location><page_5><loc_15><loc_13><loc_16><loc_14></location>H,</text>
<text><location><page_5><loc_16><loc_13><loc_17><loc_14></location>∞</text>
<text><location><page_5><loc_14><loc_12><loc_17><loc_12></location>lmω</text>
<text><location><page_5><loc_18><loc_12><loc_20><loc_13></location>=</text>
<text><location><page_5><loc_21><loc_12><loc_23><loc_13></location>R</text>
<text><location><page_5><loc_23><loc_13><loc_24><loc_14></location>H,</text>
<text><location><page_5><loc_24><loc_13><loc_26><loc_14></location>∞</text>
<text><location><page_5><loc_23><loc_12><loc_25><loc_12></location>lmω</text>
<text><location><page_5><loc_26><loc_12><loc_27><loc_13></location>(</text>
<text><location><page_5><loc_27><loc_12><loc_28><loc_13></location>A</text>
<text><location><page_5><loc_28><loc_12><loc_30><loc_13></location>nn</text>
<text><location><page_5><loc_30><loc_12><loc_30><loc_13></location>0</text>
<text><location><page_5><loc_31><loc_12><loc_32><loc_13></location>+</text>
<text><location><page_5><loc_33><loc_12><loc_34><loc_13></location>A</text>
<text><location><page_5><loc_34><loc_12><loc_35><loc_13></location>¯</text>
<text><location><page_5><loc_34><loc_12><loc_36><loc_13></location>mn</text>
<text><location><page_5><loc_36><loc_12><loc_37><loc_13></location>0</text>
<text><location><page_5><loc_37><loc_12><loc_39><loc_13></location>+</text>
<text><location><page_5><loc_39><loc_12><loc_41><loc_13></location>A</text>
<text><location><page_5><loc_41><loc_12><loc_42><loc_13></location>¯</text>
<text><location><page_5><loc_41><loc_12><loc_42><loc_13></location>m</text>
<text><location><page_5><loc_42><loc_12><loc_43><loc_13></location>¯</text>
<text><location><page_5><loc_42><loc_12><loc_43><loc_13></location>m</text>
<text><location><page_5><loc_43><loc_12><loc_44><loc_13></location>0</text>
<text><location><page_5><loc_44><loc_12><loc_45><loc_13></location>)</text>
<text><location><page_5><loc_45><loc_11><loc_47><loc_13></location>-</text>
<text><location><page_5><loc_53><loc_12><loc_54><loc_13></location>(</text>
<text><location><page_5><loc_54><loc_12><loc_55><loc_13></location>A</text>
<text><location><page_5><loc_55><loc_12><loc_56><loc_13></location>¯</text>
<text><location><page_5><loc_55><loc_12><loc_57><loc_13></location>mn</text>
<text><location><page_5><loc_57><loc_12><loc_58><loc_13></location>1</text>
<text><location><page_5><loc_58><loc_12><loc_60><loc_13></location>+</text>
<text><location><page_5><loc_60><loc_12><loc_62><loc_13></location>A</text>
<text><location><page_5><loc_62><loc_12><loc_63><loc_13></location>¯</text>
<text><location><page_5><loc_62><loc_12><loc_63><loc_13></location>m</text>
<text><location><page_5><loc_63><loc_12><loc_64><loc_13></location>¯</text>
<text><location><page_5><loc_63><loc_12><loc_64><loc_13></location>m</text>
<text><location><page_5><loc_64><loc_12><loc_65><loc_13></location>1</text>
<text><location><page_5><loc_65><loc_12><loc_68><loc_13></location>) +</text>
<text><location><page_5><loc_75><loc_12><loc_76><loc_13></location>A</text>
<text><location><page_5><loc_77><loc_12><loc_77><loc_13></location>¯</text>
<text><location><page_5><loc_76><loc_12><loc_77><loc_13></location>m</text>
<text><location><page_5><loc_78><loc_12><loc_78><loc_13></location>¯</text>
<text><location><page_5><loc_77><loc_12><loc_79><loc_13></location>m</text>
<text><location><page_5><loc_79><loc_12><loc_79><loc_13></location>2</text>
<text><location><page_5><loc_81><loc_10><loc_81><loc_11></location>r</text>
<text><location><page_5><loc_81><loc_10><loc_82><loc_11></location>0</text>
<text><location><page_5><loc_82><loc_10><loc_83><loc_11></location>,θ</text>
<text><location><page_5><loc_83><loc_10><loc_84><loc_11></location>=</text>
<text><location><page_5><loc_84><loc_10><loc_86><loc_11></location>π/</text>
<text><location><page_5><loc_86><loc_10><loc_86><loc_11></location>2</text>
<text><location><page_5><loc_85><loc_8><loc_88><loc_9></location>(14)</text>
<text><location><page_6><loc_12><loc_79><loc_88><loc_91></location>All the quantities ( A nn 0 , A ¯ mn 0 , etc.) shown in the above equations were given explicitly in [9], and the harmonic frequency is ω m = m Ω φ . The calculation of coefficients A nn 0 , A ¯ mn 0 , · · · involves the solution of the angular equation (4). There are several routes to calculate the spin weighted spheroidal function -2 S aω lm . In this paper, we adopt the method described in [9].</text>
<text><location><page_6><loc_12><loc_73><loc_88><loc_78></location>Then the amplitudes Z H, ∞ lm fully determine the fluxes of gravitational radiations to infinity and horizon,</text>
<formula><location><page_6><loc_41><loc_67><loc_88><loc_72></location>˙ E ∞ , H = ∑ lm | Z H , ∞ lm | 2 4 πω 2 m , (15)</formula>
<formula><location><page_6><loc_40><loc_62><loc_88><loc_67></location>˙ L z ∞ , H = ∑ lm m | Z H , ∞ lm | 2 4 πω 3 m , (16)</formula>
<text><location><page_6><loc_12><loc_54><loc_88><loc_61></location>where the overdot stands for d / d t , E and L z mean energy and angular momentum respectively. The notations ∞ and H on E and L z mean the fluxes to infinity and the horizon respectively. The gravitational waveform can be expressed as:</text>
<formula><location><page_6><loc_32><loc_48><loc_88><loc_52></location>h + -ih × = 2 r ∑ lm Z H lmω ω 2 m S aω m lm ( θ ) e -iω m t + imφ . (17)</formula>
<section_header_level_1><location><page_6><loc_12><loc_41><loc_88><loc_45></location>III. FITTING POLYNOMIAL FROM TEUKOLSKY-BASED ENERGY FLUXES AND WAVEFORMS</section_header_level_1>
<text><location><page_6><loc_12><loc_15><loc_88><loc_37></location>As we mentioned, the CPU expense for simulating the EMRI evolution by numerical solving the Teukolsky equation is quite huge. Adopting PN fluxes to do evolution is fast but will lost the accuracy when the small body approaching it's innermost stable circular orbit (ISCO) around the central black hole. In this section, we use the Teukolsky-based flux data of a few points to fit out a set of polynomials for replacing the PN fluxes or original Teukolsky-based fluxes. The idea is very simple: we select a few points between the initial radius and ISCO, the calculate the Teukolsky-based fluxes and waveforms at these points, and then use these data to fit out a set of polynomials. The fitted polynomials are used as fluxes and waveforms, they are functions of the radius r to the black hole.</text>
<text><location><page_6><loc_12><loc_7><loc_88><loc_14></location>We choose 7, 9, 11 and 13 points (Chebshev nodes) to fit out the 6th, 8th, 10th and 12th order polynomials and compare the results with factorized PN ones (of course, one also can use more points to fit out these polynomials). For completeness, we need four set of</text>
<text><location><page_7><loc_12><loc_87><loc_88><loc_91></location>polynomials for the orbital evolution and waveform extraction. Two of them are for fluxes down to the infinity and horizon:</text>
<formula><location><page_7><loc_36><loc_81><loc_88><loc_85></location>˙ E ∞ = n ∑ i =0 a i /r i , ˙ E H = n ∑ i =0 b i /r i , (18)</formula>
<text><location><page_7><loc_12><loc_75><loc_88><loc_79></location>where n is the order of polynomials. For waveforms, from Eq. (17), the term can be fitted by polynomials is</text>
<formula><location><page_7><loc_41><loc_70><loc_59><loc_73></location>H lm ≡ Z H lmω ω 2 m S aω m lm ( θ ) .</formula>
<text><location><page_7><loc_12><loc_64><loc_88><loc_68></location>Considering this term is a complex function, we need to use 2 × ( n +1) fitting polynomials for each ( l, m ) mode:</text>
<formula><location><page_7><loc_31><loc_58><loc_88><loc_62></location>Re[ H lm ] = n ∑ i =0 R i lm /r i , Im[ H lm ] = n ∑ i =0 I i lm /r i , (19)</formula>
<text><location><page_7><loc_12><loc_55><loc_54><loc_56></location>where R i lm and I i lm are the polynomial coefficients.</text>
<text><location><page_7><loc_12><loc_49><loc_88><loc_54></location>For reducing interpolation errors, we use the Chebshev nodes to produce the interpolating points in our model:</text>
<formula><location><page_7><loc_35><loc_44><loc_88><loc_48></location>r i = r b + r a 2 + r b -r a 2 cos 2 i +1 2 n ' +1 π , (20)</formula>
<text><location><page_7><loc_12><loc_37><loc_88><loc_43></location>where r a , r b are the boundaries of the calculation area and n ' is the total number of nodes, and i goes from 0 to n ' -1.</text>
<text><location><page_7><loc_12><loc_7><loc_88><loc_37></location>In Fig. 1 and Fig. 2, we compare the fitted polynomials of different orders with the factorized-resummation PN fluxes. We can find that the 6th polynomials do not give a good performance. And the factorized-resummation PN fluxes which are used in the EOB model perform worst. Both the 10th and 12th order polynomial can give very good fit to the numerical Teukolsky-based fluxes. It is assumed that LISA will observe gravitational waves of EMRIs at the typical frequency ∼ 10 2 Hz and the total wave cycle is about N ∼ 10 5 for 1 yr. Thus, the relative error of energy luminosity required to establish the accuracy for the cycle ∆ N ≤ 1 must be ≤ 10 -5 in circular orbit cases [41]. Therefore, the 6th polynomials cannot satisfy the requirement of accuracy. The 8th one is at the edge of this requirement. The 10th polynomials can meet the requirement well. Considering the simplification and saving CPU time, we decide to use the 10th polynomial to fit energy fluxes and waveform in this paper. We list the 10th polynomial coefficients of energy fluxes in Tab. I and Tab. II</text>
<figure>
<location><page_8><loc_19><loc_73><loc_47><loc_90></location>
</figure>
<figure>
<location><page_8><loc_51><loc_73><loc_79><loc_90></location>
</figure>
<figure>
<location><page_8><loc_19><loc_55><loc_47><loc_72></location>
</figure>
<figure>
<location><page_8><loc_51><loc_55><loc_79><loc_72></location>
<caption>FIG. 1. Comparing the 6th, 8th, 10th, 12th polynomials and factorized PN energy fluxes to infinity for a = 0 . 9 , 0 . 7 , 0 , -0 . 9 (from left to right, top to bottom). ∆ ˙ E ∞ is difference between the fitted polynomials or PN energy fluxes with the accurate numerical Teukolsky date ˙ E ∞ .</caption>
</figure>
<table>
<location><page_8><loc_12><loc_23><loc_87><loc_37></location>
<caption>TABLE I. polynomial parameters for infinity fluxes.</caption>
</table>
<text><location><page_8><loc_12><loc_15><loc_88><loc_19></location>which are used in Fig. 1 and Fig. 2. The polynomial coefficients for calculating (2, 2)-mode waveform displayed in Fig. 6 are listed in Tab. III and IV.</text>
<text><location><page_8><loc_12><loc_7><loc_88><loc_14></location>The polynomials above are obtained from the numerical data of 11 points between ISCO and 10 M + r ISCO . Actually one can choose any range which is interesting for the research to fit out the corresponding polynomials. One can also choose the number of points and</text>
<figure>
<location><page_9><loc_19><loc_73><loc_47><loc_90></location>
<caption>FIG. 2. Comparing the 6th, 8th, 10th, 12th polynomials fitted from energy fluxes to horizon for a = 0 . 9 , 0 . 7 , 0 , -0 . 9 (from left to right, top to bottom). ∆ ˙ E H is difference between the fitted polynomial energy fluxes with the accurate numerical Teukolsky data ˙ E H .</caption>
</figure>
<text><location><page_9><loc_22><loc_71><loc_22><loc_72></location>0</text>
<text><location><page_9><loc_21><loc_71><loc_22><loc_71></location>10</text>
<text><location><page_9><loc_22><loc_69><loc_22><loc_69></location>-2</text>
<text><location><page_9><loc_21><loc_69><loc_22><loc_69></location>10</text>
<text><location><page_9><loc_22><loc_66><loc_22><loc_67></location>-4</text>
<text><location><page_9><loc_21><loc_66><loc_22><loc_67></location>10</text>
<text><location><page_9><loc_22><loc_64><loc_22><loc_64></location>-6</text>
<text><location><page_9><loc_21><loc_64><loc_22><loc_64></location>10</text>
<text><location><page_9><loc_22><loc_62><loc_22><loc_62></location>-8</text>
<text><location><page_9><loc_21><loc_61><loc_22><loc_62></location>10</text>
<text><location><page_9><loc_22><loc_59><loc_22><loc_59></location>-10</text>
<text><location><page_9><loc_21><loc_59><loc_22><loc_59></location>10</text>
<text><location><page_9><loc_22><loc_57><loc_22><loc_57></location>-12</text>
<text><location><page_9><loc_21><loc_56><loc_22><loc_57></location>10</text>
<text><location><page_9><loc_19><loc_65><loc_20><loc_65></location>H</text>
<text><location><page_9><loc_19><loc_64><loc_20><loc_65></location>˙</text>
<text><location><page_9><loc_19><loc_64><loc_20><loc_65></location>E</text>
<text><location><page_9><loc_19><loc_64><loc_20><loc_64></location>/</text>
<text><location><page_9><loc_19><loc_63><loc_20><loc_64></location>H</text>
<text><location><page_9><loc_19><loc_63><loc_20><loc_63></location>˙</text>
<text><location><page_9><loc_19><loc_62><loc_20><loc_63></location>∆</text>
<text><location><page_9><loc_19><loc_63><loc_20><loc_63></location>E</text>
<text><location><page_9><loc_22><loc_56><loc_23><loc_56></location>6</text>
<text><location><page_9><loc_25><loc_56><loc_25><loc_56></location>7</text>
<text><location><page_9><loc_27><loc_56><loc_28><loc_56></location>8</text>
<text><location><page_9><loc_30><loc_56><loc_30><loc_56></location>9</text>
<text><location><page_9><loc_32><loc_56><loc_33><loc_56></location>10</text>
<text><location><page_9><loc_34><loc_56><loc_35><loc_56></location>11</text>
<text><location><page_9><loc_37><loc_56><loc_37><loc_56></location>12</text>
<text><location><page_9><loc_39><loc_56><loc_40><loc_56></location>13</text>
<text><location><page_9><loc_41><loc_56><loc_42><loc_56></location>14</text>
<text><location><page_9><loc_44><loc_56><loc_45><loc_56></location>15</text>
<text><location><page_9><loc_46><loc_56><loc_47><loc_56></location>16</text>
<text><location><page_9><loc_34><loc_55><loc_35><loc_56></location>r/M</text>
<text><location><page_9><loc_54><loc_74><loc_55><loc_75></location>2</text>
<text><location><page_9><loc_58><loc_74><loc_59><loc_75></location>4</text>
<text><location><page_9><loc_62><loc_74><loc_63><loc_75></location>6</text>
<text><location><page_9><loc_66><loc_74><loc_67><loc_75></location>8</text>
<text><location><page_9><loc_70><loc_74><loc_71><loc_75></location>10</text>
<text><location><page_9><loc_74><loc_74><loc_75><loc_75></location>12</text>
<text><location><page_9><loc_78><loc_74><loc_79><loc_75></location>14</text>
<text><location><page_9><loc_54><loc_56><loc_55><loc_56></location>8</text>
<text><location><page_9><loc_58><loc_56><loc_59><loc_56></location>10</text>
<text><location><page_9><loc_62><loc_56><loc_63><loc_56></location>12</text>
<text><location><page_9><loc_66><loc_56><loc_67><loc_56></location>14</text>
<text><location><page_9><loc_70><loc_56><loc_71><loc_56></location>16</text>
<text><location><page_9><loc_74><loc_56><loc_75><loc_56></location>18</text>
<text><location><page_9><loc_78><loc_56><loc_79><loc_56></location>20</text>
<text><location><page_9><loc_66><loc_55><loc_67><loc_56></location>r/M</text>
<table>
<location><page_9><loc_12><loc_23><loc_87><loc_37></location>
<caption>TABLE II. polynomial parameters for horizon fluxes.</caption>
</table>
<text><location><page_9><loc_12><loc_7><loc_88><loc_19></location>the order of polynomials based on the accuracy requirement. However, the polynomials we obtained may be used directly to the neighbor area (except for the area inside the ISCO). In Fig. 3, we show the validity of our polynomials fitted from the data between r ISCO and r ISCO +10 M ) in the further area. We can see that the polynomials for ˙ E ∞ fitted from the numerical data can be used directly to the further area. At the same time, the polynomials</text>
<text><location><page_9><loc_66><loc_73><loc_67><loc_74></location>r/M</text>
<text><location><page_9><loc_58><loc_59><loc_59><loc_60></location>6th</text>
<text><location><page_9><loc_58><loc_59><loc_59><loc_59></location>8th</text>
<text><location><page_9><loc_58><loc_58><loc_59><loc_58></location>10th</text>
<text><location><page_9><loc_58><loc_57><loc_59><loc_58></location>12th</text>
<text><location><page_9><loc_26><loc_59><loc_27><loc_60></location>6th</text>
<text><location><page_9><loc_26><loc_59><loc_27><loc_59></location>8th</text>
<text><location><page_9><loc_26><loc_58><loc_28><loc_58></location>10th</text>
<text><location><page_9><loc_26><loc_57><loc_28><loc_58></location>12th</text>
<text><location><page_9><loc_51><loc_83><loc_52><loc_84></location>H</text>
<text><location><page_9><loc_51><loc_83><loc_52><loc_83></location>˙</text>
<text><location><page_9><loc_51><loc_82><loc_52><loc_83></location>E</text>
<text><location><page_9><loc_51><loc_82><loc_52><loc_82></location>/</text>
<text><location><page_9><loc_51><loc_82><loc_52><loc_82></location>H</text>
<text><location><page_9><loc_51><loc_81><loc_52><loc_81></location>˙</text>
<text><location><page_9><loc_51><loc_81><loc_52><loc_81></location>∆</text>
<text><location><page_9><loc_51><loc_81><loc_52><loc_82></location>E</text>
<text><location><page_9><loc_51><loc_65><loc_52><loc_65></location>H</text>
<text><location><page_9><loc_51><loc_64><loc_52><loc_65></location>˙</text>
<text><location><page_9><loc_51><loc_64><loc_52><loc_65></location>E</text>
<text><location><page_9><loc_51><loc_64><loc_52><loc_64></location>/</text>
<text><location><page_9><loc_51><loc_63><loc_52><loc_64></location>H</text>
<text><location><page_9><loc_51><loc_63><loc_52><loc_63></location>˙</text>
<text><location><page_9><loc_51><loc_62><loc_52><loc_63></location>∆</text>
<text><location><page_9><loc_51><loc_63><loc_52><loc_63></location>E</text>
<text><location><page_9><loc_53><loc_90><loc_54><loc_90></location>0</text>
<text><location><page_9><loc_53><loc_89><loc_53><loc_90></location>10</text>
<text><location><page_9><loc_53><loc_88><loc_54><loc_88></location>-2</text>
<text><location><page_9><loc_53><loc_87><loc_53><loc_88></location>10</text>
<text><location><page_9><loc_53><loc_85><loc_54><loc_86></location>-4</text>
<text><location><page_9><loc_53><loc_85><loc_53><loc_86></location>10</text>
<text><location><page_9><loc_53><loc_83><loc_54><loc_84></location>-6</text>
<text><location><page_9><loc_53><loc_83><loc_53><loc_83></location>10</text>
<text><location><page_9><loc_53><loc_81><loc_54><loc_82></location>-8</text>
<text><location><page_9><loc_53><loc_81><loc_53><loc_81></location>10</text>
<text><location><page_9><loc_53><loc_79><loc_54><loc_79></location>-10</text>
<text><location><page_9><loc_53><loc_79><loc_53><loc_79></location>10</text>
<text><location><page_9><loc_53><loc_77><loc_54><loc_77></location>-12</text>
<text><location><page_9><loc_53><loc_77><loc_53><loc_77></location>10</text>
<text><location><page_9><loc_53><loc_75><loc_54><loc_75></location>-14</text>
<text><location><page_9><loc_53><loc_75><loc_53><loc_75></location>10</text>
<text><location><page_9><loc_53><loc_71><loc_54><loc_72></location>0</text>
<text><location><page_9><loc_53><loc_71><loc_53><loc_71></location>10</text>
<text><location><page_9><loc_53><loc_69><loc_54><loc_70></location>-2</text>
<text><location><page_9><loc_53><loc_69><loc_53><loc_69></location>10</text>
<text><location><page_9><loc_53><loc_67><loc_54><loc_67></location>-4</text>
<text><location><page_9><loc_53><loc_67><loc_53><loc_67></location>10</text>
<text><location><page_9><loc_53><loc_65><loc_54><loc_65></location>-6</text>
<text><location><page_9><loc_53><loc_65><loc_53><loc_65></location>10</text>
<text><location><page_9><loc_53><loc_63><loc_54><loc_63></location>-8</text>
<text><location><page_9><loc_53><loc_63><loc_53><loc_63></location>10</text>
<text><location><page_9><loc_53><loc_61><loc_54><loc_61></location>-10</text>
<text><location><page_9><loc_53><loc_60><loc_53><loc_61></location>10</text>
<text><location><page_9><loc_53><loc_59><loc_54><loc_59></location>-12</text>
<text><location><page_9><loc_53><loc_58><loc_53><loc_59></location>10</text>
<text><location><page_9><loc_53><loc_57><loc_54><loc_57></location>-14</text>
<text><location><page_9><loc_53><loc_56><loc_53><loc_57></location>10</text>
<text><location><page_9><loc_58><loc_78><loc_59><loc_78></location>6th</text>
<text><location><page_9><loc_58><loc_77><loc_59><loc_78></location>8th</text>
<text><location><page_9><loc_58><loc_77><loc_60><loc_77></location>10th</text>
<text><location><page_9><loc_58><loc_76><loc_60><loc_76></location>12th</text>
<table>
<location><page_10><loc_12><loc_73><loc_87><loc_88></location>
<caption>TABLE III. polynomial coefficients for waveform (2,2) mode: real part.TABLE IV. polynomial coefficients for waveform (2,2) mode: imaginary part.</caption>
</table>
<table>
<location><page_10><loc_12><loc_52><loc_87><loc_67></location>
</table>
<text><location><page_10><loc_12><loc_42><loc_88><loc_48></location>for the energy flux down to the horizon do not perform very well in the further area. However, ˙ E H is much less than ˙ E ∞ (only 10 -5 of the latter). In many cases, one can just simply omit ˙ E H .</text>
<text><location><page_10><loc_12><loc_33><loc_88><loc_40></location>In addition, one can also fit the energy fluxes and waveforms by the polynomials with postNewtonian parameter x ≡ ( v/c ) 2 = ( GM Ω /c 3 ) 2 / 3 , then Eqs. (18) and (19) are transferred to</text>
<formula><location><page_10><loc_35><loc_25><loc_88><loc_30></location>˙ E ∞ = n ∑ i =0 a ' i x i , ˙ E H = n ∑ i =0 b ' i x i , (21)</formula>
<formula><location><page_10><loc_31><loc_20><loc_88><loc_25></location>Re[ H lm ] = n ∑ i =0 R ' i lm x i , Im[ H lm ] = n ∑ i =0 I ' i lm x i . (22)</formula>
<text><location><page_10><loc_12><loc_7><loc_88><loc_16></location>The Eqs. (18 - 22) are essentially post-Newtonian expansions. However, all coefficients are obtain from numerical fitting of the Teukolsky-based data to guarantee the accuracy, in contrast to the analytical expressions of the post-Newtonian approximation like as the factorized-resummation ones.</text>
<figure>
<location><page_11><loc_19><loc_73><loc_47><loc_90></location>
</figure>
<figure>
<location><page_11><loc_51><loc_73><loc_79><loc_90></location>
</figure>
<figure>
<location><page_11><loc_19><loc_55><loc_47><loc_72></location>
</figure>
<figure>
<location><page_11><loc_51><loc_55><loc_79><loc_72></location>
<caption>FIG. 3. Reliability of the polynomials listed in Tab. I and Tab. II when they are extendedly used to the further area from the central black hole.</caption>
</figure>
<section_header_level_1><location><page_11><loc_12><loc_41><loc_58><loc_42></location>IV. ORBITAL EVOLUTION AND WAVEFORM</section_header_level_1>
<text><location><page_11><loc_12><loc_18><loc_88><loc_38></location>During the inspiralling process, the orbit of small body is semi-circular, and the frequencydomain Teukolsky based waveform is highly accurate in this process. As discussed in the last section, we use the 10th polynomials to replace the original numerical Teukolsky fluxes and waveforms. For producing the 10th polynomials, firstly we need flux and waveform data of the 11 points during the evolution. These data are calculated by the Teukolsky equation. Once the evolution area is decided, the 11 interpolation points are generated by the Chebyshev nodes in this work. Calculating the Teukolsky-based fluxes and waveforms at 11 points numerically only takes few seconds by a desktop.</text>
<text><location><page_11><loc_12><loc_7><loc_88><loc_16></location>Using the data on these 11 points, we can give out the 10th-order polynomials immediately. With the flux-polynomials at hand, we can use the EOB dynamics to evolve the orbits very fastly. The well-known EOB formalism was first introduced by Buonanno and Damour more than ten years ago to model comparable-mass black hole binaries [42, 43], and was</text>
<text><location><page_12><loc_12><loc_87><loc_88><loc_91></location>also applied in small mass-ratio systems [15, 18-20, 44, 45]. The EOB dynamical evolution equations under radiation reaction for a quasi-circular orbit can be given as [46, 47]</text>
<formula><location><page_12><loc_41><loc_82><loc_88><loc_85></location>˙ r = ∂H EOB ∂p r , (23)</formula>
<formula><location><page_12><loc_40><loc_78><loc_88><loc_81></location>˙ φ = ∂H EOB ∂p φ , (24)</formula>
<formula><location><page_12><loc_40><loc_74><loc_88><loc_77></location>˙ p r = -∂H EOB ∂r + F φ p r p φ , , (25)</formula>
<formula><location><page_12><loc_40><loc_70><loc_88><loc_72></location>˙ p φ = F φ , (26)</formula>
<text><location><page_12><loc_12><loc_64><loc_88><loc_69></location>where F φ = ˙ E/ ˙ φ , and ˙ E is the energy-flux of gravitational radiation. For a non-spinning test particle, the Hamiltonian is</text>
<formula><location><page_12><loc_37><loc_60><loc_88><loc_62></location>H NS = β i p i + α √ µ 2 + γ ij p i p j , (27)</formula>
<text><location><page_12><loc_12><loc_56><loc_88><loc_58></location>where µ = m 1 m 2 /M , m 1 , m 2 are the masses of two bodies respectively, M = m 1 + m 2 and</text>
<formula><location><page_12><loc_44><loc_50><loc_88><loc_55></location>α = 1 √ -g tt , (28)</formula>
<formula><location><page_12><loc_43><loc_47><loc_88><loc_51></location>β i = g ti g tt , (29)</formula>
<formula><location><page_12><loc_43><loc_43><loc_88><loc_47></location>γ ij = g ij -g ti g tj g tt , (30)</formula>
<text><location><page_12><loc_12><loc_40><loc_37><loc_42></location>g µν is the inverse Kerr metric.</text>
<text><location><page_12><loc_12><loc_14><loc_88><loc_39></location>In our previous ET codes, the energy fluxes are obtained from Eq. (15) by calculating the Teukolsky equation numerically at every time step. Then, the fluxes are sourced to the EOB dynamical equations (23-26) to drive the particle inspirals into the central black hole. This method will cost a lot of CPU time on the calculation of the Teukolsky equation. Now we use the flux-polynomials (18) or (21) as the source in the EOB dynamical equation. For getting the coefficients of (18) or (21), we need also to calculate the Teukolsky equation but only at a few points (11 points are chosen in this paper, calculation was finished in few seconds). With the flux-polynomials at hand, the EOB dynamical equations then can be driven in a very fast way. At every time step, we calculate the waveforms by polynomials (19) or (22). We list our numerical algorithm here for clarity:</text>
<unordered_list>
<list_item><location><page_12><loc_12><loc_7><loc_88><loc_11></location>(1) determine the calculating area of the EMRIs by distance ( r 0 , r end ) or GW frequency ( f GW 0 , f GW end );</list_item>
</unordered_list>
<figure>
<location><page_13><loc_19><loc_73><loc_47><loc_90></location>
</figure>
<figure>
<location><page_13><loc_51><loc_73><loc_79><loc_90></location>
<caption>FIG. 4. The difference of the practical orbital frequency and the circular orbital one (the left panel); The radial velocity (the right panel).</caption>
</figure>
<unordered_list>
<list_item><location><page_13><loc_12><loc_54><loc_88><loc_61></location>(2) choose n ' points (Chebyshev nodes) in this area to calculate the Teukolsky-based energy fluxes and waveform, then to fit out n order flux- and waveform- polynomials based on the accuracy requirement ( n ' > n ).</list_item>
<list_item><location><page_13><loc_12><loc_47><loc_88><loc_51></location>(3) calculate the dynamics of the small body by solving the EOB dynamical Eqs. (23)-(26) with the source-term obtained in (2) from the initial point ( r 0 , φ 0 , p r 0 , p φ 0 );</list_item>
<list_item><location><page_13><loc_12><loc_40><loc_88><loc_44></location>(4) compute the waveforms by the polynomials obtain in (2) at every evolution temporal point until to the end.</list_item>
<list_item><location><page_13><loc_12><loc_31><loc_88><loc_37></location>(5) if necessary, we need an iteration method for the correction of non-circular orbit: take the evolved data Ω φ , v r at the N ' points instead of the original Ω circ φ , v circ r = 0 into the step (2) and repeat all the remainning steps.</list_item>
</unordered_list>
<text><location><page_13><loc_12><loc_12><loc_88><loc_30></location>We use the orbital frequency Ω circ φ from circular orbital condition to calculate the Teukolsky equation. However, during the evolution process, the small body has radial velocity and the orbit is not exactly circular. Therefore the practical orbital frequency calculated by (24) is different from Ω circ φ . This is why we need an iteration procedure in the step (5). For massratio µ/M glyph[lessorsimilar] 10 -3 , the iteration is not necessary, but for the one of 10 -2 , we recommend to do the iteration. Please see Fig. 4 for the comparison of the orbital frequency and radial velocity.</text>
<text><location><page_13><loc_12><loc_7><loc_88><loc_11></location>In Fig. 5, as an example, we show the orbital evolution of EMRIs with µ/M = 10 -3 for a = 0. Because the plunge process is dominated by the conservation dynamics, the plunge</text>
<figure>
<location><page_14><loc_20><loc_73><loc_47><loc_90></location>
<caption>FIG. 5. Orbital evolution of the EMRIs (mass ratio 1/1000) for a = 0; the right panel shows the details of the orbit evolution at the final time.</caption>
</figure>
<figure>
<location><page_14><loc_53><loc_73><loc_79><loc_90></location>
</figure>
<text><location><page_14><loc_52><loc_82><loc_53><loc_82></location>y</text>
<text><location><page_14><loc_12><loc_60><loc_78><loc_61></location>orbits passed the ISCO are archive here just by ignoring the radiation-reaction.</text>
<text><location><page_14><loc_12><loc_25><loc_88><loc_59></location>The corresponding waveforms are demonstrated in Fig. 6. All these evolutions are archive in 1 second by one CPU of a desktop. In the final year of the inspiral, an EMRI waveform has 10 5 circles [30]. The computation time of such waveform is less than 300 seconds (single CPU) for an EMRI. However, the complexity of EMRI waveforms makes this procedure challenging. The inspiral waveform depends on 14 different parameters [31]. Based on the analysis of [30], if we assume that only about eight of these 14 parameters affect the phase evolution, it will need 300 × (10 5 ) 8 s to produce all waveform templates with a single CPU! Only when we assume there are two to three parameters, this computation time becomes acceptable by using thousands of CPUs with parallel technology. If we use the standard Teukolsky techniques, it will take about a few tens of days to compute one years evolution by using our codes with one CPU! It is much longer than the new procedure developed in this paper. In this sense, our new technique makes great progress in saving CPU time, though it is still far from practical demands.</text>
<text><location><page_14><loc_12><loc_7><loc_88><loc_24></location>To confirm the validation of our polynomial waveforms, we compare the fitting polynomials with the numerical Teukolsky waveform near the ISCO. We find that for the mass-ratio 1:1000, the match of our model with the numerical waveforms looks quite good (see the left panel of figure 6). For confirmation, based on the matched-filter technology (an optimal method when searching for known signals in noisy data), we use the spectral noise density of LISA [32] to calculate the overlap between our polynomial waveforms and the direct Teukolsky ones. According to the basic set-up in matched filtering, the overlap O a,b between ?two</text>
<figure>
<location><page_15><loc_19><loc_73><loc_47><loc_90></location>
</figure>
<figure>
<location><page_15><loc_51><loc_73><loc_79><loc_90></location>
<caption>FIG. 6. Waveforms (left panel: plus-polarized part of h 22 ) of the orbital evolution in figure 5 for a = 0; right panel: a part of the waveform - the solid blue line is the polynomial waveforms, and the red circles represent the numerical Teukolsky ones. The time-coordinate t = 0 means the moment when the small body arrives at the ISCO.</caption>
</figure>
<text><location><page_15><loc_12><loc_53><loc_47><loc_55></location>time series of signals a and b is defined as,</text>
<formula><location><page_15><loc_41><loc_45><loc_88><loc_50></location>O a,b = ( a | b ) √ ( a | a )( b | b ) , (31)</formula>
<text><location><page_15><loc_12><loc_41><loc_41><loc_43></location>where the product ( a | b ) is given as</text>
<formula><location><page_15><loc_34><loc_35><loc_88><loc_39></location>( a | b ) = 2 ∫ ∞ 0 ˜ a ( f ) ˜ b ∗ ( f ) + ˜ a ∗ ( f ) ˜ b ( f ) S n ( f ) df . (32)</formula>
<text><location><page_15><loc_12><loc_7><loc_88><loc_32></location>The overhead tildes stand for the Fourier transform and the star stands for a complex conjugation. The quantity Sn(f) is the spectral noise density curve taken from [32]. From these equations, the overlaps of the two kinds of waveforms (our waveform polynomials and the Teukolsky one) are 99 . 95% , 97 . 01% , 99 . 76% and 100% for a = 0 . 9 , 0 . 7 , 0 and -0 . 9 respectively. This means that our polynomial waveforms are faithful during the whole evolution process. This also confirm our previous claim: for the mass-ratio glyph[lessorsimilar] 10 -3 , we need not an iteration to correct the quasi-circular approximation. The results of comparison are plotted in Fig. ?? . However, as we have claimed, for mass-ratio around 1:100, we suggest that one should take the step (5) to obtain the more accurate waveforms because of the large radial velocity.</text>
<section_header_level_1><location><page_16><loc_12><loc_89><loc_51><loc_91></location>V. CONCLUSIONS AND DISCUSSIONS</section_header_level_1>
<text><location><page_16><loc_12><loc_77><loc_88><loc_86></location>In this paper, we use the Teukolsky-based fluxes and waveforms at a few points on the evolution route of the EMRI to fit out a set of polynomials for fluxes and waveforms. A circular orbital condition is adopted to solve the orbital parameters for numerical calculation of the Teukolsky equation in a frequency domain.</text>
<text><location><page_16><loc_12><loc_58><loc_88><loc_75></location>Essentially, these flux and waveform polynomials are also a kind of PN expansion but all coefficients are obtained from the fitting of numerical data. Usually the PN coefficients are calculated from analytical expressions, such as the resummation PN waveform and Fujitas 11th PN results for EMRIs [33]. As we can see from Fig. 1, our 10th or 12th-order polynomials are much better than the resummation PN fluxes especially for the spin black hole cases. Comparing the results shown in Fig. 4 of [33], we can find that our results are also more accurate than the 11th PN analytical fluxes which have very long expressions.</text>
<text><location><page_16><loc_12><loc_36><loc_88><loc_56></location>Yunes et al used the numerical Teukolsky-based fluxes to calibrate higher-order PN coefficients and add them to resummed PN analytical expressions [27, 28]. Though their results are quite good, the accuracy of our flux polynomials is a little better than theirs. Furthermore, the fitting of the coefficients of polynomials costs less CPU time than their calibration method. Solving the Teukolsky equation to give a few sets of fluxes and wave- forms by the semi-analytical method introduced in section 2 just needs a few seconds, and the fitting process almost does not add extra CPU time. However, Yunes has mentioned that they need O (10) minutes to complete calibration [28].</text>
<text><location><page_16><loc_12><loc_7><loc_88><loc_35></location>Just for demonstration, in the present paper we use a total of 11 points to fit out a set of 10th-order polynomials for fluxes and waveforms. However, using more points can fit out more accurate polynomials. For examples, 10th or 12th-order polynomials obtained from 20 points will give out better fluxes and waveforms. This will only add a little CPU time (twice of the case of 11 points). With these polynomials at hand, EMRIs can be evolved in a very fast way and the waveforms can be extracted at the same time. The parameters listed in tables (I-III) can be used directly for the GW data analysis. In the present paper, we only fit out the one- dimensional polynomials of r. In principle, one can try to fit out two-dimensional polynomials of both r and a. Unfortunately we find that the coefficients of some polynomials vary suddenly while a just changes a little. We will leave this to future work. For many different a of Kerr black holes, principally we can build a database of flux</text>
<text><location><page_17><loc_12><loc_81><loc_88><loc_91></location>and waveform polynomials by scanning a large number of different Kerr parameters (each a only takes a few seconds to produce the polynomials). Our fast and accurate polynomial model could be useful in the waveform-template calculations for future GW detectors such as eLISA, Taiji [48] and Tianqin [49].</text>
<section_header_level_1><location><page_17><loc_14><loc_75><loc_37><loc_76></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_17><loc_14><loc_70><loc_73><loc_72></location>This work was supported by the NSFC (No. U1431120, No.11273045).</text>
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2024arXiv241016347S | https://arxiv.org/pdf/2410.16347.pdf | <document>
<section_header_level_1><location><page_1><loc_20><loc_83><loc_80><loc_87></location>Domain-Adaptive Neural Posterior Estimation for Strong Gravitational Lens Analysis</section_header_level_1>
<text><location><page_1><loc_22><loc_76><loc_33><loc_77></location>Paxson Swierc 1</text>
<text><location><page_1><loc_19><loc_75><loc_36><loc_76></location>[email protected]</text>
<text><location><page_1><loc_39><loc_76><loc_59><loc_77></location>Marcos Tamargo-Arizmendi 2</text>
<text><location><page_1><loc_38><loc_75><loc_60><loc_76></location>[email protected]</text>
<text><location><page_1><loc_62><loc_76><loc_81><loc_77></location>Aleksandra ´ Ciprijanovi´c 1 , 2</text>
<text><location><page_1><loc_64><loc_75><loc_79><loc_76></location>[email protected]</text>
<text><location><page_1><loc_54><loc_72><loc_56><loc_72></location>1 , 2 ,</text>
<text><location><page_1><loc_56><loc_72><loc_56><loc_72></location>3</text>
<text><location><page_1><loc_44><loc_71><loc_54><loc_72></location>Brian D. Nord</text>
<text><location><page_1><loc_44><loc_70><loc_56><loc_71></location>[email protected]</text>
<text><location><page_1><loc_21><loc_67><loc_79><loc_68></location>1 Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637</text>
<text><location><page_1><loc_22><loc_64><loc_78><loc_67></location>2 Fermi National Accelerator Laboratory, Batavia, IL 60510 3 Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637</text>
<section_header_level_1><location><page_1><loc_46><loc_59><loc_54><loc_60></location>Abstract</section_header_level_1>
<text><location><page_1><loc_24><loc_39><loc_77><loc_57></location>Modeling strong gravitational lenses is prohibitively expensive for modern and nextgeneration cosmic survey data. Neural posterior estimation (NPE), a simulationbased inference (SBI) approach, has been studied as an avenue for efficient analysis of strong lensing data. However, NPE has not been demonstrated to perform well on out-of-domain target data - e.g., when trained on simulated data and then applied to real, observational data. In this work, we perform the first study of the efficacy of NPE in combination with unsupervised domain adaptation (UDA). The source domain is noiseless, and the target domain has noise mimicking modern cosmology surveys. We find that combining UDA and NPE improves the accuracy of the inference by 1-2 orders of magnitude and significantly improves the posterior coverage over an NPE model without UDA. We anticipate that this combination of approaches will help enable future applications of NPE models to real observational data.</text>
<section_header_level_1><location><page_1><loc_18><loc_35><loc_47><loc_36></location>1 Introduction and Related Work</section_header_level_1>
<text><location><page_1><loc_18><loc_9><loc_83><loc_33></location>Galaxy-scale strong gravitational lensing is a cosmic probe that provides key information about dark energy, dark matter, and galaxy evolution [4, 96, 95, 66, 49, 34, 94, 86]. Modern and future cosmic survey experiments - e.g., the Dark Energy Survey (DES) [16, 30], Hyper Suprime-Cam [3, 70], the Kilo-Degree Survey (KiDS) [22, 58], the Rubin Observatory Legacy Survey of Space and Time [46], Euclid [27], JWST [83, 13], and the Nancy Grace Roman Telescope [26, 59, 103] are expected to contain 10 3 - 10 5 lensing systems [74, 87, 17]. Traditional techniques for lens modeling have relied heavily on analytic likelihood-fitting, which is computationally expensive and human-time intensive [61]. Additionally, due to simplifying assumptions in designing the likelihoods, these techniques often lack the capability of modeling non-Gaussian likelihoods and posteriors [61]. However, these techniques have advanced notably in automation and speed [73, 39, 31, 88]. Supervised deep learning-based inference techniques - including neural network regression and the recently reinvigorated simulation-based inference (SBI) [20, 38, 1, 24, 108, 35, 45] like neural posterior estimation (NPE) [75, 36, 102, 107] - have been studied in applications on a wide variety of physics and cosmology topics [21, 51, 79, 80, 8, 41, 64], including strong lensing [47]. Once these models are trained (aka, 'amortized'), these methods are very fast compared to traditional modeling methods [20]. In many areas of cosmology, including strong lensing, when there isn't enough real observational data for training deep learning-based models, realistic simulations are used [93, 71, 9, 11]. Nevertheless, these simulated data can differ significantly from real, observational data - i.e.,</text>
<text><location><page_2><loc_18><loc_73><loc_83><loc_91></location>observational noise, astrophysics, and cosmology. The differences between the simulated training data (source domain) and the real observational data used for analysis (target domain) constitute domain shifts between data distributions that can cause models to favor the source domain [90, 99, 67]. Studies of model misspecification due to domain shift [104, 101, 15, 84] have shown this to be a significant limitation of SBI and its application to out-of-domain data. Domain adaptation (DA) is a class of deep learning techniques that help neural networks adapt to domain shifts so that the feature spaces of the source and target domains align during training [106, 97, 111, 109, 56, 60]. Unsupervised domain adaptation (UDA) does not require labels on the target data [105, 57, 85, 110]. This has been studied as an approach to ameliorate biases due to domain shifts for neural network-based analyses in many problems, including cosmology and strong lensing [14, 113, 23, 112, 91, 52, 29, 114, 81, 25, 33, 53]. In this work, we advance the state of the art by combining NPE and UDA and comparing the performance of NPE-UDA and NPE-only models on strong lensing data in two different domains, which are distinguished by the noise in the images.</text>
<section_header_level_1><location><page_2><loc_18><loc_69><loc_78><loc_70></location>2 Methods: Lensing, Neural Posterior Estimation, Domain Adaptation</section_header_level_1>
<text><location><page_2><loc_17><loc_52><loc_83><loc_67></location>Physics of strong gravitational lensing: When light from a background object encounters a sufficiently massive lensing object on its way to an observer, the image of the background object is significantly magnified and distorted [72, 100]. This warped image is the primary observable data (see Fig. 1(b) for example images). In parametric lens modeling, one can consider > 10 parameters from the background object and the lens that could be inferred from the imaging data [61, 50, 77, 10]. In this study, we infer only five parameters related to the lens: Einstein radius θ E , relative angular positions between the background object and lens ( x , y ), and lens eccentricity moduli ( e l , 1 , e l , 2 ). Like all astronomical data, strong lensing images are subject to observational noise from multiple sources -e.g., atmosphere, sky brightness, CCD gain, number of exposures, exposure time, CCD readout, and photon counting. These noises can add values to pixel counts or cause blurring in the images; they need to be accounted for in model building to avoid systematic bias and large error bars.</text>
<text><location><page_2><loc_18><loc_43><loc_83><loc_51></location>Neural Posterior Estimation (NPE): To infer parameter posterior densities, we employ NPE [36], which uses a CNN-based embedding network to summarize images into features, which are then passed to a Masked Autoregressive Flow (MAF), a combination of an autoregressive model and a normalizing flow [75], to estimate posterior densities. MAF can estimate posterior distributions of arbitrary shape (i.e., non-Gaussian). In the standard NPE-only approach, there is a single loss function L NPE that takes the form of the negative log posterior volume [36].</text>
<text><location><page_2><loc_18><loc_27><loc_83><loc_42></location>Unsupervised Domain Adaptation (UDA): In UDA methods, the source domain data have labels, and the target domain data do not have labels. Common UDA approaches include adversarial methods [91, 69, 40, 54, 48, 32] and distance-based methods [105, 28]. In distance-based methods, the loss is defined as a multi-dimensional distance between latent features from the source and target domain data. In this work, we use distance-based methods, for which the UDA loss function L UDA is the Maximum Mean Discrepancy (MMD) [37]. MMD is a method that calculates the distance between distributions: when applied to the latent feature space, it can be used as a loss function. In [84], MMD was used as a metric to quantify the NPE model misspecification. When MMD is included as a loss during training, it is intended to cause the network to align latent feature spaces for the source and target data; this leads to the extraction of domain-invariant features and enables the model to work well on both.</text>
<text><location><page_2><loc_18><loc_20><loc_82><loc_26></location>Combining NPE and UDA: We combine NPE and UDA methods via their losses. The UDA loss L UDA is calculated using the source and target domain latent features (without labels) at the end of the embedding network. The NPE loss L NPE is calculated using the source data (with labels) at the end of the MAF. The total loss function L Tot = L NPE + β UDA ∗ L UDA is used with gradient descent to update all weights; β UDA is a hyperparameter weighting the MMD loss.</text>
<section_header_level_1><location><page_2><loc_18><loc_16><loc_31><loc_17></location>3 Experiments</section_header_level_1>
<text><location><page_2><loc_18><loc_9><loc_83><loc_14></location>Data: We use the deeplenstronomy [71] software, which is built on lenstronomy [9, 11], to generate simulations of galaxy-scale strong lensing images as if observed in a ground-based survey. We use a single photometric band ( g ), which is sufficient for producing morphological features of a lensing system. Images have a pixel scale of 0 . 263 '' /pixel to match that of DES [30, 2]. During</text>
<table>
<location><page_3><loc_18><loc_75><loc_82><loc_85></location>
<caption>Table 1: Distributions and results for each lensing parameter. The parameters ('params'); (a) prior distributions for training and test sets; (b) the residuals for the NPE-only and the NPE-UDA models applied to the target domain data; (c) the mean residuals for the NPE-UDA model applied to the source domain data.</caption>
</table>
<text><location><page_3><loc_18><loc_51><loc_82><loc_64></location>simulation, the surface brightness of the lensing galaxy is omitted from the images: this exclusion represents a part of the typical lens modeling process in which lens light is removed before the lensed background image is modeled [62]. We use empirically and theoretically motivated uniform priors for distributions of physics parameters of the background object and the lens object. For the background object parameters, which we don't infer, we use the following: Sérsic index n ∼ U (2 , 4) , scale radius R ∼ U (0 . 5 '' , 1 '' ) , two-dimensional eccentricity { e s , 1 , e s , 2 } ∼ U ( -0 . 2 , 0 . 2) ; two-dimensional external shear { γ 1 , γ 2 } ∼ U ( -0 . 05 , 0 . 05) [17, 12]. The apparent magnitude of the background object has a distribution m ∼ U (22 . 5 , 23) , which is faint enough that the noise will be apparent. For the lens object parameters that we infer ( θ E , x , y , e l , 1 , and e l , 2 ), the prior distributions are shown in Table 1.</text>
<text><location><page_3><loc_18><loc_38><loc_82><loc_50></location>Weincur a shift in the domain between the source and the target in terms of image noise characteristics only - i.e., not for the physics parameters. The source data has noise characteristics that represent a relatively noiseless image: read noise is zero e -, CCD gain is 6.083 e -/count, exposure time is 90 seconds (typical modern optical cosmic surveys), number of exposures is 10, magnitude zero point is 30, sky brightness is 23.5 magnitude/arcsec 2 (dimmer than the source light profile), and seeing is 0.9 '' (moderate for modern optical cosmic surveys). In contrast, the target data has noise characteristics that mimic those of the DES: read noise is 7.0 e -, and exposure time, number of exposures, magnitude zero point, sky brightness, and seeing are sampled from empirical distributions [2]; these distributions are encoded in the deeplenstronomy package.</text>
<text><location><page_3><loc_18><loc_25><loc_83><loc_37></location>The training set contains 200,000 images in each domain - source and target. These are drawn from the training priors. The validation and test sets each contain 1,000 images in each of the domains. These are drawn from the test priors. The sbi Macke package holds out 10% of the training data for validation during training and early stopping; that validation set is independent of the one we created. For the lens parameters that we infer, the prior distributions for training are wider than the prior distributions for testing to mitigate biases near the edges of the test distribution (see Table 1). The test set is used for all results and metrics in this paper. Sample lenses from source and target data are shown in Fig. 1(b). All images are 32 × 32 pixels in shape. The data set uses ∼ 3 . 5 GB of storage space. The data used in this project can be provided upon request.</text>
<text><location><page_3><loc_18><loc_9><loc_82><loc_24></location>Model Optimization: We use the sbi Macke package [92], which utilizes PyTorch [76] to perform NPE analyses. For the NPE model, we use an embedding network to summarize the image data before input to the MAF. The embedding network architecture has six convolution blocks (each with a convolution, max-pooling, and batch normalization layer) followed by one dropout (rate is 0.5) and one dense layer with 20 nodes. The MAF has 20 transformation blocks, with 400 hidden features in each. This NPE architecture was introduced in [78]. We experimented with a variety of hyperparameter choices and data sets. We determined that the sbi Macke package defaults most clearly show the models' performances: the batch size, learning rate, optimizer, and early stopping epochs are 50, 0.0005, Adam optimizer [55], and 20, respectively. We set β UDA = 1 . 0 . We discuss computational costs for model training in Appendix F. The code for this work can be provided upon request.</text>
<text><location><page_4><loc_7><loc_60><loc_17><loc_62></location>9HUVLRQglyph<c=3,font=/SFMUHK+font000000002c84aadb>glyph<c=25,font=/SFMUHK+font000000002c84aadb>glyph<c=29,font=/SFMUHK+font000000002c84aadb>glyph<c=3,font=/SFMUHK+font000000002c84aadb></text>
<text><location><page_4><loc_7><loc_59><loc_14><loc_61></location>$XJglyph<c=3,font=/SFMUHK+font000000002c84aadb>glyph<c=22,font=/SFMUHK+font000000002c84aadb>glyph<c=19,font=/SFMUHK+font000000002c84aadb></text>
<section_header_level_1><location><page_4><loc_18><loc_89><loc_75><loc_91></location>4 Results: UDA improves NPE performance on target domain data</section_header_level_1>
<text><location><page_4><loc_18><loc_65><loc_83><loc_87></location>First, we check that the addition of UDA to NPE does not lead to a significant deterioration in model performance on source data compared to NPE alone. For all parameters, the NPE-UDA model is slightly more accurate when applied to the source data than when applied to the target data (Fig. 1(a) and Table 1(c)). Also, the NPE-UDA model has nearly the same degree of calibration on source data as on target data (Fig. 1(d)). Next, the demonstration of performance relies on comparing the NPE-only and NPE-UDA models on target data. The NPE-only model has an average residual (i.e., bias) of 0 . 26 '' for the Einstein radius. This bias is far outside the state-of-the-art uncertainties for traditional modeling techniques, which produce uncertainties at the level of ∼ 0 . 01 '' [65] or, more generally, at ∼ 1 -5% [82, 89]. In contrast, for the NPE-UDA model, the accuracy improves (the average residual reduces) by approximately 88%, 88%, 99%, 98%, and 93% for all five parameters θ E , e l , 1 , e l , 2 , x , y , respectively. Also, in applications to the target domain data, the parameter uncertainties for the NPE-UDA model are very well-calibrated, while those for the NPE-only model are highly overconfident (Fig. 1(d)). This reflects the NPE-only model's bias toward the source domain when applied to the target domain data. Finally, the feature spaces for the NPE-UDA model applied to source and target data are overlapping but not when the NPE-only model is applied to data from those domains (Fig. 1(b) left and right, respectively).</text>
<figure>
<location><page_4><loc_18><loc_29><loc_82><loc_62></location>
<caption>Figure 1: (a) : Two example images from the source (without noise; left) and target (with noise; right) domains. (b) : Feature spaces of the embedding network when models are applied to the source (points are filled circles; all points encompassed by a blue circle) and target (points are filled triangles; all points encompassed by an orange circle) domain data for the NPE-only (left) and NPE-UDA (right) models, respectively. (c) : Residuals on the five lens parameters ( θ E , x , y , e l , 1 , e l , 2 ) for the NPE-only model applied to target data (orange), the NPE-UDA model applied to target data (blue), and the NPE-UDA model applied to source data (pink). Contours show the 68th- and 95th-percentile confidence regions, and the dashed lines show zero residuals. (d) : Posterior coverage on the five lens parameters for the NPE-only model for the NPE-UDA model applied to target data (dashed, color), the NPE-only model applied to target data (solid, color), and the NPE-UDA model applied to source data (solid, black). The boundary between underconfident (upper) and overconfident (lower) is marked by a dotted gray line.</caption>
</figure>
<section_header_level_1><location><page_5><loc_18><loc_89><loc_40><loc_91></location>5 Summary and Outlook</section_header_level_1>
<text><location><page_5><loc_18><loc_75><loc_82><loc_88></location>We show for the first time that (unsupervised) domain adaptation (UDA) enhances simulation-based inference (SBI) models when applied to unlabeled target domain data. We used neural posterior estimation (NPE) to infer five parameters of lensing systems from single-band imaging data. We compare NPE models that have UDA (NPE-UDA) to NPE models that don't have UDA (NPE-only) (§2). We incurred a domain shift between the source and target domains: the source images are nearly noiseless, and the target images have the same noise characteristics as DES (§3). When applied to the target domain, the NPE-UDA model is 1-2 orders of magnitude more accurate than the NPE-only model for all five lens parameters (Fig. 1(c) and Table 1(b)). Similar approaches may significantly improve the accuracy of SBI/NPE models when they are applied to real observational data.</text>
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<section_header_level_1><location><page_18><loc_18><loc_89><loc_55><loc_91></location>Acknowledgments and Disclosure of Funding</section_header_level_1>
<section_header_level_1><location><page_18><loc_18><loc_86><loc_28><loc_87></location>A Funding</section_header_level_1>
<text><location><page_18><loc_18><loc_80><loc_82><loc_84></location>We acknowledge the Deep Skies Lab as a community of multi-domain experts and collaborators who've facilitated an environment of open discussion, idea generation, and collaboration. This community was important for the development of this project.</text>
<text><location><page_18><loc_18><loc_71><loc_83><loc_79></location>Work supported by the Fermi National Accelerator Laboratory, managed and operated by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy. The U.S. Government retains and the publisher, by accepting the article for publication, acknowledges that the U.S. Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for U.S. Government purposes.</text>
<text><location><page_18><loc_18><loc_67><loc_83><loc_69></location>This material is based upon work supported by the Department of Energy under grant No. FNAL 21-25.</text>
<section_header_level_1><location><page_18><loc_18><loc_60><loc_39><loc_61></location>B Author Contributions</section_header_level_1>
<text><location><page_18><loc_18><loc_56><loc_82><loc_58></location>Swierc: Conceptualization, Methodology, Formal analysis, Software, Validation, Investigation, Data Curation, Writing - Original Draft</text>
<text><location><page_18><loc_18><loc_52><loc_83><loc_54></location>Tamargo-Arizmendi: Methodology, Formal analysis, Software, Validation, Investigation, Writing - Review & Editing</text>
<text><location><page_18><loc_18><loc_47><loc_83><loc_50></location>' Ciprijanovi'c: Conceptualization, Methodology, Formal analysis, Writing - Review & Editing, Supervision, Project administration</text>
<text><location><page_18><loc_18><loc_43><loc_82><loc_46></location>Nord: Conceptualization, Methodology, Formal analysis, Resources, Writing - Original Draft, Writing - Review & Editing, Supervision, Project administration, Funding acquisition</text>
<text><location><page_18><loc_18><loc_37><loc_82><loc_40></location>We thank the following colleagues for their insights and discussions during the development of this work: Jason Poh.</text>
<section_header_level_1><location><page_18><loc_18><loc_33><loc_61><loc_35></location>C Attributions: Software and Computing Facilities</section_header_level_1>
<text><location><page_18><loc_18><loc_26><loc_83><loc_32></location>We used the following software packages: Astropy [7, 5, 6], deeplenstronomy [71], Elastic Analysis Facility (EAF) [43], Getdist [63], H5py [18], lenstronomy [9, 11], Matplotlib [44], Numpy [42], Python [98], PyTorch [76], sbi Macke [92], Torch [19], Torchvision [68].</text>
<section_header_level_1><location><page_18><loc_18><loc_23><loc_38><loc_24></location>D Embedding Network</section_header_level_1>
<text><location><page_18><loc_18><loc_18><loc_82><loc_21></location>We use an embedding network to reduce the feature space of the imaging data before the MAF uses it. See Table 2 for the architecture setup, including hyperparameters.</text>
<section_header_level_1><location><page_18><loc_18><loc_15><loc_51><loc_16></location>E Additional Feature Space Inspection</section_header_level_1>
<text><location><page_18><loc_18><loc_9><loc_82><loc_13></location>In the main text, we showed that the feature spaces for the NPE-UDA model on the source and target domains are well-aligned (§4 and Fig. 1(b)) for Einstein radius, and there is a clear monotonic correlation between the feature space and the Einstein radius magnitude. Here, we further inspect</text>
<text><location><page_19><loc_4><loc_42><loc_14><loc_43></location>Version 2:</text>
<text><location><page_19><loc_4><loc_40><loc_11><loc_42></location>Sep 03</text>
<table>
<location><page_19><loc_33><loc_46><loc_67><loc_82></location>
<caption>Table 2: The architecture of the embedding network used in the NPE to compress the image data into summary features. The first column lists the layer type, the second column lists the dimensionality of the output from that layer, and the third column lists the parameters of that layer; k is the kernel size, and s is the stride. The final layer outputs the summary features.</caption>
</table>
<figure>
<location><page_19><loc_19><loc_17><loc_82><loc_40></location>
<caption>Figure 2: Latent space of the embedding network when NPE is applied to the source and target domain test set data for the NPE-only (left) and NPE-UDA (right) models, respectively. This is applied to parameters x (a) , y (b) , e l , 1 (c) , e l , 2 (d) .</caption>
</figure>
<text><location><page_20><loc_18><loc_80><loc_82><loc_91></location>the feature spaces for the NPE-only and NPE-UDA models on the relative positions x and y , and on the lens eccentricity moduli e 1 , 1 and e l , 2 . A non-monotonic correlation exists between the feature space and the parameter of interest for x and y , but not for the lens eccentricities. We experimented with the isomap hyperparameter for the number of neighbors (default is five), and we found that values greater than 20 did not change the visualization. We speculate that while the NPE-UDA model requires the feature spaces to overlap, it prioritizes the correlation with the Einstein radius over other parameters. Additionally, the other parameters are more subtly represented in the images and thus may be more difficult to learn.</text>
<section_header_level_1><location><page_20><loc_18><loc_76><loc_52><loc_77></location>F Computational costs for experiments</section_header_level_1>
<text><location><page_20><loc_18><loc_68><loc_82><loc_74></location>All computing was executed on an NVIDIA A100 GPU with 10GB memory. These computations were performed on the Fermilab Elastic Analysis Facility [EAF; 43]. Training without UDA requires ∼ 4 . 0 hours, while training with UDA requires ∼ 6 . 5 hours. This additional time is primarily due to a) calculating the additional loss function for UDA and b) using twice the amount of data by including the target domain data.</text>
</document> | [] |
2020PhRvD.101l4044C | https://arxiv.org/pdf/2006.05086.pdf | <document>
<section_header_level_1><location><page_1><loc_13><loc_92><loc_87><loc_93></location>Geodesic Congruences and a Collapsing Stellar Distribution in f ( T ) Theories</section_header_level_1>
<text><location><page_1><loc_37><loc_85><loc_62><loc_90></location>Soumya Chakrabarti ∗ Theory Division, Saha Institute of Nuclear Physics, Kolkata 700064, West Bengal, India</text>
<section_header_level_1><location><page_1><loc_43><loc_82><loc_57><loc_83></location>Jackson Levi Said †</section_header_level_1>
<text><location><page_1><loc_25><loc_79><loc_76><loc_82></location>Institute of Space Sciences and Astronomy, University of Malta, Malta and Department of Physics, University of Malta, Malta</text>
<text><location><page_1><loc_43><loc_78><loc_58><loc_79></location>(Dated: June 11, 2020)</text>
<text><location><page_1><loc_18><loc_64><loc_83><loc_77></location>Teleparallel Gravity (TG) describes gravitation as a torsional- rather than curvature-based effect. As in curvature-based constructions of gravity, several different formulations can be proposed, one of which is the Teleparallel equivalent of General Relativity (TEGR) which is dynamically equivalent to GR. In this work, we explore the evolution of a spatially homogeneous collapsing stellar body in the context of two important modifications to TEGR, namely f ( T ) gravity which is the TG analogue of f ( R ) gravity, and a nonminimal coupling with a scalar field which has become popular in TG for its effects in cosmology. We explore the role of geodesic deviation to study the congruence of nearby particles in lieu of the Raychaudhuri equation. We find f ( T ) models that satisfy the null energy condition and describe interesting collapse profiles. In the case of a nonminimally coupled scalar field, we also find potential collapse models with intriguing scalar field evolution profiles.</text>
<section_header_level_1><location><page_1><loc_20><loc_60><loc_37><loc_61></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_39><loc_49><loc_58></location>The ΛCDM cosmological model is demonstrated by overwhelming observational evidence in describing the evolution of the Universe at all scales [1, 2] which is achieved by the inclusion of matter beyond the standard model of particle physics. This appears as dark matter which stabilizes galactic structures [3, 4] in the form of cold dark matter particles, and dark energy which is represented by the cosmological constant [5, 6] and produces late-time accelerated cosmic expansion [7, 8]. On the other hand, despite great efforts, there continue to remain outstanding internal problems in the cosmological constant [9], as well as no direct observations of dark matter particles [10].</text>
<text><location><page_1><loc_9><loc_18><loc_49><loc_39></location>In addition to these issues, the effectiveness of the ΛCDM model has also been called into question in recent years. Primarily, the core critique is rooted in the so-called H 0 tension problem which quantifies the inconsistency between the measured [11, 12] and predicted [13, 14] values of H 0 between early- and late-time observations. Measurements made on the tip of the red giant branch (TRGB, Carnegie-Chicago Hubble Program) have reported a lower tension [15], but ultimately the problem may be clarified by future observations from more exotic sources such as gravitational wave astronomy with observatories such as the LISA mission [16, 17] which have already shown an ability to tackle these measurements [18, 19].</text>
<text><location><page_1><loc_9><loc_15><loc_49><loc_18></location>At its core, the ΛCDM model is made up of modifications to the matter section. However, modifications to</text>
<text><location><page_1><loc_52><loc_37><loc_92><loc_61></location>the gravitational section may also provide a suitable explanation to some of the outstanding problems in modern cosmology. This has come in several forms with modifications to general relativity (GR) (see Ref.[2, 20] and references therein) being the main flavor in which exotic gravity enters cosmology such as in extended theories of gravity [20-22]. Collectively, these models of gravity are bourne out of GR through the common mechanism by which gravitation is expressed, i.e. the curvature associated with the Levi-Civita connection [1]. While the metric quantifies the amount of geometric deformation that gravity produces, its the connection which selects curvature as the property over which this is expressed [23, 24]. This is not the only choice in this regard, while retaining the metricity condition, torsion has become an increasingly popular choice for constructing cosmologically motivated theories of gravity [25-27].</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_36></location>Teleparallel Gravity (TG) embodies the collection of theories of gravity in which gravity is expressed as geometric torsion through the Weitzenbock connection [28]. This connection is torsion-ful and curvatureless, whereas the Levi-Civita connection is curvature-ful and torsion-less. All curvature quantities calculated using the Weitzenbock connection (instead of the Levi-Civita connection) naturally vanishes irrespective of the metric components. Immediately, we can confront the EinsteinHilbert action whose Lagrangian is simply the Ricci scalar, ˚ R (over-circles represent quantities calculated with the Levi-Civita connection), which produces the GR field equations. The identical dynamical equations can be arrived at in TG by replacing this Lagrangian with its so-called torsion scalar, T , counterpart. This is the so-called Teleparallel equivalent of General Relativity (TEGR), and differs from GR only at the level of Lagrangian by a total divergence quantity, B (boundary term).</text>
<text><location><page_2><loc_9><loc_64><loc_49><loc_93></location>In TEGR, the boundary term encapsulates the fourthorder corrections which appear in the action to result in a covariant theory (due to the second-order derivatives in the Einstein-Hilbert action). The impact of this feature is that extensions to TEGR will have a meaningful difference to their Levi-Civita connection counterparts. Principally this will mean that TG will have a broader range of modified theories in which the dynamical equations are second-order rather than the limits imposed by the Lovelock theorem in theories of gravity based on the Levi-Civita connection [29-31]. As an aside, TG also has several interesting properties such as its likeness to Yangmills theory [25] giving it an added particle physics dimension, its possible definition of a gravitational energymomentum tensor [32, 33], and that it does not require the introduction of a Gibbons-Hawking-York boundary term in order to produce a well-defined Hamiltonian formulation [26] making it more regular than GR. Moreover, TG can be constructed without the necessity of the weak equivalence principle [34] unlike GR.</text>
<text><location><page_2><loc_9><loc_36><loc_49><loc_63></location>Keeping to the same reasoning as f ( R ) gravity [20-22], TEGR can be arbitrarily generalized to produce f ( T ) gravity [35-40]. Due to the weakened Lovelock theorem in TG, this will be a generally second-order theory of gravity which is a notable difference to its f ( ˚ R ) gravity analogue. f ( T ) gravity has shown a number of promising results in recent years, in terms of cosmology [26, 41, 42], galactic physics [43] as well as in solar system scale phenomenology [44-47]. However, to fully embrace the possibility of limiting to f ( ˚ R ) cosmological models, we must consider the fuller f ( T, B ) theory of gravity [4853, 53, 54], in which f ( ˚ R ) = f ( -T + B ). f ( T, B ) gravity is an interesting theory due to the decoupling between the second-order torsion scalar and fourth-order boundary term contributions. On the other hand, extensions of TG in which matter is nonminimally coupled have also gained in popularity in recent years [55-62]. These have produced interesting results in cosmology and for compact objects.</text>
<text><location><page_2><loc_9><loc_9><loc_49><loc_34></location>In this work, we consider the geodesic deviation of test particles in these modifications of TG. This kind of study can be very informative for investigating particular models in these extensions of TEGR. We also study the Raychaudhuri equation with a focus on the energy conditions that result from these models. Finally, we apply the results of this work to the homogeneous collapse of stellar matter. The paper is broken into the following section, first the Raychaudhuri equation is reviewed in § . II while TG is briefly discussed in § . III. In § . IV, we explore a collapsing stellar mass distribution and the associated energy conditions for the models being investigated, while we do this again for a particularly interesting nonmimally coupled scalar-tensor model in § . V. Finally in § . VI we discuss the main results and conclusions from this work. In all that follows, Latin indices are used to refer to tangent space coordinates, while Greek indices refer to general manifold coordinates.</text>
<section_header_level_1><location><page_2><loc_55><loc_92><loc_89><loc_93></location>II. THE RAYCHAUDHURI EQUATIONS</section_header_level_1>
<text><location><page_2><loc_52><loc_64><loc_92><loc_90></location>The Raychaudhuri equation [63] offers an efficient avenue by which the tendency for nearby geodesics to converge in a gravitational system can be concisely described. One of the most notable impacts of this scheme was in the focusing theorem. Geodesic focusing [64] is a natural consequence of the Raychaudhuri equation and is a core element of the Hawking-Penrose singularity theorems in GR. In this way, the Raychaudhuri equation essentially implicates the idea that a singularity can simply be a byproduct of symmetries present in the matter distribution under consideration. The equation is a geometrical relation which governs the dynamics of mean separation between a congruence of curves. The equation and it's generalizations have found significant application in gravitational physics, for instance, validation of singularity theorems, gravitational lensing, cracking of self-gravitating compact objects, derivation of the equations of thermodynamics of spacetime.</text>
<text><location><page_2><loc_52><loc_55><loc_92><loc_62></location>In a gravitational system, the proper acceleration represented by d 2 x µ /dτ 2 is an observer dependent quantity and not covariant, and so may vanish for some observers and not for others. For this reason, one must take the acceleration as</text>
<formula><location><page_2><loc_57><loc_51><loc_92><loc_54></location>a µ = u α ; β u β = D 2 x µ Dτ 2 = d 2 x µ dτ 2 +Γ µ νσ u ν u σ , (1)</formula>
<text><location><page_2><loc_52><loc_41><loc_92><loc_50></location>for it to be covariant in nature, where D represents a covariant derivative and u ν = dx µ /dτ represents the four-velocity of a particle in the system. For a congruence of curves x µ ( τ ; λ ), where λ parametrizes the paths that satisfy Eq.(1), the four-velocity u µ ( τ, λ ) can be interpreted as one of the tangent fields together with n µ = dx µ /dλ ( τ, λ ) [65].</text>
<text><location><page_2><loc_52><loc_32><loc_92><loc_39></location>The Raychaudhuri equation is derived by considering points on infinitely close geodesics corresponding to the parametrization values λ and λ + δλ . Within this determination, the trace expansion, rotation tensor and shear are respectively defined as</text>
<formula><location><page_2><loc_62><loc_29><loc_92><loc_31></location>θ = u µ ; µ , (2)</formula>
<formula><location><page_2><loc_60><loc_26><loc_92><loc_28></location>ω αβ = ∇ [ µ u ν ] -˙ u [ µ u ν ] , (3)</formula>
<formula><location><page_2><loc_60><loc_24><loc_92><loc_27></location>σ αβ = ∇ ( α u β ) -1 3 θh αβ -˙ u ( α u ν ) , (4)</formula>
<text><location><page_2><loc_52><loc_9><loc_92><loc_23></location>where the projection tensor is defined as h µν = g µν -u µ u ν . In this scenario, the expansion scalar is defined as the fractional rate of change of volume of a matter distribution measured by a comoving observer defined as the fractional rate of change of volume of a matter distribution measured by a comoving observer. If this derivative is negative along some worldline then the matter distribution must be collapsing. The shear and the rotation tensor measures the distortion and rotation of an initially spherical matter distribution. Now, by considering the</text>
<text><location><page_3><loc_9><loc_90><loc_49><loc_93></location>congruence of timelike geodesics leads to the Raychaudhuri equation which reads as [1, 63]</text>
<formula><location><page_3><loc_10><loc_86><loc_49><loc_89></location>dθ dτ = -1 3 θ 2 + ∇ α a α -σ αβ σ αβ + ω αβ ω αβ -· R αβ u α u β , (5)</formula>
<text><location><page_3><loc_9><loc_69><loc_49><loc_85></location>where · R αβ is the Ricci tensor (determined with the Levi-Civita connection). It is important to note that the appearance of the Levi-Civita connection in the Raychaudhuri equation does not emerge from it being the connection for GR but due to the way that the equation is derived. For this reason, the Ricci tensor continues to be derived using the Levi-Civita connection within TG. Moreover, the general theorems regarding Riemann manifolds continue to hold and so the appearance of the standard gravity Ricci tensor does not cause any consistency conflicts within this regime.</text>
<text><location><page_3><loc_9><loc_59><loc_49><loc_67></location>The evolution equation for the expansion of a congruence of null geodesics defined by a null vector field k µ ( k µ k µ = 0) has a similar form as the Raychaudhuri equation in Eq.(5), but with a factor 1 / 2 rather than 1 / 3, and -R µν k µ k ν instead of -R µν u µ u ν as the last term. Thus, it reads as</text>
<formula><location><page_3><loc_12><loc_55><loc_49><loc_58></location>dθ dτ = -1 2 θ 2 -σ µν σ µν + ω µν ω µν -R µν k µ k ν , (6)</formula>
<text><location><page_3><loc_9><loc_42><loc_49><loc_54></location>where the kinematical quantities θ , σ µν and ω µν are now clearly associated with the congruence of null geodesics. An important point to be emphasized is that Eqs. (5) and (6) are purely geometric statements, and as such they make no reference to any theory of gravitation in that they are general results for Riemann manifolds, and thus would be applicable to all other connection constructs within this framework.</text>
<section_header_level_1><location><page_3><loc_11><loc_37><loc_47><loc_39></location>III. TELEPARALLEL GRAVITY AND ITS EXTENSIONS</section_header_level_1>
<text><location><page_3><loc_9><loc_27><loc_49><loc_35></location>TG is a novel reformulation of gravitation in that the curvature associated with the manifestation of gravity is exchanged with torsion [26, 27, 66] through the replacement of the Levi-Civita connection, ˚ Γ σ µν , with the socalled Weitzenbock connection expressed through [25, 28]</text>
<formula><location><page_3><loc_17><loc_24><loc_49><loc_26></location>Γ σ µν := e σ a ∂ µ e a ν + e σ a ω a bµ e b ν , (7)</formula>
<text><location><page_3><loc_9><loc_9><loc_49><loc_23></location>where e a ρ is the tetrad field ( e µ a being the transpose), and ω a bµ the spin connection (over-circles are used on all quantities calculated with the Levi-Civita connection). In fact, there exists a trinity of possible ways to express gravity through geometry with the third being based on nonmetricity rather than curvature or torsion [23]. In all these cases there exists a limit in which these formulations limit to GR in that they produce the identical dynamical equations (despite having different actions due to the appearance of a boundary term).</text>
<text><location><page_3><loc_52><loc_69><loc_92><loc_93></location>The Weitzenbock connection is the most general linear affine connection that is both curvatureless and satisfies the metricity condition [66]. The exact expression of the Weitzenbock connection depends on the tetrad, e a ρ , and the inertial spin connection, ω a bµ . The tetrad acts as a soldering agent between the general manifold and the tangent (inertial) space which are represented by Greek and Latin indices respectively. The spin connection sustains the invariance of the field equations under local Lorentz transformations (LLTs) [67]. The spin connection is a crucial ingredient which must appear in the field equations due to use of tetrads since they have one inertial index, rather than being an extra degree of freedom of the theory. Together the tetrad and spin connection describe spacetime in TG in the same way that the metric tensor does so in GR, and are thus the fundamental dynamical object of the theory.</text>
<text><location><page_3><loc_52><loc_64><loc_92><loc_68></location>Considering the full breadth of possible LLTs (boosts and rotations), Λ a b , the tetrads are transformed on the tangent space by</text>
<formula><location><page_3><loc_67><loc_61><loc_92><loc_63></location>e ' a µ = Λ a b e b µ , (8)</formula>
<text><location><page_3><loc_52><loc_59><loc_86><loc_60></location>whereas the spin connection transformed as [68]</text>
<formula><location><page_3><loc_65><loc_56><loc_92><loc_57></location>ω a bµ = Λ a c ∂ µ Λ c b , (9)</formula>
<text><location><page_3><loc_52><loc_45><loc_92><loc_54></location>which together preserve the LLTs of the theory as a whole. On the other hand, there also exist so-called good tetrads which organically produce vanishing spin connection components [69, 70]. However, given the LLT of the theory, all consistent tetrad and spin connection pairs will be dynamically equivalent in terms of the field equations they produce.</text>
<text><location><page_3><loc_52><loc_37><loc_92><loc_44></location>In TG, the tetrad embodies the effect of gravity in a similar way as the metric tensor expresses geometric deformation in curvature-based theories of gravity [26, 27]. For consistency, the tetrads observe the relations [25]</text>
<formula><location><page_3><loc_62><loc_34><loc_92><loc_36></location>e a µ e µ b = δ a b , e a µ e ν a = δ ν µ , (10)</formula>
<text><location><page_3><loc_52><loc_26><loc_92><loc_33></location>which form the orthogonality conditions of the tetrad fields. Since the effect of tetrads is to connect the general manifold and its Minkowski space, this can be used to transform between these spaces. One example of this is with the Minkowski metric which transforms as</text>
<formula><location><page_3><loc_59><loc_23><loc_92><loc_24></location>g µν = e a µ e b ν η ab , η ab = e µ a e ν b g µν . (11)</formula>
<text><location><page_3><loc_52><loc_19><loc_92><loc_21></location>The position dependence of these relations is being suppressed for brevity's sake.</text>
<text><location><page_3><loc_52><loc_9><loc_92><loc_19></location>TG is fundamentally distinct from curvature-based descriptions of gravity in that the exchange of the LeviCivita with its analog Weitzenbock connection means that all measures of curvature (such as the Riemann tensor and Ricci scalar) will organically vanish for the torsional case [66]. Thus, TG requires a wholly different formulation on which to quantify the effect of gravity. In</text>
<text><location><page_4><loc_9><loc_89><loc_49><loc_93></location>this setting, torsion is measured through the torsion tensor which is represented as an antisymmetric operation [70]</text>
<formula><location><page_4><loc_23><loc_86><loc_49><loc_88></location>T σ µν := 2Γ σ [ µν ] , (12)</formula>
<text><location><page_4><loc_9><loc_77><loc_49><loc_85></location>which also serves as the field strength of gravitation (square brackets represent the anti-symmetric operator A [ µν ] = 1 2 ( A µν -A νµ )). The torsion tensor transforms covariantly under both diffeomorphisms and LLTs, and observes the anti-symmetry T σ µν = -T σ νµ .</text>
<text><location><page_4><loc_9><loc_69><loc_49><loc_78></location>The torsion tensor, analogous to the Riemann tensor, is a measure of torsion for a gravitational field. However, other important and useful quantities exist in TG. The contorsion tensor is one such quantity measure; this is determined as the difference between the Levi-Civita and Weitzenbock connections [26, 71]</text>
<formula><location><page_4><loc_10><loc_64><loc_49><loc_68></location>K σ µν := Γ σ µν -˚ Γ σ µν = 1 2 ( T σ µ ν + T σ ν µ -T σ µν ) . (13)</formula>
<text><location><page_4><loc_9><loc_59><loc_49><loc_64></location>The contorsion tensor is crucial to relating TG with its curvature-based analogues. Along a similar vein, the socalled superpotential is another TG tensor of central importance which is defined as [25]</text>
<formula><location><page_4><loc_13><loc_53><loc_49><loc_58></location>S µν a := 1 2 ( K µν a -h ν a T αµ α + h µ a T αν α ) . (14)</formula>
<text><location><page_4><loc_9><loc_45><loc_49><loc_48></location>Contracting the torsion and the superpotetial tensors produces the torsion scalar through [27]</text>
<text><location><page_4><loc_9><loc_47><loc_49><loc_54></location>The superpotential may play a critical role in reformulating TG as a gauge current for a gravitational energymomentum tensor [72, 73]. Also, the superpotential observes the anti-symmetry S µν a = -S νµ a .</text>
<formula><location><page_4><loc_23><loc_42><loc_49><loc_44></location>T := S µν a T a µν , (15)</formula>
<text><location><page_4><loc_9><loc_34><loc_49><loc_41></location>which is determined solely by the Weitzenbock connection, in the same way that the Ricci scalar is determined by the Levi-Civita connection. Interestingly, it coincidentally turns out that the Ricci and torsion scalars are equal up to a total divergence term [48, 55]</text>
<formula><location><page_4><loc_18><loc_29><loc_49><loc_33></location>R = ˚ R + T -2 e ∂ µ ( eT σ µ σ ) = 0 , (16)</formula>
<text><location><page_4><loc_9><loc_25><loc_49><loc_29></location>where R is the Ricci scalar as calculated with the Weitzenbock connection which naturally vanishes. Thus, it follows that</text>
<formula><location><page_4><loc_16><loc_20><loc_49><loc_24></location>˚ R = -T + 2 e ∂ µ ( eT σ µ σ ) := -T + B, (17)</formula>
<text><location><page_4><loc_9><loc_9><loc_49><loc_20></location>where ˚ R is the Ricci scalar as determined using the LeviCivita connection, and e is the determinant of the tetrad field, e = det ( e a µ ) = √ -g . Here, B embodies the boundary term. This equivalency alone guarantees that the variation of the torsion and Ricci scalars produce the same dynamical equations. Also, this means that the second- and fourth-order contributions to the Ricci scalar can be decoupled from each other in TG, which</text>
<text><location><page_4><loc_52><loc_90><loc_92><loc_93></location>may have important consequences for producing a more natural generalization of f ( R ) gravity [2, 20, 21].</text>
<text><location><page_4><loc_52><loc_88><loc_92><loc_90></location>Another natural consequence of this equivalency is that the TEGR action can be defined directly as [70]</text>
<formula><location><page_4><loc_56><loc_82><loc_92><loc_86></location>S TEGR = -1 2 κ 2 ∫ d 4 x eT + ∫ d 4 x e L m , (18)</formula>
<text><location><page_4><loc_52><loc_76><loc_92><loc_82></location>where κ 2 = 8 πG , and L m represents the Lagrangian for matter. Despite being described through the tetrad and spin connection, TEGR will produce identical dynamical equations as GR, namely</text>
<formula><location><page_4><loc_58><loc_70><loc_92><loc_75></location>˚ G µν ≡ e -1 e a µ g νρ ∂ σ ( eS ρσ a ) -S σ b ν T b σµ + 1 4 Tg µν -e a µ ω b aσ S σ bν = κ 2 Ψ µν , (19)</formula>
<text><location><page_4><loc_52><loc_59><loc_92><loc_69></location>where Ψ µν is the energy-momentum tensor [1] given by Ψ µ a := δ L m /δe a µ , and ˚ G µν is the Einstein tensor calculated with the Levi-Civita connection. Also, it is important to point out that while the Weitzenbock connection is used in th gravity sector, the Levi-Civita connection continues to feature in the coupling prescription of matter [25, 27].</text>
<text><location><page_4><loc_52><loc_53><loc_92><loc_59></location>Taking the same path of modification as in f ( ˚ R ) gravity, the TEGR Lagrangian can be generalized to f ( T ) gravity [35-39], namely L f ( T ) = ef ( T ). By taking a variation with the tetrad, this results in field equations</text>
<formula><location><page_4><loc_54><loc_45><loc_92><loc_52></location>e -1 ∂ ν ( ee ρ a S µν ρ ) f T -e λ a T ρ νλ S νµ ρ f T + 1 4 e µ a f ( T ) + e ρ a S µν ρ f TT ∂ ν ( T ) + e λ b ω b aν S νµ λ f T = κ 2 e ρ a Ψ µ ρ , (20)</formula>
<text><location><page_4><loc_52><loc_16><loc_92><loc_43></location>which can be contracted with the tetrad or metric tensor depending on whether the index in question is a tangent space or general manifold label. These field equations are generically second-order in nature [26], as well as a number of other similarities to GR such as their associated gravitational waves exhibiting identical polarizations [52, 74]. However, to incorporate a framework on which to compare results with their f ( ˚ R ) gravity analog, we must consider f ( T, B ) gravity [48-53, 53, 54, 75] in which the decoupled second- and fourth-order contributions appear in the torsion scalar and boundary term respectively. In these cases, the limit to f ( ˚ R ) gravity occurs for the consideration f ( T, B ) = f ( -T + B ) = f ( ˚ R ) gravity. Another interesting avenue on which to construct modified teleparallel theories of gravity is to consider nonminimal couplings with matter [55-62]. Given the organically lower-order nature of the torsion scalar means that such modification to gravity may produce novel observational consequences.</text>
<text><location><page_4><loc_52><loc_10><loc_92><loc_16></location>In what follows, we choose frames where the spin connection is allowed to vanish. Since a frame always exists where this is possible, we do not overly limit the applicability of this work. Also, we take units where κ 2 = 1.</text>
<text><location><page_5><loc_9><loc_89><loc_49><loc_93></location>We now focus on the f ( T ) field equations written in Eq.(20) which can be written using only general manifold indices to give</text>
<formula><location><page_5><loc_12><loc_83><loc_49><loc_88></location>e -1 g µσ e a ν ∂ γ ( ee ρ a S σγ ρ ) f T -T ρ γν S γσ ρ g σµ f T + 1 4 g µν f ( T ) + g σµ S σγ ν f TT ∂ γ T = Ψ µν , (21)</formula>
<text><location><page_5><loc_9><loc_79><loc_49><loc_82></location>which follows by raising the inertial index with an inverse tetrad. This gives a trace equation</text>
<formula><location><page_5><loc_15><loc_75><loc_49><loc_78></location>e -1 e a σ ∂ γ ( ee ρ a S σγ ρ ) f T -Tf T + f ( T ) + S σγ σ f TT ∂ γ T = Ψ . (22)</formula>
<text><location><page_5><loc_9><loc_70><loc_49><loc_74></location>Using the TEGR field equations in Eq.(19), we can also write these field equations down using the standard Einstein tensor as</text>
<formula><location><page_5><loc_9><loc_64><loc_49><loc_69></location>˚ G µν f T + 1 4 g µν ( f ( T ) -Tf T ) + g σµ S σγ ν f TT ∂ γ T = Ψ µν , (23)</formula>
<text><location><page_5><loc_9><loc_62><loc_37><loc_63></location>which has an interesting trace equation</text>
<formula><location><page_5><loc_13><loc_57><loc_49><loc_61></location>f ( T ) -( ˚ R + T ) f T + S σγ σ f TT ∂ γ T = Ψ , (24)</formula>
<text><location><page_5><loc_9><loc_54><loc_49><loc_57></location>where the term in parenthesis turns out to be ˚ R + T = B using Eq.(17).</text>
<text><location><page_5><loc_9><loc_50><loc_49><loc_54></location>Given the Einstein tensor definition, namely ˚ G µν := ˚ R µν -1 2 g µν ˚ R , the Ricci tensor dependency on the f ( T ) Lagrangian can be expressed as</text>
<formula><location><page_5><loc_16><loc_37><loc_49><loc_49></location>˚ R µν = 1 2 g µν ( -T + B ) + 1 f T [ Ψ µν -1 4 g µν ( f ( T ) -Tf T ) -g σµ S σγ ν f TT ∂ γ T ] , (25)</formula>
<text><location><page_5><loc_9><loc_32><loc_49><loc_36></location>which can now be used with the Raychaudhuri equations in Eq.(6) to determine the effect of f ( T ) gravity on the congruence of null geodesics.</text>
<section_header_level_1><location><page_5><loc_11><loc_27><loc_47><loc_29></location>IV. A COLLAPSING SPHERICAL STAR IN f ( T ) GRAVITY</section_header_level_1>
<text><location><page_5><loc_9><loc_22><loc_49><loc_25></location>We consider a spatially homogeneous collapsing stellar distribution whose interior is described by the metric</text>
<formula><location><page_5><loc_16><loc_17><loc_49><loc_21></location>ds 2 = dt 2 -a ( t ) 2 [ dx 2 + dy 2 + dz 2 ] , (26)</formula>
<text><location><page_5><loc_9><loc_11><loc_49><loc_18></location>where a ( t ) is the physical radius of the collapsing system. The tetrad choice e a µ = diag(1 , a ( t ) , a ( t ) , a ( t )) is compatible with a vanishing spin connection [68]. On the other hand, we take the energy-momentum contribution to be that of a perfect fluid described by</text>
<formula><location><page_5><loc_10><loc_8><loc_49><loc_10></location>Ψ µν = ( ρ + p ) u µ u ν -p g µν with u µ = (1 , 0 , 0 , 0) , (27)</formula>
<text><location><page_5><loc_52><loc_89><loc_92><loc_93></location>where k µ = (1 , a, 0 , 0) for the null vector field in Eq.(6), p is the fluid pressure and ρ its energy density. Thus, the Raychaudhuri equation can be written as</text>
<formula><location><page_5><loc_53><loc_85><loc_92><loc_88></location>dθ dτ = -θ 2 3 -1 2 f T ( ρ +3 p + f -Tf T -72 H 2 ˙ Hf TT ) , (28)</formula>
<text><location><page_5><loc_52><loc_78><loc_92><loc_84></location>which describes the congruence of neighbouring particle geodesics, and where H = ˙ a/a , and dots refer to derivatives with respect to time, t . The field equations can then be written as</text>
<formula><location><page_5><loc_65><loc_75><loc_92><loc_78></location>ρ = 6 H 2 f T + f 2 , (29)</formula>
<formula><location><page_5><loc_61><loc_72><loc_92><loc_74></location>( ρ + p ) = 24 H 2 ˙ Hf TT -2 ˙ Hf T . (30)</formula>
<text><location><page_5><loc_52><loc_69><loc_92><loc_72></location>Using Eqs. (29) in Eq. (28), and putting θ = 3 ˙ a a , we write</text>
<formula><location><page_5><loc_61><loc_65><loc_92><loc_68></location>dθ dτ = 3 ˙ H = -3( ρ + p ) 2(2 Tf TT + f T ) . (31)</formula>
<text><location><page_5><loc_52><loc_48><loc_92><loc_64></location>Eq. (31) governs the evolution of the time-like congruence, depending on the positivity or negativity of dθ dτ . A negative dθ dτ indicates a throughout collapsing system until θ reaches -∞ , indicating a zero proper volume singularity. However, if dθ dτ changes signature to positive over the course of it's evolution then a collapse of the congruence is halted and the geodesics start to move away from each other. Therefore, the formation of a zero proper volume singularity may be avoided. The onus of avoiding a singularity therefore lies on the behavior of the RHS of Eq. (31).</text>
<text><location><page_5><loc_52><loc_26><loc_92><loc_47></location>If we assume that both the energy density ρ and the isotropic pressure p are positive in nature, the evolution of the congruence and the predictibility of the collapse depends entirely on the nature of (2 Tf TT + f T ), i.e., explicitly dependent on the choice of f ( T ) one makes. If (2 Tf TT + f T ) > 0, the congruence is collapsing and if (2 Tf TT + f T ) < 0, the congruence is expanding. Therefore the predictibility of a collapsing stellar distribution in f ( T ) theories depends on the choice of f ( T ) as a congruence of time-like geodesics would suggest. Using the definition of the torsion scalar in Eq.(15), it follows that T = -6 H 2 , throughout which one can write the RHS of Eq. (31) in terms of f = f ( H ) which simplifies this equation into</text>
<formula><location><page_5><loc_64><loc_22><loc_92><loc_25></location>dθ dτ = 3 ˙ H = 18( ρ + p ) f HH . (32)</formula>
<text><location><page_5><loc_52><loc_13><loc_92><loc_21></location>Thus, the evolution of the congruence depends on the nature of f HH . If f H is a decreasing function of H , the geodesics are imploding towards one another until a singularity is formed. If during the evolution, f H becomes an increasing function of H , the collapse halts and the geodesics start to move apart from each other.</text>
<text><location><page_5><loc_52><loc_9><loc_92><loc_11></location>Positivity of both the energy density and pressure implies that ( ρ + p ) > 0, which is the usual Null Energy</text>
<text><location><page_6><loc_9><loc_90><loc_49><loc_93></location>Condition (NEC) in the context of GR. In the context of an f ( T ) theory the NEC can be written from Eq. (6) as</text>
<formula><location><page_6><loc_24><loc_87><loc_49><loc_89></location>· R µν k µ k ν ≥ 0 . (33)</formula>
<text><location><page_6><loc_9><loc_73><loc_49><loc_86></location>The NEC is a general result of Riemann manifolds rather than GR which is why it continues to be expressed in terms of th standard gravity Ricci tensor. It is for this reason that it retains its dependence on the Levi-Civita connection rather than the Weitzenock connection. This form of the NEC statement is essentially a coordinateinvariant way for an unfixed geometrical theory of gravitation. For a general f ( T ) gravity this can be written as</text>
<formula><location><page_6><loc_19><loc_69><loc_49><loc_72></location>1 f T ( ρ + p -24 H 2 ˙ Hf TT ) ≥ 0 , (34)</formula>
<text><location><page_6><loc_9><loc_48><loc_49><loc_68></location>where the positivity of both the energy density and pressure helps one to ensure that the NEC is also satisfied throughout the evolution. In the following works, we plot the LHS of the NEC in a simple example to discuss the evolution of the matter distribution for a simple collapsing exact solution. It is quite natural in a study of gravitational collapse to plot the NEC as a function of time as was first done in the vintage paper of Kolassis, Santos and Tsoubelis [76]. The idea is to write the LHS of NEC as a function of time for different collapsing shells labelled by different values of radial distance r. In case of a spatially homogeneous meric as in our case, this makes the LHS of NEC a function of time (See also [77]).</text>
<text><location><page_6><loc_9><loc_34><loc_49><loc_46></location>Since we are completely avoiding the rigorous avenue of finding an exact solution, more analysis relies heavily on the amount of information that can be extracted from the Raychaudhuri Eq. (31). There are a few different ways of analysing further, for instance, one can choose a certain behavior of dθ dτ and solve the resulting equation. As the simplest possible example, let us assume that (2 Tf TT + f T ) = δ , where δ is a constant. This can be solved straightaway to write</text>
<formula><location><page_6><loc_19><loc_31><loc_49><loc_32></location>f ( T ) = δT +2 C 1 T 1 / 2 + C 2 . (35)</formula>
<text><location><page_6><loc_9><loc_24><loc_49><loc_30></location>Thus, depending on a positive or a negative δ , a time-like congruence under the scope of an f ( T ) theory (Eq. (35)) avoids a formation of zero proper volume singularity or not.</text>
<text><location><page_6><loc_9><loc_15><loc_49><loc_22></location>The second method is to choose a particular viable f ( T ) model from literature, and explore what constraints te Raychaudhuri equation enforces upon the model parameters. To this end, we choose two popular models of f ( T ) gravity. For a power law f ( T ) theory [37, 41]</text>
<formula><location><page_6><loc_21><loc_12><loc_49><loc_14></location>f ( T ) = T + α ( -T ) n . (36)</formula>
<text><location><page_6><loc_9><loc_9><loc_49><loc_11></location>From Eqs. (31) and (36), we find the relation governing collapsing or expanding nature of the geodesic con-</text>
<text><location><page_6><loc_52><loc_90><loc_92><loc_93></location>gruence. An initially collapsing congruence remains collapsing if</text>
<formula><location><page_6><loc_59><loc_87><loc_92><loc_89></location>1 -[2 αn ( n -1) + αn ]( -T ) n -1 > 0 , (37)</formula>
<text><location><page_6><loc_52><loc_85><loc_90><loc_86></location>and it changes nature from collapsing to expanding if</text>
<formula><location><page_6><loc_59><loc_82><loc_92><loc_84></location>1 -[2 αn ( n -1) + αn ]( -T ) n -1 < 0 . (38)</formula>
<text><location><page_6><loc_52><loc_80><loc_91><loc_81></location>Similarly, for an exponential f ( T ) model, given by [38]</text>
<formula><location><page_6><loc_61><loc_75><loc_92><loc_79></location>f ( T ) = T + αT 0 [ 1 -e -p √ T T 0 ] , (39)</formula>
<text><location><page_6><loc_52><loc_71><loc_92><loc_74></location>we find the relation governing collapsing or expanding nature of the geodesic congruence using Eq. (31). If</text>
<formula><location><page_6><loc_64><loc_67><loc_92><loc_70></location>1 -αp 2 2 e -p √ T T 0 > 0 , (40)</formula>
<text><location><page_6><loc_52><loc_61><loc_92><loc_66></location>the congruence is collapsing towards a formation of singularity. If during the course of it's evolution, the LHS of the above equation changes signature and satisfies</text>
<formula><location><page_6><loc_64><loc_57><loc_92><loc_60></location>1 -αp 2 2 e -p √ T T 0 < 0 , (41)</formula>
<text><location><page_6><loc_52><loc_54><loc_92><loc_56></location>the congruence becomes expandng, completely avoiding any formation of zero proper volume.</text>
<text><location><page_6><loc_52><loc_46><loc_92><loc_52></location>To elaborate a little more rigorously, we also try to explore from a different point of view and propose that an initially collapsing congruence can be described by a spatially homogeneous metric with a scale factor</text>
<formula><location><page_6><loc_62><loc_44><loc_92><loc_45></location>a ( t ) = δ 1 e αf ( t ) + δ 2 e -αf ( t ) . (42)</formula>
<text><location><page_6><loc_52><loc_22><loc_92><loc_42></location>The form in Eq. (42) defines a kind of parametrization and we argue that this form can give a general evolution of all the possible outcomes of an initially collapsing stellar distribution for different form of the function f ( t ) and for different values of the parameters δ 1 , δ 2 and α . f ( t ) is a continuous and differentiable function of time, which can be exactly determined by writing an exact solution, if possible, from the modified field equations. However, finding an exact solution of the field equations may be extremely non-trivial and not part of the purpose of this work. We mean to comment on the restrictions one must impose on the function f ( t ) and the parameters such that different time evolution paths are described. Using Eq. (42) in Eq. (31) we can write</text>
<formula><location><page_6><loc_55><loc_17><loc_92><loc_21></location>dθ dτ = 3 [ α f ( 1 -4 δ 1 δ 2 a 2 ) 1 / 2 + 4 δ 1 δ 2 α 2 ˙ f 2 a 2 ] . (43)</formula>
<text><location><page_6><loc_52><loc_9><loc_92><loc_16></location>Any chance of a bounce of the initially collapsing star depends on the existence of zero-s of the RHS of Eq. (43). There may be one or more of such zeros where the evolution changes nature to expanding from collapsing and vice-versa. Depending on the number of zeros,</text>
<text><location><page_7><loc_9><loc_85><loc_49><loc_93></location>an initially collapsing star can continue to collapse, it can bounce after a finite time, or it can suffer a multiple of collapse-and-bounce segments and eventually become oscillatory in nature (See Fig. 1 for graphical representations for different choices of f ( t )). Putting dθ dτ = 0 in Eq. (43) therefore yields a 'critical condition' written as</text>
<formula><location><page_7><loc_20><loc_79><loc_49><loc_83></location>f = -˙ f 2 4 αδ 1 δ 2 ( a 2 -4 δ 1 δ 2 ) 1 / 2 . (44)</formula>
<text><location><page_7><loc_9><loc_76><loc_49><loc_79></location>We can summarize the possible outcomes of the collapse from the critical condition Eq. (44) as follows</text>
<unordered_list>
<list_item><location><page_7><loc_11><loc_69><loc_49><loc_75></location>1. The value of f crossing the limit of -˙ f 2 4 αδ 1 δ 2 ( a 2 -4 δ 1 δ 2 ) 1 / 2 is a signature of change from collapse to a probable expansion/dispersal or vice versa for a collapsing star parametrized by Eq. (42).</list_item>
<list_item><location><page_7><loc_11><loc_61><loc_49><loc_68></location>2. Since ˙ f and f are real functions of time, the RHS of Eq. (44) must also be real. If any one of δ 1 or δ 2 is negative, ( a 2 -4 δ 1 δ 2 ) 1 / 2 is always real and a ( t ) can evolve on to a zero proper volume, forming a zero proper volume singularity.</list_item>
<list_item><location><page_7><loc_11><loc_41><loc_49><loc_59></location>3. However, if δ 1 and δ 2 are both positive, RHS of Eq. (44) is real if and only if ( a 2 -4 δ 1 δ 2 ) > 0. This predicts a different outcome of the collapse and notes it's sensitivity on the choice of initial parameters. In such a case, the minimum allowed value of a ( t ) is a critical = 2 δ 1 / 2 1 δ 1 / 2 2 . Beyond this critical radius no more shrinking of the congruence is allowed and a bounce/dispersal must take place. Thus, the effective modification of GR due to the non-conservation of energy-momentum distribution opens up more possibilities regarding the end-state of gravitational collapse as compared to standard GR.</list_item>
</unordered_list>
<text><location><page_7><loc_9><loc_31><loc_49><loc_39></location>As an example, we present a particular exact solution of the field equations in Eq. (29) for f ( T ) = T + α ( -T ) 2 . This is a special case of a more general theory given by f ( T ) = T + α ( -T ) n . For a perfect fluid given by p = ωρ describing the collapsing fluid distribution, manipulating the field equations we write</text>
<formula><location><page_7><loc_11><loc_25><loc_49><loc_30></location>˙ H { 48 αH 2 -2 + 24 αH 2 } -6(1 + ω ) H 2 ( 1 -12 αH 2 ) +3 H 2 (1 + ω ) -18(1 + ω ) αH 4 = 0 . (45)</formula>
<text><location><page_7><loc_9><loc_21><loc_49><loc_25></location>During the final phases of the collapse, ˙ a →∞ and a → 0, therefore H >> 1 α . In this limit the above equation can be written as</text>
<formula><location><page_7><loc_17><loc_17><loc_49><loc_20></location>˙ H = -3 4 (1 + ω ) H 2 + 1 24 α (1 + ω ) . (46)</formula>
<text><location><page_7><loc_9><loc_13><loc_49><loc_16></location>A first integral of the above equation can be calculated as</text>
<formula><location><page_7><loc_17><loc_8><loc_49><loc_12></location>˙ a = -{ a 0 a -m + (1 + ω ) a 2 24 α ( m +2) } 1 2 . (47)</formula>
<figure>
<location><page_7><loc_55><loc_59><loc_89><loc_94></location>
<caption>FIG. 1: Evolution of the radius of two-sphere as given in Eq. (48) : (i) Top curve α = 0 . 01, C = 1 and a 0 = 100 ; (ii) Bottom curve α = 0 . 01, C = 1 and a 0 = -100.</caption>
</figure>
<text><location><page_7><loc_52><loc_44><loc_92><loc_49></location>a 0 is a parameter related to the initial value of ˙ a and m = 3 2 (1 + ω ) -2. As a simple example we solve the first integral equation for ω = 1, which implies m = 1. For this we write the evolution of the radius of two-sphere as</text>
<formula><location><page_7><loc_59><loc_37><loc_92><loc_42></location>a ( t ) = e 6 C -t 6 √ α 2 2 / 3 [ e 3 C √ α -36 a 0 αe t 2 √ α ] 2 3 . (48)</formula>
<text><location><page_7><loc_52><loc_13><loc_92><loc_37></location>In Fig. 1 we plot the evolution as a function of time. The graph on top of the figure shows a plot for a 0 > 0. It is clear to note that the collapsing fluid reaches a zero proper volume at a finite future. The time of formation of this zero proper volume singularity may vary depending on the choice of the functional form of the theory, i.e., α , however, the qualitatiove behavior remains the same. In the graph below, the evolution is shown for a 0 < 0. It is clear that there is no formation of zero proper volume singularity in this case, as the collapsing fluid bounces indefinitely after reaching a minimum cutoff volume. The parameter a 0 is a critical parameter of the system whose signature determines the fate of the collapsing system. Using the equation for the NEC, as in Eq. (34), we check if the collaping fluid satisfies the NEC. This essentially ensures a positive energy density and that the speed of energy flow of matter is less than the speed of light.</text>
<text><location><page_7><loc_52><loc_9><loc_92><loc_13></location>In Fig. 2, we plot the NEC as a function of time, using the exact solution in Eq. (48), for two different initial conditions leading to collapse (top graph) and bounce</text>
<figure>
<location><page_8><loc_12><loc_60><loc_46><loc_94></location>
<caption>FIG. 2: Evolution of the NEC as given in Eq. (34) : (i) Top curve shows the evolution for collapse, i.e., α = 0 . 01, C = 1 and a 0 = 100 ; (ii) Bottom curve shows the evolution for bounce, i.e., α = 0 . 01, C = 1 and a 0 = -100.</caption>
</figure>
<text><location><page_8><loc_9><loc_39><loc_49><loc_49></location>(bottom graph). While the NEC is perfectly satisfied for a collapse to zero proper volume singularity, usually an indefinite bounce is associated with a violation of NEC. Eventually the bounce leads to a complete dispersal of all the matter distribution inside, as shown in the bottom graph, when ρ + p ∼ 0. This collapse and dispersal is extremely suggestive of a critical behavior in the system.</text>
<section_header_level_1><location><page_8><loc_9><loc_35><loc_48><loc_36></location>V. NON-MINIMAL TELEPARALLEL GRAVITY</section_header_level_1>
<text><location><page_8><loc_9><loc_29><loc_49><loc_33></location>In this section we take a modified teleparallel action where we introduce a nonminimal coupling with a scalar field as follows</text>
<formula><location><page_8><loc_9><loc_23><loc_49><loc_28></location>S = ∫ d 4 xe [ T 2 κ 2 + 1 2 ( ∂ µ φ∂ µ φ + ξTφ 2 ) -V ( φ ) + L m ] . (49)</formula>
<text><location><page_8><loc_9><loc_9><loc_49><loc_21></location>Minimally coupled scalar fields are also popular in the context of TG and such models have been studied quite extensively in the context of cosmic acceleration and reconstruction [31, 55, 78-83]. However, it is already established that a minimally coupled scalar field endowed with an interaction potential essentially serves as a fluid distribution, and will therefore do no significant change in outcome as far as the Raychaudhury equation is concerned. A non-minimal coupling on the other hand,</text>
<text><location><page_8><loc_52><loc_82><loc_92><loc_93></location>inspires an analogy of scalar-geometry interaction in strong gravity limit. Although in the nonminimal case one could use a generalized function of the torsion scalar, we keep the standard T for simplicity. We also note that the action in Eq. (49) with the torsion formulation of GR is similar to the standard non-minimal quintessence models of cosmology where the scalar field couples to the Ricci scalar.</text>
<text><location><page_8><loc_52><loc_77><loc_92><loc_80></location>Variation of action in Eq. (49) with respect to the tetrad fields yields the equation of motion</text>
<formula><location><page_8><loc_52><loc_67><loc_92><loc_76></location>( 2 κ 2 +2 ξφ 2 )[ e -1 ∂ µ ( ee ρ A S ρ µν ) -e λ A T ρ µλ S ρ νµ -1 4 e ν A T ] -e ν A [ 1 2 ∂ µ φ∂ µ φ -V ( φ ) ] + e µ A ∂ ν φ∂ µ φ +4 ξe ρ A S ρ µν φ ( ∂ µ φ ) = e ρ A Ψ ρ ν . (50)</formula>
<text><location><page_8><loc_52><loc_63><loc_92><loc_66></location>We now impose the spatially homogeneous geometry of the form (26) and write the field equations as</text>
<formula><location><page_8><loc_60><loc_58><loc_92><loc_62></location>H 2 = κ 2 3 ( ρ φ + ρ m ) , (51)</formula>
<formula><location><page_8><loc_61><loc_54><loc_92><loc_59></location>˙ H = -κ 2 2 ( ρ φ + p φ + ρ m + p m ) , (52)</formula>
<text><location><page_8><loc_52><loc_52><loc_92><loc_55></location>where the scalar field energy density and pressure is given by</text>
<formula><location><page_8><loc_53><loc_43><loc_92><loc_51></location>ρ φ = 1 2 ˙ φ 2 + V ( φ ) -3 ξH 2 φ 2 , (53) p φ = 1 2 ˙ φ 2 -V ( φ ) + 4 ξHφ ˙ φ + ξ ( 3 H 2 +2 ˙ H ) φ 2 . (54)</formula>
<text><location><page_8><loc_52><loc_41><loc_92><loc_43></location>Using Eqs. (53) and (54) in Eq. (28), putting θ = 3 ˙ a a , we write the modified Raychaudhuri equation as</text>
<formula><location><page_8><loc_56><loc_36><loc_92><loc_39></location>dθ dτ = 3 ˙ H = -3( ρ m + p m + ˙ φ 2 +4 ξH ˙ φφ ) 2(1 + ξφ 2 ) . (55)</formula>
<text><location><page_8><loc_52><loc_20><loc_92><loc_34></location>Eq. (55) governs the evolution of the time-like congruence, depending on the signature of dθ dτ . A negative dθ dτ indicates a collapsing system until θ reaches -∞ , where a zero proper volume singularity forms. However, if dθ dτ changes signature and becomes positive over the course of it's evolution then the collapse of the congruence halts and the geodesics start to move away from each other. Similar to the last section, the onus of avoiding a singularity therefore lies on the behavior of the RHS of Eq. (55).</text>
<text><location><page_8><loc_52><loc_9><loc_92><loc_19></location>If we assume that both the energy density ρ and the isotropic pressure p are positive in nature, the nature of the evolution and the predictability depends on the nature of ξ and ˙ φ . Moreover, ˙ φ 2 is always positive. If the scalar field increases as a function of time, ˙ φ > 0 as well. Thus, depending on the signature of ξ the congruence behaves accordingly, for instance if ξ < 0 such that</text>
<text><location><page_9><loc_9><loc_87><loc_49><loc_93></location>2(1 + ξφ 2 ) < 0 but 3( ρ m + p m + ˙ φ 2 +4 ξH ˙ φφ ) > 0, somewhere during the collapse, ˙ H = dθ dτ > 0, which means a formation of singularity is avoided. Otherwise there is a formation of zero proper volume singularity.</text>
<text><location><page_9><loc_9><loc_82><loc_49><loc_87></location>For a general coupling function ξ 0 ξ ( φ ) replacing the non-minimal coupling ξφ 2 in the action, one may generalize the Raychaudhuri equation to write the condition in Eq. (55) as</text>
<formula><location><page_9><loc_15><loc_75><loc_49><loc_79></location>dθ dτ = -3( ρ m + p m + ˙ φ 2 +2 ξ 0 H ˙ φ dξ dφ ) 2(1 + ξ 0 ξ ( φ )) . (56)</formula>
<text><location><page_9><loc_9><loc_70><loc_49><loc_74></location>As an example we present a particular exact solution of the system given by the field equations in Eqs. (51), (52) and the scalar field evolution equation given by</text>
<formula><location><page_9><loc_18><loc_66><loc_49><loc_69></location>¨ φ +3 H ˙ φ +6 ξH 2 φ + dV dφ = 0 . (57)</formula>
<text><location><page_9><loc_9><loc_56><loc_49><loc_64></location>We solve Eq. (57) by using a theorem on the invertability of these equations [84]. The property involves point transforming the equations into an integrable form and is derived from the symmetry analysis of a general classical anharmonic oscillator equation system. The general equation is written as</text>
<formula><location><page_9><loc_17><loc_53><loc_49><loc_55></location>¨ φ + f 1 ( t ) ˙ φ + f 2 ( t ) φ + f 3 ( t ) φ n = 0 . (58)</formula>
<text><location><page_9><loc_9><loc_43><loc_49><loc_52></location>f 1 , f 2 and f 3 are unknown functions of some variable, of t at this point, n is a constant. A transformation of this equation into an integrable form requires a pair of point transformations and the condition n / ∈ {-3 , -1 , 0 , 1 } to be satisfied. Moreover, the coefficients must satisfy the condition</text>
<formula><location><page_9><loc_14><loc_31><loc_49><loc_42></location>1 ( n +3) 1 f 3 ( t ) d 2 f 3 dt 2 -( n +4) ( n +3) 2 [ 1 f 3 ( t ) df 3 dt ] 2 + ( n -1) ( n +3) 2 [ 1 f 3 ( t ) df 3 dt ] f 1 ( t ) + 2 ( n +3) df 1 dt + 2 ( n +1) ( n +3) 2 f 2 1 ( t ) = f 2 ( t ) . (59)</formula>
<text><location><page_9><loc_9><loc_29><loc_38><loc_30></location>The point transformations are written as</text>
<formula><location><page_9><loc_12><loc_21><loc_49><loc_27></location>Φ( T ) = Cφ ( t ) f 1 n +3 3 ( t ) e 2 n +3 ∫ t f 1 ( x ) dx , (60) T ( φ, t ) = C 1 -n 2 ∫ t f 2 n +3 3 ( ξ ) e ( 1 -n n +3 ) ∫ ξ f 1 ( x ) dx dξ ,</formula>
<text><location><page_9><loc_46><loc_20><loc_49><loc_21></location>(61)</text>
<text><location><page_9><loc_9><loc_9><loc_49><loc_19></location>where C is a constant. Using this property, we solve the scalar field evolution equation assuming it's integrability at the outset. However, this assumption by no means produces unphysical solutions. The scope of this approach has been discussed at length quite recently, in the context of simple scalar field collapse, scalarGauss-bonnet gravity and cosmological reconstruction</text>
<figure>
<location><page_9><loc_55><loc_60><loc_89><loc_94></location>
<caption>FIG. 3: Evolution of the radius of two-sphere as given in Eq. (66) : (i) Top curve m = 1, C = 1 and a 0 = 1 ; (ii) Bottom curve m = 1, C = 1 and a 0 = -1.</caption>
</figure>
<text><location><page_9><loc_52><loc_48><loc_74><loc_50></location>of modified theories of gravity.</text>
<text><location><page_9><loc_52><loc_44><loc_92><loc_47></location>We assume the potential to be a sum of quadratic and quartic terms of the scalar field, written as</text>
<formula><location><page_9><loc_62><loc_39><loc_92><loc_43></location>V ( φ ) = 1 2 m 2 φ 2 -αm 2 24 f 2 φ 4 , (62)</formula>
<text><location><page_9><loc_52><loc_34><loc_92><loc_38></location>which is extremely suggestive of a Higgs Potential or an axion dark matter Potential. Using the exact form of the potential, the scalar evolution equation takes the form of</text>
<formula><location><page_9><loc_55><loc_28><loc_92><loc_33></location>¨ φ +3 H ˙ φ + ( 6 ξH 2 + m 2 ) φ -αm 2 6 f 2 φ 3 = 0 . (63)</formula>
<text><location><page_9><loc_52><loc_22><loc_92><loc_28></location>A quick comparison reveals the coefficients to be written as f 1 = 3 ˙ a a , f 2 = ( 6 ξH 2 + m 2 ) , αm 2 6 f 2 and n = 3. Using this, the integrability criterion produces the evolution equation for the radius of the two sphere as</text>
<formula><location><page_9><loc_63><loc_18><loc_92><loc_21></location>a a +(1 -6 ξ ) ˙ a 2 a 2 -m 2 = 0 . (64)</formula>
<text><location><page_9><loc_52><loc_14><loc_92><loc_16></location>The first integral of the above equation can be written as</text>
<formula><location><page_9><loc_60><loc_8><loc_92><loc_12></location>˙ a = -{ a 0 a 2(6 ξ -1) + m 2 a 2 2 -6 ξ } 1 2 . (65)</formula>
<figure>
<location><page_10><loc_12><loc_59><loc_46><loc_94></location>
<caption>FIG. 4: Evolution of the scalar field from Eq. (63) : (i) Top curve shows the evolution of the scalar for collapse, i.e., m = 1, C = 1 and a 0 = 1 ; (ii) Bottom curve shows the evolution of the scalar for bounce, i.e., m = 1, C = 1 and a 0 = -1.</caption>
</figure>
<text><location><page_10><loc_9><loc_43><loc_49><loc_47></location>As a simple example, we solve the above equation for ξ = 1 12 . This produces the solution for the radius of the two sphere as</text>
<formula><location><page_10><loc_10><loc_37><loc_49><loc_42></location>a ( t ) = e -√ 2 m ( √ 3 t +3 C ) 3 (2 m ) 4 / 3 [ e 3 √ 2 mC -6 a 0 m 2 e √ 6 mt ] 2 3 . (66)</formula>
<text><location><page_10><loc_9><loc_17><loc_49><loc_37></location>In Fig. 3, we plot the evolution as a function of time. The graph on top of the figure shows a plot for a 0 > 0. It is clear to note that the collapsing fluid reaches a zero proper volume at a finite future. The time of formation of this zero proper volume singularity may vary depending on the choice of the functional form of the theory, i.e., α , however, the qualitatiove behavior remains the same. In the graph below, the evolution is shown for a 0 < 0. It is clear that there is no formation of zero proper volume singularity in this case, as the collapsing fluid bounces indefinitely after reaching a minimum cutoff volume. The parameter a 0 is a critical parameter of the system whose signature determines the fate of the collapsing system.</text>
<text><location><page_10><loc_9><loc_9><loc_49><loc_16></location>We study numerically the evolution of the scalar field of the scalar non-minimal coupling using the Klein Gordon equation as in Eq. (63). The evolution of the scalar field with respect to time is given in Fig. 4. It is evident from the figure that, when the sphere collapses onto a</text>
<figure>
<location><page_10><loc_55><loc_60><loc_89><loc_94></location>
<caption>FIG. 5: Evolution of the NEC as given in Eq. (34) : (i) Top curve shows the evolution for collapse, i.e., m = 1, C = 1 and a 0 = 1 ; (ii) Bottom curve shows the evolution for bounce, i.e., m = 1, C = 1 and a 0 = -1.</caption>
</figure>
<text><location><page_10><loc_52><loc_33><loc_92><loc_49></location>zero proper volume, the scalar field diverges around the time of formation of singularity as well. However, from the bottom graph we note that, the collapse and bounce of the sphere is associated with a dispersal of the scalar field to zero value. It may involve radiating or exploding away the strength of scalar field during the indefinite bounce. Using the equation for the NEC, as in Eq. (34), we also check if the collaping fluid satisfies the NEC. This essentially ensures a positive energy density and that the speed of energy flow of matter is less than the speed of light.</text>
<text><location><page_10><loc_52><loc_18><loc_92><loc_33></location>In Fig. 5, we plot the NEC as a function of time, using the exact solution in Eq. (66), for two different initial conditions leading to collapse (top graph) and bounce (bottom graph). While the NEC is perfectly satisfied for a collapse to zero proper volume singularity, usually an indefinite bounce is associated with a violation of NEC. Eventually the bounce leads to a complete dispersal of all the matter distribution inside, as shown in the bottom graph, when ρ + p ∼ 0. This collapse and dispersal is extremely suggestive of a critical behavior in the system.</text>
<section_header_level_1><location><page_10><loc_56><loc_13><loc_88><loc_14></location>VI. DISCUSSION AND CONCLUSION</section_header_level_1>
<text><location><page_10><loc_52><loc_9><loc_92><loc_11></location>In this work explore TG within the context of stellar collapse through the Raychaudhuri equation and the</text>
<text><location><page_11><loc_9><loc_72><loc_49><loc_93></location>NEC. TG explores the possibility of replacing the LeviCivita connection with its Weitzenbock analogue. This has the effect of producing a generically lower-order framework of gravity in which the metric is exchanged with the tetrad in terms of the fundamental dynamical object of the theory. We use the relations between TG and standard gravity to relate the components of the Ricci tensor with their theory-dependent teleparallel analogues through Eq.(25). The effect of this is that the Raychaudhuri equation in Eq.(5) can be used to determine the congruence of neighbouring particle geodesics. The Raychaudhuri equation is a general result for Riemann manifolds which is why we can use it in this context. We then use the NEC to determine which of these solutions indeed produces collapsing models.</text>
<text><location><page_11><loc_9><loc_59><loc_49><loc_71></location>The scenario of a spatially homogeneous collapsing stellar interior is investigated in § . IV within the f ( T ) gravity extension to TEGR. Here, we assume the same rational as the widely popular f ( ˚ R ) gravity framework. By probing this scenario of TG with a perfect fluid, we find the condition for stellar collapse, both in terms of the strightforward torsion scalar but also as a function of H ( t ). Then by using the NEC, we determine trial solutions that satisfy this condition. As we show in this</text>
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<text><location><page_11><loc_52><loc_88><loc_92><loc_93></location>section, these models are consistent with a number of literature proposals, and moreover are instrumental in determining general conditions for collapse within f ( T ) gravity.</text>
<text><location><page_11><loc_52><loc_68><loc_92><loc_87></location>We again consider this scenario in § .V, but in this case we take consider an interesting nonminimally coupled scalar field which is added to the TEGR Lagrangian in the action in Eq.(49). In this case, we need to also use a very intrigueing theorem within calculus on solutions of anharmonic oscillator systems in Eq.(58). In this setup, we can then find solutions to the evolution equations. In this part of the work, we plot the NEC to determine when this is satisfied in Fig.(5). Scalar-tensor theories are very interesting in TG due to its organically lower-order nature which produces a much wider range of models that remain second-order than their standard gravity analogues.</text>
<text><location><page_11><loc_52><loc_59><loc_92><loc_67></location>TG has been mainly studies in cosmology and so works in stellar systems can reveal a lot about the physically viable theories from cosmology. Collapse models provide an intriguing test bed in which to perform these studies and may elucidate several literature models within their strong field regime.</text>
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2020PhRvD.102h4020W | https://arxiv.org/pdf/2008.09774.pdf | <document>
<section_header_level_1><location><page_1><loc_24><loc_92><loc_77><loc_93></location>Investigating the Linearized Second Law in Horndeski Gravity</section_header_level_1>
<text><location><page_1><loc_38><loc_89><loc_62><loc_90></location>Xin-Yang Wang 1, * and Jie Jiang 2, †</text>
<text><location><page_1><loc_24><loc_88><loc_24><loc_88></location>1</text>
<text><location><page_1><loc_24><loc_86><loc_77><loc_88></location>College of Education for the Future, Beijing Normal University, Zhuhai 519087, China 2 Department of Physics, Beijing Normal University, Beijing 100875, China</text>
<text><location><page_1><loc_43><loc_84><loc_58><loc_86></location>(Dated: August 25, 2020)</text>
<text><location><page_1><loc_18><loc_69><loc_83><loc_83></location>Since the entropy of stationary black holes in Horndeski gravity will be modified by the non-minimally coupling scalar field, a significant issue of whether the Wald entropy still obeys the linearized second law of black hole thermodynamics can be proposed. To investigate this issue, a physical process that the black hole in Horndeski gravity is perturbed by the accreting matter fields and finally settles down to a stationary state is considered. According to the two assumptions that there is a regular bifurcation surface in the background spacetime and that the matter fields always satisfy the null energy condition, one can show that the Wald entropy monotonically increases along the future event horizon under the linear order approximation without any specific expression of the metric. It illustrates that the Wald entropy of black holes in Horndeski gravitational theory still obeys the requirement of the linearized second law. Our work strengthens the physical explanation of Wald entropy in Horndeski gravity and takes a step towards studying the area increase theorem in the gravitational theories with non-minimal coupled matter fields.</text>
<section_header_level_1><location><page_1><loc_22><loc_65><loc_36><loc_66></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_30><loc_49><loc_63></location>In general relativity, the area theorem of black holes which states that the area of the event horizon will never decrease during physical processes has been demonstrated by Hawking [1]. From the theorem, Bekenstien [2] proposed a conjecture firstly that the area of the event horizon may be identical to the entropy of black holes. Subsequently, Hawking [3] first proved the temperature of black holes is proportional to the surface gravity, while the entropy can be written as S BH = A / 4, which is called Bekenstein-Hawking entropy. It implies that black holes can be regarded as an adiabatic system in the thermodynamics. Furthermore, the four laws of mechanics for black holes have been constructed [4, 5]. The two profound laws in the four laws of mechanics are the first and the second law. The first law can be expressed as dE = TdS for stationary black holes, where E is the Killing energy which can be measured in corotating coordinates, T is Hawking temperature, and S is the entropy of black holes. The second law is stated that the entropy of the black hole will increase irreversibly. For the generalized second law of black holes, the entropy does not only contain Bekenstein-Hawking entropy, it demands that the sum of the entropies of the horizon and the matter outside black holes, S = S BH + S out, will always increase with physical processes [6].</text>
<text><location><page_1><loc_9><loc_14><loc_49><loc_30></location>However, for any generally covariant theory of gravity, a question of whether black holes in the corresponding gravitational theory can be considered as a thermodynamic system as well will be naturally raised. From this question, the modified first law for the equilibrium state of black holes in any diffeomorphism invariant gravitational theory has been established based on the Iyer-Wald formalism [7, 8]. In the modified first law, the entropy is called the Wald entropy, and it is no longer proportional to the area of the event horizon. If we recognize that the gravitational dynamics of black holes strongly connects to the thermodynamics for any diffeomorphism invariant</text>
<text><location><page_1><loc_52><loc_36><loc_92><loc_66></location>gravitational theory, the validity of the four laws of mechanics should be examined again, especially the second law. Following this perspective, the generalized second law for other kinds of black holes has been investigated. Using the method of the field redefinition, it is shown that the Wald entropy for black holes in the f ( R ) gravity obeys the requirement of the second law [9, 10]. However, there is still existing the situation that the entropy of black hole does not obey the second law of black hole thermodynamics. For the Lagrangian which contains higher-order curvature terms, the second law can be violated in the case of the two black holes merging [11]. However, Ref. [12] has been argued that if the quantum corrections of the black hole is involved, it is sufficient to examine the second law under the first order approximation in an adiabatic process. When only concerning the first-order approximation, the second law of black holes in Gauss-Bonnet and Lovelock gravitational theories have been investigated [13, 14]. Subsequently, a general proof of the linearized second law in higher curvature gravity has been proposed by Wall [15], and the expression of the entropy which satisfies the linearized second of black holes in F ( Riemann ) gravity has been obtained.</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_36></location>Although the general relativity is the most successful theory to describe the interaction of the gravity, it cannot provide a satisfactory interpretation of some cosmological phenomena, such as the origin of the early Universe, the accelerated expansion, and the present of the dark matter and the dark energy. However, the scalar field is considered as a suitable candidate to solve these phenomena [16]. From this perspective, several cosmological models which contain the correction of scalar fields have been proposed [17-20]. Recently, the Horndeski gravitational theory, which contains a non-minimally coupled axionic scalar field, has received much attention through their application to cosmology in Galileon theories [21]. Although the Lagrangian involves more than two derivatives, the field equations and the stress-energy tensor involve no higher than second derivatives of the fields [22, 23]. On the other hand, since the scalar field couples to the Riemann curvature in Horndeski gravity, the formalism of the Wald entropy will be influenced by the scalar field. Therefore, it is natural to ask whether the Wald entropy of black holes in Horndeski grav-</text>
<text><location><page_2><loc_9><loc_88><loc_49><loc_93></location>ity still obeys the second law of black hole thermodynamics. Following the line of through in Ref. [14], we would like to examine the linearized second law of the Wald entropy in Horndeski gravity under the linear-order approximation.</text>
<text><location><page_2><loc_9><loc_73><loc_49><loc_87></location>The organization of the paper is as follows. In Sec. II, we introduce the Horndeski gravitational theory and the definition of the Wald entropy. In Sec. III, considering the matter fields perturbation, while according to the two assumptions that a regular bifurcation surface exists in the background spacetime and that the matter fields satisfy the null energy condition, we investigate whether the Wald entropy for black holes in Horndeski gravitational theory obeys the second law under the linear order approximation. The paper ends with discussions and conclusions in Sec. IV.</text>
<section_header_level_1><location><page_2><loc_9><loc_68><loc_49><loc_70></location>II. HORNDESKI GRAVITATIONAL THEORY AND WALD ENTROPY</section_header_level_1>
<text><location><page_2><loc_9><loc_60><loc_49><loc_66></location>We consider the ( n + 2 ) -dimensional Horndeski gravitational theory minimally coupling to some additional matter fields which satisfy the null energy condition. The action is given by</text>
<formula><location><page_2><loc_24><loc_58><loc_49><loc_59></location>I = I Horn + I mt (1)</formula>
<text><location><page_2><loc_9><loc_55><loc_12><loc_56></location>with</text>
<formula><location><page_2><loc_9><loc_50><loc_49><loc_54></location>I Horn = 1 16 π ∫ d n + 2 x √ -g [ R -1 2 ( β g ab -α G ab ) ∇ a χ∇ b χ ] , (2)</formula>
<text><location><page_2><loc_9><loc_43><loc_49><loc_50></location>where I mt is the action of the additional matter fields, G ab = R ab -( 1 / 2 ) Rg ab is the Einstein tensor, χ represents the scalar field, while α and β are the coupling constants between the gravity and the scalar field. The equation of motion of the gravitational part is given by</text>
<formula><location><page_2><loc_21><loc_39><loc_49><loc_42></location>H ab = 8 π ( T ( β ) ab + T mt ab ) , (3)</formula>
<text><location><page_2><loc_9><loc_36><loc_49><loc_39></location>in which T mt ab is the stress-energy tensor of the additional matter fields, and we have denoted</text>
<formula><location><page_2><loc_15><loc_32><loc_49><loc_35></location>T ( β ) ab = β 16 π ( ∇ a χ∇ b χ -1 2 g ab ∇ c χ∇ c χ ) (4)</formula>
<text><location><page_2><loc_9><loc_30><loc_11><loc_31></location>and</text>
<formula><location><page_2><loc_9><loc_18><loc_49><loc_29></location>H ab = G ab -α 2 [ R 2 ∇ a χ∇ b χ -2 ∇ c χ∇ ( a χ R c b ) + ∇ a ∇ b χ∇ 2 χ -R acbd ∇ c χ∇ d χ -∇ a ∇ c χ∇ b ∇ c χ + 1 2 G ab ( ∇χ ) 2 ] -α 4 g ab [( ∇ c ∇ d χ ) ( ∇ c ∇ d χ ) -( ∇ 2 χ ) 2 + 2 R cd ∇ c χ∇ d χ ] . (5)</formula>
<text><location><page_2><loc_9><loc_9><loc_49><loc_18></location>From the above expressions, we can see that T ( β ) ab is exactly the stress-energy tensor of the minimally coupled scalar field, while it is required that the scalar field should satisfy the null energy condition, i.e., T ( β ) ab k a k b ≥ 0 for any null vector along the future direction k a . Therefore, the term which contains the minimally coupled scalar field can be collected into the</text>
<text><location><page_2><loc_52><loc_90><loc_92><loc_93></location>additional matter fields, and the equation of motion can be rewritten as</text>
<formula><location><page_2><loc_67><loc_88><loc_92><loc_89></location>H ab = 8 π T ab , (6)</formula>
<text><location><page_2><loc_52><loc_86><loc_56><loc_87></location>where</text>
<formula><location><page_2><loc_66><loc_83><loc_92><loc_85></location>T ab = T ( β ) ab + T mt ab (7)</formula>
<text><location><page_2><loc_52><loc_78><loc_92><loc_82></location>represents the total stress-energy tensor, and it is further required that the total stress-energy tensor should satisfy the null energy condition as well.</text>
<text><location><page_2><loc_52><loc_75><loc_92><loc_78></location>For stationary black holes, the entropy is given by the Wald entropy</text>
<formula><location><page_2><loc_61><loc_71><loc_92><loc_74></location>S = -2 π ∫ d n y √ γ ∂ L ∂ R abcd /epsilon1 ab /epsilon1 cd , (8)</formula>
<text><location><page_2><loc_52><loc_60><loc_92><loc_70></location>where L is the Lagrangian of the Horndeski gravitational theory, /epsilon1 ab is the binormal on a specific slice of the event horizon, √ γ is the volume element of any slice of the horizon, and y labels the transverse coordinate of the cross section of the horizon. According to the definition of the Wald entropy and Eq. (2), the Wald entropy of black holes in Horndeski gravitational theory is obtained as</text>
<formula><location><page_2><loc_60><loc_55><loc_92><loc_59></location>S = 1 4 ∫ s d n y √ γ ( 1 -α 4 Da χ D a χ ) , (9)</formula>
<text><location><page_2><loc_52><loc_39><loc_92><loc_55></location>where the symbol s represents the cross section of the event horizon, and DaXa 1 a 2 ··· = γ b a γ b 1 a 1 γ b 2 ··· a 2 ··· ∇ b X b 1 b 2 ··· for any spatial tenor Xa 1 a 2 ··· is denoted as the spatial operator of the covariant derivative which is compatible with the induced metric γ ab , i.e., Dc γ ab = 0. According to the expression of the Wald entropy, we can see that the entropy of the stationary black hole is modified by the scalar field and is no longer proportional to the area of the event horizon. Furthermore, a meaningful question of whether the Wald entropy in Horndeski gravity still obeys the second law of the thermodynamics can be naturally proposed.</text>
<section_header_level_1><location><page_2><loc_52><loc_33><loc_91><loc_36></location>III. EXAMINING THE LINEARIZED SECOND LAW FOR BLACK HOLES IN HORNDESKI GRAVITY</section_header_level_1>
<text><location><page_2><loc_52><loc_11><loc_92><loc_31></location>In the following, a slow accreting process that the matter fields pass through the event horizon to perturb the black hole is considered. For the process, it should be required that the spacetime geometry of the black hole finally settles down to a stationary state after the perturbation, and the matter fields satisfy the null energy condition as well. The event horizon is denoted as H , which is a n -dimensional null hypersurface and can be generated by the null vector field k a =( ∂ / ∂λ ) a . If λ can be chosen as an affine parameter, the null vector field k a obeys the geodesic equation k b ∇ b k a = 0. For any cross section on the event horizon, a basis with the null vector fields { k a , l a , y a } can be constructed, where l a is a second null vector. Since l a and k a are both null vectors, they should satisfy the following relation</text>
<formula><location><page_2><loc_61><loc_8><loc_92><loc_10></location>k a ka = l a la = 0 , k a la = -1 . (10)</formula>
<text><location><page_3><loc_9><loc_89><loc_49><loc_93></location>Using the two null vectors, the binormal of the cross section is given by /epsilon1 ab = 2 k [ a l b ] , and the induced metric on any cross section of the future event horizon is defined as</text>
<formula><location><page_3><loc_22><loc_86><loc_49><loc_88></location>γ ab = g ab + 2 k ( a l b ) . (11)</formula>
<text><location><page_3><loc_9><loc_82><loc_49><loc_85></location>The relationship between the null vectors and the induced metric can be expressed as k a γ ab = l a γ ab = 0.</text>
<text><location><page_3><loc_9><loc_79><loc_49><loc_82></location>Since the extrinsic curvature of the event horizon H is defined as</text>
<formula><location><page_3><loc_23><loc_77><loc_49><loc_78></location>B ab = γ c a γ d b ∇ ckd , (12)</formula>
<text><location><page_3><loc_9><loc_73><loc_49><loc_75></location>the evolution of the induced metric along the future event horizon can be given as [14]</text>
<formula><location><page_3><loc_16><loc_68><loc_49><loc_71></location>γ c a γ d b L k γ cd = 2 ( σ ab + θ n γ ab ) = 2 B ab , (13)</formula>
<text><location><page_3><loc_9><loc_63><loc_49><loc_67></location>where σ ab and θ represents the shear and the expansion of the event horizon respectively. The evolution of the extrinsic curvature along the horizon can also be obtained as [24]</text>
<formula><location><page_3><loc_15><loc_60><loc_49><loc_62></location>γ c a γ d b L k B cd = BacB c b -γ c a γ d b R ecf d k e k f . (14)</formula>
<text><location><page_3><loc_9><loc_58><loc_48><loc_59></location>From this result, the Raychaudhuri equation can be given by</text>
<formula><location><page_3><loc_20><loc_53><loc_49><loc_57></location>d θ d λ = -θ 2 n -σ ab σ ab -R kk , (15)</formula>
<text><location><page_3><loc_9><loc_50><loc_49><loc_53></location>where we have used the convention A kk = A ab k a k b for any tensor A ab .</text>
<text><location><page_3><loc_9><loc_40><loc_49><loc_49></location>To describe the perturbation of the dynamical fields, we introduce a sufficient small parameter ε , which represents the order of the approximation of the perturbation. From the small parameter, we assume that B ab ∼ θ ∼ σ ab ∼ ∂ λ χ ∼ ∂ 2 λ χ ∼ O ( ε ) , where we have denoted ∂ λ = k a ∂ a = ∂ / ∂λ . In the following, the symbol ' /similarequal ' will be used to represent the identity under the linear order approximation.</text>
<text><location><page_3><loc_9><loc_35><loc_49><loc_39></location>Under the first-order approximation of the perturbation, the linear version of the Raychaudhuri equation can be approximately written as</text>
<formula><location><page_3><loc_25><loc_31><loc_49><loc_34></location>d θ d λ /similarequal -R kk . (16)</formula>
<text><location><page_3><loc_9><loc_27><loc_49><loc_30></location>While Eq. (14) under the first-order approximation can also be simplified as</text>
<formula><location><page_3><loc_18><loc_24><loc_49><loc_26></location>γ c a γ d b L k B cd /similarequal -γ c a γ d b R ecf d k e k f . (17)</formula>
<text><location><page_3><loc_9><loc_18><loc_49><loc_23></location>In the following, we will examine the linearized second law of the Wald entropy for Horndeski gravity in the above dynamical geometry. The expression of the Wald entropy can be simplified as</text>
<formula><location><page_3><loc_21><loc_14><loc_49><loc_17></location>S = 1 4 ∫ s d n y √ γ ( 1 + ρ ) , (18)</formula>
<text><location><page_3><loc_9><loc_12><loc_13><loc_13></location>where</text>
<formula><location><page_3><loc_23><loc_8><loc_49><loc_11></location>ρ = -α 4 D a χ Da χ (19)</formula>
<text><location><page_3><loc_52><loc_88><loc_92><loc_93></location>is defined as the entropy density that comes from the interaction between the gravity and the non-minimally coupled scalar field. The rate of change of the entropy along the future horizon is defined by</text>
<formula><location><page_3><loc_65><loc_83><loc_92><loc_86></location>dS d λ = 1 4 ∫ s d n y √ γ Θ , (20)</formula>
<text><location><page_3><loc_52><loc_78><loc_92><loc_82></location>where Θ represents the generalized expansion of the event horizon. According to Eq. (18), the change rate of the entropy can also be expressed as</text>
<formula><location><page_3><loc_60><loc_74><loc_92><loc_77></location>dS d λ = 1 4 ∫ s d n y √ γ [ L k ρ + θ ( 1 + ρ )] , (21)</formula>
<text><location><page_3><loc_52><loc_67><loc_92><loc_72></location>where the Lie derivative L k is commutative to the induced covariant derivative Da because the entropy density ρ is an intrinsic quantity on the hypersurface s . For the first term on the right-hand side of Eq. (21), it can be calculated as</text>
<formula><location><page_3><loc_52><loc_58><loc_92><loc_66></location>∫ s d n y √ γ L k ρ = α 2 ∫ s d n y √ γ [ B ab Da χ D b χ -D a χ Da ( ∂ λ χ ) ] = α 2 ∫ s d n y √ γ [ B ab Da χ D b χ + ∂ λ χ D 2 χ -Da ( D a χ∂ λ χ ) ] , (22)</formula>
<text><location><page_3><loc_52><loc_48><loc_92><loc_57></location>where we have used the definition of the extrinsic curvature in the first step. Using the Stokes' theorem, one can see that the last term only contributes a boundary term. If we assume that the cross section of the event horizon is compact, the boundary term can be neglected directly. Therefore, the result of Eq. (22) can be simplified as</text>
<formula><location><page_3><loc_54><loc_43><loc_92><loc_47></location>∫ s d n y √ γ d ρ d λ = α 2 ∫ s d n y √ γ ( B ab Da χ D b χ + ∂ λ χ D 2 χ ) . (23)</formula>
<text><location><page_3><loc_52><loc_40><loc_92><loc_43></location>According to the definition, one can obtain the expression of the generalized expansion Θ as</text>
<formula><location><page_3><loc_55><loc_35><loc_92><loc_39></location>Θ = α 2 ( B ab Da χ D b χ + ∂ λ χ D 2 χ ) + θ ( 1 + ρ ) . (24)</formula>
<text><location><page_3><loc_52><loc_32><loc_92><loc_35></location>Under the first-order approximation, the rate of change of the generalized expansion can be expressed as</text>
<formula><location><page_3><loc_54><loc_24><loc_92><loc_31></location>d Θ d λ /similarequal α 2 [ ( L k B ab ) D a χ D b χ + ∂ 2 λ χ D 2 χ ] + d θ d λ ( 1 + ρ ) /similarequal α 2 [ ∂ 2 λ χ D 2 χ -k c k d R acbd D a χ D b χ ] -( 1 + ρ ) R kk , (25)</formula>
<text><location><page_3><loc_52><loc_17><loc_92><loc_22></location>where the linear version of the Raychaudhuri equation and Eq. (17) have been used in the last step. Following a similar considering in Ref. [12], the rate of change of the generalized expansion can be written as</text>
<formula><location><page_3><loc_65><loc_12><loc_92><loc_15></location>d Θ d λ /similarequal -8 π T kk + E kk . (26)</formula>
<text><location><page_3><loc_52><loc_9><loc_92><loc_11></location>To investigate whether the Wald entropy of black holes in Horndeski gravity obeys the linearized second law, we just</text>
<text><location><page_4><loc_9><loc_88><loc_49><loc_93></location>need to demonstrate that E kk can be neglected under the firstorder approximation. It is because when considering the null energy condition T kk ≥ 0, Eq. (26) can reduce to</text>
<formula><location><page_4><loc_26><loc_85><loc_49><loc_88></location>d Θ d λ ≤ 0 (27)</formula>
<text><location><page_4><loc_9><loc_70><loc_49><loc_84></location>under the first-order approximation. Since the black hole will become a stationary state after the matter fields perturbation as mentioned above, Θ should vanish in the asymptotic future. It implies that the value of Θ must be positive in the future null direction, and the entropy of the black hole will increase along the future horizon under the first-order approximation. According to the above discussions, to examine whether the Wald entropy for black holes in Horndeski gravity obeys the second law, we should only demonstrate that E kk vanishes under the first-order approximation of the perturbation.</text>
<text><location><page_4><loc_9><loc_65><loc_49><loc_69></location>From Eq. (25), the specific expression of E kk in Horndeski gravitational theory under the first-order approximation can be expressed as</text>
<formula><location><page_4><loc_9><loc_60><loc_49><loc_64></location>E kk /similarequal H kk -( 1 + ρ ) R kk + α 2 [ ∂ 2 λ χ D 2 χ -k c k d R acbd D a χ D b χ ] . (28)</formula>
<text><location><page_4><loc_9><loc_56><loc_49><loc_60></location>According to the specific expression of the equation of motion, the first two terms on the right-hand side of Eq. (28) can be written as</text>
<formula><location><page_4><loc_10><loc_48><loc_49><loc_55></location>H kk -( 1 + ρ ) R kk /similarequal -α 2 ∂ 2 λ χ∇ 2 χ + α 2 k c k d R acbd ∇ a χ∇ b χ + α k b R bc ∇ c χ∂ λ χ + α 2 k a k b ( ∇ a ∇ c χ )( ∇ b ∇ c χ ) . (29)</formula>
<text><location><page_4><loc_9><loc_43><loc_49><loc_47></location>For the first term in Eq. (29), using the definition of the induced metric on the cross section of the event horizon, ∇ 2 χ can be decomposed as</text>
<formula><location><page_4><loc_19><loc_39><loc_49><loc_42></location>∇ 2 χ = -2 k a l b ∇ a ∇ b χ + D 2 χ , (30)</formula>
<text><location><page_4><loc_9><loc_38><loc_49><loc_39></location>and the first term in the result of Eq. (29) can be expressed as</text>
<formula><location><page_4><loc_12><loc_33><loc_49><loc_37></location>-α 2 ∂ 2 λ χ∇ 2 χ = -α 2 ∂ 2 λ χ ( D 2 χ -2 k a l b ∇ a ∇ b χ ) . (31)</formula>
<text><location><page_4><loc_9><loc_29><loc_49><loc_33></location>For the second term on the right-hand side of Eq. (29), according to the definition of the induced metric as well, the covariant derivative of the scalar fields can be decomposed as</text>
<formula><location><page_4><loc_17><loc_25><loc_49><loc_27></location>∇ a χ = -l a ∂ λ χ -k a l b ∇ b χ + D a χ . (32)</formula>
<text><location><page_4><loc_9><loc_22><loc_49><loc_24></location>Using Eq. (32), the second term under the first-order approximation can be given as</text>
<formula><location><page_4><loc_11><loc_15><loc_49><loc_21></location>α 2 k c k d R acbd ∇ a χ∇ b χ /similarequal α 2 k c k d R acbd D a χ D b χ -α l a k c k d R acbd ∂ λ χ D b χ . (33)</formula>
<text><location><page_4><loc_9><loc_9><loc_49><loc_14></location>In the following, we also assume that the background spacetime has a regular bifurcation surface. This additional assumption implies that the spacetime geometry on the bifurcation surface satisfy the boost symmetry, which demands</text>
<text><location><page_4><loc_52><loc_85><loc_92><loc_93></location>that the number of k a in the expression should be equal to the number of l a . For the second term on the right-hand side of Eq. (33), since ∂ λ χ is proportional to a first-order quantity, l a k c k d R acbd γ b e should be evaluated on the background spacetime. According to the dimensional analysis, one can find that this quantity satisfy the following relation</text>
<formula><location><page_4><loc_65><loc_80><loc_92><loc_83></location>l a k c k d R acbd γ b e ∝ 1 λ . (34)</formula>
<text><location><page_4><loc_52><loc_71><loc_92><loc_79></location>From the above expression, we can clearly see that this quantity will diverge at λ = 0. To ensure the regularity of the bifurcation surface in the background spacetime and satisfy the boost symmetry, the second term on the right-hand side of Eq. (33) can be neglected directly. Therefore, Eq. (33) can be simplified as</text>
<formula><location><page_4><loc_58><loc_67><loc_92><loc_70></location>k c k d R acbd ∇ a χ∇ b χ /similarequal k c k d R acbd D a χ D b χ . (35)</formula>
<text><location><page_4><loc_52><loc_64><loc_92><loc_67></location>Based on the decomposition of Eq. (32), the third term on the right-hand side of Eq. (29) can be written as</text>
<formula><location><page_4><loc_52><loc_60><loc_92><loc_63></location>α k b R bc ∇ c χ∂ λ χ /similarequal -α R kk l a ∇ a χ∂ λ χ + α k b R bc D c χ∂ λ χ /similarequal 0 , (36)</formula>
<text><location><page_4><loc_52><loc_53><loc_92><loc_60></location>under the first-order approximation, where we have used the fact R kk ∼ ∂ λ χ ∼ O ( ε ) , and k a Rac γ c b vanishes in the background spacetime because it does not satisfy the requirement of the boost symmetry as well. For the last term of (29), we have</text>
<formula><location><page_4><loc_53><loc_43><loc_92><loc_52></location>α 2 k a k b ( ∇ a ∇ c χ )( ∇ b ∇ c χ ) = α 2 ( k a ∇ a ∇ c χ ) γ cd ( k b ∇ b ∇ d χ ) -α ( k a k c ∇ a ∇ c χ )( k b l d ∇ b ∇ d χ ) = α 2 ( k a ∇ a ∇ c χ ) γ cd ( k b ∇ b ∇ d χ ) -α∂ 2 λ χ ( k b l d ∇ b ∇ d χ ) . (37)</formula>
<text><location><page_4><loc_52><loc_40><loc_76><loc_41></location>For the first term in Eq. (37), we have</text>
<formula><location><page_4><loc_56><loc_21><loc_92><loc_39></location>( k a ∇ a ∇ c χ ) γ cd = γ cd ∇ c ( ∂ λ χ ) -γ cd ( ∇ a χ ) ∇ cka = γ cd ∇ c ( ∂ λ χ ) -γ cd ( ∇ cka ) D a χ + γ cd l a ∇ cka ∂ λ χ + ( l b ∇ b χ ) k a ( ∇ cka ) γ cd = γ cd ∇ c ( ∂ λ χ ) -γ cd ( D a χ ) Bca + γ cd l a ∇ cka ∂ λ χ + 1 2 ( l b ∇ b χ ) Dc ( k a ka ) γ cd = γ cd ∇ c ( ∂ λ χ ) -γ cd ( D a χ ) Bca + γ cd l a ∇ cka ∂ λ χ ∼ O ( ε ) , (38)</formula>
<text><location><page_4><loc_52><loc_12><loc_92><loc_20></location>According to the result of Eq. (38), we can see that the first term in the result of Eq. (37) is a second-order quantity. Since we only consider the first-order approximation, the first term on the right-hand side of Eq. (37) can be neglected directly in our research. Therefore, the last term in the result of Eq. (29) can be reduced as</text>
<formula><location><page_4><loc_53><loc_8><loc_92><loc_11></location>α 2 k a k b ( ∇ a ∇ c χ )( ∇ b ∇ c χ ) /similarequal -α∂ 2 λ χ ( k b l d ∇ b ∇ d χ ) . (39)</formula>
<text><location><page_5><loc_9><loc_90><loc_49><loc_93></location>Substituting the results of Eq. (30), Eq. (35), Eq. (36), and Eq. (39) into Eq. (29), we can further obtain</text>
<formula><location><page_5><loc_16><loc_84><loc_49><loc_88></location>H kk -( 1 + ρ ) R kk /similarequal -α 2 ∂ 2 λ χ D 2 χ + α 2 k c k d R acbd D a χ D b χ . (40)</formula>
<text><location><page_5><loc_9><loc_81><loc_46><loc_82></location>Finally, substituting Eq. (40) into Eq. (28), we can obtain</text>
<formula><location><page_5><loc_26><loc_76><loc_49><loc_79></location>E kk /similarequal 0 . (41)</formula>
<text><location><page_5><loc_9><loc_70><loc_49><loc_75></location>According to the above discussion, this result illustrates that the Wald entropy of black holes in Horndeski gravity obeys the second law of the black hole thermodynamics at the linear order approximation.</text>
<section_header_level_1><location><page_5><loc_22><loc_64><loc_36><loc_65></location>IV. CONCLUSIONS</section_header_level_1>
<text><location><page_5><loc_9><loc_53><loc_49><loc_61></location>The entropy of stationary black holes is modified by adding a scalar field term in Horndeski gravity because of the existence of non-minimally coupled scalar field, where the expression of the entropy is given by the Wald entropy. Based on it, an issue of whether the Wald entropy of black holes in Horndeski gravitational theory can still obey the linearized</text>
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<list_item><location><page_5><loc_9><loc_10><loc_49><loc_14></location>[12] S. Bhattacharjee, S. Sarkar and A. C. Wall, 'Holographic entropy increases in quadratic curvature gravity,' Phys. Rev. D 92 , no. 6, 064006 (2015).</list_item>
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<text><location><page_5><loc_52><loc_63><loc_92><loc_93></location>second law of black hole thermodynamics is naturally proposed. In order to investigate this issue, a physical process that the black hole in Horndeski gravity is perturbed by the accreting matter fields and finally settles down to a stationary state is considered. Subsequently, two assumptions are suggested, which state that the matter fields always satisfy the null energy condition, and there is a regular bifurcation surface in the background spacetime. According to the Raychaudhuri equation, we demonstrate that the change rate of the generalized expansion along the future horizon is negative. Since the final state of the black hole after the perturbation is demanded as a stationary state, the value of the generalized expansion must be positive according to the result. It indicates that the entropy of the black holes in Horndeski gravitational theory continuously increases along the future event horizon, and the Wald entropy of black holes in Horndeski gravity satisfies the linearized second law. This result reinforces the physical explanation of the Wald entropy in Horndeski gravity and takes a step towards discussing the second law of thermodynamics in the gravitational theories with non-minimal coupling matter fields.</text>
<section_header_level_1><location><page_5><loc_64><loc_59><loc_80><loc_60></location>ACKNOWLEDGEMENT</section_header_level_1>
<text><location><page_5><loc_52><loc_53><loc_92><loc_57></location>This research was supported by National Natural Science Foundation of China (NSFC) with Grants No. 11775022 and 11873044.</text>
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<list_item><location><page_5><loc_52><loc_43><loc_92><loc_47></location>[13] A. Chatterjee and S. Sarkar, 'Physical process first law and increase of horizon entropy for black holes in Einstein-GaussBonnet gravity,' Phys. Rev. Lett. 108 , 091301 (2012).</list_item>
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<list_item><location><page_5><loc_52><loc_19><loc_92><loc_23></location>[20] E. N. Saridakis and S. V. Sushkov, 'Quintessence and phantom cosmology with non-minimal derivative coupling,' Phys. Rev. D 81 , 083510 (2010).</list_item>
<list_item><location><page_5><loc_52><loc_17><loc_92><loc_19></location>[21] A. Nicolis, R. Rattazzi and E. Trincherini, 'The Galileon as a local modification of gravity,' Phys. Rev. D 79 , 064036 (2009).</list_item>
<list_item><location><page_5><loc_52><loc_13><loc_92><loc_16></location>[22] A. Cisterna and C. Erices, 'Asymptotically locally AdS and flat black holes in the presence of an electric field in the Horndeski scenario,' Phys. Rev. D 89 , 084038 (2014).</list_item>
<list_item><location><page_5><loc_52><loc_9><loc_92><loc_12></location>[23] W. J. Jiang, H. S. Liu, H. Lu and C. N. Pope, 'DC Conductivities with Momentum Dissipation in Horndeski Theories,' JHEP 1707 , 084 (2017).</list_item>
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</document> | [] |
2017JPhCS.831a2001H | https://arxiv.org/pdf/1704.04386.pdf | <document>
<section_header_level_1><location><page_1><loc_18><loc_68><loc_82><loc_77></location>THE THEORY OF GRAVITATION: A TALE OF MANY QUESTIONS AND FEW ANSWERS</section_header_level_1>
<text><location><page_1><loc_45><loc_64><loc_56><loc_66></location>L. Herrera ∗</text>
<text><location><page_1><loc_19><loc_59><loc_81><loc_64></location>Instituto Universitario de F'ısica Fundamental y Matem'aticas, Universidad de Salamanca, Salamanca 37007, Spain</text>
<text><location><page_1><loc_41><loc_56><loc_59><loc_58></location>November 8, 2021</text>
<section_header_level_1><location><page_1><loc_46><loc_50><loc_54><loc_51></location>Abstract</section_header_level_1>
<text><location><page_1><loc_23><loc_30><loc_77><loc_48></location>We discuss on different issues pertaining the theory of gravity, which pose some unresolved fundamental questions. First we tackle the problem of observers in general relativity, with particular emphasis in tilted observers. We explain why these observers may detect dissipative processes in systems which appear isentropic to comoving observers. Next we analyze the strange relationship between vorticity and radiation, and underline the potential observational consequences of such a link. Finally we summarize all the results that have been obtained so far on the physical properties of the sources of gravitational radiation. We conclude with a list of open questions which we believe deserve further attention.</text>
<section_header_level_1><location><page_1><loc_18><loc_24><loc_40><loc_27></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_18><loc_19><loc_82><loc_23></location>'There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact - Mark Twain</text>
<text><location><page_2><loc_18><loc_75><loc_82><loc_82></location>In this manuscript I would like to elaborate on some issues that I raised recently, at the occasion of the '70&70 Gravitation Fest', which took place in Cartagena, Colombia on September 28-30, 2016, to celebrate the 70th birthday of Rodolfo Gambini and myself.</text>
<text><location><page_2><loc_18><loc_70><loc_82><loc_75></location>The text is heavily based on the conference I delivered at that occasion, but contains some new aspects of the analyzed problems, which were absent in my conference.</text>
<text><location><page_2><loc_18><loc_63><loc_82><loc_70></location>I shall focus on three issues, recurrently appearing in the theory of gravity (in any theory of gravity, not only general relativity), which seem to me particularly appealing and endowed with potential for further development. These are:</text>
<unordered_list>
<list_item><location><page_2><loc_21><loc_56><loc_82><loc_61></location>· What is the role played by the observers in general relativity? What can we learn from the study of tilted spacetimes? How observer dependent is the concept of irreversibility?</list_item>
<list_item><location><page_2><loc_21><loc_49><loc_82><loc_54></location>· What is the link between vorticity and gravitational radiation? Is it a cause-effect relationship? Or are they concomitant? What are the possible observational consequences of such a link?</list_item>
<list_item><location><page_2><loc_21><loc_37><loc_82><loc_48></location>· How to relate the gravitational radiation with the physical properties of its source? What constraints should be impossed on the source to supress the emission of gravitational radiation? What is the spacetime outside a source of gravitational radiation? How does the emission of gravitational radiation start (as seen from the source)? How does this process cease? Are there wave tails?</list_item>
</unordered_list>
<section_header_level_1><location><page_2><loc_18><loc_29><loc_82><loc_34></location>2 Tilted and comoving observers, and the definition of irreversibility (entropy)</section_header_level_1>
<text><location><page_2><loc_18><loc_24><loc_82><loc_27></location>'Irreversibility is a consequence of the explicit introduction of ignorance into the fundamental laws.' M. Born</text>
<text><location><page_2><loc_18><loc_16><loc_82><loc_23></location>In general relativity there exist an ambiguity in the description of the source, related to the arbitrariness in the choice of the four-velocity in terms of which the energy momentum tensor is split. Thus, for example, one possible interpretation of a given spacetime may correspond to a congruence</text>
<text><location><page_3><loc_18><loc_72><loc_82><loc_82></location>of observers comoving with the fluid, whereas the other corresponds to the observer congruence which has been (Lorentz) boosted, with respect to the former. In such a case, both, the general properties of the source and the kinematic properties of the congruence would be different. Many examples of this kind have been analyzed in the past (see for example [1]-[15] and references therein)</text>
<text><location><page_3><loc_18><loc_57><loc_82><loc_72></location>For example, in the case of the zero curvature FRW model, we have a perfect fluid solution for observers at rest with respect to the timelike congruence defined by the eigenvectors of the Ricci tensor, whereas for observers moving relative to the previously mentioned congruence of observers, it can also be interpreted as the exact solution for a viscous dissipative fluid [3]. It is worth noticing that the relative ('tilting') velocity between the two congruences may be related to a physical phenomenon such as the observed motion of our galaxy relative to the microwave background radiation [7].</text>
<text><location><page_3><loc_18><loc_48><loc_82><loc_57></location>Thus, zero curvature FRW models as described by 'tilted' observers will detect a dissipative fluid and energy-density inhomogeneity, as well as different values for the expansion scalar and the shear tensor, among other differences, with respect to the 'standard' (comoving) observers (see [3] for a comprehensive discussion on this example).</text>
<text><location><page_3><loc_18><loc_37><loc_82><loc_48></location>Now, the question arises: is the heat flux vector observed by tilted observers, associated to irreversible processes or not? The rationale behind such a question resides in the fact that in the past some authors have argued that dissipative fluids (understood as fluids whose energy-momentum tensors present a non-vanishing heat flux contribution), are not necessarily incompatible with reversible processes (e.g see [16]-[18]).</text>
<text><location><page_3><loc_18><loc_23><loc_82><loc_37></location>More specifically, in the context of the standard Eckart theory [19], a necessary condition for the compatibility of an imperfect fluid with vanishing entropy production (in the absence of bulk viscosity) is the existence of a conformal Killing vector field (CKV) χ α such that χ α = V α T where V α is the four-velocity of the fluid and T denotes the temperature. In the context of causal dissipative theories, e.g. [20]-[25], the existence of such CKV is also necessary for an imperfect fluid to be compatible with vanishing entropy production (see [8]).</text>
<text><location><page_3><loc_18><loc_15><loc_82><loc_22></location>However, such a claim should not worry us, for two reasons. On the one hand, a carefull analysis of the problem readily shows, that the compatibility of reversible processes with the existence of dissipative fluxes becomes trivial if a constitutive transport equation is adopted. Indeed, in this latter case</text>
<text><location><page_4><loc_18><loc_65><loc_82><loc_82></location>such compatibility forces the heat flux vector to vanish as well. In other words, even if a non-vanishing heat flow vector is assumed to exist, the imposition of the CKV and the vanishing entropy production condition, cancel the heat flux, once a transport equation is assumed (see [26] for a detailed discussion on this point). In other words, in the presence of a CKV of the kind mentioned before, the assumption of a transport equation whether in the context of the Eckart-Landau theory, or a causal theory, implies that a vanishing entropy production leads to a vanishing heat flux vector. Therefore, under the conditions above, the system is not only reversible but also non dissipative.</text>
<text><location><page_4><loc_18><loc_57><loc_82><loc_64></location>On the other hand, neither LTB [27, 28, 29] nor the Szekeres spacetimes [30, 31] admit a CKV, accordingly we may safely conclude that the heat flux vector appearing in these cases (at least), is associated to truly (entropy producing) dissipative processes [8, 9].</text>
<text><location><page_4><loc_18><loc_52><loc_82><loc_57></location>Thus, an intriguing question arises, namely: how is it possible that tilted observers may detect irreversible processes, whereas comoving observers describe an isentropic situation ?</text>
<text><location><page_4><loc_18><loc_46><loc_82><loc_51></location>As we shall see below, the answer to the above question is closely related to definition of entropy, which is highly observer dependent, as illustrated, for example, by the Gibbs paradox [32].</text>
<text><location><page_4><loc_18><loc_41><loc_82><loc_46></location>Indeed, entropy is a measure of how much is not known (uncertainty). The fact that physical objects do not have an intrinsic uncertainty (entropy) has been illustrated in great detail in [33, 34].</text>
<text><location><page_4><loc_18><loc_30><loc_82><loc_41></location>The 'antropomorphic' nature of the concept of entropy makes itself evident in the Gibbs paradox. In its simplest form, the paradox appears from the consideration of a box divided by a wall in two identical parts, each of which is filled with an ideal gas (at the same pressure and temperature). Then if the partition wall is removed, the gases of both parts of the box will mix.</text>
<text><location><page_4><loc_18><loc_15><loc_82><loc_30></location>If the gases from both sides are distinguishable, the entropy of the system will rise, while there is no increase in entropy if they are identical. This leads to the striking conclusion that irreversibility (and thereby entropy), depends on the ability of the observer to distinguish, or not, the gases from both sides of the box. In other words, irreversibility would depend on our knowledge of physics [35], confirming thereby our previous statement that physical objects are deprived of intrinsic entropy. It can only be defined after the number of states that are accesible by the system, is established.</text>
<text><location><page_5><loc_18><loc_74><loc_82><loc_82></location>Now, if a given physical system is studied by a congruence of comoving observers, then the three-velocity of any given fluid element is automatically assumed to vanish, whereas for the tilted observers this variable represents an additional degree of freedom. Therefore, the number of possible states in the latter case is much larger than in the former one.</text>
<text><location><page_5><loc_18><loc_57><loc_82><loc_73></location>In orther to obtain the tilted congruence (from the comoving one), we have to submit locally Minkowskian comoving observers to a Lorentz boost. Such an operation, of course, is performed locally, and no global transformation exists linking both congruences. The point is that, passing from the tilted congruence to the comoving one, we usually overlook the fact that both congruences of observers store different amounts of information. Here resides the clue to resolve the quandary mentioned above, about the presence or not of dissipative processes, depending on the congruence of observers that carry out the analysis of the system.</text>
<text><location><page_5><loc_18><loc_50><loc_82><loc_57></location>Thus, since for the comoving observers the system is dissipationless, it is clear that the increasing of entropy, when passing to the tilted congruence, should imply the presence of dissipative (entropy producing) fluxes, in the latter.</text>
<text><location><page_5><loc_18><loc_39><loc_82><loc_50></location>It is instructive to take a look on this issue from a different perspective. Thus, let us consider the transition from the tilted congruence to the comoving one. According to the Landauer principle, [36] (also referred to as the Brillouin principle [37]-[41]), the erasure of one bit of information stored in a system requires the dissipation into the environment of a minimal amount of energy, whose lower bound is given by</text>
<formula><location><page_5><loc_44><loc_36><loc_82><loc_37></location>/triangle E = kT ln 2 , (1)</formula>
<text><location><page_5><loc_18><loc_31><loc_82><loc_34></location>where k is the Boltzmann constant and T denotes the temperature of the environment.</text>
<text><location><page_5><loc_18><loc_20><loc_82><loc_30></location>In the theory of information, erasure, is just a reset operation restoring the system to a specific state, and is achieved by means of an external agent. In other words, one can decrease the entropy of the system by doing work on it, but then one has to increase the entropy of another system (or the environment). So to speak, Landauer principle is an expression of the fact that logical irreversibility necessarily implies thermodynamical irreversibility.</text>
<text><location><page_5><loc_18><loc_16><loc_82><loc_19></location>Thus, transforming to the comoving congruence, we reset the value of the three-velocity (of any fluid element) to zero, which implies that the</text>
<text><location><page_6><loc_18><loc_74><loc_82><loc_82></location>information has been erased, and a decrease of entropy occurs, but we have not any external agent (since we are considering self-gravitating systems), and therefore such a decrease of entropy is accounted by the dissipative flux observed in the tilted congruence (remember that in the comoving congruence the system is dissipationless).</text>
<text><location><page_6><loc_18><loc_70><loc_82><loc_73></location>We may summarize the main issues addressed in this section, in the following points:</text>
<unordered_list>
<list_item><location><page_6><loc_21><loc_67><loc_72><loc_69></location>· Uncertainty (entropy) is highly dependent on the observer.</list_item>
<list_item><location><page_6><loc_21><loc_65><loc_82><loc_66></location>· Comoving and tilted observers, store different amounts of information.</list_item>
<list_item><location><page_6><loc_21><loc_60><loc_82><loc_63></location>· According to the Landauer principle, erasure of information is always accompanied by dissipation (you have to pay to forgetting).</list_item>
<list_item><location><page_6><loc_21><loc_55><loc_82><loc_58></location>· The quandary mentioned above, is resolved at the light of the three previous comments.</list_item>
</unordered_list>
<section_header_level_1><location><page_6><loc_18><loc_50><loc_55><loc_52></location>3 Radiation and vorticity</section_header_level_1>
<text><location><page_6><loc_18><loc_43><loc_82><loc_48></location>'How often have I said to you that when you have eliminated the impossible, whatever remains, however improbable, must be the truth?' - Sherlock Holmes in: The sign of four.</text>
<text><location><page_6><loc_18><loc_32><loc_82><loc_42></location>The first theoretical evidence relating gravitational radiation to vorticity, appeared when it was established that in an expansion of inverse powers of r (where r denotes the null Bondi coordinate [42]), the coefficient of the vorticity at order 1 /r will vanish if and only if there are no news (no gravitational radiation) [43], implying that a frame dragging effect is associated with gravitational radiation. This result was further confirmed in [44]-[47].</text>
<text><location><page_6><loc_18><loc_26><loc_82><loc_31></location>The intriguing fact is that these two (very different) phenomena are related. More specifically, we wonder if there is any physical reason behind the link between vorticity and radiation.</text>
<text><location><page_6><loc_18><loc_21><loc_82><loc_26></location>The hint to solve this quandary comes from an idea put forward by Bonnor in order to explain the appearance of vorticity in the spacetime generated by a charged magnetic dipole [48].</text>
<text><location><page_6><loc_18><loc_15><loc_82><loc_20></location>For such a system, as was observed by Bonnor, there exists a nonvanishing component of the Poynting vector, describing a flow of electromagnetic energy round in circles [49]. He then suggested that such a circular</text>
<text><location><page_7><loc_18><loc_70><loc_82><loc_82></location>flow of energy affects inertial frames by producing vorticity of congruences of particles, relative to the compass of inertia. This conjecture has been shown to be valid for a general axially symmetric stationary electrovacuum metric [50]. It is worth mentioning that this 'circular' flow of electromagnetic energy is endowed with a solid physical meaning, since it is absolutely necessary in order to preserve the conservation of angular momentum, in the well known 'paradox' of the rotating disk with charges and a solenoid [49].</text>
<text><location><page_7><loc_18><loc_57><loc_82><loc_70></location>Then it came to our minds [46], the idea that a similar mechanism might be at the origin of vorticity in the gravitational case, i.e. that a circular flow of gravitational energy would produce vorticity. However the nonexistence of a local and invariant definition of gravitational energy, rose at that time the question about what expression for the 'gravitational' Poynting vector should be used. In [47] we tried with the super-Poynting vector based on the Bel-Robinson tensor [51]-[53].</text>
<text><location><page_7><loc_18><loc_37><loc_82><loc_57></location>Doing so, we succeded to establish the link between gravitational radiation and vorticity, invoking a mechanism similar to that proposed by Bonnor for the charged magnetic dipole. It was shown that the resulting vorticity is always associated to a circular flow of superenergy on the plane orthogonal to the vorticity vector. It was later shown that the vorticity appearing in stationary vacuum spacetimes also depends on the existence of a flow of superenergy on the plane orthogonal to the vorticity vector [54]. Furthermore in [55] it was shown that not only gravitational but also electromagnetic radiation produces vorticity. In this latter case we were able to isolate contributions from, both, the electromagnetic Poynting vector as well as from the super-Poynting vector.</text>
<text><location><page_7><loc_18><loc_28><loc_82><loc_37></location>It is worth noticing that for unbounded configurations (e.g. cylindrically symmetric sytems), radiation does not produce vorticity if there are no circular flows of superenergy on any plane of the 3-space (e.g. Einstein-Rosen spacetime), which reinforces further the role of the circular flow of superenergy as the link between radiation and vorticity.</text>
<text><location><page_7><loc_18><loc_15><loc_82><loc_28></location>Besides the evident theoretical interest of the issue discused in this section, we would like to emphasize the very important potential observational consequences of the association of vorticity with radiation. Indeed, the direct experimental evidence of the existence of the Lense-Thirring effect [56]-[58] brings out the high degree of development achieved in the technology required to measure rotations. In the same direction point recent proposals to detect frame dragging by means of ring lasers [59]-[63]. Also it is worth</text>
<text><location><page_8><loc_18><loc_77><loc_82><loc_82></location>mentioning the possible use of atom interferometers [64]- [66], atom lasers [67], anomalous spin-precession experiments [68] and matter wave Sagnac interferometers [69], to measure vorticity.</text>
<text><location><page_8><loc_21><loc_75><loc_77><loc_77></location>Le us close this section with a remark about an important feature.</text>
<text><location><page_8><loc_18><loc_57><loc_82><loc_75></location>At order (1 /r 2 ) there are contributions to the vorticity with a time dependent term not involving news (i.e. not associated with gravitational radiation). This last term represents the class of non-radiative motions discussed by Bondi [42] and may be thought to correspond to the tail of the wave, appearing after the radiation process has ended [70]. Thus, the obtained expression allows for 'measuring' (in a gedanken experiment, at least) the wave-tail field. This in turn implies that observing the gyroscope, for a period of time from an initial static situation until after the vanishing of the news, should allow for an unambiguous identification of a gravitational radiation process.</text>
<text><location><page_8><loc_18><loc_44><loc_82><loc_57></location>However, as it has been recently shown (see the subsection 4.6 below) the transition from a state of radiation (gravitational) to an equilibrium state, is not forbidden, after a small time interval of the order of magnitude of the hydrostatic time, the relaxation time, and the thermal adjustment time. This result is in contradiction with previous results [42], [70]-[77], that suggest that such a transition is forbidden, due to the appearance of the wave tails mentioned above.</text>
<text><location><page_8><loc_18><loc_41><loc_82><loc_44></location>Therefore the detection (or not) of the wave tail, would have very important theoretical consequences.</text>
<section_header_level_1><location><page_8><loc_18><loc_36><loc_75><loc_38></location>4 Gravitational radiation and its source</section_header_level_1>
<text><location><page_8><loc_18><loc_29><loc_82><loc_34></location>'I have no data yet. It is a capital mistake to theorize before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts' .-Sherlock Holmes in: A scandal in Bohemia.</text>
<text><location><page_8><loc_18><loc_19><loc_82><loc_28></location>In the sixties, there was a blossom of very powerful methods to study gravitational radiation, beyond the well known linear approximation, [42, 78, 79, 80]. All of them focused on the behaviour of the field very far from the source, whose specific properties did not enter into the discussion, to avoid the appearance of caustics and similar pathologies.</text>
<text><location><page_8><loc_18><loc_15><loc_82><loc_19></location>Besides the many fundamental results obtained from these methods, their main merit consists in including non-linear effects, which are known to play a</text>
<text><location><page_9><loc_18><loc_74><loc_82><loc_82></location>very important role in general relativity. The absence of a detailed discussion on the behaviour of the sources of gravitational radiation, was the triggering motivation to undertake the task to develop a general formalism that could provide a description of the effects of gravitational radiation on the physical properties of the source, and viceversa.</text>
<text><location><page_9><loc_18><loc_70><loc_82><loc_73></location>In other words we searched to establish the relationship between gravitational radiation and source properties.</text>
<text><location><page_9><loc_18><loc_55><loc_82><loc_70></location>The fact that we want to describe gravitational radiation, force us to depart from the spherical symmetry. On the other hand, since we are willing to provide an analytical description of the problem, avoiding as much as possible to resort to numerical methods, we need to impose additional restrictions. Thus, we shall rule out cylindrical symmetry on physical grounds, but we shall assume axial and reflection symmetry, which as shown in [81] is the highest degree of symmetry of the Bondi metric [42], which does not prevent the emission of gravitational radiation.</text>
<text><location><page_9><loc_18><loc_46><loc_82><loc_55></location>The general formalism mentioned before was presented in [82]. It was obtained by using the 1 + 3 formalism [83, 84, 85], in a given coordinate system, and resorting to a set of scalar functions known as Structure Scalars [86], which have been shown to be very useful in the description of selfgravitating systems [87]-[96].</text>
<text><location><page_9><loc_18><loc_41><loc_82><loc_46></location>A detailed description of this general formalism may be found in [82], here we shall focus on the main results obtained so far on this issue, and the open questions which we believe deserve further attention.</text>
<section_header_level_1><location><page_9><loc_18><loc_36><loc_47><loc_38></location>4.1 The shear-free case</section_header_level_1>
<text><location><page_9><loc_18><loc_19><loc_82><loc_35></location>We started by analyzying the shear free case [97]. Under such a condition we have found that for a general dissipative and anisotropic fluid, vanishing vorticity, is a necessary and sufficient condition for the magnetic part of the Weyl tensor to vanish, thereby providing a generalization of the same result for perfect fluids obtained in [98, 99, 100]. This result, in turn, implies that vorticity should necessarily appear if the system radiates gravitationally. We stress that this result is not restricted to the axially (and reflection) symmetric case. This further reinforces the well established link between radiation and vorticity discussed in the previous section.</text>
<text><location><page_9><loc_18><loc_15><loc_82><loc_19></location>If besides the shear-free condition, we assume further that the fluid is geodesic, then the vorticity vanishes (and thereoff the magnetic part of the</text>
<text><location><page_10><loc_18><loc_74><loc_82><loc_82></location>Weyl tensor) and no gravitational radiation is produced. In this case our study generalizes the models studied by Coley and McMannus [101, 102]. A similar result is obtained for the cylindrically symmetric case [103], suggesting a link between the shear of the source and the generation of gravitational radiation.</text>
<text><location><page_10><loc_18><loc_65><loc_82><loc_73></location>Finally, in the geodesic case, we also observe that, in the non-dissipative case, the models do not need to be FRW (as already stressed in [101]), however the system tends to the FRW spacetime (if Θ is positive). In presence of dissipative fluxes, such a tendency does not appear, which illustrates further how relevant, the dissipative processes may be.</text>
<section_header_level_1><location><page_10><loc_18><loc_60><loc_55><loc_62></location>4.2 The perfect, geodesic fluid</section_header_level_1>
<text><location><page_10><loc_18><loc_46><loc_82><loc_59></location>Next we have considered the the restricted case, when the fluid is perfect and geodesic, without assuming ab initio the shear-free condition [104]. As the result of such study we have found that: All possible models compatible with our metric, and the perfect fluid plus the geodesic condition, are FRW, and accordingly non-radiating (gravitationally). It seems that, both, the geodesic and the non-dissipative conditions, are quite restrictive, when looking for a source of gravitational waves.</text>
<text><location><page_10><loc_18><loc_32><loc_82><loc_46></location>It is worth noticing that, not only in the case of dust, but also in the absence of dissipation in a perfect fluid, the system is not expected to radiate (gravitationally) due to the reversibility of the equation of state. Indeed, radiation is an irreversible process, a fact that emerges at once if absorption is taken into account and/or Sommerfeld type conditions, which eliminate inward traveling waves, are imposed. Therefore, the irreversibility of the process of emission of gravitational waves, must be reflected in the equation of state through an entropy increasing (dissipative) factor.</text>
<text><location><page_10><loc_18><loc_25><loc_82><loc_32></location>We should recall that geodesic fluids not belonging to the class considered here (Szekeres) have also been shown not to produce gravitational radiation. This strengthens further the case of the non-radiative character of pure dust distributions.</text>
<section_header_level_1><location><page_10><loc_18><loc_20><loc_59><loc_22></location>4.3 The dissipative, geodesic fluid</section_header_level_1>
<text><location><page_10><loc_18><loc_16><loc_82><loc_19></location>Since dissipation should be present in any scenario where gravitational radition is produced, we decided to study the simplest possible fluid distribution</text>
<text><location><page_11><loc_18><loc_79><loc_82><loc_82></location>which we might expect to be compatible with a gravitational radiation, i.e. a dissipative dust under the geodesic condition [105].</text>
<text><location><page_11><loc_18><loc_75><loc_82><loc_79></location>Two possible subcases were considered separately, namely: the fluid distribution is assumed, from the beginning, to be vorticity-free, or not.</text>
<text><location><page_11><loc_18><loc_70><loc_82><loc_75></location>In the former case, it was shown that the vanishing of the vorticity implies the vanishing of the heat flux vector, and therefore, the resulting spacetime is FRW.</text>
<text><location><page_11><loc_18><loc_65><loc_82><loc_70></location>In the latter case, it is shown that the enforcement of the regularity conditions at the center, implies the vanishing of the dissipative flux, leading also to a FRW spacetime.</text>
<text><location><page_11><loc_18><loc_57><loc_82><loc_64></location>Thus all possible models, sourced by a dissipative geodesic dust fluid, belonging to the family of the line element considered here, do not radiate gravitational waves during their evolution, unless regularity conditions at the center of the distribution are relaxed.</text>
<text><location><page_11><loc_18><loc_48><loc_82><loc_57></location>In other words physically acceptable models require the inclusion of, both, dissipative and anisotropic stresses terms, i.e. the geodesic condition must be abandoned. In this case, purely analytical methods are unlikely to be sufficient to arrive at a full description of the source, and one has to resort to numerical methods.</text>
<section_header_level_1><location><page_11><loc_18><loc_41><loc_82><loc_45></location>4.4 The space-time outside the source of gravitational radiation</section_header_level_1>
<text><location><page_11><loc_18><loc_28><loc_82><loc_40></location>Based on the fact that the process of gravitational radiation is an irreversible one, and therefore must entail dissipative processes within the source, we should conclude that there should be an incoherent radiation (null fluid) at the outside of the source, produced by those dissipative processes. Keeping this fact in mind, we should remark that the Bondi-Sachs metric [42], [78], should be regarded as an approximation to the space-time outside the source, when the null fluid produced by the dissipative processes is neglected.</text>
<text><location><page_11><loc_18><loc_17><loc_82><loc_27></location>Starting with the description of this null fluid, we apply the formalism developped in [82], to study some of the properties of such a null fluid [106]. As the main result of our study we found that the absence of vorticity implies that the exterior spacetime is either static or spherically symmetric (Vaidya). Reinforcing thereby the fundamental role of vorticity in any process involving production of gravitational radiation, already stressed.</text>
<text><location><page_12><loc_18><loc_79><loc_82><loc_82></location>The spherically symmetric case (Vaidya) was, asymptotically, recovered within the context of the 1+3 formalism [106].</text>
<section_header_level_1><location><page_12><loc_18><loc_75><loc_52><loc_77></location>4.5 Leaving the equilibrium</section_header_level_1>
<text><location><page_12><loc_18><loc_59><loc_82><loc_73></location>Next, as an application of our general method [82], we analyzed the situation, just after its departure from hydrostatic and thermal equilibrium, at the smallest time scale at which the first signs of dynamic evolution appear [107]. Such a time scale is smaller than the thermal relaxation time, the thermal adjustment time and the hydrostatic time. Specifically we were able to answer to the following questions: 1) what are the first signs of non-equilibrium? 2) which physical variables do exhibit such signs? 3) what does control the onset of the dynamic regime, from an equilibrium initial configuration?</text>
<text><location><page_12><loc_18><loc_39><loc_82><loc_59></location>It was obtained that the onset of non-equilibrium will critically depend on a single function directly related to the time derivative of the vorticity. Among all fluid variables (at the time scale under consideration), only the tetrad component of the anisotropic tensor, in the subspace orthogonal to the four-velocity and the Killing vector of axial symmetry, shows signs of dynamic evolution. Also, the first step towards a dissipative regime begins with a non-vanishing time derivative of the heat flux component along the meridional direction. The magnetic part of the Weyl tensor vanishes (not so its time derivative), indicating that the emission of gravitational radiation will occur at later times. Finally, the decreasing of the effective inertial mass density, associated to thermal effects, was clearly illustrated [107].</text>
<section_header_level_1><location><page_12><loc_18><loc_35><loc_53><loc_36></location>4.6 Reaching the equilibrium</section_header_level_1>
<text><location><page_12><loc_18><loc_16><loc_82><loc_33></location>Next, we described the transition of a gravitationally radiating, axially and reflection symmetric dissipative fluid, to a non-radiating state [108]. What we wanted to elucidate was if, very shortly after the end of the radiating regime, at a time scale of the order of the thermal relaxation time, the thermal adjustment time and the hydrostatic time (whichever is larger), the system reaches the equilibrium state. We recall that in all the studies carried out in the past, on gravitational radiation outside the source, such a transition to a static case, is forbidden. However, neither of these studies include the physical properties of the source, giving rise the possibility of a quite different result.</text>
<text><location><page_13><loc_18><loc_59><loc_82><loc_82></location>Using the general method presented in [82], we were able to prove that the system does in fact reach the equilibrium state. This result is at variance with previous results mentioned above, implying that such a transition is forbidden. The reason for such a discrepancy resides in the fact that some elementary, but essential, physical properties of the source, have been overlooked in these latter studies. Our result strengths further the relevance of the physical properties of the source, in any discussion about the physical properties of the field. Also, it emphasizes the need to resort to global solutions, whenever important aspects about the behaviour of the gravitational field are discussed. In other words, the coupling between the source and the external field may introduce important constraints on the physical behaviour of the system, implying that details of the source fluid cannot be left out, because they may be relevant to distant GW scattering.</text>
<section_header_level_1><location><page_13><loc_18><loc_55><loc_52><loc_56></location>4.7 The quasi-static regime</section_header_level_1>
<text><location><page_13><loc_18><loc_48><loc_82><loc_53></location>As it is well known, in the study of self-gravitating fluids we may consider three different possible regimes of evolution, namely: the static, the quasistatic and the dynamic.</text>
<text><location><page_13><loc_18><loc_39><loc_82><loc_48></location>In the static case, the spacetime admits a timelike, hypersurface orthogonal, Killing vector. Thus, a coordinate system can always be choosen, such that all metric and physical variables are independent on the time like coordinate. The static case, for axially and reflection symmetric spacetimes, was studied in [109].</text>
<text><location><page_13><loc_18><loc_32><loc_82><loc_39></location>Next, we have the full dynamic case where the system is considered to be out of equilibrium (thermal and dynamic), the general formalism to analyze this situation, for axially and reflection symmetric spacetimes was developed in [82].</text>
<text><location><page_13><loc_18><loc_28><loc_82><loc_32></location>In between the two regimes described above, we have the quasi-static evolution.</text>
<text><location><page_13><loc_18><loc_17><loc_82><loc_28></location>In this regime the system is assumed to evolve, although sufficiently slow, so that it can be considered to be in equilibrium at each moment. This means that the system changes slowly, on a time scale that is very long compared to the typical time in which the fluid reacts to a slight perturbation of hydrostatic equilibrium. This typical time scale is called hydrostatic time scale (sometimes this time scale is also referred to as dynamical time scale).</text>
<text><location><page_14><loc_18><loc_77><loc_82><loc_82></location>Thus, in this regime the system is always very close to hydrostatic equilibrium and its evolution may be described as a sequence of equilibrium configurations.</text>
<text><location><page_14><loc_18><loc_72><loc_82><loc_77></location>In other words, we may assume safely the quasi-static approximation (QSA) whenever all the relevant characteristic times of the system under consideration, are much larger than the hydrostatic time.</text>
<text><location><page_14><loc_18><loc_59><loc_82><loc_72></location>This assumption is very sensible because the hydrostatic time scale is very small for many phases of the life of the star. It is of the order of 27 minutes for the Sun, 4 . 5 seconds for a white dwarf and 10 -4 seconds for a neutron star of one solar mass and 10 Km radius. It is a fact that any of the stellar configurations mentioned above, generally (but not always), changes on a time scale that is very long compared to their respective hydrostatic time scales.</text>
<text><location><page_14><loc_18><loc_54><loc_82><loc_59></location>Motivated by the comments above, we applied the framework developped in [82], to carry out a study of axially and reflection symmetric fluids in the quasi-static regime [110].</text>
<text><location><page_14><loc_18><loc_50><loc_82><loc_53></location>For doing that we needed to introduce different invariantly defined 'velocities', in terms of which the QSA is expressed.</text>
<text><location><page_14><loc_18><loc_44><loc_82><loc_50></location>It was obtained that the shear and the vorticity of the fluid, as well as the dissipative fluxes, may affect the (slow) evolution of the configuration, as to produce 'splittings' within the fluid distribution.</text>
<text><location><page_14><loc_18><loc_32><loc_82><loc_44></location>Also it was shown that in the QSA, the contributions of the gravitational radiation to the components of the super-Poynting vector do not necessarily vanish. However, it appears that if at any given time, the magnetic part of the Weyl tensor vanishes, then it vanishes at any other time afterwards. Thus it is not reasonable to expect gravitational radiation from a physically meaningful system, radiating for a finite period of time (in a given time interval) in the QSA.</text>
<section_header_level_1><location><page_14><loc_18><loc_27><loc_44><loc_29></location>5 OPEN ISSUES</section_header_level_1>
<text><location><page_14><loc_18><loc_20><loc_82><loc_25></location>As the reader should easily understand, the three issues considered here, still present a great deal of unanswered questions. Below we display a partial list of problems which we believe deserve some attention:</text>
<unordered_list>
<list_item><location><page_14><loc_21><loc_16><loc_82><loc_18></location>· How could one describe the 'cracking' (splitting) of the configurations,</list_item>
</unordered_list>
<text><location><page_15><loc_23><loc_79><loc_82><loc_82></location>in the context of this formalism ? How the appearance of such crackings would affect the emission of gravitational radiation?</text>
<unordered_list>
<list_item><location><page_15><loc_21><loc_69><loc_82><loc_78></location>· We do not have an exact solution (written down in closed analytical form) describing gravitational radiation in vacuum, from bounded sources. Accordingly, any specific modeling of a source, and its matching to an exterior, should be done numerically. How such a matching could be done?</list_item>
<list_item><location><page_15><loc_21><loc_60><loc_82><loc_67></location>· It should be useful to introduce the concept of the mass function, similar to the one existing in the spherically symmetric case (an extension of the Bondi mass for the interior of the source). This could be relevant, in particular, in the matching of the source to a specific exterior.</list_item>
<list_item><location><page_15><loc_21><loc_52><loc_82><loc_59></location>· How does the system analyzed in [82] look for tilted observers? Is it possible that tilted observers detect gravitational radiation from systems that are non-radiating (gravitationally), as seen by comoving observers?</list_item>
<list_item><location><page_15><loc_21><loc_45><loc_82><loc_50></location>· What could we learn by imposing different kind of symmetries (more general that isometries) on the axially symmetric dissipative fluids studied in [82].</list_item>
<list_item><location><page_15><loc_21><loc_35><loc_82><loc_43></location>· What is the threshold of sensitivity in the measure of vorticity, reached by the present technology? What are the expected values for the potential future developments of different experiments aiming the detection of rotations? Are these values within the range expected for realistic sources of gravitational radiation?</list_item>
</unordered_list>
<section_header_level_1><location><page_15><loc_18><loc_29><loc_48><loc_32></location>6 Acknowledgments</section_header_level_1>
<text><location><page_15><loc_18><loc_17><loc_82><loc_28></location>It is a real pleasure to thanks the organizying committee of '70&70 Gravitation Fest': Antonio C. Guti'errez-Pi˜neres (Universidad Tecnol'ogica de Bol'ıvar); Edison Montoya (Universidad Industrial de Santander); Jorge Mu˜niz (Universidad Tecnol'ogica de Bol'ıvar) and Luis A. N'u˜nez (Universidad Industrial de Santander), for their generosity and the great effort deployed to make the event possible.</text>
<section_header_level_1><location><page_16><loc_18><loc_81><loc_33><loc_83></location>References</section_header_level_1>
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2014A&A...572A..77S | https://arxiv.org/pdf/1409.8429.pdf | <document>
<section_header_level_1><location><page_1><loc_9><loc_82><loc_91><loc_87></location>Vertical shear instability in accretion disc models with radiation transport</section_header_level_1>
<text><location><page_1><loc_36><loc_80><loc_64><loc_81></location>Moritz H. R. Stoll and Wilhelm Kley</text>
<text><location><page_1><loc_10><loc_75><loc_82><loc_78></location>Institut für Astronomie und Astrophysik, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany e-mail: [email protected],[email protected]</text>
<text><location><page_1><loc_10><loc_72><loc_36><loc_73></location>Received 2014-05-05; accepted 2014-09-25</text>
<section_header_level_1><location><page_1><loc_46><loc_69><loc_54><loc_70></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_10><loc_64><loc_90><loc_68></location>Context. The origin of turbulence in accretion discs is still not fully understood. While the magneto-rotational instability is considered to operate in su GLYPH<14> ciently ionized discs, its role in the poorly ionized protoplanetary disc is questionable. Recently, the vertical shear instability (VSI) has been suggested as a possible alternative.</text>
<text><location><page_1><loc_10><loc_62><loc_90><loc_65></location>Aims. Our goal is to study the characteristics of this instability and the e GLYPH<14> ciency of angular momentum transport, in extended discs, under the influence of radiative transport and irradiation from the central star.</text>
<text><location><page_1><loc_10><loc_59><loc_90><loc_62></location>Methods. We use multi-dimensional hydrodynamic simulations to model a larger section of an accretion disc. First we study inviscid and weakly viscous discs using a fixed radial temperature profile in two and three spatial dimensions. The simulations are then extended to include radiative transport and irradiation from the central star.</text>
<text><location><page_1><loc_10><loc_54><loc_90><loc_59></location>Results. In agreement with previous studies we find for the isothermal disc a sustained unstable state with a weak positive angular momentum transport of the order of GLYPH<11> GLYPH<25> 10 GLYPH<0> 4 . Under the inclusion of radiative transport the disc cools o GLYPH<11> and the turbulence terminates. For discs irradiated from the central star we find again a persistent instability with a similar GLYPH<11> value as for the isothermal case.</text>
<text><location><page_1><loc_10><loc_51><loc_90><loc_54></location>Conclusions. We find that the VSI can indeed generate sustained turbulence in discs albeit at a relatively low level with GLYPH<11> about few times 10 GLYPH<0> 4 .</text>
<text><location><page_1><loc_10><loc_50><loc_57><loc_51></location>Key words. instability - hydrodynamics - accretion discs - radiative transfer</text>
<section_header_level_1><location><page_1><loc_6><loc_45><loc_18><loc_46></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_6><loc_10><loc_49><loc_44></location>The origin of the angular momentum transport in accretion discs is still not fully understood. Observationally it is well confirmed that the molecular viscosity is by many orders of magnitude too small to explain the e GLYPH<11> ective mass and angular momentum transport in discs (Pringle 1981). This can be inferred for example from time variations of the disc luminosity in close binary systems, or by correlating the infrared-excess caused by discs around young stars with the age of the system. As a consequence it is assumed that discs are driven by some kind of turbulent transport whose cause is still not known. Despite its unknown origin the e GLYPH<14> ciency of the turbulence is usually parameterized in terms of the dimensionless parameter, GLYPH<11> , as introduced by Shakura & Sunyaev (1973). Observationally, values of a few times 10 GLYPH<0> 3 as in protostellar discs to 10 GLYPH<0> 1 for discs in close binary stars are suggested. For su GLYPH<14> ciently well ionized discs the magnetorotational instability (MRI) is certainly the most promising candidate to provide the transport (Balbus 2003). While this may be true for the hot discs in close binary systems or in active galactic nuclei, there is the important class of protostellar discs where at least the thermal ionisation levels are too low to provide a su GLYPH<14> cient number of charged particles that can support the MRI (Armitage 2011). In such discs the turbulence plays an important role in several aspects. Not only does it determine the lifetime of an accretion disc, but it also influences where and how planets can form and evolve in the disc. A variety of sources such as stellar X-rays, cosmic rays or collisions with beta par-</text>
<text><location><page_1><loc_51><loc_35><loc_94><loc_47></location>ticles from radioactive nuclei have been invoked to provide the required ionization levels but recent studies indicate the presence of an extended 'dead zone' where, due to the lack of ionization, no magnetically driven instability may operate. Additionally, recent studies on the origins of turbulence in protostellar discs that include non-ideal magnetohydrodynamical (MHD) e GLYPH<11> ects such as ambipolar di GLYPH<11> usion or the Hall e GLYPH<11> ect, indicate that the MRI may even be suppressed strongly in these discs, see the review by Turner et al. (2014) and references therein.</text>
<text><location><page_1><loc_51><loc_10><loc_94><loc_34></location>As a consequence alternative mechanisms to provide turbulence are actively discussed. Typical examples for nonmagnetized discs are the convective instability (Ruden et al. 1988), the gravitational instability (Lin & Pringle 1987), or the baroclinic instability (Klahr & Bodenheimer 2003), for further references see Nelson et al. (2013). While any of these may operate under special conditions in the disc, e.g. suitable radial entropy gradients or a su GLYPH<14> ciently high disc mass, none seems to have general applicability. Searching for alternatives the vertical shear instability (VSI) has attracted recent interest. Here, the instability is caused by a vertical gradient of the angular velocity, GLYPH<10> , in the disc. Through linear analysis it has been shown that for a su GLYPH<14> ciently strong vertical shear there are always modes that can overcome the stabilizing angular momentum gradient (Rayleigh-criterion) and generate instability (Urpin & Brandenburg 1998; Urpin 2003). This instability is related to the Goldreich-Schubert-Fricke instability that can occur in differentially rotating stars (Goldreich & Schubert 1967; Fricke 1968).</text>
<text><location><page_2><loc_6><loc_59><loc_49><loc_93></location>Concerning its e GLYPH<11> ectiveness with respect to angular momentum transport numerical simulations were performed by Arlt & Urpin (2004) and Nelson et al. (2013). The first authors analysed the instability for globally isothermal discs and found that the instability in this case could only be triggered by applying finite initial perturbation because the equilibrium state of the disc (being strictly isothermal) did not contain a shear in GLYPH<10> . The maximum values of GLYPH<11> obtained by Arlt & Urpin (2004) were around 6 GLYPH<1> 10 GLYPH<0> 6 but the turbulence was decaying in the long run. Nelson et al. (2013) extended these simulations and performed high resolution simulations of the VSI for so called locally isothermal discs that contain a radial temperature gradient but are vertically isothermal. Under these conditions the equilibrium state has a vertical gradient in the shear and indeed an instability sets in. As shown by Nelson et al. (2013) the instability has two distinct growth phases, it starts from the surface layers of the disc where the shear is strongest and then protrudes towards the midplane. In the final state the vertical motions in the disc are antisymmetric with respect to the discs's midplane, such that the gas elements cross the midplane, a feature found for the vertical convective motions in discs as well (Kley et al. 1993). For the e GLYPH<14> ciency of the VSI induced turbulence Nelson et al. (2013) found a weak angular momentum transport with GLYPH<11> = 6 GLYPH<1> 10 GLYPH<0> 4 . They also showed that under the presence of a small viscosity or thermal relaxation the instability is weaker and can easily be quenched.</text>
<text><location><page_2><loc_6><loc_48><loc_49><loc_59></location>It is not clear what influence radiation transport will have on this instability. Without external heat sources one might expect that, because of radiative cooling and the dependence of the instability on temperature, the instability will die out. Here, we evaluate the evolution of the instability for a radiative discs and an ideal equation of state. Additionally, we extend the radial domain and include irradiation from the central star. We perform two-dimensional and three-dimensional hydrodynamical simulations including radiative transport.</text>
<text><location><page_2><loc_6><loc_41><loc_49><loc_48></location>This paper is organized as follows. In Section 2, we present the physical setup of our disc models and in Section 3 the numerical approach. The isothermal results are presented in Section 4, followed by the radiative cases in Section 5. Stellar irradiation is considered in Section 6 and in Section 7 we conclude.</text>
<section_header_level_1><location><page_2><loc_6><loc_37><loc_20><loc_38></location>2. Physical setup</section_header_level_1>
<text><location><page_2><loc_6><loc_31><loc_49><loc_36></location>In order to study the VSI of the disc in the presence of radiative transport we construct numerical models solving the hydrodynamical equations for a section of the accretion disc in two and three spatial dimensions.</text>
<section_header_level_1><location><page_2><loc_6><loc_28><loc_16><loc_29></location>2.1. Equations</section_header_level_1>
<text><location><page_2><loc_6><loc_22><loc_49><loc_27></location>The basis our studies are the Euler equations (1 - 3) describing the motion of an ideal gas. These are coupled to radiation transport (4) for which we use the two temperature approximation applying flux-limited di GLYPH<11> usion. The equations then read</text>
<formula><location><page_2><loc_6><loc_9><loc_49><loc_20></location>@GLYPH<26> @ t + r ( GLYPH<26> u ) = 0 (1) @ @ t GLYPH<26> u + r ( GLYPH<26> uu ) + r p = GLYPH<26> a ext (2) @ @ t e + r [( e + p ) u ] = GLYPH<26> ua ext GLYPH<0> GLYPH<20> P GLYPH<26> c ( aRT 4 GLYPH<0> E ) (3) @ @ t E + r F = GLYPH<20> P GLYPH<26> c GLYPH<16> aRT 4 GLYPH<0> E GLYPH<17> : (4)</formula>
<text><location><page_2><loc_6><loc_7><loc_24><loc_9></location>Article number, page 2 of 12</text>
<text><location><page_2><loc_51><loc_81><loc_94><loc_93></location>Here GLYPH<26> is the density, u the velocity, e the total energy density (kinetic and thermal) of the gas. p denotes the gas pressure, and the acceleration due to external forces, such as the gravitational force exerted by the central star is given by a ext . E and F are the energy density and the flux of the radiation. The last terms on the r.h.s. of eqs. (3) and (4) refer to the coupling of gas and radiation, i.e. the heating / cooling terms. Here, c stands for the speed of light, aR is the radiation constant, and GLYPH<20> P the Plank mean opacity.</text>
<text><location><page_2><loc_53><loc_80><loc_92><loc_82></location>We close the equations with the ideal gas equation of state</text>
<formula><location><page_2><loc_51><loc_78><loc_94><loc_79></location>p = ( GLYPH<13> GLYPH<0> 1) eth ; (5)</formula>
<text><location><page_2><loc_51><loc_74><loc_94><loc_77></location>where eth = e GLYPH<0> 1 = 2 GLYPH<26> u 2 is the thermal energy density. The temperature of the gas is then calculated from</text>
<formula><location><page_2><loc_51><loc_70><loc_94><loc_73></location>p = GLYPH<26> kBT GLYPH<22> mH ; (6)</formula>
<text><location><page_2><loc_51><loc_54><loc_94><loc_70></location>where GLYPH<22> is the mean molecular weight, kB the Boltzmann constant and mH the mass of the hydrogen atom. In our simulations with radiation transport we use GLYPH<13> = 1 : 4 and GLYPH<22> = 2 : 35. To compare to previous studies we performed additional isothermal simulations where we use GLYPH<13> = 1 : 001 and additionally reset to the original temperature profile in every step. This procedure corresponds to an isothermal simulation but allows for an arbitrary temperature profile. It also allows to use the feature of slowly relaxing to a given original temperature such as used for example in Nelson et al. (2013). Note that without resetting the temperature the gas remains adiabatic, and the perturbation will die out for our setup.</text>
<text><location><page_2><loc_51><loc_51><loc_94><loc_54></location>The radiation flux in the flux-limited di GLYPH<11> usion (FLD) approximation (Levermore & Pomraning 1981) is given by</text>
<formula><location><page_2><loc_51><loc_48><loc_94><loc_51></location>F = GLYPH<0> GLYPH<21> c GLYPH<20> R GLYPH<26> r E ; (7)</formula>
<text><location><page_2><loc_51><loc_40><loc_94><loc_47></location>where GLYPH<20> R is the Rosseland mean opacity and GLYPH<21> is the flux-limiter, for which we use the description of Minerbo (1978). For the Rosseland mean opacity we apply the model of Bell & Lin (1994). For simplicity we use in this initial study the same value for the Plank mean opacity, see also Bitsch et al. (2013).</text>
<text><location><page_2><loc_51><loc_36><loc_94><loc_40></location>In some of our studies we add viscosity and stellar irradiation to the momentum and energy equations. This will be pointed out below in the appropriate sections.</text>
<section_header_level_1><location><page_2><loc_51><loc_33><loc_62><loc_34></location>2.2. Disc model</section_header_level_1>
<text><location><page_2><loc_51><loc_26><loc_94><loc_32></location>To be able to study the onset of the instability we start with a reference model in equilibrium. For this purpose, we follow Nelson et al. (2013) and use a locally isothermal disc in force equilibrium, where for the midplane density we assume a power law behaviour</text>
<formula><location><page_2><loc_51><loc_22><loc_94><loc_25></location>GLYPH<26> ( R ; Z = 0) = GLYPH<26> 0 R R 0 ! p ; (8)</formula>
<text><location><page_2><loc_51><loc_20><loc_80><loc_21></location>and the temperature is constant on cylinders</text>
<formula><location><page_2><loc_51><loc_16><loc_94><loc_19></location>T ( R ; Z ) = T 0 R R 0 ! q : (9)</formula>
<text><location><page_2><loc_51><loc_10><loc_94><loc_15></location>To specify the equilibrium state we have used a cylindrical coordinate system ( R ; Z ; GLYPH<30> ). However, our simulations will be performed in spherical polar coordinates ( r ; GLYPH<18>; GLYPH<30> ) because they are better adapted to the geometry of an accretion disc. In eqs. (8)</text>
<text><location><page_3><loc_6><loc_81><loc_49><loc_93></location>and (9), GLYPH<26> 0 and T 0 are suitably chosen constants that determine the total mass content in the disc and its temperature. The exponents p and q give the radial steepness of the profiles, and typically we choose p = GLYPH<0> 3 = 2 and q = GLYPH<0> 1. Assuming that in the initial state there are no motions in the meridional plane and the flow is purely toroidal, force balance in the radial and vertical directions then leads to the equilibrium density and angular velocity profiles that we use for the initial setup (Nelson et al. 2013)</text>
<formula><location><page_3><loc_6><loc_77><loc_49><loc_81></location>GLYPH<26> ( R ; Z ) = GLYPH<26> 0 R R 0 ! p exp " GM c 2 s 1 p R 2 + Z 2 GLYPH<0> 1 R !# ; (10)</formula>
<text><location><page_3><loc_6><loc_76><loc_9><loc_77></location>and</text>
<formula><location><page_3><loc_6><loc_71><loc_49><loc_75></location>GLYPH<10> ( R ; Z ) = GLYPH<10> K " ( p + q ) GLYPH<18> H R GLYPH<19> 2 + (1 + q ) GLYPH<0> qR p R 2 + Z 2 # 1 2 : (11)</formula>
<text><location><page_3><loc_6><loc_56><loc_49><loc_70></location>Here, cs = p p =GLYPH<26> denotes the isothermal sound speed, GLYPH<10> K = p GM GLYPH<12> = R 3 the Keplerian angular velocity, and H = cs = GLYPH<10> K is the local pressure scale height of the accretion disc. We note that the Z dependence of GLYPH<10> in the equilibrium state is the origin of the VSI, because the vertical shear provides the opportunity for fluid perturbations with a wavenumber ratio kR = kZ above a threshold to tap into a negative gradient in the angular momentum as the perturbed fluid elements move away from the rotation axis (Nelson et al. 2013). The angular velocity given by eq. (11) is also used to calculate the Reynolds stress tensor, for details see below.</text>
<section_header_level_1><location><page_3><loc_6><loc_52><loc_15><loc_54></location>2.3. Stability</section_header_level_1>
<text><location><page_3><loc_6><loc_44><loc_49><loc_52></location>Nelson et al. (2013) repeated the original analysis in Goldreich & Schubert (1967) for a locally isothermal and compressive gas for an accretion disc using the local shearing sheet approximation at a reference radius r 0. They derived the same stability criterion as Urpin (2003) and obtained the following growth rate of the instability</text>
<formula><location><page_3><loc_6><loc_39><loc_49><loc_43></location>GLYPH<27> 2 = GLYPH<0> GLYPH<20> 2 0 ( c 2 0 k 2 Z + N 2 0 ) + 2 GLYPH<10> 0 c 2 0 kRkZ @ ¯ V @ z c 2 0 ( k 2 Z + k 2 R ) + GLYPH<20> 2 0 + N 2 0 ; (12)</formula>
<text><location><page_3><loc_6><loc_32><loc_49><loc_39></location>where GLYPH<20> 0 is the epicyclic frequency, c 0 the sound speed, and N 0 is the Brunt-Vaisaila frequency at the radius r 0. ¯ V denotes the mean deviation from of the Keplerian azimuthal velocity profile, and kR , kZ are the radial and vertical wavenumbers of the perturbations in the local coordinates.</text>
<text><location><page_3><loc_6><loc_30><loc_49><loc_32></location>For negligible N 0, small H 0 = R 0, and kZ = kR GLYPH<24> O ( qH 0 = R 0), as seen in their numerical simulations, Nelson et al. (2013) find</text>
<formula><location><page_3><loc_6><loc_26><loc_49><loc_29></location>GLYPH<27> GLYPH<24> q GLYPH<10> H R ; (13)</formula>
<text><location><page_3><loc_6><loc_20><loc_49><loc_26></location>which implies that the growth rate per local orbit to first order depends on the temperature gradient as given by q and on the absolute temperature, due to H = R . We will compare our numerical results with these estimates.</text>
<section_header_level_1><location><page_3><loc_6><loc_17><loc_22><loc_18></location>3. Numerical Model</section_header_level_1>
<text><location><page_3><loc_6><loc_10><loc_49><loc_16></location>To study the VSI in the presence of radiative transport we perform numerical simulations of a section of an accretion disc in two and three spatial dimensions using spherical polar coordinates ( r ; GLYPH<18>; GLYPH<30> ), and a grid which is logarithmic in radial direction, keeping the cells squared. We solve eqs. (1) to (4) with a</text>
<text><location><page_3><loc_51><loc_88><loc_94><loc_93></location>grid based method, where we use the PLUTO code from Mignone et al. (2007) that utilizes a second-order Godunov scheme, together with our radiation transport (Kolb et al. 2013) in the FLD approximation, see eq. (7).</text>
<text><location><page_3><loc_51><loc_57><loc_94><loc_88></location>The simulations span a region in radius from r = 2 GLYPH<0> 10AU, this is the range where the dead zone can be expected (Armitage 2011; Flaig et al. 2012). Here, we use a larger radial domain as Nelson et al. (2013) because we intend to study the global properties of the instability over a wider range of distances. Additionally, this larger range is useful, because we need some additional space (typically GLYPH<25> 1AU) to damp possible large scale vortices in the meridional plane that show up at the inner radial boundary of the domain (see below). The origin of these vortices is possibly that the instability moves material along cylindrically shaped shells, a motion that is not adapted to the used spherical coordinates, such that the midplane is cut out at the inner boundary. Vortices can also arise if the viscosity changes apruptly, a situation mimicking a boundary. Additionally, in some cases the wavelengths are large, such that the coupling between di GLYPH<11> erent modes cannot be captured in a small domain. Also we use a wide range because with radiation transport the growth rates are expected to depend on the opacity, which is a function of GLYPH<26> and T and thus of the radius. In the meridional direction ( GLYPH<18> ) we go up to GLYPH<6> 5 scale heights above and below the equator in the isothermal case, and we use the same extension for the radiative simulation, where it corresponds to more scale heights. For the 3D simulations we used in the azimuthal direction ( GLYPH<30> ) a quarter circle, from 0 to GLYPH<25>= 2.</text>
<text><location><page_3><loc_51><loc_42><loc_94><loc_56></location>We use reflective boundaries in the radial direction. In the meridional direction we use outflow conditions for the flow out of the domain and reflective conditions otherwise. For the radiation transport solver we set the temperature of the meridional boundary to 10K, which allows the radiation to escape freely. We use damping of the velocity near the inner radial boundary within 2 GLYPH<0> 3AU to prevent the creation of strong vortices arising through the interaction with the reflecting boundary, which can destroy the simulation. This is done by adding a small viscosity of GLYPH<23> = 2 GLYPH<1> 10 GLYPH<0> 7 with a linear decrease to zero from 2AU to 3AU (similar to the damping used in de Val-Borro et al. (2006)).</text>
<text><location><page_3><loc_51><loc_34><loc_94><loc_42></location>Weassume that the disc orbits a solar mass star and we apply a density of GLYPH<26> 0 = 10 GLYPH<0> 10 g / cm 3 at 1AU. Due to the surface density decaying with r GLYPH<0> 0 : 5 we get a surface density GLYPH<6> = 80g / cm 2 at 5AU. To study the mass dependence we vary GLYPH<26> 0 for the radiative models. To seed the instability we add a small perturbation of up to 1% of the sound speed to the equilibrium velocity, see eq. (11).</text>
<text><location><page_3><loc_51><loc_29><loc_94><loc_34></location>Because our radiation transport solver is only implemented in full 3D (Kolb et al. 2013), we use 2 grid cells in the azimuthal direction for the 2D axisymmetric simulations using radiation transport.</text>
<section_header_level_1><location><page_3><loc_51><loc_24><loc_67><loc_25></location>4. Isothermal discs</section_header_level_1>
<text><location><page_3><loc_51><loc_17><loc_94><loc_23></location>Before studying full radiative discs we first perform isothermal 2D simulations to compare our results and growth rates to those of Nelson et al. (2013). Then we will extend the simulation to full 3D using a quarter of a disc and discuss the dependence on resolution and viscosity.</text>
<section_header_level_1><location><page_3><loc_51><loc_13><loc_63><loc_14></location>4.1. Growth rates</section_header_level_1>
<text><location><page_3><loc_51><loc_10><loc_94><loc_12></location>To analyse the possible growth and instability of the initial equilibrium state, we analyse the time evolution of the kinetic energy</text>
<text><location><page_4><loc_12><loc_93><loc_12><loc_94></location>6</text>
<figure>
<location><page_4><loc_6><loc_75><loc_48><loc_93></location>
<caption>Fig. 1. The kinetic energy of the motion in the meridional plane at di GLYPH<11> erent radii in an inviscid disc. The kinetic energy at the di GLYPH<11> erent locations is in each case averaged over a radial interval with length 0.5AU. We note that the unit of time is given in local periods at the center of the specified interval. Hence, it is di GLYPH<11> erent for each curve but this allows for easy comparison.</caption>
</figure>
<text><location><page_4><loc_6><loc_62><loc_22><loc_64></location>in the meridional plane</text>
<formula><location><page_4><loc_6><loc_58><loc_49><loc_61></location>ekin = 1 2 GLYPH<26> ( u 2 r + u 2 GLYPH<18> ) ; (14)</formula>
<text><location><page_4><loc_6><loc_40><loc_49><loc_58></location>at di GLYPH<11> erent radii. The obtained growth of ekin of a run with q = GLYPH<0> 1 and p = GLYPH<0> 3 = 2 is displayed at di GLYPH<11> erent radii in Fig. 1 for an inviscid disc model with a grid resolution of 2048 GLYPH<2> 512. Note that the time is measured in local orbits (2 GLYPH<25>= GLYPH<10> ( ri )) at the corresponding centers of the intervals, ri . We measure a mean growth rate of 0 : 38 per orbit for the kinetic energy (light blue line in Fig. 1), which is twice the growth rate ( GLYPH<27> ) of the velocity. We calculate the growth rate by averaging the kinetic energy at the di GLYPH<11> erent ri over an interval with length 0 : 5AU. Our results compare favourably with the growth rates from Nelson et al. (2013) who obtained 0 : 25 per orbit averaged over 1 GLYPH<0> 2AU for q = GLYPH<0> 1. Averaging over this larger range leads to a reduced growth because the rate at 2AU, measured in orbits at 1AU, is smaller by a factor of 2 1 : 5 = 2 : 8, and so their result is a slight underestimate.</text>
<text><location><page_4><loc_6><loc_15><loc_49><loc_40></location>A closer look at figure 1 reveals two distinct growth phases. An initial strong linear growth phase with a rate of 0.38 per orbit lasting about 20 local orbits, and a slower second phase with a rate of 0.10 per orbit (gray line in Fig. 1). To understand these regimes, we present in Fig. 2 the velocity in the meridional direction, u GLYPH<18> , in 2D contour plots at di GLYPH<11> erent times. The top panel reveals that the first phase corresponds to symmetric (mirror symmetry with respect to the equatorial plane) disturbances that grow from the top and bottom surface layers of the disc. Here, the gas does not cross the midplane of the disc. When those meet in the disc's midplane they develop an anti-symmetric phase with lower growth rates where the gas flow crosses the midplane of the disc as displayed in the middle panel. The converged phase shown in the lower panel then shows the fully saturated global flow. Figure 2 indicates that in the top panel the whole domain is still in the anti-symmetric growth phase, in the middle panel only the smaller radii show symmetric growth, while in the lower panel the whole domain has reached the final equilibrium, in accordance with Fig. 1.</text>
<text><location><page_4><loc_6><loc_10><loc_49><loc_15></location>We point out that the growth rate per local orbit ( GLYPH<24> GLYPH<27>= GLYPH<10> ) is independent of radius in good agreement with the relation (13), for constant H = R . We will show later that the growth rate is also independent of resolution.</text>
<figure>
<location><page_4><loc_51><loc_40><loc_93><loc_94></location>
<caption>Fig. 2. Velocity in the meridional direction, u GLYPH<18> , in units of local Kepler velocity for an isothermal run without viscosity. The panels refer to snapshots taken at time 100, 210 and 750 (top to bottom), measured in orbital periods at 1AU. In units of local orbits at (2.5, 3.5, 4.5)AU this refers to (25, 15, 10) (53, 32, 22) (190, 115, 79) orbits, from top to bottom.</caption>
</figure>
<section_header_level_1><location><page_4><loc_51><loc_25><loc_87><loc_26></location>4.2. Comparison to 3D results and Reynolds stress</section_header_level_1>
<text><location><page_4><loc_51><loc_13><loc_94><loc_24></location>In addition to the 2D simulation we ran an equivalent 3D case using a quarter of a disc with a resolution of 512 GLYPH<2> 128 GLYPH<2> 128 grid cells. We will use this to discuss the validity of the 2D results, in particular the estimates on the turbulent e GLYPH<14> ciency factor GLYPH<11> . In Fig. 3 we compare the growth of the meridional kinetic energy for the 3D and the 2D simulation. After a slower start, the 3D simulation shows very similar growth and reaches the same final saturation level.</text>
<text><location><page_4><loc_51><loc_10><loc_94><loc_12></location>To estimate any possible angular momentum transfer caused by the turbulent motions induced by the instability we calculate</text>
<text><location><page_5><loc_12><loc_93><loc_12><loc_94></location>6</text>
<figure>
<location><page_5><loc_6><loc_75><loc_48><loc_93></location>
</figure>
<figure>
<location><page_5><loc_52><loc_73><loc_88><loc_93></location>
<caption>Fig. 3. The growth of the kinetic energy for the quarter of a disc and the 2D equivalent. The kinetic energy is averaged from 4AU to 5.5AU.</caption>
</figure>
<text><location><page_5><loc_6><loc_67><loc_39><loc_68></location>the corresponding Reynolds stress (Balbus 2003)</text>
<formula><location><page_5><loc_6><loc_63><loc_49><loc_66></location>Tr GLYPH<30> = R GLYPH<26>GLYPH<14> ur GLYPH<14> u GLYPH<30> dV GLYPH<1> V = < GLYPH<26>GLYPH<14> ur GLYPH<14> u GLYPH<30> >; (15)</formula>
<text><location><page_5><loc_6><loc_36><loc_49><loc_62></location>where GLYPH<14> ur and GLYPH<14> u GLYPH<30> are defined as the fluctuations of the velocity field from the mean flow and GLYPH<1> V is the volume of the integrated domain. To calculate a coordinate dependent stress we integrate only over thin slices with a thickness of one cell in the apropriate direction. While GLYPH<14> ur is just the radial velocity, ur , at the point of interest because the initial ur was zero, GLYPH<14> u GLYPH<30> is di GLYPH<14> cult to calculate, as one has to subtract the mean background rotational velocity. Armitage (2011) defines it as the di GLYPH<11> erence to the Kepler rotation, while strictly speaking it is the deviation from the unperturbed equilibrium state that is not Keplerian in our case, see eq. (11). In 3D simulations it is mostly calculated by averaging over the azimuthal direction (Flock et al. 2011; Fromang &Nelson 2006). But this instability is nearly axisymmetric (see Fig. 4), so this is not appropriate here and the correct way is to average over time to obtain the steady state velocity. However, this is computationally inconvenient, because this time average is not known a priori. In Fig. 5 we show that the time averaging method leads to the same results as the equilibrium method using the analytic equation (11), and we use the latter for our subsequent simulations.</text>
<text><location><page_5><loc_6><loc_23><loc_49><loc_36></location>To calculate the dimensionless GLYPH<11> -parameter, Tr GLYPH<30> has to be divided by the pressure. To show the radial and vertical dependence of GLYPH<11> it is useful to use di GLYPH<11> erent normalizations. We divide the Reynolds stress in Eq. (15) by the midplane pressure to illustrate the dependence on the meridional (vertical) coordinate, thus making it independent of the number of scale heights of the domain. The stress as a function of the radius, Tr GLYPH<30> ( R ), is divided by the vertical averaged pressure, making it again independent of the numbers of scale heights. This procedure corresponds to a density weighted height integration (Balbus 2003).</text>
<text><location><page_5><loc_6><loc_10><loc_49><loc_23></location>In figure 5 we present the di GLYPH<11> erent methods for calculating the Reynolds stress, Tr GLYPH<30> , for the simulation of a quarter of a disc with a resolution of 512x128x128 and the same initial conditions as in the 2D case. We can see that indeed the axisymmetric property of the instability leads to wrong results if one only averages over the azimuthal direction. All further results for the isothermal discs are calculated with the equilibrium method. This allows us to approximate the Reynolds stress even in a transient disc and calculate the stress continuously during the whole runtime of the simulation, strongly reducing the amount of data</text>
<figure>
<location><page_5><loc_51><loc_47><loc_93><loc_65></location>
<caption>Fig. 4. The vertical velocity in the midplane of the disk for the 3D model after 4000 orbits. The nearly axisymmetric property of the instability is clearly visible.Fig. 5. The Reynolds stress (code units) from 3-10AU averaged over 41 time steps, each step 100 orbits apart beginning with orbit 1000, calculated with di GLYPH<11> erent averaging methods. For 'mean time' the steady state ¯ u GLYPH<30> = u GLYPH<30> GLYPH<0> GLYPH<14> u GLYPH<30> , needed to calculate the Reynolds stress at each step, was calculated through averaging over the 41 time steps. For 'mean phi' the steady state velocity was calculated by averaging over the azimuthal direction at each time step and for the last one, 'equilibrium', ¯ u GLYPH<30> is calculated analytically by using the equilibrium equation (11) at each step. For the 2D model we have used the equilibrium method as well.</caption>
</figure>
<text><location><page_5><loc_51><loc_22><loc_94><loc_32></location>needed to be written to the hard drive, because the Reynolds stress can now be calculated independent of the other time steps. The computations show in addition that the stresses of the reduced 2D simulations yield stresses comparable to the full 3D case and can be used as a proxy for the full 3D case. In Fig. 4 we display the vertical velocity in the midplane of the disk for the 3D model. As shown, the motions are only very weakly nonaxisymmetric.</text>
<section_header_level_1><location><page_5><loc_51><loc_18><loc_62><loc_19></location>4.3. Resolution</section_header_level_1>
<text><location><page_5><loc_51><loc_10><loc_94><loc_18></location>In this section we take a look at the e GLYPH<11> ect of resolution. We start with a resolution of 256 GLYPH<2> 64, where the instability exists but clearly is not resolved and go, by doubling the resolution in several steps, up to a resolution of 2048 GLYPH<2> 512, where the computations start to be expensive. In Fig. 6 we show on the left the Reynolds stress divided by the midplane pressure as a function</text>
<figure>
<location><page_6><loc_6><loc_75><loc_48><loc_93></location>
<caption>Fig. 6. Radial and vertical distribution of the Reynolds stress. Left: The Reynolds stress divided by the midplane pressure over the vertical direction. Right: Reynolds stress divided by the mean pressure over the radius for di GLYPH<11> erent resolutions. Both are averaged over 4001 time steps from orbit 1000 to 5000. The model res2048 corresponds to the results shown in Fig. 1.</caption>
</figure>
<figure>
<location><page_6><loc_7><loc_35><loc_49><loc_64></location>
<caption>Fig. 7. The mean wavenumber of the instability over the radius for different numerical resolutions in the saturated phase. Upper panel: inviscid case with GLYPH<23> = 0, lower panel: viscous case with GLYPH<23> = 5 GLYPH<1> 10 GLYPH<0> 7 (dimensionless).</caption>
</figure>
<text><location><page_6><loc_6><loc_20><loc_49><loc_26></location>of vertical distance. This is then averaged over the radius from 3 GLYPH<0> 8AU. On the right we plot the Reynolds stress divided by the pressure, where both, pressure and stress, have been averaged over the meridional direction.</text>
<text><location><page_6><loc_6><loc_12><loc_49><loc_20></location>From this plot it is not clear if the values for GLYPH<11> converge to a specific level for higher resolution. But nevertheless it gives a first impression on the strength of turbulent viscosity caused by this instability being relatively weak with GLYPH<11> -values a few times 10 GLYPH<0> 4 , which is slightly smaller than the value of 6 GLYPH<1> 10 GLYPH<0> 4 as found by Nelson et al. (2013).</text>
<text><location><page_6><loc_6><loc_10><loc_49><loc_12></location>In Fig. 7 we display the wavelength of the perturbation as a function of radius for di GLYPH<11> erent numerical resolutions, where</text>
<figure>
<location><page_6><loc_51><loc_75><loc_93><loc_94></location>
<caption>Fig. 8. Histogram: Color coded is the logarithm of the probability for the occurrence of a wavelength at a radius normalised at each radius by the sum of all wavelengths for the specific radius. The black lines are proportional to the radius to the power of 2 : 5 and the lines are a factor of 2 apart from each other. The dashed line has linear slope. One can see that the instability jumps successively between di GLYPH<11> erent modes for the wavelength with corresponding jumps in frequency at the same radius.</caption>
</figure>
<text><location><page_6><loc_51><loc_36><loc_94><loc_62></location>the wavelength has been estimated by measuring the distance between two sucsessive changes of the sign of the vertically averaged vertical momentum after the instability is saturated (see Fig. 2, third panel or Fig. 10 along the radius axis, beginning with orbit 1000). This does, of course, not reveal the full spectrum, but at this point we are more interested in the characteristic mean wavelength. Note that the wavelength in the growth phase can be smaller. In all shown resolutions one wavelength is resolved with 15-50 grid cells, while larger radii are better resolved. Despite the variation with radius one notices in Fig. 7 that the wavelength clearly depends on the numerical resolution. One possible cause for this is the lack of physical viscosity. Because the (intrinsic) numerical viscosity of the code decreases with increasing resolution, this may explain the missing convergence, in particular since the growth rates depend of the wavenumbers of the disturbances, see eq. (12). We repeated the run with an intermediate resolution of 1440 GLYPH<2> 360 with reduced precision by using a first order instead of a second order spatial interpolation. This clearly increased the wavelength (by about 40%) indicating that the problem is caused by the numerical viscosity.</text>
<text><location><page_6><loc_51><loc_16><loc_94><loc_36></location>Fig. 7 indicates a strong reduction of the wavenumber with radius. To further explore this dependence of the wavelength on the radius, we performed an additional simulation with an extended radial domain from 2AU to 50AU. Again, we estimate the wavelength by measuring the distance between two sign changes in the vertical averaged vertical momentum. This time we show all the wavelengths that were detected by this method in Fig. 8, where we display how often a certain wavelength was captured, normalised to the specific radius where it was measured. An interesting behavior can be observed. While the global radial wavelength does indeed depend linearly on the radius, locally it clearly deviates from this dependence and instead depends on the radius to the power of 2.5. This can also be seen in the simulation with smaller domain, but there it can not be clearly distinguished from the interaction with the boundary.</text>
<text><location><page_6><loc_51><loc_10><loc_94><loc_16></location>This supplies us with an explanation for the resolution dependence of the instability. Since the modes can not become arbitrarily small, because of the finite grid, or large, because of the limited vertical scale hight, there will be jumps between di GLYPH<11> erent modes. The viscosity and the Kelvin-Helmholtz instability,</text>
<figure>
<location><page_7><loc_6><loc_76><loc_48><loc_93></location>
<caption>Fig. 9. Fourier power spectrum of the temporal evolutionof the instability after saturation (see figure 10). Analysed is the averaged meridional momentum of the simulation without viscosity and resolution 1024 x 256. Color coded is the logarithm of amplitude of the frequency.</caption>
</figure>
<text><location><page_7><loc_6><loc_63><loc_49><loc_67></location>which can be observed in the simulations with high resolution, are the candidates for a physical cause for this cut o GLYPH<11> at small wavelengths.</text>
<text><location><page_7><loc_6><loc_37><loc_49><loc_63></location>Due to the radius dependence of the wavenumber a spatial Fourier transform is not applicable. Additionally, as we show bellow, the wavelength in radial direction is not constant in time and also phase jumps can occur. However, to obtain nevertheless more insight into the dynamics of the system we display in Fig. 9 the results of a Fourier analysis in time of the vertical momentum of the simulation with resolution of 1024 GLYPH<2> 256 (Fig. 10, along the time axis). To reduce the problems that phase jumps (see below) pose for the analysis, we step through the data with a Hanning window over 1000 orbits and then average over those 5% of the resulting spectra that display the highest amplitude. We can see a dominant frequency at 0 : 022 GLYPH<10> K at the inner region, this frequency is halved at the outer region beginning at about 5AU. These jumps in the frequency domain coincide with the jumps in wavenumber. When the wavenumber jumps up, the frequency jumps down, indicating an inverse relationship. On each branch the frequency is constant, while the wave number varies as / r GLYPH<0> 2 : 5 . We can understand this relationship starting from eq. (12) from which one obtains for stable inertial oscillations (see eq. 36 in Nelson et al. 2013)</text>
<formula><location><page_7><loc_6><loc_32><loc_49><loc_36></location>GLYPH<27> 2 GLYPH<24> GLYPH<0> GLYPH<10> 2 k 2 Z k 2 R : (16)</formula>
<text><location><page_7><loc_6><loc_27><loc_49><loc_32></location>The vertical scale is given by the local disk's scale height H GLYPH<24> r and hence kZ GLYPH<24> r GLYPH<0> 1 . In the quasi stationary phase, we observed kR GLYPH<24> r GLYPH<0> 2 : 5 (cf. Fig. 8), leading to an oscillation frequency independent of the radius, which we also observed (cf. Fig. 9).</text>
<text><location><page_7><loc_6><loc_10><loc_49><loc_27></location>To obtain further insight into the spatio-temporal behaviour of the flow dynamics we display in Fig. 10 the vertically averaged momentum in the vertical direction as a function of space and time. In this global overview we observe waves that appear to travel slowly from larger to smaller radii. As noticed already in the Fourier analysis in Fig. 9, there exists a transition between 4-5AU with a change in wavelength of the perturbations and occasional phase jumps. Coupled to this is a change in the typical inward speed of the waves. They move slower when farther away from the star. As inferred roughly from Fig. 10, the wave speed at r = 6AU is about 0 : 5AU per 250 orbits, while at 4AU it is about 1AU. However, there is some dependence of this speed on time and space.</text>
<text><location><page_7><loc_59><loc_93><loc_59><loc_94></location>4</text>
<figure>
<location><page_7><loc_51><loc_64><loc_93><loc_93></location>
<caption>Fig. 11. The vertically averaged Reynolds stress divided by the vertically averaged pressure in the saturated phase. Upper panel: For a viscosity of 5 GLYPH<1> 10 GLYPH<0> 7 at di GLYPH<11> erent numerical resolutions. Lower panel: Fixed resolution of 1440 GLYPH<2> 360 and di GLYPH<11> erent viscosities. Both are averaged from orbits 1000 to 3000.</caption>
</figure>
<text><location><page_7><loc_51><loc_44><loc_94><loc_54></location>Near the outer boundary we see sometimes a region with standing waves, indicating that the radial domain should not be too small. This region is mostly only a few wavelengths in size (less then 1AU), but can sometimes also reach a few AU into the domain. Reflections with the outer boundary play a role here as well as can be seen in Fig. 10 for example at t GLYPH<25> 500 or 3600. We note that in contrast to our treatment at the inner radial boundary, we did not apply a damping region at the outer boundary.</text>
<text><location><page_7><loc_51><loc_16><loc_94><loc_37></location>With the wavelength fixed, the Reynolds stress also shows no strong dependence on the resolution as can be seen Fig. 11 top panel. The inner region is strongly suppressed because we also increased the damping from 2 GLYPH<0> 3AU. With that we conclude that a small viscosity is necessary to introduce a physical lengthscale for the smallest unstable wavelength. To further explore the role of viscosity we repeat the simulation for di GLYPH<11> erent viscosities. This is done with a resolution of 1440 GLYPH<2> 360. The growth rate is then calculated by fitting an linear function to the logarithm of the kinetic energy, which was at each point averaged over 100 grid cells. The results in the lower panel of Fig. 11 indicate that the for the two lowest viscosities (10 GLYPH<0> 8 and 10 GLYPH<0> 7 ) the stresses are given by the numerical viscosity. For the intermediate case (10 GLYPH<0> 7 ) the stresses are larger while for very large values the effect of the increased damping near the inner boundary influences the results.</text>
<text><location><page_7><loc_51><loc_37><loc_94><loc_44></location>To check if indeed the viscosity is important for the wavelength we add a small viscosity of GLYPH<23> = 5 GLYPH<1> 10 GLYPH<0> 7 . As expected this leads to a wavenumber that is independent of the resolution, as shown in the bottom panel of Fig. 7. The wavelength is in the order of 0 : 2AU at a radius of 4AU after the instability is saturated.</text>
<section_header_level_1><location><page_7><loc_51><loc_13><loc_78><loc_14></location>5. Discs with radiation transport</section_header_level_1>
<text><location><page_7><loc_51><loc_10><loc_94><loc_12></location>The isothermal discs discussed above do not capture the full physics, most importantly the transport of energy is missing.</text>
<figure>
<location><page_8><loc_8><loc_67><loc_92><loc_93></location>
<caption>Fig. 10. Illustration of the large scale time development of the instability. Displayed is the vertically averaged momentum in the meridional direction for the inviscid isothermal simulation with a resolution of 1024 GLYPH<2> 256 (red curve in top panel of Fig. 7).</caption>
</figure>
<figure>
<location><page_8><loc_6><loc_32><loc_49><loc_59></location>
<caption>Fig. 12. Discs with radiation transport for two di GLYPH<11> erent densities, GLYPH<26> 0. Upper panel: The midplane temperature at 4-5AU as a function of time. Lower panel: Kinetic energy in the meridional flow in the discs.</caption>
</figure>
<text><location><page_8><loc_6><loc_15><loc_49><loc_23></location>In this section we include radiative transport and the heating / cooling interaction of the gas with the radiation. In the first set of models we start from the isothermal models as described above and switch on the radiation according to eqs. (3) and (4), in a second series of models (in Sect. 6) we include irradiation from the central star.</text>
<text><location><page_8><loc_6><loc_10><loc_49><loc_15></location>For the simulations with radiative transport we use a resolution of 1024 GLYPH<2> 256 and the same spatial extent and initial conditions as in the isothermal case. In Fig. 12 we show the midplane temperature averaged from 4-5AU and the meridional ki-</text>
<text><location><page_8><loc_51><loc_43><loc_94><loc_59></location>netic energy when radiation is included, for two di GLYPH<11> erent values of the disc density GLYPH<26> 0. In both cases the kinetic energy has initially larger amplitudes than in the previous isothermal simulations because now the disc is no longer in hydrostatic equilibrium initially, and small motions in the meridional plane set in (lower panel in Fig. 12). For the same disc density as before, GLYPH<26> 0 = 10 GLYPH<0> 10 , the disc cools o GLYPH<11> quickly as soon as the instability begins to be active, at around t = 10. The reason lies in the e GLYPH<14> cient radiative cooling in this case, in particular near the surface layers where the optical depth is small and the instability most active. Hence, any turbulent heating will be radiated away quickly.</text>
<text><location><page_8><loc_51><loc_29><loc_94><loc_44></location>We repeated the simulation with a higher density, GLYPH<26> 0 = 10 GLYPH<0> 9 at 1AU, to increase the optical thickness. Now the disc does not cool so e GLYPH<14> ciently such that the instability begins to set in between t = 10 and t = 20 orbits, very similar to the isothermal models. But then radiative cooling eventually leads again to a cooling of the disc and the instability dies out. From these results it is clear that the instability does not produce enough heat and cannot survive without an external source of heat, for the typical opacities and densities expected in protoplanetary discs. This potential problem was pointed out already by Nelson et al. (2013).</text>
<section_header_level_1><location><page_8><loc_51><loc_25><loc_65><loc_26></location>6. Irradiated Disc</section_header_level_1>
<text><location><page_8><loc_51><loc_19><loc_94><loc_24></location>Here, we extend our models and include irradiation from the central star as an external heat source. Of course there are also other sources possible, for example, the inner region of the disc where the MRI is still active could be important.</text>
<section_header_level_1><location><page_8><loc_51><loc_16><loc_69><loc_17></location>6.1. Method of irradiation</section_header_level_1>
<text><location><page_8><loc_51><loc_10><loc_94><loc_15></location>We use a simple model for the external heating and consider vertical irradiation from above and below the disc, where the energy flux, F irr , depends on radius. This procedure evades the problem of finding a self-consistent solution for the flaring of</text>
<figure>
<location><page_9><loc_6><loc_75><loc_49><loc_93></location>
<caption>Fig. 13. Growth rates for models with radiation transport and irradiation (irr) and isothermal models (iso) that have the same mean temperature, for comparison. For each radius the kinetic energy was smoothed over a range of 10% of its radius before fitting it to an exponential growth.</caption>
</figure>
<text><location><page_9><loc_6><loc_64><loc_49><loc_66></location>the disc, as done for an irradiated and internally heated disc by Bitsch et al. (2013).</text>
<text><location><page_9><loc_6><loc_54><loc_49><loc_64></location>To obtain a first approximation for the flux in the meridional direction we assume that the angle of incidence of the flux is approximately R GLYPH<12> = r , where R GLYPH<12> is the star's radius. This applies to an infinitely flat disc as well as to the upper and lower surfaces our computational grid in spherical polar coordinates because all three represent planes that cross the center of the central star. We obtain for the meridional component of the flux</text>
<formula><location><page_9><loc_6><loc_50><loc_49><loc_53></location>F irr GLYPH<18> = Fr R GLYPH<12> R ; (17)</formula>
<text><location><page_9><loc_6><loc_40><loc_49><loc_49></location>where Fr = F GLYPH<12> ( R GLYPH<12> = r ) 2 is the radial flux from the star at a distance r . Applying this impinging vertical irradiation to the disc leads to a radial temperature profile exponent in the disc of q = GLYPH<0> 0 : 55, in good agreement with the models of Chiang & Goldreich (1997). Our procedure does not allow for self-shadowing e GLYPH<11> ects (Bitsch et al. 2013) but should give a physically realistic estimate of the expected temperatures in the disc.</text>
<text><location><page_9><loc_6><loc_31><loc_49><loc_40></location>For the irradiation opacity we use here, to simplify the calculations and obtain a first order estimate of the e GLYPH<11> ect, the same Rosseland opacity of Bell & Lin (1994). Hence, in the simulations we use presently the same opacity for the irradiation, Rosseland and Planck opacity (Bitsch et al. 2013). Numerically, we perform a ray-tracing method to calculate the energy deposited in each cell of the computational grid (Kolb et al. 2013).</text>
<section_header_level_1><location><page_9><loc_6><loc_28><loc_18><loc_29></location>6.2. Growth rate</section_header_level_1>
<text><location><page_9><loc_6><loc_18><loc_49><loc_27></location>To measure the growth rates of the instability for discs with radiation transport and irradiation we ran models for with zero viscosity and a higher resolution case with GLYPH<23> = 10 GLYPH<0> 7 . To be able to compare the growth rates with the previous isothermal cases, we performed additional isothermal simulations using the temperature profile from the simulations with radiation transport and irradiation.</text>
<text><location><page_9><loc_6><loc_10><loc_49><loc_18></location>The growth rates for the instability in combination with radiation transport are di GLYPH<14> cult to capture, because the simulation can not be started in hydrostatic equilibrium because the equilibrium vertical profile is unknown. We use strong damping for the first 10 orbits to remove the disturbance caused by the transition to the new density and temperature profile.</text>
<figure>
<location><page_9><loc_51><loc_76><loc_93><loc_93></location>
<caption>Fig. 14. Growth of the kinetic energy in the 2D-plane with radiation transport and isothermal for comparison.</caption>
</figure>
<text><location><page_9><loc_90><loc_75><loc_91><loc_77></location>×</text>
<figure>
<location><page_9><loc_51><loc_59><loc_93><loc_75></location>
</figure>
<text><location><page_9><loc_90><loc_58><loc_91><loc_60></location>×</text>
<text><location><page_9><loc_51><loc_33><loc_94><loc_52></location>The results are shown in Fig. 13. We note that this time the growth rates should depend on radius, because the growth depends on H = R which is not constant in the radiative cases. From Fig. 13 it is clear that the growth rates for the isothermal models are now lower than in the cases presented above, because the temperature is first lower and second the radial profile is flatter as before, and both are important for growth. For the irradiated models the growth is again lower, with 0 : 1 GLYPH<0> 0 : 2 per local orbit around half the value for the isothermal case. In Fig. 14 we display the evolution of the kinetic energy for the irradiated and corresponding isothermal model. For the inviscid case the final saturation level agree very well with each other while for the viscous disc with GLYPH<23> = 10 GLYPH<0> 7 the instability is weaker in the inner regions of the disc (see below).</text>
<section_header_level_1><location><page_9><loc_51><loc_30><loc_70><loc_31></location>6.3. Quasistationary phase</section_header_level_1>
<text><location><page_9><loc_51><loc_10><loc_94><loc_29></location>In the top panel of Fig. 15 we display the vertical temperature distribution for the saturated state at di GLYPH<11> erent radii in the disc for the model without viscosity at a resolution of 1024 GLYPH<2> 256. The other models look very similar. In the bulk part of the disc the profile is quite flat with a slight drop towards the midplane. Alower central temperature is to be expected for irradiated dics, detailed radiative transfer models indicate an even larger temperature drop towards the midplane (Dullemond et al. 2002). In the upper layers the temperature falls o GLYPH<11> because the disc is optically thin and the energy can freely leave the system. This drop of the temperatures towards the surface despite the irradiation is a result of the identical irradiation and Rosseland opacity. If more radiation is allowed to be absorbed in the disk by increasing the irradiation opacity then one can obtain hotter surface layers. For a ten times larger value we find a hot corona similar to Flock</text>
<text><location><page_10><loc_10><loc_92><loc_12><loc_94></location>180</text>
<figure>
<location><page_10><loc_7><loc_66><loc_49><loc_93></location>
<caption>Fig. 15. Upper Panel: The temperature profile for an irradiated disc in the saturated phase without viscosity and a resolution of 1024 GLYPH<2> 256. The dotted line is a run without hydrodynamics, only solving for the radiation energy. Lower Panel: The vertically integrated optical depth.</caption>
</figure>
<text><location><page_10><loc_6><loc_42><loc_49><loc_56></location>et al. (2013) and a cooler midplane. First results seem to indicate a reduction in the Reynolds stress in this case, probably due to the lower temperature in the bulk of the disc. At this point we leave the details to subsequent studies. The dotted line in Fig. 15 shows the profile for a simulation where we only solve for the radiation energy and disable the hydrodynamic solver. We can infer from this that the flat profile is a result of the combination of turbulent heating and vertical motion. A test simulation with a passive tracer added in the midplane of the disk in the saturated state showed indeed rapid spreading over the whole vertical extent of the disk.</text>
<text><location><page_10><loc_6><loc_28><loc_49><loc_42></location>The vertically integrated optical depth is shown in the lower panel, starting from very small values at the disc surfaces it reaches 30-100 at the di GLYPH<11> erent radii. The nearly constant vertical temperature within the disc motivates us to use the equilibrium azimuthal velocity for the corresponding isothermal model of the steady state to calculate the Reynolds stress. In Fig. 16 we can see that it is still a good approximation. Note that this time the comparison is done with a 2D-simulation. Also shown is the Reynolds stress calculated with the Kepler velocity instead of the equilibrium velocity.</text>
<text><location><page_10><loc_6><loc_20><loc_49><loc_28></location>While the growth rates are weaker than in the isothermal case, the kinetic energy in the meridional plane for stable saturated phase in Fig. 17 reaches the same level with radiation transport. The values for GLYPH<11> are again between 0 : 5 GLYPH<1> 10 GLYPH<0> 4 and 2 GLYPH<1> 10 GLYPH<0> 4 , depending on the wavelength and thus viscosity, but independent on radiation transport.</text>
<text><location><page_10><loc_6><loc_10><loc_49><loc_20></location>Di GLYPH<11> erent is of course the strength of the instability measured in terms of the value of the viscosity under which it still survives. Here, in the irradiated case, even a low viscosity of 10 GLYPH<0> 7 does suppress the instability in the inner regions of the disc. This is not only a result of the radiation transport, but also of the flat temperature profile. The details will depend on the source of the heating and the opacity, but nevertheless the stability will be weaker than in the purely isothermal case.</text>
<figure>
<location><page_10><loc_51><loc_75><loc_94><loc_93></location>
<caption>Fig. 16. Irradiated run: The Reynolds stress was averaged over 41 time steps, from orbit 1000 to 5000 each step 100 orbits apart, calculated with di GLYPH<11> erent averaging methods. For 'mean time' the mean u GLYPH<30> was calculated through averaging over 40 time steps. For 'Kepler' the velocity was calculated by subtracting the Kepler velocity and for the last one u GLYPH<30> is calculated analytically by using the equilibrium equation (11). Spatial averages are taken from 4AU to 10AU.</caption>
</figure>
<figure>
<location><page_10><loc_51><loc_45><loc_93><loc_64></location>
<caption>Fig. 17. Comparison of the Reynolds stress divided by the mean pressure. 'irr' stands for the irradiated disc, 'iso' for the isothermal disc with analogous initial conditions. Averaged over 2000 orbits, beginning with orbit 2000.</caption>
</figure>
<section_header_level_1><location><page_10><loc_51><loc_36><loc_62><loc_37></location>6.4. Discussion</section_header_level_1>
<text><location><page_10><loc_51><loc_23><loc_94><loc_35></location>As we have shown in the previous sections for an irradiated disc there exists the possiblity of generating an e GLYPH<11> ective turbulence through the vertical shear instability. As pointed out in Nelson et al. (2013) the instability can only be sustained if the di GLYPH<11> usion (local relaxation) time is a fraction of the local orbital period. To investigate how this condition is fulfilled in our simulations we analyse for the equilibrium irradiated disc models the timescale for radiative di GLYPH<11> usion. In units of the local orbital period this is given by</text>
<formula><location><page_10><loc_51><loc_19><loc_94><loc_22></location>t di GLYPH<11> = GLYPH<1> x 2 cP GLYPH<26> 2 GLYPH<20> R 4 GLYPH<21> acT 3 GLYPH<1> GLYPH<10> 2 GLYPH<25> (18)</formula>
<text><location><page_10><loc_51><loc_10><loc_94><loc_19></location>where GLYPH<1> x is the characteristic wavelength of the perturbation. In our case the radial di GLYPH<11> usion is relevant (Nelson et al. 2013) and we choose here GLYPH<1> x = 0 : 05 r , which is a typical radial wavelength at r = 3AU. Using eq. (18) and the results from the simulation we calculate for the optical thin region a very small cooling time per orbit of t di GLYPH<11> = 10 GLYPH<0> 10 as expected. For the optical thick region we obtain t di GLYPH<11> = 0 : 11 for our standard density, which is indeed a</text>
<text><location><page_11><loc_12><loc_93><loc_12><loc_94></location>1</text>
<figure>
<location><page_11><loc_6><loc_75><loc_49><loc_93></location>
<caption>Fig. 18. The radiative di GLYPH<11> usion time per Orbit at 4AU for a lengthscale of 0 : 1AU</caption>
</figure>
<text><location><page_11><loc_6><loc_62><loc_49><loc_69></location>small fraction of the orbital period as required for the instability to operate, see Fig. 18. The cooling time in the vertical direction is longer, about a few orbital periods as implied by the vertical optical depth (see Fig. 15) but this will keep the disc nearly isothermal, again as required for instability.</text>
<text><location><page_11><loc_6><loc_37><loc_49><loc_49></location>As seen in Fig. 15 an increase in the density leads to higher optical depths and longer di GLYPH<11> usion times, and consequently to a weaker instability. While doubling the density in a simulation with resolution 2048 GLYPH<2> 512 has no clear influence on the kinetic energy and the cooling times in the optical thin regions, the Reynolds stress was clearly weaker by a factor of around 1.5 in the simulation with doubled density (the model in the middle of Fig. 18). In addition the wavelength of the perturbations is decreased.</text>
<text><location><page_11><loc_6><loc_49><loc_49><loc_62></location>In Fig. 19 we illustrate that the instability still resembles closely the locally isothermal case except that the small scale perturbations are missing, even in the optical thin region, where we have very small cooling times. For comparison, Nelson et al. (2013) found that the instability was completely suppressed with relaxation times of t relax = 0 : 1, which is the timescale for the flow to relax to the initial isothermal profile. We take this as an indication that radiative di GLYPH<11> usion plus irradiation behaves physically di GLYPH<11> erent from a simple model of temperature relaxation as used in Nelson et al. (2013).</text>
<text><location><page_11><loc_6><loc_26><loc_49><loc_37></location>A further increase of the density leads also to a strong decrease in the kinetic energy, with again a smaller wavelength. This raises the question whether the simulation with resolution of 2048 GLYPH<2> 512 is su GLYPH<14> ciently resolved. These results indicate that in very massive discs with long di GLYPH<11> usion times (vertical and radial) the disc will behave more adiabatically, and the instability will be quenched. The minimum solar mass nebula corresponds at 5 AU approximately to our model with 2 GLYPH<26> 0 and the instability might just be operative.</text>
<section_header_level_1><location><page_11><loc_6><loc_22><loc_30><loc_23></location>7. Summary and conclusions</section_header_level_1>
<text><location><page_11><loc_6><loc_14><loc_49><loc_21></location>We have studied the vertical shear instability (VSI) as a source of turbulence in protoplanetary discs. For that purpose we have performed numerical simulations solving the equations of hydrodynamics for a grid section in spherical polar coordinates. To study the global behaviour of the instability we have used an large radial extension of the grid ranging from 2 to 10 AU.</text>
<text><location><page_11><loc_6><loc_10><loc_49><loc_14></location>In a first set of simulations we show that the instability occurs for locally isothermal discs where the radial temperature gradient is a given function of radius. Our results on the growth</text>
<figure>
<location><page_11><loc_51><loc_57><loc_92><loc_93></location>
<caption>Fig. 19. Velocity in the meridional direction, u GLYPH<18> , in units of local Kepler velocity for an irradiated run without viscosity at resolution 1024 GLYPH<2> 256 and below with resolution 2048 GLYPH<2> 512. Compare with Fig. 2 which has a spatial resolution of 2048 GLYPH<2> 512.</caption>
</figure>
<text><location><page_11><loc_51><loc_38><loc_94><loc_48></location>rates for the instability are in good agreement with the theoretical estimates by Urpin & Brandenburg (1998); Urpin (2003), and we find two basic growth regimes for the asymmetric and antisymmetric modes as seen by Nelson et al. (2013). After 20 to 30 local orbits the instability saturates and is dominated by the vertical motions, which cover the whole vertical extent of the disc.</text>
<text><location><page_11><loc_51><loc_27><loc_94><loc_38></location>Interestingly, we find that the local radial wavelength of the perturbations scales approximately with GLYPH<21> / r 2 : 5 in the saturated state with a constant frequency. However, on a global scale several jumps occur where the wavelengths are halved, such that the global scaling follows ¯ GLYPH<21> / r with ¯ GLYPH<21>= r = 0 : 03. We suspect that the instability has the tendency to generate global modes that show the observed wavelengths behaviour according to eq. (16). Due to the radial stratification of the disc jumps have to occur at some locations.</text>
<text><location><page_11><loc_51><loc_10><loc_94><loc_27></location>The waves approximately keep their shape and travel slowly inwards. The two- and three dimensional simulations yield essentially the same results concerning the growth rates and saturation levels of the instability because of its axisymmetric property. The motions give rise to a finite level of turbulence and we calculate the associated e GLYPH<14> ciency, measured in terms of GLYPH<11> . We first show that, caused be the two-dimensionality, GLYPH<11> can be measured directly from the two-dimensional simulations using the proper equilibrium state of the disc. We find that the angular momentum associated with the turbulence is positive and reaches GLYPH<11> -values of a few 10 GLYPH<0> 4 . For the isothermal simulations we find that upon higher numerical resolution GLYPH<11> becomes smaller but viscous simulations indicate a saturation at a level of about GLYPH<11> = 10 GLYPH<0> 4</text>
<text><location><page_12><loc_6><loc_91><loc_49><loc_93></location>even for very small underlying viscosities that are equivalent to GLYPH<11> < 10 GLYPH<0> 6 .</text>
<text><location><page_12><loc_6><loc_72><loc_49><loc_82></location>In summary, our simulations indicate that the VSI can indeed generate turbulence in discs albeit at a relatively low level of about few times 10 GLYPH<0> 4 . This implies that even in (magnetically) dead zones the e GLYPH<11> ective viscosity in discs will never fall below this level. Our results indicate that in fully 3D simulations the transport may be marginally larger, but further simulations will have to be performed to clarify this point.</text>
<text><location><page_12><loc_6><loc_81><loc_49><loc_91></location>Adding radiative transport leads to a cooling from the disc surfaces and the instability dies out subsequently. We then constructed models where the disc is irradiated from above and below which leads to a nearly constant vertical temperature profile within the disc. This leads again to a turbulent saturated state with a similar transport e GLYPH<14> ciency as the purely isothermal simulations, possibly slightly higher (see Fig. 17).</text>
<text><location><page_12><loc_6><loc_61><loc_49><loc_72></location>Acknowledgements. Moritz Stoll received financial support from the Landesgraduiertenförderung of the state of Baden-Württemberg. Wilhelm Kley acknowledges the support of the German Research Foundation (DFG) through grant KL 650 / 8-2 within the Collaborative Research Group FOR 759: The formation of Planets: The Critical First Growth Phase. Some simulations were performed on the bwGRiD cluster in Tübingen, which is funded by the Ministry for Education and Research of Germany and the Ministry for Science, Research and Arts of the state Baden-Württemberg, and the cluster of the Forschergruppe FOR 759 'The Formation of Planets: The Critical First Growth Phase' funded by the DFG.</text>
<section_header_level_1><location><page_12><loc_6><loc_57><loc_16><loc_58></location>References</section_header_level_1>
<text><location><page_12><loc_6><loc_24><loc_49><loc_56></location>Arlt, R. & Urpin, V. 2004, A&A, 426, 755 Armitage, P. J. 2011, ARA&A, 49, 195 Balbus, S. A. 2003, ARA&A, 41, 555 Bell, K. R. & Lin, D. N. C. 1994, ApJ, 427, 987 Bitsch, B., Crida, A., Morbidelli, A., Kley, W., & Dobbs-Dixon, I. 2013, A&A, 549, A124 Chiang, E. I. & Goldreich, P. 1997, ApJ, 490, 368 de Val-Borro, M., Edgar, R. G., Artymowicz, P., et al. 2006, MNRAS, 370, 529 Dullemond, C. P., van Zadelho GLYPH<11> , G. J., & Natta, A. 2002, A&A, 389, 464 Flaig, M., Ruo GLYPH<11> , P., Kley, W., & Kissmann, R. 2012, MNRAS, 420, 2419 Flock, M., Dzyurkevich, N., Klahr, H., Turner, N. J., & Henning, T. 2011, ApJ, 735, 122 Flock, M., Fromang, S., González, M., & Commerçon, B. 2013, A&A, 560, A43 Fricke, K. 1968, ZAp, 68, 317 Fromang, S. & Nelson, R. P. 2006, A&A, 457, 343 Goldreich, P. & Schubert, G. 1967, ApJ, 150, 571 Klahr, H. H. & Bodenheimer, P. 2003, ApJ, 582, 869 Kley, W., Papaloizou, J. C. B., & Lin, D. N. C. 1993, ApJ, 416, 679 Kolb, S. M., Stute, M., Kley, W., & Mignone, A. 2013, A&A, 559, A80 Levermore, C. D. & Pomraning, G. C. 1981, ApJ, 248, 321 Lin, D. N. C. & Pringle, J. E. 1987, MNRAS, 225, 607 Mignone, A., Bodo, G., Massaglia, S., et al. 2007, ApJS, 170, 228 Minerbo, G. N. 1978, J. Quant. Spec. Radiat. Transf., 20, 541 Nelson, R. P., Gressel, O., & Umurhan, O. M. 2013, MNRAS, 435, 2610 Pringle, J. E. 1981, ARA&A, 19, 137 Ruden, S. P., Papaloizou, J. C. B., & Lin, D. N. C. 1988, ApJ, 329, 739 Shakura, N. & Sunyaev, R. 1973, A&A, 24, 337 Turner, N. J., Fromang, S., Gammie, C., et al. 2014, ArXiv e-prints Urpin, V. 2003, A&A, 404, 397 Urpin, V. & Brandenburg, A. 1998, MNRAS, 294, 399</text>
</document> | [] |
2015ApJ...804...32G | https://arxiv.org/pdf/1502.07024.pdf | <document>
<section_header_level_1><location><page_1><loc_11><loc_81><loc_89><loc_84></location>HIDING IN PLAIN SIGHT: AN ABUNDANCE OF COMPACT MASSIVE SPHEROIDS IN THE LOCAL UNIVERSE</section_header_level_1>
<text><location><page_1><loc_41><loc_80><loc_42><loc_80></location>1</text>
<text><location><page_1><loc_15><loc_77><loc_85><loc_80></location>Alister W. Graham , Bililign T. Dullo, Giulia A.D. Savorgnan Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Hawthorn, VIC, 3122, Australia. Draft version September 2, 2018</text>
<section_header_level_1><location><page_1><loc_45><loc_74><loc_55><loc_75></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_14><loc_50><loc_86><loc_74></location>It has been widely remarked that compact, massive, elliptical-like galaxies are abundant at high redshifts but exceedingly rare in the Universe today, implying significant evolution such that their sizes at z ∼ 2 ± 0 . 6 have increased by factors of 3 to 6 to become today's massive elliptical galaxies. These claims have been based on studies which measured the half-light radii of galaxies as though they are all single component systems. Here we identify 21 spheroidal stellar systems within 90 Mpc that have half-light, major-axis radii R e /lessorsimilar 2 kpc, stellar masses 0 . 7 × 10 11 < M ∗ /M /circledot < 1 . 4 × 10 11 , and S'ersic indices typically around a value of n = 2 to 3. This abundance of compact, massive spheroids in our own backyard - with a number density of 6 . 9 × 10 -6 Mpc -3 (or 3.5 × 10 -5 Mpc -3 per unit dex in stellar mass) - and with the same physical properties as the high-redshift galaxies, had been over-looked because they are encased in stellar disks which usually result in galaxy sizes notably larger than 2 kpc. Moreover, this number density is a lower limit because it has not come from a volume-limited sample. The actual density may be closer to 10 -4 , although further work is required to confirm this. We therefore conclude that not all massive 'spheroids' have undergone dramatic structural and size evolution since z ∼ 2 ± 0 . 6. Given that the bulges of local early-type disk galaxies are known to consist of predominantly old stars which existed at z ∼ 2, it seems likely that some of the observed high redshift spheroids did not increase in size by building (3D) triaxial envelopes as commonly advocated, and that the growth of (2D) disks has also been important over the past 9-11 billion years.</text>
<text><location><page_1><loc_14><loc_47><loc_86><loc_49></location>Subject headings: galaxies: bulges - galaxies: fundamental parameters - galaxies: evolution - galaxies: formation - galaxies: high-redshift</text>
<section_header_level_1><location><page_1><loc_21><loc_43><loc_36><loc_45></location>1. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_7><loc_48><loc_43></location>A little over a decade ago it was advocated that the mass-size relation for massive galaxies ( M ∗ > 0 . 2 × 10 11 h 2 70 M /circledot ) had evolved little from z ∼ 2 . 5 to today (Trujillo et al. 2004). However this view was quickly challenged by Daddi et al. (2005), who had detected seven passively evolving galaxies at z > 1 . 4, with stellar masses /greaterorsimilar 10 11 h 2 71 M /circledot and morphologies typical of elliptical/early-type galaxies. They questioned the lack of evolution because some of their galaxies had much smaller effective half light radii than local elliptical galaxies of the same mass (see also Papovich et al. 2005). They suggested this difference was either a real sign of galaxy size evolution, or artificial because of biasing active galactic nuclei light or morphological K -corrections due to blue cores in the highz sample. Addressing the latter uncertainty, Kriek et al. (2006, 2008) found that nearly half of their distant massive galaxies had old stellar populations and van Dokkum et al. (2008) subsequently showed that half of those had sizes less than ∼ 2 kpc. Following Daddi et al. (2005), Trujillo et al. (2006a) reported galaxy sizes ranging from 1 to 5 kpc for ten massive ( ∼ 5 × 10 11 h 2 70 M /circledot ) galaxies at 1 . 2 < z < 1 . 7. From this Trujillo et al. (2006a) concluded that (i) the sizes were at least a factor of 3 to 6 times lower than 'the local counterparts' of similar mass, (ii) the structural properties of these highz objects are therefore rapidly changing, (iii) the data disagree with a scenario where</text>
<text><location><page_1><loc_52><loc_35><loc_92><loc_45></location>the more massive and passive galaxies are fully assembled by 1 . 2 < z < 1 . 7 (i.e. a monolithic scenario), and (iv) they suggested that a dry merger scenario (no new star formation) was responsible for the subsequent evolution of these galaxies (see also Trujillo et al. 2007; Toft et al. 2007; Zirm et al. 2007; Buitrago et al. 2008; van Dokkum et al. 2008; Damjanov et al. 2009).</text>
<text><location><page_1><loc_52><loc_11><loc_92><loc_35></location>While not denying the occurrence of dry mergers, nor claiming that early-type galaxies had to have formed in a monolithic collapse (see Dekel & Burkert 2014 and Zolotov et al. 2015 for an alternative), given the data under investigation we are questioning the universality of the above conclusions because they rest on the assumption as to what the 'local counterparts' to the compact, massive highz galaxies actually are. Popular opinion has been that these distant objects are very rare today (e.g. Cimatti et al. 2008, and references therein), with Trujillo et al. (2009) reporting a local number density of 10 -7 Mpc -3 , and Taylor et al. (2010) finding none and therefore concluding that the distant spheroidal stellar systems have experienced considerable size evolution to form today's massive elliptical galaxies. However, there are many independent lines of reasoning to think that this paradigm of early-type galaxy evolution may not be correct.</text>
<text><location><page_1><loc_52><loc_3><loc_92><loc_11></location>Key to much of this is that Trujillo et al. (2006a,b, 2007) used the sizes of z ∼ 0 . 1 Sloan Digital Sky Survey ( SDSS , York et al. 2000) early-type, i.e. elliptical and lenticular, galaxies (with S'ersic index n > 2 . 5), to claim that there has been significant size, and surfacemass density, evolution of the distant spheroids (see also</text>
<text><location><page_2><loc_8><loc_59><loc_48><loc_88></location>Taylor et al. 2010; Newman et al. 2012; McLure et al. 2013; Damjanov et al. 2014; Fang et al. 2015). In addition to early-type galaxies, there is however another type of spheroidal stellar system in the local universe, namely the bulges of disk galaxies, including the lenticular galaxies. Furthermore, there has been a slowly growing realization over the past three decades that most 2 early-type galaxies infact consist of a bulge and a rotating stellar disk (e.g. Carter 1987; Capaccioli 1987, 1990; Nieto et al. 1988; D'Onofrio et al. 1995; Graham et al. 1998; Emsellem et al. 2011; Scott et al. 2014), arguing against the treatment of lenticular galaxies as single component systems. While many of these galaxies do not have massive bulges, and some contain pseudobulges built out of disk material (Kormendy & Kennicutt 2004; see also Graham 2014 for cautionary remarks regarding their identification), our interest lies with some of the more massive lenticular and spiral galaxies that do have massive bulges. The most massive local bulges ( M ∗ > 2 × 10 11 M /circledot ) have sizes in excess of 2.5 kpc are therefore not considered 'compact'; as such they have been excluded from this study.</text>
<text><location><page_2><loc_8><loc_26><loc_48><loc_59></location>Here we effectively explore what evolutionary conclusion might have been drawn a decade ago for the compact massive spheroids at z ≈ 2 if the size-mass comparison had of been performed with the massive bulges of present-day early-type disk galaxies rather than with the combined bulge+disk system. In so doing we can answer the question, Have the highz spheroidal stellar systems truly evolved to near extinction, or are they perhaps hiding in plain sight around us today? While we do not deny that there has been galaxy size evolution, we are suggesting a fundamentally different formation model which presents a dramatic shift in our way of thinking about how the compact massive objects at z = 2 . 0 ± 0 . 6 have actually evolved in our Universe. It is such that they have not expanded 3-6 times in size, nor accreted, through dry minor mergers, a three-dimesnional (3D) envelope of a similarly large dimension, to become today's massive elliptical galaxies (e.g. Hopkins et al. 2009; Carrasco et al. 2010; Cimatti et al. 2012; Fan et al. 2013b, De et al. 2014). Instead, we advocate the view presented by Graham (2013) that they have grown two-dimensional (2D) planar disks and are thus effectively hidden today until bulge-to-disk decompositions are performed. It turns out that there are actually many reasons to favour this scenario.</text>
<text><location><page_2><loc_8><loc_9><loc_48><loc_26></location>There is no widely accepted solution as to how the distant objects could have grown into much larger 'spheroids' by today. For instance, major dry mergers can not account for the size evolution; they move galaxies along the mass-size relation rather than off it (e.g. Ciotti & van Albada 2001; Boylan-Kolchin et al. 2006; see also Bundy et al. 2009; Nipoti et al. 2009; Nair et al. 2011). In addition, based on observations of galaxy pairs, there have been insufficient major (i.e. near equal mass ratio) merger events to explain the removal of every distant, compact massive galaxy (e.g. Man et al. 2015; see also Shih & Stockton 2011). Similarly, there are not enough satellites observed around massive galaxies for</text>
<text><location><page_2><loc_8><loc_4><loc_48><loc_7></location>2 The exceptions are brightest cluster galaxies and luminous ( M B /lessorsimilar -20 . 5 ± 0 . 75 mag) elliptical galaxies built via major dry merger events, and very faint dwarf elliptical galaxies.</text>
<text><location><page_2><loc_52><loc_80><loc_92><loc_88></location>minor mergers (e.g. Khochfar & Burkert 2006; Maller et al. 2006; Naab et al. 2009; Hopkins et al. 2009; McLure et al. 2013) to be the sole explanation (Trujillo 2013, and references therein), and even in ΛCDM simulations there are not enough satellites to have transformed all of the distant spheroids.</text>
<text><location><page_2><loc_52><loc_52><loc_92><loc_80></location>It is worth noting that 'normal-sized' (i.e. not compact), massive elliptical-like galaxies are observed at these high-redshifts, co-existing with the 'compact', massive galaxies (e.g. Trujillo et al. 2006b; Mancini et al. 2010; Newman et al. 2010; Bruce et al. 2012; Ferreras et al. 2012; Fan et al. 2013a). Furthermore, new galaxies are known to appear on the 'red sequence' (e.g. Bell et al. 2004; Faber et al. 2007; Schawinski et al. 2007, 2014; van Dokkum et al. 2008; Martig et al. 2009; van der Wel et al. 2009; L'opez-Sanjuan et al. 2012; Carollo et al. 2013; Cassata et al. 2013; Keating et al. 2015; Belli et al. 2015). For example, Carollo et al. (2015) detail how past quenching can transform star-forming galaxies and shift them to the early-type sequence. Known as 'progenitor bias', studies which attempt to measure the evolution of the massive quiescent galaxies (e.g. Trujillo et al. 2006b; Ryan et al. 2012; Chang et al. 2013; van der Wel et al. 2014) can suffer from sample confusion because of the emerging population onto the red sequence; such studies are effectively sampling different objects as a function of redshift.</text>
<text><location><page_2><loc_52><loc_23><loc_92><loc_52></location>In mid-2011, Graham (2013, see also Driver et al. 2013) suggested that some of the distant objects are likely to be the bulges of today's massive early-type disk galaxies. Indeed, Graham (2013) showed that the distant compact massive objects and massive local bulges occupy the same region in the size-mass and size-density diagram. Dullo & Graham (2013) additionally showed that they possess the same radial concentration of light, as traced by the S'ersic index (S'ersic 1968, see Graham & Driver 2005 for a modern review in English). Moreover, massive bulges today are old. Their mass (not to be confused by their luminosity-weighted age) is dominated by old stars (MacArthur et al. 2009) and therefore such luminous bulges should be visible in images of the early universe. That is, some fraction of the compact massive objects reported at z ∼ 2 ± 0 . 6 are expected to be today's compact massive bulges. By calculating the number density of compact massive bulges at z ∼ 0, done for the first time in this work, we are able to address whether this fraction is low or instead if it might equal 1 and thus fully account for the fate of the distant compact massive objects.</text>
<text><location><page_2><loc_52><loc_4><loc_92><loc_23></location>Rather than some great mystery of unexplained galaxy growth, we are exploring the relatively mundane alternative idea that Daddi et al. (2005) and Trujillo et al. (2006a,b) simply observed the bulges of today's bright early-type galaxies at highz but did not know it. To address this, we have performed a preliminary search for nearby ( /lessorsimilar 100 Mpc), early-type disk galaxies with bulges having R e /lessorsimilar 2 kpc and M ∗ ≈ 10 11 M /circledot . This was done by checking on the galaxies presented in a handful of published papers listed in Table 1. This proved to be sufficient to illustrate that compact, massive spheroids exist in substantial numbers around us today. Section 2 of this paper presents the galaxies found, and the structural properties of their spheroidal component are also</text>
<text><location><page_3><loc_8><loc_84><loc_48><loc_88></location>provided there. Section 3 provides a general discussion of these findings and the growth of disks in our Universe. Section 4 finishes by re-iterating our main conclusions.</text>
<section_header_level_1><location><page_3><loc_21><loc_82><loc_35><loc_83></location>2. DATA SAMPLE</section_header_level_1>
<text><location><page_3><loc_8><loc_44><loc_48><loc_81></location>Five survey papers were consulted to check for the existence of compact massive bulges in the nearby Universe. Our criteria was that the stellar mass be greater than 0 . 7 × 10 11 M /circledot and the major-axis half-light radius be less than 2.5 kpc 3 . The first paper we examined was Dullo & Graham (2013) which has already broached this topic, and the second was Savorgnan et al. (2015, in preparation) which has carefully modeled the different structural components for 66 galaxies from a sample of 75 (Graham & Scott 2013; Scott et al. 2013) having directly measured black hole masses. Building on this, we checked two samples dominated by lenticular galaxies. First we inspected the sample of 175 early-type disk galaxies modeled by Laurikainen et al. (2010), and then the ATLAS 3D sample (Cappellari et al. 2011) of 260 nearby (D < 42 Mpc), predominantly northern hemisphere, early-type galaxies which have had the stellar bulge-to-total flux ratios derived by Krajnovi'c et al. (2013). The fifth catalog paper that we inspected was the spiral dominated compendium of Graham & Worley (2008), which resulted in the identification of targets in four additional papers (see Table 1). Having checked these papers, our Table 1 includes 20 known bulges along with the known compact massive galaxy NGC 1277 (van den Bosch et al. 2012), plus one additional galaxy , as opposed to a bulge, which also meets the above size and mass criteira. All but three of the 22 systems have R e < 2 kpc. All but 4 have M ∗ ≥ 0 . 9 × 10 11 M /circledot .</text>
<text><location><page_3><loc_8><loc_22><loc_48><loc_44></location>The major-axis, half-light radii of the spheroids are presented in Table 1, based on the galaxy distances which are also listed there. The circularized half-light radii would be even smaller, by a factor of √ b/a where b/a is the minor-to-major axis ratio of the bulge. This information was not readily available for many of our bulges, and so we have used the more conservative, larger radii. The published S'ersic indices have additionally been collated and given in Table 1 for ease of reference. Our preference has been for near-infrared data sets because the magnitudes are more reliable: they are less affected by dust and possible low level star formation. The absolute magnitudes were then corrected for redshift dimming (5 log[1 + z ]), foreground Galactic extinction (Schlafly & Finkbeiner 2011), and K -corrected (1 . 5 z , Poggianti 1997) which resulted in insignificant changes of typically 0.01-0.02 mag.</text>
<text><location><page_3><loc_8><loc_12><loc_48><loc_21></location>In the case of NGC 5493 (Krajnovi'c et al. 2013), the whole galaxy is rather compact ( R e = 2 . 46 kpc) - although this is the largest system in our sample - and massive ( M ∗ = 10 11 M /circledot ) 4 . This galaxy is not unique in the nearby universe. The intermediate-scale disk in NGC 1332, which does not dominate at large radii like large-scale disks do, is such that the size of this galaxy is</text>
<text><location><page_3><loc_8><loc_8><loc_48><loc_11></location>3 It should be noted that studies of the high' z galaxies typically report the smaller, circularized half-light radii rather than the major-axis half-light radii. Our approach is thus conservative.</text>
<unordered_list>
<list_item><location><page_3><loc_8><loc_4><loc_48><loc_8></location>4 Krajnovi'c et al. (2013) reported that their bulge/disk decomposition for NGC 5493 differed markedly from Laurikainen et al. (2010). Given this uncertainty, we use the whole galaxy rather than a result from a decomposition.</list_item>
</unordered_list>
<text><location><page_3><loc_52><loc_71><loc_92><loc_88></location>only slightly larger than 2 kpc (Savorgnan et al. 2015, in preparation). However here we include the bulge rather than the galaxy parameters for NGC 1332. A third and possible fourth example is NGC 1277 ( M ∗ = 1 . 2 × 10 11 M /circledot ) and NGC 5845 (excluded here because M ∗ = 0 . 5 × 10 11 M /circledot ), as already pointed out by van den Bosch et al. (2012) and Jiang et al. (2012), respectively. However, if the disks of these galaxies were to continue to grow, until they resembled the more stereotypical S0 galaxies today, with B/D flux ratios of 1/3 and R e , bulge /h disk ≈ 0 . 2 ( R e , bulge /R e , disk ≈ 0 . 12), then their galaxy sizes would likely exceed 2 kpc while the spheroidal component remained compact.</text>
<text><location><page_3><loc_52><loc_65><loc_92><loc_71></location>For reference, it was observed that the bulges with stellar masses M ∗ /greaterorsimilar 2 × 10 11 M /circledot tend to have major-axis, half-light radii R e > 2 . 5 kpc, and as such were not included here.</text>
<text><location><page_3><loc_52><loc_55><loc_92><loc_65></location>We elected not to apply any internal dust correction to the lenticular galaxies, but only to the spiral galaxies. The most massive lenticular galaxies are not representative of the typical dusty disk galaxy from which the dust corrections stem, and they likely contain little wide-spread dust. The following K -band dust correction from Driver et al. (2008) was applied to the bulges of the spiral galaxies.</text>
<formula><location><page_3><loc_55><loc_52><loc_92><loc_54></location>M corr bulge = M bulge -0 . 11 -0 . 79[1 -cos( i )] 2 . 77 , (1)</formula>
<text><location><page_3><loc_52><loc_23><loc_92><loc_51></location>where i is the inclination of the disk such that i = 90 deg corresponds to an edge-on orientation. For the Dullo & Graham (2013) sample of five lenticular galaxies, the stellar masses derived from the V -band magnitudes (RC3, de Vaucouleurs et al. 1991) agree with those derived from the K s -band magnitudes when no dust correction is applied and M/L K s = 0 . 8 and M/L V = 2 . 5 is used. While the Sombrero lenticular galaxy contains wide-spread dust in its disk, as do other lenticular galaxies (e.g. Temi et al. 2007), it may be that ion sputtering (e.g. Draine 2003) from a hot X-ray gas halo has destroyed the dust in the massive lenticular galaxies, although we have not checked for such X-ray halos in our sample. However, if we are mistaken and dust is present, it will mean that our estimated stellar masses should be increased. As shown in Driver et al. (2008), the average correction due to dust in the thousands of disk galaxies modeled as a part of the Millennium Galaxy Catalog (Liske et al. 2003; Allen et al. 2006) is < 0.14 mag if the inclination is < 45 o , and 0.18, 0.28 and 0.45 mag if the inclination is 55, 65 and 75 degrees, respectively.</text>
<text><location><page_3><loc_52><loc_3><loc_92><loc_23></location>Some of the near-infrared magnitudes in Table 1 were derived from Two Micron All-Sky Survey (2MASS) images (Skrutskie et al. 2006). As noted by Schombert & Smith (2012), the 2MASS total magnitudes are known to not capture all of a galaxy's light; they miss flux at large radii. However, for most S0s this appears to be contained to, on average, a tenth of a magnitude (Scott et al. 2013, their Figure 2), no doubt due to the rapid decline of the S0 galaxy's outer exponential light profile. For S0s with intermediate (i.e. not large) scale disks that do not dominate their galaxy's light at large radii, and if the S'ersic index of the bulge is large, then several tenths of a mag may be missed, as with the elliptical galaxies. The (Galactic extinction)-corrected B -K s colors (via NED but from the RC3, 2MASS and Schlafly & Finkbeiner</text>
<table>
<location><page_4><loc_11><loc_39><loc_88><loc_84></location>
<caption>TABLE 1 Galaxy Sample.</caption>
</table>
<text><location><page_4><loc_11><loc_26><loc_88><loc_39></location>Note . - Column 1: Galaxy identification. * The galaxy rather than bulge parameters are presented. † NGC 4649 (M60) is a particularly difficult galaxy to decompose and as such its bulge parameters may not be reliable. Column 2: Morphological Type. Column 3: Disk inclination such that 90 degrees corresponds to an edge-on disk. Column 4: Redshift taken from the NASA/IPAC Extragalactic Database (NED). Column 5: Distance from corresponding paper unless otherwise specified: NED = (Virgo + GA + Shapley) distance from NED using H 0 = 73 km s -1 Mpc -1 ); B09 = Blakeslee et al. (2009); Ton = Tonry et al. (2001) and corrected according to Blakeslee et al. (2002). Column 6: S'ersic index. Column 7: Major-axis, effective half-light radius of the galaxy's bulge component. Column 8: Near-infrared magnitude. Notes: * = galaxy rather than bulge parameters; ** 3.6 µ m magnitude rather than K - or K s -band. Column 9: Magnitude corrected for Galactic dust extinction (Schlafly & Finkbeiner 2011), (1 + z ) 2 cosmological dimming, and K -corrected using +1 . 5 z (Poggianti 1997). Column 10: Magnitude additionally corrected for internal dust using the correction from Driver et al. (2008). Column 11: Stellar mass-to-light ratio (assumes a 12 Gyr old population of solar metallicity and a Chabrier (2003) initial mass function: Baldry et al. 2008, their Figure A1). Column 12: Stellar mass derived using column 9 for the S0s and column 10 for the spirals, together with column 11.</text>
<text><location><page_4><loc_8><loc_16><loc_48><loc_24></location>2011) range from about 3.65 to 4.05. Given the expected color range of 3.85 to 4.03 for a 6 to 12.5 Gyr old population in model S0 galaxies (Buzzoni 2005), perhaps a couple of tenths of mag are missing from some 2MASS magnitudes. If one was to correct for this, it too would act to make the bulge masses bigger than reported here.</text>
<text><location><page_4><loc_8><loc_4><loc_48><loc_16></location>A Hubble constant of H 0 = 73 km s -1 Mpc -1 (e.g. WMAP 3-year; Riess et al. 2011) was used. However if it is smaller, for example 67.3 km s -1 Mpc -1 (Planck Collaboration 2014), then the absolute magnitudes will brighten by 0.18 mag and the stellar masses will increase by 18%. Based on SNe Ia data, Rigault et al. (2015) have also downwardly revised H 0 to 70 . 6 ± 2 . 6 km s -1 Mpc -1 . A mass increase of 18% would result in 18 of the 22 systems in Table 1 having stellar masses ≥ 10 11 M /circledot .</text>
<section_header_level_1><location><page_4><loc_60><loc_23><loc_84><loc_24></location>2.1. Example profile: NGC 5419</section_header_level_1>
<text><location><page_4><loc_52><loc_4><loc_92><loc_22></location>While information about the surface brightness profile decompositions can be found in the papers mentioned in Table 1, we felt that it may be instructive to include an example of how the outer exponential profile can increase the galaxy size over the bulge size. We have somewhat randomly chosen NGC 5419 from Laurikainen et al. (2010). This was chosen because a) the Laurikainen et al. paper simply contains the greatest number of compact massive spheroids in our Table 1, and b) this particular galaxy was modelled by Dullo & Graham (2014) as an elliptical galaxy while they noted that a disk might be present. Sandage & Tammann (1981) did not support the original elliptical galaxy classification in the Second Reference Catalogue of de Vaucouleurs et al. (1976), but</text>
<figure>
<location><page_5><loc_8><loc_54><loc_46><loc_89></location>
<caption>Fig. 1.Spitzer Space Telescope 3.6 µ m image of NGC 5419.</caption>
</figure>
<text><location><page_5><loc_8><loc_46><loc_48><loc_50></location>instead considered NGC 5419 to be a lenticular galaxy, as did Laurikainen et al. (2010) based on their near-infrared K -band analysis.</text>
<text><location><page_5><loc_8><loc_37><loc_48><loc_46></location>Therefore, we re-investigate this galaxy using a Spitzer Space Telescope 3.6 µ m image (see Figure 1), which was reduced following the procedures described in Savorgnan et al. (2015, in prep.). The light profile was extracted by allowing the ellipticity and position angle to vary about a fixed center using the IRAF task Ellipse (Jedrzejewski et al. 1987) and is shown in Figure 2.</text>
<text><location><page_5><loc_8><loc_15><loc_48><loc_37></location>Our modelling of the 3.6 µ m light profile within 60 '' matches the solution to the K -band data in Laurikainen et al. (2010). We also find that a single S'ersic function is inadequate; the curvature of the light profile, and the residual light profile, reveal the presence of additional components. Figure 2 shows how the addition of an outer exponential, and a very faint lens (fit with a Ferrers function), provides a better fit. These were the components employed by Laurikainen et al. (2010). From this, one can see that the half-light radius of the galaxy ( R e , gal = 53 '' . 6) is much greater than the half light radius of the inner spheroid ( R e , sph = 8 '' . 4). Laurikainen et al. (2010) reported that their bulge+lens+disk model had a bulge S'ersic index n = 1 . 4 and half light radius R e = 6 '' . 5. They reported a disk scale length h = 32 '' . 3, while our exponential model has h = 33 '' . 1.</text>
<text><location><page_5><loc_8><loc_3><loc_48><loc_15></location>One may wonder if there is also a contribution from an extended envelope around this galaxy due its location at the centre of the poor cluster Abell S753. Indeed, Sandage & Bedke (1994) referred to this galaxy as having an extended outer envelope, and Seigar et al. (2007, see also Pierini et al. 2008) established that intracluster light tends to have an exponential profile, i.e. the same radial decline as seen in disks. However, Bettoni et al. (2001) have reported a rotational velocity reaching 90 km s -1 by</text>
<figure>
<location><page_5><loc_54><loc_73><loc_92><loc_86></location>
<caption>Fig. 2.Left panel: Single S'ersic fit to the 3.6 µ m, major-axis light profile of NGC 5419. Right panel: As in Laurikainen et al. (2010), a S'ersic bulge (red), plus exponential function (dark blue), plus Ferrers function for a lens (cyan) has been fit. The root mean square of the deviations of the surface brightness data, µ , about the model is given by ∆.</caption>
</figure>
<text><location><page_5><loc_52><loc_44><loc_92><loc_55></location>the inner 5 '' , betraying the presence of at least an inner disk. In an extreme scenario, one may speculate that this galaxy has an intermediate-scale disk (the 'lens' component in Figure 2, which shows up in both the ellipticity and position angle profiles) plus intracluster stars that produce the near-exponential light profile at larger radii. More extended kinematics would be helpful in discriminating between these options.</text>
<text><location><page_5><loc_52><loc_27><loc_92><loc_44></location>The cluster-centric location of NGC 5419 is interesting, and it further prompted us to pursue the suggestion by L.Cortese (2015, priv. comm.) to check on the environment (e.g. field, group, cluster) of our sample. Table 2 shows this information, along with the R e /h size and B/T flux ratios obtained by the papers given in Table 1. While a few of the galaxies are the brightest of their small galaxy group, NGC 1316 (the interacting galaxy Fornax A) is the only other member of a substantialsized group. This rules out the idea that the compact massive spheroids might be encased in an exponentiallike 3D envelope of intracluster light rather than a 2D disk.</text>
<section_header_level_1><location><page_5><loc_67><loc_24><loc_77><loc_25></location>3. RESULTS</section_header_level_1>
<section_header_level_1><location><page_5><loc_54><loc_22><loc_91><loc_24></location>3.1. The mass-size and size-concentration diagrams</section_header_level_1>
<text><location><page_5><loc_52><loc_3><loc_92><loc_22></location>In Figure 3a we do not compare our data with the sizemass relation for early-type galaxies as given by Shen et al. (2003) because of the biases in their data which are explained in Graham & Worley (2008). Lange et al. (2015) have however fit the double power-law model from Shen et al. (2003) to the galaxy size-mass data from several thousand nearby (0 . 01 ≤ z ≤ 0 . 1) earlytype (morphologically-identified diskless) galaxies taken from the Galaxy And Mass Assembly (GAMA) survey (Driver et al. 2011). The galaxy magnitudes had been converted into stellar masses by Taylor et al. (2011), and the effective half light galaxy sizes derived by Kelvin et al. (2012) who fit single S'ersic models to images in ten bands ( ugrizZY JHK s ). We have taken the K s -band re-</text>
<table>
<location><page_6><loc_8><loc_59><loc_48><loc_84></location>
<caption>TABLE 2 Galaxy properties.</caption>
</table>
<table>
<location><page_6><loc_8><loc_47><loc_48><loc_59></location>
</table>
<text><location><page_6><loc_8><loc_28><loc_48><loc_46></location>Note . - Column 1: Galaxy identification. * The galaxy rather than bulge parameters were used. † Bulge/disk parameters may not be reliable. ‡ NGC 1332 has had its disk fit with a S'ersic model having n = 0 . 5, for which the scale length h = R e , disk / 0 . 82 (rather than R e , disk / 1 . 68 as is the case when n = 1). Column 2: Velocity dispersion (km s -1 ) from HyperLeda (Makarov et al. 2014), except for NGC 1277 (van den Bosh et al. 2012). Columns 3 and 4: Bulgeto-disk size ratio and bulge-to-total flux ratio, respectively. For the lenticular galaxies (and NGC 6646), values were taken from the respective papers. For the spiral galaxies, the dust-corrected values were derived in Graham & Worley (2008). Column 5: Environment of the galaxy. BCG = Brightest Cluster Galaxy. BGG = Brightest Group Galaxy. N represents the number of galaxies with known radial velocities in the group (Makarov & Karachentsev 2011; Tully et al. 2013). Field? = No group known to authors, and the galaxy appears to be in the field from looking at Digitized Sky Survey images available via NED.</text>
<text><location><page_6><loc_8><loc_5><loc_48><loc_24></location>sults from Lange et al. (2015) and show their mass-size relation in Figure 3a. Given that Figure 6 from Lange et al. (2015) reveals that the double power-law cannot fully capture the curvature in the GAMA data at M ∗ > 12 × 10 11 M /circledot - where galaxies have larger radii than the double power-law model - we have therefore additionally included the mass-size relation from Graham et al. (2006, see Eq.2.11 in Graham 2013). While Figure 3a reveals that the mass-size relation of local early-type galaxies have larger radii at a given mass than the distant spheroids, it also reveals that the masses and sizes of our local bulges overlap with those of the distant spheroids (Damjanov et al. 2011). Our data appears more clustered in Figure 3a simply because we have been stricter with the mass and size limit for our local bulge sample.</text>
<text><location><page_6><loc_10><loc_3><loc_48><loc_5></location>In Figure 3b we show the sizes and S'ersic indices of</text>
<text><location><page_6><loc_52><loc_70><loc_92><loc_88></location>our local compact massive systems, and the overlap with galaxies in the redshift interval 1.4 < z < 2.7, as given by Damjanov et al. (2011, their Table 2). We have excluded from Damjanov et al. those galaxies with no reported S'ersic index and those fit with an R 1 /n surface brightness model having a fixed value of n , such as 4. As with our bulge data, the highz galaxy data in Damjanov et al. has also been taken from a compilation of different surveys: 17 in their case. Their galaxy selection criteria is thus varied, and it can be seen in Figure 3 to contain a much larger range of sizes and masses than our data. Comparing the data points, we conclude that not all compact, massive spheroids need to have undergone significant structural and size evolution since z /lessorsimilar 2 . 5.</text>
<section_header_level_1><location><page_6><loc_54><loc_66><loc_90><loc_68></location>3.2. The number density of local, compact massive spheroids</section_header_level_1>
<text><location><page_6><loc_52><loc_43><loc_92><loc_65></location>Twenty one of the 22 systems (20 bulges plus 2 galaxies) are within 90 Mpc, with the additional system at 103.6 Mpc. Excluding this furthest bulge gives a number density of 6 . 9 ± 1 . 5 × 10 -6 Mpc -3 for the sample of 21. All of these systems have stellar masses in the range 0 . 7 × 10 11 < M ∗ /M /circledot < 1 . 4 × 10 11 . Ten of the 13 systems with M ∗ ≥ 10 11 M /circledot are within 70 Mpc, giving a similar number density 5 of 7 . 0 ± 2 . 2 × 10 -6 Mpc -3 . It needs to be remembered that these densities are a lower limit because we have not conducted a volume-limited, all-sky survey for compact massive spheroids. However, the ATLAS 3D survey did sample all of the bright galaxies over half the sky to a depth of 42 Mpc. Given that it contains three compact, massive systems, this corresponds to a number density of (1 . 9 ± 1 . 1) × 10 -5 Mpc -3 . This value is 2.75( ± 58%) times higher, but there is a larger uncertainty assuming Poissonian errors.</text>
<text><location><page_6><loc_52><loc_10><loc_92><loc_42></location>While Taylor et al. (2010) reported a reduction, at fixed size and mass, of at least 5000 in the co-moving number density of compact massive galaxies from z ∼ 2 . 3 to z ∼ 0 . 1, we find that there is a roughly comparable (within a factor of a few) number density of compact massive systems at z = 0 and z ∼ 2 . 5. Bezanson et al. (2009) report a number density of (3 ± 1) × 10 -5 Mpc -3 at this higher redshift, for systems with stellar mass densities greater than one billion solar masses within their innermost sphere of radius 1 kpc (or M ∗ > 10 11 M /circledot , R e < 2 . 88 kpc). Muzzin et al. (2013, their Figure 5) report a number density of ∼ 5 × 10 -5 Mpc -3 at 2 < z < 3 for all (compact and extended) quiescent galaxies with M ∗ ∼ 10 11 M /circledot , in fair agreement with the value of ∼ 4 × 10 -5 Mpc -3 for the quiescent galaxies at z = 3 ± 0 . 5 with M ∗ > 0 . 4 × 10 11 M /circledot reported by Straatman et al. (2014). At these high redshifts, it is speculated here that the bulk of the disk formation (which will remove 'galaxies' from satisfying the compactness criteria), may be yet to occur and thus one may have a cleaner sample of 'naked-bulges' with which to make a comparison with the number density of bulges in the local universe. At yet higher redshifts the 'naked-bulges' may likely still be developing themselves (e.g. Dekel & Burkert 2014).</text>
<text><location><page_6><loc_53><loc_9><loc_92><loc_10></location>Barro et al. (2013) report a similar number density as</text>
<text><location><page_6><loc_52><loc_4><loc_92><loc_8></location>5 The usual dependence on the Hubble constant - in this case h 3 73 -has been omitted given that the spherical volumes used have not been exactly matched to any particular outer-most galaxy's distance.</text>
<figure>
<location><page_7><loc_8><loc_58><loc_92><loc_90></location>
<caption>Fig. 3.Panel a) Stellar mass-size diagram, where the sizes of the local bulges are represented here by the major-axis effective half light radius R e . The curves show the parameterized fit to local 'elliptical' galaxy data from Graham et al. (2006, solid line) and Lange et al. (2015, specifically their Figure A9j and Eq.3, and their final K -band entry in Table 3, dashed line). The z = 0 bulge data is from Table 1 while the 1 . 4 < z < 2 . 7 data is from Damjanov et al. (2011). Local bulges with M ∗ /greaterorsimilar 2 × 10 11 M /circledot have R e /greaterorsimilar 2 . 5 kpc and were thus not included here because they are not considered to be compact galaxies. Panel b) Size-concentration diagram, where the concentration is quantified by the S'ersic index n .</caption>
</figure>
<text><location><page_7><loc_8><loc_25><loc_48><loc_49></location>Bezanson et al. (2009) at z = 2 . 5, even though Barro et al. used a lower stellar mass limit of 0.1 × 10 11 M /circledot for their sample (cf. 1.0 × 10 11 M /circledot used by Bezanson et al. 2009). Barro et al. reported that their number density increased as the redshift dropped, peaking at about 2.3 × 10 -4 Mpc -3 by z = 1 . 2, before dropping to lower densities as the redshift decreased further. Taken together, this suggests that the most massive, quiescent spheroids were in place first, with the less massive spheroids appearing later at lower redshifts (possibly connecting with the luminous blue compact galaxies forming at z < 1 . 4, Guzm'an et al. 1997). The above peak density at z = 1 . 2 is, at least in part, larger than the value we have found because Barro et al. included galaxies having a much broader range in stellar mass than we did. In a separate study, van der Wel et al. (2014) used a stellar mass limit > 0 . 5 × 10 11 M /circledot and found a peak density at z ∼ 1 . 2 of 1 × 10 -4 Mpc -3 .</text>
<text><location><page_7><loc_8><loc_3><loc_48><loc_25></location>The mass range sampled in our study only spans a factor of two, from 0.7 to 1.4 × 10 11 M /circledot . Our number density per unit dex in mass, as recorded in mass functions, is therefore five times higher, giving 3.5 × 10 -5 Mpc -3 dex -1 (or ≈ 10 -4 Mpc -3 dex -1 if using the volumelimited ATLAS 3D results) at M ∗ ≈ 10 11 M /circledot . These values are roughly 2-6 times lower than that from Barro et al. (2013), whose galaxy mass range exceeded 1 dex, and roughly 1-3 times lower than that from van der Wel et al. (2014), whose mass range was less than 1 dex. A proper comparison is however complicated because the local bulge mass function is not quite flat from 0.1 to 1.4 × 10 11 M /circledot , but increases as one moves to the lowermass end (e.g. Driver et al. 2007). For this reason, our number density for local compact massive bulges, as estimated above, is expected to be less than the actual</text>
<text><location><page_7><loc_52><loc_32><loc_92><loc_49></location>number density in the decade-wide mass range from say 0.14 to 1.4 × 10 11 M /circledot . Using a constant stellar M/L ratio across this mass range, to convert the bulge luminosity function in Driver et al. (2007) into a mass function, the actual number density in this decade range might be ∼ 40% higher, although predicting this properly requires knowledge of the slope of the mass function for local compact bulges. We hope to perform a more complete, volume-limited investigation of compact massive bulges within 100 Mpc in a forthcoming paper, enabling us to construct the mass function of local compact bulges with M ∗ > 10 10 M /circledot . This can then be better compared with the mass function at higher redshifts.</text>
<text><location><page_7><loc_52><loc_6><loc_92><loc_32></location>The WIde-field Nearby Galaxy-cluster Survey (WINGS) has reported the existence of many compact massive galaxies in nearby (0 . 04 < z < 0 . 07) clusters. Including substantially lower mass galaxies than us, Valentinuzzi et al. (2010) reported that 22% of the WINGS galaxies with 0 . 3 × 10 11 < M ∗ /M /circledot < 4 × 10 11 are compact, with a median size of 1 . 61 ± 0 . 29 kpc (and a median S'ersic index of 3 ± 0 . 6). For their cluster galaxy sample they derived a lower limit (because they excluded field galaxies) for the number density of (1 . 31 ± 0 . 09) × 10 -5 Mpc -3 within the co-moving volume between z = 0 . 04 and z = 0 . 07. This density drops to (0 . 46 ± 0 . 05) × 10 -5 Mpc -3 when they increase their lower mass limit from 0.3 × 10 11 M /circledot to 0.8 × 10 11 M /circledot (their Table 1). They observed that the bulk of their compact galaxies are lenticular galaxies, which are known to have ∼ 30% smaller disk sizes in galaxy cluster environments (e.g. Guti'errez et al. 2004; Head et al. 2014).</text>
<text><location><page_7><loc_52><loc_3><loc_92><loc_6></location>Using the Padova Millennium Galaxy and Group Catalogue (PM2GC) spanning 0 . 03 < z < 0 . 11, Poggianti et</text>
<text><location><page_8><loc_8><loc_76><loc_48><loc_88></location>al. (2013a,b) explored outside of clusters - most galaxies reside in the field or group environment - and concluded that 4.4% of local field and group galaxies with stellar masses 0 . 3 × 10 11 < M ∗ /M /circledot < 4 × 10 11 are compact, and that this 4.4% - most of which are lenticular galaxies - has a median mass-weighted age of 9.2 Gyr, and corresponds to a number density of (4 . 3) × 10 -4 Mpc -3 , or basically 3 . 2 × 10 -4 Mpc -3 dex -1 given their mass range sampled.</text>
<text><location><page_8><loc_8><loc_51><loc_48><loc_76></location>Although the above two studies treated the lenticular galaxies as single component systems when measuring both their ages and their circularized half-light radii, we suspect that it is the bulges of these galaxies which are the primary, compact massive spheroidal component. In a sense, we may therefore be dealing with the same type of galaxy as they are. While we do not try and resolve the question as to why their compact, massive galaxies were not found by others who searched in the /tie SDSS database, we do provide a couple of comments. Disk galaxies viewed edge-on will of course have circularized half-light radii which are notably smaller than the values obtained if they are viewed face-on. It may be of interest to see if the Valentinuzzi et al. (2010) and Poggianti et al. (2013a,b) detections are dominated by massive, edge-on disk galaxies. It may additionally be interesting to see the results (sizes, masses, number densities) after careful bulge/disk decompositions have been performed for these WINGS and PM2GC samples.</text>
<text><location><page_8><loc_8><loc_45><loc_48><loc_51></location>To recap our findings, we measure a lower limit of 3.5 × 10 -5 Mpc -3 dex -1 (or ≈ 10 -4 Mpc -3 dex -1 if using the volume-limited ATLAS 3D results) for compact ( R e /lessorsimilar 2 kpc) bulges with M ∗ ≈ 10 11 M /circledot .</text>
<section_header_level_1><location><page_8><loc_22><loc_43><loc_34><loc_44></location>4. DISCUSSION</section_header_level_1>
<section_header_level_1><location><page_8><loc_21><loc_41><loc_36><loc_42></location>4.1. The rise of disks</section_header_level_1>
<text><location><page_8><loc_8><loc_7><loc_48><loc_40></location>As noted in the Introduction, within the literature, minor mergers are the overwhelmingly preferred, albeit still problematic, solution to try and transform the compact, massive spheroids at high redshifts into larger spheroids by today. While our discovery of numerous compact massive spheroids at z = 0 suggests that this evolutionary path did not always transpire, we can still ask about minor mergers. If (wet or dry) minor mergers had built a large fraction of each disk around today's compact massive spheroids, then it may be telling us that there is something not quite right with ΛCDM simulations 6 . This is because the simulations contain somewhat random orientations of satellite galaxies which would not result in the formation of a disk, although see Tempel et al. (2015). However if minor mergers have built the disks, the observations may then be telling us that most central galaxies have had preferred 'Great Planes' on which their minor neighbors were located prior to infall and disk (rather than bulge) building. Some support for this idea can be found in the disks-of-satellites around the Milky Way and Andromeda (Kroupa et al. 2005; Metz et al. 2007; Ibata et al. 2013; Pawlowski et al. 2014). What is of course different in our scenario is that these Great Planes are not just a recent, local, phenomenon but would need to have always been present over the</text>
<text><location><page_8><loc_52><loc_87><loc_92><loc_88></location>past 10-13 Gyrs (see Goerdt, Burkert & Ceverino 2013).</text>
<text><location><page_8><loc_52><loc_63><loc_92><loc_87></location>Hammer et al. (2005, 2009) have reasoned that many rotating stellar disks could have been built in major gasrich mergers, with the disk forming due to the net angular momentum of the gas in the merger event (e.g. Barnes 2002; Robertson et al. 2006). Even in gas-poor (dry) mergers, Naab & Burkert (2003) report that disks can form if the net angular momentum is not canceled out (Fall 1979). Given that rotating spiral galaxies at z ≈ 0 . 65 are half as abundant as they are today (Neichel et al. 2008), disk growth since z = 2 ± 0 . 6 has obviously occurred, and the outer regions of galaxies with prominant bulges are known to be younger (e.g. P'erez et al. 2013; Li et al. 2015). Arnold et al. (2011) additionally remarked that lenticular galaxies might form through a two-phase inside-out assembly with the inner regions built early via a violent major merger, and wet minor mergers (rather than gas accretion) subsequently contributing to their outer parts.</text>
<text><location><page_8><loc_52><loc_34><loc_92><loc_63></location>It is recognized that there has not been enough galaxy merger events to transform the distant compact massive galaxies into today's ordinary -sized elliptical galaxies (e.g. Man et al. 2015), and therefore the above mechanisms cannot, on their own, be invoked to fully explain the growth of disks around the compact massive galaxies seen at highz such that they have become today's lenticular galaxies with typical disk-to-bulge flux ratios of 3 (i.e. bulge/total = 1 / 4). Observing higher gas fractions of lower metallicity at higher redshifts, B'ethermin et al. (2015) favour large gas reservoirs over major mergers as the cause of the intense star formation observed in massive galaxies at high redshifts. Following the idea of cyclical galaxy metamorphosis (White & Rees 1978; White & Frenk 1991; Navarro & Benz 1991; Steinmetz & Navarro 2002; see also Bournard & Combes 2002), Graham (2013) advocated earlier suggestions that 'cold' gas flows may have contributed to the development of these disks, either via spherical accretion of ∼ 10 4 -10 5 K gas (e.g. Birnboim & Dekel 2003; Birnboim et al. 2007) or streams (e.g. Kereˇs et al. 2005; Dekel et al. 2009; Kereˇs & Hernquist 2009; Danovich et al. 2014).</text>
<text><location><page_8><loc_52><loc_3><loc_92><loc_34></location>This alternative process, which can operate in parallel with mergers (e.g. Welker et al. 2015), involves the accretion of gas from not just the halo (e.g. Kauffmann et al. 1993, their section 2.7; 1999, their section 4.5; Genzel et al. 2006), and possible molecular gas reservoirs for some galaxies (e.g. Tacconi et al. 2008; Santini et al. 2014; Chapman et al. 2015), but also cold filamentary flows from the cosmic web (e.g. Ceverino et al. 2010, 2012; Goerdt et al. 2012; Rubin et al. 2012; Stewart et al. 2013). Although such cold gas accretion is yet to be convincingly demonstrated as a common phenomenon, it has the power to subsequently build a disk that transforms into stars. Indeed, it has been noted for well over a decade that the mass of a galaxy could double, due to gas accretion, in just a few Gyr (e.g. Katz et al. 1996; Bournaud & Combes 2002). The discovery of very low metallicity gas (0.02 solar) 37 kpc from a subL ∗ galaxy with solar metallicity by Ribaudo et al. (2011) revealed the presence of such gas, and we now know that there is lots of low metallicity gas in the halos of galaxies (e.g. Lehner et al. 2013; Prochaska et al. 2014). Furthermore, Bouch'e et al. (2013) have revealed the inflow (as opposed to just presence) of this material around a z = 2 . 3 galaxy</text>
<text><location><page_9><loc_8><loc_63><loc_48><loc_88></location>that has kinematics, metallicity and star formation typical of a rotationally supported disk, and it is such that the gas accretion rate is well-matched to the star formation rate (see also Conselice et al. 2013). It has also been established that there is a high spatial covering fraction of Lymanα gas clouds within 300 kpc of all galaxy types at highz (Wakker & Savage 2009; Prochaska et al. 2011; Thom et al. 2011; Stocke et al. 2013; Tumlinson et al. 2013), and a review of cool gas in high z galaxies can be found in Carilli & Walter (2013). In particular, even quiescent elliptical galaxies can have a massive cool reservoir around them (Thom et al. 2012; Tumlinson et al. 2013; Zhu et al. 2014; O'Sullivan et al. 2015). Although not surrounding a pre-existing compact, massive bulge, Prescott et al. (2015) report on the rotation of an 80 kpc Lymanα nebula at z ∼ 1 . 67 with an implied total mass of 3 × 10 11 M /circledot within a 20 kpc radius. While this is a developing field, extensive evidence for gas accretion at many redshifts is reviewed by Combes (2014, 2015b).</text>
<text><location><page_9><loc_8><loc_22><loc_48><loc_63></location>It seems likely that the compact massive galaxies would act as natural gravitational seeds around which filaments or streams of cold gas would flow inward and build disks that form stars. The existence of early-type galaxies with dual, large-scale counter-rotating stellar disks (e.g. NGC 4550, Rubin et al. 1992; NGC 3032, Young et al. 2008) supports the idea of disk growth via the accretion of external gas clouds (e.g. Coccato et al. 2014, and references therein), although it may also be due to minor mergers. Presumably feeding is usually from the same direction, given the low frequency of significant (by mass) counter-rotation, and after settling to the mid-plane the rotation is aligned. Small counter-rotating stellar disks and kinematically decoupled cores do however reveal that this is not always the case, and gas accretion can form interesting features such as warps, gas-star misalignments and polar rings (e.g. Briggs 1990; Jog & Combes 2009; Davis et al. 2011; Mapelli et al. 2015). However rather than only random accretion orientations, Pichon et al. (2011) explains how cold streams can build a disk in a coherent planer manner (see also Danovich et al. 2012, 2014; Prieto et al. 2013; Stewart et al. 2013; Cen 2014; Wang et al. 2014), which is of key importance. A second key aspect is the rapid formation of some of these disks (e.g. Agertz et al. 2009; Brooks et al. 2009) which would naturally explain the older ages of many lenticular galaxy disks today. It also implies significant galaxy growth independent of the merging of distinct entities such as dark matter halos. This process of gas accretion is still observed today in massive early-type galaxies (e.g. Davis et al. 2011).</text>
<text><location><page_9><loc_8><loc_6><loc_48><loc_22></location>Disk growth in early-type galaxies is an on-going phenomenon (e.g. Yi et al. 2005; Kaviraj et al. 2007; Fabricius et al. 2014), albeit at lower levels today as less gas is available (e.g. Combes et al. 2007; Sage et al. 2007; Huang et al. 2012; Catinella et al. 2013). The molecular gas usually resides in kpc-scale, rotating disks (e.g. Inoue et al. 1996; Wiklind et al. 1997; Okuda et al. 2005; Das et al. 2005), as does the HI gas (e.g. Serra 2012, 2014, and references therein). Young et al. (2008), for example, detail the slow ongoing growth of the disk in NGC 4526, a galaxy whose bulge has a half light radius equal to 1.43 kpc (Krajnovi'c et al. 2013) and a stellar mass 7 of</text>
<text><location><page_9><loc_52><loc_83><loc_92><loc_88></location>0 . 55 × 10 11 (which, along with other bulges, was not massive enough to be included in our sample). Perseus A is yet another example which is also still accreting cold gas (Salome et al. 2006).</text>
<text><location><page_9><loc_52><loc_56><loc_92><loc_83></location>As noted in Section 2, NGC 1277, NGC 1332, NGC 5493 and NGC 5845 are examples of local earlytype galaxies where the disk does not dominate the light at large radii, as in proto-typical lenticular galaxies. These disks are an order of magnitude larger than nuclear disks, and referred to as intermediate-sized disks. Given that the amount of gas accretion can vary, it would be natural for such disks to exist. They are not a new phenomenon (e.g. Scorza & van den Bosch 1998, and references therein), but some readers may not be familiar with their existence. At z ≈ 2, the lower mass spheroids, with less gravitational pull, and which likely formed from a smaller over-density in the early universe, may naturally experience a smaller subsequent supply of gas from cold streams and build their disks more gradually. Freefloating, 'compact elliptical' dwarf galaxies (e.g. Huxor et al. 2013; Paudel et al. 2014) might represent spheroids which never acquired a significant disk, while those in the vicinity of much larger neighbours are thought to have been largely stripped of their disks.</text>
<text><location><page_9><loc_52><loc_23><loc_92><loc_56></location>The theoretical work of Steinmetz & Navarro (2002, and references therein) suggested that a galaxy's morphology is a transient phenomenon. Galaxies do not simply progress from the 'blue cloud' to the 'red sequence' (Faber et al. 2007) but can move in the opposite direction. While the latter pathway may not be traversed in full today, because less gas is available, there are fledged examples of galaxies in the 'green valley' (e.g. Cortese & Hughes 2009; Marino et al. 2011). Elliptical galaxies may initially be built through major mergers of disk galaxies (and perhaps from a violent disk instability, Ceverino et al. 2015), and then proceed to grow a new stellar disk through gas accretion (White & Frenk 1991, p.77), which remains intact until the next significant merger (see Salim et al. 2012 and Conselice et al. 2013 for supporting arguments, and Fang et al. 2012 and Bresolin 2013 for caution in some cases). The number of cycles may however be low (i.e. 1 or 2) rather than several (3 to 5). Evidence for a bulge-then-disk scenario may exist within the Milky Way (e.g. Zoccali et al. 2006). Furthermore, the hierarchical models from Khochfar & Silk (2006), for example, present diskless galaxies at z = 2 which evolve into disk galaxies with a bulge-to-total mass ratio equal to 0.2 by z = 0 due to gas accretion from the halo, cold streams, and minor mergers.</text>
<text><location><page_9><loc_52><loc_6><loc_92><loc_23></location>There have recently been reports of infant, premature disks detected in some of the high-redshift, compact massive galaxies (Chevance et al. 2012, and references therein). In a study of 14 compact massive galaxies at z = 2 . 0 ± 0 . 5, van der Wel et al. (2011) reported that 65( ± 15)% are disk-dominated, appearing highly flattened on the sky and having disks with a median half-light radius of 2.5 kpc. This corresponds to a median scale length of 1.5 kpc, which is half the size of disks today, e.g. Graham & Worley (2008, their Figure 3). These highz disks were observed to harbor compact central components, i.e. bulges. Although most early-type galaxies around us today are known to be lenticular disk</text>
<text><location><page_10><loc_8><loc_75><loc_48><loc_88></location>galaxies - thanks to studies such as ATLAS3D 3D (Emsellem et al. 2011) - there are of course still some massive, slow or non-rotating, elliptical galaxies. Some of these may be the 'ordinary-sized' elliptical galaxies 8 , observed at highz , while others likely formed from more recent, major merger events. Some of the 'ordinary-sized' elliptical galaxies at highz may have also acquired disks, but they would appear as intermediate-sized disks if insufficient gas was accreted to build a larger scale disk around an already large galaxy.</text>
<text><location><page_10><loc_8><loc_63><loc_48><loc_75></location>Following the deep observations by Szomoru et al. (2010) to verify the compactness of a galaxy at z = 1 . 91 found by Daddi et al. (2005), we note that future work should be mindful that shallow surveys at any redshift could miss the outer disks of galaxies to varying degrees and may largely just recover the inner bulge if they are particularly shallow. If that was to occur, one would effectively, although accidentally, manage to identify higher number densities of compact massive systems.</text>
<section_header_level_1><location><page_10><loc_15><loc_60><loc_43><loc_62></location>4.2. Stellar Ages, and stellar mass loss</section_header_level_1>
<text><location><page_10><loc_8><loc_23><loc_48><loc_60></location>The above scenario for disk growth around compact 'bulges' requires the massive bulges of nearby disk galaxies to be old. Although beyond the scope of the current investigation, it will be of interest to explore the ages of the bulges and disks in the current sample of galaxies. However MacArthur, Gonz'alez & Courteau (2009) have already shown that bulges in both early- and late-type disk galaxies do indeed have old mass-weighted ages, with less than 25 per cent by mass of the stars being young, second or third generation stars built from metal enriched gas. Based on stellar populations and radial gradients, MacArthur et al. (see also Fisher et al. 1996) concluded that early-formation processes are common to bulges and that secular processes or 'rejuvenated' star formation generally contributes minimally to the stellar mass budget of bulges (see also Moorthy & Holtzman 2006; Thomas & Davies 2006; and Jablonka et al. 2007), yet it has biased luminosity-weighted age estimates in the past. Such 'frostings' of young stars, of up to 25 per cent by mass, can give the impression of positive luminosity-weighted age gradients, i.e. bulges are younger at their centres, and have misled some studies (which assumed a single stellar population) into missing the fact that the bulk of the stellar mass in most bulges is old 9 . This follows Kuntschner & Davies (1998) who revealed that the obvious lenticular galaxies in the Fornax cluster have ages which are younger than the more spheroid-dominated galaxies in the cluster.</text>
<text><location><page_10><loc_8><loc_19><loc_48><loc_23></location>These works imply that the bulk of the stellar mass in today's massive bulges already existed at high redshifts, and therefore these stellar systems should be what</text>
<text><location><page_10><loc_8><loc_14><loc_48><loc_17></location>8 If so, one may need a mechanism to curtail disk growth, such as more efficient AGN feedback due to their lower stellar density, or hot X-ray halos.</text>
<text><location><page_10><loc_8><loc_4><loc_48><loc_14></location>9 Acaveat is that disks are built both around and within 'bulges'. The surface mass density of disks is higher in their centers than their outskirts. Therefore, disks which are relatively younger than their bulges will contribute younger stars where the bulge resides, but the dominant stellar population will be old at the center of the system. It is of course also known that when disks do form, they may generate bars which can in turn further build on the'bulge' via secular processes. At the same time, strong bars can stall the inward accretion of gas, restricting it to the outer parts of the disk until the bar weakens (Bournaud & Combes 2002).</text>
<text><location><page_10><loc_52><loc_64><loc_92><loc_88></location>we are observing in deep images of our young Universe. Curiously, Saracco et al. (2009) reported that there are two kinds of early-type galaxy at z ≈ 1 . 5: an old ( ∼ 3.5 Gyr) population which needs to experience a factor of 2.5-3 size evolution to have sizes equal to today's earlytype galaxies, and a young ( ∼ 1 Gyr) population ( perhaps those which have recently acquired their disks) which already have sizes consistent with today's population of early-type galaxies. In this regard, it will be interesting to know if their old population is best described with a single S'ersic model, while their young population is better described by a bulge+disk model rather than a singlecomponent S'ersic model. Furthermore, if a younger disk has formed, or bulges developed at different epochs, they may have different initial stellar mass functions (IMFs), resulting in differing composite IMFs for the early-type galaxies today (e.g. Dutton et al. 2013, McDermid 2015, and references therein).</text>
<text><location><page_10><loc_52><loc_30><loc_92><loc_64></location>Gradual stellar mass loss due to stellar winds is a part of the ageing process for passively evolving spheroids (e.g. de Jager et al. 1988; Ciotti et al. 1991; Jungwiert et al. 2001), as is the conversion of visible stars into dark remnants such as neutron stars and stellar mass black holes. If a fraction x of the initial mass is lost from a galactic system due to stellar winds, or Type Ia supernovae clearing out gas, then there will be an adiabatic expansion because the galaxy is no longer as tightly bound and it will therefore reach a new equilibrium such that its size has increased by 1 / (1 -x ) (e.g. Jeans 1961; Hills 1980, their eq.6). Of course if any stellar ejecta remains in a galaxy - which seems likely in massive galaxies with strong potential wells -, perhaps eventually ending up as hot X-ray gas, then the expansion of the galaxy will be reduced depending on the radial expanse of this gas. After the young stars ( < 1 Gyr) have evolved, stellar winds likely account for just a /lessorsimilar 10% reduction to the stellar mass of the passive, compact massive galaxies seen at z ∼ 2 (Damjanov et al. 2009, but see Poggianti et al. 2013b). Fan et al. (2008) had previously suggested that AGN feedback may blow out the gas in distant massive galaxies and cause them to expand by factors of 3 or more. Of course if this had happened, then we would be left having to explain where all of the compact massive bulges in the Universe today came from.</text>
<section_header_level_1><location><page_10><loc_65><loc_28><loc_79><loc_29></location>4.3. Stellar density</section_header_level_1>
<text><location><page_10><loc_52><loc_3><loc_92><loc_27></location>Given the smaller half-light radii that early-type galaxies had at higher redshifts, before they acquired their disks, the stellar densities within those half-light radii were 1 to 2 orders of magnitude greater than the stellar densities inside the half-light radii of today's large early-type galaxies. This led Zirm et al. (2007) to conclude that it is a problem for models of early-type galaxy formation and evolution. Buitrago et al. (2008) claimed that within the inner 1 kpc of the highz galaxies, they have densities equal to globular clusters. This led them to advocate a scenario in which globular clusters and distant compact massive galaxies may have a similar origin, while at the same time suggesting that the more massive halos (not the globular clusters) started collapsing earlier and dragged along a larger amount of baryonic matter that later formed stars. However they overestimated the typical globular cluster size by a factor of 10/3, and thus under-estimated a typical globular cluster's density by a</text>
<text><location><page_11><loc_8><loc_87><loc_39><loc_88></location>factor of 37, undermining their conclusions.</text>
<text><location><page_11><loc_8><loc_63><loc_48><loc_87></location>Bezanson et al. (2009) showed that the inner regions ( < 1 kpc) of today's early-type galaxies have stellar densities which are 2-3 times lower than those of the distant compact massive galaxies of the same mass. However, the majority of galaxies in our sample have bulge-tototal ratios of 1/2-1/4 (Table 2), while Laurikainen et al. (2010) found an average ratio of 1/4 for their sample of lenticular galaxies. That is, the local massive bulges are 2-4 times less massive than the galaxy within which they reside. In comparing galaxies of the same mass in Figure 2d from Bezanson et al. (2009), one is effectively comparing the high redshift galaxies with local bulges that are 2-4 times less massive, possibly accounting for the lower density observed in these lower-mass bulges. If one compares the quiescent, compact massive galaxies at high-redshift with today's bulges of the same mass, one may find that they have the same density within the inner kpc.</text>
<text><location><page_11><loc_8><loc_48><loc_48><loc_63></location>Hopkins et al. (2009) reported that there is no central ( < 1 kpc) density mismatch (see also van Dokkum et al. 2014). They suggested that 'the entire population of compact, high-redshift red galaxies may be the progenitors of the high-density cores of present-day ellipticals', by which they mean spheroids. We however consider the evolution to have been very different, such that a 2D disk is built within and around the spheroid that undergoes no substantial size-evolution, rather than the continual development of a 3D envelope around a spheroid that effectively undergoes significant size growth.</text>
<section_header_level_1><location><page_11><loc_20><loc_45><loc_37><loc_46></location>4.4. Velocity Dispersions</section_header_level_1>
<text><location><page_11><loc_8><loc_26><loc_48><loc_45></location>If there is indeed no evolution of the distant spheroids, then their velocity dispersion should remain the same. Now, if a stellar disk builds within and around them, then the galaxy mass today will have increased. As a result, if one was to compare the velocity dispersions of the distant compact spheroids with the velocity dispersions of z = 0 disk galaxies having the same total stellar mass, one would actually be sampling local bulges which are less massive than the distant spheroids. The local galaxies of the same mass, containing compact bulges of 2-4 × lower mass, are naturally expected to have lower velocity dispersions. Indeed, this general behavior has been observed (e.g. Cenarro & Trujillo 2009; Cappellari et al. 2009; Newman et al. 2010).</text>
<text><location><page_11><loc_8><loc_3><loc_48><loc_26></location>If one assumes M ∗ ∝ σ 4 , then a 2-4 × difference in stellar mass will be associated with a ∼ 1.2-1.4 × difference in velocity dispersion. If one attempts to bypass the photometric stellar mass estimate and use a dynamical / virial mass estimate ( σ 2 R e , gal ) for the lenticular galaxies, the same issue occurs. This is because of the enhanced R e , gal value due to the presence of the disk component. If a galaxy's half-light radius is 3-6 times larger than the half-light radius of its central bulge, then the velocity dispersion of the bulge coupled with the galaxy size will produce a galaxy virial mass which is 3-6 times larger than what one would obtain for the bulge. This will act to artificially separate the compact highz spheroids from local massive bulges in a diagram of dynamical mass versus velocity dispersion. A note to keep in mind is that disk growth, as with envelope growth, may lead to the compression of the original inner bulge and a slight en-</text>
<text><location><page_11><loc_52><loc_86><loc_92><loc_88></location>in the central velocity dispersion (Andredakis 1998; Debattista et al. 2013).</text>
<section_header_level_1><location><page_11><loc_65><loc_83><loc_79><loc_84></location>4.5. Depleted cores</section_header_level_1>
<text><location><page_11><loc_52><loc_44><loc_92><loc_83></location>The presence of partially depleted cores in massive spheroids is thought to be a result of major, dry merger events (e.g. Begelman et al. 1980; Faber et al. 1997; Merritt et al. 2007). The large elliptical galaxies with sizes several times that of the high' z , compact massive spheroids, and with a deficit of stars over their inner tens of parsecs to a few hundred parsecs, likely formed this way. The orbital decay of the binary supermassive black hole, created by the galaxy merger, proceeds by slingshotting the central stars out of the core of the newly formed galaxy. However, as noted by Dullo & Graham (2013), the presence of such partially depleted cores in the bulges of massive disk galaxies (e.g. Dullo & Graham 2013) presents a conundrum because these disk galaxies may be unlikely to have formed from such major merger events 10 . One potential solution is that the bulges may have formed first, and the disks were subsequently accreted and grew (Graham 2013; Driver et al. 2013). It may therefore be interesting to check for partially depleted cores in the current sample of disk galaxies in Table 1. Application of the core-S'ersic model (Graham et al. 2003) to Hubble Space Telescope images - due to their superior spatial resolution over a sufficiently wide field of view - will reveal which bulges have an inner deficit of flux relative to the inward extrapolation of their outer S'ersic light profile. Knowledge of which bulges have partially depleted cores, coupled with information about their stellar age, should shed even further light on the formation history.</text>
<section_header_level_1><location><page_11><loc_53><loc_40><loc_91><loc_43></location>4.6. Formation paths and a cautionary remark on the n = 2 or n = 2 . 5 division of bulges</section_header_level_1>
<text><location><page_11><loc_52><loc_23><loc_92><loc_40></location>Surveying predominantly compact, passive, spheroidal galaxies at 1 . 4 < z < 2 . 0, with 10 10 < M ∗ /M /circledot < 10 11 , Cimatti et al. (2008, see van Dokkum et al. 2008 for higher redshifts and higher masses) noted that the galaxy sizes are typically such that R e /lessorsimilar 1 kpc, and are thus much smaller than early-type galaxies of comparable mass in the present-day Universe. However, the bulges of most disk galaxies at z ≈ 0 have compact sizes with R e /lessorsimilar 2 kpc and many bulges with 10 10 < M ∗ /M /circledot < 10 11 have R e /lessorsimilar 1 kpc (e.g. Graham & Worley 2008; Graham 2013, his Figure 1). The growth of disks in and around the compact, highz spheroids could therefore explain their apparent disappearance by today.</text>
<text><location><page_11><loc_52><loc_8><loc_92><loc_23></location>This possible explanation may at first glance appear somewhat at odds with the claim in the interesting paper by van Dokkum et al. (2013) that disk models, in which bulges were fully assembled at high redshift and disks gradually formed around them, can be ruled out. However the first distinction in these two remarks is of course that disks do not simply form around spheroids but are additionally embedded within them; which is why bulge/disk decompositions continue the disk all the way into the centers of galaxies. It would therefore be of interest to perform a bulge/disk decomposition of the aver-</text>
<text><location><page_12><loc_8><loc_54><loc_48><loc_88></location>age surface density profiles, at different redshift intervals, that were constructed by van Dokkum et al. (2013, their Figure 3). This would allow one to check how well their z = 0 (Milky Way)-like galaxy profile that they built resembles the Milky Way, and thus to know if their stacked profiles are reliable or perhaps effected by dust or some other issue 11 . A reliable decomposition of an evolving population (free from 'progenitor bias') would of course enable one to quantify how much the disk and bulge (including pseudobulge) components have changed between the different redshifts. The second point of distinction is that van Dokkum et al. (2013) were referring to the inner 2 kpc radius of (Milky Way)-mass spiral galaxies, rather than 10 11 M /circledot mass galaxies. The Milky Way has a bulge mass of 0 . 91 × 10 10 M /circledot and a bulge-to-total stellar mass ratio of 0.15 (Licquia et al. 2014); it also has a half-light radius of 0.66 kpc (Graham & Driver 2007). Therefore, within an inner radius of 2 kpc, the mass of (Milky Way)like galaxies are dominated by the disk rather than the bulge. Due to the greater prominence of bulges in earlytype disk galaxies than in late-type disk galaxies (see Figure 21 in Graham 2001), the inner regions of today's massive galaxies - which host the once distant, compact massive spheroids - are expected to have changed less over time, as already observed by van Dokkum et al. (2010, their Figure 6).</text>
<text><location><page_12><loc_8><loc_40><loc_48><loc_54></location>Within the bulge-then-disk growth scenario, the existence of the distant massive spheroids having n < 2 implies that local galaxies with bulges having n < 2 need not be pseudobulges formed from the secular evolution of a stellar disk. This topic is reviewed in Graham (2013, 2014) where it is noted that bulges with S'ersic n < 2 need not be pseudobulges even if they are also rotating (e.g. Eliche-Moral et al. 2011; Scannapieco et al. 2011; Keselman & Nusser 2012; Saha et al. 2012; dos Anjos & da Silva 2013; Graham 2014; Querejeta et al. 2015).</text>
<text><location><page_12><loc_8><loc_18><loc_48><loc_40></location>In passing we remark that formation paths in galaxy clusters in which the disks of spiral galaxies fade and lose their pattern, and possibly end up with relatively brighter bulges (e.g. Johnston et al. 2012, and references therein), to become lenticular galaxies can still operate. That is, lenticular galaxies likely have more than one evolutionary path, as can their bulges (e.g. Cole et al. 2000; Springel et al. 2005), and the presence of both classical and pseudobulges in the same lenticular galaxy would seem to support this (e.g. Erwin et al. 2003). At the same time, the preservation, i.e. the lack of evolution, of compact massive spheroids from z = 2 ± 0 . 6 until today has implications for studies of the evolution of the (black hole)-bulge mass scaling relations (reviewed in Graham 2015, and references therein). The M bh /M bulge ratio would not be expected to decrease in these particular systems since approx 1 . 4.</text>
<text><location><page_12><loc_8><loc_12><loc_48><loc_18></location>McLure et al. (2013) reported that 44 ± 12% of their z = 1 . 4 ± 0 . 1 sample of relatively passive galaxies (specific star formation rate, sSFR ≤ 0 . 1 Gyr -1 ) with M ∗ ≥ 0 . 6 × 10 11 M /circledot have a disk-like morphology. While at face</text>
<text><location><page_12><loc_8><loc_4><loc_48><loc_10></location>11 The lack of any central bulge within the inner kpc of the z = 0 profile is at odds with (Milky Way)-like disk galaxies, and the high average S'ersic index ( n = 2 . 6 ± 0 . 4) also suggests that their average profile is not representative of (Milky Way)-like galaxies, but rather an intermediate-luminosity early-type galaxy which have a very different bulge-to-total mass ratio.</text>
<text><location><page_12><loc_52><loc_67><loc_92><loc_88></location>value this claim may appear to support the notion that flattened 2D disks have developed, their morphological (disk-like) claim was based on a S'ersic index divide at n = 2 . 5 such that they labeled galaxies with n < 2 . 5 to be late-type galaxies. They adopted this divide because Shen et al. (2003) had used it to separate the bright, local galaxy population. Shen et al. (2003) could do this because their SDSS sample contained relatively few dwarf early-type galaxies which have n < 2 . 5. However bulges can have n < 2 . 5. Therefore, some/many(?) of the galaxies in the sample of McLure et al. (2013) with n < 2 . 5 may still be rather 'naked bulges', rather than disk-dominated galaxies. Indeed, the similar distributions for their passive sample with n < 2 . 5 and n ≥ 2 . 5 in the size-mass diagram (their Figure 8) would suggest this.</text>
<text><location><page_12><loc_52><loc_58><loc_92><loc_67></location>The practice of considering all highz , compact, massive objects to be disk galaxies if their S'ersic index is less than 2-2.5 is unfortunately rather common (e.g. Hathi et al. 2008; Fathi et al. 2012; Chevance et al. 2012; Muzzin et al. 2012) and may have effectively mis-led many researchers. For this reason we have provided this cautionary text against this practice.</text>
<section_header_level_1><location><page_12><loc_65><loc_56><loc_79><loc_57></location>5. CONCLUSIONS</section_header_level_1>
<text><location><page_12><loc_52><loc_36><loc_92><loc_55></location>The compact massive galaxies at redshifts z ≈ 2 ± 0 . 6 have similar stellar masses and sizes (and thus stellar densities), and radial concentrations of light, as bright bulges in local disk galaxies. Moreover, the number density is not different by hundreds or thousands but is within a factor of a few. We have identified 21 compact ( R e /lessorsimilar 2 kpc), massive (0 . 7 × 10 11 < M ∗ /M /circledot < 1 . 4 × 10 11 ) spheroids within 90 Mpc, giving a number density of 6 . 9 × 10 -6 Mpc -3 (or 3.5 × 10 -5 Mpc -3 per unit dex in stellar mass). This is however a lower limit because we have not performed a volume-limited search. Based on a sub-sample taken from a smaller volume-limited sample, this density may be 2.75( ± 58%) times higher at around × 10 -4 Mpc -3 dex -1 .</text>
<text><location><page_12><loc_52><loc_17><loc_92><loc_36></location>This observation eliminates the need to grow all of the highz compact massive spheroids by a factor of 3 to 6 by z = 0, a challenge which had remained unexplained for the past decade. Rather, many of these highz spheroids need to remain largely unchanged in order to match the massive bulges in today's early-type disk galaxies. Therefore, any study which may have claimed to have accounted for the size growth of quiescent galaxies would have inadvertently, and most likely unknowingly, introduced an equally puzzling problem. If all the distant, compact massive spheroids had evolved, we would then be faced with a second unexplained mystery, namely, where did the compact massive bulges in the local universe come from and why are they not observed in deep images of the z = 2 . 0 ± 0 . 6 universe.</text>
<text><location><page_12><loc_52><loc_3><loc_92><loc_16></location>We conclude that stellar disks have since grown around many compact massive spheroids observed at z = 2 . 0 ± 0 . 6 and in so doing transformed their morphological type from elliptical to early-type disk galaxy (i.e. lenticular galaxies and early-type spiral galaxies). Following Graham (2013), we again speculate that some of the less massive compact spheroids at high-redshifts may now reside at the centres of late-type spiral galaxies, and/or are today's compact dwarf elliptical galaxies - which were either largely stripped of their disk or never acquired one.</text>
<text><location><page_13><loc_8><loc_78><loc_48><loc_88></location>The authors thank Francoise Combes, Luca Cortese, Karl Glazebrook and Barbara Catinella for kindly reading and providing helpful comments on this manuscript. A.G. thanks the organizers and participants of the enlightening conference 'The Role of Hydrogen in the Evolution of Galaxies' 15-19 September 2014, Kuching, Malaysia (Borneo). This research was supported by the Australian Research Council through funding grant</text>
<section_header_level_1><location><page_13><loc_52><loc_87><loc_62><loc_88></location>FT110100263.</section_header_level_1>
<text><location><page_13><loc_52><loc_76><loc_92><loc_87></location>This work is based on observations made with the IRAC instrument (Fazio et al. 2004) on board the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. This research made use of the NASA/IPAC Extragalactic Database (NED: http://ned.ipac.caltech.edu) and the HyperLeda database (http://leda.univ-lyon1.fr).</text>
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2015PhRvD..91k2017B | https://arxiv.org/pdf/1503.07136.pdf | <document>
<section_header_level_1><location><page_1><loc_12><loc_86><loc_88><loc_91></location>The cosmic ray proton plus helium energy spectrum measured by the ARGO-YBJ experiment in the energy range 3-300 TeV</section_header_level_1>
<text><location><page_1><loc_12><loc_54><loc_88><loc_83></location>B. Bartoli, 1, 2 P. Bernardini, 3, 4 X.J. Bi, 5 Z. Cao, 5 S. Catalanotti, 1, 2 S.Z. Chen, 5 T.L. Chen, 6 S.W. Cui, 7 B.Z. Dai, 8 A. D'Amone, 3, 4 Danzengluobu, 6 I. De Mitri, 3, 4 B. D'Ettorre Piazzoli, 1, 2 T. Di Girolamo, 1, 2 G. Di Sciascio, 9 C.F. Feng, 10 Zhaoyang Feng, 5 Zhenyong Feng, 11 Q.B. Gou, 5 Y.Q. Guo, 5 H.H. He, 5 Haibing Hu, 6 Hongbo Hu, 5 M. Iacovacci, 1, 2 R. Iuppa, 9, 12 H.Y. Jia, 11 Labaciren, 6 H.J. Li, 6 C. Liu, 5 J. Liu, 8 M.Y. Liu, 6 H. Lu, 5 L.L. Ma, 5 X.H. Ma, 5 G. Mancarella, 3, 4 S.M. Mari, 13, 14, ∗ G. Marsella, 3, 4 S. Mastroianni, 2 P. Montini, 13, 14, † C.C. Ning, 6 L. Perrone, 3, 4 P. Pistilli, 13, 14 P. Salvini, 15 R. Santonico, 9, 12 G. Settanta, 13 P.R. Shen, 5 X.D. Sheng, 5 F. Shi, 5 A. Surdo, 4 Y.H. Tan, 5 P. Vallania, 16, 17 S. Vernetto, 16, 17 C. Vigorito, 16, 17 H. Wang, 5 C.Y. Wu, 5 H.R. Wu, 5 L. Xue, 10 Q.Y. Yang, 8 X.C. Yang, 8 Z.G. Yao, 5 A.F. Yuan, 6 M. Zha, 5 H.M. Zhang, 5 L. Zhang, 8 X.Y. Zhang, 10 Y. Zhang, 5 J. Zhao, 5 Zhaxiciren, 6 Zhaxisangzhu, 6 X.X. Zhou, 11 F.R. Zhu, 11 and Q.Q. Zhu 5</text>
<section_header_level_1><location><page_1><loc_38><loc_50><loc_62><loc_51></location>(ARGO-YBJ Collaboration)</section_header_level_1>
<text><location><page_1><loc_16><loc_7><loc_84><loc_48></location>1 Dipartimento di Fisica dell'Universit'a di Napoli 'Federico II', Complesso Universitario di Monte Sant'Angelo, via Cinthia, 80126 Napoli, Italy. 2 Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Complesso Universitario di Monte Sant'Angelo, via Cinthia, 80126 Napoli, Italy. 3 Dipartimento Matematica e Fisica 'Ennio De Giorgi', Universit'a del Salento, via per Arnesano, 73100 Lecce, Italy. 4 Istituto Nazionale di Fisica Nucleare, Sezione di Lecce, via per Arnesano, 73100 Lecce, Italy. 5 Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, P.O. Box 918, 100049 Beijing, P.R. China. 6 Tibet University, 850000 Lhasa, Xizang, P.R. China. 7 Hebei Normal University, Shijiazhuang 050016, Hebei, P.R. China. 8 Yunnan University, 2 North Cuihu Rd., 650091 Kunming, Yunnan, P.R. China. 9 Istituto Nazionale di Fisica Nucleare, Sezione di Roma Tor Vergata,</text>
<text><location><page_2><loc_19><loc_85><loc_20><loc_86></location>11</text>
<text><location><page_2><loc_20><loc_52><loc_81><loc_91></location>via della Ricerca Scientifica 1, 00133 Roma, Italy. 10 Shandong University, 250100 Jinan, Shandong, P.R. China. Southwest Jiaotong University, 610031 Chengdu, Sichuan, P.R. China. 12 Dipartimento di Fisica dell'Universit'a di Roma 'Tor Vergata', via della Ricerca Scientifica 1, 00133 Roma, Italy. 13 Dipartimento di Matematica e Fisica dell'Universit'a 'Roma Tre', via della Vasca Navale 84, 00146 Roma, Italy. 14 Istituto Nazionale di Fisica Nucleare, Sezione di Roma Tre, via della Vasca Navale 84, 00146 Roma, Italy. 15 Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, via Bassi 6, 27100 Pavia, Italy. 16 Istituto Nazionale di Fisica Nucleare, Sezione di Torino, via P. Giuria 1, 10125 Torino, Italy. 17 Dipartimento di Fisica dell'Universit'a di Torino, via P. Giuria 1, 10125 Torino, Italy.</text>
<text><location><page_2><loc_39><loc_50><loc_60><loc_51></location>(Dated: October 13, 2018)</text>
<section_header_level_1><location><page_2><loc_45><loc_46><loc_54><loc_48></location>Abstract</section_header_level_1>
<text><location><page_2><loc_12><loc_19><loc_88><loc_45></location>The ARGO-YBJ experiment is a full-coverage air shower detector located at the Yangbajing Cosmic Ray Observatory (Tibet, People's Republic of China, 4300 m a.s.l.). The high altitude, combined with the full-coverage technique, allows the detection of extensive air showers in a wide energy range and offer the possibility of measuring the cosmic ray proton plus helium spectrum down to the TeV region, where direct balloon/space-borne measurements are available. The detector has been in stable data taking in its full configuration from November 2007 to February 2013. In this paper the measurement of the cosmic ray proton plus helium energy spectrum is presented in the region 3 -300 TeV by analyzing the full collected data sample. The resulting spectral index is γ = -2 . 64 ± 0 . 01. These results demonstrate the possibility of performing an accurate measurement of the spectrum of light elements with a ground based air shower detector.</text>
<section_header_level_1><location><page_3><loc_12><loc_89><loc_32><loc_91></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_3><loc_12><loc_12><loc_88><loc_85></location>Cosmic rays are ionized nuclei reaching the Earth from outside the solar system. Many experimental efforts have been devoted to the study of cosmic ray properties. In the last decades many experiments were focused on the identification of cosmic ray sources and on the understanding of their acceleration and propagation mechanisms. Despite a very large amount of data collected so far, the origin and propagation of cosmic rays are still under discussion. Supernova remnants (SNRs) are commonly identified as the source of galactic cosmic rays since they could provide the amount of energy needed in order to accelerate particles up to the highest energies in the Galaxy. The measurement of the diffuse gammaray radiation in the energy range 1 -100 GeV supports these hypotheses on the origin and propagation of cosmic rays [1]. Moreover the TeV gamma-ray emission from SNRs, detected by ground-based experiment, can be related to the acceleration of particles up to ∼ 100 TeV [2, 3]. A very detailed measurement of the energy spectrum and composition of primary cosmic rays will lead to a deeper knowledge of the acceleration and propagation mechanisms. Since the energy spectrum spans a huge energy interval, experiments dedicated to the study of cosmic ray properties are essentially divided into two broad classes. Direct experiments operating on satellites or balloons are able to measure the energy spectrum and the isotopic composition of cosmic rays on top of the atmosphere. Due to their reduced detector active surface and the limited exposure time the maximum detectable energy is limited up to few TeV. New generation instruments, capable of long balloon flights, have extended the energy measurements up to ∼ 100 TeV. All the information concerning cosmic rays above 100 TeV is provided by ground-based air shower experiments. Air shower experiments are able to observe the cascade of particles produced by the interaction between cosmic rays and the Earth's atmosphere. Ground based experiments detect extensive air showers produced by primaries with energies up to 10 20 eV, however they do not allow an easy determination of the abundances of individual elements and the measurement of the composition is therefore limited only to the main elemental groups. Moreover, due to a lack of a model-independent energy calibration, the determination of the primary energy relies on the hadronic interaction model used in the description of the shower's development.</text>
<text><location><page_3><loc_12><loc_7><loc_88><loc_11></location>The ARGO-YBJ experiment is a high-altitude full-coverage air shower detector which was in full and stable data taking from November 2007 up to February 2013. As described in</text>
<text><location><page_4><loc_12><loc_63><loc_88><loc_91></location>section II, the detector is equipped with a digital and an analog readout systems working independently in order to study the cosmic ray properties in the energy range 1 -10 4 TeV, which is one of the main physics goals of the ARGO-YBJ experiment. The high spacetime resolution of the digital readout system allows the detection of showers produced by primaries down to few TeV, where balloon-borne measurements are available. The analog readout system was designed and built in order to detect showers in a very wide range of particle density at ground level and to explore the cosmic ray spectrum up to the PeV region. In 2012 a first measurement of the cosmic ray proton plus helium (light component) spectrum obtained by analyzing a small sample collected during the first period of data taking with the detector in its full configuration (by using the digital readout information only) has been presented [4].</text>
<text><location><page_4><loc_12><loc_47><loc_88><loc_62></location>In this paper we report the analysis of the full data sample collected by the ARGO-YBJ experiment in the period from January 2008 to December 2012 and the measurement of the light component energy spectrum of cosmic rays in the energy range 3 -300 TeV by applying an unfolding procedure based on the bayesian probabilities. The analysis of the analog readout data and the corresponding cosmic ray spectrum up to the PeV energy region is in progress and will be addressed in a future paper.</text>
<section_header_level_1><location><page_4><loc_12><loc_41><loc_47><loc_42></location>II. THE ARGO-YBJ EXPERIMENT</section_header_level_1>
<text><location><page_4><loc_12><loc_7><loc_88><loc_37></location>The ARGO-YBJ experiment (Yangbajing Cosmic Ray Observatory, Tibet, P.R. China. 4300 m a.s.l.) is a full-coverage detector made of a single layer of Resistive Plate Chambers (RPCs) with ∼ 93% active area [5, 6], surrounded by a partially instrumented guard ring designed to improve the event reconstruction. The detector is made of 1836 RPCs, arranged in 153 clusters each made of 12 chambers. The digital readout consists of 18360 pads each segmented in 8 strips. A dedicated procedure was implemented to calibrate the detector in order to achieve high pointing accuracy [7]. The angular and core reconstruction resolution are respectively 0 . 4 · and 5 m for events with at least 500 fired pads [8, 9]. The installation of the central carpet was completed in June 2006. The guard ring was completed during spring 2007 and connected to the data acquisition system [10] in November 2007. A simple trigger logic based on the coincidence between the pad signals was implemented. The detector has been in stable data taking in its full configuration for more than five years with a</text>
<text><location><page_5><loc_12><loc_60><loc_88><loc_91></location>trigger threshold N pad = 20, corresponding to a trigger rate of about 3 . 6 kHz and a dead time of 4%. The high granularity and time resolution of the detector provide a detailed three-dimensional reconstruction of the shower front. The high altitude location and the segmentation of the experiment offer the possibility to detect showers produced by charged cosmic rays with energies down to few TeV. The digital readout of the pad system allows reconstruction of showers with a particle density at ground level up to about 23 particles/m 2 , which correspond to primaries up to a few hundreds of TeV. In order to extend the detector operating range and investigate energies up to the PeV region each RPC has been equipped with two large size electrodes called Big Pads [11]. Each Big Pad provide a signal whose amplitude is proportional to the number of particles impinging the detector surface. The analog readout system allows a detailed measurement of showers with particle density at ground up to more than 10 4 particles / m 2 .</text>
<section_header_level_1><location><page_5><loc_12><loc_55><loc_34><loc_56></location>III. DATA ANALYSIS</section_header_level_1>
<section_header_level_1><location><page_5><loc_14><loc_50><loc_59><loc_51></location>A. Unfolding of the cosmic ray energy spectrum</section_header_level_1>
<text><location><page_5><loc_12><loc_30><loc_88><loc_47></location>As widely described in [4, 12], the determination of the cosmic ray spectrum starting from the measured space-time distribution of charged particles at ground level is a classical unfolding problem that can be dealt by using the bayesian technique[13]. In this framework the detector response is represented by the probability P ( M j | E i ) of measuring a multiplicity M j due to a shower produced by a primary of energy E i . The estimated number of events in a certain energy bin E i is therefore related to the number of events measured in a multiplicity bin M j by the equation</text>
<formula><location><page_5><loc_37><loc_26><loc_88><loc_29></location>ˆ N ( E i ) ∝ ∑ j N ( M j ) P ( E i | M j ) (1)</formula>
<text><location><page_5><loc_12><loc_23><loc_56><loc_24></location>where η ij is constructed by using the Bayes theorem</text>
<formula><location><page_5><loc_36><loc_18><loc_88><loc_21></location>P ( E i | M j ) = P ( M j | E i ) P ( E i ) ∑ k P ( M j | E k ) P ( E k ) . (2)</formula>
<text><location><page_5><loc_12><loc_7><loc_88><loc_16></location>The values of the probability P ( M j | E i ) are evaluated by means of a Monte Carlo simulation of the development of the shower and of the detector response. The quantity P ( E i | M j ) represents the probability that a shower detected with multiplicity M j has been produced by a primary of energy E i . The values of P ( M j | E i ) are evaluated by means of an iterative</text>
<text><location><page_6><loc_12><loc_87><loc_88><loc_91></location>procedure starting from a prior value P (0) ( E i ), in which in the n -th step P ( n ) ( E i ) is replaced by the updated value</text>
<formula><location><page_6><loc_38><loc_82><loc_88><loc_86></location>P ( n +1) ( E i ) = ˆ N ( n ) ( E i ) ∑ k ˆ N ( n ) ( E k ) , (3)</formula>
<text><location><page_6><loc_12><loc_68><loc_88><loc_81></location>where ˆ N ( n ) ( E i ) is evaluated in the n -th step according to eq. 1. As initial prior P (0) ( E i ) ∼ E -2 . 5 was chosen, the effect of using different prior distributions has been evaluated as negligible. The iterative procedure ends when the variation of all ˆ N ( E i ) in two consecutive steps are evaluated as negligible, namely less than 0.1 %. Typically the convergence is reached after 3 iterations.</text>
<section_header_level_1><location><page_6><loc_14><loc_62><loc_51><loc_63></location>B. Air shower and detector simulations</section_header_level_1>
<text><location><page_6><loc_12><loc_28><loc_88><loc_59></location>The development of the shower in the Earth's atmosphere has been simulated by using the CORSIKA (v. 6980) code [14]. The electromagnetic component are described by the EGS4 code [15, 16], while the high energy hadronic interactions are reproduced by QGSJETII.03 model [17, 18]. Low energy hadronic interactions are described by the FLUKA package [19, 20]. Showers produced by Protons, Helium, CNO nuclei and Iron have been generated with a spectral index γ = -1 in the energy range (0 . 316 -3 . 16 × 10 4 ) TeV. About 5 × 10 7 showers have been generated in the zenith angle range 0-45 degrees and azimuth angle range 0-360 degrees. Showers were sampled at the Yangbajing altitude and the shower core was randomly distributed over an area of (250 × 250) m 2 centered on the detector. The resulting CORSIKA showers have been processed by a GEANT3 [21] based code in order to reproduce the detector response, including the effects of time resolution, RPC efficiency, trigger logic, accidental background produced by each pad and electronic noise.</text>
<section_header_level_1><location><page_6><loc_14><loc_22><loc_32><loc_23></location>C. Event selection</section_header_level_1>
<text><location><page_6><loc_12><loc_7><loc_88><loc_19></location>The ARGO-YBJ experiment was in stable data taking in its full configuration for more than five years: more than 5 × 10 11 events have been recorded and reconstructed. Several tools have been implemented in order to monitor the detector operation and reconstruction quality. The detector control system (DCS) [22] continuously monitors the RPC current, the high voltage distribution, the gas mixture and the environmental conditions (temperature,</text>
<text><location><page_7><loc_12><loc_74><loc_88><loc_91></location>pressure, humidity). In this work the analysis of events collected during the period 20082012 is presented. Data and simulated events have been selected according to a multi-step procedure in order to obtain high quality events and to ensure a reliable and unbiased evaluation of the bayesian probabilities. The first step concerns the run selection: in order to obtain a sample of high-quality runs, the working condition of the detector and the quality of the reconstruction procedure have been analyzed by using the criteria described below.</text>
<unordered_list>
<list_item><location><page_7><loc_15><loc_61><loc_88><loc_70></location>· At least 128 clusters out of 130 must be active and connected to the DAQ and trigger systems. This criterium selects runs taken with almost the whole apparatus in data taking, discarding the runs that can bias the analysis because of the switched-off clusters.</list_item>
<list_item><location><page_7><loc_15><loc_51><loc_88><loc_58></location>· Only runs with a duration T glyph[greaterorequalslant] 1800 s have been considered. The runs with a short duration are generally produced when a problem in the apparatus occurs. These runs have been removed from the analysis.</list_item>
<list_item><location><page_7><loc_15><loc_42><loc_88><loc_49></location>· The value of the trigger rate for each run must stay within the range 3 . 2 -3 . 7 kHz. A trigger rate outside this range usually indicates that the detector was not operated standardly. These runs have been discarded.</list_item>
<list_item><location><page_7><loc_15><loc_32><loc_88><loc_39></location>· To monitor the quality of the event reconstruction the mean value of the unnormalised χ 2 obtained by fitting the shower front must be less than 135 ns 2 (see figure 1). Nearly all runs that have ¯ χ 2 > 135 ns 2 encountered some sort of problems.</list_item>
</unordered_list>
<text><location><page_7><loc_12><loc_22><loc_88><loc_29></location>In figure 1 the distribution of the trigger rate and the ¯ χ 2 of the reconstruction procedure are reported. The procedure described above selects a data sample of about 3 × 10 11 events, corresponding to a live time of about 24000 hours.</text>
<text><location><page_7><loc_12><loc_14><loc_88><loc_21></location>The following selection criteria (fiducial cuts) have been applied to both Monte Carlo and experimental data in order to improve the quality of the reconstruction and to obtain the best estimation of the bayesian probabilities.</text>
<unordered_list>
<list_item><location><page_7><loc_15><loc_7><loc_88><loc_11></location>· Only events with reconstructed zenith angles ϑ R glyph[lessorequalslant] 35 · have been considered. The resulting solid angle Ω is about 1.13 sr.</list_item>
</unordered_list>
<figure>
<location><page_8><loc_34><loc_74><loc_64><loc_90></location>
</figure>
<text><location><page_8><loc_36><loc_71><loc_38><loc_72></location>7000</text>
<figure>
<location><page_8><loc_34><loc_55><loc_63><loc_72></location>
<caption>FIG. 1. Distribution of the trigger rate (top) and of the unnormalised ¯ χ 2 (bottom) of all the runs collected by the ARGO-YBJ experiment (black lines). The resulting 2008-2012 sample selected according to the criteria described in section III is also reported (dashed red lines)</caption>
</figure>
<unordered_list>
<list_item><location><page_8><loc_15><loc_29><loc_88><loc_41></location>· The measured shower multiplicity M had to be in the range 150 glyph[lessorequalslant] M glyph[lessorequalslant] 5 × 10 4 . This selection cut was introduced in order to reduce bias effects in the estimation of the bayesian probabilities that are mainly located at the edges of the simulated energy range. Moreover the highest multiplicity cut avoid saturation effects of the digital readout system.</list_item>
<list_item><location><page_8><loc_15><loc_20><loc_88><loc_27></location>· The cluster with the highest multiplicity had to be contained within an area of about 40 × 40 m 2 centered on the detector. This cut was applied in order to reject events with their true shower core position located outside the detector surface.</list_item>
</unordered_list>
<text><location><page_8><loc_12><loc_13><loc_88><loc_17></location>In order to select showers induced by proton and helium nuclei the following criterium has been used.</text>
<unordered_list>
<list_item><location><page_8><loc_15><loc_7><loc_88><loc_11></location>· Density cut: the average particle density ( ρ in ) measured by the central area (20 inner clusters) of the detector must be higher than the particle density ( ρ out ) measured by</list_item>
</unordered_list>
<text><location><page_9><loc_64><loc_86><loc_65><loc_89></location>entries/bin</text>
<figure>
<location><page_9><loc_35><loc_71><loc_65><loc_90></location>
<caption>FIG. 2. Distribution of reconstructed core positions of showers selected by applying the criteria described in section III C. The boxes represent the clusters layout.</caption>
</figure>
<figure>
<location><page_9><loc_35><loc_39><loc_62><loc_59></location>
<caption>FIG. 3. Energy distribution of all Monte Carlo events (black) and of those surviving the fiducial cuts (blue) and the density cut (green and red) described in section III C according to the Horandel model [23].</caption>
</figure>
<text><location><page_9><loc_17><loc_9><loc_88><loc_26></location>the outermost area (42 outer clusters): ( ρ in > 1 . 25 ρ out ). This selection criteria based on the lateral particle distribution was introduced in order to discard events produced by nuclei heavier than helium. In fact, in showers induced by heavy primaries the lateral distribution is wider than in light-induced ones. By applying this criterion on events with the core located in a narrow area around the detector center, showers mainly produced by light primaries have been selected. The contamination of elements heavier than helium does not exceed few %, as discussed in section IV A 4.</text>
<figure>
<location><page_10><loc_20><loc_57><loc_80><loc_89></location>
<caption>FIG. 4. The light component spectrum measured by the ARGO-YBJ experiment by using data taken in each year of the period 2008-2012 and the full 2008-2012 data sample. The power-law fit of each spectrum is also reported (red lines).</caption>
</figure>
<text><location><page_10><loc_12><loc_26><loc_88><loc_43></location>In figure 2 the coordinates of the reconstructed core position of the events surviving the selection criteria described above are reported. The plot shows that the contribution of events located outside an area of 40 × 40 m 2 is negligible. In figure 3 the event rate obtained by using the Horandel model for input spectra and isotopic composition [23] and surviving the selection criteria described above is reported as a function of energy for both proton plus helium (light component) and heavier elements (heavy component). The plot shows that the selected sample is essentially made of light nuclei.</text>
<section_header_level_1><location><page_10><loc_12><loc_19><loc_55><loc_21></location>IV. THE LIGHT COMPONENT SPECTRUM</section_header_level_1>
<text><location><page_10><loc_12><loc_7><loc_88><loc_16></location>The analysis was performed on the sample selected by the criteria described in section III. Simulated events have been sorted in 16 multiplicity bins and 13 energy bins in order to minimize the statistical error and to reduce bin migration effects. The Monte Carlo data sample was analyzed in order to evaluate the probability distribution P ( M | E ) and the</text>
<text><location><page_11><loc_12><loc_55><loc_88><loc_91></location>energy resolution which turns out to be about 10% for energies below 10 TeV and of the order of 5% at energies of about 100 TeV. The multiplicity distribution extracted from data has been unfolded according to the procedure described in section III A. Results are reported in figure 4 for each year of data taking and also for the full sample. In order to investigate the stability of the detector over a long period the analysis was performed separately on the data samples collected during each solar year in the period 2008-2012. The values of the proton plus helium flux measured at 50 TeV are reported in table I. A power-law fit has been performed on the measured spectrum of each year and of the full data sample, the resulting spectral indices are reported in figure 5. Both the spectral indices and the flux values are in very good agreement between them, demonstrating the long-period reliability and the stability of the detector. The spectral index γ = -2 . 64 ± 0 . 01, obtained by analyzing the full data sample, is in good agreement with the one measured by using a smaller data sample collected in the first months of 2008 [4] which was not corrected by the contamination from heavier nuclei (see section IV A 4).</text>
<text><location><page_11><loc_12><loc_23><loc_88><loc_54></location>In table II and figure 6 the flux obtained by analyzing the full data sample is reported. The spectrum covers a wide energy range, spanning about two orders of magnitude and is in excellent agreement with the previous ARGO-YBJ measurement. Statistical errors are of the order of 1 % , more than 10 5 events have been selected in the highest energy region, while at the lowest energies more than 10 7 events have been selected. Systematic errors are discussed in the next section. The ARGO-YBJ data are in good agreement with the CREAM proton plus helium spectrum [24]. At energies around 10 TeV and 50 TeV the fluxes differ by about 10% and 20% respectively. This means that the absolute energy scale difference of the two experiments is within 4% and 6%. The uncertainty on the absolute energy scale has been evaluated by exploiting the Moon shadow tool at a level of 10% for energies below 30 TeV [8]. At present the ARGO-YBJ experiment is the only ground-based detector able to investigate the cosmic ray energy spectrum in this energy region.</text>
<section_header_level_1><location><page_11><loc_14><loc_17><loc_40><loc_18></location>A. Systematic uncertainties</section_header_level_1>
<text><location><page_11><loc_12><loc_7><loc_88><loc_14></location>A study of possible systematic effects has been performed. Four main sources of systematic uncertainties on the flux measurement have been considered in this work: variation of the selection cuts, reliability of the detector simulation, different interaction models, con-</text>
<figure>
<location><page_12><loc_38><loc_68><loc_61><loc_90></location>
<caption>FIG. 5. Spectral indices of the power-law fit of the light component spectrum measured by analyzing the data sample collected in the period 2008-2012. The spectral index obtained in a previous analysis of the ARGO-YBJ data is shown as 2008* [4] . The error bars represent the total uncertainty.</caption>
</figure>
<text><location><page_12><loc_49><loc_47><loc_50><loc_48></location>.</text>
<table>
<location><page_12><loc_12><loc_30><loc_88><loc_48></location>
<caption>TABLE I. Proton plus helium flux measured at 5 . 0 × 10 4 GeV.</caption>
</table>
<text><location><page_12><loc_12><loc_26><loc_37><loc_27></location>tamination by heavy elements.</text>
<section_header_level_1><location><page_12><loc_14><loc_19><loc_31><loc_21></location>1. Selection criteria</section_header_level_1>
<text><location><page_12><loc_12><loc_7><loc_88><loc_16></location>The fiducial selection criteria have been fine tuned in order to obtain an unbiased evaluation of the bayesian probabilities, leading to the best estimation of the cosmic ray proton plus helium energy spectrum. A possible source of systematic error is related to the values of the fiducial cuts on observables used in the event selection procedure. The uncertainty</text>
<table>
<location><page_13><loc_12><loc_43><loc_88><loc_85></location>
<caption>TABLE II. Light component energy spectrum measured by the ARGO-YBJ experiment by using the full 2008-2012 data sample in each energy bin.</caption>
</table>
<text><location><page_13><loc_12><loc_31><loc_88><loc_40></location>on the measured spectrum has been estimated by applying large variations (about 50 %) to the fiducial cuts and turns out to be of about 3%. The bins located at the edges of the measured energy range are affected by an uncertainty of about ± 5%. A variation of the quality cuts does not give a significative contribution to the total systematic uncertainty.</text>
<section_header_level_1><location><page_13><loc_14><loc_25><loc_46><loc_26></location>2. Reliability of the detector simulation</section_header_level_1>
<text><location><page_13><loc_12><loc_7><loc_88><loc_21></location>A systematic effect could arise from inaccuracies in the simulation of the detector response. The quality of the simulated events has been estimated by comparing the distribution of the observables obtained by applying the same selection criteria to Monte Carlo simulations and the data sample collected in each different year. As an example in figure 7 the multiplicity distribution obtained from the Monte Carlo events is reported with the multiplicity distribution of the data. The ratio between the two distributions is also reported</text>
<figure>
<location><page_14><loc_21><loc_65><loc_77><loc_89></location>
<caption>FIG. 6. The proton plus helium spectrum measured by the ARGO-YBJ experiment using the full 2008-2012 data sample. The error bars represent the total uncertainty. Previous measurement performed by ARGO-YBJ in a narrower energy range by analyzing a smaller data sample is also reported (blue squares) [4]. The green inverted triangles represent the sum of the proton and helium spectra measured by the CREAM experiment [24]. The proton (stars) and helium (empty stars) spectra measured by the PAMELA experiment [25] are also shown. The light component spectra according to the Gaisser-Stanev-Tilav (dashed-dotted line) [26] and Horandel (dashed line) [23] models are also shown.</caption>
</figure>
<text><location><page_14><loc_12><loc_31><loc_88><loc_38></location>showing a good agreement between the two distributions. The contribution to the total systematic uncertainty due to the reliability of the detector simulation has been evaluated by using the unfolding probabilities and turns out to be about ± 6%.</text>
<section_header_level_1><location><page_14><loc_14><loc_25><loc_39><loc_26></location>3. Hadronic interaction models</section_header_level_1>
<text><location><page_14><loc_12><loc_7><loc_88><loc_21></location>In order to estimate effects due to the particular choice of the high energy hadronic interaction model in Monte Carlo simulations, a dataset has been generated by using the SIBYLL 2.1 model [27, 28]. These data have been compared with the QGSJET dataset used in this analysis. In figure 8 the ratio between the multiplicity distributions obtained by using QGSJET model and the one obtained by using SIBYLL is reported as a function of primary energy. The plot shows that the variation of the multiplicity distributions obtained</text>
<figure>
<location><page_15><loc_35><loc_61><loc_62><loc_89></location>
<caption>FIG. 7. Multiplicity distributions of the events selected by the criteria described in section III C. Values for data and Monte Carlo (black solid line) are reported. The ratio between data and Monte Carlo is shown in the lower panel.</caption>
</figure>
<figure>
<location><page_15><loc_35><loc_27><loc_64><loc_41></location>
<caption>FIG. 8. Ratio between the multiplicity distributions obtained from QGSJET and SIBYLL based Monte Carlo simulations.</caption>
</figure>
<text><location><page_15><loc_12><loc_7><loc_88><loc_11></location>with the two hadronic models is of few percents, giving a negligible effect on the measured flux.</text>
<section_header_level_1><location><page_16><loc_14><loc_89><loc_45><loc_91></location>4. Contamination of heavier elements</section_header_level_1>
<text><location><page_16><loc_12><loc_61><loc_88><loc_86></location>A possible systematic effect relies in the contamination of elements heavier than Helium. The selection criterion based on the particle density rejects a large fraction of showers produced by heavy primaries, as shown in figure 3. The fraction of heavier elements, estimated by using the QGSJET-based simulations according to the Horandel model [23], is reduced and can be considered as negligible at energies up to 100 TeV. In the lower energy bins the contamination is about 1%, whereas in the bins below 100 TeV the contamination does not exceed few % and in the higher energy bins it is about 10%. The unfolding procedure has been set up in order to take into account the amount of heavier nuclei passing the selection criteria. The contribution of this effect is therefore not included in the total systematic uncertainty.</text>
<section_header_level_1><location><page_16><loc_14><loc_56><loc_41><loc_57></location>5. Summary of systematic errors</section_header_level_1>
<text><location><page_16><loc_12><loc_43><loc_88><loc_53></location>The total systematic uncertainty was determined by quadratically adding the individual contributions. The results are affected by a systematic uncertainty of the order of ± 5% in the central bins, while the edge bins are affected by a larger systematic uncertainty less than ± 10%.</text>
<section_header_level_1><location><page_16><loc_12><loc_38><loc_31><loc_39></location>V. CONCLUSIONS</section_header_level_1>
<text><location><page_16><loc_12><loc_7><loc_88><loc_35></location>The ARGO-YBJ experiment was in operation in its full and stable configuration for more than five years: a huge amount of data has been recorded and reconstructed. The peculiar characteristics of the detector, like the full-coverage technique, high altitude operation and high segmentation and spacetime resolution, allow the detection of showers produced by primaries in a wide energy range from a few TeV up to a few hundreds of TeV. Showers detected by ARGO-YBJ in the multiplicity range 150 -50000 strips are mainly produced by primaries in the (3 -300 TeV) energy range. The relation between the shower size spectrum and the cosmic ray energy spectrum has been established by using an unfolding method based on the Bayes theorem. The unfolding procedure has been performed on the data collected during each year and on the full data sample. The resulting energy spectrum spans the energy range 3 -300 TeV, giving a spectral index γ = -2 . 64 ± 0 . 01, which is in very</text>
<text><location><page_17><loc_12><loc_68><loc_88><loc_91></location>good agreement with the spectral indices obtained by analyzing the sample collected during each year, therefore demonstrating the excellent stability of the detector over a long period. The resulting spectral indices are also in good agreement with the one obtained by analyzing the first data taken with the detector in its full configuration [4]. Special care was devoted to the determination of the uncertainties affecting the measured spectrum. The uncertainty on the results is due to systematic effects of the order of ± 5% in the central energy bins. This measurement demonstrates the possibility to explore the cosmic ray properties down to the TeV region with a ground-based experiment, giving at present one of the most accurate measurement of the cosmic ray proton plus helium energy spectrum in the multi-TeV region.</text>
<section_header_level_1><location><page_17><loc_14><loc_61><loc_37><loc_62></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_17><loc_12><loc_37><loc_88><loc_57></location>This work is supported in China by NSFC (Contract No. 101201307940), the Chinese Ministry of Science and Technology, the Chinese Academy of Science, the Key Laboratory of Particle Astrophysics, CAS, and in Italy by the Istituto Nazionale di Fisica Nucleare (INFN), and Ministero dell'Istruzione, dell'Universit'a e della Ricerca (MIUR). We also acknowledge the essential support of W.Y. Chen, G. Yang, X.F. Yuan, C.Y. Zhao, R. Assiro, B. Biondo, S. Bricola, F. Budano, A. Corvaglia, B. D'Aquino, R. Esposito, A. Innocente, A. Mangano, E. Pastori, C. Pinto, E. Reali, F. Taurino and A. Zerbini, in the installation, debugging, and maintenance of the detector.</text>
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</document> | [{"title": "The cosmic ray proton plus helium energy spectrum measured by the ARGO-YBJ experiment in the energy range 3-300 TeV", "content": "B. Bartoli, 1, 2 P. Bernardini, 3, 4 X.J. Bi, 5 Z. Cao, 5 S. Catalanotti, 1, 2 S.Z. Chen, 5 T.L. Chen, 6 S.W. Cui, 7 B.Z. Dai, 8 A. D'Amone, 3, 4 Danzengluobu, 6 I. De Mitri, 3, 4 B. D'Ettorre Piazzoli, 1, 2 T. Di Girolamo, 1, 2 G. Di Sciascio, 9 C.F. Feng, 10 Zhaoyang Feng, 5 Zhenyong Feng, 11 Q.B. Gou, 5 Y.Q. Guo, 5 H.H. He, 5 Haibing Hu, 6 Hongbo Hu, 5 M. Iacovacci, 1, 2 R. Iuppa, 9, 12 H.Y. Jia, 11 Labaciren, 6 H.J. Li, 6 C. Liu, 5 J. Liu, 8 M.Y. Liu, 6 H. Lu, 5 L.L. Ma, 5 X.H. Ma, 5 G. Mancarella, 3, 4 S.M. Mari, 13, 14, \u2217 G. Marsella, 3, 4 S. Mastroianni, 2 P. Montini, 13, 14, \u2020 C.C. Ning, 6 L. Perrone, 3, 4 P. Pistilli, 13, 14 P. Salvini, 15 R. Santonico, 9, 12 G. Settanta, 13 P.R. Shen, 5 X.D. Sheng, 5 F. Shi, 5 A. Surdo, 4 Y.H. Tan, 5 P. Vallania, 16, 17 S. Vernetto, 16, 17 C. Vigorito, 16, 17 H. Wang, 5 C.Y. Wu, 5 H.R. Wu, 5 L. Xue, 10 Q.Y. Yang, 8 X.C. Yang, 8 Z.G. Yao, 5 A.F. Yuan, 6 M. Zha, 5 H.M. Zhang, 5 L. Zhang, 8 X.Y. Zhang, 10 Y. Zhang, 5 J. Zhao, 5 Zhaxiciren, 6 Zhaxisangzhu, 6 X.X. Zhou, 11 F.R. Zhu, 11 and Q.Q. Zhu 5", "pages": [1]}, {"title": "(ARGO-YBJ Collaboration)", "content": "1 Dipartimento di Fisica dell'Universit'a di Napoli 'Federico II', Complesso Universitario di Monte Sant'Angelo, via Cinthia, 80126 Napoli, Italy. 2 Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Complesso Universitario di Monte Sant'Angelo, via Cinthia, 80126 Napoli, Italy. 3 Dipartimento Matematica e Fisica 'Ennio De Giorgi', Universit'a del Salento, via per Arnesano, 73100 Lecce, Italy. 4 Istituto Nazionale di Fisica Nucleare, Sezione di Lecce, via per Arnesano, 73100 Lecce, Italy. 5 Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, P.O. Box 918, 100049 Beijing, P.R. China. 6 Tibet University, 850000 Lhasa, Xizang, P.R. China. 7 Hebei Normal University, Shijiazhuang 050016, Hebei, P.R. China. 8 Yunnan University, 2 North Cuihu Rd., 650091 Kunming, Yunnan, P.R. China. 9 Istituto Nazionale di Fisica Nucleare, Sezione di Roma Tor Vergata, 11 via della Ricerca Scientifica 1, 00133 Roma, Italy. 10 Shandong University, 250100 Jinan, Shandong, P.R. China. Southwest Jiaotong University, 610031 Chengdu, Sichuan, P.R. China. 12 Dipartimento di Fisica dell'Universit'a di Roma 'Tor Vergata', via della Ricerca Scientifica 1, 00133 Roma, Italy. 13 Dipartimento di Matematica e Fisica dell'Universit'a 'Roma Tre', via della Vasca Navale 84, 00146 Roma, Italy. 14 Istituto Nazionale di Fisica Nucleare, Sezione di Roma Tre, via della Vasca Navale 84, 00146 Roma, Italy. 15 Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, via Bassi 6, 27100 Pavia, Italy. 16 Istituto Nazionale di Fisica Nucleare, Sezione di Torino, via P. Giuria 1, 10125 Torino, Italy. 17 Dipartimento di Fisica dell'Universit'a di Torino, via P. Giuria 1, 10125 Torino, Italy. (Dated: October 13, 2018)", "pages": [1, 2]}, {"title": "Abstract", "content": "The ARGO-YBJ experiment is a full-coverage air shower detector located at the Yangbajing Cosmic Ray Observatory (Tibet, People's Republic of China, 4300 m a.s.l.). The high altitude, combined with the full-coverage technique, allows the detection of extensive air showers in a wide energy range and offer the possibility of measuring the cosmic ray proton plus helium spectrum down to the TeV region, where direct balloon/space-borne measurements are available. The detector has been in stable data taking in its full configuration from November 2007 to February 2013. In this paper the measurement of the cosmic ray proton plus helium energy spectrum is presented in the region 3 -300 TeV by analyzing the full collected data sample. The resulting spectral index is \u03b3 = -2 . 64 \u00b1 0 . 01. These results demonstrate the possibility of performing an accurate measurement of the spectrum of light elements with a ground based air shower detector.", "pages": [2]}, {"title": "I. INTRODUCTION", "content": "Cosmic rays are ionized nuclei reaching the Earth from outside the solar system. Many experimental efforts have been devoted to the study of cosmic ray properties. In the last decades many experiments were focused on the identification of cosmic ray sources and on the understanding of their acceleration and propagation mechanisms. Despite a very large amount of data collected so far, the origin and propagation of cosmic rays are still under discussion. Supernova remnants (SNRs) are commonly identified as the source of galactic cosmic rays since they could provide the amount of energy needed in order to accelerate particles up to the highest energies in the Galaxy. The measurement of the diffuse gammaray radiation in the energy range 1 -100 GeV supports these hypotheses on the origin and propagation of cosmic rays [1]. Moreover the TeV gamma-ray emission from SNRs, detected by ground-based experiment, can be related to the acceleration of particles up to \u223c 100 TeV [2, 3]. A very detailed measurement of the energy spectrum and composition of primary cosmic rays will lead to a deeper knowledge of the acceleration and propagation mechanisms. Since the energy spectrum spans a huge energy interval, experiments dedicated to the study of cosmic ray properties are essentially divided into two broad classes. Direct experiments operating on satellites or balloons are able to measure the energy spectrum and the isotopic composition of cosmic rays on top of the atmosphere. Due to their reduced detector active surface and the limited exposure time the maximum detectable energy is limited up to few TeV. New generation instruments, capable of long balloon flights, have extended the energy measurements up to \u223c 100 TeV. All the information concerning cosmic rays above 100 TeV is provided by ground-based air shower experiments. Air shower experiments are able to observe the cascade of particles produced by the interaction between cosmic rays and the Earth's atmosphere. Ground based experiments detect extensive air showers produced by primaries with energies up to 10 20 eV, however they do not allow an easy determination of the abundances of individual elements and the measurement of the composition is therefore limited only to the main elemental groups. Moreover, due to a lack of a model-independent energy calibration, the determination of the primary energy relies on the hadronic interaction model used in the description of the shower's development. The ARGO-YBJ experiment is a high-altitude full-coverage air shower detector which was in full and stable data taking from November 2007 up to February 2013. As described in section II, the detector is equipped with a digital and an analog readout systems working independently in order to study the cosmic ray properties in the energy range 1 -10 4 TeV, which is one of the main physics goals of the ARGO-YBJ experiment. The high spacetime resolution of the digital readout system allows the detection of showers produced by primaries down to few TeV, where balloon-borne measurements are available. The analog readout system was designed and built in order to detect showers in a very wide range of particle density at ground level and to explore the cosmic ray spectrum up to the PeV region. In 2012 a first measurement of the cosmic ray proton plus helium (light component) spectrum obtained by analyzing a small sample collected during the first period of data taking with the detector in its full configuration (by using the digital readout information only) has been presented [4]. In this paper we report the analysis of the full data sample collected by the ARGO-YBJ experiment in the period from January 2008 to December 2012 and the measurement of the light component energy spectrum of cosmic rays in the energy range 3 -300 TeV by applying an unfolding procedure based on the bayesian probabilities. The analysis of the analog readout data and the corresponding cosmic ray spectrum up to the PeV energy region is in progress and will be addressed in a future paper.", "pages": [3, 4]}, {"title": "II. THE ARGO-YBJ EXPERIMENT", "content": "The ARGO-YBJ experiment (Yangbajing Cosmic Ray Observatory, Tibet, P.R. China. 4300 m a.s.l.) is a full-coverage detector made of a single layer of Resistive Plate Chambers (RPCs) with \u223c 93% active area [5, 6], surrounded by a partially instrumented guard ring designed to improve the event reconstruction. The detector is made of 1836 RPCs, arranged in 153 clusters each made of 12 chambers. The digital readout consists of 18360 pads each segmented in 8 strips. A dedicated procedure was implemented to calibrate the detector in order to achieve high pointing accuracy [7]. The angular and core reconstruction resolution are respectively 0 . 4 \u00b7 and 5 m for events with at least 500 fired pads [8, 9]. The installation of the central carpet was completed in June 2006. The guard ring was completed during spring 2007 and connected to the data acquisition system [10] in November 2007. A simple trigger logic based on the coincidence between the pad signals was implemented. The detector has been in stable data taking in its full configuration for more than five years with a trigger threshold N pad = 20, corresponding to a trigger rate of about 3 . 6 kHz and a dead time of 4%. The high granularity and time resolution of the detector provide a detailed three-dimensional reconstruction of the shower front. The high altitude location and the segmentation of the experiment offer the possibility to detect showers produced by charged cosmic rays with energies down to few TeV. The digital readout of the pad system allows reconstruction of showers with a particle density at ground level up to about 23 particles/m 2 , which correspond to primaries up to a few hundreds of TeV. In order to extend the detector operating range and investigate energies up to the PeV region each RPC has been equipped with two large size electrodes called Big Pads [11]. Each Big Pad provide a signal whose amplitude is proportional to the number of particles impinging the detector surface. The analog readout system allows a detailed measurement of showers with particle density at ground up to more than 10 4 particles / m 2 .", "pages": [4, 5]}, {"title": "A. Unfolding of the cosmic ray energy spectrum", "content": "As widely described in [4, 12], the determination of the cosmic ray spectrum starting from the measured space-time distribution of charged particles at ground level is a classical unfolding problem that can be dealt by using the bayesian technique[13]. In this framework the detector response is represented by the probability P ( M j | E i ) of measuring a multiplicity M j due to a shower produced by a primary of energy E i . The estimated number of events in a certain energy bin E i is therefore related to the number of events measured in a multiplicity bin M j by the equation where \u03b7 ij is constructed by using the Bayes theorem The values of the probability P ( M j | E i ) are evaluated by means of a Monte Carlo simulation of the development of the shower and of the detector response. The quantity P ( E i | M j ) represents the probability that a shower detected with multiplicity M j has been produced by a primary of energy E i . The values of P ( M j | E i ) are evaluated by means of an iterative procedure starting from a prior value P (0) ( E i ), in which in the n -th step P ( n ) ( E i ) is replaced by the updated value where \u02c6 N ( n ) ( E i ) is evaluated in the n -th step according to eq. 1. As initial prior P (0) ( E i ) \u223c E -2 . 5 was chosen, the effect of using different prior distributions has been evaluated as negligible. The iterative procedure ends when the variation of all \u02c6 N ( E i ) in two consecutive steps are evaluated as negligible, namely less than 0.1 %. Typically the convergence is reached after 3 iterations.", "pages": [5, 6]}, {"title": "B. Air shower and detector simulations", "content": "The development of the shower in the Earth's atmosphere has been simulated by using the CORSIKA (v. 6980) code [14]. The electromagnetic component are described by the EGS4 code [15, 16], while the high energy hadronic interactions are reproduced by QGSJETII.03 model [17, 18]. Low energy hadronic interactions are described by the FLUKA package [19, 20]. Showers produced by Protons, Helium, CNO nuclei and Iron have been generated with a spectral index \u03b3 = -1 in the energy range (0 . 316 -3 . 16 \u00d7 10 4 ) TeV. About 5 \u00d7 10 7 showers have been generated in the zenith angle range 0-45 degrees and azimuth angle range 0-360 degrees. Showers were sampled at the Yangbajing altitude and the shower core was randomly distributed over an area of (250 \u00d7 250) m 2 centered on the detector. The resulting CORSIKA showers have been processed by a GEANT3 [21] based code in order to reproduce the detector response, including the effects of time resolution, RPC efficiency, trigger logic, accidental background produced by each pad and electronic noise.", "pages": [6]}, {"title": "C. Event selection", "content": "The ARGO-YBJ experiment was in stable data taking in its full configuration for more than five years: more than 5 \u00d7 10 11 events have been recorded and reconstructed. Several tools have been implemented in order to monitor the detector operation and reconstruction quality. The detector control system (DCS) [22] continuously monitors the RPC current, the high voltage distribution, the gas mixture and the environmental conditions (temperature, pressure, humidity). In this work the analysis of events collected during the period 20082012 is presented. Data and simulated events have been selected according to a multi-step procedure in order to obtain high quality events and to ensure a reliable and unbiased evaluation of the bayesian probabilities. The first step concerns the run selection: in order to obtain a sample of high-quality runs, the working condition of the detector and the quality of the reconstruction procedure have been analyzed by using the criteria described below. In figure 1 the distribution of the trigger rate and the \u00af \u03c7 2 of the reconstruction procedure are reported. The procedure described above selects a data sample of about 3 \u00d7 10 11 events, corresponding to a live time of about 24000 hours. The following selection criteria (fiducial cuts) have been applied to both Monte Carlo and experimental data in order to improve the quality of the reconstruction and to obtain the best estimation of the bayesian probabilities. 7000 In order to select showers induced by proton and helium nuclei the following criterium has been used. entries/bin the outermost area (42 outer clusters): ( \u03c1 in > 1 . 25 \u03c1 out ). This selection criteria based on the lateral particle distribution was introduced in order to discard events produced by nuclei heavier than helium. In fact, in showers induced by heavy primaries the lateral distribution is wider than in light-induced ones. By applying this criterion on events with the core located in a narrow area around the detector center, showers mainly produced by light primaries have been selected. The contamination of elements heavier than helium does not exceed few %, as discussed in section IV A 4. In figure 2 the coordinates of the reconstructed core position of the events surviving the selection criteria described above are reported. The plot shows that the contribution of events located outside an area of 40 \u00d7 40 m 2 is negligible. In figure 3 the event rate obtained by using the Horandel model for input spectra and isotopic composition [23] and surviving the selection criteria described above is reported as a function of energy for both proton plus helium (light component) and heavier elements (heavy component). The plot shows that the selected sample is essentially made of light nuclei.", "pages": [6, 7, 8, 9, 10]}, {"title": "IV. THE LIGHT COMPONENT SPECTRUM", "content": "The analysis was performed on the sample selected by the criteria described in section III. Simulated events have been sorted in 16 multiplicity bins and 13 energy bins in order to minimize the statistical error and to reduce bin migration effects. The Monte Carlo data sample was analyzed in order to evaluate the probability distribution P ( M | E ) and the energy resolution which turns out to be about 10% for energies below 10 TeV and of the order of 5% at energies of about 100 TeV. The multiplicity distribution extracted from data has been unfolded according to the procedure described in section III A. Results are reported in figure 4 for each year of data taking and also for the full sample. In order to investigate the stability of the detector over a long period the analysis was performed separately on the data samples collected during each solar year in the period 2008-2012. The values of the proton plus helium flux measured at 50 TeV are reported in table I. A power-law fit has been performed on the measured spectrum of each year and of the full data sample, the resulting spectral indices are reported in figure 5. Both the spectral indices and the flux values are in very good agreement between them, demonstrating the long-period reliability and the stability of the detector. The spectral index \u03b3 = -2 . 64 \u00b1 0 . 01, obtained by analyzing the full data sample, is in good agreement with the one measured by using a smaller data sample collected in the first months of 2008 [4] which was not corrected by the contamination from heavier nuclei (see section IV A 4). In table II and figure 6 the flux obtained by analyzing the full data sample is reported. The spectrum covers a wide energy range, spanning about two orders of magnitude and is in excellent agreement with the previous ARGO-YBJ measurement. Statistical errors are of the order of 1 % , more than 10 5 events have been selected in the highest energy region, while at the lowest energies more than 10 7 events have been selected. Systematic errors are discussed in the next section. The ARGO-YBJ data are in good agreement with the CREAM proton plus helium spectrum [24]. At energies around 10 TeV and 50 TeV the fluxes differ by about 10% and 20% respectively. This means that the absolute energy scale difference of the two experiments is within 4% and 6%. The uncertainty on the absolute energy scale has been evaluated by exploiting the Moon shadow tool at a level of 10% for energies below 30 TeV [8]. At present the ARGO-YBJ experiment is the only ground-based detector able to investigate the cosmic ray energy spectrum in this energy region.", "pages": [10, 11]}, {"title": "A. Systematic uncertainties", "content": "A study of possible systematic effects has been performed. Four main sources of systematic uncertainties on the flux measurement have been considered in this work: variation of the selection cuts, reliability of the detector simulation, different interaction models, con- . tamination by heavy elements.", "pages": [11, 12]}, {"title": "1. Selection criteria", "content": "The fiducial selection criteria have been fine tuned in order to obtain an unbiased evaluation of the bayesian probabilities, leading to the best estimation of the cosmic ray proton plus helium energy spectrum. A possible source of systematic error is related to the values of the fiducial cuts on observables used in the event selection procedure. The uncertainty on the measured spectrum has been estimated by applying large variations (about 50 %) to the fiducial cuts and turns out to be of about 3%. The bins located at the edges of the measured energy range are affected by an uncertainty of about \u00b1 5%. A variation of the quality cuts does not give a significative contribution to the total systematic uncertainty.", "pages": [12, 13]}, {"title": "2. Reliability of the detector simulation", "content": "A systematic effect could arise from inaccuracies in the simulation of the detector response. The quality of the simulated events has been estimated by comparing the distribution of the observables obtained by applying the same selection criteria to Monte Carlo simulations and the data sample collected in each different year. As an example in figure 7 the multiplicity distribution obtained from the Monte Carlo events is reported with the multiplicity distribution of the data. The ratio between the two distributions is also reported showing a good agreement between the two distributions. The contribution to the total systematic uncertainty due to the reliability of the detector simulation has been evaluated by using the unfolding probabilities and turns out to be about \u00b1 6%.", "pages": [13, 14]}, {"title": "3. Hadronic interaction models", "content": "In order to estimate effects due to the particular choice of the high energy hadronic interaction model in Monte Carlo simulations, a dataset has been generated by using the SIBYLL 2.1 model [27, 28]. These data have been compared with the QGSJET dataset used in this analysis. In figure 8 the ratio between the multiplicity distributions obtained by using QGSJET model and the one obtained by using SIBYLL is reported as a function of primary energy. The plot shows that the variation of the multiplicity distributions obtained with the two hadronic models is of few percents, giving a negligible effect on the measured flux.", "pages": [14, 15]}, {"title": "4. Contamination of heavier elements", "content": "A possible systematic effect relies in the contamination of elements heavier than Helium. The selection criterion based on the particle density rejects a large fraction of showers produced by heavy primaries, as shown in figure 3. The fraction of heavier elements, estimated by using the QGSJET-based simulations according to the Horandel model [23], is reduced and can be considered as negligible at energies up to 100 TeV. In the lower energy bins the contamination is about 1%, whereas in the bins below 100 TeV the contamination does not exceed few % and in the higher energy bins it is about 10%. The unfolding procedure has been set up in order to take into account the amount of heavier nuclei passing the selection criteria. The contribution of this effect is therefore not included in the total systematic uncertainty.", "pages": [16]}, {"title": "5. Summary of systematic errors", "content": "The total systematic uncertainty was determined by quadratically adding the individual contributions. The results are affected by a systematic uncertainty of the order of \u00b1 5% in the central bins, while the edge bins are affected by a larger systematic uncertainty less than \u00b1 10%.", "pages": [16]}, {"title": "V. CONCLUSIONS", "content": "The ARGO-YBJ experiment was in operation in its full and stable configuration for more than five years: a huge amount of data has been recorded and reconstructed. The peculiar characteristics of the detector, like the full-coverage technique, high altitude operation and high segmentation and spacetime resolution, allow the detection of showers produced by primaries in a wide energy range from a few TeV up to a few hundreds of TeV. Showers detected by ARGO-YBJ in the multiplicity range 150 -50000 strips are mainly produced by primaries in the (3 -300 TeV) energy range. The relation between the shower size spectrum and the cosmic ray energy spectrum has been established by using an unfolding method based on the Bayes theorem. The unfolding procedure has been performed on the data collected during each year and on the full data sample. The resulting energy spectrum spans the energy range 3 -300 TeV, giving a spectral index \u03b3 = -2 . 64 \u00b1 0 . 01, which is in very good agreement with the spectral indices obtained by analyzing the sample collected during each year, therefore demonstrating the excellent stability of the detector over a long period. The resulting spectral indices are also in good agreement with the one obtained by analyzing the first data taken with the detector in its full configuration [4]. Special care was devoted to the determination of the uncertainties affecting the measured spectrum. The uncertainty on the results is due to systematic effects of the order of \u00b1 5% in the central energy bins. This measurement demonstrates the possibility to explore the cosmic ray properties down to the TeV region with a ground-based experiment, giving at present one of the most accurate measurement of the cosmic ray proton plus helium energy spectrum in the multi-TeV region.", "pages": [16, 17]}, {"title": "ACKNOWLEDGMENTS", "content": "This work is supported in China by NSFC (Contract No. 101201307940), the Chinese Ministry of Science and Technology, the Chinese Academy of Science, the Key Laboratory of Particle Astrophysics, CAS, and in Italy by the Istituto Nazionale di Fisica Nucleare (INFN), and Ministero dell'Istruzione, dell'Universit'a e della Ricerca (MIUR). We also acknowledge the essential support of W.Y. Chen, G. Yang, X.F. Yuan, C.Y. Zhao, R. Assiro, B. Biondo, S. Bricola, F. Budano, A. Corvaglia, B. D'Aquino, R. Esposito, A. Innocente, A. Mangano, E. Pastori, C. Pinto, E. Reali, F. Taurino and A. Zerbini, in the installation, debugging, and maintenance of the detector.", "pages": [17]}] |
2022EPJWC.25700021H | https://arxiv.org/pdf/2111.01720.pdf | <document>
<section_header_level_1><location><page_1><loc_12><loc_77><loc_88><loc_81></location>Generalised scalar-tensor theories of gravity and pressure profiles of galaxy clusters</section_header_level_1>
<text><location><page_1><loc_12><loc_71><loc_88><loc_74></location>Balakrishna S. Haridasu 1 ; 2 ; 3 ; GLYPH<3> , Purnendu Karmakar 4 , Marco De Petris 4 ; 5 ; 6 , Vincenzo F. Cardone 5 ; 6 , and Roberto Maoli 4 ; 5</text>
<text><location><page_1><loc_12><loc_69><loc_12><loc_70></location>1</text>
<text><location><page_1><loc_12><loc_67><loc_80><loc_70></location>SISSA-International School for Advanced Studies, Via Bonomea 265, 34136 Trieste, Italy 2 INFN, Sezione di Trieste, Via Valerio 2, I-34127 Trieste, Italy</text>
<unordered_list>
<list_item><location><page_1><loc_12><loc_66><loc_80><loc_67></location>3 IFPU, Institute for Fundamental Physics of the Universe, via Beirut 2, 34151 Trieste, Italy</list_item>
<list_item><location><page_1><loc_12><loc_64><loc_81><loc_65></location>4 Dipartimento di Fisica, Sapienza Universitá di Roma, P.le Aldo Moro 2, 00185, Roma, Italy</list_item>
<list_item><location><page_1><loc_12><loc_62><loc_88><loc_64></location>5 INAF - Osservatorio Astronomico di Roma, Via Frascati 33, 00040, Monteporzio Catone, Roma, Italy</list_item>
<list_item><location><page_1><loc_12><loc_61><loc_12><loc_62></location>6</list_item>
<list_item><location><page_1><loc_12><loc_61><loc_61><loc_62></location>INFN- Sezione di Roma 1, P.le Aldo Moro 2, 00185, Roma, Italy</list_item>
</unordered_list>
<text><location><page_1><loc_21><loc_35><loc_79><loc_59></location>Abstract. In the current proceedings, we summarise the results presented during the mm Universe@NIKA2 conference, taken from our main results in [1]. We test the Degenerate higher-order scalar-tensor(DHOST) theory as a generalised platform for scalar-tensor theory at galaxy cluster scales to predict in such static systems small scale modification to the gravitational potential. DHOST theory is not only a good alternative to GLYPH<3> CDM for the background evolution but also predicts small-scale modification to the gravitational potential in static systems such as galaxy clusters. With a sample of 12 clusters with accurate Xray Intra Cluster Medium (ICM) data (X-COP project) and Sunyaev-Zel'dovich (SZ) ICM pressure (Planck satellite), we place preliminary constraints on the DHOST parameters defining the deviation from GR. Moreover, we also collect a few supplementary analyses we have performed during the course: i) Gaussian process reconstruction without parametric assumptions, ii) P SZ-only data analysis not aided by the X-ray data. Finally, we present possible extensions to the current work which may benefit from future high sensitivity and spatial resolution observations such as the ongoing NIKA2 camera.</text>
<section_header_level_1><location><page_1><loc_12><loc_28><loc_28><loc_29></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_12><loc_15><loc_88><loc_26></location>Einstein developed the standard general relativity (GR) by considering gravity as warps and curves in the fabric of geometric space-time, and that geometry of space-time is described only by a metric. GR can explain the majority of the cosmological observations, but challenged to explain the acceleration in the cosmological distance. Therefore, if we plan to keep the basic metric structure because of the large success of GR, and add one scalar degrees of freedom to explain the acceleration, then the degenerate higher-order scalar-tensor (DHOST) theory is the best possible platform to test the largest classes of scalar-tensor theories [2, 3].</text>
<text><location><page_1><loc_12><loc_8><loc_88><loc_14></location>The additional degree of freedom must be suppressed in within the small scales to explain the observations, is the so-called screening mechanism [4]. The modified Newtonian potential for the DHOST theory in the galaxy cluster, which produces the gravitational waves of the velocity ( c g ) equal to the velocity of light ( c ), is</text>
<formula><location><page_2><loc_32><loc_86><loc_88><loc_90></location>d GLYPH<8> ( r ) dr = G e GLYPH<11> N M HSE( r ) r 2 + GLYPH<4> 1 G e GLYPH<11> N d 2 M HSE( r ) dr 2 ; (1)</formula>
<text><location><page_2><loc_12><loc_82><loc_88><loc_86></location>while assuming the hydrostatic equilibrium (HSE), where M HSE( r ) is the total mass within the radial distance r . Both, GLYPH<4> 1 and G e GLYPH<11> N parameters track the departure of DHOST theory from GR, which modify the ICM thermal pressure profile as</text>
<formula><location><page_2><loc_35><loc_77><loc_88><loc_81></location>P th ( r ) = P th (0) GLYPH<0> Z r 0 GLYPH<26> gas(˜ r ) d GLYPH<8> (˜ r ) dr d˜ r ; (2)</formula>
<text><location><page_2><loc_12><loc_67><loc_88><loc_77></location>where GLYPH<26> gas( r ) is the gas density. Our formalism is a straight-up implementation of the so called forward -method, where the pressure profile is computed while assuming empirical profiles for the mass, M HSE( r ), and the electron density, ne ( r ), radial profiles which in our case are the standard NFW [5] and simplified Vikhlinin parametric model [6], respectively. In this context, several previous works have implemented similar approach either using stacked clusters and / or having complementary weak lensing data [7-9].</text>
<text><location><page_2><loc_12><loc_62><loc_88><loc_67></location>Throughout the current discourse we assume H 0 = 70 km / s Mpc GLYPH<0> 1 and GLYPH<10> m = 0 : 3. R 500 and M 500 carry usual definitions, i.e, total mass M GLYPH<1> within the radius R GLYPH<1> , with a mean mass density GLYPH<1> times the critical density ( GLYPH<26> c( z ) = 3 H 2 ( z ) = 8 GLYPH<25> G N).</text>
<section_header_level_1><location><page_2><loc_12><loc_58><loc_29><loc_59></location>2 Main Results</section_header_level_1>
<text><location><page_2><loc_12><loc_39><loc_88><loc_57></location>Firstly, we figure that only 8 clusters out of the 12 available clusters in the dataset prefer the NFWmass profile in a comparison through information criteria (see Fig 2, of [10]). We refer to the other 4 clusters (A644, A1644, A2319 and A2255) as non-NFW clusters for brevity. As a consequence of which, we find that the non-NFW clusters fit the data within the DHOST scenario with statistical preferences reaching GLYPH<1> log( B ) >> 25 (further details in [1] and [10]), which imply a very high preference yet providing very low values of the parameter GLYPH<4> 1, in complete disagreement with the theoretical limits [11-13]. In Figure 1, we show the fits of the GR and DHOST scenarios for A644 (non-NFW) and A1795 cluster and the 1 GLYPH<27> dispersion to provide a comparison. We show the posteriors for GLYPH<4> 1 obtained using the individual cluster and the joint constraint in left panel of Figure 2 and a comparison with earlier constraints in the right panel.</text>
<text><location><page_2><loc_12><loc_18><loc_88><loc_38></location>As the main result of our analysis we updated the limits on the modified gravity parameter GLYPH<4> 1 = GLYPH<0> 0 : 030 GLYPH<6> 0 : 043, which is a much more stringent constrain in comparison to the earlier result of GLYPH<4> 1 GLYPH<24> GLYPH<0> 0 : 028 + 0 : 23 GLYPH<0> 0 : 17 , reported in [7] using a similar formalism with stacked X-ray cluster profile. In a more conservative case where we utilise only 4 clusters (A1795, A3158, RXC1825 and ZW1215), we find a less stringent GLYPH<4> 1 = GLYPH<0> 0 : 061 GLYPH<6> 0 : 074, yet twice as tighter constraint than the earlier analysis. These 4 conservative clusters are restricted within a redshift range of 0 : 0597 < z < 0 : 0766. In the right panel of Figure 2, we show a comparison of the earlier constraints, our results and the theoretical limits. As can be seen, our results are in excellent agreement with the latter and an improvement for the constraints on GLYPH<4> 1. The lower theoretical limit is marked by the requirement of stable static solutions of non-relativistic stars [11, 14] and the upper limits are obtained on the based on the minimum mass required for hydrogen burning in low mass red dwarfs [12, 13].</text>
<text><location><page_2><loc_12><loc_5><loc_88><loc_18></location>An added advantage of our current analysis is that having individual clusters spread over a redshift range of 0 : 04 < z < 0 : 1, we are able to present for the first time the necessary assessment of time-evolution of the parameter GLYPH<4> 1( a GLYPH<17> 1 = (1 + z )). We utilise a simple first order Taylor expansion of GLYPH<4> 1( a ) as the scale factor, a ! 1 and perform a post-MCMC analysis on the inferred posteriors of the GLYPH<4> 1 obtained for individual clusters. Using 8 clusters we find a mild deviation from a constant behaviour for GLYPH<4> 1, at present is however driven only by two clusters (A85 and A2142)and the advantage of extending the analysis over a larger dataset that covers a wider redshift range is clear.</text>
<figure>
<location><page_3><loc_13><loc_70><loc_49><loc_91></location>
</figure>
<figure>
<location><page_3><loc_51><loc_70><loc_88><loc_91></location>
<caption>Figure 1. Mass ( upper ) and pressure ( lower ) radial profiles for the cluster A644 ( Left ) and A1795 ( Right ) in the cases of GR (blue) and DHOST (red), normalised to GR. The former cluster is an example case of the 4 non-NFW clusters, while the latter is one of the 4 clusters we retain in estimating our final conservative result. The vertical dashed (GR) and dot-dashed (DHOST) lines show R 500. The 3 inner P SZ data points (black) are excluded in the main analysis.</caption>
</figure>
<figure>
<location><page_3><loc_13><loc_40><loc_48><loc_56></location>
</figure>
<figure>
<location><page_3><loc_50><loc_40><loc_87><loc_55></location>
<caption>Figure 2. Left : Posterior distribution of the parameter GLYPH<4> 1, over-plotted by smooth Gaussian kernel density profiles. The vertical dashed line marks the GR ( GLYPH<4> 1 = 0) case. The black curve represents our joint constraint using 8 clusters. Right : Constraints on the parameter GLYPH<4> 1, from an earlier analysis of 58 stacked clusters (blue) [7] and our results with 8 clusters (green) and conservative 4 clusters (orange). The grey shaded region shows the allowed region for the same parameter obtained from theoretical limits [11-13].</caption>
</figure>
<section_header_level_1><location><page_3><loc_12><loc_23><loc_41><loc_25></location>3 Supplementary Results</section_header_level_1>
<text><location><page_3><loc_12><loc_19><loc_88><loc_22></location>In this section we present a few supplementary analyses we have performed, also constituting possible future extension of the current work.</text>
<section_header_level_1><location><page_3><loc_12><loc_15><loc_33><loc_16></location>3.1 Gaussian Process</section_header_level_1>
<text><location><page_3><loc_12><loc_5><loc_88><loc_13></location>In addition to the parametric analysis, we also perform a non-parametric model-independent analysis based on the Gaussian Process (GP) formalism, please see [15] and references therein for details on the method. In essence, once could reconstruct the underlying functional form [ f ( xi )] for a collection of Gaussian data points at x GLYPH<17> f x 1 ; x 2 ; :: xN g , assumed belonging to the same process, which now represents a GP. This is implemented by modelling a</text>
<figure>
<location><page_4><loc_13><loc_75><loc_50><loc_91></location>
</figure>
<figure>
<location><page_4><loc_52><loc_75><loc_89><loc_91></location>
<caption>Figure 3. Mass radial profiles with 1 GLYPH<27> dispersion for clusters A644 ( left ) and A1795 ( right ) for the GR (blue), DHOST (red) and GP (green). A644 and A1795 represent one of the non-NFW cluster and one of the 4 conservative clusters, respectively. The vertical dashed, dot-dashed and dotted lines show the R 500 in GR, DHOST and GP, respectively. The R 500 from the GP case coincides very well with that in GR case. The mass profiles obtained utilising only the P SZ data for the GR case are shown in orange. Here the GR case corresponds to ˜ GLYPH<13> N = 1.</caption>
</figure>
<text><location><page_4><loc_12><loc_44><loc_88><loc_58></location>covariance K ( xi ; xj ), amongst the data points, instead of functional form for the process itself. For this purpose we independently reconstruct the electron density and the P SZ pressure profiles assuming the squared-exponential (SE) 1 kernel. Note that here we cannot extrapolate the profile beyond the range of the available data, as GP tends to retrieve broad prior regions, with no constrain on the posterior. We intend the analysis with GP as an alternate verification for the fits based on the profile themselves as the model-independent method is agnostic with no assumptions made for the electron density profile and the mass profile. Therefore, a parametric method based on assumed empirical profiles is expected to serve better if it is in a good agreement with the GP method.</text>
<text><location><page_4><loc_12><loc_20><loc_88><loc_43></location>In Figure 3, we show the mass reconstructions obtained using the NFW profile in both the GR (blue) and DHOST (red) scenarios and the GP (green) reconstructed result. For the GP method we firstly find a lowering of the mass profile in the inner regions, which is an immediate consequence of not excluding the inner most 3 data-points of P SZ data (see [10, 16]). However, being a non-parametric estimation and that it only follows the data points, we find the GPto be in better agreement towards the outskirts of the mass profile. In the case of the A644, a non-NFW cluster, we find that the model-independent reconstruction is in better agreement with the GR case than with the DHOST scenario, which is a clear indication of our results for the latter being biased when utilising the NFW mass profile. The R 500 = 1 : 15 GLYPH<6> 0 : 06 [Mpc] and M 500 = 4 : 57 + 0 : 73 GLYPH<0> 0 : 65 [10 14 M GLYPH<12> ] estimated using GP here are in excellent agreement with our constraints in the GR case. In the case of A1795 however, the deviation between the GR and the DHOST cases is minimal and is also reflected in the GP reconstruction. Once again the GP based R 500 = 1 : 12 GLYPH<6> 0 : 04 [Mpc] and M 500 = 4 : 24 + 0 : 045 GLYPH<0> 0 : 040 [10 14 M GLYPH<12> ] are in excellent agreement with the parametric method assuming the NFW profile.</text>
<section_header_level_1><location><page_4><loc_12><loc_16><loc_32><loc_17></location>3.2 P SZ only analysis</section_header_level_1>
<text><location><page_4><loc_12><loc_9><loc_88><loc_14></location>ndeed, one of the major advantages in our analysis is that we perform the joint-fit to the T X( or P X) and the P SZ data, which provides a better estimation of the mass profile to the outskirts ( R GLYPH<24> 1 : 2 Mpc) of the clusters. In the screening mechanism, the GR is recovered</text>
<text><location><page_5><loc_12><loc_85><loc_88><loc_91></location>in the innermost region of the galaxy cluster, while the gravity gradually deviates from GR in the outskirt of the cluster. Given that, the inclusion of the P SZ data actually provides a very suitable opportunity to asses the mass profile in a larger radial range and hence the modifications to gravity.</text>
<text><location><page_5><loc_12><loc_66><loc_88><loc_84></location>To contrast our constraints against the combined analysis of P SZ + P X, we also perform an MCMC analysis using only the P SZ data (without excluding the 3 inner most data points). We also show the inferred mass profiles (orange) of the same in Figure 3. One can immediately notice that for the cluster A644, all the 4 contrasted scenarios are almost discordant at the face-value. This in turn indicates that a more careful assessment has to be made when combining the data sets and / or trying to assess the modifications to physics using this cluster. On the other hand, the A1795 cluster shows that all the approaches are are in very good agreement, especially in the outskirts of the cluster. However, comparing the P SZ parametric fit and the GP based reconstruction we find that the former tends to show higher masses, also in agreement with the P X + P SZ analysis. The results for the posterior distributions on GLYPH<4> 1 are summarised in Appendix C. of [1].</text>
<text><location><page_5><loc_12><loc_47><loc_88><loc_65></location>Within the joint analysis however, one needs to check internally for a consistency between the X-ray and SZ data, see Appendix A of [16] for more details. Therefore in a similar approach we introduce an intrinsic dispersion ( GLYPH<27> P ; int) as a free parameter which is scaled by the modelled pressure ( GLYPH<27> P ; int GLYPH<24> P model GLYPH<27> int) (Appendix D of [17]), added in quadrature to the error and inferred through the MCMC analysis. Alongside the dispersion, we also include a rescaling parameter GLYPH<17> , such that P data X ! GLYPH<17> GLYPH<2> P data X . We show the posterior distributions of these two consistency parameters in Figure 4 for the A644 and A1795 clusters. Clearly, both clusters show a good agreement between P SZ and P X data, while A644 shows a larger dispersion within the P X data points. This in turn provides additional correlations with M 500 and increases the error-bars, however note also that we fix GLYPH<17> = 1 in the main analysis. Please refer to original analysis in [10, 16, 17], where these arguments are elaborated.</text>
<figure>
<location><page_5><loc_27><loc_13><loc_73><loc_45></location>
<caption>Figure 4. We compare here the posteriors for the intrinsic dispersion in the X-ray data and the scaling parameter. The dashed grey line marks the case of GLYPH<17> = 1, which implies good agreement between the P X and P SZ data.</caption>
</figure>
<text><location><page_6><loc_12><loc_83><loc_88><loc_91></location>In summary, as we have shown here, having the galaxy cluster physics well-mapped in a larger radial range and having several of these objects also distributed along the redshift will be crucial to assess the constraints on the GLYPH<4> 1 in individual clusters and its time evolution. We intend to explore the same with soon to be available future data. For this goal, NIKA2 SZ observations could be helpful to extend this analysis.</text>
<section_header_level_1><location><page_6><loc_12><loc_79><loc_34><loc_80></location>Acknowledgements</section_header_level_1>
<text><location><page_6><loc_12><loc_66><loc_88><loc_77></location>Authors are grateful to the 'mm Universe @NIKA2' conference for providing an oppurtunity to present the work. Authors acknowledge receiving useful comments and feedback on the use of data from Stefano Ettori and Dominique Eckert. BSH is supported by the INFN INDARK grant. PK, MDP, VC,RM acknowledge support from INFN / Euclid Sezione di Roma. PK, MDP and RM also acknowledge support from Sapienza Universitá di Roma thanks to Progetti di Ricerca Medi 2018, prot. RM118164365E40D9 and 2019, prot. RM11916B7540DD8D.</text>
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2019MNRAS.483.1178M | https://arxiv.org/pdf/1811.09502.pdf | <document>
<section_header_level_1><location><page_1><loc_7><loc_80><loc_82><loc_84></location>Implications on accretion flow dynamics from spectral study of Swift J1357.2-0933</section_header_level_1>
<section_header_level_1><location><page_1><loc_7><loc_75><loc_54><loc_77></location>Santanu Mondal /star 1 , 2 and Sandip K. Chakrabarti † 2 , 3</section_header_level_1>
<text><location><page_1><loc_7><loc_71><loc_80><loc_75></location>1 Instituto de F'ısica y Astronom'ıa, Facultad de Ciencias, Universidad de Valpara'ıso, Gran Bretana N 1111, Playa Ancha, Valpara'ıso, Chile 2 Indian Centre for Space Physics, Chalantika 43, Garia Station Rd., Kolkata, 700084, India 3 S. N. Bose National Centre for Basic Sciences, Kolkata, 700098, India</text>
<section_header_level_1><location><page_1><loc_28><loc_64><loc_36><loc_65></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_28><loc_43><loc_89><loc_64></location>We report a detailed spectral study of Swift J1357.2-0933 low-mass X-ray binary during its 2017 outburst using Swift and NuSTAR observations. We fit the data with two component advective flow (TCAF) model and power-law model. We observe that the source is in hard state during the outburst, where the size of the Compton cloud changes significantly with disc accretion rate. The typical disc accretion rate for this source is ∼ 1 . 5 -2 . 0 % of the Eddington accretion rate ( ˙ MEdd ). The model fitted intermediate shock compression ratio gives an indication of the presence of jet, which is reported in the literature in di ff erent energy bands. We also split NuSTAR data into three equal segments and fit with the model. We check spectral stability using color-color diagram and accretion rate ratio (ARR) vs. intensity diagram using di ff erent segments of the light curve but do not find any significant variation in the hardness ratio or in the accretion rate ratio. To estimate the mass of the candidate, we use an important characteristics of TCAF that the the model normalization always remains a constant. We found that the mass comes out to be in the range of 4 . 0 -6 . 8 M /circledot . From the model fitted results, we study the disc geometry and di ff erent physical parameters of the flow in each observation. The count rate of the source appears to decay in a time scale of ∼ 45 day .</text>
<text><location><page_1><loc_28><loc_39><loc_89><loc_42></location>Key words: black hole physics, accretion, accretion discs, binaries: close, stars: individual (Swift J1357.2-0933)</text>
<section_header_level_1><location><page_1><loc_7><loc_33><loc_21><loc_34></location>1 INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_7><loc_8><loc_46><loc_32></location>In a black hole binary, matter falls onto the compact object while transporting angular momentum outwards and mass inwards converting half of its gravitational potential energy into thermal energy and radiation energy. Thus, it is interesting to study the accretion dynamics both in temporal and spatial domains. Unlike a persistent source, where accretion rates could be steady for a long time, in an outburst source, rates are supposed to be varying. Outbursting black hole candidates (BHCs) spend most of the times in the quiescence state and occasionally undergo bright X-ray outbursts, which are definitely due to a huge increase of accretion rate. One of the most natural ways to achieve this is by varying viscosity. It is possible that an outburst may be triggered by enhancement of viscosity at the piling radius of matter. Chakrabarti (1990, 1996) suggested that when the rise of viscous process increases the viscosity parameter above a critical value, the low angular momentum flow can acquire a Keplerian distribution which becomes a SS73-like standard disk in presence of cooling (Shakura & Sunyaev, 1973). Based on this Chakrabarti (1997) concluded that the regular rise and fall of the</text>
<unordered_list>
<list_item><location><page_1><loc_7><loc_2><loc_20><loc_3></location>/star [email protected]</list_item>
<list_item><location><page_1><loc_7><loc_1><loc_20><loc_2></location>† [email protected]</list_item>
</unordered_list>
<text><location><page_1><loc_50><loc_32><loc_89><loc_34></location>accretion rate is due to the enhancement and reduction of viscosity at the outer regions of the disc.</text>
<text><location><page_1><loc_50><loc_10><loc_89><loc_32></location>Transient BHCs show several components in their spectra, namely, blackbody, power-law and an iron line at around 6.4 keV. These distinct components are primarily from the optically thick and thin flow components. It is observed that the spectra change their shape during the outburst following a cycle from hard to soft through intermediate states i.e., see, the hardness intensity diagram (HID, Homan & Belloni 2005; Nandi et al. 2012). Several attempts were made to explain changes in the spectral shape and its variation (Remillard & McClintock 2006 for a review). However, there was a lack of physical understanding behind this. Most of the studies are qualitative and phenomenological. It is believed that changes in the accretion rate may be responsible for changes in spectral states (Maccarone & Coppi 2003; Meyer-Hofmeister et al. 2004 and references therein). However, causes of the change in accretion rate on a daily basis, the origin of corona and its temperature, optical depth etc. were not very clear.</text>
<text><location><page_1><loc_50><loc_1><loc_89><loc_9></location>The problems were satisfactorily resolved when proper usage of the solution of transonic flows in presence of viscosity was used. In a Two Component Advective Flow (TCAF, Chakrabarti & Titarchuk 1995, hereafter, CT95) solution, a Keplerian disc which arises out of higher viscosity is immersed inside a hot subKeplerian flow of lower viscosity. This sub-Keplerian, low angu-</text>
<text><location><page_2><loc_7><loc_60><loc_46><loc_87></location>lar momentum hot matter forms an axisymmetric shock due to the dominance of the centrifugal force (Chakrabarti 1990; Molteni, Lanzafame & Chakrabarti 1994). The subsonic region between the shock boundary to the inner sonic point is hot and pu ff ed up and behaves as the Compton cloud, which upscatters intercepted soft photons from the standard disc. This region also supplies matter to the jet and outflow (Chakrabarti 1999a, hereafter C99a). This is called the CENtrifugal barrier dominated BOundary Layer or CENBOL. This region could be oscillatory when its cooling time scale roughly matches with the infall (compression) timescale inside CENBOL (MSC96; Chakrabarti & Manickam 2000; Chakrabarti et al. 2015). Recently, Chakrabarti et al. (2015) applied the resonance condition for H 1743-322 black hole candidate and showed that the low frequency quasi-periodic oscillations (LFQPOs) are produced when cooling time scale roughly matches with the infall time scale. Transonic solution by Mondal & Chakrabarti (2013; CT95) shows that cooling mechanism is also responsible for the change in spectral states. The presence of two components as in CT95 is established by many other authors in the literature (Smith et al. 2001, 2002; Wu et al. 2002; Ghosh & Chakrabarti, 2018).</text>
<text><location><page_2><loc_7><loc_38><loc_46><loc_59></location>After the implementation of TCAF in XSPEC (Debnath et al. 2014) and fitting data of several black hole candidates, one obtains physical parameters of the underlying inflow, such as the accretion rates of the disk and halo components, the shock location and the shock strength. If the mass is unknown, this will also be found out from the spectral fit (e.g., Molla et al. 2016; Bhatterjee et al. 2017). A plot of photon count variation with accretion rate ratio (ARR) gives the so call ARR intensity diagram (ARRID, Mondal et al. 2014b; Jana et al. 2016) and directly shows why the spectral state changes. The changes in accretion rates on a daily basis is due to changes in viscosity parameter during the outburst (Mondal et al. 2017). The time scale of the changes can also be estimated from the model fitted accretion rates (Jana et al. 2016). From the spectral fits, QPO frequencies can be predicted as well (Debnath et al. 2014; Chakrabarti et al. 2015; Chatterjee et al. 2016).</text>
<text><location><page_2><loc_7><loc_13><loc_46><loc_37></location>Till date, many faint X-ray binaries have been discovered and with even fainter companions. Swift J1357.2-0933 has one of the shortest orbital periods and is a very faint black hole X-ray transient. The source was first detected in 2011 by the Swift Burst Alert Telescope (Barthelmy et al. 2005; Krimm et al. 2011). The distance to the source is not confirmed. This can range from ∼ 1 . 5 - 6.3 kpc (Rau et al. 2011; Shahbaz et al. 2013). There is also a large discrepancy in mass measurement of the source. The mass of the black hole is estimated to be > 3 . 6 M /circledot by Corral-Santana et al. (2013) and > 9 . 3 M /circledot by Mata S'anchez et al. (2015). Corral-Santana et al. (2013) also estimated the orbital period to be 2 . 8 ± 0 . 3 hrs from the time-resolved optical spectroscopy. They observed recurring dips on 2-8 min time-scales in the optical lightcurve, although the RXTE and XMM-Newton data do not show any of the above evidences (Armas Padilla et al. 2014). The observed broad, double-peaked H α profile supports a high orbital inclination (Torres et al. 2015). Very recently, Russell et al. (2018) found an evolving jet synchrotron emission using long term optical monitoring of the source.</text>
<text><location><page_2><loc_7><loc_1><loc_46><loc_12></location>In the earlier outburst during 2011, the source had a variable accretion and showed very regular temporal and spectral evolution. The detailed multiwavelengh lightcurve is studied by Weng & Zhang (2015). Recently, Swift J1357.2-0933 showed renewed activity on 2017 April 20 (Drake et al. 2017) and April 21 (Sivako ff et al. 2017, observed by Swift / XRT). Very recently, Stiele & Kong (2018) observed the source by NuSTAR and there is a simultaneous observation in Swift / XRT as well. Thus the present outburst covers</text>
<text><location><page_2><loc_50><loc_84><loc_89><loc_87></location>a broadband energy range. In this paper, we use the above data to study the flow dynamics of the source during its 2017 outburst.</text>
<text><location><page_2><loc_50><loc_75><loc_89><loc_84></location>The paper is organized as follows: in the next Section, we present the observation and data analysis procedure. In § 3, we discuss about the model fitted results and geometry of the disc during the outburst. We also estimate the mass from the model fit. In § 4, we calculate various physical quantities of the disc from model fitted parameters to infer about the disc properties. Finally, in § 5, we draw our brief concluding remarks.</text>
<section_header_level_1><location><page_2><loc_50><loc_70><loc_78><loc_71></location>2 OBSERVATION AND DATA ANALYSIS</section_header_level_1>
<text><location><page_2><loc_50><loc_65><loc_89><loc_69></location>In the present manuscript, we analyze both Swift / XRT (Gehrel et al., 2004) and NuSTAR (Harrison et al. 2013) satellite observations of the BHC Swift J1357.2-0933 during its 2017 outburst.</text>
<section_header_level_1><location><page_2><loc_50><loc_62><loc_56><loc_63></location>2.1 Swift</section_header_level_1>
<text><location><page_2><loc_50><loc_43><loc_89><loc_61></location>In our present analysis, we use 0.5-7.0 keV Swift / XRT observation of Swift J1357.2-0933 during 2017 outburst, the timings of which overlap with the NuSTAR observations presented below. The observation IDs are 00088094002 (Photon Counting mode, PC) and 00031918066 (Windowed Timing mode, WT). We use xrtpipeline v0.13.2 task to extract the event fits file from the raw XRT data. All filtering tasks are done by FTOOLS . To generate source and background spectra we run xselect task. We use swxpc0to4s6 20130101v014.rmf and swxwt0to2s6 20130101v015.rmf file for the response matrix. The grppha task is used to group the data. We use 10 bins in each group. Weuse same binning for both the observations and also in NuSTAR data, which we discuss in the next section.</text>
<section_header_level_1><location><page_2><loc_50><loc_39><loc_59><loc_40></location>2.2 NuSTAR</section_header_level_1>
<text><location><page_2><loc_50><loc_1><loc_89><loc_38></location>We analyze two NuSTAR observations with observation IDs: 90201057002 (hereafter O1, combined with 00088094002) and 90301005002 (hereafter O2, combined with 00031918066) of BHC Swift J1357.2-0933 with energy range 2.0-70 keV. NuSTAR data were extracted using the standard NUSTARDAS v1.3.1 software. We run 'nupipeline' task to produce cleaned event lists and 'nuproducts' for spectral file generation. We use 30 '' radius region for the source extraction and 45 '' for the background using 'ds9'. The data is grouped by 'grppha' command, where we group the whole energy bin with 10 bins in each group. We choose the same binning and fitting criteria for both the observations. However, the data quality of O2 is not good and above ∼ 20 keV it is highly noisy. Thus the count at di ff erent energy ranges in O2 is not similar to O1. We split the NuSTAR observations into three di ff erent time ranges (S1, S2, and S3) each of which contains ∼ 24 ksec (for O1) and ∼ 15 ksec (for O2) data. For that purpose, first we make our own 'GTI' files for each time range using 'gtibuild' task in SAS environment and use those GTI files during data extraction. For spectral fitting of the data we use XSPEC (Arnaud, 1996) version 12.8.1. We fit the data using 1) Power-law (PL), and 2) TCAF models. The detailed spectral fitting with other phenomenological models and with reflection model (relxill) are discussed in Stiele & Kong (2018). Using relxill model and assuming high inclination, they found that the disc is truncated close to the black hole independent of spin parameter. Here, we mainly focus on the TCAF model fitted parameters to study the physical properties of the disc and its geometry during the outburst. To fit the spectra with the</text>
<text><location><page_3><loc_7><loc_73><loc_46><loc_87></location>TCAF model in XSPEC, we have a TCAF model generated fits file (Debnath et al. 2014). We follow the same analysis procedure for TCAF fitting with Swift and NuSTAR data as discussed in Mondal et al. (2016). We use the absorption model TBabs (Wilms et al. 2000) with hydrogen column density fixed at 1.3 × 10 21 atoms cm -2 throughout the analysis. In the above absorption model solar abundance vector set to 'wilm', which includes cosmic absorption with grains and H 2 and those absorptions which are not included in the paper are set to zero. We keep the mass as a free parameter in order to estimate it from the TCAF itself.</text>
<section_header_level_1><location><page_3><loc_7><loc_68><loc_44><loc_69></location>3 MODELFITTED RESULTS AND DISC GEOMETRY</section_header_level_1>
<text><location><page_3><loc_7><loc_25><loc_46><loc_67></location>We study the BHC Swift J1357.2-0933 using Swift / XRT and NuSTAR observations with PL model. PL model fitted photon index is ∼ 1 . 7 -1 . 8. Thus the object is in a hard state. We also fit the data with the TCAF solution based fits file which uses five physical parameters: The parameters are as follows (i) Mass of the black hole, (ii) disc accretion rate, (iii) halo accretion rate, (iv) location of the shock, and (v) shock compression ratio. Parameters (ii) to (v) collectively give the electron density and temperature, photon spectrum and density, the fraction of photons intercepted by the CENBOL from the disk, as well as the reflection of hard photons from the CENBOL by the disc. All of these depend on the mass of the black hole. According to CT95, if one increases the halo rate keeping other parameters frozen, the model will produce a hard spectrum. Similarly, increasing the disc rate leaving other parameters frozen, will produce a soft spectrum. When the location of the shock is increased keeping other parameters fixed, spectrum will be harder. A similar e ff ect is seen for compression ratio also (see also, Chakrabarti 1997). In general, in an outburst, all the parameters will change smoothly in a multidimensional space. The model fitted results for both the observations are given in Table 1. In Fig. 1(a-b), we present the TCAF model fitted spectra. The model fitted parameters show that the disc rate was higher on the second day. Opposite is true for the halo rate. In both the observations, the ratio of the halo rate and disc rate is larger than unity, i.e., the flow is dominated by the sub-Keplerian rate. This is an indication of the hard state. At some point in time, before the second observation day, viscosity may have started to go up, and the Keplerian disc rate also started to increase. However, this was not enough so as to change the spectral state (as a minimum viscosity is required for such changes (Mondal et al. 2017). The PL model fitted photon index also indicates a hard spectral state.</text>
<text><location><page_3><loc_7><loc_1><loc_46><loc_24></location>As the shock is the outer boundary of the Compton cloud, the movement of the shock shows a change in size of the Compton cloud. On the second day of observation, the shock moved closer to the black hole as compared to the first day from Xs = 81 . 00 rg to ∼ 36 . 66 rg (where, Schwarzschild radius rg = 2 GMBH / c 2 ). This type of behavior is observed routinely in all the black holes during the rising phase of the outburst. The behavior of advancing inner edge of the disc was studied by several authors for various outbursting candidates (Esin et al. 1997; Tomsick et al. 2009; Dutta & Chakrabarti 2010; Shidatsu et al. 2011; Nandi et al. 2012). The shock compression ratio (the ratio of the post-shock to pre-shock flow density experienced by the low angular momentum component) is always observed to be higher than unity and is generally of intermediate strength. In this case, the jets and outflows are expected to be strong (C99a; Chakrabarti 1999b). For the compression ratio given in the Table, the expected outflow / inflow ratio will be 3 . 4 -4 . 2 %. This jet may appear and disappear during the transi-</text>
<figure>
<location><page_3><loc_55><loc_68><loc_86><loc_86></location>
</figure>
<figure>
<location><page_3><loc_55><loc_48><loc_86><loc_66></location>
<caption>Figure 1. Swift / XRT and NuSTAR (focal plane module A, FPMA) data from 0.5-70.0 keV energy range are fitted with an absorbed TCAF model. The model fitted parameters are given in Table 1.</caption>
</figure>
<text><location><page_3><loc_50><loc_21><loc_89><loc_40></location>on phase of the outburst, although the actual interrelation between the jet properties and the X-ray spectral state evolution is still debate (Kalemci et al. 2005; Dincer et al. 2014). Several attempts have been taken (Tomsick et al. 2009; Petrucci et al. 2014 and references therein) to understand the evolution of the Compton cloud and spectral state transitions as a function of luminosity during the outburst. Recent transonic solution of Mondal et al. (2014a) following the same flow geometry and jet configuration of C99a, showed that since the jet removes a significant amount of matter from CENBOL, it is easier to cool the Compton cloud. Thus, a change in spectral states in presence of jet is expected to be faster. Following the above model understanding and the values of the model fitted parameters, we conclude that the source was in rising hard state of the outburst during the observations.</text>
<text><location><page_3><loc_50><loc_10><loc_89><loc_20></location>In Fig. 2, we show the hardness intensity diagram (HID) and accretion rate ratio intensity diagram (ARRID) after splitting the data in three segments of equal time interval. The HID shows that the hardness ratio varied from 1.7 to 2.2. Thus one can say that the source is moderately variable. After fitting the data segments with TCAF model, we see that the ARR in ARRID varies by a factor of about 2. From both the diagrams, we note that the flow was not rapidly evolving.</text>
<text><location><page_3><loc_50><loc_1><loc_89><loc_9></location>In a series of papers with TCAF model fits, constant normalization was used for a given object observed by a given instrument as it is a conversion factor between the observed flux and the model flux. In Fig. 1, to fit the data we obtained model normalization 1 . 12 for NuSTAR observations, whereas Swift observations the fitting procedure produces the normalization values of 0 . 31 for O1 and</text>
<section_header_level_1><location><page_4><loc_7><loc_89><loc_37><loc_90></location>4 S. Mondal and S. K. Chakrabarti</section_header_level_1>
<table>
<location><page_4><loc_10><loc_66><loc_43><loc_85></location>
<caption>Table 1. PL and TCAF model fitted parameters of combined Swift / XRT and NuSTAR observationMBH is in unit of M /circledot and Mass accretion rates are in ˙ MEdd unit. Xs is in 2 GMBH / c 2 unit.</caption>
</table>
<figure>
<location><page_4><loc_7><loc_36><loc_50><loc_61></location>
<caption>Figure 2. (a) Hardness ratio (10.0-70.0 keV / 2.0-10.0 keV) of NuSTAR data for both the observations after splitting the data in three intervals mentioned in Sec. 2. Total flux is in 1.0E-10 ergs / cm 2 / s unit. (b) Variation of total flux with ARR. It is to be noted that the hardness ratio varies moderately between 1.7-2.2 and ARR by a factor of 2.</caption>
</figure>
<text><location><page_4><loc_39><loc_36><loc_39><loc_37></location>/</text>
<text><location><page_4><loc_7><loc_1><loc_46><loc_26></location>1 . 13 for O2. The freedom in absorption does not improve the fit significantly, rather gives a factor of 10 di ff erence in NH between O1 and O2. As the normalization is not chosen by hand and it comes out as a fitted parameter, the ratio of the normalizations is important. The relative normalization for the two instruments can be calculated from the ratio of the model normalizations obtained from the fits of each instrument's spectrum. There could be several reasons for getting di ff erent normalizations in Swift data in O1 and O2: 1) if the Swift and NuSTAR data are not strictly simultaneous, 2) there could be a pile-up issue, and 3) if the source is close to the PC / WTswitch point, then it may be relatively faint for WT. As the source was not highly variable during that period, the point 1 cannot be a reason for the di ff erence in normalization. The pile-up could be an issue as the count rate is above 0 . 5 cnts / sec. We should mention that we have performed the analysis using pile-up corrected spectrum files for PC mode generated by online Swift / XRTproduct generator (Evans et al. 2009). However, the di ff erence in normalization persists. The model fitted parameter values also remained</text>
<text><location><page_4><loc_50><loc_82><loc_89><loc_87></location>unchanged with an acceptable reduced χ 2 ( ∼ 1 . 0) keeping the mass of the source within 4 . 0 M /circledot -6 . 8 M /circledot . The third point should not be the reason for di ff erence in normalization as the source is su ffi -ciently bright for WT mode.</text>
<text><location><page_4><loc_50><loc_71><loc_89><loc_81></location>These leads us to conclude that the di ff erence in normalization could be due to other physical processes inside the accretion disc which were not incorporated in TCAF. As discussed in Jana et al. (2016) and Molla et al. (2017), the presence of jet is a reason for the di ff erence in normalization. During the present outburst period, activity in jet was observed for this source. Thus the presence of jet / outflow could be a reason behind the variation of normalization constant.</text>
<section_header_level_1><location><page_4><loc_50><loc_65><loc_81><loc_67></location>4 ESTIMATION OF DIFFERENT PHYSICAL QUANTITIES OF THE DISC</section_header_level_1>
<text><location><page_4><loc_50><loc_58><loc_89><loc_63></location>In this Section, we estimate some physical parameters of the disc using TCAF model fitted parameters. From Kepler's law, one can derive a relation between orbital period ( P ) and orbital separation a as follows:</text>
<formula><location><page_4><loc_59><loc_56><loc_89><loc_57></location>a = 3 . 5 × 10 10 m 1 / 3 2 (1 + q ) 1 / 3 P ( hr ) 2 / 3 , (1)</formula>
<text><location><page_4><loc_50><loc_51><loc_89><loc_55></location>where q ( = m 1 / m 2) is the mass ratio of the component stars. Outer disc radius ( rout ) of the primary star is calculated from the Roche lobe radius of the primary following Eggleton (1983) as,</text>
<formula><location><page_4><loc_60><loc_47><loc_89><loc_50></location>rRl , 1 = 0 . 49 × a q 2 / 3 0 . 6 × q 2 / 3 + ln [1 + q 1 / 3 ] . (2)</formula>
<text><location><page_4><loc_50><loc_30><loc_89><loc_46></location>In this work, we consider that the outer edge of the disc ( rout ) is 70% of the Roche lobe radius. We use the values of P ( = 2.8 hrs) and companion mass ( = 0 . 17 M /circledot ) in Eq. 2, already derived by Corral-Santana et al. (2013), to estimate Roche lobe radius, which appears to be rout = 0 . 68 × 10 11 . Using the above derived outer disc radius and model fitted accretion rate, we calculate kinematic viscosity ( ν ) and surface density ( Σ ) of the disc. It is to be noted that the disc accretion rate which we are using from TCAF fit is constant throughout the disc, thus we assume that the accretion at rout is same as the model fit value. We use standard disc equations (SS73) below to derive the above two physical parameters ( ν and Σ ):</text>
<formula><location><page_4><loc_67><loc_27><loc_89><loc_29></location>Σ /similarequal ˙ Md 3 πν , (3)</formula>
<text><location><page_4><loc_50><loc_1><loc_89><loc_26></location>where, ν is calculated using S S 73 α disc model ( ν = α CsH ). Here C 2 s ( = kT µ mp ) and H ( = Cs Ω K = 0 . 024) are the sound speed at the outer radius of the disc, height of the disc and Ω K is the Keplerian angular velocity at that point respectively. The temperature of the disc is calculated from the disc accretion rate. The estimated values of Σ and ν are ∼ 49 . 1 gm / cm 2 and ∼ 4 . 0 × 10 14 cm 2 / sec respectively. During our calculation, we consider that the disc is not self-gravitating i.e., vertical hydrostatic equilibrium is maintained against the pull of the gravity. The computed low surface density and kinematic viscosity are indicating a stable disc. In the context of disc stability, Weng & Zhang (2015) mentioned that the high ratio of near UV luminosity to X-ray luminosity indicates that the irradiation is unimportant in this outburst, while the near-exponential decay profile and the long decay time-scale conflicts with the disc thermal-viscous instability model. Hence they suggested that the disc is thermally stable during the outburst. Armas Padilla et al. (2013) also found that the correlation between Swift / UVOTv-band and XRT data is consistent with a non-irradiated accretion disc.</text>
<figure>
<location><page_5><loc_9><loc_62><loc_46><loc_85></location>
<caption>Figure 3. Variation of count rate with time (day). Black solid line shows the model fit and filled circles are observed count rates. Data are fitted with H / R = 0 . 024 (from above), P = 2 . 8 hrs, q = 0 . 03 and SS73 α = 0 . 25. Count rate of Swift lightcurve is adopted from Stiele & Kong (2018).</caption>
</figure>
<text><location><page_5><loc_7><loc_35><loc_46><loc_53></location>Here, we present the chain of logical steps used to fit the observed lightcurve: (i) we have mass and disc accretion rate from TCAF fit, (ii) using (i) we estimate disc temperature thus the sound speed, (iii) we also have orbital period ( P ) and mass function ( q ) from literature, (iv) outer disc radius is estimated using Eq. 1 from the parameters of (iii), (v) once (ii) and (iv) are known one can estimate height of the disc, Keplerian angular frequency, and disc kinematic viscosity to estimate the viscous time scale, and (vi) finally, we extract SS73α parameter for which the derived and the fitted τ values are consistent. Here, α takes the value 0.25 to give a decay timescale ( τ ) of ∼ 45 days. To fit the observed count rate using decay timescale, we use an exponential decay function, which is given by:</text>
<formula><location><page_5><loc_22><loc_33><loc_46><loc_34></location>f = A exp ( -t /τ ) , (4)</formula>
<text><location><page_5><loc_7><loc_20><loc_46><loc_32></location>where A is a normalization constant, which takes the value of 3 . 91 ± 0 . 16 with exponential decay timescale ( τ ) around 45 . 28 ± 4 . 78 days . The estimated SS73 disc viscosity parameter ( α = 0 . 25) becomes same order as those obtained for other observed candidates (Nagarkoti & Chakrabarti 2016; Mondal et al. 2017). Here we consider A as a constant, however it should depend on the source distance and the physical properties of the disc. The detailed calculation is beyond the scope of this paper. In Fig. 3, we show exponential decay function fitted with the observed count rate.</text>
<section_header_level_1><location><page_5><loc_7><loc_14><loc_32><loc_15></location>5 SUMMARYAND CONCLUSIONS</section_header_level_1>
<text><location><page_5><loc_7><loc_1><loc_46><loc_13></location>In this paper, we analyzed Swift / XRT and NuSTAR spectra of a known Galactic stellar mass black hole source Swift J1357.2-0933 during its 2017 outburst using a phenomenological PL model and a physical TCAF solution. We find that on both the observation dates, the sub-Keplerian halo accretion rate is higher than the Keplerian accretion rate. In the second observed day, the disc rate is increased as compared to first observed day and opposite variation is seen in the sub-Keplerian rate. The object was in hard state on both the days. As the halo rate is higher than the disc rate and the shock</text>
<text><location><page_5><loc_50><loc_58><loc_89><loc_87></location>compression ratio is always greater than unity, the shock moved inward due to cooling. This could be the signature of the hard state in the rising phase of the outburst. The shock was seen to move at ∼ 0 . 15 m / s which is similar to the shock velocity in other outbursts (Debnath et al. 2010 for GX 339-4; Mondal et al. 2015 for H1743-322 and references therein). This indicates that the outburst probably remained in the rising phase and no other spectral state has been missed in between these ∼ 40 days . It is to be noted that the companion of this candidate is a star evolved through nuclear fusion (Shahbaz et al. 2013) with an initial mass ∼ 1 . 5 M /circledot , which has evolved to 0 . 17 M /circledot . Thus there is a possibility that the accretion is mostly dominated by companion winds and thus the halo rate is always higher. Our model fitted disc rate is ∼ 1 . 5 -2% of ˙ MEdd . As the disc rate increases and the shock location decreased in ∼ 40 days, viscosity must have gone up since the first observation. As the shock compression ratio is in intermediate strength, in this case, the jets and outflows are expected to be strong with outflow / inflow rate ratio 3.4-4.2%. From TCAF model fit, we estimate the mass range for this black hole candidate to be 4 . 0 -6 . 8 M /circledot . However, a few more observations would have reduced the errorbar significantly.</text>
<text><location><page_5><loc_50><loc_43><loc_89><loc_58></location>We also study di ff erent physical parameters of the disc. For that, we calculate the surface density, kinematic viscosity and disc aspect ratio etc. of the disc using the model fitted parameters. We find that the disc surface density is not high enough signifying that the disc is stable in nature. The estimated surface density is also reasonable to produce a consistent α value to study the decay of the lightcurve. We find that the lightcurve fits with exponential decay function with the decay time scale of ∼ 45 day , which is consistent with the derived decay time scale when α = 0.25. Thus from the model fit we can study the spectra, disc properties, lightcurve decay and estimate viscosity parameter at a time.</text>
<section_header_level_1><location><page_5><loc_50><loc_38><loc_67><loc_39></location>6 ACKNOWLEDGMENT</section_header_level_1>
<text><location><page_5><loc_50><loc_19><loc_89><loc_37></location>We thank anonymous referee for useful comments on the manuscript. SM acknowledges Swift team (especially Kim Page) for useful discussions on Swift data fitting and Patricia Ar'evalo for commenting on the preliminary version of the paper. SM acknowledges FONDECYT fellowship grand (# 3160350) for this work. This research has made use of the NuSTAR Data Analysis Software (NuSTARDAS) jointly developed by the ASI Science Data Center (ASDC, Italy) and the California Institute of Technology (Caltech, USA), and the XRT Data Analysis Software (XRTDAS) developed under the responsibility of the ASI Science Data Center (ASDC), Italy. This research has made use of data obtained through the High Energy Astrophysics Science Archive Research Center Online Service, provided by the NASA / Goddard Space Flight Center.</text>
<section_header_level_1><location><page_5><loc_50><loc_14><loc_60><loc_15></location>REFERENCES</section_header_level_1>
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<text><location><page_5><loc_51><loc_8><loc_89><loc_11></location>Armas Padilla, M., Wijnands, R., Altamirano, D., et al. 2014, MNRAS, 439, 3908</text>
<text><location><page_5><loc_51><loc_3><loc_89><loc_8></location>Arnaud, K.A., ASP Conf. Ser., Astronomical Data Analysis Software and Systems V, ed. G.H. Jacoby & J. Barnes, 101, 17 (1996) Bhattacharjee, A., Banerjee, I., Banerjee, A., et al. 2017, MNRAS, 466, 1372</text>
<text><location><page_5><loc_51><loc_1><loc_75><loc_2></location>Chakrabarti, S. K. 1989, MNRAS, 240, 7</text>
<section_header_level_1><location><page_6><loc_11><loc_89><loc_37><loc_90></location>S. Mondal and S. K. Chakrabarti</section_header_level_1>
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<text><location><page_6><loc_51><loc_51><loc_89><loc_86></location>A&A, 542, 56 Petrucci, P. O., Cabanac, C., Corbel, S., et al., 2014, A&A, 564, A37 Rau, A., Greiner, J., & Filgas, R. 2011, The Astronomers Telegram, 3140 Russell, D. M., Qasim, A. A., Bernardini, E., et al. 2018, ApJ, 852, 90 Shakura, N. I., & Sunyaev, R. A. 1973, A&A, 24, 337 Shahbaz, T., Russell, D. M., Zurita, C., et al. 2013, MNRAS, 434, 2696 Shidatsu, M., et al., 2011, PASJ, 63, 785 Sivako ff , G. R., Tetarenko, B. E., Shaw, A. W., & Bahramian, A. 2017, The Astronomers Telegram, No. 10314, 314 Smith, D. M., Heindl, W. A., Markwardt, C. B., et al., 2001, ApJL, 554, L41 Smith, D. M., Heindl, W. A., & Swank, J. H., 2002, ApJ, 569, 362 Stiele, H., & Kong, A. K. H. 2018, ApJ, 852, 34 Tomsick, J. A., Yamaoka, K., Corbel, S., et al. 2009, ApJ, 707, L87 Torres, M. A. P., Jonker, P. G., Miller-Jones, J. C. A., et al. 2015, MNRAS, 450, 4292 Weng, S.-S., & Zhang, S.-N., 2015, MNRAS, 447, 486 Wilms, J., Allen, A., & McCray, R. 2000, ApJ, 542, 914 Wu, K., Soria, R., Campbell-Wilson, D., et al., 2002, ApJ, 565, 1161</text>
</document> | [] |
2022EPJP..137..557S | https://arxiv.org/pdf/2111.08495.pdf | <document>
<section_header_level_1><location><page_1><loc_12><loc_92><loc_88><loc_93></location>Gravitational Collapse of Anisotropic Compact Stars in Modified f ( R ) Gravity</section_header_level_1>
<text><location><page_1><loc_46><loc_89><loc_55><loc_90></location>Jay Solanki ∗</text>
<text><location><page_1><loc_33><loc_86><loc_68><loc_89></location>Sardar Vallabhbhai National Institute of Technology, Surat - 395007, Gujarat, India</text>
<text><location><page_1><loc_18><loc_69><loc_83><loc_85></location>The physically realistic model of compact stars undergoing gravitational collapse in f ( R ) gravity has been developed. We consider a more general model R + f ( R ) = R + kR m and describe the interior space-time of gravitationally collapsing stars with separable-form of metric admitting homothetic killing vector. We then investigate the junction conditions to match the interior space-time with exterior space-time. Considering all junction conditions, we find analytical solutions describing interior space-time metric, energy density, pressures, and heat flux density of the compact stars undergoing gravitational collapse. We impose the energy conditions to the model for describing the realistic collapse of physically possible matter distribution for particular models of GR , R + kR 2 and R + kR 4 gravity. The comprehensive graphical analysis of all energy conditions show that the model is physically acceptable and realistic. We additionally investigate the physical properties of collapsing stars which are useful to decipher the inherent nature of such gravitationally collapsing stars.</text>
<text><location><page_1><loc_18><loc_67><loc_69><loc_68></location>Keywords: Gravitational collapse, f(R) gravity, junction conditions, Anisotropic stars</text>
<section_header_level_1><location><page_1><loc_42><loc_61><loc_59><loc_62></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_46><loc_92><loc_59></location>Einstein's theory of general relativity has been extensively used to describe various astronomical phenomena such as perihelion of mercury, various properties of compact stars and their gravitational collapse, formation of black holes, Etc. General relativity is found to be very successful in describing such phenomena with great accuracy. However, it fails to describe the observations of galaxy rotation curves and expansion of the universe-like phenomena. Such phenomena can be described with general relativity by introducing dark matter and dark energy[1-5]. Another possible solution is to modify the theory of general relativity.[6-14] One possible modification in general relativity can be achieved by modifying the Einstein-Hilbert action with additional non-linear form f ( R ) of the Ricci scalar R . Thus, the Lagrangian density will become R + f ( R ), apart from the Lagrangian density for the matter distribution. Here in this paper, we consider the general non-linear form of f ( R ) = kR m giving modified gravity model R + f ( R ) = R + kR m .</text>
<text><location><page_1><loc_9><loc_22><loc_92><loc_46></location>Modified f ( R ) gravity has been extensively used to describe various astronomical phenomena such as stable neutron stars and strange stars, the gravitational collapse of such compact stars and wormholes, galaxy rotation curves, and expansion of the universe. Many authors have investigated useful forms of f ( R ) and their validity to pass solar tests and cosmological bounds in describing various astronomical phenomena[15-18]. There have been also huge efforts made by various authors to investigate compact stars and their gravitational collapse in f ( R ) gravity[19-23]. The authors of the paper [24] have investigated the mass-radius relationship of neutron stars in f ( R ) gravity by specifying the equation of state relating the isotropic matter pressure and density. Extreme neutron stars have been investigated in the extended theory of gravity in the reference [25, 26]. Recently authors have studied super-massive neutron stars, and the causal limit of neutron star maximum mass in View of GW 190814 [27, 28]. The authors of the paper [29] have investigated novel stellar astrophysics from extended gravity. Neutron stars in scalar-tensor gravity with Higgs scalar potential are studied in the paper [30]. The gravitational collapse in modified f ( R ) gravity has also been studied in the context of cosmology by the authors of the paper[31]. The spherical collapse of compact stars in f ( R ) gravity and the Belinskii-Khalatnikov-Lifshitz conjecture has been investigated in the paper[32]. Spherically symmetric collapse in modified f ( R ) gravity and a generalization to conformally flat stars are studied in [33, 34]. Gravitational collapse in generalized teleparallel gravity has been investigated in [35]. Dynamics of the Charged Radiating and dissipative Collapses in Modified Gauss-Bonnet Gravity have been studied in these papers [36, 37]. Authors of the paper [38] have studied dynamical conditions and causal transport of dissipative spherical collapse in f(R,T) gravity.</text>
<text><location><page_1><loc_9><loc_16><loc_92><loc_21></location>Many authors have also investigated the phenomena of gravitational collapse in f ( R ) gravity and the stability of polytropic fluid in f ( R ) gravity[39, 40]. Gravitational collapse in repulsive R + µ 4 /R gravity has been studied in the paper [41]. The authors of the paper [42] have investigated the gravitational collapse of compact stars for charged adiabatic LTB configuration. Higher-dimensional charged LTB collapse in f ( R ) gravity has been studied in reference</text>
<text><location><page_2><loc_9><loc_77><loc_92><loc_93></location>[43]. Recently, authors have investigated three different f ( R ) gravity models to study the collapsing phenomena as well as the nature of central singularity[44]. In the paper [45], authors have studied radiating gravitational collapse considering the effect of shear viscosity. The gravitational collapse in f ( R ) gravity and method of R matching has been investigated in the paper [46]. In that paper, it was found that the separable form of the Lemaitre-Tolman-Bondi metric is not very suitable for describing gravitational collapse in f ( R ) gravity due to violations of some junction conditions. However, recently it was found that a particular metric form admitting homothetic killing vector and non-separable of LTB metric satisfies all necessary junction conditions to describe realistic gravitational collapse. However, such detailed studies have been performed only for some restricted class of f ( R ) models like modified R 2 gravity. It is useful to investigate such physically important scenarios for a more general f ( R ) model to understand the inherent nature of gravitationally collapsing stars described in modified f ( R ) gravity. Thus, this paper studies the gravitationally collapsing stars in a more general R + f ( R ) = R + kR m model.</text>
<text><location><page_2><loc_9><loc_59><loc_92><loc_77></location>To investigate the gravitational collapse of compact stars in modified f ( R ) gravity, firstly, it is necessary to formulate field equations of f ( R ) gravity. Varying the modified Einstein-Hilbert action in terms of modified Lagrangian density R + f ( R ) + L matter , the field equations of modified f ( R ) gravity is being obtained. Then the field equations have to be solved for particular forms of interior space-time metric representing the gravitational collapse of compact stars. While solving field equations of modified f ( R ) gravity, junction conditions must be considered for smooth matching of interior and exterior space-time. In general relativity, these boundary conditions are known as the DarmoisIsrael conditions. These conditions require the first and second fundamental forms to be matched on the boundary of collapsing stars. However, in f ( R ) gravity, extra junction conditions have to be applied to smoothly match the interior and exterior space-time[47, 48]. These extra junction conditions admitting strong boundary conditions greatly restrict the physically possible solutions of the problem. After solving field equations of gravitational collapse in f ( R ) gravity, the solutions have to be satisfied the energy conditions for the description of physically possible matter distribution undergoing gravitational collapse. Finally, important physical properties of compact stars undergoing gravitational collapse can be investigated from the generated physically well-behaved model.</text>
<text><location><page_2><loc_9><loc_40><loc_92><loc_58></location>This paper is organized as follows. In section II, we describe the field equations of gravitationally collapsing stars in f ( R ) gravity for the specific interior metric admitting homothetic killing vector. Then in section III, we develop junction conditions for the form of interior space-time under consideration and exterior space-time described by the Vaidya metric. We investigate the conditions for smooth matching of interior and exterior metrics, their exterior derivatives, and Ricci scalars and their derivatives. In section IV, we solve field equations by using the method of R matching to find exact solutions describing gravitationally collapsing stars and their physical quantities for more general model f ( R ) = kR m . Then in section V, we describe necessary energy conditions to be imposed for obtaining a physically acceptable model of gravitational collapse in f ( R ) gravity. In VI, we do physical analysis of solutions for particular models of GR by choosing parameter k = 0, R + f ( R ) = R + kR 2 by choosing parameter m = 2 and R + f ( R ) = R + kR 4 gravity by choosing parameter m = 4. We find model parameters such that the models satisfy all energy conditions, and we also plot all energy conditions for each model. The graphs of all energy conditions plotted for each model show that models are physically acceptable. In section VII we investigate the physical properties of gravitationally collapsing stars and summarize all the results in the Discussion section VIII.</text>
<section_header_level_1><location><page_2><loc_19><loc_36><loc_82><loc_37></location>II. FIELD EQUATIONS OF GRAVITATIONAL COLLAPSE IN f ( R ) GRAVITY</section_header_level_1>
<text><location><page_2><loc_9><loc_28><loc_92><loc_34></location>The field equations of f ( R ) gravity can be obtained by modifying the Einstein-Hilbert action. The Einstein-Hilbert action can be modified by introducing modification in terms of the additional function of curvature f ( R ). Thus, the Lagrangian density for modified f ( R ) gravity becomes R + f ( R ) + L matter . Variation of the action written in terms of modified Lagrangian density yields the following field equation of modified f ( R ) gravity</text>
<formula><location><page_2><loc_26><loc_22><loc_92><loc_25></location>G αβ = 1 1 + f R ( T αβ + ∇ α ∇ β f R -g αβ ∇ µ ∇ µ f R + 1 2 g αβ ( f -Rf R ) ) (1)</formula>
<text><location><page_2><loc_9><loc_18><loc_92><loc_21></location>Where G αβ = R αβ -1 2 g αβ R is the well known Einstein tensor. T αβ is the energy-momentum tensor and f R = df ( R ) dR . The equation (1) can be written in the following form</text>
<formula><location><page_2><loc_41><loc_13><loc_92><loc_16></location>G αβ = 1 1 + f R ( T m αβ + T c αβ ) (2)</formula>
<text><location><page_2><loc_9><loc_9><loc_92><loc_12></location>Where the T m αβ is the energy-momentum tensor for the matter and the T c αβ is known as effective energy-momentum tensor written in terms of the function of Ricci scalar and its derivatives. Effective energy-momentum tensor is</text>
<text><location><page_3><loc_9><loc_89><loc_92><loc_93></location>the additional source of the curvature in the modified f ( R ) gravity, and thus it is of purely geometrical origin. To study the spherical gravitationally collapsing stars, we consider the following form of energy-momentum tensor for anisotropic matter distribution</text>
<formula><location><page_3><loc_31><loc_85><loc_92><loc_86></location>T m αβ = ( ρ + p t ) u α u β + p t g αβ -( p t -p r ) v α v β +2 qu ( α v β ) (3)</formula>
<text><location><page_3><loc_9><loc_78><loc_92><loc_83></location>Where ρ , p r , p t and q denotes the energy density of matter, radial and tangential pressure and heat flux density respectively. u α and v α denotes the four velocity and vector in radial direction respectively. The vector representing heat flux can be written as q α = qv α . All these three vectors follows the normalisation conditions given by u α u α = -1 , v α v α = 1 and u α q α = 0.</text>
<text><location><page_3><loc_10><loc_76><loc_63><loc_78></location>From equation (1), the effective energy-momentum tensor T C αβ is given by</text>
<formula><location><page_3><loc_33><loc_71><loc_92><loc_74></location>T c αβ = ∇ α ∇ β f R -g αβ ∇ µ ∇ µ f R + 1 2 g αβ ( f -Rf R ) (4)</formula>
<text><location><page_3><loc_9><loc_67><loc_92><loc_70></location>To study the gravitational collapse of spherically symmetric compact stars in f ( R ) gravity, we use the following form of space-time metric</text>
<formula><location><page_3><loc_26><loc_63><loc_92><loc_66></location>ds 2 = -y ( r ) 2 dt 2 +2 c ( t ) 2 ( dy dr ) 2 dr 2 + c ( t ) 2 y ( r ) 2 ( dθ 2 +( sin 2 θ ) dφ 2 ) (5)</formula>
<text><location><page_3><loc_9><loc_59><loc_92><loc_61></location>From equation (1), we obtain independent field equations of gravitational collapse in f ( R ) gravity for metric (1) as follow, (in units of c = 8 πG = 1)</text>
<formula><location><page_3><loc_19><loc_53><loc_92><loc_56></location>ρy ( r ) 2 = ( 1 + 6˙ c 2 2 c 2 ) (1 + f R ) + 1 2 y 2 ( f -Rf R ) -y 2 f '' R y ' +(2 yy ' 2 -y 2 y '' ) f ' R -6 c ˙ c ˙ f R y ' 3 2 c 2 y ' 3 (6)</formula>
<formula><location><page_3><loc_18><loc_47><loc_92><loc_50></location>p r 2 c ( t ) 2 y ' 2 = y ' 2 y 2 ( 1 -2˙ c 2 -4 c c ) (1 + f R ) -4 c ˙ c ˙ f R y ' 2 +2 c 2 f R y ' 2 -3 yf ' R y ' y 2 -c 2 y ' 2 ( f -Rf R ) (7)</formula>
<formula><location><page_3><loc_10><loc_41><loc_92><loc_44></location>p t c ( t ) 2 y ( r ) 2 = ( 1 2 -2 c c -˙ c 2 ) (1 + f R ) -1 2 c 2 y 2 ( f -Rf R ) -4 c ˙ c ˙ f R y ' 3 +2 c 2 f R y ' 3 -y 2 f '' R y ' -(2 yy ' 2 -y 2 y '' ) f ' R 2 y ' 3 (8)</formula>
<formula><location><page_3><loc_29><loc_35><loc_92><loc_38></location>-q √ 2 c ( t ) y ( r ) y ' = ( 2˙ cy ' cy ) (1 + f R ) -cy ˙ f R ' -y ˙ cf ' R -c ˙ f R y ' cy (9)</formula>
<text><location><page_3><loc_9><loc_29><loc_92><loc_33></location>Where overhead dot indicates differentiation with respect to time coordinate t and prime indicates differentiation with respect to radial coordinate r . Thus, equations (6) to (9) represents the field equations of gravitationally collapsing spherical stars in f ( R ) gravity govern by the interior metric (5).</text>
<section_header_level_1><location><page_3><loc_15><loc_25><loc_85><loc_26></location>III. JUNCTION CONDITIONS FOR GRAVITATIONAL COLLAPSE IN f ( R ) GRAVITY</section_header_level_1>
<text><location><page_3><loc_9><loc_16><loc_92><loc_23></location>While solving field equations (6) to (9), we have to consider the junction conditions so that interior metric (5) and exterior space-time smoothly match at the boundary of traditionally collapsing compact stars. Junction conditions also imply the continuity of exterior derivatives and Ricci scalar and its derivatives across the matching hypersurface. Thus, in this section, we derive the junction conditions for gravitationally collapsing stars in f ( R ) gravity governed by interior metric (5), denoting interior metric by the ds 2 -,</text>
<formula><location><page_3><loc_26><loc_10><loc_92><loc_13></location>ds 2 -= -y ( r ) 2 dt 2 +2 c ( t ) 2 ( dy dr ) 2 dr 2 + c ( t ) 2 y ( r ) 2 ( dθ 2 +( sin 2 θ ) dφ 2 ) (10)</formula>
<text><location><page_4><loc_9><loc_90><loc_92><loc_93></location>Now we denote the space-time interval at the boundary of the star as ds 2 Σ . Thus, in co-moving coordinates, we write the ds 2 Σ as follows</text>
<formula><location><page_4><loc_32><loc_86><loc_92><loc_88></location>ds 2 Σ = g ij dξ i dξ j = -dτ 2 + R 2 ( τ )( dθ 2 +( sin 2 θ ) dφ 2 ) (11)</formula>
<text><location><page_4><loc_9><loc_82><loc_92><loc_85></location>Where Σ accounts for the hypersurface dividing interior and exterior space-time. Co-moving coordinates ξ i = τ , θ and φ describes the hypersurface Σ. We consider the exterior space-time governed by the Vaidya metric as follows</text>
<formula><location><page_4><loc_29><loc_77><loc_92><loc_80></location>ds 2 + = -( 1 -2 m ( v ) ˜ r ) dv 2 -2 dvd ˜ r + ˜ r 2 ( dθ 2 +( sin 2 θ ) dφ 2 ) (12)</formula>
<text><location><page_4><loc_9><loc_73><loc_92><loc_75></location>The first junction condition implies that metrics describing interior, boundary and exterior space-time must be continuous at matching hypersurface,</text>
<formula><location><page_4><loc_44><loc_68><loc_92><loc_70></location>ds 2 -= ds 2 Σ = ds 2 + (13)</formula>
<text><location><page_4><loc_10><loc_66><loc_59><loc_67></location>Now, the boundary equation for the interior metric (10) is given by</text>
<formula><location><page_4><loc_43><loc_63><loc_92><loc_64></location>f ( t, r ) = r -r Σ = 0 (14)</formula>
<text><location><page_4><loc_9><loc_60><loc_66><loc_61></location>Where r Σ is constant. Thus, the normal vector to that hypersurface is given by</text>
<formula><location><page_4><loc_40><loc_56><loc_92><loc_59></location>n -α = { 0 , √ 2 c ( t ) ( dy dr ) Σ , 0 , 0 } (15)</formula>
<text><location><page_4><loc_9><loc_53><loc_72><loc_55></location>Putting dr = 0 in equation (10) and comparing components of ds 2 -and ds 2 Σ at Σ, we get</text>
<formula><location><page_4><loc_47><loc_49><loc_92><loc_50></location>y ( r Σ ) ˙ t = 1 (16)</formula>
<formula><location><page_4><loc_44><loc_44><loc_92><loc_45></location>c ( t ) y ( r Σ ) = R ( τ ) (17)</formula>
<text><location><page_4><loc_10><loc_41><loc_77><loc_42></location>Where, the dot stands for d dτ . Now, boundary equation for the exterior metric (12) is given by</text>
<formula><location><page_4><loc_42><loc_37><loc_92><loc_38></location>f ( v, ˜ r ) = ˜ r -˜ r Σ ( v ) = 0 (18)</formula>
<text><location><page_4><loc_9><loc_34><loc_56><loc_35></location>Which yields the following normal vector to the hypersurface (18)</text>
<formula><location><page_4><loc_33><loc_28><loc_92><loc_32></location>n + α = ( 2 d ˜ r Σ ( v ) dv +1 -2 m ˜ r Σ ) -1 2 ( -d ˜ r Σ ( v ) dv , 1 , 0 , 0 ) (19)</formula>
<text><location><page_4><loc_10><loc_26><loc_59><loc_27></location>The junction condition (11) for (11) and (13) leads to the conditions</text>
<formula><location><page_4><loc_46><loc_21><loc_92><loc_23></location>˜ r Σ ( v ) = R ( τ ) (20)</formula>
<formula><location><page_4><loc_39><loc_14><loc_92><loc_18></location>( 2 d ˜ r ( v ) dv +1 -2 m ˜ r ) Σ = ( 1 ˙ v 2 ) Σ (21)</formula>
<text><location><page_4><loc_10><loc_12><loc_92><loc_13></location>The second junction condition implies that extrinsic curvature K ij must be continuous across the hypersurface Σ.</text>
<formula><location><page_5><loc_42><loc_89><loc_92><loc_91></location>[ K ij ] = K + ij -K -ij = 0 (22)</formula>
<text><location><page_5><loc_10><loc_87><loc_40><loc_88></location>Where the extrinsic curvature is given by</text>
<formula><location><page_5><loc_37><loc_81><loc_92><loc_84></location>K ± ij = -n ± α ∂ 2 x α ± ∂ξ i ∂ξ j -n ± α Γ α µν ∂x µ ± ∂ξ i ∂x ν ± ∂ξ j (23)</formula>
<text><location><page_5><loc_10><loc_78><loc_92><loc_80></location>Now this junction condition has been developed for the following general time-dependant metric by many authors</text>
<formula><location><page_5><loc_29><loc_75><loc_92><loc_77></location>ds 2 -= -e 2 ν ( t,r ) dt 2 + e 2 ψ ( t,r ) dr 2 + Q 2 ( t, r )( dθ 2 + sin 2 θdφ 2 ) (24)</formula>
<text><location><page_5><loc_10><loc_73><loc_75><loc_74></location>Junction condition (22) for the interior metric (24) and exterior metric (12) is found to be</text>
<formula><location><page_5><loc_33><loc_67><loc_92><loc_70></location>M ( t, r ) = Q 2 [ 1 -e -2 ψ ( dQ dr ) 2 + e -2 ν ( dQ dt ) 2 ] (25)</formula>
<formula><location><page_5><loc_11><loc_60><loc_92><loc_64></location>Q 2 e -( ν + ψ ) ( 2 ˙ Q ' Q -2 ˙ Q Q ˙ ψ ψ -2 ν ' ν ˙ Q Q ) + Q 2 e -2 ν ( 2 Q Q -2 ˙ Q Q ˙ ν ν + e 2 ν Q 2 + ˙ Q 2 Q 2 -e 2( ν -ψ ) ( Q ' 2 Q 2 -2 ν ' ν Q ' Q ) ) ∣ ∣ ∣ ∣ Σ = 0 (26)</formula>
<text><location><page_5><loc_9><loc_56><loc_92><loc_59></location>By comparing the interior metric of our case (10) with generalized interior metric, (24) we can derive the junction condition corresponding to equation (22) for the metric (10) as follow</text>
<formula><location><page_5><loc_42><loc_51><loc_92><loc_53></location>M ( t, r ) = cy 2 ( 1 2 + ˙ c 2 ) (27)</formula>
<formula><location><page_5><loc_15><loc_44><loc_92><loc_48></location>1 2 √ 2 y ' ( 2˙ cy ' cy -2 ˙ c 2 c 2 1 log ( √ 2 cy ' ) -2˙ c cylog ( y ) + 2 √ 2cy ' y + √ 2 y ' cy + √ 2 y ' ˙ c 2 cy -y ' √ 2 cy + √ 2 cylog ( y ) ) ∣ ∣ ∣ ∣ Σ = 0 (28)</formula>
<text><location><page_5><loc_9><loc_40><loc_92><loc_42></location>Where M ( t, r ) denotes the Misner-Sharp mass function equals to the Schwarzschild mass when evaluated at the boundary Σ.</text>
<text><location><page_5><loc_9><loc_37><loc_92><loc_39></location>Final two junction conditions implies that the Ricci scalar and its derivative must be continuous across the hypersurface Σ,</text>
<formula><location><page_5><loc_40><loc_33><loc_92><loc_34></location>[ R ] = 0 and n α [ ∂ α R ] = 0 (29)</formula>
<text><location><page_5><loc_10><loc_30><loc_51><loc_31></location>where Ricci scalar for the interior metric (10) is given by</text>
<formula><location><page_5><loc_44><loc_24><loc_92><loc_27></location>R = 6˙ c 2 +6 c c -1 c 2 y 2 (30)</formula>
<section_header_level_1><location><page_5><loc_22><loc_20><loc_78><loc_22></location>IV. GRAVITATIONAL COLLAPSE MODEL IN f ( R ) = kR m GRAVITY</section_header_level_1>
<text><location><page_5><loc_9><loc_11><loc_92><loc_18></location>Equations (6) to (9) shows that there are six unknowns ( ρ , p r , p t , q , c and y ) but only four equations. Thus we can choose any two unknowns to solve field equations (6) to (9). Here we will choose the forms of y ( r ) and c ( t ) such that model obeys the junction conditions presented in the previous section. It is straightforward to see that the following metric form can obey all junction conditions, including the continuity of the Ricci scalar and its derivatives across the boundary</text>
<formula><location><page_6><loc_44><loc_88><loc_92><loc_91></location>y ( r ) = ( 1 -r a ) -n (31)</formula>
<text><location><page_6><loc_9><loc_83><loc_92><loc_87></location>Where a denotes the constant value of radius of the collapsing star in co-moving coordinates and n is the constant having condition n ≥ 1. Now we choose the form of c ( t ) such that model simplifies considerably such that 6˙ c 2 +6 c c = 0 holds. The condition gives the following form of c ( t ),</text>
<formula><location><page_6><loc_45><loc_79><loc_92><loc_81></location>c ( t ) = √ 1 -bt (32)</formula>
<text><location><page_6><loc_9><loc_72><loc_92><loc_77></location>where b is the integration constant. Putting the values of the y ( r ) and c ( t ) from equations (31) and (32) in equations (6) to (9), we calculate the exact solutions describing physical quantities like energy density, two pressures and heat flux density of the gravitationally collapsing stars. Here we choose n = 2 in equation (31) and use general form of c ( t ) as given in (32). Thus, we solve equations (6) to (9) for the following physically reliable forms of y ( r ) and c ( t ),</text>
<formula><location><page_6><loc_36><loc_66><loc_92><loc_69></location>y ( r ) = ( 1 -r a ) -2 and c ( t ) = √ 1 -bt (33)</formula>
<text><location><page_6><loc_10><loc_64><loc_74><loc_65></location>From equation (30), we calculate Ricci scalar R for the metric components (33) as follow</text>
<formula><location><page_6><loc_45><loc_58><loc_92><loc_61></location>R = -(1 -r a ) 4 1 -bt (34)</formula>
<text><location><page_6><loc_9><loc_53><loc_92><loc_57></location>Here we consider the model R + f ( R ) = R + kR m , which gives the identities f ( R ) = kR m and f R = kmR m -1 . Putting all these values in equations (6) to (9), we get the following values of energy density, two pressures and heat flux density for gravitationally collapsing stars</text>
<formula><location><page_6><loc_10><loc_47><loc_45><loc_50></location>ρ ( t, r ) = (1 -r a ) 4 4(1 -bt ) m +1 [(2 -2 bt +3 b 2 )(1 -bt ) m -1</formula>
<formula><location><page_6><loc_31><loc_43><loc_92><loc_46></location>+ k ( -1) m +1 ( 1 -r a ) 4 m -4 { ( -8 m 3 +20 m 2 -8 m -2)(1 -bt ) + b 2 (9 m -6 m 2 ) } ] (35)</formula>
<formula><location><page_6><loc_10><loc_34><loc_92><loc_40></location>p r ( t, r ) = (1 -r a ) 4 4(1 -bt ) m +1 [(2 -2 bt + b 2 )(1 -bt ) m -1 + k ( -1) m +1 ( 1 -r a ) 4 m -4 { ( -12 m 2 +12 m +2)(1 -bt ) + ( -4 b 2 m 3 +4 b ( b +1) m 2 -b (4 -b ) m ) } ] (36)</formula>
<formula><location><page_6><loc_10><loc_24><loc_92><loc_30></location>p t ( t, r ) = (1 -r a ) 4 4(1 -bt ) m +1 [(2 -2 bt + b 2 )(1 -bt ) m -1 + k ( -1) m +1 ( 1 -r a ) 4 m -4 { (8 m 3 -20 m 2 +12 m +2)(1 -bt ) -mb 2 (4( m -1) 2 -1) } ] (37)</formula>
<formula><location><page_6><loc_20><loc_18><loc_92><loc_21></location>q ( t, r ) = (1 -r a ) 4 √ 2(1 -bt ) m + 1 2 [ b (1 -bt ) m -1 + k ( -1) m +1 ( 1 -r a ) 4 m -4 { bm -2 m 2 b ( m -1) } ] (38)</formula>
<text><location><page_6><loc_9><loc_14><loc_92><loc_16></location>Equations (35) to (38) are analytical solutions describing physical quantities of gravitationally collapsing stars in modified f ( R ) = kR m gravity.</text>
<text><location><page_7><loc_9><loc_59><loc_30><loc_60></location>where, ∆ = √ ( ρ + p r ) 2 -4 q 2 .</text>
<section_header_level_1><location><page_7><loc_40><loc_55><loc_61><loc_56></location>B. Weak energy condition</section_header_level_1>
<text><location><page_7><loc_9><loc_49><loc_92><loc_53></location>The weak energy condition must be obeyed for a physically acceptable model of gravitational collapse. The weak energy condition states that if λ 0 signifies the eigenvalue corresponding to the time-like eigenvector, then -λ 0 ≥ 0 must be true. As a result, the shear-less fluid's weak energy condition becomes</text>
<formula><location><page_7><loc_45><loc_44><loc_92><loc_46></location>ρ -p r +∆ ≥ 0 (41)</formula>
<section_header_level_1><location><page_7><loc_38><loc_40><loc_63><loc_41></location>C. Dominant energy conditions</section_header_level_1>
<text><location><page_7><loc_9><loc_34><loc_92><loc_38></location>Any physically acceptable model must also obey the dominant energy conditions. If λ i signifies the eigenvalues corresponding to the space-like eigenvectors, then it must follow the condition given by λ 0 ≤ λ i ≤ λ 0 . Thus, the dominant energy conditions for shear-less fluid can be written in terms of physical quantities of the star as follow</text>
<formula><location><page_7><loc_43><loc_30><loc_92><loc_31></location>ρ -p r -2 p t +∆ ≥ 0 (42)</formula>
<formula><location><page_7><loc_47><loc_24><loc_92><loc_26></location>ρ -p r ≥ 0 (43)</formula>
<section_header_level_1><location><page_7><loc_39><loc_20><loc_61><loc_21></location>D. Strong energy condition</section_header_level_1>
<text><location><page_7><loc_9><loc_14><loc_92><loc_18></location>By imposing the last strong energy condition, a physically acceptable model can be found. The presence of a strong energy condition indicates that λ 0 + ∑ i λ i ≥ 0. Which implies that physical quantities of the star mast obey following condition</text>
<formula><location><page_7><loc_46><loc_10><loc_92><loc_11></location>2 p t +∆ ≥ 0 (44)</formula>
<section_header_level_1><location><page_7><loc_19><loc_92><loc_82><loc_93></location>V. ENERGY CONDITIONS FOR GRAVITATIONALLY COLLAPSING STARS</section_header_level_1>
<text><location><page_7><loc_9><loc_84><loc_92><loc_90></location>To describe the realistic compact stars undergoing gravitational collapse, the model of gravitational collapse must obey certain energy conditions. Thus, in this section we describe the energy conditions to be applied so that model become physically acceptable. We consider the shear-less matter distribution, which is essential the case for separable form of interior space-time metric (5). Thus, we apply the following energy conditions on the model.</text>
<section_header_level_1><location><page_7><loc_28><loc_80><loc_73><loc_81></location>A. Eigenvalues of energy-momentum tensor must be real</section_header_level_1>
<text><location><page_7><loc_9><loc_73><loc_92><loc_78></location>The eigenvalues of the energy-momentum tensor must be real for a physically acceptable model of any compact star. The eigenvalues of the energy-momentum tensor can be found by equation | T αβ -λg αβ | = 0. The detailed calculation performed is in the reference [45]. From that it can be derived that for shear-less matter distribution, these conditions can be written as</text>
<formula><location><page_7><loc_44><loc_68><loc_92><loc_70></location>| ρ + p r | -2 | q | ≥ 0 (39)</formula>
<formula><location><page_7><loc_43><loc_63><loc_92><loc_64></location>ρ -p r +2 p t +∆ ≥ 0 (40)</formula>
<text><location><page_8><loc_9><loc_90><loc_92><loc_93></location>Thus, physically acceptable model of stars undergoing gravitational collapse can be found by imposing all energy conditions from (39) to (44).</text>
<section_header_level_1><location><page_8><loc_32><loc_86><loc_69><loc_87></location>VI. PHYSICAL ANALYSIS OF THE MODEL</section_header_level_1>
<text><location><page_8><loc_9><loc_80><loc_92><loc_84></location>In this section we present physical analysis for the different values of k and m . Equations (35) to (38) represents analytical solutions for general case of R + f ( R ) = R + kR m . Here we choose different values of k and m and apply energy conditions for each case to generate physically acceptable models of gravitational collapse.</text>
<section_header_level_1><location><page_8><loc_38><loc_76><loc_63><loc_77></location>A. Physical analysis with k = 0</section_header_level_1>
<text><location><page_8><loc_9><loc_70><loc_92><loc_74></location>Firstly, we choose the value of k = 0. So that the field equations becomes essentially of General Relativity. Thus, with k = 0, equations (35) to (38) describes gravitational collapse in General Relativity. By putting k = 0 in them we get the following analytical forms of physical quantities by choosing a = b = 1</text>
<formula><location><page_8><loc_41><loc_64><loc_92><loc_67></location>ρ ( t, r ) = (1 -r ) 4 4(1 -t ) 2 (5 -2 t ) (45)</formula>
<formula><location><page_8><loc_41><loc_58><loc_92><loc_61></location>p r ( t, r ) = (1 -r ) 4 4(1 -t ) 2 (3 -2 t ) (46)</formula>
<formula><location><page_8><loc_41><loc_52><loc_92><loc_55></location>p t ( t, r ) = (1 -r ) 4 4(1 -t ) 2 (3 -2 t ) (47)</formula>
<formula><location><page_8><loc_43><loc_46><loc_92><loc_49></location>q ( t, r ) = (1 -r ) 4 √ 2(1 -t ) 3 2 (48)</formula>
<text><location><page_8><loc_9><loc_36><loc_92><loc_45></location>Equations (45) to (48) shows that gravitational collapse starts at t = 0 and ends up in singularity t = 1 with blowing up all physical quantities. It can be seen from the equations (46) and (47) that anisotropy of collapsing stars for k = 0 identically vanishes for all time at all interior points of the star. However, it is not the case for the non-zero value of k , providing the model in modified f ( R ) gravity. Also, we plot all the energy conditions described in section V. All the graphs are presented in FIG 1 show that this model satisfies all the energy conditions, and thus it is physically acceptable.</text>
<section_header_level_1><location><page_8><loc_33><loc_32><loc_68><loc_33></location>B. Physical analysis of the model f ( R ) = kR 2</section_header_level_1>
<text><location><page_8><loc_9><loc_27><loc_92><loc_30></location>Now we choose the value of m = 2. Thus, the model becomes R + f ( R ) = R + kR 2 . We put the values of m = 2, k = 10 -3 , and a = b = 1 in equations (35) to (38) and thus, the equations becomes</text>
<formula><location><page_8><loc_31><loc_22><loc_92><loc_25></location>ρ ( t, r ) = (1 -r ) 4 4(1 -t ) 3 [(5 -2 t )(1 -t ) + k (1 -r ) 4 (8 -2 t )] (49)</formula>
<formula><location><page_8><loc_30><loc_16><loc_92><loc_19></location>p r ( t, r ) = (1 -r ) 4 4(1 -t ) 3 [(3 -2 t )(1 -t ) + k (1 -r ) 4 (28 -22 t )] (50)</formula>
<formula><location><page_8><loc_30><loc_10><loc_92><loc_13></location>p t ( t, r ) = (1 -r ) 4 4(1 -t ) 3 [(3 -2 t )(1 -t ) -k (1 -r ) 4 (4 -10 t )] (51)</formula>
<figure>
<location><page_9><loc_9><loc_80><loc_34><loc_92></location>
</figure>
<figure>
<location><page_9><loc_65><loc_80><loc_89><loc_92></location>
</figure>
<figure>
<location><page_9><loc_9><loc_65><loc_34><loc_78></location>
</figure>
<figure>
<location><page_9><loc_37><loc_65><loc_62><loc_78></location>
</figure>
<figure>
<location><page_9><loc_65><loc_65><loc_90><loc_78></location>
<caption>FIG. 1: Energy conditions for the model of gravitational collapse with k = 0</caption>
</figure>
<formula><location><page_9><loc_36><loc_54><loc_92><loc_57></location>q ( t, r ) = (1 -r ) 4 √ 2(1 -t ) 5 2 [1 -t +6 k (1 -r ) 4 ] (52)</formula>
<text><location><page_9><loc_9><loc_44><loc_92><loc_52></location>Equations (49) to (52) represents analytical solutions of gravitational collapse in R 2 corrected f ( R ) gravity. Equations (49) to (52) shows that for this model also collapse starts at t = 0 and it ends up in singularity at t = 1, due to the choice of b = 1. Equations (50) and (51) shows that anisotropy doesn't vanish at the interior of star, unlike vanishing anisotropy in case of collapse governed by general relativity. We plot all the energy conditions for this model in FIG 2. All the graphs presented in FIG 2. shows that this model satisfy all the energy conditions and thus it is physically acceptable.</text>
<figure>
<location><page_9><loc_10><loc_28><loc_34><loc_41></location>
</figure>
<figure>
<location><page_9><loc_37><loc_28><loc_62><loc_41></location>
</figure>
<figure>
<location><page_9><loc_65><loc_28><loc_89><loc_41></location>
</figure>
<figure>
<location><page_9><loc_9><loc_14><loc_34><loc_27></location>
</figure>
<figure>
<location><page_9><loc_65><loc_14><loc_89><loc_27></location>
</figure>
<figure>
<location><page_9><loc_37><loc_14><loc_62><loc_26></location>
<caption>FIG. 2: Energy conditions for the model of gravitational collapse with f ( R ) = kR 2</caption>
</figure>
<figure>
<location><page_9><loc_37><loc_80><loc_62><loc_92></location>
</figure>
<section_header_level_1><location><page_10><loc_33><loc_92><loc_68><loc_93></location>C. Physical analysis of the model f ( R ) = kR 4</section_header_level_1>
<text><location><page_10><loc_9><loc_87><loc_92><loc_90></location>Finally we choose R + f ( R ) = R + kR 4 model to describe gravitational collapse of compact stars. We put the values of m = 4, k = 10 -7 , and a = b = 1 in equations (35) to (38) and thus, the equations becomes</text>
<formula><location><page_10><loc_29><loc_82><loc_92><loc_85></location>ρ ( t, r ) = (1 -r ) 4 4(1 -t ) 5 [(5 -2 t )(1 -t ) 3 + k (1 -r ) 12 (286 -226 t )] (53)</formula>
<formula><location><page_10><loc_28><loc_76><loc_92><loc_79></location>p r ( t, r ) = (1 -r ) 4 4(1 -t ) 5 [(3 -2 t )(1 -t ) 3 + k (1 -r ) 12 (282 -142 t )] (54)</formula>
<formula><location><page_10><loc_29><loc_70><loc_92><loc_73></location>p t ( t, r ) = (1 -r ) 4 4(1 -t ) 5 [(3 -2 t )(1 -t ) 3 -k (1 -r ) 12 (102 -242 t )] (55)</formula>
<formula><location><page_10><loc_34><loc_64><loc_92><loc_67></location>q ( t, r ) = (1 -r ) 4 √ 2(1 -t ) 9 2 [(1 -t ) 3 +92 k (1 -r ) 12 ] (56)</formula>
<text><location><page_10><loc_9><loc_49><loc_92><loc_62></location>Equations (53) to (56) represents analytical solutions of gravitational collapse in R 4 corrected f ( R ) gravity. The gravitational collapse starts at t = 0 and due to choice of b = 1, this model also governs the end of collapse in singularity at t = 1. However, due to higher-order correction in the gravitational model, the stronger gravitational pull crunches matter with more power. This can be seen by comparing scales of physical quantities in R 2 corrected and R 4 corrected gravity. Equations (54) and (55) show that anisotropy does not vanish for this case also, as was the case for R 2 corrected gravity. In the following section, we show that anisotropy has to be taken into account for describing the gravitational collapse of stars in the general f ( R ) gravity model. We also plot all the energy conditions for this model in FIG 3. The graphs show that this model satisfies all the energy conditions, and thus it is physically acceptable.</text>
<figure>
<location><page_10><loc_10><loc_34><loc_34><loc_46></location>
</figure>
<figure>
<location><page_10><loc_65><loc_34><loc_89><loc_46></location>
</figure>
<figure>
<location><page_10><loc_10><loc_20><loc_34><loc_32></location>
</figure>
<figure>
<location><page_10><loc_65><loc_20><loc_89><loc_32></location>
</figure>
<figure>
<location><page_10><loc_38><loc_34><loc_62><loc_46></location>
</figure>
<figure>
<location><page_10><loc_37><loc_20><loc_62><loc_32></location>
<caption>FIG. 3: Energy conditions for the model of gravitational collapse with f ( R ) = kR 4</caption>
</figure>
<section_header_level_1><location><page_11><loc_18><loc_92><loc_83><loc_93></location>VII. PHYSICAL PROPERTIES OF GRAVITATIONALLY COLLAPSING STARS</section_header_level_1>
<text><location><page_11><loc_9><loc_80><loc_92><loc_90></location>In section IV, we have computed the energy density, pressures, and heat flux density of gravitationally collapsing stars. The gravitational collapse in f ( R ) gravity also exhibits other useful properties, which are useful to study the inherent nature of such gravitationally collapsing compact stars. Thus, we will study the other useful properties of gravitationally collapsing stars such as shear and anisotropic behavior of matter undergoing gravitational collapse and formation of the apparent horizon by calculating various quantities like 4-velocity, 4-acceleration and expansion parameter of matter, the projection tensor, and shearing tensor. The 4-velocity and the radial vector used in (3) for the space-time metric (5) and having functions y ( r ) and c ( t ) given by (33) are given by</text>
<formula><location><page_11><loc_28><loc_74><loc_92><loc_77></location>u α = ( ( 1 -r a ) 2 , 0 , 0 , 0 ) and v α = ( 0 , a ( 1 -r a ) 3 2 √ 2 √ 1 -bt , 0 , 0 ) (57)</formula>
<text><location><page_11><loc_10><loc_71><loc_65><loc_72></location>The 4-acceleration of the fluid can be computed from the following equation</text>
<formula><location><page_11><loc_45><loc_68><loc_92><loc_70></location>a α = u β ( ∇ β u α ) (58)</formula>
<text><location><page_11><loc_10><loc_66><loc_47><loc_67></location>which has the following non-zero radial component</text>
<formula><location><page_11><loc_45><loc_60><loc_92><loc_63></location>a 1 = a ( 1 -r a ) 5 4(1 -bt ) (59)</formula>
<text><location><page_11><loc_10><loc_57><loc_48><loc_58></location>Now we computes the expansion parameter as follow</text>
<formula><location><page_11><loc_41><loc_52><loc_92><loc_56></location>Θ = ∇ α u α = -3 b ( 1 -r a ) 2 2(1 -bt ) (60)</formula>
<text><location><page_11><loc_10><loc_50><loc_55><loc_51></location>The shear tensor can be computed from the following identity</text>
<formula><location><page_11><loc_36><loc_46><loc_92><loc_49></location>σ αβ = u ( α ; β ) + a ( α u β ) -1 3 Θ( g αβ + u α u β ) (61)</formula>
<text><location><page_11><loc_9><loc_42><loc_92><loc_44></location>where h αβ = g αβ + u α u β is known as projection tensor. From equation (61), we compute the components of shearing tensor as follow</text>
<formula><location><page_11><loc_33><loc_35><loc_92><loc_39></location>σ 00 = -b ( 1 -r a ) 2 2(1 -bt ) ( -1 ( 1 -r a ) 4 + 1 ( 1 -r a ) 4 ) = 0 (62)</formula>
<formula><location><page_11><loc_36><loc_29><loc_92><loc_33></location>σ 01 = σ 10 = 1 a ( 1 -r a ) 3 -1 a ( 1 -r a ) 3 = 0 (63)</formula>
<formula><location><page_11><loc_31><loc_23><loc_92><loc_27></location>σ 11 = 2( -b 2 )( 4 a 2 ) ( 1 -r a ) 4 -( 1 -r a ) 2 ( -b 2 ) (1 -bt ) ( 2(1 -bt ) 4 a 2 ( 1 -r a ) 6 ) = 0 (64)</formula>
<formula><location><page_11><loc_32><loc_16><loc_92><loc_20></location>σ 22 = ( -b 2 ) ( 1 -r a ) 2 -( -b 2 ) ( 1 -r a ) 2 1 -bt ( 1 -bt ( 1 -r a ) 4 ) = 0 (65)</formula>
<formula><location><page_11><loc_29><loc_9><loc_92><loc_13></location>σ 33 = ( -b 2 ) ( 1 -r a ) 2 sin 2 θ -( -b 2 ) ( 1 -r a ) 2 1 -bt ( 1 -bt ( 1 -r a ) 4 ) sin 2 θ = 0 (66)</formula>
<text><location><page_12><loc_9><loc_83><loc_92><loc_93></location>Other components of the shearing tensor (61) also vanish due to the vanishing of individual parts. Thus, all components including (62) to (66) vanishes. Thus, the prior assumption of vanishing shear components holds for the study. It is important to note that the vanishing shear is only possible if gravitationally collapsing stars are being studied in the separable form of the interior space-time metric. The gravitational collapse of compact stars described by the non-separable form of interior metric involves the shear, and thus it has a non-zero shearing tensor. The detailed study of the gravitational collapse in the non-separable form of interior metric considering shear and its evolution is given in the reference [45].</text>
<text><location><page_12><loc_9><loc_80><loc_92><loc_83></location>Now, we compute the anisotropy ( S ) of the stars undergoing the gravitational collapse described by the modified R + f ( R ) = R + kR m gravity model under consideration. From equation (7) and (8) we have the following identity,</text>
<formula><location><page_12><loc_37><loc_75><loc_92><loc_78></location>S = p r -p t = f ' R ( y ' 2 + yy '' ) 2 yy ' 3 c 2 -f '' R 2 c 2 y ' 2 (67)</formula>
<text><location><page_12><loc_10><loc_72><loc_56><loc_74></location>For the model f ( R ) = kR m , the values of f ' R and f '' R is given by</text>
<formula><location><page_12><loc_40><loc_67><loc_92><loc_70></location>f ' R = -k (2 m )( m -1) R m -1 y ' y (68)</formula>
<formula><location><page_12><loc_31><loc_61><loc_92><loc_64></location>f '' R = kR m -1 [ 2 m ( m -1)(2 m -1) y ' 2 y 2 -2 m ( m -1) y '' y ] (69)</formula>
<text><location><page_12><loc_10><loc_58><loc_71><loc_59></location>Using equations (68) and (69) in equation (67), we compute the anisotropy as follow</text>
<formula><location><page_12><loc_39><loc_54><loc_92><loc_55></location>S = p r -p t = k (2 m 2 )( m -1) R m (70)</formula>
<text><location><page_12><loc_10><loc_51><loc_89><loc_52></location>The anisotropy S can be written in terms of space-time coordinate by utilizing equation (34) in (70) as follow</text>
<formula><location><page_12><loc_33><loc_45><loc_92><loc_49></location>S = p r -p t = k ( -1) m (2 m 2 )( m -1) ( 1 -r a ) 4 m (1 -bt ) m (71)</formula>
<text><location><page_12><loc_9><loc_34><loc_92><loc_44></location>Equation (70) shows that the anisotropy can vanish for k = 0 only, in which case the solutions will reduce to GR. This fact can also be observed by equations (45) to (48). Thus, for the case of modified f ( R ) gravity, we have to consider the anisotropic behavior of gravitationally collapsing stars described by the interior metric (5). Equation (71) shows that anisotropy does not vanish at the center of the gravitationally collapsing star, instead of vanishing central anisotropy for the stable, compact stars. Thus, it shows that extra curvature source (4) in f ( R ) gravity forces the matter distribution to be anisotropic in nature for gravitationally collapsing stars described by the interior space-time (5).</text>
<text><location><page_12><loc_9><loc_24><loc_92><loc_34></location>Now we investigate the formation of the apparent horizon and singularity formation for gravitational collapse of compact stars with the model of f ( R ) gravity under consideration. From equations (33) and (34), it can be seen that the gravitational collapse ends in singularity at t = 1 b when c ( t ) = 0 and Ricci scalar diverges at all co-moving radii. Now the question is that whether the singularity is naked or the horizon is formed before the observers can see the formation of a singularity. In the second case, there will be a black hole when the horizon is formed while gravitational collapse continues to the formation of a singularity. Whether the singularity will be naked or not can be investigated by solving the equation of formation of an apparent horizon as follow</text>
<formula><location><page_12><loc_38><loc_20><loc_92><loc_21></location>g αβ ∂ α ( c ( t ) 2 y ( r ) 2 ) ∂ β ( c ( t ) 2 y ( r ) 2 ) = 0 (72)</formula>
<text><location><page_12><loc_10><loc_17><loc_85><loc_18></location>We use the general forms of y ( r ) and c ( t ) as obtained in equation (33) and solve equation (72) as follow</text>
<formula><location><page_12><loc_46><loc_11><loc_92><loc_14></location>t h = 4 -2 b 2 4 b (73)</formula>
<text><location><page_12><loc_10><loc_9><loc_92><loc_10></location>Where t h is the time at which horizon is formed. For b = 1, the time when horizon is formed is t h = 1 2 . Thus, for</text>
<text><location><page_13><loc_9><loc_79><loc_92><loc_93></location>the particular solution (33) of interior space-time metric describing gravitational collapse with b = 1, the horizon is formed at t = 1 2 . The particular solution with b = 1 is of interest because it describes completely physical collapse satisfying all energy conditions as well as investigated in section 6 for gravitational collapse model GR , R + kR 2 and R + kR 4 gravity. In that case, all observers will see the formation of the horizon before the singularity forms, and thus singularity will be hidden behind the horizon. Thus, observers for those particular cases of gravitational collapse will see that gravitational collapse is started at t = 0 and horizon is formed at t = 1 2 , before the formation of singularity at t = 1. However, it should be noted that it is not the only possible physically acceptable solution. We have chosen this class of solutions with b = 1. However, for different classes of solutions satisfying all energy conditions with different values of b is maybe possible. For different solutions, the horizon and singularity will be formed at different times or gravitational collapse may also end in naked singularity as can be investigated from equation (73).</text>
<section_header_level_1><location><page_13><loc_42><loc_75><loc_59><loc_76></location>VIII. DISCUSSION</section_header_level_1>
<text><location><page_13><loc_9><loc_64><loc_92><loc_73></location>In this paper, we obtained exact solutions describing gravitational collapse of compact stars in R + f ( R ) = R + kR m gravity. We solved field equations of gravitational collapse in f ( R ) gravity by considering junction conditions and implementing the method of R matching to ensure the continuity of the Ricci scalar and its derivatives across the matching hypersurface. These extra junction conditions restrict the physically possible solutions greatly, and thus, there are not many exact solutions are available for the problem in the literature. Recently some exact solutions have been developed by some authors, but the problem is considered only for the restricted class of f ( R ) model.</text>
<text><location><page_13><loc_9><loc_51><loc_92><loc_64></location>Thus, in this paper, we formulated the exact solutions describing the gravitational collapse of compact stars in general and physically important model of f ( R ) gravity. We then investigated the values of model parameters such that they satisfy all necessary energy conditions. We have demonstrated the model's validity for GR , R + kR 2 and R + kR 4 gravity. However, the model is utterly physical for any order of corrections in f ( R ) = kR m gravity. We have presented a comprehensive graphical analysis of energy conditions for each case and explained all case and their consequences. We also explained how the gravitational collapse scenario is different for GR and lower and higher-order correction in f ( R ) gravity. For example, in the GR case, the anisotropy of gravitationally collapsing stars vanishes identically for all time at different radii, in the case of the particular interior metric considered here. However, for modified f ( R ) gravity, the anisotropy of stars has to be considered.</text>
<text><location><page_13><loc_9><loc_38><loc_92><loc_51></location>After obtaining a physically acceptable and well-behaved gravitational collapse model, we presented some important physical properties of gravitationally collapsing stars. We derived various quantities like 4-velocity, 4-acceleration, and the expansion parameter of the matter undergoing gravitational collapse. Then we calculated the shearing tensor and anisotropy of compact stars described by the particular interior metric considered in this paper. From that, we concluded that separable-form of interior metric leads to vanishing shear inside the collapsing stars. However, anisotropy has to be considered for the model of f ( R ) gravity investigated in this paper. Finally, we studied the formation of singularity and horizon for stars undergoing gravitational collapse. From that, we concluded that the model studied in this paper leads the gravitational collapse of compact stars to end in a black hole by forming a horizon before the formation of singularity at the end of the collapse.</text>
<section_header_level_1><location><page_13><loc_43><loc_34><loc_57><loc_35></location>Acknowledgement</section_header_level_1>
<text><location><page_13><loc_9><loc_29><loc_92><loc_32></location>The author is very grateful to Professor S. Odintsov for his comments and insightful suggestions throughout the research work.</text>
<section_header_level_1><location><page_13><loc_46><loc_25><loc_54><loc_26></location>References</section_header_level_1>
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2024SPIE13092E..5AA | https://arxiv.org/pdf/2411.03144.pdf | <document>
<section_header_level_1><location><page_1><loc_10><loc_86><loc_90><loc_90></location>A comparison of solar and stellar coronagraphs that make use of external occulters</section_header_level_1>
<text><location><page_1><loc_24><loc_83><loc_76><loc_84></location>Claude Aime, C´eline Theys, Simon Prunet, and Andr´e Ferrari</text>
<text><location><page_1><loc_18><loc_80><loc_82><loc_81></location>Universit´e Cˆote d'Azur, Observatoire de la Cˆote d'Azur, CNRS, Nice, France</text>
<section_header_level_1><location><page_1><loc_44><loc_75><loc_56><loc_76></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_10><loc_65><loc_90><loc_74></location>Solar and stellar externally occulted coronagraphs share similar concepts, but are actually very different because of geometric characteristics. Solar occulters were first developed with a simple geometric model of diffraction perpendicular to the occulter edges. We apply this mere approach to starshades, and introduce a simple shifted circular integral of the occulter which allows to illustrate the influence of the number of petals on the extent of the deep central dark zone. We illustrate the reasons for the presence of an internal coronagraph in the solar case and its absence in the exoplanet case.</text>
<text><location><page_1><loc_10><loc_63><loc_53><loc_64></location>Keywords: Starshade, exoplanets, external solar occulters</text>
<section_header_level_1><location><page_1><loc_40><loc_59><loc_60><loc_61></location>1. INTRODUCTION</section_header_level_1>
<figure>
<location><page_1><loc_19><loc_41><loc_81><loc_57></location>
<caption>Figure 1. Illustration of characteristic dimensions for solar and stellar coronagraphs. Distances are from 1 AU to 10 pc.</caption>
</figure>
<text><location><page_1><loc_10><loc_14><loc_90><loc_35></location>Astronomical objectives for solar and stellar coronagraphs are the study of the solar corona and the detection of exoplanets. We consider here coronagraphs using external occulters on the model of the occultation of the Sun by the Moon. These coronagraphs share similar concepts but become different because of geometric characteristics. In round numbers, distances between the objects to be occulted by these coronagraphs are as 1 astronomical unit (AU) for the solar corona vs 10 parsecs or more for exoplanets, a factor of 2 million or so. A very schematic comparative representation of the observational parameters is given in Fig. 1 and a list of experimental parameters is given in Table 1. From a near-Earth orbit, the Sun is seen at an angle of about half a degree. We seek to see the solar corona as close as possible to the solar disk, and the external occulter is just greater than the apparent diameter of the solar photosphere. The corona is observed over a few solar radii. For exoplanets, considering an extra-solar system at 10 parsecs, the angle between the exo-Sun and the exo-Earth is 100 milliarcsec, and the stellar diameter would be seen at an angle of 1 mas. Starshades, with all the experimental constraints, do not have the possibility to just obscure the 1 mas star, and consider much larger obstruction angles, 50 to 100 times as large, just to allow the detection of an exo-Earth. A heuristic and historical presentation of the ideas leading to the diversity of external solar occulters is made in Section 2. A</text>
<text><location><page_2><loc_10><loc_86><loc_90><loc_91></location>similar presentation for exoplanets is given in Section 3, which gives the opportunity to introduce a function which can be considered as a shifted apodization profile. An empirical justification of the usefulness of an internal coronagraph in the solar case and its low efficiency in the case of exoplanets is given in Section 4.</text>
<table>
<location><page_2><loc_11><loc_54><loc_89><loc_80></location>
<caption>Table 1. Comparison between solar and stellar coronagraphs. In addition to the orders of magnitude between dimensions, a main point is that the solar disk is a huge extended object versus the almost point source star. Values are for a typical extra-solar system seen at 10 parsecs.</caption>
</table>
<section_header_level_1><location><page_2><loc_33><loc_49><loc_67><loc_50></location>2. SOLAR EXTERNAL OCCULTERS</section_header_level_1>
<text><location><page_2><loc_10><loc_39><loc_90><loc_48></location>The use of external occulters to shade the bright solar photosphere to observe the faint solar corona without natural eclipses has a long history. Solar astronomers considered first that Lyot's coronagraph, 1 with the rather low quality of optical systems of their time, cannot be improved enough to observe the outer corona. 2, 3 Then, with reference to Evans's photometer, 4 an external occulter is set in front of the Lyot coronagraph in almost all space-born coronagraphs, as described by Koutchmy. 5 In overall, the effects of the external occulter provide the same rejection rate as that of the Lyot coronagraph, and these effects multiply.</text>
<text><location><page_2><loc_10><loc_31><loc_90><loc_38></location>Observers were very inventive. In 1963, Tousey, 6 used a serrated-edge external-occulter and obtained the first observations of the outer solar corona in the absence of an eclipse. Purcell and Koomen 7 proposed to use a system of multiple circular disks. A decisive step was made with the coronagraph SOHO/LASCO, 8 in operation since 1995. In their experimental study of new generations of external occulters for LASCO-C2, they 9 found that conic occulters gave performances superior to multiple discs occulters.</text>
<text><location><page_2><loc_10><loc_17><loc_90><loc_30></location>The idea that leads to serrated external occulters for solar astronomy is of geometric origin. It is assumed that wave diffraction occurs over planes perpendicular to the edges of the screen. Triangular teeth are placed on the edge of the circular occulters to move light away from the center of the optical axis. Boivin 10 shown that a dark disk of radius R cos( α ) is sheltered from the rays diffracted by the edges, where R is the occulter radius and α is the exterior angle of the half-teeth triangle as shown in the right drawing of Fig.2. This effect was recently numerically verified using the Maggi-Rubinowicz boundary integral. 11 Serrated occulters are a simplified version of petal occulters. A gain of several orders of magnitude in rejection could be obtained if the triangular teeth would be replaced by a petal having a more studied shape, for example a Sonine function, 12 but no experiment of this type is currently envisaged.</text>
<text><location><page_2><loc_10><loc_12><loc_90><loc_16></location>The use of several disk occulters followed another reasoning. The idea is that the Arago spot exists because the edge of the circular occulter seen from the telescope is bright. The solution found by solar astronomers of the sixteen is to darken this edge. Tousay 6 considered that the best arrangement was a system of several discs,</text>
<figure>
<location><page_3><loc_19><loc_18><loc_48><loc_35></location>
<caption>Figure 4. Illustration with the approach of Boivin using rays diffracting perpendicular to the edges of the petals for an occulter with 6 petals for which the profile is a mere cosine arch (left) and an occulter with 24 petals (right) corresponding to NW2 of SISTER which gives off a larger circular dark central area.</caption>
</figure>
<figure>
<location><page_3><loc_51><loc_18><loc_80><loc_35></location>
</figure>
<text><location><page_3><loc_98><loc_70><loc_99><loc_72></location>'</text>
<text><location><page_3><loc_96><loc_69><loc_99><loc_70></location>Arago spot</text>
<text><location><page_3><loc_10><loc_57><loc_100><loc_68></location>so that each disk casts a shadow on the edge of the next disk. Fig.3 gives an illustration for 2 discs at the distances z and d of the entrance aperture. The second disc of diameter ω must be in the shadow of the first disc of diameter Ω, with the mandatory requirement that from the telescope the first disc is not visible, so that Ω > ω > d/z Ω. Analytical expressions exist for 2 and 3 disks but are difficult to evaluate numerically. We can generalize these calculations to more disks and obtain the ideal shape for a multi-disk occulter by continuity. 13 The exact calculation of the Fresnel diffraction of a true 3D occulter has not been carried out, to the limit of knowledge of the authors of this paper. The Arago spot: z=0.82 m, W /2=0.015968, l =0.55 ∝ m Figure 17: Schematic illustration leading to the establishment of the radius HP = y ( x ) is seen from the point A (Arago spot) at the distance to the numerical result shown in Fig. 14, the next circular occulter sho θ/ 2 towards the axis. The tangent to the curve representing the envelo the point P is then y ' ( x ) = -HP / HB , from where Eq. 22 is derived. Fig. 16 gives a surprising result for which one can wonder whet calculation errors. The intensity of the Arago spot is computed for 2 d the expected width for the occulter of ASPIICS. Calculating the inte di ffi cult in this case. We note that rejection appears to be optimal for</text>
<text><location><page_3><loc_70><loc_56><loc_100><loc_57></location>smaller than those predicted by the simple model (Vertical red dashed li</text>
<text><location><page_3><loc_70><loc_55><loc_100><loc_56></location>The blue dashed line corresponds to the optimal value of Eq. 21). O</text>
<text><location><page_3><loc_70><loc_54><loc_100><loc_55></location>towards very high factors (several tens), as if the two occulters seems t</text>
<text><location><page_3><loc_75><loc_53><loc_100><loc_54></location>cult to believe that result, additional work is needed to dete</text>
<text><location><page_3><loc_73><loc_50><loc_95><loc_51></location>The optimal 3D envelop for multiple occulters</text>
<text><location><page_3><loc_70><loc_49><loc_88><loc_50></location>In the former section, we have seen that if</text>
<text><location><page_3><loc_89><loc_49><loc_90><loc_50></location>ω <</text>
<text><location><page_3><loc_91><loc_49><loc_91><loc_50></location>d</text>
<text><location><page_3><loc_91><loc_49><loc_91><loc_49></location>z</text>
<text><location><page_3><loc_93><loc_49><loc_93><loc_49></location>Ω</text>
<text><location><page_3><loc_93><loc_49><loc_100><loc_50></location>, the first occul</text>
<text><location><page_3><loc_81><loc_48><loc_100><loc_49></location>, only the second occulter with a bright edge i</text>
<text><location><page_3><loc_90><loc_46><loc_90><loc_47></location>d</text>
<text><location><page_3><loc_90><loc_45><loc_90><loc_46></location>z</text>
<text><location><page_3><loc_91><loc_45><loc_91><loc_46></location>×</text>
<text><location><page_3><loc_92><loc_48><loc_92><loc_50></location>×</text>
<text><location><page_3><loc_92><loc_45><loc_92><loc_46></location>Ω</text>
<text><location><page_3><loc_93><loc_45><loc_95><loc_46></location>< ω <</text>
<text><location><page_3><loc_96><loc_45><loc_96><loc_46></location>Ω</text>
<text><location><page_3><loc_70><loc_44><loc_100><loc_45></location>Moreover, numerical results of Fig. 14 and Fig. 15 favors a position co</text>
<figure>
<location><page_3><loc_18><loc_44><loc_81><loc_57></location>
</figure>
<text><location><page_3><loc_67><loc_44><loc_69><loc_44></location>Distance z</text>
<text><location><page_3><loc_70><loc_44><loc_70><loc_44></location>-</text>
<text><location><page_3><loc_70><loc_44><loc_70><loc_44></location>d</text>
<text><location><page_3><loc_70><loc_43><loc_77><loc_44></location>these two limits:</text>
<text><location><page_3><loc_83><loc_43><loc_87><loc_43></location>Distance z-d</text>
<paragraph><location><page_3><loc_10><loc_38><loc_100><loc_43></location>23 ω = Ω ( 2 z ) From that is possible to establish a di ff erential equation that can give th follow. The reasoning is illustrated in Fig. 17. Given an N th occulter of be placed in the direction that satisfies Eq. 21. Figure 3. Left, principle diagram: two disks #1 and #2 of diameters Ω and ω are at distances z and d from the entrance aperture of the telescope. Right: a substantial light reduction is obtained provided that ω is half-way between the conditions ω ≤ Ω and ω ≥ d/z Ω.</paragraph>
<text><location><page_3><loc_84><loc_36><loc_84><loc_37></location>y</text>
<text><location><page_3><loc_84><loc_36><loc_84><loc_37></location>'</text>
<text><location><page_3><loc_84><loc_36><loc_85><loc_37></location>(</text>
<text><location><page_3><loc_85><loc_36><loc_85><loc_37></location>x</text>
<text><location><page_3><loc_85><loc_36><loc_86><loc_37></location>)</text>
<text><location><page_3><loc_84><loc_34><loc_84><loc_35></location>y</text>
<text><location><page_3><loc_84><loc_34><loc_86><loc_35></location>(0)</text>
<text><location><page_3><loc_95><loc_42><loc_96><loc_43></location>d</text>
<text><location><page_3><loc_95><loc_34><loc_96><loc_37></location>√</text>
<text><location><page_3><loc_86><loc_36><loc_87><loc_37></location>=</text>
<text><location><page_3><loc_87><loc_35><loc_88><loc_37></location>-</text>
<text><location><page_3><loc_86><loc_34><loc_88><loc_35></location>= Ω</text>
<text><location><page_3><loc_88><loc_34><loc_88><loc_35></location>/</text>
<text><location><page_3><loc_88><loc_34><loc_89><loc_35></location>2</text>
<text><location><page_3><loc_70><loc_32><loc_100><loc_33></location>where we have used the fact that PAB is an isosceles triangle and expli</text>
<text><location><page_3><loc_80><loc_31><loc_100><loc_32></location>erential equation does not seem to have a formal</text>
<text><location><page_3><loc_77><loc_30><loc_82><loc_31></location>Mathematica</text>
<text><location><page_3><loc_82><loc_30><loc_100><loc_31></location>), and it has to be computed numerically (</text>
<text><location><page_3><loc_80><loc_29><loc_90><loc_30></location>) that numerical solution.</text>
<text><location><page_3><loc_88><loc_36><loc_89><loc_37></location>HP</text>
<text><location><page_3><loc_88><loc_35><loc_89><loc_36></location>HB</text>
<text><location><page_3><loc_90><loc_36><loc_90><loc_37></location>=</text>
<text><location><page_3><loc_90><loc_35><loc_91><loc_37></location>-</text>
<text><location><page_3><loc_96><loc_36><loc_97><loc_37></location>y</text>
<text><location><page_3><loc_97><loc_36><loc_97><loc_37></location>(</text>
<text><location><page_3><loc_97><loc_36><loc_97><loc_37></location>x</text>
<text><location><page_3><loc_97><loc_36><loc_98><loc_37></location>)</text>
<text><location><page_3><loc_96><loc_35><loc_96><loc_36></location>(</text>
<text><location><page_3><loc_96><loc_35><loc_97><loc_36></location>z</text>
<text><location><page_3><loc_97><loc_35><loc_97><loc_36></location>0</text>
<text><location><page_3><loc_97><loc_35><loc_98><loc_36></location>-</text>
<text><location><page_3><loc_93><loc_42><loc_94><loc_43></location>z</text>
<text><location><page_3><loc_93><loc_35><loc_94><loc_36></location>x</text>
<text><location><page_3><loc_94><loc_42><loc_95><loc_43></location>+</text>
<text><location><page_3><loc_94><loc_35><loc_95><loc_36></location>+</text>
<text><location><page_3><loc_80><loc_79><loc_86><loc_80></location>Occulter profile</text>
<text><location><page_3><loc_82><loc_77><loc_83><loc_78></location>y</text>
<text><location><page_3><loc_83><loc_77><loc_84><loc_78></location>'(x</text>
<text><location><page_3><loc_84><loc_76><loc_85><loc_78></location>)</text>
<text><location><page_3><loc_87><loc_76><loc_87><loc_77></location>q</text>
<text><location><page_3><loc_87><loc_76><loc_88><loc_77></location>/2</text>
<figure>
<location><page_3><loc_19><loc_77><loc_82><loc_91></location>
<caption>Figure 2. Illustration of the Boivin radius R cos( α ) as the envelop of rays perpendicular to the edges of the teeth. The angle α must be as small as possible so that the Boivin radius approaches that of the central disk, which requires a very large number of short teeth not to mask the inner corona. x y(x) ' z0-x</caption>
</figure>
<text><location><page_3><loc_77><loc_75><loc_78><loc_76></location>W</text>
<text><location><page_3><loc_78><loc_75><loc_78><loc_76></location>/2</text>
<text><location><page_3><loc_78><loc_70><loc_78><loc_71></location>O</text>
<text><location><page_3><loc_92><loc_70><loc_93><loc_71></location>A</text>
<text><location><page_3><loc_85><loc_75><loc_86><loc_76></location>q</text>
<text><location><page_3><loc_86><loc_75><loc_87><loc_76></location>/2</text>
<text><location><page_3><loc_91><loc_70><loc_91><loc_71></location>z0</text>
<text><location><page_3><loc_91><loc_35><loc_92><loc_36></location>z</text>
<text><location><page_3><loc_92><loc_35><loc_92><loc_36></location>0</text>
<text><location><page_3><loc_92><loc_35><loc_93><loc_36></location>-</text>
<text><location><page_3><loc_93><loc_26><loc_94><loc_27></location>20</text>
<text><location><page_3><loc_98><loc_35><loc_99><loc_36></location>x</text>
<text><location><page_3><loc_99><loc_35><loc_99><loc_36></location>)</text>
<text><location><page_3><loc_99><loc_35><loc_99><loc_36></location>2</text>
<section_header_level_1><location><page_4><loc_24><loc_89><loc_76><loc_91></location>3. STELLAR EXTERNAL OCCULTERS (STARSHADE)</section_header_level_1>
<text><location><page_4><loc_10><loc_77><loc_90><loc_89></location>For stellar external occulters, the prevailing philosophy is to define an optimal apodized transmission 14 which makes it possible to obtain a very dark central zone with an intensity less than 10 -10 allowing to see exoplanets, 15 using the smallest possible shaped-occulter, 16 , 17 . 18 Considering that these immense occulters cannot be manufactured with variable transmission, shaped occulters 19 are used as the alternative to produce the central black zone. Figure 4 shows an illustration of Boivin's empirical approach to these petal occulters. Here also as for the solar serrated occulter case, a dark central zone sheltered from the rays diffracting perpendicular to the edges of the petals is evidenced. A 24-petal occulter with a very studied profile is clearly more effective than a 6-petal occulter with a simpler profile.</text>
<figure>
<location><page_4><loc_20><loc_45><loc_80><loc_74></location>
<caption>Figure 5. The function g ( ρ, X 0 ) of Eq.3 corresponds to the mean value along concentric circles centered at x = X 0 of the occulter transmission. In these figures, an illustration of the quadratic phase of Eq.2 is given behind. Top panels are for X 0 = 0. The two left occulters correspond to the same value g ( ρ, 0) = p ( ρ ) that the apodized occulter at right, and therefore the same complex amplitude Ψ(0 , 0) at the center of the Fresnel diffraction pattern. Bottom panels are for X 0 = 5 m, 6-petals and 24-petals occulters and the apodized one. A drawing of g ( ρ, X 0 ) for different X 0 values is given in Fig.6.</caption>
</figure>
<text><location><page_4><loc_10><loc_30><loc_90><loc_33></location>For a more detailed analysis of the effects of the number of petals, consider the equation giving the Fresnel diffraction of an occulter f ( x, y ) at the distance z and at the wavelength λ in the form of a convolution product:</text>
<formula><location><page_4><loc_35><loc_26><loc_90><loc_29></location>Ψ( x, y ) = 1 -f ( x, y ) ∗ 1 iλz exp( iπ x 2 + y 2 λz ) . (1)</formula>
<text><location><page_4><loc_10><loc_23><loc_49><loc_25></location>For x = X 0 , y = 0, this convolution can be written as:</text>
<formula><location><page_4><loc_15><loc_19><loc_90><loc_22></location>Ψ( X 0 , 0) = 1 -1 iλz ∫ ∫ exp( iπ ξ 2 + η 2 λz ) f ( X 0 -ξ, 0 -η ) dξdη = 1 -2 π iλz ∫ g ( ρ, X 0 ) exp( iπ ρ 2 λz ) ρ 2 dρ (2)</formula>
<text><location><page_4><loc_10><loc_17><loc_72><loc_18></location>where a change of Cartesian to radial coordinates was made to introduce the function</text>
<formula><location><page_4><loc_33><loc_12><loc_90><loc_15></location>g ( ρ, X 0 ) = 1 2 πρ ∫ 2 π 0 f ( ρ cos( ϕ ) + X 0 , ρ sin( ϕ )) dϕ (3)</formula>
<text><location><page_5><loc_24><loc_62><loc_26><loc_62></location>meter</text>
<figure>
<location><page_5><loc_13><loc_62><loc_87><loc_91></location>
<caption>Figure 6. Top panels, the functions g ( ρ, X 0 ), for apodized, 6-petals and 24-petals occulters, for X 0 =0, 10 and 15m. As expected all functions merge with the profile p ( ρ ) of NW2 of SISTER 20 for X 0 = 0. Bottom panels correspond to the illustration X 0 = 5m given in Fig. 5. Left, the functions g ( ρ, 5), center, the differences g 6 ( ρ, 5) -g a ( ρ, 5) and g 24 ( ρ, 5) -g a ( ρ, 5), and right the intensity diffraction patterns (log10 scale) for the apodized, 6-petals and 24-petals occulters computed using the model of Vanderbei et al. 16 The vertical green line is at the position X 0 = 5m from center.</caption>
</figure>
<text><location><page_5><loc_10><loc_47><loc_90><loc_51></location>that is the integral of the occulter f ( x, y ) along concentric circles centered at the point ( X 0 , 0), taken as the new origin of the axes. An illustration of the integration is given in Fig. 5. The result of the integral is finally divided by 2 πρ to obtain the average transmission on the circles.</text>
<text><location><page_5><loc_10><loc_37><loc_90><loc_46></location>For X 0 = 0, the petal-occulters are constructed such that g ( ρ, 0) = p ( ρ ), the optimal apodized transmission. 15 In other words, the average of the all-or-nothing petal transmissions (1 or 0) on the concentric circles recovers the ideal variable transmission, here the profile NW2, 20 which cannot be physically realized otherwise. The three occulters of the top panels of Fig. 5, the 6-petals occulter (left), a spiral version of it obtained using a twist of the petals (middle), complete the request and give the same mean circularly integrated value that the apodized occulter (right). Many shaped occulters may fit this requirement.</text>
<text><location><page_5><loc_19><loc_35><loc_19><loc_36></location≯</text>
<text><location><page_5><loc_10><loc_32><loc_90><loc_36></location>The X 0 = 0 case is illustrated in the bottom panels of Fig. 5 for X 0 = 5m. There we have considered the integrals for a 6-petals and a 24-petals occulter, and for the apodized occulter of NW2. The circular averages are then all different.</text>
<text><location><page_5><loc_10><loc_19><loc_90><loc_31></location>An illustration of g ( ρ, X 0 ) for X 0 = 0 , 5 , 10 and 15 m is given in Fig.6 for the 6 and 24 shaped occulters g 6 ( ρ, 5) , g 24 ( ρ, 5) and the apodized occulter g a ( ρ, 5). As a reference p ( ρ ) is drawn in these figures. The function g ( ρ, X 0 ) departs from p ( ρ ), and spread out as X 0 increases. The calculations are made in the direction of a tip of the petals, the curves are different in the direction of a hollow, but without changing the conclusion for the comparison between occulters. It appears in any case that g 24 ( ρ, X 0 ) is much closer to g a ( ρ, X 0 ) than g 6 ( ρ, X 0 ), and so are the diffraction patterns as expected by the theory of Vanderbei et al. 16 This is well evidenced by differences between curves. We can consider a priory that the best shaped occulter is the one whose function g ( ρ, X 0 ) is as close as possible to g a ( ρ, X 0 ).</text>
<section_header_level_1><location><page_5><loc_21><loc_15><loc_79><loc_16></location>4. BENEFIT OF ADDING AN INTERNAL CORONAGRAPH</section_header_level_1>
<text><location><page_5><loc_10><loc_12><loc_90><loc_14></location>For the observation of the solar corona, a complementary Lyot coronagraph is used. Figure 7 gives the schematic diagram of a Lyot coronagraph coupled to an external occulter. The original Lyot coronagraph is slightly modified</text>
<text><location><page_5><loc_75><loc_91><loc_75><loc_91></location>X</text>
<text><location><page_5><loc_76><loc_91><loc_77><loc_91></location>=15</text>
<figure>
<location><page_6><loc_13><loc_70><loc_87><loc_91></location>
<caption>Figure 7. Schematic diagram of a Lyot coronagraph coupled to an external occulter. Planes A, B, C, D are the classic planes of the Lyot coronagraph. The Lyot mask is in plane B, the Lyot stop in plane D. The presence of the external occulter (O) modifies the device such that the internal occulter must be optimally placed on the image (O') of the occulter. For exoplanets, we cannot differentiate O' from B. Results</caption>
</figure>
<text><location><page_6><loc_31><loc_63><loc_32><loc_63></location>0</text>
<figure>
<location><page_6><loc_29><loc_38><loc_71><loc_63></location>
<caption>Figure 8. Illustration of the effect of the Lyot coronagraph on the rejection of stray light for the ASPIICS solar coronagraph. 21 The black curve is the direct image of the solar photosphere (plane B) with a straylight at 10 -4 , the blue curve gives the effect of the external occulter alone (plane O'), straylight at 10 -8 , the red curve the effect of the complete system (plane D), straylight at 10 -10 . The internal mask has a central hole for satellites positioning purposes.</caption>
</figure>
<text><location><page_6><loc_27><loc_46><loc_28><loc_56></location>Normalized amplitude</text>
<text><location><page_6><loc_10><loc_23><loc_90><loc_30></location>in order to set the mask (called the internal occulter) on the image of the external occulter. To simplify numerical calculations it is interesting to add a converging lens in plane A so as to reject the occulter to infinity and to compensate it with a diverging lens of the same negative power in C. The propagation of the wave is then done using three Fourier transforms, from A to O', O' to C, and C to D. Figure 8 shows that the coupling external occulter and Lyot coronagraph is very effective, and results in a stray light below 10 -10 , for perfect optics.</text>
<text><location><page_6><loc_10><loc_12><loc_90><loc_22></location>For the detection of exoplanets, projects are either with an external occulter alone or with an internal coronagraph alone. The schematic diagram of Fig.7 would still describe a system using a Lyot coronagraph after the starshade. The same calculation procedure can be made. In this case, given the very large distance where the external occulter is located, it will no longer be possible to differentiate experimentally plane O' from plane B. We give in Fig.9 the effect of a Lyot coronagraph with a small mask of size the first ring, and with a large mask of size the second ring, using a very strong Lyot stop. The small mask will not affect the observation of exoplanets, but its effect on straylight is weak. The large mask is more effective in terms of straylight but it</text>
<figure>
<location><page_7><loc_27><loc_70><loc_75><loc_90></location>
<caption>Figure 10. Illustration of the lack of efficiency of a 4-quadrant system with a Starshade. The top left figure corresponds to the Airy pattern (real part of the amplitude) given by a circular aperture with the ± 1 transmission of the 4 quadrant mask. The top right figure is the corresponding intensity in plane C. A total rejection of the starlight in plane D can be obtained using a Lyot stop in plane C (see Fig.7 for planes denomination). Bottom figures are for the same planes with the external occulter. There is no more an Airy pattern in the plane B and the wave remains unfortunately inside the aperture image in plane C.</caption>
</figure>
<text><location><page_7><loc_76><loc_82><loc_76><loc_82></location>1</text>
<text><location><page_7><loc_76><loc_81><loc_77><loc_81></location>0.9</text>
<text><location><page_7><loc_76><loc_80><loc_77><loc_80></location>0.8</text>
<text><location><page_7><loc_76><loc_79><loc_77><loc_79></location>0.7</text>
<text><location><page_7><loc_76><loc_78><loc_77><loc_78></location>0.6</text>
<text><location><page_7><loc_76><loc_77><loc_77><loc_77></location>0.5</text>
<text><location><page_7><loc_76><loc_76><loc_77><loc_76></location>0.4</text>
<text><location><page_7><loc_76><loc_75><loc_77><loc_75></location>0.3</text>
<text><location><page_7><loc_76><loc_74><loc_77><loc_74></location>0.2</text>
<text><location><page_7><loc_76><loc_73><loc_77><loc_73></location>0.1</text>
<figure>
<location><page_7><loc_30><loc_48><loc_48><loc_61></location>
<caption>Figure 9. Left: residual image of the star (plane O', in black) with two Lyot masks removing the first ring (in blue) or the second ring (in red). Middle: corresponding effects in the final plane D using a severe Lyot stop (diameter reduction 0.8). Only the largest mask allows notable rejection but at the expenses of image reconstruction in the vignetting zone. The vertical lines give the diameters of the Lyot masks. The figure on the right is a hybrid 2D representation that shows the two straylight rings and occulter Fresnel diffraction projected onto the sky.</caption>
</figure>
<figure>
<location><page_7><loc_54><loc_48><loc_70><loc_61></location>
</figure>
<figure>
<location><page_7><loc_30><loc_34><loc_48><loc_47></location>
</figure>
<figure>
<location><page_7><loc_54><loc_34><loc_70><loc_47></location>
</figure>
<text><location><page_7><loc_62><loc_34><loc_65><loc_35></location>meter</text>
<text><location><page_7><loc_10><loc_21><loc_77><loc_22></location>prevents the detection of exoplanets in the vignetting zone corresponding to the petals edges.</text>
<text><location><page_7><loc_10><loc_17><loc_90><loc_20></location>Finally, one might wonder whether a phase filter system would be more efficient than a Lyot coronagraph. Fig.10 illustrates that a 4 quadrant phase mask placed at the focus of a telescope 22 is not efficient either.</text>
<section_header_level_1><location><page_8><loc_43><loc_89><loc_57><loc_91></location>REFERENCES</section_header_level_1>
<unordered_list>
<list_item><location><page_8><loc_11><loc_86><loc_90><loc_88></location>[1] Lyot, B., 'The study of the solar corona without an eclipse (with plate v),' Journal of the Royal Astronomical Society of Canada 27 , 265 (1933).</list_item>
<list_item><location><page_8><loc_11><loc_83><loc_90><loc_85></location>[2] Newkirk, Jr., G. and Bohlin, D., 'Reduction of scattered light in the coronagraph,' Appl. Optics 2 , 131 (Feb. 1963).</list_item>
<list_item><location><page_8><loc_11><loc_78><loc_90><loc_82></location>[3] Newkirk, Jr., G. and Bohlin, J. D., 'Coronascope II : observation of the white light corona from a stratospheric balloon,' in [ Astronomical Observations from Space Vehicles ], Steinberg, J.-L., ed., IAU Symposium 23 , 287 (1965).</list_item>
<list_item><location><page_8><loc_11><loc_75><loc_90><loc_78></location>[4] Evans, J. W., 'Photometer for measurement of sky brightness near the sun,' J. Opt. Soc. Am. 38 , 1083 (Dec. 1948).</list_item>
<list_item><location><page_8><loc_11><loc_73><loc_77><loc_74></location>[5] Koutchmy, S., 'Space-borne coronagraphy,' Space science reviews 47 (1), 95-143 (1988).</list_item>
<list_item><location><page_8><loc_11><loc_70><loc_90><loc_73></location>[6] Tousey, R., 'Observations of the white light corona by rocket,' Annales d'Astrophysique 28 , 600 (Feb. 1965).</list_item>
<list_item><location><page_8><loc_11><loc_67><loc_90><loc_70></location>[7] Purcell, J. and Koomen, M., 'Coronagraph with improved scattered-light properties,' J. Opt. Soc. Am. 52 , 596-597 (1962).</list_item>
<list_item><location><page_8><loc_11><loc_62><loc_90><loc_66></location>[8] Brueckner, G., Howard, R., Koomen, M., Korendyke, C., Michels, D., Moses, J., Socker, D., Dere, K., Lamy, P., Llebaria, A., et al., 'The large angle spectroscopic coronagraph (lasco) visible light coronal imaging and spectroscopy,' The SOHO mission , 357-402 (1995).</list_item>
<list_item><location><page_8><loc_11><loc_59><loc_90><loc_62></location>[9] Bout, M., Lamy, P., Maucherat, A., Colin, C., and Llebaria, A., 'Experimental study of external occulters for the large angle and spectrometric coronagraph 2: Lasco-c2,' Appl. Opt. 39 , 3955-3962 (Aug 2000).</list_item>
<list_item><location><page_8><loc_10><loc_56><loc_90><loc_59></location>[10] Boivin, L., 'Reduction of diffraction errors in radiometry by means of toothed apertures,' Applied Optics 17 (20), 3323-3328 (1978).</list_item>
<list_item><location><page_8><loc_10><loc_53><loc_90><loc_56></location>[11] Rougeot, R. and Aime, C., 'Theoretical performance of serrated external occulters for solar coronagraphyapplication to aspiics,' Astronomy & Astrophysics 612 , A80 (2018).</list_item>
<list_item><location><page_8><loc_10><loc_50><loc_90><loc_52></location>[12] Aime, C., 'Theoretical performance of solar coronagraphs using sharp-edged or apodized circular external occulters,' Astronomy & Astrophysics 558 , A138 (2013).</list_item>
<list_item><location><page_8><loc_10><loc_47><loc_90><loc_49></location>[13] Aime, C., 'Fresnel diffraction of multiple disks on axis-application to coronagraphy,' Astronomy & Astrophysics 637 , A16 (2020).</list_item>
<list_item><location><page_8><loc_10><loc_43><loc_90><loc_46></location>[14] Cady, E., 'Boundary diffraction wave integrals for diffraction modeling of external occulters,' Opt. Expr. 20 (14), 15196-15208 (2012).</list_item>
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<list_item><location><page_8><loc_10><loc_26><loc_90><loc_29></location>[19] Marchal, C., 'Concept of a space telescope able to see the planets and even the satellites around the nearest stars,' Acta Astronautica 12 (3), 195-201 (1985).</list_item>
<list_item><location><page_8><loc_10><loc_22><loc_90><loc_26></location>[20] Hildebrandt, S. R., Shaklan, S. B., Cady, E. J., and Turnbull, M. C., 'Starshade imaging simulation toolkit for exoplanet reconnaissance,' Journal of Astronomical Telescopes, Instruments, and Systems 7 (2), 021217 (2021).</list_item>
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</document> | [] |
2021PhRvD.104f4048M | https://arxiv.org/pdf/2012.11209.pdf | <document>
<section_header_level_1><location><page_1><loc_29><loc_92><loc_72><loc_93></location>Spherically symmetric black holes in metric gravity</section_header_level_1>
<text><location><page_1><loc_35><loc_89><loc_65><loc_90></location>Sebastian Murk 1, 2, ∗ and Daniel R. Terno 1, †</text>
<text><location><page_1><loc_19><loc_86><loc_82><loc_88></location>1 Department of Physics and Astronomy, Macquarie University, Sydney, New South Wales 2109, Australia 2 Sydney Quantum Academy, Sydney, New South Wales 2006, Australia</text>
<text><location><page_1><loc_18><loc_76><loc_83><loc_85></location>The existence of black holes is one of the key predictions of general relativity (GR) and therefore a basic consistency test for modified theories of gravity. In the case of spherical symmetry in GR the existence of an apparent horizon and its regularity is consistent with only two distinct classes of physical black holes. Here we derive constraints that any self-consistent modified theory of gravity must satisfy to be compatible with their existence. We analyze their properties and illustrate characteristic features using the Starobinsky model. Both of the GR solutions can be regarded as zeroth-order terms in perturbative solutions of this model. We also show how to construct nonperturbative solutions without a well-defined GR limit.</text>
<section_header_level_1><location><page_1><loc_22><loc_72><loc_36><loc_73></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_53><loc_49><loc_70></location>General Relativity (GR), one of the two pillars of modern physics, is the simplest member of the family of metric theories of gravity. It is the only theory that is derived from an invariant that is linear in second derivatives of the metric. However, interpretations of astrophysical and cosmological data as well as theoretical considerations [1, 2] encourage us to consider GR as the low-energy limit of some effective theory of quantum gravity [3-5]. Extended theories of gravity, such as metric theories that involve higher-order invariants of the Riemann tensor, metric-affine theories, and theories with torsion, include additional terms in the action functional. Here we focus on metric modified theories of gravity (MTG).</text>
<text><location><page_1><loc_9><loc_28><loc_49><loc_52></location>A prerequisite for the validity of any proposed generalization of GR is that it must be compatible with current astrophysical and cosmological data. In particular, a viable candidate theory must provide a model to describe the observed astrophysical black hole candidates. Popular contemporary models describe them as ultra-compact objects with or without a horizon [6]. While there is a considerable diversity of opinions on what exactly constitutes a black hole, the presence of a trapped region - a domain of spacetime from which nothing can escape - is its most commonly accepted characteristic [7]. A trapped spacetime region that is externally bounded by an apparent horizon is referred to as physical black hole (PBH) [8]. A PBH may contain other features of black hole solutions of classical GR, such as an event horizon or singularity, or it may be a singularity-free regular black hole. To be of physical relevance, the apparent horizon must form in finite time according to a distant observer [9].</text>
<text><location><page_1><loc_9><loc_16><loc_49><loc_27></location>It is commonly accepted that curvature invariants, such as the Ricci and Kretschmann scalar, are finite at the apparent horizon. When expressed mathematically, the requirements of regularity and finite formation time provide the basis for a self-consistent analysis of black holes. In spherical symmetry (to which we restrict our considerations here), this allows for a comprehensive classification of the near-horizon geometries. There are only two classes of solutions labeled</text>
<text><location><page_1><loc_52><loc_63><loc_92><loc_73></location>by k = 0 and k = 1 , where the value of k reflects the scaling behavior of particular functions of the components of the energy-momentum tensor (EMT) near the apparent horizon. The properties of the near-horizon geometry lead to the identification of a unique scenario for black hole formation [9, 10] that involves both types of PBH solutions. We summarize its main results in Sec. III.</text>
<text><location><page_1><loc_52><loc_50><loc_92><loc_63></location>Understanding the true nature of the observed ultracompact objects requires detailed knowledge of the black hole models, their alternatives, as well as the observational signatures of both classes of solutions in GR and extended theories of gravity [6, 11]. Vacuum black hole solutions exist in a variety of MTG [1, 2, 12]. On the other hand, these theories are also used to construct models of horizonless ultra-compact objects. A generic property among some of them is the absence of horizon formation in the final stage of the collapse [13].</text>
<text><location><page_1><loc_52><loc_27><loc_92><loc_50></location>Even the simplest MTG require perturbative treatment due to the mathematical complexity inherent to the higher-order nature of the equations [2, 14, 15]. We briefly review the relevant formalism and its relationship to the self-consistent approach in Sec. II. In Sec. IV, we derive a set of conditions necessary for the existence of a PBH in an arbitrary metric MTG. The solutions are presented as expansions in the coordinate distance from the apparent horizon and do not require a GR solution as the zeroth-order perturbative solution of a MTG. Using the Starobinsky model [2, 16] (Sec. V) we demonstrate the application of the general results, illustrating the well-known features of matching solutions of systems of partial differential equations of different orders [2, 14, 15]: we find that the two classes of GR solutions can be regarded as zeroth-order perturbative solutions of this MTG, and identify a MTG solution without a well-defined GR limit.</text>
<section_header_level_1><location><page_1><loc_55><loc_21><loc_89><loc_23></location>II. MODIFIED GRAVITY FIELD EQUATIONS IN SPHERICAL SYMMETRY</section_header_level_1>
<section_header_level_1><location><page_1><loc_63><loc_18><loc_80><loc_19></location>A. General considerations</section_header_level_1>
<text><location><page_1><loc_52><loc_8><loc_92><loc_15></location>We work in the framework of semiclassical gravity, use classical notions (e.g. metric, horizons, trajectories), and describe dynamics via the modified Einstein equations. We do not make any assumptions about the underlying reason for modifications of the bulk part of the gravitational Lagrangian</text>
<text><location><page_2><loc_9><loc_89><loc_49><loc_93></location>density, but organize it according to powers of derivatives of the metric as commonly done in effective field theories [3, 4, 17], i.e.</text>
<formula><location><page_2><loc_9><loc_80><loc_49><loc_88></location>L g √ -g = M 2 P 16 π ( R + λ F ( g µν , R µνρσ ) ) = M 2 P 16 π R + a 1 R µν R µν + a 2 R 2 + a 3 R µνρσ R µνρσ + (1)</formula>
<text><location><page_2><loc_9><loc_67><loc_49><loc_79></location>where M P is the Planck mass that we set to one in what follows, the cosmological constant was omitted, and the coefficients a 1 , a 2 , a 3 are dimensionless. The dimensionless parameter λ is used to organize the perturbative analysis and set to one at the end of the calculations. Many popular models belong to the class of f ( R ) theories, where L g √ -g = f ( R ) . The prototypical example is the Starobinsky model with F = ςR 2 , ς = 16 πa 2 /M 2 P .</text>
<text><location><page_2><loc_10><loc_66><loc_37><loc_67></location>Varying the gravitational action results in</text>
<formula><location><page_2><loc_21><loc_63><loc_49><loc_64></location>G µν + λ E µν = 8 πT µν , (2)</formula>
<text><location><page_2><loc_9><loc_55><loc_49><loc_61></location>where G µν is the Einstein tensor, the terms E µν result from the variation of F ( g µν , R µνρσ ) , and T µν ≡ 〈 ˆ T µν 〉 ω denotes the expectation value of the renormalized EMT. We do not make any specific assumptions about the state ω .</text>
<text><location><page_2><loc_9><loc_47><loc_49><loc_55></location>In fact, apart from imposing spherical symmetry, we assume only that (i) an apparent horizon is formed in finite time of a distant observer; (ii) it is regular, i.e. the scalars T := T µ µ = R/ 8 π + O ( λ ) and T := T µν T µν = R µν R µν / 64 π 2 + O ( λ 2 ) are finite at the horizon.</text>
<text><location><page_2><loc_9><loc_45><loc_49><loc_47></location>A general spherically symmetric metric in Schwarzschild coordinates is given by</text>
<formula><location><page_2><loc_11><loc_41><loc_49><loc_43></location>ds 2 = -e 2 h ( t,r ) f ( t, r ) dt 2 + f ( t, r ) -1 dr 2 + r 2 d Ω , (3)</formula>
<text><location><page_2><loc_9><loc_37><loc_49><loc_40></location>where r denotes the areal radius. The Misner-Sharp mass [18, 19] C ( t, r ) is invariantly defined via</text>
<formula><location><page_2><loc_20><loc_34><loc_49><loc_36></location>1 -C ( t, r ) /r := ∂ µ r∂ µ r, (4)</formula>
<text><location><page_2><loc_9><loc_23><loc_49><loc_33></location>and thus the function f ( t, r ) = 1 -C ( t, r ) /r is invariant under general coordinate transformations. For a Schwarzschild black hole C = 2 M . We use the definition of Eq. (4) for consistency with the description of solutions in higherdimensional versions of GR. The apparent horizon is located at the Schwarzschild radius r g ( t ) that is the largest root of f ( t, r ) = 0 [19].</text>
<text><location><page_2><loc_10><loc_22><loc_47><loc_23></location>The Misner-Sharp mass of a PBH can be represented as</text>
<formula><location><page_2><loc_20><loc_18><loc_49><loc_20></location>C = r g ( t ) + W ( t, r -r g ) , (5)</formula>
<text><location><page_2><loc_9><loc_16><loc_43><loc_17></location>where the definition of the apparent horizon implies</text>
<formula><location><page_2><loc_19><loc_13><loc_49><loc_14></location>W ( t, 0) = 0 , W ( t, x ) < x, (6)</formula>
<text><location><page_2><loc_9><loc_9><loc_49><loc_11></location>and x := r -r g is the coordinate distance from the apparent horizon.</text>
<text><location><page_2><loc_50><loc_81><loc_52><loc_83></location>· · ·</text>
<text><location><page_2><loc_52><loc_82><loc_53><loc_83></location>,</text>
<text><location><page_2><loc_53><loc_92><loc_84><loc_93></location>The modified Einstein equations take the form</text>
<formula><location><page_2><loc_58><loc_89><loc_92><loc_91></location>fr -2 e 2 h ∂ r C + λ E tt = 8 πT tt , (7)</formula>
<formula><location><page_2><loc_58><loc_87><loc_92><loc_89></location>r -2 ∂ t C + λ E r t = 8 πT r t , (8)</formula>
<formula><location><page_2><loc_58><loc_84><loc_92><loc_87></location>2 f 2 r -1 ∂ r h -fr -2 ∂ r C + λ E rr = 8 πT rr . (9)</formula>
<text><location><page_2><loc_53><loc_83><loc_62><loc_84></location>The notation</text>
<formula><location><page_2><loc_55><loc_80><loc_92><loc_81></location>τ t := e -2 h T tt , τ r t := e -h T r t , τ r := T rr (10)</formula>
<text><location><page_2><loc_52><loc_77><loc_89><loc_79></location>is useful in dealing with equations in both GR and MTG.</text>
<text><location><page_2><loc_52><loc_70><loc_92><loc_77></location>Regularity of the apparent horizon is expressed as a set of conditions on the potentially divergent parts of the scalars T and T . In spherical symmetry T θ θ ≡ T φ φ and we assume that it is finite as in GR [9]. The constraints can therefore be represented mathematically as</text>
<formula><location><page_2><loc_54><loc_67><loc_92><loc_69></location>T = ( τ r -τ t ) /f → g 1 ( t ) f k 1 , (11)</formula>
<text><location><page_2><loc_52><loc_58><loc_92><loc_64></location>for some g 1 , 2 ( t ) and k 1 , 2 /greaterorequalslant 0 . There are a priori infinitely many solutions that satisfy these constraints. After reviewing the special case of GR and presenting the two admissible solutions we discuss this behavior in Sec. IV.</text>
<formula><location><page_2><loc_54><loc_63><loc_92><loc_67></location>T = ( ( τ t ) 2 -2( τ r t ) 2 +( τ r ) 2 ) /f 2 → g 2 ( t ) f k 2 , (12)</formula>
<text><location><page_2><loc_52><loc_51><loc_92><loc_58></location>Many useful results can be obtained by means of comparison of various quantities written in Schwarzschild coordinates ( t, r ) with their counterpart expressions written using the ingoing v or outgoing u null coordinate and the same areal radius r . Using ( v, r ) coordinates,</text>
<formula><location><page_2><loc_63><loc_48><loc_92><loc_50></location>dt = e -h ( e h + dv -f -1 dr ) , (13)</formula>
<text><location><page_2><loc_52><loc_45><loc_92><loc_47></location>is particularly fruitful. EMT components in ( v, r ) and ( t, r ) coordinates are related via</text>
<formula><location><page_2><loc_60><loc_42><loc_92><loc_43></location>θ v := e -2 h + Θ vv = τ t , (14)</formula>
<formula><location><page_2><loc_60><loc_39><loc_92><loc_41></location>θ vr := e -h + Θ vr = ( τ r t -τ t ) /f, (15)</formula>
<formula><location><page_2><loc_60><loc_37><loc_92><loc_39></location>θ r := Θ rr = ( τ r + τ t -2 τ r t ) /f 2 , (16)</formula>
<text><location><page_2><loc_52><loc_35><loc_89><loc_36></location>where Θ µν labels EMT components in ( v, r ) coordinates.</text>
<section_header_level_1><location><page_2><loc_63><loc_31><loc_81><loc_32></location>B. Perturbative expansion</section_header_level_1>
<text><location><page_2><loc_52><loc_26><loc_92><loc_29></location>From a formal perspective the pure GR case can be described as a system of field equations [20]</text>
<formula><location><page_2><loc_68><loc_24><loc_92><loc_25></location>E (¯ g , ¯ T ) = 0 , (17)</formula>
<text><location><page_2><loc_52><loc_18><loc_92><loc_23></location>where the EMT ¯ T and metric ¯ gnear the apparent horizon are described in a spherically symmetric setting in Sec. III. It is then usually assumed that any solution</text>
<formula><location><page_2><loc_67><loc_15><loc_92><loc_17></location>E λ ( g λ , T λ ) = 0 (18)</formula>
<text><location><page_2><loc_52><loc_9><loc_92><loc_14></location>of the MTG belongs to a one-parameter family of analytic solutions [14, 15]. The EMT T λ depends on λ through the metric g λ , and potentially also through effective corrections resulting from perturbative corrections to the modified field</text>
<text><location><page_3><loc_9><loc_83><loc_49><loc_93></location>equations Eqs. (7)-(9). The self-consistent approach is based on the assumption of at least continuity of the curvature invariants, but uses the Schwarzschild coordinate system where the metric is discontinuous [9, 10]. Imposing the requirement of regularity then allows to identify the valid black hole solutions, whose analytic properties become apparent once they are written in their 'natural' coordinate system [21].</text>
<text><location><page_3><loc_9><loc_72><loc_49><loc_83></location>The field equations are supplemented by a set of initial and boundary conditions or constraints. Higher-order terms in the action lead to higher-order equations. Even f ( R ) theories already result in systems with fourth-order metric derivatives. However, it is worth pointing out that the unperturbed solution may not satisfy the boundary conditions since its corresponding equations do not involve the higher-order derivatives [15, 22].</text>
<text><location><page_3><loc_9><loc_57><loc_49><loc_71></location>For our purposes it suffices to restrict all considerations to first-order perturbation theory. In any given theory higherorder contributions can be successfully evaluated. There are methods to produce a consistent hierarchy of the higher-order terms and deal with additional degrees of freedom that result from the presence of derivatives of order higher than two. Nevertheless, including terms of order O ( λ 2 ) and higher may not be justified without detailed knowledge of the relative importance of all possible terms in the effective Lagrangian and the cut-off scale that is used to derive it.</text>
<text><location><page_3><loc_9><loc_50><loc_49><loc_57></location>Spherical symmetry prescribes the form of the metric for all values of λ . We assume that there is a solution of Eq. (2) with the two metric functions C λ and h λ . To avoid spurious divergences we use the physical value of r g ( t ) that corresponds to the perturbed metric g λ , C λ ( r g , t ) = r g. We set</text>
<formula><location><page_3><loc_17><loc_47><loc_49><loc_49></location>C λ =: r g ( t ) + ¯ W ( t, r ) + λ Σ( t, r ) , (19)</formula>
<formula><location><page_3><loc_17><loc_45><loc_49><loc_47></location>h λ =: ¯ h ( t, r ) + λ Ω( t, r ) , (20)</formula>
<text><location><page_3><loc_9><loc_41><loc_49><loc_44></location>and define ¯ C := r g + ¯ W . Similarly, the EMT T λ ≡ T is decomposed as</text>
<formula><location><page_3><loc_22><loc_38><loc_49><loc_40></location>T µν =: ¯ T µν + λ ˜ T, (21)</formula>
<text><location><page_3><loc_9><loc_33><loc_49><loc_37></location>where ¯ T is extracted from E ( ¯ g [ r g , ¯ W, ¯ h ] , ¯ T ) = 0 . The perturbative terms must satisfy the boundary conditions</text>
<formula><location><page_3><loc_20><loc_30><loc_49><loc_31></location>Σ( t, 0) = 0 , (22)</formula>
<formula><location><page_3><loc_19><loc_27><loc_49><loc_30></location>lim r → r g Ω( t, r ) / ¯ h ( t, r ) = O (1) , (23)</formula>
<text><location><page_3><loc_9><loc_19><loc_49><loc_26></location>where the first condition follows from the definition of the Schwarzschild radius, and the perturbation can be treated as small only if the divergence of Ω is not stronger than that of ¯ h . Substituting C λ and h λ into Eq. (2) and keeping only the first-order terms in λ results in</text>
<formula><location><page_3><loc_13><loc_14><loc_49><loc_18></location>¯ G µν + λ ˜ G µν + λ ¯ E µν = 8 π ( ¯ T µν + λ ˜ T µν ) , (24)</formula>
<text><location><page_3><loc_9><loc_9><loc_49><loc_15></location>where ¯ G µν ≡ G µν [ r g , ¯ W, ¯ h ] , ˜ G µν is the first-order term in the Taylor expansion in λ where each monomial involves either Σ or Ω , and ¯ E µν ≡ E µν [ r g , ¯ W, ¯ h ] , i.e. the modified gravity terms are functions of the unperturbed solutions.</text>
<text><location><page_3><loc_52><loc_90><loc_92><loc_93></location>The explicit form of the equations can be obtained as follows. First note that</text>
<formula><location><page_3><loc_61><loc_87><loc_92><loc_90></location>e 2 h = e 2 ¯ h (1 + 2 λ Ω) + O ( λ 2 ) . (25)</formula>
<text><location><page_3><loc_52><loc_84><loc_92><loc_87></location>We introduce the splitting τ = ¯ τ + λ ˜ τ such that, for instance, the EMT terms of the tt equation can be written as</text>
<formula><location><page_3><loc_54><loc_81><loc_92><loc_83></location>¯ T tt + λ ˜ T tt = e 2 ¯ h (1 + 2 λ Ω)(¯ τ t + λ ˜ τ t ) (26)</formula>
<formula><location><page_3><loc_62><loc_77><loc_92><loc_81></location>= e 2 ¯ h ( ¯ τ t + λ (2Ω¯ τ t + ˜ τ t ) ) + O ( λ 2 ) , (27)</formula>
<text><location><page_3><loc_52><loc_72><loc_92><loc_78></location>with T r t and T rr expanded analogously. The regularity conditions Eqs. (11) and (12) imply that ˜ τ terms should either have the same behavior as their ¯ τ counterparts when r → r g, or go to zero faster.</text>
<text><location><page_3><loc_53><loc_71><loc_85><loc_72></location>Consequently, the schematic of Eq. (24) implies</text>
<formula><location><page_3><loc_54><loc_65><loc_92><loc_70></location>¯ G tt = e 2 ¯ h r 3 ( r -¯ C ) ∂ r ¯ C, (28)</formula>
<text><location><page_3><loc_52><loc_61><loc_77><loc_62></location>and thus the explicit form of Eq. (7) is</text>
<formula><location><page_3><loc_54><loc_61><loc_92><loc_66></location>˜ G tt = e 2 ¯ h r 3 [ -Σ ∂ r ¯ C + ( r -¯ C ) ( 2Ω ∂ r ¯ C + ∂ r Σ )] , (29)</formula>
<formula><location><page_3><loc_52><loc_55><loc_92><loc_60></location>-Σ ∂ r ¯ C + ( r -¯ C ) ∂ r Σ+ r 3 e -2 ¯ h ¯ E tt = 8 πr 3 ˜ τ t . (30) Similarly, Eqs. (8) and (9) can be written explicitly as</formula>
<formula><location><page_3><loc_57><loc_52><loc_92><loc_54></location>∂ t Σ+ r 2 ¯ E r t = 8 πr 2 e ¯ h (Ω¯ τ r t + ˜ τ r t ) , (31)</formula>
<formula><location><page_3><loc_57><loc_51><loc_79><loc_52></location>Σ ∂ r ¯ C ( r ¯ C )(4Σ ∂ r ¯ h + ∂ r Σ)</formula>
<formula><location><page_3><loc_62><loc_48><loc_92><loc_52></location>--+2( r -¯ C ) 2 ∂ r Ω+ r 3 ¯ E rr = 8 πr 3 ˜ τ r . (32)</formula>
<section_header_level_1><location><page_3><loc_57><loc_45><loc_87><loc_46></location>III. SELF-CONSISTENT SOLUTIONS IN GR</section_header_level_1>
<text><location><page_3><loc_52><loc_33><loc_92><loc_42></location>Here we give a brief summary of the relevant properties of the self-consistent solutions in GR [9, 10, 21]. In accord with the previous section (and in anticipation of the notation we use in Sec. IV), we label functions of pure classical GR (i.e. λ = 0 ) with a bar, e.g. the metric functions ¯ C and ¯ h . The Einstein field equations for ¯ G tt , ¯ G r t , and ¯ G rr are expressed in terms of the metric functions ¯ C and ¯ h as follows:</text>
<formula><location><page_3><loc_63><loc_30><loc_92><loc_32></location>∂ r ¯ C = 8 πr 2 ¯ τ t / ¯ f, (33)</formula>
<formula><location><page_3><loc_63><loc_28><loc_92><loc_30></location>∂ t ¯ C = 8 πr 2 e ¯ h ¯ τ r t , (34)</formula>
<formula><location><page_3><loc_63><loc_26><loc_92><loc_27></location>∂ r ¯ h = 4 πr (¯ τ t + ¯ τ r ) / ¯ f 2 . (35)</formula>
<text><location><page_3><loc_52><loc_20><loc_92><loc_24></location>Only two distinct classes of dynamic solutions are possible [21]. With respect to the regularity conditions of Eqs. (11) and (12), they correspond to the values k = 0 and k = 1 .</text>
<section_header_level_1><location><page_3><loc_63><loc_16><loc_81><loc_17></location>A. k = 0 class of solutions</section_header_level_1>
<text><location><page_3><loc_52><loc_12><loc_92><loc_14></location>In the k = 0 class of solutions, the limiting form of the reduced EMT components is given by</text>
<formula><location><page_3><loc_53><loc_8><loc_92><loc_11></location>¯ τ t →-¯ Υ 2 ( t ) , ¯ τ r →-¯ Υ 2 ( t ) , ¯ τ r t →± ¯ Υ 2 ( t ) , (36)</formula>
<text><location><page_4><loc_9><loc_90><loc_49><loc_93></location>for some function ¯ Υ( t ) . The leading terms of the metric functions are</text>
<formula><location><page_4><loc_18><loc_87><loc_49><loc_90></location>¯ C = r g -4 πr 3 / 2 g ¯ Υ √ x + O ( x ) , (37)</formula>
<formula><location><page_4><loc_19><loc_84><loc_49><loc_87></location>¯ h = -1 2 ln x ¯ ξ + O ( √ x ) , (38)</formula>
<text><location><page_4><loc_9><loc_77><loc_49><loc_83></location>where ¯ ξ ( t ) is determined by the asymptotic properties of the solution. Higher-order terms depend on the higher-order terms in the EMT expansion and will be discussed in Sec. IV. Consistency of the Einstein equations implies</text>
<formula><location><page_4><loc_23><loc_72><loc_49><loc_76></location>r ' g = ± 4 ¯ Υ √ πr g ¯ ξ. (39)</formula>
<text><location><page_4><loc_9><loc_67><loc_49><loc_73></location>The null energy condition requires T µν l µ l ν /greaterorequalslant 0 for all null vectors l µ [20, 23]. It is violated by radial vectors l ˆ a = (1 , ∓ 1 , 0 , 0) for both the evaporating and accreting solutions, respectively.</text>
<text><location><page_4><loc_9><loc_54><loc_49><loc_67></location>The accreting solution r ' g ( t ) > 0 leads to a firewall: energy density, pressure and flux experienced by an infalling observer diverge at the apparent horizon [21]. The resulting averaged negative energy density in the reference frame of a geodesic observer violates a particular quantum energy inequality [23, 24]. Unless we accept that semiclassical physics breaks down already at the horizon scale, this contradiction implies that a PBH cannot grow after its formation [21]. Hence we consider only evaporating r ' g ( t ) < 0 PBHs in what follows.</text>
<text><location><page_4><loc_9><loc_48><loc_49><loc_54></location>Matching our results with the standard semiclassical results on black hole evaporation (and accepting that the metric is sufficiently close to the ingoing Vaidya metric with decreasing mass, see [25] for details) results in</text>
<formula><location><page_4><loc_26><loc_44><loc_49><loc_48></location>¯ ξ ∼ α r g , (40)</formula>
<text><location><page_4><loc_9><loc_37><loc_49><loc_43></location>where the black hole evaporates according to r ' g ( t ) = -α/r 2 g [20, 26]. Outside of the apparent horizon the geometry differs from the Schwarzschild metric at least on the scale r -r g =: x ∼ ¯ ξ .</text>
<section_header_level_1><location><page_4><loc_23><loc_34><loc_35><loc_35></location>B. k = 1 solution</section_header_level_1>
<text><location><page_4><loc_9><loc_21><loc_49><loc_31></location>In the second class of solutions k = 1 and the limiting form of the EMT expansion is given by functions ¯ τ a ∝ ¯ f . Again, accretion leads to a firewall and thus we will consider only evaporating solutions. It has been shown that dynamic solutions are consistent only in a single case [10], where in the Schwarzschild frame the energy density ρ ( r g ) = ¯ E and pressure p ( r g ) = ¯ P at the apparent horizon are given by</text>
<formula><location><page_4><loc_21><loc_18><loc_49><loc_20></location>¯ E = -¯ P = 1 / (8 πr 2 g ) . (41)</formula>
<text><location><page_4><loc_9><loc_15><loc_49><loc_18></location>Since this is their maximal possible value this k = 1 solution is referred to as extreme [10]. The k = 1 metric functions are</text>
<formula><location><page_4><loc_20><loc_11><loc_49><loc_14></location>¯ C = r -c 32 x 3 / 2 + O ( x 2 ) , (42)</formula>
<formula><location><page_4><loc_20><loc_9><loc_49><loc_12></location>¯ h = -3 2 ln x ¯ ξ + O ( √ x ) . (43)</formula>
<text><location><page_4><loc_52><loc_92><loc_85><loc_93></location>Consistency of the Einstein equations then implies</text>
<formula><location><page_4><loc_66><loc_88><loc_92><loc_91></location>r ' g = -c 32 ¯ ξ 3 / 2 /r g . (44)</formula>
<text><location><page_4><loc_52><loc_85><loc_92><loc_88></location>For future reference we note here that for the k = 1 solution the Ricci scalar is given by</text>
<formula><location><page_4><loc_66><loc_81><loc_92><loc_84></location>¯ R = 2 /r 2 g + O ( x ) . (45)</formula>
<text><location><page_4><loc_52><loc_64><loc_92><loc_81></location>Evaporating black holes are conveniently represented in ( v, r ) coordinates, and the limiting form of the k = 0 solution as r → r g is a Vaidya metric with decreasing Misner-Sharp mass C + ( v ) ' < 0 [25]. Using ( v, r ) coordinates to describe geometry at the formation of the first marginally trapped surface reveals how the two classes of solutions are connected (see Ref. [10] for details): at its formation, a PBH is described by a k = 1 solution with ¯ E = -¯ P = 1 / (8 πr 2 g ) . It immediately switches to the k = 0 solution. However, the abrupt transition from f 1 to f 0 behavior does not lead to discontinuities in the curvature scalars or other physical quantities that could potentially be measured by a local or quasilocal observer.</text>
<section_header_level_1><location><page_4><loc_56><loc_60><loc_88><loc_61></location>IV. SELF-CONSISTENT SOLUTIONS IN MTG</section_header_level_1>
<text><location><page_4><loc_52><loc_39><loc_92><loc_58></location>To describe perturbative PBH solutions in MTG the equations must satisfy the same consistency relations as their GR counterparts. Taking the GR solutions as the zeroth-order approximation, we express the functions describing the MTG metric g λ = ¯ g + λ ˜ g and thus represent the modified Einstein equations as series in integer and half-integer powers of x := r -r g. Their order-by-order solution results in formal expressions for Σ( t, r ) and Ω( t, r ) . However, we also obtain a number of consistency conditions that must be satisfied identically in order for a given theory to admit formation of a PBH. The GR solutions with k ∈ { 0 , 1 } are sufficiently different to merit a separate treatment provided in Subsec. IV A and IV B, respectively.</text>
<text><location><page_4><loc_52><loc_30><loc_92><loc_39></location>In both instances, power expansions in various expression have to match up to allow for self-consistent solutions of the modified Einstein equations. Moreover, the relations between the EMT components that are given by Eqs. (14)-(16) must hold separately for both the unperturbed terms and the perturbations.</text>
<text><location><page_4><loc_52><loc_9><loc_92><loc_30></location>For a given MTG (that is defined by the set of parameters { a 1 , a 2 , a 3 , · · · } in Eq. (1)) these constraints may conceivably lead to several outcomes: first, it is possible that some of the terms in the Lagrangian Eq. (1) contribute terms to ¯ E µν such that their expansions around x = 0 lead to terms that diverge stronger than any other terms in Eqs. (30)-(32). If only one higher-order curvature term is responsible for such behavior, then such a theory cannot produce perturbative PBH solutions, and only nonperturbative solutions may be possible or the corresponding coefficient a i ≡ 0 . If the divergences originate from several terms, they can either cancel if a particular relationship exists between their coefficients a i , a i ' , . . . , or not. In the former case the existence of perturbative PBH solutions imposes a constraint, not on the form of the available terms, but on the relationships between their coefficients.</text>
<text><location><page_5><loc_9><loc_76><loc_49><loc_93></location>It is also possible that, as it happens in the Starobinsky model (Sec. V), divergences of the terms ¯ E µν match the divergences of the GR terms. The constraints can then be satisfied (i) identically (providing us with no additional information); (ii) only for a particular combination of the coefficients a i , thereby constraining the possible classes of MTG; (iii) only in the presence of particular higher-order terms, irrespective of the coefficients, and only for certain unperturbed solutions. In the last scenario, where only certain GR solutions are consistent with a small perturbation, this should be interpreted as an argument against the presence of that particular term in the Lagrangian of Eq. (1).</text>
<text><location><page_5><loc_9><loc_66><loc_49><loc_76></location>There is a priori no reason why ¯ g µν /greatermuch λ ˜ g µν should hold in some boundary layer around r g [15, 22]. If this condition is not satisfied, then the classification scheme of the GR solutions and a mandatory violation of the null energy condition are not necessarily true. We discuss some of the properties of the solutions without a GR limit and derive the necessary conditions for their existence in Sec. IV C.</text>
<text><location><page_5><loc_9><loc_56><loc_49><loc_66></location>Throughout this section we use the letter j ∈ Z 1 2 to label integer and half-integer coefficients and powers of x in series expansions and /lscript to refer to generic coefficients. Since we give explicit expressions only for the first few terms in each expression, we write c 12 instead of c 1 / 2 , h 12 instead of h 1 / 2 , and similarly for higher orders and coefficients of the EMT expansion.</text>
<section_header_level_1><location><page_5><loc_18><loc_51><loc_40><loc_52></location>A. Black holes of the k = 0 type</section_header_level_1>
<text><location><page_5><loc_9><loc_44><loc_49><loc_49></location>For the k = 0 class of solutions the leading terms in the metric functions of classical GR are given as series in powers of x := r -r g as</text>
<formula><location><page_5><loc_17><loc_39><loc_38><loc_44></location>¯ C = r g -c 12 √ x + ∞ ∑ 1 /lessorequalslant j ∈ Z 1 2 c j x j</formula>
<formula><location><page_5><loc_17><loc_29><loc_49><loc_41></location>= r g -c 12 √ x + c 1 x + O ( x 3 / 2 ) , (46) ¯ h = -1 2 ln x ¯ ξ + ∞ ∑ 1 2 /lessorequalslant j ∈ Z 1 2 h j x j = -1 2 ln x ¯ ξ + h 12 √ x + O ( x ) , (47)</formula>
<text><location><page_5><loc_9><loc_27><loc_13><loc_28></location>where</text>
<formula><location><page_5><loc_12><loc_22><loc_49><loc_26></location>c 12 = 4 √ πr 3 / 2 g ¯ Υ , c 1 = 1 3 + 4 √ πr 3 / 2 g (¯ τ t ) 12 3 ¯ Υ , (48)</formula>
<formula><location><page_5><loc_12><loc_18><loc_49><loc_23></location>h 12 = 2 ¯ Υ+ √ πr 3 / 2 g (3(¯ τ r ) 12 -(¯ τ t ) 12 ) 6 √ πr 3 / 2 g ¯ Υ 2 , (49)</formula>
<text><location><page_5><loc_9><loc_14><loc_49><loc_17></location>and higher-order coefficients of the metric functions are related to higher-order terms in the EMT expansion</text>
<formula><location><page_5><loc_19><loc_8><loc_49><loc_13></location>¯ τ a = -¯ Υ 2 + ∞ ∑ 1 2 /lessorequalslant j ∈ Z 1 2 (¯ τ a ) j x j , (50)</formula>
<text><location><page_5><loc_52><loc_90><loc_92><loc_93></location>where a ∈ { t , r t , r } ≡ { tt , r t , rr } . We omit the explicit specification j ∈ Z 1 2 from the summation range in what follows.</text>
<text><location><page_5><loc_52><loc_85><loc_92><loc_90></location>Regularity of the metric at the apparent horizon and consistency of the Einstein equations establish algebraic and differential relations between various coefficients. In particular, using Eqs. (14)-(16) and Eq. (34), we find</text>
<formula><location><page_5><loc_63><loc_82><loc_92><loc_84></location>(¯ τ t ) 12 +(¯ τ r ) 12 = 2(¯ τ r t ) 12 , (51)</formula>
<text><location><page_5><loc_52><loc_80><loc_54><loc_81></location>and</text>
<formula><location><page_5><loc_66><loc_75><loc_92><loc_78></location>r ' g = -4 ¯ Υ √ πr g ¯ ξ. (52)</formula>
<text><location><page_5><loc_52><loc_71><loc_92><loc_75></location>The expansion of e 2 h that is given by Eq. (25) is obtained as follows: separating the logarithmically divergent part of h ( t, x ) from the rest, Eq. (23) allows to write</text>
<formula><location><page_5><loc_64><loc_66><loc_92><loc_70></location>e 2 h = ¯ ξ + λ ˜ ξ x e 2¯ χ +2 λω (53)</formula>
<text><location><page_5><loc_52><loc_61><loc_92><loc_65></location>for some ˜ ξ ( t ) , where ¯ χ = ∑ j h j x j and ω = ∑ j ω j x j are convergent functions. First-order expansion in λ then leads to Eq. (25) with</text>
<formula><location><page_5><loc_67><loc_56><loc_92><loc_60></location>Ω = ˜ ξ 2 ¯ ξ + ω. (54)</formula>
<text><location><page_5><loc_52><loc_53><loc_92><loc_55></location>Therefore, the first-order corrections of Eqs. (19)-(20) to the metric functions of Eqs. (46)-(47) are given by the series</text>
<formula><location><page_5><loc_55><loc_47><loc_92><loc_52></location>Σ = ∞ ∑ j /greaterorequalslant 1 2 σ j x j = σ 12 x 1 / 2 + σ 1 x + O ( x 3 / 2 ) , (55)</formula>
<formula><location><page_5><loc_55><loc_42><loc_92><loc_47></location>Ω = ˜ ξ 2 ¯ ξ + ∞ ∑ j /greaterorequalslant 1 2 ω j x j = ˜ ξ 2 ¯ ξ + ω 12 x 1 / 2 + O ( x ) . (56)</formula>
<text><location><page_5><loc_52><loc_29><loc_92><loc_41></location>These two functions can be expressed in terms of the unperturbed solution and corrections ˜ τ a to the EMT from the series expansion of Eqs. (30)-(32). These equations contain various divergent expressions. For example, the term Σ ∂ r ¯ C as well as all other terms apart from e -2 ¯ h ¯ E tt in Eq. (30) are finite when x → 0 . Then Eq. (47) implies that the series expansion of ¯ E tt starts with a term that is proportional to 1 /x . Performing the same analysis for the two remaining Einstein equations Eqs. (31)-(32) yields the decompositions</text>
<formula><location><page_5><loc_59><loc_23><loc_92><loc_28></location>¯ E tt = æ ¯ 1 x + æ 12 √ x +æ 0 x 0 + ∞ ∑ j /greaterorequalslant 1 2 æ j x j , (57)</formula>
<formula><location><page_5><loc_59><loc_18><loc_92><loc_23></location>¯ E r t = œ 12 √ x +œ 0 x 0 + ∞ ∑ j /greaterorequalslant 1 2 œ j x j , (58)</formula>
<formula><location><page_5><loc_59><loc_13><loc_92><loc_18></location>¯ E rr = ø 0 + ∞ ∑ j /greaterorequalslant 1 2 ø j x j , (59)</formula>
<text><location><page_5><loc_52><loc_9><loc_92><loc_13></location>of the modified gravity terms that should hold for any F ( g µν , R µνρσ ) , where indices of coefficients of negative exponents of x are labeled by a bar.</text>
<text><location><page_6><loc_9><loc_90><loc_49><loc_93></location>From the requirement that the Ricci scalar R [ g λ ] be finite at the horizon, we obtain the condition</text>
<formula><location><page_6><loc_18><loc_86><loc_49><loc_90></location>σ 12 | R = ˜ ξ 2 ¯ ξ c 12 = 2 √ πr 3 / 2 g ˜ ξ ¯ Υ ¯ ξ . (60)</formula>
<text><location><page_6><loc_9><loc_81><loc_49><loc_85></location>We use additional subscripts (e.g. ' | R ' in the expression above) to indicate what equation was used to derive the explicit expression.</text>
<text><location><page_6><loc_9><loc_68><loc_49><loc_81></location>The perturbative contributions Eqs. (55)-(56) to the metric functions Eqs. (46)-(47) are obtained order-by-order from the series solutions of Eqs. (30)-(32). Expressions for every expansion coefficient can be obtained separately from each equation. Matching of the expressions then allows to identify the coefficients æ /lscript , œ /lscript , ø /lscript of the modified gravity terms Eqs. (57)-(59). Expressions for σ 12 for instance are obtained from the lowest-order coefficients of Eqs. (30)-(32). As a result, we obtain three independent constraints</text>
<formula><location><page_6><loc_18><loc_65><loc_49><loc_67></location>σ 12 | R = σ 12 | tt = σ 12 | tr = σ 12 | rr . (61)</formula>
<text><location><page_6><loc_9><loc_63><loc_45><loc_64></location>They are simultaneously satisfied (see Appendix A 1) if</text>
<formula><location><page_6><loc_10><loc_56><loc_49><loc_62></location>æ ¯ 1 = -8 π ˜ ξ ¯ Υ 2 , œ 12 = -8 π ˜ ξ ¯ Υ 2 √ ¯ ξ , ø 0 = -8 π ˜ ξ ¯ Υ 2 ¯ ξ . (62)</formula>
<text><location><page_6><loc_9><loc_53><loc_49><loc_57></location>These three equations not only identify the function ˜ ξ ( t ) in terms of unperturbed quantities, but also establish the two relations</text>
<formula><location><page_6><loc_22><loc_48><loc_49><loc_52></location>æ ¯ 1 = √ ¯ ξ œ 12 = ¯ ξ ø 0 (63)</formula>
<text><location><page_6><loc_9><loc_43><loc_49><loc_49></location>between the leading expansion coefficients of the MTG terms. Similarly, the next-highest order coefficients of Eqs. (30)-(32) allow to obtain expressions for σ 1 , see Appendix A 1. Comparison of</text>
<formula><location><page_6><loc_21><loc_41><loc_49><loc_42></location>σ 1 | tr ( ω 12 ) = σ 1 | rr ( ω 12 ) (64)</formula>
<text><location><page_6><loc_9><loc_35><loc_49><loc_40></location>gives an expression for ω 12 . Substitution into Eq. (64) and subsequent comparison of σ 1 | tr = σ 1 | rr with σ 1 | tt gives a relation between the next-highest order coefficients, namely</text>
<formula><location><page_6><loc_21><loc_31><loc_49><loc_34></location>æ 12 = 2 √ ¯ ξ œ 0 -¯ ξ ø 12 . (65)</formula>
<text><location><page_6><loc_9><loc_19><loc_49><loc_31></location>The modified gravity terms ¯ E µν must adhere to the expansion structures in Eqs. (57)-(59), and the relations Eqs. (63) and (65) between their coefficients must be satisfied identically. Otherwise the MTG solutions do not exist. For terms in the metric functions of order O ( x 3 / 2 ) and higher the three equations Eqs. (30)-(32) for the tt , tr , and rr component contain three additional independent variables (¯ τ a ) j /greaterorequalslant 3 / 2 and will therefore not lead to any additional constraints.</text>
<text><location><page_6><loc_9><loc_15><loc_49><loc_19></location>It is worth pointing out that the analogs of Eqs. (63) and (65) are also satisfied by the coefficients of the corresponding metric tensor and Ricci tensor components themselves, i.e.</text>
<formula><location><page_6><loc_12><loc_10><loc_49><loc_14></location>(¯ g tt ) ¯ 1 = √ ¯ ξ (¯ g r t ) 12 = ¯ ξ (¯ g rr ) 0 = 0 , (66)</formula>
<formula><location><page_6><loc_11><loc_7><loc_49><loc_11></location>( ¯ R tt ) ¯ 1 = √ ¯ ξ ( ¯ R r t ) 12 = ¯ ξ ( ¯ R rr ) 0 = -c 2 12 ¯ ξ/ (2 r 3 g ) , (67)</formula>
<text><location><page_6><loc_52><loc_92><loc_54><loc_93></location>and</text>
<text><location><page_6><loc_63><loc_19><loc_63><loc_20></location>2</text>
<formula><location><page_6><loc_55><loc_87><loc_92><loc_91></location>(¯ g tt ) 12 = 2 √ ¯ ξ (¯ g r t ) 0 -¯ ξ (¯ g rr ) 12 = -c 12 ¯ ξ/r g , (68)</formula>
<text><location><page_6><loc_52><loc_78><loc_92><loc_85></location>where Eq. (66) is satisfied trivially and Eq. (68) simplifies to (¯ g tt ) 12 = -¯ ξ (¯ g rr ) 12 due to the diagonal form of the metric tensor ¯ g r t = 0 (see Eq. (3)). Explicit expressions for the coefficients of the Ricci tensor components in Eq. (69) are provided in Appendix B, see Eqs. (B1)-(B3).</text>
<formula><location><page_6><loc_55><loc_84><loc_92><loc_88></location>( ¯ R tt ) 12 = 2 √ ¯ ξ ( ¯ R r t ) 0 -¯ ξ ( ¯ R rr ) 12 , (69)</formula>
<section_header_level_1><location><page_6><loc_61><loc_74><loc_83><loc_75></location>B. Black holes of the k = 1 type</section_header_level_1>
<text><location><page_6><loc_52><loc_68><loc_92><loc_72></location>The EMT expansion for the k = 1 solution is given in terms of x := r -r g by</text>
<formula><location><page_6><loc_56><loc_64><loc_92><loc_68></location>τ t = ¯ τ t + λ ˜ τ t = ¯ f ( ¯ E + λ ˜ E ) + ∑ j /greaterorequalslant 2 e j x j , (70)</formula>
<formula><location><page_6><loc_56><loc_56><loc_92><loc_60></location>τ r = ¯ τ r + λ ˜ τ r = ¯ f ( ¯ P + λ ˜ P ) + ∑ j /greaterorequalslant 2 p j x j , (72)</formula>
<formula><location><page_6><loc_55><loc_60><loc_92><loc_64></location>τ r t = ¯ τ r t + λ ˜ τ r t = ¯ f ( ¯ Φ+ λ ˜ Φ ) + ∑ j /greaterorequalslant 2 φ j x j , (71)</formula>
<text><location><page_6><loc_52><loc_47><loc_92><loc_56></location>where ¯ E = -¯ P = 1 / (8 πr 2 g ) and ¯ Φ = 0 . To improve readability and clarify the connection to physical quantities (energy, pressure, flux) we set (¯ τ t ) j =: ¯ e j , (¯ τ r ) j =: ¯ p j , and (¯ τ r t ) j =: ¯ φ j , and analogously for the perturbative coefficients (˜ τ a ) j . Additional relations between the coefficients are obtained from Eqs. (14)-(15), i.e.</text>
<formula><location><page_6><loc_60><loc_44><loc_92><loc_46></location>˜ E + ˜ P = 2 ˜ Φ , (73)</formula>
<formula><location><page_6><loc_60><loc_42><loc_92><loc_44></location>¯ e 2 = ¯ p 2 = ¯ φ 2 , ¯ e 52 + ¯ p 52 = 2 ¯ φ 52 , (74)</formula>
<formula><location><page_6><loc_60><loc_40><loc_92><loc_42></location>˜ e 2 + ˜ p 2 = 2 ˜ φ 2 , ˜ e 52 + ˜ p 52 = 2 ˜ φ 52 , (75)</formula>
<text><location><page_6><loc_52><loc_34><loc_92><loc_39></location>for the two next-highest orders j = 2 , 5 2 . Recall that (cf. Eqs. (42)-(43)) the leading terms in the metric functions of classical GR are given as series in powers of x := r -r g as</text>
<formula><location><page_6><loc_61><loc_31><loc_92><loc_34></location>¯ C = r g + x -c 32 x 3 / 2 + O ( x 2 ) , (76)</formula>
<formula><location><page_6><loc_61><loc_28><loc_92><loc_32></location>¯ h = -3 2 ln x ¯ ξ + h 12 √ x + O ( x ) , (77)</formula>
<text><location><page_6><loc_52><loc_26><loc_63><loc_27></location>with coefficients</text>
<formula><location><page_6><loc_56><loc_21><loc_92><loc_25></location>c 32 = 4 r 3 / 2 g √ -π ¯ e 2 / 3 , (78) 3 4 3¯ e 2 /π</formula>
<text><location><page_6><loc_57><loc_20><loc_58><loc_21></location>12</text>
<text><location><page_6><loc_60><loc_20><loc_63><loc_21></location>14¯</text>
<text><location><page_6><loc_62><loc_20><loc_63><loc_21></location>e</text>
<text><location><page_6><loc_64><loc_19><loc_65><loc_23></location>(</text>
<text><location><page_6><loc_66><loc_19><loc_68><loc_23></location>√</text>
<text><location><page_6><loc_68><loc_20><loc_69><loc_23></location>-</text>
<text><location><page_6><loc_68><loc_19><loc_69><loc_20></location>r</text>
<text><location><page_6><loc_69><loc_20><loc_69><loc_21></location>5</text>
<text><location><page_6><loc_69><loc_20><loc_70><loc_21></location>/</text>
<text><location><page_6><loc_70><loc_20><loc_71><loc_21></location>2</text>
<text><location><page_6><loc_69><loc_19><loc_69><loc_20></location>g</text>
<text><location><page_6><loc_79><loc_19><loc_80><loc_22></location>-</text>
<text><location><page_6><loc_83><loc_19><loc_85><loc_23></location>)</text>
<text><location><page_6><loc_52><loc_14><loc_92><loc_18></location>Higher-order coefficients are obtained from higher-order terms of the EMT expansion using consistency of the Einstein equations, e.g. the next-highest order coefficient of Eq. (76) is</text>
<formula><location><page_6><loc_61><loc_8><loc_92><loc_13></location>c 2 = 4 7 r g ( 1 + r 5 / 2 g √ 3 π ¯ e 52 √ -¯ e 2 ) . (80)</formula>
<text><location><page_6><loc_77><loc_20><loc_78><loc_21></location>52</text>
<text><location><page_6><loc_82><loc_20><loc_83><loc_21></location>52</text>
<text><location><page_6><loc_56><loc_20><loc_57><loc_22></location>h</text>
<text><location><page_6><loc_58><loc_20><loc_60><loc_22></location>=</text>
<text><location><page_6><loc_74><loc_20><loc_77><loc_22></location>+5¯</text>
<text><location><page_6><loc_76><loc_20><loc_77><loc_22></location>e</text>
<text><location><page_6><loc_80><loc_20><loc_82><loc_22></location>7¯</text>
<text><location><page_6><loc_81><loc_20><loc_82><loc_22></location>p</text>
<text><location><page_6><loc_85><loc_20><loc_85><loc_22></location>.</text>
<text><location><page_6><loc_89><loc_20><loc_92><loc_22></location>(79)</text>
<text><location><page_7><loc_9><loc_90><loc_49><loc_93></location>In addition, consistency of the Einstein equations requires Eq. (44) and</text>
<formula><location><page_7><loc_21><loc_86><loc_49><loc_90></location>¯ p 52 = 2 √ -¯ e 2 √ 3 πr 5 / 2 g + ¯ e 52 . (81)</formula>
<text><location><page_7><loc_9><loc_84><loc_46><loc_85></location>Substituting Eq. (81) into Eq. (79) we obtain the identity</text>
<formula><location><page_7><loc_25><loc_81><loc_49><loc_82></location>c 2 = c 32 h 12 , (82)</formula>
<text><location><page_7><loc_9><loc_74><loc_49><loc_80></location>which leads to many simplifying cancellations, e.g. the absence of the √ x term in the Ricci scalar ¯ R (cf. Eq. (45)) due to R 12 ∝ c 32 h 12 -c 2 , where R 12 denotes the √ x coefficient of ¯ R .</text>
<text><location><page_7><loc_9><loc_70><loc_49><loc_74></location>Again, the expansion of e 2 h that is given by Eq. (25) is obtained by separating the logarithmic part of h ( t, x ) from the rest. From the expansion</text>
<formula><location><page_7><loc_19><loc_65><loc_49><loc_69></location>e 2 h = ( ¯ ξ + λ ˜ ξ x ) 3 e 2¯ χ +2 λω , (83)</formula>
<text><location><page_7><loc_9><loc_63><loc_27><loc_64></location>we then obtain Eq. (25) with</text>
<formula><location><page_7><loc_24><loc_58><loc_49><loc_62></location>Ω = 3 ˜ ξ 2 ¯ ξ + ω. (84)</formula>
<text><location><page_7><loc_9><loc_54><loc_49><loc_57></location>The series expansions of the perturbative corrections of Eqs. (19)-(20) are therefore given by the power series</text>
<formula><location><page_7><loc_10><loc_49><loc_49><loc_53></location>Σ = ∞ ∑ j /greaterorequalslant 3 2 σ j x j = σ 32 x 3 / 2 + σ 2 x 2 + O ( x 5 / 2 ) , (85)</formula>
<formula><location><page_7><loc_10><loc_43><loc_49><loc_49></location>Ω = 3 ˜ ξ 2 ¯ ξ + ∞ ∑ j /greaterorequalslant 1 2 ω j x j = 3 ˜ ξ 2 ¯ ξ + ω 12 √ x + ω 1 x + O ( x 3 / 2 ) . (86)</formula>
<text><location><page_7><loc_9><loc_39><loc_49><loc_42></location>Finiteness of the Ricci scalar at the horizon requires Eq. (44) and</text>
<formula><location><page_7><loc_17><loc_35><loc_49><loc_39></location>σ 32 | R = 3 ˜ ξ 2 ¯ ξ c 32 = 2 r 3 / 2 g ˜ ξ √ -3 π ¯ e 2 ¯ ξ . (87)</formula>
<text><location><page_7><loc_9><loc_31><loc_49><loc_34></location>The expansion structure of the modified gravity terms ¯ E µν is obtained analogous to Sec. IV A. We find</text>
<formula><location><page_7><loc_13><loc_25><loc_49><loc_30></location>¯ E tt = æ 32 x 3 / 2 + æ ¯ 1 x + æ 12 √ x +æ 0 + ∞ ∑ j /greaterorequalslant 1 2 æ j x j , (88)</formula>
<formula><location><page_7><loc_13><loc_21><loc_49><loc_25></location>¯ E r t = œ 0 + ∞ ∑ j /greaterorequalslant 1 2 œ j x j , (89)</formula>
<formula><location><page_7><loc_13><loc_16><loc_49><loc_21></location>¯ E rr = ∞ ∑ j /greaterorequalslant 3 2 ø j x j . (90)</formula>
<text><location><page_7><loc_9><loc_13><loc_49><loc_15></location>The equation for the x 0 coefficient of the tr component Eq. (31) allows to identify</text>
<formula><location><page_7><loc_23><loc_9><loc_49><loc_12></location>˜ E | tr = œ 0 r g 8 π ¯ ξ 3 / 2 c 32 . (91)</formula>
<text><location><page_7><loc_52><loc_40><loc_54><loc_41></location>and</text>
<text><location><page_7><loc_52><loc_89><loc_92><loc_93></location>Substitution of Eq. (91) into the expression σ 32 | tt ( ˜ E ) obtained from Eq. (30) and subsequent comparison with the expression σ 32 | rr obtained from Eq. (32) establishes the relation</text>
<formula><location><page_7><loc_64><loc_85><loc_92><loc_88></location>æ 32 = 2 ¯ ξ 3 / 2 œ 0 -¯ ξ 3 ø 32 (92)</formula>
<text><location><page_7><loc_52><loc_73><loc_92><loc_84></location>between the lowest-order coefficients of the MTG terms, see Appendix A 2. Similarly, by substituting Eq. (91) into the expression for σ 2 | tt , and ˜ ξ | tr obtained from the √ x coefficient of Eq. (31) into the expression Eq. (87) for σ 32 | R , we can derive two distinct expressions for the sum ˜ e 2 +˜ p 2 by comparison of σ 2 | tt and σ 2 | rr obtained from Eqs. (30) and (32), respectively, as well as comparison of σ 32 | R and σ 32 | tt . Their identification establishes the additional relation</text>
<formula><location><page_7><loc_54><loc_69><loc_92><loc_72></location>æ ¯ 1 = 2 ¯ ξ 3 / 2 ( h 12 œ 0 +œ 12 ) -¯ ξ 3 (2 h 12 ø 32 +ø 2 ) (93)</formula>
<text><location><page_7><loc_52><loc_53><loc_92><loc_69></location>between the modified gravity coefficients of Eqs. (88)-(90). A detailed derivation with explicit expressions is provided in Appendix A 2. Analogous to the class of k = 0 black hole solutions discussed in Sec. IV A, the modified gravity terms ¯ E µν of any self-consistent MTG must follow the expansion structures prescribed by Eqs. (88)-(90) and identically satisfy the two relations Eqs. (92)-(93) to be compatible with black hole solutions of the k = 1 type. Again, consideration of higher-order coefficients in Eqs. (30)-(32) introduces new independent variables and will thus not yield any additional constraints.</text>
<text><location><page_7><loc_52><loc_47><loc_92><loc_53></location>Once more, the analogs of the MTG coefficient relations Eqs. (92)-(93) are also satisfied by the coefficients of the corresponding metric tensor and Ricci tensor components themselves, i.e.</text>
<formula><location><page_7><loc_59><loc_44><loc_92><loc_46></location>(¯ g tt ) 32 = -¯ ξ 3 (¯ g rr ) 32 = -c 32 ¯ ξ 3 /r g , (94)</formula>
<formula><location><page_7><loc_58><loc_42><loc_92><loc_44></location>( ¯ R tt ) 32 = 2 ¯ ξ 3 / 2 ( ¯ R r t ) 0 -¯ ξ 3 ( ¯ R rr ) 32 = 0 , (95)</formula>
<formula><location><page_7><loc_58><loc_34><loc_92><loc_39></location>(¯ g tt ) ¯ 1 = -¯ ξ 3 ( 2 h 12 (¯ g rr ) 32 +(¯ g rr ) 2 ) = -c 32 h 12 ¯ ξ 3 /r g , (96)</formula>
<text><location><page_7><loc_52><loc_22><loc_92><loc_27></location>where Eq. (82) was used to simplify the expressions, ( ¯ R tt ) 32 = ( ¯ R r t ) 0 = ( ¯ R rr ) 32 = 0 , ( ¯ R rr ) 2 = -3 c 2 32 / (2 r 3 g ) , and Eqs. (94) and (96) simplify due to the diagonal form of the metric tensor ¯ g r t = 0 (see Eq. (3)).</text>
<formula><location><page_7><loc_58><loc_27><loc_92><loc_35></location>( ¯ R tt ) ¯ 1 = 2 ¯ ξ 3 / 2 ( h 12 ( ¯ R r t ) 0 +( ¯ R r t ) 12 ) -¯ ξ 3 ( 2 h 12 ( ¯ R rr ) 32 +( ¯ R rr ) 2 ) = -3 c 2 32 ¯ ξ 3 / ( 2 r 3 g ) , (97)</formula>
<section_header_level_1><location><page_7><loc_64><loc_18><loc_80><loc_19></location>C. λ -expanded solutions</section_header_level_1>
<text><location><page_7><loc_52><loc_9><loc_92><loc_16></location>We now consider solutions where the leading reduced components of the EMT are not dominated by terms of order O ( λ 0 ) . To obtain mathematically consistent expressions we have to extend the expansion to terms of order O ( λ 2 ) as higher-order terms, if needed, are obtained analogously.</text>
<text><location><page_8><loc_9><loc_90><loc_49><loc_93></location>The k = 0 solution without GR limit has the following properties: the EMT expansion</text>
<formula><location><page_8><loc_12><loc_81><loc_49><loc_88></location>τ a = λ ˜ Ξ + λ 2 ˜ Ξ (2) + ∞ ∑ j /greaterorequalslant 1 2 [ (¯ τ a ) j + λ (˜ τ a ) j + λ 2 ( ˜ τ (2) a ) j ] x j (98)</formula>
<text><location><page_8><loc_9><loc_74><loc_49><loc_80></location>corresponds to the case where as r → r g, lim τ t = lim τ r = lim τ r t . The equations below are trivially extendable to the case where the leading term in τ r t = -λ ˜ Ξ .</text>
<text><location><page_8><loc_10><loc_73><loc_41><loc_74></location>In either case, the metric functions are given by</text>
<formula><location><page_8><loc_11><loc_66><loc_49><loc_71></location>C = r g -λσ 12 √ x + ∞ ∑ j /greaterorequalslant 1 2 ( ζ j + λσ j + λ 2 σ (2) j ) x j , (99)</formula>
<formula><location><page_8><loc_11><loc_62><loc_49><loc_66></location>h = -1 2 ln x ξ + ∞ ∑ j /greaterorequalslant 1 2 ( η j + λω j + λ 2 ω (2) j ) x j , (100)</formula>
<text><location><page_8><loc_9><loc_52><loc_49><loc_60></location>similar to the k = 0 perturbative solution. Here, the structure of the metric function h was simplified by redefining the time, and the coefficient c 12 = ζ 12 + λσ 12 + λ 2 σ (2) 12 was simplified by taking into account the requirement that the Ricci scalar must be finite at the apparent horizon, i.e.</text>
<formula><location><page_8><loc_21><loc_47><loc_49><loc_50></location>c 12 → λσ 12 = -r ' g r g √ ξ . (101)</formula>
<text><location><page_8><loc_9><loc_34><loc_49><loc_44></location>Unlike in GR, the sign of ˜ Ξ (and ˜ Ξ (2) ) cannot be determined solely from the requirements of existence and consistency of the modified Einstein equations. It is therefore unclear whether or not violation of the null energy condition is a prerequisite for the formation of a PBH. This is in contrast to GR, where such a violation has been shown to be mandatory in a variety of settings [9, 21, 26, 27].</text>
<text><location><page_8><loc_9><loc_29><loc_49><loc_33></location>The expansion structure of the non-GR terms E µν remains the same as in the perturbative k = 0 scenario discussed in Subsec. IV A, that is</text>
<formula><location><page_8><loc_14><loc_22><loc_49><loc_27></location>E tt = æ ¯ 1 x + æ 12 √ x +æ 0 x 0 + ∞ ∑ j /greaterorequalslant 1 2 æ j x j , (102)</formula>
<formula><location><page_8><loc_14><loc_18><loc_49><loc_23></location>E r t = œ 12 √ x +œ 0 x 0 + ∞ ∑ j /greaterorequalslant 1 2 œ j x j , (103)</formula>
<formula><location><page_8><loc_14><loc_13><loc_49><loc_18></location>E rr = ø 0 + ∞ ∑ j /greaterorequalslant 1 2 ø j x j , (104)</formula>
<text><location><page_8><loc_9><loc_8><loc_49><loc_11></location>where æ /lscript := ¯æ /lscript + λ ˜ æ /lscript , and similarly for the coefficients œ /lscript , ø /lscript of the non-GR terms E r t and E rr . Substitution into the</text>
<text><location><page_8><loc_52><loc_92><loc_87><loc_93></location>generic modified Einstein equations Eqs. (7)-(9) gives</text>
<formula><location><page_8><loc_52><loc_80><loc_92><loc_91></location>O ( x 0 ) terms of Eq. (7) ( ¯ æ ¯ 1 ξ -8 π ˜ Ξ ) λ = 0 , (105) ( ˜ æ ¯ 1 ξ +8 π ˜ Ξ (2) -σ 2 12 2 r 3 g ) λ 2 = 0 , (106)</formula>
<formula><location><page_8><loc_54><loc_73><loc_73><loc_77></location> x 0 ¯ ø 0 8 π ˜ Ξ λ = 0 ,</formula>
<formula><location><page_8><loc_52><loc_74><loc_92><loc_83></location>O ( x -1 / 2 ) terms of Eq. (8) ( ¯ œ 12 -8 π √ ξ ˜ Ξ ) λ = 0 , (107) [ ˜ œ 12 + √ ξ 2 ( 16 π ˜ Ξ (2) -σ 2 12 r 3 g )] λ 2 = 0 , (108)</formula>
<text><location><page_8><loc_52><loc_59><loc_92><loc_70></location> at the respective leading orders of x , and leads to the following constraints: first, the expansion coefficients ( τ a ) 12 = 0 ∀ a , where ( τ a ) /lscript := (¯ τ a ) /lscript + λ (˜ τ a ) /lscript + λ 2 (˜ τ (2) a ) /lscript , see Eq. (98). At the leading expansion order (which is O ( λ ) since the non-GR terms appear as λ E µν in Eqs. (7)-(9)) the lowest-order x coefficients satisfy</text>
<formula><location><page_8><loc_52><loc_66><loc_92><loc_75></location>O ( ) terms of Eq. (9) ( -) (109) ( ˜ ø 0 +8 π ˜ Ξ (2) -σ 2 12 2 r 3 g ) λ 2 = 0 , (110)</formula>
<formula><location><page_8><loc_62><loc_54><loc_92><loc_57></location>¯ æ ¯ 1 = √ ξ ¯ œ 12 = ξ ¯ ø 0 = 8 π ˜ Ξ ξ, (111)</formula>
<text><location><page_8><loc_52><loc_50><loc_92><loc_55></location>which is analogous to Eq. (63), but in this case identifies the leading reduced term in the EMT. Similarly, the next-order O ( λ 2 ) expansion coefficients satisfy</text>
<formula><location><page_8><loc_56><loc_45><loc_92><loc_50></location>˜ æ ¯ 1 = √ ξ ˜ œ 12 = ξ ˜ ø 0 = -8 π ˜ Ξ (2) ξ + σ 2 12 ξ 2 r 3 g . (112)</formula>
<section_header_level_1><location><page_8><loc_54><loc_42><loc_90><loc_43></location>V. BLACK HOLES IN THE STAROBINSKY MODEL</section_header_level_1>
<text><location><page_8><loc_52><loc_22><loc_92><loc_40></location>Numerous modifications of GR have been proposed, including theories that involve higher-order curvature invariants. Apopular class among these are so-called f ( R ) theories [2], in which the gravitational Lagrangian density L g is an arbitrary function of the Ricci scalar R . In this section, we consider the Starobinsky model [16] with F = ςR 2 , ς = 16 πa 2 /M 2 P (see Eq. (1)). It is a straightforward extension of GR with quadratic corrections in the Ricci scalar that is of relevance in cosmological contexts. In particular, it is the first selfconsistent model of inflation. New horizonless solutions in this model have been identified recently in an analysis [28] of static, spherically symmetric, and asymptotically flat vacuum solutions.</text>
<section_header_level_1><location><page_8><loc_62><loc_18><loc_82><loc_19></location>A. Modified Einstein equations</section_header_level_1>
<text><location><page_8><loc_52><loc_13><loc_92><loc_15></location>In f ( R ) theories, the relevant equations have a relatively simple form. For the action</text>
<formula><location><page_8><loc_56><loc_7><loc_92><loc_12></location>S = 1 16 π ∫ ( f ( R ) + L m ) √ -g d 4 x + S b , (113)</formula>
<text><location><page_9><loc_9><loc_89><loc_49><loc_93></location>where the gravitational Lagrangian L g = f ( R ) , the matter Lagrangian is represented by L m , and S b denotes the boundary term, the field equations for the metric g µν are given by</text>
<formula><location><page_9><loc_11><loc_83><loc_49><loc_88></location>f ' R µν -1 2 f g µν + ( g µν /square -∇ µ ∇ ν ) f ' = 8 πT µν , (114)</formula>
<text><location><page_9><loc_9><loc_80><loc_49><loc_85></location>where f ' := ∂ f ( R ) /∂R and /square := g µν ∇ µ ∇ ν . It is convenient to set f ( R ) =: R + λ F ( R ) . The modified Einstein equations are then</text>
<formula><location><page_9><loc_9><loc_75><loc_51><loc_80></location>G µν + λ ( F ' R µν -1 2 F g µν + ( g µν /square -∇ µ ∇ ν ) F ' ) = 8 πT µν . (115)</formula>
<text><location><page_9><loc_9><loc_70><loc_49><loc_74></location>Performing the expansion in λ and only keeping terms up to the first order we obtain expressions for the modified gravity terms ¯ E µν , i.e.</text>
<formula><location><page_9><loc_11><loc_65><loc_49><loc_69></location>¯ E µν = F ' ¯ R µν -1 2 F ¯ g µν + ( ¯ g µν ¯ /square -¯ ∇ µ ¯ ∇ ν ) F ' , (116)</formula>
<formula><location><page_9><loc_11><loc_57><loc_49><loc_61></location>/square F ' = [ ∂ t ∂ t + ∂ r ∂ r +( ∂ t h ) ∂ t +( ∂ r h +2 /r ) ∂ r ] F ' . (117)</formula>
<text><location><page_9><loc_9><loc_60><loc_49><loc_65></location>where all objects labeled by the bar are evaluated with respect to the unperturbed metric ¯ g, and F ≡ F ( ¯ R ) . In spherical symmetry the d'Alembertian is given by</text>
<text><location><page_9><loc_9><loc_54><loc_49><loc_56></location>Second-order covariant derivatives of a scalar function can be expressed in terms of partial derivatives, i.e.</text>
<formula><location><page_9><loc_18><loc_51><loc_49><loc_53></location>∇ µ ∇ ν F ' = ∂ µ ∂ ν F ' -Γ ζ µν ∂ ζ F ' . (118)</formula>
<text><location><page_9><loc_9><loc_48><loc_49><loc_51></location>In the Starobinsky model F ( R ) = ς ¯ R 2 + O ( λ ) and Eqs. (30)(32) become</text>
<formula><location><page_9><loc_10><loc_39><loc_49><loc_47></location>¯ E tt / ( λς ) = 2 ¯ R ¯ R tt -1 2 ¯ R 2 ¯ g tt +2 [ ¯ g tt ( ∂ t ∂ t + ∂ r ∂ r +( ∂ t ¯ h ) ∂ t +( ∂ r ¯ h +2 r -1 ) ∂ r ) -∂ t ∂ t +Γ t tt ∂ t +Γ r tt ∂ r ] ¯ R, (119) ¯ E r t / ( λς ) = 2 ¯ R ¯ R r t -2 ∂ t ∂ r +Γ r tt ∂ t +Γ r tr ∂ r ¯ R, (120)</formula>
<formula><location><page_9><loc_10><loc_33><loc_49><loc_41></location>( ) ¯ E rr / ( λς ) = 2 ¯ R ¯ R rr -1 2 ¯ R 2 ¯ g rr +2¯ g rr [ ∂ t ∂ t +( ∂ t ¯ h -Γ r rt ) ∂ t +( ∂ r ¯ h +2 r -1 -Γ r rr ) ∂ r ] ¯ R. (121)</formula>
<section_header_level_1><location><page_9><loc_9><loc_31><loc_48><loc_32></location>B. Compatibility with the k = 0 class of black hole solutions</section_header_level_1>
<text><location><page_9><loc_9><loc_18><loc_49><loc_29></location>With the k = 0 metric functions Eqs. (46)-(47), the constraint Eq. (52) that is obtained from the requirement that the Ricci scalar be non-divergent leads to cancellations in the Ricci tensor components ¯ R tt and ¯ R rr which ensures that the MTG terms Eqs. (119)-(121) of the ˜ f ( ¯ R ) = ς ¯ R 2 Starobinsky model conform to the structures of Eqs. (57)-(59). We find that both of the two constraints posed by Eq. (63) are satisfied, i.e.</text>
<formula><location><page_9><loc_10><loc_9><loc_49><loc_17></location>æ ¯ 1 = √ ¯ ξ œ 12 = ¯ ξ ø 0 = c 2 12 ¯ ξ ( -2 ( R 0 + r g R 1 ) + h 12 r g R 12 ) -c 12 r 2 g √ ¯ ξR ' 0 2 r 3 g , (122)</formula>
<text><location><page_9><loc_52><loc_87><loc_92><loc_93></location>where R j is used to denote coefficients of the Ricci scalar ¯ R = ∑ j R j x j = R 0 + R 12 √ x + R 1 x + O ( x 3 / 2 ) . Similarly, the next-highest order coefficients satisfy the constraint of Eq. (65), see App. B, Eqs. (B4)-(B6).</text>
<section_header_level_1><location><page_9><loc_58><loc_83><loc_85><loc_84></location>C. Compatibility with the k = 1 solution</section_header_level_1>
<text><location><page_9><loc_52><loc_70><loc_92><loc_81></location>Similar to the k = 0 case, the k = 1 constraint on the evolution of the horizon radius Eq. (44) that is required to ensure consistency of the Einstein equations and finiteness of the Ricci scalar leads to cancellations in ¯ R tt , and the k = 1 Starobinsky MTG terms of Eqs. (119)-(121) follow the structures prescribed by Eqs. (88)-(90). Using the k = 1 metric functions Eqs. (76)-(77) we obtain the lowest-order coefficients</text>
<formula><location><page_9><loc_54><loc_67><loc_92><loc_68></location>æ 32 = 2 c 32 ¯ ξ 3 /r 5 g , (123)</formula>
<formula><location><page_9><loc_55><loc_62><loc_92><loc_66></location>æ ¯ 1 = -c 32 ¯ ξ 3 ( -2 h 12 +3 c 32 ( 4 + r 3 g R 1 )) /r 5 g , (124)</formula>
<text><location><page_9><loc_52><loc_61><loc_90><loc_63></location>of ¯ E tt from Eq. (119). Similarly, we obtain the coefficients</text>
<formula><location><page_9><loc_55><loc_56><loc_92><loc_60></location>œ 0 = 0 , œ 12 = -3 c 2 32 ¯ ξ 3 / 2 ( 4 + r 3 g R 1 ) /r 5 g , (125) ø 32 = -2 c 32 /r 5 g , (126)</formula>
<text><location><page_9><loc_52><loc_45><loc_92><loc_53></location>of ¯ E r t and ¯ E rr from Eqs. (120)-(121), where R 1 denotes the x coefficient of the Ricci scalar ¯ R = ∑ j R j x j = 2 /r 2 g + R 1 x + O ( x 3 / 2 ) and R 12 = 0 , cf. Eq. (45). With the expressions given in Eqs. (123)-(127), it is easy to verify that both k = 1 constraints Eq. (92) and Eq. (93) are satisfied identically.</text>
<formula><location><page_9><loc_56><loc_52><loc_92><loc_56></location>ø 2 = c 32 ( 2 h 12 -3 c 32 ( 4 + r 3 g R 1 ) /r 5 g ) . (127)</formula>
<section_header_level_1><location><page_9><loc_53><loc_40><loc_91><loc_42></location>D. Compatibility with the λ -expanded k = 0 class of black hole solutions</section_header_level_1>
<text><location><page_9><loc_52><loc_21><loc_92><loc_38></location>Equality of the coefficients in Eqs. (111) and (112) follows in exactly the same fashion as in Sec. V B. Explicit calculation confirms that the coefficients of the MTG terms in the Starobinsky model Eqs. (119)-(121) obtained using the EMT expansion of Eq. (98) and metric functions Eqs. (99)-(100) coincide with those of Eq. (122) at the leading expansion order O ( λ ) . Terms of order O ( λ 0 ) vanish in accordance with Eq. (122) (note that c 12 ∝ O ( λ ) ). This confirms that the Starobinsky solution is consistent with the generic form of PBH solutions. However, since ˜ Ξ (2) is undetermined in the self-consistent approach, Eq. (112) does not impose any constraints on the function ξ .</text>
<section_header_level_1><location><page_9><loc_66><loc_16><loc_78><loc_17></location>VI. DISCUSSION</section_header_level_1>
<text><location><page_9><loc_52><loc_9><loc_92><loc_14></location>We have analyzed the properties of metric MTG and derived several constraints that they must satisfy to be compatible with the existence of an apparent horizon. Since we have not specified the origin of the deviations from GR, the results</text>
<text><location><page_10><loc_9><loc_90><loc_49><loc_93></location>presented here are generic and apply to all conceivable selfconsistent metric MTG.</text>
<text><location><page_10><loc_9><loc_66><loc_49><loc_90></location>Constraints on a perturbative solution in a particular metric MTG arise from two sources: first, the series expansions of the modified gravity terms ¯ E µν in terms of the distance x := r -r g from the horizon must follow a particular structure that is prescribed by the modified Einstein equations with terms that diverge in the limit r → r g. Second, a general spherically symmetric metric allows for two independent functions C and h that must satisfy three Einstein equations. The resulting relations between coefficients σ /lscript , ω /lscript of their perturbative corrections translate into relationships between the coefficients c /lscript and h /lscript , and eventually components of the unperturbed EMT. These constraints must be satisfied identically. Otherwise, a valid solution of GR cannot be perturbatively extended to a solution of a MTG. Identities that must be satisfied for the existence of the perturbative k = 0 solutions are given by Eqs. (63) and (65), and for the k = 1 solution by Eqs. (92)-(93).</text>
<text><location><page_10><loc_9><loc_58><loc_49><loc_65></location>On the other hand, there are nonperturbative solutions that do not have a well-defined GR limit. In this case, the constraints on a MTG that are imposed by the existence of a regular apparent horizon formed in finite time of a distant observer are given by Eqs. (111)-(112).</text>
<text><location><page_10><loc_9><loc_43><loc_49><loc_58></location>Using the Starobinsky R 2 model, arguably the simplest possible MTG, we identify both perturbative and nonperturbative solutions. However, this is not the only theory that should be investigated: in a future article [29], we will consider generic f ( R ) theories of the form f ( R ) = R + λ F ( R ) , where F ( R ) = ςR q and q, ς ∈ R . In particular, this includes the case q = 1 / 2 (i.e. f ( R ) = R + λς √ R ) considered in Ref. [30], as well as the case of negative exponents q < 0 considered in Ref. [31]. More general MTG (e.g. those involving higher-order curvature invariants) will also be considered.</text>
<section_header_level_1><location><page_10><loc_21><loc_37><loc_37><loc_38></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_10><loc_9><loc_23><loc_49><loc_34></location>We thank Eleni Kontou, Robert Mann, Shin'ichi Nojiri, Vasilis Oikonomou, and Christian Steinwachs for useful discussions and helpful comments. SM is supported by an International Macquarie University Research Excellence Scholarship and a Sydney Quantum Academy Scholarship. The work of DRT was supported in part by the Southern University of Science and Technology, Shenzhen, China, and by the ARC Discovery project grant DP210101279.</text>
<section_header_level_1><location><page_10><loc_12><loc_17><loc_45><loc_18></location>Appendix A: Coefficients of perturbative corrections</section_header_level_1>
<section_header_level_1><location><page_10><loc_19><loc_14><loc_39><loc_15></location>1. k = 0 black hole solutions</section_header_level_1>
<text><location><page_10><loc_9><loc_9><loc_49><loc_11></location>With the metric functions Eqs. (46)-(47) of the k = 0 solutions we obtain the following coefficients for the perturbative</text>
<text><location><page_10><loc_52><loc_92><loc_81><loc_93></location>correction Σ of Eq. (55) from Eqs. (30)-(32):</text>
<formula><location><page_10><loc_53><loc_87><loc_92><loc_90></location>σ 12 | tt = -r 3 / 2 g æ ¯ 1 4 √ π ¯ ξ ¯ Υ , (A1)</formula>
<formula><location><page_10><loc_53><loc_82><loc_92><loc_86></location>σ 12 | tr = -r 3 / 2 g ( 4 π ˜ ξ ¯ Υ 2 + √ ¯ ξ œ 12 ) 2 √ π ¯ ξ ¯ Υ , (A2)</formula>
<formula><location><page_10><loc_53><loc_78><loc_92><loc_82></location>σ 12 | rr = -r 3 / 2 g ø 0 4 √ π ¯ Υ , (A3)</formula>
<formula><location><page_10><loc_54><loc_70><loc_92><loc_78></location>σ 1 | tt = 4 ¯ Υæ ¯ 1 +6 √ πr 3 / 2 g ¯ Υ 2 36 π ¯ ξ ¯ Υ 3 ( -æ 12 +8 π ¯ ξ (˜ τ t ) 12 + √ πr 3 / 2 g æ ¯ 1 ( 6(¯ τ r ) 12 -5(¯ τ t ) 12 ) ) , (A4)</formula>
<formula><location><page_10><loc_54><loc_63><loc_93><loc_71></location>σ 1 | tr = -1 12 √ π ¯ ξ ¯ Υ [ ˜ ξ ( -4 √ π ¯ Υ+8 πr 3 / 2 g (¯ τ t ) 12 ) +3 r 3 / 2 g × √ ¯ ξ ( -œ 0 +4 π √ ¯ ξ ( (˜ τ t ) 12 +(˜ τ r ) 12 -2 ¯ Υ 2 ω 12 ) ) ] , (A5)</formula>
<formula><location><page_10><loc_54><loc_55><loc_94><loc_63></location>σ 1 | rr = 1 12 π ¯ Υ 3 ( -2 ¯ Υø 0 +6 √ πr 3 / 2 g ¯ Υ 2 ( -ø 12 +8 π (˜ τ r ) 12 ) + √ πr 3 / 2 g ø 0 ( (¯ τ t ) 12 -6(¯ τ r ) 12 ) -96 π 3 / 2 r 3 / 2 g ¯ Υ 4 ω 12 ) . (A6)</formula>
<text><location><page_10><loc_52><loc_51><loc_92><loc_54></location>Via comparison of Eqs. (A5) and (A6) we can identify the coefficient</text>
<formula><location><page_10><loc_52><loc_41><loc_92><loc_50></location>ω 12 = 1 72 π ¯ ξ ¯ Υ 2 r 3 / 2 g ( 20 √ π ˜ ξ ¯ Υ+3 r 3 / 2 g √ ¯ ξ œ 0 -6 r 3 / 2 g ¯ ξ ( ø 12 +2 π (˜ τ t ) 12 -6 π (˜ τ r ) 12 ) +16 r 3 / 2 g π ˜ ξ ( 3(¯ τ r ) 12 -(¯ τ t ) 12 ) ) (A7)</formula>
<text><location><page_10><loc_52><loc_39><loc_81><loc_40></location>for the perturbative correction Ω of Eq. (56).</text>
<text><location><page_10><loc_53><loc_37><loc_92><loc_38></location>Substitution of Eq. (A7) into Eqs. (A5) and (A6) then yields</text>
<formula><location><page_10><loc_53><loc_24><loc_92><loc_35></location>σ 1 | tr = σ 1 | rr = 1 18 √ π ¯ ξ ¯ Υ [ 3 r 3 / 2 g √ ¯ ξ ( -2œ 0 + √ ¯ ξ ( ø 12 +8 π (˜ τ t ) 12 ) ) -4 ˜ ξ ( 4 √ π ¯ Υ+ πr 3 / 2 g ( 6(¯ τ r ) 12 -5(¯ τ t ) 12 ) ) ] . (A8)</formula>
<text><location><page_10><loc_52><loc_19><loc_92><loc_23></location>Subsequent comparison of Eq. (A8) and (A4) establishes the relation Eq. (65) between the coefficients æ 12 , œ 0 , and ø 12 .</text>
<section_header_level_1><location><page_10><loc_62><loc_14><loc_81><loc_15></location>2. k = 1 black hole solution</section_header_level_1>
<text><location><page_10><loc_52><loc_9><loc_92><loc_11></location>With the metric functions Eqs. (76)-(77) of the k = 1 solution we obtain the following coefficients for the perturbative</text>
<text><location><page_11><loc_9><loc_92><loc_41><loc_93></location>correction Σ of Eq. (85) from Eqs. (30) and (32):</text>
<formula><location><page_11><loc_10><loc_87><loc_49><loc_91></location>σ 32 | tt = r 2 g ( æ 32 r g ¯ ξ 3 -8 πc 32 ˜ E ) , (A9)</formula>
<formula><location><page_11><loc_10><loc_85><loc_49><loc_87></location>σ 32 | rr = -ø 32 r 3 g +8 πc 32 r 2 g ˜ P, (A10)</formula>
<formula><location><page_11><loc_11><loc_82><loc_37><loc_85></location>σ 2 | tt = r g ¯ ξ 3 æ 32 r g (3 c 32 -2 h 12 ) + æ ¯ 1 r g</formula>
<formula><location><page_11><loc_10><loc_71><loc_49><loc_78></location>σ 2 | rr = -r 2 g ( ø 2 r g -c 32 ( 3ø 32 r g -8 π (3 c 32 + h 12 ) ˜ P ) -8 πr g ˜ p 2 ) . (A12)</formula>
<formula><location><page_11><loc_17><loc_77><loc_49><loc_85></location>2 [ -8 π ¯ ξ 3 ( c 32 (3 c 32 -h 12 ) ˜ E + r g ˜ e 2 )] , (A11)</formula>
<text><location><page_11><loc_10><loc_71><loc_46><loc_73></location>From the x 0 and √ x coefficients of Eq. (31) we obtain</text>
<formula><location><page_11><loc_11><loc_67><loc_49><loc_70></location>˜ E | tr = œ 0 r g 8 πc 32 ¯ ξ 3 / 2 , (A13)</formula>
<formula><location><page_11><loc_11><loc_60><loc_49><loc_66></location>˜ ξ | tr = -2 ¯ ξσ 32 3 c 32 -4 r 3 g 9 c 2 32 √ ¯ ξ ( œ 12 -8 π ¯ ξ 3 / 2 ˜ p 2 ) , (A14)</formula>
<text><location><page_11><loc_9><loc_57><loc_49><loc_61></location>where ˜ Φ = ( ˜ E + ˜ P ) / 2 and ˜ φ 2 = (˜ e 2 + ˜ p 2 ) / 2 , see Eqs. (73) and (75). By substitution of Eq. (A13) into Eq. (A11) we obtain</text>
<formula><location><page_11><loc_10><loc_48><loc_49><loc_56></location>˜ e 2 + ˜ p 2 = 6 c 2 32 ˜ Φ r g + 1 8 π ¯ ξ 3 [ æ ¯ 1 +æ 32 (3 c 32 -2 h 12 ) + ¯ ξ 3 (ø 2 -3 c 32 ø 32 ) + 2 ¯ ξ 3 / 2 œ 0 ( h 12 -3 c 32 ) ] . (A15)</formula>
<text><location><page_11><loc_52><loc_88><loc_92><loc_93></location>from the comparison σ 2 | tt -σ 2 | rr = 0 . Similarly, substitution of Eq. (A14) into Eq. (87) and subsequent comparison of σ 32 | R -σ 32 | tt = 0 yields</text>
<formula><location><page_11><loc_54><loc_79><loc_92><loc_83></location>˜ e 2 + ˜ p 2 = 6 c 2 32 ˜ Φ r g + 3æ 32 c 32 -¯ ξ 3 / 2 (6 c 32 -œ 0 œ 12 ) 4 π ¯ ξ 3 . (A16)</formula>
<text><location><page_11><loc_52><loc_70><loc_92><loc_73></location>Subtracting Eq. (A15) from Eq. (A16) and subsequent multiplication by 8 π ¯ ξ 3 r g yields</text>
<formula><location><page_11><loc_52><loc_58><loc_92><loc_64></location>-r g [ æ ¯ 1 -æ 32 (3 c 32 +2 h 12 ) + ¯ ξ 3 / 2 ( ¯ ξ 3 / 2 (ø 2 -3 c 32 ø 32 ) +6 c 32 œ 0 +2 h 12 œ 0 -2œ 12 )] = 0 . (A17)</formula>
<text><location><page_11><loc_52><loc_50><loc_92><loc_52></location>Lastly, substituting æ 32 from Eq. (92) into (A17) and rearranging gives Eq. (93).</text>
<paragraph><location><page_11><loc_27><loc_45><loc_74><loc_46></location>Appendix B: Additional explicit expressions for k = 0 black hole solutions</paragraph>
<text><location><page_11><loc_10><loc_40><loc_47><loc_41></location>Explicit expressions for the individual terms in Eq. (69):</text>
<formula><location><page_11><loc_16><loc_23><loc_92><loc_37></location>( ¯ R tt ) 12 = 2 √ ¯ ξ ( ¯ R r t ) 0 -¯ ξ ( ¯ R rr ) 12 = -1 24 c 12 r 3 g [ -2 c 3 12 ¯ ξ ( h 3 12 r g +6 h 32 r g + h 12 (9 h 1 r g -6) ) 6 r g √ ¯ ξ ( c 1 -1) ( √ ¯ ξ ( c 1 -1) 2 -2 r g c ' 12 ) × [ 4 c 32 r g ¯ ξ ( c 1 -1) + r g √ ¯ ξ ( 2 r g c ' 1 -h 12 ( √ ¯ ξ ( c 1 -1) 2 -2 r g c ' 12 ))] +6 c 2 12 ( ¯ ξ ( 3 -2 h 1 r g + c 32 h 12 r g -h 2 12 r g + c 1 ( -3 + 2 h 1 r g + h 2 12 r g ) ) + r 2 g √ ¯ ξh ' 12 ) ] . (B1)</formula>
<formula><location><page_11><loc_15><loc_9><loc_92><loc_24></location>( ¯ R r t ) 0 = c 12 √ ¯ ξ ( c 1 -1) /r 3 g . (B2) ( ¯ R rr ) 12 = 1 24 c 12 r 3 g √ ¯ ξ [ -2 c 3 12 √ ¯ ξ ( h 3 12 r g +6 h 32 r g + h 12 (9 h 1 r g -6) ) +6 r g ( c 1 -1) ( √ ¯ ξ ( c 1 -1) 2 -2 r g c ' 12 ) +3 c 12 r g [ 4 c 32 √ ¯ ξ ( c 1 -1) + 2 r g c ' 1 -h 12 ( √ ¯ ξ ( c 1 -1) -2 r g c ' 12 )] +6 c 2 12 [ √ ¯ ξ ( h 12 c 32 r g -2 h 1 r g -h 2 12 r g + c 1 ( 5 + 2 h 1 r g + h 2 12 r g ) -5 ) + r 2 g h ' 12 ] ] . (B3)</formula>
<text><location><page_12><loc_10><loc_92><loc_92><loc_93></location>Explicit expressions for the MTG coefficients æ 12 , œ 0 , ø 12 in the Starobinsky model of the k = 0 solution (see Subsec. V B):</text>
<formula><location><page_12><loc_13><loc_71><loc_92><loc_89></location>æ 12 = 1 12 c 12 r 5 g [ -12 r g ¯ ξR 0 ( c 1 -1) 3 +2 c 3 12 ¯ ξ ( 2 R 0 ( -6 h 12 + r g (9 h 12 h 1 + h 3 12 +6 h 32 ) ) -6 h 12 r 3 g R 1 +3 r 2 g R 12 × ( -6 + r g ( h 2 12 + h 1 ) ) +24 r 2 g R 0 √ ¯ ξc ' 12 ( c 1 -1) -3 c 12 r g √ ¯ ξ [ 8 c 32 √ ¯ ξR 0 ( c 1 -1) + 2 h 12 R 0 × ( 2 r g c ' 12 √ ¯ ξ ( c 1 -1) 2 ) + r g ( 4 R 0 c ' 1 +2 r 2 g R 12 c ' 12 -4 R ' 0 ( c 1 -1) -r g √ ¯ ξR 12 ( c 1 -1) 2 ) ] +6 c 2 12 [ -2 ¯ ξR 0 ( 5 + r g ( 2 h 1 ( c 1 -1) + h 12 ( c 32 + h 12 ( c 1 -1) ) ) -5 c 1 ) +4 ¯ ξR 2 0 + r 3 g ¯ ξ ( R 12 ( h 12 -h 12 c 1 + c 32 ) + 4 R 1 ( c 1 -1) ) -2 r 2 g √ ¯ ξR 0 h ' 12 -r 4 g √ ¯ ξR ' 12 ] ] . (B4)</formula>
<formula><location><page_12><loc_13><loc_46><loc_92><loc_64></location>ø 12 = 1 12 c 12 r 5 g √ ¯ ξ [ -2 c 3 12 √ ¯ ξ ( 2 R 0 ( r g ( 9 h 12 h 1 + h 3 12 +6 h 32 ) -30 h 12 ) -18 h 12 r 3 g R 1 +3 ( -3 h 1 + h 2 12 ) ) +12 r g R 1 ( c 1 -1) ( √ ¯ ξ ( c 1 -1) 2 -2 r g c ' 12 ) -3 c 12 r g [ -8 c 32 √ ¯ ξR 0 ( c 1 -1) + 2 h 12 R 0 × ( √ ¯ ξ ( c 1 -1) 2 -2 r g c ' 12 ) r g ( r g √ ¯ ξR 12 ( c 1 -1) 2 -4 R 0 c ' 1 +2 r 2 g R 12 c ' 12 +4 R ' 0 ( c 1 -1) )] -6 c 2 12 [ -2 √ ¯ ξR 0 ( 3 ( c 1 -1) ) + r g ( 2 h 1 ( c 1 -1) + h 12 ( c 32 + h 12 ( c 1 -1) ) ) +4 √ ¯ ξR 2 0 -2 r 2 g R 0 h ' 12 + r 2 g ( c 32 r g √ ¯ ξR 12 + h 12 ( r g √ ¯ ξR 12 ( c 1 -1) -8 R ' 0 ) +5 r 2 g R ' 12 )] ] . (B6)</formula>
<formula><location><page_12><loc_13><loc_63><loc_92><loc_71></location>œ 0 = 1 2 r 5 g [ c 2 12 √ ¯ ξ ( 2 h 12 ( 4 R 0 + r 3 g R 1 ) + r 2 g (2 h 1 r g -3) ) -r 4 g R 12 c ' 12 + c 12 ( 8 √ ¯ ξR 0 ( c 1 -1) + r 2 g ( r g √ ¯ ξ ( c 1 -1) (2 R 1 -h 12 R 12 ) + 4 h 12 R ' 0 -3 r 2 g R 2 12 ) ) ] . (B5)</formula>
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2015MPLA...3050042M | https://arxiv.org/pdf/1505.02188.pdf | <document>
<section_header_level_1><location><page_1><loc_24><loc_92><loc_77><loc_93></location>Spherically symmetric solutions in a FRW background</section_header_level_1>
<text><location><page_1><loc_38><loc_89><loc_62><loc_90></location>H. Moradpour 1 ∗ and N. Riazi 2 †</text>
<text><location><page_1><loc_13><loc_86><loc_88><loc_89></location>1 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran, 2 Physics Department, Shahid Beheshti University, Evin, Tehran 19839, Iran.</text>
<text><location><page_1><loc_18><loc_77><loc_83><loc_85></location>We impose perfect fluid concept along with slow expansion approximation to derive new solutions which, considering non-static spherically symmetric metrics, can be treated as Black Holes. We will refer to these solutions as Quasi Black Holes. Mathematical and physical features such as Killing vectors, singularities, and mass have been studied. Their horizons and thermodynamic properties have also been investigated. In addition, relationship with other related works (including mcVittie's) are described.</text>
<section_header_level_1><location><page_1><loc_42><loc_69><loc_59><loc_71></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_10><loc_66><loc_60><loc_67></location>The Universe expansion can be modeled by the so called FRW metric</text>
<formula><location><page_1><loc_30><loc_61><loc_92><loc_65></location>ds 2 = -dt 2 + a ( t ) 2 [ dr 2 (1 -kr 2 ) + r 2 dθ 2 + r 2 sin ( θ ) 2 dφ 2 ] , (1)</formula>
<text><location><page_1><loc_9><loc_57><loc_92><loc_61></location>where k = 0 , +1 , -1 are curvature scalars which represent the flat, closed and open universes, respectively. The WMAP data confirms a flat ( k = 0) universe [1]. a ( t ) is the scale factor and for a background which is filled by a perfect fluid with equation of state p = ωρ , there are three classes of expanding solutions. These three solutions are</text>
<formula><location><page_1><loc_44><loc_54><loc_92><loc_55></location>a ( t ) = a 0 t 2 3( ω +1) (2)</formula>
<text><location><page_1><loc_9><loc_50><loc_30><loc_52></location>for ω = 0 when -1 < ω and ,</text>
<text><location><page_1><loc_13><loc_50><loc_13><loc_52></location>/negationslash</text>
<formula><location><page_1><loc_45><loc_48><loc_92><loc_50></location>a ( t ) = a 0 e Ht (3)</formula>
<text><location><page_1><loc_9><loc_45><loc_57><loc_47></location>for ω = -1 (dark energy), and for the Phantom regime ( ω < -1) is</text>
<formula><location><page_1><loc_41><loc_42><loc_92><loc_45></location>a ( t ) = a 0 ( t 0 -t ) 2 3( ω +1) , (4)</formula>
<text><location><page_1><loc_9><loc_40><loc_81><loc_41></location>where t 0 is the big rip singularity time and will be available, if the universe is in the phantom regime.</text>
<text><location><page_1><loc_10><loc_38><loc_85><loc_40></location>In Eq. (3), H ( ≡ ˙ a ( t ) a ( t ) ) is the Hubble parameter and the current estimates are H = 73 +4 -3 kms -1 Mpc -1 [1].</text>
<text><location><page_1><loc_9><loc_35><loc_92><loc_38></location>Note that, at the end of the Phantom regime, everything will decompose into its fundamental constituents [2]. In addition, this spacetime can be classified as a subgroup of the Godel-type spacetime with σ = m = 0 and k ' = 1 [3].</text>
<text><location><page_1><loc_9><loc_32><loc_92><loc_35></location>A signal which was emitted at the time t 0 by a co-moving source and absorbed by a co-moving observer at a later time t is affected by a redshift ( z ) as</text>
<formula><location><page_1><loc_44><loc_28><loc_92><loc_31></location>1 + z = a ( t ) a ( t 0 ) . (5)</formula>
<text><location><page_1><loc_9><loc_26><loc_59><loc_27></location>The apparent horizon as a marginally trapped surface, is defined as [4]</text>
<formula><location><page_1><loc_44><loc_23><loc_92><loc_24></location>g µν ∂ µ ξ∂ ν ξ = 0 , (6)</formula>
<text><location><page_1><loc_9><loc_20><loc_53><loc_22></location>which for the physical radius of ξ = a ( t ) r , the solution will be:</text>
<formula><location><page_1><loc_43><loc_15><loc_92><loc_19></location>ξ = 1 √ H 2 + k a ( t ) 2 . (7)</formula>
<text><location><page_2><loc_9><loc_92><loc_55><loc_93></location>The surface gravity of the apparent horizon can be evaluated by:</text>
<formula><location><page_2><loc_40><loc_87><loc_92><loc_91></location>κ = 1 2 √ -h ∂ a ( √ -hh ab ∂ b ξ ) . (8)</formula>
<text><location><page_2><loc_9><loc_82><loc_92><loc_87></location>Where the two dimensional induced metric is h ab = diag ( -1 , a ( t ) (1 -kr 2 ) ). It was shown that the first law of thermodynamics is satisfied on the apparent horizon [5-8]. The special case of ω = -1 is called the dark energy, and by a suitable change of variables one can rewrite this case in the static form [9]:</text>
<formula><location><page_2><loc_33><loc_77><loc_92><loc_81></location>ds 2 = -(1 -H 2 r 2 ) dt 2 + dr 2 (1 -H 2 r 2 ) + r 2 d Ω 2 . (9)</formula>
<text><location><page_2><loc_9><loc_74><loc_92><loc_76></location>This metric belongs to a more general class of spherically symmetric, static metrics. For these class of spherically symmetric static metrics, the line element can be written in the form of:</text>
<formula><location><page_2><loc_38><loc_69><loc_92><loc_73></location>ds 2 = -f ( r ) dt 2 + dr 2 f ( r ) + r 2 d Ω 2 , (10)</formula>
<text><location><page_2><loc_9><loc_67><loc_33><loc_68></location>where the general form of f ( r ) is:</text>
<formula><location><page_2><loc_39><loc_63><loc_92><loc_66></location>f ( r ) = 1 -2 m r + Q 2 r 2 -H 2 r 2 . (11)</formula>
<text><location><page_2><loc_9><loc_61><loc_92><loc_62></location>In the above expression, m and Q represent mass and charge, respectively. For this metric, one can evaluate redshift:</text>
<formula><location><page_2><loc_37><loc_55><loc_92><loc_59></location>1 + z = ( 1 -2 m r + Q 2 r 2 -H 2 r 2 1 -2 m r 0 + Q 2 r 2 0 -H 2 r 2 0 ) 1 2 . (12)</formula>
<text><location><page_2><loc_9><loc_51><loc_92><loc_54></location>Where, r 0 and r are radial coordinates at the emission and the absorption points. For the horizons, the radius and the surface gravity can be found using equations</text>
<formula><location><page_2><loc_42><loc_45><loc_92><loc_50></location>g tt = f ( r ) = 0 -→ r h (13) κ = f ' ( r ) 2 | r h ,</formula>
<text><location><page_2><loc_9><loc_42><loc_92><loc_45></location>where ( ' ) denotes derivative with respect to the coordinate r [9]. From the thermodynamic laws of Black Holes (BHs) we know</text>
<formula><location><page_2><loc_46><loc_38><loc_92><loc_41></location>T = κ 2 π , (14)</formula>
<text><location><page_2><loc_9><loc_35><loc_92><loc_37></location>which T is the temperature on the horizon [9]. Validity of the first law of the thermodynamics on the static horizons for the static spherically symmetric spacetime has been shown [10, 11].</text>
<text><location><page_2><loc_9><loc_23><loc_92><loc_34></location>The BHs with the FRW dynamic background has motivated many investigations. The first approach, which is named Swiss Cheese, includes efforts in order to find the effects of the expansion of the Universe on the gravitational field of the stars [12], introduced originally by Einstein and Straus (1945) [13]. In these models, authors tried to join the Schwarzschild metric to the FRW metric by satisfying the junction conditions on the boundary, which is an expanding timelike hypersurface. The inner spacetime is described by the Schwarzschild metric, while the FRW metric explains the outer spacetime. These models don't contain dynamical BHs, Because the inner spacetime is in the Schwarzschild coordinate, hence, is static [14]. In addition, the Swiss Cheese models can be classified as a subclass of inhomogeneous Lemabitre-Tolman-Bondi models [15, 16].</text>
<text><location><page_2><loc_9><loc_9><loc_92><loc_23></location>Looking for dynamical BHs, some authors used the conformal transformation of the Schwarzschild BH, where the conformal factor is the scale factor of the famous FRW model. Originally, Thakurta (1981) have used this technique and obtained a dynamical version of the Schwarzschild BH [17]. Since the Thakurta spacetime is a conformal transformation of the Schwarzschild metric, it is now accepted that its redshift radii points to the co-moving radii of the event horizon of BH [16, 18, 19]. By considering asymptotic behavior of the gravitational lagrangian (Ricci scalar), one can classify the Thakurta BH and its extension to the charged BH into the same class of solutions [18, 19]. The Thakurta spacetime sustains an inward flow, which leads to an increase in the mass of BH [18-20]. This ingoing flow comes from the back-reaction effect and can be neglected in a low density background [20]. In fact, for the low density background, the mass will be decreased in the Phantom regime [21]. Also, the radius of event horizon increases with the scale factor when its temperature decreases by the inverse of scale factor [18, 19].</text>
<text><location><page_3><loc_9><loc_82><loc_92><loc_93></location>Using the Eddington-Finkelstein form of the Schwarzschild metric and the conformal transformation, Sultana and Dyer (2005) have constructed their metric and studied its properties [22]. In addition, unlike the Thakurta spacetime, the curvature scalars do not diverge at the redshift singularity radii (event horizon) of the Sultana and Dyer spacetimes. Since the Sultana and Dyer spacetimes is conformal transformation of the Schwarzschild metric, it is now accepted that the Sultana and Dyer spacetimes include dynamic BHs [16]. Various examples can be found in [16, 23-25]. Among these conformal BHs, only the solutions by M c Clure et al. and Thakurta can satisfy the energy conditions [16, 19]. Static charged BHs which are confined into the FRW spacetime and the dynamic, charged BHs were studied in [26-33]. The Brane solutions can be found in [34-36].</text>
<text><location><page_3><loc_9><loc_70><loc_92><loc_82></location>In another approach, mcVittie found new solutions including contracting BHs in the coordinates co-moving with the universe's expansion [37]. Its generalization to the arbitrary dimensions and to the charged BHs can be found in [38, 39]. In these solutions, it is easy to check that the curvature scalars diverge at the redshift singularities. In this approach, authors have used the isotropic form of the FRW metric along as the perfect fluid concept and could find their solutions which can contain BHs [40]. The mass and the charge of their BHs seem to be decreased with the scale factor. Also, it seems that the redshift singularities does not point to a dynamic event horizon [41-44]. Unlike the Swiss Cheese models, the energy conditions are violated by these solutions [16]. These solutions can be considered as Models for cosmological inhomogeneities [15].</text>
<text><location><page_3><loc_9><loc_61><loc_92><loc_70></location>This paper is organized as follows: in the next section, we consider the conformal transformation of a non-static spherically symmetric metric, where conformal factor has only time dependency. In addition, we derive the general possible form of metric by using perfect fluid concept. In section 3, slow time varying approximation is used in order to find the physical meaning of the parameters of metric. In continue, the mcVittie like solution and its thermodynamic properties are addressed. In section 4, we generalized our debates to the charged spacetime, when the effects of the dark energy are considerable. In section 5, we summarize and conclude the results.</text>
<section_header_level_1><location><page_3><loc_22><loc_57><loc_78><loc_58></location>II. METRIC, GENERAL PROPERTIES AND BASIC ASSUMPTIONS</section_header_level_1>
<text><location><page_3><loc_10><loc_54><loc_32><loc_55></location>Let us begin with this metric:</text>
<formula><location><page_3><loc_25><loc_49><loc_92><loc_53></location>ds 2 = a ( τ ) 2 [ -f ( τ, r ) dτ 2 + dr 2 (1 -kr 2 ) f ( τ, r ) + r 2 dθ 2 + r 2 sin ( θ ) 2 dφ 2 ] . (15)</formula>
<text><location><page_3><loc_9><loc_48><loc_77><loc_49></location>Where a ( τ ) is the arbitrary function of time coordinate τ . This metric has three Killing vectors</text>
<formula><location><page_3><loc_28><loc_45><loc_92><loc_47></location>∂ φ , sin φ ∂ θ +cot θ cos φ ∂ φ and cos φ ∂ θ -cot θ sin φ ∂ φ . (16)</formula>
<text><location><page_3><loc_9><loc_43><loc_38><loc_45></location>Now, if we define new time coordinate as</text>
<formula><location><page_3><loc_43><loc_40><loc_92><loc_43></location>τ → t = ∫ a ( τ ) dτ, (17)</formula>
<text><location><page_3><loc_9><loc_38><loc_16><loc_39></location>we will get</text>
<formula><location><page_3><loc_26><loc_33><loc_92><loc_37></location>ds 2 = -f ( t, r ) dt 2 + a ( t ) 2 [ dr 2 (1 -kr 2 ) f ( t, r ) + r 2 dθ 2 + r 2 sin ( θ ) 2 dφ 2 ] , (18)</formula>
<text><location><page_3><loc_9><loc_29><loc_92><loc_33></location>which possesses symmetries like as Eq. (16). From now, it is assumed that a ( t ) is the cosmic scale factor similar to the FRW's. For f ( t, r ) = 1, Eq. (18) is reduced to the FRW metric (1). Also, conformal BHs can be achieved by choosing f ( t, r ) = f ( r ) where, the general form of f ( r ) is [18]:</text>
<formula><location><page_3><loc_39><loc_24><loc_92><loc_28></location>f ( r ) = 1 -2 m r + Q 2 r 2 -Λ r 2 3 . (19)</formula>
<text><location><page_3><loc_9><loc_21><loc_92><loc_24></location>Therefore, conformal BHs can be classified as a special subclass of metric (18). n α = δ r α is normal to the hypersurface r = const and yields</text>
<formula><location><page_3><loc_38><loc_17><loc_92><loc_20></location>n α n α = g rr = (1 -kr 2 ) f ( t, r ) a ( t ) 2 , (20)</formula>
<text><location><page_3><loc_9><loc_12><loc_92><loc_16></location>which is timelike when (1 -kr 2 ) f ( t, r ) < 0, null for (1 -kr 2 ) f ( t, r ) = 0 and spacelike if we have (1 -kr 2 ) f ( t, r ) > 0. For an emitted signal at the coordinates t 0 and r 0 , when it is absorbed at coordinates t and r simple calculations lead to</text>
<formula><location><page_3><loc_38><loc_8><loc_92><loc_11></location>1 + z = λ λ 0 = a ( t ) a ( t 0 ) ( f ( t, r ) f ( t 0 , r 0 ) ) 1 2 , (21)</formula>
<text><location><page_4><loc_9><loc_86><loc_92><loc_93></location>as induced redshift due to the universe expansion and factor f ( t, r ). Redshift will diverge when f ( t 0 , r 0 ) goes to zero or 1 + z -→ ∞ . This divergence as the signal of singularity is independent of the curvature scalar ( k ), unlike the Mcvittie's solution and its various generalizations [38, 39], which shows that our solutions are compatible with the FRW background. As a desired expectation, it is obvious that the FRW result is covered when f ( t 0 , r 0 ) = f ( t, r ) = 1. The only non-diagonal term of the Einstein tensor is</text>
<formula><location><page_4><loc_32><loc_82><loc_92><loc_85></location>G tr = -1 -kr 2 f ( t, r ) a ( t ) 3 r ( a ( t ) ˙ f ( t, r ) -f ' ( t, r )˙ a ( t ) r ) , (22)</formula>
<text><location><page_4><loc_9><loc_79><loc_83><loc_80></location>which (˙) and ( ' ) are derivatives with respect to time and radius, respectively. Using ∂f ∂t = ˙ a ∂f ∂a , one gets</text>
<formula><location><page_4><loc_31><loc_74><loc_92><loc_77></location>G tr = -(1 -kr 2 )˙ a ( t ) f ( a ( t ) , r ) a ( t ) 3 r ( a ( t ) ˜ f ( a ( t ) , r ) -f ' ( a ( t ) , r ) r ) , (23)</formula>
<text><location><page_4><loc_9><loc_71><loc_83><loc_73></location>where ˜ f ( a ( t ) , r ) = ∂f ∂a . In order to get perfect fluid solutions, we impose condition G tr = 0 and reach to</text>
<formula><location><page_4><loc_37><loc_67><loc_92><loc_70></location>f ( t, r ) = f ( a ( t ) r ) = ∑ n b n ( a ( t ) r ) n . (24)</formula>
<text><location><page_4><loc_9><loc_57><loc_92><loc_66></location>Although Eq. (24) includes numerous terms, but the slow expansion approximation helps us to attribute physical meaning to the certain coefficients b n . Since G tr = 0, we should stress that here that there is no redial flow and thus, the backreaction effect is zero [19, 20], which means that there is no energy accretion in these solutions [45]. Finally and briefly, we see that the perfect fluid concept is in line with the no energy accretion condition. The only answer which is independent of the rate of expansion can be obtained by condition b n = δ n 0 which is yielding the FRW solution.</text>
<section_header_level_1><location><page_4><loc_24><loc_53><loc_77><loc_54></location>III. MCVITTIE LIKE SOLUTION IN THE FRW BACKGROUND</section_header_level_1>
<text><location><page_4><loc_10><loc_50><loc_64><loc_51></location>The mcVittie's solution in the flat FRW background can be written as [16]</text>
<formula><location><page_4><loc_28><loc_44><loc_92><loc_48></location>ds 2 = -( 1 -M 2 a ( t )˜ r 1 + M 2 a ( t )˜ r ) 2 dt 2 + a ( t ) 2 (1 + M 2 a ( t )˜ r ) 4 [ d ˜ r 2 + ˜ r 2 d Ω 2 ] . (25)</formula>
<text><location><page_4><loc_9><loc_42><loc_69><loc_43></location>This metric possess symmetries same as metric (18). ˜ r is isotropic radius defined by:</text>
<formula><location><page_4><loc_44><loc_38><loc_92><loc_40></location>r = ˜ r (1 + M 2˜ r ) 2 . (26)</formula>
<text><location><page_4><loc_9><loc_33><loc_92><loc_37></location>There is a redshift singularity at radii ˜ r h = M 2 a ( t ) which yields the radius r h = M 2 a ( t ) (1 + a ( t )) 2 [51]. In addition, ˜ r h is a spacelike hypersurface, and can not point to an event horizon [45].</text>
<text><location><page_4><loc_10><loc_31><loc_70><loc_33></location>Consider f ( a ( t ) r ) = 1 -2 b -1 a ( t ) r . This assumption satisfies condition (24) and leads to</text>
<formula><location><page_4><loc_28><loc_26><loc_92><loc_30></location>ds 2 = -(1 -2 b -1 a ( t ) r ) dt 2 + a ( t ) 2 [ dr 2 (1 -kr 2 )(1 -2 b -1 a ( t ) r ) + r 2 d Ω 2 ] . (27)</formula>
<text><location><page_4><loc_15><loc_23><loc_15><loc_25></location>/negationslash</text>
<text><location><page_4><loc_9><loc_22><loc_92><loc_25></location>For b -1 = 0, this metric will converge to the FRW metric when r -→ ∞ . The Schwarzschild metric is obtainable by putting a ( t ) = 1, b -1 = M and k = 0. Metric suffers from three singularities at a ( t ) = 0 (big bang), r = 0 and</text>
<formula><location><page_4><loc_39><loc_19><loc_92><loc_21></location>f ( a ( t ) r ) = 0 ⇒ a ( t ) r h = 2 b -1 . (28)</formula>
<text><location><page_4><loc_9><loc_10><loc_92><loc_18></location>Third singularity exists if b -1 > 0. In this manner, Eq. (21) will diverge at r 0 = r h . In addition and in contrast to the Gao's solutions, the radii of the redshift singularity ( r h ) in our solutions is independent of the background curvature ( k ), while for the flat case our radius is compatible with the previous works [16, 37, 39]. Also, metric changes its sign at r = r h just the same as the schwarzschild spacetime. In addition, curvature scalars diverge at this radius as well as the mcVittie spacetime. Accordingly, this singularity point to a naked singularity which can be considered as alternatives for BHs [46, 47]. In continue, we will point to the some physical and mathematical properties of this</text>
<text><location><page_5><loc_9><loc_90><loc_92><loc_93></location>singularity which has the same behaviors as event horizon if one considers slow expansion approximation. The surface area integration at this radius leads to</text>
<formula><location><page_5><loc_34><loc_86><loc_92><loc_89></location>A = ∫ √ σdθdφ = 4 πr 2 h a ( t ) 2 = 16 π ( b -1 ) 2 . (29)</formula>
<text><location><page_5><loc_9><loc_82><loc_92><loc_85></location>The main questions that arise here are: what is the nature of b -1 ? and can we better clarify the meaning of r h ? For these purposes, we consider the slow expansion approximation ( a ( t ) ≈ c ), define new coordinate η = cr and get</text>
<formula><location><page_5><loc_24><loc_77><loc_92><loc_81></location>ds 2 ≈ -(1 -2 b -1 η ) dt 2 + dη 2 (1 -k ' η 2 )(1 -2 b -1 η ) + η 2 dθ 2 + η 2 sin ( θ ) 2 dφ 2 , (30)</formula>
<text><location><page_5><loc_9><loc_66><loc_92><loc_77></location>where k ' = k c 2 . In these new coordinates, ( t, η, θ, φ ), and from Eq. (20) it is apparent that for b -1 > 0, hypersurface with equation η = η h = 2 b -1 is a null hypersurface. When our approximation is broken, then η h may not be actually a null hypersurface, despite its resemblance to that. We call this null hypersurface a quasi event horizon which is signalling us an object like a BH and we refer to that as a quasi BH. From now, we assume b -1 > 0, the reason of this option will be more clear later, when we debate mass. Therefore by the slow expansion approximation, r h (= 2 b -1 c ) plays the role of the co-moving radius of event horizon and it is decreased with time. In order to find an answer to the first question about the physical meaning of b -1 , we use Komar mass:</text>
<formula><location><page_5><loc_40><loc_62><loc_92><loc_65></location>M = 1 4 π ∫ S n α σ β /triangleinv α ξ β t dA, (31)</formula>
<text><location><page_5><loc_9><loc_57><loc_92><loc_61></location>where ξ β t is the timelike Killing vector of spacetime. Since the Komar mass is only definable for the stationary and asymptotically flat spacetimes [48], one should consider the flat case ( k = 0) and then by bearing the spirit of the stationary limit in mind (the slow expansion approximation) tries to evaluate Eq. (31).</text>
<text><location><page_5><loc_10><loc_53><loc_92><loc_56></location>Consider n α = √ 1 -2 b -1 a ( t ) r δ t α and σ β = a ( t ) √ 1 -2 b -1 a ( t ) r δ r β as the unit timelike and unit spacelike four-vectors, respectively.</text>
<text><location><page_5><loc_9><loc_52><loc_78><loc_53></location>Now using Eq. (31) and bearing the spirit of the slow expansion approximation in mind, one gets</text>
<formula><location><page_5><loc_38><loc_47><loc_92><loc_51></location>M = 1 4 π ∫ S n α σ β Γ β αt dA = b -1 , (32)</formula>
<text><location><page_5><loc_9><loc_41><loc_92><loc_46></location>which is compatible with the no energy accretion condition ( G tr = 0). In addition, we will find the same result as Eq. (32), if we considered the flat case ( k = 0) of metric (30) and use n α = √ 1 -2 b -1 η δ t α and σ β = 1 √ 1 -2 b -1 η δ η β . Since</text>
<text><location><page_5><loc_9><loc_33><loc_92><loc_42></location>the integrand is independent of the scale factor ( a ( t )), the slow expansion approximation does not change the result of integral. But, the accessibility of the slow expansion approximation is necessary if one wants to evaluate the Komar mass for dynamical spacetimes [48]. Indeed, this situation is the same as what we have in the quasi-equilibrium thermodynamical systems, where the accessibility of the quasi-equilibrium condition lets us use the equilibrium formulation for the vast thermodynamical systems [49]. It is obvious that for avoiding negative mass, we should have b -1 > 0. Relation to the Komar mass of the mcVittie's solution can be written as [16, 39]</text>
<formula><location><page_5><loc_43><loc_29><loc_92><loc_32></location>M mcVittie = M a ( t ) . (33)</formula>
<text><location><page_5><loc_9><loc_21><loc_92><loc_28></location>In addition, some studies show that the Komar mass is just a metric parameter in the mcVittie spacetime [41, 42, 45]. Indeed, Hawking-Hayward quasi-local mass satisfies ˙ M = 0, which is compatible with G rt = 0 and indicates that there is no redial flow and thus the backreaction effect, in the mcVittie's solution [19-21, 45]. In order to clarify the mass notion in the mcVittie spacetime, we consider the slow expansion approximation of the mcVittie spacetime which yields</text>
<formula><location><page_5><loc_32><loc_16><loc_92><loc_20></location>ds 2 ≈ -( 1 -M 2 η 1 + M 2 η ) 2 dt 2 +(1 + M 2 η ) 4 [ dη 2 + η 2 d Ω 2 ] . (34)</formula>
<text><location><page_5><loc_9><loc_12><loc_92><loc_15></location>This metric is signalling us that the M may play the role of the mass in the mcVittie spacetime. In addition, by defining new radii R as</text>
<formula><location><page_5><loc_41><loc_8><loc_92><loc_11></location>R ( t, r ) = a ( t )˜ r (1 + M 2˜ r ) 2 , (35)</formula>
<text><location><page_6><loc_9><loc_92><loc_47><loc_93></location>one can rewrite the mcVittie spacetime in the form of</text>
<formula><location><page_6><loc_25><loc_86><loc_92><loc_91></location>ds 2 = -(1 -2 M R -H 2 R 2 ) dt 2 -2 HR √ 1 -2 M R dtdR + dR 2 1 -2 M R + R 2 d Ω 2 , (36)</formula>
<text><location><page_6><loc_9><loc_78><loc_92><loc_85></location>where H = ˙ a a [50]. This form of the mcVittie spacetime indicates these facts that the Komar mass is a metric parameter and M is the physical mass in this spacetime [50]. Finally, we see that the results of the slow expansion approximation (Eq. (34)) and Eq. (36) are in line with the result of the study of the Hawking-Hayward quasi-local mass in the mcVittie spacetime [41, 42, 45, 50]. For the flat case ( k = 0) of our spacetime (Eq. (27)), by considering Eq. (33) and following the slow expansion approximation, we reach at</text>
<formula><location><page_6><loc_29><loc_73><loc_92><loc_77></location>ds 2 ≈ -(1 -2 M η ) dt 2 + dη 2 (1 -2 M η ) + η 2 dθ 2 + η 2 sin ( θ ) 2 dφ 2 . (37)</formula>
<text><location><page_6><loc_9><loc_71><loc_33><loc_72></location>Also, if we define new radius R as</text>
<formula><location><page_6><loc_43><loc_67><loc_92><loc_70></location>r = R a (1 + M 2 R ) 2 , (38)</formula>
<text><location><page_6><loc_9><loc_65><loc_16><loc_66></location>we obtain</text>
<formula><location><page_6><loc_27><loc_57><loc_92><loc_64></location>ds 2 = -( (1 -M 2 R ) 2 (1 + M 2 R ) 2 -R 2 H 2 (1 + M 2 R ) 6 (1 -M 2 R ) 2 ) dt 2 -2 RH (1 + M 2 R ) 5 (1 -M 2 R ) dtdR (39) + (1 + M 2 R ) 4 [ dR 2 + R 2 d Ω 2 ] .</formula>
<text><location><page_6><loc_9><loc_46><loc_92><loc_56></location>Both of the equations (37) and (39) as well as the no energy accretion condition suggest that, unlike the mcVittie's spacetime, the Komar mass may play the role of the mass in our solution. From Eq. (39) it is apparent that R = M 2 points to the spacelike hypersurface where, in the metric (36), R = 2 M points to the null hypersurface. In the next subsection and when we debate thermodynamics, we will derive the same result for the mass notion in our spacetime.Only in the a ( t ) = 1 limit (the Schwarzschild limit), Eqs. (39) and (25) will be compatible which shows that our spacetime is different with the mcVittie's. Let us note that the obtained metric (Eq. (39)) is consistent with Eq. (36), provided we take M = 0 (the FRW limit).</text>
<section_header_level_1><location><page_6><loc_36><loc_42><loc_65><loc_43></location>Horizons, energy and thermodynamics</section_header_level_1>
<text><location><page_6><loc_10><loc_39><loc_88><loc_40></location>There is an apparent horizon in accordance with the FRW background which can be evaluated from Eq. (6):</text>
<formula><location><page_6><loc_35><loc_35><loc_92><loc_38></location>(1 -kr 2 ap )(1 -2 M a ( t ) r ap ) 2 -r 2 ap ˙ a ( t ) 2 = 0 . (40)</formula>
<text><location><page_6><loc_9><loc_28><loc_92><loc_33></location>This equation covers the FRW results in the limit of M -→ 0 ( see Eq. (7)). In addition, one can get the Schwarzschild radius by considering ˙ a ( t ) = 0, which supports our previous definition for b -1 . Calculations for the flat case yield four solutions. The only solution which is in full agreement with the limiting situation of the FRW metric (in the limit of zero M ) is</text>
<formula><location><page_6><loc_41><loc_23><loc_92><loc_27></location>r ap = 1 + √ 1 -8 HM 2˙ a . (41)</formula>
<text><location><page_6><loc_9><loc_21><loc_57><loc_22></location>Therefore, the physical radius of apparent horizon ( ξ ap = a ( t ) r ap ) is</text>
<formula><location><page_6><loc_41><loc_17><loc_92><loc_21></location>ξ ap = 1 + √ 1 -8 HM 2 H , (42)</formula>
<text><location><page_6><loc_9><loc_13><loc_92><loc_16></location>which is similar to the conformal BHs [19]. It is obvious that in the limit of M -→ 0, the radius for the apparent horizon of the flat FRW is recovered. For the surface gravity of apparent horizon, one can use Eq. (8) and gets:</text>
<formula><location><page_6><loc_27><loc_8><loc_92><loc_12></location>κ = κ FRW (1 -2 M a ( t ) r ap ) 2 + M a ( t ) 2 [ 1 r 2 ap + 1 (1 -2 M a ( t ) r ap ) 2 (a ( t ) + 2 ˙ a ( t ) 2 a ( t ) )] , (43)</formula>
<text><location><page_7><loc_9><loc_89><loc_92><loc_93></location>where h ab = diag ( -1 1 -2 M a ( t ) r , 1 -2 M a ( t ) r a ( t ) 2 ), r ap is the apparent horizon co-moving radius (41) and κ FRW is the surface gravity of the flat FRW manifold</text>
<formula><location><page_7><loc_40><loc_85><loc_92><loc_88></location>κ FRW = -˙ a ( t ) 2 + a ( t )a ( t ) 2 a ( t )˙ a ( t ) . (44)</formula>
<text><location><page_7><loc_9><loc_79><loc_92><loc_84></location>The schwarzschild limit ( κ = 1 4 M ) is obtainable by inserting a ( t ) = 1 in Eq. (43). In the limiting case M -→ 0, Eq. (43) is reduced to the surface gravity of the flat FRW spacetime, as a desired result. The Misner-Sharp mass inside radius ξ for this spherically symmetric spacetime is defined as [52]:</text>
<formula><location><page_7><loc_40><loc_75><loc_92><loc_78></location>M MS = ξ 2 (1 -h ab ∂ a ξ∂ b ξ ) . (45)</formula>
<text><location><page_7><loc_9><loc_71><loc_92><loc_74></location>Because this definition does not yield true results in some theories such as the Brans-Dicke and scalar-tensor gravities, we are pointing to the Gong-Wang definition of mass [53]:</text>
<formula><location><page_7><loc_40><loc_67><loc_92><loc_70></location>M GW = ξ 2 (1 + h ab ∂ a ξ∂ b ξ ) . (46)</formula>
<text><location><page_7><loc_9><loc_54><loc_92><loc_66></location>It is apparent that, for the apparent horizon, Eqs. (45) and (46) yield the same result as M GW = M MS = ξ ap 2 . In the limit of M -→ 0, the FRW's results are recovered and we reach to M GW = M MS = ρV as a desired result [10]. Using Eqs. (45) and (46) and taking the slow expansion approximation into account, we reach to M GW = M MS /similarequal M as the mass of quasi BH. Also, this result supports our previous guess about the Komar mass as the physical mass in our solution, and is in line with the result of Eqs. (37) and (39). For the Mcvittie metric, Eqs. (45) and (46) yield M GW = M MS /similarequal M 4 as the confined mass to radius ξ h = a ( t )˜ r h = M 2 . Also, Eqs. (32), (45) and (46) leads to the same result in the Schwarzschild's limit ( M = M GW = M MS = M ). For the flat background, using metric (30), Eq. (13) and inserting results into Eq. (14), one gets</text>
<formula><location><page_7><loc_45><loc_50><loc_92><loc_53></location>T /similarequal 1 8 πM , (47)</formula>
<text><location><page_7><loc_9><loc_47><loc_92><loc_49></location>for the temperature on the surface of quasi horizon. The same calculations yield similar results, as the temperature on the horizon of the Mcvittie's solution. For the conformal Schwarzschild BH, the same analysis leads to</text>
<formula><location><page_7><loc_44><loc_42><loc_92><loc_45></location>T /similarequal 1 8 πa ( t ) M , (48)</formula>
<text><location><page_7><loc_9><loc_37><loc_92><loc_41></location>which shows that the a ( t ) M plays the role of mass, and is compatible with the energy accretion in the conformal BHs [19-21, 51]. Again, we see that the temperature analysis can support our expectation from M as the physical mass in our solutions. For the area of quasi horizon, we have</text>
<formula><location><page_7><loc_35><loc_33><loc_92><loc_36></location>A = ∫ √ σdθdφ = 4 πa ( t ) 2 r 2 h = 16 πM 2 . (49)</formula>
<text><location><page_7><loc_9><loc_28><loc_92><loc_31></location>In the mcVittie spacetime, this integral leads to A = 16 πM 2 . In order to vindicate our approximation, we consider S = A 4 for the entropy of quasi BH. In continue and from Eq. (47), we get</text>
<formula><location><page_7><loc_43><loc_25><loc_92><loc_27></location>TdS /similarequal dM = dE. (50)</formula>
<text><location><page_7><loc_34><loc_22><loc_34><loc_24></location>/negationslash</text>
<text><location><page_7><loc_9><loc_9><loc_92><loc_24></location>Whereas, we reach to TdS /similarequal dM = dE for the mcVittie spacetime. In the coordinates ( t, η, θ, φ ), we should remind that, unlike the mcVittie spacetime, E = M GW = M MS /similarequal M is valid for quasi BH and the work term can be neglected as the result of slow expansion approximation ( dW ∼ 0) [51]. Finally and unlike the mcVittie's horizon, we see that TdS /similarequal dE is valid on the quasi event horizon. This result points us to this fact that the first law of the BH thermodynamics on quasi event horizon will be satisfied if we use either the Gong-Wang or the Misner-Sharp definitions for the energy of quasi BH. TdS /similarequal dE is valid for the conformal Schwarzschild BH, too [51]. For the flat background, we see that the surface area at redshift singularity in our spacetime is equal to the mcVittie metric which is equal to the Schwarzschild metric. In continue and by bearing the slow expansion approximation in mind, we saw that the temperature on quasi horizon is like the Schwarzschild spacetime [19]. In addition, we saw that the quality of the validity of the first law of the BH thermodynamics on quasi event horizon is like the conformal Schwarzschild BH's and differs from the mcVittie's solution.</text>
<text><location><page_8><loc_9><loc_90><loc_92><loc_93></location>In another approach and for the mcVittie spacetime, if we use the Hawking-Hayward definition of mass as the total confined energy to the hypersurface ˜ r = M 2 a ( t ) , we reach to</text>
<formula><location><page_8><loc_43><loc_87><loc_92><loc_89></location>TdS /similarequal dM = dE, (51)</formula>
<text><location><page_8><loc_9><loc_80><loc_92><loc_86></location>where we have considered the slow expansion approximation. In addition, Eq. (51) will be not valid, if one uses the Komar mass (33). Finally, we saw that the first law of thermodynamics will be approximately valid in the mcVittie's solution, if one uses the Hawking-Hayward definition of energy. Also, none of the Komar, Misner-Sharp and Gong-Wang masses can not satisfy the first law of thermodynamics on the mcVittie's horizon.</text>
<section_header_level_1><location><page_8><loc_39><loc_76><loc_62><loc_77></location>IV. OTHER POSSIBILITIES</section_header_level_1>
<text><location><page_8><loc_9><loc_70><loc_92><loc_74></location>According to what we have said, it is obvious that there are two other meaningful sentences in expansion (24). The first term is due to n = -2 and points to the charge, where the second term comes from n = 2 and it is related to the cosmological constant. Therefore, the more general form of f ( t, r ) can be written as:</text>
<formula><location><page_8><loc_34><loc_66><loc_92><loc_69></location>f ( t, r ) = 1 -2 M a ( t ) r + Q 2 ( a ( t ) r ) 2 -1 3 Λ( a ( t ) r ) 2 , (52)</formula>
<text><location><page_8><loc_9><loc_52><loc_92><loc_64></location>where we have considered the slow expansion approximation and used these definitions b -2 ≡ Q 2 and b 2 ≡ -1 3 Λ. Imaginary charge ( b -2 < 0) and the anti De-Sitter (Λ < 0) solutions are allowed by this scheme, but these possibilities are removed by the other parts of physics. Consider Eq. (52) when Λ = 0, there are two horizons located at r + = M + √ M 2 -Q 2 a ( t ) and r -= M -√ M 2 -Q 2 a ( t ) . These radiuses are same as the Gao's flat case [39]. In the low expansion regime ( a ( t ) ∼ c ), these radiuses point to the event and the Coushy horizons, as the Riessner-Nordstorm metric [9]. Hence, we refer to them as quasi event and quasi Coushy horizons. The case with Q = 0, M = 0 and Λ > 0 has attractive properties. Because in the low expansion regime ( a ( t ) /similarequal c ), one can rewrite this case as</text>
<formula><location><page_8><loc_34><loc_48><loc_92><loc_52></location>ds 2 ≈ -(1 -Λ 3 η 2 ) dt 2 + dη 2 (1 -Λ 3 η 2 ) + η 2 d Ω 2 . (53)</formula>
<text><location><page_8><loc_9><loc_45><loc_92><loc_47></location>This is nothing but the De-Sitter spacetime with cosmological constant Λ, which points to the current acceleration era.</text>
<section_header_level_1><location><page_8><loc_40><loc_40><loc_60><loc_41></location>Horizons and temperature</section_header_level_1>
<text><location><page_8><loc_9><loc_36><loc_92><loc_38></location>Different f ( t, r ) yield apparent horizons with different locations, and one can use Eqs. (6) and (8) in order to find the location and the temperature of apparent horizon. For every f ( t, r ), using the slow expansion regime, we get:</text>
<formula><location><page_8><loc_38><loc_31><loc_92><loc_34></location>ds 2 ≈ -f ( η ) dt 2 + dη 2 f ( η ) + η 2 d Ω 2 . (54)</formula>
<text><location><page_8><loc_9><loc_27><loc_92><loc_30></location>Now, the location of horizons and their surface gravity can be evaluated by using Eq. (13). Their temperature is approximately equal to Eq. (14), or briefly:</text>
<formula><location><page_8><loc_44><loc_23><loc_92><loc_26></location>T i /similarequal f ' ( η ) 4 π | η hi , (55)</formula>
<text><location><page_8><loc_9><loc_20><loc_65><loc_22></location>where ( ' ) is derivative with respect to radii η and η hi is the radii of i th horizon.</text>
<section_header_level_1><location><page_8><loc_42><loc_16><loc_58><loc_17></location>V. CONCLUSIONS</section_header_level_1>
<text><location><page_8><loc_9><loc_9><loc_92><loc_14></location>We considered the conformal form of the special group of the non-static spherically symmetric metrics, where it was assumed that the time dependence of the conformal factor is like as the FRW's. We saw that the conformal BHs can be classified as a special subgroup of these metrics. In order to derive the new solutions of the Einstein equations, we have imposed perfect fluid concept and used slow expansion approximation which helps us to clarify the physical</text>
<text><location><page_9><loc_14><loc_84><loc_14><loc_86></location>/negationslash</text>
<text><location><page_9><loc_9><loc_79><loc_92><loc_93></location>meaning of the parameters of metric. Since the Einstein tensor is diagonal, there is no energy accretion and thus the backreaction effect is zero. This imply that the energy (mass) should be constant in our solutions. These new solutions have similarities with earlier metrics that have been presented by others [37-39]. A related metric which is similar to the special class of our solutions was introduced by mcVittie [37, 39]. These similarities are explicit in the flat case (temperature and entropy at the redshift singularity), but the differences will be more clear in the non-flat case ( k = 0), and we pointed to the one of them, when we debate the redshift. In addition and in the flat case, we tried to clear the some of differences between our solution and the mcVittie's. We did it by pointing to the behavior of the redshift singularity in the various coordinates, the mass notion, and thermodynamics. Meanwhile, when our slow expansion approximation is broken then there is no horizon for our solutions. Indeed, these objects can be classified as naked singularities which can be considered as alternatives for BHs [46, 47].</text>
<text><location><page_9><loc_9><loc_70><loc_92><loc_79></location>For the our solutions and similar with earlier works [37-39], the co-moving radiuses of the redshift singularities are decreased by the expansion of universe. Also, unlike the previous works [37-39], the redshift singularities in our solutions are independent of the background curvature. By considering the slow expansion approximation, we were able to find out BH's like behavior of these singularities. We pointed to these objects and their surfaces as quasi BHs and the quasi horizons, respectively. In continue, we introduced the apparent horizon for our spacetime which should be evaluated by considering the FRW background.</text>
<text><location><page_9><loc_9><loc_59><loc_92><loc_70></location>In order to compare the mcVittie's solution with our mcVittie's like solution, we have used the three existing definitions of mass including the Komar mass, the Misner-Sharp mass ( M MS ) and the Gong-Wang mass ( M GW ). We saw that the notion of the Komar mass of quasi BH differs from the mcVittie's solution. Also, in our spacetime, we showed that the M MS and M GW masses yield the same result on the apparent horizon and cover the FRW's result in the limiting situations. In addition, using the slow expansion approximation, we evaluated M MS and M GW on the quasi event horizon of our mcVittie's like solution, which leads to the same result as the Komar mass. In addition, we should express that, the same as the mcVittie spacetime, the energy conditions are not satisfied near the quasi horizon.</text>
<text><location><page_9><loc_9><loc_37><loc_92><loc_58></location>In addition, we have proved that, unlike the mcVittie's solution, the first law of thermodynamics may be satisfied on the quasi event horizon of our mcVittie's like solution, if we use the Komar mass or either M MS or M GW as the confined mass and consider the slow expansion approximation. This result is consistent with previous studies about the conformal BHs [51], which shows that the thermodynamics of our solutions is similar to the conformal BHs'. In order to clarify the mass notion, we think that the full analysis of the Hawking-Hayward mass for our solution is needed, which is out of the scope of this letter and should be considered as another work, but our resolution makes this feeling that the predictions by either the slow expansion approximation or using the suitable coordinates for describing the metric for mass, may be in line with the Hawking-Hayward definition of energy, and have reasonable accordance with the Komar, M MS and M GW masses of our solutions. Indeed, this final remark can be supported by the thermodynamics considerations and the no energy accretion condition ( G tr = 0). Moreover, we think that, in dynamic spacetimes, the thermodynamic considerations along as the slow time varying approximation can help us to get the reasonable assumptions for energy and thus mass. Finally, we saw that the first law of thermodynamics will be approximately valid in the mcVittie's and our solutions if we use the Hawking-Hayward definition of the mass in the mcVittie spacetime and the Komar mass as the physical mass in our solution, respectively. In continue, the more general solutions such as the charged quasi BHs and the some of their properties have been addressed.</text>
<text><location><page_9><loc_9><loc_31><loc_92><loc_37></location>Results obtained in this paper may help achieving a better understanding of black holes in a dynamical background. From a phenomenological point of view, this issue is important since after all, any local astrophysical object lives in an expanding cosmological background. Finally, we tried to explore the concepts of mass, entropy and temperature in a dynamic spacetime.</text>
<section_header_level_1><location><page_9><loc_39><loc_27><loc_62><loc_28></location>VI. ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_9><loc_9><loc_22><loc_92><loc_25></location>We are grateful to referee for appreciable comments which led to sensible improvements in this manuscript. This work has been supported financially by Research Institute for Astronomy & Astrophysics of Maragha (RIAAM).</text>
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</document> | [] |
2024arXiv241112834H | https://arxiv.org/pdf/2411.12834.pdf | <document>
<section_header_level_1><location><page_1><loc_9><loc_91><loc_91><loc_96></location>Bringing together African & European research communities with an inclusive astronomy conference</section_header_level_1>
<text><location><page_1><loc_9><loc_88><loc_81><loc_90></location>Chris M. Harrison 1, ⋆ and Leah Morabito 2, † , on behalf of the Organising Committees</text>
<unordered_list>
<list_item><location><page_1><loc_9><loc_86><loc_71><loc_87></location>1 School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK</list_item>
<list_item><location><page_1><loc_9><loc_85><loc_68><loc_86></location>2 Centre for Extragalactic Astronomy, Department of Physics, Durham University, Durham DH1 3LE, UK</list_item>
<list_item><location><page_1><loc_9><loc_83><loc_60><loc_85></location>⋆ Email : [email protected], † Email : [email protected]</list_item>
</unordered_list>
<section_header_level_1><location><page_1><loc_10><loc_80><loc_17><loc_81></location>Abstract</section_header_level_1>
<text><location><page_1><loc_10><loc_60><loc_90><loc_80></location>We report on an international scientific conference, where we brought together the African and European academic astronomy communities. This conference aimed to bridge the gap between those in position of privilege, with ease of access to international networking events (i.e., the typical experience of those affiliated with Western institutions), with those who have been historically excluded (affecting the majority of African scientists/institutions). We describe how we designed the conference around cutting-edge problems in the research field, but with a large focus on building networking and professional relationships. Significant effort went into: (1) ensuring a diverse representation of participants; (2) practically and financially supporting those who may have never attended an international conference and; (3) creating an inclusive and supportive environment through a careful programme of activities, both before and during the event. Throughout this process maintaining scientific integrity was a core commitment. We summarise some of the successes, challenges, and lessons learnt from organising this conference. We also present feedback obtained from participants, which demonstrates an overall achievement of our objectives. This is all combined to provide some key recommendations for any groups, from any research field, who wishes to lead similar initiatives.</text>
<section_header_level_1><location><page_1><loc_15><loc_54><loc_43><loc_55></location>Motivation & Scientific Context</section_header_level_1>
<text><location><page_1><loc_9><loc_17><loc_49><loc_53></location>Astronomy and space science can be used as an important tool for development and for achieving the United Nations (UN) Sustainable Development Goals (SDGs) 1 through: education; socio-economic growth via advances in science and technology; and promoting international peace and diplomacy [1, 2]. Partly driven by these development goals, astronomy and space science research is seeing significant growth on the continent of Africa. In recent years, a major contributing factor is the planning and development of the revolutionary radio observatory, the Square Kilometre Array (SKA) 2 . This ambitious international project will be situated across Africa and Australia. Other major astronomy projects on the African continent include the South African Astronomical Observatory (SAAO) and the High Energy Stereoscopic System (H.E.S.S) observatory in Namibia 3 [more examples in 2]. In parallel to these projects, the continent is seeing growth in postgraduate astronomy programmes, and international initiatives to support early-career African astronomers, such as the 'Development in Africa with Radio Astronomy' project [DARA, 3] and other strategic partnerships (e.g., between the United Kingdom's Science and Technology Facilities Council [STFC], and the South African National Research Foundation 4 ).</text>
<text><location><page_1><loc_12><loc_16><loc_49><loc_17></location>Despite this positive growth of projects and initiatives,</text>
<text><location><page_1><loc_51><loc_42><loc_91><loc_56></location>developing astronomy research in Africa faces many on-going challenges. These are shared across many research topics and institutions across the continent, and for people of African origin (but may be working in other countries). These relate to: limited opportunities and resources; the limited retention of young people into higher education; issues with equity and inclusion (including challenges around travel visas); and access to the latest international knowledge, facilities, and networks [4, 5].</text>
<text><location><page_1><loc_51><loc_21><loc_91><loc_42></location>Motivated by addressing some of these these challenges, in July 2024, Durham University and Newcastle University jointly hosted an international astronomy conference: 'AGN Populations Across Continents and Cosmic Time'. The main goal of this conference was to hold a scientific meeting, discussing timely scientific problems, but with a broader impact to integrate African researchers into the international community. Additionally, European researchers can often operate in their own continental spheres, and this conference offered the opportunity for them to broaden their networks. Therefore, this workshop aimed to facilitate networking that will strengthen ties between the European and African scientific communities.</text>
<section_header_level_1><location><page_1><loc_51><loc_17><loc_65><loc_18></location>Scientific context</section_header_level_1>
<text><location><page_1><loc_51><loc_8><loc_91><loc_16></location>Supermassive black holes, with masses of millions to billions times that of the Sun, are located at the nuclei of galaxies. These black holes grow by the accretion of gas. As this gas falls towards the black holes, it becomes extremely bright. Furthermore, jets of charged particles can be launched in the vicinity of these accreting black</text>
<text><location><page_2><loc_9><loc_84><loc_49><loc_92></location>holes, and these jets can extend over vast distances within the galaxies and beyond. When these accretion events are detected with astronomical observatories on Earth or in space, they are known as 'Active Galactic Nuclei' (AGN).</text>
<text><location><page_2><loc_9><loc_57><loc_49><loc_84></location>Both observational studies and simulations have shown that the energy released by AGN is important for galaxy evolution. However, there continue to be major research challenges in the field, including: (1) obtaining a complete census of AGN events; (2) understanding the detailed physical structure of the material associated with AGN; (3) establishing the properties of AGN host galaxies and the details of how AGN regulate galaxy growth, and; (4) identifying the key techniques and datasets required to make significant progress in answering these scientific questions over the coming decade [review in e.g., 6]. The scientific scope of our conference was to cover these four main themes. 5 Due to the focus on the African scientific community, the workshop looked to particularly showcase work on this scientific topic that uses SKA pathfinder observatories in addition to other astronomical facilities based on the African continent.</text>
<section_header_level_1><location><page_2><loc_10><loc_53><loc_48><loc_54></location>Selecting and Supporting the Participants</section_header_level_1>
<text><location><page_2><loc_9><loc_38><loc_49><loc_51></location>Careful planning went into selecting participants. During selection, our primary objectives were: (1) ensuring a scientifically productive meeting, with participants working on the relevant research problems, balanced across the four key themes; and (2) having significant participation from those of African institutions and/or with African origin (aiming for ≈ 30-50%). We detail the processes of selecting the organisational committees, choosing invited speakers, and the selection of participants.</text>
<section_header_level_1><location><page_2><loc_9><loc_35><loc_44><loc_36></location>Selecting the LOC, SOC and Invited Speakers</section_header_level_1>
<text><location><page_2><loc_9><loc_12><loc_49><loc_34></location>Our Local Organisation Committee (LOC) was put together primarily from the research teams (i.e., postdoctoral researchers and PhD students) of the workshop co-chairs (i.e., Chris Harrison, based at Newcastle University and Leah Morabito, based at Durham University). Therefore, they had a direct scientific interest in the workshop. The LOC were responsible for the typical activities of organising a conference (e.g., coordinating with participants, helping with administrative tasks, supporting the activities during the week etc.). Within the LOC there were 4 researchers with an African origin (representing four different African countries), although they were based at UK institutions at the time. Their perspective was crucial for helping to identify - and overcome - many of the challenges that African participants would face</text>
<text><location><page_2><loc_51><loc_86><loc_91><loc_92></location>in travelling to the UK, as well as for developing a programme of activities to ensure participants felt included, supported, and comfortable to contribute. A photograph of the LOC is shown in the top panel of Figure A.1.</text>
<text><location><page_2><loc_51><loc_54><loc_91><loc_85></location>The Scientific Organising Committee (SOC) were responsible for designing the scientific programme, choosing invited speakers, and selecting the conference participants. We ensured that we had significant representation of scientists at African institutions as well as representatives from the UK and wider Europe. Of the 9 SOC members, 4 were based at African institutions (from 3 separate countries). Furthermore, the selection of the SOC was made to ensure representation of established experts covering a diverse range of the scientific topics that were the focus of the conference. Consequently, the SOC not only provided crucial scientific input, but additionally were able to help design a participant selection process that was suitable for African-affiliated researchers (especially early-career researchers). Furthermore, they suggested activities for the programme that would be inclusive to those who have little-to-no experience of international conferences. Importantly, these SOC members could locally advertise the conference across the African community. A photograph of the SOC is shown in the bottom panel of Figure A.1.</text>
<text><location><page_2><loc_51><loc_30><loc_91><loc_54></location>The SOC chose to select only a small number of invited speakers; enough to highlight the programme with some internationally-known experts in the field, but not too many as to take significant space in the programme away from contributions by regular participants. The SOC's aims of the invited speaker selection were to select four established experts who would be able to provide an introductory review talk across the four main themes of the conference. Further requirements were to include at least one speaker of African origin and affiliation, having representation of more than one gender across all the speakers, ensuring the speakers would provide a talk at an accessible level to a broad range of career stage participants, and only selecting speakers willing to actively engage with the broader development and networking goals of the conference.</text>
<text><location><page_2><loc_53><loc_20><loc_89><loc_27></location>Key Recommendation: a cross-community perspective should be considered a requirement of any LOC and SOC for any conferences with similar goals to ours, relating to bringing together research communities with diverse cultures and experiences.</text>
<section_header_level_1><location><page_2><loc_51><loc_15><loc_68><loc_17></location>Selecting participants</section_header_level_1>
<text><location><page_2><loc_51><loc_8><loc_91><loc_15></location>We planned for ≈ 110 conference participants. This number was small enough to facilitate productive discussions and effective networking, but large enough to have a number of participants from across different continents and career stages. We aimed for a fully in-person con-</text>
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<caption>Figure 1. Map showing the countries that were represented by the participants' affiliations and origin.</caption>
</figure>
<text><location><page_3><loc_9><loc_57><loc_49><loc_64></location>rence, to maximise the benefits of social interaction, building relationships, and ad-hoc informal discussions. Partial online participation was made available for participants who had challenges and could not travel (e.g., personal circumstances or visa complications).</text>
<text><location><page_3><loc_9><loc_42><loc_49><loc_56></location>To achieve our objective of a high fraction of participation from African astronomers, it was crucial to provide full funding to those who do not have access to any travel funds (the majority of the African-affiliated astronomers). We estimated an average cost per fullyfunded participant travelling from Africa, to be ≈ £1900, to cover: travel, visa costs, accommodation, travel insurance, and local subsistence (including all meals). Our available budget limited us to fully-fund ≈ 35-40 participants, depending on exact final costs per participant.</text>
<text><location><page_3><loc_9><loc_26><loc_49><loc_41></location>Throughout the selection process we kept track of both the required funding to support African participants and the participant demographics. For the demographics, we focused on a proactive approach to ensure a high level of representation of those who are African affiliated and/or have African origin. We also kept track of the distribution of career stages and gender; however, our selection approach naturally led to good representation across these two characteristics, and required no positive actions to address the diversity.</text>
<text><location><page_3><loc_9><loc_14><loc_49><loc_26></location>Along with a dedicated website, we released an application form for conference participation 10 months before the actual conference. The deadline for submitting an application was ≈ 7 months before the conference dates. This long lead time was critical to ensure sufficient time to support travel and visa applications, as well as plan and to conduct pre-conference activities (described later).</text>
<text><location><page_3><loc_9><loc_8><loc_49><loc_13></location>Weaimed to include participants who were dedicated to contribute to, or benefit from, the wider networking aspects of the conference. Therefore, we did not exclusively request abstracts for talks/posters in the applica-</text>
<text><location><page_3><loc_51><loc_54><loc_91><loc_64></location>tion form. We concluded that requiring a high-quality scientific contribution (talk/poster) abstract to attend could bias against those who had so far had limited opportunities to work on significant, internationally cutting-edge research projects. Therefore, we had three separate categories in the application form, which were used for selecting participants. These were:</text>
<unordered_list>
<list_item><location><page_3><loc_53><loc_50><loc_91><loc_53></location>1. A standard scientific abstract for a contributed talk and/or poster.</list_item>
<list_item><location><page_3><loc_53><loc_46><loc_91><loc_49></location>2. Proposal for a collaborative research project, which could be discussed during the conference.</list_item>
<list_item><location><page_3><loc_53><loc_43><loc_79><loc_45></location>3. A statement of interest to attend.</list_item>
</unordered_list>
<text><location><page_3><loc_51><loc_30><loc_91><loc_42></location>This final category was designed to capture those who may not yet have established research outputs. For example, PhD students who were at the beginning their academic studies but were strongly motivated by the opportunities of the conference. Applicants could apply using one, two, or three of these categories. These three categories were each treated separately. In effect, every applicant had up to three possibilities to be selected.</text>
<text><location><page_3><loc_51><loc_24><loc_91><loc_30></location>We received a total of 249 applications (once removing clearly inapplicable or duplicate applications). Of these, 75 were from people with an African origin and 74 were from African affiliations (not mutually exclusive).</text>
<text><location><page_3><loc_51><loc_12><loc_91><loc_24></location>The SOC scored the submitted applications with a blinded approach (i.e., with no information on the applicants' names, affiliations etc.). Every SOC member assessed every submission for application categories 2. and 3. from the above list. However, depending on the scientific expertise of the SOC member, they only assessed abstracts for talks/posters based on the most applicable scientific theme for their own expertise.</text>
<text><location><page_3><loc_51><loc_8><loc_91><loc_12></location>Each SOC member scored the submissions with an integer grade from 1 (low) to 5 (high). The scores for each submission were then collated and averaged.</text>
<text><location><page_4><loc_9><loc_53><loc_49><loc_92></location>The first pass of participant selection was to take a simple cut in the average score and select everyone with a score above some fixed value. These values were chosen to select the number of contributed talks we aimed for in the final programme (i.e., 47), and then fill the remaining participant places with those who scored highly in the other two submission categories. However, this completely blind and uniform approach to selection was insufficient to obtain the desired representation of African participants. On average, the African submissions scored ≈ 0.6 points lower than other submissions. This gap is unsurprising due to the generally lower levels of available mentoring and peer support (including for tasks such as improving conference abstract drafts) compared to more deeply established research environments in places like the UK. We therefore decided to apply a positive action by up-weighting scores from South African submissions by 0.5 and other African submissions by 0.8. This divide within the African submissions into South Africa and other countries, was because the South African astronomy community is rapidly growing and applicants there tend to have larger peer-to-peer support groups and more mentoring, likely contributing to the typically higher submission scores. This approach ensured good geographical representation from across the continent.</text>
<text><location><page_4><loc_9><loc_28><loc_49><loc_52></location>We had capacity for 120 participants and an ideal target of ≈ 110. The final number of participants attending the conference was 107, of which 9 could only join online due to not being able to travel after unforeseen circumstances. Figure 1 shows a world map highlighting the countries of origin and affiliation that were represented across the conference participants. Overall, 43% of the participants were either from an African institution and/or of African origin. There were 20 different affiliation countries represented, of which 11 were from Africa. The break down of career stage was: 31% senior academic/lecturer; 26% postdoc/fellow; 31% PhD student; and 12% Masters students. For gender, the final breakdown was 49% Male, 49% Female, and 2% Non-binary. A conference photo, including the majority of participants, is shown in Figure A.2.</text>
<text><location><page_4><loc_11><loc_16><loc_47><loc_26></location>Key Recommendations: A long lead time for opening applications is crucial. Multiple categories for applications to attend (not just scientific abstracts) can help increase the diversity of the participants. Initial selection should be blind (anonymous) and based on a scoring rubric but positive action might be required to achieve the objective diversity.</text>
<section_header_level_1><location><page_4><loc_9><loc_11><loc_45><loc_12></location>Financial and practical support of participants</section_header_level_1>
<text><location><page_4><loc_9><loc_8><loc_49><loc_11></location>Offers for conference participation were sent out ≈ 6 months before the conference itself. This long lead-time</text>
<text><location><page_4><loc_51><loc_78><loc_91><loc_92></location>was chosen to ensure sufficient time to help participants who needed to apply for visas, and/or whose travel we needed to support. Indeed, visa applications can take a significant amount of time. For example, one participant had a visa issued only two days before travel. For similar reasons, it was necessary to book travel for fully-funded participants before visa applications were completed, otherwise flight costs could have risen to unaffordable rates as the travel dates approached.</text>
<text><location><page_4><loc_51><loc_55><loc_91><loc_78></location>Of all participants, 38 received full funding from our budget, with a small additional number of participants receiving part funding based on a case-by-case assessment of need (e.g., fee waivers for early-career researchers from any institution/country without large funding resources). As many African-affiliated participants had no means to make significant upfront costs, it was necessary for the conference host university to book the travel and the accommodation directly on their behalf. Due to the nature of the accommodation we used (university colleges), it was also possible to purchase all evening meals for the fully-funded participants in advance of the conference. Fully-funded participants also had their posters printed (when relevant) directly at the conference venue, at no cost to themselves.</text>
<text><location><page_4><loc_51><loc_34><loc_91><loc_55></location>The only costs that fully-funded participants were required to pay for themselves up-front, and later obtain a reimbursement, were the visa costs, travel insurance (as often not provided by the home universities), and local travel costs. We note that costs associated with obtaining visas were sometimes quite high, due to participants needing to travel to consulates (sometimes multiple times). Although this process generally went well, some applicants had no access to personal bank accounts and arrangement had to be made with a third party to make the reimbursement because cash payments were not possible. In some cases, we were not aware of it until after it was too late to help with alternative methods of payment beforehand.</text>
<text><location><page_4><loc_51><loc_17><loc_91><loc_33></location>We supported visa applications by providing the necessary invitation letters to attend the conference. We provided ad-hoc help for some participants on how to complete visa applications. However, with hindsight, we should have provided more specific guidance for those who had not previously applied for visas. For example, some visas were initially rejected on the grounds of not providing sufficient evidence that they had reasons to return home after the conference (e.g., by providing letters from their home universities to prove employment or registration on PhD programmes).</text>
<text><location><page_4><loc_51><loc_8><loc_91><loc_16></location>The participants applied for and received standard visitor visas. These are valid for 6 months, and some participants organised other work visits while they were in the UK. This required some extra coordination, but we were happy to support it. We had some requests during the meeting or after for other work visits, and it</text>
<text><location><page_5><loc_9><loc_75><loc_49><loc_92></location>would have been useful to highlight this opportunity to participants beforehand. This could have allowed participants to make the best use of their visas while they were valid. However, we caution that while visits to the UK for conferences and meeting do not require approval from the Academic Technology Approval Scheme (ATAS), research visits of any length do require approval, at the time of writing. If people wish to join a visit to carry out research along with this, we recommend they apply for ATAS certification at the same time or even before the visa.</text>
<text><location><page_5><loc_9><loc_59><loc_49><loc_75></location>Although the LOC communication to participants directed them to 'contact us if you have any questions or problems', we found that some did not do so until after they completed a task. In some cases (e.g., reimbursement to those who did not have bank accounts), this made fixing the problem more challenging. This may be attributable to the lack of experience or confidence of the participants. Therefore, we would strongly encourage participants to be very communicative at each stage of their diverse visa and travel processes, to give organisers a chance to assist as early as possible</text>
<text><location><page_5><loc_11><loc_37><loc_47><loc_56></location>Key Recommendations: To aid successful visa applications, ensure a long lead time and provide extensive supporting documents and guidelines for participants. Reimbursement factors should carefully be planned with local administrators, and prepayment of all the funded participant's major expenses are crucial. Be very explicit with participants which costs they can, or can not, claim back in advance of any payments being made (e.g., if taxis can be used). Plan for extra costs associated with visa applications and travel insurance. Throughout, clear communication is critical between participants and organisers.</text>
<section_header_level_1><location><page_5><loc_14><loc_30><loc_44><loc_33></location>Programme of Activities & their Evaluation</section_header_level_1>
<text><location><page_5><loc_9><loc_8><loc_49><loc_29></location>We conducted a variety of activities before, and during, the conference to aid the sharing and discussion of research outputs, as well as facilitate networking and collaborations. To assess the success of the activities, we conducted a participant evaluation questionnaire. Participants were asked to complete this during the final plenary session itself, to help increase completion rates. Overall, 73/107 participants (68%) completed the evaluation form. This was a representative sub-sample of all attendees, with ≈ 30% from each of the groups of PhD students, Postdocs/Fellows, and Lecturers, and with ≈ 12% from Masters students. Furthermore, 47% completing the survey were from an African institute and/or African origin. We provide some key results of</text>
<text><location><page_5><loc_51><loc_83><loc_91><loc_92></location>this evaluation alongside the relevant activity description. For the quantitative questions, we split the results by all participants and then just the African affiliated/origin participants, to assess if there were any deviations for this specific group for whom there was a focus on providing an inclusive, welcoming and collaborative environment.</text>
<text><location><page_5><loc_51><loc_78><loc_91><loc_82></location>The following sub-sections describe the activities and their impact, as much as we are able to assess four months after the event.</text>
<section_header_level_1><location><page_5><loc_51><loc_75><loc_87><loc_76></location>Pre-conference communications and activities</section_header_level_1>
<text><location><page_5><loc_51><loc_67><loc_91><loc_74></location>As is increasingly common practice for scientific workshops, we created a Code of Conduct. All participants were made aware that this applied to all activities before and during the workshop, and this was posted on our website.</text>
<text><location><page_5><loc_51><loc_40><loc_91><loc_67></location>Around 1 month before the workshop, we created a collaborative online Slack space, to allow participants to start introducing themselves to each other. Additionally, to encourage participants getting to know each other before the conference, we asked participants to create 2-minute introductory videos. These videos were shared on a private YouTube playlist in the month leading up to the conference. Participants were provided with clear guidelines on how to record these videos on a phone, and what these videos should include (i.e., their name, affiliation, research interests, and what they hope to get out of the conference). Only 35/107 participants created these videos; with mostly the more junior participants participating (the LOC and SOC were all compelled to make a video). Nonetheless, we had several positive comments during the feedback questionnaire about the introductory videos being a good idea, and useful to start to get to know fellow participants.</text>
<text><location><page_5><loc_51><loc_17><loc_91><loc_39></location>The most successful pre-conference networking activity was the 'Buddy' system that was created to pair all African-affiliated early-career researchers with an non African-affiliated early-career researcher. Although this was an opt-in activity, nearly all early-career researchers participated. Buddys were put into contact and encouraged to email/call before the conference to get to know each other, and then to meet each other at the start of the conference. They were also encouraged to provide feedback to each other on their scientific contributions. The comments we received demonstrated this was extremely helpful for feeling prepared for the conference and to make participants feel more confident to integrate and participate in discussions, because they already knew somebody attending the conference.</text>
<text><location><page_5><loc_51><loc_8><loc_91><loc_16></location>Due to the wide range of experience of the participants, where some may not have presented at, nor even attended, an international conferences before, we decided to provide extensive support and guidelines for preparing high-quality scientific contributions. This included: (1) an online workshop on preparing a good</text>
<text><location><page_6><loc_9><loc_75><loc_49><loc_92></location>scientific talk; (2) guidelines and template examples on preparing a good scientific poster; (3) guidelines and examples for preparing a 90 second 'sparkler' talk (more details below). The scientific talk workshop had over 30 attendees (with a similar number watching the recording later on) and the feedback from participants highlighted that the guidelines, examples and templates, were appreciated and extensively used. We believe this preworkshop guidance and training significantly contributed to the consistently high-quality scientific contributions delivered during the conference itself.</text>
<text><location><page_6><loc_11><loc_64><loc_47><loc_73></location>Key Recommendations: Pre-conference networking activities and training/guidance are extremely valuable activities. These help ensure an effective meeting where participants immediately feel confident and welcome, and deliver high-quality scientific contributions.</text>
<section_header_level_1><location><page_6><loc_9><loc_59><loc_29><loc_60></location>Day 1 Networking activity</section_header_level_1>
<text><location><page_6><loc_9><loc_47><loc_49><loc_59></location>After the first welcome and introductory talks on the first day of the conference, we conducted a 1.5 hour networking/ice-breaker activity. We felt this important to create a welcoming atmosphere where people immediately felt comfortable to approach others for later scientific discussion. Therefore, the focus was on getting to know each other, and less on discussing scientific topics.</text>
<text><location><page_6><loc_9><loc_11><loc_49><loc_47></location>We split into groups of ≈ 25, across separate rooms. These groups were then re-shuffled half way through the session. Before the activity all participants had been assigned their groups/rooms so that they knew where to be. Two activities were used: (1) 'Find 10 things in common' and (2) 'Queen Bee: Find the Answer to my Questions'. The former activity had each group split into 2-3 sub-groups and then take turns to briefly introduce themselves to each other. They then had 10 minutes to try and find 10 things in common across the whole group. This could be related to research tools used, hobbies, favourite types of food etc. The second activity involved sub-groups of ≈ 3-5. One person in each group is nominated the 'Queen Bee', who gives their 'worker bees' a question that they want the answer to. This could be something scientific (e.g., 'who could I talk to about processing these type of astronomical data'), or anything else (e.g., 'what's the best restaurant to get good Indian in Newcastle'). All the worker bees, from each group, then spend ≈ 3 minutes speaking to everyone else in the room to get the answers, and report back to the Queen Bee. The Queen Bees could then report back to the whole room what they found out, before somebody else takes a turn as Queen Bee.</text>
<text><location><page_6><loc_9><loc_8><loc_49><loc_10></location>The top-left panel of Figure 2 shows the results of asking participants 'How did you find the networking</text>
<text><location><page_6><loc_51><loc_69><loc_91><loc_92></location>session on Monday?', with answers between 1 for poor and 5 for excellent. It can be seen that the activity was extremely popular across all participants, with 91% scoring 4 or 5. In Figure 2 we also split the responses for only the African affiliated/origin participants, and found equal levels of positive responses for this specific group. The qualitative comments suggested the activity was very effective for helping early-career researchers feel more relaxed and able to approach people for conversations throughout the week. It was especially noticed feeling more comfortable approaching senior academics after this activity. Some other comments suggested the activity was a bit tiring, so it could have been reduced in length and/or more rest time built in to the programme after this intense activity.</text>
<text><location><page_6><loc_53><loc_60><loc_89><loc_67></location>Key Recommendations: A networking / icebreaking session can be very effective to enable participants to feel relaxed and more able to approach other participants to initiate discussions throughout the conference.</text>
<section_header_level_1><location><page_6><loc_51><loc_55><loc_68><loc_56></location>Scientific programme</section_header_level_1>
<text><location><page_6><loc_51><loc_41><loc_91><loc_55></location>The scientific programme included a mixture of participant contributions and break-out discussion sessions. The programme also included lengthy coffee/tea breaks (typically 45 minutes) and lunch breaks (typically 90 minutes), which helped ease any time pressure in the schedule and allowed people to spend more time networking. Quite a few of the qualitative responses to our feedback survey indicated that people appreciated this chance to talk to other participants in a leisurely way.</text>
<section_header_level_1><location><page_6><loc_51><loc_38><loc_67><loc_39></location>Science contributions</section_header_level_1>
<text><location><page_6><loc_51><loc_21><loc_91><loc_38></location>There were three types of scientific contributions during the workshop: (1) long-form talks (4 invited talks with 30 minute slots and 47 contributed talks with 15 minute slots); (2) posters and (3) sparkler talks. The invited and contributed talks were selected during the conference selection process, described earlier in this report. All other participants were invited to present a poster and/or sparkler talk. The posters were displayed throughout the conference in the same space as coffee and lunch breaks, which was effective for increasing activity and discussions around the posters.</text>
<text><location><page_6><loc_51><loc_8><loc_91><loc_21></location>Sparkler talks were 90 second presentations (with one slide only). These were done over 2 scheduled blocks on the first two days. The purpose of these were to allow participants to introduce themselves, their main research interest and/or one key scientific result, and to advertise a poster (if relevant). As described above, participants were given examples and training on how to effectively put these together, with a focus on avoiding participants trying to give a full scientific talk very quickly!</text>
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<caption>Experience of break-out discussions</caption>
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<figure>
<location><page_7><loc_58><loc_48><loc_81><loc_67></location>
<caption>Figure 2. Results from four of the rating questions of the feedback questionnaire, where 5 is high and 1 is low. The outer circles are for all participants, and the inner circles for those with African affiliation/origin. The questions were: a. 'How did you find the networking session on Monday?' ; b. 'Rate your overall experience of participants' science contributions part of the conference (contributed talks, sparkler talks, posters)' ; c. 'How did you find the break-out/discussion sessions throughout the week '; d. 'How comfortable did you feel to have discussions/conversations with new people during the conference (e.g., during coffee breaks, lunchtimes...)' .</caption>
</figure>
<text><location><page_7><loc_9><loc_19><loc_49><loc_34></location>The SOC selected chairs for the contribution sessions, to be more senior people, but keeping the geographical representation the same as the overall conference. We provided the chairs with moderator guidelines, which most chairs adhered to. These included information about keeping to time, encouraging a diversity of participants to ask questions, and ensuring the code of conduct is upheld. The LOC provided technical support for each session, with specific LOC members identified for each session.</text>
<text><location><page_7><loc_9><loc_8><loc_49><loc_18></location>To facilitate early career engagement, we handed out a conference sticker to any early-career researcher who asked a question to a speaker during any of the sessions. A prize was promised to anyone who earned more than 10 stickers, and at the end of the week there were two PhD students who had won a prize. This was a very effective way to help shift the question and answer</text>
<text><location><page_7><loc_51><loc_32><loc_91><loc_34></location>sessions more towards early-career researchers, and we highly recommend doing this at any conference.</text>
<text><location><page_7><loc_51><loc_13><loc_91><loc_31></location>The top-right panel of Figure 2 shows the results of the participant feedback to the question: 'Rate your overall experience of participants science contributions part of the conference (contributed talks, sparkler talks, posters)', ranging from 5 (Excellent) to 1 (Poor). The results are very positive with 71% ranking 5 and 23% ranking 4 across all participants. For only the African affiliated/origin, the results were 88% ranking 5 and 12% ranking 4. This highlights the high-quality scientific contributions and that we achieved our objective to put on a high-quality scientific meeting, addressing relevant problems of the field.</text>
<section_header_level_1><location><page_7><loc_51><loc_11><loc_67><loc_12></location>Break-out discussions</section_header_level_1>
<text><location><page_7><loc_51><loc_8><loc_91><loc_10></location>During the week we organised three break-out discussion sessions, of 90 minutes each. We had four rooms</text>
<paragraph><location><page_7><loc_57><loc_67><loc_80><loc_69></location>d. Comfort having conversations</paragraph>
<paragraph><location><page_7><loc_57><loc_90><loc_81><loc_92></location>b Rating of science contributions</paragraph>
<text><location><page_8><loc_9><loc_68><loc_49><loc_92></location>available for parallel discussion sessions. On the first session, we had a pre-planned 45 minute panel discussion with all participants. However, the rest of the discussion time was not pre-planned. Instead, participants were encouraged to choose topics of interest that they wanted to discuss and self-organise discussion sessions across the available rooms. The proposals for collaboration research topics that were submitted at application were also shared among participants at the beginning of the week, to help people find researchers with similar interests. This organisation was done via a collaborative, online document, so people could assign discussion topics to different rooms and/or decide which discussion topics to join. Online participants were also encouraged to participate in a hybrid format; however, we found limited engagement from online participants in this activity.</text>
<text><location><page_8><loc_9><loc_40><loc_49><loc_67></location>The bottom-left panel of Figure 2 shows the results of the participant feedback to the question: 'How did you find the break-out/discussion sessions throughout the week', ranging from 5 (Excellent) to 1 (Poor). The feedback was broadly positive with 54% of participants scoring 5 and 33% scoring 4. The qualitative comments imply that the overall concept and freedom to self-organise was appreciated. It was also appreciated that significant time was scheduled for discussion sessions, rather than the programme being dominated by presentations. However, the comments also implied that an improvement would have been to find methods for the less confident participants to participate in the discussions themselves (which were often dominated by more senior academics, often of non-African affiliation/origin). An example might have been using online voting tools, or collaborative documents, for people to provide their input into the discussions, but in a non verbal way.</text>
<text><location><page_8><loc_11><loc_26><loc_47><loc_38></location>Key Recommendation: Engaging early with all participants to offer training for giving talks can be key to ensuring a high-quality scientific meeting. Time for discussion is valuable, with the freedom to let participants chose their own topics and groups. It is important to find methods, including incentives, for early-career researchers to more actively participate in discussions.</text>
<section_header_level_1><location><page_8><loc_9><loc_22><loc_29><loc_23></location>Social events programme</section_header_level_1>
<text><location><page_8><loc_9><loc_8><loc_49><loc_21></location>In addition to the scientific programme, we had a social programme of: (1) a welcome drinks reception on the first evening; (2) a conference dinner; and (3) a mid-week, half-day sightseeing in the city of Newcastle. These activities proved to be very popular, and a highlight was encouraging participants to wear dress relevant to their cultural background for the conference dinner. Many participants did wear cultural dress, which sparked plenty of discussion and a mini-photoshoot!</text>
<text><location><page_8><loc_51><loc_74><loc_91><loc_92></location>Another point of note was that alcohol was not provided during the social events (although a bar was available during the conference dinner). This decision was made to remove the pressure of the social events evolving around drinking alcohol, which might not be comfortable for some participants - as one person said in their feedback: 'Although I like alcohol, I thought it was really nice that none of the socials centred around alcohol. That way everyone felt included but those who wanted to drink could choose to.' There were hardly any negative comments aside from a note about the poor British weather during the sight-seeing!</text>
<text><location><page_8><loc_53><loc_61><loc_89><loc_72></location>Key Recommendation: A good social programme allows for some decompression from the intense scientific programme. Allowing the diversity of cultures to be shared and showcased during the social events can be a popular highlight and another way to build relationships and understanding among communities.</text>
<section_header_level_1><location><page_8><loc_51><loc_58><loc_85><loc_59></location>Overall participant experience and feedback</section_header_level_1>
<text><location><page_8><loc_51><loc_38><loc_91><loc_57></location>The feedback we received indicates that we achieved our objective of creating a collaborative environment where people 'across continents' could come together to network and share in scientific discussion. The bottom-right panel of Figure 2 shows the results of the participant feedback to the question: 'How comfortable did you feel to have discussions/conversations with new people during the conference (e.g., during coffee breaks, lunchtimes...)', ranging from 5 (Very) to 1 (Not at all). The results are overall positive with 55% scoring 5 and 36% scoring 4, with a similar positive response for only the African affiliated/origin groups (62% scoring 5 and 36% scoring 4).</text>
<text><location><page_8><loc_51><loc_30><loc_91><loc_38></location>The quotes we received from participants during the feedback process further highlight that we achieved our objectives to put on a high-quality scientific programme, whilst simultaneously creating a comfortable and collaborative environment. Some example quotes are:</text>
<unordered_list>
<list_item><location><page_8><loc_54><loc_25><loc_91><loc_29></location>· 'I loved the conference, it was a perfect balance of presentations and time to talk and get to know people.'</list_item>
<list_item><location><page_8><loc_54><loc_15><loc_91><loc_24></location>· 'I am a senior researcher based neither in UK nor Africa and not yet maintaining connections with Africa. I attended this conference because of my interest in the subject, and thoroughly enjoyed the strong scientific programme and meeting African colleagues on the PhD and professor levels.'</list_item>
<list_item><location><page_8><loc_54><loc_12><loc_91><loc_14></location>· 'So many opportunities for talking to people in a relaxed environment.'</list_item>
<list_item><location><page_8><loc_54><loc_8><loc_91><loc_10></location>· 'This is probably the most pleasant atmosphere in a conference I have ever experienced.'</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_9><loc_12><loc_87><loc_49><loc_92></location>· 'What I liked the most is interacting with people. All the conference attendees are unexpectedly nice and helpful and comfortable to talk to.'</list_item>
<list_item><location><page_9><loc_12><loc_80><loc_49><loc_86></location>· 'I liked how it aimed to address the inequalities that those in African countries face and bring them to a conference like this. The organisation and communication was also outstanding.'</list_item>
</unordered_list>
<text><location><page_9><loc_9><loc_67><loc_49><loc_79></location>We received little feedback from the small number of online participants. However, we know that there they mostly only participated during the scientific contributions part of the conference. Our focus was always on generating an effective in-person conference experience. Future consideration would be needed to make a more effective hybrid conference format, and/or to consider an online-only equivalent.</text>
<section_header_level_1><location><page_9><loc_16><loc_63><loc_42><loc_64></location>Legacy, future and follow-up</section_header_level_1>
<figure>
<location><page_9><loc_12><loc_41><loc_46><loc_60></location>
<caption>Figure 3. The results of the feedback questionnaire question: Would you be interested in attending a conference like this in a future? , where the outer ring is for all participants, and the inner ring is for those with African affiliation or origin.</caption>
</figure>
<text><location><page_9><loc_9><loc_16><loc_49><loc_29></location>Only four months after the conference, we are not yet able to establish its long-term impact. However, initial signs are encouraging. For example, when we asked 'Would you be interested in attending a conference like this in a future?', (see Figure 3), 94% of all participants answered 'yes' (100% of the African affiliated/origin participants). There was enthusiasm to run another similar meeting, potentially on the continent of Africa next time.</text>
<text><location><page_9><loc_9><loc_11><loc_49><loc_16></location>To provide a legacy and citable location of the scientific contributions, in particular for the early-career participants, we created a Zenodo Community 6 . For those</text>
<text><location><page_9><loc_51><loc_84><loc_91><loc_92></location>who wished, we added the posters and contributed talk slides to this community. 48 contributions have been added to this community. We hope that this is a useful point of reference for participants who want to cite their contribution during the conference.</text>
<text><location><page_9><loc_51><loc_68><loc_91><loc_84></location>We already have informal knowledge of several new collaborations which have resulted from the conference. For example, a Large Programme to study AGN with the Southern African Large Telescope, led by African scientists, including non-African partnerships that were fostered during the workshop. Other smaller examples include two PhD students (one of African origin) deciding to initiate a new project combining the data of one student with the methods of the other student, and a student visit to Durham to start a radio project (combined with a workshop).</text>
<text><location><page_9><loc_51><loc_62><loc_91><loc_67></location>We are also aware that many participants who participated in the Buddy Scheme, are still in contact with their buddies, further highlighting the success in forming new networks and working relationships.</text>
<text><location><page_9><loc_51><loc_54><loc_91><loc_61></location>We plan to follow up with participants to ask them about collaborations they formed during the conference, in the 6-12 months after the conference. These results will be helpful in providing evidence of any long-term impact that our approach for this conference has had.</text>
<section_header_level_1><location><page_9><loc_67><loc_50><loc_75><loc_51></location>Summary</section_header_level_1>
<text><location><page_9><loc_51><loc_9><loc_91><loc_48></location>The main goal of this conference was to hold a scientific meeting, discussing timely scientific problems, but with a broader impact to integrate African researchers into the international community and to strengthen ties between the European and African scientific communities. According to the feedback from one participant, they reached their scientific goals for the conference, along with the bonus of '1. unplanned research insights and collaboration directions ...; 2. appreciating the responsibility that comes with the privilege of being Westaffiliated; 3. getting to know African astronomers who are great role models for everyone' From a different perspective, 'Having greater representation of Africans really made people much more approachable.' The social activities allowed everyone to bond, while many people noted that 'The quality of the participants' science contributions has been really impressive.' The feedback has been overwhelmingly positive, and there is appetite for another conference. We seem to have struck the balance between science and social aspects about right: as one participant noted the conference was 'really well thought out in terms of inclusivity, time table, networking opportunities, making people (especially students) feel comfortable to attend and talk to more senior researchers, super positive and supportive atmosphere, extremely interesting and high-quality science.'</text>
<text><location><page_9><loc_54><loc_8><loc_91><loc_9></location>We have learned a lot from this entire process, from</text>
<text><location><page_10><loc_9><loc_72><loc_49><loc_92></location>simple things like: what do people actually have to pay for if applying for a visa? and what are the logistical considerations we take for granted that may be different in other countries? Even the convention of putting a first name before a family name may be different across countries. The diversity of the LOC and SOC were fundamental in identifying potential issues before they arose, and problem-solving when unexpected challenges came up. The biggest challenges revolved around ensuring our budget would stretch to cover all of the costs we promised, and our key recommendation here is that funding requests start early, and that as much as possible should be paid for directly by the host organisations.</text>
<text><location><page_10><loc_9><loc_56><loc_49><loc_72></location>The funding for this conference pulled together from several sources. It started from the fact that the co-chairs had small amounts from their fellowships for modest science meetings, which could be combined for a single, more impactful meeting. Other funding was necessary, and it is crucial to start looking for funding sources early ( ≈ 18-24 months before). While we plan on following up the conference participants in future to track longterm collaborations, we would also like to start planning another conference to follow this one. The timeline for this is not yet set, and will likely depend on funding.</text>
<text><location><page_10><loc_9><loc_40><loc_49><loc_55></location>Overall, this meeting exceeded our expectations. The LOC and SOC did an incredible job to enact the vision, and to ensure that every small aspect was considered and addressed. Before and during the conference, the engagement from all participants was absolutely amazing. It was wonderful to see senior and junior researchers from Africa, Europe, and beyond, interacting and discussing science. We hope that the collaborations that have started will bring our communities together for a more global approach to science and inclusivity.</text>
<section_header_level_1><location><page_10><loc_20><loc_36><loc_38><loc_37></location>Acknowledgments</section_header_level_1>
<text><location><page_10><loc_9><loc_8><loc_49><loc_35></location>Funding for the conference was supplied through: two United Kingdom Research and Innovation grants (MR/V022830/1 and MR/T042842/1); the Future Leaders Fellowship Development Network Fund (PF-020); the Science Technology Facilities Council; a Royal Astronomical Society (RAS) conference/Meeting travel subsistence fund; a RAS-Astro4dev mobility grant (funded via RAS and the International Astronomical Union Office of Astronomy for Development); the Centre of Extragalactic Astronomy at Durham University and the School of Mathematics, Statistics and Physics at Newcastle University. The conference would not have been possible without the incredible efforts of the Local Organising Committee: Ivan Almeida; Emmanuel Bempong-Manful; Emmy Escott; Houda Haidar; Ann Njeri; James Petley; Shufei Rowe; and Nicole Thomas. We thank Event Durham, and in particular Thomas Ludlow, for help with all of the logistics of the conference and travel arrangements. We</text>
<text><location><page_10><loc_51><loc_84><loc_91><loc_92></location>are grateful for the crucial role of the Scientific Organising Committee in designing and planning the scientific programme, made up of: James Aird; James Chibueze; Eli Kasai; Mirjana Povi'c; Cristina Ramos Almeida; Zara Randriamanakoto; and Brooke Simmons.</text>
<section_header_level_1><location><page_10><loc_66><loc_80><loc_76><loc_82></location>References</section_header_level_1>
<unordered_list>
<list_item><location><page_10><loc_51><loc_74><loc_91><loc_80></location>[1] V. McBride, R. Venugopal, M. Hoosain, T. Chingozha, and K. Govender. The potential of astronomy for socioeconomic development in Africa. Nature Astronomy , 2:511-514, July 2018.</list_item>
<list_item><location><page_10><loc_51><loc_64><loc_91><loc_73></location>[2] M. Povi'c. Development of astronomy research and education in Africa and Ethiopia. In Rosa M. Ros, Beatriz García, Steven R. Gullberg, Javier Moldón, and Patricio Rojo, editors, Education and Heritage in the Era of Big Data in Astronomy , volume 367 of IAU Symposium , pages 24-27, January 2021.</list_item>
<list_item><location><page_10><loc_51><loc_60><loc_91><loc_62></location>[3] Melvin G. Hoare. UK aid for African radio astronomy. Nature Astronomy , 2:505-506, July 2018.</list_item>
<list_item><location><page_10><loc_51><loc_56><loc_91><loc_58></location>[4] C. Woolston. Meeting the challenges of research across Africa. Nature , 572(7767):143-145, 2019.</list_item>
<list_item><location><page_10><loc_51><loc_50><loc_91><loc_54></location>[5] M. Makoni. African researchers' work is being overlooked - here's how to change that. Nature , October 2023.</list_item>
<list_item><location><page_10><loc_51><loc_43><loc_91><loc_49></location>[6] Chris M. Harrison and Cristina Ramos Almeida. Observational Tests of Active Galactic Nuclei Feedback: An Overview of Approaches and Interpretation. Galaxies , 12(2):17, April 2024.</list_item>
</unordered_list>
<section_header_level_1><location><page_10><loc_64><loc_38><loc_78><loc_39></location>A. Photographs</section_header_level_1>
<figure>
<location><page_11><loc_13><loc_12><loc_87><loc_92></location>
<caption>Figure A.1. Top: Local Organising Committee. From left-to-right: Chris Harrison, Shufei Rowe, Ann Njeri, Emmanuel Bempong-Manful, Houda Haidar, Emmy Escott, James Petley, Nicole Thomas, Ivan Almeida and Leah Morabito. Bottom: Scientific Organising Committee. From left-to-right: James Chibueze, Mirjana Povi'c, Chris Harrison, James Aird, Leah Morabito, Brooke Simmons, Cristina Ramos-Almeida and Eli Kasai (photo missing Zara Randriamanakoto).</caption>
</figure>
<figure>
<location><page_12><loc_12><loc_8><loc_85><loc_92></location>
</figure>
</document> | [{"title": "Bringing together African & European research communities with an inclusive astronomy conference", "content": "Chris M. Harrison 1, \u22c6 and Leah Morabito 2, \u2020 , on behalf of the Organising Committees", "pages": [1]}, {"title": "Abstract", "content": "We report on an international scientific conference, where we brought together the African and European academic astronomy communities. This conference aimed to bridge the gap between those in position of privilege, with ease of access to international networking events (i.e., the typical experience of those affiliated with Western institutions), with those who have been historically excluded (affecting the majority of African scientists/institutions). We describe how we designed the conference around cutting-edge problems in the research field, but with a large focus on building networking and professional relationships. Significant effort went into: (1) ensuring a diverse representation of participants; (2) practically and financially supporting those who may have never attended an international conference and; (3) creating an inclusive and supportive environment through a careful programme of activities, both before and during the event. Throughout this process maintaining scientific integrity was a core commitment. We summarise some of the successes, challenges, and lessons learnt from organising this conference. We also present feedback obtained from participants, which demonstrates an overall achievement of our objectives. This is all combined to provide some key recommendations for any groups, from any research field, who wishes to lead similar initiatives.", "pages": [1]}, {"title": "Motivation & Scientific Context", "content": "Astronomy and space science can be used as an important tool for development and for achieving the United Nations (UN) Sustainable Development Goals (SDGs) 1 through: education; socio-economic growth via advances in science and technology; and promoting international peace and diplomacy [1, 2]. Partly driven by these development goals, astronomy and space science research is seeing significant growth on the continent of Africa. In recent years, a major contributing factor is the planning and development of the revolutionary radio observatory, the Square Kilometre Array (SKA) 2 . This ambitious international project will be situated across Africa and Australia. Other major astronomy projects on the African continent include the South African Astronomical Observatory (SAAO) and the High Energy Stereoscopic System (H.E.S.S) observatory in Namibia 3 [more examples in 2]. In parallel to these projects, the continent is seeing growth in postgraduate astronomy programmes, and international initiatives to support early-career African astronomers, such as the 'Development in Africa with Radio Astronomy' project [DARA, 3] and other strategic partnerships (e.g., between the United Kingdom's Science and Technology Facilities Council [STFC], and the South African National Research Foundation 4 ). Despite this positive growth of projects and initiatives, developing astronomy research in Africa faces many on-going challenges. These are shared across many research topics and institutions across the continent, and for people of African origin (but may be working in other countries). These relate to: limited opportunities and resources; the limited retention of young people into higher education; issues with equity and inclusion (including challenges around travel visas); and access to the latest international knowledge, facilities, and networks [4, 5]. Motivated by addressing some of these these challenges, in July 2024, Durham University and Newcastle University jointly hosted an international astronomy conference: 'AGN Populations Across Continents and Cosmic Time'. The main goal of this conference was to hold a scientific meeting, discussing timely scientific problems, but with a broader impact to integrate African researchers into the international community. Additionally, European researchers can often operate in their own continental spheres, and this conference offered the opportunity for them to broaden their networks. Therefore, this workshop aimed to facilitate networking that will strengthen ties between the European and African scientific communities.", "pages": [1]}, {"title": "Scientific context", "content": "Supermassive black holes, with masses of millions to billions times that of the Sun, are located at the nuclei of galaxies. These black holes grow by the accretion of gas. As this gas falls towards the black holes, it becomes extremely bright. Furthermore, jets of charged particles can be launched in the vicinity of these accreting black holes, and these jets can extend over vast distances within the galaxies and beyond. When these accretion events are detected with astronomical observatories on Earth or in space, they are known as 'Active Galactic Nuclei' (AGN). Both observational studies and simulations have shown that the energy released by AGN is important for galaxy evolution. However, there continue to be major research challenges in the field, including: (1) obtaining a complete census of AGN events; (2) understanding the detailed physical structure of the material associated with AGN; (3) establishing the properties of AGN host galaxies and the details of how AGN regulate galaxy growth, and; (4) identifying the key techniques and datasets required to make significant progress in answering these scientific questions over the coming decade [review in e.g., 6]. The scientific scope of our conference was to cover these four main themes. 5 Due to the focus on the African scientific community, the workshop looked to particularly showcase work on this scientific topic that uses SKA pathfinder observatories in addition to other astronomical facilities based on the African continent.", "pages": [1, 2]}, {"title": "Selecting and Supporting the Participants", "content": "Careful planning went into selecting participants. During selection, our primary objectives were: (1) ensuring a scientifically productive meeting, with participants working on the relevant research problems, balanced across the four key themes; and (2) having significant participation from those of African institutions and/or with African origin (aiming for \u2248 30-50%). We detail the processes of selecting the organisational committees, choosing invited speakers, and the selection of participants.", "pages": [2]}, {"title": "Selecting the LOC, SOC and Invited Speakers", "content": "Our Local Organisation Committee (LOC) was put together primarily from the research teams (i.e., postdoctoral researchers and PhD students) of the workshop co-chairs (i.e., Chris Harrison, based at Newcastle University and Leah Morabito, based at Durham University). Therefore, they had a direct scientific interest in the workshop. The LOC were responsible for the typical activities of organising a conference (e.g., coordinating with participants, helping with administrative tasks, supporting the activities during the week etc.). Within the LOC there were 4 researchers with an African origin (representing four different African countries), although they were based at UK institutions at the time. Their perspective was crucial for helping to identify - and overcome - many of the challenges that African participants would face in travelling to the UK, as well as for developing a programme of activities to ensure participants felt included, supported, and comfortable to contribute. A photograph of the LOC is shown in the top panel of Figure A.1. The Scientific Organising Committee (SOC) were responsible for designing the scientific programme, choosing invited speakers, and selecting the conference participants. We ensured that we had significant representation of scientists at African institutions as well as representatives from the UK and wider Europe. Of the 9 SOC members, 4 were based at African institutions (from 3 separate countries). Furthermore, the selection of the SOC was made to ensure representation of established experts covering a diverse range of the scientific topics that were the focus of the conference. Consequently, the SOC not only provided crucial scientific input, but additionally were able to help design a participant selection process that was suitable for African-affiliated researchers (especially early-career researchers). Furthermore, they suggested activities for the programme that would be inclusive to those who have little-to-no experience of international conferences. Importantly, these SOC members could locally advertise the conference across the African community. A photograph of the SOC is shown in the bottom panel of Figure A.1. The SOC chose to select only a small number of invited speakers; enough to highlight the programme with some internationally-known experts in the field, but not too many as to take significant space in the programme away from contributions by regular participants. The SOC's aims of the invited speaker selection were to select four established experts who would be able to provide an introductory review talk across the four main themes of the conference. Further requirements were to include at least one speaker of African origin and affiliation, having representation of more than one gender across all the speakers, ensuring the speakers would provide a talk at an accessible level to a broad range of career stage participants, and only selecting speakers willing to actively engage with the broader development and networking goals of the conference. Key Recommendation: a cross-community perspective should be considered a requirement of any LOC and SOC for any conferences with similar goals to ours, relating to bringing together research communities with diverse cultures and experiences.", "pages": [2]}, {"title": "Selecting participants", "content": "We planned for \u2248 110 conference participants. This number was small enough to facilitate productive discussions and effective networking, but large enough to have a number of participants from across different continents and career stages. We aimed for a fully in-person con- rence, to maximise the benefits of social interaction, building relationships, and ad-hoc informal discussions. Partial online participation was made available for participants who had challenges and could not travel (e.g., personal circumstances or visa complications). To achieve our objective of a high fraction of participation from African astronomers, it was crucial to provide full funding to those who do not have access to any travel funds (the majority of the African-affiliated astronomers). We estimated an average cost per fullyfunded participant travelling from Africa, to be \u2248 \u00a31900, to cover: travel, visa costs, accommodation, travel insurance, and local subsistence (including all meals). Our available budget limited us to fully-fund \u2248 35-40 participants, depending on exact final costs per participant. Throughout the selection process we kept track of both the required funding to support African participants and the participant demographics. For the demographics, we focused on a proactive approach to ensure a high level of representation of those who are African affiliated and/or have African origin. We also kept track of the distribution of career stages and gender; however, our selection approach naturally led to good representation across these two characteristics, and required no positive actions to address the diversity. Along with a dedicated website, we released an application form for conference participation 10 months before the actual conference. The deadline for submitting an application was \u2248 7 months before the conference dates. This long lead time was critical to ensure sufficient time to support travel and visa applications, as well as plan and to conduct pre-conference activities (described later). Weaimed to include participants who were dedicated to contribute to, or benefit from, the wider networking aspects of the conference. Therefore, we did not exclusively request abstracts for talks/posters in the applica- tion form. We concluded that requiring a high-quality scientific contribution (talk/poster) abstract to attend could bias against those who had so far had limited opportunities to work on significant, internationally cutting-edge research projects. Therefore, we had three separate categories in the application form, which were used for selecting participants. These were: This final category was designed to capture those who may not yet have established research outputs. For example, PhD students who were at the beginning their academic studies but were strongly motivated by the opportunities of the conference. Applicants could apply using one, two, or three of these categories. These three categories were each treated separately. In effect, every applicant had up to three possibilities to be selected. We received a total of 249 applications (once removing clearly inapplicable or duplicate applications). Of these, 75 were from people with an African origin and 74 were from African affiliations (not mutually exclusive). The SOC scored the submitted applications with a blinded approach (i.e., with no information on the applicants' names, affiliations etc.). Every SOC member assessed every submission for application categories 2. and 3. from the above list. However, depending on the scientific expertise of the SOC member, they only assessed abstracts for talks/posters based on the most applicable scientific theme for their own expertise. Each SOC member scored the submissions with an integer grade from 1 (low) to 5 (high). The scores for each submission were then collated and averaged. The first pass of participant selection was to take a simple cut in the average score and select everyone with a score above some fixed value. These values were chosen to select the number of contributed talks we aimed for in the final programme (i.e., 47), and then fill the remaining participant places with those who scored highly in the other two submission categories. However, this completely blind and uniform approach to selection was insufficient to obtain the desired representation of African participants. On average, the African submissions scored \u2248 0.6 points lower than other submissions. This gap is unsurprising due to the generally lower levels of available mentoring and peer support (including for tasks such as improving conference abstract drafts) compared to more deeply established research environments in places like the UK. We therefore decided to apply a positive action by up-weighting scores from South African submissions by 0.5 and other African submissions by 0.8. This divide within the African submissions into South Africa and other countries, was because the South African astronomy community is rapidly growing and applicants there tend to have larger peer-to-peer support groups and more mentoring, likely contributing to the typically higher submission scores. This approach ensured good geographical representation from across the continent. We had capacity for 120 participants and an ideal target of \u2248 110. The final number of participants attending the conference was 107, of which 9 could only join online due to not being able to travel after unforeseen circumstances. Figure 1 shows a world map highlighting the countries of origin and affiliation that were represented across the conference participants. Overall, 43% of the participants were either from an African institution and/or of African origin. There were 20 different affiliation countries represented, of which 11 were from Africa. The break down of career stage was: 31% senior academic/lecturer; 26% postdoc/fellow; 31% PhD student; and 12% Masters students. For gender, the final breakdown was 49% Male, 49% Female, and 2% Non-binary. A conference photo, including the majority of participants, is shown in Figure A.2. Key Recommendations: A long lead time for opening applications is crucial. Multiple categories for applications to attend (not just scientific abstracts) can help increase the diversity of the participants. Initial selection should be blind (anonymous) and based on a scoring rubric but positive action might be required to achieve the objective diversity.", "pages": [2, 3, 4]}, {"title": "Financial and practical support of participants", "content": "Offers for conference participation were sent out \u2248 6 months before the conference itself. This long lead-time was chosen to ensure sufficient time to help participants who needed to apply for visas, and/or whose travel we needed to support. Indeed, visa applications can take a significant amount of time. For example, one participant had a visa issued only two days before travel. For similar reasons, it was necessary to book travel for fully-funded participants before visa applications were completed, otherwise flight costs could have risen to unaffordable rates as the travel dates approached. Of all participants, 38 received full funding from our budget, with a small additional number of participants receiving part funding based on a case-by-case assessment of need (e.g., fee waivers for early-career researchers from any institution/country without large funding resources). As many African-affiliated participants had no means to make significant upfront costs, it was necessary for the conference host university to book the travel and the accommodation directly on their behalf. Due to the nature of the accommodation we used (university colleges), it was also possible to purchase all evening meals for the fully-funded participants in advance of the conference. Fully-funded participants also had their posters printed (when relevant) directly at the conference venue, at no cost to themselves. The only costs that fully-funded participants were required to pay for themselves up-front, and later obtain a reimbursement, were the visa costs, travel insurance (as often not provided by the home universities), and local travel costs. We note that costs associated with obtaining visas were sometimes quite high, due to participants needing to travel to consulates (sometimes multiple times). Although this process generally went well, some applicants had no access to personal bank accounts and arrangement had to be made with a third party to make the reimbursement because cash payments were not possible. In some cases, we were not aware of it until after it was too late to help with alternative methods of payment beforehand. We supported visa applications by providing the necessary invitation letters to attend the conference. We provided ad-hoc help for some participants on how to complete visa applications. However, with hindsight, we should have provided more specific guidance for those who had not previously applied for visas. For example, some visas were initially rejected on the grounds of not providing sufficient evidence that they had reasons to return home after the conference (e.g., by providing letters from their home universities to prove employment or registration on PhD programmes). The participants applied for and received standard visitor visas. These are valid for 6 months, and some participants organised other work visits while they were in the UK. This required some extra coordination, but we were happy to support it. We had some requests during the meeting or after for other work visits, and it would have been useful to highlight this opportunity to participants beforehand. This could have allowed participants to make the best use of their visas while they were valid. However, we caution that while visits to the UK for conferences and meeting do not require approval from the Academic Technology Approval Scheme (ATAS), research visits of any length do require approval, at the time of writing. If people wish to join a visit to carry out research along with this, we recommend they apply for ATAS certification at the same time or even before the visa. Although the LOC communication to participants directed them to 'contact us if you have any questions or problems', we found that some did not do so until after they completed a task. In some cases (e.g., reimbursement to those who did not have bank accounts), this made fixing the problem more challenging. This may be attributable to the lack of experience or confidence of the participants. Therefore, we would strongly encourage participants to be very communicative at each stage of their diverse visa and travel processes, to give organisers a chance to assist as early as possible Key Recommendations: To aid successful visa applications, ensure a long lead time and provide extensive supporting documents and guidelines for participants. Reimbursement factors should carefully be planned with local administrators, and prepayment of all the funded participant's major expenses are crucial. Be very explicit with participants which costs they can, or can not, claim back in advance of any payments being made (e.g., if taxis can be used). Plan for extra costs associated with visa applications and travel insurance. Throughout, clear communication is critical between participants and organisers.", "pages": [4, 5]}, {"title": "Programme of Activities & their Evaluation", "content": "We conducted a variety of activities before, and during, the conference to aid the sharing and discussion of research outputs, as well as facilitate networking and collaborations. To assess the success of the activities, we conducted a participant evaluation questionnaire. Participants were asked to complete this during the final plenary session itself, to help increase completion rates. Overall, 73/107 participants (68%) completed the evaluation form. This was a representative sub-sample of all attendees, with \u2248 30% from each of the groups of PhD students, Postdocs/Fellows, and Lecturers, and with \u2248 12% from Masters students. Furthermore, 47% completing the survey were from an African institute and/or African origin. We provide some key results of this evaluation alongside the relevant activity description. For the quantitative questions, we split the results by all participants and then just the African affiliated/origin participants, to assess if there were any deviations for this specific group for whom there was a focus on providing an inclusive, welcoming and collaborative environment. The following sub-sections describe the activities and their impact, as much as we are able to assess four months after the event.", "pages": [5]}, {"title": "Pre-conference communications and activities", "content": "As is increasingly common practice for scientific workshops, we created a Code of Conduct. All participants were made aware that this applied to all activities before and during the workshop, and this was posted on our website. Around 1 month before the workshop, we created a collaborative online Slack space, to allow participants to start introducing themselves to each other. Additionally, to encourage participants getting to know each other before the conference, we asked participants to create 2-minute introductory videos. These videos were shared on a private YouTube playlist in the month leading up to the conference. Participants were provided with clear guidelines on how to record these videos on a phone, and what these videos should include (i.e., their name, affiliation, research interests, and what they hope to get out of the conference). Only 35/107 participants created these videos; with mostly the more junior participants participating (the LOC and SOC were all compelled to make a video). Nonetheless, we had several positive comments during the feedback questionnaire about the introductory videos being a good idea, and useful to start to get to know fellow participants. The most successful pre-conference networking activity was the 'Buddy' system that was created to pair all African-affiliated early-career researchers with an non African-affiliated early-career researcher. Although this was an opt-in activity, nearly all early-career researchers participated. Buddys were put into contact and encouraged to email/call before the conference to get to know each other, and then to meet each other at the start of the conference. They were also encouraged to provide feedback to each other on their scientific contributions. The comments we received demonstrated this was extremely helpful for feeling prepared for the conference and to make participants feel more confident to integrate and participate in discussions, because they already knew somebody attending the conference. Due to the wide range of experience of the participants, where some may not have presented at, nor even attended, an international conferences before, we decided to provide extensive support and guidelines for preparing high-quality scientific contributions. This included: (1) an online workshop on preparing a good scientific talk; (2) guidelines and template examples on preparing a good scientific poster; (3) guidelines and examples for preparing a 90 second 'sparkler' talk (more details below). The scientific talk workshop had over 30 attendees (with a similar number watching the recording later on) and the feedback from participants highlighted that the guidelines, examples and templates, were appreciated and extensively used. We believe this preworkshop guidance and training significantly contributed to the consistently high-quality scientific contributions delivered during the conference itself. Key Recommendations: Pre-conference networking activities and training/guidance are extremely valuable activities. These help ensure an effective meeting where participants immediately feel confident and welcome, and deliver high-quality scientific contributions.", "pages": [5, 6]}, {"title": "Day 1 Networking activity", "content": "After the first welcome and introductory talks on the first day of the conference, we conducted a 1.5 hour networking/ice-breaker activity. We felt this important to create a welcoming atmosphere where people immediately felt comfortable to approach others for later scientific discussion. Therefore, the focus was on getting to know each other, and less on discussing scientific topics. We split into groups of \u2248 25, across separate rooms. These groups were then re-shuffled half way through the session. Before the activity all participants had been assigned their groups/rooms so that they knew where to be. Two activities were used: (1) 'Find 10 things in common' and (2) 'Queen Bee: Find the Answer to my Questions'. The former activity had each group split into 2-3 sub-groups and then take turns to briefly introduce themselves to each other. They then had 10 minutes to try and find 10 things in common across the whole group. This could be related to research tools used, hobbies, favourite types of food etc. The second activity involved sub-groups of \u2248 3-5. One person in each group is nominated the 'Queen Bee', who gives their 'worker bees' a question that they want the answer to. This could be something scientific (e.g., 'who could I talk to about processing these type of astronomical data'), or anything else (e.g., 'what's the best restaurant to get good Indian in Newcastle'). All the worker bees, from each group, then spend \u2248 3 minutes speaking to everyone else in the room to get the answers, and report back to the Queen Bee. The Queen Bees could then report back to the whole room what they found out, before somebody else takes a turn as Queen Bee. The top-left panel of Figure 2 shows the results of asking participants 'How did you find the networking session on Monday?', with answers between 1 for poor and 5 for excellent. It can be seen that the activity was extremely popular across all participants, with 91% scoring 4 or 5. In Figure 2 we also split the responses for only the African affiliated/origin participants, and found equal levels of positive responses for this specific group. The qualitative comments suggested the activity was very effective for helping early-career researchers feel more relaxed and able to approach people for conversations throughout the week. It was especially noticed feeling more comfortable approaching senior academics after this activity. Some other comments suggested the activity was a bit tiring, so it could have been reduced in length and/or more rest time built in to the programme after this intense activity. Key Recommendations: A networking / icebreaking session can be very effective to enable participants to feel relaxed and more able to approach other participants to initiate discussions throughout the conference.", "pages": [6]}, {"title": "Scientific programme", "content": "The scientific programme included a mixture of participant contributions and break-out discussion sessions. The programme also included lengthy coffee/tea breaks (typically 45 minutes) and lunch breaks (typically 90 minutes), which helped ease any time pressure in the schedule and allowed people to spend more time networking. Quite a few of the qualitative responses to our feedback survey indicated that people appreciated this chance to talk to other participants in a leisurely way.", "pages": [6]}, {"title": "Science contributions", "content": "There were three types of scientific contributions during the workshop: (1) long-form talks (4 invited talks with 30 minute slots and 47 contributed talks with 15 minute slots); (2) posters and (3) sparkler talks. The invited and contributed talks were selected during the conference selection process, described earlier in this report. All other participants were invited to present a poster and/or sparkler talk. The posters were displayed throughout the conference in the same space as coffee and lunch breaks, which was effective for increasing activity and discussions around the posters. Sparkler talks were 90 second presentations (with one slide only). These were done over 2 scheduled blocks on the first two days. The purpose of these were to allow participants to introduce themselves, their main research interest and/or one key scientific result, and to advertise a poster (if relevant). As described above, participants were given examples and training on how to effectively put these together, with a focus on avoiding participants trying to give a full scientific talk very quickly! The SOC selected chairs for the contribution sessions, to be more senior people, but keeping the geographical representation the same as the overall conference. We provided the chairs with moderator guidelines, which most chairs adhered to. These included information about keeping to time, encouraging a diversity of participants to ask questions, and ensuring the code of conduct is upheld. The LOC provided technical support for each session, with specific LOC members identified for each session. To facilitate early career engagement, we handed out a conference sticker to any early-career researcher who asked a question to a speaker during any of the sessions. A prize was promised to anyone who earned more than 10 stickers, and at the end of the week there were two PhD students who had won a prize. This was a very effective way to help shift the question and answer sessions more towards early-career researchers, and we highly recommend doing this at any conference. The top-right panel of Figure 2 shows the results of the participant feedback to the question: 'Rate your overall experience of participants science contributions part of the conference (contributed talks, sparkler talks, posters)', ranging from 5 (Excellent) to 1 (Poor). The results are very positive with 71% ranking 5 and 23% ranking 4 across all participants. For only the African affiliated/origin, the results were 88% ranking 5 and 12% ranking 4. This highlights the high-quality scientific contributions and that we achieved our objective to put on a high-quality scientific meeting, addressing relevant problems of the field.", "pages": [6, 7]}, {"title": "Break-out discussions", "content": "During the week we organised three break-out discussion sessions, of 90 minutes each. We had four rooms available for parallel discussion sessions. On the first session, we had a pre-planned 45 minute panel discussion with all participants. However, the rest of the discussion time was not pre-planned. Instead, participants were encouraged to choose topics of interest that they wanted to discuss and self-organise discussion sessions across the available rooms. The proposals for collaboration research topics that were submitted at application were also shared among participants at the beginning of the week, to help people find researchers with similar interests. This organisation was done via a collaborative, online document, so people could assign discussion topics to different rooms and/or decide which discussion topics to join. Online participants were also encouraged to participate in a hybrid format; however, we found limited engagement from online participants in this activity. The bottom-left panel of Figure 2 shows the results of the participant feedback to the question: 'How did you find the break-out/discussion sessions throughout the week', ranging from 5 (Excellent) to 1 (Poor). The feedback was broadly positive with 54% of participants scoring 5 and 33% scoring 4. The qualitative comments imply that the overall concept and freedom to self-organise was appreciated. It was also appreciated that significant time was scheduled for discussion sessions, rather than the programme being dominated by presentations. However, the comments also implied that an improvement would have been to find methods for the less confident participants to participate in the discussions themselves (which were often dominated by more senior academics, often of non-African affiliation/origin). An example might have been using online voting tools, or collaborative documents, for people to provide their input into the discussions, but in a non verbal way. Key Recommendation: Engaging early with all participants to offer training for giving talks can be key to ensuring a high-quality scientific meeting. Time for discussion is valuable, with the freedom to let participants chose their own topics and groups. It is important to find methods, including incentives, for early-career researchers to more actively participate in discussions.", "pages": [7, 8]}, {"title": "Social events programme", "content": "In addition to the scientific programme, we had a social programme of: (1) a welcome drinks reception on the first evening; (2) a conference dinner; and (3) a mid-week, half-day sightseeing in the city of Newcastle. These activities proved to be very popular, and a highlight was encouraging participants to wear dress relevant to their cultural background for the conference dinner. Many participants did wear cultural dress, which sparked plenty of discussion and a mini-photoshoot! Another point of note was that alcohol was not provided during the social events (although a bar was available during the conference dinner). This decision was made to remove the pressure of the social events evolving around drinking alcohol, which might not be comfortable for some participants - as one person said in their feedback: 'Although I like alcohol, I thought it was really nice that none of the socials centred around alcohol. That way everyone felt included but those who wanted to drink could choose to.' There were hardly any negative comments aside from a note about the poor British weather during the sight-seeing! Key Recommendation: A good social programme allows for some decompression from the intense scientific programme. Allowing the diversity of cultures to be shared and showcased during the social events can be a popular highlight and another way to build relationships and understanding among communities.", "pages": [8]}, {"title": "Overall participant experience and feedback", "content": "The feedback we received indicates that we achieved our objective of creating a collaborative environment where people 'across continents' could come together to network and share in scientific discussion. The bottom-right panel of Figure 2 shows the results of the participant feedback to the question: 'How comfortable did you feel to have discussions/conversations with new people during the conference (e.g., during coffee breaks, lunchtimes...)', ranging from 5 (Very) to 1 (Not at all). The results are overall positive with 55% scoring 5 and 36% scoring 4, with a similar positive response for only the African affiliated/origin groups (62% scoring 5 and 36% scoring 4). The quotes we received from participants during the feedback process further highlight that we achieved our objectives to put on a high-quality scientific programme, whilst simultaneously creating a comfortable and collaborative environment. Some example quotes are: We received little feedback from the small number of online participants. However, we know that there they mostly only participated during the scientific contributions part of the conference. Our focus was always on generating an effective in-person conference experience. Future consideration would be needed to make a more effective hybrid conference format, and/or to consider an online-only equivalent.", "pages": [8, 9]}, {"title": "Legacy, future and follow-up", "content": "Only four months after the conference, we are not yet able to establish its long-term impact. However, initial signs are encouraging. For example, when we asked 'Would you be interested in attending a conference like this in a future?', (see Figure 3), 94% of all participants answered 'yes' (100% of the African affiliated/origin participants). There was enthusiasm to run another similar meeting, potentially on the continent of Africa next time. To provide a legacy and citable location of the scientific contributions, in particular for the early-career participants, we created a Zenodo Community 6 . For those who wished, we added the posters and contributed talk slides to this community. 48 contributions have been added to this community. We hope that this is a useful point of reference for participants who want to cite their contribution during the conference. We already have informal knowledge of several new collaborations which have resulted from the conference. For example, a Large Programme to study AGN with the Southern African Large Telescope, led by African scientists, including non-African partnerships that were fostered during the workshop. Other smaller examples include two PhD students (one of African origin) deciding to initiate a new project combining the data of one student with the methods of the other student, and a student visit to Durham to start a radio project (combined with a workshop). We are also aware that many participants who participated in the Buddy Scheme, are still in contact with their buddies, further highlighting the success in forming new networks and working relationships. We plan to follow up with participants to ask them about collaborations they formed during the conference, in the 6-12 months after the conference. These results will be helpful in providing evidence of any long-term impact that our approach for this conference has had.", "pages": [9]}, {"title": "Summary", "content": "The main goal of this conference was to hold a scientific meeting, discussing timely scientific problems, but with a broader impact to integrate African researchers into the international community and to strengthen ties between the European and African scientific communities. According to the feedback from one participant, they reached their scientific goals for the conference, along with the bonus of '1. unplanned research insights and collaboration directions ...; 2. appreciating the responsibility that comes with the privilege of being Westaffiliated; 3. getting to know African astronomers who are great role models for everyone' From a different perspective, 'Having greater representation of Africans really made people much more approachable.' The social activities allowed everyone to bond, while many people noted that 'The quality of the participants' science contributions has been really impressive.' The feedback has been overwhelmingly positive, and there is appetite for another conference. We seem to have struck the balance between science and social aspects about right: as one participant noted the conference was 'really well thought out in terms of inclusivity, time table, networking opportunities, making people (especially students) feel comfortable to attend and talk to more senior researchers, super positive and supportive atmosphere, extremely interesting and high-quality science.' We have learned a lot from this entire process, from simple things like: what do people actually have to pay for if applying for a visa? and what are the logistical considerations we take for granted that may be different in other countries? Even the convention of putting a first name before a family name may be different across countries. The diversity of the LOC and SOC were fundamental in identifying potential issues before they arose, and problem-solving when unexpected challenges came up. The biggest challenges revolved around ensuring our budget would stretch to cover all of the costs we promised, and our key recommendation here is that funding requests start early, and that as much as possible should be paid for directly by the host organisations. The funding for this conference pulled together from several sources. It started from the fact that the co-chairs had small amounts from their fellowships for modest science meetings, which could be combined for a single, more impactful meeting. Other funding was necessary, and it is crucial to start looking for funding sources early ( \u2248 18-24 months before). While we plan on following up the conference participants in future to track longterm collaborations, we would also like to start planning another conference to follow this one. The timeline for this is not yet set, and will likely depend on funding. Overall, this meeting exceeded our expectations. The LOC and SOC did an incredible job to enact the vision, and to ensure that every small aspect was considered and addressed. Before and during the conference, the engagement from all participants was absolutely amazing. It was wonderful to see senior and junior researchers from Africa, Europe, and beyond, interacting and discussing science. We hope that the collaborations that have started will bring our communities together for a more global approach to science and inclusivity.", "pages": [9, 10]}, {"title": "Acknowledgments", "content": "Funding for the conference was supplied through: two United Kingdom Research and Innovation grants (MR/V022830/1 and MR/T042842/1); the Future Leaders Fellowship Development Network Fund (PF-020); the Science Technology Facilities Council; a Royal Astronomical Society (RAS) conference/Meeting travel subsistence fund; a RAS-Astro4dev mobility grant (funded via RAS and the International Astronomical Union Office of Astronomy for Development); the Centre of Extragalactic Astronomy at Durham University and the School of Mathematics, Statistics and Physics at Newcastle University. The conference would not have been possible without the incredible efforts of the Local Organising Committee: Ivan Almeida; Emmanuel Bempong-Manful; Emmy Escott; Houda Haidar; Ann Njeri; James Petley; Shufei Rowe; and Nicole Thomas. We thank Event Durham, and in particular Thomas Ludlow, for help with all of the logistics of the conference and travel arrangements. We are grateful for the crucial role of the Scientific Organising Committee in designing and planning the scientific programme, made up of: James Aird; James Chibueze; Eli Kasai; Mirjana Povi'c; Cristina Ramos Almeida; Zara Randriamanakoto; and Brooke Simmons.", "pages": [10]}] |
2015MNRAS.450.1538W | https://arxiv.org/pdf/1503.08927.pdf | <document>
<section_header_level_1><location><page_1><loc_7><loc_82><loc_79><loc_84></location>Radio-AGN Feedback: When the Little Ones were Monsters</section_header_level_1>
<section_header_level_1><location><page_1><loc_7><loc_77><loc_47><loc_79></location>W. L. Williams glyph[star] 1 , 2 and H. J. A. Rottgering 1</section_header_level_1>
<text><location><page_1><loc_7><loc_75><loc_62><loc_77></location>1 Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands 2 Netherlands Institute for Radio Astronomy (ASTRON), P.O. Box 2, 7990AA Dwingeloo, The Netherlands</text>
<text><location><page_1><loc_7><loc_71><loc_17><loc_72></location>12 November 2021</text>
<section_header_level_1><location><page_1><loc_28><loc_67><loc_36><loc_68></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_28><loc_48><loc_89><loc_67></location>We present a study of the evolution of the fraction of radio-loud active galactic nuclei (AGN) as a function of their host stellar mass. We make use of two samples of radio galaxies: one in the local universe, 0 . 01 < z glyph[lessorequalslant] 0 . 3 , using a combined SDSS-NVSS sample and one at higher redshifts, 0 . 5 < z glyph[lessorequalslant] 2 , constructed from the VLA-COSMOS DEEP Radio Survey at 1 . 4 GHz and a K s -selected catalogue of the COSMOS/UltraVISTA field. We observe an increase of more than an order of magnitude in the fraction of lower mass galaxies ( M ∗ < 10 10 . 75 M glyph[circledot] ) which host Radio-Loud AGN with radio powers P 1 . 4 GHz > 10 24 WHz -1 at z ∼ 1 -2 while the radio-loud fraction for higher mass galaxies ( M ∗ > 10 11 . 25 M glyph[circledot] ) remains the same. We argue that this increase is driven largely by the increase in cold or radiative mode accretion with increasing cold gas supply at earlier epochs. The increasing population of low mass Radio-Loud AGN can thus explain the upturn in the Radio Luminosity Function (RLF) at high redshift which is important for understanding the impact of AGN feedback in galaxy evolution.</text>
<text><location><page_1><loc_28><loc_45><loc_89><loc_48></location>Key words: galaxies: active - radio continuum: galaxies - galaxies: evolution - galaxies: jets - galaxies: luminosity function, mass function - accretion, accretion discs</text>
<section_header_level_1><location><page_1><loc_7><loc_39><loc_21><loc_40></location>1 INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_7><loc_5><loc_46><loc_38></location>During recent years it has become increasingly apparent that the radio jets of radio-loud Active Galactic Nuclei (RL AGN or radioAGN) play a crucial role in the process of galaxy formation and evolution via 'AGN feedback' (see e.g. Best et al. 2006, 2007; Bower et al. 2006; Croton et al. 2006; Fabian et al. 2006). This feedback can occur because the enormous energy output of the AGN can be injected into the surrounding medium, possibly also the fuel source of the AGN, via ionizing radiation and/or relativistic jets, thereby providing enough energy to affect star formation in the host galaxy. Radio galaxies have been shown to comprise two different populations: high and low excitation (Best et al. 2005; Tasse et al. 2008; Hickox et al. 2009), each of which may have a separate and different effect of feedback, the exact nature and evolution of which is still debated. The first population of RL AGN is associated with the classic optical 'quasars'. These sources radiate across the electromagentic spectrum and are consistent with the unified models of quasars where emission is obscured at some wavelengths when the source is viewed edge-on (e.g. Barthel 1989; Antonucci 1993; Urry & Padovani 1995). In this 'high-excitation', 'cold mode' or 'radiative mode', accretion is postulated to occur via an accretion disc in a radiatively efficient manner (e.g. Shakura & Sunyaev 1973). These high excitation radio galaxies (HERGs) are typically hosted by lower mass, bluer galaxies in less dense environments (e.g. Tasse et al. 2008; Janssen et al. 2012). The second</text>
<text><location><page_1><loc_50><loc_14><loc_89><loc_40></location>mode of radio activity was first noted by their lack of strong highexcitation narrow-line optical excitation expected from the 'quasar' mode (Hine & Longair 1979; Laing et al. 1994; Jackson & Rawlings 1997). Moreover they show no evidence of mid-infrared emission from dusty tori (Whysong & Antonucci 2004; Ogle et al. 2006) and no evidence of accretion-related X-ray emission (Hardcastle et al. 2006; Evans et al. 2006). Hardcastle et al. (2007) first suggested that this mode, known as the 'low-excitation', 'radio mode', 'hot mode' or 'jet mode' occurs when hot gas is accreted directly onto the supermassive black hole in a radiatively inefficient manner via advection dominated accretion flows (ADAFs, e.g. Narayan & Yi 1995). Best et al. (2005) showed that these low excitation radio galaxies (LERGs) are hosted by fundamentally different galaxies: higher mass, redder and occurring in more dense environments. This mode in particular provides a direct feedback connection between the AGN and its hot gas fuel supply in the manner of work done by the expanding radio lobes on the hot intra-cluster gas. For a more detailed review of the HERG versus LERG dichotomy see Heckman & Best (2014) and references therein.</text>
<text><location><page_1><loc_50><loc_1><loc_89><loc_13></location>In order to understand the relative significance of the different types of radio-AGN feedback we need to understand the cosmic evolution of radio sources in detail. It has been known for several decades that the comoving number density of powerful radio sources is two to three orders of magnitude greater at a redshift of two to three compared to the local Universe (e.g. Schmidt 1968; Sandage 1972; Osmer 1982; Peacock 1985; Schmidt et al. 1988; Dunlop & Peacock 1990; Rigby et al. 2011). Similarly, the space density of optically selected quasars (QSOs) peaks at red-</text>
<text><location><page_2><loc_7><loc_69><loc_46><loc_87></location>shift 2 < z < 3 (e.g. Boyle et al. 1988; Hewett et al. 1993; Warren et al. 1994). It is also well known that, within the local universe ( z glyph[lessorsimilar] 0 . 3 ), the fraction of galaxies which host a radio source, i.e. the radio-loud fraction, is a very steep function of host galaxy stellar mass ( f radio-loud ∝ M 2 . 5 ∗ , Best et al. 2005), increasing to > 30 per cent at stellar masses above 5 × 10 11 M glyph[circledot] for radio luminosities > 10 23 WHz -1 . This pervasiveness of radio-loudness among high mass galaxies in the local Universe, combined with the dramatic increase in density of radio-loud sources at earlier times suggests that there must be an increase in the prevalence of radio activity among galaxies of lower mass. To test this, we investigate the fraction of radio-loud sources out to redshift ∼ 2 as a function of their host stellar mass.</text>
<text><location><page_2><loc_7><loc_54><loc_46><loc_69></location>This paper is structured as follows. In Section 2 we describe the construction of the Radio-Loud AGN samples and the catalogues from which they are selected. In Section 3 the luminosity functions for these samples are constructed and binned by host galaxy stellar mass. The radio-loud fraction as a function of host stellar mass is also determined. We discuss the results and their implications in Section 5 and conclude in Section 6. Throughout this paper, we use the following cosmological parameters: H 0 = 70 kms -1 Mpc -1 , Ω m = 0 . 3 and Ω Λ = 0 . 7 . The spectral index, α , is defined as S ν ∝ ν -α and unless otherwise specified, we adopt a default value of 0 . 8 .</text>
<section_header_level_1><location><page_2><loc_7><loc_49><loc_30><loc_50></location>2 RADIO-LOUD AGN SAMPLES</section_header_level_1>
<text><location><page_2><loc_7><loc_34><loc_46><loc_48></location>To investigate the evolution of the Radio Luminosity Function (RLF) for different stellar-masses over a similar luminosity range several samples with good ancillary and derived data are needed, to provide both the required statistics at low redshift and the sensitivity at high redshift. In this work we combine one already existing matched radio-optical dataset in the local universe using the SDSS-NVSS sample, described in Section 2.1, and one at redshifts 0 . 5 < z < 2 , which we have constructed using the VLA-COSMOS DEEP Radio Survey at 1 . 4 GHz and a K s -selected catalogue of the COSMOS/UltraVISTA field, which is described in Section 2.2.</text>
<section_header_level_1><location><page_2><loc_7><loc_30><loc_23><loc_31></location>2.1 Local SDSS Sample</section_header_level_1>
<text><location><page_2><loc_7><loc_1><loc_46><loc_29></location>For the local radio source sample we use the catalogue compiled by Best & Heckman (2012), which was constructed by cross-matching optical galaxies from the seventh data release (DR7; Abazajian et al. 2009) of the Sloan Digital Sky Survey (SDSS) spectroscopic sample with radio sources in the NRAO Very Large Array (VLA) Sky Survey (NVSS; Condon et al. 1998) and the Faint Images of the Radio Sky at Twenty centimetres (FIRST; Becker et al. 1995). The parent optical sample consists of all galaxies in the valueadded spectroscopic catalogues (VASC) created by the Max Plank Institute for Astrophysics and Johns Hopkins University (MPAJHU) group (see Brinchmann et al. 2004, available at http:// www.mpa-garching.mpg.de/SDSS/ ). The cross-matching was done for all radio sources with flux densities > 5 mJy, which corresponds to radio luminosities of P 1 . 4 GHz glyph[greaterorsimilar] 10 23 WHz -1 at redshift z = 0 . 1 and P 1 . 4 GHz glyph[greaterorsimilar] 10 24 WHz -1 at redshift z = 0 . 3 . The combined radio-optical area covered is 2 . 17 str (Best & Heckman 2012). Of the 927 522 galaxies in the VASC, Best & Heckman (2012) selected a magnitude-limited sample of 18 286 radio sources, which they showed to be 95 per cent complete and 99 per cent reliable (Best et al. 2005). The sample was restricted to</text>
<text><location><page_2><loc_50><loc_72><loc_89><loc_87></location>the 'main galaxy sample' (Strauss et al. 2002), comprising galaxies within the magnitude range 14 . 5 glyph[lessorequalslant] r < 17 . 7 mag and the redshift range 0 . 01 < z glyph[lessorequalslant] 0 . 3 . This local radio-optical sample consists of 9168 radio sources. We note that, being based on the SDSS main galaxy sample, this local matched radio-optical sample excludes both radio-loud quasars and broad-line radio galaxies. However, Best et al. (2014) show that this is only problematic at radio powers above P 1 . 4 GHz glyph[greaterorsimilar] 10 26 WHz -1 . Since our LFs do not probe those high powers we make no correction for this bias. Moreover, we know that the dominant population of radio sources in this sample are not quasars.</text>
<text><location><page_2><loc_50><loc_58><loc_89><loc_72></location>Properties of the host galaxies are taken from the VASCs which, for each source, provide several basic measured parameters from the imaging data such as magnitudes, colours and sizes (York et al. 2000), as well as derived properties including, most importantly for this work, the stellar mass M ∗ (Kauffmann et al. 2003). For their matched radio sample, Best & Heckman (2012) also separated the sources into star-forming galaxies and RL AGN ( 7302 sources), which are further sub-divided into high-excitation (HERG) and low-excitation (LERG) sources, based on their optical photometric and spectroscopic parameters.</text>
<text><location><page_2><loc_50><loc_51><loc_89><loc_58></location>The left panels in Fig. 1 show the radio power and host stellar mass as a function of redshift for the radio sources in the SDSS-NVSS sample. All the radio sources are plotted in black, and sources in the restricted sub-sample within the redshift bin which defines our local sample are plotted in yellow.</text>
<section_header_level_1><location><page_2><loc_50><loc_48><loc_73><loc_49></location>2.2 Distant VLA-COSMOS Sample</section_header_level_1>
<text><location><page_2><loc_50><loc_34><loc_89><loc_47></location>The VLA-COSMOS Large Project covered the 2 deg 2 of the COSMOSField at 1 . 4 GHz with observations by the VLA in the A configuration. This survey, extensively described in Schinnerer et al. (2004, 2007), provides continuum radio observations with a resolution (half-power beam width) of 1 . 5 × 1 . 4 arcsec and a mean 1 σ sensitivity of about 10 . 5 µ Jy in the innermost 1 deg 2 region and of about 15 µ Jy in the outer parts. From the VLA-COSMOS catalogue we selected sources above 50 µ Jy, which corresponds to sources > 5 σ over 50 per cent of the survey area.</text>
<text><location><page_2><loc_81><loc_23><loc_81><loc_24></location>glyph[negationslash]</text>
<text><location><page_2><loc_50><loc_5><loc_89><loc_34></location>The optical data for this sample comes from the K s -selected catalogue of the COSMOS/UltraVISTA field (Muzzin et al. 2013) which contains PSF-matched photometry in 30 photometric bands covering the wavelength range 0 . 15 µ m → 24 µ m. The entire region overlaps with the radio survey so the combined area is that of the COSMOS/UltraVISTA data, 1 . 62 deg 2 . Following the recommended criteria of Muzzin et al. (2013), we selected a 'clean' sample from the K s catalogue of sources with flags: 'star' = 1 , 'K flag' glyph[lessorequalslant] 4 , 'contamination' = 1 , and 'nan contam' glyph[lessorequalslant] 3 (these flags relate to stars and saturated sources and the quality of the photometry for nearby sources). Finally we selected sources brighter than the 90 per cent completeness limit of K s glyph[lessorequalslant] 23 . 4 mag. The redshifts provided are determined from SED fitting to the broadband photometry using the EAZY code (Brammer et al. 2008) and for source z < 1 . 5 they quote an rms error of ( δz/ (1 + z )) = 0 . 013 and a catastrophic outlier fraction of 1 . 56 per cent. While the quoted outlier fraction is very low, we do expect that many of the outliers will be quasars (i.e. HERGs). It is well known that photometric redshifts determined for quasars are less reliable than those obtained for galaxies (e.g. Richards et al. 2001; Babbedge et al. 2004; Mobasher et al. 2004; Polletta et al. 2007).</text>
<text><location><page_2><loc_63><loc_22><loc_63><loc_23></location>glyph[negationslash]</text>
<text><location><page_2><loc_50><loc_1><loc_89><loc_5></location>Within this sample we defined three 'high-redshift' subsamples in the redshift ranges 0 . 5 < z glyph[lessorequalslant] 1 . 0 , 1 . 0 < z glyph[lessorequalslant] 1 . 5 and 1 . 5 < z glyph[lessorequalslant] 2 . 0 , chosen such that each contains ∼ 300 -700</text>
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<location><page_3><loc_13><loc_67><loc_46><loc_86></location>
<caption>Figure 1. (left) Full SDSS-NVSS Sample and the selected 'local' 0 . 01 < z glyph[lessorequalslant] 0 . 3 sample : radio power vs z and stellar mass, M ∗ , vs z. (right) Full VLACOSMOS Sample and the three selected 'high redshift' 0 . 5 < z glyph[lessorequalslant] 1 and 1 < z glyph[lessorequalslant] 2 samples : radio power vs z and stellar mass, M ∗ , vs z . The grey shaded regions show the regions in mass-space, redshift-space and radio power-space which demarcate the redshift samples, mass bins and power bins for the Luminosity Functions. The selected clean mass-complete radio samples are shown in yellow, orange and red symbols (these colours denote these redshift bins thought this paper).</caption>
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<text><location><page_3><loc_68><loc_67><loc_69><loc_68></location>z</text>
<text><location><page_3><loc_68><loc_46><loc_69><loc_47></location>z</text>
<text><location><page_3><loc_7><loc_28><loc_46><loc_35></location>galaxies. The right panels of Fig. 1 show the radio power and host stellar mass as a function of redshift for the radio sources in the VLA-COSMOS sample. Sources in the restricted ranges which define our high redshift samples are plotted in orange and red, and all the remaining sources are shown in black.</text>
<text><location><page_3><loc_7><loc_18><loc_46><loc_28></location>Note that at radio powers P 1 . 4 GHz glyph[greaterorsimilar] 10 23 WHz -1 , the samples should consist almost entirely of RL AGN - we expect very little contamination from star-forming galaxies. We confirmed this using the spectroscopy-based AGN/SF separation in the local sample (Best et al. 2005). Moreover, the star formation rate for this radio power is in excess of 25 M glyph[circledot] yr -1 (Condon 1992) so we expect AGN to continue to dominate even at higher redshifts.</text>
<section_header_level_1><location><page_3><loc_7><loc_12><loc_43><loc_15></location>3 THE STELLAR-MASS DEPENDENT LUMINOSITY FUNCTION</section_header_level_1>
<section_header_level_1><location><page_3><loc_7><loc_10><loc_26><loc_11></location>3.1 The Luminosity Function</section_header_level_1>
<text><location><page_3><loc_7><loc_1><loc_46><loc_9></location>Astandard technique for quantifying the rate of evolution of a population of galaxies is to compare their luminosity functions (LFs) at two different epochs. In this section, we therefore determine the evolution of the Radio-Loud AGN population among host galaxies of different masses by comparing the luminosity functions of the SDSS-NVSS and VLA-COSMOS samples. Determining the radio</text>
<text><location><page_3><loc_50><loc_32><loc_89><loc_35></location>LFs for the full samples first serves to confirm our sample selection and methods.</text>
<text><location><page_3><loc_50><loc_1><loc_89><loc_31></location>These radio luminosity functions were calculated in the standard way, using ρ = Σ i 1 /V i method (Schmidt 1968; Condon 1989), where V i is the volume in which a given source could be detected. This volume is determined by both the minimum and maximum distance at which a given source would be included in the sample as a result of the selection criteria: V i = V max -V min , where V max and V min are the volumes enclosed within the observed sky area out to the maximum and minimum distances respectively. The minimum accessible volume is a result of the optical cut-off on the bright end ( > 14 . 5 mag for SDSS-NVSS, while the VLA-COSMOS sample has no bright flux limit). The maximum accessible volume is determined by the flux limits of both the optical ( < 17 . 77 mag for SDSS-NVSS and < 23 . 4 mag for VLACOSMOS)and radio data ( > 5 mJy for SDSS-NVSS and > 50 µ Jy for the VLA-COSMOS sample). In practice, the maximum redshift is dominated by the radio flux limits. The normalization of the luminosity function requires knowledge of the precise intersection area of all input surveys. The sky area for the SDSS-NVSS sample is taken to be 2 . 17 sr (Best et al. 2005). For the VLA-COSMOS sample, the sky area is taken to be 1 . 62 deg 2 for all sources above 100 µ Jy. For the faintest sources in the VLA-COSMOS sample, the area in which the sources could be detected is smaller due to</text>
<table>
<location><page_4><loc_7><loc_73><loc_46><loc_83></location>
<caption>Table 1. Number of sources in each stellar mass-redshift bin.</caption>
</table>
<text><location><page_4><loc_7><loc_43><loc_46><loc_69></location>the non-uniform rms noise level in the VLA-COSMOS mosaic. We therefore weight each source by the inverse of the area in which it can be detected, which also accounts for the varying detection area within a given luminosity bin. Uncertainties are calculated as the statistical Poissonian errors with a contribution of cosmic variance where appropriate; in some luminosity/mass bins, these errors are so small that the error will be dominated by systematic effects. The area covered by VLA-COSMOS sample of 1 . 62 deg 2 is small enough that the effects of cosmic variance are not negligible for very massive galaxies. The total area covered by COSMOS/UltraVISTA is approximately 1.5 square degrees in one single field, making the effects of cosmic variance not negligible for very massive galaxies. The contribution of cosmic variance to the total error budget was estimated through the recipe of (Moster et al. 2011). The average uncertainties due to this effect for the least massive galaxies vary from 6 to 7 per cent and for the most massive galaxies vary from 10 to 12 per cent across the three highest redshift bins. These values were added in quadrature to the Poissonian error of the LFs.</text>
<text><location><page_4><loc_7><loc_34><loc_46><loc_43></location>Wehave compared our derived RLFs for radio-loud sources of all stellar masses with those in the literature. For the full samples, our local RLF agrees with that of Best et al. (2005) and our RLF for the high redshift VLA-COSMOS sample agrees with that of Smolˇci'c et al. (2009). The agreement is not unexpected as we are using the same data, but it serves to validate out sample selections and method.</text>
<text><location><page_4><loc_7><loc_12><loc_46><loc_33></location>We next split the samples into four bins of different host galaxy stellar masses: 10 . 0 < log( M ∗ /M glyph[circledot] ) glyph[lessorequalslant] 10 . 75 , 10 . 75 < log( M ∗ /M glyph[circledot] ) glyph[lessorequalslant] 11 . 0 , 11 . 0 < log( M ∗ /M glyph[circledot] ) glyph[lessorequalslant] 11 . 25 , and 11 . 25 < log( M ∗ /M glyph[circledot] ) glyph[lessorequalslant] 12 . 0 . These bins are chosen to sample the stellar masses well with similar numbers of sources in each bin (see Fig. 1). The RLFs derived in each of these stellar mass bins in the four redshift bins are plotted in Fig. 2. Note that in the low redshift sample we exclude the points below P 1 . 4 GHz glyph[lessorsimilar] 10 23 WHz -1 for the highest stellar mass bin as these sources are only detectable out to z glyph[lessorsimilar] 0 . 1 and this sample is incomplete for high mass sources at these redshifts. Table 1 shows the number of sources in each stellar mass-redshift bin. From these LFs we see that in the local universe and for P 1 . 4 GHz glyph[greaterorsimilar] 10 23 WHz -1 , the number density of the highest host stellar mass bin is the greatest, while in the higher redshift bins the number density within all the stellar mass bins is becoming increasingly similar.</text>
<section_header_level_1><location><page_4><loc_7><loc_8><loc_25><loc_9></location>3.2 Space Density Evolution</section_header_level_1>
<text><location><page_4><loc_7><loc_1><loc_46><loc_6></location>Now, to quantify the disproportional increase in radio activity among galaxies of different mass at higher redshifts, we investigate the relative comoving space density of radio-loud sources with respect to the local comoving space density. We do this by dividing</text>
<figure>
<location><page_4><loc_51><loc_16><loc_88><loc_86></location>
<caption>Figure 2. The comoving space density of radio-loud sources. This is the RLF in redshift bins 0 . 01 < z glyph[lessorequalslant] 0 . 3 (top panel), 0 . 5 < z glyph[lessorequalslant] 1 . 0 (topmiddle panel), 1 . 0 < z glyph[lessorequalslant] 1 . 5 (bottom-middle panel) and 1 . 5 < z glyph[lessorequalslant] 2 . 0 (bottom panel) for the four host stellar mass bins (plotted in colour).</caption>
</figure>
<table>
<location><page_5><loc_8><loc_72><loc_44><loc_82></location>
<caption>Table 2. Relative space density of radio-loud sources with respect to the local density as a function of redshift for all sources with radio powers greater than the cut-off P 1 . 4 GHz > 10 24 WHz -1 .</caption>
</table>
<text><location><page_5><loc_7><loc_56><loc_46><loc_69></location>the stellar mass dependent RLFs in redshift bins 0 . 5 < z glyph[lessorequalslant] 1 . 0 , 1 . 0 < z glyph[lessorequalslant] 1 . 5 and 1 . 5 < z glyph[lessorequalslant] 2 . 0 by the local RLF ( 0 . 01 < z glyph[lessorequalslant] 0 . 3 ). These relative RLFs are plotted in Fig. 3, from which it is clear that there is a difference between the relative comoving space density of low mass galaxies hosting radio sources and that of high mass hosts. We derive the relative comoving space density of radioloud sources with respect to the local comoving space density for all sources with radio powers greater than a cut-off luminosity of P 1 . 4 GHz > 10 24 WHz -1 . This is plotted as a function of redshift in Fig. 4 and the values listed in Table 2.</text>
<text><location><page_5><loc_7><loc_35><loc_46><loc_56></location>At z ∼ 1 . 5 -2 the space density of the least massive galaxies hosting Radio-Loud AGN above P 1 . 4 GHz > 10 24 WHz -1 is 45 ± 11 times greater than the local space density. Even the sources with stellar masses in the range 10 . 75 < log( M ∗ /M glyph[circledot] ) glyph[lessorequalslant] 11 . 0 are 17 . 3 ± 2 . 6 times more prevalent. We note that there is a slight decrement in the space density of the most massive galaxies hosting Radio-Loud AGN at both radio power cuts going out to z < 1 . 5 . If we consider the slightly less powerful sources, P 1 . 4 GHz > 10 23 . 5 WHz -1 , and only go out to redshift z < 1 . 5 (not plotted here), the effect is only really seen in the lowest mass bin where the increase in space density increases more rapidly with redshift: the space density of the least massive galaxies hosting Radio-Loud AGN is 40 . 8 ± 6 . 2 times greater than locally compared to the 27 . 6 ± 6 . 9 -fold increase for P 1 . 4 GHz > 10 24 WHz -1 sources in the 1 . 0 < z glyph[lessorequalslant] 1 . 5 bin.</text>
<section_header_level_1><location><page_5><loc_7><loc_31><loc_30><loc_32></location>4 THE RADIO-LOUD FRACTION</section_header_level_1>
<text><location><page_5><loc_7><loc_16><loc_46><loc_30></location>Another way of looking at the increase in prevalence of AGN in lower mass hosts is to consider the fraction of sources which are radio-loud as a function of the host stellar mass in our four redshift bins. The mass-dependence of the radio-loud fraction can be an indicator of the accretion mode of the radio-AGN largely because of the different dependence of the fuelling source (hot vs. cold gas) on stellar mass (Best et al. 2006). The radio-loud fraction can be easily calculated by dividing the stellar mass function (SMF) for radio-loud sources, ρ Rad (for sources above a given radio power limit), by the SMF for all galaxies, ρ Opt :</text>
<text><location><page_5><loc_7><loc_14><loc_23><loc_15></location>f radio-loud = ρ Rad /ρ Opt .</text>
<section_header_level_1><location><page_5><loc_7><loc_10><loc_27><loc_11></location>4.1 The Stellar Mass Function</section_header_level_1>
<text><location><page_5><loc_7><loc_1><loc_46><loc_9></location>In order to calculate the radio-loud fraction we first derived the SMFs for all galaxies and for the radio source hosts by using the 1 /V max estimator (Schmidt 1968) as previously described for the LFs, which corrects for the fact that the samples are magnitude limited. In order to construct the SMFs, the redshift-dependent limiting M ∗ above which the magnitude-limited sample is complete needs</text>
<figure>
<location><page_5><loc_51><loc_39><loc_88><loc_87></location>
<caption>Figure 3. The relative comoving space density of radio-loud sources with respect to the local comoving space density. This is the RLF in redshift bins 0 . 5 < z glyph[lessorequalslant] 1 . 0 (top panel), 1 . 0 < z glyph[lessorequalslant] 1 . 5 (middle panel) and 1 . 5 < z glyph[lessorequalslant] 2 . 0 (bottom panel) by the local, 0 . 01 < z glyph[lessorequalslant] 0 . 3 , RLF for the four host stellar mass bins (plotted in colour).</caption>
</figure>
<text><location><page_5><loc_50><loc_22><loc_89><loc_26></location>to be known. For the COSMOS/UltraVISTA sample we use the empirical 95 per cent completeness limit calculated by Muzzin et al. (2013).</text>
<text><location><page_5><loc_50><loc_1><loc_89><loc_22></location>The SMFs for radio-loud galaxies and all galaxies are plotted in Fig. 5 for a radio-power cut-off of P 1 . 4 GHz > 10 24 WHz -1 . We choose this limit because the highest redshift bin is only able to probe radio powers greater than this (see also Fig. 1). The SMFs we derive for our optical galaxy samples (for both the SDSS and COSMOS samples) are consistent with those of Muzzin et al. (2013) who use a more sophisticated maximum-likelihood analysis to derive the SMFs in several redshift bins. As expected (Best et al. 2005), ρ Rad differs significantly from ρ Opt - the hosts of radio sources are biased towards more massive systems. Interestingly, while the comoving number density of all galaxies decreases with redshift, that of the radio source hosts with log( M ∗ /M glyph[circledot] ) glyph[lessorsimilar] 11 . 0 increases. This is consistent with the results of Tasse et al. (2008) and shows that the radio-loud galaxy population evolves differently from the population of galaxies as a whole.</text>
<figure>
<location><page_6><loc_9><loc_65><loc_46><loc_87></location>
<caption>Figure 4. Relative space density of radio-loud sources with respect to the local density as a function of redshift for all sources with radio powers greater than the cut-off P 1 . 4 GHz > 10 24 WHz -1 .</caption>
</figure>
<figure>
<location><page_6><loc_8><loc_34><loc_45><loc_56></location>
<caption>Figure 7. The fraction of galaxies hosting a radio source (radio-loud fraction) relative to the local fraction for a radio-power cut-off of P 1 . 4 GHz > 10 24 WHz -1 in the three higher redshift bins.</caption>
</figure>
<text><location><page_6><loc_28><loc_34><loc_28><loc_35></location>∗</text>
<text><location><page_6><loc_32><loc_34><loc_33><loc_35></location>/circledot</text>
<paragraph><location><page_6><loc_7><loc_29><loc_46><loc_32></location>Figure 5. SMFs for all galaxies, ρ Opt , (shaded lines) and radio-loud galaxies, ρ Rad , (lines with points) for a radio-power cut-off P 1 . 4 GHz > 10 24 WHz -1 in the four redshift bins.</paragraph>
<section_header_level_1><location><page_6><loc_7><loc_24><loc_26><loc_25></location>4.2 The Radio-Loud Fraction</section_header_level_1>
<text><location><page_6><loc_7><loc_1><loc_46><loc_23></location>Having computed the relevant SMFs we can calculate the radioloud fraction as described above. Figure 6 shows the radio-loud fraction for a radio-power cut-off of P 1 . 4 GHz > 10 24 WHz -1 . The radio-loud fraction clearly increases with redshift. Moreover, the slope of the mass dependence becomes shallower, showing that the fraction of lower mass galaxies hosting radio sources increases more with redshift than the fraction for high mass galaxies. The slopes are shallower at higher redshifts, for P 1 . 4 GHz > 10 24 WHz -1 going from f RL ∝ M 2 . 7 ± 0 . 2 ∗ in the local sample, flattening to f RL ∝ M 1 . 7 ± 0 . 1 ∗ , f RL ∝ M 1 . 5 ± 0 . 1 ∗ and f RL ∝ M 1 . 0 ± 0 . 1 ∗ in the higher redshift bins. This is highlighted in the plot of the radio-loud fraction relative to the local redshift bin (fig. 7), where the relative radio-loud fraction is up to two orders of magnitude greater at the low mass end. We note that the flattening is quicker with redshift when the lower power P 1 . 4 GHz > 10 23 . 5 WHz -1 sources are considered (in the first three redshift</text>
<figure>
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<text><location><page_6><loc_71><loc_65><loc_71><loc_66></location>∗</text>
<text><location><page_6><loc_75><loc_65><loc_76><loc_66></location>/circledot</text>
<figure>
<location><page_6><loc_51><loc_32><loc_88><loc_54></location>
<caption>Figure 6. The fraction of galaxies hosting a radio source (radio-loud fraction) for a radio-power cut-off of P 1 . 4 GHz > 10 24 WHz -1 in the four redshift bins. The coloured lines show a linear fit over the stellar mass range 10 < log( M ∗ /M glyph[circledot] ) < 11 . 5 . The slopes of these fits are 2 . 7 ± 0 . 2 , 1 . 7 ± 0 . 2 , 1 . 5 ± 0 . 1 and 1 . 0 ± 0 . 1 from the lowest to highest redshift bins.</caption>
</figure>
<text><location><page_6><loc_71><loc_32><loc_71><loc_33></location>∗</text>
<text><location><page_6><loc_75><loc_32><loc_76><loc_33></location>/circledot</text>
<text><location><page_6><loc_50><loc_21><loc_89><loc_24></location>bins only, not plotted here): f RL ∝ M 2 . 7 ± 0 . 1 ∗ , f RL ∝ M 1 . 3 ± 0 . 1 ∗ and f RL ∝ M 1 . 1 ± 0 . 1 ∗ .</text>
<section_header_level_1><location><page_6><loc_50><loc_16><loc_65><loc_17></location>5 INTERPRETATION</section_header_level_1>
<text><location><page_6><loc_50><loc_11><loc_89><loc_15></location>In this section we aim to interpret and explain our results within the context of the HERG-LERG population dichotomy and their differential cosmic evolution and mass dependence.</text>
<text><location><page_6><loc_50><loc_1><loc_89><loc_11></location>In the local Universe, the density of high mass Radio-Loud AGN is an order of magnitude higher than that of low mass RadioLoud AGN at all radio powers (cf. top panel Fig. 2). In the more distant Universe, up to z < 2 we see a sharp increase in the number density of Radio-Loud AGN hosted by lower mass galaxies, while the number density of high mass Radio-Loud AGN remains constant.</text>
<text><location><page_7><loc_7><loc_58><loc_46><loc_87></location>This large increase in the prevalence of radio activity among galaxies of lower mass at higher redshifts (cf. Fig. 4) shows that it is the lower mass galaxies which are the cause of the upturn in the observed RLFs (e.g. Dunlop & Peacock 1990; Rigby et al. 2011). Moreover we suggest that this upturn is likely due to an increasing population of cold mode accretors at earlier epochs. From Best et al. (2005) we know that locally, despite the wide distributions in host stellar mass of both HERGs and LERGs, LERGs have a strong preference to be hosted by galaxies with higher masses, while HERGs are hosted by galaxies with a lower median stellar mass but with a broader distribution. Assuming this still holds at higher redshifts, it implies that this strongly evolving population of lower mass Radio-Loud AGN are HERGs. We also expect the highest mass, most powerful sources to peak in space density at higher redshifts ( z ∼ 2 -3 Rigby et al. 2011) in line with the cosmic downsizing picture where the most massive black holes have formed by z ∼ 4 . Indeed, results from many of the earlier radio surveys show that the most powerful ( P 1 . 4 GHz glyph[greaterorsimilar] 10 26 WHz -1 ) radio galaxies at z glyph[greaterorsimilar] 1 (e.g. Eales et al. 1997; Jarvis et al. 2001; Seymour et al. 2007; Fernandes et al. 2015) are predominantly HERGs hosted by the most massive galaxies.</text>
<text><location><page_7><loc_7><loc_44><loc_46><loc_58></location>Indications from studies out to z glyph[lessorsimilar] 1 show that the HERGs are indeed evolving more strongly with redshift than the LERG population (Best et al. 2014) such that within the redshift ranges of this study the radio-AGN population should be dominated by HERGs, opposite to that within the local universe. The mass dependence that we observe in this study supports this idea. Moreover, the evolution in the optical quasar luminosity function (i.e. that corresponding to radio-quiet cold mode accretion; Hasinger et al. 2005; Hopkins et al. 2007; Croom et al. 2009) is comparable to the kind of increase we observe for the low mass galaxies.</text>
<text><location><page_7><loc_7><loc_28><loc_46><loc_44></location>Furthermore, the slope of the radio-loud fraction (cf. Fig. 6) we observe at the higher redshifts, f RL ∝ M ∼ 1 . 3 ∗ , is more consistent with the slope of the radio-loud fraction found in the local universe for HERGs only (Janssen et al. 2012). On the other hand, in the local sample, our derived slope of f RL ∝ M ∼ 2 . 5 ∗ matches that which Janssen et al. (2012) found for only LERGs and matches the theoretical value for the accretion of hot gas from a halo (Best et al. 2006). In this case we know that the dominant population of all local radio sources is that of the LERGs. This lends support to the idea that there is an increase in the prevalence of HERG activity or cold mode accretion and that this mode is becoming the dominant population out to redshifts of 0 . 5 < z glyph[lessorequalslant] 2 .</text>
<text><location><page_7><loc_7><loc_18><loc_46><loc_27></location>Lastly, as Heckman & Best (2014) state, the crucial requirement for cold mode accretion is the abundant central supply of cold dense gas. And as reflected in the increase in cosmic star formation rate density which has increased tenfold out to z ∼ 2 and threefold out to z ∼ 0 . 5 (e.g. Sobral et al. 2013; Madau & Dickinson 2014, and references therein), there is significantly more cold gas fuelling star formation at these epochs.</text>
<text><location><page_7><loc_7><loc_1><loc_46><loc_17></location>Acloser inspection of Fig. 3 reveals some interesting features, most particularly in highest mass bin, 11 . 25 < log( M ∗ /M glyph[circledot] ) < 12 , in the highest redshift interval, 1 < z glyph[lessorequalslant] 1 . 5 . We suggest that while the radio-AGN sample as a whole at these redshifts is becoming dominated by cold mode accretors, the highest mass, intermediate power ( 10 24 < P 1 . 4 GHz < 10 25 WHz -1 ) sources could still be hot mode/LERG sources because of their high mass. We observe a slight decrease in the number density of these specific sources consistent with the modelling results of Rigby et al. where the ∼ 10 24 WHz -1 population peaks at z ∼ 1 and is likely dominated by LERGs. This could be significant in the context of the models of LERG evolution presented by Best et al. (2014). If these</text>
<text><location><page_7><loc_50><loc_75><loc_89><loc_87></location>are indeed LERGs the decrease in number density does not match the order of magnitude decrease predicted by the preferred model of Best et al. (2014) extrapolated out to z ∼ 1 . 5 from fits to data out to z ∼ 0 . 7 . This model includes a time delay of ∼ 1 . 5 Gyr between the formation of massive quiescent galaxies and when they are able to produce hot mode AGN, and at redshifts above ∼ 1 the available population of host galaxies is declining so rapidly that such a delay in the onset of hot mode AGN activity would imply a drastic fall in the number densities of these sources above z ∼ 1 .</text>
<section_header_level_1><location><page_7><loc_50><loc_69><loc_74><loc_70></location>6 SUMMARY AND CONCLUSIONS</section_header_level_1>
<text><location><page_7><loc_50><loc_52><loc_89><loc_68></location>In this paper we have used the SDSS value-added spectroscopic sample of radio-loud galaxies (Best & Heckman 2012) and the VLA-COSMOS radio sample (Schinnerer et al. 2004, 2007) matched to a K s -selected catalogue of the COSMOS/UltraVISTA field (Muzzin et al. 2013) to compile two samples of Radio-Loud AGNgoing out to z = 2 . The samples are of sufficiently high radio power that they are dominated by RL AGN. Using these samples we have constructed radio luminosity functions for four host stellar mass bins between log( M ∗ /M glyph[circledot] ) = 10 . 0 and log( M ∗ /M glyph[circledot] ) = 12 . 0 , in four redshift bins between z = 0 . 01 and z = 2 . We have also investigated the radio-loud fraction as a function of stellar mass in these redshift bins. Together, we found the following:</text>
<unordered_list>
<list_item><location><page_7><loc_50><loc_37><loc_89><loc_51></location>(i) Radio activity among galaxies of different mass increases differently towards higher redshifts. By considering the relative comoving space density of radio-loud sources with respect to the local comoving space density, we showed that at 1 . 5 < z < 2 the space density of galaxies with stellar masses in the range 10 . 00 < log( M ∗ /M glyph[circledot] ) glyph[lessorequalslant] 10 . 75 and 10 . 75 < log( M ∗ /M glyph[circledot] ) glyph[lessorequalslant] 11 . 0 hosting Radio-Loud AGN with P 1 . 4 GHz > 10 24 WHz -1 is respectively 45 ± 11 and 17 . 3 ± 2 . 6 times greater relative to the local space density while that of higher mass galaxies hosting RadioLoud AGN remains the same.</list_item>
<list_item><location><page_7><loc_50><loc_30><loc_89><loc_37></location>(ii) The fraction of galaxies which host Radio-Loud AGN with P 1 . 4 GHz > 10 24 WHz -1 as a function of stellar mass shows a clear increase with redshift and a flattening with mass with the mass dependence evolving from f RL ∝ M 2 . 7 ∗ in the local sample to f RL ∝ M 1 . 0 ∗ at 1 . 5 < z < 2 .</list_item>
</unordered_list>
<text><location><page_7><loc_50><loc_16><loc_89><loc_29></location>We have argued that this increase in the prevalence of radio activity among galaxies of lower mass at higher redshifts is largely due to a rising contribution of AGN accreting in the radiative mode (HERGs). With this data we cannot yet conclusively show the evolution of the different accretion modes as a function of their host stellar mass because we lack the information on the excitation state of these sources at higher redshifts. However, future work combining this and other radio-optical samples will allow more detailed studies of optical hosts of the high-redshift ( 1 glyph[lessorsimilar] z glyph[lessorsimilar] 2 ) population of radio-AGN.</text>
<section_header_level_1><location><page_7><loc_50><loc_10><loc_67><loc_11></location>ACKNOWLEDGEMENTS</section_header_level_1>
<text><location><page_7><loc_50><loc_4><loc_89><loc_9></location>The authors thank Philip Best, George Miley and Emma Rigby for useful discussions which contributed to the interpretation in this paper. The authors thank the anonymous referee for useful comments which have improved this manuscript.</text>
<text><location><page_7><loc_50><loc_1><loc_89><loc_4></location>This study uses a K s -selected catalogue of the COSMOS/UltraVISTA field from Muzzin et al. (2013). The catalogue</text>
<text><location><page_8><loc_7><loc_75><loc_46><loc_87></location>contains PSF-matched photometry in 30 photometric bands covering the wavelength range 0 . 15 µ m → 24 µ mand includes the available GALEX (Martin et al. 2005), CFHT/Subaru (Capak et al. 2007), UltraVISTA (McCracken et al. 2012), S-COSMOS (Sanders et al. 2007), and zCOSMOS (Lilly et al. 2009) datasets. This work uses the catalogue compiled by Best & Heckman (2012), which combines data from the VASC (Brinchmann et al. 2004) of the SDSSDR7(Abazajian et al. 2009) with NVSS (Condon et al. 1998) and FIRST (Becker et al. 1995).</text>
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</document> | [] |
2022CQGra..39b5012C | https://arxiv.org/pdf/2106.09751.pdf | <document>
<section_header_level_1><location><page_1><loc_33><loc_74><loc_67><loc_79></location>Midisuperspace foam and the cosmological constant</section_header_level_1>
<text><location><page_1><loc_42><loc_64><loc_58><loc_71></location>S. C arlip ∗ Department of Physics University of California Davis, CA 95616</text>
<text><location><page_1><loc_48><loc_62><loc_52><loc_64></location>USA</text>
<section_header_level_1><location><page_1><loc_46><loc_53><loc_54><loc_54></location>Abstract</section_header_level_1>
<text><location><page_1><loc_24><loc_33><loc_76><loc_52></location>Wheeler's conjectured 'spacetime foam'-large quantum fluctuations of spacetime at the Planck scale-could have important implications for quantum gravity, perhaps even explaining why the cosmological constant seems so small. Here I explore this problem in a midisuperspace model consisting of metrics with local spherical symmetry. Classically, an infinite class of 'foamy' initial data can be constructed, in which cancellations between expanding and contracting regions lead to a small average expansion even if Λ is large. Quantum mechanically, the model admits corresponding stationary states, for which the probability current is also nearly zero. These states appear to describe a self-reproducing spacetime foam with very small average expansion, effectively hiding the cosmological constant.</text>
<section_header_level_1><location><page_2><loc_13><loc_89><loc_61><loc_90></location>1 Spacetime foam and the cosmological constant</section_header_level_1>
<text><location><page_2><loc_13><loc_77><loc_87><loc_87></location>More than 65 years ago, Wheeler suggested that quantum uncertainties in the metric should be of order one at the Planck scale, leading to large fluctuations in geometry and (perhaps) topology, which he called 'spacetime foam' [1]. While this idea has continued to generate interest, it has been notoriously hard to investigate quantitatively. My goal here is to develop a model simple enough to allow calculations, but still complex enough to describe a large assortment of foam-like quantum states.</text>
<text><location><page_2><loc_13><loc_63><loc_87><loc_77></location>More specifically, one aim is to test a proposal that spacetime foam might help solve the cosmological constant problem. In [2], I suggested that our Universe might in fact have a large cosmological constant Λ, whose effects are 'hidden' in Planck scale fluctuations. I showed that for a large class of initial data, the expansion and shear-the observational signature of a cosmological constant-average to zero over small regions even if Λ is large. A slightly oversimplified picture is that a cosmological constant can cause either expansion or contraction; using a construction of Chrusciel, Isenberg, and Pollack [3,4], one can sew together Planck-scale regions with random initial data in a way that naturally leads to large cancellations.</text>
<text><location><page_2><loc_13><loc_55><loc_87><loc_63></location>A key question is whether such cancellations persist during evolution. This is fundamentally a matter for quantum gravity, and a full treatment lies beyond our current capabilities. There is, however, a partial step: we can explore mini- and midisuperspace models, ∗ in which many of the degrees of freedom are frozen out. Such models have been used to study a range of issues in quantum gravity, and while the results are rarely conclusive, they can be strongly suggestive.</text>
<text><location><page_2><loc_13><loc_43><loc_87><loc_54></location>Here I will focus on a midisuperspace consisting of geometries with local spherical symmetry. When combined with the 'dust time' of Brown and Kuchaˇr [8-10], the Wheeler-DeWitt equation takes a Schrodinger-like form, and can be solved by WKB methods. We shall see that there are stationary states in which a foamlike structure and small average expansion are preserved in time. Moreover, while regularization ambiguities exist, 'foamy' spacetimes seem to occur with high probabilities. In this setting, at least, a cosmological constant may indeed be hidden in spacetime foam.</text>
<section_header_level_1><location><page_2><loc_13><loc_39><loc_41><loc_40></location>2 Local spherical symmetry</section_header_level_1>
<text><location><page_2><loc_13><loc_24><loc_87><loc_37></location>The setting for locally spherically symmetric midisuperspace has been studied extensively by Morrow-Jones, Witt, and Schleich [5-7]. At first sight, this symmetry requirement may seem too strong: it is widely believed that spherical symmetry with Λ > 0 leads inevitably to Schwarzschild-de Sitter space. But while Birkhoff's theorem implies that a spherically symmetric vacuum spacetime must be locally isometric to some region of Schwarzschild-de Sitter space, local patches can be sewn together to form a spacetime that looks drastically different [7]. In particular, we will be able to construct explicit initial data containing both expanding and contracting regions.</text>
<text><location><page_2><loc_13><loc_19><loc_87><loc_23></location>Here, for simplicity, I will assume a positive cosmological constant, and specialize to the spatial topology S 1 × S 2 . Recall that to construct a manifold with this topology, we start with</text>
<text><location><page_3><loc_13><loc_83><loc_87><loc_90></location>a solid three-ball; cut out a ball from its center to form a manifold [0 , 1] × S 2 ; and identify the boundaries { 0 }× S 2 and { 1 }× S 2 . To build 'foamy' spacetimes, we will take this construction a step further, splitting space into an onion-like sequence of concentric shells, each with its own geometry.</text>
<text><location><page_3><loc_13><loc_78><loc_87><loc_83></location>Following [5], let us start with the initial value formalism, with geometric data consisting of a spatial metric q ij and its conjugate momentum π i j . These cannot be chosen arbitrarily, but must solve the momentum and Hamiltonian constraints</text>
<formula><location><page_3><loc_30><loc_74><loc_87><loc_77></location>P j = (3) ∇ i π i j = 0 (2.1a)</formula>
<text><location><page_3><loc_13><loc_66><loc_87><loc_70></location>(I use the conventions of [11]; in particular, κ 2 = 8 πG .) On a hypersurface of constant time, a general locally spherically symmetric metric may be written in the form</text>
<formula><location><page_3><loc_30><loc_69><loc_87><loc_75></location>H = 2 κ 2 √ q ( π i j π j i -1 2 π 2 ) -1 2 κ 2 √ q ( (3) R -2Λ ) = 0 . (2.1b)</formula>
<formula><location><page_3><loc_36><loc_63><loc_87><loc_65></location>ds 2 = h 2 dψ 2 + f 2 ( dθ 2 +sin 2 θ dϕ 2 ) (2.2)</formula>
<text><location><page_3><loc_13><loc_60><loc_69><loc_62></location>where h and f are functions of ψ and t . The spatial scalar curvature is</text>
<formula><location><page_3><loc_36><loc_54><loc_87><loc_59></location>(3) R = 1 f 2 [ -4 f h ( f ' h ) ' -2 f ' 2 h 2 +2 ] , (2.3)</formula>
<text><location><page_3><loc_13><loc_49><loc_87><loc_54></location>where a prime denotes a derivative with respect to ψ . Ref. [5] makes an added coordinate choice h ( ψ, t ) = a ( t ), f ( ψ, t ) = a ( t ) ˜ f ( ψ ); this simplifies the classical solutions, but complicates the quantum treatment by gauge-fixing one of the diffeomorphism constraints.</text>
<text><location><page_3><loc_16><loc_47><loc_48><loc_49></location>The nonvanishing canonical momenta are</text>
<formula><location><page_3><loc_36><loc_44><loc_87><loc_46></location>π θ θ = π ϕ ϕ = sin θ Q, π ψ ψ = sin θ P, (2.4)</formula>
<text><location><page_3><loc_13><loc_38><loc_87><loc_42></location>where P and Q are functions of ψ and t . (The sin θ factors appear because the momenta are tensor densities). Geometrically, P and Q are extrinsic curvatures, fractional rates of expansion; explicitly,</text>
<formula><location><page_3><loc_32><loc_32><loc_87><loc_36></location>Q = 1 2 κ 2 hf 2 ( K θ θ + K ψ ψ ) , P = 1 κ 2 hf 2 K θ θ . (2.5)</formula>
<text><location><page_3><loc_13><loc_31><loc_48><loc_32></location>The only nontrivial momentum constraint is</text>
<formula><location><page_3><loc_29><loc_25><loc_87><loc_29></location>P = P ' -h ' h P -2 f ' f Q = 0 ⇒ Q = f 2 f ' ( P ' -h ' h P ) , (2.6)</formula>
<text><location><page_3><loc_13><loc_23><loc_63><loc_24></location>while the Hamiltonian constraint becomes, after a little algebra,</text>
<formula><location><page_3><loc_29><loc_17><loc_87><loc_22></location>H = h f ' d dψ [ -κ 2 P 2 fh 2 + 1 κ 2 ( ff ' 2 h 2 -f ) + Λ 3 κ 2 f 3 ] = 0 . (2.7)</formula>
<text><location><page_3><loc_13><loc_15><loc_59><loc_17></location>This simple total derivative structure was first noted in [5].</text>
<text><location><page_4><loc_16><loc_89><loc_56><loc_90></location>Given a metric ( h, f ), we can now solve (2.7) for P ,</text>
<formula><location><page_4><loc_31><loc_82><loc_87><loc_87></location>P = ± 1 κ 2 [ f 2 h 2 ( f ' 2 h 2 -1 ) + Λ 3 f 4 h 2 + γfh 2 ] 1 / 2 (2.8)</formula>
<text><location><page_4><loc_13><loc_79><loc_87><loc_82></location>where γ is an integration constant. As also noted in [5], this constant is a mass: if we look at a slice P = 0 and gauge fix to f = ψ (allowed locally, though not globally), we see that</text>
<formula><location><page_4><loc_40><loc_73><loc_87><loc_77></location>h 2 = ( 1 -γ ψ -Λ 3 ψ 2 ) -1 , (2.9)</formula>
<text><location><page_4><loc_13><loc_71><loc_65><loc_72></location>a piece of the usual Schwarzschild-de Sitter metric with γ = 2 Gm .</text>
<text><location><page_4><loc_13><loc_67><loc_87><loc_70></location>Note that the value of the integration constant γ can have a drastic effect on the spacetime geometry [12,13]. Here I will assume that</text>
<formula><location><page_4><loc_44><loc_63><loc_87><loc_66></location>0 < γ < 2 3 √ Λ . (2.10)</formula>
<text><location><page_4><loc_13><loc_53><loc_87><loc_61></location>For the complete Schwarzschild-de Sitter metric, this is the condition for the existence of two horizons, that is, for a black hole that is smaller than the cosmological horizon. As we will see shortly, it is also necessary for the existence of a Cauchy surface with vanishing extrinsic curvature, a requirement for attaching expanding and contracting regions to realize the 'spacetime foam' of [2].</text>
<text><location><page_4><loc_13><loc_46><loc_87><loc_53></location>By standard existence theorems, the initial data ( h, f, P, Q ) can now be evolved to form a maximal globally hyperbolic spacetime. As expected, the system has a time reversal symmetry: if the data ( h, f, P, Q ) are admissible, so are ( h, f, -P, -Q ). Since P and Q determine the expansion and shear, for any expanding solution there is a corresponding contracting one.</text>
<text><location><page_4><loc_13><loc_39><loc_87><loc_46></location>Of course, there is no guarantee that the resulting spacetimes are geodesically complete. On the contrary, regions of the initial surface can be isometric to regions of the interior of a Schwarzschild-de Sitter black hole, so one might generically expect singularities to form, as happens in the more general setting [14]. Evolution thus requires a quantum treatment.</text>
<section_header_level_1><location><page_4><loc_13><loc_36><loc_23><loc_37></location>3 Sewing</section_header_level_1>
<text><location><page_4><loc_13><loc_25><loc_87><loc_34></location>Ref. [2] focused on manifolds formed by 'sewing' elementary pieces to build a model of spacetime foam. An almost identical procedure exists in the spherically symmetric setting [5]. Choose coordinates in which h is constant in a small region around ψ = ψ 0 . Pick two manifolds M 1 and M 2 , with arbitrary functions f 1 ( ψ ) and f 2 ( ψ ). Let u /epsilon1 be a smoothed step function, interpolating between u /epsilon1 ( ψ ) = 0 when ψ < -/epsilon1 and u /epsilon1 ( ψ ) = 1 when ψ > /epsilon1 .</text>
<text><location><page_4><loc_16><loc_24><loc_25><loc_25></location>Now define</text>
<formula><location><page_4><loc_31><loc_20><loc_87><loc_22></location>f ( ψ ) = [1 -u /epsilon1 ( ψ -ψ 0 )] f 1 ( ψ ) + u /epsilon1 ( ψ -ψ 0 ) f 2 ( ψ ) . (3.1)</formula>
<text><location><page_4><loc_13><loc_16><loc_87><loc_19></location>This 'sewn' f looks like f 1 for ψ ∈ ( -∞ , ψ 0 -/epsilon1 ) and like f 2 for ψ ∈ ( ψ 0 + /epsilon1, ∞ ), with a smooth interpolation between. Since the momenta P and Q are determined locally from f and h , the</text>
<text><location><page_5><loc_13><loc_87><loc_87><loc_90></location>full initial data will sew together in the same way. Topologically, this process is a connected sum M 1 # M 2 ; geometrically, it is very close to the general construction of [3,4].</text>
<text><location><page_5><loc_13><loc_80><loc_87><loc_87></location>This isn't yet quite good enough. The functions f 1 and f 2 each determine two sets of data, ( h, f 1 , ± P 1 , ± Q 1 ) and ( h, f 2 , ± P 2 , ± Q 2 ). We would like to sew any combination, for instance attaching an expanding region to a contracting region. But for P and Q to change sign, they must go through zero, a condition that is not automatic.</text>
<text><location><page_5><loc_13><loc_73><loc_87><loc_80></location>To allow such a sign change, we construct an extra 'neck' N , an intermediate manifold in which P and Q go to zero on a central two-sphere. We can then sew M 1 and M 2 via this neck, M 1 # N # M 2 , with either choice of momenta on each side. We specify N , say with ψ ∈ ( -δ, δ ), by demanding that</text>
<formula><location><page_5><loc_35><loc_70><loc_87><loc_72></location>f ' (0) = 0 , f '' (0) > 0 , P (0) = Q (0) = 0 . (3.2)</formula>
<text><location><page_5><loc_16><loc_68><loc_66><loc_69></location>The condition P (0) = 0 is easy: by (2.8), it simply requires that</text>
<formula><location><page_5><loc_44><loc_63><loc_87><loc_66></location>Λ 3 f 0 3 = f 0 -γ, (3.3)</formula>
<text><location><page_5><loc_13><loc_56><loc_87><loc_62></location>where f 0 = f (0). Note, though, that (3.3) has real, positive roots only if the inequality (2.10) is satisfied [12]. If the inequality is violated, P can never go through zero, and it becomes impossible to join expanding and contracting regions, at least within the constraint of local spherical symmetry.</text>
<text><location><page_5><loc_13><loc_51><loc_87><loc_56></location>Setting Q (0) = 0 is harder: eqn. (2.6) for Q has a factor f ' in the denominator, which goes to zero at the center of the neck. To cancel this term, we must demand that P ( ψ ) be of order ψ 3 near ψ = 0. A straightforward calculation yields the requirement</text>
<formula><location><page_5><loc_18><loc_45><loc_87><loc_50></location>f ( ψ ) = f 0 + ( 1 -Λ f 2 0 ) f 0 ( h 2 ψ 2 4 f 0 2 )( 1 -h 2 ψ 2 12 f 0 2 ) + O ( ψ 6 ) with 1 -Λ f 2 0 > 0 . (3.4)</formula>
<text><location><page_5><loc_13><loc_38><loc_87><loc_41></location>By the constraint (2.1b), the scalar curvature (3) R is large at ψ = 0, since the extrinsic curvature vanishes there. But the curvature falls off quickly away from the center of the neck:</text>
<text><location><page_5><loc_13><loc_41><loc_87><loc_45></location>This restriction is not quite as stringent as it may seem: the O (1) and O ( ψ 2 ) terms are already fixed by the demand that Q remain finite at f ' = 0.</text>
<formula><location><page_5><loc_15><loc_28><loc_81><loc_37></location>(3) R = 2Λ -8 κ 4 h 2 f 4 ( PQ -1 4 P 2 ) = 2Λ -2 κ 4 3 h 2 1 f 3 0 f '' (0) P 3 2 ψ 4 + O ( ψ 5 ) where P = 1 6 P 3 ψ 3 + O ( ψ 4 ) .</formula>
<formula><location><page_5><loc_83><loc_30><loc_87><loc_32></location>(3.5)</formula>
<text><location><page_5><loc_13><loc_15><loc_87><loc_29></location>We can now construct the promised 'onion' initial data, with a mix of expanding and contracting layers. Choose a K -layer sequence ψ 0 < ψ 1 < · · · < ψ K , where ψ K will be identified with ψ 0 to create a closed S 1 × S 2 . In each layer ψ ∈ ( ψ n , ψ n +1 ), choose functions ( h n , f n ), and randomly pick the sign of the corresponding ± ( P n , Q n ). Sew successive layers along necks as described above. For a configuration with enough layers, the random choice of signs means that P and Q will typically average to near zero, even if Λ is large. Thus, as in [2], we have hidden the initial cosmological constant, but now in a setting in which we can address the quantum theory much more concretely.</text>
<section_header_level_1><location><page_6><loc_13><loc_89><loc_36><loc_90></location>4 The problem of time</section_header_level_1>
<text><location><page_6><loc_13><loc_80><loc_87><loc_87></location>To investigate the quantum evolution of this system, we must first confront a basic puzzle, the notorious 'problem of time' [15, 16]. Quantum gravity is a relational theory, and time evolution can only be described relative to other degrees of freedom. To describe evolution, we must first choose a 'clock.'</text>
<text><location><page_6><loc_13><loc_67><loc_87><loc_80></location>In some expanding minisuperspace models [17], and perhaps more generally [18], one can use spatial volume as a clock. Here, though, our spacetimes have some regions that expand while others contract. One can often use 'York time' [19], in which the trace of the extrinsic curvature-the local Hubble constant-serves as a clock. But here, again, we are interested in cases in which the extrinsic curvature doesn't even have a fixed sign at constant time. A more general choice of time, the mean curvature flow [20], fails for the same reason. In some cosmological spacetimes, 'time since the big bang' can serve as a clock [21], but it is not obvious that the spacetimes considered here have a suitable initial big bang singularity.</text>
<text><location><page_6><loc_13><loc_56><loc_87><loc_66></location>An alternative, developed by Brown and Kuchaˇr [8, 9], uses congruences of test particles, 'dust,' as a reference system. † This is not ideal-it would be preferable to describe the evolution purely in terms of a gravitational variable-but in the present setting I don't know how to define such a 'clock.' Here I will use a slightly simpler version of dust time, due to Husain and Paw/suppresslowski [10] (see also [22]), in which a space-filling congruence of irrotational dust is used to specify time evolution. The 'cloud of clocks' is introduced with an action</text>
<formula><location><page_6><loc_34><loc_51><loc_87><loc_55></location>I dust = 1 2 ∫ d 4 x √ -g ρ ( g ab ∂ a T∂ b T -1 ) , (4.1)</formula>
<text><location><page_6><loc_13><loc_46><loc_87><loc_51></location>where ρ is a Lagrange multiplier. It is easy to see that the variation of I dust yields the usual stress-energy tensor for noninteracting irrotational matter. Carrying out a standard canonical decomposition of I grav + I dust , one finds constraints</text>
<formula><location><page_6><loc_40><loc_43><loc_87><loc_45></location>P i = P grav i + P dust i = 0 (4.2a)</formula>
<formula><location><page_6><loc_40><loc_41><loc_87><loc_43></location>H = H grav + H dust = 0 , (4.2b)</formula>
<text><location><page_6><loc_13><loc_38><loc_56><loc_40></location>where H grav and P grav are given by (2.1a)-(2.1b) and</text>
<formula><location><page_6><loc_36><loc_34><loc_87><loc_37></location>P dust i = -p T ∂ i T, (4.3a)</formula>
<text><location><page_6><loc_13><loc_30><loc_47><loc_31></location>where p T is the momentum conjugate to T .</text>
<formula><location><page_6><loc_36><loc_30><loc_87><loc_35></location>H dust = ( p T 2 + q ij P dust i P dust j ) 1 / 2 , (4.3b)</formula>
<text><location><page_6><loc_13><loc_23><loc_87><loc_30></location>As shown in [8], one can use these constraints in a 'many-fingered time' quantization, in which the wave function evolves as a functional of T . But one can also gauge fix time by choosing T = t . Physically, this amounts to using proper time along the dust worldlines as a time coordinate. With this choice, P dust i = 0, and the constraints reduce to</text>
<formula><location><page_6><loc_45><loc_20><loc_87><loc_22></location>P grav i = 0 (4.4a)</formula>
<formula><location><page_6><loc_45><loc_18><loc_87><loc_20></location>p T = H grav . (4.4b)</formula>
<text><location><page_6><loc_29><loc_38><loc_29><loc_39></location>i</text>
<text><location><page_7><loc_13><loc_82><loc_87><loc_90></location>Eqn. (4.4b) has a slightly peculiar structure. The left-hand side is independent of position, since we have chosen a gauge in which T depends only on time, but the right-hand side is certainly position-dependent. This is actually sensible: the zero mode of H grav is now a true Hamiltonian, while the remaining position-dependent modes continue to act as constraints, much like the situation in unimodular gravity [23].</text>
<section_header_level_1><location><page_7><loc_13><loc_78><loc_45><loc_79></location>5 Wheeler-DeWitt quantization</section_header_level_1>
<text><location><page_7><loc_13><loc_71><loc_87><loc_76></location>To quantize this system a la Wheeler and DeWitt [25], we rewrite the momentum and Hamiltonian constraints as operators, with canonical momenta replaced by (functional) derivatives. To find the appropriate conjugates, we start with the symplectic current</text>
<formula><location><page_7><loc_31><loc_66><loc_87><loc_70></location>ω = ∫ d 3 xπ ij δq ij = 8 π ∫ dψ ( 2 f Qδf + 1 h P δh ) , (5.1)</formula>
<text><location><page_7><loc_13><loc_64><loc_34><loc_66></location>from which can we read off</text>
<formula><location><page_7><loc_39><loc_60><loc_87><loc_63></location>P = i 8 π h δ δh , Q = i 16 π f δ δf . (5.2a)</formula>
<text><location><page_7><loc_13><loc_57><loc_87><loc_59></location>The spatial volume ˆ V and the mean curvature ˆ K (the trace of the extrinsic curvature) become</text>
<formula><location><page_7><loc_34><loc_52><loc_87><loc_56></location>ˆ V = ∫ d 3 x √ q = 4 π ∫ dψ hf 2 (5.3a)</formula>
<formula><location><page_7><loc_34><loc_48><loc_87><loc_52></location>ˆ K = -κ 2 √ q ˆ π = -iκ 2 8 π 1 hf 2 ( h δ δh + f δ δf ) , (5.3b)</formula>
<text><location><page_7><loc_13><loc_44><loc_37><loc_47></location>and satisfy [ ˆ K, ˆ V ] = -3 iκ 2 / 2 .</text>
<text><location><page_7><loc_16><loc_44><loc_52><loc_45></location>The momentum constraint (2.6) now becomes</text>
<formula><location><page_7><loc_39><loc_38><loc_87><loc_42></location>ˆ P = i 8 π ( h∂ ψ δ δh -f ' δ δf ) . (5.4)</formula>
<text><location><page_7><loc_13><loc_34><loc_87><loc_38></location>Acting on h and f , ˆ P generates spatial diffeomorphisms, and invariant wave functions can be built from integrals of the form</text>
<formula><location><page_7><loc_29><loc_29><loc_87><loc_33></location>F [ h, f ] = ∫ dψ hL [ f, Df, D 2 f, . . . ] with D = 1 h d dψ . (5.5)</formula>
<text><location><page_7><loc_13><loc_27><loc_85><loc_29></location>The Hamiltonian constraint (2.7) is a bit more complicated, but can be reduced to the form</text>
<formula><location><page_7><loc_29><loc_22><loc_87><loc_26></location>ˆ H = h f ' d dψ [ κ 2 64 π 2 1 f δ 2 δh 2 + 1 κ 2 f ( f ' 2 h 2 -1 ) + Λ 3 κ 2 f 3 ] , (5.6)</formula>
<text><location><page_7><loc_13><loc_20><loc_47><loc_21></location>while the momentum p T of (4.4b) becomes</text>
<formula><location><page_7><loc_45><loc_15><loc_87><loc_19></location>p T = ihf 2 d dT (5.7)</formula>
<text><location><page_8><loc_13><loc_87><loc_87><loc_91></location>( hf 2 is the volume measure √ q ). The Wheeler-DeWitt equation-the operator version of (4.4b), acting on a wave function Ψ-is thus</text>
<formula><location><page_8><loc_27><loc_81><loc_87><loc_86></location>if 2 f ' d Ψ dT = d dψ [ κ 2 64 π 2 1 f δ 2 δh 2 + 1 κ 2 f ( f ' 2 h 2 -1 ) + Λ 3 κ 2 f 3 ] Ψ . (5.8)</formula>
<text><location><page_8><loc_13><loc_76><loc_87><loc_81></location>As in the classical case, (5.8) can be integrated. The derivative d Ψ /dT is independent of position-Ψ is a functional of h and f , not a function of ψ -so the left-hand side of (5.8) is a total derivative. Hence</text>
<formula><location><page_8><loc_23><loc_70><loc_87><loc_75></location>i d Ψ dT = [ 3 κ 2 64 π 2 ( 1 f 2 δ δh ) 2 + 3 κ 2 1 f 2 ( f ' 2 h 2 -1 ) + Λ κ 2 + 3 γ κ 2 f 3 ] Ψ = ˆ H Ψ , (5.9)</formula>
<text><location><page_8><loc_13><loc_66><loc_87><loc_69></location>where γ is again an integration constant. This is the fundamental equation we must solve to understand quantum midisuperspace.</text>
<section_header_level_1><location><page_8><loc_13><loc_62><loc_37><loc_64></location>5.1. The wave function</section_header_level_1>
<text><location><page_8><loc_13><loc_57><loc_87><loc_61></location>Eqn. (5.9) is a Schrodinger-type equation, and there is no difficulty in principle in interpreting | Ψ | 2 as a probability density on midisuperspace [15]. In particular, the time derivative</text>
<formula><location><page_8><loc_32><loc_52><loc_87><loc_56></location>d (Ψ ∗ Ψ) dT = -3 iκ 2 64 π 2 δ δh [ 1 f 4 ( Ψ ∗ δ Ψ δh -Ψ δ Ψ ∗ δh )] (5.10)</formula>
<text><location><page_8><loc_13><loc_50><loc_56><loc_51></location>is a total (functional) derivative, so formally, the norm</text>
<formula><location><page_8><loc_45><loc_45><loc_55><loc_49></location>∫ [ df ][ dh ]Ψ ∗ Ψ</formula>
<text><location><page_8><loc_13><loc_43><loc_71><loc_44></location>is conserved, at least for suitable boundary conditions for midisuperspace.</text>
<text><location><page_8><loc_13><loc_29><loc_87><loc_42></location>There is a subtlety, though. The wave function Ψ should really be defined on 'reduced midisuperspace,' the space of symmetric metrics modulo diffeomorphisms. To properly define a norm on this quotient space, one must gauge fix the inner product to avoid overcounting diffeomorphism-equivalent configurations [24]. This is a difficult task, since one must account for not only the spatial diffeomorphisms, but also the symmetries generated by the non-zero modes of the Hamiltonian constraint. Partial results are given in the Appendix; for now, I will avoid this issue, and merely use | Ψ | 2 to determine relative probabilities, with the understanding that the inner product measure may give corrections.</text>
<text><location><page_8><loc_13><loc_19><loc_87><loc_29></location>It is worth reiterating that Ψ is a functional of the metric, not a function of space. Its role is to determine probabilities of configurations ( h, f ). Of course, the probability of any given metric occurring will depend on its spatial form, and we shall see that a wave function can imply 'a high probability for metrics with spatial characteristic X.' But this will only make sense if the 'spatial characteristic X' is expressed in a diffeomorphism-invariant way, one that does not refer to any particular values of the coordinates.</text>
<section_header_level_1><location><page_9><loc_13><loc_89><loc_42><loc_90></location>5.2. The probability current</section_header_level_1>
<text><location><page_9><loc_13><loc_79><loc_87><loc_87></location>As a Schrodinger-type equation, (5.9) should also admit a probability current. Recall that in the WKB approximation in ordinary quantum mechanics, the probability current distinguishes genuinely time-independent configurations (e.g., bound states) from steady state descriptions of a secretly time-dependent configurations (e.g., plane waves scattering off a potential barrier). We shall see that the same is true here.</text>
<text><location><page_9><loc_16><loc_77><loc_41><loc_79></location>It is easy to see that the current</text>
<formula><location><page_9><loc_33><loc_72><loc_87><loc_77></location>J [ f, h ; x ] = 3 iκ 2 64 π 2 ( Ψ ∗ 1 f 2 δ Ψ δh -Ψ 1 f 2 δ Ψ ∗ δh ) . (5.11)</formula>
<text><location><page_9><loc_13><loc_71><loc_56><loc_73></location>obeys a sort of continuity equation on midisuperspace,</text>
<formula><location><page_9><loc_42><loc_67><loc_87><loc_71></location>d (Ψ ∗ Ψ) dT + 1 f 2 δJ δh = 0 . (5.12)</formula>
<text><location><page_9><loc_13><loc_61><loc_87><loc_67></location>But J is not an observable: it is not annihilated by the momentum constraint ˆ P . We can project J onto reduced midisuperspace by taking a volume average, or equivalently forming a group average over the spatial diffeomorphisms [26],</text>
<formula><location><page_9><loc_42><loc_56><loc_87><loc_61></location>〈 J 〉 = 1 V ∫ d 3 xhf 2 J, (5.13)</formula>
<text><location><page_9><loc_13><loc_55><loc_64><loc_57></location>but this average no longer obeys an obvious continuity equation.</text>
<text><location><page_9><loc_16><loc_54><loc_44><loc_55></location>To do better, consider the functions</text>
<formula><location><page_9><loc_37><loc_49><loc_87><loc_53></location>ϕ n ( ψ ) = exp { 8 π 2 in V ∫ ψ f 2 hdψ ' } . (5.14)</formula>
<text><location><page_9><loc_13><loc_47><loc_41><loc_49></location>These are orthogonal and complete:</text>
<text><location><page_9><loc_13><loc_37><loc_27><loc_38></location>If we now project</text>
<formula><location><page_9><loc_36><loc_38><loc_87><loc_47></location>∫ hf 2 ϕ ∗ m ϕ n dψ = V 4 π δ mn , ∑ n ϕ ∗ n ( ψ ' ) ϕ n ( ψ ) = V 4 πhf 2 δ ( ψ -ψ ' ) . (5.15)</formula>
<formula><location><page_9><loc_29><loc_32><loc_87><loc_36></location>D n = 1 V ∫ d 3 xhf 2 ϕ ∗ n ( 1 f 2 δ δh ) , J n = ∫ d 3 xhf 2 ϕ n J, (5.16)</formula>
<text><location><page_9><loc_13><loc_31><loc_32><loc_32></location>a bit of work shows that</text>
<formula><location><page_9><loc_31><loc_25><loc_87><loc_30></location>∑ n D n J n = 1 V ∫ d 3 xhf 2 ( 1 f 2 δJ δh ) = -d (Ψ ∗ Ψ) dT , (5.17)</formula>
<text><location><page_9><loc_13><loc_19><loc_87><loc_26></location>where the last equality comes from (5.12). D n and J n are spatial invariants, that is, that they are annihilated by the momentum constraint. ‡ The D n may be viewed as the components of the gradient in our (infinite-dimensional) midisuperspace, making (5.17) a standard continuity equation.</text>
<section_header_level_1><location><page_10><loc_13><loc_89><loc_62><loc_90></location>6 Stationary states and the WKB approximation</section_header_level_1>
<text><location><page_10><loc_16><loc_85><loc_43><loc_87></location>We next look for stationary states</text>
<formula><location><page_10><loc_40><loc_82><loc_87><loc_84></location>Ψ[ f, h ; T ] = ˜ Ψ[ f, h ] e -iET . (6.1)</formula>
<text><location><page_10><loc_13><loc_79><loc_66><loc_81></location>For states of this form, the Wheeler-DeWitt equation (5.9) becomes</text>
<formula><location><page_10><loc_27><loc_73><loc_87><loc_78></location>[ κ 2 64 π 2 ( 1 f 2 δ δh ) 2 + 1 κ 2 1 f 2 ( f ' 2 h 2 -1 ) + ˜ Λ 3 κ 2 + γ κ 2 f 3 ] ˜ Ψ = 0 , (6.2)</formula>
<text><location><page_10><loc_13><loc_71><loc_18><loc_72></location>where</text>
<formula><location><page_10><loc_45><loc_67><loc_87><loc_70></location>˜ Λ = Λ -κ 2 E. (6.3)</formula>
<text><location><page_10><loc_13><loc_58><loc_87><loc_66></location>This equation is identical to the original gravitational Hamiltonian constraint, except that the cosmological constant is shifted by the energy, again reminiscent of unimodular gravity [23]. This shift in Λ is physical-it is a backreaction of our 'cloud of clocks' on the spacetime-and to see purely gravitational properties, we should limit ourselves to states with relatively small energies. For a Planck-scale Λ, though, this is a rather mild restriction.</text>
<text><location><page_10><loc_16><loc_56><loc_74><loc_58></location>To better understand these states, let us consider a WKB approximation,</text>
<formula><location><page_10><loc_46><loc_53><loc_87><loc_55></location>˜ Ψ = Ae iS . (6.4)</formula>
<text><location><page_10><loc_13><loc_50><loc_37><loc_52></location>The first order WKB equation,</text>
<formula><location><page_10><loc_31><loc_44><loc_87><loc_49></location>κ 2 64 π 2 ( δS δh ) 2 = 1 κ 2 f 2 ( f ' 2 h 2 -1 ) + ˜ Λ 3 κ 2 f 4 + γ κ 2 f, (6.5)</formula>
<text><location><page_10><loc_13><loc_42><loc_30><loc_44></location>can be solved exactly:</text>
<text><location><page_10><loc_13><loc_35><loc_18><loc_37></location>where</text>
<formula><location><page_10><loc_27><loc_36><loc_87><loc_41></location>S = 8 π κ 2 ∫ dψ σ [ h, f ; ψ ] ff ' { √ 1 + βh 2 -tanh -1 √ 1 + βh 2 } , (6.6)</formula>
<formula><location><page_10><loc_40><loc_30><loc_87><loc_34></location>β = f 2 f ' 2 ( ˜ Λ 3 -1 f 2 + γ f 3 ) (6.7)</formula>
<text><location><page_10><loc_13><loc_27><loc_62><loc_29></location>and σ is a functional of h and f and a function of ψ such that</text>
<formula><location><page_10><loc_36><loc_22><loc_87><loc_26></location>{ σ 2 = 1 almost everywhere ∂ ψ σ = 0 unless 1 + βh 2 = 0 . (6.8)</formula>
<text><location><page_10><loc_13><loc_15><loc_87><loc_20></location>The factor σ requires a bit of explanation. As always in the WKB approximation, the phase S is determined only up to sign. But eqn. (6.5) holds pointwise, and thus admits solutions in which this sign, σ , can vary with f , h , and ψ . The choice is not completely free, though: we must</text>
<text><location><page_11><loc_13><loc_82><loc_87><loc_90></location>still require that S be annihilated by the momentum constraint. If the position dependence of σ only arose implicitly from its dependence on h and f , then S would be of the form (5.5), and we would automatically have ˆ P S = 0. But any explicit dependence of σ on ψ introduces a new term in ˆ P S , proportional to (1+ βh 2 ) 1 / 2 ∂ ψ σ . The constraints require this term to vanish, leading to the second condition in (6.8).</text>
<text><location><page_11><loc_16><loc_80><loc_79><loc_81></location>To understand the physical significance of this condition, we can rewrite (6.7) as</text>
<formula><location><page_11><loc_43><loc_75><loc_87><loc_79></location>1 + βh 2 = κ 4 ˜ P 2 f 2 f ' 2 , (6.9)</formula>
<text><location><page_11><loc_13><loc_72><loc_18><loc_74></location>where</text>
<formula><location><page_11><loc_31><loc_66><loc_87><loc_71></location>˜ P = 1 κ 2 [ f 2 h 2 ( f ' 2 h 2 -1 ) + ˜ Λ 3 f 4 h 2 + γfh 2 ] 1 / 2 . (6.10)</formula>
<text><location><page_11><loc_13><loc_55><loc_87><loc_66></location>Comparing to (2.8), we see that ˜ P is essentially the classical momentum π ψ ψ , though with a shifted cosmological constant. Hence the sign of S can change only when the ˜ P goes through zero, the quantum version of the the classical sewing condition of section 3. There is also an analog of the inequality (2.10) for γ . For large f , β is positive, while for ˜ P to go through zero, β must be negative. But (6.7) involves the same cubic as (3.3), and β can change sign only if the inequality (2.10) is satisfied, albeit again with a shifted Λ.</text>
<text><location><page_11><loc_13><loc_45><loc_87><loc_55></location>This means, in particular, that for suitable γ , the quantum theory allows such sewing. More precisely, let ( h, f ) be a metric configuration that, as a classical metric, describes layers joined by necks in which the extrinsic curvature goes through zero. A choice of σ for this configuration is then equivalent to a choice of the sign of the extrinsic curvature in each layer. Just as this sign is not determined classically by the constraints, it is not determined quantum mechanically by the Wheeler-DeWitt equation: different choices of σ give different wave functions Ψ σ .</text>
<text><location><page_11><loc_13><loc_37><loc_87><loc_45></location>We can now explore our WKB wave functions in various regions of midisuperspace. First, consider the case of large f -that is, specialize to metrics in which f is large in some region of space, and look at the contribution of that spatial region to the integral (6.6). In such a region, ˜ P /greatermuch 0, so the sign σ is fixed, and β is dominated by the cosmological constant, yielding</text>
<formula><location><page_11><loc_42><loc_32><loc_87><loc_37></location>S ∼ 2 κ 2 σ ( ˜ Λ 3 ) 1 / 2 V. (6.11)</formula>
<text><location><page_11><loc_13><loc_30><loc_85><loc_32></location>The mean curvature operator (5.3b)-the local Hubble constant-then has a simple action,</text>
<formula><location><page_11><loc_41><loc_24><loc_87><loc_29></location>ˆ Ke iS ≈ 3 σ ( ˜ Λ 3 ) 1 / 2 e iS . (6.12)</formula>
<text><location><page_11><loc_13><loc_21><loc_87><loc_24></location>This is ordinary de Sitter behavior, and, as expected, the sign σ determines whether a spatial region is expanding or contracting. The probability current (5.16) provides similar information:</text>
<formula><location><page_11><loc_41><loc_15><loc_87><loc_19></location>J n = -3 V 4 π σ ( ˜ Λ 3 ) 1 / 2 δ n 0 , (6.13)</formula>
<text><location><page_12><loc_13><loc_85><loc_87><loc_90></location>so σ determines the direction of flow of probability. This is not unlike the WKB approximation in ordinary quantum mechanics, where a plane wave e ikx is formally a stationary state, but the probability current reveals a hidden dynamics.</text>
<text><location><page_12><loc_13><loc_80><loc_87><loc_85></location>But the wave function does not have its support only on such de Sitter-like regions. Consider the contribution of a multilayered 'foamy' region. From (5.3b) and (5.4), the mean curvature acts on diffeomorphism-invariant states as</text>
<formula><location><page_12><loc_40><loc_74><loc_87><loc_79></location>ˆ K = iκ 2 8 π 1 f 2 f ' ∂ ψ ( f δ δh ) , (6.14)</formula>
<text><location><page_12><loc_13><loc_73><loc_19><loc_74></location>yielding</text>
<formula><location><page_12><loc_26><loc_66><loc_87><loc_71></location>ˆ Ke iS = 1 f 2 f ' ∂ ψ ( f 2 f ' h σ √ 1 + βh 2 ) e iS = κ 2 f 2 f ' ∂ ψ ( f h σ ˜ P ) e iS . (6.15)</formula>
<text><location><page_12><loc_13><loc_63><loc_87><loc_67></location>As expected, the WKB wave function is not an eigenfunction of ˆ K . But in for foamy regions of midisuperspace, it is almost an eigenfunction of the spatially averaged mean curvature. Indeed,</text>
<formula><location><page_12><loc_29><loc_57><loc_87><loc_62></location>( 1 V ∫ d 3 x √ q ˆ K ) e iS = -4 πκ 2 V ∫ dψ f h ( h f ' ) ' σ ˜ P e iS . (6.16)</formula>
<text><location><page_12><loc_13><loc_50><loc_87><loc_57></location>For a typical wave function, evaluated at a typical multilayered metric, the contribution of each layer will come with a random sign σ . Hence for a configuration with many layers, we expect extensive cancellation-just as in the classical case, the averaged mean curvature will be much smaller than it would be for a single de Sitter region.</text>
<text><location><page_12><loc_13><loc_47><loc_87><loc_50></location>The evaluation of the probability current leads to an identical conclusion, perhaps even more clearly. The current is</text>
<formula><location><page_12><loc_41><loc_41><loc_87><loc_46></location>J n = -3 κ 2 4 ∫ d 3 xσ ˜ Pϕ n , (6.17)</formula>
<text><location><page_12><loc_13><loc_35><loc_87><loc_41></location>and for foamy metrics the right-hand side will again average to a very small number. Our wave functions are thus stationary not only in the sense that they are independent of T , but also in the sense that there is very little flow of probability within midisuperspace, so initial foamy structures will tend to be preserved.</text>
<text><location><page_12><loc_13><loc_29><loc_87><loc_34></location>Of course, this leaves open the question of how common such multilayered foamy configurations are. An answer requires the next order WKB approximation, which we will turn to shortly. Meanwhile, there are a few other features of (6.6) that deserve future exploration:</text>
<unordered_list>
<list_item><location><page_12><loc_15><loc_22><loc_87><loc_28></location>1. The integrand in (6.6) diverges when βh 2 = 0. Comparing (6.7) to (2.9), we might suspect this to occur at some sort of horizon. This is correct: the vanishing of βh 2 marks the location of a marginally trapped sphere, a midisuperspace version of the trapped surfaces that appear in the more general connected sum analysis [14].</list_item>
<list_item><location><page_12><loc_15><loc_15><loc_87><loc_20></location>2. The integrand vanishes at 1 + βh 2 = 0, the classical neck. But it also has zeroes at 1 + βh 2 = -z 2 n , where z n = tan z n . These are in a classically forbidden region, but in principle they are additional sites at which expanding and contracting regions can join.</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_13><loc_15><loc_86><loc_87><loc_90></location>3. When 1+ βh 2 > 0, tanh -1 √ 1 + βh 2 has a constant imaginary part. In a region between two successive zeroes of 1 + βh 2 , though, σ is constant, and the imaginary part of S is</list_item>
</unordered_list>
<formula><location><page_13><loc_35><loc_81><loc_69><loc_86></location>4 π 2 i κ 2 ∫ ψ 2 ψ 1 dψ σff ' = 2 π 2 i κ 2 σ [ f 2 ( ψ 2 ) -f 2 ( ψ 1 )]</formula>
<text><location><page_13><loc_18><loc_75><loc_87><loc_81></location>with 1 + βh 2 = 0 at ψ 1 and ψ 2 . We saw earlier that at a 'classical' zero of 1 + βh 2 , the value of f is fixed by (3.3), so the contributions of the two endpoints cancel. It is less clear what happens at the nonclassical zeroes 1 + βh 2 = -z 2 n .</text>
<section_header_level_1><location><page_13><loc_13><loc_72><loc_33><loc_73></location>7 Next order WKB</section_header_level_1>
<text><location><page_13><loc_13><loc_67><loc_87><loc_70></location>To say more about probabilities for 'foamy' spacetimes, we will need to go to the next order in the WKB approximation (6.4),</text>
<formula><location><page_13><loc_42><loc_62><loc_87><loc_66></location>2 δS δh δA δh + A δ 2 S δh 2 = 0 . (7.1)</formula>
<text><location><page_13><loc_13><loc_58><loc_87><loc_61></location>This equation involves two functional derivatives at a single point, leading to a well known divergence. For our midisuperspace model, we can write</text>
<formula><location><page_13><loc_45><loc_53><loc_87><loc_57></location>S = ∫ dψ L, (7.2)</formula>
<text><location><page_13><loc_13><loc_51><loc_44><loc_53></location>where L depends on h but not h ' . Then</text>
<formula><location><page_13><loc_30><loc_46><loc_87><loc_50></location>δS δh ( ψ ) = ∂L ∂h ( ψ ) , δ 2 S δh ( ψ ) δh ( ψ ' ) = ∂ 2 L ∂h 2 ( ψ ) δ ( ψ -ψ ' ) , (7.3)</formula>
<text><location><page_13><loc_13><loc_44><loc_37><loc_45></location>giving a factor δ (0) at ψ = ψ ' .</text>
<text><location><page_13><loc_13><loc_36><loc_87><loc_43></location>There are several proposals for regulating this infinite factor. In the original formulation of the Wheeler-DeWitt equation [25], DeWitt suggested that such δ (0) factors should be set to zero. Here, that choice would make the first order WKB approximation exact. A 'volume average regularization' [27], applied to this simple case, replaces δ ( ψ -ψ ' ) by</text>
<formula><location><page_13><loc_43><loc_31><loc_57><loc_35></location>h /lscript ∫ dψ ' δ ( ψ -ψ ' )</formula>
<text><location><page_13><loc_13><loc_22><loc_87><loc_31></location>where /lscript = ∫ dψ h is the length of the spatial S 1 factor. Heat kernel regularization [28] replaces the factor δ ( ψ -ψ ' ) by a heat kernel K ( ψ, ψ ' ; s ). In the s → 0 limit, K ( ψ, ψ ' ; s ) becomes a delta function, but if one first takes ψ → ψ ' , the divergences appear as a set of terms involving inverse powers of s 1 / 2 , which can be individually regularized. In the present context, the outcome is almost the same as the volume average: δ ( ψ -ψ ' ) becomes</text>
<formula><location><page_13><loc_43><loc_17><loc_57><loc_21></location>αh ∫ dψ ' δ ( ψ -ψ ' )</formula>
<text><location><page_13><loc_13><loc_15><loc_69><loc_17></location>where α is an undetermined constant with dimensions of inverse length.</text>
<section_header_level_1><location><page_14><loc_13><loc_89><loc_44><loc_90></location>7.1. Heat kernel regularization</section_header_level_1>
<text><location><page_14><loc_13><loc_86><loc_83><loc_87></location>Let us first consider heat kernel regularization. The second order WKB equation becomes</text>
<formula><location><page_14><loc_39><loc_80><loc_87><loc_85></location>1 A δA δh = -αh 2 ( ∂L ∂h ) -1 ∂ 2 L ∂h 2 , (7.4)</formula>
<text><location><page_14><loc_13><loc_78><loc_30><loc_80></location>or with S as in (6.6),</text>
<formula><location><page_14><loc_43><loc_74><loc_87><loc_77></location>1 A δA δh = α 2 1 1 + βh 2 . (7.5)</formula>
<text><location><page_14><loc_13><loc_71><loc_40><loc_73></location>This can be solved in closed form:</text>
<formula><location><page_14><loc_35><loc_65><loc_87><loc_70></location>A = exp { α 2 ∫ dψ 1 √ β tan -1 ( √ βh ) } . (7.6)</formula>
<text><location><page_14><loc_13><loc_62><loc_87><loc_66></location>We can now look at relative probabilities for behaviors of the metric-that is, at features of the functions ( h, f ) that lead to comparatively small or large vales of the amplitude A .</text>
<unordered_list>
<list_item><location><page_14><loc_16><loc_56><loc_87><loc_62></location>· When βh 2 is large, the integrand in (7.6) is small, since tan -1 ( √ βh ) is bounded by π/ 2. By (6.9), this implies that metrics with large regions of high extrinsic curvature are suppressed. If we abbreviate</list_item>
</unordered_list>
<text><location><page_14><loc_18><loc_48><loc_41><loc_49></location>then for a region of length /lscript 1 ,</text>
<formula><location><page_14><loc_43><loc_47><loc_87><loc_55></location>∣ ∣ ∣ ∣ ∣ κ 2 ˜ P ff ' ∣ ∣ ∣ ∣ ∣ = √ 1 + βh 2 = ∆ (7.7)</formula>
<formula><location><page_14><loc_39><loc_42><loc_87><loc_47></location>ln A ∣ ∆ /greatermuch 1 ∼ απ 4 〈 ∆ -1 〉 /lscript 1 /lessmuch α/lscript 1 2 , (7.8)</formula>
<text><location><page_14><loc_18><loc_41><loc_57><loc_45></location>∣ where the angle brackets denote a spatial average.</text>
<unordered_list>
<list_item><location><page_14><loc_16><loc_38><loc_79><loc_41></location>· When βh 2 is small but positive, it is still the case that tan -1 ( √ βh ) < √ βh , so</list_item>
<list_item><location><page_14><loc_16><loc_31><loc_30><loc_32></location>· When βh 2 < 0,</list_item>
</unordered_list>
<formula><location><page_14><loc_46><loc_31><loc_87><loc_37></location>ln A ∣ ∣ ∆ ∼ 1 /lessorsimilar α/lscript 1 2 . (7.9)</formula>
<formula><location><page_14><loc_30><loc_23><loc_87><loc_29></location>α 2 ∫ dψ 1 √ β tan -1 ( √ βh ) = α 4 ∫ dψ 1 √ | β | ln ( 1 + √ | β | h 1 -√ | β | h ) , (7.10)</formula>
<text><location><page_14><loc_18><loc_21><loc_87><loc_24></location>and the integral receives large contributions when 1 + βh 2 ∼ 0, that is, from the 'necks' of the preceding section. Again by (6.9), in regions of small ˜ P ,</text>
<formula><location><page_14><loc_33><loc_14><loc_87><loc_19></location>α 2 ∫ dψ 1 √ β tan -1 ( √ βh ) ≈ α 2 ∫ dψ h ln ( 2 ff ' κ 2 | ˜ P | ) . (7.11)</formula>
<text><location><page_15><loc_18><loc_84><loc_87><loc_90></location>Classically, we saw in section 3 that ˜ P ∼ ψ 3 near a neck, while f ' ∼ ψ , so the integrand goes as h ln( a/hψ ) for some parameter a . The integral is maximum when integrated over a region of length /lscript 1 ∼ a , and yields</text>
<formula><location><page_15><loc_45><loc_78><loc_87><loc_84></location>ln A ∣ ∣ ∆ /lessmuch 1 /lessorsimilar α/lscript 1 2 . (7.12)</formula>
<text><location><page_15><loc_18><loc_78><loc_78><loc_79></location>(The dependence on the parameter a imposes no further limits. By section 3,</text>
<formula><location><page_15><loc_44><loc_72><loc_61><loc_77></location>a ∼ ( f 2 D 3 K ) -1 / 2 ∣ ψ =0</formula>
<text><location><page_15><loc_18><loc_63><loc_87><loc_76></location>∣ where K is a component of the extrinsic curvature, D is the invariant derivative (5.5), and f (0) is fixed by (3.3). If the next order of the expansion (3.4) involves only constants of order one, then a ∼ f 0 , the characteristic size of the neck. But f 0 itself depends on the integration constant γ , and can range from 0 to √ 3 / Λ. Moreover, there is no strong reason to demand that the expansion (3.4) have coefficients of order one; if extrinsic curvatures remain small near a neck, a can be large.)</text>
<text><location><page_15><loc_13><loc_48><loc_87><loc_61></location>Combining these results, we see that the wave function strongly disfavors spaces with large regions of high extrinsic curvature (large ∆), while giving roughly equal weight to metrics with regions of relatively small intrinsic curvature near the start of a 'neck' (trapped surfaces ∆ = 1) and regions of very small extrinsic curvature where the sign of the expansion can change (∆ = 0). Note that at this order, there is no limit to the proper length /lscript . Since the integrand in (7.6) is positive, this suggests an infrared divergence. As observed in section 5, though, we have not yet accounted for the measure in the inner product, so it is premature to draw too firm a conclusion.</text>
<text><location><page_15><loc_13><loc_41><loc_87><loc_48></location>We can also ask how this higher order WKB term affects the probability current. For βh 2 > -1, A is real, so it merely multiplies the lowest order current (5.11) by A 2 . For βh 2 < -1, the exponent (7.10) becomes complex, but the imaginary part is independent of h , and does not contribute to the current. The moments (6.17) thus become</text>
<formula><location><page_15><loc_39><loc_35><loc_87><loc_40></location>J n = -3 κ 2 4 ∫ d 3 xσ | A | 2 ˜ Pϕ n . (7.13)</formula>
<section_header_level_1><location><page_15><loc_13><loc_33><loc_40><loc_34></location>7.2. Volume regularization</section_header_level_1>
<text><location><page_15><loc_13><loc_30><loc_78><loc_32></location>We can now repeat the argument using volume regularization. Eqn. (7.5) becomes</text>
<formula><location><page_15><loc_42><loc_25><loc_87><loc_29></location>1 A δA δh = 1 2 /lscript 1 1 + βh 2 , (7.14)</formula>
<text><location><page_15><loc_13><loc_23><loc_42><loc_24></location>and it is tempting to guess a solution</text>
<formula><location><page_15><loc_35><loc_16><loc_87><loc_21></location>A = exp { 1 2 /lscript ∫ dψ 1 √ β tan -1 ( √ βh ) } . (7.15)</formula>
<text><location><page_16><loc_13><loc_89><loc_70><loc_90></location>This doesn't quite work, though: because of the metric dependence of /lscript ,</text>
<formula><location><page_16><loc_31><loc_83><loc_87><loc_87></location>1 A δA δh = 1 2 /lscript 1 1 + βh 2 -1 2 /lscript 2 ∫ dψ 1 √ β tan -1 ( √ βh ) . (7.16)</formula>
<text><location><page_16><loc_13><loc_81><loc_58><loc_83></location>In fact, (7.14) is not integrable: for any putative solution,</text>
<formula><location><page_16><loc_39><loc_77><loc_61><loc_80></location>δ 2 A δh ( ψ ) δh ( ψ ' ) = δ 2 A δh ( ψ ' ) δh ( ψ ) .</formula>
<text><location><page_16><loc_49><loc_76><loc_49><loc_79></location>/negationslash</text>
<text><location><page_16><loc_13><loc_68><loc_87><loc_75></location>The amplitude (7.15) is, however, a good approximate solution, since it follows from the preceding section that the extra term in (7.16) is of order 1 //lscript . It might be possible to view this extra piece as a counterterm in the WKB equation (7.1)-it amounts to adding a term proportional to δS δh to the divergent δ 2 S δh 2 -but it would be a rather peculiar one.</text>
<text><location><page_16><loc_13><loc_63><loc_87><loc_68></location>If we ignore this issue, the conclusions from volume regularization are almost identical to those from heat kernel regularization. The only difference is that the prefactor α now becomes 1 //lscript , providing a natural infrared cutoff.</text>
<section_header_level_1><location><page_16><loc_13><loc_60><loc_43><loc_61></location>8 Conclusions and next steps</section_header_level_1>
<text><location><page_16><loc_13><loc_48><loc_87><loc_58></location>The space of locally spherically symmetric three-geometries is surprisingly rich. It includes configurations that exhibit spacetime foam: spaces with multiple layers with different geometries, joined by connected sums. The constraints determine the extrinsic curvature only up to a sign, and a typical multilayered configuration includes both expanding and contracting regions. While the these spacetimes are certainly not realistic models of our Universe, they are qualitatively very similar to the more general foamy geometries discussed in [2].</text>
<text><location><page_16><loc_13><loc_36><loc_87><loc_48></location>This structure persists in the quantum theory. By using the ''dust time' of Brown and Kuchaˇr, in which physical time is proper time along a congruence of timelike geodesics, we can reduce the Wheeler-DeWitt equation to a Schrodinger-type equation, and search for stationary states. These states have a sign ambiguity in their phase that closely mimics the classical ambiguity in the sign of the extrinsic curvature. When evaluated on a multilayer metric, this sign σ can change at precisely the locations that the sign of the extrinsic curvature can change classically.</text>
<text><location><page_16><loc_13><loc_22><loc_87><loc_36></location>As we know from quantum mechanics, some care must be taken in interpreting such states. A stationary state may imply time independence, but it may instead signify a steady state flow of probability, as in a scattering state. Indeed, here one can construct a state in which the sign σ in (6.6) is the same for every layer. The probability current is then large, proportional to the de Sitter expansion √ Λ / 3. But for a typical state, the current (7.13), evaluated at a typical multilayered geometry, will be very small: the foamy structure will tend to reproduce itself in a nearly steady state. This lends support to the proposal of [2] that a cosmological constant might be 'hidden' in spacetime foam.</text>
<text><location><page_16><loc_13><loc_15><loc_87><loc_22></location>How common are multilayered foamy configurations? The second order WKB approximation has regularization ambiguities, but standard choices such as heat kernel regularization indicate that they are at least fairly probable. Indeed, geometries with large regions of high extrinsic curvature are strongly disfavored, while necks that occur in connected sums of layers</text>
<text><location><page_17><loc_13><loc_87><loc_87><loc_90></location>are favored. Note, though, that the quantum version of the inequality (2.10) depends, via (6.3), on the state, so for fixed γ the prevalence of foamy configurations is state dependent.</text>
<text><location><page_17><loc_13><loc_71><loc_87><loc_87></location>There are, of course, further questions that must be answered before we can be confident in these conclusions. Perhaps the most important is the problem of the measure for the inner product on midisuperspace. I have been using the 'naive Schrodinger interpretation' for | Ψ | 2 [15], which implicitly assumes that the correct inner product is just a functional integral over h and f . But this functional integral must be gauge fixed to account for the invariances generated by the constraints, and the resulting Faddeev-Popov determinants may affect probabilities [24]. Partial results in this direction are described in the Appendix. Crucially, the Faddeev-Popov determinant is independent of the sign of the extrinsic curvature, and will not change the cancellation between expanding and contracting regions.</text>
<text><location><page_17><loc_13><loc_58><loc_87><loc_71></location>It would also be useful to move beyond stationary states and look at the behavior of wave packets. Our choice of time makes this a bit difficult: our 'cloud of clocks' back-reacts on the geometry, and for this to be unimportant we must restrict the energy to be small (on the scale set by Λ). This means wave packets cannot be too sharply peaked. Ideally, one would avoid this by choosing a purely gravitational 'internal time,' but such a parameter seems very hard to find in a spacetime containing both expanding and contracting regions. A more careful study of the role of the integration constant γ would also be valuable. In particular, it is not currently clear whether different choices of γ form superselection sectors.</text>
<text><location><page_17><loc_13><loc_46><loc_87><loc_57></location>A number of straightforward extensions of this work should be possible. One could apply the same techniques to a spatial topology S 3 , by again starting with a manifold [0 , 1] × S 2 but now shrinking each boundary to a point. Beyond that, though, Morrow-Jones and Witt have shown that local spherical symmetry allows a huge variety of topologies [5], although a detailed analysis would require much more complicated boundary conditions. An extension to Λ < 0 should also be easy; most of the computations will be unchanged, though the qualitative results are likely to be quite different.</text>
<text><location><page_17><loc_13><loc_29><loc_87><loc_45></location>It would also be interesting to look at the 'reduced phase space' quantization of this model, in which the reparametrization invariance of ψ is gauge fixed and the momentum constraint is eliminated classically. Whether such a quantization is equivalent to the Dirac quantization method used here is an open question, involving subtleties in operator ordering and the choice of inner product [29, 30], but it is certainly worth exploring. Unfortunately, it is also a bit harder than it might appear. Following [5], we can nearly fix the reparametrization invariance by redefining ψ to make h constant on the initial slice. But the integral ∫ hdψ is invariant, and leaves us with an extra variable (which is also classically time dependent). The resulting Hamiltonian constraint then contains a mixture of ordinary and functional derivatives, and becomes technically quite complicated.</text>
<text><location><page_17><loc_13><loc_17><loc_87><loc_28></location>Finally, there may be another somewhat orthogonal approach to these questions. One way to view spacetime foam in a universe with large Λ is to consider the nucleation of contracting bubbles in expanding regions, and expanding bubbles in contracting regions. It is then natural to ask whether there are instantons that mediate such processes. In the context of our locally spherically symmetric midisuperspace, the question is whether there are Euclidean solutions joining a space with n layers to one with n +1 layers. Some very preliminary calculations show no obstruction to such solutions, but much more work is needed.</text>
<section_header_level_1><location><page_18><loc_13><loc_89><loc_30><loc_90></location>Acknowledgments</section_header_level_1>
<text><location><page_18><loc_13><loc_82><loc_87><loc_87></location>This work was supported in part by Department of Energy grant DE-FG02-91ER40674. I would also like to thank the Quantum Gravity Unit of the Okinawa Institute of Science and Technology (OIST), where part of the work was completed, for their hospitality.</text>
<section_header_level_1><location><page_18><loc_13><loc_78><loc_53><loc_80></location>Appendix. A note on the inner product</section_header_level_1>
<text><location><page_18><loc_13><loc_69><loc_87><loc_76></location>As noted in the conclusion, a crucial remaining problem is to find the correct inner product on midisuperspace. Our wave functions are functions of f and h , and the naive inner product is simply a functional integral ∫ [ df ][ dh ]. This is not quite right, though, for two reasons.</text>
<text><location><page_18><loc_13><loc_62><loc_87><loc_70></location>First, our symmetry-reduced midisuperspace still has a residual diffeomorphism invariance, reparametrizations of ( t, ψ ). The t reparametrizations are fixed by the dust time gauge t = T of section 4, but the ψ reparametrizations remain, and the integral will count identical configurations an infinite number of times. Second, these gauge fixing of the residual diffeomorphisms will induce a Faddeev-Popov determinant, which must also be taken into account.</text>
<text><location><page_18><loc_13><loc_55><loc_87><loc_62></location>Locally, the ψ reparametrizations can be fixed by setting ∂ ψ h = 0, as was done in [5]. Note that the total length ∫ dψ h is diffeomorphism invariant, so the constant value of h is fixed. Whether this choice can be made globally over the whole midisuperspace-that is, whether there are Gribov problems [31,32]-is a more difficult question.</text>
<text><location><page_18><loc_13><loc_48><loc_87><loc_55></location>The Faddeev-Popov determinants are best understood as Jacobians in the functional integral, as described in [33, 34]. (For an early version, see [35].) Let φ be a generic field, and define an inner product 〈 δφ, δφ 〉 on the tangent space to the space of fields. The standard normalization for the path integral is</text>
<formula><location><page_18><loc_43><loc_43><loc_87><loc_47></location>∫ [ dφ ] e -〈 δφ,δφ 〉 = 1 . (A.1)</formula>
<text><location><page_18><loc_13><loc_36><loc_87><loc_43></location>But now suppose φ has a gauge symmetry φ → η φ , labeled by some parameter η . We can change variables in the functional integral to ( ¯ φ, η ), where ¯ φ is a gauge-fixed field. Geometrically, η parametrizes gauge orbits, while ¯ φ parametrizes a cross section that intersects each orbit once. (Whether such a cross section exists or not is the Gribov problem.) Then</text>
<formula><location><page_18><loc_22><loc_31><loc_87><loc_35></location>∫ [ dφ ] e -〈 δφ,δφ 〉 = ∫ [ d ¯ φ ][ dη ] Je -〈 δ ¯ φ,δ ¯ φ 〉 e -〈 δ η φ,δ η φ 〉 = ∫ [ dη ] Je -〈 δ η φ,δ η φ 〉 = 1 , (A.2)</formula>
<text><location><page_18><loc_13><loc_27><loc_87><loc_30></location>where J is a Jacobian coming from the change of variables. This Jacobian is the Faddeev-Popov determinant.</text>
<text><location><page_18><loc_16><loc_25><loc_84><loc_27></location>For us, the fields are the metric q ij and the 'dust' field T , with natural inner products</text>
<formula><location><page_18><loc_22><loc_16><loc_22><loc_18></location>〈</formula>
<formula><location><page_18><loc_22><loc_15><loc_87><loc_24></location>〈 δg, δg 〉 = ∫ d 3 x √ q q ij q k/lscript δq ik δq j/lscript = 4 π ∫ dψ hf 2 [ 4 ( δh h ) 2 +8 ( δf f ) 2 ] δT, δT 〉 = ∫ dx 3 √ q ( δT ) 2 = 4 π ∫ dψ hf 2 ( δT ) 2 . (A.3)</formula>
<text><location><page_19><loc_13><loc_85><loc_87><loc_90></location>The transformations generated by the constraints (2.1a)-(2.1b) involve two parameters ( ξ, ξ ⊥ ), a ψ reparametrization and a 'surface deformation' (basically a t reparametrization). It is straightforward to check that</text>
<formula><location><page_19><loc_26><loc_80><loc_87><loc_84></location>δ η h h = ξ ' + h ' h ξ + K ψ ψ ξ ⊥ , δ η f f = f ' f ξ + K θ θ ξ ⊥ , δ η T = ξ ⊥ , (A.4)</formula>
<text><location><page_19><loc_13><loc_76><loc_87><loc_79></location>where K ψ ψ and K θ θ are the extrinsic curvatures (2.5). Thus the inner product 〈 δ η φ, δ η φ 〉 becomes</text>
<formula><location><page_19><loc_16><loc_70><loc_87><loc_75></location>〈 δ η φ, δ η φ 〉 = 4 π ∫ dψ hf 2 [ 4 ( ξ ' + h ' h ξ + K ψ ψ ξ ⊥ ) 2 +8 ( f ' f ξ + K θ θ ξ ⊥ ) 2 +( ξ ⊥ ) 2 ] . (A.5)</formula>
<text><location><page_19><loc_13><loc_68><loc_23><loc_69></location>Let us define</text>
<formula><location><page_19><loc_38><loc_66><loc_62><loc_68></location>˜ K 2 = K i j K j i = K ψ ψ 2 +2 K θ θ 2</formula>
<text><location><page_19><loc_13><loc_63><loc_46><loc_65></location>Then the integrand in (A.5) has the form</text>
<formula><location><page_19><loc_27><loc_58><loc_73><loc_62></location>(1 + 4 ˜ K 2 )( ξ ⊥ ) 2 + [ 8 ( ξ ' + h ' h ξ ) K ψ ψ +16 f ' f ξK θ θ ] ξ ⊥ + . . .</formula>
<text><location><page_19><loc_13><loc_56><loc_85><loc_57></location>where the remaining terms are independent of ξ ⊥ . We can now complete the square, setting</text>
<formula><location><page_19><loc_28><loc_50><loc_72><loc_54></location>¯ ξ ⊥ = ξ ⊥ +4(1 + 4 ˜ K 2 ) -1 [( ξ ' + h ' h ξ ) K ψ ψ +2 f ' f ξK θ θ ] .</formula>
<text><location><page_19><loc_13><loc_48><loc_68><loc_49></location>After some integrations by parts, we obtain an expression of the form</text>
<formula><location><page_19><loc_30><loc_42><loc_87><loc_47></location>〈 δ η φ, δ η φ 〉 = 4 π ∫ dψ hf 2 [ ( ¯ ξ ⊥ ) 2 + Bξ ( D 2 + V ) ξ ] , (A.6)</formula>
<text><location><page_19><loc_13><loc_36><loc_87><loc_42></location>where D is the derivative (5.5). The coefficients B and V are complicated functions of f and h , but they are independent of ¯ ξ ⊥ . Crucially, they are also quadratic in the extrinsic curvature, and hence invariant under K i j →-K i j . The Jacobian (A.2) thus takes the form</text>
<formula><location><page_19><loc_41><loc_33><loc_87><loc_36></location>J = det 1 / 2 | B ( D 2 + V ) | , (A.7)</formula>
<text><location><page_19><loc_13><loc_27><loc_87><loc_32></location>and is independent of the sign of the extrinsic curvature. Hence the inner product does not break the time-reversal symmetry that produces cancellations between regions with different signs σ .</text>
<text><location><page_19><loc_13><loc_22><loc_87><loc_27></location>It may be possible to obtain further information about J through zeta function methods. In particular, the scaling behavior could be important in understanding the possible infrared divergences discussed in section 7.1. For now, though, this lies beyond the scope of this paper.</text>
<section_header_level_1><location><page_20><loc_13><loc_89><loc_24><loc_90></location>References</section_header_level_1>
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</document> | [] |
2019arXiv190509464C | https://arxiv.org/pdf/1905.09464.pdf | <document>
<section_header_level_1><location><page_1><loc_35><loc_77><loc_64><loc_79></location>Hyperbolic vacuum decay</section_header_level_1>
<text><location><page_1><loc_28><loc_68><loc_72><loc_75></location>Hristu Culetu, Ovidius University, Dept.of Physics and Electronics, B-dul Mamaia 124, 900527 Constanta, Romania, e-mail : [email protected]</text>
<text><location><page_1><loc_44><loc_66><loc_56><loc_67></location>June 21, 2019</text>
<section_header_level_1><location><page_1><loc_47><loc_61><loc_53><loc_62></location>Abstract</section_header_level_1>
<text><location><page_1><loc_26><loc_48><loc_74><loc_60></location>The properties of an hyperbolically-expanding wormhole are studied. Using a particular equation of state for the fluid on the wormhole throat, we reached an equation of motion for the throat that leads to a constant surface energy density σ . The Lagrangean leading to the above equation of motion contains the 'rest mass' of the expanding particle as a potential energy. The associated Hamiltonian corresponds to a relativistic free particle of a total Planck energy E P . When the wormhole is embedded in de Sitter space, we found that the cosmological constant is of Planck order but hidden at very tiny scales, in accordance with Carlip's recipe.</text>
<section_header_level_1><location><page_1><loc_22><loc_44><loc_40><loc_45></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_22><loc_27><loc_78><loc_42></location>As everything gravitates, we have to keep track of the vacuum energy, the energy of quantum fluctuations of empty space. Its gravitational effects should manifest as a cosmological constant Λ [1]. The problem is that simple evaluations give a cosmological constant (CC) which is 120 orders of magnitude larger than what observes at the cosmological scale (for example, from the accelerated expansion of the Universe). As Carlip [2] (see also [3]) has noticed, the trouble comes from the mixing of scales. Λ is generated at Planck scale but observed at cosmological scale. He proposed a new, simple alternative: our Universe does have a CC of the order of Planck's value, due to the 'spacetime foam' picture [4], which is not, of course, homogeneous, so that the model is not ruled out by observations.</text>
<text><location><page_1><loc_22><loc_18><loc_78><loc_27></location>The nontrivial topological structure of the spacetime at the Planck scale enforces the presence of wormholes (WH), i.e. hypersurfaces that connect two asymptoticall-flat spacetimes [5, 6, 7, 8, 9]. Redmount and Suen [10, 11] considered a 'Minkowski WH geometry' as a model for topological fluctuations within the Planck scale spacetime foam. Their simple WH geometry seems to be unstable against grouth to macroscopic size.</text>
<text><location><page_1><loc_22><loc_15><loc_78><loc_18></location>We investigate in this paper a slightly modified version of the dynamic RS wormhole, taking a different equation of state of the boundary fluid and show</text>
<text><location><page_2><loc_22><loc_77><loc_78><loc_84></location>that its throat evolves hyperbolically. The equation of motion of the expanding 'bubble' of the time-dependent mass is a hyperbola which macroscopically becomes the Minkowski light cone. We built the gravitational action from which the equation of motion may be derived and found the corresponding Hamiltonian which proves to be a constant (Planck energy).</text>
<text><location><page_2><loc_22><loc_74><loc_78><loc_76></location>Throughout the paper geometrical units G = c = /planckover2pi1 = 1 are used, excepting otherwise specified.</text>
<section_header_level_1><location><page_2><loc_22><loc_69><loc_64><loc_71></location>2 Equation of motion of the throat</section_header_level_1>
<text><location><page_2><loc_22><loc_47><loc_78><loc_68></location>Although at macroscopic scales the spacetime appears smooth and simply connected, on Planck length scales it fluctuates quantum-mechanically, developing all kinds of topological structures, including WHs. A microscopic WH may be extracted from the foam to give birth to a macroscopic traversable WH. Redmount and Suen [10, 11] see Lorentzian spacetime filled with many microscopic WHs. They found those WHs are quantum-mechanically unstable, like a classical stable black hole which however undergoes quantum Hawking evaporation. RS constructed a spherically-symmetric 'Minkowski wormhole' by excising a sphere of radius r = R ( t ) ( t - the Minkowski time coordinate) from two copies of the Minkowski space, identifying the two boundary surfaces r = R ( t ). To obey Einstein's equations a surface stress tensor on the boundary Σ was introduced. Outside the boundary both exterior spacetimes are flat. The boundary plays the role of the WH throat and the Einstein equations are equivalent with the Lanczos equations [7]</text>
<formula><location><page_2><loc_41><loc_43><loc_78><loc_47></location>-8 πS i j = [ K i j -δ i j K l l ] (2.1)</formula>
<text><location><page_2><loc_22><loc_37><loc_78><loc_40></location>Let us find now the extrinsic curvature tensor of the surface F ≡ r -R ( t ) = 0. The spacetime metric is Minkowskian and the geometry on Σ can be written as</text>
<text><location><page_2><loc_22><loc_39><loc_78><loc_45></location>with S i j the surface stress tensor (here i, j = τ, θ, φ ), K l l - the trace of the extrinsic curvature of the boundary Σ and [ ... ] stands for the jump of K i j when the boundary is crossed; namely, [ K ij ] = K + ij -K -ij = 2 K + ij in our situation.</text>
<formula><location><page_2><loc_42><loc_34><loc_78><loc_36></location>ds 2 Σ = -dτ 2 + R 2 d Ω 2 (2.2)</formula>
<text><location><page_2><loc_22><loc_30><loc_78><loc_34></location>where dτ = √ 1 -˙ R 2 dt , τ is the proper time on Σ and ˙ R = dR/dt . The velocity 4-vector is</text>
<text><location><page_2><loc_22><loc_25><loc_78><loc_28></location>with u b u b = -1. The unit normal to Σ may be found from (2.3) and the relations n b n b = 1 and n b u b = 0. The velocity u b from (2.3) yields</text>
<text><location><page_2><loc_22><loc_18><loc_27><loc_19></location>whence</text>
<formula><location><page_2><loc_40><loc_27><loc_78><loc_31></location>u b = dx b dτ = ( dt dτ , dR dτ , 0 , 0 ) (2.3)</formula>
<formula><location><page_2><loc_38><loc_18><loc_78><loc_24></location>u b = ( 1 √ 1 -˙ R 2 , ˙ R √ 1 -˙ R 2 , 0 , 0 ) (2.4)</formula>
<formula><location><page_2><loc_37><loc_12><loc_78><loc_18></location>n b = ( -˙ R √ 1 -˙ R 2 , 1 √ 1 -˙ R 2 , 0 , 0 ) (2.5)</formula>
<text><location><page_3><loc_22><loc_83><loc_66><loc_84></location>The second fundamental form of Σ may be obtained from [12]</text>
<formula><location><page_3><loc_42><loc_78><loc_78><loc_81></location>K ij = ∂x a ∂ξ i ∂x b ∂ξ j ∇ a n b , (2.6)</formula>
<text><location><page_3><loc_22><loc_74><loc_78><loc_77></location>where ξ i = ( τ, θ, φ ) are the coordinates on Σ and the operator 'nabla' is applied in Minkowski space in four dimensions.</text>
<text><location><page_3><loc_24><loc_73><loc_77><loc_74></location>Eq. (2.6) gives the following components of the second fundamental form</text>
<text><location><page_3><loc_22><loc_65><loc_32><loc_66></location>with the trace</text>
<formula><location><page_3><loc_32><loc_65><loc_78><loc_71></location>K ττ = -R (1 -˙ R 2 ) 3 / 2 , K θθ = R √ 1 -˙ R 2 = K φφ sin 2 θ , (2.7)</formula>
<text><location><page_3><loc_22><loc_60><loc_50><loc_61></location>Using (2.7) and (2.8), Eqs. (2.1) give us</text>
<formula><location><page_3><loc_36><loc_60><loc_78><loc_65></location>K ≡ K i i = R (1 -˙ R 2 ) 3 / 2 + 2 R √ 1 -˙ R 2 . (2.8)</formula>
<formula><location><page_3><loc_23><loc_53><loc_78><loc_58></location>S ττ = -1 2 πR √ 1 -˙ R 2 , S θθ = R 4 π √ 1 -˙ R 2 + R 2 R 4 π (1 -˙ R 2 ) 3 / 2 = S φφ sin 2 θ . (2.9)</formula>
<text><location><page_3><loc_22><loc_52><loc_67><loc_54></location>Supposing that S ij on the throat corresponds to a perfect fluid</text>
<formula><location><page_3><loc_41><loc_49><loc_78><loc_51></location>S ij = ( p s + σ ) u i u j + p s h ij (2.10)</formula>
<text><location><page_3><loc_22><loc_45><loc_78><loc_48></location>where h ij = diag ( -1 , R 2 , R 2 sin 2 θ ) is the metric on the boundary, we have σ = S ττ for the surface energy density and p s = S θθ /R 2 for the surface pressure.</text>
<text><location><page_3><loc_22><loc_35><loc_78><loc_45></location>To find the equation of motion for the throat, we need now an equation of state relating σ and p s . RS chose σ = -4 p s as equation of state but in this case the action integral (whence the equation of motion was obtained) has a complicate 'kinetic term'. Our choice for the equation of state is simply p s = -σ , as for a domain wall [13] because one seems to be the most appropriate conjecture for a Lorentz-invariant vacuum. That choice leads to the equation of motion</text>
<formula><location><page_3><loc_44><loc_32><loc_78><loc_35></location>R R + ˙ R 2 -1 = 0 , (2.11)</formula>
<formula><location><page_3><loc_44><loc_27><loc_78><loc_31></location>R ( t ) = √ t 2 + b 2 , (2.12)</formula>
<text><location><page_3><loc_22><loc_31><loc_38><loc_32></location>which has the solution</text>
<text><location><page_3><loc_22><loc_26><loc_78><loc_28></location>using appropriate initial conditions ( R min = b , which is taken to be the Planck length).</text>
<section_header_level_1><location><page_3><loc_22><loc_21><loc_49><loc_23></location>3 Free particle energy</section_header_level_1>
<text><location><page_3><loc_22><loc_19><loc_61><loc_20></location>By means of (2.9), the expression (2.12) for R ( t ) yields</text>
<formula><location><page_3><loc_41><loc_15><loc_78><loc_18></location>σ = -p s = -1 2 πb = const. (3.1)</formula>
<text><location><page_4><loc_24><loc_83><loc_42><loc_84></location>From (2.12) we also have</text>
<formula><location><page_4><loc_41><loc_78><loc_78><loc_82></location>1 -˙ R 2 = b 2 R 2 , R = b 2 R 3 . (3.2)</formula>
<text><location><page_4><loc_22><loc_75><loc_78><loc_78></location>Therefore, the component of the acceleration of the throat, normal to Σ will be [7]</text>
<formula><location><page_4><loc_36><loc_72><loc_78><loc_75></location>A ⊥ ≡ n b A b = -K ττ = 1 b = -2 πσ > 0 , (3.3)</formula>
<text><location><page_4><loc_22><loc_65><loc_78><loc_72></location>where A b is built with u b from (2.4). So we obtained the same evolution of the WH throat as Ipser and Sikivie for their domain wall which in Minkowskian coordinates is not a plane at all but rather an accelerating sphere, expanding with the acceleration 2 π | σ | .</text>
<text><location><page_4><loc_22><loc_52><loc_78><loc_65></location>A remark is in order here. The radial null geodesics (6.4) from [9] are similar with the equation of motion (2.12) of the dynamic WH throat. Note that the spacetime (2.2) from [9] is curved and the region r < b is absent from the manifold. We identify the two processes and assume that actually the null particles are carried by the WH throat during their propagation (see also [14]). In other words, the throat turns out to play the role of a de Broglie pilot wave, dragging the null particles with it. The observed absence of macroscopic WHs may be due to their very fast expansion ( R ( t ) ≈ t for t >> b ) and their energy is spread out on larger and larger volumes.</text>
<text><location><page_4><loc_22><loc_49><loc_78><loc_52></location>The gravitational action corresponding to (2.12), obtained from the integral of the scalar curvature plus the surface term, may be written as</text>
<formula><location><page_4><loc_42><loc_44><loc_78><loc_48></location>S = -∫ b 2 R √ 1 -˙ R 2 dt, (3.4)</formula>
<text><location><page_4><loc_22><loc_43><loc_47><loc_44></location>whence the Lagrangean is given by</text>
<formula><location><page_4><loc_44><loc_37><loc_78><loc_42></location>L = -b 2 R √ 1 -˙ R 2 (3.5)</formula>
<text><location><page_4><loc_24><loc_30><loc_66><loc_32></location>It is worth noting that, when ˙ R << 1, L acquires the form</text>
<text><location><page_4><loc_22><loc_31><loc_78><loc_38></location>(the factor b 2 is necessary for L to get units of length). The action (3.4) gives a model with features like those of a relativistic free particle. Moreover, when S - which is invariant - is expressed in terms of the proper time, with 0 ≤ τ < ∞ , one obtains π /planckover2pi1 / 2 ≥ S > 0.</text>
<formula><location><page_4><loc_38><loc_26><loc_78><loc_29></location>L ≈ -b 2 R (1 -˙ R 2 2 ) = Mv 2 2 -Mc 2 , (3.6)</formula>
<text><location><page_4><loc_22><loc_20><loc_78><loc_25></location>where v ( t ) = ˙ R and M ( t ) = b 2 R ( t ) . We observe that the second term from the r.h.s. of (3.6) plays the role of a time dependent 'rest' (potential) energy of the expanding 'particle'.</text>
<text><location><page_4><loc_24><loc_19><loc_48><loc_20></location>The canonical momentum will be</text>
<formula><location><page_4><loc_38><loc_12><loc_78><loc_18></location>p = ∂ L ∂ ˙ R = b 2 ˙ R R √ 1 -˙ R 2 = Mv √ 1 -v 2 (3.7)</formula>
<text><location><page_5><loc_22><loc_83><loc_43><loc_84></location>which yields the Hamiltonian</text>
<formula><location><page_5><loc_36><loc_76><loc_78><loc_81></location>H = p ˙ R -L = b 2 R √ 1 -˙ R 2 = M √ 1 -v 2 (3.8)</formula>
<text><location><page_5><loc_22><loc_74><loc_78><loc_77></location>To find the direct relation between p and H we get rid of ˙ R from the last two equations to obtain</text>
<formula><location><page_5><loc_43><loc_69><loc_78><loc_74></location>H = √ p 2 + ( b 2 R ) 2 . (3.9)</formula>
<text><location><page_5><loc_22><loc_48><loc_78><loc_69></location>Inserting all fundamental constants, we again see that b 2 /R = /planckover2pi1 /cR plays the role of a mass M ( t ) of the 'particle' (expanding WH throat in our case), namely M ( t ) = /planckover2pi1 /cR ( t ). So R ( t ) = /planckover2pi1 /M ( t ) c appears to be the Compton wavelength associated to the mass M ( t ). For t >> b , R ( t ) ≈ t so that Mc 2 t = /planckover2pi1 , which looks like an uncertainty relation. One could also see from (3.8) that H has the form of a Lorentz-boosted energy. This simple WH geometry seems to represent a spacetime foam structure unstable against growth to macroscopic size [10]. When (2.12) is used in the expression for the expanding throat energy (3.8), we get H = b = E P , where E P is the Planck energy. In other words, the total energy of the expanding 'bubble' remains constant, although the mass M and p are time-dependent. That is possible because, while M decreases with time, p increases and so there is a perfect compensation between them. When the action (3.4) is expressed in terms of the proper time and by means of (2.12), one obtains</text>
<formula><location><page_5><loc_35><loc_44><loc_78><loc_48></location>S = -∫ b cosh τ b dτ = b 2 arcsin ( 1 cosh τ b ) , (3.10)</formula>
<text><location><page_5><loc_22><loc_41><loc_78><loc_44></location>with 0 ≤ τ < ∞ . We get that π /planckover2pi1 / 2 > S > 0. In addition, as a function of variable τ only, S is an invariant.</text>
<section_header_level_1><location><page_5><loc_22><loc_37><loc_56><loc_39></location>4 The cosmological constant</section_header_level_1>
<text><location><page_5><loc_22><loc_27><loc_78><loc_36></location>Let us consider now the previous WH embedded in a de Sitter (dS) spacetime. We are no longer dealing with a 'Minkowski wormhole'. The 'dS wormhole' is obtained by excising a sphere of radius r = R ( t ) from two copies of the dS space. Outside the boundary both exterior spaces are dS , with the same horizon radius. Due to the horizon, some restrictions will be imposed on R ( τ ), where τ is the proper time on the throat.</text>
<text><location><page_5><loc_22><loc_16><loc_78><loc_27></location>The junction conditions for the system de Sitter - Schwarzschild have been studied, among others, by Blau, Guendelman and Guth [15], so that we will use their model and adopt the calculations in our situation. As we stated previously, we have [ K ij ] = K + ij -K -ij = 2 K + ij in our conditions, because of the symmetry (see, for example, [16]). Keeping in mind that we study the motion of the WH throat (a domain wall with p s = -σ > 0), we get from the junction condition for K θθ</text>
<formula><location><page_5><loc_39><loc_13><loc_78><loc_16></location>2 √ 1 -χ 2 R 2 + R ' 2 = -4 πσR, (4.1)</formula>
<text><location><page_6><loc_22><loc_80><loc_78><loc_84></location>where R ' = dR/dτ, χ 2 = Λ / 3 and Λ > 0 is the cosmological constant. We notice that (4.1) has R ( τ ) = b cosh ( τ/b ) as a solution (with R (0) = b as initial condition) provided</text>
<formula><location><page_6><loc_44><loc_78><loc_78><loc_80></location>b 2 ( χ 2 + g 2 ) = 1 , (4.2)</formula>
<text><location><page_6><loc_22><loc_75><loc_70><loc_77></location>with the acceleration g = 2 π | σ | , χ < 1 /b (or Λ < 3 /b 2 ) and g < 1 /b .</text>
<text><location><page_6><loc_24><loc_74><loc_61><loc_76></location>The metric on the WH throat may be written now</text>
<formula><location><page_6><loc_40><loc_71><loc_78><loc_74></location>ds 2 Σ = -dτ 2 + b 2 cosh 2 τ b d Ω 2 , (4.3)</formula>
<text><location><page_6><loc_22><loc_62><loc_78><loc_70></location>which is the closed dS space in three dimensions. It is worth noting that three parameters are encountered here: χ, g and b . Dealing with WHs expanding from the Planck world, b has been chosen of the order of the Planck length. From (4.2) we get χ = (1 /b ) √ 1 -b 2 g 2 whence one sees that Λ reaches its Planck value when g << 1 /b .</text>
<text><location><page_6><loc_22><loc_58><loc_78><loc_62></location>Let us express now the expanding WH throat radius in terms of the coordinate time t (the equivalent of Eq.2.12). From the static dS metric [15] and the equation of the surface Σ , r = R ( τ ), one obtains that</text>
<formula><location><page_6><loc_42><loc_52><loc_78><loc_56></location>dt dτ = √ 1 -χ 2 R 2 + R ' 2 1 -χ 2 R 2 (4.4)</formula>
<formula><location><page_6><loc_42><loc_47><loc_78><loc_51></location>dt dτ = √ 1 -χ 2 b 2 cosh τ b 1 -χ 2 b 2 cosh 2 τ b (4.5)</formula>
<text><location><page_6><loc_22><loc_51><loc_27><loc_52></location>whence</text>
<text><location><page_6><loc_22><loc_45><loc_70><loc_47></location>with cosh τ b < 1 /bχ . Eq. 4.5 could be easily integrated and gives us</text>
<formula><location><page_6><loc_37><loc_40><loc_78><loc_44></location>( α + βtanh τ 2 b ) ( β -αtanh τ 2 b ) ( α -βtanh τ 2 b ) ( β + αtanh τ 2 b ) = e 2 χt , (4.6)</formula>
<formula><location><page_6><loc_43><loc_34><loc_78><loc_37></location>χ g sinh τ b = tanh χt, (4.7)</formula>
<text><location><page_6><loc_22><loc_37><loc_78><loc_40></location>with tanh ( τ/ 2 b ) < α/β, τ (0) = 0 , α = √ 1 -bχ and β = √ 1 + bχ . We have further</text>
<text><location><page_6><loc_22><loc_32><loc_29><loc_34></location>that yields</text>
<formula><location><page_6><loc_40><loc_28><loc_78><loc_32></location>R ( t ) = b √ 1 + g 2 χ 2 tanh 2 χt. (4.8)</formula>
<text><location><page_6><loc_22><loc_24><loc_78><loc_28></location>As a consistency check, we take above χ = 0 (the Minkowski case). Having 0 / 0 in (4.8), we ought to consider the limit when χ → 0 and get</text>
<formula><location><page_6><loc_43><loc_20><loc_78><loc_24></location>R ( t ) = b √ 1 + g 2 t 2 . (4.9)</formula>
<text><location><page_6><loc_22><loc_18><loc_78><loc_21></location>But χ = 0 gives g = 1 /b and so the equation of motion (2.12) is recovered, as expected.</text>
<text><location><page_6><loc_22><loc_15><loc_78><loc_18></location>We know that the condition R < 1 /χ must be obeyed. In addition, χ < 1 /b and, therefore, one obtains R max = (1 + bχ ) / 2 χ . In the asymptotically flat</text>
<text><location><page_7><loc_22><loc_80><loc_78><loc_84></location>situation ( χ = 0), we get R max →∞ , as it should be according to (2.12), when t →∞ .</text>
<text><location><page_7><loc_22><loc_77><loc_78><loc_81></location>Let us check now the energy constraint equation[17], obtained from the 3+1 decomposition of Einstein's equations, when the matter contribution is overlooked w.r.t. Λ</text>
<formula><location><page_7><loc_40><loc_73><loc_78><loc_75></location>3 R -K 2 + K i j K j i -2Λ = 0 , (4.10)</formula>
<text><location><page_7><loc_22><loc_69><loc_78><loc_72></location>where 3 R stands for the curvature scalar of Σ with g ij from (2.2). Thanks to the equation R ( τ ) = b cosh τ b , one obtains</text>
<formula><location><page_7><loc_26><loc_62><loc_78><loc_68></location>K τ τ = R '' -χ 2 R √ 1 -χ 2 R 2 + R ' 2 = g, K θ θ = √ 1 -χ 2 R 2 + R ' 2 R = g, (4.11)</formula>
<formula><location><page_7><loc_43><loc_56><loc_78><loc_59></location>σ i j = K i j -1 3 δ i j K (4.12)</formula>
<text><location><page_7><loc_22><loc_59><loc_78><loc_63></location>whence K = 3 g and K i j K j i = 3 g 2 . From (2.2) we have 3 R = 6 /b 2 . Keeping in mind that Λ = 3 /b 2 -3 g 2 , one finds that Eq.4.10 is observed. As far as the shear tensor</text>
<text><location><page_7><loc_22><loc_51><loc_78><loc_55></location>is concerned, using the components of the extrinsic curvature tensor one finds that σ i j is vanishing. In contrast, the expansion scalar of the fluid on Σ is constant, with Θ = K = 3 g .</text>
<text><location><page_7><loc_22><loc_42><loc_78><loc_51></location>We have seen above that Λ takes Planck order values. To measure it, we have to perform experiments at Planck scale, using an apparatus of the same order of magnitude; or, the duration of the measurements to be alike. Similar difficulties arise when one intends to measure the difference between Eq. (2.12) and the standard equation of the light cone R ( t ) = t in Minkowski space: b is too small and the hyperbola is very close to its asymptotes.</text>
<section_header_level_1><location><page_7><loc_22><loc_38><loc_39><loc_40></location>5 Conclusions</section_header_level_1>
<text><location><page_7><loc_22><loc_22><loc_78><loc_36></location>We investigated in this paper a particular dynamic Lorenzian WH. Using a different equation of state compared to Redmount and Suen, we found that the dynamic WH expands hyperbolically in a fashion similar with the Coleman and de Luccia bubble [18] or Ipser and Sikivie domain wall, i.e. a Lorentz-invariant expansion. In addition, the Hamiltonian of the system equals the Planck energy b and corresponds to a relativistic free particle of a time-dependent mass M = /planckover2pi1 /cR ( t ) and a time-dependent momentum p = bt/R ( t ), in spite of the fact that its energy is constant. When | σ | is small w.r.t. 1 / 2 πb , a Planck-valued Λ emerges, in the case our WH is embedded in dS spacetime, which is in accordance with Carlip's conjecture that the CC is huge but hidden at very tiny scales.</text>
<section_header_level_1><location><page_7><loc_22><loc_17><loc_34><loc_19></location>References</section_header_level_1>
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<list_item><location><page_7><loc_23><loc_15><loc_46><loc_16></location>[1] S. Carlip, arXiv: 1905.05216.</list_item>
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<list_item><location><page_8><loc_23><loc_83><loc_46><loc_84></location>[2] S. Carlip, arXiv: 1809.08277.</list_item>
<list_item><location><page_8><loc_23><loc_80><loc_58><loc_81></location>[3] C. Pagani and M. Reuter, arXiv: 1906.02507.</list_item>
<list_item><location><page_8><loc_23><loc_78><loc_55><loc_79></location>[4] J. A. Wheeler, Phys. Rev. 97, 511 (1955).</list_item>
<list_item><location><page_8><loc_23><loc_75><loc_70><loc_76></location>[5] M. S. Morris and K. S. Thorne, Amer. J. Phys. 56, 395 (1988).</list_item>
<list_item><location><page_8><loc_23><loc_71><loc_78><loc_74></location>[6] M. S. Morris, K. S. Thorne and V. Yurtsever, Phys. Rev. Lett. 61, 1446 (1988).</list_item>
<list_item><location><page_8><loc_23><loc_69><loc_54><loc_70></location>[7] M. Visser, Phys. Rev. D39, 3182 (1989).</list_item>
<list_item><location><page_8><loc_23><loc_66><loc_54><loc_67></location>[8] M. Visser, Nucl. Phys. B328, 203 (1989).</list_item>
<list_item><location><page_8><loc_23><loc_64><loc_74><loc_65></location>[9] H. Culetu, Phys. Scripta 90, (8), 12 (2015); arXiv: gr-qc/1407.3588.</list_item>
<list_item><location><page_8><loc_22><loc_60><loc_78><loc_62></location>[10] I. H. Redmount and W.-M. Suen, Phys. Rev. D47, R2163 (1993) (arXiv: gr-qc/9210017).</list_item>
<list_item><location><page_8><loc_22><loc_56><loc_78><loc_58></location>[11] I. H. Redmount and W.-M. Suen, Phys. Rev. D49, 5199 (1994) (arXiv: gr-qc/9309017).</list_item>
<list_item><location><page_8><loc_22><loc_51><loc_78><loc_54></location>[12] J. P. S. Lemos and F. S. N. Lobo, Phys. Rev. D78, 044030 (2008) (arXiv: 0806.4459 [gr-qc]).</list_item>
<list_item><location><page_8><loc_22><loc_49><loc_62><loc_50></location>[13] J. Ipser and P. Sikivie, Phys. Rev. D30, 712 (1984).</list_item>
<list_item><location><page_8><loc_22><loc_46><loc_78><loc_48></location>[14] H. Culetu, J. Korean Phys. Soc. 57, 419 (2010) (arXiv: 0905.3474 [hep-th]).</list_item>
<list_item><location><page_8><loc_22><loc_42><loc_78><loc_45></location>[15] S. K.Blau, E. I. Guendelman and A. H. Guth, Phys. Rev. D35 (6), 1747 (1987).</list_item>
<list_item><location><page_8><loc_22><loc_38><loc_78><loc_41></location>[16] N. M. Garcia, F. S. N. Lobo and M. Visser, Phys. Rev. D86, 044026 (2012); arXiv: 1112.2057.</list_item>
<list_item><location><page_8><loc_22><loc_34><loc_78><loc_37></location>[17] P. M.-Moruno, N. M. Garcia, F. S. N. Lobo and M. Visser, JCAP 03 (2012) 034; arXiv: 1112.5253.</list_item>
<list_item><location><page_8><loc_22><loc_32><loc_68><loc_33></location>[18] S. Coleman and F. de Luccia, Phys. Rev. D21, 3305 (1980).</list_item>
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</document> | [{"title": "Hyperbolic vacuum decay", "content": "Hristu Culetu, Ovidius University, Dept.of Physics and Electronics, B-dul Mamaia 124, 900527 Constanta, Romania, e-mail : [email protected] June 21, 2019", "pages": [1]}, {"title": "Abstract", "content": "The properties of an hyperbolically-expanding wormhole are studied. Using a particular equation of state for the fluid on the wormhole throat, we reached an equation of motion for the throat that leads to a constant surface energy density \u03c3 . The Lagrangean leading to the above equation of motion contains the 'rest mass' of the expanding particle as a potential energy. The associated Hamiltonian corresponds to a relativistic free particle of a total Planck energy E P . When the wormhole is embedded in de Sitter space, we found that the cosmological constant is of Planck order but hidden at very tiny scales, in accordance with Carlip's recipe.", "pages": [1]}, {"title": "1 Introduction", "content": "As everything gravitates, we have to keep track of the vacuum energy, the energy of quantum fluctuations of empty space. Its gravitational effects should manifest as a cosmological constant \u039b [1]. The problem is that simple evaluations give a cosmological constant (CC) which is 120 orders of magnitude larger than what observes at the cosmological scale (for example, from the accelerated expansion of the Universe). As Carlip [2] (see also [3]) has noticed, the trouble comes from the mixing of scales. \u039b is generated at Planck scale but observed at cosmological scale. He proposed a new, simple alternative: our Universe does have a CC of the order of Planck's value, due to the 'spacetime foam' picture [4], which is not, of course, homogeneous, so that the model is not ruled out by observations. The nontrivial topological structure of the spacetime at the Planck scale enforces the presence of wormholes (WH), i.e. hypersurfaces that connect two asymptoticall-flat spacetimes [5, 6, 7, 8, 9]. Redmount and Suen [10, 11] considered a 'Minkowski WH geometry' as a model for topological fluctuations within the Planck scale spacetime foam. Their simple WH geometry seems to be unstable against grouth to macroscopic size. We investigate in this paper a slightly modified version of the dynamic RS wormhole, taking a different equation of state of the boundary fluid and show that its throat evolves hyperbolically. The equation of motion of the expanding 'bubble' of the time-dependent mass is a hyperbola which macroscopically becomes the Minkowski light cone. We built the gravitational action from which the equation of motion may be derived and found the corresponding Hamiltonian which proves to be a constant (Planck energy). Throughout the paper geometrical units G = c = /planckover2pi1 = 1 are used, excepting otherwise specified.", "pages": [1, 2]}, {"title": "2 Equation of motion of the throat", "content": "Although at macroscopic scales the spacetime appears smooth and simply connected, on Planck length scales it fluctuates quantum-mechanically, developing all kinds of topological structures, including WHs. A microscopic WH may be extracted from the foam to give birth to a macroscopic traversable WH. Redmount and Suen [10, 11] see Lorentzian spacetime filled with many microscopic WHs. They found those WHs are quantum-mechanically unstable, like a classical stable black hole which however undergoes quantum Hawking evaporation. RS constructed a spherically-symmetric 'Minkowski wormhole' by excising a sphere of radius r = R ( t ) ( t - the Minkowski time coordinate) from two copies of the Minkowski space, identifying the two boundary surfaces r = R ( t ). To obey Einstein's equations a surface stress tensor on the boundary \u03a3 was introduced. Outside the boundary both exterior spacetimes are flat. The boundary plays the role of the WH throat and the Einstein equations are equivalent with the Lanczos equations [7] Let us find now the extrinsic curvature tensor of the surface F \u2261 r -R ( t ) = 0. The spacetime metric is Minkowskian and the geometry on \u03a3 can be written as with S i j the surface stress tensor (here i, j = \u03c4, \u03b8, \u03c6 ), K l l - the trace of the extrinsic curvature of the boundary \u03a3 and [ ... ] stands for the jump of K i j when the boundary is crossed; namely, [ K ij ] = K + ij -K -ij = 2 K + ij in our situation. where d\u03c4 = \u221a 1 -\u02d9 R 2 dt , \u03c4 is the proper time on \u03a3 and \u02d9 R = dR/dt . The velocity 4-vector is with u b u b = -1. The unit normal to \u03a3 may be found from (2.3) and the relations n b n b = 1 and n b u b = 0. The velocity u b from (2.3) yields whence The second fundamental form of \u03a3 may be obtained from [12] where \u03be i = ( \u03c4, \u03b8, \u03c6 ) are the coordinates on \u03a3 and the operator 'nabla' is applied in Minkowski space in four dimensions. Eq. (2.6) gives the following components of the second fundamental form with the trace Using (2.7) and (2.8), Eqs. (2.1) give us Supposing that S ij on the throat corresponds to a perfect fluid where h ij = diag ( -1 , R 2 , R 2 sin 2 \u03b8 ) is the metric on the boundary, we have \u03c3 = S \u03c4\u03c4 for the surface energy density and p s = S \u03b8\u03b8 /R 2 for the surface pressure. To find the equation of motion for the throat, we need now an equation of state relating \u03c3 and p s . RS chose \u03c3 = -4 p s as equation of state but in this case the action integral (whence the equation of motion was obtained) has a complicate 'kinetic term'. Our choice for the equation of state is simply p s = -\u03c3 , as for a domain wall [13] because one seems to be the most appropriate conjecture for a Lorentz-invariant vacuum. That choice leads to the equation of motion which has the solution using appropriate initial conditions ( R min = b , which is taken to be the Planck length).", "pages": [2, 3]}, {"title": "3 Free particle energy", "content": "By means of (2.9), the expression (2.12) for R ( t ) yields From (2.12) we also have Therefore, the component of the acceleration of the throat, normal to \u03a3 will be [7] where A b is built with u b from (2.4). So we obtained the same evolution of the WH throat as Ipser and Sikivie for their domain wall which in Minkowskian coordinates is not a plane at all but rather an accelerating sphere, expanding with the acceleration 2 \u03c0 | \u03c3 | . A remark is in order here. The radial null geodesics (6.4) from [9] are similar with the equation of motion (2.12) of the dynamic WH throat. Note that the spacetime (2.2) from [9] is curved and the region r < b is absent from the manifold. We identify the two processes and assume that actually the null particles are carried by the WH throat during their propagation (see also [14]). In other words, the throat turns out to play the role of a de Broglie pilot wave, dragging the null particles with it. The observed absence of macroscopic WHs may be due to their very fast expansion ( R ( t ) \u2248 t for t >> b ) and their energy is spread out on larger and larger volumes. The gravitational action corresponding to (2.12), obtained from the integral of the scalar curvature plus the surface term, may be written as whence the Lagrangean is given by It is worth noting that, when \u02d9 R << 1, L acquires the form (the factor b 2 is necessary for L to get units of length). The action (3.4) gives a model with features like those of a relativistic free particle. Moreover, when S - which is invariant - is expressed in terms of the proper time, with 0 \u2264 \u03c4 < \u221e , one obtains \u03c0 /planckover2pi1 / 2 \u2265 S > 0. where v ( t ) = \u02d9 R and M ( t ) = b 2 R ( t ) . We observe that the second term from the r.h.s. of (3.6) plays the role of a time dependent 'rest' (potential) energy of the expanding 'particle'. The canonical momentum will be which yields the Hamiltonian To find the direct relation between p and H we get rid of \u02d9 R from the last two equations to obtain Inserting all fundamental constants, we again see that b 2 /R = /planckover2pi1 /cR plays the role of a mass M ( t ) of the 'particle' (expanding WH throat in our case), namely M ( t ) = /planckover2pi1 /cR ( t ). So R ( t ) = /planckover2pi1 /M ( t ) c appears to be the Compton wavelength associated to the mass M ( t ). For t >> b , R ( t ) \u2248 t so that Mc 2 t = /planckover2pi1 , which looks like an uncertainty relation. One could also see from (3.8) that H has the form of a Lorentz-boosted energy. This simple WH geometry seems to represent a spacetime foam structure unstable against growth to macroscopic size [10]. When (2.12) is used in the expression for the expanding throat energy (3.8), we get H = b = E P , where E P is the Planck energy. In other words, the total energy of the expanding 'bubble' remains constant, although the mass M and p are time-dependent. That is possible because, while M decreases with time, p increases and so there is a perfect compensation between them. When the action (3.4) is expressed in terms of the proper time and by means of (2.12), one obtains with 0 \u2264 \u03c4 < \u221e . We get that \u03c0 /planckover2pi1 / 2 > S > 0. In addition, as a function of variable \u03c4 only, S is an invariant.", "pages": [3, 4, 5]}, {"title": "4 The cosmological constant", "content": "Let us consider now the previous WH embedded in a de Sitter (dS) spacetime. We are no longer dealing with a 'Minkowski wormhole'. The 'dS wormhole' is obtained by excising a sphere of radius r = R ( t ) from two copies of the dS space. Outside the boundary both exterior spaces are dS , with the same horizon radius. Due to the horizon, some restrictions will be imposed on R ( \u03c4 ), where \u03c4 is the proper time on the throat. The junction conditions for the system de Sitter - Schwarzschild have been studied, among others, by Blau, Guendelman and Guth [15], so that we will use their model and adopt the calculations in our situation. As we stated previously, we have [ K ij ] = K + ij -K -ij = 2 K + ij in our conditions, because of the symmetry (see, for example, [16]). Keeping in mind that we study the motion of the WH throat (a domain wall with p s = -\u03c3 > 0), we get from the junction condition for K \u03b8\u03b8 where R ' = dR/d\u03c4, \u03c7 2 = \u039b / 3 and \u039b > 0 is the cosmological constant. We notice that (4.1) has R ( \u03c4 ) = b cosh ( \u03c4/b ) as a solution (with R (0) = b as initial condition) provided with the acceleration g = 2 \u03c0 | \u03c3 | , \u03c7 < 1 /b (or \u039b < 3 /b 2 ) and g < 1 /b . The metric on the WH throat may be written now which is the closed dS space in three dimensions. It is worth noting that three parameters are encountered here: \u03c7, g and b . Dealing with WHs expanding from the Planck world, b has been chosen of the order of the Planck length. From (4.2) we get \u03c7 = (1 /b ) \u221a 1 -b 2 g 2 whence one sees that \u039b reaches its Planck value when g << 1 /b . Let us express now the expanding WH throat radius in terms of the coordinate time t (the equivalent of Eq.2.12). From the static dS metric [15] and the equation of the surface \u03a3 , r = R ( \u03c4 ), one obtains that whence with cosh \u03c4 b < 1 /b\u03c7 . Eq. 4.5 could be easily integrated and gives us with tanh ( \u03c4/ 2 b ) < \u03b1/\u03b2, \u03c4 (0) = 0 , \u03b1 = \u221a 1 -b\u03c7 and \u03b2 = \u221a 1 + b\u03c7 . We have further that yields As a consistency check, we take above \u03c7 = 0 (the Minkowski case). Having 0 / 0 in (4.8), we ought to consider the limit when \u03c7 \u2192 0 and get But \u03c7 = 0 gives g = 1 /b and so the equation of motion (2.12) is recovered, as expected. We know that the condition R < 1 /\u03c7 must be obeyed. In addition, \u03c7 < 1 /b and, therefore, one obtains R max = (1 + b\u03c7 ) / 2 \u03c7 . In the asymptotically flat situation ( \u03c7 = 0), we get R max \u2192\u221e , as it should be according to (2.12), when t \u2192\u221e . Let us check now the energy constraint equation[17], obtained from the 3+1 decomposition of Einstein's equations, when the matter contribution is overlooked w.r.t. \u039b where 3 R stands for the curvature scalar of \u03a3 with g ij from (2.2). Thanks to the equation R ( \u03c4 ) = b cosh \u03c4 b , one obtains whence K = 3 g and K i j K j i = 3 g 2 . From (2.2) we have 3 R = 6 /b 2 . Keeping in mind that \u039b = 3 /b 2 -3 g 2 , one finds that Eq.4.10 is observed. As far as the shear tensor is concerned, using the components of the extrinsic curvature tensor one finds that \u03c3 i j is vanishing. In contrast, the expansion scalar of the fluid on \u03a3 is constant, with \u0398 = K = 3 g . We have seen above that \u039b takes Planck order values. To measure it, we have to perform experiments at Planck scale, using an apparatus of the same order of magnitude; or, the duration of the measurements to be alike. Similar difficulties arise when one intends to measure the difference between Eq. (2.12) and the standard equation of the light cone R ( t ) = t in Minkowski space: b is too small and the hyperbola is very close to its asymptotes.", "pages": [5, 6, 7]}, {"title": "5 Conclusions", "content": "We investigated in this paper a particular dynamic Lorenzian WH. Using a different equation of state compared to Redmount and Suen, we found that the dynamic WH expands hyperbolically in a fashion similar with the Coleman and de Luccia bubble [18] or Ipser and Sikivie domain wall, i.e. a Lorentz-invariant expansion. In addition, the Hamiltonian of the system equals the Planck energy b and corresponds to a relativistic free particle of a time-dependent mass M = /planckover2pi1 /cR ( t ) and a time-dependent momentum p = bt/R ( t ), in spite of the fact that its energy is constant. When | \u03c3 | is small w.r.t. 1 / 2 \u03c0b , a Planck-valued \u039b emerges, in the case our WH is embedded in dS spacetime, which is in accordance with Carlip's conjecture that the CC is huge but hidden at very tiny scales.", "pages": [7]}] |
2022arXiv221108052F | https://arxiv.org/pdf/2211.08052.pdf | <document>
<section_header_level_1><location><page_1><loc_15><loc_83><loc_85><loc_86></location>COSMIC CENSORSHIP NEAR FLRW SPACETIMES WITH NEGATIVE SPATIAL CURVATURE</section_header_level_1>
<text><location><page_1><loc_39><loc_79><loc_62><loc_80></location>DAVID FAJMAN, LIAM URBAN</text>
<section_header_level_1><location><page_1><loc_47><loc_74><loc_54><loc_75></location>Abstract</section_header_level_1>
<text><location><page_1><loc_12><loc_60><loc_88><loc_73></location>We consider general initial data for the Einstein scalar-field system on a closed 3-manifold ( M,γ ) which is close to data for a Friedman-Lemaˆıtre-Robertson-Walker solution with homogeneous scalar field matter and a negative Einstein metric γ as spatial geometry. We prove that the maximal globally hyperbolic development of such initial data in the Einstein scalar-field system is past incomplete in the contracting direction and exhibits stable collapse into a Big Bang curvature singularity. Under an additional condition on the first positive eigenvalue of -∆ γ satisfied, for example, by closed hyperbolic 3-manifolds of small diameter, we prove that the data evolves to a future complete spacetime in the expanding direction which asymptotes to a vacuum Friedman solution with ( M,γ ) as the expansion normalized spatial geometry. In particular, the Strong Cosmic Censorship conjecture holds for this class of solutions in the C 2 -sense.</text>
<section_header_level_1><location><page_1><loc_43><loc_55><loc_57><loc_56></location>1. Introduction</section_header_level_1>
<section_header_level_1><location><page_1><loc_12><loc_52><loc_73><loc_54></location>1.1. Setting and main results. We consider the Einstein scalar-field system</section_header_level_1>
<formula><location><page_1><loc_12><loc_48><loc_57><loc_51></location>Ric[ g ] µν -1 2 R [ g ] g µν =8 πT µν [ g, φ ] (1.1a)</formula>
<formula><location><page_1><loc_12><loc_45><loc_69><loc_48></location>T µν = ∇ µ φ ∇ ν φ -1 2 g µν ∇ α φ ∇ α φ (1.1b)</formula>
<formula><location><page_1><loc_12><loc_43><loc_50><loc_44></location>/square g φ =0 (1.1c)</formula>
<text><location><page_1><loc_12><loc_35><loc_88><loc_42></location>with initial data ( g 0 , k 0 , π 0 , ψ 0 ) on a closed 3-manifold M that admits a negative Einstein metric γ . 1 In this paper, we determine the maximal globally hyperbolic development emanating from such initial data given that it is sufficiently close to the initial data of a homogeneous solution with a non-trivial scalar field.</text>
<text><location><page_1><loc_12><loc_21><loc_88><loc_35></location>In the collapsing direction, we prove a stable Big Bang formation and curvature blow-up result, which requires the presence of a non-trivial scalar field. The results complement those in [RS18b, Spe18], which cover flat and spherical spatial geometry. In the expanding direction, we prove a nonlinear future stability result of the corresponding vacuum background solution, which is the Milne model, under a mild condition for the first positive eigenvalue of ∆ γ (see Definition 9.2). As discussed in more details in Remark 9.3, numerical studies (see [CS99, Ino01]) show that this condition holds for an analogue of Weeks space, and suggest that this may hold for all closed hyperbolic 3-manifolds with sectional curvature -1 9 .</text>
<text><location><page_1><loc_12><loc_17><loc_88><loc_20></location>Connecting the two regions, we prove the global stability (i.e., past and future stability) of the spacetime</text>
<formula><location><page_1><loc_12><loc_12><loc_61><loc_17></location>(1.2a) ( [0 , ∞ ) × M, -dt 2 + a ( t ) 2 γ ) ,</formula>
<text><location><page_2><loc_12><loc_88><loc_86><loc_90></location>given a negative Einstein manifold ( M,γ ) obeying the aforementioned spectral condition, with</text>
<text><location><page_2><loc_12><loc_82><loc_59><loc_84></location>for some given constant C > 0, and the scalar field given by</text>
<formula><location><page_2><loc_12><loc_83><loc_62><loc_88></location>(1.2b) a (0) = 0 , ˙ a = √ 1 9 + 4 π 3 C 2 a -4</formula>
<formula><location><page_2><loc_12><loc_79><loc_59><loc_82></location>(1.2c) ∂ t φ = Ca -3 , ∇ φ = 0 .</formula>
<text><location><page_2><loc_12><loc_78><loc_70><loc_79></location>The scale factor consequently exhibits the following asymptotic behaviour:</text>
<formula><location><page_2><loc_12><loc_74><loc_66><loc_77></location>(1.2d) a ( t ) /similarequal t 1 3 as t ↘ 0 and a ( t ) /similarequal t as t ↗∞</formula>
<text><location><page_2><loc_12><loc_71><loc_46><loc_73></location>The main result can be split into two parts:</text>
<text><location><page_2><loc_12><loc_63><loc_88><loc_70></location>Theorem 1.1 (Big Bang stability - rough version) . Let ( M,g 0 , k 0 , π 0 , ψ 0 ) be initial data for the Einstein scalar-field system that is sufficiently close to ( M,a ( t 0 ) 2 γ, -˙ a ( t 0 ) a ( t 0 ) γ, 0 , Ca ( t 0 ) -3 ) , where C > 0 and ( M,γ ) is a closed Riemannian 3-manifold with Ric[ γ ] = -2 9 γ (i.e., a closed negative Einstein manifold with scalar curvature -2 3 ).</text>
<text><location><page_2><loc_12><loc_54><loc_88><loc_62></location>Then, the past maximal globally hyperbolic development ((0 , t 0 ] × M,g,φ ) of the initial data within the Einstein scalar-field system (1.1a) -(1.1c) admits a foliation by CMC hypersurfaces Σ s = t -1 ( { s } ) with zero shift. This development remains close to the FLRW solution described in (1.2a) -(1.2c) in the past of the initial data slice Σ t 0 . In particular, the solution exhibits curvature blow-up of order t -4 and every causal geodesic becomes incomplete as t approaches 0 .</text>
<text><location><page_2><loc_12><loc_47><loc_88><loc_52></location>Theorem 1.2 (Global stability) . Let ( M,g 0 , k 0 , π 0 , ψ 0 ) be initial data as in Theorem 1.1. In addition, we suppose that the smallest positive eigenvalue of -∆ γ acting on scalar functions is strictly greater than 1 9 .</text>
<text><location><page_2><loc_12><loc_40><loc_88><loc_47></location>Then, the initial data admits a maximal globally hyperbolic development ((0 , ∞ ) × M,g,φ ) solving the Einstein scalar-field system that, in addition to the results of Theorem 1.1, is future (causally) complete. As t ↗∞ , the solution is attracted by Milne spacetime in the sense that the expansion normalized variables ( g , k , ∇ φ, φ ' ) (see Definition 9.4) converge toward ( γ, 1 3 γ, 0 , 0) .</text>
<text><location><page_2><loc_12><loc_35><loc_88><loc_40></location>A more detailed statement of Theorem 1.1 is provided in Theorem 8.2. The additional spectral condition in Theorem 1.2 is discussed at the end of Subsection 1.3, and the statement itself is proven in Section 10 to be an extension of the Milne future stability result in Theorem 9.1.</text>
<unordered_list>
<list_item><location><page_2><loc_12><loc_31><loc_88><loc_34></location>1.2. Background material. We now provide context for the previously discussed setting and the results in Theorems 1.1-1.2:</list_item>
<list_item><location><page_2><loc_12><loc_22><loc_88><loc_30></location>1.2.1. Initial data to the Einstein scalar-field equations. It is well known that the Einstein equations can, via the the 3+1 decomposition, be viewed as an elliptic-hyperbolic system of PDEs (see, for example, [AM03]). This reduces solving the Einstein equations to two problems: finding admissible Einstein initial data in physical space, and then solving the corresponding initial value problem. Regarding the former, initial data to the Einstein scalar-field system takes the form</list_item>
</unordered_list>
<formula><location><page_2><loc_45><loc_20><loc_56><loc_21></location>( M, ˚ g, ˚ k, ˚ π, ˚ ψ ) ,</formula>
<text><location><page_2><loc_12><loc_14><loc_89><loc_19></location>where ˚ g and ˚ k are symmetric (0 , 2)-tensors on M , ˚ π is a (0 , 1)-tensor (corresponding to ∇ φ ) and ˚ ψ is a scalar function (corresponding to the future directed normal derivative ∂ 0 φ of the scalar field). The initial data must satisfy the Hamiltonian and momentum constraints</text>
<formula><location><page_2><loc_12><loc_9><loc_67><loc_14></location>R[˚ g ] + ( ˚ k a a ) 2 -( ˚ k a b ˚ k b a ) =8 π [ | ˚ ψ | 2 + | ˚ π | 2 ˚ g ] , (1.3a)</formula>
<formula><location><page_2><loc_12><loc_7><loc_63><loc_10></location>div ˚ g ˚ k = -8 π · ˚ π · ˚ ψ (1.3b)</formula>
<text><location><page_3><loc_48><loc_89><loc_50><loc_90></location>˚</text>
<text><location><page_3><loc_12><loc_83><loc_88><loc_90></location>(see (2.16a) and (2.16b)), where the indices of k in the first line are raised with respect to ˚ g . We note that, in our argument, we will additionally assume that our initial data has constant mean curvature so that our gauges can be satisfied initially - this is enforced on the level of initial data by requiring</text>
<formula><location><page_3><loc_46><loc_81><loc_54><loc_83></location>tr ˚ g ˚ k = -1</formula>
<text><location><page_3><loc_12><loc_76><loc_88><loc_81></location>which, after embedding, will lead to tr g | Σ t k = τ = -3 ˙ a a (see (2.10)). We will argue in Remark 8.1 why the initial data being near-FLRW allows us to assume the initial hypersurface to be CMC without loss of generality.</text>
<text><location><page_3><loc_12><loc_70><loc_88><loc_75></location>The results of [FB52, CBG69] show that there exists an embedding 2 ι : M ↪ → ι ( M ) ⊂ M and a solution ( M,g, ∇ φ, ∂ 0 φ ) to the Einstein scalar-field equations such that ι ( M ) = Σ t 0 is a Cauchy hypersurface and such that</text>
<formula><location><page_3><loc_33><loc_67><loc_67><loc_69></location>ι ∗ g =˚ g, ι ∗ k = ˚ k, ι ∗ π =˚ π, and ι ∗ ∂ 0 φ = ˚ ψ 0 .</formula>
<text><location><page_3><loc_12><loc_63><loc_88><loc_66></location>We will perturb around initial data ( M,γ, -γ, 0 , C ) which, after embedding into FLRW spacetime, corresponds to data</text>
<formula><location><page_3><loc_33><loc_61><loc_67><loc_64></location>( M ∼ = Σ t 0 , a ( t 0 ) 2 γ, -˙ a ( t 0 ) a ( t 0 ) γ, 0 , Ca ( t 0 ) -3 )</formula>
<text><location><page_3><loc_12><loc_55><loc_88><loc_61></location>at initial time t 0 . Furthermore, the maximal globally hyperbolic development (MGHD) is unique (up to diffeomorphism), and thus we can assume ( M,g, ∇ φ, ∂ 0 φ ) to be globally hyperbolic. However, these statements provide little information on the properties of the MGHD in the future and past of the initial data slice.</text>
<text><location><page_3><loc_12><loc_31><loc_88><loc_54></location>1.2.2. Strong Cosmic Censorship. In their groundbreaking papers [Haw67, Pen65] on singularity theorems, Hawking and Penrose established very general criteria for the MGHD of spacetimes to become causally geodesically incomplete. Many spacetimes of physical relevance satisfy these criteria, including the spacetimes considered in this article. While giving us more information on the MGHD than the existence and uniqueness results mentioned above, a key issue in the application of this mathematical result to General Relativity is that no statement is made on how precisely the singularity comes about: In particular, such incompleteness (within a given regularity class) could either mean that the geodesic is inextendible - which must be caused by the blow-up of some geometric quantity - or that there exist multiple inequivalent extensions. While the latter behaviour is exhibited even for some cosmological spacetimes (see, for example, the Taub solutions discussed in [CI93]), such behaviour is usually considered to be unphysical since it would imply a breakdown of determinism. The Strong Cosmic Censorship Conjecture (SCCC) posits in its most general form that, for generic solutions to the Einstein equations, this incompleteness instead manifests as inextendibility at a given level of regularity (e.g., C 0 , C 2 , C ∞ , . . . ).</text>
<text><location><page_3><loc_12><loc_20><loc_88><loc_29></location>In certain frameworks in the homogeneous cosmological setting - i.e. for homogeneous initial data on a closed spatial hypersurface -, it was shown in fundamental works by Chrusciel-Rendall [CR95] and Ringstrom [Rin09] that the so called Kretschmann scalar R αβγδ R αβγδ is unbounded where incompleteness manifests. Thus, it is the driving force behind geodesic incompleteness in these cases, forcing C 2 -inextendibility of the MGHD. For the purposes of analyzing cosmologically relevant spacetimes, the SCCC is hence often rephrased as follows:</text>
<text><location><page_3><loc_12><loc_16><loc_88><loc_19></location>Conjecture 1.3 (Cosmological SCCC) . (See e.g. [Rin09, Chapter 17] ) For generic initial data, the Kretschmann scalar is unbounded where causal geodesics become incomplete.</text>
<text><location><page_3><loc_12><loc_10><loc_88><loc_15></location>Theorem 1.1, in short, shows that this conjecture is rigorously supported in the case of FLRW spacetimes with negative spatial curvature. More precisely, the past asymptotics of such spacetimes, determined by initial data on Σ t 0 as discussed above, are generic in the following sense: There exists</text>
<text><location><page_4><loc_12><loc_78><loc_88><loc_90></location>an open neighbourhood of said FLRW data within the set of Einstein scalar-field initial data such that the solutions past direct causal geodesics become incomplete, and the incompleteness is driven by blow-up of Kretschmann scalar with the same asymptotics as the FLRW solution. The global result in Theorem 1.2 portrays the other side of Cosmic Censorship - as with the past evolution, near-FLRW data fully determines the future of the spacetime in the sense that the MGHD is future complete, again showing that this feature of FLRW spacetimes with negative spatial sectional curvature is generic.</text>
<text><location><page_4><loc_12><loc_69><loc_88><loc_77></location>1.2.3. FLRW and generalized Kasner spacetimes with scalar fields. On a large scale, the universe is often viewed as spatially homogenous and isotropic, i.e., no point in space and no direction are distinguishable from any other point and direction. In 1935, it was shown by Robertson and Walker that, under a few very natural additional assumptions, this restricts the class of potential spacetimes to the FLRW class</text>
<formula><location><page_4><loc_36><loc_63><loc_64><loc_69></location>( I × ˜ M, ˜ g FLRW = -dt 2 + a ( t ) 2 ˜ γ ) ,</formula>
<text><location><page_4><loc_12><loc_49><loc_88><loc_54></location>Spatially flat FLRW spacetimes are a subclass of the closely related generalized Kasner spacetimes , which are still spatially homogeneous but anisotropic in general. For scalar field matter, the spacetime metric is given by</text>
<text><location><page_4><loc_12><loc_51><loc_88><loc_64></location>where ( ˜ M, ˜ γ ) is a manifold of constant sectional curvature κ and where the scale factor a depends smoothly on t . This holds before taking the Einstein equations into consideration - when doing so, the matter model determines how space expands within the cosmological model via a . We refer to Lemma 2.3 for the scalar-field solution for κ = -1 9 , but note that the scale factor behaves like t 1 3 for scalar-field matter, regardless of spatial geometry, and that the Kretschmann scalar blows up at order O ( t -4 ) toward the Big Bang ( t ↓ 0).</text>
<formula><location><page_4><loc_15><loc_43><loc_85><loc_48></location>g Kasner = -dt 2 + D ∑ i =1 t 2 p i dx i ⊗ dx i , D ∑ i =1 p i = 1 , D ∑ i =1 p 2 i = 1 -A 2 , φ Kasner ( t ) = A log( t ) .</formula>
<text><location><page_4><loc_12><loc_35><loc_88><loc_43></location>The standard Kasner family is obtained by considering the vacuum case ( A = 0), and the spatially flat FLRW spacetime by setting D = 3 , p i = 1 3 , A = √ 2 3 . If more than one of the Kasner exponents is non-zero, the generalized Kasner family satisfies the SCCC, also by exhibiting Kretschmann scalar blow-up of order t -4 as t ↓ 0 (see [RS18b, (1.8)]).</text>
<text><location><page_4><loc_12><loc_14><loc_88><loc_34></location>Kasner spacetimes are of particular relevance to cosmology due to their relationship with the BKL conjecture : Heuristically, this conjecture states that the dynamics of cosmological spacetimes generically exhibit chaotic and highly oscillatory behaviour, often referred to as 'Mixmaster' behaviour. This behaviour is driven by velocity terms within the Einstein equations and is locally comparable to that of (vacuum) Kasner solutions. However, even if the BKL picture is to be believed in general, scalar-field (or, more generally, stiff-fluid) solutions seem to form an exception to it: They have a dampening effect on said oscillations, thus generating Big Bang stability as shown rigorously in [RS18b, FRS23] for Kasner spacetimes (for more details, see Section 1.3). This scenario, often referred to as quiescient cosmology , was studied in, for example, in [BK73, Bar78, AR01]. With this in mind, both the aforementioned Kasner results and the results within this article, along with the prior FLRW results [RS18b, Spe18], confirm this quiescent effect of scalar fields in cosmology, in contrast to the BKL picture.</text>
<text><location><page_4><loc_12><loc_8><loc_88><loc_12></location>We note that one can view this as a scalar field ensuring a specific scenario in the very early universe given a class of initial data, namely matching the asymptotic behaviour of the Big Bang singularity. This fits into the recent use of nonlinear scalar fields in string cosmology, where specific</text>
<text><location><page_5><loc_12><loc_87><loc_88><loc_90></location>choices of field are made to specific behaviours (e.g., inflation) in the early universe. For a recent review, we refer to [CCM + 23].</text>
<unordered_list>
<list_item><location><page_5><loc_12><loc_74><loc_88><loc_84></location>1.3. Relation to previous work. Theorem 1.2 is the first theorem about the full global structure of FLRW spacetimes with negatively curved spatial geometry. For such solutions, existing results exclusively concern future stability, which we further discuss below. Besides [Spe18] covering the S 3 -case, it is the only open set of initial data for cosmological spacetimes (i.e., without symmetry assumptions) with Λ = 0 and in absence of accelerated expansion for which the global (future and past) dynamics are now fully understood. 3</list_item>
</unordered_list>
<text><location><page_5><loc_12><loc_56><loc_88><loc_72></location>Scalar field matter (and, more generally, semilinear wave equations or fluid matter) and their asymptotic behaviour on fixed cosmological backgrounds have been studied extensively, for example in [AR10, AFF19, Bac19, Rin19, BO20, Rin20, Rin21, Wan21]. While many of the results, in particular [Rin20], manage to analyze very general classes of equations and spacetime geometries, including the wave equation on the FLRW backgrounds studied in [AFF19, FU22], the methods used are often difficult to apply to the full Einstein scalar-field system. In [FU22], we extended the approach of [AFF19] to be able to deal with various warped product spacetimes, and in particular FLRW spacetimes with negatively curved spatial geometry, by using the spatial Laplace operator to control high order derivatives. The perturbation-adapted analogue of this strategy is at the basis of the energy method in this paper.</text>
<text><location><page_5><loc_12><loc_43><loc_88><loc_54></location>We also note that, by the results of [GaNS19], there are non-trivial waves on fixed FLRW backgrounds that converge toward the Big Bang singularity, even if, as demonstrated in [AFF19, FU22], this behaviour is non-generic. Such waves can give rise to convergent asymptotics on cosmological backgrounds as studied in [Rin20]. Thus, it will likely be difficult to replace (1.2c) with an arbitrary non-trivial reference wave while keeping past stability intact. However, by restricting to an open neighbourhood near the solution described in (1.2a)-(1.2c), we exclude this non-generic class of solutions.</text>
<text><location><page_5><loc_12><loc_24><loc_88><loc_41></location>Theorem 1.1 forms the counterpart to the pioneering works by Rodnianski-Speck [RS18a, RS18b] and Speck [Spe18], which cover nonlinear Big Bang stability for FLRW spacetimes with spatial geometry T 3 and S 3 respectively. These results were extended to Kasner spacetimes in [RS22] with | q i | < 1 6 , and to the full subcritical regime in [FRS23], i.e., (generalized) Kasner spacetimes as discussed in Section 1.2.3 with max i,j,k =1 ,...,D p i + p j -p k < 1. The former necessitates considering 1 + D -dimensional Kasner spacetimes with D ≥ 38, while the latter result also can be satisfied in D = 3 for generalized Kasner spacetimes. Recall that this means, in contrast to our setting, that the reference spacetime can be anisotropic, even if it the conditions on Kasner exponents rule out extremely anisotropic regimes. As a result, the analysis therein becomes significantly more involved, especially at top order, since approximately monotonic energy identities as used in our work as well as in [RS18b, Spe18] have not been found in these anisotropic settings.</text>
<text><location><page_5><loc_12><loc_15><loc_88><loc_23></location>We note that the argument in [FRS23] relies on identifying an almost-diagonal structure for the asymptotics of (combined) connection coefficients for an adapted frame that is carried along by Fermi-Walker transport; this is precisely where subcriticality enters. Given that these no longer can vanish in a reference frame adapted to near-hyperbolic spatial geometry, it is unclear whether this structure is sufficiently maintained.</text>
<text><location><page_6><loc_12><loc_69><loc_88><loc_90></location>Furthermore, Beyer and Oliynyk have recently shown in [BO21] that over T 3 , the Big Bang formation can be localized in the sense that data given solely on a ball within the initial hypersurface must also cause stable blow-up on a (smaller) ball on the Big Bang hypersurface. While this result further indicates that blow-up behaviour of near-FLRW spacetimes might be, at least, independent of global geometric properties as it seems to be a localizeable, we note that proof of localized stability crucially relies on the flatness of the conformal reference spacetime. To be more precise, the proof relies on extending the local initial data to global data for a Fuchsian system of metric and matter quantities as well as, again, connection coefficients for an adapted, Fermi-Walker transported frame. However, the derivation of the system for the former explicitly seems to use flat spatial geometry to obtain the necessary Fuchsian form. This forms seems to similarly be broken as soon as the connection coefficients are not perturbed around 0, since this would lead to inhomogeneous error terms of order t -1 for the rescaled variables which are stronger than what the method, so far, accounts for.</text>
<text><location><page_6><loc_12><loc_47><loc_88><loc_67></location>By contrast, in [RS18b, Spe18], the reference frame itself is used in the commutator method to obtain the necessary energy identities at high orders. In all of these works, it hence is a priori unclear how one could extend these methods to the negative spatial Einstein geometry of ( M,γ ). We provide an alternative approach that, besides establishing the complementary stability result to [RS18b, Spe18], does not rely on any information on the spatial geometry of the reference manifold in its methodology (although it is of course relevant in determing the FLRW reference solution that we are studying). Instead, we rely on differential operators adapted to the evolved spatial metric. Hence, we believe that our approach may also prove useful for stability problems in spatially inhomogeneous (and hence also anisotropic) settings. In light of [RS22, FRS23] in particular, the main challenge in achieving this would either be to find approximately monotonic energy identities with our Bel-Robinson approach that have not been observed previously, or to also find ways to circumvent the lack thereof.</text>
<text><location><page_6><loc_12><loc_28><loc_88><loc_46></location>To obtain Theorem 1.1, we use the Laplace-Beltrami-operator (acting, respectively, on scalar functions and tensor fields) with respect to the (rescaled) evolved metric as our commutating operator instead of a fixed reference frame. This, in turn, leads us to replacing the wave-like system for metric and second fundamental form exploited in [RS18b, Spe18] by an evolutionary system in the second fundamental form and Bel-Robinson variables. The latter technique to show stability of near-vacuum solutions dates back to [AM04] and even to the pioneering proof of global stability of Minkowski space in [CK93], which both cover the Einstein vacuum system, as well as to [CK90], where Bel-Robinson variables were introduced to analyse field equations on Minkowski space. It has recently also been applied to the future stability of Milne spacetimes in the massive Einstein Klein-Gordon system by Wang in [Wan19]. As far as we are aware, this method has not yet been applied to solutions that are not near-vacuum or in the context of Big Bang singularity formation.</text>
<text><location><page_6><loc_12><loc_8><loc_88><loc_26></location>Toward the Big Bang, the solutions exhibit asymptotically velocity dominated (AVTD) behaviour in the sense that they behave, to leading order, like solutions to the Einstein scalar-field equations in CMC gauge with zero shift with all terms involving spatial derivatives set to zero (the 'velocity term dominated' (VTD) equations). This behaviour also matches results obtained by studying high regularity solutions (e.g., [AR01]), or related works using Fuchsian methods that prescribe a behaviour at the singularity and then develop it locally, often under additional symmetry assumptions (e.g., [DHRW02, CBIM04, IM02, FL20]). In particular, this asymptotic behaviour leads to the same types of 'Kasner footprint states' as in [RS18a, RS18b]: As one approaches the Big Bang, the rescaled variables converge toward tensor fields on the Big Bang hypersurface that precisely solve the truncated VTD equations. Further, the distance between the footprints of the FLRW and the perturbed solution are controlled by the initial data. For example, the rescaled</text>
<text><location><page_7><loc_12><loc_86><loc_88><loc_90></location>Weingarten map a 3 k a b converges to ( K Bang ) a b on the Big Bang hypersurface, which is close to √ 4 π 3 C I a b , the rescaled FLRW footprint (see (8.3e) and (8.5c)).</text>
<text><location><page_7><loc_12><loc_67><loc_88><loc_84></location>What remains to be considered to obtain Theorem 1.2 is future stability, which we can reduce to future stability of the vacuum solution in the Einstein scalar-field system. This solution, called the Milne spacetime, has been shown to be stable within the set of vacuum solutions - see [AM11] - and a range of other Einstein systems - see, for example , [Wan19, AF20, FW21, FOW21, BF20, BFK19] and related work in lower dimensions, e.g. [AMT97, Mon08, Faj17, Faj20, Mon20]. As such, our contribution to the study of future stability of Milne spacetimes is that we deal with the massless scalar field matter via corrected energy estimates which are inspired by work of Choquet-Bruhat and Moncrief in [CBM01] for vacuum Einstein equations with U (1)-symmetry. Out of the works listed above, only [Wan19, FW21] deal with scalar field matter at all, namely the massive case. These fields exhibit stronger decay toward the future, making the matter components easier to deal with than in our analysis.</text>
<text><location><page_7><loc_12><loc_58><loc_88><loc_66></location>The additional spectral condition is needed to ensure coercivity of the corrected scalar field energy. Numerical work, e.g. [CS99, Ino01], does not suggest that this condition is violated by any closed 3-manifold with constant sectional curvature κ = -1 9 , and verifies that is is satisfied, for example, by an analogue of Weeks space in which the metric is appropriately scaled to have the required sectional curvature. We refer to Remark 9.3 where this discussed in more detail.</text>
<unordered_list>
<list_item><location><page_7><loc_12><loc_52><loc_88><loc_57></location>1.4. Challenges in the proof. The contracting and expanding regimes of near-FLRW spacetime are analyzed in two separate and methodologically independent parts. Before providing an overview of both arguments, we summarize the challenges that arise:</list_item>
<list_item><location><page_7><loc_12><loc_50><loc_87><loc_51></location>1.4.1. Big Bang stability. The main difficulties in establishing Big Bang stability are three-fold:</list_item>
</unordered_list>
<text><location><page_7><loc_12><loc_40><loc_88><loc_48></location>Firstly, we have to expect that the solutions are asymptotically velocity term dominated (as argued in Remark 8.3, we end up proving that this is the case), and thus that rescaled variables at best exhibit the same asymptotic behaviour as their counterparts in FLRW spacetime, up to a small perturbation in the asymptotic footprint. For example, note that, in the reference FLRW spacetime, one has</text>
<formula><location><page_7><loc_39><loc_36><loc_62><loc_40></location>( k FLRW ) i j = -3 ˙ a a I i j ≈ -1 t I i j .</formula>
<text><location><page_7><loc_12><loc_28><loc_88><loc_36></location>At best, the shear ˆ k i j of the perturbed solution then behaves like ε t . In fact, we show that this is the case in (4.2b). This implies that the contraction rescaled metric G ij = a -2 g ij can only be controlled up to O ( t -c √ ε ) (see (4.4c)), since one has ∂ t g ij ≈ -2 g il k l j and thus</text>
<formula><location><page_7><loc_41><loc_26><loc_59><loc_30></location>∂ t G ij ≈ G il ˆ k i j ≈ ε t ∗ G.</formula>
<text><location><page_7><loc_12><loc_18><loc_88><loc_26></location>However, to be able to use the structure of the evolution equations to cancel terms in our energy arguments, we have to work with adapted quantities. For example, we need to use integration by parts with respect to (Σ t , G t ) to cancel high order scalar field terms with help of the (rescaled) wave equation that contains ∆ G , or to obtain elliptic estimates from the lapse equation via the operator ∆ G or from the adapted div-curl-system for Σ arising from the constraint equations.</text>
<text><location><page_7><loc_12><loc_8><loc_88><loc_17></location>As a result, even the rescaled solution variables will diverge at order O ( t -c √ ε ) toward the singularity, so we need to track and control their rate of divergence within the bootstrap argument. This significantly complicates dealing with nonlinear terms, where the bootstrap assumptions often cannot be inserted naively. This in turn makes coercivity of the energies more involved to establish (see Lemma 4.5 and Remark 4.6), since this only holds up to curvature errors that also diverge and</text>
<text><location><page_8><loc_12><loc_88><loc_38><loc_90></location>thus need to be carefully tracked.</text>
<text><location><page_8><loc_12><loc_78><loc_88><loc_86></location>Secondly, and in contrast to [RS18b, Spe18], replacing the wave structure of the geometric evolution in the Einstein equations with our less geometry dependent Bel-Robinson framework seems to lose regularity at first glance: The evolution system for the scalar field energy and the geometric energies can be caricatured as follows, where the superscript for each energy term denotes the number of derivatives:</text>
<formula><location><page_8><loc_14><loc_69><loc_86><loc_78></location>-d dt E ( L ) ( φ, · ) /lessorsimilar ε 1 8 t [ E ( L ) ( φ, · ) + E ( L ) (Σ , · ) ] + . . . -d dt [ E ( L ) (Σ , · ) + E ( L ) ( W, · ) ] + · · · /lessorsimilar ε 1 8 t [ E ( L ) (Σ , · ) + E ( L ) ( W, · ) ] + ε -1 8 t · a 4 E ( L +1) ( φ, · ) + . . .</formula>
<text><location><page_8><loc_12><loc_63><loc_88><loc_68></location>Thus, it seems that we lose derivatives in the scalar field and aren't able to close the argument. This is remedied using the div-curl-system in Σ, see (2.36a) and (2.36b), which yields a weak estimate of the form</text>
<formula><location><page_8><loc_28><loc_61><loc_72><loc_63></location>a 4 E ( L +1) (Σ , · ) /lessorsimilar E ( L ) ( φ, · ) + E ( L ) ( W, · ) + E ( L ) (Σ , · ) + . . . .</formula>
<text><location><page_8><loc_12><loc_58><loc_88><loc_61></location>Combining these estimates to improve the bootstrap assumptions then necessitates an intricately constructed total energy to balance these different types of estimates against one another.</text>
<text><location><page_8><loc_12><loc_45><loc_88><loc_56></location>Finally, given (1.2c), the rescaled time derivative of the scalar field is not small and does not become so toward the Big Bang. This leads to various terms within the core linearized evolutionary system of both matter and geometry that, if estimated naively, could lead to exponential blow-up toward the singularity. When such terms occur in the scalar field energy evolution, this can be dealt with along similar lines as in [RS18b, Spe18], but we incur additional large terms in our geometric evolution that only cancel using the explicit form of the Friedman equations, which we highlight in Lemma 7.1 and its proof.</text>
<text><location><page_8><loc_12><loc_40><loc_88><loc_43></location>1.4.2. Future and global stability. For Milne stability, the canonical Sobolev energies for the scalar field variables, i.e.,</text>
<text><location><page_8><loc_12><loc_33><loc_88><loc_37></location>and higher order analogues, do not obey useful energy estimates. This can be overcome by adding an indefinite correction term of the type</text>
<formula><location><page_8><loc_42><loc_36><loc_58><loc_41></location>∫ M | φ ' | 2 g + |∇ φ | 2 g vol g</formula>
<formula><location><page_8><loc_44><loc_29><loc_57><loc_34></location>∫ M φ ' ( φ -φ )vol g</formula>
<text><location><page_8><loc_12><loc_18><loc_88><loc_29></location>to the canonical energy, see Definition 9.6. This is similar to what was done in [CBM01] in a 2 + 1-dimensional setting, as well as similar to the indefinite terms we introduce in our geometric energy to control the wave system in the metric variables, as in previous work on Milne stability in different matter models, including [AF20, FW21]. That this corrected energy controls Sobolev norms relies on the aforementioned spectral condition. As a result, and unlike for past stability, the specific spatial geometry is crucial in generating decay from energy estimates, even before considering the geometric evolution.</text>
<text><location><page_8><loc_12><loc_8><loc_88><loc_16></location>Moreover, we need to transition from the near-FLRW data used to analyze the contracting regime to data in the expanding regime on a distant enough future hypersurface such that it is near-Milne and the future stability result applies. This requires as a gauge switch from CMC gauge with zero shift to CMCSH gauge, as well as careful control of the solution variables over a finite time interval using continuous dependence on initial data. For the former, close inspection of [FK20]</text>
<text><location><page_9><loc_12><loc_85><loc_88><loc_90></location>gives us a diffeomorphism close to the identity that maps the initial data for the metric to new data satisfying the spatially harmonic gauge condition, thus allowing us to switch gauges without losing proximity to the reference solution. This is discussed in detail in Section 10.</text>
<section_header_level_1><location><page_9><loc_12><loc_82><loc_28><loc_84></location>1.5. Proof outline.</section_header_level_1>
<section_header_level_1><location><page_9><loc_12><loc_80><loc_31><loc_81></location>1.5.1. Big Bang stability.</section_header_level_1>
<text><location><page_9><loc_12><loc_59><loc_88><loc_78></location>The big picture. The key argument in our Big Bang stability proof is a hierarchized series of energy estimates that establishes the asymptotic behaviour of solution variables toward the singularity. We rely on a bootstrap argument which establishes that energies E ( L ) (see Definition 3.9) for the scalar field, the rescaled shear, the Bel-Robinson variables, the lapse and the curvature at worst only diverge slightly. Here, 0 ≤ L ≤ 18 denotes the order of derivatives considered. To this end, we make a bootstrap assumption on the solution norms C (see Definition 3.6) which controls the distance of these rescaled variables as well as the metric itself to their FLRW counterparts in terms of supremum norms with respect to G , where G = a -2 g is the rescaled adapted spatial metric (see Definition 2.9). We refer to Assumption 3.16 and Remark 3.19 for the detailed bootstrap assumptions and improvements, as well as to Lemma 3.14 for the underpinning local well-posedness result. That this bootstrap argument implies Theorem 1.1 follows from a straightfoward adaptation of the arguments in [RS18b, Theorem 15.1].</text>
<text><location><page_9><loc_12><loc_34><loc_88><loc_57></location>We work with evolution-adapted norms even though G ( t, x ) degenerates toward the Big Bang singularity. Indeed, since we need to exploit the structure of the evolutionary equations, it is more convenient to have these adapted quantities controlled by the solution norms H and C directly instead of having to perform changes of metric at that point. Once the improved energy estimates are shown, a (time-scaled) coercivity notion (see Lemma 4.5 and the proof of Corollary 7.3) and Sobolev embeddings with respect to the reference metric γ then ensure that these improved estimates translate to H and C . This then closes the bootstrap. To actually achieve this improved energy behaviour, we derive elliptic energy estimates or integral-type estimates that, once suitably combined and scaled, yield the desired improvements by straightforwardly applying the Gronwall lemma. Additionally, note that we assume that the initial data is close to FLRW data not just in H , which contains precisely the norms needed to control C by Sobolev embedding, but also scaled smallness assumptions at one order higher, contained in the top order semi-norm H top (see Assumption 3.10). This is needed to ensure that the top order energy is small initially, and thus to close the bootstrap.</text>
<text><location><page_9><loc_12><loc_19><loc_88><loc_32></location>Scale factor a(t). The precise structure of the Friedman equations (2.3)-(2.4) is crucial not only to control time integral quantities up to the Big Bang hypersurface (see Lemma 2.4), but also to ensure that certain terms in the evolution that would otherwise cause large divergences contribute with favourable sign (see the arguments in Lemma 6.2 as well as Lemma 7.1). It turns out that the sectional curvature entering the Friedman equations actually is not of key importance to large parts of the Big Bang stability analysis given that the leading order of the scale factor toward the Big Bang singularity is determined via the Friedman equation (1.2b) by the matter term, not the sectional curvature. This indicates that our method might extend to different settings.</text>
<text><location><page_9><loc_12><loc_7><loc_88><loc_18></location>Gauge choice, commutation method and Bel-Robinson variables. We commute the resulting elliptic-hyperbolic Einstein system with the Laplace-Beltrami operator ∆ G with respect to the rescaled evolved spatial metric G ( t, x ) to obtain higher order energy control. Commuting with this operator has the advantage of leaving many integration-by-parts identities intact. These are needed to provide specific cancellations, e.g., to cancel ∆ L 2 +1 φ -terms arising from the wave equation when computing ∂ t E ( L ) ( φ, · ). We also note that the only feature of the adapted metric we</text>
<text><location><page_10><loc_12><loc_85><loc_88><loc_90></location>use is that it is close to γ , and don't use any further information on the geometry, e.g., by choosing a specific reference frame in our commutation method. Further, we employ CMC gauge with zero shift to avoid badly behaved shift terms (see Remark 1.4).</text>
<text><location><page_10><loc_12><loc_70><loc_88><loc_83></location>We still, however, need to deal with the Ricci term in the evolution equation for the second fundamental form. To this end, we consider the Bel-Robinson variables E and B which are Σ t -tangent symmetric tracefree (0 , 2)-tensors and contain all information of the spacetime Weyl tensor W [ g ] (see Subsection 2.4). Suitably projecting the Gauss-Codazzi equations admits additional constraint equations in terms of E and B that allow us to replace the Ricci tensor at the 'cost' of introducing Bel-Robinson energies into the formalism, see (2.24a) and the rescaled version (2.29c). Further, E and B satisfy a Maxwell-type system (see Lemma 2.7) that can be exploited to obtain energy estimates and, as with the other evolution equations, is well adapted to commutation with ∆ G .</text>
<text><location><page_10><loc_12><loc_47><loc_88><loc_68></location>A priori low order C G -control. By applying the bootstrap assumptions on C to the evolution equations, we can immediately deduce improved low order estimates in C l G for l ≥ 10 for the solution variables by inserting them into the respective evolution equations (see Lemma 4.3), as well as via the maximum principle for the lapse (see Lemma 4.1). These usually still diverge slightly, mostly due to the asymptotic behaviour of G . However and crucially to our argument, at order 0, the renormalized time derivative Ψ of the wave, the rescaled tracefree part Σ of the second fundamental form and the rescaled Bel-Robinson variable E are in fact Kε -small in C 0 G on the bootstrap interval (see Lemma 4.2). If these estimates would not hold, this would lead to terms that diverge at order O ( a -3 -c √ ε ) in the differential inequalities, and thus causing exponential energy blow-up of order O ( e a -c √ ε ) that we could no longer control. This behaviour is closely related to the fact that Ψ and Σ converge toward footprint states on the Big Bang hypersurface that remain Kε -small (see (8.3c) and (8.3e)), and then pass this convergence on to E (see (8.8a).</text>
<text><location><page_10><loc_76><loc_46><loc_79><loc_49></location>| | G</text>
<text><location><page_10><loc_12><loc_42><loc_88><loc_45></location>Energy estimates and hierarchy. The main part of the analysis is establishing various energy estimates.</text>
<unordered_list>
<list_item><location><page_10><loc_16><loc_23><loc_88><loc_40></location>· For the lapse (see Section 5), the relevant estimates are direct results of the elliptic lapse equations (2.30a)-(2.30b). The non-lapse terms on the right hand side only diverge slightly toward the Big Bang, in contrast to the divergence at oder a -4 in (2.30a), and thus allows one to show that, at lower derivative order, the lapse converges to 1. However, since the right hand side of (2.30b) contains the scalar curvature of G , this estimate loses derivatives. On the other hand, (2.30a) does not lose derivatives, and the elliptic nature in fact allows one to lapse energies of order L +2 by energies in Σ and the scalar field of oder L . This makes it possible to control the higher order lapse term occuring, for example, in (2.28c), without losing regularity. Conversely, both of these gains in regularity are at the cost of losing powers of a . In short, (2.30b) is needed to establish the asymptotic behaviour of the lapse, and (2.30a) to obtain improved energy bounds as a whole.</list_item>
<list_item><location><page_10><loc_16><loc_16><loc_88><loc_22></location>· The core matter energy estimate (see Lemma 6.2) relies on delicate cancellations when computing the time derivative of E ( L ) ( φ, · ). While we derive this in a fashion that differs from the energy flux method used in [Spe18], the necessary cancellations to arrive at Lemma 6.2 are similar.</list_item>
<list_item><location><page_10><loc_16><loc_8><loc_88><loc_16></location>· The (rescaled) tracefree component of the second fundamental form Σ (see Lemma 6.8) and the (rescaled) Bel-Robinson variables E and B (see Lemma 6.6) need to be treated simultaneously to deal with the leading curvature term in the evolution of the former by inserting a constraint equation in which E occurs as the leading term (see (2.36d)). However, the matter terms within the evolution of E and B contain, firstly, terms where we again</list_item>
</unordered_list>
<text><location><page_11><loc_18><loc_87><loc_88><loc_90></location>need very precise estimates to show that they do not contribute large a -3 -divergences, and, secondly, matter terms that lose one order of derivatives.</text>
<text><location><page_11><loc_18><loc_82><loc_88><loc_86></location>This order of regularity can be regained using the momentum constraint equation (2.36a) and its Bel-Robinson counterpart (2.36b) containing B , which leads to a div-curl-system for Σ (see Lemma 6.10). This is, again, at the cost of losing powers of a .</text>
<unordered_list>
<list_item><location><page_11><loc_16><loc_70><loc_88><loc_81></location>· As a result, the core Gronwall argument performed in Proposition 7.2 combines energies for the matter variables, Σ and the Bel-Robinson variables, as well as energies for Ric[ G ] and additional scalar field quantities to control borderline error terms. In particular, the curvature energies are necessary to handle commutation errors within the energy estimates, and improved bounds on them need to be obtained to apply the coercivity results in Lemma 4.5 - else, none of energy improvements would extend to improved Sobolev norm bounds and the bootstrap argument would not close.</list_item>
<list_item><location><page_11><loc_18><loc_57><loc_88><loc_70></location>As many of the a priori C G -norm estimates add small additional divergences, it is necessary to perform an induction over derivative orders within this mechanism to deal with lower order error terms. Since ∆ G is elliptic, it is sufficient to perform this for even orders. Along with energies at order L ∈ 2 N 0 , the total energy also includes the energy controlling Σ as well as the scalar field and curvature energies at order L +1, appropriately scaled to account for the degenerate elliptic estimate for Σ from Lemma 6.10. This remedies the derivative loss in the Bel-Robinson energy and allows one to improve the total energy at each order until reaching L = 18, at which point the bootstrap argument can be closed.</list_item>
<list_item><location><page_11><loc_16><loc_42><loc_88><loc_57></location>· Note that the metric itself does not enter the core energy mechanism . In fact, trying to replace control of the Ricci tensor by control of G is likely too imprecise in dealing with high order curvature errors. Instead, control of G -γ and Γ[ G ] -ˆ Γ[ γ ] is a consequence of a simple integral energy inequality and the improvements achieved for Σ and matter variables (see Lemma 6.14 and Corollary 7.3). Since we cannot utilize any additional structure in dealing with the metric, we have to construct our argument carefully to allow for the metric control to be weaker than what one gets for the core variables, while still being sufficiently strong to constitute an improvement and allowing to switch between H G and H γ (and, respectively, C G and C γ ) norms.</list_item>
</unordered_list>
<text><location><page_11><loc_12><loc_37><loc_88><loc_41></location>We also point to Remark 6.1 for a more detailed sketch of how the integral inequalities for the core Gronwall argument are structured and how this leads to the bootstrap improvement for the energies.</text>
<unordered_list>
<list_item><location><page_11><loc_12><loc_18><loc_88><loc_34></location>1.5.2. Future stability and connecting the regions. We follow similar lines as in [AF20, FW21] to prove that near-FLRW spacetimes in negative spatial geometry are future stable. Since ∂ t φ decays like a -3 /similarequal t -3 in the reference spacetime, the sectional curvature becomes dominant in the Friedman equations and the scale factor approaches that of Milne spacetime as t approaches ∞ . Hence, if one moves sufficiently far to future, choosing near-FLRW data with a homogeneous scalar field is equivalent to choosing near-vacuum data. Thus, what we prove first in Section 9 is future stability of near-Milne spacetimes under the Einstein scalar-field system. Once this is established, we argue in Section 10 how early near-FLRW initial data evolves to data that is sufficiently close to Milne for large enough times, which is essentially a consequence of the scale factor and the (physical) mean curvature approaching that of Milne, up to a multiplicative constant.</list_item>
</unordered_list>
<text><location><page_11><loc_12><loc_11><loc_88><loc_16></location>In terms of dealing with geometric and elliptic estimates, we can essentially carry over the results of [AF20], as was also done in [FW21], by working in CMCSH gauge and verifying that the matter components are indeed only perturbative terms within the geometric evolution.</text>
<text><location><page_11><loc_12><loc_8><loc_88><loc_11></location>This leaves only the scalar field to be examined. Here, we introduce corrective terms to the energies (see Definition 9.6) which yield decay estimates for the corrected scalar field energy (see</text>
<text><location><page_12><loc_12><loc_87><loc_88><loc_90></location>Lemma 9.16 and Lemma 9.17). That these energies are coercive (see Lemma 9.12 and Lemma 9.13) requires the aforementioned lower bound for the first positive eigenvalue of ∆ γ .</text>
<text><location><page_12><loc_12><loc_74><loc_88><loc_86></location>Remark 1.4 (Why not use CMCSH gauge to prove Big Bang stability?) . One might consider applying this gauge to Big Bang stability as well since this is precisely the choice of gauge turning the geometric evolution into a wave-like system in ( g, k ), which seems simpler than our chosen approach in CMC gauge with zero shift. In particular, this would also not rely on any choice of reference frame, and keep the wave structure of the geometric evolution intact, unlike when using Bel-Robinson variables. However, the issue with this approach lies in the shift equation, which would take the following form for the rescaled shift vector X = a 3 ˜ X :</text>
<text><location><page_12><loc_12><loc_52><loc_88><loc_65></location>As a result, the first term has to be expected to diverge at the same rate as the metric, i.e., we expect even low order norms of ˜ X to behave like a -3 -c √ ε at best up to small prefactors. However, computing the time derivative of an integral over | G -γ | 2 G (or derivatives thereof) becomes the integral over the ( ∂ t - L ˜ X )-derivative of this quantity, and hence we get explicit terms of the form L ˜ X γ which always exist at highest order and diverge worse than t -1 . In short, the fact that the metric cannot be expected to converge to a footprint state leads to leading order terms in the differential energy estimates to carry strongly divergent pre-factors in CMCSH gauge. This obstructs improvements in a tentative bootstrap argument.</text>
<formula><location><page_12><loc_12><loc_65><loc_77><loc_75></location>∆ G X l +Ric[ G ] l m X m = -2( N +1)( G -1 ) im ( G -1 ) jn Σ ij ( Γ l mn -ˆ Γ l mn ) (1.4) +2( G -1 ) im ∇ i X n ( Γ l mn -ˆ Γ l mn ) + 〈 error terms in lapse and matter 〉</formula>
<section_header_level_1><location><page_12><loc_12><loc_49><loc_28><loc_51></location>1.6. Paper outline.</section_header_level_1>
<unordered_list>
<list_item><location><page_12><loc_16><loc_46><loc_57><loc_49></location>· Sections 2-8 cover the proof of Big Bang stability:</list_item>
<list_item><location><page_12><loc_20><loc_44><loc_88><loc_47></location>-In Section 2, we introduce notation and provide the necessary information on the FLRW background solution as well as the equations relevant to the subsequent analysis.</list_item>
<list_item><location><page_12><loc_20><loc_41><loc_88><loc_44></location>-Then, in Section 3, we discuss the solution norms and energies and state the initial data and bootstrap assumptions.</list_item>
<list_item><location><page_12><loc_20><loc_36><loc_88><loc_40></location>-In Section 4, improved low order C G -norm estimates that follow directly from the bootstrap assumptions are established, along with additional formulas and a priori estimates.</list_item>
<list_item><location><page_12><loc_20><loc_34><loc_63><loc_36></location>-Section 5 concerns the elliptic estimates for the lapse.</list_item>
<list_item><location><page_12><loc_20><loc_28><loc_88><loc_34></location>-In Section 6, we discuss the energy and Sobolev norm estimates for all other variables, all of which are integral estimates except for the aforementioned elliptic estimate for Σ, as well as a norm bound for ∇ φ that is not needed for the energy improvement.</list_item>
<list_item><location><page_12><loc_20><loc_23><loc_88><loc_26></location>-In Section 8, we show how this bootstrap argument implies the main Big Bang stability result (see Theorem 8.2, which is the formal version of Theorem 1.1).</list_item>
<list_item><location><page_12><loc_20><loc_25><loc_88><loc_29></location>-These are all combined in Section 7 to improve the bootstrap assumptions - first for the energies, then for H and finally C .</list_item>
<list_item><location><page_12><loc_16><loc_20><loc_63><loc_22></location>· Section 9 contains the proof of near-Milne future stability.</list_item>
<list_item><location><page_12><loc_16><loc_18><loc_88><loc_21></location>· In Section 10, we show that this is sufficient for future stability of near-FLRW spacetimes, proving Theorem 1.2.</list_item>
<list_item><location><page_12><loc_16><loc_15><loc_88><loc_18></location>· The appendices (Sections 11-12) collect various basic formulas and commutator expressions as well as error terms and how these can be estimated.</list_item>
</unordered_list>
<text><location><page_12><loc_12><loc_8><loc_88><loc_14></location>Acknowledgements. D.F. acknowledges support by the Austrian Science Fund through the project Relativistic Fluids in cosmology (project number P34313). L.U. acknowledges the support by the START-Project Isoperimetric study of initial data for the Einstein equations (project number Y963-N35), also by the Austrian Science Fund. L.U. is a recipient of a DOC Fellowship of the</text>
<text><location><page_13><loc_12><loc_82><loc_88><loc_90></location>Austrian Academy of Sciences at the Faculty of Mathematics at the University of Vienna. L.U. also thanks the German Academic Scholarship Foundation (Studienstiftung des deutschen Volkes) for their scholarship. The authors thank Klaus Kroncke and Roman Prosanov for their help in seeking out numerical evidence for the spectral condition used in Section 9, and Michael Eichmair and the anonymous referees for their extensive feedback on a previous version of this manuscript.</text>
<section_header_level_1><location><page_13><loc_34><loc_77><loc_67><loc_79></location>2. Big Bang stability: Preliminaries</section_header_level_1>
<section_header_level_1><location><page_13><loc_12><loc_75><loc_24><loc_76></location>2.1. Notation.</section_header_level_1>
<text><location><page_13><loc_12><loc_60><loc_88><loc_72></location>2.1.1. Foliations. On a spacetime manifold ( M,g ), we assume the existence of a spacelike Cauchy hypersurface Σ t 0 that is diffeomorphic to M . As argued in Remark 8.1, we can assume without loss of generality that it has constant mean curvature. We will ultimately show that there exists a time function t such that the past of Σ t 0 = t -1 ( t 0 ) can be foliated by Σ s = t -1 ( s ) for s ∈ (0 , t 0 ), and that where the solution exists, this is at least possible up to some T ∈ (0 , t 0 ). These constant time surfaces are then also spacelike Cauchy hypersurfaces diffeomorphic to M and CMC. We will use this notation throughout with little comment and often simply view Σ s as { s } × M .</text>
<text><location><page_13><loc_12><loc_58><loc_66><loc_59></location>2.1.2. Metrics. The spacetime metric g on M takes the general form</text>
<formula><location><page_13><loc_41><loc_54><loc_59><loc_56></location>g = -n 2 dt 2 + g ab dx a dx b</formula>
<text><location><page_13><loc_12><loc_44><loc_88><loc_53></location>where n ≡ n ( t, x ) is the lapse function and g | Σ t ≡ g | Σ t ( t, x ) is a Riemannian metric on Σ t . We will often simply denote the spatial metric by g . Furthermore, we denote the rescaled spatial metric byG ij = a -2 g ij (see Definition 2.9) and the tensor-field induced by the matrix inverse of ( G ij ) by G -1 . Similarly, det g and det G are also meant as the determinants in the matrix sense. Finally, we define vol g and µ g as the volume form and volume element with regards to g , and the same for γ and G .</text>
<text><location><page_13><loc_12><loc_27><loc_88><loc_41></location>2.1.3. Indices and coordinates. Greek indices α, β, . . . , µ, ν, . . . run from 0 to 3, lowercase latin indices a, b, . . . , i, j, . . . from 1 to 3. The spatial indices on some coordinate neighbourhood V ⊆ M are always with regards to the local frame induced by coordinates ( x 1 , x 2 , x 3 ) on M , applied to each V ∩ Σ t by the standard embedding where this intersection is non-empty. The index 0 always denotes components relative to ∂ 0 = n -1 ∂ t , where ∂ t is the derivative associate to the time function t . The Levi-Civita connections associated to g , respectively g and G , are denoted by ∇ , respectively ∇ . 4 Additionally, for the hyperbolic spatial reference metric γ on M (see Definition 2.1), we write the Levi-Civita connection as ˆ ∇ .</text>
<text><location><page_13><loc_12><loc_20><loc_88><loc_27></location>Whenever we raise or lower Greek (resp. Latin) indices without additional notation, it is with regards to g (resp. g ). When we raise indices of a tensor T with regards to the rescaled spatial metric G , we flag this by writing T /sharp . We never raise or lower with respect to γ . Refer to Section 2.1.9 as to how we distinguish taking multiple covariant derivatives from index raising.</text>
<text><location><page_13><loc_12><loc_13><loc_88><loc_18></location>2.1.4. Σ t -tangent tensors. For any Σ t -tangent tensor ξ α 1 ...α r β 1 ...β s , we write ξ ( t ) a 1 ...a r b 1 ...b r for the g -orthogonal projection of ξ onto the hypersurface Σ t . When clear from context, we will drop the time dependency in notation.</text>
<text><location><page_14><loc_12><loc_87><loc_88><loc_90></location>2.1.5. Sign conventions. Within this paper, the second fundamental form with regards to Σ t is defined as the (0 , 2)-tensor k given by</text>
<formula><location><page_14><loc_40><loc_83><loc_60><loc_86></location>k ( X,Y ) = -g ( ∇ X ∂ 0 , Y ) ,</formula>
<text><location><page_14><loc_12><loc_82><loc_82><loc_83></location>where X and Y are Σ t -tangent vectors. The Riemann curvature tensor of g is taken to be</text>
<formula><location><page_14><loc_35><loc_78><loc_65><loc_81></location>∇ α ∇ β Z γ -∇ β ∇ α Z γ = Riem[ g ] αβγ δ Z δ</formula>
<text><location><page_14><loc_12><loc_75><loc_88><loc_78></location>for the covariant vector field ( Z µ ), and the analogous convention holds for all other Riemann curvature tensors that appear.</text>
<text><location><page_14><loc_12><loc_66><loc_88><loc_74></location>2.1.6. Constants. For two nonnegative scalar functions ζ 1 , ζ 2 , we write ζ 1 /lessorsimilar ζ 2 if and only if there exists a constant K > 0 such that ζ 1 ≤ Kζ 2 . This implicit constant may depend on information from the FLRW reference solution at the starting point of the evolution (in particular on γ and a ( t 0 ), see Definition 2.1) and combinatorial quantities. We extend this notation to a real function ζ ' 1 by</text>
<text><location><page_14><loc_12><loc_61><loc_65><loc_63></location>Additionally, we write ζ 1 /similarequal ζ 2 if and only if ζ 1 /lessorsimilar ζ 2 /lessorsimilar ζ 1 is satisfied.</text>
<formula><location><page_14><loc_39><loc_61><loc_61><loc_66></location>ζ ' 1 /lessorsimilar ζ 2 : ⇔ max ( ζ ' 1 , 0 ) /lessorsimilar ζ 2 .</formula>
<text><location><page_14><loc_12><loc_52><loc_88><loc_59></location>2.1.7. Tensor contractions. We denote by ε αβγδ the Levi Civita tensor with regards to g and define Levi Civita tensor on spatial hypersurfaces Σ t by ε [ g ] ijk = ε 0 ijk . Notice that this corresponds to the Levi Civita tensor associated to g . Further, ε [ G ] ijk = a -3 ε [ g ] ijk is the Levi Civita tensor with respect to the rescaled metric G (see (2.27a)).</text>
<text><location><page_14><loc_12><loc_49><loc_88><loc_52></location>For Σ t -tangent (0 , 2)-tensors A, ˜ A and vector field v , we define the following objects as in [AM04, Section A.2]:</text>
<formula><location><page_14><loc_38><loc_47><loc_52><loc_48></location>˜ a ˜ b ˜</formula>
<formula><location><page_14><loc_22><loc_30><loc_78><loc_48></location>A · A = A b A a = 〈 A, A 〉 g ( A /circledot g ˜ A ) ij = A ik ˜ A k j ( A ∧ ˜ A ) i = ε i jp A j q ˜ A qp ( v ∧ A ) ab = ε a cd v c A db + ε b cd v c A ad ( A × ˜ A ) ij = ε i ab ε j pq A ap ˜ A bq + 1 3 ( A · ˜ A ) g ij -1 3 ( tr g A · tr g ˜ A ) g ij (curl A ) ij = (curl g A ) ij = 1 2 [ ε i cd ∇ d A cj + ε j cd ∇ d A ci ] (div g A ) i = ∇ b A ib</formula>
<text><location><page_14><loc_12><loc_26><loc_88><loc_29></location>The operations /circledot G , 〈· , ·〉 G and div G are defined analogously, with all indices raised and lowered by G instead of g . Finally, for two (0 , 1)-tensors π, ˜ π , we denote their symmetrized product by</text>
<formula><location><page_14><loc_39><loc_22><loc_61><loc_26></location>( π ⊗ ˜ π ) ij = 1 2 ( π i ˜ π j + π j ˜ π i ) .</formula>
<text><location><page_14><loc_12><loc_21><loc_62><loc_22></location>For pointwise estimates of these quantities, refer to Lemma 11.3.</text>
<text><location><page_14><loc_12><loc_13><loc_88><loc_19></location>2.1.8. Schematic term notation. We will denote as T 1 ∗· · ·∗ T l , where T i are Σ t -tangent tensors, any multiple of ( T i ), with regards to the rescaled adapted spatial metric G or as standard multiplication if no summation over indices occurs between factors. Constant prefactors and contractions with regards to G are also suppressed in this notation.</text>
<text><location><page_14><loc_12><loc_7><loc_88><loc_13></location>When working with terms where such notation is used, we will estimate these inner products by /lessorsimilar ∏ l i =1 | T i | G , making any constant in front irrelevant, and further we can view any contraction with regards to G as a product of the non-contracted tensor T with G or G -1 , and estimate that</text>
<text><location><page_15><loc_12><loc_86><loc_62><loc_91></location>up to constant by | G | G | T | G , where the first factor is simply √ 3. For similar products with respect to γ , we denote them by ∗ γ .</text>
<unordered_list>
<list_item><location><page_15><loc_12><loc_79><loc_88><loc_85></location>2.1.9. On multiple derivatives of variables. For a scalar function ζ , an ( r, s )-tensor field T and capitalized integers I, J, . . . ∈ N 0 , we denote by ∇ I ζ and ∇ I T the tensors ∇ l 1 . . . ∇ l I ζ and ∇ l 1 . . . ∇ l I T i 1 ...i r j 1 ...j s . We extend this notation to other covariant derivatives analogously. To avoid potential ambiguity with an index raised by g , we will apply the following convention:</list_item>
<list_item><location><page_15><loc_16><loc_75><loc_88><loc_78></location>· If a covariant derivative carries an uppercase letter, a formula with more than one symbol or a positive integer in its superscript, this refers to taking a derivative of that order.</list_item>
<list_item><location><page_15><loc_16><loc_72><loc_88><loc_75></location>· If a covariant derivative carries a lowercase letter or 0 in its superscript, this refers to an index.</list_item>
</unordered_list>
<text><location><page_15><loc_12><loc_68><loc_88><loc_71></location>Further, we will only apply this notation where the precise distribution of indices is not important (e.g. in schematic notation, see Section 2.1.8).</text>
<unordered_list>
<list_item><location><page_15><loc_12><loc_61><loc_88><loc_67></location>2.2. FLRW spacetimes and the Friedman equations. Herein, we collect the properties of the reference FLRW solution to the Einstein scalar-field system in CMC-transported coordinates. Our main focus will lie on the behaviour of the scale factor as determined by the Friedman equations. Before moving on to that, we collect the information on the spatial geometry we will need:</list_item>
</unordered_list>
<text><location><page_15><loc_12><loc_54><loc_88><loc_59></location>Definition 2.1 (Hyperbolic reference geometry) . ( M,γ ) is a three-dimensional, connected, closed, orientable Riemannian manifold with constant sectional curvature -1 9 , hence Ricci tensor Ric[ γ ] = -2 9 γ ij and scalar curvature R [ γ ] = -2 3 .</text>
<text><location><page_15><loc_12><loc_46><loc_88><loc_52></location>Remark 2.2 (Orientability is not a restriction) . We assume that M is orientable for the sake of simplicity. If M should be non-orientable, we may pass the initial data to the oriented double cover and solve the problem there. Since the result is equivariant with respect to the double covering map, this then solves the original problem.</text>
<text><location><page_15><loc_12><loc_42><loc_88><loc_45></location>With this in hand, we can express our classical family of solutions to the Einstein scalar-field system as follows:</text>
<text><location><page_15><loc_12><loc_37><loc_88><loc_41></location>Lemma 2.3 (FLRW solutions and Friedman equations) . Consider FLRW spacetimes ( M,g FLRW ) with M = (0 , ∞ ) × M , where ( M,γ ) is as in Definition 2.1 and where</text>
<text><location><page_15><loc_12><loc_31><loc_88><loc_34></location>holds for some a ∈ C ∞ ((0 , ∞ )) , with the conventions a (0) = 0 and ˙ a ( T ) > 0 for some arbitrary T > 0 . Further, choose a (smooth) scalar function φ FLRW such that the following holds:</text>
<formula><location><page_15><loc_12><loc_34><loc_59><loc_37></location>(2.1) g FLRW = -dt 2 + a ( t ) 2 γ</formula>
<formula><location><page_15><loc_12><loc_28><loc_74><loc_30></location>(2.2) ∂ t φ FLRW = C · a ( t ) -3 , ∇ φ FLRW = 0 , /square g FLRW φ FLRW = 0</formula>
<text><location><page_15><loc_12><loc_25><loc_88><loc_28></location>Such a pair ( g FLRW , φ FLRW ) solves the Einstein scalar-field system (1.1a) -(1.1b) on M if and only if a satisfies the Friedman equation</text>
<formula><location><page_15><loc_12><loc_20><loc_58><loc_25></location>(2.3) ˙ a = √ 1 9 + 4 π 3 C 2 a -4 .</formula>
<text><location><page_15><loc_12><loc_18><loc_29><loc_20></location>In particular, one has</text>
<formula><location><page_15><loc_12><loc_14><loc_57><loc_18></location>(2.4) a = -8 π 3 C 2 a -5 .</formula>
<text><location><page_15><loc_12><loc_8><loc_88><loc_11></location>Proof. The first statement follows from explicitly computing Ric[ g ] as in [O'N83, p.345]. That (2.3) implies (2.4) follows simply by computing the derivative of ˙ a 2 . /square</text>
<text><location><page_16><loc_12><loc_87><loc_88><loc_90></location>In the subsequent analysis, the following properties of a that follow from (2.3) will be crucial for our analysis:</text>
<text><location><page_16><loc_12><loc_78><loc_88><loc_85></location>Lemma 2.4 (Scale factor analysis) . Let a solve (2.3) with a (0) = 0 . Then a is analytic on (0 , ∞ ) and extends to a continuous function on [0 , ∞ ) with a ( t ) /similarequal t 1 3 being satisfied near t = 0 . Further, for any p > 0 , there exist constants c > 0 and K p > 0 , where c is independent of p and K p depends analytically on p , such that, for any t ∈ ( t, t 0 ] , one has</text>
<formula><location><page_16><loc_12><loc_73><loc_64><loc_79></location>(2.5) exp ( p ∫ t 0 t a ( s ) -3 ds ) ≤ K p a ( t ) -cp</formula>
<text><location><page_16><loc_12><loc_72><loc_15><loc_73></location>and</text>
<formula><location><page_16><loc_12><loc_67><loc_70><loc_72></location>(2.6) ∫ t 0 t a ( s ) -3 -p ds ≤ 1 p a ( t ) -p , ∫ t 0 t a ( s ) -3+ p ds /lessorsimilar 1 p .</formula>
<text><location><page_16><loc_12><loc_64><loc_88><loc_67></location>Moreover, for any t ∈ (0 , t 0 ] and any q > 0 , there exist constants c > 0 and K > 0 which both are independent of q such that one has</text>
<formula><location><page_16><loc_12><loc_58><loc_61><loc_63></location>(2.7) ∫ t 0 t a ( s ) -3 ds ≤ K q a ( t ) -cq .</formula>
<text><location><page_16><loc_12><loc_57><loc_32><loc_58></location>Finally, (2.3) also implies</text>
<formula><location><page_16><loc_12><loc_51><loc_56><loc_57></location>(2.8) √ 4 π 3 Ca -2 ≤ ˙ a</formula>
<text><location><page_16><loc_12><loc_47><loc_88><loc_50></location>Remark 2.5. We will use the estimates in Lemma 2.4 where p is a positive power of ε up to algebraic constants. Then, we can simply replace K p in (2.5) by a uniform constant.</text>
<text><location><page_16><loc_12><loc_43><loc_88><loc_46></location>Proof. For the first statement, we refer to [FU22, Lemma 2.1] with γ = 2. We also collect from there 5 that, for t < t 0 ,</text>
<text><location><page_16><loc_12><loc_35><loc_88><loc_38></location>is satisfied. Hence, there exists some c ' > 0 such that the following holds (with t < 1 without loss of generality, see Footnote 5):</text>
<formula><location><page_16><loc_40><loc_38><loc_61><loc_43></location>∫ t 0 t a ( s ) -3 ds /lessorsimilar 1 + | log( t ) |</formula>
<formula><location><page_16><loc_25><loc_30><loc_75><loc_35></location>exp ( p ∫ t 0 t a ( s ) -3 ds ) ≤ exp( c ' · p log( t 0 )) · exp( -c ' · p ) ≤ K p t -c ' p</formula>
<text><location><page_16><loc_12><loc_27><loc_75><loc_30></location>(2.5) follows by applying a ( t ) /similarequal t 1 3 . Noting that a -3 /similarequal ˙ a / a holds, one further has</text>
<text><location><page_16><loc_12><loc_18><loc_88><loc_22></location>and the other inequality in (2.6) follows analogously. Finally, (2.7) follows directly from (2.6) when assuming without loss of generality that a | ( t,t 0 ) only takes values in (0 , 1).</text>
<text><location><page_16><loc_87><loc_17><loc_88><loc_19></location>/square</text>
<formula><location><page_16><loc_12><loc_21><loc_77><loc_27></location>(2.9) ∫ t 0 t a ( s ) -3 -p ds /lessorsimilar ∫ a ( t 0 ) a ( t ) y -1 -p dy = 1 p ( a ( t ) -p -a ( t 0 ) -p ) ≤ 1 p a ( t ) -p ,</formula>
<text><location><page_17><loc_12><loc_87><loc_88><loc_90></location>2.3. Solutions to the Einstein scalar-field equations in CMC gauge. From here on out, we impose the CMC condition 6</text>
<formula><location><page_17><loc_12><loc_82><loc_60><loc_86></location>(2.10) k l l ( t, · ) = τ ( t ) = -3 ˙ a ( t ) a ( t ) .</formula>
<text><location><page_17><loc_12><loc_80><loc_74><loc_81></location>We use (2.3) and (2.4) to collect the following formulas for the mean curvature:</text>
<formula><location><page_17><loc_12><loc_76><loc_57><loc_79></location>∂ t τ = 12 πC 2 a -6 + 1 3 a -2 (2.11)</formula>
<formula><location><page_17><loc_12><loc_72><loc_62><loc_76></location>τ 2 = 9 ˙ a 2 a 2 = 12 πC 2 a -6 + a -2 . (2.12)</formula>
<text><location><page_17><loc_12><loc_70><loc_57><loc_72></location>We consequently define the trace-free component ˆ k of k as</text>
<formula><location><page_17><loc_12><loc_66><loc_56><loc_70></location>(2.13) ˆ k ij = k ij -τ 3 g ij</formula>
<text><location><page_17><loc_12><loc_64><loc_71><loc_66></location>and recall that the future directed unit normal to our foliation is written as</text>
<formula><location><page_17><loc_12><loc_62><loc_55><loc_64></location>(2.14) ∂ 0 = n -1 ∂ t .</formula>
<text><location><page_17><loc_15><loc_59><loc_81><loc_61></location>With this, we can express the Einstein scalar-field equations in our gauge as follows:</text>
<text><location><page_17><loc_12><loc_53><loc_88><loc_58></location>Proposition 2.6 (The Einstein scalar-field system in CMC gauge) . A pair ( g, φ ) solves the Einstein scalar-field equations (1.1a) -(1.1c) on I × M in CMC gauge (2.10) for some interval I ⊆ (0 , t 0 ] , where the scale factor satisfies (2.3) , if and only if the following equations are satisfied on I × M :</text>
<text><location><page_17><loc_12><loc_52><loc_39><loc_53></location>The metric evolution equations</text>
<formula><location><page_17><loc_12><loc_48><loc_55><loc_51></location>∂ t g ij = -2 nk ij = -2 n ˆ k ij +2 n ˙ a a g ij , (2.15a)</formula>
<formula><location><page_17><loc_34><loc_41><loc_67><loc_44></location>+4 πC 2 a -6 ( n -1) g ij + 1 9 (3 n -1) a -2 g ij ,</formula>
<formula><location><page_17><loc_12><loc_43><loc_74><loc_49></location>∂ t ˆ k ij = -∇ i ∇ j n + n [ Ric[ g ] ij -˙ a a ˆ k ij -2 ˆ k il ˆ k l j -8 π ∇ i φ ∇ j φ ] (2.15b)</formula>
<text><location><page_17><loc_12><loc_39><loc_59><loc_40></location>the Hamiltonian and momentum constraint equations</text>
<formula><location><page_17><loc_12><loc_35><loc_67><loc_37></location>R [ g ] + τ 2 ˆ k, ˆ k g =8 π ∂ 0 φ 2 + φ 2 g , (2.16a)</formula>
<text><location><page_17><loc_12><loc_33><loc_17><loc_34></location>(2.16b)</text>
<text><location><page_17><loc_12><loc_30><loc_27><loc_32></location>the lapse equation</text>
<formula><location><page_17><loc_12><loc_25><loc_81><loc_30></location>∆ g n = -12 πC 2 a -6 -1 3 a -2 + n [ 1 3 a -2 +4 πC 2 a -6 + 〈 ˆ k, ˆ k 〉 g +8 π | ∂ 0 φ | 2 ] , (2.17a)</formula>
<text><location><page_17><loc_12><loc_24><loc_32><loc_25></location>or equivalently by (2.16a)</text>
<formula><location><page_17><loc_12><loc_18><loc_77><loc_23></location>(2.17b) ∆ g n = -12 πC 2 a -6 -1 3 a -2 + n [ R [ g ] -8 π |∇ φ | 2 g +12 πC 2 a -6 + a -2 ] ,</formula>
<text><location><page_17><loc_12><loc_18><loc_31><loc_19></location>and the wave equation</text>
<formula><location><page_17><loc_12><loc_14><loc_70><loc_17></location>(2.18) /square g φ = -∂ 2 0 φ + n -1 g ij ∇ i n ∇ j φ +∆ g φ + τ∂ 0 φ = 0 .</formula>
<text><location><page_17><loc_12><loc_11><loc_88><loc_14></location>Proof. These are standard equations that follow from [Ren08, Chapter 2.3] and applying (2.10)(2.12). /square</text>
<formula><location><page_17><loc_45><loc_32><loc_63><loc_35></location>∇ l ˆ k lj = -8 π ∇ j φ · ∂ 0 φ,</formula>
<formula><location><page_17><loc_39><loc_33><loc_66><loc_38></location>2 3 -〈 〉 [ | | |∇ | ]</formula>
<text><location><page_18><loc_12><loc_87><loc_88><loc_90></location>2.4. Bel-Robinson variables. In this subsection, we briefly (re-)establish Bel-Robinson variables and how they behave within the Einstein scalar-field system.</text>
<text><location><page_18><loc_12><loc_82><loc_88><loc_85></location>Recall that the Weyl tensor W ≡ W [ g ] is the trace-free component of the spacetime curvature and, in the Einstein-scalar-field system, takes the form</text>
<formula><location><page_18><loc_12><loc_68><loc_87><loc_81></location>W αβγδ =Riem[ g ] αβγδ -P [ g ] αβγδ , P [ g ] αβγδ = 1 2 ( g αγ Ric[ g ] βδ -g βγ Ric[ g ] αδ -g αδ Ric[ g ] γβ + g βδ Ric[ g ] αγ ) -1 6 R [ g ] ( g αγ g βδ -g αδ g βγ ) =4 π ( g αγ ∇ β φ ∇ δ φ -g βγ ∇ α φ ∇ δ φ -g αδ ∇ β φ ∇ γ φ + g βδ ∇ α φ ∇ γ φ ) -4 π 3 ( ∇ ρ φ ∇ ρ φ ) ( g αγ g βδ -g αδ g βγ ) . We define the dual W ∗ of the Weyl tensor as</formula>
<formula><location><page_18><loc_41><loc_64><loc_60><loc_68></location>W ∗ αβγδ = 1 2 ε αβµν W µν γδ .</formula>
<text><location><page_18><loc_12><loc_61><loc_88><loc_64></location>The electric and magnetic components of the Weyl tensor, referred to as the Bel-Robinson variables from here on, are now defined as</text>
<formula><location><page_18><loc_24><loc_58><loc_76><loc_60></location>E ( W ) αβ = W αµβν ∂ µ 0 ∂ ν 0 = W α 0 β 0 , B ( W ) αβ = W ∗ αµβν ∂ µ 0 ∂ ν 0 = W ∗ α 0 β 0 .</formula>
<text><location><page_18><loc_12><loc_55><loc_88><loc_58></location>We note that, conversely, the Weyl tensor can be fully reconstructed from E and B since the following identities hold:</text>
<formula><location><page_18><loc_12><loc_51><loc_73><loc_54></location>(2.19) W a 0 c 0 = E ac , W abc 0 = -ε ab m B mc , W abcd = -ε abi ε cdj E ij</formula>
<text><location><page_18><loc_12><loc_47><loc_88><loc_51></location>By the symmetries of the Weyl tensor as a whole, E and B are symmetric and one has E 0 β = 0 = B 0 β . Hence, E and B are symmetric, tracefree Σ t -tangent (0 , 2)-tensors which we shall simply denote as E ij and B ij .</text>
<text><location><page_18><loc_12><loc_45><loc_26><loc_47></location>Further, we define</text>
<text><location><page_18><loc_12><loc_43><loc_16><loc_44></location>(2.20)</text>
<formula><location><page_18><loc_35><loc_42><loc_65><loc_45></location>J βγδ := ∇ α W αβγδ , J ∗ βγδ := ∇ α W ∗ αβγδ .</formula>
<text><location><page_18><loc_12><loc_40><loc_72><loc_42></location>By applying the Bianchi identity for Riem[ g ], we gain the explicit expression</text>
<formula><location><page_18><loc_12><loc_35><loc_77><loc_40></location>(2.21) J βγδ = 1 2 ( ∇ γ Ric[ g ] δβ -∇ δ Ric[ g ] γβ ) -1 12 ( g βδ ∇ γ R [ g ] -g βγ ∇ δ R [ g ] ) . Using (1.1a), we collect:</formula>
<text><location><page_18><loc_12><loc_19><loc_88><loc_24></location>Note that expressions containing ∇ α φ ∇ α φ can be ignored throughout our analysis since they are either pure trace or antisymmetric and thus will cancel in inner products with E , B , ˆ k and their rescaled analogues.</text>
<formula><location><page_18><loc_12><loc_23><loc_76><loc_35></location>J i 0 j =4 π [ ∇ i ( ∂ 0 φ ) ∇ j φ + k l i ∇ l φ ∇ j φ -∂ 0 φ ∇ i ∇ j φ (2.22) -k ij ( ∂ 0 φ ) 2 -n -1 ∇ i n · ∇ j φ · ∂ 0 φ ] -2 π 3 [ ∂ 0 ( ∇ α φ ∇ α φ ) ] g ij J ∗ i 0 j =4 π ε lmj ( ∇ l ∇ i φ + k l i ∂ 0 φ ) ∇ m φ + 2 π 3 ε imj ∇ m ( ∇ α φ ∇ α φ ) (2.23)</formula>
<text><location><page_18><loc_12><loc_17><loc_52><loc_19></location>The Bel-Robinson variables then behave as follows:</text>
<text><location><page_18><loc_12><loc_12><loc_88><loc_16></location>Lemma 2.7 (Constraint and evolution equations for Bel-Robinson variables) . If ( g, φ ) is a classical solution to the Einstein scalar-field system (1.1a) -(1.1b) in CMC gauge (see (2.10) ), E and B satisfy the following constraint equations:</text>
<formula><location><page_18><loc_12><loc_7><loc_83><loc_12></location>E = Ric[ g ] + 2 9 τ 2 g + τ 3 ˆ k -ˆ k /circledot g ˆ k -4 π ( ∇ φ ⊗∇ φ ) -( 8 π 3 | ∂ 0 φ | 2 + 4 π 3 |∇ φ | 2 g ) g (2.24a)</formula>
<formula><location><page_19><loc_12><loc_87><loc_32><loc_90></location>B = -curl ˆ k (2.24b)</formula>
<text><location><page_19><loc_12><loc_86><loc_54><loc_87></location>Further, they satisfy the following evolution equations:</text>
<text><location><page_19><loc_12><loc_83><loc_17><loc_85></location>(2.25a)</text>
<formula><location><page_19><loc_15><loc_80><loc_84><loc_83></location>∂ t E ij = n curl B ij -( ∇ n ∧ B ) ij -5 2 n ( E × k ) ij -2 3 n ( E · k ) g ij -τ 2 n · E ij -n 2 ( J i 0 j + J j 0 i )</formula>
<text><location><page_19><loc_12><loc_78><loc_17><loc_80></location>(2.25b)</text>
<formula><location><page_19><loc_15><loc_73><loc_85><loc_78></location>∂ t B ij = -n curl E ij +( ∇ n ∧ E ) ij -5 2 n ( B × k ) ij -2 3 n ( B · k ) g ij -τ 2 n · B ij -n 2 ( J ∗ i 0 j + J ∗ j 0 i )</formula>
<text><location><page_19><loc_12><loc_71><loc_89><loc_74></location>Proof. For (2.25a)-(2.25b), we refer to [AM04, (3.11a)-(3.11b)]. 7 (2.24a)-(2.24b) follow as in [Wan19, (3.63a)-(3.63b)] from contracting the Gauss-Codazzi constraints. /square</text>
<text><location><page_19><loc_12><loc_62><loc_88><loc_69></location>Remark 2.8 (Inital data for Bel-Robinson variables) . Since the Weyl tensor vanishes over FLRW spacetimes, so do E ( W [ g FLRW ]) and B ( W [ g FLRW ]). Furthermore, note that given initial data ( M, ˚ g, ˚ k, ˚ π, ˚ ψ ) on Σ t 0 in the sense discussed in Section 1.2.1, and defining ˆ ˚ k = ˚ k -τ 3 ˚ g , we can use (2.24a) and (2.24b) to define the following (0 , 2)-tensors:</text>
<formula><location><page_19><loc_12><loc_55><loc_81><loc_62></location>˚ E =Ric[˚ g ] + 2 9 τ 2 ˚ g + τ 3 ˆ ˚ k -ˆ ˚ k /circledot ˚ g ˆ ˚ k -4 π (˚ π ⊗ ˚ π ) -( 8 π 3 ∣ ∣ ∣ ˚ ψ ∣ ∣ ∣ 2 ˚ g + 4 π 3 | ˚ π | 2 ˚ g ) ˚ g (2.26a)</formula>
<formula><location><page_19><loc_12><loc_54><loc_36><loc_57></location>˚ B = -curl ˚ g ˆ ˚ k (2.26b)</formula>
<text><location><page_19><loc_12><loc_47><loc_88><loc_54></location>These are easily seen to be symmetric, and the constraints (1.3a) and (1.3b) on the initial data ensure that they are also tracefree. Hence, any choice of initial data for the Cauchy problem immediately also contains a unique choice of initial data for the Bel-Robinson variables that is consistent with solutions to the Einstein scalar-field equations in CMC gauge.</text>
<unordered_list>
<list_item><location><page_19><loc_12><loc_38><loc_88><loc_45></location>2.5. Rescaled variables and equations. It will be more convenient to work the rescaled and shifted solution variables to measure their distance from the FLRW reference solution. In this subsection, we introduce the renormalized solution variables and restate the Einstein scalar-field system in terms of these variables.</list_item>
</unordered_list>
<text><location><page_19><loc_12><loc_36><loc_88><loc_37></location>Definition 2.9 (Rescaled variables for Big Bang stability) . We will consider the rescaled variables</text>
<formula><location><page_19><loc_12><loc_33><loc_67><loc_35></location>G ij = a -2 g ij , ( G -1 ) ij = a 2 g ij , Σ ij = a ˆ k ij (2.27a)</formula>
<formula><location><page_19><loc_12><loc_30><loc_54><loc_32></location>N = n -1 (2.27b)</formula>
<formula><location><page_19><loc_12><loc_25><loc_56><loc_28></location>Ψ = a 3 ∂ 0 φ -C, (2.27d)</formula>
<formula><location><page_19><loc_12><loc_28><loc_62><loc_30></location>E ij = a 4 · E ij , B ij = a 4 · B ij (2.27c)</formula>
<text><location><page_19><loc_12><loc_17><loc_88><loc_25></location>We note that the scaling of B in (2.27c) is not the asymptotic rescaling of B - in fact, we expect B to have (approximate) leading order a -2 as one can see in (4.4g). However, keeping this scaling parallel to that of E makes the structurally very similar evolution equations significantly easier to deal with. We also don't rescale N asymptotically, unlike [RS18b, Spe18], but note that N converges to 0 at an order slightly below a 4 at low orders (see (3.17h) and (8.3a)).</text>
<text><location><page_19><loc_12><loc_12><loc_88><loc_16></location>Proposition 2.10 (The rescaled Einstein scalar-field system) . The Einstein scalar-field system in CMC gauge as in Propopsition 2.6 are solved by ( g, ˆ k, n, ∇ φ, ∂ 0 φ ) if the rescaled variables</text>
<text><location><page_20><loc_12><loc_87><loc_88><loc_90></location>( G, Σ , N, ∇ φ, Ψ , E , B ) as in Definition 2.9 solve the following set of equations: 8 The rescaled metric evolution equations</text>
<formula><location><page_20><loc_25><loc_84><loc_54><loc_86></location>∂ t G ij = 2( N +1) a -3 Σ ij +2 N ˙ a G ij</formula>
<formula><location><page_20><loc_12><loc_66><loc_84><loc_85></location>-a (2.28a) ∂ t ( G -1 ) ij =2( N +1) a -3 (Σ /sharp ) ij -2 N ˙ a a ( G -1 ) ij (2.28b) ∂ t Σ ij = -a ∇ i ∇ j N +( N +1) [ a Ric[ G ] ij -2 a -3 (Σ /circledot G Σ) ij -8 πa ∇ i φ ∇ j φ ] (2.28c) +4 πC 2 a -3 · NG ij + 1 9 (3 N +2) aG ij + N ˙ a a Σ ij ∂ t (Σ /sharp ) a b = τN (Σ /sharp ) a b -a ∇ /sharpa ∇ b N +( N +1) a [ ( Ric[ G ] /sharp ) a b + 2 9 I a b ] (2.28d) -8 π ( N +1) a ∇ /sharpa φ ∇ b φ + N ( 4 πC 2 a -3 + 1 9 a ) I a b ,</formula>
<text><location><page_20><loc_12><loc_64><loc_59><loc_65></location>the rescaled Hamiltonian and momentum constraints</text>
<formula><location><page_20><loc_12><loc_59><loc_72><loc_64></location>R [ G ] + 2 3 -a -4 〈 Σ , Σ 〉 G = 8 π [ a -4 Ψ 2 +2 Ca -4 Ψ+ |∇ φ | 2 G ] (2.29a)</formula>
<formula><location><page_20><loc_12><loc_58><loc_61><loc_60></location>∇ /sharpm Σ ml = -8 π ∇ l φ (Ψ + C ) (2.29b)</formula>
<text><location><page_20><loc_12><loc_56><loc_38><loc_58></location>with their Bel-Robinson analogues</text>
<formula><location><page_20><loc_12><loc_45><loc_76><loc_56></location>E ij = a 4 ( Ric[ G ] ij + 2 9 G ij ) + τ 3 a 3 Σ ij -(Σ /circledot G Σ) ij -4 πa 4 ∇ i φ ∇ j φ (2.29c) -[ 4 π 3 a 4 |∇ φ | 2 G + 8 π 3 Ψ 2 + 16 π 3 C Ψ ] G ij B ij = -a 2 curl G Σ ij , (2.29d)</formula>
<text><location><page_20><loc_12><loc_44><loc_35><loc_45></location>the rescaled lapse equation</text>
<formula><location><page_20><loc_12><loc_38><loc_77><loc_44></location>(2.30a) ∆ N = ( 12 πC 2 a -4 + 1 3 ) N +( N +1) a -4 [ 〈 Σ , Σ 〉 G +8 π Ψ 2 +16 πC Ψ ]</formula>
<text><location><page_20><loc_12><loc_37><loc_23><loc_39></location>or equivalently</text>
<formula><location><page_20><loc_12><loc_32><loc_75><loc_37></location>(2.30b) ∆ N = ( 12 πC 2 a -4 + 1 3 ) N +( N +1) [ R [ G ] + 2 3 -8 π |∇ φ | 2 G ] ,</formula>
<text><location><page_20><loc_12><loc_31><loc_69><loc_32></location>the rescaled evolution equations for the Bel-Robinson variables 9</text>
<formula><location><page_20><loc_12><loc_12><loc_87><loc_30></location>∂ t E ij = (3 -N ) ˙ a a E ij -a -1 ( ∇ N ∧ G B ) ij +( N +1) a -1 curl G B ij (2.31a) -( N +1) [ 5 2 a -3 ( E × G Σ) ij + 2 3 a -3 〈 E , Σ 〉 G G ij ] +4 π ( N +1) a -3 (Ψ + C ) 2 Σ ij -4 π ( N +1)˙ aa 3 ∇ i φ ∇ j φ +4 πa ∇ ( i N ∇ j ) φ (Ψ + C ) -4 πa ( N +1) [ ∇ i Ψ ∇ j φ + ∇ j Ψ ∇ i φ +(Σ /sharp ) l ( i ∇ j ) φ ∇ l φ -(Ψ + C ) ∇ i ∇ j φ ] +( N +1) [ 2 π 3 a 6 ∂ 0 ( a -6 (Ψ + C ) 2 + a -2 |∇ φ | 2 G ) +4 π ˙ a a (Ψ + C ) 2 ] G ij</formula>
<formula><location><page_21><loc_12><loc_79><loc_76><loc_90></location>∂ t B ij = ˙ a a (3 -N ) B ij + a -1 ( ∇ N ∧ G E ) ij -( N +1) a -1 curl G E ij (2.31b) -( N +1) [ 5 2 a -3 ( B × G Σ) ij + 2 3 a -3 〈 B , Σ 〉 G G ij ] -4 π ( N +1) ε [ G ] lmj ( a 3 ∇ /sharpl ∇ j φ ∇ /sharpm φ + a -1 (Σ /sharp ) l i ∇ /sharpm φ (Ψ + C ) )</formula>
<text><location><page_21><loc_12><loc_78><loc_38><loc_79></location>and the rescaled wave equation</text>
<formula><location><page_21><loc_12><loc_74><loc_70><loc_77></location>(2.32a) ∂ t Ψ = a 〈∇ N, ∇ φ 〉 G + a ( N +1)∆ φ -3 ˙ a a N (Ψ + C )</formula>
<text><location><page_21><loc_12><loc_72><loc_38><loc_73></location>along with the evolution equation</text>
<formula><location><page_21><loc_12><loc_68><loc_67><loc_71></location>(2.32b) ∂ t ∇ φ = a -3 ( N +1) ∇ Ψ+ a -3 (Ψ + C ) ∇ N .</formula>
<text><location><page_21><loc_12><loc_67><loc_59><loc_68></location>Finally, we collect the rescaled Ricci evolution equation</text>
<formula><location><page_21><loc_12><loc_54><loc_82><loc_66></location>∂ t Ric[ G ] ab = a -3 ( N +1)(∆ G Σ ab -∇ /sharpd ∇ a Σ db -∇ /sharpd ∇ b Σ da ) (2.33) + a -3 ∇ /sharpd N (2 ∇ d Σ ab -∇ a Σ db -∇ b Σ da ) -a -3 ( ∇ a N (div G Σ) b + ∇ b (div G Σ) a ) + ∆ G N ( a -3 Σ ab + τ 3 G ab ) -a -3 ( ∇ /sharpd ∇ a N · Σ db + ∇ /sharpd ∇ b N · Σ da ) + τ 3 ∇ a ∇ b N</formula>
<text><location><page_21><loc_12><loc_53><loc_71><loc_54></location>as well as (in a coordinate neighbourhood) the Christoffel evolution equation</text>
<formula><location><page_21><loc_12><loc_38><loc_73><loc_52></location>∂ t Γ k ij [ G ] = 1 2 ( G -1 ) kl ( ∇ i ( ∂ t G jl ) + ∇ j ( ∂ t G il ) -∇ l ( ∂ t G ij )) (2.34) = -( N +1) a -3 [ ∇ i (Σ /sharp ) k j + ∇ j (Σ /sharp ) k i -∇ /sharpk Σ ij ] -a -3 [ ∇ i N (Σ /sharp ) k j + ∇ j N (Σ /sharp ) k i -∇ /sharpk N Σ ij ] + ˙ a a [ ∇ i N · I k j + ∇ j N · I k i -∇ /sharpk N · G ij ]</formula>
<text><location><page_21><loc_12><loc_22><loc_88><loc_38></location>Proof. For the first identity in (2.34), we refer to [CBCLN06, Lemma 2.27] and insert the evolution equation (2.28a). Otherwise, all equations simply follow by computing the effects of rescaling on the equations from Proposition 2.6 (respectively the Ricci evolution equation as in [Ren08, Chapter 2.3, (2.32)]) as well as the Bel-Robinson evolution equations (2.25a)-(2.25b) and constraint equations (2.24a)-(2.24b). Notice that one already finds a solution to the system in Proposition (2.6) with the rescaled variables excluding (2.31a), (2.31b), (2.29c) and (2.29d). Conversely, all of the rescaled equations are satisfied by a solution to Proposition 2.6 at sufficiently high regularity. Hence, solving the full rescaled system is always sufficient to solve the Einstein system in Proposition 2.6 and they are equivalent if the initial data is regular enough to ensure sufficiently high regularity of solutions. /square</text>
<text><location><page_21><loc_12><loc_13><loc_88><loc_21></location>2.6. Commuted equations. We collect Laplace-commuted versions of the equations for the rescaled variables in Proposition 2.10 in this subsection. For the sake of brevity, we won't state all possible commutations for every equation, but restrict ourselves to the ones we actually need within the bootstrap argument. We also refer to Appendix 11.2 for expressions for commutators of spatial differential operators with each other and with ∂ t .</text>
<text><location><page_21><loc_12><loc_8><loc_88><loc_12></location>The terms written down explicity in Lemma 2.11 are ones that dominate the evolution behaviour or that are the largest higher order terms, both of which require careful treatment within the bootstrap argument. The error terms are broadly categorized into three groups:</text>
<unordered_list>
<list_item><location><page_22><loc_16><loc_83><loc_88><loc_90></location>· 'Borderline' terms are terms that critically contribute to the fact that the energies diverge toward the Big Bang singularity. This almost always takes the form of adding energy terms at the same order as the evolved variable scaled by factors of the type εa -3 or εa -3 -c √ ε , which causes the energies to slightly diverge since a -3 is barely not integrable (see (2.6)).</list_item>
<list_item><location><page_22><loc_16><loc_80><loc_88><loc_83></location>· 'Junk' terms are terms that are subcritical in the sense that they lead to integrable error terms, or terms that only contain lower order derivatives of the solution variables.</list_item>
<list_item><location><page_22><loc_16><loc_77><loc_88><loc_80></location>· 'Top' order terms (which only appear in (2.38a) and (2.38b)) are terms that are terms that are junk terms for low order energies, but become borderline terms at top order.</list_item>
</unordered_list>
<text><location><page_22><loc_12><loc_72><loc_88><loc_76></location>All of these error terms are tracked schematically in Section 11.3. Since we will only need L 2 G -bounds on these error terms, which are given in Section 11.4, we will treat them as notational 'black boxes' outside of the appendix.</text>
<text><location><page_22><loc_12><loc_66><loc_88><loc_70></location>Lemma 2.11 (Laplace-commuted rescaled equations) . Let L ∈ 2 N , L ≥ 2 . With error terms as defined in Appendix 11.3, the system in Proposition 2.10 leads to the following Laplace-commuted equations:</text>
<text><location><page_22><loc_12><loc_64><loc_88><loc_66></location>The Laplace-commuted rescaled evolution equations for the second fundamental form</text>
<formula><location><page_22><loc_12><loc_61><loc_76><loc_64></location>(2.35) ∂ t ∆ L 2 Σ = -a ∇ 2 ∆ L 2 N + a ( N +1)∆ L 2 Ric[ G ] + S L,Border + S L,Junk ,</formula>
<formula><location><page_22><loc_12><loc_55><loc_87><loc_60></location>(2.36a) div G ∆ L 2 Σ = -8 π (Ψ + C ) [ ∇ ∆ L 2 φ +∆ L 2 -1 Ric[ G ] ∗ ∇ φ ] + ∇ ∆ L 2 -1 Ric[ G ] ∗ Σ+ M L,Junk</formula>
<text><location><page_22><loc_12><loc_59><loc_68><loc_61></location>the Laplace-commuted rescaled momentum constraint equations</text>
<text><location><page_22><loc_12><loc_54><loc_15><loc_56></location>and</text>
<formula><location><page_22><loc_12><loc_51><loc_75><loc_54></location>(2.36b) curl G ∆ L 2 Σ = -a -2 ∆ L 2 B + ε [ G ] ∗ ∇ ∆ L 2 -1 Ric[ G ] ∗ Σ+ ˜ M L,Junk ,</formula>
<formula><location><page_22><loc_12><loc_45><loc_54><loc_50></location>∆ L 2 R [ G ] + a -4 ∑ I 1 + I 2 = L ∇ I 1 Σ ∗ ∇ I 2 Σ (2.36c)</formula>
<text><location><page_22><loc_12><loc_50><loc_70><loc_51></location>the Laplace-commuted rescaled Hamiltonian constraint equations</text>
<text><location><page_22><loc_12><loc_40><loc_15><loc_41></location>and</text>
<formula><location><page_22><loc_25><loc_41><loc_76><loc_46></location>=16 πCa -4 ∆ L 2 Ψ+ a -4 ∑ I 1 + I 2 = L [ ∇ I 1 Ψ ∗ ∇ I 2 Ψ+ ∇ I 1 +1 φ ∗ ∇ I 2 +1 φ ]</formula>
<formula><location><page_22><loc_12><loc_37><loc_73><loc_40></location>(2.36d) ∆ L 2 Ric[ G ] = a -4 ∆ L 2 E -τ 3 a -1 ∆ L 2 Σ+ H L,Border + H L,Junk ,</formula>
<text><location><page_22><loc_12><loc_35><loc_53><loc_36></location>the Laplace-commuted rescaled lapse equations</text>
<formula><location><page_22><loc_12><loc_26><loc_87><loc_31></location>∇ ∆ L 2 +1 N = ( 12 πC 2 a -4 + 1 3 ) ∇ ∆ L 2 N +16 πCa -4 · ∇ ∆ L 2 Ψ+ N L +1 ,Border + N L +1 ,Junk , (2.37b)</formula>
<formula><location><page_22><loc_12><loc_30><loc_81><loc_35></location>∆ L 2 +1 N = ( 12 πC 2 a -4 + 1 3 ) ∆ L 2 N +16 πCa -4 · ∆ L 2 Ψ+ N L,Border + N L,Junk , (2.37a)</formula>
<text><location><page_22><loc_12><loc_25><loc_69><loc_26></location>the Laplace-commuted rescaled Bel-Robinson evolution equations</text>
<formula><location><page_22><loc_12><loc_8><loc_81><loc_25></location>∂ t ∆ L 2 E = ˙ a a (3 -N ) ∆ L 2 E +( N +1) a -1 curl G ∆ L 2 B -a -1 ∇ ∆ L 2 N ∧ G B (2.38a) +4 πC 2 a -3 ( N +1)∆ L 2 Σ+4 πa (Ψ + C ) ∇ ∆ L 2 N ⊗∇ φ +4 πa (Ψ + C )( N +1) ∇ 2 ∆ L 2 φ -8 πa ( N +1) ( ∇ φ ⊗∇ ∆ L 2 Ψ ) + E L,Border + E L,top + E L,Junk ∂ t ∆ L 2 B = ˙ a a (3 -N )∆ L 2 B -( N +1) a -1 curl G ∆ L 2 E + a -1 ∇ ∆ L 2 N ∧ G E (2.38b) + a 3 ε [ G ] 2 ∆ L 2 φ φ + B L,Border + B L,top + B L,Junk ,</formula>
<formula><location><page_22><loc_41><loc_7><loc_51><loc_10></location>∗ ∇ ∗ ∇</formula>
<section_header_level_1><location><page_23><loc_12><loc_88><loc_63><loc_90></location>the Laplace-commuted rescaled matter evolution equations</section_header_level_1>
<formula><location><page_23><loc_12><loc_84><loc_85><loc_88></location>∂ t ∆ L 2 Ψ = a 〈∇ ∆ L 2 N, ∇ φ 〉 G + a ( N +1)∆ L 2 +1 φ -3 C ˙ a a ∆ L 2 N + P L,Border + P L,Junk (2.39a)</formula>
<formula><location><page_23><loc_12><loc_81><loc_74><loc_84></location>∂ t ∇ ∆ L 2 φ = a -3 ( N +1) ∇ ∆ L 2 Ψ+ Ca -3 ∇ ∆ L 2 N + Q L,Border + Q L,Junk (2.39b)</formula>
<text><location><page_23><loc_12><loc_80><loc_46><loc_81></location>as well as (also allowing L = 0 for (2.39d) )</text>
<formula><location><page_23><loc_12><loc_76><loc_75><loc_79></location>∂ t ∇ l ∆ L 2 Ψ = a ∇ l ∇ /sharpj ∆ L 2 N ∇ j φ + a ( N +1) ∇ l ∆ L 2 +1 φ -3 C ˙ a a ∇ l ∆ L 2 N (2.39c)</formula>
<formula><location><page_23><loc_12><loc_72><loc_58><loc_76></location>+( P L +1 ,Border ) l +( P L +1 ,Junk ) l (2.39d)</formula>
<formula><location><page_23><loc_24><loc_72><loc_82><loc_74></location>∂ t ∆ L 2 +1 φ = a -3 ( N +1)∆ L 2 +1 Ψ+ Ca -3 ∆ L 2 +1 N + Q L +1 ,Border + Q L +1 ,Junk</formula>
<text><location><page_23><loc_12><loc_69><loc_65><loc_71></location>and the Laplace-commuted rescaled Ricci evolution equations</text>
<formula><location><page_23><loc_12><loc_54><loc_85><loc_69></location>∂ t ∆ L 2 Ric[ G ] ij = a -3 ( ∆ L 2 +1 Σ ij -2 ∇ /sharpm ∇ ( i ∆ L 2 Σ j ) m ) (2.40a) -˙ a a ( ∇ i ∇ j ∆ L 2 N +∆ L 2 +1 N · G ij ) +( R L,Border ) ij +( R L,Junk ) ij ∂ t ∇ k ∆ L 2 Ric[ G ] ij = a -3 ( ∇ k ∆ L 2 +1 Σ ij -2 ∇ k ∇ /sharpm ∇ ( i ∆ L 2 Σ j ) m ) (2.40b) -˙ a a ( ∇ k ∇ i ∇ j ∆ L 2 N + ∇ k ∆ L 2 +1 N · G ij ) +( R L +1 ,Border ) ijk +( R L +1 ,Junk ) ijk .</formula>
<text><location><page_23><loc_12><loc_40><loc_88><loc_53></location>Proof. The equations (2.36c),(2.36d) and (2.37a) are obtained by simply applying ∆ L 2 on both sides of (2.29a),(2.29c) and (2.30a) respectively. For (2.36a) and (2.36b), we additionally use the commutator formulas (11.7e) and (11.7f), while for the evolution equations, we apply the respective commutator of ∂ t and spatial derivatives as collected in Lemma 11.7 and commute Laplacians past ∇ and curl where needed (see the commutators in Lemma 11.5). The commutators with ∂ t only cause additional borderline and junk terms that don't substantially influence the behaviour, while the spatial commutators often lead to high order curvature terms, for example the Ricci terms in (2.36a), that need to be more carefully tracked. /square</text>
<text><location><page_23><loc_12><loc_29><loc_88><loc_39></location>Remark 2.12 (Simplified junk term notation) . For junk terms that occur in an inner product with a tracefree symmetric tensor, any terms that are pure trace will immediately cancel and thus don't need to be taken into consideration for the following estimates, even if they have to be written down in the junk terms. Hence, we will denote with a superscript ' ‖ ' (for example H ‖ L,Junk ) on a schematic error term the expressions that arise when dropping all terms of the form ζ · G for some scalar function ζ that occur in this term's definition (see, for example, (11.11d)).</text>
<section_header_level_1><location><page_23><loc_20><loc_24><loc_81><loc_26></location>3. Big Bang stability: Norms, energies and bootstrap assumptions</section_header_level_1>
<text><location><page_23><loc_12><loc_14><loc_88><loc_23></location>Herein, we state the norms and energies we use to control the solution variables. These will allow us to state our initial data and bootstrap assumptions, and we then provide which improvement we aim to achieve for the latter. Note that we don't provide the coerciveness of our energies immediately (and actually cannot, at least not in a manner useful to our analysis), but will establish Sobolev norm control in the proof of Corollary 7.3, the key ingredient being Lemma 4.5. Furthermore, we collect a local well-posedness statement from previous work in Section 3.4.</text>
<text><location><page_23><loc_12><loc_8><loc_88><loc_12></location>3.1. Norms. Recall that γ is the hyperbolic spatial reference metric on M introduced in Definition 2.1, which we view as a metric on any foliation hypersurface Σ t (see Section 1.2.1), and G is the rescaled spatial metric arising from the evolution (see Definition 2.9).</text>
<text><location><page_24><loc_12><loc_82><loc_88><loc_90></location>Definition 3.1 (Pointwise norms and volume forms) . We denote by |·| γ (resp. |·| G ( t, · ) ) the pointwise norm with regards to γ (resp. G ( t, · )). For the sake of simplicity, we define | ζ | γ = | ζ | G ( t, · ) = | ζ ( t, · ) | for any scalar function ζ on Σ t . The volume forms on Σ t with respect to γ and G ( t, · ) are written as vol γ and vol G ( t, · ) (or just vol G ).</text>
<text><location><page_24><loc_12><loc_80><loc_88><loc_82></location>Definition 3.2 ( L 2 -norms) . Let T be a Σ t -tangent ( r, s )-tensor field (for r, s ≥ 0). Then, we define:</text>
<formula><location><page_24><loc_12><loc_72><loc_71><loc_77></location>‖ T ‖ 2 L 2 G (Σ t ) = ‖ T ( t, · ) ‖ 2 L 2 G (Σ t ) := ∫ M | T ( t, · ) | 2 G ( t, · ) vol G ( t, · ) (3.1b)</formula>
<formula><location><page_24><loc_12><loc_76><loc_66><loc_81></location>‖ T ‖ 2 L 2 γ (Σ t ) = ‖ T ( t, · ) ‖ 2 L 2 γ (Σ t ) := ∫ M | T ( t, · ) | 2 γ vol γ , (3.1a)</formula>
<text><location><page_24><loc_12><loc_70><loc_72><loc_72></location>Definition 3.3 (Sobolev norms) . Let T be as above and J ∈ N 0 . We define:</text>
<formula><location><page_24><loc_12><loc_61><loc_71><loc_67></location>‖ T ‖ 2 ˙ H J G (Σ t ) = ‖ T ( t, · ) ‖ 2 ˙ H J G = ∫ Σ t ∣ ∣ ∇ J T ( t, · ) ∣ ∣ 2 G ( t, · ) vol G ( t, · ) (3.2b)</formula>
<formula><location><page_24><loc_12><loc_64><loc_66><loc_71></location>‖ T ‖ 2 ˙ H J γ (Σ t ) = ‖ T ( t, · ) ‖ 2 ˙ H J γ = ∫ Σ t ∣ ∣ ∣ ˆ ∇ J T ( t, · ) ∣ ∣ ∣ 2 γ vol γ (3.2a)</formula>
<formula><location><page_24><loc_12><loc_57><loc_61><loc_62></location>‖ T ‖ 2 H J γ (Σ t ) = ‖ T ( t, · ) ‖ 2 H J γ = J ∑ k =0 ‖ T ‖ 2 ˙ H k γ (Σ t ) (3.2c)</formula>
<formula><location><page_24><loc_12><loc_52><loc_61><loc_57></location>‖ T ‖ 2 H J G (Σ t ) = ‖ T ( t, · ) ‖ 2 H J G = J ∑ k =0 ‖ T ‖ 2 ˙ H k G (Σ t ) (3.2d)</formula>
<text><location><page_24><loc_12><loc_50><loc_69><loc_52></location>Definition 3.4 (Supremum norms) . For T as above and J ∈ N 0 , we set:</text>
<formula><location><page_24><loc_12><loc_39><loc_83><loc_45></location>‖ T ‖ ˙ C J G (Σ t ) = ‖ T ( t, · ) ‖ ˙ C J G = sup p ∈ Σ t ∣ ∣ ∇ J T ( t, · ) ∣ ∣ G ( t, · ) , ‖ T ‖ C J G (Σ t ) = J ∑ k =0 ‖ T ‖ ˙ C k G (Σ t ) (3.3b)</formula>
<formula><location><page_24><loc_12><loc_43><loc_83><loc_50></location>‖ T ‖ ˙ C J γ (Σ t ) = ‖ T ( t, · ) ‖ ˙ C J γ = sup p ∈ Σ t ∣ ∣ ∣ ˆ ∇ J T ( t, · ) ∣ ∣ ∣ γ , ‖ T ‖ C J γ (Σ t ) = J ∑ k =0 ‖ T ‖ ˙ C k γ (Σ t ) (3.3a)</formula>
<text><location><page_24><loc_12><loc_33><loc_88><loc_40></location>Remark 3.5 (Time dependence is suppressed in notation) . When the choice of t and Σ t is clear from context, we will often drop time dependences of G , |·| G , vol G and T , surpress the hypersurface Σ t in the Sobolev and supremum norms, and simply write ∫ M instead of ∫ Σ t . For example, we write</text>
<formula><location><page_24><loc_41><loc_30><loc_59><loc_35></location>‖ T ‖ 2 L 2 G = ∫ M | T | 2 G vol G .</formula>
<text><location><page_24><loc_12><loc_27><loc_88><loc_30></location>Definition 3.6 (Solution norms) . We define the following norms to measure the size of near-FLRW solutions:</text>
<formula><location><page_24><loc_12><loc_7><loc_84><loc_26></location>H = ‖ Ψ ‖ H 18 G + ‖∇ φ ‖ H 17 G + a 2 ‖∇ φ ‖ ˙ H 18 G (3.4a) + ‖ Σ ‖ H 18 G + ‖ E ‖ H 18 G + ‖ B ‖ H 18 G + ‖ G -γ ‖ H 18 G + ‖ Ric[ G ] + 2 9 G ‖ H 16 G + a -4 ‖ N ‖ H 16 G + a -2 ‖ N ‖ ˙ H 17 G + ‖ N ‖ ˙ H 18 G H top = a 2 ‖ Ψ ‖ ˙ H 19 G + a 4 ‖∇ φ ‖ ˙ H 19 G + a 2 ‖ Σ ‖ ˙ H 19 G + a 2 ‖ Ric[ G ] + 2 9 G ‖ ˙ H 17 G (3.4b) C = ‖ Ψ ‖ C 16 G + ‖∇ φ ‖ C 15 G + ‖ Σ ‖ C 16 G + ‖ E ‖ C 16 G + ‖ B ‖ C 16 G (3.4c) + ‖ G -γ ‖ C 16 G + ‖ Ric[ G ] + 2 9 G ‖ C 14 G + a -4 ‖ N ‖ C 14 G + a -2 ‖ N ‖ ˙ C 15 G + ‖ N ‖ ˙ C 16 G C γ = ‖ Ψ ‖ C 16 γ + ‖∇ φ ‖ C 15 γ + ‖ Σ ‖ C 16 γ + ‖ E ‖ C 16 γ + ‖ B ‖ C 16 γ (3.4d)</formula>
<formula><location><page_25><loc_27><loc_87><loc_83><loc_90></location>+ ‖ G -γ ‖ C 16 γ + ‖ Ric[ G ] + 2 9 G ‖ C 14 γ + a -4 ‖ N ‖ C 14 γ + a -2 ‖ N ‖ ˙ C 15 γ + ‖ N ‖ ˙ C 16 γ</formula>
<text><location><page_25><loc_12><loc_75><loc_88><loc_86></location>Remark 3.7 (Choice of metric and controlling Christoffel symbols) . We could equivalently also phrase H in terms of γ -norms, or predominately use C γ instead of C , since we include the norms on G -γ and Ric[ G ] + 2 9 G = (Ric[ G ] + 2 9 γ ) + 2 9 ( G -γ ). We will demonstrate in Lemma 7.4 how H G and C γ norms can be used to control H γ and C G norms. We will also indicate how the initial data and bootstrap assumptions for C γ and C are equivalent in Remarks 3.11 and 3.18. The main reason for this is that, by successively replacing local coordinates in the expressions of Γ -ˆ Γ by ˆ ∇ , one has</text>
<formula><location><page_25><loc_12><loc_71><loc_72><loc_74></location>(3.5) ‖ Γ -ˆ Γ ‖ C l -1 γ ( M ) /lessorsimilar P l ( ‖ G -γ ‖ C l γ ( M ) , ‖ G -1 -γ -1 ‖ C l γ ( M ) ) .</formula>
<text><location><page_25><loc_12><loc_66><loc_88><loc_71></location>We choose to work predominately with norms in terms of the rescaled metric since quantities appearing in the Einstein system are naturally contracted by G , not γ , and we commute with differential operators associated with G .</text>
<text><location><page_25><loc_12><loc_57><loc_88><loc_65></location>Remark 3.8 (Redundancies in the solution norms) . The solution norms H , C and C γ aren't 'optimal' in the sense that controlling the norms of Ψ , ∇ φ, Σ and G -γ is entirely sufficient to gain the claimed control (up to constant) on N via the lapse equation, E and B via to the constraint equations and Ric[ G ] + 2 9 G via local coordinates. We choose to include all variables in the norms and subsequent assumptions mainly for the sake of convenience.</text>
<text><location><page_25><loc_12><loc_52><loc_88><loc_55></location>3.2. Energies. The fundamental objects used to control the solution variables are the energies that take the following form:</text>
<text><location><page_25><loc_12><loc_49><loc_52><loc_51></location>Definition 3.9 (Energies) . Let l ∈ N 0 . We define:</text>
<formula><location><page_25><loc_12><loc_33><loc_53><loc_38></location>E ( l ) (Σ , t ) = ( -1) l ∫ M 〈 Σ , ∆ l Σ 〉 G vol G (3.6c)</formula>
<formula><location><page_25><loc_12><loc_36><loc_71><loc_50></location>E ( l ) ( φ, t ) = ( -1) l ∫ M Ψ∆ l Ψ -a 4 φ ∆ l +1 φ vol G (3.6a) = { ∫ M | ∆ l 2 Ψ | 2 + a 4 |∇ ∆ l 2 φ | 2 G vol G l even ∫ M |∇ ∆ l -1 2 Ψ | 2 G + a 4 | ∆ l +1 2 φ | 2 G vol G l odd E ( l ) ( W,t ) = ( -1) l ∫ M 〈 E , ∆ l E 〉 G + 〈 B , ∆ l B 〉 G vol G (3.6b)</formula>
<formula><location><page_25><loc_12><loc_29><loc_75><loc_34></location>E ( l ) (Ric , · ) = ( -1) l ∫ M 〈 Ric[ G ] + 2 9 G, ∆ l ( Ric[ G ] + 2 9 G )〉 G vol G (3.6d)</formula>
<formula><location><page_25><loc_12><loc_25><loc_54><loc_30></location>E ( l ) ( N, · ) = ( -1) l ∫ M 〈 N, ∆ l N 〉 G vol G (3.6e)</formula>
<text><location><page_25><loc_12><loc_22><loc_88><loc_25></location>Usually, we will use integration by parts to distribute derivatives within the energies as in (3.6b). Further, we introduce the notation</text>
<formula><location><page_25><loc_12><loc_16><loc_56><loc_21></location>E ( ≤ l ) = l ∑ i =0 E ( i ) (3.7)</formula>
<text><location><page_25><loc_12><loc_14><loc_35><loc_16></location>for any of the energies above.</text>
<text><location><page_25><loc_12><loc_11><loc_65><loc_13></location>For any l ∈ N 0 and any smooth functions f 1 , f 2 ∈ C ∞ ( R + ), we have</text>
<formula><location><page_25><loc_12><loc_7><loc_65><loc_11></location>(3.8) f 1 · f 2 · E (2 l +1) ≤ f 2 1 2 E (2 l ) + f 2 2 2 E (2 l +2) .</formula>
<text><location><page_26><loc_12><loc_88><loc_57><loc_90></location>Performing the calculation for Σ as an example, we have:</text>
<text><location><page_26><loc_12><loc_82><loc_77><loc_83></location>Now, (3.8) then follows from the Young inequality. As a consequence, we also have</text>
<formula><location><page_26><loc_13><loc_83><loc_88><loc_88></location>E (2 l +1) (Σ , · ) = -∫ M 〈 ∆ l Σ , ∆ l +1 Σ 〉 G vol G ≤ ∫ M | ∆ l Σ | G | ∆ l +1 Σ | G vol G ≤ √ E (2 l ) (Σ , · ) √ E (2 l +2) (Σ , · )</formula>
<formula><location><page_26><loc_12><loc_76><loc_66><loc_81></location>(3.9) E ( ≤ 2 l ) /lessorsimilar l ∑ m =0 E (2 m ) , E ( ≤ 2 l +1) /lessorsimilar l +1 ∑ m =0 E (2 m ) .</formula>
<text><location><page_26><loc_12><loc_73><loc_88><loc_76></location>This allows us to largely restrict our analysis to even order energies, outside of how we close the bootstrap argument at top order.</text>
<unordered_list>
<list_item><location><page_26><loc_12><loc_69><loc_88><loc_72></location>3.3. Assumptions on the initial data. With the necessary solution norms and energies now defined, we can now state what we assume near-FLRW initial data to satisfy:</list_item>
</unordered_list>
<text><location><page_26><loc_12><loc_63><loc_88><loc_67></location>Assumption 3.10 (Near-FLRW initial data) . For some small enough ε ∈ (0 , 1) and the solution norms H , H top and C as in Definition 3.6, we assume the rescaled inital data to be close to that of the FLRW solution in Lemma 2.3 in the following sense:</text>
<formula><location><page_26><loc_12><loc_59><loc_61><loc_62></location>(3.10) H ( t 0 ) + H top ( t 0 ) + C ( t 0 ) /lessorsimilar ε 2</formula>
<text><location><page_26><loc_15><loc_57><loc_58><loc_59></location>The assumptions on H + H top also imply the following:</text>
<formula><location><page_26><loc_12><loc_54><loc_75><loc_57></location>E ( ≤ 18) ( φ, t 0 ) + E ( ≤ 18) (Σ , t 0 ) + E ( ≤ 18) ( W,t 0 ) + E ( ≤ 16) (Ric , t 0 ) (3.11)</formula>
<formula><location><page_26><loc_30><loc_49><loc_78><loc_52></location>a ( t 0 ) 4 E (19) ( φ, t 0 ) + a ( t 0 ) 4 E (19) (Σ , t 0 ) + a ( t 0 ) 4 E (17) (Ric , t 0 ) /lessorsimilar ε</formula>
<formula><location><page_26><loc_21><loc_50><loc_79><loc_54></location>+ ‖∇ φ ‖ 2 H 18 G + E (18) ( N,t 0 ) + a ( t 0 ) -4 E (17) ( N,t 0 ) + a ( t 0 ) -8 E ( ≤ 16) ( N,t 0 ) + 4</formula>
<text><location><page_26><loc_12><loc_47><loc_78><loc_49></location>Remark 3.11 (Initial data size in C γ ( t 0 )) . Notice that by (3.5), (3.10) implies that</text>
<formula><location><page_26><loc_12><loc_45><loc_58><loc_47></location>(3.12a) Γ ˆ Γ C 15 (Σ t ) /lessorsimilar ε 4 ,</formula>
<formula><location><page_26><loc_42><loc_44><loc_53><loc_46></location>‖ -‖ G 0</formula>
<text><location><page_26><loc_12><loc_41><loc_63><loc_44></location>and arguing along similar lines and using L 2 -L ∞ -estimates, also</text>
<formula><location><page_26><loc_12><loc_39><loc_57><loc_41></location>(3.12b) ‖ Γ Γ H 17 (Σ t ) /lessorsimilar ε</formula>
<formula><location><page_26><loc_44><loc_39><loc_59><loc_41></location>-ˆ ‖ G 0 4 .</formula>
<text><location><page_26><loc_12><loc_34><loc_88><loc_38></location>In particular, since moving from C l γ to C l G only requires control on Christoffel symbols to order l -1 for general tensors and l -2 for scalar functions, as well as zero order control on G -γ , it follows from (3.10) that</text>
<formula><location><page_26><loc_54><loc_32><loc_54><loc_33></location>2</formula>
<formula><location><page_26><loc_12><loc_31><loc_54><loc_33></location>(3.13) C γ ( t 0 ) /lessorsimilar ε</formula>
<text><location><page_26><loc_12><loc_27><loc_88><loc_30></location>We refer to the proof of Lemma 7.4 for a more detailed term analysis and how a similar argument also applies to the Sobolev norms.</text>
<text><location><page_26><loc_12><loc_17><loc_88><loc_25></location>Remark 3.12 (Redundancies in the initial data assumptions) . Similar to Remark 3.8, one could also reduce the initial data assumptions in (3.10), especially at top order. In particular, we highlight that the Bel-Robinson energy can be entirely controlled by the other terms that occur due to the additional scaled Σ-energy at order 19, or vice versa we could drop the latter in favour of the former. This will be reflected in Lemma 6.10.</text>
<text><location><page_26><loc_12><loc_9><loc_88><loc_15></location>Remark 3.13 (The volume form) . Let µ G and µ γ denote the volume elements of G and γ respectively. Since the determinant is a smooth map on invertible matrices, the initial data assumptions on G -γ also imply</text>
<formula><location><page_26><loc_12><loc_8><loc_67><loc_9></location>(3.14) µ G µ γ C 0 (Σ t ) = µ G µ γ C 0 (Σ t ) /lessorsimilar ε .</formula>
<formula><location><page_26><loc_33><loc_7><loc_66><loc_10></location>‖ -‖ G 0 ‖ -‖ γ 0 2</formula>
<text><location><page_27><loc_12><loc_88><loc_29><loc_90></location>Consequently, we have</text>
<formula><location><page_27><loc_24><loc_85><loc_76><loc_88></location>‖ vol G -vol γ ‖ C 0 γ (Σ t 0 ) = µ -1 γ ‖ vol γ ‖ C 0 γ (Σ t 0 ) ‖ µ G ( t 0 , · ) -µ γ ‖ C 0 γ (Σ t 0 ) /lessorsimilar ε 2</formula>
<text><location><page_27><loc_12><loc_81><loc_88><loc_85></location>and, since ‖ G -1 -γ -1 ‖ C 0 γ (Σ t 0 ) /lessorsimilar ε 2 also follows by a von Neumann series argument from the initial data assumption on G -γ ,</text>
<formula><location><page_27><loc_12><loc_78><loc_60><loc_81></location>(3.15) ‖ vol G -vol γ ‖ C 0 G (Σ t 0 ) /lessorsimilar ε 2 .</formula>
<text><location><page_27><loc_12><loc_67><loc_88><loc_78></location>3.4. Local well-posedness and continuation criteria. For everything that follows, we need to establish that the initial data assumptions above also ensure local well-posedness. For the core system, we translate the local well-posedness result for stiff fluids in [RS18b] to the subcase of the scalar field system. While statement and proof there are for vanishing spatial sectional curvature and what corresponds to choosing C = √ 2 / 3 , the arguments for our setting are completely analogous.</text>
<text><location><page_27><loc_12><loc_61><loc_88><loc_67></location>Lemma 3.14 (Local well-posedness and continuation criteria for the Einstein scalar-field system (Big Bang version), see [RS18b, Theorem 14.1]) . Let N ≥ 4 be an integer and ( M, ˚ g, ˚ k, ˚ π, ˚ ψ ) be geometric initial data to the Einstein scalar-field system (see Section 1.2.1) and assume that one has</text>
<text><location><page_27><loc_12><loc_55><loc_87><loc_60></location>‖ ˚ g -a ( t 0 ) 2 γ ‖ H N +1 γ ( M ) + ‖ ˚ k + τ ( t 0 ) 3 · a ( t 0 ) 2 γ ‖ H N γ ( M ) + ‖ ˚ π ‖ H N γ ( M ) + ‖ ˚ ψ -Ca ( t 0 ) -3 ‖ H N γ ( M ) < ∞ as well as, for some sufficiently small η ' > 0 ,</text>
<formula><location><page_27><loc_40><loc_52><loc_61><loc_54></location>‖ ˚ ψ -Ca ( t 0 ) -3 ‖ C 0 γ ( M ) ≤ η ' .</formula>
<text><location><page_27><loc_12><loc_43><loc_88><loc_52></location>Then, the CMC-transported Einstein scalar-field system (respectively the rescaled system) is locally well-posed in the following sense: The initial data (˚ g, ˚ k, ˚ π, ˚ ψ ) launches a unique classical solution ( g, k, n, ∇ φ, ∂ t φ ) to (2.16a) -(2.16b) , (2.15a) -(2.15b) , (2.18) and (2.17a) on [ t 1 , t 0 ] × M for some t 1 ∈ (0 , t 0 ) that satisfies k l l = -3 ˙ a a and n > 0 , launches a solution to the Einstein scalar-field system and such that the variables enjoy the following regularity:</text>
<formula><location><page_27><loc_31><loc_30><loc_69><loc_43></location>g ∈ C N -1 dt 2 + γ ([ t 1 , t 0 ] × M ) ∩ C 0 ([ t 1 , t 0 ] , H N +1 γ ( M )) k ∈ C N -2 dt 2 + γ ([ t 1 , t 0 ] × M ) ∩ C 0 ([ t 1 , t 0 ] , H N γ ( M )) ∇ φ ∈ C N -2 dt 2 + γ ([ t 1 , t 0 ] × M ) ∩ C 0 ([ t 1 , t 0 ] , H N γ ( M )) ∂ t φ ∈ C N -2 dt 2 + γ ([ t 1 , t 0 ] × M ) ∩ C 0 ([ t 1 , t 0 ] , H N γ ( M )) n ∈ C N dt 2 + γ ([ t 1 , t 0 ] × M ) ∩ C 0 ([ t 1 , t 0 ] , H N +2 γ ( M ))</formula>
<text><location><page_27><loc_12><loc_25><loc_88><loc_30></location>The rescaled variables ( G, Σ , N, ∇ φ, Ψ) enjoy the analogous regularity. If ( t , t 0 ] is the maximal interval on which the above statements hold, then one either has t = 0 or one of the following blow-up criteria are satisfied:</text>
<unordered_list>
<list_item><location><page_27><loc_15><loc_21><loc_88><loc_25></location>(1) The smallest eigenvalue of g ( t m , · ) converges to 0 for some sequence ( t m , x m ) ⊆ ( t , t 0 ] × M with t m ↓ t .</list_item>
<list_item><location><page_27><loc_15><loc_19><loc_79><loc_21></location>(2) n ( t m , x m ) converges to 0 for some sequence ( t m , x m ) ⊆ ( t , t 0 ] × M with t m ↓ t .</list_item>
<list_item><location><page_27><loc_15><loc_16><loc_88><loc_21></location>(3) ( | ∂ 0 φ | 2 + |∇ φ | 2 g ) ( t m , x m ) converges to 0 for some sequence ( t m , x m ) ⊆ ( t , t 0 ] × M with t m ↓ t . (4) s ∈ ( t , t 0 ] ↦→ ‖ g ‖ C 2 γ (Σ s ) + ‖ k ‖ C 1 γ (Σ s ) + ‖ n ‖ C 2 γ (Σ s ) + ‖ ∂ t φ ‖ C 1 γ (Σ s ) + ‖∇ φ ‖ C 1 γ (Σ s ) is unbounded.</list_item>
</unordered_list>
<text><location><page_27><loc_12><loc_7><loc_88><loc_15></location>A note on the proof. Note that the additional initial data requirement in the stiff-fluid setting that the pressure is strictly positive is covered by the smallness assumption on ˚ ψ -Ca ( t 0 ) 2 , since the pressure corresponds to | ˚ ψ | 2 + | ˚ π | 2 ˚ g and the assumptions on ∂ t φ and ∇ φ ensure that (after embedding) this quantity behaves like C 2 a ( t 0 ) -6 + O ( η ' ) at Σ t 0 . /square</text>
<text><location><page_28><loc_12><loc_83><loc_88><loc_90></location>Corollary 3.15 (Local well-posedness for the Bel-Robinson variables) . Under the assumptions of Lemma 3.14, the Bel-Robinson variables E and B corresponding to the Lorentzian metric g = -n 2 dt 2 + g satisfy the equations (2.24a) -(2.24b) , are the unique classical solutions to the evolution equations (2.25a) -(2.25b) , and satisfy</text>
<formula><location><page_28><loc_31><loc_80><loc_69><loc_83></location>E,B ∈ C N -3 ([ t 1 , t 0 ] × M ) ∩ C ([ t 1 , t 0 ] , H N -1 γ ( M ))</formula>
<text><location><page_28><loc_12><loc_72><loc_88><loc_80></location>Proof. That E and B satisfy the constraint equations, solve the evolution equations and have the stated regularity on the interval of existence is a direct consequence of Lemma 3.14 and the computations in Section 2.4. Furthermore, with initial data derived from the constraint equations as in Remark 2.8, the hyperbolic system (2.24a)-(2.24b) launches a unique solution satisfying the regularity above that must then be ( E,B ). /square</text>
<text><location><page_28><loc_15><loc_68><loc_62><loc_70></location>For sufficiently regular initial data ( N ≥ 21), it follows that</text>
<formula><location><page_28><loc_20><loc_65><loc_80><loc_68></location>E ( ≤ 19) ( φ, · ) , E ( ≤ 18) ( W, · ) , E ( ≤ 19) (Σ , · ) , E ( ≤ 17) (Ric , · ) , ‖ G -γ ‖ H 18 G ∈ C 1 ([ t 1 , t 0 ]) ,</formula>
<text><location><page_28><loc_12><loc_56><loc_88><loc_65></location>and similarly the square of any supremum norm occuring in C is continuously differentiable on [ t 1 , t 0 ]. Strictly speaking, we would need to assume this additional regularity on our initial data for the computations in the following sections (especially Section 6) to hold. However, since smooth functions are dense in H l ( M ) for any l ∈ N 0 , any bounds on H ( t ) and C ( t ) that we prove assuming sufficient regularity at Σ t 0 then immediately extend to data only satisfying the regularity implied by (3.10).</text>
<text><location><page_28><loc_12><loc_51><loc_88><loc_55></location>Thus, from here on out, we will assume without loss of genearlity that all energies and squared norms are continuously differentiable on the domain of existence, and similarly all variables are continuously differentiable for the lower order C G -norm improvements in Section 4.2.</text>
<unordered_list>
<list_item><location><page_28><loc_12><loc_46><loc_88><loc_49></location>3.5. Bootstrap assumption. To keep an overview of the entire bootstrap argument, we state all of the assumptions and comprehensively list how we intend to improve them.</list_item>
</unordered_list>
<text><location><page_28><loc_12><loc_40><loc_88><loc_45></location>Assumption 3.16 (Bootstrap assumption) . Fix some t Boot ∈ [0 , t 0 ) . Further, let c 0 > 0 , let σ ∈ ( ε 1 8 , 1] be suitably small such that c 0 σ < 1 , and K 0 > 0 a suitable constant. For any t ∈ ( t Boot , t 0 ] , we assume</text>
<formula><location><page_28><loc_12><loc_37><loc_58><loc_39></location>(3.16) C ( t ) ≤ K 0 εa ( t ) -c 0 σ .</formula>
<text><location><page_28><loc_12><loc_35><loc_47><loc_37></location>Remark 3.17. More explicitly, (3.16) means</text>
<text><location><page_28><loc_52><loc_32><loc_67><loc_34></location>‖ Ψ ‖ C 16 G ≤ K 0 εa - c 0 σ</text>
<text><location><page_28><loc_12><loc_33><loc_17><loc_34></location>(3.17a)</text>
<text><location><page_28><loc_51><loc_29><loc_67><loc_32></location>‖∇ φ ‖ C 15 G ≤ K 0 εa - c 0 σ</text>
<text><location><page_28><loc_12><loc_30><loc_17><loc_32></location>(3.17b)</text>
<text><location><page_28><loc_52><loc_27><loc_67><loc_30></location>‖ Σ ‖ C 16 G ≤ K 0 εa - c 0 σ</text>
<text><location><page_28><loc_12><loc_28><loc_17><loc_29></location>(3.17c)</text>
<text><location><page_28><loc_52><loc_24><loc_67><loc_27></location>‖ E ‖ C 16 G ≤ K 0 εa - c 0 σ</text>
<text><location><page_28><loc_12><loc_25><loc_17><loc_27></location>(3.17d)</text>
<text><location><page_28><loc_52><loc_22><loc_67><loc_25></location>‖ B ‖ C 16 G ≤ K 0 εa - c 0 σ</text>
<text><location><page_28><loc_12><loc_23><loc_17><loc_24></location>(3.17e)</text>
<text><location><page_28><loc_44><loc_19><loc_44><loc_21></location>‖</text>
<text><location><page_28><loc_44><loc_20><loc_48><loc_21></location>Ric[</text>
<text><location><page_28><loc_48><loc_20><loc_49><loc_21></location>G</text>
<text><location><page_28><loc_49><loc_20><loc_51><loc_21></location>] +</text>
<text><location><page_28><loc_52><loc_21><loc_53><loc_22></location>2</text>
<text><location><page_28><loc_52><loc_19><loc_53><loc_20></location>9</text>
<text><location><page_28><loc_53><loc_20><loc_54><loc_21></location>G</text>
<text><location><page_28><loc_54><loc_19><loc_55><loc_21></location>‖</text>
<text><location><page_28><loc_55><loc_20><loc_56><loc_21></location>C</text>
<text><location><page_28><loc_56><loc_20><loc_57><loc_21></location>14</text>
<text><location><page_28><loc_56><loc_19><loc_57><loc_20></location>G</text>
<text><location><page_28><loc_58><loc_19><loc_60><loc_21></location>≤</text>
<text><location><page_28><loc_60><loc_20><loc_61><loc_21></location>K</text>
<text><location><page_28><loc_61><loc_20><loc_62><loc_21></location>0</text>
<text><location><page_28><loc_62><loc_20><loc_64><loc_21></location>εa</text>
<text><location><page_28><loc_64><loc_20><loc_65><loc_22></location>-</text>
<text><location><page_28><loc_65><loc_21><loc_66><loc_22></location>c</text>
<text><location><page_28><loc_66><loc_21><loc_66><loc_21></location>0</text>
<text><location><page_28><loc_66><loc_21><loc_67><loc_22></location>σ</text>
<text><location><page_28><loc_12><loc_20><loc_17><loc_21></location>(3.17f)</text>
<text><location><page_28><loc_49><loc_16><loc_67><loc_19></location>‖ G - γ ‖ C 16 G ≤ K 0 εa - c 0 σ</text>
<text><location><page_28><loc_12><loc_17><loc_17><loc_19></location>(3.17g)</text>
<formula><location><page_28><loc_12><loc_14><loc_68><loc_16></location>‖ N ‖ C 14 G + a 2 ‖ N ‖ ˙ C 15 G + a 4 ‖ N ‖ ˙ C 16 G ≤ K 0 εa 4 - c 0 σ (3.17h)</formula>
<text><location><page_28><loc_50><loc_12><loc_51><loc_14></location>Γ</text>
<text><location><page_28><loc_53><loc_13><loc_54><loc_14></location>ˆ</text>
<text><location><page_28><loc_53><loc_12><loc_54><loc_14></location>Γ</text>
<text><location><page_28><loc_55><loc_12><loc_56><loc_13></location>C</text>
<text><location><page_28><loc_56><loc_12><loc_57><loc_13></location>15</text>
<text><location><page_28><loc_60><loc_12><loc_61><loc_14></location>K</text>
<text><location><page_28><loc_61><loc_12><loc_62><loc_13></location>0</text>
<text><location><page_28><loc_62><loc_12><loc_64><loc_14></location>εa</text>
<text><location><page_28><loc_64><loc_12><loc_65><loc_14></location>-</text>
<text><location><page_28><loc_65><loc_13><loc_66><loc_14></location>c</text>
<text><location><page_28><loc_66><loc_13><loc_66><loc_14></location>0</text>
<text><location><page_28><loc_66><loc_13><loc_67><loc_14></location>σ</text>
<text><location><page_28><loc_12><loc_12><loc_17><loc_14></location>(3.17i)</text>
<text><location><page_28><loc_49><loc_11><loc_50><loc_14></location>‖</text>
<text><location><page_28><loc_51><loc_11><loc_53><loc_14></location>-</text>
<text><location><page_28><loc_54><loc_11><loc_55><loc_14></location>‖</text>
<text><location><page_28><loc_56><loc_12><loc_57><loc_12></location>G</text>
<text><location><page_28><loc_58><loc_11><loc_60><loc_14></location>≤</text>
<text><location><page_28><loc_12><loc_8><loc_88><loc_11></location>Remark 3.18 (Bootstrap assumptions with respect to γ ) . Note again that we could equivalently make the above bootstrap assumptions with respect to H γ - and C γ -norms: For example, the</text>
<text><location><page_29><loc_12><loc_88><loc_42><loc_90></location>assumptions (3.17i) and (3.17g) imply</text>
<formula><location><page_29><loc_20><loc_85><loc_80><loc_88></location>‖ ζ ‖ C l γ /lessorsimilar a -cσ ‖ ζ ‖ C l G + ‖ ζ ‖ C /ceilingleft l -1 2 /ceilingright γ εa -cσ , ‖ T ‖ C l γ /lessorsimilar a -cσ ‖ T ‖ C l G + ‖ T ‖ C /ceilingleft l 2 /ceilingright γ εa -cσ</formula>
<text><location><page_29><loc_12><loc_79><loc_88><loc_85></location>for any smooth function ζ ∈ C ∞ (Σ t ), any Σ t -tangent tensor T and a constant c > 0. This is essentially a direct consequence of (3.5), and we will prove an improved version of this rigorously in Lemma 7.4. Applying this to each norm in C , we get</text>
<text><location><page_29><loc_12><loc_74><loc_38><loc_77></location>for some updated constant c ≥ c 0 .</text>
<formula><location><page_29><loc_12><loc_76><loc_54><loc_79></location>(3.18) C γ /lessorsimilar εa -cσ</formula>
<text><location><page_29><loc_12><loc_71><loc_88><loc_74></location>Remark 3.19 (Strategy for the bootstrap improvement) . Our goal is to improve the C -norm estimate to</text>
<text><location><page_29><loc_12><loc_57><loc_88><loc_69></location>where c 1 , K 1 > 0 are positive constants independent of σ and ε . Notice how this is actually an improvement if we choose σ suitably and then choose ε sufficiently small: Any update between K 0 and K 1 can be balanced out since we gain at least the additional prefactor ε 1 8 in each estimate, which we can then choose to have been suitably small. Similarly, we improve the power of a if we have ε 1 8 · σ -1 < c 0 c 1 . If we then retroactively choose σ large enough compared to ε but small overall - for example σ = ε 1 16 - and then ensure that max { c 0 , c 1 } ε 1 16 < 1 as well as c 1 ε 1 16 < c 0 are satisfied by choosing ε to have been small enough, we have strictly improved the bootstrap assumptions.</text>
<formula><location><page_29><loc_43><loc_68><loc_57><loc_71></location>C ≤ K 1 ε 9 8 a -c 1 ε 1 8 ,</formula>
<text><location><page_29><loc_12><loc_42><loc_88><loc_56></location>Remark 3.20 (Conventions within the bootstrap argument) . Throughout the rest of the argument, we tacitly assume t ∈ ( t Boot , t 0 ] if not stated otherwise, and we assume ε and σ to be sufficiently small. In the proof of Theorem 8.2, we will choose σ = ε 1 16 , but this explicit choice will not be used or needed up to that point. Finally, we allow c ≥ c 0 be a constant that we may update from line to line, and will similarly deal with prefactors by ' /lessorsimilar '-notation where the constant may change in each line. These updates will always be independent of σ and ε , but may depend on t 0 , and the quantities arising from the FLRW reference solution. Hence, we not only assume c 0 σ < 1, but cσ < 1 throughout the argument.</text>
<section_header_level_1><location><page_29><loc_31><loc_40><loc_69><loc_41></location>4. Big Bang stability: A priori estimates</section_header_level_1>
<text><location><page_29><loc_12><loc_27><loc_88><loc_39></location>In this section, we collect strong low order C G -norm estimates that follow as an immediate consequence from the bootstrap assumptions, starting with key estimates at the base level and followed by weaker, but still improved estimates at higher levels. Finally, we collect a differentiation formula for integrals with respect to vol G as well as a Sobolev estimate that lays the groundwork for energy coercivity. In particular, using the strong C G -norm estimates, said estimate proves that moving between energies and norms at most an error involving lower order energies of the controlled variable and curvature energies, scaled by a -c √ ε .</text>
<text><location><page_29><loc_12><loc_23><loc_88><loc_26></location>4.1. Strong C 0 G -estimates. First, we establish a pointwise bound on the lapse that actually holds irrespective of the bootstrap assumptions:</text>
<text><location><page_29><loc_12><loc_19><loc_88><loc_22></location>Lemma 4.1 (Maximum principle for the lapse) . The lapse remains positive and bounded throughout the evolution:</text>
<formula><location><page_29><loc_12><loc_16><loc_57><loc_18></location>(4.1) n = N +1 ∈ (0 , 3]</formula>
<text><location><page_29><loc_12><loc_11><loc_88><loc_16></location>Proof. Let t ∈ R + be arbitrary and let n min be the minimum of n over Σ t at ( t, x min ). Then, (∆ g n )( t, x min ) > 0 holds. If n min were nonpositive, (2.17a) would lead to the following contradiction:</text>
<formula><location><page_29><loc_14><loc_7><loc_86><loc_12></location>0 ≥ -12 πC 2 a -6 -1 3 a -2 + n min [ 1 3 a -2 +4 πC 2 a -6 + 〈 ˆ k, ˆ k 〉 g +8 π | ∂ 0 φ | 2 ] = ∆ g n ( t, x min ) > 0</formula>
<text><location><page_30><loc_12><loc_88><loc_59><loc_90></location>This shows n > 0, and the upper bound follows analogously.</text>
<text><location><page_30><loc_87><loc_88><loc_88><loc_90></location>/square</text>
<text><location><page_30><loc_12><loc_84><loc_88><loc_87></location>The following estimate will be essential in dealing with borderline terms throughout the bootstrap argument:</text>
<text><location><page_30><loc_12><loc_81><loc_64><loc_83></location>Lemma 4.2 (Strong C 0 G estimates) . The following estimates hold:</text>
<formula><location><page_30><loc_12><loc_78><loc_54><loc_81></location>‖ Ψ ‖ C 0 G /lessorsimilar ε (4.2a)</formula>
<formula><location><page_30><loc_12><loc_77><loc_54><loc_78></location>Σ C 0 /lessorsimilar ε (4.2b)</formula>
<text><location><page_30><loc_46><loc_76><loc_51><loc_78></location>‖ ‖ G</text>
<formula><location><page_30><loc_12><loc_74><loc_53><loc_76></location>‖ E ‖ C 0 G /lessorsimilar (4.2c)</formula>
<formula><location><page_30><loc_53><loc_75><loc_54><loc_76></location>ε</formula>
<text><location><page_30><loc_12><loc_69><loc_88><loc_73></location>Proof. (4.2a): From (2.32a), we obtain the following using Lemma 4.1 for n , the bootstrap assumptions (3.17h) and (3.17b) and that ˙ a /similarequal a 2 by (2.3):</text>
<formula><location><page_30><loc_32><loc_67><loc_68><loc_69></location>| ∂ t Ψ | /lessorsimilar εa 5 -cσ + εa 1 -cσ + εa 1 -cσ | Ψ | + εa 1 -cσ</formula>
<text><location><page_30><loc_12><loc_65><loc_70><loc_66></location>After integration, we thus obtain using the initial data assumption (3.10):</text>
<formula><location><page_30><loc_27><loc_56><loc_73><loc_65></location>| Ψ( t ) | /lessorsimilar | Ψ( t 0 ) | + ∫ t 0 t εa ( s ) 1 -cσ ds + ∫ t 0 t εa ( s ) 1 -cσ | Ψ( s ) | ds /lessorsimilar ε ( 1 + ∫ t 0 t a ( s ) 1 -cσ ds ) + ∫ t 0 t εa ( s ) 1 -cσ | Ψ( s ) | ds</formula>
<text><location><page_30><loc_12><loc_54><loc_88><loc_56></location>By (2.6), the integral over a 1 -cσ is bounded since cσ < 1, so the Gronwall lemma now yields (4.2a).</text>
<formula><location><page_30><loc_15><loc_51><loc_30><loc_53></location>(4.2b): Notice that</formula>
<formula><location><page_30><loc_12><loc_48><loc_81><loc_50></location>(4.3) ∂ t | Σ | 2 G = ( ∂ t Σ /sharp ) l m (Σ /sharp ) m l +(Σ /sharp ) l m ( ∂ t Σ /sharp ) m l = 2( ∂ t Σ /sharp ) l m (Σ /sharp ) m l ≤ 2 | ∂ t Σ /sharp | G | Σ | G .</formula>
<text><location><page_30><loc_12><loc_46><loc_87><loc_48></location>Now, we consider (2.28d) and, using the bootstrap assumptions (3.17h), (3.17f) and (3.17b), get:</text>
<formula><location><page_30><loc_24><loc_35><loc_47><loc_42></location>+ √ 3 | N | · ( 4 πC 2 a -3 + 1 9 a ) /lessorsimilar εa 1 -cσ | Σ | G + εa 1 -cσ</formula>
<formula><location><page_30><loc_18><loc_39><loc_83><loc_46></location>| ∂ t Σ /sharp | G /lessorsimilar τ | N || Σ /sharp | G + |∇ /sharp ∇ N | G a + | N +1 | a ∣ ∣ ∣ ∣ Ric[ G ] /sharp + 2 9 G /sharp ∣ ∣ ∣ ∣ G + | N +1 | a |∇ /sharp φ ∇ φ | G</formula>
<text><location><page_30><loc_12><loc_32><loc_82><loc_35></location>We can now apply Lemma 11.2 with f = | Σ | 2 G , and thus have along with (3.10) and (4.3):</text>
<formula><location><page_30><loc_34><loc_28><loc_66><loc_33></location>| Σ | G ( t ) ≤ | Σ | G ( t 0 ) + ∫ t 0 t | ∂ t Σ /sharp | G ( s ) ds /lessorsimilar ε</formula>
<text><location><page_30><loc_15><loc_26><loc_81><loc_28></location>(4.2c): Using the constraint equation (2.29c) and that 〈 G, E 〉 G = tr G E = 0, one sees</text>
<formula><location><page_30><loc_24><loc_22><loc_77><loc_27></location>| E | 2 G = 〈 a 4 ( Ric[ G ] + 2 9 G ) -˙ aa 2 Σ -Σ /circledot G Σ -4 πa 4 ∇ φ ∇ φ, E 〉 G .</formula>
<text><location><page_30><loc_12><loc_15><loc_88><loc_22></location>Then, applying the bootstrap assumptions (3.17f) and (3.17b) shows the Ricci and matter terms are bounded by εa 4 -cσ | E | G , and the a priori estimate (4.2b) along with ˙ aa 2 /similarequal 1 by (2.3) bounds the remaining terms by ε | E | G . The statement then follows by dividing by | E | G and taking the supremum. /square</text>
<text><location><page_30><loc_12><loc_8><loc_88><loc_14></location>Note that, in the proof of (4.2b), it was essential that we used (2.28d) instead of (2.28c), since using the latter would incur terms of the type | ∂ t G | G | Σ | 2 G and a -3 | Σ | 3 G when computing the time derivative of | Σ | 2 G , which, at this point, behave like εa -3 -cσ | Σ | 2 G , and thus not yield the sharp estimate (or even an improved estimate) that we will need to control borderline terms.</text>
<text><location><page_31><loc_12><loc_87><loc_88><loc_90></location>4.2. Strong low order C G -norm estimates. Now, we can prove the main supremum norm estimates in this section:</text>
<text><location><page_31><loc_12><loc_84><loc_77><loc_85></location>Lemma 4.3 (Strong low order C G -norm estimates) . The following estimates hold:</text>
<formula><location><page_31><loc_12><loc_80><loc_60><loc_84></location>‖ Ψ ‖ C 13 G /lessorsimilar εa -c √ ε (4.4a)</formula>
<formula><location><page_31><loc_12><loc_75><loc_61><loc_79></location>‖ G -γ ‖ C 12 G /lessorsimilar √ εa -c √ ε (4.4c)</formula>
<formula><location><page_31><loc_12><loc_78><loc_60><loc_81></location>‖ Σ ‖ C 12 G /lessorsimilar εa -c √ ε (4.4b)</formula>
<formula><location><page_31><loc_12><loc_73><loc_61><loc_76></location>‖ G -1 -γ -1 ‖ C 12 G /lessorsimilar √ εa -c √ ε (4.4d)</formula>
<formula><location><page_31><loc_12><loc_67><loc_62><loc_70></location>‖ Ric[ G ] + 2 9 G ‖ C 10 G /lessorsimilar √ εa -c √ ε (4.4f)</formula>
<formula><location><page_31><loc_12><loc_70><loc_62><loc_73></location>‖∇ φ ‖ C 12 G /lessorsimilar √ εa -c √ ε (4.4e)</formula>
<formula><location><page_31><loc_12><loc_64><loc_61><loc_67></location>‖ B ‖ C 11 G /lessorsimilar εa 2 -c √ ε (4.4g)</formula>
<formula><location><page_31><loc_12><loc_61><loc_60><loc_65></location>‖ E ‖ C 12 G /lessorsimilar εa -c √ ε (4.4h)</formula>
<text><location><page_31><loc_12><loc_58><loc_88><loc_61></location>Proof. Before going into the individual estimates, we collect the following commutator term estimates from the expressions in (11.10a)-(11.10b):</text>
<formula><location><page_31><loc_12><loc_54><loc_72><loc_57></location>‖ [ ∂ t , ∇ J ] ζ ‖ C 0 G /lessorsimilar a -3 ‖ N +1 ‖ C J -1 G ‖ Σ ‖ C J -1 G ‖ ζ ‖ C J -1 G + ˙ a a ‖ N ‖ C J -1 G ‖ ζ ‖ C J -1 G (4.5)</formula>
<text><location><page_31><loc_12><loc_42><loc_88><loc_50></location>With this in hand, we will prove each estimate by iterating over the derivative order as long as the bootstrap assumptions can be applied. In each step, we use the previously obtained estimates at lower order to control the commutator term (with some additional care for T = Σ which we need to consider first), while we can use similar arguments to those at order 0 to control the 'core' of the evolution equations.</text>
<formula><location><page_31><loc_12><loc_49><loc_87><loc_54></location>‖ [ ∂ t , ∇ J ] T ‖ C 0 G /lessorsimilar a -3 ‖ N +1 ‖ C J G ( ‖∇ J Σ ‖ C 0 G ‖ T ‖ C 0 G + ‖ Σ ‖ C J -1 G ‖ T ‖ C J -1 G ) + ˙ a a ‖ N ‖ C J G ‖ T ‖ C J -1 G (4.6)</formula>
<text><location><page_31><loc_12><loc_39><loc_88><loc_42></location>To start out, we apply (4.2b) on Σ and the bootstrap assumption (3.17h) on N to the rescaled evolution equations (2.28a)-(2.28b) and deduce</text>
<formula><location><page_31><loc_12><loc_36><loc_71><loc_38></location>(4.7) | ∂ t G ± 1 | G = | ∂ t ( G ± 1 -γ ± 1 ) | G /lessorsimilar εa -3 + εa 1 -cσ /lessorsimilar εa -3 .</formula>
<text><location><page_31><loc_15><loc_34><loc_30><loc_35></location>(4.4b): We assume</text>
<formula><location><page_31><loc_12><loc_30><loc_57><loc_34></location>(4.8) ‖ Σ ‖ C J -1 G /lessorsimilar εa -c √ ε</formula>
<text><location><page_31><loc_12><loc_28><loc_88><loc_30></location>to be satisfied for some J 1 , . . . , 12 (For J = 1, this is true by (4.2b)). Observe the following:</text>
<formula><location><page_31><loc_30><loc_25><loc_68><loc_28></location>∂ t |∇ Σ | G = 2 〈 ∂ t ∇ Σ , ∇ Σ 〉 G + ∂ t G -∗ ∇ Σ ∗ ∇</formula>
<text><location><page_31><loc_32><loc_26><loc_71><loc_30></location>∈ { } J 2 J J 1 J J Σ</text>
<text><location><page_31><loc_12><loc_22><loc_88><loc_25></location>Now, we commute (2.28d) with ∇ J : As before, ∇ J ∂ t Σ is bounded by εa -cσ for any admissible J . Hence and using (4.7),</text>
<formula><location><page_31><loc_26><loc_17><loc_74><loc_22></location>∂ t |∇ J Σ | 2 G /lessorsimilar εa -3 |∇ J Σ | 2 G + ( εa 1 -cσ + ‖ [ ∂ t , ∇ J ]Σ ‖ C 0 G ) |∇ J Σ | G</formula>
<text><location><page_31><loc_12><loc_16><loc_76><loc_18></location>is satisfied. Looking at the commutator term using (4.6), we have with (4.2b) that</text>
<formula><location><page_31><loc_25><loc_13><loc_75><loc_15></location>‖ [ ∂ t , ∇ J ]Σ ‖ C 0 G /lessorsimilar εa -3 ‖ Σ ‖ ˙ C J G + a -3 · ‖ Σ ‖ 2 C J -1 G + εa 1 -cσ ‖ Σ ‖ C J -1 G .</formula>
<text><location><page_31><loc_12><loc_11><loc_29><loc_12></location>Altogether, we obtain</text>
<formula><location><page_31><loc_28><loc_6><loc_72><loc_11></location>∂ t |∇ J Σ | 2 G /lessorsimilar ( εa -3 ‖ Σ ‖ ˙ C J G + εa -cσ + ε 2 a -3 -c √ ε ) |∇ J Σ | G .</formula>
<text><location><page_32><loc_12><loc_87><loc_88><loc_90></location>With Lemma 11.2 as well as the initial data assumption (3.10) and the integral formula (2.6) with p = c √ ε , this implies</text>
<formula><location><page_32><loc_29><loc_81><loc_72><loc_87></location>|∇ J Σ | G ( t ) /lessorsimilar ∫ t 0 t εa -3 ‖ Σ ‖ ˙ C J G (Σ s ) ds + ε ( 1 + √ εa -c √ ε )</formula>
<text><location><page_32><loc_12><loc_80><loc_83><loc_82></location>and consequently, after taking the supremum on the left and applying the Gronwall lemma,</text>
<formula><location><page_32><loc_42><loc_77><loc_58><loc_80></location>‖ Σ ‖ ˙ C J G (Σ s ) /lessorsimilar εa -c √ ε .</formula>
<text><location><page_32><loc_12><loc_73><loc_88><loc_76></location>Combining this with (4.8) proves the statement up to order J , and hence shows (4.4b) by iterating the argument up to J = 12.</text>
<text><location><page_32><loc_15><loc_70><loc_38><loc_71></location>(4.4a): We again assume that</text>
<formula><location><page_32><loc_55><loc_68><loc_57><loc_70></location>√</formula>
<formula><location><page_32><loc_12><loc_66><loc_57><loc_69></location>(4.9) ‖ Ψ ‖ C J -1 G /lessorsimilar εa -c ε</formula>
<text><location><page_32><loc_12><loc_63><loc_44><loc_66></location>holds for J ∈ { 1 , 2 , . . . , 13 } . Observe that</text>
<formula><location><page_32><loc_19><loc_60><loc_81><loc_64></location>| ∂ t ∇ J Ψ | G /lessorsimilar a ‖ N +1 ‖ C J +1 G ‖∇ φ ‖ C J +1 G + ˙ a a ‖∇ N ‖ C J G (1 + ‖ Ψ ‖ C J G ) + ‖ [ ∂ t , ∇ J ]Ψ ‖ C 0 G</formula>
<text><location><page_32><loc_12><loc_55><loc_88><loc_60></location>By (3.17h), (3.17b) and (3.17a), the first two summands can be bounded (up to constant) by εa 1 -cσ . By (4.9), (4.4b) and (3.17h) and using (4.5), the commutator term is bounded (up to constant) by ε 2 a -3 -c √ ε . Altogether,</text>
<text><location><page_32><loc_12><loc_51><loc_40><loc_52></location>follows. Inserting this and (4.7) into</text>
<text><location><page_32><loc_12><loc_46><loc_33><loc_47></location>implies, with Lemma 11.2,</text>
<formula><location><page_32><loc_20><loc_37><loc_80><loc_46></location>|∇ J Ψ | G ( t ) ≤|∇ J Ψ | ( t 0 ) + ∫ t 0 t ( 1 2 | ∂ t G -1 ||∇ J Ψ | G + | ∂ t ∇ J Ψ | G ) ( s ) ds /lessorsimilar ε 2 + ∫ t 0 t ( εa ( s ) -3 |∇ J Ψ( s, · ) | G + εa ( s ) 1 -cσ + ε 2 a ( s ) -3 -c √ ε ) ds .</formula>
<formula><location><page_32><loc_30><loc_31><loc_70><loc_36></location>|∇ J Ψ | G ( t ) /lessorsimilar εa ( t ) -c √ ε + ∫ t 0 t εa ( s ) -3 |∇ J Ψ( s, · ) | G ds</formula>
<text><location><page_32><loc_12><loc_27><loc_88><loc_31></location>The Gronwall lemma, applying (2.5) and taking the supremum over Σ t then implies |∇ J Ψ | ˙ C J G /lessorsimilar εa -c √ ε . This proves (4.2a) by iterating over J and adding up the individual seminorms.</text>
<text><location><page_32><loc_15><loc_24><loc_68><loc_25></location>(4.4c)-(4.4d): Note that (4.7) implies (4.4c) at order 0 since one has</text>
<formula><location><page_32><loc_15><loc_20><loc_85><loc_23></location>| ∂ t ( | G -γ | G ) 2 | /lessorsimilar | ∂ t G -1 | G | G -γ | 2 G + | ∂ t ( G -γ ) | G | G -γ | G /lessorsimilar εa -3 (1 + | G -γ | G ) | G -γ | G</formula>
<text><location><page_32><loc_12><loc_17><loc_88><loc_20></location>which we can apply the Gronwall lemma to after integrating, along with (2.7) for the error term, as in the proof of (4.4b).</text>
<text><location><page_32><loc_12><loc_14><loc_80><loc_17></location>For higher orders, commuting (2.28a) with ∇ J and inserting (4.4b) and (3.17h) implies</text>
<text><location><page_32><loc_12><loc_10><loc_15><loc_11></location>with</text>
<formula><location><page_32><loc_26><loc_12><loc_74><loc_15></location>‖ ∂ t ∇ J ( G -γ ) ‖ C 0 G /lessorsimilar εa -3 -c √ ε + εa 1 -cσ + ‖ [ ∂ t , ∇ J ]( G -γ ) ‖ C 0 G</formula>
<formula><location><page_32><loc_27><loc_6><loc_73><loc_11></location>‖ [ ∂ t , ∇ J ]( G -γ ) ‖ C 0 G /lessorsimilar ( εa -3 -c √ ε + εa 1 -cσ ) ‖ G -γ ‖ C J -1 G .</formula>
<formula><location><page_32><loc_38><loc_52><loc_63><loc_55></location>| ∂ t ∇ J Ψ | G /lessorsimilar εa 1 -cσ + ε 2 a -3 -c √ ε</formula>
<formula><location><page_32><loc_28><loc_46><loc_72><loc_51></location>| ∂ t ( |∇ J Ψ | 2 G ) | ≤ | ∂ t G -1 | G |∇ J Ψ | 2 G +2 | ∂ t ∇ J Ψ | G · |∇ J Ψ | G</formula>
<text><location><page_32><loc_12><loc_36><loc_29><loc_37></location>We obtain using (2.6):</text>
<text><location><page_33><loc_12><loc_86><loc_88><loc_90></location>Once again doing the same iterative argument over J ≤ 12 and assuming the estimate to hold up to J -1, this altogether becomes</text>
<formula><location><page_33><loc_38><loc_83><loc_62><loc_86></location>‖ ∂ t ∇ J ( G -γ ) ‖ C 0 G /lessorsimilar εa -3 -c √ ε ,</formula>
<text><location><page_33><loc_12><loc_81><loc_27><loc_83></location>implying with (2.6)</text>
<formula><location><page_33><loc_28><loc_76><loc_72><loc_81></location>‖∇ J ( G -γ ) ‖ C 0 G /lessorsimilar ε 2 + ε ∫ t 0 t a ( s ) -3 -c √ ε ds /lessorsimilar √ εa -c √ ε .</formula>
<text><location><page_33><loc_12><loc_74><loc_54><loc_76></location>The argument for G -1 -γ -1 is completely analogous.</text>
<text><location><page_33><loc_12><loc_70><loc_88><loc_73></location>(4.4e): We only prove the statement for C 0 G , the full estimate extends from there by the same iterative arguments as above. Considering (2.32b), Lemma 4.1, (4.2a) and (2.3), we have</text>
<formula><location><page_33><loc_37><loc_66><loc_63><loc_69></location>| ∂ t ∇ φ | G /lessorsimilar a -3 ( |∇ Ψ | G + |∇ N | G )</formula>
<text><location><page_33><loc_12><loc_65><loc_59><loc_66></location>and thus, with (4.4a) and the bootstrap assumption (3.17h),</text>
<formula><location><page_33><loc_42><loc_61><loc_58><loc_65></location>| ∂ t ∇ φ | G /lessorsimilar εa -3 -c √ ε .</formula>
<text><location><page_33><loc_12><loc_60><loc_30><loc_61></location>With (4.7), this implies</text>
<text><location><page_33><loc_12><loc_56><loc_84><loc_57></location>and the statement follows as usual by applying Lemma 11.2, (2.7) and the Gronwall lemma.</text>
<formula><location><page_33><loc_34><loc_57><loc_66><loc_60></location>| ∂ t |∇ φ | G | 2 /lessorsimilar εa -3 |∇ φ | 2 G + εa -3 -c √ ε |∇ φ | G</formula>
<unordered_list>
<list_item><location><page_33><loc_12><loc_51><loc_88><loc_54></location>(4.4f): This follows as in the proof of (4.4c) using (2.33) and (2.28a) and their commuted analogues.</list_item>
</unordered_list>
<text><location><page_33><loc_12><loc_44><loc_88><loc_51></location>Once again, for C 0 G , we have (4.4g): This is obtained immediately from commuting (2.29d) with ∇ J and applying (4.4b). Notice that the Levi-Civita tensor can be absorbed into the implicit constants since | ε [ G ] | = √ 6 holds (see (11.2c)).</text>
<text><location><page_33><loc_12><loc_39><loc_88><loc_44></location>(4.4h): This follows like in the proof of (4.2c) from applying (3.17f), (3.17b) and (4.4b) to the constraint equation (2.29c) commuted with ∇ J .</text>
<text><location><page_33><loc_87><loc_39><loc_88><loc_40></location>/square</text>
<unordered_list>
<list_item><location><page_33><loc_12><loc_33><loc_88><loc_36></location>4.3. Other useful a priori observations. Before moving on to the energy estimates, we collect a differentiation identity and lay the groundwork for energy coercivity:</list_item>
</unordered_list>
<text><location><page_33><loc_12><loc_29><loc_88><loc_33></location>Lemma 4.4 (The volume form and differentiation of integrals) . Let µ G = √ det G denote the volume element with regards to G . It satisfies</text>
<formula><location><page_33><loc_12><loc_25><loc_65><loc_28></location>(4.10) ∂ t µ G = 1 2 µ G ( G -1 ) ij ∂ t G ij = -Nτµ G ,</formula>
<text><location><page_33><loc_12><loc_23><loc_26><loc_24></location>and hence one has</text>
<formula><location><page_33><loc_12><loc_20><loc_57><loc_22></location>(4.11) ‖ µ G -µ γ ‖ C 0 G /lessorsimilar ε .</formula>
<text><location><page_33><loc_12><loc_17><loc_65><loc_19></location>on (Σ t ) t ∈ ( t Boot ,t 0 ] . Further, for any differentiable function ζ , one has</text>
<formula><location><page_33><loc_12><loc_13><loc_67><loc_18></location>(4.12) ∂ t ∫ M ζ vol G = ∫ M ∂ t ζ vol G -∫ M Nτ · ζ vol G</formula>
<text><location><page_33><loc_12><loc_10><loc_71><loc_13></location>Proof. From (2.28a), we obtain ( G -1 ) ij ∂ t G ij = -2 Nτ , and (4.10) follows by</text>
<formula><location><page_33><loc_34><loc_7><loc_66><loc_11></location>∂ t µ G = 1 2 √ det G ( G -1 ) ij ∂ t G ij = -Nτµ G .</formula>
<text><location><page_34><loc_12><loc_88><loc_67><loc_90></location>Hence, we have using (3.17h) and the initial data estimate (3.14) that</text>
<formula><location><page_34><loc_31><loc_83><loc_70><loc_88></location>| µ G -µ γ | ( t, · ) /lessorsimilar ε + ∫ t 0 t εa ( s ) 1 -cσ | µ G -µ γ | ( s. · ) ds</formula>
<text><location><page_34><loc_46><loc_80><loc_48><loc_81></location>µ γ</text>
<text><location><page_34><loc_12><loc_80><loc_88><loc_83></location>holds, and thus (4.11) after applying the Gronwall lemma. Finally, we obtain (4.12) by writing vol G = µ G vol γ and inserting (4.10) /square</text>
<text><location><page_34><loc_12><loc_74><loc_88><loc_79></location>Lemma 4.5 (Preliminary Sobolev norm estimates) . Let ζ be a scalar function and T be a symmetric Σ t -tangent (0 , 2) -tensor, and let l ∈ { 1 , . . . , 9 } . Then, on ( t Boot , t 0 ] , the following estimates are satisfied: For l > 5 , one has:</text>
<formula><location><page_34><loc_12><loc_71><loc_51><loc_74></location>‖∇ 2 ζ ‖ 2 L 2 G /lessorsimilar ‖ ∆ ζ ‖ 2 L 2 G + a -c √ ε ‖∇ ζ ‖ 2 L 2 G (4.13a)</formula>
<formula><location><page_34><loc_12><loc_61><loc_84><loc_67></location>2 l +1 ∑ m =1 ‖ ζ ‖ 2 ˙ H m G /lessorsimilar ‖∇ ∆ l ζ ‖ 2 L 2 G + a -c √ ε ( l -1 ∑ m =0 ‖∇ ∆ m ζ ‖ 2 L 2 G + ‖∇ ζ ‖ 2 C 2 l -12 G E ( ≤ 2 l -2) (Ric , · ) ) (4.13c)</formula>
<formula><location><page_34><loc_12><loc_66><loc_80><loc_72></location>‖ ζ ‖ 2 H 2 l G /lessorsimilar ‖ ∆ l ζ ‖ 2 L 2 G + a -c √ ε ( l -1 ∑ m =0 ‖ ∆ m ζ ‖ 2 L 2 G + ‖ ζ ‖ 2 C 2 l -12 G E ( ≤ 2 l -3) (Ric , · ) ) (4.13b)</formula>
<formula><location><page_34><loc_12><loc_56><loc_86><loc_62></location>2 l +1 ∑ m =1 ‖∇ ζ ‖ 2 ˙ H m G /lessorsimilar ‖∇ 2 ∆ l ζ ‖ 2 L 2 G + a -c √ ε ( l -1 ∑ m =0 ‖∇ 2 ∆ m ζ ‖ 2 L 2 G + ‖∇ ζ ‖ 2 C 2 l -11 G E ( ≤ 2 l -1) (Ric , · ) ) (4.13d)</formula>
<formula><location><page_34><loc_12><loc_49><loc_81><loc_55></location>‖ T ‖ 2 H 2 l G /lessorsimilar ‖ ∆ l T ‖ 2 L 2 G + a -c √ ε ( l -1 ∑ m =0 ‖ ∆ m T ‖ 2 L 2 G + ‖ T ‖ 2 C 2 l -11 G E ( ≤ 2 l -2) (Ric , · ) ) (4.14a)</formula>
<text><location><page_34><loc_12><loc_55><loc_15><loc_56></location>and</text>
<formula><location><page_34><loc_12><loc_44><loc_84><loc_50></location>2 l +1 ∑ m =1 ‖ T ‖ 2 ˙ H m G /lessorsimilar ‖∇ ∆ l T ‖ 2 L 2 G + a -c √ ε ( l -1 ∑ m =0 ‖∇ ∆ m T ‖ 2 L 2 G + ‖ T ‖ 2 C 2 l -10 G E ( ≤ 2 l -1) (Ric , · ) ) (4.14b)</formula>
<text><location><page_34><loc_12><loc_42><loc_84><loc_44></location>More precisely, the Ricci energy terms can be dropped in all of the above estimates for l ≤ 5 .</text>
<text><location><page_34><loc_12><loc_41><loc_87><loc_42></location>Remark 4.6. We stress that Lemma 4.5 is crucial for everything that follows in multiple ways:</text>
<text><location><page_34><loc_12><loc_30><loc_88><loc_39></location>Firstly, the L 2 G -norms containing ζ and T on the right hand sides above except (4.13d) are in precisely the form the energies in Definition 3.9 take. Hence, this is what will actually yield near-coercivity of our energies since the C G -norms can be controlled by a -c √ ε or better using the a priori estimates from Lemma 4.3, as well as (3.17h) for the lapse.This will be shown more explicitly as an intermediary step in improving the bootstrap assumptions for C (see proof of Corollary 7.3).</text>
<text><location><page_34><loc_12><loc_21><loc_88><loc_29></location>Secondly, a downside of using ∆ as the main differential operator to commute with the Einstein scalar-field system is that it creates error terms that we can only bound by Sobolev norms and not directly express as energies. Thus, we need a way to translate this information back to energies to formulate energy inequalities. A lot of this is done 'under the hood' in the error term estimates in subsection 11.4.</text>
<text><location><page_34><loc_12><loc_9><loc_88><loc_19></location>Finally, some top order terms also do not appear in a way that their L 2 -norm is directly the square root of an energy (see, for example, the term a ∇ 2 ∆ L 2 N in (2.35)), and some borderline terms would lead to nonintegrable divergences if we were to incur additional divergences in estimation (see, for example, the first term in (11.14a)). Lemma 4.5 precisely provides a way to relate these terms to energies . Additionally, by applying these estimates for terms of the form ∆ L 2 ζ and ∆ L 2 T , one can avoid high order curvature energies that run risk of breaking the energy hierarchy.</text>
<text><location><page_35><loc_12><loc_87><loc_88><loc_90></location>Proof. Since the arguments for all of the inequalities above are very similar, we only prove (4.14a) in full and then briefly adress the other estimates.</text>
<text><location><page_35><loc_12><loc_83><loc_88><loc_87></location>Denoting ˜ T i 1 ...i 2 l k 1 k 2 = ∇ i 1 . . . ∇ i 2 l T k 1 k 2 , we compute with the commutator formula (11.6c) and strong C G -norm estimate (4.4f):</text>
<formula><location><page_35><loc_16><loc_67><loc_84><loc_83></location>∫ M |∇ 2 ˜ T | 2 G = -∫ M 〈∇ ˜ T , ∆ ∇ ˜ T 〉 G vol G = -∫ M 〈∇ ˜ T , ∇ ∆ ˜ T 〉 G vol G + ∫ M ∇ ˜ T ∗ [ ∇ Ric[ G ] ∗ ˜ T +Ric[ G ] ∗ ∇ ˜ T ] vol G /lessorsimilar ∫ M | ∆ ˜ T | 2 G vol G +(1 + ‖ Ric[ G ] + 2 G ‖ C 1 G ) · [∫ M | ˜ T | 2 G vol G + ∫ M |∇ ˜ T | 2 G vol G ] /lessorsimilar ∫ M | ∆ ˜ T | 2 G + a -c √ ε ∫ M | ˜ T | 2 G vol G</formula>
<text><location><page_35><loc_12><loc_65><loc_55><loc_67></location>In the final step, we used integration by parts to obtain</text>
<formula><location><page_35><loc_17><loc_60><loc_83><loc_65></location>a -c √ ε ∫ M |∇ ˜ T | 2 G vol G ≤ ∫ M | ∆ ˜ T | G · a -c √ ε | ˜ T | G vol G /lessorsimilar ∫ M ( | ∆ ˜ T | 2 G + a -2 c √ ε | ˜ T | 2 G ) vol G</formula>
<text><location><page_35><loc_12><loc_56><loc_88><loc_60></location>and updated c . This already shows (4.14a) for l = 1. Assume now that (4.14a) holds up to some l ∈ N , l ≤ 9 and any symmetric Σ t -tangent (0 , 2)-tensor field. By applying (11.6d), we have</text>
<formula><location><page_35><loc_12><loc_49><loc_75><loc_56></location>∆ ˜ T = ∇ 2 l ∆ T +[∆ , ∇ 2 ] ∇ 2 l -2 T + · · · + ∇ 2 l -2 [∆ , ∇ 2 ] T = ∇ 2 l ∆ T + ∑ I Ric + I T =2 l ∇ I Ric (Ric[ G ] + 2 9 G ) ∗ ∇ I T T + G ∗ ∇ 2 l T (4.15)</formula>
<text><location><page_35><loc_12><loc_46><loc_88><loc_49></location>Subsequently, we have for l > 5 using the strong C G -norm estimate (4.4f) for any Ricci term of order 10 or lower that</text>
<text><location><page_35><loc_12><loc_37><loc_88><loc_41></location>and get the same estimate without the final term for l ≤ 5 . By assumption, we can estimate ‖ ∆ T ‖ 2 H 2 l G , ‖ T ‖ 2 H 2 l G and ‖ Ric[ G ] + 2 G ‖ 2 H 2 l G as in (4.14a), and get the following for l > 5:</text>
<formula><location><page_35><loc_12><loc_40><loc_88><loc_46></location>‖ T ‖ 2 ˙ H 2( l +1) G = ∫ M |∇ 2 ˜ T | 2 G vol G /lessorsimilar ‖ ∆ T ‖ 2 ˙ H 2 l G + ( 1 + √ εa -c √ ε ) ‖ T ‖ 2 H 2 l G + ‖ T ‖ 2 C 2 l -11 G ‖ Ric[ G ] + 2 9 G ‖ 2 H 2 l G ,</formula>
<formula><location><page_35><loc_12><loc_18><loc_89><loc_37></location>∫ M |∇ 2 l +2 T | 2 G vol G /lessorsimilar [ ‖ ∆ l ∆ T ‖ 2 L 2 G + a -c √ ε l -1 ∑ m =0 ‖ ∆ m +1 T ‖ 2 L 2 G + a -c √ ε ‖ ∆ T ‖ 2 C 2 l -12 G E ( ≤ 2 l -2) (Ric , · ) ] + [ a -c √ ε ‖ ∆ l T ‖ 2 L 2 G + a -c √ ε l -1 ∑ m =0 ‖ ∆ m T ‖ 2 L 2 G + a -c √ ε ‖ T ‖ 2 C 2 l -12 G E ( ≤ 2 l -2) (Ric , · ) ] + a -c √ ε ‖ T ‖ 2 C 2 l -11 G [ E (2 l ) (Ric , · ) + ( E ( ≤ 2 l -2) (Ric , · ) + εa -c √ ε E ( ≤ 2 l -2) (Ric , · ) )] /lessorsimilar ‖ ∆ l +1 T ‖ 2 L 2 G + a -c √ ε ( l ∑ m =0 ‖ ∆ m T ‖ 2 L 2 G + ‖ T ‖ 2 C 2 l -10 G E ( ≤ 2 l ) (Ric , · ) )</formula>
<text><location><page_35><loc_12><loc_15><loc_88><loc_18></location>For l = 5, we get analogous estimates dropping the Ricci energies in the first two lines, and for l = 4, the same with all curvature terms dropped.</text>
<text><location><page_35><loc_12><loc_11><loc_88><loc_15></location>To prove the statement for l +1, it now remains to be shown that ‖ T ‖ 2 ˙ H 2 l +1 G can be bounded by the same right hand side (up to constant) as above. By integration by parts, one has</text>
<formula><location><page_35><loc_34><loc_7><loc_66><loc_10></location>‖∇ 2 l +1 T ‖ 2 L 2 G /lessorsimilar ‖∇ 2 l T ‖ 2 L 2 G + ‖ ∆ ∇ 2 l T ‖ 2 L 2 G ,</formula>
<text><location><page_36><loc_12><loc_87><loc_88><loc_90></location>where the latter tensor is precisely ∆ ˜ T which we just treated, and the former is covered by the induction assumption at order 2 l . So, (4.14a) now follows for l +1, and thus by iteration up to l = 10.</text>
<text><location><page_36><loc_12><loc_80><loc_88><loc_85></location>The proof of (4.14b) is analogous - we note that since we actually only needed a strong estimate on ‖ Ric[ G ] + 2 G ‖ C 9 G for the previous inequality, but (4.4f) holds at C 10 G , this gives enough room to extend the argument in full despite the extra derivative order.</text>
<text><location><page_36><loc_12><loc_72><loc_88><loc_80></location>For both, note that we only need to estimate the Ricci terms in the L 2 G -norm if one can't apply the a priori estimate (4.4f) to all ∇ I Ric Ric[ G ] that occur in (4.15), and thus we could easily adjust the proof such that the Ricci energy does not occur in any of the proofs as long as 2 l -1 ≤ 10 is satisfied, so for l ≤ 5.</text>
<text><location><page_36><loc_12><loc_65><loc_88><loc_71></location>The estimates (4.13b)-(4.13d) are proved identically, the only difference being that one order of curvature less enters in the commutator terms in (4.15), leading to one order less in curvature in total. For (4.13a), we note that we can avoid incurring any L 2 -norm by carefully repeating the argument we made for ˜ T using (11.6a):</text>
<formula><location><page_36><loc_25><loc_55><loc_88><loc_65></location>∫ M |∇ 2 ζ | 2 G vol G = ∫ M -〈∇ ζ, ∇ ∆ ζ 〉 G + ∫ M Ric[ G ] ∗ ∇ ζ ∗ ∇ ζ vol G /lessorsimilar ∫ M | ∆ ζ | 2 G + a -c √ ε ∫ M |∇ ζ | 2 G vol G /square</formula>
<section_header_level_1><location><page_36><loc_29><loc_50><loc_72><loc_52></location>5. Big Bang stability: Elliptic lapse estimates</section_header_level_1>
<text><location><page_36><loc_12><loc_45><loc_88><loc_49></location>In this section, we study the elliptic structure of the equations (2.30a)-(2.30b), which admit estimates controlling (time-scaled) lapse energies by other energy quantities. To this end, we recast these equations as follows:</text>
<text><location><page_36><loc_12><loc_41><loc_88><loc_44></location>Definition 5.1 (Elliptic operators) . For any (sufficiently regular) scalar function ζ on Σ t , we define the following differential operators:</text>
<text><location><page_36><loc_12><loc_27><loc_82><loc_40></location>L ζ = a 4 ∆ ζ -f · ζ, f = 1 3 a 4 +12 πC 2 + 〈 Σ , Σ 〉 G +8 π Ψ 2 +16 πC Ψ ︸ ︷︷ ︸ =: F (5.1a) ˜ L ζ = a 4 ∆ ζ -˜ f · ζ, ˜ f = 1 3 a 4 +12 πC 2 + a 4 [ R [ G ] + 2 3 -8 π |∇ φ | 2 G ] ︸ ︷︷ ︸ = ˜ F (5.1b) Note that the lapse equations (2.30a), respectively (2.30b), now read</text>
<formula><location><page_36><loc_56><loc_25><loc_62><loc_26></location>˜ ˜ F.</formula>
<formula><location><page_36><loc_12><loc_24><loc_60><loc_26></location>(5.2) L N = F, respectively L N =</formula>
<text><location><page_36><loc_12><loc_22><loc_32><loc_23></location>Furthermore, observe that</text>
<formula><location><page_36><loc_12><loc_19><loc_72><loc_21></location>[ L , ∆] ζ =∆ f · ζ +2 〈∇ f, ∇ ζ 〉 G = ∆ F · ζ +2 〈∇ F, ∇ ζ 〉 G , (5.3a)</formula>
<formula><location><page_36><loc_12><loc_16><loc_54><loc_19></location>[ ˜ L , ∆] ζ =∆ ˜ F · ζ +2 〈∇ ˜ F, ∇ ζ 〉 G . (5.3b)</formula>
<text><location><page_36><loc_12><loc_8><loc_88><loc_16></location>5.1. Elliptic lapse estimates with L . We first study the elliptic operator L , which will admit weak lapse energy estimates in terms of scalar field quantities and Σ, up to curvature errors, that can in particular be utilized at high orders without having to resort to higher derivative levels. Before moving on to the estimates themselves, we collect a couple of inequalities we can deduce from the bootstrap assumptions and strong C G -norm estimates.</text>
<text><location><page_37><loc_12><loc_87><loc_88><loc_90></location>Remark 5.2. There exists a constant K > 0 such that, for ε > 0 small enough, the following estimates hold:</text>
<unordered_list>
<list_item><location><page_37><loc_16><loc_81><loc_88><loc_86></location>· F ≥ -Kε , and equivalently f ≥ 12 πC 2 -Kε . This is ensured by (4.2b) and (4.2a). In particular, we can assume ε to have been small enough such that f -6 πC 2 can be bounded from below by a positive constant that is independent of ε (for example 3 πC 2 ).</list_item>
<list_item><location><page_37><loc_16><loc_79><loc_67><loc_82></location>· |∇ f | G = |∇ F | G ≤ Kεa -c √ ε . This is given by (4.4b) and (4.4a).</list_item>
</unordered_list>
<text><location><page_37><loc_12><loc_76><loc_82><loc_79></location>Lemma 5.3 (Elliptic estimates with L ) . Consider scalar functions ζ, Z on Σ t that satisfy</text>
<text><location><page_37><loc_12><loc_73><loc_16><loc_74></location>Then,</text>
<formula><location><page_37><loc_12><loc_74><loc_53><loc_76></location>(5.4) L ζ = Z .</formula>
<formula><location><page_37><loc_12><loc_70><loc_66><loc_72></location>(5.5) a 4 ‖ ∆ ζ ‖ L 2 G + a 2 ‖∇ ζ ‖ L 2 G + ‖ ζ ‖ L 2 G /lessorsimilar ‖ Z ‖ L 2 G</formula>
<text><location><page_37><loc_12><loc_65><loc_88><loc_69></location>Proof. The proof follows along the same lines as that of [Spe18, Lemma 16.5]: First, we obtain the following by multiplying (5.4) with -ζ and integrating: and using that f -6 πC 2 is bounded from below by a positive constant (see the first point in Remark 5.2):</text>
<text><location><page_37><loc_12><loc_59><loc_48><loc_60></location>Next, we multiply (5.4) with a 4 ∆ ζ and obtain</text>
<formula><location><page_37><loc_37><loc_60><loc_63><loc_65></location>∫ M ( a 4 |∇ ζ | 2 G + | ζ | 2 )vol G /lessorsimilar ‖ Z ‖ 2 L 2 G</formula>
<formula><location><page_37><loc_14><loc_53><loc_86><loc_59></location>∫ M ( a 8 | ∆ ζ | 2 + a 4 |∇ ζ | 2 G f ) vol G ≤ ∫ M 1 2 | Z | 2 + a 8 2 | ∆ ζ | 2 + ( 1 2 | ζ | 2 + a 4 2 |∇ ζ | 2 G ) a 2 ‖∇ f ‖ L ∞ G vol G</formula>
<formula><location><page_37><loc_32><loc_47><loc_68><loc_53></location>∫ M ( a 8 | ∆ ζ | 2 + a 4 |∇ ζ | 2 G ) vol G /lessorsimilar (1 + ε ) ‖ Z ‖ 2 L 2 G</formula>
<text><location><page_37><loc_12><loc_53><loc_81><loc_54></location>Using the second point in Remark 5.2 as well as the previous step, we can now conclude</text>
<text><location><page_37><loc_12><loc_47><loc_44><loc_48></location>and thus the statement after rearranging.</text>
<text><location><page_37><loc_87><loc_47><loc_88><loc_48></location>/square</text>
<text><location><page_37><loc_12><loc_41><loc_88><loc_45></location>Corollary 5.4 (Intermediary elliptic lapse estimate with L ) . The following estimates hold for any l ∈ { 0 , . . . , 10 } :</text>
<formula><location><page_37><loc_27><loc_31><loc_73><loc_41></location>a 4 ‖ ∆ l +1 N ‖ L 2 G + a 2 ‖∇ ∆ l N ‖ L 2 G + ‖ ∆ l N ‖ L 2 G /lessorsimilar ‖ ∆ l F ‖ L 2 G + εa -c √ ε ‖ F ‖ H 2( l -1) G ︸ ︷︷ ︸ not present for l =0 + ε 2 a 4 -cσ √ E ( ≤ 2 l -4) (Ric , · ) ︸ ︷︷ ︸ not present for l ≤ 1</formula>
<text><location><page_37><loc_12><loc_26><loc_88><loc_31></location>Proof. We prove the statement by induction over l ∈ N : For l = 0, the estimates immediately follow from (5.2) and Lemma 5.3. Assume the statement to be satisfied up to l -1 for some l ∈ N 0 , l ≤ 11. We get applying (5.3a) iteratively:</text>
<formula><location><page_37><loc_33><loc_20><loc_67><loc_25></location>L ∆ l N = 2 l -1 ∑ I =1 ∇ I F ∗ ∇ 2 l -I N +( N +1)∆ l F</formula>
<text><location><page_37><loc_12><loc_19><loc_60><loc_20></location>Applying Lemma 5.3 to ζ = ∆ l N as well as Lemma 4.1 yields</text>
<formula><location><page_37><loc_18><loc_13><loc_82><loc_18></location>a 4 ‖ ∆ l +1 N ‖ L 2 G + a 2 ‖∇ ∆ l N ‖ L 2 G + ‖ ∆ l N ‖ L 2 G /lessorsimilar 2 l -1 ∑ I =1 ‖∇ I F ∗ ∇ 2 l -I N ‖ L 2 G + ‖ ∆ l F ‖ L 2 G</formula>
<text><location><page_37><loc_12><loc_10><loc_88><loc_13></location>Hence, using (4.13c) (replacing l with l -1) and (3.17h), (4.4a) and (4.4b) to estimate low order terms, we get</text>
<formula><location><page_37><loc_25><loc_7><loc_58><loc_10></location>a 4 ‖ ∆ l +1 N ‖ L 2 G + a 2 ‖∇ ∆ l N ‖ L 2 G + ‖ ∆ l N ‖ L 2 G</formula>
<formula><location><page_38><loc_23><loc_81><loc_77><loc_91></location>/lessorsimilar ‖ ∆ l F ‖ L 2 G + εa -c √ ε ( l -1 ∑ m =0 ‖∇ ∆ m N ‖ L 2 G + εa 4 -cσ √ E ( ≤ 2 l -4) (Ric , · ) ) + εa 4 -cσ ( ‖ F ‖ H 2( l -1) G + ‖∇ ∆ l -1 F ‖ L 2 G + εa -c √ ε √ E ( ≤ 2 l -4) (Ric , · ) )</formula>
<text><location><page_38><loc_12><loc_79><loc_77><loc_81></location>For the top order lapse term, we can redistribute the divergent prefactor as follows:</text>
<formula><location><page_38><loc_28><loc_76><loc_72><loc_79></location>εa -c √ ε ‖∇ ∆ l -1 N ‖ L 2 G /lessorsimilar ε ‖ ∆ l N ‖ L 2 G + εa -2 c √ ε ‖ ∆ l -1 N ‖ L 2 G</formula>
<text><location><page_38><loc_12><loc_71><loc_88><loc_76></location>The lower order lapse terms as well as ‖∇ ∆ l -1 F ‖ L 2 G can be estimated similarly, just without having to redistribute the prefactor. Updating c > 0 and rearranging then yields the statement at order l for suitably small ε > 0, and thus the entire statement after iteration. /square</text>
<text><location><page_38><loc_12><loc_67><loc_75><loc_69></location>Corollary 5.5 (Lapse energy estimates with L ) . For any l ∈ { 0 , . . . , 9 } , one has</text>
<formula><location><page_38><loc_23><loc_55><loc_77><loc_67></location>a 8 E (2( l +1)) ( N, · ) + a 4 E (2 l +1) ( N, · ) + E (2 l ) ( N, · ) /lessorsimilar ε 2 E (2 l ) (Σ , · ) + E (2 l ) ( φ, · ) + ε 2 a -c √ ε [ E ( ≤ 2( l -1)) (Σ , · ) + E ( ≤ 2( l -1)) ( φ, · ) ] ︸ ︷︷ ︸ not present for l =0 ( √ )</formula>
<text><location><page_38><loc_12><loc_51><loc_88><loc_54></location>Proof. Note that, by Corollary 5.4, all that needs to be done is to relate all Sobolev norms of F that occur to the respective energies. Schematically, we have</text>
<formula><location><page_38><loc_25><loc_52><loc_55><loc_59></location>+ ε 4 a -c ε + ε 2 a 8 -cσ E ( ≤ 2 l -3) (Ric , · ) ︸ ︷︷ ︸ not present for l ≤ 1</formula>
<formula><location><page_38><loc_18><loc_45><loc_82><loc_50></location>∆ l F = 16 π (∆ l Ψ)(Ψ + C ) + 2 〈 ∆ l Σ , Σ 〉 G + 2 l -1 ∑ I =1 ( ∇ I Ψ ∗ ∇ 2 l -I Ψ+ ∇ I Σ ∗ ∇ 2 l -I Σ ) .</formula>
<text><location><page_38><loc_12><loc_35><loc_88><loc_45></location>For the first two terms, we can use (4.2a) and (4.2b) to bound | Σ | G and | Ψ+ C | by ε and 1 up to constant, respectively. For the remaining terms, we similarly always bound the lower order in L ∞ G with (4.4a)-(4.4b) and bound the higher order with the energy estimates in Lemma 4.5. Further, we can use (3.8) to redistribute divergent prefactors onto energies of order l -2 and lower. This already incurs the terms on the right hand side of the claimed estimate, and the lower order norms of F only incur at equivalent or weaker error terms. /square</text>
<text><location><page_38><loc_12><loc_26><loc_88><loc_34></location>5.2. Elliptic lapse estimates with ˜ L . While the estimates in the previous subsection are useful at high orders, they aren't enough to close the bootstrap assumptions for N . This can be achieved by deriving estimates in terms of ˜ L - however, due to the explicit presence of Ricci terms in this version of the lapse equation, we use this to bound N at lower orders. Since the arguments are largely identical to the ones above, we only sketch the proofs.</text>
<text><location><page_38><loc_12><loc_17><loc_88><loc_25></location>Remark 5.6. Note that, when replacing f by ˜ f and F by ˜ F in Remark 5.2, the same statements hold for a suitable constant K . In fact, the bootstrap assumptions on Ric[ G ] and ∇ φ even imply ‖ ˜ F ‖ C 1 G /lessorsimilar εa 4 -cσ , noting | R [ G ] + 2 / 3 | G ≤ | G -1 | G | Ric[ G ] + 2 / 9 G | G /lessorsimilar | Ric[ G ] + 2 / 9 G | G and |∇ R [ G ] | G = |∇ ( R [ G ] + 2 / 3 ) | G /lessorsimilar | Ric[ G ] + 2 / 9 G | G .</text>
<text><location><page_38><loc_12><loc_15><loc_54><loc_17></location>Lemma 5.7. Any scalar functions ζ and Z such that</text>
<formula><location><page_38><loc_47><loc_12><loc_53><loc_15></location>˜ L ζ = Z</formula>
<text><location><page_38><loc_12><loc_10><loc_31><loc_12></location>holds satisfy the estimate</text>
<formula><location><page_38><loc_34><loc_7><loc_67><loc_10></location>a 4 ‖ ∆ ζ ‖ L 2 G + a 2 ‖∇ ζ ‖ L 2 G + ‖ ζ ‖ L 2 G /lessorsimilar ‖ Z ‖ L 2 G .</formula>
<text><location><page_39><loc_12><loc_86><loc_88><loc_90></location>Proof. The proof follows identically to Lemma 5.3 since all tools relating to f and F used in proving these statements were collected in Remark 5.2, and these extend to ˜ L by Remark 5.6. /square</text>
<text><location><page_39><loc_12><loc_83><loc_55><loc_86></location>Corollary 5.8. For l ∈ { 0 , . . . , 8 } , the following holds:</text>
<formula><location><page_39><loc_18><loc_81><loc_82><loc_84></location>a 8 E (2( l +1)) ( N, · ) + a 4 E (2 l +1) ( N, · ) + E (2 l ) ( N, · ) /lessorsimilar a 8 E ( ≤ 2 l ) (Ric , · ) + εa 8 -c √ ε ‖∇ φ ‖ 2 H 2 l G</formula>
<text><location><page_39><loc_12><loc_77><loc_88><loc_80></location>Proof. As in the proof of Corollary 5.4, this follows by commuting ˜ L with ∆ l iteratively and applying (4.4e) and (4.4f) to bound lower order terms within the nonlinearities. /square</text>
<section_header_level_1><location><page_39><loc_27><loc_74><loc_73><loc_75></location>6. Big Bang stability: Energy and norm estimates</section_header_level_1>
<text><location><page_39><loc_12><loc_63><loc_88><loc_73></location>In this section, we derive energy estimates for matter variables and the geometric quantities as well as Sobolev norm estimates for spatial derivatives of φ and for metric quantities. To derive all of the inequalities in this section beside the elliptic inequality in Lemma 6.10 and the bound on ∇ φ in Lemma 6.5, we will use the same basic strategy. Hence, we give a brief overview on the form our integral inequalities are going to take and how we intend to obtain improved energy bounds from there:</text>
<text><location><page_39><loc_12><loc_53><loc_88><loc_62></location>Remark 6.1 (Integral inequalities and the Gronwall argument) . Let F L denote an energy or a squared Sobolev(-type) norm at derivative level L ∈ 2 N , for example E ( L ) ( φ, · ). To derive an integral inequality for F L , we will take its time derivative, apply the respective commuted evolution equations in the integrand, estimate the resulting terms and integrate that inequality. Schematically, the resulting integral inequalities for F L then take the following form:</text>
<formula><location><page_39><loc_83><loc_39><loc_85><loc_41></location>ds</formula>
<text><location><page_39><loc_12><loc_34><loc_88><loc_38></location>For some inequalities, we won't be able to derive any benefitial ε -prefactors in the penultimate line. For example, for E ( L ) (Σ , · ), linear lapse terms in the evolution of Σ incur a term of the form</text>
<formula><location><page_39><loc_16><loc_37><loc_82><loc_54></location>F L ( t ) + ∫ t 0 t 〈 ultimately nonnegative contributions 〉 ds /lessorsimilar F L ( t 0 ) + ∫ t 0 t ( ε 1 8 a ( s ) -3 + a ( s ) -1 -c √ ε ) F L ( s ) ds + ∫ t 0 t a ( s ) -3 〈 other energies/squared Sobolev norms at same derivative level 〉 ds + ∫ t 0 t a ( s ) -3 -c √ ε 〈 energies/squared Sobolev norms at derivative levels up to L -2 〉</formula>
<formula><location><page_39><loc_33><loc_29><loc_67><loc_35></location>∫ t 0 t a ( s ) -3 · a ( s ) 4 ‖ ∆ L 2 N ‖ ˙ H 2 G · √ E ( L ) (Σ , s ) ds</formula>
<text><location><page_39><loc_12><loc_22><loc_88><loc_30></location>on the right hand side, which after applying lapse energy estimates creates ε -1 8 E ( L ) ( φ, · ) on the right. However, combining the respective inequalities for the core energy mechanism at each derivative level with appropriate scaling, this will then combine to an inequality of the following form for a total energy, which we informally denote by F total ,L :</text>
<formula><location><page_39><loc_12><loc_18><loc_16><loc_20></location>(6.1)</formula>
<formula><location><page_39><loc_27><loc_5><loc_73><loc_22></location>F total ,L ( t ) + ∫ t 0 t 〈 nonnegative quantity 〉 ds /lessorsimilar F total ,L ( t 0 ) + ∫ t 0 t ( ε 1 8 a ( s ) -3 + a ( s ) -1 -c √ ε ) F total ,L ( s ) ds + ∫ t 0 t ε 1 8 a ( s ) -3 -cε 1 8 〈 already improved terms 〉 ds ︸ ︷︷ ︸ not present for L =0</formula>
<formula><location><page_40><loc_30><loc_83><loc_68><loc_91></location>+ ε 1 4 F total ,L ( t ) + √ ε · 〈 small lower order terms 〉 ( t ) ︸ ︷︷ ︸ not present for L =0</formula>
<text><location><page_40><loc_12><loc_82><loc_88><loc_85></location>In the mentioned example, scaling E ( L ) (Σ , · ) by ε 1 4 turns the offending term int ε 1 8 a ( s ) -3 E ( L ) ( φ, s ), which can be absorbed into the first line.</text>
<text><location><page_40><loc_12><loc_76><loc_88><loc_80></location>Applying the Gronwall lemma (see Lemma 11.1) and the initial data assumption (which implies F total ,L /lessorsimilar ε 4 ) then yields</text>
<formula><location><page_40><loc_24><loc_58><loc_76><loc_76></location>F total ,L ( t ) /lessorsimilar ( ε 4 + ∫ t 0 t ε 1 8 a ( s ) -3 -cε 1 8 〈 already improved terms 〉 ds ︸ ︷︷ ︸ not present for L =0 ) · exp ( K · ∫ t 0 t ε 1 8 a ( s ) -3 + a ( s ) -1 -c √ ε ds ) + ε 1 4 F total,L ( t ) + √ ε · 〈 small lower order terms 〉 ( t ) ︸ ︷︷ ︸ not present for L =0</formula>
<text><location><page_40><loc_12><loc_55><loc_88><loc_60></location>for some constant K > 0. By (2.7) and (2.6), the exponential factor can be bounded by a -cε 1 8 , up to constant and updating c > 0. Furthermore, ε 1 4 F total,L ( t ) in the final line can be absorbed by the left hand side after updating the implicit constant in ' /lessorsimilar '.</text>
<text><location><page_40><loc_48><loc_54><loc_49><loc_55></location>1</text>
<text><location><page_40><loc_12><loc_43><loc_88><loc_54></location>Hence, for L = 0, this implies F total , 0 /lessorsimilar ε 4 a -cε 8 , and thus leads to improved bounds for base level energy quantities (see Remark 3.19 for the precise scaling hierarchy that will achieve). By iterating this argument for L > 0, the already improved terms will then be bounded (at worst) by ε 4 a -cε 1 8 , and (2.6) shows that the first line can be bounded by ε 4 a -cε 1 8 after updating c . This allows us to bound F total ,L by ε 4 a -cε 1 8 for any L up to and including top order.</text>
<text><location><page_40><loc_12><loc_30><loc_88><loc_42></location>Finally, we mention that, to control energies at order L , we need to consider scaled energies at order L +1 within F total,L - this arises since the scalar field occurs at first order in the evolution equations for E and B . We avoid losing derivatives by employing the div-curl-estimate in Lemma 6.10 at order L + 1, which allows us to control a 4 E ( L +1) (Σ , · ) by quantities at order L . This is precisely what generates the non-integral terms in the schematics above. We note that it is crucial that the scalar field occurs at no worse scaling than a -1 in (2.38a)-(2.38b) - else, moving to these time-scaled estimates at order L + 1 would lose too many powers of a and lead to exponentially divergent terms after applying the Gronwall argument.</text>
<text><location><page_40><loc_12><loc_22><loc_88><loc_29></location>Recall that L 2 G -norm estimates for error terms arising in the Laplace-commuted equations in Lemma 2.11 are collected in Section 11.4. Low order estimates (in particular estimates for L = 2) could often be improved if needed by more carefully avoiding curvature error terms, but we refrain from doing so where it is not necessary to keep estimates as unified as possible.</text>
<section_header_level_1><location><page_40><loc_12><loc_20><loc_60><loc_21></location>6.1. Integral and energy estimates for the scalar field.</section_header_level_1>
<text><location><page_40><loc_12><loc_14><loc_88><loc_19></location>6.1.1. Scalar field energy estimates. Over the following two lemmas, we prove the core energy estimates to control the matter variables, which are immediately prepared differently at base, intermediate and top order for the total energy estimates in Section 7.</text>
<text><location><page_40><loc_12><loc_11><loc_81><loc_13></location>Lemma 6.2. [Even order scalar field energy estimates] Let t ∈ ( t Boot , t 0 ] . Then, one has</text>
<formula><location><page_40><loc_12><loc_7><loc_72><loc_12></location>E (0) ( φ, t ) + ∫ t 0 t ˙ a ( s ) a ( s ) 3 E (1) ( N,s ) + ˙ a ( s ) a ( s ) E (0) ( N,s ) ds (6.2)</formula>
<formula><location><page_41><loc_12><loc_60><loc_76><loc_91></location>/lessorsimilar E (0) ( φ, t 0 ) + ∫ t 0 t εa ( s ) -3 E (0) ( φ, s ) + εa -3 E (0) (Σ , s ) ds . Further, for any L ∈ 2 N , 2 ≤ L ≤ 18 , the following estimate is satisfied: E ( L ) ( φ, t ) + ∫ t 0 t ˙ a ( s ) a ( s ) 3 E ( L +1) ( N,s ) + ˙ a ( s ) a ( s ) E ( L ) ( N,s ) ds (6.3) /lessorsimilar E ( L ) ( φ, t 0 ) + ∫ t 0 t ( εa ( s ) -3 + a ( s ) -1 -c √ ε ) E ( L ) ( φ, s ) ds + ∫ t 0 t εa ( s ) -3 E ( L ) (Σ , s ) + ε 3 2 a ( s ) -3 E ( L -2) (Ric , · ) ds + ∫ t 0 t √ εa ( s ) -3 -c √ ε E ( ≤ L -2) ( φ, s ) + εa ( s ) -3 -c √ ε E ( ≤ L -2) (Σ , s ) ds + ∫ t 0 t ε 3 2 a ( s ) -3 -c √ ε E ( ≤ L -4) (Ric , s ) ds ︸ ︷︷ ︸ if L ≥ 4</formula>
<text><location><page_41><loc_12><loc_49><loc_88><loc_62></location>Remark 6.3. This proof relies on two mechanisms: Firstly, we use the structure of the wave equation and integration by parts to cancel the highest order scalar field derivative terms. Getting this cancellation is what necessitates scaling the potential term in the scalar field energy by a 4 . Secondly, we deal with the highest order lapse terms using the elliptic structure of the (Laplacecommuted) lapse equation - both in an indirect way by invoking the elliptic energy estimate in Corollary 5.5 as well as by directly inserting (2.37a) to cancel some ill-behaved terms. While the framework significantly differs from the scalar field energy estimates [Spe18], these two core mechanisms also appear there and play similarly crucial roles.</text>
<text><location><page_41><loc_12><loc_42><loc_88><loc_48></location>Proof. Since the arguments are essentially the same, we will only write down the proof for L ≥ 2 in full and make short comments throughout the argument which terms do not occur for L = 0. We use the evolution equations (2.39a) and (2.39b) and Lemma 4.4 to compute for L ≥ 2:</text>
<formula><location><page_41><loc_12><loc_29><loc_73><loc_32></location>-2 a ( N +1) 〈∇ ∆ L 2 Ψ , ∇ ∆ L 2 φ 〉 G -2 Ca 〈∇ ∆ L 2 N, ∇ ∆ L 2 φ 〉 G (6.4b)</formula>
<formula><location><page_41><loc_12><loc_31><loc_86><loc_43></location>-∂ t E ( L ) ( φ, · ) = ∫ M -2 ∂ t ∆ L 2 Ψ · ∆ L 2 Ψ -2 a 4 〈 ∂ t ∇ ∆ L 2 φ, ∇ ∆ L 2 φ 〉 G -a 4 ( ∂ t G -1 ) ij ∇ i ∆ L 2 φ ∇ j ∆ L 2 φ -3 N ˙ a a [ | ∆ L 2 Ψ | 2 + a 4 |∇ ∆ L 2 φ | 2 G ] -4 ˙ a a · a 4 |∇ ∆ L 2 φ | 2 G vol G = ∫ M ( -2 a ( N +1)∆ L 2 +1 φ -2 a 〈∇ ∆ L 2 N, ∇ φ 〉 G +6 C ˙ a a ∆ L 2 N ) · (∆ L 2 Ψ) (6.4a)</formula>
<formula><location><page_41><loc_12><loc_26><loc_84><loc_29></location>-2( P L,Border + P L,Junk ) · ∆ L 2 Ψ -2 a 4 〈 Q L,Border + Q L,Junk , ∇ ∆ L 2 φ 〉 G (6.4c)</formula>
<formula><location><page_41><loc_12><loc_19><loc_73><loc_24></location>-3 N ˙ a a [ | ∆ L 2 Ψ | 2 + a 4 |∇ ∆ L 2 φ | 2 G ] -4 ˙ a a · a 4 |∇ ∆ L 2 φ | 2 G vol G (6.4e)</formula>
<formula><location><page_41><loc_12><loc_23><loc_73><loc_27></location>+2( N +1) a · (Σ /sharp ) ij ∇ i ∆ L 2 φ ∇ j ∆ L 2 φ -2 N ˙ a a · a 4 |∇ ∆ L 2 φ | 2 G (6.4d)</formula>
<text><location><page_41><loc_12><loc_13><loc_88><loc_18></location>Note that, for L = 0, the equivalent equality holds where the borderline and junk terms are replaced by -2 a 4 Ψ 〈∇ N, ∇ φ 〉 G (to verify this, insert (2.32a) and (2.32b) instead of (2.39a) and (2.39b)). We now go through (6.4a)-(6.4e) term by term:</text>
<text><location><page_41><loc_12><loc_11><loc_55><loc_13></location>After integrating by parts, the first term in (6.4a) reads</text>
<formula><location><page_41><loc_12><loc_7><loc_75><loc_12></location>(6.5) ∫ M 2 a ( N +1) 〈∇ ∆ L 2 φ, ∇ ∆ L 2 Ψ 〉 G +2 a 〈∇ N, ∇ ∆ L 2 φ 〉 G · ∆ L 2 Ψvol G .</formula>
<text><location><page_42><loc_12><loc_87><loc_88><loc_90></location>The first term precisely cancels the first term in (6.4b), while we can use the bootstrap assumption (3.17h) to estimate the other term in (6.5) up to constant by</text>
<formula><location><page_42><loc_30><loc_83><loc_70><loc_86></location>εa 3 -cσ · a 2 ‖∇ ∆ L 2 φ ‖ L 2 G ‖ ∆ L 2 Ψ ‖ L 2 G /lessorsimilar εa 3 -cσ E ( L ) ( φ, · ) .</formula>
<text><location><page_42><loc_12><loc_80><loc_88><loc_83></location>For the second term in (6.4a), we use (4.4e) to estimate ∇ φ and Corollary 5.5 at order L to deal with the lapse, getting</text>
<formula><location><page_42><loc_12><loc_61><loc_84><loc_80></location>∣ ∣ ∣ ∣ ∫ M 2 a 〈∇ ∆ L 2 N, ∇ φ 〉 G · ∆ L 2 Ψvol G ∣ ∣ ∣ ∣ /lessorsimilar √ εa -1 -c √ ε √ a 4 E ( L +1) ( N, · ) √ E ( L ) ( φ, · ) /lessorsimilar εa -1 · a 4 E ( L +1) ( N, · ) + a -1 -c √ ε E ( L ) ( φ, · ) /lessorsimilar ε 3 a -1 E ( L ) (Σ , · ) + a -1 -c √ ε E ( L ) ( φ, · ) + ε 2 a -1 -c √ ε E ( ≤ L -2) (Σ , · ) + ε 2 a -1 -c √ ε E ( ≤ L -2) ( φ, · ) + ε 2 a -1 -c √ ε E ( ≤ L -3) (Ric , · ) ︸ ︷︷ ︸ not present for L =2 . Repeating this argument for L = 0, the last two lines do not appear.</formula>
<text><location><page_42><loc_12><loc_56><loc_88><loc_59></location>To deal with the remaining term in (6.4a), we can insert the following zero on the right hand side of the differential equality, where the equality (6.6) holds due to (2.37a):</text>
<formula><location><page_42><loc_12><loc_41><loc_77><loc_56></location>0 = -3 8 π ˙ aa 3 ∫ M div G ( ∇ ∆ L 2 N · ∆ L 2 N ) vol G = -3 8 π ˙ aa 3 ∫ M ∆ L 2 +1 N · ∆ L 2 N + |∇ ∆ L 2 N | 2 G vol G = ∫ M -3 8 π ˙ aa 3 |∇ ∆ L 2 N | 2 G -3 8 π ( 1 3 ˙ aa 3 +12 πC 2 ˙ a a ) | ∆ L 2 N | 2 (6.6) -6 C ˙ a a ∆ L 2 N · ∆ L 2 Ψ -3 8 π ˙ aa 3 [ N L,Border + N L,Junk ] · ∆ L 2 N vol G</formula>
<text><location><page_42><loc_12><loc_32><loc_88><loc_40></location>Note that the first line has a negative sign, so (after absorbing a few terms into it without changing the sign, see namely lapse quantities in (6.7) and (6.8a)), we pull it to the left hand side of the differential equality. Further, the first term in the second line of (6.6) precisely cancels the third term in (6.4a). That leaves the borderline and junk terms in (6.6), for which we use (11.17b) and (11.19d) (along with ˙ a /similarequal a -2 due to (2.3)) to get for L ≥ 4:</text>
<text><location><page_42><loc_12><loc_16><loc_70><loc_17></location>Again, the same estimate holds for L = 0 with the last two lines dropped.</text>
<text><location><page_42><loc_12><loc_12><loc_88><loc_16></location>From (6.4a)-(6.4b), only the term -2 Ca 〈∇ ∆ L 2 N, ∇ ∆ L 2 φ 〉 G still needs to be handled: Using the inequality (2.8) arising from the Friedman equation, we can estimate this by</text>
<formula><location><page_42><loc_26><loc_15><loc_74><loc_33></location>3 8 π ˙ aa 3 ∣ ∣ ∣ ∣ ∫ M [ N L,Border + N L,Junk ] · ∆ L 2 N vol G ∣ ∣ ∣ ∣ /lessorsimilar εa -3 [ E ( L ) ( φ, · ) + E ( L ) (Σ , · ) + E ( L ) ( N, · ) ] + εa -3 -c √ ε [ E ( ≤ L -2) ( φ, · ) + E ( ≤ L -2) (Σ , · ) + E ( ≤ L -2) ( N, · ) ] + ε 3 a -3 E ( ≤ L -2) (Ric , · ) + ε 3 a -3 -c √ ε E ( ≤ L -3) (Ric , · ) ︸ ︷︷ ︸ not present for L =2</formula>
<formula><location><page_42><loc_12><loc_7><loc_85><loc_12></location>(6.7) ∫ M 2 √ 3 4 π ˙ aa 3 |∇ ∆ L 2 N | G |∇ ∆ L 2 φ | G vol G ≤ ∫ M 4˙ aa 3 |∇ ∆ L 2 φ | 2 G + 3 16 π ˙ aa 3 |∇ ∆ L 2 N | 2 G vol G .</formula>
<text><location><page_43><loc_12><loc_87><loc_88><loc_90></location>Note that the first term precisely cancels the final term in (6.4e), while the second term can be absorbed into the first term in (6.6) while preserving that term's sign.</text>
<text><location><page_43><loc_12><loc_78><loc_88><loc_85></location>To bound the error terms in (6.4c), we insert the the borderline term estimates (11.17d) and (11.17e) as well as the junk term estimates (11.19f) and (11.19h), where (3.8) is used to estimate odd order by even order energies where needed. Furthermore, observe that we can estimate the Q L -terms as</text>
<text><location><page_43><loc_12><loc_72><loc_88><loc_75></location>so all borderline and junk terms arising from it, beside the scalar field energies, are dominated by terms occurring elsewhere.</text>
<formula><location><page_43><loc_40><loc_74><loc_60><loc_79></location>( a 2 ‖ Q L ‖ L 2 G ) · √ E ( L ) ( φ, · ) ,</formula>
<text><location><page_43><loc_12><loc_66><loc_88><loc_72></location>Finally, all terms that remain, namely (6.4d) and the first term in (6.4e), can be bounded by εa -3 E ( L ) ( φ, · ) due to the strong base level estimate (4.2b) and (3.17h). In summary, and always only keeping the worst terms for each energy and squared norm, this yields for L ≥ 4:</text>
<formula><location><page_43><loc_12><loc_52><loc_85><loc_65></location>-E · E · a E · ) /lessorsimilar ( εa -3 + a -1 -c √ ε ) E ( L ) ( φ, · ) + ( εa -3 + √ εa -1 -c √ ε )( a 4 E ( L +1) ( N, · ) + E ( L ) ( N, · ) ) (6.8a) + εa -3 E ( L ) (Σ , · ) + ε 3 2 a -3 E ( L -2) (Ric , · ) + εa -3 -c √ ε E ( ≤ L -2) ( φ, · ) (6.8b) + εa -3 -c √ ε E ( ≤ L -2) (Σ , · ) + [ εa -3 -c √ ε + √ εa -1 -c √ ε ] E ( ≤ L -2) ( N, · ) (6.8c) 3 √ (6.8d)</formula>
<formula><location><page_43><loc_23><loc_64><loc_55><loc_66></location>∂ t ( L ) ( φ, ) + ˙ aa 3 ( L +1) ( N, ) + ˙ a ( L ) ( N,</formula>
<formula><location><page_43><loc_22><loc_48><loc_43><loc_54></location>+ ε 2 a -3 -c ε E ( ≤ L -4) (Ric , · ) ︸ ︷︷ ︸ not present for L =2</formula>
<text><location><page_43><loc_12><loc_43><loc_88><loc_49></location>The lapse energies in (6.8a) can now also be absorbed into those on the left hand side of the inequality by updating the implicit constant in ' /lessorsimilar '. We can treat the lower order lapse energies in (6.8c) with Corollary 5.5 and see that the resulting terms are all dominated by terms we already have on the right hand side of the inequality above.</text>
<text><location><page_43><loc_12><loc_38><loc_88><loc_41></location>Inserting these estimates and integrating over ( t, t 0 ] then yields (6.3) for L ≥ 4, and the statement for L = 2 is obtained completely analogously.</text>
<text><location><page_43><loc_12><loc_36><loc_68><loc_38></location>As mentioned earlier, (6.4c) is replaced by the following term for L = 0:</text>
<formula><location><page_43><loc_13><loc_31><loc_87><loc_36></location>∫ M -2 a Ψ 〈∇ N, ∇ φ 〉 G vol G /lessorsimilar εa -3 ∫ M a 2 |∇ N | G · a 2 |∇ φ | G vol G /lessorsimilar ε ˙ aa 3 E (1) ( N, · ) + εa -3 E (0) ( φ, · ) ,</formula>
<text><location><page_43><loc_12><loc_27><loc_88><loc_31></location>Here, we applied (4.2a) and (2.3). Both of these terms can be absorbed into terms that are already present, and (6.2) then follows by dealing with terms in ∂ t E (0) ( φ, · ) as described and integrating. /square</text>
<text><location><page_43><loc_12><loc_24><loc_88><loc_27></location>To close the argument, we will need a scaled scalar field energy estimate at the odd orders L +1, which is not covered by the previous lemma and we hence establish separately:</text>
<text><location><page_43><loc_12><loc_20><loc_82><loc_23></location>Lemma 6.4 (Odd order scalar field energy estimate) . For L ∈ 2 N , 2 ≤ L ≤ 18 , we have:</text>
<text><location><page_43><loc_12><loc_19><loc_16><loc_20></location>(6.9)</text>
<formula><location><page_43><loc_14><loc_6><loc_86><loc_19></location>a ( t ) 4 E ( L +1) ( φ, t ) + ∫ t 0 t { ˙ a ( s ) a ( s ) 7 E ( L +2) ( N,s ) + ˙ a ( s ) a ( s ) 3 E ( L +1) ( N,s ) } ds /lessorsimilar a ( t 0 ) 4 E ( L +1) ( φ, t 0 ) + ∫ t 0 t ( εa ( s ) -3 + a ( s ) -1 -c √ ε ) · a ( s ) 4 E ( L +1) ( φ, s ) ds + ∫ t 0 t { εa ( s ) -3 · a ( s ) 4 E ( L +1) (Σ , s ) + ( εa ( s ) -3 + a ( s ) -1 -c √ ε ) E ( L ) ( φ, s ) + εa ( s ) -3 E ( L ) (Σ , s )</formula>
<text><location><page_44><loc_12><loc_74><loc_80><loc_90></location>+ εa ( s ) -1 -c √ ε · a ( s ) 4 E ( L -1) (Ric , s ) + ( ε 3 a -3 + εa -1 -c √ ε ) E ( L -2) (Ric , s ) + εa ( s ) -3 -c √ ε ( E ( ≤ L -2) ( φ, s ) + E ( ≤ L -2) (Σ , s ) ) + ( ε 3 a ( s ) -3 -c √ ε + ε 2 a ( s ) -1 -c √ ε ) E ( ≤ L -4) (Ric , s ) } ds ︸ ︷︷ ︸ not present for L =2 At order 1 , the analogous estimate holds where the last three lines of (6.9) are dropped.</text>
<text><location><page_44><loc_12><loc_66><loc_88><loc_73></location>Proof. These estimates follow completely analogously to Lemma 6.2, with the exception that high order lapse terms can now be estimated at order L + 2 due to the the scalar field energy being scaled by a 4 . In particular, we note that to deal with the analogous term to (6.4a), one now inserts the following zero on the right and applies the commuted lapse equation (2.37b):</text>
<formula><location><page_44><loc_16><loc_54><loc_85><loc_67></location>0 = -3 8 π ˙ aa 7 ∫ M div G ( ∇ ∆ L 2 N · ∆ L 2 +1 N ) vol G = ∫ M { -3 8 π ˙ aa 7 | ∆ L 2 +1 N | 2 -3 8 π ( 1 3 ˙ aa 3 +12 πC 2 ˙ a a ) · a 4 |∇ ∆ L 2 N | 2 G -6 C ˙ a a · a 4 〈∇ ∆ L 2 N, ∇ ∆ L 2 Ψ 〉 -3 8 π ˙ aa 7 〈 N L +1 ,Border + N L +1 ,Junk , ∇ ∆ L 2 N 〉 G } vol G</formula>
<text><location><page_44><loc_12><loc_49><loc_88><loc_54></location>For L = 0, the argument is again the same as at higher orders with less complicated junk terms. We briefly highlight some specific junk terms: The term analogous to (6.4c) is now estimated as follows using (4.13a):</text>
<formula><location><page_44><loc_23><loc_42><loc_77><loc_49></location>a 4 · ∫ M -2 a 〈∇ Ψ ∇ N, ∇ 2 φ 〉 G /lessorsimilar ε ∫ M a 1 2 |∇ N | G · a 1 2 -c √ ε · a 4 |∇ φ | G /lessorsimilar ε ˙ aa 3 E (1) ( N, · ) + εa 1 -c √ ε · a 4 E ( ≤ 1) ( φ, · )</formula>
<text><location><page_44><loc_12><loc_41><loc_77><loc_42></location>Further, note that, by the commutator formula (11.8c) and applying (4.4e), one has</text>
<formula><location><page_44><loc_16><loc_31><loc_88><loc_41></location>∣ ∣ ∣ ∣ ∫ M a 8 [ ∂ t , ∆] φ · ∆ φ vol G ∣ ∣ ∣ ∣ /lessorsimilar εa 5 -c √ ε ( ‖∇ Σ ‖ L 2 G + ‖∇ N ‖ L 2 G ) ‖ ∆ φ ‖ L 2 G /lessorsimilar εa -1 -c √ ε ( a 4 E (1) ( φ, · ) + a 4 E (1) (Σ , · ) ) + εa 6 -cσ · ˙ aa 3 E (1) ( N, · ) . /square</formula>
<text><location><page_44><loc_12><loc_25><loc_88><loc_29></location>6.1.2. Sobolev norm estimate for ∇ φ . To improve the bootstrap assumptions on ∇ φ , we will need sharper bounds than those on a 4 ‖∇ φ ‖ 2 H L that will follow from bounds on E ( L ) ( φ, · ):</text>
<text><location><page_44><loc_12><loc_22><loc_80><loc_25></location>Lemma 6.5. Let l ∈ ( t Boot , t 0 ] . Then, for l ∈ Z + , l ≤ 17 , the following estimate holds:</text>
<formula><location><page_44><loc_25><loc_20><loc_75><loc_23></location>‖∇ φ ‖ H l G (Σ t ) /lessorsimilar (1 + εa ( t ) -c √ ε ) ‖ Σ ‖ H l +1 G (Σ t ) + εa ( t ) -c √ ε ‖ Ψ ‖ H l G (Σ t )</formula>
<text><location><page_44><loc_12><loc_16><loc_88><loc_19></location>Proof. By (4.2a), Ψ + C > C 2 holds if ε is chosen small enough. Consequently, we can rearrange (2.29b) and apply the product rule to obtain</text>
<text><location><page_44><loc_12><loc_9><loc_51><loc_10></location>The statement then follows by integrating over Σ t</text>
<text><location><page_44><loc_52><loc_9><loc_88><loc_10></location>and applying (4.2a) and (4.2b). /square</text>
<formula><location><page_44><loc_26><loc_9><loc_74><loc_16></location>|∇ l ∇ φ | G = 1 8 π ∣ ∣ ∣ ∣ ∇ l ( div G Σ Ψ+ C )∣ ∣ ∣ ∣ G /lessorsimilar ∑ I Σ + I Ψ = l |∇ I Σ +1 Σ | G |∇ I Ψ Ψ | G .</formula>
<text><location><page_45><loc_12><loc_87><loc_88><loc_90></location>6.2. Energy estimates for the Bel-Robinson variables. In this subsection, we collect the energy estimates for the Bel-Robinson variables:</text>
<text><location><page_45><loc_12><loc_83><loc_74><loc_85></location>Lemma 6.6 (Bel-Robinson energy estimates) . Let t ∈ ( t Boot , t 0 ] . Then one has</text>
<formula><location><page_45><loc_12><loc_71><loc_85><loc_84></location>E (0) ( W,t ) + ∫ t 0 t ∫ M [ 8 πC 2 a ( s ) -3 ( N +1) 〈 Σ , E 〉 G +6 ˙ a ( s ) a ( s ) ( N +1) | E | 2 G ] vol G ds (6.10) /lessorsimilar E (0) ( W,t 0 ) + ∫ t 0 t ( εa ( s ) -3 + a ( s ) -1 -c √ ε ) E (0) ( W,s ) + ε -1 8 a ( s ) -3 · a ( s ) 4 E (1) ( φ, s ) ds + ∫ t 0 t εa ( s ) -1 -c √ ε E (0) ( φ, s ) + εa ( s ) -3 E (0) (Σ , s ) ds</formula>
<text><location><page_45><loc_12><loc_68><loc_39><loc_70></location>as well as, for L ∈ 2 N , 2 ≤ L ≤ 18 ,</text>
<text><location><page_45><loc_12><loc_66><loc_16><loc_68></location>(6.11)</text>
<formula><location><page_45><loc_14><loc_37><loc_85><loc_66></location>E ( L ) ( W,t ) + ∫ t 0 t ∫ M [ 8 πC 2 a ( s ) -3 ( N +1) 〈 ∆ L 2 Σ , ∆ L 2 E 〉 G +6 ˙ a ( s ) a ( s ) | ∆ L 2 E | 2 G ] vol G ds /lessorsimilar E ( L ) ( W,t 0 ) + ∫ t 0 t ( ε 1 8 a ( s ) -3 + a ( s ) -1 -c √ ε ) E ( L ) ( W,s ) ds + ∫ t 0 t { ε -1 8 a ( s ) -3 · a ( s ) 4 E ( L +1) ( φ, s ) + ( ε 1 8 a ( s ) -3 + a ( s ) -1 ) E ( L ) ( φ, s ) + εa ( s ) -3 E ( L ) (Σ , s ) + ε 7 8 a ( s ) -3 · a ( s ) 4 E ( L -1) (Ric , s ) + ε 31 8 a ( s ) -3 E ( ≤ L -2) (Ric , s ) + ( ε 15 8 a ( s ) -3 -c √ ε + a ( s ) -1 -c √ ε ) E ( ≤ L -2) ( φ, s ) + ε 15 8 a ( s ) -3 -c √ ε ( E ( ≤ L -2) (Σ , s ) + E ( ≤ L -2) ( W,s ) ) + ε 15 8 a ( s ) -3 -c √ ε E ( ≤ L -4) (Ric , s ) ds ︸ ︷︷ ︸ not present for L =2 </formula>
<formula><location><page_45><loc_86><loc_42><loc_86><loc_43></location>.</formula>
<text><location><page_45><loc_12><loc_31><loc_88><loc_36></location>Remark 6.7. We preemptively note that the error terms on the left hand side, once combined with the similar terms on the left hand side in Lemma 6.8 and given suitable weights, will turn out to have positive sign, even if they do not have definite sign in isolation.</text>
<text><location><page_45><loc_12><loc_21><loc_88><loc_29></location>The main idea in deriving this inequality is that we can use the algebraic identity (11.3d) and integration by parts to exploit the Maxwell system that lies at the core of the Bel-Robinson evolution equations. As a result, we avoid having higher order energies of the Bel-Robinson variables on the right hand side of the integral energy inequalities (which would break the bootstrap argument), then only having to deal with scalar field and Ricci energies at the next derivative level.</text>
<text><location><page_45><loc_12><loc_15><loc_88><loc_18></location>Proof. We first prove (6.11), and then explain how the same ideas lead to the simpler estimate (6.10). To this end, we start out by taking the time derivative of the energy as usual:</text>
<formula><location><page_45><loc_14><loc_6><loc_87><loc_15></location>-∂ t E ( L ) ( W, · ) = ∫ M -3 N ˙ a a [ | ∆ L 2 E | 2 G + | ∆ L 2 B | 2 G ] -2 ( 〈 ∂ t ∆ L 2 E , ∆ L 2 E 〉 G + 〈 ∂ t ∆ L 2 B , ∆ L 2 B 〉 G ) -2( ∂ t G -1 ) i 1 j 1 ( G -1 ) i 2 j 2 [ ∆ L 2 E i 1 i 2 ∆ L 2 E j 1 j 2 +∆ L 2 B i 1 i 2 ∆ L 2 B j 1 j 2 ] vol G</formula>
<text><location><page_46><loc_12><loc_85><loc_88><loc_90></location>E and B are symmetric and tracefree, thus symmetrizations become redundant, and any scalar product with a tensor that is pure trace or with an antisymmetric tensor can be dropped. 10 With this in hand, we get, inserting (2.38a) and (2.38b):</text>
<text><location><page_46><loc_12><loc_81><loc_17><loc_82></location>(6.12a)</text>
<formula><location><page_46><loc_12><loc_72><loc_79><loc_78></location>+2( N +1) a -1 ( 〈 curl G ∆ L 2 E , ∆ L 2 B 〉 G -〈 curl G ∆ L 2 B , ∆ L 2 E 〉 G ) (6.12b)</formula>
<formula><location><page_46><loc_15><loc_76><loc_65><loc_82></location>-∂ t E ( L ) ( W, · ) = ∫ M { ˙ a a ( -6( N +1) + 9 N ) ( | ∆ L 2 E | 2 G + | ∆ L 2 B | 2 G )</formula>
<formula><location><page_46><loc_12><loc_69><loc_78><loc_74></location>+2 a -1 ( 〈∇ ∆ L 2 N ∧ G B , ∆ L 2 E 〉 G -〈∇ ∆ L 2 N ∧ G E , ∆ L 2 B 〉 G ) (6.12c)</formula>
<formula><location><page_46><loc_12><loc_66><loc_85><loc_69></location>-8 πC 2 a -3 ( N +1) 〈 ∆ L 2 Σ , ∆ L 2 E 〉 G -8 πa (Ψ + C ) 〈∇ ∆ L 2 N ∇ φ, ∆ L 2 E 〉 G (6.12d)</formula>
<formula><location><page_46><loc_12><loc_63><loc_62><loc_66></location>-8 πa (Ψ + C )( N +1) 〈∇ 2 ∆ L 2 φ, ∆ L 2 E 〉 G (6.12e)</formula>
<formula><location><page_46><loc_12><loc_59><loc_58><loc_62></location>+16 πa ( N +1) 〈∇ φ ∇ ∆ L 2 Ψ , ∆ L 2 E 〉 G (6.12f)</formula>
<formula><location><page_46><loc_12><loc_55><loc_56><loc_58></location>+ a 3 ε [ G ] ∗ ∇ φ ∗ ∇ 2 ∆ L 2 φ ∗ ∆ L 2 B (6.12g)</formula>
<formula><location><page_46><loc_12><loc_48><loc_63><loc_51></location>-2 〈 E L,Border + E L,top + E ‖ L,Junk , ∆ L 2 E 〉 G (6.12i)</formula>
<formula><location><page_46><loc_12><loc_51><loc_70><loc_56></location>+( N +1) a -3 Σ ∗ ( ∆ L 2 E ∗ ∆ L 2 E +∆ L 2 B ∗ ∆ L 2 B ) (6.12h)</formula>
<formula><location><page_46><loc_12><loc_43><loc_69><loc_49></location>-2 〈 B L,Border + B L,top + B ‖ L,Junk , ∆ L 2 B 〉 G } vol G (6.12j)</formula>
<text><location><page_46><loc_12><loc_40><loc_61><loc_43></location>For (6.12a), we pull 6( N +1) ˙ a a | ∆ L 2 E | 2 G to the left. This leaves</text>
<formula><location><page_46><loc_29><loc_36><loc_71><loc_41></location>∫ M -6 ˙ a a | ∆ L 2 B | 2 G +3 N ˙ a a | ∆ L 2 B | 2 G +9 N ˙ a a | ∆ L 2 E | 2 G vol G ,</formula>
<text><location><page_46><loc_12><loc_32><loc_88><loc_35></location>where we can estimate the last two terms up to constant by εa 1 -cσ E ( L ) ( W, · ) by (3.17h) and can drop the first term since it is nonpositive.</text>
<text><location><page_46><loc_12><loc_31><loc_41><loc_32></location>Regarding (6.12b), note that we have</text>
<formula><location><page_46><loc_17><loc_25><loc_83><loc_30></location>a -1 ( 〈 curl G ∆ L 2 E , ∆ L 2 B 〉 G - 〈 curl G ∆ L 2 B , ∆ L 2 E 〉 G ) = -a -1 div G ( ∆ L 2 E ∧ G ∆ L 2 B ) .</formula>
<text><location><page_46><loc_12><loc_22><loc_88><loc_25></location>Hence, the absolute value of (6.12b), using (11.4c) for the wedge product and (3.17h), can be bounded by:</text>
<formula><location><page_46><loc_16><loc_11><loc_84><loc_22></location>∣ ∣ ∣ ∣ ∫ M 2 a -1 ( N +1)div G ( ∆ L 2 E ∧ G ∆ L 2 B ) vol G ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∫ M 2 a -1 〈∇ N, ∆ L 2 E ∧ G ∆ L 2 B 〉 G vol G ∣ ∣ ∣ ∣ /lessorsimilar ∫ M a -1 |∇ N | G | ∆ L 2 E | G | ∆ L 2 B | G vol G /lessorsimilar εa 3 -cσ E ( L ) ( W, · )</formula>
<text><location><page_47><loc_12><loc_87><loc_88><loc_90></location>For (6.12c), we use the pointwise wedge product estimate (11.4d) and a priori estimates (4.2c) and (4.4g) to bound it as follows:</text>
<formula><location><page_47><loc_26><loc_76><loc_74><loc_87></location>| (6.12c) | ≤ 2 a -1 |∇ ∆ L 2 N | G ( | B | G · | ∆ L 2 E | G + | E | G · | ∆ L 2 B | G ) /lessorsimilar εa -3 √ a 4 E ( L +1) ( N, · ) √ E ( L ) ( W, · ) /lessorsimilar εa -3 ( E ( L ) ( W, · ) + a 4 E ( L +1) ( N, · ) )</formula>
<text><location><page_47><loc_12><loc_73><loc_88><loc_76></location>We pull the first term of (6.12d) to the left as well, and estimate the second using the strong C G -norm estimates (4.2a) and (4.4e) by</text>
<formula><location><page_47><loc_17><loc_68><loc_83><loc_73></location>√ εa -1 -c √ ε √ a 4 E ( L +1) ( N, · ) √ E ( L ) ( W, · ) /lessorsimilar a -1 -c √ ε E ( L ) ( W, · ) + εa -1 · a 4 E ( L +1) ( N, · ) .</formula>
<text><location><page_47><loc_12><loc_67><loc_83><loc_69></location>Moving on to (6.12e)-(6.12g), we see [using (4.2a), (4.4e), (4.13a) with ζ = ∆ L 2 φ and (3.8)]:</text>
<formula><location><page_47><loc_16><loc_58><loc_84><loc_67></location>| (6.12e) | /lessorsimilar ( a -3 √ a 4 E ( L +1) ( φ, · ) + a -1 √ E ( L ) ( φ, · ) + a -1 -c √ ε √ E ( L -2) ( φ, · ) ) √ E ( L ) ( W, · ) /lessorsimilar ( ε 1 8 a -3 + a -1 -c √ ε ) E ( M ) ( W, · ) + ε -1 8 a -3 · a 4 E ( M +1) ( φ, · ) + a -1 E ( ≤ L ) ( φ, · )</formula>
<formula><location><page_47><loc_30><loc_53><loc_53><loc_55></location>E · E ·</formula>
<formula><location><page_47><loc_16><loc_54><loc_55><loc_60></location>| (6.12f) | /lessorsimilar √ εa 1 -c √ ε √ E ( L +1) ( φ, · ) √ E ( L ) ( W, · ) /lessorsimilar a 1 -c √ ε ( L ) ( W, ) + εa 1 -c √ ε ( L +1) ( φ, ) ,</formula>
<formula><location><page_47><loc_16><loc_42><loc_77><loc_53></location>| (6.12g) | /lessorsimilar √ εa 1 -c √ ε · a 2 ‖∇ 2 ∆ L 2 φ ‖ L 2 G · √ E ( L ) ( W, · ) /lessorsimilar √ εa 1 -c √ ε ( √ E ( L +1) ( φ, · ) + a -c √ ε √ E ( L -1) ( φ, · ) ) · √ E ( L ) ( W, · ) /lessorsimilar a 1 -c √ ε E ( L ) ( W, · ) + εa 1 -c √ ε [ E ( L +1) ( φ, · ) + E ( L ) ( φ, · ) + E ( ≤ L -2) ( φ, · ) ]</formula>
<formula><location><page_47><loc_15><loc_23><loc_85><loc_41></location>-∂ t E ( L ) ( W, · ) + 8 πC 2 a -3 ∫ M ( N +1) 〈 ∆ L 2 Σ , ∆ L 2 E 〉 G vol G +6 ˙ a a ∫ M ( N +1) | ∆ L 2 E | 2 G vol G /lessorsimilar ( εa -3 + a -1 -c √ ε ) E ( L ) ( W, · ) + a -1 E ( L +1) ( φ, · ) + a -1 E ( L ) ( φ, · ) + εa -3 · a 4 E ( L +1) ( N, · ) + a -1 E ( ≤ L -2) ( φ, · ) [ ‖ E L,Border ‖ L 2 G + ‖ E L,top ‖ L 2 G + ‖ E ‖ L,Junk ‖ L 2 G + ‖ B L,Border ‖ L 2 G + ‖ B L,top ‖ L 2 G + ‖ B ‖ L,Junk ‖ L 2 G ] √ E ( L ) ( W, · )</formula>
<text><location><page_47><loc_12><loc_40><loc_82><loc_42></location>We can estimate (6.12h) by εa -3 E ( L ) ( W, · ) as usual, and obtain the following in summary:</text>
<text><location><page_47><loc_12><loc_16><loc_88><loc_24></location>We can now apply Corollary 5.5 for 2 l = L to estimate the lapse energy in the second line (leading to borderline scalar field energy and Σ-energy contributions as well as junk terms), and insert the borderline (see (11.17k)), top (see (11.18a) and (11.18b)) and junk estimates (see (11.19n)), dealing with the lapse energies there analogously. In particular, the top order curvature terms arise as follows:</text>
<formula><location><page_47><loc_21><loc_6><loc_79><loc_17></location>‖ E M,top ‖ L 2 G √ E ( M ) ( W, · ) /lessorsimilar √ εa -1 -c √ ε √ a 4 E ( M -1) (Ric , · ) √ E ( M ) ( W, · ) /lessorsimilar ε 1 8 a -1 -c √ ε E ( M ) ( W, · ) + ε 7 8 a -1 · a 4 E ( M -1) (Ric , · ) ‖ B M,top ‖ √ E ( M ) ( W, · ) /lessorsimilar εa -3 √ a 4 E ( M -1) (Ric , · ) √ E ( M ) ( W, · )</formula>
<formula><location><page_48><loc_40><loc_87><loc_76><loc_90></location>/lessorsimilar ε 1 8 a -3 E ( M ) ( W, · ) + ε 15 8 a -3 · a 4 E ( M -1) (Ric , · )</formula>
<text><location><page_48><loc_12><loc_84><loc_77><loc_87></location>Hence, both top order curvature terms can be bounded by ε 7 8 a -3 · a 4 E ( M -1) (Ric , · ). Integrating the inequality yields (6.11).</text>
<text><location><page_48><loc_12><loc_79><loc_88><loc_82></location>For (6.10), we get applying (2.31a) and (2.31b) and again using that E and B are symmetric and tracefree:</text>
<text><location><page_48><loc_12><loc_44><loc_88><loc_49></location>The first two lines are treated as in the general case. For the third line, we get εa 3 -cσ E (0) ( W, · ) with (11.4d) and (3.17h), while the fourth term is bounded by εa -3 E (0) ( W, · ) with (4.2b). This leaves the surviving matter terms in the final four lines.</text>
<formula><location><page_48><loc_18><loc_48><loc_82><loc_79></location>-∂ t E (0) ( W, · ) = ∫ M { ˙ a a ( -6( N +1) + 9 N ) ( | E | 2 G + | B | 2 G ) +2( N +1)( 〈 curl E , B 〉 G -〈 curl B , E 〉 G ) +2( 〈∇ N ∧ B , E 〉 G -〈∇ N ∧ E , B 〉 G ) +( N +1) a -3 Σ ∗ ( E ∗ E + B ∗ B ) -8 πa -3 ( N +1)(Ψ + C ) 2 〈 Σ , E 〉 G + [ ˙ aa 3 ∇ φ ∗ ∇ φ ∗ E + a (Ψ + C ) · ∇ N ∗ ∇ φ ] ∗ E + a ( N +1) [ ∇ φ ∗ ∇ Ψ+Σ ∗ ∇ φ ∗ ∇ φ +(Ψ+ C ) ∇ 2 φ ] ∗ E +( N +1) ε [ G ] ∗ ( a 3 ∇ 2 φ ∗ ∇ φ + a -1 (Ψ + C )Σ ∗ ∇ φ ) ∗ B } vol G</formula>
<text><location><page_48><loc_12><loc_39><loc_88><loc_44></location>We pull ∫ M 8 πa -3 ( N + 1) C 2 〈 Σ , E 〉 G vol G to the left as before. For the remaining terms, we can apply a priori estimates (4.2a), (4.4a) and (4.4e), the bootstrap assumption (3.17h) and Lemma 4.1 for N , which yields the following bound up to constant remaining terms in the last three lines:</text>
<text><location><page_48><loc_12><loc_27><loc_88><loc_30></location>Applying (4.13a) to the scalar field norm and then (3.8), this leads to (6.10) along with the previous observations. /square</text>
<formula><location><page_48><loc_29><loc_30><loc_71><loc_39></location>√ E (0) ( W, · ) · [ a -1 · a 2 ‖∇ 2 φ ‖ L 2 G + √ εa -1 -c √ ε √ E (0) ( φ, · ) + ( εa -3 + √ εa -1 -c √ ε ) √ E (0) (Σ , · ) ]</formula>
<text><location><page_48><loc_12><loc_22><loc_88><loc_25></location>6.3. Energy estimates for the second fundamental form. For the energy estimates for Σ, we again first derive even order integral estimates:</text>
<text><location><page_48><loc_12><loc_18><loc_88><loc_21></location>Lemma 6.8 (Energy estimates for the second fundamental form for even orders) . Let t ∈ ( t Boot , t 0 ] . Then, one has:</text>
<formula><location><page_48><loc_12><loc_9><loc_82><loc_18></location>E (0) (Σ , t ) + 2 ∫ t 0 t ∫ M [ a ( s ) -3 ( N +1) 〈 E , Σ 〉 G + ˙ a ( s ) a ( s ) ( N +1) | ∆ L 2 Σ | 2 G ] vol G ds (6.13) /lessorsimilar E (0) (Σ , t 0 ) + ∫ t 0 t ε 1 8 a ( s ) -3 E (0) (Σ , s ) ds + ∫ t 0 t ε -1 8 a ( s ) -3 E (0) ( φ, s ) ds</formula>
<text><location><page_49><loc_12><loc_87><loc_43><loc_90></location>For L ∈ 2 N , L ≤ 18 , the following holds:</text>
<text><location><page_49><loc_12><loc_86><loc_16><loc_87></location>(6.14)</text>
<text><location><page_49><loc_12><loc_56><loc_88><loc_64></location>Remark 6.9. The main hurdle of dealing with the second fundamental form is that a high order curvature term occurs in the evolution equation. It is to precisely this end that the Bel-Robinson variables needed to be introduced, since (2.36d) is what facilitates controlling said term without having to use E ( L ) (Ric , · ) or similar high order metric energies. Again, the resulting leading terms will turn out to have definite sign when combined with the Bel-Robinson energy estimates above.</text>
<formula><location><page_49><loc_15><loc_63><loc_86><loc_86></location>E ( L ) (Σ , t ) + 2 ∫ t 0 t ∫ M [ a ( s ) -3 ( N +1) 〈 ∆ L 2 E , ∆ L 2 Σ 〉 G vol G ds + ˙ a ( s ) a ( s ) ( N +1) | ∆ L 2 Σ | 2 G ] vol G ds /lessorsimilar E ( L ) (Σ , t 0 ) + ∫ t 0 t ε 1 8 a ( s ) -3 E ( L ) (Σ , s ) ds + ∫ t 0 t { ε -1 8 a ( s ) -3 E ( L ) ( φ, s ) + ε 15 8 a ( s ) 5 -cσ E ( L -1) (Ric , s ) + ε 2 a ( s ) -3 E ( L -2) (Ric , s ) + ε 2 a ( s ) -3 -c √ ε E ( ≤ L -2) (Σ , s ) + ( ε 15 8 a ( s ) -3 -c √ ε + εa ( s ) -1 -c √ ε ) E ( ≤ L -2) ( φ, s ) + ε 2 a ( s ) -3 -c √ ε E ( ≤ L -4) (Ric , s ) ︸ ︷︷ ︸ not present for L =2 } ds</formula>
<text><location><page_49><loc_12><loc_50><loc_88><loc_54></location>Proof. Here, we omit the proof for the inequality at order zero since is completely analogous in structure to the one for orders 2 and higher and the only differences that arise are that lower order error terms do not occur.</text>
<text><location><page_49><loc_12><loc_47><loc_68><loc_50></location>Once again, we start out by differentiating -E ( L ) (Σ , · ) and insert (2.35):</text>
<formula><location><page_49><loc_14><loc_31><loc_86><loc_47></location>-∂ t E ( L ) (Σ , · ) = ∫ M -2 〈 ∂ t ∆ L 2 Σ , ∆ L 2 Σ 〉 G +( ∂ t G -1 ) ∗ G -1 ∗ ∆ L 2 Σ ∗ ∆ L 2 Σ -3 N ˙ a a | ∆ L 2 Σ | 2 G vol G = ∫ M { 2 a 〈∇ 2 ∆ L 2 N, ∆ L 2 Σ 〉 G -2 a ( N +1) 〈 ∆ L 2 Ric[ G ] , ∆ L 2 Σ 〉 G + +( ∂ t G -1 ) ∗ G -1 ∗ ∆ L 2 Σ ∗ ∆ L 2 Σ -3 N ˙ a a | ∆ L 2 Σ | 2 G -2 〈 S L,Border , ∆ L 2 Σ 〉 G -2 〈 S ‖ L,Junk , ∆ L 2 Σ 〉 G } vol G</formula>
<text><location><page_49><loc_12><loc_28><loc_88><loc_31></location>For the first term, one can use (4.13a), Corollary 5.5 at order L and (3.8) to bound its absolute value by the following:</text>
<formula><location><page_49><loc_15><loc_6><loc_85><loc_27></location>/lessorsimilar a ‖ ∆ L 2 N ‖ ˙ H 2 G √ E ( L ) (Σ , · ) /lessorsimilar [ a -3 √ a 8 E ( L +2) ( N, · ) + a 1 -c √ ε √ E ( L ) ( N, · ) ] √ E ( L ) (Σ , · ) /lessorsimilar [ εa -3 √ E ( L ) (Σ , · ) + a -3 E ( L ) ( φ, · ) + εa -3 -c √ ε ( √ E ( ≤ L -2) (Σ , · ) + √ E ( ≤ L -2) ( φ, · ) ) + ( ε 2 a -3 -c √ ε + εa 1 -cσ ) √ E ( ≤ L -3) (Ric , · ) ︸ ︷︷ ︸ not present for L =2 ] √ E ( L ) (Σ , · ) /lessorsimilar ( ε 1 8 a -3 + a 1 -cσ ) E ( L ) (Σ , · ) + ε -1 8 a -3 E ( L ) ( φ, · ) + εa -3 -c √ ε [ E ( ≤ L -2) (Σ , · ) + E ( ≤ L -2) ( φ, · ) ]</formula>
<formula><location><page_50><loc_17><loc_86><loc_77><loc_91></location>+ ( ε 31 8 a -3 + ε 2 a 1 -cσ ) E ( L -2) (Ric , · ) + ( ε 31 8 a -3 -c √ ε + ε 2 a 1 -cσ ) E ( ≤ L -4) (Ric , · )</formula>
<formula><location><page_50><loc_14><loc_73><loc_87><loc_82></location>∫ M -2 a ( N +1) 〈 ∆ L 2 Ric[ G ] , ∆ L 2 Σ 〉 G vol G = ∫ M -2( N +1) a -3 〈 ∆ L 2 E , ∆ L 2 Σ 〉 G -2( N +1) ˙ a a | ∆ L 2 Σ | 2 G + 〈 H L,Border + H ‖ L,Junk , ∆ L 2 Σ 〉 G vol G</formula>
<text><location><page_50><loc_12><loc_81><loc_88><loc_88></location>︸ ︷︷ ︸ not present for L =2 Next, we replace the high order curvature term as follows, using the commuted rescaled Hamiltonian constraint equation (2.36d) that ∆ L 2 Σ is tracefree and symmetric:</text>
<text><location><page_50><loc_12><loc_67><loc_88><loc_73></location>We pull the first two terms to left, only keeping the error terms on the right. After inserting the borderline and junk term estimates for the Hamiltonian constraint equations ((11.17a) and (11.19c)) and the evolution equation itself ((11.17h) and (11.19k)), as well as bounding | ∂ t G -1 | /lessorsimilar εa -3 and inserting (3.17h) as usual, we obtain (6.14) by integrating. /square</text>
<text><location><page_50><loc_12><loc_56><loc_88><loc_66></location>Additionally, we can exploit the structure of the momentum constraint equations to gain an elliptic estimate for E ( L +1) (Σ , · ). Crucially, the upper bound only depends on Σ-, scalar field and BelRobinson energies up to order L , and appropriately small and time-scaled curvature contributions up to order L -1. This will allow us to close the argument since we do not need to consider the Bel-Robinson energy at order L +1 to control Σ at that order, which would require higher order scalar field and curvature energies.</text>
<text><location><page_50><loc_12><loc_51><loc_88><loc_55></location>Lemma 6.10 (Odd order energy estimate for the second fundamental form) . For any L ∈ 2 Z + , 2 ≤ L ≤ 18 , we have</text>
<formula><location><page_50><loc_15><loc_44><loc_85><loc_50></location>a 4 E ( L +1) (Σ , · ) /lessorsimilar ( a 4 -c √ ε + εa 2 -c √ ε ) E ( L ) (Σ , · ) + E ( L ) ( φ, · ) + E ( L ) ( W, · ) + ε 2 a 4 E ( L -1) (Ric , · ) + εa -c √ ε E ( ≤ L -2) ( φ, · ) + a 2 -c √ ε E ( ≤ L -2) (Σ , · ) + εa 2 -c √ ε E ( ≤ L -2) (Ric , · ) .</formula>
<text><location><page_50><loc_12><loc_50><loc_16><loc_51></location>(6.15)</text>
<text><location><page_50><loc_12><loc_42><loc_36><loc_44></location>For L = 0 , one analogously has</text>
<formula><location><page_50><loc_12><loc_38><loc_77><loc_43></location>(6.16) a 4 E (1) (Σ , · ) /lessorsimilar ( a 4 -c √ ε + εa 2 -c √ ε ) E (0) (Σ , · ) + E (0) ( φ, · ) + E (0) ( W, · ) .</formula>
<text><location><page_50><loc_12><loc_31><loc_81><loc_36></location>Proof. We prove the statement for L = 2, since the proof of (6.16) is entirely analogous. By [AM04, (A.22)], since (Σ t , g t ( t , t ], any tracefree (0 , 2) tensor U on (Σ , g ) satisfies</text>
<formula><location><page_50><loc_12><loc_26><loc_79><loc_31></location>(6.17) ∫ Σ t |∇ U | 2 g +3Ric[ g ] · U · U -R [ g ] 2 | U | 2 g vol g = ∫ Σ t | curl U | 2 g + 3 2 | div g U | 2 g vol g .</formula>
<text><location><page_50><loc_13><loc_30><loc_88><loc_34></location>) is a three-dimensional compact Riemannian manifold for any ∈ Boot 0 ij t</text>
<text><location><page_50><loc_12><loc_24><loc_58><loc_26></location>In particular, for U = ∆ L 2 Σ and after rescaling, this reads:</text>
<text><location><page_50><loc_12><loc_11><loc_88><loc_16></location>The last two terms on the left hand side can be estimated by (1 + √ εa -c √ ε ) E ( L ) (Σ , · ) in absolute value using the strong C G -norm estimate (4.4f). Thus, inserting the Laplace-commuted rescaled momentum constraint equations (2.36a) and (2.36b), we obtain for a suitable constant K > 0:</text>
<formula><location><page_50><loc_22><loc_16><loc_79><loc_24></location>∫ M ∣ ∣ ∣ ∇ ∆ L 2 Σ ∣ ∣ ∣ 2 G +3 ( Ric[ G ] /sharp ) i j ( ∆ L 2 Σ /sharp ) j l ( ∆ L 2 Σ /sharp ) l i -R [ G ] 2 | ∆ L 2 Σ | 2 G vol G = ∫ M 3 2 · | div G ∆ L 2 Σ | 2 G + a 2 | curl∆ L 2 Σ | 2 G vol G</formula>
<formula><location><page_50><loc_15><loc_6><loc_48><loc_11></location>E ( L +1) (Σ , · ) -K ( 1 + √ εa -c √ ε ) E ( L ) (Σ , · )</formula>
<formula><location><page_51><loc_14><loc_82><loc_86><loc_91></location>/lessorsimilar ∫ M { | Ψ+ C | 2 ∣ ∣ ∣ ∇ ∆ L 2 φ ∣ ∣ ∣ 2 G + |∇ φ | 2 G ∣ ∣ ∣ ∆ L 2 -1 Ric[ G ] ∣ ∣ ∣ 2 G + | Σ | 2 G ∣ ∣ ∣ ∇ ∆ L 2 -1 Ric[ G ] ∣ ∣ ∣ 2 G + | M L,Junk | 2 G ∣ ∣ ∣ ∣ ∣ ∣ } G</formula>
<formula><location><page_51><loc_18><loc_80><loc_85><loc_86></location>+ a -4 | ∆ L 2 B | 2 G + | Σ | 2 G ∣ ∣ ∇ ∆ L 2 -1 Ric[ G ] ∣ ∣ 2 G + |∇ Σ | 2 G ∣ ∣ ∇ 2 ∆ L 2 -2 Ric[ G ] ∣ ∣ 2 G + ∣ ∣ ˜ M L,Junk ∣ ∣ 2 G vol</formula>
<formula><location><page_51><loc_16><loc_73><loc_85><loc_79></location>a 4 E ( L +1) (Σ , · ) /lessorsimilar ( 1 + √ εa -c √ ε ) a 4 E ( L ) (Σ , · ) + E ( L ) ( φ, · ) + E ( L ) ( W, · ) + ε 2 a 4 E ( L -1) (Ric , · ) + ε 2 a 4 -c √ ε E ( L -2) (Ric , · ) + a 4 ‖ M L,Junk ‖ 2 L 2 G + a 4 ‖ ˜ M L,Junk ‖ 2 L 2 G .</formula>
<text><location><page_51><loc_12><loc_79><loc_88><loc_82></location>After rearranging, using the strong C G -norm estimates (4.2a), (4.4e), (4.2b) and (4.4b) and multiplying by a 4 on both sides, we get</text>
<text><location><page_51><loc_12><loc_71><loc_65><loc_72></location>The statement follows inserting the estimates (11.19a) and (11.19b).</text>
<text><location><page_51><loc_87><loc_71><loc_88><loc_72></location>/square</text>
<unordered_list>
<list_item><location><page_51><loc_12><loc_60><loc_88><loc_69></location>6.4. Energy estimates for the curvature. To control commutator errors, we will also need some additional estimates on curvature energies. Unlike the other energies, these inequalities do not rely on any delicate structure within the equations and instead just rely on pointwise estimates, the Young inequality and near-coercivity of energies in the sense of Lemma 4.5. For the sake of convenience, we phrase these estimates for E ( L -2) (Ric , · ) since this is the order needed when improving behaviour of the total energy at order L .</list_item>
</unordered_list>
<text><location><page_51><loc_12><loc_55><loc_88><loc_59></location>Lemma 6.11 (Curvature energy estimates at even orders) . Let L ∈ 2 Z , 4 ≤ L ≤ 16 and t ∈ ( t Boot , t 0 ] . Then, one has</text>
<text><location><page_51><loc_12><loc_38><loc_37><loc_39></location>Additionally, the following holds:</text>
<formula><location><page_51><loc_12><loc_39><loc_82><loc_56></location>E ( L -2) (Ric , t ) /lessorsimilar E ( L -2) (Ric , t 0 ) + ∫ t 0 t ( ε 1 8 a ( s ) -3 + a ( s ) 8 -cσ ) E ( L -2) (Ric , s ) ds (6.18) + ∫ t 0 t { ε -1 8 a ( s ) -3 ( E ( L ) ( φ, s ) + E ( L ) (Σ , s ) ) + ε -1 8 a ( s ) -3 -c √ ε ( E ( ≤ L -2) ( φ, s ) + E ( ≤ L -2) (Σ , s ) ) + ε 7 8 a ( s ) -3 -c √ ε E ( ≤ L -4) (Ric , s ) } ds .</formula>
<formula><location><page_51><loc_12><loc_31><loc_73><loc_38></location>E (0) (Ric , t ) /lessorsimilar E (0) (Ric , t 0 ) + ∫ t 0 t ε 1 8 a ( s ) -3 E (0) (Ric , s ) ds (6.19) t ds</formula>
<text><location><page_51><loc_12><loc_28><loc_32><loc_29></location>Proof. First, we note that</text>
<formula><location><page_51><loc_39><loc_29><loc_71><loc_34></location>+ ∫ 0 t ε -1 8 a ( s ) -3 ( E (0) ( φ, s ) + E (0) (Σ , s ) )</formula>
<formula><location><page_51><loc_22><loc_23><loc_78><loc_28></location>‖ div /sharp G ∇ ∆ L 2 -1 Σ ‖ L 2 G /lessorsimilar ‖∇ 2 ∆ L 2 -1 Σ ‖ L 2 G /lessorsimilar ‖ ∆ L 2 Σ ‖ L 2 G + a -c √ ε √ E ( L -2) (Σ , · )</formula>
<formula><location><page_51><loc_30><loc_17><loc_70><loc_22></location>‖∇ 2 ∆ L 2 -1 N ‖ L 2 G /lessorsimilar ‖ ∆ L 2 N ‖ L 2 G + a -c √ ε √ E ( L -2) ( N, · )</formula>
<text><location><page_51><loc_12><loc_22><loc_78><loc_24></location>holds using the low order version of (4.14a) with T = ∆ L 2 -1 Σ for l = 2, and similarly</text>
<text><location><page_51><loc_12><loc_15><loc_88><loc_18></location>using (4.13b) at order 2. Now, using ∆ L 2 -1 G = 0 for L ≥ 4, we continue as usual by applying (2.40a) to the expression below:</text>
<formula><location><page_51><loc_17><loc_6><loc_72><loc_15></location>-∂ t E ( L -2) (Ric , · ) /lessorsimilar ∫ M { a -3 ( | ∆ L 2 Σ | G + |∇ 2 ∆ L 2 -1 Σ | G ) | ∆ L 2 -1 Ric[ G ] | G + ˙ a a ( |∇ 2 ∆ L 2 -1 N | G + | ∆ L 2 N | G ) | ∆ L 2 -1 Ric[ G ] | G</formula>
<formula><location><page_52><loc_33><loc_83><loc_83><loc_90></location>+( | R L -2 ,Border | G + | R L -2 ,Junk | G ) · | ∆ L 2 -1 Ric[ G ] | G + a -3 Σ ∗ ∆ L 2 -1 Ric[ G ] ∗ ∆ L 2 -1 Ric[ G ] + N ˙ a a | ∆ L 2 -1 Ric[ G ] | 2 G } vol G</formula>
<formula><location><page_52><loc_18><loc_68><loc_82><loc_83></location>-∂ t E ( L -2) (Ric , · ) /lessorsimilar a -3 [ √ E ( L ) (Σ , · ) + √ E ( L ) ( N, · ) ] √ E ( L -2) (Ric , · ) + a -3 -c √ ε [ √ E ( ≤ L -2) (Σ , · ) + √ E ( ≤ L -2) ( N, · ) ] √ E ( L -2) (Ric , · ) + ( ‖ R L -2 ,Border ‖ L 2 G + ‖ R L -2 ,Junk ‖ L 2 G ) √ E ( L -2) (Ric , · ) + εa -3 E ( L -2) (Ric , · ) .</formula>
<text><location><page_52><loc_12><loc_82><loc_66><loc_84></location>Due to the estimates above as well as (4.2b) and (3.17h), this implies</text>
<text><location><page_52><loc_12><loc_65><loc_88><loc_68></location>Using Corollary 5.5 at order L and distributing terms containing E ( L -3) (Ric , · ) with (3.8) as usual, we get</text>
<unordered_list>
<list_item><location><page_52><loc_12><loc_49><loc_88><loc_52></location>(6.18) now follows inserting the borderline and junk term estimates (11.17i) and (11.19l) and applying the lapse energy estimates from Corollary 5.5.</list_item>
</unordered_list>
<formula><location><page_52><loc_18><loc_51><loc_83><loc_65></location>-∂ t E ( L -2) (Ric , · ) /lessorsimilar [ ε 1 8 a -3 + a 8 -cσ ] E ( L -2) (Ric , · ) + ε -1 8 a -3 [ E ( L ) (Σ , · ) + E ( L ) ( φ, · ) ] + [ ε 31 8 a -3 -c √ ε + ε 2 a 8 -cσ ] E ( ≤ L -4) (Ric , · ) + ε -1 8 a -3 -c √ ε [ E ( ≤ L -2) (Σ , · ) + E ( ≤ L -2) ( φ, · ) ] + ( ‖ R L -2 ,Border ‖ L 2 G + ‖ R L -2 ,Junk ‖ L 2 G ) √ E ( L -2) (Ric , · ) .</formula>
<text><location><page_52><loc_12><loc_45><loc_88><loc_49></location>(6.19) follows almost identically by inserting (2.33) instead of (2.40a) as well as (2.28a) for the additional ∂ t G ∗ (Ric[ G ] + 2 / 9 G )-terms. These can be estimated as</text>
<text><location><page_52><loc_12><loc_36><loc_44><loc_37></location>which can be treated as at higher orders.</text>
<text><location><page_52><loc_87><loc_36><loc_88><loc_37></location>/square</text>
<formula><location><page_52><loc_27><loc_37><loc_73><loc_46></location>/lessorsimilar ∫ M a -3 Σ ∗ (Ric[ G ] + 2 9 G ) + ˙ a a N · G ∗ (Ric[ G ] + 2 9 G ) vol G /lessorsimilar a -3 ( √ E (0) (Σ , · ) + √ E (0) ( N, · ) ) √ E (0) (Ric , · ) ,</formula>
<text><location><page_52><loc_12><loc_30><loc_88><loc_35></location>Lemma 6.12 (Odd order curvature energy estimate) . For L ∈ 2 N , 4 ≤ L ≤ 18 and t ∈ ( t Boot , t 0 ] , the following holds: (6.20)</text>
<formula><location><page_52><loc_14><loc_6><loc_86><loc_29></location>a ( t ) 4 E ( L -1) (Ric , t ) /lessorsimilar a ( t 0 ) 4 E ( L -1) (Ric , t 0 ) + ∫ t 0 t ( ε 1 8 a ( s ) -3 + a ( s ) -1 -c √ ε )( a ( s ) 4 E ( L -1) (Ric , s ) ) ds + ∫ t 0 t ε -1 8 a ( s ) -3 · a ( s ) 4 E ( L +1) (Σ , s ) ds + ∫ t 0 t { ε -1 8 a ( s ) -3 E ( L ) ( φ, s ) + ( ε 15 8 a ( s ) -3 + a ( s ) -1 -c √ ε ) E ( L ) (Σ , s ) + ( ε 15 8 a ( s ) -3 -c √ ε + a ( s ) -1 -c √ ε )( E ( ≤ L -2) ( φ, s ) + E ( ≤ L -2) (Σ , s ) ) + ε 15 8 a ( s ) -3 E ( L -2) (Ric , · ) + ε 15 8 a -3 -c √ ε E ( ≤ L -4) (Ric , s ) } ds</formula>
<text><location><page_53><loc_12><loc_81><loc_88><loc_90></location>Proof. The proof is very similar to that of Lemma 6.11 since we didn't exploit any structure within (2.40a) that doesn't equally occur in (2.40b), and thus we omit the details. As in the proof of Lemma 6.4, we note that the differences within the estimate come from how top order lapse terms are treated: The scaling of the top order energy allows one to estimate a 4 E ( L +1) ( N, · ) by scalar field energies and Σ-energies of up to order L and curvature energies up to order L -3. /square</text>
<unordered_list>
<list_item><location><page_53><loc_12><loc_76><loc_88><loc_81></location>6.5. Sobolev norm estimates for metric objects. To close the bootstrap argument, we need to improve the behaviour of metric quantities in addition to the energy formalism, both to capture the intrinsic behaviour of the metric and to relate energies to supremum norms.</list_item>
</unordered_list>
<text><location><page_53><loc_12><loc_70><loc_88><loc_75></location>Lemma 6.13 (Sobolev norm estimates for Christoffel symbols) . Let U be a coordinate neighbourhood on M , viewed as a coordinate neighbourhood on Σ t for t ∈ ( t Boot , t 0 ] . For any l ∈ N , l ≤ 15 , the following Sobolev estimate then holds:</text>
<formula><location><page_53><loc_12><loc_65><loc_78><loc_71></location>(6.21) ‖ Γ -ˆ Γ ‖ 2 H l G ( U ) /lessorsimilar a -cε 1 8 ( ε 4 + ε -1 4 sup s ∈ ( t,t 0 ) ( ‖ N ‖ 2 H l +1 G (Σ s ) + ‖ Σ ‖ 2 H l +1 G (Σ s ) ) )</formula>
<text><location><page_53><loc_12><loc_62><loc_78><loc_65></location>Proof. Commuting the evolution equation (2.34) with ∇ J , we get for J ∈ N , J ≤ 17:</text>
<formula><location><page_53><loc_33><loc_56><loc_74><loc_58></location>+( G -1 ) ∗ · · · ∗ ( G -1 ) ∗ ∂ t G ∗ ∇ J (Γ -ˆ Γ) ∗ ∇ J (Γ -ˆ Γ)</formula>
<formula><location><page_53><loc_16><loc_58><loc_76><loc_63></location>-∂ t ‖ Γ -ˆ Γ ‖ 2 ˙ H J G = ∫ U [ ( ∂ t G -1 ) ∗ G -1 ∗ · · · ∗ G -1 ∗ G ∗ ∇ J (Γ -ˆ Γ) ∗ ∇ J (Γ -ˆ Γ)</formula>
<formula><location><page_53><loc_33><loc_45><loc_84><loc_56></location>+ a -3 ∑ I N + I Σ = J +1 ∇ I N ( N +1) ∗ ∇ I Σ Σ+ ˙ a a ∇ J +1 N ∗ ∇ J (Γ -ˆ Γ) ∣ ∣ ] G</formula>
<formula><location><page_53><loc_41><loc_41><loc_59><loc_44></location>‖ Γ -ˆ Γ ‖ C 11 G /lessorsimilar √ εa -c √ ε</formula>
<text><location><page_53><loc_12><loc_44><loc_34><loc_46></location>We recall that (4.4c) implies</text>
<formula><location><page_53><loc_12><loc_42><loc_16><loc_43></location>(6.22)</formula>
<text><location><page_53><loc_12><loc_40><loc_84><loc_41></location>by (3.5). It follows from inserting this in (11.10b) along with (4.2b), (4.4b) and (3.17h) that</text>
<formula><location><page_53><loc_18><loc_36><loc_82><loc_40></location>‖ [ ∂ t , ∇ J ](Γ -ˆ Γ) ‖ L 2 G /lessorsimilar √ εa -3 -c √ ε ‖ Σ ‖ H J G + √ εa -3 -c √ ε ‖ N ‖ H J G + εa -3 ‖ Γ -ˆ Γ ‖ H J -1 G</formula>
<text><location><page_53><loc_12><loc_33><loc_88><loc_36></location>is satisfied. Consequently and using the same strong C G -norm bounds along with Lemma 4.1, the differential inequality becomes</text>
<formula><location><page_53><loc_24><loc_24><loc_77><loc_33></location>-∂ t ‖ Γ -ˆ Γ ‖ 2 ˙ H J G /lessorsimilar ε 1 8 a -3 ‖ Γ -ˆ Γ ‖ 2 ˙ H J G + ε -1 8 a -3 ( ‖ N ‖ 2 H J +1 G + ‖ Σ ‖ 2 H J +1 G ) + ε 7 8 a -3 -c √ ε ‖ Σ ‖ 2 H J G + ε 7 8 a -3 -c √ ε ‖ N ‖ 2 H J G + ε 15 8 a -3 -c √ ε ‖ Γ -ˆ Γ ‖ 2 H J -1 G .</formula>
<text><location><page_53><loc_12><loc_22><loc_33><loc_23></location>Further, we analogously get</text>
<text><location><page_53><loc_12><loc_17><loc_48><loc_18></location>and thus, with the Gronwall lemma and (2.7),</text>
<formula><location><page_53><loc_25><loc_17><loc_75><loc_22></location>-∂ t ‖ Γ -ˆ Γ ‖ 2 L 2 G /lessorsimilar ε 1 8 a -3 ‖ Γ -ˆ Γ ‖ 2 L 2 G + ε -1 8 a -3 ( ‖ Σ ‖ 2 H 1 G + ‖ N ‖ 2 H 1 G )</formula>
<formula><location><page_53><loc_21><loc_7><loc_80><loc_17></location>‖ Γ -ˆ Γ ‖ 2 L 2 G (Σ t ) /lessorsimilar a -cε 1 8 ( ε 4 + ∫ t 0 t ε -1 8 a ( s ) -3 ( ‖ Σ ‖ 2 H 1 G (Σ s ) + ‖ N ‖ 2 H 1 G (Σ s ) ) ds ) /lessorsimilar a -cε 1 8 ( ε 4 + ε -1 4 sup s ∈ ( t,t 0 ) ( ‖ Σ ‖ 2 H 1 G (Σ s ) + ‖ N ‖ 2 H 1 G (Σ s ) ) ) .</formula>
<formula><location><page_53><loc_33><loc_44><loc_79><loc_50></location>+2 〈 [ ∂ t , ∇ J ](Γ -ˆ Γ) , ∇ J (Γ -ˆ Γ) 〉 G -3 N ˙ a a ∣ ∣ ∇ J (Γ -ˆ Γ) ∣ ∣ 2 G vol</formula>
<text><location><page_54><loc_12><loc_85><loc_88><loc_90></location>This proves (6.21) for l = 0, and we assume for an iterative argument that the statement has been proved for l = J -1. Then, we obtain (estimating the error terms in Σ and N by their supremum immediately):</text>
<formula><location><page_54><loc_17><loc_76><loc_83><loc_85></location>-∂ t ‖ Γ -ˆ Γ ‖ 2 ˙ H J G /lessorsimilar ε 1 8 a -3 ‖ Γ -ˆ Γ ‖ 2 ˙ H J G + ε -1 8 a -3 ( ‖ N ‖ 2 H J G + ‖ Σ ‖ 2 H J G ) + ε 7 8 +4 a -3 -cε 1 8 + ε 7 8 a -3 -cε 1 8 sup s ∈ ( · ,t 0 ) ( ‖ N ‖ 2 H J -1 (Σ s ) + ‖ Σ ‖ 2 H J -1 (Σ s ) ) .</formula>
<text><location><page_54><loc_12><loc_74><loc_87><loc_76></location>After integrating, applying the Gronwall lemma and dealing with the first line as before, we get</text>
<formula><location><page_54><loc_21><loc_65><loc_79><loc_74></location>‖ Γ -ˆ Γ ‖ 2 ˙ H J G /lessorsimilar ε 4 a -cε 1 8 + ε -1 4 a -cε 1 8 sup s ∈ ( · ,t 0 ] ( ‖ N ‖ 2 H J G (Σ s ) + ‖ Σ ‖ 2 H J G (Σ s ) ) + ε 46 8 a -cε 1 8 + ε 6 8 a -cε 1 8 sup s ∈ ( · ,t 0 ) ( ‖ N ‖ 2 H J -1 (Σ s ) + ‖ Σ ‖ 2 H J -1 (Σ s ) ) ,</formula>
<text><location><page_54><loc_12><loc_60><loc_88><loc_64></location>where the second line can obviously be absorbed into the first up to constant. Combining this with the assumption yields (6.21) for l = J and thus iteratively for all l ≤ 17. /square</text>
<text><location><page_54><loc_12><loc_57><loc_88><loc_60></location>Lemma 6.14 (Sobolev norm estimates for the metric) . For any t ∈ ( t Boot , t 0 ] and any l ∈ N , l ≤ 18 , we have:</text>
<formula><location><page_54><loc_12><loc_51><loc_77><loc_57></location>(6.23) ‖ G -γ ‖ 2 H l G (Σ t ) /lessorsimilar a -cε 1 8 ( ε 4 + ε -1 4 sup s ∈ ( t,t 0 ) ( ‖ N ‖ 2 H l G (Σ s ) + ‖ Σ ‖ 2 H l G (Σ s ) ) )</formula>
<text><location><page_54><loc_12><loc_49><loc_66><loc_51></location>Proof. For l = 0, we compute the following using (2.28a) and (2.28b):</text>
<formula><location><page_54><loc_17><loc_32><loc_84><loc_49></location>-∂ t ‖ G -γ ‖ 2 L 2 G = ∫ M { -2( ∂ t G -1 ) i 1 j 1 ( G -1 ) i 2 j 2 ( G -γ ) i 1 i 2 ( G -γ ) j 1 j 2 -2 〈 ∂ t G,G -γ 〉 G -3 N ˙ a a | G -γ | 2 G } vol G = ∫ M { ( N +1) a -3 [Σ ∗ ( G -γ ) + Σ] ∗ ( G -γ ) + N ˙ a a | G -γ | 2 G -4 N ˙ a a 〈 G,G -γ 〉 G } vol G</formula>
<text><location><page_54><loc_12><loc_31><loc_41><loc_32></location>We apply (4.2b) and (3.17h) and get</text>
<formula><location><page_54><loc_23><loc_22><loc_77><loc_31></location>-∂ t ‖ G -γ ‖ 2 L 2 G /lessorsimilar εa -3 ‖ G -γ ‖ 2 L 2 G + a -3 ( ‖ Σ ‖ L 2 G + ‖ N ‖ L 2 G ) ‖ G -γ ‖ L 2 G /lessorsimilar ε 1 8 a -3 ‖ G -γ ‖ 2 L 2 G + ε -1 8 a -3 ( ‖ Σ ‖ 2 L 2 G + ‖ N ‖ 2 L 2 G ) .</formula>
<text><location><page_54><loc_12><loc_20><loc_88><loc_23></location>After integrating and applying the Gronwall lemma (as well as the initial data assumption), we obtain</text>
<formula><location><page_54><loc_20><loc_14><loc_80><loc_20></location>‖ G -γ ‖ 2 L 2 G (Σ t ) /lessorsimilar a -cε 1 8 ( ε 4 + ε -1 8 ∫ t 0 t a ( s ) -3 ( ‖ Σ ‖ 2 L 2 G (Σ s ) + ‖ N ‖ 2 L 2 G (Σ s ) ) ds ) .</formula>
<text><location><page_54><loc_12><loc_11><loc_88><loc_14></location>The statement for l = 0 now follows taking the supremum over the norms under the integral and applying (2.7). This extends to higher orders via the same iteration argument as in Lemma 6.13.</text>
<section_header_level_1><location><page_55><loc_23><loc_88><loc_78><loc_90></location>7. Big Bang stability: Improving the bootstrap assumptions</section_header_level_1>
<text><location><page_55><loc_12><loc_81><loc_88><loc_87></location>In this section, we combine the energy estimates obtained in the last two sections to improve the boostrap assumptions for the energies themselves, and then show how this improves the behaviour of the solution norms. For an outline of the energy improvement arguments that we perform in Section 7.1, we refer back to Remark 6.1.</text>
<text><location><page_55><loc_12><loc_78><loc_88><loc_81></location>Before carrying out the improvements themselves, we quickly collect an estimate that shows that combining Lemma 6.6 and Lemma 6.8 yields sufficient control on the energies themselves:</text>
<text><location><page_55><loc_12><loc_74><loc_65><loc_77></location>Lemma 7.1. Let L ∈ 2 N . Then, the following estimate is satisfied:</text>
<formula><location><page_55><loc_12><loc_71><loc_85><loc_75></location>(7.1) 16 πC 2 a -3 ( N +1) 〈 ∆ L 2 E , ∆ L 2 Σ 〉 G +8 πC 2 ˙ a a ( N +1) | ∆ L 2 Σ | 2 G +6 ˙ a a ( N +1) | ∆ L 2 E | 2 G ≥ 0</formula>
<text><location><page_55><loc_12><loc_68><loc_88><loc_71></location>Proof. First, we recall that N +1 > 0 holds by Lemma 4.1. Additionally, we can apply (2.8) and the Young inequality and get the following:</text>
<formula><location><page_55><loc_17><loc_55><loc_88><loc_68></location>∣ ∣ ∣ 16 πC 2 a -3 ( N +1) 〈 ∆ L 2 E , ∆ L 2 Σ 〉 G ∣ ∣ ∣ ≤ 16 πC 2 · √ 3 4 πC 2 ˙ a a · ( N +1) | ∆ L 2 E | G | ∆ L 2 Σ | G ≤ 4( N +1) ˙ a a · ( √ 3 · | ∆ L 2 E | G ) · ( √ 4 πC 2 | ∆ L 2 Σ | G ) ≤ 6 ˙ a a ( N +1) | ∆ L 2 E | 2 G +8 πC 2 ˙ a a ( N +1) | ∆ L 2 Σ | 2 G . /square</formula>
<text><location><page_55><loc_12><loc_50><loc_88><loc_54></location>This shows that E ( L ) ( W, · ) + 4 πC 2 E ( L ) (Σ , · ) is controlled by the sum of the left hand sides of the inequalities in Lemma 6.6 and 6.8 for L ∈ 2 N , 0 ≤ L ≤ 18.</text>
<section_header_level_1><location><page_55><loc_12><loc_48><loc_39><loc_50></location>7.1. Improving energy bounds.</section_header_level_1>
<text><location><page_55><loc_12><loc_44><loc_88><loc_47></location>Proposition 7.2 (Improved energy bounds) . Under the bootstrap assumptions (see Assumption 3.16) and the initial data assumptions (3.10) , the following improved estimates hold on ( t Boot , t 0 ] :</text>
<formula><location><page_55><loc_12><loc_40><loc_70><loc_43></location>E ( ≤ 18) ( φ, · ) /lessorsimilar ε 4 a -cε 1 8 (7.2a)</formula>
<formula><location><page_55><loc_12><loc_34><loc_70><loc_38></location>E ( ≤ 18) ( W, · ) /lessorsimilar ε 15 4 a -cε 1 8 (7.2c)</formula>
<formula><location><page_55><loc_12><loc_37><loc_70><loc_40></location>E ( ≤ 18) (Σ , · ) /lessorsimilar ε 15 4 a -cε 1 8 (7.2b)</formula>
<formula><location><page_55><loc_12><loc_31><loc_70><loc_35></location>E ( ≤ 16) (Ric , · ) /lessorsimilar ε 7 2 a -cε 1 8 (7.2d)</formula>
<formula><location><page_55><loc_12><loc_29><loc_71><loc_32></location>E ( ≤ 16) ( N, · ) + a 4 E (17) ( N, · ) + a 8 E (18) ( N, · ) /lessorsimilar ε 7 2 a 8 -cε 1 8 (7.2e)</formula>
<text><location><page_55><loc_12><loc_25><loc_88><loc_28></location>Proof. We prove this estimate by performing an induction over even energy orders. Starting at order 0, we first observe that by Lemma 7.1, we can bound the (base level) total energy</text>
<text><location><page_55><loc_12><loc_15><loc_88><loc_21></location>by the sum of the left hand side of (6.2), the left hand side of (6.10) weighted by ε 1 4 and the left hand side of (6.13) weighted by 4 πC 2 · ε 1 4 , and the left hand sides of (6.9) and ε 1 2 · (6.15) extended to L = 0. 11 Combining said estimates and inserting the initial data assumption from (3.11), the</text>
<formula><location><page_55><loc_19><loc_20><loc_81><loc_25></location>E (0) total := E (0) ( φ, · ) + ε 1 4 ( E (0) ( W, · ) + 4 πC 2 E (0) (Σ , · ) ) + a 4 E (1) ( φ, · ) + ε 1 2 E (1) (Σ , · )</formula>
<text><location><page_56><loc_12><loc_88><loc_30><loc_90></location>following holds in total:</text>
<formula><location><page_56><loc_12><loc_83><loc_72><loc_88></location>(7.3) E (0) total ( t ) /lessorsimilar ε 4 + ∫ t 0 t ( ε 1 8 a ( s ) -3 + a ( s ) -1 -c √ ε ) E (0) total ( s ) ds</formula>
<formula><location><page_56><loc_26><loc_77><loc_74><loc_82></location>E (0) total ( t ) /lessorsimilar ε 4 exp ( c ' ∫ t 0 t ε 1 8 a ( s ) -3 + a ( s ) -1 -c √ ε ds ) /lessorsimilar ε 4 a -c ' ε 1 8</formula>
<text><location><page_56><loc_12><loc_82><loc_88><loc_84></location>Applying the Gronwall lemma (see Lemma 11.1) to (7.3), we get for some suitable constant c ' > 0:</text>
<text><location><page_56><loc_12><loc_75><loc_63><loc_77></location>Now assume that, for L ∈ 2 N , 2 ≤ L ≤ 18, we have already shown</text>
<text><location><page_56><loc_12><loc_68><loc_88><loc_72></location>on ( t Boot , t 0 ]. Note that (7.1) means this holds true for L = 2 after updating c > 0. Further, if L ≥ 4 holds, we assume:</text>
<formula><location><page_56><loc_12><loc_71><loc_73><loc_76></location>E ( ≤ L -2) ( φ, · ) + ε 1 4 ( E ( ≤ L -2) (Σ , · ) + E ( ≤ L -2) ( W, · ) ) /lessorsimilar ε 4 a -cε 1 8 (7.4a)</formula>
<formula><location><page_56><loc_12><loc_65><loc_60><loc_68></location>E ( ≤ L -4) (Ric , · ) /lessorsimilar ε 7 2 a -cε 1 8 (7.4b)</formula>
<text><location><page_56><loc_12><loc_61><loc_88><loc_65></location>We will show that these assumptions hold at L = 4 after having shown the induction step for L = 2. We define, for 2 ≤ L ≤ 18,</text>
<text><location><page_56><loc_12><loc_49><loc_88><loc_56></location>We combine the respective energy estimates with the appropriate scalings 12 , namely (in the listed order) (6.3), (6.11), (6.14), (6.9), (6.15), (6.18) and (6.20). Observe that the sum of these scaled left hand sides control E ( L ) total by Lemma 7.1. Combining all of these estimates and inserting the initial data assumption (3.11), we get the following estimate:</text>
<formula><location><page_56><loc_17><loc_56><loc_84><loc_62></location>E ( L ) total := E ( L ) ( φ, · ) + ε 1 4 ( E ( L ) ( W, · ) + 4 πC 2 E ( L ) (Σ , · ) ) + a 4 E ( L +1) ( φ, · ) + ε 1 2 a 4 E (21) (Σ , · ) + ε 1 2 E ( L -2) (Ric , · ) + ε 3 4 a 4 E ( L -1) (Ric , · ) .</formula>
<formula><location><page_56><loc_12><loc_42><loc_61><loc_48></location>E ( L ) total ( t ) /lessorsimilar ε 4 + ∫ t 0 t ( ε 1 8 a ( s ) -3 + a ( s ) -1 -c √ ε ) E ( L ) total ( s ) ds (7.5a)</formula>
<formula><location><page_56><loc_12><loc_32><loc_75><loc_39></location>+ ε 17 8 a ( s ) -3 -c √ ε E ( ≤ L -2) ( W,s ) + ε 11 8 E ( ≤ L -4) (Ric , s ) ︸ ︷︷ ︸ if L =4 } ds (7.5c)</formula>
<formula><location><page_56><loc_12><loc_38><loc_68><loc_44></location>+ ∫ t 0 t { ε 3 8 a ( s ) -3 -c √ ε [ E ( ≤ L -2) ( φ, s ) + E ( ≤ L -2) (Σ , s ) ] (7.5b)</formula>
<formula><location><page_56><loc_12><loc_30><loc_87><loc_35></location>+ ε 1 4 ( a ( t ) 4 -c √ ε + εa ( t ) 2 -c √ ε ) · ε 1 2 E ( L ) (Σ , t ) + ε 1 2 E ( L ) ( φ, t ) + ε 1 4 · ε 1 4 E ( L ) ( W,t ) (7.5d)</formula>
<formula><location><page_56><loc_12><loc_26><loc_47><loc_29></location>+ ε 3 2 a 2 -c √ ε E ( ≤ L -2) (Ric , t ) (7.5f)</formula>
<formula><location><page_56><loc_12><loc_29><loc_83><loc_32></location>+ ε 7 4 · ε 3 4 a 4 E ( L -1) (Ric , t ) + ε 5 2 a -c √ ε E ( ≤ L -2) ( φ, t ) + ε 1 2 a 2 -c √ ε E ( ≤ L -2) (Σ , t ) (7.5e)</formula>
<text><location><page_56><loc_12><loc_14><loc_88><loc_26></location>We briefly summarize which inequalities contain the listed error term bounds as explicit terms: The first two terms in (7.5b) come from (6.18) and the latter from (6.11), those in (7.5c) from (6.3) and (6.18), and finally the last three lines are precisely the scaled right hand side of (6.15). Regarding the curvature energies in the various individual energy estimates, any summand with E ( L -3) (Ric , · ) can be split using (3.8), the resulting summands containing E ( L -2) (Ric , · ) can always be absorbed in to the total energy term in the first line, and anything with E ( ≤ L -4) (Ric , · ) is tracked in 〈 Err 〉 L for L ≥ 4.</text>
<text><location><page_57><loc_12><loc_86><loc_88><loc_90></location>Inserting (7.4a)-(7.4b), (7.5b)-(7.5c) can be bounded up to constant by ε 33 8 a -3 -cε 1 8 . Here, the error term dominating all others arises from</text>
<formula><location><page_57><loc_39><loc_83><loc_61><loc_86></location>ε 3 8 a ( s ) -3 -c √ ε E ( ≤ L -2) (Σ , s ) .</formula>
<text><location><page_57><loc_12><loc_77><loc_88><loc_83></location>Regarding (7.5d)-(7.5f), notice that the first four summands can be bounded from above by ε 1 4 E ( L ) total ( t ) up to constant. For the remaining three terms, we can again insert the induction assumptions (7.4a)-(7.4b), bounding them by ε 17 4 a ( t ) -cε 1 8 .</text>
<text><location><page_57><loc_12><loc_74><loc_76><loc_75></location>In summary and after rearranging, (7.5a)-(7.5f) becomes for some constant K > 0:</text>
<formula><location><page_57><loc_15><loc_62><loc_85><loc_74></location>(1 -Kε 1 4 ) E ( L ) total ( t ) /lessorsimilar ε 4 + ∫ t 0 t ( ε 1 8 a ( s ) -3 + a ( s ) -1 -c √ ε ) E ( L ) total ( s ) ds + ∫ t 0 t ε 33 8 a ( s ) -3 -cε 1 8 ds + ε 17 4 a ( t ) -cε 1 8 /lessorsimilar ε 4 a ( t ) -cε 1 8 + ∫ t 0 t ( ε 1 8 a ( s ) -3 + a ( s ) -1 -c √ ε ) E ( L ) total ( s ) ds</formula>
<text><location><page_57><loc_12><loc_59><loc_88><loc_62></location>The prefactor on the left hand side is positive for small enough ε > 0, and the Gronwall lemma then yields</text>
<formula><location><page_57><loc_12><loc_55><loc_58><loc_59></location>(7.6) E ( L ) total ( t ) /lessorsimilar ε 4 a -cε 1 8 .</formula>
<text><location><page_57><loc_12><loc_47><loc_88><loc_55></location>In particular, this directly implies that the induction assumptions (7.4a) and (7.4b), using (3.9) to cover the skipped odd order, hold at order L , completing the induction step, and clearly also that (7.4b) holds for L -2 = 2 using to (7.6) at order 2. This completes the induction argument, proving (7.2a)-(7.2d). Finally, applying the obtained improved estimates for ∇ φ and Ric[ G ] to Corollary 5.8, we also get (7.2e). /square</text>
<unordered_list>
<list_item><location><page_57><loc_12><loc_41><loc_88><loc_45></location>7.2. Improving solution norm control. To close the bootstrap argument, it now remains to show that the improved energy bounds also imply improved bounds for H and C . The former follows almost directly using Lemma 4.5:</list_item>
</unordered_list>
<text><location><page_57><loc_12><loc_38><loc_85><loc_40></location>Corollary 7.3 (Improved Sobolev norm bounds) . On ( t Boot , t 0 ] , the following estimates hold:</text>
<formula><location><page_57><loc_12><loc_34><loc_57><loc_38></location>H /lessorsimilar ε 7 4 a -cε 1 8 (7.7a)</formula>
<formula><location><page_57><loc_12><loc_31><loc_57><loc_35></location>‖ Σ ‖ 2 H 18 G /lessorsimilar ε 15 4 a -cε 1 8 (7.7b)</formula>
<formula><location><page_57><loc_12><loc_28><loc_56><loc_31></location>‖ N ‖ 2 H 18 G /lessorsimilar ε 4 a -cε 1 8 (7.7c)</formula>
<text><location><page_57><loc_12><loc_21><loc_88><loc_26></location>Proof. First, we apply the improved energy estimates from Proposition 7.2 as well as the strong C G -norm bounds from Lemma 4.3 to the near-coercivity estimates in Lemma 4.5. With this, we directly obtain the following Sobolev norm estimates (updating c ):</text>
<formula><location><page_57><loc_27><loc_8><loc_73><loc_20></location>‖ Ψ ‖ 2 H 18 G /lessorsimilar ε 4 a -cε 1 8 + εa -cε 1 8 · ε 15 4 a -cε 1 8 /lessorsimilar ε 4 a -cε 1 8 ‖ Σ ‖ 2 H 18 G /lessorsimilar ε 15 4 a -cε 1 8 ‖ Ric[ G ] + 2 9 G ‖ 2 H 16 G /lessorsimilar ε 7 2 a -cε 1 8 ‖ E ‖ 2 H 18 G + ‖ B ‖ 2 H 18 G /lessorsimilar ε 15 4 a -cε 1 8</formula>
<text><location><page_58><loc_12><loc_88><loc_34><loc_90></location>By Lemma 6.5, we also have</text>
<formula><location><page_58><loc_12><loc_83><loc_72><loc_88></location>(7.8) ‖∇ φ ‖ H 18 G /lessorsimilar ( 1 + εa -c √ ε ) ‖ Σ ‖ H 19 G + ε ‖ Ψ ‖ H 19 G /lessorsimilar ε 15 4 a -cε 1 8 .</formula>
<text><location><page_58><loc_12><loc_78><loc_88><loc_84></location>We take particular care in showing that the improved bound holds for a 2 ‖∇ φ ‖ ˙ H 19 G : First, note that (7.2d) implies E ( ≤ 17) (Ric , · ) /lessorsimilar ε 7 2 a -cε 1 8 . Applying this along with (4.4e) to (4.13d), as well as (4.13a) in the second line and (7.2a) as well as (7.8) in the final step, we obtain:</text>
<formula><location><page_58><loc_20><loc_63><loc_80><loc_77></location>a 4 ‖∇ φ ‖ 2 H 19 G /lessorsimilar a 4 ‖∇ φ ‖ 2 L 2 G + a 4 -c √ ε 9 ∑ m =0 ‖∇ 2 ∆ m φ ‖ 2 L 2 G + εa 4 -c √ ε · E ( ≤ 17) (Ric , · ) /lessorsimilar a 4 -c √ ε ( ‖ ∆ 9 φ ‖ 2 L 2 G + ‖ ∆ 10 φ ‖ 2 L 2 G + ‖∇ φ ‖ 2 H 17 G ) + ε 9 2 a -cε 1 8 /lessorsimilar a -c √ ε · E ( ≤ 19) ( φ, · ) + a 4 -c √ ε ‖∇ φ ‖ 2 H 17 G + ε 9 2 a -cε 1 8 /lessorsimilar ε 15 4 a -cε 1 8</formula>
<text><location><page_58><loc_12><loc_61><loc_66><loc_62></location>Further, inserting (7.2a), (7.2b) and (7.2d) into Corollary 5.5 implies</text>
<formula><location><page_58><loc_27><loc_55><loc_73><loc_60></location>a 8 ‖ ∆ 10 N ‖ 2 L 2 G + a 4 ‖∇ ∆ 9 N ‖ 2 L 2 G + 9 ∑ m =0 ‖ ∆ m N ‖ 2 L 2 G /lessorsimilar ε 11 4 a -cε 1 8</formula>
<text><location><page_58><loc_12><loc_53><loc_50><loc_55></location>and subsequently, applying Lemma 4.5 as before,</text>
<formula><location><page_58><loc_33><loc_49><loc_68><loc_52></location>a 8 ‖ N ‖ 2 ˙ H 20 G + a 4 ‖ N ‖ 2 ˙ H 19 G + ‖ N ‖ 2 H 18 G /lessorsimilar ε 4 a -cε 1 8 .</formula>
<text><location><page_58><loc_12><loc_47><loc_75><loc_49></location>Finally, having now shown (7.7b) and (7.7c), we can apply these to (6.23) to get</text>
<text><location><page_58><loc_12><loc_41><loc_23><loc_42></location>proving (7.7a).</text>
<formula><location><page_58><loc_28><loc_41><loc_88><loc_47></location>‖ G -γ ‖ 2 H 18 G /lessorsimilar a -cε 1 8 ( ε 4 + ε -1 4 + 15 4 + ε -1 4 +4 ) /lessorsimilar ε 7 2 a -cε 1 8 , /square</formula>
<text><location><page_58><loc_12><loc_32><loc_88><loc_40></location>Intuitively, the bounds on C should now follow from H by the standard Sobolev embedding. However, since both of these norms are with respect to G , the embedding constant may be time dependent. To circumvent this issue, we need to switch between norms with repsect to G and γ and then apply the embedding with respect to C γ and H γ . The following lemma ensures that we still obtain bootstrap improvements after performing these norm switches:</text>
<text><location><page_58><loc_12><loc_25><loc_88><loc_30></location>Lemma 7.4 (Moving between norms) . Let l ∈ N , l ≤ 18 , ζ be a scalar field, T be an arbitrary Σ t -tangent tensor. Then, for some multivariate polynomial P l with P l (0 , 0) = 0 , we have (7.9a)</text>
<text><location><page_58><loc_12><loc_20><loc_17><loc_21></location>(7.9b)</text>
<formula><location><page_58><loc_13><loc_21><loc_89><loc_26></location>‖ ζ ‖ C l G ( U ) /lessorsimilar a -c √ ε ‖ ζ ‖ C l γ ( M ) + a -c √ ε ‖ ζ ‖ C max { 0 , /floorleft l -1 2 /floorright } γ ( M ) · P l ( ‖ G -γ ‖ C l -1 γ ( M ) , ‖ G -1 -γ -1 ‖ C l -1 γ ( M ) )</formula>
<formula><location><page_58><loc_12><loc_15><loc_87><loc_20></location>‖ T ‖ C l G ( M ) /lessorsimilar a -c √ ε ‖ T ‖ C l γ ( M ) + a -c √ ε ‖ T ‖ C max { 0 , /floorleft l -1 2 /floorright } γ ( M ) · P l ( ‖ G -γ ‖ C l γ ( M ) , ‖ G -1 -γ -1 ‖ C l γ ( M ) )</formula>
<text><location><page_58><loc_12><loc_13><loc_86><loc_15></location>as well as the same inequalities with the roles of G and γ reversed. For l ≤ 12 , this reduces to:</text>
<formula><location><page_58><loc_12><loc_7><loc_68><loc_11></location>a c √ ε ‖ T ‖ C l γ ( M ) /lessorsimilar ‖ T ‖ C l G ( M ) /lessorsimilar a -c √ ε ‖ T ‖ C l γ ( M ) (7.10b)</formula>
<formula><location><page_58><loc_12><loc_10><loc_68><loc_14></location>a c √ ε ‖ ζ ‖ C l γ ( M ) /lessorsimilar ‖ ζ ‖ C l G ( M ) /lessorsimilar a -c √ ε ‖ ζ ‖ C l γ ( M ) (7.10a)</formula>
<text><location><page_59><loc_12><loc_88><loc_25><loc_90></location>Further, one has</text>
<text><location><page_59><loc_12><loc_86><loc_17><loc_87></location>(7.11a)</text>
<text><location><page_59><loc_12><loc_80><loc_17><loc_81></location>(7.11b)</text>
<formula><location><page_59><loc_12><loc_81><loc_90><loc_87></location>‖ ζ ‖ H l γ ( M ) 2 /lessorsimilar a -c √ ε ‖ ζ ‖ 2 H l G ( M ) + a -cε 1 8 ‖ ζ ‖ 2 C /ceilingleft l -1 2 /ceilingright G (Σ t ) ( ε 4 + ε -1 4 sup s ∈ ( t,t 0 ) ( ‖ N ‖ 2 H l -1 G (Σ s ) + ‖ Σ ‖ 2 H l -1 G (Σ s ) ) ) ,</formula>
<formula><location><page_59><loc_12><loc_74><loc_87><loc_80></location>‖ T ‖ 2 H l γ ( M ) /lessorsimilar a -c √ ε ‖ T ‖ 2 H l G ( M ) + a -cε 1 8 ‖ T ‖ 2 C /ceilingleft l -1 2 /ceilingright G (Σ t ) ( ε 4 + ε -1 4 sup s ∈ ( t,t 0 ) ( ‖ N ‖ 2 H l G (Σ s ) + ‖ Σ ‖ 2 H l G (Σ s ) ) )</formula>
<text><location><page_59><loc_12><loc_69><loc_88><loc_74></location>Remark 7.5. While we only need the tensorial inequalities for gradient vector fields and (0 , 2)tensors when applied to norms in H and C , the proof is more simple when considering tensors of arbitrary rank.</text>
<text><location><page_59><loc_12><loc_62><loc_88><loc_68></location>Proof. We restrict ourselves to proving the tensorial statements; the scalar field analogues follow analagously except for the fact that, since ∇ i ζ = ˆ ∇ i ζ = ∂ i ζ , error terms caused by Christoffel symbols always enter at one order less. Thus, it remains to show (7.9b), (7.10b) and (7.11b) by iterating over derivative order.</text>
<text><location><page_59><loc_12><loc_60><loc_65><loc_62></location>Starting with the base level estimates, we have if T is of rank ( r, s ):</text>
<formula><location><page_59><loc_16><loc_54><loc_85><loc_60></location>| T | 2 G -| T | 2 γ = [ G i 1 j 1 . . . G i r j r ( G -1 ) p 1 q 1 . . . ( G -1 ) p s q s -γ i 1 j 1 . . . γ i r j r ( γ -1 ) p 1 q 1 . . . ( γ -1 ) p s q s ] · · T i 1 ...i r p 1 ...p s T j 1 ...j r q 1 ...q s</formula>
<text><location><page_59><loc_12><loc_50><loc_88><loc_53></location>We successively replace G ± 1 by ( G ± 1 -γ ± 1 ) + γ ± 1 , take the |·| γ -norm of each factor and use (4.4c)-(4.4d). This yields</text>
<text><location><page_59><loc_12><loc_44><loc_81><loc_51></location>∣ ∣ | T | 2 G -| T | 2 γ ∣ ∣ /lessorsimilar √ εa -c √ ε | T | 2 γ , implying (7.9b) (and (7.10b)) for l = 0 after rearranging and taking supremums suitably. To show (7.11b) at base level, consider</text>
<formula><location><page_59><loc_31><loc_35><loc_69><loc_44></location>∫ M | T | 2 G vol G -∫ M | T | 2 γ vol γ = ∫ M ( | T | 2 G -| T | 2 γ ) vol G + ∫ M | T | 2 γ µ G -µ γ µ γ vol γ .</formula>
<formula><location><page_59><loc_35><loc_29><loc_65><loc_33></location>(1 -Kε ) ‖ T ‖ 2 L 2 γ /lessorsimilar (1 + √ εa -c √ ε ) ‖ T ‖ 2 L 2 G</formula>
<text><location><page_59><loc_12><loc_32><loc_88><loc_35></location>We can control the first summand on the right hand side as before, while we have | µ G -µ γ | /lessorsimilar ε by (4.11). Hence,</text>
<text><location><page_59><loc_12><loc_28><loc_79><loc_29></location>follows for a suitable constant K > 0, implying the statement for small enough ε > 0.</text>
<text><location><page_59><loc_12><loc_23><loc_88><loc_27></location>Next, we perform the iteration for (7.9b), assuming the statement and the analogue with γ and G reversed to hold up to order l -1. As above, note that</text>
<text><location><page_59><loc_12><loc_18><loc_44><loc_19></location>where we can rewrite the second term as</text>
<formula><location><page_59><loc_20><loc_17><loc_79><loc_24></location>∣ ∣ ∣ ∣ ∣ ∣ ∇ J T ∣ ∣ 2 G -∣ ∣ ∣ ˆ ∇ J T ∣ ∣ ∣ 2 γ ∣ ∣ ∣ ∣ /lessorsimilar √ εa -c √ ε ∣ ∣ ∣ ˆ ∇ J T ∣ ∣ ∣ 2 γ +(1 + √ εa -c √ ε ) ∣ ∣ ∣ ∣ ∣ ∣ ∇ J T ∣ ∣ 2 γ -∣ ∣ ∣ ˆ ∇ J T ∣ ∣ ∣ 2 γ ∣ ∣ ∣ ∣</formula>
<text><location><page_59><loc_12><loc_12><loc_61><loc_14></location>and hence obtain (moving between pointwise norms as before)</text>
<formula><location><page_59><loc_26><loc_5><loc_74><loc_12></location>∣ ∣ ∣ ∣ ∣ ∣ ∇ J T ∣ ∣ 2 G -∣ ∣ ∣ ˆ ∇ J T ∣ ∣ ∣ 2 γ ∣ ∣ ∣ ∣ /lessorsimilar a -c √ ε ∣ ∣ ∣ ˆ ∇ J T ∣ ∣ ∣ 2 γ + a -c √ ε ∣ ∣ ∣ ∇ J T -ˆ ∇ J T ∣ ∣ ∣ 2 γ .</formula>
<formula><location><page_59><loc_35><loc_11><loc_65><loc_18></location>∣ ∣ ∣ 2 〈 ˆ ∇ J T -∇ J T , ˆ ∇ T 〉 γ -|∇ J T -ˆ ∇ J T | 2 γ ∣ ∣ ∣</formula>
<text><location><page_60><loc_12><loc_87><loc_69><loc_90></location>Regarding ∇ J T -ˆ ∇ J T , we have the following schematic decomposition :</text>
<text><location><page_60><loc_42><loc_80><loc_69><loc_83></location>+ 〈 at least cubic nonlinear terms 〉 ,</text>
<formula><location><page_60><loc_12><loc_82><loc_70><loc_88></location>∇ J T -ˆ ∇ J T = J -1 ∑ I =0 ˆ ∇ J -I -1 (Γ -ˆ Γ) ∗ γ ( ∇ I T + ˆ ∇ I T ) (7.12)</formula>
<text><location><page_60><loc_12><loc_75><loc_88><loc_80></location>Here, ∗ γ encodes the analogous schematic product notation with regards to γ (see subsection 2.1.8). Regarding the Christoffel symbols, notice (7.9b) with roles of γ and G reversed holding up to l -1 implies that, for any m ∈ { 0 , . . . , l -1 } and some multivariate polynomial ˜ P m , we have</text>
<formula><location><page_60><loc_18><loc_72><loc_82><loc_76></location>‖ Γ -ˆ Γ ‖ C m γ ( M ) /lessorsimilar a -c √ ε ˜ P m ( ‖ Γ -ˆ Γ ‖ C m G ( M ) , ‖ G -γ ‖ C m G ( M ) , ‖ G -1 -γ -1 ‖ C m G ( M ) ) .</formula>
<text><location><page_60><loc_12><loc_69><loc_88><loc_72></location>As explained in Remark 3.7, we can bound ‖ Γ -ˆ Γ ‖ C m G ( M ) by a polynomial in ‖ G -γ ‖ C m +1 G ( M ) . Hence, we can apply (4.4c) to obtain</text>
<formula><location><page_60><loc_12><loc_65><loc_61><loc_69></location>(7.13) ‖ Γ -ˆ Γ ‖ C l -1 γ ( M ) /lessorsimilar √ εa -c √ ε .</formula>
<text><location><page_60><loc_12><loc_64><loc_83><loc_65></location>Moving back to (7.12) and just considering the first line for now, this implies the following:</text>
<formula><location><page_60><loc_12><loc_60><loc_16><loc_61></location>(7.14)</formula>
<text><location><page_60><loc_12><loc_42><loc_88><loc_51></location>We can rewrite ∇ m T -norms in C γ as ones in C G up to a -c √ ε as before. Then, we can apply the already obtained estimates up to order l -1 show that the first two lines of the right hands side can be estimated by the right hand side of (7.9b). The highly nonlinear terms can be dealt with similarly, closing the induction over admissible l . (7.10b) immediately follows by applying (4.4c)-(4.4d) and (7.13).</text>
<formula><location><page_60><loc_18><loc_50><loc_86><loc_64></location>∣ ∣ ∣ ‖ T ‖ ˙ C l G ( M ) 2 -‖ T ‖ 2 ˙ C l γ ( M ) ∣ ∣ ∣ /lessorsimilar a -c √ ε ( ‖ T ‖ C l γ ( M ) 2 + l -1 ∑ m =0 ‖∇ m T ‖ C 0 γ ( M ) 2 ) + ‖ T ‖ C /ceilingleft l -1 2 /ceilingright γ ( M ) 2 + /ceilingleft l -1 2 /ceilingright ∑ m =0 ‖∇ m T ‖ C 0 γ ( M ) 2 ‖ Γ -ˆ Γ ‖ C l -1 γ ( M ) 2 + 〈 at least cubic nonlinear terms 〉</formula>
<text><location><page_60><loc_12><loc_38><loc_88><loc_41></location>Now, assume (7.11b) to be proven up to order J -1. By analogous arguments as at order zero, we get after rearranging:</text>
<text><location><page_60><loc_12><loc_31><loc_88><loc_36></location>∣ ∣ so we only need to concern ourselves with the first summand. Reversing roles of G and γ compared to the proof of (7.9b), we get</text>
<formula><location><page_60><loc_24><loc_32><loc_76><loc_39></location>∫ M | ˆ ∇ J T | 2 γ vol γ /lessorsimilar ∣ ∣ ∣ ∫ M ( |∇ J T | 2 G -| ˆ ∇ J T | 2 γ ) vol G ∣ ∣ ∣ + ∫ M |∇ J T | 2 G vol G ,</formula>
<formula><location><page_60><loc_12><loc_23><loc_85><loc_31></location>∣ ∣ ∣ ∣ ∣ ∣ ∇ J T ∣ ∣ 2 G -∣ ∣ ∣ ˆ ∇ J T ∣ ∣ ∣ 2 γ ∣ ∣ ∣ ∣ /lessorsimilar √ εa -c √ ε ∣ ∣ ∇ J T ∣ ∣ 2 G + a -c √ ε ∣ ∣ ∣ 2 〈∇ J T -ˆ ∇ J T , ∇ T 〉 G -|∇ J T -ˆ ∇ J T | 2 G ∣ ∣ ∣ , and have the following, applying Lemma 6.13 immediately to estimate ‖ Γ -ˆ Γ ‖ H l -1 G :</formula>
<formula><location><page_60><loc_28><loc_7><loc_73><loc_24></location>∣ ∣ ∣ ∣ ∫ V { 2 〈∇ l T -ˆ ∇ l T , ∇ T 〉 G -|∇ l T -ˆ ∇ l T | 2 G } vol G ∣ ∣ ∣ ∣ /lessorsimilar a -c √ ε ( ‖ T ‖ 2 H l G ( M ) + ‖ ˆ ∇ ≤ l -1 T ‖ 2 H 0 G ( M ) ) + ( ‖ T ‖ 2 C /ceilingleft l -1 2 /ceilingright G (Σ t ) + ‖ ˆ ∇ ≤/ceilingleft l -1 2 /ceilingright T ‖ 2 C 0 G (Σ t ) ) · · a -cε 1 8 ( ε 4 + ε -1 4 sup s ∈ ( t,t 0 ) ( ‖ N ‖ 2 H l G (Σ s ) + ‖ Σ ‖ 2 H l G (Σ s ) ) )</formula>
<text><location><page_61><loc_12><loc_86><loc_88><loc_91></location>By the same arguments as earlier, we have ‖ ˆ ∇ ≤ l -1 T ‖ H l -1 G ( M ) /lessorsimilar a -c √ ε ‖ T ‖ H l -1 γ ( M ) and can then apply the induction hypothesis. This proves (7.11b). /square</text>
<text><location><page_61><loc_12><loc_82><loc_85><loc_84></location>Corollary 7.6 (Improved C -norm bounds) . On ( t Boot , t 0 ] , the following estimate is satisfied:</text>
<formula><location><page_61><loc_12><loc_78><loc_57><loc_81></location>(7.15) C + C γ /lessorsimilar ε 7 2 a -cε 1 8</formula>
<text><location><page_61><loc_12><loc_72><loc_88><loc_75></location>Proof. We first apply the Sobolev norm estimates in Lemma 7.4 to (7.7a), to then control C γ via the standard Sobolev embedding H l +2 ( M ) ↪ C l ( M ), and finally control with (7.9a)-(7.9b).</text>
<text><location><page_61><loc_12><loc_66><loc_88><loc_74></location>γ → C Note that by Lemma 4.3, we can control the C G -norm up to order 10 of every quantity occurring in H beside the lapse by at worst √ εa -c √ ε , while the bootstrap assumption already implies better behaviour for the lapse. Thus, we can apply (7.11a)-(7.11b) to every norm appearing in H , and obtain by applying (7.7b) and (7.7c) in the second line:</text>
<formula><location><page_61><loc_19><loc_54><loc_81><loc_66></location>C 2 γ /lessorsimilar a -c √ ε · H 2 + εa -c √ ε · a -cε 1 8 ( ε 4 + ε -1 4 sup s ∈ ( t,t 0 ) ( ‖ N ‖ 2 H 18 G (Σ s ) + ‖ Σ ‖ 2 H 18 G (Σ s ) ) ) /lessorsimilar ε 7 2 a -cε 1 8 + εa -cε 1 8 ( ε 4 + ε 7 2 ) /lessorsimilar ε 7 2 · a -cε 1 8</formula>
<text><location><page_61><loc_12><loc_52><loc_43><loc_53></location>In particular, we can update c such that</text>
<formula><location><page_61><loc_32><loc_48><loc_68><loc_51></location>| P ( ‖ G -γ ‖ C 16 γ (Σ t ) , ‖ G -γ ‖ C 16 γ (Σ t ) ) | /lessorsimilar ε 7 2 a -cε 1 8</formula>
<text><location><page_61><loc_12><loc_43><loc_88><loc_47></location>holds for any multivariate polynomial P that appears when applying (7.9a)-(7.9b). Again using the strong C G -norm estimates from Lemma 4.3, this then implies C /lessorsimilar ε 7 2 a -cε 1 8 . /square</text>
<section_header_level_1><location><page_61><loc_31><loc_39><loc_69><loc_40></location>8. Big Bang stability: The main theorem</section_header_level_1>
<text><location><page_61><loc_12><loc_33><loc_88><loc_38></location>In this section, we provide the proof of the first main result, Theorem 1.1, which we state in more detail in Theorem 8.2 below. As in [RS18b, Spe18], most of the work has already been done by establishing the necessary bounds on solution norms.</text>
<text><location><page_61><loc_12><loc_21><loc_88><loc_32></location>Remark 8.1 (Existence of a CMC hypersurface) . As mentioned in Section 1.2.1, it may seem that the generality of the results in Theorem 8.2 is restricted by taking the initial data on Σ t 0 to be CMC. However, as long as one remains close enough to a constant time hypersurface of the FLRW reference metric (which is CMC), one can locally evolve the perturbed data in harmonic gauge to a nearby hypersurface that is CMC and remains close to the FLRW reference solution. To make this a bit more precise, and also since this it is a little less involved than the arguments in [RS18b], we will briefly sketch how the arguments from [FK20, Section 2.5] extend to our setting.</text>
<text><location><page_61><loc_12><loc_11><loc_88><loc_20></location>First, we once again assume without loss of generality that our initial data is sufficiently regular. Note that we can locally evolve our data within harmonic gauge to get a C 17 -regular family of metrics with near-FLRW initial data (for well-posedness, consider the analogue of [RS18b, Proposition 14.1]). Consider the Banach manifold M 17 formed by the set of C 17 Lorentz metrics on I × M for an open interval I around t 0 such that the surfaces of constant time are Riemannian, endowed with the norm</text>
<formula><location><page_61><loc_27><loc_7><loc_74><loc_10></location>‖ ˜ g ‖ = ‖ ˜ n 2 ‖ C 17 dt 2 + γ ( I × M ) + ‖ ˜ X ‖ C 17 dt 2 + γ ( I × M ) + ‖ ˜ g t ‖ C 17 dt 2 + γ ( I × M ) ,</formula>
<text><location><page_62><loc_12><loc_86><loc_88><loc_90></location>where ˜ g ∈ M 17 has lapse ˜ n , shift ˜ X and spatial metrics (˜ g t ) t ∈ I . Further, for any f ∈ C 17 ( M,I ), we define the embedding ι f : M ↪ → M by x ↦→ ( f ( x ) , x ), and subsequently define the smooth map</text>
<formula><location><page_62><loc_23><loc_81><loc_77><loc_86></location>H 0 : D := { (˜ g, f ) ∈ M 17 × C 17 ( M,I ) | ι ∗ f ˜ g is Riemannian } -→ C 16 ( M ) (˜ g, f ) ↦→ mean curvature of ( M, ˜ g t ) embedded along ι f .</formula>
<text><location><page_62><loc_12><loc_73><loc_88><loc_81></location>One easily checks that ( g FLRW , t 0 ) is a regular point of H 0 . By the implicit function theorem for Banach manifolds, this means there is a (unique) smooth function F that maps an open neighbourhood of g FLRW in M 17 to an open neighbourbood of the constant function x ↦→ t 0 in C 16 ( M ) such that H 0 ( · , F ( · )) = τ ( t 0 ) holds in that neighbourhood.</text>
<text><location><page_62><loc_12><loc_68><loc_88><loc_74></location>Thus, we can choose a surface Σ ' with mean curvature τ ( t 0 ) near the original Σ t 0 . Furthermore, for small enough ε > 0, the initial data on Σ ' remains close to the FLRW initial data in the sense of Assumption 3.10, using similar arguments to control Sobolev norms. Thus, we can replace Σ t 0 by Σ ' without loss of generality, proving that the CMC assumption (2.10) is not a true restriction.</text>
<text><location><page_62><loc_12><loc_57><loc_88><loc_66></location>Theorem 8.2 (Stability of Big Bang formation) . Let ( M, ˚ g, ˚ k, ˚ π, ˚ ψ ) be initial data to the Einstein scalar-field system as discussed in Section 1.2.1. Further, let the data be embedded into a timeoriented 4-manifold such that it induces initial data for the rescaled solution variables (see Definition 2.9) at the initial hypersurface Σ t 0 . We also assume this rescaled initial data is close to that of the FLRW reference solution (see (2.1) and (2.2) ) in the sense that</text>
<formula><location><page_62><loc_12><loc_54><loc_61><loc_56></location>(8.1) H ( t 0 ) + H top ( t 0 ) + C ( t 0 ) ≤ ε 2</formula>
<text><location><page_62><loc_12><loc_51><loc_51><loc_54></location>is satisfied (with H and C as in Definition 3.6). 13</text>
<text><location><page_62><loc_12><loc_45><loc_88><loc_50></location>Then, the past maximal globally hyperbolic development ((0 , t 0 ] × M,g,φ ) of this data within the Einstein scalar-field system (1.1a) -(1.1c) in CMC gauge (2.10) with zero shift is foliated by the CMC hypersurfaces Σ s = t -1 ( { s } ) , and one has</text>
<formula><location><page_62><loc_12><loc_41><loc_63><loc_45></location>(8.2) H ( t ) + C ( t ) + C γ ( t ) /lessorsimilar ε 7 4 a ( t ) -cε 1 8</formula>
<text><location><page_62><loc_12><loc_39><loc_79><loc_41></location>for some c > 0 and any t ∈ (0 , t 0 ] . In particular, this implies the following statements:</text>
<text><location><page_62><loc_12><loc_27><loc_88><loc_36></location>Asymptotic behaviour of solution variables: We denote the solution metric as g = -n 2 dt 2 + g , the second fundamental form (viewed as a (1 , 1) -tensor) with respect to Σ t as k and the volume form with regards to g on Σ t by vol g . There exist a smooth function Ψ Bang ∈ C 15 γ ( M ) , a (1 , 1) -tensor field K Bang ∈ C 15 γ ( M ) and a volume form vol Bang ∈ C 15 γ ( M ) such that the following estimates hold for any t ∈ (0 , t 0 ] :</text>
<formula><location><page_62><loc_12><loc_15><loc_73><loc_25></location> ∥ ∥ a -3 vol g -vol Bang ∥ ∥ C l γ (Σ t ) /lessorsimilar εa ( t ) 4 -cε 1 8 l ≤ 14 εa ( t ) 2 -cε 1 8 l = 15 (8.3b)</formula>
<formula><location><page_62><loc_12><loc_22><loc_73><loc_28></location>‖ n -1 ‖ C l γ (Σ t ) /lessorsimilar εa ( t ) 4 -cε 1 8 l ≤ 14 εa ( t ) 2 -cε 1 8 l = 15 (8.3a)</formula>
<formula><location><page_62><loc_12><loc_8><loc_73><loc_19></location> ∥ ∥ a 3 ∂ t φ -(Ψ Bang + C ) ∥ ∥ C l γ (Σ t ) /lessorsimilar εa ( t ) 4 -cε 1 8 l ≤ 14 εa ( t ) 2 -cε 1 8 l = 15 (8.3c)</formula>
<formula><location><page_63><loc_12><loc_83><loc_77><loc_91></location>∥ ∥ ∥ ∥ φ -∫ t 0 t a ( s ) -3 ds · (Ψ Bang + C ) ∥ ∥ ∥ ∥ ˙ C l γ (Σ t ) /lessorsimilar εa ( t ) 4 -cε 1 8 1 ≤ l ≤ 14 εa ( t ) 2 -cε 1 8 l = 15 (8.3d)</formula>
<formula><location><page_63><loc_12><loc_74><loc_71><loc_77></location>( K Bang ) a a = -√ 12 πC , (8.4a)</formula>
<formula><location><page_63><loc_12><loc_77><loc_73><loc_87></location> ∥ ∥ a 3 k -K Bang ∥ ∥ C l γ (Σ t ) /lessorsimilar εa ( t ) 4 -cε 1 8 l ≤ 14 εa ( t ) 2 -cε 1 8 l = 15 (8.3e) Further, these footprint states satisfy the equations 14</formula>
<formula><location><page_63><loc_12><loc_72><loc_67><loc_74></location>8 π (Ψ Bang + C ) 2 +( K Bang ) a b ( K Bang ) b a =12 πC 2 (8.4b)</formula>
<text><location><page_63><loc_12><loc_69><loc_88><loc_72></location>and remain close to the data of the reference solution in the following sense, where I denotes the Kronecker symbol:</text>
<formula><location><page_63><loc_12><loc_65><loc_62><loc_68></location>‖ vol γ -vol Bang ‖ C 15 γ ( M ) /lessorsimilar ε (8.5a)</formula>
<formula><location><page_63><loc_12><loc_63><loc_62><loc_65></location>‖ Ψ Bang ‖ C 15 γ ( M ) /lessorsimilar ε (8.5b)</formula>
<text><location><page_63><loc_12><loc_55><loc_69><loc_58></location>Additionally, there exists a (0 , 2) -tensor field M Bang ∈ C 15 γ ( M ) satisfying</text>
<formula><location><page_63><loc_12><loc_55><loc_62><loc_64></location>∥ ∥ ∥ ∥ ∥ K Bang + √ 4 π 3 C I ∥ ∥ ∥ ∥ ∥ C 15 γ ( M ) /lessorsimilar ε (8.5c)</formula>
<formula><location><page_63><loc_12><loc_53><loc_59><loc_55></location>(8.6) ‖ M Bang -γ ‖ C 15 γ ( M ) /lessorsimilar ε</formula>
<text><location><page_63><loc_12><loc_50><loc_84><loc_52></location>and, with /circledot and exp meant in the matrix product and exponential sense respectively, one has</text>
<formula><location><page_63><loc_12><loc_40><loc_54><loc_43></location>‖ E ‖ C 16 γ (Σ t ) /lessorsimilar εa -4 -cε 1 8 (8.8a)</formula>
<formula><location><page_63><loc_12><loc_43><loc_83><loc_51></location>(8.7) ∥ ∥ ∥ ∥ g /circledot exp [( -2 ∫ t 0 t a ( s ) -3 ds ) · K Bang ] -M Bang ∥ ∥ ∥ ∥ C l γ (Σ t ) /lessorsimilar εa ( t ) 4 -cε 1 8 l ≤ 14 εa ( t ) 2 -cε 1 8 l = 15 . Moreover, the Bel-Robinson variables E and B satisfy the following estimates:</formula>
<formula><location><page_63><loc_12><loc_32><loc_63><loc_41></location>‖ B ‖ C l γ (Σ t ) /lessorsimilar εa -2 -cε 1 8 l ≤ 15 εa -4 -cε 1 8 l ≤ 16 (8.8b)</formula>
<formula><location><page_63><loc_12><loc_24><loc_71><loc_30></location>(8.9) L [ α ] = ∫ s max s 1 √ ( γ ab ) α ( s ) ˙ α a ( s ) ˙ α b ( s ) ds ≤ K a ( t ) 2 -cε 1 8 ,</formula>
<text><location><page_63><loc_12><loc_28><loc_88><loc_34></location>Causal disconnectedness: Let α be a past directed causal curve on ((0 , t ] × M,g ) for t ≤ t 0 with domain [ s 1 , s max ) such that α ( s 1 ) ∈ Σ t and s max is maximal. Then, there exists a constant K > 0 that does not depend on α such that one has</text>
<text><location><page_63><loc_12><loc_23><loc_77><loc_25></location>where γ is the negative Einstein spatial reference metric on M (see Definition 2.1).</text>
<text><location><page_63><loc_12><loc_17><loc_88><loc_21></location>Geodesic incompleteness: Let α be a past directed, affinely parametrized causal geodesic emanating from Σ t 0 with parameter time A normalized to A ( t 0 ) = 0 . Then,</text>
<text><location><page_63><loc_12><loc_11><loc_88><loc_14></location>holds for suitable constants K 1 , K 2 > 0 that are independent of α , and thus any such geodesic crashes into the Big Bang hypersurface in finite affine parameter time. Hence, for sufficiently</text>
<formula><location><page_63><loc_12><loc_13><loc_72><loc_18></location>(8.10) A (0) ≤ K 1 · |A ' ( t 0 ) | · a ( t 0 ) 1+ K 2 ε ∫ t 0 0 a ( s ) -1 -K 2 ε ds < ∞ ,</formula>
<text><location><page_64><loc_12><loc_87><loc_83><loc_90></location>points p, q ∈ Σ t with dist γ ( p, q ) > 2 K a ( t ) 2 -cε 1 8 , the causal pasts of p and q cannot intersect.</text>
<text><location><page_64><loc_15><loc_84><loc_77><loc_86></location>Blow-up: The norm | k | g behaves toward the Big Bang hypersurface as follows:</text>
<formula><location><page_64><loc_12><loc_71><loc_72><loc_79></location>(8.11b) ∥ ∥ ∥ ∥ a 12 P αβγδ P αβγδ -5 3 · (8 π ) 2 (Ψ Bang + C ) 4 ∥ ∥ ∥ ∥ C 0 γ ( M ) /lessorsimilar εa 4 -cε 1 8</formula>
<formula><location><page_64><loc_12><loc_78><loc_72><loc_85></location>(8.11a) ∥ ∥ ∥ a 6 | k | 2 g -( K Bang ) i j ( K Bang ) j i ∥ ∥ ∥ C 0 γ (Σ t ) /lessorsimilar εa 4 -cε 1 8 Further, with W [ g ] denoting the Weyl curvature and P [ g ] = Riem[ g ] -W [ g ] ,</formula>
<text><location><page_64><loc_12><loc_71><loc_79><loc_73></location>is satisfied, whereas there exists a scalar footprint W Bang ∈ C 15 γ ( M ) such that one has</text>
<text><location><page_64><loc_17><loc_65><loc_87><loc_70></location>∥ ∥ W Bang is a fourth order polynomial in ˆ K Bang = K Bang + 4 π 3 C I and Ψ Bang and satisfies</text>
<formula><location><page_64><loc_12><loc_66><loc_68><loc_72></location>(8.11c) ∥ ∥ a 12 W αβγδ W αβγδ -W Bang ∥ ∥ C 0 ( M ) /lessorsimilar εa 2 -cε 1 8 .</formula>
<formula><location><page_64><loc_42><loc_62><loc_56><loc_64></location>‖ W Bang ‖ C 15 γ ( M ) /lessorsimilar</formula>
<text><location><page_64><loc_12><loc_63><loc_63><loc_68></location>Here, √ (8.11d) ε .</text>
<text><location><page_64><loc_12><loc_58><loc_88><loc_62></location>Finally, the scalar curvature R [ g ] and the Ricci curvature invariant Ric[ g ] αβ Ric[ g ] αβ blow-up with the asymptotics</text>
<formula><location><page_64><loc_12><loc_55><loc_73><loc_58></location>‖ a 6 R [ g ] -8 π (Ψ Bang + C ) 2 ‖ C 0 ( M ) /lessorsimilar εa 4 -cε 1 8 and (8.11e)</formula>
<formula><location><page_64><loc_12><loc_52><loc_70><loc_55></location>‖ a 12 Ric[ g ] αβ Ric[ g ] αβ -(8 π ) 2 (Ψ Bang + C ) 4 ‖ /lessorsimilar εa 4 -cε 1 8 , (8.11f)</formula>
<text><location><page_64><loc_12><loc_49><loc_88><loc_52></location>and the Kretschmann scalar K = Riem[ g ] αβγδ Riem[ g ] αβγδ exhibits stable blow-up in the following sense:</text>
<formula><location><page_64><loc_12><loc_42><loc_72><loc_49></location>(8.11g) ∥ ∥ ∥ ∥ a 12 K5 3 · (8 π ) 2 (Ψ Bang + C ) 4 -W Bang ∥ ∥ ∥ ∥ C 0 ( M ) /lessorsimilar εa 2 -cε 1 8</formula>
<text><location><page_64><loc_12><loc_34><loc_88><loc_42></location>Remark 8.3 (The solution variables exhibit AVTD behaviour) . The estimates (8.3a)-(8.3e) and (8.7) imply that the solution is asymptotically velocity term dominated (AVTD) in the sense that, toward the Big Bang singularity, they behave at leading order like solutions to the (formal) velocity term dominated equations. These arise by dropping any terms containing spatial derivatives in the decomposed Einstein system, i.e. in (2.15a), (2.15b), (2.17a) and (2.18).</text>
<text><location><page_64><loc_12><loc_21><loc_88><loc_33></location>Proof. As argued at the end of Section 3.4, we can assume without loss of generality that our initial data is sufficiently regular. Hence, the local existence statement in Lemma 3.14 and the initial data requirements (8.1) ensure that there exists a local solution to the Einstein scalar-field system on [ t 1 , t 0 ] × M and that the bootstrap assumption (see Assumption 3.16) holds on [ t 1 , t 0 ] × M with t 1 ∈ (0 , t 0 ) and σ = ε 1 16 Let t ∈ (0 , t 0 ) be such that ( t , t 0 ] × M is the maximal domain on which the solution variables exist and satisfy the bootstrap assumptions. For contradiction, we now assume that t > 0 were to hold.</text>
<text><location><page_64><loc_12><loc_15><loc_88><loc_19></location>Due to Corollary 7.6, there exist (summarizing all updates) constants c 1 , K 1 > 0 such that, for any t ∈ ( t , t 0 ],</text>
<text><location><page_64><loc_12><loc_8><loc_88><loc_13></location>If ε is small enough such that K 1 ε 1 8 < K 0 and c 1 ε 1 8 < c 0 σ hold, this is a strict improvement of the bootstrap assumption. Furthermore, argued exactly as in the proof of [RS18b, Theorem 15.1], above improvement ensures none of the blow-up criteria of Lemma 3.14 are satisfied if t > 0 were</text>
<formula><location><page_64><loc_12><loc_13><loc_58><loc_16></location>(8.12) C ( t ) ≤ K 1 ε 7 2 a ( t ) -c 1 ε 1 8</formula>
<text><location><page_65><loc_12><loc_83><loc_88><loc_90></location>to hold, essentially as a direct consequence of (8.12). Hence, the solution could be classically extended to a CMC hypersurface Σ t diffeomorphic to M while satisfying the improved estimates by continuity, and further to an interval ( t ' , t 0 ] for some 0 < t ' < t on which the bootstrap assumptions must then be satisfied, also by continuity. This contradicts the maximality of ( t , t 0 ].</text>
<text><location><page_65><loc_12><loc_77><loc_88><loc_81></location>Thus, the rescaled solution variables induce a unique solution to the Einstein scalar-field system on (0 , t 0 ] × M such that (8.12) is satisfied for any t ∈ (0 , t 0 ]. The core estimate (8.2) follows since Corollary 7.3 and Corollary 7.6 now hold on (0 , t 0 ].</text>
<text><location><page_65><loc_12><loc_69><loc_88><loc_75></location>From (8.2), the asymptotic behaviour in (8.3a)-(8.3e) and (8.7) is established as in [RS18b, Theorem 15.1], which we briefly outline: First, we note that (8.3a) follows directly from (8.2). For the remaining estimates, the arguments are similar, so consider for example ∂ t φ : By the rescaled wave equation (2.32a) and (8.3a), we have that</text>
<formula><location><page_65><loc_37><loc_60><loc_63><loc_69></location>‖ ∂ t Ψ ‖ C l γ (Σ t ) /lessorsimilar εa 1 -cε 1 8 l ≤ 14 εa -1 -cε 1 8 l = 15</formula>
<text><location><page_65><loc_12><loc_59><loc_86><loc_61></location>Hence, for an arbitrary decreasing sequence ( t m ) m ∈ N , on (0 , t 0 ] that converges to zero, we have</text>
<text><location><page_65><loc_12><loc_49><loc_88><loc_53></location>for any m 1 , m 2 ∈ N , m 1 < m 2 by (2.6). This shows that Ψ( t m 1 , · ) is a Cauchy sequence in C 15 γ ( M ) and hence there exists a limit function Ψ Bang ∈ C 15 γ ( M ) that satisfies</text>
<formula><location><page_65><loc_29><loc_51><loc_71><loc_60></location>‖ Ψ( t m 1 , · ) -Ψ( t m 2 , · ) ‖ C l γ ( M ) /lessorsimilar εa ( t m 1 ) 4 -cε 1 8 l ≤ 14 εa ( t m 1 ) 2 -cε 1 8 l = 15</formula>
<formula><location><page_65><loc_29><loc_40><loc_71><loc_49></location>‖ Ψ( t, · ) -(Ψ Bang -C ) ‖ C l γ ( M ) /lessorsimilar εa ( t ) 4 -cε 1 8 l ≤ 14 εa ( t ) 2 -cε 1 8 l = 15</formula>
<text><location><page_65><loc_12><loc_38><loc_88><loc_42></location>for any t ∈ (0 , t 0 ]. Since Ψ = a 3 n -1 ∂ t φ -C holds by definition, (8.3c) now follows by examining the Taylor expansion of n -1 -1 at 0 using (8.3a).</text>
<text><location><page_65><loc_12><loc_29><loc_88><loc_37></location>(8.4a) follows directly from the CMC condition (2.10), the asymptotic behaviour (8.3e) of a 3 k and the Friedman equation (2.3). (8.4b) follows from asymptotic limit of the Hamiltonian constraint (2.16a) with (2.3), (8.3a), (8.3c) and (8.3e) as well as (8.2) for lower order terms. (8.7) follows exactly as in [RS18b, Theorem 15.1], and (8.5a)-(8.6) are a direct result of the initial data assumptions and applying the respective asymptotic estimates to t = t 0 .</text>
<text><location><page_65><loc_12><loc_28><loc_76><loc_29></location>For the first estimate in (8.8b), we apply the momentum constraint (2.29d) to get</text>
<formula><location><page_65><loc_27><loc_24><loc_73><loc_27></location>|∇ J B | G = a -4 |∇ J B | G = a -2 |∇ J curl G Σ | G /lessorsimilar a -2 |∇ J +1 Σ | G</formula>
<text><location><page_65><loc_12><loc_22><loc_60><loc_24></location>and consequently, with Lemma 7.4 as well as (4.4g) and (8.2),</text>
<formula><location><page_65><loc_25><loc_14><loc_76><loc_22></location>‖ B ‖ C 15 γ (Σ t ) /lessorsimilar a -c √ ε ‖ B ‖ C 15 G (Σ t ) + εa -2 -c √ ε · P 15 ( ‖ G -γ ‖ C 15 γ (Σ t ) ) /lessorsimilar a -2 -c √ ε ‖ Σ ‖ C 16 G (Σ t ) + εa -2 -c √ ε · P 15 ( ‖ G -γ ‖ C 15 γ (Σ t ) ) /lessorsimilar εa -2 -cε 1 8 .</formula>
<text><location><page_65><loc_12><loc_8><loc_88><loc_12></location>The remaining estimates in (8.8a) and (8.8b) are contained in (8.2). The results (8.9) and (8.10) follow as in the proofs of (15.6) and (15.7) in [RS18b, Theorem 15.1] from the asymptotic behaviour of the solution variables in (8.3a)-(8.3e) and (8.7). We briefly sketch the proof of (8.10): Consider</text>
<text><location><page_66><loc_12><loc_86><loc_88><loc_90></location>a geodesic α affinely parametrized by A as in the statement. The geodesic equations then lead to the following estimate for some suitable K > 0:</text>
<formula><location><page_66><loc_26><loc_82><loc_74><loc_87></location>|A '' | ≤ ˙ a a |A ' | + K [ ˙ a a | N | + n -1 | ∂ t N | + n -1 |∇ N | g + n | ˆ k | g ] |A ' | .</formula>
<text><location><page_66><loc_12><loc_77><loc_88><loc_82></location>The leading term is hereby arises from the mean curvature condition. Arguing as with the elliptic estimates in Section 5, one can show that | ∂ t N | /lessorsimilar εa -1 -cε 1 8 . Thus, along with the other pointwise bounds on n , g and ˆ k , one obtains</text>
<formula><location><page_66><loc_42><loc_73><loc_58><loc_76></location>|A '' | ≤ ˙ a a (1 + cε ) |A ' |</formula>
<text><location><page_66><loc_12><loc_70><loc_87><loc_72></location>and consequently |A ' ( t ) | ≤ |A ' ( t 0 ) | a -1 -cε by the Gronwall lemma. (8.10) follows by integrating.</text>
<text><location><page_66><loc_12><loc_64><loc_88><loc_69></location>Turning to the blow-up behaviour of geometric invariants, observe (8.11a) is a direct consequence of (8.3e). Regarding (8.11c), we first compute using (2.19) and standard algebraic manipulations that</text>
<text><location><page_66><loc_12><loc_60><loc_53><loc_62></location>By the rescaled constraint equation (2.29c), we have</text>
<formula><location><page_66><loc_28><loc_60><loc_72><loc_65></location>a 12 W αβγδ W αβγδ = a 12 ( 8 | E | 2 g +8 | B | 2 g ) = 8 | E | 2 G +8 | B | 2 G .</formula>
<text><location><page_66><loc_12><loc_47><loc_88><loc_56></location>for t ↓ 0. Further, by expanding (2.3) around a = 0, we have ˙ aa 2 = √ 4 π 3 C + O ( a 2 ) . Since Σ /sharp and Ψ converge to footprint states ˆ K Bang = K Bang + √ 4 πC 3 I and Ψ Bang in C 15 γ ( M ) respectively, this shows that W Bang ∈ C 15 γ ( M ) exists, is a fourth order polynomial in ˆ K Bang and Ψ Bang , and satisfies</text>
<formula><location><page_66><loc_23><loc_55><loc_77><loc_60></location>E ij = -˙ aa 2 Σ ij +(Σ /circledot Σ) ij -[ 8 π 3 Ψ 2 + 16 π 3 C Ψ ] G ij + O ( εa 4 -cε 1 8 ) ,</formula>
<formula><location><page_66><loc_36><loc_42><loc_63><loc_48></location>∥ ∥ ∥ | E | 2 G -1 8 W Bang ∥ ∥ ∥ C 0 ( M ) /lessorsimilar εa 2 -cε 1 8</formula>
<text><location><page_66><loc_12><loc_39><loc_88><loc_46></location>∥ ∥ as well as where (8.11d). Due to (8.8b), the | B | 2 G -term in the Weyl curvature scalar is negligible in comparison, and thus (8.11c) immediately follows. Furthermore, one has</text>
<formula><location><page_66><loc_34><loc_35><loc_66><loc_38></location>P αβγδ P αβγδ =2Ric[ g ] αβ Ric[ g ] αβ -2 9 R [ g ] 2 ,</formula>
<text><location><page_66><loc_12><loc_33><loc_88><loc_34></location>and the statement follows once more with (8.3c), (8.3a) as well as (8.2) for lower orders. /square</text>
<section_header_level_1><location><page_66><loc_41><loc_29><loc_59><loc_30></location>9. Future stability</section_header_level_1>
<text><location><page_66><loc_12><loc_26><loc_57><loc_28></location>The goal of this section is to show the following theorem:</text>
<text><location><page_66><loc_12><loc_16><loc_88><loc_25></location>Theorem 9.1 (Future stability of Milne spacetime) . Let the rescaled initial data ( g , k , ∇ φ, φ ' ) on M be sufficiently close to ( γ, 1 3 γ, 0 , 0) in H 5 × H 4 × H 4 × H 4 on some initial hypersurface Σ τ = τ 0 (see Definition 9.4 and Assumption 9.7). Then, its maximal globally hyperbolic development ( M,g,φ ) within the Einstein scalar-field system in CMCSH gauge is foliated by the CMC Cauchy hypersurfaces (Σ τ ) τ ∈ [ τ 0 , 0) , is future (causally) complete and exhibits the following asymptotic behaviour:</text>
<formula><location><page_66><loc_34><loc_13><loc_66><loc_16></location>( g , k , φ ' , ∇ φ )( τ ) -→ ( γ, 1 3 γ, 0 , 0) as τ ↑ 0</formula>
<text><location><page_66><loc_12><loc_8><loc_88><loc_12></location>Since the control of geometric perturbations uses the same arguments as in [AF20], the focus in this section will lie on dealing with the the scalar field. The key idea is controlling decay of the scalar field using an indefinite corrective term on top of the canonical energy (see Definition 9.6).</text>
<section_header_level_1><location><page_67><loc_12><loc_88><loc_28><loc_90></location>9.1. Preliminaries.</section_header_level_1>
<text><location><page_67><loc_12><loc_83><loc_88><loc_86></location>9.1.1. Notation, gauge and spatial reference geometry. Within this section, we will decompose the Lorentzian metric as follows:</text>
<formula><location><page_67><loc_12><loc_79><loc_66><loc_82></location>(9.1a) g = -n 2 dt 2 + g ab ( dx a + X a )( dx b + X b dt )</formula>
<text><location><page_67><loc_12><loc_78><loc_46><loc_79></location>We impose CMCSH gauge (see [AM04]) via</text>
<formula><location><page_67><loc_12><loc_74><loc_60><loc_77></location>(9.1b) t = τ, g ij (Γ a ij -ˆ Γ a ij ) = 0 ,</formula>
<text><location><page_67><loc_17><loc_72><loc_18><loc_74></location>ˆ</text>
<text><location><page_67><loc_12><loc_62><loc_88><loc_73></location>where Γ refers to the Christoffel symbols with regards to the spatial reference metric γ . We extend the notation from the Big Bang stability analysis regarding foliations, derivatives, indices and schematic term notation to this setting (see Section 2.1). In particular, Σ T and Σ τ will refer to spatial hypersurfaces along which the logarithmic time T (see (9.2c)) and the mean curvature τ are constant (see (9.2c) on why these are interchangeable), and we will write for example Σ T =0 when inserting a specific value to avoid potential ambiguity. We use similar notation for scalar functions and tensors that depend on T or, respectively, τ .</text>
<text><location><page_67><loc_12><loc_57><loc_88><loc_60></location>For the extent of the future stability analysis, we have to introduce an additional condition for the spatial geometry beyond Definition 2.1:</text>
<text><location><page_67><loc_12><loc_51><loc_88><loc_56></location>Definition 9.2 (Spectral condition for the Laplacian of the spatial reference manifold) . Let µ 0 ( γ ) to be the smallest positive eigenvalue of the Laplace operator -∆ γ = ( γ -1 ) ab ∇ a ∇ b acting on scalar functions, where ( M,γ ) is as in Definition 2.1. ( M,γ ) additionally is assumed to satisfy</text>
<formula><location><page_67><loc_46><loc_47><loc_54><loc_50></location>µ 0 ( γ ) > 1 9 .</formula>
<text><location><page_67><loc_12><loc_41><loc_88><loc_46></location>Remark 9.3 (Manifolds that satisfy Definition 9.2) . The existing literature on spectra of ∆ γ usually considers hyperbolic manifolds with sectional curvature κ = -1, and thus one needs to verify that µ 0 is strictly greater than 1 to verify the analogue of Definition 9.2 after rescaling.</text>
<text><location><page_67><loc_12><loc_28><loc_88><loc_39></location>Numerical work, e.g., [CS99, Ino01], provides evidence for over 250 compact hyperbolic 3-manifolds to satisfy this spectral bound, many of which are closed. In particular, both [CS99] 15 and [Ino01] consider the smallest closed orientable hyperbolic 3-manifold, the Weeks space m003(-3,1), and compute that it falls under Definition 9.2 with µ 0 ≈ 27 , 8 in [CS99, Table IV] and 26 /lessorapproxeql µ 0 /lessorapproxeql 27 , 8 in [Ino01, p. 639, Table 2]. Moreover, as demonstrated in [Ino01, p. 642, Figure 6], many manifolds with small enough diameter d satisfy this condition. In fact, this is ensured by, for example, the analytical bound</text>
<formula><location><page_67><loc_32><loc_23><loc_67><loc_28></location>µ 0 ≥ max { π 2 2 d 2 -1 2 , √ π 4 d 4 + 1 4 -3 2 , π 2 d 2 e -d }</formula>
<text><location><page_67><loc_12><loc_13><loc_88><loc_22></location>(cf. [CZ95, Theorem 1.1-1.2] with L = 2) which implies µ 0 > 10 for Weeks space, which has diameter d ≈ 0 . 843. Furthermore, [Ino01] finds no closed hyperbolic manifold violating this bound. We also note that it is conjectured that, for any arithmetic hyperbolic 3-manifold, one at least has µ 0 ≥ 1 (see [Ber03, Conjecture 2.3]). In fact, this is tied to the Ramanujan conjecture for automorphic forms. Finally, one can construct compact hyperbolic manifolds where µ 0 becomes arbitrarily small, see [Cal94, Corollary 4.4].</text>
<text><location><page_68><loc_12><loc_87><loc_88><loc_90></location>9.1.2. Rescaled variables and Einstein equations. We will use the standard rescaling of the solution variables by τ :</text>
<text><location><page_68><loc_12><loc_84><loc_56><loc_86></location>Definition 9.4 (Rescaled variables for future stability) .</text>
<formula><location><page_68><loc_12><loc_82><loc_66><loc_84></location>g ij = τ 2 g ij , ( g -1 ) ij = τ -2 g ij , Σ ij = τ ˆ k ij (9.2a)</formula>
<formula><location><page_68><loc_12><loc_79><loc_63><loc_81></location>n = τ 2 n, ˆ n = 1 , X a = τX a (9.2b)</formula>
<formula><location><page_68><loc_49><loc_78><loc_52><loc_81></location>n 3 -</formula>
<text><location><page_68><loc_12><loc_77><loc_49><loc_78></location>Furthermore, we introduce the logarithmic time</text>
<text><location><page_68><loc_12><loc_68><loc_88><loc_72></location>which satisfies ∂ T = -τ∂ τ . Toward the future, τ increases from τ 0 toward 0, and thus T increases from 0 to ∞ . We additionally introduce:</text>
<formula><location><page_68><loc_12><loc_72><loc_62><loc_77></location>(9.2c) T = -log ( τ τ 0 ) ⇔ τ = τ 0 e -T</formula>
<text><location><page_68><loc_12><loc_62><loc_87><loc_63></location>Moreover, for any scalar function ζ , we denote by ζ the mean integral with regards to (Σ T , g T ).</text>
<formula><location><page_68><loc_12><loc_61><loc_65><loc_68></location>˜ ∂ 0 = ∂ T + L X = -τ ( ∂ τ -L X ) (9.2d) φ ' = n -1 ˜ ∂ 0 φ = n -1 ( -τ ) -1 ( ∂ τ -L X ) φ (9.2e)</formula>
<text><location><page_68><loc_12><loc_59><loc_74><loc_61></location>For symmetric (0 , 2)-tensors h , we define the perturbed Lichnerowicz Laplacian</text>
<formula><location><page_68><loc_12><loc_54><loc_78><loc_59></location>(9.3) L g ,γ h ab = -1 µ g ˆ ∇ k ( ( g -1 ) kl µ g ˆ ∇ l h ab ) -2Riem[ γ ] akbl ( g -1 ) kk ' ( g -1 ) ll ' h k ' l ' .</formula>
<text><location><page_68><loc_12><loc_53><loc_29><loc_55></location>This operator satisfies</text>
<formula><location><page_68><loc_12><loc_50><loc_77><loc_53></location>(9.4) (Ric[ g ] -Ric[ γ ]) ij = 1 2 L g ,γ ( g -γ ) ij + J ij , ‖ J ij ‖ H l -1 /lessorsimilar ‖ g -γ ‖ H l ,</formula>
<text><location><page_68><loc_12><loc_45><loc_88><loc_49></location>see [AM03, Pf. of Theorem 3.1]. Under our conditions for the reference geometry, [Kro15] implies that the smallest positive eigenvalue of L γ,γ , denoted by λ 0 , satisfies λ 0 ≥ 1 9 , and that L γ,γ has trivial kernel. The spectral condition in Definition 9.2 is not necessary for this to hold true.</text>
<text><location><page_68><loc_12><loc_40><loc_88><loc_43></location>We now collect the 3+1-decomposition of the Einstein scalar-field equations in CMCSH gauge with the help of [AF20, (2.13)-(2.18)]:</text>
<text><location><page_68><loc_12><loc_36><loc_88><loc_39></location>Lemma 9.5 (Rescaled CMCSH equations) . The rescaled CMCSH Einstein scalar-field equations take the following form: The constraint equations</text>
<formula><location><page_68><loc_12><loc_30><loc_74><loc_35></location>(9.5a) R [ g ] -| Σ | 2 g -2 3 = 8 π [ | φ ' | 2 + |∇ φ | 2 g ] , div g Σ a = 8 πτ 3 φ ' ∇ b φ, the elliptic lapse and shift equations</formula>
<formula><location><page_68><loc_12><loc_21><loc_87><loc_26></location>∆ g X a +( g -1 ) ab Ric[ g ] bm X m =2( g -1 ) am ( g -1 ) bn ∇ b n · Σ mn -( g -1 ) ab ∇ b ˆ n +8 π n τ 3 φ ' ∇ b φ (9.5c) -2( g -1 ) bk (( g -1 ) cl n · Σ bc -∇ b X l )(Γ a kl -ˆ Γ a kl ) ,</formula>
<formula><location><page_68><loc_12><loc_25><loc_68><loc_31></location>( ∆ g -1 3 ) n = n ( | Σ | 2 g +4 π [ | φ ' | 2 + |∇ φ | 2 g ]) -1 , (9.5b)</formula>
<text><location><page_68><loc_12><loc_20><loc_38><loc_21></location>the geometric evolution equations</text>
<formula><location><page_68><loc_12><loc_14><loc_62><loc_19></location>˜ ∂ 0 g ab =2 n Σ ab +2ˆ ng ab , (9.5d) ∂ 0 ( g -1 ) ab = -2 n ( g -1 ) ac ( g -1 ) bd Σ cd -2ˆ n ( g -1 ) ab , (9.5e)</formula>
<formula><location><page_68><loc_33><loc_7><loc_77><loc_10></location>+2 n · ( g -1 ) mn Σ am Σ bn -1 3 ˆ ng ab -ˆ n Σ ab -8 π n ∇ a φ ∇ b φ</formula>
<formula><location><page_68><loc_12><loc_8><loc_65><loc_17></location>˜ ˜ ∂ 0 Σ ab = -2 Σ ab -n ( Ric[ g ] ab + 2 9 g ab ) + ∇ a ∇ b n (9.5f)</formula>
<text><location><page_69><loc_12><loc_88><loc_29><loc_90></location>and the wave equation</text>
<formula><location><page_69><loc_12><loc_82><loc_66><loc_88></location>(9.5g) ˜ ∂ 0 φ ' = 〈∇ n , ∇ φ 〉 g + n ∆ g φ +(1 -n ) φ ' .</formula>
<text><location><page_69><loc_12><loc_81><loc_88><loc_84></location>9.1.3. Energies and data assumptions. The proof will rely on the following corrected energy quantities:</text>
<text><location><page_69><loc_12><loc_78><loc_48><loc_79></location>Definition 9.6 (Energies for future stability) .</text>
<formula><location><page_69><loc_12><loc_68><loc_45><loc_74></location>E SF = 4 ∑ m =0 ( E ( m ) SF + 2 3 C ( m ) SF ) (9.6b)</formula>
<formula><location><page_69><loc_12><loc_73><loc_83><loc_78></location>E ( l ) SF =( -1) l ∫ M [ φ ' ∆ l g φ ' -φ ∆ l +1 g φ ] vol g , C ( l ) SF = ( -1) l ∫ M ( φ -φ )∆ l g φ ' vol g (9.6a)</formula>
<formula><location><page_69><loc_21><loc_65><loc_79><loc_68></location>E geom = 9 2 M 〈 g -γ, L m g ,γ ( g -γ ) 〉 g vol g + 1 2 M 〈 6 Σ , L m -1 g ,γ (6 Σ ) 〉 g vol g</formula>
<formula><location><page_69><loc_12><loc_59><loc_61><loc_69></location>5 ∑ m =1 ( ∫ ∫ (9.6c) + c E ∫ M 〈 6 Σ , L m -1 g ,γ ( g -γ ) 〉 g vol g )</formula>
<text><location><page_69><loc_12><loc_57><loc_34><loc_59></location>The constant c E is given by</text>
<formula><location><page_69><loc_12><loc_53><loc_61><loc_56></location>(9.7) c E = 1 λ 0 > 9 9( λ 0 δ ' ) λ 0 = 1 ,</formula>
<formula><location><page_69><loc_44><loc_52><loc_60><loc_57></location>{ 1 -9</formula>
<text><location><page_69><loc_12><loc_50><loc_62><loc_52></location>where δ ' > 0 is chosen to be small enough within the argument.</text>
<text><location><page_69><loc_12><loc_41><loc_88><loc_49></location>The Sobolev norms H l g and C l g are defined analogously to Definition 3.3 and 3.4, with similar conventions on surpressing time dependence in notation whereever possible. Since norms with respect to g and γ are equivalent under the bootstrap assumption (and consequently throughout the entire argument), we will simply denote the norms by H l and C l throughout unless the specific metric is crucial.</text>
<text><location><page_69><loc_12><loc_37><loc_88><loc_40></location>Assumption 9.7 (Initial data assumption) . The initial data on the spatial hypersurface Σ T =0 is assumed to be small in the following sense:</text>
<formula><location><page_69><loc_12><loc_31><loc_76><loc_36></location>‖ g -γ ‖ C 3 + ‖ Σ ‖ C 2 + ‖ ˆ n ‖ C 4 + ‖ X ‖ C 4 + ‖ φ ' ‖ C 2 + ‖∇ φ ‖ C 2 (9.8) + ‖ g -γ ‖ H 5 + ‖ Σ ‖ H 4 + ‖ ˆ n ‖ H 6 + ‖ X ‖ H 6 + ‖ φ ' ‖ H 4 + ‖∇ φ ‖ H 4 ≤ δ 2</formula>
<text><location><page_69><loc_12><loc_17><loc_88><loc_31></location>Remark 9.8 (Local well-posedness toward the future) . Under the above initial data assumption, local well-posedness is satisfied by analogizing the arguments for local well-posedness in the vacuum setting (see [AM03, Theorem 3.1]) with the matter coupling added. Since this only consists of adding another wave equation to the hyperbolic system, the argument is structurally unchanged given appropriate smallness assumptions on φ ' and ∇ φ (where φ itself does not enter into the Einstein system). As before, we can without loss of generality assume that the initial is sufficiently regular to ensure that E geom , E ( l ) SF and C ( l ) SF initially are continuously differentiable (in time) for any l ≤ 4.</text>
<text><location><page_69><loc_12><loc_13><loc_89><loc_16></location>Assumption 9.9 (Bootstrap assumption) . We assume that, on the bootstrap interval T ∈ [0 , T Boot ) ,</text>
<text><location><page_69><loc_12><loc_13><loc_18><loc_14></location>one has</text>
<formula><location><page_69><loc_12><loc_7><loc_78><loc_12></location>‖ g -γ ‖ C 3 + ‖ Σ ‖ C 2 + ‖ ˆ n ‖ C 4 + ‖ X ‖ C 4 + ‖ φ ' ‖ C 2 + ‖∇ φ ‖ C 2 (9.9) + ‖ g -γ ‖ H 5 + ‖ Σ ‖ H 4 + ‖ ˆ n ‖ H 6 + ‖ X ‖ H 6 + ‖ φ ' ‖ H 4 + ‖∇ φ ‖ H 4 ≤ δe -T 2 .</formula>
<text><location><page_70><loc_12><loc_82><loc_88><loc_90></location>We only choose not to use ' /lessorsimilar '-notation in the above assumptions for notational convenience in some technical computations. As before, δ can be chosen to have been sufficiently small for the following estimates to hold and for the decay estimates we derive from the bootstrap assumptions to be strict improvements. Moreover, note that (9.9) is satisfied since all of the norms are continuous in time (see Remark 9.8)</text>
<text><location><page_70><loc_12><loc_77><loc_88><loc_80></location>Before moving on to the energy estimates, we quickly collect the following immediate consequence of the bootstrap assumptions:</text>
<text><location><page_70><loc_12><loc_73><loc_88><loc_75></location>Lemma 9.10 (Sobolev estimate for the curvature) . The following estimate holds for any l ∈ N 0 :</text>
<formula><location><page_70><loc_12><loc_66><loc_68><loc_73></location>(9.10a) ∥ ∥ ∥ ∥ Ric[ g ] + 2 9 g ∥ ∥ ∥ ∥ H l /lessorsimilar ‖ g -γ ‖ H l +2 + ‖ g -γ ‖ 2 H l +1 Under the bootstrap assumptions, this implies</formula>
<formula><location><page_70><loc_12><loc_58><loc_67><loc_66></location>(9.10b) ∥ ∥ ∥ ∥ Ric[ g ] + 2 9 g ∥ ∥ ∥ ∥ C 1 + ∥ ∥ ∥ ∥ Ric[ g ] + 2 9 g ∥ ∥ ∥ ∥ H 3 /lessorsimilar δe -T 2 Proof. By (9.4), one has</formula>
<formula><location><page_70><loc_29><loc_51><loc_71><loc_58></location>∥ ∥ ∥ Ric[ g ] + 2 9 g ∥ ∥ ∥ ∥ H l ≤ 1 2 ‖L g ,γ ( g -γ ) ‖ H l + K ‖ g -γ ‖ 2 H l +1</formula>
<text><location><page_70><loc_12><loc_48><loc_88><loc_56></location>∥ for some suitably large K > 0 and that L g ,γ is elliptic. This implies the first inequality, while the latter follows from directly from the bootstrap assumption (9.9) and by applying the standard Sobolev embedding. /square</text>
<section_header_level_1><location><page_70><loc_12><loc_42><loc_78><loc_44></location>9.2. Elliptic estimates. We briefly collect the elliptic estimates for lapse and shift:</section_header_level_1>
<text><location><page_70><loc_12><loc_37><loc_88><loc_41></location>Lemma 9.11 (Elliptic estimates for lapse and shift) . Let l ∈ { 3 , 4 , 5 , 6 } . Then, one has n ∈ (0 , 3) (thus ˆ n ∈ ( -1 , 0) ) and the following estimates hold:</text>
<formula><location><page_70><loc_12><loc_32><loc_82><loc_37></location>‖ ˆ n ‖ H l /lessorsimilar δe -T 2 ‖ Σ ‖ H l -2 + δ 2 e -T ‖ g -γ ‖ H l -2 + δe -T 2 [ ‖ φ ' ‖ H l -2 + ‖∇ φ ‖ H l -2 ] (9.11a)</formula>
<text><location><page_70><loc_12><loc_27><loc_88><loc_30></location>Proof. The pointwise bounds on n follow via (9.5b) and the maximum principle as in Lemma 4.1. For the remaining estimates, applying elliptic regularity theory to (9.5b) and (9.5c) implies:</text>
<formula><location><page_70><loc_12><loc_29><loc_82><loc_34></location>‖ X ‖ H l /lessorsimilar δe -T 2 ‖ Σ ‖ H l -2 + δe -T 2 ‖ g -γ ‖ H l -1 + δe -T 2 [ ‖ φ ' ‖ H l -2 + ‖∇ φ ‖ H l -2 ] (9.11b)</formula>
<formula><location><page_70><loc_17><loc_14><loc_83><loc_26></location>‖ ˆ n ‖ H l /lessorsimilar ‖ Σ ‖ C /floorleft l -2 2 /floorright ‖ Σ ‖ H l -2 + ‖∇ φ ‖ 2 C 2 ‖ g -γ ‖ H l -2 + [ ‖∇ φ ‖ C 2 (1 + ‖ g -γ ‖ C 2 ) + ‖ φ ' ‖ C 2 ] [ ‖ φ ' ‖ H l -2 + ‖∇ φ ‖ H l -2 ] ‖ X ‖ H l /lessorsimilar ‖ Σ ‖ C /floorleft l -2 2 /floorright ‖ Σ ‖ H l -2 + ‖ g -γ ‖ 2 H l -1 + ‖∇ φ ‖ C 1 ‖ g -γ ‖ H l -3 + [ ‖∇ φ ‖ 2 C 2 ( 1 + ‖ g -γ ‖ C 2 ) + ‖ φ ' ‖ C 2 )] [1 + ‖ ˆ n ‖ C 2 ] [ ‖ φ ' ‖ H l -2 + ‖∇ φ ‖ H l -2 ]</formula>
<text><location><page_70><loc_12><loc_14><loc_48><loc_15></location>The statement then follows by inserting (9.9).</text>
<text><location><page_70><loc_87><loc_13><loc_88><loc_15></location>/square</text>
<section_header_level_1><location><page_70><loc_12><loc_8><loc_42><loc_9></location>9.3. Scalar field energy estimates.</section_header_level_1>
<text><location><page_71><loc_12><loc_85><loc_88><loc_90></location>9.3.1. Near-coercivity of E SF . We will be able to prove a decay estimate via a Gronwall argument only for the corrected energy E SF . Hence, we first need to verify that this energy controls the solution norms, for which we first show that it controls the 'canonical' scalar field energies:</text>
<text><location><page_71><loc_12><loc_83><loc_62><loc_84></location>Lemma 9.12 (Positivity of corrected scalar field energies) . Let</text>
<formula><location><page_71><loc_33><loc_78><loc_67><loc_83></location>Q = √ 1 + 9 q -1 √ 1 + 9 q with q = 1 2 ( µ 0 ( γ ) -1 9 ) .</formula>
<text><location><page_71><loc_12><loc_75><loc_62><loc_78></location>Then, for any l ∈ { 0 , 1 , 2 , 3 , 4 } and δ > 0 small enough, one has</text>
<formula><location><page_71><loc_12><loc_70><loc_70><loc_75></location>(9.12) Q E ( l ) SF ≤ E ( l ) SF + 2 3 C ( l ) SF , hence Q 4 ∑ m =0 E ( l ) SF ≤ E SF</formula>
<text><location><page_71><loc_12><loc_66><loc_88><loc_70></location>Proof. We denote the smallest positive eigenvalue of ∆ g acting on scalar functions on Σ T as µ 0 ( g T ). By the bootstrap assumption (9.9) and since µ 0 depends continuously on the metric, we obtain the following for small enough δ > 0:</text>
<formula><location><page_71><loc_34><loc_61><loc_66><loc_66></location>µ 0 ( g T ) ≥ µ 0 ( γ ) -1 2 ( µ 0 ( γ ) -1 9 ) ≥ 1 9 + q</formula>
<text><location><page_71><loc_12><loc_57><loc_88><loc_61></location>By the Poincar'e inequality applied on (Σ T , g T ) (see [CBM01, p.1037]), above spectral bound implies the following for any ζ ∈ H 1 (Σ T ):</text>
<text><location><page_71><loc_12><loc_51><loc_28><loc_53></location>For l = 0, this means</text>
<formula><location><page_71><loc_12><loc_53><loc_75><loc_58></location>(9.13) ‖ ζ -ζ ‖ 2 L 2 g (Σ T ) ≤ µ 0 ( g T ) -1 ‖∇ ζ ‖ 2 L 2 g (Σ T ) ≤ ( 1 9 + q ) -1 ‖∇ ζ ‖ 2 L 2 g (Σ T )</formula>
<formula><location><page_71><loc_26><loc_41><loc_74><loc_51></location>E (0) SF + 2 3 C (0) SF ≥‖ φ ' ‖ 2 L 2 g + ‖∇ φ ‖ 2 L 2 g -2 3 ‖ φ -φ ‖ L 2 g ‖ φ ' ‖ L 2 g ≥‖ φ ' ‖ 2 L 2 g + ‖∇ φ ‖ 2 L 2 g -2(1 + 9 q ) -1 2 ‖∇ φ ‖ L 2 g ‖ φ ' ‖ L 2 g ≥ √ 1 + 9 q -1 √ 1 + 9 q E (0) SF .</formula>
<text><location><page_71><loc_12><loc_38><loc_53><loc_40></location>For l = 1, notice that we can rewrite C (1) SF as follows:</text>
<text><location><page_71><loc_12><loc_32><loc_42><loc_34></location>Hence, applying (9.13) to ζ = φ ' yields</text>
<formula><location><page_71><loc_18><loc_33><loc_82><loc_39></location>C (1) SF = ∫ M 〈∇ φ, ∇ φ ' 〉 g vol g = ∫ M 〈 ∇ φ, ∇ ( φ ' -φ ' )〉 g vol g = -∫ M ( φ ' -φ ' ) ∆ g φ vol g</formula>
<formula><location><page_71><loc_22><loc_28><loc_79><loc_33></location>E (1) SF + 2 3 C (1) SF ≥ E (1) SF -2(1 + 9 q ) -1 2 ‖∇ φ ' ‖ L 2 g ‖ ∆ g φ ‖ L 2 g ≥ √ 1 + 9 q -1 √ 1 + 9 q E (1) SF .</formula>
<text><location><page_71><loc_12><loc_24><loc_88><loc_28></location>For l = 2 , 3 , 4, notice ∆ g φ = ∆ g φ ' = ∆ 2 g φ = 0 holds due to the divergence theorem, hence the argument proceeds as in l = 0 , 1. /square</text>
<text><location><page_71><loc_12><loc_19><loc_88><loc_23></location>Lemma 9.13 (Near-coercivity of corrected scalar field energy) . For any scalar function ζ and k ∈ { 1 , 2 } , one has the following under the bootstrap assumptions:</text>
<formula><location><page_71><loc_16><loc_5><loc_84><loc_20></location>∫ M |∇ 2 ζ | 2 g vol g /lessorsimilar ∫ M | ∆ g ζ | 2 g + |∇ ζ | 2 g vol g ‖ ζ ‖ 2 ˙ H 2 k /lessorsimilar ‖ ∆ k g ζ ‖ 2 L 2 + ( ‖ ζ ‖ 2 ˙ H 2 k -1 + ‖ ζ ‖ 2 ˙ H 2 k -2 ) + ‖∇ ζ ‖ 2 C 1 ∥ ∥ ∥ ∥ Ric[ g ] + 2 9 g ∥ ∥ ∥ ∥ 2 H 2 k -2 ‖∇ ζ ‖ 2 ˙ H 2 k /lessorsimilar ‖∇ ∆ k g ζ ‖ 2 L 2 + ( ‖∇ ζ ‖ 2 ˙ H 2 k -1 + ‖∇ ζ ‖ 2 ˙ H 2 k -2 ) + ‖∇ ζ ‖ 2 C 2 ∥ ∥ ∥ ∥ Ric[ g ] + 2 9 g ∥ ∥ ∥ ∥ 2 H 2 k -2</formula>
<text><location><page_72><loc_12><loc_88><loc_45><loc_90></location>Consequently, the following estimate holds:</text>
<text><location><page_72><loc_12><loc_78><loc_88><loc_86></location>∥ ∥ Proof. The inequalities for ζ follows from the same arguments as Lemma 4.5, except that we have ‖ Ric[ g ] ‖ C 1 g /lessorsimilar 1+ δ /lessorsimilar 1 by Lemma 9.10. The final estimate then follows by applying these estimates to ζ = φ ' and ζ = φ and applying Lemma 9.12. /square</text>
<formula><location><page_72><loc_12><loc_82><loc_75><loc_88></location>‖ φ ' ‖ 2 H 4 + ‖∇ φ ‖ 2 H 4 /lessorsimilar E (4) SF + ( ‖ φ ' ‖ 2 C 2 + ‖∇ φ ‖ 2 C 2 ) ∥ ∥ ∥ Ric[ g ] + 2 9 g ∥ ∥ ∥ 2 H 2 (9.14)</formula>
<text><location><page_72><loc_12><loc_73><loc_88><loc_77></location>9.3.2. Preparations for energy estimates. Before proving the energy estimate, we need to establish two technical lemmas: First, we collect a formula to differentiate integrals, and then some estimates needed to deal with the mean value of φ in the base level correction term.</text>
<text><location><page_72><loc_12><loc_69><loc_88><loc_72></location>Lemma 9.14 (Differentiation of integrals, future stability version) . For any diffentiable function ζ , one has</text>
<formula><location><page_72><loc_29><loc_50><loc_71><loc_63></location>∂ T ∫ M ζ vol g = ∫ M ∂ T ζ + ∂ T µ g µ g ζ vol g = ∫ M ∂ T ζ +3ˆ n ζ -1 2 ( g -1 ) ab L X g ab ζ vol g = ∫ M ∂ T ζ +3ˆ n ζ -div g X · ζ vol g</formula>
<text><location><page_72><loc_12><loc_62><loc_65><loc_69></location>(9.15) ∂ T ∫ M ζ vol g = ∫ M ( ˜ ∂ 0 ζ +3ˆ n ζ ) vol g . Proof. As in the proof of (4.12), we obtain</text>
<text><location><page_72><loc_12><loc_49><loc_88><loc_51></location>The statement now follows by applying Stokes' theorem to the final term and rearranging. /square</text>
<text><location><page_72><loc_12><loc_46><loc_83><loc_47></location>Lemma 9.15 (Decay estimate for the integrated time derivative) . For any T > 0 , we have</text>
<text><location><page_72><loc_12><loc_40><loc_48><loc_41></location>Consequently, the bootstrap assumptions imply</text>
<formula><location><page_72><loc_12><loc_41><loc_65><loc_46></location>(9.16) ∫ Σ T φ ' vol g = (∫ Σ T =0 φ ' vol g ) · e -2 T .</formula>
<text><location><page_72><loc_12><loc_37><loc_16><loc_38></location>(9.17)</text>
<text><location><page_72><loc_12><loc_34><loc_30><loc_35></location>for δ > 0 small enough.</text>
<formula><location><page_72><loc_39><loc_33><loc_61><loc_40></location>∣ ∣ ∣ ∣ ∫ Σ T ˜ ∂ 0 φ · φ ' vol g ∣ ∣ ∣ ∣ /lessorsimilar δ 3 e -5 2 T</formula>
<text><location><page_72><loc_12><loc_30><loc_64><loc_33></location>Proof. Using that the integral of div g ( n ∇ φ ) vanishes, we compute:</text>
<text><location><page_72><loc_12><loc_24><loc_88><loc_27></location>Hence, (9.16) precisely describes the solution to this ODE ( f ' = -2 f ) with prescribed initial value at T = 0, and the initial data assumption (9.8) implies</text>
<formula><location><page_72><loc_12><loc_25><loc_88><loc_32></location>∂ T (∫ M φ ' vol g ) = ∫ M ( ˜ ∂ 0 φ ' +3ˆ n φ ' ) vol g = ∫ M [ (1 -n ) φ ' +( n -3) φ ' ] vol g = -2 (∫ M φ ' vol g )</formula>
<text><location><page_72><loc_12><loc_18><loc_40><loc_19></location>Furthermore, one has by (9.15) that</text>
<formula><location><page_72><loc_29><loc_16><loc_71><loc_24></location>∣ ∣ ∣ ∣ ∫ M φ ' vol g ∣ ∣ ∣ ∣ ≤ ‖ φ ' ‖ C 0 (Σ T =0 ) vol g (Σ T =0 ) e -2 T ≤ δ 2 e -2 T .</formula>
<formula><location><page_72><loc_40><loc_15><loc_60><loc_16></location>∂ T vol g (Σ T ) = 3ˆ n vol g</formula>
<formula><location><page_72><loc_12><loc_13><loc_54><loc_18></location>(9.18) ∫ Σ T</formula>
<formula><location><page_72><loc_14><loc_5><loc_86><loc_12></location>˜ ∂ 0 φ = [ -∂ T vol g (Σ T ) vol g (Σ T ) · φ + 1 vol g (Σ T ) ∫ M ( ˜ ∂ 0 φ +3ˆ n φ ) vol g ] = ∫ M ( n φ ' +3ˆ n ( φ -φ ) ) vol g .</formula>
<text><location><page_72><loc_12><loc_12><loc_29><loc_13></location>Consequently, one has</text>
<text><location><page_73><loc_12><loc_87><loc_88><loc_90></location>By applying | n | < 3, the adapted Poincare inequality (9.13) and the bootstrap assumptions (9.9), this implies</text>
<text><location><page_73><loc_12><loc_83><loc_50><loc_84></location>(9.17) now follows by combining this with (9.16).</text>
<text><location><page_73><loc_87><loc_83><loc_88><loc_84></location>/square</text>
<formula><location><page_73><loc_34><loc_81><loc_66><loc_87></location>| ˜ ∂ 0 φ | /lessorsimilar ‖ φ ' ‖ L 2 + ‖∇ φ ‖ L 2 ‖ ˆ n ‖ L 2 g /lessorsimilar δe -T 2 .</formula>
<text><location><page_73><loc_12><loc_78><loc_88><loc_81></location>9.3.3. Energy estimates. Now, we can collect the following estimates for the corrected scalar field energies:</text>
<text><location><page_73><loc_12><loc_74><loc_88><loc_77></location>Lemma 9.16 (Base level estimate for the corrected scalar field energy) . Under the bootstrap assumptions, the following estimate holds for some K > 0 :</text>
<formula><location><page_73><loc_12><loc_68><loc_78><loc_74></location>(9.19) ∂ T E (0) SF ≤ -2 + E (0) SF + Kδe -T √ E (0) SF √ [ ‖ Σ ‖ 2 L 2 + ‖ g -γ ‖ 2 L 2 ] + Kδ 3 e -5 2 T</formula>
<formula><location><page_73><loc_12><loc_53><loc_85><loc_67></location>Proof. We compute, using [ ˜ ∂ 0 , ∇ ] φ = 0, ˜ ∂ 0 φ = n φ ' and the rescaled wave equation (9.5g): ∂ T E (0) SF = ∫ M [ 2 ˜ ∂ 0 φ ' · φ ' +2 〈 ∇ φ, ∇ ˜ ∂ 0 φ 〉 g + ( ˜ ∂ 0 g -1 ) ab ∇ a φ ∇ b φ +3ˆ n ( | φ ' | 2 + |∇ φ | 2 g ) ] vol g = ∫ M [ 2 ( 〈∇ n , ∇ φ 〉 g + n ∆ g φ +(1 -n ) φ ' ) φ ' -2( n φ ' ) · ∆ g φ -2 n 〈 Σ , ∇ φ ∇ φ 〉 g +3ˆ n | φ ' | 2 + ˆ n |∇ φ | 2 g ] vol g</formula>
<formula><location><page_73><loc_19><loc_41><loc_81><loc_50></location>∂ T E (0) SF ≤ ∫ M -4 | φ ' | 2 g vol g + K [ ‖∇ φ ‖ C 0 ‖ ˆ n ‖ H 1 √ E (0) SF +( ‖ Σ ‖ C 0 + ‖ ˆ n ‖ C 0 ) E (0) SF ] ≤ ∫ M -4 | φ ' | 2 vol g + Kδe -T 2 √ E (0) SF √ ‖ ˆ n ‖ H 1</formula>
<text><location><page_73><loc_12><loc_50><loc_88><loc_53></location>With 2(1 -n ) = -4 -6ˆ n , integration by parts and using the bootstrap assumption (9.9) on C -norms, we get for some constant K > 0 that we update from line to line:</text>
<text><location><page_73><loc_12><loc_40><loc_59><loc_42></location>Similarly and using the same evolution equations, we obtain:</text>
<formula><location><page_73><loc_12><loc_27><loc_82><loc_41></location>∂ T C (0) SF = ∫ M [ ˜ ∂ 0 φ · φ ' -˜ ∂ 0 φ · φ ' +( φ -φ ) ˜ ∂ 0 φ ' +3ˆ n ( φ -φ ) φ ' ] vol g = ∫ M [ 3 | φ ' | 2 +3ˆ n | φ ' | 2 + ( φ -φ ) · div g ( n ∇ φ ) -2 ( φ -φ ) φ ' -˜ ∂ 0 φ · φ ' ] vol g ≤ -2 C (0) SF + ∫ M 3 [ | φ ' | 2 -3 |∇ φ | 2 g ] vol g +3 ‖ ˆ n ‖ C 0 E (0) SF -∫ M ( ˜ ∂ 0 φ · φ ' ) vol g Applying Lemma 9.15 to the last term, we get:</formula>
<formula><location><page_73><loc_33><loc_24><loc_67><loc_26></location>∂ T C (0) SF ≤-2 C (0) SF + Kδe -T 2 E (0) SF + Kδ 3 e -5 2 T</formula>
<text><location><page_73><loc_12><loc_22><loc_82><loc_24></location>Combining these two estimates, inserting (9.11a) and (9.12) as well as updating K yields:</text>
<formula><location><page_73><loc_20><loc_7><loc_80><loc_22></location>∂ T E (0) SF = ∂ T E (0) SF + 2 3 ∂ T C (0) SF = ∫ M [( -4 + 2 3 · 3 ) | φ ' | 2 -2 3 · 3 |∇ φ | 2 g ] vol g -2 · 2 3 C 0 SF + Kδe -T 2 √ E (0) SF √ ‖ ˆ n ‖ H 1 + Kδ 3 e -5 2 T ≤ -2 E (0) SF + Kδe -T √ E (0) SF ( ‖ Σ ‖ L 2 + ‖ g -γ ‖ L 2 + √ E (0) SF ) + Kδ 3 e -5 2 T</formula>
<text><location><page_74><loc_87><loc_88><loc_88><loc_90></location>/square</text>
<text><location><page_74><loc_12><loc_84><loc_88><loc_87></location>Lemma 9.17 (Higher order estimates for the corrected scalar field energy) . For any l ∈ { 1 , . . . , 4 } , the following estimate holds:</text>
<formula><location><page_74><loc_23><loc_78><loc_76><loc_84></location>∂ T ( E ( l ) SF + 2 3 C ( l ) SF ) ≤ -2 ( E ( l ) SF + 2 3 C ( l ) SF ) + Kδe -T 2 ( l ∑ m =0 √ E ( m ) SF ) ·</formula>
<text><location><page_74><loc_12><loc_73><loc_54><loc_76></location>Proof. Starting with l = 2 k, k ∈ { 1 , 2 } , one calculates:</text>
<formula><location><page_74><loc_44><loc_74><loc_77><loc_79></location>· ( ‖ φ ' ‖ H l + ‖∇ φ ‖ H l + ‖ Σ ‖ H l + ‖ g -γ ‖ H l )</formula>
<formula><location><page_74><loc_12><loc_68><loc_66><loc_74></location>∂ T E (2 k ) SF = ∫ M [ 2∆ k g ˜ ∂ 0 φ ' · ∆ k g φ ' +2 〈∇ ∆ k g φ, ∇ ∆ k g ˜ ∂ 0 φ 〉 g (9.20a)</formula>
<formula><location><page_74><loc_12><loc_60><loc_76><loc_67></location>+2[ ˜ ∂ 0 , ∆ k g ] φ ' · ∆ k g φ ' +2 〈 [ ˜ ∂ 0 , ∇ ∆ k g ] φ, ∇ ∆ k g φ 〉 g ] vol g (9.20c)</formula>
<formula><location><page_74><loc_12><loc_64><loc_81><loc_70></location>+( ˜ ∂ 0 g -1 ) ab · ∇ a ∆ k g φ · ∇ b ∆ k g φ +3ˆ n ( | ∆ k φ ' | 2 g + |∇ ∆ k φ | 2 g ) (9.20b)</formula>
<text><location><page_74><loc_12><loc_57><loc_88><loc_62></location>We insert the rescaled wave equation (9.5g) and ˜ ∂ 0 φ = n φ ' into the right hand side of (9.20a) and obtain for some constant K > 0:</text>
<text><location><page_74><loc_12><loc_35><loc_88><loc_38></location>For (9.20b), we use (9.5e) and the bootstrap assumption (9.9) to bound it by Kδe -T 2 E (2 k ) SF . Regarding (9.20c), the commutator formulas (12.1a)-(12.1b) imply the following:</text>
<formula><location><page_74><loc_16><loc_37><loc_85><loc_59></location>(9.20a) ≤ ∫ M [ -4 | ∆ k g φ ' | 2 -6ˆ n | ∆ k g φ ' | 2 + n ∆ k +1 g φ · ∆ k g φ ' ] vol g + K ‖ ∆ k φ ' ‖ L 2 ( ‖ ˆ n ‖ H 2 k +1 ‖∇ φ ‖ C 0 + ‖∇ φ ‖ H 2 k ‖ ˆ n ‖ C 2 k ) + ∫ M [ -n ∆ k g φ ' · ∆ k +1 g ∇ φ -3 〈∇ ˆ n , ∇ ∆ k g φ 〉 g · ∆ k g φ ' ] vol g + K ‖∇ ∆ k g φ ‖ L 2 ( ‖∇ φ ‖ H 2 k +1 ‖ φ ' ‖ C 0 + ‖ ˆ n ‖ C 2 k ‖ φ ' ‖ H 2 k ) ≤ ∫ M -4 | ∆ k g φ ' | 2 vol g + K √ E (2 k ) SF · [ ( ‖∇ φ ‖ C 0 + ‖ φ ' ‖ C 0 ) · ‖ ˆ n ‖ H 2 k +1 + ( ‖∇ φ ‖ H 2 k + ‖ φ ' ‖ H 2 k ) · ‖ ˆ n ‖ C 2 k ]</formula>
<formula><location><page_74><loc_25><loc_27><loc_76><loc_35></location>‖ [ ˜ ∂ 0 , ∆ k g ] φ ' ‖ L 2 /lessorsimilar ‖ n ‖ C 2 k -1 ( ‖ φ ' ‖ C 1 ‖ Σ ‖ ˙ H 2 k -1 + ‖ Σ ‖ C 2 k -2 ‖ φ ' ‖ H 2 k ) ‖ [ ˜ ∂ 0 , ∇ ∆ k g ] φ ‖ L 2 /lessorsimilar ‖ n ‖ C 2 k ( ‖∇ φ ‖ C 1 ‖ Σ ‖ H 2 k + ‖ Σ ‖ C 2 k -2 ‖∇ φ ‖ H 2 k )</formula>
<formula><location><page_74><loc_31><loc_12><loc_69><loc_24></location>∂ T E (2 k ) SF ≤ ∫ M -4 | ∆ k g φ ' | 2 vol g + Kδe -T 2 E (2 k ) SF + Kδe -T 2 √ E (2 k ) SF ( ‖ φ ' ‖ H 2 k + ‖∇ φ ‖ H 2 k ) + Kδe -T 2 √ E (2 k ) SF ( ‖ ˆ n ‖ H 2 k +1 + ‖ Σ ‖ H 2 k )</formula>
<text><location><page_74><loc_12><loc_24><loc_88><loc_27></location>Summarizing, inserting the C -norm bounds from the bootstrap assumption (9.9) and updating K , this implies</text>
<text><location><page_74><loc_12><loc_12><loc_49><loc_13></location>Moving on to the corrective term, we compute:</text>
<formula><location><page_74><loc_12><loc_5><loc_58><loc_12></location>∂ T C (2 k ) SF = ∫ M [ ∆ k g ˜ ∂ 0 φ · ∆ k g φ ' +∆ k g φ · ∆ k g ˜ ∂ 0 φ ' (9.21a)</formula>
<formula><location><page_75><loc_12><loc_84><loc_81><loc_91></location>+3ˆ n · ∆ k g φ · ∆ k g φ ' +[ ˜ ∂ 0 , ∆ k g ] φ · ∆ g φ ' +∆ k g φ · [ ˜ ∂ 0 , ∆ k g ] φ ' ] vol g (9.21b)</formula>
<text><location><page_75><loc_12><loc_84><loc_86><loc_85></location>Inserting the evolution equations into the right hand side of (9.21a), we can bound that line by</text>
<formula><location><page_75><loc_18><loc_71><loc_82><loc_84></location>≤ ∫ M [ 3 | ∆ k g φ ' | 2 +3ˆ n | ∆ k g φ ' | 2 ] vol g + K ‖ ˆ n ‖ C 2 k ‖ φ ' ‖ H 2 k -1 ‖ ∆ k φ ' ‖ L 2 + ∫ M [ -2∆ k g φ · ∆ k g φ ' +3ˆ n ∆ k g φ · ∆ k g φ ' +3∆ k g φ · ∆ k +1 g φ +3ˆ n ∆ k g φ · ∆ k +1 g φ ] vol g + K [ ‖ ˆ n ‖ C 2 k ( ‖∇ φ ‖ H 2 k + ‖ φ ' ‖ H 2 k -1 ) + ( ‖∇ φ ‖ C 0 + ‖ φ ' ‖ C 0 ) ‖ ˆ n ‖ H 2 k +1 ] ‖ ∆ k g φ ‖ L 2</formula>
<text><location><page_75><loc_12><loc_71><loc_85><loc_72></location>Note that, after integrating by parts, the last two terms in the second line can be bounded by</text>
<formula><location><page_75><loc_25><loc_65><loc_75><loc_70></location>∫ M -3 |∇ ∆ g φ | 2 g vol g + ‖ ˆ n ‖ C 1 ( ‖∇ ∆ k φ ‖ L 2 + ‖ ∆ k φ ‖ L 2 ) ‖∇ ∆ k φ ‖ L 2 .</formula>
<text><location><page_75><loc_12><loc_61><loc_88><loc_66></location>For the terms in (9.21b), notice that the first term can be bounded by δe -T 2 ‖∇ φ ‖ H 2 k -1 √ E (2 k ) SF , while the commutator terms can be estimated as before, with</text>
<text><location><page_75><loc_12><loc_55><loc_39><loc_57></location>Combining all of the above, we get</text>
<formula><location><page_75><loc_25><loc_43><loc_75><loc_55></location>∂ T C (2 k ) SF ≤ -2 C (2 k ) SF + ∫ M [ 3 | ∆ k g φ ' | -3 |∇ ∆ k g φ | g ] vol g + Kδe -T 2 [ ‖ φ ' ‖ H 2 k + ‖∇ φ ‖ H 2 k + ‖ ˆ n ‖ H 2 k +1 + ‖ Σ ‖ H 2 k ] · (√ E (2 k ) SF + √ E (2 k -1) SF )</formula>
<formula><location><page_75><loc_23><loc_55><loc_77><loc_60></location>‖ [ ˜ ∂ 0 , ∆ k g ] φ ‖ L 2 /lessorsimilar ‖∇ φ ‖ C 0 ‖ n ‖ C 0 ‖ Σ ‖ ˙ H 2 k -1 + ‖ n ‖ C 2 k ‖ Σ ‖ C 2 k -2 ‖∇ φ ‖ H 2 k -1</formula>
<text><location><page_75><loc_12><loc_39><loc_88><loc_43></location>Finally, combining both differential estimates yields the statement for l = 2 k . For l = 2 k -1 , k ∈ { 1 , 2 } , the argument is completely analogous and hence omitted. /square</text>
<unordered_list>
<list_item><location><page_75><loc_12><loc_35><loc_88><loc_38></location>9.4. Geometric variables. We can take the following results from prior literature, where we additionally apply the elliptic estimates in Lemma 9.11:</list_item>
</unordered_list>
<text><location><page_75><loc_12><loc_30><loc_88><loc_33></location>Lemma 9.18 (Coercivity of geometric energies, [AM11, Lemma 7.4]) . For sufficiently small δ > 0 , the following estimate holds:</text>
<formula><location><page_75><loc_12><loc_26><loc_61><loc_29></location>(9.22) ‖ g -γ ‖ 2 H 5 + ‖ Σ ‖ 2 H 4 /lessorsimilar E geom</formula>
<text><location><page_75><loc_12><loc_23><loc_88><loc_26></location>Lemma 9.19 (Geometric energy estimate, [AF20, Lemma 20]) . Let δ > 0 be chosen appropriately small, and let</text>
<formula><location><page_75><loc_12><loc_17><loc_60><loc_23></location>(9.23) α = { 1 λ 0 > 1 9 1 -3 √ δ ' λ 0 = 1 9 ,</formula>
<text><location><page_75><loc_12><loc_13><loc_88><loc_16></location>where δ ' > 0 is the same as in (9.7) , in particular suitably small. Then, there exists some constant K > 0 such that the following estimate holds:</text>
<formula><location><page_75><loc_12><loc_7><loc_78><loc_12></location>∂ T E geom ≤ -2 αE geom + KE 3 2 geom + Kδe -T 2 √ E geom [ ‖ φ ' ‖ H 4 + ‖∇ φ ‖ H 4 ] (9.24)</formula>
<text><location><page_76><loc_12><loc_87><loc_88><loc_90></location>9.5. Closing the bootstrap. Now, we can collect our estimates to improve the bootstrap assumptions:</text>
<text><location><page_76><loc_12><loc_79><loc_88><loc_85></location>Proposition 9.20 (Improved bounds for future stability) . Let the bootstrap assumption (see Assumption 9.9) be satisfied for T ∈ [0 , T Boot ) and assume the initial data holds at T = 0 (see Assumption 9.7). For δ > 0 sufficiently small and α as in (9.23) with δ ' > 0 sufficiently small, the following estimates hold:</text>
<formula><location><page_76><loc_12><loc_76><loc_70><loc_78></location>‖ φ ' ‖ C 2 + ‖∇ φ ‖ C 2 + ‖ φ ' ‖ H 4 + ‖∇ φ ‖ H 4 /lessorsimilar δ 3 2 e -αT (9.25a)</formula>
<formula><location><page_76><loc_12><loc_71><loc_70><loc_73></location>‖ ˆ n ‖ C 4 + ‖ X ‖ C 4 + ‖ ˆ n ‖ H 6 + ‖ X ‖ H 6 /lessorsimilar δ 3 e -2 αT (9.25c)</formula>
<formula><location><page_76><loc_12><loc_73><loc_70><loc_76></location>‖ g -γ ‖ C 3 + ‖ Σ ‖ C 2 + ‖ g -γ ‖ H 5 + ‖ Σ ‖ H 4 /lessorsimilar δ 3 2 e -αT (9.25b)</formula>
<text><location><page_76><loc_12><loc_64><loc_88><loc_71></location>Proof. In the following, the positive constant K that may be updated from line to line. Combining the estimate from Lemma 9.16 as well as those from Lemma 9.17 at each level with Lemma 9.19 and applying the (near)-coercivity estimates (9.14) and (9.22) to the right hand sides, we obtain:</text>
<formula><location><page_76><loc_30><loc_49><loc_87><loc_57></location>-2 αE geom + KE 3 2 geom + Kδe -T 2 √ E geom √ E SF + δ 2 e -T ∥ ∥ ∥ Ric[ g ] + 2 9 g ∥ ∥ ∥ 2 H 2</formula>
<formula><location><page_76><loc_13><loc_56><loc_87><loc_65></location>∂ T ( E (4) SF + E geom ) ≤ -2 E SF + Kδe -T 2 √ E SF √ E SF + δ 2 e -T ∥ ∥ ∥ ∥ Ric[ g ] + 2 9 g ∥ ∥ ∥ ∥ 2 H 2 + √ E geom + Kδ 3 e -5 2 T</formula>
<text><location><page_76><loc_12><loc_45><loc_88><loc_53></location>∥ ∥ Applying (9.10a) to the curvature norms, as well as (9.22) to the resulting norms on g -γ and (9.9) (which implies √ E geom /lessorsimilar δe -T 2 ), this becomes</text>
<text><location><page_76><loc_12><loc_41><loc_36><loc_43></location>and consequently, since α ≤ 1,</text>
<formula><location><page_76><loc_19><loc_42><loc_81><loc_47></location>∂ T ( E (4) SF + E geom ) ≤ -2 α ( E (4) SF + E geom ) + Kδe -T 2 ( E geom + E 4 SF ) + Kδ 3 e -5 2 T .</formula>
<formula><location><page_76><loc_25><loc_37><loc_75><loc_42></location>∂ T [ e 2 αT ( E (4) SF + E geom )] /lessorsimilar δe -T 2 · e 2 αT ( E (4) SF + E geom ) + δ 3 e -T 2</formula>
<text><location><page_76><loc_12><loc_36><loc_74><loc_37></location>The Gronwall lemma, along with the initial data assumption (9.8), now implies</text>
<formula><location><page_76><loc_12><loc_33><loc_60><loc_35></location>(9.26) E (4) SF + E geom /lessorsimilar δ 3 e -2 αT .</formula>
<text><location><page_76><loc_12><loc_31><loc_87><loc_32></location>Lemma 9.18 and the standard Sobolev embedding then imply (9.25b). In particular, this means</text>
<formula><location><page_76><loc_12><loc_24><loc_60><loc_31></location>(9.27) ∥ ∥ ∥ Ric[ g ] + 2 9 g ∥ ∥ ∥ H 2 /lessorsimilar δ 3 2 e -αT</formula>
<text><location><page_76><loc_12><loc_19><loc_88><loc_29></location>∥ ∥ due to Lemma 9.10, and for δ ' > 0 small enough, inserting (9.26) and (9.27) into (9.14) shows (9.25a). This in particular improves the C -norms we just used the bootstrap assumption for, and reinserting this improvement into (9.14) improves the bound to (9.26). Moreover, (9.25c) follows directly from Lemma 9.11 and the already obtained improvements. /square</text>
<text><location><page_76><loc_12><loc_7><loc_88><loc_17></location>Proof of Theorem 9.1. The problem is locally well-posed as outlined in Remark 9.8. There then is some maximal interval [0 , T Boot ) for the logarithmic time T - or, equivalently, some maximal time interval [ τ 0 , τ Boot ) - on which the solution exists and the bootstrap assumptions (see Assumption 9.9) are satisfied. By the analogous argument to the proof of Theorem 8.2, the decay estimates in Proposition 9.20 are strictly stronger than the bootstrap assumptions for small enough δ, δ ' > 0. This implies T Boot = ∞ (resp. τ Boot = 0) since we could else extend the solution strictly beyond</text>
<text><location><page_77><loc_12><loc_87><loc_88><loc_90></location>T Boot while also satisfying the bootstrap assumptions. This proves the convergence statement in Theorem 9.1.</text>
<text><location><page_77><loc_12><loc_80><loc_88><loc_87></location>Finally, the decay estimates imply that |∇ n | g , respectively | k | g , are bounded by τ α -1 , respectively τ α +1 , up to constant on [ τ 0 , τ ). Since α is at worst slightly smaller than 1, both functions are integrable on [ τ 0 , 0) for suitably small δ ' > 0 . By [CBC02], this means the spacetime is future complete. /square</text>
<section_header_level_1><location><page_77><loc_41><loc_75><loc_59><loc_77></location>10. Global stability</section_header_level_1>
<text><location><page_77><loc_12><loc_66><loc_88><loc_74></location>To prove Theorem 1.2, what still needs to be shown is that initial data as in Theorem 1.1 develops from Σ t 0 to some hypersurface Σ t 1 ≡ Σ τ ( t 1 ) in its future such that the data in Σ t 1 is near-Milne in the sense of Assumption 9.7 and in CMCSH gauge. From there, near-Milne stability yields the behaviour in the future of Σ τ ( t 1 ) , and hence future stability of near-FLRW spacetimes as in Theorem 1.2.</text>
<text><location><page_77><loc_12><loc_60><loc_88><loc_64></location>Proof of Theorem 1.2. Within this proof, t will denote the 'physical' time coordinate used throughout the Big Bang stability analysis, while τ denotes the mean curvature time used within CMCSH gauge.</text>
<text><location><page_77><loc_12><loc_54><loc_88><loc_59></location>Consider initial data ( g, k, ∇ φ, ∂ 0 φ ) induced on the CMC hypersurface Σ t 0 within M such that the rescaled variables are close to FLRW reference data in the sense of Theorem 8.2. Moreover, let (˚ g, ˚ k, ˚ π, ˚ ψ ) be the geometric initial data on M that induce it via the embedding ι : M ↪ → M .</text>
<text><location><page_77><loc_15><loc_53><loc_24><loc_54></location>Notice that</text>
<formula><location><page_77><loc_20><loc_49><loc_80><loc_52></location>P : H 20 γ ( M ) → H 18 γ ( M ) , Y i ↦→ ∆ γ Y i +( γ -1 ) il Ric[ γ ] lj Y j = ∆ γ Y i -2 9 Y i = 0</formula>
<text><location><page_77><loc_12><loc_44><loc_88><loc_48></location>is an isomorphism since ∆ γ has no positive eigenvalues. Hence, using [FK20, Theorem 2.5, Remark 2.6], there is an metric ˚ g ' isometric to ˚ g that remains H 18 γ ( M )-close to γ and satisfies</text>
<formula><location><page_77><loc_38><loc_39><loc_62><loc_44></location>((˚ g ' ) -1 ) ij ( Γ[˚ g ' ] k ij -ˆ Γ[ γ ] k ij ) = 0 .</formula>
<text><location><page_77><loc_12><loc_29><loc_88><loc_39></location>Let θ ∈ Diff( M ) be the diffeomorphism such that θ ∗ ˚ g =˚ g ' , then the proof of [FK20, Theorem 2.5] implies that θ can be chosen close to the identity map within H 18 (Diff( M )), and consequently that θ ∗ ˚ k = ˚ k ' , θ ∗ ˚ π =˚ π ' and θ ∗ ˚ ψ = ˚ ψ ' remain close to 1 3 γ, 0 and C in H 18 γ ( M ). By the same argument as in Remark 8.1, we can now evolve this data locally and obtain a new initial hypersurface Σ ' close to Σ t 0 that is in CMCSH gauge and that ( g, k, ∇ φ, ∂ 0 φ ) is close to the reference data in the sense of Assumption 3.10, exchanging the initial time t 0 by some close time t ' 0 .</text>
<text><location><page_77><loc_12><loc_21><loc_88><loc_27></location>Since τ is strictly increasing, t ≡ t ( τ ) exists and we can interchangeably view a as a function in t or τ with some abuse of notation. The Friedman equation (2.3) implies ∂ t a ≥ 1 9 and thus a ( t ) ≥ 1 9 t on (0 , ∞ ), as well as</text>
<formula><location><page_77><loc_26><loc_18><loc_74><loc_21></location>-τ = 3 ˙ a a = 1 a + 〈 lower order terms 〉 as t →∞ (resp. τ → 0) .</formula>
<text><location><page_77><loc_12><loc_13><loc_88><loc_17></location>We choose t 1 > max { 1 , t ' 0 } large enough (resp. τ ( t 1 ) ≡ τ 0 small enough) that the following estimates hold for some small χ ∈ (0 , 1 2 ) that depends only on δ :</text>
<formula><location><page_77><loc_12><loc_9><loc_54><loc_12></location>Ca ( t 1 ) -3 τ ( t 1 ) -1 ≤ χ (10.1)</formula>
<formula><location><page_77><loc_12><loc_7><loc_63><loc_10></location>-τ ( t 1 ) · a ( t 1 ) ∈ [1 -χ, 1 + χ ] (10.2)</formula>
<text><location><page_78><loc_12><loc_80><loc_88><loc_90></location>As the solution is Cauchy stable, i.e., it and its maximal time of existence depend continuously upon the initial data, 16 one can choose ε > 0 in the analogue of Assumption 3.10 small enough to ensure the following: The solution exists until t 1 > t ' 0 and ( a -2 g, a ˆ k, ∇ φ, a 3 ˜ ∂ 0 φ ) remain Kε -close to ( γ, 0 , 0 , C ) in H 6 γ × H 5 γ × H 5 γ × H 5 γ for some suitable K > 0 along the slab ∪ s ∈ [ t ' 0 ,t 1 ] Σ s . What now remains to be shown is that this implies Assumption 9.7 in the sense that, if ε is small enough, δ can be made as small as necessary for Theorem 9.1 to apply.</text>
<text><location><page_78><loc_15><loc_77><loc_68><loc_78></location>Note that the scalings in Definition 9.4 can be rewritten as follows:</text>
<formula><location><page_78><loc_26><loc_69><loc_74><loc_76></location>g -γ = ( τ · a ) 2 · ( a -2 g -γ ) + ( τ 2 · a 2 -1) γ, Σ = τ a ( a ˆ k ) φ ' = C ( -τ -1 · a -3 ) + ( -τ -1 · a -3 ) · ( a 3 n -1 ( ∂ τ -L X ) φ -C )</formula>
<text><location><page_78><loc_12><loc_54><loc_88><loc_65></location>For the normal derivative of the wave, notice that | C ( -τ -1 · a -3 ) | is bounded by χ due to (10.1), and that -τ -1 a -3 is equivalent to a -2 by (10.2). Hence, we can similarly ensure that φ ' is bounded in H 5 by δ 3 . Since ∇ φ is not changed in either rescaling, and bounds on lapse and shift (up to constant) follow from the elliptic estimates in Lemma 9.11, it follows each individual norm in Assumption 9.7 can be bounded by δ 3 up to constants that depend only on γ , and hence the initial data assumption itself can be satisfied for suitably small δ > 0.</text>
<text><location><page_78><loc_12><loc_64><loc_88><loc_70></location>Since (10.2) implies τ · a is close to -1 at t 1 , ‖ ( τ · a ) 2 ( a -2 g -γ ) ‖ H 6 can be bounded by δ 3 2 for small enough ε . Choosing χ < δ 3 2 then implies ‖ g -γ ‖ H 6 (Σ τ 0 ) < δ 3 . That ‖ Σ ‖ H 5 can be made smaller than δ 3 for small enough ε > 0 follows since τ a behaves like 1 a 2 up to a constant by (10.2).</text>
<text><location><page_78><loc_12><loc_49><loc_88><loc_54></location>This proves that we can develop from initial data for the Big Bang stability proof to near-Milne initial data within a CMCSH foliation, and thus we obtain Theorem 1.2 from Theorem 1.1 and Theorem 9.1. /square</text>
<section_header_level_1><location><page_78><loc_35><loc_45><loc_66><loc_46></location>11. Appendix - Big Bang Stability</section_header_level_1>
<section_header_level_1><location><page_78><loc_12><loc_43><loc_43><loc_44></location>11.1. Basic formulas and estimates.</section_header_level_1>
<text><location><page_78><loc_12><loc_40><loc_42><loc_41></location>11.1.1. Tools from elementary calculus.</text>
<text><location><page_78><loc_12><loc_35><loc_88><loc_39></location>Lemma 11.1 (A Gronwall lemma) . Let f, χ, ξ : [ a, b ] -→ R be continuous functions such that χ ≥ 0 , ξ is decreasing and, for any s ∈ [ a, b ] ,</text>
<formula><location><page_78><loc_39><loc_30><loc_61><loc_35></location>f ( s ) ≤ ∫ b s χ ( r ) f ( r ) dr + ξ ( s )</formula>
<text><location><page_78><loc_12><loc_28><loc_46><loc_30></location>is satisfied. Then, for any t ∈ [ a, b ] , we have</text>
<text><location><page_78><loc_12><loc_22><loc_17><loc_23></location>Proof.</text>
<text><location><page_78><loc_17><loc_22><loc_88><loc_23></location>This follows by standard arguments as in [Dra03, Corollary 2-3]. /square</text>
<text><location><page_78><loc_12><loc_17><loc_88><loc_21></location>Lemma 11.2 (A weak fundamental theorem of calculus for square roots) . Let f : (0 , t 0 ] -→ R + 0 be a C 1 -function. Then, we have for any t ∈ (0 , t 0 ] :</text>
<formula><location><page_78><loc_12><loc_11><loc_64><loc_18></location>(11.1) √ f ( t ) ≤ √ f ( t 0 ) + ∫ t 0 t | f ' ( s ) | 2 √ f ( s ) ds</formula>
<formula><location><page_78><loc_38><loc_24><loc_62><loc_29></location>f ( t ) ≤ ξ ( t ) exp (∫ b t χ ( r ) dr ) .</formula>
<text><location><page_79><loc_12><loc_84><loc_88><loc_90></location>Proof. This follows from a straightforward application of the monotone convergence theorem to g n = √ f + 1 n . /square</text>
<text><location><page_79><loc_12><loc_81><loc_88><loc_84></location>11.1.2. Levi-Civita tensor identites. Herein, we collect some basic identities for the Levi-Civita tensor ε [ g ]: Firstly, it satisfies the contraction identities, where I a b denotes the Kronecker-symbol:</text>
<formula><location><page_79><loc_12><loc_77><loc_60><loc_80></location>ε ai 1 i 2 ε aj 1 j 2 = I i 1 j 1 I i 2 j 2 -I i 1 j 2 I i 2 j 1 (11.2a)</formula>
<formula><location><page_79><loc_12><loc_76><loc_52><loc_77></location>ε abi ε abj 2 =2 I i j (11.2b)</formula>
<formula><location><page_79><loc_12><loc_73><loc_51><loc_75></location>ε abc ε abc =6 (11.2c)</formula>
<formula><location><page_79><loc_12><loc_70><loc_51><loc_73></location>∇ ε =0 (11.2d)</formula>
<text><location><page_79><loc_12><loc_65><loc_88><loc_70></location>The analogous formulas hold for ε [ G ] when raising indices with regards to G instead of g . For a tracefree and symmetric Σ t -tangent (0 , 2)-tensor T and a Σ t -tangent (0 , 2)-tensor A , the following simplified identites hold:</text>
<formula><location><page_79><loc_12><loc_61><loc_66><loc_64></location>( T × A ) ij = ε i ab ε j pq T ap A bq + 1 3 ( T · A ) g ij (11.3a)</formula>
<formula><location><page_79><loc_12><loc_59><loc_48><loc_61></location>( T × g ) ij = -T ij (11.3b)</formula>
<formula><location><page_79><loc_12><loc_56><loc_58><loc_59></location>( T × k ) ij = -τ 3 T ij +( T × ˆ k ) ij (11.3c)</formula>
<text><location><page_79><loc_12><loc_52><loc_88><loc_55></location>Further, note the following formulas (for ˜ T as T , ˜ A as A and any Σ t -tangent (0 , 1)-tensor ξ ) (see [AM04, p.30]):</text>
<formula><location><page_79><loc_12><loc_48><loc_64><loc_51></location>div g ( A ∧ ˜ A ) = -curl A · ˜ A + A · curl ˜ A (11.3d)</formula>
<formula><location><page_79><loc_12><loc_44><loc_56><loc_46></location>T · ( A × ˜ T ) = ( T × A ) · ˜ T (11.3f)</formula>
<formula><location><page_79><loc_12><loc_46><loc_58><loc_49></location>A · ( ξ ∧ ˜ A ) = -2 ξ · ( A ∧ ˜ A ) (11.3e)</formula>
<paragraph><location><page_79><loc_12><loc_41><loc_43><loc_43></location>11.1.3. Estimates on contracted tensors.</paragraph>
<text><location><page_79><loc_12><loc_35><loc_88><loc_40></location>Lemma 11.3. Let S , T be traceless and symmetric Σ t -tangent (0 , 2) -tensors, M , N symmetric Σ t -tangent (0 , 2) -tensors and ξ a Σ t -tangent (0 , 1) -tensor. We define G,G -1 and |·| G via (2.27a) ). Then:</text>
<formula><location><page_79><loc_12><loc_32><loc_68><loc_34></location>| M /circledot G N | G ≤ | M | G | N | G , M /circledot g N = a -2 M /circledot G N (11.4a)</formula>
<formula><location><page_79><loc_12><loc_27><loc_70><loc_30></location>| S ∧ G T | G ≤ | S | G | T | G , ( S ∧ T ) l = a -3 ( S ∧ G T ) (11.4c)</formula>
<formula><location><page_79><loc_12><loc_29><loc_71><loc_32></location>| S × G T | G /lessorsimilar | S | G | T | G , ( S × T ) ij = a -3 ( S × G T ) ij (11.4b)</formula>
<formula><location><page_79><loc_12><loc_25><loc_70><loc_28></location>| ξ ∧ G T | G ≤ | ξ | G | T | G , ( ξ ∧ T ) ij = a -1 ( ξ ∧ G T ) ij (11.4d)</formula>
<formula><location><page_79><loc_12><loc_23><loc_68><loc_25></location>| curl G M | G /lessorsimilar |∇ M | G , curl M ij = a -1 curl G M ij (11.4e)</formula>
<text><location><page_79><loc_12><loc_15><loc_88><loc_20></location>Proof. The estimates with respect to the unrescaled metric are direct consequences of the contraction identities (11.2a)-(11.2c) replacing g with G , and the scalings follow simply by tracking the effects of the rescaling in Definition 2.9. In particular, note</text>
<formula><location><page_79><loc_12><loc_9><loc_82><loc_15></location>(11.5) ε [ g ] i cd = g cj g dk ε [ g ] ijk = ( a -2 ( G -1 ) cj ) ( a -2 ( G -1 ) dk ) a 3 ε [ G ] ijk = a -1 ε [ G ] i /sharpcd .</formula>
<text><location><page_80><loc_12><loc_82><loc_88><loc_90></location>11.2. Commutators. Herein, we collect a variety of commutators of spatial derivative operators with each other as well as with time derivatives. While these mostly follow by standard computations, we use the fact that our spatial hypersurfaces are three-dimensional to significantly simplify the spatial commutator formulas, and need to apply the rescaled equations from Proposition 2.10 for the time derivative formulas.</text>
<text><location><page_80><loc_12><loc_73><loc_88><loc_81></location>For higher order commutators, we denote by J terms within the commutator formula that contribute junk terms at any point where this commutator formula is used. Furthermore, in the following, ζ denotes a scalar function on M and T denotes a Σ t -tangent, symmetric (0 , 2)-tensor, always with sufficient regularity for the equations to make sense. Moreover, recall the schematic ∗ -notation as introduced in subsection 2.1.8.</text>
<text><location><page_80><loc_12><loc_69><loc_88><loc_72></location>Corollary 11.4 (Schematic first order spatial commutators) . For ζ and T as above, the following identities hold:</text>
<formula><location><page_80><loc_12><loc_55><loc_73><loc_68></location>[∆ , ∇ ] ζ =Ric[ G ] ∗ ∇ ζ (11.6a) [∆ , ∇ 2 ] ζ =Ric[ G ] ∗ ∇ 2 ζ + ∇ Ric[ G ] ∗ ∇ ζ (11.6b) [∆ , ∇ ] T =Ric[ G ] ∗ ∇ T + ∇ Ric[ G ] ∗ T (11.6c) [∆ , ∇ 2 ] T =Ric[ G ] ∗ ∇ 2 T + ∇ Ric[ G ] ∗ ∇ T + ∇ 2 Ric[ G ] ∗ T (11.6d) [∆ , div G ] T =Ric[ G ] ∗ ∇ T + ∇ Ric[ G ] ∗ T (11.6e) [∆ , curl G ] = ε [ G ] ∗ (Ric[ G ] ∗ ∇ T + ∇ Ric[ G ] ∗ T ) (11.6f)</formula>
<text><location><page_80><loc_12><loc_53><loc_78><loc_55></location>Proof. Since we are working in three spatial dimensions, the following identity holds:</text>
<formula><location><page_80><loc_24><loc_47><loc_76><loc_52></location>Riem[ G ] ijkl = G ik Ric[ G ] jl -G il Ric[ G ] jk + G jl Ric[ G ] ik -G jk Ric[ G ] il -1 2 ( G -1 ) mn Ric[ G ] mn ( G ik G jl -G il G jk )</formula>
<text><location><page_80><loc_12><loc_40><loc_88><loc_46></location>Hence, for any I ∈ N 0 , any ∇ I Riem[ G ]-term reduces to a sum of products and contractions of ∇ I Ric[ G ] with various metric tensors that are all surpressed in schematic notation. With this in mind, the above statements are simply direct consequences of standard commutation cormulas and (11.5). /square</text>
<text><location><page_80><loc_12><loc_35><loc_88><loc_39></location>Lemma 11.5 (Higher order spatial commutators) . For l ∈ N , l ≥ 2 , the following formulas hold (and extend to l = 1 when dropping any term involving ∆ l -2 ):</text>
<formula><location><page_80><loc_12><loc_32><loc_73><loc_35></location>[∆ l , ∇ ] ζ =∆ l -1 Ric[ G ] ∗ ∇ ζ + ∇ ∆ l -2 Ric[ G ] ∗ ∇ 2 ζ + J ([∆ l , ∇ ] ζ ) (11.7a)</formula>
<text><location><page_80><loc_12><loc_31><loc_17><loc_32></location>(11.7b)</text>
<text><location><page_80><loc_24><loc_31><loc_25><loc_32></location>[∆</text>
<text><location><page_80><loc_25><loc_31><loc_26><loc_32></location>l</text>
<text><location><page_80><loc_26><loc_31><loc_26><loc_32></location>,</text>
<text><location><page_80><loc_28><loc_31><loc_29><loc_32></location>2</text>
<text><location><page_80><loc_29><loc_31><loc_30><loc_32></location>]</text>
<text><location><page_80><loc_30><loc_31><loc_30><loc_32></location>ζ</text>
<text><location><page_80><loc_31><loc_31><loc_32><loc_32></location>=</text>
<text><location><page_80><loc_34><loc_31><loc_36><loc_32></location>∆</text>
<text><location><page_80><loc_36><loc_31><loc_36><loc_32></location>l</text>
<text><location><page_80><loc_36><loc_31><loc_37><loc_32></location>-</text>
<text><location><page_80><loc_37><loc_31><loc_38><loc_32></location>1</text>
<text><location><page_80><loc_38><loc_31><loc_41><loc_32></location>Ric[</text>
<text><location><page_80><loc_41><loc_31><loc_42><loc_32></location>G</text>
<text><location><page_80><loc_42><loc_31><loc_43><loc_32></location>]</text>
<text><location><page_80><loc_46><loc_31><loc_47><loc_32></location>ζ</text>
<text><location><page_80><loc_47><loc_31><loc_49><loc_32></location>+</text>
<text><location><page_80><loc_51><loc_31><loc_51><loc_32></location>2</text>
<text><location><page_80><loc_51><loc_31><loc_53><loc_32></location>∆</text>
<text><location><page_80><loc_53><loc_31><loc_53><loc_32></location>l</text>
<text><location><page_80><loc_53><loc_31><loc_54><loc_32></location>-</text>
<text><location><page_80><loc_54><loc_31><loc_55><loc_32></location>2</text>
<text><location><page_80><loc_55><loc_31><loc_58><loc_32></location>Ric[</text>
<text><location><page_80><loc_58><loc_31><loc_60><loc_32></location>G</text>
<text><location><page_80><loc_60><loc_31><loc_60><loc_32></location>]</text>
<text><location><page_80><loc_63><loc_31><loc_64><loc_32></location>2</text>
<text><location><page_80><loc_64><loc_31><loc_65><loc_32></location>ζ</text>
<text><location><page_80><loc_65><loc_31><loc_67><loc_32></location>+</text>
<text><location><page_80><loc_67><loc_31><loc_68><loc_32></location>J</text>
<text><location><page_80><loc_68><loc_31><loc_71><loc_32></location>([∆</text>
<text><location><page_80><loc_71><loc_31><loc_71><loc_32></location>l</text>
<text><location><page_80><loc_71><loc_31><loc_72><loc_32></location>,</text>
<text><location><page_80><loc_74><loc_31><loc_74><loc_32></location>2</text>
<text><location><page_80><loc_74><loc_31><loc_75><loc_32></location>]</text>
<text><location><page_80><loc_75><loc_31><loc_76><loc_32></location>ζ</text>
<text><location><page_80><loc_76><loc_31><loc_76><loc_32></location>)</text>
<formula><location><page_80><loc_12><loc_27><loc_75><loc_30></location>[∆ l , ∇ ] T = ∇ ∆ l -1 Ric[ G ] ∗ T + ∇ 2 ∆ l -2 Ric[ G ] ∗ ∇ T + J ([∆ l , ∇ ] T ) , (11.7c)</formula>
<text><location><page_80><loc_27><loc_30><loc_28><loc_32></location>∇</text>
<text><location><page_80><loc_33><loc_30><loc_34><loc_32></location>∇</text>
<text><location><page_80><loc_43><loc_30><loc_46><loc_32></location>∗ ∇</text>
<text><location><page_80><loc_49><loc_30><loc_51><loc_32></location>∇</text>
<text><location><page_80><loc_61><loc_30><loc_63><loc_32></location>∗ ∇</text>
<text><location><page_80><loc_72><loc_30><loc_74><loc_32></location>∇</text>
<formula><location><page_80><loc_12><loc_25><loc_77><loc_28></location>[∆ l , ∇ 2 ] T = ∇ 2 ∆ l -1 Ric[ G ] ∗ T + ∇ 3 ∆ l -2 Ric[ G ] ∗ ∇ T + J ([∆ l , ∇ 2 ] T ) , (11.7d)</formula>
<formula><location><page_80><loc_12><loc_19><loc_84><loc_24></location>[∆ l , curl G ] T = ε [ G ] ∗ ( ∇ ∆ l -1 Ric[ G ] ∗ T + ∇ 2 ∆ l -2 Ric[ G ] ∗ ∇ T ) + J ([∆ l , curl G ] T ) (11.7f)</formula>
<formula><location><page_80><loc_12><loc_23><loc_77><loc_25></location>[∆ l , div G ] T = ∇ ∆ l -1 Ric[ G ] ∗ T + ∇ 2 ∆ l -2 Ric[ G ] ∗ ∇ T + J ([∆ l , div G ] T ) , (11.7e)</formula>
<text><location><page_80><loc_12><loc_18><loc_24><loc_20></location>with junk terms</text>
<formula><location><page_80><loc_15><loc_12><loc_85><loc_17></location>J ([∆ l , ∇ ] ζ ) = ∑ I 1 + I ζ =2( l -1) , I ζ ≥ 2 ∇ I 1 Ric[ G ] ∗ ∇ I ζ +1 ζ + l -2 ∑ m =0 ∑ I 1 + ··· + I l -m + I ζ =2 m ∇ I 1 Ric[ G ] ∗ · · · ∗ ∇ I l -m Ric[ G ] ∗ ∇ I ζ +1 ζ</formula>
<formula><location><page_80><loc_14><loc_7><loc_88><loc_12></location>J ([∆ l , ∇ 2 ] ζ ) = ∑ I 1 + I ζ =2( l -1)+1 , I 1 ,I ζ ≥ 2 ∇ I 1 Ric[ G ] ∗ ∇ I ζ +1 ζ + l -2 ∑ m =0 ∑ I 1 + ··· + I l -m + I ζ =2 m +1 ∇ I 1 Ric[ G ] ∗ · · · ∗ ∇ I l -m Ric[ G ] ∗ ∇ I ζ +1 ζ</formula>
<formula><location><page_81><loc_15><loc_85><loc_86><loc_90></location>J ([∆ l , ∇ ] T ) = ∑ I 1 + I T =2( l -1)+1 , I T ≥ 2 ∇ I 1 Ric[ G ] ∗ ∇ I T T + l -2 ∑ m =0 ∑ I 1 + ··· + I l -m + I T =2 m +1 ∇ I 1 Ric[ G ] ∗ · · · ∗ ∇ I l -m Ric[ G ] ∗ ∇ I T T</formula>
<formula><location><page_81><loc_14><loc_80><loc_82><loc_85></location>J ([∆ l , ∇ 2 ] T ) = ∑ I 1 + I T =2 l, I T ≥ 2 ∇ I 1 Ric[ G ] ∗ ∇ I T T + l -2 ∑ m =0 ∑ I 1 + ··· + I l -m + I T =2 m +2 ∇ I 1 Ric[ G ] ∗ · · · ∗ ∇ I l -m Ric[ G ] ∗ ∇ I T T</formula>
<formula><location><page_81><loc_13><loc_75><loc_86><loc_80></location>J ([∆ l , div G ] T ) = ∑ I 1 + I T =2( l -1)+1 , I T ≥ 2 ∇ I 1 Ric[ G ] ∗ ∇ I T T + l -2 ∑ m =0 ∑ I 1 + ··· + I l -m + I T =2 m +1 ∇ I 1 Ric[ G ] ∗ · · · ∗ ∇ I l -m Ric[ G ] ∗ ∇ I T T</formula>
<formula><location><page_81><loc_12><loc_69><loc_86><loc_76></location>J ([∆ l , curl G ] T ) = ε [ G ] ∗ ∑ I 1 + I T =2( l -1)+1 , I T ≥ 2 ∇ I 1 Ric[ G ] ∗ ∇ I T T + l -2 ∑ m =0 ∑ I 1 + ··· + I l -m + + I T =2 m +1 ∇ I 1 Ric[ G ] ∗ · · · ∗ ∇ I l -m Ric[ G ] ∗ ∇ I T T </formula>
<text><location><page_81><loc_12><loc_68><loc_77><loc_69></location>Proof. The formulas follow by applying the formulas from Lemma 11.4 inductively.</text>
<text><location><page_81><loc_87><loc_68><loc_88><loc_69></location>/square</text>
<text><location><page_81><loc_12><loc_63><loc_88><loc_67></location>Lemma 11.6 (Time derivative commutators) . With respect to a solution to the Einstein scalar field system as in Proposition 2.6, the following commutator formulas hold:</text>
<formula><location><page_81><loc_12><loc_60><loc_32><loc_63></location>[ ∂ t , ∇ i ] ζ =0 (11.8a)</formula>
<formula><location><page_81><loc_12><loc_35><loc_85><loc_60></location>∇ ∇ -a ∇ (11.8b) [ ∂ t , ∆] ζ =2( N +1) a -3 〈 Σ , ∇ 2 ζ 〉 G -2 N ˙ a a ∆ ζ (11.8c) -2( N +1) a -3 〈 div G Σ , ∇ ζ 〉 G -2 a -3 〈 Σ , ∇ N ∇ ζ 〉 G + ˙ a a 〈∇ N, ∇ ζ 〉 G [ ∂ t , ∇ i ] T kl = { 2( N +1) a -3 [ ∇ i Σ m ( k + ∇ ( k Σ /sharpm i -∇ m Σ i ( k ] (11.8d) +2 a -3 [ ∇ i N Σ /sharpm ( k +Σ /sharpm i ∇ ( k N -∇ /sharpm N Σ i ( k ]} T l ) m -˙ a a [ ∇ i N T kl + ∇ ( k N T l ) i -∇ /sharpm NG i ( k T l ) m ] [ ∂ t , ∆] T kl = a -3 ( N +1)Σ ∗ ∇ 2 T + ˙ a a N ∆ T + a -3 ∇ (( N +1)Σ) ∗ ∇ T + ˙ a a ∇ N ∗ ∇ T (11.8e) + a -3 ∇ 2 (( N +1)Σ) ∗ T -˙ a a ∇ 2 N ∗ T</formula>
<formula><location><page_81><loc_22><loc_58><loc_56><loc_61></location>[ ∂ t , /sharpi ] ζ =2( N +1) a -3 Σ /sharpij j ζ 2 N ˙ a /sharpi ζ</formula>
<text><location><page_81><loc_12><loc_31><loc_88><loc_34></location>Proof. (11.8a) is simply that coordinate derivatives commute, and (11.8b) follows by applying (2.28b) and the product rule.</text>
<text><location><page_81><loc_12><loc_26><loc_88><loc_31></location>For the commutators (11.8c), (11.8d) and (11.8e), we write out the covariant derivatives in local coordinates, apply the product rule, and then the evolution equations (2.28b) and (2.34) for the inverse metric and Christoffel symbols. /square</text>
<text><location><page_81><loc_12><loc_21><loc_88><loc_24></location>Lemma 11.7 (High order time derivative commutators) . For l ∈ N , l ≥ 2 , the time derivative commutators take the following form:</text>
<text><location><page_81><loc_12><loc_19><loc_17><loc_20></location>(11.9a)</text>
<formula><location><page_81><loc_15><loc_14><loc_87><loc_18></location>[ ∂ t , ∆ l ] ζ =2 a -3 ( N +1) 〈 Σ , ∇ 2 ∆ l -1 ζ 〉 G + a -3 ∇ Σ ∗ ∇ 3 ∆ l -2 ζ 2( N +1) a -3 div G ∆ l -1 Σ , ζ G +( N +1) a -3 2 l -3 Ric Σ ζ + J ([ ∂ t , ∆ l ] ζ ) ,</formula>
<formula><location><page_81><loc_12><loc_7><loc_74><loc_16></location>-〈 ∇ 〉 ∇ ∗ ∗ ∇ [ ∂ t , ∇ ∆ l ] ζ =2 a -3 ( N +1) 〈 Σ , ∇ 3 ∆ l -1 ζ 〉 G + a -3 ( N +1) ∇ Σ ∗ ∇ 2 l ζ (11.9b) -2( N +1) a -3 〈∇ div G ∆ l -1 Σ , ∇ ζ 〉 G</formula>
<text><location><page_82><loc_12><loc_83><loc_17><loc_84></location>(11.9c)</text>
<formula><location><page_82><loc_24><loc_84><loc_68><loc_90></location>+ ˙ a a 〈∇ 2 ∆ l -1 N, ∇ ζ 〉 G +( N +1) a -3 ∇ 2 l -2 Ric[ G ] ∗ Σ ∗ ∇ ζ + J ([ ∂ t , ∇ ∆ l ] ζ )</formula>
<formula><location><page_82><loc_15><loc_73><loc_81><loc_83></location>[ ∂ t , ∆ l ] T = a -3 ( Σ ∗ ∇ 2 ∆ l -1 T + ∇ Σ ∗ ∇ 3 ∆ l -2 T + ∇ T ∗ ∇ ∆ l -1 Σ+ T ∗ ∆ l Σ ) + a -3 ( ( N +1)Σ ∗ T ∗ ∇ 2 ∆ l -2 Ric[ G ] + ∇ (( N +1)Σ ∗ T ) ∗ ∇ 2 l -3 Ric[ G ] ) + ˙ a a ∆ l N · T + ˙ a a ∇ ∆ l -1 N ∗ ∇ T + J ([ ∂ t , ∆ l ]) T ,</formula>
<text><location><page_82><loc_12><loc_72><loc_17><loc_73></location>(11.9d)</text>
<formula><location><page_82><loc_14><loc_63><loc_78><loc_71></location>[ ∂ t , ∇ ∆ l ] T = a -3 ∇ Σ ∗ ∆ l T + a -3 ( N +1)Σ ∗ ∇ 3 ∆ l -1 T + a -3 ( N +1) T ∗ ∇ ∆ l Σ + ˙ a a ∇ ∆ l N ∗ T + ˙ a a ∇ 2 ∆ l -1 N ∗ ∇ T + a -3 ( N +1)Σ ∗ ∇ 3 ∆ l -2 Ric[ G ] ∗ T + J ([ ∂ t , ∇ ∆ l ] T ) ,</formula>
<text><location><page_82><loc_12><loc_60><loc_44><loc_62></location>where the junk terms take are as follows:</text>
<text><location><page_82><loc_38><loc_38><loc_38><loc_39></location>/negationslash</text>
<formula><location><page_82><loc_12><loc_9><loc_87><loc_58></location>J ([ ∂ t , ∆ l ] ζ ) = a -3 ∑ I N + I Σ + I ζ =2( l -1) I ζ ≤ 2( l -2) ∇ I N ( N +1) ∗ ∇ I Σ Σ ∗ ∇ I ζ +2 ζ (11.9e) + ˙ a a ∑ I N + I ζ =2 l I ζ ≥ 2 ∇ I N N ∗ ∇ I ζ ζ + a -3 l -2 ∑ m =0 ∑ I N + I Σ + I ζ + ∑ l -m -1 i =1 I i =2 m ∇ I N ( N +1) ∗ ∇ I Σ Σ ∗ ∇ I 1 Ric[ G ] ∗ · · · ∗ ∇ I l -m -1 Ric[ G ] ∗ ∇ I ζ +2 ζ + a -3 l -2 ∑ m =0 ∑ I N + I Σ + I ζ + ∑ l -m -1 i =1 I i =2 m I 1 =2 l -4 ∇ I N ( N +1) ∗ ∇ I Σ Σ ∗ ∇ I 1 +1 Ric[ G ] ∗ · · · ∗ ∇ I l -m -1 Ric[ G ] ∗ ∇ I ζ +1 ζ + ˙ a a l -1 ∑ m =0 ∑ I N + ∑ l -m -1 i =1 I i + I ζ =2 m -1 I ζ =2( l -1) ∇ I N N ∗ ∇ I 1 Ric[ G ] ∗ · · · ∗ ∇ I l -m -1 Ric[ G ] ∗ ∇ I ζ +1 ζ J ([ ∂ t , ∇ ∆ l ] ζ ) = ˙ a a ∑ I N + I ζ =2 l I ζ =0 ∇ I N N ∗ ∇ I ζ +1 ζ + a -3 ∑ I N + I Σ + I ζ =2( l -1)+1 ( I Σ ,I ζ ) =(0 , 2( l -1)+1) , (1 , 2( l -1)) ∇ I N ( N +1) ∗ ∇ I Σ Σ ∗ ∇ I ζ +1 ζ (11.9f) + a -3 l -2 ∑ m =0 ∑ I N + I Σ + I ζ + ∑ l -m -1 i =1 I i =2 m +1 ∇ I N ( N +1) ∗ ∇ I Σ Σ ∗ ∇ I 1 Ric[ G ] ∗ · · · ∗ ∇ I l -m -1 Ric[ G ] ∗ ∇ I ζ +2 ζ + a -3 l -2 ∑ m =0 ∑ I N + I Σ + I ζ + ∑ l -m -1 i =1 I i =2 m +1 I 1 =2 l -3 ∇ I N ( N +1) ∗ ∇ I Σ Σ ∗ ∇ I 1 +1 Ric[ G ] ∗ · · · ∗ ∇ I l -m -1 Ric[ G ] ∗ ∇ I ζ +1 ζ + ˙ a a l -1 ∑ m =0 ∑ I N + ∑ l -m -1 i =1 I i + I ζ =2 m I ζ =2( l -1) ∇ I N N ∗ ∇ I 1 Ric[ G ] ∗ · · · ∗ ∇ I l -m -1 Ric[ G ] ∗ ∇ I ζ +1 ζ ∑ ∑ T</formula>
<text><location><page_82><loc_31><loc_28><loc_31><loc_29></location>/negationslash</text>
<text><location><page_82><loc_53><loc_28><loc_53><loc_29></location>/negationslash</text>
<text><location><page_82><loc_38><loc_11><loc_38><loc_13></location>/negationslash</text>
<formula><location><page_82><loc_12><loc_7><loc_70><loc_11></location>J ([ ∂ t , ∆ l ]) T = a -3 I N + I Σ + I T =2 l ∇ I N N ∗ ∇ I Σ Σ ∗ ∇ I T T + a -3 I Σ + I T =2 l I Σ ,I T ≥ 2 ∇ I Σ Σ ∗ ∇ I T (11.9g)</formula>
<text><location><page_82><loc_39><loc_16><loc_39><loc_18></location>/negationslash</text>
<text><location><page_82><loc_39><loc_33><loc_39><loc_34></location>/negationslash</text>
<formula><location><page_83><loc_28><loc_81><loc_83><loc_90></location>+ a -3 l -1 ∑ m =0 ∑ I N + I Σ + I T + ∑ l -m -1 i =1 I i =2 m I 1 < 2 l -3 ∇ I N ( N +1) ∗ ∇ I Σ Σ ∗ ∇ I 1 Ric[ G ] ∗ · · · ∗ ∇ I l -m -1 Ric[ G ] ∗ ∇ I T T + ˙ a a ∑ I + I T =2 l, I T 2 ∇ I N N ∗ ∇ I T T</formula>
<formula><location><page_83><loc_31><loc_81><loc_39><loc_82></location>N ≥</formula>
<text><location><page_83><loc_12><loc_78><loc_72><loc_81></location>J ([ ∂ t , ∇ ∆ l ] T ) = a -3 Σ ∗ ∇ N ∗ ∆ l Ric[ G ] + a -3 N ∗ ∇ Σ ∗ ∆ l T + ˙ a a ∇ N ∗ ∆ l T + ∇ J ([ ∂ t , ∆ l ] T ) (11.9h)</text>
<text><location><page_83><loc_12><loc_75><loc_88><loc_78></location>We can extend the formulas to l = 1 by dropping any term which would contain negative powers of ∆ or a multiindex of negative order.</text>
<formula><location><page_83><loc_12><loc_72><loc_88><loc_74></location>Proof. This follows by iteratively applying the commutators in Lemma 11.6. /square</formula>
<text><location><page_83><loc_12><loc_68><loc_88><loc_71></location>While all of the above commutators will be essential for the mainline argument, the a priori estimates require the following commutators:</text>
<text><location><page_83><loc_12><loc_65><loc_65><loc_67></location>Lemma 11.8 (Auxiliary commutators) . Let J ∈ N . Then, we have:</text>
<formula><location><page_83><loc_12><loc_48><loc_79><loc_66></location>[ ∂ t , ∇ J ] ζ = a -3 ∑ I N + I Σ + I ζ = l -1 , I ζ <J -1 ∇ I N ( N +1) ∗ ∇ I Σ Σ ∗ ∇ I ζ +1 ζ (11.10a) + ˙ a a ∑ I N + I ζ = J -1 ,I N > 0 ∇ I N N ∗ ∇ I ζ +1 ζ [ ∂ t , ∇ J ] T = a -3 ∑ I N + I Σ + I T = J, I T <J ∇ I N ( N +1) ∗ ∇ I Σ Σ ∗ ∇ I T T (11.10b) + ˙ a a ∑ I N + I T = J,I N > 0 ∇ I N N ∗ ∇ I T T</formula>
<text><location><page_83><loc_12><loc_41><loc_88><loc_46></location>Proof. For J = 1, this has already been shown in (11.8a) and (11.8d). For higher orders, the formulas follow from a straightforward induction argument using that, in local coordinates, we schematically have</text>
<text><location><page_83><loc_12><loc_37><loc_39><loc_38></location>and analogously replacing ζ with T</text>
<formula><location><page_83><loc_21><loc_36><loc_88><loc_40></location>[ ∂ t , ∇ J ] ζ = [ ∂ t , ∇ ] ∇ J -1 ζ + ∇ [ ∂ t , ∇ J -1 ] ζ = ( ∂ t Γ[ G ]) ∗ ∇ J -1 ζ + ∇ [ ∂ t , ∇ J -1 ] ζ . /square</formula>
<section_header_level_1><location><page_83><loc_12><loc_34><loc_41><loc_35></location>11.3. Borderline and junk terms.</section_header_level_1>
<text><location><page_83><loc_12><loc_30><loc_88><loc_33></location>Definition 11.9 (Error terms) . Let L ∈ 2 N , L ≥ 2. Then, the error terms in the Laplace commuted equations stated in Lemma 2.11 take the following form:</text>
<text><location><page_83><loc_72><loc_23><loc_72><loc_24></location>/negationslash</text>
<formula><location><page_83><loc_12><loc_8><loc_78><loc_28></location>For the constraint equations, we have M L,Junk = -8 π (Ψ + C ) ∇ ∆ L 2 -2 Ric[ G ] ∗ ∇ 2 φ + ∇ L -2 Ric[ G ] ∗ ∇ Σ+ ∇ L -3 Ric[ G ] ∗ ∇ 2 Σ ︸ ︷︷ ︸ if L =2 (11.11a) + ∑ I Ψ + I φ = L, I Ψ =0 ∇ I Ψ Ψ ∗ ∇ I φ +1 φ +8 π (Ψ + C ) J ([∆ L 2 , ∇ ] φ ) -J ([∆ L 2 , div G ]Σ) ˜ M L,Junk = -ε [ G ] ∗ ∇ L -3 Ric[ G ] ∗ ∇ Σ ︸ ︷︷ ︸ if L =2 -J ([∆ L 2 , curl G ]Σ) (11.11b) H L,Border = a -4 [ Σ ∗ ∆ L 2 Σ+ ∇ Σ ∗ ∇ L -1 Σ ] (11.11c) H L,Junk = ∑ I 1 + I 2 = L ∇ I 1 +1 φ ∗ ∇ I 2 +1 φ + a -4 ∑ I 1 + I 2 = L,I i ≥ 2 ∇ I 1 Σ ∗ ∇ I 2 Σ (11.11d) +∆ L 2 [ 4 π 3 |∇ φ | 2 G + 8 π 3 a -4 Ψ 2 + 16 π 3 Ca -4 Ψ ] · G</formula>
<text><location><page_83><loc_39><loc_16><loc_39><loc_17></location>/negationslash</text>
<text><location><page_83><loc_39><loc_20><loc_39><loc_21></location>/negationslash</text>
<text><location><page_84><loc_12><loc_88><loc_40><loc_90></location>The lapse equation error terms are</text>
<formula><location><page_84><loc_12><loc_76><loc_78><loc_88></location>N L,Border = a -4 ( N +1) ( Σ ∗ ∆ L 2 Σ+ ∇ Σ ∗ ∇ L -1 Σ+Ψ ∗ ∆ L 2 Ψ+ ∇ Ψ ∗ ∇ L -1 Ψ ) (11.12a) + a -4 [ | Σ | 2 G +Ψ 2 +Ψ ] ∗ ∆ L 2 N + a -4 ∇ [ | Σ | 2 G +Ψ 2 +Ψ ] ∗ ∇ L -1 N N L,Junk = a -4 ∑ I N + I 1 + I 2 = L ; I N ≤ L -2; I N > 0 or I 1 ≤ I 2 ≤ L -2 ∇ I N ( N +1) ∗ ( ∇ I 1 Σ ∗ ∇ I 2 Σ+ ∇ I 1 Ψ ∗ ∇ I 2 Ψ ) (11.12b) + a -4 N ∗ ∆ L 2 Ψ+ a -4 ∑ I N + I Ψ = L ; I Ψ ≥ 2 , I N ≥ 1 ∇ I N N ∗ ∇ I Ψ Ψ ,</formula>
<formula><location><page_84><loc_12><loc_61><loc_78><loc_75></location>as well as N L +1 ,Border = a -4 ( N +1) ( Σ ∗ ∇ ∆ L 2 Σ+ ∇ Σ ∗ ∇ L Σ+ ∇ 2 Σ ∗ ∇ L -1 Σ+Ψ ∗ ∇ ∆ L 2 Ψ (11.12c) + ∇ Ψ ∗ ∇ L Ψ+ ∇ 2 Ψ ∗ ∇ L -1 Ψ ) + a -4 [ | Σ | 2 G +Ψ 2 +Ψ ] ∗ ∇ ∆ L 2 N + ∇ Ψ ∗ ∇ L N + ∇ 2 Ψ ∗ ∇ L -1 Ψ N L +1 ,Junk = a -4 ∑ I N + I 1 + I 2 = L ; I N <L +1; I N > 0 or I 1 ≥ I 2 > 2 ∇ I N ( N +1) ∗ ( ∇ I 1 Σ ∗ ∇ I 2 Σ+ ∇ I 1 Ψ ∗ ∇ I 2 Ψ ) (11.12d) + L Ψ</formula>
<formula><location><page_84><loc_33><loc_60><loc_68><loc_63></location>a -4 N ∗ ∇ ∆ 2 Ψ+ a -4 ∑ I N + I Ψ = L +1; I N ,I Ψ > 2 ∇ I N N ∗ ∇ I Ψ</formula>
<text><location><page_84><loc_12><loc_58><loc_44><loc_59></location>whereas the scalar field error terms read</text>
<formula><location><page_84><loc_12><loc_52><loc_87><loc_57></location>P L,Border = -3Ψ ˙ a a ∆ L 2 N + ˙ a a ∇ Ψ ∗ ∇ L -1 N +2 a -3 ( N +1) 〈 Σ , ∇ 2 ∆ L 2 -1 Ψ 〉 G +2 a -3 ( N +1) ∇ L -3 Ric ∗ Σ ∗ ∇ Ψ (11.13a) -2 a -3 ( N +1) 〈 div G ∆ L 2 -1 Σ , ∇ Ψ 〉 G + a -3 ( N +1) ∇ Σ ∗ ∇ 3 ∆ L 2 -2 Ψ</formula>
<formula><location><page_84><loc_12><loc_49><loc_82><loc_52></location>P L,Junk = ˙ a a ∑ I N + I Ψ = L, I Ψ ≥ 2 ∇ I N N ∗ ∇ I Ψ Ψ+ a ∑ I N + I φ = L +1 , I N ,I φ =0 ∇ I N N ∗ ∇ I φ +1 φ + J ([ ∂ t , ∆ L 2 ]Ψ) (11.13b)</formula>
<text><location><page_84><loc_60><loc_49><loc_60><loc_50></location>/negationslash</text>
<formula><location><page_84><loc_12><loc_47><loc_67><loc_49></location>Q L,Border = a -3 Ψ ∇ ∆ L 2 N + a -3 ( N +1)Σ ∗ ∇ 3 ∆ L 2 -1 φ + a -3 ( N +1) ∇ L φ ∗ ∇ Σ (11.13c)</formula>
<text><location><page_84><loc_12><loc_45><loc_17><loc_46></location>(11.13d)</text>
<text><location><page_84><loc_19><loc_45><loc_21><loc_46></location>Q</text>
<text><location><page_84><loc_25><loc_45><loc_26><loc_46></location>=</text>
<text><location><page_84><loc_26><loc_45><loc_27><loc_46></location>a</text>
<text><location><page_84><loc_27><loc_45><loc_28><loc_46></location>-</text>
<text><location><page_84><loc_27><loc_42><loc_28><loc_43></location>+</text>
<text><location><page_84><loc_28><loc_42><loc_29><loc_43></location>a</text>
<text><location><page_84><loc_29><loc_44><loc_30><loc_45></location>I</text>
<text><location><page_84><loc_30><loc_44><loc_30><loc_45></location>N</text>
<text><location><page_84><loc_29><loc_42><loc_30><loc_43></location>-</text>
<text><location><page_84><loc_30><loc_43><loc_30><loc_43></location>3</text>
<text><location><page_84><loc_31><loc_44><loc_32><loc_45></location>+</text>
<text><location><page_84><loc_32><loc_44><loc_32><loc_45></location>I</text>
<text><location><page_84><loc_30><loc_42><loc_31><loc_43></location>(</text>
<text><location><page_84><loc_31><loc_42><loc_32><loc_43></location>N</text>
<text><location><page_84><loc_33><loc_42><loc_35><loc_43></location>+1)</text>
<text><location><page_84><loc_34><loc_44><loc_36><loc_47></location>∑ +1</text>
<text><location><page_84><loc_33><loc_44><loc_34><loc_45></location>=</text>
<text><location><page_84><loc_34><loc_44><loc_35><loc_45></location>L</text>
<text><location><page_84><loc_36><loc_44><loc_37><loc_45></location>, I</text>
<text><location><page_84><loc_35><loc_41><loc_36><loc_43></location>∇</text>
<text><location><page_84><loc_36><loc_43><loc_37><loc_43></location>2</text>
<text><location><page_84><loc_37><loc_44><loc_38><loc_45></location>N</text>
<text><location><page_84><loc_37><loc_42><loc_38><loc_43></location>∆</text>
<text><location><page_84><loc_38><loc_44><loc_39><loc_45></location>,I</text>
<text><location><page_84><loc_39><loc_44><loc_40><loc_45></location>Ψ</text>
<text><location><page_84><loc_38><loc_43><loc_39><loc_43></location>L</text>
<text><location><page_84><loc_38><loc_42><loc_39><loc_43></location>2</text>
<text><location><page_84><loc_40><loc_44><loc_40><loc_45></location>/negationslash</text>
<text><location><page_84><loc_40><loc_44><loc_42><loc_45></location>=0</text>
<text><location><page_84><loc_39><loc_42><loc_40><loc_43></location>-</text>
<text><location><page_84><loc_40><loc_43><loc_41><loc_43></location>1</text>
<text><location><page_84><loc_42><loc_45><loc_43><loc_46></location>∇</text>
<text><location><page_84><loc_41><loc_42><loc_42><loc_43></location>Σ</text>
<text><location><page_84><loc_43><loc_46><loc_44><loc_46></location>I</text>
<text><location><page_84><loc_44><loc_46><loc_44><loc_46></location>N</text>
<text><location><page_84><loc_42><loc_41><loc_44><loc_43></location>∗ ∇</text>
<text><location><page_84><loc_45><loc_45><loc_46><loc_46></location>N</text>
<text><location><page_84><loc_44><loc_42><loc_45><loc_43></location>φ</text>
<text><location><page_84><loc_46><loc_42><loc_47><loc_43></location>+(</text>
<text><location><page_84><loc_47><loc_42><loc_49><loc_43></location>N</text>
<text><location><page_84><loc_49><loc_42><loc_52><loc_43></location>+1)</text>
<text><location><page_84><loc_52><loc_42><loc_52><loc_43></location>a</text>
<text><location><page_84><loc_50><loc_45><loc_52><loc_46></location>Ψ+</text>
<text><location><page_84><loc_53><loc_45><loc_53><loc_46></location>a</text>
<text><location><page_84><loc_52><loc_42><loc_53><loc_43></location>-</text>
<text><location><page_84><loc_53><loc_45><loc_54><loc_46></location>-</text>
<text><location><page_84><loc_53><loc_43><loc_54><loc_43></location>3</text>
<text><location><page_84><loc_54><loc_46><loc_55><loc_46></location>3</text>
<text><location><page_84><loc_55><loc_45><loc_56><loc_46></location>∇</text>
<text><location><page_84><loc_54><loc_41><loc_55><loc_43></location>∇</text>
<text><location><page_84><loc_56><loc_45><loc_57><loc_46></location>∆</text>
<text><location><page_84><loc_55><loc_42><loc_56><loc_43></location>∆</text>
<text><location><page_84><loc_56><loc_43><loc_57><loc_43></location>L</text>
<text><location><page_84><loc_57><loc_42><loc_57><loc_43></location>2</text>
<text><location><page_84><loc_58><loc_46><loc_58><loc_47></location>L</text>
<text><location><page_84><loc_58><loc_46><loc_58><loc_46></location>2</text>
<text><location><page_84><loc_57><loc_42><loc_58><loc_43></location>-</text>
<text><location><page_84><loc_59><loc_45><loc_60><loc_47></location>-</text>
<text><location><page_84><loc_58><loc_43><loc_59><loc_43></location>1</text>
<text><location><page_84><loc_60><loc_46><loc_60><loc_47></location>1</text>
<text><location><page_84><loc_60><loc_45><loc_61><loc_46></location>N</text>
<text><location><page_84><loc_59><loc_42><loc_61><loc_43></location>Ric[</text>
<text><location><page_84><loc_61><loc_42><loc_63><loc_43></location>G</text>
<text><location><page_84><loc_63><loc_42><loc_63><loc_43></location>]</text>
<text><location><page_84><loc_62><loc_45><loc_64><loc_46></location>∗ ∇</text>
<text><location><page_84><loc_64><loc_46><loc_64><loc_46></location>2</text>
<text><location><page_84><loc_63><loc_41><loc_64><loc_43></location>∗</text>
<text><location><page_84><loc_65><loc_45><loc_65><loc_46></location>φ</text>
<text><location><page_84><loc_64><loc_42><loc_65><loc_43></location>Σ</text>
<text><location><page_84><loc_66><loc_45><loc_66><loc_46></location>∗</text>
<text><location><page_84><loc_67><loc_45><loc_68><loc_46></location>Σ</text>
<text><location><page_84><loc_66><loc_41><loc_68><loc_43></location>∗ ∇</text>
<text><location><page_84><loc_68><loc_42><loc_69><loc_43></location>φ</text>
<formula><location><page_84><loc_27><loc_39><loc_76><loc_41></location>+ a -3 ∇ L -2 Ric[ G ] ∗ (( N +1) ∗ Σ ∗ ∇ φ ) + ˙ a a 〈∇ 2 ∆ L 2 -1 N, ∇ φ 〉 G + J ([ ∂ t , ∇ ∆ L 2 ] φ )</formula>
<text><location><page_84><loc_12><loc_37><loc_15><loc_38></location>and</text>
<formula><location><page_84><loc_12><loc_32><loc_71><loc_36></location>P L +1 ,Border = -3Ψ ˙ a a ∇ ∆ L 2 N + ˙ a a ∇ Ψ ∗ ∇ 2 ∆ L 2 -1 N +2 a -3 〈 Σ , ∇ 3 ∆ L 2 -1 Ψ 〉 G (11.13e) + a -3 ( N +1) ∇ Σ ∗ ∇ L Ψ+2 a -3 ∇ L -2 Ric ∗ Σ ∗ ∇ Ψ</formula>
<text><location><page_84><loc_34><loc_30><loc_35><loc_31></location>+</text>
<text><location><page_84><loc_35><loc_30><loc_36><loc_31></location>a</text>
<text><location><page_84><loc_36><loc_30><loc_37><loc_32></location>-</text>
<text><location><page_84><loc_37><loc_31><loc_37><loc_32></location>3</text>
<text><location><page_84><loc_37><loc_30><loc_38><loc_31></location>(</text>
<text><location><page_84><loc_38><loc_30><loc_39><loc_31></location>N</text>
<text><location><page_84><loc_39><loc_30><loc_42><loc_31></location>+1)</text>
<text><location><page_84><loc_42><loc_30><loc_43><loc_31></location>∇</text>
<text><location><page_84><loc_43><loc_31><loc_44><loc_32></location>2</text>
<text><location><page_84><loc_44><loc_30><loc_45><loc_31></location>∆</text>
<text><location><page_84><loc_45><loc_31><loc_46><loc_32></location>L</text>
<text><location><page_84><loc_45><loc_31><loc_46><loc_31></location>2</text>
<text><location><page_84><loc_46><loc_30><loc_47><loc_32></location>-</text>
<text><location><page_84><loc_47><loc_31><loc_48><loc_32></location>1</text>
<text><location><page_84><loc_48><loc_30><loc_49><loc_31></location>Σ</text>
<text><location><page_84><loc_49><loc_30><loc_51><loc_31></location>∗ ∇</text>
<text><location><page_84><loc_51><loc_30><loc_52><loc_31></location>Ψ</text>
<formula><location><page_84><loc_12><loc_26><loc_80><loc_30></location>P L +1 ,Junk = ˙ a a ∑ I N + I Ψ = L +1 , I Ψ ≥ 2 ∇ I N N ∗ ∇ I Ψ Ψ+ a ∑ I N + I φ = L +2 , I N ,I φ =0 ∇ I N N ∗ ∇ I φ +1 φ (11.13f)</formula>
<text><location><page_84><loc_34><loc_25><loc_35><loc_26></location>+</text>
<text><location><page_84><loc_35><loc_25><loc_36><loc_26></location>J</text>
<text><location><page_84><loc_36><loc_25><loc_37><loc_26></location>([</text>
<text><location><page_84><loc_37><loc_25><loc_37><loc_26></location>∂</text>
<text><location><page_84><loc_37><loc_25><loc_38><loc_25></location>t</text>
<text><location><page_84><loc_38><loc_25><loc_38><loc_26></location>,</text>
<text><location><page_84><loc_39><loc_24><loc_40><loc_26></location>∇</text>
<text><location><page_84><loc_40><loc_25><loc_41><loc_26></location>∆</text>
<text><location><page_84><loc_41><loc_25><loc_42><loc_26></location>L</text>
<text><location><page_84><loc_41><loc_25><loc_42><loc_25></location>2</text>
<text><location><page_84><loc_42><loc_25><loc_44><loc_26></location>]Ψ)</text>
<formula><location><page_84><loc_12><loc_22><loc_81><loc_24></location>Q L +1 ,Border = a -3 Ψ∆ L 2 +1 N + a -3 Ψ ∆ L 2 N + a -3 ( N +1)Σ 2 ∆ L 2 φ + ˙ a ∆ L 2 N φ (11.13g)</formula>
<formula><location><page_84><loc_12><loc_14><loc_81><loc_19></location>Q L +1 ,Junk = a -3 ∑ I N + I Ψ = L +2 , 2 ≤ I Ψ ≤ L +1 ∇ I N N ∗ ∇ I Ψ Ψ+ a -3 ( N +1) ∇ L -2 Ric[ G ] ∗ Σ ∗ ∇ φ + a -3 ( N +1) ∇ 2 ∆ L 2 -1 Σ ∗ ∇ φ + J ([ ∂ t , ∆ L 2 +1 ] φ ) (11.13h)</formula>
<formula><location><page_84><loc_33><loc_19><loc_80><loc_23></location>∇ ∗ ∇ ∗ ∇ a ∇ ∗ ∇ + a -3 ( N +1) ∇ Σ ∗ ∇ 2 ∆ L 2 -1 φ</formula>
<text><location><page_84><loc_12><loc_12><loc_20><loc_14></location>as well as</text>
<formula><location><page_84><loc_12><loc_8><loc_70><loc_11></location>Q 1 ,Border = a -3 Ψ∆ N + a -3 ( N +1)Σ ∗ ∇ 2 φ (11.13i) Q 1 ,Junk = a -3 ∇ Ψ ∗ ∇ N + a -3 ( N +1) ∇ Σ ∗ ∇ φ + J ([ ∂ t , ∆] φ ) . (11.13j)</formula>
<text><location><page_84><loc_28><loc_46><loc_29><loc_46></location>3</text>
<text><location><page_84><loc_46><loc_45><loc_48><loc_46></location>∗ ∇</text>
<text><location><page_84><loc_48><loc_46><loc_49><loc_46></location>I</text>
<text><location><page_84><loc_21><loc_45><loc_25><loc_46></location>L,Junk</text>
<text><location><page_84><loc_32><loc_44><loc_33><loc_45></location>Ψ</text>
<text><location><page_84><loc_49><loc_46><loc_50><loc_46></location>Ψ</text>
<text><location><page_84><loc_68><loc_26><loc_68><loc_28></location>/negationslash</text>
<text><location><page_85><loc_12><loc_88><loc_66><loc_90></location>The commuted rescaled evolution equation for Σ has the error terms</text>
<formula><location><page_85><loc_12><loc_80><loc_86><loc_88></location>S L,Border = a -3 ( N +1) ( Σ ∗ ∇ 2 ∆ L 2 -1 Σ+ ∇ Σ ∗ ∇ 3 ∆ L 2 -2 Σ ) + a -3 ( ∆ L 2 N · (Σ ∗ Σ) + ∇ ∆ L 2 -1 N ∗ ∇ Σ ∗ Σ ) (11.14a) + a -3 ( N +1)Σ ∗ Σ ∗ ∇ 2 ∆ L 2 -2 Ric[ G ] + ˙ a a ∆ L 2 N ∗ Σ+ ˙ a a ∇ ∆ L 2 -1 N ∗ ∇ Σ + a -3 [( N +1) ∇ Σ ∗ Σ+ ∇ N ∗ Σ ∗ Σ] ∗ ∇ L -3 Ric[ G ]</formula>
<formula><location><page_85><loc_29><loc_78><loc_59><loc_81></location>︸ ︷︷ ︸ not present for L =2</formula>
<text><location><page_85><loc_48><loc_76><loc_48><loc_77></location>/negationslash</text>
<formula><location><page_85><loc_12><loc_69><loc_86><loc_79></location>S L,Junk = -a [∆ L 2 , ∇ 2 ] N + a ∑ I N + I Ric = L,I N =0 ∇ I N N ∗ ∇ I Ric Ric[ G ] + ˙ a a ∑ I N + I Σ = L ∇ I N N ∗ ∇ I Σ Σ (11.14b) + a -3 ∑ I 1 + I 2 = L, I i > 0 ∇ I 1 Σ ∗ ∇ I 2 Σ+ a -3 ∑ I N + I 1 + I 2 = L, I N <L ∇ I N N ∗ ∇ I 1 Σ ∗ ∇ I 2 Σ + a ∑ I N + I 1 + I 2 = L ∇ I N ( N +1) ∗ ∇ I 1 +1 φ ∗ ∇ I 2 +1 φ + ( 4 πC 2 a -3 + 1 3 a ) ∆ L 2 N · G + J ([ ∂ t , ∆ L 2 ]Σ)</formula>
<text><location><page_85><loc_12><loc_67><loc_67><loc_68></location>while the commuted Ricci tensor evolution equations have error terms</text>
<formula><location><page_85><loc_12><loc_61><loc_76><loc_66></location>R L,Border = a -3 [ ∇ L +2 N · Σ+ ∇ L +1 N ∗ ∇ Σ+Σ ∗ ∇ 2 ∆ L 2 -1 Ric[ G ] + ∇ Σ ∗ ∇ L -1 Ric[ G ] ] (11.15a) R L +1 ,Border = a -3 [ ∇ L +3 N · Σ+ ∇ L +2 N ∗ ∇ Σ+Σ ∗ ∇ 3 ∆ L 2 -1 Ric[ G ] + ∇ Σ ∗ ∇ L Ric[ G ] ] (11.15b)</formula>
<text><location><page_85><loc_12><loc_60><loc_17><loc_61></location>(11.15c)</text>
<text><location><page_85><loc_12><loc_45><loc_17><loc_46></location>(11.15d)</text>
<text><location><page_85><loc_22><loc_60><loc_23><loc_61></location>R</text>
<text><location><page_85><loc_21><loc_45><loc_22><loc_46></location>L</text>
<text><location><page_85><loc_22><loc_45><loc_24><loc_46></location>+1</text>
<text><location><page_85><loc_24><loc_45><loc_27><loc_46></location>,Junk</text>
<text><location><page_85><loc_27><loc_60><loc_29><loc_61></location>=</text>
<text><location><page_85><loc_29><loc_60><loc_29><loc_61></location>a</text>
<text><location><page_85><loc_27><loc_45><loc_29><loc_46></location>=</text>
<text><location><page_85><loc_29><loc_45><loc_29><loc_46></location>a</text>
<text><location><page_85><loc_29><loc_60><loc_30><loc_61></location>-</text>
<text><location><page_85><loc_29><loc_57><loc_30><loc_58></location>+</text>
<text><location><page_85><loc_30><loc_57><loc_31><loc_58></location>a</text>
<text><location><page_85><loc_30><loc_61><loc_31><loc_61></location>3</text>
<text><location><page_85><loc_42><loc_59><loc_43><loc_61></location>∇</text>
<text><location><page_85><loc_42><loc_56><loc_43><loc_57></location>=</text>
<text><location><page_85><loc_43><loc_56><loc_44><loc_57></location>L</text>
<text><location><page_85><loc_43><loc_61><loc_44><loc_61></location>I</text>
<text><location><page_85><loc_44><loc_60><loc_45><loc_61></location>N</text>
<text><location><page_85><loc_44><loc_54><loc_45><loc_56></location>-</text>
<text><location><page_85><loc_33><loc_59><loc_34><loc_60></location>+</text>
<text><location><page_85><loc_34><loc_59><loc_34><loc_60></location>I</text>
<text><location><page_85><loc_33><loc_55><loc_33><loc_56></location>(</text>
<text><location><page_85><loc_33><loc_55><loc_34><loc_56></location>I</text>
<text><location><page_85><loc_34><loc_55><loc_35><loc_55></location>Σ</text>
<text><location><page_85><loc_34><loc_59><loc_35><loc_59></location>Σ</text>
<text><location><page_85><loc_35><loc_59><loc_36><loc_60></location>=</text>
<text><location><page_85><loc_37><loc_59><loc_39><loc_60></location>+2</text>
<text><location><page_85><loc_39><loc_59><loc_40><loc_60></location>, I</text>
<text><location><page_85><loc_35><loc_56><loc_36><loc_57></location>I</text>
<text><location><page_85><loc_36><loc_59><loc_38><loc_62></location>∑ L</text>
<text><location><page_85><loc_36><loc_56><loc_36><loc_56></location>N</text>
<text><location><page_85><loc_35><loc_55><loc_36><loc_56></location>,I</text>
<text><location><page_85><loc_37><loc_56><loc_38><loc_57></location>+</text>
<text><location><page_85><loc_38><loc_56><loc_38><loc_57></location>I</text>
<text><location><page_85><loc_36><loc_55><loc_37><loc_55></location>Ric</text>
<text><location><page_85><loc_37><loc_55><loc_38><loc_56></location>)</text>
<text><location><page_85><loc_38><loc_54><loc_38><loc_56></location>/negationslash</text>
<text><location><page_85><loc_40><loc_59><loc_40><loc_59></location>Σ</text>
<text><location><page_85><loc_38><loc_56><loc_40><loc_59></location>∑</text>
<text><location><page_85><loc_38><loc_56><loc_39><loc_56></location>Σ</text>
<text><location><page_85><loc_39><loc_56><loc_40><loc_57></location>+</text>
<text><location><page_85><loc_40><loc_56><loc_40><loc_57></location>I</text>
<text><location><page_85><loc_38><loc_55><loc_40><loc_56></location>=(0</text>
<text><location><page_85><loc_40><loc_55><loc_41><loc_56></location>,L</text>
<text><location><page_85><loc_41><loc_55><loc_41><loc_56></location>)</text>
<text><location><page_85><loc_41><loc_55><loc_42><loc_56></location>,</text>
<text><location><page_85><loc_42><loc_55><loc_43><loc_56></location>(1</text>
<text><location><page_85><loc_43><loc_55><loc_44><loc_56></location>,L</text>
<text><location><page_85><loc_45><loc_60><loc_46><loc_61></location>N</text>
<text><location><page_85><loc_45><loc_55><loc_46><loc_56></location>1)</text>
<formula><location><page_85><loc_29><loc_50><loc_86><loc_54></location>+ a -3 L 2 -1 ∑ m =0 ∑ I N + I Σ + ∑ L/ 2 -m +1 i =1 I i =2 m ∇ I N ( N +1) ∗ ∇ I Σ Σ ∗ ∇ I 1 Ric[ G ] ∗ · · · ∗ ∇ I L 2 -m +1 Ric[ G ]</formula>
<formula><location><page_85><loc_29><loc_47><loc_76><loc_49></location>+ ˙ a a ( [∆ L 2 , ∇ 2 ] N +∆ L 2 N ∗ Ric[ G ] + ∇ L -1 N ∗ ∇ Ric[ G ] ) + J ([ ∂ t , ∆ L 2 ]Ric[ G ])</formula>
<text><location><page_85><loc_29><loc_45><loc_30><loc_47></location>-</text>
<text><location><page_85><loc_30><loc_46><loc_31><loc_47></location>3</text>
<text><location><page_85><loc_42><loc_45><loc_43><loc_46></location>∇</text>
<text><location><page_85><loc_43><loc_46><loc_44><loc_47></location>I</text>
<text><location><page_85><loc_44><loc_46><loc_45><loc_46></location>N</text>
<text><location><page_85><loc_45><loc_45><loc_46><loc_46></location>N</text>
<text><location><page_85><loc_47><loc_45><loc_49><loc_46></location>∗ ∇</text>
<text><location><page_85><loc_49><loc_46><loc_49><loc_47></location>I</text>
<text><location><page_85><loc_49><loc_46><loc_50><loc_46></location>Σ</text>
<text><location><page_85><loc_50><loc_45><loc_51><loc_46></location>Σ</text>
<text><location><page_85><loc_36><loc_44><loc_38><loc_47></location>∑</text>
<text><location><page_85><loc_31><loc_44><loc_32><loc_45></location>I</text>
<text><location><page_85><loc_32><loc_44><loc_33><loc_45></location>N</text>
<text><location><page_85><loc_33><loc_44><loc_34><loc_45></location>+</text>
<text><location><page_85><loc_34><loc_44><loc_34><loc_45></location>I</text>
<text><location><page_85><loc_34><loc_44><loc_35><loc_45></location>Σ</text>
<text><location><page_85><loc_35><loc_44><loc_36><loc_45></location>=</text>
<text><location><page_85><loc_36><loc_44><loc_37><loc_45></location>L</text>
<text><location><page_85><loc_37><loc_44><loc_39><loc_45></location>+3</text>
<text><location><page_85><loc_39><loc_44><loc_40><loc_45></location>, I</text>
<text><location><page_85><loc_40><loc_44><loc_40><loc_45></location>Σ</text>
<text><location><page_85><loc_41><loc_44><loc_42><loc_45></location>2</text>
<formula><location><page_85><loc_29><loc_40><loc_67><loc_44></location>+ a -3 ∑ I N + I Σ + I Ric = L ( I Σ ,I Ric ) =(0 ,L +1) , (1 ,L ) ∇ I N ( N +1) ∗ ∇ I Σ Σ ∗ ∇ I Ric Ric[ G ]</formula>
<text><location><page_85><loc_38><loc_40><loc_38><loc_41></location>/negationslash</text>
<formula><location><page_85><loc_29><loc_35><loc_86><loc_39></location>+ a -3 L 2 -1 ∑ m =0 ∑ I + I + ∑ L/ 2 -m +1 I =2 m +1 ∇ I N ( N +1) ∗ ∇ I Σ Σ ∗ ∇ I 1 Ric[ G ] ∗ . . . ∇ I L 2 -m +1 Ric[ G ]</formula>
<formula><location><page_85><loc_29><loc_32><loc_82><loc_36></location>N Σ i =1 i + ˙ a a ( ∇ [∆ L 2 , ∇ 2 ] N + ∇ ∆ L 2 N ∗ Ric[ G ] + ∇ 2 ∆ L 2 -1 N ∗ ∇ Ric[ G ] ) + J ([ ∂ t , ∇ ∆ L 2 ]Ric[ G ])</formula>
<text><location><page_85><loc_15><loc_28><loc_55><loc_30></location>Finally, the Bel-Robinson evolution error terms are</text>
<text><location><page_85><loc_63><loc_14><loc_63><loc_15></location>/negationslash</text>
<formula><location><page_85><loc_12><loc_7><loc_77><loc_28></location>E L,Border = τ 3 ( ∆ L 2 N · E + ∇ L -1 N ∗ ∇ E ) -a -1 ( ∆ L 2 E × Σ+ E × ∆ L 2 Σ ) (11.16a) + a -3 ε [ G ] ∗ ε [ G ] ∗ ( ∇ L -1 E ∗ ∇ Σ+ ∇ E ∗ ∇ L -1 Σ ) + a -3 ∆ L 2 N · ( E ∗ Σ) + a -3 ∇ ∆ L 2 -1 N ∗ [ ∇ E ∗ Σ+ E ∗ ∇ Σ] + a -3 ( Σ ∗ ∇ 2 ∆ L 2 -1 E + ∇ Σ ∗ ∇ 3 ∆ L 2 -2 E + ∇ E ∗ ∇ ∆ L 2 -1 Σ+ E ∗ ∆ L 2 Σ ) + a -3 ( N +1)Σ ∗ E ∗ ∇ 2 ∆ L 2 -2 Ric[ G ] + ∇ (( N +1) ∗ Σ ∗ E ) ∗ ∇ L -3 Ric[ G ] ︸ ︷︷ ︸ if L =2 +4 πa -3 (Ψ + C ) 2 ∆ L 2 N · Σ + 4 πa -3 ∇ L -1 N ∗ [ (Ψ + C ) 2 ∇ Σ+2(Ψ+ C ) ∗ ∇ Ψ ∗ Σ ] +4 πa -3 ( N +1) [ (Ψ 2 +2 C Ψ)∆ L 2 Σ+2(Ψ+ C )∆ L 2 Ψ · Σ ] +4 πa -3 ∇ L -1 Σ ∗ [ (Ψ + C ) 2 ∇ N +2( N +1)(Ψ + C ) ∇ Ψ ]</formula>
<text><location><page_85><loc_47><loc_59><loc_49><loc_61></location>∗ ∇</text>
<text><location><page_85><loc_46><loc_56><loc_47><loc_58></location>∇</text>
<text><location><page_85><loc_47><loc_58><loc_48><loc_58></location>I</text>
<text><location><page_85><loc_49><loc_61><loc_49><loc_61></location>I</text>
<text><location><page_85><loc_48><loc_57><loc_49><loc_58></location>N</text>
<text><location><page_85><loc_49><loc_61><loc_50><loc_61></location>Σ</text>
<text><location><page_85><loc_49><loc_57><loc_50><loc_58></location>(</text>
<text><location><page_85><loc_50><loc_57><loc_51><loc_58></location>N</text>
<text><location><page_85><loc_51><loc_57><loc_54><loc_58></location>+1)</text>
<text><location><page_85><loc_50><loc_60><loc_51><loc_61></location>Σ</text>
<text><location><page_85><loc_20><loc_45><loc_21><loc_46></location>R</text>
<text><location><page_85><loc_23><loc_60><loc_27><loc_61></location>L,Junk</text>
<text><location><page_85><loc_31><loc_59><loc_32><loc_60></location>I</text>
<text><location><page_85><loc_32><loc_59><loc_33><loc_59></location>N</text>
<text><location><page_85><loc_31><loc_57><loc_32><loc_58></location>-</text>
<text><location><page_85><loc_32><loc_57><loc_32><loc_58></location>3</text>
<text><location><page_85><loc_40><loc_58><loc_41><loc_60></location>≥</text>
<text><location><page_85><loc_41><loc_59><loc_42><loc_60></location>2</text>
<text><location><page_85><loc_40><loc_56><loc_42><loc_56></location>Ric</text>
<text><location><page_85><loc_40><loc_44><loc_41><loc_45></location>≥</text>
<text><location><page_85><loc_54><loc_56><loc_56><loc_58></location>∗ ∇</text>
<text><location><page_85><loc_56><loc_58><loc_57><loc_58></location>I</text>
<text><location><page_85><loc_57><loc_57><loc_58><loc_58></location>Σ</text>
<text><location><page_85><loc_58><loc_57><loc_59><loc_58></location>Σ</text>
<text><location><page_85><loc_59><loc_56><loc_61><loc_58></location>∗ ∇</text>
<text><location><page_85><loc_61><loc_58><loc_62><loc_58></location>I</text>
<text><location><page_85><loc_62><loc_57><loc_63><loc_58></location>Ric</text>
<text><location><page_85><loc_63><loc_57><loc_66><loc_58></location>Ric[</text>
<text><location><page_85><loc_66><loc_57><loc_67><loc_58></location>G</text>
<text><location><page_85><loc_67><loc_57><loc_67><loc_58></location>]</text>
<formula><location><page_86><loc_27><loc_88><loc_54><loc_90></location>+4 πa -3 (Ψ + C ) L -1 Ψ [( N +1) Σ+ N</formula>
<formula><location><page_86><loc_37><loc_88><loc_57><loc_89></location>∇ ∗ ∇ ∇ ∗ Σ]</formula>
<text><location><page_86><loc_12><loc_86><loc_73><loc_88></location>E L,top = a -1 ( N +1) ε [ G ] B ∆ L 2 -1 Ric[ G ] + a ( N +1)(Ψ + C ) ∆ L 2 -1 Ric[ G ] φ (11.16b)</text>
<formula><location><page_86><loc_12><loc_83><loc_49><loc_85></location>E L,Junk = a I + I E = L, I L 2 ∇ I N N ∗ ∇ I E E (11.16c)</formula>
<text><location><page_86><loc_35><loc_70><loc_35><loc_71></location>/negationslash</text>
<formula><location><page_86><loc_27><loc_52><loc_87><loc_87></location>∗ ∗ ∇ ∇ ∗ ∇ ˙ a ∑ N N ≤ -+ a -1 ε [ G ] ∗ ∑ I N + I B = L +1 , I N ,I B ≤ L ∇ I N N ∗ ∇ I B B +( N +1) ∇ 2 ∆ L 2 -2 Ric[ G ] ∗ ∇ B + a -3 ε [ G ] ∗ ε [ G ] ∗ ∑ I N + I E + I Σ = L, I N ≤ L -2; I N > 0 or I E ,I Σ ≥ 2 ∇ I N N ∗ ∇ I E E ∗ ∇ I Σ Σ + a ∑ I N + I Ψ + I φ = L +1 I N ,I Ψ ,I φ = L +1 ∇ I N ( N +1) ∗ ∇ I Ψ (Ψ + C ) ∗ ∇ I φ +1 φ + a -3 ∑ I N + I Σ + I 1 + I 2 = L I N ,I Σ ,I i ≤ L -2 ∇ I N ( N +1) ∗ ∇ I Σ Σ ∗ ∇ I 1 (Ψ + C ) ∗ ∇ I 2 (Ψ + C ) + ˙ aa 3 ∑ I N + I 1 + I 2 = L ∇ I N ( N +1) ∗ ∇ I 1 +1 φ ∗ ∇ I 2 +1 φ + a ∑ I N + I Σ + I 1 + I 2 = L ∇ I N ( N +1) ∗ ∇ I Σ Σ ∗ ∇ I 1 +1 φ ∗ ∇ I 2 +1 φ + a -1 ε [ G ] ∗ B ∗ [∆ L 2 , ∇ ] N +4 πa ( N +1)(Ψ + C ) [ ∇ L -2 Ric[ G ] ∗ ∇ 2 φ + J ([∆ L 2 , ∇ 2 ] φ ) ] + a { (Ψ + C )[∆ L 2 , ∇ ] N +( N +1)[∆ L 2 , ∇ ]Ψ } ∗ ∇ φ +( N +1) a -1 J ([∆ L 2 , curl G ] B ) + J ([ ∂ t , ∆ L 2 ] E ) +∆ L 2 [ a -3 ( N +1) E ∗ Σ+ 2 π 3 a 6 ( N +1) ( ∂ 0 ( a -6 (Ψ + C ) 2 + a -2 |∇ φ | 2 G ) +4 π ˙ a a (Ψ + C ) 2 )] · G</formula>
<text><location><page_86><loc_61><loc_36><loc_61><loc_37></location>/negationslash</text>
<text><location><page_86><loc_28><loc_34><loc_79><loc_36></location>a -1 ( N +1)(Ψ + C ) ε [ G ] ∆ L 2 φ Σ+ a -1 ε [ G ] 2 L φ (( N +1)(Ψ + C )Σ)</text>
<formula><location><page_86><loc_12><loc_35><loc_73><loc_50></location>B L,Border = τ 3 ( ∆ L 2 N · B + ∇ L -1 N ∗ ∇ B ) -a -1 ( ∆ L 2 B × Σ+ B ∗ ∆ L 2 Σ ) (11.16d) + a -3 ε [ G ] ∗ ε [ G ] ∗ ( ∇ L -1 B ∗ ∇ Σ+ ∇ B ∗ ∇ L -1 Σ ) + a -3 ∆ L 2 N · ( B ∗ Σ) + a -3 ∇ ∆ L 2 -1 N · [ ∇ B ∗ Σ+ B ∗ ∇ Σ] + a -3 ( Σ ∗ ∇ 2 ∆ L 2 -1 B + ∇ Σ ∗ ∇ 3 ∆ L 2 -2 B + ∇ B ∗ ∇ ∆ L 2 -1 Σ+ B ∗ ∆ L 2 Σ ) + a -3 ( N +1)Σ ∗ B ∗ ∇ L -2 Ric[ G ] + ∇ (( N +1) ∗ Σ ∗ B ) ∗ ∇ L -3 Ric[ G ] ︸ ︷︷ ︸ if L =2 </formula>
<text><location><page_86><loc_12><loc_32><loc_72><loc_33></location>B L,top = a 3 ( N +1) ε [ G ] ∆ 2 -1 Ric[ G ] φ φ + a -1 ε [ G ] E ∆ 2 -1 Ric[ G ] (11.16e)</text>
<text><location><page_86><loc_27><loc_32><loc_67><loc_35></location>+ · ∗ ∇ ∗ ∗ ∇ ∇ ∗ ∇ ∗ ∇ L ∗ ∇ ∗ ∇ ∗ ∗ ∇ L</text>
<formula><location><page_86><loc_12><loc_7><loc_74><loc_32></location>B L,Junk = ˙ a a ∑ I N + I B = L,I N ≤ L -2 ∇ I N N ∗ ∇ I B B (11.16f) + a -1 ε [ G ] ∗ ∑ I N + I E = L +1 ,I N ,I E ≤ L ∇ I N N ∗ ∇ I E E + ∇ 2 ∆ L 2 -2 Ric[ G ] ∗ ∇ E + a -3 ε [ G ] ∗ ε [ G ] ∗ ∑ I N + I E + I Σ = L, I N ≤ L -2; I N > 0 or I B ,I Σ ≥ 2 ∇ I N ( N +1) ∗ ∇ I B B ∗ ∇ I Σ Σ + a 3 ε [ G ] ∗ ∑ I N + I 1 + I 2 = L, I N > 0 or I 2 <L ∇ I N ( N +1) ∗ ∇ I 1 +1 φ ∗ ∇ I 2 +2 φ + a -1 ε [ G ] ∗ ∑ I N + I Ψ + I φ + I Σ = L I φ ≤ L -2 ∇ I N ( N +1) ∗ ∇ I Ψ (Ψ + C ) ∗ ∇ I φ +1 φ ∗ ∇ I Σ Σ + a -1 ε [ G ] ∗ E ∗ [∆ L 2 , ∇ ] N + a -1 ( N +1)(Ψ + C ) · ε [ G ] ∗ Σ ∗ [ ∆ L 2 , ∇ ] φ</formula>
<formula><location><page_87><loc_27><loc_82><loc_78><loc_90></location>+ a 3 ( N +1) ε [ G ] ∗ ∇ φ ∗ ( ∇ 2 ∆ L 2 -2 Ric[ G ] ∗ ∇ 2 φ + J ([∆ L 2 , ∇ ] φ ) ) -a -1 ( N +1) J ([∆ L 2 , curl G ] E ) + J ([ ∂ t , ∆ L 2 ] B ) + ∆ L 2 [ a -3 ( N +1) B ∗ Σ ] · G +∆ L 2 [ 4 πa 2 ∇ /sharpm φ (Ψ + C ) + 2 π 3 a 5 ∆ L 2 ∇ /sharpm ( a -6 (Ψ + C ) 2 + a -2 |∇ φ | 2 G ) ] ε [ G ] ( · ) m ( · )</formula>
<text><location><page_87><loc_12><loc_70><loc_88><loc_81></location>11.4. L 2 G error term estimates. In this subsection, we collect how the error terms can be controlled in terms of energies as well as homogenous Sobolev norms of φ . We don't claim that these estimates are optimal - in particular, we note that at low order (like L = 2), many of the curvature errors that appear in the estimates below could be avoided entirely: These arise as a result of applying the general estimates in Lemma 4.5 where the Ricci tensor doesn't naturally occur in the respective equations, and can be avoided at low orders by applying (4.4f) on all curvature terms that occur.</text>
<text><location><page_87><loc_12><loc_63><loc_88><loc_69></location>Instead of optimality, we try to keep both notation and form of the error term estimates as simple as possible and the energy estimates between base and top level as unified as possible. In particular, we track the 'worst' curvature energy occurring at high orders for all estimates below, even if these terms are added in artificially for low orders.</text>
<text><location><page_87><loc_12><loc_58><loc_88><loc_61></location>Lemma 11.10 (Estimates for borderline error terms) . Let L ∈ 2 Z + , L ≤ 20 . Then, the following estimates hold:</text>
<text><location><page_87><loc_12><loc_55><loc_18><loc_56></location>(11.17a)</text>
<formula><location><page_87><loc_17><loc_5><loc_85><loc_19></location>P L,Border ‖ L 2 G /lessorsimilar εa -3 E ( L ) ( φ, · ) + εa -3 E ( L ) ( N, · ) + εa -3 -c √ ε E ( ≤ L -2) ( N, · ) + εa -3 -c √ ε √ E ( ≤ L -2) ( φ, · ) + εa -3 √ E ( L ) (Σ , · ) + εa -3 -c √ ε √ E ( L -2) (Σ , · ) + ε 2 a -3 -c √ ε √ E ( ≤ L -3) (Ric , · ) ︸ ︷︷ ︸ not present for L =2</formula>
<formula><location><page_87><loc_12><loc_14><loc_86><loc_55></location>‖ H L,Border ‖ L 2 G /lessorsimilar εa -4 √ E ( L ) (Σ , · ) + εa -4 -c √ ε √ E ( ≤ L -2) (Σ , · ) + ε 2 a -4 -c √ ε √ E ( ≤ L -2) (Ric , · ) ‖ N L,Border ‖ L 2 G /lessorsimilar εa -4 [ √ E ( L ) ( φ, · ) + √ E ( L ) (Σ , · ) ] + εa -4 √ E ( L ) ( N, · ) (11.17b) + εa -4 -c √ ε [ √ E ( ≤ L -2) ( φ, · ) + √ E ( ≤ L -2) (Σ , · ) + √ E ( ≤ L -2) ( N, · ) ] + ε 2 a -4 √ E ( ≤ L -2) (Ric , · ) + εa -4 -c √ ε √ E ( ≤ L -4) (Ric , · ) ︸ ︷︷ ︸ not present for L =2 ‖ N L +1 ,Border ‖ L 2 G /lessorsimilar εa -6 [ √ a 4 E ( L +1) ( φ, · ) + √ a 4 E ( L +1) (Σ , · ) ] + εa -6 √ a 4 E ( L +1) ( N, · ) (11.17c) + εa -4 [ √ E ( L ) ( φ, · ) + √ E ( L ) (Σ , · ) + √ E ( L ) ( N, · ) ] + εa -4 -c √ ε [ √ E ( ≤ L -2) ( φ, · ) + √ E ( ≤ L -2) (Σ , · ) + √ E ( ≤ L -2) ( N, · ) ] + ε 2 a -4 -c √ ε √ E ( ≤ L -2) (Ric , · ) ‖ √ √ √ (11.17d)</formula>
<text><location><page_88><loc_12><loc_89><loc_18><loc_90></location>(11.17e)</text>
<text><location><page_88><loc_12><loc_81><loc_18><loc_82></location>(11.17f)</text>
<formula><location><page_88><loc_16><loc_81><loc_61><loc_89></location>‖ Q L,Border ‖ L 2 G /lessorsimilar εa -3 √ E ( L +1) ( N, · ) + εa -3 √ a -4 E ( L ) ( φ, · ) + εa -3 -c √ ε √ a -4 E ( ≤ L -2) ( φ, · )</formula>
<text><location><page_88><loc_12><loc_70><loc_18><loc_71></location>(11.17g)</text>
<formula><location><page_88><loc_14><loc_70><loc_86><loc_81></location>‖ P L +1 ,Border ‖ L 2 G /lessorsimilar εa -3 √ E ( L +1) ( φ, · ) + εa -3 √ E ( L +1) ( N, · ) + εa -3 -c √ ε √ E ( ≤ L -1) ( N, · ) + εa -3 -c √ ε √ E ( ≤ L -1) ( φ, · ) + εa -3 √ E ( L +1) (Σ , · ) + εa -3 -c √ ε √ E ( L -1) (Σ , · ) + ε 2 a -3 -c √ ε √ E ( ≤ L -2) (Ric , · )</formula>
<text><location><page_88><loc_12><loc_62><loc_18><loc_63></location>(11.17h)</text>
<formula><location><page_88><loc_14><loc_62><loc_72><loc_70></location>‖ Q L +1 ,Border ‖ L 2 G /lessorsimilar εa -3 √ E ( L +2) ( N, · ) + εa -3 -c √ ε √ E ( L +1) ( N, · ) + εa -3 √ a -4 E ( L +1) ( φ, · ) + εa -3 -c √ ε √ a -4 E ( ≤ L -1) ( φ, · )</formula>
<text><location><page_88><loc_12><loc_49><loc_18><loc_50></location>(11.17i)</text>
<formula><location><page_88><loc_16><loc_48><loc_75><loc_62></location>‖ S L,Border ‖ L 2 G /lessorsimilar εa -3 √ E ( L ) (Σ , · ) + εa -3 √ E ( L ) ( N, · ) + ε 2 a -3 √ E ( L -2) (Ric , · ) + εa -3 -c √ ε √ E ( ≤ L -2) (Σ , · ) + εa -3 -c √ ε √ E ( ≤ L -2) ( N, · ) + ε 2 a -3 -c √ ε √ E ( ≤ L -4) (Ric , · ) ︸ ︷︷ ︸ not present for L =2</formula>
<text><location><page_88><loc_12><loc_41><loc_18><loc_42></location>(11.17j)</text>
<formula><location><page_88><loc_16><loc_41><loc_67><loc_49></location>‖ R L,Border ‖ L 2 G /lessorsimilar εa -3 √ E ( L +2) ( N, · ) + εa -3 -c √ ε √ E ( ≤ L ) ( N, · ) + εa -3 √ E ( L ) (Ric , · ) + εa -3 -c √ ε √ E ( ≤ L -2) (Ric , · )</formula>
<formula><location><page_88><loc_14><loc_33><loc_69><loc_42></location>‖ R L +1 ,Border ‖ L 2 G /lessorsimilar εa -3 √ E ( L +3) ( N, · ) + εa -3 -c √ ε √ E ( ≤ L +1) ( N, · ) + εa -3 √ E ( L +1) (Ric , · ) + εa -3 -c √ ε √ E ( ≤ L -1) (Ric , · )</formula>
<text><location><page_88><loc_12><loc_32><loc_18><loc_33></location>(11.17k)</text>
<text><location><page_88><loc_12><loc_7><loc_88><loc_14></location>Proof. All of these estimates follow from applying L 2 G -L ∞ G -type Holder estimates to the individual nonlinear terms. The lower order terms are either controlled by the zero order estimates in subsection 4.1 or the a priori estimates in Lemma 4.3. Furthermore, we apply Lemma 4.5, along with again Lemma 4.3, to translate L 2 G -norms into energies up to additional curvature energy terms.</text>
<formula><location><page_88><loc_26><loc_13><loc_80><loc_33></location>‖ E L,Border ‖ L 2 G + ‖ B L,Border ‖ L 2 G /lessorsimilar εa -3 √ E ( L ) ( φ, · ) + εa -3 -c √ ε √ E ( ≤ L -2) ( φ, · ) + εa -3 ( √ E ( L ) ( N, · ) + √ E ( L ) (Σ , · ) ) + εa -3 √ E ( L ) ( W, · ) + εa -3 -c √ ε ( √ E ( ≤ L -2) ( N, · ) + √ E ( ≤ L -2) (Σ , · ) + √ E ( ≤ L -2) ( W, · ) ) + ε 2 a -3 √ E ( L -2) (Ric , · ) + ε 2 a -3 -c √ ε √ E ( ≤ L -4) (Ric , · ) ︸ ︷︷ ︸ not present for L =2</formula>
<text><location><page_89><loc_12><loc_85><loc_88><loc_90></location>For the sake of simplicity, we always estimate ˙ a a by a -3 up to constant (see (2.3)), and liberally apply (3.8) to deal with odd order energies and to distribute a -c √ ε factors to lower orders while updating c > 0 whereever this is convenient. /square</text>
<text><location><page_89><loc_12><loc_82><loc_54><loc_84></location>Lemma 11.11 (Estimates for top order error terms) .</text>
<formula><location><page_89><loc_12><loc_74><loc_80><loc_83></location>‖ E L,top ‖ L 2 G /lessorsimilar √ εa 1 -c √ ε √ E ( L -1) (Ric , · ) = √ εa -1 -c √ ε √ a 4 E ( L -1) (Ric , · (11.18a) ‖ B L,top ‖ L 2 G /lessorsimilar εa -1 √ E ( L -1) (Ric , · ) = εa -3 √ a 4 E ( L -1) (Ric , · ) (11.18b)</formula>
<formula><location><page_89><loc_80><loc_79><loc_81><loc_81></location>)</formula>
<text><location><page_89><loc_12><loc_74><loc_85><loc_75></location>Proof. This follows directly using (4.2c) and (4.4g) for the Bel-Robinson terms as well as (4.4e).</text>
<text><location><page_89><loc_87><loc_74><loc_88><loc_75></location>/square</text>
<text><location><page_89><loc_12><loc_70><loc_83><loc_73></location>Lemma 11.12 (Junk terms) . Recalling ‖ -notation from Remark 2.12, the following holds:</text>
<formula><location><page_89><loc_12><loc_10><loc_85><loc_41></location>‖ N L +1 ,Junk ‖ L 2 G /lessorsimilar εa -4 -c √ ε √ E ( ≤ L -1) ( N, · ) + εa -cσ ( √ E ( L +1) ( φ, · ) + √ E ( L +1) (Σ , · ) ) + εa -4 -c √ ε [ √ E ( ≤ L -1) ( φ, · ) + √ E ( ≤ L -1) (Σ , · ) ] + ε 2 a -4 √ E ( ≤ L -1) (Ric , · ) + ε 2 a -4 -c √ ε √ E ( ≤ L -3) (Ric , · ) ︸ ︷︷ ︸ not present for L =2 ‖ P L,Junk ‖ L 2 G /lessorsimilar εa 1 -cσ √ E ( L ) ( φ, · ) + εa -3 -c √ ε √ E ( ≤ L -2) ( φ, · ) (11.19f) + εa -3 -c √ ε √ E ( ≤ L -2) (Σ , · ) + √ εa 1 -c √ ε √ E ( L ) ( N, · ) + [ εa -3 -c √ ε + √ εa -1 -c √ ε ] √ E ( ≤ L -2) ( N, · ) + ε 2 a 1 -cσ √ E ( ≤ L -2) (Ric , · ) + ε 2 a -3 -c √ ε √ E ( ≤ L -3) (Ric , · ) ︸ ︷︷ ︸ not present for L =2 (11.19g)</formula>
<text><location><page_89><loc_12><loc_40><loc_74><loc_72></location>‖ M L,Junk ‖ L 2 G /lessorsimilar εa -2 -c √ ε √ E ( ≤ L -1) ( φ, · ) + a -2 -c √ ε √ E ( ≤ L -2) ( φ, · ) (11.19a) + a -c √ ε √ E ( ≤ L -1) (Σ , · ) + √ εa -c √ ε √ E ( ≤ L -2) (Ric , · ) ‖ ˜ M L,Junk ‖ L 2 G /lessorsimilar εa -1 -c √ ε √ E ( ≤ L -2) (Ric , · ) + a -1 -c √ ε √ E ( ≤ L -1) (Σ , · ) (11.19b) ‖ H ‖ L,Junk ‖ L 2 G /lessorsimilar εa -4 -c √ ε √ E ( ≤ L -2) (Σ , · ) + √ εa -2 -c √ ε √ E ( ≤ L ) ( φ, · ) (11.19c) + εa -c √ ε √ E ( ≤ L -2) (Ric , · ) ‖ N L,Junk ‖ L 2 G /lessorsimilar εa -4 -c √ ε √ E ( ≤ L -2) ( N, · ) + εa -cσ √ E ( L ) ( φ, · ) (11.19d) + εa -4 -c √ ε [ √ E ( ≤ L -2) ( φ, · ) + √ E ( ≤ L -2) (Σ , · ) ] + εa -4 √ E ( ≤ L -2) (Ric , · ) + εa -4 -c √ ε √ E ( ≤ L -4) (Ric , · ) ︸ ︷︷ ︸ not present for L =2 (11.19e)</text>
<formula><location><page_89><loc_19><loc_6><loc_69><loc_11></location>‖ P L +1 ,Junk ‖ L 2 G /lessorsimilar εa 1 -cσ √ E ( L +1) ( φ, · ) + εa -3 -c √ ε √ E ( ≤ L -1) ( φ, · )</formula>
<formula><location><page_90><loc_12><loc_64><loc_75><loc_91></location>+ εa -3 -c √ ε √ E ( ≤ L -1) (Σ , · ) + √ εa 1 -c √ ε √ E ( L +1) ( N, · ) + [ εa -3 -c √ ε + √ εa -1 -c √ ε ] √ E ( ≤ L -1) ( N, · ) + ε 2 a -1 -cσ √ a 4 E ( L -1) (Ric , · ) + ( ε 2 a -1 -cσ + ε 2 a -3 -c √ ε ) √ E ( ≤ L -2) (Ric , · ) ‖ Q L,Junk ‖ L 2 G /lessorsimilar εa -1 -c √ ε √ a -4 E ( L ) ( φ, · ) + εa -3 -c √ ε E ( ≤ L -2) ( φ, · ) (11.19h) + √ εa -3 -c √ ε √ E ( ≤ L ) (Σ , · ) + √ εa -3 -c √ ε √ E ( ≤ L ) ( N, · ) + εa -3 -c √ ε √ E ( ≤ L -2) (Ric , · ) ︸ ︷︷ ︸ not present for L =2</formula>
<text><location><page_90><loc_12><loc_61><loc_18><loc_62></location>(11.19j)</text>
<text><location><page_90><loc_19><loc_60><loc_20><loc_62></location>‖</text>
<text><location><page_90><loc_20><loc_61><loc_21><loc_62></location>Q</text>
<formula><location><page_90><loc_12><loc_62><loc_84><loc_67></location>‖ Q 1 ,Junk ‖ /lessorsimilar εa -3 -c √ ε E (1) ( N, · ) + ε 3 2 a -3 -c √ ε ‖∇ φ ‖ L 2 G + √ εa -3 -c √ ε √ E ( ≤ 1) (Σ , · ) (11.19i)</formula>
<text><location><page_90><loc_21><loc_61><loc_22><loc_61></location>L</text>
<text><location><page_90><loc_22><loc_61><loc_24><loc_62></location>+1</text>
<text><location><page_90><loc_24><loc_61><loc_28><loc_61></location>,Junk</text>
<text><location><page_90><loc_28><loc_60><loc_29><loc_62></location>‖</text>
<text><location><page_90><loc_29><loc_60><loc_30><loc_61></location>L</text>
<text><location><page_90><loc_30><loc_61><loc_30><loc_62></location>2</text>
<text><location><page_90><loc_30><loc_60><loc_30><loc_61></location>G</text>
<text><location><page_90><loc_31><loc_61><loc_33><loc_62></location>/lessorsimilar</text>
<text><location><page_90><loc_33><loc_61><loc_35><loc_62></location>εa</text>
<formula><location><page_90><loc_33><loc_52><loc_55><loc_58></location>+ εa -3 -c √ ε √ E ( ≤ L -1) (Ric , · )</formula>
<formula><location><page_90><loc_33><loc_56><loc_78><loc_64></location>-1 -c √ ε √ a -4 E ( L +1) ( φ, · ) + -3 c E ( ≤ L · + √ εa -3 -c √ ε √ E ( ≤ L +1) (Σ , · ) + √ εa -3 -c √ ε √ E ( ≤ L +1) ( N, · )</formula>
<text><location><page_90><loc_55><loc_61><loc_57><loc_62></location>εa</text>
<text><location><page_90><loc_59><loc_61><loc_60><loc_62></location>-</text>
<text><location><page_90><loc_66><loc_61><loc_67><loc_62></location>-</text>
<text><location><page_90><loc_68><loc_61><loc_69><loc_62></location>(</text>
<text><location><page_90><loc_69><loc_61><loc_71><loc_62></location>φ,</text>
<text><location><page_90><loc_71><loc_61><loc_72><loc_62></location>)</text>
<formula><location><page_90><loc_12><loc_46><loc_87><loc_54></location>‖ S ‖ L,Junk ‖ L 2 G /lessorsimilar εa 1 -cσ √ E ( L ) (Σ , · ) + εa -3 -c √ ε √ E ( ≤ L -2) (Σ , · ) + √ εa -1 -c √ ε √ E ( L ) ( φ, · ) (11.19k) + ( εa -3 + a 1 -c √ ε ) √ E ( ≤ L ) ( N, · ) + εa 5 -cσ √ E ( ≤ L -1) (Ric , · )</formula>
<text><location><page_90><loc_33><loc_45><loc_35><loc_46></location>+</text>
<text><location><page_90><loc_35><loc_45><loc_37><loc_46></location>εa</text>
<text><location><page_90><loc_37><loc_45><loc_38><loc_47></location>-</text>
<text><location><page_90><loc_38><loc_46><loc_39><loc_47></location>3</text>
<text><location><page_90><loc_40><loc_46><loc_40><loc_47></location>(</text>
<text><location><page_90><loc_40><loc_46><loc_41><loc_47></location>L</text>
<text><location><page_90><loc_41><loc_45><loc_42><loc_47></location>-</text>
<text><location><page_90><loc_42><loc_46><loc_43><loc_47></location>2)</text>
<text><location><page_90><loc_44><loc_45><loc_47><loc_46></location>(Ric</text>
<text><location><page_90><loc_47><loc_45><loc_47><loc_46></location>,</text>
<text><location><page_90><loc_48><loc_45><loc_51><loc_46></location>) +</text>
<text><location><page_90><loc_51><loc_45><loc_53><loc_46></location>εa</text>
<text><location><page_90><loc_53><loc_45><loc_54><loc_47></location>-</text>
<text><location><page_90><loc_54><loc_46><loc_55><loc_47></location>3</text>
<text><location><page_90><loc_55><loc_45><loc_56><loc_47></location>-</text>
<text><location><page_90><loc_56><loc_46><loc_56><loc_47></location>c</text>
<text><location><page_90><loc_56><loc_46><loc_57><loc_47></location>√</text>
<text><location><page_90><loc_57><loc_46><loc_58><loc_47></location>ε</text>
<text><location><page_90><loc_61><loc_61><loc_62><loc_63></location>√</text>
<text><location><page_90><loc_60><loc_44><loc_61><loc_46></location>E</text>
<text><location><page_90><loc_39><loc_44><loc_40><loc_46></location>E</text>
<text><location><page_90><loc_48><loc_44><loc_48><loc_46></location>·</text>
<text><location><page_90><loc_58><loc_43><loc_60><loc_48></location>√</text>
<text><location><page_90><loc_61><loc_46><loc_62><loc_47></location>(</text>
<text><location><page_90><loc_63><loc_46><loc_64><loc_47></location>L</text>
<text><location><page_90><loc_65><loc_46><loc_66><loc_47></location>4)</text>
<text><location><page_90><loc_67><loc_61><loc_68><loc_62></location>1)</text>
<text><location><page_90><loc_66><loc_45><loc_69><loc_46></location>(Ric</text>
<text><location><page_90><loc_69><loc_45><loc_70><loc_46></location>,</text>
<text><location><page_90><loc_71><loc_45><loc_71><loc_46></location>)</text>
<formula><location><page_90><loc_12><loc_31><loc_74><loc_45></location>︸ ︷︷ ︸ not present for L =2 ‖ R L,Junk ‖ L 2 G /lessorsimilar ε 2 a 1 -cσ √ E ( ≤ L -1) (Ric , · ) + εa -3 -c √ ε √ E ( ≤ L -2) (Ric , · ) (11.19l) + εa 1 -cσ √ E ( ≤ L +2) (Σ , · ) + a -3 -c √ ε √ E ( ≤ L ) (Σ , · ) + a -3 -c √ ε √ E ( ≤ L ) ( N, · ) (11.19m)</formula>
<text><location><page_90><loc_12><loc_20><loc_18><loc_21></location>(11.19n)</text>
<text><location><page_90><loc_62><loc_61><loc_62><loc_62></location>ε</text>
<text><location><page_90><loc_62><loc_45><loc_63><loc_47></location>≤</text>
<text><location><page_90><loc_64><loc_45><loc_65><loc_47></location>-</text>
<text><location><page_90><loc_70><loc_44><loc_71><loc_46></location>·</text>
<formula><location><page_90><loc_19><loc_21><loc_72><loc_32></location>‖ R L +1 ,Junk ‖ L 2 G /lessorsimilar ε 2 a 1 -cσ √ E ( ≤ L ) (Ric , · ) + εa -3 -c √ ε √ E ( ≤ L -1) (Ric , · ) + εa 1 -cσ √ E ( ≤ L +3) (Σ , · ) + a -3 -c √ ε √ E ( ≤ L +1) (Σ , · ) + a -3 -c √ ε √ E ( ≤ L +1) ( N, · )</formula>
<formula><location><page_90><loc_13><loc_6><loc_87><loc_20></location>‖ E ‖ L,Junk ‖ L 2 G + ‖ B ‖ L,Junk ‖ L 2 G /lessorsimilar εa -1 -c √ ε √ E ( ≤ L ) ( W, · ) + εa -3 -c √ ε √ E ( ≤ L -2) ( W, · ) + εa -1 -c √ ε √ E ( L ) ( φ, · ) + ( εa -3 -c √ ε + a -1 -c √ ε ) √ E ( ≤ L -2) ( φ, · ) + √ εa 1 -c √ ε √ E ( ≤ L ) ( N, · ) + εa -3 -c √ ε √ E ( ≤ L -2) ( N, · ) + εa -1 -cσ √ E ( L ) (Σ , · ) + εa -3 -c √ ε √ E ( ≤ L -2) (Σ , · )</formula>
<formula><location><page_91><loc_38><loc_83><loc_78><loc_91></location>+ εa -3 √ E ( L -2) (Ric , · ) + εa -3 -c √ ε √ E ( ≤ L -4) (Ric , · ) ︸ ︷︷ ︸ not present for L =2</formula>
<text><location><page_91><loc_12><loc_70><loc_88><loc_77></location>Recognizing that every low order curvature term can be estimated up to constant by a -c √ ε at worst (see (4.4f)), we also note that any of the highly nonlinear curvature terms in J -expressions turn out to be negligible after updating c compared to Ricci energies arising from applying Lemma 4.5 or compared to junk terms in which Ric[ G ] is tracked explicitly. /square</text>
<text><location><page_91><loc_12><loc_75><loc_88><loc_84></location>Proof. Once again, this follows by applying the a priori estimates from subsection 4.1 and Lemma 4.3, as well as the bootstrap assumption (3.17h) for the lapse, to deal with the lower order terms in the nonlinearities, and then applying Lemma 4.5 as well as (3.8) whereever this is necessary. Further, especially in (11.19n), it is often more convenient to use the bootstrap assumption for ‖∇ φ ‖ C G instead of the a priori estimate (4.4e) to gain higher powers of ε in prefactors.</text>
<section_header_level_1><location><page_91><loc_36><loc_66><loc_65><loc_68></location>12. Appendix - Future stability</section_header_level_1>
<text><location><page_91><loc_12><loc_62><loc_88><loc_65></location>Here, we collect the commutators in CMCSH gauge necessary to study the commuted scalarfield equations:</text>
<text><location><page_91><loc_12><loc_58><loc_88><loc_61></location>Lemma 12.1 (Commutator formulas for future stabilty) . Let ζ be a scalar function on Σ T . Then, the following formulas hold:</text>
<text><location><page_91><loc_12><loc_50><loc_41><loc_52></location>Schematically, for k ∈ N , this implies</text>
<formula><location><page_91><loc_12><loc_48><loc_17><loc_50></location>(12.1a)</formula>
<formula><location><page_91><loc_12><loc_41><loc_69><loc_47></location>[ ˜ ∂ 0 , ∇ ∆ k g ] ζ = ∑ I n + I Σ + I ζ =2 k ∇ I n n ∗ g ∇ I Σ Σ ∗ g ∇ I ζ +1 ζ (12.1b)</formula>
<formula><location><page_91><loc_31><loc_45><loc_71><loc_51></location>[ ˜ ∂ 0 , ∆ k g ] ζ = ∑ I n + I Σ + I ζ =2 k -1 ∇ I n n ∗ g ∇ I Σ Σ ∗ g ∇ I ζ +1 ζ</formula>
<text><location><page_91><loc_12><loc_34><loc_88><loc_37></location>Proof. This follows from straightfoward computations, similar to Lemma 11.6 for the low order commutators and to Lemma 11.7 for higher orders. /square</text>
<section_header_level_1><location><page_91><loc_45><loc_31><loc_55><loc_32></location>References</section_header_level_1>
<text><location><page_91><loc_12><loc_27><loc_88><loc_30></location>[AF20] Lars Andersson and David Fajman. Nonlinear stability of the Milne model with matter. Comm. Math. Phys., 378(1):261-298, 2020.</text>
<text><location><page_91><loc_12><loc_24><loc_88><loc_27></location>[AFF19] Artur Alho, Grigorios Fournodavlos, and Anne T. Franzen. The wave equation near flat Friedmann- Lemaˆıtre-Robertson-Walker and Kasner Big Bang singularities. J. Hyperbolic Differ. Equ., 16(2):379-</text>
<text><location><page_91><loc_20><loc_23><loc_27><loc_24></location>400, 2019.</text>
<text><location><page_91><loc_12><loc_22><loc_88><loc_23></location>[AM03] Lars Andersson and Vincent Moncrief. Elliptic-hyperbolic systems and the Einstein equations. Ann.</text>
<text><location><page_91><loc_20><loc_20><loc_41><loc_21></location>Henri Poincar´e, 4(1):1-34, 2003.</text>
<text><location><page_91><loc_12><loc_16><loc_88><loc_20></location>[AM04] Lars Andersson and Vincent Moncrief. Future complete vacuum spacetimes. In Piotr T. Chru´sciel and Helmut Friedrich, editors, The Einstein Equations and the Large Scale Behavior of Gravitational Fields, pages 299-330, Basel, 2004. Birkh¨auser Basel.</text>
<text><location><page_91><loc_12><loc_15><loc_17><loc_16></location>[AM11]</text>
<text><location><page_91><loc_20><loc_13><loc_88><loc_16></location>Lars Andersson and Vincent Moncrief. Einstein spaces as attractors for the Einstein flow. J. Differential Geom., 89(1):1-47, 2011.</text>
<text><location><page_91><loc_12><loc_12><loc_88><loc_13></location>[AMT97] Lars Andersson, Vincent Moncrief, and Anthony J. Tromba. On the global evolution problem in 2 + 1</text>
<text><location><page_91><loc_20><loc_11><loc_51><loc_12></location>gravity. J. Geom. Phys., 23(3-4):191-205, 1997.</text>
<text><location><page_91><loc_12><loc_8><loc_88><loc_10></location>[AR01] Lars Andersson and Alan D. Rendall. Quiescent cosmological singularities. Comm. Math. Phys., 218(3):479-511, 2001.</text>
<formula><location><page_91><loc_24><loc_50><loc_76><loc_57></location>[ ˜ ∂ 0 , ∇ ] ζ =0 [ ˜ ∂ 0 , ∆ g ] ζ =( ˜ ∂ 0 ( g -1 ) ab ) ∇ a ∇ b ζ -2( g -1 ) ab (div g ( n Σ ) a -2 ∇ a n ) ∇ b ζ</formula>
<section_header_level_1><location><page_92><loc_42><loc_91><loc_58><loc_92></location>D. FAJMAN, L. URBAN</section_header_level_1>
<table>
<location><page_92><loc_12><loc_8><loc_89><loc_90></location>
</table>
<text><location><page_92><loc_20><loc_8><loc_33><loc_9></location>arXiv:2107.00457v1.</text>
<table>
<location><page_93><loc_12><loc_16><loc_89><loc_90></location>
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</document> | [] |
2019EPJC...79..678C | https://arxiv.org/pdf/1908.02595.pdf | <document>
<section_header_level_1><location><page_1><loc_7><loc_82><loc_73><loc_84></location>Spacetime mappings of the Brown-York quasilocal energy</section_header_level_1>
<text><location><page_1><loc_7><loc_78><loc_54><loc_80></location>Jeremy Cˆot'e a,1 , Marianne Lapierre-L'eonard b,1 , Valerio Faraoni c,1</text>
<text><location><page_1><loc_7><loc_76><loc_76><loc_78></location>1 Department of Physics & Astronomy, Bishop's University, 2600 College Street, Sherbrooke, Qu'ebec, Canada J1M 1Z7</text>
<text><location><page_1><loc_7><loc_66><loc_25><loc_67></location>Received: date / Accepted: date</text>
<text><location><page_1><loc_7><loc_55><loc_48><loc_63></location>Abstract In several areas of theoretical physics it is useful to know how a quasilocal energy transforms under conformal rescalings or generalized Kerr-Schild mappings. We derive the transformation properties of the Brown-York quasilocal energy in spherical symmetry and we contrast them with those of the Misner-Sharp-Hernandez energy.</text>
<text><location><page_1><loc_7><loc_50><loc_44><loc_54></location>Keywords Brown-York quasilocal energy · conformal transformation · Kerr-Schild transformation</text>
<section_header_level_1><location><page_1><loc_7><loc_46><loc_18><loc_47></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_7><loc_15><loc_47><loc_45></location>The mass of a non-isolated system in General Relativity (GR) has been the subject of intense study but there is no agreement as to what the mass-energy should be. Due to the equivalence principle, the energy of the gravitational field cannot be localized and the mass-energy of a self-gravitating system includes also this energy. Unless the geometry reduces asymptotically to Minkowski (in which case the ADM energy [1] is appropriate), one resorts to quasilocal energy definitions. There are several quasilocal constructs in the literature, which differ from each other (see [2] for a recent review). Overall, quasilocal energy has been studied in the domain of formal relativity, but one ought to do better. First, the mass of a gravitating system is one of its most basic properties in astrophysics and a mass-energy definition is ultimately of no use if it cannot be employed in practical calculations (for example, in astrophysics and/or in cosmology). Second, various authors are already using, implicitly, the Hawking-Hayward quasilocal energy [3,4] (usually in its Misner-Sharp-Hernandezform defined for spherical symmetry [5]) in black hole thermodynamics [6], in which this</text>
<text><location><page_1><loc_50><loc_61><loc_90><loc_63></location>quasilocal energy plays the role of the internal energy of the system.</text>
<text><location><page_1><loc_50><loc_31><loc_91><loc_60></location>Black hole thermodynamics (especially the thermodynamics of time-dependent apparent horizons) is usually studied in the context of GR and most often in spherical symmetry, where the Misner-Sharp-Hernandez mass is adopted almost universally [6] (see, however, Ref. [7] for an analogous study using the Brown-York mass). The Misner-SharpHernandezmass is also the quasilocal construct used in spherical fluid dynamics and in black hole collapse [5] and is the Noether charge associated with the covariant conservation of the Kodama energy current [11]. But there are several other definitions of quasilocal energy [2] and one wonders what changes the use of another quasilocal construct, for example the Brown-York energy, would bring. When a black hole is dynamical, it is difficult to calculate its temperature unambiguously and the recent literature on the thermodynamics of dynamical black holes focuses on this quantity. If the definition of internal energy is also uncertain, the problems accumulate. Quasilocal energies have been used also in the now rather broad field of thermodynamics of spacetime [8].</text>
<text><location><page_1><loc_50><loc_10><loc_92><loc_31></location>A full discussion of which quasilocal mass should be used, and why, requires more insight on quasilocal energies than is presently available. Here we consider a particular aspect, more related to tool-building than to core issues, which has been discussed recently in the literature. Since several analytic solutions of the Einstein field equations which describe dynamical black holes are generated by using the Schwarzschild (or another static black hole) solution as a seed and performing a conformal or a KerrSchild transformation [9], the transformation properties of the Misner-Sharp-Hernandezmass under these spacetime mappings were discussed [10]. Later, relinquishing the simplifying assumption of spherical symmetry, the transformation properties of the Hawking-Hayward quasilocal energy [3,</text>
<text><location><page_2><loc_7><loc_72><loc_47><loc_89></location>4] (which reduces to the Misner-Sharp-Hernandez prescription [5] in spherical symmetry [11]) were also obtained [12]. Generalizations of the Hawking-Hayward energy to scalartensor gravity have also been introduced ([13,14], see also [15], and [16] for the case of Lovelock gravity), following earlier generalizations of the Brown-York mass to these theories [17]. A useful trick consists of remembering that these theories admit a representation in the Einstein conformal frame which is formally very similar to GR. If one chooses a different quasilocal energy, it becomes important to establish how this construct transforms under these spacetime mappings.</text>
<text><location><page_2><loc_7><loc_39><loc_48><loc_71></location>There are also other motivations for studying the transformation properties of quasilocal energies. As noted above, these quantities are defined rather formally and are not yet used in practical calculations in astrophysics and cosmology, with the exception of the recent works [18,19,20]. There, the Hawking-Haywardquasilocal construction was employed in a new approach to cosmological problems in which the expansion of the universe competes with the local dynamics of inhomogeneities, namely the Newtonian simulations of large scale structure formation in the early universe [18], the turnaround radius in the present accelerated universe [19], and lensing by the cosmological constant or by dark energy [20]. To first order in the metric perturbations present in these problems, the Brown-York energy yields the same results as the Hawking-Hayward energy, provided that an appropriate gauge is chosen for the gauge-dependent BrownYork energy in the comparison [21]. Conformal transformations were used in these works as a mere calculational tool, not for any conceptual reason. This is one more reason to establish how the Brown-York mass behaves under spacetime mappings, if it was going to replace the HawkingHayward/Misner-Sharp-Hernandez construct.</text>
<text><location><page_2><loc_7><loc_26><loc_47><loc_39></location>In this work we restrict to spherical symmetry and, correspondingly, to a line element expressed in the gauge [22] ds 2 = -A ( t , R ) dt 2 + B ( t , R ) dR 2 + R 2 d Ω 2 ( 2 ) , (1) where R is the areal radius, a well-defined geometric invariant once spherical symmetry is assumed, and d Ω 2 ( 2 ) ≡ d θ 2 + sin 2 θ d ϕ 2 is the line element on the unit 2-sphere. It is well known that, in this gauge , the Brown-York mass is given by [24,25,26]</text>
<formula><location><page_2><loc_7><loc_22><loc_47><loc_26></location>MBY = R ( 1 -1 √ B ) . (2)</formula>
<text><location><page_2><loc_7><loc_13><loc_47><loc_23></location>The general definition of Brown-York mass is based on an integral of the extrinsic curvature of a 3-surface in the real space minus the same quantity evaluated on the same Riemannian surface but with a Riemannian 3-space as a reference [24]. It is clear from the definition that the Brown-York mass is gauge-dependent. By contrast, the Misner-SharpHernandez mass M MSH is given by the scalar equation</text>
<formula><location><page_2><loc_7><loc_10><loc_47><loc_13></location>1 -2 M MSH ( R ) R = ∇ c R ∇ cR (3)</formula>
<text><location><page_2><loc_50><loc_86><loc_90><loc_89></location>and is, therefore, gauge-independent, which is a significant practical advantage over the Brown-York mass.</text>
<text><location><page_2><loc_50><loc_75><loc_90><loc_86></location>Since the Brown-York mass is gauge-dependent,it makes sense to derive its transformation properties under conformal and Kerr-Schild spacetime mappings only when a certain gauge is preserved by the map. This is what we do in the following sections. Since both the expression and the value of the Brown-York mass are very different in different gauges, it is meaningless to compare them in these different gauges.</text>
<section_header_level_1><location><page_2><loc_50><loc_70><loc_71><loc_71></location>2 Conformal transformations</section_header_level_1>
<text><location><page_2><loc_50><loc_66><loc_92><loc_68></location>Aconformaltransformation of the metric is the point-dependent rescaling</text>
<formula><location><page_2><loc_50><loc_63><loc_90><loc_65></location>g ab → ˜ g ab = Ω 2 g ab , (4)</formula>
<text><location><page_2><loc_50><loc_57><loc_90><loc_62></location>where the conformal factor Ω is a smooth positive function of the spacetime point. We require the conformal factor to respect the spherical symmetry, Ω = Ω ( t , R ) . Under such a mapping, the line element (1) becomes</text>
<formula><location><page_2><loc_50><loc_54><loc_83><loc_56></location>d ˜ s 2 = Ω 2 ds 2 = Ω 2 Adt 2 + Ω 2 BdR 2 + ˜ R 2 d Ω 2 2 ,</formula>
<formula><location><page_2><loc_61><loc_53><loc_90><loc_56></location>-( ) (5)</formula>
<text><location><page_2><loc_50><loc_48><loc_90><loc_52></location>where the 'new' areal radius is ˜ R = Ω R . The line element (5) is not in the form (1). To bring it to this form with tilded quantities, i.e. ,</text>
<formula><location><page_2><loc_50><loc_44><loc_90><loc_47></location>d ˜ s 2 = -˜ Ad ˜ t 2 + ˜ Bd ˜ R 2 + ˜ R 2 d Ω 2 ( 2 ) , (6)</formula>
<text><location><page_2><loc_50><loc_41><loc_90><loc_44></location>one has to introduce a new time coordinate. Begin by substituting the differential</text>
<formula><location><page_2><loc_50><loc_38><loc_90><loc_41></location>dR = d ˜ R -Ω , t Rdt Ω , RR + Ω (7)</formula>
<text><location><page_2><loc_50><loc_36><loc_72><loc_37></location>in the line element (1), obtaining</text>
<formula><location><page_2><loc_50><loc_27><loc_90><loc_35></location>d ˜ s 2 = -Ω 2 [ A -Ω 2 , t R 2 B ( Ω , RR + Ω ) 2 ] dt 2 + Ω 2 B ( Ω , RR + Ω ) 2 d ˜ R 2 -2 Ω 2 Ω , t BR ( Ω , RR + Ω ) 2 dtd ˜ R + ˜ R 2 d Ω 2 ( 2 ) . (8)</formula>
<text><location><page_2><loc_50><loc_22><loc_90><loc_26></location>In order to bring this line element back to the gauge (1), one eliminates the term proportional to dt d ˜ R by changing the time coordinate to ˜ t ( t , ˜ R ) defined by</text>
<formula><location><page_2><loc_50><loc_17><loc_90><loc_21></location>d ˜ t = 1 F ( dt + β d ˜ R ) , (9)</formula>
<text><location><page_2><loc_50><loc_14><loc_90><loc_18></location>where β ( t , R ) is a function to be determined in such a way that the dtd ˜ R term disappears and F ( t , ˜ R ) is a (non-unique) integrating factor satisfying</text>
<formula><location><page_2><loc_50><loc_9><loc_90><loc_13></location>∂ ∂ ˜ R ( 1 F ) = ∂ ∂ t ( β F ) (10)</formula>
<text><location><page_3><loc_7><loc_86><loc_47><loc_89></location>to guarantee that d ˜ t is an exact differential. Using dt = Fd ˜ t -β d ˜ R in the line element gives</text>
<formula><location><page_3><loc_7><loc_67><loc_45><loc_86></location>d ˜ s 2 = -Ω 2 [ A -Ω 2 , t R 2 B ( Ω , RR + Ω ) 2 ] F 2 d ˜ t 2 + { -β 2 [ Ω 2 A -Ω 2 , t R 2 Ω 2 B ( Ω , RR + Ω ) 2 ] + Ω 2 B ( Ω , RR + Ω ) 2 + 2 βΩ , t Ω 2 BR ( Ω , RR + Ω ) 2 } d ˜ R 2 + 2 F Ω 2 · · { β [ A -Ω 2 , t R 2 B ( Ω , RR + Ω ) 2 ] -Ω , t RB ( Ω , RR + Ω ) 2 } d ˜ t d ˜ R ˜ 2 2</formula>
<text><location><page_3><loc_12><loc_66><loc_47><loc_68></location>+ R d Ω 2 . (11)</text>
<text><location><page_3><loc_7><loc_65><loc_19><loc_66></location>By imposing that</text>
<formula><location><page_3><loc_7><loc_59><loc_47><loc_64></location>β ( t , R ) = Ω , t BR [ A -Ω 2 , t R 2 B ( Ω , R R + Ω ) 2 ] ( Ω , RR + Ω ) 2 (12)</formula>
<text><location><page_3><loc_7><loc_58><loc_24><loc_59></location>the line element becomes</text>
<formula><location><page_3><loc_7><loc_54><loc_33><loc_57></location>d ˜ s 2 = -Ω 2 [ A -Ω 2 , t BR 2 ( Ω , RR + Ω ) 2 ] F 2 d ˜ t 2</formula>
<formula><location><page_3><loc_12><loc_49><loc_47><loc_53></location>+ AB Ω 2 A ( Ω , RR + Ω ) 2 -Ω 2 , t BR 2 d ˜ R 2 + ˜ R 2 d Ω 2 2 . (13)</formula>
<text><location><page_3><loc_7><loc_48><loc_20><loc_49></location>Therefore, we have</text>
<formula><location><page_3><loc_7><loc_43><loc_47><loc_47></location>˜ B = AB Ω 2 A ( Ω , RR + Ω ) 2 -Ω 2 , t BR 2 . (14)</formula>
<text><location><page_3><loc_7><loc_41><loc_47><loc_43></location>Using this expression, Eq. (2) gives the Brown-York mass in the conformally rescaled world and in the chosen gauge</text>
<formula><location><page_3><loc_7><loc_34><loc_47><loc_40></location>˜ MBY = Ω R 1 -√ A ( Ω , RR + Ω ) 2 -Ω 2 , t BR 2 √ AB Ω . (15)</formula>
<text><location><page_3><loc_7><loc_28><loc_48><loc_35></location>In general, there is no simple expression of the 'new' BrownYork mass in terms of the 'old' one plus a simple correction, analogous to the one previously obtained for the MisnerSharp-Hernandezmass [10]. One could, for example, rewrite Eq. (15) as</text>
<formula><location><page_3><loc_7><loc_26><loc_17><loc_27></location>˜ MBY = Ω MBY</formula>
<formula><location><page_3><loc_11><loc_20><loc_47><loc_24></location>+ R √ AB ( Ω √ A -√ A ( Ω , RR + Ω ) 2 -Ω 2 , t BR 2 ) (16)</formula>
<text><location><page_3><loc_7><loc_13><loc_47><loc_20></location>but this decomposition is arbitrary and not particularly enlightening anyway, even in the simplest situations in which the scale factor depends only on one of the variables ( t , R ) . For comparison, the transformation property of the MisnerSharp-Hernandez mass under a conformal rescaling is [10]</text>
<formula><location><page_3><loc_7><loc_10><loc_47><loc_13></location>˜ M MSH = Ω M MSH -R 3 2 Ω ∇ a Ω∇ a Ω -R 2 ∇ a Ω∇ aR . (17)</formula>
<text><location><page_3><loc_50><loc_75><loc_90><loc_89></location>The first term Ω MBY in Eq. (16) can be interpreted by introducing Newton's constant and remembering that, in a simple intepretation (dating back to Dicke) lengths and times scale with Ω , while masses scale with Ω -1 [28]. However, the quasilocal energy is a complicated construct and cannot be expected to scale in a simple way under conformal transformations. As a consequence, the second term in the right hand side of Eq. (16) defies simple physical interpretation (the same can be said for the transformation property (17) of the Misner-Sharp-Hernandez mass).</text>
<text><location><page_3><loc_50><loc_69><loc_90><loc_74></location>The effect of the transformation (16) on black hole thermodynamics is difficult to interpret. A Smarr relation was derived in Ref. [7] for vacuum, static, spherical black holes of the form (1):</text>
<formula><location><page_3><loc_50><loc_66><loc_90><loc_67></location>2 TS = MBY + 2 pA , (18)</formula>
<text><location><page_3><loc_50><loc_62><loc_90><loc_65></location>where T and S are the temperature and area of the event horizon, S is the entropy, and</text>
<formula><location><page_3><loc_50><loc_58><loc_90><loc_62></location>p = 1 8 π ( A ' H 2 AH √ BH + 1 rH √ BH -1 rH ) , (19)</formula>
<text><location><page_3><loc_50><loc_52><loc_90><loc_58></location>while the subscript H denotes quantities evaluated at the horizon. Assuming that, under a conformal transformation, ˜ T /similarequal T / Ω in an adiabatic approximation (as argued in [29]), S = A / 4, and ˜ A = Ω 2 A , Eq. (19) would yield</text>
<formula><location><page_3><loc_50><loc_47><loc_90><loc_51></location>2 ˜ T ˜ S = ˜ MBY -R √ AB ( Ω √ A -√ A ( Ω , RR + Ω ) 2 -Ω , t BR 2 )</formula>
<formula><location><page_3><loc_55><loc_44><loc_91><loc_47></location>+ 2 p ˜ A Ω (20)</formula>
<text><location><page_3><loc_87><loc_36><loc_87><loc_38></location>/negationslash</text>
<text><location><page_3><loc_50><loc_26><loc_90><loc_44></location>in the tilded world. Not much should be construed from this complicated relation between tilded quantities: a conformal transformation with Ω = Ω ( t , r ) preserving the spherical symmetry has changed the situation in which Eq. (19) was derived [7]. Vacuum is no longer vacuum and, if Ω , t = 0, the black hole is not static, the event horizon is no longer present, and the notion of black hole is now defined by an apparent (instead of event) horizon, which is not null [30]. The time dependence of the (apparent) horizon must be taken into account even in an adiabatic approximation [29]. Therefore, simple statements on the effect of the conformal transformation on thermodunamics cannot be made.</text>
<section_header_level_1><location><page_3><loc_50><loc_22><loc_72><loc_23></location>3 Kerr-Schild transformations</section_header_level_1>
<text><location><page_3><loc_50><loc_19><loc_87><loc_20></location>A generalized Kerr-Schild transformation has the form</text>
<formula><location><page_3><loc_50><loc_16><loc_90><loc_18></location>g ab → ¯ g ab = g ab + 2 λ lal b , (21)</formula>
<text><location><page_3><loc_50><loc_13><loc_90><loc_15></location>where λ is a positive function and l a is a null and geodesic vector field of g ab , that is,</text>
<formula><location><page_3><loc_50><loc_10><loc_90><loc_12></location>g ab l a l b = 0 , l a ∇ al b = 0 . (22)</formula>
<text><location><page_4><loc_7><loc_86><loc_47><loc_89></location>It is easy to see that l a is null and geodesic also with respect to ¯ g ab :</text>
<formula><location><page_4><loc_7><loc_84><loc_47><loc_85></location>¯ g ab l a l b = g ab l a l b + 2 λ ( lcl c ) 2 = 0 , (23)</formula>
<formula><location><page_4><loc_7><loc_81><loc_47><loc_82></location>¯ g ab l a ∇ b l c = 0 , (24)</formula>
<text><location><page_4><loc_7><loc_75><loc_47><loc_80></location>and that the inverse metric of ¯ g ab is ¯ g ab = g ab -2 λ l a l b since ¯ g µν ¯ g να = δ µ α [10]. In order to respect the spherical symmetry of the geometry (1) we require that λ = λ ( t , R ) and</text>
<formula><location><page_4><loc_7><loc_71><loc_47><loc_74></location>l µ ( t , R ) = ( l 0 , l 1 , 0 , 0 ) (25)</formula>
<text><location><page_4><loc_7><loc_69><loc_48><loc_71></location>in this gauge. The generalized Kerr-Schild transformation (21) gives</text>
<formula><location><page_4><loc_7><loc_66><loc_25><loc_68></location>d ¯ s 2 = ds 2 + 2 λ lal b dx a dx b</formula>
<formula><location><page_4><loc_10><loc_61><loc_48><loc_64></location>= -[ A -2 λ ( l 0 ) 2 ] dt 2 + [ B + 2 λ ( l 1 ) 2 ] dR 2 + 4 λ l 0 l 1 dtdR</formula>
<formula><location><page_4><loc_12><loc_59><loc_48><loc_61></location>+ R 2 d Ω 2 ( 2 ) . (26)</formula>
<text><location><page_4><loc_7><loc_54><loc_47><loc_58></location>We now repeat the procedure of Ref. [10] in order to eliminate the cross-term in dtdR . To this end, it is necessary to introduce a new time coordinate T defined by</text>
<formula><location><page_4><loc_7><loc_50><loc_47><loc_53></location>dT = 1 F ( dt + β dR ) , (27)</formula>
<text><location><page_4><loc_7><loc_45><loc_47><loc_49></location>where β ( t , R ) is a function to be determined and F ( t , R ) is an integrating factor. The substitution of dt = FdT -β dR into the line element yields</text>
<formula><location><page_4><loc_7><loc_40><loc_27><loc_44></location>d ¯ s 2 = -[ A -2 λ ( l 0 ) 2 ] F 2 dT 2</formula>
<formula><location><page_4><loc_12><loc_37><loc_46><loc_40></location>+ { B + 2 λ ( l 1 ) 2 -β 2 [ A -2 λ ( l 0 ) 2 ] -4 λ l 0 l 1 β } dR 2</formula>
<formula><location><page_4><loc_12><loc_33><loc_47><loc_37></location>+ 2 F { β [ A -2 λ ( l 0 ) 2 ] + 2 λ l 0 l 1 } dTdR + R 2 d Ω 2 ( 2 ) , (28)</formula>
<text><location><page_4><loc_7><loc_29><loc_47><loc_32></location>from which one deduces that the required form of the function β is</text>
<formula><location><page_4><loc_7><loc_25><loc_47><loc_28></location>β ( t , R ) = -2 λ l 0 l 1 A -2 λ ( l 0 ) 2 . (29)</formula>
<text><location><page_4><loc_7><loc_23><loc_47><loc_24></location>With this choice, the metric is brought back to the gauge (1),</text>
<formula><location><page_4><loc_7><loc_19><loc_47><loc_22></location>d ¯ s 2 = -[ A -2 λ ( l 0 ) 2 ] F 2 dT 2 (30)</formula>
<formula><location><page_4><loc_12><loc_15><loc_46><loc_19></location>+ { B + 2 λ ( l 1 ) 2 + 4 λ 2 ( l 0 ) 2 ( l 1 ) 2 A -2 λ ( l 0 ) 2 } dR 2 + R 2 d Ω 2 ( 2 ) ,</formula>
<text><location><page_4><loc_7><loc_14><loc_20><loc_15></location>where we note that</text>
<formula><location><page_4><loc_7><loc_9><loc_47><loc_13></location>¯ B = B + 2 λ A ( l 1 ) 2 A -2 λ ( l 0 ) 2 (31)</formula>
<text><location><page_4><loc_50><loc_85><loc_91><loc_89></location>and there is residual gauge freedom due to the non-uniqueness of the integrating factor F . The Brown-York mass of the barred spacetime is then given by the expression (2) as</text>
<formula><location><page_4><loc_50><loc_77><loc_90><loc_84></location>¯ MBY = R 1 -1 √ B + 2 λ A ( l 1 ) 2 A -2 λ ( l 0 ) 2 . (32)</formula>
<text><location><page_4><loc_50><loc_73><loc_90><loc_77></location>Because of the normalization lcl c = 0 of the null vector l a , it is possible to rescale its components so that, say, l 0 = -1. Then</text>
<formula><location><page_4><loc_50><loc_69><loc_90><loc_73></location>l µ = ( 1 A , ± 1 √ AB , 0 , 0 ) (33)</formula>
<text><location><page_4><loc_50><loc_68><loc_67><loc_69></location>and Eq. (32) simplifies to</text>
<formula><location><page_4><loc_50><loc_59><loc_90><loc_68></location>¯ MBY = R ( 1 -√ A -2 λ √ AB ) = MBY + R √ B ( 1 -√ 1 -2 λ A ) . (34)</formula>
<text><location><page_4><loc_50><loc_54><loc_90><loc_58></location>For comparison, the transformation property of the MisnerSharp-Hernandezmass under a generalized Kerr-Schild map is [10]</text>
<formula><location><page_4><loc_50><loc_51><loc_90><loc_53></location>¯ M MSH = M MSH + λ R AB . (35)</formula>
<text><location><page_4><loc_50><loc_44><loc_90><loc_50></location>Again, the action of the Kerr-Schild transformation arising from a nonvanishing λ cannot be given a simple interpretation due to the fact that quasilocal energies are rather complicated constructs.</text>
<section_header_level_1><location><page_4><loc_50><loc_40><loc_58><loc_41></location>4 Examples</section_header_level_1>
<text><location><page_4><loc_50><loc_36><loc_91><loc_38></location>Here we present examples illustrating the transformation properties of the Brown-York mass.</text>
<section_header_level_1><location><page_4><loc_50><loc_31><loc_70><loc_33></location>4.1 Conformal transformation</section_header_level_1>
<text><location><page_4><loc_50><loc_29><loc_89><loc_30></location>Consider the Minkowski line element in polar coordinates</text>
<formula><location><page_4><loc_50><loc_25><loc_90><loc_28></location>ds 2 = -d η 2 + dr 2 + r 2 d Ω 2 ( 2 ) , (36)</formula>
<text><location><page_4><loc_50><loc_23><loc_90><loc_25></location>where r = R is trivially the areal radius. The spatially flat Friedmann-Lemaˆıtre-Robertson-Walke(FLRW) line element</text>
<formula><location><page_4><loc_50><loc_18><loc_90><loc_22></location>d ˜ s 2 = a 2 ( η ) ( -d η 2 + dr 2 + r 2 d Ω 2 ( 2 ) ) (37)</formula>
<text><location><page_4><loc_50><loc_9><loc_90><loc_19></location>where η is the conformal time, is manifestly conformally flat. The conformal transformation d ˜ s 2 = Ω 2 ds 2 relating (36) and (37) has conformal factor Ω = a ( η ) , the scale factor of the universe. The FLRW areal radius is ˜ R = a ( η ) r and the Hubble parameters in comoving time t (given by dt = ad η ) and conformal time η are, respectively, H ≡ ˙ a / a and H =</text>
<text><location><page_5><loc_7><loc_86><loc_47><loc_89></location>a η / a = aH (an overdot denoting differentiation with respect to t ).</text>
<text><location><page_5><loc_7><loc_83><loc_47><loc_86></location>To write the FLRW line element (37) in Schwarzschildlike coordinates, we introduce the new time T by</text>
<formula><location><page_5><loc_7><loc_79><loc_47><loc_83></location>dT = 1 F ( dt + β d ˜ R ) , (38) which transforms (37) to</formula>
<formula><location><page_5><loc_7><loc_75><loc_26><loc_79></location>d ˜ s 2 = -( 1 -H 2 ˜ R 2 ) F 2 dT 2</formula>
<formula><location><page_5><loc_12><loc_71><loc_36><loc_75></location>-2 F [ -( 1 -H 2 ˜ R 2 ) β + H ˜ R ] dTd ˜ R</formula>
<formula><location><page_5><loc_12><loc_68><loc_47><loc_71></location>+ [ -( 1 -H 2 ˜ R 2 ) β 2 + 2 H ˜ R β + 1 ] d ˜ R 2 + ˜ R 2 d Ω 2 ( 2 ) . (39)</formula>
<text><location><page_5><loc_7><loc_66><loc_14><loc_67></location>The choice</text>
<formula><location><page_5><loc_7><loc_62><loc_47><loc_66></location>β ( t , ˜ R ) = H ˜ R 1 -H 2 ˜ R 2 (40)</formula>
<text><location><page_5><loc_7><loc_57><loc_48><loc_63></location>reduces the line element to the Schwarzschild-like gauge (6) with ˜ A = ( 1 -H 2 ˜ R 2 ) F 2 , ˜ B = ( 1 -H 2 ˜ R 2 ) -1 . As a consequence, the expression (16) of the Brown-York mass in spherical symmetry yields</text>
<formula><location><page_5><loc_7><loc_46><loc_47><loc_57></location>˜ M ( FLRW ) BY = ˜ R ( 1 -√ 1 -H 2 ˜ R 2 ) = ˜ R ( 1 -√ 1 -8 π 3 ρ ˜ R 2 ) = ˜ R 1 -√ 1 -2 M ( FLRW ) MSH ˜ R , (41)</formula>
<text><location><page_5><loc_7><loc_38><loc_49><loc_47></location>where M ( FLRW ) MSH = H 2 ˜ R 2 / 2 = 4 π 3 ρ ˜ R 3 is the Misner-SharpHernandez mass of FLRW space and we used the Friedmann equation H 2 = 8 πρ / 3, with ρ the energy density of the cosmic fluid. The relation (41) between the two quasilocal masses in this example mirrors that holding in the Schwarzschild geometry</text>
<formula><location><page_5><loc_7><loc_34><loc_47><loc_38></location>ds 2 = -( 1 -2 m r ) dt 2 + dr 2 1 -2 m / r + r 2 d Ω 2 ( 2 ) (42)</formula>
<text><location><page_5><loc_7><loc_30><loc_47><loc_35></location>with (constant) mass m . The Misner-Sharp-Hernandez mass contained in a sphere of radius r is M ( Schw ) MSH = m for any value of r > 2 m and the Brown-York mass is</text>
<formula><location><page_5><loc_7><loc_26><loc_47><loc_30></location>M ( Schw ) BY = r ( 1 -√ 1 -2 m r ) , (43)</formula>
<text><location><page_5><loc_7><loc_23><loc_47><loc_26></location>and it asymptotes to m as r → + ∞ . However, on the Schwarzschild event horizon r = 2 m , it is M ( Schw ) BY = 2 M ( Schw ) MSH .</text>
<section_header_level_1><location><page_5><loc_7><loc_19><loc_28><loc_21></location>4.2 Kerr-Schild transformation</section_header_level_1>
<text><location><page_5><loc_7><loc_14><loc_48><loc_18></location>As an example of Kerr-Schild transformation, consider the mapbetween the Minkowski geometry (36) and the ReissnerNordstrom one</text>
<formula><location><page_5><loc_7><loc_9><loc_45><loc_13></location>d ˜ s 2 = -( 1 -2 m r + q 2 r 2 ) dt 2 + dr 2 1 -2 m r + q 2 r 2 + r 2 d Ω 2 ( 2 ) ,</formula>
<text><location><page_5><loc_87><loc_88><loc_90><loc_89></location>(44)</text>
<text><location><page_5><loc_50><loc_85><loc_64><loc_86></location>which corresponds to</text>
<formula><location><page_5><loc_50><loc_81><loc_90><loc_84></location>λ ( t , r ) = m r -q 2 r 2 , l µ =( 1 , -1 , 0 , 0 ) (45)</formula>
<text><location><page_5><loc_50><loc_78><loc_76><loc_80></location>and to the time redefinition t → T with</text>
<formula><location><page_5><loc_50><loc_73><loc_90><loc_78></location>dT = dt -( 2 m r + q 2 r 2 ) 1 -2 m r + q 2 r 2 dr . (46)</formula>
<text><location><page_5><loc_50><loc_68><loc_90><loc_72></location>Minkowski space has vanishing Brown-York mass and the transformation property (34) of the Brown-York mass under Kerr-Schild transformations yields</text>
<formula><location><page_5><loc_50><loc_63><loc_90><loc_67></location>¯ MBY = r ( 1 -√ 1 -2 m r + q 2 r 2 ) , (47)</formula>
<text><location><page_5><loc_50><loc_60><loc_90><loc_62></location>which coincides with the well known expression calculated directly using Eq. (2).</text>
<section_header_level_1><location><page_5><loc_50><loc_54><loc_60><loc_55></location>5 Conclusions</section_header_level_1>
<text><location><page_5><loc_50><loc_17><loc_90><loc_52></location>There are several reasons to derive the transformation properties of a quasilocal energy under conformal or (generalized) Kerr-Schild transformations. This procedure is part of the tool-building process useful in various areas of theoretical gravity (black hole thermodynamics, analytical solutions of GR describing dynamical black holes, spacetime thermodynamics, etc. ). The relativity community seems to have concentrated on the Hawking-Hayward/Misner-SharpHernandez quasilocal energy (see, however, Refs. [26,27, 7]) but the Brown-York energy is also interesting in principle because it is based on the Hamilton-Jacobi formulation of GR. However, contrary to the Misner-Sharp-Hernandez mass, the Brown-York constructs suffers from a daunting gauge-dependence even in spherical symmetry. For this reason, the comparison of the 'new' Brown-York energy after a spacetime mapping with the 'old' one is meaningful only after restoring the gauge which is altered by the spacetime mapping. Having done this and having obtained the 'new' Brown-York mass in terms of the 'old' one and of the geometry, the result cannot be encapsulated in a simple formula analogous to Eqs. (17) or (35) obtained for the Misner-Sharp-Hernandez mass in [10]. From a pragmatic point of view, the Misner-Sharp-Hernandez construct looks definitely more attractive than the Brown-York one.</text>
<text><location><page_5><loc_50><loc_10><loc_90><loc_15></location>Acknowledgements Weare grateful to a referee for constructing comments. This work is supported, in part, by the Natural Sciences and Engineering Research Council of Canada (Grant No. 2016-03803 to V.F.) and by Bishop's University.</text>
<section_header_level_1><location><page_6><loc_7><loc_88><loc_15><loc_89></location>References</section_header_level_1>
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</document> | [{"title": "Spacetime mappings of the Brown-York quasilocal energy", "content": "Jeremy C\u02c6ot'e a,1 , Marianne Lapierre-L'eonard b,1 , Valerio Faraoni c,1 1 Department of Physics & Astronomy, Bishop's University, 2600 College Street, Sherbrooke, Qu'ebec, Canada J1M 1Z7 Received: date / Accepted: date Abstract In several areas of theoretical physics it is useful to know how a quasilocal energy transforms under conformal rescalings or generalized Kerr-Schild mappings. We derive the transformation properties of the Brown-York quasilocal energy in spherical symmetry and we contrast them with those of the Misner-Sharp-Hernandez energy. Keywords Brown-York quasilocal energy \u00b7 conformal transformation \u00b7 Kerr-Schild transformation", "pages": [1]}, {"title": "1 Introduction", "content": "The mass of a non-isolated system in General Relativity (GR) has been the subject of intense study but there is no agreement as to what the mass-energy should be. Due to the equivalence principle, the energy of the gravitational field cannot be localized and the mass-energy of a self-gravitating system includes also this energy. Unless the geometry reduces asymptotically to Minkowski (in which case the ADM energy [1] is appropriate), one resorts to quasilocal energy definitions. There are several quasilocal constructs in the literature, which differ from each other (see [2] for a recent review). Overall, quasilocal energy has been studied in the domain of formal relativity, but one ought to do better. First, the mass of a gravitating system is one of its most basic properties in astrophysics and a mass-energy definition is ultimately of no use if it cannot be employed in practical calculations (for example, in astrophysics and/or in cosmology). Second, various authors are already using, implicitly, the Hawking-Hayward quasilocal energy [3,4] (usually in its Misner-Sharp-Hernandezform defined for spherical symmetry [5]) in black hole thermodynamics [6], in which this quasilocal energy plays the role of the internal energy of the system. Black hole thermodynamics (especially the thermodynamics of time-dependent apparent horizons) is usually studied in the context of GR and most often in spherical symmetry, where the Misner-Sharp-Hernandez mass is adopted almost universally [6] (see, however, Ref. [7] for an analogous study using the Brown-York mass). The Misner-SharpHernandezmass is also the quasilocal construct used in spherical fluid dynamics and in black hole collapse [5] and is the Noether charge associated with the covariant conservation of the Kodama energy current [11]. But there are several other definitions of quasilocal energy [2] and one wonders what changes the use of another quasilocal construct, for example the Brown-York energy, would bring. When a black hole is dynamical, it is difficult to calculate its temperature unambiguously and the recent literature on the thermodynamics of dynamical black holes focuses on this quantity. If the definition of internal energy is also uncertain, the problems accumulate. Quasilocal energies have been used also in the now rather broad field of thermodynamics of spacetime [8]. A full discussion of which quasilocal mass should be used, and why, requires more insight on quasilocal energies than is presently available. Here we consider a particular aspect, more related to tool-building than to core issues, which has been discussed recently in the literature. Since several analytic solutions of the Einstein field equations which describe dynamical black holes are generated by using the Schwarzschild (or another static black hole) solution as a seed and performing a conformal or a KerrSchild transformation [9], the transformation properties of the Misner-Sharp-Hernandezmass under these spacetime mappings were discussed [10]. Later, relinquishing the simplifying assumption of spherical symmetry, the transformation properties of the Hawking-Hayward quasilocal energy [3, 4] (which reduces to the Misner-Sharp-Hernandez prescription [5] in spherical symmetry [11]) were also obtained [12]. Generalizations of the Hawking-Hayward energy to scalartensor gravity have also been introduced ([13,14], see also [15], and [16] for the case of Lovelock gravity), following earlier generalizations of the Brown-York mass to these theories [17]. A useful trick consists of remembering that these theories admit a representation in the Einstein conformal frame which is formally very similar to GR. If one chooses a different quasilocal energy, it becomes important to establish how this construct transforms under these spacetime mappings. There are also other motivations for studying the transformation properties of quasilocal energies. As noted above, these quantities are defined rather formally and are not yet used in practical calculations in astrophysics and cosmology, with the exception of the recent works [18,19,20]. There, the Hawking-Haywardquasilocal construction was employed in a new approach to cosmological problems in which the expansion of the universe competes with the local dynamics of inhomogeneities, namely the Newtonian simulations of large scale structure formation in the early universe [18], the turnaround radius in the present accelerated universe [19], and lensing by the cosmological constant or by dark energy [20]. To first order in the metric perturbations present in these problems, the Brown-York energy yields the same results as the Hawking-Hayward energy, provided that an appropriate gauge is chosen for the gauge-dependent BrownYork energy in the comparison [21]. Conformal transformations were used in these works as a mere calculational tool, not for any conceptual reason. This is one more reason to establish how the Brown-York mass behaves under spacetime mappings, if it was going to replace the HawkingHayward/Misner-Sharp-Hernandez construct. In this work we restrict to spherical symmetry and, correspondingly, to a line element expressed in the gauge [22] ds 2 = -A ( t , R ) dt 2 + B ( t , R ) dR 2 + R 2 d \u2126 2 ( 2 ) , (1) where R is the areal radius, a well-defined geometric invariant once spherical symmetry is assumed, and d \u2126 2 ( 2 ) \u2261 d \u03b8 2 + sin 2 \u03b8 d \u03d5 2 is the line element on the unit 2-sphere. It is well known that, in this gauge , the Brown-York mass is given by [24,25,26] The general definition of Brown-York mass is based on an integral of the extrinsic curvature of a 3-surface in the real space minus the same quantity evaluated on the same Riemannian surface but with a Riemannian 3-space as a reference [24]. It is clear from the definition that the Brown-York mass is gauge-dependent. By contrast, the Misner-SharpHernandez mass M MSH is given by the scalar equation and is, therefore, gauge-independent, which is a significant practical advantage over the Brown-York mass. Since the Brown-York mass is gauge-dependent,it makes sense to derive its transformation properties under conformal and Kerr-Schild spacetime mappings only when a certain gauge is preserved by the map. This is what we do in the following sections. Since both the expression and the value of the Brown-York mass are very different in different gauges, it is meaningless to compare them in these different gauges.", "pages": [1, 2]}, {"title": "2 Conformal transformations", "content": "Aconformaltransformation of the metric is the point-dependent rescaling where the conformal factor \u2126 is a smooth positive function of the spacetime point. We require the conformal factor to respect the spherical symmetry, \u2126 = \u2126 ( t , R ) . Under such a mapping, the line element (1) becomes where the 'new' areal radius is \u02dc R = \u2126 R . The line element (5) is not in the form (1). To bring it to this form with tilded quantities, i.e. , one has to introduce a new time coordinate. Begin by substituting the differential in the line element (1), obtaining In order to bring this line element back to the gauge (1), one eliminates the term proportional to dt d \u02dc R by changing the time coordinate to \u02dc t ( t , \u02dc R ) defined by where \u03b2 ( t , R ) is a function to be determined in such a way that the dtd \u02dc R term disappears and F ( t , \u02dc R ) is a (non-unique) integrating factor satisfying to guarantee that d \u02dc t is an exact differential. Using dt = Fd \u02dc t -\u03b2 d \u02dc R in the line element gives + R d \u2126 2 . (11) By imposing that the line element becomes Therefore, we have Using this expression, Eq. (2) gives the Brown-York mass in the conformally rescaled world and in the chosen gauge In general, there is no simple expression of the 'new' BrownYork mass in terms of the 'old' one plus a simple correction, analogous to the one previously obtained for the MisnerSharp-Hernandezmass [10]. One could, for example, rewrite Eq. (15) as but this decomposition is arbitrary and not particularly enlightening anyway, even in the simplest situations in which the scale factor depends only on one of the variables ( t , R ) . For comparison, the transformation property of the MisnerSharp-Hernandez mass under a conformal rescaling is [10] The first term \u2126 MBY in Eq. (16) can be interpreted by introducing Newton's constant and remembering that, in a simple intepretation (dating back to Dicke) lengths and times scale with \u2126 , while masses scale with \u2126 -1 [28]. However, the quasilocal energy is a complicated construct and cannot be expected to scale in a simple way under conformal transformations. As a consequence, the second term in the right hand side of Eq. (16) defies simple physical interpretation (the same can be said for the transformation property (17) of the Misner-Sharp-Hernandez mass). The effect of the transformation (16) on black hole thermodynamics is difficult to interpret. A Smarr relation was derived in Ref. [7] for vacuum, static, spherical black holes of the form (1): where T and S are the temperature and area of the event horizon, S is the entropy, and while the subscript H denotes quantities evaluated at the horizon. Assuming that, under a conformal transformation, \u02dc T /similarequal T / \u2126 in an adiabatic approximation (as argued in [29]), S = A / 4, and \u02dc A = \u2126 2 A , Eq. (19) would yield /negationslash in the tilded world. Not much should be construed from this complicated relation between tilded quantities: a conformal transformation with \u2126 = \u2126 ( t , r ) preserving the spherical symmetry has changed the situation in which Eq. (19) was derived [7]. Vacuum is no longer vacuum and, if \u2126 , t = 0, the black hole is not static, the event horizon is no longer present, and the notion of black hole is now defined by an apparent (instead of event) horizon, which is not null [30]. The time dependence of the (apparent) horizon must be taken into account even in an adiabatic approximation [29]. Therefore, simple statements on the effect of the conformal transformation on thermodunamics cannot be made.", "pages": [2, 3]}, {"title": "3 Kerr-Schild transformations", "content": "A generalized Kerr-Schild transformation has the form where \u03bb is a positive function and l a is a null and geodesic vector field of g ab , that is, It is easy to see that l a is null and geodesic also with respect to \u00af g ab : and that the inverse metric of \u00af g ab is \u00af g ab = g ab -2 \u03bb l a l b since \u00af g \u00b5\u03bd \u00af g \u03bd\u03b1 = \u03b4 \u00b5 \u03b1 [10]. In order to respect the spherical symmetry of the geometry (1) we require that \u03bb = \u03bb ( t , R ) and in this gauge. The generalized Kerr-Schild transformation (21) gives We now repeat the procedure of Ref. [10] in order to eliminate the cross-term in dtdR . To this end, it is necessary to introduce a new time coordinate T defined by where \u03b2 ( t , R ) is a function to be determined and F ( t , R ) is an integrating factor. The substitution of dt = FdT -\u03b2 dR into the line element yields from which one deduces that the required form of the function \u03b2 is With this choice, the metric is brought back to the gauge (1), where we note that and there is residual gauge freedom due to the non-uniqueness of the integrating factor F . The Brown-York mass of the barred spacetime is then given by the expression (2) as Because of the normalization lcl c = 0 of the null vector l a , it is possible to rescale its components so that, say, l 0 = -1. Then and Eq. (32) simplifies to For comparison, the transformation property of the MisnerSharp-Hernandezmass under a generalized Kerr-Schild map is [10] Again, the action of the Kerr-Schild transformation arising from a nonvanishing \u03bb cannot be given a simple interpretation due to the fact that quasilocal energies are rather complicated constructs.", "pages": [3, 4]}, {"title": "4 Examples", "content": "Here we present examples illustrating the transformation properties of the Brown-York mass.", "pages": [4]}, {"title": "4.1 Conformal transformation", "content": "Consider the Minkowski line element in polar coordinates where r = R is trivially the areal radius. The spatially flat Friedmann-Lema\u02c6\u0131tre-Robertson-Walke(FLRW) line element where \u03b7 is the conformal time, is manifestly conformally flat. The conformal transformation d \u02dc s 2 = \u2126 2 ds 2 relating (36) and (37) has conformal factor \u2126 = a ( \u03b7 ) , the scale factor of the universe. The FLRW areal radius is \u02dc R = a ( \u03b7 ) r and the Hubble parameters in comoving time t (given by dt = ad \u03b7 ) and conformal time \u03b7 are, respectively, H \u2261 \u02d9 a / a and H = a \u03b7 / a = aH (an overdot denoting differentiation with respect to t ). To write the FLRW line element (37) in Schwarzschildlike coordinates, we introduce the new time T by The choice reduces the line element to the Schwarzschild-like gauge (6) with \u02dc A = ( 1 -H 2 \u02dc R 2 ) F 2 , \u02dc B = ( 1 -H 2 \u02dc R 2 ) -1 . As a consequence, the expression (16) of the Brown-York mass in spherical symmetry yields where M ( FLRW ) MSH = H 2 \u02dc R 2 / 2 = 4 \u03c0 3 \u03c1 \u02dc R 3 is the Misner-SharpHernandez mass of FLRW space and we used the Friedmann equation H 2 = 8 \u03c0\u03c1 / 3, with \u03c1 the energy density of the cosmic fluid. The relation (41) between the two quasilocal masses in this example mirrors that holding in the Schwarzschild geometry with (constant) mass m . The Misner-Sharp-Hernandez mass contained in a sphere of radius r is M ( Schw ) MSH = m for any value of r > 2 m and the Brown-York mass is and it asymptotes to m as r \u2192 + \u221e . However, on the Schwarzschild event horizon r = 2 m , it is M ( Schw ) BY = 2 M ( Schw ) MSH .", "pages": [4, 5]}, {"title": "4.2 Kerr-Schild transformation", "content": "As an example of Kerr-Schild transformation, consider the mapbetween the Minkowski geometry (36) and the ReissnerNordstrom one (44) which corresponds to and to the time redefinition t \u2192 T with Minkowski space has vanishing Brown-York mass and the transformation property (34) of the Brown-York mass under Kerr-Schild transformations yields which coincides with the well known expression calculated directly using Eq. (2).", "pages": [5]}, {"title": "5 Conclusions", "content": "There are several reasons to derive the transformation properties of a quasilocal energy under conformal or (generalized) Kerr-Schild transformations. This procedure is part of the tool-building process useful in various areas of theoretical gravity (black hole thermodynamics, analytical solutions of GR describing dynamical black holes, spacetime thermodynamics, etc. ). The relativity community seems to have concentrated on the Hawking-Hayward/Misner-SharpHernandez quasilocal energy (see, however, Refs. [26,27, 7]) but the Brown-York energy is also interesting in principle because it is based on the Hamilton-Jacobi formulation of GR. However, contrary to the Misner-Sharp-Hernandez mass, the Brown-York constructs suffers from a daunting gauge-dependence even in spherical symmetry. For this reason, the comparison of the 'new' Brown-York energy after a spacetime mapping with the 'old' one is meaningful only after restoring the gauge which is altered by the spacetime mapping. Having done this and having obtained the 'new' Brown-York mass in terms of the 'old' one and of the geometry, the result cannot be encapsulated in a simple formula analogous to Eqs. (17) or (35) obtained for the Misner-Sharp-Hernandez mass in [10]. From a pragmatic point of view, the Misner-Sharp-Hernandez construct looks definitely more attractive than the Brown-York one. Acknowledgements Weare grateful to a referee for constructing comments. This work is supported, in part, by the Natural Sciences and Engineering Research Council of Canada (Grant No. 2016-03803 to V.F.) and by Bishop's University.", "pages": [5]}] |
2022IJMPA..3750065M | https://arxiv.org/pdf/2108.05723.pdf | <document>
<text><location><page_1><loc_28><loc_84><loc_72><loc_87></location>Dyonic black holes in nonlinear electrodynamics from Kaluza-Klein theory with a Gauss-Bonnet term</text>
<section_header_level_1><location><page_1><loc_45><loc_75><loc_55><loc_76></location>S. Mignemi †</section_header_level_1>
<text><location><page_1><loc_32><loc_69><loc_68><loc_74></location>Dipartimento di Matematica, Universit'a di Cagliari via Ospedale 72, 09124 Cagliari, Italy and</text>
<text><location><page_1><loc_32><loc_66><loc_68><loc_69></location>INFN, Sezione di Cagliari, Cittadella Universitaria, 09042 Monserrato, Italy</text>
<section_header_level_1><location><page_1><loc_46><loc_55><loc_54><loc_56></location>Abstract</section_header_level_1>
<text><location><page_1><loc_12><loc_43><loc_88><loc_54></location>Five-dimensional Kaluza-Klein theory with an Einstein-Gauss-Bonnet Lagrangian induces nonlinear corrections to the four-dimensional Maxwell equations, which however remain second order. Although these corrections do not have effect on the purely electric or magnetic monopole solutions for pointlike charges, they affect the dyonic solutions, smoothing the electric field at the origin for positive values of the GaussBonnet coupling constant. We investigate these solutions in flat space, and then extend them in the presence of a minimal coupling to gravity, obtaining exact charged black hole solutions that generalize the ReissnerNordstrom metric.</text>
<section_header_level_1><location><page_2><loc_12><loc_89><loc_24><loc_91></location>1. Introduction</section_header_level_1>
<text><location><page_2><loc_12><loc_81><loc_88><loc_88></location>It is well known that Maxwell equations can be generalized in a non-linear way, adding to the Lagrangian higher powers of the invariants constructed from the electromagnetic field. Well-known examples are the corrections due to quantum electrodynamics that were proposed by Heisenberg and Euler [1] or the highly non-linear Born-Infeld Lagrangian [2] and their generalizations by Pleba'nski [3]. These generalizations still yield second order field equations, but can give rise to solutions with regular electric or magnetic field [3].</text>
<text><location><page_2><loc_12><loc_73><loc_88><loc_80></location>Nonminimal coupling of the Maxwell equations to the gravitational field is instead more difficult, if one requires that the field equations remain second order and linear in the second derivatives of the electromagnetic potential and of the metric tensor. This problem has been studied in general in [4]. It is notable that a simple example of a model obeying this property can be obtained by dimensional reduction of a Kaluza-Klein (KK) theory containing a Gauss-Bonnet (GB) contribution [5-8].</text>
<text><location><page_2><loc_12><loc_62><loc_88><loc_73></location>We recall that KK theories [9,10] provide a unification of general relativity with electromagnetism based on the assumption that spacetime is five-dimensional and the fifth dimension is not observable because it is curled in an extremely small circle. However, in higher dimensions the Einstein-Hilbert Lagrangian is not unique, and one may add to it a GB term, which would not be effective in four dimension, since in that case it reduces to a total derivative. GB terms were shown in [11] to give the most general corrections to higher-dimensional gravity leading to second order field equations and compatible with some natural assumptions.</text>
<text><location><page_2><loc_12><loc_45><loc_88><loc_62></location>The introduction of this term in the five-dimensional Lagrangian permits to obtain by dimensional reduction to four dimensions a model whose predictions differ from those of the Einstein-Maxwell (EM) theory, giving rise to the possibility of an indirect evidence of the existence of a fifth dimension. In fact, the dimensionally reduced theory contains corrections to the EM coupling that are of the kind discussed in [4]. Moreover, they provide nonlinear modifications of the pure electromagnetic lagrangian, that give rise to corrections of the standard electrodynamics [6]. Although these corrections can be considered as a special case of Pleba'nski's nonlinear electrodynamics [3], and in particular of its simplified version proposed in [12], their properties are rather peculiar, due to the particular combination of coefficients in the Lagrangian. For example, the purely electric or magnetic solutions of the Maxwell equations are not modified. It follows that, although regular solutions can be obtained for more general quadratic electrodynamics coupled to gravity [13-15], this is not the case for this model.</text>
<text><location><page_2><loc_12><loc_36><loc_88><loc_45></location>These facts are particularly relevant in relation with uniqueness and no-hair theorems for black holes. These theorems state that the only spherically symmetric asymptotically flat solution of the EM theory is the RN metric [16]. However, if nonlinear electromagnetic terms, like those of Born-Infeld [17] or Pleba'nski [13-14], or extra fields with nonminimal coupling, like the dilaton [18-19], are added to the standard EM action, the theory will exhibit different solutions. Also the generalization to Yang-Mills fields can give rise to nontrivial solutions [20].</text>
<text><location><page_2><loc_12><loc_22><loc_88><loc_36></location>In fact, the solutions of the five-dimensional Einstein-GB theory have been studied from a higherdimensional point of view in ref. [8], where it was shown that the effect of the GB term is only detectable through the coupling of electrodynamics to the gravitational field. However, the case of dyonic solutions was disregarded in that paper. As we shall see, dyonic solutions of the standard Maxwell equation are modified, even in the absence of the gravitational field, due to the nonlinear terms present in the field equations. Dyons were introduced in ref. [21] and have found many application in grand unified theories. Especially interesting are also their implications on the properties of charged black holes, in particular in relation with uniqueness and no-hair theorems. The coupling of our dyonic solution with gravity of course modifies the standard RN black holes, giving another example of the failure of the uniqueness theorems in case of nontrivial couplings.</text>
<text><location><page_2><loc_12><loc_14><loc_88><loc_22></location>In this paper, we describe exact flat-space dyonic solutions of the nonlinear Maxwell equations derived from GB-KK theory and show that solutions with everywhere regular electric field are possible. We then investigate the effect of these configurations on the black hole solutions of general relativity if the electromagnetic field is minimally coupled, obtaining an exact solution. We briefly discuss its properties and thermodynamical parameters.</text>
<text><location><page_2><loc_12><loc_10><loc_88><loc_14></location>However, we shall not consider the nonminimal couplings with the gravitational field arising from the dimensional reduction of the GB Lagrangian, since this problem is more involved and will therefore be studied separately [22].</text>
<section_header_level_1><location><page_3><loc_12><loc_89><loc_41><loc_91></location>2. The dyonic solution in flat space</section_header_level_1>
<text><location><page_3><loc_15><loc_87><loc_68><loc_89></location>We consider a five-dimensional Einstein-Gauss-Bonnet theory, with action</text>
<formula><location><page_3><loc_41><loc_82><loc_88><loc_86></location>I = ∫ √ -g d 5 x ( R + αS ) , (1)</formula>
<text><location><page_3><loc_12><loc_78><loc_88><loc_82></location>where α is a coupling constant of dimension (lenght) 2 , R is the Ricci scalar and S the Gauss-Bonnet term, S = R µνρσ R µνρσ -4 R µν R µν + R 2 .</text>
<text><location><page_3><loc_15><loc_78><loc_36><loc_79></location>We use the simple ansatz † [5]</text>
<formula><location><page_3><loc_39><loc_72><loc_88><loc_76></location>g µν = ( g ij + g 2 A i A j gA i gA j 1 ) , (2)</formula>
<text><location><page_3><loc_12><loc_69><loc_88><loc_72></location>where A i is the Maxwell potential and g a coupling constant. Discarding total derivatives, the action (1) reduces to [1-3]</text>
<formula><location><page_3><loc_21><loc_63><loc_88><loc_68></location>I = ∫ √ -g d 4 x [ R -g 2 4 F ij F ij + 3 αg 4 16 [ ( F ij F ij ) 2 -2 F ij F jk F kl F li ] -αg 2 2 L int ] , (3)</formula>
<text><location><page_3><loc_12><loc_62><loc_16><loc_63></location>where</text>
<formula><location><page_3><loc_35><loc_59><loc_88><loc_62></location>L int = F ij F kl ( R ijkl -4 R ik δ jl + Rδ ik δ jl ) , (4)</formula>
<text><location><page_3><loc_12><loc_57><loc_88><loc_59></location>and F ij = ∂ i A j -∂ j A i . This model of electromagnetism modified with nonlinear terms has been previously considered in ref. [4]. The Einstein-Maxwell coupling (4) has also been investigated in ref. [8].</text>
<text><location><page_3><loc_12><loc_52><loc_88><loc_56></location>In this section we shall consider only the electromagnetic field in flat spacetime, neglecting gravity, since we are mainly interested in the nonlinear modifications of the Maxwell theory. The solutions of the EM theory (3) will be discussed in the following section.</text>
<text><location><page_3><loc_15><loc_51><loc_60><loc_52></location>The electromagnetic sector of the action (3) then reduces to [2]</text>
<formula><location><page_3><loc_26><loc_45><loc_88><loc_49></location>I em = ∫ d 4 x ( -g 2 4 F ij F ij + 3 αg 4 16 [ ( F ij F ij ) 2 -2 F ij F jk F kl F li ] ) . (5)</formula>
<text><location><page_3><loc_12><loc_43><loc_41><loc_45></location>The field equations derived from (4) read</text>
<formula><location><page_3><loc_33><loc_38><loc_88><loc_42></location>( 1 -3 αg 2 2 F 2 ) ∂ j F ji +3 αg 2 ∂ j ( F jk F kl F li ) = 0 . (6)</formula>
<text><location><page_3><loc_12><loc_35><loc_88><loc_38></location>and contain derivatives of the potential A i not higher than second order. Of course, the field F ij also satisfies the Bianchi identities, ∂ ( i F jk ) = 0.</text>
<text><location><page_3><loc_12><loc_30><loc_88><loc_35></location>It is easy to see that for purely electric or magnetic solutions the terms coming from the GB correction give no contribution [2,4], in contrast with most nonlinear models of electrodynamics [3]. However, let us consider a spherically symmetric dyonic solution, whose potential is given in spherical coordinates by</text>
<formula><location><page_3><loc_41><loc_27><loc_88><loc_29></location>A = a ( r ) dt + P cos θ dφ. (7)</formula>
<text><location><page_3><loc_12><loc_25><loc_34><loc_26></location>In an orthogonal frame one has</text>
<formula><location><page_3><loc_41><loc_22><loc_88><loc_25></location>F 01 = a ' ( r ) , F 23 = P r 2 , (8)</formula>
<text><location><page_3><loc_12><loc_20><loc_41><loc_21></location>where ' = d/dr . Clearly, F 23 satisfies (6).</text>
<text><location><page_3><loc_12><loc_17><loc_88><loc_20></location>To find the solution for the electric potential a ( r ), it is convenient to write the action in terms of it and perform the variation. After integration on the angular variables, the action is proportional to</text>
<formula><location><page_3><loc_36><loc_12><loc_88><loc_16></location>I em = ∫ r 2 dr [ a ' 2 -P 2 r 4 +3 αg 2 P 2 a ' 2 r 4 ] , (9)</formula>
<text><location><page_4><loc_12><loc_89><loc_28><loc_91></location>and its variation gives</text>
<text><location><page_4><loc_12><loc_84><loc_19><loc_85></location>Therefore,</text>
<text><location><page_4><loc_12><loc_77><loc_88><loc_80></location>with Q an integration constant, that can be identified with the electric charge. The potential can be obtained by integration.</text>
<formula><location><page_4><loc_40><loc_85><loc_88><loc_89></location>[ r 2 ( 1 + 3 αg 2 P 2 r 4 ) a ' ] ' = 0 . (10)</formula>
<formula><location><page_4><loc_33><loc_78><loc_88><loc_84></location>a ' = F 01 = Q r 2 ( 1 + 3 αg 2 P 2 r 4 ) = r 4 r 4 +3 αg 2 P 2 Q r 2 , (11)</formula>
<text><location><page_4><loc_12><loc_71><loc_88><loc_77></location>It follows that in this model the electric field of a point charge is distorted in the presence of a magnetic monopole; in particular, for α < 0 it diverges at r = (3 | α | g 2 P 2 ) 1 / 4 , while for α > 0 it is regular everywhere. However, the magnetic field (7) is still singular at the origin. In the limits P = 0 or Q = 0 one recovers the standard solutions.</text>
<section_header_level_1><location><page_4><loc_12><loc_68><loc_33><loc_69></location>3. The coupling to gravity</section_header_level_1>
<text><location><page_4><loc_12><loc_59><loc_88><loc_67></location>We now introduce gravity, to see the effects of nonlinear electromagnetism on spherically symmetric black hole solutions. However, we neglect the nonminimal EM coupling (4), since in this way we can obtain exact solutions. The inclusion of this term will be investigated elsewhere [22]. Moreover, contrary to ref. [4], we consider the solutions of the four-dimensional effective theory, rather than those of the five-dimensional one, since they admit a more transparent interpretation. In the following, in order to obtain the standard normalization, we shall set g 2 = 4 and define ¯ α = α/ 4, so that αg 2 = ¯ α .</text>
<text><location><page_4><loc_15><loc_57><loc_55><loc_58></location>We seek for spherically symmetric solutions of the form</text>
<formula><location><page_4><loc_38><loc_54><loc_88><loc_56></location>ds 2 = -e 2 ν dt 2 + e 2 µ dr 2 + e 2 ρ d Ω 2 , (12)</formula>
<formula><location><page_4><loc_41><loc_52><loc_88><loc_53></location>A = a ( r ) dt + P cos θ dφ. (13)</formula>
<text><location><page_4><loc_12><loc_48><loc_88><loc_51></location>Like in the flat case, we calculate the field equations by substituting this ansatz into the action and performing the variation. We obtain</text>
<formula><location><page_4><loc_18><loc_42><loc_88><loc_47></location>I = 2 ∫ dr [ (2 ν ' ρ ' + ρ ' 2 ) /epsilon1 ν -µ +2 ρ + e ν + µ + a ' 2 e -ν -µ +2 ρ -P 2 e ν + µ -2 ρ -3¯ αP 2 a ' 2 e -µ -ν -2 ρ ] . (14)</formula>
<text><location><page_4><loc_12><loc_41><loc_58><loc_43></location>In the gauge e ρ = r the field equations stemming from (14) read</text>
<formula><location><page_4><loc_30><loc_37><loc_88><loc_40></location>2 ν ' r + 1 r 2 -e 2 µ r 2 + a ' 2 e -2 ν + P 2 r 4 e 2 µ +3¯ αa ' 2 P 2 r 4 e -2 ν = 0 , (15)</formula>
<formula><location><page_4><loc_29><loc_33><loc_88><loc_36></location>-2 µ ' r + 1 r 2 -e 2 µ r 2 + a ' 2 e -2 ν + P 2 r 4 e 2 µ +3¯ αa ' 2 P 2 r 4 e -2 ν = 0 , (16)</formula>
<formula><location><page_4><loc_38><loc_28><loc_88><loc_33></location>[ r 2 e -ν -µ ( 1 + 3¯ α P 2 r 4 ) a ' ] ' = 0 . (17)</formula>
<text><location><page_4><loc_12><loc_27><loc_35><loc_28></location>Combining (15) and (16), we get</text>
<formula><location><page_4><loc_46><loc_25><loc_88><loc_27></location>ν ' + µ ' = 0 (18) .</formula>
<text><location><page_4><loc_12><loc_22><loc_62><loc_24></location>Hence, for asymptotically flat solutions, µ = -ν and, integrating (17),</text>
<formula><location><page_4><loc_44><loc_19><loc_88><loc_22></location>a ' = Qr 2 r 4 +3¯ αP 2 , (19)</formula>
<text><location><page_4><loc_12><loc_16><loc_65><loc_18></location>with Q an integration constant. Substituting in (15), one can rearrange as</text>
<formula><location><page_4><loc_27><loc_12><loc_88><loc_15></location>( re 2 ν ) ' = 1 -P 2 r 2 -Q 2 r 2 r 4 +3¯ αP 2 ≈ 1 -P 2 + Q 2 r 2 -3¯ αP 2 Q 2 r 6 + . . . (20)</formula>
<text><location><page_4><loc_12><loc_10><loc_78><loc_11></location>which displays order-¯ α corrections to the corresponding equation for the RN metric function.</text>
<figure>
<location><page_5><loc_20><loc_77><loc_80><loc_91></location>
<caption>Fig. 1: The metric function e 2 ν for generic (left panel) and near-extremal black holes (right panel). In the right panel, we have chosen values of the parameters that are extremal for the RN black hole. The continuous lines show the RN solution, the dashed line the α> 0 solution (21) and the dotted line the α< 0 solution (31). The singularity at r = r 0 occurring when α< 0 is clearly visible.</caption>
</figure>
<text><location><page_5><loc_12><loc_67><loc_88><loc_71></location>Eq. (20) can be solved exactly. Let us first consider the case α > 0. Setting γ = √ 3¯ αP 2 and choosing suitable boundary conditions, the solution is</text>
<formula><location><page_5><loc_13><loc_60><loc_88><loc_66></location>e 2 ν = 1 -2 M r + P 2 r 2 + Q 2 2 √ 2 γ r [ π +arctan ( 1 -√ 2 r √ γ ) -arctan ( 1 + √ 2 r √ γ ) + 1 2 log r 2 -√ 2 γ r + γ r 2 + √ 2 γ r + γ ] . (21)</formula>
<text><location><page_5><loc_12><loc_54><loc_88><loc_60></location>This metric exhibits some similarity with the so-called geon solution of the Born-Infeld nonlinear electromagnetism coupled to gravity [11]. ‡ For small γ , it gives a slight correction to the RN solution, which however is relevant for the uniqueness theorems. In Fig. 1 some solutions are depicted together with the corresponding RN metric function.</text>
<text><location><page_5><loc_15><loc_53><loc_63><loc_54></location>The asymptotic behavior of (21) reproduces that of the RN metric,</text>
<formula><location><page_5><loc_37><loc_47><loc_88><loc_51></location>e 2 ν = 1 -2 M r + P 2 + Q 2 r 2 + o ( 1 r 3 ) , (22)</formula>
<text><location><page_5><loc_12><loc_43><loc_88><loc_46></location>and one can identify M with the mass, Q and P with the electric and magnetic charge, respectively. Instead for r → 0 the behavior is different from that of RN,</text>
<formula><location><page_5><loc_36><loc_38><loc_88><loc_42></location>e 2 ν ∼ P 2 r 2 -( 2 M -πQ 2 2 √ 2 γ ) 1 r + o (1) . (23)</formula>
<text><location><page_5><loc_12><loc_29><loc_88><loc_37></location>In particular, the 1 r term becomes repulsive near the origin for M < πQ 2 4 √ 2 γ . The departure from the RN behavior are therefore greater for small r . The term proportional to 1 √ γ arises because we have fixed the boundary conditions so that M is the mass of the solution. It can be useful to define an effective mass near the singularity as m = M -πQ 2 4 √ 2 γ .</text>
<text><location><page_5><loc_12><loc_26><loc_88><loc_30></location>The curvature scalar is given by R = 4 γ 2 Q 2 ( r 4 + γ 2 ) 2 and is regular everywhere, but, in contrast with the RN solution does not vanish. However also in our case a curvature singularity occurs at the origin, since</text>
<formula><location><page_5><loc_33><loc_21><loc_88><loc_25></location>R ijkl R ijkl ∼ 56 P 4 r 8 -96 mP 2 r 7 + 48 m 2 r 6 + o ( 1 r 5 ) . (24)</formula>
<text><location><page_5><loc_12><loc_19><loc_58><loc_20></location>The leading order term in (24) depends only on P and not on Q .</text>
<text><location><page_5><loc_12><loc_13><loc_88><loc_18></location>The causal structure depends on the values of the parameters M , P and Q that characterize the solution. Due to the nontrivial form of the metric, a general discussion can be made only numerically. However, if γ /lessmuch 1, as arguable on physical grounds, the solution should not differ much from the RN metric and one can</text>
<text><location><page_6><loc_12><loc_88><loc_88><loc_91></location>resort to a perturbative expansion in γ . However, one must be careful, because this fails at small r , since, as follows from (23), in this regime √ γ appears at the denominator.</text>
<text><location><page_6><loc_15><loc_86><loc_74><loc_88></location>The RN metric is known to exhibit a singularity at the origin and two horizons at</text>
<formula><location><page_6><loc_40><loc_81><loc_88><loc_85></location>˜ r ± = M ± √ M 2 -P 2 -Q 2 . (25)</formula>
<text><location><page_6><loc_12><loc_79><loc_88><loc_82></location>Unfortunately, it is not possible to obtain an exact expression for the location of the horizons of the metric (21). We can obtain an approximation by expanding in the small parameter γ ,</text>
<formula><location><page_6><loc_35><loc_75><loc_88><loc_78></location>e 2 ν = 1 -2 M r + P 2 + Q 2 r 2 -γ 2 Q 2 5 r 6 + o ( γ 4 ) . (26)</formula>
<text><location><page_6><loc_12><loc_70><loc_88><loc_73></location>The leading-order corrections are proportional to γ 2 , and we can compute the zeroes of the metric as r ± = ¯ r ± + γ 2 ∆ r ± + o ( γ ), where</text>
<formula><location><page_6><loc_41><loc_66><loc_88><loc_71></location>∆ r ± = ± Q 2 5 ¯ r 4 ± (¯ r + -¯ r -) . (27)</formula>
<text><location><page_6><loc_12><loc_63><loc_88><loc_66></location>Hence, the two horizons are farther than in the RN case. From (27) follows that at leading order the condition of extremality r + = r -is, recalling that γ 2 = 3¯ αP 2 ,</text>
<formula><location><page_6><loc_39><loc_59><loc_88><loc_62></location>M 2 ≈ Q 2 + P 2 + 3¯ αP 2 Q 2 5( P 2 + Q 2 ) 2 . (28)</formula>
<text><location><page_6><loc_12><loc_52><loc_88><loc_58></location>The causal structure is analogous to that of RN: for M greater than its extremal value, one has two horizons, while for M smaller than the extremal value a naked singularity occurs. Although these results are obtained for small γ , the qualitative behavior of the solution remains the same also for generic values of the coupling constant.</text>
<text><location><page_6><loc_12><loc_49><loc_88><loc_52></location>The thermodynamical quantities can be computed in the standard way: the area of the external horizon is usually identified with the entropy, therefore</text>
<formula><location><page_6><loc_36><loc_44><loc_88><loc_48></location>S = 4 π ( ¯ r 2 + + 6¯ αP 2 Q 2 5 r 3 + (¯ r + -¯ r -) ) + o (¯ α 2 ) , (29)</formula>
<text><location><page_6><loc_12><loc_42><loc_42><loc_43></location>while the temperature can be calculated as</text>
<formula><location><page_6><loc_27><loc_35><loc_88><loc_40></location>T = 1 4 π e 2 ν dr ∣ ∣ ∣ r = r + = 1 4 π ( ¯ r + -¯ r -¯ r 2 + + 6¯ αP 2 Q 2 5¯ r 7 + 2¯ r + -¯ r -¯ r + -¯ r -) + o (¯ α 2 ) . (30)</formula>
<text><location><page_6><loc_12><loc_33><loc_88><loc_36></location>For extremal black holes the temperature vanishes. Both temperature and entropy are increased with respect to the Reissner-Nordstrom black hole.</text>
<text><location><page_6><loc_15><loc_32><loc_74><loc_33></location>Let us briefly comment on the case α < 0. The metric function has a simpler form,</text>
<formula><location><page_6><loc_27><loc_26><loc_88><loc_31></location>e 2 ν = 1 -2 M r + P 2 r 2 + Q 2 2 √ γ r [ π 2 -arctan r √ γ -1 2 log r -√ γ r + √ γ ] , (31)</formula>
<text><location><page_6><loc_12><loc_19><loc_88><loc_26></location>where now γ = √ 3 | ¯ α | P 2 , but has the same asymptotic behavior (22) as the previous solution. Also the expansion for small γ has the same form as (26) except for the sign of the term proportional to γ 2 . The curvature scalar is R = 4 γ 2 Q 2 ( r 4 -γ 2 ) 2 . Now a curvature singularity occurs at the surface r 0 = √ γ , while the horizons are located at</text>
<formula><location><page_6><loc_41><loc_15><loc_88><loc_19></location>r ± ≈ ¯ r ± ± 3¯ αP 2 Q 2 5 ¯ r ± (¯ r + -¯ r -) . (32)</formula>
<text><location><page_6><loc_12><loc_10><loc_88><loc_15></location>If r 0 > r -, a single horizon is present and the causal structure is similar to that of the Schwarzschild solutions. Otherwise, the properties are analogous to those of the solution with positive α and all the previous formulas still hold, taking into account that ¯ α has opposite sign. This is true in particular for the thermodynamical quantities.</text>
<section_header_level_1><location><page_7><loc_12><loc_89><loc_23><loc_91></location>4. Conclusion</section_header_level_1>
<text><location><page_7><loc_12><loc_83><loc_88><loc_89></location>We have considered the effect of the nonlinearity of the electrodynamics induced by a five-dimensional KK model with Einstein-GB lagrangian on the dyonic solutions with a pointlike source. While it is well known that purely electric or magnetic solutions are not modified in this model, we have shown that the dyonic solutions differ from those of the Maxwell theory, and the electric field can be regular everywhere.</text>
<text><location><page_7><loc_12><loc_75><loc_88><loc_83></location>In our model, the field equations contain at most cubic terms in A i , but the model can be generalized to higher powers by increasing the number of the internal dimensions and adding higher-order GB terms. Also in this case the pure electric or magnetic fields of pointlike sources maintain the standard form, but the dyonic solutions are modified and for suitable ranges of values of the coupling constants the singularity of the electric field is suppressed.</text>
<text><location><page_7><loc_12><loc_66><loc_88><loc_75></location>We have also examined the coupling with gravity and have found a new class of solutions that modify the RN metric, with a Maxwell field identical to the flat space solution and a metric that deforms the RN solution. The solutions still depend on the three parameters M , Q and P , but are no longer dual for the interchange of Q and P . For positive α they exhibit a pointlike singularity, while for negative α the singularity is spherical. The horizon structure is similar to that of RN, with two horizon, but for some values of the parameters it can present one or no horizons.</text>
<text><location><page_7><loc_12><loc_62><loc_88><loc_66></location>Our result is notable since it shows that the introduction of nonlinear equations for the electromagnetic fields affects the results of the black hole uniqueness theorems also in case of minimal coupling to gravity, analogously to what happens in more general models [13-15].</text>
<text><location><page_7><loc_12><loc_59><loc_88><loc_62></location>Going to higher dimensions also allows the introduction of Yang-Mills fields through the Kaluza-Klein mechanism. Of course, in this case more complicated solutions are expected.</text>
<text><location><page_7><loc_12><loc_53><loc_88><loc_59></location>In this paper we have not considered the nonminimal coupling between gravity and electromagnetism induced by the KK-GB model. Such coupling spoils the possibility of solving the field equations analytically, but the main properties of the solutions should be preserved. We plan to investigate this topic in a future publication [22].</text>
<section_header_level_1><location><page_7><loc_46><loc_48><loc_54><loc_49></location>References</section_header_level_1>
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<list_item><location><page_7><loc_15><loc_41><loc_78><loc_43></location>[3] J. Pleba'nski, Lectures on non-linear electrodynamics , NORDITA, Copenhagen, 1968.</list_item>
<list_item><location><page_7><loc_15><loc_39><loc_53><loc_40></location>[4] G.W. Horndeski, J. Math. Phys. 17 , 1980 (1976).</list_item>
<list_item><location><page_7><loc_15><loc_37><loc_47><loc_38></location>[5] H.A. Buchdal, J. Phys. A12 , 1037 (1979).</list_item>
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<list_item><location><page_7><loc_15><loc_26><loc_42><loc_27></location>[10] O. Klein, Z. Phys. 37 , 895 (1926).</list_item>
<list_item><location><page_7><loc_15><loc_23><loc_49><loc_25></location>[11] D. Lovelock, J. Math. Phys. 12 , 498 (1971).</list_item>
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<list_item><location><page_7><loc_15><loc_19><loc_63><loc_20></location>[13] R. Pellicer and R. J. Torrence, J. Math. Phys. 19 , 1718 (1969).</list_item>
<list_item><location><page_7><loc_15><loc_16><loc_59><loc_18></location>[14] H.P. de Oliveira, Class. Quantum Grav. 11 , 1469 (1994).</list_item>
<list_item><location><page_7><loc_15><loc_14><loc_82><loc_15></location>[15] A. Garcia, E. Hackmann, C. Lammerzahl and A. Mac'ıas, Phys. Rev. D86 , 024037 (2012).</list_item>
<list_item><location><page_7><loc_15><loc_12><loc_52><loc_13></location>[16] W. Israel, Commun. Math. Phys. 8 , 245 (1968).</list_item>
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</document> | [] |
2013CQGra..30f5021I | https://arxiv.org/pdf/1205.1349.pdf | <document>
<section_header_level_1><location><page_1><loc_21><loc_69><loc_76><loc_74></location>New Compactifications of Eleven Dimensional Supergravity</section_header_level_1>
<section_header_level_1><location><page_1><loc_43><loc_57><loc_54><loc_58></location>Ali Imaanpur</section_header_level_1>
<text><location><page_1><loc_22><loc_49><loc_76><loc_53></location>Department of Physics, School of Sciences Tarbiat Modares University, P.O.Box 14155-4838, Tehran, Iran</text>
<section_header_level_1><location><page_1><loc_45><loc_43><loc_53><loc_44></location>Abstract</section_header_level_1>
<text><location><page_1><loc_19><loc_28><loc_79><loc_41></location>Using canonical forms on S 7 , viewed as an SU (2) bundle over S 4 , we introduce consistent ansatze for the 4-form field strength of eleven-dimensional supergravity and rederive the known squashed, stretched, and the Englert solutions. Further, by rewriting the metric of S 7 as a U (1) bundle over CP 3 , we present yet more general ansatze. As a result, we find a new compactifying solution of the type AdS 5 × CP 3 , where CP 3 is stretched along its S 2 fiber. We also find a new solution of AdS 2 × H 2 × S 7 type in Euclidean space.</text>
<section_header_level_1><location><page_2><loc_14><loc_84><loc_36><loc_86></location>1 Introduction</section_header_level_1>
<text><location><page_2><loc_14><loc_66><loc_84><loc_82></location>Eleven-dimensional supergravity solutions have been extensively studied in 1980's. Among these, the Freund-Rubin solution [1] was the simplest one as it included a 4form field strength with components only along the AdS direction. Then, attentions were turned to possible solutions with nonvanishing components along the compact directions. Englert was the first to construct such a solution; AdS 4 × S 7 with the round metric on S 7 [2]. Later, the so-called squashed solutions with non-standard Einstein metric on S 7 were found [3, 4]. Here, S 7 is considered as an SU (2) bundle over S 4 and squashing corresponds to rescaling the metric along the fiber. For a specific value of the squashing parameter the metric turns out to be Einstein.</text>
<text><location><page_2><loc_14><loc_55><loc_84><loc_66></location>In constructing the Englert type solutions Killing spinors play a significant role. Killing spinors are also required for having supersymmetric solutions [5, 7, 8, 9, 10]. Alternatively, on compact manifolds with a bundle structure on a Kahler base, one can use the holomorphic top form and the Kahler form to write consistent ansatze for the 4-form field strength [11]. Algebraic approaches have also been used to study the supergravity solutions [12].</text>
<text><location><page_2><loc_14><loc_22><loc_84><loc_55></location>In the present work, however, instead of looking for Killing spinors we directly use canonical forms on S 7 to write a consistent ansatz for the 4-form field strength. In particular, this allows us to rederive the squashed, stretched, and the Englert solutions in a unified scheme. There are some independent earlier works which also use canonical geometric methods [13, 14]. In Sec. 2, we consider S 7 as an S 3 bundle over S 4 , and identify a natural basis of such forms in terms of the volume forms of the fiber and the base. We will see that a linear combination of these forms provides a suitable ansatz for the Maxwell equation, so that the field equations reduce to algebraic equations for the parameters of the ansatz. In Sec. 3, we rewrite the squashed metric of S 7 as a U (1) bundle over CP 3 , where it appears as an S 2 bundle over S 4 with rescaled fibers. Moreover, in this form, we can introduce a different rescaling parameter for the U (1) fibers. This enables us to provide more general ansatze. In Sec. 4, we consider a direct product of a 5 and 6-dimensional spaces and find a new compactifying solution of AdS 5 × CP 3 , in which CP 3 is stretched along its S 2 fiber. In Sec. 5, we discuss solutions in which the eleven dimensional space has a Euclidean signature and is a direct product of two 2-dimensional spaces and S 7 . We find a solution of AdS 2 × H 2 × S 7 type, in which H 2 is a hyperbolic surface, and S 7 is stretched along its U (1) fiber by a factor of 2.</text>
<section_header_level_1><location><page_2><loc_14><loc_17><loc_57><loc_19></location>2 Squashed solution revisited</section_header_level_1>
<text><location><page_2><loc_14><loc_12><loc_84><loc_15></location>Let us start our discussion with the Freund-Rubin solution, for which the 4-form field strength has components only along the four dimensions</text>
<formula><location><page_2><loc_44><loc_7><loc_84><loc_11></location>F 4 = 3 8 R 3 /epsilon1 4 , (1)</formula>
<text><location><page_3><loc_14><loc_84><loc_31><loc_86></location>and the metric reads</text>
<formula><location><page_3><loc_38><loc_81><loc_84><loc_84></location>ds 2 = R 2 ( 1 4 ds 2 AdS 4 + ds 2 S 7 ) . (2)</formula>
<text><location><page_3><loc_14><loc_79><loc_77><loc_80></location>The round metric on S 7 can be written as an SU (2) bundle over S 4 [3, 15]</text>
<formula><location><page_3><loc_28><loc_74><loc_84><loc_77></location>ds 2 S 7 = 1 4 ( dµ 2 + 1 4 sin 2 µ Σ 2 i +( σ i -cos 2 µ/ 2 Σ i ) 2 ) , (3)</formula>
<text><location><page_3><loc_14><loc_70><loc_75><loc_72></location>with 0 ≤ µ ≤ π , and Σ i 's and σ i 's are two sets of left-invariant one-forms</text>
<formula><location><page_3><loc_37><loc_63><loc_63><loc_69></location>Σ 1 = cos γ dα +sin γ sin αdβ , Σ 2 = -sin γ dα +cos γ sin αdβ , Σ 3 = dγ +cos αdβ ,</formula>
<text><location><page_3><loc_14><loc_58><loc_84><loc_61></location>where 0 ≤ γ ≤ 4 π, 0 ≤ α ≤ π, 0 ≤ β ≤ 2 π , and with a similar expression for σ i 's. They satisfy the SU (2) algebra</text>
<formula><location><page_3><loc_30><loc_53><loc_84><loc_57></location>d Σ i = -1 2 /epsilon1 ijk Σ j ∧ Σ k , dσ i = -1 2 /epsilon1 ijk σ j ∧ σ k , (4)</formula>
<text><location><page_3><loc_14><loc_50><loc_33><loc_52></location>with i, j, k, . . . = 1 , 2 , 3.</text>
<text><location><page_3><loc_17><loc_49><loc_76><loc_50></location>Squashing corresponds to modifying the round metric on S 7 as follows</text>
<formula><location><page_3><loc_27><loc_44><loc_84><loc_47></location>ds 2 S 7 = 1 4 ( dµ 2 + 1 4 sin 2 µ Σ 2 i + λ 2 ( σ i -cos 2 µ/ 2 Σ i ) 2 ) , (5)</formula>
<text><location><page_3><loc_14><loc_39><loc_84><loc_42></location>with λ the squashing parameter. So, let us take the following ansatz for the 11d metric:</text>
<formula><location><page_3><loc_23><loc_34><loc_84><loc_38></location>ds 2 = R 2 4 ( ds 2 4 + dµ 2 + 1 4 sin 2 µ Σ 2 i + λ 2 ( σ i -cos 2 µ/ 2 Σ i ) 2 ) , (6)</formula>
<text><location><page_3><loc_14><loc_31><loc_55><loc_33></location>and choose the orthonormal basis of vielbeins as</text>
<formula><location><page_3><loc_26><loc_27><loc_84><loc_30></location>e 0 = dµ, e i = 1 2 sin µ Σ i , ˆ e i = λ ( σ i -cos 2 µ/ 2 Σ i ) . (7)</formula>
<text><location><page_3><loc_14><loc_22><loc_84><loc_26></location>Further, in order to construct our ansatz in the next section we need to evaluate the exterior derivatives of the vielbeins</text>
<formula><location><page_3><loc_20><loc_18><loc_84><loc_21></location>de i = cot µe 0 ∧ e i -1 sin µ /epsilon1 ijk e j ∧ e k (8)</formula>
<formula><location><page_3><loc_20><loc_13><loc_84><loc_17></location>d ˆ e i = λe 0 ∧ e i + 1 2 /epsilon1 ijk ( λe j ∧ e k -1 λ ˆ e j ∧ ˆ e k -2 ( 1 + cos µ sin µ ) e j ∧ ˆ e k ) , (9)</formula>
<text><location><page_3><loc_14><loc_10><loc_41><loc_12></location>where use has been made of (4).</text>
<section_header_level_1><location><page_4><loc_14><loc_84><loc_33><loc_86></location>2.1 The ansatz</section_header_level_1>
<text><location><page_4><loc_14><loc_81><loc_65><loc_83></location>Let us now introduce ω 3 , the volume element of the fiber S 3 :</text>
<formula><location><page_4><loc_42><loc_77><loc_84><loc_80></location>ω 3 = ˆ e 1 ∧ ˆ e 2 ∧ ˆ e 3 , (10)</formula>
<text><location><page_4><loc_14><loc_75><loc_58><loc_77></location>taking the derivative along with using (9), we obtain</text>
<formula><location><page_4><loc_31><loc_69><loc_84><loc_74></location>dω 3 = λ 2 ( /epsilon1 ijk e 0 ∧ e i ∧ ˆ e j ∧ ˆ e k + e i ∧ e j ∧ ˆ e i ∧ ˆ e j ) . (11)</formula>
<text><location><page_4><loc_14><loc_68><loc_33><loc_69></location>The Hodge dual reads</text>
<formula><location><page_4><loc_34><loc_63><loc_84><loc_67></location>∗ dω 3 = λ ˆ e i ∧ ( e 0 ∧ e i + 1 2 /epsilon1 ijk e j ∧ e k ) , (12)</formula>
<text><location><page_4><loc_14><loc_61><loc_44><loc_62></location>so that using (8) and (9), we derive</text>
<formula><location><page_4><loc_38><loc_56><loc_84><loc_60></location>d ∗ dω 3 = 6 λ 2 ω 4 -1 λ dω 3 , (13)</formula>
<text><location><page_4><loc_14><loc_54><loc_19><loc_55></location>where</text>
<formula><location><page_4><loc_41><loc_51><loc_84><loc_53></location>ω 4 = e 0 ∧ e 1 ∧ e 2 ∧ e 3 , (14)</formula>
<text><location><page_4><loc_14><loc_46><loc_84><loc_51></location>is the volume element of the base. Note that ω 4 is closed; dω 4 = 0. Further, since d ∗ ω 4 = dω 3 , for a linear combination of these two forms we have</text>
<formula><location><page_4><loc_30><loc_43><loc_84><loc_46></location>d ∗ ( αω 4 + β dω 3 ) = 6 λ 2 β ω 4 +( α -β/λ ) dω 3 (15)</formula>
<text><location><page_4><loc_14><loc_40><loc_84><loc_43></location>namely, the subspace with a basis of ω 4 and dω 3 is closed under d ∗ operation. This is exactly what we need to construct a consistent ansatz for the 4-form field strength.</text>
<text><location><page_4><loc_17><loc_38><loc_71><loc_39></location>The above analysis shows that we can take the following ansatz:</text>
<formula><location><page_4><loc_38><loc_35><loc_84><loc_36></location>F 4 = N/epsilon1 4 + αω 4 + β dω 3 , (16)</formula>
<text><location><page_4><loc_14><loc_30><loc_84><loc_33></location>with N , α , and β constant parameters to be determined by field equations, also note that dF 4 = 0. Substituting this into the field equation 1</text>
<formula><location><page_4><loc_40><loc_25><loc_84><loc_29></location>d ∗ 11 F 4 = -1 2 F 4 ∧ F 4 , (17)</formula>
<text><location><page_4><loc_14><loc_23><loc_19><loc_24></location>we get</text>
<formula><location><page_4><loc_22><loc_18><loc_84><loc_22></location>R 3 8 d ( N ω 3 ∧ ω 4 + α/epsilon1 4 ∧ ω 3 + β/epsilon1 4 ∧ ∗ dω 3 ) = -N/epsilon1 4 ∧ ( αω 4 + βdω 3 ) , (18)</formula>
<text><location><page_4><loc_14><loc_15><loc_44><loc_17></location>therefore, using (13), we must have</text>
<formula><location><page_4><loc_32><loc_11><loc_84><loc_14></location>6 λ 2 β = -8 N R 3 α, α -β λ = -8 N R 3 β . (19)</formula>
<text><location><page_5><loc_14><loc_84><loc_39><loc_86></location>A nontrivial solution exists if</text>
<formula><location><page_5><loc_37><loc_79><loc_84><loc_84></location>λ ( 8 N R 3 ) 2 -8 N R 3 -6 λ 3 = 0 . (20)</formula>
<text><location><page_5><loc_14><loc_78><loc_73><loc_79></location>We will return to this equation after discussing the Einstein equations.</text>
<text><location><page_5><loc_17><loc_76><loc_52><loc_78></location>Now let us turn to the Einstein equations:</text>
<formula><location><page_5><loc_26><loc_71><loc_84><loc_75></location>R MN = 1 12 F MPQR F PQR N -1 3 · 48 g MN F PQRS F PQRS , (21)</formula>
<text><location><page_5><loc_14><loc_68><loc_84><loc_71></location>where M,N,P,... = 0 , 1 , . . . , 10. With ansatz (16), we can calculate the right hand side of the above equations:</text>
<formula><location><page_5><loc_20><loc_63><loc_84><loc_67></location>R µν = ( 4 R 2 ) 4 ( -3! 12 N 2 -4! 3 · 48 ( -N 2 + α 2 +6 λ 2 β 2 ) ) g µν , (22)</formula>
<formula><location><page_5><loc_20><loc_54><loc_84><loc_59></location>R ˆ α ˆ β = ( 4 R 2 ) 4 ( 3! 12 (4 λ 2 β 2 ) -4! 3 · 48 ( -N 2 + α 2 +6 λ 2 β 2 ) ) δ ˆ α ˆ β , (24)</formula>
<formula><location><page_5><loc_20><loc_58><loc_84><loc_63></location>R αβ = ( 4 R 2 ) 4 ( 3! 12 ( α 2 +3 λ 2 β 2 ) -4! 3 · 48 ( -N 2 + α 2 +6 λ 2 β 2 ) ) δ αβ , (23)</formula>
<text><location><page_5><loc_14><loc_49><loc_84><loc_54></location>with µ, ν = 0 , . . . , 3 , α, β = 4 , . . . 7 , and ˆ α, ˆ β = 8 , 9 , 10. Notice that different terms in our ansatz (16) do not contract into each other. For the left hand side, on the other hand, the Ricci tensor of metric (6) becomes</text>
<formula><location><page_5><loc_35><loc_40><loc_84><loc_48></location>R αβ = ( 4 R 2 ) ( 3(2 -λ 2 ) 2 ) δ αβ , R ˆ α ˆ β = ( 4 R 2 ) ( 1 + 2 λ 4 2 λ 2 ) δ ˆ α ˆ β , (25)</formula>
<text><location><page_5><loc_14><loc_38><loc_69><loc_39></location>these are to be substituted on the left hand side of (23) and (24).</text>
<text><location><page_5><loc_14><loc_32><loc_84><loc_37></location>We can now solve (19) and (20) for β and N , and then plug it into (23) and (24). The two resulting equations can be solved for λ and α . We get two types of solutions. Those with no internal flux:</text>
<formula><location><page_5><loc_44><loc_30><loc_84><loc_31></location>α = β = 0 , (26)</formula>
<text><location><page_5><loc_14><loc_25><loc_84><loc_29></location>together with λ 2 = 1, which is the round sphere. Or, we can have λ 2 = 1 / 5, which corresponds to the squashed sphere solution. We also get solutions with fluxes</text>
<formula><location><page_5><loc_35><loc_23><loc_84><loc_24></location>α 2 = 9 / 5 , β 2 = 9 , λ 2 = 1 / 5 . (27)</formula>
<text><location><page_5><loc_14><loc_18><loc_84><loc_23></location>For λ = 1 / √ 5, α = -3 / √ 5 , β = 3, and N = 3 R 3 / (4 √ 5) we have a non-zero 4-form field strength along S 7 , and it represents the squashed S 7 with Einstein metric</text>
<formula><location><page_5><loc_30><loc_14><loc_84><loc_18></location>R αβ = ( 4 R 2 ) 27 10 δ αβ , R ˆ α ˆ β = ( 4 R 2 ) 27 10 δ ˆ α ˆ β . (28)</formula>
<text><location><page_5><loc_14><loc_7><loc_84><loc_14></location>The above solution, the so-called squashed solution with torsion, was obtained in 1980's using the covariantly constant spinors of the squashed sphere without torsion [5, 10]. We can also take λ = -1 / √ 5, α = -3 / √ 5 , β = -3, and N = -3 R 3 / (4 √ 5) instead, this is the skew-whiffed squashed solution.</text>
<section_header_level_1><location><page_6><loc_14><loc_83><loc_58><loc_86></location>3 CP 3 as an S 2 bundle over S 4</section_header_level_1>
<text><location><page_6><loc_14><loc_71><loc_84><loc_82></location>In the previous section the metric of S 7 was written as an S 3 bundle over S 4 . It is also possible to write the metric as a U (1) bundle over CP 3 . On the other hand, it is observed that CP 3 itself can be written as an S 2 bundle over S 4 . In this form one can construct a family of homogeneous metrics by rescaling the fibers. In fact, we can see that the metric (3) can be rewritten as a U (1) bundle over such a deformed CP 3 [16, 17]. First note that 2</text>
<formula><location><page_6><loc_19><loc_60><loc_84><loc_70></location>ds 2 S 7 = dµ 2 + 1 4 sin 2 µ Σ 2 i + λ 2 ( σ i -cos 2 µ/ 2 Σ i ) 2 = dµ 2 + 1 4 sin 2 µ Σ 2 i + λ 2 ( dτ -A ) 2 + λ 2 ( dθ -sin φA 1 +cos φA 2 ) 2 + λ 2 sin 2 θ ( dφ -cot θ (cos φA 1 +sin φA 2 ) + A 3 ) 2 , (29)</formula>
<text><location><page_6><loc_14><loc_59><loc_19><loc_60></location>where</text>
<formula><location><page_6><loc_41><loc_57><loc_84><loc_58></location>A i = cos 2 µ/ 2 Σ i , (30)</formula>
<text><location><page_6><loc_14><loc_54><loc_18><loc_56></location>and,</text>
<formula><location><page_6><loc_28><loc_53><loc_84><loc_54></location>A = cos θ dφ +sin θ (cos φA 1 +sin φA 2 ) + cos θA 3 . (31)</formula>
<text><location><page_6><loc_14><loc_50><loc_63><loc_52></location>σ i 's are left-invariant one-forms that are chosen as follows:</text>
<formula><location><page_6><loc_37><loc_42><loc_63><loc_49></location>σ 1 = sin φdθ +sin θ cos φdτ , σ 2 = -cos φdθ +sin θ sin φdτ , σ 3 = -dφ +cos θ dτ .</formula>
<text><location><page_6><loc_14><loc_37><loc_84><loc_42></location>In the new form of the metric (29), we can further rescale the U (1) fibers so that the Ricci tensor (in a basis we introduce shortly) is still diagonal. Hence, we take the metric to be</text>
<formula><location><page_6><loc_19><loc_29><loc_84><loc_36></location>ds 2 S 7 = dµ 2 + 1 4 sin 2 µ Σ 2 i + λ 2 ( dθ -sin φA 1 +cos φA 2 ) 2 + λ 2 sin 2 θ ( dφ -cot θ (cos φA 1 +sin φA 2 ) + A 3 ) 2 + ˜ λ 2 ( dτ -A ) 2 , (32)</formula>
<text><location><page_6><loc_14><loc_28><loc_39><loc_29></location>and choose the following basis</text>
<formula><location><page_6><loc_30><loc_16><loc_84><loc_27></location>e 0 = dµ, e i = 1 2 sin µ Σ i , e 5 = λ ( dθ -sin φA 1 +cos φA 2 ) , e 6 = λ sin θ ( dφ -cot θ (cos φA 1 +sin φA 2 ) + A 3 ) , e 7 = ˜ λ ( dτ -A ) . (33)</formula>
<text><location><page_6><loc_14><loc_14><loc_56><loc_16></location>In this basis the Ricci tensor is diagonal and reads</text>
<formula><location><page_6><loc_27><loc_8><loc_84><loc_14></location>R 00 = R 11 = R 22 = R 33 = 3 -λ 2 -˜ λ 2 / 2 , R 55 = R 66 = λ 2 +1 /λ 2 -˜ λ 2 / 2 λ 4 , R 77 = ˜ λ 2 + ˜ λ 2 / 2 λ 4 . (34)</formula>
<section_header_level_1><location><page_7><loc_14><loc_84><loc_33><loc_86></location>3.1 The ansatz</section_header_level_1>
<text><location><page_7><loc_14><loc_79><loc_84><loc_83></location>As in the previous section, a natural 3-form to begin with is ω 3 = e 567 . To proceed, however, it proves useful to define the following forms</text>
<formula><location><page_7><loc_20><loc_72><loc_84><loc_78></location>R 1 = sin φ ( e 01 + e 23 ) -cos φ ( e 02 + e 31 ) , R 2 = cos θ cos φ ( e 01 + e 23 ) + cos θ sin φ ( e 02 + e 31 ) -sin θ ( e 03 + e 12 ) , K = sin θ cos φ ( e 01 + e 23 ) + sin θ sin φ ( e 02 + e 31 ) + cos θ ( e 03 + e 12 ) . (35)</formula>
<text><location><page_7><loc_14><loc_66><loc_84><loc_70></location>The key feature of this definition, that we will use frequently in this paper, is that these three forms are orthogonal to each other, i.e.,</text>
<formula><location><page_7><loc_36><loc_62><loc_84><loc_65></location>R 1 ∧ R 2 = K ∧ R 1 = K ∧ R 2 = 0 . (36)</formula>
<text><location><page_7><loc_14><loc_60><loc_29><loc_61></location>Let us also define,</text>
<formula><location><page_7><loc_27><loc_55><loc_84><loc_58></location>ReΩ = R 1 ∧ e 5 + R 2 ∧ e 6 , ImΩ = R 1 ∧ e 6 -R 2 ∧ e 5 , (37)</formula>
<text><location><page_7><loc_14><loc_53><loc_77><loc_55></location>we will further need to work out the exterior derivatives of the above forms</text>
<formula><location><page_7><loc_32><loc_48><loc_84><loc_52></location>d ReΩ = 4 λω 4 -2 λ e 56 ∧ K, d ImΩ = 0 , (38)</formula>
<text><location><page_7><loc_14><loc_46><loc_40><loc_47></location>for dω 3 in the new basis we get</text>
<formula><location><page_7><loc_37><loc_41><loc_84><loc_44></location>dω 3 = λ ImΩ ∧ e 7 -˜ λe 56 ∧ F , (39)</formula>
<text><location><page_7><loc_14><loc_39><loc_18><loc_41></location>with</text>
<text><location><page_7><loc_17><loc_35><loc_30><loc_36></location>Note that since</text>
<formula><location><page_7><loc_31><loc_30><loc_84><loc_33></location>d ImΩ = 0 , ImΩ ∧ F = -ImΩ ∧ K = 0 , (41)</formula>
<text><location><page_7><loc_14><loc_23><loc_84><loc_30></location>we have three independent 4-forms ω 4 , e 7 ∧ ImΩ, and e 56 ∧ K , which are closed and do not contract into each other. Furthermore, the set of these 4-forms is closed under d ∗ operation, and hence a suitable ansatz for F 4 is as follows</text>
<formula><location><page_7><loc_32><loc_20><loc_84><loc_23></location>F 4 = N/epsilon1 4 + αω 4 + β e 7 ∧ ImΩ + γ K ∧ e 56 , (42)</formula>
<text><location><page_7><loc_14><loc_18><loc_71><loc_19></location>for α , β , and γ three real constants. Taking the Hodge dual we have</text>
<formula><location><page_7><loc_28><loc_13><loc_84><loc_16></location>∗ 11 F 4 = Nω 3 ∧ ω 4 + /epsilon1 4 ∧ ( αω 3 -β ReΩ + γ K ∧ e 7 ) . (43)</formula>
<text><location><page_7><loc_14><loc_11><loc_19><loc_13></location>Using</text>
<formula><location><page_7><loc_38><loc_36><loc_84><loc_39></location>F = dA = -K -e 56 /λ 2 . (40)</formula>
<formula><location><page_7><loc_35><loc_8><loc_84><loc_11></location>de 56 = λ ImΩ , dK = -1 λ ImΩ , (44)</formula>
<text><location><page_8><loc_14><loc_84><loc_60><loc_86></location>and (38), we see that Maxwell equations (17) reduce to</text>
<formula><location><page_8><loc_37><loc_75><loc_84><loc_82></location>-αλ 2 + Nλβ + γ = 0 , α ˜ λ +2 β/λ +( ˜ λ/λ 2 + N ) γ = 0 , Nα -4 λβ +2 ˜ λγ = 0 . (45)</formula>
<text><location><page_8><loc_14><loc_73><loc_75><loc_74></location>As for the Einstein equations, we use (34) and the ansatz (42) to obtain</text>
<formula><location><page_8><loc_31><loc_60><loc_84><loc_71></location>3 -λ 2 -˜ λ 2 2 = 1 3 ( α 2 + β 2 + 1 2 γ 2 + 1 2 N 2 ) , λ 2 + 1 λ 2 -˜ λ 2 2 λ 4 = 1 3 ( -α 2 2 + β 2 +2 γ 2 + 1 2 N 2 ) , ˜ λ 2 + ˜ λ 2 2 λ 4 = 1 3 ( -α 2 2 +4 β 2 -γ 2 + 1 2 N 2 ) . (46)</formula>
<text><location><page_8><loc_14><loc_51><loc_84><loc_58></location>In general, it is not easy to solve set of coupled equations (45) and (46). In fact, apart from the known solutions, we have found no real (i.e., real coefficients for F 4 ) solutions. In especial cases, though, we can reduce the equations further and find solutions. Let us start by assuming</text>
<formula><location><page_8><loc_46><loc_48><loc_52><loc_50></location>λ = ˜ λ,</formula>
<text><location><page_8><loc_14><loc_40><loc_84><loc_46></location>then by the Einstein equations we must have β 2 = γ 2 . Taking β = -γ yields λ = ˜ λ = 1 / √ 5, N = -6 / √ 5, and α 2 = β 2 = γ 2 = 9 / 5 which is the squashed solution (with torsion) of the previous section with R µν = -45 / 10 g µν .</text>
<text><location><page_8><loc_14><loc_34><loc_84><loc_41></location>For β = γ , we get λ = ˜ λ = 1, N = -2, and α 2 = β 2 = γ 2 = 1; this is an Englert type solution with R µν = -5 / 2 g µν . This has the same four-dimensional Ricci tensor as the original solution found by Englert in [2] using parallelizing torsions on the 7-sphere, and later by [6] and [11] using Killing spinors.</text>
<section_header_level_1><location><page_8><loc_14><loc_29><loc_45><loc_31></location>3.2 Pope-Warner solution</section_header_level_1>
<text><location><page_8><loc_14><loc_25><loc_84><loc_28></location>In this section we rederive the Pope-Warner ansatz and the solution [11] using the canonical forms language. Let us then begin by defining</text>
<formula><location><page_8><loc_26><loc_20><loc_84><loc_23></location>Re L = -R 1 ∧ e 5 + R 2 ∧ e 6 , Im L = R 1 ∧ e 6 + R 2 ∧ e 5 . (47)</formula>
<text><location><page_8><loc_14><loc_18><loc_71><loc_20></location>We note that in the vielbein basis (33), A in (31) can be written as</text>
<formula><location><page_8><loc_31><loc_13><loc_84><loc_17></location>A = cot θ e 6 λ + cot µ/ 2 sin θ (cos φe 1 +sin φe 2 ) , (48)</formula>
<text><location><page_8><loc_14><loc_10><loc_78><loc_12></location>which, together with (35), allows us to write de 5 and de 6 more compactly as</text>
<formula><location><page_8><loc_30><loc_6><loc_84><loc_8></location>de 5 = -e 6 ∧ A + λR 1 , de 6 = e 5 ∧ A + λR 2 . (49)</formula>
<text><location><page_9><loc_14><loc_84><loc_53><loc_86></location>Taking the exterior derivative once more yields</text>
<formula><location><page_9><loc_25><loc_80><loc_84><loc_83></location>λdR 1 = λR 2 ∧ A + e 6 ∧ K, λdR 2 = -λR 1 ∧ A -e 5 ∧ K. (50)</formula>
<text><location><page_9><loc_14><loc_79><loc_63><loc_80></location>Having derived (49) and (50), it is now easy to prove that</text>
<formula><location><page_9><loc_30><loc_75><loc_84><loc_78></location>d Re L = -2 A ∧ Im L, d Im L = 2 A ∧ Re L. (51)</formula>
<text><location><page_9><loc_17><loc_74><loc_62><loc_75></location>To absorb A into e 7 in the above equations, we define</text>
<formula><location><page_9><loc_44><loc_71><loc_84><loc_73></location>P = e -2 iτ L, (52)</formula>
<text><location><page_9><loc_14><loc_68><loc_40><loc_70></location>by using eqs. (51), we see that</text>
<text><location><page_9><loc_14><loc_62><loc_39><loc_63></location>On the other hand, note that</text>
<text><location><page_9><loc_14><loc_58><loc_33><loc_59></location>so we can write (53) as</text>
<formula><location><page_9><loc_42><loc_64><loc_84><loc_68></location>dP = -2 i ˜ λ e 7 ∧ P . (53)</formula>
<formula><location><page_9><loc_44><loc_59><loc_84><loc_62></location>∗ L = iL ∧ e 7 , (54)</formula>
<formula><location><page_9><loc_43><loc_54><loc_84><loc_58></location>dP = 2 ˜ λ ∗ P . (55)</formula>
<text><location><page_9><loc_14><loc_52><loc_63><loc_54></location>This implies that for the 4-form field strength we can take</text>
<formula><location><page_9><loc_30><loc_49><loc_84><loc_51></location>F 4 = N/epsilon1 4 + η e 7 ∧ (sin 2 τ Re L -cos 2 τ Im L ) , (56)</formula>
<text><location><page_9><loc_14><loc_43><loc_84><loc_49></location>with η a real constant. Maxwell eq. (17) then requires N = -2 / ˜ λ , whereas, the Einstein equations imply λ 2 = 1, and ˜ λ 2 = 2, together with η 2 = 2. Note that in this solution the U (1) fibers of S 7 are stretched by a factor of 2.</text>
<text><location><page_9><loc_82><loc_37><loc_82><loc_39></location>/negationslash</text>
<text><location><page_9><loc_14><loc_36><loc_84><loc_43></location>We can construct another consistent ansatz by taking a linear combination of Pope-Warner ansatz and the one introduced in the previous section. However, by this we get non-zero off diagonal components of energy-momentum tensor, i.e., T 56 = 0, unless we set β = 0. Let us then set</text>
<formula><location><page_9><loc_22><loc_32><loc_84><loc_35></location>F 4 = N/epsilon1 4 + αω 4 + γ K ∧ e 56 + η e 7 ∧ (sin 2 τ Re L -cos 2 τ Im L ) , (57)</formula>
<text><location><page_9><loc_14><loc_31><loc_48><loc_32></location>Maxwell eqs. (17) and (45) then require</text>
<formula><location><page_9><loc_33><loc_27><loc_84><loc_30></location>N = -2 / ˜ λ, λ 2 = ˜ λ 2 = 1 , α = γ , (58)</formula>
<text><location><page_9><loc_14><loc_26><loc_44><loc_27></location>while, the Einstein equations imply</text>
<formula><location><page_9><loc_40><loc_23><loc_84><loc_25></location>α 2 = γ 2 = η 2 = 1 , (59)</formula>
<text><location><page_9><loc_14><loc_19><loc_84><loc_22></location>which is the Englert solution with R µν = -5 / 2 g µν . Note that here we have α = γ = 1, hence the second and the third terms in (57) combine to</text>
<formula><location><page_9><loc_39><loc_14><loc_84><loc_18></location>ω 4 + K ∧ e 56 = 1 2 F ∧ F , (60)</formula>
<text><location><page_9><loc_14><loc_7><loc_84><loc_14></location>with F the Kahler form defined in (40). We can now recognize (57) as exactly the Englert solution of [11]. The F ∧ F term and the term proportional to η are each invariant under an SU (4) symmetry, but with the given values of the constant coefficients, α, β , and γ , the symmetry enhances to SO (7).</text>
<section_header_level_1><location><page_10><loc_14><loc_82><loc_68><loc_86></location>4 A new AdS 5 × CP 3 compactification</section_header_level_1>
<text><location><page_10><loc_14><loc_73><loc_84><loc_82></location>With the ansatz introduced in Sec. 3.1, we can think of eleven dimensional metrics which are direct product of 5 and 6-dimensional spaces with F 4 given by (42) setting N and β equal to zero. By this, apart from the result of [18] we derive a new solution of AdS 5 × CP 3 so that the CP 3 factor is stretched along its S 2 fiber by a factor of 2.</text>
<text><location><page_10><loc_14><loc_69><loc_84><loc_73></location>Let us then take the eleven dimensional spacetime to be the direct product of a 5 and 6-dimensional spaces,</text>
<formula><location><page_10><loc_42><loc_68><loc_84><loc_69></location>ds 2 11 = ds 2 5 + ds 2 6 . (61)</formula>
<text><location><page_10><loc_14><loc_63><loc_84><loc_67></location>For the 6-dimensional space we take the same metric that appeared in S 7 description in (29):</text>
<formula><location><page_10><loc_25><loc_55><loc_84><loc_62></location>ds 2 6 = dµ 2 + 1 4 sin 2 µ Σ 2 i + λ 2 ( dθ -sin φA 1 +cos φA 2 ) 2 + λ 2 sin 2 θ ( dφ -cot θ (cos φA 1 +sin φA 2 ) + A 3 ) 2 , (62)</formula>
<text><location><page_10><loc_14><loc_50><loc_84><loc_55></location>as mentioned before, this is an S 2 bundle over S 4 , and for λ 2 = 1 we get the FubiniStudy metric on CP 3 . By taking the basis e 0 , . . . , e 6 as in (33) the Ricci tensor reads</text>
<formula><location><page_10><loc_36><loc_44><loc_84><loc_48></location>R 00 = R 11 = R 22 = R 33 = 3 -λ 2 , R 55 = R 66 = λ 2 +1 /λ 2 . (63)</formula>
<text><location><page_10><loc_17><loc_41><loc_51><loc_42></location>As for F 4 , we choose the following ansatz</text>
<formula><location><page_10><loc_40><loc_37><loc_84><loc_39></location>F 4 = αω 4 + γ K ∧ e 56 , (64)</formula>
<text><location><page_10><loc_14><loc_34><loc_54><loc_36></location>which is closed. Taking the Hodge dual we have</text>
<formula><location><page_10><loc_38><loc_30><loc_84><loc_33></location>∗ 11 F 4 = /epsilon1 5 ∧ ( αe 56 + γ K ) . (65)</formula>
<text><location><page_10><loc_14><loc_27><loc_60><loc_30></location>As F 4 ∧ F 4 = 0, in this case the Maxwell equation reads</text>
<formula><location><page_10><loc_33><loc_24><loc_84><loc_26></location>d ∗ 11 F 4 = -( αλ -γ /λ ) /epsilon1 5 ∧ ImΩ = 0 , (66)</formula>
<text><location><page_10><loc_14><loc_22><loc_57><loc_23></location>where use has been made of (44). So, we must have</text>
<formula><location><page_10><loc_45><loc_18><loc_84><loc_20></location>αλ 2 = γ . (67)</formula>
<text><location><page_10><loc_17><loc_15><loc_82><loc_17></location>The Einstein equations along compact 6 dimensions, on the other hand, imply</text>
<formula><location><page_10><loc_32><loc_6><loc_84><loc_14></location>3 -λ 2 = 1 3 ( α 2 + 1 2 γ 2 ) = 1 3 (1 + λ 4 2 ) α 2 , λ 2 + 1 λ 2 = 1 3 ( -α 2 2 +2 γ 2 ) = 1 3 ( -1 2 +2 λ 4 ) α 2 , (68)</formula>
<text><location><page_11><loc_14><loc_82><loc_84><loc_86></location>where we used (67) in the last equalities. From the above equations we get two solutions:</text>
<formula><location><page_11><loc_39><loc_80><loc_84><loc_82></location>λ 2 = 1 , α 2 = γ 2 = 4 , (69)</formula>
<text><location><page_11><loc_14><loc_76><loc_84><loc_79></location>for which the metric is the standard Fubini-Study metric of CP 3 . The 5d Ricci tensor becomes</text>
<formula><location><page_11><loc_43><loc_73><loc_84><loc_76></location>R µν = -2 g µν , (70)</formula>
<text><location><page_11><loc_14><loc_70><loc_84><loc_73></location>with µ, ν = 0 , . . . , 4. Therefore the 5-dimensional spacetime is anti-de Sitter. This solution was first derived in [18].</text>
<text><location><page_11><loc_17><loc_68><loc_44><loc_69></location>For the second solution we have</text>
<formula><location><page_11><loc_37><loc_64><loc_84><loc_66></location>λ 2 = 2 , α 2 = 1 , γ 2 = 4 , (71)</formula>
<text><location><page_11><loc_14><loc_61><loc_35><loc_63></location>with the 5d Ricci tensor;</text>
<formula><location><page_11><loc_43><loc_58><loc_84><loc_61></location>R µν = -3 2 g µν . (72)</formula>
<text><location><page_11><loc_14><loc_46><loc_84><loc_57></location>This new solution corresponds to an stretched CP 3 , in which the S 2 fibers are stretched by a factor of 2. Note that, for this solution the 6-dimensional metric is no longer Einstein. Also, note that according to our discussion at the end of the previous section the first solution, (69), has an SU (4) symmetry, whereas in the new solution, (71), this symmetry is reduced to SO (3) × SO (5), i.e, to the direct product of the symmetry subgroups of the fiber and the base.</text>
<section_header_level_1><location><page_11><loc_14><loc_40><loc_62><loc_44></location>5 AdS 2 × H 2 × S 7 compactification</section_header_level_1>
<text><location><page_11><loc_14><loc_29><loc_84><loc_40></location>With metric (32) for the S 7 , we can take yet another ansatz for the metric and F 4 and come up with a new compactification. In fact, in this section we obtain a new solution of type AdS 2 × H 2 × S 7 , with H 2 a hyperbolic surface. As we will see, this solution exists only in 11-dimensional space with Euclidean signature, and like the Pope-Warner solution the S 7 metric gets stretched along its U (1) fibers by a factor of 2.</text>
<text><location><page_11><loc_14><loc_25><loc_84><loc_29></location>Let the eleven dimensional spacetime to be the direct product of two 2-dimensional spaces and S 7 ,</text>
<formula><location><page_11><loc_38><loc_23><loc_84><loc_25></location>ds 2 11 = ds 2 A + ds 2 2 + ds 2 S 7 , (73)</formula>
<text><location><page_11><loc_14><loc_20><loc_53><loc_22></location>where ds 2 S 7 is the same as (32). For F 4 we take</text>
<formula><location><page_11><loc_14><loc_16><loc_84><loc_19></location>F 4 = N/epsilon1 A 2 ∧ /epsilon1 2 + αω 4 + β e 7 ∧ ImΩ+ γ K ∧ e 56 + /epsilon1 2 ∧ ( ξ 1 K + η 1 e 56 )+ /epsilon1 A 2 ∧ ( ξ 2 K + η 2 e 56 ) , (74)</formula>
<text><location><page_11><loc_14><loc_8><loc_84><loc_15></location>note that the first four terms are the same as those appeared in (42). ξ 1 , ξ 2 , η 1 , and η 2 are constant parameters. We take the 4-dimensional space to be the direct product of two Euclidean subspaces with /epsilon1 A 2 and /epsilon1 2 as their 2-dimensional volume elements.</text>
<text><location><page_12><loc_17><loc_84><loc_45><loc_86></location>The Bianchi identity requires that</text>
<formula><location><page_12><loc_38><loc_81><loc_84><loc_82></location>ξ 1 = λ 2 η 1 , ξ 2 = λ 2 η 2 , (75)</formula>
<text><location><page_12><loc_14><loc_74><loc_84><loc_79></location>For the Maxwell equation, first note that in Euclidean 11-dimensional space we need to account for an extra i factor coming from the Chern-Simons term so that (17) is replaced by</text>
<formula><location><page_12><loc_40><loc_70><loc_84><loc_74></location>d ∗ 11 F 4 = -i 2 F 4 ∧ F 4 , (76)</formula>
<text><location><page_12><loc_14><loc_66><loc_84><loc_70></location>therefore, with our ansatze (32) and (74) the Maxwell equations reduce to the following algebraic equations</text>
<formula><location><page_12><loc_31><loc_53><loc_84><loc_65></location>˜ λ (2 ξ 1 + η 1 /λ 2 ) = -i (2 ξ 2 γ + αη 2 ) , ˜ λ (2 ξ 2 + η 2 /λ 2 ) = -i (2 ξ 1 γ + αη 1 ) , α ˜ λ +2 β/λ + γ ( ˜ λ/λ 2 + iN ) = -i ( ξ 1 η 2 + η 1 ξ 2 ) , iNα -4 λβ +2 ˜ λγ = -2 iξ 1 ξ 2 , -αλ 2 + iNλβ + γ = 0 . (77)</formula>
<text><location><page_12><loc_14><loc_51><loc_54><loc_52></location>Using (75), the first two equations above imply</text>
<formula><location><page_12><loc_40><loc_48><loc_84><loc_49></location>ξ 2 1 = ξ 2 2 , η 2 1 = η 2 2 . (78)</formula>
<text><location><page_12><loc_14><loc_35><loc_84><loc_46></location>Had we chosen a Lorentzian signature metric for the 4-dimensional space, since ∗ /epsilon1 A 2 = -/epsilon1 2 and ∗ /epsilon1 2 = /epsilon1 A 2 we would have obtained ξ 2 1 = -ξ 2 2 , with no real solution. On the other hand, in Euclidean 11-dimensional space eq. (76) implies that whenever the RHS is nonvanishing F 4 is necessarily complex valued, and so there is no restriction on the coefficients of F 4 to be real. However, for having a well-defined metric we still require that λ and ˜ λ to be real.</text>
<text><location><page_12><loc_17><loc_32><loc_80><loc_35></location>To carry on, we set ξ 1 = ξ 2 ≡ ξ without loss of generality, and (77) becomes</text>
<formula><location><page_12><loc_34><loc_22><loc_84><loc_32></location>αλ 2 +2 λ 4 γ -i ˜ λ (1 + 2 λ 4 ) = 0 , ( ˜ λλ 2 -iN ) α +6 λβ +( iNλ 2 -˜ λ ) γ = 0 , ˜ λγ -2 λβ + iNα/ 2 + iξ 2 = 0 , -αλ 2 + iNλβ + γ = 0 , (79)</formula>
<text><location><page_12><loc_27><loc_17><loc_27><loc_20></location>/negationslash</text>
<text><location><page_12><loc_14><loc_15><loc_84><loc_22></location>where the second equation is obtained by dividing the third and fourth equations in (77). For β = 0, we have found no solution of (79) for which λ and ˜ λ are both real. Let us then discuss the case with β = 0. In this case, the second and the forth equations above imply</text>
<formula><location><page_12><loc_41><loc_13><loc_84><loc_14></location>λ 2 = 1 , α = γ . (80)</formula>
<text><location><page_12><loc_14><loc_10><loc_51><loc_12></location>Plugging this into the first equation we have</text>
<formula><location><page_12><loc_44><loc_7><loc_84><loc_9></location>α = γ = i ˜ λ, (81)</formula>
<text><location><page_13><loc_14><loc_84><loc_43><loc_86></location>and finally the third equation gives</text>
<formula><location><page_13><loc_41><loc_79><loc_84><loc_83></location>ξ 2 = -˜ λ 2 -i 2 ˜ λN . (82)</formula>
<text><location><page_13><loc_14><loc_74><loc_84><loc_78></location>Let us now look at the Einstein equations along S 7 . Taking into account λ 2 = 1 and α = γ , they read</text>
<formula><location><page_13><loc_38><loc_65><loc_84><loc_73></location>2 -˜ λ 2 2 = 1 2 α 2 -1 6 N 2 -1 2 ξ 2 , 3 2 ˜ λ 2 = -1 2 α 2 -1 6 N 2 -1 2 ξ 2 , (83)</formula>
<text><location><page_13><loc_14><loc_61><loc_84><loc_64></location>note the sign change of N 2 as a result of using the Riemannian signature (compare with (46)). Using (81), we can solve for ˜ λ :</text>
<formula><location><page_13><loc_46><loc_58><loc_84><loc_59></location>˜ λ 2 = 2 , (84)</formula>
<text><location><page_13><loc_14><loc_54><loc_18><loc_56></location>and,</text>
<formula><location><page_13><loc_41><loc_52><loc_84><loc_54></location>N 2 +3 ξ 2 +12 = 0 , (85)</formula>
<text><location><page_13><loc_14><loc_50><loc_71><loc_51></location>this last equation together with (82) can be solved to give N and ξ .</text>
<text><location><page_13><loc_17><loc_48><loc_62><loc_49></location>The Ricci tensor along two 2-dimensional spaces reads</text>
<formula><location><page_13><loc_26><loc_39><loc_84><loc_47></location>R ab = ( ξ 2 2 + 1 2 η 2 2 + 1 2 N 2 -1 6 ( N 2 + α 2 +2 γ 2 ) -1 2 ξ 2 ) g ab , R a ' b ' = ( ξ 2 1 + 1 2 η 2 1 + 1 2 N 2 -1 6 ( N 2 + α 2 +2 γ 2 ) -1 2 ξ 2 ) g a ' b ' , (86)</formula>
<text><location><page_13><loc_14><loc_37><loc_75><loc_38></location>with a, b = 0 , 1, and a ' , b ' = 2 , 3. Now, using (81), (84), and (85) , we get</text>
<formula><location><page_13><loc_44><loc_30><loc_84><loc_35></location>R ab = -3 g ab , R a ' b ' = -3 g a ' b ' . (87)</formula>
<text><location><page_13><loc_14><loc_19><loc_84><loc_29></location>Therefore, the 4-dimensional space is a direct product of a Euclidean AdS 2 and a 2-dimensional hyperbolic surface. Interestingly, this solution has some common features with the Pope-Warner and the Freund-Rubin solutions. As in the PopeWarner solution, here the metric of S 7 is stretched by a factor of 2 along its U (1) fiber with SU (4) isometry group. And, on the other hand, the 4-dimensional Ricci tensor is equal to that of the Freund-Rubin solution.</text>
<section_header_level_1><location><page_13><loc_14><loc_14><loc_35><loc_16></location>6 Conclusions</section_header_level_1>
<text><location><page_13><loc_14><loc_7><loc_84><loc_12></location>In this paper we provided a unified approach to study the squashed, stretched, and the Englert type solutions of 11-dimensional supergravity, especially when there are fluxes in the compact direction. With the special form of the metric (32), we</text>
<text><location><page_14><loc_14><loc_71><loc_84><loc_86></location>were able to construct more general ansatze by bringing together the earlier known ones and those constructed in Sec. 3. We then used the ansatz to reduce the field equations to algebraic ones and rederive the known solutions. Further, using these ansatze we were able to find new compactifying solutions to 5 and 4 dimensions. In compactifying to 5 dimensions, we derived a solution of AdS 5 × CP 3 type with the CP 3 factor stretched. We also derived a solution of AdS 2 × H 2 × S 7 type compactifying to Euclidean 4 dimensions. In this solution, the compact space was a stretched S 7 .</text>
<text><location><page_14><loc_14><loc_66><loc_84><loc_71></location>Having derived the above solutions, the next important issue to address is that of stability. It is also worth studying the new solutions in the context of holographic superconductivity in M-theory [19].</text>
<section_header_level_1><location><page_14><loc_14><loc_60><loc_29><loc_62></location>References</section_header_level_1>
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<list_item><location><page_14><loc_16><loc_50><loc_84><loc_53></location>[2] F. Englert, Spontaneous Compactification Of Eleven-Dimensional Supergravity , Phys. Lett. B 119 , 339 (1982).</list_item>
<list_item><location><page_14><loc_16><loc_45><loc_84><loc_48></location>[3] M. A. Awada, M. J. Duff and C. N. Pope, N = 8 supergravity breaks down to N = 1 , Phys. Rev. Lett. 50 , 294 (1983).</list_item>
<list_item><location><page_14><loc_16><loc_40><loc_84><loc_43></location>[4] M. J. Duff, B. E. W. Nilsson and C. N. Pope, Spontaneous Supersymmetry Breaking By The Squashed Seven Sphere , Phys. Rev. Lett. 50 , 2043 (1983).</list_item>
<list_item><location><page_14><loc_16><loc_33><loc_84><loc_38></location>[5] F. A. Bais, H. Nicolai and P. van Nieuwenhuizen, Geometry Of Coset Spaces And Massless Modes Of The Squashed Seven Sphere In Supergravity , Nucl. Phys. B 228 , 333 (1983).</list_item>
<list_item><location><page_14><loc_16><loc_28><loc_84><loc_32></location>[6] M. J. Duff, Supergravity, The Seven Sphere, And Spontaneous Symmetry Breaking , Nucl. Phys. B 219 , 389 (1983).</list_item>
<list_item><location><page_14><loc_16><loc_23><loc_84><loc_27></location>[7] L. Castellani and L. J. Romans, N=3 And N=1 Supersymmetry In A New Class Of Solutions For D = 11 Supergravity , Nucl. Phys. B 238 , 683 (1984).</list_item>
<list_item><location><page_14><loc_16><loc_18><loc_84><loc_22></location>[8] D. V. Volkov, D. P. Sorokin and V. I. Tkach, Supersymmetry Vacuum Configurations in D = 11 Supergravity , JETP Lett. 40 , 1162 (1984).</list_item>
<list_item><location><page_14><loc_16><loc_12><loc_84><loc_17></location>[9] D. P. Sorokin, V. I. Tkach and D. V. Volkov, On the Relationship between Compactified Vacua of D = 11 and D = 10 Supergravities , Phys. Lett. B 161 , 301 (1985).</list_item>
<list_item><location><page_14><loc_15><loc_7><loc_84><loc_10></location>[10] F. Englert, M. Rooman and P. Spindel, Supersymmetry Breaking By Torsion And The Ricci Flat Squashed Seven Spheres , Phys. Lett. B 127 , 47 (1983).</list_item>
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<list_item><location><page_15><loc_15><loc_82><loc_84><loc_86></location>[11] C. N. Pope and N. P. Warner, An SU(4) Invariant Compactification Of D = 11 Supergravity On A Stretched Seven Sphere , Phys. Lett. B 150 , 352 (1985).</list_item>
<list_item><location><page_15><loc_15><loc_77><loc_84><loc_81></location>[12] V. D. Lyakhovsky and D. V. Vassilevich, Algebraic Approach to Kaluza-Klein Models , Lett. Math. Phys. 17 , 109 (1989).</list_item>
<list_item><location><page_15><loc_15><loc_71><loc_84><loc_76></location>[13] J. P. Gauntlett, S. Kim, O. Varela and D. Waldram, Consistent supersymmetric Kaluza-Klein truncations with massive modes , JHEP 0904 , 102 (2009) [arXiv:0901.0676 [hep-th]].</list_item>
<list_item><location><page_15><loc_15><loc_66><loc_84><loc_69></location>[14] D. Cassani and P. Koerber, Tri-Sasakian consistent reduction , JHEP 1201 , 086 (2012) [arXiv:1110.5327 [hep-th]].</list_item>
<list_item><location><page_15><loc_15><loc_61><loc_84><loc_64></location>[15] M. J. Duff, B. E. W. Nilsson and C. N. Pope, Kaluza-Klein Supergravity , Phys. Rept. 130 , 1 (1986).</list_item>
<list_item><location><page_15><loc_15><loc_56><loc_84><loc_59></location>[16] B. E. W. Nilsson and C. N. Pope, Hopf Fibration Of Eleven-Dimensional Supergravity , Class. Quant. Grav. 1 , 499 (1984).</list_item>
<list_item><location><page_15><loc_15><loc_51><loc_84><loc_54></location>[17] G. Aldazabal and A. Font, A Second look at N=1 supersymmetric AdS(4) vacua of type IIA supergravity , JHEP 0802 , 086 (2008), [arXiv:0712.1021].</list_item>
<list_item><location><page_15><loc_15><loc_46><loc_84><loc_49></location>[18] C. N. Pope and P. van Nieuwenhuizen, Compactifications of d = 11 Supergravity on Kahler Manifolds , Commun. Math. Phys. 122 , 281 (1989).</list_item>
<list_item><location><page_15><loc_15><loc_39><loc_84><loc_44></location>[19] J. P. Gauntlett, J. Sonner and T. Wiseman, Quantum Criticality and Holographic Superconductors in M-theory , JHEP 1002 , 060 (2010), [arXiv:0912.0512 [hep-th]].</list_item>
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</document> | [] |
2022CQGra..39g5012G | https://arxiv.org/pdf/2108.07210.pdf | <document>
<section_header_level_1><location><page_1><loc_9><loc_89><loc_85><loc_91></location>Bianchi IX geometry and the Einstein-Maxwell theory</section_header_level_1>
<text><location><page_1><loc_27><loc_71><loc_67><loc_78></location>A. M. Ghezelbash 1 Department of Physics and Engineering Physics, University of Saskatchewan, Saskatoon, Saskatchewan S7N 5E2, Canada</text>
<section_header_level_1><location><page_1><loc_43><loc_57><loc_51><loc_58></location>Abstract</section_header_level_1>
<text><location><page_1><loc_14><loc_41><loc_81><loc_56></location>We construct numerical solutions to the higher-dimensional Einstein-Maxwell theory. The solutions are based on embedding the four dimensional Bianchi type IX space in the theory. We find the solutions as superposition of two functions, which one of them can be found numerically. We show that the solutions in any dimensions, are almost regular everywhere, except a singular point. We find that the solutions interpolate between the two exact analytical solutions to the higher dimensional Einstein-Maxwell theory, which are based on Eguchi-Hanson type I and II geometries. Moreover, we construct the exact cosmological solutions to the theory, and study the properties of the solutions.</text>
<section_header_level_1><location><page_2><loc_9><loc_89><loc_31><loc_91></location>1 Introduction</section_header_level_1>
<text><location><page_2><loc_9><loc_53><loc_85><loc_87></location>Exploring the different aspects of gravitational physics is possible through finding the new solutions to gravity, especially coupled to the other fields, such as the electromagnetic field. Moreover, the possibility of extending the known solutions in asymptotically flat spacetime to the asymptotically de-Sitter and anti de-Sitter solutions, is crucial and important in high energy physics. These extended solutions provide better understanding of the holographic proposals between the extended theories of gravity and the conformal field theories in different dimensions [1]-[2]. The constructed and explored solutions also include different solutions with different charges, such as NUT charges [3], dyonic charges [4], as well as different matter fields [5], axion field [6] and skyrmions [7]. The theories of gravity coupled to the other matter fields are useful to study and explore the rotating black holes [8], topological charged hairy black holes [9], cosmic censorship [10], gravitational radiation [11] and hyper-scaling violation [12]. Moreover, finding new solutions to the higher dimensional gravitational theories reveals new phenomena and possibilities, which may not exist in four dimensions. The discovery of the black hole solutions in five dimensions with squashed 3-sphere horizon [13], the black rings with S 2 × S 1 horizon [14] and black lenses [15] are just some of the rich variety of the black objects in five and higher dimensions. Other relevant solutions in higher dimensional gravity coupled to the matter fields, are the dyonic solutions [16], the solitonic solutions [17], supergravity solutions [18, 19], braneworld cosmologies [20], and string theory extended solutions [21, 22].</text>
<text><location><page_2><loc_9><loc_22><loc_85><loc_52></location>The black hole solutions with different type of topologies for the horizon also were constructed and explored in [23]. Moreover, the references [24, 25] include the other solutions to extended theories of gravity with different type of matter fields in different dimensions. The class of solutions to Einstein-Maxwell-dilaton theory, in which the dilaton field couples to the cosmological constant and the Maxwell field, was considered in [26]. These solutions are relevant to the generalization of the Freund-Rubin compactification of M-theory [27]. Moreover, in interesting papers [28], the authors constructed and explored the convolution-like solutions for the fully localized type IIA D2 branes intersecting with the D6 branes. The type IIA solutions are obtained by compactifying the corresponding convolution-like M2 brane solutions, over a circle of transverse self-dual geometries including the Bianchi type IX geometry. The solutions consist of analytically convolution-like integrals of two functions, which depend on the transverse directions to the branes. The solutions preserve eight supersymmetries and are valid everywhere; near and far from the core of D6 branes. Due to the self-duality of the transverse geometries in the constructed solutions, the compactified solutions are supersymmetric. We also mention that one interesting feature of the solutions is that, the solutions are expressed completely in terms of convolution integrals, that is a result of taking special ansatzes for the solutions, and separability of the field equations.</text>
<text><location><page_2><loc_9><loc_15><loc_85><loc_21></location>The motivations for this article come from several works on finding exact solutions in different theories of gravity. Inspired by the convolution-like solutions in M-theory [28], in this article, we construct similar convolution-like solutions in six and higher dimensional Einstein-Maxwell theory based on Bianchi type IX geometry. According to our knowledge,</text>
<text><location><page_3><loc_9><loc_74><loc_85><loc_90></location>there are not any known convolution-like solutions in six dimensional or higher dimensional Einstein-Maxwell theories based on the Binachi geometries. Moreover, we are inspired with the papers [29], in which the authors constructed charged black hole and black string solutions in five dimensional Einstein-Maxwell theory. The solutions are based on embedding some known four-dimensional geometries, like Kasner space into five dimensions by using appropriate ansatzes for the metric and the Maxwell field. We note that to have non-trivial convolution-like solutions, the minimal dimensionality of Einstein-Maxwell theory should be six. Moreover, we consider the Einstein-Maxwell theories with positive cosmological constant in six and higher dimensions and find the exact cosmological convoluted solutions.</text>
<text><location><page_3><loc_9><loc_62><loc_85><loc_74></location>To find the new solutions to the higher-dimensional Einstein-Maxwell theory, we consider the Bianchi type IX space, which is an exact solution to the four-dimensional Einstein's equations. Different types of solutions to the Einstein-Maxwell theory were constructed, such as solutions with the NUT charge [30], solitonic and dyonic solutions [16], as well as braneworld solutions [20]. Moreover, solutions to the extension of Einstein-Maxwell theory with the axion field and Chern-Simons term, were constructed and studied extensively in [31].</text>
<text><location><page_3><loc_9><loc_38><loc_85><loc_61></location>We organize the article as follows: In section 2, we consider the physics of Bianchi type IX spaces. In section 3, we present some numerical solutions to the Einstein-Maxwell theory in six and higher dimensions, where the metric function can be written as the convolution-like integral of two functions. In section 4, we present the second class of numerical solutions to the Einstein-Maxwell theory in six and higher dimensions. The second class of solutions is completely independent of the solutions in section 3. In section 5, we use the results of sections 3 and 4, and explicitly construct some cosmological solutions to the Einstein-Maxwell theory with positive cosmological constant, in six and higher dimensions. In section 6, we consider the Bianchi type IX space with the special cases of the Bianchi parameter as k = 0 and k = 1. We construct some exact solutions to the Einstein-Maxwell theory in six and higher dimensions, where the radial function involves the Heun-C functions. We discuss the physical properties of the solutions. We wrap up the article by the concluding remarks and four appendices in section 7.</text>
<section_header_level_1><location><page_3><loc_9><loc_33><loc_46><loc_35></location>2 The Bianchi geometries</section_header_level_1>
<text><location><page_3><loc_9><loc_19><loc_85><loc_31></location>The classification of the homogeneous and isotropic spaces, is crucial to understand the cosmological models of the universe, as well as finding the theoretical models, which are consistent with the experimental data. The first classification of the homogeneous spaces was done long time ago by Bianchi [32]. We know now that there are eleven different homogeneous spaces [33], called Bianchi type I , · · · , VI (class A or B), VII (class A or B), VIII and IX. The Bianchi type IX has been used mainly in cosmological models [34], supergravity theories [35] and extensions of gravity [36].</text>
<text><location><page_3><loc_12><loc_17><loc_83><loc_18></location>The four-dimensional Bianchi type IX metric, is given locally by the line element [37]</text>
<formula><location><page_3><loc_22><loc_13><loc_85><loc_15></location>ds 2 = e 2( A ( η )+ B ( η )+ C ( η )) dη 2 + e 2 A ( η ) σ 2 1 + e 2 B ( η ) σ 2 2 ++ e 2 C ( η ) σ 2 3 , (2.1)</formula>
<text><location><page_4><loc_9><loc_87><loc_85><loc_90></location>where σ i , i = 1 , 2 , 3 are three Maurer-Cartan forms, and A, B and C are three functions of the coordinate η . The Maurer-Cartan forms are given by</text>
<formula><location><page_4><loc_33><loc_84><loc_85><loc_85></location>σ 1 = dψ +cos θ dφ, (2.2)</formula>
<formula><location><page_4><loc_33><loc_82><loc_85><loc_83></location>σ 2 = cos ψdθ +sin ψ sin θ dφ, (2.3)</formula>
<formula><location><page_4><loc_33><loc_79><loc_85><loc_81></location>σ 3 = -sin ψdθ +cos ψ sin θ dφ, (2.4)</formula>
<text><location><page_4><loc_9><loc_72><loc_85><loc_78></location>in terms of three coordinates θ, φ, and ψ of a unit S 3 . The line element (2.1) has an SU (2) isometry group. The metric satisfies exactly the vacuum Einstein's equations, provided the functions A ( η ) , B ( η ) and C ( η ) satisfy</text>
<formula><location><page_4><loc_30><loc_67><loc_85><loc_71></location>2 d 2 A ( η ) dη 2 = e 4 A ( η ) -( e 2 B ( η ) -e 2 C ( η ) ) 2 , (2.5)</formula>
<formula><location><page_4><loc_30><loc_63><loc_85><loc_67></location>2 d 2 B ( η ) dη 2 = e 4 B ( η ) -( e 2 C ( η ) -e 2 A ( η ) ) 2 , (2.6)</formula>
<formula><location><page_4><loc_30><loc_59><loc_85><loc_63></location>2 d 2 C ( η ) dη 2 = e 4 C ( η ) -( e 2 A ( η ) -e 2 B ( η ) ) 2 , (2.7)</formula>
<text><location><page_4><loc_9><loc_56><loc_17><loc_57></location>as well as</text>
<formula><location><page_4><loc_9><loc_49><loc_88><loc_55></location>4 dA ( η ) dη dB ( η ) dη +4 dB ( η ) dη dC ( η ) dη +4 dC ( η ) dη dA ( η ) dη = 2( e 2 A ( η )+2 B ( η ) + e 2 B ( η )+2 C ( η ) + e 2 C ( η )+2 A ( η ) ) -( e 4 A ( η ) + e 4 B ( η ) + e 4 C ( η ) ) . (2.8)</formula>
<text><location><page_4><loc_9><loc_42><loc_85><loc_47></location>We should notice equation (2.8) is the first integral of (2.5)-(2.7). All Bianchi type IX solutions are self-dual geometries, which leads to the following first order differential equations for the metric functions A ( η ) , B ( η ) and C ( η )</text>
<formula><location><page_4><loc_25><loc_37><loc_85><loc_40></location>2 dA ( η ) dη = -e 2 A ( η ) + e 2 B ( η ) + e 2 C ( η ) -β 1 e B ( η )+ C ( η ) , (2.9)</formula>
<formula><location><page_4><loc_25><loc_33><loc_85><loc_36></location>2 dB ( η ) dη = -e 2 B ( η ) + e 2 C ( η ) + e 2 A ( η ) -β 2 e C ( η )+ A ( η ) , (2.10)</formula>
<formula><location><page_4><loc_25><loc_29><loc_85><loc_32></location>2 dC ( η ) dη = -e 2 C ( η ) + e 2 A ( η ) + e 2 B ( η ) -β 3 e A ( η )+ B ( η ) , (2.11)</formula>
<text><location><page_4><loc_9><loc_26><loc_77><loc_27></location>where β i , i = 1 , 2 , 3 are three integration constants, which satisfy β 2 i = 0 or 4 and</text>
<formula><location><page_4><loc_41><loc_22><loc_85><loc_24></location>β i β j = 2 glyph[epsilon1] ijk β k . (2.12)</formula>
<text><location><page_4><loc_9><loc_14><loc_85><loc_21></location>The solutions to equation (2.12) are given by ( β 1 , β 2 , β 3 ) = (0 , 0 , 0) , ( β 1 , β 2 , β 3 ) = (2 , 2 , 2) , ( β 1 , β 2 , β 3 ) = (2 , -2 , -2) , ( β 1 , β 2 , β 3 ) = ( -2 , 2 , -2) , ( β 1 , β 2 , β 3 ) = ( -2 , -2 , 2). In appendix A, we show that we can not construct the exact solutions to the higher dimensional EinsteinMaxwell theory, based on embedding the four dimensional solutions where ( β 1 , β 2 , β 3 ) =</text>
<text><location><page_5><loc_9><loc_87><loc_85><loc_90></location>(2 , 2 , 2). We also show that the other three cases ( β 1 , β 2 , β 3 ) = (2 , -2 , -2) , ( β 1 , β 2 , β 3 ) = ( -2 , 2 , -2) , ( β 1 , β 2 , β 3 ) = ( -2 , -2 , 2) are equivalent to ( β 1 , β 2 , β 3 ) = (2 , 2 , 2).</text>
<text><location><page_5><loc_9><loc_82><loc_85><loc_87></location>Hence, the only viable and interesting solution which we consider is ( β 1 , β 2 , β 3 ) = (0 , 0 , 0). In fact, we can solve the set of differential equations (2.9)-(2.11) exactly, where ( β 1 , β 2 , β 3 ) = (0 , 0 , 0). We find the solutions as</text>
<formula><location><page_5><loc_32><loc_76><loc_85><loc_80></location>e 2 A ( η ) = c 2 cn ( c 2 η, k 2 ) dn ( c 2 η, k 2 ) sn ( -c 2 η, k 2 ) , (2.13)</formula>
<formula><location><page_5><loc_31><loc_72><loc_85><loc_76></location>e 2 B ( η ) = c 2 cn ( c 2 η, k 2 ) dn ( c 2 η, k 2 ) sn ( -c 2 η, k 2 ) , (2.14)</formula>
<formula><location><page_5><loc_32><loc_68><loc_85><loc_72></location>e 2 C ( η ) = c 2 dn ( c 2 η, k 2 ) cn ( c 2 η, k 2 ) sn ( -c 2 η, k 2 ) , (2.15)</formula>
<text><location><page_5><loc_9><loc_56><loc_85><loc_66></location>where c and k are integration constants, and sn ( z, l ), cn ( z, l ) and dn ( z, l ) are the Jacobi elliptic functions with the variable z and the parameter 0 ≤ l ≤ 1. For completeness, we present the explicit forms of the Jacobi elliptic functions sn ( z, l ), cn ( z, l ) and dn ( z, l ) in appendix B. By a straightforward calculation, we also find that the exact solutions (2.13)(2.15), indeed satisfy the other field equations (2.5)-(2.8). We change the coordinate η in the metric (2.1) to the new coordinate r , which is given by</text>
<formula><location><page_5><loc_40><loc_51><loc_85><loc_54></location>r = 2 c √ sn ( c 2 η, k 2 ) . (2.16)</formula>
<text><location><page_5><loc_9><loc_40><loc_85><loc_49></location>For simplicity, we choose coordinate η in the range [0 , α ( c )( k )(1) ] where α ( c )( k )( m ) is the mth positive root of sn ( c 2 η, k 2 ). We can equivalently consider any other range of the form [ α ( c )( k )(2 n ) , α ( c )( k )(2 n +1) ] with n = 1 , 2 , 3 , · · · or [ -α ( c )( k )(2 n ) , -α ( c )( k )(2 n -1) ] for the coordinate η . In figure 2.1, we show the typical behaviour of the new coordinate r versus η , where η ∈ [0 , α (1)( 1 2 )(1) ]. Note that α (1)( 1 2 )(1) glyph[similarequal] 3 . 193.</text>
<figure>
<location><page_6><loc_33><loc_67><loc_62><loc_90></location>
<caption>Figure 2.1: The radial coordinate r versus η , where we set c = and k = 1 / 2. The coordinate η ∈ [0 , α (1)( 1 2 )(1) ], where α (1)( 1 2 )(1) glyph[similarequal] 3 . 193.</caption>
</figure>
<text><location><page_6><loc_9><loc_56><loc_85><loc_59></location>After the change of coordinate (2.16), we find the metric (2.1) changes to the triaxial Bianchi type IX form, which is given by [38]</text>
<formula><location><page_6><loc_23><loc_50><loc_85><loc_54></location>ds 2 BIX = dr 2 √ F ( r ) + r 2 √ F ( r ) 4 ( σ 2 1 1 -a 4 1 r 4 + σ 2 2 1 -a 4 2 r 4 + σ 2 3 1 -a 4 3 r 4 ) , (2.17)</formula>
<text><location><page_6><loc_9><loc_47><loc_85><loc_49></location>where a i , i = 1 , 2 , 3 are three integration constants, and the metric function F ( r ) is given by</text>
<formula><location><page_6><loc_39><loc_41><loc_85><loc_46></location>F ( r ) = 3 ∏ i =1 (1 -a 4 i r 4 ) . (2.18)</formula>
<text><location><page_6><loc_9><loc_31><loc_85><loc_40></location>We note that the three integration constants in (2.17), are given by a 1 = 0 , a 2 = 2 kc and a 3 = 2 c , in terms of the variables and the parameters of the Jacobi elliptic functions 0 ≤ k ≤ 1 and c > 0. We also note that a 1 ≤ a 2 ≤ a 3 . The metric (2.17) is regular for all values of the radial coordinate r > 2 c . The Ricci scalar for the Bianchi type IX space (2.17) is zero and the Kretschmann invariant is given by</text>
<formula><location><page_6><loc_12><loc_18><loc_85><loc_30></location>K = 1649267441664 c 8 (2 ck -r ) 3 (8 c 3 k 3 +4 c 2 k 2 r +2 ckr 2 + r 3 ) 3 (4 c 2 + r 2 ) 3 (4 c 2 -r 2 ) 3 r 12 × (( k 8 -k 4 +1 16777216 ) r 24 -3 c 4 k 4 ( k 4 +1) r 20 1048576 + 15 c 8 k 8 r 16 65536 -5 c 12 k 8 ( k 4 +1) r 12 2048 + 3 c 16 k 8 ( k 8 +3 k 4 +1) r 8 256 -3 16 c 20 k 12 ( k 4 +1 ) r 4 + c 24 k 16 ) . (2.19)</formula>
<text><location><page_6><loc_9><loc_13><loc_85><loc_16></location>We notice the Kretschmann invariant (2.19) is regular everywhere, since r > 2 c . Moreover, all components of the Ricci tensor are regular, too.</text>
<section_header_level_1><location><page_7><loc_9><loc_86><loc_85><loc_91></location>3 Embedding the Bianchi type IX space in D ≥ 6 -dimensional Einstein-Maxwell theory</section_header_level_1>
<text><location><page_7><loc_9><loc_83><loc_56><loc_84></location>We consider the D -dimensional Einstein-Maxwell theory</text>
<formula><location><page_7><loc_36><loc_78><loc_85><loc_82></location>S = ∫ d D x √ -g ( R -1 4 F 2 ) , (3.1)</formula>
<text><location><page_7><loc_9><loc_75><loc_85><loc_77></location>where F µν = ∂ ν A µ -∂ µ A ν . We consider the D -dimensional ansatz for the metric, as in [39]</text>
<formula><location><page_7><loc_22><loc_70><loc_85><loc_73></location>ds 2 D = -dt 2 H D ( r, x ) 2 + H D ( r, x ) 2 D -3 ( dx 2 + x 2 d Ω 2 D -6 + ds 2 BIX ) , (3.2)</formula>
<text><location><page_7><loc_9><loc_65><loc_85><loc_68></location>where ds 2 BIX is given by (2.17) and d Ω 2 D -6 is the metric on a unit sphere S D -6 , where D ≥ 6. We take the components of the F µν as in [39]</text>
<formula><location><page_7><loc_34><loc_60><loc_85><loc_63></location>F tr = -α H D ( r, x ) 2 ∂H D ( r, x ) ∂r , (3.3)</formula>
<formula><location><page_7><loc_34><loc_56><loc_85><loc_59></location>F tx = -α H D ( r, x ) 2 ∂H D ( r, x ) ∂x , (3.4)</formula>
<text><location><page_7><loc_9><loc_51><loc_85><loc_54></location>where α is a constant. We note that (3.3) and (3.4) correspond to the potential A t = α H D ( r,x ) , where all the other components are zero, A µ = t = 0.</text>
<text><location><page_7><loc_46><loc_51><loc_46><loc_52></location>glyph[negationslash]</text>
<text><location><page_7><loc_9><loc_47><loc_85><loc_50></location>We show in appendix C that all the Einstein's and Maxwell's field equations are satisfied, if the metric function H ( r, x ) obeys the partial differential equation</text>
<formula><location><page_7><loc_10><loc_38><loc_97><loc_46></location>( r 9 256 -1 16 c 4 ( k 4 +1 ) r 5 + c 8 k 4 r )( F ( r ) ∂ 2 ∂r 2 H D ( r, x ) + √ F ( r ) ∂ 2 ∂x 2 H D ( r, x ) + F ' ( r ) ∂ ∂r H D ( r, x ) ) + 7 F ( r ) ( 3 r 8 1792 -5 c 4 ( k 4 +1) r 4 112 + c 8 k 4 ) ∂ ∂r H D ( r, x ) = 0 , (3.5)</formula>
<text><location><page_7><loc_9><loc_33><loc_85><loc_36></location>and the constant α in (3.3) and (3.4) is given by α 2 = D -2 D -3 . To solve the partial differential equation (3.5), we consider</text>
<formula><location><page_7><loc_36><loc_31><loc_85><loc_32></location>H D ( r, x ) = 1 + βR ( r ) X ( x ) , (3.6)</formula>
<text><location><page_7><loc_9><loc_23><loc_85><loc_30></location>where two functions R ( r ) and X ( x ) describe the separation of coordinates, and β is a constant. Plugging the equation (3.6) into equation (3.5), we find two ordinary differential equations for the functions R ( r ) and X ( x ). The differential equation for the function X ( x ) is</text>
<formula><location><page_7><loc_28><loc_20><loc_85><loc_23></location>x d 2 dx 2 X ( x ) + ( D -6) d dx X ( x ) -g 2 xX ( x ) = 0 , (3.7)</formula>
<text><location><page_7><loc_9><loc_18><loc_79><loc_19></location>where g denote the separation constant. We find the solutions to (3.7) are given by</text>
<formula><location><page_7><loc_34><loc_12><loc_85><loc_16></location>X ( x ) = x 1 I N ( gx ) x N + x 2 K N ( gx ) x N , (3.8)</formula>
<text><location><page_8><loc_9><loc_87><loc_85><loc_90></location>where I N and K N are the modified Bessel functions of the first and second kind, respectively, and x 1 and x 2 are the integration constants and</text>
<formula><location><page_8><loc_42><loc_83><loc_85><loc_86></location>N = D -7 2 . (3.9)</formula>
<text><location><page_8><loc_9><loc_81><loc_65><loc_82></location>Moreover, we find the differential equation for the function R ( r ) as</text>
<formula><location><page_8><loc_25><loc_71><loc_85><loc_80></location>( -256 c 8 k 4 +16 c 4 k 4 r 4 +16 c 4 r 4 -r 8 ) r d 2 d r 2 R ( r ) + ( 256 c 8 k 4 +16 c 4 k 4 r 4 +16 c 4 r 4 -3 r 8 ) d d r R ( r ) -g 2 r 5 √ 256 c 8 k 4 -16 c 4 k 4 r 4 -16 c 4 r 4 + r 8 R ( r ) = 0 . (3.10)</formula>
<text><location><page_8><loc_9><loc_63><loc_85><loc_69></location>We note that the radial differential equation (3.10) is independent of the dimension D of the spacetime. Tough we can't find any analytic solutions for the equation (3.10), however we try to find the analytic solutions to the differential equation (3.10) in asymptotic region r →∞ . In the limit of r →∞ , the equation (3.10) reduces to</text>
<formula><location><page_8><loc_32><loc_58><loc_85><loc_62></location>r d 2 d r 2 R ( r ) + 3 d d r R ( r ) + rg 2 R ( r ) = 0 . (3.11)</formula>
<text><location><page_8><loc_9><loc_56><loc_52><loc_57></location>The exact solutions to equation (3.11) are given by</text>
<formula><location><page_8><loc_35><loc_51><loc_85><loc_55></location>R ( r ) = r 1 J 1 ( gr ) r + r 2 Y 1 ( gr ) r , (3.12)</formula>
<text><location><page_8><loc_9><loc_44><loc_85><loc_51></location>for r → ∞ , where J 1 and Y 1 are the Bessel functions of the first and second kind, respectively. In figure 3.1 and 3.2, we plot the behaviour of the R ( r ) where r → ∞ . We notice the asymptotic radial function monotonically and periodically approaches zero, as r →∞ , independent of the sign of g .</text>
<figure>
<location><page_8><loc_18><loc_19><loc_77><loc_42></location>
<caption>Figure 3.1: The radial function R ( r ) for large values of r , where we set r 1 = 1 , r 2 = 0 (left), and r 1 = 0 , r 2 = 1 (right) and g = 1.</caption>
</figure>
<figure>
<location><page_9><loc_18><loc_67><loc_77><loc_91></location>
<caption>Figure 3.2: The radial function R ( r ) for large values of r , where we set r 1 = 1 , r 2 = 0 (left), and r 1 = 0 , r 2 = 1 (right) and g = -1.</caption>
</figure>
<text><location><page_9><loc_9><loc_45><loc_85><loc_59></location>Furnished by the asymptotic behaviour of the radial function, we solve numerically the radial differential equation (3.10) for 0 < k < 1. In figure 3.3, we plot the numerical solutions for the radial function R ( r ), where we set k = 1 2 , c = 1 and g = ± 2. We should notice that considering g = -2 in the radial differential equation (3.10) leads to the same radial differential equation with g = 2. So, we find that the numerical solutions for g = ± 2 are exactly identical, as long as we use the same initial conditions in numerical integration of the differential equation. Hence, in this section, we consider only positive values for the separation constant</text>
<formula><location><page_9><loc_45><loc_43><loc_85><loc_45></location>g ≥ 0 , (3.13)</formula>
<text><location><page_9><loc_9><loc_37><loc_85><loc_42></location>without loosing any generality. We notice from figure 3.3 that the radial function becomes divergent as r → 2, and decays rapidly as r →∞ , in agreement with the asymptotic solutions (3.12) and figures 3.1 and 3.2.</text>
<figure>
<location><page_10><loc_33><loc_67><loc_62><loc_90></location>
<caption>Figure 3.3: The numerical solution for the radial function R ( r ), where we set k = 1 2 , c = 1 and g = ± 2.</caption>
</figure>
<text><location><page_10><loc_9><loc_54><loc_85><loc_59></location>The general structure of the radial function is the same for other values of the Bianchi parameter c . The divergent behaviour of the radial function happens at r → 2 c and the radial function decays rapidly for r →∞ .</text>
<text><location><page_10><loc_9><loc_50><loc_85><loc_54></location>Moreover, in figure 3.4, we plot the numerical solutions for the radial function R ( r ), where we set k = 1 4 and k = 3 4 with c = 1 and g = 2.</text>
<figure>
<location><page_10><loc_18><loc_25><loc_77><loc_48></location>
<caption>Figure 3.4: The numerical solutions for the radial function R ( r ), where we set k = 1 4 (left) and k = 3 4 (right) with c = 1 , g = ± 2.</caption>
</figure>
<text><location><page_10><loc_9><loc_14><loc_85><loc_17></location>Tough the figures 3.3 and 3.4 are quite similar, however they have subtle dependence on the Bianchi parameter k . In figure 3.5, we plot three radial functions, over a small</text>
<text><location><page_11><loc_9><loc_81><loc_85><loc_91></location>interval of r , for k = 1 4 , 1 2 and 3 4 . As we notice from figure 3.5, the radial function R ( r ), in general, slightly increases with increasing the Bianchi parameter k . Changing the separation constant, in general, keeps the overall structure of the radial function. However, increasing the separation constant g leads to more oscillatory behaviour. In figure 3.6, we plot the radial functions, for k = 1 2 and two other separation constants g = 6 and g = 12.</text>
<text><location><page_11><loc_9><loc_76><loc_85><loc_81></location>Superimposing all the different solutions with the different separation constants g , we can write the most general solutions to the partial differential equation (3.5) in D -dimensions, as</text>
<formula><location><page_11><loc_22><loc_71><loc_85><loc_75></location>H D ( r, x ) = 1+ ∫ ∞ 0 dg x N ( P ( g ) I N ( gx ) + Q ( g ) K N ( gx ) ) R ( r ) , (3.14)</formula>
<text><location><page_11><loc_9><loc_67><loc_85><loc_70></location>where P ( g ) and Q ( g ) stand for the integration constants, for a specific value of the separation constant g , and N is given by (3.9).</text>
<figure>
<location><page_11><loc_33><loc_41><loc_62><loc_64></location>
<caption>Figure 3.5: The numerical solutions for the radial function R ( r ), where k = 3 4 (up), k = 1 2 (middle) and k = 1 4 (down) with c = 1 , g = 2.</caption>
</figure>
<figure>
<location><page_12><loc_18><loc_67><loc_77><loc_91></location>
<caption>Figure 3.6: The numerical solutions for the radial function R ( r ), where we set g = 6 (left) and g = 12 (right) with k = 1 2 , c = 1.</caption>
</figure>
<text><location><page_12><loc_9><loc_54><loc_85><loc_59></location>To find the functions P ( g ) and Q ( g ), we may compare the general solutions (3.14), with another related exact solutions to the theory. In fact, if we consider the large values for the radial coordinate r →∞ , then the Bianchi type IX metric (2.17) changes to</text>
<formula><location><page_12><loc_26><loc_49><loc_85><loc_53></location>d S 2 = dr 2 + r 2 4 ( dθ 2 + dφ 2 + dψ 2 ) + r 2 cos θ 2 dφdψ, (3.15)</formula>
<text><location><page_12><loc_9><loc_42><loc_85><loc_48></location>which is the metric on R 4 . Using the asymptotic metric (3.15) for the Bianchi type IX geometry, we find an exact solutions to the Einstein-Maxwell theory in D -dimensions, where the gravity is described by the metric</text>
<formula><location><page_12><loc_23><loc_37><loc_85><loc_41></location>d S 2 D = -dt 2 H D ( r, x ) 2 + H D ( r, x ) 2 D -3 ( dx 2 + x 2 d Ω 2 D -6 + d S 2 ) , (3.16)</formula>
<text><location><page_12><loc_9><loc_34><loc_43><loc_36></location>together with the Maxwell's field F µν , as</text>
<formula><location><page_12><loc_34><loc_29><loc_85><loc_32></location>F tr = -α H D ( r, x ) 2 ∂ H D ( r, x ) ∂r , (3.17)</formula>
<formula><location><page_12><loc_34><loc_25><loc_85><loc_28></location>F tx = -α H D ( r, x ) 2 ∂ H D ( r, x ) ∂x . (3.18)</formula>
<text><location><page_12><loc_9><loc_20><loc_85><loc_23></location>By solving all the Einstein's and Maxwell's field equations, we find that the metric function H D in D -dimensions, is given by the exact form</text>
<formula><location><page_12><loc_35><loc_15><loc_85><loc_19></location>H D ( r, x ) = 1 + γ ( r 2 + x 2 ) N +2 , (3.19)</formula>
<text><location><page_13><loc_9><loc_85><loc_85><loc_91></location>where γ is a constant and α 2 = D -2 D -3 . The D -dimensional asymptotic solution (3.16) with the metric function (3.19) describe a charged spacetime. The Ricci scalar and the Kretschmann invaraint for the solutions (3.16) are finite at r = x = 0. The invariants are finite on</text>
<formula><location><page_13><loc_41><loc_82><loc_85><loc_83></location>H D ( r, x ) = 0 , (3.20)</formula>
<text><location><page_13><loc_9><loc_77><loc_85><loc_80></location>as long as we consider the constant γ > 0 in (3.19). In fact, the Ricci scalar of the solutions (3.16) at r = x = 0 is given by</text>
<formula><location><page_13><loc_36><loc_71><loc_85><loc_75></location>R D = ξ D ( D -3)( D -4) γ 2 D -3 , (3.21)</formula>
<text><location><page_13><loc_9><loc_66><loc_85><loc_70></location>where ξ 6 = +1 , ξ D> 6 = -1. Similarly the Kretschmann invariant of the solutions (3.16) at r = x = 0 is given by</text>
<formula><location><page_13><loc_42><loc_63><loc_85><loc_67></location>K D = η D γ 4 D -3 , (3.22)</formula>
<text><location><page_13><loc_9><loc_59><loc_85><loc_62></location>where η D is a constant; η 6 = 348 , η 7 = 1064 , η 8 = 2560 , · · · . The Ricci scalar and the Kretschmann invariant, are given by</text>
<formula><location><page_13><loc_33><loc_53><loc_85><loc_57></location>R D = β D γ 2 (( r 2 + x 2 ) N +2 + γ ) 2( D -2) D -3 , (3.23)</formula>
<formula><location><page_13><loc_33><loc_48><loc_85><loc_53></location>K D = f D ( r, x, γ ) γ 2 (( r 2 + x 2 ) N +2 + γ ) 4( D -2) D -3 , (3.24)</formula>
<text><location><page_13><loc_9><loc_45><loc_72><loc_47></location>where β 6 = 6 , β 7 = -12 , β 8 = -20 , · · · and some of f D ( r, x, γ ) are given by</text>
<formula><location><page_13><loc_9><loc_39><loc_95><loc_44></location>f 6 ( r, x, γ ) = 12 ( 80 r 6 +240 r 4 x 2 +240 r 2 x 4 +80 x 6 -64 γ r 2 √ r 2 + x 2 -64 γ x 2 √ r 2 + x 2 +29 γ 2 ) , (3.25)</formula>
<formula><location><page_13><loc_9><loc_35><loc_95><loc_38></location>f 7 ( r, x, γ ) = 8 ( 300 r 8 +1200 r 6 x 2 +1800 r 4 x 4 +1200 r 2 x 6 +300 x 8 -300 γ r 4 -600 γ r 2 x 2 -300 γ x 4 + 133 γ 2 ) , (3.26)</formula>
<formula><location><page_13><loc_9><loc_29><loc_95><loc_35></location>f 8 ( r, x, γ ) = 80 ( 63 r 10 +315 r 8 x 2 +630 r 6 x 4 +630 r 4 x 6 +315 r 2 x 8 +63 x 10 -72 γ r 4 √ r 2 + x 2 -144 γ r 2 x 2 √ r 2 + x 2 -72 γ x 4 √ r 2 + x 2 +32 γ 2 ) . (3.27)</formula>
<text><location><page_13><loc_9><loc_21><loc_85><loc_26></location>As we notice, the Ricci scalar and the Kretschmann invariant are finite on H D ( r, x ) = 0, as long as we choose positive values for γ , where there are no solutions for H D ( r, x ) = 0. The electric charge of the black hole solutions (3.16) is given by</text>
<formula><location><page_13><loc_37><loc_16><loc_85><loc_19></location>Q D = 1 2 ∫ ∂ Σ D -1 ds µν F µν , (3.28)</formula>
<text><location><page_14><loc_9><loc_87><loc_85><loc_90></location>where Σ D -1 is a ( D -1)-dimensional spacelike hypersurface and ∂ Σ D -1 is its boundary. We find the components (3.17) and (3.18) are given by</text>
<formula><location><page_14><loc_29><loc_81><loc_85><loc_86></location>F tr = 2 αγ ( N +2) ( r 2 + x 2 ) N +1 r ( ( r 2 + x 2 ) N +2 + γ ) 2 , (3.29)</formula>
<formula><location><page_14><loc_29><loc_75><loc_85><loc_80></location>F tx = 2 αγ ( N +2) ( r 2 + x 2 ) N +1 x ( ( r 2 + x 2 ) N +2 + γ ) 2 . (3.30)</formula>
<text><location><page_14><loc_9><loc_71><loc_44><loc_73></location>Calculating the integral in (3.28), we find</text>
<formula><location><page_14><loc_41><loc_68><loc_85><loc_69></location>Q D = αγ Ω D -6 . (3.31)</formula>
<text><location><page_14><loc_9><loc_65><loc_78><loc_66></location>We note that Ω D -6 in (3.31) is the volume of a unit sphere S D -6 which is given by</text>
<formula><location><page_14><loc_38><loc_59><loc_85><loc_63></location>Ω D -6 = 2(2 π ) ( D -6) / 2 ( D -5)!! , (3.32)</formula>
<text><location><page_14><loc_9><loc_54><loc_85><loc_58></location>where n !! is equal to 1 · 3 · 5 · · · · · (2 k -1) · (2 k +1) for an odd n = 2 k +1, and is equal to 2 ( k +1 / 2) k ! √ π for an even n = 2 k .</text>
<text><location><page_14><loc_9><loc_50><loc_85><loc_54></location>Finding the exact solutions (3.16) with (3.19), enable us to find an integral equation for the functions P and Q in (3.14). In fact, we have the integral equation</text>
<formula><location><page_14><loc_19><loc_45><loc_85><loc_49></location>∫ ∞ 0 dg x N ( P ( g ) I N ( gx ) + Q ( g ) K N ( gx ) ) lim r →∞ R ( r ) = γ ( r 2 + x 2 ) N +2 . (3.33)</formula>
<text><location><page_14><loc_9><loc_40><loc_85><loc_44></location>However, we know lim r →∞ R ( r ) is given by (3.12). Considering r 1 = 1 g , r 2 = 0, we solve the integral equation (3.33) and find</text>
<formula><location><page_14><loc_31><loc_35><loc_85><loc_39></location>P ( g ) = 0 , Q ( g ) = γ 2 N +1 Γ( N +2) g N +3 , (3.34)</formula>
<text><location><page_14><loc_9><loc_29><loc_85><loc_34></location>where Γ( N +2) is the gamma function. Moreover, considering the other possibility r 1 = 0, does not lead to consistent solutions for the functions P and Q in the integral equation (3.33). Summarizing the results, we find the metric function in D -dimensions, as</text>
<formula><location><page_14><loc_24><loc_23><loc_85><loc_27></location>H D ( r, x ) = 1 + γ 2 N +1 Γ( N +2) ∫ ∞ 0 dg x N g N +3 K N ( gx ) R ( r ) . (3.35)</formula>
<text><location><page_14><loc_9><loc_13><loc_85><loc_22></location>We should note that the radial function R ( r ) as a part of integrand in (3.35), indeed depends on integration parameter g . Though it is not feasible to find explicitly the Ricci scalar and the Kretschmann invariant of the spacetime (3.2), as functions of the coordinates (due to the semi-analytic metric function (3.35)), however, we may expect that the solutions are regular everywhere (outside of an event horizon with a singularity at r = 2 c ), as they approach</text>
<text><location><page_15><loc_9><loc_71><loc_85><loc_90></location>smoothly to their asymptotic limit (3.16) with the metric function (3.19). We also expect the electric charge of the spacetime (3.2) is the same as equation (3.31). As the metric ansatz (3.2) and (3.3) and (3.4) are similar to [29], we also expect the spacetime (3.2) can describe the coalescence of the extremal charged black holes in D ≥ 6-dimensions, where the spatial section of the black holes consists a copy of the four-dimensional Bianchi type IX. The extremality is coming from noting that the only free parameter in the metric function (3.35) is γ . Hence the total mass of the gravitational system should be a multiple of γ . If the electric charge of the spacetime is given by (3.31), then we find that the total mass and the electric charge of the solution are proportional to each other. Of course, it is not feasible to find and verify explicitly the extremality of the solutions, due to the semi-analytic metric function (3.35).</text>
<section_header_level_1><location><page_15><loc_9><loc_63><loc_85><loc_68></location>4 The second class of solutions in D ≥ 6 -dimensional Einstein-Maxwell theory</section_header_level_1>
<text><location><page_15><loc_9><loc_54><loc_85><loc_61></location>In this section, we present the second independent class of solutions for the metric function which satisfies the partial differential equation (3.5). In separation of the coordinates, we replace g → ig in the differential equation (3.7). The solutions of the differential equation are</text>
<formula><location><page_15><loc_34><loc_51><loc_85><loc_54></location>˜ X ( x ) = ˜ x 1 J N ( gx ) x N + ˜ x 2 Y N ( gx ) x N , (4.1)</formula>
<text><location><page_15><loc_9><loc_47><loc_85><loc_50></location>in terms of the Bessel functions, where ˜ x 1 and ˜ x 2 are constants of integration, and N is given by (3.9). Moreover the radial differential equation becomes</text>
<formula><location><page_15><loc_10><loc_40><loc_95><loc_45></location>( -256 c 8 k 4 +16 c 4 k 4 r 4 +16 c 4 r 4 -r 8 ) r d 2 d r 2 ˜ R ( r ) + ( 256 c 8 k 4 +16 c 4 k 4 r 4 +16 c 4 r 4 -3 r 8 ) d d r ˜ R ( r ) + g 2 r 5 √ 256 c 8 k 4 -16 c 4 k 4 r 4 -16 c 4 r 4 + r 8 ˜ R ( r ) = 0 . (4.2)</formula>
<text><location><page_15><loc_9><loc_35><loc_85><loc_38></location>We present the numerical solutions to equation (4.2) for 0 < k < 1, where its asymptotic form is given by</text>
<formula><location><page_15><loc_32><loc_31><loc_85><loc_35></location>r d 2 d r 2 ˜ R ( r ) + 3 d d r ˜ R ( r ) -rg 2 ˜ R ( r ) = 0 . (4.3)</formula>
<text><location><page_15><loc_9><loc_29><loc_51><loc_30></location>The exact solutions to equation (4.3) are given by</text>
<formula><location><page_15><loc_35><loc_24><loc_85><loc_27></location>˜ R ( r ) = ˜ r 1 I 1 ( gr ) r + ˜ r 2 K 1 ( gr ) r , (4.4)</formula>
<text><location><page_15><loc_9><loc_14><loc_85><loc_22></location>where I 1 and K 1 are the modified Bessel functions of the first and second kind, respectively, and ˜ r 1 and ˜ r 2 are constants. In figure 4.1, we plot the behaviour of the ˜ R ( r ) where r →∞ . We notice two very different behaviours for the asymptotic radial function, as r → ∞ . Moreover, the plots for ˜ R ( r ) where r → ∞ , with g = -1 are exactly the same as in figure 4.1, since the modified Bessel functions I 1 and K 1 are even functions of gr .</text>
<figure>
<location><page_16><loc_18><loc_68><loc_77><loc_91></location>
<caption>Figure 4.1: The radial function ˜ R ( r ) for large values of r , where we set ˜ r 1 = 1 , ˜ r 2 = 0 (left), and ˜ r 1 = 0 , ˜ r 2 = 1 (right) and g = ± 1.</caption>
</figure>
<text><location><page_16><loc_9><loc_44><loc_85><loc_60></location>In figure 4.2, we plot the numerical solutions for the radial function ˜ R ( r ), where we set k = 1 2 , c = 1 and g = ± 2. We should notice that considering g = -2 in the radial differential equation (4.2) leads to the same radial differential equation with g = 2. So, we find that the numerical solutions for g = ± 2 are exactly identical, as long as we use the same initial conditions in numerical integration of the differential equation. Hence, in this section, we consider only positive values for the separation constant, as in (3.13), without loosing any generality. We notice from figure 4.2 that the radial function approaches zero for r → 2 c , and becomes large, as r → ∞ , in agreement with the asymptotic solutions (4.4), where ˜ r 1 = 1 , ˜ r 2 = 0, and the left plot in figure 4.1.</text>
<figure>
<location><page_16><loc_33><loc_19><loc_62><loc_42></location>
<caption>Figure 4.2: The numerical solution for the radial function ˜ R ( r ), where we set k = 1 2 , c = 1 and g = ± 2.</caption>
</figure>
<text><location><page_17><loc_9><loc_85><loc_85><loc_90></location>The general structure of the radial function is the same for the other values of the Bianchi parameter c . The divergent behaviour of the radial function happens at r → ∞ and the radial function approaches zero, for r → 2 c .</text>
<text><location><page_17><loc_9><loc_74><loc_85><loc_85></location>Moreover, in figure 4.3, we plot the numerical solutions for the radial function ˜ R ( r ), where we set k = 1 4 and k = 3 4 with c = 1 and g = 2. Tough the figures 4.2 and 4.3 are quite similar, however they have subtle dependence on the Bianchi parameter k . In figure 4.4, we plot three radial functions, over a small interval of r , for k = 1 4 , 1 2 and 3 4 . As we notice from figure 4.4, the radial function ˜ R ( r ), in general, slightly increases with increasing the Bianchi parameter k , very similar to the situation for the first radial solutions.</text>
<text><location><page_17><loc_9><loc_67><loc_85><loc_74></location>Changing the separation constant, in general, keeps the overall structure of the radial function. However, increasing the separation constant g leads to a slower decaying behaviour for the radial function. In figure 4.5, we plot the radial functions, for k = 1 2 and two other separation constants g = 6 and g = 12.</text>
<text><location><page_17><loc_9><loc_63><loc_85><loc_66></location>Hence, we find the second most general solutions for the metric function in D -dimensions, as given by</text>
<formula><location><page_17><loc_22><loc_58><loc_85><loc_62></location>˜ H D ( r, x ) = 1+ ∫ ∞ 0 dg x N ( ˜ P ( g ) J N ( gx ) + ˜ Q ( g ) Y N ( gx ) ) ˜ R ( r ) , (4.5)</formula>
<text><location><page_17><loc_9><loc_53><loc_85><loc_57></location>where ˜ P ( g ) and ˜ Q ( g ) stand for the integration constants, for a specific value of the separation constant g and N is given by (3.9).</text>
<figure>
<location><page_17><loc_18><loc_28><loc_78><loc_51></location>
<caption>Figure 4.3: The numerical solutions for the radial function ˜ R ( r ), where we set k = 1 4 (left) and k = 3 4 (right) with c = 1 , g = 2.</caption>
</figure>
<figure>
<location><page_18><loc_33><loc_68><loc_62><loc_91></location>
<caption>Figure 4.4: The numerical solutions for the radial function ˜ R ( r ), where k = 3 4 (up), k = 1 2 (middle) and k = 1 4 (down) with c = 1 , g = 2.</caption>
</figure>
<figure>
<location><page_18><loc_18><loc_34><loc_78><loc_58></location>
<caption>Figure 4.5: The numerical solutions for the radial function ˜ R ( r ), where we set g = 6 (left) and g = 12 (right) with k = 1 2 , c = 1.</caption>
</figure>
<text><location><page_18><loc_9><loc_23><loc_85><loc_27></location>Requiring the metric function (4.5) in the limit of r →∞ , reduces to the exact solutions (3.19), we find an integral equation for the functions ˜ P ( g ) and ˜ Q ( g ), as</text>
<formula><location><page_18><loc_19><loc_18><loc_85><loc_22></location>∫ ∞ 0 dg x N ( ˜ P ( g ) J N ( gx ) + ˜ Q ( g ) Y N ( gx ) ) lim r →∞ ˜ R ( r ) = γ ( r 2 + x 2 ) N +2 . (4.6)</formula>
<text><location><page_18><loc_9><loc_14><loc_85><loc_17></location>Using equation (4.4) for lim r →∞ ˜ R ( r ) with ˜ r 1 = 1 g , ˜ r 2 = 0, we can solve the integral equation</text>
<text><location><page_19><loc_9><loc_89><loc_20><loc_90></location>(4.6) and find</text>
<formula><location><page_19><loc_32><loc_85><loc_85><loc_89></location>˜ P ( g ) = γ Γ( -N ) 2 N +2 ( N +1) g N +3 , ˜ Q ( g ) = 0 . (4.7)</formula>
<text><location><page_19><loc_9><loc_80><loc_85><loc_84></location>Moreover, considering the other possibility ˜ r 1 = 0, does not lead to consistent solutions for the functions ˜ P and ˜ Q in the integral equation (4.6). Summarizing the results, we find the second metric function in D -dimensions, as</text>
<formula><location><page_19><loc_24><loc_75><loc_85><loc_79></location>˜ H D ( r, x ) = 1 + γ Γ( -N ) 2 N +2 ( N +1) ∫ ∞ 0 dg x N g N +3 J N ( gx ) ˜ R ( r ) . (4.8)</formula>
<section_header_level_1><location><page_19><loc_9><loc_68><loc_85><loc_73></location>5 The solutions in D ≥ 6 dimensional Einstein-Maxwell theory with cosmological constant</section_header_level_1>
<text><location><page_19><loc_9><loc_59><loc_85><loc_66></location>In this section, we consider the D ≥ 6 Einstein-Maxwell theory with a cosmological constant Λ. We show that there are non-trivial solutions to the D ≥ 6 dimensional Einstein-Maxwell theory with the cosmological constant. We start by considering the D ≥ 6 dimensional metric, as</text>
<formula><location><page_19><loc_19><loc_56><loc_85><loc_58></location>ds 2 D = -H D ( t, r, x ) -2 dt 2 + H D ( t, r, x ) 1 N +2 ( dx 2 + x 2 d Ω D -6 + ds 2 BIX ) , (5.1)</formula>
<text><location><page_19><loc_9><loc_39><loc_85><loc_55></location>where the metric function depends on the coordinate t , as well as the coordinates r and x . The components of the Maxwell's field, are given by equations (3.3) and (3.4), after substitution H D ( r, x ) to H D ( t, r, x ). We should note considering the ansatzes (3.2)-(3.4) with metric function H depending on more spatial directions lead to inconsistencies (appendix D). Inspired with the well-known time-dependent metric functions in the spacetimes with cosmological constant (such as expanding/contracting patches of the de-Sitter spacetime or FLRWspacetime), the above-mentioned inconsistencies could be resolved only by considering a time dependent metric function in the metric, as in (5.1). Moreover, the components of the Maxwell's field, are given by</text>
<formula><location><page_19><loc_32><loc_34><loc_85><loc_38></location>F tr = -α H D ( t, r, x ) 2 ∂H D ( t, r, x ) ∂r , (5.2)</formula>
<formula><location><page_19><loc_32><loc_30><loc_85><loc_33></location>F tx = -α H D ( t, r, x ) 2 ∂H D ( t, r, x ) ∂x . (5.3)</formula>
<text><location><page_19><loc_9><loc_26><loc_85><loc_29></location>A lengthy calculation shows that all the Einstein's and Maxwell's field equations are satisfied by taking the separation of variables in the metric function, as</text>
<formula><location><page_19><loc_34><loc_23><loc_85><loc_24></location>H D ( t, r, x ) = T ( t ) + R ( r ) X ( x ) , (5.4)</formula>
<text><location><page_19><loc_9><loc_16><loc_85><loc_21></location>where the functions R ( r ) and X ( x ) satisfy exactly equations (3.10) and (3.7) for the first class of solutions, respectively, and α 2 = D -2 D -3 . Of course the analytical continuation g → ig , yields the corresponding equations for ˜ R ( r ) and ˜ X ( x ). Moreover, we find</text>
<formula><location><page_19><loc_41><loc_13><loc_85><loc_15></location>T ( t ) = 1 + λt, (5.5)</formula>
<text><location><page_20><loc_9><loc_89><loc_14><loc_90></location>where</text>
<formula><location><page_20><loc_34><loc_85><loc_85><loc_89></location>λ = ( D -3) √ 2Λ ( D -2)( D -1) . (5.6)</formula>
<text><location><page_20><loc_9><loc_79><loc_85><loc_84></location>Of course, we should consider only the positive cosmological constant Λ to have a real λ . To summarize, we find two classes of the cosmological solutions, where the metric functions are given by</text>
<text><location><page_20><loc_9><loc_73><loc_12><loc_74></location>and</text>
<formula><location><page_20><loc_21><loc_75><loc_85><loc_79></location>H D ( t, r, x ) = 1 + λt + γ 2 N +1 Γ( N +2) ∫ ∞ 0 dg x N g N +3 K N ( gx ) R ( r ) , (5.7)</formula>
<formula><location><page_20><loc_22><loc_69><loc_85><loc_73></location>˜ H D ( t, r, x ) = 1 + λt + γ Γ( -N ) 2 N +2 ( N +1) ∫ ∞ 0 dg x N g N +3 J N ( gx ) ˜ R ( r ) . (5.8)</formula>
<text><location><page_20><loc_9><loc_65><loc_85><loc_69></location>We note that the two metric functions (5.7) and (5.8) definitely approach to the exact metric function</text>
<formula><location><page_20><loc_32><loc_63><loc_85><loc_66></location>H D ( t, r, x ) = 1 + λt + γ ( r 2 + x 2 ) N +2 , (5.9)</formula>
<text><location><page_20><loc_9><loc_58><loc_85><loc_62></location>in the limit of large radial coordinate. The metric function (5.9) is an exact solution to the Einstein's and Maxwell's field equations with a cosmological constant Λ, where</text>
<formula><location><page_20><loc_21><loc_53><loc_85><loc_57></location>d S 2 D = -dt 2 H D ( t, r, x ) 2 + H D ( t, r, x ) 2 D -3 ( dx 2 + x 2 d Ω 2 D -6 + d S 2 ) , (5.10)</formula>
<text><location><page_20><loc_9><loc_51><loc_31><loc_52></location>and d S 2 is given by (3.15).</text>
<text><location><page_20><loc_12><loc_49><loc_85><loc_50></location>To get a glimpse of the solution (5.1), we consider the very late time coordinate, and find</text>
<formula><location><page_20><loc_28><loc_45><loc_85><loc_47></location>ds 2 D = -dτ 2 + e λ N +2 τ ( dx 2 + x 2 d Ω D -6 + ds 2 BIX ) , (5.11)</formula>
<text><location><page_20><loc_9><loc_42><loc_14><loc_44></location>where</text>
<formula><location><page_20><loc_37><loc_39><loc_85><loc_42></location>τ = ln(1 + λt ) λ glyph[similarequal] ln( λt ) λ . (5.12)</formula>
<text><location><page_20><loc_9><loc_32><loc_85><loc_38></location>The constantτ hypersurface of the metric (5.11) describes the foliation of ( D -1)-dimensional de-Sitter space, by flat spacelike Bianchi type IX. The volume of such a hypersurface reaches its minimum at τ = 0, and increases exponentially with τ . The Hubble parameter H D for the D -dimensional metric (5.11) is equal to</text>
<formula><location><page_20><loc_31><loc_26><loc_85><loc_30></location>H D = λ 2( N +2) = √ 2Λ ( D -2)( D -1) , (5.13)</formula>
<text><location><page_20><loc_9><loc_22><loc_82><loc_24></location>and the cosmic volume of the spacetime (5.11) is given by ∫ ∞ 0 V D ( τ, r, x ) dτdrdx , where</text>
<formula><location><page_20><loc_32><loc_18><loc_85><loc_21></location>V D ( τ, r, x ) = 1 4 e ( D -1) λ 2( N +2) τ r 3 x D -6 Ω D -6 . (5.14)</formula>
<text><location><page_20><loc_9><loc_13><loc_85><loc_17></location>We note that Ω D -6 in (5.14) is the volume of a unit sphere S D -6 which is given by equation (3.32). We also find the cosmic volume of the constantτ hypersurface of the metric (5.11) is</text>
<figure>
<location><page_21><loc_33><loc_63><loc_62><loc_85></location>
<caption>equal to ∫ ∞ 0 V D ( τ = constant , r, x ) drdx . As we notice from figure 5.1, the Hubble parameter (5.13) reduces rapidly with increasing the dimensionality of the spacetime (5.11).Figure 5.1: The Hubble parameter H D versus the cosmological dimension, where we set Λ = 1.</caption>
</figure>
<text><location><page_21><loc_9><loc_25><loc_85><loc_54></location>We note that the metric (5.11) describes the big bang patch. This patch covers half the dS spacetime from a big bang initiated at past horizon. The bang continues into the Bianchi foliations at future infinity. The other half of dS spacetime is covered by a big crunch patch. This patch initiates at past infinity with the Bianchi foliations. The crush continues towards the future horizon. The contracting patch of solutions at future infinity implies that black hole solutions based on Bianchi type IX space can describe the coalescence of the black holes in asymptotically dS spacetimes. Moreover, there is a correspondence between the phenomena occurring near the boundary (or in the deep interior) of asymptotically dS (or AdS) spacetime and the ultraviolet (infrared) physics in the dual CFT [40]. As a result, any solutions in asymptotically (locally) dS spacetimes lead to interpretation in terms of renormalization group flows, and the associated generalized c-theorem. The renormalization group flows toward the infrared in any contracting patch of dS spacetimes. Moreover the renormalization group flows toward the ultraviolet in any expanding patch of dS spacetimes. A useful quantity to represent the dS metric with different boundary geometries, such as direct products of flat space, the sphere and hyperbolic space, is the c-function. The cfunction for the D -dimensional asymptotically dS spacetime (5.11) is given by [41]</text>
<formula><location><page_21><loc_38><loc_20><loc_85><loc_24></location>c ∼ 1 ( G ij n i n j ) ( D -2) / 2 , (5.15)</formula>
<text><location><page_21><loc_9><loc_13><loc_85><loc_19></location>where n i is the unit normal vector to the hypersurface which gives the Hamiltonian constraint and G is the Einstein tensor [41]. The c-function should show an increasing (decreasing) behaviour versus time for any expanding (contracting) patch of the spacetime.</text>
<text><location><page_22><loc_9><loc_82><loc_85><loc_90></location>In figure 5.2, we plot the behaviour of the c -function (5.15 ) for two different spacetime dimensions. Both figures show increasing behaviour for the c -function which shows expansion of the constantτ hypersurface with the cosmological time τ . Moreover, we notice that increasing the dimension of spacetime leads to deceasing the value of the c -function. So, the constantτ hypersurfaces expand slower in the higher dimensions.</text>
<figure>
<location><page_22><loc_18><loc_57><loc_77><loc_80></location>
<caption>Figure 5.2: The c -function of the spacetime (5.11) versus the cosmological time coordinate τ for D = 6 (left) and D = 7 (right), where we set k = 1 2 , c = 1 , Λ = 4 , r = 3.</caption>
</figure>
<section_header_level_1><location><page_22><loc_9><loc_43><loc_85><loc_48></location>6 The D ≥ 6 solutions with k = 0 and k = 1 in EinsteinMaxwell theory</section_header_level_1>
<text><location><page_22><loc_9><loc_36><loc_85><loc_41></location>In previous sections, we constructed solutions to D ≥ 6-dimensional Einstein-Maxwell theory, with and without cosmological constant, based on Bianchi type IX geometry with 0 < k < 1. In this section, we consider such solutions, where k = 0 and k = 1.</text>
<section_header_level_1><location><page_22><loc_9><loc_32><loc_20><loc_33></location>6.1 k = 0</section_header_level_1>
<text><location><page_22><loc_9><loc_27><loc_85><loc_30></location>First we consider the Bianchi type IX metric (2.17) with k = 0. The metric (2.17) reduces to √</text>
<formula><location><page_22><loc_25><loc_22><loc_85><loc_27></location>ds 2 k =0 = dr 2 √ 1 -a 4 r 4 + r 4 -a 4 4 ( σ 2 1 + σ 2 2 ) + r 2 4 σ 2 3 √ 1 -a 4 r 4 , (6.1)</formula>
<text><location><page_22><loc_9><loc_16><loc_85><loc_21></location>where we set a = 2 c and the radial coordinate r ≥ a . The metric (6.1) is the metric for the Eguchi-Hanson type I geometry [42]. The Ricci scalar for (6.1) is identically zero, and the Kretschmann invariant is</text>
<formula><location><page_22><loc_36><loc_13><loc_85><loc_16></location>K = 384 a 8 ( r 2 + a 2 ) 3 ( r 2 -a 2 ) 3 , (6.2)</formula>
<text><location><page_23><loc_9><loc_89><loc_84><loc_90></location>which indicates r = a is a singularity. The radial differential equation for R ( r ) is given by</text>
<formula><location><page_23><loc_20><loc_84><loc_85><loc_88></location>r ( a 4 -r 4 ) d 2 d r 2 R ( r ) + ( a 4 -3 r 4 ) d d r R ( r ) -g 2 r 3 √ r 4 -a 4 R ( r ) = 0 , (6.3)</formula>
<text><location><page_23><loc_9><loc_79><loc_85><loc_82></location>while the differential equation for X ( x ) is the same as (3.7) with the solutions (3.8). We find the real analytical solutions for (6.3), which is</text>
<formula><location><page_23><loc_24><loc_70><loc_85><loc_78></location>R ( r ) = 1 2 H C ( 0 , 0 , 0 , ig 2 a 2 2 , -ig 2 a 2 4 , a 2 -i √ r 4 -a 4 2 a 2 ) + 1 2 H ∗ C ( 0 , 0 , 0 , ig 2 a 2 2 , -ig 2 a 2 4 , a 2 -i √ r 4 -a 4 2 a 2 ) , (6.4)</formula>
<text><location><page_23><loc_9><loc_62><loc_85><loc_68></location>where H C is the Heun-C function. In figure 6.1, we plot the radial function for different values of the Eguchi-Hanson parameter a . We note that the radial function has the oscillatory behaviour similar to cases where 0 < k < 1, however the radial function remains finite at r = a .</text>
<figure>
<location><page_23><loc_17><loc_37><loc_77><loc_60></location>
<caption>Figure 6.1: The radial function R ( r ) for k = 0, where we set a = 1 (left), and a = 2 (right) and g = 2.</caption>
</figure>
<text><location><page_23><loc_9><loc_25><loc_85><loc_29></location>Combining the different solutions for the radial function R ( r ) and X ( x ), we find the most general solution for the metric function H D ( r, x ) in D -dimensions, where k = 0, as</text>
<formula><location><page_23><loc_22><loc_15><loc_85><loc_23></location>H D ( r, x ) = 1+ ∫ ∞ 0 dg x N ( P 0 ( g ) I N ( gx ) + Q 0 ( g ) K N ( gx ) ) × glyph[Rfractur] ( H C ( 0 , 0 , 0 , ig 2 a 2 2 , -ig 2 a 2 4 , a 2 -i √ r 4 -a 4 2 a 2 )) , (6.5)</formula>
<text><location><page_24><loc_9><loc_85><loc_85><loc_90></location>where P 0 ( g ) and Q 0 ( g ) stand for the integration constants, for a specific value of the separation constant g . To find the functions P 0 ( g ) and Q 0 ( g ), we consider the limit r →∞ , where the Eguchi-Hanson type I space (6.1) becomes</text>
<formula><location><page_24><loc_34><loc_80><loc_85><loc_84></location>ds 2 k =0 = dr 2 + r 2 4 ( σ 2 1 + σ 2 2 + σ 2 3 ) . (6.6)</formula>
<text><location><page_24><loc_9><loc_74><loc_85><loc_80></location>The asymptotic line element (6.6) describes R 4 , and embedding it in the D -dimensional theory, leads to the exact solution (3.16)-(3.18) with the metric function (3.19). We find the integral equation</text>
<formula><location><page_24><loc_10><loc_66><loc_88><loc_74></location>∫ ∞ 0 dg x N ( P 0 ( g ) I N ( gx ) + Q 0 ( g ) K N ( gx ) ) lim r →∞ glyph[Rfractur] ( H C ( 0 , 0 , 0 , ig 2 a 2 2 , -ig 2 a 2 4 , a 2 -i √ r 4 -a 4 2 a 2 )) = γ ( r 2 + x 2 ) N +2 , (6.7)</formula>
<text><location><page_24><loc_9><loc_63><loc_53><loc_64></location>for the functions P 0 ( g ) and Q 0 ( g ). Moreover, we find</text>
<formula><location><page_24><loc_22><loc_58><loc_85><loc_63></location>lim r →∞ ( H C ( 0 , 0 , 0 , ig 2 a 2 2 , -ig 2 a 2 4 , a 2 -i √ r 4 -a 4 2 a 2 )) = 2 gr J 1 ( gr ) . (6.8)</formula>
<text><location><page_24><loc_9><loc_55><loc_79><loc_57></location>After a very long calculation, we find the solutions to the integral equation (6.7), as</text>
<formula><location><page_24><loc_31><loc_50><loc_85><loc_54></location>P 0 ( g ) = γ Γ( -N ) 2 N +3 ( N +1) g N +3 , Q 0 ( g ) = 0 . (6.9)</formula>
<text><location><page_24><loc_9><loc_44><loc_85><loc_49></location>Hence, we find the exact solutions for the metric function H ( r, x ) in D -dimensional EinsteinMaxwell theory with an embedded four-dimensional Eguchi-Hanson type I geometry (6.1) in the spatial section of the spacetime, as</text>
<formula><location><page_24><loc_9><loc_37><loc_91><loc_44></location>H D ( r, x ) = 1+ γ Γ( -N ) 2 N +3 ( N +1) ∫ ∞ 0 dg x N g N +3 I N ( gx ) glyph[Rfractur] ( H C ( 0 , 0 , 0 , ig 2 a 2 2 , -ig 2 a 2 4 , a 2 -i √ r 4 -a 4 2 a 2 )) . (6.10)</formula>
<text><location><page_24><loc_9><loc_32><loc_85><loc_35></location>We also verify that our numerical solutions to the differential equation (3.10), where k = 0, represent exactly the profile of the Heun-C function in equation (6.10).</text>
<text><location><page_24><loc_9><loc_29><loc_85><loc_32></location>Changing the separation constant g → ig generates the second class of solutions, where k = 0. We find the radial equation</text>
<formula><location><page_24><loc_20><loc_24><loc_85><loc_27></location>r ( a 4 -r 4 ) d 2 d r 2 ˜ R ( r ) + ( a 4 -3 r 4 ) d d r ˜ R ( r ) + g 2 r 3 √ r 4 -a 4 ˜ R ( r ) = 0 , (6.11)</formula>
<text><location><page_24><loc_9><loc_21><loc_41><loc_23></location>where the exact solutions are given by</text>
<formula><location><page_24><loc_24><loc_12><loc_85><loc_21></location>˜ R ( r ) = 1 2 H C ( 0 , 0 , 0 , -ig 2 a 2 2 , ig 2 a 2 4 , a 2 -i √ r 4 -a 4 2 a 2 ) + 1 2 H ∗ C ( 0 , 0 , 0 , -ig 2 a 2 2 , ig 2 a 2 4 , a 2 -i √ r 4 -a 4 2 a 2 ) . (6.12)</formula>
<text><location><page_25><loc_9><loc_87><loc_85><loc_91></location>In figure 6.2, we plot the radial function ˜ R for different values of the Eguchi-Hanson parameter a .</text>
<figure>
<location><page_25><loc_18><loc_62><loc_78><loc_85></location>
<caption>Figure 6.2: The radial function ˜ R ( r ) for k = 0, where we set a = 1 (left), and a = 2 (right) and g = 2.</caption>
</figure>
<text><location><page_25><loc_9><loc_49><loc_85><loc_55></location>We then superimpose the different solutions for the radial function ˜ R ( r ) and ˜ X ( x ), as given by (4.1), to find the second most general solution for the metric function ˜ H D ( r, x ) in D -dimensions, where k = 0, as</text>
<formula><location><page_25><loc_22><loc_38><loc_85><loc_46></location>˜ H D ( r, x ) = 1+ ∫ ∞ 0 dg x N ( ˜ P 0 ( g ) J N ( gx ) + ˜ Q 0 ( g ) Y N ( gx ) ) × glyph[Rfractur] ( H C ( 0 , 0 , 0 , -ig 2 a 2 2 , ig 2 a 2 4 , a 2 -i √ r 4 -a 4 2 a 2 )) , (6.13)</formula>
<text><location><page_25><loc_9><loc_32><loc_85><loc_37></location>where ˜ P 0 ( g ) and ˜ Q 0 ( g ) stand for the integration constants, for a specific value of the separation constant g . To find the functions ˜ P 0 ( g ) and ˜ Q 0 ( g ), we consider the limit r →∞ , where the Eguchi-Hanson type I space (6.1) becomes (6.6). Hence we find the integral equation</text>
<formula><location><page_25><loc_10><loc_22><loc_88><loc_30></location>∫ ∞ 0 dg x N ( ˜ P 0 ( g ) J N ( gx ) + ˜ Q 0 ( g ) Y N ( gx ) ) lim r →∞ glyph[Rfractur] ( H C ( 0 , 0 , 0 , -ig 2 a 2 2 , ig 2 a 2 4 , a 2 -i √ r 4 -a 4 2 a 2 )) = γ ( r 2 + x 2 ) N +2 , (6.14)</formula>
<text><location><page_25><loc_9><loc_18><loc_45><loc_20></location>for the functions ˜ P 0 ( g ) and ˜ Q 0 ( g ). We find</text>
<formula><location><page_25><loc_22><loc_13><loc_85><loc_17></location>lim r →∞ ( H C ( 0 , 0 , 0 , -ig 2 a 2 2 , ig 2 a 2 4 , a 2 -i √ r 4 -a 4 2 a 2 )) = 2 gr I 1 ( gr ) . (6.15)</formula>
<text><location><page_26><loc_9><loc_89><loc_80><loc_90></location>After a very long calculation, we find the solutions to the integral equation (6.14), as</text>
<formula><location><page_26><loc_28><loc_84><loc_85><loc_88></location>˜ P 0 ( g ) = ( -1) N +1 γ Γ( -N -1) 2 N +3 g N +3 , ˜ Q 0 ( g ) = 0 . (6.16)</formula>
<text><location><page_26><loc_9><loc_77><loc_85><loc_83></location>Hence, we find the second exact solutions for the metric function ˜ H ( r, x ) in D -dimensional Einstein-Maxwell theory with an embedded four-dimensional Eguchi-Hanson type I geometry (6.1) in the spatial section of the spacetime, as</text>
<formula><location><page_26><loc_22><loc_68><loc_85><loc_76></location>˜ H D ( r, x ) = 1+ ( -1) N +1 γ Γ( -N -1) 2 N +3 ∫ ∞ 0 dg x N g N +3 J N ( gx ) × glyph[Rfractur] ( H C ( 0 , 0 , 0 , -ig 2 a 2 2 , ig 2 a 2 4 , a 2 -i √ r 4 -a 4 2 a 2 )) . (6.17)</formula>
<text><location><page_26><loc_9><loc_64><loc_85><loc_67></location>We also verify that our numerical solutions to the differential equation (4.2), where k = 0, represent exactly the profile of the Heun-C function in equation (6.17).</text>
<section_header_level_1><location><page_26><loc_9><loc_59><loc_20><loc_61></location>6.2 k = 1</section_header_level_1>
<text><location><page_26><loc_9><loc_55><loc_85><loc_58></location>In this section, we consider the Bianchi type IX metric (2.17) with k = 1. The metric (2.17) reduces to</text>
<formula><location><page_26><loc_29><loc_51><loc_85><loc_55></location>ds 2 k =1 = dr 2 1 -a 4 r 4 + r 4 -a 4 4 r 2 σ 2 1 + r 2 4 ( σ 2 2 + σ 2 3 ) , (6.18)</formula>
<text><location><page_26><loc_9><loc_47><loc_85><loc_50></location>which is the known metric for the Eguchi-Hanson type II space, where r ≥ a [42]. The Ricci scalar for (6.18) is identically zero, and the Kretschmann invariant is given by</text>
<formula><location><page_26><loc_42><loc_42><loc_85><loc_45></location>K = 384 a 8 r 12 , (6.19)</formula>
<text><location><page_26><loc_9><loc_39><loc_40><loc_41></location>which is regular and finite, for r ≥ a .</text>
<text><location><page_26><loc_9><loc_34><loc_85><loc_39></location>The exact solutions for the metric function H ( r, x ) in D -dimensional Einstein-Maxwell theory with an embedded four-dimensional Eguchi-Hanson type II geometry (6.18) in the spatial section of the spacetime, is given by [43]</text>
<formula><location><page_26><loc_15><loc_27><loc_85><loc_33></location>H D ( r, x ) = 1+ γ 2 ξ D ∫ ∞ 0 dg x N g N +3 K N ( gx ) H C ( 0 , 0 , 0 , -g 2 a 2 2 , g 2 a 2 4 , a 2 -r 2 2 a 2 ) , (6.20)</formula>
<text><location><page_26><loc_9><loc_20><loc_85><loc_25></location>where ξ 6+2 n = √ π 2 (2 n + 1)!! , n = 0 , 1 , 2 , · · · for even dimensions D = 6 + 2 n and ξ 7+2 n = (2 n + 2)!! , n = 0 , 1 , 2 , · · · for odd dimensions D = 7 + 2 n . Moreover, the second class of solutions, is given by the metric function</text>
<formula><location><page_26><loc_14><loc_13><loc_85><loc_19></location>˜ H D ( r, x ) = 1+ γπ ( -) D 4 ξ D ∫ ∞ 0 dg x N g N +3 J N ( gx ) H C ( 0 , 0 , 0 , g 2 a 2 2 , -g 2 a 2 4 , a 2 -r 2 2 a 2 ) . (6.21)</formula>
<text><location><page_27><loc_9><loc_74><loc_85><loc_90></location>We also verify that our numerical solutions to the differential equation (3.10), where k = 1, represent exactly the profile of the Heun-C functions in equations (6.20) and (6.21). We should note that the exact semi-analytical results (3.35), (4.8), (5.7), (5.8), as well as the exact analytical solutions (6.10) and (6.17) are all novel results. Moreover, the exact solutions for asymptotic metrics (3.19) and (5.9) are novel results. These results for the metric function H ( r, x ) furnish new exact solutions to the Einstein-Maxwell theory and its cosmological extension with a continuous parameter 0 < k < 1 in any dimension D greater than or equal to six. Moreover, we find new exact analytical solutions to the Einstein-Maxwell theory with k = 0, where the Bianchi space reduces to the Eguchi-Hanson type I space.</text>
<section_header_level_1><location><page_27><loc_9><loc_69><loc_42><loc_71></location>7 Concluding Remarks</section_header_level_1>
<text><location><page_27><loc_9><loc_24><loc_85><loc_67></location>We construct a class of exact solutions to the Einstein-Maxwell theory with a continuous parameter k . We find the metric function for the solutions in any dimensions D ≥ 6 uniquely, as a superposition of all radial solutions with their corresponding solutions in the other spatial direction. We solve and present numerical solutions to the radial equation, as we can't find any analytical solutions to the radial field equation. To find the weight functions in the superposition integral, we present another exact solutions to the Einstein-Maxwell theory, such that the superposition integral approaches to the exact metric function of the another solutions, in an appropriate limit. We find complicated integral equations for the weight functions, that we solve and find the unique solutions for the weight functions. We also consider the positive cosmological constant, and show the field equations for the EinsteinMaxwell theory with positive cosmological constant, can be separated. We find the exact solutions to the field equations and study the properties of the cosmological solutions. We notice that the exact semi-analytical results (3.35), (4.8), (5.7) and (5.8) are the novel results in this article. Moreover, the exact analytical solutions (6.10), (6.17), (3.19) and (5.9) are also novel results. We also consider the special case, where the Bianchi type IX parameter k is zero. We show that the Bianchi type IX geometry reduces to a less-known EguchiHanson type I geometry. We find real analytical solutions to the radial equation, in any dimensions, in terms of the Heun-C functions. We also find the weight functions, such that the superimposed solutions reduce to the known exact solutions, in an appropriate limit. We notice that the exact analytical results (6.10) and (6.17) are the novel results in this article. As the ansatzes for the metric and the Maxwell's field are similar to [29], we expect the exact solutions can describe the coalescence of the extremal charged black holes in D ≥ 6-dimensions, where the spatial section of the black holes consists a copy of the four-dimensional Bianchi type IX.</text>
<section_header_level_1><location><page_27><loc_12><loc_21><loc_32><loc_23></location>Acknowledgments</section_header_level_1>
<text><location><page_27><loc_9><loc_15><loc_85><loc_20></location>The author would like to express his sincere gratitude to the anonymous referees for their interesting comments and suggestions, to improve the quality of the article. This work was supported by the Natural Sciences and Engineering Research Council of Canada.</text>
<section_header_level_1><location><page_28><loc_9><loc_89><loc_82><loc_91></location>A Bianchi type IX solutions with ( β 1 , β 2 , β 3 ) = (2 , 2 , 2)</section_header_level_1>
<text><location><page_28><loc_9><loc_84><loc_85><loc_87></location>We can find the exact solutions to equations (2.9)-(2.11) where ( β 1 , β 2 , β 3 ) = (2 , 2 , 2). In fact, we find the solutions are given by [44]</text>
<formula><location><page_28><loc_33><loc_78><loc_85><loc_82></location>e 2 A ( η ) = 2 π ϑ 2 ( iη ) ϑ ' 3 ( iη ) ϑ ' 4 ( iη ) ϑ ' 2 ( iη ) ϑ 3 ( iη ) ϑ 4 ( iη ) , (A.1)</formula>
<formula><location><page_28><loc_33><loc_74><loc_85><loc_78></location>e 2 B ( η ) = 2 π ϑ ' 2 ( iη ) ϑ 3 ( iη ) ϑ ' 4 ( iη ) ϑ 2 ( iη ) ϑ ' 3 ( iη ) ϑ 4 ( iη ) , (A.2)</formula>
<formula><location><page_28><loc_33><loc_69><loc_85><loc_74></location>e 2 C ( η ) = 2 π ϑ ' 2 ( iη ) ϑ ' 3 ( iη ) ϑ 4 ( iη ) ϑ 2 ( iη ) ϑ 3 ( iη ) ϑ ' 4 ( iη ) , (A.3)</formula>
<text><location><page_28><loc_9><loc_67><loc_49><loc_68></location>where the ϑ 's denote the Jacobi Theta functions</text>
<formula><location><page_28><loc_40><loc_63><loc_85><loc_65></location>ϑ i ( iη ) = ϑ i (0 | iη ) , (A.4)</formula>
<text><location><page_28><loc_9><loc_58><loc_85><loc_62></location>and ' = d dη . We recall that in general, the Jacobi Theta functions ϑ i , i = 1 , 2 , 3 , 4 are given by</text>
<formula><location><page_28><loc_37><loc_55><loc_85><loc_56></location>ϑ 1 ( ν | τ ) = ϑ [ 1 1 ] ( ν | τ ) , (A.5)</formula>
<formula><location><page_28><loc_37><loc_52><loc_85><loc_54></location>ϑ 2 ( ν | τ ) = ϑ [ 1 0 ] ( ν | τ ) , (A.6)</formula>
<formula><location><page_28><loc_37><loc_50><loc_85><loc_52></location>ϑ 3 ( ν | τ ) = ϑ [ 0 0 ] ( ν | τ ) , (A.7)</formula>
<formula><location><page_28><loc_37><loc_48><loc_85><loc_50></location>ϑ 4 ( ν | τ ) = ϑ [ 0 1 ] ( ν | τ ) , (A.8)</formula>
<formula><location><page_28><loc_31><loc_41><loc_85><loc_45></location>ϑ [ a b ] ( ν | τ ) = ∑ n ∈ Z e iπ ( n -a 2 ) { τ ( n -a 2 )+2( ν -b 2 ) } , (A.9)</formula>
<text><location><page_28><loc_9><loc_28><loc_85><loc_40></location>where a and b are two real numbers. The metric (2.1) with the metric functions (A.1)(A.3) describes the Atiyah-Hitchin space. The other three possible cases ( β 1 , β 2 , β 3 ) = (2 , -2 , -2) , ( β 1 , β 2 , β 3 ) = ( -2 , 2 , -2) , ( β 1 , β 2 , β 3 ) = ( -2 , -2 , 2) are related to ( β 1 , β 2 , β 3 ) = (2 , 2 , 2). In fact the metric functions for the solutions( β 1 , β 2 , β 3 ) = (2 , -2 , -2) , ( β 1 , β 2 , β 3 ) = ( -2 , 2 , -2) and ( β 1 , β 2 , β 3 ) = ( -2 , -2 , 2) are obtained by the replacements e A to -e A , e B to -e B , and e C to -e C , respectively in (2.1). Upon embedding the Atiyah-Hitchin space ds 2 AH , into the higher-dimensional ansatz (3.2)</text>
<formula><location><page_28><loc_24><loc_23><loc_85><loc_26></location>ds 2 D = -dt 2 H ( r, x ) 2 + H ( r, x ) 2 D -3 ( dx 2 + x 2 d Ω 2 D -6 + ds 2 AH ) , (A.10)</formula>
<text><location><page_28><loc_9><loc_18><loc_85><loc_21></location>we find a differential equation which can not be separated, unlike the separable differential equation (3.5) for embedding the Bianchi type IX.</text>
<text><location><page_28><loc_9><loc_45><loc_14><loc_46></location>where</text>
<section_header_level_1><location><page_29><loc_9><loc_89><loc_54><loc_91></location>B The Jacobi elliptic functions</section_header_level_1>
<text><location><page_29><loc_9><loc_84><loc_85><loc_87></location>The Jacobi elliptic functions sn ( z, l ), cn ( z, l ) and dn ( z, l ) with the variable z and the parameter l , are related to the Jacobi elliptic function am ( z, l ) by</text>
<formula><location><page_29><loc_35><loc_81><loc_85><loc_82></location>sn ( z, l ) = sin( am ( z, l )) , (B.1)</formula>
<formula><location><page_29><loc_35><loc_79><loc_85><loc_80></location>cn ( z, l ) = cos( am ( z, l )) , (B.2)</formula>
<formula><location><page_29><loc_35><loc_76><loc_85><loc_78></location>dn ( z, l ) = √ 1 -l 2 sn 2 ( z, l ) . (B.3)</formula>
<text><location><page_29><loc_9><loc_71><loc_85><loc_74></location>The Jacobi elliptic function am ( z, l ) is the inverse of the trigonometric form of the elliptic integral of the first kind f ( ϕ, l ). In other words</text>
<formula><location><page_29><loc_39><loc_68><loc_85><loc_70></location>am ( f (sin φ, l ) , l ) = φ. (B.4)</formula>
<text><location><page_29><loc_9><loc_65><loc_84><loc_66></location>We note that the elliptic integral of the first kind f ( ϕ, k ) is given by the following integral</text>
<formula><location><page_29><loc_33><loc_59><loc_85><loc_64></location>f ( ϕ, l ) = ∫ sin -1 ( ϕ ) 0 dθ √ 1 -l 2 sin 2 θ . (B.5)</formula>
<section_header_level_1><location><page_29><loc_9><loc_55><loc_74><loc_57></location>C The Einstein and Maxwell's field equations</section_header_level_1>
<text><location><page_29><loc_9><loc_52><loc_44><loc_53></location>The Einstein's field equations are given by</text>
<formula><location><page_29><loc_38><loc_48><loc_85><loc_50></location>glyph[epsilon1] µν def = G µν -T µν = 0 , (C.1)</formula>
<text><location><page_29><loc_9><loc_45><loc_37><loc_47></location>while the Maxwell's equations are</text>
<formula><location><page_29><loc_41><loc_43><loc_85><loc_45></location>ϕ ν def = F µν ; µ = 0 , (C.2)</formula>
<text><location><page_29><loc_9><loc_39><loc_85><loc_42></location>where G µν is the Einstein tensor, F µν is the Maxwell tensor. The T µν is the energy-momentum tensor for the electromagnetic field, which is given by</text>
<formula><location><page_29><loc_37><loc_34><loc_85><loc_38></location>T µν = F λ µ F νλ -1 4 F 2 g µν . (C.3)</formula>
<text><location><page_29><loc_9><loc_32><loc_65><loc_33></location>First, we consider D = 6, in which the metric ansatz (3.2) becomes</text>
<formula><location><page_29><loc_29><loc_27><loc_85><loc_30></location>ds 2 = -dt 2 H ( r, x ) 2 + H ( r, x ) 2 3 ( dx 2 + ds 2 BIX ) . (C.4)</formula>
<text><location><page_29><loc_9><loc_22><loc_85><loc_25></location>The non-zero components of the Maxwell tensor also are given by (3.3) and (3.4). Equation (C.2) leads to</text>
<formula><location><page_29><loc_9><loc_13><loc_96><loc_21></location>ϕ t = ( r 9 256 -1 16 c 4 ( k 4 +1 ) r 5 + c 8 k 4 r )( F ( r ) ∂ 2 ∂r 2 H ( r, x ) + √ F ( r ) ∂ 2 ∂x 2 H ( r, x ) + F ' ( r ) ∂ ∂r H ( r, x ) ) + 7 F ( r ) ( 3 r 8 1792 -5 c 4 ( k 4 +1) r 4 112 + c 8 k 4 ) ∂ ∂r H ( r, x ) , (C.5)</formula>
<text><location><page_30><loc_9><loc_87><loc_85><loc_91></location>and all the other components of ϕ ν are identically zero. Equation (C.1) leads to some long expressions for the components of glyph[epsilon1] µν . For example,</text>
<text><location><page_30><loc_9><loc_83><loc_10><loc_84></location>glyph[epsilon1]</text>
<formula><location><page_30><loc_10><loc_48><loc_100><loc_85></location>rr = -458752 3 ( F ( r )) 5 / 2 H ( r, x ) (16 k 4 c 4 -r 4 ) 2 (16 c 4 -r 4 ) 2 r 2 ( 1 / 7 ( c 2 k 2 +1 / 4 r 2 ) ( ck + r/ 2) ( c -r/ 2) × (( 9 r 9 1024 -9 c 4 ( k 4 +1) r 5 64 +9 / 4 c 8 k 4 r ) H ( r, x ) d 2 d r 2 F ( r ) + d d r F ( r ) (( r 9 256 -1 / 16 c 4 ( k 4 +1 ) r 5 + c 8 k 4 r ) ∂ ∂r H ( r, x ) + 63 H ( r, x ) 4 ( 3 r 8 1792 -5 c 4 ( k 4 +1) r 4 112 + c 8 k 4 ))) r ( c 2 +1 / 4 r 2 ) ( ck -r/ 2) ( c + r/ 2) ( F ( r )) 3 / 2 + ( 1 / 7 ( c 2 k 2 +1 / 4 r 2 ) 2 ( ck + r/ 2) 2 ( c -r/ 2) 2 r 2 ( c 2 +1 / 4 r 2 ) 2 ( ck -r/ 2) 2 ( c + r/ 2) 2 ∂ 2 ∂r 2 H ( r, x ) + ( c 2 k 2 +1 / 4 r 2 ) ( c 2 k 2 -r 2 / 4 ) ( c 2 -r 2 / 4 ) ( 3 r 8 1792 -5 c 4 ( k 4 +1) r 4 112 + c 8 k 4 ) r ( c 2 +1 / 4 r 2 ) ∂ ∂r H ( r, x ) + 36 c 4 H ( r, x ) 7 (( k 4 8192 + 1 8192 ) r 12 + c 4 ( k 8 -4 k 4 +1) r 8 512 -1 32 c 8 k 4 ( k 4 +1 ) r 4 + c 12 k 8 )) F ( r ) 5 / 2 -9 ( ck -r/ 2) 2 ( c 2 k 2 +1 / 4 r 2 ) 2 ( c + r/ 2) 2 ( ck + r/ 2) 2 ( c -r/ 2) 2 r 2 ( c 2 +1 / 4 r 2 ) 2 56 × ( ( d d r F ( r ) ) 2 H ( r, x ) √ F ( r ) -8 ( F ( r )) 2 ∂ 2 ∂x 2 H ( r, x ) 9 )) , (C.6)</formula>
<text><location><page_31><loc_9><loc_89><loc_12><loc_90></location>and</text>
<formula><location><page_31><loc_12><loc_50><loc_85><loc_88></location>glyph[epsilon1] φψ = 32768 cos ( θ ) r 8 (16 k 4 c 4 -r 4 ) (16 c 4 -r 4 ) H ( r, x ) × (( -r 18 393216 + c 4 ( k 4 +1) r 14 24576 -c 8 k 4 r 10 1536 ) F ( r ) ∂ 2 ∂r 2 H ( r, x ) + ( -r 18 393216 + c 4 ( k 4 +1) r 14 24576 -c 8 k 4 r 10 1536 ) √ F ( r ) ∂ 2 ∂x 2 H ( r, x ) + ( -r 18 524288 + c 4 ( k 4 +1) r 14 32768 -c 8 k 4 r 10 2048 ) H ( r, x ) d 2 d r 2 F ( r ) -r 9 ∂ ∂r H ( r, x ) 1536 (( r 9 256 -1 / 16 c 4 ( k 4 +1 ) r 5 + c 8 k 4 r ) d d r F ( r ) + 7 F ( r ) ( 3 r 8 1792 -5 c 4 ( k 4 +1) r 4 112 + c 8 k 4 )) + H ( r, x ) ( -11 r 9 d d r F ( r ) 2048 ( 7 r 8 2816 -9 c 4 ( k 4 +1) r 4 176 + c 8 k 4 ) -3 r 8 F ( r ) 256 ( r 8 768 -1 / 24 c 4 ( k 4 +1 ) r 4 + c 8 k 4 ) + ( c 8 k 4 -1 / 8 c 4 k 4 r 4 + r 8 256 ) × ( c 8 k 4 -1 / 8 c 4 r 4 + r 8 256 ))) . (C.7)</formula>
<text><location><page_31><loc_9><loc_47><loc_53><loc_48></location>Moreover, the off-diagonal elements of (C.1), such as</text>
<formula><location><page_31><loc_35><loc_41><loc_85><loc_45></location>glyph[epsilon1] rx = ( α 2 -4 3 ) ∂H ( r,x ) ∂x ∂H ( r,x ) ∂r H 2 ( r, x ) , (C.8)</formula>
<text><location><page_31><loc_9><loc_33><loc_85><loc_39></location>lead to α 2 = 4 3 . We algebraically solve equation ϕ t = 0 for ∂ 2 H ( r,x ) ∂r 2 , and find it in terms of other derivatives of H . Upon substituting the algebraic result for ∂ 2 H ( r,x ) ∂r 2 in all components of the Einstein equations (C.1) (for example (C.6) and (C.7)), we find</text>
<formula><location><page_31><loc_44><loc_30><loc_85><loc_32></location>glyph[epsilon1] µν = 0 , (C.9)</formula>
<text><location><page_31><loc_9><loc_17><loc_85><loc_28></location>exactly. We also note that equation ϕ t = 0 is exactly (3.5). Similar calculations for D = 7 , 8 , · · · show that all the Einstein's equations are exactly satisfied, upon substitution ∂ 2 H ( r,x ) ∂r 2 which is obtained by solving algebraically the only non-zero component ϕ t of the Maxwell's field equations (C.2). Moreover, we find α 2 = D -2 D -3 . It is worth to note that the differential equation (C.5) does not depend on dimensionality of the spacetime. We always find (C.5), for the metric ansatz (3.2), in all different dimensions D ≥ 6.</text>
<section_header_level_1><location><page_32><loc_9><loc_89><loc_87><loc_91></location>D Time-dependent solutions with cosmological constant</section_header_level_1>
<text><location><page_32><loc_9><loc_82><loc_85><loc_87></location>In this appendix, we show that considering ansatzes (3.2)-(3.4) with more dependence on the spatial coordinates in H , in D ≥ 6 Einstein-Maxwell theory with a cosmological constant Λ leads to inconsistencies. We consider D = 6 with the metric ansatz</text>
<formula><location><page_32><loc_27><loc_77><loc_85><loc_81></location>ds 2 6 = -dt 2 H ( r, x, θ ) 2 + H ( r, x, θ ) 2 3 ( dx 2 + ds 2 BIX ) , (D.1)</formula>
<text><location><page_32><loc_9><loc_73><loc_85><loc_76></location>where the metric function depends on three spatial coordinates r, x and θ . We also consider the components of the F µν , as</text>
<formula><location><page_32><loc_33><loc_68><loc_85><loc_71></location>F tr = -α H ( r, x, θ ) 2 ∂H ( r, x, θ ) ∂r , (D.2)</formula>
<formula><location><page_32><loc_33><loc_64><loc_85><loc_67></location>F tx = -α H ( r, x, θ ) 2 ∂H ( r, x, θ ) ∂x , (D.3)</formula>
<formula><location><page_32><loc_33><loc_60><loc_85><loc_63></location>F tθ = -α H ( r, x, θ ) 2 ∂H ( r, x, θ ) ∂θ , (D.4)</formula>
<text><location><page_32><loc_9><loc_57><loc_64><loc_58></location>where α is a constant. The Einstein's field equations are given by</text>
<formula><location><page_32><loc_35><loc_53><loc_85><loc_55></location>glyph[epsilon1] µν def = G µν +Λ g µν -T µν = 0 , (D.5)</formula>
<text><location><page_32><loc_9><loc_50><loc_37><loc_52></location>while the Maxwell's equations are</text>
<formula><location><page_32><loc_41><loc_48><loc_85><loc_50></location>ϕ ν def = F µν ; µ = 0 , (D.6)</formula>
<text><location><page_32><loc_9><loc_44><loc_85><loc_47></location>where Λ is the cosmological constant. We find the only non-zero component of the Maxwell's equations as</text>
<formula><location><page_32><loc_31><loc_42><loc_85><loc_44></location>ϕ t = L 1 ( r, x, θ ) + cos 2 ψ L 2 ( r, x, θ ) = 0 , (D.7)</formula>
<text><location><page_32><loc_9><loc_39><loc_14><loc_41></location>where</text>
<formula><location><page_32><loc_9><loc_13><loc_94><loc_38></location>L 1 ( x, r, θ ) = ( 4 r 12 -64 c 4 ( k 4 +1 ) r 8 +1024 c 8 r 4 k 4 ) sin ( θ ) ∂ 2 ∂θ 2 H ( r, x, θ ) + ( r 14 -16 c 4 ( k 4 +1 ) r 10 +256 c 8 k 4 r 6 ) F ( r ) sin ( θ ) ∂ 2 ∂r 2 H ( r, x, θ ) + ( r 14 -16 c 4 ( k 4 +1 ) r 10 +256 c 8 k 4 r 6 ) sin ( θ ) √ F ( r ) ∂ 2 ∂x 2 H ( r, x, θ ) + 256 ( ∂ ∂r H ( r, x, θ ) ) r 5 sin ( θ ) (( r 9 256 -1 / 16 c 4 ( k 4 +1 ) r 5 + c 8 k 4 r ) d d r F ( r ) + 7 ( 3 r 8 1792 -5 c 4 ( k 4 +1) r 4 112 + c 8 k 4 ) F ( r ) ) -16384 ( c 2 +1 / 4 r 2 ) ( ∂ ∂θ H ( r, x, θ ) ) ( c 2 k 2 +1 / 4 r 2 ) 2 ( c 2 k 2 -r 2 / 4 ) 2 ( c 2 -r 2 / 4 ) cos ( θ ) , (D.8)</formula>
<text><location><page_33><loc_9><loc_89><loc_12><loc_90></location>and</text>
<formula><location><page_33><loc_10><loc_70><loc_85><loc_88></location>L 2 ( x, r, θ ) = 16384 cos( θ ) ( ∂ ∂θ H ( r, x, θ ) ) c 12 k 8 -16384 sin ( θ ) ( ∂ 2 ∂θ 2 H ( r, x, θ ) ) c 12 k 8 -1024 cos ( θ ) ( ∂ ∂θ H ( r, x, θ ) ) c 8 k 4 r 4 -1024 cos ( θ ) ( ∂ ∂θ H ( r, x, θ ) ) c 8 k 8 r 4 + 64 cos( θ ) ( ∂ ∂θ H ( r, x, θ ) ) c 4 k 4 r 8 +1024 sin ( θ ) ( ∂ 2 ∂θ 2 H ( r, x, θ ) ) c 8 k 8 r 4 -64 sin ( θ ) ( ∂ 2 ∂θ 2 H ( r, x, θ ) ) c 4 k 4 r 8 +1024 sin ( θ ) ( ∂ 2 ∂θ 2 H ( r, x, θ ) ) c 8 k 4 r 4 . (D.9)</formula>
<text><location><page_33><loc_9><loc_67><loc_82><loc_69></location>Using the separation of variables H ( t, r, θ ) = Θ( θ ) + R ( r ) X ( x ), we find L 2 = 0 leads to</text>
<formula><location><page_33><loc_39><loc_65><loc_85><loc_66></location>Θ( θ ) = ξ cos θ + ζ, (D.10)</formula>
<text><location><page_33><loc_9><loc_62><loc_58><loc_64></location>where ξ and ζ are two constants. Equation L 1 = 0 leads to</text>
<formula><location><page_33><loc_13><loc_58><loc_85><loc_61></location>P 1 ( r ) 1 R ( r ) d 2 R ( r ) dr 2 + P 2 ( r ) 1 R ( r ) dR ( r ) dr + Q 1 1 X ( x ) d 2 X ( x ) dx 2 + Q 2 ( r, x, θ ) R ( r ) X ( x ) = 0 , (D.11)</formula>
<text><location><page_33><loc_9><loc_55><loc_14><loc_56></location>where</text>
<formula><location><page_33><loc_25><loc_51><loc_85><loc_55></location>P 1 ( r ) = ( r 14 -16 c 4 ( k 4 +1) r 10 +256 c 8 k 4 r 6 ) √ F ( r ) 16384( r 14 -16 c 4 ( k 4 +1) r 10 +256 c 8 k 4 r 6 ) , (D.12)</formula>
<formula><location><page_33><loc_9><loc_42><loc_91><loc_50></location>P 2 ( r ) = r 5 ( ( r 9 256 -1 / 16 c 4 ( k 4 +1) r 5 + c 8 k 4 r ) d d r F ( r ) + 7 F ( r ) ( 3 r 8 1792 -5 c 4 ( k 4 +1 ) r 4 112 + c 8 k 4 )) 64 √ F ( r )( r 14 -16 c 4 ( k 4 +1) r 10 +256 c 8 k 4 r 6 ) , (D.13)</formula>
<formula><location><page_33><loc_42><loc_38><loc_85><loc_41></location>Q 1 = 1 16384 , (D.14)</formula>
<text><location><page_33><loc_9><loc_36><loc_12><loc_37></location>and</text>
<formula><location><page_33><loc_12><loc_30><loc_85><loc_35></location>Q 2 ( r, x, θ ) = ξ (2048 c 12 k 8 -128 k 4 r 4 ( k 4 +3) c 8 +24 r 8 ( k 4 +2 / 3) c 4 -r 12 ) cos ( θ ) 2048 √ F ( r ) X ( x ) R ( r ) ( r 14 -16 c 4 ( k 4 +1) r 10 +256 c 8 k 4 r 6 ) . (D.15)</formula>
<text><location><page_33><loc_9><loc_13><loc_87><loc_29></location>Clearly the differential equation (D.11) is not separable, due to the non-zero function Q 2 ( r, x, θ ). This term is zero only if ξ = 0, which points that the metric function H can't be a function of θ . Of course, proceeding with H ( r, θ ) to solve the field equations, leads to the already known solutions, as given in (3.35), along with Λ = 0. We also considered the metric function depending on other spatial directions. However, we always find that we can consistently solve all the field equations only if H is a function of r and x , which in turn leads to Λ = 0. Moreover, we considered other choices for the ansatz (3.2) with arbitrary exponents for the metric function. However the field equations imply that the exponents should be as -2 and 2 D -3 , as we considered in all ansatzes through the article.</text>
<section_header_level_1><location><page_34><loc_9><loc_89><loc_24><loc_91></location>References</section_header_level_1>
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</document> | [{"title": "Bianchi IX geometry and the Einstein-Maxwell theory", "content": "A. M. Ghezelbash 1 Department of Physics and Engineering Physics, University of Saskatchewan, Saskatoon, Saskatchewan S7N 5E2, Canada", "pages": [1]}, {"title": "Abstract", "content": "We construct numerical solutions to the higher-dimensional Einstein-Maxwell theory. The solutions are based on embedding the four dimensional Bianchi type IX space in the theory. We find the solutions as superposition of two functions, which one of them can be found numerically. We show that the solutions in any dimensions, are almost regular everywhere, except a singular point. We find that the solutions interpolate between the two exact analytical solutions to the higher dimensional Einstein-Maxwell theory, which are based on Eguchi-Hanson type I and II geometries. Moreover, we construct the exact cosmological solutions to the theory, and study the properties of the solutions.", "pages": [1]}, {"title": "1 Introduction", "content": "Exploring the different aspects of gravitational physics is possible through finding the new solutions to gravity, especially coupled to the other fields, such as the electromagnetic field. Moreover, the possibility of extending the known solutions in asymptotically flat spacetime to the asymptotically de-Sitter and anti de-Sitter solutions, is crucial and important in high energy physics. These extended solutions provide better understanding of the holographic proposals between the extended theories of gravity and the conformal field theories in different dimensions [1]-[2]. The constructed and explored solutions also include different solutions with different charges, such as NUT charges [3], dyonic charges [4], as well as different matter fields [5], axion field [6] and skyrmions [7]. The theories of gravity coupled to the other matter fields are useful to study and explore the rotating black holes [8], topological charged hairy black holes [9], cosmic censorship [10], gravitational radiation [11] and hyper-scaling violation [12]. Moreover, finding new solutions to the higher dimensional gravitational theories reveals new phenomena and possibilities, which may not exist in four dimensions. The discovery of the black hole solutions in five dimensions with squashed 3-sphere horizon [13], the black rings with S 2 \u00d7 S 1 horizon [14] and black lenses [15] are just some of the rich variety of the black objects in five and higher dimensions. Other relevant solutions in higher dimensional gravity coupled to the matter fields, are the dyonic solutions [16], the solitonic solutions [17], supergravity solutions [18, 19], braneworld cosmologies [20], and string theory extended solutions [21, 22]. The black hole solutions with different type of topologies for the horizon also were constructed and explored in [23]. Moreover, the references [24, 25] include the other solutions to extended theories of gravity with different type of matter fields in different dimensions. The class of solutions to Einstein-Maxwell-dilaton theory, in which the dilaton field couples to the cosmological constant and the Maxwell field, was considered in [26]. These solutions are relevant to the generalization of the Freund-Rubin compactification of M-theory [27]. Moreover, in interesting papers [28], the authors constructed and explored the convolution-like solutions for the fully localized type IIA D2 branes intersecting with the D6 branes. The type IIA solutions are obtained by compactifying the corresponding convolution-like M2 brane solutions, over a circle of transverse self-dual geometries including the Bianchi type IX geometry. The solutions consist of analytically convolution-like integrals of two functions, which depend on the transverse directions to the branes. The solutions preserve eight supersymmetries and are valid everywhere; near and far from the core of D6 branes. Due to the self-duality of the transverse geometries in the constructed solutions, the compactified solutions are supersymmetric. We also mention that one interesting feature of the solutions is that, the solutions are expressed completely in terms of convolution integrals, that is a result of taking special ansatzes for the solutions, and separability of the field equations. The motivations for this article come from several works on finding exact solutions in different theories of gravity. Inspired by the convolution-like solutions in M-theory [28], in this article, we construct similar convolution-like solutions in six and higher dimensional Einstein-Maxwell theory based on Bianchi type IX geometry. According to our knowledge, there are not any known convolution-like solutions in six dimensional or higher dimensional Einstein-Maxwell theories based on the Binachi geometries. Moreover, we are inspired with the papers [29], in which the authors constructed charged black hole and black string solutions in five dimensional Einstein-Maxwell theory. The solutions are based on embedding some known four-dimensional geometries, like Kasner space into five dimensions by using appropriate ansatzes for the metric and the Maxwell field. We note that to have non-trivial convolution-like solutions, the minimal dimensionality of Einstein-Maxwell theory should be six. Moreover, we consider the Einstein-Maxwell theories with positive cosmological constant in six and higher dimensions and find the exact cosmological convoluted solutions. To find the new solutions to the higher-dimensional Einstein-Maxwell theory, we consider the Bianchi type IX space, which is an exact solution to the four-dimensional Einstein's equations. Different types of solutions to the Einstein-Maxwell theory were constructed, such as solutions with the NUT charge [30], solitonic and dyonic solutions [16], as well as braneworld solutions [20]. Moreover, solutions to the extension of Einstein-Maxwell theory with the axion field and Chern-Simons term, were constructed and studied extensively in [31]. We organize the article as follows: In section 2, we consider the physics of Bianchi type IX spaces. In section 3, we present some numerical solutions to the Einstein-Maxwell theory in six and higher dimensions, where the metric function can be written as the convolution-like integral of two functions. In section 4, we present the second class of numerical solutions to the Einstein-Maxwell theory in six and higher dimensions. The second class of solutions is completely independent of the solutions in section 3. In section 5, we use the results of sections 3 and 4, and explicitly construct some cosmological solutions to the Einstein-Maxwell theory with positive cosmological constant, in six and higher dimensions. In section 6, we consider the Bianchi type IX space with the special cases of the Bianchi parameter as k = 0 and k = 1. We construct some exact solutions to the Einstein-Maxwell theory in six and higher dimensions, where the radial function involves the Heun-C functions. We discuss the physical properties of the solutions. We wrap up the article by the concluding remarks and four appendices in section 7.", "pages": [2, 3]}, {"title": "2 The Bianchi geometries", "content": "The classification of the homogeneous and isotropic spaces, is crucial to understand the cosmological models of the universe, as well as finding the theoretical models, which are consistent with the experimental data. The first classification of the homogeneous spaces was done long time ago by Bianchi [32]. We know now that there are eleven different homogeneous spaces [33], called Bianchi type I , \u00b7 \u00b7 \u00b7 , VI (class A or B), VII (class A or B), VIII and IX. The Bianchi type IX has been used mainly in cosmological models [34], supergravity theories [35] and extensions of gravity [36]. The four-dimensional Bianchi type IX metric, is given locally by the line element [37] where \u03c3 i , i = 1 , 2 , 3 are three Maurer-Cartan forms, and A, B and C are three functions of the coordinate \u03b7 . The Maurer-Cartan forms are given by in terms of three coordinates \u03b8, \u03c6, and \u03c8 of a unit S 3 . The line element (2.1) has an SU (2) isometry group. The metric satisfies exactly the vacuum Einstein's equations, provided the functions A ( \u03b7 ) , B ( \u03b7 ) and C ( \u03b7 ) satisfy as well as We should notice equation (2.8) is the first integral of (2.5)-(2.7). All Bianchi type IX solutions are self-dual geometries, which leads to the following first order differential equations for the metric functions A ( \u03b7 ) , B ( \u03b7 ) and C ( \u03b7 ) where \u03b2 i , i = 1 , 2 , 3 are three integration constants, which satisfy \u03b2 2 i = 0 or 4 and The solutions to equation (2.12) are given by ( \u03b2 1 , \u03b2 2 , \u03b2 3 ) = (0 , 0 , 0) , ( \u03b2 1 , \u03b2 2 , \u03b2 3 ) = (2 , 2 , 2) , ( \u03b2 1 , \u03b2 2 , \u03b2 3 ) = (2 , -2 , -2) , ( \u03b2 1 , \u03b2 2 , \u03b2 3 ) = ( -2 , 2 , -2) , ( \u03b2 1 , \u03b2 2 , \u03b2 3 ) = ( -2 , -2 , 2). In appendix A, we show that we can not construct the exact solutions to the higher dimensional EinsteinMaxwell theory, based on embedding the four dimensional solutions where ( \u03b2 1 , \u03b2 2 , \u03b2 3 ) = (2 , 2 , 2). We also show that the other three cases ( \u03b2 1 , \u03b2 2 , \u03b2 3 ) = (2 , -2 , -2) , ( \u03b2 1 , \u03b2 2 , \u03b2 3 ) = ( -2 , 2 , -2) , ( \u03b2 1 , \u03b2 2 , \u03b2 3 ) = ( -2 , -2 , 2) are equivalent to ( \u03b2 1 , \u03b2 2 , \u03b2 3 ) = (2 , 2 , 2). Hence, the only viable and interesting solution which we consider is ( \u03b2 1 , \u03b2 2 , \u03b2 3 ) = (0 , 0 , 0). In fact, we can solve the set of differential equations (2.9)-(2.11) exactly, where ( \u03b2 1 , \u03b2 2 , \u03b2 3 ) = (0 , 0 , 0). We find the solutions as where c and k are integration constants, and sn ( z, l ), cn ( z, l ) and dn ( z, l ) are the Jacobi elliptic functions with the variable z and the parameter 0 \u2264 l \u2264 1. For completeness, we present the explicit forms of the Jacobi elliptic functions sn ( z, l ), cn ( z, l ) and dn ( z, l ) in appendix B. By a straightforward calculation, we also find that the exact solutions (2.13)(2.15), indeed satisfy the other field equations (2.5)-(2.8). We change the coordinate \u03b7 in the metric (2.1) to the new coordinate r , which is given by For simplicity, we choose coordinate \u03b7 in the range [0 , \u03b1 ( c )( k )(1) ] where \u03b1 ( c )( k )( m ) is the mth positive root of sn ( c 2 \u03b7, k 2 ). We can equivalently consider any other range of the form [ \u03b1 ( c )( k )(2 n ) , \u03b1 ( c )( k )(2 n +1) ] with n = 1 , 2 , 3 , \u00b7 \u00b7 \u00b7 or [ -\u03b1 ( c )( k )(2 n ) , -\u03b1 ( c )( k )(2 n -1) ] for the coordinate \u03b7 . In figure 2.1, we show the typical behaviour of the new coordinate r versus \u03b7 , where \u03b7 \u2208 [0 , \u03b1 (1)( 1 2 )(1) ]. Note that \u03b1 (1)( 1 2 )(1) glyph[similarequal] 3 . 193. After the change of coordinate (2.16), we find the metric (2.1) changes to the triaxial Bianchi type IX form, which is given by [38] where a i , i = 1 , 2 , 3 are three integration constants, and the metric function F ( r ) is given by We note that the three integration constants in (2.17), are given by a 1 = 0 , a 2 = 2 kc and a 3 = 2 c , in terms of the variables and the parameters of the Jacobi elliptic functions 0 \u2264 k \u2264 1 and c > 0. We also note that a 1 \u2264 a 2 \u2264 a 3 . The metric (2.17) is regular for all values of the radial coordinate r > 2 c . The Ricci scalar for the Bianchi type IX space (2.17) is zero and the Kretschmann invariant is given by We notice the Kretschmann invariant (2.19) is regular everywhere, since r > 2 c . Moreover, all components of the Ricci tensor are regular, too.", "pages": [3, 4, 5, 6]}, {"title": "3 Embedding the Bianchi type IX space in D \u2265 6 -dimensional Einstein-Maxwell theory", "content": "We consider the D -dimensional Einstein-Maxwell theory where F \u00b5\u03bd = \u2202 \u03bd A \u00b5 -\u2202 \u00b5 A \u03bd . We consider the D -dimensional ansatz for the metric, as in [39] where ds 2 BIX is given by (2.17) and d \u2126 2 D -6 is the metric on a unit sphere S D -6 , where D \u2265 6. We take the components of the F \u00b5\u03bd as in [39] where \u03b1 is a constant. We note that (3.3) and (3.4) correspond to the potential A t = \u03b1 H D ( r,x ) , where all the other components are zero, A \u00b5 = t = 0. glyph[negationslash] We show in appendix C that all the Einstein's and Maxwell's field equations are satisfied, if the metric function H ( r, x ) obeys the partial differential equation and the constant \u03b1 in (3.3) and (3.4) is given by \u03b1 2 = D -2 D -3 . To solve the partial differential equation (3.5), we consider where two functions R ( r ) and X ( x ) describe the separation of coordinates, and \u03b2 is a constant. Plugging the equation (3.6) into equation (3.5), we find two ordinary differential equations for the functions R ( r ) and X ( x ). The differential equation for the function X ( x ) is where g denote the separation constant. We find the solutions to (3.7) are given by where I N and K N are the modified Bessel functions of the first and second kind, respectively, and x 1 and x 2 are the integration constants and Moreover, we find the differential equation for the function R ( r ) as We note that the radial differential equation (3.10) is independent of the dimension D of the spacetime. Tough we can't find any analytic solutions for the equation (3.10), however we try to find the analytic solutions to the differential equation (3.10) in asymptotic region r \u2192\u221e . In the limit of r \u2192\u221e , the equation (3.10) reduces to The exact solutions to equation (3.11) are given by for r \u2192 \u221e , where J 1 and Y 1 are the Bessel functions of the first and second kind, respectively. In figure 3.1 and 3.2, we plot the behaviour of the R ( r ) where r \u2192 \u221e . We notice the asymptotic radial function monotonically and periodically approaches zero, as r \u2192\u221e , independent of the sign of g . Furnished by the asymptotic behaviour of the radial function, we solve numerically the radial differential equation (3.10) for 0 < k < 1. In figure 3.3, we plot the numerical solutions for the radial function R ( r ), where we set k = 1 2 , c = 1 and g = \u00b1 2. We should notice that considering g = -2 in the radial differential equation (3.10) leads to the same radial differential equation with g = 2. So, we find that the numerical solutions for g = \u00b1 2 are exactly identical, as long as we use the same initial conditions in numerical integration of the differential equation. Hence, in this section, we consider only positive values for the separation constant without loosing any generality. We notice from figure 3.3 that the radial function becomes divergent as r \u2192 2, and decays rapidly as r \u2192\u221e , in agreement with the asymptotic solutions (3.12) and figures 3.1 and 3.2. The general structure of the radial function is the same for other values of the Bianchi parameter c . The divergent behaviour of the radial function happens at r \u2192 2 c and the radial function decays rapidly for r \u2192\u221e . Moreover, in figure 3.4, we plot the numerical solutions for the radial function R ( r ), where we set k = 1 4 and k = 3 4 with c = 1 and g = 2. Tough the figures 3.3 and 3.4 are quite similar, however they have subtle dependence on the Bianchi parameter k . In figure 3.5, we plot three radial functions, over a small interval of r , for k = 1 4 , 1 2 and 3 4 . As we notice from figure 3.5, the radial function R ( r ), in general, slightly increases with increasing the Bianchi parameter k . Changing the separation constant, in general, keeps the overall structure of the radial function. However, increasing the separation constant g leads to more oscillatory behaviour. In figure 3.6, we plot the radial functions, for k = 1 2 and two other separation constants g = 6 and g = 12. Superimposing all the different solutions with the different separation constants g , we can write the most general solutions to the partial differential equation (3.5) in D -dimensions, as where P ( g ) and Q ( g ) stand for the integration constants, for a specific value of the separation constant g , and N is given by (3.9). To find the functions P ( g ) and Q ( g ), we may compare the general solutions (3.14), with another related exact solutions to the theory. In fact, if we consider the large values for the radial coordinate r \u2192\u221e , then the Bianchi type IX metric (2.17) changes to which is the metric on R 4 . Using the asymptotic metric (3.15) for the Bianchi type IX geometry, we find an exact solutions to the Einstein-Maxwell theory in D -dimensions, where the gravity is described by the metric together with the Maxwell's field F \u00b5\u03bd , as By solving all the Einstein's and Maxwell's field equations, we find that the metric function H D in D -dimensions, is given by the exact form where \u03b3 is a constant and \u03b1 2 = D -2 D -3 . The D -dimensional asymptotic solution (3.16) with the metric function (3.19) describe a charged spacetime. The Ricci scalar and the Kretschmann invaraint for the solutions (3.16) are finite at r = x = 0. The invariants are finite on as long as we consider the constant \u03b3 > 0 in (3.19). In fact, the Ricci scalar of the solutions (3.16) at r = x = 0 is given by where \u03be 6 = +1 , \u03be D> 6 = -1. Similarly the Kretschmann invariant of the solutions (3.16) at r = x = 0 is given by where \u03b7 D is a constant; \u03b7 6 = 348 , \u03b7 7 = 1064 , \u03b7 8 = 2560 , \u00b7 \u00b7 \u00b7 . The Ricci scalar and the Kretschmann invariant, are given by where \u03b2 6 = 6 , \u03b2 7 = -12 , \u03b2 8 = -20 , \u00b7 \u00b7 \u00b7 and some of f D ( r, x, \u03b3 ) are given by As we notice, the Ricci scalar and the Kretschmann invariant are finite on H D ( r, x ) = 0, as long as we choose positive values for \u03b3 , where there are no solutions for H D ( r, x ) = 0. The electric charge of the black hole solutions (3.16) is given by where \u03a3 D -1 is a ( D -1)-dimensional spacelike hypersurface and \u2202 \u03a3 D -1 is its boundary. We find the components (3.17) and (3.18) are given by Calculating the integral in (3.28), we find We note that \u2126 D -6 in (3.31) is the volume of a unit sphere S D -6 which is given by where n !! is equal to 1 \u00b7 3 \u00b7 5 \u00b7 \u00b7 \u00b7 \u00b7 \u00b7 (2 k -1) \u00b7 (2 k +1) for an odd n = 2 k +1, and is equal to 2 ( k +1 / 2) k ! \u221a \u03c0 for an even n = 2 k . Finding the exact solutions (3.16) with (3.19), enable us to find an integral equation for the functions P and Q in (3.14). In fact, we have the integral equation However, we know lim r \u2192\u221e R ( r ) is given by (3.12). Considering r 1 = 1 g , r 2 = 0, we solve the integral equation (3.33) and find where \u0393( N +2) is the gamma function. Moreover, considering the other possibility r 1 = 0, does not lead to consistent solutions for the functions P and Q in the integral equation (3.33). Summarizing the results, we find the metric function in D -dimensions, as We should note that the radial function R ( r ) as a part of integrand in (3.35), indeed depends on integration parameter g . Though it is not feasible to find explicitly the Ricci scalar and the Kretschmann invariant of the spacetime (3.2), as functions of the coordinates (due to the semi-analytic metric function (3.35)), however, we may expect that the solutions are regular everywhere (outside of an event horizon with a singularity at r = 2 c ), as they approach smoothly to their asymptotic limit (3.16) with the metric function (3.19). We also expect the electric charge of the spacetime (3.2) is the same as equation (3.31). As the metric ansatz (3.2) and (3.3) and (3.4) are similar to [29], we also expect the spacetime (3.2) can describe the coalescence of the extremal charged black holes in D \u2265 6-dimensions, where the spatial section of the black holes consists a copy of the four-dimensional Bianchi type IX. The extremality is coming from noting that the only free parameter in the metric function (3.35) is \u03b3 . Hence the total mass of the gravitational system should be a multiple of \u03b3 . If the electric charge of the spacetime is given by (3.31), then we find that the total mass and the electric charge of the solution are proportional to each other. Of course, it is not feasible to find and verify explicitly the extremality of the solutions, due to the semi-analytic metric function (3.35).", "pages": [7, 8, 9, 10, 11, 12, 13, 14, 15]}, {"title": "4 The second class of solutions in D \u2265 6 -dimensional Einstein-Maxwell theory", "content": "In this section, we present the second independent class of solutions for the metric function which satisfies the partial differential equation (3.5). In separation of the coordinates, we replace g \u2192 ig in the differential equation (3.7). The solutions of the differential equation are in terms of the Bessel functions, where \u02dc x 1 and \u02dc x 2 are constants of integration, and N is given by (3.9). Moreover the radial differential equation becomes We present the numerical solutions to equation (4.2) for 0 < k < 1, where its asymptotic form is given by The exact solutions to equation (4.3) are given by where I 1 and K 1 are the modified Bessel functions of the first and second kind, respectively, and \u02dc r 1 and \u02dc r 2 are constants. In figure 4.1, we plot the behaviour of the \u02dc R ( r ) where r \u2192\u221e . We notice two very different behaviours for the asymptotic radial function, as r \u2192 \u221e . Moreover, the plots for \u02dc R ( r ) where r \u2192 \u221e , with g = -1 are exactly the same as in figure 4.1, since the modified Bessel functions I 1 and K 1 are even functions of gr . In figure 4.2, we plot the numerical solutions for the radial function \u02dc R ( r ), where we set k = 1 2 , c = 1 and g = \u00b1 2. We should notice that considering g = -2 in the radial differential equation (4.2) leads to the same radial differential equation with g = 2. So, we find that the numerical solutions for g = \u00b1 2 are exactly identical, as long as we use the same initial conditions in numerical integration of the differential equation. Hence, in this section, we consider only positive values for the separation constant, as in (3.13), without loosing any generality. We notice from figure 4.2 that the radial function approaches zero for r \u2192 2 c , and becomes large, as r \u2192 \u221e , in agreement with the asymptotic solutions (4.4), where \u02dc r 1 = 1 , \u02dc r 2 = 0, and the left plot in figure 4.1. The general structure of the radial function is the same for the other values of the Bianchi parameter c . The divergent behaviour of the radial function happens at r \u2192 \u221e and the radial function approaches zero, for r \u2192 2 c . Moreover, in figure 4.3, we plot the numerical solutions for the radial function \u02dc R ( r ), where we set k = 1 4 and k = 3 4 with c = 1 and g = 2. Tough the figures 4.2 and 4.3 are quite similar, however they have subtle dependence on the Bianchi parameter k . In figure 4.4, we plot three radial functions, over a small interval of r , for k = 1 4 , 1 2 and 3 4 . As we notice from figure 4.4, the radial function \u02dc R ( r ), in general, slightly increases with increasing the Bianchi parameter k , very similar to the situation for the first radial solutions. Changing the separation constant, in general, keeps the overall structure of the radial function. However, increasing the separation constant g leads to a slower decaying behaviour for the radial function. In figure 4.5, we plot the radial functions, for k = 1 2 and two other separation constants g = 6 and g = 12. Hence, we find the second most general solutions for the metric function in D -dimensions, as given by where \u02dc P ( g ) and \u02dc Q ( g ) stand for the integration constants, for a specific value of the separation constant g and N is given by (3.9). Requiring the metric function (4.5) in the limit of r \u2192\u221e , reduces to the exact solutions (3.19), we find an integral equation for the functions \u02dc P ( g ) and \u02dc Q ( g ), as Using equation (4.4) for lim r \u2192\u221e \u02dc R ( r ) with \u02dc r 1 = 1 g , \u02dc r 2 = 0, we can solve the integral equation (4.6) and find Moreover, considering the other possibility \u02dc r 1 = 0, does not lead to consistent solutions for the functions \u02dc P and \u02dc Q in the integral equation (4.6). Summarizing the results, we find the second metric function in D -dimensions, as", "pages": [15, 16, 17, 18, 19]}, {"title": "5 The solutions in D \u2265 6 dimensional Einstein-Maxwell theory with cosmological constant", "content": "In this section, we consider the D \u2265 6 Einstein-Maxwell theory with a cosmological constant \u039b. We show that there are non-trivial solutions to the D \u2265 6 dimensional Einstein-Maxwell theory with the cosmological constant. We start by considering the D \u2265 6 dimensional metric, as where the metric function depends on the coordinate t , as well as the coordinates r and x . The components of the Maxwell's field, are given by equations (3.3) and (3.4), after substitution H D ( r, x ) to H D ( t, r, x ). We should note considering the ansatzes (3.2)-(3.4) with metric function H depending on more spatial directions lead to inconsistencies (appendix D). Inspired with the well-known time-dependent metric functions in the spacetimes with cosmological constant (such as expanding/contracting patches of the de-Sitter spacetime or FLRWspacetime), the above-mentioned inconsistencies could be resolved only by considering a time dependent metric function in the metric, as in (5.1). Moreover, the components of the Maxwell's field, are given by A lengthy calculation shows that all the Einstein's and Maxwell's field equations are satisfied by taking the separation of variables in the metric function, as where the functions R ( r ) and X ( x ) satisfy exactly equations (3.10) and (3.7) for the first class of solutions, respectively, and \u03b1 2 = D -2 D -3 . Of course the analytical continuation g \u2192 ig , yields the corresponding equations for \u02dc R ( r ) and \u02dc X ( x ). Moreover, we find where Of course, we should consider only the positive cosmological constant \u039b to have a real \u03bb . To summarize, we find two classes of the cosmological solutions, where the metric functions are given by and We note that the two metric functions (5.7) and (5.8) definitely approach to the exact metric function in the limit of large radial coordinate. The metric function (5.9) is an exact solution to the Einstein's and Maxwell's field equations with a cosmological constant \u039b, where and d S 2 is given by (3.15). To get a glimpse of the solution (5.1), we consider the very late time coordinate, and find where The constant\u03c4 hypersurface of the metric (5.11) describes the foliation of ( D -1)-dimensional de-Sitter space, by flat spacelike Bianchi type IX. The volume of such a hypersurface reaches its minimum at \u03c4 = 0, and increases exponentially with \u03c4 . The Hubble parameter H D for the D -dimensional metric (5.11) is equal to and the cosmic volume of the spacetime (5.11) is given by \u222b \u221e 0 V D ( \u03c4, r, x ) d\u03c4drdx , where We note that \u2126 D -6 in (5.14) is the volume of a unit sphere S D -6 which is given by equation (3.32). We also find the cosmic volume of the constant\u03c4 hypersurface of the metric (5.11) is We note that the metric (5.11) describes the big bang patch. This patch covers half the dS spacetime from a big bang initiated at past horizon. The bang continues into the Bianchi foliations at future infinity. The other half of dS spacetime is covered by a big crunch patch. This patch initiates at past infinity with the Bianchi foliations. The crush continues towards the future horizon. The contracting patch of solutions at future infinity implies that black hole solutions based on Bianchi type IX space can describe the coalescence of the black holes in asymptotically dS spacetimes. Moreover, there is a correspondence between the phenomena occurring near the boundary (or in the deep interior) of asymptotically dS (or AdS) spacetime and the ultraviolet (infrared) physics in the dual CFT [40]. As a result, any solutions in asymptotically (locally) dS spacetimes lead to interpretation in terms of renormalization group flows, and the associated generalized c-theorem. The renormalization group flows toward the infrared in any contracting patch of dS spacetimes. Moreover the renormalization group flows toward the ultraviolet in any expanding patch of dS spacetimes. A useful quantity to represent the dS metric with different boundary geometries, such as direct products of flat space, the sphere and hyperbolic space, is the c-function. The cfunction for the D -dimensional asymptotically dS spacetime (5.11) is given by [41] where n i is the unit normal vector to the hypersurface which gives the Hamiltonian constraint and G is the Einstein tensor [41]. The c-function should show an increasing (decreasing) behaviour versus time for any expanding (contracting) patch of the spacetime. In figure 5.2, we plot the behaviour of the c -function (5.15 ) for two different spacetime dimensions. Both figures show increasing behaviour for the c -function which shows expansion of the constant\u03c4 hypersurface with the cosmological time \u03c4 . Moreover, we notice that increasing the dimension of spacetime leads to deceasing the value of the c -function. So, the constant\u03c4 hypersurfaces expand slower in the higher dimensions.", "pages": [19, 20, 21, 22]}, {"title": "6 The D \u2265 6 solutions with k = 0 and k = 1 in EinsteinMaxwell theory", "content": "In previous sections, we constructed solutions to D \u2265 6-dimensional Einstein-Maxwell theory, with and without cosmological constant, based on Bianchi type IX geometry with 0 < k < 1. In this section, we consider such solutions, where k = 0 and k = 1.", "pages": [22]}, {"title": "6.1 k = 0", "content": "First we consider the Bianchi type IX metric (2.17) with k = 0. The metric (2.17) reduces to \u221a where we set a = 2 c and the radial coordinate r \u2265 a . The metric (6.1) is the metric for the Eguchi-Hanson type I geometry [42]. The Ricci scalar for (6.1) is identically zero, and the Kretschmann invariant is which indicates r = a is a singularity. The radial differential equation for R ( r ) is given by while the differential equation for X ( x ) is the same as (3.7) with the solutions (3.8). We find the real analytical solutions for (6.3), which is where H C is the Heun-C function. In figure 6.1, we plot the radial function for different values of the Eguchi-Hanson parameter a . We note that the radial function has the oscillatory behaviour similar to cases where 0 < k < 1, however the radial function remains finite at r = a . Combining the different solutions for the radial function R ( r ) and X ( x ), we find the most general solution for the metric function H D ( r, x ) in D -dimensions, where k = 0, as where P 0 ( g ) and Q 0 ( g ) stand for the integration constants, for a specific value of the separation constant g . To find the functions P 0 ( g ) and Q 0 ( g ), we consider the limit r \u2192\u221e , where the Eguchi-Hanson type I space (6.1) becomes The asymptotic line element (6.6) describes R 4 , and embedding it in the D -dimensional theory, leads to the exact solution (3.16)-(3.18) with the metric function (3.19). We find the integral equation for the functions P 0 ( g ) and Q 0 ( g ). Moreover, we find After a very long calculation, we find the solutions to the integral equation (6.7), as Hence, we find the exact solutions for the metric function H ( r, x ) in D -dimensional EinsteinMaxwell theory with an embedded four-dimensional Eguchi-Hanson type I geometry (6.1) in the spatial section of the spacetime, as We also verify that our numerical solutions to the differential equation (3.10), where k = 0, represent exactly the profile of the Heun-C function in equation (6.10). Changing the separation constant g \u2192 ig generates the second class of solutions, where k = 0. We find the radial equation where the exact solutions are given by In figure 6.2, we plot the radial function \u02dc R for different values of the Eguchi-Hanson parameter a . We then superimpose the different solutions for the radial function \u02dc R ( r ) and \u02dc X ( x ), as given by (4.1), to find the second most general solution for the metric function \u02dc H D ( r, x ) in D -dimensions, where k = 0, as where \u02dc P 0 ( g ) and \u02dc Q 0 ( g ) stand for the integration constants, for a specific value of the separation constant g . To find the functions \u02dc P 0 ( g ) and \u02dc Q 0 ( g ), we consider the limit r \u2192\u221e , where the Eguchi-Hanson type I space (6.1) becomes (6.6). Hence we find the integral equation for the functions \u02dc P 0 ( g ) and \u02dc Q 0 ( g ). We find After a very long calculation, we find the solutions to the integral equation (6.14), as Hence, we find the second exact solutions for the metric function \u02dc H ( r, x ) in D -dimensional Einstein-Maxwell theory with an embedded four-dimensional Eguchi-Hanson type I geometry (6.1) in the spatial section of the spacetime, as We also verify that our numerical solutions to the differential equation (4.2), where k = 0, represent exactly the profile of the Heun-C function in equation (6.17).", "pages": [22, 23, 24, 25, 26]}, {"title": "6.2 k = 1", "content": "In this section, we consider the Bianchi type IX metric (2.17) with k = 1. The metric (2.17) reduces to which is the known metric for the Eguchi-Hanson type II space, where r \u2265 a [42]. The Ricci scalar for (6.18) is identically zero, and the Kretschmann invariant is given by which is regular and finite, for r \u2265 a . The exact solutions for the metric function H ( r, x ) in D -dimensional Einstein-Maxwell theory with an embedded four-dimensional Eguchi-Hanson type II geometry (6.18) in the spatial section of the spacetime, is given by [43] where \u03be 6+2 n = \u221a \u03c0 2 (2 n + 1)!! , n = 0 , 1 , 2 , \u00b7 \u00b7 \u00b7 for even dimensions D = 6 + 2 n and \u03be 7+2 n = (2 n + 2)!! , n = 0 , 1 , 2 , \u00b7 \u00b7 \u00b7 for odd dimensions D = 7 + 2 n . Moreover, the second class of solutions, is given by the metric function We also verify that our numerical solutions to the differential equation (3.10), where k = 1, represent exactly the profile of the Heun-C functions in equations (6.20) and (6.21). We should note that the exact semi-analytical results (3.35), (4.8), (5.7), (5.8), as well as the exact analytical solutions (6.10) and (6.17) are all novel results. Moreover, the exact solutions for asymptotic metrics (3.19) and (5.9) are novel results. These results for the metric function H ( r, x ) furnish new exact solutions to the Einstein-Maxwell theory and its cosmological extension with a continuous parameter 0 < k < 1 in any dimension D greater than or equal to six. Moreover, we find new exact analytical solutions to the Einstein-Maxwell theory with k = 0, where the Bianchi space reduces to the Eguchi-Hanson type I space.", "pages": [26, 27]}, {"title": "7 Concluding Remarks", "content": "We construct a class of exact solutions to the Einstein-Maxwell theory with a continuous parameter k . We find the metric function for the solutions in any dimensions D \u2265 6 uniquely, as a superposition of all radial solutions with their corresponding solutions in the other spatial direction. We solve and present numerical solutions to the radial equation, as we can't find any analytical solutions to the radial field equation. To find the weight functions in the superposition integral, we present another exact solutions to the Einstein-Maxwell theory, such that the superposition integral approaches to the exact metric function of the another solutions, in an appropriate limit. We find complicated integral equations for the weight functions, that we solve and find the unique solutions for the weight functions. We also consider the positive cosmological constant, and show the field equations for the EinsteinMaxwell theory with positive cosmological constant, can be separated. We find the exact solutions to the field equations and study the properties of the cosmological solutions. We notice that the exact semi-analytical results (3.35), (4.8), (5.7) and (5.8) are the novel results in this article. Moreover, the exact analytical solutions (6.10), (6.17), (3.19) and (5.9) are also novel results. We also consider the special case, where the Bianchi type IX parameter k is zero. We show that the Bianchi type IX geometry reduces to a less-known EguchiHanson type I geometry. We find real analytical solutions to the radial equation, in any dimensions, in terms of the Heun-C functions. We also find the weight functions, such that the superimposed solutions reduce to the known exact solutions, in an appropriate limit. We notice that the exact analytical results (6.10) and (6.17) are the novel results in this article. As the ansatzes for the metric and the Maxwell's field are similar to [29], we expect the exact solutions can describe the coalescence of the extremal charged black holes in D \u2265 6-dimensions, where the spatial section of the black holes consists a copy of the four-dimensional Bianchi type IX.", "pages": [27]}, {"title": "Acknowledgments", "content": "The author would like to express his sincere gratitude to the anonymous referees for their interesting comments and suggestions, to improve the quality of the article. This work was supported by the Natural Sciences and Engineering Research Council of Canada.", "pages": [27]}, {"title": "A Bianchi type IX solutions with ( \u03b2 1 , \u03b2 2 , \u03b2 3 ) = (2 , 2 , 2)", "content": "We can find the exact solutions to equations (2.9)-(2.11) where ( \u03b2 1 , \u03b2 2 , \u03b2 3 ) = (2 , 2 , 2). In fact, we find the solutions are given by [44] where the \u03d1 's denote the Jacobi Theta functions and ' = d d\u03b7 . We recall that in general, the Jacobi Theta functions \u03d1 i , i = 1 , 2 , 3 , 4 are given by where a and b are two real numbers. The metric (2.1) with the metric functions (A.1)(A.3) describes the Atiyah-Hitchin space. The other three possible cases ( \u03b2 1 , \u03b2 2 , \u03b2 3 ) = (2 , -2 , -2) , ( \u03b2 1 , \u03b2 2 , \u03b2 3 ) = ( -2 , 2 , -2) , ( \u03b2 1 , \u03b2 2 , \u03b2 3 ) = ( -2 , -2 , 2) are related to ( \u03b2 1 , \u03b2 2 , \u03b2 3 ) = (2 , 2 , 2). In fact the metric functions for the solutions( \u03b2 1 , \u03b2 2 , \u03b2 3 ) = (2 , -2 , -2) , ( \u03b2 1 , \u03b2 2 , \u03b2 3 ) = ( -2 , 2 , -2) and ( \u03b2 1 , \u03b2 2 , \u03b2 3 ) = ( -2 , -2 , 2) are obtained by the replacements e A to -e A , e B to -e B , and e C to -e C , respectively in (2.1). Upon embedding the Atiyah-Hitchin space ds 2 AH , into the higher-dimensional ansatz (3.2) we find a differential equation which can not be separated, unlike the separable differential equation (3.5) for embedding the Bianchi type IX. where", "pages": [28]}, {"title": "B The Jacobi elliptic functions", "content": "The Jacobi elliptic functions sn ( z, l ), cn ( z, l ) and dn ( z, l ) with the variable z and the parameter l , are related to the Jacobi elliptic function am ( z, l ) by The Jacobi elliptic function am ( z, l ) is the inverse of the trigonometric form of the elliptic integral of the first kind f ( \u03d5, l ). In other words We note that the elliptic integral of the first kind f ( \u03d5, k ) is given by the following integral", "pages": [29]}, {"title": "C The Einstein and Maxwell's field equations", "content": "The Einstein's field equations are given by while the Maxwell's equations are where G \u00b5\u03bd is the Einstein tensor, F \u00b5\u03bd is the Maxwell tensor. The T \u00b5\u03bd is the energy-momentum tensor for the electromagnetic field, which is given by First, we consider D = 6, in which the metric ansatz (3.2) becomes The non-zero components of the Maxwell tensor also are given by (3.3) and (3.4). Equation (C.2) leads to and all the other components of \u03d5 \u03bd are identically zero. Equation (C.1) leads to some long expressions for the components of glyph[epsilon1] \u00b5\u03bd . For example, glyph[epsilon1] and Moreover, the off-diagonal elements of (C.1), such as lead to \u03b1 2 = 4 3 . We algebraically solve equation \u03d5 t = 0 for \u2202 2 H ( r,x ) \u2202r 2 , and find it in terms of other derivatives of H . Upon substituting the algebraic result for \u2202 2 H ( r,x ) \u2202r 2 in all components of the Einstein equations (C.1) (for example (C.6) and (C.7)), we find exactly. We also note that equation \u03d5 t = 0 is exactly (3.5). Similar calculations for D = 7 , 8 , \u00b7 \u00b7 \u00b7 show that all the Einstein's equations are exactly satisfied, upon substitution \u2202 2 H ( r,x ) \u2202r 2 which is obtained by solving algebraically the only non-zero component \u03d5 t of the Maxwell's field equations (C.2). Moreover, we find \u03b1 2 = D -2 D -3 . It is worth to note that the differential equation (C.5) does not depend on dimensionality of the spacetime. We always find (C.5), for the metric ansatz (3.2), in all different dimensions D \u2265 6.", "pages": [29, 30, 31]}, {"title": "D Time-dependent solutions with cosmological constant", "content": "In this appendix, we show that considering ansatzes (3.2)-(3.4) with more dependence on the spatial coordinates in H , in D \u2265 6 Einstein-Maxwell theory with a cosmological constant \u039b leads to inconsistencies. We consider D = 6 with the metric ansatz where the metric function depends on three spatial coordinates r, x and \u03b8 . We also consider the components of the F \u00b5\u03bd , as where \u03b1 is a constant. The Einstein's field equations are given by while the Maxwell's equations are where \u039b is the cosmological constant. We find the only non-zero component of the Maxwell's equations as where and Using the separation of variables H ( t, r, \u03b8 ) = \u0398( \u03b8 ) + R ( r ) X ( x ), we find L 2 = 0 leads to where \u03be and \u03b6 are two constants. Equation L 1 = 0 leads to where and Clearly the differential equation (D.11) is not separable, due to the non-zero function Q 2 ( r, x, \u03b8 ). This term is zero only if \u03be = 0, which points that the metric function H can't be a function of \u03b8 . Of course, proceeding with H ( r, \u03b8 ) to solve the field equations, leads to the already known solutions, as given in (3.35), along with \u039b = 0. We also considered the metric function depending on other spatial directions. However, we always find that we can consistently solve all the field equations only if H is a function of r and x , which in turn leads to \u039b = 0. Moreover, we considered other choices for the ansatz (3.2) with arbitrary exponents for the metric function. However the field equations imply that the exponents should be as -2 and 2 D -3 , as we considered in all ansatzes through the article.", "pages": [32, 33]}, {"title": "References", "content": "Grunau, Phys. Rev. D92 (2015) 104027; A. B. Balakin, Phys. Rev. D94 (2016) 024021; A. B. Balakin and A. E. Zayats, Euro. Phys. J. C77 (2017) 519. Rev. D 99, 123531 (2019); Phys. Rev. D99 (2019) 123531; J. H. Bae, Class. Quantum Grav. 3 2 (2015) 075006; E. J. Kim and S. Kawai, Phys. Rev. D87 (2013) 083517; A. Corichi and E. Montoya, Class. Quantum Grav. 3 4 (2017) 054001; P. Sundell and T. Koivisto, Phys. Rev. D92 (2015) 123259; Y. Misonoh, K. Maeda and T. Kobayashi, Phys. Rev. D84 (2011) 064030.", "pages": [35, 39]}] |
2016CQGra..33l5009C | https://arxiv.org/pdf/1512.08198.pdf | <document>
<section_header_level_1><location><page_1><loc_33><loc_89><loc_66><loc_91></location>Vorticity in analogue gravity</section_header_level_1>
<text><location><page_1><loc_43><loc_86><loc_56><loc_88></location>Bethan Cropp ∗</text>
<text><location><page_1><loc_22><loc_79><loc_78><loc_86></location>SISSA, Via Bonomea 265, 34136 Trieste, Italy, INFN sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy and School of Physics, Indian Institute of Science Education and Research Thiruvananthapuram (IISER-TVM), Trivandrum 695016, India.</text>
<text><location><page_1><loc_25><loc_72><loc_74><loc_77></location>Stefano Liberati † and Rodrigo Turcati ‡ SISSA, Via Bonomea 265, 34136 Trieste, Italy and INFN sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy.</text>
<text><location><page_1><loc_17><loc_49><loc_82><loc_70></location>In the analogue gravity framework, the acoustic disturbances in a moving fluid can be described by an equation of motion identical to a relativistic scalar massless field propagating in a curved spacetime. This description is possible only when the fluid under consideration is barotropic, inviscid and irrotational. In this case, the propagation of the perturbations is governed by an acoustic metric which depends algebrically on the local speed of sound, density and the background flow velocity, the latter assumed to be vorticity free. In this work we provide an straightforward extension in order to go beyond the irrotational constraint. Using a charged relativistic and non-relativistic - Bose-Einstein condensate as a physical system, we show that in the low momentum limit and performing the eikonal approximation we can derive a d'Alembertian equation of motion for the charged phonons where the emergent acoustic metric depends on a flow velocity in the presence of vorticity.</text>
<section_header_level_1><location><page_2><loc_44><loc_90><loc_56><loc_91></location>CONTENTS</section_header_level_1>
<table>
<location><page_2><loc_12><loc_62><loc_88><loc_87></location>
</table>
<section_header_level_1><location><page_2><loc_40><loc_57><loc_60><loc_58></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_2><loc_12><loc_46><loc_88><loc_54></location>Since the seminal work of Unruh [1], and its subsequent development [2, 3], the analogue gravity framework has become a field of intense investigation over the last years. Issues such as Hawking radiation [1, 4], cosmological particle production [6, 7], emergence of spacetime and the fate of Lorentz invariance at short distances have acquired new insights within the analogue gravity field (for a general review, see [5]).</text>
<text><location><page_2><loc_12><loc_35><loc_88><loc_45></location>Analogue gravity was born out of the realization that acoustic disturbances in flowing fluids can, under suitable conditions, be described in the formalism of curved spacetime. In fact, when taking into account linearized perturbations over barotropic irrotational moving fluids, it can be demonstrated that the phonon excitations propagate under an effective acoustic metric, which is fully characterized by the background quantities of the flow. This remarkable result is valid both in relativistic and non-relativistic cases [1, 3, 8-10].</text>
<text><location><page_2><loc_12><loc_21><loc_88><loc_35></location>Among the several condensed-matter and optical systems considered in analogue gravity, Bose-Einstein condensates (BECs) [4, 11-14] are of particular interest as they provide simple low temperature systems with a high level of quantum coherence. The phonons in the BECs are treated as quantum perturbations over a classical background condensate, so providing an attractive scenario to describe semiclassical gravity phenomena, such as acoustic Hawking radiation [1, 15, 16]. BECs also have inspired many interesting analogue models for the emergence of symmetries, spacetime and more recently a mechanism of emergent dynamics [17].</text>
<text><location><page_2><loc_12><loc_7><loc_88><loc_21></location>It is worth noticing that all these analogue systems leading to these striking results, are characterized by the requirement of zero vorticity, i.e., in their thermodynamic limit they all provide irrotational fluids, for instance, in section 2.3 of [5]. In the case of BECs this is automatically imposed by the fact that the associated current is the derivative of a scalar, i.e., the phase of wavefunction of the condensate. Some works have tried to introduce acoustic metrics for different classes of fluid flows. An important contribution in this subject was made in [18] and we shall discuss the relative difference between our and their finds later on.</text>
<text><location><page_3><loc_12><loc_79><loc_88><loc_91></location>Our aim in this paper is demonstrate that we can, in a charged condensate, for sufficiently low momenta, find a wave equation to describe the propagation of phonons under an effective acoustic metric in the presence of vorticity. First, we investigate the behavior of the mode excitations in a charged condensate. Analyzing the charged fluid we conclude that the electromagnetic minimal coupling introduces vorticity in the system. In the appropriate limit, we demonstrate that it is possible to build an acoustic metric which takes into account vorticity. We perform this analysis in the relativistic and non-relativistic cases.</text>
<text><location><page_3><loc_12><loc_70><loc_88><loc_78></location>The paper is arranged as follows: In section II we describe how the condensation of a charged relativistic gas can be formed. In section III, we analyze the perturbations over the top of the charged condensate. The relativistic acoustic metric in the presence of vorticity is derived in section IV. In section V, we show that the same results appear in the nonrelativistic condensate. Our conclusions and remarks are present in section VI.</text>
<text><location><page_3><loc_14><loc_68><loc_64><loc_70></location>In our conventions the signature of the metric is (-,+,+,+).</text>
<section_header_level_1><location><page_3><loc_18><loc_64><loc_81><loc_65></location>II. CHARGED RELATIVISTIC BOSE-EINSTEIN CONDENSATE</section_header_level_1>
<text><location><page_3><loc_12><loc_55><loc_88><loc_61></location>Let us start by considering the general theory of Bose-Einstein condensation of a charged relativistic ideal gas (for a general review, see [19]). The U (1) gauge invariant Lagrangian describing the interaction of the complex scalar field φ and the electromagnetic field A µ can be written as</text>
<formula><location><page_3><loc_25><loc_50><loc_88><loc_54></location>L = -η µν ( D µ φ ) ∗ ( D ν φ ) -m 2 c 2 /planckover2pi1 2 φ ∗ φ -λ ( φ ∗ φ ) 2 -1 4 F µν F µν , (1)</formula>
<text><location><page_3><loc_12><loc_44><loc_88><loc_49></location>where F µν ( ≡ ∂ µ A ν -∂ ν A µ ) is the electromagnetic field strength tensor, m is the mass of the bosons, λ is a coupling constant and the covariant derivative is defined by D µ = ∂ µ + iq c /planckover2pi1 A µ . The Noether theorem leads to a locally conserved current j µ which is given by</text>
<formula><location><page_3><loc_37><loc_40><loc_88><loc_43></location>j µ = i 2 [ φ ( D µ φ ) ∗ -φ ∗ ( D µ φ )] . (2)</formula>
<text><location><page_3><loc_12><loc_37><loc_61><loc_39></location>The conserved charge associated to local U (1) symmetry is</text>
<formula><location><page_3><loc_35><loc_31><loc_88><loc_36></location>Q = i ∫ d 3 x [ φ ( D 0 φ ) ∗ -φ ∗ ( D 0 φ )] , (3)</formula>
<text><location><page_3><loc_12><loc_28><loc_88><loc_31></location>which has an associated bosonic chemical potential, µ . The momentum canonically conjugate to φ is</text>
<formula><location><page_3><loc_39><loc_24><loc_88><loc_28></location>π = ∂ L ∂ ˙ φ = ˙ φ ∗ + i q /planckover2pi1 c A 0 φ ∗ . (4)</formula>
<text><location><page_3><loc_12><loc_20><loc_88><loc_23></location>Since we are treating the fields φ and φ ∗ independently, the Hamiltonian density will be given by</text>
<formula><location><page_3><loc_41><loc_17><loc_88><loc_20></location>H = π ˙ φ + π ∗ ˙ φ ∗ -L . (5)</formula>
<text><location><page_3><loc_14><loc_16><loc_44><loc_17></location>The partition function is defined by</text>
<formula><location><page_3><loc_13><loc_6><loc_88><loc_15></location>Z = N ∫ ( DA ) ( Dπ ) ∗ ( Dπ ) ( Dφ ) ∗ ( Dφ ) exp {∫ β 0 dτ ∫ d 3 x [ π ˙ φ + π ∗ ˙ φ ∗ -( HµQ ) ] } × × det ( ∂F ∂ω ) δF. (6)</formula>
<text><location><page_4><loc_14><loc_89><loc_51><loc_91></location>Integrating the momenta away, we arrive at</text>
<formula><location><page_4><loc_21><loc_84><loc_88><loc_89></location>Z = N ∫ ( DA ) ( Dφ ) ∗ ( Dφ ) exp [∫ β 0 dτ ∫ d 3 x L eff ] × det ( ∂F ∂ω ) δF, (7)</formula>
<formula><location><page_4><loc_14><loc_74><loc_88><loc_82></location>L eff = -[ ∂ µ -iq c /planckover2pi1 ( A µ + µ q η µ 0 )] φ ∗ [ ∂ µ + iq c /planckover2pi1 ( A µ + µ q η µ 0 )] φ -m 2 c 2 /planckover2pi1 2 φ ∗ φ -λ ( φ ∗ φ ) 2 -1 4 F µν F µν . (8)</formula>
<text><location><page_4><loc_12><loc_81><loc_54><loc_84></location>where the effective Lagrangian L eff of the theory is</text>
<text><location><page_4><loc_14><loc_72><loc_88><loc_74></location>Before going on, a careful analysis in the effective Lagrangian (8) indicate that the shift</text>
<formula><location><page_4><loc_42><loc_68><loc_88><loc_72></location>A µ → A µ -µ q η µ 0 , (9)</formula>
<text><location><page_4><loc_12><loc_54><loc_88><loc_67></location>will provide an µ -independent partition function. However, if the Lagrangian (8) does not have any µ -dependence, the charged condensate will not be formed because the spontaneous symmetry breaking cannot occur since we are assuming m 2 > 0. To circumvent this problem, we can add to the effective Lagrangian a term like q J µ A µ , where J µ ( ≡ J 0 η µ 0 ) is a constant background charge density. Physically, it means that this extra term will compensate the charge density of the scalar field, making the system electrically neutral, so thermodynamic equilibrium can be achieved and the charged condensate can emerge (For a further discussion, see [19]).</text>
<text><location><page_4><loc_14><loc_51><loc_65><loc_53></location>It follows that the effective Lagrangian L eff can be written as</text>
<formula><location><page_4><loc_22><loc_44><loc_88><loc_51></location>L eff = -( D µ φ ) ∗ ( D µ φ ) + iµ /planckover2pi1 c ( φ∂ 0 φ ∗ -φ ∗ ∂ 0 φ ) + 2 qµ /planckover2pi1 2 c 2 A 0 φ ∗ φ -V ( φ ) -1 4 F µν F µν -q /planckover2pi1 c J µ A µ , (10)</formula>
<text><location><page_4><loc_12><loc_42><loc_17><loc_44></location>where</text>
<formula><location><page_4><loc_33><loc_37><loc_88><loc_42></location>V ( φ ) = -[ µ 2 /planckover2pi1 2 c 2 -m 2 c 2 /planckover2pi1 2 ] φ ∗ φ + λ ( φ ∗ φ ) 2 (11)</formula>
<text><location><page_4><loc_12><loc_18><loc_88><loc_37></location>is the effective potential. It is clear from the effective potential V ( φ ) that the charged relativistic gas will form a condensate only when µ 2 / ( /planckover2pi1 c ) 2 > m 2 c 2 / /planckover2pi1 2 . We would like to remark that besides the many similarities between Bose-Einstein condensation and spontaneous symmetry breaking mechanism, the physical and mathematical details concerning these phenomena are not completely equivalent. For instance, if we assume m 2 < 0 in the Lagrangian density (1), the vacuum undergoes spontaneous symmetry breaking and the system ends up with a real scalar field and a massive real vector field. This is the usual Higgs mechanism. However, in the Higgs case, we cannot construct an acoustic geometric description for the perturbations since the conserved current for the real scalar field is identically null and the continuity equation is trivial. Nevertheless, as explained above, when we introduce the chemical potential µ , the possibility of condensation exists [19].</text>
<text><location><page_4><loc_12><loc_13><loc_88><loc_18></location>Now, one can apply the mean-field prescription by performing the substitution φ → ϕ , where ϕ is the classical condensate field. So, performing the variation of ϕ ∗ in the effective Lagrangian (10), the equation of motion for ϕ assumes the form</text>
<formula><location><page_4><loc_12><loc_7><loc_90><loc_12></location>[ /square -m 2 c 2 /planckover2pi1 2 +2 iµ /planckover2pi1 c ∂ 0 -µ 2 /planckover2pi1 2 c 2 +2 iq c /planckover2pi1 A µ ∂ µ + iq c /planckover2pi1 ∂ µ A µ -q 2 c 2 /planckover2pi1 2 A µ A µ -2 qµ /planckover2pi1 2 c 2 A 0 -U ' ( /rho1 ; λ ) ] ϕ = 0 , (12)</formula>
<text><location><page_5><loc_12><loc_87><loc_88><loc_91></location>where /rho1 = ϕ ∗ ϕ and U ≡ λ ( ϕ ∗ ϕ ) 2 is the self-interaction term. Making a shift A µ → A µ -µ q η µ 0 , we can factor out the chemical potential dependence and express the field equation as</text>
<formula><location><page_5><loc_23><loc_81><loc_88><loc_86></location>[ /square -m 2 c 2 /planckover2pi1 2 +2 iq c /planckover2pi1 A µ ∂ µ + iq c /planckover2pi1 ∂ µ A µ -q 2 c 2 /planckover2pi1 2 A µ A µ -U ' ( /rho1 ; λ ) ] ϕ = 0 , (13)</formula>
<text><location><page_5><loc_12><loc_79><loc_65><loc_81></location>which describes a charged relativistic Bose-Einstein condensate.</text>
<section_header_level_1><location><page_5><loc_30><loc_75><loc_69><loc_76></location>III. DYNAMICS OF PERTURBATIONS</section_header_level_1>
<text><location><page_5><loc_12><loc_62><loc_88><loc_73></location>We are interested in analyzing the dynamics of the linearized perturbation of the condensate. In a dilute gas, i.e., when the correlations in the gas can be neglected, we can use the mean-field approximation to compute the perturbations. So, applying the decomposition φ = ϕ (1 + ψ ) in the modified Klein-Gordon equation (13), where the classical condensate field ϕ satisfies equation (13) and ψ is the relative quantum (i.e. of order /planckover2pi1 ) field fluctuation, we show that the linearized perturbations obey, at linear order of ψ ,</text>
<formula><location><page_5><loc_24><loc_57><loc_88><loc_61></location>/square ψ +2 η µν ( ∂ µ lnϕ ) ∂ ν ψ +2 iq c /planckover2pi1 A µ ∂ µ ψ -/rho1U '' ( /rho1 ; λ i )( ψ + ψ ∗ ) = 0 . (14)</formula>
<text><location><page_5><loc_12><loc_52><loc_88><loc_57></location>It is very convenient to decompose the degrees of freedom of the complex mean-field ϕ through the Madelung representation, given by ϕ = √ /rho1e iθ , where /rho1 is the density and θ is the phase of the condensate. With that, we can define:</text>
<formula><location><page_5><loc_30><loc_47><loc_88><loc_51></location>u µ ≡ /planckover2pi1 m η µν ∂ ν θ, (15)</formula>
<formula><location><page_5><loc_30><loc_43><loc_88><loc_47></location>c 2 0 ≡ /planckover2pi1 2 2 m 2 /rho1U '' ( /rho1 ; λ i ) , (16)</formula>
<formula><location><page_5><loc_30><loc_40><loc_88><loc_43></location>T /rho1 ≡ -/planckover2pi1 2 2 m ( /square + η µν ∂ µ ln/rho1∂ ν ) = -/planckover2pi1 2 2 m/rho1 η µν ∂ µ /rho1∂ ν , (17)</formula>
<text><location><page_5><loc_12><loc_35><loc_88><loc_38></location>where c 0 encodes the strength of the interactions and has dimensions of velocity and T /rho1 is a generalized kinetic operator that in the non-relativistic limit and for constant ρ reduces to</text>
<formula><location><page_5><loc_37><loc_30><loc_88><loc_34></location>T /rho1 →-/planckover2pi1 2 2 m/rho1 ∇ /rho1 ∇ = -/planckover2pi1 2 2 m ∇ 2 , (18)</formula>
<text><location><page_5><loc_12><loc_27><loc_48><loc_29></location>which is the usual kinetic operator [14-16].</text>
<text><location><page_5><loc_12><loc_24><loc_88><loc_27></location>In the phase-density decomposition, the locally conserved current and the condensate classical wave assume the form</text>
<formula><location><page_5><loc_36><loc_21><loc_88><loc_22></location>∂ µ ( /rho1f µ ) = 0 , (19)</formula>
<formula><location><page_5><loc_37><loc_16><loc_88><loc_21></location>-f µ f µ = c 2 + /planckover2pi1 2 m 2 [ U ' -/square √ /rho1 √ /rho1 ] , (20)</formula>
<text><location><page_5><loc_12><loc_10><loc_88><loc_15></location>where f µ ( ≡ u µ + q mc A µ ) is the timelike gauge invariant four-velocity of the condensate. Using the above definitions, it is easy to show, following the steps of [20], that the equation describing the propagation of charged perturbations is</text>
<formula><location><page_5><loc_36><loc_4><loc_88><loc_9></location>[ i /planckover2pi1 f µ ∂ µ -T /rho1 -mc 2 0 ] ψ = mc 2 0 ψ ∗ . (21)</formula>
<text><location><page_6><loc_12><loc_86><loc_88><loc_91></location>It is instructive to obtain a single equation for the quantum field ψ . This can be done taking the Hermitian conjugate of the above equation and using the result to eliminate ψ ∗ . After some manipulation, we find</text>
<formula><location><page_6><loc_27><loc_80><loc_88><loc_85></location>{ [ i /planckover2pi1 f µ ∂ µ + T /rho1 ] 1 c 2 0 [ -i /planckover2pi1 f ν ∂ ν + T /rho1 ] -/planckover2pi1 2 /rho1 η µν ∂ µ /rho1∂ ν } ψ = 0 . (22)</formula>
<text><location><page_6><loc_12><loc_68><loc_88><loc_80></location>The equation (22) is a relativistic equation of motion describing the propagation of charged linearized perturbations on top of a charged rBEC. We call attention to the fact that the relative fluctuation ψ does not change under a gauge transformation, which assures us that equation (22) is indeed gauge invariant as expected. We also remark that the electromagnetic field under consideration is endowed with a small magnitude. This assumption is very convenient since we intend to neglect background effects coming from electromagnetic fluctuations in the phonon propagation.</text>
<section_header_level_1><location><page_6><loc_27><loc_64><loc_73><loc_65></location>IV. THE RELATIVISTIC ACOUSTIC METRIC</section_header_level_1>
<text><location><page_6><loc_12><loc_53><loc_88><loc_62></location>One of the usual assumptions in analogue gravity is the flow be locally irrotational, i.e., vorticity free. Nevertheless, when the condensate couples with the electromagnetic field, the situation changes and vorticity appears in the system. It can be seen explicity by inspection of the vorticity tensor w µν contracted with the charged four-velocity f µ . Introducing the projection tensor h µ ν = δ µ ν + f µ f ν /f 2 , where f 2 = f α f α , this is given by</text>
<formula><location><page_6><loc_29><loc_49><loc_88><loc_53></location>w µν = h α µ h β ν ∇ [ α f β ] = h α µ h β ν q mc ( ∂ α A β -∂ β A α ) . (23)</formula>
<text><location><page_6><loc_12><loc_44><loc_88><loc_49></location>One can easily see that the vorticity tensor w µν cannot be set generically equal zero since the field strength tensor F µν is generically non zero and also the components of its projection on the spacelike hypersurface orthogonal to f µ .</text>
<text><location><page_6><loc_12><loc_39><loc_88><loc_44></location>The question that arises is the following: Is it still possible describe charged linearized perturbations in the formalism of curved spacetime in the presence of vorticity? Remarkably, the answer is yes. Let us specify carefully the conditions under which this can be achieved.</text>
<text><location><page_6><loc_12><loc_31><loc_88><loc_38></location>First of all, we note that the charged rBECs lead to a excitation spectra similar to the uncharged case analyzed by [20]. The difference lies in the fact that the electromagnetic minimal coupling modifies the uncoupled conserved current due the presence of the gauge field, which introduces a shift in the four-velocity u µ , namely,</text>
<formula><location><page_6><loc_36><loc_26><loc_88><loc_31></location>j µ = ρ m /planckover2pi1 u µ → ρ m /planckover2pi1 ( u µ + q mc A µ ) . (24)</formula>
<text><location><page_6><loc_12><loc_16><loc_88><loc_27></location>Despite having a complicated excitation spectra, for sufficiently low momenta the phonon propagation described in equation (22) leads to two quasiparticle modes, a massive and a massless, as in uncoupled case. Bearing in mind that we want to find an energy regime where we can apply the analogue gravity framework to describe phonon propagation in the presence of vorticity, we will focus on the gapless modes. The appropriate limit which the aforementioned framework can be acquired needs to satisfy basically two conditions:</text>
<unordered_list>
<list_item><location><page_6><loc_13><loc_12><loc_88><loc_16></location>(i) one needs to be within the so called the phononic regime, the low momenta range for the gapless excitation, i.e. [20]</list_item>
</unordered_list>
<formula><location><page_6><loc_41><loc_6><loc_88><loc_11></location>| k | /lessmuch 2 mc 0 /planckover2pi1 [ 1 + ( c 0 f 0 ) 2 ] , (25)</formula>
<text><location><page_7><loc_16><loc_88><loc_88><loc_91></location>where the term on the right is a relativistic generalization of the inverse of the healing length for charged condensates, and</text>
<unordered_list>
<list_item><location><page_7><loc_13><loc_79><loc_88><loc_86></location>(ii) one should be able to neglect the quantum potential T /rho1 in equation (22), which can be achieved assuming that the background quantities varies slowly in space and time on scales comparable with the wavelength w of the perturbations, conditions that can be written as</list_item>
</unordered_list>
<text><location><page_7><loc_14><loc_72><loc_58><loc_77></location>∣ ∣ ∣ ∣ The previous considerations reduce equation (22) to</text>
<formula><location><page_7><loc_32><loc_72><loc_88><loc_80></location>∣ ∣ ∣ ∂ t ρ ρ ∣ ∣ ∣ /lessmuch w, ∣ ∣ ∣ ∂ t c 0 c 0 ∣ ∣ ∣ /lessmuch w, ∣ ∣ ∣ ∣ ∂ t f µ f µ ∣ ∣ ∣ ∣ /lessmuch w. (26)</formula>
<formula><location><page_7><loc_35><loc_67><loc_88><loc_71></location>[ f µ ∂ µ 1 c 2 0 f ν ∂ ν -1 /rho1 η µν ∂ µ /rho1∂ ν ] ψ = 0 . (27)</formula>
<text><location><page_7><loc_14><loc_65><loc_73><loc_67></location>Multiplying (27) by /rho1 and using the continuity equation (19), we have</text>
<formula><location><page_7><loc_38><loc_59><loc_88><loc_64></location>∂ µ [ /rho1 c 2 0 f µ f ν -/rho1η µν ] ∂ ν ψ = 0 . (28)</formula>
<text><location><page_7><loc_14><loc_57><loc_65><loc_59></location>It is obvious that one can express the above equation (28) as</text>
<formula><location><page_7><loc_43><loc_54><loc_88><loc_56></location>∂ µ ( γ µν ∂ ν ψ ) = 0 , (29)</formula>
<text><location><page_7><loc_12><loc_51><loc_22><loc_53></location>where γ µν is</text>
<formula><location><page_7><loc_34><loc_45><loc_66><loc_50></location>γ µν = ρ c 2 0 [ -c 2 0 -( f 0 ) 2 -f 0 f j -f 0 f i c 2 0 δ ij -f i f j ] .</formula>
<text><location><page_7><loc_14><loc_42><loc_45><loc_46></location>If one identifies γ µν = √ -gg µν , then</text>
<text><location><page_7><loc_12><loc_36><loc_15><loc_37></location>and</text>
<formula><location><page_7><loc_38><loc_37><loc_88><loc_42></location>√ -g = ρ 2 √ 1 -f α f α /c 2 0 , (30)</formula>
<formula><location><page_7><loc_29><loc_30><loc_70><loc_35></location>g µν = 1 ρc 2 0 √ 1 -f α f α /c 2 0 [ -c 2 0 -( f 0 ) 2 -f 0 f j -f 0 f i c 2 0 δ ij -f i f j ] .</formula>
<text><location><page_7><loc_14><loc_29><loc_55><loc_30></location>Therefore, equation (28) can be cast in the form</text>
<formula><location><page_7><loc_37><loc_22><loc_88><loc_27></location>/triangle ψ ≡ 1 √ -g ∂ µ ( √ -gg µν ∂ ν ψ ) , (31)</formula>
<text><location><page_7><loc_12><loc_12><loc_88><loc_22></location>which is a d'Alembertian in a curved background. From the above equation, we promptly realize that the quasiparticle propagation in a nonhomogeneous fluid can be described by a relativistic equation of motion in a curved acoustic spacetime, where the emergent geometry is determined by the acoustic metric g µν . Inverting g µν , one can then see that the acoustic metric g µν for phonons propagation in a (3+1)D relativistic, barotropic, rotational fluid flow is given by</text>
<formula><location><page_7><loc_30><loc_4><loc_88><loc_11></location>g µν = /rho1 √ 1 -f α f α /c 2 0 [ η µν ( 1 -f α f α c 2 0 ) + f µ f ν c 2 0 ] . (32)</formula>
<text><location><page_8><loc_12><loc_88><loc_88><loc_91></location>We remark that the acoustic metric (32) is gauge invariant. Another useful way to express the relativistic acoustic metric (32) is using the definitions</text>
<formula><location><page_8><loc_35><loc_81><loc_88><loc_87></location>v µ = c f µ || f || , || f || = √ -η µν f µ f ν , (33)</formula>
<text><location><page_8><loc_12><loc_79><loc_88><loc_82></location>where v µ is the normalized four-velocity and || f || is the normalization factor. With that, the relativistic acoustic metric in the presence of vorticity assumes the form</text>
<formula><location><page_8><loc_36><loc_73><loc_88><loc_78></location>g µν = /rho1c c s [ η µν + ( 1 -c 2 s c 2 ) v µ v ν c 2 ] , (34)</formula>
<text><location><page_8><loc_12><loc_70><loc_88><loc_73></location>which is disformally related to the background Minkowski spacetime and the speed of sound c s is defined as</text>
<formula><location><page_8><loc_42><loc_64><loc_88><loc_69></location>c 2 s = c 2 c 2 0 / || f || 2 1 + c 2 0 / || f || 2 . (35)</formula>
<text><location><page_8><loc_12><loc_49><loc_88><loc_64></location>The relativistic acoustic metric (34) describes in a simple fashion way perturbations in a charged rBEC which include vorticity. It is important to mention that this is not the first attempt to incorporate vorticity in the context of analogue gravity. An approach through the use of Clebsch potentials can be found in [18]. Nevertheless, the wave equation generated is more complicated than a simple d'Alembertian and the construction significantly more difficult. We further stress that at the level of geometrical acoustics one may incorporate viscosity. However, geometrical acoustics is not enough for many purposes, and an irrotational fluid is a crucial assumption in deriving the wave equation on the effective curved metric.</text>
<section_header_level_1><location><page_8><loc_13><loc_44><loc_86><loc_45></location>V. THE NON-RELATIVISTIC CHARGED BOSE-EINSTEIN CONDENSATE</section_header_level_1>
<text><location><page_8><loc_12><loc_32><loc_88><loc_42></location>We have shown in the previous section that linearized perturbations over a relativistic charged condensate can, under suitable assumptions, be described in the same way as a scalar field propagating in a curved spacetime. Now, we can therefore ask if this description can be achieved when the charged condensate is non-relativistic. To see if such a description can indeed be done, we will start considering the non-relativistic limit of the relativistic equation of motion (22).</text>
<text><location><page_8><loc_12><loc_23><loc_88><loc_32></location>To begin with, in the non-relativistic regime, the external interaction qA 0 is much smaller then the atomic's rest energy mc 2 , namely qA 0 /lessmuch mc 2 . Moreover, the self-interaction between the atoms in the condensate must be weak, which means that c 0 /lessmuch c . It is also easy to see from equation (20) that in the non-relativistic regime f 0 → c . In addition, the speed of sound c s defined by relation (35) reduces to c 0 .</text>
<text><location><page_8><loc_14><loc_21><loc_83><loc_23></location>Assuming that these conditions are satisfied, the condition (25) is in turn given by</text>
<formula><location><page_8><loc_45><loc_17><loc_88><loc_20></location>| k | /lessmuch 2 mc 0 /planckover2pi1 , (36)</formula>
<text><location><page_8><loc_12><loc_15><loc_81><loc_16></location>which determine the momenta scale where the acoustic description can be applied.</text>
<text><location><page_8><loc_12><loc_11><loc_88><loc_14></location>Taking into account the previous considerations in the equation of motion (22) we arrive at</text>
<formula><location><page_8><loc_21><loc_5><loc_88><loc_10></location>{ [ i /planckover2pi1 ( ∂ t + f i ∂ i ) + T NR ] 1 c 2 0 [ -i /planckover2pi1 ( ∂ t + f j ∂ j ) + T NR ] -/planckover2pi1 2 ρ ∇ ρ ∇ } ψ = 0 , (37)</formula>
<text><location><page_9><loc_12><loc_86><loc_88><loc_91></location>where f i ( ≡ v i + q mc A i ) is the charged 3-velocity the condensate, v i is the velocity of the condensate in the uncharged case and the standard quantum potential in the non-relativistic limit T NR is</text>
<formula><location><page_9><loc_42><loc_82><loc_88><loc_86></location>T NR ≡ -/planckover2pi1 2 2 mρ ∇ ρ ∇ , (38)</formula>
<text><location><page_9><loc_12><loc_80><loc_35><loc_82></location>where ρ is the mass density.</text>
<text><location><page_9><loc_12><loc_75><loc_88><loc_80></location>The phonon propagation under an effective acoustic metric can be done only when one can neglect the quantum potential T NR , which can be achieved under the assumptions (26). In this case, the equation (37) reduces to</text>
<formula><location><page_9><loc_32><loc_68><loc_88><loc_74></location>[ ( ∂ t + f i ∂ i ) 1 c 2 0 ( ∂ t + f j ∂ j ) -1 ρ ∇ ρ ∇ ] ψ = 0 . (39)</formula>
<text><location><page_9><loc_14><loc_67><loc_76><loc_69></location>Multiplying (39) by the mass density ρ and using the continuity equation</text>
<text><location><page_9><loc_12><loc_61><loc_30><loc_63></location>we promptly arrive at</text>
<formula><location><page_9><loc_42><loc_62><loc_88><loc_66></location>∂ t ρ + ∂ i ( ρf i ) = 0 , (40)</formula>
<formula><location><page_9><loc_24><loc_55><loc_88><loc_60></location>-∂ t [ ρ c 2 0 ( ∂ t ψ + f j ∂ j ψ ) ] + ∂ i [ ρ∂ i ψ -ρ c 2 0 f i ( ∂ t ψ + f j ∂ j ψ ) ] = 0 , (41)</formula>
<text><location><page_9><loc_12><loc_51><loc_88><loc_56></location>which has the same structure that the wave equation describing the propagation of acoustic disturbances in irrotational fluids [See section 2.3 of [5]]. Following the standard methodology of analogue gravity, we find the usual non-relativistic acoustic metric</text>
<formula><location><page_9><loc_38><loc_43><loc_88><loc_47></location>g µν = ρ c s [ -( c 2 s -f 2 ) -f j -f i δ ij ] , (42)</formula>
<text><location><page_9><loc_12><loc_40><loc_60><loc_42></location>where f 2 = f i f i is the squared 3 velocity of the fluid flow.</text>
<text><location><page_9><loc_12><loc_32><loc_88><loc_40></location>In non-relativistic fluids the locally vorticity free condition over the velocity flow vector v is assured by the requirement ∇ × v = 0 . In our prescription, we have derived the non-relativistic acoustic metric without making use of the vorticity free assumption. To see explicity that the system is endowed with vorticity, we note that the rotational of the charged 3-velocity f is</text>
<formula><location><page_9><loc_38><loc_29><loc_88><loc_32></location>∇× f = q mc ∇× A = q mc B , (43)</formula>
<text><location><page_9><loc_12><loc_27><loc_75><loc_28></location>which is clearly non zero and implies the existence of vorticity in the BEC.</text>
<text><location><page_9><loc_12><loc_18><loc_88><loc_27></location>A point that deserves a careful attention is the absence of the A 0 component as a background quantity in the non-relativistic acoustic metric (42). In the framework of hydrodinamical systems, external potentials do not influence the description of the linearized perturbations. As a physical consequence in the BEC, even when the charged condensate is under the action of a static electric field, the phonons propagation are insensitive to it.</text>
<text><location><page_9><loc_12><loc_11><loc_88><loc_18></location>Now, in order to check that our derivation of the non-relativistic acoustic metric in the presence of vorticity is correct, one can take the alternative route to impose the electromagnetic minimal coupling directly to the equation that describes the condensate in the non-relativistic limit, namely,</text>
<formula><location><page_9><loc_34><loc_6><loc_88><loc_10></location>i /planckover2pi1 ∂ t φ = 1 2 m ( -i /planckover2pi1 ∂ i ) 2 φ + V ext + g | φ | 2 φ, (44)</formula>
<text><location><page_10><loc_12><loc_79><loc_88><loc_91></location>where V ext is an external potential, m is the mass of the bosons, | φ | 2 is the atomic density and g is the effective coupling constant which describes locally the scattering of atoms. The equation (44) emerges quite naturally in the analysis of BEC up to a first order approach, and it is formally equivalent to the Schrodinger equation with a nonlinear term g | φ | 2 . It is worth noting that the Ginzburg-Landau theory of superconductivity [21] is a particular case of (44). Performing the electromagnetic minimal coupling, the equation (44) assumes the form</text>
<formula><location><page_10><loc_30><loc_73><loc_88><loc_78></location>i /planckover2pi1 ∂ t φ = 1 2 m ( -i /planckover2pi1 ∂ i + q c A i ) 2 φ + qA 0 φ + g | φ | 2 φ. (45)</formula>
<text><location><page_10><loc_12><loc_70><loc_88><loc_73></location>Proceeding exactly the same way as in the relativistic case in order to obtain the equation for ψ , we insert φ = ϕ (1 + ψ ) in equation (45) and get, at the linearized level,</text>
<formula><location><page_10><loc_22><loc_64><loc_88><loc_69></location>[ i /planckover2pi1 ∂ t + /planckover2pi1 2 2 m ∇ 2 + /planckover2pi1 2 m ( ∂ i lnϕ ) ∂ i + i /planckover2pi1 q mc A i ∂ i -mc 2 sNR ] ψ = mc 2 sNR ψ ∗ , (46)</formula>
<text><location><page_10><loc_14><loc_60><loc_88><loc_62></location>Again, decomposing the condensate wave function as ϕ = ρe , where ρ is the mass</text>
<text><location><page_10><loc_12><loc_58><loc_70><loc_64></location>where c 2 sNR = gρ/m is the speed of sound in the non-relativistic BEC. √ iθ density, and rewriting to obtain a single equation to ψ , we get</text>
<formula><location><page_10><loc_20><loc_52><loc_88><loc_57></location>{ [ i /planckover2pi1 ( ∂ t + f i ∂ i ) + T NR ] 1 c 2 sNR [ -i /planckover2pi1 ( ∂ t + f j ∂ j ) + T NR ] -/planckover2pi1 2 ρ ∇ ρ ∇ } ψ = 0 , (47)</formula>
<text><location><page_10><loc_12><loc_47><loc_88><loc_52></location>which is exactly the same as equation (37). Employing the same conditions that previously discussed and following the usual steps, we arrive at the non-relativistic acoustic metric (42), so lending support to our previous derivation.</text>
<section_header_level_1><location><page_10><loc_40><loc_42><loc_60><loc_43></location>VI. CONCLUSIONS</section_header_level_1>
<text><location><page_10><loc_12><loc_31><loc_88><loc_40></location>One standard limitation with the procedures of analogue gravity is that the derivation of the wave function for the acoustic disturbances is possible only when we have a barotropic and inviscid fluid and the flow is irrotational. Under these assumptions, one can derive a d'Alembertian equation of motion to describe the linearized perturbations, which is identical to a relativistic scalar massless field propagating in a curved spacetime.</text>
<text><location><page_10><loc_12><loc_10><loc_88><loc_31></location>In this work we have shown that is possible to overcome the irrotational constraint in moving fluids and incorporate vorticity in the description of sound propogation in condensates. Performing the electromagnetic minimal coupling to the BEC we found that the conserved current related to gauge U (1)-symmetry depends on the complex scalar and gauge fields. Therefore the requirement that the gauge invariant charged flow velocity be locally irrotational is no longer imposed as the vorticity tensor is generically non-zero. So, taking into account the low momentum limit of the charged BEC and in the regime where one can neglect the quantum potential, the propogation of the charged quasiparticles in the presence of vorticity can be described by the formalism of the quantum field theory in a curved spacetime. Without these assumptions, the charged phonons propagation are described by a complicated differential wave equation. We emphasize that both cases (relativistic and non-relativistic) have gauge invariant equations for perturbations.</text>
<text><location><page_10><loc_12><loc_7><loc_88><loc_10></location>We also would like to remark that the condition on the momenta (36) which defines the phononic regime in the non-relativistic limit is exactly the same as required for the uncharged</text>
<text><location><page_11><loc_12><loc_77><loc_88><loc_91></location>BEC [20]. As in the uncoupled case, the momenta scale in which the analogue framework can be applied depends on the inverse of the healing length mc 0 / /planckover2pi1 . This coincidence can be easily comprehended by noting that according to (25), which determines the phononic regime in the relativistic range, when one takes into account the non-relativistic limit, u 0 ≈ c and A 0 becomes a negligible term, implying that ( c 0 /f 0 ) 2 → c 2 0 /c 2 /lessmuch 1. In this case we see that the relativistic phononic regime (25) reduces to (36). As previously discussed in section V, in the non-relativistic limit, static electric fields contributes to the dynamics of the background equations of motion, but does not influence the quasiparticles' propagation.</text>
<text><location><page_11><loc_12><loc_68><loc_88><loc_77></location>The cheif value of this work, with respect to the previous attempts to incorporate vorticity is the simplicity, both in the technical details of the construction, and in the physical picture of vorticity arising from the action of a magnetic field on a charged flow. As such, it is a clear demonstration that it is possible to easily incorporate at least some systems with vorticity into the analogue framework.</text>
<section_header_level_1><location><page_11><loc_38><loc_64><loc_61><loc_64></location>ACKNOWLEDGMENTS</section_header_level_1>
<text><location><page_11><loc_12><loc_49><loc_88><loc_61></location>The authors are grateful to Matt Visser for illuminating discussions and useful comments on the manuscript. The authors are also grateful to S. Shankaranarayanan for comments on an earlier version of the manuscript. Bethan Cropp is supported by Max Planck-India Partner Group on Gravity and Cosmology. Rodrigo Turcati is very grateful to CNPq for financial support. This publication was made possible through the support of a grant from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.</text>
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</document> | [{"title": "Vorticity in analogue gravity", "content": "Bethan Cropp \u2217 SISSA, Via Bonomea 265, 34136 Trieste, Italy, INFN sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy and School of Physics, Indian Institute of Science Education and Research Thiruvananthapuram (IISER-TVM), Trivandrum 695016, India. Stefano Liberati \u2020 and Rodrigo Turcati \u2021 SISSA, Via Bonomea 265, 34136 Trieste, Italy and INFN sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy. In the analogue gravity framework, the acoustic disturbances in a moving fluid can be described by an equation of motion identical to a relativistic scalar massless field propagating in a curved spacetime. This description is possible only when the fluid under consideration is barotropic, inviscid and irrotational. In this case, the propagation of the perturbations is governed by an acoustic metric which depends algebrically on the local speed of sound, density and the background flow velocity, the latter assumed to be vorticity free. In this work we provide an straightforward extension in order to go beyond the irrotational constraint. Using a charged relativistic and non-relativistic - Bose-Einstein condensate as a physical system, we show that in the low momentum limit and performing the eikonal approximation we can derive a d'Alembertian equation of motion for the charged phonons where the emergent acoustic metric depends on a flow velocity in the presence of vorticity.", "pages": [1]}, {"title": "I. INTRODUCTION", "content": "Since the seminal work of Unruh [1], and its subsequent development [2, 3], the analogue gravity framework has become a field of intense investigation over the last years. Issues such as Hawking radiation [1, 4], cosmological particle production [6, 7], emergence of spacetime and the fate of Lorentz invariance at short distances have acquired new insights within the analogue gravity field (for a general review, see [5]). Analogue gravity was born out of the realization that acoustic disturbances in flowing fluids can, under suitable conditions, be described in the formalism of curved spacetime. In fact, when taking into account linearized perturbations over barotropic irrotational moving fluids, it can be demonstrated that the phonon excitations propagate under an effective acoustic metric, which is fully characterized by the background quantities of the flow. This remarkable result is valid both in relativistic and non-relativistic cases [1, 3, 8-10]. Among the several condensed-matter and optical systems considered in analogue gravity, Bose-Einstein condensates (BECs) [4, 11-14] are of particular interest as they provide simple low temperature systems with a high level of quantum coherence. The phonons in the BECs are treated as quantum perturbations over a classical background condensate, so providing an attractive scenario to describe semiclassical gravity phenomena, such as acoustic Hawking radiation [1, 15, 16]. BECs also have inspired many interesting analogue models for the emergence of symmetries, spacetime and more recently a mechanism of emergent dynamics [17]. It is worth noticing that all these analogue systems leading to these striking results, are characterized by the requirement of zero vorticity, i.e., in their thermodynamic limit they all provide irrotational fluids, for instance, in section 2.3 of [5]. In the case of BECs this is automatically imposed by the fact that the associated current is the derivative of a scalar, i.e., the phase of wavefunction of the condensate. Some works have tried to introduce acoustic metrics for different classes of fluid flows. An important contribution in this subject was made in [18] and we shall discuss the relative difference between our and their finds later on. Our aim in this paper is demonstrate that we can, in a charged condensate, for sufficiently low momenta, find a wave equation to describe the propagation of phonons under an effective acoustic metric in the presence of vorticity. First, we investigate the behavior of the mode excitations in a charged condensate. Analyzing the charged fluid we conclude that the electromagnetic minimal coupling introduces vorticity in the system. In the appropriate limit, we demonstrate that it is possible to build an acoustic metric which takes into account vorticity. We perform this analysis in the relativistic and non-relativistic cases. The paper is arranged as follows: In section II we describe how the condensation of a charged relativistic gas can be formed. In section III, we analyze the perturbations over the top of the charged condensate. The relativistic acoustic metric in the presence of vorticity is derived in section IV. In section V, we show that the same results appear in the nonrelativistic condensate. Our conclusions and remarks are present in section VI. In our conventions the signature of the metric is (-,+,+,+).", "pages": [2, 3]}, {"title": "II. CHARGED RELATIVISTIC BOSE-EINSTEIN CONDENSATE", "content": "Let us start by considering the general theory of Bose-Einstein condensation of a charged relativistic ideal gas (for a general review, see [19]). The U (1) gauge invariant Lagrangian describing the interaction of the complex scalar field \u03c6 and the electromagnetic field A \u00b5 can be written as where F \u00b5\u03bd ( \u2261 \u2202 \u00b5 A \u03bd -\u2202 \u03bd A \u00b5 ) is the electromagnetic field strength tensor, m is the mass of the bosons, \u03bb is a coupling constant and the covariant derivative is defined by D \u00b5 = \u2202 \u00b5 + iq c /planckover2pi1 A \u00b5 . The Noether theorem leads to a locally conserved current j \u00b5 which is given by The conserved charge associated to local U (1) symmetry is which has an associated bosonic chemical potential, \u00b5 . The momentum canonically conjugate to \u03c6 is Since we are treating the fields \u03c6 and \u03c6 \u2217 independently, the Hamiltonian density will be given by The partition function is defined by Integrating the momenta away, we arrive at where the effective Lagrangian L eff of the theory is Before going on, a careful analysis in the effective Lagrangian (8) indicate that the shift will provide an \u00b5 -independent partition function. However, if the Lagrangian (8) does not have any \u00b5 -dependence, the charged condensate will not be formed because the spontaneous symmetry breaking cannot occur since we are assuming m 2 > 0. To circumvent this problem, we can add to the effective Lagrangian a term like q J \u00b5 A \u00b5 , where J \u00b5 ( \u2261 J 0 \u03b7 \u00b5 0 ) is a constant background charge density. Physically, it means that this extra term will compensate the charge density of the scalar field, making the system electrically neutral, so thermodynamic equilibrium can be achieved and the charged condensate can emerge (For a further discussion, see [19]). It follows that the effective Lagrangian L eff can be written as where is the effective potential. It is clear from the effective potential V ( \u03c6 ) that the charged relativistic gas will form a condensate only when \u00b5 2 / ( /planckover2pi1 c ) 2 > m 2 c 2 / /planckover2pi1 2 . We would like to remark that besides the many similarities between Bose-Einstein condensation and spontaneous symmetry breaking mechanism, the physical and mathematical details concerning these phenomena are not completely equivalent. For instance, if we assume m 2 < 0 in the Lagrangian density (1), the vacuum undergoes spontaneous symmetry breaking and the system ends up with a real scalar field and a massive real vector field. This is the usual Higgs mechanism. However, in the Higgs case, we cannot construct an acoustic geometric description for the perturbations since the conserved current for the real scalar field is identically null and the continuity equation is trivial. Nevertheless, as explained above, when we introduce the chemical potential \u00b5 , the possibility of condensation exists [19]. Now, one can apply the mean-field prescription by performing the substitution \u03c6 \u2192 \u03d5 , where \u03d5 is the classical condensate field. So, performing the variation of \u03d5 \u2217 in the effective Lagrangian (10), the equation of motion for \u03d5 assumes the form where /rho1 = \u03d5 \u2217 \u03d5 and U \u2261 \u03bb ( \u03d5 \u2217 \u03d5 ) 2 is the self-interaction term. Making a shift A \u00b5 \u2192 A \u00b5 -\u00b5 q \u03b7 \u00b5 0 , we can factor out the chemical potential dependence and express the field equation as which describes a charged relativistic Bose-Einstein condensate.", "pages": [3, 4, 5]}, {"title": "III. DYNAMICS OF PERTURBATIONS", "content": "We are interested in analyzing the dynamics of the linearized perturbation of the condensate. In a dilute gas, i.e., when the correlations in the gas can be neglected, we can use the mean-field approximation to compute the perturbations. So, applying the decomposition \u03c6 = \u03d5 (1 + \u03c8 ) in the modified Klein-Gordon equation (13), where the classical condensate field \u03d5 satisfies equation (13) and \u03c8 is the relative quantum (i.e. of order /planckover2pi1 ) field fluctuation, we show that the linearized perturbations obey, at linear order of \u03c8 , It is very convenient to decompose the degrees of freedom of the complex mean-field \u03d5 through the Madelung representation, given by \u03d5 = \u221a /rho1e i\u03b8 , where /rho1 is the density and \u03b8 is the phase of the condensate. With that, we can define: where c 0 encodes the strength of the interactions and has dimensions of velocity and T /rho1 is a generalized kinetic operator that in the non-relativistic limit and for constant \u03c1 reduces to which is the usual kinetic operator [14-16]. In the phase-density decomposition, the locally conserved current and the condensate classical wave assume the form where f \u00b5 ( \u2261 u \u00b5 + q mc A \u00b5 ) is the timelike gauge invariant four-velocity of the condensate. Using the above definitions, it is easy to show, following the steps of [20], that the equation describing the propagation of charged perturbations is It is instructive to obtain a single equation for the quantum field \u03c8 . This can be done taking the Hermitian conjugate of the above equation and using the result to eliminate \u03c8 \u2217 . After some manipulation, we find The equation (22) is a relativistic equation of motion describing the propagation of charged linearized perturbations on top of a charged rBEC. We call attention to the fact that the relative fluctuation \u03c8 does not change under a gauge transformation, which assures us that equation (22) is indeed gauge invariant as expected. We also remark that the electromagnetic field under consideration is endowed with a small magnitude. This assumption is very convenient since we intend to neglect background effects coming from electromagnetic fluctuations in the phonon propagation.", "pages": [5, 6]}, {"title": "IV. THE RELATIVISTIC ACOUSTIC METRIC", "content": "One of the usual assumptions in analogue gravity is the flow be locally irrotational, i.e., vorticity free. Nevertheless, when the condensate couples with the electromagnetic field, the situation changes and vorticity appears in the system. It can be seen explicity by inspection of the vorticity tensor w \u00b5\u03bd contracted with the charged four-velocity f \u00b5 . Introducing the projection tensor h \u00b5 \u03bd = \u03b4 \u00b5 \u03bd + f \u00b5 f \u03bd /f 2 , where f 2 = f \u03b1 f \u03b1 , this is given by One can easily see that the vorticity tensor w \u00b5\u03bd cannot be set generically equal zero since the field strength tensor F \u00b5\u03bd is generically non zero and also the components of its projection on the spacelike hypersurface orthogonal to f \u00b5 . The question that arises is the following: Is it still possible describe charged linearized perturbations in the formalism of curved spacetime in the presence of vorticity? Remarkably, the answer is yes. Let us specify carefully the conditions under which this can be achieved. First of all, we note that the charged rBECs lead to a excitation spectra similar to the uncharged case analyzed by [20]. The difference lies in the fact that the electromagnetic minimal coupling modifies the uncoupled conserved current due the presence of the gauge field, which introduces a shift in the four-velocity u \u00b5 , namely, Despite having a complicated excitation spectra, for sufficiently low momenta the phonon propagation described in equation (22) leads to two quasiparticle modes, a massive and a massless, as in uncoupled case. Bearing in mind that we want to find an energy regime where we can apply the analogue gravity framework to describe phonon propagation in the presence of vorticity, we will focus on the gapless modes. The appropriate limit which the aforementioned framework can be acquired needs to satisfy basically two conditions: where the term on the right is a relativistic generalization of the inverse of the healing length for charged condensates, and \u2223 \u2223 \u2223 \u2223 The previous considerations reduce equation (22) to Multiplying (27) by /rho1 and using the continuity equation (19), we have It is obvious that one can express the above equation (28) as where \u03b3 \u00b5\u03bd is If one identifies \u03b3 \u00b5\u03bd = \u221a -gg \u00b5\u03bd , then and Therefore, equation (28) can be cast in the form which is a d'Alembertian in a curved background. From the above equation, we promptly realize that the quasiparticle propagation in a nonhomogeneous fluid can be described by a relativistic equation of motion in a curved acoustic spacetime, where the emergent geometry is determined by the acoustic metric g \u00b5\u03bd . Inverting g \u00b5\u03bd , one can then see that the acoustic metric g \u00b5\u03bd for phonons propagation in a (3+1)D relativistic, barotropic, rotational fluid flow is given by We remark that the acoustic metric (32) is gauge invariant. Another useful way to express the relativistic acoustic metric (32) is using the definitions where v \u00b5 is the normalized four-velocity and || f || is the normalization factor. With that, the relativistic acoustic metric in the presence of vorticity assumes the form which is disformally related to the background Minkowski spacetime and the speed of sound c s is defined as The relativistic acoustic metric (34) describes in a simple fashion way perturbations in a charged rBEC which include vorticity. It is important to mention that this is not the first attempt to incorporate vorticity in the context of analogue gravity. An approach through the use of Clebsch potentials can be found in [18]. Nevertheless, the wave equation generated is more complicated than a simple d'Alembertian and the construction significantly more difficult. We further stress that at the level of geometrical acoustics one may incorporate viscosity. However, geometrical acoustics is not enough for many purposes, and an irrotational fluid is a crucial assumption in deriving the wave equation on the effective curved metric.", "pages": [6, 7, 8]}, {"title": "V. THE NON-RELATIVISTIC CHARGED BOSE-EINSTEIN CONDENSATE", "content": "We have shown in the previous section that linearized perturbations over a relativistic charged condensate can, under suitable assumptions, be described in the same way as a scalar field propagating in a curved spacetime. Now, we can therefore ask if this description can be achieved when the charged condensate is non-relativistic. To see if such a description can indeed be done, we will start considering the non-relativistic limit of the relativistic equation of motion (22). To begin with, in the non-relativistic regime, the external interaction qA 0 is much smaller then the atomic's rest energy mc 2 , namely qA 0 /lessmuch mc 2 . Moreover, the self-interaction between the atoms in the condensate must be weak, which means that c 0 /lessmuch c . It is also easy to see from equation (20) that in the non-relativistic regime f 0 \u2192 c . In addition, the speed of sound c s defined by relation (35) reduces to c 0 . Assuming that these conditions are satisfied, the condition (25) is in turn given by which determine the momenta scale where the acoustic description can be applied. Taking into account the previous considerations in the equation of motion (22) we arrive at where f i ( \u2261 v i + q mc A i ) is the charged 3-velocity the condensate, v i is the velocity of the condensate in the uncharged case and the standard quantum potential in the non-relativistic limit T NR is where \u03c1 is the mass density. The phonon propagation under an effective acoustic metric can be done only when one can neglect the quantum potential T NR , which can be achieved under the assumptions (26). In this case, the equation (37) reduces to Multiplying (39) by the mass density \u03c1 and using the continuity equation we promptly arrive at which has the same structure that the wave equation describing the propagation of acoustic disturbances in irrotational fluids [See section 2.3 of [5]]. Following the standard methodology of analogue gravity, we find the usual non-relativistic acoustic metric where f 2 = f i f i is the squared 3 velocity of the fluid flow. In non-relativistic fluids the locally vorticity free condition over the velocity flow vector v is assured by the requirement \u2207 \u00d7 v = 0 . In our prescription, we have derived the non-relativistic acoustic metric without making use of the vorticity free assumption. To see explicity that the system is endowed with vorticity, we note that the rotational of the charged 3-velocity f is which is clearly non zero and implies the existence of vorticity in the BEC. A point that deserves a careful attention is the absence of the A 0 component as a background quantity in the non-relativistic acoustic metric (42). In the framework of hydrodinamical systems, external potentials do not influence the description of the linearized perturbations. As a physical consequence in the BEC, even when the charged condensate is under the action of a static electric field, the phonons propagation are insensitive to it. Now, in order to check that our derivation of the non-relativistic acoustic metric in the presence of vorticity is correct, one can take the alternative route to impose the electromagnetic minimal coupling directly to the equation that describes the condensate in the non-relativistic limit, namely, where V ext is an external potential, m is the mass of the bosons, | \u03c6 | 2 is the atomic density and g is the effective coupling constant which describes locally the scattering of atoms. The equation (44) emerges quite naturally in the analysis of BEC up to a first order approach, and it is formally equivalent to the Schrodinger equation with a nonlinear term g | \u03c6 | 2 . It is worth noting that the Ginzburg-Landau theory of superconductivity [21] is a particular case of (44). Performing the electromagnetic minimal coupling, the equation (44) assumes the form Proceeding exactly the same way as in the relativistic case in order to obtain the equation for \u03c8 , we insert \u03c6 = \u03d5 (1 + \u03c8 ) in equation (45) and get, at the linearized level, Again, decomposing the condensate wave function as \u03d5 = \u03c1e , where \u03c1 is the mass where c 2 sNR = g\u03c1/m is the speed of sound in the non-relativistic BEC. \u221a i\u03b8 density, and rewriting to obtain a single equation to \u03c8 , we get which is exactly the same as equation (37). Employing the same conditions that previously discussed and following the usual steps, we arrive at the non-relativistic acoustic metric (42), so lending support to our previous derivation.", "pages": [8, 9, 10]}, {"title": "VI. CONCLUSIONS", "content": "One standard limitation with the procedures of analogue gravity is that the derivation of the wave function for the acoustic disturbances is possible only when we have a barotropic and inviscid fluid and the flow is irrotational. Under these assumptions, one can derive a d'Alembertian equation of motion to describe the linearized perturbations, which is identical to a relativistic scalar massless field propagating in a curved spacetime. In this work we have shown that is possible to overcome the irrotational constraint in moving fluids and incorporate vorticity in the description of sound propogation in condensates. Performing the electromagnetic minimal coupling to the BEC we found that the conserved current related to gauge U (1)-symmetry depends on the complex scalar and gauge fields. Therefore the requirement that the gauge invariant charged flow velocity be locally irrotational is no longer imposed as the vorticity tensor is generically non-zero. So, taking into account the low momentum limit of the charged BEC and in the regime where one can neglect the quantum potential, the propogation of the charged quasiparticles in the presence of vorticity can be described by the formalism of the quantum field theory in a curved spacetime. Without these assumptions, the charged phonons propagation are described by a complicated differential wave equation. We emphasize that both cases (relativistic and non-relativistic) have gauge invariant equations for perturbations. We also would like to remark that the condition on the momenta (36) which defines the phononic regime in the non-relativistic limit is exactly the same as required for the uncharged BEC [20]. As in the uncoupled case, the momenta scale in which the analogue framework can be applied depends on the inverse of the healing length mc 0 / /planckover2pi1 . This coincidence can be easily comprehended by noting that according to (25), which determines the phononic regime in the relativistic range, when one takes into account the non-relativistic limit, u 0 \u2248 c and A 0 becomes a negligible term, implying that ( c 0 /f 0 ) 2 \u2192 c 2 0 /c 2 /lessmuch 1. In this case we see that the relativistic phononic regime (25) reduces to (36). As previously discussed in section V, in the non-relativistic limit, static electric fields contributes to the dynamics of the background equations of motion, but does not influence the quasiparticles' propagation. The cheif value of this work, with respect to the previous attempts to incorporate vorticity is the simplicity, both in the technical details of the construction, and in the physical picture of vorticity arising from the action of a magnetic field on a charged flow. As such, it is a clear demonstration that it is possible to easily incorporate at least some systems with vorticity into the analogue framework.", "pages": [10, 11]}, {"title": "ACKNOWLEDGMENTS", "content": "The authors are grateful to Matt Visser for illuminating discussions and useful comments on the manuscript. The authors are also grateful to S. Shankaranarayanan for comments on an earlier version of the manuscript. Bethan Cropp is supported by Max Planck-India Partner Group on Gravity and Cosmology. Rodrigo Turcati is very grateful to CNPq for financial support. This publication was made possible through the support of a grant from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.", "pages": [11]}] |
2019arXiv191012311C | https://arxiv.org/pdf/1910.12311.pdf | <document>
<section_header_level_1><location><page_1><loc_8><loc_91><loc_21><loc_92></location>ORIGINAL PAPER</section_header_level_1>
<section_header_level_1><location><page_1><loc_8><loc_87><loc_50><loc_89></location>Extreme Primordial Black Holes</section_header_level_1>
<text><location><page_1><loc_8><loc_82><loc_58><loc_85></location>Siri Chongchitnan 1,2 | Teeraparb Chantavat 3 | Jenna Zunder 4</text>
<text><location><page_1><loc_8><loc_64><loc_31><loc_80></location>1 Mathematics Institute, University of Warwick, Zeeman Building, CV4 7AL, Coventry, United Kingdom 2 E. A. Milne Centre for Astrophysics, University of Hull, Cottingham Rd., HU6 7RX Hull, United Kingdom 3 Institute for Fundamental Study, Naresuan University, 65000 Phitsanulok, Thailand 4 Department of Mathematics, University of York, Heslington, YO10 5DDYork, United Kingdom</text>
<section_header_level_1><location><page_1><loc_8><loc_62><loc_17><loc_63></location>Correspondence</section_header_level_1>
<text><location><page_1><loc_7><loc_58><loc_30><loc_62></location>*Siri Chongchitnan, Mathematics Institute, University of Warwick. Email: [email protected]</text>
<section_header_level_1><location><page_1><loc_8><loc_49><loc_27><loc_50></location>1 INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_7><loc_31><loc_49><loc_48></location>Primordial black holes (PBHs) originate from large inflationary perturbations that subsequently collapse into black holes in the early Universe (for reviews, see Carr, Kühnel, & Sandstad (2016); García-Bellido (2017); Kashlinsky, A. et al. (2019); Sasaki, Suyama, Tanaka, & Yokoyama (2018)). LIGO gravitational wave events 1 over the past few years have given rise to the resurgence of PBHs not only as a viable dark matter candidate, but also as potential pregenitors of massive black holes ( GLYPH<192> 30 M GLYPH<230> ) that can typically give rise to the observed amplitude of gravitational waves.</text>
<text><location><page_1><loc_8><loc_11><loc_49><loc_20></location>The framework discussed is based on previous work by one of us (Chongchitnan, 2015; Chongchitnan & Hunt, 2017) in the context of extreme cosmic voids, as well as previous work by Harrison & Coles (2011, 2012) on extreme galaxy clusters. Our main result will be the probability density function (pdf) for the</text>
<text><location><page_1><loc_7><loc_19><loc_49><loc_31></location>Given an inflationary scenario, it would be useful to predict the mass of the most massive PBHs expected within a given observational volume. Such a calculation would serve as an additional observational test of competing inflationary theories. The primary aim of this work is to present such a framework, whilst demonstrating the method for a particular model of PBH formation.</text>
<section_header_level_1><location><page_1><loc_35><loc_77><loc_41><loc_78></location>Abstract</section_header_level_1>
<text><location><page_1><loc_35><loc_65><loc_91><loc_77></location>We present a formalism for calculating the probability distribution of the most massive primordial black holes (PBHs) expected within an observational volume. We show how current observational upper bounds on the fraction of PBHs in dark matter translate to constraints on extreme masses of primordial black holes. We demonstrate the power of our formalism via a case study, and argue that our formalism can be used to produce extreme-value distributions for a wide range of PBH formation theories.</text>
<section_header_level_1><location><page_1><loc_35><loc_62><loc_44><loc_63></location>KEYWORDS:</section_header_level_1>
<text><location><page_1><loc_35><loc_61><loc_54><loc_63></location>cosmology; primordial black holes</text>
<text><location><page_1><loc_51><loc_46><loc_92><loc_51></location>most massive PBHs expected in an observational volume. We will apply the framework to a simple model of PBH formation and demonstrate the soundness of the calculations.</text>
<text><location><page_1><loc_51><loc_42><loc_93><loc_46></location>Throughout this work we will use the cosmological parameters for the LCDM model from Planck (Collaboration, 2018).</text>
<section_header_level_1><location><page_1><loc_51><loc_37><loc_92><loc_39></location>2 FRACTIONOFTHEUNIVERSEINPBHS</section_header_level_1>
<text><location><page_1><loc_51><loc_32><loc_92><loc_36></location>In this section we will derive an expression for the cosmological abundance of PBHs, namely</text>
<text><location><page_1><loc_51><loc_10><loc_93><loc_29></location>where GLYPH<26> PBH is the mean cosmic density in PBHs, and GLYPH<26> crit is the critical density. Typically we will be interested in the abundance of PBHs within a certain mass range (say, GLYPH<10> pbh . > M / , i.e. the fraction of the Universe in PBHs of mass greater than M ). The PBH abundance naturally depends on how primordial perturbations were generated ( e.g. the shape of the primordial power spectrum of curvature perturbations), details of the PBH collapse mechanism ( e.g. structure formation theory), and thermodynamical conditions during the radiation era when PBHs were formed. We will obtain an expression for GLYPH<10> PBH that depends on all these factors.</text>
<formula><location><page_1><loc_66><loc_28><loc_92><loc_33></location>GLYPH<10> PBH = GLYPH<26> PBH GLYPH<26> crit ; (1)</formula>
<text><location><page_1><loc_51><loc_7><loc_93><loc_11></location>One viable approach to begin modelling the PBH abundance is to use the Press-Schechter (PS) theory (Press & Schechter,</text>
<text><location><page_2><loc_7><loc_81><loc_49><loc_94></location>1974), but with a modification of the collapse threshold. This approach has been widely used in previous work to model PBH abundances ( e.g. Ballesteros & Taoso (2018); Byrnes, Hindmarsh, Young, & Hawkins (2018); Chongchitnan & Efstathiou (2007a); Wang, Terada, & Kohri (2019); Young, Byrnes, & Sasaki (2014)). We present the key equations below. In section 6.1, we present a different approach based on Peak Theory.</text>
<formula><location><page_2><loc_18><loc_71><loc_49><loc_75></location>P . GLYPH<14> / dGLYPH<14> = 1 ø 2 GLYPH<25> 1 GLYPH<27> e * GLYPH<14> 2 _2 GLYPH<27> 2 d GLYPH<14> ; (2)</formula>
<text><location><page_2><loc_8><loc_75><loc_49><loc_82></location>In the PS formalism, the probability that a region within a window function of size R , containing mass M , has density contrast in the range [ GLYPH<14> ; GLYPH<14> + dGLYPH<14> ] is given by the Gaussian distribution</text>
<text><location><page_2><loc_7><loc_61><loc_49><loc_71></location>where GLYPH<27> is the variance of the primordial density perturbations GLYPH<14> smoothed on scale R . Assuming that PBHs originate from Fourier modes that re-entered the Hubble radius shortly after inflation ends ( i.e. during radiation era, when R becomes comparable to k *1 = . aH / *1 ), GLYPH<27> can be expressed as Liddle & Lyth (2000)</text>
<formula><location><page_2><loc_8><loc_51><loc_49><loc_61></location>GLYPH<27> 2 . k / = GLYPH<216> ˚ *GLYPH<216> W 2 . qk *1 / P GLYPH<14> . q / d ln q (3) = GLYPH<216> ˚ *GLYPH<216> 16 81 W 2 . qk *1 /. qk *1 / 4 T 2 . q; k *1 / P R . q / d ln q: (4)</formula>
<text><location><page_2><loc_8><loc_43><loc_49><loc_51></location>In the above equations, P GLYPH<14> and P R are, respectively, the primordial density and curvature power spectra; W is the Fourierspace window function chosen to be Gaussian 2 ( W . x / = e * x 2 _2 ); T is the transfer function given by:</text>
<formula><location><page_2><loc_14><loc_40><loc_49><loc_43></location>T . q; GLYPH<28> / = 3 y 3 .sin y * y cos y / ; y GLYPH<146> qGLYPH<28> ø 3 : (5)</formula>
<formula><location><page_2><loc_21><loc_27><loc_49><loc_29></location>M = K . GLYPH<14> * GLYPH<14> c / GLYPH<13> M H : (6)</formula>
<text><location><page_2><loc_7><loc_29><loc_49><loc_40></location>Numerical simulations suggest that the initial mass, M , of a PBH formed when density perturbation of wavenumber k reenters the Hubble radius, is known to be a fraction of the total mass, M H , within the Hubble volume ( M H is usually called the 'horizon mass'). In this work, we follow Musco & Miller (2013) in modelling M as</text>
<text><location><page_2><loc_7><loc_20><loc_49><loc_27></location>where we take GLYPH<14> c = 0 : 45 (the threshold overdensity for collapse into a PBH during radiation era), with K = 3 : 3 and GLYPH<13> = 0 : 36 . (See section 6.2 for further discussion of the values of GLYPH<14> c and K .)</text>
<text><location><page_2><loc_51><loc_89><loc_92><loc_94></location>For a given Hubble volume with horizon mass M H , the corresponding temperature, T , satisfies the equation</text>
<formula><location><page_2><loc_58><loc_85><loc_92><loc_90></location>M H = 12 H m Pl ø 8 GLYPH<25> I 3 0 10 g < ;GLYPH<26> . T / 1 1_2 T *2 ; (7)</formula>
<text><location><page_2><loc_51><loc_74><loc_93><loc_84></location>(Wang et al., 2019) where m Pl is the Planck mass, and the effective degree of freedom g < ;GLYPH<26> . T / , corresponding to energy density GLYPH<26> , can be numerically obtained as described in Saikawa & Shirai (2018). The latter reference also gave the fitting function for the effective degree of freedom g < ;s . T / corresponding to entropy s , which we will also need.</text>
<text><location><page_2><loc_51><loc_67><loc_93><loc_74></location>Using the extended PS formalism, one obtains the following expression for GLYPH<12> M H , the fraction of PBHs within a Hubble volume containing mass M H (Byrnes, Hindmarsh, et al., 2018; Niemeyer & Jedamzik, 1998)</text>
<formula><location><page_2><loc_62><loc_56><loc_92><loc_66></location>GLYPH<12> M H = 2 GLYPH<216> ˚ GLYPH<14> c M M H P . GLYPH<14> / d GLYPH<14> = GLYPH<216> ˚ *GLYPH<216> B M H . M / d ln M: (8)</formula>
<text><location><page_2><loc_51><loc_45><loc_93><loc_56></location>The factor of 2 is the usual Press-Schechter correction stemming from the possibility of PBHs formed through a cloud-in-cloud collapse (Bond, Cole, Efstathiou, & Kaiser, 1991). The integrand B M H . M / can be interpreted as the probability density function (pdf) for PBH masses on logarithmic scale at formation time. Using (2) and (6), one finds</text>
<formula><location><page_2><loc_51><loc_39><loc_95><loc_45></location>B M H . M / = K ø 2 GLYPH<25> GLYPH<13> GLYPH<27> . k H / GLYPH<22> 1+1_ GLYPH<13> exp 0 * 1 2 GLYPH<27> 2 . k H / GLYPH<0> GLYPH<14> c + GLYPH<22> 1_ GLYPH<13> GLYPH<1> 2 1 ; (9)</formula>
<formula><location><page_2><loc_56><loc_36><loc_92><loc_39></location>GLYPH<22> GLYPH<146> M KM H ; (10)</formula>
<formula><location><page_2><loc_53><loc_28><loc_92><loc_36></location>k H Mpc *1 = 3 : 745 GLYPH<157> 10 6 0 M H M GLYPH<230> 1 *1_2 4 g < ;GLYPH<26> . T . M H // 106 : 75 5 1_4 § GLYPH<157> 4 g < ;s . T . M H // 106 : 75 5 *1_3 : (11)</formula>
<text><location><page_2><loc_51><loc_20><loc_93><loc_28></location>We next consider an important quantity f . M / , the presentday fraction of dark matter in the form of PBHs of mass M . For our purposes, the expression for f . M / can be expressed as Byrnes, Cole, & Patil (2018):</text>
<formula><location><page_2><loc_54><loc_7><loc_92><loc_11></location>GLYPH<28> . M H / GLYPH<146> g < ;GLYPH<26> . T . M H // g < ;GLYPH<26> . T eq / g < ;s . T eq / g < ;s . T . M H // T . M H / T eq ; (13)</formula>
<formula><location><page_2><loc_55><loc_11><loc_92><loc_21></location>f . M / GLYPH<146> 1 GLYPH<10> CDM d GLYPH<10> PBH d log M = GLYPH<10> m GLYPH<10> CDM GLYPH<216> ˚ *GLYPH<216> GLYPH<28> . M H / B M H . M / d log M H ; (12)</formula>
<text><location><page_3><loc_7><loc_70><loc_49><loc_94></location>where GLYPH<10> CDM and GLYPH<10> m are the cosmic density parameters for cold dark matter and total matter (CDM+baryons) respectively. The first integral (12) is an integration over all horizon masses. Since a range of PBH masses are expected to form within a given horizon mass M H , this integral picks out the fraction of those PBHs with mass M . The thermodynamical factor, GLYPH<28> . M H / , relates the formation-time variables to present-day observables. Using standard expressions for the evolution of cosmic densities during the matter and radiation era, and the fact that GLYPH<26> PBH _ GLYPH<26> GLYPH<237> T *1 up to matter-radiation equality, we can intuitively understand the appearance of temperature at matterradiation equality ( T eq ) and at formation time ( T . M H / ) in (13). See Inomata, Kawasaki, Mukaida, Tada, & Yanagida (2017) for a detailed derivation.</text>
<text><location><page_3><loc_8><loc_65><loc_49><loc_70></location>Once we have calculated the PBH fraction, f . M / , the total present-day fraction of PBHs in dark matter can be calculated by integrating over all PBH masses,</text>
<formula><location><page_3><loc_17><loc_58><loc_49><loc_64></location>f PBH = GLYPH<216> ˚ log M min f . M / d log M; (14)</formula>
<text><location><page_3><loc_7><loc_53><loc_49><loc_59></location>(we will discuss M min in the next section). Finally, the fraction of the Universe in PBHs of mass > M can simply be integrated as</text>
<formula><location><page_3><loc_12><loc_48><loc_49><loc_53></location>GLYPH<10> PBH . > M / = GLYPH<10> CDM GLYPH<216> ˚ log M f . M ¤ / d log M ¤ : (15)</formula>
<section_header_level_1><location><page_3><loc_8><loc_43><loc_33><loc_44></location>3 PBH NUMBER COUNT</section_header_level_1>
<text><location><page_3><loc_8><loc_34><loc_49><loc_42></location>In analogy with the abundance of massive galaxy clusters (see e.g. Mo, van den Bosch, & White (2010) for a pedagogical treatment), the differential number density of PBHs at present time ( i.e. the PBH 'mass function') can be expressed as:</text>
<text><location><page_3><loc_7><loc_25><loc_49><loc_31></location>where GLYPH<132> GLYPH<26> is the present-day mean cosmic density. In an observational volume covering the fraction f sky of the sky up to redshift z , we would find the total number of PBHs to be</text>
<formula><location><page_3><loc_10><loc_30><loc_49><loc_35></location>d n d log M = * GLYPH<132> GLYPH<26> M d GLYPH<10> PBH . > M / d log M = GLYPH<132> GLYPH<26> M GLYPH<10> CDM f . M / ; (16)</formula>
<formula><location><page_3><loc_8><loc_18><loc_50><loc_25></location>N tot . z / = f sky z ˚ 0 .1 + z ¤ / 3 d z ¤ GLYPH<216> ˚ log M min . z ¤ / d log M d V d z ¤ d n d log M ; (17)</formula>
<text><location><page_3><loc_7><loc_15><loc_43><loc_18></location>where d V _ d z is the Hubble volume element given by</text>
<formula><location><page_3><loc_12><loc_9><loc_49><loc_15></location>d V d z = 4 GLYPH<25> H . z / ' r r p z ˚ 0 d z ¤ H . z ¤ / a s s q 2 ; (18)</formula>
<formula><location><page_3><loc_11><loc_7><loc_49><loc_10></location>H . z / ø H 0 GLYPH<4> GLYPH<10> m .1 + z / 3 +GLYPH<10> r .1 + z / 4 +GLYPH<10> GLYPH<3> GLYPH<5> 1_2 ; (19)</formula>
<text><location><page_3><loc_51><loc_89><loc_92><loc_94></location>where the cosmic densities GLYPH<10> i have their usual meaning. In this work, we will assume that f sky = 1 .</text>
<text><location><page_3><loc_51><loc_83><loc_92><loc_90></location>The lower bound in the d log M integration in Eq. (17) is the minimum PBH mass (at formation time) below which a PBH would have evaporated by redshift z . For z = 0 , it is well known that</text>
<formula><location><page_3><loc_54><loc_80><loc_92><loc_83></location>M min . z = 0/ = 5 : 1 GLYPH<157> 10 14 g ø 2 : 6 GLYPH<157> 10 *19 M GLYPH<230> ; (20)</formula>
<text><location><page_3><loc_51><loc_65><loc_93><loc_80></location>(Hawking, 1974). At higher redshifts, the minimum initial mass can be estimated assuming some basic properties of black holes. Weoutline the calculations in Appendix A. We found that M min remains within the same order of magnitude for a wide range of redshift (see Fig. 1 therein). Therefore, for models which generate an observationally interesting abundance of PBHs, it is sufficient to make the approximation M min . z / ø M min .0/ in Eq. 17. We have checked that this makes no numerical difference for the models studied in this work.</text>
<section_header_level_1><location><page_3><loc_51><loc_60><loc_88><loc_61></location>4 PBH FORMATION: A CASE STUDY</section_header_level_1>
<section_header_level_1><location><page_3><loc_51><loc_56><loc_70><loc_58></location>4.1 The logGLYPH<14> model</section_header_level_1>
<text><location><page_3><loc_51><loc_34><loc_93><loc_56></location>It is well known that the simplest models of single-field slowroll inflation cannot produce observable abundance of PBHs unless the primordial power spectrum is very blue (although this has firmly been ruled out by CMB constraints). Viable inflation models which generate an interesting density of PBHs are potentials that typically produce sharp features in the primordial power spectrum, so as to generate power at small scales (see Drees & Erfani (2011); García-Bellido & Ruiz Morales (2017); Kawasaki, Sugiyama, & Yanagida (1998); Mishra & Sahni (2019); Pi, Zhang, Huang, & Sasaki (2018) for some theoretical models). In this work, we will represent a generic primordial power spectrum with a sharp feature using a delta function spike in ln k , i.e.</text>
<formula><location><page_3><loc_63><loc_31><loc_92><loc_34></location>P R . k / = AGLYPH<14> D .ln k *ln k 0 / ; (21)</formula>
<text><location><page_3><loc_51><loc_22><loc_92><loc_31></location>where GLYPH<14> D is the Dirac delta function. The constants A and k 0 parametrize the amplitude and location of the spike in the resulting matter power spectrum. This log GLYPH<14> -function model was previously studied in Wang et al. (2019) in the context of gravitational wave production by PBHs.</text>
<section_header_level_1><location><page_3><loc_51><loc_18><loc_79><loc_19></location>4.2 Observational constraints</section_header_level_1>
<text><location><page_3><loc_51><loc_7><loc_93><loc_18></location>A range of observational constraints, including CMB anisotropies (Aloni, Blum, & Flauger, 2017; Poulter, AliHaïmoud, Hamann, White, & Williams, 2019) and microlensing observations (Green, 2016; Niikura et al., 2019), have placed upper bounds on f . M / , i.e. the PBH fraction in CDM (see, for example, Carr, Raidal, Tenkanen, Vaskonen, & Veermäe</text>
<figure>
<location><page_4><loc_9><loc_70><loc_48><loc_92></location>
<caption>FIGURE1 The dashed lines show the theoretical values of the PBH to CDM ratio, f PBH (Eq. 14) as a function of the parameters A and M H in the logGLYPH<14> model. The thick line indicates the observational upper bound f PBH,max (Eq. 22) converted from monochromatic constraints in the literature.</caption>
</figure>
<text><location><page_4><loc_35><loc_69><loc_36><loc_71></location>/circledot</text>
<text><location><page_4><loc_7><loc_44><loc_49><loc_55></location>(2017); Carr, Kohri, Sendouda, & Yokoyama (2010)). Nevertheless, the published bounds assume that all PBHs have the same mass. These so-called monochromatic constraints on f . M / were traditionally the main quantity of interest in the literature, as there is a wide range of observational techniques that can place upper bounds on f . M / over several decades of M .</text>
<text><location><page_4><loc_8><loc_26><loc_49><loc_32></location>The upshot from these studies is that the corrected upper bound for the total PBH fraction in CDM, f PBH,max , is given by</text>
<text><location><page_4><loc_8><loc_31><loc_49><loc_45></location>If we now assume that PBHs are formed across a spectrum of masses, the monochromatic upper bounds, denoted f mono max . M / , must be corrected using procedures such as those previously presented in Azhar & Loeb (2018); Carr et al. (2017); Kühnel & Freese (2017); Lehmann, Profumo, & Yant (2018). These studies have only relatively recently gained traction, but are nevertheless indispensable if PBHs were to be taken as a serious candidate for dark matter and GW sources.</text>
<formula><location><page_4><loc_13><loc_20><loc_49><loc_25></location>f PBH,max = 0 ˚ f . M / f mono max . M / d log M 1 *1 ; (22)</formula>
<text><location><page_4><loc_7><loc_7><loc_49><loc_19></location>Carr et al. (2017). The result from applying this correction to monochromatic constraints on the logGLYPH<14> model is shown in Fig. 1 . The figure shows the contour lines of constant PBH fraction f PBH (Eq. 14), as a function of model parameter A (vertical axis) and k 0 (horizontal axis, converted to the corresponding horizon mass through Eq. (11)). The thick line shows the corrected upper bound f PBH,max . In other words, the region below</text>
<text><location><page_4><loc_51><loc_89><loc_92><loc_94></location>the thick line is the allowed parameter space for the logGLYPH<14> model given current observations 3 .</text>
<text><location><page_4><loc_51><loc_59><loc_93><loc_82></location>Another interesting observation from the figure is the values of f pbh along the thick line. The maximum occurs when the spike is at M H = 10 *8 M GLYPH<230> , with f PBH ø 0 : 46 , and the minimum at M H = 10 3 M GLYPH<230> , with f PBH ø 1 : 6 GLYPH<157> 10 *3 . This means that present constraints allows the logGLYPH<14> model to consolidate almost half of all dark matter into PBHs. However, this comes from imposing a spike at very small scales where the additional nonlinear effects (which have been unaccounted for) become significant. These small-scale effects include large PBH velocity dispersion, accretion and clustering effects seen in previous numerical investigations ( e.g. Hütsi, Raidal, & Veermäe (2019); Inman & Ali-Haïmoud (2019)). These effects will weaken the validity of the upper bounds on f pbh at such small scales.</text>
<text><location><page_4><loc_51><loc_81><loc_93><loc_90></location>The upper bound is increasing in the domain shown, until M H GLYPH<237> 10 2 M GLYPH<230> , where the dip corresponds to the more stringent constraint from the CMB anisotropies, since PBH accretion effects can significantly alter the ionization and thermal history of the Universe (Ricotti, Ostriker, & Mack, 2008).</text>
<section_header_level_1><location><page_4><loc_51><loc_55><loc_71><loc_56></location>5 EXTREME PBHS</section_header_level_1>
<text><location><page_4><loc_51><loc_40><loc_93><loc_54></location>Having established a method to calculate the PBH number count and mass function, we now set out to derive the probability distribution of the most massive PBHs expected in an observational volume. Our calculation is based on the exact extreme-value formalism previously used in the context of massive galaxy clusters (Harrison & Coles, 2011, 2012) and cosmic voids (Chongchitnan, 2015; Chongchitnan & Hunt, 2017). We summarise the key concepts and equations in this section.</text>
<text><location><page_4><loc_51><loc_16><loc_93><loc_41></location>Using the most massive PBHs to constrain their cosmological origin is motivated by the same reasons that the most massive galaxy clusters and the largest cosmic voids can be used to constrain cosmology: the largest and most massive structures can typically be observed more easily whilst smaller objects are more dynamic and their observation typically suffers from larger systematic errors. In terms of PBHs, extreme-value probabilities can, at least, constrain the parameters of the underlying inflationary theory, or shed light on their merger history, or, at best, rule out the formation theory altogether. Whilst a complete mass distribution of PBHs within an observational volume would be an even more powerful discriminant of PBH formation theories, in practice it would be extremely challenging to determine with certainty which black holes are primordial and</text>
<text><location><page_5><loc_7><loc_88><loc_49><loc_94></location>which are formed through a stellar collapse or a series of mergers (see Chen & Huang (2019); García-Bellido (2017) for some novel methods).</text>
<section_header_level_1><location><page_5><loc_8><loc_84><loc_40><loc_85></location>5.1 Exact extreme-value formalism</section_header_level_1>
<text><location><page_5><loc_7><loc_76><loc_49><loc_83></location>From the PBH number count (17), we can construct the probability density function (pdf) for the mass distribution of PBHs with mass in the interval [log M; log M + d log M ] within the redshift range [0 ; z ] as</text>
<formula><location><page_5><loc_16><loc_71><loc_49><loc_76></location>f <z . M / = f sky N tot z ˚ 0 d z d V d z d n d log M : (23)</formula>
<text><location><page_5><loc_7><loc_67><loc_49><loc_71></location>To verify that this function behaves like a pdf, one can see that by comparing (17) and (23), we have the correct normalization</text>
<formula><location><page_5><loc_19><loc_62><loc_37><loc_67></location>GLYPH<216> ˚ *GLYPH<216> f <z . M / d log M = 1 :</formula>
<text><location><page_5><loc_8><loc_58><loc_49><loc_62></location>The cumulative probability distribution (cdf), F . M / , can then be constructed by integrating the pdf as usual:</text>
<formula><location><page_5><loc_17><loc_52><loc_49><loc_57></location>F . M / = log M ˚ log M min f <z . m / d log m: (24)</formula>
<text><location><page_5><loc_7><loc_43><loc_49><loc_52></location>This gives the probability that an observed PBH has mass f M . Now consider N observations of PBHs drawn from a probability distribution with cdf F . M / . We can ask: what is the probability that the observed PBH will all have mass f M < ? The required probability, GLYPH<8> , is simply the product of the cdfs:</text>
<formula><location><page_5><loc_13><loc_39><loc_49><loc_43></location>GLYPH<8>. M < ; N / = N ˙ i =1 F i . M f M < / = F N . M < / ; (25)</formula>
<formula><location><page_5><loc_8><loc_28><loc_49><loc_34></location>GLYPH<30> . M < ; N / = d d log M < F N . M < / = Nf <z . M < /[ F . M < /] N *1 : (26)</formula>
<text><location><page_5><loc_8><loc_33><loc_49><loc_39></location>assuming that PBH masses are independent, identically distributed variables. As GLYPH<8> is another cdf, the pdf of extreme-mass PBH can be obtained by differentiation:</text>
<text><location><page_5><loc_7><loc_24><loc_49><loc_29></location>It is also useful to note that the peak of the extreme-value pdf (the turning point of GLYPH<30> ) is attained at the zero of the function</text>
<text><location><page_5><loc_8><loc_19><loc_37><loc_21></location>as can be seen by setting d GLYPH<30> _ d log M < = 0 .</text>
<formula><location><page_5><loc_17><loc_21><loc_49><loc_25></location>X . M / = . N *1/ f 2 <z + F d f <z d log M ; (27)</formula>
<text><location><page_5><loc_8><loc_15><loc_49><loc_20></location>In summary, starting with the PBH mass function, one can derive the extreme-value pdf for PBHs using Eq. 26.</text>
<section_header_level_1><location><page_5><loc_8><loc_11><loc_39><loc_13></location>5.2 Application to the logGLYPH<14> model</section_header_level_1>
<text><location><page_5><loc_8><loc_7><loc_49><loc_11></location>Figure 2 shows the pdfs of extreme-mass PBHs given for N = 10 2 ; 10 3 and 10 4 observations up z = 0 : 2 (this figure</text>
<figure>
<location><page_5><loc_51><loc_70><loc_93><loc_93></location>
<caption>FIGURE2 The extreme-value probability density function for PBHs assuming the logGLYPH<14> model with spike at M H = 10 2 M GLYPH<230> , assuming that N = 10 , 10 2 , 10 3 observations up to z = 0 : 2 .</caption>
</figure>
<figure>
<location><page_5><loc_51><loc_39><loc_93><loc_60></location>
<caption>FIGURE 3 Profile of the extreme-value pdf peaks (solid line) and the 5th/95th percentiles (dashed lines) for the logGLYPH<14> model as the spike location is varied (horizontal axis), assuming 100 observations of PBHs up to z = 0 : 2 . The vertical axis shows the location of the peaks. The relationship is linear to a good approximation (Eq. 28).</caption>
</figure>
<text><location><page_5><loc_51><loc_12><loc_93><loc_24></location>summarises the key results of this work). We assume the logGLYPH<14> model with the power-spectrum spike at M H = 10 2 M GLYPH<230> . The pdfs are not symmetric but have a positive skewness, consistent with previous derivations of extreme-value pdfs Chongchitnan (2015); Chongchitnan & Hunt (2017). As N increases, the peaks of the pdf naturally shifts towards higher values of M < , with increasing kurtosis ( i.e. more sharply peaked).</text>
<text><location><page_5><loc_51><loc_7><loc_92><loc_13></location>When we vary the location of the spike (whilst keeping N fixed, and using values of A that saturate the upper bound shown in Fig. 1 ), we obtain an almost linear variation as shown in</text>
<figure>
<location><page_6><loc_8><loc_70><loc_49><loc_92></location>
<caption>FIGURE 4 Extreme-value pdfs for the logGLYPH<14> model with M H = 10 3 M GLYPH<230> given current constraints (thick line) and futuristic constraints. The pair of vertical dotted lines on each pdf indicate the 5th and 95th percentiles. If observational constraints on f pbh were tightened to 50% the current values, the peak of the extreme-value pdf would shift downwards by 20~ , whilst the inter-percentile distance would shrink by GLYPH<237> 30~ .</caption>
</figure>
<text><location><page_6><loc_8><loc_44><loc_49><loc_55></location>Fig. 3 (in which N = 100 ). Each vertical slice of this figure can be regarded as the profile of the extreme-value pdf, with the peak of the pdf being along the solid line, whilst the 5th and 95th percentiles are shown in dashed lines. The band is linear to a good approximation, with the peak M < peak satisfying the relation</text>
<formula><location><page_6><loc_22><loc_41><loc_49><loc_44></location>M < peak ø 2 : 3 M H : (28)</formula>
<text><location><page_6><loc_7><loc_13><loc_49><loc_29></location>It is also interesting to consider how tightening observation bounds will affect the extreme-value pdfs. Fig. 4 shows what happens in this situation in the model with M H = 10 3 M GLYPH<230> (with N = 100 ), supposing that the upper bound on f pbh is tightened to 50% of the current values (a realistic prospects for future experiments such as Euclid (Habouzit et al., 2019)). We see that, in line with expectation, the pdf shifts to smaller masses by GLYPH<237> 20~ , whilst the distance between the 5% and 95% percentiles shrinks by GLYPH<237> 30~ .</text>
<text><location><page_6><loc_7><loc_28><loc_49><loc_41></location>The percentile band spans a narrow range of logarithmic masses. We see that the logGLYPH<14> model can produce massive PBHs with masses of order GLYPH<237> 30 M GLYPH<230> , using spikes at M H GLYPH<237> 10 M GLYPH<230> (the former being within the 5th and 95th percentile band). It is possible to integrate the extreme-value pdfs in figure 2 to calculate the probability that the extreme-mass PBH at redshift 0.2 is, say, > 30 M GLYPH<230> .</text>
<text><location><page_6><loc_8><loc_7><loc_49><loc_14></location>Finally, one might ask what value of N should be used in this kind of study. Although from a statistical point of view, N is defined as the number of distinct samples drawn the pdf f . M / , in practice it is unclear how to quantify the true number</text>
<text><location><page_6><loc_51><loc_78><loc_93><loc_94></location>of PBH observations and especially given the additional complication of various selection biases. Individually identified PBHs can be detected using different probes such as microlensing and gravitational-wave emission, but the observable number of PBHs detectable by one particular method is much smaller than the total number that would be theoretically observable. Thus, to minimise the effect of selection bias, it is more precise to define N as the observed observable number of PBHs from a particular choice of observation.</text>
<section_header_level_1><location><page_6><loc_51><loc_71><loc_82><loc_74></location>6 DISCUSSION ON RECENT THEORETICAL DEVELOPMENT</section_header_level_1>
<section_header_level_1><location><page_6><loc_51><loc_68><loc_67><loc_70></location>6.1 Peak Theory</section_header_level_1>
<text><location><page_6><loc_51><loc_44><loc_93><loc_68></location>To complete our investigation, we consider an alternative to calculating PBH abundances using Peak Theory which postulates that PBHs result from peaks in the primordial overdensity field exceeding a threshold value (see Bardeen, Bond, Kaiser, & Szalay (1986); Green, Liddle, Malik, & Sasaki (2004) for reviews). It is well known that the PS and Peak Theory do not agree, although previous authors have suggested that Peak Theory is grounded on a firmer theoretical footing, and is more sensitive to the shape of the inflationary power spectrum (Germani & Musco, 2019; Kalaja et al., 2019; Young, Musco, & Byrnes, 2019). Nevertheless, there are still conceptual issues with both PS and Peak Theory, with a number of extensions having been recently proposed (Germani & Sheth, 2020; Suyama &Yokoyama, 2020).</text>
<text><location><page_6><loc_51><loc_36><loc_92><loc_45></location>In this section, we recalculate the extreme-value distribution GLYPH<30> . M / shown previously in Figure 2 using Peak Theory in the formulation proposed by Young, Musco and Byrnes Young et al. (2019). Using their formalism, we found the PBH fraction f . M / (see Eq. 16) for the logGLYPH<14> model to be 4</text>
<formula><location><page_6><loc_55><loc_29><loc_92><loc_36></location>f peak . M / = GLYPH<10> m GLYPH<10> CDM GLYPH<216> ˚ a . M / GLYPH<28> . M H / B peak M H . M / d log M H (29)</formula>
<formula><location><page_6><loc_61><loc_19><loc_92><loc_25></location>GLYPH<23> = GLYPH<14> . M / GLYPH<27> . M H / = GLYPH<24> M KM H GLYPH<25> 1_ GLYPH<13> + GLYPH<14> c GLYPH<27> . M H / (31) H I</formula>
<formula><location><page_6><loc_53><loc_25><loc_92><loc_29></location>B peak M H . M / = = M 3 GLYPH<25> M H 0 k 0 aH 1 3 GLYPH<23> 3 e * GLYPH<23> 2 _2 ; (30)</formula>
<formula><location><page_6><loc_58><loc_15><loc_92><loc_19></location>a . M / = log M K . 2 3 * GLYPH<14> c / GLYPH<13> ; (32)</formula>
<text><location><page_6><loc_51><loc_13><loc_74><loc_16></location>where GLYPH<28> . M H / is given in Eq. (13).</text>
<figure>
<location><page_7><loc_8><loc_70><loc_48><loc_92></location>
<caption>FIGURE 5 The extreme-value probability density function from Peak Theory, assuming the logGLYPH<14> model with spike at M H = 10 2 M GLYPH<230> , assuming that N = 10 , 10 2 , 10 3 observations up to z = 0 : 2 . In contrast with the pdfs in Fig. 2 (obtained using PS theory), the Peak-Theory pdfs attain maxima at lower M < , and have skewness of the opposite sign.</caption>
</figure>
<text><location><page_7><loc_8><loc_49><loc_49><loc_55></location>Some extreme-value pdfs from Peak Theory are shown in Fig. 5 . This should be compared with the same distributions calculated using PS formalism in Fig. 2 .</text>
<text><location><page_7><loc_8><loc_37><loc_49><loc_50></location>With N GLYPH<192> 100 observations, the extreme-value pdfs from both formalisms attain similar profiles. We observed that the Peak Theory pdfs attain maxima at a slightly lower M < values compared to the Press-Schechter pdfs. With N = 100 , a similar numerical analysis of the relation between the maxima of the extreme-value-pdf ( M < peak-PT) ) and the location of the spike is found to be:</text>
<formula><location><page_7><loc_21><loc_33><loc_49><loc_36></location>M < peak-PT ø M H : (33)</formula>
<text><location><page_7><loc_7><loc_20><loc_49><loc_33></location>(compare with Eq. 28). Although we did observe that the two formalisms predict total PBH number counts that are different by a few orders of magnitude (as corroborated by previous studies), the extreme-value pdfs are not so drastically different. This is because the extreme pdfs are integrated over redshifts and masses, thus when the pdfs are normalised, the effects from large differences in N tot are suppressed.</text>
<text><location><page_7><loc_7><loc_7><loc_49><loc_21></location>It is also interesting to note that the different functional forms of B M H lead to different skewness in the extreme-value pdfs (where the skewness is calculated on semilog scale as shown). The Peak-Theory pdfs are negatively skewed whilst the PS pdfs are positively skewed. See the Appendix B for an analytic explanation. It may be possible to use this property to distinguish between the peak-theory and Press-Schechter-like formalism of PBH formation.</text>
<figure>
<location><page_7><loc_51><loc_70><loc_93><loc_92></location>
<caption>FIGURE 6 The effect of changing K and GLYPH<14> c on the extremevalue probability density function from Peak Theory, assuming the logGLYPH<14> model with spike at M H = 10 2 M GLYPH<230> , assuming that N = 10 3 observations up to z = 0 : 2 . Changing . K;GLYPH<14> c / from .3 : 3 ; 0 : 45/ to .4 ; 0 : 55/ (solid and dashed lines respectively) shifts the pdf to higher M < by a few percent.</caption>
</figure>
<section_header_level_1><location><page_7><loc_51><loc_54><loc_76><loc_56></location>6.2 The overdensity profile</section_header_level_1>
<text><location><page_7><loc_51><loc_32><loc_93><loc_54></location>The shape and height of the profile of density peaks are governed by the constant K and the critical density GLYPH<14> c in equation 6. Both quantities can vary depending on typical profiles of the density perturbation (see Musco (2019) for a comprehensive theoretical study). To this end, we re-evaluate the Peak Theory pdfs using the values K = 4 and GLYPH<14> c = 0 : 55 as proposed by Young et al. (2019), instead of the fiducial values K = 3 : 3 and GLYPH<14> c = 0 : 45 . Figure 6 shows the comparison between the two sets of parameters for the model with M H = 10 2 M GLYPH<230> assuming N = 10 3 . It appears that the extreme-value pdfs only depend weakly to changes in K and GLYPH<14> c : Increasing these parameters (by GLYPH<237> 30~ ) only results in a few-percent shift of the pdf to higher logarithmic masses.</text>
<section_header_level_1><location><page_7><loc_51><loc_27><loc_68><loc_28></location>7 CONCLUSION</section_header_level_1>
<text><location><page_7><loc_51><loc_18><loc_92><loc_26></location>In this work, we have established a framework to calculate the mass distribution of most massive PBHs expected within a given observational volume. The calculations were based mainly on four main ingredients:</text>
<unordered_list>
<list_item><location><page_7><loc_53><loc_14><loc_92><loc_18></location>· the PBH formation mechanism ( e.g. details of inflation or the shape of P R . k / )</list_item>
<list_item><location><page_7><loc_53><loc_10><loc_92><loc_14></location>· the abundance of massive objects ( e.g. Press-Schechter or Peak Theory)</list_item>
<list_item><location><page_7><loc_53><loc_7><loc_78><loc_9></location>· the exact extreme-value formalism</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_8><loc_9><loc_89><loc_49><loc_94></location>· the observational constraints on f pbh (the PBH fraction in CDM).</list_item>
</unordered_list>
<text><location><page_8><loc_7><loc_69><loc_49><loc_89></location>We applied our formalism to the logGLYPH<14> model, a prototype of models with a spike in the power spectrum. Such spikes are generically associated with inflationary models that produce interesting densities of PBH ( e.g. via a phase transition in the early Universe). Our main results are the extreme-value pdf shown in Fig. 2 and 5 . The fact that the location of the powerspectrum spike is close to the peak of the resulting extremevalue pdf gives assurance that our calculations are sound, and can thus be applied to many inflationary models known to produce PBHs. In future work, we will present a survey of extreme-value pdfs for a range of inflationary scenarios.</text>
<text><location><page_8><loc_7><loc_46><loc_49><loc_70></location>Some avenues for further investigation include studying the effect of changing the mass function (for example, extending the Sheth-Tormen mass function to PBHs (Chongchitnan & Efstathiou, 2007b)), as well as understanding the role of PBH clustering and merger (Kohri & Terada, 2018; Raidal, Spethmann, Vaskonen, & Veermäe, 2019; Raidal, Vaskonen, &Veermäe, 2017; Tada & Yokoyama, 2015; Young & Byrnes, 2019) which will serve to strengthen the validity of the extremevalue formalism presented here. We envisage that there are other uses for the extreme-value pdfs that the formalism presented can be adapted, for instance, to quantify distribution of the most massive intermediate-mass black holes that could subsequently seed supermassive black holes at galactic centres (Dolgov, 2020).</text>
<section_header_level_1><location><page_8><loc_8><loc_41><loc_29><loc_42></location>7.1 Acknowledgements</section_header_level_1>
<text><location><page_8><loc_7><loc_30><loc_49><loc_41></location>Wethank the referee for helpful comments, and Sai Wang for his help in the early stages of this paper. We acknowledge further helpful comments and feedback from Joe Silk, Kazunori Kohri, Ying-li Zhang, Cristiano Germani and Hardi Veermae. Many of our numerical computations were performed on the University of Hull's VIPER supercomputer.</text>
<section_header_level_1><location><page_8><loc_8><loc_17><loc_44><loc_22></location>APPENDIX A: THE MINIMUM INITIAL MASS OF AN UNEVAPORATED BLACK HOLE AT REDSHIFT Z .</section_header_level_1>
<text><location><page_8><loc_8><loc_7><loc_49><loc_16></location>Consider a Schwarzschild black hole. Its decay rate depends on three variables, namely, 1) the spin ( s ) of the particles it decays into, 2) the energy ( E ) of those particles, and 3) the instantaneous mass . M / of the black hole. By summing over all the emitted particles, the decay rate of a black hole can be</text>
<text><location><page_8><loc_51><loc_91><loc_59><loc_94></location>expressed as</text>
<text><location><page_8><loc_51><loc_79><loc_92><loc_87></location>where the sum is taken over all emitted particle species. The integral is taken over .0 ; GLYPH<216>/ for massless particles, or . GLYPH<22> j ; GLYPH<216>/ for massive particles with rest energy GLYPH<22> j . GLYPH<0> j is the dimensionless absorption probability, and s j is the spin of the j th species.</text>
<formula><location><page_8><loc_51><loc_86><loc_92><loc_91></location>d M d t = * 1 2 GLYPH<25> =c 2 GLYPH<201> j GLYPH<0> j ˚ d E E exp GLYPH<0> 8 GLYPH<25> GE M _ =c 3 GLYPH<1> *.*1/ 2 s j ; (A1)</formula>
<text><location><page_8><loc_51><loc_76><loc_92><loc_80></location>MacGibbon (MacGibbon, 1991) showed that Eq. (A1) can be written as</text>
<text><location><page_8><loc_51><loc_66><loc_93><loc_74></location>The function f emit . M / is given in a rather complicated piecewise form in Eq. (7) in MacGibbon (1991). Eq. (A2) can be inverted and integrated to yield the evaporation timescale, GLYPH<28> evap , as</text>
<formula><location><page_8><loc_56><loc_73><loc_92><loc_76></location>d M d t = *5 : 34 GLYPH<157> 10 22 f emit . M / M *2 kg s *1 : (A2)</formula>
<formula><location><page_8><loc_52><loc_61><loc_92><loc_66></location>GLYPH<28> evap = GLYPH<0> 1 : 87266 GLYPH<157> 10 *23 s kg *1 GLYPH<1> M i ˚ M f d Mf . M / *1 M 2 ; (A3)</formula>
<formula><location><page_8><loc_62><loc_51><loc_92><loc_55></location>GLYPH<28> evap . z / GLYPH<243> GLYPH<243> GLYPH<243> GLYPH<243> M f =0 ;M i = M < = t univ . z / ; (A4)</formula>
<text><location><page_8><loc_51><loc_55><loc_92><loc_61></location>where M i is the initial mass of the black hole and M f is the final mass. Letting M f = 0 and M i = M < , we can obtain M < as a function of z by solving the non-linear equation:</text>
<text><location><page_8><loc_51><loc_43><loc_92><loc_52></location>where t univ is the age of the universe at redshift z . The minimum initial black hole mass as a function of redshift is shown in Figure 1 . The figure closely resembles Fig. 1 of MacGibbon (1991), although we believe that the labelling of the two curves in that figure should be exchanged.</text>
<section_header_level_1><location><page_8><loc_51><loc_35><loc_91><loc_39></location>APPENDIX B: SKEWNESS OF THE EXTREME-VALUE DISTRIBUTION (PRESS-SCHECHTER VS. PEAK-THEORY).</section_header_level_1>
<text><location><page_8><loc_51><loc_20><loc_93><loc_34></location>Let us explore a simple analytic approximation of the extremevalue pdf in order to show that the skewnesses (on log mass scale) calculated using the above theories are of opposite signs. To this end, we will study the simplest case N = 1 , in which case the extreme-value pdf is GLYPH<30> . M / = f <z . M / GLYPH<237> f . M /_ M (using Eqs. 16, 23 and 26). Here f . M / are the PBH fractions given in 12 and 29 for the two formalisms respectively. We will focus on the evolution in M only.</text>
<formula><location><page_8><loc_53><loc_7><loc_91><loc_12></location>f PS . M / GLYPH<237> M 4 ˚ x *13_3 exp H * x 4_3 4 GLYPH<14> c + M 3 x 3 5 2 I d x:</formula>
<text><location><page_8><loc_51><loc_11><loc_93><loc_20></location>Press-Schechter : We make the approximation for the variance of PBH-forming overdensity: GLYPH<27> GLYPH<237> R *2 / where R is the scale of the window function. Thus, in terms of mass within the filter, GLYPH<27> GLYPH<237> M *2_3 . The mass-dependent terms in the mass fraction 12 can then be expressed as</text>
<figure>
<location><page_9><loc_9><loc_69><loc_49><loc_92></location>
<caption>FIGURE 1 Minimum initial mass of a black hole which have not evaporated, observed at redshift z . The solid line is the result assuming the LCDM universe with Planck 2018 parameters. For comparison, the dashed line assumes the Einstein-de Sitter (EdS) cosmology .GLYPH<10> m = 1 ; GLYPH<10> GLYPH<3> = 0/ .</caption>
</figure>
<text><location><page_9><loc_8><loc_46><loc_49><loc_56></location>We approximate the integral in two regimes. When M is small, the M dependence in the integral is negligible, so f PS GLYPH<237> M 4 . When M is large, the M dependence can be removed from the integral via a substitution t = M 6 x *14_3 ; leaving a gamma function and an overall M dependence of f PS GLYPH<237> M *2_7 . In summary, we find</text>
<formula><location><page_9><loc_17><loc_41><loc_39><loc_46></location>GLYPH<30> PS . M / GLYPH<237> T M 3 ; M ~ 1 ; M *9_7 ; M , 1 :</formula>
<text><location><page_9><loc_8><loc_34><loc_49><loc_41></location>Peak-Theory : A similar approximation scheme [but this time retaining the integral due to the M dependence in the lower limit of (29)] shows that the mass fraction can be approximated by an incomplete gamma function. One then obtains:</text>
<text><location><page_9><loc_8><loc_23><loc_49><loc_31></location>The functions GLYPH<30> PS and GLYPH<30> peak are plotted in figure (2 ) below on semilog scale. Evidently their skewnesses differs in sign, as can be confirmed numerically. For larger N , the extreme-value pdfs retain their respective sign as seen in the main text.</text>
<formula><location><page_9><loc_17><loc_30><loc_40><loc_34></location>GLYPH<30> peak . M / GLYPH<237> M 3_2 GLYPH<0> GLYPH<24> 25 8 ; M 4_3 GLYPH<25> :</formula>
<section_header_level_1><location><page_9><loc_8><loc_18><loc_21><loc_19></location>REFERENCES</section_header_level_1>
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<caption>FIGURE 2 The simplified extreme-value pdf for the PressSchechter and Peak-Theory formalisms, normalised so that the maximum is at GLYPH<30> = 1 . Their skewnesses (on log mass scale) are evidently of opposite signs.</caption>
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</document> | [] |
2013IJMPD..2242014C | https://arxiv.org/pdf/1308.6155.pdf | <document>
<section_header_level_1><location><page_1><loc_25><loc_88><loc_76><loc_89></location>Effective theory for quantum gravity</section_header_level_1>
<section_header_level_1><location><page_1><loc_43><loc_80><loc_58><loc_82></location>Xavier Calmet 1</section_header_level_1>
<text><location><page_1><loc_18><loc_76><loc_83><loc_77></location>Physics & Astronomy, University of Sussex, Falmer, Brighton, BN1 9QH, UK</text>
<text><location><page_1><loc_36><loc_66><loc_64><loc_67></location>Submission date: March 14, 2013</text>
<section_header_level_1><location><page_1><loc_46><loc_42><loc_54><loc_43></location>Abstract</section_header_level_1>
<text><location><page_1><loc_17><loc_24><loc_84><loc_39></location>In this paper, we discuss an effective theory for quantum gravity and discuss the bounds on the parameters of this effective action. In particular we show that measurement in pulsars binary systems are unlikely to improve the bounds on the coefficients of the R 2 and R µν R µν terms obtained from probes of Newton's potential performed on Earth. Furthermore, we argue that if the coefficients of these terms are induced by quantum gravity, they should be at most of order unity since R 2 and R µν R µν are dimension four operators. The same applies to the non-minimal coupling of the Higgs boson to the Ricci scalar.</text>
<text><location><page_1><loc_12><loc_16><loc_89><loc_17></location>Essay written for the Gravity Research Foundation 2013 Awards for Essays on Gravitation.</text>
<text><location><page_2><loc_12><loc_70><loc_89><loc_89></location>Finding a quantum formulation of general relativity [1] is notoriously difficult. Despite several interesting proposals see e.g. [2] for a review, we are still far away from having a satisfactory quantum mechanical description of gravity. It would thus be helpful to have some guidance from nature. Most experiments designed to date to test quantum gravitational effects rely on the hope that some basic symmetry of nature is violated by quantum gravitational effects or that the dispersion relation of light is modified. Here we will describe a framework which enables one to probe quantum gravitational effects directly without having to make speculative assumptions. The only hypothesis is that general covariance is the correct symmetry of gravity at the quantum level as well.</text>
<text><location><page_2><loc_12><loc_47><loc_89><loc_69></location>When a fundamental theory is not known, the notion of effective field theory can be useful. Indeed it has proven to be extremely successful in different branches of physics ranging from particle physics to condensed matter physics. Effective theories are appropriate when one considers experiments at energies well below the scale at which the full underlying physics is expected to become apparent. Clearly this is the case of quantum gravity. Physics experiments can be performed typically at center of mass energies going up to 300 TeV if we think of high energetic cosmic rays, while quantum gravity is expected to become important at some energy scale M /star traditionally identified with the reduced Planck scale M P = 2 . 4335 × 10 18 GeV. The hierarchy between these two scales is the reason why it is so tough to probe quantum gravity experimentally.</text>
<text><location><page_2><loc_12><loc_33><loc_89><loc_46></location>While general relativity is very successful on macroscopic scales, it is well known that its quantization is not straightforward. Indeed, general relativity is not renormalizable, at least in perturbation theory. But, assuming that we know the fundamental symmetry of nature, namely diffeomorphism invariance, and that gravity can be described by a massless spin 2 particle, one can formulate an effective theory for quantum general relativity [3-5] valid up to the energy scale at which quantum gravitational effects become strong M /star .</text>
<text><location><page_2><loc_12><loc_26><loc_89><loc_32></location>The effective field theory containing the metric g µν , which describes the graviton if linearized, a cosmological constant and the standard model of particle physics L SM (Higgs doublet H included) is given by</text>
<formula><location><page_2><loc_14><loc_21><loc_89><loc_25></location>S = ∫ d 4 x √ -g [( 1 2 M 2 + ξH † H ) RΛ 4 C + c 1 R 2 + c 2 R µν R µν + L SM + O ( M -2 /star ) ] (1)</formula>
<text><location><page_2><loc_12><loc_15><loc_89><loc_21></location>This effective action is an expansion in space-time curvature. The Higgs boson has a nonzero vacuum expectation value, v = 246 GeV. The parameters M and ξ are then determined by</text>
<formula><location><page_2><loc_41><loc_12><loc_89><loc_13></location>( M 2 + ξv 2 ) = M 2 P . (2)</formula>
<text><location><page_2><loc_12><loc_7><loc_88><loc_10></location>This effective field theory contains several energy scales. There is the reduced Planck scale M P of the order of 10 18 GeV (equivalently Newton's constant), the cosmological constant</text>
<text><location><page_3><loc_12><loc_83><loc_89><loc_89></location>Λ C of order of 10 -3 eV and the M /star which is traditionally identified with M P but this needs not to be the case if we do not trust this effective theory up to that energy scale. In the language of effective field theory, the coefficients c 1 / 2 and ξ are called Wilson coefficients.</text>
<text><location><page_3><loc_12><loc_49><loc_89><loc_82></location>As we know so little about quantum gravity, it is not clear how many of the Wilson coefficients of this effective action are fundamental parameters of nature, i.e. new coupling constants, or calculable using other parameters of the effective action. This distinction is important. For example, the Wilson coefficients of dimension four operators are expected to be very tiny if they are generated by some new physics at a scale Λ NP . Indeed, in the limit where Λ NP →∞ they must disappear. Hence they must be of the form exp ( -λ/ Λ NP ) where λ is identified with the ultra violet cutoff of the theory. Here we need to be more careful. When one expands g µν around a constant background metric η µν using g µν = η µν + h µν /M P , one finds that the dimension four operator ξH † H R is actually a dimension 6 operator of the type ξH † Hh /square h/M 2 P where h is the graviton. The remaining dimension 4 operators of the type R 2 are actually dimension 8 operators h /square hh /square h/M 4 P . One thus expects that ξ , c 1 and c 2 are of order unity. On the other hand they might be arbitrarily large if they are genuinely new fundamental parameters of nature. The Wilson coefficient of higher order terms in R , if induced by quantum gravity, are proportional to (1 /M P ) n ( λ/M P ) m and thus expected to be small unless again they are fundamental parameters not calculable in terms of M P .</text>
<text><location><page_3><loc_12><loc_29><loc_89><loc_48></location>One should stress that it is not clear what is the cutoff λ for this effective theory, it might be M P , an energy scale corresponding to the cosmological constant scale or maybe even some other particle physics scale such as the weak scale or some other scale of grand unification. The smallness of the observed cosmological constant could be the sign that the cutoff should be of the order of the cosmological constant scale. However, the fact that we see no sign of new physics beyond usual general relativity at this energy scale could be interpreted as failure of the naturalness argument just like the discovery of a single Higgs boson without any sign of new physics so far, seems to indicate that naturalness is not a helpful argument in our quest for physics beyond the standard model.</text>
<text><location><page_3><loc_12><loc_19><loc_89><loc_27></location>We shall now describe the current bounds on the values of the parameters of the quantum gravitational action given in Eq. (1) and discuss whether future experiments are likely to improve them or not. Stelle has shown [6] that the terms c 1 R 2 and c 2 R µν R µν lead to Yukawa-like corrections to Newton's potential of a point mass m :</text>
<formula><location><page_3><loc_33><loc_14><loc_89><loc_18></location>Φ( r ) = -Gm r ( 1 + 1 3 e -m 0 r -4 3 e -m 2 r ) (3)</formula>
<text><location><page_3><loc_12><loc_11><loc_16><loc_13></location>with</text>
<formula><location><page_3><loc_38><loc_5><loc_89><loc_9></location>m -1 0 = √ 32 πG (3 c 1 -c 2 ) (4)</formula>
<text><location><page_4><loc_12><loc_88><loc_15><loc_89></location>and</text>
<formula><location><page_4><loc_41><loc_82><loc_89><loc_86></location>m -1 2 = √ 16 πGc 2 . (5)</formula>
<text><location><page_4><loc_12><loc_76><loc_88><loc_82></location>Sub-millimeter tests of Newton's law [7] using sophisticated pendulums are used to bound c 1 and c 2 . One finds that, in the absence of accidental fine cancellations between both Yukawa terms, they are constrained to be less than 10 61 [8] .</text>
<text><location><page_4><loc_12><loc_49><loc_89><loc_74></location>Astrophysical observations lead to bounds on these terms [9]. Binary systems of pulsars are the most promising environment to probe gravity at high curvature. However, a back of the envelop estimate quickly reveals that astrophysical observations will not be able to compete with Earth-based tests of Newton's law. Let us approximate the Ricci scalar in the binary system of pulsars by GM/ ( r 3 c 2 ) where M is the mass of the pulsar and r is the distance to the center of the pulsar. Clearly, if the distance is larger than the radius of the pulsar, then the Ricci scalar vanishes, so we are doing a rather crude estimate. Let us be optimistic and assume we could probe gravity at the surface of the pulsar, we thus take r = 13 . 1 km and M=2 solar masses. One then requests that the R 2 term should become comparable to the leading order Einstein-Hilbert term 1 2 M 2 P R and finds that one could reach only bounds of the order of 10 78 on c 1 . Similar weak bounds are expected on c 2 . Such limits are obviously much weaker that those obtained on Earth.</text>
<text><location><page_4><loc_12><loc_30><loc_89><loc_47></location>We now consider the bounds on the non-minimal coupling of the Higgs boson to the Ricci scalar which is the third dimension four operator of the effective action. The fact that the boson discovered at CERN behaves very much like what is expected of the Higgs boson of the standard model of particle physics, enables one to set a limit on the non-minimal coupling as for large ξ the Higgs boson would decouple from the standard model. One finds | ξ | > 2 . 6 × 10 15 is excluded at the 95% C.L. [10]. Future colliders will not be able to improve this bound much and only potentially by one order of magnitude at a hypothetical future linear collider.</text>
<text><location><page_4><loc_12><loc_23><loc_88><loc_29></location>There is no bound to date on the coefficients of higher dimensional operators. Because they are suppressed by powers of the Planck scale, these terms are expected to be completely irrelevant unless their Wilson coefficients are unnaturally large.</text>
<text><location><page_4><loc_12><loc_7><loc_89><loc_21></location>Any progress towards measuring the parameters of the effective action will clearly require to be very creative. While this is clearly a difficult task, it may be the only way to probe directly quantum gravity and to differentiate empirically between different frameworks to quantize gravity. We stress that although the parameters of the quantum gravitational effective action are expected to be small if they are generated via quantum gravitational corrections, they could be large if they are truly independent parameters and not calculable in terms of the Planck mass. Indeed, who could have guessed without experimental guidance</text>
<text><location><page_5><loc_12><loc_66><loc_89><loc_89></location>that gravity would be that much weaker that the weak interactions? The reason for this phenomenon is that the Planck mass is so large. Why wouldn't the other coefficients of the effective action be large as well? It is thus important to continue the ongoing experimental and observational efforts to measure these parameters. One should also keep in mind that one of the major theoretical development of the last 15 years has been the realization that M /star , could be much below the traditional 2 . 4335 × 10 18 GeV if there are extra-dimensions with a large volume [11,12] or even in four space-time dimensions if there is a large hidden sector of particles [8]. Clearly the LHC has been leading the way by setting some of the tightest limits to date on the experimental value of the Planck scale which we now know is above a few TeVs, but more creative ideas could lead to stronger bounds on this parameter of the quantum gravitational effective action.</text>
<section_header_level_1><location><page_5><loc_12><loc_58><loc_27><loc_60></location>References</section_header_level_1>
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<list_item><location><page_5><loc_13><loc_51><loc_74><loc_53></location>[2] C. Kiefer, 'Quantum gravity,' Int. Ser. Monogr. Phys. 124 , 1 (2004).</list_item>
<list_item><location><page_5><loc_13><loc_49><loc_69><loc_50></location>[3] J. F. Donoghue, Phys. Rev. D 50 , 3874 (1994) [gr-qc/9405057].</list_item>
<list_item><location><page_5><loc_13><loc_46><loc_79><loc_47></location>[4] J. F. Donoghue, AIP Conf. Proc. 1483 , 73 (2012) [arXiv:1209.3511 [gr-qc]].</list_item>
<list_item><location><page_5><loc_13><loc_43><loc_66><loc_44></location>[5] C. P. Burgess, Living Rev. Rel. 7 , 5 (2004) [gr-qc/0311082].</list_item>
<list_item><location><page_5><loc_13><loc_40><loc_52><loc_41></location>[6] K. S. Stelle, Gen. Rel. Grav. 9 , 353 (1978).</list_item>
<list_item><location><page_5><loc_13><loc_35><loc_88><loc_38></location>[7] C. D. Hoyle, D. J. Kapner, B. R. Heckel, E. G. Adelberger, J. H. Gundlach, U. Schmidt and H. E. Swanson, Phys. Rev. D 70 , 042004 (2004) [hep-ph/0405262].</list_item>
<list_item><location><page_5><loc_13><loc_31><loc_89><loc_34></location>[8] X. Calmet, S. D. H. Hsu and D. Reeb, Phys. Rev. D 77 , 125015 (2008) [arXiv:0803.1836 [hep-th]].</list_item>
<list_item><location><page_5><loc_13><loc_28><loc_80><loc_29></location>[9] D. Psaltis, Living Rev. Relativity 11, (2008), 9. [arXiv:0806.1531 [astro-ph]].</list_item>
<list_item><location><page_5><loc_12><loc_23><loc_88><loc_27></location>[10] M. Atkins and X. Calmet, Phys. Rev. Lett 110, 051301 (2013) arXiv:1211.0281 [hepph].</list_item>
<list_item><location><page_5><loc_12><loc_17><loc_89><loc_22></location>[11] N. Arkani-Hamed, S. Dimopoulos, G. R. Dvali, Phys. Lett. B429 , 263-272 (1998) [hep-ph/9803315]; I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos et al. , Phys. Lett. B436 , 257-263 (1998) [hep-ph/9804398].</list_item>
<list_item><location><page_5><loc_12><loc_14><loc_85><loc_16></location>[12] L. Randall, R. Sundrum, Phys. Rev. Lett. 83 , 3370-3373 (1999) [hep-ph/9905221].</list_item>
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</document> | [{"title": "Xavier Calmet 1", "content": "Physics & Astronomy, University of Sussex, Falmer, Brighton, BN1 9QH, UK Submission date: March 14, 2013", "pages": [1]}, {"title": "Abstract", "content": "In this paper, we discuss an effective theory for quantum gravity and discuss the bounds on the parameters of this effective action. In particular we show that measurement in pulsars binary systems are unlikely to improve the bounds on the coefficients of the R 2 and R \u00b5\u03bd R \u00b5\u03bd terms obtained from probes of Newton's potential performed on Earth. Furthermore, we argue that if the coefficients of these terms are induced by quantum gravity, they should be at most of order unity since R 2 and R \u00b5\u03bd R \u00b5\u03bd are dimension four operators. The same applies to the non-minimal coupling of the Higgs boson to the Ricci scalar. Essay written for the Gravity Research Foundation 2013 Awards for Essays on Gravitation. Finding a quantum formulation of general relativity [1] is notoriously difficult. Despite several interesting proposals see e.g. [2] for a review, we are still far away from having a satisfactory quantum mechanical description of gravity. It would thus be helpful to have some guidance from nature. Most experiments designed to date to test quantum gravitational effects rely on the hope that some basic symmetry of nature is violated by quantum gravitational effects or that the dispersion relation of light is modified. Here we will describe a framework which enables one to probe quantum gravitational effects directly without having to make speculative assumptions. The only hypothesis is that general covariance is the correct symmetry of gravity at the quantum level as well. When a fundamental theory is not known, the notion of effective field theory can be useful. Indeed it has proven to be extremely successful in different branches of physics ranging from particle physics to condensed matter physics. Effective theories are appropriate when one considers experiments at energies well below the scale at which the full underlying physics is expected to become apparent. Clearly this is the case of quantum gravity. Physics experiments can be performed typically at center of mass energies going up to 300 TeV if we think of high energetic cosmic rays, while quantum gravity is expected to become important at some energy scale M /star traditionally identified with the reduced Planck scale M P = 2 . 4335 \u00d7 10 18 GeV. The hierarchy between these two scales is the reason why it is so tough to probe quantum gravity experimentally. While general relativity is very successful on macroscopic scales, it is well known that its quantization is not straightforward. Indeed, general relativity is not renormalizable, at least in perturbation theory. But, assuming that we know the fundamental symmetry of nature, namely diffeomorphism invariance, and that gravity can be described by a massless spin 2 particle, one can formulate an effective theory for quantum general relativity [3-5] valid up to the energy scale at which quantum gravitational effects become strong M /star . The effective field theory containing the metric g \u00b5\u03bd , which describes the graviton if linearized, a cosmological constant and the standard model of particle physics L SM (Higgs doublet H included) is given by This effective action is an expansion in space-time curvature. The Higgs boson has a nonzero vacuum expectation value, v = 246 GeV. The parameters M and \u03be are then determined by This effective field theory contains several energy scales. There is the reduced Planck scale M P of the order of 10 18 GeV (equivalently Newton's constant), the cosmological constant \u039b C of order of 10 -3 eV and the M /star which is traditionally identified with M P but this needs not to be the case if we do not trust this effective theory up to that energy scale. In the language of effective field theory, the coefficients c 1 / 2 and \u03be are called Wilson coefficients. As we know so little about quantum gravity, it is not clear how many of the Wilson coefficients of this effective action are fundamental parameters of nature, i.e. new coupling constants, or calculable using other parameters of the effective action. This distinction is important. For example, the Wilson coefficients of dimension four operators are expected to be very tiny if they are generated by some new physics at a scale \u039b NP . Indeed, in the limit where \u039b NP \u2192\u221e they must disappear. Hence they must be of the form exp ( -\u03bb/ \u039b NP ) where \u03bb is identified with the ultra violet cutoff of the theory. Here we need to be more careful. When one expands g \u00b5\u03bd around a constant background metric \u03b7 \u00b5\u03bd using g \u00b5\u03bd = \u03b7 \u00b5\u03bd + h \u00b5\u03bd /M P , one finds that the dimension four operator \u03beH \u2020 H R is actually a dimension 6 operator of the type \u03beH \u2020 Hh /square h/M 2 P where h is the graviton. The remaining dimension 4 operators of the type R 2 are actually dimension 8 operators h /square hh /square h/M 4 P . One thus expects that \u03be , c 1 and c 2 are of order unity. On the other hand they might be arbitrarily large if they are genuinely new fundamental parameters of nature. The Wilson coefficient of higher order terms in R , if induced by quantum gravity, are proportional to (1 /M P ) n ( \u03bb/M P ) m and thus expected to be small unless again they are fundamental parameters not calculable in terms of M P . One should stress that it is not clear what is the cutoff \u03bb for this effective theory, it might be M P , an energy scale corresponding to the cosmological constant scale or maybe even some other particle physics scale such as the weak scale or some other scale of grand unification. The smallness of the observed cosmological constant could be the sign that the cutoff should be of the order of the cosmological constant scale. However, the fact that we see no sign of new physics beyond usual general relativity at this energy scale could be interpreted as failure of the naturalness argument just like the discovery of a single Higgs boson without any sign of new physics so far, seems to indicate that naturalness is not a helpful argument in our quest for physics beyond the standard model. We shall now describe the current bounds on the values of the parameters of the quantum gravitational action given in Eq. (1) and discuss whether future experiments are likely to improve them or not. Stelle has shown [6] that the terms c 1 R 2 and c 2 R \u00b5\u03bd R \u00b5\u03bd lead to Yukawa-like corrections to Newton's potential of a point mass m : with and Sub-millimeter tests of Newton's law [7] using sophisticated pendulums are used to bound c 1 and c 2 . One finds that, in the absence of accidental fine cancellations between both Yukawa terms, they are constrained to be less than 10 61 [8] . Astrophysical observations lead to bounds on these terms [9]. Binary systems of pulsars are the most promising environment to probe gravity at high curvature. However, a back of the envelop estimate quickly reveals that astrophysical observations will not be able to compete with Earth-based tests of Newton's law. Let us approximate the Ricci scalar in the binary system of pulsars by GM/ ( r 3 c 2 ) where M is the mass of the pulsar and r is the distance to the center of the pulsar. Clearly, if the distance is larger than the radius of the pulsar, then the Ricci scalar vanishes, so we are doing a rather crude estimate. Let us be optimistic and assume we could probe gravity at the surface of the pulsar, we thus take r = 13 . 1 km and M=2 solar masses. One then requests that the R 2 term should become comparable to the leading order Einstein-Hilbert term 1 2 M 2 P R and finds that one could reach only bounds of the order of 10 78 on c 1 . Similar weak bounds are expected on c 2 . Such limits are obviously much weaker that those obtained on Earth. We now consider the bounds on the non-minimal coupling of the Higgs boson to the Ricci scalar which is the third dimension four operator of the effective action. The fact that the boson discovered at CERN behaves very much like what is expected of the Higgs boson of the standard model of particle physics, enables one to set a limit on the non-minimal coupling as for large \u03be the Higgs boson would decouple from the standard model. One finds | \u03be | > 2 . 6 \u00d7 10 15 is excluded at the 95% C.L. [10]. Future colliders will not be able to improve this bound much and only potentially by one order of magnitude at a hypothetical future linear collider. There is no bound to date on the coefficients of higher dimensional operators. Because they are suppressed by powers of the Planck scale, these terms are expected to be completely irrelevant unless their Wilson coefficients are unnaturally large. Any progress towards measuring the parameters of the effective action will clearly require to be very creative. While this is clearly a difficult task, it may be the only way to probe directly quantum gravity and to differentiate empirically between different frameworks to quantize gravity. We stress that although the parameters of the quantum gravitational effective action are expected to be small if they are generated via quantum gravitational corrections, they could be large if they are truly independent parameters and not calculable in terms of the Planck mass. Indeed, who could have guessed without experimental guidance that gravity would be that much weaker that the weak interactions? The reason for this phenomenon is that the Planck mass is so large. Why wouldn't the other coefficients of the effective action be large as well? It is thus important to continue the ongoing experimental and observational efforts to measure these parameters. One should also keep in mind that one of the major theoretical development of the last 15 years has been the realization that M /star , could be much below the traditional 2 . 4335 \u00d7 10 18 GeV if there are extra-dimensions with a large volume [11,12] or even in four space-time dimensions if there is a large hidden sector of particles [8]. Clearly the LHC has been leading the way by setting some of the tightest limits to date on the experimental value of the Planck scale which we now know is above a few TeVs, but more creative ideas could lead to stronger bounds on this parameter of the quantum gravitational effective action.", "pages": [1, 2, 3, 4, 5]}] |
2016MNRAS.457.2516M | https://arxiv.org/pdf/1601.05478.pdf | <document>
<section_header_level_1><location><page_1><loc_7><loc_81><loc_91><loc_85></location>Hydrodynamical simulations of the tidal stripping of binary stars by massive black holes</section_header_level_1>
<section_header_level_1><location><page_1><loc_7><loc_76><loc_87><loc_78></location>Deborah Mainetti 1 , 2 , 3 glyph[star] , Alessandro Lupi 3 , 4 , Sergio Campana 2 and Monica Colpi 1 , 3</section_header_level_1>
<unordered_list>
<list_item><location><page_1><loc_7><loc_75><loc_76><loc_76></location>1 Dipartimento di Fisica G. Occhialini, Università degli Studi di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy</list_item>
<list_item><location><page_1><loc_7><loc_74><loc_56><loc_75></location>2 INAF, Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807, Merate (LC), Italy</list_item>
<list_item><location><page_1><loc_7><loc_73><loc_51><loc_74></location>3 INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy</list_item>
<list_item><location><page_1><loc_7><loc_71><loc_83><loc_72></location>4 Institut d'Astrophysique de Paris, Sorbonne Universitès, UPMC Univ Paris 6 et CNRS, UMR 7095, 98 bis bd Arago, F-75014 Paris, France</list_item>
</unordered_list>
<text><location><page_1><loc_7><loc_67><loc_56><loc_68></location>Accepted 2016 January 20. Received 2016 January 19; in original form 2015 December 10</text>
<section_header_level_1><location><page_1><loc_29><loc_63><loc_37><loc_64></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_29><loc_41><loc_91><loc_63></location>In a galactic nucleus, a star on a low angular momentum orbit around the central massive black hole can be fully or partially disrupted by the black hole tidal field, lighting up the compact object via gas accretion. This phenomenon can repeat if the star, not fully disrupted, is on a closed orbit. Because of the multiplicity of stars in binary systems, also binary stars may experience in pairs such a fate, immediately after being tidally separated. The consumption of both the binary components by the black hole is expected to power a double-peaked flare. In this paper, we perform for the first time, with GADGET2, a suite of smoothed particle hydrodynamics simulations of binary stars around a galactic central black hole in the Newtonian regime. We show that accretion luminosity light curves from double tidal disruptions reveal a more prominent knee, rather than a double peak, when decreasing the impact parameter of the encounter and when elevating the difference between the mass of the star which leaves the system after binary separation and the mass of the companion. The detection of a knee can anticipate the onset of periodic accretion luminosity flares if one of the stars, only partially disrupted, remains bound to the black hole after binary separation. Thus knees could be precursors of periodic flares, which can then be predicted, followed up and better modelled. Analytical estimates in the black hole mass range 10 5 -10 8 M glyph[circledot] show that the knee signature is enhanced in the case of black holes of mass 10 6 -10 7 M glyph[circledot] .</text>
<text><location><page_1><loc_29><loc_38><loc_91><loc_40></location>Key words: hydrodynamics - methods: numerical - binaries: close - galaxies: kinematics and dynamics - galaxies: nuclei</text>
<section_header_level_1><location><page_1><loc_7><loc_32><loc_21><loc_33></location>1 INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_7><loc_12><loc_47><loc_31></location>Supermassive black holes (BHs) are ubiquitous in the centre of massive galaxies. For most of the time they are in a quiescent state, but sometimes they can accrete matter from the surroundings and power an active galactic nucleus (AGN; Ho 2008). Stars orbiting around the central BH of a galaxy interact with each other, increasing the probability for one of them to be scattered on a low angular momentum orbit (Alexander 2012). A tidal disruption event (TDE) could thus occur contributing to the BH flaring on timescales of months or years (e.g. Rees 1988; Phinney 1989). For solar mass stars, this occurs when the (non-spinning) BH mass is less than about 10 8 M glyph[circledot] . For heavier BHs, these stars cross the horizon and are fully swallowed before being tidally disrupted (Macleod, Ramirez-Ruiz & Guillochon 2012). As a consequence, TDEs contribute to the detection of otherwise quiescent BHs in inactive (or weakly active) galaxies in a mass interval somewhat complemen-</text>
<text><location><page_1><loc_51><loc_31><loc_91><loc_33></location>tary to that probed in surveys of bright AGNs and QSOs (Vestergaard & Osmer 2009).</text>
<text><location><page_1><loc_51><loc_26><loc_91><loc_31></location>The total tidal disruption of a single star of mass M ∗ and radius R ∗ moving on a parabolic orbit around the central BH of a galaxy of mass M BH occurs when its pericentre radius r p is less than about the so-called BH tidal radius</text>
<formula><location><page_1><loc_51><loc_22><loc_91><loc_25></location>r t = R ∗ ( M BH M ∗ ) 1 / 3 ∼ 10 2 R glyph[circledot] ( R ∗ 1R glyph[circledot] )( M BH 10 6 M glyph[circledot] 1M glyph[circledot] M ∗ ) 1 / 3 , (1)</formula>
<text><location><page_1><loc_51><loc_7><loc_91><loc_21></location>corresponding to the distance where the BH tidal force overcomes the star self-gravity at its surface (Hills 1975; Frank & Rees 1976). On the contrary, if r p glyph[greaterorsimilar] r t , the star undergoes less distortion and suffers only partial disruption. The value of the impact parameter β = r t /r p defines how deep the disruption is (Guillochon & Ramirez-Ruiz 2013, 2015a). Roughly, only about half of the produced stellar debris remains bound to the BH and accretes on to it, powering the emission of a characteristic flare (e.g. Rees 1988; Phinney 1989). In the regime of partial TDEs, the star, if on a bound orbit, could transfer a fraction of its mass to the BH every time it passes through the pericentre of its orbit, thus powering one flare</text>
<text><location><page_2><loc_7><loc_85><loc_47><loc_88></location>for every orbital period and maybe 'spoon-feeding' the quiescent luminosity of weakly active galaxies (MacLeod et al. 2013).</text>
<text><location><page_2><loc_7><loc_44><loc_47><loc_85></location>TDEs are quite rare events, with estimated rates of ∼ 10 -5 galaxy -1 yr -1 (e.g. Donley et al. 2002). Despite this and sparse observations, a few TDEs have been observed mainly in the opticalUV (Renzini et al. 1995; Gezari et al. 2006, 2008, 2009, 2012; Komossa et al. 2008; van Velzen et al. 2011; Wang et al. 2011, 2012; Cenko et al. 2012a; Gezari 2012; Arcavi et al. 2014; Chornock et al. 2014; Holoien et al. 2014; Vinko et al. 2015) and soft X-ray bands (Bade, Komossa & Dahlem 1996; Komossa & Bade 1999; Komossa & Greiner 1999; Grupe, Thomas & Leighly 1999; Greiner et al. 2000; Li, Ramesh & Kristen 2002; Halpern, Gezari & Komossa 2004; Komossa 2004, 2012, 2015; Komossa et al. 2004; Esquej et al. 2007, 2008; Cappelluti et al. 2009; Maksym, Ulmer & Eracleous 2010; Lin et al. 2011, 2015; Saxton et al. 2012, 2015; Maksym et al. 2013; Donato et al. 2014; Khabibullin & Sazonov 2014; Maksym, Lin & Irwin 2014), but also in the radio and hard X-ray bands (Bloom et al. 2011; Burrows et al. 2011; Levan et al. 2011; Zauderer et al. 2011; Cenko et al. 2012b; Hryniewicz & Walter 2016; Lei et al. 2016). Many theoretical studies have been carried out to understand the physics of TDEs and model their accretion luminosity light curves (hereafter just light curves or flares), considering stars approaching the BH on a variety of orbits, from parabolic to bound (Nolthenius & Katz 1982; Bicknell & Gingold 1983; Carter & Luminet 1985; Luminet & Marck 1985; Luminet & Carter 1986; Rees 1988; Evans & Kochanek 1989; Phinney 1989; Khokhlov, Novikov & Pethick 1993a, b; Laguna et al. 1993; Diener et al. 1995, 1997; Ivanov & Novikov 2001; Kobayashi et al. 2004; Rosswog, Ramirez-Ruiz & His 2008, 2009; Guillochon et al. 2009; Lodato, King & Pringle 2009; Ramirez-Ruiz & Rosswog 2009; Strubbe & Quataert 2009; Kasen & Ramirez-Ruiz 2010; Lodato & Rossi 2010; Amaro-Seoane, Miller & Kennedy 2012; MacLeod et al. 2012, 2013; Guillochon & Ramirez-Ruiz 2013, 2015a; Hayasaki, Stone & Loeb 2013).</text>
<text><location><page_2><loc_7><loc_31><loc_47><loc_43></location>So far, only single-star TDEs have been taken into account. However, most of the stars in the field are in binaries (Duquennoy & Mayor 1991b; Fischer & Marcy 1992); hence, it is worth also studying close encounters between binaries and galactic central BHs which can lead to the disruption of both the binary members. The topic was first discussed by Mandel & Levin (2015), suggesting that in a binary-BH encounter under certain conditions both binary components may undergo tidal disruption in sequence immediately after the tidal binary break-up. A double-peaked flare is expected to occur, signature of such a peculiar event.</text>
<text><location><page_2><loc_7><loc_14><loc_47><loc_30></location>In this paper, we present for the first time the results of a series of smoothed particle hydrodynamics (SPH) simulations performed using the GADGET2 code (Springel 2005; the code can be freely downloaded from http://wwwmpa.mpa-garching.mpg.de/gadget/) in the aim at studying the physics of double tidal disruptions and at characterizing the expected light curves. As a first exploratory study, we consider parabolic encounters of binaries with galactic central BHs in the Newtonian regime, in order to explore which are the most favourable conditions for the occurrence of double-peaked flares. In particular, we address the following questions. Are all simulated encounters leading to double-peaked light curves or are there cases of single-peaked light curves? How can we disentangle the different outcomes? How prominent are the double peaks?</text>
<text><location><page_2><loc_7><loc_7><loc_47><loc_13></location>The paper is organized as follows. In Section 2, we resume the conditions required for double TDEs and the associated space of binary parameters (Mandel & Levin 2015). In Section 3, we initialize low-resolution SPH simulations of binary-BH encounters with different r p values of the centre of mass (CM) of the binaries around</text>
<text><location><page_2><loc_51><loc_82><loc_91><loc_88></location>the BH. Not all encounters can lead to double TDEs, and in Section 4 we introduce a classification of the obtained outcomes. In Section 5, we show the results of a selected sample of high-resolution simulations and the light curves directly inferred from them. Section 6 sums up results and conclusions.</text>
<section_header_level_1><location><page_2><loc_51><loc_77><loc_85><loc_78></location>2 BASICS FOR DOUBLE TIDAL DISRUPTIONS</section_header_level_1>
<text><location><page_2><loc_51><loc_73><loc_91><loc_76></location>We are here interested in identifying the set conditions for the sequential tidal disruption of binary stars around galactic central BHs, following Mandel & Levin (2015).</text>
<text><location><page_2><loc_51><loc_70><loc_91><loc_72></location>Tidal break-up of a binary on a parabolic orbit around a BH occurs if the binary CM around the BH enters a sphere of radius</text>
<formula><location><page_2><loc_51><loc_65><loc_91><loc_69></location>r tb = a bin ( M BH M bin ) 1 / 3 ∼ 10 3 R glyph[circledot] ( a bin 10R glyph[circledot] )( M BH 10 6 M glyph[circledot] 1M glyph[circledot] M ∗ ) 1 / 3 , (2)</formula>
<text><location><page_2><loc_51><loc_57><loc_91><loc_64></location>where a bin and M bin are the binary semimajor axis and total mass (Miller et al. 2005; Sesana, Madau & Haardt 2009). We notice that binary break-up comes before single-star tidal disruptions, given that r tb glyph[greaterorsimilar] r t (see equations 1 and 2). Tidal break-up occurs when the specific angular momentum (in modulus) of the binary CM at pericentre becomes less than</text>
<formula><location><page_2><loc_51><loc_53><loc_91><loc_56></location>l CM ( r tb ) ∼ √ GM BH a bin ( M BH M bin ) 1 / 3 . (3)</formula>
<text><location><page_2><loc_51><loc_47><loc_91><loc_53></location>Orbits which allow tidal binary break-up are called loss cone orbits (Merritt 2013). A binary on a loss cone orbit is broken up after one pericentre passage, over a time-scale T ∼ 2 π √ r 3 /GM BH , corresponding to the orbital period of a binary on a circular orbit at the same distance from the BH.</text>
<text><location><page_2><loc_51><loc_39><loc_91><loc_46></location>Both stars of a binary can undergo a sequential tidal disruption immediately after the tidal binary break-up only if the specific angular momentum of the binary CM around the BH at the closest approach, defined as l CM ( r p ) ∼ √ GM BH r p , instantly changes from being greater than l CM ( r tb ) to becoming less than l CM ( r t ) , where</text>
<formula><location><page_2><loc_51><loc_35><loc_91><loc_38></location>l CM ( r t ) ∼ √ GM BH R ∗ ( M BH M ∗ ) 1 / 3 . (4)</formula>
<text><location><page_2><loc_51><loc_21><loc_91><loc_35></location>In this way, the binary enters intact the region of single-star TDEs. This occurs if the binary experiences a large enough change ∆ L , at least of the order of l CM ( r tb ) , in the specific circular angular momentum l circ ( r ) ∼ √ GM BH r , over a time-scale T . Interactions with surrounding stars and/or massive perturbers can promote such a change (Perets, Hopman & Alexander 2007; Alexander 2012). We consider empty the portion of the loss cone, in phase space, corresponding to binaries that break up before entering the region of single-star TDEs, and full the portion of the loss cone corresponding to binaries which can enter intact the region of single-star TDEs (Merritt 2013).</text>
<text><location><page_2><loc_51><loc_9><loc_91><loc_21></location>In order to evaluate the distribution of the binary parameters associated with double disruptions, it is useful to determine r min , defined as the distance of the binary from the BH before experiencing the change ∆ L in l circ , separating the two regimes. Considering two-body relaxation over a time-scale t r = 0 . 065( GM BH /r ) 3 / 2 / [ G 2 M 2 ∗ n ( r ) ln Λ] (Spitzer & Hart 1971) as the main mechanism which drives changes in angular momentum, it is known that the change in specific circular angular momentum l circ ( r ) over a period T is of the order of</text>
<formula><location><page_2><loc_51><loc_7><loc_91><loc_8></location>∆ L ∼ ( T/t r ) 1 / 2 l circ (5)</formula>
<text><location><page_3><loc_7><loc_83><loc_47><loc_88></location>(Merritt 2013). Thus, the critical condition ∆ L ∼ l CM ( r tb ) enables us to infer r min . If the binary is orbiting inside a Bahcall-Wolf density profile n ( r ) = n 0 ( r/r 0 ) -7 / 4 (Bahcall & Wolf 1976), r min reads</text>
<formula><location><page_3><loc_7><loc_72><loc_47><loc_81></location>r min ∼ [ 0 . 065 2 πn 0 r 7 / 4 0 ln Λ ] 4 / 9 ( M BH M ∗ ) 28 / 27 a 4 / 9 bin ∼ 10 7 R glyph[circledot] ( 1 . 3 × 10 6 pc -3 n 0 ) 4 / 9 ( 0 . 3pc r 0 ) 7 / 9 ( 10 ln Λ ) 4 / 9 × ( M BH 10 6 M glyph[circledot] 1M glyph[circledot] M ∗ ) 28 / 27 ( a bin 10R glyph[circledot] ) 4 / 9 , (6)</formula>
<text><location><page_3><loc_7><loc_69><loc_47><loc_71></location>taking n 0 and r 0 as for the Milky Way (Merritt 2010). We note that r min is comparable to the radius of gravitational influence of a BH</text>
<formula><location><page_3><loc_7><loc_66><loc_47><loc_68></location>r h = GM BH σ 2 ∼ 5 × 10 7 R glyph[circledot] ( M BH 10 6 M glyph[circledot] ) ( 65km / s σ ) 2 (7)</formula>
<text><location><page_3><loc_7><loc_64><loc_25><loc_65></location>(Peebles 1972; Merritt 2000).</text>
<text><location><page_3><loc_7><loc_54><loc_47><loc_63></location>A binary carries internal degrees of freedom, and in particular the relative velocity of the two binary components, √ GM bin /a bin , is clearly smaller than the orbital velocity of the binary CM relative to the BH, √ GM BH /r . The velocity of the two stars relative to the centre of mass of the stellar binary gives then a small contribution to the specific angular momentum of each binary star relative to the BH at r tb that approximately is</text>
<formula><location><page_3><loc_7><loc_51><loc_47><loc_54></location>δl ∼ √ GM bin a bin ( M BH M bin ) 1 / 3 . (8)</formula>
<text><location><page_3><loc_7><loc_45><loc_47><loc_50></location>Sequential disruptions are expected to be favoured when δl is small. Indeed, the smaller δl is, the more each binary component has an orbit around the BH similar to the one of the binary CM, i.e. a similar pericentre passage. Thus, we require</text>
<formula><location><page_3><loc_7><loc_42><loc_47><loc_44></location>δl l CM ( r t ) ∼ √ a bin R ∗ ( M ∗ M BH ) 1 / 3 glyph[lessmuch] 1 , (9)</formula>
<text><location><page_3><loc_7><loc_33><loc_47><loc_41></location>where we approximated M bin ∼ M ∗ . For M ∗ = 1M glyph[circledot] , R ∗ = 1R glyph[circledot] , M BH = 10 6 M glyph[circledot] we need a bin glyph[lessmuch] 10 4 R glyph[circledot] . Hence, the second condition for double TDEs, which joins the condition on ∆ L , is the involvement of close binaries. Furthermore, very close binaries are required in order to avoid their evaporation due to interactions with field stars before tidal binary break-up (Merritt 2013).</text>
<text><location><page_3><loc_7><loc_28><loc_47><loc_33></location>In the full loss cone regime, the parameter space of binaries that can undergo double TDEs can be inferred from the rate of binary entrance in the region of stellar TDEs per unit of r and a bin as found in Mandel & Levin (2015):</text>
<formula><location><page_3><loc_7><loc_25><loc_47><loc_27></location>d 3 N ( a bin , r ) drda bin dt ∼ ( l CM ( r t ) l CM ( r tb ) ) 2 4 πr 2 n ( r ) ξ ( a bin ) T , (10)</formula>
<text><location><page_3><loc_7><loc_15><loc_47><loc_24></location>where ( l CM ( r t ) /l CM ( r tb ) ) 2 is the probability for a binary to enter directly the single TDE region (Merritt 2013) and ξ ( a bin ) = [ ln ( a max /a min )] -1 a -1 bin is the distribution function for a bin given in Opik (1924), with a max and a min being the maximum and the minimum semimajor axes of stellar binaries in a generic galactic field.</text>
<text><location><page_3><loc_7><loc_10><loc_47><loc_15></location>Integration of equation 10 over r , between r min and + ∞ , enables us to evaluate the number of binaries that may undergo sequential tidal disruption of their components per unit of time and unit of a bin . The resulting integral scales as</text>
<formula><location><page_3><loc_7><loc_7><loc_47><loc_9></location>d 2 N ( a bin ) da bin dt ∝ [ ln ( a max a min )] -1 R ∗ a -19 / 9 bin . (11)</formula>
<text><location><page_3><loc_51><loc_84><loc_91><loc_88></location>From Kepler's law, we can connect a bin with the internal orbital period of the stellar binaries P bin to infer the number of events per unit of time and unit of P bin . The resulting rate is</text>
<formula><location><page_3><loc_51><loc_81><loc_91><loc_83></location>d 2 N ( P bin ) dP bin dt ∝ [ ln ( a max a min )] -1 R ∗ P -47 / 27 bin . (12)</formula>
<text><location><page_3><loc_51><loc_78><loc_91><loc_80></location>We use this scaling to extract the initial conditions of our SPH simulations.</text>
<text><location><page_3><loc_51><loc_63><loc_91><loc_77></location>Thus, in the case of solar mass stars (i.e. R ∗ = 1R glyph[circledot] ), the contribution of double TDEs to all TDEs could be approximately estimated by integrating equation 11 over a bin between 1 and 10 4 R glyph[circledot] and dividing it by the corresponding integral obtained after integration over r of equation 10, with R ∗ in place of a bin (also in equation 6) 1 . This ratio scales as (19 / 9) [ ln ( a max /a min )] -1 , which gives a maximum of ∼ 20 per cent assuming a max = 10 4 R glyph[circledot] and a min = 1R glyph[circledot] and considering that the multiplicity of stars is single:double ∼ 50:50 for 100 solar-type stars (Duquennoy & Mayor 1991b), disregarding uncertainties in the number of very close binaries.</text>
<text><location><page_3><loc_51><loc_55><loc_91><loc_63></location>The definition of the parameter space of binaries that may be double tidally disrupted is fundamental to guide us to sensibly define the initial conditions of a small number of representative lowresolution simulations aimed at checking different outcomes from different initial parameters, and particularly from different pericentre radii of the binary CM.</text>
<section_header_level_1><location><page_3><loc_51><loc_49><loc_84><loc_52></location>3 GENERAL PARAMETER DEFINITION FOR LOW-RESOLUTION SPH SIMULATIONS</section_header_level_1>
<text><location><page_3><loc_51><loc_24><loc_91><loc_48></location>The simulations in this paper are performed using the TreeSPH code GADGET2 (Springel 2005). In SPH codes, a star is represented by a set of gas particles. Each particle is characterized by a spatial distance, the smoothing length, over which its properties are 'smoothed' by its kernel function, i.e. evaluated by summing the properties of particles in the range of the kernel according to the kernel itself (Price 2005). In particular, in GADGET2 the smoothing length of each particle is defined so that its kernel volume contains a constant mass, and is allowed to vary with time, thus adapting to the local conditions. The kernel adopted here is the one used most commonly and is based on cubic splines (Monaghan & Lattanzio 1985). On the other hand, gravitational interactions between particles are computed through a hierarchical oct-tree algorithm, which significantly reduces the number of pair interactions needed to be computed. The definition of a gravitational softening length glyph[epsilon1] ∼ 0 . 1 R ∗ / ( N part ) 1 / 3 , where N part is the total number of particles, prevents particle overlapping. GADGET2 enables us to follow the temporal evolution of single particle properties and to infer from them TDE light curves (see Section 5.2).</text>
<text><location><page_3><loc_51><loc_12><loc_91><loc_24></location>We run 14 low-resolution simulations of parabolic encounters between equal-mass binaries and BHs (LE runs) to test the nature of the outcomes for different initial conditions, varying binary parameters, M BH and r p . The stellar binaries are first evolved in isolation for several dynamical times to ensure their stability. The BH force is implemented in the code analytically, as a Newtonian potential, and particles which fall below the innermost stable circular orbit radius R ISCO are excised from simulations. We consider equal solar mass stars modelled as polytropes of index 5/3 and we</text>
<text><location><page_3><loc_51><loc_7><loc_91><loc_9></location>1 Note that substituting R ∗ to a bin in equation 10, d 3 N ( R ∗ , r ) /drdR ∗ dt ∼ 4 πr 2 n ( r ) /T .</text>
<section_header_level_1><location><page_4><loc_7><loc_89><loc_8><loc_91></location>4</section_header_level_1>
<table>
<location><page_4><loc_8><loc_66><loc_46><loc_78></location>
<caption>Table 1. Outcomes of our low-resolution SPH simulations of parabolic binary-BH encounters ( M ∗ = 1M glyph[circledot] , R ∗ = 1R glyph[circledot] ) as a function of a bin and r p . Here M BH = 10 6 M glyph[circledot] , r t = 100 . 0R glyph[circledot] , r tb ( a bin = 4 . 9R glyph[circledot] ) = 390 . 0R glyph[circledot] , r tb ( a bin = 9 . 8R glyph[circledot] ) = 780 . 0R glyph[circledot] . TD-TDE stands for total double TDE, ATD-TDE for almost total double TDE (i.e. more than ∼ 70% of stellar mass lost), PD-TDE for partial double TDE, MG for merger, BBK for binary break-up without stellar disruptions.</caption>
</table>
<text><location><page_4><loc_7><loc_46><loc_47><loc_64></location>sample each of them with 10 3 particles. Some correspondent highresolution simulations are presented in Section 5.2. The initial binary internal orbital periods P bin and semimajor axes a bin are extracted according to the distributions described in Section 2. Based on the work of Duquennoy & Mayor (1991a), we consider binaries with 0 . 1 d ( a bin ∼ 1R glyph[circledot] ) < P bin < 10 d ( a bin ∼ 10R glyph[circledot] ) to be circular, binaries with 10 d ≤ P bin ≤ 1000 d ( a bin ∼ 500R glyph[circledot] ) to have internal eccentricities distributed according to a Gaussian of mean 0.3 and standard deviation 0.15 and binaries with 1000 d < P bin < 1000 yr ( a bin ∼ 10 4 R glyph[circledot] ) to have internal eccentricities which follow a thermal distribution p ( e bin ) ∼ 2 e bin . In order to avoid immediate collisions between the binary components, the initial pericentre radius of the internal binaries (i.e. a bin (1 -e bin ) ) is set greater than twice the sum of the stellar radii, which are</text>
<formula><location><page_4><loc_7><loc_42><loc_47><loc_45></location>R ∗ = ( M ∗ M glyph[circledot] ) k R glyph[circledot] , (13)</formula>
<text><location><page_4><loc_7><loc_26><loc_47><loc_41></location>with k = 0 . 8 for M ∗ < 1M glyph[circledot] and k = 0 . 6 for M ∗ > 1M glyph[circledot] , according to Kippenhahn & Weigert (1994), R ∗ = 1R glyph[circledot] for M ∗ = 1M glyph[circledot] . Binaries are then placed on parabolic orbits around the BH at an initial distance 10 times greater than the tidal binary break-up radius r tb , thus preventing initial tidal distortions from the BH. BHs of masses 10 5 and 10 6 M glyph[circledot] are considered. The nominal pericentre distances r p are generated between 1 and 300R glyph[circledot] (Mandel & Levin 2015). Stars are placed on Keplerian orbits, and their positions and velocities relative to their binary centre of mass and to the BH are assigned accordingly. The initial internal binary plane is set, arbitrarily, perpendicular to the orbital plane around the BH. The results of these simulations are shown in Section 4.</text>
<section_header_level_1><location><page_4><loc_7><loc_20><loc_38><loc_23></location>4 OUTCOMES OF LOW-RESOLUTION SPH SIMULATIONS</section_header_level_1>
<text><location><page_4><loc_7><loc_14><loc_47><loc_19></location>Tables 1 and 2 summarize the results of our low-resolution simulations as a function of M BH , a bin and r p . Several outcomes from binary-BH encounters are possible, including the results of our simulations:</text>
<unordered_list>
<list_item><location><page_4><loc_9><loc_11><loc_29><loc_12></location>(i) PD-TDE: partial double TDE,</list_item>
</unordered_list>
<text><location><page_4><loc_7><loc_8><loc_47><loc_11></location>(ii) ATD-TDE: almost total double TDE, i.e. more than ∼ 70% of stellar mass is lost,</text>
<text><location><page_4><loc_9><loc_7><loc_40><loc_8></location>(iii) P&T-TDE: single partial plus single total TDE,</text>
<table>
<location><page_4><loc_52><loc_69><loc_90><loc_82></location>
<caption>Table 2. Same as Table 1, with M BH = 10 5 M glyph[circledot] , r t = 50 . 0R glyph[circledot] , r tb ( a bin = 4 . 9R glyph[circledot] ) = 180 . 0R glyph[circledot] , r tb ( a bin = 9 . 8R glyph[circledot] ) = 360 . 0R glyph[circledot] . PD-TDE stands for partial double TDE, MG for merger, BBK for binary break-up without stellar disruptions, UN for undisturbed binary.</caption>
</table>
<text><location><page_4><loc_53><loc_66><loc_73><loc_67></location>(iv) TD-TDE: total double TDE,</text>
<unordered_list>
<list_item><location><page_4><loc_53><loc_65><loc_79><loc_66></location>(v) MG: merger of the binary components,</list_item>
<list_item><location><page_4><loc_53><loc_64><loc_91><loc_65></location>(vi) BBK: tidal binary break-up without stellar tidal disruptions,</list_item>
</unordered_list>
<text><location><page_4><loc_53><loc_62><loc_71><loc_63></location>(vii) UN: undisturbed binary.</text>
<text><location><page_4><loc_51><loc_42><loc_91><loc_61></location>The intensity of the disruptions, i.e. the morphology of the resulting objects, is estimated from our simulation results based on the tidal deformation, the extent of stellar mass loss and possible orbital changes of the binary stars with pericentre passage. After closest approach, the orbital evolution of the binary stars around the BH is computed using an N -body Hermite code (e.g. Hut & Makino 1995; the code can be freely downloaded from https://www.ids.ias.edu/ ∼ piet/act/comp/algorithms/starter/), knowing the current position and velocity of the centre of mass of each binary component from SPH simulations (see Section 5.2 for the recipe used to infer the position and velocity of the centres of mass). The use of the Hermite code enables us to overcome the high computational time required by SPH simulations to track the dynamics of stars when the bulk of the hydrodynamical processes have subsided.</text>
<text><location><page_4><loc_51><loc_7><loc_91><loc_42></location>Appendix A contains an inventory of representative orbits according to the classification highlighted above. Tables A1 and A2, respectively, refer to the simulations described in Tables 1 and 2. There, we show the orbital evolution of the binary components around the BH, for each simulation, in the ( x, y ) and ( y, z ) planes, starting from (0,0), (0,0). Units are in R glyph[circledot] . Blue curves represent the initial parabolic orbits of the binary CM around the BH, each inferred from the position of the BH and the pericentre radius r p . Red curves trace the orbital evolution of the binary components as inferred from SPH simulations, while green curves trace the orbital evolution of the stars as computed using the Hermite code. Black dots indicate the position of the BH. Mergers (MGs; LE5, LE7) are found when the two binary components progressively reduce their relative separation starting from just before the pericentre passage around the BH, without being tidally separated. The MG product, which is represented by stars at a fixed minimum distance in simulations performed using the Hermite code, follows an orbit which overlaps the initial parabolic one of the binary CM. In the UN case (LE8, LE14), the binary keeps its internal and external orbits unchanged, even after pericentre passage. D-TDEs (LE1, LE2, LE3, LE4, LE6, LE9, LE10, LE12) are preceded by tidal binary separation, which can also occur without stellar disruptions (BBK; LE11, LE13). Binary break-up leads one star to get bound to the BH and the other to remain unbound. In the case of BBKs or partial disruptions, the latter may leave the system at a high velocity, becoming a hypervelocity star (Hills 1988; Antonini, Lombardi & Merritt 2011).</text>
<table>
<location><page_5><loc_13><loc_78><loc_41><loc_84></location>
<caption>Table 3. Same as Table 1 for our high resolution simulations involving equal-mass binaries.</caption>
</table>
<figure>
<location><page_5><loc_9><loc_66><loc_25><loc_75></location>
<caption>Fig 2 shows representative snapshots of the SPH particle distribution, projected in the ( x, y ) plane and in fractions of pericentre time, depicting the dynamics of simulations HEp50 (left column), HEp100 (central column) and HEp143 (right column). Panels are in R glyph[circledot] . In each simulation, black particles originally shape the star</caption>
</figure>
<figure>
<location><page_5><loc_29><loc_66><loc_46><loc_75></location>
</figure>
<figure>
<location><page_5><loc_9><loc_56><loc_25><loc_65></location>
</figure>
<figure>
<location><page_5><loc_29><loc_56><loc_45><loc_65></location>
<caption>Figure 1. Orbital evolution of the binary stars, starting from (0,0) in the ( x, y ) plane, as inferred from the corresponding low- (red curves) and high-resolution (green curves) SPH simulations for LE2/HEp100 and LE3/HEp143 (upper panels) and LU2/HUp70a and LU3/HUp70b (bottom panels). We do not consider simulations LE1/HEp50 and LU1/HUp42 given that both the binary stars are totally disrupted when approaching the BH. Black dots indicate the position of the BH. Units are in R glyph[circledot] .</caption>
</figure>
<section_header_level_1><location><page_5><loc_7><loc_42><loc_38><loc_43></location>5 HIGH-RESOLUTION SPH SIMULATIONS</section_header_level_1>
<section_header_level_1><location><page_5><loc_7><loc_40><loc_34><loc_41></location>5.1 A glimpse to simulated double TDEs</section_header_level_1>
<text><location><page_5><loc_7><loc_30><loc_47><loc_39></location>The low-resolution simulations described in Sections 3 and 4 serve as guide for the selection of three higher resolution SPH simulations, with an increased number of particles per star equal to 10 5 . Anumber of particles per star of 10 6 would require too much computational time. Indeed, the computational cost in GADGET2 scales as N part log( N part ) , which is a factor of 12 higher in the case of N part = 10 6 with respect to N part = 10 5 .</text>
<text><location><page_5><loc_7><loc_14><loc_47><loc_30></location>Our goal is to infer directly from simulations the light curves associated with double TDEs of different intensities. For this reason, we set the initial conditions for an almost total, a partial and a total double disruption event, following simulations LE2 and LE3 for the not fully disruptive events and simulation LE1 in order to obtain a total double disruption. Table 3 summarizes the outcomes which come out from these three high-resolution simulations (HE runs) as a function of a bin and r p . These results are the same as expected from the corresponding low-resolution simulations (see Table 1). Furthermore, Fig. 1 (upper panels) points out that the orbits of the binary stars follow the same evolution in corresponding low- (red curves) and high-resolution (green curves) SPH simulations after pericentre passage, assuring numerical convergence.</text>
<text><location><page_5><loc_51><loc_70><loc_91><loc_88></location>which will get bound to the BH after binary separation, whereas red particles initially belong to the one which will unbind. The remnant of the binary components after disruption is clearly visible in the almost total (HEp100) and partial double (HEp143) TDE cases. Forward in time, the distribution of the particles which leave the stars once tidally disrupted visibly spreads, and particles originally associated with the two different stars tend to mix, preventing their by-eye distinction. For this reason, snapshots of the SPH particle distribution are introduced in place of snapshots of the SPH particle density, which are shown for the first time in Fig. 3 (in log scale), projected in the ( x, y ) plane, only at 0 . 0004yr ( ∼ 0 . 15d ) after pericentre passage for the simulated total double (HEp50) and partial double (HEp143) TDE. Again, the remnant of the binary components is clearly visible in the partially disruptive encounter.</text>
<text><location><page_5><loc_51><loc_63><loc_91><loc_69></location>The selection of the stellar debris associated with a specific star is possible thanks to a detailed analysis of the snapshots. This enables us to extract the light curves associated with each singlestar disruption and then to infer the composite light curves associated with double disruptions. We discuss this in Section 5.2.</text>
<section_header_level_1><location><page_5><loc_51><loc_60><loc_91><loc_61></location>5.2 Double TDE light curves: the case of equal-mass binaries</section_header_level_1>
<text><location><page_5><loc_51><loc_47><loc_91><loc_59></location>The basic (simplifying) assumption when inferring the light curves associated with TDEs is that the accretion rate on to the BH has close correspondence to the rate of stellar debris which returns to pericentre after disruption. Indeed, if the viscous time (Li et al. 2002) driving the fallback of stellar debris on to the BH is negligible compared to the returning time at pericentre of the most bound material since the time of stellar disruption (which is generally the case in our simulations), then the rate of debris returning at pericentre</text>
<formula><location><page_5><loc_51><loc_44><loc_91><loc_47></location>˙ M ( t ) = ( 2 πGM BH ) 2 / 3 3 dM dE t -5 / 3 , (14)</formula>
<text><location><page_5><loc_51><loc_40><loc_91><loc_43></location>coincides to first approximation to the rate of accretion on to the BH. Inferring ˙ M ( t ) is thus equivalent to computing the luminosity L ( t ) associated with a TDE</text>
<formula><location><page_5><loc_51><loc_38><loc_91><loc_39></location>L ( t ) = η ˙ M ( t ) c 2 , (15)</formula>
<text><location><page_5><loc_51><loc_36><loc_74><loc_37></location>assuming an appropriate efficiency η .</text>
<text><location><page_5><loc_51><loc_26><loc_91><loc_35></location>In equation 14, dM/dE is the distribution of the stellar debris per unit energy as a function of E , the specific binding energy relative to the BH. Generally, such a distribution is neither flat nor constant in time (e.g. Lodato et al. 2009; Guillochon & RamirezRuiz 2013), allowing ˙ M ( t ) to deviate from the classically assumed t -5 / 3 trend, inferred from equation 14 when taking a uniform distribution in E (e.g. Rees 1988; Phinney 1989).</text>
<text><location><page_5><loc_51><loc_15><loc_91><loc_26></location>Here we compute dM/dE as a function of time for each binary component directly from our simulations, following the recipe from Guillochon & Ramirez-Ruiz (2013). The position and velocity of the centre of mass of each star around the BH are computed through an iterative approach. The initial reference point is the particle with the highest local density. Particles within 2R glyph[circledot] from it (a bit more than R ∗ ) are considered to be still bound to the star and their total mass is denoted as M B . The specific binding energy of the i -th particle relative to the star is calculated as</text>
<formula><location><page_5><loc_51><loc_11><loc_91><loc_14></location>E ∗ i = 1 2 | v i -v peak | 2 -GM B | r i -r peak | , (16)</formula>
<text><location><page_5><loc_51><loc_7><loc_91><loc_11></location>where v i -v peak and r i -r peak are the velocity and position of the i -th particle relative to the reference particle. Velocity and position of the temporary centre of mass are determined through the standard</text>
<section_header_level_1><location><page_6><loc_7><loc_89><loc_22><loc_91></location>6 Mainetti et al.</section_header_level_1>
<figure>
<location><page_6><loc_11><loc_75><loc_32><loc_88></location>
</figure>
<text><location><page_6><loc_13><loc_68><loc_14><loc_68></location>y</text>
<text><location><page_6><loc_23><loc_75><loc_24><loc_75></location>x</text>
<figure>
<location><page_6><loc_13><loc_61><loc_30><loc_74></location>
</figure>
<figure>
<location><page_6><loc_9><loc_50><loc_34><loc_58></location>
</figure>
<figure>
<location><page_6><loc_37><loc_50><loc_62><loc_58></location>
</figure>
<figure>
<location><page_6><loc_64><loc_50><loc_89><loc_58></location>
<caption>Figure 2. Representative high-resolution snapshots of the SPH particle distribution, respectively, in simulations HEp50 (left column), HEp100 (central column) and HEp143 (right column), projected in the ( x, y ) plane. Positional units are in R glyph[circledot] and times are in fractions of pericentre time. Black particles originally belong to the star which will get bound to the BH after tidal binary break-up and red particles depict its companion. The BH is at position ( x, y )=(-3779.62,875.17) (simulation HEp50), ( x, y )=(-3679.62,-1229.57) (simulation HEp100), ( x, y )=(-3594.48,-1459.84) (simulation HEp143). The survived binary components are clearly visible in the almost total (HEp100) and partial (HEp143) TDE cases, whereas stars are fully disrupted after pericentre passage in the total TDE case (HEp50).</caption>
</figure>
<text><location><page_6><loc_7><loc_27><loc_47><loc_36></location>formulae by considering only particles with E ∗ i < 0 . Equation 16 is then re-evaluated with the new velocity and position of the centre of mass in place of v peak and r peak . This process is re-iterated until the velocity of the centre of mass converges to a constant value, to less than 10 -5 R glyph[circledot] yr -1 . Particles with E ∗ i > 0 , i.e. unbound from the star, are then selected in the aim at evaluating their specific binding energy relative to the BH</text>
<formula><location><page_6><loc_7><loc_24><loc_47><loc_27></location>E i = 1 2 | v i | 2 -GM BH | r i -r BH | , (17)</formula>
<text><location><page_6><loc_7><loc_14><loc_47><loc_24></location>where v i and r i -r BH are the velocity and position of the i -th particle relative to the BH. Particles with E i > 0 are unbound from the BH, whereas particles with E i < 0 form the stream of debris bound to the BH. Data are then binned in E , i.e. the specific binding energies E i < 0 are grouped in bins and the correspondent particles fill this histogram. dM/dE as a function of E (i.e. time) is obtained dividing the total mass of particles in each bin by the bin amplitude.</text>
<text><location><page_6><loc_7><loc_8><loc_47><loc_13></location>The time t in equation 14 is the time since disruption, which is coincident with the first pericentre passage for our purposes. Thus, only material with orbital periods P orb = 2 πGM BH / (2 E ) 3 / 2 around the BH less than t contributes to the accretion till that time.</text>
<text><location><page_6><loc_10><loc_7><loc_47><loc_8></location>To build the composite light curves, we need to compute the</text>
<text><location><page_6><loc_51><loc_29><loc_91><loc_36></location>light curve for each star by interpolating the data coming from different snapshots, and then we sum the results of interpolations, point to point. Green and blue curves in Fig. 4 are associated with the disruption of the single binary components, while red curves represent the point-wise sum of the green and blue curves. Panels on the top-right corners show logarithmic plots.</text>
<text><location><page_6><loc_51><loc_19><loc_91><loc_28></location>The light curves associated with TDEs are described by characteristic parameters, which can be assessed directly from the light curves and also analytically, in order to check the reliability of the recipe that we have followed. The first characteristic parameter is t most , the returning time at pericentre of the most bound stellar debris since disruption. For a star on a parabolic orbit around a BH, it can be evaluated as</text>
<formula><location><page_6><loc_51><loc_16><loc_91><loc_18></location>˜ t most = π √ 2 GM BH E 3 / 2 ∼ π √ 2 1 √ G M 1 / 2 BH M ∗ R 3 / 2 ∗ , (18)</formula>
<text><location><page_6><loc_51><loc_7><loc_91><loc_15></location>where E is the specific energy spread caused by the disruption ∆ E ∼ GM BH R ∗ /r 2 t , given that the orbital energy associated with a parabolic orbit is zero. In our simulations, the binary CM is set on a parabolic orbit around the BH but the binary components are a bit out of it. Moreover, after the tidal binary separation, they follow new orbits: an ellipse for the bound star and hyperbola for the</text>
<figure>
<location><page_6><loc_39><loc_75><loc_59><loc_88></location>
</figure>
<text><location><page_6><loc_51><loc_75><loc_52><loc_75></location>x</text>
<figure>
<location><page_6><loc_42><loc_61><loc_57><loc_74></location>
</figure>
<figure>
<location><page_6><loc_70><loc_61><loc_84><loc_74></location>
</figure>
<figure>
<location><page_6><loc_67><loc_75><loc_87><loc_88></location>
</figure>
<text><location><page_6><loc_79><loc_75><loc_79><loc_75></location>x</text>
<figure>
<location><page_7><loc_8><loc_70><loc_43><loc_88></location>
</figure>
<figure>
<location><page_7><loc_56><loc_68><loc_89><loc_86></location>
</figure>
<figure>
<location><page_7><loc_8><loc_51><loc_43><loc_69></location>
<caption>Figure 3. Snapshots of the SPH particle density (in log scale) for the simulated total double (HEp50; upper panel) and partial double (HEp143; bottom panel) TDE at t = 0 . 0004yr ( ∼ 0 . 15d ) after pericentre passage, projected in the ( x, y ) plane. The remnant binary components are clearly visible in the partial double disruption case.</caption>
</figure>
<text><location><page_7><loc_7><loc_35><loc_47><loc_39></location>unbound star. Thus, the returning time associated with each binary component is not simply ˜ t most , as it requires knowledge of the new orbits of the separated stars. Hereafter, we denote with subscript 1 (2) the bound (unbound) binary component.</text>
<text><location><page_7><loc_10><loc_33><loc_45><loc_34></location>For the bound star, the returning time can be evaluated as</text>
<formula><location><page_7><loc_7><loc_26><loc_47><loc_32></location>t most 1 = π √ 2 GM BH E 3 / 2 ∼ ˜ t most ( M ∗ M BH ) 1 / 2 × 1 ( β 1 (1 -e 1 )) 3 / 2 ( 1 2 + ( M ∗ /M BH ) 1 / 3 β 1 (1 -e 1 ) ) -3 / 2 , (19)</formula>
<text><location><page_7><loc_42><loc_21><loc_42><loc_22></location>glyph[negationslash]</text>
<text><location><page_7><loc_7><loc_7><loc_47><loc_25></location>where e 1 is the eccentricity of its new orbit (computed through the Hermite code), β 1 the impact parameter of its centre of mass and E ∼ E orb +∆ E , with E orb ∼ GM BH β 1 (1 -e 1 ) / (2 r t ) = 0 . We infer this time also from our simulations, considering as 'mostly bound' the first returned particles after disruption associated with the bound star. As minimum of significance we assume 10 particles out of the set of particles, associated with the bound star, bound to the BH. If the impact parameters of both the binary components, β 1 and β 2 , are close to unity, the two estimates of t most 1 are in good agreement. In this case, we infer the returning time for the unbound star, t most 2 , directly from our simulations. On the contrary, the more β 1 and β 2 depart from unity, the worse the agreement is. In this case, we introduce a correction factor between the two estimates of t most 1 , and we use it to correct t most 2 as inferred from</text>
<figure>
<location><page_7><loc_56><loc_46><loc_89><loc_64></location>
</figure>
<figure>
<location><page_7><loc_56><loc_24><loc_90><loc_42></location>
<caption>Figure 4. Light curves [ ˙ M versus time; see equation 15 in Section 5.2 to convert accretion rates into luminosities] inferred from our high-resolution simulations of parabolic equal-mass binary-BH encounters, depicting a fully disruptive encounter (simulation HEp50), an almost total double disruption (simulation HEp100) and a partial double TDE (simulation HEp143). Green and blue curves are associated with the disruption of the binary components; red curves reproduce the point-wise sum of the green and blue curves. On the top-right corners, we show the same plots in logarithmic scale. A knee in the red curve is somehow visible in simulation HEp143, especially in the logarithmic plot, and it decays more steeply than the classically assumed power law of index -5/3.</caption>
</figure>
<table>
<location><page_8><loc_10><loc_51><loc_44><loc_71></location>
<caption>Table 4. Characteristic parameters of the light curves inferred from our high-resolution simulations of equal-mass binary-BH encounters, as analytically estimated (see Section 5.2). Simulations HEp50, HEp100 and HEp143, respectively, correspond to the ones in Fig. 4. t most is the returning time at pericentre of the most bound stellar debris since disruption and t peak the rise time from stellar disruption to accretion rate peak, ˙ M peak . Tilded values are evaluated setting the binary components on parabolic orbits corresponding to the initial one of the binary CM; untilded values consider the effective orbits of the binary stars. The 1 (2) subscript denote the BH bound (unbound) star. ∆ t peak and ∆ ˙ M peak are the differences in rise times and accretion rate peaks between the two 'humps' expected in the composite light curves associated with double TDEs, actually visible only in the partially disruptive encounter (simulation HEp143; see Fig. 4).</caption>
</table>
<text><location><page_8><loc_7><loc_47><loc_47><loc_49></location>simulations. ˜ t most , t most 1 and t most 2 are reported in Table 4 for our three high-resolution simulations.</text>
<text><location><page_8><loc_7><loc_36><loc_47><loc_46></location>Corrections for the new orbits of the separated stars also involve the second characteristic parameter of TDE light curves, t peak , that is the rise time between the time of stellar disruption and the time at which the accretion rate peaks. If the two binary components were on parabolic orbits corresponding to the initial one of their binary CM, the rise time for each star would be denoted as ˜ t peak (1,2) and could be evaluated following Guillochon &Ramirez-Ruiz (2013, 2015a). Corrected values come out to be</text>
<formula><location><page_8><loc_7><loc_33><loc_47><loc_35></location>t peak ∼ ˜ t peak t most ˜ t most , (20)</formula>
<text><location><page_8><loc_7><loc_29><loc_47><loc_32></location>assuming that t most and t peak change proportionally. Table 4 collects ˜ t peak (1,2) and t peak (1,2) for our three high-resolution simulations.</text>
<text><location><page_8><loc_7><loc_23><loc_47><loc_28></location>The last characteristic parameter of TDE light curves is the peak of accretion rate, ˙ M peak . According to MacLeod et al. (2013), this parameter is linked to the mass of the debris which binds to the BH M bound BH and to the rise time t peak through the relation</text>
<formula><location><page_8><loc_7><loc_20><loc_47><loc_23></location>˙ M peak ∼ 2 3 M bound BH t peak . (21)</formula>
<text><location><page_8><loc_7><loc_7><loc_47><loc_19></location>Values for stars on parabolic orbits, ˜ ˙ M peak (1,2), can be evaluated considering M bound BH to be half the mass lost from each star (e.g. Rees 1988) and t peak ≡ ˜ t peak . Corrected values require M bound BH as inferred from our simulations and t peak from equation 20. Given that standard assumptions work for β ∼ 1 , we estimate ˜ ˙ M peak (1,2) and ˙ M peak (1,2) as just mentioned for simulation HEp100 (see Section 5.1), and then we convert them in the corresponding values for the other two simulations, based on the dependence of ˙ M peak from the impact parameter β reported in Guillochon & Ramirez-Ruiz</text>
<table>
<location><page_8><loc_61><loc_75><loc_82><loc_84></location>
<caption>Table 5. Same as Table 1 for our high-resolution simulations involving unequal-mass binaries.</caption>
</table>
<text><location><page_8><loc_51><loc_60><loc_91><loc_73></location>(2013, 2015a). Indeed, the only difference among our simulations is the value of the pericentre radius, i.e. β 2 . However, recall that the relation between ˙ M peak and β works for parabolic orbits. Consequently, some differences between the values assessed from the inferred light curves (Fig. 4) and our analytical estimates are to be expected. Values of ˜ ˙ M peak (1,2) and ˙ M peak for our three simulations are reported in Table 4. Good agreement is found between light-curve parameters inferred from Fig. 4 and analytical evaluations, motivating the recipe we have followed in the aim to derive TDE light curves.</text>
<text><location><page_8><loc_51><loc_33><loc_91><loc_60></location>As previously said in this section, the composite light curves associated with double TDEs are obtained by summing the light curves associated with the disruption of the single binary components. Given that the binary components have different returning and rising times, one should expect to observe a double peak in their composite light curve. In Table 4, we collect, where possible, the values of ∆ t peak and ∆ ˙ M peak as inferred from Fig. 4, which are the differences in rise times and accretion rate peaks between the two 'humps' in the composite light curves. From Table 4 and Fig. 4, we see that only in simulation HEp143, which corresponds to a grazing encounter, the composite light curve shows not exactly a double peak, as predicted, but anyway a knee. In this case, the single-star light curves are distinguishable enough to be both glimpsed in the composite light curve. As shown in hydrodynamical simulations of single TDEs of Guillochon & RamirezRuiz (2013), grazing encounters give rise to steep light curves (i.e. steeper than -5/3) immediately after the peak and, in the context of double disruptions, this favours the visibility of the knee in the composite light curves. Therefore, in the case of double TDEs of equal-mass binaries, only grazing encounters can produce a knee in the composite light curve.</text>
<section_header_level_1><location><page_8><loc_51><loc_27><loc_87><loc_30></location>5.3 Double TDE light curves: the case of unequal-mass binaries</section_header_level_1>
<text><location><page_8><loc_51><loc_14><loc_91><loc_26></location>What happens in the case of deeper encounters if the binary components have unequal masses? Using the same procedure described in Section 5.2, we carry on and analyse three high-resolution SPH simulations of unequal-mass binaries on parabolic orbits around a BH ( M BH = 10 6 M glyph[circledot] ) (HU runs). Table 5 collects the outcomes of these simulations as a function of a bin and r p . We also perform the correspondent low-resolution SPH simulations (LU runs), respectively, denoted as LU1, LU2 and LU3, finding out the same outcomes and the same orbital evolution of the binary components (Fig. 1, bottom panels).</text>
<text><location><page_8><loc_54><loc_12><loc_91><loc_13></location>In particular, simulations LU1/HUp42 consider M 1 =</text>
<figure>
<location><page_9><loc_8><loc_51><loc_43><loc_88></location>
<caption>Figure 5. Zoom in the SPH particle density (in log scale) for simulations HUp70a (upper panel) and HUp70b (bottom panel) in Section 5.3, projected in the ( x, y ) plane, at t = 0 . 0034yr ( ∼ 1 . 2d ) after pericentre passage. The remnant less massive star is clearly visible in both the simulations.</caption>
</figure>
<text><location><page_9><loc_7><loc_18><loc_47><loc_41></location>0 . 4M glyph[circledot] , r t 1 ∼ 65 . 2R glyph[circledot] , M 2 = 0 . 27M glyph[circledot] , r t 2 = 54 . 3R glyph[circledot] , simulations LU2/HUp70a: M 1 = 0 . 5M glyph[circledot] , r t 1 ∼ 72 . 4R glyph[circledot] , M 2 = 1M glyph[circledot] , r t 2 = 100 . 0R glyph[circledot] , simulations LU3/HUp70b: M 1 = 1 . M glyph[circledot] , r t 1 ∼ 100 . R glyph[circledot] , M 2 = 0 . 5M glyph[circledot] , r t 2 = 72 . 4R glyph[circledot] . The initial conditions of simulations LU1/HUp42 are those considered in Mandel & Levin (2015). With simulations HUp70a and HUp70b, we explore the dependence of the visibility of a double peak on the mass difference between the binary components and on the mass of the captured star, whether it is the less or the more massive of the two. Indeed, simulations HUp70a and HUp70b only differ in that they are out of phase by 180 · . In the high-resolution regime, stars denoted as 1, which remain bound to the BH after binary separation, are modelled respectively with 4 × 10 4 , 10 5 and 2 × 10 5 particles and stars 2, which unbind from the BH, with 2 . 7 × 10 4 , 2 × 10 5 and 10 5 particles. Fig. 5 shows a zoom in the SPH particle density (in log scale), projected in the ( x, y ) plane, at t = 0 . 0034yr ( ∼ 1 . 2d ) after pericentre passage for simulations HUp70a and HUp70b. The remnant less massive star is clearly visible in both the simulations.</text>
<text><location><page_9><loc_7><loc_7><loc_47><loc_17></location>Table 6 collects the characteristic parameters of the light curves inferred from simulations HUp42, HUp70a and HUp70b, respectively, as analytically estimated following Section 5.2. Fig. 6 shows single-star and composite light curves inferred from simulations HUp42, HUp70a and HUp70b following the recipe described in Section 5.2. Not exactly a double peak, but a knee in the composite light curve is observed when the mass difference between the two stars is increased and when the star which gets bound to</text>
<table>
<location><page_9><loc_53><loc_63><loc_89><loc_84></location>
<caption>Table 6. Same as Table 4 for simulations HUp42, HUp70a and HUp70b in Section 5.3.</caption>
</table>
<text><location><page_9><loc_51><loc_54><loc_91><loc_60></location>the BH is the less massive of the two (simulation HUp70a). This is because a low-mass star is less compact than a higher-mass star (the compactness parameter is ∝ M ∗ /R ∗ ), and this leads to an increased difference between the narrow peak of the low-mass star light curve and the broader peak of the higher-mass star light curve.</text>
<section_header_level_1><location><page_9><loc_51><loc_49><loc_77><loc_50></location>6 SUMMARY AND CONCLUSIONS</section_header_level_1>
<text><location><page_9><loc_51><loc_36><loc_91><loc_48></location>A stellar tidal disruption occurs when a star passes close enough to the central BH of a galaxy to experience the BH tidal field. The star can be fully or partially torn apart, according to the distance of closest approach (Guillochon & Ramirez-Ruiz 2013, 2015a). The stellar debris which accretes on to the BH powers a long-lasting single flare (e.g. Rees 1988; Phinney 1989) or even periodic flares, if the star, partially disrupted, keeps on orbiting around the BH (MacLeod et al. 2013). Such events contribute to detect otherwise quiescent BHs of masses complementary to that probed in bright AGN and QSO surveys (Vestergaard & Osmer 2009).</text>
<text><location><page_9><loc_51><loc_7><loc_91><loc_35></location>Given the high number of field stars in binary systems (Duquennoy & Mayor 1991b; Fischer & Marcy 1992), encounters with a galactic central BH can involve stellar binaries instead of single stars. The high central densities and velocity dispersions present in galactic nuclei reduce the number of binaries. Indeed, most binaries are 'soft', i.e. the relative velocity of their components is much smaller than the velocities of the field stars. Thus, soft binaries can be separated via close encounters with other stars over the galaxy lifetime (Merritt 2013). However, some binaries are 'hard' enough, which also means close enough, to survive encounters with field stars for a longer time. The members of these binaries, under certain conditions, when approaching the central BH can experience total or partial tidal disruption immediately after the tidal binary break-up. From an encounter of this kind, a double-peaked flare is expected to blaze up (Mandel & Levin 2015). Generally, after binary break-up one star leaves the system while the other binds to the BH (e.g. Antonini et al. 2011). In the case of partial double disruptions, the bound star can be thus repeatedly disrupted, lighting up periodic ( ∼ 1 -10 yr) single-peaked flares. Hence, we argue that this channel could be one of the most likely mechanisms that allow stars to become bound to central galactic BHs and undergo periodic TDEs, as suggested for IC3599 (Campana et al. 2015). Periodicity</text>
<figure>
<location><page_10><loc_12><loc_68><loc_45><loc_86></location>
</figure>
<figure>
<location><page_10><loc_12><loc_46><loc_45><loc_64></location>
</figure>
<figure>
<location><page_10><loc_12><loc_24><loc_45><loc_42></location>
<caption>Figure 6. Light curves [ ˙ M versus time; see equation 15 in Section 5.2] inferred from the high-resolution simulations HUp42, HUp70a and HUp70b in Section 5.3. Green and blue curves are associated with single-star disruptions; the composite light curves are the red ones. On the top-right corners, we show the same plots in logarithmic scale. A knee in the composite light curve is visible in simulation HUp70a.</caption>
</figure>
<text><location><page_10><loc_51><loc_83><loc_91><loc_88></location>increases the chance of observing and modelling TDE flares, and it could be predicted if a double peak were detected. This is rare but not impossible, given that double TDEs should contribute up to about the 20 per cent of all TDEs.</text>
<text><location><page_10><loc_51><loc_67><loc_91><loc_82></location>This is the first paper that explores the process of double tidal disruption through hydrodynamical simulations, in the aim at detailing the dynamics of the binary-BH interaction (see Figs. 2, 3 and 5) and the shape of the outcoming light curve. Based on the results of a set of 14 low-resolution SPH simulations of parabolic equal-mass binary-BH encounters, we set the initial conditions of three high-resolution SPH simulations in order to explore double TDEs of different intensities. For twin stars of equal masses, we found that a knee, rather than a double peak, in the composite light curve is observed only in the case of grazing double TDEs. Otherwise, flares without knees can be observed, indistinguishably from single-star tidal disruptions (see Fig. 4).</text>
<text><location><page_10><loc_51><loc_45><loc_91><loc_67></location>We also explored the case of unequal-mass binaries experiencing double TDEs, running three additional high-resolution simulations. We found that the most favourable conditions for the visibility of a knee in the composite light curves occur when the difference in mass between the binary components is increased and the star fated to bind to the BH is lighter than the star fated to leave the system (see Fig. 6). Indeed, the knee becomes more and more defined when the difference in the peak width between the two single-star light curves increases. The less massive star, which is less compact, generates a light curve that is rising and declining on a shorter time-scale. Varying the binary semimajor axis, internal eccentricity and internal orbital plane inclination with respect to the binary CM orbital plane around the BH affects less the shape of the double TDE light curves. These parameters mainly act on the single-star impact parameters, but even if these are different to the maximum degree, they cannot be so much different, otherwise double TDEs are inhibited.</text>
<text><location><page_10><loc_51><loc_32><loc_91><loc_45></location>Starting from the light curve which shows a knee in the case of unequal-mass binaries (Fig. 6, middle panel), we estimated analytically how much the light curve would change when changing the BH mass, M BH . We considered the interval between 10 5 and 10 8 M glyph[circledot] and follow the dependence on M BH of single times and peak accretion rates as reported by Guillochon & Ramirez-Ruiz (2013, 2015a). We found that ∆ t peak tends to increase whereas ∆ ˙ M peak tends to decrease increasing M BH to the point that intermediate values of M BH (i.e. 10 6 -10 7 M glyph[circledot] ) are more favourable to the observation of the knee in the composite light curve.</text>
<text><location><page_10><loc_51><loc_14><loc_91><loc_32></location>It is worth noting that relativistic effects should also be taken into account in future studies on double TDEs, especially in the case of deep encounters, given that they could cause deviations of the debris evolution from the one assumed here. Lens-Thirring effects can warp the accretion disc which forms around a spinning BH, powering quasi-periodic oscillations (Franchini, Lodato & Facchini 2016). In-plane relativistic precession leads the stream of debris to self-cross (Shiokawa et al. 2015), speeding up the circularization process (Bonnerot et al. 2016), but nodal precession which arises from the BH spin can deflect debris out of its original orbital plane, delaying self-intersection and then circularization, which however depends on the efficiency of radiative cooling (Hayasaki, Stone & Loeb 2015), and flaring (Guillochon & Ramirez-Ruiz 2015b).</text>
<text><location><page_10><loc_51><loc_7><loc_91><loc_13></location>Up to now, candidate TDE observations have been too widely spaced in time to allow the notice of a possible knee. The challenge for the future will be to find a way to get more detailed light curves from observations (e.g. Holoien et al. 2016), particularly in the region of peak emission, as well as in the late-time decay. In this way,</text>
<text><location><page_11><loc_7><loc_80><loc_47><loc_88></location>it will be possible to distinguish between light curves which show or not a knee, opening the opportunity to predict and follow up periodic flares, and to separate TDEs from other phenomena which nowadays could be misinterpreted due to the scarcity of data. The advent of new telescopes, such as LSST (http://www.lsst.org/lsst), may contribute to such a purpose.</text>
<section_header_level_1><location><page_11><loc_7><loc_76><loc_25><loc_77></location>ACKNOWLEDGEMENTS</section_header_level_1>
<text><location><page_11><loc_7><loc_70><loc_47><loc_75></location>We thank the ISCRA staff for allowing us to perform our simulations on the Cineca Supercomputing Cluster GALILEO. We also thank the anonymous referee for valuable comments on the manuscript and constructive suggestions.</text>
<section_header_level_1><location><page_11><loc_7><loc_66><loc_17><loc_67></location>REFERENCES</section_header_level_1>
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<section_header_level_1><location><page_12><loc_7><loc_61><loc_46><loc_65></location>APPENDIX A: BINARY STAR ORBITS FROM LOW-RESOLUTION SPH SIMULATIONS AND N -BODY INTEGRATOR</section_header_level_1>
<text><location><page_12><loc_7><loc_40><loc_47><loc_60></location>This appendix shows the collection of orbits associated with our low-resolution simulations of binary-BH encounters. Tables A1 and A2, respectively, refer to the simulations presented in Tables 1 and 2 in Section 4 and include figures which represent the orbital evolution of the binary components around the BH projected in the ( x, y ) and ( y, z ) planes. Evolutions start at (0,0), (0,0). Units are in R glyph[circledot] . Blue curves reproduce the initial parabolic orbits of the binary CM around the BH, each inferred from the position of the BH, marked in figures with a black dot, and the pericentre radius r p . Red and green curves represent the early and late orbital evolution around the BH of each binary component, respectively, inferred from SPH simulations (see also Section 5.2) and computed through an N -body Hermite code (e.g. Hut & Makino 1995; see Section 4). The usage of an N -body code in drawing advanced orbits enables us to overcome the high computational time required by SPH simulations to track them.</text>
<text><location><page_12><loc_7><loc_24><loc_47><loc_39></location>MGs occur when the binary components progressively reduce their relative separation without being tidally separated, till merging in a single product. This MG product, which corresponds to having the binary components at a fixed minimum distance in our N -body simulations, follows the initial parabolic orbit of the binary CMaround the BH. Undisturbed binaries (UNs) keep their internal and external orbits unchanged. Double disruptions (D-TDEs) are immediately preceded by binary separation, which can still also occur without stellar disruptions (BBKs). Binary break-up gets one star bound to the BH and leaves the other unbound. The unbound component, if not totally disrupted, may exit the system as hypervelocity star (Hills 1988; Antonini et al. 2011).</text>
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<caption>SPH simulations of binary TDEs Figures as 780 . 0 LE11: BBK</caption>
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<text><location><page_14><loc_7><loc_89><loc_9><loc_91></location>14</text>
<text><location><page_14><loc_12><loc_90><loc_23><loc_91></location>Mainetti et al.</text>
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<text><location><page_14><loc_77><loc_5><loc_82><loc_5></location>MNRAS</text>
<text><location><page_14><loc_82><loc_5><loc_84><loc_5></location>000</text>
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</document> | [] |
2016MNRAS.463.4287C | https://arxiv.org/pdf/1503.08218.pdf | <document>
<section_header_level_1><location><page_1><loc_7><loc_82><loc_86><loc_86></location>Reconstruction of small-scale galaxy cluster substructure with lensing flexion</section_header_level_1>
<section_header_level_1><location><page_1><loc_7><loc_77><loc_66><loc_79></location>Benjamin Cain 1 glyph[star] , Maruˇsa Bradaˇc 1 , and Rebecca Levinson 2</section_header_level_1>
<text><location><page_1><loc_7><loc_76><loc_64><loc_77></location>1 University of California Davis, Department of Physics, One Shields Ave., Davis, CA 95616</text>
<text><location><page_1><loc_7><loc_74><loc_75><loc_75></location>2 Massachusetts Institute of Technology, Department of Physics, 77 Massachusetts Ave., Cambridge, MA 02139</text>
<text><location><page_1><loc_7><loc_70><loc_15><loc_71></location>27 May 2022</text>
<section_header_level_1><location><page_1><loc_28><loc_66><loc_38><loc_67></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_28><loc_48><loc_89><loc_66></location>We present a reconstructions of galaxy-cluster-scale mass distributions from simulated gravitational lensing data sets including strong lensing, weak lensing shear, and measurements of quadratic image distortions - flexion. The lensing data is constructed to make a direct comparison between mass reconstructions with and without flexion. We show that in the absence of flexion measurements, significant galaxy-group scale substructure can remain undetected in the reconstructed mass profiles, and that the resulting profiles underestimate the aperture mass in the substructure regions by ∼ 25 -40%. When flexion is included, subhaloes down to a mass of ∼ 3 × 10 12 M glyph[circledot] can be detected at an angular resolution smaller than 10 '' . Aperture masses from profiles reconstructed with flexion match the input distribution values to within an error of ∼ 13%, including both statistical error and scatter. This demonstrates the important constraint that flexion measurements place on substructure in galaxy clusters and its utility for producing high-fidelity mass reconstructions.</text>
<text><location><page_1><loc_28><loc_44><loc_89><loc_47></location>Key words: gravitational lensing: strong, gravitational lensing: weak, galaxies: clusters: general, dark matter, methods: data analysis</text>
<section_header_level_1><location><page_1><loc_7><loc_38><loc_24><loc_39></location>1 INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_7><loc_8><loc_46><loc_37></location>The distribution of mass structure in the Universe provides a fundamental test for any plausible cosmological model. Empirical data from multiple experiments have robustly constrained cosmology to be consistent with a flat ΛCDM model (Hinshaw et al. 2013; Kowalski et al. 2008; Planck Collaboration et al. 2013; Eisenstein et al. 2005). But to further understand the constituents of this cosmology, in particular the dark matter, we must investigate the properties of the structures which form from the primordial mass distribution at many scales. To understand the interaction between dark matter and baryonic matter, or to understand the selfinteractions between dark matter particles we must measure the distribution of structures on relatively small scales, meaning measuring structures down to galaxy scales where the properties of both baryons and dark matter influence the evolution of structure. As interactions between dark matter and baryons are more important to the structure of haloes for more non-linear overdensities, measurements of substructures inside of larger halos (e.g., substructures within galaxy clusters) provide an important window into the nature of dark matter.</text>
<text><location><page_1><loc_10><loc_7><loc_46><loc_8></location>For a ΛCDM cosmology, the particular initial condi-</text>
<unordered_list>
<list_item><location><page_1><loc_7><loc_3><loc_26><loc_4></location>glyph[star] E-mail: [email protected]</list_item>
</unordered_list>
<text><location><page_1><loc_50><loc_9><loc_89><loc_39></location>tions (e.g., the primordial matter power spectrum) as well as the particular properties of the dark matter - both its self-interactions and its interaction with baryons, if any will affect the structures that are formed. This includes the mass function of virialized dark matter haloes (Tinker et al. 2008) and the spatial and mass distributions of subhaloes (Nagai & Kravtsov 2005; Kravtsov et al. 2004). The existence of roughly self-similar haloes at all scales is a robust prediction of ΛCDM (Moore et al. 1999), making the detection of galaxy cluster substructures an important cosmological test. The fraction of a cluster's mass which is associated with substructures as well as the other subhalo properties (e.g., density profiles, ellipticities, etc.) can be compared to subhaloes produced in cosmological N -body simulations to further constrain dark matter properties. Recent cosmological simulations of clusters with self-interacting dark matter (SIDM) have found that dark matter interactions affect both the shape of haloes, with more spherical profiles and flatter halo cores as a result of a higher interaction cross-section for galaxy-scale haloes, as well as reducing the number of subhaloes of a given mass (Shaw et al. 2006; Rocha et al. 2013; Peter et al. 2013).</text>
<text><location><page_1><loc_50><loc_3><loc_89><loc_8></location>To measure the mass distribution of substructures within galaxy clusters we cannot use statistical measures of cluster properties, such as empirical scaling relations (see, e.g., Giodini et al. 2013, for a review of these relations).</text>
<text><location><page_2><loc_7><loc_74><loc_46><loc_89></location>These relationships are useful for large samples of clusters but the intrinsic scatter between individual clusters limits their predictive power. Neither can we use baryonic tracers of structure due to uncertainty in the interactions between baryons and dark matter. However, gravitational lensing, and particularly combining strong lensing deflections and linear image distortions (weak lensing shear and convergence) has been successfully shown to produce detailed mass reconstructions of galaxy clusters and resolved complicated, asymmetric morphologies (Clowe et al. 2006; Natarajan et al. 2007, 2009).</text>
<text><location><page_2><loc_7><loc_57><loc_46><loc_73></location>For typical cluster imaging datasets, the correlation of spatially-separated galaxy shapes with uncertain redshifts produces an effective smoothing length of glyph[greaterorsimilar] 10 '' away from the central region of the cluster which is constrained by strong lensing. Simulations indicate that at least half of dark matter subhaloes will exist outside of the Einstein radius of a the main halo (Nagai & Kravtsov 2005). In this large portion of the cluster, at least 99% of the projected area within the cluster virial radius, substructures cannot be efficiently detected in galaxy clusters without additional information or the investment of prohibitively large amounts of telescope time.</text>
<text><location><page_2><loc_7><loc_23><loc_46><loc_57></location>In order to overcome these limitations, to improve the resolution of reconstructed mass distributions, and thus to probe small-scale mass structure, we includie higher-order lensing effects. In regions where the reduced lensing shear g = γ/ (1 -κ ) is a significant fraction of unity, as is the case within a radius of ∼ 60 '' from the cluster center for a typical galaxy cluster, nonlinear image distortions become significant and the linear distortion approximation becomes increasingly less valid as the shear increases (see Bartelmann & Schneider 2001, for a review of linear weak lensing), which also degrades the reliability of galaxy ellipticity as an estimator for shear with standard approaches (Melchior et al. 2010). Quadratic image distortions due to nonnegligible third derivatives of the lensing potential, called flexion, can be observed and quantitatively measured in this regime. Flexion distortions cause intrinsically elliptical galaxy images to be bent by into 'banana-shaped' arclets, augmenting the image distortions due to linear shear and magnification. The two lensing fields which cause flexion distortions are sourced by the gradients of the convergence and shear fields. This has the practical effect of making flexion a sensitive probe of small-scale variations in the surface mass density. Limits on the measurable flexion also constrain the amount of possible substructure in a region where no flexion is significantly detected.</text>
<text><location><page_2><loc_7><loc_6><loc_46><loc_22></location>In this paper we present a method for reconstructing a lensing mass distribution using multiple image positions (strong lensing), ellipticity measurements (weak lensing shear), and flexion measurements. This significantly improves the sensitivity of the reconstruction to small mass substructures; we are able to resolve structures to a significantly finer spatial resolution and a lower mass threshold than with strong lensing and shear alone. We are able to directly detect structures with mass ∼ 9 × 10 12 M glyph[circledot] which are not detected without the flexion measurements, even in the case where they are located far from the strong lensing regime.</text>
<text><location><page_2><loc_7><loc_3><loc_46><loc_5></location>Throughout we employ a standard flat cosmological model with Ω m = 0 . 3, Ω Λ = 0 . 7, and H 0 = 70 km/s/Mpc.</text>
<text><location><page_2><loc_50><loc_85><loc_89><loc_89></location>We also employ standard complex notation for lensing fields, with ∇ = ∂ 1 + i∂ 2 being the derivative operator acting in the image plane.</text>
<section_header_level_1><location><page_2><loc_50><loc_79><loc_83><loc_80></location>2 MASS RECONSTRUCTION METHOD</section_header_level_1>
<text><location><page_2><loc_50><loc_51><loc_89><loc_78></location>Our mass reconstruction method is an expansion of the method described in Bradaˇc et al. (2009), which itself was an improvement of previous methods (Bradaˇc et al. 2005, 2006). We review the basic concept of the method here to place our expansion in context. We include five complex lensing fields, all of which yield measurable lensing distortions which are inputs into our reconstruction. All five fields arise as linear combinations of the first through third derivatives of a single, real-valued lensing potential ψ sourced by the lens mass distribution. These lensing fields are the deflection α = ∇ ψ , the convergence κ = 1 2 ∇∇ ∗ ψ , the shear γ = 1 2 ∇ 2 ψ , 1-flexion (comatic flexion) F = ∇ κ , and 3-flexion (trefoil flexion) G = ∇ γ . Here we have used complex notation for both the fields and the derivative operators, and the star operator denotes the complex conjugate. The positions of the multiple images in a strong lensing system constrain the deflection, and the ellipticity of lensed images of background galaxies constrains the shear and the convergence (see Bartelmann & Schneider 2001, for a review of strong lensing and shear constraints on lens mass distributions).</text>
<text><location><page_2><loc_50><loc_27><loc_89><loc_50></location>Both 1-flexion and 3-flexion are constrained by measurements of non-linear distortions due to gradients in the convergence and shear. Since the convergence is the surface mass density of the lens scaled to the critical lensing density, measurements of image distortions due to flexion provide an important direct probe of mass gradients, and therefore small-scale mass structures. The flexion fields can be measured using several methods available in the literature (Goldberg & Bacon 2005; Goldberg & Leonard 2007; Okura & Futamase 2009; Schneider & Er 2008). In our mass reconstruction we adopt the flexion formalism as defined by Cain et al. (2011, hereafter CSB). Our reconstruction method assumes measurements of the reduced flexion fields Ψ 1 = F / (1 -κ ) and Ψ 3 = G / (1 -κ ) from lensed source images using the Analytic Image Model (AIM) approach, rather than calculating flexion estimates from image characteristics like moments or shapelet weights.</text>
<text><location><page_2><loc_50><loc_7><loc_89><loc_27></location>Using the estimators for each of the lensing fields obtained from analyzing cluster data images, we constrain a model lensing potential and the mass distribution which corresponds to that potential. Our model is defined as lens potential values ψ k on a 'mesh' of points θ k to be constrained by the data. The value of the derivatives of the model potential at an arbitrary point θ j are calculated using a generalized finite differencing method using appropriate linear combination of nearby potential values on the mesh to give the desired derivative. The coefficients of this linear transformation are determined using a singular-value decomposition (SVD) as in Bradaˇc et al. (2009) to yield a system of equations relating the lensing potential values in the neighborhood of an arbitrary point in the field to the derivatives of the lensing potential:</text>
<formula><location><page_2><loc_59><loc_3><loc_89><loc_5></location>∂ n 1 ∂ m 2 ψ ( θ j ) = ∑ k a jk ( n, m ) ψ k , (1)</formula>
<text><location><page_2><loc_67><loc_0><loc_68><loc_1></location>c ©</text>
<text><location><page_2><loc_68><loc_0><loc_80><loc_1></location>0000 RAS, MNRAS</text>
<text><location><page_2><loc_81><loc_0><loc_83><loc_1></location>000</text>
<text><location><page_2><loc_83><loc_0><loc_89><loc_1></location>, 000-000</text>
<text><location><page_3><loc_7><loc_72><loc_46><loc_89></location>where n and m are the derivative orders for each dimension, k indexes the mesh points and a jk ( n, m ) is the SVD matrix coefficient for the weight of the potential value contribution of the k th mesh point to the n, m derivative of the potential at θ j . As the value for each lensing field is a linear combination of potential field derivatives, this scheme calculates each lensing field at any point in the field as a linear combination of model potential values from neighboring mesh points to θ j . To constrain the model at the location of a lensed image, the model value for the lensing field(s) which have estimators derived from the image can be calculated and compared to the model.</text>
<text><location><page_3><loc_7><loc_56><loc_46><loc_72></location>We then determine the values of ψ k which best fit the data by minimizing a model penalty function ( χ 2 ) which includes contributions from weak lensing shear, strong lensing, and flexion to penalize models which do not produce the measured lensing distortions. The form of the strong and weak lensing penalty functions are identical to those of Bradaˇc et al. (2009). The value of the flexion penalty function is determined by the reduced flexion fields Ψ i 1 and Ψ i 3 (where i indexes the N F galaxies with measured flexion fields), and the model values for the flexion and convergence fields F i , G i , and κ i calculated for the location of that galaxy:</text>
<formula><location><page_3><loc_8><loc_50><loc_46><loc_55></location>χ 2 FL = N F ∑ i [ 1 σ 2 F ,i ∣ ∣ ∣ ∣ Ψ i 1 -F i 1 -κ i ∣ ∣ ∣ ∣ 2 + 1 σ 2 G ,i ∣ ∣ ∣ ∣ Ψ i 3 -G i 1 -κ i ∣ ∣ ∣ ∣ 2 ] . (2)</formula>
<text><location><page_3><loc_7><loc_38><loc_46><loc_50></location>Here, N F is the number of galaxy images with flexion measurements. As described above, the model flexion and convergence values are determined from a linear combination of mesh potential values. The uncertainties σ F ,i and σ G ,i include both intrinsic flexion scatter and measurement error. This is the same basic form for the penalty function as described in Er et al. (2010), but we explicitly formulate it in terms of the reduced flexion estimators Ψ 1 and Ψ 3 from CSB.</text>
<text><location><page_3><loc_7><loc_6><loc_46><loc_37></location>As in Bradaˇc et al. (2009), we include in our full penalty function regularization terms, which prevent the highlynonlinear nature of the penalty function's dependence on the mesh point potential values from driving the model to assume values wildly discrepant from the initial conditions during the reconstruction. This regularization is particularly important for mesh points in regions that are weakly constrained by the data and therefore more susceptible to numerical effects in the reconstruction. We choose to regularize only on the values of convergence and shear, meaning that we penalize model potential values at each mesh point which require large pixel-to-pixel variations in the lensing fields fields relative to a reference value. Note that this does not mean that the model cannot vary from its initial conditions! Instead, we are requiring that the deviation from the initial conditions be driven by the data, rather than numerical effects. We iterate the reconstruction process and reset the reference value to the output of the previous iteration. This causes the model to step smoothly away from the initial guess in a controlled and converging way. Without regularization, the highly nonlinear dependence of the reduced lensing fields prevents the reconstruction process from finding a solution which minimizes the penalty function.</text>
<text><location><page_3><loc_7><loc_3><loc_46><loc_5></location>Our regularization penalty function for convergence is defined over as a sum over the N mesh mesh points located</text>
<text><location><page_3><loc_50><loc_88><loc_55><loc_89></location>at θ k as</text>
<formula><location><page_3><loc_56><loc_83><loc_89><loc_87></location>χ 2 reg,κ = N mesh ∑ k η ( κ ) ( κ ( θ k ) -κ (0) ( θ k ) ) 2 , (3)</formula>
<text><location><page_3><loc_50><loc_63><loc_89><loc_82></location>with similar terms for each component of the shear field. The value of the regularization weights η ( κ ) , η ( γ 1 ) , and η ( γ 2 ) , are determined empirically. We test a range of values for each reconstruction to not only find an appropriate value which prevents spurious nonlinear numerical artifacts while still allowing the reconstruction to deviate from initial conditions as indicated by data, but also to ensure that any reconstruction is stable with respect to variations the regularization coefficients. We also select values which place the total regularization penalty per mesh point (as a proxy for the regularization degrees of freedom) to be approximately equal that of the penalty per degree of freedom for each data portion of the full penalty function (i.e., strong, weak, or flexion lensing).</text>
<section_header_level_1><location><page_3><loc_50><loc_58><loc_82><loc_60></location>3 TESTING THE RECONSTRUCTION METHOD</section_header_level_1>
<text><location><page_3><loc_50><loc_37><loc_89><loc_56></location>With flexion included into the mass reconstruction, we test our method using simulated measurement catalogs. We generate realistic lensing estimator values, including realistic measurement errors, from an underlying mass model which includes substructure. We then reconstruct the mass distribution using all the available data, and compare that reconstruction to a modified reconstruction which incorporates the same data for strong and weak lensing shear, but omits the flexion data. This allows us to make direct comparisons between reconstructions and evaluate the contribution that flexion makes to the fidelity of the final mass reconstructions. In particular, we are interested in finding substructure in the mass distribution which would be unobserved without the inclusion of flexion data.</text>
<text><location><page_3><loc_50><loc_26><loc_89><loc_37></location>This latter point is important for how we construct our input simulated lensing measurements. It has already been well established that strong lensing is a sensitive probe of substructures. The goal of the mass reconstructions using our simulated measurement catalogs is to test whether realistic substructures in cluster lenses can exist which are detectable using flexion but not without. As we will show, this is indeed the case.</text>
<section_header_level_1><location><page_3><loc_50><loc_23><loc_79><loc_24></location>3.1 Input Lensing Mass Distribution</section_header_level_1>
<text><location><page_3><loc_50><loc_7><loc_89><loc_21></location>Our input mass distribution is created to mimic a galaxycluster sized halo with a small number of substructures, each simulated using a nonsingular isothermal ellipse model. We set the scale of the large central halo to be comparable with a galaxy cluster velocity dispersion of approximately 1500 km/s. This is a large cluster, similar to those in the CLASH (Postman et al. 2012) and Hubble Frontier Fields initiative 1 (PI Lotz: Program ID 13495) samples. We also include two substructures in the lens within a radius of 1.5 arcminutes of the cluster center. These subhaloes are also nonsingular isothermal ellipses with varying masses from group scale</text>
<text><location><page_4><loc_7><loc_85><loc_46><loc_89></location>down to large-galaxy scale (see Appendix A for tabulated subhalo masses). We construct six different data sets corresponding to substructured cluster haloes.</text>
<text><location><page_4><loc_7><loc_59><loc_46><loc_85></location>The positioning of the halo mass substructures was not done in a fine-tuned way, i.e., we did not select the location or masses of the substructures in any way to enhance the effect of the substructures on the resultant lensing morphology. Instead, we use a conservative approach and restrict the possible locations such that the strong lensing morphology does not initially indicate that there is significant substructure in the lens. The intent of this work is to characterize the differential effect that flexion makes on the resulting mass distribution. Thus, we restrict the substructure halo masses and positions to be outside the approximate radius of the critical curve so that the additional substructure is not easily detected using the strong lensing data alone. Substructures which are too massive or near the critical curves create significant deviations in the location, morphology, and magnification of strong lensing images. This can be an extremely sensitive probe of substructure (e.g. Vegetti et al. 2012), however our goal is to make a total census of the substructure in galaxy clusters.</text>
<section_header_level_1><location><page_4><loc_7><loc_54><loc_34><loc_55></location>3.2 Simulated Lensing Estimators</section_header_level_1>
<text><location><page_4><loc_7><loc_31><loc_46><loc_53></location>We simulate lensing measurement catalogs to test the mass reconstruction method and the effect of including flexion. Using the main halo and smaller substructures we define as above, we analytically calculate the appropriate lensing field estimators from the total mass profile using the full lens equation and its derivatives. We assume a single, moderate redshift of z lens = 0 . 3 for the lens halo (including its substructures) in all of the simulated datasets. Independent random noise is added to each simulated measurement with scatter chosen to match observational constraints from single orbit Hubble Space Telescope Advanced Camera for Surveys ( HST /ACS) cluster observations. For the simulations presented here, we employ an input lensing mass distribution consisting of three non-singular isothermal ellipse (NIE) halo models for computational efficiency in constructing the simulated measurements and a simple comparison model.</text>
<text><location><page_4><loc_7><loc_21><loc_46><loc_31></location>To determine our simulated measurement values based on the lensing mass distribution, the source-lens geometry must be determined along the line of sight in addition to the plane of the sky. This is because the lensing fields are all dependent on the redshift of the lensing mass as well as the redshift of the source object. We randomly select the redshift of each lensed source from a gamma distribution</text>
<formula><location><page_4><loc_19><loc_17><loc_46><loc_20></location>p ( z ) = z 2 2 z 3 0 exp( -z/z 0 ) (4)</formula>
<text><location><page_4><loc_7><loc_3><loc_46><loc_17></location>with z 0 = 2 / 3. This is the same distribution as used in previous work to emulate the true distribution of sources (Brainerd et al. 1996; Schrabback et al. 2007; Bradaˇc et al. 2004). We draw a new random redshift for each flexion and weak lensing source image, and each strong lensing image system has its own redshift as well. Redshift errors on the weak lensing/flexion sources are added with σ z = 0 . 05(1+ z ). The strong lensing systems do not have redshift errors, to reflect the spectroscopic redshifts which are available for many multiply imaged galaxies. We describe the measure-</text>
<text><location><page_4><loc_50><loc_86><loc_89><loc_89></location>nt generation procedure for each type of catalog (flexion, weak lensing, and strong lensing) below.</text>
<text><location><page_4><loc_50><loc_71><loc_89><loc_85></location>To generate our flexion measurement catalog we select a random distribution of source positions for the flexion and leak lensing shear measurements. The positions are selected uniformly across a square field 3 . ' 5 on a side to a density of 80 galaxies/arcmin 2 . For the flexion measurements we calculate the reduced flexion fields Ψ 1 and Ψ 3 as defined in CSB at the position of each of the sources. We add Gaussian noise to each component of the flexion measurements with a standard deviation of 0.03 arcsec -1 . These noise levels are motivated by previous studies of intrinsic flexion (Goldberg & Leonard 2007, CSB).</text>
<text><location><page_4><loc_50><loc_30><loc_89><loc_70></location>We then remove from the flexion dataset any sources where the magnitude of the measured flexion exceeds 1.0 arcsec -1 . This ceiling on the measured flexion reflects how the quadratic expansion of the lens equation which defines the flexion fields is only valid when the flexion correction is small relative to the shear distortion. In a real dataset, images with flexion values in these regimes would either be identifiable as strongly-lensed arcs and thus in the strong lensing dataset instead of the flexion catalog, or would be removed during image fitting as in CSB. There the authors found that for flexion values this large the AIM image fitting did not produce accurate results, but the properties of the AIM covariance matrices allowed for efficient removal of these catastrophic outlier images. Our limit on the flexion values is a uniform approximation of the limit that would occur in real data: the directly observable distortion in a lensed image due to flexion is quantified by the unitless product of the image scale (such as the half-light radius) and the reduced flexion field, rather than the more physically relevant reduced flexion field alone. Because flexion is a quadratic effect, measuring flexion requires that the lensed galaxy image be larger than is required for ellipticity - in practice, this means 2-3 times the point-spread function size along the long axis of the image. In order to fully include varying image size and its effect on flexion estimation and the resulting mass reconstruction, a better understanding of intrinsic flexion and its correlation to galaxy size and magnitude must be achieved. However, the simpler cut on the value of the flexion field is a sufficient approximation for our purpose.</text>
<text><location><page_4><loc_50><loc_3><loc_89><loc_29></location>The weak lensing shear and strong lensing datasets are constructed from the reduced shear and deflections fields. The source locations for the shear are randomly distributed through the simulated field and locations with reduced shears nearing unity are removed (for the same reason as large flexion values are removed). The selection of the source position for the strong lensing systems is not completely random, however. For each system we initially draw a random position from a small field in the source plane centered on the main halo core and solve the lens equation to find the image positions. We then perform a visual inspection and remove systems if they show obvious signs of substructure. With some source positions relative to the distribution of mass in the lens, multiple image systems will form around the smaller substructure. We reject these systems, as the utility of strong lensing for substructure detection is already well established. Instead, we select only image systems that are representative of a single, central halo and do not have obvious substructure included.</text>
<section_header_level_1><location><page_5><loc_7><loc_88><loc_39><loc_89></location>3.3 Initial Conditions and Regularization</section_header_level_1>
<text><location><page_5><loc_7><loc_67><loc_46><loc_86></location>Our approach to reconstructing the mass distribution from the simulated measurements is the same approach we would take using real data. We determine initial conditions for our model fitting, set a mesh of points where the model potential will be defined, and reconstruct the lensing potential all using data-driven conditions. We perform several reconstructions with different values for some of the lessstrongly-constrained input parameters (e.g., the regularization weights). We evaluate the output reconstructions, and in particular we evaluate the behavior of the penalty function values as the reconstruction converges, to determine the best values for these parameters. We do not see a strong dependence on the particular values - small changes do not significantly affect the output reconstruction.</text>
<text><location><page_5><loc_7><loc_41><loc_46><loc_67></location>The first step in our reconstruction is to assign initial conditions for the reconstruction which are motivated by the data. We use an analytic model to set the initial model lensing potential values which, in turn, determine the initial value of the penalty function and its gradient. As was shown previously (Bradaˇc et al. 2004; Bradaˇc et al. 2005), the final reconstruction using this method is not strongly dependent the particular initial model used provided that the regularization is chosen appropriately. We employ a simple, parametric, axisymmetric model as our initial guess. We take a single, nonsingular isothermal sphere as our initial model, locating the center of the NIS at the average position of the images in the strong lensing data set. The mean of the distances from these strong lensing images from their average position defines an approximate Einstein radius for the overall lensing system. Using the approximated Einstein radius estimate we construct our initial model from a NIS profile. These simple, reasonable approximations provide datadriven initial conditions.</text>
<text><location><page_5><loc_7><loc_7><loc_46><loc_40></location>Another important input parameter to the mass reconstruction is the weight of each component of the regularization in the penalty function ( η κ , etc., from Eq. 3), which defines the regularization weight relative to the strong lensing, weak lensing shear, and flexion contributions. As noted in § 2, too little weight given to regularization (small η values) yields reconstructions with unrealistically large amounts of small-scale spatial variation as well as aphysical lensing potential values, such as negative convergence. These numerical artifacts of the nonlinear lens inversion prevent the optimization algorithm from finding the appropriate minimum as it iterates. Excessive weight to the regularization (large η ) causes the reconstruction to simply return a model which is minimally different from the initial mass model, preventing the data from informing the reconstructed mass distribution. Because it is not possible to determine the most appropriate η values a priori , and because the most appropriate specific value for the regularization weight will depend not only on the details of the data available but also on the particular mesh geometry chosen, we determine η empirically by reconstructing with a range of values. By comparing the χ 2 values for each part of the penalty function, we are able to separate successful reconstructions from those which are over- or under-regularized.</text>
<text><location><page_5><loc_7><loc_3><loc_46><loc_7></location>By evaluating the χ 2 values output by these reconstructions we determine the approximate best combination of values for σ NIS and regularization weight to use for the given</text>
<text><location><page_5><loc_50><loc_61><loc_89><loc_89></location>dataset. To compare different reconstructions, we consider both the total χ 2 value, as an indicator of the overall fitness of the reconstruction, and the individual contributions of the different datasets. Reconstructions which yield values of χ 2 / d.o.f. ≈ 1 for each of the three datasets are preferred, even in cases where that increases both the overall and regularization χ 2 values. Furthermore, we prefer models which reproduce the strong lensing data more precisely above those which minimize the weak lensing or flexion penalty functions, provided that there are no obvious signs of significant pixel to pixel variations. This is because strong lensing data have very small measurement errors, in fact smaller than the error we use in the strong lensing penalty function which we inflate in order to more easily locate the penalty function minimum. Thus if the strong lensing image systems are robustly identified in a given cluster, we can reasonably require that the solution have a strong lensing penalty value of effectively zero. The reconstruction which meets these criteria while minimizing the overall χ 2 value is the reconstruction which we select as our solution.</text>
<text><location><page_5><loc_50><loc_45><loc_89><loc_61></location>In the presence of a significantly substructured mass distribution, the reconstruction will not converge with a single halo initial model, and will plateau to χ 2 values of ∼ 5-10 (or more) per degree of freedom after several iterations. This is an indication of an initial guess which is far from the solution, and the highly non-linear nature of the reconstruction process requires and update to the initial conditions based off this information. We update the initial conditions iteratively, occasionally adding additional haloes at the location of and sizes of substructures detected in the non-converged reconstruction in order to arrive at a good fit. See § 4 for a discussion of our substructure detection method.</text>
<section_header_level_1><location><page_5><loc_50><loc_41><loc_73><loc_42></location>3.3.1 Reconstruction Model Mesh</section_header_level_1>
<text><location><page_5><loc_50><loc_22><loc_89><loc_40></location>We distribute mesh points over the 'observed field', which in this case is the full area around the mean of the strong lensing image locations in a square region 3 . ' 5 on a side (mimicking the HST/ACS field of view). These mesh points are where the model lensing potential and its derived lensing fields are defined. The 'base' mesh is defined by a rectangular, regular grid of points evenly defined across our field. We refine this grid in a circular region centered on the field center-point. The radius of this circular region is 1.5 times the mean distance of the strong lensing images from the average strong lensing image location. Within this radius, we distribute mesh points in a rectilinear grid at twice the base density.</text>
<text><location><page_5><loc_50><loc_10><loc_89><loc_22></location>We further refine in regions within a 3 '' radius of a strong lensing image position with four times the base density. This refinement is necessary to accurately constrain the source positions for the strong lensing systems. Numerical error in the calculation of the gradient of the model lensing potential due to a paucity of mesh points near strong lensing images can dominate the strong lensing contribution to the penalty function, which will preclude convergence and accurate reconstruction.</text>
<text><location><page_5><loc_50><loc_3><loc_89><loc_10></location>This nested mesh structure places increasing numbers of mesh points where the mass distribution is expected to be varying more rapidly, as well as where we have tighter constraints from the lensing data, and therefore a higher density of mesh points for the finite difference derivatives</text>
<text><location><page_6><loc_7><loc_86><loc_46><loc_89></location>used to calculate the model lensing field values enhances the accuracy of the reconstruction output.</text>
<text><location><page_6><loc_7><loc_75><loc_46><loc_86></location>In the reconstructions presented here, we begin with a base mesh density of roughly 70 mesh points/arcmin 2 , corresponding to about one galaxy image per mesh point, with the density of mesh points higher in the refinement regions. The optimal configuration will have increased mesh point density wherever the potential is changing rapidly, and will not have any changes in refinement level of more than a single step.</text>
<section_header_level_1><location><page_6><loc_7><loc_71><loc_17><loc_72></location>4 RESULTS</section_header_level_1>
<text><location><page_6><loc_7><loc_55><loc_46><loc_69></location>Using the methods described above, we reconstruct the mass distribution for each of the simulated cluster datasets, and bootstrap the simulation 1 catalogs and reconstruct each of those. Here we present our results, first in a detailed investigation of simulation 1 and the statistics of the reconstructions after bootstrap-resampling of the lensing catalogs, then more broadly for the rest of the cluster data sets. In discussing these results, we will refer to locations within the reconstructed field by celestial directions (north being up and east being to the left), as we would were these actual cluster fields and not arbitrarily oriented simulations.</text>
<section_header_level_1><location><page_6><loc_7><loc_51><loc_20><loc_52></location>4.1 Simulation 1</section_header_level_1>
<text><location><page_6><loc_7><loc_35><loc_46><loc_50></location>Figure 1 shows a comparison of the input mass distribution, the reconstruction without flexion, and the reconstruction with flexion. In both the without-flexion reconstruction and in the with-flexion case, the strong lensing data is extremely well fit by the reconstruction, essentially perfectly reproducing the multiple image positions for each of the sources. The final weak lensing penalty function values in each reconstruction are very similar as well. Both reconstructions fit the strong and weak lensing data equally well within statistical uncertainty (we discuss the statistics of these reconstructions below).</text>
<text><location><page_6><loc_7><loc_15><loc_46><loc_34></location>However, the reconstruction with flexion includes additional substructure undetected in the flexion-less reconstruction. One of the two subhaloes (to the northeast) is clearly detected by eye, and the second is apparent as an extension to the southwest. To quantify substructure detections, we employ Source Extractor (SExtractor, Bertin & Arnouts 1996) to detect 'objects' in our mass maps. This is similar in some ways to detecting objects in a crowded field, though with an unusually fine pixel scale. We require a large minimum area (an area of ∼ 200 arcsec 2 ) above the detection threshold to identify an 'object'. This prevents SExtractor from identifying the small, local variations in the convergence map due to the mesh point spacings as individual objects.</text>
<text><location><page_6><loc_7><loc_3><loc_46><loc_15></location>Because cluster subhaloes are not as discrete as galaxies in a typical field and instead are a contiguous distribution of matter with significant blending between haloes, and because estimating the 'background' convergence level in the environment of a parent halo is problematic, we do not discount morphological evidence for substructure in our reconstructions. While the southern substructure is not individually detected by SExtractor, it instead appears as an elongation of the source in that direction. Combining both</text>
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<caption>Figure 1. With flexion measurements we detect substructure in the mass distribution of Simulation 1. From top to bottom, the input convergence (surface mass density) map, the convergence map reconstructed without flexion, and the map reconstructed including flexion data. The innermost contour levels are the same in all three cases beginning at a convergence of 1.5 and decrease by 0.375 with each step, for a source at infinite distance.</caption>
</figure>
<text><location><page_7><loc_7><loc_84><loc_46><loc_89></location>the SExtractor detections and the visible morphology we conclude a strong detection of the northern subhalo and a likely detection of the second, though it blends partially into the main halo.</text>
<section_header_level_1><location><page_7><loc_7><loc_79><loc_32><loc_80></location>4.2 Bootstrap Error Estimation</section_header_level_1>
<text><location><page_7><loc_7><loc_53><loc_46><loc_77></location>We estimate the uncertainty in our mass reconstructions via a bootstrap resampling of the weak lensing shear and flexion catalogs. We randomly select measurements from the two catalogs uniformly, allowing entries to be selected multiple times, to form a new pair of resampled catalogs. From these catalogs we perform a mass reconstruction using the same initial conditions model and reconstruction parameters as used in the second step reconstruction of the original dataset. We repeat this process 100 times, each with a different realization of the resampled catalog, to generate a distribution of mass reconstructions which we then compare to the input convergence map. We hold the strong lensing sources constant in the bootstrapping process. This is acceptable given that the identification of multiple images in real data allow us to more securely identify the real systems that the simulated strong lensing catalogs represent and the likelihood of mistakenly including interloper galaxies which are not strongly lensed is relatively low.</text>
<text><location><page_7><loc_7><loc_39><loc_46><loc_53></location>The bootstrap reconstructions produce a large datacube of information, with a distribution of model values for each of the lensing fields: ψ , α i , κ , γ i , F i , and G i . Ten in all, eleven if the magnification µ is included. To characterize these distributions in a comprehensible way, we produce an average convergence map and the RMS variation of the convergence in the 100 reconstructions, which is displayed in Figure 2. These maps show that the substructure detected with flexion is significant relative to the scale of statistical variations in the catalog data.</text>
<text><location><page_7><loc_7><loc_20><loc_46><loc_39></location>We also map both the RMS variation of the bootstrapped convergence maps relative to the primary reconstruction, and also the difference between the primary and average convergence maps, presented in Figure 3. These give us a metric of bias in the primary reconstruction, i.e., how much of an outlier is the primary reconstruction. Were we to see a large amount of bias in these maps, meaning a large difference between the RMS variation about the average and the RMS variation about the primary, or were we to see a large difference between the primary and the average convergence maps, this would suggest that the primary reconstruction were a statistical outlier rather than a robust minimum solution to reconstructing the mass distribution. We do not see any evidence that this is the case.</text>
<text><location><page_7><loc_7><loc_3><loc_46><loc_19></location>We see the expected small number of outliers among the reconstructions, but overall the bootstrapped catalog reconstructions should not vary drastically from the primary reconstruction if they are valid estimators of the error in the reconstruction. A few reconstructions ( glyph[lessorsimilar] 5) have hugely discrepant penalty function values, universally due to a poor fit to the the strong lensing data and thus an incorrectly located critical curve. These outliers are easily removed and discarded from our statistical analyses, though leaving them in does not significantly affect the results as the lensing field maps, such as the convergence, are not catastrophically deviant from the primary solution even with these outlier so-</text>
<text><location><page_7><loc_50><loc_86><loc_89><loc_89></location>lutions since they depend less on the exact position of the critical curve.</text>
<text><location><page_7><loc_50><loc_53><loc_89><loc_86></location>The reconstruction method is computationally intensive, requiring the inversion of large sparse matrices used for the finite-differences calculation of the gradient of the penalty function with respect to the lensing potential values on the mesh, so we perform this bootstrap analysis on only one of our simulations (Simulation 1), both with and without flexion measurements included, to estimate the amount of variation to expect due to the statistical uncertainty in our lensing field estimator measurements. As with real data, the variation depends not only on the accuracy and density of the weak lensing and flexion estimators across the field, but also on the amount of specific constraint provided by the particular set of strong lensing images available. The number and, importantly, the location of the multiple images have an effect on the spatially varying amount of freedom that the mass model can change within the statistical errors of the measurements. A more full statistical analysis of the cluster lensing mass reconstruction made including flexion allowing a broader variation of possible halo and subhalo configurations is a large undertaking, and beyond the scope of this work. However we can use our bootstraps to begin to estimate the errors in our mass maps, and compare the difference between the input mass distribution and the reconstructed mass map to these error estimates.</text>
<text><location><page_7><loc_50><loc_36><loc_89><loc_53></location>Using the bootstrap-resampled catalog reconstructions, we can understand the statistical variation in our mass maps due to the catalog sampling. One immediate feature to note in the non-flexion RMS map (Figure 2) is that the variation is small and is relatively uniform across the field. This means that despite the catalog resampling, a consistent mass distribution is preferred by the data. The input subhaloes are larger than any of the statistical variations by quite a large margin, the non-detection of the subhaloes is not due to noise or a statistical fluctuation. Instead, it is due to incomplete information in the lensing inputs, namely the lack of flexion information measuring the mass and shear gradients.</text>
<text><location><page_7><loc_50><loc_3><loc_89><loc_36></location>The RMS map for the flexion reconstruction is less uniform than the non-flexion RMS map, which can be understood considering how flexion measurements inform the mass reconstruction. Because the flexion field is sourced very locally by the gradient of the mass distribution, and since the flexion measurement is not dominated by statistical noise in the same way that shear measurements are, excluding certain individual flexion measurements from the catalog can have a more significant effect on the resulting mass map than happens with weak lensing shear measurements that, by their nature, must be correlated together to extract the lensing signal. Thus we expect that bootstrapping the flexion measurement catalogs in addition to the shear catalogs will increase the RMS values near where the flexion measurements are most informative and constraining. This includes the locations of substructures, as can be seen in the RMS maps in Figures 2-3, but also where the flexion field is strong but reliable measurements are relatively sparse (near the main halo center) as well as where flexion measurements are particularly near a strong lensing image. The RMS map shows that neither substructure detection can be attributed to the statistics of the flexion information, with one certain detection (north) and a second, less significant detection (south).</text>
<figure>
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<caption>Figure 2. Flexion-detected substructure is robust to catalog resampling. On the left, the average convergence map for the set of reconstructions using the bootstrapped catalogs (above) and the RMS variation about that average (below) for reconstructions of data from simulation 1 without flexion. On the right, the same two plots for simulation 1 reconstructions including flexion. Contours for the top two plots are as in Figure 1. For the bottom two plots, the innermost contours begin at 0.2 and decrease by 0.025 with each step, also assuming a source at infinite distance.</caption>
</figure>
<text><location><page_8><loc_7><loc_13><loc_46><loc_28></location>As noted above in comparing the average reconstruction to the primary reconstruction, both with and without flexion, we see no significant overall bias (Figure 3). For reconstructions with flexion, the average reconstruction underestimates the mass in the substructures as is expected since the bootstrap reconstructions include catalogs where some of the flexion measurements nearest the substructures are not included which will reduce the lensing signal from that substructure. The features in these maps are what would be expected given the information available (and not available) to each reconstruction.</text>
<text><location><page_8><loc_7><loc_3><loc_46><loc_12></location>These results, the primary mass reconstruction together with the properties of the reconstructions from bootstrap resampled catalogs, show that the inclusion of flexion measurements can detect substructures which are otherwise undetectable in strong lensing and weak lensing shear data alone. Furthermore, these detections are statistically significant and robust to the variations of bootstrapped catalogs,</text>
<text><location><page_8><loc_50><loc_21><loc_89><loc_28></location>i.e., they do not depend solely on a few fortuitously-located images, though there is a more marked dependence of the resulting mass map on the location of the flexion measurements relative to any substructures than is seen for weak lensing shear measurements.</text>
<section_header_level_1><location><page_8><loc_50><loc_18><loc_72><loc_19></location>4.3 Simulations 2 through 6</section_header_level_1>
<text><location><page_8><loc_50><loc_6><loc_89><loc_17></location>Figures 4, 5, and 6 each show comparisons between the reconstructed mass distributions (on the left) and the input mass distribution (right) the eight simulations. These, along with Table A1 show the varying substructure sensitivity enhancement that including flexion yields. Simulations 1 and 2 each have two equal mass subhaloes, though the positions differ between the simulations. For simulations 3-6, we vary the masses of the subhaloes.</text>
<text><location><page_8><loc_50><loc_3><loc_89><loc_5></location>A few trends emerge from these simulations. As would be expected, more massive substructures are reliably de-</text>
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<caption>Figure 3. The RMS variation of the bootstraps about the primary reconstruction (top) and the difference between the primary and the average bootstrapped reconstructions (bottom) for simulation 1. The left column is without flexion, the right with flexion. Green contours indicate positive values, with the innermost contours at 0.2 decreasing in steps of 0.025 to zero. Blue contours indicate negative values, with the innermost value at -0.2 and the same step size.</caption>
</figure>
<text><location><page_9><loc_7><loc_17><loc_46><loc_29></location>tected, particularly if they are well separated from the core of the main halo. The one instance where a larger subhalo is not detected (simulation 7) is one where the subhalo is blended into the main halo. In many configurations, this type of subhalo (in terms of mass and location) would be detectable from strong lensing data alone, as we know from the set of halo+substructure configurations we generated but rejected for this study due to their obviously substructured strong lensing observations.</text>
<text><location><page_9><loc_7><loc_3><loc_46><loc_17></location>For smaller substructures, the detectability decreased with distance from the main halo. This is also expected. The value of the non-reduced flexion fields (i.e., F and G ) is dominated by the gradients in the convergence and shear field whose distortions depend on the proximity to the subhalo mass distribution. The total convergence (which factors into the reduced flexion) is dominated by the main halo mass. At larger radii from the main halo center, the total convergence falls away from unity, the flexion decreases quickly away from the subhalo, and both effects cause the measur-</text>
<text><location><page_9><loc_50><loc_16><loc_89><loc_29></location>able reduced flexion signal to decrease. The least massive subhalo we detect (in simulation 4), though we only detect it marginally, was located quite close to the main halo center where the total convergence is much closer to unity. More radially distant subhalos were either undetected or more massive. As we discuss in detail in § 4.4, the inclusion of flexion does more than detect substructure and improve the reconstructed ellipticity. Flexion also improves the overall fidelity of the mass reconstruction independent of the detectability of individual structures.</text>
<section_header_level_1><location><page_9><loc_50><loc_11><loc_67><loc_12></location>4.4 Aperture Masses</section_header_level_1>
<text><location><page_9><loc_50><loc_3><loc_89><loc_10></location>In addition to detestability of substructure in terms of identifying individual haloes, we can also quantify the fidelity of our mass reconstructions in terms of how how well the mass maps compare to the input mass distribution by comparing aperture masses. Figure 7 plots these results. We</text>
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<caption>Figure 4. A comparison of input mass distributions (left) and the reconstructions including flexion (right) for simulations 1 (top) and 2 (bottom). Contours are as in Figure 1. In these plots, and in Figures 5-6, the reconstruction field displayed is centered on the strong lensing image centroid, and therefore are slightly offset from the true halo center.</caption>
</figure>
<text><location><page_10><loc_7><loc_20><loc_46><loc_30></location>compare aperture masses rather than point-by-point masses since we expect relationship between the input and reconstructed mass distribution to not only include noise propagated from the lensing field estimators, but also an effective convolution kernel that will vary somewhat across the field based on the density of lensing field measurements and the type of measurements in that region (i.e., strong lensing, shear, or flexion).</text>
<text><location><page_10><loc_7><loc_3><loc_46><loc_18></location>There is a moderate tightening of the scatter (in dex) as the aperture radius increases, which is consistent with the idea of the reconstruction method imposing and effective convolution kernel just as larger apertures in photometry will omit fewer photons dispersed by a non-trivial pointspread function. Though the measurements for a single halo location with multiple aperture radii are certainly correlated, there is similarly good agreement between the input mass distribution and the reconstructed masses in each of these apertures, confirming that we are in fact reconstructing the substructures on these small scales, as the fraction of</text>
<text><location><page_10><loc_50><loc_28><loc_89><loc_30></location>the mass enclosed due to the substructure increases as the aperture shrinks.</text>
<text><location><page_10><loc_50><loc_11><loc_89><loc_28></location>The measurements scatter about the unity line by 13% for reconstructions with flexion, which together with the RMS scatter between the different bootstrapped catalog realizations which have an RMS variation of ∼ 10% suggest an intrinsic scatter of approximately 8%. Without flexion, on the other hand, the reconstructed aperture masses are systematically biased low at the location of the substructures, as we expect given that the substructures are not detected without flexion, by 25-40%. This underestimate is also seen in aperture measurements of the main halo masses, showing that the improvement from adding flexion is not only a substructure phenomenon.</text>
<section_header_level_1><location><page_10><loc_50><loc_7><loc_70><loc_8></location>4.5 Simulation Summary</section_header_level_1>
<text><location><page_10><loc_50><loc_3><loc_89><loc_5></location>We see from these simulations the feasibility for detecting and identifying substructures depends both on their mass</text>
<figure>
<location><page_11><loc_10><loc_38><loc_85><loc_89></location>
<caption>Figure 5. As in Figure 4, for simulations 3 (top) and 4 (bottom).</caption>
</figure>
<text><location><page_11><loc_7><loc_11><loc_46><loc_33></location>and position relative to the cluster halo. We do not have many detections of substructure at small radii because we select 'un-substructured' strong lensing image configurations and are therefore artificially enhancing the likelihood that substructure near the Einstein radius of the cluster will remain undetected. More centrally located subhaloes are also less likely to be detected because of the reduced number of flexion measurements - again, these substructures are those that would typically be visible in the strong lensing data but do not appear in our data because we select against the configurations that make them apparent. However, independent of subhalo identification, the masses we measure in apertures as small as 10 '' are accurate to an error (scatter plus statistical) of 13% across the reconstructed field. This is much smaller than the observed systematic offset from not including flexion in the reconstruction.</text>
<text><location><page_11><loc_7><loc_3><loc_46><loc_10></location>Flexion data included into the mass reconstructions fill an important information gap in the full lensing data set. With a mass limit that depends both on the angular size of the subhalo Einstein radius (which is a proxy for the subhalo mass) and its distance from the main cluster halo</text>
<text><location><page_11><loc_50><loc_19><loc_89><loc_33></location>center, flexion can resolve cluster substructures which are otherwise unobservable from the strong and weak lensing data alone. Though the specific approach to appropriately identify an individual subhalo in these mass reconstructions has room for optimization, the aperture mass results show that the accuracy of the reconstruction is high, and that reconstructions without flexion systematically underestimate these aperture masses. This is true even away from the subhalo locations, where flexion information better constrains the overall main halo shape as well.</text>
<text><location><page_11><loc_50><loc_3><loc_89><loc_12></location>The three-dimensional halo mass of the substructures at the margins of our detection threshold in this simulation sample have masses of 2-3 × 10 12 M glyph[circledot] within an aperture of 10 '' , though there is a dependence on the radial position of the subhalo in its detectability. This demonstrates the efficacy of flexion as a probe of small-scale galaxy cluster substructure.</text>
<figure>
<location><page_12><loc_10><loc_38><loc_85><loc_89></location>
<caption>Figure 6. As in Figure 4, for simulations 5 (top) and 6 (bottom).</caption>
</figure>
<section_header_level_1><location><page_12><loc_7><loc_32><loc_20><loc_33></location>5 DISCUSSION</section_header_level_1>
<text><location><page_12><loc_7><loc_16><loc_46><loc_30></location>We have shown that including flexion into the mass reconstruction enhances the sensitivity of the mass reconstruction to substructure a galaxy-cluster-scale lens. Significantly massive subhaloes which otherwise are undetected in data sets including only multiple image systems and ellipticity measurements are made detectable by including flexion data measured to a precision achievable using current techniques (e.g., CSB). The additional information from flexion requires no additional observational investment - single-orbit HST observations are sufficient for including flexion into the lensing analysis and the mass reconstruction for a typical cluster.</text>
<text><location><page_12><loc_7><loc_3><loc_46><loc_15></location>This result indicates that the addition of flexion into a lensing mass reconstruction better constrains the formation and structure of galaxy clusters and the subhalo mass function. The work here shows the distinct possibility that dark subhaloes in galaxy clusters of significant mass, such as those included in the simulated measurements we present here, are not detected in lensing mass maps produced without flexion. Constraints on the subhalo mass function depend on the number density and mass of the detected subhaloes within a</text>
<text><location><page_12><loc_50><loc_19><loc_89><loc_33></location>cluster, as well as a thorough understanding of the detection limits for subhaloes of a given mass. By not including flexion, the number density of detected substructures will only be a lower limit on the true substructure density. With such large structures (up to roughly 10% the mass of the main halo) it is unlikely, though not impossible, that a cluster would have that much mass in substructure (Giocoli et al. 2010). Given that substructures this large were not detectable without flexion, to constrain smaller amounts of substructure with confidence requires that flexion data be included.</text>
<text><location><page_12><loc_50><loc_3><loc_89><loc_18></location>The possibility of having significant substructure in a galaxy cluster field which is otherwise undetected by strong or weak lensing analyses also has important implications on the inferences drawn from the lensing mass distributions. For example, recent work using galaxy clusters as 'cosmic telescopes' to select and study high-redshift ( z glyph[greaterorsimilar] 7) galaxies (e.g. Hall et al. 2011; Postman et al. 2012; Zheng et al. 2012; Zitrin et al. 2012) and constraining the properties galaxy population formed at or shortly after the epoch of reionization requires an accurate magnification map of each cluster field. The overall cluster halo amplifies the effect of sub-</text>
<figure>
<location><page_13><loc_7><loc_64><loc_47><loc_89></location>
<caption>Figure 7. The input aperture mass is accurately recovered by the flexion reconstructions. Color indicate different aperture radii, with blue, green, and red for 10 '' , 15 '' , and 20 '' , respectively. The line indicates unity, not a fit to the data. Points marked with an X are measured from reconstructions without flexion, and systematically underestimate the input mass. The RMS variation of bootstrapped catalog reconstructions indicate a typical aperture mass error of 10%.</caption>
</figure>
<text><location><page_13><loc_7><loc_40><loc_46><loc_48></location>structures on the resulting magnification map, as much of the cluster is typically very near the lensing critical density. By including flexion into the reconstruction and more accurately constraining the lensing potential, the systematic error in the inferred intrinsic properties of the high-redshift galaxy population from unresolved substructures is reduced.</text>
<text><location><page_13><loc_7><loc_25><loc_46><loc_40></location>Using Simulation 1 as an example, as compared to an identical main halo without substructure, a HST/ACS field containing the substructured halo lens would probe only ∼ 88% of the solid angle that would be inferred to have been probed if the single halo were the only one reconstructed. Though the magnitude of this systematic error in each case is dependent on the exact position and mass of the substructures, it is not unreasonable to estimate that for any clusters analyzed without flexion, there is a 10-15% systematic uncertainty in the solid angle, and therefore the differential volume, probed at any given source redshift.</text>
<text><location><page_13><loc_7><loc_9><loc_46><loc_25></location>Furthermore, the intrinsic luminosity determined for any detected high-redshift objects is dependent on the local absolute-value of the magnification determined by the lens reconstruction. Again referring to the comparison between our simulated lens with substructure and the halo without substructure, the average absolute magnification across a HST/ACS field for an infinitely distant source is ∼ 50% higher for the substructured lens, which would correspond to an average shift of 0.45 magnitude. Locally the variations between the two lens models is much larger, and in some regions will magnify while in other demagnify a source if the substructures are detected.</text>
<text><location><page_13><loc_7><loc_3><loc_46><loc_8></location>The distribution of local variations in absolute magnification has a long tail corresponding to deviations in the critical curves which would most likely only affect strongly lensed/multiply imaged sources in the z glyph[greaterorsimilar] 7 population. In</text>
<text><location><page_13><loc_50><loc_54><loc_89><loc_89></location>the rest of the field, where a large number of the observed high-redshift galaxy sources are likely to be observed, the ratio of the substructured lens magnification to the singlehalo lens is well-correlated to the substructured lens magnification ( ρ ≈ 0 . 75). This means that in regions where the magnification is more enhanced by the substructure, i.e., where magnification maps without flexion will more likely be erroneous, non-detection of substructure and the systematic error in the magnification map produced by that nondetection will create a larger error in the inferred intrinsic luminosity of the object. Typical magnification ratios in regions near the substructures are | µ sub /µ single | ∼ 5, which would create a shift of ∼ 1 . 75 magnitude in the intrinsic source magnitude inferred without substructure. That said, there is a significant spread to be expected in these magnification ratios based on the image position relative to the critical curve. The Einstein radius of the substructure imbedded in the main halo in our simulations vary from about 3 '' -10 '' , meaning that individual detections can be even more than the typical factor and, for lensed image positions near the centers larger substructures the inferred magnification could in fact be lower than expected from a non-substructures lens model. This type of systematic error would significantly change the resulting luminosity function, as well as the inferred properties of the lensed sources.</text>
<text><location><page_13><loc_50><loc_27><loc_89><loc_51></location>Lensing can also determine cluster mass distributions to be compared with other mass estimators. In particular, the Sunyaev-Zel'dovich decrement and the X-ray surface brightness of a cluster, as functions of position, also constrain the mass distribution of the cluster under the assumption of hydrostatic equilibrium for the intracluster medium. Nbody simulations show that non-thermal pressure support, from cosmic rays and bulk gas motion, plays a significant role in the pressure budget of the cluster, accounting for up to half of the total pressure supporting the cluster (Nelson et al. 2014). While these effects can be accounted for statistically for large cluster samples, doing so on the level of substructures within the cluster requires additional information. Cluster mass profiles, with the substructure appropriately constrained using flexion, provide an essential test for the hydrostatic assumption, allowing the fractional non-thermal pressure support to be directly quantified for individual clusters.</text>
<text><location><page_13><loc_50><loc_3><loc_89><loc_23></location>Flexion is an important addition to the variety of lensing information available from detailed imaging data. The results presented here strongly motivate the application of this reconstruction technique to simulated data sets with more realistic mass distributions than the simple halo profiles we employ, and additionally applying it to real data. And while there remains important work to be done to make flexion as robust as the more mature lensing techniques (strong lensing and weak lensing shear), e.g. improving our measurement of intrinsic shape scatter in flexion measurements, better understanding the effects of different source selection on the flexion noise properties, etc., we have shown that this work is well worth the undertaking for the resulting enhanced accuracy of reconstructed mass and magnification maps that come from including flexion.</text>
<section_header_level_1><location><page_14><loc_7><loc_88><loc_27><loc_89></location>ACKNOWLEDGEMENTS</section_header_level_1>
<text><location><page_14><loc_7><loc_74><loc_46><loc_87></location>Based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555 and NNX08AD79G. Support for this work was provided by NASA through HST -AR-13238 and HST -AR13235 from STScI. The authors gratefully acknowledge Paul Schechter for important suggestions and mentorship leading to this work.</text>
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<section_header_level_1><location><page_14><loc_50><loc_45><loc_88><loc_48></location>APPENDIX A: EXTENDED APERTURE MASS RESULTS</section_header_level_1>
<text><location><page_14><loc_50><loc_29><loc_89><loc_44></location>Table A1 shows input and measured masses for each subhalo, along with the distance from each subhalo to the cluster halo center. We include a three-dimensional mass for the subhalo alone, as well as the mass within the aperture for the full input mass distribution, and the reconstructed mass distribution. Note that the aperture masses include contribution from both the subhalo and the main halo. Values are tabulated at different aperture radii - 10 '' , 15 '' , and 20 '' . We also include aperture mass measurements for the locations of the subhaloes in Simulation 1 as reconstructed without flexion.</text>
<section_header_level_1><location><page_15><loc_6><loc_3><loc_7><loc_89></location>T able A 1: Masses for eac h sim ulation subhalo at three differen t radii: the three dimensional input mass for the subhalo alone, along with input and measured ap erture</section_header_level_1>
<text><location><page_15><loc_7><loc_25><loc_9><loc_29></location>flexion)</text>
<text><location><page_15><loc_7><loc_21><loc_9><loc_25></location>without</text>
<text><location><page_15><loc_7><loc_15><loc_9><loc_20></location>structions</text>
<text><location><page_15><loc_7><loc_13><loc_9><loc_15></location>recon</text>
<text><location><page_15><loc_7><loc_8><loc_9><loc_12></location>denotes</text>
<text><location><page_15><loc_7><loc_7><loc_9><loc_8></location>*</text>
<text><location><page_15><loc_7><loc_3><loc_9><loc_7></location>masses.</text>
<table>
<location><page_15><loc_11><loc_3><loc_43><loc_89></location>
</table>
<figure>
<location><page_15><loc_17><loc_12><loc_42><loc_17></location>
</figure>
</document> | [{"title": "ABSTRACT", "content": "We present a reconstructions of galaxy-cluster-scale mass distributions from simulated gravitational lensing data sets including strong lensing, weak lensing shear, and measurements of quadratic image distortions - flexion. The lensing data is constructed to make a direct comparison between mass reconstructions with and without flexion. We show that in the absence of flexion measurements, significant galaxy-group scale substructure can remain undetected in the reconstructed mass profiles, and that the resulting profiles underestimate the aperture mass in the substructure regions by \u223c 25 -40%. When flexion is included, subhaloes down to a mass of \u223c 3 \u00d7 10 12 M glyph[circledot] can be detected at an angular resolution smaller than 10 '' . Aperture masses from profiles reconstructed with flexion match the input distribution values to within an error of \u223c 13%, including both statistical error and scatter. This demonstrates the important constraint that flexion measurements place on substructure in galaxy clusters and its utility for producing high-fidelity mass reconstructions. Key words: gravitational lensing: strong, gravitational lensing: weak, galaxies: clusters: general, dark matter, methods: data analysis", "pages": [1]}, {"title": "Benjamin Cain 1 glyph[star] , Maru\u02c7sa Brada\u02c7c 1 , and Rebecca Levinson 2", "content": "1 University of California Davis, Department of Physics, One Shields Ave., Davis, CA 95616 2 Massachusetts Institute of Technology, Department of Physics, 77 Massachusetts Ave., Cambridge, MA 02139 27 May 2022", "pages": [1]}, {"title": "1 INTRODUCTION", "content": "The distribution of mass structure in the Universe provides a fundamental test for any plausible cosmological model. Empirical data from multiple experiments have robustly constrained cosmology to be consistent with a flat \u039bCDM model (Hinshaw et al. 2013; Kowalski et al. 2008; Planck Collaboration et al. 2013; Eisenstein et al. 2005). But to further understand the constituents of this cosmology, in particular the dark matter, we must investigate the properties of the structures which form from the primordial mass distribution at many scales. To understand the interaction between dark matter and baryonic matter, or to understand the selfinteractions between dark matter particles we must measure the distribution of structures on relatively small scales, meaning measuring structures down to galaxy scales where the properties of both baryons and dark matter influence the evolution of structure. As interactions between dark matter and baryons are more important to the structure of haloes for more non-linear overdensities, measurements of substructures inside of larger halos (e.g., substructures within galaxy clusters) provide an important window into the nature of dark matter. For a \u039bCDM cosmology, the particular initial condi- tions (e.g., the primordial matter power spectrum) as well as the particular properties of the dark matter - both its self-interactions and its interaction with baryons, if any will affect the structures that are formed. This includes the mass function of virialized dark matter haloes (Tinker et al. 2008) and the spatial and mass distributions of subhaloes (Nagai & Kravtsov 2005; Kravtsov et al. 2004). The existence of roughly self-similar haloes at all scales is a robust prediction of \u039bCDM (Moore et al. 1999), making the detection of galaxy cluster substructures an important cosmological test. The fraction of a cluster's mass which is associated with substructures as well as the other subhalo properties (e.g., density profiles, ellipticities, etc.) can be compared to subhaloes produced in cosmological N -body simulations to further constrain dark matter properties. Recent cosmological simulations of clusters with self-interacting dark matter (SIDM) have found that dark matter interactions affect both the shape of haloes, with more spherical profiles and flatter halo cores as a result of a higher interaction cross-section for galaxy-scale haloes, as well as reducing the number of subhaloes of a given mass (Shaw et al. 2006; Rocha et al. 2013; Peter et al. 2013). To measure the mass distribution of substructures within galaxy clusters we cannot use statistical measures of cluster properties, such as empirical scaling relations (see, e.g., Giodini et al. 2013, for a review of these relations). These relationships are useful for large samples of clusters but the intrinsic scatter between individual clusters limits their predictive power. Neither can we use baryonic tracers of structure due to uncertainty in the interactions between baryons and dark matter. However, gravitational lensing, and particularly combining strong lensing deflections and linear image distortions (weak lensing shear and convergence) has been successfully shown to produce detailed mass reconstructions of galaxy clusters and resolved complicated, asymmetric morphologies (Clowe et al. 2006; Natarajan et al. 2007, 2009). For typical cluster imaging datasets, the correlation of spatially-separated galaxy shapes with uncertain redshifts produces an effective smoothing length of glyph[greaterorsimilar] 10 '' away from the central region of the cluster which is constrained by strong lensing. Simulations indicate that at least half of dark matter subhaloes will exist outside of the Einstein radius of a the main halo (Nagai & Kravtsov 2005). In this large portion of the cluster, at least 99% of the projected area within the cluster virial radius, substructures cannot be efficiently detected in galaxy clusters without additional information or the investment of prohibitively large amounts of telescope time. In order to overcome these limitations, to improve the resolution of reconstructed mass distributions, and thus to probe small-scale mass structure, we includie higher-order lensing effects. In regions where the reduced lensing shear g = \u03b3/ (1 -\u03ba ) is a significant fraction of unity, as is the case within a radius of \u223c 60 '' from the cluster center for a typical galaxy cluster, nonlinear image distortions become significant and the linear distortion approximation becomes increasingly less valid as the shear increases (see Bartelmann & Schneider 2001, for a review of linear weak lensing), which also degrades the reliability of galaxy ellipticity as an estimator for shear with standard approaches (Melchior et al. 2010). Quadratic image distortions due to nonnegligible third derivatives of the lensing potential, called flexion, can be observed and quantitatively measured in this regime. Flexion distortions cause intrinsically elliptical galaxy images to be bent by into 'banana-shaped' arclets, augmenting the image distortions due to linear shear and magnification. The two lensing fields which cause flexion distortions are sourced by the gradients of the convergence and shear fields. This has the practical effect of making flexion a sensitive probe of small-scale variations in the surface mass density. Limits on the measurable flexion also constrain the amount of possible substructure in a region where no flexion is significantly detected. In this paper we present a method for reconstructing a lensing mass distribution using multiple image positions (strong lensing), ellipticity measurements (weak lensing shear), and flexion measurements. This significantly improves the sensitivity of the reconstruction to small mass substructures; we are able to resolve structures to a significantly finer spatial resolution and a lower mass threshold than with strong lensing and shear alone. We are able to directly detect structures with mass \u223c 9 \u00d7 10 12 M glyph[circledot] which are not detected without the flexion measurements, even in the case where they are located far from the strong lensing regime. Throughout we employ a standard flat cosmological model with \u2126 m = 0 . 3, \u2126 \u039b = 0 . 7, and H 0 = 70 km/s/Mpc. We also employ standard complex notation for lensing fields, with \u2207 = \u2202 1 + i\u2202 2 being the derivative operator acting in the image plane.", "pages": [1, 2]}, {"title": "2 MASS RECONSTRUCTION METHOD", "content": "Our mass reconstruction method is an expansion of the method described in Brada\u02c7c et al. (2009), which itself was an improvement of previous methods (Brada\u02c7c et al. 2005, 2006). We review the basic concept of the method here to place our expansion in context. We include five complex lensing fields, all of which yield measurable lensing distortions which are inputs into our reconstruction. All five fields arise as linear combinations of the first through third derivatives of a single, real-valued lensing potential \u03c8 sourced by the lens mass distribution. These lensing fields are the deflection \u03b1 = \u2207 \u03c8 , the convergence \u03ba = 1 2 \u2207\u2207 \u2217 \u03c8 , the shear \u03b3 = 1 2 \u2207 2 \u03c8 , 1-flexion (comatic flexion) F = \u2207 \u03ba , and 3-flexion (trefoil flexion) G = \u2207 \u03b3 . Here we have used complex notation for both the fields and the derivative operators, and the star operator denotes the complex conjugate. The positions of the multiple images in a strong lensing system constrain the deflection, and the ellipticity of lensed images of background galaxies constrains the shear and the convergence (see Bartelmann & Schneider 2001, for a review of strong lensing and shear constraints on lens mass distributions). Both 1-flexion and 3-flexion are constrained by measurements of non-linear distortions due to gradients in the convergence and shear. Since the convergence is the surface mass density of the lens scaled to the critical lensing density, measurements of image distortions due to flexion provide an important direct probe of mass gradients, and therefore small-scale mass structures. The flexion fields can be measured using several methods available in the literature (Goldberg & Bacon 2005; Goldberg & Leonard 2007; Okura & Futamase 2009; Schneider & Er 2008). In our mass reconstruction we adopt the flexion formalism as defined by Cain et al. (2011, hereafter CSB). Our reconstruction method assumes measurements of the reduced flexion fields \u03a8 1 = F / (1 -\u03ba ) and \u03a8 3 = G / (1 -\u03ba ) from lensed source images using the Analytic Image Model (AIM) approach, rather than calculating flexion estimates from image characteristics like moments or shapelet weights. Using the estimators for each of the lensing fields obtained from analyzing cluster data images, we constrain a model lensing potential and the mass distribution which corresponds to that potential. Our model is defined as lens potential values \u03c8 k on a 'mesh' of points \u03b8 k to be constrained by the data. The value of the derivatives of the model potential at an arbitrary point \u03b8 j are calculated using a generalized finite differencing method using appropriate linear combination of nearby potential values on the mesh to give the desired derivative. The coefficients of this linear transformation are determined using a singular-value decomposition (SVD) as in Brada\u02c7c et al. (2009) to yield a system of equations relating the lensing potential values in the neighborhood of an arbitrary point in the field to the derivatives of the lensing potential: c \u00a9 0000 RAS, MNRAS 000 , 000-000 where n and m are the derivative orders for each dimension, k indexes the mesh points and a jk ( n, m ) is the SVD matrix coefficient for the weight of the potential value contribution of the k th mesh point to the n, m derivative of the potential at \u03b8 j . As the value for each lensing field is a linear combination of potential field derivatives, this scheme calculates each lensing field at any point in the field as a linear combination of model potential values from neighboring mesh points to \u03b8 j . To constrain the model at the location of a lensed image, the model value for the lensing field(s) which have estimators derived from the image can be calculated and compared to the model. We then determine the values of \u03c8 k which best fit the data by minimizing a model penalty function ( \u03c7 2 ) which includes contributions from weak lensing shear, strong lensing, and flexion to penalize models which do not produce the measured lensing distortions. The form of the strong and weak lensing penalty functions are identical to those of Brada\u02c7c et al. (2009). The value of the flexion penalty function is determined by the reduced flexion fields \u03a8 i 1 and \u03a8 i 3 (where i indexes the N F galaxies with measured flexion fields), and the model values for the flexion and convergence fields F i , G i , and \u03ba i calculated for the location of that galaxy: Here, N F is the number of galaxy images with flexion measurements. As described above, the model flexion and convergence values are determined from a linear combination of mesh potential values. The uncertainties \u03c3 F ,i and \u03c3 G ,i include both intrinsic flexion scatter and measurement error. This is the same basic form for the penalty function as described in Er et al. (2010), but we explicitly formulate it in terms of the reduced flexion estimators \u03a8 1 and \u03a8 3 from CSB. As in Brada\u02c7c et al. (2009), we include in our full penalty function regularization terms, which prevent the highlynonlinear nature of the penalty function's dependence on the mesh point potential values from driving the model to assume values wildly discrepant from the initial conditions during the reconstruction. This regularization is particularly important for mesh points in regions that are weakly constrained by the data and therefore more susceptible to numerical effects in the reconstruction. We choose to regularize only on the values of convergence and shear, meaning that we penalize model potential values at each mesh point which require large pixel-to-pixel variations in the lensing fields fields relative to a reference value. Note that this does not mean that the model cannot vary from its initial conditions! Instead, we are requiring that the deviation from the initial conditions be driven by the data, rather than numerical effects. We iterate the reconstruction process and reset the reference value to the output of the previous iteration. This causes the model to step smoothly away from the initial guess in a controlled and converging way. Without regularization, the highly nonlinear dependence of the reduced lensing fields prevents the reconstruction process from finding a solution which minimizes the penalty function. Our regularization penalty function for convergence is defined over as a sum over the N mesh mesh points located at \u03b8 k as with similar terms for each component of the shear field. The value of the regularization weights \u03b7 ( \u03ba ) , \u03b7 ( \u03b3 1 ) , and \u03b7 ( \u03b3 2 ) , are determined empirically. We test a range of values for each reconstruction to not only find an appropriate value which prevents spurious nonlinear numerical artifacts while still allowing the reconstruction to deviate from initial conditions as indicated by data, but also to ensure that any reconstruction is stable with respect to variations the regularization coefficients. We also select values which place the total regularization penalty per mesh point (as a proxy for the regularization degrees of freedom) to be approximately equal that of the penalty per degree of freedom for each data portion of the full penalty function (i.e., strong, weak, or flexion lensing).", "pages": [2, 3]}, {"title": "3 TESTING THE RECONSTRUCTION METHOD", "content": "With flexion included into the mass reconstruction, we test our method using simulated measurement catalogs. We generate realistic lensing estimator values, including realistic measurement errors, from an underlying mass model which includes substructure. We then reconstruct the mass distribution using all the available data, and compare that reconstruction to a modified reconstruction which incorporates the same data for strong and weak lensing shear, but omits the flexion data. This allows us to make direct comparisons between reconstructions and evaluate the contribution that flexion makes to the fidelity of the final mass reconstructions. In particular, we are interested in finding substructure in the mass distribution which would be unobserved without the inclusion of flexion data. This latter point is important for how we construct our input simulated lensing measurements. It has already been well established that strong lensing is a sensitive probe of substructures. The goal of the mass reconstructions using our simulated measurement catalogs is to test whether realistic substructures in cluster lenses can exist which are detectable using flexion but not without. As we will show, this is indeed the case.", "pages": [3]}, {"title": "3.1 Input Lensing Mass Distribution", "content": "Our input mass distribution is created to mimic a galaxycluster sized halo with a small number of substructures, each simulated using a nonsingular isothermal ellipse model. We set the scale of the large central halo to be comparable with a galaxy cluster velocity dispersion of approximately 1500 km/s. This is a large cluster, similar to those in the CLASH (Postman et al. 2012) and Hubble Frontier Fields initiative 1 (PI Lotz: Program ID 13495) samples. We also include two substructures in the lens within a radius of 1.5 arcminutes of the cluster center. These subhaloes are also nonsingular isothermal ellipses with varying masses from group scale down to large-galaxy scale (see Appendix A for tabulated subhalo masses). We construct six different data sets corresponding to substructured cluster haloes. The positioning of the halo mass substructures was not done in a fine-tuned way, i.e., we did not select the location or masses of the substructures in any way to enhance the effect of the substructures on the resultant lensing morphology. Instead, we use a conservative approach and restrict the possible locations such that the strong lensing morphology does not initially indicate that there is significant substructure in the lens. The intent of this work is to characterize the differential effect that flexion makes on the resulting mass distribution. Thus, we restrict the substructure halo masses and positions to be outside the approximate radius of the critical curve so that the additional substructure is not easily detected using the strong lensing data alone. Substructures which are too massive or near the critical curves create significant deviations in the location, morphology, and magnification of strong lensing images. This can be an extremely sensitive probe of substructure (e.g. Vegetti et al. 2012), however our goal is to make a total census of the substructure in galaxy clusters.", "pages": [3, 4]}, {"title": "3.2 Simulated Lensing Estimators", "content": "We simulate lensing measurement catalogs to test the mass reconstruction method and the effect of including flexion. Using the main halo and smaller substructures we define as above, we analytically calculate the appropriate lensing field estimators from the total mass profile using the full lens equation and its derivatives. We assume a single, moderate redshift of z lens = 0 . 3 for the lens halo (including its substructures) in all of the simulated datasets. Independent random noise is added to each simulated measurement with scatter chosen to match observational constraints from single orbit Hubble Space Telescope Advanced Camera for Surveys ( HST /ACS) cluster observations. For the simulations presented here, we employ an input lensing mass distribution consisting of three non-singular isothermal ellipse (NIE) halo models for computational efficiency in constructing the simulated measurements and a simple comparison model. To determine our simulated measurement values based on the lensing mass distribution, the source-lens geometry must be determined along the line of sight in addition to the plane of the sky. This is because the lensing fields are all dependent on the redshift of the lensing mass as well as the redshift of the source object. We randomly select the redshift of each lensed source from a gamma distribution with z 0 = 2 / 3. This is the same distribution as used in previous work to emulate the true distribution of sources (Brainerd et al. 1996; Schrabback et al. 2007; Brada\u02c7c et al. 2004). We draw a new random redshift for each flexion and weak lensing source image, and each strong lensing image system has its own redshift as well. Redshift errors on the weak lensing/flexion sources are added with \u03c3 z = 0 . 05(1+ z ). The strong lensing systems do not have redshift errors, to reflect the spectroscopic redshifts which are available for many multiply imaged galaxies. We describe the measure- nt generation procedure for each type of catalog (flexion, weak lensing, and strong lensing) below. To generate our flexion measurement catalog we select a random distribution of source positions for the flexion and leak lensing shear measurements. The positions are selected uniformly across a square field 3 . ' 5 on a side to a density of 80 galaxies/arcmin 2 . For the flexion measurements we calculate the reduced flexion fields \u03a8 1 and \u03a8 3 as defined in CSB at the position of each of the sources. We add Gaussian noise to each component of the flexion measurements with a standard deviation of 0.03 arcsec -1 . These noise levels are motivated by previous studies of intrinsic flexion (Goldberg & Leonard 2007, CSB). We then remove from the flexion dataset any sources where the magnitude of the measured flexion exceeds 1.0 arcsec -1 . This ceiling on the measured flexion reflects how the quadratic expansion of the lens equation which defines the flexion fields is only valid when the flexion correction is small relative to the shear distortion. In a real dataset, images with flexion values in these regimes would either be identifiable as strongly-lensed arcs and thus in the strong lensing dataset instead of the flexion catalog, or would be removed during image fitting as in CSB. There the authors found that for flexion values this large the AIM image fitting did not produce accurate results, but the properties of the AIM covariance matrices allowed for efficient removal of these catastrophic outlier images. Our limit on the flexion values is a uniform approximation of the limit that would occur in real data: the directly observable distortion in a lensed image due to flexion is quantified by the unitless product of the image scale (such as the half-light radius) and the reduced flexion field, rather than the more physically relevant reduced flexion field alone. Because flexion is a quadratic effect, measuring flexion requires that the lensed galaxy image be larger than is required for ellipticity - in practice, this means 2-3 times the point-spread function size along the long axis of the image. In order to fully include varying image size and its effect on flexion estimation and the resulting mass reconstruction, a better understanding of intrinsic flexion and its correlation to galaxy size and magnitude must be achieved. However, the simpler cut on the value of the flexion field is a sufficient approximation for our purpose. The weak lensing shear and strong lensing datasets are constructed from the reduced shear and deflections fields. The source locations for the shear are randomly distributed through the simulated field and locations with reduced shears nearing unity are removed (for the same reason as large flexion values are removed). The selection of the source position for the strong lensing systems is not completely random, however. For each system we initially draw a random position from a small field in the source plane centered on the main halo core and solve the lens equation to find the image positions. We then perform a visual inspection and remove systems if they show obvious signs of substructure. With some source positions relative to the distribution of mass in the lens, multiple image systems will form around the smaller substructure. We reject these systems, as the utility of strong lensing for substructure detection is already well established. Instead, we select only image systems that are representative of a single, central halo and do not have obvious substructure included.", "pages": [4]}, {"title": "3.3 Initial Conditions and Regularization", "content": "Our approach to reconstructing the mass distribution from the simulated measurements is the same approach we would take using real data. We determine initial conditions for our model fitting, set a mesh of points where the model potential will be defined, and reconstruct the lensing potential all using data-driven conditions. We perform several reconstructions with different values for some of the lessstrongly-constrained input parameters (e.g., the regularization weights). We evaluate the output reconstructions, and in particular we evaluate the behavior of the penalty function values as the reconstruction converges, to determine the best values for these parameters. We do not see a strong dependence on the particular values - small changes do not significantly affect the output reconstruction. The first step in our reconstruction is to assign initial conditions for the reconstruction which are motivated by the data. We use an analytic model to set the initial model lensing potential values which, in turn, determine the initial value of the penalty function and its gradient. As was shown previously (Brada\u02c7c et al. 2004; Brada\u02c7c et al. 2005), the final reconstruction using this method is not strongly dependent the particular initial model used provided that the regularization is chosen appropriately. We employ a simple, parametric, axisymmetric model as our initial guess. We take a single, nonsingular isothermal sphere as our initial model, locating the center of the NIS at the average position of the images in the strong lensing data set. The mean of the distances from these strong lensing images from their average position defines an approximate Einstein radius for the overall lensing system. Using the approximated Einstein radius estimate we construct our initial model from a NIS profile. These simple, reasonable approximations provide datadriven initial conditions. Another important input parameter to the mass reconstruction is the weight of each component of the regularization in the penalty function ( \u03b7 \u03ba , etc., from Eq. 3), which defines the regularization weight relative to the strong lensing, weak lensing shear, and flexion contributions. As noted in \u00a7 2, too little weight given to regularization (small \u03b7 values) yields reconstructions with unrealistically large amounts of small-scale spatial variation as well as aphysical lensing potential values, such as negative convergence. These numerical artifacts of the nonlinear lens inversion prevent the optimization algorithm from finding the appropriate minimum as it iterates. Excessive weight to the regularization (large \u03b7 ) causes the reconstruction to simply return a model which is minimally different from the initial mass model, preventing the data from informing the reconstructed mass distribution. Because it is not possible to determine the most appropriate \u03b7 values a priori , and because the most appropriate specific value for the regularization weight will depend not only on the details of the data available but also on the particular mesh geometry chosen, we determine \u03b7 empirically by reconstructing with a range of values. By comparing the \u03c7 2 values for each part of the penalty function, we are able to separate successful reconstructions from those which are over- or under-regularized. By evaluating the \u03c7 2 values output by these reconstructions we determine the approximate best combination of values for \u03c3 NIS and regularization weight to use for the given dataset. To compare different reconstructions, we consider both the total \u03c7 2 value, as an indicator of the overall fitness of the reconstruction, and the individual contributions of the different datasets. Reconstructions which yield values of \u03c7 2 / d.o.f. \u2248 1 for each of the three datasets are preferred, even in cases where that increases both the overall and regularization \u03c7 2 values. Furthermore, we prefer models which reproduce the strong lensing data more precisely above those which minimize the weak lensing or flexion penalty functions, provided that there are no obvious signs of significant pixel to pixel variations. This is because strong lensing data have very small measurement errors, in fact smaller than the error we use in the strong lensing penalty function which we inflate in order to more easily locate the penalty function minimum. Thus if the strong lensing image systems are robustly identified in a given cluster, we can reasonably require that the solution have a strong lensing penalty value of effectively zero. The reconstruction which meets these criteria while minimizing the overall \u03c7 2 value is the reconstruction which we select as our solution. In the presence of a significantly substructured mass distribution, the reconstruction will not converge with a single halo initial model, and will plateau to \u03c7 2 values of \u223c 5-10 (or more) per degree of freedom after several iterations. This is an indication of an initial guess which is far from the solution, and the highly non-linear nature of the reconstruction process requires and update to the initial conditions based off this information. We update the initial conditions iteratively, occasionally adding additional haloes at the location of and sizes of substructures detected in the non-converged reconstruction in order to arrive at a good fit. See \u00a7 4 for a discussion of our substructure detection method.", "pages": [5]}, {"title": "3.3.1 Reconstruction Model Mesh", "content": "We distribute mesh points over the 'observed field', which in this case is the full area around the mean of the strong lensing image locations in a square region 3 . ' 5 on a side (mimicking the HST/ACS field of view). These mesh points are where the model lensing potential and its derived lensing fields are defined. The 'base' mesh is defined by a rectangular, regular grid of points evenly defined across our field. We refine this grid in a circular region centered on the field center-point. The radius of this circular region is 1.5 times the mean distance of the strong lensing images from the average strong lensing image location. Within this radius, we distribute mesh points in a rectilinear grid at twice the base density. We further refine in regions within a 3 '' radius of a strong lensing image position with four times the base density. This refinement is necessary to accurately constrain the source positions for the strong lensing systems. Numerical error in the calculation of the gradient of the model lensing potential due to a paucity of mesh points near strong lensing images can dominate the strong lensing contribution to the penalty function, which will preclude convergence and accurate reconstruction. This nested mesh structure places increasing numbers of mesh points where the mass distribution is expected to be varying more rapidly, as well as where we have tighter constraints from the lensing data, and therefore a higher density of mesh points for the finite difference derivatives used to calculate the model lensing field values enhances the accuracy of the reconstruction output. In the reconstructions presented here, we begin with a base mesh density of roughly 70 mesh points/arcmin 2 , corresponding to about one galaxy image per mesh point, with the density of mesh points higher in the refinement regions. The optimal configuration will have increased mesh point density wherever the potential is changing rapidly, and will not have any changes in refinement level of more than a single step.", "pages": [5, 6]}, {"title": "4 RESULTS", "content": "Using the methods described above, we reconstruct the mass distribution for each of the simulated cluster datasets, and bootstrap the simulation 1 catalogs and reconstruct each of those. Here we present our results, first in a detailed investigation of simulation 1 and the statistics of the reconstructions after bootstrap-resampling of the lensing catalogs, then more broadly for the rest of the cluster data sets. In discussing these results, we will refer to locations within the reconstructed field by celestial directions (north being up and east being to the left), as we would were these actual cluster fields and not arbitrarily oriented simulations.", "pages": [6]}, {"title": "4.1 Simulation 1", "content": "Figure 1 shows a comparison of the input mass distribution, the reconstruction without flexion, and the reconstruction with flexion. In both the without-flexion reconstruction and in the with-flexion case, the strong lensing data is extremely well fit by the reconstruction, essentially perfectly reproducing the multiple image positions for each of the sources. The final weak lensing penalty function values in each reconstruction are very similar as well. Both reconstructions fit the strong and weak lensing data equally well within statistical uncertainty (we discuss the statistics of these reconstructions below). However, the reconstruction with flexion includes additional substructure undetected in the flexion-less reconstruction. One of the two subhaloes (to the northeast) is clearly detected by eye, and the second is apparent as an extension to the southwest. To quantify substructure detections, we employ Source Extractor (SExtractor, Bertin & Arnouts 1996) to detect 'objects' in our mass maps. This is similar in some ways to detecting objects in a crowded field, though with an unusually fine pixel scale. We require a large minimum area (an area of \u223c 200 arcsec 2 ) above the detection threshold to identify an 'object'. This prevents SExtractor from identifying the small, local variations in the convergence map due to the mesh point spacings as individual objects. Because cluster subhaloes are not as discrete as galaxies in a typical field and instead are a contiguous distribution of matter with significant blending between haloes, and because estimating the 'background' convergence level in the environment of a parent halo is problematic, we do not discount morphological evidence for substructure in our reconstructions. While the southern substructure is not individually detected by SExtractor, it instead appears as an elongation of the source in that direction. Combining both the SExtractor detections and the visible morphology we conclude a strong detection of the northern subhalo and a likely detection of the second, though it blends partially into the main halo.", "pages": [6, 7]}, {"title": "4.2 Bootstrap Error Estimation", "content": "We estimate the uncertainty in our mass reconstructions via a bootstrap resampling of the weak lensing shear and flexion catalogs. We randomly select measurements from the two catalogs uniformly, allowing entries to be selected multiple times, to form a new pair of resampled catalogs. From these catalogs we perform a mass reconstruction using the same initial conditions model and reconstruction parameters as used in the second step reconstruction of the original dataset. We repeat this process 100 times, each with a different realization of the resampled catalog, to generate a distribution of mass reconstructions which we then compare to the input convergence map. We hold the strong lensing sources constant in the bootstrapping process. This is acceptable given that the identification of multiple images in real data allow us to more securely identify the real systems that the simulated strong lensing catalogs represent and the likelihood of mistakenly including interloper galaxies which are not strongly lensed is relatively low. The bootstrap reconstructions produce a large datacube of information, with a distribution of model values for each of the lensing fields: \u03c8 , \u03b1 i , \u03ba , \u03b3 i , F i , and G i . Ten in all, eleven if the magnification \u00b5 is included. To characterize these distributions in a comprehensible way, we produce an average convergence map and the RMS variation of the convergence in the 100 reconstructions, which is displayed in Figure 2. These maps show that the substructure detected with flexion is significant relative to the scale of statistical variations in the catalog data. We also map both the RMS variation of the bootstrapped convergence maps relative to the primary reconstruction, and also the difference between the primary and average convergence maps, presented in Figure 3. These give us a metric of bias in the primary reconstruction, i.e., how much of an outlier is the primary reconstruction. Were we to see a large amount of bias in these maps, meaning a large difference between the RMS variation about the average and the RMS variation about the primary, or were we to see a large difference between the primary and the average convergence maps, this would suggest that the primary reconstruction were a statistical outlier rather than a robust minimum solution to reconstructing the mass distribution. We do not see any evidence that this is the case. We see the expected small number of outliers among the reconstructions, but overall the bootstrapped catalog reconstructions should not vary drastically from the primary reconstruction if they are valid estimators of the error in the reconstruction. A few reconstructions ( glyph[lessorsimilar] 5) have hugely discrepant penalty function values, universally due to a poor fit to the the strong lensing data and thus an incorrectly located critical curve. These outliers are easily removed and discarded from our statistical analyses, though leaving them in does not significantly affect the results as the lensing field maps, such as the convergence, are not catastrophically deviant from the primary solution even with these outlier so- lutions since they depend less on the exact position of the critical curve. The reconstruction method is computationally intensive, requiring the inversion of large sparse matrices used for the finite-differences calculation of the gradient of the penalty function with respect to the lensing potential values on the mesh, so we perform this bootstrap analysis on only one of our simulations (Simulation 1), both with and without flexion measurements included, to estimate the amount of variation to expect due to the statistical uncertainty in our lensing field estimator measurements. As with real data, the variation depends not only on the accuracy and density of the weak lensing and flexion estimators across the field, but also on the amount of specific constraint provided by the particular set of strong lensing images available. The number and, importantly, the location of the multiple images have an effect on the spatially varying amount of freedom that the mass model can change within the statistical errors of the measurements. A more full statistical analysis of the cluster lensing mass reconstruction made including flexion allowing a broader variation of possible halo and subhalo configurations is a large undertaking, and beyond the scope of this work. However we can use our bootstraps to begin to estimate the errors in our mass maps, and compare the difference between the input mass distribution and the reconstructed mass map to these error estimates. Using the bootstrap-resampled catalog reconstructions, we can understand the statistical variation in our mass maps due to the catalog sampling. One immediate feature to note in the non-flexion RMS map (Figure 2) is that the variation is small and is relatively uniform across the field. This means that despite the catalog resampling, a consistent mass distribution is preferred by the data. The input subhaloes are larger than any of the statistical variations by quite a large margin, the non-detection of the subhaloes is not due to noise or a statistical fluctuation. Instead, it is due to incomplete information in the lensing inputs, namely the lack of flexion information measuring the mass and shear gradients. The RMS map for the flexion reconstruction is less uniform than the non-flexion RMS map, which can be understood considering how flexion measurements inform the mass reconstruction. Because the flexion field is sourced very locally by the gradient of the mass distribution, and since the flexion measurement is not dominated by statistical noise in the same way that shear measurements are, excluding certain individual flexion measurements from the catalog can have a more significant effect on the resulting mass map than happens with weak lensing shear measurements that, by their nature, must be correlated together to extract the lensing signal. Thus we expect that bootstrapping the flexion measurement catalogs in addition to the shear catalogs will increase the RMS values near where the flexion measurements are most informative and constraining. This includes the locations of substructures, as can be seen in the RMS maps in Figures 2-3, but also where the flexion field is strong but reliable measurements are relatively sparse (near the main halo center) as well as where flexion measurements are particularly near a strong lensing image. The RMS map shows that neither substructure detection can be attributed to the statistics of the flexion information, with one certain detection (north) and a second, less significant detection (south). As noted above in comparing the average reconstruction to the primary reconstruction, both with and without flexion, we see no significant overall bias (Figure 3). For reconstructions with flexion, the average reconstruction underestimates the mass in the substructures as is expected since the bootstrap reconstructions include catalogs where some of the flexion measurements nearest the substructures are not included which will reduce the lensing signal from that substructure. The features in these maps are what would be expected given the information available (and not available) to each reconstruction. These results, the primary mass reconstruction together with the properties of the reconstructions from bootstrap resampled catalogs, show that the inclusion of flexion measurements can detect substructures which are otherwise undetectable in strong lensing and weak lensing shear data alone. Furthermore, these detections are statistically significant and robust to the variations of bootstrapped catalogs, i.e., they do not depend solely on a few fortuitously-located images, though there is a more marked dependence of the resulting mass map on the location of the flexion measurements relative to any substructures than is seen for weak lensing shear measurements.", "pages": [7, 8]}, {"title": "4.3 Simulations 2 through 6", "content": "Figures 4, 5, and 6 each show comparisons between the reconstructed mass distributions (on the left) and the input mass distribution (right) the eight simulations. These, along with Table A1 show the varying substructure sensitivity enhancement that including flexion yields. Simulations 1 and 2 each have two equal mass subhaloes, though the positions differ between the simulations. For simulations 3-6, we vary the masses of the subhaloes. A few trends emerge from these simulations. As would be expected, more massive substructures are reliably de- tected, particularly if they are well separated from the core of the main halo. The one instance where a larger subhalo is not detected (simulation 7) is one where the subhalo is blended into the main halo. In many configurations, this type of subhalo (in terms of mass and location) would be detectable from strong lensing data alone, as we know from the set of halo+substructure configurations we generated but rejected for this study due to their obviously substructured strong lensing observations. For smaller substructures, the detectability decreased with distance from the main halo. This is also expected. The value of the non-reduced flexion fields (i.e., F and G ) is dominated by the gradients in the convergence and shear field whose distortions depend on the proximity to the subhalo mass distribution. The total convergence (which factors into the reduced flexion) is dominated by the main halo mass. At larger radii from the main halo center, the total convergence falls away from unity, the flexion decreases quickly away from the subhalo, and both effects cause the measur- able reduced flexion signal to decrease. The least massive subhalo we detect (in simulation 4), though we only detect it marginally, was located quite close to the main halo center where the total convergence is much closer to unity. More radially distant subhalos were either undetected or more massive. As we discuss in detail in \u00a7 4.4, the inclusion of flexion does more than detect substructure and improve the reconstructed ellipticity. Flexion also improves the overall fidelity of the mass reconstruction independent of the detectability of individual structures.", "pages": [8, 9]}, {"title": "4.4 Aperture Masses", "content": "In addition to detestability of substructure in terms of identifying individual haloes, we can also quantify the fidelity of our mass reconstructions in terms of how how well the mass maps compare to the input mass distribution by comparing aperture masses. Figure 7 plots these results. We compare aperture masses rather than point-by-point masses since we expect relationship between the input and reconstructed mass distribution to not only include noise propagated from the lensing field estimators, but also an effective convolution kernel that will vary somewhat across the field based on the density of lensing field measurements and the type of measurements in that region (i.e., strong lensing, shear, or flexion). There is a moderate tightening of the scatter (in dex) as the aperture radius increases, which is consistent with the idea of the reconstruction method imposing and effective convolution kernel just as larger apertures in photometry will omit fewer photons dispersed by a non-trivial pointspread function. Though the measurements for a single halo location with multiple aperture radii are certainly correlated, there is similarly good agreement between the input mass distribution and the reconstructed masses in each of these apertures, confirming that we are in fact reconstructing the substructures on these small scales, as the fraction of the mass enclosed due to the substructure increases as the aperture shrinks. The measurements scatter about the unity line by 13% for reconstructions with flexion, which together with the RMS scatter between the different bootstrapped catalog realizations which have an RMS variation of \u223c 10% suggest an intrinsic scatter of approximately 8%. Without flexion, on the other hand, the reconstructed aperture masses are systematically biased low at the location of the substructures, as we expect given that the substructures are not detected without flexion, by 25-40%. This underestimate is also seen in aperture measurements of the main halo masses, showing that the improvement from adding flexion is not only a substructure phenomenon.", "pages": [9, 10]}, {"title": "4.5 Simulation Summary", "content": "We see from these simulations the feasibility for detecting and identifying substructures depends both on their mass and position relative to the cluster halo. We do not have many detections of substructure at small radii because we select 'un-substructured' strong lensing image configurations and are therefore artificially enhancing the likelihood that substructure near the Einstein radius of the cluster will remain undetected. More centrally located subhaloes are also less likely to be detected because of the reduced number of flexion measurements - again, these substructures are those that would typically be visible in the strong lensing data but do not appear in our data because we select against the configurations that make them apparent. However, independent of subhalo identification, the masses we measure in apertures as small as 10 '' are accurate to an error (scatter plus statistical) of 13% across the reconstructed field. This is much smaller than the observed systematic offset from not including flexion in the reconstruction. Flexion data included into the mass reconstructions fill an important information gap in the full lensing data set. With a mass limit that depends both on the angular size of the subhalo Einstein radius (which is a proxy for the subhalo mass) and its distance from the main cluster halo center, flexion can resolve cluster substructures which are otherwise unobservable from the strong and weak lensing data alone. Though the specific approach to appropriately identify an individual subhalo in these mass reconstructions has room for optimization, the aperture mass results show that the accuracy of the reconstruction is high, and that reconstructions without flexion systematically underestimate these aperture masses. This is true even away from the subhalo locations, where flexion information better constrains the overall main halo shape as well. The three-dimensional halo mass of the substructures at the margins of our detection threshold in this simulation sample have masses of 2-3 \u00d7 10 12 M glyph[circledot] within an aperture of 10 '' , though there is a dependence on the radial position of the subhalo in its detectability. This demonstrates the efficacy of flexion as a probe of small-scale galaxy cluster substructure.", "pages": [10, 11]}, {"title": "5 DISCUSSION", "content": "We have shown that including flexion into the mass reconstruction enhances the sensitivity of the mass reconstruction to substructure a galaxy-cluster-scale lens. Significantly massive subhaloes which otherwise are undetected in data sets including only multiple image systems and ellipticity measurements are made detectable by including flexion data measured to a precision achievable using current techniques (e.g., CSB). The additional information from flexion requires no additional observational investment - single-orbit HST observations are sufficient for including flexion into the lensing analysis and the mass reconstruction for a typical cluster. This result indicates that the addition of flexion into a lensing mass reconstruction better constrains the formation and structure of galaxy clusters and the subhalo mass function. The work here shows the distinct possibility that dark subhaloes in galaxy clusters of significant mass, such as those included in the simulated measurements we present here, are not detected in lensing mass maps produced without flexion. Constraints on the subhalo mass function depend on the number density and mass of the detected subhaloes within a cluster, as well as a thorough understanding of the detection limits for subhaloes of a given mass. By not including flexion, the number density of detected substructures will only be a lower limit on the true substructure density. With such large structures (up to roughly 10% the mass of the main halo) it is unlikely, though not impossible, that a cluster would have that much mass in substructure (Giocoli et al. 2010). Given that substructures this large were not detectable without flexion, to constrain smaller amounts of substructure with confidence requires that flexion data be included. The possibility of having significant substructure in a galaxy cluster field which is otherwise undetected by strong or weak lensing analyses also has important implications on the inferences drawn from the lensing mass distributions. For example, recent work using galaxy clusters as 'cosmic telescopes' to select and study high-redshift ( z glyph[greaterorsimilar] 7) galaxies (e.g. Hall et al. 2011; Postman et al. 2012; Zheng et al. 2012; Zitrin et al. 2012) and constraining the properties galaxy population formed at or shortly after the epoch of reionization requires an accurate magnification map of each cluster field. The overall cluster halo amplifies the effect of sub- structures on the resulting magnification map, as much of the cluster is typically very near the lensing critical density. By including flexion into the reconstruction and more accurately constraining the lensing potential, the systematic error in the inferred intrinsic properties of the high-redshift galaxy population from unresolved substructures is reduced. Using Simulation 1 as an example, as compared to an identical main halo without substructure, a HST/ACS field containing the substructured halo lens would probe only \u223c 88% of the solid angle that would be inferred to have been probed if the single halo were the only one reconstructed. Though the magnitude of this systematic error in each case is dependent on the exact position and mass of the substructures, it is not unreasonable to estimate that for any clusters analyzed without flexion, there is a 10-15% systematic uncertainty in the solid angle, and therefore the differential volume, probed at any given source redshift. Furthermore, the intrinsic luminosity determined for any detected high-redshift objects is dependent on the local absolute-value of the magnification determined by the lens reconstruction. Again referring to the comparison between our simulated lens with substructure and the halo without substructure, the average absolute magnification across a HST/ACS field for an infinitely distant source is \u223c 50% higher for the substructured lens, which would correspond to an average shift of 0.45 magnitude. Locally the variations between the two lens models is much larger, and in some regions will magnify while in other demagnify a source if the substructures are detected. The distribution of local variations in absolute magnification has a long tail corresponding to deviations in the critical curves which would most likely only affect strongly lensed/multiply imaged sources in the z glyph[greaterorsimilar] 7 population. In the rest of the field, where a large number of the observed high-redshift galaxy sources are likely to be observed, the ratio of the substructured lens magnification to the singlehalo lens is well-correlated to the substructured lens magnification ( \u03c1 \u2248 0 . 75). This means that in regions where the magnification is more enhanced by the substructure, i.e., where magnification maps without flexion will more likely be erroneous, non-detection of substructure and the systematic error in the magnification map produced by that nondetection will create a larger error in the inferred intrinsic luminosity of the object. Typical magnification ratios in regions near the substructures are | \u00b5 sub /\u00b5 single | \u223c 5, which would create a shift of \u223c 1 . 75 magnitude in the intrinsic source magnitude inferred without substructure. That said, there is a significant spread to be expected in these magnification ratios based on the image position relative to the critical curve. The Einstein radius of the substructure imbedded in the main halo in our simulations vary from about 3 '' -10 '' , meaning that individual detections can be even more than the typical factor and, for lensed image positions near the centers larger substructures the inferred magnification could in fact be lower than expected from a non-substructures lens model. This type of systematic error would significantly change the resulting luminosity function, as well as the inferred properties of the lensed sources. Lensing can also determine cluster mass distributions to be compared with other mass estimators. In particular, the Sunyaev-Zel'dovich decrement and the X-ray surface brightness of a cluster, as functions of position, also constrain the mass distribution of the cluster under the assumption of hydrostatic equilibrium for the intracluster medium. Nbody simulations show that non-thermal pressure support, from cosmic rays and bulk gas motion, plays a significant role in the pressure budget of the cluster, accounting for up to half of the total pressure supporting the cluster (Nelson et al. 2014). While these effects can be accounted for statistically for large cluster samples, doing so on the level of substructures within the cluster requires additional information. Cluster mass profiles, with the substructure appropriately constrained using flexion, provide an essential test for the hydrostatic assumption, allowing the fractional non-thermal pressure support to be directly quantified for individual clusters. Flexion is an important addition to the variety of lensing information available from detailed imaging data. The results presented here strongly motivate the application of this reconstruction technique to simulated data sets with more realistic mass distributions than the simple halo profiles we employ, and additionally applying it to real data. And while there remains important work to be done to make flexion as robust as the more mature lensing techniques (strong lensing and weak lensing shear), e.g. improving our measurement of intrinsic shape scatter in flexion measurements, better understanding the effects of different source selection on the flexion noise properties, etc., we have shown that this work is well worth the undertaking for the resulting enhanced accuracy of reconstructed mass and magnification maps that come from including flexion.", "pages": [12, 13]}, {"title": "ACKNOWLEDGEMENTS", "content": "Based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555 and NNX08AD79G. Support for this work was provided by NASA through HST -AR-13238 and HST -AR13235 from STScI. The authors gratefully acknowledge Paul Schechter for important suggestions and mentorship leading to this work.", "pages": [14]}, {"title": "REFERENCES", "content": "Bartelmann M., Schneider P., 2001, Physics Reports, 340, 291 Bertin E., Arnouts S., 1996, A&AS, 117, 393 Brada\u02c7c M., et al., 2005, Astronomy and Astrophysics, 437, 49 Brada\u02c7c M., Schneider P., Lombardi M., Erben T., 2005, A&A, 437, 39 Eisenstein D. J., et al., 2005, ApJ, 633, 560 Er X., Li G., Schneider P., 2010, arXiv.org Giocoli C., Tormen G., Sheth R. K., van den Bosch F. C., 2010, Monthly Notices of the Royal Astronomical Society, 404, 502 Giodini S., Lovisari L., Pointecouteau E., Ettori S., Reiprich T. H., Hoekstra H., 2013, Space Sci. Rev., 177, 247 Goldberg D. M., Bacon D. J., 2005, The Astrophysical Journal, 619, 741 Hall N., et al., 2011, arXiv.org Hinshaw G., et al., 2013, The Astrophysical Journal Supplement, 208, 19 Kowalski M., et al., 2008, ApJ, 686, 749 Kravtsov A. V., Berlind A. A., Wechsler R. H., Klypin A. A., Gottlober S., Allgood B., Primack J. R., 2004, ApJ, 609, 35 Moore B., Ghigna S., Governato F., Lake G., Quinn T., Stadel J., Tozzi P., 1999, ApJ, 524, L19 Nagai D., Kravtsov A. V., 2005, ApJ, 618, 557 Natarajan P., Kneib J.-P., Smail I., Treu T., Ellis R., Moran S., Limousin M., Czoske O., 2009, ApJ, 693, 970 Nelson K., Lau E. T., Nagai D., 2014, ApJ, 792, 25 Okura Y., Futamase T., 2009, The Astrophysical Journal, Rocha M., Peter A. H. G., Bullock J. S., Kaplinghat M., Garrison-Kimmel S., O\u02dcnorbe J., Moustakas L. A., 2013, Monthly Notices of the Royal Astronomical Society, 430, 81 Zheng W., et al., 2012, Nature, 489, 406 Zitrin A., et al., 2012, ApJ, 747, L9", "pages": [14]}, {"title": "APPENDIX A: EXTENDED APERTURE MASS RESULTS", "content": "Table A1 shows input and measured masses for each subhalo, along with the distance from each subhalo to the cluster halo center. We include a three-dimensional mass for the subhalo alone, as well as the mass within the aperture for the full input mass distribution, and the reconstructed mass distribution. Note that the aperture masses include contribution from both the subhalo and the main halo. Values are tabulated at different aperture radii - 10 '' , 15 '' , and 20 '' . We also include aperture mass measurements for the locations of the subhaloes in Simulation 1 as reconstructed without flexion.", "pages": [14]}, {"title": "T able A 1: Masses for eac h sim ulation subhalo at three differen t radii: the three dimensional input mass for the subhalo alone, along with input and measured ap erture", "content": "flexion) without structions recon denotes * masses.", "pages": [15]}] |
2015MNRAS.446.1895S | https://arxiv.org/pdf/1409.1220.pdf | <document>
<section_header_level_1><location><page_1><loc_8><loc_79><loc_92><loc_84></location>The cold mode: A phenomenological model for the evolution of density perturbations in the intracluster medium</section_header_level_1>
<section_header_level_1><location><page_1><loc_8><loc_74><loc_43><loc_76></location>Ashmeet Singh † , Prateek Sharma ‡</section_header_level_1>
<text><location><page_1><loc_9><loc_72><loc_9><loc_74></location>† ‡</text>
<unordered_list>
<list_item><location><page_1><loc_10><loc_73><loc_85><loc_74></location>Department of Physics, Indian Institute of Technology Roorkee, Roorkee 247667, Uttarakhand, India ([email protected])</list_item>
<list_item><location><page_1><loc_9><loc_72><loc_94><loc_73></location>Department of Physics and Joint Astronomy Program, Indian Institute of Science, Bangalore, India 560012 ([email protected])</list_item>
</unordered_list>
<text><location><page_1><loc_8><loc_68><loc_16><loc_69></location>1 March 2022</text>
<section_header_level_1><location><page_1><loc_30><loc_64><loc_39><loc_65></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_30><loc_42><loc_92><loc_64></location>Cool cluster cores are in global thermal equilibrium but are locally thermally unstable. We study a nonlinear phenomenological model for the evolution of density perturbations in the ICM due to local thermal instability and gravity. We have analyzed and extended a model for the evolution of an over-dense blob in the ICM. We find two regimes in which the over-dense blobs can cool to thermally stable low temperatures. One for large t cool /t ff ( t cool is the cooling time and t ff is the free fall time), where a large initial over-density is required for thermal runaway to occur; this is the regime which was previously analyzed in detail. We discover a second regime for t cool /t ff ∼ < 1 (in agreement with Cartesian simulations of local thermal instability in an external gravitational field), where runaway cooling happens for arbitrarily small amplitudes. Numerical simulations have shown that cold gas condenses out more easily in a spherical geometry. We extend the analysis to include geometrical compression in weakly stratified atmospheres such as the ICM. With a single parameter, analogous to the mixing length, we are able to reproduce the results from numerical simulations; namely, small density perturbations lead to the condensation of extended cold filaments only if t cool /t ff ∼ < 10.</text>
<text><location><page_1><loc_30><loc_39><loc_92><loc_42></location>Key words: galaxies: clusters: intracluster medium; galaxies: haloes; hydrodynamics; instabilities</text>
<section_header_level_1><location><page_1><loc_8><loc_33><loc_25><loc_34></location>1 INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_8><loc_7><loc_48><loc_32></location>The intracluster medium (ICM) of galaxy clusters is composed of ionized plasma at the virial temperature ( ∼ 10 7 -10 8 K). The ICM is heated due to the release of gravitational energy which is converted into thermal energy at the accretion shock. Given the ICM temperature (Mohr 1999), the lighter elements like H and He are fully ionized. The typical density in cluster cores lines in the range of 0 . 001 -0 . 1 cm -3 (e.g., Cavagnolo et al. 2009). The hot ionized ICM emits X-Rays due to free-free and bound-free/bound-bound emission, which is enhanced due to the high metallicity ( ∼ 0 . 3 solar) of the ICM (Rybicki, G. B. and Lightman, A. P. 2004). The cooling loss rate per unit volume is proportional to the square of the particle number density. Since the inner ICM is denser than the outer regions, the gas near the center is expected to cool faster than the gas in the outskirts. This relatively fast cooling of the gas near the center of the cluster should reduce the thermal support it provides to the overlying layers. Hence, the dilute outer layers are expected to flow into the inner cooling</text>
<text><location><page_1><loc_52><loc_30><loc_92><loc_34></location>region, resulting in a subsonic flow of the ICM plasma called the cooling flow. The cooling flow, being subsonic, maintains quasi-hydrostatic equilibrium in the ICM (Fabian 1994).</text>
<text><location><page_1><loc_52><loc_12><loc_92><loc_29></location>Observations of galaxy clusters by X-Ray satellites such as Chandra and XMM-Newton (e.g., Peterson et al. 2003) show a dramatic lack of cooling. Similarly, signs of cold gas and star-formation are missing (e.g., Edge 2001; O'Dea et al. 2008), contrary to the predictions of the cooling flow model. It is therefore indicative, that some form of heating in the ICM balances cooling and explains the lack of a cooling flow. The ICM, thus, stays in a state of global thermal equilibrium where heating, on average, balances radiative cooling. Of the various possibilities discussed, heating due to mechanical energy injection by jets and bubbles driven by the central Active Galactic Nuclei (AGN) appears to be the most promising (see McNamara & Nulsen 2007 for a review).</text>
<text><location><page_1><loc_52><loc_7><loc_92><loc_11></location>Pizzolato & Soker (2005), and later Pizzolato & Soker (2010), have argued that over-dense blobs of gas cool faster than the rest of the ICM. These fast-cooling blobs become</text>
<text><location><page_2><loc_8><loc_75><loc_48><loc_86></location>heavier than the background ICM, sink and feed the central Active Galactic Nucleus (AGN). This cold feedback from nonlinear perturbations triggers winds and/or jets from the AGN which heat the ICM and maintain the global thermal equilibrium, as mentioned earlier. The mechanical energy input creates density perturbations in the ICM, which if the ICM is sufficiently dense, can lead to the next episode of multiphase cooling and feedback heating.</text>
<text><location><page_2><loc_8><loc_60><loc_48><loc_75></location>While Pizzolato & Soker 2005 focus on the evolution of density perturbations, the aim of Pizzolato & Soker 2010 is to show that the cold blobs can lose angular momentum sufficiently fast due to drag and cloud-cloud collisions. In the current paper we are not concerned about the angular momentum problem but about the condensation of cold gas from almost spherical, non-rotating ICM. Later works, with various degrees of realism (e.g., Sharma et al. 2012b; Gaspari, Ruszkowski & Sharma 2012; Li & Bryan 2013), have vindicated this basic picture in which condensation and accretion of cold gas plays a key role in closing the feedback loop.</text>
<text><location><page_2><loc_8><loc_30><loc_48><loc_60></location>Pizzolato & Soker 2005 (hereafter PS05) consider a Cartesian setup, not accounting for geometric compression due to radial gravity. We expect geometrical compression to be important if over-dense blobs travel a distance comparable to the location of their birth. The differences between plane-parallel and spherical geometries have been highlighted by McCourt et al. 2012 and Sharma et al. 2012b. 1 They show that the cold blobs seeded by small perturbations saturate at large densities (and thermally stable temperatures) if t cool /t ff ∼ < 1 in a plane-parallel atmosphere and if t cool /t ff ∼ < 10 in a spherical atmosphere; i.e., cold gas condenses more easily in a spherical atmosphere. The ratio t cool /t ff of the background ICM (more precisely, t TI /t ff , the ratio of thermal instability timescale and the free-fall time) is an important parameter governing the evolution of small density perturbations. For small t cool /t ff the cooling time is short and a slightly over-dense blob cools to the stable temperature and falls in ballistically. On the other hand, for large t cool /t ff , the over-dense blob responds to gravity as it is cooling; the shear generated due to the motion of the over-dense blob relative to the ICM leads to mixing of the cooling blob and it cannot cool to the stable temperature.</text>
<text><location><page_2><loc_8><loc_16><loc_48><loc_30></location>In this paper we extend PS05's analysis to account for spherical compression as the blob travels large distances. In order that spherical compression does not over-compress blobs, we also include a model to account for the mixing of blobs. With a single adjustable parameter, we are able to reproduce the results of Sharma et al. (2012b) - that cold gas condenses with tiny perturbations if t cool /t ff ∼ < 10 - for a wide range of cluster parameters. In addition to this, we also highlight that there are two regimes for the evolution of over-dense blobs, in both Cartesian and spherical geometries: for t cool /t ff</text>
<text><location><page_2><loc_8><loc_7><loc_48><loc_13></location>1 The plane-parallel and spherical atmospheres are physically distinct; they are not the same initial conditions solved using different coordinate systems. In a plane-parallel atmosphere gravitational field vectors are everywhere parallel but in a spherical atmosphere the gravitational field points toward a single point.</text>
<text><location><page_2><loc_52><loc_73><loc_92><loc_86></location>smaller than a critical value ( ≈ 1 in a plane-parallel and ≈ 10 in a spherical atmosphere) the over-dense blobs cool in a runaway fashion; for t cool /t ff greater than the critical value a finite amplitude is needed (a larger amplitude is required for a higher t cool /t ff ) for runaway cooling. PS05 have highlighted the second point and have argued that nonlinear amplitudes are required for producing cold gas in cluster cores. However, as McCourt et al. (2012); Sharma et al. (2012b) show, with spherical compression even small amplitude perturbations should give cold gas in cool-core clusters.</text>
<text><location><page_2><loc_52><loc_53><loc_92><loc_72></location>We emphasize that our paper presents a simple analytic model for the evolution of density perturbations in the ICM. We are interested in the qualitative physics of the evolution of over-dense blobs in cluster cores and in comparing analytic models with numerical simulations. Quantitative results with accurate modeling of nonlinear processes such as mixing can only be obtained via numerical simulations. Moreover, our analytic models neglect effects such as thermal conduction, cosmic rays, magnetic fields, turbulence. Examples of recent simulations of the interplay of cooling and AGN feedback, including some of these effects, are Dubois et al. (2011); Sharma et al. (2012b); Gaspari, Ruszkowski & Sharma (2012); Li & Bryan (2013); Wagh, Sharma & McCourt (2014); Banerjee & Sharma (2014).</text>
<text><location><page_2><loc_52><loc_39><loc_92><loc_53></location>The paper is organized as follows. In section 2 we present the 1-D profiles of the ICM (density, temperature, etc.) in hydrostatic equilibrium with a fixed dark matter halo. In section 3 we describe the equations governing the evolution of an over-dense blob in the ICM. In section 4 we present our results, starting with our phenomenological model to account for geometrical compression. We also discuss blob evolution for different cluster and blob parameters, contrasting the evolution in Cartesian and spherical atmospheres. In section 5 we conclude with astrophysical implications.</text>
<section_header_level_1><location><page_2><loc_52><loc_33><loc_87><loc_36></location>2 1D MODEL FOR THE INTRACLUSTER MEDIUM</section_header_level_1>
<text><location><page_2><loc_52><loc_27><loc_92><loc_32></location>Hydrostatic equilibrium (HSE) in the ICM is a good approximation for the relaxed, cool-core clusters (for observational constraints on turbulent pressure support, see Churazov et al. 2008; Werner et al. 2009). Thus,</text>
<formula><location><page_2><loc_66><loc_24><loc_92><loc_26></location>dP dr = -ρ ( r ) g ( r ) , (1)</formula>
<text><location><page_2><loc_52><loc_19><loc_92><loc_23></location>where P is the gas pressure, ρ is the gas mass density and g is the acceleration due to gravity in the cluster at a distance r from the center.</text>
<text><location><page_2><loc_52><loc_12><loc_92><loc_19></location>The Navarro-Frenk-White (Navarro, Frenk & White 1996) (NFW) profile provides a good description of the dark matter (DM) distribution in relaxed halos such as galaxy clusters. We have used a spherically symmetric NFW profile for the dark matter as:</text>
<formula><location><page_2><loc_63><loc_9><loc_92><loc_12></location>ρ DM ( r ) ρ crit = ∆ c r/r s (1 + [ r/r s ] 2 ) , (2)</formula>
<text><location><page_2><loc_52><loc_7><loc_92><loc_8></location>where ∆ c and r s are the characteristic density parameter and</text>
<text><location><page_3><loc_8><loc_78><loc_48><loc_86></location>the scale radius, and ρ crit is the critical density of the universe. Generally the size of the dark matter halo is taken to be r 200 , the virial radius around the cluster center within which the average dark matter density is 200 times the critical density of the universe. The NFW parameters can be recast in terms of the concentration parameter c = r 200 /r s , where</text>
<formula><location><page_3><loc_18><loc_75><loc_48><loc_77></location>∆ c = 200 3 c 3 ln(1 + c ) -c/ (1 + c ) . (3)</formula>
<text><location><page_3><loc_8><loc_73><loc_48><loc_74></location>The corresponding acceleration due to gravity in the ICM is</text>
<formula><location><page_3><loc_19><loc_69><loc_48><loc_72></location>g NFW ( r ) = GM DM , encl ( r ) r 2 , (4)</formula>
<text><location><page_3><loc_8><loc_63><loc_48><loc_69></location>where we ignore the gravity due to baryons as their mass fraction is small and they are more diffuse compared to dark matter. The cluster mass is given by M DM , 200 , the dark matter mass enclosed within the sphere of radius r 200 .</text>
<text><location><page_3><loc_8><loc_51><loc_48><loc_63></location>The ionized plasma (most of the baryonic matter) in clusters is confined by the gravitational potential of the dark matter. To study the evolution of over-dense blobs in a generic ICM, we numerically compute the ICM profile by solving hydrostatic equilibrium in the NFW gravitational potential (Eq. 4) and by specifying the entropy profile of the ICM. The Xray entropy profile for the ICM, K (in keV cm 2 ) is defined to be the adiabatic invariant (for temperature in keV T keV , electron number density n e , adiabatic index γ = 5 3 )</text>
<formula><location><page_3><loc_24><loc_47><loc_48><loc_50></location>K = T keV n γ -1 e . (5)</formula>
<text><location><page_3><loc_8><loc_44><loc_48><loc_47></location>The entropy profile of most galaxy clusters is reasonably well fit by a model given by,</text>
<formula><location><page_3><loc_18><loc_40><loc_48><loc_43></location>K = K 0 + K 100 ( r 100kpc ) α , (6)</formula>
<text><location><page_3><loc_8><loc_34><loc_48><loc_40></location>where the parameters K 0 , K 100 and α are the parameters introduced by Cavagnolo et al. (2009). Combining this with the hydrostatic equilibrium equation yields for the total particle number density n ,</text>
<formula><location><page_3><loc_14><loc_31><loc_48><loc_34></location>dn dr = -n γ K [ µ e γ -1 m p g NFW n 1 -γ keV k B µ γ -2 + dK dr ] , (7)</formula>
<text><location><page_3><loc_8><loc_11><loc_48><loc_30></location>where keV=1 . 16 × 10 7 (the conversion of keV to Kelvin), m p is the proton mass, and µ / µ e / µ i are the mean mass per particle/electron/ion, governed by ρ = nµm p = n e µ e m p = n i µ i m p . The total number density is the sum of electron and ion number densities n = n e + n i . We choose a constant ICM metallicity corresponding to the standard value of a third of the solar value (Leccardi & Molendi 2008); thus, µ = 0 . 62 and µ e = 1 . 17. From Eq. 6, we can get dK/dr and by specifying the number density at a point in the ICM as a boundary condition, ICM profiles for pressure, density, etc. can be solved for. The entropy profile of the ICM from Eq. 6 is a monotonically increasing function of r and hence the ICM is convectively stable according to the Schwarzschild criterion for adiabatic atmospheres. 2 An adiabatic blob will oscillate</text>
<text><location><page_3><loc_8><loc_4><loc_10><loc_5></location>©</text>
<text><location><page_3><loc_9><loc_4><loc_9><loc_5></location>c</text>
<figure>
<location><page_3><loc_54><loc_63><loc_90><loc_86></location>
<caption>Figure 1. Various profiles for the fiducial cluster (see Table 1): normalized gas number density ( n/n 200 ), temperature T in keV, entropy K (keVcm 2 ) and the ratio of the cooling time and the free-fall time ( t cool /t ff ).</caption>
</figure>
<text><location><page_3><loc_52><loc_49><loc_92><loc_54></location>in the atmosphere with the local Brunt-Vaisala oscillation frequency (c.f. Eq. 19). However, as we show later, spherical compression is important for weakly stratified medium such as the ICM.</text>
<text><location><page_3><loc_52><loc_31><loc_92><loc_48></location>In Figure 1, we plot the profiles of different variables with NFW parameters c = 3 . 3 and M 200 = 5 . 24 × 10 14 M /circledot , and with entropy parameters K 0 = 8 keVcm 2 , K 100 = 80 keVcm 2 , α = 1 . 1 and n ( r 200 ) = 10 -4 cm -3 . These parameters correspond to the fiducial run listed in Table 1. Our fiducial cluster has a virial radius r 200 ≈ 1603kpc and M gas /M 200 (hot gas to DM mass ratio) = 0.18. While we use the same NFW halo parameters throughout the paper (except in Fig. 9, where we investigate the blob evolution in a 10 12 M /circledot halo), we use different entropy parameters for investigating the effects of various parameters (such as the ratio of the cooling time to the free-fall time, t cool /t ff ) on the evolution of overdense blobs.</text>
<text><location><page_3><loc_52><loc_21><loc_92><loc_30></location>The entropy parameters and density boundary conditions for the model ICMs used in our study are stated in Table 1. The entropy parameters span a range covered by cool clusters in the ACCEPT sample (Cavagnolo et al. 2009). The minimum value of the parameter t cool /t ff , which plays a very important role in governing the fate of an over-dense blob in the ICM, is also shown.</text>
<text><location><page_3><loc_55><loc_20><loc_92><loc_21></location>We use the parameters included in Table 1 for our models</text>
<text><location><page_3><loc_52><loc_7><loc_92><loc_17></location>ICM plasma with anisotropic conduction, and the ICM core is unstable to the heat-flux driven buoyancy instability (Quataert 2008). However, this instability saturates by reorienting field lines perpendicular to the radial direction and in the saturated state the response of a perturbed blob is stable, qualitatively similar to a convectively stable adiabatic atmosphere (Sharma et al. 2009). Therefore, we ignore the effects of thermal conduction in this paper.</text>
<section_header_level_1><location><page_4><loc_8><loc_89><loc_30><loc_91></location>4 A. Singh, P. Sharma</section_header_level_1>
<table>
<location><page_4><loc_9><loc_75><loc_48><loc_83></location>
<caption>Table 1. Entropy parameters and boundary density for our ICM profiles ( c = 3 . 3 , M 200 = 5 . 24 × 10 14 M /circledot )</caption>
</table>
<section_header_level_1><location><page_4><loc_8><loc_74><loc_12><loc_75></location>Notes</section_header_level_1>
<text><location><page_4><loc_8><loc_72><loc_21><loc_74></location>† The fiducial profile.</text>
<text><location><page_4><loc_8><loc_69><loc_48><loc_72></location>‡ The β parameter (c.f. Eq. 22) determined for blob-sizes of 10, 50, 100 pc to form cold gas at radii within which t cool /t ff falls below 10. All our runs use the average β value of 0.0244.</text>
<text><location><page_4><loc_8><loc_51><loc_48><loc_66></location>with spherical compression. However, as we show in section 4.2.1, the nature of the evolution of the blob over-density changes if the ratio t cool /t ff is below a critical value, for both Cartesian and spherical geometries. This critical value for Cartesian setups is 1, typically not reached by even the coolest cluster cores (see Table 1). Therefore, in order to elucidate the physics of the evolution of over-density in a Cartesian setup we artificially increase the cooling function Λ for all Cartesian runs by a factor of 10; this way we can achieve t cool /t ff ∼ < 1 required to study the runaway cooling of over-dense blobs in a plane-parallel atmosphere.</text>
<section_header_level_1><location><page_4><loc_8><loc_45><loc_43><loc_47></location>3 LOCAL THERMAL INSTABILITY IN A GLOBALLY STABLE ICM</section_header_level_1>
<text><location><page_4><loc_8><loc_41><loc_48><loc_44></location>In the ICM the rate of energy loss due to free-free/boundfree/bound-bound radiation can be modeled as</text>
<formula><location><page_4><loc_19><loc_38><loc_48><loc_40></location>P loss = dE dtdV = -n e n i Λ( T ) , (8)</formula>
<text><location><page_4><loc_8><loc_22><loc_48><loc_37></location>where n e and n i represent the electron and ion number densities and Λ( T ) is the cooling function. We use the cooling function from Tozzi & Norman 2001 based on Sutherland & Dopita 1993, as adapted by Sharma, Parrish & Quataert 2010 (solid line in their Fig. 1). In case of thermal bremsstrahlung (which dominates above 10 7 K), the cooling function Λ( T ) ∝ T 1 / 2 and hence P loss ∝ n 2 T 1 / 2 . Due to the observational lack of cooling flows, and given other hints of global thermal equilibrium in the ICM, we use the model of Sharma et al. (2012b) in which heating ( H ) balances average cooling at every radius in cluster cores,</text>
<formula><location><page_4><loc_22><loc_20><loc_48><loc_21></location>H = < n e n i Λ( T ) > . (9)</formula>
<text><location><page_4><loc_8><loc_8><loc_48><loc_19></location>While this clearly is an idealization, since AGN heating is expected to be intermittent with large spatial and temporal fluctuations around thermal equilibrium, jet simulations agree well with the simulations based on the idealized model (Eq. 9; for comparison of jet simulations with the idealized model see Gaspari, Ruszkowski & Sharma 2012; Li & Bryan 2013). More importantly for this paper, our heating term is analytically tractable and captures the basic thermal state of the ICM.</text>
<text><location><page_4><loc_11><loc_7><loc_48><loc_8></location>An important parameter, which governs the evolution of</text>
<text><location><page_4><loc_52><loc_84><loc_92><loc_86></location>linear over-dense blobs, is the ratio of the ICM cooling time t cool (more precisely, the thermal instability time),</text>
<formula><location><page_4><loc_64><loc_80><loc_92><loc_83></location>t cool = 3 2 n 0 k B T 0 n e, 0 n i, 0 Λ( T 0 ) (10)</formula>
<text><location><page_4><loc_52><loc_79><loc_68><loc_80></location>and the free fall time t ff ,</text>
<formula><location><page_4><loc_67><loc_75><loc_92><loc_78></location>t ff = ( 2 r 0 g 0 ) 1 / 2 (11)</formula>
<text><location><page_4><loc_52><loc_67><loc_92><loc_74></location>in the background ICM at the location of the blob (as highlighted by McCourt et al. 2012; Sharma et al. 2012b), where the quantities subscripted with '0' stand for their unperturbed background values at the radius under consideration. The t cool /t ff profile for the fiducial cluster is included in Figure 1.</text>
<section_header_level_1><location><page_4><loc_52><loc_64><loc_78><loc_65></location>3.1 Evolution of a spherical blob</section_header_level_1>
<text><location><page_4><loc_52><loc_49><loc_92><loc_63></location>We base our study on the phenomenological model of PS05 and Pizzolato & Soker 2010, who propose that over-dense blobs of gas in the ICM cool faster than their surroundings, become heavier and sink to the center to feed the AGN. Both these papers consider the evolution of blobs in a background cooling flow with some simplifications. In this paper we make the more realistic assumption that the core is in rough thermal balance without any equilibrium flow. This implicitly assumes that heating keeps up with otherwise catastrophic cooling.</text>
<text><location><page_4><loc_52><loc_34><loc_92><loc_49></location>We derive the conditions in which the over-dense blobs cool to the stable atomic phase ( T < 10 4 K ), leading to a multiphase core. The conditions for the formation of the cold phase depend on the various parameters of both the blob and the ICM. We show that for a large t cool /t ff , cold gas condensation requires a finite over-density. For a sufficiently small t cool /t ff , however, cold gas condenses out of the ICM for even a tiny over-density. Moreover, we extend the PS05 formalism to account for spherical compression that makes it easier for cold gas to condense out in a spherical geometry as compared to a plane-parallel atmosphere.</text>
<text><location><page_4><loc_52><loc_23><loc_92><loc_34></location>Linear thermal instability analysis has been done in the Appendix section of Sharma, Parrish & Quataert 2010 (the original reference is Field 1965) in the limit t cool /greatermuch t sound (the sound crossing time for the modes). In such a scenario, neglecting gravity, we have isobaric conditions such that the perturbation always remains in pressure equilibrium with its surroundings and the linear growth rate σ of the instability is (in absence of thermal conduction),</text>
<formula><location><page_4><loc_61><loc_19><loc_92><loc_22></location>σ = -i ω = -d ln(Λ /T 2 ) d ln T 1 γ t cool . (12)</formula>
<text><location><page_4><loc_52><loc_12><loc_92><loc_19></location>As done in PS05, we consider an over-dense spherical blob of radius a , whose parameters will be represented with primed quantities ( n ' , T ' ), having a density contrast (over-density) with respect to the ICM (unprimed quantities represent the ambient ICM):</text>
<formula><location><page_4><loc_68><loc_9><loc_92><loc_11></location>δ ≡ n ' -n n . (13)</formula>
<text><location><page_4><loc_52><loc_7><loc_92><loc_8></location>The blob is assumed to remain in pressure equilibrium at all</text>
<text><location><page_5><loc_8><loc_78><loc_48><loc_86></location>times (i.e. P ' ≡ P ) with the ICM in the isobaric limit of t cool /greatermuch t sound (the blob evolves isochorically in the opposite limit, which may happen for a short time when the blob temperature is close to the peak of the cooling function; e.g., see Burkert & Lin 2000; the isobaric assumption should not affect our results qualitatively), and thus,</text>
<formula><location><page_5><loc_24><loc_75><loc_48><loc_78></location>T ' = T 1 + δ . (14)</formula>
<text><location><page_5><loc_8><loc_72><loc_48><loc_74></location>In a one-dimensional analysis, we re-write from PS05 the basic equations governing the blob evolution:</text>
<formula><location><page_5><loc_26><loc_69><loc_48><loc_71></location>dr dt = v , (15)</formula>
<formula><location><page_5><loc_16><loc_65><loc_48><loc_67></location>dv dt = -g δ 1 + δ -sign( v ) 3 C 8 a v 2 1 + δ , (16)</formula>
<text><location><page_5><loc_8><loc_53><loc_48><loc_64></location>where r is the radial coordinate of the blob, v is the blob's radial velocity, a is the blob radius, C is the dimensionless drag coefficient, which for most part of the paper has been taken as 0.75 (see, e.g., Churazov et al. 2001), and sign( v ) ensures that the drag force always points against the velocity. The energy equation for the over-dense blob can be written, assuming global thermal balance with a heating term (Eq. 9), 3</text>
<formula><location><page_5><loc_10><loc_50><loc_48><loc_52></location>P γ -1 d dt ln [ P ( n ' ) γ ] = -n ' e n ' i Λ( T ' ) + n e n i Λ( T ) . (17)</formula>
<text><location><page_5><loc_8><loc_41><loc_48><loc_49></location>The primed term in the RHS of Eq. 17 is the cooling of the blob and the unprimed term represent the volume averaged heating of the ICM (and the blob). Following Pizzolato & Soker 2010 and substituting d/dt by vd/dr for the background quantities, this can be recast to obtain an equation for the over-density of the blob (assuming γ = 5 / 3):</text>
<formula><location><page_5><loc_9><loc_38><loc_48><loc_40></location>dδ dt = 2 (1 + δ ) 5 n k B T [ n ' e n ' i Λ( T ' ) -n e n i Λ( T )]+ (1 + δ ) g vN 2 , (18)</formula>
<text><location><page_5><loc_8><loc_33><loc_48><loc_37></location>where N 2 is the square of the Brunt Vaisala frequency and represents the linear response of an over-dense blob in a stably-stratified ICM,</text>
<formula><location><page_5><loc_20><loc_29><loc_48><loc_32></location>N 2 = g γ d dr ln ( T n γ -1 ) . (19)</formula>
<text><location><page_5><loc_8><loc_18><loc_48><loc_29></location>In absence of cooling and heating we obtain stable Brunt Vaisala oscillations. These oscillations are damped in time because of the drag term in Eq. 16. Although Eq. 18 is identical to Eq. 12 in Pizzolato & Soker (2010), the interpretations are slightly different because Pizzolato & Soker (2010) assume cooling of the background ICM but ignore the background inflow, whereas we assume global thermal balance with no net inflow of the hot gas.</text>
<text><location><page_5><loc_8><loc_7><loc_48><loc_16></location>3 Here we are assuming that the background ICM profile is not changing with time. In reality, the background profile changes because cold gas condenses out of the hot phase and falls in, reducing the density of the remaining hot gas. In this paper our focus is on whether cold gas can condense out in global thermal equilibrium for a given ICM profile, rather than on the effect of condensation on the background profile.</text>
<text><location><page_5><loc_8><loc_4><loc_10><loc_5></location>c ©</text>
<text><location><page_5><loc_52><loc_82><loc_92><loc_86></location>The cooling and heating terms on the RHS of Eq. 18 in the linear ( δ /lessmuch 1) isobaric ( n ' T ' = nT ) regime can be reduced to,</text>
<formula><location><page_5><loc_53><loc_79><loc_91><loc_81></location>2(1 + δ ) 5 nk B T n e n i T 2 [ Λ( T ' ) T ' 2 -Λ( T ) T 2 ] = 1 γt cool ( 2 -d ln Λ d ln T ) δ,</formula>
<text><location><page_5><loc_52><loc_70><loc_92><loc_78></location>which when plugged in Eq. 18 gives the correct growth rate for the thermal instability in the isobaric regime (Eq. 12). Linearized versions of Eqs. 16 & 18, using above, give the expression for thermal instability (more precisely, over-stability) in a stably stratified atmosphere (e.g., see Binney, Nipoti & Fraternali 2009),</text>
<formula><location><page_5><loc_65><loc_66><loc_92><loc_69></location>d 2 δ dt 2 + N 2 δ = 1 t TI dδ dt , (20)</formula>
<text><location><page_5><loc_52><loc_40><loc_92><loc_65></location>where t TI = γt cool / (2 -d ln Λ /d ln T ). 4 For a heating rate per unit volume proportional to density (e.g., as is the case for photoelectric heating), the heating term in Eq. 17 is (1 + δ ) n e n i Λ( T ) and the corresponding linear analysis gives t TI = γt cool / (1 -d ln Λ /d ln T ), the correct analytic result (e.g., see McCourt et al. 2012). Thus, strictly speaking, the cold gas over-density grows at the thermal instability timescale ( t TI ) and not at the cooling time ( t cool ); this is indeed verified by the simulations of McCourt et al. 2012. We use t cool instead of t TI throughout the paper because t TI = (10 / 9) t cool ≈ t cool for the case of a constant heating rate per unit volume. Local thermal instability as the source of extended cold gas in cluster cores, with a constant heating rate per unit volume, is supported by observations (Fig. 11 in McCourt et al. 2012). Similar scaling of the heating rate with the local density is supported by the simulations of turbulent heating/mixing in equilibrium with cooling (Banerjee & Sharma 2014).</text>
<text><location><page_5><loc_52><loc_37><loc_92><loc_40></location>During its evolution, the blob mass should be conserved. For the spherical blob of mass M , we have,</text>
<formula><location><page_5><loc_53><loc_34><loc_92><loc_36></location>M = 4 π 3 a 3 n (1 + δ ) µm p = 4 π 3 a 0 3 n 0 (1 + δ 0 ) µm p , (21)</formula>
<text><location><page_5><loc_52><loc_30><loc_92><loc_33></location>where quantities with subscript '0' represent their initial/background values.</text>
<text><location><page_5><loc_52><loc_15><loc_92><loc_30></location>Eqs. 15, 16, 18 and 21 represent a system of 4 equations in 4 variables ( r [ t ] , v [ t ] , δ [ t ] , a [ t ]) that can be solved numerically as an initial value problem to study the evolution of blobs in the ICM. We specify the initial values r 0 , δ 0 and a 0 ; initial velocity is chosen to be zero because cold gas is condensing out of the gas in hydrostatic equilibrium. In absence of the drag term, Eqs. (15), (16) and (18) do not depend on the blob-size, and blobs of different size evolve in a similar fashion. Drag slows down smaller blobs and give them more time to cool before they can fall in. Drag also damps stable oscillations.</text>
<section_header_level_1><location><page_6><loc_8><loc_85><loc_19><loc_86></location>4 RESULTS</section_header_level_1>
<text><location><page_6><loc_8><loc_62><loc_48><loc_84></location>In this section we present a phenomenological model to include the effects of geometrical compression in a spherical ICM. Later, we present the results on the evolution of overdense blobs in both Cartesian and spherical geometries. The evolution and saturation of the over-density ( δ ) of the blob is a sensitive function of the background t cool /t ff , the initial over density δ 0 and the spherical compression term; there is much weaker dependence on the initial blob radius a 0 , the drag coefficient and entropy stratification (because it is weak in the ICM, as we discuss later). We construct models in which runaway cooling occurs starting with tiny amplitude of perturbations only if the background t cool /t ff ∼ < 1 , 10 in Cartesian and spherical atmospheres, respectively. If t cool /t ff is larger than the critical value for runaway at tiny amplitudes, a finite amplitude of the over-density is required for runaway.</text>
<text><location><page_6><loc_8><loc_47><loc_48><loc_62></location>McCourt et al. (2012) and Sharma et al. (2012b) have carried out idealized simulations of hot atmospheres in thermal and hydrostatic equilibrium, using Cartesian and spherical geometries, respectively. Somewhat surprisingly, they find that it is much easier for cold gas to condense out of the hot phase in a spherical geometry. This, they attribute to spherical compression that an over-dense blob undergoes as it moves toward the centre in a spherical geometry. More quantitatively, they find that cold gas can condense out from small initial perturbations if t cool /t ff ∼ < 1 in Cartesian geometry and if t cool /t ff ∼ < 10 for a spherical setup.</text>
<text><location><page_6><loc_8><loc_30><loc_48><loc_47></location>PS05 do not consider any influence of geometry in their analysis, and therefore their results do not depend on whether the blob is falling in a plane-parallel atmosphere or a spherical one. Unlike in a Cartesian atmosphere, in spherical geometry the gravitational forces at the diametrically opposite ends of a spherical blob (at the same height) are not parallel. The radial component of gravity pointing toward cluster center will compress the blob and make it further over-dense, in addition to the effects already accounted for in Eq. 18. Thus, an over-dense blob in a spherical geometry is compressed more than the one in a plane-parallel atmosphere, and forms cold gas more easily.</text>
<section_header_level_1><location><page_6><loc_8><loc_26><loc_37><loc_27></location>4.1 Modeling geometric compression</section_header_level_1>
<text><location><page_6><loc_8><loc_7><loc_48><loc_25></location>In this section, we present a simple phenomenological prescription that allows us to parametrically model the blob's compression in a spherical geometry. Considering only geometrical compression, the transverse cross-section of the blob (which subtends a constant solid angle at the centre as it falls in) decreases as it falls in toward the centre, and the blob over-density increases such that (1 + δ ) r 2 = constant ( r is the radial coordinate of the centre of the blob). The corresponding expressions for cylindrical and Cartesian geometries are (1+ δ ) R = constant ( R is the cylindrical radius) and (1+ δ ) = constant. This implies that there is no geometrical compression in the Cartesian geometry. The compression in cylindrical geometry is smaller than in a spherical atmo-</text>
<figure>
<location><page_6><loc_54><loc_63><loc_90><loc_86></location>
<caption>Figure 2. The location of the over-dense blob ( top panel ) and the over-density ( bottom panel ) as a function of time for fiducial models with and without spherical compression, and without the modulation of the spherical compression term (i.e., β = 0). The β parameter (Eq. 22) for some runs is chosen ( β = 0 . 0244) such that small amplitude perturbations outside t cool /t ff = 10 do not condense out. The initial blob-size is a 0 = 50 pc; results are only weakly dependent on a 0 . The blob evolution is also shown for the spherical compression model without cooling.</caption>
</figure>
<text><location><page_6><loc_52><loc_30><loc_92><loc_48></location>ere because over-dense blobs are compressed to a line and not a point as they fall in. Numerical simulations in cylindrical geometry indeed show that the threshold t cool /t ff for the production of multiphase gas with tiny initial amplitude lies in between the results from Cartesian and spherical setups (M. McCourt, private communication). Our model fine-tuned for spherical profiles shows that the critical value of t cool /t ff for condensation of cold gas in a cylindrical geometry, starting from tiny amplitudes, ranges from 2.5 to 5 (as compared to the Cartesian and spherical cases, this value is somewhat sensitive to the background entropy profile; the analogous values for Cartesian and spherical atmospheres are 1 and 10, respectively; c. f. Figs. 4, 5).</text>
<text><location><page_6><loc_52><loc_28><loc_92><loc_30></location>Therefore, in order to model spherical compression, we should add a term like</text>
<formula><location><page_6><loc_58><loc_24><loc_85><loc_27></location>( dδ dt ) sph = -2 (1 + δ ) r dr dt = -2 (1 + δ ) v r ,</formula>
<text><location><page_6><loc_52><loc_18><loc_92><loc_23></location>to the right hand side of Eq. 18. This term is negligible if the entropy scale height (1 / [ d ln K/dr ]) is much smaller than r ; or in other words, if the atmosphere is strongly stratified, as we show shortly. Hence, in spherical geometry,</text>
<formula><location><page_6><loc_52><loc_12><loc_92><loc_17></location>dδ dt = 2 (1 + δ ) 5 n k B T [ n ' e n ' i Λ( T ' ) -n e n i Λ( T )] + (1 + δ ) g v N 2 -2 (1 + δ ) v r e ( -βM encr /M ) , (22)</formula>
<text><location><page_6><loc_52><loc_7><loc_92><loc_11></location>where we have added a phenomenological modulation term (similar in spirit to the widely-used mixing-length models) to the compression term, which suppresses geometrical compres-</text>
<text><location><page_7><loc_8><loc_71><loc_48><loc_86></location>sion when the mass encountered by the blob in the ICM becomes comparable to the blob's mass. This term is motivated by the fact that hydrodynamic effects, such as distortion due to ram pressure and the loss of sphericity of the blob, are expected to become dominant over geometrical compression, once the mass encountered is of order the blob mass. The β parameter will be chosen so that the model gives results consistent with Sharma et al. (2012b); namely, the over-dense blobs with tiny amplitudes run away to the stable temperature if and only if t cool /t ff ∼ < 10. The mass encountered by the blob is given by integrating</text>
<formula><location><page_7><loc_18><loc_67><loc_48><loc_71></location>dM encr dt = π a 2 ( t ) ρ ( r ( t )) ∣ ∣ v ( t ) ∣ ∣ , (23)</formula>
<text><location><page_7><loc_8><loc_59><loc_48><loc_67></location>where ∣ ∣ v ( t ) ∣ ∣ ensures that the mass encountered by the blob increases monotonically; this is required because hydrodynamic distortion happens irrespective of the direction of blob motion. Thus, in spherical geometry we need to solve 5 equations, namely Eqs. 15, 16, 21, 22, and 23 for 5 unknowns (the additional unknown compared to PS05 being M encr ).</text>
<text><location><page_7><loc_8><loc_55><loc_48><loc_59></location>The spherical compression term (the last term in Eq. 22) can be linearized as -2 v/r . The linearized equation governing the evolution of δ , including spherical compression, becomes</text>
<formula><location><page_7><loc_18><loc_51><loc_48><loc_54></location>d 2 δ dt 2 + ( N 2 -2 g r ) δ = 1 t TI dδ dt , (24)</formula>
<text><location><page_7><loc_8><loc_26><loc_48><loc_50></location>where we have used the linearized equation of motion dv/dt = -gδ . From Eq. 19, N 2 = g/ ( γH ), where H = 1 / ( d ln K/d ln r ) is the entropy scale height. It is clear from Eq. 24 that the spherical compression term leads to linear instability if H > r/ (2 γ ), even in the absence of thermal instability. As expected, the local plane-parallel approximation, with a negligible effect of spherical compression, holds in the limit H /lessmuch r . The entropy scale height of the ICM is comparable to r , and hence the amplitude of over-dense blobs grows linearly because of spherical compression; the amplitude saturates nonlinearly when the blob encounters its own mass in the ICM and the spherical compression term in Eq. 22 is suppressed (see the line with square markers in Fig. 2 which shows the evolution with spherical compression in the absence of cooling). When t cool /t ff ∼ > 10 over-density never becomes large or becomes large only when blob falls to the very center. However, with t cool /t ff ∼ < 10 local thermal instability can lead to extended cold gas.</text>
<text><location><page_7><loc_8><loc_13><loc_48><loc_25></location>The t cool /t ff profiles for cool-core clusters have a characteristic shape with a minimum in t cool /t ff at ∼ 10 kpc (see Fig. 1). There are two points where t cool /t ff = 10, on either side of the minimum. We find out the value of β for each cluster such that the blob released at the outer point where t cool /t ff ≈ 10 (lets call it r 10 ) would just form cold gas on reaching 1 kpc. 5 For such a value of β , no cold gas forms when r 0 > r 10 , i.e., the region where t cool /t ff > 10. We chose the outer r 10 to determine β because the outer regions</text>
<text><location><page_7><loc_52><loc_80><loc_92><loc_86></location>with t cool /t ff ∼ < 10 are most likely to form extended cold gas. For starting radii well within the bottom of the t cool /t ff 'cup' and for small over-densities the infall time is shorter than the cooling time, and runaway happens very close to the center (at r < 1 kpc).</text>
<text><location><page_7><loc_52><loc_67><loc_92><loc_79></location>For the analytic model to be useful, the value of β should not vary for different clusters and blob sizes. We list the β values for various ICM profiles and blob parameters in Table 1. Note that the smaller blobs require a smaller β ; i.e., higher compression, to form extended cold gas. This is because a smaller blob encounters its own mass in the ICM faster than a bigger one. For all our runs we use the average of all β s listed in Table 1; i.e., β = 0 . 0244. As we show, our results are not very sensitive to the exact value of β .</text>
<text><location><page_7><loc_52><loc_46><loc_92><loc_67></location>Figure 2 shows the evolution of 50 pc blobs with a small initial over-density ( δ 0 = 10 -5 ), released from initial radii of 10 and 40 kpc in the fiducial ICM with spherical compression. The evolution of the blobs is qualitatively similar to those in PS05, except that cold gas condenses for a longer cooling time because of spherical compression. If the cooling time is long, the over-density grows (because of spherical compression) but saturates with a small amplitude, and the blob undergoes stable oscillations (dotted lines); the blob falls in from its initial position by a substantial distance because it becomes further over-dense due to the spherical compression term. The blob released at the location (10 kpc) with a smaller t cool /t ff ( < 10; see Fig. 1) increases its over-density to a large value ( ∼ > 10 4 ) and falls in toward the center at an accelerated rate (solid line).</text>
<text><location><page_7><loc_52><loc_22><loc_92><loc_46></location>In Figure 2 the dotted and dashed lines correspond to the blobs released at the same radius (40 kpc) but with and without the β modulation term (Eq. 22). The t cool /t ff ratio is 13.4 at the release radius of 40 kpc; the β parameter is chosen such that the blobs released outside of the t cool /t ff = 10 region do not run away. For β = 0, spherical compression acts even at smaller radii and makes the blob cool in a runaway fashion even if t cool /t ff > 10, but β = 0 . 0244 recovers the result from numerical simulations; i.e., runaway cooling for small initial over-densities happens only if t cool /t ff < 10. Therefore, the inclusion of the modulation term in Eq. 22 is necessary to get a quantitative match with numerical simulations. The value of β = 0 . 0244 is the mean of β s that were obtained by requiring cold gas ( δ ∼ > 10 4 ) to form only if t cool /t ff < 10 for the profiles in Table 1 (for blob-sizes of 10, 50, 100 pc); the β range is narrow, from 0.021 to 0.028 for a blob-size of 50 pc, and does not vary systematically for a given blob-size.</text>
<text><location><page_7><loc_52><loc_7><loc_92><loc_22></location>Figure 3 shows the blob location and the over-density as a function of time, with different initial radii, for the fiducial profile and for the profile with t cool /t ff > 10 everywhere. As expected, with β adjusted such that extended cold gas is produced only if t cool /t ff < 10, all initial radii for the fiducial profile except for 30 kpc (at which the background t cool /t ff > 10) lead to the formation of multiphase gas. The blob released at 5 kpc cools to the stable temperature very close to the center. Even in the most thermally unstable cases, multiphase gas appears only at r ∼ < 5 kpc. For the high entropy model shown in Figure 3 none of the blobs released at radii > 5 kpc</text>
<figure>
<location><page_8><loc_10><loc_63><loc_48><loc_87></location>
<caption>Figure 3. The evolution of the blob location (upper panel) and the blob over-density (lower panel) as a function of time for the blobs released at 5, 10, 15, 20, 25, 30 kpc: solid lines correspond to the fiducial model with K 0 = 8 keV cm 2 ; dashed lines correspond to the model with K 0 = 37 . 9 keV cm 2 . The lower panel shows that thermal runaway for higher K 0 occurs only for a release radius of 5 kpc; moreover, runaway occurs as the blob reaches the very center ( /lessmuch 1 kpc). Runaway cooling occurs for all release radii except r 0 = 30 kpc (where t cool /t ff > 10) for the fiducial ( K 0 = 8 keV cm 2 ) model. The blob-size is 50 pc and we include spherical compression with β = 0 . 0244 (see Eq. 22).</caption>
</figure>
<text><location><page_8><loc_8><loc_29><loc_48><loc_45></location>produce cold gas, even when they fall in toward the center; the cooling time is longer than the inflow time. We can relate the cold gas condensing out at large radii ( > 1 kpc) and at small radii ( < 1 kpc) to the observational appearance of extended atomic filaments and centrally concentrated cold gas, respectively, as observed by McDonald et al. 2010. Thus, the results of our phenomenological model are consistent with the observational and computational results, which show that extended cold gas condenses out only if t cool /t ff ∼ < 10. Cold gas condenses out much farther out if the core density is high and t cool /t ff ∼ < 10 close to 100 kpc.</text>
<section_header_level_1><location><page_8><loc_8><loc_24><loc_48><loc_27></location>4.2 Extended cold filaments with tiny over-density: the critical t cool /t ff</section_header_level_1>
<text><location><page_8><loc_8><loc_7><loc_48><loc_23></location>There are three ways in which an over-dense blob evolves in the ICM: it can saturate at low amplitude ( δ < 1) away from the cluster center or becomes dense only at small radii ( < 1 kpc) as it falls in; it can cool to a large over-density far from the center ( > 1 kpc) and then fall in; it can become under-dense and overheated such that δ → 1 as it moves out. We expect thermal runaway to happen because the ICM is locally thermally unstable. Only below 10 4 K, below which the cooling function decreases rapidly ( ∝ T 6 ), is the blob expected to stop cooling further. In isobaric conditions the blob temperature T ' and the over-density ( δ ) have an inverse proportionality, the cold gas can reach a maximum over density</text>
<text><location><page_8><loc_52><loc_73><loc_92><loc_86></location>δ ∼ > 100 (positive runaway) 6 due to cooling. On the other hand, if the blob starts becoming under-dense with time, it rushes away from the ICM center due to its lower density, and in this case δ →-1, implying a negligibly small density of the blob. In both these extreme cases of runaway, the blob either forms hot or very cold gas compared to the background ICM. These runaway cases are required for the formation of extended multiphase gas in the ICM. We define a 'multiphase flag' (mp) to assess the formation of extended multiphase gas in our models as follows:</text>
<formula><location><page_8><loc_52><loc_64><loc_92><loc_72></location>mp = +1 ; positive runaway, δ ∼ > 100 outside r = 1 kpc 0 ; no runaway outside 1 kpc and/or stable oscillations -1 ; negative runaway, δ →-1 (25)</formula>
<text><location><page_8><loc_52><loc_59><loc_92><loc_64></location>We focus on positive runaways in which there is formation of cold gas ( T ' < 10 4 K ), which sinks and feeds the central black hole required for the feedback cycle to close, as discussed earlier.</text>
<text><location><page_8><loc_52><loc_43><loc_92><loc_58></location>As already mentioned, McCourt et al. (2012) and Sharma et al. (2012b), in their Cartesian and spherical simulations of locally unstable hydrostatic gas in global thermal balance, find that runaway cooling of even slightly over-dense blobs occurs if the ratio of the cooling time to the free-fall time ( t cool /t ff ) is smaller than a critical value. This critical value, ( t cool /t ff ) crit , is dependent on the geometry of the gravitational potential in a weakly stratified atmosphere such as the ICM. Namely, multiphase gas condenses out of the ICM if t cool /t ff ∼ < 10 in spherical potential and if t cool /t ff ∼ < 1 in Cartesian geometry.</text>
<section_header_level_1><location><page_8><loc_52><loc_40><loc_87><loc_41></location>4.2.1 Critical t cool /t ff in a plane-parallel atmosphere</section_header_level_1>
<text><location><page_8><loc_52><loc_17><loc_92><loc_39></location>We use the same set of equations as PS05 to study the evolution of over-dense blobs in a plane-parallel atmosphere. PS05 emphasized that the ICM required nonlinear density perturbations in order for multiphase gas to condense out of the ICM. However, they missed the significance of spherical compression brought to the fore by the idealized simulations of McCourt et al. (2012); Sharma et al. (2012b), and included in this paper phenomenologically in Eq. 22. As shown by numerical simulations, and as we show later, this new compression term can lead to the condensation of cold gas from arbitrarily small perturbations in cluster cores if t cool /t ff ∼ < 10. Additionally, and understandably, because galaxy cluster cores have t cool /t ff ∼ > 10, PS05 overlooked that there was another regime for the local thermal instability where arbitrarily small perturbations can lead to the condensation of cold gas even in a plane-parallel atmosphere. We find, in agreement with</text>
<text><location><page_8><loc_52><loc_7><loc_92><loc_14></location>6 Once the over-dense blob reaches δ ∼ 10, it cools very rapidly to the stable temperature (see Fig. 3). Therefore, an over-density of 100 is reached at almost the same radius at which the blob cools to T ' ≈ 10 4 K. Moreover, a different δ corresponds to the temperature of the stable phase; e.g., δ corresponding to the stable phase in a 1 keV group is ≈ 10 3 and in a 10 6 K galactic halo is ≈ 100.</text>
<text><location><page_9><loc_8><loc_82><loc_48><loc_86></location>McCourt et al. (2012), that in a plane-parallel atmosphere cold gas condenses out starting from a small amplitude if t cool /t ff ∼ < 1.</text>
<text><location><page_9><loc_8><loc_60><loc_48><loc_82></location>As already mentioned, we achieve t cool /t ff ∼ < 1 in the ICM by artificially increasing the cooling function Λ by a factor of 10 for plane-parallel models. We start with a tiny over-density ( δ 0 = 10 -5 ), and initialize the blobs at multiple radii r 0 , and integrate the blob evolution equations for 5 Gyr for different 1-D ICM profiles given in Table 1. In Figure 4 we plot the multiphase flag as a function of t cool /t ff (at the initial location where the blob is released) in a Cartesian potential, starting with a small over-density. It is evident that for t cool /t ff > 1, there are no runaways seen for small δ 0 . The critical t cool /t ff in a Cartesian gravity setup using the simple model of PS05 is ( t cool /t ff ) crit = 1. The critical value of t cool /t ff is fairly insensitive to the blob size, the initial over-density and the drag coefficient.For t cool /t ff ∼ < 1 thermal instability forms multiphase gas irrespective of the blob parameters, as seen in the simulations of McCourt et al. (2012).</text>
<section_header_level_1><location><page_9><loc_8><loc_56><loc_36><loc_58></location>4.2.2 Critical t cool /t ff in a spherical ICM</section_header_level_1>
<text><location><page_9><loc_8><loc_45><loc_48><loc_56></location>We now incorporate the geometrical compression model discussed earlier in section 4.1, to study blob evolution in a spherical setup and determine the critical t cool /t ff for runaway to occur with tiny over-densities. Using spherical simulations Sharma et al. 2012b find the critical value for the condensation of cold gas to be ( t cool /t ff ) crit ≈ 10. This critical criterion agrees with the observations of cool cluster cores which show extended cold filaments (Fig. 11 in McCourt et al. 2012).</text>
<text><location><page_9><loc_8><loc_24><loc_48><loc_44></location>Our compression model includes a free parameter β , which we adjust to match the critical t cool /t ff from simulations and observations, as discussed in section 4.1. For all our spherical models we use β = 0 . 0244, the average of β s obtained for various blob and ICM parameters. Clusters with K 0 ∼ > 30 keVcm 2 , have t cool /t ff > 10 everywhere. Hence, extended cold phase is not expected in such clusters for small amplitudes. Our models with spherical compression show a similar trend. Figure 5 shows the results of our model by plotting the multiphase flag of Eq. 25 against t cool /t ff for over-dense blobs starting at various positions in some of our clusters in Table 1. It is evident that multiphase condensation takes place for arbitrarily small amplitudes only when t cool /t ff ∼ < 10, as found in numerical simulations by Sharma et al. 2012b.</text>
<section_header_level_1><location><page_9><loc_8><loc_20><loc_42><loc_21></location>4.3 Multiphase gas with long cooling times</section_header_level_1>
<text><location><page_9><loc_8><loc_7><loc_48><loc_19></location>In cases where the background t cool /t ff is higher than the critical value, there is a competition between cooling and freefall, and cooling to low temperatures occurs only with a finite initial over-density ( δ 0 ). This should be contrasted with the cases where t cool /t ff ∼ < t cool /t ff , crit , and cold gas condenses out even with a tiny over-density. We release stationary blobs at varying initial radii, r 0 (ranging from 10 to 100 kpc), to achieve a spectrum of t cool /t ff > t cool /t ff , crit ) and track their evolution for 5 Gyr in both Cartesian and spherical potentials.</text>
<figure>
<location><page_9><loc_8><loc_4><loc_10><loc_5></location>
<caption>Figure 7 shows the critical over-density ( δ 0 ,c ) required for runaway cooling in a spherical ICM. We include the geometrical compression term with modulation due to the encountered mass (the last term in Eq. 22). For larger blobs the compression term acts for a longer time and tends to compress it more because it takes longer to sweep up a bigger blob's own mass in the ICM. On the other hand, like in a plane-parallel atmosphere, the lower drag makes it fall faster. These two effects oppose each other, and we see that the critical δ 0 required to form cold phase in spherical geometry is insensitive to the blob size at higher t cool /t ff . In both Cartesian and spherical atmospheres, the important point to note is the sharp fall of δ 0 , c near the critical t cool /t ff . It is evident that below the critical value of t cool /t ff , any arbitrary small value of δ 0 will form cold phase. This shows the existence of the critical t cool /t ff</caption>
</figure>
<figure>
<location><page_9><loc_54><loc_66><loc_89><loc_87></location>
</figure>
<text><location><page_9><loc_89><loc_86><loc_90><loc_87></location>2</text>
<paragraph><location><page_9><loc_52><loc_59><loc_92><loc_65></location>Figure 6. The critical value of the initial over-density ( δ 0 ) required for runaway cooling as a function of t cool /t ff in a plane-parallel atmosphere. Runaway cooling happens only if the over-density is larger than this critical value; for a smaller δ 0 , the blob undergoes linear oscillations.</paragraph>
<text><location><page_9><loc_52><loc_42><loc_92><loc_54></location>In both Cartesian and spherical atmospheres, the trends seen for the critical initial over-density ( δ 0 ,c ) required to form extended cold phase outside of 1 kpc are similar. This is intuitively expected; as t cool /t ff increases, cooling becomes less effective and the blob needs a higher density contrast to cool faster than it can fall in to form extended cold gas. The geometrical compression term in Eq. 22 increases the critical value of t cool /t ff , above which a finite large over-density is required to form cold phase.</text>
<text><location><page_9><loc_52><loc_28><loc_92><loc_41></location>In Figure 6, we show the dependence of δ 0 , c on the background t cool /t ff for the fiducial and K 0 = 1 keV cm 2 profiles using a variety of blob sizes. It is seen that in the absence of geometrical compression, smaller blobs cool more easily. The blob-size enters the equations via the drag term in the equation of motion (Eq. 16). A smaller blob (for a given overdensity), or equivalently a larger drag term, implies that the blob is slowed down more as it is falling in, and therefore has a longer time to cool compared to a bigger blob. This explains a smaller critical δ 0 for smaller blobs.</text>
<figure>
<location><page_10><loc_22><loc_52><loc_80><loc_86></location>
<caption>Figure 4. Multiphase flag (labelled with the initial blob size) as a function of t cool /t ff at the release radius in a plane-parallel ICM for three different profiles. Runaway cooling of the blobs with small initial over-densities ( δ = 10 -5 is chosen) happens only if the ratio t cool /t ff ∼ < 1; the results are not sensitively dependent on the blob-size or the drag-coefficient ( C in Eq. 22). Note that for the same t cool /t ff and blob-size we can have a positive or a negative runaway; this is because the blobs with release radii on either side of min( t cool /t ff ) evolve differently (see Fig. 8). Since observed clusters have t cool /t ff ∼ > 5, we have artificially increased the cooling function (Λ[ T ]) by a factor of 10 to attain t cool /t ff ∼ < 1 in our plane-parallel runs.</caption>
</figure>
<figure>
<location><page_10><loc_8><loc_21><loc_48><loc_41></location>
<caption>Figure 7. Critical Initial over-density (blob-size is labelled) δ 0 , c for different t cool /t ff for a spherical ICM. Notice that the larger blobs form extended cold gas for a slightly higher t cool /t ff .</caption>
</figure>
<text><location><page_10><loc_8><loc_10><loc_48><loc_12></location>and the domination of thermal instability growth for a small t cool /t ff .</text>
<text><location><page_10><loc_8><loc_7><loc_48><loc_10></location>Figure 8 shows the critical over-density required for runaway cooling as a function of t cool /t ff in spherical and Carte-</text>
<text><location><page_10><loc_52><loc_18><loc_92><loc_40></location>sian geometries for the highest entropy model in Table 1. The critical over-density has a characteristic shape for both Cartesian and spherical atmospheres. Also, note that the minimum over-density required for runaway cooling does not correspond to the release radius with minimum t cool /t ff ; it corresponds to a radius slightly inward of the minimum. The asymmetry in the response of over-dense blobs inward and outward of the minimum is likely responsible for the presence of extended cold filaments in some cool cluster cores (McDonald et al. 2010). Somewhat counter-intuitively, the blobs farther away from the center (with a slightly longer cooling time than at the center) are more likely to result in cold gas because the blobs at inner radii fall in before they can cool to the stable temperatures. This leads to spatially extended cold gas over 10s of kpc, rather distinct from the centrally concentrated cold gas.</text>
<section_header_level_1><location><page_10><loc_52><loc_13><loc_80><loc_14></location>5 DISCUSSION & CONCLUSIONS</section_header_level_1>
<text><location><page_10><loc_52><loc_7><loc_92><loc_12></location>In this paper we have presented a phenomenological model for the evolution of over-dense blobs in the ICM. We have extended the model of Pizzolato & Soker (2005) to include the important influence of geometrical compression. A com-</text>
<text><location><page_11><loc_46><loc_53><loc_48><loc_54></location>10</text>
<figure>
<location><page_11><loc_23><loc_53><loc_79><loc_86></location>
</figure>
<text><location><page_11><loc_52><loc_53><loc_54><loc_54></location>cool</text>
<text><location><page_11><loc_55><loc_53><loc_55><loc_54></location>ff</text>
<figure>
<location><page_11><loc_11><loc_19><loc_48><loc_44></location>
<caption>Figure 5. Multiphase flag (labelled with the initial blob size) as a function of t cool /t ff in spherical gravity for δ 0 = 10 -5 . Runaway cooling in spherical geometry happens for a higher t cool /t ff ( ∼ < 10); i.e., in comparison to a plane-parallel atmosphere, cold gas condenses out more easily in a spherical atmosphere. Also note, in comparison to the plane-parallel atmospheres (Fig. 4), there are no hot runaways in spherical geometry because of the geometrical compression term in Eq. 22.Figure 8. The critical value of the over-density as a function of t cool /t ff for the high entropy model in Cartesian (upper panel) and spherical (lower panel) geometries. For the same value of t cool /t ff the blobs starting farther away from the minimum of t cool /t ff require a smaller over-density for runaway cooling as compared to the blobs released at smaller radii. Note that the vertical scale is logarithmic (linear) for the plane-parallel (spherical) atmosphere.</caption>
</figure>
<text><location><page_11><loc_8><loc_4><loc_10><loc_5></location>©</text>
<text><location><page_11><loc_9><loc_4><loc_9><loc_5></location>c</text>
<text><location><page_11><loc_52><loc_27><loc_92><loc_44></location>parison of idealized simulations in spherical (Sharma et al. 2012b) and plane-parallel (McCourt et al. 2012) atmospheres show that it is much easier for multiphase gas to condense in the presence of spherical compression. We have incorporated a phenomenological spherical compression term in the model of PS05; this increases the over-density as the blob falls in toward the center. With a single adjustable parameter ( β in Eq. 22), which is analogous to the mixing length, we are able to obtain the key result of Sharma et al. (2012b); i.e., cold gas condenses out at large radii starting from tiny perturbations when the ratio of the cooling time and the local free-fall time ( t cool /t ff ) in the ICM is ∼ < 10.</text>
<text><location><page_11><loc_52><loc_25><loc_92><loc_27></location>In the following we discuss various astrophysical implications of our results.</text>
<text><location><page_11><loc_52><loc_13><loc_92><loc_23></location>(i) Robustness of the phenomenological model: Our phenomenological model, which extends PS05's treatment to account for spherical compression, agrees well with the results of numerical simulations with just a single adjustable parameter ( β ; see Eq. 22) analogous to the mixing length. Moreover, the parameter is fairly insensitive to various parameters such as the blob-size, release radius, entropy profiles, halo mass, etc.</text>
<text><location><page_11><loc_52><loc_7><loc_92><loc_12></location>We have also applied our model (with the same β parameter = 0.0244) to a much lower mass Milky-Way-like halo with the halo mass of 10 12 M /circledot . Unlike clusters, for such halos cooling is important even at the virial radius and the hot</text>
<text><location><page_12><loc_13><loc_86><loc_14><loc_87></location>4</text>
<figure>
<location><page_12><loc_11><loc_63><loc_48><loc_87></location>
<caption>Figure 9. The top panel shows the entropy (in keV cm 2 ) and t cool /t ff profiles as a function of radius for two lower mass halo (10 12 M /circledot ) models accounting for spherical compression ( β = 0 . 0244; see Eq. 22): one with a constant entropy core ( K = 80 keV cm 2 ) and another with a constant t cool /t ff = 7 'core'. The bottom panel shows the location of the blob (with initial radius 50 pc and initial over-density δ = 10 -5 ) as a function of time for different initial release radii for these two different profiles. While the constant t cool /t ff profile gives extended multiphase gas for all release radii, extended cold gas condenses out only at radii larger than 80 kpc for the constant entropy profile. The blob starts to fall in much faster after it cools to the stable temperature; this leads to a characteristic 'knee' in cases with runaway cooling (e.g., see Figs. 2 & 3).</caption>
</figure>
<text><location><page_12><loc_8><loc_15><loc_48><loc_40></location>gas density is expected to be much lower (e.g., see Fig. 1 of Sharma et al. 2012a). We have followed the evolution of slightly over-dense ( δ = 10 -5 ) 50 pc blobs starting at rest from various initial radii for two plausible hot gas profiles a constant entropy ( K = 80 keV cm 2 ) core and a constant t cool /t ff = 7 'core'. The top panel of Figure 9 shows the entropy and t cool /t ff profiles that we have used. While the constant t cool /t ff profile is expected to form extended multiphase gas irrespective of the initial radius, the constant entropy profile should give runaway cooling only at large radii ( ∼ > 80 kpc) where t cool /t ff ∼ < 10. For the constant entropy profile t cool /t ff at 80 kpc is 12.5 and at 60 kpc it is 20; therefore, runaway cooling occurs only if t cool /t ff slightly exceeds 10. This shows the robustness of our model (which uses the same value of the β parameter that was fine-tuned for the massive cluster) which gives the threshold of t cool /t ff ∼ < 10 for runaway cooling, even if the halo mass and the ICM profiles are changed drastically.</text>
<text><location><page_12><loc_8><loc_7><loc_48><loc_15></location>(ii) Interplay of local thermal instability and gravity: Cold gas condenses out much closer in than the location of the minimum of t cool /t ff . This is seen in the idealized simulations (e.g., Fig. 4 in Sharma et al. 2012b shows gold gas only extended out to 5 kpc but t cool /t ff < 10 out to 20 kpc for the corresponding ICM profile) and in our analytic models (e.g.,</text>
<text><location><page_12><loc_52><loc_61><loc_92><loc_86></location>Fig. 3 shows that the blob released at 20 kpc cools to the stable phase only within 5 kpc). This result has implications for the location of high velocity clouds in Milky Way-like galaxies, the farthest of which are believed to have condensed out of the hot halo gas. While t cool /t ff ∼ < 10 even out to the virial radius for 10 12 M /circledot halos (e.g., Sharma et al. 2012a), the cold gas (emitting 21 cm line) may exist well within the halo (see also Fig. 9). This is consistent with the high velocity clouds detected out to 50 kpc from the center of the Andromeda galaxy (Thilker et al. 2004). Note that the cooling time close to the virial radius in Figure 9 can be comparable to the Hubble time and the approximation of a constant halo mass, etc. break down. However, the qualitative results of our model are expected to hold. Cold gas in galaxy clusters and groups can be pushed farther out compared to the predictions of our analytic models and idealized simulations with thermal balance because of the large velocities generated by AGN-driven bubbles.</text>
<text><location><page_12><loc_52><loc_33><loc_92><loc_61></location>Malagoli, Rosner & Bodo (1987), Balbus & Soker (1989) and Binney, Nipoti & Fraternali (2009), among others, have investigated the interplay of local thermal instability and gravity. These works have highlighted the importance of stable stratification in thermally unstable atmospheres, because of which the instability is an over-stability with fast oscillations. Using a Lagrangian analysis, Balbus & Soker (1989) have shown that the thermal instability is suppressed in presence of a background cooling flow because the background conditions change over the same timescale as the instability itself. This was verified in numerical simulations of cooling flows with finite density perturbations (see the Appendix section of Sharma et al. 2012b) which show that large amplitude density perturbations are required for multiphase gas to condense out of the cooling flows. However, in the state of thermal balance (which is strongly favored by observations over the past decade) there is no cooling flow and the background state is quasi-static. In such a state even tiny density perturbations can become nonlinear if the cooling time of the background ICM is short enough (i.e., t cool /t ff ∼ < 10).</text>
<text><location><page_12><loc_52><loc_14><loc_92><loc_33></location>The linear response of a strongly-stratified atmosphere in thermal and hydrostatic balance is quite simple (Eq. 20) and shows over-stability (see the line marked by circles in Fig. 2) damped nonlinearly by the drag term. In a weakly-stratified spherical ICM, however, a slightly over-dense blob becomes further over-dense because of the spherical compression term (see Eq. 24; spherical compression is unimportant if the entropy scale height is much smaller than the radius) and the heavier blob falls inward with a large velocity, at which point the drag force is large and stops the blob from falling in further. Moreover, the spherical compression term in Eq. 22 becomes weaker as the blob encounters a mass comparable to its own mass. Thus, spherical compression is crucial in making it easier for the over-dense blobs to condense out of the ICM.</text>
<text><location><page_12><loc_52><loc_7><loc_92><loc_14></location>By constructing strongly stratified artificial ICM profiles, we have verified that the spherical compression term is not effective when the entropy scale height is much smaller than the blob release radius (i.e., if the atmosphere is strongly stratified; see Eq. 24). We can also understand the dependence</text>
<text><location><page_13><loc_8><loc_70><loc_48><loc_86></location>of the critical t cool /t ff on the background entropy profile in cylindrical geometry from Eq. 24 (as mentioned in section 4.1, the critical value varies from 2.5 to 5 for typical cool cluster entropy profiles). For weak entropy stratification of the ICM the spherical compression term in Eq. 24 overwhelms the oscillatory N 2 term. For cylindrical geometry, however, the compression term is half of its spherical value ( -g/r instead of -2 g/r in Eq. 24) and even weak stratification of the ICM decreases the critical t cool /t ff for condensation. Therefore, in cylindrical geometry the critical t cool /t ff is larger if the ICM is weakly stratified (e.g., K ∝ r 0 ) and smaller if it is even slightly stratified (e.g., K ∝ r 1 . 1 ).</text>
<text><location><page_13><loc_8><loc_60><loc_48><loc_70></location>Stratification is not expected to be very strong for astrophysical coronae, which by definition, are close to the virial temperature. Thus, our t cool /t ff ∼ < 10 criterion for the condensation of extended cold gas is expected to be valid in most astrophysical coronae, ranging from the solar corona to hot accretion flows (e.g., see Sharma 2013; Das & Sharma 2013; Gaspari, Ruszkowski & Oh 2013).</text>
<text><location><page_13><loc_8><loc_37><loc_48><loc_58></location>(iii) Extended multiphase gas and cold feedback: The accretion of cold gas by the central massive black holes in cluster cores is essential for AGN feedback to close the globally stable feedback loop. Hot accretion rate via Bondi accretion is small and not as sensitively dependent on the ICM density ( ∝ n ) as radiative cooling ( ∝ n 2 ). Therefore, accretion in the hot phase alone seems incapable of globally balancing radiative cooling. Moreover, the Bondi accretion rate is about two orders of magnitude smaller than the multiphase mass cooling rate close to 10 kpc, which should be of order the black hole accretion rate (e.g., see Gaspari, Ruszkowski & Oh 2013). Moreover, as discussed in detail in section 5.1 of Sharma et al. (2012b), the cold feedback model may also naturally account for the observed correlation between the estimated Bondi accretion rate and the jet power.</text>
<text><location><page_13><loc_8><loc_7><loc_48><loc_37></location>Our phenomenological model tries to provide a physical basis for the condensation of cold gas in the ICM. In particular, it tries to explain the large quantitative difference in the condition for the condensation of multiphase gas in a planeparallel and a spherical atmosphere; namely, t cool /t ff ∼ < 1 versus 10 for the condensation of cold gas starting from small perturbations. Thus, we have provided a firmer footing to the cold feedback paradigm by extending PS05's model to agree with observations and numerical simulations. We also argue that the characteristic shape of t cool /t ff profile with a minimum at ∼ 10 kpc (rather than right at the center) is very crucial. Because of this, a large amount of gas far away from the sphere of influence of the supermassive black hole (unlike Bondi accretion) can condense out and episodically boost the accretion rate (due to cold gas) by a large amount. This can rather effectively stop the cooling flow and make t cool /t ff > 10 throughout, and the condensation of cold gas is suppressed. The core can form again because of a feeble accretion rate in the hot mode, and once t cool /t ff ∼ < 10 again, the core can suffer heating due to AGN jets/cavities and the global cycle continues. The presence of extended cold gas also agrees with the observations of McDonald et al. (2010) and others, which</text>
<text><location><page_13><loc_8><loc_4><loc_10><loc_5></location>©</text>
<text><location><page_13><loc_9><loc_4><loc_9><loc_5></location>c</text>
<text><location><page_13><loc_52><loc_84><loc_92><loc_86></location>show extended cold gas at 10s of kpc where t cool /t ff ∼ < 10 (e.g., see Fig. 11 of McCourt et al. 2012).</text>
<section_header_level_1><location><page_13><loc_52><loc_79><loc_72><loc_80></location>ACKNOWLEDGEMENTS</section_header_level_1>
<text><location><page_13><loc_52><loc_70><loc_92><loc_78></location>The authors thank Noam Soker for his comments on the paper. AS thanks the Indian Academy of Sciences, Indian National Science Academy and the National Academy of Sciences India for awarding the Summer Research Fellowship 2012 with grant number PHYS1231. This work is partly supported by the DST-India grant no. Sr/S2/HEP-048/2012.</text>
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<text><location><page_13><loc_53><loc_8><loc_92><loc_65></location>Balbus S. A., Soker N., 1989, ApJ, 341, 611 Banerjee N., Sharma P., 2014, MNRAS, 443, 687 Binney J., Nipoti C., Fraternali F., 2009, MNRAS, 397, 1804 Burkert A., Lin D. N. C., 2000, ApJ, 537, 270 Cavagnolo K. W., Donahue M., Voit G. M., Sun M., 2009, ApJS, 182, 12 Churazov E., Bruggen M., Kaiser C. R., Bohringer H., Forman W., 2001, ApJ, 554, 261 Churazov E., Forman W., Vikhlinin A., Tremaine S., Gerhard O., Jones C., 2008, MNRAS, 388, 1062 Das U., Sharma P., 2013, MNRAS, 435, 2431 Dubois Y., Devriendt J., Teyssier R., Slyz A., 2011, MNRAS, 417, 1853 Edge A., 2001, Monthly Notices of the Royal Astronomical Society, 328, 762 Fabian A. C., 1994, Ann. Rev. Astr. Astr., 32, 277 Field G. B., 1965, ApJ, 142, 531 Gaspari M., Ruszkowski M., Oh S. P., 2013, MNRAS, 432, 3401 Gaspari M., Ruszkowski M., Sharma P., 2012, ApJ, 746, 94 Leccardi A., Molendi S., 2008, A&A, 487, 461 Li Y., Bryan G. L., 2013, ArXiv e-prints Malagoli A., Rosner R., Bodo G., 1987, ApJ, 319, 632 McCourt M., Sharma P., Parrish I. J., Quataert E., 2012, MNRAS, 419, 3319 McDonald M., Veilleux S., Rupke D. S. N., Mushotzky R., 2010, ApJ, 721, 1262 McNamara B. R., Nulsen P. E. J., 2007, Ann. Rev. Astr. Astr., 45, 117 Mohr J. J., 1999, ApJ, 517, 627 Navarro J. F., Frenk C. S., White S. D. M., 1996, ApJ, 463, 563 O'Dea C. P. et al., 2008, The Astrophysical Journal, 681, 1035 Peterson J. R., Kahn S. M., Paerels F. B. S., Kaastra J. S., Tamura T., Bleeker J. A. M., Ferrigno C., Jernigan J. G., 2003, The Astrophysical Journal, 590, 207 Pizzolato F., Soker N., 2005, ApJ, 632, 821 Pizzolato F., Soker N., 2010, MNRAS, 408, 961 Quataert E., 2008, ApJ, 673, 758 Rybicki, G. B. and Lightman, A. P., 2004, Radiative Pro-</text>
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</document> | [{"title": "ABSTRACT", "content": "Cool cluster cores are in global thermal equilibrium but are locally thermally unstable. We study a nonlinear phenomenological model for the evolution of density perturbations in the ICM due to local thermal instability and gravity. We have analyzed and extended a model for the evolution of an over-dense blob in the ICM. We find two regimes in which the over-dense blobs can cool to thermally stable low temperatures. One for large t cool /t ff ( t cool is the cooling time and t ff is the free fall time), where a large initial over-density is required for thermal runaway to occur; this is the regime which was previously analyzed in detail. We discover a second regime for t cool /t ff \u223c < 1 (in agreement with Cartesian simulations of local thermal instability in an external gravitational field), where runaway cooling happens for arbitrarily small amplitudes. Numerical simulations have shown that cold gas condenses out more easily in a spherical geometry. We extend the analysis to include geometrical compression in weakly stratified atmospheres such as the ICM. With a single parameter, analogous to the mixing length, we are able to reproduce the results from numerical simulations; namely, small density perturbations lead to the condensation of extended cold filaments only if t cool /t ff \u223c < 10. Key words: galaxies: clusters: intracluster medium; galaxies: haloes; hydrodynamics; instabilities", "pages": [1]}, {"title": "Ashmeet Singh \u2020 , Prateek Sharma \u2021", "content": "\u2020 \u2021 1 March 2022", "pages": [1]}, {"title": "1 INTRODUCTION", "content": "The intracluster medium (ICM) of galaxy clusters is composed of ionized plasma at the virial temperature ( \u223c 10 7 -10 8 K). The ICM is heated due to the release of gravitational energy which is converted into thermal energy at the accretion shock. Given the ICM temperature (Mohr 1999), the lighter elements like H and He are fully ionized. The typical density in cluster cores lines in the range of 0 . 001 -0 . 1 cm -3 (e.g., Cavagnolo et al. 2009). The hot ionized ICM emits X-Rays due to free-free and bound-free/bound-bound emission, which is enhanced due to the high metallicity ( \u223c 0 . 3 solar) of the ICM (Rybicki, G. B. and Lightman, A. P. 2004). The cooling loss rate per unit volume is proportional to the square of the particle number density. Since the inner ICM is denser than the outer regions, the gas near the center is expected to cool faster than the gas in the outskirts. This relatively fast cooling of the gas near the center of the cluster should reduce the thermal support it provides to the overlying layers. Hence, the dilute outer layers are expected to flow into the inner cooling region, resulting in a subsonic flow of the ICM plasma called the cooling flow. The cooling flow, being subsonic, maintains quasi-hydrostatic equilibrium in the ICM (Fabian 1994). Observations of galaxy clusters by X-Ray satellites such as Chandra and XMM-Newton (e.g., Peterson et al. 2003) show a dramatic lack of cooling. Similarly, signs of cold gas and star-formation are missing (e.g., Edge 2001; O'Dea et al. 2008), contrary to the predictions of the cooling flow model. It is therefore indicative, that some form of heating in the ICM balances cooling and explains the lack of a cooling flow. The ICM, thus, stays in a state of global thermal equilibrium where heating, on average, balances radiative cooling. Of the various possibilities discussed, heating due to mechanical energy injection by jets and bubbles driven by the central Active Galactic Nuclei (AGN) appears to be the most promising (see McNamara & Nulsen 2007 for a review). Pizzolato & Soker (2005), and later Pizzolato & Soker (2010), have argued that over-dense blobs of gas cool faster than the rest of the ICM. These fast-cooling blobs become heavier than the background ICM, sink and feed the central Active Galactic Nucleus (AGN). This cold feedback from nonlinear perturbations triggers winds and/or jets from the AGN which heat the ICM and maintain the global thermal equilibrium, as mentioned earlier. The mechanical energy input creates density perturbations in the ICM, which if the ICM is sufficiently dense, can lead to the next episode of multiphase cooling and feedback heating. While Pizzolato & Soker 2005 focus on the evolution of density perturbations, the aim of Pizzolato & Soker 2010 is to show that the cold blobs can lose angular momentum sufficiently fast due to drag and cloud-cloud collisions. In the current paper we are not concerned about the angular momentum problem but about the condensation of cold gas from almost spherical, non-rotating ICM. Later works, with various degrees of realism (e.g., Sharma et al. 2012b; Gaspari, Ruszkowski & Sharma 2012; Li & Bryan 2013), have vindicated this basic picture in which condensation and accretion of cold gas plays a key role in closing the feedback loop. Pizzolato & Soker 2005 (hereafter PS05) consider a Cartesian setup, not accounting for geometric compression due to radial gravity. We expect geometrical compression to be important if over-dense blobs travel a distance comparable to the location of their birth. The differences between plane-parallel and spherical geometries have been highlighted by McCourt et al. 2012 and Sharma et al. 2012b. 1 They show that the cold blobs seeded by small perturbations saturate at large densities (and thermally stable temperatures) if t cool /t ff \u223c < 1 in a plane-parallel atmosphere and if t cool /t ff \u223c < 10 in a spherical atmosphere; i.e., cold gas condenses more easily in a spherical atmosphere. The ratio t cool /t ff of the background ICM (more precisely, t TI /t ff , the ratio of thermal instability timescale and the free-fall time) is an important parameter governing the evolution of small density perturbations. For small t cool /t ff the cooling time is short and a slightly over-dense blob cools to the stable temperature and falls in ballistically. On the other hand, for large t cool /t ff , the over-dense blob responds to gravity as it is cooling; the shear generated due to the motion of the over-dense blob relative to the ICM leads to mixing of the cooling blob and it cannot cool to the stable temperature. In this paper we extend PS05's analysis to account for spherical compression as the blob travels large distances. In order that spherical compression does not over-compress blobs, we also include a model to account for the mixing of blobs. With a single adjustable parameter, we are able to reproduce the results of Sharma et al. (2012b) - that cold gas condenses with tiny perturbations if t cool /t ff \u223c < 10 - for a wide range of cluster parameters. In addition to this, we also highlight that there are two regimes for the evolution of over-dense blobs, in both Cartesian and spherical geometries: for t cool /t ff 1 The plane-parallel and spherical atmospheres are physically distinct; they are not the same initial conditions solved using different coordinate systems. In a plane-parallel atmosphere gravitational field vectors are everywhere parallel but in a spherical atmosphere the gravitational field points toward a single point. smaller than a critical value ( \u2248 1 in a plane-parallel and \u2248 10 in a spherical atmosphere) the over-dense blobs cool in a runaway fashion; for t cool /t ff greater than the critical value a finite amplitude is needed (a larger amplitude is required for a higher t cool /t ff ) for runaway cooling. PS05 have highlighted the second point and have argued that nonlinear amplitudes are required for producing cold gas in cluster cores. However, as McCourt et al. (2012); Sharma et al. (2012b) show, with spherical compression even small amplitude perturbations should give cold gas in cool-core clusters. We emphasize that our paper presents a simple analytic model for the evolution of density perturbations in the ICM. We are interested in the qualitative physics of the evolution of over-dense blobs in cluster cores and in comparing analytic models with numerical simulations. Quantitative results with accurate modeling of nonlinear processes such as mixing can only be obtained via numerical simulations. Moreover, our analytic models neglect effects such as thermal conduction, cosmic rays, magnetic fields, turbulence. Examples of recent simulations of the interplay of cooling and AGN feedback, including some of these effects, are Dubois et al. (2011); Sharma et al. (2012b); Gaspari, Ruszkowski & Sharma (2012); Li & Bryan (2013); Wagh, Sharma & McCourt (2014); Banerjee & Sharma (2014). The paper is organized as follows. In section 2 we present the 1-D profiles of the ICM (density, temperature, etc.) in hydrostatic equilibrium with a fixed dark matter halo. In section 3 we describe the equations governing the evolution of an over-dense blob in the ICM. In section 4 we present our results, starting with our phenomenological model to account for geometrical compression. We also discuss blob evolution for different cluster and blob parameters, contrasting the evolution in Cartesian and spherical atmospheres. In section 5 we conclude with astrophysical implications.", "pages": [1, 2]}, {"title": "2 1D MODEL FOR THE INTRACLUSTER MEDIUM", "content": "Hydrostatic equilibrium (HSE) in the ICM is a good approximation for the relaxed, cool-core clusters (for observational constraints on turbulent pressure support, see Churazov et al. 2008; Werner et al. 2009). Thus, where P is the gas pressure, \u03c1 is the gas mass density and g is the acceleration due to gravity in the cluster at a distance r from the center. The Navarro-Frenk-White (Navarro, Frenk & White 1996) (NFW) profile provides a good description of the dark matter (DM) distribution in relaxed halos such as galaxy clusters. We have used a spherically symmetric NFW profile for the dark matter as: where \u2206 c and r s are the characteristic density parameter and the scale radius, and \u03c1 crit is the critical density of the universe. Generally the size of the dark matter halo is taken to be r 200 , the virial radius around the cluster center within which the average dark matter density is 200 times the critical density of the universe. The NFW parameters can be recast in terms of the concentration parameter c = r 200 /r s , where The corresponding acceleration due to gravity in the ICM is where we ignore the gravity due to baryons as their mass fraction is small and they are more diffuse compared to dark matter. The cluster mass is given by M DM , 200 , the dark matter mass enclosed within the sphere of radius r 200 . The ionized plasma (most of the baryonic matter) in clusters is confined by the gravitational potential of the dark matter. To study the evolution of over-dense blobs in a generic ICM, we numerically compute the ICM profile by solving hydrostatic equilibrium in the NFW gravitational potential (Eq. 4) and by specifying the entropy profile of the ICM. The Xray entropy profile for the ICM, K (in keV cm 2 ) is defined to be the adiabatic invariant (for temperature in keV T keV , electron number density n e , adiabatic index \u03b3 = 5 3 ) The entropy profile of most galaxy clusters is reasonably well fit by a model given by, where the parameters K 0 , K 100 and \u03b1 are the parameters introduced by Cavagnolo et al. (2009). Combining this with the hydrostatic equilibrium equation yields for the total particle number density n , where keV=1 . 16 \u00d7 10 7 (the conversion of keV to Kelvin), m p is the proton mass, and \u00b5 / \u00b5 e / \u00b5 i are the mean mass per particle/electron/ion, governed by \u03c1 = n\u00b5m p = n e \u00b5 e m p = n i \u00b5 i m p . The total number density is the sum of electron and ion number densities n = n e + n i . We choose a constant ICM metallicity corresponding to the standard value of a third of the solar value (Leccardi & Molendi 2008); thus, \u00b5 = 0 . 62 and \u00b5 e = 1 . 17. From Eq. 6, we can get dK/dr and by specifying the number density at a point in the ICM as a boundary condition, ICM profiles for pressure, density, etc. can be solved for. The entropy profile of the ICM from Eq. 6 is a monotonically increasing function of r and hence the ICM is convectively stable according to the Schwarzschild criterion for adiabatic atmospheres. 2 An adiabatic blob will oscillate \u00a9 c in the atmosphere with the local Brunt-Vaisala oscillation frequency (c.f. Eq. 19). However, as we show later, spherical compression is important for weakly stratified medium such as the ICM. In Figure 1, we plot the profiles of different variables with NFW parameters c = 3 . 3 and M 200 = 5 . 24 \u00d7 10 14 M /circledot , and with entropy parameters K 0 = 8 keVcm 2 , K 100 = 80 keVcm 2 , \u03b1 = 1 . 1 and n ( r 200 ) = 10 -4 cm -3 . These parameters correspond to the fiducial run listed in Table 1. Our fiducial cluster has a virial radius r 200 \u2248 1603kpc and M gas /M 200 (hot gas to DM mass ratio) = 0.18. While we use the same NFW halo parameters throughout the paper (except in Fig. 9, where we investigate the blob evolution in a 10 12 M /circledot halo), we use different entropy parameters for investigating the effects of various parameters (such as the ratio of the cooling time to the free-fall time, t cool /t ff ) on the evolution of overdense blobs. The entropy parameters and density boundary conditions for the model ICMs used in our study are stated in Table 1. The entropy parameters span a range covered by cool clusters in the ACCEPT sample (Cavagnolo et al. 2009). The minimum value of the parameter t cool /t ff , which plays a very important role in governing the fate of an over-dense blob in the ICM, is also shown. We use the parameters included in Table 1 for our models ICM plasma with anisotropic conduction, and the ICM core is unstable to the heat-flux driven buoyancy instability (Quataert 2008). However, this instability saturates by reorienting field lines perpendicular to the radial direction and in the saturated state the response of a perturbed blob is stable, qualitatively similar to a convectively stable adiabatic atmosphere (Sharma et al. 2009). Therefore, we ignore the effects of thermal conduction in this paper.", "pages": [2, 3]}, {"title": "Notes", "content": "\u2020 The fiducial profile. \u2021 The \u03b2 parameter (c.f. Eq. 22) determined for blob-sizes of 10, 50, 100 pc to form cold gas at radii within which t cool /t ff falls below 10. All our runs use the average \u03b2 value of 0.0244. with spherical compression. However, as we show in section 4.2.1, the nature of the evolution of the blob over-density changes if the ratio t cool /t ff is below a critical value, for both Cartesian and spherical geometries. This critical value for Cartesian setups is 1, typically not reached by even the coolest cluster cores (see Table 1). Therefore, in order to elucidate the physics of the evolution of over-density in a Cartesian setup we artificially increase the cooling function \u039b for all Cartesian runs by a factor of 10; this way we can achieve t cool /t ff \u223c < 1 required to study the runaway cooling of over-dense blobs in a plane-parallel atmosphere.", "pages": [4]}, {"title": "3 LOCAL THERMAL INSTABILITY IN A GLOBALLY STABLE ICM", "content": "In the ICM the rate of energy loss due to free-free/boundfree/bound-bound radiation can be modeled as where n e and n i represent the electron and ion number densities and \u039b( T ) is the cooling function. We use the cooling function from Tozzi & Norman 2001 based on Sutherland & Dopita 1993, as adapted by Sharma, Parrish & Quataert 2010 (solid line in their Fig. 1). In case of thermal bremsstrahlung (which dominates above 10 7 K), the cooling function \u039b( T ) \u221d T 1 / 2 and hence P loss \u221d n 2 T 1 / 2 . Due to the observational lack of cooling flows, and given other hints of global thermal equilibrium in the ICM, we use the model of Sharma et al. (2012b) in which heating ( H ) balances average cooling at every radius in cluster cores, While this clearly is an idealization, since AGN heating is expected to be intermittent with large spatial and temporal fluctuations around thermal equilibrium, jet simulations agree well with the simulations based on the idealized model (Eq. 9; for comparison of jet simulations with the idealized model see Gaspari, Ruszkowski & Sharma 2012; Li & Bryan 2013). More importantly for this paper, our heating term is analytically tractable and captures the basic thermal state of the ICM. An important parameter, which governs the evolution of linear over-dense blobs, is the ratio of the ICM cooling time t cool (more precisely, the thermal instability time), and the free fall time t ff , in the background ICM at the location of the blob (as highlighted by McCourt et al. 2012; Sharma et al. 2012b), where the quantities subscripted with '0' stand for their unperturbed background values at the radius under consideration. The t cool /t ff profile for the fiducial cluster is included in Figure 1.", "pages": [4]}, {"title": "3.1 Evolution of a spherical blob", "content": "We base our study on the phenomenological model of PS05 and Pizzolato & Soker 2010, who propose that over-dense blobs of gas in the ICM cool faster than their surroundings, become heavier and sink to the center to feed the AGN. Both these papers consider the evolution of blobs in a background cooling flow with some simplifications. In this paper we make the more realistic assumption that the core is in rough thermal balance without any equilibrium flow. This implicitly assumes that heating keeps up with otherwise catastrophic cooling. We derive the conditions in which the over-dense blobs cool to the stable atomic phase ( T < 10 4 K ), leading to a multiphase core. The conditions for the formation of the cold phase depend on the various parameters of both the blob and the ICM. We show that for a large t cool /t ff , cold gas condensation requires a finite over-density. For a sufficiently small t cool /t ff , however, cold gas condenses out of the ICM for even a tiny over-density. Moreover, we extend the PS05 formalism to account for spherical compression that makes it easier for cold gas to condense out in a spherical geometry as compared to a plane-parallel atmosphere. Linear thermal instability analysis has been done in the Appendix section of Sharma, Parrish & Quataert 2010 (the original reference is Field 1965) in the limit t cool /greatermuch t sound (the sound crossing time for the modes). In such a scenario, neglecting gravity, we have isobaric conditions such that the perturbation always remains in pressure equilibrium with its surroundings and the linear growth rate \u03c3 of the instability is (in absence of thermal conduction), As done in PS05, we consider an over-dense spherical blob of radius a , whose parameters will be represented with primed quantities ( n ' , T ' ), having a density contrast (over-density) with respect to the ICM (unprimed quantities represent the ambient ICM): The blob is assumed to remain in pressure equilibrium at all times (i.e. P ' \u2261 P ) with the ICM in the isobaric limit of t cool /greatermuch t sound (the blob evolves isochorically in the opposite limit, which may happen for a short time when the blob temperature is close to the peak of the cooling function; e.g., see Burkert & Lin 2000; the isobaric assumption should not affect our results qualitatively), and thus, In a one-dimensional analysis, we re-write from PS05 the basic equations governing the blob evolution: where r is the radial coordinate of the blob, v is the blob's radial velocity, a is the blob radius, C is the dimensionless drag coefficient, which for most part of the paper has been taken as 0.75 (see, e.g., Churazov et al. 2001), and sign( v ) ensures that the drag force always points against the velocity. The energy equation for the over-dense blob can be written, assuming global thermal balance with a heating term (Eq. 9), 3 The primed term in the RHS of Eq. 17 is the cooling of the blob and the unprimed term represent the volume averaged heating of the ICM (and the blob). Following Pizzolato & Soker 2010 and substituting d/dt by vd/dr for the background quantities, this can be recast to obtain an equation for the over-density of the blob (assuming \u03b3 = 5 / 3): where N 2 is the square of the Brunt Vaisala frequency and represents the linear response of an over-dense blob in a stably-stratified ICM, In absence of cooling and heating we obtain stable Brunt Vaisala oscillations. These oscillations are damped in time because of the drag term in Eq. 16. Although Eq. 18 is identical to Eq. 12 in Pizzolato & Soker (2010), the interpretations are slightly different because Pizzolato & Soker (2010) assume cooling of the background ICM but ignore the background inflow, whereas we assume global thermal balance with no net inflow of the hot gas. 3 Here we are assuming that the background ICM profile is not changing with time. In reality, the background profile changes because cold gas condenses out of the hot phase and falls in, reducing the density of the remaining hot gas. In this paper our focus is on whether cold gas can condense out in global thermal equilibrium for a given ICM profile, rather than on the effect of condensation on the background profile. c \u00a9 The cooling and heating terms on the RHS of Eq. 18 in the linear ( \u03b4 /lessmuch 1) isobaric ( n ' T ' = nT ) regime can be reduced to, which when plugged in Eq. 18 gives the correct growth rate for the thermal instability in the isobaric regime (Eq. 12). Linearized versions of Eqs. 16 & 18, using above, give the expression for thermal instability (more precisely, over-stability) in a stably stratified atmosphere (e.g., see Binney, Nipoti & Fraternali 2009), where t TI = \u03b3t cool / (2 -d ln \u039b /d ln T ). 4 For a heating rate per unit volume proportional to density (e.g., as is the case for photoelectric heating), the heating term in Eq. 17 is (1 + \u03b4 ) n e n i \u039b( T ) and the corresponding linear analysis gives t TI = \u03b3t cool / (1 -d ln \u039b /d ln T ), the correct analytic result (e.g., see McCourt et al. 2012). Thus, strictly speaking, the cold gas over-density grows at the thermal instability timescale ( t TI ) and not at the cooling time ( t cool ); this is indeed verified by the simulations of McCourt et al. 2012. We use t cool instead of t TI throughout the paper because t TI = (10 / 9) t cool \u2248 t cool for the case of a constant heating rate per unit volume. Local thermal instability as the source of extended cold gas in cluster cores, with a constant heating rate per unit volume, is supported by observations (Fig. 11 in McCourt et al. 2012). Similar scaling of the heating rate with the local density is supported by the simulations of turbulent heating/mixing in equilibrium with cooling (Banerjee & Sharma 2014). During its evolution, the blob mass should be conserved. For the spherical blob of mass M , we have, where quantities with subscript '0' represent their initial/background values. Eqs. 15, 16, 18 and 21 represent a system of 4 equations in 4 variables ( r [ t ] , v [ t ] , \u03b4 [ t ] , a [ t ]) that can be solved numerically as an initial value problem to study the evolution of blobs in the ICM. We specify the initial values r 0 , \u03b4 0 and a 0 ; initial velocity is chosen to be zero because cold gas is condensing out of the gas in hydrostatic equilibrium. In absence of the drag term, Eqs. (15), (16) and (18) do not depend on the blob-size, and blobs of different size evolve in a similar fashion. Drag slows down smaller blobs and give them more time to cool before they can fall in. Drag also damps stable oscillations.", "pages": [4, 5]}, {"title": "4 RESULTS", "content": "In this section we present a phenomenological model to include the effects of geometrical compression in a spherical ICM. Later, we present the results on the evolution of overdense blobs in both Cartesian and spherical geometries. The evolution and saturation of the over-density ( \u03b4 ) of the blob is a sensitive function of the background t cool /t ff , the initial over density \u03b4 0 and the spherical compression term; there is much weaker dependence on the initial blob radius a 0 , the drag coefficient and entropy stratification (because it is weak in the ICM, as we discuss later). We construct models in which runaway cooling occurs starting with tiny amplitude of perturbations only if the background t cool /t ff \u223c < 1 , 10 in Cartesian and spherical atmospheres, respectively. If t cool /t ff is larger than the critical value for runaway at tiny amplitudes, a finite amplitude of the over-density is required for runaway. McCourt et al. (2012) and Sharma et al. (2012b) have carried out idealized simulations of hot atmospheres in thermal and hydrostatic equilibrium, using Cartesian and spherical geometries, respectively. Somewhat surprisingly, they find that it is much easier for cold gas to condense out of the hot phase in a spherical geometry. This, they attribute to spherical compression that an over-dense blob undergoes as it moves toward the centre in a spherical geometry. More quantitatively, they find that cold gas can condense out from small initial perturbations if t cool /t ff \u223c < 1 in Cartesian geometry and if t cool /t ff \u223c < 10 for a spherical setup. PS05 do not consider any influence of geometry in their analysis, and therefore their results do not depend on whether the blob is falling in a plane-parallel atmosphere or a spherical one. Unlike in a Cartesian atmosphere, in spherical geometry the gravitational forces at the diametrically opposite ends of a spherical blob (at the same height) are not parallel. The radial component of gravity pointing toward cluster center will compress the blob and make it further over-dense, in addition to the effects already accounted for in Eq. 18. Thus, an over-dense blob in a spherical geometry is compressed more than the one in a plane-parallel atmosphere, and forms cold gas more easily.", "pages": [6]}, {"title": "4.1 Modeling geometric compression", "content": "In this section, we present a simple phenomenological prescription that allows us to parametrically model the blob's compression in a spherical geometry. Considering only geometrical compression, the transverse cross-section of the blob (which subtends a constant solid angle at the centre as it falls in) decreases as it falls in toward the centre, and the blob over-density increases such that (1 + \u03b4 ) r 2 = constant ( r is the radial coordinate of the centre of the blob). The corresponding expressions for cylindrical and Cartesian geometries are (1+ \u03b4 ) R = constant ( R is the cylindrical radius) and (1+ \u03b4 ) = constant. This implies that there is no geometrical compression in the Cartesian geometry. The compression in cylindrical geometry is smaller than in a spherical atmo- ere because over-dense blobs are compressed to a line and not a point as they fall in. Numerical simulations in cylindrical geometry indeed show that the threshold t cool /t ff for the production of multiphase gas with tiny initial amplitude lies in between the results from Cartesian and spherical setups (M. McCourt, private communication). Our model fine-tuned for spherical profiles shows that the critical value of t cool /t ff for condensation of cold gas in a cylindrical geometry, starting from tiny amplitudes, ranges from 2.5 to 5 (as compared to the Cartesian and spherical cases, this value is somewhat sensitive to the background entropy profile; the analogous values for Cartesian and spherical atmospheres are 1 and 10, respectively; c. f. Figs. 4, 5). Therefore, in order to model spherical compression, we should add a term like to the right hand side of Eq. 18. This term is negligible if the entropy scale height (1 / [ d ln K/dr ]) is much smaller than r ; or in other words, if the atmosphere is strongly stratified, as we show shortly. Hence, in spherical geometry, where we have added a phenomenological modulation term (similar in spirit to the widely-used mixing-length models) to the compression term, which suppresses geometrical compres- sion when the mass encountered by the blob in the ICM becomes comparable to the blob's mass. This term is motivated by the fact that hydrodynamic effects, such as distortion due to ram pressure and the loss of sphericity of the blob, are expected to become dominant over geometrical compression, once the mass encountered is of order the blob mass. The \u03b2 parameter will be chosen so that the model gives results consistent with Sharma et al. (2012b); namely, the over-dense blobs with tiny amplitudes run away to the stable temperature if and only if t cool /t ff \u223c < 10. The mass encountered by the blob is given by integrating where \u2223 \u2223 v ( t ) \u2223 \u2223 ensures that the mass encountered by the blob increases monotonically; this is required because hydrodynamic distortion happens irrespective of the direction of blob motion. Thus, in spherical geometry we need to solve 5 equations, namely Eqs. 15, 16, 21, 22, and 23 for 5 unknowns (the additional unknown compared to PS05 being M encr ). The spherical compression term (the last term in Eq. 22) can be linearized as -2 v/r . The linearized equation governing the evolution of \u03b4 , including spherical compression, becomes where we have used the linearized equation of motion dv/dt = -g\u03b4 . From Eq. 19, N 2 = g/ ( \u03b3H ), where H = 1 / ( d ln K/d ln r ) is the entropy scale height. It is clear from Eq. 24 that the spherical compression term leads to linear instability if H > r/ (2 \u03b3 ), even in the absence of thermal instability. As expected, the local plane-parallel approximation, with a negligible effect of spherical compression, holds in the limit H /lessmuch r . The entropy scale height of the ICM is comparable to r , and hence the amplitude of over-dense blobs grows linearly because of spherical compression; the amplitude saturates nonlinearly when the blob encounters its own mass in the ICM and the spherical compression term in Eq. 22 is suppressed (see the line with square markers in Fig. 2 which shows the evolution with spherical compression in the absence of cooling). When t cool /t ff \u223c > 10 over-density never becomes large or becomes large only when blob falls to the very center. However, with t cool /t ff \u223c < 10 local thermal instability can lead to extended cold gas. The t cool /t ff profiles for cool-core clusters have a characteristic shape with a minimum in t cool /t ff at \u223c 10 kpc (see Fig. 1). There are two points where t cool /t ff = 10, on either side of the minimum. We find out the value of \u03b2 for each cluster such that the blob released at the outer point where t cool /t ff \u2248 10 (lets call it r 10 ) would just form cold gas on reaching 1 kpc. 5 For such a value of \u03b2 , no cold gas forms when r 0 > r 10 , i.e., the region where t cool /t ff > 10. We chose the outer r 10 to determine \u03b2 because the outer regions with t cool /t ff \u223c < 10 are most likely to form extended cold gas. For starting radii well within the bottom of the t cool /t ff 'cup' and for small over-densities the infall time is shorter than the cooling time, and runaway happens very close to the center (at r < 1 kpc). For the analytic model to be useful, the value of \u03b2 should not vary for different clusters and blob sizes. We list the \u03b2 values for various ICM profiles and blob parameters in Table 1. Note that the smaller blobs require a smaller \u03b2 ; i.e., higher compression, to form extended cold gas. This is because a smaller blob encounters its own mass in the ICM faster than a bigger one. For all our runs we use the average of all \u03b2 s listed in Table 1; i.e., \u03b2 = 0 . 0244. As we show, our results are not very sensitive to the exact value of \u03b2 . Figure 2 shows the evolution of 50 pc blobs with a small initial over-density ( \u03b4 0 = 10 -5 ), released from initial radii of 10 and 40 kpc in the fiducial ICM with spherical compression. The evolution of the blobs is qualitatively similar to those in PS05, except that cold gas condenses for a longer cooling time because of spherical compression. If the cooling time is long, the over-density grows (because of spherical compression) but saturates with a small amplitude, and the blob undergoes stable oscillations (dotted lines); the blob falls in from its initial position by a substantial distance because it becomes further over-dense due to the spherical compression term. The blob released at the location (10 kpc) with a smaller t cool /t ff ( < 10; see Fig. 1) increases its over-density to a large value ( \u223c > 10 4 ) and falls in toward the center at an accelerated rate (solid line). In Figure 2 the dotted and dashed lines correspond to the blobs released at the same radius (40 kpc) but with and without the \u03b2 modulation term (Eq. 22). The t cool /t ff ratio is 13.4 at the release radius of 40 kpc; the \u03b2 parameter is chosen such that the blobs released outside of the t cool /t ff = 10 region do not run away. For \u03b2 = 0, spherical compression acts even at smaller radii and makes the blob cool in a runaway fashion even if t cool /t ff > 10, but \u03b2 = 0 . 0244 recovers the result from numerical simulations; i.e., runaway cooling for small initial over-densities happens only if t cool /t ff < 10. Therefore, the inclusion of the modulation term in Eq. 22 is necessary to get a quantitative match with numerical simulations. The value of \u03b2 = 0 . 0244 is the mean of \u03b2 s that were obtained by requiring cold gas ( \u03b4 \u223c > 10 4 ) to form only if t cool /t ff < 10 for the profiles in Table 1 (for blob-sizes of 10, 50, 100 pc); the \u03b2 range is narrow, from 0.021 to 0.028 for a blob-size of 50 pc, and does not vary systematically for a given blob-size. Figure 3 shows the blob location and the over-density as a function of time, with different initial radii, for the fiducial profile and for the profile with t cool /t ff > 10 everywhere. As expected, with \u03b2 adjusted such that extended cold gas is produced only if t cool /t ff < 10, all initial radii for the fiducial profile except for 30 kpc (at which the background t cool /t ff > 10) lead to the formation of multiphase gas. The blob released at 5 kpc cools to the stable temperature very close to the center. Even in the most thermally unstable cases, multiphase gas appears only at r \u223c < 5 kpc. For the high entropy model shown in Figure 3 none of the blobs released at radii > 5 kpc produce cold gas, even when they fall in toward the center; the cooling time is longer than the inflow time. We can relate the cold gas condensing out at large radii ( > 1 kpc) and at small radii ( < 1 kpc) to the observational appearance of extended atomic filaments and centrally concentrated cold gas, respectively, as observed by McDonald et al. 2010. Thus, the results of our phenomenological model are consistent with the observational and computational results, which show that extended cold gas condenses out only if t cool /t ff \u223c < 10. Cold gas condenses out much farther out if the core density is high and t cool /t ff \u223c < 10 close to 100 kpc.", "pages": [6, 7, 8]}, {"title": "4.2 Extended cold filaments with tiny over-density: the critical t cool /t ff", "content": "There are three ways in which an over-dense blob evolves in the ICM: it can saturate at low amplitude ( \u03b4 < 1) away from the cluster center or becomes dense only at small radii ( < 1 kpc) as it falls in; it can cool to a large over-density far from the center ( > 1 kpc) and then fall in; it can become under-dense and overheated such that \u03b4 \u2192 1 as it moves out. We expect thermal runaway to happen because the ICM is locally thermally unstable. Only below 10 4 K, below which the cooling function decreases rapidly ( \u221d T 6 ), is the blob expected to stop cooling further. In isobaric conditions the blob temperature T ' and the over-density ( \u03b4 ) have an inverse proportionality, the cold gas can reach a maximum over density \u03b4 \u223c > 100 (positive runaway) 6 due to cooling. On the other hand, if the blob starts becoming under-dense with time, it rushes away from the ICM center due to its lower density, and in this case \u03b4 \u2192-1, implying a negligibly small density of the blob. In both these extreme cases of runaway, the blob either forms hot or very cold gas compared to the background ICM. These runaway cases are required for the formation of extended multiphase gas in the ICM. We define a 'multiphase flag' (mp) to assess the formation of extended multiphase gas in our models as follows: We focus on positive runaways in which there is formation of cold gas ( T ' < 10 4 K ), which sinks and feeds the central black hole required for the feedback cycle to close, as discussed earlier. As already mentioned, McCourt et al. (2012) and Sharma et al. (2012b), in their Cartesian and spherical simulations of locally unstable hydrostatic gas in global thermal balance, find that runaway cooling of even slightly over-dense blobs occurs if the ratio of the cooling time to the free-fall time ( t cool /t ff ) is smaller than a critical value. This critical value, ( t cool /t ff ) crit , is dependent on the geometry of the gravitational potential in a weakly stratified atmosphere such as the ICM. Namely, multiphase gas condenses out of the ICM if t cool /t ff \u223c < 10 in spherical potential and if t cool /t ff \u223c < 1 in Cartesian geometry.", "pages": [8]}, {"title": "4.2.1 Critical t cool /t ff in a plane-parallel atmosphere", "content": "We use the same set of equations as PS05 to study the evolution of over-dense blobs in a plane-parallel atmosphere. PS05 emphasized that the ICM required nonlinear density perturbations in order for multiphase gas to condense out of the ICM. However, they missed the significance of spherical compression brought to the fore by the idealized simulations of McCourt et al. (2012); Sharma et al. (2012b), and included in this paper phenomenologically in Eq. 22. As shown by numerical simulations, and as we show later, this new compression term can lead to the condensation of cold gas from arbitrarily small perturbations in cluster cores if t cool /t ff \u223c < 10. Additionally, and understandably, because galaxy cluster cores have t cool /t ff \u223c > 10, PS05 overlooked that there was another regime for the local thermal instability where arbitrarily small perturbations can lead to the condensation of cold gas even in a plane-parallel atmosphere. We find, in agreement with 6 Once the over-dense blob reaches \u03b4 \u223c 10, it cools very rapidly to the stable temperature (see Fig. 3). Therefore, an over-density of 100 is reached at almost the same radius at which the blob cools to T ' \u2248 10 4 K. Moreover, a different \u03b4 corresponds to the temperature of the stable phase; e.g., \u03b4 corresponding to the stable phase in a 1 keV group is \u2248 10 3 and in a 10 6 K galactic halo is \u2248 100. McCourt et al. (2012), that in a plane-parallel atmosphere cold gas condenses out starting from a small amplitude if t cool /t ff \u223c < 1. As already mentioned, we achieve t cool /t ff \u223c < 1 in the ICM by artificially increasing the cooling function \u039b by a factor of 10 for plane-parallel models. We start with a tiny over-density ( \u03b4 0 = 10 -5 ), and initialize the blobs at multiple radii r 0 , and integrate the blob evolution equations for 5 Gyr for different 1-D ICM profiles given in Table 1. In Figure 4 we plot the multiphase flag as a function of t cool /t ff (at the initial location where the blob is released) in a Cartesian potential, starting with a small over-density. It is evident that for t cool /t ff > 1, there are no runaways seen for small \u03b4 0 . The critical t cool /t ff in a Cartesian gravity setup using the simple model of PS05 is ( t cool /t ff ) crit = 1. The critical value of t cool /t ff is fairly insensitive to the blob size, the initial over-density and the drag coefficient.For t cool /t ff \u223c < 1 thermal instability forms multiphase gas irrespective of the blob parameters, as seen in the simulations of McCourt et al. (2012).", "pages": [8, 9]}, {"title": "4.2.2 Critical t cool /t ff in a spherical ICM", "content": "We now incorporate the geometrical compression model discussed earlier in section 4.1, to study blob evolution in a spherical setup and determine the critical t cool /t ff for runaway to occur with tiny over-densities. Using spherical simulations Sharma et al. 2012b find the critical value for the condensation of cold gas to be ( t cool /t ff ) crit \u2248 10. This critical criterion agrees with the observations of cool cluster cores which show extended cold filaments (Fig. 11 in McCourt et al. 2012). Our compression model includes a free parameter \u03b2 , which we adjust to match the critical t cool /t ff from simulations and observations, as discussed in section 4.1. For all our spherical models we use \u03b2 = 0 . 0244, the average of \u03b2 s obtained for various blob and ICM parameters. Clusters with K 0 \u223c > 30 keVcm 2 , have t cool /t ff > 10 everywhere. Hence, extended cold phase is not expected in such clusters for small amplitudes. Our models with spherical compression show a similar trend. Figure 5 shows the results of our model by plotting the multiphase flag of Eq. 25 against t cool /t ff for over-dense blobs starting at various positions in some of our clusters in Table 1. It is evident that multiphase condensation takes place for arbitrarily small amplitudes only when t cool /t ff \u223c < 10, as found in numerical simulations by Sharma et al. 2012b.", "pages": [9]}, {"title": "4.3 Multiphase gas with long cooling times", "content": "In cases where the background t cool /t ff is higher than the critical value, there is a competition between cooling and freefall, and cooling to low temperatures occurs only with a finite initial over-density ( \u03b4 0 ). This should be contrasted with the cases where t cool /t ff \u223c < t cool /t ff , crit , and cold gas condenses out even with a tiny over-density. We release stationary blobs at varying initial radii, r 0 (ranging from 10 to 100 kpc), to achieve a spectrum of t cool /t ff > t cool /t ff , crit ) and track their evolution for 5 Gyr in both Cartesian and spherical potentials. 2 In both Cartesian and spherical atmospheres, the trends seen for the critical initial over-density ( \u03b4 0 ,c ) required to form extended cold phase outside of 1 kpc are similar. This is intuitively expected; as t cool /t ff increases, cooling becomes less effective and the blob needs a higher density contrast to cool faster than it can fall in to form extended cold gas. The geometrical compression term in Eq. 22 increases the critical value of t cool /t ff , above which a finite large over-density is required to form cold phase. In Figure 6, we show the dependence of \u03b4 0 , c on the background t cool /t ff for the fiducial and K 0 = 1 keV cm 2 profiles using a variety of blob sizes. It is seen that in the absence of geometrical compression, smaller blobs cool more easily. The blob-size enters the equations via the drag term in the equation of motion (Eq. 16). A smaller blob (for a given overdensity), or equivalently a larger drag term, implies that the blob is slowed down more as it is falling in, and therefore has a longer time to cool compared to a bigger blob. This explains a smaller critical \u03b4 0 for smaller blobs. and the domination of thermal instability growth for a small t cool /t ff . Figure 8 shows the critical over-density required for runaway cooling as a function of t cool /t ff in spherical and Carte- sian geometries for the highest entropy model in Table 1. The critical over-density has a characteristic shape for both Cartesian and spherical atmospheres. Also, note that the minimum over-density required for runaway cooling does not correspond to the release radius with minimum t cool /t ff ; it corresponds to a radius slightly inward of the minimum. The asymmetry in the response of over-dense blobs inward and outward of the minimum is likely responsible for the presence of extended cold filaments in some cool cluster cores (McDonald et al. 2010). Somewhat counter-intuitively, the blobs farther away from the center (with a slightly longer cooling time than at the center) are more likely to result in cold gas because the blobs at inner radii fall in before they can cool to the stable temperatures. This leads to spatially extended cold gas over 10s of kpc, rather distinct from the centrally concentrated cold gas.", "pages": [9, 10]}, {"title": "5 DISCUSSION & CONCLUSIONS", "content": "In this paper we have presented a phenomenological model for the evolution of over-dense blobs in the ICM. We have extended the model of Pizzolato & Soker (2005) to include the important influence of geometrical compression. A com- 10 cool ff \u00a9 c parison of idealized simulations in spherical (Sharma et al. 2012b) and plane-parallel (McCourt et al. 2012) atmospheres show that it is much easier for multiphase gas to condense in the presence of spherical compression. We have incorporated a phenomenological spherical compression term in the model of PS05; this increases the over-density as the blob falls in toward the center. With a single adjustable parameter ( \u03b2 in Eq. 22), which is analogous to the mixing length, we are able to obtain the key result of Sharma et al. (2012b); i.e., cold gas condenses out at large radii starting from tiny perturbations when the ratio of the cooling time and the local free-fall time ( t cool /t ff ) in the ICM is \u223c < 10. In the following we discuss various astrophysical implications of our results. (i) Robustness of the phenomenological model: Our phenomenological model, which extends PS05's treatment to account for spherical compression, agrees well with the results of numerical simulations with just a single adjustable parameter ( \u03b2 ; see Eq. 22) analogous to the mixing length. Moreover, the parameter is fairly insensitive to various parameters such as the blob-size, release radius, entropy profiles, halo mass, etc. We have also applied our model (with the same \u03b2 parameter = 0.0244) to a much lower mass Milky-Way-like halo with the halo mass of 10 12 M /circledot . Unlike clusters, for such halos cooling is important even at the virial radius and the hot 4 gas density is expected to be much lower (e.g., see Fig. 1 of Sharma et al. 2012a). We have followed the evolution of slightly over-dense ( \u03b4 = 10 -5 ) 50 pc blobs starting at rest from various initial radii for two plausible hot gas profiles a constant entropy ( K = 80 keV cm 2 ) core and a constant t cool /t ff = 7 'core'. The top panel of Figure 9 shows the entropy and t cool /t ff profiles that we have used. While the constant t cool /t ff profile is expected to form extended multiphase gas irrespective of the initial radius, the constant entropy profile should give runaway cooling only at large radii ( \u223c > 80 kpc) where t cool /t ff \u223c < 10. For the constant entropy profile t cool /t ff at 80 kpc is 12.5 and at 60 kpc it is 20; therefore, runaway cooling occurs only if t cool /t ff slightly exceeds 10. This shows the robustness of our model (which uses the same value of the \u03b2 parameter that was fine-tuned for the massive cluster) which gives the threshold of t cool /t ff \u223c < 10 for runaway cooling, even if the halo mass and the ICM profiles are changed drastically. (ii) Interplay of local thermal instability and gravity: Cold gas condenses out much closer in than the location of the minimum of t cool /t ff . This is seen in the idealized simulations (e.g., Fig. 4 in Sharma et al. 2012b shows gold gas only extended out to 5 kpc but t cool /t ff < 10 out to 20 kpc for the corresponding ICM profile) and in our analytic models (e.g., Fig. 3 shows that the blob released at 20 kpc cools to the stable phase only within 5 kpc). This result has implications for the location of high velocity clouds in Milky Way-like galaxies, the farthest of which are believed to have condensed out of the hot halo gas. While t cool /t ff \u223c < 10 even out to the virial radius for 10 12 M /circledot halos (e.g., Sharma et al. 2012a), the cold gas (emitting 21 cm line) may exist well within the halo (see also Fig. 9). This is consistent with the high velocity clouds detected out to 50 kpc from the center of the Andromeda galaxy (Thilker et al. 2004). Note that the cooling time close to the virial radius in Figure 9 can be comparable to the Hubble time and the approximation of a constant halo mass, etc. break down. However, the qualitative results of our model are expected to hold. Cold gas in galaxy clusters and groups can be pushed farther out compared to the predictions of our analytic models and idealized simulations with thermal balance because of the large velocities generated by AGN-driven bubbles. Malagoli, Rosner & Bodo (1987), Balbus & Soker (1989) and Binney, Nipoti & Fraternali (2009), among others, have investigated the interplay of local thermal instability and gravity. These works have highlighted the importance of stable stratification in thermally unstable atmospheres, because of which the instability is an over-stability with fast oscillations. Using a Lagrangian analysis, Balbus & Soker (1989) have shown that the thermal instability is suppressed in presence of a background cooling flow because the background conditions change over the same timescale as the instability itself. This was verified in numerical simulations of cooling flows with finite density perturbations (see the Appendix section of Sharma et al. 2012b) which show that large amplitude density perturbations are required for multiphase gas to condense out of the cooling flows. However, in the state of thermal balance (which is strongly favored by observations over the past decade) there is no cooling flow and the background state is quasi-static. In such a state even tiny density perturbations can become nonlinear if the cooling time of the background ICM is short enough (i.e., t cool /t ff \u223c < 10). The linear response of a strongly-stratified atmosphere in thermal and hydrostatic balance is quite simple (Eq. 20) and shows over-stability (see the line marked by circles in Fig. 2) damped nonlinearly by the drag term. In a weakly-stratified spherical ICM, however, a slightly over-dense blob becomes further over-dense because of the spherical compression term (see Eq. 24; spherical compression is unimportant if the entropy scale height is much smaller than the radius) and the heavier blob falls inward with a large velocity, at which point the drag force is large and stops the blob from falling in further. Moreover, the spherical compression term in Eq. 22 becomes weaker as the blob encounters a mass comparable to its own mass. Thus, spherical compression is crucial in making it easier for the over-dense blobs to condense out of the ICM. By constructing strongly stratified artificial ICM profiles, we have verified that the spherical compression term is not effective when the entropy scale height is much smaller than the blob release radius (i.e., if the atmosphere is strongly stratified; see Eq. 24). We can also understand the dependence of the critical t cool /t ff on the background entropy profile in cylindrical geometry from Eq. 24 (as mentioned in section 4.1, the critical value varies from 2.5 to 5 for typical cool cluster entropy profiles). For weak entropy stratification of the ICM the spherical compression term in Eq. 24 overwhelms the oscillatory N 2 term. For cylindrical geometry, however, the compression term is half of its spherical value ( -g/r instead of -2 g/r in Eq. 24) and even weak stratification of the ICM decreases the critical t cool /t ff for condensation. Therefore, in cylindrical geometry the critical t cool /t ff is larger if the ICM is weakly stratified (e.g., K \u221d r 0 ) and smaller if it is even slightly stratified (e.g., K \u221d r 1 . 1 ). Stratification is not expected to be very strong for astrophysical coronae, which by definition, are close to the virial temperature. Thus, our t cool /t ff \u223c < 10 criterion for the condensation of extended cold gas is expected to be valid in most astrophysical coronae, ranging from the solar corona to hot accretion flows (e.g., see Sharma 2013; Das & Sharma 2013; Gaspari, Ruszkowski & Oh 2013). (iii) Extended multiphase gas and cold feedback: The accretion of cold gas by the central massive black holes in cluster cores is essential for AGN feedback to close the globally stable feedback loop. Hot accretion rate via Bondi accretion is small and not as sensitively dependent on the ICM density ( \u221d n ) as radiative cooling ( \u221d n 2 ). Therefore, accretion in the hot phase alone seems incapable of globally balancing radiative cooling. Moreover, the Bondi accretion rate is about two orders of magnitude smaller than the multiphase mass cooling rate close to 10 kpc, which should be of order the black hole accretion rate (e.g., see Gaspari, Ruszkowski & Oh 2013). Moreover, as discussed in detail in section 5.1 of Sharma et al. (2012b), the cold feedback model may also naturally account for the observed correlation between the estimated Bondi accretion rate and the jet power. Our phenomenological model tries to provide a physical basis for the condensation of cold gas in the ICM. In particular, it tries to explain the large quantitative difference in the condition for the condensation of multiphase gas in a planeparallel and a spherical atmosphere; namely, t cool /t ff \u223c < 1 versus 10 for the condensation of cold gas starting from small perturbations. Thus, we have provided a firmer footing to the cold feedback paradigm by extending PS05's model to agree with observations and numerical simulations. We also argue that the characteristic shape of t cool /t ff profile with a minimum at \u223c 10 kpc (rather than right at the center) is very crucial. Because of this, a large amount of gas far away from the sphere of influence of the supermassive black hole (unlike Bondi accretion) can condense out and episodically boost the accretion rate (due to cold gas) by a large amount. This can rather effectively stop the cooling flow and make t cool /t ff > 10 throughout, and the condensation of cold gas is suppressed. The core can form again because of a feeble accretion rate in the hot mode, and once t cool /t ff \u223c < 10 again, the core can suffer heating due to AGN jets/cavities and the global cycle continues. The presence of extended cold gas also agrees with the observations of McDonald et al. (2010) and others, which \u00a9 c show extended cold gas at 10s of kpc where t cool /t ff \u223c < 10 (e.g., see Fig. 11 of McCourt et al. 2012).", "pages": [10, 11, 12, 13]}, {"title": "ACKNOWLEDGEMENTS", "content": "The authors thank Noam Soker for his comments on the paper. AS thanks the Indian Academy of Sciences, Indian National Science Academy and the National Academy of Sciences India for awarding the Summer Research Fellowship 2012 with grant number PHYS1231. This work is partly supported by the DST-India grant no. Sr/S2/HEP-048/2012.", "pages": [13]}, {"title": "REFERENCES", "content": "Balbus S. A., Soker N., 1989, ApJ, 341, 611 Banerjee N., Sharma P., 2014, MNRAS, 443, 687 Binney J., Nipoti C., Fraternali F., 2009, MNRAS, 397, 1804 Burkert A., Lin D. N. C., 2000, ApJ, 537, 270 Cavagnolo K. W., Donahue M., Voit G. M., Sun M., 2009, ApJS, 182, 12 Churazov E., Bruggen M., Kaiser C. 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2024PDU....4401441L | https://arxiv.org/pdf/2308.12556.pdf | <document>
<section_header_level_1><location><page_1><loc_12><loc_87><loc_87><loc_92></location>Relic abundance of dark matter with coannihilation in non-standard cosmological scenarios</section_header_level_1>
<section_header_level_1><location><page_1><loc_34><loc_81><loc_65><loc_83></location>Fangyu Liu, Hoernisa Iminniyaz ∗</section_header_level_1>
<text><location><page_1><loc_23><loc_75><loc_76><loc_79></location>School of Physics Science and Technology, Xinjiang University, Urumqi 830017, China</text>
<section_header_level_1><location><page_1><loc_46><loc_72><loc_54><loc_73></location>Abstract</section_header_level_1>
<text><location><page_1><loc_16><loc_48><loc_84><loc_70></location>Weinvestigate the relic abundance of dark matter from coannihilation in non-standard cosmological scenarios. We explore the effect of coannihilation on the relic density of dark matter and freeze out temperature in quintessence model with kination phase and brane world cosmological scenarios. Since the Hubble expansion rate is enhanced in quintessence and brane world cosmological models, it causes the larger relic density compared to that in the standard one. On the other hand, the relic density of dark matter is decreased due to the coannihilation in the standard cosmological scenario. After including coannihilation in quintessence or brane world cosmological scenarios, we find the decrease of the relic density of dark matter is slightly slower than that in the standard cosmological scenario.</text>
<section_header_level_1><location><page_2><loc_11><loc_90><loc_33><loc_92></location>1 Introduction</section_header_level_1>
<text><location><page_2><loc_11><loc_69><loc_89><loc_88></location>Recent Planck results provide the abundance of Dark Matter (DM) as Ω DM h 2 = 0 . 120 ± 0 . 001 [1] . Although the precise value of DM abundance is known from the observation, the nature of DM is still a mystery for the scientists. The Weakly Interacting Massive Particles (WIMPs) are the most popular candidates for DM. The best motivated one is the lightest supersymmetric particle (LSP) neutralino which is stabalized due to R-Parity, we denote it by χ 1 . It is usually assumed that WIMPs were in full thermal and chemical equilibrium when the interaction rate of DM particles Γ is greater than the expansion rate H of the universe. WIMPs freeze out when Γ <H .</text>
<text><location><page_2><loc_11><loc_43><loc_89><loc_68></location>The Big-Bang nucleosynthesis (BBN) successfully predicted the abundances of light element isotopes D, 3 He, 4 He, and 7 Li, and it is known that the nucleosynthesis takes place at the temperature scale ∼ 1 Mev [2]. On the other hand, we don't have observational information of the universe prior to BBN. Therefore, it is necessary to consider scenarios that could modify the cosmic expansion rate in the period before BBN and some of which yielded significant physical implications. In non-standard cosmological scenarios, the expansion rate of the universe is larger or smaller than the standard one. The evolution of number density n of DM is described by the Boltzmann equation [3]. Solving the Boltzmann equation, we find that the non-standard expansion rate leaves its imprint on the relic density of DM [4, 5]. In order not to spoil the predictions of BBN, the non-standard cosmic expansion rate must returned to the form of the radiation-dominated era at the beginning of BBN.</text>
<text><location><page_2><loc_11><loc_8><loc_89><loc_42></location>There are several kinds of modified cosmological scenarios. One example is the quintessence model with kination phase. If the universe experience a period called 'kination', in which the energy density ρ ( a ) of the universe is dominated by the kinetic energy of the scalar field with a being the scale factor, then ρ ( a ) drops very quickly as 1 /a 6 [6, 7, 8, 9, 10, 11]. The abundance of DM can be significantly affected by this kind of fast evolution. [8, 9] discussed the consequences of kination period on the relic abundance of DM. They found the resultant relic density is increased due to the enhanced Hubble expansion rate in kination model. Another example of the modified cosmology is the brane world scenario which also predicted the Hubble expansion rate of the universe is different from the standard cosmology [12, 13, 14, 15, 16]. In brane world scenario, the ordinary matter is assumed to be confined onto a three-dimensional subspace, called brane which is embedded in a higher dimensional spacetime, named bulk. Brane world scenario is relatively different from the standard cosmology of four-dimensional universe. Extra-dimension effects may play an important role in the early universe. the DM relic density is also affected by the physics of extra dimensions. [17, 18] explored the impact of brane cosmology on the DM relic density. They conclude the relic density is considerably</text>
<text><location><page_3><loc_11><loc_90><loc_57><loc_91></location>enhanced compared to that in the standard cosmology.</text>
<text><location><page_3><loc_11><loc_23><loc_89><loc_89></location>The standard calculation of relic density of DM included the annihilations of LSP only. When the next-to-the-lightest supersymmetric particles (NLSP) are slightly heavier than the LSP, the relic abundances of LSP is determined both by the annihilation cross section of LSP and the annihilation of the heavier particles. The heavier particles later decay into the LSP [19]. This process is called 'coannihilation'. In [19], the authors considered the case where the squark's mass is slightly heavier than the LSP in the standard cosmological scenario. They found the relic abundance of the LSP is significantly reduced because of the coannihilation. In our work, we investigate the relic density of LSP with coannihilation in the non-standard cosmological scenarios, which includes quintessence model with kination phase and brane world cosmology. In these two non-standard cosmological models, we plot the ratio of the relic density of DM without coannihilation to the case including coannihilation as a function of the relative mass splitting for different modifications and cross section enhancements, here the relative mass spilitting is the ratio of the mass difference between NLSP and LSP to the mass of LSP. For comparison we also plot the ratio of the relic density as a function of the relative mass splitting in the standard cosmological scenario to the non-standard cosmological scenarios which includes coannihilation. The enhanced Hubble expansion rate leads to the earlier particle freeze out and increased relic density in kination and brane world cosmological scenarios. On the other hand, the coannihilations reduce the relic density of DM. After considering the coannihilation in the non-standard cosmology, we found the decrease of the relic density in those models is slower than the standard cosmological scenario. The effect of coannihilation on the relic density depends on the size of the modification factor in nonstandard cosmology. For larger modification, the increase of the DM relic density is sizable in kination model and brane world cosmology, therefore, after including coannihilation, there is slight decrease of the relic abundance in contrast with the minor modification of Hubble rate. In other words, compared to the increase of the relic density because of the enhanced Hubble expansion rate, the decrease of the relic density due to coannihilation of DM is insignificant for larger modification. We found the constraints on the Hubble enhancement factors and the relative mass splitting which let the coannihilation has effect on the relic density of DM.</text>
<text><location><page_3><loc_11><loc_9><loc_89><loc_22></location>We organize the paper as follows. In section 2, we briefly review the quintessence model with kination phase and brane world cosmology. In section 3, we discuss the coannihilation in the standard cosmological scenario and then extend it to the non-standard cosmological scenarios. We investigate the effect of coannihilation on the relic abundance in detail in quintessence model with kination phase and brane world cosmology. The last section is devoted to the conclusion and summary.</text>
<section_header_level_1><location><page_4><loc_11><loc_86><loc_89><loc_92></location>2 Expansion rate of the universe in kination and brane world cosmology</section_header_level_1>
<text><location><page_4><loc_14><loc_82><loc_70><loc_84></location>The expansion rate of the universe is given by Friedmann equation</text>
<formula><location><page_4><loc_40><loc_76><loc_89><loc_81></location>( d a d t ) 2 + k = 8 πGρa 2 3 , (1)</formula>
<text><location><page_4><loc_11><loc_67><loc_89><loc_74></location>where G is the Newton's gravitational constant. The contribution of non-relativistic and relativistic matter to quantity ρa 2 grows as '1 /a ' and '1 /a 2 ' respectively. At sufficiently early times, a → 0, the constant ' k ' can be neglected[6]. Then, Eq.(1) becomes</text>
<formula><location><page_4><loc_44><loc_63><loc_89><loc_67></location>H 2 = 8 πGρ 3 , (2)</formula>
<text><location><page_4><loc_11><loc_58><loc_89><loc_62></location>here H = ˙ a/a . In the standard cosmological scenario, the dominant radiation energy density ρ rad is</text>
<formula><location><page_4><loc_43><loc_54><loc_89><loc_58></location>ρ rad = π 2 30 g ∗ T 4 . (3)</formula>
<text><location><page_4><loc_11><loc_50><loc_89><loc_54></location>Here g ∗ is the effective number of relativistic degrees of freedom. Therefore, the standard expansion rate of the universe is</text>
<formula><location><page_4><loc_42><loc_45><loc_89><loc_50></location>H std = 2 πT 2 m Pl √ πg ∗ 45 (4)</formula>
<text><location><page_4><loc_14><loc_41><loc_89><loc_42></location>Quintessence is a time-varying vacuum energy that depends on one or more scalar fields</text>
<text><location><page_4><loc_11><loc_42><loc_57><loc_46></location>where m Pl ≡ 1 / √ G = 1 . 22 × 10 19 GeV is Planck mass.</text>
<text><location><page_4><loc_11><loc_38><loc_55><loc_40></location>[6]. Considering a single real scalar field, its action is</text>
<formula><location><page_4><loc_30><loc_31><loc_89><loc_37></location>I ϕ = -∫ d 4 x √ -Detg [ 1 2 g µν ∂ϕ ∂x µ ∂ϕ ∂x ν + V ( ϕ ) ] , (5)</formula>
<text><location><page_4><loc_11><loc_27><loc_89><loc_31></location>where g µν is Robertson-Walker metric and ϕ depends only on the time due to the homogeneity. Varying the field δϕ ( t ), the action is unchanged δI ϕ = 0, then one can get the field equation</text>
<formula><location><page_4><loc_36><loc_22><loc_89><loc_25></location>d 2 ϕ ( t ) d t 2 +3 H d ϕ ( t ) d t + d V ( ϕ ) d ϕ = 0 . (6)</formula>
<text><location><page_4><loc_11><loc_18><loc_60><loc_20></location>The energy density ρ ϕ = 1 2 ˙ ϕ 2 + V is derived from Eq.(6) as</text>
<formula><location><page_4><loc_30><loc_14><loc_89><loc_17></location>ρ ϕ = ρ ϕ ( t 0 ) e -∫ t t 0 6 1+ β ( t ) H ( t ) dt = ρ ( t 0 ) e -∫ a a 0 6 1+ β ( a ) da a , (7)</formula>
<text><location><page_4><loc_11><loc_8><loc_89><loc_12></location>where β is defined as β ≡ V ( ϕ ) / ( 1 2 ˙ ϕ 2 ) [7] . The energy density of a special period called 'kination' is obtained from Eq.(7). During 'kination' period, the kinetic energy 1 2 ˙ ϕ 2 dominates</text>
<text><location><page_5><loc_11><loc_87><loc_89><loc_91></location>over the potential energy V ( ϕ ) i.e. 1 2 ˙ ϕ 2 /greatermuch V and β → 0. At kination phase, performing the integration in Eq.(7), then ρ ϕ is obtained as [7]</text>
<formula><location><page_5><loc_46><loc_83><loc_89><loc_86></location>ρ ϕ ∝ 1 a 6 . (8)</formula>
<text><location><page_5><loc_11><loc_75><loc_89><loc_82></location>The expansion rate in kination model is constrained by BBN [8], which means the expansion rate in kination model returns to the standard case before BBN( ∼ 1 MeV) starts. Defining η as</text>
<formula><location><page_5><loc_44><loc_72><loc_89><loc_75></location>η = ρ ϕ ( T r ) ρ rad ( T r ) , (9)</formula>
<text><location><page_5><loc_11><loc_67><loc_89><loc_71></location>we choose T r = 10 MeV and when η /lessmuch 1, the universe is radiation dominated. Combining Eq.(3) and Eq.(9) with Eq.(8), again using the entropy conservation</text>
<formula><location><page_5><loc_36><loc_62><loc_89><loc_66></location>sa 3 = constant i.e. a ∝ 1 Tg 1 / 3 ∗ s , (10)</formula>
<text><location><page_5><loc_11><loc_57><loc_89><loc_61></location>where g ∗ s is the effective number of entropic degrees of freedom, then the expression of the energy density ρ ϕ is written as [7, 8, 9]</text>
<formula><location><page_5><loc_36><loc_51><loc_89><loc_56></location>ρ ϕ = η ρ rad ( T r ) [ g ∗ s ( T ) g ∗ s ( T r ) ] 2 ( T T r ) 6 . (11)</formula>
<text><location><page_5><loc_11><loc_47><loc_89><loc_51></location>Now, we can have the relationship between the expansion rates in the standard cosmology and kination model,</text>
<formula><location><page_5><loc_38><loc_43><loc_89><loc_47></location>H 2 que = 8 πG 3 ρ rad (1 + ρ ϕ ρ rad ) . (12)</formula>
<text><location><page_5><loc_11><loc_41><loc_59><loc_43></location>Inserting Eq.(3) and Eq.(11) into Eq.(12), one can obtain</text>
<formula><location><page_5><loc_33><loc_35><loc_89><loc_40></location>H que H std = √ 1 + η g ∗ ( T r ) g ∗ ( T ) [ g ∗ s ( T ) g ∗ s ( T r ) ] 2 ( T T r ) 2 . (13)</formula>
<text><location><page_5><loc_11><loc_30><loc_89><loc_34></location>We use the dimensionless quantity x = m 1 /T with m 1 being the mass of the DM particle χ 1 , then</text>
<text><location><page_5><loc_11><loc_9><loc_89><loc_25></location>In Ref.[12, 13, 14, 15, 16], authors provided comprehensive introduction to brane cosmology. In brane world cosmology, standard model particles are confined on a three-dimensional subspace '3-brane', embedded in a higher dimensional spacetime. RS II model is the simple and interesting brane world model which is proposed by Randall and Sundrum (RS) [13]. In this model, the 4-dimensional universe is realized on the 3-brane with a positive tension, located at the ultra-violet boundary of the five dimensional Anti de-Sitter spacetime. In this framework, the expansion rate is given by the modified Friedmann equation,</text>
<formula><location><page_5><loc_33><loc_25><loc_89><loc_30></location>H que H std = √ 1 + η g ∗ ( x r ) g ∗ ( x ) [ g ∗ s ( x ) g ∗ s ( x r ) ] 2 ( x r x ) 2 . (14)</formula>
<formula><location><page_5><loc_39><loc_5><loc_89><loc_8></location>H 2 bra = 8 πG 3 ρ rad (1 + ρ rad 2 λ ) , (15)</formula>
<text><location><page_6><loc_11><loc_90><loc_35><loc_91></location>where λ is the brane tension</text>
<formula><location><page_6><loc_44><loc_86><loc_89><loc_90></location>λ = 48 πM 6 5 m 2 Pl , (16)</formula>
<text><location><page_6><loc_11><loc_80><loc_89><loc_85></location>here M 5 is the five-dimensional Plank mass. In the brane world cosmological scenario, the radiation energy density becomes dominant at BBN ( ∼ 1 Mev) too,</text>
<formula><location><page_6><loc_37><loc_77><loc_89><loc_80></location>ρ rad 2 λ /lessmuch 1 , when T = 1 MeV , (17)</formula>
<text><location><page_6><loc_11><loc_67><loc_89><loc_75></location>i.e. M 5 /greatermuch 1 . 1 × 10 4 GeV, where g ∗ (1 MeV) = 10. In addition, the precision measurements of the gravitational law in submillimeter range give the further limit on M 5 > 10 8 GeV [12]. In this work, we consider the constraint by BBN. Therefore, the relationship between the expansion rates in standard cosmology and brane world cosmological model is</text>
<formula><location><page_6><loc_33><loc_60><loc_89><loc_65></location>H bra H std = √ 1 + ρ rad ρ 0 = √ 1 + πg ∗ m 2 Pl m 4 1 2880 M 6 5 x 4 . (18)</formula>
<section_header_level_1><location><page_6><loc_11><loc_52><loc_89><loc_58></location>3 Relic density with coannihilation in non-standard cosmology</section_header_level_1>
<text><location><page_6><loc_11><loc_36><loc_89><loc_50></location>In Refs.[19, 20, 21], authors provided detailed analysis to coannihilaiton. Here we review the Boltzmann equation obtained in Ref.[19] including the coannihilation very briefly. For a series of supersymmetric particles χ i with increasing mass m 1 < · · · m i < · · · m j < · · · m N , with internal degrees of freedom g 1 , · · · g i , · · · g j , · · · g N , the Boltzmann equation for the total number densities n = ∑ N i=1 n i of particles χ i is</text>
<formula><location><page_6><loc_33><loc_32><loc_89><loc_37></location>d n d t = -3 Hn -N ∑ i , j=1 〈 σ ij v 〉 ( n i n j -n eq i n eq j ) , (19)</formula>
<text><location><page_6><loc_11><loc_19><loc_89><loc_31></location>where 〈 σ ij v 〉 is the thermal average of the pair annihilation cross sections of the particles χ i and χ j times their relative velocity, n eq i is the equilibrium value of the number density. Since the scattering rate of supersymmetric particles off particles in the thermal background is much faster than their annihilation rate, then n i /n is well approximated by its equilibrium value n eq i /n eq , i.e. n i /n ≈ n eq i /n eq , where n eq i ≈ g i ( m i T/ 2 π ) 3 / 2 exp( -m i /T ). Defining</text>
<formula><location><page_6><loc_34><loc_14><loc_89><loc_18></location>r i ≡ n eq i n eq = g i (1 + ∆ i ) 3 / 2 exp( -x ∆ i ) g eff , (20)</formula>
<text><location><page_6><loc_11><loc_10><loc_37><loc_13></location>where ∆ i = ( m i -m 1 ) /m 1 , and</text>
<formula><location><page_6><loc_35><loc_4><loc_89><loc_10></location>g eff = N ∑ i=1 g i (1 + ∆ i ) 3 / 2 exp( -x ∆ i ) , (21)</formula>
<text><location><page_7><loc_11><loc_90><loc_74><loc_91></location>then Eq.(19) transforms into the standard form of the Boltzmann equation</text>
<formula><location><page_7><loc_36><loc_84><loc_89><loc_88></location>d n d t = -3 Hn -〈 σ eff v 〉 ( n 2 -n 2 eq ) , (22)</formula>
<text><location><page_7><loc_11><loc_82><loc_16><loc_83></location>where</text>
<formula><location><page_7><loc_21><loc_76><loc_89><loc_82></location>σ eff = N ∑ ij σ ij r i r j = N ∑ ij σ ij g i g j g 2 eff (1 + ∆ i ) 3 / 2 (1 + ∆ j ) 3 / 2 exp[ -x (∆ i +∆ j )] . (23)</formula>
<text><location><page_7><loc_11><loc_70><loc_89><loc_76></location>Now, we analyze the relic density of DM with coannihilation in non-standard cosmology. For convenience, a useful definition is applied as Y ≡ n/s . In kination and brane world models, using Eqs.(14), (18) and t = 0 . 301 g -1 / 2 ∗ m Pl x 2 /m 2 , the Boltzmann Equation (22) is written as</text>
<formula><location><page_7><loc_33><loc_65><loc_89><loc_69></location>d Y d x = -2 π 2 45 g ∗ s m 3 1 x -4 H nstd 〈 σ eff v 〉 ( Y 2 -Y 2 eq ) , (24)</formula>
<text><location><page_7><loc_11><loc_62><loc_16><loc_64></location>where</text>
<text><location><page_7><loc_11><loc_34><loc_89><loc_57></location>Here H nstd is the Hubble rate in the non-standard cosmological scenarios which includes quintessence model with kination phase and brane world cosmology. We solve the Boltzmann equation (24) by following the standard picture of the DM particle evolution [22]. DM particles are in thermal equilibrium at high temperature in the early universe. When the temperature decreases, the equilibrium number densities of the DM particles drops exponentially. In the end, the interaction rate becomes smaller than the expansion rate, then the DM particles decoupled from the equilibrium state and their number densities become almost constant from that freeze out point. Usually, the non-standard cosmic expansion rate is greater than the standard cosmic expansion rate prior to BBN. It leads the DM particles decouple earlier than in the standard cosmology.</text>
<formula><location><page_7><loc_24><loc_57><loc_89><loc_62></location>Y eq = N ∑ i=1 n eq i s = 0 . 145 x 3 / 2 e -x g ∗ s [ N ∑ i=1 g i (1 + ∆ i ) 3 / 2 exp( -x ∆ i ) ] . (25)</formula>
<text><location><page_7><loc_11><loc_28><loc_89><loc_33></location>While the Boltzmann equation (24) can be computed numerically, it is still useful to obtain the analytic solution for Eq.(24). We rewrite it in terms of δ = Y -Y eq ,</text>
<formula><location><page_7><loc_30><loc_24><loc_89><loc_28></location>d δ d x = -d Y eq d x -2 π 2 45 g ∗ s m 3 1 x -4 H nstd 〈 σ eff v 〉 δ [ δ +2 Y eq ] . (26)</formula>
<text><location><page_7><loc_11><loc_16><loc_89><loc_23></location>The solution of equation (26) is considered in two regimes. Y tracks its equilibrium value at high temperature, in this situation, δ 2 and d δ/ d x are negligible. The Boltzmann equation is simplified to</text>
<formula><location><page_7><loc_35><loc_13><loc_89><loc_17></location>d Y eq d x = -4 π 2 45 g ∗ s m 3 1 x -4 H nstd 〈 σ eff v 〉 δ Y eq . (27)</formula>
<text><location><page_7><loc_11><loc_10><loc_43><loc_12></location>The approximate solution of Eq.(27) is</text>
<formula><location><page_7><loc_37><loc_4><loc_89><loc_9></location>δ ≈ g 1 / 2 ∗ x 2 H nstd 0 . 528 g ∗ s m 1 m Pl 〈 σ eff v 〉 H std , (28)</formula>
<text><location><page_8><loc_11><loc_87><loc_89><loc_91></location>where d Y eq / d x ≈ -Y eq is applied for x /greatermuch 1. This solution for δ is used to fix the scaled freeze out temperature x f .</text>
<text><location><page_8><loc_11><loc_83><loc_89><loc_87></location>At late times ( x /greatermuch x f ), δ ≈ Y /greatermuch Y eq , therefore Y eq and d Y eq / d x can be neglected, then Eq.(26) becomes</text>
<formula><location><page_8><loc_37><loc_79><loc_89><loc_83></location>d δ d x = -2 π 2 45 g ∗ s m 3 1 x -4 H nstd 〈 σ eff v 〉 δ 2 . (29)</formula>
<text><location><page_8><loc_11><loc_73><loc_88><loc_78></location>Assuming δ ( x f ) /greatermuch δ ( ∞ ), an approximate solution is obtained by integrating Eq.(29) from x f to ∞ ,</text>
<text><location><page_8><loc_11><loc_65><loc_89><loc_69></location>The modified expansion rate and coannihilation both affect the scaled freeze out temperature x f . Freeze out occurs when δ is of the same order as the equilibrium value of Y ,</text>
<formula><location><page_8><loc_27><loc_67><loc_89><loc_74></location>Y ( ∞ ) = g 1 / 2 ∗ 0 . 264 g ∗ s m Pl m 1 1 ∫ ∞ x f ( H std /H nstd ) 〈 σ eff v 〉 /x 2 d x . (30)</formula>
<formula><location><page_8><loc_42><loc_61><loc_89><loc_63></location>δ ( x f ) = c Y eq ( x f ) , (31)</formula>
<text><location><page_8><loc_11><loc_52><loc_89><loc_60></location>where c is a constant. When c = √ 2 -1, the approximate result matches with the numerical one well [22]. Substituting Eq.(28) into Eq.(31), the scaled freeze out temperature x f including coannihilation with the modified expansion rate is obtained</text>
<formula><location><page_8><loc_30><loc_44><loc_89><loc_51></location>x f = ln ( 0 . 076 c g eff m 1 m Pl 〈 σ eff v 〉 g 1 / 2 ∗ x 1 / 2 H std H nstd )∣ ∣ ∣ ∣ x = x f , (32)</formula>
<text><location><page_8><loc_11><loc_36><loc_89><loc_47></location>∣ Considering a certain DM candidate and its pair annihilation cross section 〈 σv 〉 , the modified mechanism ' H nstd + coannihilation ' brings a significant difference on the relic abundance compared to that in the standard mechanism ' H std + pair annihilation '. The differences are as follows,</text>
<formula><location><page_8><loc_21><loc_24><loc_89><loc_31></location>Ω nstd+co = Ω std ∫ ∞ x f, std 〈 σv 〉 /x 2 d x ∫ ∞ x f, nstd+co ( H std /H nstd ) 〈 σ eff v 〉 /x 2 d x , (33a)</formula>
<text><location><page_8><loc_11><loc_10><loc_89><loc_22></location>∣ where Ω is the present-day mass density divided by the critical density, Ω ≡ ρ χ /ρ c with ρ χ = nm 1 = s 0 Y χ m 1 and the critic density ρ c = 3 H 2 0 m 2 Pl / 8 π , where s 0 /similarequal 2900 cm -3 is the present entropy density and H 0 is the Hubble constant, x f, std is the scaled freeze out temperature in standard cosmology with pair annihilation. Here</text>
<formula><location><page_8><loc_21><loc_18><loc_89><loc_26></location>x f, nstd+co = x f, std +ln ( x 1 / 2 f, std x 1 / 2 f, nstd+co H std H nstd g eff g 1 〈 σ eff v 〉 〈 σv 〉 | x = x f, std )∣ ∣ ∣ ∣ x = x f, nstd+co , (33b)</formula>
<formula><location><page_8><loc_39><loc_5><loc_89><loc_9></location>Ω std h 2 = s 0 m 1 h 2 ρ c Y std ( ∞ ) (34)</formula>
<text><location><page_9><loc_11><loc_90><loc_14><loc_91></location>and</text>
<text><location><page_9><loc_11><loc_82><loc_84><loc_85></location>here h = 0 . 674 ± 0 . 005 is the scaled Hubble constant in units of 100 km s -1 Mpc -1 [1].</text>
<formula><location><page_9><loc_32><loc_83><loc_89><loc_90></location>Y std ( ∞ ) = g 1 / 2 ∗ 0 . 264 g ∗ s m Pl m 1 1 ∫ ∞ x f, std 〈 σv 〉 /x 2 d x . (35)</formula>
<text><location><page_9><loc_11><loc_74><loc_89><loc_82></location>Due to the presence of the non-standard expansion rates and coannihilation, the relic density of DM changes significantly compared to that in the standard case. We follow the example taken by [19], for the case of two particles system, neutralino ˜ χ (denoted by χ 1 ) and squark ˜ q (denoted by χ 2 ), since</text>
<formula><location><page_9><loc_24><loc_70><loc_89><loc_73></location>σ 22 (˜ q ¯ ˜ q → gg ) ≈ ( α s /α ) σ 12 (˜ χ ˜ q → q g ) ≈ ( α s /α ) 2 σ 11 (˜ χ ˜ χ → q ¯ q ) , (36)</formula>
<text><location><page_9><loc_11><loc_68><loc_32><loc_69></location>Eqs.(23) and (21) become</text>
<formula><location><page_9><loc_31><loc_62><loc_89><loc_67></location>σ eff = σ 11 [ 1 + Aw 1 + w ] 2 , g eff = g 1 (1 + w ) , (37)</formula>
<text><location><page_9><loc_11><loc_56><loc_89><loc_62></location>where w = (1+∆) 3 / 2 exp( -x ∆) g 2 /g 1 , A = α s /α , ∆ = ( m 2 -m 1 ) /m 1 , ' g , q ' denote a gluon and quark respectively, ' α s , α ' are the strong-interaction coupling and the electroweak coupling respectively.</text>
<figure>
<location><page_9><loc_16><loc_14><loc_82><loc_54></location>
<caption>Figure 1: Evolution of the relic abundance Y of DM in quintessence model with kination phase and brane world cosmology as a function of the inverse-scaled temperature x . (a) and (c) are plotted without coannihilation; (b) and (d) are with coannihilation. Here m 1 = 100 GeV, g 1 = 2, g ∗ = 90.</caption>
</figure>
<text><location><page_10><loc_11><loc_59><loc_89><loc_91></location>Following, we discuss the effect of modified expansion rate on the relic density of DM particle with coannihilation. The relic abundance Y of DM with and without coannihilation as a function of the inverse-scaled temperature x in quintessence model with kination phase and brane world cosmology is plotted in Fig.1. We set g ∗ = g ∗ s = constant for simplicity and take m 1 = 100 GeV, T r = 10 MeV, g ∗ = g ∗ s = 90, A = 20, g 2 /g 1 = 3, x f, std = 22. Panels (a) and (c) are for the case which is not including coannihilation in kination model and brane world cosmology; (b) and (d) are plotted with coannihilation. We found the enhanced Hubble expansion rate in kination model and brane world cosmology accelerates the decoupling of DM particles and results larger values for the relic density. The size of increase depends on the modification factor η in kination model and the five-dimensional Planck mass M 5 in brane world cosmology. If we compare panel (a) with panel (b), we learn that the DM relic density is reduced considerably after including coannihilation. The same result is obtained for the case of brane world cosmology as shown in panels (c) and (d). When η = 0 and M 5 = 5 × 10 6 GeV, the standard case is recovered.</text>
<figure>
<location><page_10><loc_13><loc_35><loc_85><loc_57></location>
<caption>Figure 2: The ratio of the relic density of DM including only neutralino pair annihilation σ 11 to the case including coannihilation σ eff as a function of the relative mass spliting ∆ in kination model and brane world cosmology. Here m 1 = 100 GeV, g 1 = 2, g ∗ = 90.</caption>
</figure>
<text><location><page_10><loc_11><loc_6><loc_89><loc_22></location>Fig.2 shows the ratio of the relic density of DM as a function of relative mass splitting ∆ for different modifications in kination model and brane world cosmology without coannihilation to the case including coannihilation. Panel (a) is for the constant cross section ( s -wave) which corresponds to the freeze out temperature x f, std = 22 in kination model; panel (b) is for the brane world cosmology. Here the line with plus sign is plotted using the numerical solution of Eq.(24). This figure shows the extent of the coannihilation effect on the relic density in non-standard cosmology. The Hubble expansion rate is enhanced in kination and brane world</text>
<text><location><page_11><loc_11><loc_73><loc_89><loc_91></location>cosmological scenarios. DM particles decouple earlier than the standard case, which caused the larger relic density compared to the standard cosmology. After including coannihilation, the relic abundance is decreased due to the coannihilation. The extent of the decrease depends on the size of modification. When the modification factor is large, the increase of the DM relic density will be sizable. Therefore, after including coannihilation, there is slight decrease of the relic abundance for larger modification in contrast with the minor modification. The same result is obtained for the brane world cosmology in panel (b). The effect of coannihilation is less sizable for smaller M 5 .</text>
<figure>
<location><page_11><loc_13><loc_49><loc_85><loc_71></location>
<caption>Figure 3: The ratio of the relic density of DM as a function of relative mass spliting ∆ in standard cosmology including only neutralino pair annihilation σ 11 to the kination model and brane world cosmology including coannihilation σ eff . Here m 1 = 100 GeV, g 1 = 2, g ∗ = 90.</caption>
</figure>
<text><location><page_11><loc_11><loc_6><loc_89><loc_36></location>In Fig.3, we plot the ratio of the relic density in the standard cosmological scenario which only takes into neutralino pair annihilation to the non-standard cosmology which includes coannihilation as a function of the relative mass splitting ∆. Here, we analyze the difference of the relic abundance resulted by the combined effect of non-standard expansion and coannihilation. The decrease of relic density of DM because of coannihilation is quite mild in non-standard cosmology due to the enhancement of Hubble rate. After including the coannihilation in non-standard cosmology, the decrease of the relic density is slower than the standard cosmology. We take an example of η = 10 -3 in plot ( a ), when the relative mass splitting ∆ is 0.04, the ratio of the relic density in the standard cosmology is 35.7 and in kination model is 5.2. It means there is already sizable decrease of the relic density with coannihilation in the standard cosmolgy while the effect starts to be important in kination model. The same result is obtained for the brane world cosmology in panel (b). We also noticed that for larger modification, the effect of coannihilation is insignificant in Fig.3. Fig.4 shows the constraints</text>
<text><location><page_12><loc_11><loc_85><loc_89><loc_91></location>on η , M 5 and ∆ for which the coannihilation to be important. In the shaded regions in panel ( a ) and ( b ), Ω std / Ω nstd+co > 1. It means in those shaded regions the coannihilations let the DM relic density decrease.</text>
<text><location><page_12><loc_30><loc_81><loc_30><loc_83></location>Ω</text>
<text><location><page_12><loc_32><loc_81><loc_33><loc_83></location>Ω</text>
<text><location><page_12><loc_65><loc_81><loc_66><loc_83></location>Ω</text>
<text><location><page_12><loc_68><loc_81><loc_69><loc_83></location>Ω</text>
<figure>
<location><page_12><loc_13><loc_61><loc_85><loc_82></location>
<caption>Figure 4: Contour plots of the relative mass splitting ∆ and η ( M 5 ) when Ω std / Ω nstd+co > 1. Here A = 20, m 1 = 100 GeV, g 1 = 2, g ∗ = 90. x f, std = 22.</caption>
</figure>
<figure>
<location><page_12><loc_13><loc_27><loc_85><loc_49></location>
<caption>Figure 5: The ratio of the relic density of DM in non-standard cosmology without coannihilation to the case including coannihilation as a function of the relative mass splitting ∆ for different cross section enhancements A . Here m 1 = 100 GeV, g 1 = 2, g ∗ = 90.</caption>
</figure>
<text><location><page_12><loc_11><loc_6><loc_89><loc_15></location>The ratio of the relic density of DM in non-standard cosmology without coannihilation to the case including coannihilation as a function of the relative mass splitting ∆ for different cross section enhancements A is plotted in Fig.5. We note that the limit for relative mass splitting for which the coannihilation to be significant in non-standard cosmology is smaller</text>
<text><location><page_13><loc_11><loc_59><loc_89><loc_91></location>than the standard one. As long as ∆ < 0 . 1 for A = 10, there is sizable effect of coannihilation on the relic abundance in standard cosmology while the limit becomes ∆ < 0 . 08 in kination model for η = 10 -3 and ∆ < 0 . 05 in brane world cosmology for M 5 = 1 × 10 6 GeV. It means when the second lightest particle's mass difference between the lightest particle's mass is within about 8% for kination and 5% for brane world cosmology, the coannihilation becomes relevant. From that figures we can determine how close in mass χ 2 must be for the coannihilation effect to be important. The decrease of the relic density is larger for the larger enhancement factor A in kination and brane world model. The decrease of relic density is also slower in non-standard cosmology for different A in comparison with the standard one. The range of relative mass splitting for which the coannihilation to be important is smaller in those models than in the standard cosmology. For example when A=40, the coannihilation effect is sizable within mass differences of 15% for standard cosmology and 14% for kination, 11% for brane world cosmology in Fig.5. The enhanced expansion rate always makes the effect of coannihilation to be weak in different extents.</text>
<figure>
<location><page_13><loc_13><loc_35><loc_85><loc_57></location>
<caption>Figure 6: Ω nstd / Ω nstd+co as a function of the relative mass splitting ∆ for different g 2 /g 1 . Here m 1 = 100 GeV, g 1 = 2, g ∗ = 90.</caption>
</figure>
<text><location><page_13><loc_11><loc_16><loc_89><loc_24></location>Fig.6 is for Ω nstd / Ω nstd+co as a function of relative mass splitting ∆ when g 2 /g 1 takes different values. The effect of coannihilation is more relevant when g 2 /g 1 is larger. On the other hand, when the extent of the modification is larger in kination and brane world cosmology, the effect of coannihilation is weaker.</text>
<figure>
<location><page_14><loc_31><loc_70><loc_67><loc_91></location>
<caption>Figure 7: The scaled freeze out temperature in non-standard cosmology including coannihilation as a function of the relative mass splitting ∆. Here m 1 = 100 GeV, g 1 = 2, g ∗ = 90, x f, std = 22.</caption>
</figure>
<text><location><page_14><loc_11><loc_50><loc_89><loc_59></location>The scaled freeze out temperature in non-standard cosmology including coannihilation as a function of the relative mass splitting is shown in Fig.7. The scaled freeze out temperature is decreased in non-standard cosmology including coannihilation with respect to the case of stadard one.</text>
<section_header_level_1><location><page_14><loc_11><loc_44><loc_31><loc_46></location>4 Conclusion</section_header_level_1>
<text><location><page_14><loc_11><loc_19><loc_89><loc_42></location>We discussed the coannhilation effect on the relic density of DM in the quintessence model with kination phase and brane world cosmology. The Hubble expansion rate is modified in non-standard cosmological scenarios. It leaves its imprint on the relic density of DM particle. There is increased relic density of DM due to the enhanced Hubble expansion rate. On the other hand, the coannihilation mechanism reduces the DM relic density in the standard cosmological scenario. We found the coannihilations also decrease the relic density of DM in non-standard cosmological scenarios, while the enhanced expansion rate always makes the coannihilation effect to be weak in various extents. The reduction of relic density because of coannihilation depends on the size of the modification in non-standard cosmological scenarios. The decrease is mild for larger modification.</text>
<text><location><page_14><loc_11><loc_7><loc_89><loc_18></location>If compared the relic density of DM in standard cosmology including only LSP pair annihilation to the case of non-standard cosmology with coannihilation, we note the coannihilation effect is insignificant for larger modification. The reason is that the relic density is enhanced sizably when the modification is large in kination and brane world models. We found constraints on the modification and relative mass splitting for which the coannihilation to be</text>
<text><location><page_15><loc_11><loc_87><loc_89><loc_91></location>relevant. The effect of coannihilation on the relic density is also more relevant for the larger cross section enhancement A and g 2 /g 1 .</text>
<text><location><page_15><loc_11><loc_83><loc_89><loc_87></location>Our result is important to know what extent of the coannihilating particle's mass should be in order to the coannihilation effect is sizable in the non-standard cosmological scenarios.</text>
<section_header_level_1><location><page_15><loc_11><loc_77><loc_36><loc_79></location>Acknowledgments</section_header_level_1>
<text><location><page_15><loc_11><loc_71><loc_89><loc_74></location>The work is supported by the National Natural Science Foundation of China (2020640017, 11765021, 2022D01C52).</text>
<section_header_level_1><location><page_15><loc_11><loc_65><loc_26><loc_67></location>References</section_header_level_1>
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2013arXiv1305.6370F | https://arxiv.org/pdf/1305.6370.pdf | <document>
<section_header_level_1><location><page_1><loc_12><loc_80><loc_88><loc_86></location>Three-Dimensional MHD Simulations of Emerging Active Region Flux in a Turbulent Rotating Solar Convective Envelope: the Numerical Model and Initial Results</section_header_level_1>
<text><location><page_1><loc_34><loc_76><loc_66><loc_78></location>Y. Fan, N. Featherstone 2 , and F. Fang</text>
<text><location><page_1><loc_12><loc_71><loc_88><loc_74></location>High Altitude Observatory, National Center for Atmospheric Research 1 , 3080 Center Green Drive, Boulder, CO 80301</text>
<section_header_level_1><location><page_1><loc_44><loc_66><loc_56><loc_67></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_17><loc_21><loc_83><loc_62></location>We describe a 3D finite-difference spherical anelastic MHD (FSAM) code for modeling the subsonic dynamic processes in the solar convective envelope. A comparison of this code with the widely used global spectral anlastic MHD code, ASH (Anelastic Spherical Harmonics), shows that FSAM produces convective flows with statistical properties and mean flows similar to the ASH results. Using FSAM, we first simulate the rotating solar convection in a partial spherical shell domain and obtain a statistically steady, giant-cell convective flow with a solarlike differential rotation. We then insert buoyant toroidal flux tubes near the bottom of the convecting envelope and simulate the rise of the flux tubes in the presence of the giant cell convection. We find that for buoyant flux tubes with an initial field strength of 100 kG, the magnetic buoyancy largely determines the rise of the tubes although strong down flows produce significant undulation and distortion to the shape of the emerging Ω-shaped loops. The convective flows significantly reduce the rise time it takes for the apex of the flux tube to reach the top. For the weakly twisted and untwisted cases we simulated, the apex portion is found to rise nearly radially to the top in about a month, and produce an emerging region (at a depth of about 30 Mm below the photosphere) with an overall tilt angle consistent with the active region tilts, although the emergence pattern is more complex compared to the case without convection. Near the top boundary at a depth of about 30 Mm, the emerging flux shows a retrograde zonal flow of about 345 m/s relative to the mean flow at that latitude.</text>
<section_header_level_1><location><page_2><loc_42><loc_85><loc_58><loc_86></location>1. Introduction</section_header_level_1>
<text><location><page_2><loc_12><loc_41><loc_88><loc_82></location>If we believe that active regions on the solar surface originate from a strong toroidal magnetic field generated at the base of the convection zone by the solar dynamo mechanism, then we need to understand how active-region-scale flux tubes rise through the turbulent solar convection zone to the surface. Recently significant insight has been gained in this area from a series of work (Weber et al. 2011, 2012) conducted using a thin flux tube model driven via the drag force term by a time dependent giant-cell convective flow with a solar like differential rotation, computed separately from a 3D global convection simulation with the Anelastic Spherical Harmonic (ASH) code (Miesch et al. 2006). Because of the low computational cost for the 1-D thin flux tube model, a large number of simulations of rising flux tubes with a range of initial field strengths, fluxes, initial latitudes, and sampling different time spans of the convective flow field were carried out. Meaningful statistics on the properties of the emerging tubes in regard to the latitude of emergence, tilt angles, apparent zonal motion, and clustering in longitudes of emergence (i.e. active longitudes) is obtained. It is found that the dynamic evolution of the flux tube changes from convection dominated to magnetic buoyancy dominated as the initial field strength increases from 15 kG to 100 kG. At 100 kG, the development of Ω-shaped rising loops is mainly controlled by the growth of the magnetic buoyancy instability, with the strongest convective downdrafts capable of producing moderate undulations on the emerging loops. It is found that although helical convection promotes mean tilts towards the observed Joy's law trend, results still favor stronger fields ( > 40 kG) for the initial toroidal tubes to avoid too large a tilt angle scatter produced by convection to be consistent with the observations (Weber et al. 2012).</text>
<text><location><page_2><loc_12><loc_10><loc_88><loc_39></location>Although the thin flux tube model essentially preserves the frozen-in condition for the evolution of the flux tube and allows for a large number of simulations to achieve meaningful statistics, it is highly idealized. It ignores the 3D nature of the magnetic field evolution and assumes the tube is a cohesive object. In parallel to the thin flux tube calculations, several self-consistent 3D global MHD simulations of rising flux tubes in a rotating spherical shell of solar convection and the associated mean flows have been carried out (Jouve & Brun 2009; Jouve et al. 2013) using the ASH code. These simulations study rather large flux tubes (significantly greater than the flux contained in typical active regions) due to the limited numerical resolutions in the global scale simulations. It is found that the rise velocity and the characteristics of the emerging loops are strongly affected by the convective motions when loops of less than 10 5 G are considered. In addition, the question of how strong buoyant flux tubes can self-consistently form from dynamo generated mean field and rise to the surface is also being addressed in a set of full global convective dynamo simulations of a fast rotating stellar envelope with 3 times the solar rotation rate (Nelson et al. 2011, 2013a,b), also using the ASH code.</text>
<text><location><page_3><loc_12><loc_38><loc_88><loc_86></location>In this paper, we describe a new 3D Finite-difference Spherical Anelastic MHD (FSAM) code for modeling the subsonic dynamics of the turbulent solar convective envelope. The code uses a modified Lax-Friedrichs scheme (as described in the Appendix) for computing the upwinded fluxes in the advection terms, which allows for stable numerical integration of the anelastic MHD equation with no explicit diffusion. Of course numerical diffusion is present, but is minimized in smooth regions which helps to preserve the frozen-in condition of the magnetic field evolution in the simulations of rising flux tubes. We carry out a comparison between the FSAM code and the ASH code with a simulation of rotating convective flow in a spherical shell. We find that even with the absence of the polar region (necessary due to the polar singularity associated with the latitude-longitude grid discretization), the FSAM code can produce convective flows with similar statistical properties and mean flow properties as the fully global ASH spectral code. We then use the FSAM code to conduct a simulation of rotating solar convection in a spherical shell wedge domain driven at the lower boundary by the diffusive heat flux corresponding to the solar luminosity. We obtain a statistically steady solution of giant-cell convection with a solar-like differential rotation. Into the giantcell convective flow, we then insert buoyant toroidal flux tubes with an initial field strength of 10 5 G near the bottom of the envelope, to study how the tubes rise under the presence of convection. We find that the buoyant loops rise based on the initial magnetic buoyancy distribution and also are significantly reshaped by the strong convective downdrafts. They can rise to the surface nearly radially, and produce emerging regions with radial flux distribution of the two polarities that are consistent with the observed mean tilt angles of solar active regions. At a depth of about 30 Mm below the photosphere, the emerging flux shows a retrograde zonal motion in the midst of the prograde flow of the banana cells, with a speed of ∼ -350 m/s relative to the mean plasma zonal flow at the emerging latitude.</text>
<section_header_level_1><location><page_3><loc_37><loc_32><loc_63><loc_34></location>2. The Numerical Model</section_header_level_1>
<text><location><page_3><loc_16><loc_29><loc_79><loc_30></location>We solve the following anelastic MHD equation in a spherical shell domain:</text>
<formula><location><page_3><loc_44><loc_25><loc_88><loc_27></location>∇· ( ρ 0 v ) = 0 , (1)</formula>
<formula><location><page_3><loc_20><loc_20><loc_88><loc_24></location>ρ 0 [ ∂ v ∂t +( v · ∇ ) v ] = 2 ρ 0 v × Ω -∇ p 1 + ρ 1 g + 1 4 π ( ∇× B ) × B + ∇· D (2)</formula>
<formula><location><page_3><loc_13><loc_16><loc_88><loc_19></location>ρ 0 T 0 [ ∂s 1 ∂t +( v · ∇ )( s 0 + s 1 ) ] = ∇· ( Kρ 0 T 0 ∇ s 1 ) -( D·∇ ) · v + 1 4 π η ( ∇× B ) 2 -∇· F rad (3)</formula>
<formula><location><page_3><loc_46><loc_13><loc_88><loc_15></location>∇· B = 0 (4)</formula>
<formula><location><page_3><loc_35><loc_9><loc_88><loc_12></location>∂ B ∂t = ∇× ( v × B ) -∇× ( η ∇× B ) , (5)</formula>
<formula><location><page_4><loc_44><loc_83><loc_88><loc_86></location>ρ 1 ρ 0 = p 1 p 0 -T 1 T 0 , (6)</formula>
<formula><location><page_4><loc_42><loc_79><loc_88><loc_82></location>s 1 c p = T 1 T 0 -γ -1 γ p 1 p 0 , (7)</formula>
<text><location><page_4><loc_12><loc_62><loc_88><loc_78></location>where s 0 ( r ), p 0 ( r ), ρ 0 ( r ), T 0 ( r ), and g = -g 0 ( r )ˆ r denote the profiles of entropy, pressure, density, temperature, and the gravitational acceleration of a time-independent, reference state of hydrostatic equilibrium and nearly adiabatic stratification, c p is the specific heat capacity at constant pressure, γ is the ratio of specific heats, and v , B , s 1 , p 1 , ρ 1 , and T 1 are the dependent variables of velocity, magnetic field, entropy, pressure, density, and temperature to be solved that describe the changes from the reference state. In equation (2), Ω denotes the solid body rotation rate of the Sun and is the rotation rate of the frame of reference, where Ω = 2 . 7 × 10 -6 rad s -1 , and D is the viscous stress tensor:</text>
<formula><location><page_4><loc_37><loc_57><loc_88><loc_61></location>D ij = ρ 0 ν [ S ij -2 3 ( ∇· v ) δ ij ] , (8)</formula>
<text><location><page_4><loc_12><loc_52><loc_88><loc_55></location>where ν is the kinematic viscosity, δ ij is the unit tensor, and S ij is given by the following in spherical polar coordinates:</text>
<formula><location><page_4><loc_45><loc_48><loc_88><loc_52></location>S rr = 2 ∂v r ∂r (9)</formula>
<formula><location><page_4><loc_42><loc_44><loc_88><loc_48></location>S θθ = 2 r ∂v θ ∂θ + 2 v r r (10)</formula>
<formula><location><page_4><loc_34><loc_40><loc_88><loc_44></location>S φφ = 2 r sin θ ∂v φ ∂φ + 2 v r r + 2 v θ r sin θ cos θ (11)</formula>
<formula><location><page_4><loc_37><loc_36><loc_88><loc_39></location>S rθ = S θr = 1 r ∂v r ∂θ + r ∂ ∂r ( v θ r ) (12)</formula>
<formula><location><page_4><loc_33><loc_32><loc_88><loc_35></location>S θφ = S φθ = 1 r sin θ ∂v 2 ∂φ + sin θ r ∂ ∂θ ( v φ sin θ ) (13)</formula>
<formula><location><page_4><loc_35><loc_28><loc_88><loc_31></location>S φr = S rφ = 1 r sin θ ∂v r ∂φ + r ∂ ∂r ( v φ r ) . (14)</formula>
<text><location><page_4><loc_12><loc_21><loc_88><loc_27></location>Futhremore, K in equation (3) denotes the thermal diffusivity, and η in equations (5) and (3) denotes the magnetic diffusivity. The last term in equation (3) is a heating source term due to the radiative diffusive heat flux F rad in the solar interior, where</text>
<formula><location><page_4><loc_42><loc_16><loc_88><loc_20></location>F rad = 16 σ s T 0 3 3 κρ 0 ∇ T 0 , (15)</formula>
<text><location><page_4><loc_12><loc_13><loc_75><loc_14></location>and σ s is the Stephan-Boltzman constatn, κ is the Rosseland mean opacity.</text>
<text><location><page_5><loc_12><loc_81><loc_88><loc_86></location>Using equations (6) and (7) to express ρ 1 in terms of p 1 and s 1 in equation (2), and after some manipulations using the ideal gas law and hydrostatic balance for the reference state, we obtain</text>
<formula><location><page_5><loc_26><loc_72><loc_88><loc_79></location>ρ 0 [ ∂ v ∂t +( v · ∇ ) v ] = 2 ρ 0 v × Ω -ρ 0 ∇ ( p 1 ρ 0 ) + ρ 0 g 0 s 1 c p ˆ r + 1 4 π ( ∇× B ) × B + ∇· D . (16)</formula>
<text><location><page_5><loc_12><loc_69><loc_85><loc_71></location>Note, in deriving the above equation, we have ignored terms of higher order in δ , where</text>
<formula><location><page_5><loc_42><loc_65><loc_88><loc_68></location>δ ≡ dln T 0 dln p 0 -γ -1 γ (17)</formula>
<text><location><page_5><loc_12><loc_58><loc_88><loc_63></location>is the non-dimensional super-adiabaticity of the reference stratification, and its magnitude is glyph[lessmuch] 1 in the anelatic approximation. The super-adiabaticity δ is related to the entropy gradient of the reference state as follows:</text>
<formula><location><page_5><loc_44><loc_53><loc_88><loc_57></location>d s 0 d r = -c p δ H p 0 , (18)</formula>
<text><location><page_5><loc_12><loc_50><loc_17><loc_52></location>where</text>
<formula><location><page_5><loc_41><loc_47><loc_88><loc_51></location>H p 0 = -( dln p 0 d r ) -1 (19)</formula>
<text><location><page_5><loc_12><loc_45><loc_40><loc_46></location>denotes the pressure scale height.</text>
<text><location><page_5><loc_12><loc_38><loc_88><loc_43></location>To ensure the divergence free condition of equation (1) is satisfied, p 1 in equation (16) needs to satisfy the following linear elliptic equation, which we solve at every time step before using it in the above momentum equation to advance v :</text>
<formula><location><page_5><loc_39><loc_33><loc_88><loc_36></location>∇· [ ρ 0 ∇ ( p 1 ρ 0 )] = ∇· F (20)</formula>
<text><location><page_5><loc_12><loc_30><loc_17><loc_31></location>where</text>
<formula><location><page_5><loc_22><loc_27><loc_88><loc_30></location>F = -ρ 0 v · ∇ v +2 ρ 0 v × Ω + ρ 0 g 0 s 1 c p ˆ r + 1 4 π ( ∇× B ) × B + ∇· D . (21)</formula>
<text><location><page_5><loc_12><loc_23><loc_88><loc_26></location>Also applying the divergence free condition of equation (1), we can rewrite the entropy equation as follows:</text>
<formula><location><page_5><loc_13><loc_9><loc_88><loc_21></location>ρ 0 T 0 ∂s 1 ∂t = -∇· [ ρ 0 v T 0 ( s 1 + s 0 )] -ρ 0 v r ( s 1 + s 0 ) g 0 c p + ρ 0 ν [ S rθ 2 + S θφ 2 + S φr 2 + 1 6 (( S rr -S θθ ) 2 + ( S θθ -S φφ ) 2 +( S φφ -S rr ) 2 ) ] + η ( ∇× B ) 2 + ∇· ( Kρ 0 T 0 ∇ s 1 ) + ∇· ( 16 σ s T 0 3 3 κρ 0 ∇ T 0 ) . (22)</formula>
<text><location><page_6><loc_12><loc_85><loc_50><loc_86></location>In deriving the above equation, we have used</text>
<formula><location><page_6><loc_45><loc_80><loc_88><loc_83></location>d T 0 d r = -g 0 c p (23)</formula>
<text><location><page_6><loc_12><loc_71><loc_88><loc_79></location>where we have ignored the terms of order O ( δ ) produced by the small superadiabaticity in the reference profile of T 0 and only preserved the zeroth order term (corresponding to the adiabatic stratification). The viscous heating term, which is positive definite, has also been written out explicitly in terms of the tensor components S ij .</text>
<text><location><page_6><loc_12><loc_60><loc_88><loc_70></location>Thus, in summary, we numerically solve equations (16), (20), (22), and (5), to advance the dependent variables v , p 1 , s 1 , and B . A more detailed description of the numerical schemes used to solve these equations is given in the appendix. We further note that by summing v · equation (16), B · equation (5), and equation (22), we can also obtain the following equation for total energy conservation:</text>
<formula><location><page_6><loc_21><loc_45><loc_88><loc_59></location>∂ ∂t ( ρ 0 v 2 2 + ρ 0 T 0 s 1 + B 2 8 π ) = -∇· [( ρ 0 v 2 2 + p 1 + ρ 0 T 0 ( s 1 + s 0 ) ) v -1 4 π ( v × B ) × B ] -ρ 0 v r s 0 g 0 c p + ∇· ( v · D ) + ∇· [ -η ( ∇× B ) × B ] + ∇· ( Kρ 0 T 0 ∇ s 1 ) + ∇· ( 16 σ s T 0 3 3 κρ 0 ∇ T 0 ) . (24)</formula>
<text><location><page_6><loc_12><loc_39><loc_88><loc_44></location>Since numerically we are solving the entropy equation (22) instead of the above total energy equation explicitly in conservative form, the total energy equation can serve as an independent check on the effects of numerical dissipation.</text>
<section_header_level_1><location><page_6><loc_31><loc_32><loc_68><loc_34></location>3. A comparison of FSAM and ASH</section_header_level_1>
<text><location><page_6><loc_12><loc_15><loc_88><loc_30></location>Were FSAM to include the polar caps, we could assess the accuracy of its results by direct comparison against the hydrodynamic anelastic benchmark solution of Jones et al. (2011). The benchmark solution is most appropriate for solution domains encompassing the full sphere as it manifests as a sectoral mode of convection, localized around the equator, and propagating prograde with time. Unfortunately, we find that the absence of a polar region in FSAM alters the meridional circulations achieved in the benchmark solution, ultimately preventing FSAM from obtaining the pure spherical harmonic mode of convection achievable when the full sphere is simulated.</text>
<text><location><page_6><loc_12><loc_10><loc_88><loc_13></location>One major use of FSAM is for studies of magnetic flux emergence through a turbulent solar convection zone, and while benchmarks are of some interest, we are most concerned with</text>
<text><location><page_7><loc_12><loc_63><loc_88><loc_86></location>its ability to yield convective motions with properties similar to those thought to exist in the Sun. To this end, we have chosen to run a somewhat more turbulent simulation and compare the properties of the solution against those obtained using the Anelastic Spherical Harmonic (ASH) code. ASH solves the three-dimensional (3-D) anelastic MHD equations in deep spherical shells using a pseudospectral approach. It employs a spherical harmonic expansion in the horizontal direction, and Chebyshev polynomials or a finite-difference approach in the radial direction. ASH has been used extensively to model the solar convection zone (e.g. Brun et al. 2004; Miesch et al. 2008), and has shown good agreement with other anelastic codes when applied to the benchmark problems of Jones et al. (2011). By comparing the results of FSAM against ASH in a somewhat more turbulent regime than the weakly non-linear benchmark test in Jones et al. (2011), we anticipate that the properties of the convective flows in the bulk of the solution is less affected by the role played by the polar region.</text>
<section_header_level_1><location><page_7><loc_38><loc_56><loc_62><loc_58></location>3.1. Experimental Setup</section_header_level_1>
<text><location><page_7><loc_12><loc_43><loc_88><loc_54></location>We have constructed a comparison experiment by modeling convection in a spherical shell spanning the full depth of the solar convection zone, albeit with a much reduced density stratification relative to the Sun. We assume that the gravitational acceleration g 0 ( r ) varies as GM int /r 2 within the shell, where M int is the mass interior to the base of the convection zone and G is the gravitational constant, and use the following adiabatically stratified, polytropic atmosphere as the reference state:</text>
<formula><location><page_7><loc_29><loc_38><loc_88><loc_42></location>ρ 0 = ρ i ( ζ ζ i ) n , T 0 = T i ζ ζ i , p 0 = p i ( ζ ζ i ) n +1 , (25)</formula>
<text><location><page_7><loc_12><loc_31><loc_88><loc_37></location>where the subscript 'i' denotes the value of a quantity at the inner boundary, and n is the polytropic index. The radial variation of the reference state is captured by the variable ζ , defined as</text>
<formula><location><page_7><loc_44><loc_28><loc_88><loc_31></location>ζ = c 0 + c 1 d r , (26)</formula>
<text><location><page_7><loc_12><loc_26><loc_88><loc_28></location>where d = r o -r i is the depth of the convection zone. The constants c 0 and c 1 are given by</text>
<formula><location><page_7><loc_33><loc_22><loc_88><loc_25></location>c 0 = 2 ζ o -β -1 1 -β , c 1 = (1 + β )(1 -ζ o ) (1 -β ) 2 , (27)</formula>
<text><location><page_7><loc_12><loc_19><loc_15><loc_21></location>with</text>
<formula><location><page_7><loc_33><loc_16><loc_88><loc_19></location>ζ o = β +1 β exp( N ρ /n ) + 1 , ζ i = 1 + β -ζ o β . (28)</formula>
<text><location><page_7><loc_12><loc_10><loc_88><loc_15></location>Here ζ i and ζ o are the values of ζ on the inner and outer boundaries, β = r i /r o , and N ρ is the number of density scale heights across the shell. Further details of this model setup may be found in Jones et al. (2011).</text>
<text><location><page_8><loc_12><loc_63><loc_88><loc_86></location>We choose to employ this description for the reference state, with a density variation of one scale height across the shell. Entropy ( s 1 ) is fixed at both the upper and lower boundary with a constant entropy difference ∆ S across the domain, allowing us to specify a Rayleigh number. Our model further differs from the Sun in that radiative heating by photon diffusion (the last term on the right hand side of eq. [22]), particularly important near the base of the convection zone, is neglected. The thermal energy throughput of the system is instead entirely determined by thermal conduction (the 2nd to last term on the right hand side of eq. [22]) at the boundaries. The degree of thermal conduction may vary in time (due to the changes in ∂s 1 /∂r at the boundaries), but reaches a statistically steady state that is itself determined by the entropy gradients established by the convection. Our thermal diffusivity K and viscosity ν are taken to be constant functions of depth with a Prandtl number P r of unity. Values for the simulation parameters are provided in Table 1.</text>
<text><location><page_8><loc_12><loc_37><loc_88><loc_61></location>For the ASH simulation, we used 200 points in radius, and chose a maximum spherical harmonic degree of 170, yielding an effective resolution of 200 × 256 × 512 ( r , θ , φ ). To assess the effects of resolution and polar region removal, we chose to run three distinct FSAM simulations. These three simulations were identical in every respect except for spatial resolution and domain size. The primary simulation (case A), employed a resolution that is approximately half that of the ASH simulation, with a resolution of 96 × 128 × 256, and extended to ± 60 · in latitude. Case B extended over the same latitude range, but employed a resolution of 192 × 256 × 512 (twice that of case A). Case C extended from ± 75 · in latitude, and employed a resolution of 96 × 160 × 256 (similar to case A). With the exception of case B, each simulation was evolved for 4000 days (about seven thermal and viscous diffusion times) to ensure that a thermally and dynamically well-equilibrated state was obtained. The somewhat more expensive Case B was evolved for 1200 days (about two thermal diffusion times).</text>
<section_header_level_1><location><page_8><loc_36><loc_31><loc_64><loc_33></location>3.2. Convective Morphology</section_header_level_1>
<text><location><page_8><loc_12><loc_12><loc_88><loc_29></location>A survey of the radial flows realized in ASH and case A is illustrated in Figure 1. Here snapshots of radial velocity v r , taken at three depths from one instant in time near the end of each simulation, are shown. We have omitted the polar regions of ASH in this plot for ease of comparison. Near-surface flow structures are similar in ASH and FSAM results, with flows in both simulations achieving amplitudes of roughly 50 m s -1 at the top of the domain. Banana cell-like patterns are prominent at the equator in each simulation, but extend to somewhat higher latitudes in the ASH results, possibly due to the inclusion of the polar regions. At mid-convection zone, flows are comparable in amplitude, but the disparity in latitudinal extent of the banana cells has grown, and near the base of the simulation, the</text>
<text><location><page_9><loc_12><loc_83><loc_88><loc_86></location>solutions are markedly different. Radial flows in FSAM at this depth are both weaker than those realized in ASH and are largely confined to a much narrower equatorial region.</text>
<text><location><page_9><loc_12><loc_66><loc_88><loc_81></location>The zonal velocity v φ for each simulation is shown in Figure 2. Both simulations develop a prograde differential rotation at the equator. Flow amplitudes compare at depth in a similar manner to the radial flows, with the prograde region of differential rotation occupying a smaller latitudinal extent near the base of the convection zone in the FSAM results than in the ASH results. There are two effects that contribute to the differences in these two solutions, the most obvious of which is the absence of a polar region in FSAM. Moreover, the numerical diffusion scheme employed in FSAM, which operates in addition to the explicit diffusivities, can lead to differing results where the simulation is under-resolved.</text>
<text><location><page_9><loc_12><loc_47><loc_88><loc_64></location>These snapshots of the flow are from but one instant in each simulation. A sense of how the two solutions compare in a statistical sense may be better gained by looking at probability distribution functions (PDFs) of radial velocity. Figure 3 depicts PDFs of v r , averaged over 500 days of evolution at the end of each simulation, and shown at the same three depths as Figure 1. Case A (red) exhibits substantially higher power in the wings of its PDF near the surface than does ASH. At mid-convection zone, the two distributions are in much better agreement, although the FSAM simulation still exhibits somewhat stronger wings. Near the base of the convection zone, where the flows are noticiably different in Figure 1, the ASH flows exhibit significantly stronger wings.</text>
<text><location><page_9><loc_12><loc_21><loc_88><loc_45></location>We find that these differences in the lower convection zone are substantially diminished when the spatial resolution of the simulation is doubled. We suspect that flow structures associated with convective downdrafts impacting the impenetrable lower boundary are underresolved in case A, especially in the horizontal dimensions where the resolution is about 4 times worse than the vertical resolution. Figure 4 a depicts v r near the base of the convection zone for FSAM case B, which has twice the spatial resolution of case A. Flow amplitudes and structure sizes are comparable to the ASH case for case B. The PDF for v r at the base of the convection zone for case B (Figure 4 b , red line) is also close to that of the ASH simulation (black line). Interestingly, the high power wings present in case A in the upper portion of the domain are still present in the PDF of case B (not shown). These appear to be related instead to an overdriving of the FSAM systems relative to ASH that arises from removal of the polar regions, a subtle effect that we discuss shortly.</text>
<section_header_level_1><location><page_10><loc_24><loc_85><loc_76><loc_86></location>3.3. Mean Flows and Thermodynamics of the System</section_header_level_1>
<text><location><page_10><loc_12><loc_59><loc_88><loc_82></location>Convective flows realized in FSAM case A and ASH possess mean components that are similar in nature to one another. Figure 5 depicts the mean differential rotation and meridional circulations from each simulation. Case A exhibits a prograde differential rotation at the equator, similar to that of ASH. Meridional circulations in each case are predominantly poleward in the upper convection zone and equatorward at the base of the convection zone. Closer inspection reveals that both simulations tend to develop small counter cells of circulation in the near-surface equatorial regions. The differential rotation of case A, however, is noticiably stronger than that realized in ASH. This enhanced differential rotation realized in FSAM is consistent with its convection being driven more strongly than that of ASH, as suggested by the velocity PDFs. As convection becomes more vigorous, the resulting banana cells become more efficient at establishing a prograde equator in systems such as these where the rotational influence is strong.</text>
<text><location><page_10><loc_12><loc_38><loc_88><loc_57></location>The disparity in convective driving becomes evident when looking at the thermodynamic properties of these systems. Figure 6( a ) depicts the time-averaged, spherically symmetric entropy perturbations attained by each simulation. The profiles are similar, but convection in case A (red) tends to build steeper gradients in the boundary layers than that of ASH (black). This is more readily apparent when looking at the entropy gradient (Figure 6( b )) for the two simulations. The flows established by FSAM tend to establish an entropy gradient that is 10% stronger near both the top and bottom boundaries than that established in ASH, while maintaining a more nearly adiabatic interior throughout the bulk of the convection zone. An enhanced entropy gradient near the boundaries translates directly into an increase in the thermal energy throughput of the system.</text>
<section_header_level_1><location><page_10><loc_32><loc_32><loc_68><loc_33></location>3.4. The Role of The Polar Regions</section_header_level_1>
<text><location><page_10><loc_12><loc_14><loc_88><loc_29></location>How might the difference in convective driving between the two systems arise? For constant entropy boundary conditions, convection is allowed to set the latidudinal profile of the heat flux at the boundaries. In the presence of rotation, convective transport is preferentially more efficient at the high latitudes (e.g. Elliott et al. 2000) where Coriolis constraints on radial motions are weakest. With the polar regions absent, convection in FSAM has less freedom to establish such latitudinal asymetries, and therefore leads to stronger driving of the convective motions at the lower latitudes, and subsequently to stronger banana cell-like structures.</text>
<text><location><page_10><loc_16><loc_11><loc_88><loc_13></location>Extension of the latitudinal boundaries to ± 75 · , as we have done with case C, allows us</text>
<text><location><page_11><loc_12><loc_73><loc_88><loc_86></location>to examine this effect. We find that in this regime, FSAM results begin to converge toward those of ASH. Figure 7 depicts the mean flows for case C. Meridional circulations are very similar to those of the ASH simulation, and the strength of the differential rotation, while still stronger than in ASH, is diminished relative to case A. The wings of the velocity PDF for case C (Figure 8 a ) have come down substantially from their counterparts in case A, most noticiably so in the downflows. Moreover, the steep entropy gradients that developed near the boundaries of case A have diminished in case C relative to case A (Figure 8 b .)</text>
<text><location><page_11><loc_12><loc_54><loc_88><loc_71></location>These tests suggest that FSAM, when properly resolved, can produce convective flows in accord with those produced by the more widely used ASH code. We find reasonable convergence between the full- and partial-sphere simulations as the latitudinal extent of the simulation is increased. On the other hand, these tests suggests that we may be wellcautioned to carefully consider the luminosity we adopt for our simulations. Otherwise we may inadvertently overdrive the convection. As the level of turbulence is increased, however, convection tends to become more homogeneous in latitude (e.g. Gastine et al. 2012; Featherstone et al. 2013) and we expect the role of the absent poles to be diminished in the more turbulent regimes.</text>
<section_header_level_1><location><page_11><loc_14><loc_45><loc_86><loc_49></location>4. Buoyant Rise of Active Region Flux Tubes in a Solar Like Convective Envelope</section_header_level_1>
<section_header_level_1><location><page_11><loc_27><loc_42><loc_72><loc_43></location>4.1. A simulation of rotating solar convection</section_header_level_1>
<text><location><page_11><loc_12><loc_16><loc_88><loc_39></location>We now proceed to carry out a hydrodynamic simulation to obtain a statistically steady solution of a solar like, rotating convective flow field in a spherical shell domain with r ∈ [ r i , r o ], spanning from r i = 0 . 722 R glyph[circledot] at the base of the convection zone (CZ) to r o = 0 . 971 R glyph[circledot] at about 20 Mm below the photosphere, θ ∈ [ π/ 2 -∆ θ, π/ 2 + ∆ θ ] with ∆ θ = π/ 3, and φ ∈ [0 , 2 π ]. The domain is resolved by a grid with 96 grid points in r , 512 grid points in θ , and 768 grid points in φ . The grid is uniform in r , θ , and φ respectively. J. ChristensenDalsgaard's solar model (Christensen-Dalsgaard et al. 1996), commonly known as Model S, is used for the reference profiles of T 0 , ρ 0 , p 0 , g 0 in the simulation domain. We assumed that s 0 = 0 for the reference state, i.e. is isentropic. We also omit the heating term due to radiative diffusion ∇ · F rad in the CZ in equation (22), but instead, drive convection by imposing at the lower boundary a fixed entropy gradient ∂s 1 /∂r such that the solar luminosity L s is forced through the lower boundary as a diffusive heat flux:</text>
<formula><location><page_11><loc_40><loc_10><loc_88><loc_14></location>( Kρ 0 T 0 ∂s 1 ∂r ) r i = L s 4 πr i 2 (29)</formula>
<text><location><page_12><loc_12><loc_83><loc_88><loc_86></location>where K = 2 × 10 13 cm 2 s -1 in our simulation domain. We also impose a latitudinal variation of entropy at the lower boundary:</text>
<formula><location><page_12><loc_43><loc_77><loc_88><loc_81></location>( ∂s 1 ∂θ ) r i = ds i ( θ ) dθ (30)</formula>
<text><location><page_12><loc_12><loc_75><loc_17><loc_76></location>where</text>
<formula><location><page_12><loc_37><loc_71><loc_88><loc_75></location>s i ( θ ) = -∆ s i cos ( θ -π/ 2 ∆ θ π ) , (31)</formula>
<text><location><page_12><loc_12><loc_65><loc_88><loc_70></location>to represent the tachocline induced entropy variation that can break the Taylor-Proudman constraint in the convective envelope. In the above we set ∆ s i = 215 . 7 erg g -1 K -1 . For the initial condition, we let the initial s 1 be:</text>
<formula><location><page_12><loc_36><loc_62><loc_88><loc_63></location>s 1 | t =0 = <s 1 > t =0 + s i ( θ ) -<s i ( θ ) > (32)</formula>
<text><location><page_12><loc_12><loc_58><loc_82><loc_60></location>where <> denotes the horizontal average at a constant r , and <s 1 > t =0 is given by:</text>
<formula><location><page_12><loc_39><loc_54><loc_88><loc_57></location>Kρ 0 T 0 d <s 1 > t =0 dr = L s 4 πr 2 , (33)</formula>
<text><location><page_12><loc_12><loc_33><loc_88><loc_53></location>such that initially the constant solar luminosity is being carried through the solar convection zone by thermal diffusion. This results in an unstable initial stratification, and with a small initial velocity perturbation, convection ensues in the domain. For the upper boundary s 1 is held fixed to its initial value, while at the lower boundary the fixed gradient of ∂s 1 /∂r given by equation (29) maintains a conductive heat flux corresponding to the solar luminosity through the lower boundary. The latitudinal gradient of s 1 given by equation (30) is also imposed at the lower boundary, but the horizontally averaged value of s 1 is allowed to change with time. At the two θ boundaries, s 1 is assumed symmetric. The velocity field is nonpenetrating and stress free at the top, bottom and the two θ -boundaries. The top and bottom boundary condition for p 1 is</text>
<formula><location><page_12><loc_43><loc_28><loc_88><loc_32></location>ρ 0 ∂ ∂r ( p 1 ρ 0 ) = F r (34)</formula>
<text><location><page_12><loc_12><loc_23><loc_88><loc_27></location>at r = r i and r o , and F r is the r -component of F given in equation (21). At the two θ -boundaries</text>
<formula><location><page_12><loc_43><loc_20><loc_88><loc_24></location>ρ 0 r ∂ ∂θ ( p 1 ρ 0 ) = F θ , (35)</formula>
<text><location><page_12><loc_12><loc_10><loc_88><loc_19></location>and F θ is the θ -component of F given in equation (21). All quantities are naturally periodic at the φ boundaries. The kinematic viscosity ν = 10 12 cm 2 s -1 in the simulation domain. This gives a Prandlt number of Pr = 0 . 05 for our simulation. The reference frame rotation rate Ω in equation (16) is set to 2 . 7 × 10 -6 rad s -1 , and with respect to this frame, the initial velocity is essentially zero with a very small initial perturbation.</text>
<text><location><page_13><loc_12><loc_71><loc_88><loc_86></location>With the above setup of the simulation, we let the convection in the domain evolve to a statistical steady state, which is reached after about 6000 days. The final steady state entropy gradient reached by the rotating solar convective envelope is shown in Figure 9. The horizontally averaged entropy gradient reaches a value of about -7 . 5 × 10 -6 erg K -1 cm -1 near the top boundary at about 0 . 97 R s , which is of a similar order of magnitude as the entropy gradient ( ∼ 10 -5 erg K -1 cm -1 ) at this depth obtained by Model S (ChristensenDalsgaard et al. 1996). Figure 10 shows the steady state, horizontally integrated total heat flux due to convection:</text>
<formula><location><page_13><loc_32><loc_65><loc_88><loc_69></location>H conv = 4 πr 2 A ( r ) ∫ 2 π 0 ∫ π/ 2+∆ θ π/ 2 -∆ θ ρ 0 T 0 v r s 1 r 2 dθdφ (36)</formula>
<text><location><page_13><loc_12><loc_62><loc_25><loc_64></location>and conduction</text>
<formula><location><page_13><loc_31><loc_58><loc_88><loc_62></location>H cond = 4 πr 2 A ( r ) ∫ 2 π 0 ∫ π/ 2+∆ θ π/ 2 -∆ θ Kρ 0 T 0 ∂s 1 ∂r r 2 dθdφ (37)</formula>
<text><location><page_13><loc_12><loc_56><loc_64><loc_57></location>where A ( r ) is the total area of the spherical surface at radius r</text>
<formula><location><page_13><loc_38><loc_50><loc_88><loc_54></location>A ( r ) = ∫ 2 π 0 ∫ π/ 2+∆ θ π/ 2 -∆ θ r 2 dθdφ. (38)</formula>
<text><location><page_13><loc_12><loc_25><loc_88><loc_49></location>In the above, the total heat fluxes H conv and H cond have been scaled up to the values for the area of a full sphere so that they can be compared directly with the solar luminosity L s . Here the conductive heat flux represents the heat transport due to turbulent diffusion by unresolved convection. It can be seen from Figure 10 that with the large value of K used, the solar luminosity is mostly carried through by thermal conduction and the heat flux transported by the resolved convective flow is only a small fraction ( ∼ 20%) of the solar luminosity. In this way the convective flow speed for the resolved giant cell convection is not too high, (even with a relatively low viscosity ν = 10 12 cm 2 s -1 used), so that the convective flow is sufficiently rotationally constrained to allow the maintainance of a solarlike differential rotation with faster equator than the polar regions (e.g Featherstone & Miesch 2012). The relatively low viscocity is chosen so that the subsequent simulations of the buoyant rise of active region flux tubes are not in a too viscous regime.</text>
<text><location><page_13><loc_12><loc_10><loc_88><loc_24></location>Figure 11a shows a snapshot of the radial velocity field of the rotating solar convection at a depth of about 30 Mm below the photosphere displayed on the full sphere in Mollweide projection. It shows giant-cell convection patterns with broad upflows in the network of narrow downflow lanes, and with columnar, rotationally aligned cells (banana cells) at low latitudes. The time and azimuthally averaged mean zonal flow (Figure 11c) shows a solar-like differential rotation profile with faster rotation at the equator and slower rotation towards the poles, and more conical shaped contours of constant angular speed of rotation at mid-latitude</text>
<text><location><page_14><loc_12><loc_71><loc_88><loc_86></location>range. Figure 11d shows the time and azimuthally averaged kinetic helicity H k = v · ( ∇× v ) of the flow. It shows predominantly negative (positive) kinetic helicity in the upper 1/3 to 1/2 of the CZ in the northern (southern) hemisphere and weakly positive (negative) kinetic helicity in the deeper depths of the CZ. The depth of the upper layer with predominantly negative (positive) H k in the northern (southern) hemisphere is relatively shallow because, as can be seen in Figure 11b, the concerntrated downflow plumes do not penetrate very deep. They generally reach less than half of the total depth of the CZ before starting to diverge and leading to a reversal of the kinetic helicity.</text>
<section_header_level_1><location><page_14><loc_31><loc_64><loc_68><loc_66></location>4.2. Simulations of Rising Flux Tubes</section_header_level_1>
<section_header_level_1><location><page_14><loc_39><loc_61><loc_60><loc_62></location>4.2.1. Simulation Setup</section_header_level_1>
<text><location><page_14><loc_12><loc_51><loc_88><loc_58></location>Into the statistically-steady, rotating convective flow with a self-consistently maintained solar like differential rotation, we insert buoyant toroidal flux tubes near the bottom of the CZ to study how they rise through the CZ. The initial flux tube we insert into the convecting domain is given by the following:</text>
<formula><location><page_14><loc_36><loc_46><loc_88><loc_50></location>B = ∇× ( A ( r, θ ) r sin θ ˆ φ ) + B φ ( r, θ ) ˆ φ, (39)</formula>
<text><location><page_14><loc_12><loc_43><loc_17><loc_44></location>where</text>
<formula><location><page_14><loc_35><loc_40><loc_88><loc_43></location>A ( r, θ ) = 1 2 qa 3 B t exp ( -glyph[pi1] 2 ( r, θ ) a 2 ) , (40)</formula>
<formula><location><page_14><loc_35><loc_35><loc_88><loc_39></location>B φ ( r, θ ) = aB t r sin θ exp ( -glyph[pi1] 2 ( r, θ ) a 2 ) . (41)</formula>
<formula><location><page_14><loc_35><loc_32><loc_88><loc_34></location>glyph[pi1] = ( r 2 + r 2 0 -2 rr 0 cos( θ -θ 0 )) 1 / 2 . (42)</formula>
<text><location><page_14><loc_12><loc_10><loc_88><loc_31></location>In the above, q is the rate of twist (angle of field line rotation about the axis per unit length of the tube), a denotes the e-folding radius of the tube, r 0 and θ 0 are respectively the initial r and θ values of the tube axis. For all of the simulations of this paper, a = 6 . 7 × 10 8 cm which is about 0.12 times the pressure scale height at the base of the solar convection zone, r 0 = 5 . 2 × 10 10 cm is at approximately 0 . 757 R glyph[circledot] , θ 0 corresponds to 15 · latitude, and the initial field strength at the axis of the toroidal flux tube is B t a/ ( r 0 sin θ 0 ) = 10 5 G. Thus the total flux in the initial toroidal flux tube is 1 . 4 × 10 23 Mx, which is about a factor of 10 greater than the typical flux in a solar active region. Due to the limited numerical resolution of our global scale simulations of the convective envelope, we can only consider tubes with a rather large cross-section in order for it to be resolved by the numerical grid. In our current simulations the initial tube diameter is resolved by about 7 grid points.</text>
<text><location><page_15><loc_12><loc_81><loc_88><loc_86></location>We consider initially buoyant toroidal flux tubes, and specify the initial buoyancy along the tube in the following two ways. In one way, an initial sinusoidal variation (with an azimuthal wavelength of π/ 2 in φ ) of entropy:</text>
<formula><location><page_15><loc_22><loc_76><loc_88><loc_79></location>δs 1 = c p ( 1 -1 γ ) B 2 φ 8 πp 0 [ 1 2 ( 1 -1 γ -1 ) -1 2 ( 1 + 1 γ -1 ) cos(4 φ ) ] (43)</formula>
<text><location><page_15><loc_12><loc_61><loc_88><loc_75></location>is being added to the original s 1 of the convective flow field at the location of the toroidal tube. Thus along each π/ 2 azimuthal segment of the toroidal tube, the tube is varying from being (approximately) in thermal equilibrium with the surrounding and thus buoyant, to being approximately in neutral buoyancy. The peak buoyancy in the initial tube is approximately B 2 φ / 8 πH p 0 , corresponding to the magnetic buoyancy associated with a flux tube in thermal equilibrium with its surrounding. Another initial buoyancy state we used is to specify a uniform buoyancy along the tube, by adding</text>
<formula><location><page_15><loc_40><loc_56><loc_88><loc_60></location>δs 1 = c p ( 1 -1 γ ) B 2 φ 8 πp 0 (44)</formula>
<text><location><page_15><loc_12><loc_10><loc_88><loc_55></location>to the original s 1 of the convective flow field at the location of the toroidal tube. In this way it is uniformly buoyant along the tube with the magnetic buoyancy B 2 φ / 8 πH p 0 . We run two simulations of rising flux tubes in the convective flows with the sinusoidal initial buoyancy (eq. [43]), one with a weak initial (left-handed) twist rate of q = -0 . 15 a -1 , and the other with no initial twist q = 0. We name these runs 'SbWt' (Sinusoidal-buoyancyWeak-Twist) and 'SbZt' (Sinusoidal-buoyancy-Zero-twist) respectively. As a reference for these two simulations, we run two corresponding simulations of the same initial buoyant tubes in a quiescent rotating envelope with no convective flows, but with the same reference stratification of p 0 , ρ 0 , and T 0 . These two runs are named 'SbWt-ref' and 'SbZt-ref'. Furthermore, we run a simulation (named 'UbZt') of the uniformly buoyant initial tube (using eq. [44]) with no initial twist rising in the convective flow. A summary of these runs is given in Table 2. In this paper we only conduct these few sample runs to examine qualitatively how a solar-like rotating convective flow may influence the rise of relatively strong (100 kG) buoyant flux tubes. The peak Alfv'en speed v a in the initial flux tube is 764 m/s. Compared to the convective flow speeds shown in Figure 12, the flux tube is significantly super-equipartition with respect to the mean kinetic energy of the convective flows as reflected by the RMS velocity. However, as discussed in Fan et al. (2003), the hydrodynamic forces from the convective flows would be able to counteract the magnetic buoyancy of the flux tube if the speed of the convective flows is ∼ > ( a/H p ) 1 / 2 v a which is ∼ 265 m/s considering the initial radius a of the buoyant toroidal flux tubes in the present simulations. Figure 12 shows that the peak downflow speed exceeds that value for most of the convection zone, indicating that the downflow plumes can significantly impede the buoyant rise of the flux tube even for the 100 kG strong flux tubes considered here.</text>
<text><location><page_16><loc_12><loc_73><loc_88><loc_86></location>For the simulations of the rising flux tubes, we preserve the kinematic viscosity ν = 10 12 cm 2 s -1 used for the simulation of the rotating convective flow solution in the entire simulation domain. The thermal diffusivity used in the original convection simulation is much greater (2 × 10 13 cm 2 s -1 ). This large value is used in order to achieve a solar like differential rotation profile (fast equator, slow poles) in the rotating convection solution. For the rising flux tube simulations, we apply a magnetic field strength dependent quenching of K :</text>
<formula><location><page_16><loc_42><loc_69><loc_88><loc_73></location>K = K 0 1 + ( B/B cr ) 2 (45)</formula>
<text><location><page_16><loc_12><loc_53><loc_88><loc_68></location>where K 0 = 2 × 10 13 cm 2 s -1 is the original value of the diffusivity used in the convection simulation and B cr = 100 G represents a low threshhold field strength above which quenching of thermal diffusion begins to take place. Convection is expected to be suppressed by the strong magnetic field in the flux tube, thus K , which represents unresolved eddy diffusion, should be suppressed in the rising flux tubes. For the magnetic field, we also do not include any explicit resistivity η in the simulation, so only numerical diffusion is present. This way we minimize magnetic diffusion to preserve the frozen-in condition of the buoyant flux tube as much as possible, given the numerical resolution.</text>
<section_header_level_1><location><page_16><loc_44><loc_47><loc_56><loc_48></location>4.2.2. Results</section_header_level_1>
<text><location><page_16><loc_12><loc_23><loc_88><loc_44></location>Figures 13a and 13b show the rising flux tubes that have developed from the SbWt and SbZt simulations respectively, when an apex of the tube has reached the top boundary. For comparison, the resulting rising tubes from the corresponding reference simulations SbWtref and SbZt-ref (without convection) are shown in Figures 13c and 13d. MPEG movies of the evolution of the tube for each of the simulations are available in the online version of the paper. In the absence of convection, four identical rising loops develop due to the initial buoyancy prescription and rise to the top of the domain. Convective flows are found to produce additional undulations on the rising loops, pushing down certain portions while promoting the rise of other portions. With convection, the rise time for an apex of the tube to reach the top is significantly reduced (for example, changed from about 49 days for SbWt-ref to 26.5 days for SbWt).</text>
<text><location><page_16><loc_12><loc_10><loc_88><loc_21></location>There is little difference in the morphology of the rising tubes (at least as shown in the volume rendering of the absolute magnetic field strength) between the weakly twisted and the untwisted cases, both with and without convection. One of the reasons for this is that the twist is rather weak, about a half of the necessary twist rate required for a cohesive rise of the flux of the original flux tube as a whole, similar to the weakly twisted case studied in Fan (2008) (see the LNT case shown in Figure 8 of that paper). In other words, the</text>
<text><location><page_17><loc_12><loc_57><loc_88><loc_86></location>magnetic energy density associated with the initial twist component of the field (i.e. the B θ and B r components in the initial toroidal flux tube) is smaller than the kinetic energy density associated with the relative velocity between the tube and the surrounding plasma. As a result the initial twist does not have a great effect on maintaining the cohesion of the rising tube compared to the untwisted case. There is continued flux loss during the rise, forming a track of flux behind the rising apex, as can be seen in the meridional cross-section of B φ at the apex longitude for all the cases as shown in Figure 14. We also note that the current simulations of the rising flux tubes are in a fairly laminar regime. The Reynolds number for the rising flux tube estimated based on the tube diameter D ∼ 10 9 cm, typical rise speed attained V rise ∼ 100 m/s, and the viscosity ν = 10 12 cm 2 / s (kept the same as that used for obtaining the rotating convection solution), is R e = V rise D/ν ∼ 10. Such a low Reynolds number reduces the production of small scale features and fragmentation of the flux tube and thus generally improves the cohesion for the rising flux, especially for the untwisted case. This is also a reason for the reduced difference in the magnetic field morphology between the untwisted and weakly twisted cases.</text>
<text><location><page_17><loc_12><loc_21><loc_88><loc_55></location>In all the cases, the apex rises nearly radially, with a small poleward drift. Figure 15 shows the normal flux distribution produced by the emerging apex portion near the top boundary on a constant r surface at r = 0 . 957 R s . It can be seen that for all of the four cases the latitude of emergence is centered at a location just slightly poleward (by no more than about 3 . 5 · ) than the initial latitude of 15 · . For the cases without convection (panels (c) and (d)), the apex portion produces a simple bipolar structure with a tilt angle of 7 . 2 · clockwise for the weakly twisted (SbWt-ref) case, and 16 · clockwise for the untwisted (SbZt-ref) case. These tilts are consistent with the mean tilt of solar active regions as described by Joy's law. With convection, the additional distortion and undulation caused by the convective flows produce a more complex emergence pattern with multiple bipolar structures in the SbWt and SbZt cases as shown in Figures 15a and 15b. However, the leading (negative) polarity flux is on average equatorward and westward of the following (positive) polarity, consistent with the direction of the active region mean tilt. The tilt angle as determined by the flux weighted positions of the leading and following polarity flux concentrations is 29 . 2 · clockwise for the weakly twisted (SbWt) case and 53 . 2 · clockwise for the untwisted (SbZt) case, which are of the right sign but are of a significantly greater magnitude than the active region mean tilt.</text>
<text><location><page_17><loc_12><loc_10><loc_88><loc_20></location>Figure 16 shows 3D views of a few selected field lines traced from the apex portion in the rising flux tubes for the four cases: (a) SbWt, (b) SbZt, (c) SbWt-ref, and (d) SbZt-ref, as viewed from the pole (upper panel in each case), and from the equator (lower panel in each case). For all the cases, the apex of the rising tube is at the 6 o'clock location in the polar view and at the central meridian in the equatorial view. It can be seen that the field lines</text>
<text><location><page_18><loc_12><loc_63><loc_88><loc_86></location>at the apex are pointing southeast-ward, i.e. consistent with the sense of tilts of solar active regions. The tilt angles of the field orientation from the east-west direction are significantly bigger in the convective cases (SbWt and SbZt in Figures 16a and 16b) compared to the non-convective reference cases (SbWt-ref and SbZt-ref in Figures 16c and 16d). In these particular convective cases, the convective flows have driven additional clock-wise rotation of the fields at the rising apex. A statistical study with many more simulations of rising flux tubes, sampling different times and locations of the convective flows (as was done with the thin flux tube model in Weber et al. (e.g. 2012)) are needed to determine whether the tilt angles at the apex of the emerging flux obey Joy's law for solar active regions. Our initial simulations here show that even with a relatively strong initial magnetic field of 100 kG, a solar-like giant cell convection can significantly reshape the buoyantly rising loops and shorten the time for the apex to reach the top.</text>
<text><location><page_18><loc_12><loc_13><loc_88><loc_61></location>We have also run a simulation (case UbZt) where the initial toroidal flux tube is uniformly buoyant along the tube with the magnetic buoyancy, such that the flux tube would have risen axisymmetrically under its buoyancy had it not been for the effect of the convective flows. Thus the development of undulations or loop structures is due entirely to the convective flows. Figure 17 shows 3D volume rendering of the absolute magnetic field strength of the rising loops that develop, as viewed from 3 different perspectives, with the apex portion approaching the top boundary located at the right in all three views. An MPEG movie of the evolution of the rising flux tube viewed from the same perspectives is available in the online version. We see that loops with shorter footpoint separations form compared to the 4 major loops formed in the SbZt case. A set of selected field lines traced from the apex portion approaching the top are also shown in a polar view (Figure 18a) with the apex positioned at the 6 o'clock location, and two equatorial views (Figures 18b and 18c) with the apex positioned at the central meridian and at the west limb respectively. We can see that despite the fact that the initial buoyancy is uniform along the tube, the convective downdrafts are able to hold back portions of the buoyant tube and lead to the formation of loop structures with undulations that span up to 70% of the depth of the convection zone (based on the apex and troughs of the field lines) in a time scale of about a month. However, the troughs of the loops are not as deeply rooted as the major loops formed in the SbZt case (compare Figure 18a with the top panel of Figure 16b). All of the troughs are above the initial depth of the toroidal tube, meaning that the downdrafts are not able to completely overcome the magnetic buoyancy. Similar to the SbZt case, the convective flows have driven a significantly larger clockwise tilt from the east-west direction at the apex of the emerging loop, as can be seen in the field line orientation at the central meridian in Figure 18b. of the emerging loop</text>
<text><location><page_18><loc_16><loc_10><loc_88><loc_12></location>Figure 19 shows the normal flux distribution B r , radial velocity v r , and zonal flow v φ</text>
<text><location><page_19><loc_12><loc_63><loc_88><loc_86></location>on a constant r slice at r = 0 . 957 R s , about 30 Mm below the top boundary, at the time when the apex portion of the rising flux tube approaches the top boundary for the UbZt case. An emerging region with a large overall tilt (75 . 4 · clockwise based on the flux weighted positions of the leading and following polarity flux concentrations) of the correct sign has formed by the apex of the rising tube. The region of emerging flux corresponds to a local region of upflow (with speed reaching about 100 m/s) surrounded by narrow downflow lanes (see Figure 19b). The emerging flux also shows a retrograde zonal flow (peaks at about -200 m/s) in the midst of the prograde flows of the banana cells (see Figure 19c). It corresponds to the most retrograde portion of plasma at that latitude. Relative to the mean plasma zonal flow speed at that latitude (about 225 m/s), the emerging flux region has a relative (flux weighted) mean speed of -348 m/s. Similar results on the relative speeds of the emerging flux region are found for the SbWt and SbZt cases.</text>
<section_header_level_1><location><page_19><loc_43><loc_56><loc_57><loc_58></location>5. Discussions</section_header_level_1>
<text><location><page_19><loc_12><loc_27><loc_88><loc_54></location>We have used a finite-difference based spherical anelastic MHD code (FSAM) to simulate rotating solar convection and the buoyant rise of super-equipartition field strength flux tubes through the convective envelope in the presence of the giant-cell convection and the associated mean flows. We achieved a statistically steady solution of giant-cell convection with a solar-like differential rotation using a relatively low viscosity ν = 10 12 cm 2 s -1 , but a high value of thermal diffusion K = 2 × 10 13 cm 2 s -1 . The high thermal diffusion allows most of the solar luminosity to be carried via thermal conduction, so that the resolved giant-cell convection flow speed is not too high and the convection remains sufficiently rotationally constrained to give a solar-like differential rotation with the right amplitude. Into the giantcell convection near the bottom of the convective envelope, we insert toroidal flux tubes of 100 kG field strength and with different forms of magnetic buoyancy distribution to model their rise through the convective envelope in the presence of convection. We simulate the rise of the flux tube with no explicit magnetic diffusion η and a quenching of thermal diffusion K in the flux tube to best preserve the magnetic buoyancy of the initial flux tube.</text>
<text><location><page_19><loc_12><loc_12><loc_88><loc_25></location>The simulations show that with a strong, super-equipartition field strength of 100 kG, magnetic buoyancy dominates the rise but the strong down-flows can significantly modify the shape of the Ω-shaped emerging loops, and substantially reduce the rise time for the apex to reach the top boundary. Even if the initial tube is uniformly buoyant, it is found that convection can produce loop structures with undulations that extend most of the depth of the CZ in a time scale of about a month. For the weakly twisted and (initially) untwisted cases we simulated, the apex portion rises nearly radially and produces an emerging region</text>
<text><location><page_20><loc_12><loc_46><loc_88><loc_86></location>with an overall tilt angle consistent with the active region tilts, although there is continued and substantial loss of flux during the rise. Thus it appears that the current simulations suggest that a significant twist in the toroidal magnetic fields in the bottom of the convection zone is not required for the emergence of coherent active regions. We emphasize that the current simulations are in a rather laminar region with the Reynolds number for the rising tube estimated to be ∼ 10. This would limit the formation of small scale structures and improve the cohesion of the rising flux. However there is difficulty to significantly reduce the viscosity if one wants to also self-consistently maintain a solar-like differential rotation (e.g. Featherstone & Miesch 2012). On the other hand, the ubiquitous presence of small scale magnetic fields in a convective dynamo in the CZ may suppress the development of small scale flows via the magnetic stresses, effectively increasing the viscosity (Longcope et al. 2003), and allow a solar like differential rotation to be maintained at a substantially lower fluid viscosity (Fan 2013 in preparation). Thus the presence of the ambient small scale magnetic field may effectively improve the cohesion of the strong buoyant flux tubes with weak twists, which is indicated in the recent convective dynamo simulations in faster rotating convective envelopes by Nelson et al. (2013b). Clearly 3D convective dynamo simulations in the solar convective envelope that model both the generation of the dynamo mean field and the self-consistent formation and rise of active region flux in the midst of small scale fields are needed to obtain a complete understanding of the solar cycle dynamo and active region formation.</text>
<section_header_level_1><location><page_20><loc_30><loc_40><loc_70><loc_42></location>A. The Numerical Algorthms of FSAM</section_header_level_1>
<text><location><page_20><loc_12><loc_27><loc_88><loc_38></location>In this Appendix we describe how FSAM numerically solves equations (16), (20), (22), and (5), to advance the dependent variables v , p 1 , s 1 , and B . FSAM uses a staggered spatial discretization as described in Stone & Norman (1992a), where the vector quantities v and B are defined on the faces of each finite-volume cell of the grid, scalar quantities p 1 , s 1 , T 1 , are defined at the center of each finite-volume cell, and the v × B electric field and the current density ∇× B in the induction equation are defined on the cell edges.</text>
<text><location><page_20><loc_12><loc_11><loc_88><loc_25></location>First we define some notations to be used frequently in the rest of the Appendix. For the spherical polar coordinate system used by this code, we use subscript m = 1 , 2 , 3 to denote respectively the r , θ , φ direction or component, i.e. we have ( x 1 , x 2 , x 3 ) = ( r, θ, φ ), ( v 1 , v 2 , v 3 ) = ( v r , v θ , v φ ), ( B 1 , B 2 , B 3 ) = ( B r , B θ , B φ ). Also we make use of the following coordinate scaling coefficients defined as: g 2 = r , g 31 = r , and g 32 = sin θ (notations used in Stone & Norman (1992a)). Consider in general a row of cells in the m -direction ( m = 1 , 2 , 3), whose centers' x m coordinates are located at x m,i , i = 1 , 2 , 3 , ... , and whose cell averaged Q</text>
<text><location><page_21><loc_12><loc_33><loc_20><loc_35></location>and we let</text>
<text><location><page_21><loc_12><loc_82><loc_88><loc_86></location>values are denoted by Q i . For evaluating the various fluxes at the cell face located at x m,i -1 / 2 between the two adjacent cells centered on x m,i -1 and x m,i , we define</text>
<formula><location><page_21><loc_43><loc_79><loc_88><loc_81></location>δ m Q ≡ Q i -Q i -1 (A1)</formula>
<text><location><page_21><loc_12><loc_67><loc_88><loc_77></location>to be the simple finite difference between the two adjacent cells (in the m-direction), and we will use Q L and Q R to denote the 'left' and 'right' Q values on the cell face, evaluated through a certain reconstruction of the Q profile within the cell to the left and right of the cell face, respectively. Specifically, the assumed profile Q ( x m ) within the cell centered on x m,i is given by a linear reconstruction with a minmod slope limiter:</text>
<formula><location><page_21><loc_37><loc_64><loc_88><loc_65></location>Q ( x m ) = Q i + s m,i ( x m -x m,i ) , (A2)</formula>
<text><location><page_21><loc_12><loc_60><loc_70><loc_62></location>where s m,i is a limited slope (in the m-direction) for the cell given by</text>
<formula><location><page_21><loc_31><loc_55><loc_88><loc_59></location>s m,i = minmod ( Q i +1 -Q i x m,i +1 -x m,i , Q i -Q i -1 x m,i -x m,i -1 ) (A3)</formula>
<text><location><page_21><loc_12><loc_52><loc_44><loc_53></location>and the minmod function is defined as</text>
<formula><location><page_21><loc_27><loc_49><loc_88><loc_50></location>minmod ( y 1 , y 2) ≡ sgn ( y 1) max[0 , min( y 1 , sgn ( y 1) y 2)] . (A4)</formula>
<text><location><page_21><loc_12><loc_43><loc_88><loc_47></location>Thus the right and left values, Q R and Q L , for the cell face located at x m,i -1 / 2 , between the two neighboring cells centered at x m,i -1 and x m,i are:</text>
<formula><location><page_21><loc_34><loc_39><loc_88><loc_41></location>Q L = Q i -1 + s m,i -1 ( x m,i -1 / 2 -x m,i -1 ) , (A5)</formula>
<formula><location><page_21><loc_37><loc_36><loc_88><loc_38></location>Q R = Q i -s m,i ( x m,i -x m,i -1 / 2 ) , (A6)</formula>
<formula><location><page_21><loc_42><loc_31><loc_88><loc_33></location>∆ m Q ≡ Q R -Q L , (A7)</formula>
<text><location><page_21><loc_12><loc_26><loc_88><loc_30></location>denoting the limited difference between the right and left states at the cell face at x m,i -1 / 2 , and</text>
<formula><location><page_21><loc_42><loc_23><loc_88><loc_27></location><Q> m = Q R + Q L 2 (A8)</formula>
<text><location><page_21><loc_12><loc_21><loc_83><loc_22></location>denoting the mean of the left, right values of Q evaluated at the cell face at x m,i -1 / 2 .</text>
<text><location><page_21><loc_16><loc_18><loc_88><loc_19></location>The 1,2,3-components of the momentum equation (16), and the entropy equation (22)</text>
<text><location><page_22><loc_12><loc_85><loc_58><loc_86></location>we solve written explicitly in spherical coordinates are:</text>
<formula><location><page_22><loc_14><loc_47><loc_88><loc_83></location>∂ ∂t ( ρ 0 v 1 ) = -1 g 2 g 31 ∂ ∂x 1 [ g 2 g 31 ( ρ 0 v 1 v 1 ) ∗ ] -1 g 2 g 32 ∂ ∂x 2 [ g 32 ( ρ 0 v 2 v 1 ) ∗ ] -1 g 31 g 32 ∂ ∂x 3 ( ρ 0 v 3 v 1 ) ∗ + 1 g 2 g 32 ∂ ∂x 2 ( g 32 B 1 B 2 ) + 1 g 31 g 32 ∂ ∂x 3 ( B 1 B 3 ) + 1 g 2 2 g 31 2 ∂ ∂x 1 ( g 2 2 g 31 2 B 1 2 2 ) -1 g 31 2 ∂ ∂x 1 ( g 31 2 B 3 2 2 ) -1 g 2 2 ∂ ∂x 1 ( g 2 2 B 2 2 2 ) + ρ 0 g 0 s 1 c p -ρ 0 ∂ ∂x 1 ( p 1 ρ 0 ) + ρ 0 v 2 2 1 g 2 ∂g 2 ∂x 1 + ρ 0 v 2 3 1 g 31 ∂g 31 ∂x 1 +2Ω ρ 0 v 3 sin θ + [ 1 g 2 g 31 ∂ ∂x 1 ( g 2 g 31 ρ 0 νS 11 ) + 1 g 2 g 32 ∂ ∂x 2 ( g 32 ρ 0 νS 12 ) + 1 g 31 g 32 ∂ ∂x 3 ( ρ 0 νS 13 ) ] -ρ 0 νS 22 1 g 2 ∂g 2 ∂x 1 -ρ 0 νS 33 1 g 31 ∂g 31 ∂x 1 -2 3 ∂ ∂x 1 ( ρ 0 ν ∇· v ) , (A9)</formula>
<formula><location><page_23><loc_14><loc_48><loc_88><loc_87></location>∂ ∂t ( ρ 0 v 2 ) = -1 g 2 g 2 g 31 ∂ ∂x 1 [ g 2 g 31 g 2 2 ( ρ 0 v 1 v 2 g 2 ) ∗ ] -1 g 2 g 2 g 32 ∂ ∂x 2 [ g 32 g 2 2 ( ρ 0 v 2 v 2 g 2 ) ∗ ] -1 g 2 g 31 g 32 ∂ ∂x 3 [ g 2 2 ( ρ 0 v 3 v 2 g 2 ) ∗ ] + 1 g 2 2 g 31 ∂ ∂x 1 ( g 2 2 g 31 B 2 B 1 ) + 1 g 2 g 32 2 ∂ ∂x 2 ( g 32 2 B 2 2 2 ) + 1 g 31 g 32 ∂ ∂x 3 ( B 2 B 3 ) -1 g 2 ∂ ∂x 2 ( B 1 2 2 ) -1 g 32 2 g 2 ∂ ∂x 2 ( g 32 2 B 3 2 2 ) -ρ 0 g 2 ∂ ∂x 2 ( p 1 ρ 0 ) + ρ 0 v 2 3 1 g 32 g 2 ∂g 32 ∂x 2 +2Ω ρ 0 v 3 cos θ + [ 1 g 2 g 31 ∂ ∂x 1 ( g 2 g 31 ρ 0 νS 21 ) + 1 g 2 g 32 ∂ ∂x 2 ( g 32 ρ 0 νS 22 ) + 1 g 31 g 32 ∂ ∂x 3 ( ρ 0 νS 23 ) ] + ρ 0 νS 21 1 g 2 ∂g 2 ∂x 1 -ρ 0 νS 33 1 g 2 g 32 ∂g 32 ∂x 2 -2 3 1 g 2 ∂ ∂x 2 ( ρ 0 ν ∇· v ) , (A10)</formula>
<formula><location><page_23><loc_14><loc_12><loc_88><loc_45></location>∂ ∂t ( ρ 0 v 3 ) = -1 g 31 g 32 g 2 g 31 ∂ ∂x 1 [ g 2 g 31 g 2 31 g 2 32 (( ρ 0 v 1 v 3 g 31 g 32 ) ∗ + ρ 0 v 1 Ω )] -1 g 31 g 32 g 2 g 32 ∂ ∂x 2 [ g 32 g 2 31 g 2 32 (( ρ 0 v 2 v 3 g 31 g 32 ) ∗ + ρ 0 v 2 Ω )] -1 g 31 g 32 g 31 g 32 ∂ ∂x 3 [ g 2 31 g 2 32 (( ρ 0 v 3 v 3 g 31 g 32 ) ∗ + ρ 0 v 3 Ω )] + 1 g 31 g 32 g 2 g 31 ∂ ∂x 1 ( g 2 g 31 g 31 g 32 B 3 B 1 ) + 1 g 31 g 32 g 2 g 32 ∂ ∂x 2 ( g 32 g 31 g 32 B 3 B 2 ) + 1 g 31 g 32 ∂ ∂x 3 ( B 3 2 2 ) -1 g 31 g 32 ∂ ∂x 3 ( B 2 2 2 ) -1 g 31 g 32 ∂ ∂x 3 ( B 1 2 2 ) -ρ 0 g 31 g 32 ∂ ∂x 3 ( p 1 ρ 0 ) + [ 1 g 31 g 32 g 2 g 31 ∂ ∂x 1 ( g 2 g 31 g 31 g 32 ρ 0 νS 31 ) + 1 g 31 g 32 g 2 g 32 ∂ ∂x 2 ( g 32 g 31 g 32 ρ 0 νS 32 ) + 1 g 31 g 32 ∂ ∂x 3 ( ρ 0 νS 33 ) ] -2 3 1 g 31 g 32 ∂ ∂x 3 ( ρ 0 ν ∇· v ) , (A11)</formula>
<formula><location><page_24><loc_13><loc_62><loc_88><loc_86></location>ρ 0 T 0 ∂s 1 ∂t = -1 g 2 g 31 ∂ ∂x 1 [ g 2 g 31 ( ρ 0 T 0 v 1 ( s 1 + s 0 )) ∗ ] -1 g 2 g 32 ∂ ∂x 2 [ g 32 ( ρ 0 T 0 v 2 ( s 1 + s 0 )) ∗ ] -1 g 31 g 32 ∂ ∂x 3 ( ρ 0 T 0 v 3 ( s 1 + s 0 )) ∗ -ρ 0 v 1 ( s 1 + s 0 ) g 0 c p + ρ 0 ν ( S 2 12 + S 2 23 + S 2 31 ) + 1 6 ρ 0 ν [ ( S 11 -S 22 ) 2 +( S 22 -S 33 ) 2 +( S 33 -S 11 ) 2 ] + Q num + η ( ∇× B ) 2 + ∇· ( Kρ 0 T 0 ∇ s 1 ) + ∇· ( 16 σ s T 3 0 3 κρ 0 ∇ T 0 ) , (A12)</formula>
<text><location><page_24><loc_12><loc_59><loc_17><loc_60></location>where</text>
<formula><location><page_24><loc_45><loc_55><loc_88><loc_59></location>S 11 = 2 ∂v 1 ∂x 1 , (A13)</formula>
<formula><location><page_24><loc_40><loc_51><loc_88><loc_55></location>S 22 = 2 g 2 ∂v 2 ∂x 2 + 2 v 1 g 2 ∂g 2 ∂x 1 (A14)</formula>
<formula><location><page_24><loc_33><loc_47><loc_88><loc_50></location>S 33 = 2 g 31 g 32 ∂v 3 ∂x 3 + 2 v 1 g 31 ∂g 31 ∂x 1 + 2 v 2 g 2 g 32 ∂g 32 ∂x 2 (A15)</formula>
<formula><location><page_24><loc_36><loc_42><loc_88><loc_46></location>S 12 = S 21 = 1 g 2 ∂v 1 ∂x 2 + g 2 ∂ ∂x 1 ( v 2 g 2 ) (A16)</formula>
<formula><location><page_24><loc_33><loc_38><loc_88><loc_41></location>S 23 = S 32 = 1 g 31 g 32 ∂v 2 x 3 + g 32 g 2 ∂ ∂x 2 ( v 3 g 32 ) (A17)</formula>
<formula><location><page_24><loc_33><loc_33><loc_88><loc_37></location>S 31 = S 13 = 1 g 31 g 32 ∂v 1 ∂x 3 + g 31 ∂ ∂x 1 ( v 3 g 31 ) . (A18)</formula>
<text><location><page_24><loc_12><loc_29><loc_88><loc_32></location>Note that the 3-component of the momentum equation (A11) is written in the angular momentum conservative form.</text>
<text><location><page_24><loc_12><loc_16><loc_88><loc_27></location>For spatially discretizing equations (A9), (A10), (A11), and (A12), standard 2nd order interpolations and finite-differences are applied to all the quantities and derivatives, except for the fluxes (with superscript '*') in the first 3 advection terms on the right hand side (RHS) of each of the above equations. For evaluating these fluxes through their respective cell faces, we use a modified Lax-Friedrichs scheme (Rempel et al. 2009) to get an upwinded evaluation of the fluxes as follows.</text>
<text><location><page_24><loc_12><loc_11><loc_88><loc_14></location>For the first 3 terms on the RHS of equation (A9), the upwinded evaluation of the 1-, 2-, and 3-fluxes ρ 0 v 1 v 1 , ρ 0 v 2 v 1 , and ρ 0 v 3 v 1 through their respective cell-faces at respectively</text>
<text><location><page_25><loc_12><loc_84><loc_52><loc_86></location>x 1 = x 1 ,i -1 / 2 , x 2 = x 2 ,i -1 / 2 , and x 3 = x 3 ,i -1 / 2 are</text>
<formula><location><page_25><loc_23><loc_78><loc_88><loc_83></location>( ρ 0 v 1 v 1 ) ∗ i -1 / 2 = ( ρ 0 v 1 ) i -1 / 2 <v 1 > 1 -( ρ 0 | v 1 | + c a q l 1 ,v 1 2 ) i -1 / 2 ∆ 1 v 1 , (A19)</formula>
<formula><location><page_25><loc_23><loc_72><loc_88><loc_76></location>( ρ 0 v 2 v 1 ) ∗ i -1 / 2 = ( ρ 0 v 2 ) i -1 / 2 <v 1 > 2 -( ρ 0 | v 2 | + c a q l 2 ,v 1 2 ) i -1 / 2 ∆ 2 v 1 , (A20)</formula>
<formula><location><page_25><loc_22><loc_66><loc_88><loc_71></location>( ρ 0 v 3 v 1 ) ∗ i -1 / 2 = ( ρ 0 v 3 ) i -1 / 2 <v 1 > 3 -( ρ 0 | v 3 | + c a q l 3 ,v 1 2 ) i -1 / 2 , ∆ 3 v 1 , (A21)</formula>
<text><location><page_25><loc_12><loc_59><loc_88><loc_65></location>where <v 1 > 1 , <v 1 > 2 , <v 1 > 3 correspond to the left-right averages at the respective cellfaces as given by equation (A8), and ∆ 1 v 1 , ∆ 2 v 1 , ∆ 3 v 1 correspond to the limited differences evaluated at the respective cell-faces as given by equation (A7), and</text>
<formula><location><page_25><loc_44><loc_54><loc_88><loc_58></location>q m,v 1 = ∆ m v 1 δ m v 1 , (A22)</formula>
<text><location><page_25><loc_12><loc_35><loc_88><loc_53></location>with m = 1 , 2 , 3, and δ m v 1 given by equation (A1). Also on the RHS of equations (A19), (A20), (A21), all the other quantities in () are evaluated via standard 2nd order interpolation at the cell-faces, and c a denotes the Alfv'en speed and l = 4. The 2nd terms on the RHS of equations (A19), (A20), and (A21) correspond to a diffusive flux resulting from the upwinded evaluation (Rempel et al. 2009). It can be seen that the speed c a in the diffusive flux is scale by the smoothness factor q m,v 1 (given by equation [A22]) to the l th power. It can be shown that the limited difference ∆ m v 1 is always of the same sign and of a smaller magnitude compared to the simple finite difference δ m v 1 . The factor q l m,v 1 glyph[lessmuch] 1 when the variation of v 1 in the m -direction is smooth, and thus reduces the speed in the diffusive flux.</text>
<text><location><page_25><loc_12><loc_30><loc_88><loc_34></location>In the same way, for equation (A10), the upwinded 1-, 2-, and 3-fluxes ρ 0 v 1 ( v 2 /g 2 ), ρ 0 v 2 ( v 2 /g 2 ), and ρ 0 v 3 ( v 2 /g 2 ) through their respective cell-faces are:</text>
<formula><location><page_25><loc_18><loc_24><loc_88><loc_29></location>( ρ 0 v 1 v 2 g 2 ) ∗ i -1 / 2 = ( ρ 0 v 1 ) i -1 / 2 < v 2 g 2 > 1 -( ρ 0 | v 1 | + c a q l 1 ,v 2 2 ) i -1 / 2 ∆ 1 ( v 2 g 2 ) (A23)</formula>
<formula><location><page_25><loc_18><loc_17><loc_88><loc_22></location>( ρ 0 v 2 v 2 g 2 ) ∗ i -1 / 2 = ( ρ 0 v 2 ) i -1 / 2 < v 2 g 2 > 2 -( ρ 0 | v 2 | + c a q l 2 ,v 2 2 ) i -1 / 2 ∆ 2 ( v 2 g 2 ) (A24)</formula>
<formula><location><page_25><loc_17><loc_11><loc_88><loc_16></location>( ρ 0 v 3 v 2 g 2 ) ∗ i -1 / 2 = ( ρ 0 v 3 ) i -1 / 2 < v 2 g 2 > 3 -( ρ 0 | v 3 | + c a q l 3 ,v 2 2 ) i -1 / 2 ∆ 3 ( v 2 g 2 ) , (A25)</formula>
<text><location><page_26><loc_12><loc_85><loc_17><loc_86></location>where</text>
<formula><location><page_26><loc_42><loc_81><loc_88><loc_85></location>q m,v 2 = ∆ m ( v 2 /g 2 ) δ m ( v 2 /g 2 ) , (A26)</formula>
<text><location><page_26><loc_12><loc_77><loc_88><loc_80></location>for equation (A11), the upwinded 1-, 2-, and 3-fluxes ρ 0 v 1 ( v 3 /g 31 g 32 ), ρ 0 v 2 ( v 3 /g 31 g 32 ), and ρ 0 v 3 ( v 3 /g 31 g 32 ) through their respective cell-faces are:</text>
<formula><location><page_26><loc_13><loc_70><loc_88><loc_75></location>( ρ 0 v 1 v 3 g 31 g 32 ) ∗ i -1 / 2 = ( ρ 0 v 1 ) i -1 / 2 < v 3 g 31 g 32 > 1 -( ρ 0 | v 1 | + c a q l 1 ,v 3 2 ) i -1 / 2 ∆ 1 ( v 3 g 31 g 32 ) (A27)</formula>
<formula><location><page_26><loc_13><loc_64><loc_88><loc_69></location>( ρ 0 v 2 v 3 g 31 g 32 ) ∗ i -1 / 2 = ( ρ 0 v 2 ) i -1 / 2 < v 3 g 31 g 32 > 2 -( ρ 0 | v 2 | + c a q l 2 ,v 3 2 ) i -1 / 2 ∆ 2 ( v 3 g 31 g 32 ) (A28)</formula>
<formula><location><page_26><loc_23><loc_52><loc_88><loc_62></location>( ρ 0 v 3 v 3 g 31 g 32 ) ∗ i -1 / 2 = ( ρ 0 v 3 ) i -1 / 2 < v 3 g 31 g 32 > 3 -( ρ 0 | v 3 | + c a q l 3 ,v 3 2 ) i -1 / 2 ∆ 3 ( v 3 g 31 g 32 ) , (A29)</formula>
<text><location><page_26><loc_12><loc_49><loc_17><loc_50></location>where</text>
<formula><location><page_26><loc_40><loc_46><loc_88><loc_49></location>q m,v 3 = ∆ m ( v 3 /g 31 g 32 ) δ m ( v 3 /g 31 g 32 ) , (A30)</formula>
<text><location><page_26><loc_12><loc_41><loc_88><loc_45></location>and finally for equation (A12), the upwinded 1-, 2-, and 3-fluxes ρ 0 T 0 v 1 ( s 1 + s 0 ), ρ 0 T 0 v 2 ( s 1 + s 0 ), and ρ 0 T 0 v 3 ( s 1 + s 0 ) through their respective cell-faces are:</text>
<formula><location><page_26><loc_22><loc_32><loc_88><loc_39></location>( ρ 0 T 0 v 1 ( s 1 + s 0 )) ∗ i -1 / 2 = ( ρ 0 T 0 v 1 ) i -1 / 2 <s 1 + s 0 > 1 -( ρ 0 T 0 | v 1 | + c a q l 1 ,s 2 ) i -1 / 2 ∆ 1 ( s 1 + s 0 ) , (A31)</formula>
<formula><location><page_26><loc_22><loc_22><loc_88><loc_29></location>( ρ 0 T 0 v 2 ( s 1 + s 0 )) ∗ i -1 / 2 = ( ρ 0 T 0 v 2 ) i -1 / 2 <s 1 + s 0 > 2 -( ρ 0 T 0 | v 2 | + c a q l 2 ,s 2 ) i -1 / 2 ∆ 2 ( s 1 + s 0 ) , (A32)</formula>
<formula><location><page_26><loc_22><loc_11><loc_88><loc_19></location>( ρ 0 T 0 v 3 ( s 1 + s 0 )) ∗ i -1 / 2 = ( ρ 0 T 0 v 3 ) i -1 / 2 <s 1 + s 0 > 3 -( ρ 0 T 0 | v 3 | + c a q l 3 ,s 2 ) i -1 / 2 ∆ 3 ( s 1 + s 0 ) , (A33)</formula>
<text><location><page_27><loc_12><loc_85><loc_17><loc_86></location>where</text>
<formula><location><page_27><loc_42><loc_81><loc_88><loc_85></location>q m,s = ∆ m ( s 1 + s 0 ) δ m ( s 1 + s 0 ) . (A34)</formula>
<text><location><page_27><loc_12><loc_71><loc_88><loc_80></location>Furthermore, in the RHS of the entropy equation (A12) we have also included a numerical heating term Q num that corresponds to the dissipation of kinetic energy due to the diffusive fluxes (the 2nd term in the RHS of eqs. [A19], [A20], [A21], [A23], [A24], [A25], [A27], [A28], [A29]), by taking the dot product of the diffusive fluxes with the appropriate velocity gradients (computed via the standard centered finite difference),</text>
<formula><location><page_27><loc_12><loc_35><loc_89><loc_70></location>Q num = ( ρ 0 | v 1 | + c a q l 1 ,v 1 2 ∆ 1 v 1 ) ∂v 1 ∂x 1 + ( ρ 0 | v 2 | + c a q l 2 ,v 1 2 ∆ 2 v 1 ) 1 g 2 ∂v 1 ∂x 2 + ( ρ 0 | v 3 | + c a q l 3 ,v 1 2 ∆ 3 v 1 ) 1 g 31 g 32 ∂v 1 ∂x 3 + ( ρ 0 | v 1 | + c a q l 1 ,v 2 2 g 2 2 ∆ 1 ( v 2 g 2 ) ) ∂ ∂x 1 ( v 2 g 2 ) + ( ρ 0 | v 2 | + c a q l 2 ,v 2 2 g 2 2 ∆ 2 ( v 2 g 2 ) ) 1 g 2 ∂ ∂x 2 ( v 2 g 2 ) + ( ρ 0 | v 3 | + c a q l 3 ,v 2 2 g 2 2 ∆ 3 ( v 2 g 2 ) ) 1 g 31 g 32 ∂ ∂x 3 ( v 2 g 2 ) + ( ρ 0 | v 1 | + c a q l 1 ,v 3 2 g 2 31 g 2 32 ∆ 1 ( v 3 g 31 g 32 ) ) ∂ ∂x 1 ( v 3 g 31 g 32 ) + ( ρ 0 | v 2 | + c a q l 2 ,v 3 2 g 2 31 g 2 32 ∆ 2 ( v 3 g 31 g 32 ) ) 1 g 2 ∂ ∂x 2 ( v 3 g 31 g 32 ) + ( ρ 0 | v 3 | + c a q l 3 ,v 3 2 g 2 31 g 2 32 ∆ 3 ( v 3 g 31 g 32 ) ) 1 g 31 g 32 ∂ ∂x 3 ( v 3 g 31 g 32 ) , (A35)</formula>
<text><location><page_27><loc_12><loc_32><loc_63><loc_34></location>and then interpolating to the cell centers where s 1 is defined.</text>
<text><location><page_27><loc_16><loc_30><loc_63><loc_31></location>The pressure equation (20) we solve can be rewritten as:</text>
<formula><location><page_27><loc_27><loc_20><loc_88><loc_28></location>1 ρ 0 ∂ ∂x 1 ( g 2 g 31 ρ 0 ∂P ∂x 1 ) + 1 g 32 ∂ ∂x 2 ( g 32 ∂P ∂x 2 ) + 1 g 2 32 ∂ 2 P ∂x 2 3 = g 2 2 ρ 0 ∇· F , (A36)</formula>
<text><location><page_27><loc_12><loc_14><loc_88><loc_19></location>where P ≡ p 1 /ρ 0 . This linear equation is solved as follows. The 3-direction ( φ -direction) is periodic (for a full 2 π azimuth), so we carry out a Fourier decomposition of P in the x 3 -dimension such that:</text>
<formula><location><page_27><loc_37><loc_9><loc_88><loc_14></location>P k ≡ P | x 3 = x 3 ,k = N -1 ∑ n =0 ˆ P n e i 2 πf n x 3 ,k (A37)</formula>
<text><location><page_28><loc_12><loc_78><loc_88><loc_86></location>where x 3 ,k with k = 1 , 2 , ...N are the N uniformly spaced grid points in x 3 ∈ [0 , 2 π ], f n = n/ 2 π with n = 0 , 1 , ...N -1 denotes the discrete spatial frequency, and ˆ P n denotes the amplitude of the Fourier component with frequency f n . Then the centered finite difference evaluation of ∂ 2 P/∂x 2 3 gives:</text>
<formula><location><page_28><loc_18><loc_72><loc_88><loc_77></location>( ∂ 2 P ∂x 2 3 ) k = P k +1 -2 P k + P k -1 ( δx 3 ) 2 = N -1 ∑ n =0 ( 2 cos(2 πn/N ) -2 ( δx 3 ) 2 ) ˆ P n e i 2 πf n x 3 ,k , (A38)</formula>
<text><location><page_28><loc_12><loc_67><loc_88><loc_71></location>and equation (A36) leads to the following 2D separable linear equation for the Fourier component ˆ P n ( x 1 , x 2 ):</text>
<formula><location><page_28><loc_18><loc_59><loc_88><loc_65></location>1 ρ 0 ∂ ∂x 1 ( g 2 g 31 ρ 0 ∂ ˆ P n ∂x 1 ) + 1 g 32 ∂ ∂x 2 ( g 32 ∂ ˆ P n ∂x 2 ) -1 g 2 32 ( 2 -2 cos(2 πn/N ) ( δx 3 ) 2 ) ˆ P n = ˆ R n , (A39)</formula>
<text><location><page_28><loc_12><loc_47><loc_88><loc_57></location>where δx 3 denotes the grid spacing in x 3 , and ˆ R n is the Fourier transform of the RHS of equation (A36). Discretizing the above 2D linear equation leads to a block tridiagonal system, which is solved using the routine blktri.f in the FISHPACK math library of the National Center for Atmosphereic Research (NCAR), based on the generalized cyclic reduction scheme developed by P. Swatztrauber of NCAR.</text>
<text><location><page_28><loc_12><loc_22><loc_88><loc_46></location>For solving the induction equation (5) we use the constrained transport (CT) scheme on the staggered grid (Stone & Norman 1992b) to ensure the divergence free condition for the magnetic field (eq. [4]) is satisfied to round-off errors. The CT scheme is used in conjuction with an upwinded evaluation of both v and B based on the Alfv'en wave characteristics for computing the v × B electric field on the cell edges as described in Stone & Norman (1992b). The upwinded evaluation of the electric field would entail numerical dissipation of the magnetic field, which we did not put back as heating into the entropy equation. Thus, this is a cause of loss of conservation of total energy due to numerical dissipation in the code. We also evaluate the physical resistive electric field η ∇× B in equation (5) on the cell edges following the CT scheme, with the derivatives computed using simple second order finite differences. The Ohmic heating produced by the physical resistivity is being included in the entropy equation (A12).</text>
<text><location><page_28><loc_12><loc_11><loc_88><loc_20></location>After the RHS of all of the equations (A9), (A10), (A11), and (A12) are evaluated at the appropriated cell locations as described above, we advance the equations in time using a simple second-order predictor-corrector time stepping. The linear elliptic pressure equation (A36) is solved at every sub-timestep to obtain p 1 needed for advancing equations (A9), (A10), and (A11).</text>
<text><location><page_29><loc_12><loc_79><loc_88><loc_86></location>NCAR is sponsored by the National Science Foundation. This work is supported by the NASA SHP grant NNX10AB81G and NASA LWSCSW grant NNX13AG04A to NCAR. The numerical simulations were carried out on the Pleiades supercomputer at the NASA Advanced Supercomputing Division under project GID s1106.</text>
<section_header_level_1><location><page_29><loc_43><loc_72><loc_57><loc_74></location>REFERENCES</section_header_level_1>
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<text><location><page_29><loc_12><loc_59><loc_35><loc_61></location>Fan, Y. 2008, ApJ, 676, 680</text>
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<text><location><page_29><loc_12><loc_53><loc_67><loc_54></location>Featherstone, N., Brown, B., & Miesch, M. S. 2013, in preparation</text>
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<figure>
<location><page_31><loc_13><loc_33><loc_84><loc_78></location>
<caption>Fig. 1.- Radial velocity v r at three depths from FSAM case A and the ASH simulation, shown in Mollweide projection. Horizontal lines indicate lines of constant latitude, and curved arcs lines of constant longitude. Regions of yellow denote upflows, and regions of blue indicate downflows. Flows in FSAM are stronger near the surface and weaker near the bottom than their counterparts in ASH. Banana cell patterns in ASH tend to reach to higher latitudes than those in FSAM.</caption>
</figure>
<figure>
<location><page_32><loc_13><loc_30><loc_85><loc_75></location>
<caption>Fig. 2.- Zonal velocity v φ plotted at three depths in each simulation at the same time instant as Fig. 1. Azimuthal flows in FSAM case A are similar to those of ASH throughout most of the convection zone, becoming somewhat stronger near the surface.</caption>
</figure>
<figure>
<location><page_33><loc_33><loc_32><loc_67><loc_78></location>
<caption>Fig. 3.- Probability distribution functions (PDF) taken near (a) the top, (b) middle, and (c) bottom of each simulation, averaged in time over 10 rotation periods. PDFs for FSAM case A are shown in red. PDFs for ASH are plotted in black and have been computed over the FSAM range of latitudes. Both simulations show good agreement in the mid-convection zone. The cores of the distributions agree well in the boundary layers, though the wings are stronger for FSAM in the upper boundary layer and weaker relative to ASH in the lower boundary layer.</caption>
</figure>
<figure>
<location><page_34><loc_33><loc_38><loc_68><loc_72></location>
<caption>Fig. 4.- The effect of doubling the spatial resolution. (a) Radial velocity v r from FSAM case B near the bottom of the convection zone (0.72R). Flows are noticiably stronger than case A and extend to higher latitudes when the spatial resolution is doubled. (b) PDFs of radial velocity (at 0.72R) for FSAM case B (red), and the ASH simulation (black). The high velocity wings of the distribution have been enhanced substantially with respect to case A, reaching good agreement with those of the ASH simulation.</caption>
</figure>
<figure>
<location><page_35><loc_33><loc_30><loc_68><loc_80></location>
<caption>Fig. 5.- Mean flows averaged in longitude and over 10 rotation periods. (a) Differential rotation as realized with FSAM case A and (b) ASH. The differential rotation established in FSAM is somewhat stronger than that realized in ASH. (c) Latitudinal mass flux achieved in FSAM and (b) ASH. Blue (red) tones indicated poleward flow in the northern (southern) hemisphere, while red (blue) tones indicate equatorward flow. Individual hemispheres tend to be dominated by a large circulation cell for each simulation.</caption>
</figure>
<figure>
<location><page_36><loc_32><loc_31><loc_68><loc_76></location>
<caption>Fig. 6.- Mean entropy profiles (a) and entropy gradients (b) established in each case. ASH results are plotted in black, and FSAM case A in red. Convection in the FSAM simulations tends to build a somewhat more adiabatic interior and steeper entropy gradients near the boundaries than ASH.</caption>
</figure>
<figure>
<location><page_37><loc_33><loc_44><loc_68><loc_67></location>
<caption>Fig. 7.- Mean flows averaged in longitude and over 10 rotation periods for FSAM case C. (a) Differential rotation realized when a higher latitude range is included shows reduced amplitude relative to case A, yielding better agreement with the ASH simulation. (b) Latitudinal component of mass flux for case C. Blue (red) tones indicated poleward flow in the northern (southern) hemisphere, while red (blue) tones indicate equatorward flow. Meridional flows in case C also show good agreement with ASH.</caption>
</figure>
<figure>
<location><page_38><loc_33><loc_41><loc_68><loc_86></location>
<caption>Fig. 8.(a) PDFs of v r for FSAM case C (red) and the ASH case (black) near the top of the simulation (0.99R). The inclusion of higher latitudes yields a PDF in case C that is in better agreement with the ASH results than case A. (b) Mean entropy gradients established in case C (red) and the ASH case (black). The superadiabaticity of the boundary layers present in case A is diminished as more latitudes are included in the simulation.</caption>
</figure>
<figure>
<location><page_39><loc_16><loc_34><loc_82><loc_67></location>
<caption>Fig. 9.- The final steady state entropy gradient reached by the rotating solar convective envelope in the convection simulation described in Section 4.1</caption>
</figure>
<figure>
<location><page_40><loc_23><loc_34><loc_82><loc_66></location>
<caption>Fig. 10.- Total heat flux due to convection ( H conv ) and conduction ( H cond ) through the convective envelope when the solution has reached a statistical steady state.</caption>
</figure>
<figure>
<location><page_41><loc_13><loc_44><loc_87><loc_63></location>
<caption>Fig. 11.- (a) A snapshot of the radial velocity of the rotating solar convection at a depth of about 30 Mm below the photosphere, shown on the full sphere in Mollweide projection. (b) A meridional slice of the radial velocity of the convective flow at the same time. (c and d) Time and azimuthally averaged angular rate of rotation Ω and kinetic helicity H k in the convective envelope after it has reached the statistical steady state.</caption>
</figure>
<text><location><page_41><loc_17><loc_44><loc_20><loc_44></location>Vr (m/s)</text>
<figure>
<location><page_42><loc_26><loc_38><loc_77><loc_65></location>
<caption>Fig. 12.- Peak downflow speed, peak upflow speed, and the RMS vertical flow speed of the convective flows in the solar convective envelope as a function of depth, averaged over 30 evenly spaced temporal samples over a period of about 3 months.</caption>
</figure>
<figure>
<location><page_43><loc_19><loc_21><loc_81><loc_85></location>
<caption>Fig. 13.- 3D volume rendering of the absolute magnetic field strength of the rising flux tubes developed from simulations SbWt (a), SbZt (b), SbWt-ref (c), and SbZt-ref(d) when an apex of the tube has reached the top boundary. Animations of the evolution of the tube for each simulations are available in the online version of the paper.</caption>
</figure>
<figure>
<location><page_44><loc_19><loc_42><loc_81><loc_62></location>
<caption>Fig. 14.B φ in the meridional plane at the longitude of the apex location for cases SbWt (a), SbZt (b) , SbWt-ref (c), and SbZt-ref (d) at the same corresponding times shown in Figure 13</caption>
</figure>
<figure>
<location><page_45><loc_21><loc_55><loc_48><loc_76></location>
</figure>
<figure>
<location><page_45><loc_52><loc_55><loc_80><loc_76></location>
</figure>
<figure>
<location><page_45><loc_21><loc_31><loc_48><loc_52></location>
</figure>
<figure>
<location><page_45><loc_52><loc_31><loc_80><loc_52></location>
<caption>Fig. 15.- The normal flux distribution produced by the top of the rising tube approaching the upper boundary on a constant r surface at r = 0 . 957 R s , for the cases SbWt (a), SbZt (b), SbWt-ref (c), and SbZt-ref (d). Note in these plots, we have shifted the longitude of the apex location of the rising flux to 0 degree longitude.</caption>
</figure>
<figure>
<location><page_46><loc_29><loc_20><loc_71><loc_84></location>
<caption>Fig. 16.- Polar and equatorial views of selected field lines in the rising flux tubes for the cases SbWt (a), SbZt (b), SbWt-ref (c), and SbZt-ref (d). For each of the cases, the polar (equatorial) view is the upper (lower) panel</caption>
</figure>
<figure>
<location><page_47><loc_19><loc_44><loc_81><loc_60></location>
<caption>Fig. 17.- 3D volume rendering of the absolute magnetic field strength of the rising flux tube developed from the UbZt simulation as the apex at the right is reaching the top boundary. An MPEG animation of the evolution is available in the online version.</caption>
</figure>
<figure>
<location><page_48><loc_40><loc_28><loc_60><loc_76></location>
<caption>Fig. 18.- Selected field lines in the rising flux tube for the case UbZt as viewed from the pole with the apex located at the 6 o'clock position (a), and viewed from the equator with the apex located at the central meridian (b) and west limb (c) respectively.</caption>
</figure>
<figure>
<location><page_49><loc_13><loc_28><loc_87><loc_77></location>
<caption>Fig. 19.- Constant r slices of B r (a), v r (b) and v φ (c) at r = 0 . 957 R s (about 30 Mm below the top boundary) at the time when the the apex portion of the rising tube is reaching the top boundary for the UbZt case. White contours in (b) and (c) are contours of B r outlining the positive (solid line) and negative (dashed line) magnetic flux concentrations.</caption>
</figure>
<table>
<location><page_50><loc_34><loc_51><loc_66><loc_79></location>
<caption>Table 1. Simulation Parameters for FSAM/ASH ComparisonTable 2. Summary of the Rising Flux Tube Simulations</caption>
</table>
<table>
<location><page_50><loc_19><loc_20><loc_81><loc_36></location>
</table>
</document> | [] |
2016GrCo...22..208N | https://arxiv.org/pdf/1604.01394.pdf | <document>
<section_header_level_1><location><page_1><loc_9><loc_82><loc_82><loc_86></location>The effect of universe inhomogeneities on cosmological distance measurements</section_header_level_1>
<section_header_level_1><location><page_1><loc_9><loc_78><loc_44><loc_79></location>A. V. Nikolaev a, 1 and S. V. Chervon a,b</section_header_level_1>
<unordered_list>
<list_item><location><page_1><loc_9><loc_74><loc_65><loc_76></location>a Ilya Ulyanov State Pedagogical University, 432700 Ulyanovsk, Russia</list_item>
<list_item><location><page_1><loc_9><loc_72><loc_64><loc_74></location>b Astrophysics and Cosmology Research Unit, School of Mathematics,</list_item>
</unordered_list>
<text><location><page_1><loc_13><loc_69><loc_91><loc_72></location>Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X54 001, Durban 4000, South Africa</text>
<text><location><page_1><loc_9><loc_66><loc_30><loc_67></location>Received January 21, 2016</text>
<text><location><page_1><loc_12><loc_56><loc_88><loc_63></location>Using the focusing equation, the equation for the cosmological angular diameter distance 2 is derived, based on the ideas of Academician Zel'dovich, namely, that the distribution of matter at small angles is not homogeneous, and the light cone is close to being empty. We propose some ways of testing a method for measuring the angular diameter distances and show that the proposed method leads to results that agree better with the experimental data than those obtained by the usual methods.</text>
<text><location><page_1><loc_9><loc_28><loc_48><loc_50></location>The abundance of observational data in modern cosmology allows for testing a number of ideas put forward in the past. One of such ideas is the approach of Academician Ya. B. Zel'dovich to measurements of the cosmological angular diameter distances [1], which takes into account that null geodesics propagate in a homogeneous Friedmann universe, but the null geodesic congruence (or light cone) from the source experiences a smaller focusing than in a homogeneous universe. 3 Such an effect is possible if the density of matter inside a light cone is smaller than the mean density in the Friedmann universe .</text>
<text><location><page_1><loc_9><loc_18><loc_48><loc_28></location>Ref. [2] suggested a derivation of a generalized differential equation using such tools as the null geodesics and the ratio of longitudinal and transverse angular momentum of a photon. We will show that this equation can also be obtained on the basis of the focusing equation [6], which follows from</text>
<text><location><page_1><loc_9><loc_4><loc_48><loc_11></location>3 A light cone is understood here and henceforth as 'a cone of light rays' or a cone that bounds a beam of null geodesics, not to be confused with the light (or null) cone of relativity theory, which separates spacelike and timelike directions.</text>
<text><location><page_1><loc_52><loc_47><loc_91><loc_50></location>the Sachs equations [3] (a special case of the Raychaudhuri equations [5]):</text>
<formula><location><page_1><loc_54><loc_42><loc_91><loc_46></location>d 2 dλ 2 √ S = -( | σ | 2 + 1 2 R αβ k α k β ) √ S, (1)</formula>
<text><location><page_1><loc_52><loc_35><loc_91><loc_42></location>where S is the light cone cross-section area, λ is the affine parameter, k α is the null wave vector, Greek indices run over the values (0 , 1 , 2 , 3), and σ is the shear defined as follows:</text>
<formula><location><page_1><loc_54><loc_32><loc_91><loc_35></location>| σ | 2 = 1 2 k α ; β k α ; β -1 4 ( k α ; α ) 2 . (2)</formula>
<text><location><page_1><loc_55><loc_30><loc_89><loc_31></location>In the Friedmann-Robertson-Walker metric</text>
<formula><location><page_1><loc_54><loc_26><loc_91><loc_29></location>ds 2 = dt 2 -a ( t ) 2 [ dr 2 + f 2 ( r )( dθ 2 +sin 2 θdφ 2 )] (3)</formula>
<text><location><page_1><loc_52><loc_15><loc_91><loc_26></location>the wave vector has the following components for the arriving geodesics: k α in = ( -1 /a, 1 /a 2 , 0 , 0); the affine parameter is related to time through the scale factor: dλ = -adt . Directly calculating the covariant derivatives in the metric (3), we verify that | σ | 2 = 0:</text>
<formula><location><page_1><loc_54><loc_12><loc_91><loc_15></location>| σ | 2 = ( f ˙ a -f ' ) 2 f 2 a 4 -( f ˙ a -f ' ) 2 f 2 a 4 = 0 . (4)</formula>
<text><location><page_1><loc_52><loc_6><loc_91><loc_11></location>Expressing the light cone cross-section in terms of the linear size of the source S = πl 2 4 (see [2] for details) and substituting (4) into (1), we obtain</text>
<formula><location><page_1><loc_54><loc_2><loc_91><loc_5></location>l -˙ a a ˙ l + a 2 1 2 R αβ k α k β l = 0 . (5)</formula>
<text><location><page_2><loc_9><loc_86><loc_48><loc_89></location>Using in (5) the definition of an angular diameter distance d a = l/φ , we arrive at</text>
<formula><location><page_2><loc_11><loc_82><loc_48><loc_85></location>d a -˙ a a ˙ d a + a 2 1 2 R αβ k α k β d a = 0 . (6)</formula>
<text><location><page_2><loc_12><loc_80><loc_39><loc_81></location>Contracting the Einstein equations</text>
<formula><location><page_2><loc_11><loc_76><loc_48><loc_79></location>R αβ -1 2 Rg αβ = κT αβ (7)</formula>
<text><location><page_2><loc_9><loc_71><loc_48><loc_75></location>with k α k β (using the null nature of the wave vector, g αβ k α k β = 0), we obtain</text>
<formula><location><page_2><loc_11><loc_67><loc_48><loc_70></location>1 2 R αβ k α k β = κ 2 T αβ k α k β . (8)</formula>
<text><location><page_2><loc_9><loc_55><loc_48><loc_66></location>Here κ is the Einstein gravitational constant. It should be noted that the beam focusing is affected by only the local value of the Ricci tensor, or, due to the Einstein equations, by the local value of the energy-momentum tensor. The result (8) allows us to convert the equation for the cosmological angular diameter distance (6) to the form</text>
<formula><location><page_2><loc_11><loc_50><loc_48><loc_54></location>d a -˙ a a ˙ d a +4 πGa 2 T αβ k α k β d a = 0 , (9)</formula>
<text><location><page_2><loc_9><loc_43><loc_50><loc_50></location>where T αβ is the local value of the energy-momentum tensor inside the light cone. One can introduce the parameter α showing how much matter is there inside the cone:</text>
<formula><location><page_2><loc_11><loc_38><loc_48><loc_42></location>α = T αβ k α k β T full αβ k α k β . (10)</formula>
<text><location><page_2><loc_9><loc_32><loc_48><loc_37></location>Using the definition of the energy-momentum tensor for a perfect fluid, T β α = diag( ρ, -p, -p, -p ), we obtain:</text>
<formula><location><page_2><loc_11><loc_30><loc_48><loc_31></location>a 2 T αβ k α k β = p + ρ. (11)</formula>
<text><location><page_2><loc_9><loc_25><loc_48><loc_28></location>Making explicit the components for the ΛCDM model, we get:</text>
<formula><location><page_2><loc_11><loc_22><loc_48><loc_24></location>p + ρ = p Λ + p M + p R + ρ Λ + ρ M + ρ R . (12)</formula>
<text><location><page_2><loc_9><loc_16><loc_48><loc_21></location>Using the equations of state for baryonic matter ( p M = 0), dark energy ( p Λ = -ρ Λ ), and radiation ( p R = ρ R / 3), we convert the relation (12) to</text>
<formula><location><page_2><loc_11><loc_12><loc_48><loc_15></location>p + ρ = 4 3 ρ R + ρ M , (13)</formula>
<text><location><page_2><loc_9><loc_10><loc_14><loc_11></location>where</text>
<formula><location><page_2><loc_10><loc_1><loc_48><loc_9></location>ρ M = 3 H 2 0 Ω M 8 πG ( a 0 a ) 3 , ρ R = 3 H 2 0 Ω R 8 πG ( a 0 a ) 4 . (14)</formula>
<text><location><page_2><loc_52><loc_88><loc_88><loc_89></location>Using (11) and (13), Eq. (9) acquires the form</text>
<formula><location><page_2><loc_54><loc_82><loc_91><loc_87></location>d a -˙ a a ˙ d a +4 πG ( 4 3 ρ R + ρ M ) d a = 0 (15)</formula>
<text><location><page_2><loc_52><loc_72><loc_91><loc_82></location>Thus it has been established that dark energy does not participate in focusing of the light rays (which makes clear the question raised in [11]). Since inside the light cone, as a rule, ρ M and ρ R tend to zero, the value of d a will be larger than in Friedmann's homogeneous model [2].</text>
<text><location><page_2><loc_52><loc_62><loc_91><loc_72></location>Let us now discuss a number of tests for approaches to calculations of the angular diameter distance which follow from the data on the Sunyaev-Zel'dovich effect (SZE) for galaxy clusters [12-14]. The angular diameter distance may be expressed through the SZE data [12]:</text>
<formula><location><page_2><loc_54><loc_48><loc_91><loc_61></location>d SZE a = (∆ T 0 ) 2 S X 0 ( m e c 2 k B T e 0 ) 2 × λ eH 0 µ e /µ H 4 π 3 / 2 f 2 ( x,T e ) T 2 CMB σ 2 T (1 + z ) 4 1 θ c x × [ Γ(3 β/ 2) Γ(3 β/ 2 -1 / 2) ] 2 Γ(3 β -1 / 2) Γ(3 β ) , (16)</formula>
<text><location><page_2><loc_52><loc_30><loc_91><loc_47></location>where Γ( x ) is the gamma function, S X 0 is the Xray surface brightness of the cluster center, z is the redshift, λ eH is the cooling function of the cluster center, σ T is the total scattering cross-section, k B is the Boltzmann constant, ∆ T 0 is the SZE temperature difference, θ c is the angular size of the galactic nucleus, m e is the electron mass, f ( x, T e ) is the SZE frequency dependence, and T CMB is the temperature of the microwave background radiation.</text>
<text><location><page_2><loc_52><loc_24><loc_91><loc_30></location>Thus there emerges a test for the angular diameter distance connected with the Hubble constant H 0 . Let us write down the solution of (15) for an empty light cone in ΛCDM [2]:</text>
<formula><location><page_2><loc_54><loc_18><loc_91><loc_22></location>d empty a = 1 H 0 ∫ 1 1 1+ z dx √ Ω S (17)</formula>
<text><location><page_2><loc_52><loc_12><loc_91><loc_17></location>where Ω S = Ω Λ +Ω k x -2 +Ω M x -3 +Ω R x -4 , while for a light cone filled with matter whose density is equal to the mean density of the Universe, the</text>
<text><location><page_3><loc_9><loc_88><loc_32><loc_89></location>solution of (15) has the form</text>
<formula><location><page_3><loc_10><loc_70><loc_48><loc_87></location>d full a = 1 1 + z 1 H 0 √ Ω k sin ∫ 1 1 1+ z √ Ω k dx x 2 √ Ω S for k = 1 , d full a = 1 1 + z ∫ 1 1 1+ z dx H 0 x 2 √ Ω S for k = 0 , d full a = 1 1 + z 1 H 0 √ Ω k sinh ∫ 1 1 1+ z √ Ω k dx x 2 √ Ω S , for k = -1 . (18)</formula>
<text><location><page_3><loc_9><loc_58><loc_48><loc_69></location>This allows us to compare the values of the Hubble constant predicted by the standard solution (18) and the new formula (17). The value of the cosmological angular diameter distance is calculated directly from the SZE, which means that equating d empty a and d full a to d SZE a , we can find H 0 . Therefore, for an empty light cone we obtain</text>
<formula><location><page_3><loc_11><loc_52><loc_48><loc_56></location>H empty 0 = (∫ 1 1 1+ z dx √ Ω S )/ d ZSE a , (19)</formula>
<text><location><page_3><loc_9><loc_49><loc_29><loc_51></location>while for a full light cone</text>
<formula><location><page_3><loc_10><loc_31><loc_48><loc_48></location>H ZSE 0 = 1 (1 + z ) d ZSE a 1 √ Ω k sin ∫ 1 1 1+ z √ Ω k dx x 2 √ Ω S for k = 1 , H ZSE 0 = 1 (1 + z ) d ZSE a ∫ 1 1 1+ z dx x 2 √ Ω S for k = 0 , H ZSE 0 = 1 (1 + z ) d ZSE a 1 √ Ω k sinh ∫ 1 1 1+ z √ Ω k dx x 2 √ Ω S for k = -1 . (20)</formula>
<text><location><page_3><loc_9><loc_19><loc_48><loc_31></location>The calculation of the simple averages from the data for clusters of galaxies [12] allows us to conclude that more consistent values of the Hubble constant are given by Eqs. (19) than (20). A detailed analysis of the galactic cluster data in the context of using Eqs. (19) can serve as a material for further experimental studies.</text>
<text><location><page_3><loc_9><loc_14><loc_48><loc_19></location>The next test is connected with the duality between the cosmological angular diameter distance d a and the luminosity distance d l :</text>
<formula><location><page_3><loc_11><loc_9><loc_48><loc_12></location>η = d l d a (1 + z ) -2 = 1 , (21)</formula>
<text><location><page_3><loc_9><loc_6><loc_45><loc_8></location>which follows from the Eddington identity [4]:</text>
<formula><location><page_3><loc_11><loc_3><loc_48><loc_5></location>r 2 s = r 2 o (1 + z ) 2 , (22)</formula>
<text><location><page_3><loc_52><loc_83><loc_91><loc_89></location>where r s is the distance to the source and r o is the distance to the observer, which is determined through the solid angle and the cross-section area, dS = r 2 d Ω (for more details see [7]).</text>
<text><location><page_3><loc_52><loc_78><loc_91><loc_83></location>In [10], an attempt is undertaken to test the validity of the identity (21) on the basis of the data from galaxy clusters [12] using the formula</text>
<formula><location><page_3><loc_54><loc_72><loc_91><loc_76></location>η ( z ) = √ d Th a d data a , (23)</formula>
<text><location><page_3><loc_52><loc_61><loc_91><loc_71></location>where d Th a is obtained from theoretical calculations according to (17) or (18). An analysis [9] shows that the new method of calculations of the angular diameter distance allows one to experimentally confirm the identity (21) with a greater accuracy than the standard method.</text>
<text><location><page_3><loc_52><loc_50><loc_91><loc_60></location>Another approach to verification of the identity has been proposed in [8], using the surface brightness data in the X-ray spectrum together with SZE data [13, 14]. To assess the validity of the identity (21), one uses the mass fraction of gas in the galaxy, f = M gas /M Tot , and the ratio</text>
<formula><location><page_3><loc_54><loc_45><loc_91><loc_49></location>η ( z ) = f SZE f X -ray , (24)</formula>
<text><location><page_3><loc_52><loc_39><loc_91><loc_44></location>where f SZE is the mass fraction of gas measured with the aid of the SZE, and f X -ray is the same calculated assuming the validity of (21) [14, 15].</text>
<text><location><page_3><loc_52><loc_22><loc_91><loc_39></location>If we insert the correction connected with applying the new method of calculation of the angular diameter distance (17), an analysis shows that the z dependence of η ( z )' becomes closer to unity. This argues in favor of the new method of calculation of the angular diameter distance. The very distribution of values of η is shifted to unity, showing that the duality identity for cosmological distances holds with an accuracy of 1 σ , in contrast to the result (2 σ ) of the original work [8]..</text>
<text><location><page_3><loc_52><loc_5><loc_91><loc_22></location>In conclusion, we would like to note that, based on the aforementioned reasoning, Zel'dovich's idea receives a confirmation. In contrast to the papers [16, 17], developing the ideas of Dyer and Roeder, we obtain simpler calculation formulas which reflect the physical meaning of measurements of the cosmological angular diameter distance in the Friedmann universe taking into account the inhomogeneities. Our approach makes it possible to pass on to the stage of experimental verification.</text>
<section_header_level_1><location><page_4><loc_9><loc_88><loc_26><loc_90></location>Acknowledgments</section_header_level_1>
<text><location><page_4><loc_9><loc_74><loc_48><loc_87></location>The authors are grateful to the participants of the Zel'manov seminar on gravitation and cosmology at the Sternberg astronomical Institute of Moscow State University and the VNIIMS gravitational seminar for their constructive criticism and helpful discussions. SC and AN were partly supported by the State order of Ministry of education and science of RF number 2014/391 on the project 1670.</text>
<section_header_level_1><location><page_4><loc_9><loc_69><loc_21><loc_71></location>References</section_header_level_1>
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<list_item><location><page_4><loc_9><loc_59><loc_48><loc_62></location>[3] R. K. Sachs, Proc. Roy. Soc. London 264 , 309 (1961).</list_item>
<list_item><location><page_4><loc_9><loc_57><loc_46><loc_59></location>[4] G. F. R. Ellis, Gen. Rel. Grav. 39 , 1047 (2007).</list_item>
<list_item><location><page_4><loc_9><loc_52><loc_48><loc_56></location>[5] E. Poisson, A Relativist's Toolkit. The Mathematics of Black-Hole Mechanics (Cambridge University Press, Cambridge, 2004).</list_item>
<list_item><location><page_4><loc_9><loc_49><loc_48><loc_51></location>[6] P. Schneider, J. Ehlers, and E. E. Falco, Gravitational Lenses (Springer, Berlin, 1999).</list_item>
<list_item><location><page_4><loc_9><loc_46><loc_41><loc_48></location>[7] G. Ellis, Gen. Rel. Grav. 41 , 581 (2009).</list_item>
<list_item><location><page_4><loc_9><loc_43><loc_48><loc_46></location>[8] R. F. L. Holanda, R. S. Goncalves, and J. S. Alcaniz, JCAP 1206 , 022 (2012).</list_item>
<list_item><location><page_4><loc_9><loc_39><loc_48><loc_42></location>[9] A. V. Nikolaev and S. V. Chervon, IJMPA 31 , 1641013 (2016)</list_item>
<list_item><location><page_4><loc_9><loc_36><loc_48><loc_38></location>[10] J.-P. Uzan, N. Aghanim, and Y. Mellier, Phys. Rev. D 70 , 083533 (2004).</list_item>
<list_item><location><page_4><loc_9><loc_32><loc_48><loc_35></location>[11] M. Sereno, E. Piedipalumbo, and M. V. Sazhin, Mon. Not. Roy. Astron. Soc. 335 , 1061 (2002).</list_item>
<list_item><location><page_4><loc_9><loc_27><loc_48><loc_31></location>[12] E. D. Reese, J. E. Carlstrom, M. Joy, J. J. Mohr, L. Grego, and W. L. Holzapfel, Astroph. J. 581 , 53 (2002).</list_item>
<list_item><location><page_4><loc_9><loc_22><loc_48><loc_26></location>[13] S. J. LaRoque, M. Bonamente, J. E. Carlstrom, M. K. Joy, D. Nagai, E. D. Reese, and K. S. Dawson, Astroph. J. 652 , 917 (2006).</list_item>
<list_item><location><page_4><loc_9><loc_17><loc_48><loc_21></location>[14] S. Ettori, A. Morandi, P. Tozzi, I. Balestra, S. Borgani, P. Rosati, L. Lovisari and F. Terenziani, Astron. Astroph. 501 , 61 (2009).</list_item>
<list_item><location><page_4><loc_9><loc_11><loc_48><loc_16></location>[15] S. W. Allen, D. A. Rapetti, R. W. Schmidt, H. Ebeling, R. G. Morris, and A. C. Fabian, Mon. Not. Roy. Astron. Soc. 383 , 879 (2008).</list_item>
<list_item><location><page_4><loc_9><loc_8><loc_48><loc_11></location>[16] P. Schneider and A. Weiss, Astroph. J. 327 , 526 (1988).</list_item>
<list_item><location><page_4><loc_9><loc_4><loc_48><loc_7></location>[17] S. Seitz, P. Schneider, and J. Ehlers, Class.QuantumGrav. 11 , 2345 (1994).</list_item>
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</document> | [{"title": "A. V. Nikolaev a, 1 and S. V. Chervon a,b", "content": "Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X54 001, Durban 4000, South Africa Received January 21, 2016 Using the focusing equation, the equation for the cosmological angular diameter distance 2 is derived, based on the ideas of Academician Zel'dovich, namely, that the distribution of matter at small angles is not homogeneous, and the light cone is close to being empty. We propose some ways of testing a method for measuring the angular diameter distances and show that the proposed method leads to results that agree better with the experimental data than those obtained by the usual methods. The abundance of observational data in modern cosmology allows for testing a number of ideas put forward in the past. One of such ideas is the approach of Academician Ya. B. Zel'dovich to measurements of the cosmological angular diameter distances [1], which takes into account that null geodesics propagate in a homogeneous Friedmann universe, but the null geodesic congruence (or light cone) from the source experiences a smaller focusing than in a homogeneous universe. 3 Such an effect is possible if the density of matter inside a light cone is smaller than the mean density in the Friedmann universe . Ref. [2] suggested a derivation of a generalized differential equation using such tools as the null geodesics and the ratio of longitudinal and transverse angular momentum of a photon. We will show that this equation can also be obtained on the basis of the focusing equation [6], which follows from 3 A light cone is understood here and henceforth as 'a cone of light rays' or a cone that bounds a beam of null geodesics, not to be confused with the light (or null) cone of relativity theory, which separates spacelike and timelike directions. the Sachs equations [3] (a special case of the Raychaudhuri equations [5]): where S is the light cone cross-section area, \u03bb is the affine parameter, k \u03b1 is the null wave vector, Greek indices run over the values (0 , 1 , 2 , 3), and \u03c3 is the shear defined as follows: In the Friedmann-Robertson-Walker metric the wave vector has the following components for the arriving geodesics: k \u03b1 in = ( -1 /a, 1 /a 2 , 0 , 0); the affine parameter is related to time through the scale factor: d\u03bb = -adt . Directly calculating the covariant derivatives in the metric (3), we verify that | \u03c3 | 2 = 0: Expressing the light cone cross-section in terms of the linear size of the source S = \u03c0l 2 4 (see [2] for details) and substituting (4) into (1), we obtain Using in (5) the definition of an angular diameter distance d a = l/\u03c6 , we arrive at Contracting the Einstein equations with k \u03b1 k \u03b2 (using the null nature of the wave vector, g \u03b1\u03b2 k \u03b1 k \u03b2 = 0), we obtain Here \u03ba is the Einstein gravitational constant. It should be noted that the beam focusing is affected by only the local value of the Ricci tensor, or, due to the Einstein equations, by the local value of the energy-momentum tensor. The result (8) allows us to convert the equation for the cosmological angular diameter distance (6) to the form where T \u03b1\u03b2 is the local value of the energy-momentum tensor inside the light cone. One can introduce the parameter \u03b1 showing how much matter is there inside the cone: Using the definition of the energy-momentum tensor for a perfect fluid, T \u03b2 \u03b1 = diag( \u03c1, -p, -p, -p ), we obtain: Making explicit the components for the \u039bCDM model, we get: Using the equations of state for baryonic matter ( p M = 0), dark energy ( p \u039b = -\u03c1 \u039b ), and radiation ( p R = \u03c1 R / 3), we convert the relation (12) to where Using (11) and (13), Eq. (9) acquires the form Thus it has been established that dark energy does not participate in focusing of the light rays (which makes clear the question raised in [11]). Since inside the light cone, as a rule, \u03c1 M and \u03c1 R tend to zero, the value of d a will be larger than in Friedmann's homogeneous model [2]. Let us now discuss a number of tests for approaches to calculations of the angular diameter distance which follow from the data on the Sunyaev-Zel'dovich effect (SZE) for galaxy clusters [12-14]. The angular diameter distance may be expressed through the SZE data [12]: where \u0393( x ) is the gamma function, S X 0 is the Xray surface brightness of the cluster center, z is the redshift, \u03bb eH is the cooling function of the cluster center, \u03c3 T is the total scattering cross-section, k B is the Boltzmann constant, \u2206 T 0 is the SZE temperature difference, \u03b8 c is the angular size of the galactic nucleus, m e is the electron mass, f ( x, T e ) is the SZE frequency dependence, and T CMB is the temperature of the microwave background radiation. Thus there emerges a test for the angular diameter distance connected with the Hubble constant H 0 . Let us write down the solution of (15) for an empty light cone in \u039bCDM [2]: where \u2126 S = \u2126 \u039b +\u2126 k x -2 +\u2126 M x -3 +\u2126 R x -4 , while for a light cone filled with matter whose density is equal to the mean density of the Universe, the solution of (15) has the form This allows us to compare the values of the Hubble constant predicted by the standard solution (18) and the new formula (17). The value of the cosmological angular diameter distance is calculated directly from the SZE, which means that equating d empty a and d full a to d SZE a , we can find H 0 . Therefore, for an empty light cone we obtain while for a full light cone The calculation of the simple averages from the data for clusters of galaxies [12] allows us to conclude that more consistent values of the Hubble constant are given by Eqs. (19) than (20). A detailed analysis of the galactic cluster data in the context of using Eqs. (19) can serve as a material for further experimental studies. The next test is connected with the duality between the cosmological angular diameter distance d a and the luminosity distance d l : which follows from the Eddington identity [4]: where r s is the distance to the source and r o is the distance to the observer, which is determined through the solid angle and the cross-section area, dS = r 2 d \u2126 (for more details see [7]). In [10], an attempt is undertaken to test the validity of the identity (21) on the basis of the data from galaxy clusters [12] using the formula where d Th a is obtained from theoretical calculations according to (17) or (18). An analysis [9] shows that the new method of calculations of the angular diameter distance allows one to experimentally confirm the identity (21) with a greater accuracy than the standard method. Another approach to verification of the identity has been proposed in [8], using the surface brightness data in the X-ray spectrum together with SZE data [13, 14]. To assess the validity of the identity (21), one uses the mass fraction of gas in the galaxy, f = M gas /M Tot , and the ratio where f SZE is the mass fraction of gas measured with the aid of the SZE, and f X -ray is the same calculated assuming the validity of (21) [14, 15]. If we insert the correction connected with applying the new method of calculation of the angular diameter distance (17), an analysis shows that the z dependence of \u03b7 ( z )' becomes closer to unity. This argues in favor of the new method of calculation of the angular diameter distance. The very distribution of values of \u03b7 is shifted to unity, showing that the duality identity for cosmological distances holds with an accuracy of 1 \u03c3 , in contrast to the result (2 \u03c3 ) of the original work [8].. In conclusion, we would like to note that, based on the aforementioned reasoning, Zel'dovich's idea receives a confirmation. In contrast to the papers [16, 17], developing the ideas of Dyer and Roeder, we obtain simpler calculation formulas which reflect the physical meaning of measurements of the cosmological angular diameter distance in the Friedmann universe taking into account the inhomogeneities. Our approach makes it possible to pass on to the stage of experimental verification.", "pages": [1, 2, 3]}, {"title": "Acknowledgments", "content": "The authors are grateful to the participants of the Zel'manov seminar on gravitation and cosmology at the Sternberg astronomical Institute of Moscow State University and the VNIIMS gravitational seminar for their constructive criticism and helpful discussions. SC and AN were partly supported by the State order of Ministry of education and science of RF number 2014/391 on the project 1670.", "pages": [4]}] |
2014MNRAS.441..813V | https://arxiv.org/pdf/1307.2682.pdf | <document>
<section_header_level_1><location><page_1><loc_13><loc_83><loc_83><loc_90></location>Non-axisymmetric vertical shear and convective instabilities as a mechanism of angular momentum transport</section_header_level_1>
<section_header_level_1><location><page_1><loc_13><loc_78><loc_32><loc_79></location>Francesco Volponi /star</section_header_level_1>
<text><location><page_1><loc_13><loc_77><loc_66><loc_77></location>Graduate School of Frontier Science, The University of Tokyo, Chiba 277-8561, Japan</text>
<section_header_level_1><location><page_1><loc_34><loc_69><loc_44><loc_70></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_34><loc_57><loc_87><loc_69></location>Discs with a rotation profile depending on radius and height are subject to an axisymmetric linear instability, the vertical shear instability. Here we show that non-axisymmetric perturbations, while eventually stabilized, can sustain huge exponential amplifications with growth rate close to the axisymmetric one. Transient growths are therefore to all effects genuine instabilities. The ensuing angular momentum transport is positive. These growths occur when the product of the radial times the vertical wavenumbers (both evolving with time) is positive for a positive local vertical shear, or negative for a negative local vertical shear.</text>
<text><location><page_1><loc_34><loc_51><loc_87><loc_57></location>We studied, as well, the interaction of these vertical shear induced growths with a convective instability. The asymptotic behaviour depends on the relative strength of the axisymmetric vertical shear ( s vs ) and convective ( s c ) growth rates.</text>
<text><location><page_1><loc_34><loc_46><loc_87><loc_51></location>For s vs > s c we observed the same type of behaviour described above large growths occur with asymptotic stabilization. When s c > s vs the system is asymptotically unstable, with a growth rate which can be slightly enhanced with respect to s c .</text>
<text><location><page_1><loc_34><loc_41><loc_87><loc_45></location>The most interesting feature is the sign of the angular momentum transport. This is always positive in the phase in which the vertical shear driven transients growths occur, even in the case s c > s vs .</text>
<text><location><page_1><loc_34><loc_39><loc_87><loc_41></location>Thermal diffusion has a stabilizing influence on the convective instability, specially for short wavelengths.</text>
<text><location><page_1><loc_34><loc_35><loc_87><loc_38></location>Key words: accretion, accretion discs - hydrodynamics - convection instabilities</text>
<section_header_level_1><location><page_2><loc_13><loc_91><loc_30><loc_92></location>1 INTRODUCTION</section_header_level_1>
<text><location><page_2><loc_13><loc_76><loc_48><loc_89></location>The evolution of accretion discs is determined by the outward transport of angular momentum, which induces infall of matter toward the orbital center. It is generally assumed that the accretion rate depends on an effective viscosity (Shakura & Sunyaev 1973). In magnetized discs this mechanism is triggered by a linear axisymmetric destabilization, the magnetorotational instability, which occurs when a weak poloidal magnetic field is coupled to rotation (Balbus & Hawley 1991).</text>
<text><location><page_2><loc_13><loc_68><loc_48><loc_76></location>In hydrodynamic Keplerian discs, instead, the presence of an efficient outward transport of angular momentum is still an open question. In this context there is no linear axisymmetric mechanism of growth, since Kepler rotation is stable according to Rayleigh criterion.</text>
<text><location><page_2><loc_13><loc_48><loc_48><loc_67></location>A possibility is given by the bypass scenario (Ioannou & Kakouris 2001, Chagelishvili et al. 2003), where arbitrarily large amplifications of perturbations can occur due to the leading-trailing evolution of shearwise wavenumbers. These growths occur independently of the presence of a linear instability drive. This linear mechanism and non-linear interactions, providing a positive feedback loop, could trigger a subcritical transition into turbulence. However, Lesur & Longaretti (2005) showed that, while subcritical transition to turbulence indeed occurs in non-stratified discs, values of the Shakura-Sunyaev α parameter (Shakura & Sunyaev 1973) are much too low to give rise to an efficient turbulent transport.</text>
<text><location><page_2><loc_13><loc_24><loc_48><loc_47></location>If radial stratification is taken into account radial entropy gradients can drive non-linear subcritical baroclinic destabilization leading to positive angular momentum transport (Klahr & Bodenheimer 2003, Lesur & Papaloizou 2010) and formation of vortices. However, discs are in general radially and vertically stratified and the angular velocity is a function of both radial and vertical coordinates (Kippenhahm & Thomas 1982, Urpin 1984). Vertical velocity shear drives an axisymmetric instability (Goldreich & Schubert 1967; Urpin 2003) which can result in positive transport of angular momentum in the non-linear regime (Arlt & Urpin 2004). The linear destabilization mechanism was derived by means of a local dispersion relation in the full cylindrical geometry. Instability occurs when k r /greatermuch k z .</text>
<text><location><page_2><loc_13><loc_20><loc_48><loc_25></location>In a recent study Nelson, Gressel & Umurhan (2012) found that in non-axisymmetric non-linear simulations outward transport can occur with α ∼ 10 -4 .</text>
<text><location><page_2><loc_13><loc_6><loc_48><loc_20></location>Vertical convection is another possible path to turbulence. Ryu & Goodman (1992) showed, by means of a linear analysis, that an unstable vertical stratification is not favourable for accretion; angular momentum transport occurs, but inward. Stone & Balbus (1996) extended the analysis to non-linear regimes with analogous results. Departures from this picture are however possible as shown by Lesur & Ogilvie (2010), in the strongly non-linear regime of high Rayleigh number, and by Volponi (2010), for linear</text>
<text><location><page_2><loc_52><loc_90><loc_87><loc_92></location>non-axisymmetric Boussinesq perturbations in discs with radial and vertical stratification.</text>
<text><location><page_2><loc_52><loc_75><loc_87><loc_90></location>In the present study we will concentrate on the vertical shear and vertical convective instabilities. Precisely, first, we will extend the linear analysis of the vertical shear instability to non-axisymmetric perturbations. Results will be derived by solving the shearing sheet equations in the presence of vertical velocity shear (Knobloch & Spruit 1986) and in the short wavelength regime. We will show that, while nonaxisymmetric perturbations eventually decay, they experience transient exponential growths which are indistinguishable from a genuine instability.</text>
<text><location><page_2><loc_52><loc_59><loc_87><loc_74></location>Second, we will study the interaction between the vertical shear and convective instabilities. This is the central part of the present study. We will show that the presence of vertical shear changes the direction of the vertical convection induced angular momentum transport from inward to outward depending on the sign of the product of vertical and radial wavenumbers (i.e. positive for A z > 0 or negative for A z < 0, where A z is the local vertical shear). Thermal diffusion has a stabilizing influence on the convective instability, specially for short wavelengths.</text>
<section_header_level_1><location><page_2><loc_52><loc_55><loc_82><loc_56></location>2 EQUATIONS AND EQUILIBRIUM</section_header_level_1>
<text><location><page_2><loc_52><loc_52><loc_87><loc_54></location>In the shearing sheet approximation, the equations governing the dynamics of a 3-dimensional disc are</text>
<formula><location><page_2><loc_52><loc_49><loc_87><loc_51></location>∂ t D + ∇· D V = 0 , (1)</formula>
<formula><location><page_2><loc_52><loc_45><loc_87><loc_49></location>∂ t V + V · ∇ V = -∇ P D -2 Ω × V +2 q Ω 2 x ˆ x -Ω 2 z ˆ z , (2)</formula>
<formula><location><page_2><loc_52><loc_44><loc_87><loc_45></location>∂ t (ln S ) + V · ∇ (ln S ) = 0 , (3)</formula>
<text><location><page_2><loc_52><loc_30><loc_87><loc_43></location>where D and P are density and pressure, V is the fluid velocity, S = P D -γ is a measure of the fluid entropy, γ is the adiabatic index, Ω is the local rotation frequency and q is the shear parameter ( q = 1 . 5 for Keplerian rotation). The term -2 Ω × V is the Coriolis term, 2 q Ω 2 x ˆ x is the tidal expansion of the effective potential and -Ω 2 z ˆ z is the vertical gravitational acceleration. The equations are expressed in terms of the pseudoCartesian coordinates x = r -r 0 , y = r 0 ( φ -φ 0 ) and z ( r 0 and φ 0 are reference radius and angle).</text>
<text><location><page_2><loc_52><loc_21><loc_87><loc_30></location>The above equations are more manageable than the full equations in cylindrical geometry due to the neglect of curvature effects. The simplification holds if the length scale of radial gradients L /lessmuch r , where r is the cylindrical radial coordinate (Knobloch & Spruit 1986).</text>
<text><location><page_2><loc_52><loc_15><loc_87><loc_21></location>We consider fully stratified (i.e. radially and vertically) baroclinic discs. P e ( x, z ) and ρ e ( x, z ) are respectively the equilibrium pressure and density. Equilibrium in general requires the angular velocity to be x and z dependent.</text>
<text><location><page_2><loc_52><loc_13><loc_82><loc_14></location>The equilibrium velocity field is then given by</text>
<formula><location><page_2><loc_52><loc_10><loc_87><loc_12></location>[ -q Ω x + ∂ x P e ( x, 0) 2Ω ρ e ( x, 0) + a V z 2 ]ˆ y , (4)</formula>
<text><location><page_2><loc_52><loc_6><loc_87><loc_9></location>which is obtained from the expansion of the angular velocity profile about the midplane as given in Kley</text>
<text><location><page_3><loc_13><loc_88><loc_48><loc_92></location>& Lin (1992) (see as well Urpin 1984), in the limit L/r /lessmuch 1. The dimensional coefficient a V can be derived from equation (A23) of Kley & Lin (1992).</text>
<text><location><page_3><loc_13><loc_76><loc_48><loc_88></location>The z dependence gives rise to vertical shear at the origin of the axisymmetric vertical shear instability. Notice that at the midplane the vertical shear goes to zero, so the instability drive is present away from it. This is the most interesting region for the present analysis. Away from the midplane we can approximate the last z -quadratic term, a V z 2 , in (4) with a linear one, ¯ A z z . Therefore in the following our reference equilibrium flow will be</text>
<formula><location><page_3><loc_13><loc_73><loc_48><loc_75></location>V e ( x, z ) = [ -q Ω x + ∂ x P e ( x, 0) 2Ω ρ e ( x, 0) + ¯ A z z ]ˆ y , (5)</formula>
<text><location><page_3><loc_13><loc_65><loc_48><loc_72></location>bearing in mind that our conclusions will not hold at the midplane. The second term on the RHS of the above equation will be dealt with as in Johnson & Gammie (2005) by considering the background flow as giving an effective shear rate</text>
<formula><location><page_3><loc_13><loc_62><loc_48><loc_65></location>˜ q ( x )Ω = -dV e ( x ) dx , (6)</formula>
<text><location><page_3><loc_13><loc_60><loc_25><loc_61></location>that varies with x .</text>
<text><location><page_3><loc_13><loc_59><loc_30><loc_60></location>Vertical equilibrium gives</text>
<formula><location><page_3><loc_13><loc_56><loc_48><loc_58></location>∂ z P e ρ e = -Ω 2 z ≡ -g z . (7)</formula>
<text><location><page_3><loc_13><loc_54><loc_46><loc_55></location>By defining 1 /L P z = ∂ z P e /γP e we obtain from (7)</text>
<formula><location><page_3><loc_13><loc_50><loc_48><loc_53></location>g z = -c s 2 L Pz , (8)</formula>
<text><location><page_3><loc_13><loc_48><loc_26><loc_49></location>where c s 2 = γP e /ρ e .</text>
<section_header_level_1><location><page_3><loc_13><loc_44><loc_38><loc_45></location>3 LINEAR PERTURBATIONS</section_header_level_1>
<text><location><page_3><loc_13><loc_40><loc_48><loc_43></location>We decompose the physical variables in equilibrium and perturbation parts</text>
<formula><location><page_3><loc_13><loc_37><loc_48><loc_39></location>V = V e + v ' , D = ρ e + ρ ' , P = P e + P ' , (9)</formula>
<text><location><page_3><loc_13><loc_31><loc_48><loc_37></location>and consider the linearized equations. Localized on the x and z dependent flow (see Johnson & Gammie 2005) and on the x and z dependent density and pressure backgrounds, we consider short wavelength Eulerian perturbations of the type</text>
<formula><location><page_3><loc_13><loc_27><loc_48><loc_30></location>δ ' ( t, x, y, z ) = ˆ δ ' ( t ) e i ∫ ˜ K x ( t,x ) dx + iK y y + i ˜ K z ( t ) z , (10)</formula>
<text><location><page_3><loc_13><loc_26><loc_17><loc_27></location>where</text>
<formula><location><page_3><loc_13><loc_23><loc_48><loc_25></location>˜ K x ( t, x ) = K x +˜ q ( x )Ω K y t, ˜ K z ( t ) = K z -¯ A z K y t. (11)</formula>
<text><location><page_3><loc_13><loc_16><loc_48><loc_23></location>Notice that due to the presence of vertical shear the vertical wavenumber as well evolves with time. The x dependence in ˜ q ( x ) is very weak and in the rest of the paper we will consider the effective shear parameter constant and very close to its Keplerian value.</text>
<text><location><page_3><loc_13><loc_8><loc_48><loc_16></location>As previously stated to be consistent with the neglect of the curvature terms we consider a background with radial and vertical length scales L /lessmuch r and H /lessmuch r . The short wavelength perturbations are such that ˜ K x L /greatermuch 1 and ˜ K z H /greatermuch 1.</text>
<text><location><page_3><loc_13><loc_6><loc_48><loc_9></location>With equation (10) the evolution of linearized perturbations is given by</text>
<formula><location><page_3><loc_52><loc_90><loc_87><loc_93></location>∂ t ˆ ρ ' ρ e + ˆ v ' x L ρ x + ˆ v ' z L ρ z + i ˜ K x ˆ v ' x + iK y ˆ v ' y + i ˜ K z ˆ v ' z = 0 , (12)</formula>
<formula><location><page_3><loc_52><loc_86><loc_87><loc_89></location>∂ t ˆ v ' x = 2Ω ˆ v ' y -i ˜ K x ˆ P ' ρ e + c 2 s L Px ˆ ρ ' ρ e , (13)</formula>
<formula><location><page_3><loc_52><loc_82><loc_87><loc_85></location>∂ t ˆ v ' y = -(2 -˜ q )Ω ˆ v ' x -¯ A z ˆ v ' z -iK y ˆ P ' ρ e , (14)</formula>
<formula><location><page_3><loc_52><loc_79><loc_87><loc_82></location>∂ t ˆ v ' z = -i ˜ K z ˆ P ' ρ e -ˆ ρ ' ρ e g z , (15)</formula>
<formula><location><page_3><loc_52><loc_75><loc_87><loc_78></location>∂ t ˆ P ' ρ e -c 2 s ∂ t ˆ ρ ' ρ e + c 2 s ˆ v ' x L Sx + c 2 s ˆ v ' z L Sz = 0 . (16)</formula>
<text><location><page_3><loc_52><loc_72><loc_87><loc_74></location>The radial and vertical length scales for pressure, density and entropy are defined by</text>
<formula><location><page_3><loc_52><loc_68><loc_87><loc_71></location>1 L P x ≡ ∂ x P e γP e = 1 L ρ x + 1 L Sx ≡ ∂ x ρ e ρ e + ∂ x S e γS e , (17)</formula>
<formula><location><page_3><loc_52><loc_65><loc_87><loc_67></location>1 L P z ≡ ∂ z P e γP e = 1 L ρ z + 1 L Sz ≡ ∂ z ρ e ρ e + ∂ z S e γS e . (18)</formula>
<text><location><page_3><loc_52><loc_61><loc_87><loc_64></location>In the Boussinesq approximation equations (12) and (16) become</text>
<formula><location><page_3><loc_52><loc_59><loc_87><loc_60></location>˜ K x ˆ v ' x + K y ˆ v ' y + ˜ K z ˆ v ' z = 0 , (19)</formula>
<formula><location><page_3><loc_52><loc_55><loc_87><loc_58></location>∂ t ˆ ρ ' ρ e = ˆ v ' x L Sx + ˆ v ' z L Sz , (20)</formula>
<text><location><page_3><loc_52><loc_45><loc_87><loc_54></location>Equations (13)-(15), (19) and (20) are equations (59)(63) of Knobloch & Spruit (1986) expressed in terms of the short wavelength shearing modes (10). The only difference is given by the presence of the radial stratification term in (13). We will see in the following, however, that it has essentially no influence on the perturbations evolution.</text>
<text><location><page_3><loc_52><loc_38><loc_87><loc_45></location>Now by deriving with respect to time the incompressibility condition (19) and then, in the equation obtained, expressing ∂ t ˆ v ' x , ∂ t ˆ v ' y and ∂ t ˆ v ' z with equations (13), (14) and (15), we can express ˆ P ' in terms of ˆ ρ ' , ˆ v ' x , ˆ v ' y and ˆ v ' z</text>
<formula><location><page_3><loc_52><loc_31><loc_87><loc_37></location>i ˆ P ' ρ e = 1 ˜ K 2 [( ˜ K x c 2 s L P x -g z ˜ K z ) ˆ ρ ' ρ e + 2(˜ q -1)Ω k y ˆ v ' x +2Ω ˜ K x ˆ v ' y -2 ¯ A z K y ˆ v ' z ] , (21)</formula>
<text><location><page_3><loc_52><loc_29><loc_87><loc_32></location>where ˜ K 2 = ˜ K 2 x + K 2 y + ˜ K 2 z . By means of equation (21) we obtain the system</text>
<formula><location><page_3><loc_52><loc_22><loc_87><loc_28></location>∂ t ˆ v ' x = -2(˜ q -1)Ω K y ˜ K x ˜ K 2 ˆ v ' x +2Ω(1 -˜ K 2 x ˜ K 2 ) ˆ v ' y + 2 ¯ A z ˜ K x K y ˜ K 2 ˆ v ' z + c 2 s L Px (1 -˜ K 2 x ˜ K 2 ) ˆ ρ ' ρ e + g z ˜ K z ˜ K x ˜ K 2 ˆ ρ ' ρ e , (22)</formula>
<text><location><page_3><loc_69><loc_19><loc_70><loc_20></location>K</text>
<text><location><page_3><loc_69><loc_18><loc_70><loc_19></location>˜</text>
<text><location><page_3><loc_52><loc_18><loc_53><loc_19></location>∂</text>
<text><location><page_3><loc_53><loc_18><loc_53><loc_19></location>t</text>
<text><location><page_3><loc_53><loc_18><loc_54><loc_19></location>v</text>
<text><location><page_3><loc_54><loc_18><loc_55><loc_19></location>y</text>
<text><location><page_3><loc_55><loc_18><loc_59><loc_19></location>= Ω[˜ q</text>
<text><location><page_3><loc_59><loc_17><loc_61><loc_19></location>-</text>
<text><location><page_3><loc_61><loc_18><loc_62><loc_19></location>2</text>
<text><location><page_3><loc_62><loc_17><loc_63><loc_19></location>-</text>
<text><location><page_3><loc_63><loc_18><loc_66><loc_19></location>2(˜</text>
<text><location><page_3><loc_65><loc_18><loc_65><loc_19></location>q</text>
<text><location><page_3><loc_66><loc_17><loc_67><loc_19></location>-</text>
<text><location><page_3><loc_67><loc_18><loc_69><loc_19></location>1)</text>
<text><location><page_3><loc_70><loc_20><loc_71><loc_20></location>2</text>
<text><location><page_3><loc_70><loc_19><loc_71><loc_20></location>y</text>
<text><location><page_3><loc_69><loc_17><loc_70><loc_18></location>K</text>
<text><location><page_3><loc_70><loc_18><loc_71><loc_19></location>2</text>
<text><location><page_3><loc_71><loc_18><loc_72><loc_19></location>]</text>
<text><location><page_3><loc_72><loc_18><loc_72><loc_19></location>v</text>
<text><location><page_3><loc_72><loc_19><loc_73><loc_19></location>'</text>
<text><location><page_3><loc_72><loc_18><loc_73><loc_19></location>x</text>
<text><location><page_3><loc_73><loc_17><loc_75><loc_19></location>-</text>
<text><location><page_3><loc_75><loc_18><loc_77><loc_19></location>2Ω</text>
<text><location><page_3><loc_78><loc_19><loc_79><loc_20></location>y</text>
<text><location><page_3><loc_79><loc_19><loc_80><loc_20></location>K</text>
<text><location><page_3><loc_78><loc_18><loc_79><loc_19></location>˜</text>
<text><location><page_3><loc_77><loc_19><loc_78><loc_20></location>K</text>
<text><location><page_3><loc_78><loc_17><loc_79><loc_18></location>K</text>
<text><location><page_3><loc_79><loc_19><loc_80><loc_20></location>˜</text>
<text><location><page_3><loc_79><loc_18><loc_80><loc_19></location>2</text>
<text><location><page_3><loc_81><loc_18><loc_82><loc_19></location>v</text>
<text><location><page_3><loc_82><loc_19><loc_82><loc_19></location>'</text>
<text><location><page_3><loc_82><loc_18><loc_83><loc_19></location>y</text>
<text><location><page_3><loc_83><loc_18><loc_84><loc_19></location>+</text>
<formula><location><page_3><loc_55><loc_14><loc_88><loc_17></location>2 ¯ A z K 2 y ˜ K 2 ˆ v ' z -¯ A z ˆ v ' z -K y ˜ K 2 ( ˜ K x c 2 s L Px -g z ˜ K z ) ˆ ρ ' ρ e , (23)</formula>
<formula><location><page_3><loc_52><loc_7><loc_87><loc_13></location>∂ t ˆ v ' z = -2(˜ q -1)Ω K y ˜ K z ˜ K 2 ˆ v ' x -2Ω ˜ K x ˜ K z ˜ K 2 ˆ v ' y + 2 ¯ A z K y ˜ K z ˜ K 2 ˆ v ' z -g z (1 -˜ K 2 z ˜ K 2 ) ˆ ρ ' ρ e -c 2 s L P x ˜ K z ˜ K x ˜ K 2 ˆ ρ ' ρ e , (24)</formula>
<text><location><page_3><loc_54><loc_19><loc_54><loc_20></location>ˆ '</text>
<text><location><page_3><loc_72><loc_19><loc_73><loc_20></location>ˆ</text>
<text><location><page_3><loc_80><loc_19><loc_81><loc_20></location>x</text>
<text><location><page_3><loc_82><loc_19><loc_82><loc_20></location>ˆ</text>
<formula><location><page_4><loc_13><loc_90><loc_48><loc_93></location>∂ t ˆ ρ ' ρ e = ˆ v ' x L Sx + ˆ v ' z L Sz . (25)</formula>
<text><location><page_4><loc_13><loc_85><loc_48><loc_89></location>Normalizing time with Ω -1 , velocities with L Sz Ω and density with ρ e we obtain for the evolution of the nondimensional variables v x , v y , v z , ρ the system</text>
<formula><location><page_4><loc_13><loc_78><loc_48><loc_84></location>∂ t v x = -2(˜ q -1) k y ˜ k x ˜ k 2 v x +2(1 -˜ k 2 x ˜ k 2 ) v y + 2 A z ˜ k x k y ˜ k 2 v z + L Sx L Sz Ri x ( ˜ k 2 x ˜ k 2 -1) ρ + Ri z ˜ k z ˜ k x ˜ k 2 ρ, (26)</formula>
<formula><location><page_4><loc_13><loc_70><loc_48><loc_76></location>∂ t v y = [˜ q -2 -2(˜ q -1) k 2 y ˜ k 2 ] v x -2 k y ˜ k x ˜ k 2 v y + 2 A z k 2 y ˜ k 2 v z -A z v z + k y ˜ k 2 ( ˜ k x L Sx L Sz Ri x + Ri z ˜ k z ) ρ, (27)</formula>
<formula><location><page_4><loc_18><loc_66><loc_43><loc_69></location>∂ t v z = -2(˜ q -1) k y ˜ k z ˜ k 2 v x -2 ˜ k x ˜ k z ˜ k 2 v y +</formula>
<formula><location><page_4><loc_13><loc_63><loc_48><loc_66></location>2 A z k y ˜ k z ˜ k 2 v z + Ri z ( ˜ k 2 z ˜ k 2 -1) ρ + L Sx L Sz Ri x ˜ k z ˜ k x ˜ k 2 ρ, (28)</formula>
<formula><location><page_4><loc_13><loc_59><loc_48><loc_61></location>∂ t ρ = L Sz L Sx v x + v z , (29)</formula>
<text><location><page_4><loc_13><loc_53><loc_48><loc_58></location>where ( k x , k y , k z ) ≡ L Sz ( K x , K y , K z ), ˜ k x ≡ L Sz ˜ K x , ˜ k z ≡ L Sz ˜ K z and ˜ k 2 ≡ L 2 S z ˜ K 2 . We introduced, as well, A z ≡ ¯ A z Ω and the Richardson numbers</text>
<formula><location><page_4><loc_13><loc_50><loc_48><loc_53></location>Ri x ≡ N 2 x Ω 2 , Ri z ≡ N 2 z Ω 2 . (30)</formula>
<text><location><page_4><loc_13><loc_48><loc_43><loc_49></location>N x and N z are the Brunt-Vaisala frequencies</text>
<formula><location><page_4><loc_13><loc_44><loc_48><loc_47></location>N 2 x ≡ -c 2 s L Sx L Px , N 2 z ≡ g z L Sz = -c 2 s L Sz L Pz . (31)</formula>
<text><location><page_4><loc_13><loc_36><loc_48><loc_44></location>We notice that the above equations are scale invariant in the sense that results pertaining to wavenumbers k x , k y and k z hold as well for wavenumbers βk x , βk y and βk z , where β is a real number. This symmetry is broken if we introduce the effect of thermal diffusivity in the system above.</text>
<section_header_level_1><location><page_4><loc_13><loc_30><loc_29><loc_31></location>4 AXISYMMETRY</section_header_level_1>
<text><location><page_4><loc_13><loc_26><loc_48><loc_29></location>For axisymmetric perturbations (i.e. k y = 0) equations (26)-(29) become</text>
<formula><location><page_4><loc_13><loc_23><loc_48><loc_26></location>∂ t v x = 2 k 2 z k 2 v y -L Sx L Sz Ri x k 2 z k 2 ρ + Ri z k z k x k 2 ρ, (32)</formula>
<formula><location><page_4><loc_13><loc_20><loc_48><loc_22></location>∂ t v y = (˜ q -2) v x -A z v z , (33)</formula>
<formula><location><page_4><loc_13><loc_17><loc_48><loc_20></location>∂ t v z = -2 k x k z k 2 v y -Ri z k 2 x k 2 ρ + L Sx L Sz Ri x k z k x k 2 ρ, (34)</formula>
<formula><location><page_4><loc_13><loc_14><loc_48><loc_16></location>∂ t ρ = L Sz L Sx v x + v z . (35)</formula>
<text><location><page_4><loc_13><loc_6><loc_48><loc_13></location>In the above equations there are no time dependent coefficients and therefore we can assume an exponential form of the type e st for the velocity and density perturbation fields. Neglecting radial stratification the dispersion relation reads</text>
<formula><location><page_4><loc_52><loc_90><loc_87><loc_93></location>s 2 + 2(2 -˜ q ) k 2 z k 2 + Ri z k 2 x k 2 -2 k x k z A z k 2 = 0 . (36)</formula>
<text><location><page_4><loc_52><loc_80><loc_87><loc_89></location>This is the shearing sheet equivalent of equation (9) in Urpin (2003) in the case of zero diffusivity. For finite Ri z the vertical shear term is dominated either by the epycyclic frequency term or by the Ri z term. In the case of weak vertical stratification (i.e. Ri z /lessmuch 1) and for ˜ q = 3 / 2 the maximum growth rate of the instability occurs for</text>
<formula><location><page_4><loc_52><loc_75><loc_87><loc_79></location>k x k z = 1 A z ± √ 1 A 2 z +4 2 , (37)</formula>
<text><location><page_4><loc_52><loc_71><loc_87><loc_75></location>which is the same as equation (34) of Urpin (2003) since 1 A 2 z /greatermuch 4 and ¯ A z = r∂ z Ω. The growth rate is given by</text>
<formula><location><page_4><loc_52><loc_65><loc_87><loc_70></location>s max = √ √ √ √ 2 √ 1 + 4 A 2 z 4 + 1 A 2 z + 1 A z √ 1 A 2 z +4 . (38)</formula>
<text><location><page_4><loc_52><loc_61><loc_87><loc_65></location>For A 2 z /lessmuch 1, a condition to be expected in astrophysical discs, we have therefore s max ≈ | A z | .</text>
<text><location><page_4><loc_52><loc_58><loc_87><loc_62></location>Next we consider the opposite limit by neglecting the vertical stratification. In this case the dispersion relation becomes</text>
<formula><location><page_4><loc_52><loc_54><loc_87><loc_57></location>s 2 + k 2 z k 2 [2(2 -˜ q ) + Ri x ] -2 k x k z k 2 A z = 0 . (39)</formula>
<text><location><page_4><loc_52><loc_51><loc_87><loc_54></location>Being radial stratification typically weak, stability is not influenced.</text>
<text><location><page_4><loc_52><loc_49><loc_87><loc_51></location>When both radial and vertical gradients are present the dispersion relation reads</text>
<formula><location><page_4><loc_52><loc_42><loc_87><loc_48></location>s 2 + k 2 z k 2 [2(2 -˜ q ) + Ri x ] -k x k z k 2 (2 A z + L Sx L Sz Ri x + L Sz L Sx Ri z ) + Ri z k 2 x k 2 = 0 . (40)</formula>
<text><location><page_4><loc_52><loc_31><loc_87><loc_42></location>In this case additional destabilization is possible in principle due to the term Σ = L Sx L Sz Ri x + L Sz L Sx Ri z . However its effect will be dominated either by the stabilizing effect of the epicyclic frequency (for k z /greaterorequalslant k x ) or by the last term on the LHS of (40) (for k x /greatermuch k z ). We notice that for a barotropic equilibrium ( A z = 0) we have Σ = ± 2 √ Ri x Ri z and Ri x , Ri z are bound to have the same sign (Volponi 2010) and</text>
<formula><location><page_4><loc_52><loc_27><loc_87><loc_31></location>s 2 = -k 2 z 2(2 -˜ q ) + ( k z √ Ri x ± k x √ Ri z ) 2 k 2 (41)</formula>
<text><location><page_4><loc_52><loc_24><loc_87><loc_27></location>In the presence of diffusion the only modification of the evolution equations occurs in (20), which becomes</text>
<formula><location><page_4><loc_52><loc_20><loc_87><loc_23></location>∂ t ˆ ρ ' ρ e = ˆ v ' x L Sx + ˆ v ' z L Sz -χ ˜ K 2 ˆ ρ ' ρ e , (42)</formula>
<text><location><page_4><loc_52><loc_19><loc_80><loc_20></location>where χ is the thermal diffusion coefficient.</text>
<text><location><page_4><loc_52><loc_17><loc_80><loc_18></location>Once normalized the above equation reads</text>
<formula><location><page_4><loc_52><loc_14><loc_87><loc_16></location>∂ t ρ = L Sz L Sx v x + v z -˜ k 2 1 Pe ρ, (43)</formula>
<text><location><page_4><loc_52><loc_11><loc_78><loc_13></location>where Pe = L 2 Sz Ω χ is the Peclet number.</text>
<text><location><page_4><loc_52><loc_10><loc_86><loc_11></location>The axisymmetric dispersion relation then becomes</text>
<formula><location><page_4><loc_55><loc_6><loc_82><loc_9></location>s 3 + k 2 Pe -1 s 2 + s [ k 2 z k 2 [2(2 -˜ q ) + Ri x ] -</formula>
<formula><location><page_5><loc_13><loc_87><loc_48><loc_93></location>k x k z k 2 (2 A z + L Sx L Sz Ri x + L Sz L Sx Ri z ) + Ri z k 2 x k 2 ] + Pe -1 [2 k 2 z (2 -˜ q ) -2 k x k z A z ] = 0 . (44)</formula>
<text><location><page_5><loc_13><loc_80><loc_48><loc_87></location>The above equation is completely equivalent to the dispersion relation (7) in Urpin (2003) when neglecting the effect of viscosity and the same instability conditions derived in his paper follow. We briefly sum up his results. By casting equation (44) in the form</text>
<formula><location><page_5><loc_13><loc_78><loc_48><loc_79></location>s 3 + a 2 s 2 + a 1 s + a 0 = 0 , (45)</formula>
<text><location><page_5><loc_13><loc_76><loc_17><loc_77></location>where</text>
<formula><location><page_5><loc_13><loc_64><loc_48><loc_75></location>a 2 = k 2 Pe -1 , a 1 = [ k 2 z k 2 [2(2 -˜ q ) + Ri x ] -k x k z k 2 (2 A z + L Sx L Sz Ri x + L Sz L Sx Ri z ) + Ri z k 2 x k 2 ] , a 0 = Pe -1 [2 k 2 z (2 -˜ q ) -2 k x k z A z ] , (46)</formula>
<text><location><page_5><loc_13><loc_61><loc_48><loc_64></location>instability conditions read (see Urpin 2003 and references therein)</text>
<formula><location><page_5><loc_13><loc_59><loc_48><loc_60></location>a 0 < 0 , a 1 a 2 < a 0 , a 2 < 0 . (47)</formula>
<text><location><page_5><loc_13><loc_55><loc_48><loc_58></location>The first of the inequalities above (equation (17) in Urpin 2003),</text>
<formula><location><page_5><loc_13><loc_52><loc_48><loc_54></location>Pe -1 [2 k 2 z (2 -˜ q ) -2 k x k z A z ] < 0 , (48)</formula>
<text><location><page_5><loc_13><loc_47><loc_48><loc_52></location>shows that the presence of thermal diffusion relaxes the instability condition with respect to the ideal case. The second inequality (equation (14) in Urpin 2003) reads</text>
<formula><location><page_5><loc_13><loc_43><loc_48><loc_46></location>[ k 2 z k 2 Ri x -k x k z k 2 ( L Sx L Sz Ri x + L Sz L Sx Ri z )+ Ri z k 2 x k 2 ] < 0(49)</formula>
<text><location><page_5><loc_13><loc_40><loc_48><loc_42></location>and it is a statement about the convective stability of the disk (vertical shear does not appear in it).</text>
<text><location><page_5><loc_13><loc_24><loc_48><loc_39></location>To end this section we present the evolution of axisymmetric perturbations for a convectively unstable vertical stratification. In Fig. 1 we set k x /k z = 10, which pertains to the maximal vertical shear growth rate for A z = 0 . 1, Ri z = -0 . 2 and Pe = 10 7 . We notice that the growth rate is the convective one ( √ -Ri z ), but differently from conventional understanding W xy ≡ ( v x v y ) /v 2 > 0, where v 2 = v 2 x + v 2 y + v 2 z . To ascertain the origin of the positive sign of W xy we set A z = 0. As can be seen in Fig. 2, this results in W xy < 0.</text>
<text><location><page_5><loc_13><loc_16><loc_48><loc_24></location>We can therefore conclude that, when the vertical convective growth rate is larger than the vertical shear one, the ensuing instability is of mixed type in the sense that the growth rate is given by convection whereas the sign of W xy is determined by vertical shear.</text>
<text><location><page_5><loc_13><loc_8><loc_48><loc_16></location>Linear inviscid axisymmetric perturbations cannot transport angular momentum in hydrodynamic discs (Ruden, Papaloizou & Lin 1988), however we will see in the next section that for non-axisymmetric perturbations, which induce transport, the same type of behaviour described above holds.</text>
<text><location><page_5><loc_13><loc_6><loc_48><loc_8></location>We determined as well the effect of thermal diffusion</text>
<figure>
<location><page_5><loc_52><loc_71><loc_88><loc_93></location>
<caption>Figure 1. Evolution of velocities and normalized xy -Reynolds stress (i.e. W xy ≡ ( v x v y ) /v 2 , where v 2 = v 2 x + v 2 y + v 2 z ) for Pe = 10 7 , Ri z = -0 . 2, Ri x = 0 . 01, L Sz /L Sx = 0 . 223, A z = 0 . 1 and k x = 500, k z = 50, k y = 0.</caption>
</figure>
<figure>
<location><page_5><loc_52><loc_36><loc_88><loc_58></location>
<caption>Figure 2. Same as previous figure but with A z = 0.</caption>
</figure>
<text><location><page_5><loc_52><loc_22><loc_87><loc_28></location>on the growth rate s . In Table 1 we report the dependence of s on Pe . We notice that s is drastically reduced down to s ∼ A z . Further decreasing of Pe does not have effect on s .</text>
<table>
<location><page_5><loc_52><loc_8><loc_87><loc_12></location>
<caption>Table 1. Dependence of mixed (convective + vertical shear) axisymmetric growth rate on Peclet number (relative to the case in Fig. 1)</caption>
</table>
<figure>
<location><page_6><loc_13><loc_71><loc_50><loc_93></location>
<caption>Figure 3. Evolution of velocities and normalized xy -Reynolds stress in the case of very weak stratification for A z = 0 . 1 and k x = 1000, k z = 100, k y = 1.</caption>
</figure>
<section_header_level_1><location><page_6><loc_13><loc_61><loc_34><loc_62></location>5 NON-AXISYMMETRY</section_header_level_1>
<section_header_level_1><location><page_6><loc_13><loc_59><loc_41><loc_60></location>5.1 Case 1: only vertical shear drive</section_header_level_1>
<text><location><page_6><loc_13><loc_54><loc_48><loc_57></location>We consider here the evolution of ideal nonaxisymmetric perturbations in the presence of a convectively stable vertical stratification.</text>
<text><location><page_6><loc_13><loc_45><loc_48><loc_53></location>With a reference stratification of Ri z = 0 . 1, Ri x = 0 . 01 and a realistic A z , perturbations always decay for any value of the wavenumbers. In this case the vertical shear destabilizing drive is never able to overcome the stabilizing influence of stratification and epicyclic frequency.</text>
<text><location><page_6><loc_13><loc_26><loc_48><loc_45></location>If we allow for weaker stratification the vertical shear can drive huge transient growths in the perturbations. Fixing A z to a positive value, the interval of growth is connected to ˜ k x and ˜ k z having the same sign. We can see this in Fig. 3 for the case ˜ k x > 0 and ˜ k z > 0, where we set the k x /k z ratio to the value pertaining to the maximal axisymmetric growth rate A z . By increasing k y clearly the interval of growth is shortened and therefore the largest amplification reached diminishes. We stress that the transients are indistinguishable from a full-fledged instability. The growths are in fact exponential with growth rate close to A z . Varying the ratio k x /k z , as well, causes a decrease in the maximum amplification reached.</text>
<text><location><page_6><loc_13><loc_23><loc_48><loc_26></location>In the case A z < 0 growths occurs when ˜ k x and ˜ k z have opposite signs.</text>
<text><location><page_6><loc_13><loc_20><loc_48><loc_23></location>We notice as well that during the phase of growth, radial transport is always positive.</text>
<text><location><page_6><loc_13><loc_13><loc_48><loc_20></location>As stated before eventually all perturbations decay. Heuristically this can be shown neglecting the stratification in equations (26)-(29). By using the incompressibility condition (19) we obtain for the x and y velocity components the evolution equations</text>
<formula><location><page_6><loc_13><loc_6><loc_48><loc_13></location>∂ t v x = -2(˜ q -1 + A z ˜ k x ˜ k z ) k y ˜ k x ˜ k 2 v x + 2(1 -˜ k 2 x ˜ k 2 -A z k 2 y ˜ k x ˜ k 2 ˜ k z ) v y , (50)</formula>
<text><location><page_6><loc_53><loc_90><loc_54><loc_92></location>∂</text>
<text><location><page_6><loc_54><loc_90><loc_54><loc_91></location>t</text>
<text><location><page_6><loc_54><loc_90><loc_55><loc_92></location>v</text>
<text><location><page_6><loc_55><loc_90><loc_56><loc_91></location>y</text>
<text><location><page_6><loc_56><loc_90><loc_59><loc_92></location>= [˜</text>
<text><location><page_6><loc_58><loc_90><loc_59><loc_92></location>q</text>
<text><location><page_6><loc_59><loc_90><loc_60><loc_92></location>-</text>
<text><location><page_6><loc_62><loc_90><loc_63><loc_92></location>-</text>
<text><location><page_6><loc_66><loc_90><loc_67><loc_92></location>-</text>
<text><location><page_6><loc_69><loc_91><loc_69><loc_92></location>k</text>
<text><location><page_6><loc_70><loc_92><loc_70><loc_93></location>2</text>
<text><location><page_6><loc_69><loc_91><loc_70><loc_92></location>y</text>
<text><location><page_6><loc_69><loc_90><loc_69><loc_91></location>˜</text>
<text><location><page_6><loc_69><loc_90><loc_69><loc_91></location>k</text>
<text><location><page_6><loc_70><loc_90><loc_70><loc_91></location>2</text>
<text><location><page_6><loc_71><loc_90><loc_72><loc_92></location>-</text>
<text><location><page_6><loc_75><loc_91><loc_76><loc_92></location>k</text>
<text><location><page_6><loc_76><loc_92><loc_76><loc_93></location>2</text>
<text><location><page_6><loc_76><loc_91><loc_76><loc_92></location>y</text>
<text><location><page_6><loc_75><loc_90><loc_76><loc_91></location>˜</text>
<text><location><page_6><loc_75><loc_90><loc_76><loc_91></location>k</text>
<text><location><page_6><loc_76><loc_90><loc_76><loc_91></location>2</text>
<text><location><page_6><loc_77><loc_92><loc_77><loc_93></location>˜</text>
<text><location><page_6><loc_77><loc_91><loc_77><loc_92></location>k</text>
<text><location><page_6><loc_77><loc_91><loc_78><loc_92></location>x</text>
<text><location><page_6><loc_77><loc_90><loc_77><loc_91></location>˜</text>
<text><location><page_6><loc_77><loc_90><loc_77><loc_91></location>k</text>
<text><location><page_6><loc_77><loc_90><loc_78><loc_90></location>z</text>
<formula><location><page_6><loc_52><loc_86><loc_87><loc_89></location>A z ˜ k x ˜ k z ] v x +( A z k y ˜ k z -2 k y ˜ k x ˜ k 2 -2 A z k 3 y ˜ k 2 ˜ k z ) v y , (51)</formula>
<text><location><page_6><loc_52><loc_84><loc_75><loc_86></location>which in the limit t -→ ∞ become</text>
<formula><location><page_6><loc_52><loc_81><loc_87><loc_84></location>∂ t v x = 2 A 2 z q 2 + A 2 z v y , (52)</formula>
<text><location><page_6><loc_52><loc_78><loc_61><loc_80></location>∂ t v y = -2 v x .</text>
<text><location><page_6><loc_84><loc_79><loc_87><loc_80></location>(53)</text>
<text><location><page_6><loc_52><loc_77><loc_77><loc_78></location>These can be reduced to the equation</text>
<formula><location><page_6><loc_52><loc_74><loc_87><loc_76></location>∂ 2 t v x = -4 A 2 z q 2 + A 2 z v x , (54)</formula>
<text><location><page_6><loc_52><loc_72><loc_67><loc_73></location>which implies stability.</text>
<text><location><page_6><loc_52><loc_66><loc_87><loc_71></location>For sake of completeness, we considered as well the case of stable weak vertical and unstable strong radial stratifications. The vertical shear growth rate is only slightly enhanced.</text>
<section_header_level_1><location><page_6><loc_52><loc_61><loc_87><loc_64></location>5.2 Case 2: combined effect of vertical shear and vertical unstable convection drives</section_header_level_1>
<text><location><page_6><loc_52><loc_41><loc_87><loc_60></location>We examine here the evolution of ideal perturbations in the presence of an unstable vertical stratification. This is the central part of this study, because the interaction of vertical shear and convective drives results in evolutions which are determined by the relative strength of the vertical shear growth rate ( s vs ∼ | A z | ) and the vertical convective growth rate ( s c ∼ √ -Ri z ). For s vs > s c we observed the same type of behaviour described in Case 1 - large exponential amplifications with growth rate s vs occur with asymptotic stabilization. For s c > s vs the system is asymptotically unstable with growth rate s c . For s vs ∼ s c a variety of mixed evolutions is possible: these include transient growths, exponential instability and algebraic instability.</text>
<text><location><page_6><loc_52><loc_29><loc_87><loc_41></location>In the following we will concentrate on the case s c > s vs , for which the system is asymptotically unstable. Similarly to what we found in the axisymmetric case, the main difference with a purely convective instability is given by the sign of the angular momentum transport, which is positive exactly in the same time intervals in which transient growth occured in Case 1. In other words for A z > 0( < 0) positive transport occurs when ˜ k x ˜ k z > 0( < 0).</text>
<text><location><page_6><loc_52><loc_13><loc_87><loc_28></location>This is shown in Figs. 4 and 5 where we fixed k x /k z ∼ 10 pertaining to maximal axisymmetric vertical shear growth rate for A z = 0 . 1 ( s vs ∼ 0 . 1), and chose Ri z = -0 . 2. As can be seen, the growth rate is given by s c ∼ √ -Ri z and transport is positive in the regimes described above. This is important because, in the absence of vertical shear, convection has the tendency to transport angular momentum inward rather than outward (Ryu & Goodman 1992). To see this we switched off the vertical drive (i.e. A z = 0) and found that this results in negative transport (Fig. 6).</text>
<text><location><page_6><loc_52><loc_11><loc_87><loc_13></location>We stress that this mechanism occurs for any value of the vertical shear growth rate, however small.</text>
<text><location><page_6><loc_52><loc_6><loc_87><loc_10></location>We noticed as well that when s c > s vs , but their values are close, the growth rate of the mixed instability, s mix , is enhanced of a fraction of s c . For example,</text>
<text><location><page_6><loc_61><loc_90><loc_61><loc_92></location>2</text>
<text><location><page_6><loc_63><loc_90><loc_65><loc_92></location>2(˜</text>
<text><location><page_6><loc_65><loc_90><loc_65><loc_92></location>q</text>
<text><location><page_6><loc_67><loc_90><loc_68><loc_92></location>1)</text>
<text><location><page_6><loc_72><loc_90><loc_73><loc_92></location>2</text>
<text><location><page_6><loc_73><loc_90><loc_74><loc_92></location>A</text>
<text><location><page_6><loc_74><loc_90><loc_75><loc_91></location>z</text>
<text><location><page_6><loc_79><loc_90><loc_80><loc_92></location>+</text>
<figure>
<location><page_7><loc_13><loc_70><loc_88><loc_93></location>
<caption>Figure 4. Evolution of velocities and normalized xy -Reynolds stress (up to t = 400) for Ri z = -0 . 2, Ri x = 0 . 01, L Sz /L Sx = 0 . 223, A z = 0 . 1 and k x = 500, k z = 50, k y = 1.</caption>
</figure>
<figure>
<location><page_7><loc_13><loc_39><loc_89><loc_61></location>
<caption>Figure 6. Evolution of velocities and normalized xy -Reynolds stress (up to t = 400) for Ri z = -0 . 2, Ri x = 0 . 01, L Sz /L Sx = 0 . 223, A z = 0 and k x = 500, k z = 50, k y = 1.Figure 5. Same as previous picture up to t = 800.</caption>
</figure>
<text><location><page_7><loc_13><loc_28><loc_48><loc_33></location>in the case k x /k z ∼ 3, A z = 0 . 4 (i.e. s vs ∼ 0 . 4), Ri z = -0 . 2 (i.e. s c ∼ 0 . 45) and k y = 0 . 1 we obtained s mix ∼ 0 . 55.</text>
<section_header_level_1><location><page_7><loc_13><loc_25><loc_45><loc_26></location>5.3 Case 3: presence of thermal diffusion</section_header_level_1>
<text><location><page_7><loc_13><loc_22><loc_48><loc_24></location>Two are the main differences with respect to the ideal case.</text>
<text><location><page_7><loc_13><loc_6><loc_48><loc_21></location>The first regards the vertical shear instability and consists in the fact, predicted by the axisymmetric theory (Urpin 2003), that the condition for instability is relaxed. Indeed for non-axisymmetric perturbations as well, we noticed that the presence of a stable vertical stratification does not have influence on growth, which occurs for any value of Ri z . In this case also, instability transforms into a transient amplification with growth rate close to the axisymmetric one, as shown in Fig. 7. Again growth is connected to the signs of ˜ k x and ˜ k z exactly in the same way discussed in Case 1.</text>
<paragraph><location><page_7><loc_52><loc_32><loc_87><loc_37></location>Figure 7. Evolution of velocities and normalized xy -Reynolds stress for Pe = 10 5 , Ri z = 0 . 1, Ri x = 0 . 01, L Sz /L Sx = 0 . 316, A z = 0 . 1 and k x = 1000, k z = 100, k y = 1.</paragraph>
<text><location><page_7><loc_52><loc_6><loc_87><loc_30></location>The second difference consists in the fact that, as for the axisymmetric case, thermal diffusion has a stabilizing effect on the convective instability. This can be seen in Fig. 8, where we considered Ri z = -0 . 2, A z = 0 . 1, k x = 500, k y = 1, k z = 50, with Pe = 10 5 . Convective growth is suppressed and the growth rate in the amplification phase is the one pertaining to the vertical shear destabilization. We can however increase the growth rate toward the one pertaining to a convective instability in two ways. The first is by increasing the Peclet number as done in Fig. 9, where parameters have the same values as in Fig. 8 but with Pe = 10 6 . The second is by increasing the wavelength of the perturbations as done in Fig. 10 which is the same as Fig. 8 except that now k x = 100, k z = 10, k y = 0 . 2. It can be seen that the growth rate is strongly enhanced. We caution however that here we are reaching a bor-</text>
<figure>
<location><page_8><loc_13><loc_71><loc_89><loc_93></location>
<caption>Figure 8. Evolution of velocities and normalized xy -Reynolds stress for Pe = 10 5 , Ri z = -0 . 2, Ri x = 0 . 01, L Sz /L Sx = 0 . 223, A z = 0 . 1 and k x = 500, k z = 50, k y = 1.</caption>
</figure>
<figure>
<location><page_8><loc_13><loc_36><loc_50><loc_58></location>
<caption>Figure 9. Evolution of velocities and normalized xy -Reynolds stress for Pe = 10 6 , Ri z = -0 . 2, Ri x = 0 . 01, L Sz /L Sx = 0 . 223, A z = 0 . 1 and k x = 500, k z = 50, k y = 1.</caption>
</figure>
<text><location><page_8><loc_13><loc_22><loc_44><loc_23></location>derline regime for our short wavelength theory.</text>
<text><location><page_8><loc_13><loc_6><loc_48><loc_21></location>We can sum up the effects of thermal diffusivity as follows. Diffusivity acts as a stabilization mechanism for convection and not for the vertical shear instability. By decreasing the Peclet number the growth rate decreases until the vertical shear instability growth rate is reached. There the diffusion induced stabilization stops. On the other side if we keep Pe fixed and decrease wavenumbers, the growth rate increases toward the convective instability value ( ∼ √ -Ri z ). Therefore we can conclude that the stabilizing effect of thermal diffusion affects convection for short wavelengths.</text>
<paragraph><location><page_8><loc_52><loc_64><loc_87><loc_68></location>Figure 10. Evolution of velocity and normalized xy -Reynolds stress for Pe = 10 5 , Ri z = -0 . 2, Ri x = 0 . 01, L Sz /L Sx = 0 . 223, A z = 0 . 1 and k x = 100, k z = 10, k y = 0 . 2.</paragraph>
<section_header_level_1><location><page_8><loc_52><loc_59><loc_64><loc_60></location>6 SUMMARY</section_header_level_1>
<text><location><page_8><loc_52><loc_48><loc_87><loc_58></location>We presented the evolution of non-axisymmetric perturbations in discs whose rotation profile is x and z dependent. In the shearing sheet limit and moving to a reference frame corotating with the background flow the presence of vertical shear induces the evolution of the vertical wavenumber ˜ k z . We considered short wavelength perturbations.</text>
<text><location><page_8><loc_52><loc_18><loc_87><loc_48></location>Perturbations are always stabilized asymptotically; however, huge exponential growths can occur which are virtually indistinguishable from a fullfledged instability. The condition for the occurrence of such amplifications is ˜ k x ˜ k z > 0 for A z > 0 and ˜ k x ˜ k z < 0 for A z < 0. We summarized this in Table 2. We examine more in detail this condition. Let's consider A z > 0, k x > 0 and k z > 0. In this case ˜ k x is always positive, ˜ k z instead evolves from positive to negative values. Therefore we observe the transient exponential phase of growth when k z -A z k y t > 0 (i.e. for t < k z A z k y ≡ t g ). We stress that this is different from the usual transient growth paradigm (Chagelishvili et al. 2003) since here the growth is exponential and the physical origin is the vertical shear drive. Similar type of exponential transient growths were described by Balbus & Hawley (1992) in the context of the nonaxisymmetric magnetorotational instability, Korycansky (1992) in hydrodynamic convection and Volponi, Yoshida & Tatsuno (2000) in the stabilization of plasmas kink modes. In all these studies, though, the flow was just radially sheared.</text>
<text><location><page_8><loc_52><loc_11><loc_87><loc_17></location>For t ∼ t g our short wavelength approximation breaks down since ˜ k z -→ 0; however, huge amplification values are reached at times t well before t g , when ˜ K z /greatermuch H .</text>
<text><location><page_8><loc_52><loc_6><loc_87><loc_12></location>The growth rate is close to the axisymmetric growth rate. The transport is positive in the phase of growth. As for the axisymmetric theory the most easily destabilizable modes occur in the non-adiabatic case of fi-</text>
<table>
<location><page_9><loc_13><loc_81><loc_48><loc_90></location>
<caption>Table 2. Vertical Shear (Ideal, Ri z /lessmuch 1, or Non ideal)Table 3. Vertical Shear and Convection (Ideal, s c > s vs )</caption>
</table>
<table>
<location><page_9><loc_13><loc_66><loc_48><loc_75></location>
</table>
<text><location><page_9><loc_13><loc_58><loc_48><loc_63></location>nite thermal diffusivity. In the ideal case the presence of a stable vertical stratification prevents the occurrence of growth. Growth is possible only for small values of the vertical Richardson number.</text>
<text><location><page_9><loc_13><loc_48><loc_48><loc_57></location>This first set of results extends the axisymmetric theory to non-axisymmetric perturbations, providing as well a linear local mechanism to explain the non-axisymmetric non-linear results in Nelson et al. (2012). The linear non-axisymmetric mechanism is essentially the axisymmetric one which is still active for short to intermediate times.</text>
<text><location><page_9><loc_13><loc_42><loc_48><loc_48></location>We considered as well the concomitant action of the vertical shear and vertical convective instabilities. The asymptotic behaviour depends on the relative strength of s vs and s c .</text>
<text><location><page_9><loc_13><loc_36><loc_48><loc_42></location>For s vs > s c we observed the same type of behaviour described above - large growths occur with asymptotic stabilization. When s c > s vs the system is asymptotically unstable. For s vs ∼ s c a variety of mixed evolutions is possible.</text>
<text><location><page_9><loc_13><loc_23><loc_48><loc_35></location>The most interesting feature for the present study is connected with the sign of the angular momentum transport. In fact we observed that in the phase in which the vertical shear driven transients growths occur (i.e. ˜ k x ˜ k z > 0 for A z > 0 and ˜ k x ˜ k z < 0 for A z < 0) transport is always positive, even in the case s c > s vs . When the vertical shear drive is switched off (i.e. A z = 0), the convective transport turns negative. Table 3 summarizes these results.</text>
<text><location><page_9><loc_13><loc_11><loc_48><loc_23></location>We conclude that, for s c > s vs , the interaction of vertical shear growths and convective instability results in a destabilization whose strength (i.e. growth rate) is governed mainly by convection and whose transport by vertical shear. This mechanism holds for any value of s vs ( < s c ), however small. Again in the case s c > s vs , but when their values are close, the growth rate of the mixed instability is increased of a fraction of s c with respect to s c .</text>
<text><location><page_9><loc_13><loc_6><loc_48><loc_10></location>Thermal diffusion has a stabilizing influence on the convective instability. This is specially strong for short wavelengths. Instead it has no influence on the</text>
<text><location><page_9><loc_52><loc_83><loc_87><loc_92></location>vertical shear growth rate. In the case s c > s vs , by decreasing the Peclet number, we observed the transition from convection to vertical shear dominated evolution. By increasing the Peclet number or decreasing the wavenumbers huge growths occur. The regime of very long wavelengths is outside the scope of the present theory.</text>
<text><location><page_9><loc_52><loc_75><loc_87><loc_83></location>We can say that whereas non-axisymmetricity stabilizes the axisymmetric vertical shear instability asymptotically for t -→ ∞ (not for intermediate times though) but has weak influence on the convective one, non-adiabaticity stabilizes the convective instability but has no influence on the vertical shear growth rate.</text>
<text><location><page_9><loc_52><loc_69><loc_87><loc_74></location>This second set of results instead points at a new possible path to explain outward angular momentum transport in astrophysical discs: interaction of vertical shear and convective instabilities.</text>
<section_header_level_1><location><page_9><loc_52><loc_65><loc_73><loc_66></location>ACKNOWLEDGEMENTS</section_header_level_1>
<text><location><page_9><loc_52><loc_56><loc_87><loc_64></location>The author would like to express his gratitude to Prof. Zensho Yoshida for his suggestions, advice and continuous support and to Prof. Ryoji Matsumoto for the many discussions and encouragement. He would also like to acknowledge very stimulating conversations with Prof. Alexander Tevzadze.</text>
<section_header_level_1><location><page_9><loc_52><loc_51><loc_64><loc_52></location>REFERENCES</section_header_level_1>
<text><location><page_9><loc_53><loc_13><loc_87><loc_50></location>Arlt R., Urpin V., 2004, A&A 426 , 755 Balbus S. A., Hawley J. F., 1991, ApJ 376 , 214 Balbus S. A., Hawley J. F., 1992, ApJ 400 , 610 Chagelishvili G. D., Zahn J.-P., Tevzadze A. G., Lominadze, J. G., 2003, A&A 402 , 401 Goldreich P., Schubert G., 1967, ApJ 150 , 571 Ioannou P. J., Kakouris A., 2001, ApJ 550 , 931 Johnson B. M., Gammie C. F., 2005, ApJ 626 , 978 Kippenhahn R., Thomas H.-C., 1982, A&A 114 , 77 Klahr H. H., Bodenheimer P., 2003, ApJ 582 , 869 Kley W., Lin D.N.C., 1992, ApJ 397 , 600 Knobloch E., Spruit H. C., 1986, A&A 166 , 359 Korycansky D. G., 1992, ApJ 399 , 176 Lesur G., Longaretti, P.-Y., 2005, A&A 444 , 25 Lesur G., Papaloizou J. C. B., 2010, A&A 513 , A60 Lesur G., Ogilvie G. I., 2010, MNRAS 404 , L64 Nelson R. P., Gressel O., Umurhan, O. M., 2012, (arXiv:astro-ph/1209.2753) Ruden S. P., Papaloizou J. C. B., Lin D. N. C., 1988, ApJ 329 , 739 Ryu D., Goodman J., 1992, ApJ 388 , 438 Shakura N. I., Sunyaev R. A., 1973, A&A 24 , 337 Stone J. M., Balbus S. A., 1996, ApJ 464 , 364 Urpin V., 1984, Sov. Astronom. 28 , 50 Urpin V., 2003, A&A 404 , 397 Volponi F., Tatsuno T., Yoshida, Z., 2000, Phys. Plasmas 7 , 2314</text>
<text><location><page_9><loc_53><loc_12><loc_77><loc_13></location>Volponi, F., 2010, MNRAS 406 , 551</text>
</document> | [] |
2021CQGra..38x7001G | https://arxiv.org/pdf/2107.04889.pdf | <document>
<section_header_level_1><location><page_1><loc_12><loc_87><loc_87><loc_91></location>The Linet - Tian metrics are a restricted set of Krasi'nski's solutions of Einstein's field equations for a rotating perfect fluid.</section_header_level_1>
<text><location><page_1><loc_41><loc_84><loc_59><loc_85></location>Reinaldo J. Gleiser ∗</text>
<text><location><page_1><loc_30><loc_80><loc_70><loc_82></location>Instituto de F'ısica Enrique Gaviola and FAMAF,</text>
<text><location><page_1><loc_17><loc_79><loc_83><loc_80></location>Universidad Nacional de C'ordoba, Ciudad Universitaria, (5000) C'ordoba, Argentina</text>
<text><location><page_1><loc_17><loc_70><loc_82><loc_77></location>In this note we show that the Linet - Tian family of solutions of the vacuum Einstein equations with a cosmological constant are a restricted set of the solutions of the Einstein field equations for a rotating perfect fluid previously found by A. Krasi'nski.</text>
<text><location><page_1><loc_17><loc_67><loc_35><loc_69></location>PACS numbers: 04.20.Jb</text>
<section_header_level_1><location><page_1><loc_40><loc_61><loc_60><loc_62></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_12><loc_52><loc_88><loc_58></location>There is an interesting family of solutions of the vacuum Einstein field equations with a cosmological constant Λ, that can be positive or negative, found independently by Linet [1] and Tian [2]. The solutions are static, and contain also two other orthogonal Killing vectors. They may be written in the form [3],</text>
<formula><location><page_1><loc_20><loc_44><loc_88><loc_51></location>ds 2 = -y 1 / 3+ p 1 / 2 (1 -Λ y ) 1 / 3 -p 1 / 2 C 1 dt 2 + 1 3 y (1 -Λ y ) dy 2 (1) + y 1 / 3+ p 2 / 2 (1 -Λ y ) 1 / 3 -p 2 / 2 C 2 dz 2 + y 1 / 3+ p 3 / 2 (1 -Λ y ) 1 / 3 -p 3 / 2 C 3 dφ 2</formula>
<text><location><page_1><loc_12><loc_39><loc_88><loc_43></location>where the C i are arbitrary constants, and, for Λ > 0, y is restricted to 0 < y < 1 / Λ, while for Λ < 0 the range of y is 0 ≤ y ≤ ∞ . They satisfy Einstein's equations [4],</text>
<formula><location><page_1><loc_44><loc_36><loc_88><loc_39></location>G µν = -Λ g µν (2)</formula>
<text><location><page_1><loc_12><loc_34><loc_52><loc_36></location>provided the parameters p i satisfy the relations,</text>
<formula><location><page_1><loc_43><loc_27><loc_88><loc_33></location>p 1 + p 2 + p 3 = 0 (3) p 2 1 + p 2 2 + p 2 3 = 8 3</formula>
<text><location><page_1><loc_12><loc_23><loc_88><loc_27></location>and this, in turn implies that the allowed values of the parameters p i are restricted to the range,</text>
<formula><location><page_1><loc_43><loc_20><loc_88><loc_23></location>-4 / 3 ≤ p i ≤ 4 / 3 (4)</formula>
<text><location><page_1><loc_12><loc_14><loc_88><loc_21></location>and all values in that range are allowed. Clearly, (3) is still satisfied if we change every p i to -p i , so that, given a solution of (3), we get a new solution by simply changing the sign of every p i . However, this new solution is diffeomorphic to the original one, as can be seen by changing coordinates in (1), in accordance with,</text>
<formula><location><page_1><loc_43><loc_10><loc_88><loc_12></location>y = (1 -Λ z ) / Λ (5)</formula>
<text><location><page_2><loc_12><loc_89><loc_30><loc_91></location>which changes (1) to,</text>
<formula><location><page_2><loc_20><loc_81><loc_88><loc_88></location>ds 2 = -z 1 / 3 -p 1 / 2 (1 -Λ y ) 1 / 3+ p 1 / 2 ˜ C 1 dt 2 + 1 3 y (1 -Λ y ) dy 2 (6) + y 1 / 3 -p 2 / 2 (1 -Λ y ) 1 / 3+ p 2 / 2 C 2 dz 2 + y 1 / 3 -p 3 / 2 (1 -Λ y ) 1 / 3+ p 3 / 2 C 3 dφ 2</formula>
<text><location><page_2><loc_12><loc_74><loc_88><loc_84></location>˜ ˜ where ˜ C i = C i / Λ p i , which is identical to (1), up to a rescaling of the remaining coordinates, and the change p i →-p i . Similarly, we get the same solution, up to a rescaling of appropriate coordinates, by exchanging p 2 and p 3 . Finally we notice that we may solve (3) for p 2 and p 3 to get,</text>
<formula><location><page_2><loc_38><loc_64><loc_88><loc_73></location>p 2 = -p 1 2 + /epsilon1 √ 48 -27 p 1 2 6 (7) p 3 = -p 1 2 -/epsilon1 √ 48 -27 p 1 2 6</formula>
<text><location><page_2><loc_12><loc_60><loc_88><loc_63></location>where /epsilon1 = ± 1. This can also be seen as a proof of (4). As just discussed, we will set /epsilon1 = 1, without loss of generality. These properties will be important in the discussions that follows.</text>
<text><location><page_2><loc_12><loc_46><loc_88><loc_58></location>The properties and applications of the Linet - Tian metrics have been the subject of a number of studies. A recent review of these and similar types of metrics has been recently presented in [6]. It apparently has escaped this review, as well as most the research papers centering on these type of metrics, that the Linet - Tian solutions are contained, as a restricted set, in a large family of solutions of the vacuum Einstein equations with a cosmological constant, previously found by A. Krasi'nski [7], [8]. These metrics can be obtained from the following Ansatz [9]. If one writes the metrics in the form,</text>
<formula><location><page_2><loc_21><loc_36><loc_88><loc_44></location>ds 2 = 1 v 2 / 3 dx 0 2 +2 x 2 v 2 / 3 dx 0 dx 1 + x 2 2 -V v 2 / 3 dx 1 2 (8) -J 2 sv 2 exp ( -∫ x 2 V dx 2 ) dx 2 2 -V sv 2 / 3 exp ( -∫ x 2 V dx 2 ) dx 3 2</formula>
<text><location><page_2><loc_12><loc_32><loc_88><loc_36></location>where V = V ( x 2 ), v = v ( x 2 ), and J and s are constants, then, a necessary condition for the Einstein equations (2) to be satisfied is that V is a solution of,</text>
<formula><location><page_2><loc_45><loc_27><loc_88><loc_31></location>d 2 V dx 2 2 -2 = 0 (9)</formula>
<text><location><page_2><loc_12><loc_25><loc_30><loc_26></location>and v is a solution of,</text>
<formula><location><page_2><loc_28><loc_14><loc_88><loc_24></location>d 2 v dx 2 2 = 1 V ( dV dx 2 -x 2 ) dv dx 2 (10) + 3 4 V 2 ( V d 2 V dx 2 2 -V + x 2 dV dx 2 -( dV dx 2 ) 2 ) v</formula>
<text><location><page_2><loc_12><loc_10><loc_88><loc_13></location>Notice that (9) and (10) are independent of Λ, and, in fact, of the signature of (8). The general (real) solution of (9) can be written in form,</text>
<formula><location><page_2><loc_39><loc_6><loc_88><loc_8></location>V ( x 2 ) = ( x 2 -q 0 )( x 2 -p 0 ) (11)</formula>
<text><location><page_3><loc_12><loc_87><loc_88><loc_91></location>where p 0 , and q 0 , are constants, and we have three possibilities, namely, p 0 , and q 0 , are real and distinct, p 0 = q 0 , (both real), and p 0 = q ∗ 0 , i.e., complex conjugate of each other.</text>
<text><location><page_3><loc_12><loc_84><loc_88><loc_87></location>The Linet-Tian metrics (1) have three orthogonal Killing vectors, while this, in principle, is not the case for the Krasi'nski metrics (8). The terms in question are,</text>
<formula><location><page_3><loc_24><loc_73><loc_88><loc_82></location>dσ 2 = 1 v 2 / 3 dx 0 2 +2 x 2 v 2 / 3 dx 0 dx 1 + x 2 2 -V v 2 / 3 dx 1 2 (12) = 1 v 2 / 3 ( dx 0 2 +2 x 2 dx 0 dx 1 +(( p 0 + q 0 ) x 2 -p 0 q 0 ) dx 1 2 )</formula>
<text><location><page_3><loc_12><loc_71><loc_88><loc_74></location>implying that ∂ x 0 , and ∂ x 1 are not orthogonal. We, therefore, consider a (linear) change of coordinate basis of the form,</text>
<formula><location><page_3><loc_42><loc_66><loc_88><loc_69></location>x 0 = a 1 y 0 + a 2 y 1 (13) x 1 = b 1 y 0 + b 2 y 1 ,</formula>
<text><location><page_3><loc_12><loc_62><loc_70><loc_64></location>and find that the conditions for the orthogonality of ∂ y 0 , and ∂ y 1 are,</text>
<formula><location><page_3><loc_41><loc_58><loc_88><loc_61></location>b 1 b 2 -a 1 a 2 p 0 q 0 = 0 , (14)</formula>
<text><location><page_3><loc_12><loc_56><loc_21><loc_57></location>and either,</text>
<text><location><page_3><loc_12><loc_52><loc_14><loc_53></location>or,</text>
<formula><location><page_3><loc_44><loc_50><loc_88><loc_51></location>a 2 p 0 + b 2 = 0 . (16)</formula>
<text><location><page_3><loc_12><loc_47><loc_48><loc_49></location>Solving (14) for b 1 , and (15) for a 2 , we get,</text>
<formula><location><page_3><loc_24><loc_42><loc_88><loc_46></location>dσ 2 = a 1 2 ( x 2 -p 0 )( q 0 -p 0 ) p 0 2 v 2 / 3 dy 0 2 + b 2 2 ( q 0 -x 2 )( q 0 -p 0 ) v 2 / 3 dy 1 2 , (17)</formula>
<text><location><page_3><loc_12><loc_39><loc_58><loc_41></location>while solving (16) for a 2 , and replacing in (12), we find,</text>
<formula><location><page_3><loc_24><loc_34><loc_88><loc_38></location>dσ 2 = a 1 2 ( q 0 -x 2 )( q 0 -p 0 ) q 0 2 v 2 / 3 dy 0 2 + b 2 2 ( x 2 -p 0 )( q 0 -p 0 ) v 2 / 3 dy 1 2 . (18)</formula>
<text><location><page_3><loc_12><loc_24><loc_88><loc_33></location>We first notice that since a 1 and b 2 are arbitrary, (17) and (18) are equivalent up to irrelevant changes of names. Next we see that for complex q 0 and p 0 the transformation (13) leads to complex coefficients in dσ 2 , while for p 0 = q 0 the transformation is singular. The only acceptable case is then for q 0 and p 0 real and distinct. In what follows we assume, without loss of generality, q 0 > p 0 . The Krasi'nski metric (8) then takes the form,</text>
<formula><location><page_3><loc_21><loc_14><loc_88><loc_22></location>ds 2 = ( x 2 -p 0 ) v 2 / 3 dy 0 2 + ( q 0 -x 2 ) v 2 / 3 dy 1 2 (19) -J 2 sv 2 exp ( -∫ x 2 V dx 2 ) dx 2 2 -V sv 2 / 3 exp ( -∫ x 2 V dx 2 ) dx 3 2</formula>
<text><location><page_3><loc_12><loc_7><loc_88><loc_14></location>where, without loss of generality, we have have chosen (17), and assigned values to a 1 , and b 2 so as to simplify the resulting expressions. With this restriction we still have to consider three separate cases, namely, x 2 ≥ q 0 , q 0 ≥ x 2 ≥ p 0 , and p 0 ≥ x 2 . These are analyzed in what follows.</text>
<formula><location><page_3><loc_44><loc_54><loc_88><loc_56></location>a 2 q 0 + b 2 = 0 , (15)</formula>
<section_header_level_1><location><page_4><loc_39><loc_88><loc_61><loc_91></location>II. THE CASE x 2 ≥ q 0 .</section_header_level_1>
<section_header_level_1><location><page_4><loc_37><loc_86><loc_63><loc_87></location>A. The form of the metric.</section_header_level_1>
<text><location><page_4><loc_14><loc_81><loc_63><loc_84></location>To continue our analysis of (8) we notice that for x 2 ≥ q 0 ,</text>
<formula><location><page_4><loc_28><loc_73><loc_88><loc_81></location>exp ( -∫ x 2 V dx 2 ) = exp ( -∫ x 2 ( x 2 -p 0 )( x 2 -q 0 ) ) (20) = C ( x 2 -p 0 ) p 0 q 0 -p 0 ( x 2 -q 0 ) -q 0 q 0 -p 0</formula>
<text><location><page_4><loc_12><loc_69><loc_88><loc_72></location>where C is a constant. Since the left hand side of (20) is real and positive, we may set C = 1.</text>
<text><location><page_4><loc_14><loc_66><loc_68><loc_69></location>Again for x 2 ≥ q 0 , we may write the solution of (10) in the form,</text>
<formula><location><page_4><loc_21><loc_58><loc_88><loc_67></location>v ( x 2 ) = ( x 2 -p 0 ) p 0 -2 q 0 2( p 0 -q 0 ) -√ q 0 2 -p 0 q 0 + p 0 2 2( p 0 -q 0 ) ( x 2 -q 0 ) 2 p 0 -q 0 2( p 0 -q 0 ) -√ q 0 2 -p 0 q 0 + p 0 2 2( p 0 -q 0 ) ( Q ( x 2 -q 0 ) √ q 0 2 -p 0 q 0 + p 0 2 p 0 -q 0 + P ( x 2 -p 0 ) √ q 0 2 -p 0 q 0 + p 0 2 p 0 -q 0 ) (21)</formula>
<text><location><page_4><loc_12><loc_53><loc_88><loc_56></location>where P and Q are real constants. In what follows we restrict to P > 0, and Q > 0, to avoid singularities in V ( x 2 ).</text>
<text><location><page_4><loc_12><loc_49><loc_88><loc_53></location>Replacing (11), (20), and (21) in (8) we find that the full set of Einstein's equations (2) is satisfied if we impose,</text>
<formula><location><page_4><loc_38><loc_46><loc_88><loc_49></location>Λ = -sQP ( q 2 0 -q 0 p 0 + p 2 0 ) 3 J 2 (22)</formula>
<text><location><page_4><loc_12><loc_29><loc_88><loc_45></location>We remark again that equations (2) are satisfied independently of the signature assigned to (8), or the particular signs of J 2 or s , and, therefore, (8) provides a (possibly large) family of solutions of (2). At this point it is convenient to go back to the 'diagonal' form (19). This restricted set of Krasi'nski's metrics provides solutions of (2) with three commuting Killing vectors ( ∂ y 0 , ∂ y 1 , and ∂ x 3 ), but we still need to fix the signature of the metrics. In the case of Λ > 0, if we assume J 2 > 0, from (22) we must take s < 0. Without loss of generality we may take s = -1, and this makes the metric (19) static and with signature ( -, + , + , +), and, in accordance with (22), corresponding to Λ > 0. In more detail, the 'diagonal' metric is then given by,</text>
<formula><location><page_4><loc_24><loc_19><loc_88><loc_28></location>ds 2 = -( x 2 -q 0 ) v 3 / 2 dy 1 2 + J 2 ( x 2 -p 0 ) p 0 q 0 -p 0 ( x 2 -q 0 ) -q 0 q 0 -p 0 v 2 dx 2 2 (23) + ( x 2 -p 0 ) -q 0 q 0 -p 0 ( x 2 -q 0 ) -p 0 q 0 -p 0 v 2 / 3 dx 3 2 + ( x 2 -p 0 ) v 3 / 2 dy 0 2</formula>
<text><location><page_4><loc_12><loc_11><loc_88><loc_18></location>But, the Linet and Tian analysis shows that the static solutions of (2) with three orthogonal commuting Killing vectors are unique, up to diffeomorphisms. Therefore, the Linet - Tian metrics and Kransinski's metrics should be related by a coordinate transformation. That this is the case is shown explicitly in the next Section.</text>
<section_header_level_1><location><page_5><loc_35><loc_89><loc_65><loc_91></location>B. A coordinate transformation.</section_header_level_1>
<text><location><page_5><loc_12><loc_84><loc_88><loc_87></location>We consider again (1) and (23) and a coordinate coordinate change of the form y = y ( x 2 ) that would take (1) into (23). Under this change we should have,</text>
<text><location><page_5><loc_12><loc_76><loc_14><loc_78></location>or,</text>
<formula><location><page_5><loc_27><loc_78><loc_88><loc_83></location>1 3 y (1 -Λ y ) ( dy dx 2 ) 2 = J 2 ( x 2 -p 0 ) p 0 q 0 -p 0 ( x 2 -q 0 ) -q 0 q 0 -p 0 v ( x 2 ) 2 (24)</formula>
<formula><location><page_5><loc_23><loc_71><loc_88><loc_77></location>[ 1 3 y (1 -Λ y ) ] 1 / 2 ( dy dx 2 ) = [ J 2 ( x 2 -p 0 ) p 0 q 0 -p 0 ( x 2 -q 0 ) -q 0 q 0 -p 0 v ( x 2 ) 2 ] 1 / 2 (25)</formula>
<text><location><page_5><loc_14><loc_69><loc_36><loc_71></location>This can be integrated to,</text>
<formula><location><page_5><loc_26><loc_57><loc_88><loc_69></location>ln ( 1 -2Λ y -2 i √ Λ y √ 1 -Λ y ) (26) = ln √ Q ( x 2 -q 0 ) √ q 2 0 + q 0 p 0 + p 2 0 2( p 0 -q 0 ) -i √ P ( x 2 -p 0 ) √ q 2 0 + q 0 p 0 + p 2 0 2( p 0 -q 0 ) √ Q ( x 2 -q 0 ) √ q 2 0 + q 0 p 0 + p 2 0 2( p 0 -q 0 ) + i √ P ( x 2 -p 0 ) √ q 2 0 + q 0 p 0 + p 2 0 2( p 0 -q 0 ) </formula>
<text><location><page_5><loc_12><loc_56><loc_70><loc_57></location>where we have used (21) to eliminate Λ. Eq. (26) can be solved for y .</text>
<formula><location><page_5><loc_25><loc_46><loc_88><loc_55></location>y = P ( x 2 -p 0 ) √ q 0 2 -q 0 p 0 + p 0 2 -q 0 + p 0 [ Q ( x 2 -q 0 ) √ q 0 2 -q 0 p 0 + p 0 2 -q 0 + p 0 + P ( x 2 -p 0 ) √ q 0 2 -q 0 p 0 + p 0 2 -q 0 + p 0 ] Λ (27)</formula>
<text><location><page_5><loc_14><loc_43><loc_76><loc_45></location>The range of x 2 is ( q 0 ≤ x 2 ≤ ∞ ), and, in accordance with (27), we have,</text>
<formula><location><page_5><loc_33><loc_39><loc_88><loc_43></location>1 Λ ≥ y ≥ P Λ( P + Q ) , for q 0 ≤ x 2 ≤ ∞ . (28)</formula>
<text><location><page_5><loc_12><loc_35><loc_88><loc_38></location>We must remark that although there is a sign ambiguity in (25), we only need that (24) be satisfied, and one can check that (27) satisfies this requirement.</text>
<text><location><page_5><loc_14><loc_33><loc_62><loc_34></location>Next we consider the coefficient of dy 1 2 in (23). We have,</text>
<formula><location><page_5><loc_20><loc_23><loc_88><loc_33></location>-x 2 -q 0 v 2 / 3 = -( x 2 -q 0 ) p 0 -2 q 0 + √ q 0 2 -q 0 p 0 + p 0 2 3( -q 0 + p 0 ) ( x 2 -p 0 ) 2 q 0 -p 0 + √ q 0 2 -q 0 p 0 + p 0 2 3( -q 0 + p 0 ) ( Q ( x 2 -q 0 ) √ q 0 2 -q 0 p 0 + p 0 2 -q 0 + p 0 + P ( x 2 -p 0 ) √ q 0 2 -q 0 p 0 + p 0 2 -q 0 + p 0 ) 2 / 3 (29)</formula>
<text><location><page_5><loc_12><loc_19><loc_88><loc_22></location>On the other hand, going back to (1), if we consider the coefficient of dt 2 , and change variables in accordance with (27), to get,</text>
<formula><location><page_5><loc_20><loc_7><loc_88><loc_18></location>-C 1 y 1 3 + p 1 2 (1 -Λ y ) 1 3 -p 1 2 = (30) -C 1 P 1 3 + p 1 2 ( x 2 -p 0 ) √ q 0 2 -q 0 p 0 + p 0 2 (2+3 p 1 ) 6( -q 0 + p 0 ) ( x 2 -q 0 ) √ q 0 2 -q 0 p 0 + p 0 2 (2 -3 p 1 ) 6( -q 0 + p 0 ) Λ 1 3 + p 1 2 Q -1 3 + p 1 2 ( Q ( x 2 -q 0 ) √ q 0 2 -q 0 p 0 + p 0 2 -q 0 + p 0 + P ( x 2 -p 0 ) √ q 0 2 -q 0 p 0 + p 0 2 -q 0 + p 0 ) 2 / 3</formula>
<text><location><page_6><loc_14><loc_89><loc_60><loc_91></location>We then have that (30) will be equal to (29) if we set,</text>
<formula><location><page_6><loc_38><loc_86><loc_88><loc_88></location>C 1 = Λ 1 3 + p 1 2 P -1 3 -p 1 2 Q -1 3 + p 1 2 (31)</formula>
<text><location><page_6><loc_12><loc_83><loc_15><loc_84></location>and,</text>
<text><location><page_6><loc_14><loc_77><loc_57><loc_78></location>Similarly, for the coefficient of dy 0 2 in (23) we have,</text>
<formula><location><page_6><loc_40><loc_76><loc_88><loc_83></location>p 1 = 2(2 q 0 -p 0 ) 3 √ p 0 2 -p 0 q 0 + q 0 2 (32)</formula>
<formula><location><page_6><loc_22><loc_67><loc_88><loc_76></location>x 2 -p 0 v 2 / 3 = ( x 2 -q 0 ) q 0 -2 p 0 + √ q 0 2 -q 0 p 0 + p 0 2 3( -q 0 + p 0 ) ( x 2 -p 0 ) 2 p 0 -q 0 + √ q 0 2 -q 0 p 0 + p 0 2 3( -q 0 + p 0 ) ( Q ( x 2 -q 0 ) √ q 0 2 -q 0 p 0 + p 0 2 -q 0 + p 0 + P ( x 2 -p 0 ) √ q 0 2 -q 0 p 0 + p 0 2 -q 0 + p 0 ) 2 / 3 (33)</formula>
<text><location><page_6><loc_12><loc_64><loc_50><loc_65></location>while for the coefficient of dz 2 in (1) we have,</text>
<formula><location><page_6><loc_21><loc_52><loc_88><loc_63></location>C 2 y 1 3 + p 2 2 (1 -Λ y ) 1 3 -p 2 2 = (34) C 2 P 1 3 + p 2 2 ( x 2 -p 0 ) √ q 0 2 -q 0 p 0 + p 0 2 (2+3 p 2 ) 6( -q 0 + p 0 ) ( x 2 -q 0 ) √ q 0 2 -q 0 p 0 + p 0 2 (2 -3 p 2 ) 6( -q 0 + p 0 ) Λ 1 3 + p 1 2 Q -1 3 + p 1 2 ( Q ( x 2 -q 0 ) √ q 0 2 -q 0 p 0 + p 0 2 -q 0 + p 0 + P ( x 2 -p 0 ) √ q 0 2 -q 0 p 0 + p 0 2 -q 0 + p 0 ) 2 / 3</formula>
<text><location><page_6><loc_12><loc_49><loc_52><loc_50></location>and we have equality of (34) and (33) imposing,</text>
<formula><location><page_6><loc_38><loc_45><loc_88><loc_47></location>C 2 = Λ 1 3 + p 2 2 P -1 3 -p 2 2 Q -1 3 + p 2 2 (35)</formula>
<text><location><page_6><loc_12><loc_42><loc_15><loc_44></location>and,</text>
<text><location><page_6><loc_14><loc_36><loc_81><loc_38></location>Finally, and using the same similar procedure, for the coefficient of dφ 2 we find,</text>
<formula><location><page_6><loc_40><loc_36><loc_88><loc_42></location>p 2 = 2(2 p 0 -q 0 ) 3 √ p 0 2 -p 0 q 0 + q 0 2 (36)</formula>
<formula><location><page_6><loc_38><loc_33><loc_88><loc_35></location>C 3 = Λ 1 3 + p 3 2 P -1 3 -p 3 2 Q -1 3 + p 3 2 (37)</formula>
<text><location><page_6><loc_12><loc_30><loc_15><loc_31></location>and,</text>
<formula><location><page_6><loc_39><loc_23><loc_88><loc_30></location>p 3 = -2( p 0 + q 0 ) 3 √ p 0 2 -p 0 q 0 + q 0 2 (38)</formula>
<text><location><page_6><loc_12><loc_18><loc_88><loc_25></location>We can immediately check that the p i in (32), (36), and (38) satisfy (3). Moreover, considering again (32), (36), and (38) we have three different cases: p 0 > 0, p 0 = 0, and p 0 < 0. In the case p 0 > 0 we may set p 0 = 1, since the p i depend only on the ratio q 0 /p 0 , and then considering all q 0 > p 0 we find for the p i the ranges,</text>
<formula><location><page_6><loc_43><loc_6><loc_88><loc_17></location>2 3 ≤ p 1 ≤ 4 3 2 3 ≥ p 2 ≥ -2 3 (39) -4 3 ≤ p 3 ≤ -2 3</formula>
<text><location><page_7><loc_14><loc_89><loc_39><loc_91></location>Similarly, for p 0 < 0 we have,</text>
<formula><location><page_7><loc_43><loc_77><loc_88><loc_88></location>-2 3 ≤ p 1 ≤ 4 3 -2 3 ≥ p 2 ≥ -2 3 (40) + 4 3 ≤ p 3 ≤ -2 3</formula>
<text><location><page_7><loc_14><loc_73><loc_76><loc_76></location>The case p 0 = 0 corresponds just to p 1 = 4 / 3, p 2 = -2 / 3, and p 3 = -2 / 3.</text>
<section_header_level_1><location><page_7><loc_35><loc_70><loc_65><loc_71></location>III. SUMMARY OF RESULTS.</section_header_level_1>
<text><location><page_7><loc_12><loc_45><loc_88><loc_68></location>Eqs. (39) and (40), together with the properties already discussed of the p i imply that the full relevant range of values of the p i are covered with appropriate choices of p 0 , and q 0 . Considering the range of y , we notice that once the p i are fixed, the only relevant quantity is Λ, since the C i in (1) represent only rescalings of the coordinates. But according to (22), fixing Λ we still have a large freedom in the choices of P , Q , and J 2 . In accordance with (28), this freedom can be used to cover essentially the full range of y in (1), with the exception of the singular point y = 0, so that we have shown that the Linet-Tian metrics are effectively a restricted set of the Krasi'nski metrics (8), in the case x 2 > q 0 . But essentially similar derivations show that this is also the case for q 0 ≥ x 2 ≥ p 0 , and p 0 ≥ x 2 . In the first case the full range 0 ≤ y ≤ 1 / Λ is covered, while in the second the point y = 1 / Λ is excluded. This, in turn, shows that the three possible ranges of x 2 in (8) correspond, up to isometries, to the same solution, and also completes the proof of the equivalence of the metrics (1) and (8) in the cases where we take p 0 and q 0 in (11) as real and distinct.</text>
<section_header_level_1><location><page_7><loc_42><loc_41><loc_58><loc_42></location>Acknowledgments</section_header_level_1>
<text><location><page_7><loc_12><loc_35><loc_88><loc_38></location>I am grateful to A. Krasi'nski for bringing to my attention the references to his work and its possible relation to the Linet - Tian metrics, and also for his comments on this note.</text>
<unordered_list>
<list_item><location><page_7><loc_12><loc_28><loc_47><loc_29></location>[1] B. Linet, J. Math. Phys. 27 1817 (1986).</list_item>
<list_item><location><page_7><loc_12><loc_26><loc_47><loc_27></location>[2] Q. T. Tian, Phys. Rev. D 33 3549 (1986).</list_item>
<list_item><location><page_7><loc_12><loc_22><loc_88><loc_25></location>[3] R. J. Gleiser, Linear stability of the Linet - Tian solution with positive cosmological constant. , arXiv:1810.07296v2 [gr-qc]</list_item>
<list_item><location><page_7><loc_12><loc_19><loc_72><loc_22></location>[4] I shall use throughout the conventions of [5], with signature ( -, + , + , +).</list_item>
<list_item><location><page_7><loc_12><loc_19><loc_88><loc_20></location>[5] Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. 1973, Gravitation (Freeman, San Francisco)</list_item>
<list_item><location><page_7><loc_12><loc_17><loc_79><loc_18></location>[6] K. A. Bronnikov, N. O. Santos, A. Wang, Class. Quantum Grav. 37 ,113002 (2020)</list_item>
<list_item><location><page_7><loc_12><loc_15><loc_52><loc_16></location>[7] A. Krasi'nski, Acta Phys. Polon., B6 , 223 (1975)</list_item>
<list_item><location><page_7><loc_12><loc_13><loc_49><loc_14></location>[8] A. Krasi'nski, J. Math. Phys. 16 , 125 (1975)</list_item>
<list_item><location><page_7><loc_12><loc_9><loc_88><loc_13></location>[9] I have, without loss of generality, and for convenience in the derivations that follow, simplified the expression of the metric (8), as compared to the original form in [8].</list_item>
</unordered_list>
</document> | [] |
2017IJTP...56.1364C | https://arxiv.org/pdf/1604.05738.pdf | <document>
<section_header_level_1><location><page_1><loc_24><loc_92><loc_76><loc_93></location>If gravity is geometry, is dark energy just arithmetic?</section_header_level_1>
<text><location><page_1><loc_44><loc_89><loc_57><loc_90></location>Marek Czachor 1 , 2</text>
<text><location><page_1><loc_32><loc_84><loc_69><loc_89></location>1 Katedra Fizyki Teoretycznej i Informatyki Kwantowej, Politechnika Gda'nska, 80-233 Gda'nsk, Poland, 2 Centrum Leo Apostel (CLEA), Vrije Universiteit Brussel, 1050 Brussels, Belgium,</text>
<text><location><page_1><loc_18><loc_61><loc_83><loc_81></location>Arithmetic operations (addition, subtraction, multiplication, division), as well as the calculus they imply, are non-unique. The examples of four-dimensional spaces, R 4 + and ( -L/ 2 , L/ 2) 4 , are considered where different types of arithmetic and calculus coexist simultaneously. In all the examples there exists a non-Diophantine arithmetic that makes the space globally Minkowskian, and thus the laws of physics are formulated in terms of the corresponding calculus. However, when one switches to the 'natural' Diophantine arithmetic and calculus, the Minkowskian character of the space is lost and what one effectively obtains is a Lorentzian manifold. I discuss in more detail the problem of electromagnetic fields produced by a pointlike charge. The solution has the standard form when expressed in terms of the non-Diophantine formalism. When the 'natural' formalsm is used, the same solution looks as if the fields were created by a charge located in an expanding universe, with nontrivially accelerating expansion. The effect is clearly visible also in solutions of the Friedman equation with vanishing cosmological constant. All of this suggests that phenomena attributed to dark energy may be a manifestation of a miss-match between the arithmetic employed in mathematical modeling, and the one occurring at the level of natural laws. Arithmetic is as physical as geometry.</text>
<text><location><page_1><loc_18><loc_59><loc_45><loc_59></location>PACS numbers: 04.50.Kd, 04.20.Cv, 05.45.Df</text>
<section_header_level_1><location><page_1><loc_20><loc_55><loc_37><loc_56></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_34><loc_49><loc_52></location>The idea of relativity of arithmetic follows from the observation that the four basic arithmetic operations (addition, subtraction, multiplication, division) are fundamentally non-unique, even if one assumes commutativity and associativity of 'plus' and 'times', and distributivity of 'times' with respect to 'plus'. The ambiguity extends to calculus and algebra since even the most elementary notions, such as derivatives or matrix products, involve arithmetic operations, sometimes accompanied by limits ('to zero', say). A 'zero', the neutral element of addition, inherits its ambiguity from the ambiguity of addition. The same concerns a 'one', the neutral element of multiplication.</text>
<text><location><page_1><loc_9><loc_25><loc_49><loc_33></location>The freedom of choosing arithmetic and its corresponding calculus is a universal symmetry of any mathematical model, but we are still lacking its physical understanding. For all that, treated just as a mathematical trick, the idea has found concrete applications in fractal theory [1-4].</text>
<text><location><page_1><loc_9><loc_17><loc_49><loc_24></location>In the paper, I discuss a situation where there is a miss-match between the arithmetic employed at the level of mathematical principles, and the one that is naturally 'employed' by Nature. We will see that consequences of the miss-match can be similar to those of dark energy.</text>
<text><location><page_1><loc_9><loc_8><loc_49><loc_17></location>In natural sciences there exists at least one example of such a miss-match, and we experience it in our everyday life. This is the Weber phenomenon known from neuroscience [5, 6]. Namely, it is an experimental fact that the increment x → x + kx of intensity of a stimulus (sound, light, taste, etc.) is perceived by our nervous system as</text>
<text><location><page_1><loc_52><loc_50><loc_92><loc_56></location>being independent of x . k depends on the type of stimulus and is known as a Weber constant. The Weber law ∆ x/x = k ≈ const is valid in a wide range of stimulus parameters.</text>
<text><location><page_1><loc_52><loc_46><loc_92><loc_50></location>From a mathematical point of view the Weber law appears as the solution of the following problem [7]: Find a generalized subtraction glyph[circleminus] such that</text>
<formula><location><page_1><loc_55><loc_44><loc_92><loc_45></location>( x + kx ) glyph[circleminus] x = f -1 ( f ( x + kx ) -f ( x ) ) = δx (1)</formula>
<text><location><page_1><loc_52><loc_32><loc_92><loc_42></location>is independent of x . The solution f is unique and is given by a logarithm, as shown by G. Fechner in 1850 [8]. This is the reason why decibels correspond to a logarithmic scale. We hear, see, taste and feel the world outside of us though a logarithmic channel of our neurons, although we are typically as unaware of it, as we are unaware of experiencing curvature of space when we feel our weight.</text>
<text><location><page_1><loc_52><loc_28><loc_92><loc_32></location>The Fechner problem extends to any natural science. In order to appreciate it, consider the function f ( x ) = x 3 , and define</text>
<formula><location><page_1><loc_57><loc_26><loc_92><loc_27></location>x ⊕ y = f -1 ( f ( x ) + f ( y ) ) = 3 √ x 3 + y 3 , (2)</formula>
<formula><location><page_1><loc_57><loc_24><loc_92><loc_25></location>x glyph[circleminus] y = f -1 ( f ( x ) -f ( y ) ) = 3 √ x 3 -y 3 , (3)</formula>
<formula><location><page_1><loc_57><loc_22><loc_92><loc_23></location>x glyph[circledot] y = f -1 ( f ( x ) f ( y ) ) = xy, (4)</formula>
<formula><location><page_1><loc_57><loc_20><loc_92><loc_21></location>x glyph[circledivide] y = f -1 ( f ( x ) /f ( y ) ) = x/y. (5)</formula>
<text><location><page_1><loc_52><loc_17><loc_76><loc_18></location>Here multiplication is unchanged.</text>
<text><location><page_1><loc_52><loc_8><loc_92><loc_17></location>Mathematicians would generally say that we have introduced two types of field , related by the field isomorphism f , but I prefer the terminology of Burgin [9, 10] where formulas such as (2)-(5) occur in arithmetic contexts. A terminology involving 'fields' or 'relativity of fields' would be very confusing in the context of</text>
<text><location><page_2><loc_9><loc_85><loc_49><loc_93></location>physics, especially in relativistic field theory. An arithmetic involving a non-trivial f is termed by Burgin a nonDiophantine one, as opposed to the Diophantine case of f ( x ) = x . I will also stick to this distinction, although one should bear in mind that it is largely a matter of convention which of the two arithmetics is Diophantine.</text>
<text><location><page_2><loc_10><loc_83><loc_33><loc_84></location>The non-Diophantine derivative</text>
<formula><location><page_2><loc_15><loc_79><loc_49><loc_82></location>DA ( x ) Dx = lim h → 0 ( A ( x ⊕ h ) glyph[circleminus] A ( x ) ) glyph[circledivide] h (6)</formula>
<text><location><page_2><loc_9><loc_74><loc_49><loc_78></location>satisfies all the basic rules of differentiation (the Leibnitz rule for glyph[circledot] , the chain rule for composition of functions, linearity with respect to ⊕ ...). The solution of</text>
<formula><location><page_2><loc_18><loc_70><loc_49><loc_73></location>DA ( x ) Dx = A ( x ) , A (0) = 1 , (7)</formula>
<text><location><page_2><loc_9><loc_66><loc_49><loc_69></location>is unique, but it comes as some surprise, at least when one first encounters it, that</text>
<formula><location><page_2><loc_23><loc_64><loc_49><loc_65></location>A ( x ) = e x 3 / 3 , (8)</formula>
<text><location><page_2><loc_9><loc_56><loc_49><loc_62></location>as one can verify directly from definition (6). Had one replaced f ( x ) = x 3 by f ( x ) = x 5 , one would have found A ( x ) = e x 5 / 5 = f -1 ( e f ( x ) ), as the reader has probably already guessed.</text>
<text><location><page_2><loc_9><loc_45><loc_49><loc_56></location>Notice that the change of f is not a change of variables. The differential equation (7) remains linear in spite of non-linearity of f . The change of f cannot be regarded as a change of gauge either, with covariant derivative D/Dx , since the corresponding connection would be trivial, in spite of non-triviality of f . The change of arithmetic operates at a more primitive level than a change of variables or gauge.</text>
<text><location><page_2><loc_9><loc_42><loc_49><loc_45></location>Of course, one can denote g = f -1 and rewrite (2)-(3) as</text>
<formula><location><page_2><loc_13><loc_39><loc_49><loc_42></location>x + y = g -1 ( g ( x ) ⊕ g ( y ) ) = ( 3 √ x ⊕ 3 √ y ) 3 , (9)</formula>
<formula><location><page_2><loc_13><loc_37><loc_49><loc_39></location>x -y = g -1 ( g ( x ) glyph[circleminus] g ( y ) ) = ( 3 √ x glyph[circleminus] 3 √ y ) 3 . (10)</formula>
<text><location><page_2><loc_9><loc_27><loc_49><loc_36></location>Now comes the fundamental question: Which of the two additions, + or ⊕ , is more natural or physical? Which of them is Diophantine, and which is not? Clearly, ± and ⊕ , glyph[circleminus] coexist in the same set, and it is hard to say why (2)-(3) should be regarded as less simple, or more weird, than (9)-(10).</text>
<text><location><page_2><loc_9><loc_23><loc_49><loc_27></location>Put another way, what kind of a rule is responsible for the implicit preference of f ( x ) = x over any other one-to-one f ? The Ockham razor?</text>
<text><location><page_2><loc_9><loc_13><loc_49><loc_23></location>Expressing it yet differently, let us assume that it is indeed the 'natural' (whatever it means) arithmetic with ± etc. that we should employ in practical computations. What kind of a rule guarantees that the laws of physics (variational principles, say) are formulated in terms of the same arithmetic and calculus, and not by means of ⊕ , glyph[circledot] , D/Dx , and the like, for some unknown f ?</text>
<text><location><page_2><loc_10><loc_12><loc_33><loc_13></location>I believe the questions are open.</text>
<text><location><page_2><loc_9><loc_9><loc_49><loc_11></location>It is not completely unrelated to mention that the case of a linear f was extensively studied by Benioff [11]</text>
<text><location><page_2><loc_52><loc_69><loc_92><loc_93></location>in a somewhat different context. Here, the departure point was the observation that natural numbers N can be identified with any countable well-ordered set whose first two elements define 0 and 1. For example, the set N 2 = { 0 , 2 , 4 , . . . } of even natural numbers may be regarded as a representation of N . The value function f ( x ) = x/ 2 maps a natural number x ∈ N 2 into its value, but to make the structure self-consistent one has to redefine the multiplication: x glyph[circledot] y = f -1 ( f ( x ) f ( y ) ) = xy/ 2. So, in Benioff's approach numbers themselves are just elements of some formal axiomatic structure, while their values are determined by appropriate linear value maps, which are simultaneously used to redefine the arithmetic operations. The idea was then extended by Benioff to real and complex numbers, and generalized in many ways, including space-time dependent fields of value functions [12, 13].</text>
<text><location><page_2><loc_52><loc_37><loc_92><loc_68></location>From the Burgin perspective it is essential that one can also encounter nonlinear functions f (Fechner's logarithms, Cantor-type functions for Cantor sets [1, 2], Peano-type space-filling curves in Sierpi'nski-set cases [3]...). Whenever one tries to apply a Burgin-type generalization to a physical system, one immediately encounters the problem that nontrivial f s typically require dimensionless arguments in f ( x ). Still, physical x s are dimensional. One has to associate with a physically meaningful x a dimensionless number, and we return to the problem described by Benioff. Indeed, a dimensionless x is obtained for the price of including a physical unit (of length, say), and units can be chosen arbitrarily. The choice of units effectively introduces a value map, and a change of scale changes this map. Burgin's f , when employed in a way proposed in [1], can be an arbitrary bijection, so can be composed with any value map with no loss of bijectivity of the composition. Benioff's value maps are thus intrinsically related with Burgin's f s, but are not necessarily equivalent to them. The problem of dimensional vs. dimensionless x was discussed in more detail in [2].</text>
<text><location><page_2><loc_52><loc_26><loc_92><loc_36></location>The goal of the present paper is to illustrate these problems on concrete examples from relativistic physics, and to contemplate the possibility of detecting a non-trivial f by means of physical observations. We will see that phenomena of dark-energy variety may suggest the presence of some f between the universe of our mathematical formulas, and the physical Universe.</text>
<text><location><page_2><loc_52><loc_9><loc_92><loc_26></location>We will consider two spaces, R 4 + and ( -L/ 2 , L/ 2) 4 , which become Minkowski space-times when appropriate arithmetic is selected. We will then consider the problem of electromagnetic fields produced by pointlike sources, but the Maxwell field will be formulated in terms of this concrete arithmetic ⊕ , glyph[circleminus] , glyph[circledot] , glyph[circledivide] , which will make the space Minkowskian. However, when we switch back to the 'standard' arithmetic the geometry becomes locally Lorenzian, with a non-Minkowskian global structure. Static charges will appear moving toward event horizons of the space, thus creating an impression of an expanding universe, with accelerating expansion. All of</text>
<text><location><page_3><loc_9><loc_90><loc_49><loc_93></location>this happens in spite of the fact that the spaces are static, no matter which arithmetic one works with.</text>
<text><location><page_3><loc_9><loc_59><loc_49><loc_90></location>The paper is organized as follows. In the next section, two types of real numbers, equipped with their own arithmetic and calculus, are introduced. The two types of reals are related by a bijection f . In Section III, the two types of reals are employed in construction of two types of Minkowski spaces. Sections IV and V discuss in detail the two concrete examples of Minkowski spaces: R 4 + and ( -L/ 2 , L/ 2) 4 . Section VI discusses electromagnetic fields produced by a pointlike charge, with Maxwell equations formulated in terms of a non-Diophantine formalism defined by some f . The formalism is implicitly non-Diophantine since an observer who employs in his observations and calculations the arithmetic defined by the same f will never discover that some nontrivial f is implicitly involved. However, in case there is a missmatch, i.e. two different f s come into play, the conflict of arithmetics can have observational consequences. This is discussed on explicit examples in sections VII and VIII. Finally, in section IX, an example of a non-Diophantinecalculus Friedman equation is discussed. We observe accelerated expansion with vanishing cosmological constant.</text>
<section_header_level_1><location><page_3><loc_13><loc_55><loc_45><loc_56></location>II. 'LOWER' AND 'UPPER' REALITY</section_header_level_1>
<text><location><page_3><loc_9><loc_43><loc_49><loc_52></location>Consider real numbers R equipped with the 'standard' arithmetic operations of addition (+), subtraction ( -), multiplication ( · ), and division ( / ). Let us term these real numbers the 'lower reals', and denote them by lowercase symbols. As usual, ' · ' can be skipped: a · b = ab . Neutral elements of addition and multiplication in R are denoted by 0 and 1.</text>
<text><location><page_3><loc_9><loc_32><loc_49><loc_42></location>Now, let us assume that there exist some 'upper reals', whose set is denoted by R . By assumption, R is related to R by some bijection, f : R → R . Those upper reals are equipped with their own arithmetic and calculus. The arithmetic operations in R will be denoted by ⊕ , glyph[circleminus] , glyph[circledot] , glyph[circledivide] . We use the convention that elements of R are denoted by upper-case fonts.</text>
<text><location><page_3><loc_9><loc_30><loc_49><loc_32></location>The arithmetic in R is non-Diophantine in the sense of Burgin [9, 10],</text>
<formula><location><page_3><loc_18><loc_27><loc_49><loc_29></location>X ⊕ Y = f -1 ( f ( X ) + f ( Y ) ) , (11)</formula>
<formula><location><page_3><loc_18><loc_25><loc_49><loc_27></location>X glyph[circleminus] Y = f -1 ( f ( X ) -f ( Y ) ) , (12)</formula>
<formula><location><page_3><loc_18><loc_23><loc_49><loc_25></location>X glyph[circledot] Y = f -1 ( f ( X ) f ( X ) ) , (13)</formula>
<formula><location><page_3><loc_18><loc_21><loc_49><loc_23></location>X glyph[circledivide] Y = f -1 ( f ( X ) /f ( Y ) ) . (14)</formula>
<text><location><page_3><loc_9><loc_13><loc_49><loc_20></location>Both arithmetics are commutative, associative, and multiplications are distributive with respect to (appropriate) additions. Neutral elements of addition and multiplication in R are denoted by 0 ' and 1 ' , which implies 0 ' = f -1 (0), 1 ' = f -1 (1).</text>
<text><location><page_3><loc_9><loc_9><loc_49><loc_13></location>A negative in R is defined by glyph[circleminus] X = 0 ' glyph[circleminus] X = f -1 ( -f ( X ) ) . Multiplication by zero yields zero in both arithmetics, in particular 0 ' glyph[circledot] X = 0 ' .</text>
<text><location><page_3><loc_52><loc_45><loc_56><loc_46></location>satisfy</text>
<text><location><page_3><loc_52><loc_90><loc_92><loc_93></location>Multiplication is equivalent to repeated addition in the following sense. Let N ∈ N and X ' = f -1 ( X ) ∈ R . Then</text>
<formula><location><page_3><loc_62><loc_88><loc_92><loc_89></location>N ' ⊕ X ' = ( N + X ) ' , (15)</formula>
<formula><location><page_3><loc_62><loc_86><loc_92><loc_87></location>N ' glyph[circledot] X ' = ( NX ) ' (16)</formula>
<formula><location><page_3><loc_69><loc_82><loc_92><loc_85></location>= X ' ⊕··· ⊕ X ' ︸ ︷︷ ︸ N times . (17)</formula>
<text><location><page_3><loc_52><loc_78><loc_92><loc_81></location>A power function A ( X ) = X glyph[circledot] · · · glyph[circledot] X ( N times) will be denoted by X N ' , since</text>
<formula><location><page_3><loc_58><loc_75><loc_92><loc_77></location>X N ' glyph[circledot] X M ' = X ( N + M ) ' = X N ' ⊕ M ' . (18)</formula>
<text><location><page_3><loc_53><loc_72><loc_89><loc_73></location>A derivative of a function A : R → R is defined by</text>
<formula><location><page_3><loc_56><loc_68><loc_92><loc_71></location>DA ( X ) DX = lim H → 0 ' ( A ( X ⊕ H ) glyph[circleminus] A ( X ) ) glyph[circledivide] H, (19)</formula>
<text><location><page_3><loc_52><loc_64><loc_92><loc_67></location>as contrasted with the derivative of a function a : R → R , defined with respect to the lowercase arithmetic,</text>
<formula><location><page_3><loc_59><loc_60><loc_92><loc_63></location>da ( x ) dx = lim h → 0 ( a ( x + h ) -a ( x ) ) /h. (20)</formula>
<text><location><page_3><loc_52><loc_57><loc_74><loc_59></location>Now let A = f -1 · a · f . Then,</text>
<formula><location><page_3><loc_62><loc_52><loc_92><loc_56></location>DA ( X ) DX = f -1 ( da ( f ( X ) ) df ( X ) ) , (21)</formula>
<formula><location><page_3><loc_57><loc_48><loc_92><loc_51></location>∫ Y X A ( X ' ) DX ' = f -1 ( ∫ f ( Y ) f ( X ) a ( x ) dx ) , (22)</formula>
<formula><location><page_3><loc_58><loc_41><loc_80><loc_44></location>D DX ∫ X A ( X ' ) DX ' = A ( X ) ,</formula>
<formula><location><page_3><loc_60><loc_37><loc_92><loc_42></location>Y (23) ∫ X Y DA ( X ' ) DX ' DX ' = A ( X ) glyph[circleminus] A ( Y ) . (24)</formula>
<text><location><page_3><loc_52><loc_30><loc_92><loc_35></location>Formula (21) follows directly from the definitions of D/DX and d/dx . As stressed in the introduction, (21) is not the usual formula relating derivatives of A = f -1 · a · f with those of a . Indeed,</text>
<formula><location><page_3><loc_64><loc_26><loc_92><loc_28></location>DA DX = f -1 · da dx · f, (25)</formula>
<text><location><page_3><loc_52><loc_12><loc_92><loc_24></location>so that D/DX behaves like a covariant derivative, but with a trivial connection for any bijection f : R → R . The standard approach, employed in differential geometry or gauge theories, would employ the arithmetic of R , and one would have to assume differentiability of f and f -1 . Here bijectivity is enough since no derivatives of either f or f -1 will occur in (21) and (25). This is why this type of calculus is so useful and natural in fractal theory [1-4].</text>
<text><location><page_3><loc_52><loc_9><loc_92><loc_11></location>Partial derivatives and multidimensional integrals are defined analogously.</text>
<section_header_level_1><location><page_4><loc_12><loc_91><loc_46><loc_93></location>III. LOWER AND UPPER MINKOWSKI SPACE-TIMES</section_header_level_1>
<text><location><page_4><loc_10><loc_87><loc_49><loc_89></location>Consider a point x in Minkowski space R 4 and assume</text>
<formula><location><page_4><loc_11><loc_85><loc_49><loc_86></location>( x 0 , x 1 , x 2 , x 3 ) = ( f ( X 0 ) , f ( X 1 ) , f ( X 2 ) , f ( X 3 ) ) (26)</formula>
<text><location><page_4><loc_9><loc_79><loc_49><loc_83></location>where ( x 0 , x 1 , x 2 , x 3 ) ∈ R 4 and ( X 0 , X 1 , X 2 , X 3 ) ∈ R 4 . The two spaces are Minkowskian in the sense that the invariant quadratic forms are defined by</text>
<formula><location><page_4><loc_13><loc_77><loc_49><loc_78></location>g ab x a x b = ( x 0 ) 2 -( x 1 ) 2 -( x 2 ) 2 -( x 3 ) 2 , (27)</formula>
<formula><location><page_4><loc_11><loc_74><loc_49><loc_76></location>G ab X a X b = ( X 0 ) 2 ' glyph[circleminus] ( X 1 ) 2 ' glyph[circleminus] ( X 2 ) 2 ' glyph[circleminus] ( X 3 ) 2 ' . (28)</formula>
<text><location><page_4><loc_9><loc_72><loc_13><loc_73></location>Since</text>
<formula><location><page_4><loc_11><loc_67><loc_49><loc_71></location>f -1 ( g ab x a x b ) = f -1 ( f ( X 0 ) 2 -f ( X 1 ) 2 -f ( X 2 ) 2 -f ( X 3 ) 2 ) (29)</formula>
<formula><location><page_4><loc_21><loc_64><loc_49><loc_66></location>= f -1 ( f ( G ab ) f ( X a ) f ( X b ) ) (30)</formula>
<formula><location><page_4><loc_21><loc_61><loc_49><loc_63></location>= G ab X a X b , (31)</formula>
<text><location><page_4><loc_9><loc_56><loc_49><loc_60></location>the two quadratic forms are related by f , and g ab = f ( G ab ). Covariant and contravariant world-vectors are defined in the usual way,</text>
<formula><location><page_4><loc_16><loc_53><loc_49><loc_55></location>X a = G ab X b (32)</formula>
<formula><location><page_4><loc_19><loc_49><loc_49><loc_53></location>= f -1 ( 3 ∑ b =0 f ( G ab ) f ( X b ) ) (33)</formula>
<formula><location><page_4><loc_19><loc_45><loc_49><loc_48></location>= f -1 ( 3 ∑ b =0 g ab x b ) = f -1 ( x a ) . (34)</formula>
<text><location><page_4><loc_9><loc_41><loc_49><loc_43></location>The quadratic forms are invariant with respect to Lorentz transformations,</text>
<formula><location><page_4><loc_18><loc_36><loc_49><loc_40></location>x ' a = λ a b x b = 3 ∑ b =0 λ a b x b , (35)</formula>
<formula><location><page_4><loc_18><loc_34><loc_49><loc_35></location>X ' a = Λ a b X b = ⊕ 3 b =0 Λ a b glyph[circledot] X b (36)</formula>
<formula><location><page_4><loc_21><loc_29><loc_49><loc_33></location>= f -1 ( 3 ∑ b =0 f (Λ a b ) f ( X b ) ) . (37)</formula>
<text><location><page_4><loc_9><loc_24><loc_49><loc_28></location>This type of Lorentz transformation, but for f representing a Cantor set, was explicitly used in [1, 2] to construct fractal homogeneous spaces.</text>
<text><location><page_4><loc_10><loc_23><loc_36><loc_24></location>We will later need an explicit boost,</text>
<formula><location><page_4><loc_15><loc_16><loc_49><loc_22></location>Λ a b = Cosh φ glyph[circleminus] Sinh φ 0 ' 0 ' glyph[circleminus] Sinh φ Cosh φ 0 ' 0 ' 0 ' 0 ' 1 ' 0 ' 0 ' 0 ' 0 ' 1 ' , (38)</formula>
<text><location><page_4><loc_9><loc_12><loc_49><loc_15></location>where Sinh φ = f -1 ( sinh f ( φ ) ) , Cosh φ = f -1 ( cosh f ( φ ) ) satisfy</text>
<formula><location><page_4><loc_19><loc_9><loc_49><loc_10></location>Cosh 2 ' φ glyph[circleminus] Sinh 2 ' φ = 1 ' . (39)</formula>
<text><location><page_4><loc_52><loc_92><loc_64><loc_93></location>The four-velocity</text>
<formula><location><page_4><loc_64><loc_85><loc_92><loc_90></location>U a = Cosh φ Sinh φ 0 ' 0 ' , (40)</formula>
<text><location><page_4><loc_52><loc_82><loc_78><loc_83></location>is mapped by (38) into (1 ' , 0 ' , 0 ' , 0 ' ).</text>
<section_header_level_1><location><page_4><loc_57><loc_77><loc_86><loc_79></location>IV. MINKOWSKI SPACE-TIME R 4 +</section_header_level_1>
<text><location><page_4><loc_52><loc_68><loc_92><loc_75></location>Now let us make the analysis more explicit. Let the Fechner function f ( X ) = µ ln X + ν , µ > 0, be the bijection f : R + → R . Accordingly, R = R + . f -1 ( x ) = e ( x -ν ) /µ , and thus 0 ' = f -1 (0) = e -ν/µ , 1 ' = f -1 (1) = e (1 -ν ) /µ .</text>
<section_header_level_1><location><page_4><loc_66><loc_64><loc_78><loc_65></location>A. Arithmetic</section_header_level_1>
<text><location><page_4><loc_52><loc_59><loc_92><loc_62></location>Let us begin with the explicit form of arithmetic operations. Addition and subtraction explicitly read</text>
<formula><location><page_4><loc_61><loc_52><loc_92><loc_58></location>X ⊕ Y = f -1 ( f ( X ) + f ( Y ) ) = XYe ν/µ , (41) X glyph[circleminus] Y = f -1 ( f ( X ) -f ( Y ) )</formula>
<formula><location><page_4><loc_67><loc_50><loc_92><loc_52></location>= e -ν/µ X/Y. (42)</formula>
<text><location><page_4><loc_52><loc_43><loc_92><loc_49></location>The arithmetic operations occurring at the right sides of (41) and (42) are those from R and not from R (the latter occur at the left sides of these formulas). For example, X ⊕ 0 ' = Xe -ν/µ e ν/µ = X .</text>
<text><location><page_4><loc_52><loc_36><loc_92><loc_43></location>Note that although X > 0 in f ( X ) = µ ln X + ν , one nevertheless has a well defined negative number glyph[circleminus] X = 0 ' glyph[circleminus] X = e -2 ν/µ /X ∈ R = R + , which is positive from the point of view of the arithmetic of R . Let us cross-check the negativity of glyph[circleminus] X :</text>
<formula><location><page_4><loc_61><loc_33><loc_92><loc_34></location>glyph[circleminus] X ⊕ X = ( glyph[circleminus] X ) Xe ν/µ (43)</formula>
<formula><location><page_4><loc_68><loc_31><loc_92><loc_33></location>= ( e -2 ν/µ /X ) Xe ν/µ (44)</formula>
<formula><location><page_4><loc_68><loc_29><loc_92><loc_31></location>= e -ν/µ = 0 ' . (45)</formula>
<text><location><page_4><loc_52><loc_23><loc_92><loc_27></location>( R , ⊕ ) = ( R + , ⊕ ) is a group, as opposed to ( R + , +). In consequence, the Minkowski space R 4 + is invariant under the non-Diophantine Poincar'e group.</text>
<text><location><page_4><loc_53><loc_22><loc_85><loc_23></location>The multiplication in R is explicitly given by</text>
<formula><location><page_4><loc_56><loc_13><loc_92><loc_21></location>X glyph[circledot] Y = f -1 ( f ( X ) f ( Y ) ) = e µ ln X ln Y + ν ln X + ν ln Y + ν 2 /µ -ν/µ , (46) X glyph[circledivide] Y = f -1 ( f ( X ) /f ( Y ) ) = e (ln X + ν/µ ) / ( µ ln Y + ν ) -ν/µ . (47)</formula>
<text><location><page_4><loc_52><loc_9><loc_92><loc_11></location>Again the expressions at the right-hand sides of (46) and (47) involve the arithmetic of R .</text>
<figure>
<location><page_5><loc_13><loc_81><loc_45><loc_93></location>
<caption>FIG. 1: Light cone X a X a = 0 ' in 1+2 dimensional Minkowski space R 3 + , for µ = 10, ν = -20, and its close-up (right) in the neighborhood of (0 ' , 0 ' , 0 ' ), where 0 ' = e -ν/µ = e 2 ≈ 7 . 39.</caption>
</figure>
<section_header_level_1><location><page_5><loc_23><loc_72><loc_35><loc_73></location>B. Light cone</section_header_level_1>
<text><location><page_5><loc_10><loc_68><loc_46><loc_70></location>The light cone in R 4 + consists of vectors satisfying</text>
<formula><location><page_5><loc_15><loc_65><loc_49><loc_67></location>G ab X a X b = f -1 ( f ( G ab ) f ( X a ) f ( X b ) ) (48)</formula>
<formula><location><page_5><loc_23><loc_63><loc_49><loc_65></location>= f -1 ( g ab f ( X a ) f ( X b ) ) (49)</formula>
<formula><location><page_5><loc_23><loc_61><loc_49><loc_63></location>= 0 ' = f -1 (0) = e -ν/µ . (50)</formula>
<text><location><page_5><loc_9><loc_57><loc_49><loc_60></location>This is equivalent to f ( X 0 ) 2 = f ( X 1 ) 2 + f ( X 2 ) 2 + f ( X 3 ) 2 , i.e.</text>
<formula><location><page_5><loc_9><loc_49><loc_49><loc_56></location>X 0 = f -1 ( ± √ f ( X 1 ) 2 + f ( X 2 ) 2 + f ( X 3 ) 2 ) (51) = e ( ± √ (ln X 1 + ν/µ ) 2 +(ln X 2 + ν/µ ) 2 +(ln X 3 + ν/µ ) 2 -ν/µ ) (52)</formula>
<text><location><page_5><loc_9><loc_45><loc_49><loc_48></location>Fig. 1 shows the light cone G ab X a X b = 0 ' in 1 + 2 dimensional Minkowski space R 3 + .</text>
<text><location><page_5><loc_10><loc_43><loc_34><loc_45></location>An arbitrarily located light cone</text>
<formula><location><page_5><loc_17><loc_41><loc_49><loc_42></location>G ab ( X a glyph[circleminus] Y a )( X b glyph[circleminus] Y b ) = 0 ' (53)</formula>
<text><location><page_5><loc_9><loc_38><loc_19><loc_39></location>corresponds to</text>
<formula><location><page_5><loc_13><loc_35><loc_49><loc_37></location>g ab ( f ( X a ) -f ( Y a ) )( f ( X b ) -f ( Y b ) ) = 0 , (54)</formula>
<text><location><page_5><loc_9><loc_33><loc_13><loc_34></location>that is</text>
<formula><location><page_5><loc_11><loc_30><loc_49><loc_32></location>X 0 = Y 0 e ± √ ln 2 ( X 1 /Y 1 )+ln 2 ( X 2 /Y 2 )+ln 2 ( X 3 /Y 3 ) . (55)</formula>
<text><location><page_5><loc_9><loc_27><loc_47><loc_28></location>For Y 0 = · · · = Y 3 = 0 ' = e -ν/µ we reconstruct (52).</text>
<text><location><page_5><loc_9><loc_10><loc_49><loc_27></location>Fig. 2 shows the light cones (55) in small neighborhoods of various origins Y a . The plots suggest that a Lorentzian geometry is typical of both the standard 'lower-case' space-time R 3 , and of the 'upper-case' R 3 . Recall that the latter is also globally Minkowskian, but with respect to the non-Diophantine calculus. Interestingly, when we employ in R 3 the miss-matched formalism taken from R 3 , the formulas are locally Lorentzian. The further away from the 'walls' of R 3 + the observation is performed, the more Minkowskian the geometry appears, provided one does not observe objects that are too far away from the observer, as we shall see later.</text>
<figure>
<location><page_5><loc_52><loc_62><loc_92><loc_93></location>
<caption>FIG. 2: Minkowskian correspondence principle in 1 + 2 dimensional Fechnerian space-time. Light cone ( X a glyph[circleminus] Y a )( X a glyph[circleminus] Y a ) = 0 ' for (A) ( Y 0 , Y 1 , Y 2 ) = (10000 , 10001 , 10002), (B) ( Y 0 , Y 1 , Y 2 ) = (10000 , 5000 , 10000), (C) ( Y 0 , Y 1 , Y 2 ) = (10000 , 10000 , 5000), and (D) ( Y 0 , Y 1 , Y 2 ) = (5000 , 10000 , 10001). The further away from the boundaries of R 3 + , the more Minkowskian-looking the light cones are.</caption>
</figure>
<text><location><page_5><loc_52><loc_44><loc_92><loc_47></location>assumed to perform his analysis in the 'wrong' formalism. Then, for | glyph[epsilon1] a /Y a | glyph[lessmuch] 1,</text>
<formula><location><page_5><loc_57><loc_41><loc_92><loc_42></location>g ab f ' ( Y a ) glyph[epsilon1] a f ' ( Y b ) glyph[epsilon1] b = ˜ g ab ( Y ) glyph[epsilon1] a glyph[epsilon1] b ≈ 0 (56)</formula>
<text><location><page_5><loc_52><loc_38><loc_56><loc_39></location>where</text>
<formula><location><page_5><loc_54><loc_30><loc_92><loc_36></location>˜ g ab = g ab f ' ( Y a ) f ' ( Y b ) (no sum) (57) = diag ( ( Y 0 ) -2 , -( Y 1 ) -2 , -( Y 2 ) -2 , -( Y 3 ) -2 ) (58)</formula>
<text><location><page_5><loc_52><loc_25><loc_92><loc_28></location>is the Lorentzian metric. ˜ g ab becomes just a conformally rescaled Minkowskian g ab if Y 0 = Y 1 = Y 2 = Y 3 .</text>
<text><location><page_5><loc_70><loc_22><loc_70><loc_24></location>glyph[negationslash]</text>
<text><location><page_5><loc_52><loc_22><loc_92><loc_25></location>The same effect occurs for general hyperboloids G ab ( X a glyph[circleminus] Y a )( X b glyph[circleminus] Y b ) = 0 ' .</text>
<section_header_level_1><location><page_5><loc_54><loc_16><loc_89><loc_17></location>V. MINKOWSKI SPACE-TIME ( -L/ 2 , L/ 2) 4</section_header_level_1>
<text><location><page_5><loc_52><loc_10><loc_92><loc_14></location>Let R = ( -L/ 2 , L/ 2) and f ( X ) = tan( πX/L ), f -1 ( x ) = ( L/π ) arctan x . f : R → R is a possible bijection, with 0 ' = 0 and 1 ' = L/ 4.</text>
<figure>
<location><page_6><loc_10><loc_77><loc_48><loc_93></location>
<caption>FIG. 3: Light-cone with the origin at ( Y 0 , Y 1 , Y 2 ) = (0 , -0 . 4 , -0 . 2) in 1 + 2 dimensional Minkowski space ( -0 . 5 , 0 . 5) 3 ( L = 1). (A) The global picture, and (B) the close-up of the origin of the cone</caption>
</figure>
<figure>
<location><page_6><loc_10><loc_53><loc_47><loc_69></location>
<caption>FIG. 4: The same as in the previous figure, but with ( Y 0 , Y 1 , Y 2 ) = (0 . 01 , -0 . 02 , -0 . 03).</caption>
</figure>
<section_header_level_1><location><page_6><loc_45><loc_45><loc_56><loc_46></location>A. Light cone</section_header_level_1>
<text><location><page_6><loc_10><loc_41><loc_53><loc_43></location>The light cone G ab ( X a glyph[circleminus] Y a )( X b glyph[circleminus] Y b ) = 0 ' corresponds to</text>
<formula><location><page_6><loc_10><loc_38><loc_92><loc_40></location>X 0 = f -1 ( f ( Y 0 ) ± √ ( f ( X 1 ) -f ( Y 1 ) ) 2 + ( f ( X 2 ) -f ( Y 2 ) ) 2 + ( f ( X 3 ) -f ( Y 3 ) ) 2 ) (59)</formula>
<formula><location><page_6><loc_13><loc_30><loc_92><loc_36></location>= L π arctan tan πY 0 L ± √ ( tan πX 1 L -tan πY 1 L ) 2 + ( tan πX 2 L -tan πY 2 L ) 2 + ( tan πX 3 L -tan πY 3 L ) 2 . (60)</formula>
<text><location><page_6><loc_9><loc_27><loc_83><loc_29></location>Now let X a = Y a + glyph[epsilon1] a , where | glyph[epsilon1] a /L | glyph[lessmuch] 1. The effective Lorentzian metric in a neighborhood of Y reads</text>
<formula><location><page_6><loc_23><loc_23><loc_92><loc_26></location>˜ g ab ( Y ) = π 2 L 2 diag ( cos -4 πY 0 L , -cos -4 πY 1 L , -cos -4 πY 2 L , -cos -4 πY 3 L ) . (61)</formula>
<text><location><page_6><loc_9><loc_16><loc_49><loc_19></location>Another example of a bijection f : R → R is provided by f ( X ) = arctanh(2 X/L ), f -1 ( x ) = ( L/ 2) tanh x .</text>
<section_header_level_1><location><page_6><loc_54><loc_16><loc_90><loc_18></location>VI. FIELD PRODUCED BY A POINTLIKE CHARGE</section_header_level_1>
<text><location><page_6><loc_52><loc_9><loc_92><loc_13></location>The above two examples show that several different types of arithmetic may coexist in the same space-time. Let us try to find a phenomenon where a nontrivial f can</text>
<text><location><page_7><loc_9><loc_92><loc_17><loc_93></location>be detected.</text>
<text><location><page_7><loc_9><loc_86><loc_49><loc_92></location>I propose to concentrate on Maxwell equations, but formulated in terms of this form of arithmetic that makes the geometry globally Minkowskian. Consider the d'Alembertian</text>
<formula><location><page_7><loc_21><loc_82><loc_49><loc_85></location>glyph[square] ' = G ab D DX a D DX b (62)</formula>
<text><location><page_7><loc_9><loc_74><loc_49><loc_81></location>defined with respect to non-Diophantine partial derivatives. Let R glyph[owner] S ↦→ X a ( S ) ∈ R 4 be a world-line of a pointlike charge, and let J a ( X ) be the current associated with the world-line. The Maxwell equations can be taken in the form</text>
<formula><location><page_7><loc_21><loc_72><loc_49><loc_73></location>glyph[square] ' A a ( X ) = J a ( X ) . (63)</formula>
<text><location><page_7><loc_9><loc_66><loc_49><loc_71></location>The procedure of finding A a ( X ) is standard [14], but we only have to take care of appropriate definitions of nonDiophantine arithmetic and calculus. The end result is</text>
<formula><location><page_7><loc_11><loc_62><loc_49><loc_65></location>A a ( X ) = f -1 ( f ( C ) f ( U a ( Y ) ) g bc f ( U b ( Y ) )( f ( X c ) -f ( Y c ) ) ) . (64)</formula>
<text><location><page_7><loc_9><loc_53><loc_49><loc_60></location>Here C is a constant, U a ( X ) is the four-velocity of the charge, and X b glyph[circleminus] Y b is future-pointing and null. We say that X a ∈ R 4 is future-pointing if x a = f ( X a ) ∈ R 4 is future-pointing. The summation convention is applied unless otherwise stated.</text>
<text><location><page_7><loc_9><loc_49><loc_49><loc_53></location>The four-velocity is normalized by U a U a = 1 ' (where 1 ' = f -1 (1) is the neutral element of multiplication), which is equivalent to</text>
<formula><location><page_7><loc_18><loc_46><loc_49><loc_48></location>g ab f ( U a ( Y ) ) f ( U b ( Y ) ) = 1 . (65)</formula>
<text><location><page_7><loc_9><loc_39><loc_49><loc_45></location>The four-potential is a world-vector gauge field and thus under the action of a Lorentz transformation Λ transforms by A a ( X ) ↦→ A ' a ( X ) = (Λ A ) a (Λ -1 X ), up to a gauge transformation. In standard notation</text>
<formula><location><page_7><loc_15><loc_35><loc_49><loc_38></location>A ' a ( X ) = C Λ U a ( Y ) U b ( Y ) ( (Λ -1 X ) b -Y b ) (66)</formula>
<formula><location><page_7><loc_21><loc_31><loc_49><loc_35></location>= C Λ U a ( Y ) (Λ U ) b ( Y ) ( X b -(Λ Y ) b ) . (67)</formula>
<text><location><page_7><loc_9><loc_17><loc_49><loc_30></location>Now consider a Y -independent U a of the form (40) and let Λ a b be given by (38). To simplify further analysis let us take the point of observation at the origin X a = (0 ' , 0 ' , 0 ' , 0 ' ) ≡ 0 ' and the source at Y a = ( Y 0 , Y 1 , 0 ' , 0 ' ). Y a is null and past-pointing, which implies f ( Y 0 ) = -| f ( Y 1 ) | (notice that in our examples we assume that f is increasing and there exist various arguments why this is important for physical consistency of the formalism). Accordingly,</text>
<formula><location><page_7><loc_11><loc_15><loc_49><loc_16></location>(Λ Y ) 0 = Y 0 glyph[circledot] Cosh φ glyph[circleminus] Y 1 glyph[circledot] Sinh φ, (68)</formula>
<formula><location><page_7><loc_9><loc_11><loc_49><loc_14></location>f ( (Λ Y ) 0 ) = -| f ( Y 1 ) | ( cosh f ( φ ) + f ( Y 1 ) | f ( Y 1 ) | sinh f ( φ ) )</formula>
<formula><location><page_7><loc_17><loc_9><loc_49><loc_11></location>= -| f ( Y 1 ) | e f ( Y 1 ) | f ( Y 1 ) | f ( φ ) . (69)</formula>
<text><location><page_7><loc_52><loc_92><loc_92><loc_93></location>The only non-vanishing component of the potential reads</text>
<formula><location><page_7><loc_58><loc_89><loc_92><loc_91></location>A ' 0 (0 ' ) = C glyph[circledivide] ( glyph[circleminus] Λ Y ) 0 (70)</formula>
<formula><location><page_7><loc_64><loc_85><loc_92><loc_89></location>= f -1 ( f ( C ) f (0 ' ) -f ( (Λ Y ) 0 ) ) (71)</formula>
<formula><location><page_7><loc_64><loc_80><loc_92><loc_84></location>= f -1 f ( C ) | f ( Y 1 ) | e f ( Y 1 ) | f ( Y 1 ) | f ( φ ) . (72)</formula>
<text><location><page_7><loc_52><loc_76><loc_92><loc_78></location>In order to continue we have to make f more concrete. Let us begin with the Minkowski space ( -L/ 2 , L/ 2) 4 .</text>
<formula><location><page_7><loc_60><loc_71><loc_83><loc_72></location>VII. FIELDS IN ( -L/ 2 , L/ 2) 4</formula>
<text><location><page_7><loc_53><loc_68><loc_73><loc_69></location>f ( X ) = tan( πX/L ) implies</text>
<formula><location><page_7><loc_55><loc_62><loc_92><loc_66></location>A ' 0 (0 ' ) = L π arctan f ( C ) | tan( πY 1 /L ) | e Y 1 | Y 1 | f ( φ ) . (73)</formula>
<text><location><page_7><loc_52><loc_55><loc_92><loc_60></location>Let us first consider the case of | πY 1 /L | glyph[lessmuch] 1, but with Y 1 sufficiently far from the singularity at the origin 0 ' = 0, so that the arguments of both tan and arctan are small. The field is then approximately Coulombian</text>
<formula><location><page_7><loc_60><loc_50><loc_92><loc_53></location>A ' 0 (0 ' ) ≈ L π f ( C ) | πY 1 /L | e Y 1 | Y 1 | f ( φ ) (74)</formula>
<formula><location><page_7><loc_66><loc_46><loc_92><loc_49></location>= L 2 π 2 f ( C ) 1 | Y | e -Y 1 | Y 1 | f ( φ ) . (75)</formula>
<text><location><page_7><loc_52><loc_39><loc_92><loc_45></location>Notice that up to this point we have worked with linearized forms of f and f -1 since for small distances and small potentials it is enough if one replaces exact nonlinear f by its linear approximation.</text>
<text><location><page_7><loc_52><loc_32><loc_92><loc_39></location>Let us concentrate on a charge at rest, i.e. with f ( φ ) = 0, and denote q = L 2 f ( C ) /π 2 . We conclude that a source placed in a neighborhood of the origin produces the Coulomb field whose value at the point of observation is</text>
<formula><location><page_7><loc_66><loc_28><loc_92><loc_31></location>A ' 0 (0 ' ) ≈ q | Y | (76)</formula>
<text><location><page_7><loc_52><loc_25><loc_82><loc_27></location>where | Y | is the distance from the source.</text>
<text><location><page_7><loc_52><loc_20><loc_92><loc_25></location>Now, let us increase | Y | so that tan cannot be anymore approximated by its argument. The argument of arctan is even smaller, so here the approximation is still valid. The result is</text>
<formula><location><page_7><loc_61><loc_15><loc_92><loc_18></location>A ' 0 (0 ' ) ≈ L π f ( C ) | tan( πY 1 /L ) | (77)</formula>
<formula><location><page_7><loc_67><loc_12><loc_92><loc_15></location>= q | Y | πY 1 /L tan( πY 1 /L ) (78)</formula>
<formula><location><page_7><loc_67><loc_8><loc_92><loc_11></location>= q | Y | e -Y 1 | Y 1 | f ( φ ( Y 1 ) ) (79)</formula>
<text><location><page_8><loc_9><loc_88><loc_49><loc_93></location>So, in spite of our assumption that in the neighborhood of the origin the charge is at rest, at large distances the charge looks like moving with the four-velocity determined by certain φ ( Y 1 ), defined by</text>
<formula><location><page_8><loc_18><loc_84><loc_49><loc_87></location>e Y 1 | Y 1 | f ( φ ( Y 1 ) ) = tan( πY 1 /L ) πY 1 /L . (80)</formula>
<text><location><page_8><loc_9><loc_82><loc_33><loc_83></location>The velocity β = tanh f ( φ ( Y 1 ) ) is</text>
<formula><location><page_8><loc_18><loc_75><loc_49><loc_81></location>β = Y 1 | Y 1 | ( tan( πY 1 /L ) πY 1 /L ) 2 -1 ( tan( πY 1 /L ) πY 1 /L ) 2 +1 . (81)</formula>
<text><location><page_8><loc_9><loc_62><loc_49><loc_74></location>β is the velocity deduced by the observer located at X a = 0 ' who analyses his data on the basis of the 'standard' Diophantine arithmetic, whereas the physical non-Diophantine arithmetic, employed in Maxwell's equations, is given by glyph[circledot] , ⊕ , etc. The observer is related with the physical Universe by means of an 'information channel' f , but is unaware of it. The miss-match of mathematical structures leads to unexpected behavior of distant objects.</text>
<text><location><page_8><loc_9><loc_51><loc_49><loc_61></location>Had the observer decided to employ Einstein's general relativity, a similar miss-match would have occurred. A solution would produce an expanding universe whose behavior would be consistent with the Hubble law at small distances, but very distant objects would acquire an unexplained acceleration. We will demonstrate this explicitly in section IX.</text>
<text><location><page_8><loc_10><loc_50><loc_33><loc_51></location>As our second example consider</text>
<formula><location><page_8><loc_11><loc_45><loc_49><loc_49></location>A ' 0 (0 ' ) = L 2 tanh f ( C ) | arctanh(2 Y 1 /L ) | e Y 1 | Y 1 | f ( φ ) . (82)</formula>
<text><location><page_8><loc_9><loc_43><loc_38><loc_44></location>With the same approximations as before</text>
<formula><location><page_8><loc_17><loc_39><loc_49><loc_42></location>A ' 0 (0 ' ) ≈ L 2 f ( C ) | 2 Y 1 /L | e -Y 1 | Y 1 | f ( φ ) (83)</formula>
<formula><location><page_8><loc_23><loc_36><loc_49><loc_38></location>= q | Y | e -Y 1 | Y 1 | f ( φ ) , (84)</formula>
<text><location><page_8><loc_9><loc_31><loc_49><loc_35></location>where q = L 2 f ( C ) / 4 and Y is in a neighborhood of the origin. For Y further away from the origin and with f ( φ ) = 0 we get</text>
<formula><location><page_8><loc_17><loc_27><loc_49><loc_30></location>A ' 0 (0 ' ) ≈ L 2 f ( C ) | arctanh(2 Y 1 /L ) | (85)</formula>
<formula><location><page_8><loc_23><loc_23><loc_49><loc_26></location>= q | Y | 2 Y 1 /L arctanh(2 Y 1 /L ) . (86)</formula>
<text><location><page_8><loc_9><loc_21><loc_43><loc_22></location>Repeating the remaining calculations we obtain</text>
<formula><location><page_8><loc_17><loc_15><loc_49><loc_20></location>β = Y 1 | Y 1 | ( arctanh(2 Y 1 /L ) 2 Y 1 /L ) 2 -1 ( arctanh(2 Y 1 /L ) 2 Y 1 /L ) 2 +1 . (87)</formula>
<text><location><page_8><loc_9><loc_9><loc_49><loc_14></location>Fig. 5 compares (81) with (87). In both cases the charge looks as if it moved toward the boundaries Y 1 = ± L/ 2, where it approaches the velocity of light. The motion is non-trivially accelerated.</text>
<figure>
<location><page_8><loc_53><loc_75><loc_91><loc_93></location>
<caption>FIG. 5: The plot of (81) (full) and (87) (dashed) for L = 1. The velocity has the same sign as Y 1 so the motion is toward the horizons Y 1 = ± 1 / 2.</caption>
</figure>
<section_header_level_1><location><page_8><loc_63><loc_65><loc_80><loc_67></location>VIII. FIELDS IN R 4 +</section_header_level_1>
<text><location><page_8><loc_52><loc_57><loc_92><loc_63></location>It is instructive to consider explicitly also the case of R 4 + since 0 ' = e -ν/µ and thus certain counterintuitive elements of a non-Diophantine arithmetic and calculus become more visible. The example will make further generalizations easier to understand.</text>
<text><location><page_8><loc_52><loc_51><loc_92><loc_56></location>Here f ( X ) = µ ln X + ν , µ > 0, f -1 ( x ) = e ( x -ν ) /µ ≈ e -ν/µ (1 + x/µ ) = 0 ' + xe -ν/µ /µ . Setting Y 1 = 0 ' + r , f (0 ' + r ) = µ ln[0 ' (1 + r/ 0 ' )] + ν = µ ln(1 + r/ 0 ' ), and assuming | r | glyph[lessmuch] 1, we obtain</text>
<formula><location><page_8><loc_57><loc_45><loc_92><loc_50></location>A ' 0 (0 ' ) = f -1 f ( C ) | f ( Y 1 ) | e f ( Y 1 ) | f ( Y 1 ) | f ( φ ) (88)</formula>
<formula><location><page_8><loc_63><loc_41><loc_92><loc_45></location>≈ 0 ' + f ( C ) e -ν/µ /µ 2 | ln(1 + r/ 0 ' ) | e f ( Y 1 ) | f ( Y 1 ) | f ( φ ) (89)</formula>
<formula><location><page_8><loc_63><loc_37><loc_92><loc_40></location>≈ 0 ' + f ( C )( e -ν/µ /µ ) 2 | r | e r | r | f ( φ ) . (90)</formula>
<text><location><page_8><loc_52><loc_33><loc_92><loc_36></location>We identify q = f ( C )( e -ν/µ /µ ) 2 . For f ( φ ) = 0 and larger r</text>
<formula><location><page_8><loc_61><loc_29><loc_92><loc_32></location>A ' 0 (0 ' ) ≈ 0 ' + f ( C ) e -ν/µ /µ 2 | ln(1 + r/ 0 ' ) | (91)</formula>
<formula><location><page_8><loc_67><loc_25><loc_92><loc_28></location>= 0 ' + q | r | ∣ ∣ ∣ ln(1+ r/ 0 ' ) r/ 0 ' ∣ ∣ ∣ . (92)</formula>
<text><location><page_8><loc_52><loc_21><loc_92><loc_24></location>An observer located at 0 ' will conclude that the field is produced by a charge which satisfies</text>
<formula><location><page_8><loc_62><loc_17><loc_92><loc_20></location>e r | r | f ( φ ) = ∣ ∣ ∣ ∣ ln(1 + r/ 0 ' ) r/ 0 ' ∣ ∣ ∣ ∣ , (93)</formula>
<text><location><page_8><loc_52><loc_15><loc_56><loc_16></location>hence</text>
<formula><location><page_8><loc_62><loc_9><loc_92><loc_14></location>β = r | r | ( ln(1+ r/ 0 ' ) r/ 0 ' ) 2 -1 ( ln(1+ r/ 0 ' ) r/ 0 ' ) 2 +1 . (94)</formula>
<figure>
<location><page_9><loc_10><loc_75><loc_48><loc_93></location>
<caption>FIG. 6: The plot of (94): β as a function of r/ 0 ' .</caption>
</figure>
<text><location><page_9><loc_9><loc_61><loc_49><loc_69></location>Fig. 6 shows that β given by (94) is always negative: The charge moves toward the horizon Y 1 = 0. Fields produced by charges located between the observer and the horizon would be red-shifted. However, an observer located between the horizon and the charge would detect a blue-shifted field.</text>
<section_header_level_1><location><page_9><loc_17><loc_57><loc_41><loc_58></location>IX. FRIEDMAN EQUATION</section_header_level_1>
<text><location><page_9><loc_9><loc_48><loc_49><loc_54></location>Taylor expansions of (81), (87) in a neighborhood of Y 1 = 0 begin with third order terms ∼ ( Y 1 /L ) 3 . The effect is small. However, when we switch to nonDiophantine generalized Einstein equations, the correction should become visible at large distances.</text>
<text><location><page_9><loc_9><loc_39><loc_49><loc_47></location>So, let us consider the Friedman equation for a flat, matter dominated FRW model with exactly vanishing cosmological constant [15]. In matter dominated cosmology (Ω m = 1 , Ω r = 0), with no dark energy (Ω v = 0), the scale factor is given by a ( t ) = ( t/t 0 ) 2 / 3 . In the nonDiophantine notation the solution reads</text>
<formula><location><page_9><loc_17><loc_36><loc_49><loc_38></location>A ( T ) = ( T glyph[circledivide] T 0 ) 2 ' glyph[circledivide] 3 ' (95)</formula>
<formula><location><page_9><loc_21><loc_34><loc_49><loc_35></location>= f -1 ( ( f ( T ) /f ( T 0 ) ) 2 / 3 ) . (96)</formula>
<text><location><page_9><loc_9><loc_31><loc_49><loc_32></location>To make (96) more explicit, let us experiment with some</text>
<text><location><page_9><loc_52><loc_85><loc_92><loc_93></location>f . For example, the choice of R = ( -L/ 2 , L/ 2) and f ( X ) = tan( πX/L ), f -1 ( x ) = ( L/π ) arctan x leads to the scale factor depicted at Fig. 7, where (96) is compared with the standard ( t/t 0 ) 2 / 3 , for L = 20, T 0 = 1 ' , and t 0 chosen in a way guaranteeing a reasonable fit of the two plots.</text>
<text><location><page_9><loc_52><loc_79><loc_92><loc_84></location>The curve bends up in a characteristic way, typical of dark-energy models of accelerating Universe. The effect is of purely arithmetic origin, with no need of dark energy. Arithmetic becomes as physical as geometry.</text>
<section_header_level_1><location><page_9><loc_64><loc_74><loc_80><loc_75></location>X. CONCLUSIONS</section_header_level_1>
<text><location><page_9><loc_52><loc_69><loc_92><loc_72></location>Gravity is geometry. Is dark energy just arithmetic? Is dark energy dark, because it is always darkest under the</text>
<figure>
<location><page_9><loc_53><loc_50><loc_90><loc_69></location>
<caption>FIG. 7: The plot of (96) (full) as compared with ( t/t 0 ) 2 / 3 for f ( X ) = tan( πX/L ). Both models involve no dark energy, i.e. Ω v = 0 = 0 ' .</caption>
</figure>
<text><location><page_9><loc_52><loc_37><loc_92><loc_42></location>lantern, and we are so accustomed to 'plus' and 'times' that we overlooked the fundamental ambiguity of these operations? Is there a physical law that determines the form of arithmetic, a kind of Einstein equation for f ?</text>
<text><location><page_9><loc_52><loc_31><loc_92><loc_36></location>The questions are relatively well posed. We can change the paradigm and do physics with unspecified f , leaving determination of f to experimentalists. Perhaps one day we will understand which arithmetic is physical, and why.</text>
<text><location><page_10><loc_12><loc_92><loc_30><loc_93></location>und Hartel, Leipzig (1860).</text>
<unordered_list>
<list_item><location><page_10><loc_10><loc_88><loc_49><loc_92></location>[9] M. Burgin, Non-Diophantine Arithmetics , Ukrainian Academy of Information Sciences, Kiev (1997) (in Russian).</list_item>
<list_item><location><page_10><loc_9><loc_85><loc_49><loc_88></location>[10] M. Burgin, Introduction to projective arithmetics, arXiv:1010.3287 [math.GM] (2010).</list_item>
<list_item><location><page_10><loc_9><loc_81><loc_49><loc_85></location>[11] P. Benioff, New gauge field from extension of space time parallel transport of vector spaces to the underlying number systems, Int. J. Theor. Phys. 50 , 1887 (2011).</list_item>
<list_item><location><page_10><loc_9><loc_79><loc_49><loc_81></location>[12] P. Benioff, Fiber bundle description of number scaling in gauge theory and geometry, Quantum Stud.: Math.</list_item>
</unordered_list>
<unordered_list>
<list_item><location><page_10><loc_55><loc_92><loc_80><loc_93></location>Found. 2 , 289 (2015); arXiv:1412.1493.</list_item>
<list_item><location><page_10><loc_52><loc_87><loc_92><loc_92></location>[13] P. Benioff, Space and time dependent scaling of numbers in mathematical structures: Effects on physical and geometric quantities, Quantum Inf. Proc. 15 , 1081 (2016); arXiv:1508.01732.</list_item>
<list_item><location><page_10><loc_52><loc_84><loc_92><loc_86></location>[14] J. D. Jackson, Classical Electrodynamics , Wiley, New York (1962).</list_item>
<list_item><location><page_10><loc_52><loc_80><loc_92><loc_84></location>[15] J. B. Hartle, Gravity. An Introduction to Einstein's General Relativity , Benjamin Cummings, San Francisco (2003).</list_item>
</document> | [{"title": "If gravity is geometry, is dark energy just arithmetic?", "content": "Marek Czachor 1 , 2 1 Katedra Fizyki Teoretycznej i Informatyki Kwantowej, Politechnika Gda'nska, 80-233 Gda'nsk, Poland, 2 Centrum Leo Apostel (CLEA), Vrije Universiteit Brussel, 1050 Brussels, Belgium, Arithmetic operations (addition, subtraction, multiplication, division), as well as the calculus they imply, are non-unique. The examples of four-dimensional spaces, R 4 + and ( -L/ 2 , L/ 2) 4 , are considered where different types of arithmetic and calculus coexist simultaneously. In all the examples there exists a non-Diophantine arithmetic that makes the space globally Minkowskian, and thus the laws of physics are formulated in terms of the corresponding calculus. However, when one switches to the 'natural' Diophantine arithmetic and calculus, the Minkowskian character of the space is lost and what one effectively obtains is a Lorentzian manifold. I discuss in more detail the problem of electromagnetic fields produced by a pointlike charge. The solution has the standard form when expressed in terms of the non-Diophantine formalism. When the 'natural' formalsm is used, the same solution looks as if the fields were created by a charge located in an expanding universe, with nontrivially accelerating expansion. The effect is clearly visible also in solutions of the Friedman equation with vanishing cosmological constant. All of this suggests that phenomena attributed to dark energy may be a manifestation of a miss-match between the arithmetic employed in mathematical modeling, and the one occurring at the level of natural laws. Arithmetic is as physical as geometry. PACS numbers: 04.50.Kd, 04.20.Cv, 05.45.Df", "pages": [1]}, {"title": "I. INTRODUCTION", "content": "The idea of relativity of arithmetic follows from the observation that the four basic arithmetic operations (addition, subtraction, multiplication, division) are fundamentally non-unique, even if one assumes commutativity and associativity of 'plus' and 'times', and distributivity of 'times' with respect to 'plus'. The ambiguity extends to calculus and algebra since even the most elementary notions, such as derivatives or matrix products, involve arithmetic operations, sometimes accompanied by limits ('to zero', say). A 'zero', the neutral element of addition, inherits its ambiguity from the ambiguity of addition. The same concerns a 'one', the neutral element of multiplication. The freedom of choosing arithmetic and its corresponding calculus is a universal symmetry of any mathematical model, but we are still lacking its physical understanding. For all that, treated just as a mathematical trick, the idea has found concrete applications in fractal theory [1-4]. In the paper, I discuss a situation where there is a miss-match between the arithmetic employed at the level of mathematical principles, and the one that is naturally 'employed' by Nature. We will see that consequences of the miss-match can be similar to those of dark energy. In natural sciences there exists at least one example of such a miss-match, and we experience it in our everyday life. This is the Weber phenomenon known from neuroscience [5, 6]. Namely, it is an experimental fact that the increment x \u2192 x + kx of intensity of a stimulus (sound, light, taste, etc.) is perceived by our nervous system as being independent of x . k depends on the type of stimulus and is known as a Weber constant. The Weber law \u2206 x/x = k \u2248 const is valid in a wide range of stimulus parameters. From a mathematical point of view the Weber law appears as the solution of the following problem [7]: Find a generalized subtraction glyph[circleminus] such that is independent of x . The solution f is unique and is given by a logarithm, as shown by G. Fechner in 1850 [8]. This is the reason why decibels correspond to a logarithmic scale. We hear, see, taste and feel the world outside of us though a logarithmic channel of our neurons, although we are typically as unaware of it, as we are unaware of experiencing curvature of space when we feel our weight. The Fechner problem extends to any natural science. In order to appreciate it, consider the function f ( x ) = x 3 , and define Here multiplication is unchanged. Mathematicians would generally say that we have introduced two types of field , related by the field isomorphism f , but I prefer the terminology of Burgin [9, 10] where formulas such as (2)-(5) occur in arithmetic contexts. A terminology involving 'fields' or 'relativity of fields' would be very confusing in the context of physics, especially in relativistic field theory. An arithmetic involving a non-trivial f is termed by Burgin a nonDiophantine one, as opposed to the Diophantine case of f ( x ) = x . I will also stick to this distinction, although one should bear in mind that it is largely a matter of convention which of the two arithmetics is Diophantine. The non-Diophantine derivative satisfies all the basic rules of differentiation (the Leibnitz rule for glyph[circledot] , the chain rule for composition of functions, linearity with respect to \u2295 ...). The solution of is unique, but it comes as some surprise, at least when one first encounters it, that as one can verify directly from definition (6). Had one replaced f ( x ) = x 3 by f ( x ) = x 5 , one would have found A ( x ) = e x 5 / 5 = f -1 ( e f ( x ) ), as the reader has probably already guessed. Notice that the change of f is not a change of variables. The differential equation (7) remains linear in spite of non-linearity of f . The change of f cannot be regarded as a change of gauge either, with covariant derivative D/Dx , since the corresponding connection would be trivial, in spite of non-triviality of f . The change of arithmetic operates at a more primitive level than a change of variables or gauge. Of course, one can denote g = f -1 and rewrite (2)-(3) as Now comes the fundamental question: Which of the two additions, + or \u2295 , is more natural or physical? Which of them is Diophantine, and which is not? Clearly, \u00b1 and \u2295 , glyph[circleminus] coexist in the same set, and it is hard to say why (2)-(3) should be regarded as less simple, or more weird, than (9)-(10). Put another way, what kind of a rule is responsible for the implicit preference of f ( x ) = x over any other one-to-one f ? The Ockham razor? Expressing it yet differently, let us assume that it is indeed the 'natural' (whatever it means) arithmetic with \u00b1 etc. that we should employ in practical computations. What kind of a rule guarantees that the laws of physics (variational principles, say) are formulated in terms of the same arithmetic and calculus, and not by means of \u2295 , glyph[circledot] , D/Dx , and the like, for some unknown f ? I believe the questions are open. It is not completely unrelated to mention that the case of a linear f was extensively studied by Benioff [11] in a somewhat different context. Here, the departure point was the observation that natural numbers N can be identified with any countable well-ordered set whose first two elements define 0 and 1. For example, the set N 2 = { 0 , 2 , 4 , . . . } of even natural numbers may be regarded as a representation of N . The value function f ( x ) = x/ 2 maps a natural number x \u2208 N 2 into its value, but to make the structure self-consistent one has to redefine the multiplication: x glyph[circledot] y = f -1 ( f ( x ) f ( y ) ) = xy/ 2. So, in Benioff's approach numbers themselves are just elements of some formal axiomatic structure, while their values are determined by appropriate linear value maps, which are simultaneously used to redefine the arithmetic operations. The idea was then extended by Benioff to real and complex numbers, and generalized in many ways, including space-time dependent fields of value functions [12, 13]. From the Burgin perspective it is essential that one can also encounter nonlinear functions f (Fechner's logarithms, Cantor-type functions for Cantor sets [1, 2], Peano-type space-filling curves in Sierpi'nski-set cases [3]...). Whenever one tries to apply a Burgin-type generalization to a physical system, one immediately encounters the problem that nontrivial f s typically require dimensionless arguments in f ( x ). Still, physical x s are dimensional. One has to associate with a physically meaningful x a dimensionless number, and we return to the problem described by Benioff. Indeed, a dimensionless x is obtained for the price of including a physical unit (of length, say), and units can be chosen arbitrarily. The choice of units effectively introduces a value map, and a change of scale changes this map. Burgin's f , when employed in a way proposed in [1], can be an arbitrary bijection, so can be composed with any value map with no loss of bijectivity of the composition. Benioff's value maps are thus intrinsically related with Burgin's f s, but are not necessarily equivalent to them. The problem of dimensional vs. dimensionless x was discussed in more detail in [2]. The goal of the present paper is to illustrate these problems on concrete examples from relativistic physics, and to contemplate the possibility of detecting a non-trivial f by means of physical observations. We will see that phenomena of dark-energy variety may suggest the presence of some f between the universe of our mathematical formulas, and the physical Universe. We will consider two spaces, R 4 + and ( -L/ 2 , L/ 2) 4 , which become Minkowski space-times when appropriate arithmetic is selected. We will then consider the problem of electromagnetic fields produced by pointlike sources, but the Maxwell field will be formulated in terms of this concrete arithmetic \u2295 , glyph[circleminus] , glyph[circledot] , glyph[circledivide] , which will make the space Minkowskian. However, when we switch back to the 'standard' arithmetic the geometry becomes locally Lorenzian, with a non-Minkowskian global structure. Static charges will appear moving toward event horizons of the space, thus creating an impression of an expanding universe, with accelerating expansion. All of this happens in spite of the fact that the spaces are static, no matter which arithmetic one works with. The paper is organized as follows. In the next section, two types of real numbers, equipped with their own arithmetic and calculus, are introduced. The two types of reals are related by a bijection f . In Section III, the two types of reals are employed in construction of two types of Minkowski spaces. Sections IV and V discuss in detail the two concrete examples of Minkowski spaces: R 4 + and ( -L/ 2 , L/ 2) 4 . Section VI discusses electromagnetic fields produced by a pointlike charge, with Maxwell equations formulated in terms of a non-Diophantine formalism defined by some f . The formalism is implicitly non-Diophantine since an observer who employs in his observations and calculations the arithmetic defined by the same f will never discover that some nontrivial f is implicitly involved. However, in case there is a missmatch, i.e. two different f s come into play, the conflict of arithmetics can have observational consequences. This is discussed on explicit examples in sections VII and VIII. Finally, in section IX, an example of a non-Diophantinecalculus Friedman equation is discussed. We observe accelerated expansion with vanishing cosmological constant.", "pages": [1, 2, 3]}, {"title": "II. 'LOWER' AND 'UPPER' REALITY", "content": "Consider real numbers R equipped with the 'standard' arithmetic operations of addition (+), subtraction ( -), multiplication ( \u00b7 ), and division ( / ). Let us term these real numbers the 'lower reals', and denote them by lowercase symbols. As usual, ' \u00b7 ' can be skipped: a \u00b7 b = ab . Neutral elements of addition and multiplication in R are denoted by 0 and 1. Now, let us assume that there exist some 'upper reals', whose set is denoted by R . By assumption, R is related to R by some bijection, f : R \u2192 R . Those upper reals are equipped with their own arithmetic and calculus. The arithmetic operations in R will be denoted by \u2295 , glyph[circleminus] , glyph[circledot] , glyph[circledivide] . We use the convention that elements of R are denoted by upper-case fonts. The arithmetic in R is non-Diophantine in the sense of Burgin [9, 10], Both arithmetics are commutative, associative, and multiplications are distributive with respect to (appropriate) additions. Neutral elements of addition and multiplication in R are denoted by 0 ' and 1 ' , which implies 0 ' = f -1 (0), 1 ' = f -1 (1). A negative in R is defined by glyph[circleminus] X = 0 ' glyph[circleminus] X = f -1 ( -f ( X ) ) . Multiplication by zero yields zero in both arithmetics, in particular 0 ' glyph[circledot] X = 0 ' . satisfy Multiplication is equivalent to repeated addition in the following sense. Let N \u2208 N and X ' = f -1 ( X ) \u2208 R . Then A power function A ( X ) = X glyph[circledot] \u00b7 \u00b7 \u00b7 glyph[circledot] X ( N times) will be denoted by X N ' , since A derivative of a function A : R \u2192 R is defined by as contrasted with the derivative of a function a : R \u2192 R , defined with respect to the lowercase arithmetic, Now let A = f -1 \u00b7 a \u00b7 f . Then, Formula (21) follows directly from the definitions of D/DX and d/dx . As stressed in the introduction, (21) is not the usual formula relating derivatives of A = f -1 \u00b7 a \u00b7 f with those of a . Indeed, so that D/DX behaves like a covariant derivative, but with a trivial connection for any bijection f : R \u2192 R . The standard approach, employed in differential geometry or gauge theories, would employ the arithmetic of R , and one would have to assume differentiability of f and f -1 . Here bijectivity is enough since no derivatives of either f or f -1 will occur in (21) and (25). This is why this type of calculus is so useful and natural in fractal theory [1-4]. Partial derivatives and multidimensional integrals are defined analogously.", "pages": [3]}, {"title": "III. LOWER AND UPPER MINKOWSKI SPACE-TIMES", "content": "Consider a point x in Minkowski space R 4 and assume where ( x 0 , x 1 , x 2 , x 3 ) \u2208 R 4 and ( X 0 , X 1 , X 2 , X 3 ) \u2208 R 4 . The two spaces are Minkowskian in the sense that the invariant quadratic forms are defined by Since the two quadratic forms are related by f , and g ab = f ( G ab ). Covariant and contravariant world-vectors are defined in the usual way, The quadratic forms are invariant with respect to Lorentz transformations, This type of Lorentz transformation, but for f representing a Cantor set, was explicitly used in [1, 2] to construct fractal homogeneous spaces. We will later need an explicit boost, where Sinh \u03c6 = f -1 ( sinh f ( \u03c6 ) ) , Cosh \u03c6 = f -1 ( cosh f ( \u03c6 ) ) satisfy The four-velocity is mapped by (38) into (1 ' , 0 ' , 0 ' , 0 ' ).", "pages": [4]}, {"title": "IV. MINKOWSKI SPACE-TIME R 4 +", "content": "Now let us make the analysis more explicit. Let the Fechner function f ( X ) = \u00b5 ln X + \u03bd , \u00b5 > 0, be the bijection f : R + \u2192 R . Accordingly, R = R + . f -1 ( x ) = e ( x -\u03bd ) /\u00b5 , and thus 0 ' = f -1 (0) = e -\u03bd/\u00b5 , 1 ' = f -1 (1) = e (1 -\u03bd ) /\u00b5 .", "pages": [4]}, {"title": "A. Arithmetic", "content": "Let us begin with the explicit form of arithmetic operations. Addition and subtraction explicitly read The arithmetic operations occurring at the right sides of (41) and (42) are those from R and not from R (the latter occur at the left sides of these formulas). For example, X \u2295 0 ' = Xe -\u03bd/\u00b5 e \u03bd/\u00b5 = X . Note that although X > 0 in f ( X ) = \u00b5 ln X + \u03bd , one nevertheless has a well defined negative number glyph[circleminus] X = 0 ' glyph[circleminus] X = e -2 \u03bd/\u00b5 /X \u2208 R = R + , which is positive from the point of view of the arithmetic of R . Let us cross-check the negativity of glyph[circleminus] X : ( R , \u2295 ) = ( R + , \u2295 ) is a group, as opposed to ( R + , +). In consequence, the Minkowski space R 4 + is invariant under the non-Diophantine Poincar'e group. The multiplication in R is explicitly given by Again the expressions at the right-hand sides of (46) and (47) involve the arithmetic of R .", "pages": [4]}, {"title": "B. Light cone", "content": "The light cone in R 4 + consists of vectors satisfying This is equivalent to f ( X 0 ) 2 = f ( X 1 ) 2 + f ( X 2 ) 2 + f ( X 3 ) 2 , i.e. Fig. 1 shows the light cone G ab X a X b = 0 ' in 1 + 2 dimensional Minkowski space R 3 + . An arbitrarily located light cone corresponds to that is For Y 0 = \u00b7 \u00b7 \u00b7 = Y 3 = 0 ' = e -\u03bd/\u00b5 we reconstruct (52). Fig. 2 shows the light cones (55) in small neighborhoods of various origins Y a . The plots suggest that a Lorentzian geometry is typical of both the standard 'lower-case' space-time R 3 , and of the 'upper-case' R 3 . Recall that the latter is also globally Minkowskian, but with respect to the non-Diophantine calculus. Interestingly, when we employ in R 3 the miss-matched formalism taken from R 3 , the formulas are locally Lorentzian. The further away from the 'walls' of R 3 + the observation is performed, the more Minkowskian the geometry appears, provided one does not observe objects that are too far away from the observer, as we shall see later. assumed to perform his analysis in the 'wrong' formalism. Then, for | glyph[epsilon1] a /Y a | glyph[lessmuch] 1, where is the Lorentzian metric. \u02dc g ab becomes just a conformally rescaled Minkowskian g ab if Y 0 = Y 1 = Y 2 = Y 3 . glyph[negationslash] The same effect occurs for general hyperboloids G ab ( X a glyph[circleminus] Y a )( X b glyph[circleminus] Y b ) = 0 ' .", "pages": [5]}, {"title": "V. MINKOWSKI SPACE-TIME ( -L/ 2 , L/ 2) 4", "content": "Let R = ( -L/ 2 , L/ 2) and f ( X ) = tan( \u03c0X/L ), f -1 ( x ) = ( L/\u03c0 ) arctan x . f : R \u2192 R is a possible bijection, with 0 ' = 0 and 1 ' = L/ 4.", "pages": [5]}, {"title": "A. Light cone", "content": "The light cone G ab ( X a glyph[circleminus] Y a )( X b glyph[circleminus] Y b ) = 0 ' corresponds to Now let X a = Y a + glyph[epsilon1] a , where | glyph[epsilon1] a /L | glyph[lessmuch] 1. The effective Lorentzian metric in a neighborhood of Y reads Another example of a bijection f : R \u2192 R is provided by f ( X ) = arctanh(2 X/L ), f -1 ( x ) = ( L/ 2) tanh x .", "pages": [6]}, {"title": "VI. FIELD PRODUCED BY A POINTLIKE CHARGE", "content": "The above two examples show that several different types of arithmetic may coexist in the same space-time. Let us try to find a phenomenon where a nontrivial f can be detected. I propose to concentrate on Maxwell equations, but formulated in terms of this form of arithmetic that makes the geometry globally Minkowskian. Consider the d'Alembertian defined with respect to non-Diophantine partial derivatives. Let R glyph[owner] S \u21a6\u2192 X a ( S ) \u2208 R 4 be a world-line of a pointlike charge, and let J a ( X ) be the current associated with the world-line. The Maxwell equations can be taken in the form The procedure of finding A a ( X ) is standard [14], but we only have to take care of appropriate definitions of nonDiophantine arithmetic and calculus. The end result is Here C is a constant, U a ( X ) is the four-velocity of the charge, and X b glyph[circleminus] Y b is future-pointing and null. We say that X a \u2208 R 4 is future-pointing if x a = f ( X a ) \u2208 R 4 is future-pointing. The summation convention is applied unless otherwise stated. The four-velocity is normalized by U a U a = 1 ' (where 1 ' = f -1 (1) is the neutral element of multiplication), which is equivalent to The four-potential is a world-vector gauge field and thus under the action of a Lorentz transformation \u039b transforms by A a ( X ) \u21a6\u2192 A ' a ( X ) = (\u039b A ) a (\u039b -1 X ), up to a gauge transformation. In standard notation Now consider a Y -independent U a of the form (40) and let \u039b a b be given by (38). To simplify further analysis let us take the point of observation at the origin X a = (0 ' , 0 ' , 0 ' , 0 ' ) \u2261 0 ' and the source at Y a = ( Y 0 , Y 1 , 0 ' , 0 ' ). Y a is null and past-pointing, which implies f ( Y 0 ) = -| f ( Y 1 ) | (notice that in our examples we assume that f is increasing and there exist various arguments why this is important for physical consistency of the formalism). Accordingly, The only non-vanishing component of the potential reads In order to continue we have to make f more concrete. Let us begin with the Minkowski space ( -L/ 2 , L/ 2) 4 . f ( X ) = tan( \u03c0X/L ) implies Let us first consider the case of | \u03c0Y 1 /L | glyph[lessmuch] 1, but with Y 1 sufficiently far from the singularity at the origin 0 ' = 0, so that the arguments of both tan and arctan are small. The field is then approximately Coulombian Notice that up to this point we have worked with linearized forms of f and f -1 since for small distances and small potentials it is enough if one replaces exact nonlinear f by its linear approximation. Let us concentrate on a charge at rest, i.e. with f ( \u03c6 ) = 0, and denote q = L 2 f ( C ) /\u03c0 2 . We conclude that a source placed in a neighborhood of the origin produces the Coulomb field whose value at the point of observation is where | Y | is the distance from the source. Now, let us increase | Y | so that tan cannot be anymore approximated by its argument. The argument of arctan is even smaller, so here the approximation is still valid. The result is So, in spite of our assumption that in the neighborhood of the origin the charge is at rest, at large distances the charge looks like moving with the four-velocity determined by certain \u03c6 ( Y 1 ), defined by The velocity \u03b2 = tanh f ( \u03c6 ( Y 1 ) ) is \u03b2 is the velocity deduced by the observer located at X a = 0 ' who analyses his data on the basis of the 'standard' Diophantine arithmetic, whereas the physical non-Diophantine arithmetic, employed in Maxwell's equations, is given by glyph[circledot] , \u2295 , etc. The observer is related with the physical Universe by means of an 'information channel' f , but is unaware of it. The miss-match of mathematical structures leads to unexpected behavior of distant objects. Had the observer decided to employ Einstein's general relativity, a similar miss-match would have occurred. A solution would produce an expanding universe whose behavior would be consistent with the Hubble law at small distances, but very distant objects would acquire an unexplained acceleration. We will demonstrate this explicitly in section IX. As our second example consider With the same approximations as before where q = L 2 f ( C ) / 4 and Y is in a neighborhood of the origin. For Y further away from the origin and with f ( \u03c6 ) = 0 we get Repeating the remaining calculations we obtain Fig. 5 compares (81) with (87). In both cases the charge looks as if it moved toward the boundaries Y 1 = \u00b1 L/ 2, where it approaches the velocity of light. The motion is non-trivially accelerated.", "pages": [6, 7, 8]}, {"title": "VIII. FIELDS IN R 4 +", "content": "It is instructive to consider explicitly also the case of R 4 + since 0 ' = e -\u03bd/\u00b5 and thus certain counterintuitive elements of a non-Diophantine arithmetic and calculus become more visible. The example will make further generalizations easier to understand. Here f ( X ) = \u00b5 ln X + \u03bd , \u00b5 > 0, f -1 ( x ) = e ( x -\u03bd ) /\u00b5 \u2248 e -\u03bd/\u00b5 (1 + x/\u00b5 ) = 0 ' + xe -\u03bd/\u00b5 /\u00b5 . Setting Y 1 = 0 ' + r , f (0 ' + r ) = \u00b5 ln[0 ' (1 + r/ 0 ' )] + \u03bd = \u00b5 ln(1 + r/ 0 ' ), and assuming | r | glyph[lessmuch] 1, we obtain We identify q = f ( C )( e -\u03bd/\u00b5 /\u00b5 ) 2 . For f ( \u03c6 ) = 0 and larger r An observer located at 0 ' will conclude that the field is produced by a charge which satisfies hence Fig. 6 shows that \u03b2 given by (94) is always negative: The charge moves toward the horizon Y 1 = 0. Fields produced by charges located between the observer and the horizon would be red-shifted. However, an observer located between the horizon and the charge would detect a blue-shifted field.", "pages": [8, 9]}, {"title": "IX. FRIEDMAN EQUATION", "content": "Taylor expansions of (81), (87) in a neighborhood of Y 1 = 0 begin with third order terms \u223c ( Y 1 /L ) 3 . The effect is small. However, when we switch to nonDiophantine generalized Einstein equations, the correction should become visible at large distances. So, let us consider the Friedman equation for a flat, matter dominated FRW model with exactly vanishing cosmological constant [15]. In matter dominated cosmology (\u2126 m = 1 , \u2126 r = 0), with no dark energy (\u2126 v = 0), the scale factor is given by a ( t ) = ( t/t 0 ) 2 / 3 . In the nonDiophantine notation the solution reads To make (96) more explicit, let us experiment with some f . For example, the choice of R = ( -L/ 2 , L/ 2) and f ( X ) = tan( \u03c0X/L ), f -1 ( x ) = ( L/\u03c0 ) arctan x leads to the scale factor depicted at Fig. 7, where (96) is compared with the standard ( t/t 0 ) 2 / 3 , for L = 20, T 0 = 1 ' , and t 0 chosen in a way guaranteeing a reasonable fit of the two plots. The curve bends up in a characteristic way, typical of dark-energy models of accelerating Universe. The effect is of purely arithmetic origin, with no need of dark energy. Arithmetic becomes as physical as geometry.", "pages": [9]}, {"title": "X. CONCLUSIONS", "content": "Gravity is geometry. Is dark energy just arithmetic? Is dark energy dark, because it is always darkest under the lantern, and we are so accustomed to 'plus' and 'times' that we overlooked the fundamental ambiguity of these operations? Is there a physical law that determines the form of arithmetic, a kind of Einstein equation for f ? The questions are relatively well posed. We can change the paradigm and do physics with unspecified f , leaving determination of f to experimentalists. Perhaps one day we will understand which arithmetic is physical, and why. und Hartel, Leipzig (1860).", "pages": [9, 10]}] |
2020arXiv201005354A | https://arxiv.org/pdf/2010.05354.pdf | <document>
<section_header_level_1><location><page_1><loc_28><loc_88><loc_72><loc_89></location>Black hole or Gravastar? The GW190521 case</section_header_level_1>
<text><location><page_1><loc_43><loc_85><loc_57><loc_86></location>Ioannis Antoniou 1, ∗</text>
<text><location><page_1><loc_24><loc_82><loc_76><loc_84></location>1 Department of Physics, University of Ioannina, GR-45110, Ioannina, Greece (Dated: December 9, 2022)</text>
<text><location><page_1><loc_17><loc_66><loc_83><loc_81></location>The existence of cosmological compact objects with very strong gravity is a prediction of General Relativity and an exact solution of the Einstein equations. These objects are called black holes and recently we had the first observations of them. However, the theory of black hole formation has some disadvantages. In order to avoid these, some scientists suggest the existence of gravastars (gravitation vacuum stars), an alternative stellar model which seems to solve the problems of the black hole theory. In this work we compare black holes and gravastars using a wide range of the literature and we emphasize the properties of gravastars, which are consistent with the current cosmological observations. Also, we propose gravastars as the solution of the 'pair-instability' effect and a possible explanation for the observed masses of the compact objects, before the collapse, from the gravitational signal GW190521, since in the formation of a gravastar there aren't mass restrictions.</text>
<text><location><page_1><loc_17><loc_64><loc_45><loc_65></location>PACS numbers: 98.62.Ai, 04.20.Cv, 04.30.-w</text>
<section_header_level_1><location><page_1><loc_23><loc_60><loc_40><loc_61></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_14><loc_38><loc_49><loc_58></location>One of the most attractive concepts in General Relativity is the existence and the properties of black holes, a region of spacetime where gravity is so strong that nothing, no particles or even electromagnetic radiation such as light, can escape from it. A black hole is characterized by the charge Q , the mass M and the total angular momentum J . The mass M is in the range above 3 M ⊙ until a few decades of the mass of Sun, or a few millions or billions M ⊙ . The latter are known as supermassive black holes and exist in the centers of most galaxies. Even if the theory of black hole formation is well known, there are some abstruse and 'strange properties' such as</text>
<unordered_list>
<list_item><location><page_1><loc_17><loc_31><loc_49><loc_37></location>· the 'event horizon', a boundary in spacetime through which matter and light can pass only inward towards the mass of the black hole.</list_item>
<list_item><location><page_1><loc_17><loc_26><loc_49><loc_31></location>· the gravitational singularity at the center of a black hole, a region where the spacetime curvature becomes infinite.</list_item>
<list_item><location><page_1><loc_17><loc_19><loc_49><loc_25></location>· the information paradox, where a pure quantum state which passes over the event horizon can evolve into a mixed state during the evaporation of the black hole.</list_item>
</unordered_list>
<text><location><page_1><loc_14><loc_16><loc_49><loc_18></location>The first picture of the structure in the center of M87 [1] is a strong evidence for the existence of</text>
<text><location><page_1><loc_51><loc_33><loc_86><loc_61></location>black holes, or some other compact object with very strong gravity. Many scientists dispute the existence of black holes because if we take into account quantum effects, the gravitational collapse of objects comes to a halt and furthermore no event horizon forms [2]. Cosmologists all around the world investigate the existence of an alternative model [3], which has not the above 'strange properties' [4]. Consequently, this model will be more attractive and realistic than black holes [5, 6]. The most successful cosmological model in this direction is called gravastar configuration, an alternative endpoint of gravitational collapse of a massive star, which do not involve horizon and proposed by Mazur and Mottola [7]. The main idea is that the gravitational collapse stops at a radius greater than the radius of the horizon [8]. Mazur and Mottola also showed that their gravastars are thermodynamically stable unlike in the case of the black hole.</text>
<section_header_level_1><location><page_1><loc_57><loc_27><loc_81><loc_29></location>II. THE STRUCTURE OF A GRAVASTAR</section_header_level_1>
<text><location><page_1><loc_51><loc_10><loc_86><loc_24></location>Thin spherical shells are boundary hypersurfaces with shell radius R , surface energy density σ and surface pressure p . Gravastar is a spherical static stable thin shell configuration and an exact solution of the equations of motion in GR when the shell is spherical and infinitesimally thin [9-11]. Usually a gravastar is divided in three regions. The interior geometry is de Sitter metric with equation of state p = -ρ (false vacuum or dark energy) and the exterior is usu-</text>
<text><location><page_2><loc_14><loc_81><loc_49><loc_89></location>ly Schwarzschild metric with equation of state p = 0 = ρ (true vacuum). The exterior metric is related to the interior metric in the context of the Israel junction conditions [9]. On the horizon there is a thin shell of matter (perfect fluid) with radius R and equation of state p = ρ .</text>
<text><location><page_2><loc_14><loc_63><loc_49><loc_80></location>The concept of gravastar is related to a Riemannian manifold which is divided by a 3dimentional timelike surface in two pieces, named exterior and interior regions and are gluing on this surface. Visser [12] introduced such an approach to develop thin-shell gravastars by the junction of both spacetimes that eliminate the event horizon and singularity. The mathematical procedure used to develop the geometry of the gravastars is based (mainly) on Israel's [9] and Mazur and Mottola works [7]. The metric has the form</text>
<formula><location><page_2><loc_14><loc_57><loc_49><loc_62></location>ds 2 ± = -f ± ( r ) dt 2 ± + dr 2 ± f ± ( r ) + r 2 ± ( dθ 2 ± +sin 2 θ ± dφ 2 ± ) (2.1)</formula>
<text><location><page_2><loc_14><loc_51><loc_49><loc_57></location>where the (+) corresponds to the exterior region, while the (-) to interior region. The f ± ( r ) function determines the interior and exterior background of the thin shell.</text>
<text><location><page_2><loc_14><loc_31><loc_49><loc_51></location>The crucial question about the concept of gravastars is under which conditions these configurations are stable. Many properties affect the answer, such as the background geometry [13], radial perturbations (using static spherical models) about the equilibrium shell's radius [12, 14, 15], the function of the innermost and exterior mass [12] and the rotation of the shell [16, 17]. In every case there are values of the corresponding parameters for which the potential V ( r ) of the shell has minimum equal to zero. Actually, there is a wide range of parameters which allow stable gravastar solutions [13, 18], which means that the gravastar is a viable cosmological structure.</text>
<text><location><page_2><loc_14><loc_10><loc_49><loc_30></location>In order to construct a stable gravastar, many authors consider a shell with different cosmological backgrounds (except the Schwarzshild-de Sitter) such as a background with cosmological constant [19] or regular spacetimes (Bardeen and Bardeen-de Sitter black holes) [15]. They investigate the stability under radial perturbations [12, 13] or under slow rotation [16, 17, 20]. Rotating gravastars can affect the inertial frames and they induce the dragging effect [21, 22]. The rotation of a massive shell induces rotation of the inertial frame which tend to be equal near the Schwarzschild radius and the results of the rotation and expanding/re-collapsing shell are exam-</text>
<text><location><page_2><loc_51><loc_86><loc_86><loc_89></location>ined for their consistency with particular interpretations of Mach's principle.</text>
<text><location><page_2><loc_51><loc_65><loc_86><loc_85></location>The detection of gravitational waves has opened a new window in the Universe and a new way to observe cosmological structures and signatures of them. Many authors have investigated the behaviour of a gravitation wave in the vicinity of a compact mass. Authors of Ref. [23] found that the dust shell causes the gravitational wave to be modified both in magnitude and phase without energy transfer, while the authors of Ref. [24] found that a compact mass such as a black hole induces modification in frequency, magnitude and energy of a wave in the vicinity of the mass. The different behaviour could be a criterion to distinguish a gravastar from a black hole.</text>
<text><location><page_2><loc_51><loc_43><loc_86><loc_64></location>Another way to discriminate a gravastar from a black hole is developing in Ref. [25]. Quasinormal frequency modes (complex numbers where the imaginary part corresponds to the loss of energy) are produced from a black hole. The authors of Ref. [25] computed polar and axial oscillation modes of gravastar. They found that the quasinormal mode spectrum is completely different from that of a black hole when both have the same gravitational mass. Also, the equation of state of the matter-shell affects the polar spectrum. In Ref. [26] the authors calculated quasinormal modes of axial parity perturbations and they found that the decay rate of a black hole and a gravastar are not the same.</text>
<text><location><page_2><loc_51><loc_19><loc_86><loc_41></location>The evolution of a star with mass above 3 M ⊙ ends with a black hole, but there is an upper mass-limit. The mass must be less than 64 M ⊙ [27]. From stellar evolution in close binaries, no black holes between 52 M ⊙ and 133 M ⊙ are expected. The effect is known as 'pair-instability' [28]. The GW190521 shows that there are compact objects with mass in the above range (confidence level 99%) and the scientific community is looking for answers [29]. The mass of the initial compact objects was 85 +21 -14 M ⊙ and 66 +171 -18 M ⊙ (90% credible intervals), which falls in the mass gap predicted by pulsational pair-instability supernova theory [30]. The mass of the remnant after the collapse is 142 +28 -16 M ⊙ which is near the above limit of the mass gap.</text>
<text><location><page_2><loc_51><loc_10><loc_86><loc_17></location>In order to calculate the total mass of a gravastar, we must divide it in three regions, the inner region, the thin shell and the outer region. In the case where the interior region has negative pressure with equation of state p = -ρ (de Sitter) the</text>
<text><location><page_3><loc_14><loc_88><loc_26><loc_89></location>inner mass is [31]</text>
<formula><location><page_3><loc_17><loc_83><loc_49><loc_87></location>m -( R ) = ∫ R 0 4 πr 2 c 0 dr = 4 3 πR 3 c 0 (2.2)</formula>
<text><location><page_3><loc_14><loc_75><loc_49><loc_82></location>where c 0 is the constant matter density throughout the interior region. The parameter c 0 includes information about the constant pressure and constant density throughout the interior region.</text>
<text><location><page_3><loc_14><loc_70><loc_49><loc_75></location>The mass of the ultrarelativistic, extremely thin stiff shell [32] of radius R , which obeys the equation of state p = ρ (the density is very high) and has width glyph[epsilon1] glyph[lessmuch] 1 is</text>
<formula><location><page_3><loc_27><loc_67><loc_49><loc_68></location>m s = 4 πR 2 σ (2.3)</formula>
<text><location><page_3><loc_14><loc_63><loc_49><loc_66></location>where σ is the surface energy density of the thin shell.</text>
<text><location><page_3><loc_14><loc_54><loc_49><loc_63></location>The exterior region is a static Schwarzschild geometry with equation of state p = ρ = 0 and the parameter m + in the metric is the total mass of the stellar structure of a gravastar. Thus, the total mass [32] is the sum of the equations (2.2) and (2.3), ie</text>
<formula><location><page_3><loc_15><loc_50><loc_49><loc_53></location>M gr ≡ m + = m -+ m s = 4 πR 2 ( c 0 R 3 + σ ) (2.4)</formula>
<text><location><page_3><loc_14><loc_40><loc_49><loc_49></location>This is the mass which is observable from the Earth for this stellar object. From eq. (2.4) we conclude that the mass M gr depends on the radius of the thin shell, the matter density of the inner region and the surface energy density of the thin shell. The result could be as big as needed, to support the cosmological observations.</text>
<figure>
<location><page_3><loc_14><loc_22><loc_49><loc_38></location>
<caption>FIG. 1. The total mass of the gravastar in the range (10 M glyph[circledot] , 150 M glyph[circledot] ) as a function of its radius, when the matter density and surface energy density are stable.</caption>
</figure>
<text><location><page_3><loc_14><loc_10><loc_49><loc_15></location>For example, if the mass of the gravastar is M = 10 3 M glyph[circledot] the corresponding Schwarzschild radius is R s = 3 × 10 3 km . We consider the radius</text>
<text><location><page_3><loc_51><loc_82><loc_86><loc_89></location>of the gravastar as R = 10 4 km and we conclude that c 0 R 3 + σ glyph[similarequal] 10 23 kg/m 2 . Thus, the order of the surface energy density of the thin shell must be ∼ 10 23 kg/m 2 and the matter density of the inner region ∼ 10 19 kg/m 3 .</text>
<text><location><page_3><loc_51><loc_75><loc_86><loc_82></location>In figure 1 we have plot the mass of the gravastar (in the pair instability region of a black hole) as a function of its radius, when the matter density is c 0 = 3 × 10 19 kg/m 3 and the surface energy density σ = 10 23 kg/m 2 .</text>
<text><location><page_3><loc_51><loc_57><loc_86><loc_74></location>It is obvious that the theory of gravastars hasn't mass restrictions, so it seems that this configuration can solve the problem of the mass gap that GW190521 suffers from. This mass gap arises from the pair instability or pulsational pair instability causing mass loss or destruction of the stellar progenitor prior to the formation of any remnant. In the process of stellar evolution there could appear gravastars of masses from GW190521. Probably, this gravitational wave signal (GW190521) is a strong indication for the existence of gravastars.</text>
<section_header_level_1><location><page_3><loc_53><loc_53><loc_85><loc_54></location>III. THE HIERARCHICAL SCENARIO</section_header_level_1>
<text><location><page_3><loc_51><loc_35><loc_86><loc_50></location>GW190521 is the most massive gravitational wave event observed to date. It was reported as a binary black hole merger, in which the inferred masses of the black holes in the binary place them (and the merger remnant) in the pair instability mass gap. A hierarchical scenario [33, 34] for the formation of the binary suffices for reconciling the observed masses with the pair instability mass gap, although it raises interesting questions about the environment in which such mergers would happen.</text>
<text><location><page_3><loc_51><loc_22><loc_86><loc_34></location>Our purpose in this work is to suggest an alternative and possibly more realistic scenario than the hierarchical one, for the origin of the GW190521 and not to review the literature about possible explanations. In Ref. [33] and references therein, one can find a wide review. In this review [33], especially in subsection 4.4, for the interpretation of the GW190521 through the hierarchic scenario are required 2 conditions</text>
<unordered_list>
<list_item><location><page_3><loc_54><loc_16><loc_86><loc_20></location>· 4 black holes in the same region. From that we know until now, we haven't any observation or indication for this event.</list_item>
<list_item><location><page_3><loc_54><loc_10><loc_86><loc_15></location>· a special scenario for the collapse. In the first generation the black holes have merged two by two and in the second generation the</list_item>
</unordered_list>
<text><location><page_4><loc_18><loc_86><loc_49><loc_89></location>new black holes have merged again. This scenario is called 2g+2g [34] (higher-g)!!</text>
<text><location><page_4><loc_14><loc_76><loc_49><loc_83></location>It is obvious that these conditions restrict the possibility of this scenario and we think that is less possible than our proposal (one way of possibly circumventing the limitations from the mass gap is by having a gravastar merger).</text>
<text><location><page_4><loc_14><loc_56><loc_49><loc_75></location>In Ref. [33], the authors review several alternative explanations for the occurrence of GW190521 such as population III stars at very low-metallicity, accretion onto either stellarorigin or primordial BHs, and stellar mergers. Another speculations include exotic compact objects (beyond ΛCDM model) and dark-matter annihilation. Also, there is possible that the primary and secondary components of GW190521 are not inside the pair instability gap respectively, but above and below this gap. From all these, it is obvious that the GW190521 is an open issue in cosmology.</text>
<text><location><page_4><loc_14><loc_42><loc_49><loc_55></location>In the literature [35, 36], there are previous works which connect the gravitational waves (mainly the first signal GW150914) with the existence of gravastars. The mass of the origin of these signals is outside the range of the mass gap (pair instability). Also, LIGO's observations of gravitational waves from colliding objects have been found to be indistinguishable from ordinary black holes [36].</text>
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<section_header_level_1><location><page_4><loc_60><loc_84><loc_77><loc_85></location>IV. CONCLUSIONS</section_header_level_1>
<text><location><page_4><loc_51><loc_54><loc_86><loc_81></location>Gravastar is a suggested (hypothetical) stellar compact object alternative to a black hole. It is a kind of spherical shells, which is a stable configuration under rotation and radial perturbations [37]. Until now, there is no observational way to distinguish a black hole from a gravastar. Thus, a possible observation of a black hole can be a possible gravastar. Recent observations of gravitational wave GW190521 pointed out that the theory of black holes formation is not consistent with some observations, since there is a mass gap in their formation. From theoretical view, there is a way to avoid the pair instability effect/mass gap with gravastar formation. The GW190521 event could be an indication for the existence of gravastars, or spherical shells which might avoid the mass gap. This stellar configuration (gravastar) is a promising candidate to explain the observations and to complete the puzzle.</text>
<section_header_level_1><location><page_4><loc_57><loc_53><loc_81><loc_54></location>V. ACKNOWLEDGEMENTS</section_header_level_1>
<text><location><page_4><loc_51><loc_42><loc_86><loc_51></location>I would like to thank Professors Leandros Perivolaropoulos (University of Ioannina) and Demetrios Papadopoulos (Aristotle University) for useful discussions and indications about the gravastars, which improved the quality of this work.</text>
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<list_item><location><page_5><loc_51><loc_16><loc_86><loc_23></location>[37] I. Antoniou, D. Kazanas, D. Papadopoulos, and L. Perivolaropoulos, 'Stabilizing spherical energy shells with angular momentum in gravitational backgrounds,' Int. J. Mod. Phys. D 31 , 2250064 (2022), arXiv:2204.14003 [gr-qc].</list_item>
</document> | [{"title": "Black hole or Gravastar? The GW190521 case", "content": "Ioannis Antoniou 1, \u2217 1 Department of Physics, University of Ioannina, GR-45110, Ioannina, Greece (Dated: December 9, 2022) The existence of cosmological compact objects with very strong gravity is a prediction of General Relativity and an exact solution of the Einstein equations. These objects are called black holes and recently we had the first observations of them. However, the theory of black hole formation has some disadvantages. In order to avoid these, some scientists suggest the existence of gravastars (gravitation vacuum stars), an alternative stellar model which seems to solve the problems of the black hole theory. In this work we compare black holes and gravastars using a wide range of the literature and we emphasize the properties of gravastars, which are consistent with the current cosmological observations. Also, we propose gravastars as the solution of the 'pair-instability' effect and a possible explanation for the observed masses of the compact objects, before the collapse, from the gravitational signal GW190521, since in the formation of a gravastar there aren't mass restrictions. PACS numbers: 98.62.Ai, 04.20.Cv, 04.30.-w", "pages": [1]}, {"title": "I. INTRODUCTION", "content": "One of the most attractive concepts in General Relativity is the existence and the properties of black holes, a region of spacetime where gravity is so strong that nothing, no particles or even electromagnetic radiation such as light, can escape from it. A black hole is characterized by the charge Q , the mass M and the total angular momentum J . The mass M is in the range above 3 M \u2299 until a few decades of the mass of Sun, or a few millions or billions M \u2299 . The latter are known as supermassive black holes and exist in the centers of most galaxies. Even if the theory of black hole formation is well known, there are some abstruse and 'strange properties' such as The first picture of the structure in the center of M87 [1] is a strong evidence for the existence of black holes, or some other compact object with very strong gravity. Many scientists dispute the existence of black holes because if we take into account quantum effects, the gravitational collapse of objects comes to a halt and furthermore no event horizon forms [2]. Cosmologists all around the world investigate the existence of an alternative model [3], which has not the above 'strange properties' [4]. Consequently, this model will be more attractive and realistic than black holes [5, 6]. The most successful cosmological model in this direction is called gravastar configuration, an alternative endpoint of gravitational collapse of a massive star, which do not involve horizon and proposed by Mazur and Mottola [7]. The main idea is that the gravitational collapse stops at a radius greater than the radius of the horizon [8]. Mazur and Mottola also showed that their gravastars are thermodynamically stable unlike in the case of the black hole.", "pages": [1]}, {"title": "II. THE STRUCTURE OF A GRAVASTAR", "content": "Thin spherical shells are boundary hypersurfaces with shell radius R , surface energy density \u03c3 and surface pressure p . Gravastar is a spherical static stable thin shell configuration and an exact solution of the equations of motion in GR when the shell is spherical and infinitesimally thin [9-11]. Usually a gravastar is divided in three regions. The interior geometry is de Sitter metric with equation of state p = -\u03c1 (false vacuum or dark energy) and the exterior is usu- ly Schwarzschild metric with equation of state p = 0 = \u03c1 (true vacuum). The exterior metric is related to the interior metric in the context of the Israel junction conditions [9]. On the horizon there is a thin shell of matter (perfect fluid) with radius R and equation of state p = \u03c1 . The concept of gravastar is related to a Riemannian manifold which is divided by a 3dimentional timelike surface in two pieces, named exterior and interior regions and are gluing on this surface. Visser [12] introduced such an approach to develop thin-shell gravastars by the junction of both spacetimes that eliminate the event horizon and singularity. The mathematical procedure used to develop the geometry of the gravastars is based (mainly) on Israel's [9] and Mazur and Mottola works [7]. The metric has the form where the (+) corresponds to the exterior region, while the (-) to interior region. The f \u00b1 ( r ) function determines the interior and exterior background of the thin shell. The crucial question about the concept of gravastars is under which conditions these configurations are stable. Many properties affect the answer, such as the background geometry [13], radial perturbations (using static spherical models) about the equilibrium shell's radius [12, 14, 15], the function of the innermost and exterior mass [12] and the rotation of the shell [16, 17]. In every case there are values of the corresponding parameters for which the potential V ( r ) of the shell has minimum equal to zero. Actually, there is a wide range of parameters which allow stable gravastar solutions [13, 18], which means that the gravastar is a viable cosmological structure. In order to construct a stable gravastar, many authors consider a shell with different cosmological backgrounds (except the Schwarzshild-de Sitter) such as a background with cosmological constant [19] or regular spacetimes (Bardeen and Bardeen-de Sitter black holes) [15]. They investigate the stability under radial perturbations [12, 13] or under slow rotation [16, 17, 20]. Rotating gravastars can affect the inertial frames and they induce the dragging effect [21, 22]. The rotation of a massive shell induces rotation of the inertial frame which tend to be equal near the Schwarzschild radius and the results of the rotation and expanding/re-collapsing shell are exam- ined for their consistency with particular interpretations of Mach's principle. The detection of gravitational waves has opened a new window in the Universe and a new way to observe cosmological structures and signatures of them. Many authors have investigated the behaviour of a gravitation wave in the vicinity of a compact mass. Authors of Ref. [23] found that the dust shell causes the gravitational wave to be modified both in magnitude and phase without energy transfer, while the authors of Ref. [24] found that a compact mass such as a black hole induces modification in frequency, magnitude and energy of a wave in the vicinity of the mass. The different behaviour could be a criterion to distinguish a gravastar from a black hole. Another way to discriminate a gravastar from a black hole is developing in Ref. [25]. Quasinormal frequency modes (complex numbers where the imaginary part corresponds to the loss of energy) are produced from a black hole. The authors of Ref. [25] computed polar and axial oscillation modes of gravastar. They found that the quasinormal mode spectrum is completely different from that of a black hole when both have the same gravitational mass. Also, the equation of state of the matter-shell affects the polar spectrum. In Ref. [26] the authors calculated quasinormal modes of axial parity perturbations and they found that the decay rate of a black hole and a gravastar are not the same. The evolution of a star with mass above 3 M \u2299 ends with a black hole, but there is an upper mass-limit. The mass must be less than 64 M \u2299 [27]. From stellar evolution in close binaries, no black holes between 52 M \u2299 and 133 M \u2299 are expected. The effect is known as 'pair-instability' [28]. The GW190521 shows that there are compact objects with mass in the above range (confidence level 99%) and the scientific community is looking for answers [29]. The mass of the initial compact objects was 85 +21 -14 M \u2299 and 66 +171 -18 M \u2299 (90% credible intervals), which falls in the mass gap predicted by pulsational pair-instability supernova theory [30]. The mass of the remnant after the collapse is 142 +28 -16 M \u2299 which is near the above limit of the mass gap. In order to calculate the total mass of a gravastar, we must divide it in three regions, the inner region, the thin shell and the outer region. In the case where the interior region has negative pressure with equation of state p = -\u03c1 (de Sitter) the inner mass is [31] where c 0 is the constant matter density throughout the interior region. The parameter c 0 includes information about the constant pressure and constant density throughout the interior region. The mass of the ultrarelativistic, extremely thin stiff shell [32] of radius R , which obeys the equation of state p = \u03c1 (the density is very high) and has width glyph[epsilon1] glyph[lessmuch] 1 is where \u03c3 is the surface energy density of the thin shell. The exterior region is a static Schwarzschild geometry with equation of state p = \u03c1 = 0 and the parameter m + in the metric is the total mass of the stellar structure of a gravastar. Thus, the total mass [32] is the sum of the equations (2.2) and (2.3), ie This is the mass which is observable from the Earth for this stellar object. From eq. (2.4) we conclude that the mass M gr depends on the radius of the thin shell, the matter density of the inner region and the surface energy density of the thin shell. The result could be as big as needed, to support the cosmological observations. For example, if the mass of the gravastar is M = 10 3 M glyph[circledot] the corresponding Schwarzschild radius is R s = 3 \u00d7 10 3 km . We consider the radius of the gravastar as R = 10 4 km and we conclude that c 0 R 3 + \u03c3 glyph[similarequal] 10 23 kg/m 2 . Thus, the order of the surface energy density of the thin shell must be \u223c 10 23 kg/m 2 and the matter density of the inner region \u223c 10 19 kg/m 3 . In figure 1 we have plot the mass of the gravastar (in the pair instability region of a black hole) as a function of its radius, when the matter density is c 0 = 3 \u00d7 10 19 kg/m 3 and the surface energy density \u03c3 = 10 23 kg/m 2 . It is obvious that the theory of gravastars hasn't mass restrictions, so it seems that this configuration can solve the problem of the mass gap that GW190521 suffers from. This mass gap arises from the pair instability or pulsational pair instability causing mass loss or destruction of the stellar progenitor prior to the formation of any remnant. In the process of stellar evolution there could appear gravastars of masses from GW190521. Probably, this gravitational wave signal (GW190521) is a strong indication for the existence of gravastars.", "pages": [1, 2, 3]}, {"title": "III. THE HIERARCHICAL SCENARIO", "content": "GW190521 is the most massive gravitational wave event observed to date. It was reported as a binary black hole merger, in which the inferred masses of the black holes in the binary place them (and the merger remnant) in the pair instability mass gap. A hierarchical scenario [33, 34] for the formation of the binary suffices for reconciling the observed masses with the pair instability mass gap, although it raises interesting questions about the environment in which such mergers would happen. Our purpose in this work is to suggest an alternative and possibly more realistic scenario than the hierarchical one, for the origin of the GW190521 and not to review the literature about possible explanations. In Ref. [33] and references therein, one can find a wide review. In this review [33], especially in subsection 4.4, for the interpretation of the GW190521 through the hierarchic scenario are required 2 conditions new black holes have merged again. This scenario is called 2g+2g [34] (higher-g)!! It is obvious that these conditions restrict the possibility of this scenario and we think that is less possible than our proposal (one way of possibly circumventing the limitations from the mass gap is by having a gravastar merger). In Ref. [33], the authors review several alternative explanations for the occurrence of GW190521 such as population III stars at very low-metallicity, accretion onto either stellarorigin or primordial BHs, and stellar mergers. Another speculations include exotic compact objects (beyond \u039bCDM model) and dark-matter annihilation. Also, there is possible that the primary and secondary components of GW190521 are not inside the pair instability gap respectively, but above and below this gap. From all these, it is obvious that the GW190521 is an open issue in cosmology. In the literature [35, 36], there are previous works which connect the gravitational waves (mainly the first signal GW150914) with the existence of gravastars. The mass of the origin of these signals is outside the range of the mass gap (pair instability). Also, LIGO's observations of gravitational waves from colliding objects have been found to be indistinguishable from ordinary black holes [36].", "pages": [3, 4]}, {"title": "IV. CONCLUSIONS", "content": "Gravastar is a suggested (hypothetical) stellar compact object alternative to a black hole. It is a kind of spherical shells, which is a stable configuration under rotation and radial perturbations [37]. Until now, there is no observational way to distinguish a black hole from a gravastar. Thus, a possible observation of a black hole can be a possible gravastar. Recent observations of gravitational wave GW190521 pointed out that the theory of black holes formation is not consistent with some observations, since there is a mass gap in their formation. From theoretical view, there is a way to avoid the pair instability effect/mass gap with gravastar formation. The GW190521 event could be an indication for the existence of gravastars, or spherical shells which might avoid the mass gap. This stellar configuration (gravastar) is a promising candidate to explain the observations and to complete the puzzle.", "pages": [4]}, {"title": "V. ACKNOWLEDGEMENTS", "content": "I would like to thank Professors Leandros Perivolaropoulos (University of Ioannina) and Demetrios Papadopoulos (Aristotle University) for useful discussions and indications about the gravastars, which improved the quality of this work.", "pages": [4]}] |
2018arXiv181002484H | https://arxiv.org/pdf/1810.02484.pdf | <document>
<text><location><page_1><loc_74><loc_87><loc_85><loc_88></location>SNSN-323-63</text>
<text><location><page_1><loc_74><loc_85><loc_88><loc_86></location>October 8, 2018</text>
<section_header_level_1><location><page_1><loc_21><loc_70><loc_86><loc_74></location>The Anisotropies and Origins of Ultrahigh Energy Cosmic Rays</section_header_level_1>
<section_header_level_1><location><page_1><loc_46><loc_57><loc_61><loc_57></location>Francis Halzen</section_header_level_1>
<text><location><page_1><loc_33><loc_51><loc_74><loc_54></location>Wisconsin IceCube Particle Astrophysics Center Madison, WI USA</text>
<text><location><page_1><loc_23><loc_32><loc_84><loc_39></location>After updating the status of the measurements of the cosmic neutrino flux by the IceCube experiment, we summarize the observations of the first identified source of cosmic rays and speculate on the connection between the two observations.</text>
<section_header_level_1><location><page_2><loc_18><loc_86><loc_60><loc_88></location>1 Detecting cosmic neutrinos</section_header_level_1>
<text><location><page_2><loc_18><loc_50><loc_89><loc_84></location>Cosmic rays have been studied for more than a century. They reach energies in excess of 10 8 TeV, populating an extreme universe that is opaque to electromagnetic radiation [1, 2]. We don't yet know where or how particles are accelerated to these extreme energies, but with the observation of a distant blazar in coincidence with the direction and time of a very high energy muon neutrino, neutrino astronomy has made a breakthrough in resolving this puzzle [3, 4]. The rationale for finding cosmic ray sources by observing neutrinos is straightforward: near neutron stars and black holes, gravitational energy released in the accretion of matter can power the acceleration of protons ( p ) or heavier nuclei that subsequently interact with gas (' pp ') or ambient radiation (' pγ ') to produce neutrinos originating from the decay of pions and other secondary particles. In the case of photoproduction, both neutral and charged pion secondaries are produced in the processes p + γ bg → p + π 0 and p + γ bg → n + π + , for instance. While neutral pions decay as π 0 → γ + γ and create a flux of high-energy gamma rays, the charged pions decay into three high-energy neutrinos ( ν ) and anti-neutrinos ( ν ) via the decay chain π + → µ + + ν µ followed by µ + → e + + ν µ + ν e , and the charged-conjugate process. We refer to these photons as pionic photons. They provide the rational for multimessenger astronomy and should be distinguished from photons radiated by electrons that may be accelerated along with the cosmic rays.</text>
<text><location><page_2><loc_18><loc_34><loc_89><loc_50></location>High-energy neutrinos interact predominantly with matter via deep inelastic scattering off nucleons: the neutrino scatters off quarks in the target nucleus by the exchange of a Z or W weak boson, referred to as neutral current and charged current interactions, respectively. Whereas the neutral current interaction leaves the neutrino state intact, in a charged current interaction a charged lepton is produced that shares the initial neutrino flavor. The average relative energy fraction transferred from the neutrino to the lepton is at the level of 80% at high energies. The struck nucleus does not remain intact and its high-energy fragments typically initiate hadronic showers in the target medium.</text>
<text><location><page_2><loc_18><loc_15><loc_89><loc_33></location>Immense particle detectors are required to collect cosmic neutrinos in statistically significant numbers. Already by the 1970s, it had been understood [5] that a kilometerscale detector was needed to observe the cosmogenic neutrinos produced in the interactions of cosmic rays with background microwave photons [6]. The IceCube project has transformed one cubic kilometer of natural Antarctic ice into a Cherenkov detector. Photomultipliers embedded in the ice transform the Cherenkov light radiated by secondary particles produced in neutrino interactions into electrical signals using the photoelectric effect; see Figs. 1 and 2. Computers at the surface use this information to reconstruct the light patterns produced to infer the arrival directions, energies and flavor of the neutrinos.</text>
<text><location><page_2><loc_21><loc_14><loc_89><loc_15></location>Two patterns of Cherenkov radiation are of special interest, 'tracks' and 'cascades.'</text>
<figure>
<location><page_3><loc_19><loc_58><loc_88><loc_87></location>
<caption>Figure 1: The principal idea of neutrino telescopes from the point of view of IceCube located at the South Pole. Neutrinos dominantly interact with a nucleus in a transparent medium like water or ice and produce a muon that is detected by the wake of Cherenkov photons it leaves inside the detector. The background of high-energy muons (solid blue arrows) produced in the atmosphere can be reduced by placing the detector underground. The surviving fraction of muons is further reduced by looking for upgoing muon tracks that originate from muon neutrinos (dashed blue arrows) interacting close to the detector. This still leaves the contribution of muons generated by atmospheric muon neutrino interactions. This contribution can be separated from the diffuse cosmic neutrino emission by an analysis of the combined neutrino spectrum.</caption>
</figure>
<text><location><page_3><loc_18><loc_14><loc_89><loc_35></location>The term 'track' refers to the Cherenkov emission of a long-lived muon passing through the detector after production in a charged current interaction of a muon neutrino inside or in the vicinity of the detector. Because of the large background of muons produced by cosmic ray interactions in the atmosphere, the observation of muon neutrinos is limited to upgoing muon tracks that are produced by neutrinos that have passed through the Earth that acts as a neutrino filter. The remaining background consists of atmospheric neutrinos, which are indistinguishable from cosmic neutrinos on an event-by-event basis. However, the steeply falling spectrum ( ∝ E -3 . 7 ) of atmospheric neutrinos allows identifying diffuse astrophysical neutrino emission above a few hundred TeV by a spectral analysis. The atmospheric background is also reduced for muon neutrino observation from point-like sources, in particular transient neutrino sources.</text>
<figure>
<location><page_4><loc_20><loc_59><loc_87><loc_86></location>
<caption>Figure 2: Architecture of the IceCube observatory (left) and the schematics of a digital optical module (right) (see Ref. [7] for details).</caption>
</figure>
<text><location><page_4><loc_18><loc_34><loc_89><loc_50></location>Energetic electrons and taus produced in interactions of electron and tau neutrinos, respectively, will initiate an electromagnetic shower that develops over less than 10 meters. This shower as well as the hadronic particle shower generated by the target struck by a neutrino in the ice radiate Cherenkov photons. The light pattern is mostly spherical and referred to as a 'cascade.' The direction of the initial neutrino can only be reconstructed from the Cherenkov emission of secondary particles produced close to the neutrino interaction point, and the angular resolution is worse than for track events. On the other hand, the energy of the initial neutrino can be constructed with a better resolution than for tracks.</text>
<text><location><page_4><loc_18><loc_14><loc_89><loc_34></location>Two methods are used to identify cosmic neutrinos. Traditionally, neutrino searches have focused on the observation of muon neutrinos that interact primarily outside the detector to produce kilometer-long muon tracks passing through the instrumented volume. Although this allows the identification of neutrinos that interact outside the detector, it is necessary to use the Earth as a filter in order to remove the background of cosmic-ray muons. This limits the neutrino view to a single flavor and half the sky. An alternative method exclusively identifies high-energy neutrinos interacting inside the detector, so-called high-energy starting events (HESE). It divides the instrumented volume of ice into an outer veto shield and a ∼ 420-megaton inner fiducial volume. The advantage of focusing on neutrinos interacting inside the instrumented volume of ice is that the detector functions as a total absorption calorimeter, measuring the</text>
<figure>
<location><page_5><loc_22><loc_70><loc_52><loc_88></location>
</figure>
<figure>
<location><page_5><loc_54><loc_70><loc_85><loc_88></location>
<caption>Figure 3: Left Panel: Light pool produced in IceCube by a shower initiated by an electron or tau neutrino. The measured energy is 1 . 14 PeV, which represents a lower limit on the energy of the neutrino that initiated the shower. White dots represent sensors with no signal. For the colored dots, the color indicates arrival time, from red (early) to purple (late) following the rainbow, and size reflects the number of photons detected. Right Panel: An upgoing muon track traverses the detector at an angle of 11 · below the horizon. The deposited energy, i.e., the energy equivalent of the total Cherenkov light of all charged secondary particles inside the detector, is 2.6 PeV.</caption>
</figure>
<text><location><page_5><loc_18><loc_39><loc_89><loc_52></location>neutrino energy of cascades with a 10-15 % resolution [8]. Furthermore, with this method, neutrinos from all directions in the sky can be identified, including both muon tracks as well as secondary showers, produced by charged-current interactions of electron and tau neutrinos, and neutral current interactions of neutrinos of all flavors. For illustration, the Cherenkov patterns initiated by an electron (or tau) neutrino of about 1 PeV energy and a muon neutrino losing 2.6 PeV energy in the form of Cherenkov photons while traversing the detector are contrasted in Fig. 3.</text>
<text><location><page_5><loc_18><loc_14><loc_89><loc_39></location>In general, the arrival times of photons at the optical sensors, whose positions are known, determine the particle's trajectory, while the number of photons is a proxy for the deposited energy, i.e., the energy of all charged secondary particles from the interactions. For instance, for the cascade event shown in the left panel of Fig. 3, more than 300 digital optical modules (DOMs) report a total of more than 100,000 photoelectrons. The two abovementioned methods of separating neutrinos from the cosmic-ray muon background have complementary advantages. The long tracks produced by muon neutrinos can be pointed back to their sources with a ≤ 0 . 4 · angular resolution. In contrast, the reconstruction of the direction of cascades in the HESE analysis, in principle possible to a few degrees, is still in the development stage in IceCube [8]. Their reconstruction, originally limited to within 10 · ∼ 15 · of the direction of the incident neutrino, has now achieved resolutions closer to 5 · [9]. Determining the deposited energy from the observed light pool is, however, relatively straightforward, and a resolution of better than 15 % is possible; the same value holds</text>
<text><location><page_6><loc_18><loc_86><loc_87><loc_88></location>for the reconstruction of the energy deposited by a muon track inside the detector.</text>
<figure>
<location><page_6><loc_28><loc_47><loc_78><loc_85></location>
<caption>Figure 4: Summary of diffuse neutrino observations (per flavor) by IceCube. The black and grey data show IceCube's measurement of the atmospheric ν e + ν e [10, 11] and ν µ + ν µ [12] spectra. The magenta line and magenta-shaded area indicate the best-fit and 1 σ uncertainty range of a power-law fit to the six-year HESE data. Note that the HESE analysis vetoes atmospheric neutrinos and can probe astrophysical neutrinos below the atmospheric neutrino flux. The corresponding fit to the eight-year ν µ + ν µ analysis is shown in red. Figure from Ref. [13].</caption>
</figure>
<text><location><page_6><loc_18><loc_13><loc_89><loc_31></location>Using the Earth as a filter, a flux of neutrinos has been identified that is predominantly of atmospheric origin. IceCube has measured this flux over three orders of magnitude in energy with a result that is consistent with theoretical calculations. However, with eight years of data, an excess of events is observed at energies beyond 100TeV [14, 15, 16], which cannot be accommodated by the atmospheric flux; see Fig. 4. Allowing for large uncertainties on the extrapolation of the atmospheric component to higher energy, the statistical significance of the excess astrophysical flux is 6 . 7 σ . While IceCube measures only the energy of the secondary muon inside the detector, from Standard Model physics we can infer the energy spectrum of the parent neutrinos. The cosmic neutrino flux is well described by a power law with a</text>
<text><location><page_7><loc_18><loc_83><loc_89><loc_88></location>spectral index Γ = 2 . 19 ± 0 . 10 and a normalization at 100 TeV neutrino energy of (1 . 01 +0 . 26 -0 . 23 ) × 10 -18 GeV -1 cm -2 sr -1 [16]. The neutrino energies contributing to this power-law fit cover the range from 119 TeV to 4.8 PeV.</text>
<text><location><page_7><loc_18><loc_46><loc_89><loc_82></location>Using only two years of data, it was the alternative HESE method, which selects neutrinos interacting inside the detector, that revealed the first evidence for cosmic neutrinos [17, 18]. The segmentation of the detector into a surrounding veto and active signal region has been optimized to reduce the background of atmospheric muons and neutrinos to a handful of events per year, while keeping most of the cosmic signal. Neutrinos of atmospheric and cosmic origin can be separated not only by using their well-measured energy but also on the basis that background atmospheric neutrinos reaching us from the Southern Hemisphere can be removed because they are accompanied by particles produced in the same air shower where the neutrinos originate. A sample event with a light pool of roughly one hundred thousand photoelectrons extending over more than 500 meters is shown in the left panel of Fig. 3. With PeV energy, and no trace of accompanying muons from an atmospheric shower, these events are highly unlikely to be of atmospheric origin. The six-year data set contains a total of 82 neutrino events with deposited energies ranging from 60 TeV to 10 PeV. The data are consistent with an astrophysical component with a spectrum close to E -2 above an energy of ∼ 200TeV. In summary, IceCube has observed cosmic neutrinos using both methods for rejecting background. Based on different methods for reconstruction and energy measurement, their results agree, pointing at extragalactic sources whose flux has equilibrated in the three flavors after propagation over cosmic distances [19] with ν e : ν µ : ν τ ∼ 1 : 1 : 1.</text>
<text><location><page_7><loc_18><loc_28><loc_89><loc_46></location>An extrapolation of this high-energy flux to lower energy suggests an interesting excess of events in the 30 -100 TeV energy range over and above a single power-law fit; see Fig. 5. This conclusion is supported by a subsequent analysis that has lowered the threshold of the starting-event analysis [20] and by a variety of other analyses. The astrophysical flux measured by IceCube is not featureless; either the spectrum of cosmic accelerators cannot be described by a single power law or a second component of cosmic neutrino sources emerges in the spectrum. Because of the self-veto of atmospheric neutrinos in the HESE analysis, i.e., the veto triggered by accompanying atmospheric muons, it is very difficult to accommodate the component below 100 TeV as a feature in the atmospheric background.</text>
<text><location><page_7><loc_18><loc_13><loc_89><loc_27></location>In Figure 5 we show the arrival directions of the most energetic events in the eight-year upgoing ν µ + ν µ analysis ( glyph[circledot] ) and the six-year HESE data sets. The HESE data are separated into tracks ( ⊗ ) and cascades ( ⊕ ). The median angular resolution of the cascade events is indicated by thin circles around the best-fit position. The most energetic muons with energy E µ > 200 TeV in the upgoing ν µ + ν µ data set accumulate near the horizon in the Northern Hemisphere. Elsewhere, muon neutrinos are increasingly absorbed in the Earth causing the apparent anisotropy of the events in the Northern Hemisphere. Also HESE events with deposited energy of E dep > 100 TeV</text>
<figure>
<location><page_8><loc_20><loc_62><loc_87><loc_87></location>
<caption>Figure 5: Mollweide projection in Galactic coordinates of the arrival direction of neutrino events. We show the results of the eight-year upgoing track analysis [16] with reconstructed muon energy E µ glyph[greaterorsimilar] 200 TeV ( glyph[circledot] ). The events of the six-year high-energy starting event (HESE) analysis with deposited energy larger than 100 TeV (tracks ⊗ and cascades ⊕ ) are also shown [21, 22, 16]. The thin circles indicate the median angular resolution of the cascade events ( ⊕ ). The blue-shaded region indicates the zenith-dependent range where Earth absorption of 100 TeV neutrinos becomes important, reaching more than 90% close to the nadir. The dashed line indicates the horizon and the star ( glyph[star] ) the Galactic Center. We highlight the four most energetic events in both analyses by their deposited energy (magenta numbers) and reconstructed muon energy (red number). Figure from Ref. [13].</caption>
</figure>
<text><location><page_8><loc_18><loc_28><loc_89><loc_37></location>suffer from absorption in the Earth and are therefore mostly detected when originating in the Southern Hemisphere. After correcting for absorption, the arrival directions of cosmic neutrinos are isotropic, suggesting extragalactic sources. In fact, no correlation of the arrival directions of the highest energy events, shown in Fig. 5, with potential sources or source classes has reached the level of 3 σ [20].</text>
<section_header_level_1><location><page_8><loc_18><loc_23><loc_75><loc_25></location>2 IceCube neutrinos and Fermi photons</section_header_level_1>
<text><location><page_8><loc_18><loc_13><loc_89><loc_21></location>The most important message emerging from the IceCube measurements of the highenergy cosmic neutrino flux is not apparent yet: the prominent and surprisingly important role of protons relative to electrons in the nonthermal universe. Photons are produced in association with neutrinos when accelerated cosmic rays produce neutral and charged pions in interactions with target photons or nuclei in the vicinity</text>
<text><location><page_9><loc_18><loc_68><loc_89><loc_88></location>of the accelerator. Targets include strong radiation fields that may be associated with the accelerator as well as concentrations of matter, such as molecular clouds in their vicinity. Additionally, pions can be produced in the interaction of cosmic rays with the extragalactic background light (EBL) when propagating through the interstellar or intergalactic background. As already discussed in section 1, a high-energy flux of neutrinos is produced in the subsequent decay of charged pions via π + → µ + + ν µ followed by µ + → e + + ν e + ν µ and the charge-conjugate processes. High-energy gamma rays result from the decay of neutral pions, π 0 → γ + γ . Pionic gamma rays and neutrinos carry, on average, 1/2 and 1/4 of the energy of the parent pion, respectively. With these approximations, the neutrino production rate Q ν α (units of GeV -1 s -1 ) can be related to the one for charged pions as</text>
<formula><location><page_9><loc_37><loc_63><loc_89><loc_66></location>∑ α E ν Q ν α ( E ν ) glyph[similarequal] 3 [ E π Q π ± ( E π )] E π glyph[similarequal] 4 E ν . (1)</formula>
<text><location><page_9><loc_18><loc_58><loc_89><loc_61></location>Similarly, the production rate of pionic gamma-rays is related to the one for neutral pions as</text>
<formula><location><page_9><loc_39><loc_56><loc_89><loc_58></location>E γ Q γ ( E γ ) glyph[similarequal] 2 [ E π Q π 0 ( E π )] E π glyph[similarequal] 2 E γ . (2)</formula>
<text><location><page_9><loc_18><loc_35><loc_89><loc_55></location>Note, that the relative production rates of pionic gamma rays and neutrinos only depend on the ratio of charged-to-neutral pions produced in cosmic-ray interactions, denoted by K π = N π ± /N π 0 . Pion production by cosmic rays in interactions with photons can proceed resonantly in the processes p + γ → ∆ + → π 0 + p and p + γ → ∆ + → π + + n . These channels produce charged and neutral pions with probabilities 2/3 and 1/3, respectively. However, the additional contribution of nonresonant pion production changes this ratio to approximately 1/2 and 1/2. In contrast, cosmic rays interacting with matter, e.g., hydrogen in the Galactic disk, produce equal numbers of pions of all three charges: p + p → N π [ π 0 + π + + π -] + X , where N π is the pion multiplicity. From above arguments we have K π glyph[similarequal] 2 for cosmic ray interactions with gas ( pp ) and K π glyph[similarequal] 1 for interactions with photons ( pγ ) [13].</text>
<text><location><page_9><loc_18><loc_32><loc_89><loc_35></location>With this approximation we can combine Eqs. (1) and (2) to derive a powerful relation between the pionic gamma-ray and neutrino production rates:</text>
<formula><location><page_9><loc_35><loc_26><loc_89><loc_30></location>1 3 ∑ α E 2 ν Q ν α ( E ν ) glyph[similarequal] K π 4 [ E 2 γ Q γ ( E γ ) ] E γ =2 E ν . (3)</formula>
<text><location><page_9><loc_18><loc_21><loc_89><loc_24></location>The prefactor 1 / 4 accounts for the energy ratio 〈 E ν 〉 / 〈 E γ 〉 glyph[similarequal] 1 / 2 and the two gamma rays produced in the neutral pion decay.</text>
<text><location><page_9><loc_18><loc_14><loc_89><loc_21></location>Note that this relation relates pionic neutrinos and gamma rays without any reference to the cosmic ray beam; it simply reflects the fact that a π 0 produces two γ rays for every charged pion producing a ν µ + ν µ pair, which cannot be separated by current experiments.</text>
<text><location><page_10><loc_18><loc_77><loc_89><loc_88></location>Before applying this relation to a cosmic accelerator, we have to take into account the fact that, unlike neutrinos, gamma rays interact with photons of the cosmic microwave background before reaching Earth. The resulting electromagnetic shower subdivides the initial photon energy, resulting in multiple photons in the GeV-TeV energy range by the time the photons reach Earth. Calculating the cascaded gammaray flux accompanying IceCube neutrinos is straightforward [23, 24].</text>
<text><location><page_10><loc_18><loc_59><loc_89><loc_77></location>As an illustration, an example of γ -ray and neutrino emission is shown as blue lines in Fig. 6 assuming that the underlying π 0 / π ± production follows from cosmic-ray interactions with gas in the universe. In this way, the initial emission spectrum of γ -rays and neutrinos from pion decay is almost identical to the spectrum of cosmic rays (assumed to be a power law, E -2 . 19 as is the case for the diffuse cosmic neutrino flux above and energy of 100 TeV), after accounting for the different normalizations and energy scales. The flux of neutrinos arriving at Earth (blue dashed line) follows this initial CR emission spectrum. However, the observable flux of γ -rays (blue solid lines) is strongly attenuated above 100 GeV by interactions with extragalactic background photons [13].</text>
<text><location><page_10><loc_18><loc_44><loc_89><loc_58></location>The overall normalization of the emission is chosen in a way that the model does not exceed the isotropic γ -ray background observed by the Fermi satellite (blue data). This implies an upper limit on the neutrino flux shown as the blue dashed line. Interestingly, the neutrino data shown in Fig. 6 saturates this limit above 100 TeV. Moreover, the HESE data that extends to lower energies is only marginally consistent with the upper bound implied by the model (blue dashed line). This example shows that multimessenger studies of γ -ray and neutrino data are powerful tools to study the neutrino production mechanism and to constrain neutrino source models [25].</text>
<text><location><page_10><loc_18><loc_26><loc_89><loc_44></location>The matching energy densities of the extragalactic gamma-ray flux detected by Fermi and the high-energy neutrino flux measured by IceCube suggest that, rather than detecting some exotic sources, it is more likely that IceCube to a large extent observes the same universe conventional astronomy does. Clearly, an extreme universe modeled exclusively on the basis of electromagnetic processes is no longer realistic. The finding implies that a large fraction, possibly most, of the energy in the nonthermal universe originates in hadronic processes, indicating a larger role than previously thought. The high intensity of the neutrino flux below 100 TeV in comparison to the Fermi data might indicate that these sources are even more efficient neutrino than gamma-ray sources [26, 27].</text>
<text><location><page_10><loc_18><loc_13><loc_89><loc_26></location>Interestingly, the common energy density of photons and neutrinos is also comparable to that of the ultra-high-energy extragalactic cosmic rays (above 10 9 GeV) observed, for instance, by the Auger observatory [28] (green data). Unless accidental, this indicates a common origin of the signal and illustrates the potential of multimessenger studies. A scenario where the high-energy neutrinos observed at IceCube could actually originate in the same sources could be realized as follows: the cosmic ray sources can be embedded in environments that act as 'storage rooms' for cosmic rays</text>
<figure>
<location><page_11><loc_21><loc_63><loc_86><loc_87></location>
<caption>Figure 6: The spectral flux ( φ ) of neutrinos inferred from the eight-year upgoing track analysis (red fit) and the seven-year HESE analysis (magenta fit) compared to the flux of unresolved extragalactic γ -ray sources [31] (blue data) and ultra-high-energy cosmic rays [28] (green data). The neutrino spectra are indicated by the best-fit power-law (solid line) and 1 σ uncertainty range (shaded range). We highlight the various multimessenger interfaces: A: The joined production of charged pions ( π ± ) and neutral pions ( π 0 ) in cosmic-ray interactions leads to the emission of neutrinos (dashed blue) and γ -rays (solid blue), respectively. B: Cosmic ray emission models (solid green) of the most energetic cosmic rays imply a maximal flux (calorimetric limit) of neutrinos from the same sources (green dashed). C: The same cosmic ray model predicts the emission of cosmogenic neutrinos from the collision with cosmic background photons (GZK mechanism). Figure from Ref. [13].</caption>
</figure>
<text><location><page_11><loc_18><loc_24><loc_89><loc_36></location>with energies far below the 'ankle' ( E CR glyph[lessmuch] 1EeV). This energy-dependent trapping can be achieved via cosmic ray diffusion in magnetic fields. While these cosmic rays are trapped, they can produce γ -rays and neutrinos via collisions with gas. If the conditions are right, this mechanism can be so efficient that the total energy stored in cosmic rays below the ankle is converted to that of γ -rays and neutrinos. These 'calorimetric' conditions can be achieved, for instance, in starburst galaxies [29] or galaxy clusters [30].</text>
<text><location><page_11><loc_18><loc_13><loc_89><loc_23></location>The extragalactic γ -ray background observed by Fermi [31] has contributions from identified point-like sources on top of an isotropic γ -ray background (IGRB) shown in Fig. 6. This IGRB is expected to consist mostly of emission from the same class of γ -ray sources that are individually below Fermi's point-source detection threshold (see, e.g., Ref. [32]). A significant contribution of γ -rays associated with IceCube's neutrino observation would have the somewhat surprising implication that indeed many</text>
<text><location><page_12><loc_18><loc_75><loc_89><loc_88></location>extragalactic γ -ray sources are also neutrino emitters, while none had been detected so far. This dramatically changed when IceCube developed methods for performing realtime multiwavelength observations in cooperation with some twenty other observatories to identify the sources and build on the discovery of cosmic neutrinos to launch a new era in astronomy [33, 34]. This effort led to the identification of a distant flaring blazar as a cosmic ray accelerator in a multimessenger campaign launched by a 290 TeV energy neutrino detected from the constellation of Orion on September 22, 2017 [3].</text>
<section_header_level_1><location><page_12><loc_18><loc_70><loc_79><loc_72></location>3 The first truly multimessenger campaign</section_header_level_1>
<text><location><page_12><loc_18><loc_61><loc_89><loc_68></location>Neutrinos only originate in environments where protons are accelerated to produce pions and other particles that decay into neutrinos. Neutrinos can thus exclusively pinpoint cosmic ray accelerators, and this is exactly what one neutrino did on September 22, 2017.</text>
<text><location><page_12><loc_18><loc_41><loc_89><loc_61></location>IceCube detects muon neutrinos, a type of neutrino that leaves a well-reconstructed track in the detector roughly every five minutes. Most of them are low-energy neutrinos produced in the Earth's atmosphere, which are of interest for studying the neutrinos themselves, but are a persistent background when doing neutrino astronomy. In 2016, IceCube installed an online filter that selects from this sample, in real time, very high energy neutrinos that are likely to be of cosmic origin [34]. We reconstruct their energy and celestial coordinates, typically in less than one minute, and distribute the information automatically via the Gamma-ray Coordinate Network to a group of telescopes around the globe and in space for follow-up observations. These telescopes look for electromagnetic radiation from the arrival direction of the neutrino, searching for coincident emission that can reveal its origin.</text>
<text><location><page_12><loc_18><loc_14><loc_89><loc_41></location>The tenth such alert [35], IceCube-170922A, on September 22, 2017, reported a well-reconstructed muon neutrino with an energy of 290 TeV and, therefore, with a high probability of originating in an astronomical source. The Fermi telescope detected a flaring blazar aligned with the cosmic neutrino within 0.06 degrees. The source is a known blazar, a supermassive black hole spitting out high-energy particles in twin jets aligned with its rotation axis which is directed at Earth. This blazar, TXS 0506+056, had been relatively poorly studied until now, although it was identified as the highest energy gamma ray source detected by EGRET from any blazar with two photons above 40 GeV [36]. The set of observations triggered by the September 22 neutrino has yielded a treasure trove of multiwavelength data that will allow us to probe the physics of the first cosmic ray accelerator. An optical telescope eventually measured its distance [37], which was found to be 4 billion light-years. Its large distance points to a special galaxy, which sets it apart from the ten-times-closer blazars such as the Markarian sources that dominate the extreme gamma-ray sky observed by NASA's Fermi satellite.</text>
<text><location><page_13><loc_18><loc_74><loc_89><loc_88></location>TXS 0506+056 was originally flagged by the Fermi [38] and Swift [39] satellite telescopes. Follow-up observations with the MAGIC air Cherenkov telescope [40] identified it as a rare TeV blazar with the potential to produce the very high energy neutrino detected by IceCube. The source was subsequently scrutinized in X-ray, optical, and radio wavelengths. This is a first, truly multimessenger observation: none of the instruments could have made this breakthrough independently. In total, more than 20 telescopes observed the flaring blazar as a highly variable source in a high state [3].</text>
<text><location><page_13><loc_18><loc_61><loc_89><loc_73></location>It is important to realize that nearby blazars like the Markarian sources are at a redshift that is ten times smaller, and therefore TXS 0506+056, with a similar flux despite the greater distance, is one of the most luminous sources in the Universe. It likely belongs to a special class of blazars that accelerate proton beams as revealed by the neutrino. This is further supported by the fact that a variety of previous attempts to associate the arrival directions of cosmic neutrinos with a variety of Fermi blazar catalogues failed.</text>
<text><location><page_13><loc_18><loc_52><loc_89><loc_60></location>Informed by the multimessenger campaign, IceCube searched its archival neutrino data up to and including October 2017 in the direction of IC170922 using the likelihood routinely used in previous searches. This revealed a spectacular burst of over a dozen high-energy neutrinos in 110 days in the 2014-2015 data with a spectral index consistent with the one observed for the diffuse cosmic neutrino spectrum [4].</text>
<text><location><page_13><loc_18><loc_41><loc_89><loc_51></location>Interestingly, the AGILE collaboration, which operates an orbiting X-ray and gamma ray telescope, reported a day-long flare in the direction of a previous neutrino alert sent on Juli 31, 2016 [41]. The flare occurred more than one day before the time of the alert. In light of the rapid daily variations observed near the peak emission of the TXS 0506+056 flare at the time of IC170922A, this may well be a genuine coincidence.</text>
<section_header_level_1><location><page_13><loc_18><loc_33><loc_89><loc_37></location>4 Flaring sources and the high-energy neutrino flux</section_header_level_1>
<text><location><page_13><loc_18><loc_15><loc_89><loc_31></location>The extraordinary detection of more than a dozen cosmic neutrinos in the 2014 flare despite its 0.34 redshift further suggests that TXS 0506+056 belongs to a special class of sources that produce cosmic rays. The single neutrino flare dominates the flux of the source over the 9.5 years of archival IceCube data, leaving IC170922 as a less luminous second flare in the sample. We will show that a subset of about 5% of all blazars bursting once in 10 years at the level of TXS 0506+056 in 2014, can accommodate the diffuse flux cosmic neutrino flux observed by IceCube. We already pointed out in the previous section that the energy density corresponding to this flux is similar to the one in the highest energy cosmic rays.</text>
<text><location><page_13><loc_21><loc_13><loc_89><loc_14></location>In order to calculate the flux of high-energy neutrinos from a population of sources,</text>
<text><location><page_14><loc_18><loc_82><loc_89><loc_88></location>we follow [42] and relate the diffuse neutrino flux to the injection rate of cosmic rays and their efficiency to produce neutrinos in the source. For a class of sources with density ρ and neutrino luminosity L ν , the all-sky neutrino flux is</text>
<formula><location><page_14><loc_41><loc_77><loc_89><loc_81></location>∑ α E 2 ν dN ν dE ν = 1 4 π c H 0 ξ z L ν ρ, (4)</formula>
<text><location><page_14><loc_18><loc_70><loc_89><loc_75></location>where ξ z is a factor of order unity that parametrizes the integration over the redshift evolution of the sources. The relation can be adapted to a fraction F of sources with episodic emission of flares of duration ∆ t over a total observation time T :</text>
<formula><location><page_14><loc_38><loc_65><loc_89><loc_69></location>∑ α E 2 ν dN ν dE ν = 1 4 π c H 0 ξ z L ν ρ F ∆ t T . (5)</formula>
<text><location><page_14><loc_18><loc_60><loc_89><loc_63></location>Applying this relation to the 2014 TXS 0506+056 burst that dominates the flux over the 9.5 years of neutrino observations, yields</text>
<formula><location><page_14><loc_24><loc_51><loc_89><loc_58></location>3 × 10 -11 TeVcm -2 s -1 sr -1 = F 4 π ( R H 3 Gpc )( ξ z 0 . 7 )( L ν 1 . 2 × 10 47 erg / s ) ( ρ 1 . 5 × 10 -8 Mpc -3 )( ∆ t 110 d 10 yr T ) , (6)</formula>
<text><location><page_14><loc_18><loc_33><loc_89><loc_49></location>a relation which is satisfied for F ∼ 0 . 05. In summary, a special class of BL Lac blazars that undergo ∼ 110-day duration flares like TXS 0506+056 once every 10 years accommodates the observed diffuse flux of high-energy cosmic neutrinos. The class of such neutrino-flaring sources represents 5% of the sources. The argument implies the observation of roughly 100 muon neutrinos per year. This is exactly the flux of cosmic neutrinos that corresponds to the E -2 . 19 diffuse flux measured above 100 TeV. (Note that the majority of these neutrinos cannot be separated from the atmospheric background, leaving us with the reduced number of very high energy events discussed in the previous sections).</text>
<text><location><page_14><loc_18><loc_29><loc_89><loc_33></location>As previously discussed, he energetics of the cosmic neutrinos is matched by the energy of the highest energy cosmic rays, with [43]</text>
<formula><location><page_14><loc_34><loc_24><loc_89><loc_28></location>1 3 ∑ α E 2 ν dN ν dE ν glyph[similarequal] c 4 π ( 1 2 (1 -e -f π ) ξ z t H dE dt ) . (7)</formula>
<text><location><page_14><loc_18><loc_13><loc_89><loc_22></location>The cosmic rays' injection rate dE/dt above 10 16 eV is (1 -2) × 10 44 erg Mpc -3 yr -1 [44, 45]. From Eq. 6 it follows that the energy densities match for a pion production efficiency of the neutrino source of f π glyph[greaterorsimilar] 0 . 4. This high efficiency requirement is consistent with the premise that a special class of efficient sources is responsible for producing the high-energy cosmic neutrino flux seen by IceCube. The sources must</text>
<text><location><page_15><loc_18><loc_75><loc_89><loc_88></location>contain sufficient target density in photons, possibly protons, to generate the large value of f π . It is clear that the emission of flares producing the large number of cosmic neutrinos detected in the 2014 burst must correspond to major accretion events onto the black hole lasting a few months. The pionic photons will lose energy in the source and the neutrino emission is not accompanied by a flare as was the case for the 2017 event; the Fermi data, consistent with the scenario proposed, reveal photons with energies of tens of GeV, but no flaring activity.</text>
<text><location><page_15><loc_18><loc_61><loc_89><loc_75></location>A key question is whether the neutrino and gamma ray spectra for the 2014 neutrino burst from TXS 0506+056 satisfy the multimessenger relationship introduced in section 2. With the low statistics of the very high-energy gamma ray measurements during the burst period, the energetics is a more robust measure for evaluating the connection, especially because the source is opaque to high-energy gamma rays, as indicated by the large value of f π , and the pionic gamma rays will lose energy inside the source before cascading in the microwave photon background; for details see Ref. [43].</text>
<text><location><page_15><loc_18><loc_30><loc_89><loc_60></location>It is worth noting that this model for the diffuse neutrino flux clarifies why earlier attempts to associate it with blazars were unsuccessful; see Ref. [46, 47]. Clearly, the time-integrated studies are not applicable to time-dependent sources. Moreover, with a subclass of more energetic sources with lower density responsible for the diffuse flux, the constraints on blazars obtained from the relation between the point source limits and the diffuse flux are mitigated. Additional issues with this limit arise from the use of point source sensitivity (defined as 90% C.L.) rather than discovery potential (defined as 5 σ C.L.) as a constraint. The former does not provide a robust statistical measure. It has been argued that the limits from the nonobservation of a point source in time-integrated searches would suppress the contribution of blazars to the total neutrino emission because the time-averaged flux of sources is directly correlated with bursts [48]. However, two points are missed in this argument. First, this argument only applies if the duty cycle of the source and the duration of the flare, supplies neutrino bursts shorter than the total time that the detector has been taking data. Second, it neglects the excess buried under the atmospheric background which might not be statistically compelling yet. For instance, in the case of TXS 0506+056, the time-integrated search reports a significance of 2.1 σ only.</text>
<text><location><page_15><loc_18><loc_21><loc_89><loc_29></location>We are aware that these speculations are qualitative and that they may be premature and the hope is that multimessenger astronomy will provide us with more clues after the breakthrough event of September 22, 2017 that generated an unmatched data sample over all wavelengths of the electromagnetic spectrum on the first identified cosmic ray accelerator.</text>
<section_header_level_1><location><page_16><loc_18><loc_86><loc_36><loc_88></location>5 Summary</section_header_level_1>
<text><location><page_16><loc_18><loc_68><loc_89><loc_84></location>Getting all the elements of this puzzle to fit together is not easy, but they suggest that the blazar may contain important clues on the origin of cosmic neutrinos and cosmic rays. This breakthrough is just the beginning and raises intriguing questions. What is special about this source? Can a subclass of blazars to which it belongs accommodate the diffuse flux observed by IceCube? Are these also the sources of all high-energy cosmic rays or only of some? The TXS 0506+056 emission over the last 10 years is dominated by the single flare in 2014. If this is characteristic of the subclass of sources that it belongs to, identifying additional sources will be difficult unless more and larger neutrino telescopes yield more frequent and higher statistics neutrino alerts.</text>
<text><location><page_16><loc_18><loc_56><loc_89><loc_68></location>Unlike the previous SN1987A and GW170817 multimessenger events, this event could not have been observed with a single instrument. Without the initial coincident observation, IC170922A would be just one more of the few hundred cosmic neutrinos detected by IceCube and the accompanying radiation just one more flaring blazar observed by Fermi-LAT. Neutrino astronomy was born with a supernova in 1987. Thirty years later, this recent event involves neutrinos that are tens of millions of times more energetic and are from a source a hundred thousand times more distant.</text>
<section_header_level_1><location><page_16><loc_18><loc_50><loc_33><loc_52></location>References</section_header_level_1>
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<list_item><location><page_16><loc_19><loc_32><loc_89><loc_37></location>[3] M. Aartsen, et al., Multimessenger observations of a flaring blazar coincident with high-energy neutrino IceCube-170922A, Science 361 (6398) (2018) eaat1378. doi:10.1126/science.aat1378 .</list_item>
<list_item><location><page_16><loc_19><loc_25><loc_89><loc_30></location>[4] M. Aartsen, et al., Neutrino emission from the direction of the blazar TXS 0506+056 prior to the IceCube-170922A alert, Science 361 (6398) (2018) 147-151. doi:10.1126/science.aat2890 .</list_item>
<list_item><location><page_16><loc_19><loc_18><loc_89><loc_23></location>[5] A. Roberts, The Birth of high-energy neutrino astronomy: A Personal history of the DUMAND project, Rev.Mod.Phys. 64 (1992) 259-312. doi:10.1103/ RevModPhys.64.259 .</list_item>
<list_item><location><page_16><loc_19><loc_14><loc_89><loc_17></location>[6] V. Berezinsky, G. Zatsepin, Cosmic rays at ultrahigh-energies (neutrino?), Phys.Lett. B28 (1969) 423-424. doi:10.1016/0370-2693(69)90341-4 .</list_item>
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2015PhRvD..91h4008C | https://arxiv.org/pdf/1504.05914.pdf | <document>
<section_header_level_1><location><page_1><loc_28><loc_82><loc_68><loc_86></location>Lorenz gauge quantization in conformally flat spacetimes</section_header_level_1>
<text><location><page_1><loc_32><loc_74><loc_65><loc_76></location>Jesse C. Cresswell ∗ and Dan N. Vollick</text>
<text><location><page_1><loc_29><loc_63><loc_67><loc_72></location>Irving K. Barber School of Arts and Sciences University of British Columbia Okanagan 3333 University Way Kelowna, B.C. Canada</text>
<text><location><page_1><loc_44><loc_61><loc_52><loc_63></location>V1V 1V7</text>
<section_header_level_1><location><page_1><loc_44><loc_55><loc_53><loc_57></location>Abstract</section_header_level_1>
<text><location><page_1><loc_82><loc_40><loc_82><loc_43></location>/negationslash</text>
<text><location><page_1><loc_12><loc_38><loc_85><loc_50></location>Recently it was shown that Dirac's method of quantizing constrained dynamical systems can be used to impose the Lorenz gauge condition in a four-dimensional cosmological spacetime. In this paper we use Dirac's method to impose the Lorenz gauge condition in a general four-dimensional conformally flat spacetime and find that there is no particle production. We show that in cosmological spacetimes with dimension D = 4 there will be particle production when the scale factor changes, and we calculate the particle production due to a sudden change.</text>
<section_header_level_1><location><page_2><loc_12><loc_84><loc_34><loc_86></location>1 Introduction</section_header_level_1>
<text><location><page_2><loc_12><loc_76><loc_85><loc_83></location>Recently it was shown [1] that Dirac's method of quantizing constrained dynamical systems [2, 3] can be used to impose the Lorenz gauge condition in a four-dimensional cosmological spacetime. This method was used to show that there is no particle production agreeing with earlier results by Parker [4, 5].</text>
<text><location><page_2><loc_12><loc_68><loc_85><loc_75></location>In this paper a gauge fixed Lagrangian is introduced for the electromagnetic field in a conformally flat spacetime of arbitrary dimension. The Lorenz gauge condition is imposed as a gauge constraint and requires a secondary constraint for consistency. There are no further constraints and both constraints are first class.</text>
<text><location><page_2><loc_12><loc_57><loc_85><loc_68></location>In four spacetime dimensions the Hamiltonian simplifies greatly. Due to the constraints imposed on the wave function the Hamiltonian can be quantized using the flat spacetime procedure. As a result there is no particle production in four-dimensional conformally flat spacetimes. The Lagrangian introduced in this paper produces a simpler Hamiltonian and simpler constraints than the one used in Ref. [1] for the fourdimensional case.</text>
<text><location><page_2><loc_12><loc_46><loc_85><loc_57></location>We also use the gauge fixed Lagrangian to show that there will generally be particle production in D dimensional cosmological spacetimes due to the changing scale factor. We then calculate the particle production that occurs for a sudden change in scale factor. The Hamiltonian is not bounded under such a transition so the wave function of the system does not remain unchanged. We show that the state vector picks up a phase factor and we calculate the resulting particle production.</text>
<section_header_level_1><location><page_2><loc_12><loc_41><loc_81><loc_43></location>2 Hamiltonian and Constraints in D Dimensions</section_header_level_1>
<text><location><page_2><loc_12><loc_38><loc_68><loc_40></location>Consider a D dimensional conformally flat spacetime with a metric</text>
<formula><location><page_2><loc_31><loc_33><loc_85><loc_37></location>ds 2 = a 2 ( x µ ) [ -dt 2 + dx 2 1 + ... + dx 2 D -1 ] . (1)</formula>
<text><location><page_2><loc_12><loc_31><loc_35><loc_33></location>The gauge fixed Lagrangian</text>
<formula><location><page_2><loc_33><loc_26><loc_85><loc_30></location>L = -1 4 √ gF µν ˜ F µν -1 2 √ g ( ∇ µ ˜ A µ ) 2 , (2)</formula>
<text><location><page_2><loc_12><loc_23><loc_84><loc_25></location>where ˜ F µν = g µα g νβ F αβ and ˜ A µ = g µν A ν can, after integration by parts, be written as</text>
<formula><location><page_2><loc_30><loc_17><loc_85><loc_22></location>L = -1 2 √ g [ ( ∇ µ A ν ) ( ∇ µ ˜ A ν ) -R µν ˜ A µ ˜ A ν ] . (3)</formula>
<text><location><page_2><loc_12><loc_16><loc_73><loc_17></location>For the above metric, this becomes, after additional integration by parts,</text>
<formula><location><page_2><loc_13><loc_9><loc_85><loc_14></location>L = -1 2 b ( ∂ µ A ν ) ( ∂ µ A ν ) -2 b ( ψ µ A µ )( ∂ ν A ν ) -1 2 b [4( D -3) ψ µ ψ ν -( D -4) ∂ µ ψ ν ] A µ A ν , (4)</formula>
<text><location><page_3><loc_12><loc_84><loc_50><loc_86></location>where A µ = η µν A ν , ∂ µ = η µν ∂ ν , b = a D -4 and</text>
<formula><location><page_3><loc_43><loc_80><loc_85><loc_83></location>ψ µ = 1 a ∂ µ a. (5)</formula>
<text><location><page_3><loc_12><loc_77><loc_35><loc_79></location>The Lorenz gauge condition</text>
<text><location><page_3><loc_12><loc_73><loc_26><loc_74></location>can be written as</text>
<text><location><page_3><loc_12><loc_68><loc_35><loc_70></location>The canonical momenta are</text>
<formula><location><page_3><loc_38><loc_65><loc_85><loc_67></location>Π µ = b ˙ A µ +2 bδ µ t ( ψ α A α ) (8)</formula>
<text><location><page_3><loc_12><loc_62><loc_78><loc_64></location>and the Lorenz gauge condition, written in terms of the canonical momenta, is</text>
<formula><location><page_3><loc_32><loc_57><loc_85><loc_61></location>χ 1 = 1 b Π t + ∂ k A k +( D -4) ψ µ A µ = 0 . (9)</formula>
<text><location><page_3><loc_12><loc_55><loc_42><loc_57></location>The Hamiltonian density is given by</text>
<formula><location><page_3><loc_16><loc_50><loc_85><loc_54></location>h = 1 2 b Π µ Π µ + 1 2 b ( ∂ k A µ )( ∂ k A µ ) -1 2 b ( D -4)( ∂ µ ψ ν ) A µ A ν +2 b ( ψ α A α ) χ 1 . (10)</formula>
<text><location><page_3><loc_12><loc_48><loc_41><loc_49></location>For consistency it is necessary that</text>
<formula><location><page_3><loc_38><loc_43><loc_85><loc_47></location>˙ χ 1 = { χ 1 , H } + ∂χ 1 ∂t ≈ 0 , (11)</formula>
<text><location><page_3><loc_12><loc_38><loc_85><loc_42></location>where { } denotes the Poisson bracket, H = ∫ hd ( D -1) x , and ≈ denotes a weak equality. This condition gives a secondary constraint,</text>
<formula><location><page_3><loc_25><loc_33><loc_85><loc_37></location>χ 2 = ∂ k ( ∂ k A t + 1 b Π k ) +( D -4) ψ k ( ∂ k A t + 1 b Π k ) ≈ 0 . (12)</formula>
<text><location><page_3><loc_12><loc_26><loc_85><loc_33></location>It is interesting to note that χ 2 = 1 b ∂ k ( bF k 0 ). The condition ˙ χ 2 ≈ 0 does not produce a new constraint, so the procedure terminates here. The constraints χ 1 and χ 2 are first class since { χ 1 ( x ) , χ 2 ( y ) } = 0.</text>
<section_header_level_1><location><page_3><loc_12><loc_22><loc_62><loc_24></location>3 Quantization in Four Dimensions</section_header_level_1>
<text><location><page_3><loc_12><loc_19><loc_61><loc_21></location>In four spacetime dimensions, the Hamiltonian is given by</text>
<formula><location><page_3><loc_26><loc_14><loc_85><loc_18></location>H = ∫ [ 1 2 Π µ Π µ + 1 2 ( ∂ k A µ )( ∂ k A µ ) + 2( ψ α A α ) χ 1 ] d 3 x, (13)</formula>
<text><location><page_3><loc_12><loc_12><loc_40><loc_13></location>while the constraints are given by</text>
<formula><location><page_3><loc_42><loc_9><loc_85><loc_10></location>χ 1 = Π t + ∂ k A k (14)</formula>
<formula><location><page_3><loc_44><loc_74><loc_85><loc_77></location>∇ µ A µ = 0 (6)</formula>
<formula><location><page_3><loc_37><loc_70><loc_85><loc_73></location>∂ µ A µ +( D -2) ψ µ A µ = 0 . (7)</formula>
<text><location><page_4><loc_12><loc_84><loc_15><loc_86></location>and</text>
<text><location><page_4><loc_12><loc_78><loc_85><loc_81></location>These are identical to the expressions in flat spacetime, except for the last term in the Hamiltonian involving χ 1 .</text>
<formula><location><page_4><loc_39><loc_81><loc_85><loc_84></location>χ 2 = ∂ k ( ∂ k A t +Π k ) . (15)</formula>
<text><location><page_4><loc_12><loc_73><loc_85><loc_78></location>To quantize the theory we follow the procedure developed by Dirac [2, 3]. In the Schrodinger picture, the dynamical variables A µ and Π µ become time independent operators satisfying</text>
<formula><location><page_4><loc_34><loc_71><loc_85><loc_73></location>[ A µ ( /vectorx ) , A ν ( /vectory )] = [Π µ ( /vectorx ) , Π ν ( /vectory )] = 0 (16)</formula>
<text><location><page_4><loc_12><loc_68><loc_15><loc_70></location>and</text>
<formula><location><page_4><loc_36><loc_66><loc_85><loc_68></location>[ A µ ( /vectorx ) , Π ν ( /vectory )] = i δ ν µ δ 3 ( /vectorx, /vectory ) , (17)</formula>
<text><location><page_4><loc_12><loc_58><loc_85><loc_65></location>where [ ] denotes the commutator and we have set ¯ h = 1. Note that there is an ambiguity in the ordering of the operators in the last term in the Hamiltonian since χ 1 contains Π t and ψ α A α contains A t . We have chosen the ordering so that the Hamiltonian is given by (13).</text>
<text><location><page_4><loc_15><loc_56><loc_71><loc_58></location>A state vector is introduced that satisfies the Schrodinger equation</text>
<formula><location><page_4><loc_40><loc_51><loc_85><loc_55></location>i d dt | Ψ > = H | Ψ > . (18)</formula>
<text><location><page_4><loc_12><loc_49><loc_63><loc_50></location>The constraints are imposed on the wave function as follows:</text>
<formula><location><page_4><loc_34><loc_44><loc_85><loc_47></location>χ 1 | Ψ > = 0 and χ 2 | Ψ > = 0 . (19)</formula>
<text><location><page_4><loc_12><loc_40><loc_85><loc_44></location>The last term in the Hamiltonian will therefore not affect the equations of motion, and the theory can be quantized by following the flat spacetime procedure.</text>
<text><location><page_4><loc_12><loc_29><loc_85><loc_40></location>This generalizes the results of Ref. [1] from a four-dimensional cosmological spacetime to a general four-dimensional conformally flat spacetime. Thus, there is no particle production in four-dimensional conformally flat spacetimes, as expected based on Parker's calculation for a massless conformally coupled scalar field [6]. The Lagrangian used in this paper differs from the one used in Ref. [1] by a total derivative and gives a simpler H , χ 1 and χ 2 .</text>
<section_header_level_1><location><page_4><loc_12><loc_21><loc_87><loc_26></location>4 Quantization in D -dimensional Cosmological Spacetimes</section_header_level_1>
<text><location><page_4><loc_57><loc_17><loc_57><loc_20></location>/negationslash</text>
<text><location><page_4><loc_12><loc_16><loc_85><loc_20></location>To examine particle production in spacetimes with D = 4 we consider the case in which a depends only on t . In this case ψ k = 0, and we write ψ t = ψ .</text>
<text><location><page_4><loc_15><loc_14><loc_31><loc_16></location>The Hamiltonian is</text>
<formula><location><page_4><loc_16><loc_9><loc_85><loc_13></location>H = 1 2 ∫ [ 1 b Π µ Π µ + b ( ∂ k A µ )( ∂ k A µ ) -4 bψA t χ 1 -b ( D -4) ˙ ψA 2 t ] d ( D -1) x, (20)</formula>
<text><location><page_5><loc_12><loc_84><loc_32><loc_86></location>while the constraints are</text>
<text><location><page_5><loc_12><loc_78><loc_15><loc_79></location>and</text>
<formula><location><page_5><loc_35><loc_80><loc_85><loc_84></location>χ 1 = 1 b Π t + ∂ k A k -( D -4) ψA t (21)</formula>
<formula><location><page_5><loc_38><loc_74><loc_85><loc_78></location>χ 2 = ∂ k ( ∂ k A t + 1 b Π k ) . (22)</formula>
<text><location><page_5><loc_12><loc_69><loc_85><loc_74></location>For a consistent quantum theory we require that [ χ 1 , χ 2 ] = αχ 1 + βχ 2 , where α and β are operators that appear to the left of the constraints. This is satisfied since [ χ 1 , χ 2 ] = 0.</text>
<text><location><page_5><loc_15><loc_67><loc_72><loc_69></location>To preserve the constraints under time evolution it is necessary that</text>
<formula><location><page_5><loc_40><loc_62><loc_85><loc_66></location>∂χ k ∂t -i [ χ k , H ] ≈ 0 , (23)</formula>
<text><location><page_5><loc_12><loc_58><loc_85><loc_61></location>where, in the quantum theory, A ≈ 0 implies that A | Ψ > = 0. It is easy to show that the constraints are preserved, as they are in the classical case.</text>
<text><location><page_5><loc_12><loc_52><loc_85><loc_58></location>The constraint χ 2 can be simplified. The term ∂ m Π m involves only the longitudinal part of Π m , and this longitudinal part can be written as the gradient of a scalar U . Thus, ∂ m Π m = ∇ 2 U . The constraint χ 2 can therefore be written as</text>
<formula><location><page_5><loc_39><loc_48><loc_85><loc_52></location>χ 2 = ∇ 2 ( A t + 1 b U ) . (24)</formula>
<text><location><page_5><loc_12><loc_42><loc_85><loc_47></location>Now, ∇ 2 ( A t + 1 b U ) ≈ 0 over all space has the unique solution A t + 1 b U ≈ 0 if the fields vanish at infinity. Since we are quantizing the electromagnetic field on a fixed background spacetime we are free to make this assumption.</text>
<text><location><page_5><loc_15><loc_40><loc_85><loc_41></location>The Hamiltonian can be decomposed into transverse and longitudinal/timelike parts:</text>
<formula><location><page_5><loc_26><loc_34><loc_85><loc_39></location>H T = 1 2 ∫ [ 1 b Π m ( T ) Π ( T ) m + b ( ∂ s A ( T ) m ) ( ∂ s A m ( T ) ) ] d ( D -1) x, (25)</formula>
<formula><location><page_5><loc_15><loc_30><loc_85><loc_34></location>H (1) L = b 2 ∫ [ ∂ r ( 1 b U -A t ) ∂ r ( 1 b U + A t ) + ( ∂ m A m -1 b Π t -DψA t ) χ 1 ] d ( D -1) x, (26)</formula>
<text><location><page_5><loc_12><loc_29><loc_15><loc_30></location>and</text>
<formula><location><page_5><loc_22><loc_25><loc_85><loc_29></location>H (2) L = 1 2 ( D -4) b ∫ { 2 ψA t ( ∂ k A k ) -[( D -4) ψ 2 + ˙ ψ ] A 2 t } d ( D -1) x. (27)</formula>
<text><location><page_5><loc_12><loc_22><loc_49><loc_25></location>Note that H (1) L ≈ 0, so that H ≈ H T + H (2) L .</text>
<text><location><page_5><loc_12><loc_20><loc_85><loc_23></location>To set up a Fock space representation in the Minkowski in and out regions, where a is constant, the operators</text>
<formula><location><page_5><loc_26><loc_14><loc_85><loc_19></location>a ( λ ) /vector k = ∫ e -i /vector k · /vectorx [ k √ b/epsilon1 ( λ ) /vector kµ A µ ( /vectorx ) + i √ b /epsilon1 ( λ ) /vector kµ Π µ ( /vectorx ) ] d ( D -1) x (28)</formula>
<formula><location><page_5><loc_26><loc_8><loc_85><loc_12></location>a † ( λ ) /vector k = ∫ e i /vector k · /vectorx [ k √ b/epsilon1 ( λ ) /vector kµ A µ ( /vectorx ) -i √ b /epsilon1 ( λ ) /vector kµ Π µ ( /vectorx ) ] d ( D -1) x (29)</formula>
<text><location><page_5><loc_12><loc_12><loc_15><loc_14></location>and</text>
<text><location><page_6><loc_12><loc_77><loc_85><loc_86></location>can be introduced. Here k = | /vector k | and /epsilon1 ( λ ) /vector kµ are the standard (real) Minkowski polarization vectors. The factors of b in the above expressions can be deduced by computing these operators using the standard approach in the Heisenberg picture and then transforming them into the Schrodinger picture. These are also the unique factors of b that give the standard commutation relations</text>
<formula><location><page_6><loc_35><loc_72><loc_85><loc_75></location>[ a ( λ ) /vector k , a ( λ ' ) /vector k ' ] = [ a † ( λ ) /vector k , a † ( λ ' ) /vector k ' ] = 0 , (30)</formula>
<formula><location><page_6><loc_29><loc_68><loc_85><loc_71></location>[ a ( λ ) /vector k , a † ( λ ' ) /vector k ' ] = (2 π ) ( D -1) (2 k ) η λλ ' δ ( D -1) ( /vector k -/vector k ' ) (31)</formula>
<text><location><page_6><loc_12><loc_67><loc_39><loc_68></location>and normal ordered Hamiltonian</text>
<formula><location><page_6><loc_30><loc_61><loc_85><loc_66></location>: H T : = 1 2(2 π ) ( D -1) D -2 ∑ λ =1 ∫ a † ( λ ) /vector k a ( λ ) /vector k d ( D -1) k. (32)</formula>
<text><location><page_6><loc_12><loc_56><loc_85><loc_60></location>Note that the polarizations corresponding to λ = 1 ... ( D -2) are transverse polarizations. A vacuum state | 0 > can be introduced that satisfies</text>
<formula><location><page_6><loc_24><loc_51><loc_85><loc_55></location>[ 1 b Π t + ∂ m A m ] | 0 > = 0 , [ 1 b ∂ m Π m + ∇ 2 A t ] | 0 > = 0 , (33)</formula>
<text><location><page_6><loc_12><loc_49><loc_15><loc_50></location>and</text>
<formula><location><page_6><loc_32><loc_46><loc_85><loc_49></location>a ( λ ) /vector k | 0 > = 0 , λ = 1 ... ( D -2) . (34)</formula>
<text><location><page_6><loc_70><loc_41><loc_70><loc_43></location>/negationslash</text>
<text><location><page_6><loc_12><loc_38><loc_85><loc_46></location>The operators a ( λ ) /vector k act as annihilation operators, and the operators a † ( λ ) /vector k act as creation operators. Note that | 0 in > will not be the same as | 0 out > if b out = b in . There will therefore be particle production unless the in-vacuum state happens to evolve into the out-vacuum state.</text>
<text><location><page_6><loc_12><loc_28><loc_85><loc_38></location>As an explicit example of particle production, consider the case of a 'sudden' change from a Minkowski space with b in to one with b out . The sudden approximation cannot be used because the Hamiltonian contains terms involving ψ and ˙ ψ which do not remain bounded as the time interval over which the change takes place goes to zero. The behavior of the state vector can be determined by introducing the ket | Ψ T > ,</text>
<formula><location><page_6><loc_23><loc_24><loc_85><loc_28></location>| Ψ T > = exp { ib ∫ [ A t ( ∂ k A k ) -1 2 ( D -4) ψA 2 t ] d ( D -1) x } | Ψ >, (35)</formula>
<text><location><page_6><loc_12><loc_21><loc_43><loc_23></location>which satisfies the equation of motion</text>
<formula><location><page_6><loc_39><loc_16><loc_85><loc_20></location>i d dt | Ψ T > = H T | Ψ T > (36)</formula>
<text><location><page_6><loc_12><loc_14><loc_28><loc_15></location>and the constraints</text>
<formula><location><page_6><loc_28><loc_9><loc_85><loc_12></location>Π t | Ψ T > = 0 and ∂ k Π k | Ψ T > = 0 . (37)</formula>
<text><location><page_7><loc_12><loc_75><loc_85><loc_86></location>Since H T (25) remains bounded during the transition, the sudden approximation can be used on the evolution of | Ψ T > . This means that | Ψ T > does not change and that | Ψ > picks up a phase factor during the transition. Thus, if the initial state is the in-vacuum, then the state in the out region, just after the transition, will be the in-vacuum with a phase factor. Since the phase factor commutes with the transverse a ( λ ) (in) /vector k the state of the system will still be in the in-vacuum state.</text>
<text><location><page_7><loc_15><loc_73><loc_78><loc_75></location>The Bogolubov transformation between the in and out operators is given by</text>
<formula><location><page_7><loc_29><loc_68><loc_85><loc_72></location>a ( λ ) (out) /vector k = [ b in + b out 2 √ b in b out ] a ( λ ) (in) /vector k ± [ b out -b in 2 √ b in b out ] a † ( λ ) (in) -/vector k (38)</formula>
<text><location><page_7><loc_12><loc_62><loc_85><loc_66></location>where we have taken /epsilon1 ( λ ) ( -/vector k ) µ = ± /epsilon1 ( λ ) /vector kµ . The expectation value of N ( λ ) (out) /vector k = a † ( λ ) (out) /vector k a ( λ ) (out) /vector k in the in-vacuum state is</text>
<formula><location><page_7><loc_34><loc_59><loc_85><loc_63></location>< 0 in | N ( λ ) (out) /vector k | 0 in > = ( b out -b in ) 2 4 b in b out . (39)</formula>
<text><location><page_7><loc_12><loc_55><loc_85><loc_58></location>There will therefore be particles produced by the sudden change in the scale factor when D = 4.</text>
<text><location><page_7><loc_14><loc_54><loc_14><loc_56></location>/negationslash</text>
<section_header_level_1><location><page_7><loc_12><loc_50><loc_32><loc_52></location>5 Conclusion</section_header_level_1>
<text><location><page_7><loc_12><loc_37><loc_85><loc_48></location>In this paper we used Dirac's method of quantizing constrained dynamical systems to generalize the results of Ref. [1]. We found that in four-dimensional conformally flat spacetimes the Hamiltonian and constraints have the same form as in flat space but for an extra term in the Hamiltonian. Due to the constraints on the system, the extra term has no effect on the equations of motion, so there is no particle production in agreement with Ref. [6].</text>
<text><location><page_7><loc_60><loc_34><loc_60><loc_37></location>/negationslash</text>
<text><location><page_7><loc_12><loc_21><loc_85><loc_37></location>We also considered cosmological spacetimes with D = 4 and found that there is particle production unless the in-vacuum state happens to evolve into the out-vacuum state. For a spacetime that undergoes a sudden change in scale factor the wave function of the system picks up a phase factor because the Hamiltonian does not remain bounded. Under the sudden change, we found that if the initial state is the in-vacuum the final state will be the in-vacuum with a phase factor. A Bogolubov transformation between the in and out creation and annihilation operators showed that the expectation value of the out-number operator in the in-vacuum state is ( b out -b in ) 2 / 4 b in b out where b = a D -4 and a ( t ) is the scale factor of the spacetime.</text>
<section_header_level_1><location><page_7><loc_12><loc_16><loc_38><loc_18></location>Acknowledgements</section_header_level_1>
<text><location><page_7><loc_12><loc_11><loc_85><loc_14></location>This research was supported by the Natural Sciences and Engineering Research Council of Canada.</text>
<section_header_level_1><location><page_8><loc_12><loc_84><loc_27><loc_86></location>References</section_header_level_1>
<unordered_list>
<list_item><location><page_8><loc_13><loc_81><loc_55><loc_83></location>[1] D. N. Vollick, Phys. Rev. D 86 , 084057 (2012).</list_item>
<list_item><location><page_8><loc_13><loc_76><loc_85><loc_79></location>[2] P.A.M. Dirac, The Principles of Quantum Mechanics , 4th ed. (Oxford University Press, New York, 1958), Chap. VII.</list_item>
<list_item><location><page_8><loc_13><loc_73><loc_77><loc_75></location>[3] P.A.M. Dirac, Lectures on Quantum Mechanics (Dover, New York, 1964).</list_item>
<list_item><location><page_8><loc_13><loc_70><loc_52><loc_71></location>[4] L. Parker, Phys. Rev. Lett. 21 , 562 (1968).</list_item>
<list_item><location><page_8><loc_13><loc_67><loc_49><loc_68></location>[5] L. Parker, Phys. Rev. 183 , 1057 (1969).</list_item>
<list_item><location><page_8><loc_13><loc_64><loc_48><loc_65></location>[6] L. Parker, Phys. Rev. D 7 , 976 (1973).</list_item>
</unordered_list>
</document> | [] |
2021MNRAS.501.5697B | https://arxiv.org/pdf/2010.08257.pdf | <document>
<section_header_level_1><location><page_1><loc_7><loc_86><loc_87><loc_90></location>Vector modes in Λ CDM: the gravitomagnetic potential in dark matter haloes from relativistic N -body simulations</section_header_level_1>
<text><location><page_1><loc_8><loc_82><loc_69><loc_84></location>Cristian Barrera-Hinojosa, 1 glyph[star] Baojiu Li 1 , Marco Bruni 2 , 3 and Jian-hua He 4 , 5</text>
<text><location><page_1><loc_7><loc_81><loc_66><loc_82></location>1 Institute for Computational Cosmology, Department of Physics, Durham University, Durham DH1 3LE, UK</text>
<unordered_list>
<list_item><location><page_1><loc_7><loc_80><loc_80><loc_81></location>2 Institute of Cosmology and Gravitation, University of Portsmouth, Dennis Sciama Building, Burnaby Road, Portsmouth PO1 3FX, UK</list_item>
<list_item><location><page_1><loc_7><loc_78><loc_40><loc_79></location>3 INFN Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy</list_item>
<list_item><location><page_1><loc_7><loc_77><loc_54><loc_78></location>4 School of Astronomy and Space Science, Nanjing University, Nanjing 210023, China</list_item>
<list_item><location><page_1><loc_7><loc_76><loc_75><loc_77></location>5 Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210023, China</list_item>
</unordered_list>
<text><location><page_1><loc_7><loc_72><loc_36><loc_73></location>Accepted XXX. Received YYY; in original form ZZZ</text>
<section_header_level_1><location><page_1><loc_7><loc_67><loc_15><loc_68></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_7><loc_47><loc_92><loc_67></location>We investigate the transverse modes of the gravitational and velocity fields in Λ CDM, based on a high-resolution simulation performed using the adaptive-mesh refinement general-relativistic N -body code GRAMSES. We study the generation of vorticity in the dark matter velocity field at low redshift, providing fits to the shape and evolution of its power spectrum over a range of scales. By analysing the gravitomagnetic vector potential, which is absent in Newtonian simulations, in dark matter haloes with masses ranging from ∼ 10 12 . 5 h -1 M glyph[circledot] to ∼ 10 15 h -1 M glyph[circledot] , we find that its magnitude correlates with the halo mass, peaking in the inner regions. Nevertheless, on average, its ratio against the scalar gravitational potential remains fairly constant, below percent level, decreasing roughly linearly with redshift and showing a weak dependence on halo mass. Furthermore, we show that the gravitomagnetic acceleration in haloes peaks towards the core and reaches almost 10 -10 h cm/s 2 in the most massive halo of the simulation. However, regardless of the halo mass, the ratio between the gravitomagnetic force and the standard gravitational force is typically at around the 10 -5 level inside the haloes, again without significant radius dependence. This result confirms that the gravitomagnetic effects have negligible impact on structure formation, even for the most massive structures, although its behaviour in low density regions remains to be explored. Likewise, the impact on observations remains to be understood in the future.</text>
<text><location><page_1><loc_7><loc_45><loc_79><loc_46></location>Key words: gravitation - cosmology: theory - large-scale structure of the Universe - methods: numerical.</text>
<section_header_level_1><location><page_1><loc_7><loc_39><loc_21><loc_40></location>1 INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_7><loc_15><loc_48><loc_37></location>While the dynamics of the large-scale structure (LSS) of the universe is mainly governed by scalar perturbations, vector and tensor degrees of freedom are promising alternatives for exploring the nature of dark matter and gravity. The effects of the vector modes of the spacetime metric on matter such as frame dragging and geodetic precession have been measured in the Solar system during the last decade (Everitt et al. 2011), but there is still no cosmological signal detected. The recent observation of radio galaxies showing coherent angular velocities on scales of ∼ 20 Mpc at z = 1 reported by Taylor & Jagannathan (2016) has motivated to seek a physical interpretation in terms of vector modes, but it has not been possible to establish a clear connection so far (Cusin et al. 2017; Bonvin et al. 2018). More recently, and motivated by the accurate data provided by Gaia DR2, a simple model to explain the flat rotation curve of the Milky Way in terms of frame dragging has been proposed in Crosta et al. (2020).</text>
<text><location><page_1><loc_7><loc_10><loc_48><loc_15></location>In Λ CDM cosmology, vector modes are typically neglected. In a perfect fluid, vorticity - the covariant curl of the 4-velocity field - satisfies a homogenous nonlinear equation, hence it vanishes exactly, i.e. at all orders in perturbation theory (Lu et al. 2009), unless</text>
<text><location><page_1><loc_51><loc_26><loc_92><loc_40></location>it is either introduced by initial conditions 1 or generated by physics beyond such fluid model. Moreover, vorticity is not generated by standard inflationary scenarios, and even if it was, this type of perturbation quickly decays during the matter-dominated era. Nonetheless, vorticity is found to be generated dynamically via shell (orbit) crossing of matter, a phenomenon extremely common at late times whose modelling is beyond the grasp of the single-streaming fluid regime. Therefore, N -body simulations represent a valuable tool for the study of vorticity generation (Pueblas & Scoccimarro 2009; Hahn et al. 2015; Jelic-Cizmek et al. 2018).</text>
<text><location><page_1><loc_51><loc_13><loc_92><loc_26></location>In the Poisson gauge, generalising the longitudinal gauge to include tensor and vector perturbations (Bertschinger 1993), the latter are encoded by the non-diagonal spacetime metric components, the shift vector B i ≡ g 0 i , and represent in this gauge the gauge-invariant gravitomagnetic vector potential (Bardeen 1980). In Λ CDM, safely assuming purely scalar perturbations at first-order, the shift vector vanishes at the linear level, while at second order it satisfies a constraint equation sourced by the product of first-order density and velocity perturbations. However, it is expected that, just</text>
<text><location><page_2><loc_7><loc_90><loc_48><loc_93></location>like vorticity, the gravito-magnetic field also receives corrections from phenomena beyond the perfect fluid description.</text>
<text><location><page_2><loc_7><loc_61><loc_48><loc_90></location>The impact of vector modes on LSS observables is expected to be small relative to the scalar perturbations, both from perturbative (Lu et al. 2009) and non-perturbative analyses (Bruni et al. 2014; Adamek et al. 2016b), although it can represent a new systematic which needs to be taken into account (Bonvin et al. 2018). For instance, their effect on gravitational lensing seems to be not strong enough to be detectable by current observations (Thomas et al. 2015a; Saga et al. 2015; Gressel et al. 2019), and the imprints of the vector potential in the angular power spectrum and bispectrum of galaxies are also weak (Durrer & Tansella 2016; Jolicoeur et al. 2019), although a vector perturbation can be isolated from the full signal if it violates statistical isotropy and defines a preferred frame (see, e.g., Tansella et al. 2018). On the other hand, the vector potential power spectrum is known to peak around the equality scale (Lu et al. 2009), and its behaviour as well as impact on observables at highly nonlinear scales remains largely unexplored, although deviations from perturbation theory can be significant (Bruni et al. 2014). Furthermore, in popular f ( R ) gravity models, vector modes can have considerable deviations from GR on small scales (Thomas et al. 2015c), so these could also play a role in discriminating cosmological models.</text>
<text><location><page_2><loc_7><loc_30><loc_48><loc_61></location>The work of Pueblas & Scoccimarro (2009) provided the first insights into the generation of vorticity via shell crossing using N -body simulations, which allowed to quantify its impact on the density and velocity power spectra estimates from linear perturbation theory. In particular, vorticity was found to peak in the outskirts of virialised structures as particle velocities in inner regions are strongly aligned with density gradients, as also found later in Hahn et al. (2015) from a different set of simulations. Although - contrary to vorticity - the investigation of the gravitomagnetic vector field in principle requires a completely general-relativistic numerical framework as Newtonian simulations only model a single scalar gravitational potential, Φ , in Bruni et al. (2014) and Thomas et al. (2015b) a novel method to extract its power spectrum by post-processing the momentum density field from a Newtonian simulation was introduced. This is motivated by the fact that the leading contribution to the shift vector in post-Friedmann expansion (Milillo et al. 2015) is sourced by the transverse part of the momentum density field. Although this method neglects the feedback of the shift vector into the simulation dynamics, this approximation is well justified as perturbation theory estimates that the magnitude of the vector potential is at most one percent of the scalar gravitational potential (Lu et al. 2009).</text>
<text><location><page_2><loc_7><loc_8><loc_48><loc_30></location>Cosmological codes which are capable of simulating vector modes of the metric have been only recently developed (e.g., Adamek et al. 2016a; Adamek et al. 2016b; Mertens et al. 2016; Giblin et al. 2017; Macpherson et al. 2017; Barrera-Hinojosa & Li 2020a), and have proven robust enough to study different relativistic distortions in the large-scale structures (LSS); (see Adamek et al. 2020, for an actual comparison of frame-dragging observables in a toy universe simulated using these codes). In particular, the cross correlation between the shift vector and vorticity has been studied in Jelic-Cizmek et al. (2018) using the relativistic N -body code gevolution (Adamek et al. 2016a; Adamek et al. 2016b), showing that the vector potential is only weakly sourced by vorticity alone, which is subdominant compared with the density-dependent terms coming from the transverse projection of the full momentum field, in qualitative agreement with post-Friedmann expansion results from Bruni et al. (2014); Thomas et al. (2015b).</text>
<text><location><page_2><loc_9><loc_6><loc_48><loc_7></location>The objective of this paper is to study the vector modes of both</text>
<text><location><page_2><loc_51><loc_72><loc_92><loc_93></location>the gravitational and matter velocity fields from large sub-horizon scales down to deeply nonlinear scales using GRAMSES (BarreraHinojosa & Li 2020a,b), a recently-introduced general-relativistic N -body code based on RAMSES (Teyssier 2002). We expand on previous studies in the following ways: (i) similarly to Jelic-Cizmek et al. (2018), we provide a direct calculation of the gravitomagnetic field, represented by the shift vector, from the simulation, also relaxing the weak-field approximation in our approach; (ii) we present results for scales in the deeply nonlinear regime which have not been previously explored in this context, and which are accessible thanks to the adaptive-mesh refinement (AMR) capabilities of GRAMSES. For the first time, we explore the gravitomagnetic vector potential in dark matter haloes in a broad range of halo masses; (iii) furthermore, we quantify the gravitomagnetic acceleration inside the dark matter haloes and compare this against the standard gravitational one.</text>
<text><location><page_2><loc_51><loc_53><loc_92><loc_72></location>Wenote that, with the exception of Jelic-Cizmek et al. (2018), previous studies of vorticity use simulations that incorporate a softening length scale, a numerical parameter used to prevent divergences in the calculation of inter-particle forces which also determines the spatial resolution. In GRAMSES - similarly to gevolution - the metric components and their spatial derivatives are calculated on a Cartesian mesh. AMR codes, such as GRAMSES, are generally slower than fixed-mesh-resolution codes such as gevolution which can benefit from efficient standard libraries such as FFTW, but their adaptivelyproduced mesh structure in high-density regions allows them to be more focused on the fine details in such regions, without increasing the overall cost of the simulation substantially. Therefore, they provide complementary ways to study the vector modes from cosmological simulations.</text>
<text><location><page_2><loc_51><loc_41><loc_92><loc_52></location>The rest of this paper is organised as follows. In Section 2 we fix our notations and briefly describe the general-relativistic formalism and methods implemented in the GRAMSES code that are relevant for the vector modes. In Section 3.1 we show the results for the different power spectra of the velocity field components as well as of the gravitomagnetic potential. Then, in Section 3.2, we focus on dark matter haloes, providing comparisons of the gravitomagnetic potential and corresponding acceleration with the scalar counterparts.</text>
<text><location><page_2><loc_51><loc_36><loc_92><loc_41></location>Throughout this paper, Greek indices are used to label spacetime vectors and run over (0 , 1 , 2 , 3) , while Latin indices run over (1 , 2 , 3) . Unless otherwise stated, we follow the unit convention that the speed of light c = 1 .</text>
<section_header_level_1><location><page_2><loc_51><loc_31><loc_74><loc_32></location>2 METHOD AND DEFINITIONS</section_header_level_1>
<text><location><page_2><loc_51><loc_23><loc_92><loc_30></location>For the sake of clarity and completeness, let us briefly summarise the terminology and conventions adopted in this paper, which in some part stem from GRAMSES ' implementation itself. More details can be found in the main code paper (Barrera-Hinojosa & Li 2020a) and the references therein.</text>
<text><location><page_2><loc_51><loc_19><loc_92><loc_23></location>In order to solve the gravitational sector equations and geodesic equations, GRAMSES adopts the 3+1 formalism in which the spacetime metric takes the form</text>
<formula><location><page_2><loc_51><loc_14><loc_92><loc_18></location>d s 2 = g µν d x µ d x ν = -α 2 d t 2 + γ ij ( β i d t +d x i )( β j d t +d x j ) , (1)</formula>
<text><location><page_2><loc_51><loc_6><loc_92><loc_13></location>where α is the lapse function, β i the shift vector and γ ij the induced metric on the spatial hypersurfaces, which in the constrained formulation adopted by GRAMSES is approximated by a conformally-flat metric, γ ij = ψ 4 δ ij , with ψ being the conformal factor and δ ij the Kronecker delta.</text>
<text><location><page_3><loc_8><loc_42><loc_9><loc_43></location>i</text>
<text><location><page_3><loc_7><loc_85><loc_48><loc_93></location>In the 3 + 1 formalism n µ = ( -α, 0) is the unit timelike vector normal to the time slices, the 3-dimensional spatial hypersurfaces with metric γ ij , and Eulerian observers are those with 4-velocity n µ . The energy density ρ and momentum density S i measured by these normal observers are given by the following projections of the energy-momentum tensor T µν ,</text>
<text><location><page_3><loc_8><loc_83><loc_9><loc_84></location>ρ</text>
<text><location><page_3><loc_11><loc_83><loc_12><loc_84></location>n</text>
<text><location><page_3><loc_12><loc_83><loc_12><loc_83></location>µ</text>
<text><location><page_3><loc_12><loc_83><loc_13><loc_84></location>n</text>
<text><location><page_3><loc_13><loc_83><loc_14><loc_83></location>ν</text>
<text><location><page_3><loc_14><loc_83><loc_15><loc_84></location>T</text>
<text><location><page_3><loc_17><loc_83><loc_17><loc_84></location>,</text>
<text><location><page_3><loc_46><loc_83><loc_48><loc_84></location>(2)</text>
<formula><location><page_3><loc_7><loc_80><loc_48><loc_82></location>S i ≡ -γ iµ n ν T µν , (3)</formula>
<text><location><page_3><loc_9><loc_82><loc_10><loc_84></location>≡</text>
<text><location><page_3><loc_7><loc_73><loc_48><loc_80></location>where the action of n µ projects onto the timelike direction, while γ µν = g µν + n µ n ν projects onto the spatial hypersurface. Eq. (2) and (3) are the source terms for the Hamiltonian constraint and momentum constraint, respectively. Additionally, the spatial stress and its trace are defined as</text>
<formula><location><page_3><loc_7><loc_70><loc_48><loc_72></location>S ij ≡ γ iµ γ jν T µν , S = γ ij S ij , (4)</formula>
<text><location><page_3><loc_7><loc_60><loc_48><loc_70></location>which, in addition to ρ and S i , appear in the evolution equations for the extrinsic curvature tensor. In GRAMSES, the (dark) matter sector is represented by an ensemble of non-interacting simulation particles of rest mass m and four-velocity u µ = dx µ /dτ , where τ is an affine parameter. The equations for the gravitational sector are numerically solved based on conformal matter sources, which are scaled using γ = det( γ ij ) as</text>
<formula><location><page_3><loc_7><loc_58><loc_48><loc_60></location>s 0 ( x ) ≡ √ γρ ∝ mαu 0 , (5)</formula>
<formula><location><page_3><loc_7><loc_54><loc_48><loc_56></location>s ij ( x ) ≡ √ γS ij ∝ m u i u j αu 0 . (7)</formula>
<formula><location><page_3><loc_8><loc_56><loc_48><loc_59></location>s i ( x ) ≡ √ γS i ∝ mu i , (6)</formula>
<text><location><page_3><loc_7><loc_46><loc_48><loc_53></location>In these, x is a (discrete) position vector on the cartesian simulation grid and the proportionality symbol in each equation stands for the standard cloud-in-cell (CIC) weights used for the particle-mesh projection (Hockney & Eastwood 1988). From Eqs. (5)-(7) we have the following useful relations:</text>
<formula><location><page_3><loc_7><loc_44><loc_48><loc_45></location>s 0 = ρ Γ , (8)</formula>
<text><location><page_3><loc_8><loc_42><loc_8><loc_43></location>s</text>
<text><location><page_3><loc_9><loc_42><loc_11><loc_43></location>=</text>
<text><location><page_3><loc_12><loc_42><loc_13><loc_43></location>u</text>
<text><location><page_3><loc_13><loc_42><loc_14><loc_43></location>i</text>
<text><location><page_3><loc_14><loc_42><loc_14><loc_43></location>,</text>
<text><location><page_3><loc_46><loc_42><loc_48><loc_43></location>(9)</text>
<formula><location><page_3><loc_7><loc_39><loc_48><loc_41></location>s ij = ρ Γ 2 u i u j , = ⇒ s = ρ (1 -Γ -2 ) , (10)</formula>
<text><location><page_3><loc_11><loc_43><loc_12><loc_44></location>ρ</text>
<text><location><page_3><loc_11><loc_41><loc_12><loc_42></location>Γ</text>
<formula><location><page_3><loc_7><loc_36><loc_48><loc_39></location>u i = Γ 2 s i s 0 , (11)</formula>
<text><location><page_3><loc_7><loc_30><loc_48><loc_36></location>where Γ ≡ αu 0 = √ 1 + γ ij u i u j is the Lorentz factor. For a perfect fluid, s ≡ √ γS is proportional to pressure in linear theory, and then it vanishes for CDM (dust) in such regime. Naturally, s also vanishes in the non-relativistic limit.</text>
<text><location><page_3><loc_7><loc_26><loc_48><loc_30></location>The equations of motion for collisionless particles correspond to the geodesic equation u µ ∇ µ u ν = 0 , which in the 3 + 1 form reads</text>
<formula><location><page_3><loc_7><loc_24><loc_48><loc_26></location>du i dt = -Γ ∂ i α + u j ∂ i β j -α u j u k 2Γ ∂ i γ jk , (12)</formula>
<formula><location><page_3><loc_7><loc_21><loc_48><loc_24></location>dx i dt = α γ ij u j Γ -β i . (13)</formula>
<text><location><page_3><loc_7><loc_13><loc_48><loc_20></location>In Eq. (12), the term u j ∂ i β j corresponds to a force that is absent in both the Newtonian limit and the linear perturbation regime. In the case where β j is purely a vector-type perturbation (e.g., the Poisson gauge), this force term is known as gravitomagnetic force , in formal analogy with the magnetic Lorentz force.</text>
<section_header_level_1><location><page_3><loc_7><loc_10><loc_24><loc_11></location>2.1 Vector decomposition</section_header_level_1>
<text><location><page_3><loc_7><loc_6><loc_48><loc_9></location>Given that in this paper we are particularly interested in vector modes (transverse modes), we start by splitting a vector field V i</text>
<text><location><page_3><loc_15><loc_83><loc_17><loc_84></location>µν</text>
<text><location><page_3><loc_51><loc_92><loc_52><loc_93></location>(</text>
<text><location><page_3><loc_52><loc_92><loc_53><loc_93></location>V</text>
<text><location><page_3><loc_53><loc_92><loc_55><loc_93></location>) as</text>
<formula><location><page_3><loc_51><loc_90><loc_92><loc_91></location>V = V ‖ + V ⊥ , (14)</formula>
<text><location><page_3><loc_51><loc_86><loc_92><loc_89></location>where V ‖ and V ⊥ are respectively the scalar (irrotational) and vector (rotational) components, i.e., these satisfy</text>
<formula><location><page_3><loc_51><loc_84><loc_92><loc_86></location>∇ × V ‖ = 0 , ∇ · V ⊥ = 0 . (15)</formula>
<text><location><page_3><loc_51><loc_81><loc_92><loc_84></location>In the case of the velocity field 2 u i ( u ), we define the velocity divergence and vorticity as</text>
<formula><location><page_3><loc_51><loc_76><loc_92><loc_80></location>θ ≡ ∇ · u , (16) ω ≡ ∇ × u . (17)</formula>
<text><location><page_3><loc_51><loc_74><loc_92><loc_76></location>As usual, the power spectra of these quantities are respectively defined as</text>
<formula><location><page_3><loc_54><loc_71><loc_92><loc_73></location>〈 θ ( k ) θ ∗ ( k ' ) 〉 = δ ( k -k ' )(2 π ) 3 P θ ( k ) , (18)</formula>
<formula><location><page_3><loc_51><loc_68><loc_90><loc_71></location>〈 ω i ( k ) ω ∗ j ( k ' ) 〉 = δ ( k -k ' )(2 π ) 3 1 2 ( δ ij -k i k j k 2 ) P ω ( k ) ,</formula>
<text><location><page_3><loc_90><loc_67><loc_92><loc_68></location>(19)</text>
<text><location><page_3><loc_51><loc_65><loc_83><loc_66></location>and the velocity power spectrum satisfies the relation</text>
<formula><location><page_3><loc_51><loc_63><loc_92><loc_64></location>P | u | = k 2 ( P θ + P ω ) . (20)</formula>
<text><location><page_3><loc_51><loc_60><loc_92><loc_62></location>The power spectrum of the vector modes of the shift vector is defined in analogous way to Eq. (19).</text>
<section_header_level_1><location><page_3><loc_51><loc_56><loc_90><loc_57></location>2.2 Gauge choice and the constraint for the vector potential</section_header_level_1>
<text><location><page_3><loc_51><loc_29><loc_92><loc_55></location>For solving the gravitational and geodesic equations, GRAMSES implements a constrained formulation of GR (Bonazzola et al. 2004; Cordero-Carrión et al. 2009), in which both the tensor modes of the spatial metric and the transverse-traceless (TT) part of the extrinsic curvature are neglected during the evolution. In contrast, the scalar and vector modes of the gravitational field are treated fully nonlinearly. In order to do this in a robust way, the formalism adopts the constant-mean-curvature slicing (Smarr & York 1978b; Shibata 1999; Shibata & Sasaki 1999) and a minimal-distortion gauge condition under the conformal flatness approximation (Smarr & York 1978a). Contrary to the Poisson gauge, in this gauge the shift vector contains both scalar and vector ( 1+2 ) degrees of freedom. At linear order, the latter modes match the gauge-invariant shift vector from the Poisson gauge (Matarrese et al. 1998a; Lu et al. 2009), while the mismatch in the scalar piece reflects the fact that the time foliations are different in these two gauges. Then, in this formalism the components of the shift vector are solved from a combination of the 3 + 1 evolution equation for the extrinsic curvature, the momentum constraint and the gauge conditions (Barrera-Hinojosa & Li 2020a)</text>
<formula><location><page_3><loc_51><loc_26><loc_92><loc_28></location>(∆ L β ) i = 16 παψ -6 s i + ∂ j ( αψ -6 ) ¯ A ij L , (21)</formula>
<text><location><page_3><loc_51><loc_23><loc_92><loc_26></location>where s i = δ ij s i , (∆ L β ) i := ∂ 2 β i + ∂ i ( ∂ j β j ) / 3 denotes the flatspace vector Laplacian operator, and</text>
<formula><location><page_3><loc_51><loc_20><loc_92><loc_22></location>¯ A ij L = ∂ i W j + ∂ j W i -2 3 δ ij ∂ k W k , (22)</formula>
<text><location><page_3><loc_51><loc_16><loc_92><loc_19></location>is the longitudinal part of the traceless extrinsic curvature tensor. The auxiliary potential W i introduced in Eq. (22) is directly solved from the momentum constraint equation,</text>
<formula><location><page_3><loc_51><loc_13><loc_92><loc_15></location>(∆ L W ) i = 16 πs i . (23)</formula>
<text><location><page_3><loc_51><loc_6><loc_92><loc_11></location>2 We use u to represent the velocity u i rather than u i , as it is the former that is used in the 3 + 1 form of the geodesic equations (12) and (13) which are implemented in GRAMSES. u i is what we call 'CMC-MD-gauge velocity', and is different from u i . See Barrera-Hinojosa & Li (2020b) for more details.</text>
<text><location><page_4><loc_7><loc_85><loc_48><loc_93></location>Then, from Eq. (21) we note that, at leading order, the shift vector is sourced by the momentum field and thus β i ∝ W i by Eq. (23), while differences appear at higher order due to the extrinsic curvature tensor sourcing β i . Given that throughout this paper we will be interested in the vector modes of the shift vector, this is decomposed in the same fashion of Eq. (14), i.e.</text>
<formula><location><page_4><loc_7><loc_83><loc_48><loc_84></location>β i = B i + β i ‖ , (24)</formula>
<text><location><page_4><loc_7><loc_76><loc_48><loc_82></location>where B i ≡ β i ⊥ ( B ) is referred to as the vector potential or gravitomagnetic potential, and β i ‖ is the scalar mode of the shift. Let us note that, using Eq. (9), the curl of the conformal momentum density field s i ( s ) can be written non-perturbatively as</text>
<formula><location><page_4><loc_7><loc_73><loc_48><loc_75></location>∇ × s = Γ -1 [(1 + δ ) ω + ∇ δ × u -∇ Γ × s ] , (25)</formula>
<text><location><page_4><loc_7><loc_61><loc_48><loc_73></location>where δ = ρ/ ¯ ρ -1 is the density contrast and ¯ ρ is the mean density. Previous studies have shown that the terms δ ω and ∇ δ × u in the r.h.s. of Eq. (25) are the main sources for the vector potential (Bruni et al. 2014; Thomas et al. 2015b; Jelic-Cizmek et al. 2018), while the contribution from vorticity itself is subdominant at all scales. In the r.h.s. of Eq. (25), the last term and the overall modulation by the Lorentz Factor Γ arise due to the definition of s in Eq. (9), and both contributions vanish in the linear regime and the non-relativistic limit.</text>
<section_header_level_1><location><page_4><loc_7><loc_56><loc_15><loc_57></location>3 RESULTS</section_header_level_1>
<text><location><page_4><loc_7><loc_41><loc_48><loc_55></location>For the investigation in this paper, we have run a high-resolution simulation using GRAMSES, with a comoving box size L box = 256 h -1 Mpc and N part = 1024 3 dark-matter particles, corresponding to a particle mass resolution of 1 . 33 × 10 9 h -1 M glyph[circledot] . Because GRAMSES makes use of AMR in high-density regions, the spatial resolution is not uniform throughout the simulation volume: while the coarsest (domain) grid has N part cells, corresponding to a comoving spatial resolution of 0 . 25 h -1 Mpc , the most refined (high density) regions reach a resolution of 128 3 × N part grid elements, with corresponding spatial resolution of 2 h -1 kpc .</text>
<text><location><page_4><loc_7><loc_27><loc_48><loc_41></location>Initial conditions suitable for the relativistic simulation were generated at z = 49 with a modified version of 2LPTic code (Crocce et al. 2006) fed with the matter power spectrum obtained from a modified version of CAMB (Lewis et al. 2000) that works for the particular gauge needed for GRAMSES. More details on this can be found in Barrera-Hinojosa & Li (2020b). The cosmological parameters adopted for the simulation are { Ω Λ , Ω m , Ω K , h } = { 0 . 693 , 0 . 307 , 0 , 0 . 68 } and a primordial spectrum with amplitude A s = 2 . 1 × 10 -9 , spectral index n s = 0 . 96 and a pivot scale k pivot = 0 . 05 Mpc -1 .</text>
<text><location><page_4><loc_7><loc_6><loc_48><loc_27></location>In order to measure the velocity fields from simulation snapshots, we use the publicly-available DTFE code (Cautun & van de Weygaert 2011) which is based on the Delaunay tessellation method, although other methods have been explored in the literature during the last few years. Notably, the phase-interpolation method introduced in Abel et al. (2012) shows better performance than DTFE in shell-crossing regions (Hahn et al. 2015), where the finite-difference estimation of velocity divergence and vorticity across caustics can be problematic due to the multiply-valued nature of the velocity field. Nonetheless, the power spectra of these two fields are not strongly affected by this since the volume-weighted contribution from caustics is negligible, and both methods converge when nonlinear scales are well resolved. In addition, while the vorticity power spectrum is affected by resolution effects, this is weakly affected by finite-volume effects (Pueblas &Scoccimarro 2009; Jelic-Cizmek et al. 2018). We note that, while</text>
<text><location><page_4><loc_51><loc_79><loc_92><loc_93></location>the initial velocity field is vorticity-free by construction, spurious vorticity will be present at some degree due to the numerical errors introduced by particle-mesh projections. In addition, shell-crossing events - which source vorticity - are rare at high redshift, and its insufficient sampling restricts the possibility of estimating the velocity field robustly. Therefore, in this paper we shall focus mainly on low redshifts, z < 1 . 5 , at which vorticity results are expected to be robust. Contrary to the velocity field, the gravitomagnetic potential is already solved by the code on a Cartesian mesh so there is no need for post-processing particle-mesh projections.</text>
<text><location><page_4><loc_51><loc_57><loc_92><loc_79></location>It is worthwhile to mention that, although the GR simulations do not necessitate the specification of a cosmological background (Barrera-Hinojosa & Li 2020a), throughout this paper the notion of redshift is still used and should be understood as the standard, background one. This is achieved through the constant-mean-curvature slicing condition, which allows us to fix the trace of the extrinsic curvature of the spatial hypersurfaces as K = -3 H ( t ) , where the Hubble parameter H can be conveniently fixed via 'fiducial' Friedmann equations (Giblin et al. 2019; Barrera-Hinojosa & Li 2020a). In addition, even though in the gauge adopted by GRAMSES the scalar gravitational potentials as well as the matter fields are not gaugeinvariant quantities, gauge effects are only prominent on large scales and become strongly suppressed for modes inside the horizon. Since in this work we are mainly interested in the latter, as well as in redshifts below z = 1 . 5 (in which the horizon is already larger than the box size), we do not explore potential gauge issues further.</text>
<text><location><page_4><loc_51><loc_33><loc_92><loc_56></location>Figure 1 provides a visual representation of the density field (top left), velocity divergence (top right), the magnitude 3 of the vorticity vector field, ω ≡ | ω | = ( ω 2 x + ω 2 y + ω 2 z ) 1 / 2 (bottom left), and the vector potential magnitude, B ≡ | B | = ( B 2 x + B 2 y + B 2 z ) 1 / 2 (bottom right), across a slice of the simulation box at z = 0 . From this figure, it is clear that the density field has a similar large-scale distribution to the velocity divergence, consistently with linear perturbation theory. Since velocity divergence can take negative values, we use a linear scale on its map, with a cutoff of extreme values to help visualisation. As expected, the velocity divergence is negative in collapsing regions due to matter in-fall, and positive in voids and low-density regions. The vorticity field also shows a clear correlation with both density and velocity divergence. However, we should bear in mind that, as we have discussed before, the velocity divergence and vorticity estimated by DTFE are not completely reliable near caustics (Hahn et al. 2015), and therefore such maps only provide qualitative information and an accurate picture on large scales.</text>
<text><location><page_4><loc_51><loc_12><loc_92><loc_32></location>From the bottom right panel in Fig. 1, we observe that the magnitude of the vector potential has some degree of correlation with the structures observed in density, velocity divergence and vorticity, particularly in very high-density and low-density regions. As shown by Eqs. (21)-(25), the vector potential is not sourced by any of these components alone but is correlated with the rotational part of the full momentum density field. This panel also shows that the distribution of the vector potential magnitude is a great deal smoother than the cases of matter and velocity fields. This is expected since the vector potential components satisfy the elliptic-type equation (21), and then long-wavelength modes become dominant due to the Laplacian operator ∂ 2 . Although not included here, the same happens in the case of the conformal factor ψ which satisfies the Hamiltonian constraint (or the Poisson equation in the Newtonian limit). From the quantitative side, we note that the vector potential magnitude seems to</text>
<figure>
<location><page_5><loc_10><loc_60><loc_47><loc_91></location>
<caption>Figure 1. (Colour Online) A slice of the simulation box at z = 0 showing the density (top left), velocity divergence (top right), vorticity (bottom left) and vector potential magnitude (bottom right) fields. The velocity values shown are normalised by H f , where H ≡ aH is the conformal Hubble parameter and f is the linear growth rate in Λ CDM. The density field is normalised by its mean value in the simulation box.</caption>
</figure>
<text><location><page_5><loc_10><loc_58><loc_13><loc_60></location>10</text>
<text><location><page_5><loc_13><loc_58><loc_14><loc_60></location>-</text>
<text><location><page_5><loc_14><loc_59><loc_15><loc_60></location>1</text>
<text><location><page_5><loc_18><loc_58><loc_20><loc_60></location>10</text>
<text><location><page_5><loc_20><loc_59><loc_21><loc_60></location>0</text>
<text><location><page_5><loc_24><loc_58><loc_27><loc_60></location>10</text>
<text><location><page_5><loc_27><loc_59><loc_28><loc_60></location>1</text>
<text><location><page_5><loc_31><loc_58><loc_33><loc_60></location>10</text>
<text><location><page_5><loc_33><loc_59><loc_34><loc_60></location>2</text>
<text><location><page_5><loc_37><loc_58><loc_40><loc_60></location>10</text>
<text><location><page_5><loc_40><loc_59><loc_41><loc_60></location>3</text>
<text><location><page_5><loc_44><loc_58><loc_46><loc_60></location>10</text>
<text><location><page_5><loc_46><loc_59><loc_47><loc_60></location>4</text>
<figure>
<location><page_5><loc_9><loc_24><loc_47><loc_57></location>
</figure>
<figure>
<location><page_5><loc_51><loc_24><loc_90><loc_57></location>
</figure>
<text><location><page_5><loc_54><loc_57><loc_56><loc_60></location>-</text>
<text><location><page_5><loc_62><loc_57><loc_64><loc_60></location>-</text>
<text><location><page_5><loc_69><loc_57><loc_71><loc_60></location>-</text>
<text><location><page_5><loc_77><loc_57><loc_78><loc_60></location>-</text>
<text><location><page_5><loc_7><loc_11><loc_48><loc_14></location>typically remain between O (10 -8 ) and O (10 -7 ) , with some peaks of a few times O (10 -7 ) only in very specific regions.</text>
<text><location><page_5><loc_7><loc_6><loc_48><loc_9></location>We will explore the behaviour of the vector modes in more detail in the next sections.</text>
<section_header_level_1><location><page_5><loc_51><loc_13><loc_63><loc_14></location>3.1 Power spectra</section_header_level_1>
<text><location><page_5><loc_51><loc_6><loc_92><loc_11></location>In this section we analyse the power spectra of the velocity field and gravitomagnetic vector potential. The auto and cross spectra of matter quantities such as density, velocity divergence and vorticity (which are measured with DTFE from particle data) are calculated</text>
<figure>
<location><page_5><loc_52><loc_59><loc_90><loc_91></location>
</figure>
<text><location><page_6><loc_7><loc_75><loc_48><loc_93></location>using NBODYKIT (Hand et al. 2018). In contrast, the vector (as well as scalar) potential values are calculated and stored by GRAMSES in cells of hierarchical AMR meshes, and the spectrum is measured by a different code that is able to handle such mesh data directly and to write it on a regular grid by interpolation. While the vector potential spectrum can also be measured in the same way as the matter quantities by writing its values at the particles' positions rather than in AMR cells, which means DTFE and NBODYKIT can be used, the above method yields better results on small scales as shown in Appendix A. In all figures, we normalise the velocity power spectra by the factor ( H f ) 2 , where H = aH is the conformal Hubble parameter of the reference Friedmann universe, and f the linear growth rate in Λ CDMparameterised as (Linder 2005)</text>
<formula><location><page_6><loc_7><loc_73><loc_48><loc_74></location>f ( a ) = Ω m ( a ) 6 / 11 , (26)</formula>
<text><location><page_6><loc_7><loc_67><loc_48><loc_72></location>where Ω m ( a ) = Ω m a -3 / ( H/H 0 ) 2 . In this way, the amplitude of P θ matches that of the matter power spectrum in the linear regime, where the continuity equation δ = -θ/ ( H f ) is expected to hold.</text>
<text><location><page_6><loc_7><loc_38><loc_48><loc_68></location>Figure 2 shows the velocity divergence power spectrum (top left panel), the vorticity power spectrum (top right panel), the cross spectrum between density and velocity divergence (bottom left) and the power spectrum of two different contributions to the momentum field (bottom right) at different redshifts in the range 0 ≤ z ≤ 1 . 5 . In the case of velocity divergence, we find a very good agreement with linear theory at scales k ≤ 0 . 1 h Mpc -1 for all redshifts. Above that scale, deviations become stronger towards lower redshifts, and a localised power loss ('dip') eventually develops around k ≈ 1 . 2 h Mpc -1 . In the case of the vorticity power spectrum, we note that towards large scales this is several orders of magnitude smaller than velocity divergence, while at around k ∼ 1 h Mpc -1 the spectrum starts to peak and they become comparable. Note that, unlike the velocity divergence, there is no standard perturbation theory prediction for the vorticity as this exactly vanishes in the perfect fluid description. Interestingly, the 'dip' in the velocity divergence power spectrum is at the similar position to the peak in the vorticity power spectrum, which has been interpreted as the consequence of shell crossing occurring around that scales, where the angular momentum can be large enough to dampen the growth of structures as it forces particles to rotate around them (Jelic-Cizmek et al. 2018).</text>
<text><location><page_6><loc_7><loc_16><loc_48><loc_38></location>Note that, due to the high cost 4 of GR simulations using GRAMSES, we have not performed runs with even higher resolutions to check the convergence of the velocity and vorticity power spectra. A useful convergence test for gevolution simulations was done in Jelic-Cizmek et al. (2018) (see Fig. 6 there), which shows that the amplitude of P ω decreases as the force resolution increases. The simulations there have the same box size of L box = 256 h -1 Mpc , and the run labelled 'high resolution 1' has the same mesh resolution as our domain grid ( 1024 3 cells); while this resolution is eight times poorer than that of the run labelled 'high resolution 2', which has 2048 3 cells, the AMR nature of GRAMSES means that higher resolution can be achieved in high-density regions - with the highest resolution attained in our run being equivalent to a regular mesh with 128 3 × 1024 3 cells. Hence, since 'high resolution' 1 and 2 are already converged in Jelic-Cizmek et al. (2018), we conclude that our simulation has also converged to at least a similar level.</text>
<text><location><page_6><loc_7><loc_6><loc_48><loc_14></location>4 A GR simulation using GRAMSES takes about an order of magnitude longer than an equivalent Newtonian simulation using default RAMSES, partly due to the 10 (compared to one) GR metric potentials to be solved, and partly due to the cost of preparing the source terms for the nonlinear equations that govern the metric potentials, as well as the additional MPI communications.</text>
<text><location><page_6><loc_51><loc_79><loc_92><loc_93></location>The cross spectra P δθ is useful for detecting deviations from linear theory and provides information about shell crossing. Considering the continuity equation, the linear-theory expectation is that P θδ / ( H f ) = -P δ , but towards shell-crossing scales the initial (linear) anti-correlation of δ and θ is lost and correlations appear (Hahn et al. 2015). From the bottom left panel of Fig. 2 we find that the anti-correlation drops dramatically and flips sign at k ≈ 2 h Mpc -1 at z = 0 , which is slightly higher than the scale at which the vorticity spectrum peaks as also found in previous studies (Jelic-Cizmek et al. 2018).</text>
<text><location><page_6><loc_51><loc_68><loc_92><loc_79></location>The bottom right panel of Fig. 2 shows the power spectra of δ ω and ∇ δ × u , which are the main source terms for the metric vector potential in Eq. (25). In particular, the contribution of ω to Eq. (25) is already small compared to δ ω on nonlinear scales because δ glyph[greatermuch] 1 . We find good agreement with the z = 0 results shown in Bruni et al. (2014) based on a post-Friedmann expansion. We find that towards higher redshifts the contribution due to ∇ δ × u starts to become larger than that of δ ω at slightly larger scales.</text>
<text><location><page_6><loc_51><loc_61><loc_92><loc_68></location>Although vorticity vanishes in standard perturbation theory, the effective field theory of LSS (EFTofLSS) predicts that its power spectrum today can be characterised by a power law over a range of scales (Carrasco et al. 2014). On large scales, we can find the slope of the vorticity power spectrum by fitting a power law,</text>
<formula><location><page_6><loc_51><loc_59><loc_92><loc_60></location>P ω ( k ) = A ω k n ω , (27)</formula>
<text><location><page_6><loc_51><loc_45><loc_92><loc_58></location>where n ω is the large-scale spectral index, and A ω the amplitude that is not fixed by theory. The EFTofLSS predicts n ω = 3 . 6 for 0 . 1 h Mpc -1 glyph[lessorsimilar] k glyph[lessorsimilar] 0 . 2 h Mpc -1 and n ω = 2 . 8 for 0 . 2 h Mpc -1 glyph[lessorsimilar] k glyph[lessorsimilar] 0 . 6 h Mpc -1 (Carrasco et al. 2014). Previous N -body simulations have found n ω ≈ 2 . 5 for k glyph[lessorsimilar] 0 . 1 h Mpc -1 (Hahn et al. 2015); a similar value was found at k glyph[lessorsimilar] 0 . 4 h Mpc -1 in Jelic-Cizmek et al. (2018). Moreover, on scales k glyph[greaterorsimilar] 1 h Mpc -1 , there is partial evidence suggesting that the spectral index approaches the asymptotic value n NL ω →-1 . 5 (Hahn et al. 2015).</text>
<text><location><page_6><loc_51><loc_19><loc_92><loc_45></location>Figure 3 shows the best fits of the power law (27) to the simulation data at z = 0 on large scales (small scales) with their corresponding spectral index n ω ( n NL ω ), and the shaded region represents the interval of validity for the fit. On large sub-horizon scales, we find n ω ≈ 2 . 7 , which is slightly higher than previous simulations results in the literature, and slightly lower than the EFTofLSS prediction. Notice, however, that there is not complete overlap between the region used for the fit and the EFTofLSS prediction used for comparison as the latter extends up to k ∼ 0 . 6 h Mpc -1 but it is clear that the slope of the power spectrum already decreases at k ∼ 0 . 32 h Mpc -1 . In addition, the slope does not seem to become steeper at larger scales as predicted by the EFTofLSS, a feature also found by the previous study (Jelic-Cizmek et al. 2018), which is likely related to the large-scale cutoff imposed by the finite box of the simulation. Toward smaller scales, we find the spectral index n NL ω ≈ -1 . 4 , which is slightly less steep than that suggested in Hahn et al. (2015). However, there is a slight but clear increase in power at around k ∼ 7 h Mpc -1 which introduces an oscillatory feature not captured by a perfect power law.</text>
<text><location><page_6><loc_51><loc_15><loc_92><loc_18></location>As originally proposed in Pueblas & Scoccimarro (2009), it is also interesting to characterise the evolution of the large-scale vorticity power spectrum as</text>
<formula><location><page_6><loc_51><loc_11><loc_92><loc_14></location>P ω ( k ; z ) = ( D + ( z ) D + (0) ) γ ω P ω ( k ; z = 0) , (28)</formula>
<text><location><page_6><loc_51><loc_5><loc_92><loc_10></location>where D + ( z ) is the linear growth rate at z and γ ω a new parameter. In Pueblas & Scoccimarro (2009), the best-fit value found is γ ω = 7 ± 0 . 3 using the snapshots z = 0 , 1 , 3 , which is overall consistent</text>
<figure>
<location><page_7><loc_8><loc_39><loc_91><loc_93></location>
<caption>Figure 2. (Colour Online) Various auto and cross power spectra involving the velocity field for z = 0 (black), z = 0 . 5 (orange), z = 1 (red) and z = 1 . 5 (blue). The top left and top right panels show the velocity divergence power spectrum and vorticity power spectrum, respectively, both of which are normalised by ( H f ) 2 . Bottom left: the cross spectrum between density and velocity divergence. Since in linear theory P δθ < 0 , we plot its absolute magnitude normalised by H f . The discontinuity corresponds to the flip in sign on nonlinear scales, after which density and velocity divergence become correlated. Bottom right: the power spectrum of δ ω and ∇ δ × u , which are the main source terms for the metric vector potential, c.f. Eq. (25). These are normalised by ( H f ) 2 . In the two left panels, the solid lines denote the corresponding linear-theory predictions.</caption>
</figure>
<text><location><page_7><loc_7><loc_18><loc_48><loc_26></location>with Thomas et al. (2015b); Jelic-Cizmek et al. (2018), although the latter references suggest values γ ω ≥ 7 . Moreover, these have only considered snapshots with z ≤ 1 since the scaling breaks down at higher redshifts, which is likely related to resolution effects in the sampling of vorticity due to a lower fraction of particles undergoing shell crossing at higher redshifts.</text>
<text><location><page_7><loc_7><loc_6><loc_48><loc_17></location>The top panels of Fig. 4 show the results for the best fist of the D γ ω + scaling of Eq. (28) using several snapshots below z = 1 . 5 . The top left panel of Fig. 4 shows the power spectrum at these various redshifts scaled using ( D + ( z ) /D + (0)) 7 . 7 , while in the top right panel we select three different modes from the shaded green region of the top left panel and find the corresponding value of γ ω from a best fit to the corresponding vorticity spectra. We find that there is some scale dependence in γ ω and the amplitude of the vorticity</text>
<text><location><page_7><loc_51><loc_9><loc_92><loc_26></location>power spectrum evolves approximately with γ ≈ 7 . 7 over the scales 0 . 08 glyph[lessorsimilar] k glyph[lessorsimilar] 0 . 4 , which is higher than other simulation results in the literature (Pueblas & Scoccimarro 2009; Thomas et al. 2015b; JelicCizmek et al. 2018). However, compared to the latter two references, in the case here we are able to fit the amplitude up to z = 1 . 5 before the scaling breaks down. Besides the results from Jelic-Cizmek et al. (2018) based on the gevolution code, which works in a fixedresolution grid, previous studies of vorticity use N -body simulation codes in which a softening length scale in the force calculation determines the spatial resolution. In the case of GRAMSES, the AMR capabilities allow one to achieve high spatial resolution ( ∼ 2 h -1 kpc ) in high-density regions.</text>
<text><location><page_7><loc_51><loc_6><loc_92><loc_9></location>We can extend the previous analysis to model the time evolution of the vorticity power spectrum at nonlinear scales, in terms of a new</text>
<figure>
<location><page_8><loc_8><loc_67><loc_47><loc_93></location>
<caption>Figure 3. (Colour Online) Power-law fitting of the vorticity power spectrum at z = 0 . The solid blue and solid red lines show the best fits of the simulation data (black dots) on large and small scales, respectively, while the shaded regions represent the validity interval for each fit. As a reference, the dashed magenta line shows the EFTofLSS prediction from Carrasco et al. (2014) for the region 0 . 2 h Mpc -1 glyph[lessorsimilar] k glyph[lessorsimilar] 0 . 6 h Mpc -1 , which only has a small overlap with the fitting region used on large sub-horizon scales.</caption>
</figure>
<text><location><page_8><loc_7><loc_25><loc_48><loc_48></location>scale-independent parameter γ NL ω in Eq. (28). From Fig. 2, it is clear that the power spectrum evolves more slowly in this regime compared with large scales, and so we expect γ NL ω to be smaller than γ ω . In the bottom left panel of Fig. 4, we show the scaling of the vorticity spectra by ( D + ( z ) /D + (0)) 2 . 6 , where we find that such evolution works as a good approximation on scales k glyph[greaterorsimilar] 3 . 2 h Mpc -1 . In the bottom right panel we show the best-fit value of γ NL ω for three different k -modes. In this case, unlike in the previous fit for large subhorizon scales, we have not considered the z = 1 . 5 spectrum for the fit as from the bottom left panel it is already clear that the scaling for such spectrum (orange solid line) would deviate from the lower redshift results. This result suggests that the amplitude of the vorticity power spectrum can be actually estimated using a scale-independent parameter in the power law of Eq. (28) on deeply nonlinear scales. However, there is an obvious scale dependence in the transition between the large- and small-scale regimes which is not captured by these parameterisations and requires further investigation.</text>
<text><location><page_8><loc_7><loc_13><loc_48><loc_24></location>Let us now discuss the results for the vector potential. In Λ CDM cosmology, this appears as a second-order perturbation at its lowest order, which in the case of a perfect fluid is sourced by the product of the first-order density contrast and velocity divergence (Matarrese et al. 1998b; Lu et al. 2009). However, the single-stream fluid description of CDM breaks down at late times when shell crossing occurs, and then we expect corrections to the vector potential particularly at quasi-linear and nonlinear scales.</text>
<text><location><page_8><loc_7><loc_10><loc_48><loc_13></location>The second-order perturbation theory prediction for the dimensionless power spectrum of B ,</text>
<formula><location><page_8><loc_7><loc_6><loc_48><loc_8></location>∆ B ( k ) ≡ k 3 2 π 2 P B ( k ) , (29)</formula>
<text><location><page_8><loc_51><loc_92><loc_68><loc_93></location>is given by (Lu et al. 2009)</text>
<formula><location><page_8><loc_51><loc_85><loc_93><loc_91></location>∆ B ( k ) = 9Ω 2 m H 4 0 2 a 2 k 2 ∫ ∞ 0 d w × (30) ∫ 1+ w | 1 -w | d u Π [ ∆ δ ( ku )∆ v ( kw ) -w u ∆ δv ( ku )∆ δv ( kw ) ] ,</formula>
<text><location><page_8><loc_51><loc_37><loc_92><loc_84></location>where ∆ δ and ∆ v are the dimensionless power spectra of the density perturbation and velocity potential v , ∆ δv their cross spectrum, and Π( u, w ) = u -2 w -4 [ 4 w 2 -(1 + w 2 -u 2 ) 2 ] is an integration kernel that depends on w = k ' /k and u = √ 1 + w 2 -2 w cos ϑ , with cos ϑ defined by cos ϑ = k ' · k / ( kk ' ) . At any given scale, the convolution in Eq. (30) couples different k -modes of δ and v . Since the simulation can only access modes within a finite k -range, this is equivalent to having a large-scale ( k min ) and small-scale ( k max ) cutoffs in Eq. (30), therefore leading to a lower amplitude of P B than the true result. For instance, Adamek et al. (2016b) found that in order to get good agreement between simulation results and perturbation-theory calculations using Eq. (30), the box should be large enough to contain the matter-radiation equality scale. In practice, to account for this effect due to missing k -modes, to compare with Eq. (30), we use the large-scale cutoff k min ∼ 0 . 8 × 2 π/L , i.e. 80 percent of the fundamental mode of the box, as well as a small-scale cutoff k max = πN 1 / 3 part /L , which corresponds to the Nyquist wavenumber of the coarsest grid used by the simulation. The left panel of Fig. 5 shows the simulation measurements of the dimensionless power spectrum of the vector potential at four different redshifts, and their corresponding perturbation-theory predictions. At z ≥ 1 we see good agreement between the simulation and perturbation-theory results up to k ∼ 0 . 3 h Mpc -1 , while at z = 0 discrepancies start already at k ∼ 0 . 2 h Mpc -1 , which is qualitatively consistent with Adamek et al. (2014); Bruni et al. (2014); see also Andrianomena et al. (2014) for a prescription of the nonlinear corrections to the perturbation-theory result using HALOFIT. At highly nonlinear scales the amplitude of the spectrum measured from the simulation can be more than two orders of magnitude higher than the perturbation-theory prediction. Note that at all four redshifts the simulation spectra flatten at the largest k -mode sampled by the simulation box, which can be interpreted as a finite-box effect.</text>
<text><location><page_8><loc_51><loc_15><loc_92><loc_37></location>The right panel of Fig. 5 shows the ratio between the power spectra of vector potential B and that of the scalar potential Φ measured from the simulation, the latter defined as the fully nonlinear perturbation to the lapse function in the metric (1), i.e. Φ ≡ α -1 . At z = 0 , we find the ratio to be within 2 × 10 -5 and 4 × 10 -5 for 0 . 2 h Mpc -1 glyph[lessorsimilar] k glyph[lessorsimilar] 10 h Mpc -1 , which is in good agreement with Bruni et al. (2014). The ratio reaches a peak of 5 × 10 -5 at k ∼ 15 h Mpc -1 , after which it starts to decrease. At higher redshift the evolution of B makes the ratio larger. Our results confirm that the ratio between both potentials reach the percent-level on nonlinear scales at z = 0 . As pointed out by Bruni et al. (2014), though this ratio is close to the value found in Lu et al. (2009) for the ratio between scalar and vector modes in perturbation theory, here the fully nonlinear B , Φ fields are used. In fact, the vector potential power spectrum from the left panel of Fig. 5 can be over two orders of magnitude larger than that found in the latter reference.</text>
<section_header_level_1><location><page_8><loc_51><loc_10><loc_90><loc_12></location>3.2 The vector potential and frame-dragging acceleration in dark matter haloes</section_header_level_1>
<text><location><page_8><loc_51><loc_6><loc_92><loc_9></location>Let us further analyse the vector potential on nonlinear scales by investigating its magnitude inside the dark matter haloes from the</text>
<figure>
<location><page_9><loc_8><loc_39><loc_91><loc_92></location>
<caption>Figure 4. (Colour Online) Power-law modelling of the time evolution of the vorticity power spectrum based on Eq. (28). Top panels show results for the large scales regime and the bottom panels analogous results for nonlinear scales. Top left: vorticity power spectra at different redshifts scaled using γ ω = 7 . 7 . Shaded regions represent the interval of validity considered for the fit, and the colors { orange, red, purple, cyan, gray, blue, magenta, green, brown, yellow, brown, black } correspond to z = { 1.5, 1, 0.85, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2, 0.1, 0 } , respectively. Bottom left: similar to top left panel but for nonlinear scales. Right panels: Time evolution of the vorticity power spectrum for a set of fixed k -modes as a function of D + ( z ) (normalised by today's value of D + ). The solid lines correspond the best fit curves with the respective power-law indices γ ω and γ NL ω shown. On the bottom right panel, the data point for z = 1 . 5 has not been included for the fit, as the bottom left panel shows a clear discrepancy with lower redshifts.</caption>
</figure>
<text><location><page_9><loc_7><loc_15><loc_48><loc_24></location>above general-relativistic simulation. For this we have generated halo catalogues using the phase-space Friends-of-Friends halo finder ROCKSTAR (Behroozi et al. 2013). From this catalogue we then get their centre positions, radii R 200 c and masses M 200 c . The latter two are defined respectively as the distance from the halo centre which encloses a mean density of 200 times the critical density of the universe as a given redshift, and the mass enclosed within such a sphere.</text>
<text><location><page_9><loc_7><loc_6><loc_48><loc_14></location>Unfortunately, the inaccuracy when estimating the velocity divergence and vorticity fields on small scales using DTFE prevents us from studying their behaviour in haloes alongside the vector potential. We have tested that indeed, the velocity estimations are strongly affected by resolution and do not converge either using a resolution for the tessellation grid similar to the mean inter-particle dis-</text>
<text><location><page_9><loc_51><loc_14><loc_92><loc_24></location>tance of dark matter particles in the haloes or otherwise. The phaseinterpolation method was used in Hahn et al. (2015) to successfully estimate the vorticity in haloes in the case of warm dark matter, but still it is not possible to robustly measure this from CDM simulations either: this is related to the difficulty of resolving the perturbations up to highly nonlinear scales in the CDM case, which in warm dark matter models is not required as the spectrum truncates at some finite free-streaming scale.</text>
<text><location><page_9><loc_51><loc_5><loc_92><loc_13></location>Figure 6 shows density (left column), vector potential magnitude (middle column) and scalar gravitational potential (right column) in the vicinity of three selected dark matter haloes at z = 0 , with masses M h ≈ 6 . 5 × 10 15 h -1 M glyph[circledot] (top row), M h ≈ 3 . 0 × 10 13 h -1 M glyph[circledot] (middle row) and M h ≈ 3 . 1 × 10 12 h -1 M glyph[circledot] (bottom</text>
<figure>
<location><page_10><loc_8><loc_63><loc_91><loc_93></location>
<caption>Figure 5. (Colour Online) Left: The dimensionless power spectrum of the vector potential, ∆ B ( k ) = k 3 P B ( k ) / (2 π 2 ) . The solid lines represent the corresponding second-order perturbation theory predictions (Lu et al. 2009), in which cutoffs have been introduced in the convolution calculation to accommodate the lack of power in the simulation results on large scales due to box size. Right: The ratio between the power spectrum of the vector potential and that of the scalar gravitational potential defined as the fully nonlinear perturbation to the lapse function, i.e., Φ ≡ α -1 . In both panels, each colour corresponds to z = 0 (black), z = 0 . 5 (orange), z = 1 (red) and z = 1 . 5 (blue).</caption>
</figure>
<text><location><page_10><loc_7><loc_17><loc_48><loc_51></location>row). In all cases, the map centre is aligned with the halo centre and the width of the shown region corresponds to four times the halo radius R 200 c . As also shown in Fig 1, overall we observe some degree of correlation between the vector potential and the matter density, but clearly not at the level of the scalar potential. In particular, in the case of the most massive halo (top row) we can see that while both potentials peak towards the halo centre, unlike for the scalar potential, the global maximum of the vector potential within the shown region is actually found in the lower left part of the map, where there appears to be another, smaller, halo infalling towards the central one. Again, this qualitative difference is not surprising since the vector potential is sourced by the transverse part of the momentum density, Eq. (25), while the matter source term for the scalar potential is the density contrast itself (up to higher-order terms). As before, we can also see that both potentials are smoother than the density field owing to the elliptic-type nature of their equations (Barrera-Hinojosa & Li 2020a), in which short-wavelength modes are dominated. In addition, in the most massive halo we can observe that the scalar potential tends to be more spherically symmetric around the center than B , which displays large values in most part of the left and upper part of the map. Indeed, although the low-density (dark) regions in the bottom right and top left parts of the density map are of similar characteristics, and these are clearly correlated with the Φ map, these are not correlated with features in the B map at all.</text>
<text><location><page_10><loc_7><loc_6><loc_48><loc_17></location>For the halo shown in the middle row of Fig. 6, the density and potential contours have more similar shapes to each other than in the most massive halo. Nonetheless, the scalar potential again seems to decay more rapidly outside R 200 c than the vector potential magnitude. This also seems to be the case in the halo shown in the bottom panels, although in this case the potentials are smaller and shallower. Note that, for the halo in the middle panels, | Φ | is largest in the central region (red/orange/green), decays when one moves fur-</text>
<text><location><page_10><loc_51><loc_45><loc_92><loc_51></location>her away from the halo centre (blue), but grows again far from the halo (green); this is because this halo resides in a low-density environment, with a positive environmental contribution to the total potential so that the latter crosses zero.</text>
<text><location><page_10><loc_51><loc_22><loc_92><loc_45></location>It is important to bear in mind that, although the halo centres are approximately located at a local maximum of | Φ | , the potentials themselves are not an observable quantity: it is the gradient of the potentials that contributes as force terms in the geodesic equation (12), while the values of the potential themselves can be largely influenced by their environments. In this subsection, we are mainly interested in haloes which are isolated and therefore less affected by environments. To select such haloes, we try to split the potential at each point into two contributions: one from the halo itself and one from its environment, i.e., well beyond a distance R 200 c from its centre. Since the potentials are not necessarily spherically symmetric, as it is evident from the top row of Fig. 6, as a crude way, we shall take the spherical average in a radial bin at 2 R 200 c and subtract this from the values at smaller radii, which allows to get 'shifted' radial halo profiles for both Φ and B that vanish at 2 R 200 c . For Φ ( B ) we expect this profile to monotonically increase (decrease) to zero as r increases to 2 R 200 c , for well-isolated relaxed haloes.</text>
<text><location><page_10><loc_51><loc_6><loc_92><loc_21></location>Figure 7 shows, from the top to the bottom row, the radial profiles of density, the vector potential magnitude and its ratio against the scalar gravitational potential. All profiles have been measured from the centres of a sample of haloes in different mass ranges, for three redshifts: z = 0 (left column), z = 0 . 5 (middle column) and z = 1 (right column). For this we have selected three subsamples of haloes with O (100) haloes each based on mass cuts: we define a higher mass range M h ≥ 10 14 . 5 h -1 M glyph[circledot] , an intermediate mass range with mean mass ¯ M h = 10 13 . 5 h -1 M glyph[circledot] , and a lower mass range with mean mass ¯ M h = 10 12 . 5 h -1 M glyph[circledot] . For each halo from a given mass range, we then calculate the spherical average of the den-</text>
<text><location><page_11><loc_42><loc_21><loc_43><loc_24></location>×</text>
<text><location><page_11><loc_54><loc_21><loc_56><loc_24></location>×</text>
<text><location><page_11><loc_66><loc_21><loc_67><loc_24></location>×</text>
<paragraph><location><page_11><loc_83><loc_21><loc_91><loc_24></location>2 × 10 -5</paragraph>
<figure>
<location><page_11><loc_9><loc_22><loc_88><loc_92></location>
<caption>Figure 6. (Colour Online) Visualisation of three selected dark matter haloes at z = 0 , with masses M h = 6 . 5 × 10 14 h -1 M glyph[circledot] (top row), M h = 3 . 0 × 10 13 h -1 M glyph[circledot] (middle row) and M h = 3 . 1 × 10 12 h -1 M glyph[circledot] (bottom row). In each row, each panel shows, from left to right: matter density, magnitude of the vector potential and absolute magnitude the scalar gravitational potential (since typically Φ ≤ 0 in the inner parts of a halo). Interpolation has been used to display smoother maps. All maps are in logarithmic scale.</caption>
</figure>
<text><location><page_11><loc_88><loc_46><loc_88><loc_47></location>5</text>
<figure>
<location><page_12><loc_8><loc_33><loc_91><loc_93></location>
<caption>Figure 7. (Colour Online) Halo profiles (spherical averages) at z = 0 (left column), z = 0 . 5 (middle column) and z = 1 (right column). Each row shows, from top to bottom, density, vector potential magnitude and its ratio against the scalar gravitational potential. In the case of the potentials, their spherical-average at R 200 c has been subtracted from each individual halo profile as a way to remove their environmental contributions. The upper, middle and lower halo mass ranges are represented by red, green and blue, respectively, for which the solid line shows the mean calculated over all haloes in a given mass range, and the shaded regions are the 1 σ regions. The values of M h shown in the inset are in units of h -1 M glyph[circledot] .</caption>
</figure>
<text><location><page_12><loc_90><loc_36><loc_91><loc_37></location>0</text>
<text><location><page_12><loc_7><loc_7><loc_48><loc_21></location>sity, vector potential and scalar potential up to 2 R 200 c , and average over the full population. As mentioned in the previous paragraph, in the case of the potentials we have subtracted their average values at 2 R 200 c in the profile of each individual halo. In this process, we have discarded the haloes in which the resulting spherical average of B becomes negative for some r < R 200 c after the subtraction, which typically happens in lower mass haloes due to their shallow potentials. However, these haloes are the most abundant type and hence we retain a sample of size O (100) even at z = 1 , while the</text>
<text><location><page_12><loc_51><loc_18><loc_92><loc_21></location>number of haloes in the middle and higher mass bins is around ∼ 50 at that same redshift.</text>
<text><location><page_12><loc_51><loc_6><loc_92><loc_17></location>From Fig. 7 we find that at the 1 σ level there is a clear correlation between halo mass and the magnitude of the gravitomagnetic potential, which can differ by up to two orders of magnitude between halos with masses close to 10 12 . 5 h -1 M glyph[circledot] and those with masses larger than 10 14 . 5 h -1 M glyph[circledot] . In all cases, the vector potential flattens toward the halo centres and it decreases towards the outskirts. However, from the bottom row of Fig. 7 we find that the ratio between vector and scalar potentials is roughly constant inside haloes</text>
<figure>
<location><page_13><loc_8><loc_68><loc_47><loc_92></location>
<caption>Figure 8. (Colour Online) Evolution of the ratio between the vector potential and the scalar gravitational potential for the different halo mass ranges. At each redshift, the value shown corresponds to the average of the ratio for r ≤ R 200 c . We have only included cases where the number of haloes in a given mass range is greater than ten at a given redshift. The values of M h shown in the inset are in units of h -1 M glyph[circledot] .</caption>
</figure>
<text><location><page_13><loc_7><loc_36><loc_48><loc_53></location>across all masses and redshifts considered, and the dependence of this ratio upon halo mass is quite weak as all means lie within 1 σ of each other. At z = 0 , we find that the ratio is a few times 10 -3 , which is roughly consistent with the value inferred from the ratio of O (10 -5 ) between the power spectra of the vector and scalar potentials at k glyph[greaterorsimilar] O (0 . 1) h Mpc -1 , as shown in Fig. 5 (note that the subtraction of the environmental contributions in these potentials essentially removes the long-wavelength contributions to B/ Φ , thereby marking this comparison with Fig. 5 reasonable; but as we only look at a small fraction of the total volume, inside a sub-group of haloes, we of course should not expect an exact equality). At z = 0 . 5 and z = 1 , the picture is qualitatively the same apart from the increase in the amplitude of the vector potential.</text>
<text><location><page_13><loc_7><loc_24><loc_48><loc_35></location>In CDM simulations, it is well known that the density profile of haloes can be described by the universal Navarro-Frenk-White (NFW; Navarro et al. 1996) fitting formula, which has a corresponding analytical prediction for the Newtonian potential profiles of haloes. The constancy of B/ | Φ | inside haloes which is found here implies that it might be straightforward to derive an analytical fitting function for the B profiles in haloes, which is closely related to the NFW function, though this will not be pursued in this paper.</text>
<text><location><page_13><loc_7><loc_6><loc_48><loc_24></location>Given that Fig. 7 shows that the ratio between the vector and scalar potentials is roughly constant inside the halos - and we have checked that such constant ratio holds even above z = 1 - we can characterise this ratio by a single number at each halo mass and redshift. As an extension of the bottom row of Fig. 7, Fig. 8 shows the mean value of such ratio calculated within r < R 200 c at different redshifts. Since the number of haloes in a given mass bin decreases towards higher redshifts, here we only consider cases in which the number of haloes in a given mass range is greater than ten at a given redshift. We find that for all mass bins B/ Φ increases almost linearly with redshift. At redshift z = 2 the rate of change of this ratio with respect to redshift slows down slightly for the lowest mass range (blue line), after which it picks up again: this could be due to a lack</text>
<text><location><page_13><loc_51><loc_74><loc_92><loc_93></location>of simulation resolution at high z . Observationally, the ratio between vector and scalar potentials is particularly relevant for weak lensing, as post-Newtonian calculations show that the relative correction to the Newtonian convergence field κ is proportional to B/ Φ (Sereno 2002, 2003; Bruni et al. 2014). Therefore, Fig. 8 suggests that, in the case of dark matter haloes, the lensing convergence correction due to the gravitomagnetic potential is between the O (10 -3 ) and O (10 -2 ) level, in agreement with previous studies (Sereno 2007; Cuesta-Lazaro et al. 2018; Tang et al. 2020). Moreover, this only depends weakly on the halo mass and could be more easily detected on high-mass haloes at high redshifts. However, we note that at higher orders in the post-Newtonian expansion, new contributions from the time derivative of B appear (Bruni et al. 2014; Thomas et al. 2015a) as well, which requires further inspection.</text>
<text><location><page_13><loc_51><loc_61><loc_92><loc_73></location>Besides investigating the potentials, we can also look at the force that each of these exert on the particles according to Eq. (12), which shows that the total force is mainly composed by two contributions; the standard gravitational force arising from the gradient of the scalar potential (first term on the r.h.s.), and the gravitomagnetic force (contained in the second term on the r.h.s.) which is responsible for the frame-dragging effect. The latter is naturally not taken into account in Newtonian gravity. The third term in the r.h.s of Eq. (12) is subdominant and so we shall not explore it here.</text>
<text><location><page_13><loc_51><loc_31><loc_92><loc_60></location>Figure 9 is a visualisation of the magnitude of the gravitomagnetic acceleration (middle column) and that of the standard gravitational acceleration (right column) in units of h cm / s 2 , in the vicinity of three different dark matter haloes. These haloes have similar masses to those shown in Fig. 6. We find that the forces are correlated with the density field up to some degree, particularly in the haloes in the middle and bottom rows, although the gravitomagnetic force seems to be less smooth than the Newtonian one. For the halo in the top row, there is a clearer difference between the forces compared to the other two cases. The peaks of the gravitomagnetic acceleration seem to occur at the density peaks but the opposite is not true, and there is no clear correspondence between their amplitudes. Interestingly, in this halo the values of gravitomagnetic force around a few times 10 -13 h cm / s 2 (green region) extend around the centre and towards the left part of the map, where the density field has already decreased by various orders of magnitude. This kind of asymmetry between both kinds of maps might be due to the actual dynamical state of the particles in a given region. Even if the density is low, if the particles' velocity happens to be aligned with the gradient of the vector potential components they will contribute significantly to | u · ∂ i B | .</text>
<text><location><page_13><loc_51><loc_5><loc_92><loc_31></location>As before, we can calculate the spherical averages of the forces, which allows us to get radial profiles (although no subtraction from radial bins beyond 2 R 200 c is required this time). Figure 10 shows a comparison of the gravitomagnetic (frame-dragging) acceleration and the standard gravitational one in dark matter haloes in an analogous way to the scalar and vector potential profiles shown in Fig. 7. We find that the magnitude of the gravitomagnetic force is larger towards the inner parts of the halo, and the dependence on the halo mass is weaker than in the case of the scalar gravitational potential. As we discussed before, this can also be explained by the fact that the gravitomagnetic force not only depends on density but on the actual dynamical state of particles. Similarly to the behaviour of B/ | Φ | , from Fig. 10 we find that the ratio of the two corresponding forces also remains fairly constant inside the haloes, although in the most massive haloes it tends to increase toward the outskirts. A weak dependence on halo mass is found at all redshifts. In Adamek et al. (2016b) the maximum gravitomagnetic acceleration measured from the simulation box at z = 0 is found to be roughly 7 × 10 -12 h</text>
<figure>
<location><page_14><loc_9><loc_20><loc_90><loc_92></location>
<caption>Figure 9. (Colour Online) Visualisation of three selected dark matter haloes at z = 0 , with masses M h = 2 . 7 × 10 14 h -1 M glyph[circledot] (top row), M h = 3 . 3 × 10 13 h -1 M glyph[circledot] (middle row) and M h = 3 . 2 × 10 12 h -1 M glyph[circledot] (bottom row). In each row, each column shows, from left to right: matter density, the magnitude of the gravitomagnetic acceleration and the magnitude of the standard gravitational acceleration, the latter two in units of h cm / s 2 . Interpolation has been used to display smoother maps. All maps are in logarithmic scale.</caption>
</figure>
<text><location><page_14><loc_90><loc_68><loc_91><loc_70></location>6</text>
<figure>
<location><page_15><loc_8><loc_34><loc_91><loc_93></location>
<caption>Figure 10. (Colour Online) Halo profiles (spherical averages) at z = 0 (left column), z = 0 . 5 (middle column) and z = 1 (right column). In a given column, each row shows, from top to bottom, the gravitomagnetic (frame-dragging) acceleration, standard gravitational acceleration and their ratio. The upper, middle and lower halo mass ranges are represented by red, green and blue, respectively, for which the solid line shows the mean calculated over all haloes in a given mass range, and the shaded regions are the 1 σ regions. The values of M h shown in the inset are in units of h -1 M glyph[circledot] .</caption>
</figure>
<text><location><page_15><loc_7><loc_9><loc_48><loc_23></location>cm/s 2 for the highest resolution used ( 125 h -1 kpc ), while the value measured from lower resolution runs decreases monotonically. From Fig. 10 we find that this is comparable with our results for haloes in the upper mass range at the 1 σ level. However, we note that for the most massive halo in our simulation, we find the maximum value of the gravitomagnetic acceleration to be 7 × 10 -11 h cm/s 2 , i.e. roughly one order of magnitude higher. This difference could be explained by the fact that in our simulation the most refined regions are resolved with a resolution of 2 h -1 kpc . In addition, GRAMSES treats the vector potential non-perturbatively, although the difference due</text>
<text><location><page_15><loc_51><loc_20><loc_92><loc_22></location>to higher-order corrections is likely to be subdominant with respect to the aforementioned resolution dependence.</text>
<section_header_level_1><location><page_15><loc_51><loc_15><loc_64><loc_16></location>4 CONCLUSIONS</section_header_level_1>
<text><location><page_15><loc_51><loc_6><loc_92><loc_14></location>We have investigated the vector modes of the matter fields as well as those of the Λ CDM spacetime metric, from large sub-horizon scales to deeply nonlinear scales using a high-resolution run of the general-relativistic N -body GRAMSES code (Barrera-Hinojosa & Li 2020a,b). On the one hand, vorticity vanishes at the non-perturbative level in a perfect fluid description and yet it is generated dynamically</text>
<text><location><page_16><loc_7><loc_81><loc_48><loc_93></location>due to the collisionless nature of dark matter. On the other hand, the metric vector potential - responsible for frame-dragging - appears beyond linear order in perturbation theory and is not solved for in Newtonian simulations. Therefore, the physics behind the vector modes is highly non-trivial and numerical simulations play an important role in their study. Although the relativistic nature of the code is not particularly exploited from the point of view of vorticity, the vector potential is a prime quantity as this is not part of Newtonian gravity and therefore not implemented in Newtonian simulations.</text>
<text><location><page_16><loc_7><loc_67><loc_48><loc_80></location>To this end, we have run a high-resolution N -body simulation using GRAMSES, that employs N part = 1024 3 particles in a box of comoving size L box = 256 h -1 Mpc . In GRAMSES, the GR metric potentials - in the fully constrained formalism and conformally flat approximation - are solved on meshes in configuration space. The AMR capabilities of GRAMSES allows it to start off with a regular grid with 1024 3 cells, and hierarchically refine it in high-density regions to reach a spatial resolution of 2 h -1 kpc in the most refined places, namely dark matter haloes. This enables a quantitative analysis of the behaviour of vector modes in such regions.</text>
<text><location><page_16><loc_9><loc_65><loc_43><loc_66></location>The key findings of this paper are summarised as follows:</text>
<unordered_list>
<list_item><location><page_16><loc_7><loc_52><loc_48><loc_64></location>(i) On scales 0 . 06 h Mpc -1 glyph[lessorsimilar] k glyph[lessorsimilar] 0 . 3 h Mpc -1 , the vorticity power spectrum can be characterised by the power law in Eq. (27) with an index n ω ≈ 2 . 7 , a value that is overall consistent with recent simulation results of Hahn et al. (2015); Jelic-Cizmek et al. (2018). On nonlinear scales ( 2 . 3 h Mpc -1 glyph[lessorsimilar] k glyph[lessorsimilar] 20 h Mpc -1 ), the power spectrum can again be described by a power-law function, but the index changes to n NL ω ≈ -1 . 4 , close to the asymptotic value of -1 . 5 suggested by Hahn et al. (2015); cf. Fig. 3.</list_item>
</unordered_list>
<text><location><page_16><loc_7><loc_36><loc_48><loc_40></location>(iii) The vector potential power spectrum remains below 4 × 10 -5 relative to the scalar gravitational potential down to k = 20 h Mpc -1 ; cf. Fig. 5.</text>
<text><location><page_16><loc_7><loc_40><loc_48><loc_53></location>(ii) On scales 0 . 1 h Mpc -1 glyph[lessorsimilar] k glyph[lessorsimilar] 0 . 4 h Mpc -1 the amplitude of the vorticity power spectrum seems to evolve as ∼ [ D + ( z ) /D + (0)] 7 . 7 at z ≤ 1 . 5 , which is higher than previous values found in the literature (Thomas et al. 2015b; Jelic-Cizmek et al. 2018). Nonetheless, these references also found larger values than the scaling with the seventh power originally proposed in Pueblas & Scoccimarro (2009). On scales k glyph[greaterorsimilar] 3 . 5 h Mpc -1 , the evolution of the amplitude of the power spectrum can be similarly neatly described as ∼ [ D + ( z ) /D + (0)] 2 . 6 up to z = 1 ; cf. Fig. 4.</text>
<text><location><page_16><loc_7><loc_24><loc_48><loc_36></location>(iv) Inside dark matter haloes, the magnitude of the vector potential peaks towards the centres at ∼ 10 -7 for haloes more massive than 10 14 . 5 h -1 M glyph[circledot] , which can reduce by two orders of magnitude in haloes of masses around 10 12 . 5 h -1 M glyph[circledot] . Its ratio against the scalar gravitational potential remains typically a few times 10 -3 inside the haloes, regardless of their mass (cf. Fig. 7). The ratio B/ | Φ | remains nearly flat within the halo radius R 200 c , for the halo redshift ( z < 3 ) and mass (10 12 . 5 ∼ 10 15 h -1 M glyph[circledot] ) ranges checked, and this constant increases roughly linearly with z ; cf. Fig. 8.</text>
<text><location><page_16><loc_7><loc_11><loc_48><loc_23></location>(v) The magnitude of the gravitomagnetic acceleration also peaks at the halo centres where it can reach a few times 10 -11 h cm / s 2 in haloes above ∼ 10 14 . 5 h -1 M glyph[circledot] . Its ratio against the standard gravitational acceleration remains around ∼ 10 -5 on average, regardless of the halo mass and distance from the halo centre; cf. Fig. 10. This suggests that the effect of the gravitomagnetic force on cosmic structure formation is, even for the most massive structures, negligible however, note that we have not studied the behaviour in low-density regions, i.e., voids.</text>
<text><location><page_16><loc_7><loc_6><loc_48><loc_10></location>While we have presented a first study of the gravitomagnetic potential in dark matter haloes with general-relativistic simulations, there are several possible extensions in this direction. The analy-</text>
<text><location><page_16><loc_51><loc_76><loc_92><loc_93></location>sis of the gravitomagnetic potential and forces done in this paper could be extended to galaxies, e.g., by constructing a catalogue using certain semi-analytic models. It is then possible to calculate the gravitomagnetic accelerations of galaxies based on their coordinates and velocities. However, as we have seen above, this acceleration is much weaker than the standard gravitational acceleration, and the impact of baryons on small scales still remains to be assessed. The implementation of (magneto)hydrodynamics in the default RAMSES code could be used in conjunction with the general-relativistic implementation of GRAMSES as a first approximation to address this question, although we generally expect that uncertainties in baryonic physics should surpass GR effects.</text>
<text><location><page_16><loc_51><loc_60><loc_92><loc_76></location>A perhaps more interesting possibility is to self-consistently implement massive neutrinos and radiation in this relativistic code. In the second GRAMSES code paper (Barrera-Hinojosa & Li 2020b), we have introduced a method to generate initial conditions for GRAMSES simulations that does not require back-scaling. It is therefore natural to evolve these matter components which are neglected in traditional simulations (e.g., Adamek et al. 2017). On the same vein, a Newtonian (quasi-static) implementation of modified gravity models on GRAMSES would allow to study the gravitomagnetic potential in such type of theory. In particular, the modified gravity code ECOSMOG (Li et al. 2012; Li et al. 2013) is based on RAMSES and can be easily made compatible with GRAMSES for such purpose.</text>
<text><location><page_16><loc_51><loc_43><loc_92><loc_59></location>In this paper, we have primarily focused on the general-relativistic physical quantities that could impact cosmic structure formation, and this can ultimately only be observed by detecting photons (McDonald 2009; Croft 2013; Bonvin et al. 2014; Alam et al. 2017). Therefore, besides the gravitomagnetic force acting on massive particles, it is also important to study how vector modes, as well as other GR effects, could influence the photon trajectories on nonlinear scales, and what is the consequent impact on observables, e.g. lensing (Thomas et al. 2015a; Saga et al. 2015; Gressel et al. 2019). This requires the implementation of general-relativistic ray tracing algorithms (e.g. Barreira et al. 2016; Breton et al. 2019; Lepori et al. 2020; Reverdy 2014) and is left as a future project.</text>
<section_header_level_1><location><page_16><loc_51><loc_32><loc_69><loc_33></location>ACKNOWLEDGEMENTS</section_header_level_1>
<text><location><page_16><loc_51><loc_24><loc_92><loc_31></location>We thank Marius Cautun for assistance with the DTFE code, and Raúl Angulo for useful discussions on the vorticity estimation from N -body simulations. We are also grateful to James Mertens and to the anonymous referee for their valuable comments and observations.</text>
<text><location><page_16><loc_51><loc_16><loc_92><loc_24></location>CB-H is supported by the Chilean National Agency of Research and Development (ANID) through grant CONICYT/Becas-Chile (No. 72180214). BL is supported by the European Research Council (ERC) through ERC starting Grant No. 716532, and STFC Consolidated Grant (Nos. ST/I00162X/1, ST/P000541/1). MB is supported by UK STFC Consolidated Grant No. ST/S000550/1.</text>
<text><location><page_16><loc_51><loc_6><loc_92><loc_16></location>This work used the DiRAC@Durham facility managed by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility ( www.dirac.ac.uk ). The equipment was funded by BEIS via STFC capital grants ST/K00042X/1, ST/P002293/1, ST/R002371/1 and ST/S002502/1, Durham University and STFC operation grant ST/R000832/1. DiRAC is part of the UK National e-Infrastructure.</text>
<section_header_level_1><location><page_17><loc_7><loc_92><loc_22><loc_93></location>DATA AVAILABILITY</section_header_level_1>
<text><location><page_17><loc_7><loc_90><loc_40><loc_91></location>For access to the simulation data please contact CB-H.</text>
<section_header_level_1><location><page_17><loc_7><loc_84><loc_17><loc_85></location>REFERENCES</section_header_level_1>
<table>
<location><page_17><loc_7><loc_6><loc_48><loc_83></location>
</table>
<table>
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<section_header_level_1><location><page_17><loc_51><loc_39><loc_89><loc_42></location>APPENDIX A: COMPARISON OF POWER SPECTRUM CALCULATION METHODS</section_header_level_1>
<text><location><page_17><loc_51><loc_19><loc_92><loc_38></location>In Section 3.1, the power spectrum of density, velocity and vorticity has been measured from particle-type data using DTFE and NBODYKIT, while the spectrum of the scalar and vector potentials has been measured using a different code that is able to read their values calculated and stored by GRAMSES in cells of hierarchical AMR meshes and interpolate them to a regular grid for the power spectrum measurement. We call this method the 'AMR-FFT' method, which was introduced in He et al. (2015), where more details can be found. An alternative to using this AMR-FFT method to calculate the power spectrum of the potentials is by writing their values with GRAMSES at the particles' positions rather than in AMR cells, so that DTFE can be used to read such 'particle-type' data and interpolate this to a regular grid, where NBODYKIT can be applied to measure the spectrum. We call this method 'DTFE+NBODYKIT'.</text>
<text><location><page_17><loc_51><loc_6><loc_92><loc_18></location>Figure A1 shows the dimensionless power spectra at z = 1 of the scalar potential Φ (left panel) and the vector potential spectrum (right panel), measured by these two methods, where solid lines represent the perturbation-theory predictions. In both methods the FFT grid size is 2048 3 , as is the tessellation grid size used for DTFE. We find that both methods have good agreement on large scales, specially at k glyph[greaterorsimilar] 0 . 1 h Mpc -1 , where the effect of cosmic variance is not present. However, in the region k glyph[greaterorsimilar] 3 h Mpc -1 the AMR-FFT method has better performance than DTFE+NBODYKIT</text>
<text><location><page_18><loc_11><loc_89><loc_13><loc_91></location>10</text>
<text><location><page_18><loc_13><loc_89><loc_14><loc_91></location>-</text>
<text><location><page_18><loc_11><loc_83><loc_13><loc_85></location>10</text>
<text><location><page_18><loc_13><loc_83><loc_14><loc_85></location>-</text>
<text><location><page_18><loc_14><loc_90><loc_15><loc_91></location>10</text>
<text><location><page_18><loc_14><loc_84><loc_15><loc_85></location>12</text>
<text><location><page_18><loc_9><loc_80><loc_10><loc_81></location>Φ</text>
<text><location><page_18><loc_8><loc_79><loc_10><loc_80></location>∆</text>
<text><location><page_18><loc_11><loc_77><loc_13><loc_78></location>10</text>
<text><location><page_18><loc_13><loc_76><loc_14><loc_78></location>-</text>
<text><location><page_18><loc_11><loc_70><loc_13><loc_72></location>10</text>
<text><location><page_18><loc_13><loc_70><loc_14><loc_72></location>-</text>
<text><location><page_18><loc_14><loc_77><loc_15><loc_78></location>14</text>
<text><location><page_18><loc_14><loc_71><loc_15><loc_72></location>16</text>
<text><location><page_18><loc_21><loc_72><loc_31><loc_73></location>1st order PT</text>
<text><location><page_18><loc_21><loc_70><loc_29><loc_72></location>AMR-FFT</text>
<text><location><page_18><loc_21><loc_69><loc_25><loc_70></location>dtfe</text>
<text><location><page_18><loc_26><loc_69><loc_27><loc_70></location>+</text>
<text><location><page_18><loc_27><loc_69><loc_35><loc_70></location>nbodykit</text>
<text><location><page_18><loc_20><loc_66><loc_22><loc_67></location>10</text>
<text><location><page_18><loc_22><loc_66><loc_23><loc_68></location>-</text>
<text><location><page_18><loc_23><loc_66><loc_24><loc_68></location>1</text>
<text><location><page_18><loc_27><loc_64><loc_28><loc_65></location>k</text>
<text><location><page_18><loc_29><loc_64><loc_29><loc_65></location>[</text>
<text><location><page_18><loc_29><loc_64><loc_30><loc_65></location>h</text>
<text><location><page_18><loc_31><loc_64><loc_35><loc_65></location>Mpc</text>
<text><location><page_18><loc_35><loc_64><loc_36><loc_66></location>-</text>
<figure>
<location><page_18><loc_50><loc_64><loc_91><loc_93></location>
<caption>Figure A1. (Colour Online) Comparison of the power spectra of the scalar and vector potentials measured with the AMR-FFT method, and NBODYKIT combined with DTFE. In both methods the grid size used for the FFT is 2048 3 , and is equal to the tessellation grid size used in DTFE. Both panels show the dimensionless power spectrum ∆( k ) = k 3 P ( k ) / (2 π 2 ) of the respective field. Left : The dimensionless power spectrum of the of the scalar gravitational potential Φ defined as the fully nonlinear perturbation to the lapse function, i.e., Φ ≡ α -1 . The solid line represents the first-order perturbation theory prediction of the Bardeen potential from CAMB. Right : The dimensionless power spectrum of the vector potential B . The solid line corresponds to the second-order perturbation theory result from Eq. (30). All results are at z = 1 .</caption>
</figure>
<text><location><page_18><loc_36><loc_64><loc_37><loc_65></location>1</text>
<text><location><page_18><loc_37><loc_64><loc_38><loc_65></location>]</text>
<text><location><page_18><loc_7><loc_40><loc_48><loc_50></location>which blows up. This is because the AMR-FFT method can reach higher resolution by using the potential information in the AMR cells, and because DTFE does a volume weighted average of the field which smears out small-scale features. Therefore, the spectrum of the scalar and vector potentials from the simulation shown in Fig. 5 are measured by the AMR-FFT method, which yields robust results up to k ∼ 15 h Mpc -1 .</text>
<text><location><page_18><loc_7><loc_37><loc_46><loc_38></location>This paper has been typeset from a T E X/L A T E X file prepared by the author.</text>
<text><location><page_18><loc_32><loc_66><loc_34><loc_67></location>10</text>
<text><location><page_18><loc_34><loc_66><loc_35><loc_68></location>0</text>
<text><location><page_18><loc_41><loc_89><loc_42><loc_91></location>z</text>
<text><location><page_18><loc_43><loc_89><loc_46><loc_91></location>= 1</text>
<text><location><page_18><loc_44><loc_66><loc_46><loc_67></location>10</text>
<text><location><page_18><loc_46><loc_66><loc_46><loc_68></location>1</text>
</document> | [{"title": "ABSTRACT", "content": "We investigate the transverse modes of the gravitational and velocity fields in \u039b CDM, based on a high-resolution simulation performed using the adaptive-mesh refinement general-relativistic N -body code GRAMSES. We study the generation of vorticity in the dark matter velocity field at low redshift, providing fits to the shape and evolution of its power spectrum over a range of scales. By analysing the gravitomagnetic vector potential, which is absent in Newtonian simulations, in dark matter haloes with masses ranging from \u223c 10 12 . 5 h -1 M glyph[circledot] to \u223c 10 15 h -1 M glyph[circledot] , we find that its magnitude correlates with the halo mass, peaking in the inner regions. Nevertheless, on average, its ratio against the scalar gravitational potential remains fairly constant, below percent level, decreasing roughly linearly with redshift and showing a weak dependence on halo mass. Furthermore, we show that the gravitomagnetic acceleration in haloes peaks towards the core and reaches almost 10 -10 h cm/s 2 in the most massive halo of the simulation. However, regardless of the halo mass, the ratio between the gravitomagnetic force and the standard gravitational force is typically at around the 10 -5 level inside the haloes, again without significant radius dependence. This result confirms that the gravitomagnetic effects have negligible impact on structure formation, even for the most massive structures, although its behaviour in low density regions remains to be explored. Likewise, the impact on observations remains to be understood in the future. Key words: gravitation - cosmology: theory - large-scale structure of the Universe - methods: numerical.", "pages": [1]}, {"title": "Vector modes in \u039b CDM: the gravitomagnetic potential in dark matter haloes from relativistic N -body simulations", "content": "Cristian Barrera-Hinojosa, 1 glyph[star] Baojiu Li 1 , Marco Bruni 2 , 3 and Jian-hua He 4 , 5 1 Institute for Computational Cosmology, Department of Physics, Durham University, Durham DH1 3LE, UK Accepted XXX. Received YYY; in original form ZZZ", "pages": [1]}, {"title": "1 INTRODUCTION", "content": "While the dynamics of the large-scale structure (LSS) of the universe is mainly governed by scalar perturbations, vector and tensor degrees of freedom are promising alternatives for exploring the nature of dark matter and gravity. The effects of the vector modes of the spacetime metric on matter such as frame dragging and geodetic precession have been measured in the Solar system during the last decade (Everitt et al. 2011), but there is still no cosmological signal detected. The recent observation of radio galaxies showing coherent angular velocities on scales of \u223c 20 Mpc at z = 1 reported by Taylor & Jagannathan (2016) has motivated to seek a physical interpretation in terms of vector modes, but it has not been possible to establish a clear connection so far (Cusin et al. 2017; Bonvin et al. 2018). More recently, and motivated by the accurate data provided by Gaia DR2, a simple model to explain the flat rotation curve of the Milky Way in terms of frame dragging has been proposed in Crosta et al. (2020). In \u039b CDM cosmology, vector modes are typically neglected. In a perfect fluid, vorticity - the covariant curl of the 4-velocity field - satisfies a homogenous nonlinear equation, hence it vanishes exactly, i.e. at all orders in perturbation theory (Lu et al. 2009), unless it is either introduced by initial conditions 1 or generated by physics beyond such fluid model. Moreover, vorticity is not generated by standard inflationary scenarios, and even if it was, this type of perturbation quickly decays during the matter-dominated era. Nonetheless, vorticity is found to be generated dynamically via shell (orbit) crossing of matter, a phenomenon extremely common at late times whose modelling is beyond the grasp of the single-streaming fluid regime. Therefore, N -body simulations represent a valuable tool for the study of vorticity generation (Pueblas & Scoccimarro 2009; Hahn et al. 2015; Jelic-Cizmek et al. 2018). In the Poisson gauge, generalising the longitudinal gauge to include tensor and vector perturbations (Bertschinger 1993), the latter are encoded by the non-diagonal spacetime metric components, the shift vector B i \u2261 g 0 i , and represent in this gauge the gauge-invariant gravitomagnetic vector potential (Bardeen 1980). In \u039b CDM, safely assuming purely scalar perturbations at first-order, the shift vector vanishes at the linear level, while at second order it satisfies a constraint equation sourced by the product of first-order density and velocity perturbations. However, it is expected that, just like vorticity, the gravito-magnetic field also receives corrections from phenomena beyond the perfect fluid description. The impact of vector modes on LSS observables is expected to be small relative to the scalar perturbations, both from perturbative (Lu et al. 2009) and non-perturbative analyses (Bruni et al. 2014; Adamek et al. 2016b), although it can represent a new systematic which needs to be taken into account (Bonvin et al. 2018). For instance, their effect on gravitational lensing seems to be not strong enough to be detectable by current observations (Thomas et al. 2015a; Saga et al. 2015; Gressel et al. 2019), and the imprints of the vector potential in the angular power spectrum and bispectrum of galaxies are also weak (Durrer & Tansella 2016; Jolicoeur et al. 2019), although a vector perturbation can be isolated from the full signal if it violates statistical isotropy and defines a preferred frame (see, e.g., Tansella et al. 2018). On the other hand, the vector potential power spectrum is known to peak around the equality scale (Lu et al. 2009), and its behaviour as well as impact on observables at highly nonlinear scales remains largely unexplored, although deviations from perturbation theory can be significant (Bruni et al. 2014). Furthermore, in popular f ( R ) gravity models, vector modes can have considerable deviations from GR on small scales (Thomas et al. 2015c), so these could also play a role in discriminating cosmological models. The work of Pueblas & Scoccimarro (2009) provided the first insights into the generation of vorticity via shell crossing using N -body simulations, which allowed to quantify its impact on the density and velocity power spectra estimates from linear perturbation theory. In particular, vorticity was found to peak in the outskirts of virialised structures as particle velocities in inner regions are strongly aligned with density gradients, as also found later in Hahn et al. (2015) from a different set of simulations. Although - contrary to vorticity - the investigation of the gravitomagnetic vector field in principle requires a completely general-relativistic numerical framework as Newtonian simulations only model a single scalar gravitational potential, \u03a6 , in Bruni et al. (2014) and Thomas et al. (2015b) a novel method to extract its power spectrum by post-processing the momentum density field from a Newtonian simulation was introduced. This is motivated by the fact that the leading contribution to the shift vector in post-Friedmann expansion (Milillo et al. 2015) is sourced by the transverse part of the momentum density field. Although this method neglects the feedback of the shift vector into the simulation dynamics, this approximation is well justified as perturbation theory estimates that the magnitude of the vector potential is at most one percent of the scalar gravitational potential (Lu et al. 2009). Cosmological codes which are capable of simulating vector modes of the metric have been only recently developed (e.g., Adamek et al. 2016a; Adamek et al. 2016b; Mertens et al. 2016; Giblin et al. 2017; Macpherson et al. 2017; Barrera-Hinojosa & Li 2020a), and have proven robust enough to study different relativistic distortions in the large-scale structures (LSS); (see Adamek et al. 2020, for an actual comparison of frame-dragging observables in a toy universe simulated using these codes). In particular, the cross correlation between the shift vector and vorticity has been studied in Jelic-Cizmek et al. (2018) using the relativistic N -body code gevolution (Adamek et al. 2016a; Adamek et al. 2016b), showing that the vector potential is only weakly sourced by vorticity alone, which is subdominant compared with the density-dependent terms coming from the transverse projection of the full momentum field, in qualitative agreement with post-Friedmann expansion results from Bruni et al. (2014); Thomas et al. (2015b). The objective of this paper is to study the vector modes of both the gravitational and matter velocity fields from large sub-horizon scales down to deeply nonlinear scales using GRAMSES (BarreraHinojosa & Li 2020a,b), a recently-introduced general-relativistic N -body code based on RAMSES (Teyssier 2002). We expand on previous studies in the following ways: (i) similarly to Jelic-Cizmek et al. (2018), we provide a direct calculation of the gravitomagnetic field, represented by the shift vector, from the simulation, also relaxing the weak-field approximation in our approach; (ii) we present results for scales in the deeply nonlinear regime which have not been previously explored in this context, and which are accessible thanks to the adaptive-mesh refinement (AMR) capabilities of GRAMSES. For the first time, we explore the gravitomagnetic vector potential in dark matter haloes in a broad range of halo masses; (iii) furthermore, we quantify the gravitomagnetic acceleration inside the dark matter haloes and compare this against the standard gravitational one. Wenote that, with the exception of Jelic-Cizmek et al. (2018), previous studies of vorticity use simulations that incorporate a softening length scale, a numerical parameter used to prevent divergences in the calculation of inter-particle forces which also determines the spatial resolution. In GRAMSES - similarly to gevolution - the metric components and their spatial derivatives are calculated on a Cartesian mesh. AMR codes, such as GRAMSES, are generally slower than fixed-mesh-resolution codes such as gevolution which can benefit from efficient standard libraries such as FFTW, but their adaptivelyproduced mesh structure in high-density regions allows them to be more focused on the fine details in such regions, without increasing the overall cost of the simulation substantially. Therefore, they provide complementary ways to study the vector modes from cosmological simulations. The rest of this paper is organised as follows. In Section 2 we fix our notations and briefly describe the general-relativistic formalism and methods implemented in the GRAMSES code that are relevant for the vector modes. In Section 3.1 we show the results for the different power spectra of the velocity field components as well as of the gravitomagnetic potential. Then, in Section 3.2, we focus on dark matter haloes, providing comparisons of the gravitomagnetic potential and corresponding acceleration with the scalar counterparts. Throughout this paper, Greek indices are used to label spacetime vectors and run over (0 , 1 , 2 , 3) , while Latin indices run over (1 , 2 , 3) . Unless otherwise stated, we follow the unit convention that the speed of light c = 1 .", "pages": [1, 2]}, {"title": "2 METHOD AND DEFINITIONS", "content": "For the sake of clarity and completeness, let us briefly summarise the terminology and conventions adopted in this paper, which in some part stem from GRAMSES ' implementation itself. More details can be found in the main code paper (Barrera-Hinojosa & Li 2020a) and the references therein. In order to solve the gravitational sector equations and geodesic equations, GRAMSES adopts the 3+1 formalism in which the spacetime metric takes the form where \u03b1 is the lapse function, \u03b2 i the shift vector and \u03b3 ij the induced metric on the spatial hypersurfaces, which in the constrained formulation adopted by GRAMSES is approximated by a conformally-flat metric, \u03b3 ij = \u03c8 4 \u03b4 ij , with \u03c8 being the conformal factor and \u03b4 ij the Kronecker delta. i In the 3 + 1 formalism n \u00b5 = ( -\u03b1, 0) is the unit timelike vector normal to the time slices, the 3-dimensional spatial hypersurfaces with metric \u03b3 ij , and Eulerian observers are those with 4-velocity n \u00b5 . The energy density \u03c1 and momentum density S i measured by these normal observers are given by the following projections of the energy-momentum tensor T \u00b5\u03bd , \u03c1 n \u00b5 n \u03bd T , (2) \u2261 where the action of n \u00b5 projects onto the timelike direction, while \u03b3 \u00b5\u03bd = g \u00b5\u03bd + n \u00b5 n \u03bd projects onto the spatial hypersurface. Eq. (2) and (3) are the source terms for the Hamiltonian constraint and momentum constraint, respectively. Additionally, the spatial stress and its trace are defined as which, in addition to \u03c1 and S i , appear in the evolution equations for the extrinsic curvature tensor. In GRAMSES, the (dark) matter sector is represented by an ensemble of non-interacting simulation particles of rest mass m and four-velocity u \u00b5 = dx \u00b5 /d\u03c4 , where \u03c4 is an affine parameter. The equations for the gravitational sector are numerically solved based on conformal matter sources, which are scaled using \u03b3 = det( \u03b3 ij ) as In these, x is a (discrete) position vector on the cartesian simulation grid and the proportionality symbol in each equation stands for the standard cloud-in-cell (CIC) weights used for the particle-mesh projection (Hockney & Eastwood 1988). From Eqs. (5)-(7) we have the following useful relations: s = u i , (9) \u03c1 \u0393 where \u0393 \u2261 \u03b1u 0 = \u221a 1 + \u03b3 ij u i u j is the Lorentz factor. For a perfect fluid, s \u2261 \u221a \u03b3S is proportional to pressure in linear theory, and then it vanishes for CDM (dust) in such regime. Naturally, s also vanishes in the non-relativistic limit. The equations of motion for collisionless particles correspond to the geodesic equation u \u00b5 \u2207 \u00b5 u \u03bd = 0 , which in the 3 + 1 form reads In Eq. (12), the term u j \u2202 i \u03b2 j corresponds to a force that is absent in both the Newtonian limit and the linear perturbation regime. In the case where \u03b2 j is purely a vector-type perturbation (e.g., the Poisson gauge), this force term is known as gravitomagnetic force , in formal analogy with the magnetic Lorentz force.", "pages": [2, 3]}, {"title": "2.1 Vector decomposition", "content": "Given that in this paper we are particularly interested in vector modes (transverse modes), we start by splitting a vector field V i \u00b5\u03bd ( V ) as where V \u2016 and V \u22a5 are respectively the scalar (irrotational) and vector (rotational) components, i.e., these satisfy In the case of the velocity field 2 u i ( u ), we define the velocity divergence and vorticity as As usual, the power spectra of these quantities are respectively defined as (19) and the velocity power spectrum satisfies the relation The power spectrum of the vector modes of the shift vector is defined in analogous way to Eq. (19).", "pages": [3]}, {"title": "2.2 Gauge choice and the constraint for the vector potential", "content": "For solving the gravitational and geodesic equations, GRAMSES implements a constrained formulation of GR (Bonazzola et al. 2004; Cordero-Carri\u00f3n et al. 2009), in which both the tensor modes of the spatial metric and the transverse-traceless (TT) part of the extrinsic curvature are neglected during the evolution. In contrast, the scalar and vector modes of the gravitational field are treated fully nonlinearly. In order to do this in a robust way, the formalism adopts the constant-mean-curvature slicing (Smarr & York 1978b; Shibata 1999; Shibata & Sasaki 1999) and a minimal-distortion gauge condition under the conformal flatness approximation (Smarr & York 1978a). Contrary to the Poisson gauge, in this gauge the shift vector contains both scalar and vector ( 1+2 ) degrees of freedom. At linear order, the latter modes match the gauge-invariant shift vector from the Poisson gauge (Matarrese et al. 1998a; Lu et al. 2009), while the mismatch in the scalar piece reflects the fact that the time foliations are different in these two gauges. Then, in this formalism the components of the shift vector are solved from a combination of the 3 + 1 evolution equation for the extrinsic curvature, the momentum constraint and the gauge conditions (Barrera-Hinojosa & Li 2020a) where s i = \u03b4 ij s i , (\u2206 L \u03b2 ) i := \u2202 2 \u03b2 i + \u2202 i ( \u2202 j \u03b2 j ) / 3 denotes the flatspace vector Laplacian operator, and is the longitudinal part of the traceless extrinsic curvature tensor. The auxiliary potential W i introduced in Eq. (22) is directly solved from the momentum constraint equation, 2 We use u to represent the velocity u i rather than u i , as it is the former that is used in the 3 + 1 form of the geodesic equations (12) and (13) which are implemented in GRAMSES. u i is what we call 'CMC-MD-gauge velocity', and is different from u i . See Barrera-Hinojosa & Li (2020b) for more details. Then, from Eq. (21) we note that, at leading order, the shift vector is sourced by the momentum field and thus \u03b2 i \u221d W i by Eq. (23), while differences appear at higher order due to the extrinsic curvature tensor sourcing \u03b2 i . Given that throughout this paper we will be interested in the vector modes of the shift vector, this is decomposed in the same fashion of Eq. (14), i.e. where B i \u2261 \u03b2 i \u22a5 ( B ) is referred to as the vector potential or gravitomagnetic potential, and \u03b2 i \u2016 is the scalar mode of the shift. Let us note that, using Eq. (9), the curl of the conformal momentum density field s i ( s ) can be written non-perturbatively as where \u03b4 = \u03c1/ \u00af \u03c1 -1 is the density contrast and \u00af \u03c1 is the mean density. Previous studies have shown that the terms \u03b4 \u03c9 and \u2207 \u03b4 \u00d7 u in the r.h.s. of Eq. (25) are the main sources for the vector potential (Bruni et al. 2014; Thomas et al. 2015b; Jelic-Cizmek et al. 2018), while the contribution from vorticity itself is subdominant at all scales. In the r.h.s. of Eq. (25), the last term and the overall modulation by the Lorentz Factor \u0393 arise due to the definition of s in Eq. (9), and both contributions vanish in the linear regime and the non-relativistic limit.", "pages": [3, 4]}, {"title": "3 RESULTS", "content": "For the investigation in this paper, we have run a high-resolution simulation using GRAMSES, with a comoving box size L box = 256 h -1 Mpc and N part = 1024 3 dark-matter particles, corresponding to a particle mass resolution of 1 . 33 \u00d7 10 9 h -1 M glyph[circledot] . Because GRAMSES makes use of AMR in high-density regions, the spatial resolution is not uniform throughout the simulation volume: while the coarsest (domain) grid has N part cells, corresponding to a comoving spatial resolution of 0 . 25 h -1 Mpc , the most refined (high density) regions reach a resolution of 128 3 \u00d7 N part grid elements, with corresponding spatial resolution of 2 h -1 kpc . Initial conditions suitable for the relativistic simulation were generated at z = 49 with a modified version of 2LPTic code (Crocce et al. 2006) fed with the matter power spectrum obtained from a modified version of CAMB (Lewis et al. 2000) that works for the particular gauge needed for GRAMSES. More details on this can be found in Barrera-Hinojosa & Li (2020b). The cosmological parameters adopted for the simulation are { \u2126 \u039b , \u2126 m , \u2126 K , h } = { 0 . 693 , 0 . 307 , 0 , 0 . 68 } and a primordial spectrum with amplitude A s = 2 . 1 \u00d7 10 -9 , spectral index n s = 0 . 96 and a pivot scale k pivot = 0 . 05 Mpc -1 . In order to measure the velocity fields from simulation snapshots, we use the publicly-available DTFE code (Cautun & van de Weygaert 2011) which is based on the Delaunay tessellation method, although other methods have been explored in the literature during the last few years. Notably, the phase-interpolation method introduced in Abel et al. (2012) shows better performance than DTFE in shell-crossing regions (Hahn et al. 2015), where the finite-difference estimation of velocity divergence and vorticity across caustics can be problematic due to the multiply-valued nature of the velocity field. Nonetheless, the power spectra of these two fields are not strongly affected by this since the volume-weighted contribution from caustics is negligible, and both methods converge when nonlinear scales are well resolved. In addition, while the vorticity power spectrum is affected by resolution effects, this is weakly affected by finite-volume effects (Pueblas &Scoccimarro 2009; Jelic-Cizmek et al. 2018). We note that, while the initial velocity field is vorticity-free by construction, spurious vorticity will be present at some degree due to the numerical errors introduced by particle-mesh projections. In addition, shell-crossing events - which source vorticity - are rare at high redshift, and its insufficient sampling restricts the possibility of estimating the velocity field robustly. Therefore, in this paper we shall focus mainly on low redshifts, z < 1 . 5 , at which vorticity results are expected to be robust. Contrary to the velocity field, the gravitomagnetic potential is already solved by the code on a Cartesian mesh so there is no need for post-processing particle-mesh projections. It is worthwhile to mention that, although the GR simulations do not necessitate the specification of a cosmological background (Barrera-Hinojosa & Li 2020a), throughout this paper the notion of redshift is still used and should be understood as the standard, background one. This is achieved through the constant-mean-curvature slicing condition, which allows us to fix the trace of the extrinsic curvature of the spatial hypersurfaces as K = -3 H ( t ) , where the Hubble parameter H can be conveniently fixed via 'fiducial' Friedmann equations (Giblin et al. 2019; Barrera-Hinojosa & Li 2020a). In addition, even though in the gauge adopted by GRAMSES the scalar gravitational potentials as well as the matter fields are not gaugeinvariant quantities, gauge effects are only prominent on large scales and become strongly suppressed for modes inside the horizon. Since in this work we are mainly interested in the latter, as well as in redshifts below z = 1 . 5 (in which the horizon is already larger than the box size), we do not explore potential gauge issues further. Figure 1 provides a visual representation of the density field (top left), velocity divergence (top right), the magnitude 3 of the vorticity vector field, \u03c9 \u2261 | \u03c9 | = ( \u03c9 2 x + \u03c9 2 y + \u03c9 2 z ) 1 / 2 (bottom left), and the vector potential magnitude, B \u2261 | B | = ( B 2 x + B 2 y + B 2 z ) 1 / 2 (bottom right), across a slice of the simulation box at z = 0 . From this figure, it is clear that the density field has a similar large-scale distribution to the velocity divergence, consistently with linear perturbation theory. Since velocity divergence can take negative values, we use a linear scale on its map, with a cutoff of extreme values to help visualisation. As expected, the velocity divergence is negative in collapsing regions due to matter in-fall, and positive in voids and low-density regions. The vorticity field also shows a clear correlation with both density and velocity divergence. However, we should bear in mind that, as we have discussed before, the velocity divergence and vorticity estimated by DTFE are not completely reliable near caustics (Hahn et al. 2015), and therefore such maps only provide qualitative information and an accurate picture on large scales. From the bottom right panel in Fig. 1, we observe that the magnitude of the vector potential has some degree of correlation with the structures observed in density, velocity divergence and vorticity, particularly in very high-density and low-density regions. As shown by Eqs. (21)-(25), the vector potential is not sourced by any of these components alone but is correlated with the rotational part of the full momentum density field. This panel also shows that the distribution of the vector potential magnitude is a great deal smoother than the cases of matter and velocity fields. This is expected since the vector potential components satisfy the elliptic-type equation (21), and then long-wavelength modes become dominant due to the Laplacian operator \u2202 2 . Although not included here, the same happens in the case of the conformal factor \u03c8 which satisfies the Hamiltonian constraint (or the Poisson equation in the Newtonian limit). From the quantitative side, we note that the vector potential magnitude seems to 10 - 1 10 0 10 1 10 2 10 3 10 4 - - - - typically remain between O (10 -8 ) and O (10 -7 ) , with some peaks of a few times O (10 -7 ) only in very specific regions. We will explore the behaviour of the vector modes in more detail in the next sections.", "pages": [4, 5]}, {"title": "3.1 Power spectra", "content": "In this section we analyse the power spectra of the velocity field and gravitomagnetic vector potential. The auto and cross spectra of matter quantities such as density, velocity divergence and vorticity (which are measured with DTFE from particle data) are calculated using NBODYKIT (Hand et al. 2018). In contrast, the vector (as well as scalar) potential values are calculated and stored by GRAMSES in cells of hierarchical AMR meshes, and the spectrum is measured by a different code that is able to handle such mesh data directly and to write it on a regular grid by interpolation. While the vector potential spectrum can also be measured in the same way as the matter quantities by writing its values at the particles' positions rather than in AMR cells, which means DTFE and NBODYKIT can be used, the above method yields better results on small scales as shown in Appendix A. In all figures, we normalise the velocity power spectra by the factor ( H f ) 2 , where H = aH is the conformal Hubble parameter of the reference Friedmann universe, and f the linear growth rate in \u039b CDMparameterised as (Linder 2005) where \u2126 m ( a ) = \u2126 m a -3 / ( H/H 0 ) 2 . In this way, the amplitude of P \u03b8 matches that of the matter power spectrum in the linear regime, where the continuity equation \u03b4 = -\u03b8/ ( H f ) is expected to hold. Figure 2 shows the velocity divergence power spectrum (top left panel), the vorticity power spectrum (top right panel), the cross spectrum between density and velocity divergence (bottom left) and the power spectrum of two different contributions to the momentum field (bottom right) at different redshifts in the range 0 \u2264 z \u2264 1 . 5 . In the case of velocity divergence, we find a very good agreement with linear theory at scales k \u2264 0 . 1 h Mpc -1 for all redshifts. Above that scale, deviations become stronger towards lower redshifts, and a localised power loss ('dip') eventually develops around k \u2248 1 . 2 h Mpc -1 . In the case of the vorticity power spectrum, we note that towards large scales this is several orders of magnitude smaller than velocity divergence, while at around k \u223c 1 h Mpc -1 the spectrum starts to peak and they become comparable. Note that, unlike the velocity divergence, there is no standard perturbation theory prediction for the vorticity as this exactly vanishes in the perfect fluid description. Interestingly, the 'dip' in the velocity divergence power spectrum is at the similar position to the peak in the vorticity power spectrum, which has been interpreted as the consequence of shell crossing occurring around that scales, where the angular momentum can be large enough to dampen the growth of structures as it forces particles to rotate around them (Jelic-Cizmek et al. 2018). Note that, due to the high cost 4 of GR simulations using GRAMSES, we have not performed runs with even higher resolutions to check the convergence of the velocity and vorticity power spectra. A useful convergence test for gevolution simulations was done in Jelic-Cizmek et al. (2018) (see Fig. 6 there), which shows that the amplitude of P \u03c9 decreases as the force resolution increases. The simulations there have the same box size of L box = 256 h -1 Mpc , and the run labelled 'high resolution 1' has the same mesh resolution as our domain grid ( 1024 3 cells); while this resolution is eight times poorer than that of the run labelled 'high resolution 2', which has 2048 3 cells, the AMR nature of GRAMSES means that higher resolution can be achieved in high-density regions - with the highest resolution attained in our run being equivalent to a regular mesh with 128 3 \u00d7 1024 3 cells. Hence, since 'high resolution' 1 and 2 are already converged in Jelic-Cizmek et al. (2018), we conclude that our simulation has also converged to at least a similar level. 4 A GR simulation using GRAMSES takes about an order of magnitude longer than an equivalent Newtonian simulation using default RAMSES, partly due to the 10 (compared to one) GR metric potentials to be solved, and partly due to the cost of preparing the source terms for the nonlinear equations that govern the metric potentials, as well as the additional MPI communications. The cross spectra P \u03b4\u03b8 is useful for detecting deviations from linear theory and provides information about shell crossing. Considering the continuity equation, the linear-theory expectation is that P \u03b8\u03b4 / ( H f ) = -P \u03b4 , but towards shell-crossing scales the initial (linear) anti-correlation of \u03b4 and \u03b8 is lost and correlations appear (Hahn et al. 2015). From the bottom left panel of Fig. 2 we find that the anti-correlation drops dramatically and flips sign at k \u2248 2 h Mpc -1 at z = 0 , which is slightly higher than the scale at which the vorticity spectrum peaks as also found in previous studies (Jelic-Cizmek et al. 2018). The bottom right panel of Fig. 2 shows the power spectra of \u03b4 \u03c9 and \u2207 \u03b4 \u00d7 u , which are the main source terms for the metric vector potential in Eq. (25). In particular, the contribution of \u03c9 to Eq. (25) is already small compared to \u03b4 \u03c9 on nonlinear scales because \u03b4 glyph[greatermuch] 1 . We find good agreement with the z = 0 results shown in Bruni et al. (2014) based on a post-Friedmann expansion. We find that towards higher redshifts the contribution due to \u2207 \u03b4 \u00d7 u starts to become larger than that of \u03b4 \u03c9 at slightly larger scales. Although vorticity vanishes in standard perturbation theory, the effective field theory of LSS (EFTofLSS) predicts that its power spectrum today can be characterised by a power law over a range of scales (Carrasco et al. 2014). On large scales, we can find the slope of the vorticity power spectrum by fitting a power law, where n \u03c9 is the large-scale spectral index, and A \u03c9 the amplitude that is not fixed by theory. The EFTofLSS predicts n \u03c9 = 3 . 6 for 0 . 1 h Mpc -1 glyph[lessorsimilar] k glyph[lessorsimilar] 0 . 2 h Mpc -1 and n \u03c9 = 2 . 8 for 0 . 2 h Mpc -1 glyph[lessorsimilar] k glyph[lessorsimilar] 0 . 6 h Mpc -1 (Carrasco et al. 2014). Previous N -body simulations have found n \u03c9 \u2248 2 . 5 for k glyph[lessorsimilar] 0 . 1 h Mpc -1 (Hahn et al. 2015); a similar value was found at k glyph[lessorsimilar] 0 . 4 h Mpc -1 in Jelic-Cizmek et al. (2018). Moreover, on scales k glyph[greaterorsimilar] 1 h Mpc -1 , there is partial evidence suggesting that the spectral index approaches the asymptotic value n NL \u03c9 \u2192-1 . 5 (Hahn et al. 2015). Figure 3 shows the best fits of the power law (27) to the simulation data at z = 0 on large scales (small scales) with their corresponding spectral index n \u03c9 ( n NL \u03c9 ), and the shaded region represents the interval of validity for the fit. On large sub-horizon scales, we find n \u03c9 \u2248 2 . 7 , which is slightly higher than previous simulations results in the literature, and slightly lower than the EFTofLSS prediction. Notice, however, that there is not complete overlap between the region used for the fit and the EFTofLSS prediction used for comparison as the latter extends up to k \u223c 0 . 6 h Mpc -1 but it is clear that the slope of the power spectrum already decreases at k \u223c 0 . 32 h Mpc -1 . In addition, the slope does not seem to become steeper at larger scales as predicted by the EFTofLSS, a feature also found by the previous study (Jelic-Cizmek et al. 2018), which is likely related to the large-scale cutoff imposed by the finite box of the simulation. Toward smaller scales, we find the spectral index n NL \u03c9 \u2248 -1 . 4 , which is slightly less steep than that suggested in Hahn et al. (2015). However, there is a slight but clear increase in power at around k \u223c 7 h Mpc -1 which introduces an oscillatory feature not captured by a perfect power law. As originally proposed in Pueblas & Scoccimarro (2009), it is also interesting to characterise the evolution of the large-scale vorticity power spectrum as where D + ( z ) is the linear growth rate at z and \u03b3 \u03c9 a new parameter. In Pueblas & Scoccimarro (2009), the best-fit value found is \u03b3 \u03c9 = 7 \u00b1 0 . 3 using the snapshots z = 0 , 1 , 3 , which is overall consistent with Thomas et al. (2015b); Jelic-Cizmek et al. (2018), although the latter references suggest values \u03b3 \u03c9 \u2265 7 . Moreover, these have only considered snapshots with z \u2264 1 since the scaling breaks down at higher redshifts, which is likely related to resolution effects in the sampling of vorticity due to a lower fraction of particles undergoing shell crossing at higher redshifts. The top panels of Fig. 4 show the results for the best fist of the D \u03b3 \u03c9 + scaling of Eq. (28) using several snapshots below z = 1 . 5 . The top left panel of Fig. 4 shows the power spectrum at these various redshifts scaled using ( D + ( z ) /D + (0)) 7 . 7 , while in the top right panel we select three different modes from the shaded green region of the top left panel and find the corresponding value of \u03b3 \u03c9 from a best fit to the corresponding vorticity spectra. We find that there is some scale dependence in \u03b3 \u03c9 and the amplitude of the vorticity power spectrum evolves approximately with \u03b3 \u2248 7 . 7 over the scales 0 . 08 glyph[lessorsimilar] k glyph[lessorsimilar] 0 . 4 , which is higher than other simulation results in the literature (Pueblas & Scoccimarro 2009; Thomas et al. 2015b; JelicCizmek et al. 2018). However, compared to the latter two references, in the case here we are able to fit the amplitude up to z = 1 . 5 before the scaling breaks down. Besides the results from Jelic-Cizmek et al. (2018) based on the gevolution code, which works in a fixedresolution grid, previous studies of vorticity use N -body simulation codes in which a softening length scale in the force calculation determines the spatial resolution. In the case of GRAMSES, the AMR capabilities allow one to achieve high spatial resolution ( \u223c 2 h -1 kpc ) in high-density regions. We can extend the previous analysis to model the time evolution of the vorticity power spectrum at nonlinear scales, in terms of a new scale-independent parameter \u03b3 NL \u03c9 in Eq. (28). From Fig. 2, it is clear that the power spectrum evolves more slowly in this regime compared with large scales, and so we expect \u03b3 NL \u03c9 to be smaller than \u03b3 \u03c9 . In the bottom left panel of Fig. 4, we show the scaling of the vorticity spectra by ( D + ( z ) /D + (0)) 2 . 6 , where we find that such evolution works as a good approximation on scales k glyph[greaterorsimilar] 3 . 2 h Mpc -1 . In the bottom right panel we show the best-fit value of \u03b3 NL \u03c9 for three different k -modes. In this case, unlike in the previous fit for large subhorizon scales, we have not considered the z = 1 . 5 spectrum for the fit as from the bottom left panel it is already clear that the scaling for such spectrum (orange solid line) would deviate from the lower redshift results. This result suggests that the amplitude of the vorticity power spectrum can be actually estimated using a scale-independent parameter in the power law of Eq. (28) on deeply nonlinear scales. However, there is an obvious scale dependence in the transition between the large- and small-scale regimes which is not captured by these parameterisations and requires further investigation. Let us now discuss the results for the vector potential. In \u039b CDM cosmology, this appears as a second-order perturbation at its lowest order, which in the case of a perfect fluid is sourced by the product of the first-order density contrast and velocity divergence (Matarrese et al. 1998b; Lu et al. 2009). However, the single-stream fluid description of CDM breaks down at late times when shell crossing occurs, and then we expect corrections to the vector potential particularly at quasi-linear and nonlinear scales. The second-order perturbation theory prediction for the dimensionless power spectrum of B , is given by (Lu et al. 2009) where \u2206 \u03b4 and \u2206 v are the dimensionless power spectra of the density perturbation and velocity potential v , \u2206 \u03b4v their cross spectrum, and \u03a0( u, w ) = u -2 w -4 [ 4 w 2 -(1 + w 2 -u 2 ) 2 ] is an integration kernel that depends on w = k ' /k and u = \u221a 1 + w 2 -2 w cos \u03d1 , with cos \u03d1 defined by cos \u03d1 = k ' \u00b7 k / ( kk ' ) . At any given scale, the convolution in Eq. (30) couples different k -modes of \u03b4 and v . Since the simulation can only access modes within a finite k -range, this is equivalent to having a large-scale ( k min ) and small-scale ( k max ) cutoffs in Eq. (30), therefore leading to a lower amplitude of P B than the true result. For instance, Adamek et al. (2016b) found that in order to get good agreement between simulation results and perturbation-theory calculations using Eq. (30), the box should be large enough to contain the matter-radiation equality scale. In practice, to account for this effect due to missing k -modes, to compare with Eq. (30), we use the large-scale cutoff k min \u223c 0 . 8 \u00d7 2 \u03c0/L , i.e. 80 percent of the fundamental mode of the box, as well as a small-scale cutoff k max = \u03c0N 1 / 3 part /L , which corresponds to the Nyquist wavenumber of the coarsest grid used by the simulation. The left panel of Fig. 5 shows the simulation measurements of the dimensionless power spectrum of the vector potential at four different redshifts, and their corresponding perturbation-theory predictions. At z \u2265 1 we see good agreement between the simulation and perturbation-theory results up to k \u223c 0 . 3 h Mpc -1 , while at z = 0 discrepancies start already at k \u223c 0 . 2 h Mpc -1 , which is qualitatively consistent with Adamek et al. (2014); Bruni et al. (2014); see also Andrianomena et al. (2014) for a prescription of the nonlinear corrections to the perturbation-theory result using HALOFIT. At highly nonlinear scales the amplitude of the spectrum measured from the simulation can be more than two orders of magnitude higher than the perturbation-theory prediction. Note that at all four redshifts the simulation spectra flatten at the largest k -mode sampled by the simulation box, which can be interpreted as a finite-box effect. The right panel of Fig. 5 shows the ratio between the power spectra of vector potential B and that of the scalar potential \u03a6 measured from the simulation, the latter defined as the fully nonlinear perturbation to the lapse function in the metric (1), i.e. \u03a6 \u2261 \u03b1 -1 . At z = 0 , we find the ratio to be within 2 \u00d7 10 -5 and 4 \u00d7 10 -5 for 0 . 2 h Mpc -1 glyph[lessorsimilar] k glyph[lessorsimilar] 10 h Mpc -1 , which is in good agreement with Bruni et al. (2014). The ratio reaches a peak of 5 \u00d7 10 -5 at k \u223c 15 h Mpc -1 , after which it starts to decrease. At higher redshift the evolution of B makes the ratio larger. Our results confirm that the ratio between both potentials reach the percent-level on nonlinear scales at z = 0 . As pointed out by Bruni et al. (2014), though this ratio is close to the value found in Lu et al. (2009) for the ratio between scalar and vector modes in perturbation theory, here the fully nonlinear B , \u03a6 fields are used. In fact, the vector potential power spectrum from the left panel of Fig. 5 can be over two orders of magnitude larger than that found in the latter reference.", "pages": [5, 6, 7, 8]}, {"title": "3.2 The vector potential and frame-dragging acceleration in dark matter haloes", "content": "Let us further analyse the vector potential on nonlinear scales by investigating its magnitude inside the dark matter haloes from the above general-relativistic simulation. For this we have generated halo catalogues using the phase-space Friends-of-Friends halo finder ROCKSTAR (Behroozi et al. 2013). From this catalogue we then get their centre positions, radii R 200 c and masses M 200 c . The latter two are defined respectively as the distance from the halo centre which encloses a mean density of 200 times the critical density of the universe as a given redshift, and the mass enclosed within such a sphere. Unfortunately, the inaccuracy when estimating the velocity divergence and vorticity fields on small scales using DTFE prevents us from studying their behaviour in haloes alongside the vector potential. We have tested that indeed, the velocity estimations are strongly affected by resolution and do not converge either using a resolution for the tessellation grid similar to the mean inter-particle dis- tance of dark matter particles in the haloes or otherwise. The phaseinterpolation method was used in Hahn et al. (2015) to successfully estimate the vorticity in haloes in the case of warm dark matter, but still it is not possible to robustly measure this from CDM simulations either: this is related to the difficulty of resolving the perturbations up to highly nonlinear scales in the CDM case, which in warm dark matter models is not required as the spectrum truncates at some finite free-streaming scale. Figure 6 shows density (left column), vector potential magnitude (middle column) and scalar gravitational potential (right column) in the vicinity of three selected dark matter haloes at z = 0 , with masses M h \u2248 6 . 5 \u00d7 10 15 h -1 M glyph[circledot] (top row), M h \u2248 3 . 0 \u00d7 10 13 h -1 M glyph[circledot] (middle row) and M h \u2248 3 . 1 \u00d7 10 12 h -1 M glyph[circledot] (bottom row). In all cases, the map centre is aligned with the halo centre and the width of the shown region corresponds to four times the halo radius R 200 c . As also shown in Fig 1, overall we observe some degree of correlation between the vector potential and the matter density, but clearly not at the level of the scalar potential. In particular, in the case of the most massive halo (top row) we can see that while both potentials peak towards the halo centre, unlike for the scalar potential, the global maximum of the vector potential within the shown region is actually found in the lower left part of the map, where there appears to be another, smaller, halo infalling towards the central one. Again, this qualitative difference is not surprising since the vector potential is sourced by the transverse part of the momentum density, Eq. (25), while the matter source term for the scalar potential is the density contrast itself (up to higher-order terms). As before, we can also see that both potentials are smoother than the density field owing to the elliptic-type nature of their equations (Barrera-Hinojosa & Li 2020a), in which short-wavelength modes are dominated. In addition, in the most massive halo we can observe that the scalar potential tends to be more spherically symmetric around the center than B , which displays large values in most part of the left and upper part of the map. Indeed, although the low-density (dark) regions in the bottom right and top left parts of the density map are of similar characteristics, and these are clearly correlated with the \u03a6 map, these are not correlated with features in the B map at all. For the halo shown in the middle row of Fig. 6, the density and potential contours have more similar shapes to each other than in the most massive halo. Nonetheless, the scalar potential again seems to decay more rapidly outside R 200 c than the vector potential magnitude. This also seems to be the case in the halo shown in the bottom panels, although in this case the potentials are smaller and shallower. Note that, for the halo in the middle panels, | \u03a6 | is largest in the central region (red/orange/green), decays when one moves fur- her away from the halo centre (blue), but grows again far from the halo (green); this is because this halo resides in a low-density environment, with a positive environmental contribution to the total potential so that the latter crosses zero. It is important to bear in mind that, although the halo centres are approximately located at a local maximum of | \u03a6 | , the potentials themselves are not an observable quantity: it is the gradient of the potentials that contributes as force terms in the geodesic equation (12), while the values of the potential themselves can be largely influenced by their environments. In this subsection, we are mainly interested in haloes which are isolated and therefore less affected by environments. To select such haloes, we try to split the potential at each point into two contributions: one from the halo itself and one from its environment, i.e., well beyond a distance R 200 c from its centre. Since the potentials are not necessarily spherically symmetric, as it is evident from the top row of Fig. 6, as a crude way, we shall take the spherical average in a radial bin at 2 R 200 c and subtract this from the values at smaller radii, which allows to get 'shifted' radial halo profiles for both \u03a6 and B that vanish at 2 R 200 c . For \u03a6 ( B ) we expect this profile to monotonically increase (decrease) to zero as r increases to 2 R 200 c , for well-isolated relaxed haloes. Figure 7 shows, from the top to the bottom row, the radial profiles of density, the vector potential magnitude and its ratio against the scalar gravitational potential. All profiles have been measured from the centres of a sample of haloes in different mass ranges, for three redshifts: z = 0 (left column), z = 0 . 5 (middle column) and z = 1 (right column). For this we have selected three subsamples of haloes with O (100) haloes each based on mass cuts: we define a higher mass range M h \u2265 10 14 . 5 h -1 M glyph[circledot] , an intermediate mass range with mean mass \u00af M h = 10 13 . 5 h -1 M glyph[circledot] , and a lower mass range with mean mass \u00af M h = 10 12 . 5 h -1 M glyph[circledot] . For each halo from a given mass range, we then calculate the spherical average of the den- \u00d7 \u00d7 \u00d7 5 0 sity, vector potential and scalar potential up to 2 R 200 c , and average over the full population. As mentioned in the previous paragraph, in the case of the potentials we have subtracted their average values at 2 R 200 c in the profile of each individual halo. In this process, we have discarded the haloes in which the resulting spherical average of B becomes negative for some r < R 200 c after the subtraction, which typically happens in lower mass haloes due to their shallow potentials. However, these haloes are the most abundant type and hence we retain a sample of size O (100) even at z = 1 , while the number of haloes in the middle and higher mass bins is around \u223c 50 at that same redshift. From Fig. 7 we find that at the 1 \u03c3 level there is a clear correlation between halo mass and the magnitude of the gravitomagnetic potential, which can differ by up to two orders of magnitude between halos with masses close to 10 12 . 5 h -1 M glyph[circledot] and those with masses larger than 10 14 . 5 h -1 M glyph[circledot] . In all cases, the vector potential flattens toward the halo centres and it decreases towards the outskirts. However, from the bottom row of Fig. 7 we find that the ratio between vector and scalar potentials is roughly constant inside haloes across all masses and redshifts considered, and the dependence of this ratio upon halo mass is quite weak as all means lie within 1 \u03c3 of each other. At z = 0 , we find that the ratio is a few times 10 -3 , which is roughly consistent with the value inferred from the ratio of O (10 -5 ) between the power spectra of the vector and scalar potentials at k glyph[greaterorsimilar] O (0 . 1) h Mpc -1 , as shown in Fig. 5 (note that the subtraction of the environmental contributions in these potentials essentially removes the long-wavelength contributions to B/ \u03a6 , thereby marking this comparison with Fig. 5 reasonable; but as we only look at a small fraction of the total volume, inside a sub-group of haloes, we of course should not expect an exact equality). At z = 0 . 5 and z = 1 , the picture is qualitatively the same apart from the increase in the amplitude of the vector potential. In CDM simulations, it is well known that the density profile of haloes can be described by the universal Navarro-Frenk-White (NFW; Navarro et al. 1996) fitting formula, which has a corresponding analytical prediction for the Newtonian potential profiles of haloes. The constancy of B/ | \u03a6 | inside haloes which is found here implies that it might be straightforward to derive an analytical fitting function for the B profiles in haloes, which is closely related to the NFW function, though this will not be pursued in this paper. Given that Fig. 7 shows that the ratio between the vector and scalar potentials is roughly constant inside the halos - and we have checked that such constant ratio holds even above z = 1 - we can characterise this ratio by a single number at each halo mass and redshift. As an extension of the bottom row of Fig. 7, Fig. 8 shows the mean value of such ratio calculated within r < R 200 c at different redshifts. Since the number of haloes in a given mass bin decreases towards higher redshifts, here we only consider cases in which the number of haloes in a given mass range is greater than ten at a given redshift. We find that for all mass bins B/ \u03a6 increases almost linearly with redshift. At redshift z = 2 the rate of change of this ratio with respect to redshift slows down slightly for the lowest mass range (blue line), after which it picks up again: this could be due to a lack of simulation resolution at high z . Observationally, the ratio between vector and scalar potentials is particularly relevant for weak lensing, as post-Newtonian calculations show that the relative correction to the Newtonian convergence field \u03ba is proportional to B/ \u03a6 (Sereno 2002, 2003; Bruni et al. 2014). Therefore, Fig. 8 suggests that, in the case of dark matter haloes, the lensing convergence correction due to the gravitomagnetic potential is between the O (10 -3 ) and O (10 -2 ) level, in agreement with previous studies (Sereno 2007; Cuesta-Lazaro et al. 2018; Tang et al. 2020). Moreover, this only depends weakly on the halo mass and could be more easily detected on high-mass haloes at high redshifts. However, we note that at higher orders in the post-Newtonian expansion, new contributions from the time derivative of B appear (Bruni et al. 2014; Thomas et al. 2015a) as well, which requires further inspection. Besides investigating the potentials, we can also look at the force that each of these exert on the particles according to Eq. (12), which shows that the total force is mainly composed by two contributions; the standard gravitational force arising from the gradient of the scalar potential (first term on the r.h.s.), and the gravitomagnetic force (contained in the second term on the r.h.s.) which is responsible for the frame-dragging effect. The latter is naturally not taken into account in Newtonian gravity. The third term in the r.h.s of Eq. (12) is subdominant and so we shall not explore it here. Figure 9 is a visualisation of the magnitude of the gravitomagnetic acceleration (middle column) and that of the standard gravitational acceleration (right column) in units of h cm / s 2 , in the vicinity of three different dark matter haloes. These haloes have similar masses to those shown in Fig. 6. We find that the forces are correlated with the density field up to some degree, particularly in the haloes in the middle and bottom rows, although the gravitomagnetic force seems to be less smooth than the Newtonian one. For the halo in the top row, there is a clearer difference between the forces compared to the other two cases. The peaks of the gravitomagnetic acceleration seem to occur at the density peaks but the opposite is not true, and there is no clear correspondence between their amplitudes. Interestingly, in this halo the values of gravitomagnetic force around a few times 10 -13 h cm / s 2 (green region) extend around the centre and towards the left part of the map, where the density field has already decreased by various orders of magnitude. This kind of asymmetry between both kinds of maps might be due to the actual dynamical state of the particles in a given region. Even if the density is low, if the particles' velocity happens to be aligned with the gradient of the vector potential components they will contribute significantly to | u \u00b7 \u2202 i B | . As before, we can calculate the spherical averages of the forces, which allows us to get radial profiles (although no subtraction from radial bins beyond 2 R 200 c is required this time). Figure 10 shows a comparison of the gravitomagnetic (frame-dragging) acceleration and the standard gravitational one in dark matter haloes in an analogous way to the scalar and vector potential profiles shown in Fig. 7. We find that the magnitude of the gravitomagnetic force is larger towards the inner parts of the halo, and the dependence on the halo mass is weaker than in the case of the scalar gravitational potential. As we discussed before, this can also be explained by the fact that the gravitomagnetic force not only depends on density but on the actual dynamical state of particles. Similarly to the behaviour of B/ | \u03a6 | , from Fig. 10 we find that the ratio of the two corresponding forces also remains fairly constant inside the haloes, although in the most massive haloes it tends to increase toward the outskirts. A weak dependence on halo mass is found at all redshifts. In Adamek et al. (2016b) the maximum gravitomagnetic acceleration measured from the simulation box at z = 0 is found to be roughly 7 \u00d7 10 -12 h 6 cm/s 2 for the highest resolution used ( 125 h -1 kpc ), while the value measured from lower resolution runs decreases monotonically. From Fig. 10 we find that this is comparable with our results for haloes in the upper mass range at the 1 \u03c3 level. However, we note that for the most massive halo in our simulation, we find the maximum value of the gravitomagnetic acceleration to be 7 \u00d7 10 -11 h cm/s 2 , i.e. roughly one order of magnitude higher. This difference could be explained by the fact that in our simulation the most refined regions are resolved with a resolution of 2 h -1 kpc . In addition, GRAMSES treats the vector potential non-perturbatively, although the difference due to higher-order corrections is likely to be subdominant with respect to the aforementioned resolution dependence.", "pages": [8, 9, 10, 11, 12, 13, 14, 15]}, {"title": "4 CONCLUSIONS", "content": "We have investigated the vector modes of the matter fields as well as those of the \u039b CDM spacetime metric, from large sub-horizon scales to deeply nonlinear scales using a high-resolution run of the general-relativistic N -body GRAMSES code (Barrera-Hinojosa & Li 2020a,b). On the one hand, vorticity vanishes at the non-perturbative level in a perfect fluid description and yet it is generated dynamically due to the collisionless nature of dark matter. On the other hand, the metric vector potential - responsible for frame-dragging - appears beyond linear order in perturbation theory and is not solved for in Newtonian simulations. Therefore, the physics behind the vector modes is highly non-trivial and numerical simulations play an important role in their study. Although the relativistic nature of the code is not particularly exploited from the point of view of vorticity, the vector potential is a prime quantity as this is not part of Newtonian gravity and therefore not implemented in Newtonian simulations. To this end, we have run a high-resolution N -body simulation using GRAMSES, that employs N part = 1024 3 particles in a box of comoving size L box = 256 h -1 Mpc . In GRAMSES, the GR metric potentials - in the fully constrained formalism and conformally flat approximation - are solved on meshes in configuration space. The AMR capabilities of GRAMSES allows it to start off with a regular grid with 1024 3 cells, and hierarchically refine it in high-density regions to reach a spatial resolution of 2 h -1 kpc in the most refined places, namely dark matter haloes. This enables a quantitative analysis of the behaviour of vector modes in such regions. The key findings of this paper are summarised as follows: (iii) The vector potential power spectrum remains below 4 \u00d7 10 -5 relative to the scalar gravitational potential down to k = 20 h Mpc -1 ; cf. Fig. 5. (ii) On scales 0 . 1 h Mpc -1 glyph[lessorsimilar] k glyph[lessorsimilar] 0 . 4 h Mpc -1 the amplitude of the vorticity power spectrum seems to evolve as \u223c [ D + ( z ) /D + (0)] 7 . 7 at z \u2264 1 . 5 , which is higher than previous values found in the literature (Thomas et al. 2015b; Jelic-Cizmek et al. 2018). Nonetheless, these references also found larger values than the scaling with the seventh power originally proposed in Pueblas & Scoccimarro (2009). On scales k glyph[greaterorsimilar] 3 . 5 h Mpc -1 , the evolution of the amplitude of the power spectrum can be similarly neatly described as \u223c [ D + ( z ) /D + (0)] 2 . 6 up to z = 1 ; cf. Fig. 4. (iv) Inside dark matter haloes, the magnitude of the vector potential peaks towards the centres at \u223c 10 -7 for haloes more massive than 10 14 . 5 h -1 M glyph[circledot] , which can reduce by two orders of magnitude in haloes of masses around 10 12 . 5 h -1 M glyph[circledot] . Its ratio against the scalar gravitational potential remains typically a few times 10 -3 inside the haloes, regardless of their mass (cf. Fig. 7). The ratio B/ | \u03a6 | remains nearly flat within the halo radius R 200 c , for the halo redshift ( z < 3 ) and mass (10 12 . 5 \u223c 10 15 h -1 M glyph[circledot] ) ranges checked, and this constant increases roughly linearly with z ; cf. Fig. 8. (v) The magnitude of the gravitomagnetic acceleration also peaks at the halo centres where it can reach a few times 10 -11 h cm / s 2 in haloes above \u223c 10 14 . 5 h -1 M glyph[circledot] . Its ratio against the standard gravitational acceleration remains around \u223c 10 -5 on average, regardless of the halo mass and distance from the halo centre; cf. Fig. 10. This suggests that the effect of the gravitomagnetic force on cosmic structure formation is, even for the most massive structures, negligible however, note that we have not studied the behaviour in low-density regions, i.e., voids. While we have presented a first study of the gravitomagnetic potential in dark matter haloes with general-relativistic simulations, there are several possible extensions in this direction. The analy- sis of the gravitomagnetic potential and forces done in this paper could be extended to galaxies, e.g., by constructing a catalogue using certain semi-analytic models. It is then possible to calculate the gravitomagnetic accelerations of galaxies based on their coordinates and velocities. However, as we have seen above, this acceleration is much weaker than the standard gravitational acceleration, and the impact of baryons on small scales still remains to be assessed. The implementation of (magneto)hydrodynamics in the default RAMSES code could be used in conjunction with the general-relativistic implementation of GRAMSES as a first approximation to address this question, although we generally expect that uncertainties in baryonic physics should surpass GR effects. A perhaps more interesting possibility is to self-consistently implement massive neutrinos and radiation in this relativistic code. In the second GRAMSES code paper (Barrera-Hinojosa & Li 2020b), we have introduced a method to generate initial conditions for GRAMSES simulations that does not require back-scaling. It is therefore natural to evolve these matter components which are neglected in traditional simulations (e.g., Adamek et al. 2017). On the same vein, a Newtonian (quasi-static) implementation of modified gravity models on GRAMSES would allow to study the gravitomagnetic potential in such type of theory. In particular, the modified gravity code ECOSMOG (Li et al. 2012; Li et al. 2013) is based on RAMSES and can be easily made compatible with GRAMSES for such purpose. In this paper, we have primarily focused on the general-relativistic physical quantities that could impact cosmic structure formation, and this can ultimately only be observed by detecting photons (McDonald 2009; Croft 2013; Bonvin et al. 2014; Alam et al. 2017). Therefore, besides the gravitomagnetic force acting on massive particles, it is also important to study how vector modes, as well as other GR effects, could influence the photon trajectories on nonlinear scales, and what is the consequent impact on observables, e.g. lensing (Thomas et al. 2015a; Saga et al. 2015; Gressel et al. 2019). This requires the implementation of general-relativistic ray tracing algorithms (e.g. Barreira et al. 2016; Breton et al. 2019; Lepori et al. 2020; Reverdy 2014) and is left as a future project.", "pages": [15, 16]}, {"title": "ACKNOWLEDGEMENTS", "content": "We thank Marius Cautun for assistance with the DTFE code, and Ra\u00fal Angulo for useful discussions on the vorticity estimation from N -body simulations. We are also grateful to James Mertens and to the anonymous referee for their valuable comments and observations. CB-H is supported by the Chilean National Agency of Research and Development (ANID) through grant CONICYT/Becas-Chile (No. 72180214). BL is supported by the European Research Council (ERC) through ERC starting Grant No. 716532, and STFC Consolidated Grant (Nos. ST/I00162X/1, ST/P000541/1). MB is supported by UK STFC Consolidated Grant No. ST/S000550/1. This work used the DiRAC@Durham facility managed by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility ( www.dirac.ac.uk ). The equipment was funded by BEIS via STFC capital grants ST/K00042X/1, ST/P002293/1, ST/R002371/1 and ST/S002502/1, Durham University and STFC operation grant ST/R000832/1. DiRAC is part of the UK National e-Infrastructure.", "pages": [16]}, {"title": "DATA AVAILABILITY", "content": "For access to the simulation data please contact CB-H.", "pages": [17]}, {"title": "APPENDIX A: COMPARISON OF POWER SPECTRUM CALCULATION METHODS", "content": "In Section 3.1, the power spectrum of density, velocity and vorticity has been measured from particle-type data using DTFE and NBODYKIT, while the spectrum of the scalar and vector potentials has been measured using a different code that is able to read their values calculated and stored by GRAMSES in cells of hierarchical AMR meshes and interpolate them to a regular grid for the power spectrum measurement. We call this method the 'AMR-FFT' method, which was introduced in He et al. (2015), where more details can be found. An alternative to using this AMR-FFT method to calculate the power spectrum of the potentials is by writing their values with GRAMSES at the particles' positions rather than in AMR cells, so that DTFE can be used to read such 'particle-type' data and interpolate this to a regular grid, where NBODYKIT can be applied to measure the spectrum. We call this method 'DTFE+NBODYKIT'. Figure A1 shows the dimensionless power spectra at z = 1 of the scalar potential \u03a6 (left panel) and the vector potential spectrum (right panel), measured by these two methods, where solid lines represent the perturbation-theory predictions. In both methods the FFT grid size is 2048 3 , as is the tessellation grid size used for DTFE. We find that both methods have good agreement on large scales, specially at k glyph[greaterorsimilar] 0 . 1 h Mpc -1 , where the effect of cosmic variance is not present. However, in the region k glyph[greaterorsimilar] 3 h Mpc -1 the AMR-FFT method has better performance than DTFE+NBODYKIT 10 - 10 - 10 12 \u03a6 \u2206 10 - 10 - 14 16 1st order PT AMR-FFT dtfe + nbodykit 10 - 1 k [ h Mpc - 1 ] which blows up. This is because the AMR-FFT method can reach higher resolution by using the potential information in the AMR cells, and because DTFE does a volume weighted average of the field which smears out small-scale features. Therefore, the spectrum of the scalar and vector potentials from the simulation shown in Fig. 5 are measured by the AMR-FFT method, which yields robust results up to k \u223c 15 h Mpc -1 . This paper has been typeset from a T E X/L A T E X file prepared by the author. 10 0 z = 1 10 1", "pages": [17, 18]}] |
2024arXiv241015410D | https://arxiv.org/pdf/2410.15410.pdf | <document>
<section_header_level_1><location><page_1><loc_8><loc_85><loc_92><loc_87></location>Partial suppression of chaos in relativistic three-body problems</section_header_level_1>
<text><location><page_1><loc_28><loc_82><loc_72><loc_84></location>Pierfrancesco Di Cintio 1 , 2 , 3 and Alessandro Alberto Trani 4,</text>
<unordered_list>
<list_item><location><page_1><loc_10><loc_79><loc_85><loc_80></location>1 National Council of Research - Institute of Complex Systems, Via Madonna del piano 10, I-50019 Sesto Fiorentino, Italy</list_item>
<list_item><location><page_1><loc_10><loc_78><loc_78><loc_79></location>2 National Institute of Nuclear Physics (INFN) - Florence unit, via G. Sansone 1, I-50019 Sesto Fiorentino, Italy</list_item>
<list_item><location><page_1><loc_10><loc_76><loc_89><loc_78></location>3 National Institute of Astrophysics - Arcetri Astrophysical Observatory (INAF-OAA), Piazzale E. Fermi 5, I-50125 Firenze, Italy e-mail: [email protected]</list_item>
<list_item><location><page_1><loc_10><loc_73><loc_73><loc_75></location>4 Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark e-mail: [email protected]</list_item>
</unordered_list>
<text><location><page_1><loc_10><loc_71><loc_40><loc_72></location>Received M DD, YYYY; accepted M DD, YYYY</text>
<section_header_level_1><location><page_1><loc_46><loc_68><loc_54><loc_69></location>ABSTRACT</section_header_level_1>
<text><location><page_1><loc_10><loc_63><loc_90><loc_67></location>Context. Recent numerical results seem to suggest that, in certain regimes of typical particle velocities, when the post-Newtonian (PN) force terms are included, the gravitational N -body problem (for 3 ≤ N ≲ 10 3 ) is intrinsically less chaotic than its classical counterpart, which exhibits a slightly larger maximal Lyapunov exponent Λ max.</text>
<text><location><page_1><loc_10><loc_60><loc_90><loc_63></location>Aims. In this work, we explore the dynamics of wildly chaotic, regular and nearly regular configurations of the three-body problem with and without the PN corrective terms, with the aim being to shed light on the behaviour of the Lyapunov spectra under the e ff ect of the PN corrections.</text>
<text><location><page_1><loc_10><loc_54><loc_90><loc_60></location>Methods. Because the interaction of the tangent-space dynamics in gravitating systems -which is needed to evaluate the Lyapunov exponents- becomes rapidly computationally heavy due to the complexity of the higher-order force derivatives involving multiple powers of v / c , we introduce a technique to compute a proxy of the Lyapunov spectrum based on the time-dependent diagonalization of the inertia tensor of a cluster of trajectories in phase-space. In addition, we also compare the dynamical entropy of the classical and relativistic cases.</text>
<text><location><page_1><loc_10><loc_48><loc_90><loc_54></location>Results. We find that, for a broad range of orbital configurations, the relativistic three-body problem has a smaller Λ max than its classical counterpart starting with the exact same initial conditions. However, the other (positive) Lyapunov exponents can be either lower or larger than the corresponding classical ones, thus suggesting that the relativistic precession e ff ectively reduces chaos only along one (or a few) directions in phase-space. As a general trend, the dynamical entropy of the relativistic simulations as a function of the rescaled speed of light falls below the classical value over a broad range of values.</text>
<text><location><page_1><loc_10><loc_46><loc_90><loc_48></location>Conclusions. We observe that analyses based solely on Λ max could lead to misleading conclusions regarding the chaoticity of systems with small (and possibly large) N .</text>
<text><location><page_1><loc_10><loc_44><loc_64><loc_45></location>Key words. Chaos - Celestial mechanics - Methods: numerical - Relativistic Processes</text>
<section_header_level_1><location><page_1><loc_6><loc_39><loc_18><loc_41></location>1. Introduction</section_header_level_1>
<text><location><page_1><loc_6><loc_27><loc_49><loc_38></location>The gravitational N -body problem (for N ≥ 3) is intrinsically chaotic due to the large number of degrees of freedom with respect to its conserved quantities (Contopoulos 2002). Since the seminal work of Miller (1964, 1971), much attention has been paid to the scaling of the degree of this chaoticity, which is quantified in terms of the maximal Lyapunov exponent Λ max (e.g. see Lichtenberg & Lieberman 1992) for a total number of particles N . In numerical simulations, Λ max is usually estimated using the standard Benettin et al. (1976) method, as</text>
<formula><location><page_1><loc_6><loc_22><loc_49><loc_25></location>Λ max( t ) = 1 L ∆ t L X k = 1 ln || W ( k ∆ t ) || || W 0 || , (1)</formula>
<text><location><page_1><loc_6><loc_18><loc_49><loc_20></location>for a (large) time t = L ∆ t , where W is the 6 N -dimensional tangent-space vector,</text>
<formula><location><page_1><loc_6><loc_15><loc_49><loc_16></location>W = ( w i , ˙ w i , ..., w N , ˙ w N ) , (2)</formula>
<text><location><page_1><loc_6><loc_10><loc_49><loc_13></location>which is initialised to W 0 at time t = 0, and || ... || is the standard Euclidean norm, here applied in R 6 N to adimensionalized w i s and ˙ w i s. For classical gravitational N -body dynamics of parti-</text>
<text><location><page_1><loc_51><loc_39><loc_65><loc_40></location>les of equal mass m ,</text>
<formula><location><page_1><loc_51><loc_35><loc_94><loc_38></location>r i = -Gm N X i , j = 1 r i -r j r 3 i j , (3)</formula>
<text><location><page_1><loc_51><loc_31><loc_94><loc_33></location>the variational equations in tangent space needed to evaluate W 6 N are (see also Rein & Tamayo 2016)</text>
<formula><location><page_1><loc_51><loc_26><loc_94><loc_30></location>w i = -Gm N X i , j = 1 w i -w j r 3 i j -3( r i -r j ) ⟨ ( w i -w j )( r i -r j ) ⟩ r 5 i j , (4)</formula>
<text><location><page_1><loc_51><loc_23><loc_94><loc_25></location>where G is the usual gravitational constant, ri j = || r i -r j || , and ⟨ xy ⟩ indicates the scalar product.</text>
<text><location><page_1><loc_51><loc_11><loc_94><loc_22></location>In the continuum limit ( N → ∞ ) assumption, which is formulated in terms of one-particle phase-space distribution functions f by the collisionless Boltzmann equation (e.g. see Binney & Tremaine 2008), no chaos should be present in the sense of asymptotically diverging initially nearby trajectories (Kandrup & Smith 1991). For this reason, for a given form of the initial density and velocity profiles, Λ max -evaluated in simulations via Eq. (1)- is expected to decrease for increasing N at fixed total mass M = mN .</text>
<text><location><page_1><loc_53><loc_10><loc_94><loc_11></location>Using semi-analytical arguments, Gurzadyan & Savvidy</text>
<text><location><page_1><loc_77><loc_7><loc_94><loc_8></location>Article number, page 1 of 9</text>
<text><location><page_2><loc_6><loc_91><loc_49><loc_93></location>(1986); Gurzadyan & Kocharyan (2009); Ovod & Ossipkov (2013) estimated that</text>
<formula><location><page_2><loc_6><loc_88><loc_49><loc_89></location>Λ max ∝ N -1 / 3 . (5)</formula>
<text><location><page_2><loc_6><loc_69><loc_49><loc_86></location>Since then, several numerical studies (Goodman et al. 1993; Habib et al. 1997; Kandrup & Sideris 2001; Sideris & Kandrup 2002; Hemsendorf & Merritt 2002; Cipriani & Pettini 2003; ElZant 2002; Kandrup & Siopis 2003; El-Zant et al. 2019, and references therein) based on both single-particle integration in fixed N -body potentials and the full solution of the N -body problem seem to indicate that individual orbits become more regular (i.e. approach their counterparts propagated in the smooth mean field gravitational potential) as N increases. On the other hand, their largest Lyapunov exponents remain somewhat independent of N , while the full N -particle dynamics become less chaotic as N increases with a N -1 / 2 trend, which is at odds with the Gurzadyan &Savvidy (1986) N -1 / 3 estimate 1 .</text>
<text><location><page_2><loc_6><loc_55><loc_49><loc_69></location>In a series of papers, Di Cintio & Casetti (2019, 2020a,b) performed extensive direct numerical integrations with 10 2 ≤ N ≤ 10 5 for a wide range of density profiles and velocity distribution. These authors found that Λ max( N ) exhibits a strong dependence on the specific choice of the initial conditions, while being always bounded between the N -1 / 2 and N -1 / 3 trends. The N -scaling of the Lyapunov exponents of individual particle trajectories for a given model is strongly dependent on the initial energy and angular momentum of the particles, and the somewhat flat behaviour observed in previous studies was merely an e ff ect of the low number of particles employed.</text>
<text><location><page_2><loc_6><loc_24><loc_49><loc_55></location>More recently, Portegies Zwart et al. (2022) repeated the numerical experiments of Di Cintio & Casetti (2019) for 3 ≤ N ≤ 10 5 using the arbitrary precision integrator brutus (see Portegies Zwart & Boekholt 2014), and also including the relativistic correction to the interparticle forces at 1 / c 2 order using the EinsteinInfeld-Ho ff mann Lagrangian (EIH, Einstein et al. 1938) equations as well as the first post-Newtonian-order perturbative term (1PN, Blanchet 2024). One of the main results of the analysis presented by Portegies Zwart et al. (2022) is that, surprisingly, at typical velocities v typ of approximately 2 × 10 -3 or higher -in units of the speed of light c -, the 1PN relativistic N -body systems are less chaotic than their classical counterparts, exhibiting a smaller maximum Lyapunov exponent Λ max. Remarkably, Boekholt et al. (2021) found a similar trend in the context of the relativistic Pythagorean three-body problem (i.e. the three bodies of masses m 1 = 3, m 2 = 4, and m 3 = 5 start at rest at the vertexes of a right-angled triangle to engage in complex dynamics involving multiple encounters before expelling the lightest body m 1; see Burrau 1913) with PN corrections of up to the order 2.5. It is worth mentioning that similar studies, though not concentrating on Lyapunov exponents, have also been carried out by Valtonen et al. (1995) and Chitan et al. (2022), with the authors concluding that increasing the mass of the components favours the merger rather than the escape of m 1.</text>
<text><location><page_2><loc_6><loc_15><loc_49><loc_24></location>Kovács et al. (2011) investigated the Sitnikov (1960) threebody problem (i.e. the third body of mass m 3 ≈ 0 moves along a straight line orthogonal to the plane of the main binary and crosses its centre of mass), and found that the inclusion of the 1PN term systematically reduces the central regular phase-space region for increasing values of the parameter α = G ( m 1 + m 2) / ac 2 , where a is the semi-major axis of the</text>
<text><location><page_2><loc_51><loc_82><loc_94><loc_93></location>binary of m 1 and m 2. Moreover, Dubeibe et al. (2017a,b) analysed the Poincaré sections 2 and the Lyapunov spectra of the relativistic circular restricted three-body problem (see Maindl & Dvorak 1994). Their findings suggest that the so-called pseudoNewtonian approximation with the Fodor-Hoenselaers-Perjés potentials (see Fodor et al. 1989, see also Steklain & Letelier 2006 and references therein) enhances chaoticity in classically regular orbits, while significantly reducing chaos in some irregular orbits for specific ranges of energy and Jacobi constant.</text>
<text><location><page_2><loc_51><loc_66><loc_94><loc_81></location>A clear interpretation of this intriguing finding (i.e. perturbing a chaotic system may either reduce or enhance its degree of chaos) is still missing. Of course, one might argue that, even if the initial conditions of the numerical experiments are the same, the classical and relativistic N -body problems are described by two di ff erent Hamiltonians, H class and H rel, with potentially differently structured energy surfaces. A fair comparison of their maximal Lyapunov exponents is thus most an likely ill-posed problem, except in regimes of reasonably low values of v typ / c such that one can assume, perturbatively, that H rel = H class + ϵ H , where the small (i.e. ϵ ≪ 1) perturbation term ϵ H contains the PN corrections.</text>
<text><location><page_2><loc_51><loc_59><loc_94><loc_66></location>Aiming to shed some light on the unclear behaviour described above, in this work we explore this matter further by estimating the positive part of the Lyapunov spectrum as well as the dynamical entropy of four representative, relativistic threebody problems and their parent classical systems.</text>
<text><location><page_2><loc_51><loc_50><loc_94><loc_59></location>The rest of the paper is structured as follows: In Sect. 2 we introduce the governing equations, discuss the numerical integration, and describe the procedure to evaluate the Lyapunov exponents. In Sect. 3 we present our numerical simulations and discuss the results. Finally, in Sect. 4 we draw our conclusions and interpret our findings within the context provided by previous work.</text>
<section_header_level_1><location><page_2><loc_51><loc_45><loc_60><loc_46></location>2. Methods</section_header_level_1>
<section_header_level_1><location><page_2><loc_51><loc_42><loc_69><loc_43></location>2.1. Numerical integration</section_header_level_1>
<text><location><page_2><loc_51><loc_32><loc_94><loc_41></location>At variance with Boekholt et al. (2021), we consider the relativistic corrections up to the 2PN order (i.e. we neglect the dissipative 2.5PN terms associated with the gravitational wave emission, Blanchet 1996 and higher corrections), where the Equations of motion in Eq.3 are augmented by the extra terms P 1 and P 2 given (e.g. see Damour & Deruelle 1985; So ff el 1989; Spurzem et al. 2008) by</text>
<formula><location><page_2><loc_51><loc_21><loc_94><loc_29></location>P 1 i = G c 2 N X i , j = 1 mj r 2 i j " -v 2 i -2 v 2 j + 4 ⟨ v i v j ⟩ + 3 2 ⟨ n i j v j ⟩ + + 5 Gmi + 4 Gmj ri j ! n i j + GLYPH<16> v i -v j GLYPH<17> GLYPH<16> 4 ⟨ n i j v i ⟩ -3 ⟨ n i j v j ⟩ GLYPH<17> # (6)</formula>
<text><location><page_3><loc_6><loc_92><loc_9><loc_93></location>and</text>
<formula><location><page_3><loc_6><loc_82><loc_7><loc_83></location>+</formula>
<formula><location><page_3><loc_6><loc_65><loc_49><loc_90></location>+ + 3 2 v 2 i ⟨ n i j v j ⟩ 2 + 9 2 v 2 j ⟨ n i j v j ⟩ 2 -6 ⟨ v i v j ⟩⟨ n i j v j ⟩ 2 -15 8 ⟨ n i j v j ⟩ 4 + Gmj ri j 4 v 2 j -8 ⟨ v i v j ⟩ + 2 ⟨ n i j v i ⟩ 2 -4 ⟨ n i j v i ⟩⟨ n i j v j ⟩ -6 ⟨ n i j v j ⟩ 2 ! + + Gmi ri j -15 4 v 2 i + 5 4 v 2 j -5 2 ⟨ v i v j ⟩ + 39 2 ⟨ n i j v i ⟩ 2 + -39 ⟨ n i j v i ⟩⟨ n i j v j ⟩ + 17 2 ⟨ n i j v j ⟩ 2 !# n i j + + ( v i -v j ) " v 2 i ⟨ n i j v j ⟩ + 4 v 2 j ⟨ n i j v i ⟩ -5 v 2 j ⟨ n i j v j ⟩ -4 ⟨ v i v j ⟩⟨ n i j v i ⟩ + + 4 ⟨ v i v j ⟩⟨ n i j v j ⟩ -6 ⟨ n i j v i ⟩⟨ n i j v j ⟩ 2 + 9 2 ⟨ n i j v j ⟩ 3 + -</formula>
<formula><location><page_3><loc_11><loc_87><loc_43><loc_91></location>P 2 i = G c 4 N X i , j = 1 mj r 2 i j (" -2 v 4 j + 4 v 2 j ⟨ v i v j ⟩ -2 ⟨ v i v j ⟩ 2</formula>
<formula><location><page_3><loc_8><loc_59><loc_50><loc_67></location>Gmi ri j 63 4 ⟨ n i j v i ⟩ + 55 4 ⟨ n i j v j ⟩ ! -2 Gmj ri j ⟨ n i j v i ⟩ + ⟨ n i j v j ⟩ !#) + -G 3 mj r 4 i j " 57 4 m 2 i + 9 m 2 j + 69 2 mimj # n i j , (7)</formula>
<text><location><page_3><loc_6><loc_54><loc_49><loc_58></location>where n i j = ( r i -r j ) / || r i -r j || is short-hand notation for the direction of ri j , v i = ˙ r i are particle velocities, and we assume the general case where the particle masses mi could be di ff erent.</text>
<text><location><page_3><loc_6><loc_41><loc_49><loc_54></location>Following Boekholt & Portegies Zwart (2023), we adopt the usual N -body units (Heggie & Mathieu 1986), setting P i mi = M = 1 = G and the total (Newtonian) energy E tot = -1 / 4. By doing so, the spatial, velocity, and timescales become r ∗ = -GM / 4 E tot = 1; v ∗ = √ -2 E tot / M = 1; and t ∗ = 2 r ∗ / v ∗ = 2 √ 2, respectively. With such a choice of units, in the simulations we then have only one free parameter, namely the speed of light in units of v ∗ . It must be pointed out that, in principle, in natural units the appropriated PN order n is dictated (see Blanchet 2024) by the relation</text>
<formula><location><page_3><loc_6><loc_38><loc_49><loc_40></location>h GM / ( c 2 r ∗ ) i n = ( v ∗ / c ) 2 n . (8)</formula>
<text><location><page_3><loc_6><loc_20><loc_49><loc_37></location>In this work, our choice of normalisation essentially parametrises the extent to which the specific configuration of the three-body problem departs from scale invariance via the v ∗ / c parameter. We stress the fact that, a di ff erent choice of dimensionless quantities does not alter its meaning. For example, assuming G = M = r ∗ = 1 (where r ∗ is now a typical radius) and defining the timescale t ∗ ≡ p r 3 ∗ / GM fixes the velocity and specific energy scales v ∗ = r ∗ / t ∗ and E ∗ = -v 2 ∗ / 2, still leaving the normalised speed of light as a control parameter. Classical and relativistic simulations are performed with the same normalisation, as we are implicitly assuming that the PN terms behave as a small perturbation of the Newtonian problem (cfr. Sect. 1).</text>
<text><location><page_3><loc_6><loc_10><loc_49><loc_20></location>We propagate the equations of motion for the N = 3 gravitationally interacting particles using the fourth-order implementation of the multi-order symplectic scheme (see e.g. Yoshida 1990, 1993; Kinoshita et al. 1991 and reference therein) with a fixed time step δ t . When incorporating the 1 and 2PN corrections, where the interparticle forces depend explicitly on the relative velocities v i -v i , each velocity update step is further divided in two sub-steps where the auxiliary velocity at</text>
<figure>
<location><page_3><loc_54><loc_64><loc_93><loc_94></location>
<caption>Fig. 1. Schematic evolution in phase-space of the distribution of perturbed realisations (purple ellipses) of the dynamics starting from the initial state s 0 (yellow curve).</caption>
</figure>
<text><location><page_3><loc_74><loc_64><loc_75><loc_66></location>r</text>
<text><location><page_3><loc_51><loc_46><loc_94><loc_57></location>half step is recomputed using the acceleration at the previous step (see e.g. Mikkola & Merritt 2006; Di Cintio et al. 2018b; Trani et al. 2024), which is basically equivalent to the Hut et al. 1995 semi-implicit leapfrog with adaptive δ t . In the simulations discussed in this work, according to the specific configuration of the three-body problem, we employed 2 × 10 -7 ≤ δ t ≤ 10 -5 in N-body units. For each classical computation, we verified empirically that the total energy is preserved up one part in 10 -8 in double precision.</text>
<section_header_level_1><location><page_3><loc_51><loc_43><loc_81><loc_44></location>2.2. Evaluation of the Lyapunov exponents</section_header_level_1>
<text><location><page_3><loc_51><loc_34><loc_94><loc_42></location>For non-relativistic dynamics, the numerical maximal Lyapunov exponent is easily obtained following Benettin et al. (1976) by time-dependently computing the sum in Eq. (1), with the additional precaution of periodically re-normalising W (typically every ten iterations) to its initial magnitude (see e.g. Skokos 2010 and references therein).</text>
<text><location><page_3><loc_51><loc_26><loc_94><loc_34></location>The variational Equations (4) become increasingly more complicated when including the corrective PN terms (6) and (7) due to the higher-order derivatives of the 1 / r 3 i j and 1 / r 4 i j factors as well as those of the velocity dependence. This implies a heavier computational cost even for N as small as 3. Typically, this is overcome by substituting W in the definition (1) with</text>
<formula><location><page_3><loc_51><loc_23><loc_94><loc_25></location>˜ W = ( r i -r ' i , ˙ r i -˙ r ' i , ..., r N -r ' N , ˙ r N -˙ r ' N ) , (9)</formula>
<text><location><page_3><loc_51><loc_10><loc_94><loc_23></location>where, at each time time step, ( r ' i , ˙ r ' i ) are the phase space coordinates of trajectory whose initial condition was obtained by slightly perturbing that of the reference system with coordinates ( r i , ˙ r i ). The value of the numerical Λ max is strongly sensitive to the magnitude of the initial perturbation δ , which is at variance with what is obtained by integrating the tangent dynamics w i , as shown by Mei & Huang (2018). In particular, the reliability of the Lyapunov exponents evaluated this way is also a ff ected by the order of the integration scheme and the machine precision. Di Cintio & Casetti (2019) found empirically that in order to ob-</text>
<text><location><page_3><loc_51><loc_80><loc_54><loc_81></location>v</text>
<text><location><page_4><loc_6><loc_82><loc_49><loc_93></location>tain good agreement between the values of Λ max computed using w and ˜ w , δ should be of the order of one part in 10 -13 when using a fourth-order scheme in double precision (cfr. their Fig. 1). We note that, in principle, several equivalent ways to implement the perturbation do exist. For example, Portegies Zwart et al. (2022) shift the spatial coordinates of a single particle in the perturbed realisation (see also Boekholt et al. 2023). In this work, except where specifically stated, we apply the perturbation to each of the system's particles.</text>
<text><location><page_4><loc_6><loc_70><loc_49><loc_81></location>Evaluating the full Lyapunov spectrum for the three-body problem (i.e. 2 N = 18 exponents) in the standard way (e.g. see Quarles et al. 2011, see also Benettin et al. 1980; Wolf et al. 1985) involves computing the tangent dynamics up to a time τ n , when w becomes parallel to the Hamiltonian flow, and then orthonormalising the tangent space, for example with the usual Gram-Schmidt scheme (Wiesel 1993; Christiansen & Rugh 1997). At this point, the values of λ i ( τ n ) are obtained recursively from the orthonormalised w ' i as</text>
<formula><location><page_4><loc_6><loc_65><loc_49><loc_68></location>λ i ( τ n ) = 1 τ n λ i -1( τ n ) + log GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> w i -i X j = 1 ⟨ w i w ' j -1 ⟩ w ' j -1 GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> , (10)</formula>
<text><location><page_4><loc_6><loc_57><loc_49><loc_63></location>where we assume that the initial normalisation is 1 for all w i and impose λ 0 = w 0 = 0. This protocol is repeated until a given convergence criterion is met, which is typically when the fluctuations of the considered λ i are smaller than a fixed percentage of its median value for the whole length of a given time window.</text>
<text><location><page_4><loc_6><loc_41><loc_49><loc_56></location>In the relativistic numerical simulations presented in this work, where the perturbed trajectory scheme is used in lieu of the tangent dynamics, the Lyapunov spectra are estimated in an alternative way. For a given 18-dimensional initial state vector s 0 = ( r 01 , r 02 , r 03 , v 01 , v 02 , v 03) , we initialise N pert perturbed initial conditions s k 0 distributed homogeneously within a hypersphere S 18 of radius δ in R 18 . Assuming that the time evolution of these N pert independent trajectories s ( τ ) deforms S 18 into a hyper-ellipsoid (at least after a critical timescale τ n; see the sketch in Fig. 1), its semi-axes ai can be used to compute a proxy for the Lyapunov exponents associated with the evolution of s 0. To do so, we construct the 18 × 18 tensor,</text>
<formula><location><page_4><loc_6><loc_36><loc_49><loc_39></location>Ii j = N pert X k = 1 s k i ( τ ) s k j ( τ ); i , j = 1 ... 18 , (11)</formula>
<text><location><page_4><loc_6><loc_30><loc_49><loc_35></location>and extract its eigenvalues ei . As usual, ai = √ ei , and so the putative Lyapunov exponents (indicated hereafter with a tilde to distinguish them from the proper ones defined in Eq. 10) read</text>
<formula><location><page_4><loc_6><loc_26><loc_49><loc_29></location>˜ λ i ( τ ) = 1 τ log ai ( τ ) δ . (12)</formula>
<text><location><page_4><loc_6><loc_17><loc_49><loc_25></location>Again, as in the case above, the procedure is reiterated until the desired convergence and the resulting final λ s are sorted into decreasing order such that ˜ Λ max ≡ ˜ λ 1. We note that Kandrup et al. (2001) and Terzi'c & Kandrup (2003) also used a statistical argument to estimate the Lyapunov exponents in lower-dimensional systems based on multiple realisations of the wanted orbit.</text>
<text><location><page_4><loc_6><loc_10><loc_49><loc_17></location>Another important quantity linked to the chaoticity of trajectories is the so-called Kolmogorov-Sinai (KS, Kolmogorov 1958, 1959; Sinai 1959) entropy S KS (see Lichtenberg & Lieberman 1992), which for a given coarse graining of time and phasespace is linked to the probability that a given trajectory lies at time tn + 1 in the cell j given its history up to tn . Pesin (1977)</text>
<text><location><page_4><loc_51><loc_91><loc_94><loc_93></location>found that the upper bound of the KS entropy is always equal to the sum of the positive Lyapunov exponents,</text>
<formula><location><page_4><loc_51><loc_87><loc_94><loc_90></location>S + KS = X λ i > 0 λ i . (13)</formula>
<text><location><page_4><loc_51><loc_71><loc_94><loc_86></location>In our case of interest, the three-body problem, 10 out of 18 λ i s vanish as they are associated with the conserved quantities of the system 3 (see Boccaletti & Pucacco 1999), namely the total energy E , the three components of the angular momentum J , and the motion of the centre of mass, i.e. three coordinates X , Y , and Z and three velocity components ˙ X , ˙ Y , and ˙ Z (Roy 2005, see also Kol 2023 for an alternative formulation). The remaining 8 λ i are made up of four equal-magnitude and opposite-sign pairs (corresponding to contracting and expanding directions in phasespace), which means that, in our analysis, we need to compute only three additional λ other than Λ max.</text>
<section_header_level_1><location><page_4><loc_51><loc_68><loc_73><loc_69></location>3. Simulations and results</section_header_level_1>
<section_header_level_1><location><page_4><loc_51><loc_66><loc_68><loc_67></location>3.1. Lyapunov spectrum</section_header_level_1>
<text><location><page_4><loc_51><loc_53><loc_94><loc_65></location>We integrated a wide range of initial conditions spanning four configurations of the three-body problem. In this work, we present classical and relativistic integrations for a Pythagorean (hereafter pytha), a figure-eight choreography (hereafter fig8 (Moore 1993; Chenciner & Montgomery 2000), and two hierarchical triples, where the outer orbit is either markedly elliptical or quasi-circular with e ≈ 10 -4 , (hereafter hie1 and hie2, respectively). In Fig. 2, the three particle trajectories are shown with di ff erent colours in the classical case.</text>
<text><location><page_4><loc_51><loc_40><loc_94><loc_53></location>For all runs, we first evaluated the largest Lyapunov exponent Λ max using its discrete time definition (1). In the classical cases, Λ max was evaluated using both the tangent dynamics given in Eq. (2) and the commonly adopted perturbed trajectory subtraction. We find that, using double precision and a second-order integration scheme, the values of the largest Lyapunov exponent obtained with the two choices are in good agreement when the size of the perturbation in the second method is ≈ 5 × 10 -6 and the perturbed orbit is re-normalised every 10 ∆ t according to the Benettin et al. (1976) algorithm.</text>
<text><location><page_4><loc_51><loc_21><loc_94><loc_40></location>In Figure 3, we show the time series of Λ max( t ) for the classical (black lines) and relativistic (purple lines) calculations for the four configurations introduced above. Each value of Λ max( t ) is the median value of ten realisations. For all systems, c / v ∗ ≈ 10 2 , while the integration has been stopped when the instantaneous value of Λ max is smaller that 0 . 5% of its median for at least 5 t ∗ . In all test cases shown herein, the 2PN relativistic computation yields a maximal Lyapunov exponent that is smaller than its classical counterpart (i.e. in the limit of c / v ∗ → ∞ ), corresponding to a longer Lyapunov time, which appears to confirm what was observed by Portegies Zwart et al. (2022) for 1PN perturbations; for example, see their Figures (9) and (10) for the low N cases. A similar picture (not shown here) is also obtained when perturbing the phase-space coordinates of only one of the three initial particles.</text>
<text><location><page_4><loc_51><loc_13><loc_94><loc_21></location>However, other choices of c / v ∗ in the relativistic simulations at fixed system orbital parameters yield values of Λ max that may or may not be smaller than what is computed for the classical case. In order to obtain a clearer view of the trends of the Lyapunov exponents with the scaled speed of light, we calculated the first four approximated Lyapunov indicators ˜ λ i (corresponding</text>
<figure>
<location><page_5><loc_6><loc_77><loc_94><loc_94></location>
<caption>Fig. 2. X -Y projections of the classical test orbits in the centre of mass frame for (from left to right) a Pythagorean, a figure-eight, and two hierarchical configurations of the three-body problem.</caption>
</figure>
<figure>
<location><page_5><loc_6><loc_45><loc_49><loc_73></location>
<caption>Fig. 3. Evolution of the numerical finite maximum Lyapunov exponent over time for four orbits of the classical (black lines) and relativistic 2PN (purple lines) three-body problem with c / v ∗ ≈ 10 2 .</caption>
</figure>
<text><location><page_5><loc_6><loc_10><loc_49><loc_37></location>to the only positive exponents λ 1 , 2 , 3 , 4 for the classical three-body problem) using the expression given by Eq. (12) in the range 10 < c / v ∗ < 10 6 . The tensor Ii , j is evaluated in all cases using 10 3 independent perturbed trajectories iterated for up to 15 units of normalised time, which is much less than the roughly 150 t ∗ needed, on average, to converge the series used to evaluate Λ max. Propagating up to larger times would typically hinder the procedure as the instantaneous distribution of orbits in phase-space would most likely no longer be enclosed in a hyperellipsoid, in particular in the markedly chaotic systems. In Fig. 4 we show ˜ λ 1 ≈ Λ max for (from left to right) the pytha, fig8, hie1, and hie2 sets of numerical simulations with 10 < c / v ∗ < 10 6 . For comparison, the horizontal dashed lines mark the value obtained for the purely classical model. As a general trend, all orbital types yield values of ˜ λ 1 both larger and smaller than its classical value, with a somewhat common tendency to have larger ˜ λ 1 for c / v ∗ → 10 and smaller values for c / v ∗ → 10 6 . The fig8 and hie1 cases (second and third panel) have values of ˜ λ 1 that are systematically lower (except for a few outliers) than for the classical limit, which is indicated by the horizontal dashed line, while for the pytha and hie2 cases, the relativistic ˜ λ 1 fall below the classical</text>
<text><location><page_5><loc_51><loc_71><loc_91><loc_72></location>limit in recognisable intervals of c / v ∗ at around 10 2 and 10 3 .</text>
<text><location><page_5><loc_51><loc_53><loc_94><loc_71></location>Interestingly, a non-monotonic trend of Λ max with the strength of a perturbing term in an N -particle Hamiltonian has also been observed by Di Cintio et al. 2018a, 2019; Dhar et al. 2019 for one-dimensional non-linear lattices. In such cases, the largest Lyapunov exponent plotted as a function of ϵ H for di ff erent N s or specific energy E tot / N presented a well-defined local minimum at a specific value of the perturbation strength. The latter was parametrised there by the exponent α of a power-law extra coupling between lattice nodes (e.g. see Fig. 2 in Di Cintio et al. 2018a or Fig. 8 in Di Cintio et al. 2019). We note that, a di ff erent normalisation of the equations of motion would simply shift the position of the relative minima of λ 1, as equal values of c / v ∗ and c / v ' ∗ would correspond to di ff erent total energies in natural units.</text>
<text><location><page_5><loc_51><loc_10><loc_94><loc_53></location>The other ˜ λ i are shown in Figure 5 (filled symbols) as functions of the rescaled c against the values obtained in the classical case, which are again marked by the horizontal dashed lines. We observe that, for large values of c / v ∗ (i.e. vanishing PN corrections), all Lyapunov exponents converge to the values obtained in the simulations where the PN terms were switched o ff . On the other hand, in the limit of small c / v ∗ (corresponding to strong relativistic corrections), the values attained by ˜ λ i are typically larger than or comparable to their estimated classical values. In the intermediate regime, one or more exponents can be considerably smaller than their counterparts for the parent classical system. Remarkably, the pytha and hie2 sets of simulations (top left and bottom right panels in Fig. 5) yield a distribution of ˜ λ i showing a strongly non-monotonic trend with c / v ∗ , in particular for ˜ λ 2 and ˜ λ 3. In practice, for a given orbital configuration, accounting for the first two PN corrections does not necessarily imply an increase in chaoticity -as one would expect- associated to shorter exponentiation times of the distance of initially nearby trajectories; or at least not in every phase-space direction. When the relativistic corrections are in the regime corresponding to the scaled speed of light, 20 ≲ c / v ∗ ≲ 2 × 10 3 , the value of the first Lyapunov exponent might be smaller than the classical one, while the second, third, and fourth could instead be slightly larger. Remarkably, the second exponent ˜ λ 2 for the Pythagorean problem shows a sharp drop at around c / v ∗ ≈ 10 3 followed by an increase at c / v ∗ ≈ 300. Therefore, in some cases (cfr. again the hie2 runs in Fig. 5 for c / v ∗ ≈ 200), surprisingly, while ˜ λ 1 is larger in the PN simulations, ˜ λ 2 , 3 , 4 might be considerably smaller. This diversity in behaviour is summarised in Figure 6, where we show the ratio of the relativistic to classical positive Lyapunov spectra (top panel) as well as the individual values of ˜ λ i (bottom panel) for the same systems as in Fig. 5 and choices of c / v ∗ indicated by the colour map ranging from light</text>
<figure>
<location><page_6><loc_6><loc_79><loc_94><loc_93></location>
<caption>Fig. 4. Maximal Lyapunov exponents λ 1 as function of the c / v ∗ ratio for (from left to right) a wildly chaotic Pythagorean, a figure-eight, and two hierarchical triplets (symbols). The horizontal dashed lines mark the values for the parent classical systems.</caption>
</figure>
<text><location><page_6><loc_88><loc_29><loc_89><loc_30></location>6</text>
<figure>
<location><page_6><loc_10><loc_26><loc_89><loc_74></location>
<caption>Fig. 5. Numerical Lyapunov exponents λ 2, λ 3 , and λ 4 as functions of the c / v ∗ ratio for (clockwise top left) a wildly chaotic Pythagorean, a figureeight, and two hierarchical triplets (symbols). The horizontal dashed lines mark the values for the parent classical systems.</caption>
</figure>
<text><location><page_6><loc_6><loc_11><loc_49><loc_20></location>green (corresponding to c / v ∗ ≈ 3 . 5) to dark blue (corresponding to c / v ∗ ≈ 3 × 10 6 ). Again, it appears clear how, for many values of c / v ∗ , even the smaller exponents ˜ λ 2 , 3 , 4 might lower when the PN terms are activated. This implies that there exist systems where chaoticity can indeed be mitigated along multiple phasespace directions other than that associated with Λ max, as reported by Portegies Zwart et al. (2022) for the 1PN corrections.</text>
<text><location><page_6><loc_9><loc_10><loc_49><loc_11></location>From the point of view of the orbital structure in configu-</text>
<text><location><page_6><loc_51><loc_15><loc_94><loc_20></location>ration space, this can be interpreted as either a stabilising or a destabilising e ff ect induced by the precession caused by the nondissipative relativistic terms 4 . As an example, in Fig. 7, for the purely classical and 2PN Pythagorean problem with c / v ∗ ≈ 950 ,</text>
<text><location><page_7><loc_93><loc_61><loc_94><loc_62></location>5</text>
<figure>
<location><page_7><loc_6><loc_59><loc_94><loc_94></location>
<caption>Fig. 6. Ratio ˜ λ PN / ˜ λ (top panel, squares) of the first four estimated classical and relativistic Lyapunov exponents (bottom panel, red crosses and colour-coded circles). The colour coding indicates the c / v ∗ ratio.</caption>
</figure>
<figure>
<location><page_7><loc_10><loc_41><loc_46><loc_54></location>
<caption>Fig. 7. Time-dependent distance R from the centre of mass of the systems for the three masses in a Pythagorean problem without (left panel) and with (right panel) 2PN corrections for c / v ∗ ≈ 950.</caption>
</figure>
<text><location><page_7><loc_6><loc_19><loc_49><loc_34></location>we show the evolution of the distance from the centre of mass of the three particles m 1, m 2, and m 3. In this case (cfr. left panel in Fig. 2), the PN terms alter the outcome of the strong encounter happening at t ≈ 58, and so the triplet remains bound for at least another 60 time units. In general, chaos in lowN gravitational systems is ascribed to the superposition of resonances (Mardling 2008, see also Chirikov 1979). For example, hierarchical triplets undergoing the von Zeipel-Lidov-Kozai (ZLK) mechanism (von Zeipel 1910; Lidov 1962; Kozai 1962) are 'stabilised' (i.e. the ZLK mechanism is suppressed) by the inclusion of PN terms, because apsidal precession averages the torques on the inner binary to zero (Naoz et al. 2013).</text>
<section_header_level_1><location><page_7><loc_6><loc_16><loc_23><loc_17></location>3.2. Dynamical entropy</section_header_level_1>
<text><location><page_7><loc_6><loc_10><loc_49><loc_15></location>It appears clear that one cannot rely solely on the first (i.e. maximal) Lyapunov exponent to characterise the 'amount of chaos' of a given model under the type of perturbations explored here. In order to refine our quantification of chaos, we evaluated the</text>
<text><location><page_7><loc_51><loc_23><loc_94><loc_54></location>upper boundary of the KS dynamical entropy as the sum of the ˜ λ i s according to Eq. (13). In Figure 8, S + KS is shown as a function of c / v ∗ for the systems presented in Figs. 4 and 5 and is compared with its classical value, indicated in each of the four panels by a horizontal dashed line. The behaviour of the estimated dynamical entropy confirms that the examined three-body problems exhibit a partial suppression of their chaoticity when the 2PN corrections are turned on, that is, for several values of the normalised c when the latter is between 10 and 3 × 10 2 ; this is particularly evident for the Pythagorean problem (top left panel), where in two definite intervals of c / v ∗ , S + KS is of a factor of ≈ 1 . 2 smaller in the relativistic simulations. The hie2 case also shows a significant drop in S + KS for c / v ∗ between 10 and 10 3 . For c / v ∗ → 1 (not shown in the plot), we observe a systematic increase in S + KS that is associated with augmented chaoticity. However, we stress that in the c / v ∗ ≈ 1 limit, the perturbative approach, which is valid for the integration scheme employed here, is no longer consistent. Consequently, the PN terms cannot be considered as ϵ H -type perturbations and even the PN expansion itself fails to be a valid approximation of general relativity. Moreover, whether or not other definitions of dynamical entropy -not necessarily associated with the Lyapunov exponents (Kandrup & Mahon 1994)- could yield similar results when applied to PN dynamics remains to be determined.</text>
<section_header_level_1><location><page_7><loc_51><loc_18><loc_76><loc_19></location>4. Discussion and conclusions</section_header_level_1>
<text><location><page_7><loc_51><loc_10><loc_94><loc_17></location>Prompted by the numerical work of Portegies Zwart et al. (2022) on the relativistic N -problem, we explored the behaviour of the positive part of the Lyapunov spectrum of some configurations of the gravitational three-body problem under the e ff ect of the first and second post-Newtonian (PN) perturbative terms. Our analysis employed three di ff erent metrics of chaos, namely the maxi-</text>
<text><location><page_8><loc_17><loc_63><loc_19><loc_64></location>+</text>
<figure>
<location><page_8><loc_18><loc_52><loc_82><loc_93></location>
<caption>Fig. 8. Dynamical entropy S + KS as a function of the c / v ∗ ratio for the same systems as in Fig. 5. Again, the horizontal dashed line marks the classical case.</caption>
</figure>
<text><location><page_8><loc_82><loc_54><loc_83><loc_55></location>6</text>
<text><location><page_8><loc_6><loc_38><loc_49><loc_46></location>mal Lyapunov exponent (Figures 3 and 4), the Lyapunov spectra (Figure 5), and the Kolmogorov-Sinai entropy (Figure 8). We applied these chaos indicators to four di ff erent configurations of the three-body problem: the Pythagorean configuration (Burkert et al. 2012), the figure-eight Chenciner & Montgomery (2000), and two di ff erent configurations of hierarchical triple systems.</text>
<text><location><page_8><loc_9><loc_37><loc_49><loc_38></location>The main results of this paper can be summarised as follows:</text>
<unordered_list>
<list_item><location><page_8><loc_7><loc_32><loc_49><loc_35></location>1. Including up to second-order PN terms lowers the largest Lyapunov exponent Λ max in a broad range of values of the parameter c / v ∗ at fixed orbit configuration.</list_item>
<list_item><location><page_8><loc_7><loc_27><loc_49><loc_31></location>2. The values of the other three classical non-negative Lyapunov exponents can be either lower or higher in the relativistic case, regardless of whether Λ max is smaller or larger than its classical counterpart.</list_item>
<list_item><location><page_8><loc_7><loc_22><loc_49><loc_26></location>3. For a given orbital configuration, the dynamical KS entropy exhibits a markedly non-monotonic trend with c / v ∗ , in particular between 10 and ≈ 10 3 .</list_item>
</unordered_list>
<text><location><page_8><loc_6><loc_10><loc_49><loc_21></location>These results confirm and strengthen the findings of Portegies Zwart et al. (2022) and Boekholt et al. (2021), as it appears that, not only might the terms proportional to 1 / c 2 mitigate chaos, but also this intriguing behaviour survives, at least when the secondorder 1 / c 4 PN corrections are turned on. We conjecture that this apparent mitigation of chaos might be ascribed to the relativistic precession that, under certain conditions, softens the otherwise hard particle encounters. As a consequence, N -body models where the PN corrections become important might have a longer</text>
<text><location><page_8><loc_51><loc_29><loc_94><loc_46></location>(or shorter) chaotic instability timescale associated with the inverse of the largest Lyapunov exponent (Gurzadyan & Savvidy 1986) if the latter are accounted for in the numerical calculations. In particular, in systems dominated by a large central mass, such as the nuclear star clusters (Seth et al. 2008; Antonini 2013; Graham 2016), and where three-body encounters are likely to shape the dynamical evolution, a correct determination of the relativistic chaos suppression or enhancement is therefore needed (Dall'Amico et al. 2024). Moreover, we expect similar implications in the stellar dynamics around binary supermassive black holes, where the di ff usion of orbits -studied for example in Kandrup et al. 2003- should in principle be a ff ected by the PN corrections.</text>
<text><location><page_8><loc_51><loc_11><loc_94><loc_29></location>Nevertheless, it remains to be determined whether the behaviour of the three-(and possibly N )body problem with relativistic extra terms could be interpreted in the same way as the models reported by Di Cintio et al. 2018a, 2019, where the transport properties are both quantitatively and qualitatively changed by the specific strength and form of the perturbation (see also Iubini et al. 2018). In particular, the cases corresponding to relative minima of Λ max present energy di ff usion patterns akin to those of integrable or nearly integrable systems (cf. Fig. 1 in Di Cintio et al. 2018a). In this regard, it is worth noting that the concept of 'stabilising perturbations' was already introduced in the context of the so-called Hamiltonian control by Ciraolo et al. (2004a,b, 2005); Vittot et al. (2005) building on an earlier hypothesis put forward by Gallavotti (1982). As proposed by this</text>
<text><location><page_9><loc_6><loc_68><loc_49><loc_93></location>latter author, additional perturbative terms are added to a chaotic Hamiltonian in a way that some chaotic orbits are turned into regular or nearly regular orbits, while preserving the large scale structure of phase-space. Interestingly, the requirements of the Hamiltonian control theory are compatible with our picture of the gravitational three-body problem subjected to small relativistic corrections, in the sense that a non-integrable Hamiltonian exhibits a more regular behaviour (i.e. less chaotic) once perturbed with an extra ϵ H term. Finally, it would be worth investigating how including higher-order PN terms a ff ects the energy dependence of chaotic three-body encounters within dense stellar systems featuring a central cusp. Specifically, we aim to explore how enhanced or reduced relativistic chaos influences resonant relaxation (Rauch & Tremaine 1996; Sridhar & Touma 2016; Bar-Or & Fouvry 2018; Máthé et al. 2023). A numerical study addressing this question is currently underway. In such endeavours, it is essential to normalise the system such that the proper c / v ∗ is derived from individual orbital energies rather than being treated as a free parameter, as done in this work.</text>
<text><location><page_9><loc_6><loc_57><loc_49><loc_68></location>Acknowledgements. We thank Stefano Ru ff o, Simon Portegies-Zwart and Antonio Politi for the stimulating discussion at the initial stage of this project and the anonymous Referee for his / her comments that helped improving the presentation of our results. We acknowledge funding by the 'Fondazione Cassa di Risparmio di Firenze' under the project HIPERCRHEL for the use of high performance computing resources at the University of Firenze. AAT acknowledges support from the Horizon Europe research and innovation programs under the Marie Skłodowska-Curie grant agreement no. 101103134. PFDC wishes to thank the hospitality of the Kavli Institute of Theoretical Physics at the University of Santa Barbara where part of the present work was completed.</text>
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2019arXiv190109612B | https://arxiv.org/pdf/1901.09612.pdf | <document>
<section_header_level_1><location><page_1><loc_24><loc_82><loc_77><loc_84></location>Black holes with Lambert W function horizons</section_header_level_1>
<text><location><page_1><loc_15><loc_72><loc_89><loc_79></location>Moises Bravo Gaete ∗ , Sebastian Gomez † and Mokhtar Hassaine ‡ ∗ Facultad de Ciencias B'asicas, Universidad Cat'olica del Maule, Casilla 617, Talca, Chile. † Facultad de Ingenier'ıa, Universidad Aut'onoma de Chile, 5 poniente 1670, Talca, Chile. ‡ Instituto de Matem'atica y F'ısica, Universidad de Talca, Casilla 747, Talca, Chile.</text>
<text><location><page_1><loc_42><loc_68><loc_59><loc_70></location>September 17, 2021</text>
<section_header_level_1><location><page_1><loc_47><loc_62><loc_54><loc_63></location>Abstract</section_header_level_1>
<text><location><page_1><loc_18><loc_39><loc_83><loc_60></location>We consider Einstein gravity with a negative cosmological constant endowed with distinct matter sources. The different models analyzed here share the following two properties: (i) they admit static symmetric solutions with planar base manifold characterized by their mass and some additional Noetherian charges, and (ii) the contribution of these latter in the metric has a slower falloff to zero than the mass term, and this slowness is of logarithmic order. Under these hypothesis, it is shown that, for suitable bounds between the mass and the additional Noetherian charges, the solutions can represent black holes with two horizons whose locations are given in term of the real branches of the Lambert W functions. We present various examples of such black hole solutions with electric, dyonic or axionic charges with AdS and Lifshitz asymptotics. As an illustrative example, we construct a purely AdS magnetic black hole in five dimensions with a matter source given by three different Maxwell invariants.</text>
<section_header_level_1><location><page_1><loc_14><loc_34><loc_32><loc_36></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_14><loc_21><loc_88><loc_32></location>The AdS/CFT correspondence has been proved to be extremely useful for getting a better understanding of strongly coupled systems by studying classical gravity, and more specifically black holes. In particular, the gauge/gravity duality can be a powerful tool for analyzing finite temperature systems in presence of a background magnetic field. In such cases, from the dictionary of the correspondence, the black holes must be endowed with a magnetic charge corresponding to the external magnetic field of the CFT. In light of this constatation, it is clear that</text>
<text><location><page_2><loc_14><loc_43><loc_88><loc_89></location>dyonic black holes are of great importance in order to study the charge transport at quantum critical point, particulary for strongly coupled CFTs in presence of an external magnetic field. For example, four-dimensional dyonic black holes have been proved to be relevant for a better comprehension of planar condensed matter phenomena such as the quantum Hall effect [1], the superconductivity-superfluidity [2] or the Nernst effect [3]. The study of dyonic black holes is not only interesting in four dimensions, but also in higher dimensions where their holographic applications have been discussed in the current literature. For example, it has been shown that large dyonic AdS black holes are dual to stationary solutions of a charged fluid in presence of an external magnetic field [4]. In this last reference, the AdS/CFT correspondence was used conversely and stationary solutions of the Navier-Stokes equations were constructed corresponding to an hypothetical five-dimensional AdS dyonic rotating black string with nonvanishing momentum along the string. We can also mention that magnetic/dyonic black holes present some interest from a purely gravity point of view. Indeed, there is a wide range of contexts in which magnetic/dyonic solutions are currently studied including in particular supergravity models [5, 6], Einstein-Yang-Mills theory [7] or nonlinear electrodynamics [8]. Nevertheless, in spite of partial results, the problem of finding magnetic solutions in higher dimension is an highly nontrivial problem. For example, it is easy to demonstrate that under suitable hypothesis, magnetic solutions in odd dimensions D ≥ 5 for the Einstein-Maxwell or for the Lovelock-Maxwell theories do not exist [9, 10]. This observation is in contrast with the four-dimensional situation where static dyonic configuration can be easily constructed thanks to the electromagnetic duality which rotates the electric field into the magnetic field. In the same register, one may also suspect the lack of electromagnetic duality and of the conformal invariance in dimension D > 4 to explain the difficulty for constructing the higher-dimensional extension of the Kerr-Newmann solution.</text>
<text><location><page_2><loc_14><loc_16><loc_88><loc_42></location>The purpose of the present paper is twofold. Firstly, we would like to present a simple dyonic extension of the five-dimensional Reissner-Nordstrom solution with planar horizon. The solution will be magnetically charged by considering an electromagnetic source composed by at least three different Maxwell gauge fields. Each of these U (1) gauge fields will be sustained by one of the three different coordinates of the planar base manifold. Interestingly enough, the magnetic contribution in the metric has an asymptotically logarithmic falloff of the form ln r r 2 . Nevertheless, in spite of this slowly behavior, the thermodynamics analysis yields finite quantities even for the magnetic charge. Since we are working in five dimensions, we extend as well this dyonic solution to the case of Einstein-Gauss-Bonnet gravity. We can also mention that the causal structure of the dyonic solution can not be done analytically. Nevertheless from different simulations, one can observe that the solution has a Reissner-Nordstrom like behavior. Indeed, depending on the election of the integration constants, the solution can be a black hole with inner and outer horizons or an extremal black hole or the solution can have a naked</text>
<text><location><page_3><loc_14><loc_59><loc_88><loc_89></location>singularity located at the origin. On the other hand, we notice that the horizon structure of the purely magnetic solution can be treated analytically. More precisely, we will show that, as for the Reissner- Nordstrom solution, the absence of naked singularity can be guaranteed for a suitable bound relation between the mass and the magnetic charge. Moreover, in this case, the location of the inner and outer horizons are expressed analytically in term of the real branches of so-called Lambert W function. This latter is defined to be the multivalued inverse of the complex function f ( ω ) = ωe ω which has an infinite countable number of branches but only two of them are real-valued, see Ref. [11] for a nice review. The Lambert W functions have a wide range of applications as for example in combinatoric with the tree functions that are used in the enumeration of trees [12] or for equations with delay that have applications for biological, chemical or physical phenomena, see e. g. [13] or in the AdS/CFT correspondence as in the expression of the large-spin expansion of the energy of the Gubser-Klebanov-Polyakov string theory [14]. Just to conclude this parenthesis about the Lambert W function, we also mention that this function can be used in the case of the Schwarzschild metric as going from the Eddington-Finkelstein coordinates to the standard Schwarzschild coordinates</text>
<text><location><page_3><loc_14><loc_31><loc_88><loc_59></location>The plan of the paper is organized as follows. In the next section, we present our toy model for dyonic solutions which consists on the five-dimensional Einstein-Gauss-Bonnet action with three different Abelian gauge fields. For this model, we derive a dyonic black hole configuration as well as its GR limit. A particular attention will be devoted to the purely magnetic GR solution for which a bound relation between the mass and the magnetic charge ensures the existence of an event horizon covering the naked singularity. In this case, the inner and outer horizons are expressed in term of the two real branches of the Lambert W functions. We will establish that this mass bound is essentially due to the fact that the magnetic charge has a slower falloff of logarithmic order to zero than the mass term in the metric function. Starting from this observation, we will present in Sec. 3 various examples of black holes sharing this same feature with electric, axionic or magnetic charges and with AdS and Lifshitz asymptotics. In Sec. 4, we extend the previous solutions to general dyonic configurations with axionic charges. Finally, the last section is devoted to our conclusion and an appendix is provided where some useful properties of the Lambert W functions are given.</text>
<section_header_level_1><location><page_3><loc_14><loc_26><loc_68><loc_27></location>2 Five-dimensional dyonic black hole solution</section_header_level_1>
<text><location><page_3><loc_14><loc_16><loc_88><loc_24></location>In Refs. [9, 10], it has been proved that, under suitable hypothesis, magnetic black hole solutions for Einstein-Maxwell action in odd dimensions D ≥ 5 can not exit. As we will show below, a simple way of circumventing this obstruction is to consider more than one Maxwell gauge field. The fact of considering various Abelian fields in order to construct dyonic black holes in five</text>
<text><location><page_4><loc_14><loc_80><loc_88><loc_89></location>dimensions have already been considered, see Refs. [15] and [16]. More precisely, we will establish that the Einstein gravity eventually supplemented by the Gauss-Bonnet term since we are working in D = 5 can admit dyonic black hole solutions for an electromagnetic source given at least by three different Maxwell invariants. In order to achieve this task, we consider the following action</text>
<formula><location><page_4><loc_14><loc_74><loc_88><loc_78></location>S [ g, A I ] = ∫ d 5 x √ -g [ R -2Λ 2 + α 2 ( R 2 -4 R µν R µν + R αβµν R αβµν ) -1 4 3 ∑ I =1 F ( I ) µν F µν ( I ) ] , (2.1)</formula>
<text><location><page_4><loc_14><loc_67><loc_88><loc_73></location>where the F ( I ) µν 's are the three different Maxwell field strengths associated to the U (1) gauge fields A I for I = { 1 , 2 , 3 } and α represents the Gauss-Bonnet coupling constant. The field equations obtained by varying this action read</text>
<formula><location><page_4><loc_26><loc_59><loc_88><loc_66></location>G µν +Λ g µν + αK GB µν = 3 ∑ I =1 ( F ( I ) µσ F σ ( I ) ν ) -1 4 g µν 3 ∑ I =1 ( F ( I ) σρ F σρ ( I ) ) , ∇ µ F µν ( I ) = 0 , for I = { 1 , 2 , 3 } , (2.2)</formula>
<text><location><page_4><loc_14><loc_56><loc_59><loc_57></location>where the variation of the Gauss-Bonnet term is given by</text>
<formula><location><page_4><loc_14><loc_52><loc_87><loc_55></location>K GB µν = 2 ( RR µν -2 R µρ R ρ ν -2 R ρσ R µρνσ + R ρσγ µ R νρσγ ) -1 2 g µν ( R 2 -4 R ρσ R ρσ + R ρσλδ R ρσλδ ) .</formula>
<text><location><page_4><loc_14><loc_37><loc_88><loc_50></location>In one hand, it is known that the field equations (2.2) with one Maxwell invariant I = 1 admits electrically charged black holes [17] generalizing the solution of Boulware-Deser [18]. On the other hand, it is simple to prove that the magnetic extension of the Boulware-Deser solution can not exist [9, 10]. Nevertheless, as shown below, the presence of two extra Maxwell invariants renders possible the magnetic extension of the Boulware-Deser solution but only in the case of flat horizon. In fact a dyonic solution with flat horizon of the field equations (2.2) is found to be</text>
<formula><location><page_4><loc_27><loc_19><loc_88><loc_36></location>ds 2 = -F ( r ) dt 2 + dr 2 F ( r ) + r 2 3 ∑ i =1 dx 2 i , F ( r ) = r 2 4 α [ 1 ∓ √ 1 + 4 α Λ 3 + 16 α M 3 | Σ 3 | r 4 -4 α Q 2 e | Σ 3 | 2 r 6 + 8 α Q 2 m ln( r ) | Σ 3 | 2 r 4 ] , (2.3) A 1 = -Q e 2 | Σ 3 | r 2 dt + Q m | Σ 3 | x 2 dx 3 , A 2 = -Q e 2 | Σ 3 | r 2 dt + Q m | Σ 3 | x 3 dx 1 , A 3 = -Q e 2 | Σ 3 | r 2 dt + Q m | Σ 3 | x 1 dx 2 ,</formula>
<text><location><page_4><loc_14><loc_14><loc_88><loc_18></location>where M , Q e and Q m are three integration constants corresponding respectively to the mass, the electric and the magnetic charge and | Σ 3 | is the finite volume of the compact 3 -dimensional</text>
<text><location><page_5><loc_14><loc_76><loc_88><loc_89></location>flat base manifold. Various comments can be made concerning this dyonic solution. Firstly, in the absence of the magnetic charge Q m = 0, the solution reduces to the electrically extension of the Boulware-Deser solution [17] even if there are three different Maxwell invariants. This is because each of these three invariants contributes in the same footing for the full solution, and hence one could have switch off two of them from the very beginning. The GR limit α → 0 of the solution concerns only the upper branch of the solution and yields to the metric function given by</text>
<formula><location><page_5><loc_30><loc_71><loc_88><loc_75></location>F GR ( r ) = -r 2 Λ 6 -2 M 3 | Σ 3 | r 2 + Q 2 e 2 | Σ 3 | 2 r 4 -Q 2 m ln r | Σ 3 | 2 r 2 , (2.4)</formula>
<text><location><page_5><loc_14><loc_65><loc_88><loc_70></location>while the Abelian gauge fields remain identical. Computing the Kretschmann invariant, one notices that the dyonic solution in the Einstein-Gauss-Bonnet theory or its GR limit has a singularity located at the origin.</text>
<text><location><page_5><loc_14><loc_47><loc_88><loc_64></location>The causal structure of the dyonic solution is quite involved and can not be treated analytically as in the case of the four-dimensional Reissner-Nordstrom dyonic solution. Nevertheless, it is quite simple to see that the GR solution (2.4) with Λ < 0 and without magnetic charge, has a Reissner-Nordstrom like behavior in the sense that for M≥ 3 5 3 |Q e | 4 3 ( -Λ) 1 3 / 4 4 3 | Σ 3 | 1 3 , the solution describes a (extremal) black hole while the case M < 3 5 3 |Q e | 4 3 ( -Λ) 1 3 / 4 4 3 | Σ 3 | 1 3 will yield a naked singularity. The dyonic GR solution has also a similar behavior which can be appreciated only by means of some simulations reported in the graphics below. In the next subsection, we will see that in the purely magnetic case, the causal structure of the solution can be analyzed analytically.</text>
<text><location><page_5><loc_14><loc_40><loc_88><loc_46></location>To conclude this section, we mention that the GR dyonic solution (2.4) satisfies the dominant energy conditions. Indeed, it is simple to see that the energy density µ , the radial pressure p r and the tangential pressure p t given by</text>
<formula><location><page_5><loc_19><loc_36><loc_88><loc_39></location>µ = 3 2 | Σ 3 | 2 r 6 [ Q 2 m r 2 + Q 2 e ] , p r = -µ, p t = 1 2 | Σ 3 | 2 r 6 [ Q 2 m r 2 +3 Q 2 e ] , (2.5)</formula>
<text><location><page_5><loc_14><loc_34><loc_43><loc_35></location>verify the dominant energy conditions</text>
<formula><location><page_5><loc_32><loc_31><loc_88><loc_32></location>µ ≥ 0 , -µ ≤ p r ≤ µ, -µ ≤ p t ≤ µ. (2.6)</formula>
<section_header_level_1><location><page_5><loc_14><loc_26><loc_47><loc_27></location>2.1 Purely magnetic GR solution</section_header_level_1>
<text><location><page_5><loc_14><loc_23><loc_69><loc_24></location>For α → 0 and Q e = 0, the purely magnetic GR solution (2.4) becomes</text>
<formula><location><page_5><loc_22><loc_14><loc_88><loc_22></location>ds 2 = -F ( r ) dt 2 + dr 2 F ( r ) + r 2 3 ∑ i =1 dx 2 i , F ( r ) = -r 2 Λ 6 -2 M 3 | Σ 3 | r 2 -Q 2 m ln r | Σ 3 | 2 r 2 , A 1 = Q m | Σ 3 | x 2 dx 3 , A 2 = Q m | Σ 3 | x 3 dx 1 , A 3 = Q m | Σ 3 | x 1 dx 2 . (2.7)</formula>
<figure>
<location><page_6><loc_17><loc_64><loc_49><loc_89></location>
</figure>
<figure>
<location><page_6><loc_52><loc_65><loc_85><loc_89></location>
<caption>Figure 1: Plot of the metric function F ( r ) in the GR-limit (2.4), where the left panel corresponds to the electric solution for Λ = -1 and Q e / | Σ 3 | = 2 / √ 3, while the right panel is the dyonic situation with the same values of Λ and Q e / | Σ 3 | , together with Q m / | Σ 3 | = 0 . 1. For both cases, the naked singularity solution is represented by a blue dashed-dotted line, the extremal black holes with a black dashed line and the solution with inner and outer horizons by a red continuous line.</caption>
</figure>
<text><location><page_6><loc_14><loc_47><loc_74><loc_48></location>In order to study the variations of the metric function F , it is useful to define</text>
<formula><location><page_6><loc_20><loc_42><loc_88><loc_45></location>h ( x ) = 6 | Σ 3 | 2 r 2 F ( r ) = -Λ | Σ 3 | 2 x -3 2 Q 2 m ln x -4 M| Σ 3 | , with x = r 4 . (2.8)</formula>
<text><location><page_6><loc_14><loc_33><loc_88><loc_41></location>For negative cosmological constant Λ < 0, we have lim x →∞ h ( x ) = lim x → 0 + h ( x ) = ∞ and the function h has a global minimum at x = 3 Q 2 m -2Λ | Σ 3 | 2 . The equation for the zeros of the function h that will give as well the location of the horizons for the metric function F through (2.8) is of the form (6.2). Hence, its corresponding discriminant as defined in Eq. (6.3) is given by</text>
<formula><location><page_6><loc_40><loc_28><loc_88><loc_32></location>∆ = 2Λ | Σ 3 | 2 3 Q 2 m e -8 M| Σ 3 | 3 Q 2 m . (2.9)</formula>
<text><location><page_6><loc_14><loc_20><loc_88><loc_27></location>Since we are considering the negative cosmological constant case Λ < 0, the discriminant is negative, and as mentioned in the appendix, the equation h ( x ) = 0 will have two real roots only if ∆ ∈ ] -1 e , 0[. This condition in turn requires that the mass M must satisfy the following bound relation</text>
<formula><location><page_6><loc_33><loc_15><loc_88><loc_18></location>M > 3 Q 2 m 8 | Σ 3 | [ 1 -ln ( 3 Q 2 m -2Λ | Σ 3 | 2 )] = M 0 . (2.10)</formula>
<text><location><page_7><loc_14><loc_84><loc_88><loc_89></location>For M satisfying such bound, the metric function F has an inner (Cauchy) horizon r -and an outer (event) horizon r + whose locations are expressed in term of the two real branches of the Lambert W functions, W 0 and W -1 as</text>
<formula><location><page_7><loc_28><loc_80><loc_88><loc_82></location>r -= e -W 0 (∆) 4 -2 M| Σ 3 | 3 Q 2 m , r + = e -W -1 (∆) 4 -2 M| Σ 3 | 3 Q 2 m , (2.11)</formula>
<text><location><page_7><loc_14><loc_60><loc_88><loc_78></location>with ∆ given by (2.9). In contrast with the four-dimensional magnetic Reissner-Nordstrom solution (or even the dyonic configuration), the bound (2.10) does not restrict the mass M to be positive. In fact, for Q 2 m ≥ -2Λ | Σ 3 | 2 e/ 3, the bound M 0 ≤ 0, and hence the singularity at the origin can still be covered by an horizon even for a solution with a negative mass. On the other hand, for M saturating the bound (2.10), namely M = M 0 or equivalently ∆ = -1 e , one ends up with an extremal black hole with r + = r -. Finally, for M < M 0 , the solution will have a naked singularity. To be complete, we also mention that the energy density, the radial and tangential pressure of the purely magnetic GR solution are given by (2.5) with Q e = 0, and hence the magnetic solution satisfies as well the dominant energy conditions (2.6).</text>
<section_header_level_1><location><page_7><loc_14><loc_53><loc_88><loc_57></location>3 Other examples of black holes with Lambert W function horizons</section_header_level_1>
<text><location><page_7><loc_14><loc_25><loc_88><loc_51></location>In the previous section, we have shown that purely magnetic black holes of five-dimensional Einstein gravity with 3 different Abelian gauge fields exist provided a certain bound relation between the mass and the magnetic charge. In addition, the location of the horizons can be expressed thanks to the real branches of the Lambert W functions. In this section, we will present few examples enjoying these same features (bound for the mass and horizons expressed in term of the Lambert W functions) with different Noetherian charges (electric, dyonic, magnetic or axionic) and different asymptotics (AdS or Lifshitz). In order to achieve this task, it is clear from the previous analysis that the Noetherian charges in the metric must have a slower falloff of logarithmic order in comparison to the mass term. In what follows, we will present four different such solutions: an AdS dyonic black hole in five dimensions, an AdS electrically charged solution in odd dimension and two Lifshitz black holes with a magnetic and axionic charge in arbitrary dimension. These configurations are particular solutions of the following general D -dimensional action</text>
<formula><location><page_7><loc_18><loc_21><loc_88><loc_24></location>S [ g, φ, A, A , ψ j ] = ∫ d D x √ -g L , (3.1)</formula>
<formula><location><page_7><loc_18><loc_15><loc_87><loc_20></location>L = R -2Λ 2 -1 2 ∂ µ φ∂ µ φ -1 4 e λφ ( F µν F µν ) q -1 4 n ∑ I =1 e α I φ F ( I ) µν F µν ( I ) -1 2 D -2 ∑ j =1 e β j φ ∂ µ ψ j ∂ µ ψ j .</formula>
<text><location><page_8><loc_14><loc_61><loc_88><loc_90></location>In this action, we leave open the possibility of having a nonlinear Maxwell term ( F µν F µν ) q where F µν = ∂ µ A ν -∂ ν A µ . Such nonlinearity has been shown to be fruitful to obtain charged solutions in different gravity contexts, see e. g. [19]. As in the previous example, in order to sustain a magnetic charge, we will also add some extra Abelian gauge fields F ( I ) µν = ∂ µ A ( I ) ν -∂ ν A ( I ) µ for I = { 1 , 2 , · · · , n } . We justify the presence of axionic fields ψ j from the fact that we are looking for solutions with planar base manifold, and as shown in the fourth example or in the next section, the axionic fields perfectly accommodate an ansatz of the form ψ j = λx j where the x j are the planar coordinates of the base manifold (3.2). This particular ansatz for the axionic fields also provides a simple mechanism of momentum dissipation [20], and in this case the holographic DC conductivities (electrical, thermoelectric and thermal) can be expressed in terms of the black hole horizon data [21, 22]. For examples, the DC conductivities of dyonic black holes with axionic fields have been computed recently in Refs. [23]. Finally, we note that the model considered here (3.1) also allows a possible coupling of the electromagnetic fields A , A and the axionic fields ψ j to a dilaton field φ .</text>
<text><location><page_8><loc_16><loc_59><loc_59><loc_61></location>The field equations associated to the action (3.1) read</text>
<formula><location><page_8><loc_15><loc_52><loc_83><loc_59></location>G µν +Λ g µν = ( ∂ µ φ∂ ν φ -1 2 g µν ∂ σ φ∂ σ φ ) + q e λφ ( F σρ F σρ ) q -1 F µσ F σ ν -g µν 4 e λφ ( F σρ F σρ ) q n D -2 ,</formula>
<formula><location><page_8><loc_18><loc_51><loc_83><loc_55></location>+ ∑ I =1 e α I φ ( F ( I ) µσ F σ ( I ) ν -1 4 g µν F ( I ) σρ F σρ ( I ) ) + ∑ j =1 e β j φ ( ∂ µ ψ j ∂ ν ψ j -1 2 g µν ∂ σ ψ j ∂ σ ψ j )</formula>
<formula><location><page_8><loc_59><loc_48><loc_61><loc_50></location>∇</formula>
<formula><location><page_8><loc_61><loc_42><loc_83><loc_50></location>µ ( e λφ ( F σρ F σρ ) q -1 F µν ) = 0 , ∇ µ ( e α I φ F µν ( I ) ) = 0 , ∇ µ ( e β j φ ∇ µ ψ j ) = 0 ,</formula>
<formula><location><page_8><loc_21><loc_37><loc_83><loc_41></location>glyph[square] φ -λ 4 e λφ ( F ( i ) σρ F σρ ( i ) ) q -1 4 n ∑ I =1 α I e α I φ F ( I ) µν F µν ( I ) -1 2 D -2 ∑ j =1 β j e β j φ ∂ µ ψ j ∂ µ ψ j = 0 ,</formula>
<text><location><page_8><loc_14><loc_35><loc_70><loc_36></location>and we look for a static ansatz with a planar base manifold of the form</text>
<formula><location><page_8><loc_33><loc_29><loc_88><loc_34></location>ds 2 = -N 2 ( r ) F ( r ) dt 2 + dr 2 F ( r ) + r 2 D -2 ∑ i =1 dx 2 i . (3.2)</formula>
<text><location><page_8><loc_14><loc_25><loc_88><loc_29></location>In what follows, we will derive four classes of solutions of the previous field equations, and their analysis will only be considered in the case of a negative cosmological constant Λ < 0.</text>
<section_header_level_1><location><page_8><loc_14><loc_19><loc_88><loc_23></location>3.1 Electrically charged AdS black holes for nonlinear Maxwell theory in odd dimension</section_header_level_1>
<text><location><page_8><loc_14><loc_14><loc_88><loc_18></location>This case will correspond of setting φ = A I = ψ j = 0 in (3.1) and the Maxwell nonlinearity q is of the form q = D -1 2 . As shown in Ref. [24], there exists a purely electric solution with</text>
<text><location><page_9><loc_14><loc_84><loc_88><loc_89></location>logarithmic falloff, and this solution, in order to be real, must be restricted to odd dimension D = 2 k +1 with k ≥ 1. Hence the Maxwell nonlinearity is q = k and the metric function and the electric potential are given by</text>
<formula><location><page_9><loc_17><loc_74><loc_88><loc_83></location>F ( r ) = -r 2 Λ k (2 k -1) -2 M (2 k -1) | Σ 2 k -1 | r 2 k -2 -( -2) k -1 [ ( -1) k (2) 1 -k Q e k | Σ 2 k -1 | ] 2 k 2 k -1 ln r r 2 k -2 , N ( r ) = 1 , A 0 = -[ ( -1) k (2) 1 -k Q e k | Σ 2 k -1 | ] 1 2 k -1 ln r. (3.3)</formula>
<text><location><page_9><loc_14><loc_70><loc_88><loc_73></location>Here M is the mass, Q e is the electric charge and | Σ 2 k -1 | denotes the finite volume element of the compact (2 k -1) -dimensional base manifold.</text>
<text><location><page_9><loc_14><loc_63><loc_88><loc_69></location>As for the previous magnetic solution, the equation determining the zeros of the metric function F can be put in the form (6.2) by substituting x = r 2 k and in this case, the discriminant (6.3) is given by</text>
<formula><location><page_9><loc_15><loc_58><loc_88><loc_62></location>∆ = -| Σ 2 k -1 | Λ e 2 M k B B , with B = ( -2) k -2 (2 k -1) [ ( -1) k (2) 1 -k Q e k | Σ 2 k -1 | ] 2 k 2 k -1 | Σ 2 k -1 | . (3.4)</formula>
<text><location><page_9><loc_14><loc_49><loc_88><loc_57></location>We note that, because of the presence of the term ( -2) k -2 , the sign of the discriminant will depend on the parity of the integer k . Indeed, for even k or equivalently for odd dimensions D = 5 mod 4, the discriminant is positive, and hence the solution is a black hole for any value of the mass M , and there is a single horizon located at</text>
<formula><location><page_9><loc_44><loc_45><loc_57><loc_47></location>r h = e -W 0 (∆) 2 k -M B .</formula>
<text><location><page_9><loc_14><loc_36><loc_88><loc_43></location>Nevertheless, in this case, it is simple to see that the energy density is always negative, and consequently the energy conditions do not hold. On the other hand, for odd k or equivalently for odd dimensions D = 3 mod 4, the solution will be a black hole provided that the mass satisfies the following bound relation with the electric charge</text>
<formula><location><page_9><loc_36><loc_31><loc_88><loc_35></location>M≥B 2 k [ 1 -ln ( B Λ | Σ 2 k -1 | )] . (3.5)</formula>
<text><location><page_9><loc_14><loc_29><loc_56><loc_30></location>In this case, the inner and outer horizons are given by</text>
<formula><location><page_9><loc_33><loc_25><loc_88><loc_27></location>r -= e -W 0 (∆) 2 k -M B , r + = e -W -1 (∆) 2 k -M B , (3.6)</formula>
<text><location><page_9><loc_14><loc_22><loc_60><loc_23></location>and the dominant energy conditions (2.6) are satisfied with</text>
<formula><location><page_9><loc_14><loc_17><loc_89><loc_21></location>µ = 2 k -2 (2 k -1) r 2 k [ ( -1) k (2) 1 -k Q e k | Σ 2 k -1 | ] 2 k 2 k -1 , p r = -µ, p t = 2 k -2 r 2 k [ ( -1) k (2) 1 -k Q e k | Σ 2 k -1 | ] 2 k 2 k -1 . (3.7)</formula>
<section_header_level_1><location><page_10><loc_14><loc_86><loc_88><loc_89></location>3.2 Five-dimensional AdS dyonic black holes and particular stealth configuration</section_header_level_1>
<text><location><page_10><loc_14><loc_73><loc_88><loc_84></location>In five dimensions, the previous solution can be magnetically charged in such a way that the magnetic charge also appears in the metric function with a slower falloff of logarithmic order in comparison to the mass. The corresponding model that sustains such solution is given by the action (3.1) by setting φ = ψ j = 0 and by considering n = 3 extra gauge fields A I as well as the nonlinear Maxwell term with the exponent q = 2. In this case, the dyonic solution which can also be viewed as an electric extension of the solution (2.7) is given by the ansatz (3.2) with</text>
<formula><location><page_10><loc_19><loc_67><loc_71><loc_71></location>F ( r ) = -Λ 6 r 2 -2 M 3 | Σ 3 | r 2 -ln r r 2 [ Q 2 m | Σ 3 | 2 -2 |Q e | 4 3 (4 | Σ 3 | ) 4 3 ] , N ( r ) = 1 ,</formula>
<formula><location><page_10><loc_19><loc_62><loc_85><loc_67></location>A 1 = Q m | Σ 3 | x 2 dx 3 , A 2 = Q m | Σ 3 | x 3 dx 1 , A 3 = Q m | Σ 3 | x 1 dx 2 , A 0 = -|Q e | 1 3 (4 | Σ 3 | ) 1 3 ln r.</formula>
<text><location><page_10><loc_14><loc_60><loc_77><loc_61></location>Before proceeding as before, we would like to point out that the point defined by</text>
<formula><location><page_10><loc_42><loc_54><loc_88><loc_58></location>|Q e | = 2 5 4 |Q m | 3 2 | Σ 3 | 1 2 , (3.8)</formula>
<text><location><page_10><loc_14><loc_45><loc_88><loc_53></location>is very special in the sense that the metric function reduces to the Schwarzschild AdS metric with a flat horizon. This in turn implies that the field equations at the point (3.8) can be interpreted as a stealth configuration [25] defined on the Schwarzschild AdS background since both side (geometric and matter part) of the Einstein equations vanish separately, i. e.</text>
<formula><location><page_10><loc_25><loc_36><loc_88><loc_44></location>G µν +Λ g µν = 0 = 3 ∑ I =1 ( F ( I ) µσ F σ ( I ) ν ) -1 4 g µν 3 ∑ I =1 ( F ( I ) σρ F σρ ( I ) ) (3.9) +2 F µσ F σ ν ( F αβ F αβ ) -1 4 g µν ( F αβ F αβ ) 2 .</formula>
<text><location><page_10><loc_14><loc_29><loc_88><loc_35></location>Note that such stealth configuration but for a dyonic four-dimensional Reissner-Nordstrom black hole was known in the case of an Abelian gauge field coupled to a particular Horndeski term [26] or for a generalized Proca field theory [27].</text>
<text><location><page_10><loc_14><loc_25><loc_87><loc_29></location>Outside the stealth point, the discriminant associated to the zeros of the metric function F is given by</text>
<formula><location><page_10><loc_29><loc_20><loc_88><loc_24></location>∆ = 2Λ 3 A e -8 M 3 | Σ 3 | A , with A = Q 2 m | Σ 3 | 2 -2 |Q e | 4 3 (4 | Σ 3 | ) 4 3 . (3.10)</formula>
<text><location><page_10><loc_14><loc_17><loc_83><loc_18></location>Since, we are only considering the negative cosmological constant case, we conclude that:</text>
<section_header_level_1><location><page_11><loc_15><loc_88><loc_35><loc_89></location>(i) For A < 0, that is for</section_header_level_1>
<formula><location><page_11><loc_46><loc_84><loc_59><loc_88></location>|Q e | > 2 5 4 |Q m | 3 2 | Σ 3 | 1 2 ,</formula>
<text><location><page_11><loc_18><loc_82><loc_51><loc_83></location>the solution has a single horizon located at</text>
<formula><location><page_11><loc_45><loc_78><loc_61><loc_80></location>r h = e -W 0 (∆) 4 -2 M 3 | Σ 3 | A ,</formula>
<text><location><page_11><loc_18><loc_72><loc_88><loc_76></location>but the solution does not satisfy the dominant energy conditions neither the weak energy conditions since the energy density µ = 3 A 2 r 4 is always negative.</text>
<section_header_level_1><location><page_11><loc_15><loc_69><loc_35><loc_71></location>(ii) For A > 0, that is for</section_header_level_1>
<formula><location><page_11><loc_46><loc_65><loc_59><loc_69></location>|Q e | < 2 5 4 |Q m | 3 2 | Σ 3 | 1 2 ,</formula>
<text><location><page_11><loc_18><loc_63><loc_60><loc_64></location>the solution represents a dyonic AdS black hole only if</text>
<formula><location><page_11><loc_41><loc_58><loc_65><loc_62></location>M≥ 3 | Σ 3 | A 8 [ 1 -ln ( 3 A -2Λ )] ,</formula>
<text><location><page_11><loc_18><loc_55><loc_85><loc_57></location>and in this case, the solution is shown to satisfy the dominant energy conditions (2.6).</text>
<unordered_list>
<list_item><location><page_11><loc_14><loc_52><loc_40><loc_54></location>(iii) Finally, for A = 0 that is for</list_item>
</unordered_list>
<formula><location><page_11><loc_46><loc_48><loc_59><loc_52></location>|Q e | = 2 5 4 |Q m | 3 2 | Σ 3 | 1 2 ,</formula>
<text><location><page_11><loc_18><loc_44><loc_88><loc_47></location>the solution represents a black hole stealth dyonic configuration on the Schwarzschild AdS background where the horizon is located at</text>
<formula><location><page_11><loc_45><loc_39><loc_60><loc_42></location>r h = ( 4 M -Λ | Σ 3 | ) 1 4 .</formula>
<section_header_level_1><location><page_11><loc_14><loc_35><loc_87><loc_37></location>3.3 Purely magnetic Lifshitz black hole with dynamical exponent z = D -4</section_header_level_1>
<text><location><page_11><loc_14><loc_24><loc_88><loc_33></location>We now turn to derive examples with a different asymptotic behavior characterized by an anisotropy scale between the time and the space, the so-called Lifshitz asymptotic. This anisotropy is reflected by a dynamical exponent denoted usually by z and defined such that the case z = 1 corresponds to the AdS isotropic case. Note that Lifshitz black holes have been insensitively studied during the last decade, see for examples Refs. [28]</text>
<text><location><page_11><loc_14><loc_16><loc_88><loc_23></location>In order to obtain Lifshitz black holes, we consider the action (3.1) without axionic fields ψ j = 0 and with the standard Maxwell term q = 1. Note that the presence of the Maxwell term is mandatory to ensure the Lifshitz asymptotic of the solution. In even (resp. odd) dimension, the solution will be sustained by n = 1 (resp. n = 3) extra gauge field(s) A I . In both case, a</text>
<text><location><page_12><loc_14><loc_86><loc_88><loc_89></location>purely magnetic Lifshitz black hole with dynamical exponent z = D -4 is found through the ansatz (3.2) with</text>
<formula><location><page_12><loc_22><loc_77><loc_88><loc_85></location>F ( r ) = r 2 -2 M ( D -2) | Σ D -2 | r 2( D -4) -Q 2 m | Σ D -2 | 2 r 2( D -4) ln( r ) , N ( r ) = r D -5 , A 0 = √ D -5 2( D -3) r 2( D -3) , e φ = r √ ( D -2)( D -5) , (3.11)</formula>
<formula><location><page_12><loc_22><loc_71><loc_73><loc_76></location>A (1) = √ 2 Q m 2 | Σ D -2 | D -2 2 ∑ i =1 ( x 2 i -1 dx 2 i -x 2 i dx 2 i -1 ) , for even dimension ,</formula>
<formula><location><page_12><loc_22><loc_66><loc_81><loc_71></location>A ( I ) = Q m 2 | Σ D -2 | 3 ∑ J,K =1 glyph[epsilon1] IJK x J dx K + √ 6 6 Q m | Σ D -2 | D -3 2 ∑ i =2 ( x 2 i dx 2 i +1 -x 2 i +1 dx 2 i ) ,</formula>
<text><location><page_12><loc_14><loc_61><loc_88><loc_65></location>for odd dimension with I = { 1 , 2 , 3 } . Here | Σ D -2 | denotes the finite volume element of the compact ( D -2) -dimensional base manifold and glyph[epsilon1] IJK is defined as</text>
<formula><location><page_12><loc_30><loc_55><loc_71><loc_60></location>glyph[epsilon1] IJK = 1 for any even permutation of (1 , 2 , 3) , -1 for any odd permutation of (1 , 2 , 3) , 0 otherwise.</formula>
<text><location><page_12><loc_14><loc_52><loc_76><loc_53></location>In this case, the coupling constants of the problem must take the following form</text>
<formula><location><page_12><loc_22><loc_47><loc_80><loc_51></location>Λ = -( D -3)(2 D -7) , λ = -2 √ D -2 D -5 , α 1 = α 2 = α 3 = -2 √ D -5 D -2 .</formula>
<text><location><page_12><loc_14><loc_41><loc_88><loc_46></location>Proceeding as in the two previous examples, one notes that the discriminant is always negative, and hence the existence of horizons is again ensured provided that the mass satisfies the following bound</text>
<formula><location><page_12><loc_29><loc_37><loc_88><loc_40></location>M≥ ( D -2) Q 2 m 4( D -3) | Σ D -2 | [ 1 -ln ( Q 2 m 2( D -3) | Σ D -2 | 2 )] . (3.12)</formula>
<text><location><page_12><loc_14><loc_32><loc_88><loc_36></location>For this lifshitz solution with dynamical exponent z = D -4, the energy density and the radial/tangential pressures are given by</text>
<formula><location><page_12><loc_23><loc_20><loc_81><loc_31></location>µ = ( D -5)( D -3) + ( D -2) Q 2 m 2 | Σ D -2 | 2 r 2( D -3) + ( D -2)( D -5) 2 r 2 F ( r ) , p r = µ -1 | Σ D -2 | 2 r 2( D -3) ( 2( D -3)( D -5) | Σ D -2 | 2 r 2( D -3) +( D -2) Q 2 m ) , p t = -µ -1 | Σ D -2 | 2 r 2( D -3) ( 2( D -3)( D -5) | Σ D -2 | 2 r 2( D -3) +2 Q 2 m ) ,</formula>
<text><location><page_12><loc_14><loc_18><loc_21><loc_19></location>and since</text>
<formula><location><page_12><loc_17><loc_14><loc_84><loc_17></location>p r + µ = ( D -2)( D -5) F ( r ) r 2 , p t -µ = -( D -2)( D -5) F ( r ) r 2 -( D -4) Q 2 m | Σ D -2 | 2 r 2( D -3) ,</formula>
<text><location><page_13><loc_14><loc_86><loc_88><loc_89></location>one can notice that the dominant energy conditions (2.6) are satisfied outside the event horizon, that is for F ( r ) ≥ 0.</text>
<section_header_level_1><location><page_13><loc_14><loc_82><loc_78><loc_83></location>3.4 Axionic Lifshitz black hole with dynamical exponent z = D -2</section_header_level_1>
<text><location><page_13><loc_14><loc_75><loc_88><loc_80></location>We now consider the action (3.1) with a source only given by the axionic fields ψ j and with the standard Maxwell term q = 1 in order to sustain the Lifshitz asymptotic. In this case, an axionic Lifshitz black hole solution with dynamical exponent z = D -2 is found to be</text>
<formula><location><page_13><loc_22><loc_62><loc_88><loc_74></location>F ( r ) = r 2 -2 M ( D -2) | Σ D -2 | r 2( D -3) -Q 2 a | Σ D -2 | 2 r 2( D -3) ln( r ) , N ( r ) = r D -3 , A 0 = √ D -3 2( D -2) r 2( D -2) , e φ = r √ ( D -2)( D -3) , (3.13) ψ j ( x j ) = -Q a | Σ D -2 | x j , with j = { 1 , 2 , · · · , D -2 } ,</formula>
<text><location><page_13><loc_14><loc_59><loc_85><loc_60></location>where now Q a denotes the axionic charge. The coupling constants must be chosen such as</text>
<formula><location><page_13><loc_14><loc_54><loc_88><loc_57></location>Λ = -( D -2)(2 D -5) , λ = -2 √ D -2 D -3 , β j = -2 √ D -3 D -2 , with j = { 1 , 2 , · · · , D -2 } .</formula>
<text><location><page_13><loc_14><loc_49><loc_88><loc_53></location>As for the previous case, the discriminant is negative and the mass parameter must satisfy the following bound with respect to the axionic charge in order to avoid naked singularity</text>
<formula><location><page_13><loc_32><loc_44><loc_88><loc_48></location>M≥ Q 2 a 4 | Σ D -2 | [ 1 -ln ( Q 2 a 2( D -2) | Σ D -2 | 2 )] . (3.14)</formula>
<text><location><page_13><loc_14><loc_38><loc_88><loc_41></location>As in the previous Lifshitz case, the dominant energy conditions (2.6) are satisfied outside the event horizon with</text>
<formula><location><page_13><loc_19><loc_22><loc_86><loc_37></location>µ = ( D -2)( D -3) + ( D -2) Q 2 a 2 | Σ D -2 | 2 r 2( D -2) + ( D -2)( D -3) 2 r 2 F ( r ) , p r = µ -1 | Σ D -2 | 2 r 2( D -2) ( 2( D -3)( D -2) | Σ D -2 | 2 r 2( D -2) +( D -2) Q 2 a ) , p t = -µ -1 | Σ D -2 | 2 r 2( D -2) ( 2( D -3)( D -2) | Σ D -2 | 2 r 2( D -2) + Q 2 a ) , p r + µ = ( D -2)( D -3) F ( r ) r 2 , p t -µ = -( D -2)( D -3) F ( r ) r 2 -( D -3) Q 2 a | Σ D -2 | 2 r 2( D -2) .</formula>
<text><location><page_13><loc_14><loc_15><loc_88><loc_21></location>We now compute the DC conductivity σ DC of this solution which can be expressed in term of the black hole horizon data [21, 22] thanks to the presence of the axionic fields homogenously distributed along the coordinates of the planar base. In order to achieve this task, we will</text>
<text><location><page_14><loc_14><loc_86><loc_88><loc_89></location>follow the prescriptions as given in these last references by first turning on the following relevant perturbations 1</text>
<formula><location><page_14><loc_19><loc_82><loc_82><loc_84></location>δA 0 = -Et + a x 1 ( r ) , δg tx 1 = r 2 h tx 1 ( r ) , δg rx 1 = r 2 h rx 1 ( r ) , δψ 1 = χ 1 ( r ) ,</formula>
<text><location><page_14><loc_14><loc_75><loc_88><loc_82></location>where E is a constant. The perturbed Maxwell current given by J = √ -ge λφ F rx 1 is a conserved quantity along the radial coordinate. A straightforward computation along the same lines as those in [21, 22] yields a DC conductivity σ DC given by</text>
<formula><location><page_14><loc_30><loc_70><loc_71><loc_74></location>σ DC = ∂J ∂E ∣ ∣ ∣ r h = r -D h + 2( D -2)( D -3) | Σ D -2 | 2 r D -4 h Q 2 a .</formula>
<text><location><page_14><loc_14><loc_65><loc_88><loc_69></location>As it should be expected in the absence of the axionic charge, the expression of the DC conductivity σ DC will blows up.</text>
<section_header_level_1><location><page_14><loc_14><loc_60><loc_88><loc_62></location>4 More general dyonic-axionic solutions in arbitrary dimension</section_header_level_1>
<text><location><page_14><loc_14><loc_43><loc_88><loc_58></location>The solutions derived previously can accommodate extra Noetherian charges but in this case the location of the horizons is more involved and can not be treated as before with the help of the Lambert W functions. Nevertheless, for completeness, we report in this section more general solutions, each of them having a dyonic and an axionic charge. In order to achieve this task, we consider a slightly different action than the one defined by Eq. (3.1). Indeed, we will add an extra Maxwell term without any nonlinearity q = 1 since it is known that electrically charged Lifshitz black holes require the introduction of at least two Maxwell terms [29]. We then consider the following D -dimensional action</text>
<formula><location><page_14><loc_17><loc_33><loc_89><loc_41></location>S [ g µν , φ, A ( i ) µ , A ( I ) µ , ψ j ] = ∫ d D x √ -g L , L = R -2Λ 2 -1 2 ∂ µ φ∂ µ φ -1 4 2 ∑ i =1 e λ i φ F ( i ) µν F µν ( i ) -e αφ 1 4 n ∑ I =1 F ( I ) µν F µν ( I ) + 1 2 D -2 ∑ j =1 ∂ µ ψ j ∂ µ ψ j ,</formula>
<text><location><page_14><loc_14><loc_28><loc_88><loc_31></location>with F ( i ) µν = ∂ µ A ( i ) ν -∂ ν A ( i ) µ for i = { 1 , 2 } and n extra gauge fields that will sustain the magnetic charge, F ( I ) µν = ∂ µ A ( I ) ν -∂ ν A ( I ) µ for I = 1 , · · · n with n = 1 in even dimension and</text>
<text><location><page_15><loc_14><loc_88><loc_52><loc_89></location>n = 3 in odd dimension. The field equations read</text>
<formula><location><page_15><loc_15><loc_84><loc_16><loc_85></location>G</formula>
<formula><location><page_15><loc_16><loc_65><loc_83><loc_87></location>µν +Λ g µν = ( ∇ µ φ ∇ ν φ -1 2 g µν ∇ σ φ ∇ σ φ ) + 2 ∑ i =1 ( e λ i φ F ( i ) µσ F σ ( i ) ν -1 4 g µν e λ i φ F ( i ) σρ F σρ ( i ) ) e αφ D -2 ∑ j =1 ( ∂ µ ψ j ∂ ν ψ j -1 2 g µν ∂ σ ψ j ∂ σ ψ j ) -1 4 g µν e αφ n ∑ I =1 F ( I ) σρ F σρ ( I ) + e αφ n ∑ I =1 F ( I ) µσ F σ ( I ) ν , ∇ µ ( e λ i φ F µν ( i ) ) = 0 , ∇ µ ( e αφ F µν ( I ) ) = 0 , glyph[square] ψ j = 0 , glyph[square] φ -2 ∑ i =1 ( λ i 4 e λ i φ F ( i ) σρ F σρ ( i ) ) -αe αφ 1 4 n ∑ I =1 F ( I ) µν F µν ( I ) + 1 2 D -2 ∑ j =1 ∂ µ ψ j ∂ µ ψ j = 0 ,</formula>
<text><location><page_15><loc_14><loc_60><loc_88><loc_63></location>A general Lifshitz dyonic-axionic solution with arbitrary dynamical exponent z of the field equations is given by</text>
<formula><location><page_15><loc_14><loc_55><loc_88><loc_59></location>F ( r ) = r 2 -λ 2 ( D -2 -z ) r 2 z -2 -M r z + D -4 + Q 2 ( D -2)( z + D -4) r 2( z + D -4) -P 2 ( D -4 -z ) r 2 z , (4.2)</formula>
<text><location><page_15><loc_14><loc_53><loc_24><loc_54></location>together with</text>
<formula><location><page_15><loc_16><loc_46><loc_88><loc_52></location>N ( r ) = r z -1 , A (1) t = √ z -1 z + D -2 r z + D -2 dt, A (2) t = -Q ( z + D -4) r z + D -4 dt, ψ j ( x j ) = λx j , e φ = r √ ( D -2)( z -1) , (4.3)</formula>
<formula><location><page_15><loc_18><loc_35><loc_85><loc_45></location>A (1) = √ 2 P 2 D -2 2 ∑ i =1 ( x 2 i -1 dx 2 i -x 2 i dx 2 i -1 ) , in even dimensions , A ( I ) = P 2 3 ∑ J,K =1 glyph[epsilon1] IJK x J dx K + √ 6 6 P D -3 2 ∑ i =2 ( x 2 i dx 2 i +1 -x 2 i +1 dx 2 i ) , in odd dimensions</formula>
<text><location><page_15><loc_14><loc_32><loc_57><loc_34></location>provided that the coupling constants are tied as follows</text>
<formula><location><page_15><loc_17><loc_27><loc_84><loc_31></location>Λ = -( z + D -2)( z + D -3) 2 , λ 1 = -2 √ ( D -2 z -1 ) , λ 2 = -α = 2 √ ( z -1 D -2 ) .</formula>
<text><location><page_15><loc_69><loc_23><loc_69><loc_24></location>glyph[negationslash]</text>
<text><location><page_15><loc_14><loc_20><loc_88><loc_26></location>Note that in the AdS limit z = 1, the dilaton field φ disappears as well as the Maxwell potential A (1) t which is precisely responsible to sustain the Lifshitz asymptotic z = 1. The thermodynamical variables of the solution can be computed using the Hamiltonian formalism [30] yielding</text>
<formula><location><page_15><loc_40><loc_14><loc_88><loc_19></location>M = 1 2 ( D -2) M | Σ D -2 | , S = 2 πr D -2 h | Σ D -2 | , (4.4)</formula>
<text><location><page_16><loc_14><loc_88><loc_62><loc_89></location>with the electric, magnetic and axionic potentials and charges</text>
<formula><location><page_16><loc_22><loc_75><loc_88><loc_87></location>Q e = Q | Σ D -2 | , Φ e = ( r 4 -D -z h z -4 + D ) Q, Q m = | Σ D -2 | P, Φ m = -( D -2 D -4 -z ) r D -4 -z h P, (4.5) Q j = -| Σ D -2 | λ, ˆ Ψ j ( r h ) = r D -2 -z h λ D -2 -z , with j = { 1 , 2 , · · · , D -2 } .</formula>
<text><location><page_16><loc_14><loc_72><loc_62><loc_73></location>It is a simple exercise to check the consistency of the first law</text>
<formula><location><page_16><loc_30><loc_66><loc_88><loc_71></location>d M = Td S +Φ e d Q e +Φ m d Q m + D -2 ∑ j =1 ˆ Ψ j ( r h ) d Q j , (4.6)</formula>
<text><location><page_16><loc_14><loc_63><loc_40><loc_65></location>where the temperature is given by</text>
<formula><location><page_16><loc_19><loc_58><loc_88><loc_62></location>T = N ( r ) F ' ( r ) 4 π ∣ ∣ ∣ r = r h = 1 4 π [ ( D -2 + z ) r z h -Q 2 ( D -2) r z +2 D -6 h -P 2 r z +2 h -λ 2 r z h ] . (4.7)</formula>
<text><location><page_16><loc_14><loc_51><loc_88><loc_56></location>It is clear that from the expression of the metric function, the cases z = D -4 and z = D -2 must be treated separately. In fact, for z = D -4, one yields a metric function involving a logarithmic magnetic contribution</text>
<formula><location><page_16><loc_21><loc_46><loc_88><loc_50></location>F ( r ) = r 2 -M r 2( D -4) + Q 2 2( D -2)( D -4) r 4( D -4) -P 2 r 2( D -4) ln( r ) -λ 2 2 r 2( D -5) , (4.8)</formula>
<text><location><page_16><loc_14><loc_41><loc_88><loc_45></location>and the remaining fields are given by (4.3) with z = D -4. The thermodynamical quantities computed by means of the Euclidean method [30] read</text>
<formula><location><page_16><loc_17><loc_22><loc_88><loc_40></location>M = 1 2 ( D -2) M | Σ D -2 | , T = 1 4 π [ 2( D -3) r D -4 h -Q 2 ( D -2) r 3 D -10 h -P 2 r D -2 h -λ 2 r D -4 h ] , S = 2 πr D -2 h | Σ D -2 | , Q e = Q | Σ D -2 | , Φ e = Q 2( D -4) r 2( D -4) h , (4.9) Q m = | Σ D -2 | P, Φ m = -( D -2) ln( r h ) P, Q j = -| Σ D -2 | λ, ˆ Ψ j ( r h ) = r 2 h λ 2 , con j = { 1 , 2 , · · · , D -2 } .</formula>
<text><location><page_16><loc_14><loc_19><loc_51><loc_20></location>Finally, the solution with z = D -2 is given by</text>
<formula><location><page_16><loc_23><loc_14><loc_88><loc_18></location>F ( r ) = r 2 -M r 2( D -3) + Q 2 2( D -2)( D -3) r 4( D -3) + P 2 2 r 2( D -4) -λ 2 ln( r ) r 2( D -3) . (4.10)</formula>
<text><location><page_17><loc_14><loc_88><loc_74><loc_89></location>with (4.3), and the thermodynamic parameters associated to the solution are</text>
<formula><location><page_17><loc_18><loc_68><loc_88><loc_87></location>M = 1 2 ( D -2) M | Σ D -2 | , T = 1 4 π [ 2( D -2) r D -2 h -Q 2 ( D -2) r 3 D -8 h -P 2 r D h -λ 2 r D -2 h ] , S = 2 πr D -2 h | Σ D -2 | , Q e = Q | Σ D -2 | , Φ e = Q 2( D -3) r 2( D -3) h , (4.11) Q m = | Σ D -2 | P, Φ m = ( D -2) P 2 r 2 h , Q j = -| Σ D -2 | λ, ˆ Ψ j ( r h ) = λ ln( r h ) , con j = { 1 , 2 , · · · , D -2 } .</formula>
<text><location><page_17><loc_14><loc_65><loc_88><loc_66></location>In both cases, that is for z = D -4 and z = D -2, it is easy to see that the first law (4.6) holds.</text>
<section_header_level_1><location><page_17><loc_14><loc_60><loc_30><loc_62></location>5 Conclusion</section_header_level_1>
<text><location><page_17><loc_14><loc_16><loc_88><loc_58></location>Here, we have presented a dyonic extension of the five-dimensional Boulware-Deser solution for the Einstein-Gauss-Bonnet theory. The emergence of a magnetic charge is shown to be possible for a flat horizon and by considering at least three different Maxwell invariants. The magnetic contribution in the metric function has a logarithmic falloff but still yields to finite physical quantities. As usual, one of the two branches has a well-defined GR-limit with a magnetic logarithmic falloff term. For suitable bounds between the mass and the magnetic charge, the purely magnetic GR solution can be shown to admit an inner and outer horizons. These latter are given in terms of the two real branches of the Lambert W functions. We have noticed that this bound's mass was due to the fact that the magnetic charge in the metric has a slower falloff of logarithmic order than the mass. Exploiting this observation, we have derived other examples of solutions sharing these same properties for different models and different asymptotics. For example, we have obtained an electrically charged AdS black hole solution in odd dimension for a nonlinear Maxwell theory with a single horizon in dimensions D = 5 mod4 and with two horizons in D = 3 mod4. Interestingly enough, a dyonic configuration with logarithmic falloff of the electric and magnetic charges was also derived in five dimensions. Depending on the strength of the electric charge with respect to the magnetic charge, the solution can have one or two horizons, and in this latter case, the mass must satisfy a certain bound. Moreover, for a precise relation between the electric and the magnetic charges, the solution turns out to be a stealth dyonic configuration defined on the Schwarzschild AdS background. For the asymptotic AdS solutions, we have remarked that our black hole solutions presenting an inner and outer horizons always satisfy the dominant energy conditions (2.6) while these conditions even in their</text>
<text><location><page_18><loc_14><loc_86><loc_88><loc_89></location>weak version do not hold for our solutions with a single horizon. This can be explained by the fact that the metric functions in our set-up were of the following form</text>
<formula><location><page_18><loc_26><loc_81><loc_88><loc_85></location>F ( r ) = -2Λ r 2 ( D -1)( D -2) -2 M ( D -2) | Σ D -2 | r D -3 -N ln r r D -3 , (5.1)</formula>
<text><location><page_18><loc_14><loc_74><loc_88><loc_80></location>where N represent the additional Noetherian charge with slower falloff of logarithmic order than the mass M . The corresponding discriminant associated to the zeros of the metric function F (6.3) is given by</text>
<formula><location><page_18><loc_39><loc_71><loc_63><loc_74></location>∆ = 2Λ ( D -2) N e -2( D -1) M ( D -2) | Σ D -2 |N .</formula>
<text><location><page_18><loc_14><loc_63><loc_88><loc_70></location>Now, since we are considering the AdS case, it is clear that for N < 0, the discriminant will be positive and, hence the solution will represent a black hole with a single horizon for any value of the mass M . On the other hand, for N > 0, one has ∆ < 0 and consequently the solution will be a black hole provided that the mass satisfies the following bound</text>
<formula><location><page_18><loc_32><loc_58><loc_69><loc_61></location>M > ( D -2) | Σ D -2 |N 2( D -1) [ 1 -ln ( ( D -2) N -2Λ )] ,</formula>
<text><location><page_18><loc_14><loc_53><loc_88><loc_56></location>and in this case, the solution presents two horizons. On the other hand, the energy density, the radial and tangential pressures are given generically by</text>
<formula><location><page_18><loc_32><loc_48><loc_69><loc_52></location>µ = ( D -2) N 2 r D -1 , p r = -µ, p t = N 2 r D -1 ,</formula>
<text><location><page_18><loc_14><loc_36><loc_88><loc_47></location>and hence it is evident that the dominant energy conditions (2.6) will only be satisfied for the solutions with N > 0. It seems to be physically acceptable that solutions without any restrictions on the mass do not satisfy the dominant or the weak or even the null energy conditions. On the other hand, our examples of black holes with a bound's mass verify the dominant energy conditions. It will be interesting to explore more deeply this relation between the lack of restriction on the mass with the absence of energy conditions.</text>
<text><location><page_18><loc_14><loc_31><loc_88><loc_35></location>We also mention that a necessary condition to obtain AdS black holes with an Ansatz of the form</text>
<formula><location><page_18><loc_32><loc_28><loc_69><loc_32></location>ds 2 = -F ( r ) dt 2 + dr 2 F ( r ) + r 2 ( dx 2 1 + · · · dx 2 D -2 ) ,</formula>
<text><location><page_18><loc_14><loc_20><loc_88><loc_27></location>with a metric function given by (5.1) is that the energy momentum tensor of the matter source T µν satisfies T t t + ( D -2) T i i = 0 without summation for the planar indices i . Indeed, in this case, the consistency of the Einstein equations T t t + ( D -2) T i i + ( D -1)Λ = 0 yields to a nonhomogeneous Euler's differential equation of second-order</text>
<formula><location><page_18><loc_24><loc_15><loc_78><loc_18></location>( D -2) F '' 2 + ( D -2)(2 D -5) F ' 2 r + ( D -2)( D -3) 2 F 2 r 2 = -( D -1)Λ ,</formula>
<text><location><page_19><loc_14><loc_86><loc_88><loc_90></location>whose characteristic polynomial has a double root given by r -D +3 and hence the general solution of this Euler's equation is given by Eq. (5.1).</text>
<text><location><page_19><loc_14><loc_74><loc_88><loc_85></location>We have also presented two other examples with Lifshitz asymptotics with fixed values of the dynamical exponent with a magnetic charge and an axionic charge. The emergence of such asymptotic solutions is essentially due to the presence of dilatonic fields. Note that there also exist Lifshitz black holes with a logarithmic falloff in the case of higher-order gravity [31]. Finally, for completeness, we have extended the previous solutions to accommodate a dyonic as well an axionic charge in arbitrary dimension.</text>
<section_header_level_1><location><page_19><loc_14><loc_69><loc_61><loc_71></location>6 Appendix: The Lambert W functions</section_header_level_1>
<text><location><page_19><loc_14><loc_63><loc_88><loc_67></location>The Lambert W functions are a set of functions that represent the countably infinite number of solutions denoted by W k ( z ) of the equation</text>
<formula><location><page_19><loc_45><loc_60><loc_88><loc_62></location>We W = z, (6.1)</formula>
<text><location><page_19><loc_14><loc_51><loc_88><loc_58></location>for a given z ∈ C . There are only two real-valued branches of the Lambert W functions that are denoted, by convention, W 0 and W -1 with W 0 : [ -1 e , ∞ [ → [ -1 , ∞ [ and W -1 : [ -1 e , 0[ → ] - ∞ , -1[ with the convention that W 0 ( -1 e ) = W -1 ( -1 e ) = -1. The Lambert W functions appear for the resolution of the equations of the form</text>
<formula><location><page_19><loc_34><loc_47><loc_88><loc_48></location>ax + b ln( x ) + c = 0 , a = 0 , b = 0 . (6.2)</formula>
<text><location><page_19><loc_55><loc_47><loc_55><loc_48></location>glyph[negationslash]</text>
<text><location><page_19><loc_61><loc_47><loc_61><loc_48></location>glyph[negationslash]</text>
<text><location><page_19><loc_14><loc_40><loc_88><loc_45></location>Indeed, by defining w = ln( x ), the equation (6.2) becomes ae w + bw + c = 0, which is equivalent after some basic algebraic manipulations to (6.1) with W = -w -c b and z = ∆, where the discriminant is defined by</text>
<formula><location><page_19><loc_45><loc_35><loc_88><loc_38></location>∆ = a b e -c b . (6.3)</formula>
<text><location><page_19><loc_14><loc_33><loc_29><loc_34></location>It is then clear that</text>
<unordered_list>
<list_item><location><page_19><loc_15><loc_29><loc_80><loc_31></location>(i) If ∆ ≥ 0 or ∆ = -1 e , the equation (6.2) admits a unique solution in R given by</list_item>
</unordered_list>
<formula><location><page_19><loc_45><loc_26><loc_88><loc_28></location>x = e -W 0 (∆) -c b . (6.4)</formula>
<text><location><page_19><loc_15><loc_22><loc_68><loc_24></location>(ii) If ∆ ∈ ] -1 e , 0[, the equation (6.2) has two real solutions given by</text>
<formula><location><page_19><loc_36><loc_18><loc_88><loc_20></location>x 1 = e -W 0 (∆) -c b , x 2 = e -W -1 (∆) -c b . (6.5)</formula>
<text><location><page_19><loc_14><loc_14><loc_68><loc_16></location>(iii) Finally, if ∆ < -1 e , the equation (6.2) does not admit real roots.</text>
<text><location><page_20><loc_14><loc_77><loc_88><loc_81></location>Acknowledgments: MB is supported by grant Conicyt/ Programa Fondecyt de Iniciaci'on en Investigaci'on No. 11170037.</text>
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</document> | [{"title": "Black holes with Lambert W function horizons", "content": "Moises Bravo Gaete \u2217 , Sebastian Gomez \u2020 and Mokhtar Hassaine \u2021 \u2217 Facultad de Ciencias B'asicas, Universidad Cat'olica del Maule, Casilla 617, Talca, Chile. \u2020 Facultad de Ingenier'\u0131a, Universidad Aut'onoma de Chile, 5 poniente 1670, Talca, Chile. \u2021 Instituto de Matem'atica y F'\u0131sica, Universidad de Talca, Casilla 747, Talca, Chile. September 17, 2021", "pages": [1]}, {"title": "Abstract", "content": "We consider Einstein gravity with a negative cosmological constant endowed with distinct matter sources. The different models analyzed here share the following two properties: (i) they admit static symmetric solutions with planar base manifold characterized by their mass and some additional Noetherian charges, and (ii) the contribution of these latter in the metric has a slower falloff to zero than the mass term, and this slowness is of logarithmic order. Under these hypothesis, it is shown that, for suitable bounds between the mass and the additional Noetherian charges, the solutions can represent black holes with two horizons whose locations are given in term of the real branches of the Lambert W functions. We present various examples of such black hole solutions with electric, dyonic or axionic charges with AdS and Lifshitz asymptotics. As an illustrative example, we construct a purely AdS magnetic black hole in five dimensions with a matter source given by three different Maxwell invariants.", "pages": [1]}, {"title": "1 Introduction", "content": "The AdS/CFT correspondence has been proved to be extremely useful for getting a better understanding of strongly coupled systems by studying classical gravity, and more specifically black holes. In particular, the gauge/gravity duality can be a powerful tool for analyzing finite temperature systems in presence of a background magnetic field. In such cases, from the dictionary of the correspondence, the black holes must be endowed with a magnetic charge corresponding to the external magnetic field of the CFT. In light of this constatation, it is clear that dyonic black holes are of great importance in order to study the charge transport at quantum critical point, particulary for strongly coupled CFTs in presence of an external magnetic field. For example, four-dimensional dyonic black holes have been proved to be relevant for a better comprehension of planar condensed matter phenomena such as the quantum Hall effect [1], the superconductivity-superfluidity [2] or the Nernst effect [3]. The study of dyonic black holes is not only interesting in four dimensions, but also in higher dimensions where their holographic applications have been discussed in the current literature. For example, it has been shown that large dyonic AdS black holes are dual to stationary solutions of a charged fluid in presence of an external magnetic field [4]. In this last reference, the AdS/CFT correspondence was used conversely and stationary solutions of the Navier-Stokes equations were constructed corresponding to an hypothetical five-dimensional AdS dyonic rotating black string with nonvanishing momentum along the string. We can also mention that magnetic/dyonic black holes present some interest from a purely gravity point of view. Indeed, there is a wide range of contexts in which magnetic/dyonic solutions are currently studied including in particular supergravity models [5, 6], Einstein-Yang-Mills theory [7] or nonlinear electrodynamics [8]. Nevertheless, in spite of partial results, the problem of finding magnetic solutions in higher dimension is an highly nontrivial problem. For example, it is easy to demonstrate that under suitable hypothesis, magnetic solutions in odd dimensions D \u2265 5 for the Einstein-Maxwell or for the Lovelock-Maxwell theories do not exist [9, 10]. This observation is in contrast with the four-dimensional situation where static dyonic configuration can be easily constructed thanks to the electromagnetic duality which rotates the electric field into the magnetic field. In the same register, one may also suspect the lack of electromagnetic duality and of the conformal invariance in dimension D > 4 to explain the difficulty for constructing the higher-dimensional extension of the Kerr-Newmann solution. The purpose of the present paper is twofold. Firstly, we would like to present a simple dyonic extension of the five-dimensional Reissner-Nordstrom solution with planar horizon. The solution will be magnetically charged by considering an electromagnetic source composed by at least three different Maxwell gauge fields. Each of these U (1) gauge fields will be sustained by one of the three different coordinates of the planar base manifold. Interestingly enough, the magnetic contribution in the metric has an asymptotically logarithmic falloff of the form ln r r 2 . Nevertheless, in spite of this slowly behavior, the thermodynamics analysis yields finite quantities even for the magnetic charge. Since we are working in five dimensions, we extend as well this dyonic solution to the case of Einstein-Gauss-Bonnet gravity. We can also mention that the causal structure of the dyonic solution can not be done analytically. Nevertheless from different simulations, one can observe that the solution has a Reissner-Nordstrom like behavior. Indeed, depending on the election of the integration constants, the solution can be a black hole with inner and outer horizons or an extremal black hole or the solution can have a naked singularity located at the origin. On the other hand, we notice that the horizon structure of the purely magnetic solution can be treated analytically. More precisely, we will show that, as for the Reissner- Nordstrom solution, the absence of naked singularity can be guaranteed for a suitable bound relation between the mass and the magnetic charge. Moreover, in this case, the location of the inner and outer horizons are expressed analytically in term of the real branches of so-called Lambert W function. This latter is defined to be the multivalued inverse of the complex function f ( \u03c9 ) = \u03c9e \u03c9 which has an infinite countable number of branches but only two of them are real-valued, see Ref. [11] for a nice review. The Lambert W functions have a wide range of applications as for example in combinatoric with the tree functions that are used in the enumeration of trees [12] or for equations with delay that have applications for biological, chemical or physical phenomena, see e. g. [13] or in the AdS/CFT correspondence as in the expression of the large-spin expansion of the energy of the Gubser-Klebanov-Polyakov string theory [14]. Just to conclude this parenthesis about the Lambert W function, we also mention that this function can be used in the case of the Schwarzschild metric as going from the Eddington-Finkelstein coordinates to the standard Schwarzschild coordinates The plan of the paper is organized as follows. In the next section, we present our toy model for dyonic solutions which consists on the five-dimensional Einstein-Gauss-Bonnet action with three different Abelian gauge fields. For this model, we derive a dyonic black hole configuration as well as its GR limit. A particular attention will be devoted to the purely magnetic GR solution for which a bound relation between the mass and the magnetic charge ensures the existence of an event horizon covering the naked singularity. In this case, the inner and outer horizons are expressed in term of the two real branches of the Lambert W functions. We will establish that this mass bound is essentially due to the fact that the magnetic charge has a slower falloff of logarithmic order to zero than the mass term in the metric function. Starting from this observation, we will present in Sec. 3 various examples of black holes sharing this same feature with electric, axionic or magnetic charges and with AdS and Lifshitz asymptotics. In Sec. 4, we extend the previous solutions to general dyonic configurations with axionic charges. Finally, the last section is devoted to our conclusion and an appendix is provided where some useful properties of the Lambert W functions are given.", "pages": [1, 2, 3]}, {"title": "2 Five-dimensional dyonic black hole solution", "content": "In Refs. [9, 10], it has been proved that, under suitable hypothesis, magnetic black hole solutions for Einstein-Maxwell action in odd dimensions D \u2265 5 can not exit. As we will show below, a simple way of circumventing this obstruction is to consider more than one Maxwell gauge field. The fact of considering various Abelian fields in order to construct dyonic black holes in five dimensions have already been considered, see Refs. [15] and [16]. More precisely, we will establish that the Einstein gravity eventually supplemented by the Gauss-Bonnet term since we are working in D = 5 can admit dyonic black hole solutions for an electromagnetic source given at least by three different Maxwell invariants. In order to achieve this task, we consider the following action where the F ( I ) \u00b5\u03bd 's are the three different Maxwell field strengths associated to the U (1) gauge fields A I for I = { 1 , 2 , 3 } and \u03b1 represents the Gauss-Bonnet coupling constant. The field equations obtained by varying this action read where the variation of the Gauss-Bonnet term is given by In one hand, it is known that the field equations (2.2) with one Maxwell invariant I = 1 admits electrically charged black holes [17] generalizing the solution of Boulware-Deser [18]. On the other hand, it is simple to prove that the magnetic extension of the Boulware-Deser solution can not exist [9, 10]. Nevertheless, as shown below, the presence of two extra Maxwell invariants renders possible the magnetic extension of the Boulware-Deser solution but only in the case of flat horizon. In fact a dyonic solution with flat horizon of the field equations (2.2) is found to be where M , Q e and Q m are three integration constants corresponding respectively to the mass, the electric and the magnetic charge and | \u03a3 3 | is the finite volume of the compact 3 -dimensional flat base manifold. Various comments can be made concerning this dyonic solution. Firstly, in the absence of the magnetic charge Q m = 0, the solution reduces to the electrically extension of the Boulware-Deser solution [17] even if there are three different Maxwell invariants. This is because each of these three invariants contributes in the same footing for the full solution, and hence one could have switch off two of them from the very beginning. The GR limit \u03b1 \u2192 0 of the solution concerns only the upper branch of the solution and yields to the metric function given by while the Abelian gauge fields remain identical. Computing the Kretschmann invariant, one notices that the dyonic solution in the Einstein-Gauss-Bonnet theory or its GR limit has a singularity located at the origin. The causal structure of the dyonic solution is quite involved and can not be treated analytically as in the case of the four-dimensional Reissner-Nordstrom dyonic solution. Nevertheless, it is quite simple to see that the GR solution (2.4) with \u039b < 0 and without magnetic charge, has a Reissner-Nordstrom like behavior in the sense that for M\u2265 3 5 3 |Q e | 4 3 ( -\u039b) 1 3 / 4 4 3 | \u03a3 3 | 1 3 , the solution describes a (extremal) black hole while the case M < 3 5 3 |Q e | 4 3 ( -\u039b) 1 3 / 4 4 3 | \u03a3 3 | 1 3 will yield a naked singularity. The dyonic GR solution has also a similar behavior which can be appreciated only by means of some simulations reported in the graphics below. In the next subsection, we will see that in the purely magnetic case, the causal structure of the solution can be analyzed analytically. To conclude this section, we mention that the GR dyonic solution (2.4) satisfies the dominant energy conditions. Indeed, it is simple to see that the energy density \u00b5 , the radial pressure p r and the tangential pressure p t given by verify the dominant energy conditions", "pages": [3, 4, 5]}, {"title": "2.1 Purely magnetic GR solution", "content": "For \u03b1 \u2192 0 and Q e = 0, the purely magnetic GR solution (2.4) becomes In order to study the variations of the metric function F , it is useful to define For negative cosmological constant \u039b < 0, we have lim x \u2192\u221e h ( x ) = lim x \u2192 0 + h ( x ) = \u221e and the function h has a global minimum at x = 3 Q 2 m -2\u039b | \u03a3 3 | 2 . The equation for the zeros of the function h that will give as well the location of the horizons for the metric function F through (2.8) is of the form (6.2). Hence, its corresponding discriminant as defined in Eq. (6.3) is given by Since we are considering the negative cosmological constant case \u039b < 0, the discriminant is negative, and as mentioned in the appendix, the equation h ( x ) = 0 will have two real roots only if \u2206 \u2208 ] -1 e , 0[. This condition in turn requires that the mass M must satisfy the following bound relation For M satisfying such bound, the metric function F has an inner (Cauchy) horizon r -and an outer (event) horizon r + whose locations are expressed in term of the two real branches of the Lambert W functions, W 0 and W -1 as with \u2206 given by (2.9). In contrast with the four-dimensional magnetic Reissner-Nordstrom solution (or even the dyonic configuration), the bound (2.10) does not restrict the mass M to be positive. In fact, for Q 2 m \u2265 -2\u039b | \u03a3 3 | 2 e/ 3, the bound M 0 \u2264 0, and hence the singularity at the origin can still be covered by an horizon even for a solution with a negative mass. On the other hand, for M saturating the bound (2.10), namely M = M 0 or equivalently \u2206 = -1 e , one ends up with an extremal black hole with r + = r -. Finally, for M < M 0 , the solution will have a naked singularity. To be complete, we also mention that the energy density, the radial and tangential pressure of the purely magnetic GR solution are given by (2.5) with Q e = 0, and hence the magnetic solution satisfies as well the dominant energy conditions (2.6).", "pages": [5, 6, 7]}, {"title": "3 Other examples of black holes with Lambert W function horizons", "content": "In the previous section, we have shown that purely magnetic black holes of five-dimensional Einstein gravity with 3 different Abelian gauge fields exist provided a certain bound relation between the mass and the magnetic charge. In addition, the location of the horizons can be expressed thanks to the real branches of the Lambert W functions. In this section, we will present few examples enjoying these same features (bound for the mass and horizons expressed in term of the Lambert W functions) with different Noetherian charges (electric, dyonic, magnetic or axionic) and different asymptotics (AdS or Lifshitz). In order to achieve this task, it is clear from the previous analysis that the Noetherian charges in the metric must have a slower falloff of logarithmic order in comparison to the mass term. In what follows, we will present four different such solutions: an AdS dyonic black hole in five dimensions, an AdS electrically charged solution in odd dimension and two Lifshitz black holes with a magnetic and axionic charge in arbitrary dimension. These configurations are particular solutions of the following general D -dimensional action In this action, we leave open the possibility of having a nonlinear Maxwell term ( F \u00b5\u03bd F \u00b5\u03bd ) q where F \u00b5\u03bd = \u2202 \u00b5 A \u03bd -\u2202 \u03bd A \u00b5 . Such nonlinearity has been shown to be fruitful to obtain charged solutions in different gravity contexts, see e. g. [19]. As in the previous example, in order to sustain a magnetic charge, we will also add some extra Abelian gauge fields F ( I ) \u00b5\u03bd = \u2202 \u00b5 A ( I ) \u03bd -\u2202 \u03bd A ( I ) \u00b5 for I = { 1 , 2 , \u00b7 \u00b7 \u00b7 , n } . We justify the presence of axionic fields \u03c8 j from the fact that we are looking for solutions with planar base manifold, and as shown in the fourth example or in the next section, the axionic fields perfectly accommodate an ansatz of the form \u03c8 j = \u03bbx j where the x j are the planar coordinates of the base manifold (3.2). This particular ansatz for the axionic fields also provides a simple mechanism of momentum dissipation [20], and in this case the holographic DC conductivities (electrical, thermoelectric and thermal) can be expressed in terms of the black hole horizon data [21, 22]. For examples, the DC conductivities of dyonic black holes with axionic fields have been computed recently in Refs. [23]. Finally, we note that the model considered here (3.1) also allows a possible coupling of the electromagnetic fields A , A and the axionic fields \u03c8 j to a dilaton field \u03c6 . The field equations associated to the action (3.1) read and we look for a static ansatz with a planar base manifold of the form In what follows, we will derive four classes of solutions of the previous field equations, and their analysis will only be considered in the case of a negative cosmological constant \u039b < 0.", "pages": [7, 8]}, {"title": "3.1 Electrically charged AdS black holes for nonlinear Maxwell theory in odd dimension", "content": "This case will correspond of setting \u03c6 = A I = \u03c8 j = 0 in (3.1) and the Maxwell nonlinearity q is of the form q = D -1 2 . As shown in Ref. [24], there exists a purely electric solution with logarithmic falloff, and this solution, in order to be real, must be restricted to odd dimension D = 2 k +1 with k \u2265 1. Hence the Maxwell nonlinearity is q = k and the metric function and the electric potential are given by Here M is the mass, Q e is the electric charge and | \u03a3 2 k -1 | denotes the finite volume element of the compact (2 k -1) -dimensional base manifold. As for the previous magnetic solution, the equation determining the zeros of the metric function F can be put in the form (6.2) by substituting x = r 2 k and in this case, the discriminant (6.3) is given by We note that, because of the presence of the term ( -2) k -2 , the sign of the discriminant will depend on the parity of the integer k . Indeed, for even k or equivalently for odd dimensions D = 5 mod 4, the discriminant is positive, and hence the solution is a black hole for any value of the mass M , and there is a single horizon located at Nevertheless, in this case, it is simple to see that the energy density is always negative, and consequently the energy conditions do not hold. On the other hand, for odd k or equivalently for odd dimensions D = 3 mod 4, the solution will be a black hole provided that the mass satisfies the following bound relation with the electric charge In this case, the inner and outer horizons are given by and the dominant energy conditions (2.6) are satisfied with", "pages": [8, 9]}, {"title": "3.2 Five-dimensional AdS dyonic black holes and particular stealth configuration", "content": "In five dimensions, the previous solution can be magnetically charged in such a way that the magnetic charge also appears in the metric function with a slower falloff of logarithmic order in comparison to the mass. The corresponding model that sustains such solution is given by the action (3.1) by setting \u03c6 = \u03c8 j = 0 and by considering n = 3 extra gauge fields A I as well as the nonlinear Maxwell term with the exponent q = 2. In this case, the dyonic solution which can also be viewed as an electric extension of the solution (2.7) is given by the ansatz (3.2) with Before proceeding as before, we would like to point out that the point defined by is very special in the sense that the metric function reduces to the Schwarzschild AdS metric with a flat horizon. This in turn implies that the field equations at the point (3.8) can be interpreted as a stealth configuration [25] defined on the Schwarzschild AdS background since both side (geometric and matter part) of the Einstein equations vanish separately, i. e. Note that such stealth configuration but for a dyonic four-dimensional Reissner-Nordstrom black hole was known in the case of an Abelian gauge field coupled to a particular Horndeski term [26] or for a generalized Proca field theory [27]. Outside the stealth point, the discriminant associated to the zeros of the metric function F is given by Since, we are only considering the negative cosmological constant case, we conclude that:", "pages": [10]}, {"title": "(i) For A < 0, that is for", "content": "the solution has a single horizon located at but the solution does not satisfy the dominant energy conditions neither the weak energy conditions since the energy density \u00b5 = 3 A 2 r 4 is always negative.", "pages": [11]}, {"title": "(ii) For A > 0, that is for", "content": "the solution represents a dyonic AdS black hole only if and in this case, the solution is shown to satisfy the dominant energy conditions (2.6). the solution represents a black hole stealth dyonic configuration on the Schwarzschild AdS background where the horizon is located at", "pages": [11]}, {"title": "3.3 Purely magnetic Lifshitz black hole with dynamical exponent z = D -4", "content": "We now turn to derive examples with a different asymptotic behavior characterized by an anisotropy scale between the time and the space, the so-called Lifshitz asymptotic. This anisotropy is reflected by a dynamical exponent denoted usually by z and defined such that the case z = 1 corresponds to the AdS isotropic case. Note that Lifshitz black holes have been insensitively studied during the last decade, see for examples Refs. [28] In order to obtain Lifshitz black holes, we consider the action (3.1) without axionic fields \u03c8 j = 0 and with the standard Maxwell term q = 1. Note that the presence of the Maxwell term is mandatory to ensure the Lifshitz asymptotic of the solution. In even (resp. odd) dimension, the solution will be sustained by n = 1 (resp. n = 3) extra gauge field(s) A I . In both case, a purely magnetic Lifshitz black hole with dynamical exponent z = D -4 is found through the ansatz (3.2) with for odd dimension with I = { 1 , 2 , 3 } . Here | \u03a3 D -2 | denotes the finite volume element of the compact ( D -2) -dimensional base manifold and glyph[epsilon1] IJK is defined as In this case, the coupling constants of the problem must take the following form Proceeding as in the two previous examples, one notes that the discriminant is always negative, and hence the existence of horizons is again ensured provided that the mass satisfies the following bound For this lifshitz solution with dynamical exponent z = D -4, the energy density and the radial/tangential pressures are given by and since one can notice that the dominant energy conditions (2.6) are satisfied outside the event horizon, that is for F ( r ) \u2265 0.", "pages": [11, 12, 13]}, {"title": "3.4 Axionic Lifshitz black hole with dynamical exponent z = D -2", "content": "We now consider the action (3.1) with a source only given by the axionic fields \u03c8 j and with the standard Maxwell term q = 1 in order to sustain the Lifshitz asymptotic. In this case, an axionic Lifshitz black hole solution with dynamical exponent z = D -2 is found to be where now Q a denotes the axionic charge. The coupling constants must be chosen such as As for the previous case, the discriminant is negative and the mass parameter must satisfy the following bound with respect to the axionic charge in order to avoid naked singularity As in the previous Lifshitz case, the dominant energy conditions (2.6) are satisfied outside the event horizon with We now compute the DC conductivity \u03c3 DC of this solution which can be expressed in term of the black hole horizon data [21, 22] thanks to the presence of the axionic fields homogenously distributed along the coordinates of the planar base. In order to achieve this task, we will follow the prescriptions as given in these last references by first turning on the following relevant perturbations 1 where E is a constant. The perturbed Maxwell current given by J = \u221a -ge \u03bb\u03c6 F rx 1 is a conserved quantity along the radial coordinate. A straightforward computation along the same lines as those in [21, 22] yields a DC conductivity \u03c3 DC given by As it should be expected in the absence of the axionic charge, the expression of the DC conductivity \u03c3 DC will blows up.", "pages": [13, 14]}, {"title": "4 More general dyonic-axionic solutions in arbitrary dimension", "content": "The solutions derived previously can accommodate extra Noetherian charges but in this case the location of the horizons is more involved and can not be treated as before with the help of the Lambert W functions. Nevertheless, for completeness, we report in this section more general solutions, each of them having a dyonic and an axionic charge. In order to achieve this task, we consider a slightly different action than the one defined by Eq. (3.1). Indeed, we will add an extra Maxwell term without any nonlinearity q = 1 since it is known that electrically charged Lifshitz black holes require the introduction of at least two Maxwell terms [29]. We then consider the following D -dimensional action with F ( i ) \u00b5\u03bd = \u2202 \u00b5 A ( i ) \u03bd -\u2202 \u03bd A ( i ) \u00b5 for i = { 1 , 2 } and n extra gauge fields that will sustain the magnetic charge, F ( I ) \u00b5\u03bd = \u2202 \u00b5 A ( I ) \u03bd -\u2202 \u03bd A ( I ) \u00b5 for I = 1 , \u00b7 \u00b7 \u00b7 n with n = 1 in even dimension and n = 3 in odd dimension. The field equations read A general Lifshitz dyonic-axionic solution with arbitrary dynamical exponent z of the field equations is given by together with provided that the coupling constants are tied as follows glyph[negationslash] Note that in the AdS limit z = 1, the dilaton field \u03c6 disappears as well as the Maxwell potential A (1) t which is precisely responsible to sustain the Lifshitz asymptotic z = 1. The thermodynamical variables of the solution can be computed using the Hamiltonian formalism [30] yielding with the electric, magnetic and axionic potentials and charges It is a simple exercise to check the consistency of the first law where the temperature is given by It is clear that from the expression of the metric function, the cases z = D -4 and z = D -2 must be treated separately. In fact, for z = D -4, one yields a metric function involving a logarithmic magnetic contribution and the remaining fields are given by (4.3) with z = D -4. The thermodynamical quantities computed by means of the Euclidean method [30] read Finally, the solution with z = D -2 is given by with (4.3), and the thermodynamic parameters associated to the solution are In both cases, that is for z = D -4 and z = D -2, it is easy to see that the first law (4.6) holds.", "pages": [14, 15, 16, 17]}, {"title": "5 Conclusion", "content": "Here, we have presented a dyonic extension of the five-dimensional Boulware-Deser solution for the Einstein-Gauss-Bonnet theory. The emergence of a magnetic charge is shown to be possible for a flat horizon and by considering at least three different Maxwell invariants. The magnetic contribution in the metric function has a logarithmic falloff but still yields to finite physical quantities. As usual, one of the two branches has a well-defined GR-limit with a magnetic logarithmic falloff term. For suitable bounds between the mass and the magnetic charge, the purely magnetic GR solution can be shown to admit an inner and outer horizons. These latter are given in terms of the two real branches of the Lambert W functions. We have noticed that this bound's mass was due to the fact that the magnetic charge in the metric has a slower falloff of logarithmic order than the mass. Exploiting this observation, we have derived other examples of solutions sharing these same properties for different models and different asymptotics. For example, we have obtained an electrically charged AdS black hole solution in odd dimension for a nonlinear Maxwell theory with a single horizon in dimensions D = 5 mod4 and with two horizons in D = 3 mod4. Interestingly enough, a dyonic configuration with logarithmic falloff of the electric and magnetic charges was also derived in five dimensions. Depending on the strength of the electric charge with respect to the magnetic charge, the solution can have one or two horizons, and in this latter case, the mass must satisfy a certain bound. Moreover, for a precise relation between the electric and the magnetic charges, the solution turns out to be a stealth dyonic configuration defined on the Schwarzschild AdS background. For the asymptotic AdS solutions, we have remarked that our black hole solutions presenting an inner and outer horizons always satisfy the dominant energy conditions (2.6) while these conditions even in their weak version do not hold for our solutions with a single horizon. This can be explained by the fact that the metric functions in our set-up were of the following form where N represent the additional Noetherian charge with slower falloff of logarithmic order than the mass M . The corresponding discriminant associated to the zeros of the metric function F (6.3) is given by Now, since we are considering the AdS case, it is clear that for N < 0, the discriminant will be positive and, hence the solution will represent a black hole with a single horizon for any value of the mass M . On the other hand, for N > 0, one has \u2206 < 0 and consequently the solution will be a black hole provided that the mass satisfies the following bound and in this case, the solution presents two horizons. On the other hand, the energy density, the radial and tangential pressures are given generically by and hence it is evident that the dominant energy conditions (2.6) will only be satisfied for the solutions with N > 0. It seems to be physically acceptable that solutions without any restrictions on the mass do not satisfy the dominant or the weak or even the null energy conditions. On the other hand, our examples of black holes with a bound's mass verify the dominant energy conditions. It will be interesting to explore more deeply this relation between the lack of restriction on the mass with the absence of energy conditions. We also mention that a necessary condition to obtain AdS black holes with an Ansatz of the form with a metric function given by (5.1) is that the energy momentum tensor of the matter source T \u00b5\u03bd satisfies T t t + ( D -2) T i i = 0 without summation for the planar indices i . Indeed, in this case, the consistency of the Einstein equations T t t + ( D -2) T i i + ( D -1)\u039b = 0 yields to a nonhomogeneous Euler's differential equation of second-order whose characteristic polynomial has a double root given by r -D +3 and hence the general solution of this Euler's equation is given by Eq. (5.1). We have also presented two other examples with Lifshitz asymptotics with fixed values of the dynamical exponent with a magnetic charge and an axionic charge. The emergence of such asymptotic solutions is essentially due to the presence of dilatonic fields. Note that there also exist Lifshitz black holes with a logarithmic falloff in the case of higher-order gravity [31]. Finally, for completeness, we have extended the previous solutions to accommodate a dyonic as well an axionic charge in arbitrary dimension.", "pages": [17, 18, 19]}, {"title": "6 Appendix: The Lambert W functions", "content": "The Lambert W functions are a set of functions that represent the countably infinite number of solutions denoted by W k ( z ) of the equation for a given z \u2208 C . There are only two real-valued branches of the Lambert W functions that are denoted, by convention, W 0 and W -1 with W 0 : [ -1 e , \u221e [ \u2192 [ -1 , \u221e [ and W -1 : [ -1 e , 0[ \u2192 ] - \u221e , -1[ with the convention that W 0 ( -1 e ) = W -1 ( -1 e ) = -1. The Lambert W functions appear for the resolution of the equations of the form glyph[negationslash] glyph[negationslash] Indeed, by defining w = ln( x ), the equation (6.2) becomes ae w + bw + c = 0, which is equivalent after some basic algebraic manipulations to (6.1) with W = -w -c b and z = \u2206, where the discriminant is defined by It is then clear that (ii) If \u2206 \u2208 ] -1 e , 0[, the equation (6.2) has two real solutions given by (iii) Finally, if \u2206 < -1 e , the equation (6.2) does not admit real roots. Acknowledgments: MB is supported by grant Conicyt/ Programa Fondecyt de Iniciaci'on en Investigaci'on No. 11170037.", "pages": [19, 20]}] |
2018IJMPD..2750174B | https://arxiv.org/pdf/1612.04414.pdf | <document>
<section_header_level_1><location><page_1><loc_18><loc_83><loc_83><loc_88></location>Einstein's quadrupole formula from the kinetic-conformal Hoˇrava theory</section_header_level_1>
<text><location><page_1><loc_30><loc_76><loc_71><loc_78></location>Jorge Bellor'ın a, 1 and Alvaro Restuccia a,b, 2</text>
<unordered_list>
<list_item><location><page_1><loc_16><loc_72><loc_85><loc_74></location>a Department of Physics, Universidad de Antofagasta, 1240000 Antofagasta, Chile.</list_item>
<list_item><location><page_1><loc_16><loc_71><loc_84><loc_72></location>b Department of Physics, Universidad Sim'on Bol'ıvar, 1080-A Caracas, Venezuela.</list_item>
</unordered_list>
<text><location><page_1><loc_32><loc_68><loc_33><loc_69></location>1</text>
<text><location><page_1><loc_33><loc_67><loc_69><loc_68></location>[email protected], [email protected]</text>
<text><location><page_1><loc_55><loc_68><loc_56><loc_69></location>2</text>
<section_header_level_1><location><page_1><loc_46><loc_60><loc_55><loc_61></location>Abstract</section_header_level_1>
<text><location><page_1><loc_21><loc_27><loc_80><loc_58></location>We analyze the radiative and nonradiative linearized variables in a gravity theory within the familiy of the nonprojectable Hoˇrava theories, the Hoˇrava theory at the kinetic-conformal point. There is no extra mode in this formulation, the theory shares the same number of degrees of freedom with general relativity. The large-distance effective action, which is the one we consider, can be given in a generally-covariant form under asymptotically flat boundary conditions, the Einstein-aether theory under the condition of hypersurface orthogonality on the aether vector. In the linearized theory we find that only the transverse-traceless tensorial modes obey a sourced wave equation, as in general relativity. The rest of variables are nonradiative. The result is gauge-independent at the level of the linearized theory. For the case of a weak source, we find that the leading mode in the far zone is exactly Einstein's quadrupole formula of general relativity, if some coupling constants are properly identified. There are no monopoles nor dipoles in this formulation, in distinction to the nonprojectable Horava theory outside the kinetic-conformal point. We also discuss some constraints on the theory arising from the observational bounds on Lorentz-violating theories.</text>
<section_header_level_1><location><page_2><loc_16><loc_86><loc_38><loc_88></location>1 Introduction</section_header_level_1>
<text><location><page_2><loc_16><loc_63><loc_85><loc_84></location>Gravitational waves have recently been detected [1, 2, 3]. The detected signals fit well with the waves produced by the coalescense of binary systems of black holes, according to the predictions of General Relativity (GR). This detection constitutes one of the most important recent achievements in the study of gravitational phenomena, and it is another success of GR. On the other hand, there are motivations to study alternatives or modifications to GR. One important issue is that GR is not renormalizable under the perturbative approach, thus, at least in the perturbative scheme, it cannot be a fundamental theory by itself. There is also the issue of the dark matter, for which there has not been found any candidate in the particle experiments or space observations. Therefore, for any proposed modification of GR a question comes out inmediatly: how close to the detected wave signals and the corresponding predictions of GR is the radiation predicted by the new theory?.</text>
<text><location><page_2><loc_16><loc_17><loc_85><loc_62></location>Here we focus on the study of the production and propagation of gravitational waves at the leading order in a Lorentz-violating theory. The theory [4] belongs to the family of the nonprojectable Hoˇrava theories [5, 6]. The heart of the Hoˇrava proposal [5] is to introduce a preferred timelike direction that breaks the symmetry between space and time characteristic of relativistic theories. This is done with the aim of introducing higher order spatial derivatives in the Lagrangian that improve the renormalizability of the theory while, in principle, preserve its unitarity. The special formulation studied in Ref. [4], where only the purely gravitational theory without coupling to matter sources was analyzed, consists of setting a specific value of the kinetic coupling constant for which two additional second-class constraints emerge. The constant is usually denoted by λ and the special value in 3 + 1 dimensions is λ = 1 / 3. The additional constraints at λ = 1 / 3 eliminate the extra scalar mode that otherwise the nonprojectable Hoˇrava theory exhibits. Because of this, it is reasonable to expect that this formulation tends to stay more close to GR, at least in the low-energy regime, where the lowest order operators are the most relevant ones 1 . Since the special value λ = 1 / 3 is related to a conformal symmetry on the kinetic term of the Lagrangian [5], in Ref. [9] we called this formulation the Hoˇrava theory at the kinetic-conformal point (the KCP Hoˇrava theory, for short). We stress that the theory is not conformally invariant, only its kinetic term is. In [9] the power-counting renormalizability as well as the absence of ghosts in the theory were shown. A recent report on the status of the Hoˇrava theory, dealing with its several versions, can be found in Ref. [10]. We comment that the value λ = 1 / 3 leading to the kinetic-conformal formulation is fixed by the dynamics, it does not get quantum corrections. This is due to the second-class constraints of the theory, not to symmetries. Further discussion can be found in Ref. [11].</text>
<text><location><page_2><loc_16><loc_14><loc_85><loc_17></location>We study the gravitational waves at leading order in the large-distance effective action of the theory. This effective action is of second order in time and spatial</text>
<text><location><page_3><loc_16><loc_79><loc_85><loc_88></location>derivatives. It admits a generally-covariant version which is the Einstein-aether theory [12] under the restriction of hypersurface orthogonality on the aether vector [13, 14]. We deal with the generally-covariant formulation since it allows a more direct comparison with the standard approaches of GR devoted to gravitational waves.</text>
<text><location><page_3><loc_16><loc_59><loc_85><loc_79></location>We analyse the perturbatively linearized generally-covariant theory coupled to a generic weak matter source. We develop all the analysis in terms of gauge-invariant variables of the linearized theory. These are combinations of the metric and the aether field that remain invariant under linearized general diffeomorphisms. This formulation allows us to get totally gauge-invariant results. Once we determine what variables are related to the sources by Poisson equations, hence they are nonradiative, and what variables are governed by the wave equation, we study the generation of the waves at the leading order. We follow the standard procedure of approximating the solution by considering that it is produced by a source that at the leading order has negligible self-gravity, it is in a slow motion regime, and that the observation is made far enough from the source (at the wave zone).</text>
<text><location><page_3><loc_16><loc_39><loc_85><loc_58></location>Our study is close to Refs. [15, 16, 17]. In Ref. [15] a perturbative analysis of the unrestricted Einstein-aether theory without matter sources was done. The linearized vacuum equations of motion of the several modes, which are homogeneous wave equations for each mode with different speeds, were studied there. In Ref. [16] the lowest order multipole moments were studied for the unrestricted Eintein-aether theory coupled to a weak source. Our analysis differs from these two studies due to (besides the absence of sources in [15]) the lower number of propagating degrees of freedom we have and the fact that the equations of motion of the nonprojectable Hoˇrava theory are not equivalent to the ones obtained by substituting the hypersurface orthogonality condition in the equations of motion of the Einstein-aether theory [14].</text>
<text><location><page_3><loc_73><loc_32><loc_73><loc_35></location>/negationslash</text>
<text><location><page_3><loc_16><loc_10><loc_85><loc_38></location>In Ref. [17] the Einstein-aether theory with the condition of hypersurface orthogonality imposed at the level of the action was studied (this theory is also called the khronometric theory). The analysis is rigorously consistent for λ = 1 / 3. In [17] the wave equations with sources were found for the tensorial modes and the extra mode, as well as the Poissonian equations for the nonradiative modes. They also found the dominant modes in the multipolar expansion for a weak source. These results are affected by the presence of the extra mode. Our study differs from the one of Ref. [17] because we take the KCP theory independently with its intrinsic degrees of freedom. As we shall see, this has important consequences on the radiation formulas. In general, the KCP formulation cannot be obtained rigurously as a limit of the theory with the extra mode due to the discontinuity in the number of constraints and, in particular, in the number of propagating modes. Our approach is consistent in the λ = 1 / 3 case since we obtain the formulas of the radiation directly from the KCP theory. However, if one wants a quick comparison with the non-kinetic-conformal case, heuristically our formulas coincide with the radiation formulas of Ref. [17] in the limiting case of sending to infinity the speed of the extra</text>
<text><location><page_4><loc_16><loc_83><loc_85><loc_88></location>mode (this divergence is induced by the λ = 1 / 3 value). However, in general the reinterpretation of an hyperbolic equation (the wave equation of the extra mode) as an elliptic equation is not consistent (for example, the initial data problem).</text>
<text><location><page_4><loc_18><loc_74><loc_18><loc_77></location>/negationslash</text>
<text><location><page_4><loc_16><loc_53><loc_85><loc_82></location>In addition to the study of the gravitational radiation, here we consider some observational implications on the kinetic-conformal theory. Our aim is to highlight that some of the observational bounds applicable to the Hoˇrava theory with λ = 1 / 3 must be addressed in a different way in the kinetic-conformal case. In Hoˇrava theory observational bounds are frequently combined with theoretical restrictions needed for the consistency of the extra mode. In the kinetic-conformal theory this is not necessary since there is no extra mode. Another important issue is the cosmological-scale solutions. We have argued that they may arise in a different way in the kinetic-conformal case [11]. We further comment on this point below. In particular, here we compare with the observational bounds coming from binary pulsars found in Ref. [18, 19], since these phenomena are related to wave production. Those authors studied the Einstein-aether theory both unrestricted and with the hypersurface orthogonality condition (with λ = 1 / 3). They found stringent constraints on the space of coupling constants of these theories. Here we show how the kinetic-conformal theory stays more close to GR, in particular there is no dipolar contribution at the level of the dominant modes.</text>
<text><location><page_4><loc_67><loc_58><loc_67><loc_60></location>/negationslash</text>
<text><location><page_4><loc_16><loc_28><loc_85><loc_53></location>This paper is organized as follows: in section 2 we summarize the analysis of Ref. [14] to present the Einstein-aether theory under the restriction of hypersurface orthogonality, together with its equivalence to the second-order action of the nonprojectable Hoˇrava theory. In section 3.1 we discuss the gauge invariants of the generally-covariant theory that can be formed by combining the metric variables with the hypersurface-orthogonal aether field. In section 3.2 we present and analyze the linearized field equations, coupled to a matter source, in terms of these gauge invariants. In section 3.3 we study the leading mode for the production of waves far from the source, obtaining the quadrupole formula of Einstein. In section 4 we discuss some observational bounds. Finally we present some conclusions. Since it is also interesting to analyze the linearized field equations in the FDiff-covariant language, which is the original formulation of the Hoˇrava theory [5], we add one appendix to present the FDiff-gauge invariants and the field equations in terms of them.</text>
<section_header_level_1><location><page_4><loc_16><loc_23><loc_81><loc_25></location>2 The covariant version of the Hoˇrava theory</section_header_level_1>
<text><location><page_4><loc_16><loc_10><loc_85><loc_21></location>The Einstein-aether theory [12] is a modification of GR that incorporates an everywhere timelike unit vector field, called the aether, as a fundamental field. Since the aether is considered dynamical, the action possesses the symmetry of general diffeomorphisms that is also present in GR. However, at the level of the solutions, the presence of the aether field breaks the local Lorentz symmetry. There is a relationship [13, 14] between the Einstein-aether theory and the action of second</text>
<text><location><page_5><loc_16><loc_79><loc_85><loc_88></location>order in derivatives of the nonprojectable Hoˇrava theory, which is also a theory with a preferred frame. Throghout this paper we deal only with the second-order action (excluding the cosmological constant), since at low precision the physics of the gravitational waves can be described by it. In the following we summarize the relationship between these two theories.</text>
<text><location><page_5><loc_16><loc_64><loc_86><loc_79></location>The Hoˇrava theory [5] was originally formulated in terms of the standard ArnowittDeser-Misner (ADM) variables N , N i and g ij , in such a way that the action possesses the symmetry of the diffeomorphisms that preserve a given foliation (FDiff) along a timelike direction. The Lagrangian of the nonprojectable theory, which is the version we study here, depends on the spatial curvature and the spatial derivatives of the lapse function N . These arise in the Lagrangian in terms of the FDiff-covariant vector a i = ∂ i ln N [6]. The action of second order in derivatives, which we call the z = 1 action, is</text>
<formula><location><page_5><loc_26><loc_59><loc_85><loc_63></location>S = 1 2 κ H ∫ dtd 3 x √ (3) gN ( K ij K ij -λK 2 + β (3) R + αa i a i ) , (2.1)</formula>
<formula><location><page_5><loc_40><loc_53><loc_85><loc_58></location>K ij ≡ 1 2 N ( ˙ g ij -2 ∇ ( i N j ) ) (2.2)</formula>
<text><location><page_5><loc_16><loc_57><loc_21><loc_59></location>where</text>
<text><location><page_5><loc_16><loc_49><loc_85><loc_54></location>is the extrinsic curvature of the spacelike leaves Σ of the foliation. The dot denotes the time derivative, ˙ g ij = ∂g ij /∂t . K ≡ g ij K ij , (3) R is the scalar curvature of Σ, and κ H , λ , β and α are coupling constants.</text>
<text><location><page_5><loc_16><loc_31><loc_85><loc_49></location>In the above all the coupling constants are in principle arbitrary. Now, in the purely gravitational theory it is known that a scalar degree of freedom, additional to the transverse-traceless tensorial modes, is eliminated from the phase space if the coupling constant λ is set to the value λ = 1 / 3 [4, 9]. With this value of λ the kinetic term in (2.1) acquires a conformal invariance [5], although the full theory is not conformal since in general the terms in the potential break the conformal symmetry (except for very specific terms). For this reason the value λ = 1 / 3 was called the kinetic-conformal point in Ref. [9]. As we have mentioned, this feature raises interest in studying this special formulation of the nonprojectable Hoˇrava theory, as it is our case in this paper, since it becomes closer to GR.</text>
<text><location><page_5><loc_16><loc_16><loc_85><loc_30></location>The Einstein-aether theory [12] is physically equivalent to the z = 1 nonprojectable Hoˇrava theory (2.1) (for all λ ) if the aether vector is restricted to be hypersurface orthogonal. The total equivalence between the two theories, at the level of their Lagrangians, holds only if the restriction on the aether vector is imposed at the level of the action, i. e. before deriving the equations of motion [13, 14]. Here we take the Einstein-aether action from Ref. [14], considering also the coupling to matter sources. The full generally-covariant action is given by S Total = S EA + S Matter , where</text>
<text><location><page_5><loc_16><loc_8><loc_85><loc_12></location>is the Einstein-aether action. u µ is the aether vector, which in general is subject to the condition of being a timelike unit vector, u µ u µ = -1. κ EA is the Einstein-aether</text>
<formula><location><page_5><loc_27><loc_12><loc_85><loc_16></location>S EA [ g µν , u µ ] = 1 2 κ EA ∫ d 4 x √ -g ( R -M αβγδ ∇ α u γ ∇ β u δ ) (2.3)</formula>
<text><location><page_6><loc_16><loc_86><loc_57><loc_88></location>gravitational constant. M αβγδ is the hypermatrix</text>
<formula><location><page_6><loc_28><loc_83><loc_85><loc_85></location>M αβγδ = c 1 g αβ g γδ + c 2 g αγ g βδ + c 3 g αδ g βγ + c 4 u α u β g γδ , (2.4)</formula>
<text><location><page_6><loc_16><loc_71><loc_85><loc_82></location>where c 1 , c 2 , c 3 and c 4 are coupling constants. With the aim of minimizing Lorentzbreaking effects in the matter sector, where experimental bounds are highly restrictive, it is required that the matter sources do not couple to the aether field (see discussion in Refs. [20, 16]). Then, S Matter [ g µν , ψ ] is the action for the matter sector, with ψ representing the matter sources in a generic way. As a consequence, the equations of motion of the sources maintain the same structures they have in GR.</text>
<text><location><page_6><loc_16><loc_64><loc_85><loc_71></location>The restriction of hypersurface orthogonality on u µ is equivalent (locally) to express u µ in terms of a scalar function T = T ( t, /vectorx ) that satisfies the condition of its gradient is timelike, ∂ α T∂ α T < 0. The hypersurface-orthogonal aether vector is written in terms of T as</text>
<text><location><page_6><loc_16><loc_51><loc_85><loc_60></location>Under this restriction the functional degrees of freedom originally contained in u µ are reduced to the one of T once (2.5) has been substituted in the action (2.3). Actually, definition (2.5), which automatically implies that u µ is a timelike unit vector, depends on the norm of the gradient of T , hence the hypersurface-orthogonal u µ is a composite object made with the T field and the metric g αβ .</text>
<formula><location><page_6><loc_43><loc_59><loc_85><loc_64></location>u µ = ∂ µ T √ -∂ α T∂ α T . (2.5)</formula>
<text><location><page_6><loc_16><loc_38><loc_85><loc_51></location>The equation of motion of the T field, that is, the equation of motion derived from (2.3) by taking variations with respect to T , is implied by the Einstein equations and the matter equations of motion [14]. Since this fact is crucial for our study, let us repeat the argument that supports it. The main point is that T is a single, nonzero-gradient, scalar field coupled to gravity in a generally-covariant way. Since S Total is invariant under general diffeomorphisms, we have that, under a diffeomorphism parameterized by ζ µ ,</text>
<formula><location><page_6><loc_28><loc_33><loc_85><loc_37></location>0 = ∫ d 4 x ( δS Total δg µν L ζ g µν + δS Total δT L ζ T + δS Total δψ L ζ ψ ) . (2.6)</formula>
<text><location><page_6><loc_16><loc_29><loc_85><loc_32></location>Now suppose that this identity is evaluated on configurations that satisfy the Einstein and matter equations. Over such configurations identity (2.6) becomes</text>
<formula><location><page_6><loc_41><loc_24><loc_85><loc_27></location>0 = ∫ d 4 x δS Total δT L ζ T . (2.7)</formula>
<text><location><page_6><loc_16><loc_16><loc_85><loc_23></location>Since T cannot be constant along all possible directions and the above condition must be satisfied by all vectors ζ µ , whe have that δS Total /δT = δS EA /δT = 0 automatically for all configurations that satisfy the Einstein equations and the matter equations of motion.</text>
<text><location><page_6><loc_16><loc_12><loc_85><loc_15></location>The physical equivalence between the action (2.3), restricted by (2.5), and the action (2.1) can be seen as follows [14]. The object</text>
<formula><location><page_6><loc_43><loc_9><loc_85><loc_11></location>P α β = δ α β + u α u β (2.8)</formula>
<text><location><page_7><loc_16><loc_84><loc_85><loc_88></location>is a spatial projector, whereas P αβ is the induced metric on the spatial hypersurfaces. The extrinsic curvature and the acceleration vector are defined, respectively, by</text>
<formula><location><page_7><loc_35><loc_80><loc_85><loc_83></location>K µν = P µ α ∇ α u ν , a µ = u α ∇ α u µ . (2.9)</formula>
<text><location><page_7><loc_16><loc_74><loc_85><loc_79></location>Since u µ is hypersurface orthogonal K µν is a symmetric tensor. K µν and a µ are spatial objects, K µν u ν = a µ u µ = 0. We may decompose the covariant derivative of u µ in terms of these objects,</text>
<formula><location><page_7><loc_42><loc_70><loc_85><loc_72></location>∇ µ u ν = K µν -u µ a ν . (2.10)</formula>
<text><location><page_7><loc_16><loc_60><loc_85><loc_69></location>Now, Since the T field equation need not be imposed explicitly and this a theory with general covariance, we can take T as the time coordinate, T = t . By doing so we break the symmetry of general diffeomorphisms over the spacetime. In addition, we can write the spacetime metric in the ADM variables N , N i and g ij . With these settings we have that the aether part of the Lagrangian in (2.3) takes the form</text>
<formula><location><page_7><loc_23><loc_56><loc_85><loc_58></location>M αβγδ ∇ α u γ ∇ β u δ = ( c 1 + c 3 ) K ij K ij + c 2 K 2 -( c 1 -c 4 ) a i a i . (2.11)</formula>
<text><location><page_7><loc_16><loc_48><loc_85><loc_56></location>In addition, √ -g = √ (3) gN and the decomposition of R adds K ij K ij -K 2 + (3) R to the Lagrangian. By putting all this in the action (2.3), we have that the z = 1 Hoˇrava action (2.1) is reproduced from it if the coupling constants of both theories are identified according to</text>
<formula><location><page_7><loc_18><loc_42><loc_85><loc_46></location>β = 1 1 -c 1 -c 3 , κ H = βκ EA , λ = β (1 + c 2 ) , α = β ( c 1 -c 4 ) . (2.12)</formula>
<text><location><page_7><loc_16><loc_38><loc_85><loc_41></location>Therefore, the z = 1 Hoˇrava action (2.1) is a gauge-fixed version of the hypersurfaceorthogonal Einstein-aether action given in (2.3) and (2.5).</text>
<section_header_level_1><location><page_7><loc_16><loc_33><loc_45><loc_35></location>3 Linearized theory</section_header_level_1>
<section_header_level_1><location><page_7><loc_16><loc_29><loc_60><loc_31></location>3.1 Gauge invariants with the T field</section_header_level_1>
<text><location><page_7><loc_16><loc_19><loc_85><loc_28></location>Now we focus on the linearized generally-covariant theory. Minkowski spacetime, whose metric we denote by η αβ , is a solution of the theory in absence of matter sources and with the condition T = t , which yields a zero aether energy-momentum tensor. We introduce the perturbative variables by expanding around this solution in the way</text>
<formula><location><page_7><loc_28><loc_16><loc_85><loc_17></location>g µν ( t, /vectorx ) = η µν + /epsilon1 h µν ( t, /vectorx ) , T ( t, /vectorx ) = t + /epsilon1 τ ( t, /vectorx ) . (3.1)</formula>
<text><location><page_7><loc_16><loc_9><loc_85><loc_14></location>The dependence of h µν and τ on the spacetime coordinates is arbitrary, except for the asymptotic conditions h µν , τ → 0 as r →∞ .</text>
<text><location><page_8><loc_16><loc_83><loc_85><loc_88></location>We investigate the possible gauge invariants of the linearized theory that can be formed with the metric h µν and the τ field. Under an arbitrary diffeomorphism over the spacetime, given by</text>
<formula><location><page_8><loc_45><loc_81><loc_85><loc_82></location>δx µ = ζ µ ( x α ) , (3.2)</formula>
<text><location><page_8><loc_16><loc_78><loc_76><loc_80></location>the exact spacetime metric g µν and the exact scalar field T transform as</text>
<formula><location><page_8><loc_39><loc_74><loc_85><loc_76></location>δg µν = -ζ α ∂ α g µν -2 g α ( µ ∂ ν ) ζ α , (3.3)</formula>
<formula><location><page_8><loc_39><loc_71><loc_85><loc_74></location>δT = -ζ α ∂ α T . (3.4)</formula>
<text><location><page_8><loc_16><loc_69><loc_71><loc_71></location>On the perturbative variables these transformations take the form</text>
<formula><location><page_8><loc_45><loc_65><loc_85><loc_67></location>δh µν = -2 ∂ ( µ ζ ν ) , (3.5)</formula>
<formula><location><page_8><loc_45><loc_64><loc_85><loc_65></location>δτ = ζ 0 , (3.6)</formula>
<text><location><page_8><loc_16><loc_56><loc_85><loc_62></location>where we have used the background metric to lower the index, ζ µ = η µν ζ ν . For the compatibility with the asymptotic conditions on the field variables we require that ζ µ → 0 as r → 0.</text>
<text><location><page_8><loc_16><loc_53><loc_85><loc_56></location>Now it is convenient to introduce the transverse and longitudinal decompositions for ζ i , h 0 i and h ij . They are given by</text>
<formula><location><page_8><loc_30><loc_50><loc_85><loc_51></location>ζ i = ξ i + ∂ i χ, (3.7)</formula>
<formula><location><page_8><loc_30><loc_47><loc_85><loc_49></location>h 0 i = m i + ∂ i b , (3.8)</formula>
<formula><location><page_8><loc_30><loc_43><loc_85><loc_47></location>h ij = h TT ij + 1 2 ( δ ij -∂ ij ∂ -2 ) h T + ∂ ( i h L j ) + ∂ ij ∂ -2 h L . (3.9)</formula>
<text><location><page_8><loc_16><loc_33><loc_85><loc_42></location>The symbol ∂ ij ··· k stands for ∂ i ∂ j · · · ∂ k , ∂ 2 is the flat Euclidean Laplacian, ∂ 2 ≡ ∂ kk , and ∂ -2 is its inverse, ∂ -2 ≡ ( ∂ 2 ) -1 . The restrictions on the variables are ∂ i ξ i = ∂ i m i = ∂ i h L i = ∂ i h TT ij = h TT ii = 0. For the uniqueness of the decompositions and the compatibility with the asymptotic behavior of the original field variables, we asume the asymptotic conditions</text>
<formula><location><page_8><loc_30><loc_29><loc_85><loc_32></location>ξ i , χ, m i , b, h TT ij , h T , h L i , h L → 0 as r →∞ . (3.10)</formula>
<text><location><page_8><loc_16><loc_27><loc_84><loc_28></location>By substituting (3.7 - 3.9) in the transformation (3.5), we obtain that it becomes</text>
<formula><location><page_8><loc_46><loc_22><loc_85><loc_25></location>δh 00 = -2 ˙ ζ 0 , (3.11)</formula>
<formula><location><page_8><loc_46><loc_18><loc_85><loc_21></location>δm i = -˙ ξ i , (3.13)</formula>
<formula><location><page_8><loc_46><loc_20><loc_85><loc_23></location>δb = -˙ χ -ζ 0 , (3.12)</formula>
<formula><location><page_8><loc_46><loc_16><loc_85><loc_19></location>δh L = -2 ∂ 2 χ, (3.14)</formula>
<formula><location><page_8><loc_46><loc_12><loc_85><loc_14></location>δh T = 0 , (3.16)</formula>
<formula><location><page_8><loc_46><loc_14><loc_85><loc_16></location>δh L i = -2 ξ i , (3.15)</formula>
<formula><location><page_8><loc_46><loc_10><loc_85><loc_12></location>δh TT ij = 0 . (3.17)</formula>
<text><location><page_9><loc_16><loc_83><loc_85><loc_88></location>From these transformations we extract that h T and h TT ij are gauge invariants. By combining with (3.6), we may define three variables that are also gauge invariants, namely</text>
<formula><location><page_9><loc_42><loc_78><loc_85><loc_81></location>p ≡ h 00 +2˙ τ , (3.18)</formula>
<formula><location><page_9><loc_42><loc_74><loc_85><loc_77></location>v i ≡ ˙ h L i -2 m i . (3.20)</formula>
<formula><location><page_9><loc_42><loc_76><loc_85><loc_79></location>q ≡ ˙ h L -2 ∂ 2 b -2 ∂ 2 τ , (3.19)</formula>
<text><location><page_9><loc_16><loc_66><loc_85><loc_73></location>This approach is rather different to the standard approach of linearized GR (see, for example, [21]), since the scalar τ is involved in the definition of the gauge invariants p and q . The gauge invariant of linearized GR that depends only on the metric components is</text>
<formula><location><page_9><loc_41><loc_63><loc_85><loc_66></location>Φ ≡ ∂ 2 h 00 + h L -2 ∂ 2 ˙ b . (3.21)</formula>
<text><location><page_9><loc_16><loc_54><loc_85><loc_63></location>Here Φ is not an independent quantity since it can be obtained from a combination of p and q , Φ = ∂ 2 p + ˙ q . Therefore, in the linearized theory that depends on h µν and τ and that is generally covariant, the independent gauge invariants are h T , h TT ij , p , q and v i . In Appendix A we show that an analogous construction of gauge invariants can be done for the case of the theory formulated with the FDiff-gauge symmetry.</text>
<section_header_level_1><location><page_9><loc_16><loc_50><loc_56><loc_51></location>3.2 Linearized Einstein equations</section_header_level_1>
<text><location><page_9><loc_16><loc_42><loc_85><loc_49></location>Here we study the linearized field equations. We consider the presence of matter sources, hence there is an active energy-momentum tensor T matter µν for the matter. We define the energy-momentum tensors T aether µν and T matter µν in such a way that the Einstein equations take the form</text>
<formula><location><page_9><loc_40><loc_38><loc_85><loc_40></location>G µν = T aether µν + κ EA T matter µν , (3.22)</formula>
<text><location><page_9><loc_16><loc_34><loc_83><loc_37></location>with the usual expression G µν ≡ R µν -1 2 g µν R . We decompose T matter µν in the way</text>
<formula><location><page_9><loc_29><loc_31><loc_85><loc_33></location>T matter 00 = ρ , (3.23)</formula>
<formula><location><page_9><loc_29><loc_29><loc_85><loc_31></location>T matter 0 i = S i + ∂ i S , (3.24)</formula>
<formula><location><page_9><loc_29><loc_25><loc_85><loc_29></location>T matter ij = σ TT ij + 1 2 ( δ ij -∂ ij ∂ -2 ) σ T + ∂ ( i σ L j ) + ∂ ij ∂ -2 σ L , (3.25)</formula>
<text><location><page_9><loc_16><loc_22><loc_74><loc_24></location>where the variables are restricted by ∂ i S i = ∂ i σ L i = ∂ i σ TT ij = σ TT ii = 0.</text>
<text><location><page_9><loc_16><loc_17><loc_85><loc_22></location>The linearized Einstein equations can be completely expressed in terms of the gauge invariants defined in the previous section. Indeed, after the decompositions (3.8 - 3.9) and (3.23 - 3.25) are done and the gauge invariants (3.18 - 3.20) are</text>
<text><location><page_10><loc_16><loc_86><loc_65><loc_88></location>introduced, the linearized Einstein equations take the form</text>
<formula><location><page_10><loc_30><loc_82><loc_85><loc_85></location>β∂ 2 h T -α∂ 2 p = -2 κ H ρ , (3.26)</formula>
<formula><location><page_10><loc_30><loc_79><loc_85><loc_80></location>∂ 2 v i = 4 κ H S i , (3.28)</formula>
<formula><location><page_10><loc_30><loc_80><loc_85><loc_83></location>λ ˙ h T -(1 -λ ) q = -2 κ H S , (3.27)</formula>
<formula><location><page_10><loc_30><loc_75><loc_85><loc_78></location>λ h T -(1 -λ ) ˙ q = -2 κ H σ L , (3.29)</formula>
<formula><location><page_10><loc_30><loc_74><loc_85><loc_76></location>˙ v i = 2 κ H σ L i , (3.30)</formula>
<formula><location><page_10><loc_30><loc_71><loc_85><loc_74></location>(1 -3 λ )( h T + ˙ q ) + β∂ 2 h T -2 β∂ 2 p = 2 κ H ( σ T + σ L ) , (3.31)</formula>
<formula><location><page_10><loc_30><loc_69><loc_85><loc_72></location>h TT ij -β∂ 2 h TT ij = 2 κ H σ TT ij . (3.32)</formula>
<text><location><page_10><loc_16><loc_48><loc_85><loc_68></location>Equation (3.26) is the 00 component of the Einstein equations, Eqs. (3.27) and (3.28) come from the 0 i components and the last four equations constitute the ij components. We have used relations (2.12) to change the constants κ EA and c 1 , 2 , 3 , 4 of the generally-covariant formulation by the constants κ H , λ , β and α of the FDiff-covariant formulation since the latter are the ones that the linearized theory naturally adopts. We stress that no gauge-fixing condition has been imposed to obtain these equations. All the variables of the left-hand sides belong to the set of gauge invariants of the linearized theory. In Appendix A we show that if one uses the original FDiff-invariant formulation of the z = 1 Hoˇrava theory, then the linearized field equations can also be written purely in terms of the corresponding FDiff-gauge invariants.</text>
<text><location><page_10><loc_16><loc_45><loc_85><loc_48></location>Evidently, Eqs. (3.27), (3.28) (3.29) and (3.30) imply the following conditions on the source,</text>
<formula><location><page_10><loc_47><loc_42><loc_85><loc_44></location>σ L = ˙ S , (3.33)</formula>
<formula><location><page_10><loc_47><loc_39><loc_85><loc_41></location>∂ 2 σ L i = 2 ˙ S i . (3.34)</formula>
<text><location><page_10><loc_16><loc_31><loc_85><loc_38></location>Consequently, we drop Eqs. (3.29) and (3.30) out from the list of independent Einstein equations and impose Eqs. (3.33 - 3.34) as complementary conditions that must be satisfied by the matter source (equations (3.33 - 3.34) are independent of h µν and τ ).</text>
<text><location><page_10><loc_16><loc_9><loc_85><loc_30></location>Let us analyze the system of equations (3.26 - 3.32) as a set of equations for the gauge invariants with the matter source given and, momentarily, without imposing any restriction on the coupling constants. Equations (3.26), (3.27) and (3.28) do not depend on the second time derivative of any variable. Hence, they are constraints on the initial data. Specifically, Eq. (3.26) corresponds to the Hamiltonian constraint whereas Eqs. (3.27) and (3.28) constitute the momentum constraint (see the canonical formulation for the theory at the vaccum in [22] for the case out of the KCP and in [4] for the case at the KCP). On the other hand, Eqs. (3.31) and (3.32) do depend on the second time derivate, so they are the ones that govern the propagation of the dynamical modes. In this covariant formalism we have the ten components of the metric field and the scalar field τ , which sum up eleven field variables. Four of these must be fixed by a coordinate-system choice and the remaining seven are</text>
<text><location><page_11><loc_16><loc_73><loc_85><loc_88></location>subject to the system (3.26 - 3.32), which has just seven independent equations (we recall that Eqs. (3.29) and (3.30) have been dropped out and that Eqs. (3.28) and (3.32) yield four independent equations). Therefore, the system is closed for the field variables with the matter source given. The seven gauge-independent variables are subject to four constraints, Eqs. (3.26 - 3.28), hence in this general theory there remain three propagating physical degrees of freedom, whose evolution is governed by Eqs. (3.31) and (3.32). One can identify two of these propagating modes as the two tensorial modes of GR. The remaining mode is an extra scalar mode.</text>
<text><location><page_11><loc_16><loc_70><loc_85><loc_73></location>Now we move to the case of our interest. It is evident that Eq. (3.31) changes its character of evolution equation if we set the coupling constant λ to</text>
<formula><location><page_11><loc_47><loc_66><loc_85><loc_68></location>λ = 1 / 3 . (3.35)</formula>
<text><location><page_11><loc_16><loc_54><loc_85><loc_65></location>In this case Eq. (3.31) lacks its dependence on the second time derivative h T , hence it becomes an additional constraint. Notice that the change of evolution equations by constraints is not a smooth one. The kinetic-conformal theory is an independent theory on its own. In terms of the Einstein-aether constants this condition is c 1 + c 3 + 3 c 2 = -2. Let us write here the resulting set of independent field equations under the condition (3.35),</text>
<formula><location><page_11><loc_39><loc_50><loc_85><loc_52></location>β∂ 2 h T -α∂ 2 p = -2 κ H ρ , (3.36)</formula>
<formula><location><page_11><loc_39><loc_46><loc_85><loc_48></location>∂ 2 v i = 4 κ H S i , (3.38)</formula>
<formula><location><page_11><loc_39><loc_47><loc_85><loc_50></location>˙ h T -2 q = -6 κ H S , (3.37)</formula>
<formula><location><page_11><loc_39><loc_43><loc_85><loc_46></location>β∂ 2 h T -2 β∂ 2 p = 2 κ H ( σ T + ˙ S ) , (3.39)</formula>
<formula><location><page_11><loc_39><loc_41><loc_85><loc_44></location>h TT ij -β∂ 2 h TT ij = 2 κ H σ TT ij . (3.40)</formula>
<text><location><page_11><loc_16><loc_29><loc_85><loc_40></location>The four first equations are constraints that fix the gauge invariants h T , p , q and v i , whereas the last one is the evolution equation for the tranverse-traceless tensorial mode h TT ij . Therefore, under the condition (3.35), which defines the kineticconformal point, the extra mode is annihilated, in agreement with the vacuum theory [4, 9], and the propagating physical degrees of freedom are the same of GR, which are described by h TT ij .</text>
<text><location><page_11><loc_16><loc_22><loc_85><loc_29></location>By imposing some bounds on the coupling constants α and β , we can ensure that the constraints form a closed system of partial differential equations and that the evolution equation for h TT ij is the sourced wave equation. The conditions on the coupling constants are</text>
<formula><location><page_11><loc_42><loc_20><loc_85><loc_22></location>β > 0 , α = 2 β . (3.41)</formula>
<text><location><page_11><loc_54><loc_19><loc_54><loc_22></location>/negationslash</text>
<text><location><page_12><loc_16><loc_86><loc_82><loc_88></location>Under these conditions the formal solutions of the constraints (3.36 - 3.39) are</text>
<formula><location><page_12><loc_28><loc_80><loc_85><loc_85></location>h T = -2 k H β (2 β -α ) ∂ -2 [ 2 βρ + α ( σ T + ˙ S ) ] , (3.42)</formula>
<formula><location><page_12><loc_30><loc_72><loc_85><loc_77></location>q = -k H β (2 β -α ) ∂ -2 [ 2 β ˙ ρ + α ( ˙ σ T + S ) ] +3 k H S , (3.44)</formula>
<formula><location><page_12><loc_30><loc_76><loc_85><loc_81></location>p = -2 k H 2 β -α ∂ -2 [ ρ + σ T + ˙ S ] , (3.43)</formula>
<formula><location><page_12><loc_29><loc_71><loc_85><loc_73></location>v i = 4 k H ∂ -2 [ S i ] . (3.45)</formula>
<text><location><page_12><loc_16><loc_64><loc_85><loc_69></location>Equations (3.40) and (3.42 - 3.45) tell us that h T , p , q and v i are variables bounded to the sources whereas h TT ij is the radiative variable. Therefore, as in GR, h TT ij acquires a pure physical meanning as the only radiative field.</text>
<text><location><page_12><loc_16><loc_44><loc_85><loc_65></location>The speed of the waves h TT ij is √ β . This agrees with Ref. [15] in what concerns the h TT ij mode. However, as we have already mentioned, in the unrestricted Einsteinaether theory studied in Ref. [15] there are three additional modes propagating themselves with wave equations and with different speeds. In Ref. [16] the same linearized equations were studied with matter sources. We remark again that the total equivalence between the hypersurface orthogonal Einstein-aether theory and the nonprojectable z = 1 Hoˇrava theory holds only if the condition of hypersurface orthogonality is imposed at the level of the action. Therefore, the linearized z = 1 Hoˇrava equations at the kinetic-conformal point, whose covariant version is (3.36 - 3.40), are not obtained by direct substitution of the hypersurface orthogonality condition on the equations analyzed in Refs. [15, 16].</text>
<section_header_level_1><location><page_12><loc_16><loc_39><loc_50><loc_41></location>3.3 The quadrupole formula</section_header_level_1>
<text><location><page_12><loc_16><loc_29><loc_85><loc_38></location>In the standard perturbative scheme (post-Minkowskian approach), the exact solution is increasely approximated by the perturbative solution if the nolinear terms of the field equations are considered as sources for the fields at the order of interest. For example, the perturbative wave equation (3.40) at higher order in perturbations can be casted as</text>
<text><location><page_12><loc_16><loc_18><loc_85><loc_27></location>where h TT ij in the left-hand side is evaluated at the order of interest. t ij represents the nonlinear terms coming from the Einstein tensor and the aether energy-momentum tensor. The solutions for all the variables at lower orders must be substituted in t ij and T matter ij such that the desired order in perturbations is reached in all terms of this equation.</text>
<formula><location><page_12><loc_35><loc_26><loc_85><loc_30></location>h TT ij -β∂ 2 h TT ij = ( 2 κ H T matter ij + t ij ) TT , (3.46)</formula>
<text><location><page_12><loc_16><loc_12><loc_85><loc_17></location>In this iterative scheme the linearized field equations determine the leading contribution. Here we extract information from the solution of the linearized wave equation relevant for the physics at large distances from the source. The solution</text>
<text><location><page_13><loc_16><loc_86><loc_52><loc_88></location>of Eq. (3.40) with no incoming radiation is</text>
<formula><location><page_13><loc_32><loc_80><loc_85><loc_86></location>h TT ij = κ H 2 πβ ∫ d 3 x ' σ TT ij ( t -| /vectorx -/vectorx ' | / √ β , /vectorx ' ) | /vectorx -/vectorx ' | . (3.47)</formula>
<text><location><page_13><loc_16><loc_70><loc_85><loc_79></location>Following the outline of Ref. [23], our next steps consist of approximating this expression according to weakness criteria: the observer is far from the source, the self-gravity of the source is negligible and the motion of the source is sufficiently slow. Then, by defining ˆ n = /vectorx/r with r = | /vectorx | , solution (3.47) can be expanded in the form</text>
<formula><location><page_13><loc_20><loc_64><loc_85><loc_69></location>h TT ij = κ H 2 πβr ∞ ∑ m =0 1 m !( √ β ) m ∂ m ∂t m ∫ d 3 x ' (ˆ n · /vectorx ' ) m σ TT ij ( t -r/ √ β , /vectorx ' ) . (3.48)</formula>
<text><location><page_13><loc_16><loc_62><loc_48><loc_63></location>The leading mode of this expansion is</text>
<formula><location><page_13><loc_34><loc_56><loc_85><loc_60></location>h TT ij = κ H 2 πβr ∫ d 3 x ' σ TT ij ( t -r/ √ β , /vectorx ' ) . (3.49)</formula>
<text><location><page_13><loc_16><loc_41><loc_85><loc_55></location>Let us massage this expression following a procedure similar of the one of Ref. [16]. One can handle first the integration (and the expansion already done) in terms of the full energy-momentum tensor and then perform the projection to the transversetraceless sector. At order O (1 /r ) the projection to the transverse sector is equivalent to the algebraic projection to the plane orthogonal to ˆ n . The projector to this plane is θ ij ≡ δ ij -n i n j , and the operator that projects tensors and substracts the trace is P TT ijkl ≡ θ ik θ jl -1 2 θ ij θ kl . Then we have</text>
<formula><location><page_13><loc_31><loc_36><loc_85><loc_41></location>h TT ij = κ H 2 πβr P TT ijkl ∫ d 3 x ' T matter kl ( t -r/ √ β , /vectorx ' ) . (3.50)</formula>
<text><location><page_13><loc_16><loc_31><loc_85><loc_36></location>Similarly to the criterium of order used Ref. [16], we require that the equations of motion of the matter source, approximated to the order of interest, imply the Newtonian law of mass conservation, namely 2</text>
<formula><location><page_13><loc_47><loc_27><loc_85><loc_29></location>˙ ρ = ∂ 2 S . (3.51)</formula>
<text><location><page_13><loc_16><loc_16><loc_85><loc_25></location>This and Eqs. (3.33 - 3.34) are equivalent to the preservation of the matter energymomentum tensor, ∂ µ T µν matter = 0. Therefore, the requisite (3.51) is equivalent to demand that, at the linear order in perturbations and at the nonrelativistic limit, the equations of motion of the matter sources imply the conservation of its energymomentum tensor (actually, the spatial components ∂ µ T µi matter = 0, which are given</text>
<text><location><page_14><loc_16><loc_83><loc_85><loc_88></location>by Eqs. (3.33) and (3.34), are already implied by the Einstein equations). Following the standard approach of GR, from the conservation of the matter energymomentum tensor at the order considered one obtains the relation</text>
<formula><location><page_14><loc_22><loc_78><loc_85><loc_81></location>∫ d 3 x ' T matter ij = 1 2 ∫ d 3 x ' x ' i x ' j T matter 00 = 1 2 ∫ d 3 x ' x ' i x ' j ¨ ρ ≡ 1 2 I ij . (3.52)</formula>
<text><location><page_14><loc_16><loc_73><loc_85><loc_77></location>Therefore, the leading contribution for the generation of gravitational waves has the same structure of Einstein's quadrupole formula that arises in GR,</text>
<formula><location><page_14><loc_36><loc_68><loc_85><loc_73></location>h TT ij = κ H 4 πβr P TT ijkl d 2 I kl ( t -r/ √ β ) dt 2 . (3.53)</formula>
<text><location><page_14><loc_16><loc_60><loc_85><loc_67></location>The only difference this formula has with respect to Einstein's quadrupole formula is the presence of the coupling constants κ H and β of the Hoˇrava or Einstein-aether theory. If these constants are adjusted to the GR values κ H = 8 πG N and β = 1, then (3.53) becomes identical to Einstein's quadrupole formula.</text>
<section_header_level_1><location><page_14><loc_16><loc_55><loc_61><loc_57></location>4 On the observational bounds</section_header_level_1>
<text><location><page_14><loc_16><loc_42><loc_85><loc_53></location>In this section we discuss some observational bounds on the theory at the kineticconformal point. Some of the formulas we use can be directly deduced from the analysis previously done in the literature of the nonprojectable Horava theory with general λ . However, there are features that do not follow as a particular case of the general theory, essentially due to the discontinuity in the number of degrees of freedom.</text>
<text><location><page_14><loc_18><loc_37><loc_18><loc_40></location>/negationslash</text>
<text><location><page_14><loc_16><loc_33><loc_85><loc_42></location>We comment that in the (projectable and nonprojectable) Hoˇrava theory with λ = 1 / 3 there are theoretical restrictions on the coupling constants that are necessary for the stability of the extra mode. Although these conditions are widely used, they do not apply in the kinetic-conformal formulation due to the obvious reason that there is no extra mode. For example, there is a restriction on λ given by [6]</text>
<formula><location><page_14><loc_46><loc_27><loc_85><loc_32></location>3 λ -1 λ -1 > 0 , (4.1)</formula>
<text><location><page_14><loc_52><loc_19><loc_52><loc_21></location>/negationslash</text>
<text><location><page_14><loc_16><loc_14><loc_85><loc_27></location>necessary to avoid that the extra mode becomes a ghost at the level of the linearized theory. The interpretation in the kinetic-conformal formulation, λ = 1 / 3, is simply that there is no such bound. We highlight this point since the bounds coming from the physics of the extra mode in the case λ = 1 / 3 are frequently combined with the observational bounds when the phenomenology of the theory in under scrutiny (see, for example, [24, 19]), hence extrapoling directly the conclusions from the λ = 1 / 3 case can be misleading in the kinetic-conformal case.</text>
<text><location><page_14><loc_80><loc_15><loc_80><loc_18></location>/negationslash</text>
<text><location><page_14><loc_16><loc_9><loc_85><loc_14></location>We start with the observational bounds of the weak regime englobed in the parametrized-post-Newtonian (PPN) parameters of the solar-system tests. The PPN parameters of the kinetic-conformal theory can be obtained from the general</text>
<text><location><page_15><loc_16><loc_66><loc_85><loc_88></location>nonprojectable Hoˇrava theory with arbitrary λ since the discrepancy in the propagating degrees of freedom does not affect the PPN potentials. The PPN parameters for the nonprojectable Hoˇrava theory were computed in Ref. [17] (see also [25]), using the covariant formulation of the second-order in derivatives action, i. e., the hypersurface-orthogonal Einstein-aether theory. The PPN parameters are obtained for a weak, non-relativistic source, and the procedure is similar to the one of the Einstein-aether theory [20]. It results that the hypersurface orthogonal Einsteinaether theory (as well as the unrestricted Einstein-aether theory) reproduces the same values of the PPN constants of GR, except for the parameters α 1 and α 2 , whose nonzero values signal violations of the Lorentz symmetry. The expressions obtained in [17] for these constants, written with our conventions for the coupling constants and for general λ , are</text>
<formula><location><page_15><loc_28><loc_62><loc_85><loc_64></location>α 1 = 8( β -1) -4 α, (4.2)</formula>
<formula><location><page_15><loc_28><loc_58><loc_85><loc_62></location>α 2 = [ β (1 -λ ) + ( β -1)(1 -3 λ ) -α (1 -2 λ ) 4(2 β -α )(1 -λ ) ] α 1 . (4.3)</formula>
<text><location><page_15><loc_16><loc_55><loc_79><loc_57></location>Notice that α 1 is independent of λ . For λ = 1 / 3, the parameter α 2 becomes</text>
<formula><location><page_15><loc_46><loc_50><loc_85><loc_54></location>α 2 = 1 8 α 1 . (4.4)</formula>
<text><location><page_15><loc_16><loc_40><loc_85><loc_49></location>The current observational bounds on these parameters, which are dimensionless, are | α 1 | < 10 -4 and | α 2 | < 10 -7 [26]. In the kinetic-conformal theory the relation (4.4) demands that the strong bound, which is the one on α 2 , must be satisfied by both parameters. Taking this into account, we use relation (4.2) to solve one coupling constant in terms of the other one,</text>
<formula><location><page_15><loc_43><loc_36><loc_85><loc_39></location>α = 2( β -1) + δ , (4.5)</formula>
<text><location><page_15><loc_16><loc_30><loc_85><loc_35></location>where δ represents the narrow observational window for the α 2 parameter, i. e. | δ | < 10 -7 . With (4.5) the kinetic-conformal Hoˇrava theory satisfies all the conditions derived from the PPN analysis of the solar-system tests.</text>
<text><location><page_15><loc_16><loc_10><loc_85><loc_30></location>Nonrelativistic gravitational theories (coupled to relativistic particles) produce Cherenkov radiation if the velocities of the gravitational modes are lower than the speed of the relativistic particles [27]. The implications of this have been studied for the Einstein-aether theory in Refs. [28, 24], obtaining very stringent lower bounds on the coupling constants. In the Einstein-aether theory the lower bounds affect several coupling constants since this theory has several propagating modes, each one with a different dependence of its velocity on the coupling constants. In the kineticconformal Hoˇrava theory the lower bounds coming from the Cherenkov radiation are very simple to implement since the propagating modes are the same of GR. We have seen that, when the theory is truncated to its second-order effective action, the squared velocity of the transverse-traceless tensorial modes of the linearized</text>
<text><location><page_16><loc_16><loc_84><loc_85><loc_88></location>theory is β , hence in this case the bounds affect only to this constant. Therefore, the Cherenkov radiation puts the bound</text>
<formula><location><page_16><loc_48><loc_80><loc_85><loc_83></location>β ≥ 1 . (4.6)</formula>
<text><location><page_16><loc_16><loc_76><loc_85><loc_80></location>We comment that at higher energies the higher order operators could be relevant for this analysis, hence other coupling constants can enter in the game.</text>
<text><location><page_16><loc_16><loc_9><loc_85><loc_76></location>Now we want to make some considerations about the kinetic-conformal theory at cosmological scales, since there is an important restriction on this theory at this scale. The restriction concerns to the Lagrangian formulation of the theory, but in the Hamiltonian formalism it is still an open question [11]. In the field equations derived from the Lagrangian, if the ansatz for a full homogeneous and isotropic metric is imposed, together with a homogeneous and isotropic perfect fluid, then it turns out that the only possibility left by the field equations is that the density and pressure vanish. With full homogeneous and isotropic we mean that these conditions are imposed in all the components of the spacetime metric. This restriction can be deduced from the analysis of Ref. [6], where the extension of the nonprojectable theory and its first cosmological application were presented (the Einstein-aether theory exhibits an analogous behavior [29, 30]). There it was found that the effective cosmological gravitational constant arising in the Friedmann equations differs from the Newtonian (local) gravitational constant by a scale factor that depends on λ . At the kinetic-conformal point, λ = 1 / 3, this scale factor diverges. What really this divergence means is the vanishing of the density and the pressure at the kineticconformal point, as we have commented. We point out two issues concerning this restriction. The first one is that in the kinetic-conformal theory the equations of motion derived from the Hamiltonian admits more solutions than the Lagrangian field equations [11]. This is essentially due to the role played by the Lagrange multipliers once all the constraints have been added to the Hamiltonian. On certain configurations admissible for the Lagrange multipliers, the Legendre transformation cannot be inverted, hence the Hamiltonian lost the equivalence with the Lagrangian. The second issue is that the Hoˇrava theory is originally formulated as a theory with the symmetry of the FDiff, it is not a generally covariant theory. Indeed, the equivalence with the hypersurface orthogonal Einstein-aether theory holds only at the level of the action of second-order in derivatives and on the Lagrangian formulations. Under the reduced FDiff symmetry, the conditions of homogeneity and isotropy can be restricted to the spatial metric, whereas the laspe function and the shift vector can have a more general dependence on the time and the space. These two observations have been discussed in Ref. [11]. Moreover, the analysis of the cosmological-scale configurations in the kinetic-conformal Hoˇrava gravity requires a previous and deep analysis on the structure of the second-class constraints of the theory. Perhaps a reformulation of the theory only in terms of first-class constraints is convenient. Under such scenario, some of the original second-class constraints could be circumvented by recurring to different 'gauge fixing' conditions. Thus, the configurations of cosmological scale would require a</text>
<text><location><page_17><loc_16><loc_81><loc_85><loc_88></location>better gauge fixing than, for example, the π = 0 condition that arises originally as a second-class constraint. Therefore, we consider that the kinetic-conformal point, λ = 1 / 3, is not necessarily ruled out by the restriction about homogenous and isotropic configurations mentioned above.</text>
<text><location><page_17><loc_16><loc_48><loc_85><loc_80></location>Finally, we contrast with the bounds coming from binary pulsars, which are strong sources, that were obtained in Refs. [18, 19] on two Lorentz-violating theories: the Einstein-aether theory and the hypersurface-orthogonal Einstein aether theory without the kinetic-conformal condition (called the krhonometric theory in those references). The authors of [18, 19] arrive at very stringent constraints on the space of coupling constants of these theories after contrasting with the observations on several binary pulsars. For the khronometric theory in particular, the region of the space of parameters on which the theory can reproduce the decay rate of orbital period within the observational error excludes the kinetic-conformal point λ = 1 / 3. This is because they combine the analysis on the binary pulsar with the bounds resulting from the stability of the extra mode, the Cherenkov radiation and the cosmological-scale effect of rescaling the gravitational constant. As we have discussed, the stability of the extra mode does not apply in the kinetic-conformal theory. The Cherenkov radiation only constraint the constant β , one can put β /greaterorsimilar 1 consistently on the kinetic-conformal theory. The rescaling of the gravitational constant at cosmological scales leads to the apparent restriction at λ = 1 / 3 that we discussed above. Upon the arguments we have given we consider that it does not rule out unavoidably the kinetic-conformal formulation of the theory.</text>
<text><location><page_17><loc_16><loc_37><loc_85><loc_48></location>Indeed, it is interesting to extrapolate the formulas of the orbital evolution of the binary pulsars obtained in [18, 19] to the kinetic-conformal case. This orbital evolution is related to the effective multipoles of the theory, and we have shown in the previous sections that in the kinetic-conformal theory the dominant mode in the far zone of a weak source is the same of GR. We reproduce here the rate of change of the orbital period obtained in [19], which uses previous results of [16, 31, 17],</text>
<formula><location><page_17><loc_27><loc_28><loc_85><loc_36></location>˙ P b P b = -3 aG Æ G µm 〈{ A 1 5 ... Q ij ... Q ij + A 2 5 ... Q ij ... Q ij + A 3 5 ... Q ij ... Q ij + B 1 ... I ... I + B 2 ... I ... I + B 3 ... I ... I + C ˙ Σ i ˙ Σ i }〉 . (4.7)</formula>
<text><location><page_17><loc_16><loc_21><loc_85><loc_28></location>G is the effective gravitational constant in the binary system, G Æ is the gravitational constant of the theory (related to κ EA and κ H ), a is the semi-major axis, m ≡ m 1 + m 2 , and µ ≡ m 1 m 2 /m . The quadrupole moment Q ij is the trace-free part of the system's mass quadrupole moment I ij :</text>
<formula><location><page_17><loc_43><loc_16><loc_85><loc_19></location>I ij = ∑ A m A x i A x j A . (4.8)</formula>
<text><location><page_17><loc_16><loc_12><loc_75><loc_15></location>Q ij is the trace-free part of the rescaled mass quadrupole moment I ij :</text>
<formula><location><page_17><loc_41><loc_8><loc_85><loc_12></location>I ij = ∑ A s A m A x i A x j A , (4.9)</formula>
<text><location><page_18><loc_16><loc_83><loc_85><loc_88></location>where s A are constants associated to the sensitivities of the system, which are parameters encoding the departing of bodies' wordlines from the relativistic trayectories. The dipolar moment Σ i is</text>
<formula><location><page_18><loc_42><loc_77><loc_85><loc_81></location>Σ i = -∑ A s A m A v i A . (4.10)</formula>
<text><location><page_18><loc_16><loc_71><loc_85><loc_76></location>We have written only the dominant modes for the multipoles. Further details can be found in [19, 16, 31, 17]. The coefficients arising in (4.7), using our conventions for the coupling constants, are given by</text>
<formula><location><page_18><loc_31><loc_54><loc_85><loc_69></location>A 1 ≡ 1 c t + 3 α ( Z 1) 2 2(2 β -α ) c s , A 2 ≡ -2 β ( Z 1) (2 β -α ) c 3 s , A 3 ≡ 2 β 2 3 α (2 β -α ) c 5 s , B 1 ≡ α Z 2 4(2 β -α ) c s , B 2 ≡ -β Z 3(2 β -α ) c 2 s , B 3 ≡ β 2 9 α (2 β -α ) c 5 s , C = 4 β 2 3 α (2 β -α ) c 3 s , (4.11)</formula>
<text><location><page_18><loc_16><loc_52><loc_21><loc_53></location>where</text>
<text><location><page_18><loc_16><loc_46><loc_19><loc_48></location>and</text>
<formula><location><page_18><loc_36><loc_47><loc_85><loc_52></location>Z ≡ α 1 -2 α 2 3(2 β -α -2) = 2 α 2 2 β -α -2 , (4.12)</formula>
<formula><location><page_18><loc_39><loc_42><loc_85><loc_46></location>c 2 t = β , c 2 s = β (2 β -α ) λ α (3 λ -1) , (4.13)</formula>
<text><location><page_18><loc_16><loc_35><loc_85><loc_42></location>are the velocities of the tensorial modes and the extra scalar mode respectively. Notice that A 1 is the only coefficient with information about the propagation of the tensorial modes. In (4.7) the angled-brackets stand for an average over several wavelengths.</text>
<text><location><page_18><loc_16><loc_25><loc_85><loc_35></location>Now, it is easy to see how all the contributions of the extra mode disappear and only the quadrupole contribution remains as the dominant mode. By substituting λ = 1 / 3 in the velocity c s given in (4.13), we get that c s diverges, which is an informal way to express that the extra mode gets frozen. Then all coefficients in (4.11) vanish except for A 1 , which becomes A 1 = 1 / √ β . Equation (4.7) becomes</text>
<formula><location><page_18><loc_39><loc_19><loc_85><loc_24></location>˙ P b P b = -3 aG Æ 5 √ β G µm 〈 ... Q ij ... Q ij 〉 . (4.14)</formula>
<text><location><page_18><loc_16><loc_14><loc_85><loc_19></location>This results expresses that in the kinetic-conformal theory the quadrupole mode is the dominant radiative contribution to the rate of the orbital period decay, as in GR. Again, we may put β = 1 consistently in the kinetic-conformal theory.</text>
<section_header_level_1><location><page_19><loc_16><loc_86><loc_32><loc_88></location>Conclusions</section_header_level_1>
<text><location><page_19><loc_16><loc_66><loc_85><loc_84></location>Our central results are that, first, in the Lorentz-violating, power-counting renormalizable (once the higher order operators are considered) and unitary theory coupled to matter sources that we have considered, the only radiative degrees of freedom are the same transverse-traceless tensorial modes of GR, and that, second, at the leading order the Einstein quadrupole formula is reproduced (if two coupling constant are adjusted to their GR values). We have worked in the gauge-invariant formalism of the linearized theory, hence our results are not gauge-fixing artifacts. We have clearly identified the set of the gauge-invariant variables that are nonradiative and the ones that are radiative. The nonradiative variables are linked to the sources by Poissonian equations.</text>
<text><location><page_19><loc_16><loc_55><loc_85><loc_66></location>It is interesting that the kinetic-conformal Hoˇrava theory is able to reproduce the same leading mode of GR in what concerns the production and propagation of gravitational waves, but with a better quantum behavior. In Ref. [9] we showed the power-counting renormalizability of this theory without matter sources by analyzing the superficial degree of divergence of general one-particle irreducible diagrams, as well as its unitarity. There remains to prove its complete renormalizability.</text>
<text><location><page_19><loc_16><loc_43><loc_85><loc_55></location>Of course, a study of the next post-Newtonian orders is needed to make an exhaustive comparison between the gravitational waves produced and propagated in the kinetic-conformal Hoˇrava theory and the detected signals. Our study is the first step towards this goal since it is restricted to the leading order that can be extracted from a weak source. At higher orders, the other field variables besides h TT ij are also relevant for the wave production and propagation due to the nonlinearities of the theory.</text>
<text><location><page_19><loc_16><loc_23><loc_85><loc_42></location>We have also considered some observational bounds. The PPN analysis fixes one of the coupling constants of the action of second-order in derivatives, such that all solar-system tests are satisfied by the theory after this restriction is imposed. Cherenkov radiation puts a lower bound on other coupling constant. The λ = 1 / 3 value that defines the kinetic-conformal condition is not discarded by the bounds coming from binary pulsars showed in [18, 19], since these bounds were combined with theoretical restrictions and cosmological considerations that do not necessarily apply to our case, as we have argued. Indeed, we have shown, based on the formulas of [19], how the rate of decay of orbital period is very close to its corresponding expression in GR at the dominant level. In particular, there is no dipolar contribution to this decay, unlike the theories considered in [18, 19].</text>
<section_header_level_1><location><page_19><loc_16><loc_18><loc_41><loc_20></location>Acknowledgments</section_header_level_1>
<text><location><page_19><loc_16><loc_11><loc_85><loc_16></location>A. R. is partially supported by grant Fondecyt No. 1161192, Chile. J. B. is partially supported by the Programa MECE Educaci'on Superior of Ministerio de Educaci'on, Chile.</text>
<section_header_level_1><location><page_20><loc_16><loc_86><loc_65><loc_88></location>A Waves in the FDiff formulation</section_header_level_1>
<section_header_level_1><location><page_20><loc_16><loc_82><loc_53><loc_84></location>A.1 The FDiff gauge invariants</section_header_level_1>
<text><location><page_20><loc_16><loc_76><loc_85><loc_81></location>In this appendix we study the formulation of linearized gauge invariants when the gauge symmetry is given by the diffeomorphisms that preserve a given foliation, the FDiff.</text>
<text><location><page_20><loc_16><loc_69><loc_85><loc_76></location>Since there is a subtlety when implementing the FDiff transformations together with a prescribed asymptotic-flatness condition on the field variables, let us start by presenting the FDiff without asymptotic conditions. The diffeomorphisms that preserve a given foliation act on the coordinates ( t, /vectorx ) in the following way</text>
<formula><location><page_20><loc_38><loc_65><loc_85><loc_67></location>δt = f ( t ) , δx i = ζ i ( t, /vectorx ) . (A.1)</formula>
<text><location><page_20><loc_16><loc_62><loc_66><loc_63></location>The corresponding transformation on the ADM variables is</text>
<formula><location><page_20><loc_32><loc_53><loc_85><loc_61></location>δN = ζ k ∂ k N + f ˙ N + ˙ fN , δN i = ζ k ∂ k N i + N k ∂ i ζ k + f ˙ N i + ˙ fN i + ˙ ζ j g ij , δg ij = ζ k ∂ k g ij +2 g k ( i ∂ j ) ζ k + f ˙ g ij . (A.2)</formula>
<text><location><page_20><loc_16><loc_49><loc_85><loc_52></location>Minkowski spacetime is a vacuum solution of the theory without cosmological constant. We introduce the perturbative variables in the following way</text>
<formula><location><page_20><loc_31><loc_45><loc_85><loc_47></location>N = 1 + /epsilon1n , N i = /epsilon1n i , g ij = δ ij + /epsilon1h ij . (A.3)</formula>
<text><location><page_20><loc_16><loc_36><loc_85><loc_44></location>Since the transformation parameters f and ζ i are of linear order in perturbations, it is convenient to redefine them in the way f ( t ) → /epsilon1f ( t ) and ζ i ( t, /vectorx ) → /epsilon1ζ i ( t, /vectorx ), such that the new variables f and ζ i do not depend on the scale of perturbations /epsilon1 . The FDiff transformations of the perturbative variables are</text>
<formula><location><page_20><loc_46><loc_33><loc_85><loc_35></location>δn = ˙ f , (A.4)</formula>
<formula><location><page_20><loc_46><loc_31><loc_85><loc_33></location>δn i = ˙ ζ i , (A.5)</formula>
<formula><location><page_20><loc_46><loc_29><loc_85><loc_30></location>δh ij = 2 ∂ ( i ζ j ) . (A.6)</formula>
<text><location><page_20><loc_16><loc_13><loc_85><loc_27></location>Now, we impose the asymptotic conditions needed for asymptotic flatness. We require that n, n i , h ij → 0 as r → 0, and require that the parameters f ( t ) and ζ i ( t, /vectorx ) be compatible with these conditions. However, as we anticipated, there is a subtlety here since f ( t ) is a function only of time. If we require that f ( t ) goes to zero as r → ∞ , then necessarily f ( t ) = 0. The consequence for the linearized theory is that the perturbative variable n is actually a gauge invariant under linearized FDiff, δn = 0. Besides this, we require ζ i → 0 as r →∞ .</text>
<text><location><page_20><loc_16><loc_11><loc_85><loc_14></location>More information about the FDiff transformations (A.4 - A.6) is extracted when the vectors ζ i and n i and the tensor h ij are decomposed in transverse and longitu-</text>
<text><location><page_21><loc_16><loc_86><loc_62><loc_88></location>inal parts. We introduce the standard decompositions</text>
<formula><location><page_21><loc_30><loc_83><loc_85><loc_85></location>ζ i = ξ i + ∂ i χ, (A.7)</formula>
<formula><location><page_21><loc_30><loc_81><loc_85><loc_82></location>n i = m i + ∂ i b , (A.8)</formula>
<formula><location><page_21><loc_30><loc_76><loc_85><loc_80></location>h ij = h TT ij + 1 2 ( δ ij -∂ ij ∂ -2 ) h T + ∂ ( i h L j ) + ∂ ij ∂ -2 h L . (A.9)</formula>
<text><location><page_21><loc_16><loc_72><loc_85><loc_76></location>The vectors and tensors of the above decomposition are restricted by ∂ i ξ i = ∂ i m i = ∂ i h L i = ∂ i h TT ij = h TT kk = 0.</text>
<text><location><page_21><loc_16><loc_69><loc_85><loc_72></location>By performing the transverse and longitudinal decomposition on the transformations (A.4 - A.6), we obtain the decomposed FDiff gauge transformations, namely</text>
<formula><location><page_21><loc_47><loc_66><loc_85><loc_67></location>δn = 0 , (A.10)</formula>
<formula><location><page_21><loc_47><loc_64><loc_85><loc_65></location>δb = ˙ χ, (A.11)</formula>
<formula><location><page_21><loc_47><loc_61><loc_85><loc_63></location>δm i = ˙ ξ i , (A.12)</formula>
<formula><location><page_21><loc_47><loc_59><loc_85><loc_61></location>δh L = 2 ∂ 2 χ, (A.13)</formula>
<formula><location><page_21><loc_47><loc_57><loc_85><loc_59></location>δh L i = 2 ξ i , (A.14)</formula>
<formula><location><page_21><loc_47><loc_55><loc_85><loc_57></location>δh T = 0 , (A.15)</formula>
<formula><location><page_21><loc_47><loc_52><loc_85><loc_54></location>δh TT ij = 0 . (A.16)</formula>
<text><location><page_21><loc_16><loc_44><loc_85><loc_51></location>Actually, these transformations can be deduced directly from (3.11 - 3.17) by setting ζ 0 = 0, which we recall is required on the FDiff transformations by the asymptotic conditions. From these transformations we automatically extract that n , h T and h TT ij are FDiff-gauge invariant. The combinations</text>
<formula><location><page_21><loc_45><loc_40><loc_85><loc_43></location>Q ≡ ˙ h L -2 ∂ 2 b , (A.17)</formula>
<formula><location><page_21><loc_45><loc_37><loc_85><loc_40></location>v i ≡ ˙ h L i -2 m i (A.18)</formula>
<text><location><page_21><loc_16><loc_35><loc_37><loc_37></location>are also gauge invariants.</text>
<text><location><page_21><loc_16><loc_22><loc_85><loc_35></location>It is illustrating to contrast with the gauge invariants of linearized pure GR (without the T or aether field). In linearized GR the gauge symmetry is bigger, so we expect having less gauge invariants on the side of GR when the same field variables are used (ADM variables, in this case). Indeed, if the gauge transformation is extended to a general spacetime diffeomorfism, as in (3.11 - 3.17), then the variables h T , h TT ij and v i are still gauge invariants. n and Q are not separately invariant, but the combination of them,</text>
<formula><location><page_21><loc_43><loc_18><loc_85><loc_21></location>Φ = -2 ∂ 2 n + ˙ Q, (A.19)</formula>
<text><location><page_21><loc_16><loc_9><loc_85><loc_18></location>is. This is the invariant of GR defined in (3.21). There are seven functional degrees of freedom among the FDiff-gauge invariants h T , h TT ij , v i , n and Q , whereas there are six gauge invariants on the side of pure GR since the enhacement of the gauge symmetry leads to the lacking of one of them, only a combination of Q and n survives.</text>
<section_header_level_1><location><page_22><loc_16><loc_86><loc_57><loc_88></location>A.2 The linearized field equations</section_header_level_1>
<text><location><page_22><loc_16><loc_80><loc_85><loc_85></location>In this appendix we study the dynamics of the linearized theory in the FDiffcovariant formalism. For simplicity we consider here the pure gravity theory, without coupling to matter sources. The action of second order in derivatives is (2.1),</text>
<formula><location><page_22><loc_29><loc_74><loc_85><loc_78></location>S = 1 2 κ H ∫ dtd 3 x √ gN ( G ijkl K ij K kl + βR + αa i a i ) . (A.20)</formula>
<text><location><page_22><loc_16><loc_70><loc_85><loc_73></location>The equations of motion, obtained by taking variations with resptect to g ij , N and N i , are, respectively,</text>
<formula><location><page_22><loc_21><loc_47><loc_85><loc_69></location>1 √ g ∂ ∂t ( √ gG ijkl K kl ) + 2 G klm ( i | ∇ k ( K lm N | j ) ) -G ijkl ∇ m ( K kl N m ) +2 N ( K ik K k j -λKK ij ) -1 2 Ng ij G klmn K kl K mn + βN ( R ij -1 2 g ij R ) -β ( ∇ i ∇ j N -g ij ∇ 2 N ) + αN -1 ( ∇ i N ∇ j N -1 2 g ij ∇ k N ∇ k N ) = 0 , (A.21) G ijkl K ij K kl -βR +2 αN -2 ( N ∇ 2 N -1 2 ∇ i N ∇ i N ) = 0 , (A.22) G ijkl ∇ j K kl = 0 . (A.23)</formula>
<text><location><page_22><loc_16><loc_38><loc_85><loc_47></location>Equations (A.22) and (A.23) are constraints on the initial data since they do not contain second-order time derivatives of the field variables. These equation are the analogous of the Hamiltonian and momentum constraints of GR. The additional constraint arising at the kinetic-conformal point λ = 1 / 3 can be extracted from Eq. (A.21). Indeed, with λ = 1 / 3 the hypermatrix G ijkl becomes degenerated,</text>
<formula><location><page_22><loc_45><loc_35><loc_85><loc_36></location>g ij G ijkl = 0 , (A.24)</formula>
<text><location><page_22><loc_16><loc_24><loc_85><loc_33></location>g ij being its null eigenvector. In Eq. (A.21) the only term that has a second-order time derivative is the first one, specifically when ∂ ∂t acts on K kl . It is clear that if we take the trace of this equation with g ij , considering λ = 1 / 3, this second-order time derivative disappears due to (A.24). Thus, at λ = 1 / 3 the trace of Eq. (A.21) is another constraint of the theory. It can be written in the form</text>
<formula><location><page_22><loc_16><loc_18><loc_85><loc_23></location>NG ijkl K ij K kl -G ijkl ∇ i K jk N l +( 1 2 + λ ) ( K ij ∇ i N j -K ∇ k N k ) -( β -α 2 ) ∇ 2 N = 0 . (A.25)</formula>
<text><location><page_22><loc_16><loc_13><loc_85><loc_18></location>We have used the constraint (A.22) to bring the trace to this form. Therefore, at λ = 1 / 3 the constraints of the theory, in the Lagrangian formalism, are the Eqs. (A.22), (A.23) and (A.25).</text>
<text><location><page_22><loc_16><loc_9><loc_85><loc_12></location>Now we study the field equations (A.21), (A.22), (A.23) and (A.25) perturbatively. It turns out that they can be completely expressed and solved in terms of</text>
<text><location><page_23><loc_16><loc_84><loc_85><loc_88></location>the FDiff-gauge invariants introduced in the previous appendix. At linear order in perturbations, constraints (A.22) and (A.25) yield, respectively,</text>
<formula><location><page_23><loc_41><loc_81><loc_85><loc_83></location>β∂ 2 h T +2 α∂ 2 n = 0 , (A.26)</formula>
<formula><location><page_23><loc_43><loc_78><loc_85><loc_81></location>(2 β -α ) ∂ 2 n = 0 . (A.27)</formula>
<text><location><page_23><loc_30><loc_74><loc_30><loc_77></location>/negationslash</text>
<text><location><page_23><loc_39><loc_74><loc_39><loc_77></location>/negationslash</text>
<text><location><page_23><loc_16><loc_70><loc_85><loc_77></location>By assuming α = 2 β , β = 0 and the asymptotic behavior h T = O (1 /r ) and n = O (1 /r ) at r → ∞ , we have that these equations imply h T = n = 0 at linear order in perturbations. The perturbative version of the constraint (A.23), after substituting h T = 0 into it, takes the form</text>
<formula><location><page_23><loc_40><loc_66><loc_85><loc_68></location>∂ 2 v i +2(1 -λ ) ∂ i Q = 0 . (A.28)</formula>
<text><location><page_23><loc_16><loc_61><loc_85><loc_65></location>We recall that we are considering λ = 1 / 3. The spatial divergence of this equation yields the condition</text>
<formula><location><page_23><loc_47><loc_60><loc_85><loc_61></location>∂ 2 Q = 0 . (A.29)</formula>
<text><location><page_23><loc_16><loc_53><loc_85><loc_59></location>Assuming the asymptotic behavior Q = O (1 /r ) at r →∞ , this equations has Q = 0 as its only solution. Putting this back in Eq. (A.28) we obtain v i = 0 if v i = O (1 /r ) at spatial infinity.</text>
<text><location><page_23><loc_16><loc_48><loc_85><loc_53></location>Finally, we study the perturbative version of the field equation (A.21), droping out all the variables that we already known are zero. It yields the wave equation for h TT ij ,</text>
<formula><location><page_23><loc_42><loc_45><loc_85><loc_48></location>h TT ij -β∂ 2 h TT ij = 0 . (A.30)</formula>
<text><location><page_23><loc_16><loc_30><loc_85><loc_45></location>Summarizing, the linearized FDiff formulation also admits a representation in terms of FDiff gauge invariants. The field equations are completely analogous to the generally-covariant formulation (in the vacuum, in this case). The fundamental result is that h TT ij is the only propagating mode and that it is radiative (in the sense of the linearized theory). We remark that the vanishing of the gauge invariants h T , n , Q and v i holds only in the vacuum theory. If matter sources were present these variables were nonzero and their expressions were bounded to the sources in the sense we presented in the generally-covariant formulation.</text>
<section_header_level_1><location><page_23><loc_16><loc_25><loc_31><loc_27></location>References</section_header_level_1>
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</document> | [{"title": "Einstein's quadrupole formula from the kinetic-conformal Ho\u02c7rava theory", "content": "Jorge Bellor'\u0131n a, 1 and Alvaro Restuccia a,b, 2 1 [email protected], [email protected] 2", "pages": [1]}, {"title": "Abstract", "content": "We analyze the radiative and nonradiative linearized variables in a gravity theory within the familiy of the nonprojectable Ho\u02c7rava theories, the Ho\u02c7rava theory at the kinetic-conformal point. There is no extra mode in this formulation, the theory shares the same number of degrees of freedom with general relativity. The large-distance effective action, which is the one we consider, can be given in a generally-covariant form under asymptotically flat boundary conditions, the Einstein-aether theory under the condition of hypersurface orthogonality on the aether vector. In the linearized theory we find that only the transverse-traceless tensorial modes obey a sourced wave equation, as in general relativity. The rest of variables are nonradiative. The result is gauge-independent at the level of the linearized theory. For the case of a weak source, we find that the leading mode in the far zone is exactly Einstein's quadrupole formula of general relativity, if some coupling constants are properly identified. There are no monopoles nor dipoles in this formulation, in distinction to the nonprojectable Horava theory outside the kinetic-conformal point. We also discuss some constraints on the theory arising from the observational bounds on Lorentz-violating theories.", "pages": [1]}, {"title": "1 Introduction", "content": "Gravitational waves have recently been detected [1, 2, 3]. The detected signals fit well with the waves produced by the coalescense of binary systems of black holes, according to the predictions of General Relativity (GR). This detection constitutes one of the most important recent achievements in the study of gravitational phenomena, and it is another success of GR. On the other hand, there are motivations to study alternatives or modifications to GR. One important issue is that GR is not renormalizable under the perturbative approach, thus, at least in the perturbative scheme, it cannot be a fundamental theory by itself. There is also the issue of the dark matter, for which there has not been found any candidate in the particle experiments or space observations. Therefore, for any proposed modification of GR a question comes out inmediatly: how close to the detected wave signals and the corresponding predictions of GR is the radiation predicted by the new theory?. Here we focus on the study of the production and propagation of gravitational waves at the leading order in a Lorentz-violating theory. The theory [4] belongs to the family of the nonprojectable Ho\u02c7rava theories [5, 6]. The heart of the Ho\u02c7rava proposal [5] is to introduce a preferred timelike direction that breaks the symmetry between space and time characteristic of relativistic theories. This is done with the aim of introducing higher order spatial derivatives in the Lagrangian that improve the renormalizability of the theory while, in principle, preserve its unitarity. The special formulation studied in Ref. [4], where only the purely gravitational theory without coupling to matter sources was analyzed, consists of setting a specific value of the kinetic coupling constant for which two additional second-class constraints emerge. The constant is usually denoted by \u03bb and the special value in 3 + 1 dimensions is \u03bb = 1 / 3. The additional constraints at \u03bb = 1 / 3 eliminate the extra scalar mode that otherwise the nonprojectable Ho\u02c7rava theory exhibits. Because of this, it is reasonable to expect that this formulation tends to stay more close to GR, at least in the low-energy regime, where the lowest order operators are the most relevant ones 1 . Since the special value \u03bb = 1 / 3 is related to a conformal symmetry on the kinetic term of the Lagrangian [5], in Ref. [9] we called this formulation the Ho\u02c7rava theory at the kinetic-conformal point (the KCP Ho\u02c7rava theory, for short). We stress that the theory is not conformally invariant, only its kinetic term is. In [9] the power-counting renormalizability as well as the absence of ghosts in the theory were shown. A recent report on the status of the Ho\u02c7rava theory, dealing with its several versions, can be found in Ref. [10]. We comment that the value \u03bb = 1 / 3 leading to the kinetic-conformal formulation is fixed by the dynamics, it does not get quantum corrections. This is due to the second-class constraints of the theory, not to symmetries. Further discussion can be found in Ref. [11]. We study the gravitational waves at leading order in the large-distance effective action of the theory. This effective action is of second order in time and spatial derivatives. It admits a generally-covariant version which is the Einstein-aether theory [12] under the restriction of hypersurface orthogonality on the aether vector [13, 14]. We deal with the generally-covariant formulation since it allows a more direct comparison with the standard approaches of GR devoted to gravitational waves. We analyse the perturbatively linearized generally-covariant theory coupled to a generic weak matter source. We develop all the analysis in terms of gauge-invariant variables of the linearized theory. These are combinations of the metric and the aether field that remain invariant under linearized general diffeomorphisms. This formulation allows us to get totally gauge-invariant results. Once we determine what variables are related to the sources by Poisson equations, hence they are nonradiative, and what variables are governed by the wave equation, we study the generation of the waves at the leading order. We follow the standard procedure of approximating the solution by considering that it is produced by a source that at the leading order has negligible self-gravity, it is in a slow motion regime, and that the observation is made far enough from the source (at the wave zone). Our study is close to Refs. [15, 16, 17]. In Ref. [15] a perturbative analysis of the unrestricted Einstein-aether theory without matter sources was done. The linearized vacuum equations of motion of the several modes, which are homogeneous wave equations for each mode with different speeds, were studied there. In Ref. [16] the lowest order multipole moments were studied for the unrestricted Eintein-aether theory coupled to a weak source. Our analysis differs from these two studies due to (besides the absence of sources in [15]) the lower number of propagating degrees of freedom we have and the fact that the equations of motion of the nonprojectable Ho\u02c7rava theory are not equivalent to the ones obtained by substituting the hypersurface orthogonality condition in the equations of motion of the Einstein-aether theory [14]. /negationslash In Ref. [17] the Einstein-aether theory with the condition of hypersurface orthogonality imposed at the level of the action was studied (this theory is also called the khronometric theory). The analysis is rigorously consistent for \u03bb = 1 / 3. In [17] the wave equations with sources were found for the tensorial modes and the extra mode, as well as the Poissonian equations for the nonradiative modes. They also found the dominant modes in the multipolar expansion for a weak source. These results are affected by the presence of the extra mode. Our study differs from the one of Ref. [17] because we take the KCP theory independently with its intrinsic degrees of freedom. As we shall see, this has important consequences on the radiation formulas. In general, the KCP formulation cannot be obtained rigurously as a limit of the theory with the extra mode due to the discontinuity in the number of constraints and, in particular, in the number of propagating modes. Our approach is consistent in the \u03bb = 1 / 3 case since we obtain the formulas of the radiation directly from the KCP theory. However, if one wants a quick comparison with the non-kinetic-conformal case, heuristically our formulas coincide with the radiation formulas of Ref. [17] in the limiting case of sending to infinity the speed of the extra mode (this divergence is induced by the \u03bb = 1 / 3 value). However, in general the reinterpretation of an hyperbolic equation (the wave equation of the extra mode) as an elliptic equation is not consistent (for example, the initial data problem). /negationslash In addition to the study of the gravitational radiation, here we consider some observational implications on the kinetic-conformal theory. Our aim is to highlight that some of the observational bounds applicable to the Ho\u02c7rava theory with \u03bb = 1 / 3 must be addressed in a different way in the kinetic-conformal case. In Ho\u02c7rava theory observational bounds are frequently combined with theoretical restrictions needed for the consistency of the extra mode. In the kinetic-conformal theory this is not necessary since there is no extra mode. Another important issue is the cosmological-scale solutions. We have argued that they may arise in a different way in the kinetic-conformal case [11]. We further comment on this point below. In particular, here we compare with the observational bounds coming from binary pulsars found in Ref. [18, 19], since these phenomena are related to wave production. Those authors studied the Einstein-aether theory both unrestricted and with the hypersurface orthogonality condition (with \u03bb = 1 / 3). They found stringent constraints on the space of coupling constants of these theories. Here we show how the kinetic-conformal theory stays more close to GR, in particular there is no dipolar contribution at the level of the dominant modes. /negationslash This paper is organized as follows: in section 2 we summarize the analysis of Ref. [14] to present the Einstein-aether theory under the restriction of hypersurface orthogonality, together with its equivalence to the second-order action of the nonprojectable Ho\u02c7rava theory. In section 3.1 we discuss the gauge invariants of the generally-covariant theory that can be formed by combining the metric variables with the hypersurface-orthogonal aether field. In section 3.2 we present and analyze the linearized field equations, coupled to a matter source, in terms of these gauge invariants. In section 3.3 we study the leading mode for the production of waves far from the source, obtaining the quadrupole formula of Einstein. In section 4 we discuss some observational bounds. Finally we present some conclusions. Since it is also interesting to analyze the linearized field equations in the FDiff-covariant language, which is the original formulation of the Ho\u02c7rava theory [5], we add one appendix to present the FDiff-gauge invariants and the field equations in terms of them.", "pages": [2, 3, 4]}, {"title": "2 The covariant version of the Ho\u02c7rava theory", "content": "The Einstein-aether theory [12] is a modification of GR that incorporates an everywhere timelike unit vector field, called the aether, as a fundamental field. Since the aether is considered dynamical, the action possesses the symmetry of general diffeomorphisms that is also present in GR. However, at the level of the solutions, the presence of the aether field breaks the local Lorentz symmetry. There is a relationship [13, 14] between the Einstein-aether theory and the action of second order in derivatives of the nonprojectable Ho\u02c7rava theory, which is also a theory with a preferred frame. Throghout this paper we deal only with the second-order action (excluding the cosmological constant), since at low precision the physics of the gravitational waves can be described by it. In the following we summarize the relationship between these two theories. The Ho\u02c7rava theory [5] was originally formulated in terms of the standard ArnowittDeser-Misner (ADM) variables N , N i and g ij , in such a way that the action possesses the symmetry of the diffeomorphisms that preserve a given foliation (FDiff) along a timelike direction. The Lagrangian of the nonprojectable theory, which is the version we study here, depends on the spatial curvature and the spatial derivatives of the lapse function N . These arise in the Lagrangian in terms of the FDiff-covariant vector a i = \u2202 i ln N [6]. The action of second order in derivatives, which we call the z = 1 action, is where is the extrinsic curvature of the spacelike leaves \u03a3 of the foliation. The dot denotes the time derivative, \u02d9 g ij = \u2202g ij /\u2202t . K \u2261 g ij K ij , (3) R is the scalar curvature of \u03a3, and \u03ba H , \u03bb , \u03b2 and \u03b1 are coupling constants. In the above all the coupling constants are in principle arbitrary. Now, in the purely gravitational theory it is known that a scalar degree of freedom, additional to the transverse-traceless tensorial modes, is eliminated from the phase space if the coupling constant \u03bb is set to the value \u03bb = 1 / 3 [4, 9]. With this value of \u03bb the kinetic term in (2.1) acquires a conformal invariance [5], although the full theory is not conformal since in general the terms in the potential break the conformal symmetry (except for very specific terms). For this reason the value \u03bb = 1 / 3 was called the kinetic-conformal point in Ref. [9]. As we have mentioned, this feature raises interest in studying this special formulation of the nonprojectable Ho\u02c7rava theory, as it is our case in this paper, since it becomes closer to GR. The Einstein-aether theory [12] is physically equivalent to the z = 1 nonprojectable Ho\u02c7rava theory (2.1) (for all \u03bb ) if the aether vector is restricted to be hypersurface orthogonal. The total equivalence between the two theories, at the level of their Lagrangians, holds only if the restriction on the aether vector is imposed at the level of the action, i. e. before deriving the equations of motion [13, 14]. Here we take the Einstein-aether action from Ref. [14], considering also the coupling to matter sources. The full generally-covariant action is given by S Total = S EA + S Matter , where is the Einstein-aether action. u \u00b5 is the aether vector, which in general is subject to the condition of being a timelike unit vector, u \u00b5 u \u00b5 = -1. \u03ba EA is the Einstein-aether gravitational constant. M \u03b1\u03b2\u03b3\u03b4 is the hypermatrix where c 1 , c 2 , c 3 and c 4 are coupling constants. With the aim of minimizing Lorentzbreaking effects in the matter sector, where experimental bounds are highly restrictive, it is required that the matter sources do not couple to the aether field (see discussion in Refs. [20, 16]). Then, S Matter [ g \u00b5\u03bd , \u03c8 ] is the action for the matter sector, with \u03c8 representing the matter sources in a generic way. As a consequence, the equations of motion of the sources maintain the same structures they have in GR. The restriction of hypersurface orthogonality on u \u00b5 is equivalent (locally) to express u \u00b5 in terms of a scalar function T = T ( t, /vectorx ) that satisfies the condition of its gradient is timelike, \u2202 \u03b1 T\u2202 \u03b1 T < 0. The hypersurface-orthogonal aether vector is written in terms of T as Under this restriction the functional degrees of freedom originally contained in u \u00b5 are reduced to the one of T once (2.5) has been substituted in the action (2.3). Actually, definition (2.5), which automatically implies that u \u00b5 is a timelike unit vector, depends on the norm of the gradient of T , hence the hypersurface-orthogonal u \u00b5 is a composite object made with the T field and the metric g \u03b1\u03b2 . The equation of motion of the T field, that is, the equation of motion derived from (2.3) by taking variations with respect to T , is implied by the Einstein equations and the matter equations of motion [14]. Since this fact is crucial for our study, let us repeat the argument that supports it. The main point is that T is a single, nonzero-gradient, scalar field coupled to gravity in a generally-covariant way. Since S Total is invariant under general diffeomorphisms, we have that, under a diffeomorphism parameterized by \u03b6 \u00b5 , Now suppose that this identity is evaluated on configurations that satisfy the Einstein and matter equations. Over such configurations identity (2.6) becomes Since T cannot be constant along all possible directions and the above condition must be satisfied by all vectors \u03b6 \u00b5 , whe have that \u03b4S Total /\u03b4T = \u03b4S EA /\u03b4T = 0 automatically for all configurations that satisfy the Einstein equations and the matter equations of motion. The physical equivalence between the action (2.3), restricted by (2.5), and the action (2.1) can be seen as follows [14]. The object is a spatial projector, whereas P \u03b1\u03b2 is the induced metric on the spatial hypersurfaces. The extrinsic curvature and the acceleration vector are defined, respectively, by Since u \u00b5 is hypersurface orthogonal K \u00b5\u03bd is a symmetric tensor. K \u00b5\u03bd and a \u00b5 are spatial objects, K \u00b5\u03bd u \u03bd = a \u00b5 u \u00b5 = 0. We may decompose the covariant derivative of u \u00b5 in terms of these objects, Now, Since the T field equation need not be imposed explicitly and this a theory with general covariance, we can take T as the time coordinate, T = t . By doing so we break the symmetry of general diffeomorphisms over the spacetime. In addition, we can write the spacetime metric in the ADM variables N , N i and g ij . With these settings we have that the aether part of the Lagrangian in (2.3) takes the form In addition, \u221a -g = \u221a (3) gN and the decomposition of R adds K ij K ij -K 2 + (3) R to the Lagrangian. By putting all this in the action (2.3), we have that the z = 1 Ho\u02c7rava action (2.1) is reproduced from it if the coupling constants of both theories are identified according to Therefore, the z = 1 Ho\u02c7rava action (2.1) is a gauge-fixed version of the hypersurfaceorthogonal Einstein-aether action given in (2.3) and (2.5).", "pages": [4, 5, 6, 7]}, {"title": "3.1 Gauge invariants with the T field", "content": "Now we focus on the linearized generally-covariant theory. Minkowski spacetime, whose metric we denote by \u03b7 \u03b1\u03b2 , is a solution of the theory in absence of matter sources and with the condition T = t , which yields a zero aether energy-momentum tensor. We introduce the perturbative variables by expanding around this solution in the way The dependence of h \u00b5\u03bd and \u03c4 on the spacetime coordinates is arbitrary, except for the asymptotic conditions h \u00b5\u03bd , \u03c4 \u2192 0 as r \u2192\u221e . We investigate the possible gauge invariants of the linearized theory that can be formed with the metric h \u00b5\u03bd and the \u03c4 field. Under an arbitrary diffeomorphism over the spacetime, given by the exact spacetime metric g \u00b5\u03bd and the exact scalar field T transform as On the perturbative variables these transformations take the form where we have used the background metric to lower the index, \u03b6 \u00b5 = \u03b7 \u00b5\u03bd \u03b6 \u03bd . For the compatibility with the asymptotic conditions on the field variables we require that \u03b6 \u00b5 \u2192 0 as r \u2192 0. Now it is convenient to introduce the transverse and longitudinal decompositions for \u03b6 i , h 0 i and h ij . They are given by The symbol \u2202 ij \u00b7\u00b7\u00b7 k stands for \u2202 i \u2202 j \u00b7 \u00b7 \u00b7 \u2202 k , \u2202 2 is the flat Euclidean Laplacian, \u2202 2 \u2261 \u2202 kk , and \u2202 -2 is its inverse, \u2202 -2 \u2261 ( \u2202 2 ) -1 . The restrictions on the variables are \u2202 i \u03be i = \u2202 i m i = \u2202 i h L i = \u2202 i h TT ij = h TT ii = 0. For the uniqueness of the decompositions and the compatibility with the asymptotic behavior of the original field variables, we asume the asymptotic conditions By substituting (3.7 - 3.9) in the transformation (3.5), we obtain that it becomes From these transformations we extract that h T and h TT ij are gauge invariants. By combining with (3.6), we may define three variables that are also gauge invariants, namely This approach is rather different to the standard approach of linearized GR (see, for example, [21]), since the scalar \u03c4 is involved in the definition of the gauge invariants p and q . The gauge invariant of linearized GR that depends only on the metric components is Here \u03a6 is not an independent quantity since it can be obtained from a combination of p and q , \u03a6 = \u2202 2 p + \u02d9 q . Therefore, in the linearized theory that depends on h \u00b5\u03bd and \u03c4 and that is generally covariant, the independent gauge invariants are h T , h TT ij , p , q and v i . In Appendix A we show that an analogous construction of gauge invariants can be done for the case of the theory formulated with the FDiff-gauge symmetry.", "pages": [7, 8, 9]}, {"title": "3.2 Linearized Einstein equations", "content": "Here we study the linearized field equations. We consider the presence of matter sources, hence there is an active energy-momentum tensor T matter \u00b5\u03bd for the matter. We define the energy-momentum tensors T aether \u00b5\u03bd and T matter \u00b5\u03bd in such a way that the Einstein equations take the form with the usual expression G \u00b5\u03bd \u2261 R \u00b5\u03bd -1 2 g \u00b5\u03bd R . We decompose T matter \u00b5\u03bd in the way where the variables are restricted by \u2202 i S i = \u2202 i \u03c3 L i = \u2202 i \u03c3 TT ij = \u03c3 TT ii = 0. The linearized Einstein equations can be completely expressed in terms of the gauge invariants defined in the previous section. Indeed, after the decompositions (3.8 - 3.9) and (3.23 - 3.25) are done and the gauge invariants (3.18 - 3.20) are introduced, the linearized Einstein equations take the form Equation (3.26) is the 00 component of the Einstein equations, Eqs. (3.27) and (3.28) come from the 0 i components and the last four equations constitute the ij components. We have used relations (2.12) to change the constants \u03ba EA and c 1 , 2 , 3 , 4 of the generally-covariant formulation by the constants \u03ba H , \u03bb , \u03b2 and \u03b1 of the FDiff-covariant formulation since the latter are the ones that the linearized theory naturally adopts. We stress that no gauge-fixing condition has been imposed to obtain these equations. All the variables of the left-hand sides belong to the set of gauge invariants of the linearized theory. In Appendix A we show that if one uses the original FDiff-invariant formulation of the z = 1 Ho\u02c7rava theory, then the linearized field equations can also be written purely in terms of the corresponding FDiff-gauge invariants. Evidently, Eqs. (3.27), (3.28) (3.29) and (3.30) imply the following conditions on the source, Consequently, we drop Eqs. (3.29) and (3.30) out from the list of independent Einstein equations and impose Eqs. (3.33 - 3.34) as complementary conditions that must be satisfied by the matter source (equations (3.33 - 3.34) are independent of h \u00b5\u03bd and \u03c4 ). Let us analyze the system of equations (3.26 - 3.32) as a set of equations for the gauge invariants with the matter source given and, momentarily, without imposing any restriction on the coupling constants. Equations (3.26), (3.27) and (3.28) do not depend on the second time derivative of any variable. Hence, they are constraints on the initial data. Specifically, Eq. (3.26) corresponds to the Hamiltonian constraint whereas Eqs. (3.27) and (3.28) constitute the momentum constraint (see the canonical formulation for the theory at the vaccum in [22] for the case out of the KCP and in [4] for the case at the KCP). On the other hand, Eqs. (3.31) and (3.32) do depend on the second time derivate, so they are the ones that govern the propagation of the dynamical modes. In this covariant formalism we have the ten components of the metric field and the scalar field \u03c4 , which sum up eleven field variables. Four of these must be fixed by a coordinate-system choice and the remaining seven are subject to the system (3.26 - 3.32), which has just seven independent equations (we recall that Eqs. (3.29) and (3.30) have been dropped out and that Eqs. (3.28) and (3.32) yield four independent equations). Therefore, the system is closed for the field variables with the matter source given. The seven gauge-independent variables are subject to four constraints, Eqs. (3.26 - 3.28), hence in this general theory there remain three propagating physical degrees of freedom, whose evolution is governed by Eqs. (3.31) and (3.32). One can identify two of these propagating modes as the two tensorial modes of GR. The remaining mode is an extra scalar mode. Now we move to the case of our interest. It is evident that Eq. (3.31) changes its character of evolution equation if we set the coupling constant \u03bb to In this case Eq. (3.31) lacks its dependence on the second time derivative h T , hence it becomes an additional constraint. Notice that the change of evolution equations by constraints is not a smooth one. The kinetic-conformal theory is an independent theory on its own. In terms of the Einstein-aether constants this condition is c 1 + c 3 + 3 c 2 = -2. Let us write here the resulting set of independent field equations under the condition (3.35), The four first equations are constraints that fix the gauge invariants h T , p , q and v i , whereas the last one is the evolution equation for the tranverse-traceless tensorial mode h TT ij . Therefore, under the condition (3.35), which defines the kineticconformal point, the extra mode is annihilated, in agreement with the vacuum theory [4, 9], and the propagating physical degrees of freedom are the same of GR, which are described by h TT ij . By imposing some bounds on the coupling constants \u03b1 and \u03b2 , we can ensure that the constraints form a closed system of partial differential equations and that the evolution equation for h TT ij is the sourced wave equation. The conditions on the coupling constants are /negationslash Under these conditions the formal solutions of the constraints (3.36 - 3.39) are Equations (3.40) and (3.42 - 3.45) tell us that h T , p , q and v i are variables bounded to the sources whereas h TT ij is the radiative variable. Therefore, as in GR, h TT ij acquires a pure physical meanning as the only radiative field. The speed of the waves h TT ij is \u221a \u03b2 . This agrees with Ref. [15] in what concerns the h TT ij mode. However, as we have already mentioned, in the unrestricted Einsteinaether theory studied in Ref. [15] there are three additional modes propagating themselves with wave equations and with different speeds. In Ref. [16] the same linearized equations were studied with matter sources. We remark again that the total equivalence between the hypersurface orthogonal Einstein-aether theory and the nonprojectable z = 1 Ho\u02c7rava theory holds only if the condition of hypersurface orthogonality is imposed at the level of the action. Therefore, the linearized z = 1 Ho\u02c7rava equations at the kinetic-conformal point, whose covariant version is (3.36 - 3.40), are not obtained by direct substitution of the hypersurface orthogonality condition on the equations analyzed in Refs. [15, 16].", "pages": [9, 10, 11, 12]}, {"title": "3.3 The quadrupole formula", "content": "In the standard perturbative scheme (post-Minkowskian approach), the exact solution is increasely approximated by the perturbative solution if the nolinear terms of the field equations are considered as sources for the fields at the order of interest. For example, the perturbative wave equation (3.40) at higher order in perturbations can be casted as where h TT ij in the left-hand side is evaluated at the order of interest. t ij represents the nonlinear terms coming from the Einstein tensor and the aether energy-momentum tensor. The solutions for all the variables at lower orders must be substituted in t ij and T matter ij such that the desired order in perturbations is reached in all terms of this equation. In this iterative scheme the linearized field equations determine the leading contribution. Here we extract information from the solution of the linearized wave equation relevant for the physics at large distances from the source. The solution of Eq. (3.40) with no incoming radiation is Following the outline of Ref. [23], our next steps consist of approximating this expression according to weakness criteria: the observer is far from the source, the self-gravity of the source is negligible and the motion of the source is sufficiently slow. Then, by defining \u02c6 n = /vectorx/r with r = | /vectorx | , solution (3.47) can be expanded in the form The leading mode of this expansion is Let us massage this expression following a procedure similar of the one of Ref. [16]. One can handle first the integration (and the expansion already done) in terms of the full energy-momentum tensor and then perform the projection to the transversetraceless sector. At order O (1 /r ) the projection to the transverse sector is equivalent to the algebraic projection to the plane orthogonal to \u02c6 n . The projector to this plane is \u03b8 ij \u2261 \u03b4 ij -n i n j , and the operator that projects tensors and substracts the trace is P TT ijkl \u2261 \u03b8 ik \u03b8 jl -1 2 \u03b8 ij \u03b8 kl . Then we have Similarly to the criterium of order used Ref. [16], we require that the equations of motion of the matter source, approximated to the order of interest, imply the Newtonian law of mass conservation, namely 2 This and Eqs. (3.33 - 3.34) are equivalent to the preservation of the matter energymomentum tensor, \u2202 \u00b5 T \u00b5\u03bd matter = 0. Therefore, the requisite (3.51) is equivalent to demand that, at the linear order in perturbations and at the nonrelativistic limit, the equations of motion of the matter sources imply the conservation of its energymomentum tensor (actually, the spatial components \u2202 \u00b5 T \u00b5i matter = 0, which are given by Eqs. (3.33) and (3.34), are already implied by the Einstein equations). Following the standard approach of GR, from the conservation of the matter energymomentum tensor at the order considered one obtains the relation Therefore, the leading contribution for the generation of gravitational waves has the same structure of Einstein's quadrupole formula that arises in GR, The only difference this formula has with respect to Einstein's quadrupole formula is the presence of the coupling constants \u03ba H and \u03b2 of the Ho\u02c7rava or Einstein-aether theory. If these constants are adjusted to the GR values \u03ba H = 8 \u03c0G N and \u03b2 = 1, then (3.53) becomes identical to Einstein's quadrupole formula.", "pages": [12, 13, 14]}, {"title": "4 On the observational bounds", "content": "In this section we discuss some observational bounds on the theory at the kineticconformal point. Some of the formulas we use can be directly deduced from the analysis previously done in the literature of the nonprojectable Horava theory with general \u03bb . However, there are features that do not follow as a particular case of the general theory, essentially due to the discontinuity in the number of degrees of freedom. /negationslash We comment that in the (projectable and nonprojectable) Ho\u02c7rava theory with \u03bb = 1 / 3 there are theoretical restrictions on the coupling constants that are necessary for the stability of the extra mode. Although these conditions are widely used, they do not apply in the kinetic-conformal formulation due to the obvious reason that there is no extra mode. For example, there is a restriction on \u03bb given by [6] /negationslash necessary to avoid that the extra mode becomes a ghost at the level of the linearized theory. The interpretation in the kinetic-conformal formulation, \u03bb = 1 / 3, is simply that there is no such bound. We highlight this point since the bounds coming from the physics of the extra mode in the case \u03bb = 1 / 3 are frequently combined with the observational bounds when the phenomenology of the theory in under scrutiny (see, for example, [24, 19]), hence extrapoling directly the conclusions from the \u03bb = 1 / 3 case can be misleading in the kinetic-conformal case. /negationslash We start with the observational bounds of the weak regime englobed in the parametrized-post-Newtonian (PPN) parameters of the solar-system tests. The PPN parameters of the kinetic-conformal theory can be obtained from the general nonprojectable Ho\u02c7rava theory with arbitrary \u03bb since the discrepancy in the propagating degrees of freedom does not affect the PPN potentials. The PPN parameters for the nonprojectable Ho\u02c7rava theory were computed in Ref. [17] (see also [25]), using the covariant formulation of the second-order in derivatives action, i. e., the hypersurface-orthogonal Einstein-aether theory. The PPN parameters are obtained for a weak, non-relativistic source, and the procedure is similar to the one of the Einstein-aether theory [20]. It results that the hypersurface orthogonal Einsteinaether theory (as well as the unrestricted Einstein-aether theory) reproduces the same values of the PPN constants of GR, except for the parameters \u03b1 1 and \u03b1 2 , whose nonzero values signal violations of the Lorentz symmetry. The expressions obtained in [17] for these constants, written with our conventions for the coupling constants and for general \u03bb , are Notice that \u03b1 1 is independent of \u03bb . For \u03bb = 1 / 3, the parameter \u03b1 2 becomes The current observational bounds on these parameters, which are dimensionless, are | \u03b1 1 | < 10 -4 and | \u03b1 2 | < 10 -7 [26]. In the kinetic-conformal theory the relation (4.4) demands that the strong bound, which is the one on \u03b1 2 , must be satisfied by both parameters. Taking this into account, we use relation (4.2) to solve one coupling constant in terms of the other one, where \u03b4 represents the narrow observational window for the \u03b1 2 parameter, i. e. | \u03b4 | < 10 -7 . With (4.5) the kinetic-conformal Ho\u02c7rava theory satisfies all the conditions derived from the PPN analysis of the solar-system tests. Nonrelativistic gravitational theories (coupled to relativistic particles) produce Cherenkov radiation if the velocities of the gravitational modes are lower than the speed of the relativistic particles [27]. The implications of this have been studied for the Einstein-aether theory in Refs. [28, 24], obtaining very stringent lower bounds on the coupling constants. In the Einstein-aether theory the lower bounds affect several coupling constants since this theory has several propagating modes, each one with a different dependence of its velocity on the coupling constants. In the kineticconformal Ho\u02c7rava theory the lower bounds coming from the Cherenkov radiation are very simple to implement since the propagating modes are the same of GR. We have seen that, when the theory is truncated to its second-order effective action, the squared velocity of the transverse-traceless tensorial modes of the linearized theory is \u03b2 , hence in this case the bounds affect only to this constant. Therefore, the Cherenkov radiation puts the bound We comment that at higher energies the higher order operators could be relevant for this analysis, hence other coupling constants can enter in the game. Now we want to make some considerations about the kinetic-conformal theory at cosmological scales, since there is an important restriction on this theory at this scale. The restriction concerns to the Lagrangian formulation of the theory, but in the Hamiltonian formalism it is still an open question [11]. In the field equations derived from the Lagrangian, if the ansatz for a full homogeneous and isotropic metric is imposed, together with a homogeneous and isotropic perfect fluid, then it turns out that the only possibility left by the field equations is that the density and pressure vanish. With full homogeneous and isotropic we mean that these conditions are imposed in all the components of the spacetime metric. This restriction can be deduced from the analysis of Ref. [6], where the extension of the nonprojectable theory and its first cosmological application were presented (the Einstein-aether theory exhibits an analogous behavior [29, 30]). There it was found that the effective cosmological gravitational constant arising in the Friedmann equations differs from the Newtonian (local) gravitational constant by a scale factor that depends on \u03bb . At the kinetic-conformal point, \u03bb = 1 / 3, this scale factor diverges. What really this divergence means is the vanishing of the density and the pressure at the kineticconformal point, as we have commented. We point out two issues concerning this restriction. The first one is that in the kinetic-conformal theory the equations of motion derived from the Hamiltonian admits more solutions than the Lagrangian field equations [11]. This is essentially due to the role played by the Lagrange multipliers once all the constraints have been added to the Hamiltonian. On certain configurations admissible for the Lagrange multipliers, the Legendre transformation cannot be inverted, hence the Hamiltonian lost the equivalence with the Lagrangian. The second issue is that the Ho\u02c7rava theory is originally formulated as a theory with the symmetry of the FDiff, it is not a generally covariant theory. Indeed, the equivalence with the hypersurface orthogonal Einstein-aether theory holds only at the level of the action of second-order in derivatives and on the Lagrangian formulations. Under the reduced FDiff symmetry, the conditions of homogeneity and isotropy can be restricted to the spatial metric, whereas the laspe function and the shift vector can have a more general dependence on the time and the space. These two observations have been discussed in Ref. [11]. Moreover, the analysis of the cosmological-scale configurations in the kinetic-conformal Ho\u02c7rava gravity requires a previous and deep analysis on the structure of the second-class constraints of the theory. Perhaps a reformulation of the theory only in terms of first-class constraints is convenient. Under such scenario, some of the original second-class constraints could be circumvented by recurring to different 'gauge fixing' conditions. Thus, the configurations of cosmological scale would require a better gauge fixing than, for example, the \u03c0 = 0 condition that arises originally as a second-class constraint. Therefore, we consider that the kinetic-conformal point, \u03bb = 1 / 3, is not necessarily ruled out by the restriction about homogenous and isotropic configurations mentioned above. Finally, we contrast with the bounds coming from binary pulsars, which are strong sources, that were obtained in Refs. [18, 19] on two Lorentz-violating theories: the Einstein-aether theory and the hypersurface-orthogonal Einstein aether theory without the kinetic-conformal condition (called the krhonometric theory in those references). The authors of [18, 19] arrive at very stringent constraints on the space of coupling constants of these theories after contrasting with the observations on several binary pulsars. For the khronometric theory in particular, the region of the space of parameters on which the theory can reproduce the decay rate of orbital period within the observational error excludes the kinetic-conformal point \u03bb = 1 / 3. This is because they combine the analysis on the binary pulsar with the bounds resulting from the stability of the extra mode, the Cherenkov radiation and the cosmological-scale effect of rescaling the gravitational constant. As we have discussed, the stability of the extra mode does not apply in the kinetic-conformal theory. The Cherenkov radiation only constraint the constant \u03b2 , one can put \u03b2 /greaterorsimilar 1 consistently on the kinetic-conformal theory. The rescaling of the gravitational constant at cosmological scales leads to the apparent restriction at \u03bb = 1 / 3 that we discussed above. Upon the arguments we have given we consider that it does not rule out unavoidably the kinetic-conformal formulation of the theory. Indeed, it is interesting to extrapolate the formulas of the orbital evolution of the binary pulsars obtained in [18, 19] to the kinetic-conformal case. This orbital evolution is related to the effective multipoles of the theory, and we have shown in the previous sections that in the kinetic-conformal theory the dominant mode in the far zone of a weak source is the same of GR. We reproduce here the rate of change of the orbital period obtained in [19], which uses previous results of [16, 31, 17], G is the effective gravitational constant in the binary system, G \u00c6 is the gravitational constant of the theory (related to \u03ba EA and \u03ba H ), a is the semi-major axis, m \u2261 m 1 + m 2 , and \u00b5 \u2261 m 1 m 2 /m . The quadrupole moment Q ij is the trace-free part of the system's mass quadrupole moment I ij : Q ij is the trace-free part of the rescaled mass quadrupole moment I ij : where s A are constants associated to the sensitivities of the system, which are parameters encoding the departing of bodies' wordlines from the relativistic trayectories. The dipolar moment \u03a3 i is We have written only the dominant modes for the multipoles. Further details can be found in [19, 16, 31, 17]. The coefficients arising in (4.7), using our conventions for the coupling constants, are given by where and are the velocities of the tensorial modes and the extra scalar mode respectively. Notice that A 1 is the only coefficient with information about the propagation of the tensorial modes. In (4.7) the angled-brackets stand for an average over several wavelengths. Now, it is easy to see how all the contributions of the extra mode disappear and only the quadrupole contribution remains as the dominant mode. By substituting \u03bb = 1 / 3 in the velocity c s given in (4.13), we get that c s diverges, which is an informal way to express that the extra mode gets frozen. Then all coefficients in (4.11) vanish except for A 1 , which becomes A 1 = 1 / \u221a \u03b2 . Equation (4.7) becomes This results expresses that in the kinetic-conformal theory the quadrupole mode is the dominant radiative contribution to the rate of the orbital period decay, as in GR. Again, we may put \u03b2 = 1 consistently in the kinetic-conformal theory.", "pages": [14, 15, 16, 17, 18]}, {"title": "Conclusions", "content": "Our central results are that, first, in the Lorentz-violating, power-counting renormalizable (once the higher order operators are considered) and unitary theory coupled to matter sources that we have considered, the only radiative degrees of freedom are the same transverse-traceless tensorial modes of GR, and that, second, at the leading order the Einstein quadrupole formula is reproduced (if two coupling constant are adjusted to their GR values). We have worked in the gauge-invariant formalism of the linearized theory, hence our results are not gauge-fixing artifacts. We have clearly identified the set of the gauge-invariant variables that are nonradiative and the ones that are radiative. The nonradiative variables are linked to the sources by Poissonian equations. It is interesting that the kinetic-conformal Ho\u02c7rava theory is able to reproduce the same leading mode of GR in what concerns the production and propagation of gravitational waves, but with a better quantum behavior. In Ref. [9] we showed the power-counting renormalizability of this theory without matter sources by analyzing the superficial degree of divergence of general one-particle irreducible diagrams, as well as its unitarity. There remains to prove its complete renormalizability. Of course, a study of the next post-Newtonian orders is needed to make an exhaustive comparison between the gravitational waves produced and propagated in the kinetic-conformal Ho\u02c7rava theory and the detected signals. Our study is the first step towards this goal since it is restricted to the leading order that can be extracted from a weak source. At higher orders, the other field variables besides h TT ij are also relevant for the wave production and propagation due to the nonlinearities of the theory. We have also considered some observational bounds. The PPN analysis fixes one of the coupling constants of the action of second-order in derivatives, such that all solar-system tests are satisfied by the theory after this restriction is imposed. Cherenkov radiation puts a lower bound on other coupling constant. The \u03bb = 1 / 3 value that defines the kinetic-conformal condition is not discarded by the bounds coming from binary pulsars showed in [18, 19], since these bounds were combined with theoretical restrictions and cosmological considerations that do not necessarily apply to our case, as we have argued. Indeed, we have shown, based on the formulas of [19], how the rate of decay of orbital period is very close to its corresponding expression in GR at the dominant level. In particular, there is no dipolar contribution to this decay, unlike the theories considered in [18, 19].", "pages": [19]}, {"title": "Acknowledgments", "content": "A. R. is partially supported by grant Fondecyt No. 1161192, Chile. J. B. is partially supported by the Programa MECE Educaci'on Superior of Ministerio de Educaci'on, Chile.", "pages": [19]}, {"title": "A.1 The FDiff gauge invariants", "content": "In this appendix we study the formulation of linearized gauge invariants when the gauge symmetry is given by the diffeomorphisms that preserve a given foliation, the FDiff. Since there is a subtlety when implementing the FDiff transformations together with a prescribed asymptotic-flatness condition on the field variables, let us start by presenting the FDiff without asymptotic conditions. The diffeomorphisms that preserve a given foliation act on the coordinates ( t, /vectorx ) in the following way The corresponding transformation on the ADM variables is Minkowski spacetime is a vacuum solution of the theory without cosmological constant. We introduce the perturbative variables in the following way Since the transformation parameters f and \u03b6 i are of linear order in perturbations, it is convenient to redefine them in the way f ( t ) \u2192 /epsilon1f ( t ) and \u03b6 i ( t, /vectorx ) \u2192 /epsilon1\u03b6 i ( t, /vectorx ), such that the new variables f and \u03b6 i do not depend on the scale of perturbations /epsilon1 . The FDiff transformations of the perturbative variables are Now, we impose the asymptotic conditions needed for asymptotic flatness. We require that n, n i , h ij \u2192 0 as r \u2192 0, and require that the parameters f ( t ) and \u03b6 i ( t, /vectorx ) be compatible with these conditions. However, as we anticipated, there is a subtlety here since f ( t ) is a function only of time. If we require that f ( t ) goes to zero as r \u2192 \u221e , then necessarily f ( t ) = 0. The consequence for the linearized theory is that the perturbative variable n is actually a gauge invariant under linearized FDiff, \u03b4n = 0. Besides this, we require \u03b6 i \u2192 0 as r \u2192\u221e . More information about the FDiff transformations (A.4 - A.6) is extracted when the vectors \u03b6 i and n i and the tensor h ij are decomposed in transverse and longitu- inal parts. We introduce the standard decompositions The vectors and tensors of the above decomposition are restricted by \u2202 i \u03be i = \u2202 i m i = \u2202 i h L i = \u2202 i h TT ij = h TT kk = 0. By performing the transverse and longitudinal decomposition on the transformations (A.4 - A.6), we obtain the decomposed FDiff gauge transformations, namely Actually, these transformations can be deduced directly from (3.11 - 3.17) by setting \u03b6 0 = 0, which we recall is required on the FDiff transformations by the asymptotic conditions. From these transformations we automatically extract that n , h T and h TT ij are FDiff-gauge invariant. The combinations are also gauge invariants. It is illustrating to contrast with the gauge invariants of linearized pure GR (without the T or aether field). In linearized GR the gauge symmetry is bigger, so we expect having less gauge invariants on the side of GR when the same field variables are used (ADM variables, in this case). Indeed, if the gauge transformation is extended to a general spacetime diffeomorfism, as in (3.11 - 3.17), then the variables h T , h TT ij and v i are still gauge invariants. n and Q are not separately invariant, but the combination of them, is. This is the invariant of GR defined in (3.21). There are seven functional degrees of freedom among the FDiff-gauge invariants h T , h TT ij , v i , n and Q , whereas there are six gauge invariants on the side of pure GR since the enhacement of the gauge symmetry leads to the lacking of one of them, only a combination of Q and n survives.", "pages": [20, 21]}, {"title": "A.2 The linearized field equations", "content": "In this appendix we study the dynamics of the linearized theory in the FDiffcovariant formalism. For simplicity we consider here the pure gravity theory, without coupling to matter sources. The action of second order in derivatives is (2.1), The equations of motion, obtained by taking variations with resptect to g ij , N and N i , are, respectively, Equations (A.22) and (A.23) are constraints on the initial data since they do not contain second-order time derivatives of the field variables. These equation are the analogous of the Hamiltonian and momentum constraints of GR. The additional constraint arising at the kinetic-conformal point \u03bb = 1 / 3 can be extracted from Eq. (A.21). Indeed, with \u03bb = 1 / 3 the hypermatrix G ijkl becomes degenerated, g ij being its null eigenvector. In Eq. (A.21) the only term that has a second-order time derivative is the first one, specifically when \u2202 \u2202t acts on K kl . It is clear that if we take the trace of this equation with g ij , considering \u03bb = 1 / 3, this second-order time derivative disappears due to (A.24). Thus, at \u03bb = 1 / 3 the trace of Eq. (A.21) is another constraint of the theory. It can be written in the form We have used the constraint (A.22) to bring the trace to this form. Therefore, at \u03bb = 1 / 3 the constraints of the theory, in the Lagrangian formalism, are the Eqs. (A.22), (A.23) and (A.25). Now we study the field equations (A.21), (A.22), (A.23) and (A.25) perturbatively. It turns out that they can be completely expressed and solved in terms of the FDiff-gauge invariants introduced in the previous appendix. At linear order in perturbations, constraints (A.22) and (A.25) yield, respectively, /negationslash /negationslash By assuming \u03b1 = 2 \u03b2 , \u03b2 = 0 and the asymptotic behavior h T = O (1 /r ) and n = O (1 /r ) at r \u2192 \u221e , we have that these equations imply h T = n = 0 at linear order in perturbations. The perturbative version of the constraint (A.23), after substituting h T = 0 into it, takes the form We recall that we are considering \u03bb = 1 / 3. The spatial divergence of this equation yields the condition Assuming the asymptotic behavior Q = O (1 /r ) at r \u2192\u221e , this equations has Q = 0 as its only solution. Putting this back in Eq. (A.28) we obtain v i = 0 if v i = O (1 /r ) at spatial infinity. Finally, we study the perturbative version of the field equation (A.21), droping out all the variables that we already known are zero. It yields the wave equation for h TT ij , Summarizing, the linearized FDiff formulation also admits a representation in terms of FDiff gauge invariants. The field equations are completely analogous to the generally-covariant formulation (in the vacuum, in this case). The fundamental result is that h TT ij is the only propagating mode and that it is radiative (in the sense of the linearized theory). We remark that the vanishing of the gauge invariants h T , n , Q and v i holds only in the vacuum theory. If matter sources were present these variables were nonzero and their expressions were bounded to the sources in the sense we presented in the generally-covariant formulation.", "pages": [22, 23]}] |
2019PhRvD.100j4021V | https://arxiv.org/pdf/1906.07738.pdf | <document>
<section_header_level_1><location><page_1><loc_14><loc_91><loc_90><loc_94></location>Anisotropic fluid spheres in Hořava gravity and Einstein-æther theory with a non-static æther</section_header_level_1>
<text><location><page_1><loc_46><loc_88><loc_58><loc_89></location>Daniele Vernieri</text>
<text><location><page_1><loc_27><loc_85><loc_77><loc_88></location>Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências da Universidade de Lisboa, Campo Grande, PT1749-016 Lisboa, Portugal</text>
<text><location><page_1><loc_42><loc_84><loc_61><loc_85></location>(Dated: November 19, 2019)</text>
<text><location><page_1><loc_18><loc_76><loc_85><loc_83></location>In this paper we consider spherically symmetric interior spacetimes filled by anisotropic fluids in the context of Hořava gravity and Einstein-æther theory. We assume a specific non-static configuration of the æther vector field and show that the field equations admit a family of exact analytical solutions which can be obtained if one of the two metric coefficients is assigned. We study as an illustrative example the case in which the metric of the interior spacetime reproduces the Newtonian potential of a fluid sphere with constant density.</text>
<section_header_level_1><location><page_1><loc_43><loc_72><loc_61><loc_73></location>I. INTRODUCTION</section_header_level_1>
<text><location><page_1><loc_9><loc_31><loc_95><loc_70></location>Hořava gravity was proposed in 2009 as a power-counting renormalizable theory of quantum gravity [1, 2]. In the past years much work has been done to show that the theory is renormalizable [3-6] beyond the power-counting arguments [7-12]. Hořava gravity has also been severely constrained by means of some tests at both astrophysical [1315] and cosmological scales [16, 17], and it passes all of them with flying colors. Moreover it is also consistent with the constraint on the speed of propagation of gravitational waves coming from the near-simultaneous temporal and spatial observation of the gravitational-wave event GW170817 and the gamma-ray burst GRB 170817A [18, 19]. The theory breaks Lorentz invariance at any energy scale since a preferred direction is naturally encoded in its formulation. This locally amounts to having a timelike hypersurface-orthogonal æther vector field which is defined in each point of the spacetime. If one considers the low-energy limit of Hořava gravity in a covariant form, the latter proves to be equivalent to Einstein-æther theory [20] once the æther vector is taken to be hypersurface-orthogonal at the level of the action [21]. In spherical symmetry, any vector is automatically hypersurface-orthogonal; therefore by virtue of this fact it can be shown that the two theories share the same solutions in such a background [22]. Because of the intrinsic highly non-linear structure of the field equations, only a few analytical and numerical solutions are known both in vacuum [23-28] and inside matter [29-32]. Thus, it is really necessary to focus more effort in this direction, since many of the phenomenological implications of the theory are still unknown, even in highly symmetric spacetimes. For this purpose in the present manuscript we consider spherically symmetric interior spacetimes filled by anisotropic fluids [33-35] in the context of the low-energy limit of Hořava gravity. The approach that we undertake here is similar to the one used in Refs. [31, 32] in which the equation of state of the inner fluid is left unspecified, but after a viable solution has been found, it can be instead reconstructed a posteriori by using the same method exploited in Ref. [32]. This approach generically looks more realistic since, despite all the work that has been done until now, we still lack a proper modeling of the interior spacetime of relativistic objects (see Ref. [36] and references therein). Then it seems appropriate to leave unspecified the equation of state relating the thermodynamical quantities, whose study is anyhow out of the scope of the present paper. In Refs. [31, 32], this kind of setting has already been studied, but in the more restricted case of a static æther, which means that the æther vector is aligned with the timelike Killing vector and then has only one non-vanishing component. Here we consider instead a more general ansatz where the æther vector field indeed has two non-trivial components. We derive the corresponding field equations and find a strategy to analytically solve them by means of some choice of the æther components and an appropriate redefinition of variables. In this framework, we show that the field equations admit a family of infinite exact analytical solutions, characterized by choosing arbitrarily one of the metric coefficients of the spherically symmetric interior spacetime.</text>
<text><location><page_1><loc_9><loc_21><loc_94><loc_31></location>In Sec. II, we present the gravitational action and the field equations of Hořava gravity and discuss the equivalence of its low-energy limit to Einstein-æther theory when the æther is chosen to be hypersurface-orthogonal at the level of the action. In Sec. III, we introduce the spherically symmetric background metric, the corresponding æther vector field, and the stress-energy tensor suitable for the anisotropic fluid description. In Sec. IV, we write down explicitly the system of the independent field equations in terms of the metric, the æther components, and the thermodynamical quantities. In Sec. V we focus on a specific case in which the interior metric reproduces the Newtonian potential of a fluid sphere with constant density. In Sec. VI, the conclusions are drawn.</text>
<section_header_level_1><location><page_2><loc_27><loc_92><loc_77><loc_94></location>II. HOŘAVA GRAVITY AND EINSTEIN-ÆTHER THEORY</section_header_level_1>
<text><location><page_2><loc_11><loc_89><loc_70><loc_90></location>The action of Hořava gravity [1, 2] as written in the preferred foliation looks like</text>
<formula><location><page_2><loc_20><loc_84><loc_94><loc_88></location>S H = 1 16 πG H ∫ dTd 3 x √ -g ( K ij K ij -λK 2 + ξ R + ηa i a i + L 4 M 2 ∗ + L 6 M 4 ∗ ) + S m [ g µν , ψ ] , (1)</formula>
<text><location><page_2><loc_9><loc_76><loc_94><loc_84></location>where G H is the effective gravitational coupling constant; g is the determinant of the metric g µν ; R is the Ricci scalar of the three-dimensional constantT hypersurfaces; K ij is the extrinsic curvature and K is its trace; and a i = ∂ i ln N , where N is the lapse function and S m is the matter action for the matter fields collectively denoted by ψ . The couplings { λ, ξ, η } are dimensionless, and general relativity (GR) is identically recovered when they take the values { 1 , 1 , 0 } , respectively. Finally, L 4 and L 6 collectively denote the fourth-order and sixth-order operators that make the theory power-counting renormalizable, and M ∗ is a characteristic mass scale which suppresses them.</text>
<text><location><page_2><loc_9><loc_72><loc_94><loc_76></location>In what follows, we consider the covariantized version of the low-energy limit of Hořava gravity, named the khronometric model, that is obtained by discarding the higher-order operators in L 4 and L 6 . In order to write it in a covariant form, let us consider the action of Einstein-æther theory [20]; that is,</text>
<formula><location><page_2><loc_33><loc_67><loc_94><loc_71></location>S æ = 1 16 πG æ ∫ d 4 x √ -g ( -R + L æ ) + S m [ g µν , ψ ] , (2)</formula>
<text><location><page_2><loc_9><loc_64><loc_94><loc_67></location>where G æ is the 'bare' gravitational constant; R is the four-dimensional Ricci scalar; u a is a timelike vector field of unit norm, i.e. , g µν u µ u ν = 1 , from now on referred to as the 'æther'; and</text>
<formula><location><page_2><loc_42><loc_61><loc_94><loc_63></location>L æ = -M αβ µν ∇ α u µ ∇ β u ν , (3)</formula>
<text><location><page_2><loc_9><loc_59><loc_26><loc_60></location>with M αβ µν defined as</text>
<formula><location><page_2><loc_33><loc_56><loc_94><loc_58></location>M αβ µν = c 1 g αβ g µν + c 2 δ α µ δ β ν + c 3 δ α ν δ β µ + c 4 u α u β g µν , (4)</formula>
<text><location><page_2><loc_9><loc_54><loc_44><loc_55></location>where c i 's are dimensionless coupling constants.</text>
<text><location><page_2><loc_9><loc_51><loc_94><loc_53></location>Then, one can take the æther vector to be hypersurface-orthogonal at the level of the action, which locally amounts to choosing</text>
<formula><location><page_2><loc_44><loc_45><loc_94><loc_50></location>u α = ∂ α T √ g µν ∂ µ T∂ ν T , (5)</formula>
<text><location><page_2><loc_9><loc_42><loc_94><loc_45></location>where in the covariant formulation, the preferred time T becomes a scalar field (the khronon ) which defines the preferred foliation. Finally, the two actions in Eqs. (1) and (2) are shown to be equivalent if the parameters of the two respective theories are mapped into each other as [21]</text>
<formula><location><page_2><loc_34><loc_37><loc_94><loc_40></location>G H G æ = ξ = 1 1 -c 13 , λ ξ = 1 + c 2 , η ξ = c 14 , (6)</formula>
<text><location><page_2><loc_9><loc_34><loc_94><loc_36></location>where c ij = c i + c j . In what follows, we consider the covariant formulation of Hořava gravity in order to perform the calculations.</text>
<text><location><page_2><loc_11><loc_31><loc_76><loc_32></location>The variation of the action in Eq. (2) with respect to g αβ and T yields, respectively [15],</text>
<formula><location><page_2><loc_41><loc_26><loc_94><loc_29></location>G αβ -T æ αβ = 8 πG æ T m αβ , (7)</formula>
<formula><location><page_2><loc_41><loc_23><loc_94><loc_27></location>∂ µ ( 1 √ ∇ α T ∇ α T √ -g Æ µ ) = 0 , (8)</formula>
<text><location><page_2><loc_11><loc_20><loc_48><loc_22></location>where G αβ = R αβ -Rg αβ / 2 is the Einstein tensor,</text>
<formula><location><page_2><loc_21><loc_12><loc_94><loc_19></location>T æ αβ = ∇ µ ( J µ ( α u β ) -J µ ( α u β ) -J ( αβ ) u µ ) + c 1 [( ∇ µ u α )( ∇ µ u β ) -( ∇ α u µ )( ∇ β u µ )] + [ u ν ( ∇ µ J µν ) -c 4 ˙ u 2 ] u α u β + c 4 ˙ u α ˙ u β -1 2 L æ g αβ +2 Æ ( α u β ) (9)</formula>
<text><location><page_3><loc_9><loc_92><loc_35><loc_94></location>is the khronon stress-energy tensor,</text>
<formula><location><page_3><loc_20><loc_88><loc_94><loc_90></location>J α µ = M αβ µν ∇ β u ν , ˙ u ν = u µ ∇ µ u ν , Æ µ = ( ∇ α J αν -c 4 ˙ u α ∇ ν u α ) ( g µν -u µ u ν ) , (10)</formula>
<text><location><page_3><loc_11><loc_86><loc_50><loc_87></location>and T m αβ is the matter stress-energy tensor, defined as</text>
<formula><location><page_3><loc_45><loc_81><loc_94><loc_85></location>T m αβ = 2 √ -g δS m δg αβ . (11)</formula>
<section_header_level_1><location><page_3><loc_10><loc_79><loc_94><loc_80></location>III. SPHERICALLY SYMMETRIC METRIC, ANISOTROPIC FLUIDS, AND A NON-STATIC ÆTHER</section_header_level_1>
<text><location><page_3><loc_11><loc_75><loc_59><loc_77></location>In spherical symmetry, the most general metric can be written as</text>
<text><location><page_3><loc_9><loc_70><loc_94><loc_75></location>ds 2 = A ( r ) dt 2 -B ( r ) dr 2 -r 2 ( dθ 2 +sin 2 θdφ 2 ) . (12) Moreover, in what follows, we will consider the interior spacetime of a fluid sphere filled by an anisotropic fluid whose stress-energy tensor is given by</text>
<formula><location><page_3><loc_35><loc_66><loc_94><loc_68></location>T µν = ( ρ + p t ) v µ v ν -p t g µν +( p r -p t ) s µ s ν , (13)</formula>
<text><location><page_3><loc_9><loc_63><loc_94><loc_65></location>where ρ is the density, p r and p t are the radial and transversal pressure, respectively, v µ denotes the 4-velocity of the fluid</text>
<text><location><page_3><loc_9><loc_57><loc_38><loc_59></location>and s µ is a spacelike 4-vector defined as</text>
<formula><location><page_3><loc_43><loc_57><loc_94><loc_62></location>v µ = ( 1 √ A ( r ) , 0 , 0 , 0 ) , (14)</formula>
<formula><location><page_3><loc_43><loc_51><loc_94><loc_57></location>s µ = ( 0 , 1 √ B ( r ) , 0 , 0 ) , (15)</formula>
<text><location><page_3><loc_9><loc_50><loc_95><loc_52></location>satisfying the relations s µ s µ = -1 and s µ u µ = 0 . It can be easily shown that the components of the stress-energy tensor can be explicitly written as</text>
<formula><location><page_3><loc_41><loc_45><loc_94><loc_49></location>T ν µ = diag ( ρ, -p r , -p t , -p t ) . (16)</formula>
<text><location><page_3><loc_9><loc_44><loc_95><loc_46></location>The æther vector field, which is by definition a timelike vector of unit norm, in spherical symmetry is always hypersurface-orthogonal and takes the following general form:</text>
<formula><location><page_3><loc_38><loc_38><loc_94><loc_42></location>u α = ( F ( r ) , √ A ( r ) F ( r ) 2 -1 B ( r ) , 0 , 0 ) . (17)</formula>
<text><location><page_3><loc_9><loc_33><loc_95><loc_37></location>The independent field equations that we have to consider are the modified Einstein equations (0-0), (1-1), (1-2) and (2-2) in Eq. (7); the equation for the scalar field T in Eq. (8); and the conservation equation for the stress-energy tensor of anisotropic matter that is,</text>
<formula><location><page_3><loc_34><loc_30><loc_94><loc_33></location>p ' r ( r ) + [ ρ ( r ) + p r ( r )] A ' ( r ) 2 A ( r ) = 2 r [ p t ( r ) -p r ( r )] . (18)</formula>
<text><location><page_3><loc_9><loc_24><loc_94><loc_29></location>The expressions of the field equations are very long and awful, so they are not displayed here. However, through a direct inspection it is quite straightforward to notice that the field equations are considerably simplified by making the choice F ( r ) = q √ A ( r ) , and the æther vector in Eq. (17) then becomes</text>
<formula><location><page_3><loc_40><loc_18><loc_94><loc_24></location>u α = ( q √ A ( r ) , √ q 2 -1 B ( r ) , 0 , 0 ) . (19)</formula>
<text><location><page_3><loc_9><loc_14><loc_94><loc_19></location>Let us stress that, even by implementing this specific assumption, the æther vector field has anyhow two non-trivial components. Then, it is still more general than the æther vector widely used in literature and referred to as 'static æther' which is by definition aligned with the timelike Killing vector. The choice of a static æther is just a special case which is included in the more general framework developed here by setting q = 1 in Eq. (19).</text>
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<text><location><page_4><loc_24><loc_19><loc_25><loc_20></location>λ</text>
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<text><location><page_4><loc_56><loc_19><loc_57><loc_20></location>ξ</text>
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<text><location><page_4><loc_74><loc_22><loc_75><loc_23></location>2</text>
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<text><location><page_4><loc_81><loc_19><loc_82><loc_20></location>λ</text>
<text><location><page_4><loc_83><loc_19><loc_86><loc_20></location>+1)</text>
<text><location><page_4><loc_86><loc_19><loc_87><loc_20></location>Y</text>
<text><location><page_4><loc_87><loc_19><loc_88><loc_20></location>(</text>
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<section_header_level_1><location><page_4><loc_41><loc_92><loc_62><loc_94></location>IV. FIELD EQUATIONS</section_header_level_1>
<text><location><page_4><loc_9><loc_88><loc_94><loc_90></location>In order to write down the field equations in a more compact form, let us consider the following redefinition of variables:</text>
<formula><location><page_4><loc_39><loc_84><loc_94><loc_87></location>Y ( r ) = r A ' ( r ) A ( r ) , W ( r ) = r B ' ( r ) B ( r ) . (20)</formula>
<text><location><page_4><loc_9><loc_82><loc_69><loc_83></location>Then, the (0-0) component of the modified Einstein equations in Eq. (7) becomes</text>
<formula><location><page_4><loc_16><loc_72><loc_94><loc_79></location>1 8 ξ ( η -λ +1) r 2 B ( r ) [ 8 ξB ( r )( η -λ +1) + 8 ( -η +2 λ 2 -3 λ + q 2 ( η -λ +1)(2 λ -ξ -1) + 1 ) + Y ( r ) ( λ + q 2 ( η -λ +1) -1 ) (8( λ -ξ ) + ( -η + λ -1) Y ( r )) + 8 W ( r )( ηλ -λξ + ξ ) ] = 8 πG ae ρ ( r ) , (21)</formula>
<text><location><page_4><loc_9><loc_71><loc_26><loc_72></location>the (1-1) component is</text>
<formula><location><page_4><loc_24><loc_61><loc_94><loc_69></location>1 8 ξr 2 B ( r ) [ -8 ξB ( r ) + 16 λ +8 q 2 ( -2 λ + ξ +1) + ( λ + q 2 ( η -λ +1) -1 ) Y ( r ) 2 +8 ( λ -λq 2 + ξq 2 ) Y ( r ) -8 ] = 8 πG ae p r ( r ) , (22)</formula>
<text><location><page_4><loc_9><loc_60><loc_29><loc_62></location>and the (2-2) component is</text>
<formula><location><page_4><loc_13><loc_49><loc_94><loc_58></location>1 8 ξ ( η -λ +1) r 2 B ( r ) [ 16( λ -1) ( λ ( q 2 -1 ) -ξq 2 ) +4 W ( r ) ( -2 ηλ + η +2 λξ -3 λ + q 2 ( -ξ ( η +3 λ +1) + 2 ηλ -η +3 λ +2 ξ 2 -1 ) +1 ) + Y ( r ) ( -4 ξ ( q 2 ( η +3 λ +1) -2 λ ) +4 ( q 2 -1 ) ( η + λ (2 λ -1) + 1) + 8 ξ 2 q 2 -( η -λ +1) ( -λ + q 2 ( η + λ -2 ξ +1) -1 ) Y ( r ) ) ] = 8 πG ae p t ( r ) . (23)</formula>
<text><location><page_4><loc_9><loc_48><loc_56><loc_49></location>Moreover the modified Einstein equation (1-2) can be written as</text>
<formula><location><page_4><loc_23><loc_42><loc_94><loc_46></location>Y ' ( r ) = 8 -8 λ + W ( r ) [ -4 λ +4 ξ +( η -λ +1) Y ( r )] -2 Y ( r )( η + λ -2 ξ +1) 2 r ( η -λ +1) , (24)</formula>
<text><location><page_4><loc_9><loc_39><loc_94><loc_41></location>which has already been substituted in Eqs. (21)-(23). Finally, Eq. (8) for the scalar field and Eq. (18) for the stress-energy tensor conservation (after Eqs. (21)-(23) have been used) become, respectively,</text>
<formula><location><page_4><loc_15><loc_27><loc_94><loc_36></location>(2 W ( r ) -3 Y ( r )) [ -8 λ + W ( r )( -4 λ +4 ξ +( η -λ +1) Y ( r )) -2 Y ( r )( η + λ -2 ξ +1) + 8] r 2 -2 W ' ( r ) [ -4 λ +4 ξ +( η -λ +1) Y ( r )] r -2 Y ' ( r ) [ -4( η -ξ +1) + 3( η -λ +1) W ( r ) -3( η -λ +1) Y ( r )] r +4( η -λ +1) Y '' ( r ) = 0 , (25)</formula>
<text><location><page_4><loc_9><loc_25><loc_12><loc_26></location>and</text>
<text><location><page_4><loc_92><loc_14><loc_94><loc_15></location>(26)</text>
<text><location><page_5><loc_9><loc_84><loc_94><loc_94></location>However, by substituting Eq. (24) and its first derivative in Eqs. (25) and (26), these are identically satisfied. So, we are finally left with only four independent field equations, i.e. , Eqs. (21)-(24). This means that, by assigning one of the two metric coefficients, A ( r ) or B ( r ) , the other can be obtained by solving the differential Eq. (24), and the thermodynamical variables ρ ( r ) , p r ( r ) , and p t ( r ) can be read from Eqs. (21), (22) and (23), respectively. In this way we have obtained a family of infinite exact analytical solutions of the aforementioned system of equations if one of the two metric functions A ( r ) or B ( r ) is assigned. Notice that spherically symmetric solutions in Hořava gravity are identical to those of Einstein-æther theory; therefore, our conclusions will hold for both theories all the same [22].</text>
<section_header_level_1><location><page_5><loc_11><loc_79><loc_92><loc_82></location>V. SOLUTION WHICH REPRODUCES THE POTENTIAL FOR A CONSTANT-DENSITY FLUID SPHERE</section_header_level_1>
<text><location><page_5><loc_9><loc_75><loc_94><loc_77></location>We are now ready to work out the full system of field equations (21)-(24). Let us notice that Eq. (24) can be generically solved for W ( r ) , which turns out to be</text>
<formula><location><page_5><loc_28><loc_69><loc_94><loc_73></location>W ( r ) = 2 (4( λ -1) + r ( η -λ +1) Y ' ( r ) + Y ( r )( η + λ -2 ξ +1)) -4 λ +4 ξ +( η -λ +1) Y ( r ) , (27)</formula>
<text><location><page_5><loc_9><loc_67><loc_52><loc_68></location>and by making A ( r ) and B ( r ) , explicit the latter becomes</text>
<formula><location><page_5><loc_13><loc_62><loc_94><loc_65></location>rB ' ( r ) B ( r ) = -2 ( r 2 ( -η + λ -1) A ' ( r ) 2 + rA ( r ) ( r ( η -λ +1) A '' ( r ) + 2( η -ξ +1) A ' ( r )) + 4( λ -1) A ( r ) 2 ) A ( r ) ( r ( -η + λ -1) A ' ( r ) + 4 A ( r )( λ -ξ )) . (28)</formula>
<text><location><page_5><loc_9><loc_58><loc_94><loc_61></location>The equation above can be solved by assigning one of the two metric functions A ( r ) or B ( r ) . Then the system of field equations is closed, and a family of infinite exact and analytical solutions can be found.</text>
<text><location><page_5><loc_9><loc_53><loc_94><loc_58></location>As an illustrative example let us choose the analytic form of A ( r ) which reminds the Newtonian potential of a fluid sphere of constant density: i.e. , A ( r ) = a + br 2 , where a and b are arbitrary constants. Moreover, this choice also corresponds to the well-known Tolman IV solution in GR for an isotropic fluid [37]. Substituting in Eq. (28) the expression given above for A ( r ) , we obtain</text>
<formula><location><page_5><loc_24><loc_47><loc_94><loc_51></location>rB ' ( r ) B ( r ) = -4 a 2 ( λ -1) + 2 abr 2 ( -3 η -3 λ +2 ξ +1) -2 b 2 r 4 ( η +3 λ -2 ξ -1) ( a + br 2 ) (2 a ( λ -ξ ) -br 2 ( η -3 λ +2 ξ +1)) . (29)</formula>
<text><location><page_5><loc_9><loc_45><loc_72><loc_47></location>This is a first-order ordinary differential equation that can be easily integrated to give</text>
<formula><location><page_5><loc_26><loc_40><loc_94><loc_44></location>B ( r ) = B 0 r 2( λ -1) ξ -λ [ 2 a ( λ -ξ ) -br 2 ( η -3 λ +2 ξ +1) ] 1+ 2 η η -3 λ +2 ξ +1 + λ -1 λ -ξ ( a + br 2 ) 2 , (30)</formula>
<text><location><page_5><loc_9><loc_37><loc_36><loc_38></location>where B 0 is an integration constant.</text>
<text><location><page_5><loc_9><loc_35><loc_94><loc_37></location>It is now straightforward to get algebraically from Eqs. (21)-(23) the explicit analytical expressions for the thermodynamical quantities ρ ( r ) , p r ( r ) , and p t ( r ) that are shown below:</text>
<formula><location><page_5><loc_14><loc_13><loc_94><loc_33></location>ρ ( r ) = 1 64 πB 0 ξG æ ( η -λ +1) ( a + br 2 ) 2 r 2( ξ -1) λ -ξ ( 2 a ( λ -ξ ) -br 2 ( η -3 λ +2 ξ +1) ) -2 η η -3 λ +2 ξ +1 + 1 -λ λ -ξ -1 × [ -16( ηλ -λξ + ξ ) ( 2 a 2 ( λ -1) + abr 2 (3 η +3 λ -2 ξ -1) + b 2 r 4 ( η +3 λ -2 ξ -1) ) ( a + br 2 ) (2 a ( λ -ξ ) -br 2 ( η -3 λ +2 ξ +1)) + 8 B 0 ξ ( η -λ +1) r 2( λ -1) ξ -λ ( 2 a ( λ -ξ ) -br 2 ( η -3 λ +2 ξ +1) ) 2 η η -3 λ +2 ξ +1 + λ -1 λ -ξ +1 ( a + br 2 ) 2 -4 br 2 ( λ + q 2 ( η -λ +1) -1 ) ( 4 a ( ξ -λ ) + br 2 ( η -5 λ +4 ξ +1) ) ( a + br 2 ) 2 +8 ( -η +2 λ 2 -3 λ + q 2 ( η -λ +1)(2 λ -ξ -1) + 1 ) ] , (31)</formula>
<text><location><page_6><loc_9><loc_81><loc_12><loc_82></location>and</text>
<text><location><page_6><loc_9><loc_24><loc_12><loc_26></location>and</text>
<text><location><page_6><loc_9><loc_14><loc_94><loc_17></location>The implementation of the junction conditions to the exterior vacuum metric amounts to requiring that the radial pressure p r ( r ) in Eq. (32) vanish at r = ¯ R . Then, p r [ ¯ R ] = 0 results in</text>
<formula><location><page_6><loc_12><loc_83><loc_94><loc_94></location>p r ( r ) = 1 64 πB 0 ξG æ ( a + br 2 ) 2 r 2( ξ -1) λ -ξ ( 2 a ( λ -ξ ) -br 2 ( η -3 λ +2 ξ +1) ) -2 η η -3 λ +2 ξ +1 + 1 -λ λ -ξ -1 × [ 4 b 2 r 4 ( λ + q 2 ( η -λ +1) -1 ) ( a + br 2 ) 2 -8 B 0 ξr 2( λ -1) ξ -λ ( 2 a ( λ -ξ ) -br 2 ( η -3 λ +2 ξ +1) ) 2 η η -3 λ +2 ξ +1 + λ -1 λ -ξ +1 ( a + br 2 ) 2 + 16 br 2 ( λ -λq 2 + ξq 2 ) a + br 2 +16 λ +8 q 2 ( -2 λ + ξ +1) -8 ] , (32)</formula>
<formula><location><page_6><loc_14><loc_59><loc_94><loc_79></location>p t ( r ) = 1 64 πB 0 ξG æ ( η -λ +1) ( a + br 2 ) 2 r 2( ξ -1) λ -ξ ( 2 a ( λ -ξ ) -br 2 ( η -3 λ +2 ξ +1) ) -2 η η -3 λ +2 ξ +1 + 1 -λ λ -ξ -1 × [ -8 ( -2 ηλ + η +2 λξ -3 λ + q 2 ( η (2 λ -ξ -1) -( ξ -1)(3 λ -2 ξ -1)) + 1 ( a + br 2 ) (2 a ( λ -ξ ) -br 2 ( η -3 λ +2 ξ +1)) ) × ( 2 a 2 ( λ -1) + abr 2 (3 η +3 λ -2 ξ -1) + b 2 r 4 ( η +3 λ -2 ξ -1) ) + 2 br 2 a + br 2 ( -2 br 2 ( η -λ +1) ( -λ + q 2 ( η + λ -2 ξ +1) -1 ) a + br 2 -4 ξ ( q 2 ( η +3 λ +1) -2 λ ) +4 ( q 2 -1 ) ( η + λ (2 λ -1) + 1) + 8 ξ 2 q 2 ) +16( λ -1) ( λ ( q 2 -1 ) -ξq 2 ) ] . (33)</formula>
<text><location><page_6><loc_9><loc_56><loc_94><loc_59></location>We also calculate the limit of the above expressions for large values of the radius r /greatermuch 1 , when the constants a and b are sufficiently small:</text>
<formula><location><page_6><loc_14><loc_39><loc_94><loc_54></location>ρ ( r /greatermuch 1) = 1 64 πB 0 ξG ae ( η -λ +1) b 2 r 2( ξ -1) λ -ξ +4 ( -br 2 ( η -3 λ +2 ξ +1) ) -2 η η -3 λ +2 ξ +1 + 1 -λ λ -ξ -1 × [ 8 B 0 ξ ( η -λ +1) r 2( λ -1) ξ -λ -4 ( -br 2 ( η -3 λ +2 ξ +1) ) 2 η η -3 λ +2 ξ +1 + λ -1 λ -ξ +1 b 2 + 16( η +3 λ -2 ξ -1)( ηλ -λξ + ξ ) η -3 λ +2 ξ +1 +8 ( -η +2 λ 2 -3 λ + q 2 ( η -λ +1)(2 λ -ξ -1) + 1 ) -4( η -5 λ +4 ξ +1) ( λ + q 2 ( η -λ +1) -1 ) ] , (34)</formula>
<formula><location><page_6><loc_12><loc_26><loc_94><loc_38></location>p r ( r /greatermuch 1) = 1 64 πB 0 ξG ae b 2 r 2( ξ -1) λ -ξ +4 ( -br 2 ( η -3 λ +2 ξ +1) ) -2 η η -3 λ +2 ξ +1 + 1 -λ λ -ξ -1 × [ -8 B 0 ξr 2( λ -1) ξ -λ -4 ( -br 2 ( η -3 λ +2 ξ +1) ) 2 η η -3 λ +2 ξ +1 + λ -1 λ -ξ +1 b 2 +16 λ +4 ( λ + q 2 ( η -λ +1) -1 ) +8 q 2 ( -2 λ + ξ +1) + 16 ( λ -λq 2 + ξq 2 ) -8 ] , (35)</formula>
<formula><location><page_6><loc_18><loc_15><loc_94><loc_22></location>p t ( r /greatermuch 1) = 1 16 πB 0 ξG ae ( η -3 λ +2 ξ +1) 2 br 2( λ -1) λ -ξ ( -br 2 ( η -3 λ +2 ξ +1) ) -2 η η -3 λ +2 ξ +1 + 1 -λ λ -ξ × [ (3 λ -1)( η +9 λ -6 ξ -3) + q 2 ( η 2 + η ( -6 λ +4 ξ +2) -3( -3 λ +2 ξ +1) 2 )] . (36)</formula>
<formula><location><page_7><loc_17><loc_80><loc_94><loc_92></location>1 64 πB 0 ξG æ ( a + b ¯ R 2 ) 2 ¯ R 2( ξ -1) λ -ξ ( 2 a ( λ -ξ ) -b ¯ R 2 ( η -3 λ +2 ξ +1) ) -2 η η -3 λ +2 ξ +1 + 1 -λ λ -ξ -1 [ 4 b 2 ¯ R 4 ( λ + q 2 ( η -λ +1) -1 ) -8 B 0 ξ ¯ R 2( λ -1) ξ -λ ( 2 a ( λ -ξ ) -b ¯ R 2 ( η -3 λ +2 ξ +1) ) 2 η η -3 λ +2 ξ +1 + λ -1 λ -ξ +1 ( a + b ¯ R 2 ) 2 + 16 b ¯ R 2 ( λ -λq 2 + ξq 2 ) a + b ¯ R 2 +16 λ +8 q 2 ( -2 λ + ξ +1) -8 ] = 0 , (37)</formula>
<text><location><page_7><loc_9><loc_78><loc_22><loc_79></location>which is solved by</text>
<formula><location><page_7><loc_9><loc_71><loc_95><loc_76></location>q = ± [ 2 B 0 ξ ¯ R 2( λ -1) ξ -λ ( 2 a ( λ -ξ ) -b ¯ R 2 ( η -3 λ +2 ξ +1) ) 2 η η -3 λ +2 ξ +1 + λ -1 λ -ξ +1 -b ¯ R 2 (3 λ -1) ( 3 b ¯ R 2 +4 a ) +2 a 2 (1 -2 λ ) 4 ab ¯ R 2 ( -3 λ +2 ξ +1) + b 2 ¯ R 4 ( η -9 λ +6 ξ +3) + 2 a 2 ( -2 λ + ξ +1) ] 1 2 . (38)</formula>
<text><location><page_7><loc_9><loc_65><loc_95><loc_70></location>At this stage, we set G æ = G N (1 -η/ 2 ξ ) , where G N is the Newton's constant, which is needed to recover the Newtonian limit [2, 38]. Moreover, we also implement the constraint coming from the near-simultaneous observation of the gravitational-wave event GW170817 and the gamma-ray burst GRB 170817A [18], which consists in setting the speed of propagation of the spin-2 mode to 1, i.e. , ξ = 1 , up to an uncertainty of about 10 -15 [18, 19].</text>
<text><location><page_7><loc_9><loc_58><loc_94><loc_64></location>The outcome of such analysis is that the solution worked out above cannot be considered valid in the whole interior spacetime. Indeed, it is plagued by a singularity in the center as the curvature invariants R , R µν R µν , and R αβµν R αβµν diverge at r = 0 ; then the corresponding spacetime is also not geodetically complete because of the singularity that it inherits. For the sake of simplicity, we only write below the corresponding expression for the scalar curvature R , which is</text>
<formula><location><page_7><loc_15><loc_44><loc_94><loc_57></location>R = -2 B 0 r 2 ( 2 a ( λ -1) -br 2 ( η -3 λ +3) ) -2 η η -3 λ +3 -4 [ -2 br 4 ( br 2 ( η -3 λ +3) -2 a ( λ -1) ) ( -4 a 2 ( λ -1) -9 ab ( λ -1) r 2 + b 2 r 4 ( η -6 λ +6) ) +2 r 2 ( a + br 2 ) ( br 2 ( η -3 λ +3) -2 a ( λ -1) ) ( 2 a 2 ( λ -1) + 3 abr 2 ( η + λ -1) + b 2 r 4 ( η +3 λ -3) ) + b 2 r 6 ( br 2 ( η -3 λ +3) -2 a ( λ -1) ) 2 + B 0 ( 2 a ( λ -1) -br 2 ( η -3 λ +3) ) 2 η η -3 λ +3 +4 -r 2 ( a + br 2 ) 2 ( br 2 ( η -3 λ +3) -2 a ( λ -1) ) 2 ] . (39)</formula>
<text><location><page_7><loc_9><loc_36><loc_94><loc_44></location>It is easy to show that the expression above diverges in the center as ∼ 1 /r 2 . Moreover, the thermodynamical quantities ρ ( r ) and p r ( r ) take infinite values at r = 0 . Then, the internal spacetime described by this solution loses its physical predictability at very small scales in the interior of astrophysical objects. Nevertheless, one might still consider this kind of solution as a viable model in the context of a star with several internal shells. Indeed, in that case, the solution at hand would only hold from the surface towards a certain internal physical radius, while from the latter up to to the center of the fluid sphere a new interior metric would be needed.</text>
<text><location><page_7><loc_9><loc_25><loc_94><loc_36></location>Anyhow, this proof shows that in the context of Hořava gravity and Einstein-æther theory with a non-static æther of the form given by Eq. (19), it is not possible to construct a viable solution whose metric coefficient A ( r ) reproduces the potential of a Newtonian constant density sphere across the whole interior spherically symmetric spacetime. Similar results can be obtained for several choices of A ( r ) , which means that the specific choice considered here does not imply a loss of generality, since qualitatively the conclusions do not change by making a different ansatz. Notice that if the æther is assumed to be static instead, such a solution exists [31, 32]. One possibility to resolve this issue might be to relax the hypothesis made for the function F ( r ) and then to consider even more general configurations of the æther vector field.</text>
<section_header_level_1><location><page_7><loc_43><loc_21><loc_60><loc_22></location>VI. CONCLUSIONS</section_header_level_1>
<text><location><page_7><loc_9><loc_14><loc_94><loc_19></location>We have taken into account spherically symmetric interior spacetimes filled by anisotropic fluids in the context of Hořava gravity and Einstein-æther theory. A general setting in which the æther vector field is non-static has been implemented, which means that the æther has two non-trivial components, instead of a single one as in the case of a static æther. We have anyhow made some assumption on the component F ( r ) of the æther by means of which the</text>
<text><location><page_8><loc_9><loc_81><loc_94><loc_94></location>field equations become analytically solvable. Then, we have shown that a family of infinite exact analytical solutions exists when one of the two metric coefficients is assigned. The result is quite remarkable, since these are the first exact and analytical solutions ever found in the context of such theories in presence of anisotropic matter and with a non-static æther. As an illustrative example, we have solved the field equations by selecting the metric coefficient A ( r ) in such a way as to reproduce the Newtonian potential for a fluid sphere of constant density, which also coincides with the Tolman IV solution in GR for an isotropic fluid. Nevertheless, the resulting analytical solution that we have found for the metric coefficients and the thermodynamical quantities is plagued by an unavoidable divergence in the center. Indeed, all the curvature invariants are found to diverge at r = 0 . For simplicity, we have displayed only the expression of the scalar curvature R which diverges with a power of r -2 .</text>
<text><location><page_8><loc_9><loc_64><loc_95><loc_81></location>In the framework of a quantum gravity theory like Hořava gravity, that should also account for strong-gravity effects, the presence of any kind of singularity in the description of astrophysical objects is somehow questionable. Most likely, this fact might signal that such a solution is not admissible only in the context that we have considered here. Notice that, both in the case of static æther and isotropic fluids [29, 30], and in the case of static æther and anisotropic fluids [31, 32], such a singularity for spherically symmetric interior spacetimes is not present. As a direct consequence of this, one can infer that the main physical difference between those scenarios and the one under study in this paper is due to the choice of a non-static æther that we are performing here, and not due to the anisotropy of pressure, which indeed does not seem to play any relevant role in this respect. So, further investigations are necessary in order to understand if more general configurations of the æther vector field can solve this issue. Moreover, an alternative route could be to consider in the gravitational action also the higher-order operators which render the theory power-counting renormalizable. Those extra operators might indeed cure the pathology which plagues the solution in the center. In both cases, one would need to use numerical approaches in order to solve the resulting field equations.</text>
<text><location><page_8><loc_9><loc_57><loc_94><loc_63></location>Acknowledgments: The author would like to thank Sante Carloni for enlightening discussions that led to the initiation of this project. This research was supported by Fundação para a Ciência e a Tecnologia (FCT) through the Research Grant UID/FIS/04434/2019, and by Projects PTDC/FIS-OUT/29048/2017, COMPETE2020: POCI01-0145-FEDER-028987 & FCT: PTDC/FIS-AST/28987/2017, and IF/00852/2015 of FCT.</text>
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</document> | [{"title": "Anisotropic fluid spheres in Ho\u0159ava gravity and Einstein-\u00e6ther theory with a non-static \u00e6ther", "content": "Daniele Vernieri Instituto de Astrof\u00edsica e Ci\u00eancias do Espa\u00e7o, Faculdade de Ci\u00eancias da Universidade de Lisboa, Campo Grande, PT1749-016 Lisboa, Portugal (Dated: November 19, 2019) In this paper we consider spherically symmetric interior spacetimes filled by anisotropic fluids in the context of Ho\u0159ava gravity and Einstein-\u00e6ther theory. We assume a specific non-static configuration of the \u00e6ther vector field and show that the field equations admit a family of exact analytical solutions which can be obtained if one of the two metric coefficients is assigned. We study as an illustrative example the case in which the metric of the interior spacetime reproduces the Newtonian potential of a fluid sphere with constant density.", "pages": [1]}, {"title": "I. INTRODUCTION", "content": "Ho\u0159ava gravity was proposed in 2009 as a power-counting renormalizable theory of quantum gravity [1, 2]. In the past years much work has been done to show that the theory is renormalizable [3-6] beyond the power-counting arguments [7-12]. Ho\u0159ava gravity has also been severely constrained by means of some tests at both astrophysical [1315] and cosmological scales [16, 17], and it passes all of them with flying colors. Moreover it is also consistent with the constraint on the speed of propagation of gravitational waves coming from the near-simultaneous temporal and spatial observation of the gravitational-wave event GW170817 and the gamma-ray burst GRB 170817A [18, 19]. The theory breaks Lorentz invariance at any energy scale since a preferred direction is naturally encoded in its formulation. This locally amounts to having a timelike hypersurface-orthogonal \u00e6ther vector field which is defined in each point of the spacetime. If one considers the low-energy limit of Ho\u0159ava gravity in a covariant form, the latter proves to be equivalent to Einstein-\u00e6ther theory [20] once the \u00e6ther vector is taken to be hypersurface-orthogonal at the level of the action [21]. In spherical symmetry, any vector is automatically hypersurface-orthogonal; therefore by virtue of this fact it can be shown that the two theories share the same solutions in such a background [22]. Because of the intrinsic highly non-linear structure of the field equations, only a few analytical and numerical solutions are known both in vacuum [23-28] and inside matter [29-32]. Thus, it is really necessary to focus more effort in this direction, since many of the phenomenological implications of the theory are still unknown, even in highly symmetric spacetimes. For this purpose in the present manuscript we consider spherically symmetric interior spacetimes filled by anisotropic fluids [33-35] in the context of the low-energy limit of Ho\u0159ava gravity. The approach that we undertake here is similar to the one used in Refs. [31, 32] in which the equation of state of the inner fluid is left unspecified, but after a viable solution has been found, it can be instead reconstructed a posteriori by using the same method exploited in Ref. [32]. This approach generically looks more realistic since, despite all the work that has been done until now, we still lack a proper modeling of the interior spacetime of relativistic objects (see Ref. [36] and references therein). Then it seems appropriate to leave unspecified the equation of state relating the thermodynamical quantities, whose study is anyhow out of the scope of the present paper. In Refs. [31, 32], this kind of setting has already been studied, but in the more restricted case of a static \u00e6ther, which means that the \u00e6ther vector is aligned with the timelike Killing vector and then has only one non-vanishing component. Here we consider instead a more general ansatz where the \u00e6ther vector field indeed has two non-trivial components. We derive the corresponding field equations and find a strategy to analytically solve them by means of some choice of the \u00e6ther components and an appropriate redefinition of variables. In this framework, we show that the field equations admit a family of infinite exact analytical solutions, characterized by choosing arbitrarily one of the metric coefficients of the spherically symmetric interior spacetime. In Sec. II, we present the gravitational action and the field equations of Ho\u0159ava gravity and discuss the equivalence of its low-energy limit to Einstein-\u00e6ther theory when the \u00e6ther is chosen to be hypersurface-orthogonal at the level of the action. In Sec. III, we introduce the spherically symmetric background metric, the corresponding \u00e6ther vector field, and the stress-energy tensor suitable for the anisotropic fluid description. In Sec. IV, we write down explicitly the system of the independent field equations in terms of the metric, the \u00e6ther components, and the thermodynamical quantities. In Sec. V we focus on a specific case in which the interior metric reproduces the Newtonian potential of a fluid sphere with constant density. In Sec. VI, the conclusions are drawn.", "pages": [1]}, {"title": "II. HO\u0158AVA GRAVITY AND EINSTEIN-\u00c6THER THEORY", "content": "The action of Ho\u0159ava gravity [1, 2] as written in the preferred foliation looks like where G H is the effective gravitational coupling constant; g is the determinant of the metric g \u00b5\u03bd ; R is the Ricci scalar of the three-dimensional constantT hypersurfaces; K ij is the extrinsic curvature and K is its trace; and a i = \u2202 i ln N , where N is the lapse function and S m is the matter action for the matter fields collectively denoted by \u03c8 . The couplings { \u03bb, \u03be, \u03b7 } are dimensionless, and general relativity (GR) is identically recovered when they take the values { 1 , 1 , 0 } , respectively. Finally, L 4 and L 6 collectively denote the fourth-order and sixth-order operators that make the theory power-counting renormalizable, and M \u2217 is a characteristic mass scale which suppresses them. In what follows, we consider the covariantized version of the low-energy limit of Ho\u0159ava gravity, named the khronometric model, that is obtained by discarding the higher-order operators in L 4 and L 6 . In order to write it in a covariant form, let us consider the action of Einstein-\u00e6ther theory [20]; that is, where G \u00e6 is the 'bare' gravitational constant; R is the four-dimensional Ricci scalar; u a is a timelike vector field of unit norm, i.e. , g \u00b5\u03bd u \u00b5 u \u03bd = 1 , from now on referred to as the '\u00e6ther'; and with M \u03b1\u03b2 \u00b5\u03bd defined as where c i 's are dimensionless coupling constants. Then, one can take the \u00e6ther vector to be hypersurface-orthogonal at the level of the action, which locally amounts to choosing where in the covariant formulation, the preferred time T becomes a scalar field (the khronon ) which defines the preferred foliation. Finally, the two actions in Eqs. (1) and (2) are shown to be equivalent if the parameters of the two respective theories are mapped into each other as [21] where c ij = c i + c j . In what follows, we consider the covariant formulation of Ho\u0159ava gravity in order to perform the calculations. The variation of the action in Eq. (2) with respect to g \u03b1\u03b2 and T yields, respectively [15], where G \u03b1\u03b2 = R \u03b1\u03b2 -Rg \u03b1\u03b2 / 2 is the Einstein tensor, is the khronon stress-energy tensor, and T m \u03b1\u03b2 is the matter stress-energy tensor, defined as", "pages": [2, 3]}, {"title": "III. SPHERICALLY SYMMETRIC METRIC, ANISOTROPIC FLUIDS, AND A NON-STATIC \u00c6THER", "content": "In spherical symmetry, the most general metric can be written as ds 2 = A ( r ) dt 2 -B ( r ) dr 2 -r 2 ( d\u03b8 2 +sin 2 \u03b8d\u03c6 2 ) . (12) Moreover, in what follows, we will consider the interior spacetime of a fluid sphere filled by an anisotropic fluid whose stress-energy tensor is given by where \u03c1 is the density, p r and p t are the radial and transversal pressure, respectively, v \u00b5 denotes the 4-velocity of the fluid and s \u00b5 is a spacelike 4-vector defined as satisfying the relations s \u00b5 s \u00b5 = -1 and s \u00b5 u \u00b5 = 0 . It can be easily shown that the components of the stress-energy tensor can be explicitly written as The \u00e6ther vector field, which is by definition a timelike vector of unit norm, in spherical symmetry is always hypersurface-orthogonal and takes the following general form: The independent field equations that we have to consider are the modified Einstein equations (0-0), (1-1), (1-2) and (2-2) in Eq. (7); the equation for the scalar field T in Eq. (8); and the conservation equation for the stress-energy tensor of anisotropic matter that is, The expressions of the field equations are very long and awful, so they are not displayed here. However, through a direct inspection it is quite straightforward to notice that the field equations are considerably simplified by making the choice F ( r ) = q \u221a A ( r ) , and the \u00e6ther vector in Eq. (17) then becomes Let us stress that, even by implementing this specific assumption, the \u00e6ther vector field has anyhow two non-trivial components. Then, it is still more general than the \u00e6ther vector widely used in literature and referred to as 'static \u00e6ther' which is by definition aligned with the timelike Killing vector. The choice of a static \u00e6ther is just a special case which is included in the more general framework developed here by setting q = 1 in Eq. (19). ( W ( r ) + - W 2( \u03b7 ' ( r ) [ - Y ( r )) [8( \u03bb - \u03bb +1) Y 4 \u03bb +4 \u03be +( \u03b7 '' ( r ) = 0 . r - - 1) + W ( r )(4( \u03bb - \u03bb +1) Y ( r )] - \u03be ) + ( + r ' 2 ( r ) [ Y - \u03b7 + \u03bb - 4( \u03b7 - - 1) Y ( r )) + 2 Y ( r )( \u03b7 + \u03bb \u03be +1) + 3( \u03b7 - \u03bb +1) W ( r ) r - 2 \u03be +1)] - 2( \u03b7 - \u03bb +1) Y ( r )]", "pages": [3, 4]}, {"title": "IV. FIELD EQUATIONS", "content": "In order to write down the field equations in a more compact form, let us consider the following redefinition of variables: Then, the (0-0) component of the modified Einstein equations in Eq. (7) becomes the (1-1) component is and the (2-2) component is Moreover the modified Einstein equation (1-2) can be written as which has already been substituted in Eqs. (21)-(23). Finally, Eq. (8) for the scalar field and Eq. (18) for the stress-energy tensor conservation (after Eqs. (21)-(23) have been used) become, respectively, and (26) However, by substituting Eq. (24) and its first derivative in Eqs. (25) and (26), these are identically satisfied. So, we are finally left with only four independent field equations, i.e. , Eqs. (21)-(24). This means that, by assigning one of the two metric coefficients, A ( r ) or B ( r ) , the other can be obtained by solving the differential Eq. (24), and the thermodynamical variables \u03c1 ( r ) , p r ( r ) , and p t ( r ) can be read from Eqs. (21), (22) and (23), respectively. In this way we have obtained a family of infinite exact analytical solutions of the aforementioned system of equations if one of the two metric functions A ( r ) or B ( r ) is assigned. Notice that spherically symmetric solutions in Ho\u0159ava gravity are identical to those of Einstein-\u00e6ther theory; therefore, our conclusions will hold for both theories all the same [22].", "pages": [4, 5]}, {"title": "V. SOLUTION WHICH REPRODUCES THE POTENTIAL FOR A CONSTANT-DENSITY FLUID SPHERE", "content": "We are now ready to work out the full system of field equations (21)-(24). Let us notice that Eq. (24) can be generically solved for W ( r ) , which turns out to be and by making A ( r ) and B ( r ) , explicit the latter becomes The equation above can be solved by assigning one of the two metric functions A ( r ) or B ( r ) . Then the system of field equations is closed, and a family of infinite exact and analytical solutions can be found. As an illustrative example let us choose the analytic form of A ( r ) which reminds the Newtonian potential of a fluid sphere of constant density: i.e. , A ( r ) = a + br 2 , where a and b are arbitrary constants. Moreover, this choice also corresponds to the well-known Tolman IV solution in GR for an isotropic fluid [37]. Substituting in Eq. (28) the expression given above for A ( r ) , we obtain This is a first-order ordinary differential equation that can be easily integrated to give where B 0 is an integration constant. It is now straightforward to get algebraically from Eqs. (21)-(23) the explicit analytical expressions for the thermodynamical quantities \u03c1 ( r ) , p r ( r ) , and p t ( r ) that are shown below: and and The implementation of the junction conditions to the exterior vacuum metric amounts to requiring that the radial pressure p r ( r ) in Eq. (32) vanish at r = \u00af R . Then, p r [ \u00af R ] = 0 results in We also calculate the limit of the above expressions for large values of the radius r /greatermuch 1 , when the constants a and b are sufficiently small: which is solved by At this stage, we set G \u00e6 = G N (1 -\u03b7/ 2 \u03be ) , where G N is the Newton's constant, which is needed to recover the Newtonian limit [2, 38]. Moreover, we also implement the constraint coming from the near-simultaneous observation of the gravitational-wave event GW170817 and the gamma-ray burst GRB 170817A [18], which consists in setting the speed of propagation of the spin-2 mode to 1, i.e. , \u03be = 1 , up to an uncertainty of about 10 -15 [18, 19]. The outcome of such analysis is that the solution worked out above cannot be considered valid in the whole interior spacetime. Indeed, it is plagued by a singularity in the center as the curvature invariants R , R \u00b5\u03bd R \u00b5\u03bd , and R \u03b1\u03b2\u00b5\u03bd R \u03b1\u03b2\u00b5\u03bd diverge at r = 0 ; then the corresponding spacetime is also not geodetically complete because of the singularity that it inherits. For the sake of simplicity, we only write below the corresponding expression for the scalar curvature R , which is It is easy to show that the expression above diverges in the center as \u223c 1 /r 2 . Moreover, the thermodynamical quantities \u03c1 ( r ) and p r ( r ) take infinite values at r = 0 . Then, the internal spacetime described by this solution loses its physical predictability at very small scales in the interior of astrophysical objects. Nevertheless, one might still consider this kind of solution as a viable model in the context of a star with several internal shells. Indeed, in that case, the solution at hand would only hold from the surface towards a certain internal physical radius, while from the latter up to to the center of the fluid sphere a new interior metric would be needed. Anyhow, this proof shows that in the context of Ho\u0159ava gravity and Einstein-\u00e6ther theory with a non-static \u00e6ther of the form given by Eq. (19), it is not possible to construct a viable solution whose metric coefficient A ( r ) reproduces the potential of a Newtonian constant density sphere across the whole interior spherically symmetric spacetime. Similar results can be obtained for several choices of A ( r ) , which means that the specific choice considered here does not imply a loss of generality, since qualitatively the conclusions do not change by making a different ansatz. Notice that if the \u00e6ther is assumed to be static instead, such a solution exists [31, 32]. One possibility to resolve this issue might be to relax the hypothesis made for the function F ( r ) and then to consider even more general configurations of the \u00e6ther vector field.", "pages": [5, 6, 7]}, {"title": "VI. CONCLUSIONS", "content": "We have taken into account spherically symmetric interior spacetimes filled by anisotropic fluids in the context of Ho\u0159ava gravity and Einstein-\u00e6ther theory. A general setting in which the \u00e6ther vector field is non-static has been implemented, which means that the \u00e6ther has two non-trivial components, instead of a single one as in the case of a static \u00e6ther. We have anyhow made some assumption on the component F ( r ) of the \u00e6ther by means of which the field equations become analytically solvable. Then, we have shown that a family of infinite exact analytical solutions exists when one of the two metric coefficients is assigned. The result is quite remarkable, since these are the first exact and analytical solutions ever found in the context of such theories in presence of anisotropic matter and with a non-static \u00e6ther. As an illustrative example, we have solved the field equations by selecting the metric coefficient A ( r ) in such a way as to reproduce the Newtonian potential for a fluid sphere of constant density, which also coincides with the Tolman IV solution in GR for an isotropic fluid. Nevertheless, the resulting analytical solution that we have found for the metric coefficients and the thermodynamical quantities is plagued by an unavoidable divergence in the center. Indeed, all the curvature invariants are found to diverge at r = 0 . For simplicity, we have displayed only the expression of the scalar curvature R which diverges with a power of r -2 . In the framework of a quantum gravity theory like Ho\u0159ava gravity, that should also account for strong-gravity effects, the presence of any kind of singularity in the description of astrophysical objects is somehow questionable. Most likely, this fact might signal that such a solution is not admissible only in the context that we have considered here. Notice that, both in the case of static \u00e6ther and isotropic fluids [29, 30], and in the case of static \u00e6ther and anisotropic fluids [31, 32], such a singularity for spherically symmetric interior spacetimes is not present. As a direct consequence of this, one can infer that the main physical difference between those scenarios and the one under study in this paper is due to the choice of a non-static \u00e6ther that we are performing here, and not due to the anisotropy of pressure, which indeed does not seem to play any relevant role in this respect. So, further investigations are necessary in order to understand if more general configurations of the \u00e6ther vector field can solve this issue. Moreover, an alternative route could be to consider in the gravitational action also the higher-order operators which render the theory power-counting renormalizable. Those extra operators might indeed cure the pathology which plagues the solution in the center. In both cases, one would need to use numerical approaches in order to solve the resulting field equations. Acknowledgments: The author would like to thank Sante Carloni for enlightening discussions that led to the initiation of this project. This research was supported by Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia (FCT) through the Research Grant UID/FIS/04434/2019, and by Projects PTDC/FIS-OUT/29048/2017, COMPETE2020: POCI01-0145-FEDER-028987 & FCT: PTDC/FIS-AST/28987/2017, and IF/00852/2015 of FCT.", "pages": [7, 8]}] |
2020arXiv200104423A | https://arxiv.org/pdf/2001.04423.pdf | <document>
<section_header_level_1><location><page_1><loc_21><loc_97><loc_78><loc_98></location>A LOCAL RESOLUTION OF THE PROBLEM OF TIME</section_header_level_1>
<section_header_level_1><location><page_1><loc_34><loc_94><loc_65><loc_95></location>VIII. Assignment of Observables</section_header_level_1>
<section_header_level_1><location><page_1><loc_41><loc_91><loc_59><loc_93></location>Edward Anderson 1</section_header_level_1>
<text><location><page_1><loc_28><loc_89><loc_72><loc_90></location>based on calculations done at Peterhouse, Cambridge</text>
<section_header_level_1><location><page_1><loc_46><loc_86><loc_53><loc_87></location>Abstract</section_header_level_1>
<text><location><page_1><loc_13><loc_81><loc_87><loc_85></location>Given a state space, Assignment of Observables involves Taking Function spaces Thereover. At the classical level, the state space in question is phase space or configuration space. This assignment picks up nontrivialities when whichever combination of constraints and the quantum apply.</text>
<text><location><page_1><loc_13><loc_70><loc_87><loc_81></location>For Finite Theories, weak observables equations are inhomogeneous-linear first-order PDE systems. Their general solution thus splits into complementary function plus particular integral: strong and nontrivially-weak observables respectively. We provide a PDE analysis for each of these. In the case of single observables equations - corresponding to single constraints - the Flow Method readily applies. Finding all the observables requires free characteristic problems. For systems, this method can be applied sequentially, due to integrability conferred by Frobenius' Theorem. In each case, the first part of this approach is Lie's Integral Theory of Geometrical Invariants, or the physical counterpart thereof. The second part finds the function space thereover, giving the entire space of (local) observables.</text>
<text><location><page_1><loc_13><loc_65><loc_87><loc_70></location>We also outline the Field Theory counterpart. Here one has functional differential equations. Banach (or tame Fréchet) Calculus is however sufficiently standard for the Flow Method and free characteristic problems to still apply. These calculi support the Lie-theoretic combination of machinery that our Local Resolution of the Problem of Time requires, by which Field Theory and GR are included.</text>
<text><location><page_1><loc_11><loc_62><loc_47><loc_64></location>1 dr.e.anderson.maths.physics *at* protonmail.com</text>
<section_header_level_1><location><page_1><loc_9><loc_58><loc_27><loc_60></location>1 Introduction</section_header_level_1>
<text><location><page_1><loc_9><loc_43><loc_91><loc_57></location>This is the eighth Article [81, 82, 83, 84, 85, 86, 87] on A Local Resolution of the Problem of Time [17, 18, 15, 24, 36, 38, 39, 45, 57, 59, 61, 63, 64, 75, 72, 80] and its underlying Local Theory of Background Independence. We here expand on Sec 3 of Article III's opening account of Assignment of Observables [5, 38, 39, 41], now atop [63, 72] the triple unification of Constraint Closure [6, 9, 15, 28, 37, 72, 83, 87] with Temporal [13, 25, 59, 66, 72, 81, 85, 86] and Configurational [12, 25, 59, 72, 82, 85, 86] Relationalism. That we can treat this after completing the triple, and separately from Article IX's Constructability extension, is one of the great decouplings of Problem of Time Facets [72, 81]. Resolving the triple of facets gives a consistent phase space, P hase. Taking Function Spaces Thereover -the essence of Assigning Observables - entails addressing a subsequent mathematical problem on P hase, rather than imposing some further conditions on whether that P hase is adequate.</text>
<text><location><page_1><loc_9><loc_21><loc_91><loc_42></location>We pose concrete mathematical problems for Finite Theory's observables at the level of brackets algebras in Sec 2 and of PDEs in Secs 3 and 4. Weak observables equations are inhomogeneous-linear first-order PDE systems. Their general solution thus splits into complementary function plus particular integral: strong and nontriviallyweak observables respectively. In the case of single observables equations - corresponding to single constraints - the Flow Method readily applies. Finding all the observables requires free characteristic problems . For systems - corresponding to multiple constraints - this approach can be applied sequentially, due to integrability as conferred by Frobenius' Theorem [27, 62]. In each case, the first part of our approach is Lie's Integral Theory of Geometrical Invariants (or some physical counterpart thereof). The second part is Finding the Function space Thereover, giving the entire space of (local) observables. The more practical matter of Expression in Terms of Observables - requiring only enough observables to span phase (or configuration) space - is covered in Sec 5. Strategies for addressing the Problem of Observables are in Sec 6. We also give explicit examples of quite a number of distinct notions of observables [5, 37, 38, 39, 41, 63, 72, 76, 78] in Sec 7. Purely geometrical such moreover play a further role [76, 77] in the Foundations of Geometry [3, 4, 33, 47, 52, 76], as well as coinciding with the configuration space q restriction of P hase.</text>
<text><location><page_1><loc_9><loc_12><loc_91><loc_19></location>The Field Theory and GR counterpart is in Sec 8. This now gives FDEs - functional differential equations as its observables equations. Similar Flow Method and sequential use of free characteristic problems carry over, however, thanks to the underlying benevolence of Banach Calculus [27] (for now, or tame Fréchet Calculus [29] more generally). These calculi support the Lie-theoretic combination of machinery required by our Local Resolution of the Problem of Time, by which Field Theory and GR are included.</text>
<text><location><page_1><loc_9><loc_7><loc_91><loc_10></location>Appendix A keeps track of TRi (Temporal Relationalism implementing) modifications and other Problem of Time facet interferences in part involving Assignment of Observables.</text>
<section_header_level_1><location><page_2><loc_9><loc_97><loc_46><loc_98></location>2 Brackets-level considerations</section_header_level_1>
<section_header_level_1><location><page_2><loc_9><loc_94><loc_50><loc_95></location>2.1 Strong observables-constraints system</section_header_level_1>
<text><location><page_2><loc_9><loc_91><loc_75><loc_92></location>Structure 1 Strong observables O extend the constraint algebraic structure C p S q as follows.</text>
<formula><location><page_2><loc_43><loc_89><loc_91><loc_90></location>{ C , C } GLYPH<16> C C , (1)</formula>
<formula><location><page_2><loc_45><loc_84><loc_91><loc_86></location>{ C , O } GLYPH<16> 0 , (2)</formula>
<text><location><page_2><loc_48><loc_84><loc_49><loc_84></location>r</text>
<formula><location><page_2><loc_44><loc_81><loc_91><loc_83></location>{ O r , O r } GLYPH<16> O r O r . (3)</formula>
<text><location><page_2><loc_9><loc_79><loc_47><loc_80></location>Remark 1 This is overall [72] of direct product form,</text>
<formula><location><page_2><loc_41><loc_76><loc_91><loc_77></location>C p S q GLYPH<2> C anO bs p S q . (4)</formula>
<section_header_level_1><location><page_2><loc_9><loc_72><loc_49><loc_74></location>2.2 Weak observables-constraints system</section_header_level_1>
<text><location><page_2><loc_9><loc_70><loc_71><loc_71></location>Structure 2 Weak observables O w extend the constraint algebraic structure as follows.</text>
<formula><location><page_2><loc_43><loc_67><loc_91><loc_68></location>{ C , C } GLYPH<16> C C , (5)</formula>
<formula><location><page_2><loc_43><loc_62><loc_91><loc_65></location>{ C , O r w } GLYPH<16> W GLYPH<128> C , (6)</formula>
<formula><location><page_2><loc_43><loc_59><loc_91><loc_61></location>{ O r w , O r w } GLYPH<16> O r O r w . (7)</formula>
<text><location><page_2><loc_52><loc_81><loc_53><loc_81></location>r</text>
<text><location><page_2><loc_52><loc_80><loc_53><loc_80></location>r</text>
<text><location><page_2><loc_52><loc_58><loc_53><loc_59></location>r</text>
<text><location><page_2><loc_52><loc_58><loc_53><loc_58></location>r</text>
<text><location><page_2><loc_9><loc_55><loc_91><loc_58></location>Remark 1 The second equation signifies that C is a good C anO bs p S q -object. This is a consequence of constraints closing weakly, via the Jacobi identity.</text>
<text><location><page_2><loc_9><loc_52><loc_54><loc_53></location>Remark 2 This is now overall [72] of semidirect product form,</text>
<formula><location><page_2><loc_41><loc_49><loc_91><loc_50></location>C p S q GLYPH<11> C anO bs p S q . (8)</formula>
<text><location><page_2><loc_9><loc_42><loc_91><loc_47></location>Remark 3 Comparing (5) and (2) or (6) implies that the C are themselves in some sense observables. However, since we already knew that C GLYPH<19> 0, in studying observables we are really looking for further quantities outside of this trivial case. Let us call these other quantities proper observables ; the rest of the Series will always take 'observables' to mean this.</text>
<text><location><page_2><loc_9><loc_39><loc_79><loc_40></location>Definition 1 We term the general solution of the weak observables equation O W weak observables .</text>
<text><location><page_2><loc_9><loc_34><loc_91><loc_37></location>Remark 4 The weak observables equation is moreover an inhomogeneous counterpart of the homogeneous linear strong observables equation. So</text>
<formula><location><page_2><loc_45><loc_33><loc_91><loc_34></location>O W GLYPH<16> O GLYPH<0> O w (9)</formula>
<text><location><page_2><loc_9><loc_29><loc_91><loc_32></location>in the manner of a complementary function plus particular integral split of the general solution of an inhomogeneous linear equation,</text>
<formula><location><page_2><loc_44><loc_27><loc_91><loc_29></location>GS GLYPH<16> CF GLYPH<0> PI . (10)</formula>
<text><location><page_2><loc_9><loc_24><loc_91><loc_27></location>Definition 2 We term particular-integral solutions of the weak equation O w - weak observables which are explicitly independent of any strong observables nontrivially weak observables .</text>
<text><location><page_2><loc_9><loc_19><loc_91><loc_22></location>Remark 5 The pure configurational geometry case cannot support any nontrivially-weak observables. This is because we are in a context in which constraints have to depend on momenta</text>
<formula><location><page_2><loc_34><loc_16><loc_91><loc_18></location>C GLYPH<16> C p Q , P q with specific P dependence . (11)</formula>
<text><location><page_2><loc_9><loc_11><loc_91><loc_15></location>A fortiori, first-class linear constraints refer specifically to being linear in their momenta P , By this, the weak configurational observables equation would have a momentum-dependent inhomogeneous term right-hand side. But this is inconsistent with admitting a solution with purely Q -dependent right-hand-side.</text>
<section_header_level_1><location><page_3><loc_9><loc_97><loc_51><loc_98></location>2.3 Why not a more general C , O w system?</section_header_level_1>
<text><location><page_3><loc_9><loc_94><loc_50><loc_95></location>Naïve algebraic generality suggests the more general form</text>
<formula><location><page_3><loc_39><loc_91><loc_91><loc_93></location>{ C , C } GLYPH<16> C C GLYPH<0> D r O r w , (12)</formula>
<formula><location><page_3><loc_39><loc_86><loc_91><loc_89></location>{ C , O r w } GLYPH<16> W C GLYPH<0> X GLYPH<128> GLYPH<128> O r w , (13)</formula>
<formula><location><page_3><loc_39><loc_83><loc_91><loc_85></location>{ O r w , O r w } GLYPH<16> L C GLYPH<0> O r w O r w . (14)</formula>
<text><location><page_3><loc_48><loc_23><loc_49><loc_23></location>r</text>
<text><location><page_3><loc_49><loc_82><loc_50><loc_83></location>r</text>
<text><location><page_3><loc_49><loc_82><loc_50><loc_82></location>r</text>
<text><location><page_3><loc_55><loc_83><loc_56><loc_83></location>r</text>
<text><location><page_3><loc_55><loc_82><loc_56><loc_82></location>r</text>
<text><location><page_3><loc_9><loc_80><loc_80><loc_82></location>However, D GLYPH<16> 0 since physical systems provide the constraints C without reference to observables O .</text>
<text><location><page_3><loc_9><loc_77><loc_49><loc_79></location>X GLYPH<16> 0 preserves C 's status as a good C anO bs p S q -object.</text>
<text><location><page_3><loc_9><loc_73><loc_91><loc_76></location>L GLYPH<24> 0 gives a further sense of weak, though it prevents the observables from forming a subalgebra. Now starting with the observables would imply the constraints, but the constraints are already prescribed elsewise by the physics.</text>
<section_header_level_1><location><page_3><loc_9><loc_69><loc_60><loc_71></location>2.4 Relation between O w and W structure constants</section_header_level_1>
<section_header_level_1><location><page_3><loc_9><loc_67><loc_16><loc_68></location>Lemma 1</section_header_level_1>
<text><location><page_3><loc_9><loc_63><loc_12><loc_64></location>Proof</text>
<formula><location><page_3><loc_34><loc_65><loc_91><loc_67></location>GLYPH<1> 2 W B A r O δ P s Q GLYPH<1> δ A B O w Q OP GLYPH<9> W C BQ C C GLYPH<16> 0 (15)</formula>
<formula><location><page_3><loc_20><loc_61><loc_91><loc_63></location>0 GLYPH<16> J p C , O w , O w q GLYPH<16> GLYPH<1> W C AO δ C D δ P O GLYPH<1> W C AP δ C D δ O R GLYPH<1> O w Q OP δ A D δ Q R GLYPH<9> W E DR C E (16)</formula>
<text><location><page_3><loc_9><loc_59><loc_20><loc_60></location>and factorize. ✷</text>
<text><location><page_3><loc_9><loc_56><loc_36><loc_57></location>Corollary 1 This can be achieved by</text>
<text><location><page_3><loc_9><loc_52><loc_31><loc_53></location>returning the strong case, or by</text>
<formula><location><page_3><loc_41><loc_49><loc_91><loc_52></location>O w Q OP GLYPH<16> 2 c W A A r O δ P s Q (18)</formula>
<text><location><page_3><loc_9><loc_47><loc_60><loc_48></location>with c : GLYPH<16> dim p C q , or by a zero double-trace 'perpendicularity' condition.</text>
<text><location><page_3><loc_9><loc_44><loc_63><loc_46></location>Remark 1 On the other hand, no capacity to influence O w can be found in</text>
<formula><location><page_3><loc_45><loc_41><loc_91><loc_43></location>J p C , C , O w q GLYPH<16> 0 (19)</formula>
<section_header_level_1><location><page_3><loc_9><loc_38><loc_61><loc_39></location>2.5 Strong and weak observables considered together</section_header_level_1>
<text><location><page_3><loc_9><loc_34><loc_91><loc_37></location>Structure 3 By our CF + PI split, the strong and nontrivially-weak canonical observables algebra is consequently of the direct product form</text>
<formula><location><page_3><loc_32><loc_32><loc_91><loc_34></location>C anO bs W p S q GLYPH<16> C anO bs w p S q GLYPH<2> C anO bs p S q . (20)</formula>
<text><location><page_3><loc_9><loc_30><loc_71><loc_31></location>Structure 4 Including the constraints as well, we have the 3-block algebraic structure.</text>
<formula><location><page_3><loc_43><loc_27><loc_91><loc_29></location>{ C , C } GLYPH<16> C C , (21)</formula>
<formula><location><page_3><loc_45><loc_23><loc_91><loc_25></location>{ C , O } GLYPH<16> 0 , (22)</formula>
<formula><location><page_3><loc_47><loc_54><loc_91><loc_55></location>W GLYPH<16> 0 , (17)</formula>
<formula><location><page_3><loc_44><loc_20><loc_91><loc_22></location>{ O r , O r } GLYPH<16> O r O r , (23)</formula>
<text><location><page_3><loc_52><loc_20><loc_53><loc_20></location>r</text>
<text><location><page_3><loc_52><loc_19><loc_53><loc_19></location>r</text>
<formula><location><page_3><loc_43><loc_16><loc_91><loc_19></location>{ C , O r w } GLYPH<16> W GLYPH<128> C , (24)</formula>
<formula><location><page_3><loc_42><loc_13><loc_91><loc_15></location>{ O w GLYPH<128> , O r w } GLYPH<16> O r w O r w , (25)</formula>
<text><location><page_3><loc_52><loc_13><loc_53><loc_13></location>r</text>
<text><location><page_3><loc_52><loc_12><loc_53><loc_12></location>r</text>
<formula><location><page_3><loc_44><loc_11><loc_91><loc_12></location>{ O , O w } GLYPH<16> 0 . (26)</formula>
<text><location><page_3><loc_45><loc_10><loc_46><loc_10></location>r</text>
<text><location><page_3><loc_47><loc_10><loc_48><loc_10></location>r</text>
<text><location><page_3><loc_9><loc_9><loc_88><loc_10></location>Remark 1 This includes each of C , O and O w as subalgebras, by the first, third and fifth equations respectively.</text>
<text><location><page_4><loc_9><loc_95><loc_91><loc_98></location>Remark 2 The sixth equation's zero right-hand-side is part of the implementation of the CF to PI linear independence. This is partly enforced by</text>
<formula><location><page_4><loc_44><loc_94><loc_91><loc_95></location>J p C , O , O w q GLYPH<16> 0 , (27)</formula>
<text><location><page_4><loc_9><loc_91><loc_47><loc_93></location>which renders an initial O and O w combination pureO .</text>
<text><location><page_4><loc_9><loc_88><loc_49><loc_90></location>Remark 3 The overall algebraic structure is of the form</text>
<formula><location><page_4><loc_35><loc_86><loc_91><loc_87></location>p C p S q GLYPH<11> C anO bs w p S qq GLYPH<2> C anO bs p S q . (28)</formula>
<section_header_level_1><location><page_4><loc_9><loc_81><loc_42><loc_83></location>3 Strong observables PDEs</section_header_level_1>
<text><location><page_4><loc_9><loc_77><loc_91><loc_80></location>One can obtain explicit PDEs by writing out what the Poisson brackets definition of constrained observables means [68]. For strong observables, (2) gives</text>
<formula><location><page_4><loc_36><loc_73><loc_91><loc_76></location>0 GLYPH<16> { C , O } GLYPH<16> B C B Q B O B P GLYPH<1> B C B P B O B Q . (29)</formula>
<text><location><page_4><loc_9><loc_70><loc_24><loc_72></location>We can moreover take</text>
<formula><location><page_4><loc_39><loc_68><loc_91><loc_70></location>B C B Q and B C B P to be knowns , (30)</formula>
<text><location><page_4><loc_9><loc_66><loc_73><loc_67></location>leaving us with a homogeneous-linear first-order PDE system. I.e. a homogeneous subcase</text>
<formula><location><page_4><loc_42><loc_62><loc_91><loc_65></location>, A a A p x B , φ qB A φ GLYPH<16> 0 (31)</formula>
<text><location><page_4><loc_9><loc_58><loc_91><loc_60></location>of the claimed first-order linear form (III.87). First order their containing first-order partial derivatives B α φ and no higher. Linear refers to the unknown variables O .</text>
<text><location><page_4><loc_9><loc_55><loc_55><loc_56></location>Remark 1 As the strong case involves a homogeneous equation,</text>
<formula><location><page_4><loc_46><loc_52><loc_91><loc_53></location>O GLYPH<16> const , (32)</formula>
<text><location><page_4><loc_9><loc_48><loc_91><loc_50></location>is always a solution. We refer to this as the trivial solution . We call all other solutions of first-order homogeneous quasilinear PDEs proper solutions : a nontrivial kernel condition.</text>
<section_header_level_1><location><page_4><loc_9><loc_44><loc_47><loc_45></location>3.1 Single strong observables equation</section_header_level_1>
<text><location><page_4><loc_9><loc_42><loc_46><loc_43></location>The Flow Method [26, 62] immediately applies here.</text>
<text><location><page_4><loc_9><loc_39><loc_53><loc_40></location>Structure 5 This gives a corresponding ODE system of form</text>
<formula><location><page_4><loc_45><loc_36><loc_91><loc_37></location>9 x A GLYPH<16> a A p x q , (33)</formula>
<formula><location><page_4><loc_47><loc_33><loc_91><loc_34></location>9 O GLYPH<16> 0 . (34)</formula>
<text><location><page_4><loc_9><loc_31><loc_24><loc_32></location>Here, the dot denotes</text>
<formula><location><page_4><loc_48><loc_28><loc_91><loc_31></location>d d ν , (35)</formula>
<text><location><page_4><loc_9><loc_25><loc_91><loc_27></location>for ν a fiducial variable to be eliminated, rather than carrying any temporal (or other geometrical or physical) significance.</text>
<text><location><page_4><loc_9><loc_20><loc_91><loc_23></location>Remark 1 In the geometrical setting, the first equation here corresponds to Lie's Integral Approach to Geometrical Invariants.</text>
<text><location><page_4><loc_9><loc_12><loc_91><loc_18></location>Remark 2 The last equation uplifts this to Taking the Function Space Thereover (over configuration space in Geometry or over phase space in the canonical approach to Physics.) This involves feeding in the 'characteristic' solution u of the first equation into the last equation by eliminating ν . This u then ends up featuring as the 'functional form' that the observables depend on Q and P via,</text>
<formula><location><page_4><loc_46><loc_10><loc_91><loc_11></location>O GLYPH<16> O p u q . (36)</formula>
<text><location><page_5><loc_9><loc_94><loc_91><loc_98></location>This simple outcome reflects that, in the strong case, the last equation is just a trivial ODE. The function space in question needs to be at least once continuously differentiable. So the observables are suitably-smooth but elsewisearbitrary functions of Lie's invariants (literally in Geometry, or their phase space counterparts in Physics).</text>
<text><location><page_5><loc_9><loc_85><loc_91><loc_92></location>Remark 3 Solving for such arbitrary functions is to be contrasted with obtaining a single function by prescribing a specific boundary condition. Not prescribing such a boundary condition, on the one hand, amounts to implementing Taking a Function Space Thereover. On the other hand, its more general technical name is free alias natural [7] characteristic problem. The Characteristic Problem formulation for a single linear (or quasilinear) flow PDE is a standard prescription [7, 26].</text>
<text><location><page_5><loc_9><loc_80><loc_91><loc_83></location>Remark 4 At the geometrical level, our procedure is, given a constraint subalgebra C A , the observables equation [ C A , O ] first determining a characteristic surface</text>
<formula><location><page_5><loc_45><loc_77><loc_91><loc_78></location>χ GLYPH<16> χ p Q , P q . (37)</formula>
<text><location><page_5><loc_9><loc_72><loc_91><loc_76></location>This follows from solving all but the last equation in the equivalent flow ODE system. Secondly, the last equation in the flow ODE system is a trivial ODE solved by any suitably-smooth function thereover. This gives the observables algebra as</text>
<formula><location><page_5><loc_13><loc_69><loc_91><loc_70></location>C anO bs p S q GLYPH<16> C 8 p χ q GLYPH<16> t phase space functions whose restrictions to χ , f | χ , are smooth u . (38)</formula>
<text><location><page_5><loc_9><loc_65><loc_91><loc_67></location>Remark 5 C k for some fixed k ¥ 1 could be used instead, or some (perhaps weighted) Sobolev space [27]; we adhere to C 8 for simplicity.</text>
<section_header_level_1><location><page_5><loc_9><loc_61><loc_77><loc_62></location>3.2 N -point geometrical = purely configurational physical observables</section_header_level_1>
<text><location><page_5><loc_9><loc_59><loc_48><loc_60></location>This subproblem has been covered in e.g. [1, 2, 14, 76].</text>
<text><location><page_5><loc_9><loc_56><loc_41><loc_57></location>Definition 1 classical geometrical observables</text>
<formula><location><page_5><loc_42><loc_54><loc_91><loc_57></location>are O p Q q . (39)</formula>
<text><location><page_5><loc_9><loc_52><loc_73><loc_53></location>Structure 6 In the purely-geometrical setting, the a priori free functions O are subject to</text>
<formula><location><page_5><loc_45><loc_49><loc_91><loc_50></location>[ S , O ] GLYPH<16> 0 , (40)</formula>
<text><location><page_5><loc_9><loc_46><loc_82><loc_48></location>Here, S is the sum-over-N-points [14, 76] q I , I GLYPH<16> 1 to N of each particular generator, with components</text>
<formula><location><page_5><loc_40><loc_41><loc_91><loc_45></location>S G : GLYPH<16> N , I GLYPH<16> 1 G G b p q cI q B B q bI . (41)</formula>
<text><location><page_5><loc_9><loc_38><loc_59><loc_40></location>Remark 1 As detailed in [78], this is the pure-geometry analogue of C .</text>
<text><location><page_5><loc_9><loc_34><loc_91><loc_36></location>Remark 2 The Lie bracket equation (40) can furthermore be written out as an explicit PDE system. It should by now be clear that this PDE is moreover a subcase of that for canonical observables in Theoretical Physics,</text>
<formula><location><page_5><loc_44><loc_30><loc_91><loc_32></location>B C B P B O B Q GLYPH<16> 0 , (42)</formula>
<formula><location><page_5><loc_39><loc_25><loc_91><loc_28></location>with B C B P treated as knowns . (43)</formula>
<text><location><page_5><loc_9><loc_22><loc_91><loc_25></location>In particular, N -point geometrical observables coincide with the purely configuration space restriction of physical observables [78]; see Sec 7 for examples.</text>
<section_header_level_1><location><page_5><loc_9><loc_19><loc_50><loc_20></location>3.3 Pure-momentum physical observables</section_header_level_1>
<text><location><page_5><loc_9><loc_16><loc_56><loc_17></location>We furthermore consider the notion of pure-momentum observables</text>
<formula><location><page_5><loc_47><loc_14><loc_91><loc_15></location>O p P q . (44)</formula>
<text><location><page_5><loc_9><loc_11><loc_49><loc_12></location>These solve the pure-momentum observables PDE system</text>
<formula><location><page_5><loc_44><loc_7><loc_91><loc_9></location>B C B Q B O B P GLYPH<16> 0 , (45)</formula>
<formula><location><page_6><loc_39><loc_95><loc_91><loc_98></location>for B C B Q treated as knowns . (46)</formula>
<text><location><page_6><loc_9><loc_89><loc_91><loc_94></location>Remark 1 Configuration and momentum observables each readily represent a restriction of functions over P hase, to just over q , and to just over the space of momenta P respectively. These are, more specifically, polarization restrictions [35] since they precisely halve the number of variables. This applies at least in the quadratic theories we consider in the current Series. These are by far the simplest and most standard form for bosonic theories in Physics.</text>
<section_header_level_1><location><page_6><loc_9><loc_85><loc_41><loc_87></location>3.4 Various notions of genericity</section_header_level_1>
<text><location><page_6><loc_9><loc_83><loc_75><loc_84></location>PDE genericity From a PDE point of view, systems are more generic than single equations.</text>
<text><location><page_6><loc_9><loc_78><loc_91><loc_81></location>Finite-theory geometrical genericity However, in fixed-background finite theories of Geometry (or Physics), it is geometrically generic to have no (generalized) Killing vectors, and thus 0 or 1 observables PDEs.</text>
<text><location><page_6><loc_9><loc_74><loc_91><loc_77></location>Remark 1 This is 0 in pure geometry and in temporally-absolute finite physics, to 1 in temporally-relational finite physics: commutation with C .</text>
<text><location><page_6><loc_9><loc_71><loc_83><loc_72></location>Remark 2 For Finite Theories, PDE system genericity is in general obscured by geometrical genericity.</text>
<text><location><page_6><loc_9><loc_66><loc_91><loc_69></location>Remark 3 For Finite Theories, moreover, having 1 Killing vector is of secondary genericity between having no, and multiple, Killing vectors.</text>
<text><location><page_6><loc_9><loc_60><loc_91><loc_64></location>Remark 4 From this point of view, unconstrained observables are most generic (no observables PDEs at all), single observables PDEs are next most generic, and the more involved case of multiple observables equations is only tertiary in significance.</text>
<section_header_level_1><location><page_6><loc_9><loc_57><loc_68><loc_58></location>3.5 Nontrivial system case: determinedness and integrability</section_header_level_1>
<text><location><page_6><loc_9><loc_53><loc_91><loc_56></location>Remark 1 For nontrivial systems, multiple sequential uses of the Flow Method may apply. What needs to be checked first is determinedness [11], and, if over-determinedness occurs, integrability.</text>
<text><location><page_6><loc_9><loc_44><loc_91><loc_51></location>Remark 2 In Geometry, we have g : GLYPH<16> dim p g q constraints, and thus g observables equations. The observables carry an index O that has no a priori dependence on g . Thus, a priori, any of under-, well- or over-determinedness can occur (see also XIV.7). This conclusion transcends to the canonical approach to Physics as well. Here Temporal Relationalism and/or Constraint Closure can contribute further first-class constraints. g is thus replaced by a more general count f : GLYPH<16> dim p F q of functionally-independent first-class constraints.</text>
<text><location><page_6><loc_9><loc_38><loc_91><loc_42></location>On the one hand, generalized Killing equations' integrability conditions are not met generically ([8] or XIV.7). This signifies that there are only any proper generalized Killing vectors at all in a zero-measure subset of x M , σ y . This corresponds to the generic manifold admitting no (generalized) symmetries.</text>
<text><location><page_6><loc_9><loc_35><loc_91><loc_36></location>On the other hand, preserved equations moreover always succeed in meeting integrability, by the following Theorem.</text>
<text><location><page_6><loc_9><loc_32><loc_58><loc_33></location>Theorem 1 Classical canonical observables equations are integrable.</text>
<text><location><page_6><loc_9><loc_29><loc_76><loc_30></location>Remark 3 Consequently, classical observables always exist (subject to the following caveats).</text>
<text><location><page_6><loc_9><loc_26><loc_59><loc_27></location>Caveat 1 The current Article, and series, consider only local existence.</text>
<text><location><page_6><loc_9><loc_20><loc_91><loc_24></location>Caveat 2 Sufficiently large point number N is required in the case of finite point-particle theories. This is clear from the examples in [76, 77, 78], and corresponds to zero-dimensional reduced spaces having no coordinates left to support thereover any functions of coordinates.]</text>
<text><location><page_6><loc_9><loc_17><loc_91><loc_18></location>Remark 4 This Theorem is proven in [78], resting on the following vaguely modern version of Frobenius' Theorem.</text>
<text><location><page_6><loc_9><loc_12><loc_91><loc_15></location>Theorem 2 (A version of Frobenius' Theorem at the level of differentiable manifolds [27, 62]. A collection W of subspaces of a tangent space possesses integral submanifolds iff 1</text>
<formula><location><page_6><loc_37><loc_9><loc_91><loc_11></location>@ X, Y P W , |[ X , Y ]| P W . (47)</formula>
<section_header_level_1><location><page_7><loc_9><loc_97><loc_85><loc_98></location>3.6 Sequence of free characteristic problems for the strong observables system</section_header_level_1>
<text><location><page_7><loc_9><loc_94><loc_55><loc_95></location>We now have a more extensive ODE system of the form (33, 34).</text>
<text><location><page_7><loc_9><loc_90><loc_91><loc_92></location>Remark 1 Regardless of the single-equation to system distinction, corresponding observables ODEs are moreover autonomous (none of the functions therein depend on ν ).</text>
<text><location><page_7><loc_9><loc_87><loc_56><loc_88></location>Remark 2 We now have a first block rather than a first equation.</text>
<text><location><page_7><loc_9><loc_82><loc_91><loc_85></location>Remark 3 While the system version is not a standard prescription; study of strong observables PDE systems gets past this by of our integrability guarantee.</text>
<text><location><page_7><loc_9><loc_75><loc_91><loc_80></location>Sequential Approach . Suppose we have two equations. Solve one for its characteristics u 1 , say. Then substitute Q GLYPH<16> Q p u 1 q into the second equation to find which functional restrictions on the first solution's characteristics the second equation enforces. This procedure can moreover be applied inductively. It is independent of the choice of ordering in which the restrictions are applied by the nature of restrictions corresponding to geometrical intersections.</text>
<text><location><page_7><loc_9><loc_70><loc_91><loc_73></location>The Free Characteristic Problem posed above moreover leads to consideration of intersections of characteristic surfaces, which can moreover be conceived of in terms of restriction maps.</text>
<section_header_level_1><location><page_7><loc_9><loc_67><loc_24><loc_68></location>Theorem 3 Suppose</section_header_level_1>
<formula><location><page_7><loc_30><loc_66><loc_91><loc_67></location>V such that [ C V , V ] GLYPH<16> 0 forms characteristic surface χ V (48)</formula>
<text><location><page_7><loc_9><loc_63><loc_11><loc_65></location>and</text>
<formula><location><page_7><loc_29><loc_62><loc_91><loc_63></location>W such that [ C W , W ] GLYPH<16> 0 forms characteristic surface χ W (49)</formula>
<text><location><page_7><loc_9><loc_60><loc_48><loc_61></location>for constraint subalgebraic structures C V and C W . Then</text>
<text><location><page_7><loc_25><loc_57><loc_75><loc_58></location>O such that [ C V Y W , O ] GLYPH<16> 0 forms the characteristic surface of Fig 1 .</text>
<figure>
<location><page_7><loc_33><loc_44><loc_66><loc_53></location>
<caption>Figure 1: Characteristic surface resulting from commutation with two constraint subalgebraic structures.</caption>
</figure>
<text><location><page_7><loc_9><loc_36><loc_82><loc_37></location>Remark 4 This approach extends inductively to a finite number of equations in our flow ODE system.</text>
<text><location><page_7><loc_9><loc_33><loc_48><loc_34></location>Remark 5 See e.g. [76, 77, 78] for examples of its use.</text>
<text><location><page_7><loc_9><loc_25><loc_91><loc_31></location>Remark 6 The integrated form of the first m equations is used to eliminate t , with the other m GLYPH<1> 1 providing a basis of characteristic coordinates u ¯ a arising as constants of integration. In the geometrical case, this can still be considered to be Lie's Method of Geometric Invariants. After all, essentially all the most familiar geometries involve more than one independent condition on their integral invariants.</text>
<text><location><page_7><loc_9><loc_19><loc_91><loc_23></location>Remark 7 To elevate this to a determination of the system's observables, we then substitute these characteristic coordinates into the last equation. We thus obtain the general - and thus free alias natural problem-solving characteristic solution.</text>
<text><location><page_7><loc_9><loc_15><loc_91><loc_17></location>Remark 8 Our last equation remains a trivial ODE. It is thus solved by an suitably-smooth but elsewise arbitrary function of these characteristic coordinates, u with components u ¯ a</text>
<formula><location><page_7><loc_47><loc_12><loc_91><loc_13></location>O GLYPH<16> O p u q (50)</formula>
<text><location><page_7><loc_9><loc_8><loc_91><loc_10></location>Remark 9 The current Article (and Series) just considers a local rather than global treatment of observables equations.</text>
<section_header_level_1><location><page_8><loc_9><loc_97><loc_40><loc_98></location>4 Weak observables PDEs</section_header_level_1>
<text><location><page_8><loc_9><loc_94><loc_56><loc_95></location>Structure 7 For weak observables, the brackets equation (6) gives</text>
<formula><location><page_8><loc_35><loc_89><loc_91><loc_92></location>W GLYPH<128> C GLYPH<16> { C , O } GLYPH<16> B C B Q B O B P GLYPH<1> B C B P B O B Q . (51)</formula>
<text><location><page_8><loc_9><loc_87><loc_18><loc_88></location>We can again</text>
<formula><location><page_8><loc_37><loc_84><loc_91><loc_86></location>take B C B Q and B C B P to be knowns , (52)</formula>
<text><location><page_8><loc_9><loc_80><loc_91><loc_83></location>now alongside C being a known as well and W taking some prescribed value. This leaves us with an inhomogeneouslinear first-order PDE system.</text>
<text><location><page_8><loc_9><loc_76><loc_91><loc_78></location>Remark 1 The pure-geometry case cannot however support any properly weak observables. This is because we are in a context in which constraints have to depend on momenta</text>
<formula><location><page_8><loc_34><loc_73><loc_91><loc_74></location>C GLYPH<16> f p Q , P q with specific P dependence . (53)</formula>
<text><location><page_8><loc_9><loc_67><loc_91><loc_71></location>A fortiori, first-class linear constraints refer specifically to being linear in their momenta P , By this, the weak configurational observables equation would have a momentum-dependent inhomogeneous term right-hand side. But this is inconsistent with admitting a solution with purely Q -dependent right-hand-side.</text>
<section_header_level_1><location><page_8><loc_9><loc_64><loc_45><loc_65></location>4.1 Single weak observables equation</section_header_level_1>
<text><location><page_8><loc_9><loc_60><loc_91><loc_62></location>The weak observables PDE consists of a single inhomogeneous-linear equation. Its corresponding ODE system is now of the form</text>
<formula><location><page_8><loc_45><loc_58><loc_91><loc_59></location>9 x GLYPH<16> a p x, φ q , (54)</formula>
<formula><location><page_8><loc_42><loc_56><loc_91><loc_57></location>9 φ GLYPH<16> b p x, φ q GLYPH<16> W C 1 . (55)</formula>
<text><location><page_8><loc_9><loc_52><loc_91><loc_55></location>Remark 1 The first block is the same as before. Lie's integral invariants (in Geometry or their canonical Physics generalizations) thus still enter our expressions for observables.</text>
<text><location><page_8><loc_9><loc_48><loc_91><loc_50></location>Remark 2 The inhomogeneous term in the last equation, however, means that one has further particular-integral work to do in this weak case.</text>
<section_header_level_1><location><page_8><loc_9><loc_44><loc_48><loc_45></location>4.2 Nontrivial weak observables system</section_header_level_1>
<text><location><page_8><loc_9><loc_42><loc_64><loc_43></location>The corresponding ODE system now has inhomogeneous term b O GLYPH<16> W B AO C B .</text>
<text><location><page_8><loc_9><loc_37><loc_91><loc_40></location>Remark 1 Determinedness and integrability considerations carry over. So does sequential use of free characteristic problems on the first block.</text>
<text><location><page_8><loc_9><loc_34><loc_88><loc_35></location>Remark 2 Solving the last equation in the system is then conceptually the same as in the previous subsection.</text>
<text><location><page_8><loc_9><loc_31><loc_89><loc_32></location>Remark 3 We leave a systematic Green's function approach to weak observables equations for another occasion.</text>
<text><location><page_8><loc_9><loc_27><loc_91><loc_29></location>Remark 4 If we reduce all constraints out, the reduced formulation has strong r U being all the observables there can be.</text>
<text><location><page_8><loc_9><loc_22><loc_91><loc_25></location>Remark 5 If we reduce all constraints bar C out, the following applies since finite theory's Chronos is a single constraint equation.</text>
<text><location><page_8><loc_9><loc_18><loc_91><loc_20></location>Corollary 2 Suppose there is only one constraint. Then there is only one independent proper weak observable, and the proper weak observables algebra is abelian.</text>
<text><location><page_8><loc_9><loc_15><loc_58><loc_16></location>Proof The PI is now a single integral up to a multiplicative constant.</text>
<text><location><page_8><loc_9><loc_12><loc_46><loc_13></location>Any finite-algebra generator commutes with itself. ✷</text>
<text><location><page_8><loc_9><loc_9><loc_62><loc_10></location>Remark 6 Let us check how this comes to be consistent with Corollary 1.</text>
<text><location><page_9><loc_9><loc_95><loc_91><loc_98></location>The first part of the first factor of (15)'s antisymmetry is not supported in nontrivially-weak observables space of dimension 1. (15) thus collapses to the furtherly factorizable form</text>
<formula><location><page_9><loc_45><loc_92><loc_91><loc_94></location>O w W O C GLYPH<16> 0 . (56)</formula>
<text><location><page_9><loc_9><loc_85><loc_91><loc_91></location>C GLYPH<16> 0 is disallowed as C is a nontrivial constraint, whereas W O : GLYPH<16> W 1 1 O GLYPH<16> 0 is disallowed as O w is nontrivially-weak. The there is no space for the zero double-trace perpendicularity as each trace is now over a 1d index and the 1 matrix 1 is manifestly nondegenerate. Thus O w GLYPH<16> 0 is the only surviving possibility, coinciding with the above deduction that the nontrivially-weak observables algebra is abelian.</text>
<section_header_level_1><location><page_9><loc_9><loc_81><loc_54><loc_82></location>5 Expression in Terms of Observables</section_header_level_1>
<text><location><page_9><loc_9><loc_77><loc_91><loc_79></location>We here continue to develop Sec III.3. First recall that, by Lemma III.3, the B themselves form a closed algebraic structure. The amount of observables, is, moreover, very large due to the following Lemma.</text>
<text><location><page_9><loc_9><loc_74><loc_54><loc_75></location>Lemma 3 Functions of observables are themselves observables.</text>
<text><location><page_9><loc_9><loc_71><loc_12><loc_72></location>Proof</text>
<formula><location><page_9><loc_33><loc_68><loc_91><loc_71></location>{ C , F p O q } GLYPH<16> { C , O } B F B O GLYPH<19> 0 GLYPH<2> B F B O GLYPH<16> 0 . ✷ (57)</formula>
<text><location><page_9><loc_9><loc_61><loc_91><loc_67></location>Remark 1 Thus one is not looking for individual solutions of the observables PDEs, but a fortiori for whole algebras of solutions. Namely, it is closed among themselves and large enough to span all of a physical theory's mathematical content. I.e. enough to span P hase, with each functionally independent of the others. This renders useful the concept mentioned in Article III of finding 'basis observables'.</text>
<text><location><page_9><loc_9><loc_57><loc_91><loc_60></location>Remark 2 Only a smaller number of suitably-chosen observables moreover suffice for practical use: expression in terms of observables.</text>
<text><location><page_9><loc_9><loc_51><loc_91><loc_55></location>Remark 3 A common case is for dim(reduced P hase) GLYPH<16> 2 t k GLYPH<1> g u basis observables (of type G ) to be required. Here k : GLYPH<16> dim p q q and g : GLYPH<16> dim p g q is the total number of constraints involved, which are all gauge constraints. See e.g. Sec 7.3 for examples of 'basis observables'.</text>
<section_header_level_1><location><page_9><loc_9><loc_47><loc_61><loc_48></location>6 Strategies for the Problem of Observables</section_header_level_1>
<text><location><page_9><loc_9><loc_44><loc_59><loc_45></location>The 'bottom' alias 'zero' and 'top' alias 'unit' strategies are as follows.</text>
<text><location><page_9><loc_9><loc_41><loc_81><loc_42></location>Strategy 4.0) Use Unconstrained Observables , U [48, 50], entailing no commutation conditions at all.</text>
<text><location><page_9><loc_9><loc_38><loc_51><loc_39></location>Strategy 4.1) Insist on Constructing Dirac Observables , D .</text>
<text><location><page_9><loc_9><loc_30><loc_91><loc_36></location>Remark 1 Strategy 4.1) has the conceptual and physical advantage of employing all the information in the final algebraic structure of all-first-class constraints of the theory, F . It has the practical disadvantage that finding any Dirac observables - much less a basis set for each theory in question - can be a hard mathematical venture, especially for Gravitational Theories [5, 10, 19, 37, 38, 39, 41, 48, 50, 51, 54, 56, 63, 69, 72].</text>
<text><location><page_9><loc_9><loc_26><loc_91><loc_28></location>Remark 2 Strategy 4.2) is diametrically opposite in each of the above regards. It can moreover be used as first stepping stone toward the former.</text>
<text><location><page_9><loc_9><loc_20><loc_91><loc_24></location>Remark 3 Strategy 4.1) moreover amounts to concurrently addressing the unsplit totality of constraints (Constraint Closure facet) and Taking Function Spaces Thereover. As a four-aspect venture (Fig 3), it is unsurprisingly harder than Use Unconstrained Observables, which is single-aspect.</text>
<text><location><page_9><loc_9><loc_15><loc_91><loc_18></location>Strategy 4.K) Find Kuchař Observables K . This can entail treating Q uad distinctly from the F lin , some motivations for which were covered in VII. A further pragmatic reason is that the K are simpler to find than the D .</text>
<text><location><page_9><loc_9><loc_8><loc_91><loc_14></location>Strategy 4.G) Find g -observables , G . If one looks more closely, some of these motivations are actually tied to the G in cases in which these and the K are distinct. For instance, it is more generally G - rather than the K - which arise from the g -act, g -all construction in cases in which the candidate S huffle is confirmed as a G auge . Also, theories having either trivial Configurational Relationalism - or Best Matching resolved - have as a ready consequence a</text>
<text><location><page_10><loc_9><loc_92><loc_91><loc_98></location>known full set of classical G . We take this on board by pointing to this further distinct strategy. This takes into account the triple combination of Configurational Relationalism, Constraint Closure, and Assignmant of Observables aspects. Complementarily, finding the G (or K ) represents a timeless pursuit due to the absence of C hronos (or underlying Temporal Relationalism) from the workings in question.</text>
<text><location><page_10><loc_9><loc_85><loc_91><loc_90></location>Strategy 4.C) Find Chronos Observables , C . This takes into account the triple aspect combination of Temporal Relationalism, Constraint Closure and Taking Function Spaces Thereover. In theories with nontrivial g or some further first-class constraints, this is to be viewed as a stepping stone. It is available if C hronos indeed constitutes a constraint subalgebraic structure.</text>
<text><location><page_10><loc_9><loc_77><loc_91><loc_83></location>Remark 4 (Non)universality arguments are pertinent at this point. Using U or D is always in principle possible. The first of these follows from no restrictions being imposed. The second follows from how any theory's full set of constraints can in principle be cast as a closed algebraic structure of first-class constraints. This is by use of the Dirac bracket, or the effective method, so as to remove any second-class constraints.</text>
<text><location><page_10><loc_9><loc_67><loc_91><loc_75></location>Strategy 4.A) We finally introduce an additional universal strategy based on using some kind of A-Observables , A x , that a theory happens to possess . This corresponds to the closed subalgebraic structures of constraints which are realized by that theory. It is the general 'middling' replacement for considering K , G or C hronos observables. (None of these notions are universal over all physical theories.) While the ultimate aim is to reach the top of the lattice, it is often practically attainable to, firstly, land somewhere in the middle of the lattice. Secondly, to work one's way up by solving further DEs (or, geometrically, by further restricting constraint surfaces).</text>
<text><location><page_10><loc_9><loc_63><loc_44><loc_65></location>A second source of strategic diversity is as follows.</text>
<text><location><page_10><loc_9><loc_60><loc_82><loc_62></location>Strategy 4 1 .0) The unreduced approach : working on the unreduced P hase p S q with all the constraints.</text>
<text><location><page_10><loc_9><loc_54><loc_91><loc_59></location>Strategy 4 1 .1) The true space approach - working on T rueP hase p S q GLYPH<16> P hase p S q{ F - is much harder, due to quotienting out C hronos p S q being harder and having to be done potential by potential. If True is known, classical observables are trivial.</text>
<text><location><page_10><loc_9><loc_51><loc_78><loc_53></location>Strategy 4 1 .G) The reduced approach : working on GLYPH<131> P hase p S q GLYPH<16> P hase p S q{ g with just C hronos.</text>
<text><location><page_10><loc_9><loc_48><loc_86><loc_50></location>Strategy 4 1 .P) The partly-reduced approach : working on P hase p S , H q GLYPH<16> P hase p S q{ H for some id GLYPH<160> H GLYPH<160> g</text>
<text><location><page_10><loc_9><loc_44><loc_91><loc_46></location>Remark 5 The above primed family of strategies are moreover found to all coincide in output in the case of strong observables [78]. E.g. both unreduced and reduced approaches end up with</text>
<formula><location><page_10><loc_36><loc_41><loc_91><loc_42></location>C anG augeO bs p S q GLYPH<16> C 8 p GLYPH<131> P hase p S qq (58)</formula>
<text><location><page_10><loc_9><loc_38><loc_33><loc_39></location>or, in the purely geometrical case,</text>
<formula><location><page_10><loc_36><loc_37><loc_91><loc_38></location>G eomG augeO bs p S q GLYPH<16> C 8 p r q p S qq . (59)</formula>
<section_header_level_1><location><page_10><loc_9><loc_32><loc_80><loc_34></location>7 Explicit solutions for observables and observables algebras</section_header_level_1>
<section_header_level_1><location><page_10><loc_9><loc_29><loc_33><loc_31></location>7.1 Unreduced examples</section_header_level_1>
<text><location><page_10><loc_9><loc_27><loc_42><loc_28></location>Unreduced observables form the function space</text>
<formula><location><page_10><loc_36><loc_24><loc_91><loc_26></location>C anU nresO bs p S q GLYPH<16> C 8 p P hase p S qq (60)</formula>
<text><location><page_10><loc_9><loc_22><loc_51><loc_23></location>of suitably-smooth functions over our system's phase space.</text>
<text><location><page_10><loc_9><loc_19><loc_53><loc_20></location>Unreduced configurational observables form the function space</text>
<formula><location><page_10><loc_37><loc_16><loc_91><loc_17></location>G eomU nresO bs p S q GLYPH<16> C 8 p q p S qq (61)</formula>
<text><location><page_10><loc_9><loc_13><loc_56><loc_14></location>of suitably-smooth functions over our system's configuration space.</text>
<text><location><page_10><loc_9><loc_10><loc_54><loc_11></location>Unreduced pure-momentum observables form the function space</text>
<formula><location><page_10><loc_37><loc_7><loc_91><loc_9></location>M omU nresO bs p S q GLYPH<16> C 8 p P p S qq (62)</formula>
<formula><location><page_11><loc_48><loc_94><loc_91><loc_95></location>P p S q (63)</formula>
<text><location><page_11><loc_9><loc_92><loc_66><loc_93></location>that consists of the totality of values taken by the given model S 's momenta P .</text>
<text><location><page_11><loc_9><loc_87><loc_91><loc_90></location>Example 1 For Mechanics on R d , the configuration space is the constellation space (I.5-6), q p d, N q GLYPH<16> R dN . This being a flat space, the corresponding constellation phase space is</text>
<formula><location><page_11><loc_41><loc_85><loc_91><loc_86></location>P hase p N,d q : GLYPH<16> R 2 N d . (64)</formula>
<text><location><page_11><loc_9><loc_82><loc_29><loc_83></location>The unrestricted observables</text>
<formula><location><page_11><loc_45><loc_81><loc_91><loc_82></location>U GLYPH<16> U p q , p q (65)</formula>
<text><location><page_11><loc_9><loc_78><loc_15><loc_80></location>here form</text>
<formula><location><page_11><loc_28><loc_77><loc_91><loc_78></location>C anU nresO bs p d, N q : GLYPH<16> C 8 p P hase p M , N qq GLYPH<16> C 8 p R 2 N d q . (66)</formula>
<text><location><page_11><loc_9><loc_75><loc_85><loc_76></location>The unrestricted configurational observables U p q q , alias absolute space geometry's N -point invariants, form</text>
<formula><location><page_11><loc_30><loc_72><loc_91><loc_74></location>G eomU nresO bs p d, N q : GLYPH<16> C 8 p q p d, N qq GLYPH<16> C 8 p R dN q . (67)</formula>
<text><location><page_11><loc_9><loc_70><loc_54><loc_71></location>Finally the unrestricted pure-momentum observables U p p q form</text>
<formula><location><page_11><loc_30><loc_67><loc_91><loc_69></location>M omU nresO bs p d, N q : GLYPH<16> C 8 p p p d, N qq GLYPH<16> C 8 p R N d q . (68)</formula>
<text><location><page_11><loc_9><loc_65><loc_60><loc_66></location>These results readily generalize to RPMs over manifolds other than R n .</text>
<section_header_level_1><location><page_11><loc_9><loc_61><loc_81><loc_63></location>7.2 Nontrivial geometrical examples of Kuchař, gauge- and A-observables</section_header_level_1>
<text><location><page_11><loc_9><loc_56><loc_91><loc_60></location>Simplification 1 Suppose the constraints being taken into consideration depend at most linearly on the momenta. This holds within Aff p d q and its lattice of subgroups, covering all but the last example in the current subsection. Then in the observables equations,</text>
<formula><location><page_11><loc_29><loc_52><loc_91><loc_55></location>the cofactor of B G B Q - i.e. B C B P - is independent of P . (69)</formula>
<text><location><page_11><loc_9><loc_48><loc_91><loc_51></location>Simplification 2 Restrict attention to purely configurational such observables K p Q q GLYPH<16> G p Q q . This gives the particularly simple PDE [following on from (VII.22)]</text>
<formula><location><page_11><loc_40><loc_44><loc_91><loc_47></location>F A N p Q alone q B G B Q A GLYPH<16> 0 ; (70)</formula>
<text><location><page_11><loc_9><loc_42><loc_47><loc_43></location>we drop the Q suffix when considering pure geometry.)</text>
<text><location><page_11><loc_9><loc_37><loc_91><loc_40></location>Simplification 3 A few of the below examples have but a single observables equation (i.e. the N -index takes a single value). This is then amenable to the standard Flow Method.</text>
<text><location><page_11><loc_9><loc_28><loc_91><loc_35></location>Simplification 4 Within Aff p d q 's lattice of subgroups, passage to the centre of mass frame is available to take translations out of contention. Upon doing this, mass-weighted relative Jacobi coordinates [34, 82] furthermore furnish a widespread simplification of one's remaining equations [59, 73]. This determines ab-initio-translationless and translation-reduced actions, constraints, Hamiltonians and observables, as being of the same form but with one object less: the so-called Jacobi map [73].</text>
<text><location><page_11><loc_9><loc_22><loc_91><loc_26></location>Notation 1 We additionally provided a compact notation for the outcome of solving the preserved equations for Sim p d q and its subgroups [73, 74] in Sec III.3. This is visible within row 1 of Fig 2, which furthermore extends this notation to Aff p d q and its subgroups [67, 77].</text>
<text><location><page_11><loc_9><loc_19><loc_71><loc_21></location>Example 1) For translation-invariant geometry g GLYPH<16> Tr p d q - the observables PDE is</text>
<formula><location><page_11><loc_43><loc_15><loc_91><loc_18></location>, N I =1 B G B q I GLYPH<16> 0 . (71)</formula>
<text><location><page_11><loc_9><loc_10><loc_91><loc_14></location>The solutions of this are, immediately, the relative interparticle separation vectors. These can be reformulated as linear combinations thereof, among which the relative Jacobi coordinates ρ turn out to be particularly convenient by the Jacobi map. Our solutions form</text>
<formula><location><page_11><loc_33><loc_7><loc_91><loc_8></location>G eomG augeO bs p d, N ; Tr p d qq GLYPH<16> C 8 p R nd q : (72)</formula>
<figure>
<location><page_12><loc_8><loc_39><loc_91><loc_98></location>
</figure>
<table>
<location><page_12><loc_8><loc_39><loc_91><loc_98></location>
<caption>Figure 2: a) As a reminder of Article III's notation, the minus stands for difference, the backslash for ratio, and the dot for scalar product. The new wedge symbol stands for the 'top form' supported by the dimension in question. So e.g. area's cross product in 2d or volume's scalar triple product in 3d . The notation for this symbol is slightly different than for the others, since it is an n -ary rather than binary operation.</caption>
</table>
<text><location><page_12><loc_9><loc_27><loc_91><loc_31></location>b) The semicolon here denotes cross ratio. See [77] for what any notions of geometry mentioned in the first two rows mean. I subsequently found that Guggenheimer [14], and then that previously Lie himself [1] already derived the top result here - the 1d projective group - by a flow PDE method. So (for now at least), the top case of Example 8 is to be attributed to Lie: an early success of his Integral Approach to Geometrical Invariants.</text>
<text><location><page_12><loc_9><loc_24><loc_26><loc_25></location>c) is derived in detal in [78].</text>
<text><location><page_12><loc_9><loc_20><loc_38><loc_21></location>the smooth functions over relative space.</text>
<text><location><page_12><loc_9><loc_17><loc_64><loc_19></location>Example 2 For scale-invariant geometry g GLYPH<16> Dil - the observables PDE is</text>
<formula><location><page_12><loc_42><loc_13><loc_91><loc_16></location>, N I =1 q I GLYPH<4> B G B q I GLYPH<16> 0 . (73)</formula>
<text><location><page_12><loc_9><loc_9><loc_91><loc_11></location>This is an Euler homogeneity equation of degree zero. Its solutions are therefore ratios of components of configurations. These form</text>
<formula><location><page_12><loc_33><loc_7><loc_91><loc_9></location>G eomG augeO bs p d, N ; Dil q GLYPH<16> C 8 p S N d GLYPH<1> 1 q : (74)</formula>
<text><location><page_13><loc_9><loc_97><loc_36><loc_98></location>the smooth functions over ratio space.</text>
<text><location><page_13><loc_9><loc_94><loc_71><loc_95></location>Example 3 For rotationally-invariant geometry g GLYPH<16> Rot p d q - the observables PDE is</text>
<formula><location><page_13><loc_41><loc_90><loc_91><loc_92></location>, N I =1 q I GLYPH<2> B G B q I GLYPH<16> 0 . (75)</formula>
<text><location><page_13><loc_9><loc_85><loc_91><loc_88></location>This is solved by the dot products q I GLYPH<4> q J . Norms and angles are moreover particular cases of functionals of the above, which are an allowed extension by Lemma 3. In 2d , these form</text>
<formula><location><page_13><loc_32><loc_83><loc_91><loc_84></location>G eomG augeO bs p 2 , N ; Rot p 2 qq GLYPH<16> C 8 p C p CP n qq , (76)</formula>
<text><location><page_13><loc_9><loc_80><loc_65><loc_81></location>where C p M q denotes topological and geometrical cone over M and n : GLYPH<16> N GLYPH<1> 1.</text>
<text><location><page_13><loc_9><loc_73><loc_91><loc_79></location>Example 4 In Euclidean geometry g GLYPH<16> Eucl p d q GLYPH<16> Tr p d q GLYPH<11> Rot p d q - the observables PDEs are both (71) and (75). Sequential use of the Flow Method gives that the solutions are dots of differences of position vectors. This can be worked into the form of dots of relative Jacobi vectors. These are additionally Euclidean RPM's [25, 59] configurational Kuchař = gauge observables [68]. In 2d , they form</text>
<formula><location><page_13><loc_30><loc_70><loc_91><loc_71></location>G eomG augeO bs p 2 , N ; Eucl p 2 qq GLYPH<16> C 8 p C p CP n GLYPH<1> 1 qq : (77)</formula>
<text><location><page_13><loc_9><loc_68><loc_39><loc_69></location>the smooth functions over relational space.</text>
<text><location><page_13><loc_9><loc_63><loc_91><loc_66></location>Example 5 In rotational-and-dilational geometry g GLYPH<16> Rot p d q GLYPH<2> Dil - the observables PDEs are both (75) and (73). The geometrical G are thus ratios of dots. In 2d , these form</text>
<formula><location><page_13><loc_31><loc_60><loc_91><loc_62></location>G eomG augeO bs p 2 , N ; Rot p 2 q GLYPH<2> Dil q GLYPH<16> C 8 p CP n q . (78)</formula>
<text><location><page_13><loc_9><loc_56><loc_91><loc_59></location>Example 6 In dilatational geometry g GLYPH<16> Dilatat p d q : GLYPH<16> Tr p d q GLYPH<11> Dil - the observables PDEs are both (71) and (73). The geometrical G are thus ratios of differences. These form</text>
<formula><location><page_13><loc_31><loc_54><loc_91><loc_55></location>G eomG augeO bs p d, N ; Dilatat p d qq GLYPH<16> C 8 p S nd GLYPH<1> 1 q : (79)</formula>
<text><location><page_13><loc_9><loc_51><loc_42><loc_52></location>the smooth functions over preshape space [46].</text>
<text><location><page_13><loc_9><loc_45><loc_91><loc_50></location>Example 7 In similarity geometry g GLYPH<16> Sim p d q GLYPH<16> Tr p d q GLYPH<11> p Rot p d q GLYPH<2> Dil q - the observables PDEs are all three of (71), (75) and (73). The geometrical G are thus ratios of dots of differences. These are additionally similarity RPM's [59] configurational Kuchař observables K [68]. In 2d , these form</text>
<formula><location><page_13><loc_32><loc_43><loc_91><loc_44></location>G eomG augeO bs p 2 , N ; Sim p 2 qq GLYPH<16> C 8 p CP n GLYPH<1> 1 q : (80)</formula>
<text><location><page_13><loc_9><loc_40><loc_39><loc_41></location>the smooth functions over shape space [46].</text>
<text><location><page_13><loc_9><loc_34><loc_91><loc_39></location>Example 8 Row 1 of Fig 2 identifies geometrical observables for Affine Geometry - corresponding to g GLYPH<16> Aff p d q - and for various of its further subgroups. We need dimension d ¥ 2 for these groups to be nontrivially realized. These additionally constitute configurational observables for affine RPM [67].</text>
<text><location><page_13><loc_9><loc_26><loc_91><loc_32></location>Each of Examples 1 to 7 also constitute nontriviallyA examples (in the present context A that are not also K GLYPH<16> G ) for affine geometry, as do all other entries in the third subfigure of Fig 2.a) Each of Examples 1 to 6 perform this function for similarity geometry as well. Two-thirds of Examples 1 to 3 perform this function for each of Examples 4 to 6. Flat Geometry thus already provides copious numbers of examples of nontriviallyA observables.</text>
<text><location><page_13><loc_9><loc_19><loc_91><loc_25></location>Example 9 Row 2 of Fig 2 identifies geometrical observables for 1d Projective Geometry - corresponding to g GLYPH<16> Proj p 1 q - and various of its further subgroups. Passing to the centre of mass however ceases to function as a simplification for Projective Geometry (and Conformal Geometry). This is since translations are more intricately involved here than via Affine Geometry's semidirect product addendum.</text>
<section_header_level_1><location><page_13><loc_9><loc_15><loc_55><loc_17></location>7.3 Geometrical examples of 'basis observables'</section_header_level_1>
<text><location><page_13><loc_9><loc_13><loc_57><loc_14></location>We here provide Euclidean RPM K GLYPH<16> G observables examples of this.</text>
<text><location><page_13><loc_9><loc_7><loc_91><loc_11></location>Example 1 3 particles in 1d have the mass-weighted relative Jacobi separations ρ 1 , ρ 2 as useful basis observables (in K GLYPH<16> G sense). This extends to N particles in 1d having as basis observables a given clustering's ρ i , i GLYPH<16> 1 to n GLYPH<16> N GLYPH<1> 1.</text>
<text><location><page_14><loc_9><loc_95><loc_86><loc_96></location>Example 2 The 3 Hopf-Dragt coordinates of Sec V.5.5 are basis observables for the relational triangle [59].</text>
<text><location><page_14><loc_9><loc_92><loc_89><loc_93></location>Example 3 The 8 'Gell-Mann' coordinates detailed in [60] are basis observables for the relational quadrilateral.</text>
<text><location><page_14><loc_9><loc_79><loc_91><loc_90></location>Motivation Kinematical quantization uses a lot less classical observables than the totality of suitably smooth functions over P hase. Kinematical quantization uses, more specifically, a linear subspace thereof that the canonical group acts upon [30]. This linear subspace moreover has enough coordinates to locally characterize q , thus fitting within our looser conception of 'basis beables'. Kinematical quantization's linear subspace quantities do moreover literally form a basis for that linear space. In this way, they constitute 'more of a basis' for configuration space than the a priori concept of 'basis observables' do. This linearity does not however in general extend to kinematical quantization's corresponding momentum observables (angular momentum suffices to see this). All of our examples above are useful for kinematical quantization.</text>
<section_header_level_1><location><page_14><loc_9><loc_75><loc_76><loc_76></location>7.4 Physically nontrivial examples of strong K , G , A and C observables</section_header_level_1>
<text><location><page_14><loc_9><loc_71><loc_91><loc_74></location>Example 1 Translationally-invariant RPM's pure-momentum observables are freely specifiable. However, since the total centre of mass position is meaningless in this problem, its momentum is meaningless as well, leaving us with</text>
<formula><location><page_14><loc_40><loc_68><loc_91><loc_70></location>G GLYPH<16> G GLYPH<1> p i GLYPH<1> p N GLYPH<9> GLYPH<16> G p π i q (81)</formula>
<text><location><page_14><loc_9><loc_65><loc_43><loc_66></location>for π i the conjugate momenta to ρ i . These form</text>
<formula><location><page_14><loc_35><loc_62><loc_91><loc_63></location>M omG auge-bs p d, N ; Tr q GLYPH<16> C 8 p R nd q , (82)</formula>
<text><location><page_14><loc_9><loc_59><loc_49><loc_60></location>i.e. the smooth functions over relative momentum space.</text>
<text><location><page_14><loc_9><loc_56><loc_39><loc_57></location>The corresponding general observables are</text>
<formula><location><page_14><loc_35><loc_53><loc_91><loc_55></location>G GLYPH<16> G GLYPH<1> q i GLYPH<1> q N , p i GLYPH<1> p N GLYPH<9> GLYPH<16> G GLYPH<0> ρ i , π i GLYPH<8> . (83)</formula>
<text><location><page_14><loc_9><loc_50><loc_17><loc_51></location>These form</text>
<formula><location><page_14><loc_34><loc_48><loc_91><loc_50></location>C anG augeO bs p d, N ; Tr p d qq GLYPH<16> C 8 p R 2 nd q , (84)</formula>
<text><location><page_14><loc_9><loc_46><loc_45><loc_47></location>i.e. the smooth functions over relative phase space.</text>
<text><location><page_14><loc_9><loc_43><loc_68><loc_44></location>Example 2 Dilationally-invariant RPM's pure-momentum observables PDE is the</text>
<formula><location><page_14><loc_47><loc_40><loc_91><loc_42></location>q I GLYPH<216> p I (85)</formula>
<text><location><page_14><loc_9><loc_36><loc_91><loc_39></location>of the corresponding geometrical observables PDE system. This is thus solved by ratios of components of momenta. These form</text>
<formula><location><page_14><loc_33><loc_35><loc_91><loc_36></location>M omG auge-bs p c, N ; Tr p d qq GLYPH<16> C 8 p S N d GLYPH<1> 1 q , (86)</formula>
<text><location><page_14><loc_9><loc_32><loc_47><loc_33></location>i.e. the smooth functions over momentum ratio space.</text>
<text><location><page_14><loc_9><loc_29><loc_47><loc_30></location>The corresponding general observables PDE system is</text>
<formula><location><page_14><loc_36><loc_25><loc_91><loc_28></location>, N I =1 " B G B p I GLYPH<4> p I GLYPH<1> q I GLYPH<4> B G B q I * GLYPH<16> 0 . (87)</formula>
<text><location><page_14><loc_9><loc_21><loc_91><loc_23></location>This is also an Euler homogeneity equation of degree zero. Its solutions are therefore ratios of phase space coordinates. These now correspond not to sphere in phase space but to a quadric surface</text>
<formula><location><page_14><loc_38><loc_17><loc_91><loc_19></location>, N I =1 GLYPH<1> || p I || 2 GLYPH<1> || q I || 2 GLYPH<9> GLYPH<16> const (88)</formula>
<text><location><page_14><loc_9><loc_15><loc_71><loc_16></location>in phase space. The observables constitute the C 8 functions over this phase ratio space.</text>
<text><location><page_14><loc_9><loc_9><loc_91><loc_13></location>Example 3 Rotationally-invariant RPM's pure-momentum observables PDE system is also the (85) of the corresponding geometrical observables PDE system. This is thus is solved by suitably smooth functions of the dot product,</text>
<formula><location><page_14><loc_44><loc_7><loc_91><loc_8></location>G GLYPH<16> G p p I GLYPH<4> p J q . (89)</formula>
<text><location><page_15><loc_9><loc_97><loc_21><loc_98></location>In 2d , these form</text>
<formula><location><page_15><loc_32><loc_95><loc_91><loc_97></location>M omG auge-bs p d, N ; Rot p 2 qq GLYPH<16> C 8 p C p CP N qq . (90)</formula>
<text><location><page_15><loc_9><loc_93><loc_47><loc_94></location>The corresponding general observables PDE system is</text>
<formula><location><page_15><loc_34><loc_89><loc_91><loc_92></location>, N I =1 " B G B p I GLYPH<2> p I GLYPH<0> B G B q I GLYPH<2> q I * GLYPH<16> 0 . (91)</formula>
<text><location><page_15><loc_9><loc_86><loc_41><loc_87></location>This is solved by suitably-smooth functions of</text>
<formula><location><page_15><loc_41><loc_83><loc_91><loc_84></location>GLYPH<4> S : GLYPH<16> q I GLYPH<4> p J GLYPH<0> p I GLYPH<4> q J : (92)</formula>
<text><location><page_15><loc_9><loc_80><loc_81><loc_81></location>phase space symmetrized dot products. These are the outcome of applying the product rule to q I GLYPH<4> q J .</text>
<text><location><page_15><loc_9><loc_76><loc_91><loc_78></location>Example 4 Euclidean RPM combines the above translational and rotational equations. The Jacobi map applying, the Euclidean momentum observables solutions are suitably smooth functions</text>
<formula><location><page_15><loc_45><loc_73><loc_91><loc_74></location>G GLYPH<16> G p π i GLYPH<4> π j q (93)</formula>
<text><location><page_15><loc_9><loc_70><loc_21><loc_71></location>In 2d , these form</text>
<formula><location><page_15><loc_32><loc_69><loc_91><loc_70></location>M omG auge-bs p d, N ; Eucl p 2 qq GLYPH<16> C 8 p C p CP n qq : (94)</formula>
<text><location><page_15><loc_9><loc_66><loc_91><loc_67></location>the smooth functions over relational momentum space. The general observable solutions are suitably smooth functions</text>
<formula><location><page_15><loc_37><loc_63><loc_91><loc_65></location>G GLYPH<16> G p ρ i GLYPH<4> π j GLYPH<0> π i GLYPH<4> ρ j q GLYPH<16> G pGLYPH<1> GLYPH<4> S GLYPH<1>q , (95)</formula>
<text><location><page_15><loc_9><loc_61><loc_45><loc_62></location>i.e. relative phase space symmetrized dot products.</text>
<text><location><page_15><loc_9><loc_56><loc_91><loc_59></location>Example 5 Similarity RPM combines all three of the above translational, rotational and dilational equations. So for instance, the momentum observables solutions are suitably smooth functions</text>
<formula><location><page_15><loc_42><loc_53><loc_91><loc_55></location>G GLYPH<16> G p π i GLYPH<4> π j { π k GLYPH<4> π l q , (96)</formula>
<text><location><page_15><loc_9><loc_51><loc_50><loc_52></location>and the general observables are suitably smooth functions</text>
<formula><location><page_15><loc_42><loc_48><loc_91><loc_49></location>G GLYPH<16> G pGLYPH<1> GLYPH<4> S GLYPH<1>{ GLYPH<1> GLYPH<4> S GLYPH<1>q . (97)</formula>
<text><location><page_15><loc_9><loc_44><loc_91><loc_46></location>Example 6 If the generators are quadratic (which we know from Article III to apply in the conformal and projective cases), then (85) symmetry among observables is broken.</text>
<text><location><page_15><loc_9><loc_41><loc_65><loc_42></location>Example 7 Chronos observables C for the general p N,d q Euclidean RPM solve</text>
<formula><location><page_15><loc_41><loc_36><loc_91><loc_39></location>p GLYPH<4> B C B q GLYPH<0> B V B q B C B q GLYPH<16> 0 . (98)</formula>
<text><location><page_15><loc_9><loc_34><loc_44><loc_35></location>In the case of constant potential, this simplifies to</text>
<formula><location><page_15><loc_45><loc_30><loc_91><loc_32></location>p GLYPH<4> B C B q GLYPH<16> 0 (99)</formula>
<text><location><page_15><loc_9><loc_27><loc_21><loc_28></location>which is solved by</text>
<formula><location><page_15><loc_35><loc_26><loc_91><loc_27></location>C GLYPH<16> C GLYPH<0> q Λ p N d GLYPH<1> p Λ q N d , p Λ 2 GLYPH<1> p N d 2 GLYPH<8> . (100)</formula>
<text><location><page_15><loc_9><loc_20><loc_91><loc_25></location>Treating the N d component differently is an arbitrary choice; Λ then runs over all the other values, 1 to N d GLYPH<1> 1. For RPMs with any Configurational Relationalism, these are a further species of A observable that is not a K or G observable.</text>
<text><location><page_15><loc_9><loc_16><loc_91><loc_18></location>Remark 1 That C hronosO bs (and D iracO bs) are potential-dependent, constitutes a massive complication at the computational level.</text>
<section_header_level_1><location><page_16><loc_9><loc_97><loc_61><loc_98></location>7.5 Examples of strong nontrivially-Dirac observables</section_header_level_1>
<text><location><page_16><loc_9><loc_93><loc_91><loc_95></location>Example 1 For RPMs involving whichever combination of translations and rotations, the additional PDE to obtain strong Dirac observables is</text>
<text><location><page_16><loc_40><loc_93><loc_41><loc_93></location>#</text>
<text><location><page_16><loc_58><loc_93><loc_59><loc_93></location>+</text>
<formula><location><page_16><loc_35><loc_89><loc_91><loc_92></location>, N I =1 B V B q I GLYPH<4> B D B p I GLYPH<1> p I GLYPH<4> B D B q I '=' 0 . (101)</formula>
<text><location><page_16><loc_9><loc_85><loc_91><loc_88></location>If dilations are involved as well, one needs to divide the kinetic term contribution by the total moment of inertia to have a ratio form.</text>
<text><location><page_16><loc_9><loc_81><loc_91><loc_84></location>The above are moreover mathematically equivalent to the corresponding p d, n q Chronos problems, so e.g. in the Eucl p d q case</text>
<formula><location><page_16><loc_37><loc_79><loc_91><loc_81></location>r D GLYPH<16> r D GLYPH<0> ρ τ π N GLYPH<1> ρ N π σ , π τ 2 GLYPH<1> π N 2 GLYPH<8> , (102)</formula>
<text><location><page_16><loc_9><loc_77><loc_47><loc_78></location>with the true space index τ running over 1 to nd GLYPH<1> 1.</text>
<text><location><page_16><loc_9><loc_73><loc_91><loc_75></location>Example 2 Minisuperspace (spatially homogeneous GR) only has a H , and a single finite constraint oversimplifies the diversity of notions of observables. Here the sole strong observables brackets equation is</text>
<formula><location><page_16><loc_45><loc_70><loc_91><loc_71></location>{ H , D } GLYPH<16> 0 , (103)</formula>
<text><location><page_16><loc_9><loc_67><loc_28><loc_68></location>giving the observables PDE</text>
<formula><location><page_16><loc_40><loc_64><loc_91><loc_67></location>B H B Q B D B P GLYPH<1> B H B P B D B Q GLYPH<16> 0 (104)</formula>
<formula><location><page_16><loc_39><loc_60><loc_91><loc_63></location>for B H B Q and B H B P knowns . (105)</formula>
<text><location><page_16><loc_9><loc_57><loc_91><loc_59></location>As per Article I, simple examples include Q GLYPH<16> α, φ or α, β GLYPH<8> . Each minisuperspace's type of potential (one part fixed by GR, another part variable with the nature of appended matter physics).</text>
<text><location><page_16><loc_9><loc_54><loc_55><loc_55></location>There being just one such equation gives that U GLYPH<16> G GLYPH<16> K GLYPH<24> C GLYPH<16> D .</text>
<text><location><page_16><loc_9><loc_51><loc_69><loc_52></location>See e.g. [40] for direct construction of classical Dirac observables for Minisuperspace.</text>
<section_header_level_1><location><page_16><loc_9><loc_47><loc_42><loc_49></location>7.6 Nontrivially weak observables</section_header_level_1>
<text><location><page_16><loc_9><loc_42><loc_91><loc_46></location>Reduced versus indirect makes a clear difference here, since the indirect case has a longer string of constraints in its PI. Only at least partly indirectly formulated case has any space for nontrivially weak such: if all constraints are reduced out, no PI is left.</text>
<text><location><page_16><loc_9><loc_37><loc_91><loc_40></location>Example 1 Weak translational observables (a type of gauge = Kuchař observables). 1 particle in 1d with inhomogeneous term W P GLYPH<16> Wp supports the PI</text>
<formula><location><page_16><loc_45><loc_36><loc_91><loc_37></location>G w GLYPH<16> GLYPH<1> Wqp . (106)</formula>
<text><location><page_16><loc_9><loc_34><loc_68><loc_35></location>For 2 particles in 1d , the inhomogeneous term W P GLYPH<16> W p p 1 GLYPH<0> p 2 q supports the PI</text>
<formula><location><page_16><loc_26><loc_31><loc_91><loc_32></location>G w GLYPH<16> GLYPH<1> p aq 1 GLYPH<0> b q 2 q p p 1 GLYPH<0> p 2 q GLYPH<16> GLYPH<1> p aq 1 GLYPH<0> b q 2 q P , a GLYPH<0> b GLYPH<16> W . (107)</formula>
<text><location><page_16><loc_9><loc_27><loc_91><loc_29></location>Per fixed W , this gives a R of solutions, corresponding to viewing a as free. Considering all W , we have a R GLYPH<2> R GLYPH<6> of properly weak solutions.</text>
<text><location><page_16><loc_9><loc_16><loc_91><loc_25></location>These are now however much less numerous than the strong solutions. This is since the strong solutions now comprise the C p R 2 q of suitably-smooth functions of x 1 GLYPH<1> x 2 and p 1 GLYPH<1> p 2 . This is the general situation for enough degrees of freedom: that the weak observables' parameter space is an appendage of measure zero relative to that of the strong observables' function space. For translation-invariant mechanics, N GLYPH<16> 2 particles is minimal to exhibit this effect. Translational mathematics being the simplest nontrivial Configurational Relationalism considered in the current article, this example has further senses in which it is 'the simplest nontrivial example' of its kind.</text>
<text><location><page_16><loc_9><loc_12><loc_91><loc_14></location>Looking at general particle number and dimension, we find that our method extends. This serves to show that [78] for translation-invariant models, no configuration-geometrical or pure-momentum weak observables are supported.</text>
<text><location><page_17><loc_9><loc_97><loc_65><loc_98></location>Example 2 Weak Chronos observables. Now 1 particle in 1d supports the PI</text>
<formula><location><page_17><loc_33><loc_93><loc_91><loc_96></location>C w GLYPH<16> GLYPH<1> W x p GLYPH<2> p 2 2 GLYPH<1> k GLYPH<10> GLYPH<16> GLYPH<1> W x p C hronos . (108)</formula>
<text><location><page_17><loc_9><loc_90><loc_52><loc_91></location>This is one function per value of W , or R GLYPH<6> functions in total.</text>
<text><location><page_17><loc_9><loc_87><loc_24><loc_88></location>For 2 particles in 1d ,</text>
<formula><location><page_17><loc_17><loc_83><loc_91><loc_86></location>C w GLYPH<16> GLYPH<1> " a q 1 p 1 GLYPH<0> b q 2 p 2 *GLYPH<2> p 1 2 GLYPH<0> p 2 2 2 GLYPH<1> k GLYPH<10> GLYPH<16> GLYPH<1> " a q 1 p 1 GLYPH<0> b q 2 p 2 * C hronos , a GLYPH<0> b GLYPH<16> W . (109)</formula>
<text><location><page_17><loc_9><loc_78><loc_91><loc_82></location>The space of these again coincides with the corresponding translational problem. 2 particles in 1d is again minimal for weak observables to be of zero measure relative to strong observables. Our solution again extends to arbitrary particle number and dimension [78].</text>
<text><location><page_17><loc_9><loc_69><loc_91><loc_76></location>Example 3 Weak Dirac observables are not just weak Chronos restricted by G auge or vice versa. This is since in the D w system, firstly, the Chronos observables equation includes a G auge inhomogeneous term as well as a C hronos one. Secondly, the gauge observables equations include a C hronos inhomogeneous term as well as G auge ones. In contrast, the C hronos w system has just a C hronos inhomogeneous term, and the G auge w system has just a G auge one. This shows that weak observables systems do not involve a simple restriction hierarchy like strong observables ones do.</text>
<section_header_level_1><location><page_17><loc_9><loc_66><loc_17><loc_67></location>Example 4</section_header_level_1>
<text><location><page_17><loc_9><loc_32><loc_32><loc_33></location>giving the weak observables PDE</text>
<formula><location><page_17><loc_41><loc_64><loc_91><loc_66></location>GLYPH<131> C anO bs w GLYPH<24> C anO bs w (110)</formula>
<text><location><page_17><loc_9><loc_59><loc_91><loc_63></location>is shown to be possible in [78]. A simple argument for this is that C anO bs has more scope for PI terms than GLYPH<131> C anO bs does, by being naturally associated with a larger constraint algebra. (The number of PI terms is c 2 for c : GLYPH<16> dim p C q , since there are c weak observables equations, each of which has c inhomogeneous terms.)</text>
<text><location><page_17><loc_9><loc_56><loc_69><loc_57></location>Example 5 The reduced treatment of translational RPM for 2 particles in 1d gives</text>
<formula><location><page_17><loc_35><loc_51><loc_91><loc_54></location>r D w GLYPH<16> GLYPH<1> W x 1 GLYPH<1> x 2 p 1 GLYPH<1> p 2 GLYPH<2> p p 1 GLYPH<1> p 2 q 2 2 GLYPH<1> k GLYPH<10> . (111)</formula>
<text><location><page_17><loc_9><loc_44><loc_91><loc_50></location>This is not however enough to have weak observables be of measure zero relative to strong observables, since we now have two constraints to two degrees of freedom. We do however have a general particle number and dimension solution to both the translational and chronos problems, however. So it is not hard to give the 3-particle, 1d minimal example of this effect.</text>
<text><location><page_17><loc_9><loc_40><loc_91><loc_43></location>Remark 1 The usual relational numerology [67, 74] readily lets us pick out minimal examples for strong observables dominance for further transformations (rotations, dilations, Euclidean, similarity, affine...)</text>
<text><location><page_17><loc_9><loc_37><loc_66><loc_38></location>Example 6 Minisuperspace Here the sole weak observables brackets equation is</text>
<formula><location><page_17><loc_43><loc_34><loc_91><loc_36></location>{ H , D w } GLYPH<16> W H , (112)</formula>
<text><location><page_17><loc_46><loc_34><loc_47><loc_34></location>r</text>
<text><location><page_17><loc_53><loc_34><loc_54><loc_34></location>GLYPH<128></text>
<formula><location><page_17><loc_39><loc_29><loc_91><loc_32></location>B H B Q B D r w B P GLYPH<1> B H B P B D r w B Q GLYPH<16> W GLYPH<128> H (113)</formula>
<formula><location><page_17><loc_38><loc_25><loc_91><loc_28></location>for H , B H B Q and B H B P knowns . (114)</formula>
<text><location><page_17><loc_9><loc_20><loc_91><loc_24></location>Remark 2 Studying just minisuperspace, however, leaves one unaware of most of the diversity of types of observables, and of almost every effect described in the current section. Flat geometry and RPMs thereupon are thus rather more instructive in setting up a general theory of observables.</text>
<section_header_level_1><location><page_17><loc_9><loc_16><loc_42><loc_18></location>8 Field Theory counterpart</section_header_level_1>
<section_header_level_1><location><page_17><loc_9><loc_13><loc_35><loc_14></location>8.1 Brackets algebra level</section_header_level_1>
<text><location><page_17><loc_9><loc_11><loc_61><loc_12></location>Overall, we have the following finite-field portmanteau brackets equation,</text>
<formula><location><page_17><loc_36><loc_7><loc_91><loc_10></location>S A " δ B C δ B Q A δ B B δ B P A GLYPH<1> δ B C δ B P A δ B B δ B Q A * '=' 0 . (115)</formula>
<text><location><page_18><loc_9><loc_95><loc_91><loc_98></location>This is to be interpreted as a δ B DE system, i.e. a portmanteau of a PDE (III.87) in the finite case and an FDE (functional differential equation) 2</text>
<formula><location><page_18><loc_29><loc_90><loc_91><loc_94></location>» d n z , A # δ p C |B ξ q δ Q A p z q δ p O r |B ξ r q δ P A p z q GLYPH<1> δ p C |B ξ q δ P A p z q δ p O r |B ξ r q δ Q A p z q + '=' 0 . (116)</formula>
<section_header_level_1><location><page_18><loc_9><loc_87><loc_23><loc_88></location>8.2 FDE level</section_header_level_1>
<text><location><page_18><loc_9><loc_85><loc_33><loc_86></location>Remark 1 We can moreover take</text>
<formula><location><page_18><loc_39><loc_82><loc_91><loc_85></location>δ C δ Q and δ C δ P to be knowns , (117)</formula>
<text><location><page_18><loc_9><loc_79><loc_91><loc_81></location>leaving us with a homogeneous-linear first-order FDE system. We would also fix particular smearing functions or formally do not smear in locally posing and solving our FDE system.</text>
<text><location><page_18><loc_9><loc_76><loc_81><loc_77></location>Remark 2 The general form - analogous to (III.87) for the corresponding Finite Theory PDE case -</text>
<formula><location><page_18><loc_32><loc_73><loc_91><loc_74></location>a A r x, y B p x q , Φ r y B p x qss D y A Φ GLYPH<16> b A p x, y B p x q , Φ q (118)</formula>
<text><location><page_18><loc_9><loc_69><loc_91><loc_71></location>covers both our homogeneous strong observables FDE system p b GLYPH<16> 0 q and our inhomogeneous weak observables FDE system p b GLYPH<24> 0 q .</text>
<section_header_level_1><location><page_18><loc_9><loc_65><loc_53><loc_66></location>8.3 Flow method transcends to Banach space</section_header_level_1>
<text><location><page_18><loc_9><loc_61><loc_91><loc_64></location>Remark 1 This is as far as we detail in the current series, though transfer to the tame Fréchet space setting is also possible.</text>
<text><location><page_18><loc_9><loc_57><loc_91><loc_60></location>Structure 8 Let B be a Banach manifold and v a vector field thereupon. Curves, tangent vectors and tangent spaces remain defined on B [27].</text>
<text><location><page_18><loc_9><loc_54><loc_79><loc_55></location>Definition 1 An integral curve of B is a curve γ such that at each point b the tangent vector is v b .</text>
<text><location><page_18><loc_9><loc_51><loc_66><loc_52></location>Definition 2 The differential system on B defined by v still takes its usual form,</text>
<formula><location><page_18><loc_44><loc_48><loc_91><loc_49></location>9 φ p ν q GLYPH<16> v p φ p ν qq . (119)</formula>
<text><location><page_18><loc_9><loc_44><loc_91><loc_47></location>Remark 2 We can append a multi-index on φ and v if needs be, to cover multi-component field and multi-field versions.</text>
<text><location><page_18><loc_9><loc_41><loc_56><loc_42></location>Remark 3 DE Existence and uniqueness theorems carry over [27].</text>
<text><location><page_18><loc_9><loc_36><loc_91><loc_39></location>Remark 4 This guarantees a local flow (which is as much as the current Series' considerations cater for). As in the finite case, this provides a 1-parameter group.</text>
<text><location><page_18><loc_9><loc_32><loc_91><loc_35></location>Structure 9 Differential forms carry over to (sufficiently smooth) B . So do pullback, exterior differential operator and internal product [27].</text>
<text><location><page_18><loc_9><loc_29><loc_74><loc_30></location>The familiar 'Cartan's magic formula' for the Lie derivative is consequently available [27].</text>
<text><location><page_18><loc_9><loc_24><loc_91><loc_27></location>Thus in turn Lie dragging remains available, as does Lie correcting (toward implementing Configurational Relationalism) and the diffeomorphism interpretation of the flow.</text>
<text><location><page_18><loc_9><loc_21><loc_91><loc_22></location>Banach Lie algebras , and Banach Lie groups , are well-established [20], enabling Configurational Relationalism.</text>
<text><location><page_18><loc_9><loc_15><loc_91><loc_19></location>Frobenius' Theorem - as required for Lie's integral method for geometrical invariants, its canonical physics generalization and the uplifts of each of these to finding function spaces of observables thereover - carries over as well [42].</text>
<text><location><page_18><loc_9><loc_9><loc_91><loc_13></location>End-Remark 1 All in all, local Lie Theory, in the somewhat broader sense required for A Local Resolution of the Problem of Time is thus established in the Banach space setting which can be taken to underlie much of Field Theory.</text>
<section_header_level_1><location><page_19><loc_9><loc_97><loc_23><loc_98></location>8.4 Examples</section_header_level_1>
<text><location><page_19><loc_9><loc_93><loc_91><loc_95></location>For conventional Gauge Theory, the observables equation imposes gauge invariance at the level of configuration space based on both space and internal gauge space.</text>
<text><location><page_19><loc_9><loc_90><loc_37><loc_91></location>In each of the first two examples below,</text>
<formula><location><page_19><loc_44><loc_88><loc_91><loc_89></location>D GLYPH<16> K GLYPH<16> G GLYPH<24> U , (120)</formula>
<text><location><page_19><loc_9><loc_86><loc_64><loc_87></location>since these just have first-class linear constraints which are gauge constraints.</text>
<text><location><page_19><loc_9><loc_82><loc_91><loc_84></location>Example 1) Electromagnetism has the abelian algebra of constraints (VII.37). G GLYPH<16> K for Electromagnetism solve the brackets equation</text>
<formula><location><page_19><loc_35><loc_80><loc_91><loc_81></location>{ p G | B ξ q , p K | B χ q } '=' 0 æ the FDE (121)</formula>
<text><location><page_19><loc_43><loc_79><loc_44><loc_79></location>r</text>
<text><location><page_19><loc_44><loc_30><loc_45><loc_30></location>r</text>
<text><location><page_19><loc_46><loc_79><loc_47><loc_79></location>r</text>
<formula><location><page_19><loc_45><loc_76><loc_91><loc_79></location>B GLYPH<4> δ K δ A '=' 0 . (122)</formula>
<text><location><page_19><loc_9><loc_74><loc_44><loc_75></location>This is solved by the electric and magnetic fields,</text>
<formula><location><page_19><loc_42><loc_71><loc_91><loc_72></location>E and B GLYPH<16> B GLYPH<2> A , (123)</formula>
<text><location><page_19><loc_9><loc_68><loc_26><loc_69></location>and thus by a functional</text>
<formula><location><page_19><loc_47><loc_67><loc_91><loc_68></location>F r B , E s (124)</formula>
<text><location><page_19><loc_9><loc_63><loc_91><loc_66></location>by Lemma 3. These are not however a conjugate pair. Since this looks to be a common occurrence in further examples, let us introduce the term 'associated momenta' to describe it.</text>
<text><location><page_19><loc_9><loc_60><loc_60><loc_61></location>F r B , E s can also be written in the integrated version in terms of fluxes:</text>
<formula><location><page_19><loc_33><loc_54><loc_91><loc_59></location>F GLYPH<20> GLYPH<21> … S B GLYPH<4> dS , … S E GLYPH<4> dS GLYPH<28> GLYPH<29> GLYPH<16> F r W p γ q , Φ E S s (125)</formula>
<text><location><page_19><loc_9><loc_51><loc_35><loc_53></location>for electric flux Φ E S and loop variable</text>
<formula><location><page_19><loc_37><loc_45><loc_91><loc_50></location>W A p γ q : GLYPH<16> exp GLYPH<4> GLYPH<5> i GLYPH<190> γ d x GLYPH<4> A p x q GLYPH<12> GLYPH<13> . (126)</formula>
<text><location><page_19><loc_9><loc_40><loc_91><loc_44></location>This is by use of Stokes' Theorem with γ : GLYPH<16> B S and subsequent insertion of the exponential function subcase of Lemma 3. This ties the construct to the geometrical notion of holonomy. Moreover, these are well-known to form an over -complete set: there are so-called Mandelstam identities between them [43].</text>
<text><location><page_19><loc_9><loc_37><loc_70><loc_38></location>Example 2) Its Yang-Mills generalization has the Lie algebra of constraints (VII.39)</text>
<text><location><page_19><loc_9><loc_33><loc_49><loc_35></location>G GLYPH<16> K for Yang-Mills Theory solve the brackets equation</text>
<formula><location><page_19><loc_35><loc_31><loc_91><loc_32></location>{ p G I | ξ I q , p K | χ q } GLYPH<19> 0 æ the FDE (127)</formula>
<text><location><page_19><loc_46><loc_30><loc_47><loc_30></location>r</text>
<formula><location><page_19><loc_45><loc_26><loc_91><loc_29></location>D GLYPH<4> δ K δ A GLYPH<19> 0 . (128)</formula>
<text><location><page_19><loc_9><loc_23><loc_53><loc_25></location>This is solved by Yang-Mills Theory's generalized E and B, so</text>
<formula><location><page_19><loc_47><loc_20><loc_91><loc_21></location>F r E , B s (129)</formula>
<text><location><page_19><loc_9><loc_17><loc_47><loc_18></location>is also a solution. Once again, this can be rewritten as</text>
<formula><location><page_19><loc_45><loc_14><loc_91><loc_16></location>F r W p γ q , Φ E S s , (130)</formula>
<text><location><page_19><loc_9><loc_12><loc_25><loc_13></location>now for g -loop variable</text>
<formula><location><page_19><loc_33><loc_7><loc_91><loc_11></location>W A p γ q : GLYPH<16> Tr GLYPH<4> GLYPH<5> P exp GLYPH<4> GLYPH<5> i g GLYPH<190> γ d x A p x q g p x q GLYPH<12> GLYPH<13> GLYPH<12> GLYPH<13> . (131)</formula>
<text><location><page_20><loc_9><loc_96><loc_79><loc_98></location>g are here group generators of g YM , g is the coupling constant, and P is the path-ordering symbol.</text>
<text><location><page_20><loc_9><loc_92><loc_91><loc_95></location>Example 3 The cause cél ' e bre of canonical treatment of observables is GR. In this case, one has the spatial 3diffeomorphisms</text>
<formula><location><page_20><loc_46><loc_91><loc_91><loc_92></location>Diff p Σ q (132)</formula>
<text><location><page_20><loc_9><loc_87><loc_91><loc_90></location>momentum constraint M - linear in its momenta - and a Hamiltonian constraint H - quadratic in its momenta - to commute with.</text>
<text><location><page_20><loc_9><loc_84><loc_57><loc_85></location>Example 3) K for GR as Geometrodynamics, the brackets equation</text>
<formula><location><page_20><loc_14><loc_79><loc_91><loc_83></location>{ p M | B L q , p K r | B χ r q } ' GLYPH<16> 1 0 æ the FDE GLYPH<2> " £ B L h ij δ δ h ij GLYPH<0> £ B L p ij δ δ p ij * K r GLYPH<7> GLYPH<7> GLYPH<7> GLYPH<7> B χ r GLYPH<10> '=' 0 . (133)</formula>
<text><location><page_20><loc_9><loc_77><loc_38><loc_78></location>This corresponds to the unsmeared FDE</text>
<formula><location><page_20><loc_28><loc_73><loc_91><loc_75></location>2 h jk D i δ K δ h ij GLYPH<0> D i p lj GLYPH<1> 2 δ j i t D e p le GLYPH<0> p le D e u ( δ K δ p lj '=' 0 . (134)</formula>
<text><location><page_20><loc_9><loc_70><loc_75><loc_71></location>In the weak case, we can furthermore discard the penultimate term. The K p Q q subcase solve</text>
<formula><location><page_20><loc_43><loc_66><loc_91><loc_69></location>2 h jk D i δ K δ h ij '=' 0 . (135)</formula>
<text><location><page_20><loc_9><loc_60><loc_91><loc_64></location>These are, formally, 3-geometry quantities ' G p 3 q ' by (135) emulating (and moreover logically preceding) the 'momenta to the right' ordered quantum GR momentum constraint (IV.31). This analogy holds for the current Series' range of finite models as well (of relevance to those with nontrivial F lin ).</text>
<text><location><page_20><loc_9><loc_57><loc_56><loc_58></location>Remark 1 Explicit 'basis observables' are not known in this case.</text>
<text><location><page_20><loc_9><loc_54><loc_53><loc_55></location>On the other hand, the FDE for the K p P q (formally 'Π G p 3 q ') is</text>
<formula><location><page_20><loc_34><loc_49><loc_91><loc_52></location>D i p lj GLYPH<1> 2 δ j i t D e p le GLYPH<0> p le D e u ( δ K δ p lj '=' 0 . (136)</formula>
<text><location><page_20><loc_9><loc_47><loc_58><loc_48></location>Example 4) D's for Geometrodynamics' D require an extra FDE [68]</text>
<formula><location><page_20><loc_34><loc_44><loc_91><loc_45></location>{ p H | B J q , p D | B ζ q } '=' 0 æ the FDE (137)</formula>
<text><location><page_20><loc_43><loc_43><loc_44><loc_43></location>r</text>
<text><location><page_20><loc_46><loc_43><loc_47><loc_43></location>r</text>
<formula><location><page_20><loc_31><loc_39><loc_91><loc_42></location>GLYPH<2> " δ D r δ p G GLYPH<1> M D 2 ( GLYPH<1> δ D r δ h N p * B J GLYPH<7> GLYPH<7> GLYPH<7> GLYPH<7> B ζ r GLYPH<10> '=' 0 . (138)</formula>
<text><location><page_20><loc_9><loc_37><loc_39><loc_38></location>This features the DeWitt vector quantities</text>
<formula><location><page_20><loc_17><loc_34><loc_91><loc_36></location>G : GLYPH<16> 2 ? h p ia p a j GLYPH<1> p 2 p ij ( GLYPH<1> 1 2 ? h p ab p ab GLYPH<1> p 2 2 ( h ij GLYPH<1> ? h 2 h ij R GLYPH<1> 2 R ij ( GLYPH<0> ? h Λ h ij . (139)</formula>
<text><location><page_20><loc_9><loc_29><loc_91><loc_32></location>These are already familiar from the ADM equations of motion [12], and also D 2 with components D i D j . In unsmeared form, (138) is the FDE</text>
<formula><location><page_20><loc_39><loc_26><loc_91><loc_29></location>G GLYPH<1> M D 2 ( δ D δ p '=' 2 p N δ D δ h . (140)</formula>
<text><location><page_20><loc_9><loc_24><loc_60><loc_25></location>M and N here are the DeWitt supermetric and its inverse respectively.</text>
<section_header_level_1><location><page_20><loc_9><loc_21><loc_39><loc_22></location>8.5 Genericity in Field Theory</section_header_level_1>
<text><location><page_20><loc_9><loc_16><loc_91><loc_19></location>In Field Theory, for instance Electromagnetism has one observables equation, whereas Yang-Mills Theory has g : GLYPH<16> dim p g YM q such.</text>
<text><location><page_20><loc_9><loc_12><loc_91><loc_15></location>GR-as-Geometrodynamics has four observables equations: commutation with the 3 components of the momentum constraint and with the single Hamiltonian constraint.</text>
<text><location><page_20><loc_9><loc_8><loc_91><loc_10></location>So, on the one hand, for Yang-Mills Theory and for GR, PDE system genericity can be relevant. On the other hand, for Finite Theories, PDE system genericity is in general obscured by geometrical genericity.</text>
<section_header_level_1><location><page_21><loc_9><loc_97><loc_25><loc_98></location>9 Conclusion</section_header_level_1>
<section_header_level_1><location><page_21><loc_9><loc_94><loc_23><loc_95></location>9.1 Summary</section_header_level_1>
<text><location><page_21><loc_9><loc_76><loc_91><loc_92></location>For Finite Theories, strong observables are to be found by solving brackets equations which can be recast as homogeneous linear first-order PDE systems. Cases with a single such equation - corresponding to a constraint (sub)algebra with a single generator - are mathematically standard; the Flow Method readily applies. For nontrivial systems of such PDEs, there is no universal approach; the outcome depends, rather, on determinedness status. For observables PDE systems, moreover, over-determinedness is vanquished by integrability conditions guaranteed by Frobenius' Theorem. We furthermore provide firm grounding for a free alias natural [7] characteristic problem treatment being appropriate for observables, indeed embodying the Taking of Function Spaces Thereover. Sequential use of the Flow Method then produces the requisite strong observables. The above workings amount to a minor extension of Lie's Integral Approach to Geometrical Invariants. Firstly, the observables are not just 'the invariants' but rather any suitably smooth function of the invariants. Secondly, this extension generally plays out in phase space for us rather than in Lie's purely geometric setting.</text>
<text><location><page_21><loc_9><loc_69><loc_91><loc_74></location>Properly weak observables obey the inhomogeneous-linear counterpart of the above. Their general solution thus splits into complementary function plus particular integral, which roles are played by strong and nontrivially-weak observables respectively. For examples with enough degrees of freedom, moreover, properly weak observables give but a measure-zero extension to the space of strong observables.</text>
<text><location><page_21><loc_9><loc_61><loc_91><loc_67></location>We illustrate all of strong versus properly weak observables, of unrestricted, Dirac and middlingly restricted observables, as well as of properly weak observables' dependence on extent of reduction. Also fully reduced treatments have no room for properly weak observables. This and the above zero-measure comment are two reasons to not place too much stock in further developing the theory of properly weak observables.</text>
<text><location><page_21><loc_9><loc_55><loc_91><loc_59></location>The above summary corresponds to the first major extension of [72]'s treatment of Background Independence and the Problem of Time. This is by providing a concrete theory of solving for observables on the endpoint of the first branch of Fig 1.b), for now in the local, classical finite-theory setting.</text>
<text><location><page_21><loc_9><loc_46><loc_91><loc_53></location>For Field Theories, observables equations give instead linear first-order functional differential equation systems. Banach (or tame Fréchet) Calculus is however sufficiently standard that the Flow Method and free characteristic problems still apply. These calculi support the Lie-theoretic combination of machinery that our Local Resolution of the Problem of Time requires. In this way, Field Theory - exemplified by Electromagnetism and Yang-Mills Theory in the current Article - and GR are included in our Local Resolution of the Problem of Time's formulation.</text>
<section_header_level_1><location><page_21><loc_9><loc_43><loc_62><loc_44></location>9.2 Present and future of the pedagogy of observables</section_header_level_1>
<text><location><page_21><loc_9><loc_33><loc_91><loc_41></location>Pedagogical difficulties with presenting observables moreover abound. Many quantum treatises, as well as popular accounts [93], immediately launch into the quantum version. They do not mention the corresponding classical counterpart because of its relative simplicity. In the process, they miss the point that the classical version becomes nontrivial in the presence of constraints. This difficulty moreover recurs at the quantum level, where it is missed out again. In this way, the most advanced theory of observables - of quantum constrained observables - is not mentioned on Wikipedia [93].</text>
<text><location><page_21><loc_9><loc_19><loc_91><loc_31></location>The suggested restructuring is to start with classical unconstrained functions over phase space (and configuration space, with connections to pure geometry). Then, on the one hand, one is to consider the nontrivialities of classical constrained systems. On the other hand, one is to keep the existing quantum account as the other main complicating factor. This is now to emphasize that only small subalgebras of the classical observables algebra can be consistently quantized. Finally, these two sources of complexity are to be combined. Article X spells out further sources of improvement, along the lines of the current paragraph's canonical notions of observables having spacetime, path or histories observables analogues. The common theme is that of Function Spaces Thereover, meaning over state spaces. Be that (un)constrained classical phase space, the space of spacetimes or some quantum state space.</text>
<section_header_level_1><location><page_21><loc_9><loc_16><loc_75><loc_17></location>9.3 The theory of strong observables has at least a presheaf flavour</section_header_level_1>
<text><location><page_21><loc_9><loc_7><loc_91><loc_14></location>Taking Function Spaces Thereover models moreover multiplicity of function spaces over state spaces. This is in the sense of applying various different consistent constraint algebraic substructures. For strong observables - the generic part of observables theory for sufficient degrees of freedom - these combine in the manner of applying successive restrictions to the observables DE's characteristic surfaces as per Theorem 3. Such inter-relation by restriction maps amounts to Taking Presheaves Thereover. How widely this can be extended to Taking Sheaves Thereover remains</text>
<text><location><page_22><loc_9><loc_95><loc_91><loc_98></location>to be determined. This is an interesting question to investigate, due to sheaves possessing further localization and globalization properties; 3 it is however beyond the reach of the current Series' local treatise.</text>
<section_header_level_1><location><page_22><loc_9><loc_91><loc_32><loc_93></location>A TRi observables</section_header_level_1>
<figure>
<location><page_22><loc_8><loc_44><loc_91><loc_89></location>
<caption>Figure 3: This continuation of the 'technicolour guide' to the Problem of Time sits to the left of Fig VII.5. The new Assignment of Observables content is highlighted in orange.</caption>
</figure>
<unordered_list>
<list_item><location><page_22><loc_9><loc_37><loc_91><loc_38></location>1) Q , P and P hase are already-TRI. So are C and { , } , and thus the definition of constrained observables as well.</list_item>
<list_item><location><page_22><loc_9><loc_33><loc_91><loc_35></location>2) The current article's smearing functions are given in TRi form since first-class constraints are both trivially weak observables and TRi-smeared, pointing to all observables requiring TRi-smearing as well.</list_item>
<list_item><location><page_22><loc_9><loc_28><loc_91><loc_31></location>3) Observables algebraic structures are already-TRi in the Finite Theory case, or readily rendered TRi by adopting TRi-smearing in the Field Theory case.</list_item>
<list_item><location><page_22><loc_9><loc_21><loc_91><loc_26></location>4) Split C into Temporal and Configurational Relationalism parts has the knock-on effect (already in III) that some notions of observables are just Configurationally Relational - Kuchař observables - or just Temporally Relational - Chronos observables - in theories admitting such a split. GR permits Kuchař observables but not Chronos observables, whereas RPM supports both.</list_item>
<list_item><location><page_22><loc_9><loc_12><loc_91><loc_19></location>5) See Fig 3 for Expression in Terms of Observables' further Problem of Time facet interferences. This scantness of interaction is testimony to the great decoupling of facets alluded to in the Introduction. Further historical problems with the Problem of Observables stem from failing to distinguish between canonical and spacetime notions of observables. Discussing these must however await detailed consideration of the spacetime observables version in Article X.</list_item>
</unordered_list>
<section_header_level_1><location><page_23><loc_9><loc_97><loc_21><loc_98></location>References</section_header_level_1>
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</table>
</document> | [{"title": "Edward Anderson 1", "content": "based on calculations done at Peterhouse, Cambridge", "pages": [1]}, {"title": "Abstract", "content": "Given a state space, Assignment of Observables involves Taking Function spaces Thereover. At the classical level, the state space in question is phase space or configuration space. This assignment picks up nontrivialities when whichever combination of constraints and the quantum apply. For Finite Theories, weak observables equations are inhomogeneous-linear first-order PDE systems. Their general solution thus splits into complementary function plus particular integral: strong and nontrivially-weak observables respectively. We provide a PDE analysis for each of these. In the case of single observables equations - corresponding to single constraints - the Flow Method readily applies. Finding all the observables requires free characteristic problems. For systems, this method can be applied sequentially, due to integrability conferred by Frobenius' Theorem. In each case, the first part of this approach is Lie's Integral Theory of Geometrical Invariants, or the physical counterpart thereof. The second part finds the function space thereover, giving the entire space of (local) observables. We also outline the Field Theory counterpart. Here one has functional differential equations. Banach (or tame Fr\u00e9chet) Calculus is however sufficiently standard for the Flow Method and free characteristic problems to still apply. These calculi support the Lie-theoretic combination of machinery that our Local Resolution of the Problem of Time requires, by which Field Theory and GR are included. 1 dr.e.anderson.maths.physics *at* protonmail.com", "pages": [1]}, {"title": "1 Introduction", "content": "This is the eighth Article [81, 82, 83, 84, 85, 86, 87] on A Local Resolution of the Problem of Time [17, 18, 15, 24, 36, 38, 39, 45, 57, 59, 61, 63, 64, 75, 72, 80] and its underlying Local Theory of Background Independence. We here expand on Sec 3 of Article III's opening account of Assignment of Observables [5, 38, 39, 41], now atop [63, 72] the triple unification of Constraint Closure [6, 9, 15, 28, 37, 72, 83, 87] with Temporal [13, 25, 59, 66, 72, 81, 85, 86] and Configurational [12, 25, 59, 72, 82, 85, 86] Relationalism. That we can treat this after completing the triple, and separately from Article IX's Constructability extension, is one of the great decouplings of Problem of Time Facets [72, 81]. Resolving the triple of facets gives a consistent phase space, P hase. Taking Function Spaces Thereover -the essence of Assigning Observables - entails addressing a subsequent mathematical problem on P hase, rather than imposing some further conditions on whether that P hase is adequate. We pose concrete mathematical problems for Finite Theory's observables at the level of brackets algebras in Sec 2 and of PDEs in Secs 3 and 4. Weak observables equations are inhomogeneous-linear first-order PDE systems. Their general solution thus splits into complementary function plus particular integral: strong and nontriviallyweak observables respectively. In the case of single observables equations - corresponding to single constraints - the Flow Method readily applies. Finding all the observables requires free characteristic problems . For systems - corresponding to multiple constraints - this approach can be applied sequentially, due to integrability as conferred by Frobenius' Theorem [27, 62]. In each case, the first part of our approach is Lie's Integral Theory of Geometrical Invariants (or some physical counterpart thereof). The second part is Finding the Function space Thereover, giving the entire space of (local) observables. The more practical matter of Expression in Terms of Observables - requiring only enough observables to span phase (or configuration) space - is covered in Sec 5. Strategies for addressing the Problem of Observables are in Sec 6. We also give explicit examples of quite a number of distinct notions of observables [5, 37, 38, 39, 41, 63, 72, 76, 78] in Sec 7. Purely geometrical such moreover play a further role [76, 77] in the Foundations of Geometry [3, 4, 33, 47, 52, 76], as well as coinciding with the configuration space q restriction of P hase. The Field Theory and GR counterpart is in Sec 8. This now gives FDEs - functional differential equations as its observables equations. Similar Flow Method and sequential use of free characteristic problems carry over, however, thanks to the underlying benevolence of Banach Calculus [27] (for now, or tame Fr\u00e9chet Calculus [29] more generally). These calculi support the Lie-theoretic combination of machinery required by our Local Resolution of the Problem of Time, by which Field Theory and GR are included. Appendix A keeps track of TRi (Temporal Relationalism implementing) modifications and other Problem of Time facet interferences in part involving Assignment of Observables.", "pages": [1]}, {"title": "2.1 Strong observables-constraints system", "content": "Structure 1 Strong observables O extend the constraint algebraic structure C p S q as follows. r Remark 1 This is overall [72] of direct product form,", "pages": [2]}, {"title": "2.2 Weak observables-constraints system", "content": "Structure 2 Weak observables O w extend the constraint algebraic structure as follows. r r r r Remark 1 The second equation signifies that C is a good C anO bs p S q -object. This is a consequence of constraints closing weakly, via the Jacobi identity. Remark 2 This is now overall [72] of semidirect product form, Remark 3 Comparing (5) and (2) or (6) implies that the C are themselves in some sense observables. However, since we already knew that C GLYPH<19> 0, in studying observables we are really looking for further quantities outside of this trivial case. Let us call these other quantities proper observables ; the rest of the Series will always take 'observables' to mean this. Definition 1 We term the general solution of the weak observables equation O W weak observables . Remark 4 The weak observables equation is moreover an inhomogeneous counterpart of the homogeneous linear strong observables equation. So in the manner of a complementary function plus particular integral split of the general solution of an inhomogeneous linear equation, Definition 2 We term particular-integral solutions of the weak equation O w - weak observables which are explicitly independent of any strong observables nontrivially weak observables . Remark 5 The pure configurational geometry case cannot support any nontrivially-weak observables. This is because we are in a context in which constraints have to depend on momenta A fortiori, first-class linear constraints refer specifically to being linear in their momenta P , By this, the weak configurational observables equation would have a momentum-dependent inhomogeneous term right-hand side. But this is inconsistent with admitting a solution with purely Q -dependent right-hand-side.", "pages": [2]}, {"title": "2.3 Why not a more general C , O w system?", "content": "Na\u00efve algebraic generality suggests the more general form r r r r r However, D GLYPH<16> 0 since physical systems provide the constraints C without reference to observables O . X GLYPH<16> 0 preserves C 's status as a good C anO bs p S q -object. L GLYPH<24> 0 gives a further sense of weak, though it prevents the observables from forming a subalgebra. Now starting with the observables would imply the constraints, but the constraints are already prescribed elsewise by the physics.", "pages": [3]}, {"title": "Lemma 1", "content": "Proof and factorize. \u2737 Corollary 1 This can be achieved by returning the strong case, or by with c : GLYPH<16> dim p C q , or by a zero double-trace 'perpendicularity' condition. Remark 1 On the other hand, no capacity to influence O w can be found in", "pages": [3]}, {"title": "2.5 Strong and weak observables considered together", "content": "Structure 3 By our CF + PI split, the strong and nontrivially-weak canonical observables algebra is consequently of the direct product form Structure 4 Including the constraints as well, we have the 3-block algebraic structure. r r r r r r Remark 1 This includes each of C , O and O w as subalgebras, by the first, third and fifth equations respectively. Remark 2 The sixth equation's zero right-hand-side is part of the implementation of the CF to PI linear independence. This is partly enforced by which renders an initial O and O w combination pureO . Remark 3 The overall algebraic structure is of the form", "pages": [3, 4]}, {"title": "3 Strong observables PDEs", "content": "One can obtain explicit PDEs by writing out what the Poisson brackets definition of constrained observables means [68]. For strong observables, (2) gives We can moreover take leaving us with a homogeneous-linear first-order PDE system. I.e. a homogeneous subcase of the claimed first-order linear form (III.87). First order their containing first-order partial derivatives B \u03b1 \u03c6 and no higher. Linear refers to the unknown variables O . Remark 1 As the strong case involves a homogeneous equation, is always a solution. We refer to this as the trivial solution . We call all other solutions of first-order homogeneous quasilinear PDEs proper solutions : a nontrivial kernel condition.", "pages": [4]}, {"title": "3.1 Single strong observables equation", "content": "The Flow Method [26, 62] immediately applies here. Structure 5 This gives a corresponding ODE system of form Here, the dot denotes for \u03bd a fiducial variable to be eliminated, rather than carrying any temporal (or other geometrical or physical) significance. Remark 1 In the geometrical setting, the first equation here corresponds to Lie's Integral Approach to Geometrical Invariants. Remark 2 The last equation uplifts this to Taking the Function Space Thereover (over configuration space in Geometry or over phase space in the canonical approach to Physics.) This involves feeding in the 'characteristic' solution u of the first equation into the last equation by eliminating \u03bd . This u then ends up featuring as the 'functional form' that the observables depend on Q and P via, This simple outcome reflects that, in the strong case, the last equation is just a trivial ODE. The function space in question needs to be at least once continuously differentiable. So the observables are suitably-smooth but elsewisearbitrary functions of Lie's invariants (literally in Geometry, or their phase space counterparts in Physics). Remark 3 Solving for such arbitrary functions is to be contrasted with obtaining a single function by prescribing a specific boundary condition. Not prescribing such a boundary condition, on the one hand, amounts to implementing Taking a Function Space Thereover. On the other hand, its more general technical name is free alias natural [7] characteristic problem. The Characteristic Problem formulation for a single linear (or quasilinear) flow PDE is a standard prescription [7, 26]. Remark 4 At the geometrical level, our procedure is, given a constraint subalgebra C A , the observables equation [ C A , O ] first determining a characteristic surface This follows from solving all but the last equation in the equivalent flow ODE system. Secondly, the last equation in the flow ODE system is a trivial ODE solved by any suitably-smooth function thereover. This gives the observables algebra as Remark 5 C k for some fixed k \u00a5 1 could be used instead, or some (perhaps weighted) Sobolev space [27]; we adhere to C 8 for simplicity.", "pages": [4, 5]}, {"title": "3.2 N -point geometrical = purely configurational physical observables", "content": "This subproblem has been covered in e.g. [1, 2, 14, 76]. Definition 1 classical geometrical observables Structure 6 In the purely-geometrical setting, the a priori free functions O are subject to Here, S is the sum-over-N-points [14, 76] q I , I GLYPH<16> 1 to N of each particular generator, with components Remark 1 As detailed in [78], this is the pure-geometry analogue of C . Remark 2 The Lie bracket equation (40) can furthermore be written out as an explicit PDE system. It should by now be clear that this PDE is moreover a subcase of that for canonical observables in Theoretical Physics, In particular, N -point geometrical observables coincide with the purely configuration space restriction of physical observables [78]; see Sec 7 for examples.", "pages": [5]}, {"title": "3.3 Pure-momentum physical observables", "content": "We furthermore consider the notion of pure-momentum observables These solve the pure-momentum observables PDE system Remark 1 Configuration and momentum observables each readily represent a restriction of functions over P hase, to just over q , and to just over the space of momenta P respectively. These are, more specifically, polarization restrictions [35] since they precisely halve the number of variables. This applies at least in the quadratic theories we consider in the current Series. These are by far the simplest and most standard form for bosonic theories in Physics.", "pages": [5, 6]}, {"title": "3.4 Various notions of genericity", "content": "PDE genericity From a PDE point of view, systems are more generic than single equations. Finite-theory geometrical genericity However, in fixed-background finite theories of Geometry (or Physics), it is geometrically generic to have no (generalized) Killing vectors, and thus 0 or 1 observables PDEs. Remark 1 This is 0 in pure geometry and in temporally-absolute finite physics, to 1 in temporally-relational finite physics: commutation with C . Remark 2 For Finite Theories, PDE system genericity is in general obscured by geometrical genericity. Remark 3 For Finite Theories, moreover, having 1 Killing vector is of secondary genericity between having no, and multiple, Killing vectors. Remark 4 From this point of view, unconstrained observables are most generic (no observables PDEs at all), single observables PDEs are next most generic, and the more involved case of multiple observables equations is only tertiary in significance.", "pages": [6]}, {"title": "3.5 Nontrivial system case: determinedness and integrability", "content": "Remark 1 For nontrivial systems, multiple sequential uses of the Flow Method may apply. What needs to be checked first is determinedness [11], and, if over-determinedness occurs, integrability. Remark 2 In Geometry, we have g : GLYPH<16> dim p g q constraints, and thus g observables equations. The observables carry an index O that has no a priori dependence on g . Thus, a priori, any of under-, well- or over-determinedness can occur (see also XIV.7). This conclusion transcends to the canonical approach to Physics as well. Here Temporal Relationalism and/or Constraint Closure can contribute further first-class constraints. g is thus replaced by a more general count f : GLYPH<16> dim p F q of functionally-independent first-class constraints. On the one hand, generalized Killing equations' integrability conditions are not met generically ([8] or XIV.7). This signifies that there are only any proper generalized Killing vectors at all in a zero-measure subset of x M , \u03c3 y . This corresponds to the generic manifold admitting no (generalized) symmetries. On the other hand, preserved equations moreover always succeed in meeting integrability, by the following Theorem. Theorem 1 Classical canonical observables equations are integrable. Remark 3 Consequently, classical observables always exist (subject to the following caveats). Caveat 1 The current Article, and series, consider only local existence. Caveat 2 Sufficiently large point number N is required in the case of finite point-particle theories. This is clear from the examples in [76, 77, 78], and corresponds to zero-dimensional reduced spaces having no coordinates left to support thereover any functions of coordinates.] Remark 4 This Theorem is proven in [78], resting on the following vaguely modern version of Frobenius' Theorem. Theorem 2 (A version of Frobenius' Theorem at the level of differentiable manifolds [27, 62]. A collection W of subspaces of a tangent space possesses integral submanifolds iff 1", "pages": [6]}, {"title": "3.6 Sequence of free characteristic problems for the strong observables system", "content": "We now have a more extensive ODE system of the form (33, 34). Remark 1 Regardless of the single-equation to system distinction, corresponding observables ODEs are moreover autonomous (none of the functions therein depend on \u03bd ). Remark 2 We now have a first block rather than a first equation. Remark 3 While the system version is not a standard prescription; study of strong observables PDE systems gets past this by of our integrability guarantee. Sequential Approach . Suppose we have two equations. Solve one for its characteristics u 1 , say. Then substitute Q GLYPH<16> Q p u 1 q into the second equation to find which functional restrictions on the first solution's characteristics the second equation enforces. This procedure can moreover be applied inductively. It is independent of the choice of ordering in which the restrictions are applied by the nature of restrictions corresponding to geometrical intersections. The Free Characteristic Problem posed above moreover leads to consideration of intersections of characteristic surfaces, which can moreover be conceived of in terms of restriction maps.", "pages": [7]}, {"title": "Theorem 3 Suppose", "content": "and for constraint subalgebraic structures C V and C W . Then O such that [ C V Y W , O ] GLYPH<16> 0 forms the characteristic surface of Fig 1 . Remark 4 This approach extends inductively to a finite number of equations in our flow ODE system. Remark 5 See e.g. [76, 77, 78] for examples of its use. Remark 6 The integrated form of the first m equations is used to eliminate t , with the other m GLYPH<1> 1 providing a basis of characteristic coordinates u \u00af a arising as constants of integration. In the geometrical case, this can still be considered to be Lie's Method of Geometric Invariants. After all, essentially all the most familiar geometries involve more than one independent condition on their integral invariants. Remark 7 To elevate this to a determination of the system's observables, we then substitute these characteristic coordinates into the last equation. We thus obtain the general - and thus free alias natural problem-solving characteristic solution. Remark 8 Our last equation remains a trivial ODE. It is thus solved by an suitably-smooth but elsewise arbitrary function of these characteristic coordinates, u with components u \u00af a Remark 9 The current Article (and Series) just considers a local rather than global treatment of observables equations.", "pages": [7]}, {"title": "4 Weak observables PDEs", "content": "Structure 7 For weak observables, the brackets equation (6) gives We can again now alongside C being a known as well and W taking some prescribed value. This leaves us with an inhomogeneouslinear first-order PDE system. Remark 1 The pure-geometry case cannot however support any properly weak observables. This is because we are in a context in which constraints have to depend on momenta A fortiori, first-class linear constraints refer specifically to being linear in their momenta P , By this, the weak configurational observables equation would have a momentum-dependent inhomogeneous term right-hand side. But this is inconsistent with admitting a solution with purely Q -dependent right-hand-side.", "pages": [8]}, {"title": "4.1 Single weak observables equation", "content": "The weak observables PDE consists of a single inhomogeneous-linear equation. Its corresponding ODE system is now of the form Remark 1 The first block is the same as before. Lie's integral invariants (in Geometry or their canonical Physics generalizations) thus still enter our expressions for observables. Remark 2 The inhomogeneous term in the last equation, however, means that one has further particular-integral work to do in this weak case.", "pages": [8]}, {"title": "4.2 Nontrivial weak observables system", "content": "The corresponding ODE system now has inhomogeneous term b O GLYPH<16> W B AO C B . Remark 1 Determinedness and integrability considerations carry over. So does sequential use of free characteristic problems on the first block. Remark 2 Solving the last equation in the system is then conceptually the same as in the previous subsection. Remark 3 We leave a systematic Green's function approach to weak observables equations for another occasion. Remark 4 If we reduce all constraints out, the reduced formulation has strong r U being all the observables there can be. Remark 5 If we reduce all constraints bar C out, the following applies since finite theory's Chronos is a single constraint equation. Corollary 2 Suppose there is only one constraint. Then there is only one independent proper weak observable, and the proper weak observables algebra is abelian. Proof The PI is now a single integral up to a multiplicative constant. Any finite-algebra generator commutes with itself. \u2737 Remark 6 Let us check how this comes to be consistent with Corollary 1. The first part of the first factor of (15)'s antisymmetry is not supported in nontrivially-weak observables space of dimension 1. (15) thus collapses to the furtherly factorizable form C GLYPH<16> 0 is disallowed as C is a nontrivial constraint, whereas W O : GLYPH<16> W 1 1 O GLYPH<16> 0 is disallowed as O w is nontrivially-weak. The there is no space for the zero double-trace perpendicularity as each trace is now over a 1d index and the 1 matrix 1 is manifestly nondegenerate. Thus O w GLYPH<16> 0 is the only surviving possibility, coinciding with the above deduction that the nontrivially-weak observables algebra is abelian.", "pages": [8, 9]}, {"title": "5 Expression in Terms of Observables", "content": "We here continue to develop Sec III.3. First recall that, by Lemma III.3, the B themselves form a closed algebraic structure. The amount of observables, is, moreover, very large due to the following Lemma. Lemma 3 Functions of observables are themselves observables. Proof Remark 1 Thus one is not looking for individual solutions of the observables PDEs, but a fortiori for whole algebras of solutions. Namely, it is closed among themselves and large enough to span all of a physical theory's mathematical content. I.e. enough to span P hase, with each functionally independent of the others. This renders useful the concept mentioned in Article III of finding 'basis observables'. Remark 2 Only a smaller number of suitably-chosen observables moreover suffice for practical use: expression in terms of observables. Remark 3 A common case is for dim(reduced P hase) GLYPH<16> 2 t k GLYPH<1> g u basis observables (of type G ) to be required. Here k : GLYPH<16> dim p q q and g : GLYPH<16> dim p g q is the total number of constraints involved, which are all gauge constraints. See e.g. Sec 7.3 for examples of 'basis observables'.", "pages": [9]}, {"title": "6 Strategies for the Problem of Observables", "content": "The 'bottom' alias 'zero' and 'top' alias 'unit' strategies are as follows. Strategy 4.0) Use Unconstrained Observables , U [48, 50], entailing no commutation conditions at all. Strategy 4.1) Insist on Constructing Dirac Observables , D . Remark 1 Strategy 4.1) has the conceptual and physical advantage of employing all the information in the final algebraic structure of all-first-class constraints of the theory, F . It has the practical disadvantage that finding any Dirac observables - much less a basis set for each theory in question - can be a hard mathematical venture, especially for Gravitational Theories [5, 10, 19, 37, 38, 39, 41, 48, 50, 51, 54, 56, 63, 69, 72]. Remark 2 Strategy 4.2) is diametrically opposite in each of the above regards. It can moreover be used as first stepping stone toward the former. Remark 3 Strategy 4.1) moreover amounts to concurrently addressing the unsplit totality of constraints (Constraint Closure facet) and Taking Function Spaces Thereover. As a four-aspect venture (Fig 3), it is unsurprisingly harder than Use Unconstrained Observables, which is single-aspect. Strategy 4.K) Find Kucha\u0159 Observables K . This can entail treating Q uad distinctly from the F lin , some motivations for which were covered in VII. A further pragmatic reason is that the K are simpler to find than the D . Strategy 4.G) Find g -observables , G . If one looks more closely, some of these motivations are actually tied to the G in cases in which these and the K are distinct. For instance, it is more generally G - rather than the K - which arise from the g -act, g -all construction in cases in which the candidate S huffle is confirmed as a G auge . Also, theories having either trivial Configurational Relationalism - or Best Matching resolved - have as a ready consequence a known full set of classical G . We take this on board by pointing to this further distinct strategy. This takes into account the triple combination of Configurational Relationalism, Constraint Closure, and Assignmant of Observables aspects. Complementarily, finding the G (or K ) represents a timeless pursuit due to the absence of C hronos (or underlying Temporal Relationalism) from the workings in question. Strategy 4.C) Find Chronos Observables , C . This takes into account the triple aspect combination of Temporal Relationalism, Constraint Closure and Taking Function Spaces Thereover. In theories with nontrivial g or some further first-class constraints, this is to be viewed as a stepping stone. It is available if C hronos indeed constitutes a constraint subalgebraic structure. Remark 4 (Non)universality arguments are pertinent at this point. Using U or D is always in principle possible. The first of these follows from no restrictions being imposed. The second follows from how any theory's full set of constraints can in principle be cast as a closed algebraic structure of first-class constraints. This is by use of the Dirac bracket, or the effective method, so as to remove any second-class constraints. Strategy 4.A) We finally introduce an additional universal strategy based on using some kind of A-Observables , A x , that a theory happens to possess . This corresponds to the closed subalgebraic structures of constraints which are realized by that theory. It is the general 'middling' replacement for considering K , G or C hronos observables. (None of these notions are universal over all physical theories.) While the ultimate aim is to reach the top of the lattice, it is often practically attainable to, firstly, land somewhere in the middle of the lattice. Secondly, to work one's way up by solving further DEs (or, geometrically, by further restricting constraint surfaces). A second source of strategic diversity is as follows. Strategy 4 1 .0) The unreduced approach : working on the unreduced P hase p S q with all the constraints. Strategy 4 1 .1) The true space approach - working on T rueP hase p S q GLYPH<16> P hase p S q{ F - is much harder, due to quotienting out C hronos p S q being harder and having to be done potential by potential. If True is known, classical observables are trivial. Strategy 4 1 .G) The reduced approach : working on GLYPH<131> P hase p S q GLYPH<16> P hase p S q{ g with just C hronos. Strategy 4 1 .P) The partly-reduced approach : working on P hase p S , H q GLYPH<16> P hase p S q{ H for some id GLYPH<160> H GLYPH<160> g Remark 5 The above primed family of strategies are moreover found to all coincide in output in the case of strong observables [78]. E.g. both unreduced and reduced approaches end up with or, in the purely geometrical case,", "pages": [9, 10]}, {"title": "7.1 Unreduced examples", "content": "Unreduced observables form the function space of suitably-smooth functions over our system's phase space. Unreduced configurational observables form the function space of suitably-smooth functions over our system's configuration space. Unreduced pure-momentum observables form the function space that consists of the totality of values taken by the given model S 's momenta P . Example 1 For Mechanics on R d , the configuration space is the constellation space (I.5-6), q p d, N q GLYPH<16> R dN . This being a flat space, the corresponding constellation phase space is The unrestricted observables here form The unrestricted configurational observables U p q q , alias absolute space geometry's N -point invariants, form Finally the unrestricted pure-momentum observables U p p q form These results readily generalize to RPMs over manifolds other than R n .", "pages": [10, 11]}, {"title": "7.2 Nontrivial geometrical examples of Kucha\u0159, gauge- and A-observables", "content": "Simplification 1 Suppose the constraints being taken into consideration depend at most linearly on the momenta. This holds within Aff p d q and its lattice of subgroups, covering all but the last example in the current subsection. Then in the observables equations, Simplification 2 Restrict attention to purely configurational such observables K p Q q GLYPH<16> G p Q q . This gives the particularly simple PDE [following on from (VII.22)] we drop the Q suffix when considering pure geometry.) Simplification 3 A few of the below examples have but a single observables equation (i.e. the N -index takes a single value). This is then amenable to the standard Flow Method. Simplification 4 Within Aff p d q 's lattice of subgroups, passage to the centre of mass frame is available to take translations out of contention. Upon doing this, mass-weighted relative Jacobi coordinates [34, 82] furthermore furnish a widespread simplification of one's remaining equations [59, 73]. This determines ab-initio-translationless and translation-reduced actions, constraints, Hamiltonians and observables, as being of the same form but with one object less: the so-called Jacobi map [73]. Notation 1 We additionally provided a compact notation for the outcome of solving the preserved equations for Sim p d q and its subgroups [73, 74] in Sec III.3. This is visible within row 1 of Fig 2, which furthermore extends this notation to Aff p d q and its subgroups [67, 77]. Example 1) For translation-invariant geometry g GLYPH<16> Tr p d q - the observables PDE is The solutions of this are, immediately, the relative interparticle separation vectors. These can be reformulated as linear combinations thereof, among which the relative Jacobi coordinates \u03c1 turn out to be particularly convenient by the Jacobi map. Our solutions form b) The semicolon here denotes cross ratio. See [77] for what any notions of geometry mentioned in the first two rows mean. I subsequently found that Guggenheimer [14], and then that previously Lie himself [1] already derived the top result here - the 1d projective group - by a flow PDE method. So (for now at least), the top case of Example 8 is to be attributed to Lie: an early success of his Integral Approach to Geometrical Invariants. c) is derived in detal in [78]. the smooth functions over relative space. Example 2 For scale-invariant geometry g GLYPH<16> Dil - the observables PDE is This is an Euler homogeneity equation of degree zero. Its solutions are therefore ratios of components of configurations. These form the smooth functions over ratio space. Example 3 For rotationally-invariant geometry g GLYPH<16> Rot p d q - the observables PDE is This is solved by the dot products q I GLYPH<4> q J . Norms and angles are moreover particular cases of functionals of the above, which are an allowed extension by Lemma 3. In 2d , these form where C p M q denotes topological and geometrical cone over M and n : GLYPH<16> N GLYPH<1> 1. Example 4 In Euclidean geometry g GLYPH<16> Eucl p d q GLYPH<16> Tr p d q GLYPH<11> Rot p d q - the observables PDEs are both (71) and (75). Sequential use of the Flow Method gives that the solutions are dots of differences of position vectors. This can be worked into the form of dots of relative Jacobi vectors. These are additionally Euclidean RPM's [25, 59] configurational Kucha\u0159 = gauge observables [68]. In 2d , they form the smooth functions over relational space. Example 5 In rotational-and-dilational geometry g GLYPH<16> Rot p d q GLYPH<2> Dil - the observables PDEs are both (75) and (73). The geometrical G are thus ratios of dots. In 2d , these form Example 6 In dilatational geometry g GLYPH<16> Dilatat p d q : GLYPH<16> Tr p d q GLYPH<11> Dil - the observables PDEs are both (71) and (73). The geometrical G are thus ratios of differences. These form the smooth functions over preshape space [46]. Example 7 In similarity geometry g GLYPH<16> Sim p d q GLYPH<16> Tr p d q GLYPH<11> p Rot p d q GLYPH<2> Dil q - the observables PDEs are all three of (71), (75) and (73). The geometrical G are thus ratios of dots of differences. These are additionally similarity RPM's [59] configurational Kucha\u0159 observables K [68]. In 2d , these form the smooth functions over shape space [46]. Example 8 Row 1 of Fig 2 identifies geometrical observables for Affine Geometry - corresponding to g GLYPH<16> Aff p d q - and for various of its further subgroups. We need dimension d \u00a5 2 for these groups to be nontrivially realized. These additionally constitute configurational observables for affine RPM [67]. Each of Examples 1 to 7 also constitute nontriviallyA examples (in the present context A that are not also K GLYPH<16> G ) for affine geometry, as do all other entries in the third subfigure of Fig 2.a) Each of Examples 1 to 6 perform this function for similarity geometry as well. Two-thirds of Examples 1 to 3 perform this function for each of Examples 4 to 6. Flat Geometry thus already provides copious numbers of examples of nontriviallyA observables. Example 9 Row 2 of Fig 2 identifies geometrical observables for 1d Projective Geometry - corresponding to g GLYPH<16> Proj p 1 q - and various of its further subgroups. Passing to the centre of mass however ceases to function as a simplification for Projective Geometry (and Conformal Geometry). This is since translations are more intricately involved here than via Affine Geometry's semidirect product addendum.", "pages": [11, 12, 13]}, {"title": "7.3 Geometrical examples of 'basis observables'", "content": "We here provide Euclidean RPM K GLYPH<16> G observables examples of this. Example 1 3 particles in 1d have the mass-weighted relative Jacobi separations \u03c1 1 , \u03c1 2 as useful basis observables (in K GLYPH<16> G sense). This extends to N particles in 1d having as basis observables a given clustering's \u03c1 i , i GLYPH<16> 1 to n GLYPH<16> N GLYPH<1> 1. Example 2 The 3 Hopf-Dragt coordinates of Sec V.5.5 are basis observables for the relational triangle [59]. Example 3 The 8 'Gell-Mann' coordinates detailed in [60] are basis observables for the relational quadrilateral. Motivation Kinematical quantization uses a lot less classical observables than the totality of suitably smooth functions over P hase. Kinematical quantization uses, more specifically, a linear subspace thereof that the canonical group acts upon [30]. This linear subspace moreover has enough coordinates to locally characterize q , thus fitting within our looser conception of 'basis beables'. Kinematical quantization's linear subspace quantities do moreover literally form a basis for that linear space. In this way, they constitute 'more of a basis' for configuration space than the a priori concept of 'basis observables' do. This linearity does not however in general extend to kinematical quantization's corresponding momentum observables (angular momentum suffices to see this). All of our examples above are useful for kinematical quantization.", "pages": [13, 14]}, {"title": "7.4 Physically nontrivial examples of strong K , G , A and C observables", "content": "Example 1 Translationally-invariant RPM's pure-momentum observables are freely specifiable. However, since the total centre of mass position is meaningless in this problem, its momentum is meaningless as well, leaving us with for \u03c0 i the conjugate momenta to \u03c1 i . These form i.e. the smooth functions over relative momentum space. The corresponding general observables are These form i.e. the smooth functions over relative phase space. Example 2 Dilationally-invariant RPM's pure-momentum observables PDE is the of the corresponding geometrical observables PDE system. This is thus solved by ratios of components of momenta. These form i.e. the smooth functions over momentum ratio space. The corresponding general observables PDE system is This is also an Euler homogeneity equation of degree zero. Its solutions are therefore ratios of phase space coordinates. These now correspond not to sphere in phase space but to a quadric surface in phase space. The observables constitute the C 8 functions over this phase ratio space. Example 3 Rotationally-invariant RPM's pure-momentum observables PDE system is also the (85) of the corresponding geometrical observables PDE system. This is thus is solved by suitably smooth functions of the dot product, In 2d , these form The corresponding general observables PDE system is This is solved by suitably-smooth functions of phase space symmetrized dot products. These are the outcome of applying the product rule to q I GLYPH<4> q J . Example 4 Euclidean RPM combines the above translational and rotational equations. The Jacobi map applying, the Euclidean momentum observables solutions are suitably smooth functions In 2d , these form the smooth functions over relational momentum space. The general observable solutions are suitably smooth functions i.e. relative phase space symmetrized dot products. Example 5 Similarity RPM combines all three of the above translational, rotational and dilational equations. So for instance, the momentum observables solutions are suitably smooth functions and the general observables are suitably smooth functions Example 6 If the generators are quadratic (which we know from Article III to apply in the conformal and projective cases), then (85) symmetry among observables is broken. Example 7 Chronos observables C for the general p N,d q Euclidean RPM solve In the case of constant potential, this simplifies to which is solved by Treating the N d component differently is an arbitrary choice; \u039b then runs over all the other values, 1 to N d GLYPH<1> 1. For RPMs with any Configurational Relationalism, these are a further species of A observable that is not a K or G observable. Remark 1 That C hronosO bs (and D iracO bs) are potential-dependent, constitutes a massive complication at the computational level.", "pages": [14, 15]}, {"title": "7.5 Examples of strong nontrivially-Dirac observables", "content": "Example 1 For RPMs involving whichever combination of translations and rotations, the additional PDE to obtain strong Dirac observables is # + If dilations are involved as well, one needs to divide the kinetic term contribution by the total moment of inertia to have a ratio form. The above are moreover mathematically equivalent to the corresponding p d, n q Chronos problems, so e.g. in the Eucl p d q case with the true space index \u03c4 running over 1 to nd GLYPH<1> 1. Example 2 Minisuperspace (spatially homogeneous GR) only has a H , and a single finite constraint oversimplifies the diversity of notions of observables. Here the sole strong observables brackets equation is giving the observables PDE As per Article I, simple examples include Q GLYPH<16> \u03b1, \u03c6 or \u03b1, \u03b2 GLYPH<8> . Each minisuperspace's type of potential (one part fixed by GR, another part variable with the nature of appended matter physics). There being just one such equation gives that U GLYPH<16> G GLYPH<16> K GLYPH<24> C GLYPH<16> D . See e.g. [40] for direct construction of classical Dirac observables for Minisuperspace.", "pages": [16]}, {"title": "7.6 Nontrivially weak observables", "content": "Reduced versus indirect makes a clear difference here, since the indirect case has a longer string of constraints in its PI. Only at least partly indirectly formulated case has any space for nontrivially weak such: if all constraints are reduced out, no PI is left. Example 1 Weak translational observables (a type of gauge = Kucha\u0159 observables). 1 particle in 1d with inhomogeneous term W P GLYPH<16> Wp supports the PI For 2 particles in 1d , the inhomogeneous term W P GLYPH<16> W p p 1 GLYPH<0> p 2 q supports the PI Per fixed W , this gives a R of solutions, corresponding to viewing a as free. Considering all W , we have a R GLYPH<2> R GLYPH<6> of properly weak solutions. These are now however much less numerous than the strong solutions. This is since the strong solutions now comprise the C p R 2 q of suitably-smooth functions of x 1 GLYPH<1> x 2 and p 1 GLYPH<1> p 2 . This is the general situation for enough degrees of freedom: that the weak observables' parameter space is an appendage of measure zero relative to that of the strong observables' function space. For translation-invariant mechanics, N GLYPH<16> 2 particles is minimal to exhibit this effect. Translational mathematics being the simplest nontrivial Configurational Relationalism considered in the current article, this example has further senses in which it is 'the simplest nontrivial example' of its kind. Looking at general particle number and dimension, we find that our method extends. This serves to show that [78] for translation-invariant models, no configuration-geometrical or pure-momentum weak observables are supported. Example 2 Weak Chronos observables. Now 1 particle in 1d supports the PI This is one function per value of W , or R GLYPH<6> functions in total. For 2 particles in 1d , The space of these again coincides with the corresponding translational problem. 2 particles in 1d is again minimal for weak observables to be of zero measure relative to strong observables. Our solution again extends to arbitrary particle number and dimension [78]. Example 3 Weak Dirac observables are not just weak Chronos restricted by G auge or vice versa. This is since in the D w system, firstly, the Chronos observables equation includes a G auge inhomogeneous term as well as a C hronos one. Secondly, the gauge observables equations include a C hronos inhomogeneous term as well as G auge ones. In contrast, the C hronos w system has just a C hronos inhomogeneous term, and the G auge w system has just a G auge one. This shows that weak observables systems do not involve a simple restriction hierarchy like strong observables ones do.", "pages": [16, 17]}, {"title": "Example 4", "content": "giving the weak observables PDE is shown to be possible in [78]. A simple argument for this is that C anO bs has more scope for PI terms than GLYPH<131> C anO bs does, by being naturally associated with a larger constraint algebra. (The number of PI terms is c 2 for c : GLYPH<16> dim p C q , since there are c weak observables equations, each of which has c inhomogeneous terms.) Example 5 The reduced treatment of translational RPM for 2 particles in 1d gives This is not however enough to have weak observables be of measure zero relative to strong observables, since we now have two constraints to two degrees of freedom. We do however have a general particle number and dimension solution to both the translational and chronos problems, however. So it is not hard to give the 3-particle, 1d minimal example of this effect. Remark 1 The usual relational numerology [67, 74] readily lets us pick out minimal examples for strong observables dominance for further transformations (rotations, dilations, Euclidean, similarity, affine...) Example 6 Minisuperspace Here the sole weak observables brackets equation is r GLYPH<128> Remark 2 Studying just minisuperspace, however, leaves one unaware of most of the diversity of types of observables, and of almost every effect described in the current section. Flat geometry and RPMs thereupon are thus rather more instructive in setting up a general theory of observables.", "pages": [17]}, {"title": "8.1 Brackets algebra level", "content": "Overall, we have the following finite-field portmanteau brackets equation, This is to be interpreted as a \u03b4 B DE system, i.e. a portmanteau of a PDE (III.87) in the finite case and an FDE (functional differential equation) 2", "pages": [17, 18]}, {"title": "8.2 FDE level", "content": "Remark 1 We can moreover take leaving us with a homogeneous-linear first-order FDE system. We would also fix particular smearing functions or formally do not smear in locally posing and solving our FDE system. Remark 2 The general form - analogous to (III.87) for the corresponding Finite Theory PDE case - covers both our homogeneous strong observables FDE system p b GLYPH<16> 0 q and our inhomogeneous weak observables FDE system p b GLYPH<24> 0 q .", "pages": [18]}, {"title": "8.3 Flow method transcends to Banach space", "content": "Remark 1 This is as far as we detail in the current series, though transfer to the tame Fr\u00e9chet space setting is also possible. Structure 8 Let B be a Banach manifold and v a vector field thereupon. Curves, tangent vectors and tangent spaces remain defined on B [27]. Definition 1 An integral curve of B is a curve \u03b3 such that at each point b the tangent vector is v b . Definition 2 The differential system on B defined by v still takes its usual form, Remark 2 We can append a multi-index on \u03c6 and v if needs be, to cover multi-component field and multi-field versions. Remark 3 DE Existence and uniqueness theorems carry over [27]. Remark 4 This guarantees a local flow (which is as much as the current Series' considerations cater for). As in the finite case, this provides a 1-parameter group. Structure 9 Differential forms carry over to (sufficiently smooth) B . So do pullback, exterior differential operator and internal product [27]. The familiar 'Cartan's magic formula' for the Lie derivative is consequently available [27]. Thus in turn Lie dragging remains available, as does Lie correcting (toward implementing Configurational Relationalism) and the diffeomorphism interpretation of the flow. Banach Lie algebras , and Banach Lie groups , are well-established [20], enabling Configurational Relationalism. Frobenius' Theorem - as required for Lie's integral method for geometrical invariants, its canonical physics generalization and the uplifts of each of these to finding function spaces of observables thereover - carries over as well [42]. End-Remark 1 All in all, local Lie Theory, in the somewhat broader sense required for A Local Resolution of the Problem of Time is thus established in the Banach space setting which can be taken to underlie much of Field Theory.", "pages": [18]}, {"title": "8.4 Examples", "content": "For conventional Gauge Theory, the observables equation imposes gauge invariance at the level of configuration space based on both space and internal gauge space. In each of the first two examples below, since these just have first-class linear constraints which are gauge constraints. Example 1) Electromagnetism has the abelian algebra of constraints (VII.37). G GLYPH<16> K for Electromagnetism solve the brackets equation r r r This is solved by the electric and magnetic fields, and thus by a functional by Lemma 3. These are not however a conjugate pair. Since this looks to be a common occurrence in further examples, let us introduce the term 'associated momenta' to describe it. F r B , E s can also be written in the integrated version in terms of fluxes: for electric flux \u03a6 E S and loop variable This is by use of Stokes' Theorem with \u03b3 : GLYPH<16> B S and subsequent insertion of the exponential function subcase of Lemma 3. This ties the construct to the geometrical notion of holonomy. Moreover, these are well-known to form an over -complete set: there are so-called Mandelstam identities between them [43]. Example 2) Its Yang-Mills generalization has the Lie algebra of constraints (VII.39) G GLYPH<16> K for Yang-Mills Theory solve the brackets equation r This is solved by Yang-Mills Theory's generalized E and B, so is also a solution. Once again, this can be rewritten as now for g -loop variable g are here group generators of g YM , g is the coupling constant, and P is the path-ordering symbol. Example 3 The cause c\u00e9l ' e bre of canonical treatment of observables is GR. In this case, one has the spatial 3diffeomorphisms momentum constraint M - linear in its momenta - and a Hamiltonian constraint H - quadratic in its momenta - to commute with. Example 3) K for GR as Geometrodynamics, the brackets equation This corresponds to the unsmeared FDE In the weak case, we can furthermore discard the penultimate term. The K p Q q subcase solve These are, formally, 3-geometry quantities ' G p 3 q ' by (135) emulating (and moreover logically preceding) the 'momenta to the right' ordered quantum GR momentum constraint (IV.31). This analogy holds for the current Series' range of finite models as well (of relevance to those with nontrivial F lin ). Remark 1 Explicit 'basis observables' are not known in this case. On the other hand, the FDE for the K p P q (formally '\u03a0 G p 3 q ') is Example 4) D's for Geometrodynamics' D require an extra FDE [68] r r This features the DeWitt vector quantities These are already familiar from the ADM equations of motion [12], and also D 2 with components D i D j . In unsmeared form, (138) is the FDE M and N here are the DeWitt supermetric and its inverse respectively.", "pages": [19, 20]}, {"title": "8.5 Genericity in Field Theory", "content": "In Field Theory, for instance Electromagnetism has one observables equation, whereas Yang-Mills Theory has g : GLYPH<16> dim p g YM q such. GR-as-Geometrodynamics has four observables equations: commutation with the 3 components of the momentum constraint and with the single Hamiltonian constraint. So, on the one hand, for Yang-Mills Theory and for GR, PDE system genericity can be relevant. On the other hand, for Finite Theories, PDE system genericity is in general obscured by geometrical genericity.", "pages": [20]}, {"title": "9.1 Summary", "content": "For Finite Theories, strong observables are to be found by solving brackets equations which can be recast as homogeneous linear first-order PDE systems. Cases with a single such equation - corresponding to a constraint (sub)algebra with a single generator - are mathematically standard; the Flow Method readily applies. For nontrivial systems of such PDEs, there is no universal approach; the outcome depends, rather, on determinedness status. For observables PDE systems, moreover, over-determinedness is vanquished by integrability conditions guaranteed by Frobenius' Theorem. We furthermore provide firm grounding for a free alias natural [7] characteristic problem treatment being appropriate for observables, indeed embodying the Taking of Function Spaces Thereover. Sequential use of the Flow Method then produces the requisite strong observables. The above workings amount to a minor extension of Lie's Integral Approach to Geometrical Invariants. Firstly, the observables are not just 'the invariants' but rather any suitably smooth function of the invariants. Secondly, this extension generally plays out in phase space for us rather than in Lie's purely geometric setting. Properly weak observables obey the inhomogeneous-linear counterpart of the above. Their general solution thus splits into complementary function plus particular integral, which roles are played by strong and nontrivially-weak observables respectively. For examples with enough degrees of freedom, moreover, properly weak observables give but a measure-zero extension to the space of strong observables. We illustrate all of strong versus properly weak observables, of unrestricted, Dirac and middlingly restricted observables, as well as of properly weak observables' dependence on extent of reduction. Also fully reduced treatments have no room for properly weak observables. This and the above zero-measure comment are two reasons to not place too much stock in further developing the theory of properly weak observables. The above summary corresponds to the first major extension of [72]'s treatment of Background Independence and the Problem of Time. This is by providing a concrete theory of solving for observables on the endpoint of the first branch of Fig 1.b), for now in the local, classical finite-theory setting. For Field Theories, observables equations give instead linear first-order functional differential equation systems. Banach (or tame Fr\u00e9chet) Calculus is however sufficiently standard that the Flow Method and free characteristic problems still apply. These calculi support the Lie-theoretic combination of machinery that our Local Resolution of the Problem of Time requires. In this way, Field Theory - exemplified by Electromagnetism and Yang-Mills Theory in the current Article - and GR are included in our Local Resolution of the Problem of Time's formulation.", "pages": [21]}, {"title": "9.2 Present and future of the pedagogy of observables", "content": "Pedagogical difficulties with presenting observables moreover abound. Many quantum treatises, as well as popular accounts [93], immediately launch into the quantum version. They do not mention the corresponding classical counterpart because of its relative simplicity. In the process, they miss the point that the classical version becomes nontrivial in the presence of constraints. This difficulty moreover recurs at the quantum level, where it is missed out again. In this way, the most advanced theory of observables - of quantum constrained observables - is not mentioned on Wikipedia [93]. The suggested restructuring is to start with classical unconstrained functions over phase space (and configuration space, with connections to pure geometry). Then, on the one hand, one is to consider the nontrivialities of classical constrained systems. On the other hand, one is to keep the existing quantum account as the other main complicating factor. This is now to emphasize that only small subalgebras of the classical observables algebra can be consistently quantized. Finally, these two sources of complexity are to be combined. Article X spells out further sources of improvement, along the lines of the current paragraph's canonical notions of observables having spacetime, path or histories observables analogues. The common theme is that of Function Spaces Thereover, meaning over state spaces. Be that (un)constrained classical phase space, the space of spacetimes or some quantum state space.", "pages": [21]}, {"title": "9.3 The theory of strong observables has at least a presheaf flavour", "content": "Taking Function Spaces Thereover models moreover multiplicity of function spaces over state spaces. This is in the sense of applying various different consistent constraint algebraic substructures. For strong observables - the generic part of observables theory for sufficient degrees of freedom - these combine in the manner of applying successive restrictions to the observables DE's characteristic surfaces as per Theorem 3. Such inter-relation by restriction maps amounts to Taking Presheaves Thereover. How widely this can be extended to Taking Sheaves Thereover remains to be determined. This is an interesting question to investigate, due to sheaves possessing further localization and globalization properties; 3 it is however beyond the reach of the current Series' local treatise.", "pages": [21, 22]}] |
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